Copyright (c) 1992 Association of Mizar Users
environ
vocabulary BHSP_1, PRE_TOPC, NORMSP_1, SEQ_1, RLVECT_1, FUNCT_1, SERIES_1,
ARYTM_1, SUPINF_2, SEQ_2, ORDINAL2, SEQM_3, BHSP_3, LATTICES, ABSVALUE,
ARYTM_3, POWER, PROB_1, INT_1, METRIC_1, ARYTM, GROUP_1;
notation SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, STRUCT_0, REAL_1,
NAT_1, FUNCT_2, SEQ_1, SEQ_2, SEQM_3, SERIES_1, ABSVALUE, PRE_TOPC,
RLVECT_1, NORMSP_1, BHSP_1, BHSP_2, BHSP_3, PREPOWER, POWER, INT_1;
constructors REAL_1, NAT_1, SEQ_2, SERIES_1, BHSP_2, BHSP_3, PREPOWER,
PARTFUN1, MEMBERED, XBOOLE_0;
clusters INT_1, STRUCT_0, XREAL_0, SEQ_1, RELSET_1, MEMBERED, ZFMISC_1,
XBOOLE_0, ORDINAL2;
requirements NUMERALS, REAL, SUBSET, ARITHM;
theorems REAL_1, NAT_1, FUNCT_2, SEQ_1, SEQ_2, SEQM_3, SEQ_4, SERIES_1,
ABSVALUE, RLVECT_1, SQUARE_1, BHSP_1, BHSP_2, BHSP_3, PREPOWER, POWER,
INT_1, AXIOMS, REAL_2, NORMSP_1, XREAL_0, XCMPLX_0, XCMPLX_1;
schemes NAT_1, BHSP_1, RECDEF_1;
begin
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve a, b, r, M2 for Real;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
deffunc 0'(RealUnitarySpace) = 0.$1;
scheme Rec_Func_Ex_RUS{ X() -> RealUnitarySpace,
z() -> Point of X(),
G(Nat, Point of X()) -> Point of X()}:
ex f being Function of NAT, the carrier of X() st
f.0 = z() & for n being Element of NAT,
x being Point of X() st x = f.n holds f.(n + 1) = G(n,x)
proof
deffunc _G(Nat,Point of X()) = G($1,$2);
consider f being Function of NAT, the carrier of X() such that
A1: f.0 = z() &
for n being Element of NAT holds f.(n+1) = _G(n,f.n) from LambdaRecExD;
take f;
thus f.0 = z() by A1;
thus for n being Nat, x being Point of X() st x = f.n
holds f.(n+1) = G(n,x) by A1;
end;
definition
let X;
let seq;
func Partial_Sums(seq) -> sequence of X means
:Def1: it.0 = seq.0 &
for n holds it.(n + 1) = it.n + seq.(n + 1);
existence
proof deffunc _G(Nat,Point of X) = $2 + seq.($1 + 1);
consider f being Function of
NAT, the carrier of X such that
A1: f.0 = seq.0 &
for n being Nat, x being Point of X st x = f.n holds
f.(n + 1) = _G(n,x) from Rec_Func_Ex_RUS;
reconsider f as sequence of X by NORMSP_1:def 3;
take f;
thus thesis by A1;
end;
uniqueness
proof
let seq1,seq2;
assume that
A2: seq1.0 = seq.0 &
for n holds seq1.(n + 1) = seq1.n + seq.(n + 1) and
A3: seq2.0 = seq.0 &
for n holds seq2.(n + 1) = seq2.n + seq.(n + 1);
defpred _P[Nat] means seq1.$1 = seq2.$1;
A4: _P[0] by A2,A3;
A5: for k st _P[k] holds _P[k+1] proof let k;
assume seq1.k =seq2.k;
hence seq1.(k + 1) = seq2.k + seq.(k+1) by A2
.= seq2.(k + 1) by A3;
end;
for n holds _P[n] from Ind(A4,A5);
hence seq1 = seq2 by FUNCT_2:113;
end;
end;
theorem Th1:
Partial_Sums(seq1) + Partial_Sums(seq2) =
Partial_Sums(seq1 + seq2)
proof
set PSseq1 = Partial_Sums(seq1);
set PSseq2 = Partial_Sums(seq2);
A1: (PSseq1 + PSseq2).0 = PSseq1.0 + PSseq2.0 by NORMSP_1:def 5
.= seq1.0 + PSseq2.0 by Def1
.= seq1.0 + seq2.0 by Def1
.= (seq1 + seq2).0 by NORMSP_1:def 5;
now let n;
thus
(PSseq1 + PSseq2).(n + 1)
= PSseq1.(n + 1) + PSseq2.(n + 1) by NORMSP_1:def 5
.= PSseq1.n + seq1.(n + 1) + PSseq2.(n + 1) by Def1
.= PSseq1.n + seq1.(n + 1) + (seq2.(n + 1) + PSseq2.n) by Def1
.= PSseq1.n + seq1.(n + 1) + seq2.(n + 1) + PSseq2.n by RLVECT_1:def 6
.= PSseq1.n + (seq1.(n + 1) + seq2.(n + 1)) + PSseq2.n by RLVECT_1:def 6
.= PSseq1.n + (seq1 + seq2).(n + 1) + PSseq2.n by NORMSP_1:def 5
.= PSseq1.n + PSseq2.n + (seq1 + seq2).(n + 1) by RLVECT_1:def 6
.= (PSseq1 + PSseq2).n + (seq1 + seq2).(n + 1) by NORMSP_1:def 5;
end;
hence thesis by A1,Def1;
end;
theorem Th2:
Partial_Sums(seq1) - Partial_Sums(seq2) =
Partial_Sums(seq1 - seq2)
proof
set PSseq1 = Partial_Sums(seq1);
set PSseq2 = Partial_Sums(seq2);
A1: (PSseq1 - PSseq2).0 = (PSseq1).0 - (PSseq2).0 by NORMSP_1:def 6
.= seq1.0 - (PSseq2).0 by Def1
.= seq1.0 - seq2.0 by Def1
.= (seq1 - seq2).0 by NORMSP_1:def 6;
now let n;
thus (PSseq1 - PSseq2).(n + 1)
= PSseq1.(n + 1) - PSseq2.(n + 1) by NORMSP_1:def 6
.= (PSseq1.n + seq1.(n + 1)) - PSseq2.(n + 1) by Def1
.= (PSseq1.n + seq1.(n + 1)) - (seq2.(n + 1) + PSseq2.n) by Def1
.= ((PSseq1.n + seq1.(n + 1)) - seq2.(n + 1)) - PSseq2.n by RLVECT_1:41
.= (PSseq1.n + (seq1.(n + 1) - seq2.(n + 1))) - PSseq2.n by RLVECT_1:42
.= PSseq1.n + ((seq1.(n + 1) - seq2.(n + 1)) - PSseq2.n) by RLVECT_1:42
.= PSseq1.n + (- PSseq2.n + (seq1.(n + 1) - seq2.(n + 1))) by RLVECT_1:def 11
.= (PSseq1.n + - PSseq2.n) + (seq1.(n + 1) - seq2.(n + 1)) by RLVECT_1:def 6
.= (PSseq1.n - PSseq2.n) + (seq1.(n + 1) - seq2.(n + 1)) by RLVECT_1:def 11
.= (PSseq1 - PSseq2).n + (seq1.(n + 1) - seq2.(n + 1)) by NORMSP_1:def 6
.= (PSseq1 - PSseq2).n + (seq1 - seq2).(n + 1) by NORMSP_1:def 6;
end;
hence thesis by A1,Def1;
end;
theorem Th3:
Partial_Sums(a * seq) = a * Partial_Sums(seq)
proof
set PSseq = Partial_Sums(seq);
A1: (a * PSseq).0 = a * PSseq.0 by NORMSP_1:def 8
.= a * seq.0 by Def1
.= (a * seq).0 by NORMSP_1:def 8;
now let n;
thus (a * PSseq).(n + 1) = a * PSseq.(n + 1) by NORMSP_1:def 8
.= a * (PSseq.n + seq.(n + 1)) by Def1
.= a * PSseq.n + a * seq.(n + 1) by RLVECT_1:def 9
.= (a * PSseq).n + a * seq.(n + 1) by NORMSP_1:def 8
.= (a * PSseq).n + (a * seq).(n + 1) by NORMSP_1:def 8;
end;
hence thesis by A1,Def1;
end;
theorem
Partial_Sums(- seq) = - Partial_Sums(seq)
proof
Partial_Sums((-1) * seq) = (-1) * Partial_Sums(seq) by Th3;
then Partial_Sums(- seq) = (-1) * Partial_Sums(seq) by BHSP_1:77;
hence thesis by BHSP_1:77;
end;
theorem
a * Partial_Sums(seq1) + b * Partial_Sums(seq2) =
Partial_Sums(a * seq1 + b * seq2)
proof
thus a * Partial_Sums(seq1) + b * Partial_Sums(seq2)
= Partial_Sums(a * seq1) + b * Partial_Sums(seq2) by Th3
.= Partial_Sums(a * seq1) + Partial_Sums(b * seq2) by Th3
.= Partial_Sums(a * seq1 + b * seq2) by Th1;
end;
definition let X;
let seq;
attr seq is summable means
:Def2: Partial_Sums(seq) is convergent;
func Sum(seq) -> Point of X equals
:Def3: lim Partial_Sums(seq);
correctness;
end;
theorem
seq1 is summable & seq2 is summable implies seq1 + seq2 is summable
& Sum(seq1 + seq2) = Sum(seq1) + Sum(seq2)
proof
assume that
A1: seq1 is summable and
A2: seq2 is summable;
A3: Partial_Sums(seq1) is convergent by A1,Def2;
A4: Partial_Sums(seq2) is convergent by A2,Def2;
then Partial_Sums(seq1) + Partial_Sums(seq2) is convergent
by A3,BHSP_2:3;
then Partial_Sums(seq1 + seq2) is convergent by Th1;
hence seq1 + seq2 is summable by Def2;
thus Sum(seq1 + seq2)
= lim Partial_Sums(seq1 + seq2) by Def3
.= lim (Partial_Sums(seq1) + Partial_Sums(seq2)) by Th1
.= lim Partial_Sums(seq1) + lim Partial_Sums(seq2)
by A3,A4,BHSP_2:13
.= Sum(seq1) + lim Partial_Sums(seq2) by Def3
.= Sum(seq1) + Sum(seq2) by Def3;
end;
theorem
seq1 is summable & seq2 is summable implies seq1 - seq2 is summable
& Sum(seq1 - seq2) = Sum(seq1) - Sum(seq2)
proof
assume that
A1: seq1 is summable and
A2: seq2 is summable;
A3: Partial_Sums(seq1) is convergent by A1,Def2;
A4: Partial_Sums(seq2) is convergent by A2,Def2;
then Partial_Sums(seq1) - Partial_Sums(seq2) is convergent
by A3,BHSP_2:4;
then Partial_Sums(seq1 - seq2) is convergent by Th2;
hence seq1 - seq2 is summable by Def2;
