:: ASYMPT_1 semantic presentation
begin
Lm1: for n being Nat st n >= 2 holds
2 to_power n > n + 1
proof
defpred S1[ Nat] means 2 to_power $1 > $1 + 1;
A1: for k being Nat st k >= 2 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 2 & S1[k] implies S1[k + 1] )
assume that
k >= 2 and
A2: 2 to_power k > k + 1 ; ::_thesis: S1[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 = 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 S1[k + 1] by A3, XXREAL_0:2; ::_thesis: verum
end;
2 to_power 2 = 2 ^2 by POWER:46
.= 4 ;
then A4: S1[2] ;
for n being Nat st n >= 2 holds
S1[n] from NAT_1:sch_8(A4, A1);
hence for n being Nat st n >= 2 holds
2 to_power n > n + 1 ; ::_thesis: verum
end;
theorem :: ASYMPT_1:1
for t, t1 being Real_Sequence st t . 0 = 0 & ( for n being Element of NAT st n > 0 holds
t . n = ((((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2))) + ((log (2,n)) ^2)) + 36 ) & ( for n being Element of NAT st n > 0 holds
t1 . n = (n to_power 3) * (log (2,n)) ) holds
ex s, s1 being eventually-positive Real_Sequence st
( s = t & s1 = t1 & s in Big_Oh s1 )
proof
ex s being Real_Sequence st
( s . 0 = 0 & ( for n being Element of NAT st n > 0 holds
s . n = ((log (2,n)) ^2) + 36 ) )
proof
defpred S1[ Element of NAT , Real] means ( ( $1 = 0 implies $2 = 0 ) & ( $1 > 0 implies $2 = ((log (2,$1)) ^2) + 36 ) );
A1: for x being Element of NAT ex y being Element of REAL st S1[x,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of REAL st S1[n,y]
( n = 0 or n > 0 ) ;
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
consider h being Function of NAT,REAL such that
A2: for x being Element of NAT holds S1[x,h . x] from FUNCT_2:sch_3(A1);
take h ; ::_thesis: ( h . 0 = 0 & ( for n being Element of NAT st n > 0 holds
h . n = ((log (2,n)) ^2) + 36 ) )
thus ( h . 0 = 0 & ( for n being Element of NAT st n > 0 holds
h . n = ((log (2,n)) ^2) + 36 ) ) by A2; ::_thesis: verum
end;
then consider q being Real_Sequence such that
A3: q . 0 = 0 and
A4: for n being Element of NAT st n > 0 holds
q . n = ((log (2,n)) ^2) + 36 ;
q is eventually-positive
proof
take 1 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 1 <= b1 or not q . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 1 <= n or not q . n <= 0 )
A5: ((log (2,n)) ^2) + 36 > 0 + 0 by XREAL_1:8, XREAL_1:63;
assume n >= 1 ; ::_thesis: not q . n <= 0
hence not q . n <= 0 by A4, A5; ::_thesis: verum
end;
then reconsider q = q as eventually-positive Real_Sequence ;
let f, g be Real_Sequence; ::_thesis: ( f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = ((((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2))) + ((log (2,n)) ^2)) + 36 ) & ( for n being Element of NAT st n > 0 holds
g . n = (n to_power 3) * (log (2,n)) ) implies ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = g & s in Big_Oh s1 ) )
assume that
A6: f . 0 = 0 and
A7: for n being Element of NAT st n > 0 holds
f . n = ((((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2))) + ((log (2,n)) ^2)) + 36 and
A8: for n being Element of NAT st n > 0 holds
g . n = (n to_power 3) * (log (2,n)) ; ::_thesis: ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = g & s in Big_Oh s1 )
A9: g is eventually-positive
proof
