asympt_1.miz
begin
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 ::
ASYMPT_1:1
for t,t1 be
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)))) holds ex s,s1 be
eventually-positive
Real_Sequence st s
= t & s1
= t1 & s
in (
Big_Oh s1)
proof
ex s be
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 be
Element of
NAT holds ex y be
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 be
sequence of
REAL such that
A3: for x be
Element of
NAT holds
P[x, (h
. x)] from
FUNCT_2:sch 3(
A1);
take h;
thus thesis by
A3;
end;
then
consider q be
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,
XREAL_1: 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,
XREAL_1: 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 be
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 be
Element of
NAT holds ex y be
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 be
sequence of
REAL such that
A36: for x be
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 be
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 be
Nat such that
A44: for n be
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 be
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 be
logbase
Real, f be
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 ::
ASYMPT_1:2
for a,b be
logbase
Real, f,g be
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 be
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 be
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 be
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;
::
ASYMPT_1:def1
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 be
sequence of
REAL such that
A1: for n be
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 be
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 ::
ASYMPT_1:3
for a,b be
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 be
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;
definition
::
ASYMPT_1:def2
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 be
Element of
NAT holds ex y be
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 be
sequence of
REAL such that
A3: for x be
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;
::
ASYMPT_1:def3
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 be
Element of
NAT holds ex y be
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 be
sequence of
REAL such that
A3: for x be
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 be
Real_Sequence, n be
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 be
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 ::
ASYMPT_1:4
Th4: for f,g be
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 be
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 be
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 be
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 be
positive
Real, f be
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 be
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 be
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 be
Element of
NAT holds ex y be
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 be
sequence of
REAL such that
A3: for x be
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 be
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 be
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 ::
ASYMPT_1:5
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;
theorem ::
ASYMPT_1:6
(
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;
Lm12: for f be
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 be
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 be
Nat holds
P[N] from
NAT_1:sch 2(
A5,
A1);
hence thesis;
end;
Lm13: for f,g be
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 be
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 be
Nat holds
P[N] from
NAT_1:sch 2(
A5,
A1);
hence thesis;
end;
Lm14: for f be
Real_Sequence, b be
Real st (f
.
0 )
=
0 & (for n st n
>
0 holds (f
. n)
= b) holds for N be
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 be
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 be
Nat holds
P[N] from
NAT_1:sch 2(
A5,
A3);
hence thesis;
end;
Lm15: for f be
Real_Sequence, N,M be
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 be
Real_Sequence, M be
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 be
Nat holds
[/(n
/ 2)\]
<= n
proof
let n be
Nat;
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 be
Real_Sequence, b be
Real, N be
Element of
NAT st (f
.
0 )
=
0 & (for n st n
>
0 holds (f
. n)
= b) holds for M be
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 be
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 be
Nat holds
P[M] from
NAT_1:sch 2(
A5,
A3);
hence thesis;
end;
theorem ::
ASYMPT_1:7
for f be
Real_Sequence, k be
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 be
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 be
Element of
NAT holds ex y be
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;
thus thesis by
A5;
end;
end;
consider h be
sequence of
REAL such that
A6: for x be
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 be
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)
<= ((
- 1)
+ 1);
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 be
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 be
Element of
NAT holds ex y be
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 be
sequence of
REAL such that
A30: for x be
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 be
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;
theorem ::
ASYMPT_1:8
for f be
Real_Sequence st (for n st n
>
0 holds (f
. n)
= (n
to_power (
log (2,n)))) holds ex s be
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
consider t be
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;
definition
::$Canceled
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 be
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 ::
ASYMPT_1:9
Th9: for f be
eventually-nonnegative
Real_Sequence holds ex F be
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 be
Element of (
Funcs (
NAT ,
REAL )) such that
A1: h
= t and
A2: ex f,g be
Element of (
Funcs (
NAT ,
REAL )), N be
Element of
NAT st f
in F & g
in G & for n be
Element of
NAT st n