thus Sum(seq1 - seq2)
= lim Partial_Sums(seq1 - seq2) by Def3
.= lim (Partial_Sums(seq1) - Partial_Sums(seq2)) by Th2
.= lim Partial_Sums(seq1) - lim Partial_Sums(seq2)
by A3,A4,BHSP_2:14
.= Sum(seq1) - lim Partial_Sums(seq2) by Def3
.= Sum(seq1) - Sum(seq2) by Def3;
end;
theorem
seq is summable implies
a * seq is summable & Sum(a * seq) = a * Sum(seq)
proof
assume seq is summable;
then A1: Partial_Sums(seq) is convergent by Def2;
then a * Partial_Sums(seq) is convergent by BHSP_2:5;
then Partial_Sums(a * seq) is convergent by Th3;
hence a * seq is summable by Def2;
thus Sum(a * seq) = lim Partial_Sums(a * seq) by Def3
.= lim (a * Partial_Sums(seq)) by Th3
.= a * (lim Partial_Sums(seq)) by A1,BHSP_2:15
.= a * Sum(seq) by Def3;
end;
theorem Th9:
seq is summable implies
seq is convergent & lim seq = 0.X
proof
set PSseq = Partial_Sums(seq);
assume seq is summable;
then A1: PSseq is convergent by Def2;
then A2: PSseq ^\1 is convergent &
lim(PSseq ^\1) = lim(PSseq) by BHSP_3:44;
then A3: lim(PSseq ^\1 - PSseq)
= lim(PSseq) - lim(PSseq) by A1,BHSP_2:14
.= 0'(X) by RLVECT_1:28;
A4: seq ^\1 = PSseq ^\1 - PSseq
proof
now let n;
(PSseq).(n + 1) = (PSseq).n + seq.(n + 1) by Def1
.= (PSseq).n + (seq ^\1).n by BHSP_3:def 5;
hence
(PSseq ^\1).n = (PSseq).n + (seq ^\1).n by BHSP_3:def 5;
end;
then A5: (PSseq ^\1) = PSseq + seq ^\1 by NORMSP_1:def 5;
seq ^\1 + (PSseq - PSseq) = seq ^\1
proof
now let n;
thus (seq ^\1 + (PSseq - PSseq)).n
= (seq ^\1).n + (PSseq - PSseq).n by NORMSP_1:def 5
.= (seq ^\1).n + (PSseq.n - PSseq.n) by NORMSP_1:def 6
.= (seq ^\1).n + 0'(X) by RLVECT_1:28
.= (seq ^\1).n by RLVECT_1:10;
end;
hence thesis by FUNCT_2:113;
end;
hence thesis by A5,BHSP_1:83;
end;
then seq ^\1 is convergent by A1,A2,BHSP_2:4; hence
A6: seq is convergent by BHSP_3:46;
seq ^\1 is_subsequence_of seq by BHSP_3:43;
hence thesis by A3,A4,A6,BHSP_3:33;
end;
theorem Th10:
X is_Hilbert_space implies
( seq is summable iff ( for r st r > 0 ex k st for n, m st ( n >= k &
m >= k ) holds ||.(Partial_Sums(seq)).n - (Partial_Sums(seq)).m.|| < r ))
proof
assume X is_Hilbert_space;
then A1: X is RealUnitarySpace & X is_complete_space by BHSP_3:def 7;
set PSseq = Partial_Sums(seq);
thus
seq is summable implies ( for r st r > 0 ex k st for n, m st ( n >= k &
m >= k ) holds ||.(PSseq).n - (PSseq).m.|| < r )
proof
assume seq is summable;
then PSseq is convergent by Def2;
then PSseq is_Cauchy_sequence by BHSP_3:9;
hence thesis by BHSP_3:2;
end;
thus ( for r st r > 0 ex k st for n, m st ( n >= k & m >= k ) holds
||.(PSseq).n - (PSseq).m.|| < r )
implies seq is summable
proof
assume
for r st r > 0 ex k st for n, m st ( n >= k & m >= k )
holds ||.(PSseq).n - (PSseq).m.|| < r;
then PSseq is_Cauchy_sequence by BHSP_3:2;
then PSseq is convergent by A1,BHSP_3:def 6;
hence thesis by Def2;
end;
end;
theorem
seq is summable implies Partial_Sums(seq) is bounded
proof
assume seq is summable;
then Partial_Sums(seq) is convergent by Def2;
hence thesis by BHSP_3:24;
end;
theorem Th12:
for seq, seq1 st for n holds seq1.n = seq.0 holds
Partial_Sums(seq^\1) = (Partial_Sums(seq)^\1) - seq1
proof
let seq;
let seq1;
assume
A1: for n holds seq1.n = seq.0;
A2: ((Partial_Sums(seq)^\1) - seq1).0 = (Partial_Sums(seq)^\1).0 - seq1.0
by NORMSP_1:def 6
.= (Partial_Sums(seq)^\1).0 - seq.0 by A1
.= Partial_Sums(seq).(0 + 1) - seq.0 by BHSP_3:def 5
.= Partial_Sums(seq).0 + seq.(0 + 1) - seq.0 by Def1
.= seq.(0 + 1) + seq.0 - seq.0 by Def1
.= seq.(0 + 1) + (seq.0 - seq.0) by RLVECT_1:42
.= seq.(0 + 1) + 0'(X) by RLVECT_1:28
.= seq.(0 + 1) by RLVECT_1:10
.= (seq^\1).0 by BHSP_3:def 5;
now let n;
thus ((Partial_Sums(seq)^\1) - seq1).(n + 1)
= (Partial_Sums(seq)^\1).(n + 1) - seq1.(n + 1) by NORMSP_1:def 6
.= (Partial_Sums(seq)^\1).(n + 1) - seq.0 by A1
.= Partial_Sums(seq).(n + 1 + 1) - seq.0 by BHSP_3:def 5
.= seq.(n + 1 + 1) + Partial_Sums(seq).(n + 1) - seq.0 by Def1
.= seq.(n + 1 + 1) + Partial_Sums(seq).(n + 1) - seq1.n by A1
.= seq.(n + 1 + 1) + (Partial_Sums(seq).(n + 1) - seq1.n) by RLVECT_1:42
.= seq.(n + 1 + 1) + ((Partial_Sums(seq)^\1).n - seq1.n) by BHSP_3:def 5
.= seq.(n + 1 + 1) + ((Partial_Sums(seq)^\1) - seq1).n by NORMSP_1:def 6
.= ((Partial_Sums(seq)^\1) - seq1).n + (seq^\1).(n + 1) by BHSP_3:def 5;
end;
hence thesis by A2,Def1;
end;
theorem Th13:
seq is summable implies for k holds seq^\k is summable
proof defpred _P[Nat] means seq^\($1) is summable;
assume seq is summable;
then A1: _P[0] by BHSP_3:35;
A2: for k st _P[k] holds _P[k+1] proof let k;
assume seq^\k is summable;
then Partial_Sums(seq^\k) is convergent by Def2;
then A3: Partial_Sums(seq^\k)^\1 is convergent by BHSP_3:44;
A4: seq^\(k+1)=(seq^\k)^\1 by BHSP_3:37;
deffunc _F(Nat) = (seq^\k).0;
consider seq1 such that
A5: for m holds seq1.m = _F(m) from Ex_Seq_in_RUS;
seq1 is constant by A5,NORMSP_1:def 4;
then A6: seq1 is convergent by BHSP_2:1;
Partial_Sums(seq^\k^\1) = (Partial_Sums(seq^\k)^\1) - seq1
by A5,Th12;
then Partial_Sums(seq^\k^\1) is convergent by A3,A6,BHSP_2:4;
hence thesis by A4,Def2;
end;
thus for k holds _P[k] from Ind(A1,A2);
end;
theorem
(ex k st seq^\k is summable) implies seq is summable
proof
given k such that
A1: seq^\k is summable;
seq^\k^\1 is summable by A1,Th13;
then A2: Partial_Sums(seq^\k^\1) is convergent by Def2;
deffunc _F(Nat) = Partial_Sums(seq).k;
consider seq1 such that
A3: for n holds seq1.n = _F(n) from Ex_Seq_in_RUS;
seq1 is constant by A3,NORMSP_1:def 4;
then A4: seq1 is convergent by BHSP_2:1;
defpred _P[Nat] means
(Partial_Sums(seq)^\(k+1)).$1 = Partial_Sums(seq^\(k+1)).$1 + seq1.$1;
for m holds _P[m] proof
A5: Partial_Sums(seq^\(k+1)).0 = (seq^\(k+1)).0 by Def1
.= seq.(0+(k+1)) by BHSP_3:def 5
.= seq.(k+1);
(Partial_Sums(seq)^\(k+1)).0
= Partial_Sums(seq).(0+(k+1)) by BHSP_3:def 5
.= Partial_Sums(seq).k + seq.(k+1) by Def1
.= Partial_Sums(seq^\(k+1)).0 + seq1.0 by A3,A5;
then A6: _P[0];
A7: now let m;
assume
A8: _P[m];
Partial_Sums(seq^\(k+1)).(m+1) + seq1.(m+1)
= seq1.(m+1) + (Partial_Sums(seq^\(k+1)).m + (seq^\(k+1)).(m+1)) by Def1
.= seq1.(m+1) + Partial_Sums(seq^\(k+1)).m + (seq^\(k+1)).(m+1)
by RLVECT_1:def 6
.= Partial_Sums(seq).k + Partial_Sums(seq^\(k+1)).m + (seq^\(k+1)).(m+1)
by A3
.= (Partial_Sums(seq)^\(k+1)).m + (seq^\(k+1)).(m+1) by A3,A8
.= Partial_Sums(seq).(m+(k+1)) + (seq^\(k+1)).(m+1) by BHSP_3:def 5
.= Partial_Sums(seq).(m+(k+1)) + seq.(m+1+(k+1)) by BHSP_3:def 5
.= Partial_Sums(seq).(m+(k+1)) + seq.(m+(k+1)+1) by XCMPLX_1:1
.= Partial_Sums(seq).(m+(k+1)+1) by Def1
.= Partial_Sums(seq).(m+1+(k+1)) by XCMPLX_1:1
.= (Partial_Sums(seq)^\(k+1)).(m+1) by BHSP_3:def 5; hence
_P[m+1];
end;
thus thesis from Ind(A6,A7);
end;
then Partial_Sums(seq)^\(k+1) = Partial_Sums(seq^\(k+1)) + seq1 by NORMSP_1:
def 5
.= Partial_Sums((seq^\k)^\1) + seq1 by BHSP_3:37;
then Partial_Sums(seq)^\(k+1) is convergent by A2,A4,BHSP_2:3;
then Partial_Sums(seq) is convergent by BHSP_3:46;
hence thesis by Def2;
end;
definition
let X, seq, n;
func Sum(seq,n) -> Point of X equals
:Def4: Partial_Sums(seq).n;
correctness;
end;
canceled;
theorem Th16:
Sum(seq, 0) = seq.0
proof
thus Sum(seq, 0) = Partial_Sums(seq).0 by Def4
