:: Polynomially Bounded Sequences and Polynomial Sequences
:: by Hiroyuki Okazaki and Yuichi Futa
::
:: Received June 30, 2015
:: Copyright (c) 2015-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, REAL_1, SUBSET_1, SEQ_1, FDIFF_1, FUNCT_1, PARTFUN1,
PARTFUN3, ARYTM_1, VALUED_0, ORDINAL2, CARD_1, RELAT_1, ARYTM_3,
XXREAL_0, COMPLEX1, NAT_1, TARSKI, VALUED_1, FUNCT_2, SQUARE_1, XBOOLE_0,
ORDINAL4, LIMFUNC1, FUNCT_7, ASYMPT_1, POWER, ASYMPT_0, FINSET_1,
ORDINAL1, INT_1, CARD_3, AFINSQ_1, FINSEQ_1, TAYLOR_1, ASYMPT_2;
notations TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, ORDINAL1, RELSET_1,
PARTFUN1, FUNCT_2, FINSET_1, PARTFUN2, PARTFUN3, NUMBERS, XCMPLX_0,
XXREAL_0, XREAL_0, SQUARE_1, NAT_1, INT_1, VALUED_0, COMPLEX1, VALUED_1,
SEQ_1, RFUNCT_1, RVSUM_1, FDIFF_1, POWER, SERIES_1, LIMFUNC1, ASYMPT_0,
ASYMPT_1, AFINSQ_1, TAYLOR_1, AFINSQ_2;
constructors REAL_1, SQUARE_1, RCOMP_1, PARTFUN2, PARTFUN3, LIMFUNC1, FDIFF_1,
RELSET_1, RVSUM_1, NEWTON, PREPOWER, SERIES_1, ASYMPT_0, AFINSQ_2,
ASYMPT_1, TAYLOR_1, SIN_COS;
registrations ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0, XREAL_0, NAT_1, MEMBERED,
POWER, FCONT_3, VALUED_0, VALUED_1, SQUARE_1, AFINSQ_1, AFINSQ_2,
XBOOLE_0, RELAT_1, INT_1, ASYMPT_0, ASYMPT_1, FUNCT_1, PARTFUN3;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
begin :: Preliminaries
theorem :: ASYMPT_2:1
for m,k be Nat st 1 <= m holds 1 <= m to_power k;
theorem :: ASYMPT_2:2
for m,n be Nat holds m <= m to_power (n+1);
theorem :: ASYMPT_2:3
for m,n be Nat st 2 <= m holds n+1 <= m to_power n;
theorem :: ASYMPT_2:4
for k be Nat holds 2*k <= 2 to_power k;
theorem :: ASYMPT_2:5
for k,n be Nat st k <= n holds n+k <= 2 to_power n;
theorem :: ASYMPT_2:6
for k,m be Nat st 2*k + 1 <= m holds
2 to_power k <= (2 to_power m) / m;
theorem :: ASYMPT_2:7
for a,b,c be Real st 1 < a & 0 < b & b <= c holds
log(a,b) <= log(a,c);
theorem :: ASYMPT_2:8
for n be Nat, a be Real st 1 < a holds
a to_power n < a to_power (n+1);
theorem :: ASYMPT_2:9
for n be Nat, a be Real st 1 <= a holds
a to_power n <= a to_power (n+1);
theorem :: ASYMPT_2:10
ex g be PartFunc of REAL,REAL st
dom(g)=right_open_halfline(0) &
(for x be Real st x in right_open_halfline(0) holds g.x=log(2,x))
&
g is_differentiable_on right_open_halfline(0) &
for x be Real st x in right_open_halfline(0)
holds g is_differentiable_in x
& diff(g,x)=(log(2,number_e))/x
& 0 < diff(g,x);
theorem :: ASYMPT_2:11
ex f be PartFunc of REAL,REAL st
right_open_halfline (number_e) = dom f &
(for x be Real st x in dom f holds f.x = x/log(2,x) ) &
f is_differentiable_on right_open_halfline (number_e) &
(for x0 being Real st x0 in right_open_halfline (number_e) holds
0 <= diff (f,x0) )