take 2 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 2 <= b1 or not g . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 2 <= n or not g . n <= 0 )
assume A10: n >= 2 ; ::_thesis: not g . n <= 0
then log (2,n) >= log (2,2) by PRE_FF:10;
then A11: log (2,n) >= 1 by POWER:52;
n to_power 3 > 0 by A10, POWER:34;
then (n to_power 3) * (log (2,n)) > (n to_power 3) * 0 by A11, XREAL_1:68;
hence not g . n <= 0 by A8, A10; ::_thesis: verum
end;
4 = 2 ^2
.= 2 to_power 2 by POWER:46 ;
then A12: log (2,4) = 2 * (log (2,2)) by POWER:55
.= 2 * 1 by POWER:52
.= 2 ;
A13: for n being Element of NAT st n >= 4 holds
7 * (n ^2) > q . n
proof
defpred S1[ Nat] means 7 * ($1 ^2) > q . $1;
A14: for k being Nat st k >= 4 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 4 & S1[k] implies S1[k + 1] )
assume that
A15: k >= 4 and
A16: 7 * (k ^2) > q . k ; ::_thesis: S1[k + 1]
A17: q . (k + 1) = ((log (2,(k + 1))) ^2) + 36 by A4;
k >= 2 by A15, XXREAL_0:2;
then A18: 2 to_power k > k + 1 by Lm1;
k + 1 > k + 0 by XREAL_1:8;
then 2 to_power k > k by A18, XXREAL_0:2;
then log (2,(2 to_power k)) > log (2,k) by A15, POWER:57;
then k * (log (2,2)) > log (2,k) by POWER:55;
then A19: k * 1 > log (2,k) by POWER:52;
log (2,k) >= 2 by A12, A15, PRE_FF:10;
then 14 * k > 2 * (log (2,k)) by A19, XREAL_1:98;
then ((7 * 2) * k) + 7 > (2 * (log (2,k))) + 1 by XREAL_1:8;
then A20: ((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 A15, 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 A21: ((log (2,(k + k))) ^2) + 36 < (((log (2,k)) ^2) + ((7 * (2 * k)) + 7)) + 36 by A20, XREAL_1:6;
k >= 1 by A15, XXREAL_0:2;
then k + k >= k + 1 by XREAL_1:6;
then A22: log (2,(k + k)) >= log (2,(k + 1)) by PRE_FF:10;
k + 1 >= 4 + 0 by A15, XREAL_1:8;
then log (2,(k + 1)) >= 2 by A12, PRE_FF:10;
then (log (2,(k + k))) ^2 >= (log (2,(k + 1))) ^2 by A22, SQUARE_1:15;
then A23: q . (k + 1) <= ((log (2,(k + k))) ^2) + 36 by A17, XREAL_1:6;
7 * ((k + 1) ^2) = (7 * (k ^2)) + ((7 * (2 * k)) + (7 * 1)) ;
then A24: 7 * ((k + 1) ^2) > (q . k) + ((7 * (2 * k)) + (7 * 1)) by A16, XREAL_1:6;
k in NAT by ORDINAL1:def_12;
then q . k = ((log (2,k)) ^2) + 36 by A4, A15;
then (q . k) + ((7 * (2 * k)) + (7 * 1)) > q . (k + 1) by A23, A21, XXREAL_0:2;
hence S1[k + 1] by A24, XXREAL_0:2; ::_thesis: verum
end;
q . 4 = (2 ^2) + 36 by A4, A12
.= 40 ;
then A25: S1[4] ;
for n being Nat st n >= 4 holds
S1[n] from NAT_1:sch_8(A25, A14);
hence for n being Element of NAT st n >= 4 holds
7 * (n ^2) > q . n ; ::_thesis: verum
end;
reconsider g = g as eventually-positive Real_Sequence by A9;
f is eventually-positive
proof
log (2,3) > log (2,2) by POWER:57;
then A26: log (2,3) > 1 by POWER:52;
take 3 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 3 <= b1 or not f . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 3 <= n or not f . n <= 0 )
assume A27: n >= 3 ; ::_thesis: not f . n <= 0
then A28: n to_power 2 > 0 by POWER:34;
n > 1 by A27, XXREAL_0:2;
then A29: n to_power 3 > n to_power 2 by POWER:39;
log (2,n) >= log (2,3) by A27, PRE_FF:10;
then log (2,n) > 1 by A26, XXREAL_0:2;
then (n to_power 3) * (log (2,n)) > (n to_power 2) * 1 by A29, A28, XREAL_1:98;
then 12 * ((n to_power 3) * (log (2,n))) > 5 * (n to_power 2) by A28, 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 not f . n <= 0 by A7, A27; ::_thesis: verum