.= seq.0 by Def1;
end;
theorem Th17:
Sum(seq, 1) = Sum(seq, 0) + seq.1
proof
Partial_Sums(seq).1 = Partial_Sums(seq).0 + seq.(0 + 1) by Def1
.= Partial_Sums(seq).0 + seq.1;
hence Sum(seq, 1) = Partial_Sums(seq).0 + seq.1 by Def4
.= Sum(seq, 0) + seq.1 by Def4;
end;
theorem Th18:
Sum(seq, 1) = seq.0 + seq.1
proof
thus Sum(seq, 1) = Sum(seq, 0) + seq.1 by Th17
.= seq.0 + seq.1 by Th16;
end;
theorem Th19:
Sum(seq, n + 1) = Sum(seq, n) + seq.(n + 1)
proof
thus
Sum(seq, n + 1) = Partial_Sums(seq).(n + 1) by Def4
.= Partial_Sums(seq).n + seq.(n + 1) by Def1
.= Sum(seq, n) + seq.(n + 1) by Def4;
end;
theorem Th20:
seq.(n + 1) = Sum(seq, n + 1) - Sum(seq, n)
proof
thus
Sum(seq, n + 1) - Sum(seq, n) = ( seq.(n + 1) + Sum(seq, n) ) - Sum
(seq, n) by Th19
.= seq.(n + 1) + ( Sum(seq, n) - Sum(seq, n) ) by RLVECT_1:42
.= seq.(n + 1) + 0.X by RLVECT_1:28
.= seq.(n + 1) by RLVECT_1:10;
end;
theorem
seq.1 = Sum(seq, 1) - Sum(seq, 0)
proof
seq.(0 + 1) = Sum(seq, 0 + 1) - Sum(seq, 0) by Th20;
hence thesis;
end;
definition
let X, seq, n, m;
func Sum(seq, n, m) -> Point of X equals
:Def5: Sum(seq, n) - Sum(seq, m);
correctness;
end;
canceled;
theorem
Sum(seq, 1, 0) = seq.1
proof
Sum(seq, 1, 0) = Sum(seq, 1) - Sum(seq, 0) by Def5
.= (seq.0 + seq.1) - Sum(seq, 0) by Th18
.= (seq.1 + seq.0) - seq.0 by Th16
.= seq.1 + (seq.0 - seq.0) by RLVECT_1:42
.= seq.1 + 0'(X) by RLVECT_1:28;
hence thesis by RLVECT_1:10;
end;
theorem
Sum(seq, n+1, n) = seq.(n+1)
proof
Sum(seq, n+1, n) = Sum(seq, n+1) - Sum(seq, n) by Def5;
hence thesis by Th20;
end;
theorem Th25:
X is_Hilbert_space implies
( seq is summable iff ( for r st r > 0 ex k st for n, m st ( n >= k &
m >= k ) holds ||.Sum(seq, n) - Sum(seq, m).|| < r ))
proof
assume
A1: X is_Hilbert_space;
thus
seq is summable implies ( for r st r > 0 ex k st for n, m st ( n >= k &
m >= k ) holds ||.Sum(seq, n) - Sum(seq, m).|| < r )
proof
assume
A2: seq is summable;
now let r;
assume
r > 0;
then consider k such that
A3: for n, m st ( n >= k & m >= k ) holds
||.(Partial_Sums(seq)).n - (Partial_Sums(seq)).m.||
< r by A1,A2,Th10;
take k;
let n, m;
assume n >= k & m >= k;
then ||.(Partial_Sums(seq)).n - (Partial_Sums(seq)).m.|| < r by A3;
then ||.Sum(seq, n) - (Partial_Sums(seq)).m.|| < r by Def4; hence
||.Sum(seq, n) - Sum(seq, m).|| < r by Def4;
end;
hence ( for r st r > 0 ex k st for n, m st ( n >= k &
m >= k ) holds ||.Sum(seq, n) - Sum(seq, m).|| < r );
end;
thus ( for r st r > 0 ex k st for n, m st ( n >= k & m >= k ) holds
||.Sum(seq, n) - Sum(seq, m).|| < r ) implies seq is summable
proof
assume
A4: for r st r > 0 ex k st for n, m st ( n >= k & m >= k ) holds
||.Sum(seq, n) - Sum(seq, m).|| < r;
now let r;
assume
r > 0;
then consider k such that
A5: for n, m st ( n >= k & m >= k ) holds
||.Sum(seq, n) - Sum(seq, m).|| < r by A4;
take k;
let n, m;
assume n >= k & m >= k;
then ||.Sum(seq, n) - Sum(seq, m).|| < r by A5;
then ||.(Partial_Sums(seq)).n - Sum(seq, m).|| < r by Def4; hence
||.(Partial_Sums(seq)).n - (Partial_Sums(seq)).m.|| < r by Def4;
end;
hence seq is summable by A1,Th10;
end;
end;
theorem
X is_Hilbert_space implies
( seq is summable iff ( for r st r > 0 ex k st for n, m st ( n >= k &
m >= k ) holds ||.Sum(seq, n, m).|| < r ))
proof
assume
A1: X is_Hilbert_space;
thus
seq is summable implies ( for r st r > 0 ex k st for n, m st ( n >= k &
m >= k ) holds ||.Sum(seq, n, m).|| < r )
proof
assume
A2: seq is summable;
now let r;
assume r > 0;
then consider k such that
A3: for n, m st ( n >= k & m >= k ) holds
||.Sum(seq, n) - Sum(seq, m).|| < r by A1,A2,Th25;
take k;
let n, m;
assume
n >= k & m >= k;
then ||.Sum(seq, n) - Sum(seq, m).|| < r by A3; hence
||.Sum(seq, n, m).|| < r by Def5;
end;
hence ( for r st r > 0 ex k st for n, m st ( n >= k &
m >= k ) holds ||.Sum(seq, n, m).|| < r );
end;
thus
( for r st r > 0 ex k st for n, m st ( n >= k & m >= k ) holds
||.Sum(seq, n, m).|| < r ) implies seq is summable
proof
assume
A4: for r st r > 0 ex k st for n, m st ( n >= k & m >= k ) holds
||.Sum(seq, n, m).|| < r;
now let r;
assume r > 0;
then consider k such that
A5: for n, m st ( n >= k & m >= k ) holds
||.Sum(seq, n, m).|| < r by A4;
take k;
let n, m;
assume n >= k & m >= k;
then ||.Sum(seq, n, m).|| < r by A5; hence
||.Sum(seq, n) - Sum(seq, m).|| < r by Def5;
end;
hence seq is summable by A1,Th25;
end;
end;
definition
let Rseq, n;
func Sum(Rseq, n) -> Real equals
:Def6: Partial_Sums(Rseq).n;
correctness;
end;
definition
let Rseq, n,m;
func Sum(Rseq, n, m) -> Real equals
:Def7: Sum(Rseq, n) - Sum(Rseq, m);
correctness;
end;
definition
let X, seq;
attr seq is absolutely_summable means
:Def8: ||.seq.|| is summable;
end;
theorem
seq1 is absolutely_summable & seq2 is absolutely_summable implies
seq1 + seq2 is absolutely_summable
proof
assume that
A1: seq1 is absolutely_summable and
A2: seq2 is absolutely_summable;
A3: ||.seq1.|| is summable by A1,Def8;
||.seq2.|| is summable by A2,Def8;
then A4: ||.seq1.|| + ||.seq2.|| is summable by A3,SERIES_1:10;
for n holds ||.seq1 + seq2.||.n >= 0 &
||.seq1 + seq2.||.n <= (||.seq1.|| + ||.seq2.||).n
proof
let n;
thus ||.seq1 + seq2.||.n >= 0
proof
||.seq1 + seq2.||.n = ||.(seq1 + seq2).n.|| by BHSP_2:def 3;
hence thesis by BHSP_1:34;
end;
thus ||.seq1 + seq2.||.n <= (||.seq1.|| + ||.seq2.||).n
proof
||.seq1 + seq2.||.n = ||.(seq1 + seq2).n.|| by BHSP_2:def 3
.= ||.(seq1).n + (seq2).n.|| by NORMSP_1:def 5;
then ||.seq1 + seq2.||.n <= ||.(seq1).n.|| + ||.(seq2).n.||
by BHSP_1:36;
then ||.seq1 + seq2.||.n <= ||.seq1.||.n + ||.(seq2).n.||
by BHSP_2:def 3;
then ||.seq1 + seq2.||.n <= ||.seq1.||.n + ||.seq2.||.n by BHSP_2:def
3;
hence thesis by SEQ_1:11;
end;
end;
then ||.seq1 + seq2.|| is summable by A4,SERIES_1:24;
hence thesis by Def8;
end;
theorem
seq is absolutely_summable implies a * seq is absolutely_summable
proof
assume seq is absolutely_summable;
then ||.seq.|| is summable by Def8;
then A1: abs(a) (#) ||.seq.|| is summable by SERIES_1:13;
for n holds ||.a * seq.||.n >= 0 &
||.a * seq.||.n <= (abs(a) (#) ||.seq.||).n
proof
let n;
thus ||.a * seq.||.n >= 0
proof
||.a * seq.||.n = ||.(a * seq).n.|| by BHSP_2:def 3;
hence thesis by BHSP_1:34;
end;
thus ||.a * seq.||.n <= (abs(a) (#) ||.seq.||).n
proof
||.a * seq.||.n = ||.(a * seq).n.|| by BHSP_2:def 3
.= ||.a * seq.n.|| by NORMSP_1:def 8
.= abs(a) * ||.seq.n.|| by BHSP_1:33
.= abs(a) * ||.seq.||.n by BHSP_2:def 3
.= (abs(a) (#) ||.seq.||).n by SEQ_1:13;
hence thesis;
end;
end;
then ||.a * seq.|| is summable by A1,SERIES_1:24;
hence thesis by Def8;
end;
theorem
( for n holds ||.seq.||.n <= Rseq.n ) & Rseq is summable implies
seq is absolutely_summable
proof
assume that
A1: for n holds ||.seq.||.n <= Rseq.n and
A2: Rseq is summable;
for n holds ||.seq.||.n >= 0
proof
let n;
||.seq.||.n = ||.seq.n.|| by BHSP_2:def 3;
hence thesis by BHSP_1:34;
end;
then ||.seq.|| is summable by A1,A2,SERIES_1:24;
hence thesis by Def8;
end;
theorem
( for n holds seq.n <> 0.X &
Rseq.n = ||.seq.(n+1).||/||.seq.n.|| )
& Rseq is convergent & lim Rseq < 1 implies
seq is absolutely_summable
proof
assume that
A1: for n holds seq.n <> 0'(X) & Rseq.n = ||.seq.(n+1).||/||.seq.n.|| and
