:: Asymptotic notation. Part II: Examples and Problems
:: by Richard Krueger, Piotr Rudnicki and Paul Shelley
::
:: Received November 4, 1999
:: Copyright (c) 1999-2016 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, NAT_1, XXREAL_0, POWER, ARYTM_3,
RELAT_1, CARD_1, SQUARE_1, SEQ_1, FUNCT_1, ARYTM_1, ASYMPT_0, FUNCT_2,
INT_1, VALUED_1, TARSKI, SEQ_2, ORDINAL2, COMPLEX1, CARD_3, SERIES_1,
REALSET1, VALUED_0, NEWTON, FINSEQ_1, FINSEQ_2, ORDINAL4, XBOOLE_0,
ASYMPT_1, FUNCT_7;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0,
XREAL_0, COMPLEX1, REAL_1, FUNCT_1, FUNCT_2, FUNCT_7, SQUARE_1, INT_1,
NAT_1, VALUED_1, SEQ_1, COMSEQ_2, SEQ_2, NEWTON, POWER, SERIES_1,
FUNCOP_1, FINSEQ_1, FINSEQ_2, RVSUM_1, ASYMPT_0;
constructors PARTFUN1, DOMAIN_1, REAL_1, SQUARE_1, NAT_1, NAT_D, SEQ_2,
SEQM_3, RVSUM_1, NEWTON, PREPOWER, SERIES_1, ASYMPT_0, RELSET_1,
FUNCOP_1, BINOP_2, COMSEQ_2, FUNCT_7, SEQ_1, NUMBERS;
registrations XBOOLE_0, SUBSET_1, ORDINAL1, RELSET_1, NUMBERS, XXREAL_0,
XREAL_0, NAT_1, INT_1, MEMBERED, ASYMPT_0, SQUARE_1, VALUED_0, VALUED_1,
FUNCT_2, POWER, FINSEQ_1;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
definitions ASYMPT_0;
equalities ASYMPT_0, SQUARE_1, VALUED_1, XCMPLX_0, SEQ_1;
expansions ASYMPT_0, VALUED_1;
theorems TARSKI, FUNCT_2, INT_1, NAT_1, SEQ_1, SQUARE_1, SEQ_2, ABSVALUE,
FUNCOP_1, POWER, NEWTON, EULER_2, SERIES_1, FINSEQ_1, PRE_FF, RVSUM_1,
FINSEQ_2, ASYMPT_0, XREAL_0, XBOOLE_0, XCMPLX_0, XCMPLX_1, XREAL_1,
XXREAL_0, ORDINAL1, NAT_D, VALUED_1;
schemes FUNCT_2, SEQ_1, NAT_1;
begin :: Examples from the text
reserve c, c1, c2, d, d1, d2, e, y for Real,
k, n, m, N, n1, N0, N1, N2, N3, M for Element of NAT,
x for set;
Lm1: for n be Nat st n >= 2 holds 2 to_power n > n+1
proof
defpred P[Nat] means 2 to_power $1 > $1+1;
A1: for k be Nat st k >= 2 & P[k] holds P[k+1]
proof
let k be Nat such that
k >= 2 and
A2: 2 to_power k > k+1;
2 to_power (k+1) = (2 to_power k)*(2 to_power 1) by POWER:27
.= (2 to_power k)*2 by POWER:25
.= (2 to_power k) + (2 to_power k);
then
A3: 2 to_power (k+1) > (k+1) + (2 to_power k) by A2,XREAL_1:6;
reconsider k as Element of NAT by ORDINAL1:def 12;
2 to_power k >= 0 + 1 by INT_1:7,POWER:34;
then (k+1) + (2 to_power k) >= (k+1) + 1 by XREAL_1:6;
hence thesis by A3,XXREAL_0:2;
end;
2 to_power 2 = 2^2 by POWER:46
.= 4;
then
A4: P[2];
for n be Nat st n >= 2 holds P[n] from NAT_1:sch 8(A4, A1);
hence thesis;
end;
reconsider zz=0 as Element of REAL by XREAL_0:def 1;
theorem
for t,t1 being Real_Sequence st t.0 = 0 & (for n st n > 0 holds t.n =
12*(n to_power 3)*log(2,n) - 5*n^2 + (log(2,n))^2 +36) & (for n st n > 0 holds
t1.n = (n to_power 3)*log(2,n)) ex s,s1 being eventually-positive Real_Sequence
st s = t & s1 = t1 & s in Big_Oh(s1)
proof
ex s being Real_Sequence st s.0 = 0 & for n st n > 0 holds s.n = (log(2
,n))^2 + 36
proof
defpred P[Element of NAT,Real]
means ($1 = 0 implies $2 = 0) & ($1 > 0
implies $2 = (log(2,$1))^2 + 36);
A1: for x being Element of NAT ex y being Element of REAL st P[x,y]
proof
let n;
per cases;
suppose n= zz;
hence thesis;
end;
suppose
A2: n > 0;
(log(2,n))^2 + 36 in REAL by XREAL_0:def 1;
hence thesis by A2;
end;
end;
consider h being sequence of REAL such that
A3: for x being Element of NAT holds P[x,h.x] from FUNCT_2:sch 3(A1);
take h;
thus thesis by A3;
end;
then consider q being Real_Sequence such that
A4: q.0 = 0 and
A5: for n st n > 0 holds q.n = (log(2,n))^2 + 36;
q is eventually-positive
proof
take 1;
let n be Nat;
A6: n in NAT by ORDINAL1:def 12;
A7: (log(2,n))^2 + 36 > 0+0 by XREAL_1:8,63;
assume n >= 1;
hence thesis by A5,A7,A6;
end;
then reconsider q as eventually-positive Real_Sequence;
let f,g be Real_Sequence such that
A8: f.0 = 0 and
A9: for n st n > 0 holds f.n = 12*(n to_power 3)*log(2,n) - 5*n^2 + (
log(2,n))^2 + 36 and
A10: for n st n > 0 holds g.n = (n to_power 3)*log(2,n);
A11: g is eventually-positive
proof
take 2;
let n be Nat;
assume
A12: n >= 2;
then log(2,n) >= log(2,2) by PRE_FF:10;
then
A13: log(2,n) >= 1 by POWER:52;
A14: n in NAT by ORDINAL1:def 12;
n to_power 3 > 0 by A12,POWER:34;
then (n to_power 3)*log(2,n) > (n to_power 3)*0 by A13,XREAL_1:68;
hence thesis by A10,A12,A14;
end;
4 = 2^2 .= 2 to_power 2 by POWER:46;
then
A15: log(2,4) = 2*log(2,2) by POWER:55
.= 2*1 by POWER:52
.= 2;
A16: for n st n >= 4 holds 7*n^2 > q.n
proof
defpred P[Nat] means 7*$1^2 > q.$1;
A17: for k be Nat st k >= 4 & P[k] holds P[k+1]
proof
let k be Nat such that
A18: k >= 4 and
A19: 7*k^2 > q.k;
A20: q.(k+1) = (log(2,k+1))^2 + 36 by A5;
k >= 2 by A18,XXREAL_0:2;
then
A21: 2 to_power k > k + 1 by Lm1;
k+1 > k+0 by XREAL_1:8;
then 2 to_power k > k by A21,XXREAL_0:2;
then log(2,2 to_power k) > log(2,k) by A18,POWER:57;
then k*log(2,2) > log(2,k) by POWER:55;
then
A22: k*1 > log(2,k) by POWER:52;
log(2,k) >= 2 by A15,A18,PRE_FF:10;
then 14*k > 2*log(2,k) by A22,XREAL_1:98;
then (7*2)*k + 7 > 2*log(2,k) + 1 by XREAL_1:8;
then
A23: (log(2,k))^2 + (2*log(2,k) + 1) < (log(2,k))^2 + (7*(2*k) + 7) by
XREAL_1:6;
log(2,k+k) = log(2,2*k);
then log(2,k+k) = log(2,k) + log(2,2) by A18,POWER:53;
then (log(2,k+k))^2 = (log(2,k) + 1)^2 by POWER:52
.= (log(2,k))^2 + 2*log(2,k) + 1;
then
A24: (log(2,k+k))^2 + 36 < ((log(2,k))^2 + (7*(2*k) + 7)) + 36 by A23,
XREAL_1:6;
k >= 1 by A18,XXREAL_0:2;
then k+k >= k+1 by XREAL_1:6;
then
A25: log(2,k+k) >= log(2,k+1) by PRE_FF:10;
k+1 >= 4+0 by A18,XREAL_1:8;
then log(2,k+1) >= 2 by A15,PRE_FF:10;
then (log(2,k+k))^2 >= (log(2,k+1))^2 by A25,SQUARE_1:15;
then
A26: q.(k+1) <= (log(2,k+k))^2 + 36 by A20,XREAL_1:6;
7*(k+1)^2 = 7*k^2 + (7*(2*k) + 7*1);
then
A27: 7*(k+1)^2 > q.k + (7*(2*k) + 7*1) by A19,XREAL_1:6;
k in NAT by ORDINAL1:def 12;
then q.k = (log(2,k))^2 + 36 by A5,A18;
then q.k + (7*(2*k) + 7*1) > q.(k+1) by A26,A24,XXREAL_0:2;
hence thesis by A27,XXREAL_0:2;
end;
q.4 = 2^2 + 36 by A5,A15
.= 40;
then
A28: P[4];
for n be Nat st n >= 4 holds P[n] from NAT_1:sch 8(A28, A17);
hence thesis;
end;
reconsider g as eventually-positive Real_Sequence by A11;
f is eventually-positive
proof
log(2,3) > log(2,2) by POWER:57;
then
A29: log(2,3) > 1 by POWER:52;
take 3;
let n be Nat;
assume
A30: n >= 3;
then
A31: n to_power 2 > 0 by POWER:34;
n > 1 by A30,XXREAL_0:2;
then
A32: n to_power 3 > n to_power 2 by POWER:39;
A33: n in NAT by ORDINAL1:def 12;
log(2,n) >= log(2,3) by A30,PRE_FF:10;
then log(2,n) > 1 by A29,XXREAL_0:2;
then (n to_power 3)*log(2,n) > (n to_power 2)*1 by A32,A31,XREAL_1:98;
then 12*((n to_power 3)*log(2,n)) > 5*(n to_power 2) by A31,XREAL_1:98;
then 12*(n to_power 3)*log(2,n) > 5*n^2 + 0 by POWER:46;
then 12*(n to_power 3)*log(2,n) - 5*n^2 > 0 by XREAL_1:20;
then (12*(n to_power 3)*log(2,n)-5*n^2)+(log(2,n))^2>0+0 by XREAL_1:8,63;
then (12*(n to_power 3)*log(2,n)-5*n^2)+(log(2,n))^2+36 > 0+0;
hence thesis by A9,A30,A33;
end;
then reconsider f as eventually-positive Real_Sequence;
take f, g;
ex s being Real_Sequence st s.0 = 0 & for n st n > 0 holds s.n = 12*(n
to_power 3)*log(2,n) - 5*n^2
proof
defpred P[Element of NAT,Real] means ($1 = 0 implies $2 = 0) & ($1 > 0
implies $2 = 12*($1 to_power 3)*log(2,$1) - 5*($1)^2);
A34: for x being Element of NAT ex y being Element of REAL st P[x,y]
proof
let n;
A35: n = zz or n > 0;
12*(n to_power 3)*log(2,n) - 5*n^2 in REAL by XREAL_0:def 1;
hence thesis by A35;
end;
consider h being sequence of REAL such that
A36: for x being Element of NAT holds P[x,h.x] from FUNCT_2:sch 3(A34);
take h;
thus h.0 = 0 by A36;
let n;
thus thesis by A36;
end;
then consider p being Real_Sequence such that
A37: p.0 = 0 and
A38: for n st n > 0 holds p.n = 12*(n to_power 3)*log(2,n) - 5*n^2;
p is eventually-positive
proof
log(2,3) > log(2,2) by POWER:57;
then
A39: log(2,3) > 1 by POWER:52;
take 3;
let n be Nat;
assume
A40: n >= 3;
then
A41: n to_power 2 > 0 by POWER:34;
n > 1 by A40,XXREAL_0:2;
then
A42: n to_power 3 > n to_power 2 by POWER:39;
A43: n in NAT by ORDINAL1:def 12;
log(2,n) >= log(2,3) by A40,PRE_FF:10;
then log(2,n) > 1 by A39,XXREAL_0:2;
then (n to_power 3)*log(2,n) > (n to_power 2)*1 by A42,A41,XREAL_1:98;
then 12*((n to_power 3)*log(2,n)) > 5*(n to_power 2) by A41,XREAL_1:98;
then 12*(n to_power 3)*log(2,n) > 5*n^2 + 0 by POWER:46;
then 12*(n to_power 3)*log(2,n) - 5*n^2 > 0 by XREAL_1:20;
hence thesis by A38,A40,A43;
end;
then reconsider p as eventually-positive Real_Sequence;
set t = max(p,q);
consider N being Nat such that
A44: for n being Nat st n >= N holds t.n > 0 by ASYMPT_0:def 4;
A45: for n st n >= 4 holds p.n > 7*n^2
proof
let n;
assume
A46: n >= 4;
then n > 1 by XXREAL_0:2;
then
A47: n to_power 3 > n to_power 2 by POWER:39;
log(2,n) >= log(2,4) by A46,PRE_FF:10;
then
A48: log(2,n) > 1 by A15,XXREAL_0:2;
n to_power 2 > 0 by A46,POWER:34;
then (n to_power 3)*log(2,n) > (n to_power 2)*1 by A47,A48,XREAL_1:98;
then 12*((n to_power 3)*log(2,n)) > 12*(n to_power 2) by XREAL_1:68;
then
A49: 12*(n to_power 3)*log(2,n) > 12*n^2 by POWER:46;
p.n = 12*(n to_power 3)*log(2,n) - 5*n^2 by A38,A46;
then p.n > 12*n^2 - 5*n^2 by A49,XREAL_1:9;
hence thesis;
end;
A50: for n st n >= 4 holds p.n > q.n
proof
let n;
assume
A51: n >= 4;
then
A52: 7*n^2 > q.n by A16;
p.n > 7*n^2 by A45,A51;
hence thesis by A52,XXREAL_0:2;
end;
A53: for n st n >= 4 holds t.n = p.n
proof
let n;
assume n >= 4;
then
A54: p.n > q.n by A50;
thus t.n = max( p.n, q.n ) by ASYMPT_0:def 7
.= p.n by A54,XXREAL_0:def 10;
end;
reconsider mN = max(4,N) as Element of NAT by ORDINAL1:def 12;
A55: now
let n;
assume
A56: n >= mN;
A57: max(4,N) >= 4 by XXREAL_0:25;
then t.n = p.n by A53,A56,XXREAL_0:2
.= 12*(n to_power 3)*log(2,n) - 5*n^2 by A38,A56,A57;
then t.n <= 12*(n to_power 3)*log(2,n) - 0 by XREAL_1:13;
then t.n <= 12*((n to_power 3)*log(2,n));
hence t.n <= 12*g.n by A10,A56,A57;
max(4,N) >= N by XXREAL_0:25;
then n >= N by A56,XXREAL_0:2;
hence t.n >= 0 by A44;
end;
t is Element of Funcs(NAT, REAL) by FUNCT_2:8;
then
A58: t in Big_Oh(g) by A55;
for n being Nat holds f.n = p.n + q.n
proof
let n be Nat;
A59: n in NAT by ORDINAL1:def 12;
thus f.n = p.n + q.n
proof
per cases;
suppose
n = 0;
hence thesis by A8,A37,A4;
end;
suppose
A60: n > 0;
then p.n = 12*(n to_power 3)*log(2,n) - 5*n^2 by A38,A59;
then
p.n + q.n = (12*(n to_power 3)*log(2,n)-5*n^2)+((log(2,n))^2 +36)
by A5,A60,A59
.= 12*(n to_power 3)*log(2,n)-5*n^2+(log(2,n))^2 +36;
hence thesis by A9,A60,A59;
end;
end;
end;
then
A61: Big_Oh(f) = Big_Oh(p+q) by SEQ_1:7
.= Big_Oh(t) by ASYMPT_0:9;
f in Big_Oh(f) by ASYMPT_0:10;
hence thesis by A61,A58,ASYMPT_0:12;
end;
Lm2: for a being logbase Real, f being Real_Sequence st a > 1 & (for n st n >
0 holds f.n = log(a,n)) holds f is eventually-positive
proof
let a be logbase Real, f be Real_Sequence such that
A1: a > 1 and
A2: for n st n > 0 holds f.n = log(a,n);
set N = [/a\];
A3: [/a\] >= a by INT_1:def 7;
A4: a > 0 by ASYMPT_0:def 1;
then
A5: N > 0 by INT_1:def 7;
then reconsider N as Element of NAT by INT_1:3;
A6: a <> 1 by ASYMPT_0:def 1;
now
A7: log(a,N) >= log(a,a) by A1,A3,PRE_FF:10;
let n be Nat;
A8: n in NAT by ORDINAL1:def 12;
assume
A9: n >= N+1;
N+1 > N+0 by XREAL_1:8;
then n > N by A9,XXREAL_0:2;
then log(a,n) > log(a,N) by A1,A5,POWER:57;
then log(a,n) > 0 by A4,A6,A7,POWER:52;
hence f.n > 0 by A2,A9,A8;
end;
hence thesis;
end;
theorem
for a,b being logbase Real, f,g being Real_Sequence st a > 1 & b > 1 &
(for n st n > 0 holds f.n = log(a,n)) & (for n st n > 0 holds g.n = log(b,n))
holds ex s,s1 being eventually-positive Real_Sequence st s = f & s1 = g &
Big_Oh(s) = Big_Oh(s1)
proof
let a,b be logbase Real, f,g be Real_Sequence such that
A1: a > 1 and
A2: b > 1 and
A3: for n st n > 0 holds f.n = log(a,n) and
A4: for n st n > 0 holds g.n = log(b,n);
reconsider g as eventually-positive Real_Sequence by A2,A4,Lm2;
reconsider f as eventually-positive Real_Sequence by A1,A3,Lm2;
take f,g;
A5: a <> 1 by ASYMPT_0:def 1;
A6: b <> 1 by ASYMPT_0:def 1;
A7: b > 0 by ASYMPT_0:def 1;
A8: a > 0 by ASYMPT_0:def 1;
now
let x be object;
hereby
assume x in Big_Oh(f);
then consider t being Element of Funcs(NAT, REAL) such that
A9: x = t and
A10: ex c,N st c > 0 & for n st n >= N holds t.n <= c*f.n & t.n >= 0;
consider c,N such that
A11: c > 0 and
A12: for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A10;
A13: now
take N1 = N+1;
let n;
assume
A14: n >= N1;
then
A15: f.n = log(a,n) by A3
.= log(a,b)*log(b,n) by A8,A5,A7,A6,A14,POWER:56;
N+1 > N+0 by XREAL_1:8;
then
A16: n > N by A14,XXREAL_0:2;
then t.n <= c*f.n by A12;
then t.n <= (c*log(a,b))*log(b,n) by A15;
hence t.n <= (c*log(a,b))*g.n by A4,A14;
thus t.n >= 0 by A12,A16;
end;
log(a,b) > log(a,1) by A1,A2,POWER:57;
then log(a,b) > 0 by A8,A5,POWER:51;
then c*log(a,b) > c*0 by A11,XREAL_1:68;
hence x in Big_Oh(g) by A9,A13;
end;
assume x in Big_Oh(g);
then consider t being Element of Funcs(NAT, REAL) such that
A17: x = t and
A18: ex c,N st c > 0 & for n st n >= N holds t.n <= c*g.n & t.n >= 0;
consider c,N such that
A19: c > 0 and
A20: for n st n >= N holds t.n <= c*g.n & t.n >= 0 by A18;
A21: now
take N1 = N+1;
let n;
assume
A22: n >= N1;
then
A23: g.n = log(b,n) by A4
.= log(b,a)*log(a,n) by A8,A5,A7,A6,A22,POWER:56;
N+1 > N+0 by XREAL_1:8;
then
A24: n > N by A22,XXREAL_0:2;
then t.n <= c*g.n by A20;
then t.n <= (c*log(b,a))*log(a,n) by A23;
hence t.n <= (c*log(b,a))*f.n by A3,A22;
thus t.n >= 0 by A20,A24;
end;
log(b,a) > log(b,1) by A1,A2,POWER:57;
then log(b,a) > 0 by A7,A6,POWER:51;
then c*log(b,a) > c*0 by A19,XREAL_1:68;
hence x in Big_Oh(f) by A17,A21;
end;
hence thesis by TARSKI:2;
end;
definition
let a,b,c be Real;
func seq_a^(a,b,c) -> Real_Sequence means
:Def1:
it.n = a to_power (b*n+c);
existence
proof
deffunc F(Element of NAT) = In(a to_power (b*$1+c),REAL);
consider h being sequence of REAL such that
A1: for n being Element of NAT holds h.n = F(n) from FUNCT_2:sch 4;
take h;
let n;
thus h.n = In(a to_power (b*n+c),REAL) by A1
.= In(a to_power (b*n+c),REAL)
.= a to_power (b*n+c);
end;
uniqueness
proof
let j,k be Real_Sequence such that
A2: for n holds j.n = a to_power (b*n+c) and
A3: for n holds k.n = a to_power (b*n+c);
now
let n;
thus j.n = a to_power (b*n+c) by A2
.= k.n by A3;
end;
hence thesis by FUNCT_2:63;
end;
end;
registration
let a be positive Real, b,c be Real;
cluster seq_a^(a,b,c) -> eventually-positive;
coherence
proof
take 0;
set f = seq_a^(a,b,c);
let n be Nat;
A1: n in NAT by ORDINAL1:def 12;
assume n >= 0;
f.n = a to_power (b*n+c) by Def1,A1;
hence thesis by POWER:34;
end;
end;
Lm3: for a,b,c being Real st a > 0 & c > 0 & c <> 1
holds a to_power b = c to_power (b*log(c,a))
proof
let a,b,c be Real;
assume that
A1: a > 0 and
A2: c > 0 and
A3: c <> 1;
A4: a to_power b > 0 by A1,POWER:34;
log(c, a to_power b) = b*log(c,a) by A1,A2,A3,POWER:55;
hence thesis by A2,A3,A4,POWER:def 3;
end;
theorem
for a,b being positive Real st a < b
holds not seq_a^(b,1,0) in Big_Oh(seq_a^(a,1,0))
proof
let a,b be positive Real such that
A1: a < b;
set g = seq_a^(a,1,0);
set f = seq_a^(b,1,0);
hereby
set d = (log(2,b) - log(2,a));
assume f in Big_Oh(g);
then consider s being Element of Funcs(NAT, REAL) such that
A2: s = f and
A3: ex c,N st c > 0 & for n st n >= N holds s.n <= c*g.n & s.n >= 0;
consider c,N such that
A4: c > 0 and
A5: for n st n >= N holds s.n <= c*g.n & s.n >= 0 by A3;
set N0 = [/log(2,c) / d\];
set N1 = max( N, N0 );
A6: N1 >= N by XXREAL_0:25;
A7: N1 = N or N1 = N0 by XXREAL_0:16;
A8: N1 >= N0 by XXREAL_0:25;
reconsider N1 as Element of NAT by A6,A7,INT_1:3;
set n = N1 + 1;
set e = (2 to_power (n*log(2,a)));
A9: e > 0 by POWER:34;
A10: N0 >= log(2,c) / d by INT_1:def 7;
log(2,b) > log(2,a) + 0 by A1,POWER:57;
then
A11: d > 0 by XREAL_1:20;
A12: N1 + 1 > N1 + 0 by XREAL_1:8;
then n > N0 by A8,XXREAL_0:2;
then n > log(2,c) / d by A10,XXREAL_0:2;
then n*d > (log(2,c) / d)*d by A11,XREAL_1:68;
then n*d > log(2,c) by A11,XCMPLX_1:87;
then 2 to_power (n*d) > 2 to_power log(2,c) by POWER:39;
then 2 to_power (n*log(2,b) - n*log(2,a)) > c by A4,POWER:def 3;
then (2 to_power (n*log(2,b))) / e > c by POWER:29;
then ((2 to_power (n*log(2,b)))/e)*e > c*e by A9,XREAL_1:68;
then 2 to_power (n*log(2,b)) > c*e by A9,XCMPLX_1:87;
then b to_power n > c*(2 to_power (n*log(2,a))) by Lm3;
then
A13: b to_power n > c*(a to_power n) by Lm3;
n > N by A6,A12,XXREAL_0:2;
then f.n <= c*g.n by A2,A5;
then b to_power (1*n + 0) <= c*g.n by Def1;
hence contradiction by A13,Def1;
end;
end;
:: Example p. 84
definition
func seq_logn -> Real_Sequence means
:Def2:
it.0 = 0 & for n st n > 0 holds it.n = log(2,n);
existence
proof
defpred P[Element of NAT,Real] means ($1 = 0 implies $2 = 0) & ($1 > 0
implies $2 = log(2,$1));
A1: for x being Element of NAT ex y being Element of REAL st P[x,y]
proof
let n;
per cases;
suppose
n = zz;
hence thesis;
end;
suppose
A2: n > 0;
log(2,n) in REAL by XREAL_0:def 1;
hence thesis by A2;
end;
end;
consider h being sequence of REAL such that
A3: for x being Element of NAT holds P[x,h.x] from FUNCT_2:sch 3(A1);
take h;
thus h.0 = 0 by A3;
let n;
thus thesis by A3;
end;
uniqueness
proof
let j,k be Real_Sequence such that
A4: j.0 = 0 and
A5: for n st n > 0 holds j.n = log(2,n) and
A6: k.0 = 0 and
A7: for n st n > 0 holds k.n = log(2,n);
now
let n;
per cases;
suppose
n = 0;
hence j.n = k.n by A4,A6;
end;
suppose
A8: n > 0;
then j.n = log(2,n) by A5;
hence j.n = k.n by A7,A8;
end;
end;
hence thesis by FUNCT_2:63;
end;
end;
definition
let a be Real;
func seq_n^(a) -> Real_Sequence means
:Def3:
it.0 = 0 & for n st n > 0 holds it.n = n to_power a;
existence
proof
defpred P[Element of NAT,Real] means ($1 = 0 implies $2 = 0) & ($1 > 0
implies $2 = $1 to_power a);
A1: for x being Element of NAT ex y being Element of REAL st P[x,y]
proof
let n;
per cases;
suppose
n = zz;
hence thesis;
end;
suppose
A2: n > 0;
n to_power a in REAL by XREAL_0:def 1;
hence thesis by A2;
end;
end;
consider h being sequence of REAL such that
A3: for x being Element of NAT holds P[x,h.x] from FUNCT_2:sch 3(A1);
take h;
thus h.0 = 0 by A3;
let n;
thus thesis by A3;
end;
uniqueness
proof
let j,k be Real_Sequence such that
A4: j.0 = 0 and
A5: for n st n > 0 holds j.n = n to_power a and
A6: k.0 = 0 and
A7: for n st n > 0 holds k.n = n to_power a;
now
let n;
per cases;
suppose
n = 0;
hence j.n = k.n by A4,A6;
end;
suppose
A8: n > 0;
then j.n = n to_power a by A5;
hence j.n = k.n by A7,A8;
end;
end;
hence thesis by FUNCT_2:63;
end;
end;
registration
cluster seq_logn -> eventually-positive;
coherence
proof
take 2;
set f = seq_logn;
let n be Nat;
A1: n in NAT by ORDINAL1:def 12;
assume
A2: n >= 2;
then
A3: log(2,n) >= log(2,2) by PRE_FF:10;
f.n = log(2,n) by A2,Def2,A1;
hence thesis by A3,POWER:52;
end;
end;
registration
let a be Real;
cluster seq_n^(a) -> eventually-positive;
coherence
proof
take 1;
set f = seq_n^(a);
let n be Nat;
A1: n in NAT by ORDINAL1:def 12;
assume
A2: n >= 1;
then f.n = n to_power a by Def3,A1;
hence thesis by A2,POWER:34;
end;
end;
Lm4: for f,g being Real_Sequence, n being Nat holds (f/"g).n = f.n/
g.n
proof
let f,g be Real_Sequence, n be Nat;
thus (f/"g).n = f.n*g".n by SEQ_1:8
.= f.n*(g.n)" by VALUED_1:10
.= f.n/g.n;
end;
Lm5: for f,g being eventually-nonnegative Real_Sequence holds f in Big_Oh(g) &
g in Big_Oh(f) iff Big_Oh(f) = Big_Oh(g)
proof
let f,g be eventually-nonnegative Real_Sequence;
hereby
assume that
A1: f in Big_Oh(g) and
A2: g in Big_Oh(f);
A3: Big_Oh(g) c= Big_Oh(f) by A2,ASYMPT_0:11;
Big_Oh(f) c= Big_Oh(g) by A1,ASYMPT_0:11;
hence Big_Oh(f) = Big_Oh(g) by A3,XBOOLE_0:def 10;
end;
thus thesis by ASYMPT_0:10;
end;
theorem Th4:
for f,g being eventually-nonnegative Real_Sequence holds Big_Oh(f
) c= Big_Oh(g) & not Big_Oh(f) = Big_Oh(g) iff f in Big_Oh(g) & not f in
Big_Omega(g)
proof
let f,g be eventually-nonnegative Real_Sequence;
hereby
assume that
A1: Big_Oh(f) c= Big_Oh(g) and
A2: not Big_Oh(f) = Big_Oh(g);
A3: f in Big_Oh(f) by ASYMPT_0:10;
now
assume f in Big_Omega(g);
then g in Big_Oh(f) by ASYMPT_0:19;
hence contradiction by A1,A2,A3,Lm5;
end;
hence f in Big_Oh(g) & not f in Big_Omega(g) by A1,A3;
end;
assume that
A4: f in Big_Oh(g) and
A5: not f in Big_Omega(g);
now
let x be object;
assume x in Big_Oh(f);
then consider t being Element of Funcs(NAT, REAL) such that
A6: x = t and
A7: ex c,N st c > 0 & for n st n >= N holds t.n <= c*f.n & t.n >= 0;
consider c,N such that
c > 0 and
A8: for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A7;
now
reconsider N as Nat;
take N;
let n be Nat;
A9: n in NAT by ORDINAL1:def 12;
assume n >= N;
hence t.n >= 0 by A8,A9;
end;
then
A10: t is eventually-nonnegative;
t in Big_Oh(f) by A7;
hence x in Big_Oh(g) by A4,A6,A10,ASYMPT_0:12;
end;
hence Big_Oh(f) c= Big_Oh(g) by TARSKI:def 3;
assume Big_Oh(f) = Big_Oh(g);
then g in Big_Oh(f) by Lm5;
hence contradiction by A5,ASYMPT_0:19;
end;
Lm6: for a, b, c being Real st 0 < a & a <= b & c >= 0 holds a to_power
c <= b to_power c
proof
let a, b, c be Real;
assume that
A1: 0 < a and
A2: a <= b and
A3: c >= 0;
per cases by A3;
suppose
A4: c = 0;
then a to_power c = 1 by POWER:24;
hence thesis by A4,POWER:24;
end;
suppose
A5: c > 0;
per cases by A2,XXREAL_0:1;
suppose
a = b;
hence thesis;
end;
suppose
a < b;
hence thesis by A1,A5,POWER:37;
end;
end;
end;
Lm7: for n be Nat st n >= 4 holds 2*n + 3 < 2 to_power n
proof
defpred P[Nat] means 2*$1 + 3 < 2 to_power $1;
A1: for k be Nat st k >= 4 & P[k] holds P[k+1]
proof
let k be Nat such that
A2: k >= 4 and
A3: 2*k + 3 < 2 to_power k;
k > 1 by A2,XXREAL_0:2;
then 2 to_power k > 2 to_power 1 by POWER:39;
then 2 to_power k > 2 by POWER:25;
then
A4: (2 to_power k) + (2 to_power k) > 2 + (2 to_power k) by XREAL_1:6;
2*(k+1) + 3 = 2 + (2*k + 3);
then 2*(k+1) + 3 < 2 + (2 to_power k) by A3,XREAL_1:6;
then 2*(k+1) + 3 < 2*(2 to_power k) by A4,XXREAL_0:2;
then 2*(k+1) + 3 < (2 to_power 1)*(2 to_power k) by POWER:25;
hence thesis by POWER:27;
end;
A5: P[4] by POWER:62;
for n be Nat st n >= 4 holds P[n] from NAT_1:sch 8(A5, A1);
hence thesis;
end;
Lm8: for n st n >= 6 holds (n+1)^2 < 2 to_power n
proof
defpred P[Nat] means ($1+1)^2 < 2 to_power $1;
A1: for k be Nat st k >= 6 & P[k] holds P[k+1]
proof
let k be Nat such that
A2: k >= 6 and
A3: (k+1)^2 < 2 to_power k;
k >= 4 by A2,XXREAL_0:2;
then 2*k + 3 < 2 to_power k by Lm7;
then
A4: (k+1)^2 + (2*k + 3) < (k+1)^2 + (2 to_power k) by XREAL_1:6;
(k+1)^2 + (2 to_power k) < (2 to_power k) + (2 to_power k) by A3,XREAL_1:6;
then ((k+1)+1)^2 < 2*(2 to_power k) by A4,XXREAL_0:2;
then ((k+1)+1)^2 < (2 to_power 1)*(2 to_power k) by POWER:25;
hence thesis by POWER:27;
end;
A5: P[6] by POWER:64;
for n be Nat st n >= 6 holds P[n] from NAT_1:sch 8(A5, A1);
hence thesis;
end;
Lm9: for c being Real st c > 6 holds c^2 < 2 to_power c
proof
A1: 5 = 6-1;
let c be Real such that
A2: c > 6;
set i0 = [\c/], i1 = [/c\];
per cases;
suppose
i0 = i1;
then c is Integer by INT_1:34;
then reconsider c as Element of NAT by A2,INT_1:3;
c+0 < c+1 by XREAL_1:8;
then
A3: c^2 < (c+1)^2 by SQUARE_1:16;
(c+1)^2 < 2 to_power c by A2,Lm8;
hence thesis by A3,XXREAL_0:2;
end;
suppose
not i0 = i1;
then
A4: i0 + 1 = i1 by INT_1:41;
then
A5: i0 = i1 - 1;
A6: i1 >= c by INT_1:def 7;
then reconsider i1 as Element of NAT by A2,INT_1:3;
i1 > 6 by A2,A6,XXREAL_0:2;
then
A7: i0 > 5 by A1,A5,XREAL_1:9;
then reconsider i0 as Element of NAT by INT_1:3;
i0 <= c by INT_1:def 6;
then
A8: 2 to_power i0 <= 2 to_power c by PRE_FF:8;
i1 >= c by INT_1:def 7;
then
A9: i1^2 >= c^2 by A2,SQUARE_1:15;
i0 >= 5 + 1 by A7,INT_1:7;
then i1^2 < 2 to_power i0 by A4,Lm8;
then c^2 < 2 to_power i0 by A9,XXREAL_0:2;
hence thesis by A8,XXREAL_0:2;
end;
end;
Lm10: for e being positive Real, f being Real_Sequence
st (for n st n > 0
holds f.n = log(2,n to_power e)) holds (f /" seq_n^(e)) is convergent & lim (f
/" seq_n^(e)) = 0
proof
let e be positive Real, f be Real_Sequence such that
A1: for n st n > 0 holds f.n = log(2,n to_power e);
set g = seq_n^(e);