>= N holds (t
. n)
= ((f
. n)
to_power (g
. n));
consider f,g be
Element of (
Funcs (
NAT ,
REAL )), N0 be
Element of
NAT such that
A3: f
in F and
A4: g
in G and
A5: for n be
Element of
NAT st n
>= N0 holds (t
. n)
= ((f
. n)
to_power (g
. n)) by
A2;
consider g9 be
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 (n
to_power (g
. n))
>= (n
to_power
0 ) by
A6,
A16,
PRE_FF: 8,
A9;
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 be
Element of
NAT holds ex y be
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;
thus thesis by
A26;
end;
end;
consider g be
sequence of
REAL such that
A27: for x be
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
theorem ::
ASYMPT_1:10
for f be
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 be
Element of
NAT such that
A1: t1
= ((10
* 10)
* 10);
consider t2 be
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;
begin
theorem ::
ASYMPT_1:11
(
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 ::
ASYMPT_1:12
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 be
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 ::
ASYMPT_1:13
ex s be
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 be
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 ;
::
ASYMPT_1:def5
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 be
sequence of
REAL such that
A1: for n be
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 ::
ASYMPT_1:14
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 be
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 be
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
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 ::
ASYMPT_1:15
for f be
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 be
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
theorem ::
ASYMPT_1:16
ex s be
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 be
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
theorem ::
ASYMPT_1:17
(
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;
begin
theorem ::
ASYMPT_1:18
for f,g be
Real_Sequence st (for n holds (f
. n)
= (n
mod 2)) & (for n holds (g
. n)
= ((n
+ 1)
mod 2)) holds ex s,s1 be
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 be
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 be
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;
begin
theorem ::
ASYMPT_1:19
for f,g be
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 be
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 be
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 ::
ASYMPT_1:20
for f,g be
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 be
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 be
Nat such that
A2: for n be
Nat st n
>= N2 holds (g
. n)
>=
0 by
ASYMPT_0:def 2;
consider N1 be
Nat such that
A3: for n be
Nat st n
>= N1 holds (f
. n)
>=
0 by
ASYMPT_0:def 2;
A4: (
Big_Theta f)
= { s where s be
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 be
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 be
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;
begin
Lm21: for n holds (((n
^2 )
- n)
+ 1)
>
0
proof
defpred
P[
Nat] means ((($1
^2 )
- $1)
+ 1)
>
0 ;
A1: for k be
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 be
Nat holds
P[n] from
NAT_1:sch 2(
A2,
A1);
hence thesis;
end;
Lm22: for f,g be
Real_Sequence, N be
Element of
NAT , c be
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 be
Nat such that
A5: for n be
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 be
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 be
Nat such that
A6: for n be
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 ::
ASYMPT_1:21
for e be
Real, f be
Real_Sequence st
0
< e & (for n st n
>
0 holds (f
. n)
= (n
* (
log (2,n)))) holds ex s be
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 ::
ASYMPT_1:22
for e be
Real, g be
Real_Sequence st e
< 1 & (for n st n
> 1 holds (g
. n)
= ((n
to_power 2)
/ (
log (2,n)))) holds ex s be
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 ::
ASYMPT_1:23
for f be
Real_Sequence st (for n st n
> 1 holds (f
. n)
= ((n
to_power 2)
/ (
log (2,n)))) holds ex s be
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 ::
ASYMPT_1:24
for g be
Real_Sequence st (for n holds (g
. n)
= ((((n
^2 )
- n)
+ 1)
to_power 4)) holds ex s be
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,
POWER: 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 ::
ASYMPT_1:25
for e be
Real st
0
< e & e
< 1 holds ex s be
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
Lm26: (2
to_power 12)
= 4096
proof
thus (2
to_power 12)
= (2
to_power (6
+ 6))
.= (64
* 64) by
POWER: 27,
POWER: 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,
POWER: 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 be
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,
POWER: 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 be
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,
POWER: 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,
POWER: 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 be
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 be
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 be
Nat such that
A2: for n be
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 be
Real_Sequence st for n be
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,
XCMPLX_1: 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,
XCMPLX_1: 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 be
Real_Sequence such that
A1: for n be
Nat holds (seq
. n)
=
F(n) from
SEQ_1:sch 1;
assume
A2: n
>= 1;
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 ::
ASYMPT_1:26
for f,g be
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 be
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 ::
ASYMPT_1:27
for f be
Real_Sequence st (for n st n
>
0 holds (f
. n)
= (n
to_power (
sqrt n))) holds ex s,s1 be
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 ::
ASYMPT_1:28
ex s,s1 be
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 ::
ASYMPT_1:29
ex s,s1 be
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 ::
ASYMPT_1:30
ex s be
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 ::
ASYMPT_1:31
(
Big_Oh (
seq_n!