A2: Rseq is convergent and
A3: lim Rseq < 1;
for n holds ||.seq.||.n > 0 & Rseq.n = ||.seq.||.(n+1)/||.seq.||.n
proof
let n;
thus ||.seq.||.n > 0
proof
seq.n <> 0'(X) by A1;
then A4: ||.seq.n.|| <> 0 by BHSP_1:32;
||.seq.n.|| >= 0 by BHSP_1:34;
then ||.seq.n.|| > 0 by A4,REAL_1:def 5;
hence thesis by BHSP_2:def 3;
end;
thus Rseq.n = ||.seq.||.(n+1)/||.seq.||.n
proof
Rseq.n = ||.seq.(n+1).||/||.seq.n.|| by A1
.= ||.seq.||.(n+1)/||.seq.n.|| by BHSP_2:def 3;
hence thesis by BHSP_2:def 3;
end;
end;
then ||.seq.|| is summable by A2,A3,SERIES_1:30;
hence thesis by Def8;
end;
theorem Th31:
r > 0 & ( ex m st for n st n >= m holds ||.seq.n.|| >= r) implies
not seq is convergent or lim seq <> 0.X
proof
assume
A1: r > 0;
given m such that
A2: for n st n >= m holds ||.seq.n.|| >= r;
now per cases;
suppose not seq is convergent;
hence thesis;
suppose
A3: seq is convergent;
now assume lim seq = 0'(X);
then consider k such that
A4: for n st n >= k holds ||.seq.n - 0'(X).|| < r by A1,A3,BHSP_2:19;
now let n;
assume
A5: n >= m+k;
m+k >= m by NAT_1:29;
then A6: n >= m by A5,AXIOMS:22;
m+k >= k by NAT_1:29;
then n >= k by A5,AXIOMS:22;
then ||.seq.n - 0'(X).|| < r by A4;
then ||.seq.n.|| < r by RLVECT_1:26;
hence contradiction by A2,A6;
end;
hence contradiction;
end;
hence thesis;
end;
hence thesis;
end;
theorem Th32:
( for n holds seq.n <> 0.X ) &
( ex m st for n st n >= m holds ||.seq.(n+1).||/||.seq.n.|| >= 1 )
implies not seq is summable
proof
assume
A1: for n holds seq.n <> 0'(X);
given m such that
A2: for n st n >= m holds ||.seq.(n+1).||/||.seq.n.|| >= 1;
seq.m <> 0'(X) by A1;
then A3: ||.seq.m.|| <> 0 by BHSP_1:32;
||.seq.m.|| >= 0 by BHSP_1:34;
then A4: ||.seq.m.|| > 0 by A3,REAL_1:def 5;
now let n;
assume
A5: n >= m;
defpred _P[Nat] means ||.seq.(m+$1).|| >= ||.seq.m.||;
A6: _P[0];
A7: for k st _P[k] holds _P[k+1] proof let k;
assume
A8: ||.seq.(m+k).|| >= ||.seq.m.||;
m+k >= m by NAT_1:29;
then A9: ||.seq.(m+k+1).||/||.seq.(m+k).|| >= 1 by A2;
seq.(m+k) <> 0'(X) by A1;
then A10: ||.seq.(m+k).|| <> 0 by BHSP_1:32;
||.seq.(m+k).|| >= 0 by BHSP_1:34;
then ||.seq.(m+k).|| > 0 by A10,REAL_1:def 5;
then ||.seq.(m+k+1).|| >= ||.seq.(m+k).|| by A9,REAL_2:118;
then ||.seq.(m+k+1).|| >= ||.seq.m.|| by A8,AXIOMS:22;
hence thesis by XCMPLX_1:1;
end;
A11: for k holds _P[k] from Ind(A6,A7);
ex k st n = m + k by A5,NAT_1:28;
hence ||.seq.n.|| >= ||.seq.m.|| by A11;
end;
then not seq is convergent or lim seq <> 0'(X) by A4,Th31;
hence thesis by Th9;
end;
theorem
(for n holds seq.n <> 0.X) &
(for n holds Rseq.n = ||.seq.(n+1).||/||.seq.n.||)
& Rseq is convergent & lim Rseq > 1
implies not seq is summable
proof
assume that
A1: for n holds seq.n <> 0'(X) and
A2: for n holds Rseq.n = ||.seq.(n+1).||/||.seq.n.|| and
A3: Rseq is convergent & lim Rseq > 1;
lim Rseq - 1 > 0 by A3,SQUARE_1:11;
then consider m such that
A4: for n st n >= m holds abs(Rseq.n - lim Rseq) < lim Rseq - 1
by A3,SEQ_2:def 7;
now let n;
assume
A5: n >= m + 1;
m + 1 >= m by NAT_1:29;
then A6: n >= m by A5,AXIOMS:22;
Rseq.n = ||.seq.(n+1).||/||.seq.n.|| by A2;
then abs(||.seq.(n+1).||/||.seq.n.|| - lim Rseq) < lim Rseq - 1
by A4,A6;
then - (lim Rseq - 1) < ||.seq.(n+1).||/||.seq.n.|| - lim Rseq
by SEQ_2:9;
then 1 - lim Rseq < ||.seq.(n+1).||/||.seq.n.|| - lim Rseq
by XCMPLX_1:143;
then 1 - lim Rseq + lim Rseq < ||.seq.(n+1).||/||.seq.n.|| - lim Rseq
+
lim Rseq by REAL_1:53;
then 1 < ||.seq.(n+1).||/||.seq.n.||
- lim Rseq + lim Rseq by XCMPLX_1:27;hence
||.seq.(n+1).||/||.seq.n.|| >= 1 by XCMPLX_1:27;
end;
hence thesis by A1,Th32;
end;
theorem
( for n holds Rseq.n = n-root (||.seq.n.||) ) &
Rseq is convergent & lim Rseq < 1 implies
seq is absolutely_summable
proof
assume that
A1: for n holds Rseq.n = n-root (||.seq.n.||) and
A2: Rseq is convergent and
A3: lim Rseq < 1;
for n holds ||.seq.||.n >= 0 & Rseq.n = n-root (||.seq.||.n)
proof
let n;
thus ||.seq.||.n >= 0
proof
||.seq.||.n = ||.seq.n.|| by BHSP_2:def 3;
hence thesis by BHSP_1:34;
end;
thus Rseq.n = n-root (||.seq.||.n)
proof
Rseq.n = n-root (||.seq.n.||) by A1;
hence thesis by BHSP_2:def 3;
end;
end;
then ||.seq.|| is summable by A2,A3,SERIES_1:32;
hence thesis by Def8;
end;
theorem
(for n holds Rseq.n = n-root (||.seq.||.n))
& (ex m st for n st n >= m holds Rseq.n >= 1)
implies not seq is summable
proof
assume
A1: for n holds Rseq.n = n-root (||.seq.||.n);
given m such that
A2: for n st n >= m holds Rseq.n >= 1;
now let n;
assume
A3: n >= m+1;
m+1 >= m by NAT_1:29;
then A4: n>=m by A3,AXIOMS:22;
A5: ||.seq.n.|| >= 0 by BHSP_1:34;
Rseq.n = n-root (||.seq.||.n) by A1
.= n-root ||.seq.n.|| by BHSP_2:def 3;
then n-root ||.seq.n.|| >= 1 by A2,A4;
then A6: n-root ||.seq.n.|| |^ n >= 1 by PREPOWER:19;
m + 1 >= 1 by NAT_1:29;
then n >= 1 by A3,AXIOMS:22;
hence ||.seq.n.|| >= 1 by A5,A6,POWER:5;
end;
then not seq is convergent or lim seq <> 0'(X) by Th31;
hence thesis by Th9;
end;
theorem
(for n holds Rseq.n = n-root (||.seq.||.n))
& Rseq is convergent & lim Rseq > 1
implies not seq is summable
proof
assume that
A1: for n holds Rseq.n = n-root (||.seq.||.n) and
A2: Rseq is convergent & lim Rseq > 1;
lim Rseq - 1 > 0 by A2,SQUARE_1:11;
then consider m such that
A3: for n st n >= m holds abs(Rseq.n - lim Rseq) < lim Rseq - 1
by A2,SEQ_2:def 7;
now let n;
assume
A4: n >= m + 1;
m + 1 >= m by NAT_1:29;
then A5: n >= m by A4,AXIOMS:22;
A6: ||.seq.n.|| >= 0 by BHSP_1:34;
Rseq.n = n-root (||.seq.||.n) by A1
.= n-root ||.seq.n.|| by BHSP_2:def 3;
then abs(n-root ||.seq.n.|| - lim Rseq) < lim Rseq - 1 by A3,A5;
then - (lim Rseq - 1) < n-root ||.seq.n.|| - lim Rseq by SEQ_2:9;
then 1 - lim Rseq < n-root ||.seq.n.|| - lim Rseq by XCMPLX_1:143;
then 1 - lim Rseq + lim Rseq < n-root ||.seq.n.|| - lim Rseq + lim
Rseq
by REAL_1:53;
then 1 < n-root ||.seq.n.|| - lim Rseq + lim Rseq by XCMPLX_1:27;
then n-root ||.seq.n.|| >= 1 by XCMPLX_1:27;
then A7: n-root ||.seq.n.|| |^ n >= 1 by PREPOWER:19;
m + 1 >= 1 by NAT_1:29;
then n >= 1 by A4,AXIOMS:22;
hence ||.seq.n.|| >= 1 by A6,A7,POWER:5;
end;
then not seq is convergent or lim seq <> 0'(X) by Th31;
hence thesis by Th9;
end;
theorem Th37:
Partial_Sums(||.seq.||) is non-decreasing
proof
now let n;
||.seq.(n+1).|| >= 0 by BHSP_1:34;
then ||.seq.||.(n+1) >= 0 by BHSP_2:def 3;
then 0 + Partial_Sums(||.seq.||).n <=
||.seq.||.(n+1) + Partial_Sums(||.seq.||).n by AXIOMS:24;hence
Partial_Sums(||.seq.||).n <= Partial_Sums(||.seq.||).(n+1)
by SERIES_1:def 1;
end;
hence thesis by SEQM_3:def 13;
end;
theorem
for n holds Partial_Sums(||.seq.||).n >= 0
proof
A1: Partial_Sums(||.seq.||) is non-decreasing by Th37;
let n;
A2: Partial_Sums(||.seq.||).0 <= Partial_Sums(||.seq.||).n by A1,SEQM_3:21;
||.(seq.0).|| >= 0 by BHSP_1:34;
then ||.seq.||.0 >= 0 by BHSP_2:def 3;
then Partial_Sums(||.seq.||).0 >= 0 by SERIES_1:def 1;
hence thesis by A2,AXIOMS:22;
end;
theorem Th39:
for n holds ||.Partial_Sums(seq).n.|| <= Partial_Sums(||.seq.||).n
proof
defpred _P[Nat] means ||.Partial_Sums(seq).$1.|| <= Partial_Sums(||.seq.||).$1;
Partial_Sums(||.seq.||).0 = ||.seq.||.0 by SERIES_1:def 1
.= ||.(seq.0).|| by BHSP_2:def 3;