set h = f/"g;
A2: now
let p be Real;
reconsider p1 = p as Real;
set i0 = [/(7/p1) to_power (1/e)\];
set i1 = [/(p1 to_power -(2/e)) + 1\];
set N = max( max(i0, i1), 2 );
A3: N >= max(i0, i1) by XXREAL_0:25;
A4: N is Integer
proof
per cases by XXREAL_0:16;
suppose
N = max(i0, i1);
hence thesis by XXREAL_0:16;
end;
suppose
N = 2;
hence thesis;
end;
end;
A5: (p to_power -(2/e)) + 1 > (p to_power -(2/e)) + 0 by XREAL_1:8;
i1 >= (p to_power -(2/e)) + 1 by INT_1:def 7;
then
A6: i1 > (p to_power -(2/e)) by A5,XXREAL_0:2;
assume
A7: p > 0;
then
A8: p1 to_power 2 > 0 by POWER:34;
max(i0, i1) >= i1 by XXREAL_0:25;
then
A9: N >= i1 by A3,XXREAL_0:2;
A10: i0 >= (7/p) to_power (1/e) by INT_1:def 7;
max(i0, i1) >= i0 by XXREAL_0:25;
then
A11: N >= i0 by A3,XXREAL_0:2;
A12: N >= 2 by XXREAL_0:25;
A13: p1 to_power -(2/e) > 0 by A7,POWER:34;
A14: 7*p" > 7*0 by A7,XREAL_1:68;
then
A15: (7/p1) to_power (1/e) > 0 by POWER:34;
N in NAT by A12,A4,INT_1:3;
then reconsider N as Nat;
take N;
let n be Nat;
set c = p1*(n to_power e);
assume
A16: n >= N;
then n >= i0 by A11,XXREAL_0:2;
then n >= (7/p) to_power (1/e) by A10,XXREAL_0:2;
then n to_power e >= (7/p) to_power (1/e) to_power e by A15,Lm6;
then n to_power e >= (7/p1) to_power (e*(1/e)) by A14,POWER:33;
then n to_power e >= (7/p) to_power 1 by XCMPLX_1:106;
then n to_power e >= 7/p1 by POWER:25;
then p*(n to_power e) >= (7/p)*p by A7,XREAL_1:64;
then p*(n to_power e) >= 7 by A7,XCMPLX_1:87;
then p*(n to_power e) > 6 by XXREAL_0:2;
then
A17: (p1*(n to_power e))^2 < 2 to_power (p1*(n to_power e)) by Lm9;
n >= i1 by A9,A16,XXREAL_0:2;
then n > p to_power -(2/e) by A6,XXREAL_0:2;
then n to_power e > p to_power (-(2/e)) to_power e by A13,POWER:37;
then n to_power e > p1 to_power ((-(2/e))*e) by A7,POWER:33;
then n to_power e > p to_power -((2/e)*e);
then n to_power e > p to_power -2 by XCMPLX_1:87;
then
(p to_power 2)*(n to_power e) > (p to_power 2)*(p to_power -2) by A8,
XREAL_1:68;
then (p1 to_power 2)*(n to_power e) > p1 to_power (2+-2) by A7,POWER:27;
then (p1 to_power 2)*(n to_power e) > 1 by POWER:24;
then p1^2*(n to_power e) > 1 by POWER:46;
then
A18: 1/(p*c) < 1/1 by XREAL_1:88;
2 to_power c > 0 by POWER:34;
then
A19: (2 to_power c)/(c*p) < (2 to_power c)*1 by A18,XREAL_1:68;
A20: n to_power e > 0 by A12,A16,POWER:34;
then p*(n to_power e) > p*0 by A7,XREAL_1:68;
then c < (2 to_power c)/c by A17,XREAL_1:81;
then n to_power e < ((2 to_power c)/c)/p by A7,XREAL_1:81;
then n to_power e < (2 to_power c)/(c*p) by XCMPLX_1:78;
then n to_power e < 2 to_power c by A19,XXREAL_0:2;
then log(2,n to_power e) < log(2,2 to_power c) by A20,POWER:57;
then log(2,n to_power e) < c*log(2,2) by POWER:55;
then log(2,n to_power e) < c*1 by POWER:52;
then
A21: log(2,n to_power e)/(n to_power e) < p by A12,A16,POWER:34,XREAL_1:83;
n >= 2 by A12,A16,XXREAL_0:2;
then n > 1 by XXREAL_0:2;
then n to_power e > n to_power 0 by POWER:39;
then n to_power e > 1 by POWER:24;
then log(2,n to_power e) > log(2,1) by POWER:57;
then
A22: log(2,n to_power e) > 0 by POWER:51;
reconsider nn=n as Element of NAT by ORDINAL1:def 12;
h.n = f.n/g.nn by Lm4
.= log(2,nn to_power e)/g.n by A1,A12,A16
.= log(2,nn to_power e)/(n to_power e) by A12,A16,Def3;
hence |.h.n-0.| < p by A20,A21,A22,ABSVALUE:def 1;
end;
hence h is convergent by SEQ_2:def 6;
hence thesis by A2,SEQ_2:def 7;
end;
Lm11: for e being Real st e > 0
holds (seq_logn /" seq_n^(e)) is convergent &
lim(seq_logn /" seq_n^(e)) = 0
proof
set f = seq_logn;
let e be Real;
assume e > 0;
then reconsider e as positive Real;
set g = seq_n^(e);
set h = f/"g;
ex s being Real_Sequence st s.0 = 0 & for n st n > 0 holds s.n = log(2,n
to_power e)
proof
defpred P[Element of NAT,Real] means ($1 = 0 implies $2 = 0) & ($1 > 0
implies $2 = log(2,$1 to_power e));
A1: for x being Element of NAT ex y being Element of REAL st P[x,y]
proof
let n;
per cases;
suppose
n = zz;
hence thesis;
end;
suppose
A2: n > 0;
log(2,n to_power e) in REAL by XREAL_0:def 1;
hence thesis by A2;
end;
end;
consider h being sequence of REAL such that
A3: for x being Element of NAT holds P[x,h.x] from FUNCT_2:sch 3(A1);
take h;
thus h.0 = 0 by A3;
let n;
thus thesis by A3;
end;
then consider p being Real_Sequence such that
A4: p.0 = 0 and
A5: for n st n > 0 holds p.n = log(2,n to_power e);
set q = p/"g;
A6: q is convergent by A5,Lm10;
A7: 1 = e/e by XCMPLX_1:60
.= e*(1/e);
A8: for n being Nat holds h.n = (1/e)*q.n
proof
let n be Nat;
A9: n in NAT by ORDINAL1:def 12;
A10: h.n = f.n / g.n by Lm4;
A11: q.n = p.n / g.n by Lm4;
per cases;
suppose
A12: n = 0;
then h.n = 0 / g.n by A10,Def2
.= 0*(1/e);
hence thesis by A4,A11,A12;
end;
suppose
A13: n > 0;
then
A14: n to_power e > 0 by POWER:34;
h.n = log(2,n) / g.n by A10,A13,Def2,A9
.= log(2,n to_power (e*(1/e))) / g.n by A7,POWER:25
.= log(2,n to_power e to_power (1/e)) / g.n by A13,POWER:33
.= ((1/e)*log(2,n to_power e)) / g.n by A14,POWER:55
.= ((1/e)*log(2,n to_power e)) * (g.n)"
.= (1/e)*(log(2,n to_power e) * (g.n)")
.= (1/e)*(log(2,n to_power e) / g.n);
hence thesis by A5,A11,A13,A9;
end;
end;
then
A15: h = (1/e)(#)q by SEQ_1:9;
A16: lim q = 0 by A5,Lm10;
lim h = lim((1/e)(#)q) by A8,SEQ_1:9
.= (1/e)*0 by A6,A16,SEQ_2:8;
hence thesis by A6,A15,SEQ_2:7;
end;
theorem Th5:
Big_Oh(seq_logn) c= Big_Oh(seq_n^(1/2)) &
not Big_Oh(seq_logn) = Big_Oh(seq_n^(1/2))
proof
set g = seq_n^(1/2);
set f = seq_logn;
A1: lim (f/"g) = 0 by Lm11;
A2: f/"g is convergent by Lm11;
then not g in Big_Oh(f) by A1,ASYMPT_0:16;
then
A3: not f in Big_Omega(g) by ASYMPT_0:19;
f in Big_Oh(g) by A2,A1,ASYMPT_0:16;
hence thesis by A3,Th4;
end;
:: Example p. 86
theorem
seq_n^(1/2) in Big_Omega(seq_logn) & not seq_logn in Big_Omega(seq_n^( 1/2))
proof
seq_logn in Big_Oh(seq_n^(1/2)) by Th4,Th5;
hence thesis by Th4,Th5,ASYMPT_0:19;
end;
:: Example p. 88
Lm12: for f being Real_Sequence holds for N holds (for n st n <= N holds f.n
>= 0) implies Sum(f,N) >= 0
proof
let f be Real_Sequence;
defpred P[Nat] means
(for n st n <= $1 holds f.n >= 0) implies Sum(f,$1) >= 0;
A1: for N being Nat st P[N] holds P[N+1]
proof
let N be Nat;
assume
A2: (for n st n <= N holds f.n >= 0) implies Sum(f,N) >= 0;
assume
A3: for n st n <= (N+1) holds f.n >= 0;
A4: now
let n;
assume n <= N;
then n+0 <= N+1 by XREAL_1:7;
hence f.n >= 0 by A3;
end;
f.(N+1) >= 0 by A3;
then Sum(f,N) + f.(N+1) >= 0 + 0 by A2,A4;
then (Partial_Sums(f)).N + f.(N+1) >= 0 by SERIES_1:def 5;
then (Partial_Sums(f)).(N+1) >= 0 by SERIES_1:def 1;
hence thesis by SERIES_1:def 5;
end;
A5: P[0]
proof
assume for n st n <= 0 holds f.n >= 0;
then f.0 >= 0;
then (Partial_Sums(f)).0 >= 0 by SERIES_1:def 1;
hence thesis by SERIES_1:def 5;
end;
for N being Nat holds P[N] from NAT_1:sch 2(A5, A1);
hence thesis;
end;
Lm13: for f,g being Real_Sequence holds for N holds (for n st n <= N holds f.n
<= g.n) implies Sum(f,N) <= Sum (g,N)
proof
let f,g be Real_Sequence;
defpred P[Nat] means
(for n st n <= $1 holds f.n <= g.n) implies
Sum(f,$1) <= Sum(g,$1);
A1: for N being Nat st P[N] holds P[N+1]
proof
let N be Nat;
assume
A2: (for n st n <= N holds f.n <= g.n) implies Sum(f,N) <= Sum(g,N);
assume
A3: for n st n <= (N+1) holds f.n <= g.n;
A4: now
let n;
assume n <= N;
then n+0 <= N+1 by XREAL_1:7;
hence f.n <= g.n by A3;
end;
f.(N+1) <= g.(N+1) by A3;
then Sum(f,N) + f.(N+1) <= Sum(g,N) + g.(N+1) by A2,A4,XREAL_1:7;
then (Partial_Sums(f)).N + f.(N+1) <= Sum(g,N) + g.(N+1) by SERIES_1:def 5;
then (Partial_Sums(f)).(N+1) <= Sum(g,N) + g.(N+1) by SERIES_1:def 1;
then Sum(f,N+1) <= Sum(g,N) + g.(N+1) by SERIES_1:def 5;
then Sum(f,N+1) <= (Partial_Sums(g)).N + g.(N+1) by SERIES_1:def 5;
then Sum(f,N+1) <= (Partial_Sums(g)).(N+1) by SERIES_1:def 1;
hence thesis by SERIES_1:def 5;
end;
A5: P[0]
proof
assume for n st n <= 0 holds f.n <= g.n;
then f.0 <= g.0;
then Partial_Sums(f).0 <= g.0 by SERIES_1:def 1;
then Partial_Sums(f).0 <= Partial_Sums(g).0 by SERIES_1:def 1;
then Sum(f,0) <= Partial_Sums(g).0 by SERIES_1:def 5;
hence thesis by SERIES_1:def 5;
end;
for N being Nat holds P[N] from NAT_1:sch 2(A5, A1);
hence thesis;
end;
Lm14: for f being Real_Sequence, b being Real
st f.0 = 0 & (for n st n > 0
holds f.n = b) holds for N being Element of NAT holds Sum(f,N) = b*N
proof
let f be Real_Sequence, b be Real;
defpred P[Nat] means Sum(f,$1) = b*$1;
assume that
A1: f.0 = 0 and
A2: for n st n > 0 holds f.n = b;
A3: for N being Nat st P[N] holds P[N+1]
proof
let N be Nat;
assume
A4: Sum(f, N) = b*N;
Sum(f, N+1) = (Partial_Sums(f)).(N+1) by SERIES_1:def 5
.= (Partial_Sums(f)).N + f.(N+1) by SERIES_1:def 1
.= b*N + f.(N+1) by A4,SERIES_1:def 5
.= b*N + b*1 by A2
.= b*(N+1);
hence thesis;
end;
(Partial_Sums(f)).0 = 0 by A1,SERIES_1:def 1;
then
A5: P[0] by SERIES_1:def 5;
for N being Nat holds P[N] from NAT_1:sch 2( A5, A3 );
hence thesis;
end;
Lm15: for f being Real_Sequence, N,M being Nat holds Sum(f,N,M) + f.(N+1) =
Sum(f,N+1,M)
proof
let f be Real_Sequence, N,M be Nat;
Sum(f,N,M) + f.(N+1) = Sum(f,N) - Sum(f,M) + f.(N+1) by SERIES_1:def 6
.= Sum(f,N) + f.(N+1) + -Sum(f,M)
.= (Partial_Sums(f)).N + f.(N+1) + -Sum (f,M) by SERIES_1:def 5
.= (Partial_Sums(f)).(N+1) + -Sum(f,M) by SERIES_1:def 1
.= Sum(f,N+1) + -Sum(f,M) by SERIES_1:def 5
.= Sum(f,N+1) - Sum(f,M)
.= Sum(f,N+1,M) by SERIES_1:def 6;
hence thesis;
end;
Lm16: for f,g being Real_Sequence, M being Element of NAT holds for N st N >=
M+1 holds (for n st M+1 <= n & n <= N holds f.n <= g.n) implies Sum(f,N,M) <=
Sum (g,N,M)
proof
let f,g be Real_Sequence, M be Element of NAT;
defpred P[Nat] means (for n st M+1 <= n & n <= $1 holds f.n <= g.n) implies
Sum(f,$1,M) <= Sum(g,$1,M);
A1: for N1 be Nat st N1 >= M+1 & P[N1] holds P[N1+1]
proof
let N1 be Nat;
assume that
A2: N1 >= M+1 and
A3: (for n st M+1 <= n & n <= N1 holds f.n <= g.n) implies Sum(f,N1,M)
<= Sum(g,N1,M);
assume
A4: for n st M+1 <= n & n <= N1+1 holds f.n <= g.n;
A5: now
let n;
assume that
A6: M+1 <= n and
A7: n <= N1;
n + 0 <= N1 + 1 by A7,XREAL_1:7;
hence f.n <= g.n by A4,A6;
end;
N1+1 >= M+1+0 by A2,XREAL_1:7;
then f.(N1+1) <= g.(N1+1) by A4;
then Sum(f,N1,M) + f.(N1+1) <= g.(N1+1) + Sum(g,N1,M) by A3,A5,XREAL_1:7;
then Sum(f,N1+1,M) <= g.(N1+1) + Sum(g,N1,M) by Lm15;
hence thesis by Lm15;
end;
A8: P[M+1]
proof
A9: Sum(g,M+1,M) = Sum(g,M+1) - Sum(g,M) by SERIES_1:def 6
.= (Partial_Sums(g)).(M+1) - Sum(g,M) by SERIES_1:def 5
.= g.(M+1) + (Partial_Sums(g)).M - Sum(g,M) by SERIES_1:def 1
.= g.(M+1) + Sum(g,M) - Sum(g,M) by SERIES_1:def 5
.= g.(M+1) + 0;
A10: Sum(f,M+1,M) = Sum(f,M+1) - Sum(f,M) by SERIES_1:def 6
.= (Partial_Sums(f)).(M+1) - Sum(f,M) by SERIES_1:def 5
.= f.(M+1) + (Partial_Sums(f)).M - Sum(f,M) by SERIES_1:def 1
.= f.(M+1) + Sum(f,M) - Sum(f,M) by SERIES_1:def 5
.= f.(M+1) + 0;
assume for n st M+1 <= n & n <= M+1 holds f.n <= g.n;
hence thesis by A10,A9;
end;
for N be Nat st N >= M+1 holds P[N] from NAT_1:sch 8(A8, A1);
hence thesis;
end;
Lm17: for n holds [/n/2\] <= n
proof
let n;
per cases;
suppose
n = 0;
hence thesis by INT_1:30;
end;
suppose
n > 0;
then
A1: n >= 0+1 by NAT_1:13;
per cases by A1,XXREAL_0:1;
suppose
A2: n = 1;
now
assume [/1/2\] > 1;
then
A3: [/1/2\] >= 1+1 by INT_1:7;
[/1/2\] < 1/2 + 1 by INT_1:def 7;
hence contradiction by A3,XXREAL_0:2;
end;
hence thesis by A2;
end;
suppose
n > 1;
then
A4: n >= 1+1 by NAT_1:13;
A5: now
assume n/2 + 1 > n;
then 2*(n/2 + 1) > 2*n by XREAL_1:68;
then 2*(n/2) + 2*1 > 2*n;
then 2 > 2*n - n by XREAL_1:19;
hence contradiction by A4;
end;
[/n/2\] < n/2 + 1 by INT_1:def 7;
hence thesis by A5,XXREAL_0:2;
end;
end;
end;
Lm18: for f being Real_Sequence,
b being Real, N being Element of NAT st f.0 =
0 & (for n st n > 0 holds f.n = b) holds for M being Element of NAT holds Sum(f
, N, M) = b*(N-M)
proof
let f be Real_Sequence, b be Real, N be Element of NAT such that
A1: f.0 = 0 and
A2: for n st n > 0 holds f.n = b;
defpred P[Nat] means Sum(f, N, $1) = b*(N-$1);
A3: for M being Nat st P[M] holds P[M+1]
proof
let M be Nat;
assume
A4: Sum(f, N, M) = b*(N-M);
Sum(f, N, M+1) = Sum(f, N) - Sum(f, M+1) by SERIES_1:def 6
.= Sum(f, N) - (Partial_Sums(f)).(M+1) by SERIES_1:def 5
.= Sum (f, N) - ((Partial_Sums(f)).M + f.(M+1)) by SERIES_1:def 1
.= (Sum (f, N) - (Partial_Sums(f)).M) + -f.(M+1)
.= (Sum(f, N) - Sum(f, M)) + -f.(M+1) by SERIES_1:def 5
.= b*(N-M) + -f.(M+1) by A4,SERIES_1:def 6
.= b*(N-M) + -b by A2
.= b*(N-(M+1));
hence thesis;
end;
Sum(f, 0) = (Partial_Sums(f)).0 by SERIES_1:def 5
.= 0 by A1,SERIES_1:def 1;
then Sum(f, N, 0) = Sum(f, N) - 0 by SERIES_1:def 6
.= b*(N-0) by A1,A2,Lm14;
then
A5: P[0];
for M being Nat holds P[M] from NAT_1:sch 2(A5, A3);
hence thesis;
end;
theorem
for f being Real_Sequence, k being Element of NAT st (for n holds f.n
= Sum(seq_n^(k), n)) holds f in Big_Theta(seq_n^(k+1))
proof
let f be Real_Sequence, k be Element of NAT such that
A1: for n holds f.n = Sum(seq_n^(k), n);
set g = seq_n^(k+1);
A2: f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
A3: now
set p = seq_n^(k);
let n;
set n1 = [/n/2\];
ex s being Real_Sequence st s.0 = 0 & for m st m > 0 holds s.m = ((n/
2) to_power k)
proof
defpred P[Element of NAT,Real] means ($1 = 0 implies $2 = 0) & ($1 > 0
implies $2 = ((n/2) to_power k));
A4: for x being Element of NAT ex y being Element of REAL st P[x,y]
proof
let x be Element of NAT;
per cases;
suppose
x = zz;
hence thesis;
end;
suppose
A5: x > 0;
reconsider y = (n/2) to_power k as Element of REAL by XREAL_0:def 1;
take y;
P[x,y] by A5;
hence thesis;
end;
end;
consider h being sequence of REAL such that
A6: for x being Element of NAT holds P[x,h.x] from FUNCT_2:sch 3(
A4);
take h;
thus h.0 = 0 by A6;
let n;
thus thesis by A6;
end;
then consider q being Real_Sequence such that
A7: q.0 = 0 and
A8: for m st m > 0 holds q.m = (n/2) to_power k;
A9: [/n/2\] >= n/2 by INT_1:def 7;
then reconsider n1 as Element of NAT by INT_1:3;
set n2 = n1-1;
assume
A10: n >= 1;
then
A11: n*2" > 0*2" by XREAL_1:68;
then
A12: ((n/2) to_power k) > 0 by POWER:34;
now
assume n2 < 0;
then n1-1 <= -1 by INT_1:8;
then (n1-1)+1 <= -1+1 by XREAL_1:6;
hence contradiction by A11,INT_1:def 7;
end;
then reconsider n2 as Element of NAT by INT_1:3;
A13: now
[/n/2\] < n/2 + 1 by INT_1:def 7;
then n2 < n/2 by XREAL_1:19;
then
A14: n/2 + n2 < n/2 + n/2 by XREAL_1:6;
assume n-n2 < n/2;
hence contradiction by A14,XREAL_1:19;
end;
Sum(q,n,n2) = (n-n2)*((n/2) to_power k) by A7,A8,Lm18;
then Sum(q,n,n2) >= (n/2) * ((n/2) to_power k) by A13,A12,XREAL_1:64;
then Sum(q,n,n2) >= ((n/2) to_power 1) * ((n/2) to_power k) by POWER:25;
then Sum(q,n,n2) >= ((n/2) to_power (k+1)) by A11,POWER:27;
then
A15: Sum(q,n,n2) >= (n to_power (k+1))/(2 to_power (k+1)) by A10,POWER:31;
A16: f.n = Sum(p, n) by A1;
A17: now
let m;
assume m <= n;
per cases;
suppose
m = 0;
hence p.m >= 0 by Def3;
end;
suppose
m > 0;
then p.m = m to_power k by Def3;
hence p.m >= 0;
end;
end;
now
let m;
n1 <= n1+1 by NAT_1:11;
then
A18: n2 <= n1 by XREAL_1:20;
A19: n1 <= n by Lm17;
assume m <= n2;
then m <= n1 by A18,XXREAL_0:2;
then m <= n by A19,XXREAL_0:2;
hence p.m >= 0 by A17;
end;
then Sum(p,n2) >= 0 by Lm12;
then
A20: Sum(p,n) + Sum(p,n2) >= Sum(p,n) + 0 by XREAL_1:7;
A21: for N0 st n1 <= N0 & N0 <= n holds q.N0 <= p.N0
proof
let N0;
assume that
A22: n1 <= N0 and
N0 <= n;
A23: N0 >= n/2 by A9,A22,XXREAL_0:2;
A24: p.N0 = N0 to_power k by A11,A9,A22,Def3;
q.N0 = (n/2) to_power k by A8,A11,A9,A22;
hence thesis by A11,A24,A23,Lm6;
end;
n >= n2 + 1 by Lm17;
then Sum(p,n,n2) >= Sum(q,n,n2) by A21,Lm16;
then
A25: Sum(p,n,n2) >= (n to_power (k+1))*(2 to_power (k+1))" by A15,XXREAL_0:2;
Sum(p,n,n2) = Sum(p,n) - Sum(p,n2) by SERIES_1:def 6;
then
A26: Sum(p,n) >= Sum(p,n,n2) by A20,XREAL_1:20;
g.n = n to_power (k+1) by A10,Def3;
hence (2 to_power (k+1))"*g.n <= f.n by A16,A26,A25,XXREAL_0:2;
Sum(p,n) >= 0 by A17,Lm12;
hence f.n >= 0 by A1;
end;
now
set p = seq_n^(k);
let n;
assume
A27: n >= 1;
ex s being Real_Sequence st s.0 = 0 & for m st m > 0 holds s.m = (n
to_power k)
proof
defpred P[Element of NAT,Real] means ($1 = 0 implies $2 = 0) & ($1 > 0
implies $2 = n to_power k);
A28: for x being Element of NAT ex y being Element of REAL st P[x,y]
proof
let x be Element of NAT;
per cases;
suppose
x = zz;
hence thesis;
end;
suppose
A29: x > 0;
reconsider y = n to_power k as Element of REAL by XREAL_0:def 1;
take y;
thus thesis by A29;
end;
end;
consider h being sequence of REAL such that
A30: for x being Element of NAT holds P[x,h.x] from FUNCT_2:sch 3(
A28
);
take h;
thus h.0 = 0 by A30;
let n;
thus thesis by A30;
end;
then consider q being Real_Sequence such that
A31: q.0 = 0 and
A32: for m st m > 0 holds q.m = n to_power k;
now
let m;
assume
A33: m <= n;
per cases;
suppose
m = 0;
hence p.m <= q.m by A31,Def3;
end;
suppose
A34: m > 0;
then
A35: q.m = n to_power k by A32;
p.m = m to_power k by A34,Def3;
hence p.m <= q.m by A33,A34,A35,Lm6;
end;
end;
then
A36: Sum(p, n) <= Sum(q, n) by Lm13;
Sum(q, n) = (n to_power k)*n by A31,A32,Lm14
.= (n to_power k)*(n to_power 1) by POWER:25
.= n to_power (k+1) by A27,POWER:27
.= g.n by A27,Def3;
hence f.n <= 1*g.n by A1,A36;
A37: now
let m;
assume m <= n;
per cases;
suppose
m = 0;
hence p.m >= 0 by Def3;
end;
suppose
m > 0;
then p.m = m to_power k by Def3;
hence p.m >= 0;
end;
end;
f.n = Sum(p, n) by A1;
hence f.n >= 0 by A37,Lm12;
end;
then
A38: f in Big_Oh(g) by A2;
2 to_power (k+1) > 0 by POWER:34;
then f in Big_Omega(g) by A2,A3;
hence thesis by A38,XBOOLE_0:def 4;
end;
:: Example p. 89
theorem
for f being Real_Sequence st (for n st n > 0 holds f.n = (n to_power
log(2,n))) holds ex s being eventually-positive Real_Sequence st s = f & not s
is smooth
proof
let f be Real_Sequence such that
A1: for n st n > 0 holds f.n = (n to_power log(2,n));
A2: f is eventually-positive
proof
take 1;
let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
assume
A4: n >= 1;
then f.n = n to_power log(2,n) by A1,A3;
hence thesis by A4,POWER:34;
end;
set g = f taken_every 2;
reconsider f as eventually-positive Real_Sequence by A2;
take f;
now
assume f is smooth;
then f is_smooth_wrt 2;
then g in Big_Oh(f);
then consider t being Element of Funcs(NAT, REAL) such that
A5: t = g and
A6: ex c,N st c > 0 & for n st n >= N holds t.n <= c*f.n & t.n >= 0;
consider c,N such that
A7: c > 0 and
A8: for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A6;
A9: sqrt c > 0 by A7,SQUARE_1:25;
set N0 = [/sqrt c / sqrt 2\];
reconsider N2=max(N, N0) as Integer by XXREAL_0:16;
set N1 = max( N2, 2 );
A10: N1 >= N2 by XXREAL_0:25;
N2 >= N0 by XXREAL_0:25;
then
A11: N1 >= N0 by A10,XXREAL_0:2;
A12: N1 is Integer by XXREAL_0:16;
N2 >= N by XXREAL_0:25;
then
A13: N1 >= N by A10,XXREAL_0:2;
N1 >= 2 by XXREAL_0:25;
then reconsider N1 as Element of NAT by A12,INT_1:3;
set n = N1 + 1;
A14: n to_power log(2,n) > 0 by POWER:34;
A15: 2*n > 2*0 by XREAL_1:68;
A16: sqrt 2 <> 0 by SQUARE_1:25;
A17: sqrt 2 > 0 by SQUARE_1:25;
A18: N0 >= sqrt c / sqrt 2 by INT_1:def 7;
A19: n > N1 + 0 by XREAL_1:8;
then n > N0 by A11,XXREAL_0:2;
then n > sqrt c / sqrt 2 by A18,XXREAL_0:2;
then n*sqrt 2 > (sqrt c / sqrt 2) * sqrt 2 by A17,XREAL_1:68;
then n*sqrt 2 > sqrt c by A16,XCMPLX_1:87;
then (n*sqrt 2)^2 > (sqrt c)^2 by A9,SQUARE_1:16;
then n^2*(sqrt 2)^2 > c by A7,SQUARE_1:def 2;
then
A20: 2*n^2 > c by SQUARE_1:def 2;
(2*n^2)*(n to_power log(2,n)) = ((2*n)*n)*(n to_power log(2,n))
.= ((2*n)*(2 to_power (log(2,n))))*(n to_power log(2,n)) by POWER:def 3
.= (2*n)*((2 to_power (log(2,n)))*(n to_power log(2,n)))
.= (2*n)*((2*n) to_power log(2,n)) by POWER:30
.= ((2*n) to_power 1)*((2*n) to_power log(2,n)) by POWER:25
.= (2*n) to_power (1 + log(2,n)) by A15,POWER:27
.= (2*n) to_power (log(2,2) + log(2,n)) by POWER:52
.= (2*n) to_power log(2,2*n) by POWER:53;
then (2*n) to_power log(2,2*n) > c*(n to_power log(2,n)) by A14,A20,
XREAL_1:68;
then f.(2*n) > c*(n to_power log(2,n)) by A1,A15;
then t.n > c*(n to_power log(2,n)) by A5,ASYMPT_0:def 15;
then
A21: t.n > c*f.n by A1;
n > N by A13,A19,XXREAL_0:2;
hence contradiction by A8,A21;
end;
hence thesis;
end;
:: Example p. 92
definition
::$CD
end;
registration
cluster seq_const(1) -> eventually-nonnegative;
coherence
proof
take 0;
let n be Nat;
assume n >= 0;
thus thesis;
end;
end;
Lm19: for a,b,c being Real
holds a>1 & b>=a & c>=1 implies log(a,c) >= log(b,c)
proof
let a,b,c be Real;
assume that
A1: a > 1 and
A2: b >= a and
A3: c >= 1;
b > 1 by A1,A2,XXREAL_0:2;
then log(b,c) >= log(b,1) by A3,PRE_FF:10;
then
A4: log(b,c) >= 0 by A1,A2,POWER:51;
log(a,b) >= log(a,a) by A1,A2,PRE_FF:10;
then log(a,b) >= 1 by A1,POWER:52;
then log(a,b)*log(b,c) >= 1*log(b,c) by A4,XREAL_1:64;
hence thesis by A1,A2,A3,POWER:56;
end;
theorem Th9:
for f being eventually-nonnegative Real_Sequence holds ex F being
FUNCTION_DOMAIN of NAT,REAL st F = { seq_n^(1) } & (f in F to_power Big_Oh(
seq_const(1)) iff ex N,c,k st c>0 & for n st n >= N holds 1 <= f.n & f.n <= c*(
seq_n^(k)).n)
proof
set p = seq_const(1);
set G = Big_Oh(seq_const(1));
reconsider F = { seq_n^(1) } as FUNCTION_DOMAIN of NAT,REAL by FUNCT_2:121;
let h be eventually-nonnegative Real_Sequence;
take F;
thus F = { seq_n^(1) };
now
hereby
reconsider i = 1 as Element of NAT;
assume h in F to_power Big_Oh(seq_const(1));
then consider t being Element of Funcs(NAT,REAL) such that
A1: h = t and
A2: ex f,g being Element of Funcs(NAT,REAL), N being Element of NAT
st f in F & g in G & for n being Element of NAT st n>=N holds t.n=(f.n)
to_power (g.n);
consider f,g being Element of Funcs(NAT,REAL), N0 being Element of NAT
such that
A3: f in F and
A4: g in G and
A5: for n being Element of NAT st n>=N0 holds t.n = (f.n) to_power (
g.n) by A2;
consider g9 being Element of Funcs(NAT, REAL) such that
A6: g = g9 and
A7: ex c,N st c > 0 & for n st n >= N holds g9.n <= c*p.n & g9.n >= 0 by A4;
consider c,N1 such that
A8: c > 0 and
A9: for n st n >= N1 holds g9.n <= c*p.n & g9.n >= 0 by A7;
set k = [/c\];
A10: k > 0 by A8,INT_1:def 7;
set N = max(2,max(N0,N1));
A11: N >= max(N0,N1) by XXREAL_0:25;
max(N0,N1) >= N0 by XXREAL_0:25;
then
A12: N >= N0 by A11,XXREAL_0:2;
A13: k >= c by INT_1:def 7;
reconsider k as Element of NAT by A10,INT_1:3;
take N,i,k;
thus i > 0;
let n;
assume
A14: n >= N;
A15: N >= 2 by XXREAL_0:25;
then n >= 2 by A14,XXREAL_0:2;
then
A16: n > 1 by XXREAL_0:2;
then
A17: n to_power c <= n to_power k by A13,PRE_FF:8;
f = seq_n^(1) by A3,TARSKI:def 1;
then f.n = n to_power 1 by A15,A14,Def3
.= n by POWER:25;
then
A18: h.n = n to_power (g.n) by A1,A5,A12,A14,XXREAL_0:2;
max(N0,N1) >= N1 by XXREAL_0:25;
then N >= N1 by A11,XXREAL_0:2;
then
A19: n >= N1 by A14,XXREAL_0:2;
then g9.n >= 0 by A9;
then n to_power (g.n) >= n to_power 0 by A6,A16,PRE_FF:8;
hence 1 <= h.n by A18,POWER:24;
A20: p.n = 1 by FUNCOP_1:7;
g.n <= c*p.n by A6,A9,A19;
then h.n <= n to_power (c*1) by A20,A16,A18,PRE_FF:8;
then h.n <= n to_power k by A17,XXREAL_0:2;
hence h.n <= i*(seq_n^(k)).n by A15,A14,Def3;
end;
reconsider f = seq_n^(1) as Element of Funcs(NAT,REAL) by FUNCT_2:8;
reconsider t = h as Element of Funcs(NAT,REAL) by FUNCT_2:8;
given N0,c,k such that
c > 0 and
A21: for n st n >= N0 holds 1 <= h.n & h.n <= c*(seq_n^(k)).n;
reconsider N = max(N0,2) as Element of REAL by XREAL_0:def 1;
defpred Q[Element of NAT,Real] means ($1 < N implies $2 = 1) & ($1 >= N
implies $2 = log($1,t.$1));
A22: N >= 2 by XXREAL_0:25;
then
A23: N > 1 by XXREAL_0:2;
A24: for x being Element of NAT ex y being Element of REAL st Q[x,y]
proof
let n;
per cases;
suppose
A25: n < N;
1 in REAL by XREAL_0:def 1;
hence thesis by A25;
end;
suppose
A26: n >= N;
reconsider y=log(n,t.n) as Element of REAL by XREAL_0:def 1;
take y;
Q[n,y] by A26;
hence thesis;
end;
end;
consider g being sequence of REAL such that
A27: for x being Element of NAT holds Q[x,g.x] from FUNCT_2:sch 3(A24);
A28: N >= N0 by XXREAL_0:25;
A29: now
let n be Element of NAT;
assume
A30: n >= N;
then n >= N0 by A28,XXREAL_0:2;