& f is non-decreasing;
theorem :: ASYMPT_2:12
for x,y be Real st number_e < x & x <= y holds
x/log(2,x) <= y/log(2,y);
theorem :: ASYMPT_2:13
for k be Nat st number_e < k holds
ex N be Nat st for n be Nat st N<=n holds 2 to_power k <= n/log(2,n);
theorem :: ASYMPT_2:14
for x be Nat st 1 < x holds
ex N be Nat st for n be Nat st N<=n holds 4 < n/log(x,n);
theorem :: ASYMPT_2:15
for x be Nat st 1 < x holds
ex N,c be Nat st
for n be Nat st N <= n holds n to_power x <= c * (x to_power n);
theorem :: ASYMPT_2:16
for x be Nat st 1 < x holds
not ex N,c be Nat st
for n be Nat st N <= n holds 2 to_power n <= c * (n to_power x);
theorem :: ASYMPT_2:17
for a,b be Nat st a <= b holds
seq_n^ a in Big_Oh (seq_n^ b);
theorem :: ASYMPT_2:18
for x be Nat st 1 < x holds
not ex N,c be Nat st
for n be Nat st N <= n holds x to_power n <= c * (n to_power x);
theorem :: ASYMPT_2:19
for a be non negative Real, n be Nat st 1 <= n
holds 0 < (seq_n^(a)).n;
begin :: Polynomially Bounded Sequences
definition
let p be Real_Sequence;
attr p is polynomially-bounded means
:: ASYMPT_2:def 1
ex k be Nat st p in Big_Oh(seq_n^(k));
end;
theorem :: ASYMPT_2:20
for f be Real_Sequence st f is non polynomially-bounded holds
for k be Nat holds not f in Big_Oh(seq_n^(k));
theorem :: ASYMPT_2:21
for f be Real_Sequence st
for k be Nat holds not f in Big_Oh(seq_n^(k))
holds f is non polynomially-bounded;
theorem :: ASYMPT_2:22
for a be positive Real holds seq_a^(a,1,0) is positive;
theorem :: ASYMPT_2:23
for a be Real st 1 <= a holds seq_a^(a,1,0) is non-decreasing;
theorem :: ASYMPT_2:24
for a be Real st 1 < a holds seq_a^(a,1,0) is increasing;
theorem :: ASYMPT_2:25
for a be Nat st 1 < a holds
seq_a^(a,1,0) is non polynomially-bounded;
begin :: Polynomial Sequences
theorem :: ASYMPT_2:26
for x be XFinSequence of REAL,
y be Real_Sequence
holds x(#)y is finite Sequence of REAL
& dom (x(#)y) = dom x
& for i be object st i in dom x
holds (x(#)y).i = (x.i) * (y.i);
definition
let x be XFinSequence of REAL,
y be Real_Sequence;
redefine func x(#)y -> XFinSequence of REAL;
end;
theorem :: ASYMPT_2:27
for d be XFinSequence of REAL
holds for x,i be Nat st i in dom d holds
(d (#) seq_a^(x,1,0)).i = (d.i) * x to_power (i);
definition
let c be XFinSequence of REAL;
func seq_p(c) -> Real_Sequence means
:: ASYMPT_2:def 2
for x be Nat holds it.x = Sum(c (#) seq_a^(x,1,0));
end;
theorem :: ASYMPT_2:28
for d be XFinSequence of REAL,k be Nat st len d = k+1 holds
ex a be Real,d1 be XFinSequence of REAL,
y be Real_Sequence st len d1 = k & d1= d | k & a = d.k & d =d1^<% a %> &
seq_p(d) = seq_p(d1) + y &
for i be Nat holds y.i = a* (i to_power k);
theorem :: ASYMPT_2:29
for d be XFinSequence of REAL,k be Nat st len d = 1 holds