then A1: _P[0] by Def1;
A2: now let n;
assume
A3: _P[n];
Partial_Sums(seq).(n+1) = Partial_Sums(seq).n + seq.(n+1) by Def1;
then A4: ||.Partial_Sums(seq).(n+1).|| <=
||.Partial_Sums(seq).n.|| + ||.seq.(n+1).|| by BHSP_1:36;
||.Partial_Sums(seq).n.|| + ||.seq.(n+1).|| <=
Partial_Sums(||.seq.||).n + ||.seq.(n+1).|| by A3,AXIOMS:24;
then ||.Partial_Sums(seq).(n+1).|| <=
Partial_Sums(||.seq.||).n + ||.seq.(n+1).|| by A4,AXIOMS:22;
then ||.Partial_Sums(seq).(n+1).|| <=
Partial_Sums(||.seq.||).n + ||.seq.||.(n+1) by BHSP_2:def 3; hence
_P[n+1] by SERIES_1:def 1;
end;
thus for n holds _P[n] from Ind(A1,A2);
end;
theorem
for n holds ||.Sum(seq, n).|| <= Sum(||.seq.||, n)
proof
let n;
||.Partial_Sums(seq).n.|| <= Partial_Sums(||.seq.||).n by Th39;
then ||.Sum(seq, n).|| <= Partial_Sums(||.seq.||).n by Def4;
hence thesis by Def6;
end;
theorem Th41:
for n, m holds ||.Partial_Sums(seq).m - Partial_Sums(seq).n.|| <=
abs(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).n)
proof
let n, m;
A1: now assume
A2: n <= m;
set PSseq = Partial_Sums(seq);
set PSseq' = Partial_Sums(||.seq.||);
defpred _P[Nat] means
||.PSseq.(n+$1) - PSseq.n.|| <= abs(PSseq'.(n+$1) - PSseq'.n);
||.PSseq.(n+0) - PSseq.n.|| = ||.0'(X).|| by RLVECT_1:28
.= 0 by BHSP_1:32;
then A3: _P[0] by ABSVALUE:5;
A4: PSseq' is non-decreasing by Th37;
A5: now let k;
assume
A6: _P[k];
||.PSseq.(n+(k+1)) - PSseq.n.|| = ||.PSseq.(n+k+1) - PSseq.n.||
by XCMPLX_1:1
.= ||.seq.(n+k+1) + PSseq.(n+k) - PSseq.n.|| by Def1
.= ||.seq.(n+k+1) + (PSseq.(n+k) - PSseq.n).|| by RLVECT_1:42;
then A7: ||.PSseq.(n+(k+1)) - PSseq.n.|| <= ||.seq.(n+k+1).|| +
||.PSseq.(n+k) - PSseq.n.|| by BHSP_1:36;
||.seq.(n+k+1).|| + ||.PSseq.(n+k) - PSseq.n.|| <=
||.seq.(n+k+1).|| + abs(PSseq'.(n+k) - PSseq'.n) by A6,REAL_1:55;
then A8: ||.PSseq.(n+(k+1)) - PSseq.n.|| <= ||.seq.(n+k+1).|| +
abs(PSseq'.(n+k)-PSseq'.n) by A7,AXIOMS:22;
PSseq'.(n+k) >= PSseq'.n by A4,SEQM_3:11;
then A9: PSseq'.(n+k) - PSseq'.n >= 0 by SQUARE_1:12;
then A10: ||.PSseq.(n+(k+1)) - PSseq.n.|| <= ||.seq.(n+k+1).|| +
(PSseq'.(n+k)-PSseq'.n) by A8,ABSVALUE:def 1;
||.seq.(n+k+1).|| >= 0 by BHSP_1:34;
then A11: ||.seq.(n+k+1).|| + (PSseq'.(n+k) - PSseq'.n) >= 0 + 0 by A9,REAL_1
:55;
abs(PSseq'.(n+(k+1)) - PSseq'.n) =
abs(PSseq'.(n+k+1) - PSseq'.n) by XCMPLX_1:1
.= abs(PSseq'.(n+k) + (||.seq.||).(n+k+1) - PSseq'.n) by SERIES_1:def 1
.= abs(||.seq.(n+k+1).|| + PSseq'.(n+k) - PSseq'.n) by BHSP_2:def 3
.= abs(||.seq.(n+k+1).|| + (PSseq'.(n+k) - PSseq'.n)) by XCMPLX_1:29;
hence _P[k+1] by A10,A11,ABSVALUE:def 1;
end;
A12: for k holds _P[k] from Ind(A3,A5);
reconsider d = n, t = m as Integer;
reconsider k = t - d as Nat by A2,INT_1:18;
n + k = m by XCMPLX_1:27;
hence thesis by A12;
end;
now assume
A13: n >= m;
set PSseq = Partial_Sums(seq);
set PSseq' = Partial_Sums(||.seq.||);
defpred _P[Nat] means
||.PSseq.m - PSseq.(m+$1).|| <= abs(PSseq'.m - PSseq'.(m+$1));
||.PSseq.m - PSseq.(m+0).|| = ||.0'(X).|| by RLVECT_1:28
.= 0 by BHSP_1:32;
then A14: _P[0] by ABSVALUE:5;
A15: PSseq' is non-decreasing by Th37;
A16: now let k;
assume
A17: _P[k];
||.PSseq.m - PSseq.(m+(k+1)).|| = ||.PSseq.m - PSseq.(m+k+1).||
by XCMPLX_1:1
.= ||.PSseq.m - (PSseq.(m+k) + seq.(m+k+1)).|| by Def1
.= ||.(PSseq.m - PSseq.(m+k)) - seq.(m+k+1).|| by RLVECT_1:41
.= ||.(PSseq.m - PSseq.(m+k)) + -seq.(m+k+1).|| by RLVECT_1:def 11;
then ||.PSseq.m - PSseq.(m+(k+1)).|| <= ||.PSseq.m - PSseq.(m+k).|| +
||.-seq.(m+k+1).|| by BHSP_1:36;
then A18: ||.PSseq.m - PSseq.(m+(k+1)).|| <= ||.PSseq.m - PSseq.(m+k).|| +
||.seq.(m+k+1).|| by BHSP_1:37;
||.PSseq.m - PSseq.(m+k).|| + ||.seq.(m+k+1).|| <=
abs(PSseq'.m - PSseq'.(m+k)) + ||.seq.(m+k+1).|| by A17,AXIOMS:24;
then ||.PSseq.m - PSseq.(m+(k+1)).|| <=
abs(PSseq'.m - PSseq'.(m+k)) + ||.seq.(m+k+1).|| by A18,AXIOMS:22;
then ||.PSseq.m - PSseq.(m+(k+1)).|| <=
abs(-(PSseq'.(m+k) - PSseq'.m)) + ||.seq.(m+k+1).|| by XCMPLX_1:143;
then A19: ||.PSseq.m - PSseq.(m+(k+1)).|| <=
abs(PSseq'.(m+k) - PSseq'.m) + ||.seq.(m+k+1).|| by ABSVALUE:17;
PSseq'.(m+k) >= PSseq'.m by A15,SEQM_3:11;
then A20: PSseq'.(m+k) - PSseq'.m >= 0 by SQUARE_1:12;
then A21: ||.PSseq.m - PSseq.(m+(k+1)).|| <=
(PSseq'.(m+k) - PSseq'.m) + ||.seq.(m+k+1).|| by A19,ABSVALUE:def 1;
||.seq.(m+k+1).|| >= 0 by BHSP_1:34;
then A22: (PSseq'.(m+k) - PSseq'.m) + ||.seq.(m+k+1).|| >= 0 + 0 by A20,
REAL_1:55;
abs(PSseq'.m - PSseq'.(m+(k+1)))
= abs(-(PSseq'.(m+(k+1)) - PSseq'.m)) by XCMPLX_1:143
.= abs(PSseq'.(m+(k+1)) - PSseq'.m) by ABSVALUE:17
.= abs(PSseq'.(m+k+1) - PSseq'.m) by XCMPLX_1:1
.= abs(PSseq'.(m+k) + (||.seq.||).(m+k+1) - PSseq'.m) by SERIES_1:def 1
.= abs(||.seq.(m+k+1).|| + PSseq'.(m+k) - PSseq'.m) by BHSP_2:def 3
.= abs((PSseq'.(m+k) - PSseq'.m) + ||.seq.(m+k+1).||) by XCMPLX_1:29;
hence _P[k+1] by A21,A22,ABSVALUE:def 1;
end;
A23: for k holds _P[k] from Ind(A14,A16);
reconsider d = n, t = m as Integer;
reconsider k = d - t as Nat by A13,INT_1:18;
m + k = n by XCMPLX_1:27;
hence thesis by A23;
end;
hence thesis by A1;
end;
theorem Th42:
for n, m holds ||.Sum(seq, m) - Sum(seq, n).|| <=
abs(Sum(||.seq.||, m) - Sum(||.seq.||, n))
proof
let n, m;
||.Partial_Sums(seq).m - Partial_Sums(seq).n.|| <=
abs(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).n) by Th41;
then ||.Sum(seq, m) - Partial_Sums(seq).n.|| <=
abs(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).n) by Def4;
then ||.Sum(seq, m) - Sum(seq, n).|| <=
abs(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).n) by Def4;
then ||.Sum(seq, m) - Sum(seq, n).|| <=
abs(Sum(||.seq.||, m) - Partial_Sums(||.seq.||).n) by Def6;
hence thesis by Def6;
end;
theorem
for n, m holds ||.Sum(seq, m, n).|| <= abs(Sum(||.seq.||, m, n))
proof
let n, m;
||.Sum(seq, m) - Sum(seq, n).|| <=
abs(Sum(||.seq.||, m) - Sum(||.seq.||, n)) by Th42;
then ||.Sum(seq, m, n).|| <=
abs(Sum(||.seq.||, m) - Sum(||.seq.||, n)) by Def5;
hence thesis by Def7;
end;
theorem
X is_Hilbert_space implies
( seq is absolutely_summable implies seq is summable )
proof
assume that
A1: X is_Hilbert_space and
A2: seq is absolutely_summable;
A3: ||.seq.|| is summable by A2,Def8;
now let r;
assume r > 0;
then r/2 > 0 by SEQ_2:3;
then consider k such that
A4: for m st m >= k holds
abs(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).k) < r/2
by A3,SERIES_1:25;
take k;
now let m, n;
assume
A5: m >= k & n >= k;
then A6: abs(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).k) < r/2 by
A4;
A7: abs(Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||
).k) < r/2 by A4,A5;
abs(Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).m) =
abs((Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k) -
(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||
).k)) by XCMPLX_1:22;
then A8: abs(Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).m) <=
abs(Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k) +
abs(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).k) by ABSVALUE:19;
abs(Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k) +
abs(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).k) < r/2 + r/2
by A6,A7,REAL_1:67;