then
A31: t.n >= 1 by A21;
thus (f.n) to_power (g.n) = (n to_power 1) to_power (g.n) by A22,A30,Def3
.= n to_power (g.n) by POWER:25
.= n to_power (1*log(n,t.n)) by A27,A30
.= t.n by A23,A30,A31,POWER:def 3;
end;
set c1= max(c,2);
A32: N <> 1 by XXREAL_0:25;
set a = log(N,c1);
set b = k+a;
A33: c1 >= 2 by XXREAL_0:25;
then
A34: c1 > 1 by XXREAL_0:2;
A35: f in F by TARSKI:def 1;
A36: g is Element of Funcs(NAT,REAL) by FUNCT_2:8;
A37: N > 0 by XXREAL_0:25;
now
log(N,1) = 0 by A37,A32,POWER:51;
then a > 0 by A23,A34,POWER:57;
hence b > 0;
let n;
A38: (seq_const(1)).n = 1 by FUNCOP_1:7;
assume
A39: n >= N;
then
A40: n <> 1 by A22,XXREAL_0:2;
A41: (seq_n^(k)).n = n to_power k by A22,A39,Def3;
then
A42: c*(seq_n^(k)).n <= c1*(seq_n^(k)).n by XREAL_1:64,XXREAL_0:25;
(seq_n^(k)).n > 0 by A22,A39,A41,POWER:34;
then
A43: log(n,c1*(seq_n^(k)).n) = log(n,c1) + log(n,n to_power k) by A22,A33,A39
,A40,A41,POWER:53
.= log(n,c1) + k*log(n,n) by A22,A39,A40,POWER:55
.= log(n,c1) + k*1 by A22,A39,A40,POWER:52;
a >= log(n,c1) by A23,A34,A39,Lm19;
then
A44: log(n,c1) + k <= a + k by XREAL_1:6;
A45: n >= N0 by A28,A39,XXREAL_0:2;
then
A46: 1 <= t.n by A21;
t.n = (f.n) to_power (g.n) by A29,A39
.= (n to_power 1) to_power (g.n) by A22,A39,Def3
.= n to_power (g.n) by POWER:25;
then
A47: log(n,t.n) = g.n*log(n,n) by A22,A39,A40,POWER:55
.= g.n*1 by A22,A39,A40,POWER:52;
n >= 2 by A22,A39,XXREAL_0:2;
then
A48: n > 1 by XXREAL_0:2;
t.n <= c*(seq_n^(k)).n by A21,A45;
then t.n <= c1*(seq_n^(k)).n by A42,XXREAL_0:2;
then log(n,t.n) <= log(n,c1*(seq_n^(k)).n) by A48,A46,PRE_FF:10;
hence g.n <= b*(seq_const(1)).n by A47,A43,A44,A38,XXREAL_0:2;
g.n = log(n,t.n) by A27,A39;
then g.n >= log(n,1) by A48,A46,PRE_FF:10;
hence g.n >= 0 by A22,A39,A40,POWER:51;
end;
then g in G by A36;
hence h in F to_power Big_Oh(seq_const(1)) by A36,A29,A35;
end;
hence thesis;
end;
begin :: Problem 3.1
theorem
for f being Real_Sequence st (for n holds f.n = 3*(10 to_power 6) - 18
*(10 to_power 3)*n + 27*n^2) holds f in Big_Oh(seq_n^(2))
proof
set g = seq_n^(2);
consider t1 being Element of NAT such that
A1: t1 = 10*10*10;
consider t2 being Element of NAT such that
A2: t2 = t1*t1;
t1 = 10*10^2 by A1;
then t1 = 10*(10 to_power 2) by POWER:46;
then t1 = (10 to_power 1)*(10 to_power 2) by POWER:25;
then
A3: t1 = 10 to_power (1+2) by POWER:27;
then
A4: t2 = 10 to_power (3+3) by A2,POWER:27
.= 10 to_power 6;
A5: 10 to_power 3 = 10 to_power (2+1)
.= (10 to_power 2)*(10 to_power 1) by POWER:27
.= (10 to_power 2)*10 by POWER:25
.= (10^2)*10 by POWER:46
.= 1000;
A6: for n st n >= 400 holds 18*t1*n - 3*t2 < 27*n^2
proof
defpred P[Nat] means 18*t1*$1 - 3*t2 < 27*$1^2;
A7: for k be Nat st k >= 400 & P[k] holds P[k+1]
proof
let k be Nat such that
A8: k >= 400 and
A9: 18*t1*k - 3*t2 < 27*k^2;
54*400 <= 54*k by A8,XREAL_1:64;
then
A10: 18*t1 < 54*k by A3,A5,XXREAL_0:2;
54*k + 0 <= 54*k + 27 by XREAL_1:7;
then 18*t1 < 54*k + 27 by A10,XXREAL_0:2;
then
A11: 27*k^2 + 18*t1 < 27*k^2 + (54*k + 27) by XREAL_1:6;
18*t1*(k+1) - 3*t2 = (18*t1*k - 3*t2) + 18*t1;
then 18*t1*(k+1) - 3*t2 < 27*k^2 + 18*t1 by A9,XREAL_1:6;
hence thesis by A11,XXREAL_0:2;
end;
A12: P[400] by A2,A3,A5;
for n be Nat st n >= 400 holds P[n]from NAT_1:sch 8(A12, A7);
hence thesis;
end;
let f be Real_Sequence such that
A13: for n holds f.n = 3*(10 to_power 6) - 18*(10 to_power 3)*n + 27*n^2;
A14: for n st n >= 400 holds f.n <= 27*n^2
proof
let n such that
A15: n >= 400;
now
assume f.n > 27*n^2;
then 3*t2 - 18*(10 to_power 3)*n + 27*n^2 > 27*n^2 by A13,A4;
then 3*t2 + -18*t1*n > 27*n^2 - 27*n^2 by A3,XREAL_1:19;
then 3*t2 - 18*t1*n > 0;
then
A16: 3*t2 > 0 + 18*t1*n by XREAL_1:20;
(18*t1)*n >= (18*t1)*400 by A15,XREAL_1:64;
then 3*(10 to_power (3+3)) > t1*7200 by A4,A16,XXREAL_0:2;
then 3*((10 to_power 3)*(10 to_power 3)) > t1*7200 by POWER:27;
hence contradiction by A3,A5;
end;
hence thesis;
end;
A17: now
let n;
assume
A18: n >= 400;
then f.n <= 27*n^2 by A14;
then f.n <= 27*(n to_power 2) by POWER:46;
hence f.n <= 27*g.n by A18,Def3;
0 + (18*t1*n - 3*t2) < 27*n^2 by A6,A18;
then 0 < 27*n^2 - (18*t1*n - 3*t2) by XREAL_1:20;
then 0 < 3*(10 to_power 6) - 18*t1*n + 27*n^2 by A4;
hence f.n >= 0 by A13,A3;
end;
f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
hence thesis by A17;
end;
:: Problem 3.2 -- omitted
:: Problem 3.3 -- omitted
:: Problem 3.4 -- omitted
begin :: Problem 3.5
theorem
seq_n^(2) in Big_Oh(seq_n^(3))
proof
set g = seq_n^(3);
set f = seq_n^(2);
A1: now
let n;
assume
A2: n >= 2;
then
A3: n > 1 by XXREAL_0:2;
A4: f.n = n to_power 2 by A2,Def3;
g.n = n to_power 3 by A2,Def3;
hence f.n <= 1*g.n by A3,A4,POWER:39;
thus f.n >= 0 by A4;
end;
f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
hence thesis by A1;
end;
theorem :: Part 2
not seq_n^(2) in Big_Omega(seq_n^(3))
proof
set g = seq_n^(3);
set f = seq_n^(2);
now
assume seq_n^(2) in Big_Omega(seq_n^(3));
then consider s being Element of Funcs(NAT, REAL) such that
A1: s = f and
A2: ex d,N st d > 0 & for n st n >= N holds d*g.n <= s.n & s.n >= 0;
consider d,N such that
A3: d > 0 and
A4: for n st n >= N holds d*g.n <= s.n & s.n >= 0 by A2;
A5: N+2 > 1+0 by XREAL_1:8;
ex n st n >= N & d*g.n > s.n
proof
take n = max( N, [/(N+2)/d\] );
A6: n >= N by XXREAL_0:25;
A7: n is Integer by XXREAL_0:16;
A8: [/(N+2)/d\] >= (N+2)/d by INT_1:def 7;
(N+2)*d" > 0*d" by A3,XREAL_1:68;
then
A9: n > 0 by A8,XXREAL_0:25;
reconsider n as Element of NAT by A6,A7,INT_1:3;
A10: f.n*(n to_power -2) = (n to_power 2)*(n to_power -2) by A9,Def3
.= (n to_power (2+(-2))) by A9,POWER:27
.= 1 by POWER:24;
A11: (n to_power -2) > 0 by A9,POWER:34;
A12: d*n >= d*([/(N+2)/d\]) by A3,XREAL_1:64,XXREAL_0:25;
d*([/(N+2)/d\]) >= d*((N+2)/d) by A3,A8,XREAL_1:64;
then d*n >= ((N+2)/d)*d by A12,XXREAL_0:2;
then
A13: d*n >= N+2 by A3,XCMPLX_1:87;
(d*g.n)*(n to_power -2) = d*(n to_power 3)*(n to_power -2) by A9,Def3
.= d*((n to_power 3)*(n to_power -2))
.= d*(n to_power (3+(-2))) by A9,POWER:27
.= d*n by POWER:25;
then (d*g.n)*(n to_power -2) > f.n*(n to_power -2) by A5,A10,A13,
XXREAL_0:2;
hence thesis by A1,A11,XREAL_1:64,XXREAL_0:25;
end;
hence contradiction by A4;
end;
hence thesis;
end;
theorem :: Part 3
ex s being eventually-positive Real_Sequence st s = seq_a^(2,1,1) &
seq_a^(2,1,0) in Big_Theta(s)
proof
reconsider g = seq_a^(2,1,1) as eventually-positive Real_Sequence;
set f = seq_a^(2,1,0);
take g;
thus g = seq_a^(2,1,1);
A1: f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
A2: now
let n;
assume n >= 2;
A3: f.n = 2 to_power (1*n+0) by Def1;
A4: g.n = 2 to_power (1*n+1) by Def1;
then (2 to_power -1)*g.n = 2 to_power ((-1)+(n+1)) by POWER:27
.= f.n by A3;
hence (2 to_power -1)*g.n <= f.n;
n+0 <= n+1 by XREAL_1:7;
hence f.n <= 1*g.n by A3,A4,PRE_FF:8;
end;
A5: (2 to_power -1) > 0 by POWER:34;
Big_Theta(g) = { s where s is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*g.n <= s.n & s.n <= c*g.n } by
ASYMPT_0:27;
hence thesis by A1,A5,A2;
end;
definition
let a be Element of NAT;
func seq_n!(a) -> Real_Sequence means
:Def4:
it.n = (n+a)!;
existence
proof
deffunc F(Element of NAT) = In(($1+a)!,REAL);
consider h being sequence of REAL such that
A1: for n being Element of NAT holds h.n = F(n) from FUNCT_2:sch 4;
take h;
let n;
h.n = F(n) by A1;
hence thesis;
end;
uniqueness
proof
let j,k be Real_Sequence such that
A2: for n holds j.n = (n+a)! and
A3: for n holds k.n = (n+a)!;
now
let n;
thus j.n = (n+a)! by A2
.= k.n by A3;
end;
hence thesis by FUNCT_2:63;
end;
end;
registration
let a be Element of NAT;
cluster seq_n!(a) -> eventually-positive;
coherence
proof
take 0;
set f = seq_n!(a);
let n be Nat;
A1: n in NAT by ORDINAL1:def 12;
assume n >= 0;
f.n = (n+a)! by Def4,A1;
hence thesis by NEWTON:17;
end;
end;
theorem :: Part 4
not seq_n!(0) in Big_Theta(seq_n!(1))
proof
set g = seq_n!(1);
set f = seq_n!(0);
A1: Big_Theta(g) = { s where s is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*g.n <= s.n & s.n <= c*g.n } by
ASYMPT_0:27;
now
assume f in Big_Theta(g);
then consider s being Element of Funcs(NAT, REAL) such that
A2: s = f and
A3: ex c,d,N st c > 0 & d > 0 & for n st n >= N holds d*g.n <= s.n & s
.n <= c*g.n by A1;
consider c,d,N such that
c > 0 and
A4: d > 0 and
A5: for n st n >= N holds d*g.n <= s.n & s.n <= c*g.n by A3;
ex n st n >= N & d*g.n > f.n
proof
[/(N+1)/d\] >= (N+1)/d by INT_1:def 7;
then [/(N+1)/d\] + 1 >= (N+1)/d + 1 by XREAL_1:6;
then
A6: d*([/(N+1)/d\] + 1) >= d*((N+1)/d + 1) by A4,XREAL_1:64;
A7: N+1 >= 1+0 by XREAL_1:6;
d*(((N+1)/d) + 1) = d*((N+1)/d) + d*1 .= (N+1) + d by A4,XCMPLX_1:87;
then
A8: d*(((N+1)/d) + 1) > 1 by A4,A7,XREAL_1:8;
take n = max( N, [/(N+1)/d\] );
A9: n >= N by XXREAL_0:25;
A10: n >= [/(N+1)/d\] by XXREAL_0:25;
n is Integer by XXREAL_0:16;
then reconsider n as Element of NAT by A9,INT_1:3;
A11: n! <> 0 by NEWTON:17;
n+1 >= [/(N+1)/d\]+1 by A10,XREAL_1:6;
then d*(n+1) >= d*([/(N+1)/d\]+1) by A4,XREAL_1:64;
then
A12: d*(n+1) >= d*(((N+1)/d) + 1) by A6,XXREAL_0:2;
A13: f.n*(n!)" = (n+0)!*(n!)" by Def4
.= 1 by A11,XCMPLX_0:def 7;
(d*g.n)*(n!)" = (d*((n+1)!))*(n!)" by Def4
.= (d*((n+1)*(n!)))*(n!)" by NEWTON:15
.= (d*(n+1)*((n!)*(n!)"))
.= (d*(n+1))*1 by A11,XCMPLX_0:def 7
.= d*(n+1);
then (d*g.n)*(n!)" > 1 by A12,A8,XXREAL_0:2;
hence thesis by A13,XREAL_1:64,XXREAL_0:25;
end;
hence contradiction by A2,A5;
end;
hence thesis;
end;
begin :: Problem 3.6
Lm20: now
let a,b,c,d be Real;
assume that
A1: 0 <= b and
A2: a <= b and
A3: 0 <= c and
A4: c <= d;
A5: b*c <= b*d by A1,A4,XREAL_1:64;
a*c <= b*c by A2,A3,XREAL_1:64;
hence a*c <= b*d by A5,XXREAL_0:2;
end;
theorem
for f being Real_Sequence st f in Big_Oh(seq_n^(1)) holds f(#)f in
Big_Oh(seq_n^(2))
proof
let f be Real_Sequence;
set h = seq_n^(2);
set g = seq_n^(1);
assume f in Big_Oh(g);
then consider t being Element of Funcs(NAT, REAL) such that
A1: t = f and
A2: ex c,N st c > 0 & for n st n >= N holds t.n <= c*g.n & t.n >= 0;
consider c, N such that
A3: c > 0 and
A4: for n st n >= N holds t.n <= c*g.n & t.n >= 0 by A2;
set d = max( c, c*c );
A5: 0 to_power 1 = 0 by POWER:def 2;
A6: now
take N;
let n;
assume
A7: n >= N;
then
A8: t.n >= 0 by A4;
for n holds g.n <= h.n
proof
let n;
per cases;
suppose
A9: n = 0;
then g.n = 0 by Def3;
hence thesis by A9,Def3;
end;
suppose
n > 0;
then
A10: n >= 0+1 by NAT_1:13;
thus g.n <= h.n
proof
per cases by A10,XXREAL_0:1;
suppose
A11: n = 1;
A12: (1 to_power 2) = 1 by POWER:26;
(1 to_power 1) = 1 by POWER:26;
then g.n = (1 to_power 2) by A11,A12,Def3;
hence thesis by A11,Def3;
end;
suppose
A13: n > 1;
then (n to_power 1) < (n to_power 2) by POWER:39;
then g.n < (n to_power 2) by A13,Def3;
hence thesis by A13,Def3;
end;
end;
end;
end;
then
A14: h.n >= g.n;
g.n >= 0
proof
per cases;
suppose
n = 0;
hence thesis by Def3;
end;
suppose
n > 0;
then g.n = n to_power 1 by Def3
.= n by POWER:25;
hence thesis;
end;
end;
then
A15: (c*c)*h.n <= d*h.n by A14,XREAL_1:64,XXREAL_0:25;
A16: (n to_power 1)*(n to_power 1) = (n to_power (1+1))
proof
per cases;
suppose
n = 0;
hence thesis by A5,POWER:def 2;
end;
suppose
n > 0;
hence thesis by POWER:27;
end;
end;
A17: g.n*g.n = h.n
proof
per cases;
suppose
A18: n = 0;
hence g.n*g.n = 0*g.n by Def3
.= h.n by A18,Def3;
end;
suppose
A19: n > 0;
hence g.n*g.n = (n to_power 1)*g.n by Def3
.= (n to_power (1+1)) by A16,A19,Def3
.= h.n by A19,Def3;
end;
end;
t.n <= c*g.n by A4,A7;
then t.n*t.n <= (c*g.n)*(c*g.n) by A8,Lm20;
then t.n*t.n <= d*h.n by A17,A15,XXREAL_0:2;
hence (t(#)t).n <= d*h.n by SEQ_1:8;
t.n*t.n >= t.n*0 by A8;
hence (t(#)t).n >= 0 by SEQ_1:8;
end;
A20: t(#)t is Element of Funcs(NAT, REAL) by FUNCT_2:8;
d > 0 by A3,XXREAL_0:25;
hence thesis by A1,A20,A6;
end;
begin :: Problem 3.7
theorem
ex s being eventually-positive Real_Sequence st s = seq_a^(2,1,0) & 2
(#)seq_n^(1) in Big_Oh( seq_n^(1) ) & not seq_a^(2,2,0) in Big_Oh(s)
proof
reconsider q = seq_a^(2,1,0) as eventually-positive Real_Sequence;
set p = seq_a^(2,2,0);
set g = seq_n^(1);
set f = 2(#)seq_n^(1);
take q;
thus q = seq_a^(2,1,0);
A1: now
let n;
assume n >= 0;
thus f.n <= 2*g.n by SEQ_1:9;
A2: g.n = n
proof
per cases;
suppose
n = 0;
hence thesis by Def3;
end;
suppose
n > 0;
hence g.n = n to_power 1 by Def3
.= n by POWER:25;
end;
end;
2*n >= 2*0;
hence f.n >= 0 by A2,SEQ_1:9;
end;
f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
hence f in Big_Oh(g) by A1;
now
assume p in Big_Oh(q);
then consider t being Element of Funcs(NAT, REAL) such that
A3: t = p and
A4: ex c,N st c > 0 & for n st n >= N holds t.n <= c*q.n & t.n >= 0;
consider c,N such that
A5: c > 0 and
A6: for n st n >= N holds t.n <= c*q.n & t.n >= 0 by A4;
ex n st n >= N & t.n > c*q.n
proof
take n = max( N, [/log(2,c)+1\] );
A7: n >= N by XXREAL_0:25;
n is Integer by XXREAL_0:16;
then reconsider n as Element of NAT by A7,INT_1:3;
A8: 2 to_power n >= 2 to_power [/log(2,c)+1\] by PRE_FF:8,XXREAL_0:25;
A9: 2 to_power -n > 0 by POWER:34;
[/log(2,c)+1\] >= log(2,c)+1 by INT_1:def 7;
then
A10: 2 to_power [/log(2,c)+1\] >= 2 to_power (log(2,c)+1) by PRE_FF:8;
A11: 2 to_power (log(2,c)+1) = (2 to_power log(2,c))*(2 to_power 1) by
POWER:27
.= c*(2 to_power 1) by A5,POWER:def 3
.= c*2 by POWER:25;
(c*q.n)*(2 to_power -n) = (c*(2 to_power (1*n+0)))*(2 to_power -n)
by Def1
.= c*((2 to_power n)*(2 to_power -n))
.= c*(2 to_power (n+-n)) by POWER:27
.= c*1 by POWER:24;
then 2 to_power (log(2,c)+1) > (c*q.n)*(2 to_power -n) by A5,A11,
XREAL_1:68;
then
A12: 2 to_power [/log(2,c)+1\] > (c*q.n)*(2 to_power -n) by A10,XXREAL_0:2;
p.n*(2 to_power -n) = (2 to_power (2*n+0))*(2 to_power -n) by Def1
.= (2 to_power (2*n+(-1)*n)) by POWER:27
.= (2 to_power (1*n));
then p.n*(2 to_power -n) > (c*q.n)*(2 to_power -n) by A8,A12,XXREAL_0:2;
hence thesis by A3,A9,XREAL_1:64,XXREAL_0:25;
end;
hence contradiction by A6;
end;
hence thesis;
end;
begin :: Problem 3.8
theorem
log(2,3) < 159/100 implies seq_n^(log(2,3)) in Big_Oh(seq_n^(159/100))
& not seq_n^(log(2,3)) in Big_Omega(seq_n^(159/100)) & not seq_n^(log(2,3)) in
Big_Theta(seq_n^(159/100))
proof
set c = 159/100 - log(2,3);
set g = seq_n^(159/100);
set f = seq_n^(log(2,3));
set h = f/"g;
assume
A1: log(2,3) < 159/100;
then
A2: log(2,3) - log(2,3) < 159/100 - log(2,3) by XREAL_1:9;
A3: c/2 <> 0 by A1;
A4: now
A5: c*(1/2) < c*1 by A2,XREAL_1:68;
let p be Real such that
A6: p > 0;
reconsider p1 = p as Real;
A7: ((1/p1) to_power (1/(c/2))) > 0 by A6,POWER:34;
set N1 = max([/((1/p1) to_power (1/(c/2)))\], 2);
A8: N1 >= [/((1/p) to_power (1/(c/2)))\] by XXREAL_0:25;
A9: N1 is Integer by XXREAL_0:16;
A10: N1 >= 2 by XXREAL_0:25;
then
A11: N1 > 1 by XXREAL_0:2;
N1 in NAT by A10,A9,INT_1:3;
then reconsider N1 as Nat;
take N1;
let n be Nat;
A12: n in NAT by ORDINAL1:def 12;
A13: h.n = f.n/g.n by Lm4;
assume
A14: n >= N1;
then f.n = (n to_power log(2,3)) by A10,Def3,A12;
then
A15: h.n = (n to_power log(2,3)) / (n to_power (159/100))
by A10,A14,A13,Def3,A12
.= (n to_power (log(2,3) - (159/100))) by A10,A14,POWER:29
.= (n to_power -c);
[/((1/p) to_power (1/(c/2)))\]>= ((1/p) to_power (1/(c/2))) by INT_1:def 7;
then N1 >= ((1/p) to_power (1/(c/2))) by A8,XXREAL_0:2;
then n >= ((1/p) to_power (1/(c/2))) by A14,XXREAL_0:2;
then n to_power (c/2) >= ((1/p) to_power (1/(c/2))) to_power (c/2) by A2,A7
,Lm6;
then n to_power (c/2) >= (1/p1) to_power ((1/(c/2))*(c/2)) by A6,POWER:33;
then n to_power (c/2) >= (1/p) to_power 1 by A3,XCMPLX_1:87;
then n to_power (c/2) >= 1/p1 by POWER:25;
then 1 / (n to_power (c/2)) <= 1 / (p") by A6,XREAL_1:85;
then
A16: n to_power -(c/2) <= p by A10,A14,POWER:28;
n > 1 by A11,A14,XXREAL_0:2;
then
A17: n to_power (c/2) < n to_power c by A5,POWER:39;
n to_power (c/2) > 0 by A10,A14,POWER:34;
then 1 / (n to_power (c/2)) > 1 / (n to_power c) by A17,XREAL_1:88;
then n to_power -(c/2) > 1 / (n to_power c) by A10,A14,POWER:28;
then h.n < n to_power -(c/2) by A10,A14,A15,POWER:28;
then
A18: h.n < p by A16,XXREAL_0:2;
h.n > 0 by A10,A14,A15,POWER:34;
hence |.h.n-0.| < p by A18,ABSVALUE:def 1;
end;
then
A19: h is convergent by SEQ_2:def 6;
then
A20: lim h = 0 by A4,SEQ_2:def 7;
hence f in Big_Oh(g) by A19,ASYMPT_0:16;
A21: not g in Big_Oh(f) by A19,A20,ASYMPT_0:16;
hence not f in Big_Omega(g) by ASYMPT_0:19;
not f in Big_Omega(g) by A21,ASYMPT_0:19;
hence thesis by XBOOLE_0:def 4;
end;
:: Problem 3.9 -- Proven in theorem ASYMPT_0:10
:: Problem 3.10 -- Proven in theorem ASYMPT_0:12
begin :: Problem 3.11
theorem
for f,g being Real_Sequence st (for n holds f.n = n mod 2) & (for n
holds g.n = n+1 mod 2) holds ex s,s1 being eventually-nonnegative Real_Sequence
st s = f & s1 = g & not s in Big_Oh(s1) & not s1 in Big_Oh(s)
proof
let f,g be Real_Sequence such that
A1: for n holds f.n = n mod 2 and
A2: for n holds g.n = n+1 mod 2;
g is eventually-nonnegative
proof
take 0;
let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
assume n >= 0;
A4: g.n = n+1 mod 2 by A2,A3;
per cases by A4,NAT_D:12;
suppose
g.n = 0;
hence thesis;
end;
suppose
g.n = 1;
hence thesis;
end;
end;
then reconsider g as eventually-nonnegative Real_Sequence;
f is eventually-nonnegative
proof
take 0;
let n be Nat;
A5: n in NAT by ORDINAL1:def 12;
assume n >= 0;
A6: f.n = n mod 2 by A1,A5;
per cases by A6,NAT_D:12;
suppose
f.n = 0;
hence thesis;
end;
suppose
f.n = 1;
hence thesis;
end;
end;
then reconsider f as eventually-nonnegative Real_Sequence;
A7: now
assume g in Big_Oh(f);
then consider t being Element of Funcs(NAT, REAL) such that
A8: t = g and
A9: ex c,N st c > 0 & for n st n >= N holds t.n <= c*f.n & t.n >= 0;
consider c,N such that
c > 0 and
A10: for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A9;
ex n st n >= N & t.n > c*f.n
proof
per cases by NAT_D:12;
suppose
A11: N mod 2 = 0;
then f.N = 0 by A1;
then
A12: c*f.N = 0;
t.N = (N+1) mod 2 by A2,A8
.= (0+(1 mod 2)) mod 2 by A11,EULER_2:6
.= (0+1) mod 2 by NAT_D:14
.= 1 by NAT_D:14;
hence thesis by A12;
end;
suppose
A13: N mod 2 = 1;
f.(N+1) = (N+1) mod 2 by A1
.= (1+(1 mod 2)) mod 2 by A13,EULER_2:6
.= (1+1) mod 2 by NAT_D:14
.= 0 by NAT_D:25;
then
A14: c*f.(N+1) = 0;
A15: N+1 >= N by NAT_1:13;
t.(N+1) = ((N+1)+1) mod 2 by A2,A8
.= (N+(1+1)) mod 2
.= (1+(2 mod 2)) mod 2 by A13,EULER_2:6
.= (1+0) mod 2 by NAT_D:25
.= 1 by NAT_D:14;
hence thesis by A15,A14;
end;
end;
hence contradiction by A10;
end;
take f,g;
now
assume f in Big_Oh(g);
then consider t being Element of Funcs(NAT, REAL) such that
A16: t = f and
A17: ex c,N st c > 0 & for n st n >= N holds t.n <= c*g.n & t.n >= 0;
consider c,N such that
c > 0 and
A18: for n st n >= N holds t.n <= c*g.n & t.n >= 0 by A17;
ex n st n >= N & t.n > c*g.n
proof
per cases by NAT_D:12;
suppose
A19: N mod 2 = 0;
g.(N+1) = ((N+1)+1) mod 2 by A2
.= (N+(1+1)) mod 2
.= (0+(2 mod 2)) mod 2 by A19,EULER_2:6
.= (0+0) mod 2 by NAT_D:25
.= 0 by NAT_D:26;
then
A20: c*g.(N+1) = 0;
A21: N+1 >= N by NAT_1:13;
t.(N+1) = (N+1) mod 2 by A1,A16
.= (0+(1 mod 2)) mod 2 by A19,EULER_2:6
.= (0+1) mod 2 by NAT_D:14
.= 1 by NAT_D:14;
hence thesis by A21,A20;
end;
suppose
A22: N mod 2 = 1;
g.N = (N+1) mod 2 by A2
.= (1+(1 mod 2)) mod 2 by A22,EULER_2:6
.= (1+1) mod 2 by NAT_D:14
.= 0 by NAT_D:25;
then
A23: c*g.N = 0;
t.N = 1 by A1,A16,A22;
hence thesis by A23;
end;
end;
hence contradiction by A18;
end;
hence thesis by A7;
end;
:: Problem 3.12 -- Proven in theorems ASYMPT_0:20, ASYMPT_0:21
:: Problem 3.13 -- omitted (uses a notation that we do not support)
:: Problem 3.14 -- omitted (cannot Mizar an incorrect proof)
:: Problem 3.15 -- omitted (cannot Mizar an incorrect proof)
:: Problem 3.16 -- omitted (maximum rule over multiple functions can be
:: reduced to repeated applications of the binary
:: maximum rule)
:: Problem 3.17 -- omitted (cannot Mizar an incorrect proof)
:: Problem 3.18 -- Proven in theorems ASYMPT_0:28, ASYMPT_0:29, ASYMPT_0:30
begin :: Problem 3.19
theorem
for f,g being eventually-nonnegative Real_Sequence holds Big_Oh(f) =
Big_Oh(g) iff f in Big_Theta(g)
proof
let f,g be eventually-nonnegative Real_Sequence;
hereby
assume
A1: Big_Oh(f) = Big_Oh(g);
then g in Big_Oh(f) by ASYMPT_0:10;
then
A2: f in Big_Omega(g) by ASYMPT_0:19;
f in Big_Oh(g) by A1,ASYMPT_0:10;
hence f in Big_Theta(g) by A2,XBOOLE_0:def 4;
end;
assume
A3: f in Big_Theta(g);
now
let x be object;
hereby
assume x in Big_Oh(f);
then consider t being Element of Funcs(NAT, REAL) such that
A4: x = t and
A5: ex c,N st c > 0 & for n st n >= N holds t.n <= c*f.n & t.n >= 0;
consider c,N such that
c > 0 and
A6: for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A5;
now
reconsider N as Nat;
take N;
let n be Nat;
A7: n in NAT by ORDINAL1:def 12;
assume n >= N;
hence t.n >= 0 by A6,A7;
end;
then
A8: t is eventually-nonnegative;
A9: f in Big_Oh(g) by A3,XBOOLE_0:def 4;
t in Big_Oh(f) by A5;
hence x in Big_Oh(g) by A4,A8,A9,ASYMPT_0:12;
end;
assume x in Big_Oh(g);
then consider t being Element of Funcs(NAT, REAL) such that
A10: x = t and
A11: ex c,N st c > 0 & for n st n >= N holds t.n <= c*g.n & t.n >= 0;
consider c,N such that
c > 0 and
A12: for n st n >= N holds t.n <= c*g.n & t.n >= 0 by A11;
now
reconsider N as Nat;
take N;
let n be Nat;
A13: n in NAT by ORDINAL1:def 12;
assume n >= N;
hence t.n >= 0 by A12,A13;
end;
then
A14: t is eventually-nonnegative;
f in Big_Omega(g) by A3,XBOOLE_0:def 4;
then
A15: g in Big_Oh(f) by ASYMPT_0:19;
t in Big_Oh(g) by A11;
hence x in Big_Oh(f) by A10,A14,A15,ASYMPT_0:12;
end;
hence thesis by TARSKI:2;
end;
theorem :: Part 2
for f,g being eventually-nonnegative Real_Sequence holds f in
Big_Theta(g) iff Big_Theta(f) = Big_Theta(g)
proof
let f,g be eventually-nonnegative Real_Sequence;
A1: Big_Theta(g) = { s where s is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*g.n <= s.n & s.n <= c*g.n } by
ASYMPT_0:27;
consider N2 being Nat such that
A2: for n being Nat st n >= N2 holds g.n >= 0 by ASYMPT_0:def 2;
consider N1 being Nat such that
A3: for n being Nat st n >= N1 holds f.n >= 0 by ASYMPT_0:def 2;
A4: Big_Theta(f) = { s where s is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*f.n <= s.n & s.n <= c*f.n } by
ASYMPT_0:27;
hereby
assume
A5: f in Big_Theta(g);
now
let x be object;
A6: g in Big_Theta(f) by A5,ASYMPT_0:29;
hereby
assume x in Big_Theta(f);
then consider s being Element of Funcs(NAT, REAL) such that
A7: s = x and
A8: ex c,d,N st c > 0 & d > 0 & for n st n >= N holds d*f.n <= s.n
& s.n <= c*f.n by A4;
consider c,d,N3 such that
c > 0 and
A9: d > 0 and
A10: for n st n >= N3 holds d*f.n <= s.n & s.n <= c*f.n by A8;
reconsider N = max( N1, N3 ) as Nat by TARSKI:1;
A11: N >= N3 by XXREAL_0:25;
A12: N >= N1 by XXREAL_0:25;
now
take N;
let n be Nat;
A13: n in NAT by ORDINAL1:def 12;
assume
A14: n >= N;
then n >= N1 by A12,XXREAL_0:2;
then f.n >= 0 by A3;
then
A15: d*f.n >= d*0 by A9;
n >= N3 by A11,A14,XXREAL_0:2;
hence s.n >= 0 by A10,A15,A13;
end;
then
A16: s is eventually-nonnegative;
s in Big_Theta(f) by A4,A8;
hence x in Big_Theta(g) by A5,A7,A16,ASYMPT_0:30;
end;
assume x in Big_Theta(g);
then consider s being Element of Funcs(NAT, REAL) such that
A17: s = x and
A18: ex c,d,N st c > 0 & d > 0 & for n st n >= N holds d*g.n <= s.n
& s.n <= c*g.n by A1;
consider c,d,N3 such that
c > 0 and
A19: d > 0 and
A20: for n st n >= N3 holds d*g.n <= s.n & s.n <= c*g.n by A18;
reconsider N = max( N2, N3 ) as Nat by TARSKI:1;
A21: N >= N3 by XXREAL_0:25;
A22: N >= N2 by XXREAL_0:25;
now
take N;
let n be Nat;
A23: n in NAT by ORDINAL1:def 12;
assume
A24: n >= N;
then n >= N2 by A22,XXREAL_0:2;
then g.n >= 0 by A2;
then
A25: d*g.n >= d*0 by A19;
n >= N3 by A21,A24,XXREAL_0:2;
hence s.n >= 0 by A20,A25,A23;
end;
then
A26: s is eventually-nonnegative;
s in Big_Theta(g) by A1,A18;
hence x in Big_Theta(f) by A17,A26,A6,ASYMPT_0:30;
end;
hence Big_Theta(f) = Big_Theta(g) by TARSKI:2;
end;
assume Big_Theta(f) = Big_Theta(g);
hence thesis by ASYMPT_0:28;
end;
:: Problem 3.20 -- Proven above in theorem ASYMPT_1:?????