end;
then reconsider f = f as eventually-positive Real_Sequence ;
take f ; ::_thesis: ex s1 being eventually-positive Real_Sequence st
( f = f & s1 = g & f in Big_Oh s1 )
take g ; ::_thesis: ( f = f & g = g & f in Big_Oh g )
ex s being Real_Sequence st
( s . 0 = 0 & ( for n being Element of NAT st n > 0 holds
s . n = ((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2)) ) )
proof
defpred S1[ 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)) ) );
A30: for x being Element of NAT ex y being Element of REAL st S1[x,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of REAL st S1[n,y]
( n = 0 or n > 0 ) ;
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
consider h being Function of NAT,REAL such that
A31: for x being Element of NAT holds S1[x,h . x] from FUNCT_2:sch_3(A30);
take h ; ::_thesis: ( h . 0 = 0 & ( for n being Element of NAT st n > 0 holds
h . n = ((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2)) ) )
thus h . 0 = 0 by A31; ::_thesis: for n being Element of NAT st n > 0 holds
h . n = ((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2))
let n be Element of NAT ; ::_thesis: ( n > 0 implies h . n = ((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2)) )
thus ( n > 0 implies h . n = ((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2)) ) by A31; ::_thesis: verum
end;
then consider p being Real_Sequence such that
A32: p . 0 = 0 and
A33: for n being Element of NAT 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 A34: log (2,3) > 1 by POWER:52;
take 3 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 3 <= b1 or not p . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 3 <= n or not p . n <= 0 )
assume A35: n >= 3 ; ::_thesis: not p . n <= 0
then A36: n to_power 2 > 0 by POWER:34;
n > 1 by A35, XXREAL_0:2;
then A37: n to_power 3 > n to_power 2 by POWER:39;
log (2,n) >= log (2,3) by A35, PRE_FF:10;
then log (2,n) > 1 by A34, XXREAL_0:2;
then (n to_power 3) * (log (2,n)) > (n to_power 2) * 1 by A37, A36, XREAL_1:98;
then 12 * ((n to_power 3) * (log (2,n))) > 5 * (n to_power 2) by A36, 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 not p . n <= 0 by A33, A35; ::_thesis: verum
end;
then reconsider p = p as eventually-positive Real_Sequence ;
set t = max (p,q);
consider N being Element of NAT such that
A38: for n being Element of NAT st n >= N holds
(max (p,q)) . n > 0 by ASYMPT_0:def_4;
A39: for n being Element of NAT st n >= 4 holds
p . n > 7 * (n ^2)
proof
let n be Element of NAT ; ::_thesis: ( n >= 4 implies p . n > 7 * (n ^2) )
assume A40: n >= 4 ; ::_thesis: p . n > 7 * (n ^2)
then n > 1 by XXREAL_0:2;
then A41: n to_power 3 > n to_power 2 by POWER:39;
log (2,n) >= log (2,4) by A40, PRE_FF:10;
then A42: log (2,n) > 1 by A12, XXREAL_0:2;
n to_power 2 > 0 by A40, POWER:34;
then (n to_power 3) * (log (2,n)) > (n to_power 2) * 1 by A41, A42, XREAL_1:98;
then 12 * ((n to_power 3) * (log (2,n))) > 12 * (n to_power 2) by XREAL_1:68;
then A43: (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 A33, A40;
then p . n > (12 * (n ^2)) - (5 * (n ^2)) by A43, XREAL_1:9;
hence p . n > 7 * (n ^2) ; ::_thesis: verum
end;
A44: for n being Element of NAT st n >= 4 holds