0 ))
c= (
Big_Oh (
seq_n! 1)) & (
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 ::
ASYMPT_1:32
for g be
Real_Sequence st (for n st n
>
0 holds (g
. n)
= (n
to_power n)) holds ex s be
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 be
Real_Sequence such that
A5: for n be
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
Lm43: for k,n be
Nat st k
<= n holds (n
choose k)
>= (((n
+ 1)
choose k)
/ (n
+ 1))
proof
let k,n be
Nat;
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 ::
ASYMPT_1:33
for n st n
>= 1 holds for f be
Real_Sequence, k be
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 be
Real_Sequence, k be
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 be
Nat st n
>= 1 &
P[n] holds
P[(n
+ 1)]
proof
let n be
Nat such that n
>= 1 and
A2: for f be
Real_Sequence, k be
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 be
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
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
Lm44: for f be
Real_Sequence st (for n be
Nat holds (f
. n)
= (
log (2,(n
! )))) holds for n holds (f
. n)
= (
Sum (
seq_logn ,n))
proof
set g =
seq_logn ;
let f be
Real_Sequence such that
A1: for n be
Nat holds (f
. n)
= (
log (2,(n
! )));
defpred
P[
Nat] means (f
. $1)
= (
Sum (g,$1));
A2: for k be
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 be
Nat holds
P[n] from
NAT_1:sch 2(
A6,
A2);
hence thesis;
end;
Lm45: for n be
Nat st n
>= 4 holds (n
* (
log (2,n)))
>= (2
* n)
proof
let n be
Nat;
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 ::
ASYMPT_1:34
for f,g be
Real_Sequence st (for n st n
>
0 holds (g
. n)
= (n
* (
log (2,n)))) & (for n be
Nat holds (f
. n)
= (
log (2,(n
! )))) holds ex s be
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 be
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 be
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 be
Element of
NAT holds ex y be
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 be
sequence of
REAL such that
A10: for x be
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 be
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 be
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 be
Element of
NAT holds ex y be
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 be
sequence of
REAL such that
A28: for x be
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 be
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 be
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;
begin
theorem ::
ASYMPT_1:35
for f be
eventually-nondecreasing
eventually-nonnegative
Real_Sequence, t be
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 be
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 be
Nat such that
A3: for n be
Nat st n
>= N0 holds (f
. n)
<= (f
. (n
+ 1)) by
ASYMPT_0:def 6;
assume t
in (
Big_Theta f);
then
consider s be
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;
begin
Lm46: for n be
Nat st n
>= 2 holds
[/(n
/ 2)\]
< n
proof
let n be
Nat 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;
begin
definition
::
ASYMPT_1:def6
func
POWEROF2SET -> non
empty
Subset of
NAT equals the set of all (2
to_power n) where n be
Element of
NAT ;
coherence
proof
set IT = the set of all (2
to_power n) where n be
Element of
NAT ;
A1:
now
let x be
object;
assume x
in IT;
then ex n be
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 be
Nat 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 be
Nat st n
>= 1 holds ((2
to_power (n
+ 1))
- (2
to_power n))
> 1
proof
let n be
Nat;
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 be
Nat st n
>= 2 holds not ((2
to_power n)
- 1)
in
POWEROF2SET
proof
A1: 1
= (2
- 1);
let n be
Nat;
assume n
>= 2;
then (n
- 1)
>= 1 by
A1,
XREAL_1: 9;
then ((2
to_power ((n
+ (
- 1))
+ 1))
- (2
to_power (n
- 1)))
> 1 by
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 ::
ASYMPT_1:36
for f be
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 be
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 be
Nat such that
A12: for m be
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 be
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
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 g
is_smooth_wrt 2 by
A36,
A1;
hence g is
smooth by
ASYMPT_0: 37;
A37: 3
= (4
- 1);
hereby
assume f is
eventually-nondecreasing;
then
consider N be
Nat such that
A38: for n be
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 ::
ASYMPT_1:37
for f,g be
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)) holds ex s be
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 be
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 be
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 be
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
consider t be
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 ::
ASYMPT_1:38
for g be
Real_Sequence st (for n holds (n
in
POWEROF2SET implies (g
. n)
= n) & ( not n
in
POWEROF2SET implies (g
. n)
= (n
to_power 2))) holds ex s be
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 & not s is
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
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 be