begin :: Problem 3.21
Lm21: for n holds (n^2 - n + 1) > 0
proof
defpred P[Nat] means ($1^2 - $1 + 1) > 0;
A1: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat;
assume (k^2 - k + 1) > 0;
then (k^2 - k + 1) + 2*k > 0+0;
hence thesis;
end;
A2: P[0];
for n being Nat holds P[n] from NAT_1:sch 2(A2, A1);
hence thesis;
end;
Lm22: for f,g being Real_Sequence, N being Element of NAT, c being Real st f
is convergent & lim f = c & for n st n >= N holds f.n = g.n holds g is
convergent & lim g = c
proof
let f,g be Real_Sequence, N be Element of NAT, c be Real such that
A1: f is convergent and
A2: lim f = c and
A3: for n st n >= N holds f.n = g.n;
A4: now
let p be Real;
assume p > 0;
then consider M being Nat such that
A5: for n being Nat st n >= M holds |.f.n-c.| < p by A1,A2,SEQ_2:def 7;
reconsider N1 = max(N, M) as Nat by TARSKI:1;
A6: N1 >= N by XXREAL_0:25;
take N1;
let n be Nat;
A7: n in NAT by ORDINAL1:def 12;
assume
A8: n >= N1;
N1 >= M by XXREAL_0:25;
then n >= M by A8,XXREAL_0:2;
then |.f.n-c.| < p by A5;
hence |.g.n-c.| < p by A3,A6,A8,XXREAL_0:2,A7;
end;
hence g is convergent by SEQ_2:def 6;
hence thesis by A4,SEQ_2:def 7;
end;
Lm23: for n st n >= 1 holds (n^2 -n + 1) <= n^2
proof
let n such that
A1: n >= 1;
now
assume (n^2 -n + 1) > n^2;
then -n^2 + (n^2 + (-n + 1)) > n^2 + -n^2 by XREAL_1:6;
then 1 > 0 -(-n) by XREAL_1:19;
hence contradiction by A1;
end;
hence thesis;
end;
Lm24: for n st n >= 1 holds n^2 <= 2*(n^2 -n + 1)
proof
defpred P[Nat] means $1^2 <= 2*($1^2 -$1 + 1);
A1: for k be Nat st k>=1 & P[k] holds P[k+1]
proof
let k be Nat such that
A2: k >= 1 and
A3: k^2 <= 2*(k^2 -k + 1);
A4: k^2 + (2*k + 1) <= 2*(k^2 -k + 1) + (2*k + 1) by A3,XREAL_1:6;
2*k >= 2*1 by A2,XREAL_1:64;
then 2*k + 2 >= 2 + 2 by XREAL_1:6;
then
A5: 2*k^2 + 4 <= 2*k^2 + (2*k + 2) by XREAL_1:6;
2*k^2 + 3 <= 2*k^2 + 4 by XREAL_1:6;
then 2*(k^2 -k + 1) + (2*k + 1) <= 2*k^2 + (2*k + 2) by A5,XXREAL_0:2;
hence thesis by A4,XXREAL_0:2;
end;
A6: P[1];
for n be Nat st n >= 1 holds P[n] from NAT_1:sch 8(A6, A1);
hence thesis;
end;
Lm25: for e being Real st 0 < e & e < 1 holds ex N st for n st n >= N holds n*
log(2,1+e) - 8*log(2,n) > 8*log(2,n)
proof
set f = seq_logn;
let e be Real such that
A1: 0 < e and
A2: e < 1;
set d = log(2,1+e);
set g = seq_n^(e);
set h = f/"g;
A3: h is convergent by A1,Lm11;
A4: lim h = 0 by A1,Lm11;
0+1 < e+1 by A1,XREAL_1:6;
then log(2,1) < log(2,e+1) by POWER:57;
then
A5: d > 0 by POWER:51;
then d*(1/16) > d*0 by XREAL_1:68;
then consider N being Nat such that
A6: for n being Nat st n >= N holds |.h.n-0.| < d/16 by A3,A4,SEQ_2:def 7;
ex N st for n st n >= N holds n*log(2,1+e) - 8*log(2,n) > 8*log(2,n)
proof
reconsider N1 = max(2, N) as Element of NAT by ORDINAL1:def 12;
A7: N1 >= 2 by XXREAL_0:25;
A8: N1 >= N by XXREAL_0:25;
now
take N1;
let n;
assume
A9: n >= N1;
then
A10: n to_power e > 0 by A7,POWER:34;
A11: n >= 2 by A7,A9,XXREAL_0:2;
then log(2,2) <= log(2,n) by PRE_FF:10;
then
A12: 1 <= log(2,n) by POWER:52;
A13: h.n = f.n / g.n by Lm4
.= log(2,n) / g.n by A7,A9,Def2
.= log(2,n) / (n to_power e) by A7,A9,Def3;
n > 1 by A11,XXREAL_0:2;
then n to_power 1 > n to_power e by A2,POWER:39;
then 1 / (n to_power 1) < 1 / (n to_power e) by A10,XREAL_1:88;
then 1 / n < 1 / (n to_power e) by POWER:25;
then
A14: log(2,n) / n < log(2,n) * (1/(n to_power e)) by A12,XREAL_1:68;
n >= N by A8,A9,XXREAL_0:2;
then |.h.n-0.| < d/16 by A6;
then h.n < d/16 by A12,A13,A10,ABSVALUE:def 1;
then
A15: log(2,n) / n < d / 16 by A13,A14,XXREAL_0:2;
log(2,n)*n" > 0*n" by A7,A9,A12,XREAL_1:68;
then 1 / (log(2,n) / n) > 1 / (d / 16) by A15,XREAL_1:88;
then n / log(2,n) > 1 / (d / 16) by XCMPLX_1:57;
then n / log(2,n) > 16 / d by XCMPLX_1:57;
then d * (n / log(2,n)) > 16 / d * d by A5,XREAL_1:68;
then d * (n / log(2,n)) > 16 by A5,XCMPLX_1:87;
then d * (n / log(2,n)) * log(2,n) > 16*log(2,n) by A12,XREAL_1:68;
then d * ((n / log(2,n)) * log(2,n)) > 16*log(2,n);
then d*n > (8+8)*log(2,n) by A12,XCMPLX_1:87;
then d*n - 8*log(2,n) > (8*log(2,n) + 8*log(2,n)) - 8*log(2,n) by
XREAL_1:9;
hence n*d - 8*log(2,n) > 8*log(2,n);
end;
hence thesis;
end;
hence thesis;
end;
theorem :: (Part 1, O(nlogn) c O(n^(1+e))), slightly generalized
for e being Real, f being Real_Sequence st 0 < e & (for n st n > 0
holds f.n = n*log(2,n)) holds ex s being eventually-positive Real_Sequence st s
= f & Big_Oh(s) c= Big_Oh(seq_n^(1+e)) & not Big_Oh(s) = Big_Oh(seq_n^(1+e))
proof
set seq = seq_logn;
let e be Real, f be Real_Sequence such that
A1: 0 < e and
A2: for n st n > 0 holds f.n = n*log(2,n);
set seq1 = seq_n^(e);
set p = seq /" seq1;
A3: lim p = 0 by A1,Lm11;
f is eventually-positive
proof
take 2;
let n be Nat;
A4: n in NAT by ORDINAL1:def 12;
assume
A5: n >= 2;
then log(2,n) >= log(2,2) by PRE_FF:10;
then log(2,n) >= 1 by POWER:52;
then n*log(2,n) > n*0 by A5,XREAL_1:68;
hence thesis by A2,A5,A4;
end;
then reconsider f as eventually-positive Real_Sequence;
set g = seq_n^(1+e);
set h = f /" g;
A6: for n st n >= 1 holds h.n = p.n
proof
let n;
assume
A7: n >= 1;
h.n = f.n / g.n by Lm4
.= n*log(2,n) / g.n by A2,A7
.= n*log(2,n) / (n to_power (1+e)) by A7,Def3
.= (n to_power 1)*log(2,n) / (n to_power (1+e)) by POWER:25
.= (n to_power 1)*log(2,n) * (n to_power (1+e))"
.= log(2,n)*((n to_power 1)*(n to_power (1+e))")
.= log(2,n)*((n to_power 1)/(n to_power (1+e)))
.= log(2,n)*(n to_power (1-(1+e))) by A7,POWER:29
.= log(2,n)*(n to_power (1+(-1+-e)))
.= log(2,n)*(1/(n to_power e)) by A7,POWER:28
.= log(2,n)/(n to_power e)
.= seq.n / (n to_power e) by A7,Def2
.= seq.n / seq1.n by A7,Def3
.= p.n by Lm4;
hence thesis;
end;
A8: p is convergent by A1,Lm11;
then
A9: lim h = 0 by A3,A6,Lm22;
A10: h is convergent by A8,A3,A6,Lm22;
then not g in Big_Oh(f) by A9,ASYMPT_0:16;
then
A11: not f in Big_Omega(g) by ASYMPT_0:19;
take f;
f in Big_Oh(g) by A10,A9,ASYMPT_0:16;
hence thesis by A11,Th4;
end;
theorem
for e being Real, g being Real_Sequence st e < 1 & (for n st n > 1
holds g.n = (n to_power 2)/log(2,n)) holds ex s being eventually-positive
Real_Sequence st s = g & Big_Oh(seq_n^(1+e)) c= Big_Oh(s) & not Big_Oh(seq_n^(1
+e)) = Big_Oh(s)
proof
set seq = seq_logn;
let e be Real, g be Real_Sequence such that
A1: e < 1 and
A2: for n st n > 1 holds g.n = (n to_power 2)/log(2,n);
set seq1 = seq_n^(1-e);
set p = seq /" seq1;
set f = seq_n^(1+e);
set h = f /" g;
g is eventually-positive
proof
take 2;
let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
assume
A4: n >= 2;
then log(2,n) >= log(2,2) by PRE_FF:10;
then
A5: log(2,n) >= 1 by POWER:52;
n > 1 by A4,XXREAL_0:2;
then
A6: g.n = (n to_power 2)/log(2,n) by A2,A3
.= (n to_power 2)*(log(2,n))";
n to_power 2 > 0 by A4,POWER:34;
then (n to_power 2)*(log(2,n))" > (n to_power 2)*0 by A5,XREAL_1:68;
hence thesis by A6;
end;
then reconsider g as eventually-positive Real_Sequence;
A7: (1+e)-2 = e-1;
A8: for n st n >= 2 holds h.n = p.n
proof
let n;
assume
A9: n >= 2;
then
A10: n > 1 by XXREAL_0:2;
h.n = f.n / g.n by Lm4
.= (n to_power (1+e)) / g.n by A9,Def3
.= (n to_power (1+e)) / ((n to_power 2) / log(2,n)) by A2,A10
.= (n to_power (1+e)) * ((n to_power 2) / log(2,n))"
.= (n to_power (1+e)) * (log(2,n) / (n to_power 2)) by XCMPLX_1:213
.= (n to_power (1+e)) * (log(2,n) * (n to_power 2)")
.= ((n to_power (1+e)) * (n to_power 2)") * log(2,n)
.= ((n to_power (1+e)) / (n to_power 2)) * log(2,n)
.= (n to_power -(1-e)) * log(2,n) by A7,A9,POWER:29
.= log(2,n) * (1 / (n to_power (1-e))) by A9,POWER:28
.= log(2,n) / (n to_power (1-e))
.= seq.n / (n to_power (1-e)) by A9,Def2
.= seq.n / seq1.n by A9,Def3
.= p.n by Lm4;
hence thesis;
end;
take g;
0+e < 1 by A1;
then
A11: 0 < 1-e by XREAL_1:20;
then
A12: p is convergent by Lm11;
A13: lim p = 0 by A11,Lm11;
then
A14: lim h = 0 by A12,A8,Lm22;
A15: h is convergent by A12,A13,A8,Lm22;
then not g in Big_Oh(f) by A14,ASYMPT_0:16;
then
A16: not f in Big_Omega(g) by ASYMPT_0:19;
f in Big_Oh(g) by A15,A14,ASYMPT_0:16;
hence thesis by A16,Th4;
end;
theorem
for f being Real_Sequence st (for n st n > 1 holds f.n = (n to_power 2
)/log(2,n)) holds ex s being eventually-positive Real_Sequence st s = f &
Big_Oh(s) c= Big_Oh(seq_n^(8)) & not Big_Oh(s) = Big_Oh(seq_n^(8))
proof
set g = seq_n^(8);
let f be Real_Sequence such that
A1: for n st n > 1 holds f.n = (n to_power 2)/log(2,n);
A2: f is eventually-positive
proof
take 2;
let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
assume
A4: n >= 2;
then log(2,n) >= log(2,2) by PRE_FF:10;
then
A5: log(2,n) >= 1 by POWER:52;
n > 1 by A4,XXREAL_0:2;
then
A6: f.n = (n to_power 2)/log(2,n) by A1,A3
.= (n to_power 2)*(log(2,n))";
n to_power 2 > 0 by A4,POWER:34;
then (n to_power 2)*(log(2,n))" > (n to_power 2)*0 by A5,XREAL_1:68;
hence thesis by A6;
end;
set h = f/"g;
reconsider f as eventually-positive Real_Sequence by A2;
A7: now
A8: log(2,3) > log(2,2) by POWER:57;
let p be Real;
assume
A9: p > 0;
A10: [/(p to_power -(1/6))\] >= (p to_power -(1/6)) by INT_1:def 7;
reconsider p1 = p as Real;
set N = max(3,[/(p1 to_power -(1/6))\]);
A11: N >= 3 by XXREAL_0:25;
A12: N is Integer by XXREAL_0:16;
A13: N >= [/(p to_power -(1/6))\] by XXREAL_0:25;
N in NAT by A11,A12,INT_1:3;
then reconsider N as Nat;
take N;
let n be Nat;
A14: n in NAT by ORDINAL1:def 12;
assume
A15: n >= N;
then
A16: n >= 3 by A11,XXREAL_0:2;
then
A17: n > 1 by XXREAL_0:2;
A18: h.n = f.n/g.n by Lm4
.= ((n to_power 2)/log(2,n)) / g.n by A1,A17,A14
.= ((n to_power 2)/log(2,n)) / (n to_power 8) by A11,A15,Def3,A14
.= ((n to_power 2)*(log(2,n))") / (n to_power 8)
.= ((log(2,n))"*(n to_power 2)) * (n to_power 8)"
.= (log(2,n))"*((n to_power 2)*(n to_power 8)")
.= (log(2,n))"*((n to_power 2)/(n to_power 8))
.= (log(2,n))"*(n to_power (2-8)) by A11,A15,POWER:29
.= (log(2,n))"*(n to_power -6)
.= (log(2,n))"*(1/(n to_power 6)) by A11,A15,POWER:28
.= (1/(n to_power 6))*(1/log(2,n))
.= 1/((n to_power 6)*log(2,n)) by XCMPLX_1:102;
n >= [/(p to_power -(1/6))\] by A13,A15,XXREAL_0:2;
then
A19: n >= (p to_power -(1/6)) by A10,XXREAL_0:2;
(p1 to_power -(1/6)) > 0 by A9,POWER:34;
then n to_power 6 >= (p to_power -(1/6)) to_power 6 by A19,Lm6;
then
A20: n to_power 6 >= p1 to_power ((-(1/6))*6) by A9,POWER:33;
p1 to_power -1 > 0 by A9,POWER:34;
then 1/(n to_power 6) <= 1/(p to_power -1) by A20,XREAL_1:85;
then 1/(n to_power 6) <= 1/(1/(p1 to_power 1)) by A9,POWER:28;
then
A21: 1/(n to_power 6) <= p by POWER:25;
log(2,n) >= log(2,3) by A16,PRE_FF:10;
then log(2,n) > log(2,2) by A8,XXREAL_0:2;
then
A22: log(2,n) > 1 by POWER:52;
A23: n to_power 6 > 0 by A11,A15,POWER:34;
then (n to_power 6)*1 < (n to_power 6)*log(2,n) by A22,XREAL_1:68;
then h.n < 1/(n to_power 6) by A23,A18,XREAL_1:88;
then h.n < p by A21,XXREAL_0:2;
hence |.h.n-0.| < p by A22,A18,ABSVALUE:def 1;
end;
then
A24: h is convergent by SEQ_2:def 6;
then
A25: lim h = 0 by A7,SEQ_2:def 7;
then not g in Big_Oh(f) by A24,ASYMPT_0:16;
then
A26: not f in Big_Omega(g) by ASYMPT_0:19;
take f;
f in Big_Oh(g) by A24,A25,ASYMPT_0:16;
hence thesis by A26,Th4;
end;
theorem
for g being Real_Sequence st (for n holds g.n = (n^2 - n + 1) to_power
4) holds ex s being eventually-positive Real_Sequence st s = g & Big_Oh(seq_n^(
8)) = Big_Oh(s)
proof
let g be Real_Sequence such that
A1: for n holds g.n = (n^2 - n + 1) to_power 4;
g is eventually-positive
proof
take 0;
let n be Nat;
A2: n in NAT by ORDINAL1:def 12;
assume n >= 0;
g.n = (n^2 - n + 1) to_power 4 by A1,A2;
hence thesis by Lm21,POWER:34,A2;
end;
then reconsider g as eventually-positive Real_Sequence;
take g;
set f = seq_n^(8);
A3: now
let n;
A4: g.n = (n^2 -n + 1) to_power 4 by A1;
assume
A5: n >= 1;
then
A6: n^2 -n + 1 <= n^2 by Lm23;
f.n = n to_power (2*4) by A5,Def3
.= n to_power 2 to_power 4 by A5,POWER:33
.= n^2 to_power 4 by POWER:46;
hence g.n <= 1*f.n by A4,A6,Lm6,Lm21;
thus g.n >= 0 by A4,Lm21,POWER:34;
end;
A7: now
let n;
A8: g.n = (n^2 -n + 1) to_power 4 by A1;
A9: (n^2 -n + 1) > 0 by Lm21;
assume
A10: n >= 1;
then
A11: f.n = n to_power (2*4) by Def3
.= n to_power 2 to_power 4 by A10,POWER:33
.= n^2 to_power 4 by POWER:46;
A12: n*n > n*0 by A10,XREAL_1:68;
n^2 <= 2*(n^2 -n + 1) by A10,Lm24;
then f.n <= (2*(n^2 -n + 1)) to_power 4 by A11,A12,Lm6;
hence f.n <= 16*g.n by A8,A9,POWER:30,62;
thus f.n >= 0 by A11,A12,POWER:34;
end;
f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
then
A13: f in Big_Oh(g) by A7;
g is Element of Funcs(NAT, REAL) by FUNCT_2:8;
then g in Big_Oh(f) by A3;
hence thesis by A13,Lm5;
end;
theorem :: (Part 5, O(n^8) c O((1+e)^n))
for e being Real st 0 < e & e < 1 holds ex s being eventually-positive
Real_Sequence st s = seq_a^(1+e,1,0) & Big_Oh(seq_n^(8)) c= Big_Oh(s) & not
Big_Oh(seq_n^(8)) = Big_Oh(s)
proof
set f = seq_n^(8);
let e be Real such that
A1: 0 < e and
A2: e < 1;
consider N such that
A3: for n st n >= N holds n*log(2,1+e) - 8*log(2,n) > 8*log(2,n) by A1,A2,Lm25;
set g = seq_a^(1+e,1,0);
set h = f/"g;
reconsider g as eventually-positive Real_Sequence by A1;
take g;
thus g = seq_a^(1+e,1,0);
A4: now
let p be Real such that
A5: p > 0;
reconsider p1 = p as Real;
A6: ((1/p1) to_power (1/8)) > 0 by A5,POWER:34;
set N1 = max( N, max([/((1/p1) to_power (1/8))\], 2) );
A7: N1 >= N by XXREAL_0:25;
A8: N1 is Integer
proof
per cases by XXREAL_0:16;
suppose
N1 = N;
hence thesis;
end;
suppose
N1 = max([/((1/p) to_power (1/8))\], 2);
hence thesis by XXREAL_0:16;
end;
end;
A9: N1 >= max([/((1/p) to_power (1/8))\], 2) by XXREAL_0:25;
max([/((1/p) to_power (1/8))\], 2) >= [/((1/p) to_power (1/8))\] by
XXREAL_0:25;
then
A10: N1 >= [/((1/p) to_power (1/8))\] by A9,XXREAL_0:2;
N1 in NAT by A7,A8,INT_1:3;
then reconsider N1 as Nat;
take N1;
let n be Nat;
A11: n in NAT by ORDINAL1:def 12;
assume
A12: n >= N1;
then n >= N by A7,XXREAL_0:2;
then ((n*log(2,1+e)) - (8*log(2,n))) > (8*log(2,n)) by A3,A11;
then
A13: 2 to_power ((n*log(2,1+e)) - (8*log(2,n))) > 2 to_power (8*log(2,n))
by POWER:39;
A14: max([/((1/p) to_power (1/8))\], 2) >= 2 by XXREAL_0:25;
A15: g.n = ((1+e) to_power (1*n + 0)) by Def1;
h.n = f.n/g.n by Lm4;
then
A16: h.n = (n to_power 8) / ((1+e) to_power n) by A9,A14,A12,A15,Def3
.= (2 to_power (8*log(2,n))) / ((1+e) to_power n) by A9,A14,A12,Lm3
.= (2 to_power (8*log(2,n))) / (2 to_power (n*log(2,1+e))) by A1,Lm3
.= (2 to_power ((8*log(2,n)) - (n*log(2,1+e)))) by POWER:29
.= (2 to_power -((n*log(2,1+e)) - (8*log(2,n))));
[/((1/p) to_power (1/8))\] >= ((1/p) to_power (1/8)) by INT_1:def 7;
then N1 >= ((1/p) to_power (1/8)) by A10,XXREAL_0:2;
then n >= ((1/p) to_power (1/8)) by A12,XXREAL_0:2;
then n to_power 8 >= ((1/p) to_power (1/8)) to_power 8 by A6,Lm6;
then n to_power 8 >= (1/p1) to_power ((1/8)*8) by A5,POWER:33;
then n to_power 8 >= 1/p1 by POWER:25;
then 1 / (n to_power 8) <= 1 / (p") by A5,XREAL_1:85;
then 1 / (2 to_power (8*log(2,n))) <= p by A9,A14,A12,Lm3;
then
A17: 2 to_power -(8*log(2,n)) <= p by POWER:28;
2 to_power (8*log(2,n)) > 0 by POWER:34;
then
1 / (2 to_power ((n*log(2,1+e)) - (8*log(2,n)))) < 1 / (2 to_power (8
*log(2,n))) by A13,XREAL_1:88;
then
2 to_power -((n*log(2,1+e)) - (8*log(2,n))) < 1 / (2 to_power (8*log(
2,n))) by POWER:28;
then h.n < 2 to_power -(8*log(2,n)) by A16,POWER:28;
then
A18: h.n < p by A17,XXREAL_0:2;
h.n > 0 by A16,POWER:34;
hence |.h.n-0.| < p by A18,ABSVALUE:def 1;
end;
then
A19: h is convergent by SEQ_2:def 6;
then
A20: lim h = 0 by A4,SEQ_2:def 7;
then not g in Big_Oh(f) by A19,ASYMPT_0:16;
then
A21: not f in Big_Omega(g) by ASYMPT_0:19;
f in Big_Oh(g) by A19,A20,ASYMPT_0:16;
hence thesis by A21,Th4;
end;
begin :: Problem 3.22
Lm26: 2 to_power 12 = 4096
proof
thus 2 to_power 12 = 2 to_power (6+6) .= 64*64 by POWER:27,64
.= 4096;
end;
Lm27: for n be Nat st n >= 3 holds n^2 > 2*n + 1
proof
defpred P[Nat] means $1^2 > 2*$1 + 1;
A1: for n be Nat st n >= 3 & P[n] holds P[n+1]
proof
let n be Nat;
assume that
A2: n >= 3 and
A3: n^2 > 2*n + 1;
n > 1 by A2,XXREAL_0:2;
then n + n > 1 + 0 by XREAL_1:8;
then
A4: 2*n + 2*(n+1) > 1 + 2*(n+1) by XREAL_1:6;
n^2 + (n + (n+1)) > 2*n + 1 + (n + (n+1)) by A3,XREAL_1:6;
hence (n+1)^2 > 2*(n+1) + 1 by A4,XXREAL_0:2;
end;
A5: P[3];
for n be Nat st n >= 3 holds P[n] from NAT_1:sch 8(A5, A1);
hence thesis;
end;
Lm28: for n st n >= 10 holds 2 to_power (n-1) > (2*n)^2
proof
defpred P[Nat] means 2 to_power ($1-1) > (2*$1)^2;
A1: for n be Nat st n >= 10 & P[n] holds P[n+1]
proof
let n be Nat;
assume that
A2: n >= 10 and
A3: 2 to_power (n-1) > (2*n)^2;
A4: now
assume ((2*n)^2)*2 <= (2*2)*(n*n + 2*n + 1);
then ((2*2)*(n*n))*2 <= (2*2)*(n*n) + (2*2)*(2*n + 1);
then ((2*2)*(n*n))*2 - (2*2)*(n*n) <= (2*2)*(2*n + 1) by XREAL_1:20;
then (2*2)"*((2*2)*(n*n)) <= (2*2)"*((2*2)*(2*n + 1)) by XREAL_1:64;
then
A5: n^2 <= 2*n + 1;
n >= 3 by A2,XXREAL_0:2;
hence contradiction by A5,Lm27;
end;
2 to_power ((n+1)-1) = 2 to_power ((n+-1)+1)
.= (2 to_power (n-1))*(2 to_power 1) by POWER:27
.= (2 to_power (n-1))*2 by POWER:25;
then 2 to_power ((n+1)-1) > ((2*n)^2)*2 by A3,XREAL_1:68;
hence 2 to_power ((n+1)-1) > (2*(n+1))^2 by A4,XXREAL_0:2;
end;
2 to_power (10-1) = 2 to_power (6+3) .= 64*(2 to_power (2+1)) by POWER:27,64
.= 64*((2 to_power 2)*(2 to_power 1)) by POWER:27
.= 64*((2 to_power (1+1))*2) by POWER:25
.= 64*(((2 to_power 1)*(2 to_power 1))*2) by POWER:27
.= 64*((2*(2 to_power 1))*2) by POWER:25
.= 64*((2*2)*2) by POWER:25
.= 512;
then
A6: P[10];
for n be Nat st n >= 10 holds P[n] from NAT_1:sch 8(A6, A1);
hence thesis;
end;
Lm29: for n be Nat st n >= 9 holds (n+1) to_power 6 < 2*(n to_power 6)
proof
defpred P[Nat] means ($1+1) to_power 6 < 2*($1 to_power 6);
A1: for n be Nat st n >= 9 & P[n] holds P[n+1]
proof
let n be Nat;
assume that
A2: n >= 9 and
A3: (n+1) to_power 6 < 2*(n to_power 6);
(n+1) to_power 6 / n to_power 6 < 2 by A2,A3,POWER:34,XREAL_1:83;
then
A4: ((n+1) / n) to_power 6 < 2 by A2,POWER:31;
A5: now
assume (n+2) / (n+1) >= (n+1) / n;
then (n+2) / (n+1) * (n+1) >= (n+1) / n * (n+1) by XREAL_1:64;
then (n+2) >= (n+1) / n * (n+1) by XCMPLX_1:87;
then (n+2)*n >= (n+1) / n * (n+1) * n by XREAL_1:64;
then (n+2)*n >= ((n+1) / n * n) * (n+1);
then n^2 + 2*n >= (n+1)^2 by A2,XCMPLX_1:87;
then n^2 + 2*n >= n^2 + 2*n + 1*1;
then (n^2 + 2*n) - (n^2 + 2*n) >= 1 by XREAL_1:19;
hence contradiction;
end;
A6: (n+1) to_power 6 > 0 by POWER:34;
(n+2)*((n+1)") > 0*((n+1)") by XREAL_1:68;
then ((n+2) / (n+1)) to_power 6 < ((n+1) / n) to_power 6 by A5,POWER:37;
then ((n+2) / (n+1)) to_power 6 < 2 by A4,XXREAL_0:2;
then ((n+2) to_power 6) / ((n+1) to_power 6) < 2 by POWER:31;
then
((n+2) to_power 6) / ((n+1) to_power 6) * ((n+1) to_power 6) < 2*((n+
1) to_power 6) by A6,XREAL_1:68;
hence thesis by A6,XCMPLX_1:87;
end;
A7: P[9]
proof
A8: 9 to_power 2 = 9 to_power (1+1)
.= (9 to_power 1)*(9 to_power 1) by POWER:27
.= 9*(9 to_power 1) by POWER:25
.= 9*9 by POWER:25
.= 81;
2*(9 to_power 4) = 2*(9 to_power (2+2)) .= 2*(81*81) by A8,POWER:27
.= 13122;
then
A9: (13122*9)*9 = (2*((9 to_power 4)*9))*9
.= (2*((9 to_power 4)*(9 to_power 1)))*9 by POWER:25
.= (2*(9 to_power (4+1)))*9 by POWER:27
.= 2*((9 to_power 5)*9)
.= 2*((9 to_power 5)*(9 to_power 1)) by POWER:25
.= 2*(9 to_power (5+1)) by POWER:27
.= 2*(9 to_power 6);
consider t6 being Element of NAT such that
A10: t6 = 10*10*10*10*10*10;
A11: 10 to_power 3 = 10 to_power (2+1)
.= (10 to_power 2)*(10 to_power 1) by POWER:27
.= (10 to_power (1+1))*10 by POWER:25
.= ((10 to_power 1)*(10 to_power 1))*10 by POWER:27
.= (10*(10 to_power 1))*10 by POWER:25
.= (10*10)*10 by POWER:25;
10 to_power 6 = 10 to_power (3+3) .= (10*10*10)*(10*10*10) by A11,POWER:27
.= t6 by A10;
hence thesis by A10,A9;
end;
for n be Nat st n >= 9 holds P[n] from NAT_1:sch 8(A7, A1);
hence thesis;
end;
Lm30: for n st n >= 30 holds 2 to_power n > n to_power 6
proof
defpred P[Nat] means 2 to_power $1 > $1 to_power 6;
A1: for n be Nat st n >= 30 & P[n] holds P[n+1]
proof
let n be Nat;
assume that
A2: n >= 30 and
A3: 2 to_power n > n to_power 6;
n >= 9 by A2,XXREAL_0:2;
then
A4: (n+1) to_power 6 < 2*(n to_power 6) by Lm29;
A5: 2 to_power (n+1) = (2 to_power n)*(2 to_power 1) by POWER:27
.= (2 to_power n)*2 by POWER:25;
(2 to_power n)*2 > (n to_power 6)*2 by A3,XREAL_1:68;
hence thesis by A5,A4,XXREAL_0:2;
end;
2 to_power 30 = 2 to_power (5*6) .= 32 to_power 6 by POWER:33,63;
then
A6: P[30] by POWER:37;
for n be Nat st n >= 30 holds P[n] from NAT_1:sch 8(A6, A1);
hence thesis;
end;
Lm31: for x being Real st x > 9 holds 2 to_power x > (2*x)^2
proof
let x be Real such that
A1: x > 9;
set n = [/x\];
A2: n >= x by INT_1:def 7;
then reconsider n as Element of NAT by A1,INT_1:3;
2*n >= 2*x by A2,XREAL_1:64;
then
A3: (2*n)^2 >= (2*x)*(2*x) by A1,Lm20;
n > 9 by A1,A2,XXREAL_0:2;
then n >= 9+1 by NAT_1:13;
then
A4: 2 to_power (n-1) > (2*n)^2 by Lm28;
[/x\] - [\x/] <= 1
proof
per cases;
suppose
x is Integer;
then [\x/] = [/x\] by INT_1:34;
hence thesis;
end;
suppose
not x is Integer;
then not [\x/] = [/x\] by INT_1:34;
then [\x/] + 1 = [/x\] by INT_1:41;
hence thesis;
end;
end;
then [/x\] <= 1 + [\x/] by XREAL_1:20;
then [\x/] >= n-1 by XREAL_1:20;
then
A5: 2 to_power [\x/] >= 2 to_power (n-1) by PRE_FF:8;
x >= [\x/] by INT_1:def 6;
then 2 to_power x >= 2 to_power [\x/] by PRE_FF:8;
then 2 to_power x >= 2 to_power (n-1) by A5,XXREAL_0:2;
then 2 to_power x > (2*n)^2 by A4,XXREAL_0:2;
hence thesis by A3,XXREAL_0:2;
end;
Lm32: ex N st for n st n >= N holds sqrt n - log(2, n) > 1
proof
ex N st for n st n >= N holds n/2 > log(2,n)*log(2,n)
proof
reconsider N = 2 to_power 10 as Element of NAT;
now
take N;
let n;
set x = log(2,n);
A1: 2 to_power 9 > 0 by POWER:34;
assume
A2: n >= N;
then
A3: n > 0 by POWER:34;
2 to_power 10 > 2 to_power 0 by POWER:39;
then n > 2 to_power 0 by A2,XXREAL_0:2;
then n > 1 by POWER:24;
then log(2,n) > log(2,1) by POWER:57;
then log(2,n) > 0 by POWER:51;
then log(2,n)*log(2,n) > 0*log(2,n) by XREAL_1:68;
then
A4: 4*(log(2,n)*log(2,n)) > 2*(log(2,n)*log(2,n)) by XREAL_1:68;
2 to_power 10 > 2 to_power 1 by POWER:39;
then n > 2 to_power 1 by A2,XXREAL_0:2;
then
A5: n > 2 by POWER:25;
then
A6: 2*n > 2*2 by XREAL_1:68;
n*n > 2*n by A5,XREAL_1:68;
then n^2 > 2*2 by A6,XXREAL_0:2;
then log(2,n^2) > log(2,2^2) by POWER:57;
then log(2,n^2) > log(2,2 to_power 2) by POWER:46;
then log(2,n^2) > 2*log(2,2) by POWER:55;
then
A7: log(2,n^2) > 2*1 by POWER:52;
then
A8: (log(2,n^2))^2 > 0 by SQUARE_1:12;
2 to_power 10 > 2 to_power 9 by POWER:39;
then n > 2 to_power 9 by A2,XXREAL_0:2;
then log(2,n) > log(2,2 to_power 9) by A1,POWER:57;
then x > 9*log(2,2) by POWER:55;
then
A9: x > 9*1 by POWER:52;
then
A10: 2*x > 0*x by XREAL_1:68;
then (2*x)*(2*x) > 0*(2*x) by XREAL_1:68;
then log(2, 2 to_power x) > log(2,(2*x)^2) by A9,Lm31,POWER:57;
then x*log(2,2) > log(2,(2*x)^2) by POWER:55;
then x*1 > log(2,(2*x)^2) by POWER:52;
then x > log(2,(2*x) to_power 2) by POWER:46;
then x > 2*log(2,2*x) by A10,POWER:55;
then log(2,n) > 2*log(2,log(2,n to_power 2)) by A3,POWER:55;
then log(2,n) > 2*log(2,log(2,n^2)) by POWER:46;
then 2 to_power log(2,n) > 2 to_power (2*log(2,log(2,n^2))) by POWER:39;
then n > 2 to_power (2*log(2,log(2,n^2))) by A3,POWER:def 3;
then n > 2 to_power (log(2,log(2,n^2) to_power 2)) by A7,POWER:55;
then n > 2 to_power (log(2,(log(2,n^2))^2)) by POWER:46;
then n > (log(2,n^2))^2 by A8,POWER:def 3;
then n > (log(2,n to_power 2))^2 by POWER:46;
then n > (2*log(2,n))^2 by A3,POWER:55;
then n > 2*(log(2,n)*log(2,n)) by A4,XXREAL_0:2;
hence n/2 > log(2,n)*log(2,n) by XREAL_1:81;
end;
hence thesis;
end;
then consider N3 such that
A11: for n st n >= N3 holds n/2 > log(2,n)*log(2,n);
now
take N = 30;
let n;
assume
A12: n >= N;
then
A13: n to_power 6 > 0 by POWER:34;
2 to_power n > n to_power 6 by A12,Lm30;
then log(2,2 to_power n) > log(2,n to_power 6) by A13,POWER:57;
then n*log(2,2) > log(2,n to_power 6) by POWER:55;
then n*1 > log(2,n to_power 6) by POWER:52;
then n > (3*2)*log(2,n) by A12,POWER:55;
then n > 3*(2*log(2,n));
hence n/3 > 2*log(2,n) by XREAL_1:81;
end;
then consider N2 such that
A14: for n st n >= N2 holds n/3 > 2*log(2,n);
now
take N = 7;
let n;
assume n >= N;
then n > 6 by XXREAL_0:2;
then n/6 > 6/6 by XREAL_1:74;
hence n/6 > 1;
end;
then consider N1 such that
A15: for n st n >= N1 holds n/6 > 1;
set N = max( max(N1, 2), max( N2, N3 ) );
A16: N >= max(N1, 2) by XXREAL_0:25;
max(N1, 2) >= 2 by XXREAL_0:25;
then
A17: N >= 2 by A16,XXREAL_0:2;
A18: N >= max( N2, N3 ) by XXREAL_0:25;
max( N2, N3 ) >= N3 by XXREAL_0:25;
then
A19: N >= N3 by A18,XXREAL_0:2;
max( N2, N3 ) >= N2 by XXREAL_0:25;
then
A20: N >= N2 by A18,XXREAL_0:2;
max(N1, 2) >= N1 by XXREAL_0:25;
then
A21: N >= N1 by A16,XXREAL_0:2;
now
let n;
A22: (1 + 2*log(2,n)) + log(2,n)*log(2,n) = (1 + log(2,n))^2;
assume
A23: n >= N;
then n >= N2 by A20,XXREAL_0:2;
then
A24: n/3 > 2*log(2,n) by A14;
n >= 2 by A17,A23,XXREAL_0:2;
then log(2,n) >= log(2,2) by PRE_FF:10;
then
A25: log(2,n) >= 1 by POWER:52;
n >= N1 by A21,A23,XXREAL_0:2;
then n/6 > 1 by A15;
then
A26: n/6 + n/3 > 1 + 2*log(2,n) by A24,XREAL_1:8;
n >= N3 by A19,A23,XXREAL_0:2;
then
A27: n/2 > log(2,n)*log(2,n) by A11;
(n/6 + n/3) + n/2 = n;
then n > (1 + log(2,n))^2 by A26,A27,A22,XREAL_1:8;
then sqrt n > sqrt((1 + log(2,n))^2) by SQUARE_1:27,XREAL_1:63;
then sqrt n > 1 + log(2,n) by A25,SQUARE_1:22;
hence sqrt n - log(2,n) > 1 by XREAL_1:20;
end;
hence thesis;
end;
Lm33: 5! = 120
proof
(4+1)! = (4+1)*(4!) by NEWTON:15
.= 5*((3+1)*(3!)) by NEWTON:15
.= 5*(4*((2+1)*(2!))) by NEWTON:15
.= 120 by NEWTON:14;
hence thesis;
end;
Lm34: for n st n >= 10 holds 2 to_power (2*n) / (n!) < 1/(2 to_power (n-9))
proof
defpred P[Nat] means 2 to_power (2*$1) / ($1!) < 1/(2 to_power ($1-9));
A1: not 4096 / 14175 >= 1 / 2;
A2: 7 = 8-1;
A3: for k be Nat st k >= 10 & P[k] holds P[k+1]
proof
let k be Nat;
assume that
A4: k >= 10 and
A5: 2 to_power (2*k) / (k!) < 1/(2 to_power (k-9));
A6: 2 to_power 1 > 0 by POWER:34;
A7: now
assume ((2 to_power 2)/(k+1)) >= (1/(2 to_power 1));
then (2 to_power 1)*((2 to_power 2)*(k+1)") >= (1/(2 to_power 1))*(2
to_power 1) by XREAL_1:64;
then ((2 to_power 1)*(2 to_power 2))*(k+1)" >= (1/(2 to_power 1))*(2
to_power 1);
then (2 to_power (1+2))*(k+1)" >= (1/(2 to_power 1))*(2 to_power 1) by
POWER:27;
then 8/(k+1) >= 1 by A6,POWER:61,XCMPLX_1:106;
then 8/(k+1)*(k+1) >= 1*(k+1) by XREAL_1:64;
then 8 >= k+1 by XCMPLX_1:87;
then 7 >= k by A2,XREAL_1:19;
hence contradiction by A4,XXREAL_0:2;
end;
2 to_power -(k-9) > 0 by POWER:34;
then (1/(2 to_power (k-9))) > 0 by POWER:28;
then
A8: ((2 to_power 2)/(k+1))*(1/(2 to_power (k-9))) < (1/(2 to_power 1))*(1
/(2 to_power (k-9))) by A7,XREAL_1:68;
(2 to_power 2) > 0 by POWER:34;
then
A9: (2 to_power 2)*(k+1)" > 0*(k+1)" by XREAL_1:68;
2 to_power (2*(k+1)) / ((k+1)!) = (2 to_power (2*k + 2*1)) / ((k+1)!)