p . n > q . n
proof
let n be Element of NAT ; ::_thesis: ( n >= 4 implies p . n > q . n )
assume A45: n >= 4 ; ::_thesis: p . n > q . n
then A46: 7 * (n ^2) > q . n by A13;
p . n > 7 * (n ^2) by A39, A45;
hence p . n > q . n by A46, XXREAL_0:2; ::_thesis: verum
end;
A47: for n being Element of NAT st n >= 4 holds
(max (p,q)) . n = p . n
proof
let n be Element of NAT ; ::_thesis: ( n >= 4 implies (max (p,q)) . n = p . n )
assume n >= 4 ; ::_thesis: (max (p,q)) . n = p . n
then A48: p . n > q . n by A44;
thus (max (p,q)) . n = max ((p . n),(q . n)) by ASYMPT_0:def_7
.= p . n by A48, XXREAL_0:def_10 ; ::_thesis: verum
end;
A49: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_max_(4,N)_holds_
(_(max_(p,q))_._n_<=_12_*_(g_._n)_&_(max_(p,q))_._n_>=_0_)
let n be Element of NAT ; ::_thesis: ( n >= max (4,N) implies ( (max (p,q)) . n <= 12 * (g . n) & (max (p,q)) . n >= 0 ) )
assume A50: n >= max (4,N) ; ::_thesis: ( (max (p,q)) . n <= 12 * (g . n) & (max (p,q)) . n >= 0 )
A51: max (4,N) >= 4 by XXREAL_0:25;
then (max (p,q)) . n = p . n by A47, A50, XXREAL_0:2
.= ((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2)) by A33, A50, A51 ;
then (max (p,q)) . n <= ((12 * (n to_power 3)) * (log (2,n))) - 0 by XREAL_1:13;
then (max (p,q)) . n <= 12 * ((n to_power 3) * (log (2,n))) ;
hence (max (p,q)) . n <= 12 * (g . n) by A8, A50, A51; ::_thesis: (max (p,q)) . n >= 0
max (4,N) >= N by XXREAL_0:25;
then n >= N by A50, XXREAL_0:2;
hence (max (p,q)) . n >= 0 by A38; ::_thesis: verum
end;
max (p,q) is Element of Funcs (NAT,REAL) by FUNCT_2:8;
then A52: max (p,q) in Big_Oh g by A49;
for n being Element of NAT holds f . n = (p . n) + (q . n)
proof
let n be Element of NAT ; ::_thesis: f . n = (p . n) + (q . n)
thus f . n = (p . n) + (q . n) ::_thesis: verum
proof
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: f . n = (p . n) + (q . n)
hence f . n = (p . n) + (q . n) by A6, A32, A3; ::_thesis: verum
end;
supposeA53: n > 0 ; ::_thesis: f . n = (p . n) + (q . n)
then p . n = ((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2)) by A33;
then (p . n) + (q . n) = (((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2))) + (((log (2,n)) ^2) + 36) by A4, A53
.= ((((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2))) + ((log (2,n)) ^2)) + 36 ;
hence f . n = (p . n) + (q . n) by A7, A53; ::_thesis: verum
end;
end;
end;
end;
then A54: Big_Oh f = Big_Oh (p + q) by SEQ_1:7
.= Big_Oh (max (p,q)) by ASYMPT_0:9 ;
f in Big_Oh f by ASYMPT_0:10;
hence ( f = f & g = g & f in Big_Oh g ) by A54, A52, ASYMPT_0:12; ::_thesis: verum
end;
Lm2: for a being logbase Real
for f being Real_Sequence st a > 1 & ( for n being Element of NAT st n > 0 holds
f . n = log (a,n) ) holds
f is eventually-positive
proof
let a be logbase Real; ::_thesis: for f being Real_Sequence st a > 1 & ( for n being Element of NAT st n > 0 holds
f . n = log (a,n) ) holds
f is eventually-positive
let f be Real_Sequence; ::_thesis: ( a > 1 & ( for n being Element of NAT st n > 0 holds
f . n = log (a,n) ) implies f is eventually-positive )
assume that
A1: a > 1 and
A2: for n being Element of NAT st n > 0 holds
f . n = log (a,n) ; ::_thesis: f is eventually-positive
set N = [/a\];