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 be
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
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 be
Nat such that
A35: for n be
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;
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
Lm50: for n be
Nat 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 be
Nat st n
<= n1 holds (n
! )
<= (n1
! )
proof
defpred
P[
Nat] means for n be
Nat st n
<= $1 holds (n
! )
<= ($1
! );
A1: for k be
Nat st
P[k] holds
P[(k
+ 1)]
proof
let k be
Nat such that
A2: for n be
Nat st n
<= k holds (n
! )
<= (k
! );
let n be
Nat 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 be
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,
NEWTON: 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;
::
ASYMPT_1:def7
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 be
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 be
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 be
Nat st n
>= 3 holds (n
! )
> n
proof
let n be
Nat;
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 ::
ASYMPT_1:39
for f be
Real_Sequence st (for n holds (f
. n)
= (
Step1 n)) holds ex s be
eventually-positive
Real_Sequence st s
= f & not s is
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
consider t be
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;
begin
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;
theorem ::
ASYMPT_1:40
for F be
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 be
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 be
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 be
Element of (
Funcs (
NAT ,
REAL )) such that
A3: t
= x and
A4: ex f9,g9 be
Element of (
Funcs (
NAT ,
REAL )) st f9
in (
Big_Theta F) & g9
in (
Big_Theta g) & for n be
Element of
NAT holds (t
. n)
= ((f9
. n)
+ (g9
. n));
consider f9,g9 be
Element of (
Funcs (
NAT ,
REAL )) such that
A5: f9
in (
Big_Theta F) and
A6: g9
in (
Big_Theta g) and
A7: for n be
Element of
NAT holds (t
. n)
= ((f9
. n)
+ (g9
. n)) by
A4;
ex r be
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 be
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 be
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 be
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
theorem ::
ASYMPT_1:41
ex F be
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 be
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
theorem ::
ASYMPT_1:42
for c be non
negative
Real, x,f be
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 be
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
theorem ::
ASYMPT_1:43
(2
to_power 12)
= 4096 by
Lm26;
theorem ::
ASYMPT_1:44
for n st n
>= 3 holds (n
^2 )
> ((2
* n)
+ 1) by
Lm27;
theorem ::
ASYMPT_1:45
for n st n
>= 10 holds (2
to_power (n
- 1))
> ((2
* n)
^2 ) by
Lm28;
theorem ::
ASYMPT_1:46
for n st n
>= 9 holds ((n
+ 1)
to_power 6)
< (2
* (n
to_power 6)) by
Lm29;
theorem ::
ASYMPT_1:47
for n st n
>= 30 holds (2
to_power n)
> (n
to_power 6) by
Lm30;
theorem ::
ASYMPT_1:48
for x be
Real st x
> 9 holds (2
to_power x)
> ((2
* x)
^2 ) by
Lm31;
theorem ::
ASYMPT_1:49
ex N st for n st n
>= N holds ((
sqrt n)
- (
log (2,n)))
> 1 by
Lm32;
theorem ::
ASYMPT_1:50
for a,b,c be
Real st a
>
0 & c
>
0 & c
<> 1 holds (a
to_power b)
= (c
to_power (b
* (
log (c,a)))) by
Lm3;
theorem ::
ASYMPT_1:51
(5
! )
= 120 by
Lm33;
theorem ::
ASYMPT_1:52
(5
to_power 5)
= 3125 by
Lm36;
theorem ::
ASYMPT_1:53
(4
to_power 4)
= 256 by
Lm37;
theorem ::
ASYMPT_1:54
for n holds (((n
^2 )
- n)
+ 1)
>
0 by
Lm21;
theorem ::
ASYMPT_1:55
for n be
Nat st n
>= 2 holds (n
! )
> 1 by
Lm50;
theorem ::
ASYMPT_1:56
for n1,n be
Nat st n
<= n1 holds (n
! )
<= (n1
! ) by
Lm51;
theorem ::
ASYMPT_1:57
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 ::
ASYMPT_1:58
for n be
Nat st n
>= 2 holds
[/(n
/ 2)\]
< n by
Lm46;
theorem ::
ASYMPT_1:59
for n be
Nat st n
>= 3 holds (n
! )
> n by
Lm53;
theorem ::
ASYMPT_1:60
((
seq_n^ 1)
- (
seq_const 1)) is
eventually-positive by
Lm54;
theorem ::
ASYMPT_1:61
for n st n
>= 2 holds (2
to_power n)
> (n
+ 1) by
Lm1;
theorem ::
ASYMPT_1:62
for a be
logbase
Real, f be
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 ::
ASYMPT_1:63
for f,g be
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 ::
ASYMPT_1:64
for a,b,c be
Real st
0
< a & a
<= b & c
>=
0 holds (a
to_power c)
<= (b
to_power c) by
Lm6;
theorem ::
ASYMPT_1:65
for n st n
>= 4 holds ((2
* n)
+ 3)
< (2
to_power n) by
Lm7;
theorem ::
ASYMPT_1:66
for n st n
>= 6 holds ((n
+ 1)
^2 )
< (2
to_power n) by
Lm8;
theorem ::
ASYMPT_1:67
for c be
Real st c
> 6 holds (c
^2 )
< (2
to_power c) by
Lm9;
theorem ::
ASYMPT_1:68
for e be
positive
Real, f be
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 ::
ASYMPT_1:69
for e be
Real st e
>
0 holds (
seq_logn
/" (
seq_n^ e)) is
convergent & (
lim (
seq_logn
/" (
seq_n^ e)))
=
0 by
Lm11;
theorem ::
ASYMPT_1:70
for f be
Real_Sequence holds for N holds (for n st n
<= N holds (f
. n)
>=
0 ) implies (
Sum (f,N))
>=
0 by
Lm12;
theorem ::
ASYMPT_1:71
for f,g be
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 ::
ASYMPT_1:72
for f be
Real_Sequence, b be
Real st (f
.