.= ((2 to_power (2*k))*(2 to_power 2)) / ((k+1)!) by POWER:27
.= ((2 to_power (2*k))*(2 to_power 2)) / ((k+1)*(k!)) by NEWTON:15
.= ((2 to_power 2)/(k+1))* ((2 to_power (2*k))/(k!)) by XCMPLX_1:76;
then 2 to_power (2*(k+1)) / ((k+1)!) < ((2 to_power 2)/(k+1))*(1/(2
to_power (k-9))) by A5,A9,XREAL_1:68;
then
2 to_power (2*(k+1)) / ((k+1)!) < (1/(2 to_power 1))*(1/(2 to_power (
k-9))) by A8,XXREAL_0:2;
then
2 to_power (2*(k+1)) / ((k+1)!) < (1/((2 to_power 1)*(2 to_power (k-9
)))) by XCMPLX_1:102;
then 2 to_power (2*(k+1)) / ((k+1)!) < (1/(2 to_power (1+(k+-9)))) by
POWER:27;
hence 2 to_power (2*(k+1)) / ((k+1)!) < (1/(2 to_power ((k+1)-9)));
end;
2 to_power (2*10) / (10!) = 2 to_power 20 / ((9+1)*(9!)) by NEWTON:15
.= 2 to_power (1+19) / (10*(9!))
.= ((2 to_power 1)*(2 to_power 19)) / (10*(9!)) by POWER:27
.= (2*(2 to_power 19)) / (2*(5*(9!))) by POWER:25
.= (2 to_power 19) / (5*(9!)) by XCMPLX_1:91
.= (2 to_power 19) / (5*((8+1)*(8!))) by NEWTON:15
.= (2 to_power 19) / ((5*9)*(8!))
.= (2 to_power 19) / (45*((7+1)*(7!))) by NEWTON:15
.= (2 to_power (3+16)) / (8*(45*(7!)))
.= (8*(2 to_power 16)) / (8*(45*(7!))) by POWER:27,61
.= (2 to_power 16) / (45*(7!)) by XCMPLX_1:91
.= (2 to_power (4+12)) / (45*((6+1)*(6!))) by NEWTON:15
.= ((2 to_power (3+1))*4096) / (45*((6+1)*(6!))) by Lm26,POWER:27
.= ((8*(2 to_power 1))*4096) / (45*((6+1)*(6!))) by POWER:27,61
.= ((8*2)*4096) / (45*((6+1)*(6!))) by POWER:25
.= (16*4096) / ((45*7)*(6!))
.= (16*4096) / (315*((5+1)*(5!))) by NEWTON:15
.= 4096 / 14175 by Lm33;
then
A10: P[10] by A1,POWER:25;
for n be Nat st n >= 10 holds P[n] from NAT_1:sch 8(A10, A3);
hence thesis;
end;
Lm35: for n st n >= 3 holds 2*(n-2) >= n-1
proof
defpred P[Nat] means 2*($1-2) >= $1-1;
A1: for n be Nat st n >= 3 & P[n] holds P[n+1]
proof
let n be Nat;
assume that
n >= 3 and
A2: 2*(n-2) >= n-1;
2*(n-2) + 2 >= (n+-1) +1 by A2,XREAL_1:7;
hence 2*((n+1)-2) >= (n+1)-1;
end;
A3: P[3];
for n be Nat st n >= 3 holds P[n] from NAT_1:sch 8(A3, A1);
hence thesis;
end;
Lm36: 5 to_power 5 = 3125
proof
5 to_power 5 = 5 to_power (4+1) .= (5 to_power 4)*(5 to_power 1) by POWER:27
.= (5 to_power (3+1))*5 by POWER:25
.= ((5 to_power 3)*(5 to_power 1))*5 by POWER:27
.= ((5 to_power (2+1))*5)*5 by POWER:25
.= (((5 to_power 2)*(5 to_power 1))*5)*5 by POWER:27
.= (((5 to_power (1+1))*5)*5)*5 by POWER:25
.= ((((5 to_power 1)*(5 to_power 1))*5)*5)*5 by POWER:27
.= ((((5 to_power 1)*5)*5)*5)*5 by POWER:25
.= (((5*5)*5)*5)*5 by POWER:25
.= 3125;
hence thesis;
end;
Lm37: 4 to_power 4 = 256
proof
4 to_power 4 = 4 to_power (3+1) .= (4 to_power 3)*(4 to_power 1) by POWER:27
.= (4 to_power (2+1))*4 by POWER:25
.= ((4 to_power 2)*(4 to_power 1))*4 by POWER:27
.= ((4 to_power (1+1))*4)*4 by POWER:25
.= (((4 to_power 1)*(4 to_power 1))*4)*4 by POWER:27
.= (((4 to_power 1)*4)*4)*4 by POWER:25
.= ((4*4)*4)*4 by POWER:25
.= 256;
hence thesis;
end;
Lm38: for a,b,d,e being Real holds (a/b) / (d/e) = (a/d) * (e/b)
proof
let a,b,d,e be Real;
thus (a/b) / (d/e) = (a*e)/(b*d) by XCMPLX_1:84
.= (a/d) * (e/b) by XCMPLX_1:76;
end;
Lm39: for x being Real st x>=0 holds sqrt x = x to_power (1/2)
proof
let x be Real;
assume
A1: x >= 0;
per cases by A1;
suppose
x = 0;
hence thesis by POWER:def 2,SQUARE_1:17;
end;
suppose
A2: x > 0;
then
A3: x to_power (1/2)>0 by POWER:34;
(x to_power (1/2))^2=(x to_power (1/2)) to_power 2 by POWER:46
.=x to_power ((1/2)*2) by A2,POWER:33
.=x by POWER:25;
hence thesis by A3,SQUARE_1:22;
end;
end;
Lm40: ex N st for n st n >= N holds n - sqrt n*log(2,n) > n/2
proof
set seq1 = seq_n^(1/2);
set seq = seq_logn;
set p = seq /" seq1;
A1: lim p = 0 by Lm11;
p is convergent by Lm11;
then consider N being Nat such that
A2: for n being Nat st n >= N holds |.p.n-0.| < 1/2 by A1,SEQ_2:def 7;
reconsider N as Element of NAT by ORDINAL1:def 12;
set N1 = max(2, N);
A3: N1 >= 2 by XXREAL_0:25;
A4: N1 >= N by XXREAL_0:25;
now
let n;
assume
A5: n >= N1;
then
A6: sqrt n > 0 by A3,SQUARE_1:25;
n >= N by A4,A5,XXREAL_0:2;
then |.p.n-0.| < 1/2 by A2;
then
A7: p.n < 1/2 by ABSVALUE:def 1;
A8: sqrt n <> 0 by A3,A5,SQUARE_1:25;
p.n = seq.n / seq1.n by Lm4
.= log(2,n) / seq1.n by A3,A5,Def2
.= log(2,n) / (n to_power (1/2)) by A3,A5,Def3
.= log(2,n) / sqrt n by Lm39;
then log(2,n) / sqrt n * sqrt n < sqrt n * (1/2) by A6,A7,XREAL_1:68;
then log(2,n) < sqrt n * (1/2) by A8,XCMPLX_1:87;
then sqrt n*log(2,n) < sqrt n * (sqrt n * (1/2)) by A6,XREAL_1:68;
then sqrt n*log(2,n) < ((sqrt n)^2) * (1/2);
then sqrt n*log(2,n) < n * (1/2) by SQUARE_1:def 2;
then n/2 + sqrt n*log(2,n) < n/2 + n/2 by XREAL_1:6;
hence n/2 < n - sqrt n*log(2,n) by XREAL_1:20;
end;
hence thesis;
end;
Lm41: for s being Real_Sequence
st for n being Nat holds s.n = (1 + 1/(n+1)) to_power (n+1)
holds s is non-decreasing
proof
let s be Real_Sequence such that
A1: for n be Nat holds s.n = (1 + 1/(n+1)) to_power (n+1);
now
let n be Nat;
A2: (1 + 1/(n+1+1))/(1 + 1/(n+1)) = ((1*(n+1+1) + 1)/(n+1+1))/(1 + 1/(n+1)
) by XCMPLX_1:113
.= ((n+1+1+1)/(n+1+1))/((1*(n+1) + 1)/(n+1)) by XCMPLX_1:113
.= ((n+(1+1)+1)*(n+1))/((n+2)*(n+2)) by XCMPLX_1:84
.= (n*n+n*2+2*n+3+1-1)/((n+2)*(n+2))
.= ((n+2)*(n+2))/((n+2)*(n+2)) - 1/((n+2)*(n+2))
.= 1 - 1/((n+2)*(n+2)) by XCMPLX_1:6,60;
n+1+1>0+1 by XREAL_1:6;
then (n+2)*(n+2)>1 by XREAL_1:161;
then 1/((n+2)*(n+2))<1 by XREAL_1:212;
then - 1/((n+2)*(n+2)) > -1 by XREAL_1:24;
then
(1 + -1/((n+2)*(n+2))) to_power (n+1+1) >= 1 + (n+1+1)*(-1/((n+2)*(n+
2) )) by POWER:49;
then (1 - 1/((n+2)*(n+2))) to_power (n+1+1) >= 1 - (n+2)*1/((n+2)*(n+2));
then (1 - 1/((n+2)*(n+2))) to_power (n+1+1) >= 1 - ((n+2)/(n+2)*1)/(n+2)
by XCMPLX_1:83;
then
A3: (1 - 1/((n+2)*(n+2))) to_power (n+1+1) >= 1 - (1*1)/(n+2) by XCMPLX_1:60;
s.(n+1)/s.n =(1 + 1/(n+1+1)) to_power (n+1+1) / s.n by A1
.=(1 + 1/(n+1+1)) to_power (n+1+1)/(1+1/(n+1)) to_power (n+1)*1
by A1
.=(1 + 1/(n+1+1)) to_power (n+1+1)/(1 + 1/(n+1)) to_power (n+1) * ((1+
1/(n+1))/(1+1/(n+1))) by XCMPLX_1:60
.=(1+1/(n+1)) * (1 + 1/(n+1+1)) to_power (n+1+1) / ((1 + 1/(n+1))
to_power (n+1) * (1+1/(n+1))) by XCMPLX_1:76
.=(1+1/(n+1)) * (1 + 1/(n+1+1)) to_power (n+1+1) / ((1 + 1/(n+1))
to_power (n+1) * (1+1/(n+1)) to_power 1) by POWER:25
.=(1+1/(n+1)) * (1 + 1/(n+1+1)) to_power (n+1+1) / (1 + 1/(n+1))
to_power (n+1+1) by POWER:27
.=(1+1/(n+1)) * ((1 + 1/(n+1+1)) to_power (n+1+1) / (1 + 1/(n+1))
to_power (n+1+1))
.=(1+1/(n+1)) * ((1 + 1/(n+1+1))/(1 + 1/(n+1))) to_power (n+1+1) by
POWER:31;
then s.(n+1)/s.n >= (1 + 1/(n+1)) * (1 - 1/(n+2)) by A2,A3,XREAL_1:64;
then s.(n+1)/s.n >= ((1*(n+1) + 1)/(n+1)) * (1 - 1/(n+2)) by XCMPLX_1:113;
then s.(n+1)/s.n >= ((n+2)/(n+1)) * ((1*(n+2) - 1)/(n+2)) by XCMPLX_1:127;
then s.(n+1)/s.n >= ((n+1)*(n+2))/((n+1)*(n+2)) by XCMPLX_1:76;
then
A4: s.(n+1)/s.n >= 1 by XCMPLX_1:6,60;
(1+1/(n+1)) to_power (n+1) > 0 by POWER:34;
then s.n>0 by A1;
hence s.(n+1)>=s.n by A4,XREAL_1:191;
end;
hence thesis;
end;
Lm42: for n st n >= 1 holds ((n+1)/n) to_power n <= ((n+2)/(n+1)) to_power (n+
1)
proof
deffunc F(Nat) = (1 + 1/($1+1)) to_power ($1+1);
let n;
consider seq being Real_Sequence such that
A1: for n being Nat holds seq.n = F(n) from SEQ_1:sch 1;
assume
A2: n >= 1;
then n >= 0+1;
then n-1 >= 0;
then reconsider m = n-1 as Element of NAT by INT_1:3;
seq is non-decreasing by A1,Lm41;
then seq.m <= seq.(m+1);
then (1 + 1/(m+1)) to_power (m+1) <= seq.(m+1) by A1;
then (1 + 1/n) to_power n <= (1 + 1/(n+1)) to_power (n+1) by A1;
then (n/n + 1/n) to_power n <= (1 + 1/(n+1)) to_power (n+1) by A2,XCMPLX_1:60
;
then ((n+1)/n) to_power n <= ((n+1)/(n+1) + 1/(n+1)) to_power (n+1) by
XCMPLX_1:60;
hence thesis;
end;
theorem :: (Part 1, O(n^logn) c O(n^sqrt n))
for f,g being Real_Sequence st (for n st n > 0 holds f.n = (n to_power
log(2,n))) & (for n st n > 0 holds g.n = (n to_power sqrt n)) holds ex s,s1
being eventually-positive Real_Sequence st s = f & s1 = g & Big_Oh(s) c= Big_Oh
(s1) & not Big_Oh(s) = Big_Oh(s1)
proof
let f,g be Real_Sequence such that
A1: for n st n > 0 holds f.n = (n to_power log(2,n)) and
A2: for n st n > 0 holds g.n = (n to_power sqrt n);
set h = f/"g;
g is eventually-positive
proof
take 1;
let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
assume
A4: n >= 1;
then g.n = n to_power sqrt n by A2,A3;
hence thesis by A4,POWER:34;
end;
then reconsider g as eventually-positive Real_Sequence;
f is eventually-positive
proof
take 1;
let n be Nat;
A5: n in NAT by ORDINAL1:def 12;
assume
A6: n >= 1;
then f.n = n to_power log(2,n) by A1,A5;
hence thesis by A6,POWER:34;
end;
then reconsider f as eventually-positive Real_Sequence;
take f,g;
consider N such that
A7: for n st n >= N holds sqrt n - log(2, n) > 1 by Lm32;
A8: now
let p be Real such that
A9: p > 0;
set N1 = max( N, max([/1/p\], 2) );
A10: N1 >= N by XXREAL_0:25;
A11: N1 is Integer
proof
per cases by XXREAL_0:16;
suppose
N1 = N;
hence thesis;
end;
suppose
N1 = max([/1/p\], 2);
hence thesis by XXREAL_0:16;
end;
end;
A12: N1 >= max([/1/p\], 2) by XXREAL_0:25;
max([/1/p\], 2) >= [/1/p\] by XXREAL_0:25;
then
A13: N1 >= [/1/p\] by A12,XXREAL_0:2;
A14: max([/1/p\], 2) >= 2 by XXREAL_0:25;
then N1 >= 2 by A12,XXREAL_0:2;
then
A15: N1 > 1 by XXREAL_0:2;
N1 in NAT by A10,A11,INT_1:3;
then reconsider N1 as Nat;
take N1;
let n be Nat;
A16: n in NAT by ORDINAL1:def 12;
A17: h.n = f.n/g.n by Lm4;
assume
A18: n >= N1;
then f.n = n to_power log(2,n) by A1,A12,A14,A16;
then
A19: h.n = (n to_power log(2,n)) / (n to_power sqrt n)
by A2,A12,A14,A18,A17,A16
.= n to_power (log(2,n) - sqrt n) by A12,A14,A18,POWER:29
.= n to_power -(sqrt n - log(2,n));
then
A20: h.n > 0 by A12,A14,A18,POWER:34;
n >= N by A10,A18,XXREAL_0:2;
then sqrt n - log(2, n) > 1 by A7,A16;
then
A21: (-1)*(sqrt n - log(2, n)) < (-1)*1 by XREAL_1:69;
n > 1 by A15,A18,XXREAL_0:2;
then
A22: n to_power -(sqrt n - log(2,n)) < n to_power -1 by A21,POWER:39;
[/1/p\] >= 1/p by INT_1:def 7;
then N1 >= 1/p by A13,XXREAL_0:2;
then n >= 1/p by A18,XXREAL_0:2;
then
A23: 1/n <= 1/(1/p) by A9,XREAL_1:85;
n to_power -1 = 1/(n to_power 1) by A12,A14,A18,POWER:28
.= 1/n by POWER:25;
then h.n < p by A19,A22,A23,XXREAL_0:2;
hence |.h.n-0.| < p by A20,ABSVALUE:def 1;
end;
then
A24: h is convergent by SEQ_2:def 6;
then
A25: lim h = 0 by A8,SEQ_2:def 7;
then not g in Big_Oh(f) by A24,ASYMPT_0:16;
then
A26: not f in Big_Omega(g) by ASYMPT_0:19;
f in Big_Oh(g) by A24,A25,ASYMPT_0:16;
hence thesis by A26,Th4;
end;
theorem
for f being Real_Sequence st (for n st n > 0 holds f.n = (n to_power
sqrt n)) holds ex s,s1 being eventually-positive Real_Sequence st s = f & s1 =
seq_a^(2,1,0) & Big_Oh(s) c= Big_Oh(s1) & not Big_Oh(s) = Big_Oh(s1)
proof
set g = seq_a^(2,1,0);
let f be Real_Sequence such that
A1: for n st n > 0 holds f.n = (n to_power sqrt n);
A2: f is eventually-positive
proof
take 1;
let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
assume
A4: n >= 1;
then f.n = n to_power sqrt n by A1,A3;
hence thesis by A4,POWER:34;
end;
set h = f/"g;
reconsider f as eventually-positive Real_Sequence by A2;
reconsider g as eventually-positive Real_Sequence;
take f,g;
consider N such that
A5: for n st n >= N holds n - sqrt n*log(2,n) > n/2 by Lm40;
A6: now
let p be Real;
assume
A7: p > 0;
set N1 = max( N, max(2*[/log(2,1/p)\], 2) );
A8: N1 >= N by XXREAL_0:25;
A9: N1 is Integer
proof
per cases by XXREAL_0:16;
suppose
N1 = N;
hence thesis;
end;
suppose
N1 = max(2*[/log(2,1/p)\], 2);
hence thesis by XXREAL_0:16;
end;
end;
A10: N1 >= max(2*[/log(2,1/p)\], 2) by XXREAL_0:25;
max(2*[/log(2,1/p)\], 2) >= 2*[/log(2,1/p)\] by XXREAL_0:25;
then
A11: N1 >= 2*[/log(2,1/p)\] by A10,XXREAL_0:2;
N1 in NAT by A8,A9,INT_1:3;
then reconsider N1 as Nat;
take N1;
let n be Nat;
A12: n in NAT by ORDINAL1:def 12;
A13: h.n = f.n/g.n by Lm4;
A14: [/log(2,1/p)\] >= log(2,1/p) by INT_1:def 7;
assume
A15: n >= N1;
then n >= 2*[/log(2,1/p)\] by A11,XXREAL_0:2;
then n/2 >= [/log(2,1/p)\] by XREAL_1:77;
then n/2 >= log(2,1/p) by A14,XXREAL_0:2;
then -(n/2) <= -log(2,1/p) by XREAL_1:24;
then 2 to_power -(n/2) <= 2 to_power -log(2,1/p) by PRE_FF:8;
then 2 to_power -(n/2) <= 1/(2 to_power log(2,1/p)) by POWER:28;
then
A16: 2 to_power -(n/2) <= 1/(1/p) by A7,POWER:def 3;
A17: g.n = 2 to_power (1*n+0) by Def1
.= 2 to_power n;
A18: max(2*[/log(2,1/p)\], 2) >= 2 by XXREAL_0:25;
then f.n = n to_power sqrt n by A1,A10,A15,A12
.= 2 to_power (sqrt n*log(2,n)) by A10,A18,A15,Lm3;
then
A19: h.n = 2 to_power ((sqrt n*log(2,n)) - n) by A13,A17,POWER:29
.= 2 to_power -(n - sqrt n*log(2,n));
then
A20: h.n > 0 by POWER:34;
n >= N by A8,A15,XXREAL_0:2;
then n - sqrt n*log(2,n) > n/2 by A5,A12;
then -(n - sqrt n*log(2,n)) < -(n/2) by XREAL_1:24;
then 2 to_power -(n - sqrt n*log(2,n)) < 2 to_power -(n/2) by POWER:39;
then h.n < p by A19,A16,XXREAL_0:2;
hence |.h.n-0.| < p by A20,ABSVALUE:def 1;
end;
then
A21: h is convergent by SEQ_2:def 6;
then
A22: lim h = 0 by A6,SEQ_2:def 7;
then not g in Big_Oh(f) by A21,ASYMPT_0:16;
then
A23: not f in Big_Omega(g) by ASYMPT_0:19;
f in Big_Oh(g) by A21,A22,ASYMPT_0:16;
hence thesis by A23,Th4;
end;
theorem
ex s, s1 being eventually-positive Real_Sequence st s = seq_a^(2,1,0)
& s1 = seq_a^(2,1,1) & Big_Oh(s) = Big_Oh(s1)
proof
set g = seq_a^(2,1,1);
set f = seq_a^(2,1,0);
set h = f/"g;
reconsider f as eventually-positive Real_Sequence;
reconsider g as eventually-positive Real_Sequence;
take f,g;
thus f = seq_a^(2,1,0) & g = seq_a^(2,1,1);
A1: now
let n;
A2: g.n = 2 to_power (1*n + 1) by Def1;
f.n = 2 to_power (1*n + 0) by Def1;
then h.n = (2 to_power n)/g.n by Lm4
.= 2 to_power (n-(n+1)) by A2,POWER:29
.= 2 to_power (0+-1)
.= 1 / 2 to_power 1 by POWER:28
.= 1 / 2 by POWER:25
.= 2";
hence h.n = 2";
end;
A3: now
let p be Real such that
A4: p > 0;
reconsider N = 0 as Nat;
take N;
let n be Nat;
A5: n in NAT by ORDINAL1:def 12;
assume n >= N;
|.h.n-2".| = |.2"-2".| by A1,A5
.= 0 by ABSVALUE:2;
hence |.h.n-2".| < p by A4;
end;
then
A6: h is convergent by SEQ_2:def 6;
then lim h > 0 by A3,SEQ_2:def 7;
hence thesis by A6,ASYMPT_0:15;
end;
theorem :: (Part 4, O(2^n) c O(2^2n))
ex s, s1 being eventually-positive Real_Sequence st s = seq_a^(2,1,0)
& s1 = seq_a^(2,2,0) & Big_Oh(s) c= Big_Oh(s1) & not Big_Oh(s) = Big_Oh(s1)
proof
reconsider g = seq_a^(2,2,0) as eventually-positive Real_Sequence;
reconsider f = seq_a^(2,1,0) as eventually-positive Real_Sequence;
take f,g;
thus f = seq_a^(2,1,0) & g = seq_a^(2,2,0);
set h = f/"g;
A1: for n holds h.n = (2 to_power -n)
proof
let n;
h.n = f.n/g.n by Lm4
.= (2 to_power (1*n+0))/g.n by Def1
.= (2 to_power (1*n))/(2 to_power (2*n+0)) by Def1
.= 2 to_power (1*n-(2*n)) by POWER:29
.= 2 to_power -n;
hence thesis;
end;
A2: now
let p be Real;
set N = max(1,[/log(2,1/p)\] + 1);
A3: N >= 1 by XXREAL_0:25;
A4: N is Integer by XXREAL_0:16;
A5: [/log(2,1/p)\] >= log(2,1/p) by INT_1:def 7;
[/log(2,1/p)\] + 1 > [/log(2,1/p)\] by XREAL_1:29;
then [/log(2,1/p)\] + 1 > log(2,1/p) by A5,XXREAL_0:2;
then
A6: 2 to_power ([/log(2,1/p)\]+1) > 2 to_power log(2,1/p) by POWER:39;
N in NAT by A3,A4,INT_1:3;
then reconsider N as Nat;
assume
A7: p > 0;
take N;
let n be Nat;
A8: n in NAT by ORDINAL1:def 12;
[/log(2,1/p)\] + 1 <= N by XXREAL_0:25; then
2 to_power N >= 2 to_power ([/log(2,1/p)\] + 1) by PRE_FF:8;
then
A9: 2 to_power N > 2 to_power log(2,1/p) by A6,XXREAL_0:2;
assume n >= N;
then 2 to_power n >= 2 to_power N by PRE_FF:8;
then 2 to_power n > 2 to_power log(2,1/p) by A9,XXREAL_0:2;
then 2 to_power n > 1/p by A7,POWER:def 3;
then (2 to_power n)*p > 1/p*p by A7,XREAL_1:68;
then
A10: p*(2 to_power n) > 1 by A7,XCMPLX_1:87;
2 to_power n > 0 by POWER:34;
then (p*(2 to_power n))*(2 to_power n)" > 1*(2 to_power n)" by A10,
XREAL_1:68;
then
A11: p*((2 to_power n)*(2 to_power n)") > (2 to_power n)";
2 to_power n <> 0 by POWER:34;
then p*1 > (2 to_power n)" by A11,XCMPLX_0:def 7;
then
A12: p > 1/(2 to_power n);
A13: (2 to_power -n) > 0 by POWER:34;
|.h.n-0.| = |.(2 to_power -n).| by A1,A8;
then |.h.n-0.| = (2 to_power -n) by A13,ABSVALUE:def 1;
hence |.h.n-0.| < p by A12,POWER:28;
end;
then
A14: h is convergent by SEQ_2:def 6;
then
A15: lim h = 0 by A2,SEQ_2:def 7;
then not g in Big_Oh(f) by A14,ASYMPT_0:16;
then
A16: not f in Big_Omega(g) by ASYMPT_0:19;
f in Big_Oh(g) by A14,A15,ASYMPT_0:16;
hence thesis by A16,Th4;
end;
theorem :: (Part 5, O(2^2n) c O(n!))
ex s being eventually-positive Real_Sequence st s = seq_a^(2,2,0) &
Big_Oh(s) c= Big_Oh(seq_n!(0)) & not Big_Oh(s) = Big_Oh(seq_n!(0))
proof
reconsider f = seq_a^(2,2,0)as eventually-positive Real_Sequence;
set g = seq_n!(0);
take f;
thus f = seq_a^(2,2,0);
set h = f/"g;
A1: now
let p be Real;
assume
A2: p > 0;
set N = max(10,[/9+log(2,1/p)\]);
A3: N >= 10 by XXREAL_0:25;
A4: N is Integer by XXREAL_0:16;
A5: N >= [/9+log(2,1/p)\] by XXREAL_0:25;
N in NAT by A3,A4,INT_1:3;
then reconsider N as Nat;
take N;
let n be Nat;
A6: n in NAT by ORDINAL1:def 12;
A7: [/9+log(2,1/p)\] >= 9+log(2,1/p) by INT_1:def 7;
assume
A8: n >= N;
then n >= [/9+log(2,1/p)\] by A5,XXREAL_0:2;
then n >= 9+log(2,1/p) by A7,XXREAL_0:2;
then n-9 >= log(2,1/p) by XREAL_1:19;
then 2 to_power (n-9) >= 2 to_power log(2,1/p) by PRE_FF:8;
then 2 to_power (n-9) >= 1/p by A2,POWER:def 3;
then
A9: 1 / (2 to_power (n-9)) <= 1/(1/p) by A2,XREAL_1:85;
A10: h.n = f.n/g.n by Lm4
.= (2 to_power (2*n+0)) / g.n by Def1,A6
.= (2 to_power (2*n+0)) / ((n+0)!) by Def4
.= (2 to_power (2*n)) / (n!);
n >= 10 by A3,A8,XXREAL_0:2;
then h.n < 1 / (2 to_power (n-9)) by A10,Lm34,A6;
then h.n < p by A9,XXREAL_0:2;
hence |.h.n-0.| < p by A10,ABSVALUE:def 1;
end;
then
A11: h is convergent by SEQ_2:def 6;
then
A12: lim h = 0 by A1,SEQ_2:def 7;
then not g in Big_Oh(f) by A11,ASYMPT_0:16;
then
A13: not f in Big_Omega(g) by ASYMPT_0:19;
f in Big_Oh(g) by A11,A12,ASYMPT_0:16;
hence thesis by A13,Th4;
end;
theorem :: (Part 6, O(n!) c O((n+1)!))