ex a be Real st a = d.0 &
for x be Nat holds (seq_p(d)).x = a;
theorem :: ASYMPT_2:30
for d be XFinSequence of REAL,k be Nat
st len d = 1 & d is nonnegative-yielding holds
seq_p(d) in Big_Oh( seq_n^(k) );
theorem :: ASYMPT_2:31
for k be Nat,a be Real,y be Real_Sequence
st 0<=a &
for i be Nat holds
y.i = a* (i to_power k) holds y in Big_Oh( seq_n^k );
theorem :: ASYMPT_2:32
for k,n be Nat st k <= n
holds Big_Oh( seq_n^k ) c= Big_Oh( seq_n^(n) );
theorem :: ASYMPT_2:33
for k be Nat,
c be nonnegative-yielding XFinSequence of REAL
st len c = k+1
holds seq_p(c) in Big_Oh( seq_n^(k) );
theorem :: ASYMPT_2:34
for k be Nat,
c be XFinSequence of REAL holds
ex d be XFinSequence of REAL st len d = len c &
for i be Nat st i in dom d holds d.i = |. c.i .|;
theorem :: ASYMPT_2:35
for c be XFinSequence of REAL,
d be XFinSequence of REAL
st len d = len c &
for i be Nat st i in dom d holds d.i = |. c.i .| holds
for n be Nat holds ( seq_p(c) ).n <= ( seq_p(d) ).n;
theorem :: ASYMPT_2:36
for k be Nat,
c be XFinSequence of REAL
st len c = k+1 & seq_p(c) is eventually-nonnegative
holds seq_p(c) in Big_Oh( seq_n^(k) );
theorem :: ASYMPT_2:37
for k,n be Nat st 0 < n holds
n*(seq_n^(k)).n = ((seq_n^(k+1)).n);
theorem :: ASYMPT_2:38
for c be XFinSequence of REAL st len c = 0 holds
for x be Nat holds (seq_p(c)).x = 0;
theorem :: ASYMPT_2:39
for f be eventually-nonnegative Real_Sequence,
k be Nat st
f in Big_Oh(seq_n^(k)) holds
ex N be Nat st
for n be Nat st N <= n holds f.n <= (seq_n^(k+1)).n;
theorem :: ASYMPT_2:40
for c be XFinSequence of REAL holds
ex absc be XFinSequence of REAL st
absc = |. c .| &
for n be Nat holds (seq_p(c)).n <= (seq_p(absc)).n;
theorem :: ASYMPT_2:41
for c,absc be XFinSequence of REAL st absc = |. c .|
holds for n be Nat holds |.(seq_p(c)).n .| <= (seq_p(absc)).n;
theorem :: ASYMPT_2:42
for a be Real st 0 < a holds
for k be Nat, d be nonnegative-yielding XFinSequence of REAL
st len d = k holds
ex N be Nat st
for x be Nat st N <=x holds
for i be Nat st i in dom d
holds (d.i) * (x to_power i)*k <= a* (x to_power k);
theorem :: ASYMPT_2:43
for k be Nat,
d be XFinSequence of REAL,
a be Real,
y be Real_Sequence
st 0 < a
& len d = k
& for x be Nat holds y.x = a * (x to_power k)
holds
ex N be Nat st
for x be Nat st N <= x holds |.(seq_p(d)).x .| <= y.x;
theorem :: ASYMPT_2:44
for k be Nat, d be XFinSequence of REAL
st len d = k+1 & 0 < d.k holds
seq_p(d) is eventually-nonnegative;
theorem :: ASYMPT_2:45
for k be Nat,
c be XFinSequence of REAL
st len c = k+1 & 0 < c.k
holds seq_p(c) in Big_Oh (seq_n^(k));
theorem :: ASYMPT_2:46
for k be Nat,
c be XFinSequence of REAL
st len c = k+1 & 0 < c.k holds seq_p(c) is polynomially-bounded;