A3: [/a\] >= a by INT_1:def_7;
A4: a > 0 by ASYMPT_0:def_1;
then A5: [/a\] > 0 by INT_1:def_7;
then reconsider N = [/a\] as Element of NAT by INT_1:3;
A6: a <> 1 by ASYMPT_0:def_1;
now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_N_+_1_holds_
f_._n_>_0
A7: log (a,N) >= log (a,a) by A1, A3, PRE_FF:10;
let n be Element of NAT ; ::_thesis: ( n >= N + 1 implies f . n > 0 )
assume A8: n >= N + 1 ; ::_thesis: f . n > 0
N + 1 > N + 0 by XREAL_1:8;
then n > N by A8, 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, A8; ::_thesis: verum
end;
hence f is eventually-positive by ASYMPT_0:def_4; ::_thesis: verum
end;
theorem :: ASYMPT_1:2
for a, b being logbase Real
for f, g being Real_Sequence st a > 1 & b > 1 & ( for n being Element of NAT st n > 0 holds
f . n = log (a,n) ) & ( for n being Element of NAT st n > 0 holds
g . n = log (b,n) ) holds
ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = g & Big_Oh s = Big_Oh s1 )
proof
let a, b be logbase Real; ::_thesis: for f, g being Real_Sequence st a > 1 & b > 1 & ( for n being Element of NAT st n > 0 holds
f . n = log (a,n) ) & ( for n being Element of NAT st n > 0 holds
g . n = log (b,n) ) holds
ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = g & Big_Oh s = Big_Oh s1 )
let f, g be Real_Sequence; ::_thesis: ( a > 1 & b > 1 & ( for n being Element of NAT st n > 0 holds
f . n = log (a,n) ) & ( for n being Element of NAT st n > 0 holds
g . n = log (b,n) ) implies ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = g & Big_Oh s = Big_Oh s1 ) )
assume that
A1: a > 1 and
A2: b > 1 and
A3: for n being Element of NAT st n > 0 holds
f . n = log (a,n) and
A4: for n being Element of NAT st n > 0 holds
g . n = log (b,n) ; ::_thesis: ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = g & Big_Oh s = Big_Oh s1 )
reconsider g = g as eventually-positive Real_Sequence by A2, A4, Lm2;
reconsider f = f as eventually-positive Real_Sequence by A1, A3, Lm2;
take f ; ::_thesis: ex s1 being eventually-positive Real_Sequence st
( f = f & s1 = g & Big_Oh f = Big_Oh s1 )
take g ; ::_thesis: ( f = f & g = g & Big_Oh f = Big_Oh 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__::_thesis:_for_x_being_set_holds_
(_(_x_in_Big_Oh_f_implies_x_in_Big_Oh_g_)_&_(_x_in_Big_Oh_g_implies_x_in_Big_Oh_f_)_)
let x be set ; ::_thesis: ( ( x in Big_Oh f implies x in Big_Oh g ) & ( x in Big_Oh g implies x in Big_Oh f ) )
hereby ::_thesis: ( x in Big_Oh g implies x in Big_Oh f )
assume x in Big_Oh f ; ::_thesis: x in Big_Oh g
then consider t being Element of Funcs (NAT,REAL) such that
A9: x = t and
A10: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
A11: c > 0 and
A12: for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) by A10;
A13: now__::_thesis:_ex_N1_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N1_holds_
(_t_._n_<=_(c_*_(log_(a,b)))_*_(g_._n)_&_t_._n_>=_0_)
take N1 = N + 1; ::_thesis: for n being Element of NAT st n >= N1 holds
( t . n <= (c * (log (a,b))) * (g . n) & t . n >= 0 )
let n be Element of NAT ; ::_thesis: ( n >= N1 implies ( t . n <= (c * (log (a,b))) * (g . n) & t . n >= 0 ) )
assume A14: n >= N1 ; ::_thesis: ( t . n <= (c * (log (a,b))) * (g . n) & t . n >= 0 )
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; ::_thesis: t . n >= 0