then abs(Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k) +
abs(Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).k) < r
by XCMPLX_1:66;
then A9: abs(Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).m) < r
by A8,AXIOMS:22;
||.Partial_Sums(seq).n - Partial_Sums(seq).m.|| <=
abs(Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).m) by Th41; hence
||.Partial_Sums(seq).n - Partial_Sums(seq).m.|| < r by A9,AXIOMS:22;
end;
hence for n, m st ( n >= k & m >= k ) holds
||.Partial_Sums(seq).n - Partial_Sums(seq).m.|| < r;
end;
hence thesis by A1,Th10;
end;
definition
let X, seq, Rseq;
func Rseq * seq -> sequence of X means
:Def9: for n holds it.n = Rseq.n * seq.n;
existence proof
deffunc _F(Nat) = Rseq.$1 * seq.$1;
thus ex M being sequence of X st for n holds M.n = _F(n) from Ex_Seq_in_RUS;
end;
uniqueness
proof
let seq1, seq2;
assume that
A1: for n holds seq1.n = Rseq.n * seq.n and
A2: for n holds seq2.n = Rseq.n * seq.n;
now let n;
seq1.n = Rseq.n * seq.n by A1;
hence seq1.n = seq2.n by A2;
end;
hence seq1 = seq2 by FUNCT_2:113;
end;
end;
theorem
Rseq * (seq1 + seq2) = Rseq * seq1 + Rseq * seq2
proof
now let n;
thus (Rseq * (seq1 + seq2)).n = Rseq.n * (seq1 + seq2).n by Def9
.= Rseq.n * (seq1.n + seq2.n) by NORMSP_1:def 5
.= Rseq.n * seq1.n + Rseq.n * seq2.n by RLVECT_1:def 9
.= (Rseq * seq1).n + Rseq.n * seq2.n by Def9
.= (Rseq * seq1).n + (Rseq * seq2).n by Def9
.= (Rseq * seq1 + Rseq * seq2).n by NORMSP_1:def 5;
end;
hence thesis by FUNCT_2:113;
end;
theorem
(Rseq1 + Rseq2) * seq = Rseq1 * seq + Rseq2 * seq
proof
now let n;
thus ((Rseq1 + Rseq2) * seq).n = (Rseq1 + Rseq2).n * seq.n by Def9
.= (Rseq1.n + Rseq2.n) * seq.n by SEQ_1:11
.= Rseq1.n * seq.n + Rseq2.n * seq.n by RLVECT_1:def 9
.= (Rseq1 * seq).n + Rseq2.n * seq.n by Def9
.= (Rseq1 * seq).n + (Rseq2 * seq).n by Def9
.= (Rseq1 * seq + Rseq2 * seq).n by NORMSP_1:def 5;
end;
hence thesis by FUNCT_2:113;
end;
theorem
(Rseq1 (#) Rseq2) * seq = Rseq1 * (Rseq2 * seq)
proof
now let n;
thus ((Rseq1 (#) Rseq2) * seq).n = (Rseq1 (#) Rseq2).n * seq.n by Def9
.= (Rseq1.n * Rseq2.n) * seq.n by SEQ_1:12
.= Rseq1.n * (Rseq2.n * seq.n) by RLVECT_1:def 9
.= Rseq1.n * (Rseq2 * seq).n by Def9
.= (Rseq1 * (Rseq2 * seq)).n by Def9;
end;
hence thesis by FUNCT_2:113;
end;
theorem Th48:
(a (#) Rseq) * seq = a * (Rseq * seq)
proof
now let n;
thus ((a (#) Rseq) * seq).n = (a (#) Rseq).n * seq.n by Def9
.= (a * Rseq.n) * seq.n by SEQ_1:13
.= a * (Rseq.n * seq.n) by RLVECT_1:def 9
.= a * (Rseq * seq).n by Def9
.= (a * (Rseq * seq)).n by NORMSP_1:def 8;
end;
hence thesis by FUNCT_2:113;
end;
theorem
Rseq * (- seq) = (- Rseq) * seq
proof
now let n;
thus (Rseq * (- seq)).n = Rseq.n * (-seq).n by Def9
.= Rseq.n * (-(seq.n)) by BHSP_1:def 10
.= (-(Rseq.n)) * seq.n by RLVECT_1:38
.= (- Rseq).n * seq.n by SEQ_1:14
.= ((- Rseq) * seq).n by Def9;
end;
hence thesis by FUNCT_2:113;
end;
theorem Th50:
Rseq is convergent & seq is convergent
implies Rseq * seq is convergent
proof
assume that
A1: Rseq is convergent and
A2: seq is convergent;
consider p being real number such that
A3: for r being real number st r > 0 ex m st for n st n >= m holds
abs(Rseq.n - p) < r by A1,SEQ_2:def 6;
reconsider p as Real by XREAL_0:def 1;
consider g such that
A4: for r st r > 0 ex m st for n st n >= m holds
||.seq.n - g.|| < r by A2,BHSP_2:9;
now take h = p * g;
Rseq is bounded by A1,SEQ_2:27;
then consider b being real number such that
A5: b > 0 and
A6: for n holds abs(Rseq.n) < b by SEQ_2:15;
reconsider b as Real by XREAL_0:def 1;
A7: ||.g.|| >= 0 by BHSP_1:34;
then A8: b + ||.g.|| > 0 + 0 by A5,REAL_1:67;
let r;
assume r > 0;
then A9: r/(b + ||.g.||) > 0 by A8,REAL_2:127;
then consider m1 such that
A10: for n st n >= m1 holds abs(Rseq.n - p) < r/(b + ||.g.||) by A3;
consider m2 such that
A11: for n st n >= m2 holds ||.seq.n - g.|| < r/(b + ||.g.||) by A4,A9;
take m = m1 + m2;
let n such that
A12: n >= m;
m1 + m2 >= m1 by NAT_1:37;
then n >= m1 by A12,AXIOMS:22;
then A13: abs(Rseq.n - p) <= r/(b + ||.g.||) by A10;
A14: abs(Rseq.n) < b by A6;
m >= m2 by NAT_1:37;
then n >= m2 by A12,AXIOMS:22;
then A15: ||.seq.n - g.|| < r/(b + ||.g.||) by A11;
||.(Rseq * seq).n - p * g.|| = ||.Rseq.n * seq.n - p * g.|| by Def9
.= ||.(Rseq.n * seq.n - p * g) + 0'(X).|| by RLVECT_1:10
.= ||.(Rseq.n * seq.n - p * g) + (Rseq.n * g - Rseq.n * g).||
by RLVECT_1:28
.= ||.Rseq.n * seq.n - (p * g - (Rseq.n * g - Rseq.n * g)).||
by RLVECT_1:43
.= ||.Rseq.n * seq.n - (Rseq.n * g + (p * g - Rseq.n * g)).|| by RLVECT_1:43
.= ||.(Rseq.n * seq.n - Rseq.n * g) - (p * g - Rseq.n * g).||
by RLVECT_1:41
.= ||.(Rseq.n * seq.n - Rseq.n * g) + -(p * g - Rseq.n * g).||
by RLVECT_1:def 11
.= ||.(Rseq.n * seq.n - Rseq.n * g) + (Rseq.n * g + - p * g).||
by RLVECT_1:47
.= ||.(Rseq.n * seq.n - Rseq.n * g) + (Rseq.n * g - p * g).||
by RLVECT_1:def 11;
then ||.(Rseq * seq).n - p * g.|| <= ||.Rseq.n * seq.n - Rseq.n * g.|| +
||.Rseq.n * g - p * g.|| by BHSP_1:36;
then ||.(Rseq * seq).n - p * g.|| <= ||.Rseq.n * (seq.n - g).|| +
||.Rseq.n * g - p * g.|| by RLVECT_1:48;
then ||.(Rseq * seq).n - p * g.|| <= ||.Rseq.n * (seq.n - g).|| +
||.(Rseq.n - p) * g.|| by RLVECT_1:49;
then ||.(Rseq * seq).n - p * g.|| <= abs(Rseq.n) * ||.seq.n - g.|| +
||.(Rseq.n - p) * g.|| by BHSP_1:33;
then A16: ||.(Rseq * seq).n - p * g.|| <= abs(Rseq.n) * ||.seq.n - g.|| +
||.g.|| * abs(Rseq.n - p) by BHSP_1:33;
A17: abs(Rseq.n) >= 0 by ABSVALUE:5;
||.seq.n - g.|| >= 0 by BHSP_1:34;
then A18: abs(Rseq.n) * ||.seq.n - g.|| < b * (r/(b + ||.g.||))
by A14,A15,A17,SEQ_2:7;
||.g.|| * abs(Rseq.n - p) <= ||.g.|| * (r/(b + ||.g.||))
by A7,A13,AXIOMS:25;
then abs(Rseq.n) * ||.seq.n - g.|| + ||.g.|| * abs(Rseq.n - p) <
b * (r/(b + ||.g.||)) + ||.g.|| * (r/(b + ||.g.||)) by A18,REAL_1:67;
then abs(Rseq.n) * ||.seq.n - g.|| + ||.g.|| * abs(Rseq.n - p) <
(b * r)/(b + ||.g.||) + ||.g.|| * (r/(b + ||.g.||)) by XCMPLX_1:75;
then abs(Rseq.n) * ||.seq.n - g.|| + ||.g.|| * abs(Rseq.n - p) <
(b * r)/(b + ||.g.||) + (||.g.|| * r)/(b + ||.g.||) by XCMPLX_1:75;
then abs(Rseq.n) * ||.seq.n - g.|| + ||.g.|| * abs(Rseq.n - p) <
(b * r + ||.g.|| * r)/(b + ||.g.||) by XCMPLX_1:63;
then abs(Rseq.n) * ||.seq.n - g.|| + ||.g.|| * abs(Rseq.n - p) <
((b + ||.g.||) * r)/(b + ||.g.||) by XCMPLX_1:8;
then abs(Rseq.n) * ||.seq.n - g.|| + ||.g.|| * abs(Rseq.n - p) < r
by A8,XCMPLX_1:90;
hence ||.(Rseq * seq).n - h.|| < r by A16,AXIOMS:22;
end;
hence thesis by BHSP_2:9;
end;
theorem
Rseq is bounded & seq is bounded implies Rseq * seq is bounded
proof
assume that
A1: Rseq is bounded and
A2: seq is bounded;
consider M1 being real number such that
A3: M1 > 0 and
A4: for n holds abs(Rseq.n) < M1 by A1,SEQ_2:15;
consider M2 such that
A5: M2 > 0 and
A6: for n holds ||.seq.n.|| <= M2 by A2,BHSP_3:def 3;
now
reconsider M = M1 * M2 as Real by XREAL_0:def 1;
take M;
now let n;
A7: abs(Rseq.n) <= M1 by A4;
A8: ||.seq.n.|| <= M2 by A6;
abs(Rseq.n) >= 0 by ABSVALUE:5;
then A9: abs(Rseq.n) * ||.seq.n.|| <= abs(Rseq.n) * M2 by A8,AXIOMS:25;
A10: abs(Rseq.n) * M2 <= M1 * M2 by A5,A7,AXIOMS:25;
||.(Rseq * seq).n.|| = ||.Rseq.n * seq.n.|| by Def9