Big_Oh(seq_n!(0)) c= Big_Oh(seq_n!(1)) & not Big_Oh(seq_n!(0)) =
Big_Oh(seq_n!(1))
proof
set g = seq_n!(1);
set f = seq_n!(0);
set h = f/"g;
A1: for n holds h.n = 1/(n+1)
proof
let n;
A2: n! <> 0 by NEWTON:17;
h.n = f.n/g.n by Lm4
.= (n+0)!/g.n by Def4
.= (n!)/((n+1)!) by Def4
.= ((n!)*1)/((n+1)*(n!)) by NEWTON:15
.= (1/(n+1))*((n!)/(n!)) by XCMPLX_1:76
.= (1/(n+1))*1 by A2,XCMPLX_1:60;
hence thesis;
end;
A3: now
let p be Real;
assume
A4: p > 0;
set N = max( 1, [/1/p\] );
A5: N >= 1 by XXREAL_0:25;
A6: N >= ([/1/p\]) by XXREAL_0:25;
N is Integer by XXREAL_0:16;
then reconsider N as Element of NAT by A5,INT_1:3;
[/1/p\] >= 1/p by INT_1:def 7;
then
A7: N >= 1/p by A6,XXREAL_0:2;
reconsider N as Nat;
take N;
let n be Nat;
A8: n in NAT by ORDINAL1:def 12;
assume n >= N;
then n+1 > N by NAT_1:13;
then (n+1) > 1/p by A7,XXREAL_0:2;
then 1/(n+1) < 1/(1/p) by A4,XREAL_1:88;
then
A9: h.n < p by A1,A8;
A10: 0 < 1/(n+1);
-p < -0 by A4,XREAL_1:24;
then
A11: -p < h.n by A1,A10,A8;
|.h.n.| < p
proof
per cases;
suppose
h.n >= 0;
hence thesis by A9,ABSVALUE:def 1;
end;
suppose
A12: h.n < 0;
A13: (-1)*(-p) > (-1)*h.n by A11,XREAL_1:69;
|.h.n.| = -h.n by A12,ABSVALUE:def 1;
hence thesis by A13;
end;
end;
hence |.h.n-0.| < p;
end;
then
A14: h is convergent by SEQ_2:def 6;
then
A15: lim h = 0 by A3,SEQ_2:def 7;
then not g in Big_Oh(f) by A14,ASYMPT_0:16;
then
A16: not f in Big_Omega(g) by ASYMPT_0:19;
f in Big_Oh(g) by A14,A15,ASYMPT_0:16;
hence thesis by A16,Th4;
end;
theorem :: (Part 7, O((n+1)!) c O(n^n))
for g being Real_Sequence st (for n st n > 0 holds g.n = (n to_power n
)) holds ex s being eventually-positive Real_Sequence st s = g & Big_Oh(seq_n!(
1)) c= Big_Oh(s) & not Big_Oh(seq_n!(1)) = Big_Oh(s)
proof
set f = seq_n!(1);
let g be Real_Sequence such that
A1: for n st n > 0 holds g.n = (n to_power n);
A2: g is eventually-positive
proof
take 1;
let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
assume
A4: n >= 1;
then g.n = n to_power n by A1,A3;
hence thesis by A4,POWER:34;
end;
set h = f/"g;
reconsider g as eventually-positive Real_Sequence by A2;
deffunc F(Nat) = h.$1 / h.($1+1);
consider p being Real_Sequence such that
A5: for n being Nat holds p.n = F(n) from SEQ_1:sch 1;
defpred P1[Nat] means p.$1 > 2;
A6: for n st n > 0 holds p.n = ((n+1)/(n+2)) * (((n+1)/n) to_power n)
proof
let n;
assume
A7: n > 0;
A8: (n+1)! > 0 by NEWTON:17;
p.n = h.n / h.(n+1) by A5
.= (f.n / g.n) / h.(n+1) by Lm4
.= ((n+1)! / g.n) / h.(n+1) by Def4
.= ((n+1)! / (n to_power n)) / h.(n+1) by A1,A7
.= ((n+1)! / (n to_power n)) / (f.(n+1) / g.(n+1)) by Lm4
.= ((n+1)! / (n to_power n)) / (((n+1)+1)! / g.(n+1)) by Def4
.= ((n+1)! / (n to_power n)) / (((n+1)+1)! / ((n+1) to_power (n+1)))
by A1
.= ((n+1)! / (((n+1)+1)!)) * (((n+1) to_power (n+1)) / (n to_power n))
by Lm38
.= ((n+1)! / (((n+1)+1)*((n+1)!))) * (((n+1) to_power (n+1)) / (n
to_power n)) by NEWTON:15
.= (1/((n+1)+1) * ((n+1)! / ((n+1)!))) * (((n+1) to_power (n+1)) / (n
to_power n)) by XCMPLX_1:103
.= ((1/((n+1)+1)) * 1) * (((n+1) to_power (n+1)) / (n to_power n)) by A8,
XCMPLX_1:60
.= (1/(n+2)) * ((((n+1) to_power n)*((n+1) to_power 1)) / (n to_power
n)) by POWER:27
.= (1/(n+2)) * ((((n+1) to_power n)*(n+1)) / (n to_power n)) by POWER:25
.= (1/(n+2)) * ((((n+1) to_power n)*(n+1)) * ((n to_power n)"))
.= (1/(n+2)) * ((((n+1) to_power n)*((n to_power n)"))*(n+1))
.= (1/(n+2)) * ((((n+1) to_power n)/(n to_power n))*(n+1))
.= (1/(n+2)) * ((((n+1)/n) to_power n)*(n+1)) by A7,POWER:31
.= ((n+1)*(1/(n+2))) * (((n+1)/n) to_power n)
.= ((n+1)/(n+2)) * (((n+1)/n) to_power n);
hence thesis;
end;
A9: for k be Nat st k >= 4 & P1[k] holds P1[k+1]
proof
let k be Nat;
assume that
A10: k >= 4 and
A11: p.k > 2;
(k+2)*(k+1)" > 0*(k+1)" by XREAL_1:68;
then
A12: (((k+2)/(k+1)) to_power (k+1)) > 0 by POWER:34;
(k+1)*k" > 0*k" by A10,XREAL_1:68;
then (((k+1)/k) to_power k) > 0 by POWER:34;
then
A13: (((k+1)/k) to_power k)*(((k+2)/(k+1)) to_power (k+1))" > 0*(((k+2)/(k
+1)) to_power (k+1))" by A12,XREAL_1:68;
A14: k in NAT by ORDINAL1:def 12;
A15: now
assume ((k+1)*(k+3)) >= ((k+2)*(k+2));
then (k*k + 4*k) + 3 >= k*k + 2*(2*k) + 2^2;
hence contradiction by XREAL_1:6;
end;
then ((k+1)*(k+3)) < 1*((k+2)*(k+2));
then
A16: (((k+1)*(k+3)) / ((k+2)*(k+2))) < 1 by XREAL_1:83;
(k+1)*(k+3) > 0*(k+3) by XREAL_1:68;
then
A17: (((k+1)*(k+3)) * ((k+2)*(k+2))") > 0*((k+2)*(k+2))" by A15,XREAL_1:68;
k >= 1 by A10,XXREAL_0:2;
then
(((k+1)/k) to_power k) <= 1*(((k+2)/(k+1)) to_power (k+1)) by A14,Lm42;
then (((k+1)/k) to_power k) / (((k+2)/(k+1)) to_power (k+1)) <= 1 by A12,
XREAL_1:79;
then
A18: (((k+1)*(k+3)) / ((k+2)*(k+2))) * ((((k+1)/k) to_power k) / (((k+2)/(
k+1)) to_power (k+1))) < 1*1 by A13,A17,A16,XREAL_1:98;
(k+2)*(k+3)" > 0*(k+3)" by XREAL_1:68;
then
A19: ((k+2)/(k+3))*(((k+2)/(k+1)) to_power (k+1)) > ((k+2)/(k+3))*0 by A12,
XREAL_1:68;
A20: p.(k+1) = (((k+1)+1)/((k+1)+2)*(((k+(1+1))/(k+1)) to_power (k+1))) by A6
.= ((k+2)/(k+3)*(((k+2)/(k+1)) to_power (k+1)));
then
p.k / p.(k+1) = ((k+1)/(k+2)) * (((k+1)/k) to_power k) / ((k+2)/(k+3)
*(((k+2)/(k+1)) to_power (k+1))) by A6,A10,A14
.= (((k+1)/(k+2)) / ((k+2)/(k+3))) * ((((k+1)/k) to_power k) / (((k+2)
/(k+1)) to_power (k+1))) by XCMPLX_1:76
.= (((k+1)*(k+3)) / ((k+2)*(k+2))) * ((((k+1)/k) to_power k) / (((k+2)
/(k+1)) to_power (k+1))) by XCMPLX_1:84;
then p.k / p.(k+1) * p.(k+1) < 1*p.(k+1) by A20,A19,A18,XREAL_1:68;
then p.(k+1) > p.k by A20,A19,XCMPLX_1:87;
hence thesis by A11,XXREAL_0:2;
end;
defpred P[Nat] means h.$1 < 1/($1-2);
take g;
A21: for n st n >= 1 holds h.n > 0
proof
let n;
A22: (n+1)! > 0 by NEWTON:17;
assume
A23: n >= 1;
then n to_power n > 0 by POWER:34;
then
A24: (n+1)!*(1/(n to_power n)) > (n+1)!*0 by A22,XREAL_1:68;
h.n = f.n/g.n by Lm4
.= (n+1)! / g.n by Def4
.= (n+1)! / (n to_power n) by A1,A23;
hence thesis by A24;
end;
p.4 = ((4+1) / (4+2)) * (((4+1)/4) to_power 4) by A6
.= (5 / 6) * ((5 to_power 4) / 256) by Lm37,POWER:31
.= (5 * (5 to_power 4)) / (6 * 256)
.= ((5 to_power 1)*(5 to_power 4)) / 1536 by POWER:25
.= (5 to_power (4+1)) / 1536 by POWER:27
.= 3125 / 1536 by Lm36;
then
A25: P1[4];
A26: for n be Nat st n >= 4 holds P1[n] from NAT_1:sch 8(A25, A9);
A27: 3 = 4-1;
A28: for k be Nat st k >= 4 & P[k] holds P[k+1]
proof
let k be Nat;
assume that
A29: k >= 4 and
A30: h.k < 1/(k-2);
A31: k in NAT by ORDINAL1:def 12;
A32: h.(k+1) > 0 by A21,NAT_1:11;
p.k > 2 by A26,A29;
then h.k / h.(k+1) > 2 by A5;
then (h.k / h.(k+1)) * h.(k+1) > 2*h.(k+1) by A32,XREAL_1:68;
then h.k > 2*h.(k+1) by A32,XCMPLX_1:87;
then
A33: h.k/2 > h.(k+1) by XREAL_1:81;
A34: k-1 >= 3 by A27,A29,XREAL_1:9;
k >= 3 by A29,XXREAL_0:2;
then 2*(k-2) >= k-1 by A31,Lm35;
then
A35: 1/(2*(k-2)) <= 1/(k-1) by A34,XREAL_1:85;
h.k*(1/2) < (1/(k-2))*(1/2) by A30,XREAL_1:68;
then h.k/2 < 1/(2*(k-2)) by XCMPLX_1:102;
then h.k/2 < 1/(k-1) by A35,XXREAL_0:2;
hence thesis by A33,XXREAL_0:2;
end;
h.4 = f.4/g.4 by Lm4
.= (4+1)!/g.4 by Def4
.= 120/256 by A1,Lm33,Lm37;
then
A36: P[4];
A37: for n be Nat st n >= 4 holds P[n] from NAT_1:sch 8(A36, A28);
A38: now
let p be Real;
set N = [/1/p + 4\];
A39: [/1/p + 4\] >= 1/p + 4 by INT_1:def 7;
assume
A40: p > 0;
then
A41: 4 + 1/p > 4 by XREAL_1:29;
then
A42: N >= 4 by A39,XXREAL_0:2;
N in NAT by A39,A41,INT_1:3;
then reconsider N as Nat;
take N;
let n be Nat;
A43: n in NAT by ORDINAL1:def 12;
assume
A44: n >= N;
then
A45: n >= 4 by A42,XXREAL_0:2;
then n >= 1 by XXREAL_0:2;
then
A46: h.n > 0 by A21,A43;
A47: 1/p + 2 > 1/p by XREAL_1:29;
n >= (1/p + 2) + 2 by A39,A44,XXREAL_0:2;
then n - 2 >= 1/p + 2 by XREAL_1:19;
then n - 2 > 1/p by A47,XXREAL_0:2;
then
A48: 1/(n-2) < 1/(1/p) by A40,XREAL_1:88;
h.n < 1/(n-2) by A37,A45;
then h.n < p by A48,XXREAL_0:2;
hence |.h.n-0.| < p by A46,ABSVALUE:def 1;
end;
then
A49: h is convergent by SEQ_2:def 6;
then
A50: lim h = 0 by A38,SEQ_2:def 7;
then not g in Big_Oh(f) by A49,ASYMPT_0:16;
then
A51: not f in Big_Omega(g) by ASYMPT_0:19;
f in Big_Oh(g) by A49,A50,ASYMPT_0:16;
hence thesis by A51,Th4;
end;
begin :: Problem 3.23
Lm43: for k,n st k <= n holds n choose k >= ((n+1) choose k) / (n+1)
proof
let k,n;
set n1 = n+1;
assume
A1: k <= n;
then reconsider l = n-k as Element of NAT by INT_1:5;
set l1 = l+1;
A2: l1 = n1-k;
0+1 <= l+1 by XREAL_1:6;
then 1/1 >= 1/l1 by XREAL_1:85;
then
A3: (n choose k) * (1/1) >= (n choose k) * (1/l1) by XREAL_1:64;
n + 0 <= n + 1 by XREAL_1:6;
then k <= n+1 by A1,XXREAL_0:2;
then (n1 choose k) / n1 = ((n1!)/((k!) * (l1!))) / n1 by A2,NEWTON:def 3
.= (n1*(n!)/((k!) * (l1!))) / (n1*1) by NEWTON:15
.= (n1*(n!)*((k!) * (l1!))") / (n1*1)
.= (n1*((n!)*((k!) * (l1!))")) / (n1*1)
.= (n1*((n!)/((k!) * (l1!)))) / (n1*1)
.= (n1/n1) * (((n!)/((k!) * (l1!)))/1)
.= 1 * (((n!)/((k!) * (l1!)))/1) by XCMPLX_1:60
.= (n!)/((k!) * ((l!) * l1)) by NEWTON:15
.= (n!)*1/((k!) * (l!) * l1)
.= (n!)/((k!) * (l!)) * (1/l1) by XCMPLX_1:76
.= (n choose k) * (1/l1) by A1,NEWTON:def 3;
hence thesis by A3;
end;
theorem
for n st n >= 1 holds for f being Real_Sequence, k being Element of
NAT st (for n holds f.n = Sum(seq_n^(k), n)) holds f.n >= (n to_power (k+1)) /
(k+1)
proof
defpred P[Nat] means for f being Real_Sequence, k being Element of NAT st (
for n holds f.n = Sum(seq_n^(k), n)) holds f.$1 >= ($1 to_power (k+1)) / (k+1);
A1: for n being Nat st n >= 1 & P[n] holds P[n+1]
proof
let n be Nat such that
n >= 1 and
A2: for f being Real_Sequence, k being Element of NAT st (for n holds
f.n = Sum(seq_n^(k), n)) holds f.n >= (n to_power (k+1)) / (k+1);
reconsider n as Element of NAT by ORDINAL1:def 12;
let f be Real_Sequence, k be Element of NAT such that
A3: for n holds f.n = Sum(seq_n^(k), n);
set R3 = ((n,1) In_Power (k+1));
len R3 = (k+1)+1 by NEWTON:def 4
.= k+2;
then reconsider R3 as Element of (k+2)-tuples_on REAL by FINSEQ_2:92;
set R2 = ((k+1)")*((n,1) In_Power (k+1));
len R2 = len ((n,1) In_Power (k+1)) by NEWTON:2
.= (k+1)+1 by NEWTON:def 4
.= k+2;
then reconsider R2 as Element of (k+2)-tuples_on REAL by FINSEQ_2:92;
reconsider nk = (n to_power (k+1)) / (k+1) as Element of REAL
by XREAL_0:def 1;
set R1 = <*nk*>^((n,1) In_Power k);
A4: len <*(n to_power (k+1)) / (k+1)*> = 1 by FINSEQ_1:40;
set g = seq_n^(k);
f.n >= (n to_power (k+1)) / (k+1) by A2,A3;
then Sum(g,n) >= (n to_power (k+1)) / (k+1) by A3;
then
A5: (Partial_Sums(g)).n >= (n to_power (k+1)) / (k+1) by SERIES_1:def 5;
reconsider nk =(n to_power (k+1)) / (k+1) as Element of REAL
by XREAL_0:def 1;
g.(n+1) = (n+1) to_power k by Def3
.= Sum((n,1) In_Power k) by NEWTON:30;
then
A6: (n to_power (k+1)) / (k+1) + g.(n+1)
= Sum (<*nk*>^((n,1) In_Power k)) by RVSUM_1:76;
len ((n,1) In_Power k) = k+1 by NEWTON:def 4;
then
A7: len R1 = (k+1)+1 by A4,FINSEQ_1:22
.= k+2;
then reconsider R1 as Element of (k+2)-tuples_on REAL
by FINSEQ_2:92;
A8: for i being Nat st i in Seg (k+2) holds R2.i<=R1.i
proof
set k1 = (k+1)";
let i be Nat such that
A9: i in Seg (k+2);
A10: 1 <= i by A9,FINSEQ_1:1;
set r2 = R2.i, r1 = R1.i;
A11: i <= (k+2) by A9,FINSEQ_1:1;
per cases by A10,XXREAL_0:1;
suppose
A12: i = 1;
n|^(k+1) = R3.1 by NEWTON:28;
then r2 = k1*(n|^(k+1)) by A12,RVSUM_1:45
.= (n to_power (k+1)) / (k+1);
hence thesis by A12,FINSEQ_1:41;
end;
suppose
A13: i > 1;
set i0 = i-1;
set m = i0-1;
A14: i-1 > 1-1 by A13,XREAL_1:9;
then reconsider i0 as Element of NAT by INT_1:3;
set l = k-m;
A15: i0 >= 0+1 by A14,INT_1:7;
then m >= 0;
then reconsider m as Element of NAT by INT_1:3;
set i3 = (k+1)-i0;
len (((n,1) In_Power k)) = k+1 by NEWTON:def 4;
then
A16: dom(((n,1) In_Power k)) = Seg(k+1) by FINSEQ_1:def 3;
i-1 <= (k+2)-1 by A11,XREAL_1:9;
then
A17: i0 in dom(((n,1) In_Power k)) by A15,A16,FINSEQ_1:1;
m = i-2;
then
A18: k >= m+0 by A11,XREAL_1:20;
then l >= 0 by XREAL_1:19;
then reconsider l as Element of NAT by INT_1:3;
A19: i3 = l;
then
A20: i0 + 0 <= (k+1) by XREAL_1:19;
reconsider i3 as Element of NAT by A19;
len(((n,1) In_Power (k+1))) = (k+1)+1 by NEWTON:def 4;
then dom(((n,1) In_Power (k+1))) = Seg(k+2) by FINSEQ_1:def 3;
then R3.i = ((k+1) choose i0)*(n|^i3)*(1|^i0) by A9,NEWTON:def 4;
then
A21: r2 = k1*(((k+1) choose i0)*(n|^i3)*(1|^i0)) by RVSUM_1:45
.= k1*(((k+1) choose l)*(n|^l)*(1|^i0)) by A20,NEWTON:20
.= k1*(((k+1) choose l)*(n|^l)*1) by NEWTON:10
.= k1*((k+1) choose l)*(n to_power l);
k-m <= k-0 by XREAL_1:13;
then
A22: ((k+1) choose l) / (k+1) <= (k choose l) by Lm43;
r1 = ((n,1) In_Power k).i0 by A4,A7,A11,A13,FINSEQ_1:24
.= (k choose m)*(n|^l)*(1|^m) by A17,NEWTON:def 4
.= (k choose l)*(n|^l)*(1|^m) by A18,NEWTON:20
.= (k choose l)*(n|^l)*1 by NEWTON:10
.= (k choose l)*(n to_power l);
hence thesis by A21,A22,XREAL_1:64;
end;
end;
((n+1) to_power (k+1)) / (k+1) = ((n+1)|^(k+1))*(k+1)"
.= Sum((n,1) In_Power (k+1))*(k+1)" by NEWTON:30
.= Sum(((k+1)")*((n,1) In_Power (k+1))) by RVSUM_1:87;
then
A23: ((n+1) to_power (k+1)) / (k+1) <= Sum R1 by A8,RVSUM_1:82;
f.(n+1) = Sum(g,n+1) by A3
.= (Partial_Sums(g)).(n+1) by SERIES_1:def 5
.= (Partial_Sums(g)).n + g.(n+1) by SERIES_1:def 1;
then f.(n+1) >= (n to_power (k+1)) / (k+1) + g.(n+1) by A5,XREAL_1:6;
hence thesis by A6,A23,XXREAL_0:2;
end;
A24: P[1]
proof
let f be Real_Sequence, k be Element of NAT such that
A25: for n holds f.n = Sum(seq_n^(k), n);
set g = seq_n^(k);
A26: (1 to_power (k+1)) / (k+1) = 1 / (k+1) by POWER:26;
A27: 0+1 <= k+1 by XREAL_1:6;
f.1 = Sum(g,1) by A25
.= (Partial_Sums(g)).(0+1) by SERIES_1:def 5
.= (Partial_Sums(g)).0 + g.1 by SERIES_1:def 1
.= g.1 + g.0 by SERIES_1:def 1
.= (1 to_power k) + g.0 by Def3
.= 1 + g.0 by POWER:26
.= 1 + 0 by Def3
.= 1/1;
hence thesis by A26,A27,XREAL_1:85;
end;
for n be Nat st n >= 1 holds P[n] from NAT_1:sch 8(A24, A1);
hence thesis;
end;
begin :: Problem 3.24
Lm44: for f being Real_Sequence st (for n being Nat holds f.n = log(2,n!))
for n holds f.n = Sum(seq_logn, n)
proof
set g = seq_logn;
let f be Real_Sequence such that
A1: for n being Nat holds f.n = log(2,n!);
defpred P[Nat] means f.$1 = Sum(g, $1);
A2: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat such that
A3: f.k = Sum(g, k);
A4: k! > 0 by NEWTON:17;
f.(k+1) = log(2,(k+1)!) by A1
.= log(2,(k+1)*(k!)) by NEWTON:15
.= log(2,k+1) + log(2,k!) by A4,POWER:53
.= log(2,k+1) + Sum(g, k) by A1,A3
.= g.(k+1) + Sum(g, k) by Def2
.= g.(k+1) + Partial_Sums(g).k by SERIES_1:def 5
.= Partial_Sums(g).(k+1) by SERIES_1:def 1
.= Sum(g, k+1) by SERIES_1:def 5;
hence thesis;
end;
A5: Sum(g, 0) = Partial_Sums(g).0 by SERIES_1:def 5
.= g.0 by SERIES_1:def 1
.= 0 by Def2;
f.0 = log(2,1) by A1,NEWTON:12
.= 0 by POWER:51;
then
A6: P[0] by A5;
for n being Nat holds P[n] from NAT_1:sch 2(A6, A2);
hence thesis;
end;
Lm45: for n st n >= 4 holds n*log(2,n) >= 2*n
proof
let n;
assume n >= 4;
then log(2,n) >= log(2,2^2) by PRE_FF:10;
then log(2,n) >= log(2,2 to_power 2) by POWER:46;
then log(2,n) >= 2*log(2,2) by POWER:55;
then log(2,n) >= 2*1 by POWER:52;
hence thesis by XREAL_1:64;
end;
theorem
for f,g being Real_Sequence st (for n st n > 0 holds g.n = n*log(2,n))
& (for n being Nat holds f.n = log(2,n!))
ex s being eventually-nonnegative Real_Sequence st s = g & f in Big_Theta(s)
proof
set h = seq_logn;
let f,g be Real_Sequence such that
A1: for n st n > 0 holds g.n = n*log(2,n) and
A2: for n being Nat holds f.n = log(2,n!);
g is eventually-positive
proof
take 2;
let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
assume
A4: n >= 2;
then log(2,n) >= log(2,2) by PRE_FF:10;
then log(2,n) >= 1 by POWER:52;
then n*log(2,n) > n*0 by A4,XREAL_1:68;
hence thesis by A1,A4,A3;
end;
then reconsider g as eventually-positive Real_Sequence;
A5: now
let n;
set n1 = [/n/2\];
assume
A6: n >= 4;
then
A7: (n/2)*log(2,n/2) = (n/2)*(log(2,n) - log(2,2)) by POWER:54
.= (n/2)*(log(2,n) - 1) by POWER:52
.= (n*log(2,n))/2 - (n/2);
ex s being Real_Sequence st s.0 = 0 & for m st m > 0 holds s.m = (log
(2,n/2))
proof
defpred P[Element of NAT,Real] means ($1 = 0 implies $2 = 0) & ($1 > 0
implies $2 = log(2,n/2));
A8: for x being Element of NAT ex y being Element of REAL st P[x,y]
proof
let x be Element of NAT;
per cases;
suppose
x = zz;
hence thesis;
end;
suppose
A9: x > 0;
log(2,n/2) in REAL by XREAL_0:def 1;
hence thesis by A9;
end;
end;
consider h being sequence of REAL such that
A10: for x being Element of NAT holds P[x,h.x] from FUNCT_2:sch 3(
A8);
take h;
thus h.0 = 0 by A10;
let n;
thus thesis by A10;
end;
then consider p being Real_Sequence such that
A11: p.0 = 0 and
A12: for m st m > 0 holds p.m = log(2,n/2);
A13: [/n/2\] >= n/2 by INT_1:def 7;
then reconsider n1 as Element of NAT by INT_1:3;
set n2 = n1-1;
A14: n*2" > 0*2" by A6,XREAL_1:68;
A15: now
assume n2 < 0;
then n1-1 <= -1 by INT_1:8;
then (n1-1)+1 <= -1+1 by XREAL_1:6;
hence contradiction by A14,INT_1:def 7;
end;
(n*log(2,n))*4" >= (2*n)*4" by A6,Lm45,XREAL_1:64;
then (n*log(2,n)) / 2 - (n*log(2,n)) / 4 >= n/2;
then (n*log(2,n)) / 2 >= n/2 + (n*log(2,n)) / 4 by XREAL_1:19;
then
A16: (n*log(2,n))/2 - (n/2) >= n*log(2,n) / 4 by XREAL_1:19;
2*2 <= n by A6;
then 2 <= n/2 by XREAL_1:77;
then log(2,2) <= log(2,n/2) by PRE_FF:10;
then
A17: 1 <= log(2,n/2) by POWER:52;
reconsider n2 as Element of NAT by A15,INT_1:3;
A18: for k st n2+1 <= k & k <= n holds p.k <= h.k
proof
let k such that
A19: n2+1 <= k and
k <= n;
n/2 <= k by A13,A19,XXREAL_0:2;
then log(2,n/2) <= log(2,k) by A14,PRE_FF:10;
then p.k <= log(2,k) by A12,A19;
hence thesis by A19,Def2;
end;
n >= n1 by Lm17;
then
A20: Sum(h, n, n2) >= Sum(p, n, n2) by A18,Lm16;
A21: now
[/n/2\] < n/2 + 1 by INT_1:def 7;
then n2 < n/2 by XREAL_1:19;
then
A22: n/2 + n2 < n/2 + n/2 by XREAL_1:6;
assume n-n2 < n/2;
hence contradiction by A22,XREAL_1:19;
end;
for k st k <= n2 holds h.k >= 0
proof
let k such that
k <= n2;
per cases;
suppose
k = 0;
hence thesis by Def2;
end;
suppose
A23: k > 0;
then k >= 0+1 by NAT_1:13;
then log(2,k) >= log(2,1) by PRE_FF:10;
then log(2,k) >= 0 by POWER:51;
hence thesis by A23,Def2;
end;
end;
then Sum(h,n2) >= 0 by Lm12;
then Sum(h, n) + Sum(h,n2) >= Sum(h, n) + 0 by XREAL_1:6;
then Sum(h, n) >= Sum(h, n) - Sum(h,n2) by XREAL_1:20;
then
A24: Sum(h, n) >= Sum(h, n, n2) by SERIES_1:def 6;
Sum(p, n, n2) = (n - n2)*log(2,n/2) by A11,A12,Lm18;
then
A25: Sum(p, n, n2) >= (n/2)*log(2,n/2) by A21,A17,XREAL_1:64;
n*log(2,n) / 4 = g.n / 4 by A1,A6
.= (1/4)*g.n;
then Sum(p, n, n2) >= (1/4)*g.n by A25,A7,A16,XXREAL_0:2;
then Sum(h, n, n2) >= (1/4)*g.n by A20,XXREAL_0:2;
then Sum(h, n) >= (1/4)*g.n by A24,XXREAL_0:2;
hence (1/4)*g.n <= f.n by A2,Lm44;
ex s being Real_Sequence st s.0 = 0 & for m st m > 0 holds s.m = (log
(2,n))
proof
defpred P[Element of NAT,Real] means ($1 = 0 implies $2 = 0) & ($1 > 0
implies $2 = log(2,n));
A26: for x being Element of NAT ex y being Element of REAL st P[x,y]
proof
let x be Element of NAT;
per cases;
suppose
x = zz;
hence thesis;
end;
suppose
A27: x > 0;
log(2,n) in REAL by XREAL_0:def 1;
hence thesis by A27;
end;
end;
consider h being sequence of REAL such that
A28: for x being Element of NAT holds P[x,h.x] from FUNCT_2:sch 3(
A26);
take h;
thus h.0 = 0 by A28;
let n;
thus thesis by A28;
end;
then consider q being Real_Sequence such that
A29: q.0 = 0 and
A30: for m st m > 0 holds q.m = log(2,n);
A31: Sum(q, n) = n*log(2,n) by A29,A30,Lm14;
for k st k <= n holds h.k <= q.k
proof
let k such that
A32: k <= n;
per cases;
suppose
k = 0;
hence thesis by A29,Def2;
end;
suppose
A33: k > 0;
then log(2,k) <= log(2,n) by A32,PRE_FF:10;
then h.k <= log(2,n) by A33,Def2;
hence thesis by A30,A33;
end;
end;
then
A34: Sum(h, n) <= Sum(q, n) by Lm13;
log(2,n!) = f.n by A2
.= Sum(h, n) by A2,Lm44;
then log(2,n!) <= 1*g.n by A1,A6,A34,A31;
hence f.n <= 1*g.n by A2;
end;
take g;
A35: f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
Big_Theta(g) = { s where s is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*g.n <= s.n & s.n <= c*g.n } by
ASYMPT_0:27;
hence thesis by A35,A5;
end;
:: Problem 3.25 -- Proven in theorem ASYMPT_0:1
begin :: Problem 3.26
:: For a "natural" example of an algorithm that would take time in
:: O(t(n)), consider a naive algorithm that tries to find at least one
:: factor of a given Nat. For all even numbers, the task is trivial,
:: and for all odd numbers, the algorithm might try dividing the number by all
:: numbers smaller than it, giving a running time of n.