0 )
=
0 & (for n st n
>
0 holds (f
. n)
= b) holds for N be
Element of
NAT holds (
Sum (f,N))
= (b
* N) by
Lm14;
theorem ::
ASYMPT_1:73
for f be
Real_Sequence, N,M be
Element of
NAT holds ((
Sum (f,N,M))
+ (f
. (N
+ 1)))
= (
Sum (f,(N
+ 1),M)) by
Lm15;
theorem ::
ASYMPT_1:74
for f,g be
Real_Sequence, M be
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 ::
ASYMPT_1:75
for n holds
[/(n
/ 2)\]
<= n by
Lm17;
theorem ::
ASYMPT_1:76
for f be
Real_Sequence, b be
Real, N be
Element of
NAT st (f
.
0 )
=
0 & (for n st n
>
0 holds (f
. n)
= b) holds for M be
Element of
NAT holds (
Sum (f,N,M))
= (b
* (N
- M)) by
Lm18;
theorem ::
ASYMPT_1:77
for f,g be
Real_Sequence, N be
Element of
NAT , c be
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 ::
ASYMPT_1:78
for n st n
>= 1 holds (((n
^2 )
- n)
+ 1)
<= (n
^2 ) by
Lm23;
theorem ::
ASYMPT_1:79
for n st n
>= 1 holds (n
^2 )
<= (2
* (((n
^2 )
- n)
+ 1)) by
Lm24;
theorem ::
ASYMPT_1:80
for e be
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 ::
ASYMPT_1:81
for n st n
>= 10 holds ((2
to_power (2
* n))
/ (n
! ))
< (1
/ (2
to_power (n
- 9))) by
Lm34;
theorem ::
ASYMPT_1:82
for n st n
>= 3 holds (2
* (n
- 2))
>= (n
- 1) by
Lm35;
theorem ::
ASYMPT_1:83
for c be
Real st c
>=
0 holds (c
to_power (1
/ 2))
= (
sqrt c) by
Lm39;
theorem ::
ASYMPT_1:84
ex N st for n st n
>= N holds (n
- ((
sqrt n)
* (
log (2,n))))
> (n
/ 2) by
Lm40;
theorem ::
ASYMPT_1:85
for s be
Real_Sequence st for n be
Nat holds (s
. n)
= ((1
+ (1
/ (n
+ 1)))
to_power (n
+ 1)) holds s is
non-decreasing by
Lm41;
theorem ::
ASYMPT_1:86
for n st n
>= 1 holds (((n
+ 1)
/ n)
to_power n)
<= (((n
+ 2)
/ (n
+ 1))
to_power (n
+ 1)) by
Lm42;
theorem ::
ASYMPT_1:87
for k,n be
Nat st k
<= n holds (n
choose k)
>= (((n
+ 1)
choose k)
/ (n
+ 1)) by
Lm43;
theorem ::
ASYMPT_1:88
for f be
Real_Sequence st (for n be
Nat holds (f
. n)
= (
log (2,(n
! )))) holds for n holds (f
. n)
= (
Sum (
seq_logn ,n)) by
Lm44;
theorem ::
ASYMPT_1:89
for n be
Nat st n
>= 4 holds (n
* (
log (2,n)))
>= (2
* n) by
Lm45;
theorem ::
ASYMPT_1:90
for n be
Nat st n
>= 2 holds (n
^2 )
> (n
+ 1) by
Lm47;
theorem ::
ASYMPT_1:91
for n be
Nat st n
>= 1 holds ((2
to_power (n
+ 1))
- (2
to_power n))
> 1 by
Lm48;
theorem ::
ASYMPT_1:92
for n be
Nat st n
>= 2 holds not ((2
to_power n)
- 1)
in
POWEROF2SET by
Lm49;
theorem ::
ASYMPT_1:93
for n, k st k
>= 1 & (n
! )
<= k & k
< ((n
+ 1)
! ) holds (
Step1 k)
= (n
! ) by
Def6;
theorem ::
ASYMPT_1:94
for a,b,c be
Real st a
> 1 & b
>= a & c
>= 1 holds (
log (a,c))
>= (
log (b,c)) by
Lm19;