.= abs(Rseq.n) * ||.seq.n.|| by BHSP_1:33;
hence ||.(Rseq * seq).n.|| <= M by A9,A10,AXIOMS:22;
end;
hence M > 0 & for n holds ||.(Rseq * seq).n.|| <= M
by A3,A5,REAL_2:122;
end;
hence thesis by BHSP_3:def 3;
end;
theorem
Rseq is convergent & seq is convergent implies
Rseq * seq is convergent & lim (Rseq * seq) = lim Rseq * lim seq
proof
assume that
A1: Rseq is convergent and
A2: seq is convergent;
A3: Rseq * seq is convergent by A1,A2,Th50;
Rseq is bounded by A1,SEQ_2:27;
then consider b being real number such that
A4: b > 0 and
A5: for n holds abs(Rseq.n) < b by SEQ_2:15;
reconsider b as Real by XREAL_0:def 1;
A6: ||.lim seq.|| >= 0 by BHSP_1:34;
then A7: b + ||.lim seq.|| > 0 + 0 by A4,REAL_1:67;
now let r;
assume
r > 0;
then A8: r/(b + ||.lim seq.||) > 0 by A7,REAL_2:127;
then consider m1 such that
A9: for n st n >= m1 holds
abs(Rseq.n - lim Rseq) < r/(b + ||.lim seq.||) by A1,SEQ_2:def 7;
consider m2 such that
A10: for n st n >= m2 holds
dist(seq.n, lim seq) < r/(b + ||.lim seq.||) by A2,A8,BHSP_2:def 2;
take m = m1 + m2;
let n such that
A11: n >= m;
m1 + m2 >= m1 by NAT_1:37;
then n >= m1 by A11,AXIOMS:22;
then A12: abs(Rseq.n - lim Rseq) <= r/(b + ||.lim seq.||) by A9;
A13: abs(Rseq.n) < b by A5;
m >= m2 by NAT_1:37;
then n >= m2 by A11,AXIOMS:22;
then dist(seq.n, lim seq) < r/(b + ||.lim seq.||) by A10;
then A14: ||.seq.n - lim seq.|| < r/(b + ||.lim seq.||) by BHSP_1:def 5;
||.(Rseq * seq).n - lim Rseq * lim seq.||
= ||.Rseq.n * seq.n - lim Rseq * lim seq.|| by Def9
.= ||.(Rseq.n * seq.n - lim Rseq * lim seq) + 0'(X).|| by RLVECT_1:10
.= ||.(Rseq.n * seq.n - lim Rseq * lim seq) +
(Rseq.n * lim seq - Rseq.n * lim seq).|| by RLVECT_1:28
.= ||.Rseq.n * seq.n - (lim Rseq * lim seq -
(Rseq.n * lim seq - Rseq.n * lim seq)).|| by RLVECT_1:43
.= ||.Rseq.n * seq.n - (Rseq.n * lim seq +
(lim Rseq * lim seq - Rseq.n * lim seq)).|| by RLVECT_1:43
.= ||.(Rseq.n * seq.n - Rseq.n * lim seq) -
(lim Rseq * lim seq - Rseq.n * lim seq).|| by RLVECT_1:41
.= ||.(Rseq.n * seq.n - Rseq.n * lim seq) +
-(lim Rseq * lim seq - Rseq.n * lim seq).|| by RLVECT_1:def 11
.= ||.(Rseq.n * seq.n - Rseq.n * lim seq) +
(Rseq.n * lim seq + - lim Rseq * lim seq).|| by RLVECT_1:47
.= ||.(Rseq.n * seq.n - Rseq.n * lim seq) +
(Rseq.n * lim seq - lim Rseq * lim seq).|| by RLVECT_1:def 11;
then ||.(Rseq * seq).n - lim Rseq * lim seq.|| <=
||.Rseq.n * seq.n - Rseq.n * lim seq.|| +
||.Rseq.n * lim seq - lim Rseq * lim seq.|| by BHSP_1:36;
then ||.(Rseq * seq).n - lim Rseq * lim seq.|| <=
||.Rseq.n * (seq.n - lim seq).|| +
||.Rseq.n * lim seq - lim Rseq * lim seq.|| by RLVECT_1:48;
then ||.(Rseq * seq).n - lim Rseq * lim seq.|| <=
||.Rseq.n * (seq.n - lim seq).|| +
||.(Rseq.n - lim Rseq) * lim seq.|| by RLVECT_1:49;
then ||.(Rseq * seq).n - lim Rseq * lim seq.|| <=
abs(Rseq.n) * ||.seq.n - lim seq.|| +
||.(Rseq.n - lim Rseq) * lim seq.|| by BHSP_1:33;
then A15: ||.(Rseq * seq).n - lim Rseq * lim seq.|| <=
abs(Rseq.n) * ||.seq.n - lim seq.|| +
||.lim seq.|| * abs(Rseq.n - lim Rseq) by BHSP_1:33;
A16: abs(Rseq.n) >= 0 by ABSVALUE:5;
||.seq.n - lim seq.|| >= 0 by BHSP_1:34;
then A17: abs(Rseq.n) * ||.seq.n - lim seq.|| < b * (r/(b + ||.lim seq.||))
by A13,A14,A16,SEQ_2:7;
||.lim seq.|| * abs(Rseq.n - lim Rseq) <= ||.lim seq.|| *
(r/(b + ||.lim seq.||)) by A6,A12,AXIOMS:25;
then abs(Rseq.n) * ||.seq.n - lim seq.|| +
||.lim seq.|| * abs(Rseq.n - lim Rseq) <
b * (r/(b + ||.lim seq.||)) +
||.lim seq.|| * (r/(b + ||.lim seq.||)) by A17,REAL_1:67;
then abs(Rseq.n) * ||.seq.n - lim seq.|| +
||.lim seq.|| * abs(Rseq.n - lim Rseq) <
(b * r)/(b + ||.lim seq.||) +
||.lim seq.|| * (r/(b + ||.lim seq.||)) by XCMPLX_1:75;
then abs(Rseq.n) * ||.seq.n - lim seq.|| +
||.lim seq.|| * abs(Rseq.n - lim Rseq) <
(b * r)/(b + ||.lim seq.||) +
(||.lim seq.|| * r)/(b + ||.lim seq.||) by XCMPLX_1:75;
then abs(Rseq.n) * ||.seq.n - lim seq.|| +
||.lim seq.|| * abs(Rseq.n - lim Rseq) <
(b * r + ||.lim seq.|| * r)/(b + ||.lim seq.||) by XCMPLX_1:63;
then abs(Rseq.n) * ||.seq.n - lim seq.|| +
||.lim seq.|| * abs(Rseq.n - lim Rseq) <
((b + ||.lim seq.||) * r)/(b + ||.lim seq.||) by XCMPLX_1:8;
then abs(Rseq.n) * ||.seq.n - lim seq.|| +
||.lim seq.|| * abs(Rseq.n - lim Rseq) < r
by A7,XCMPLX_1:90;
then ||.(Rseq * seq).n - lim Rseq * lim seq.|| < r by A15,AXIOMS:22;
hence dist((Rseq * seq).n, (lim Rseq * lim seq)) < r by BHSP_1:def 5;
end;
hence thesis by A3,BHSP_2:def 2;
end;
definition
let Rseq;
attr Rseq is Cauchy means
:Def10: for r st r > 0 ex k st for n, m st ( n >= k & m >= k )
holds abs((Rseq.n - Rseq.m)) < r;
synonym Rseq is_Cauchy_sequence;
end;
theorem
X is_Hilbert_space implies
( seq is_Cauchy_sequence & Rseq is_Cauchy_sequence implies
Rseq * seq is_Cauchy_sequence )
proof
assume that
A1: X is_Hilbert_space and
A2: seq is_Cauchy_sequence and
A3: Rseq is_Cauchy_sequence;
X is_complete_space by A1,BHSP_3:def 7;
then A4: seq is convergent by A2,BHSP_3:def 6;
now let r be real number;
A5: r is Real by XREAL_0:def 1;
assume
r > 0;
then consider k such that
A6: for n, m st ( n >= k & m >= k ) holds
abs((Rseq.n - Rseq.m)) < r by A3,A5,Def10;
take k;
thus for n st n >= k holds
abs((Rseq.n - Rseq.k)) < r by A6;
end;
then Rseq is convergent by SEQ_4:58;
then Rseq * seq is convergent by A4,Th50;
hence thesis by BHSP_3:9;
end;
theorem Th54:
for n holds
Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
= Partial_Sums(Rseq * seq).(n+1)
- (Rseq * Partial_Sums(seq)).(n+1)
proof
defpred _P[Nat] means Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).$1
= Partial_Sums(Rseq * seq).($1+1)
- (Rseq * Partial_Sums(seq)).($1+1);
A1: Partial_Sums(Rseq * seq).(0+1)
= Partial_Sums(Rseq * seq).0 + (Rseq * seq).(0+1) by Def1
.= (Rseq * seq).0 + (Rseq * seq).1 by Def1
.= Rseq.0 * seq.0 + (Rseq * seq).1 by Def9
.= Rseq.0 * seq.0 + Rseq.1 * seq.1 by Def9;
A2: Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).0
= ((Rseq - Rseq^\1) * Partial_Sums(seq)).0 by Def1
.= (Rseq - Rseq^\1).0 * Partial_Sums(seq).0 by Def9
.= (Rseq - Rseq^\1).0 * seq.0 by Def1
.= (Rseq + -Rseq^\1).0 * seq.0 by SEQ_1:15
.= (Rseq.0 + (-Rseq^\1).0) * seq.0 by SEQ_1:11
.= (Rseq.0 + -(Rseq^\1).0) * seq.0 by SEQ_1:14
.= (Rseq.0 - (Rseq^\1).0) * seq.0 by XCMPLX_0:def 8
.= (Rseq.0 - Rseq.(0+1)) * seq.0 by SEQM_3:def 9
.= Rseq.0 * seq.0 - Rseq.1 * seq.0 by RLVECT_1:49;
A3: (Rseq * Partial_Sums(seq)).(0+1)
= Rseq.(0+1) * Partial_Sums(seq).(0+1) by Def9
.= Rseq.(0+1) * (Partial_Sums(seq).0 + seq.(0+1)) by Def1
.= Rseq.1 * (seq.0 + seq.1) by Def1
.= Rseq.1 * seq.0 + Rseq.1 * seq.1 by RLVECT_1:def 9;
Partial_Sums(Rseq * seq).(0+1)
= (Rseq.0 * seq.0 + 0'(X)) + Rseq.1 * seq.1 by A1,RLVECT_1:10
.= (Rseq.0 * seq.0 + (Rseq.1 * seq.0 - Rseq.1 * seq.0))
+ Rseq.1 * seq.1 by RLVECT_1:28
.= (Rseq.0 * seq.0 + (-(Rseq.1 * seq.0) + Rseq.1 * seq.0))
+ Rseq.1 * seq.1 by RLVECT_1:def 11
.= ((Rseq.0 * seq.0 + -(Rseq.1 * seq.0)) + Rseq.1 * seq.0)
+ Rseq.1 * seq.1 by RLVECT_1:def 6
.= ((Rseq.0 * seq.0 - Rseq.1 * seq.0) + Rseq.1 * seq.0)
+ Rseq.1 * seq.1 by RLVECT_1:def 11
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).0 +
(Rseq * Partial_Sums(seq)).(0+1) by A2,A3,RLVECT_1:def 6;