theorem
for f being eventually-nondecreasing eventually-nonnegative
Real_Sequence , t being Real_Sequence st (for n holds (n mod 2 = 0 implies t.n
= 1) & (n mod 2 = 1 implies t.n = n)) holds not t in Big_Theta(f)
proof
let f be eventually-nondecreasing eventually-nonnegative Real_Sequence, t be
Real_Sequence such that
A1: for n holds (n mod 2 = 0 implies t.n = 1) & (n mod 2 = 1 implies t.n = n);
A2: Big_Theta(f) = { s where s is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*f.n <= s.n & s.n <= c*f.n } by
ASYMPT_0:27;
hereby
consider N0 being Nat such that
A3: for n being Nat st n >= N0 holds f.n <= f.(n+1) by ASYMPT_0:def 6;
assume t in Big_Theta(f);
then consider s being Element of Funcs(NAT, REAL) such that
A4: s = t and
A5: ex c,d,N st c > 0 & d > 0 & for n st n >= N holds d*f.n <= s.n & s
.n <= c*f.n by A2;
consider c,d,N such that
A6: c > 0 and
A7: d > 0 and
A8: for n st n >= N holds d*f.n <= s.n & s.n <= c*f.n by A5;
set N1 = max([/c/d\] + 1, max(N, N0));
A9: N1 >= [/c/d\] + 1 by XXREAL_0:25;
A10: N1 is Integer by XXREAL_0:16;
A11: N1 >= max(N, N0) by XXREAL_0:25;
max(N, N0) >= N0 by XXREAL_0:25;
then
A12: N1 >= N0 by A11,XXREAL_0:2;
max(N, N0) >= N by XXREAL_0:25;
then
A13: N1 >= N by A11,XXREAL_0:2;
reconsider N1 as Element of NAT by A11,A10,INT_1:3;
thus contradiction
proof
per cases by NAT_D:12;
suppose
A14: N1 mod 2 = 1;
A15: [/c/d\] >= c/d by INT_1:def 7;
[/c/d\] + 1 > [/c/d\] + 0 by XREAL_1:8;
then [/c/d\] + 1 > c/d by A15,XXREAL_0:2;
then N1 > c/d by A9,XXREAL_0:2;
then N1*c" > c"*(c/d) by A6,XREAL_1:68;
then N1/c > (c"*c)*(1/d);
then
A16: N1/c > 1*(1/d) by A6,XCMPLX_0:def 7;
A17: f.(N1+1) >= f.(N1) by A3,A12;
s.N1 = N1 by A1,A4,A14;
then N1 <= c*f.N1 by A8,A13;
then N1/c <= f.N1 by A6,XREAL_1:79;
then f.N1 > 1/d by A16,XXREAL_0:2;
then f.(N1+1) > 1/d by A17,XXREAL_0:2;
then
A18: d*(1/d) < d*f.(N1+1) by A7,XREAL_1:68;
N1+1 > N1+0 by XREAL_1:8;
then
A19: N1+1 > N by A13,XXREAL_0:2;
(N1+1) mod 2 = (1+(1 mod 2)) mod 2 by A14,EULER_2:6
.= (1+1) mod 2 by NAT_D:14
.= 0 by NAT_D:25;
then t.(N1+1) = 1 by A1;
then d*f.(N1+1) <= 1 by A4,A8,A19;
hence thesis by A7,A18,XCMPLX_1:106;
end;
suppose
A20: N1 mod 2 = 0;
then (N1+1) mod 2 = (0+(1 mod 2)) mod 2 by EULER_2:6
.= (0+1) mod 2 by NAT_D:14
.= 1 by NAT_D:14;
then
A21: s.(N1+1) = N1+1 by A1,A4;
A22: [/c/d\] >= c/d by INT_1:def 7;
A23: N1+1 > N1+0 by XREAL_1:8;
then N1+1 > N0 by A12,XXREAL_0:2;
then
A24: f.((N1+1)+1) >= f.(N1+1) by A3;
[/c/d\] + 1 > [/c/d\] + 0 by XREAL_1:8;
then [/c/d\] + 1 > c/d by A22,XXREAL_0:2;
then N1 > c/d by A9,XXREAL_0:2;
then (N1+1) > c/d by A23,XXREAL_0:2;
then (N1+1)*c" > c"*(c/d) by A6,XREAL_1:68;
then (N1+1)/c > (c"*c)*(1/d);
then
A25: (N1+1)/c > 1*(1/d) by A6,XCMPLX_0:def 7;
N1+1 > N by A13,A23,XXREAL_0:2;
then N1+1 <= c*f.(N1+1) by A8,A21;
then (N1+1)/c <= f.(N1+1) by A6,XREAL_1:79;
then f.(N1+1) > 1/d by A25,XXREAL_0:2;
then f.(N1+2) > 1/d by A24,XXREAL_0:2;
then
A26: d*(1/d) < d*f.(N1+2) by A7,XREAL_1:68;
N1+2 > N1+0 by XREAL_1:8;
then
A27: N1+2 > N by A13,XXREAL_0:2;
(N1+2) mod 2 = (0+(2 mod 2)) mod 2 by A20,EULER_2:6
.= (0+0) mod 2 by NAT_D:25
.= 0 by NAT_D:26;
then t.(N1+2) = 1 by A1;
then d*f.(N1+2) <= 1 by A4,A8,A27;
hence thesis by A7,A26,XCMPLX_1:106;
end;
end;
end;
end;
:: Problem 3.27 -- Proven as part of theorem ASYMPT_0:37
begin :: Problem 3.28
Lm46: for n st n >= 2 holds [/n/2\] < n
proof
let n such that
A1: n >= 2;
A2: now
assume n/2 + 1 > n;
then 2*(n/2 + 1) > 2*n by XREAL_1:68;
then 2*(n/2) + 2*1 > 2*n;
then 2 > 2*n - n by XREAL_1:19;
hence contradiction by A1;
end;
[/n/2\] < n/2 + 1 by INT_1:def 7;
hence thesis by A2,XXREAL_0:2;
end;
:: Problem 3.29 -- Proven in theorems ASYMPT_0:39
begin :: Problem 3.30
definition
func POWEROF2SET -> non empty Subset of NAT equals
the set of all 2 to_power n where n is Element of NAT ;
coherence
proof
set IT = the set of all 2 to_power n where n is Element of NAT ;
A1: now
let x be object;
assume x in IT;
then ex n being Element of NAT st 2 to_power n = x;
hence x in NAT;
end;
2 to_power 1 in IT;
hence thesis by A1,TARSKI:def 3;
end;
end;
Lm47: for n st n >= 2 holds n^2 > n+1
proof
defpred P[Nat] means $1^2 > $1+1;
A1: for k be Nat st k >= 2 & P[k] holds P[k+1]
proof
let k be Nat such that
A2: k >= 2 and
A3: k^2 > k+1;
2*k > 2*0 by A2,XREAL_1:68;
then 2*k + 1 > 0 + 1 by XREAL_1:6;
then
A4: (k+1) + (2*k + 1) > (k+1) + 1 by XREAL_1:6;
k^2 + (2*k + 1) > (k+1) + (2*k + 1) by A3,XREAL_1:6;
hence (k+1)^2 > (k+1) + 1 by A4,XXREAL_0:2;
end;
A5: P[2];
for n be Nat st n >= 2 holds P[n] from NAT_1:sch 8(A5, A1);
hence thesis;
end;
Lm48: for n st n >= 1 holds (2 to_power (n+1)) - (2 to_power n) > 1
proof
let n;
assume n >= 1;
then 2 to_power n >= 2 to_power 1 by PRE_FF:8;
then (2 to_power n)*1 >= 2 by POWER:25;
then (2 to_power n)*2 - (2 to_power n)*1 > 1 by XXREAL_0:2;
then (2 to_power n)*(2 to_power 1) - (2 to_power n) > 1 by POWER:25;
hence thesis by POWER:27;
end;
Lm49: for n st n >= 2 holds not ((2 to_power n) - 1) in POWEROF2SET
proof
A1: 1 = 2-1;
let n;
assume n >= 2;
then
A2: n-1 >= 1 by A1,XREAL_1:9;
then n-1 is Element of NAT by INT_1:3;
then (2 to_power ((n+-1)+1)) - (2 to_power (n-1)) > 1 by A2,Lm48;
then (2 to_power n) > 1 + (2 to_power (n-1)) by XREAL_1:20;
then
A3: (2 to_power n) - 1 > (2 to_power (n-1)) by XREAL_1:20;
assume (2 to_power n) - 1 in POWEROF2SET;
then consider m such that
A4: 2 to_power m = (2 to_power n) - 1;
now
assume m >= n;
then
A5: 2 to_power m >= 2 to_power n by PRE_FF:8;
(2 to_power n) + 1 > (2 to_power n) + 0 by XREAL_1:6;
hence contradiction by A4,A5,XREAL_1:19;
end;
then m+1 <= n by INT_1:7;
then
A6: m <= n-1 by XREAL_1:19;
m >= n-1 by A4,A3,POWER:39;
hence contradiction by A4,A3,A6,XXREAL_0:1;
end;
theorem :: Part a)
for f being Real_Sequence st (for n holds (n in POWEROF2SET implies f.
n = n) & (not n in POWEROF2SET implies f.n = 2 to_power n)) holds f in
Big_Theta(seq_n^(1), POWEROF2SET) & not f in Big_Theta(seq_n^(1)) & seq_n^(1)
is smooth & not f is eventually-nondecreasing
proof
set X = POWEROF2SET;
set p = seq_logn;
set g = seq_n^(1);
set h = (g taken_every 2);
set q = p/"g;
A1: now
let n;
assume
A2: n >= 1;
then
A3: 2*n > 2*0 by XREAL_1:68;
A4: h.n = g.(2*n) by ASYMPT_0:def 15
.= (2*n) to_power 1 by A3,Def3
.= 2*n by POWER:25;
g.n = n to_power 1 by A2,Def3
.= n by POWER:25;
hence h.n <= 2*g.n by A4;
thus h.n >= 0 by A4;
end;
let f be Real_Sequence such that
A5: for n holds (n in POWEROF2SET implies f.n = n) & (not n in
POWEROF2SET implies f.n = 2 to_power n);
A6: now
let n such that
A7: n >= 1 and
A8: n in X;
A9: g.n = n to_power 1 by A7,Def3
.= n by POWER:25;
hence 1*g.n <= f.n by A5,A8;
thus f.n <= 1*g.n by A5,A8,A9;
end;
f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
hence f in Big_Theta(g,X) by A6;
A10: Big_Theta(g) = { t where t is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*g.n <= t.n & t.n <= c*g.n } by
ASYMPT_0:27;
hereby
A11: lim q = 0 by Lm11;
q is convergent by Lm11;
then consider N0 being Nat such that
A12: for m being Nat st m >= N0 holds |.q.m-0.| < 1/2 by A11,SEQ_2:def 7;
assume f in Big_Theta(g);
then consider t being Element of Funcs(NAT, REAL) such that
A13: t = f and
A14: ex c,d,N st c > 0 & d > 0 & for n st n >= N holds d*g.n <= t.n &
t.n <= c*g.n by A10;
consider c,d,N such that
A15: c > 0 and
d > 0 and
A16: for n st n >= N holds d*g.n <= t.n & t.n <= c*g.n by A14;
set N2 = max( max(N0, N), max([/c\], 2) );
A17: N2 >= max(N0, N) by XXREAL_0:25;
A18: N2 is Integer
proof
per cases by XXREAL_0:16;
suppose
N2 = max(N0, N);
hence thesis;
end;
suppose
N2 = max([/c\],2);
hence thesis by XXREAL_0:16;
end;
end;
max(N0, N) >= N0 by XXREAL_0:25;
then
A19: N2 >= N0 by A17,XXREAL_0:2;
A20: N2 >= max([/c\],2) by XXREAL_0:25;
max([/c\],2) >= [/c\] by XXREAL_0:25;
then
A21: N2 >= [/c\] by A20,XXREAL_0:2;
A22: max([/c\],2) >= 2 by XXREAL_0:25;
then
A23: N2 >= 2 by A20,XXREAL_0:2;
max(N0, N) >= N by XXREAL_0:25;
then
A24: N2 >= N by A17,XXREAL_0:2;
A25: [/c\] >= c by INT_1:def 7;
reconsider N2 as Element of NAT by A17,A18,INT_1:3;
set N3 = (2 to_power N2) - 1;
2 to_power N2 > 0 by POWER:34;
then 2 to_power N2 >= 0 + 1 by NAT_1:13;
then N3 >= 0;
then reconsider N3 as Element of NAT by INT_1:3;
A26: 2 to_power N3 > 0 by POWER:34;
not N3 in POWEROF2SET by A20,A22,Lm49,XXREAL_0:2;
then
A27: t.N3 = 2 to_power N3 by A5,A13;
2 to_power N2 > N2 + 1 by A23,Lm1;
then
A28: N3 > N2 by XREAL_1:20;
then
A29: g.N3 = N3 to_power 1 by Def3
.= N3 by POWER:25;
N3 >= N by A24,A28,XXREAL_0:2;
then 2 to_power N3 <= c*N3 by A16,A27,A29;
then log(2,2 to_power N3) <= log(2,c*N3) by A26,PRE_FF:10;
then N3*log(2,2) <= log(2,c*N3) by POWER:55;
then N3*1 <= log(2,c*N3) by POWER:52;
then
A30: N3 <= log(2,c) + log(2,N3) by A15,A28,POWER:53;
N3 >= [/c\] by A21,A28,XXREAL_0:2;
then N3 >= c by A25,XXREAL_0:2;
then log(2,N3) >= log(2,c) by A15,PRE_FF:10;
then log(2,N3) + log(2,N3) >= log(2,c) + log(2,N3) by XREAL_1:6;
then N3 <= 2*log(2,N3) by A30,XXREAL_0:2;
then N3 / 2 <= log(2,N3) by XREAL_1:79;
then N3" * (N3 * (1/2)) <= log(2,N3) * N3" by XREAL_1:64;
then (N3" * N3) * (1/2) <= log(2,N3) * N3";
then
A31: log(2,N3) / N3 >= 1/2 by A28,XCMPLX_0:def 7;
N3 >= N0 by A19,A28,XXREAL_0:2;
then
A32: |.q.N3-0.| < 1/2 by A12;
q.N3 = p.N3 / g.N3 by Lm4
.= log(2,N3) / g.N3 by A28,Def2
.= log(2,N3) / (N3 to_power 1) by A28,Def3
.= log(2,N3) / N3 by POWER:25;
hence contradiction by A31,A32,ABSVALUE:def 1;
end;
now
let n be Nat;
A33: n in NAT by ORDINAL1:def 12;
assume n >= 0;
A34: n+0 <= n+1 by XREAL_1:6;
A35: g.n = n
proof
per cases;
suppose
n = 0;
hence thesis by Def3;
end;
suppose
n > 0;
hence g.n = n to_power 1 by Def3,A33
.= n by POWER:25;
end;
end;
g.(n+1) = (n+1) to_power 1 by Def3
.= n+1 by POWER:25;
hence g.n <= g.(n+1) by A35,A34;
end;
then
A36: g is eventually-nondecreasing;
h is Element of Funcs(NAT, REAL) by FUNCT_2:8;
then h in Big_Oh(g) by A1;
then g is_smooth_wrt 2 by A36;
hence g is smooth by ASYMPT_0:37;
A37: 3 = 4-1;
hereby
assume f is eventually-nondecreasing;
then consider N being Nat such that
A38: for n being Nat st n >= N holds f.n <= f.(n+1);
set N1 = (2 to_power (N+2)) - 1;
A39: 2 to_power 2 = 2^2 by POWER:46
.= 4;
A40: N+2 >= 0+2 by XREAL_1:6;
then 2 to_power (N+2) >= 2 to_power 2 by PRE_FF:8;
then N1 >= 3 by A37,A39,XREAL_1:9;
then reconsider N1 as Element of NAT by INT_1:3;
2 to_power (N+2) > (N+2) + 1 by A40,Lm1;
then
A41: N1 > (N+2) by XREAL_1:20;
N+2 >= N+0 by XREAL_1:6;
then
A42: N1 >= N by A41,XXREAL_0:2;
N1+1 in POWEROF2SET;
then
A43: f.(N1+1) = N1+1 by A5;
not N1 in POWEROF2SET by A40,Lm49;
then f.N1 = 2 to_power N1 by A5;
then f.N1 > f.(N1+1) by A43,A41,POWER:39;
hence contradiction by A38,A42;
end;
end;
theorem :: Part b)
for f,g being Real_Sequence st (for n st n > 0 holds f.n = (n to_power
(2 to_power [\log(2,n)/] ))) & (for n st n > 0 holds g.n = (n to_power n)) ex s
being eventually-positive Real_Sequence st s = g & f in Big_Theta(s,
POWEROF2SET) & not f in Big_Theta(s) & f is eventually-nondecreasing & s is
eventually-nondecreasing & not s is_smooth_wrt 2
proof
set X = POWEROF2SET;
let f,g be Real_Sequence such that
A1: for n st n > 0 holds f.n = (n to_power (2 to_power [\log(2,n)/] )) and
A2: for n st n > 0 holds g.n = (n to_power n);
A3: g is eventually-positive
proof
take 1;
let n be Nat;
A4: n in NAT by ORDINAL1:def 12;
assume
A5: n >= 1;
then g.n = n to_power n by A2,A4;
hence thesis by A5,POWER:34;
end;
set h = (g taken_every 2);
reconsider g as eventually-positive Real_Sequence by A3;
A6: now
let n such that
A7: n >= 1 and
A8: n in X;
consider n1 being Element of NAT such that
A9: n = 2 to_power n1 by A8;
A10: f.n = n to_power (2 to_power [\log(2,n)/]) by A1,A7;
log(2,n) = n1*log(2,2) by A9,POWER:55
.= n1*1 by POWER:52;
then
A11: f.n = n to_power n by A10,A9,INT_1:25;
hence 1*g.n <= f.n by A2,A7;
thus f.n <= 1*g.n by A2,A7,A11;
end;
A12: Big_Theta(g) = { t where t is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*g.n <= t.n & t.n <= c*g.n } by
ASYMPT_0:27;
A13: now
assume f in Big_Theta(g);
then consider t being Element of Funcs(NAT, REAL) such that
A14: t = f and
A15: ex c,d,N st c > 0 & d > 0 & for n st n >= N holds d*g.n <= t.n &
t.n <= c*g.n by A12;
consider c,d,N such that
c > 0 and
A16: d > 0 and
A17: for n st n >= N holds d*g.n <= t.n & t.n <= c*g.n by A15;
set N1 = max([/1/d\], max(N, 2));
A18: N1 >= [/1/d\] by XXREAL_0:25;
A19: N1 is Integer by XXREAL_0:16;
A20: N1 >= max(N, 2) by XXREAL_0:25;
max(N, 2) >= N by XXREAL_0:25;
then
A21: N1 >= N by A20,XXREAL_0:2;
max(N, 2) >= 2 by XXREAL_0:25;
then
A22: N1 >= 2 by A20,XXREAL_0:2;
reconsider N1 as Element of NAT by A20,A19,INT_1:3;
reconsider N2 = 2 to_power N1 as Element of NAT;
A23: N2 > N1 + 1 by A22,Lm1;
N1 > 1 by A22,XXREAL_0:2;
then 2 to_power (N1+1) - 2 to_power N1 > 1 by Lm48;
then 2 to_power (N1+1) > N2 + 1 by XREAL_1:20;
then log(2, 2 to_power (N1+1)) > log(2, N2+1) by POWER:57;
then (N1+1)*log(2,2) > log(2, N2+1) by POWER:55;
then
A24: (N1+1)*1 > log(2, N2+1) by POWER:52;
A25: now
assume [\log(2,N2+1)/] > N1;
then
A26: [\log(2,N2+1)/] >= N1 + 1 by INT_1:7;
log(2,N2+1) >= [\log(2,N2+1)/] by INT_1:def 6;
hence contradiction by A24,A26,XXREAL_0:2;
end;
A27: g.(N2+1) = (N2+1) to_power (N2+1) by A2;
then
A28: g.(N2+1) > 0 by POWER:34;
N1+1 > N1+0 by XREAL_1:8;
then
A29: N2 > N1 by A23,XXREAL_0:2;
A30: N2+1 > N2+0 by XREAL_1:8;
then N2+1 > N1 by A29,XXREAL_0:2;
then N2+1 > N by A21,XXREAL_0:2;
then
A31: d*g.(N2+1) <= t.(N2+1) by A17;
[/1/d\] >= 1/d by INT_1:def 7;
then N1 >= 1/d by A18,XXREAL_0:2;
then N2 >= 1/d by A29,XXREAL_0:2;
then
A32: N2+1 > 1/d + 0 by XREAL_1:8;
log(2,N2) = N1*log(2,2) by POWER:55
.= N1*1 by POWER:52;
then log(2,N2+1) > N1 by A23,A30,POWER:57;
then [\log(2,N2+1)/] >= [\N1/] by PRE_FF:9;
then
A33: [\log(2,N2+1)/] >= N1 by INT_1:25;
t.(N2+1) = (N2+1) to_power (2 to_power [\log(2,N2+1)/]) by A1,A14;
then g.(N2+1) / t.(N2+1) = (N2+1) to_power (N2+1) / (N2+1) to_power N2 by
A27,A33,A25,XXREAL_0:1
.= (N2+1) to_power ((N2+1) - N2) by POWER:29
.= N2+1 by POWER:25;
then 1 / (g.(N2+1) / t.(N2+1)) < 1/(1/d) by A16,A32,XREAL_1:88;
then t.(N2+1) / g.(N2+1) < d by XCMPLX_1:57;
then t.(N2+1) / g.(N2+1) * g.(N2+1) < d*g.(N2+1) by A28,XREAL_1:68;
hence contradiction by A31,A28,XCMPLX_1:87;
end;
A34: now
assume g is_smooth_wrt 2;
then h in Big_Oh(g);
then consider t being Element of Funcs(NAT, REAL) such that
A35: t = h and
A36: ex c,N st c > 0 & for n st n >= N holds t.n <= c*g.n & t.n >= 0;
consider c,N such that
c > 0 and
A37: for n st n >= N holds t.n <= c*g.n & t.n >= 0 by A36;
set N0 = max( [/c\], max(N, 2) );
A38: N0 >= [/c\] by XXREAL_0:25;
A39: N0 is Integer by XXREAL_0:16;
A40: N0 >= max(N, 2) by XXREAL_0:25;
max(N, 2) >= N by XXREAL_0:25;
then
A41: N0 >= N by A40,XXREAL_0:2;
A42: max(N, 2) >= 2 by XXREAL_0:25;
then
A43: 2*N0 > 1*N0 by A40,XREAL_1:68;
A44: N0 >= 2 by A40,A42,XXREAL_0:2;
then
A45: N0 > 1 by XXREAL_0:2;
reconsider N0 as Element of NAT by A40,A39,INT_1:3;
[/c\] >= c by INT_1:def 7;
then
A46: N0 >= c by A38,XXREAL_0:2;
N0 >= 1 by A44,XXREAL_0:2;
then N0 + N0 >= N0 + 1 by XREAL_1:6;
then
A47: N0 to_power (2*N0) >= N0 to_power (N0+1) by A45,PRE_FF:8;
N0 to_power (N0+1) = (N0 to_power N0)*(N0 to_power 1) by A40,A42,POWER:27
.= (N0 to_power N0)*N0 by POWER:25;
then
A48: N0 to_power (N0+1) >= c*(N0 to_power N0) by A46,XREAL_1:64;
h.N0 = g.(2*N0) by ASYMPT_0:def 15
.= (2*N0) to_power (2*N0) by A2,A43;
then h.N0 > N0 to_power (2*N0) by A40,A42,A43,POWER:37;
then h.N0 > N0 to_power (N0+1) by A47,XXREAL_0:2;
then h.N0 > c*(N0 to_power N0) by A48,XXREAL_0:2;
then h.N0 > c*g.N0 by A2,A40,A42;
hence contradiction by A35,A37,A41;
end;
A49: now
let n be Nat;
A50: n in NAT by ORDINAL1:def 12;
A51: f.(n+1) = (n+1) to_power (2 to_power [\log(2,(n+1))/]) by A1;
assume
A52: n >= 2;
then
A53: f.n = n to_power (2 to_power [\log(2,n)/]) by A1,A50;
A54: n + 1 > n + 0 by XREAL_1:8;
then log(2,n) <= log(2,(n+1)) by A52,POWER:57;
then [\log(2,n)/] <= [\log(2,(n+1))/] by PRE_FF:9;
then
A55: 2 to_power [\log(2,n)/] <= 2 to_power [\log(2,(n+1))/] by PRE_FF:8;
n+1 > 0+1 by A52,XREAL_1:8;
then
A56: (n+1) to_power (2 to_power [\log(2,n)/]) <= (n+1) to_power (2
to_power [\log(2,(n+1))/]) by A55,PRE_FF:8;
log(2,n) >= log(2,2) by A52,PRE_FF:10;
then log(2,n) >= 1 by POWER:52;
then [\log(2,n)/] >= [\1/] by PRE_FF:9;
then [\log(2,n)/] >= 1 by INT_1:25;
then 2 to_power [\log(2,n)/] > 2 to_power 0 by POWER:39;
then
n to_power (2 to_power [\log(2,n)/]) <= (n+1) to_power (2 to_power [\
log(2,n)/]) by A52,A54,POWER:37;
hence f.n <= f.(n+1) by A53,A51,A56,XXREAL_0:2;
end;
A57: now
let n be Nat;
A58: n in NAT by ORDINAL1:def 12;
assume
A59: n >= 1;
A60: n + 1 > n + 0 by XREAL_1:8;
then
A61: n to_power n < (n+1) to_power n by A59,POWER:37;
n+1 >= 1+1 by A59,XREAL_1:6;
then n+1 > 1 by XXREAL_0:2;
then
A62: (n+1) to_power n < (n+1) to_power (n+1) by A60,POWER:39;
A63: g.(n+1) = (n+1) to_power (n+1) by A2;
g.n = n to_power n by A2,A59,A58;
hence g.n <= g.(n+1) by A63,A62,A61,XXREAL_0:2;
end;
take g;
f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
hence thesis by A6,A13,A49,A57,A34;
end;
theorem
for g being Real_Sequence st (for n holds (n in POWEROF2SET implies g.
n = n) & (not n in POWEROF2SET implies g.n = n to_power 2))
ex s being
eventually-positive Real_Sequence st s = g & seq_n^(1) in Big_Theta(s,
POWEROF2SET) & not seq_n^(1) in Big_Theta(s) & s taken_every 2 in Big_Oh(s) &
seq_n^(1) is eventually-nondecreasing & s is not eventually-nondecreasing
proof
let g be Real_Sequence such that
A1: for n holds (n in POWEROF2SET implies g.n = n) & (not n in
POWEROF2SET implies g.n = n to_power 2);
A2: g is eventually-positive
proof
take 1;
let n be Nat;
A3: n in NAT by ORDINAL1:def 12;
assume
A4: n >= 1;
thus g.n > 0
proof
per cases;
suppose
n in POWEROF2SET;
hence thesis by A1,A4;
end;
suppose
not n in POWEROF2SET;
then g.n = n to_power 2 by A1,A3;
hence thesis by A4,POWER:34;
end;
end;
end;
set h = (g taken_every 2);
reconsider s=g as eventually-positive Real_Sequence by A2;
take s;
thus s = g;
set X = POWEROF2SET;
set f = seq_n^(1);
A5: h is Element of Funcs(NAT, REAL) by FUNCT_2:8;
A6: now
let n;
assume n >= 0;
A7: h.n = g.(2*n) by ASYMPT_0:def 15;
thus h.n <= 4*g.n
proof
per cases;
suppose
A8: n in POWEROF2SET;
then consider m such that
A9: n = 2 to_power m;
2*n = (2 to_power 1)*(2 to_power m) by A9,POWER:25
.= 2 to_power (m+1) by POWER:27;
then 2*n in POWEROF2SET;
then
A10: g.(2*n) = 2*n by A1;
g.n = n by A1,A8;
hence thesis by A7,A10,XREAL_1:64;
end;
suppose
A11: not n in POWEROF2SET;
now
assume 2*n in POWEROF2SET;
then consider m such that
A12: 2*n = 2 to_power m;
thus contradiction
proof
per cases;
suppose
A13: m = 0;
A14: now
assume 1/2 is Element of NAT;
then 0+1 <= 1/2 by NAT_1:13;
hence contradiction;
end;
(n*2)*2" = 1*2" by A12,A13,POWER:24;
hence thesis by A14;
end;
suppose
m > 0;
then m >= 0+1 by NAT_1:13;
then m-1 >= 0;
then
A15: m-1 is Element of NAT by INT_1:3;
2*n = 2 to_power ((m+-1)+1) by A12
.= (2 to_power (m-1))*(2 to_power 1) by POWER:27
.= (2 to_power (m-1))*2 by POWER:25;
hence thesis by A11,A15;
end;
end;
end;
then
A16: g.(2*n) = (2*n) to_power 2 by A1
.= (2*n)^2 by POWER:46
.= 4*n^2;
g.n = n to_power 2 by A1,A11
.= n^2 by POWER:46;
hence thesis by A16,ASYMPT_0:def 15;
end;
end;
thus h.n >= 0
proof
per cases;
suppose
(2*n) in POWEROF2SET;
hence thesis by A1,A7;
end;
suppose
not (2*n) in POWEROF2SET;
then g.(2*n) = (2*n) to_power 2 by A1;
hence thesis by ASYMPT_0:def 15;
end;
end;
end;
A17: Big_Theta s = { t where t is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*g.n <= t.n & t.n <= c*g.n } by
ASYMPT_0:27;
A18: now
assume f in Big_Theta s;
then consider t being Element of Funcs(NAT, REAL) such that
A19: t = f and
A20: ex c,d,N st c > 0 & d > 0 & for n st n >= N holds d*g.n <= t.n &
t.n <= c*g.n by A17;
consider c,d,N such that
c > 0 and
A21: d > 0 and
A22: for n st n >= N holds d*g.n <= t.n & t.n <= c*g.n by A20;
set N0 = max(max(N,2), [/1/d\]);
A23: N0 >= max(N,2) by XXREAL_0:25;
max(N,2) >= N by XXREAL_0:25;
then
A24: N0 >= N by A23,XXREAL_0:2;
A25: N0 >= [/1/d\] by XXREAL_0:25;
A26: max(N,2) >= 2 by XXREAL_0:25;
then
A27: N0 >= 2 by A23,XXREAL_0:2;
N0 is Integer by XXREAL_0:16;
then reconsider N0 as Element of NAT by A23,INT_1:3;
set N1 = (2 to_power N0) - 1;
2 to_power N0 > 0 by POWER:34;
then 2 to_power N0 >= 0 + 1 by NAT_1:13;
then N1 >= 0;
then reconsider N1 as Element of NAT by INT_1:3;
A28: [/1/d\] >= 1/d by INT_1:def 7;
not N1 in POWEROF2SET by A23,A26,Lm49,XXREAL_0:2;
then
A29: g.N1 = N1 to_power 2 by A1
.= N1^2 by POWER:46;
2 to_power N0 > N0 + 1 by A27,Lm1;
then
A30: N1 > N0 by XREAL_1:20;
then
A31: N1 >= N by A24,XXREAL_0:2;
N1 > [/1/d\] by A25,A30,XXREAL_0:2;
then N1 > 1/d by A28,XXREAL_0:2;
then N1*N1 > N1*(1/d) by A30,XREAL_1:68;
then d*N1^2 > (N1*(1/d))*d by A21,XREAL_1:68;
then d*N1^2 > N1*(1/d*d);
then
A32: d*N1^2 > N1*1 by A21,XCMPLX_1:87;
t.N1 = N1 to_power 1 by A19,A30,Def3
.= N1 by POWER:25;
hence contradiction by A22,A31,A32,A29;
end;
A33: 3 = 4-1;
A34: now
assume g is eventually-nondecreasing;
then consider N being Nat such that
A35: for n being Nat st n >= N holds g.n <= g.(n+1);
set N0 = max(N, 1);
set N1 = (2 to_power (2*N0)) - 1;
A36: N0 >= N by XXREAL_0:25;
2 to_power (2*N0) >= 2 to_power 0 by PRE_FF:8;
then 2 to_power (2*N0) >= 1 by POWER:24;
then (2 to_power (2*N0)) - 1 >= 1 - 1 by XREAL_1:9;
then reconsider N1 as Element of NAT by INT_1:3;
A37: 2*N0 >= 2*1 by XREAL_1:64,XXREAL_0:25;
then 2 to_power (2*N0) > 2*N0 + 1 by Lm1;
then
A38: N1 > 2*N0 by XREAL_1:20;
2 to_power (2*N0) >= 2 to_power 2 by A37,PRE_FF:8;
then 2 to_power (2*N0) >= 2^2 by POWER:46;
then N1 >= 3 by A33,XREAL_1:9;
then N1 >= 2 by XXREAL_0:2;
then
A39: N1^2 > N1+1 by Lm47;
A40: 2*N0 in NAT by ORDINAL1:def 12;
2*N0 >= 2*1 by XREAL_1:64,XXREAL_0:25;
then not N1 in POWEROF2SET by Lm49,A40;
then
A41: g.N1 = N1 to_power 2 by A1;
2*N0 >= 1*N0 by XREAL_1:64;
then N1 >= N0 by A38,XXREAL_0:2;
then
A42: N1 >= N by A36,XXREAL_0:2;
N1+1 in POWEROF2SET by A40;
then g.(N1+1) = N1+1 by A1;
then g.N1 > g.(N1+1) by A41,A39,POWER:46;
hence contradiction by A35,A42;
end;
A43: now
let n be Nat;
A44: n in NAT by ORDINAL1:def 12;
assume n >= 0;
A45: n+0 <= n+1 by XREAL_1:6;
A46: f.n = n
proof
per cases;
suppose
n = 0;
hence thesis by Def3;
end;
suppose
n > 0;
hence f.n = n to_power 1 by Def3,A44
.= n by POWER:25;
end;
end;
f.(n+1) = (n+1) to_power 1 by Def3
.= n+1 by POWER:25;
hence f.n <= f.(n+1) by A46,A45;
end;
reconsider jj = 1 as Real;
reconsider j = 1 as Element of NAT;
A47: now
let n such that
A48: n >= j and
A49: n in X;
A50: f.n = n to_power 1 by A48,Def3
.= n by POWER:25;
hence jj*s.n <= f.n by A1,A49;
thus f.n <= jj*s.n by A1,A49,A50;
end;
f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
hence seq_n^(1) in Big_Theta(s,POWEROF2SET) by A47;
thus thesis by A18,A5,A6,A43,A34;
end;
begin :: Problem 3.31
Lm50: for n st n >= 2 holds n! > 1
proof
defpred P[Nat] means $1! > 1;
A1: for k be Nat st k >= 2 & P[k] holds P[k+1]
proof
let k be Nat such that
A2: k >= 2 and
A3: k! > 1;
A4: k+1 > 0+1 by A2,XREAL_1:6;
(k+1)*(k!) > (k+1)*1 by A3,XREAL_1:68;
then (k+1)*(k!) > 1 by A4,XXREAL_0:2;
hence thesis by NEWTON:15;
end;
A5: P[2] by NEWTON:14;
for n be Nat st n >= 2 holds P[n] from NAT_1:sch 8(A5, A1);
hence thesis;
end;
Lm51: for n1, n st n <= n1 holds n! <= n1!