then Partial_Sums(Rseq * seq).(0+1) - (Rseq * Partial_Sums(seq)).(0+1)
= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).0 +
((Rseq * Partial_Sums(seq)).(0+1) - (Rseq * Partial_Sums(seq)).(0+1))
by RLVECT_1:42;
then Partial_Sums(Rseq * seq).(0+1) - (Rseq * Partial_Sums(seq)).(0+1)
= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).0 + 0'(X)
by RLVECT_1:28;
then A4: _P[0] by RLVECT_1:10;
A5: now let n;
assume _P[n];
then Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n +
(Rseq * Partial_Sums(seq)).(n+1) =
Partial_Sums(Rseq * seq).(n+1) - ((Rseq * Partial_Sums(seq)).(n+1)
- (Rseq * Partial_Sums(seq)).(n+1)) by RLVECT_1:43;
then A6: Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n +
(Rseq * Partial_Sums(seq)).(n+1) =
Partial_Sums(Rseq * seq).(n+1) - 0'(X) by RLVECT_1:28;
Partial_Sums(Rseq * seq).((n+1)+1)
= Partial_Sums(Rseq * seq).(n+1) + (Rseq * seq).((n+1)+1) by Def1
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (Rseq * Partial_Sums(seq)).(n+1)
+ (Rseq * seq).((n+1)+1) by A6,RLVECT_1:26
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ ((Rseq * Partial_Sums(seq)).(n+1)
+ (Rseq * seq).((n+1)+1)) by RLVECT_1:def 6
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (Rseq.(n+1) * Partial_Sums(seq).(n+1)
+ (Rseq * seq).((n+1)+1)) by Def9
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (Rseq.(n+1) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by Def9
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (((Rseq.(n+1) - Rseq.((n+1)+1)) + Rseq.((n+1)+1))
* Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by XCMPLX_1:27
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (((Rseq.(n+1) - Rseq.((n+1)+1)) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * Partial_Sums(seq).(n+1))
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by RLVECT_1:def 9
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (((Rseq.(n+1) - (Rseq^\1).(n+1)) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * Partial_Sums(seq).(n+1))
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by SEQM_3:def 9
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (((Rseq.(n+1) + -(Rseq^\1).(n+1)) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * Partial_Sums(seq).(n+1))
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by XCMPLX_0:def 8
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (((Rseq.(n+1) + (-Rseq^\1).(n+1)) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * Partial_Sums(seq).(n+1))
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by SEQ_1:14
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (((Rseq + -(Rseq^\1)).(n+1) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * Partial_Sums(seq).(n+1))
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by SEQ_1:11
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (((Rseq - (Rseq^\1)).(n+1) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * Partial_Sums(seq).(n+1))
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by SEQ_1:15
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ ((Rseq - (Rseq^\1)).(n+1) * Partial_Sums(seq).(n+1)
+ (Rseq.((n+1)+1) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * seq.((n+1)+1))) by RLVECT_1:def 6
.= (Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ (Rseq - (Rseq^\1)).(n+1) * Partial_Sums(seq).(n+1))
+ (Rseq.((n+1)+1) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by RLVECT_1:def 6
.= (Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n
+ ((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1))
+ (Rseq.((n+1)+1) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by Def9
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1)
+ (Rseq.((n+1)+1) * Partial_Sums(seq).(n+1)
+ Rseq.((n+1)+1) * seq.((n+1)+1)) by Def1
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1)
+ (Rseq.((n+1)+1) *
(Partial_Sums(seq).(n+1) + seq.((n+1)+1))) by RLVECT_1:def 9
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1)
+ (Rseq.((n+1)+1) * Partial_Sums(seq).((n+1)+1)) by Def1
.= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1)
+ (Rseq * Partial_Sums(seq)).((n+1)+1) by Def9;
then Partial_Sums(Rseq * seq).((n+1)+1) - (Rseq * Partial_Sums(seq)).((n+1)
+1)
= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1)
+ ((Rseq * Partial_Sums(seq)).((n+1)+1) -
(Rseq * Partial_Sums(seq)).((n+1)+1)) by RLVECT_1:42;
then Partial_Sums(Rseq * seq).((n+1)+1) - (Rseq * Partial_Sums(seq)).((n+1)
+1)
= Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).(n+1)
+ 0'(X) by RLVECT_1:28;
hence _P[n+1] by RLVECT_1:10;
end;
thus for n holds _P[n] from Ind(A4,A5);
end;
theorem Th55:
for n holds
Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) -
Partial_Sums((Rseq^\1 - Rseq) * Partial_Sums(seq)).n
proof
let n;
Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n +
(Rseq * Partial_Sums(seq)).(n+1)
= (Partial_Sums(Rseq * seq).(n+1)
- (Rseq * Partial_Sums(seq)).(n+1))
+ (Rseq * Partial_Sums(seq)).(n+1) by Th54;
then Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n +
(Rseq * Partial_Sums(seq)).(n+1)
= Partial_Sums(Rseq * seq).(n+1)
- ((Rseq * Partial_Sums(seq)).(n+1)
- (Rseq * Partial_Sums(seq)).(n+1)) by RLVECT_1:43;
then Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n +
(Rseq * Partial_Sums(seq)).(n+1)
= Partial_Sums(Rseq * seq).(n+1) - 0'(X) by RLVECT_1:28;
then Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) +
Partial_Sums((Rseq - Rseq^\1) * Partial_Sums(seq)).n by RLVECT_1:26;
then Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) +
Partial_Sums((Rseq + -(Rseq^\1)) * Partial_Sums(seq)).n by SEQ_1:15;
then Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) +
Partial_Sums((-(Rseq^\1) - -Rseq) * Partial_Sums(seq)).n by SEQ_1:37;
then Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) +
Partial_Sums(((-1) (#) (Rseq^\1) - -Rseq) * Partial_Sums(seq)).n
by SEQ_1:25;
then Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) +
Partial_Sums(((-1) (#) (Rseq^\1) - (-1) (#) Rseq) * Partial_Sums(seq)).n
by SEQ_1:25;
then Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) +
Partial_Sums(((-1) (#) (Rseq^\1 - Rseq)) * Partial_Sums(seq)).n
by SEQ_1:32;
then Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) +
Partial_Sums((-1) * ((Rseq^\1 - Rseq) * Partial_Sums(seq))).n
by Th48;
then Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) +
((-1) * Partial_Sums((Rseq^\1 - Rseq) * Partial_Sums(seq))).n
by Th3;
then Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) +
(-1) * Partial_Sums((Rseq^\1 - Rseq) * Partial_Sums(seq)).n
by NORMSP_1:def 8;
then Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) +
- Partial_Sums((Rseq^\1 - Rseq) * Partial_Sums(seq)).n
by RLVECT_1:29;
hence thesis by RLVECT_1:def 11;
end;
theorem
for n holds
Sum(Rseq * seq, n+1) =
(Rseq * Partial_Sums(seq)).(n+1) -
Sum((Rseq^\1 - Rseq) * Partial_Sums(seq), n)
proof
let n;
Partial_Sums(Rseq * seq).(n+1) =
(Rseq * Partial_Sums(seq)).(n+1) -
Partial_Sums((Rseq^\1 - Rseq) * Partial_Sums(seq)).n by Th55;
then Sum(Rseq * seq, n+1) =
(Rseq * Partial_Sums(seq)).(n+1) -
Partial_Sums((Rseq^\1 - Rseq) * Partial_Sums(seq)).n by Def4;
hence thesis by Def4;
end;