proof
defpred P[Nat] means for n st n <= $1 holds n! <= ($1)!;
A1: for k being Nat st P[k] holds P[k+1]
proof
let k be Nat such that
A2: for n st n <= k holds n! <= k!;
let n such that
A3: n <= k+1;
per cases by A3,NAT_1:8;
suppose
A4: n <= k;
k+1 >= 0+1 by XREAL_1:6;
then (k+1)*(k!) >= 1*(k!) by XREAL_1:64;
then
A5: (k+1)! >= k! by NEWTON:15;
n! <= k! by A2,A4;
hence thesis by A5,XXREAL_0:2;
end;
suppose
n = k+1;
hence thesis;
end;
end;
A6: P[0];
for n1 being Nat holds P[n1] from NAT_1:sch 2(A6, A1);
hence thesis;
end;
Lm52: for k st k >= 1 holds
ex n st (n! <= k & k < (n+1)! & for m st m! <= k & k < (m+1)! holds m = n)
proof
defpred P[Nat] means ex n st (n! <= $1 & $1 < (n+1)! & for m st m! <= $1 &
$1 < (m+1)! holds m = n);
A1: for k be Nat st k >= 1 & P[k] holds P[k+1]
proof
let k be Nat;
assume that
k >= 1 and
A2: ex n st (n! <= k & k < (n+1)! & for m st m! <= k & k < (m+1)! holds m = n);
consider n such that
A3: n! <= k and
A4: k < (n+1)! and
for m st m! <= k & k < (m+1)! holds m = n by A2;
A5: k+1 <= (n+1)! by A4,INT_1:7;
per cases by A5,XXREAL_0:1;
suppose
A6: k+1 < (n+1)!;
take n;
k+0 <= k+1 by XREAL_1:6;
hence n! <= k+1 by A3,XXREAL_0:2;
thus k+1 < (n+1)! by A6;
let m;
assume that
A7: m! <= k+1 and
A8: k+1 < (m+1)!;
now
assume
A9: m <> n;
thus contradiction
proof
per cases by A9,XXREAL_0:1;
suppose
m > n;
then m >= n+1 by NAT_1:13;
then m! >= (n+1)! by Lm51;
hence thesis by A6,A7,XXREAL_0:2;
end;
suppose
m < n;
then m+1 <= n by NAT_1:13;
then (m+1)! <= n! by Lm51;
then
A10: (m+1)! <= k by A3,XXREAL_0:2;
k <= k+1 by NAT_1:11;
hence thesis by A8,A10,XXREAL_0:2;
end;
end;
end;
hence thesis;
end;
suppose
A11: k+1 = (n+1)!;
take N = n+1;
thus N! <= k+1 by A11;
A12: N! > 0 by NEWTON:17;
N+1 > 0+1 by XREAL_1:6;
then (N+1)*(N!) > 1*(N!) by A12,XREAL_1:68;
hence k+1 < (N+1)! by A11,NEWTON:15;
let m;
assume that
A13: m! <= k+1 and
A14: k+1 < (m+1)!;
now
assume
A15: m <> N;
thus contradiction
proof
per cases by A15,XXREAL_0:1;
suppose
m > N;
then m >= N+1 by NAT_1:13;
then m! >= (N+1)! by Lm51;
then
A16: k+1 >= (N+1)! by A13,XXREAL_0:2;
n + 2 >= 0 + 2 by XREAL_1:6;
then
A17: N+1 > 1 by XXREAL_0:2;
N! > 0 by NEWTON:17;
then (N+1)*(N!) > 1*(N!) by A17,XREAL_1:68;
hence thesis by A11,A16,NEWTON:15;
end;
suppose
m < N;
then m+1 <= N by NAT_1:13;
hence thesis by A11,A14,Lm51;
end;
end;
end;
hence thesis;
end;
end;
A18: P[1]
proof
take 1;
thus 1! <= 1 & 1 < (1+1)! by NEWTON:13,14;
let m;
assume that
A19: m! <= 1 and
A20: 1 < (m+1)!;
A21: now
assume m > 1;
then m >= 1+1 by NAT_1:13;
hence contradiction by A19,Lm50;
end;
m <> 0 by A20,NEWTON:13;
then m >= 0+1 by NAT_1:13;
hence thesis by A21,XXREAL_0:1;
end;
for k be Nat st k >= 1 holds P[k] from NAT_1:sch 8(A18,A1);
hence thesis;
end;
definition
let x be Nat;
func Step1(x) -> Element of NAT means
:Def6:
ex n st n! <= x & x < (n+1)! & it = n! if x <> 0 otherwise it = 0;
consistency;
existence
proof
A1: x in NAT by ORDINAL1:def 12;
hereby
assume x <> 0;
then x >= 0+1 by NAT_1:13;
then consider k being Element of NAT such that
A2: k! <= x and
A3: x < (k+1)! and
for m st m! <= x & x < (m+1)! holds m = k by Lm52,A1;
consider k1 being Real such that
A4: k1 = k!;
reconsider k1 as Element of NAT by A4;
take k1;
thus ex m st m! <= x & x < (m+1)! & k1 = m! by A2,A3,A4;
end;
thus thesis;
end;
uniqueness
proof
let n1, n2 be Element of NAT;
now
assume that
A5: ex n st n! <= x & x < (n+1)! & n1 = n! and
A6: ex n st n! <= x & x < (n+1)! & n2 = n!;
consider n such that
A7: n! <= x and
A8: x < (n+1)! and
A9: n1 = n! by A5;
consider m such that
A10: m! <= x and
A11: x < (m+1)! and
A12: n2 = m! by A6;
now
assume
A13: m <> n;
thus contradiction
proof
per cases by A13,XXREAL_0:1;
suppose
m > n;
then m >= n+1 by INT_1:7;
then m! >= (n+1)! by Lm51;
hence thesis by A8,A10,XXREAL_0:2;
end;
suppose
m < n;
then m+1 <= n by INT_1:7;
then (m+1)! <= n! by Lm51;
hence thesis by A7,A11,XXREAL_0:2;
end;
end;
end;
hence n1 = n2 by A9,A12;
end;
hence thesis;
end;
end;
Lm53: for n st n >= 3 holds n! > n
proof
let n;
assume
A1: n >= 3;
set n1 = n-1;
2 = 3-1;
then
A2: n1 >= 2 by A1,XREAL_1:9;
then reconsider n1 as Element of NAT by INT_1:3;
n1! >= 2 by A2,Lm51,NEWTON:14;
then n1! > 1 by XXREAL_0:2;
then
A3: n*(n1!) > n*1 by A1,XREAL_1:68;
n1+1 = n;
hence thesis by A3,NEWTON:15;
end;
theorem
for f being Real_Sequence st (for n holds f.n = Step1(n))
ex s being eventually-positive Real_Sequence st s = f
& s is not smooth & (for n holds f.n <= (seq_n^(1)).n)
& f is eventually-nondecreasing
proof
set g = seq_n^(1);
let f be Real_Sequence such that
A1: for n holds f.n = Step1(n);
f is eventually-positive
proof
take 1;
let n be Nat;
A2: n in NAT by ORDINAL1:def 12;
assume n >= 1;
then
A3: ex m st m! <= n & n < (m+1)! & Step1(n) = m! by Def6;
f.n = Step1(n) by A1,A2;
hence thesis by A3,NEWTON:17;
end;
then reconsider s=f as eventually-positive Real_Sequence;
take s;
thus s=f;
now
let k;
thus f.k <= f.(k+1)
proof
per cases;
suppose
A4: k = 0;
A5: f.(0+1) = Step1(1) by A1;
f.0 = Step1(0) by A1
.= 0 by Def6;
hence thesis by A4,A5;
end;
suppose
k > 0;
then consider n1 such that
A6: n1! <= k and
A7: k < (n1+1)! and
A8: Step1(k) = n1! by Def6;
A9: k+1 <= (n1+1)! by A7,INT_1:7;
A10: k <= k+1 by NAT_1:11;
A11: f.k = n1! by A1,A8;
per cases by A9,XXREAL_0:1;
suppose
A12: k+1 < (n1+1)!;
n1! <= k+1 by A10,A6,XXREAL_0:2;
then Step1(k+1) = n1! by A12,Def6;
hence thesis by A1,A11;
end;
suppose
A13: k+1 = (n1+1)!;
A14: (n1+1)! > 0 by NEWTON:17;
n1 + 2 > 0 + 1 by XREAL_1:8;
then 1*((n1+1)!) < (n1+2)*((n1+1)!) by A14,XREAL_1:68;
then
A15: k+1 < ((n1+1)+1)! by A13,NEWTON:15;
f.(k+1) = Step1(k+1) by A1
.= (n1+1)! by A13,A15,Def6;
hence thesis by A11,Lm51,NAT_1:11;
end;
end;
end;
end;
then
A16: for k st k >= 0 holds f.k <= f.(k+1);
A17: 1 = 2-1;
hereby
set h = (f taken_every 2);
assume s is smooth;
then s is_smooth_wrt 2;
then h in Big_Oh s;
then consider t being Element of Funcs(NAT, REAL) such that
A18: t = h and
A19: ex c,N st c > 0 & for n st n >= N holds t.n <= c*f.n & t.n >= 0;
consider c,N such that
c > 0 and
A20: for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A19;
set n2 = max(max(N,3), [/c\]+1);
A21: n2 >= max(N,3) by XXREAL_0:25;
max(N,3) >= N by XXREAL_0:25;
then
A22: n2 >= N by A21,XXREAL_0:2;
A23: n2 >= [/c\]+1 by XXREAL_0:25;
A24: n2 is Integer by XXREAL_0:16;
A25: max(N,3) >= 3 by XXREAL_0:25;
then
A26: n2 >= 3 by A21,XXREAL_0:2;
reconsider n2 as Element of NAT by A21,A24,INT_1:3;
set n1 = n2!-1;
A27: n2 > 2 by A26,XXREAL_0:2;
then
A28: n2! >= 2 by Lm51,NEWTON:14;
then
A29: n1 >= 1 by A17,XREAL_1:9;
set n3 = n2-1;
1+1 <= n2 by A26,XXREAL_0:2;
then
A30: 1 <= n2-1 by XREAL_1:19;
A31: n3 >= 1 by A17,A27,XREAL_1:9;
reconsider n1 as Element of NAT by A29,INT_1:3;
A32: t.n1 = f.(2*n1) by A18,ASYMPT_0:def 15;
n2! > n2 by A26,Lm53;
then n2! >= n2 + 1 by INT_1:7;
then n1 >= n2 by XREAL_1:19;
then n1 >= N by A22,XXREAL_0:2;
then
A33: t.n1 <= c*f.n1 by A20;
n2 < n2+1 by NAT_1:13;
then n2*(n2!) < (n2+1)*(n2!) by A28,XREAL_1:68;
then
A34: n2*(n2!) < (n2+1)! by NEWTON:15;
n2! + 2 <= n2! + n2! by A28,XREAL_1:6;
then
A35: n2! <= 2*(n2!) - 2*1 by XREAL_1:19;
A36: n2!-1 < n2!-0 by XREAL_1:15;
then
A37: 2*n1 < 2*(n2!) by XREAL_1:68;
reconsider n3 as Element of NAT by A31,INT_1:3;
n3! >= 1 by A31,Lm51,NEWTON:13;
then 1*1 <= (n2-1)*(n3!) by A30,Lm20;
then n2*1 <= (n2-1)*(n3!)*n2 by XREAL_1:64;
then n2 <= (n2-1)*((n3!)*(n3+1));
then n2 <= (n2-1)*(n2!) by NEWTON:15;
then
A38: n2! + n2 <= (n2!)*1 + (n2-1)*(n2!) by XREAL_1:6;
A39: n3+1 = n2+0;
then n2*(n3!) = n2! by NEWTON:15;
then n2*(n3!) <= n2*(n2!) - n2 by A38,XREAL_1:19;
then n3! <= (n2*(n2!-1))/(n2*1) by A21,A25,XREAL_1:77;
then
A40: n3! <= (n2! - 1)/1 by A21,A25,XCMPLX_1:91;
A41: [/c\] >= c by INT_1:def 7;
[/c\] + 1 > [/c\] + 0 by XREAL_1:8;
then [/c\] + 1 > c by A41,XXREAL_0:2;
then
A42: n2 > c by A23,XXREAL_0:2;
A43: n3! > 0 by NEWTON:17;
2*(n2!) <= n2*(n2!) by A27,XREAL_1:64;
then 2*(n2!) < (n2+1)! by A34,XXREAL_0:2;
then
A44: 2*n1 < (n2+1)! by A37,XXREAL_0:2;
A45: f.(2*n1) = Step1(2*n1) by A1
.= (n3+1)! by A28,A44,A35,Def6
.= n2*(n3!) by NEWTON:15;
f.n1 = Step1(n1) by A1
.= n3! by A29,A36,A39,A40,Def6;
hence contradiction by A33,A32,A45,A42,A43,XREAL_1:68;
end;
hereby
let n;
thus f.n <= g.n
proof
per cases;
suppose
A46: n = 0;
f.0 = Step1(0) by A1
.= 0 by Def6;
hence thesis by A46,Def3;
end;
suppose
A47: n > 0;
then
A48: g.n = n to_power 1 by Def3
.= n by POWER:25;
ex n1 st n1! <= n & n < (n1+1)! & Step1(n) = n1! by A47,Def6;
hence thesis by A1,A48;
end;
end;
end;
reconsider zz=0 as Nat;
take zz;
let n be Nat;
n in NAT by ORDINAL1:def 12;
hence thesis by A16;
end;
:: Problem 3.32 -- omitted (corresponding theory section omitted)
:: Problem 3.33 -- Proven in theorems ASYMPT_0:41, ASYMPT_0:42
begin :: Problem 3.34
Lm54: seq_n^(1) - seq_const(1) is eventually-positive
proof
take 2;
set g = seq_const(1);
set f = seq_n^(1);
let n be Nat;
A1: n in NAT by ORDINAL1:def 12;
A2: g.n = 1 by FUNCOP_1:7,A1;
assume
A3: n >= 2;
then
A4: n > 1+0 by XXREAL_0:2;
A5: f.n = n to_power 1 by A3,Def3,A1
.= n by POWER:25;
(f-g).n = f.n + (-g).n by SEQ_1:7
.= n + -1 by A5,A2,SEQ_1:10
.= n-1;
hence thesis by A4,XREAL_1:20;
end;
:: This is to show that Big_Theta(n-1) + Big_Theta(n) = Big_Theta(n).
:: Note that it is not true that Big_Theta(n) = Big_Theta(n) - Big_Theta(n-1).
:: Consider n and n. The function n - n = 0, which is not in Big_Theta(n).
theorem
for F being eventually-nonnegative Real_Sequence st F = seq_n^(1) -
seq_const(1) holds Big_Theta(F) + Big_Theta(seq_n^(1)) = Big_Theta(seq_n^(1))
proof
set q = seq_const(1);
set p = seq_n^(1);
set f = p - q;
set g = p;
A1: Big_Theta(g) = { t where t is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*g.n <= t.n & t.n <= c*g.n } by
ASYMPT_0:27;
let F be eventually-nonnegative Real_Sequence;
assume F = seq_n^(1) - seq_const(1);
then
A2: Big_Theta(F) = { t where t is Element of Funcs(NAT, REAL) : ex c,d,N st
c > 0 & d > 0 & for n st n >= N holds d*f.n <= t.n & t.n <= c*f.n } by
ASYMPT_0:27;
now
let x be object;
hereby
assume x in Big_Theta(F) + Big_Theta(g);
then consider t being Element of Funcs(NAT, REAL) such that
A3: t = x and
A4: ex f9,g9 being Element of Funcs(NAT,REAL) st f9 in Big_Theta(F)
& g9 in Big_Theta(g) & for n being Element of NAT holds t.n = f9.n + g9.n
;
consider f9,g9 being Element of Funcs(NAT,REAL) such that
A5: f9 in Big_Theta(F) and
A6: g9 in Big_Theta(g) and
A7: for n being Element of NAT holds t.n = f9.n + g9.n by A4;
ex r being Element of Funcs(NAT,REAL) st r = f9 & ex c,d,N st c
> 0 & d > 0 & for n st n >= N holds d*f.n <= r.n & r.n <= c*f.n by A2,A5;
then consider c1,d1,N1 such that
A8: c1 > 0 and
A9: d1 > 0 and
A10: for n st n >= N1 holds d1*f.n <= f9.n & f9.n <= c1*f.n;
ex s being Element of Funcs(NAT,REAL) st s = g9 & ex c,d,N st c
> 0 & d > 0 & for n st n >= N holds d*g.n <= s.n & s.n <= c*g.n by A1,A6;
then consider c2,d2,N2 such that
A11: c2 > 0 and
A12: d2 > 0 and
A13: for n st n >= N2 holds d2*g.n <= g9.n & g9.n <= c2*g.n;
set d = d2, c = c1+c2;
set N = max(1,max(N1,N2));
A14: N >= 1 by XXREAL_0:25;
A15: N>=max(N1,N2) by XXREAL_0:25;
max(N1,N2)>= N2 by XXREAL_0:25;
then
A16: N >= N2 by A15,XXREAL_0:2;
max(N1,N2)>=N1 by XXREAL_0:25;
then
A17: N >= N1 by A15,XXREAL_0:2;
now
let n;
A18: (seq_const(1)).n = 1 by FUNCOP_1:7;
assume
A19: n >= N;
then
A20: g.n = (n to_power 1) by A14,Def3
.= n by POWER:25;
n >= 1 by A14,A19,XXREAL_0:2;
then -n <= -1 by XREAL_1:24;
then (-n)*d1 <= (-1)*d1 by A9,XREAL_1:64;
then
A21: n*(-d1) + (d1+d2)*n <= -d1 + (d1+d2)*n by XREAL_1:6;
A22: f.n = (seq_n^(1)).n + (-seq_const(1)).n by SEQ_1:7
.= (seq_n^(1)).n + -((seq_const(1)).n) by SEQ_1:10
.= (n to_power 1) + -((seq_const(1)).n) by A14,A19,Def3
.= n + -1 by A18,POWER:25;
A23: n >= N2 by A16,A19,XXREAL_0:2;
then d2*g.n <= g9.n by A13;
then
A24: d1*f.n + d2*g.n <= d1*f.n + g9.n by XREAL_1:6;
g9.n <= c2*g.n by A13,A23;
then
A25: c1*f.n+g9.n<=c1*f.n+c2*g.n by XREAL_1:6;
A26: n >= N1 by A17,A19,XXREAL_0:2;
then f9.n <= c1*f.n by A10;
then f9.n+g9.n<=c1*f.n+g9.n by XREAL_1:6;
then
A27: f9.n + g9.n <= c1*f.n + c2*g.n by A25,XXREAL_0:2;
d1*f.n <= f9.n by A10,A26;
then d1*f.n + g9.n <= f9.n + g9.n by XREAL_1:6;
then d1*f.n + d2*g.n <= f9.n + g9.n by A24,XXREAL_0:2;
then d2*n <= f9.n + g9.n by A20,A22,A21,XXREAL_0:2;
hence d*g.n <= t.n by A7,A20;
-c1 + (c1+c2)*n <= 0 + (c1+c2)*n by A8,XREAL_1:6;
then f9.n + g9.n <= (c1+c2)*n by A20,A22,A27,XXREAL_0:2;
hence t.n <= c*g.n by A7,A20;
end;
hence x in Big_Theta(g) by A1,A3,A8,A11,A12;
end;
assume x in Big_Theta(g);
then consider t being Element of Funcs(NAT, REAL) such that
A28: t = x and
A29: ex c,d,N st c > 0 & d > 0 & for n st n >= N holds d*g.n <= t.n &
t.n <= c*g.n by A1;
consider c,d,N such that
A30: c > 0 and
A31: d > 0 and
A32: for n st n >= N holds d*g.n <= t.n & t.n <= c*g.n by A29;
set f9 = 2"(#)t, g9 = 2"(#)t;
A33: f9 is Element of Funcs(NAT,REAL) by FUNCT_2:8;
A34: for n being Element of NAT holds t.n = f9.n + g9.n
proof
let n be Element of NAT;
f9.n = 2"*t.n by SEQ_1:9;
hence thesis;
end;
A35: 2"*d > 2"*0 by A31,XREAL_1:68;
set N0 = max(N, 2);
A36: N0 >= N by XXREAL_0:25;
A37: N0 >= 2 by XXREAL_0:25;
reconsider N0 as Element of NAT;
A38: now
let n;
assume n >= N0;
then
A39: n >= N by A36,XXREAL_0:2;
then
A40: 2"*t.n <= 2"*(c*g.n) by A32,XREAL_1:64;
2"*(d*g.n) <= 2"*t.n by A32,A39,XREAL_1:64;
hence 2"*d*g.n <= g9.n & g9.n <= 2"*c*g.n by A40,SEQ_1:9;
end;
2"*c > 2"*0 by A30,XREAL_1:68;
then
A41: g9 in Big_Theta(g) by A1,A35,A33,A38;
now
let n;
A42: q.n = 1 by FUNCOP_1:7;
assume
A43: n >= N0;
then
A44: g.n = (n to_power 1) - 0 by A37,Def3
.= n-0 by POWER:25;
n >= 2 by A37,A43,XXREAL_0:2;
then n + 2 <= n + n by XREAL_1:6;
then n <= 2*n - 2*1 by XREAL_1:19;
then 2"*n <= 2"*(2*(n-1)) by XREAL_1:64;
then
A45: c*(2"*n) <= c*(n-1) by A30,XREAL_1:64;
A46: n >= N by A36,A43,XXREAL_0:2;
then
A47: 2"*(d*g.n) <= 2"*t.n by A32,XREAL_1:64;
A48: f.n = p.n + (-q).n by SEQ_1:7
.= p.n + -1 by A42,SEQ_1:10
.= p.n - 1
.= (n to_power 1) - 1 by A37,A43,Def3
.= n-1 by POWER:25;
then f.n <= g.n by A44,XREAL_1:13;
then 2"*d*f.n <= 2"*d*g.n by A31,XREAL_1:64;
then 2"*d*f.n <= 2"*t.n by A47,XXREAL_0:2;
hence (2"*d)*f.n <= f9.n by SEQ_1:9;
2"*t.n <= 2"*(c*g.n) by A32,A46,XREAL_1:64;
then 2"*t.n <= c*f.n by A48,A44,A45,XXREAL_0:2;
hence f9.n <= c*f.n by SEQ_1:9;
end;
then f9 in Big_Theta(F) by A2,A30,A35,A33;
hence x in Big_Theta(F) + Big_Theta(g) by A28,A33,A41,A34;
end;
hence thesis by TARSKI:2;
end;
begin :: Problem 3.35
theorem
ex F being FUNCTION_DOMAIN of NAT,REAL st F = { seq_n^(1) } & (for n
holds (seq_n^(-1)).n <= (seq_n^(1)).n) & not seq_n^(-1) in F to_power Big_Oh(
seq_const(1))
proof
set t = seq_n^(-1);
reconsider F = { seq_n^(1) } as FUNCTION_DOMAIN of NAT,REAL by FUNCT_2:121;
take F;
thus F = { seq_n^(1) };
A1: now
let n;
per cases;
suppose
A2: n = 0;
then (seq_n^(-1)).n = 0 by Def3;
hence (seq_n^(-1)).n <= (seq_n^(1)).n by A2,Def3;
end;
suppose
A3: n > 0;
then
A4: n >= 0+1 by INT_1:7;
A5: n to_power (-1) <= n to_power 1
proof
per cases by A4,XXREAL_0:1;
suppose
A6: n = 1;
then n to_power (-1) = 1 by POWER:26;
hence thesis by A6,POWER:26;
end;
suppose
n > 1;
hence thesis by PRE_FF:8;
end;
end;
(seq_n^(-1)).n=n to_power (-1) by A3,Def3;
hence (seq_n^(-1)).n <= (seq_n^(1)).n by A3,A5,Def3;
end;
end;
now
assume
A7: t in F to_power Big_Oh(seq_const(1));
ex H being FUNCTION_DOMAIN of NAT,REAL st H = F &( t in H to_power
Big_Oh(seq_const(1)) iff ex N,c,k st c>0 & for n st n >= N holds 1 <= t.n & t.n
<= c*(seq_n^(k)).n) by Th9;
then consider N0,c,k such that
c>0 and
A8: for n st n >= N0 holds 1 <= t.n & t.n <= c*(seq_n^(k)).n by A7;
set N = max(N0,2);
A9: N >= 2 by XXREAL_0:25;
A10: N >= N0 by XXREAL_0:25;
now
let n;
assume
A11: n >= N;
then n >= 2 by A9,XXREAL_0:2;
then
A12: n > 1 by XXREAL_0:2;
n >= N0 by A10,A11,XXREAL_0:2;
then
A13: t.n >= 1 by A8;
t.n = n to_power (-1) by A9,A11,Def3;
hence contradiction by A13,A12,POWER:36;
end;
hence contradiction;
end;
hence thesis by A1;
end;
begin :: Addition
:: In theorem ASYMPT_0:14, if we restrict our attentions to functions that
:: do not have a subsequence that converges to 0, then the reverse implication
:: is true.
theorem
for c being non negative Real, x,f being eventually-nonnegative
Real_Sequence st ex e,N st e > 0 & for n st n >= N holds f.n >= e holds x in
Big_Oh(c+f) implies x in Big_Oh(f)
proof
let c be non negative Real, x,f be eventually-nonnegative Real_Sequence;
given e,N0 such that
A1: e > 0 and
A2: for n st n >= N0 holds f.n >= e;
assume x in Big_Oh(c+f);
then consider t being Element of Funcs(NAT, REAL) such that
A3: x = t and
A4: ex d,N st d > 0 & for n st n >= N holds t.n <= d*(c+f).n & t.n >= 0;
consider d,N1 such that
A5: d > 0 and
A6: for n st n >= N1 holds t.n <= d*(c+f).n & t.n >= 0 by A4;
set b = max(2*d, (2*d*c)/e);
2*d > 2*0 by A5,XREAL_1:68;
then
A7: b > 0 by XXREAL_0:25;
set N = max(N0, N1);
A8: N >= N1 by XXREAL_0:25;
A9: N >= N0 by XXREAL_0:25;
now
let n;
assume
A10: n >= N;
then
A11: n >= N1 by A8,XXREAL_0:2;
then t.n <= d*(c+f).n by A6;
then
A12: t.n <= d*(c+f.n) by VALUED_1:2;
A13: n >= N0 by A9,A10,XXREAL_0:2;
thus t.n <= b*f.n
proof
per cases;
suppose
c >= f.n;
then d*c >= d*f.n by A5,XREAL_1:64;
then d*c + d*c >= d*c + d*f.n by XREAL_1:6;
then t.n <= (2*(d*c))*1 by A12,XXREAL_0:2;
then
A14: t.n <= (2*(d*c))*((1/e)*e) by A1,XCMPLX_1:106;
b*e >= ((2*d*c)/e)*e by A1,XREAL_1:64,XXREAL_0:25;
then
A15: t.n <= b*e by A14,XXREAL_0:2;
b*f.n >= b*e by A2,A7,A13,XREAL_1:64;
hence thesis by A15,XXREAL_0:2;
end;
suppose
c < f.n;
then d*c < d*f.n by A5,XREAL_1:68;
then d*c + d*f.n < d*f.n + d*f.n by XREAL_1:6;
then
A16: t.n < 2*(d*f.n) by A12,XXREAL_0:2;
f.n > 0 by A1,A2,A13;
then b*f.n >= (2*d)*f.n by XREAL_1:64,XXREAL_0:25;
hence thesis by A16,XXREAL_0:2;
end;
end;
thus t.n >= 0 by A6,A11;
end;
hence thesis by A3,A7;
end;
begin :: Potentially Useful Facts
theorem
2 to_power 12 = 4096 by Lm26;
theorem
for n st n >= 3 holds n^2 > 2*n + 1 by Lm27;
theorem
for n st n >= 10 holds 2 to_power (n-1) > (2*n)^2 by Lm28;
theorem
for n st n >= 9 holds (n+1) to_power 6 < 2*(n to_power 6) by Lm29;
theorem
for n st n >= 30 holds 2 to_power n > n to_power 6 by Lm30;
theorem
for x being Real st x > 9 holds 2 to_power x > (2*x)^2 by Lm31;
theorem
ex N st for n st n >= N holds sqrt n - log(2, n) > 1 by Lm32;
theorem
for a,b,c being Real st a > 0 & c > 0 & c <> 1 holds a to_power b = c
to_power (b*log(c,a)) by Lm3;
theorem
5! = 120 by Lm33;
theorem
5 to_power 5 = 3125 by Lm36;
theorem
4 to_power 4 = 256 by Lm37;
theorem
for n holds (n^2 - n + 1) > 0 by Lm21;
theorem
for n st n >= 2 holds n! > 1 by Lm50;
theorem
for n1, n st n <= n1 holds n! <= n1! by Lm51;
theorem
for k st k >= 1 holds ex n st (n! <= k & k < (n+1)! & for m st m! <= k
& k < (m+1)! holds m = n) by Lm52;
theorem
for n st n >= 2 holds [/n/2\] < n by Lm46;
theorem
for n st n >= 3 holds n! > n by Lm53;
theorem
seq_n^(1) - seq_const(1) is eventually-positive by Lm54;
theorem
for n st n >= 2 holds 2 to_power n > n+1 by Lm1;
theorem
for a being logbase Real, f being Real_Sequence st a > 1 & f.0 = 0 & (
for n st n > 0 holds f.n = log(a,n)) holds f is eventually-positive by Lm2;
theorem
for f,g being eventually-nonnegative Real_Sequence holds f in Big_Oh(g
) & g in Big_Oh(f) iff Big_Oh(f) = Big_Oh(g) by Lm5;
theorem
for a, b, c being Real st 0 < a & a <= b & c >= 0 holds a to_power c
<= b to_power c by Lm6;
theorem
for n st n >= 4 holds 2*n + 3 < 2 to_power n by Lm7;
theorem
for n st n >= 6 holds (n+1)^2 < 2 to_power n by Lm8;
theorem
for c being Real st c > 6 holds c^2 < 2 to_power c by Lm9;
theorem
for e being positive Real, f being Real_Sequence st f.0 = 0 & (for n
st n > 0 holds f.n = log(2,n to_power e)) holds (f /" seq_n^(e)) is convergent
& lim (f /" seq_n^(e)) = 0 by Lm10;
theorem
for e being Real st e > 0 holds (seq_logn/"seq_n^(e)) is convergent &
lim(seq_logn/"seq_n^(e)) = 0 by Lm11;
theorem
for f being Real_Sequence holds for N holds (for n st n <= N holds f.n
>= 0) implies Sum(f,N) >= 0 by Lm12;
theorem
for f,g being Real_Sequence holds for N holds (for n st n <= N holds f
.n <= g.n) implies Sum(f,N) <= Sum (g,N) by Lm13;
theorem
for f being Real_Sequence, b being Real st f.0 = 0 & (for n st n > 0
holds f.n = b) holds for N being Element of NAT holds Sum(f,N) = b*N by Lm14;
theorem
for f being Real_Sequence, N,M being Element of NAT holds Sum(f,N,M) +
f.(N+1) = Sum(f,N+1,M) by Lm15;
theorem
for f,g being Real_Sequence, M being Element of NAT holds for N st N
>= M+1 holds (for n st M+1 <= n & n <= N holds f.n <= g.n) implies Sum(f,N,M)
<= Sum (g,N,M) by Lm16;
theorem
for n holds [/n/2\] <= n by Lm17;
theorem
for f being Real_Sequence, b being Real, N being Element of NAT st f.0
= 0 & (for n st n > 0 holds f.n = b) holds for M being Element of NAT holds Sum
(f, N, M) = b*(N-M) by Lm18;
theorem
for f,g being Real_Sequence, N being Element of NAT, c being Real st f
is convergent & lim f = c & for n st n >= N holds f.n = g.n holds g is
convergent & lim g = c by Lm22;
theorem
for n st n >= 1 holds (n^2 -n + 1) <= n^2 by Lm23;
theorem
for n st n >= 1 holds n^2 <= 2*(n^2 -n + 1) by Lm24;
theorem
for e being Real st 0 < e & e < 1 holds ex N st for n st n >= N holds
n*log(2,1+e) - 8*log(2,n) > 8*log(2,n) by Lm25;
theorem
for n st n >= 10 holds 2 to_power (2*n) / (n!) < 1/(2 to_power (n-9))
by Lm34;
theorem
for n st n >= 3 holds 2*(n-2) >= n-1 by Lm35;
theorem
for c being Real st c >= 0 holds c to_power (1/2) = sqrt c by Lm39;
theorem
ex N st for n st n >= N holds n - sqrt n*log(2,n) > n/2 by Lm40;
:: The proof of this theorem has been taken directly from the
:: article POWER.miz (with very slight modifications to fit this new theorem).
theorem
for s being Real_Sequence st
for n being Nat holds s.n = (1 + 1/(n+1)) to_power
(n+1) holds s is non-decreasing by Lm41;
:: (1 + 1/n)^n is non-decreasing
theorem
for n st n >= 1 holds ((n+1)/n) to_power n <= ((n+2)/(n+1)) to_power (
n+1) by Lm42;
theorem
for k,n st k <= n holds n choose k >= ((n+1) choose k) / (n+1) by Lm43;
theorem
for f being Real_Sequence st (for n being Nat holds f.n = log(2,n!))
for n holds f.n = Sum(seq_logn, n) by Lm44;
theorem
for n st n >= 4 holds n*log(2,n) >= 2*n by Lm45;
theorem
for n st n >= 2 holds n^2 > n+1 by Lm47;
theorem
for n st n >= 1 holds (2 to_power (n+1)) - (2 to_power n) > 1 by Lm48;
theorem
for n st n >= 2 holds not ((2 to_power n) - 1) in POWEROF2SET by Lm49;
theorem
for n,k st k >= 1 & n! <= k & k < (n+1)! holds Step1(k) = n! by Def6;
theorem
for a,b,c being Real holds a>1 & b>=a & c>=1 implies log(a,c) >= log(b
,c) by Lm19;