thus t . n >= 0 by A12, A16; ::_thesis: verum
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; ::_thesis: verum
end;
assume x in Big_Oh g ; ::_thesis: x in Big_Oh f
then consider t being Element of Funcs (NAT,REAL) such that
A17: x = t and
A18: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (g . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
A19: c > 0 and
A20: for n being Element of NAT st n >= N holds
( t . n <= c * (g . n) & t . n >= 0 ) by A18;
A21: now__::_thesis:_ex_N1_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N1_holds_
(_t_._n_<=_(c_*_(log_(b,a)))_*_(f_._n)_&_t_._n_>=_0_)
take N1 = N + 1; ::_thesis: for n being Element of NAT st n >= N1 holds
( t . n <= (c * (log (b,a))) * (f . n) & t . n >= 0 )
let n be Element of NAT ; ::_thesis: ( n >= N1 implies ( t . n <= (c * (log (b,a))) * (f . n) & t . n >= 0 ) )
assume A22: n >= N1 ; ::_thesis: ( t . n <= (c * (log (b,a))) * (f . n) & t . n >= 0 )
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; ::_thesis: t . n >= 0
thus t . n >= 0 by A20, A24; ::_thesis: verum
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; ::_thesis: verum
end;
hence ( f = f & g = g & Big_Oh f = Big_Oh g ) by TARSKI:1; ::_thesis: verum
end;
definition
let a, b, c be Real;
func seq_a^ (a,b,c) -> Real_Sequence means :Def1: :: ASYMPT_1:def 1
for n being Element of NAT holds it . n = a to_power ((b * n) + c);
existence
ex b1 being Real_Sequence st
for n being Element of NAT holds b1 . n = a to_power ((b * n) + c)
proof
deffunc H1( Element of NAT ) -> Element of REAL = a to_power ((b * $1) + c);
consider h being Function of NAT,REAL such that
A1: for n being Element of NAT holds h . n = H1(n) from FUNCT_2:sch_4();
take h ; ::_thesis: for n being Element of NAT holds h . n = a to_power ((b * n) + c)
let n be Element of NAT ; ::_thesis: h . n = a to_power ((b * n) + c)
thus h . n = a to_power ((b * n) + c) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = a to_power ((b * n) + c) ) & ( for n being Element of NAT holds b2 . n = a to_power ((b * n) + c) ) holds
b1 = b2
proof
let j, k be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds j . n = a to_power ((b * n) + c) ) & ( for n being Element of NAT holds k . n = a to_power ((b * n) + c) ) implies j = k )
assume that
A2: for n being Element of NAT holds j . n = a to_power ((b * n) + c) and
A3: for n being Element of NAT holds k . n = a to_power ((b * n) + c) ; ::_thesis: j = k
now__::_thesis:_for_n_being_Element_of_NAT_holds_j_._n_=_k_._n
let n be Element of NAT ; ::_thesis: j . n = k . n
thus j . n = a to_power ((b * n) + c) by A2
.= k . n by A3 ; ::_thesis: verum
end;
hence j = k by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines seq_a^ ASYMPT_1:def_1_:_
for a, b, c being Real
for b4 being Real_Sequence holds
( b4 = seq_a^ (a,b,c) iff for n being Element of NAT holds b4 . n = a to_power ((b * n) + c) );
registration
let a be positive Real;
let b, c be Real;
cluster seq_a^ (a,b,c) -> eventually-positive ;
coherence
seq_a^ (a,b,c) is eventually-positive
proof
take 0 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 0 <= b1 or not (seq_a^ (a,b,c)) . b1 <= 0 )
set f = seq_a^ (a,b,c);
let n be Element of NAT ; ::_thesis: ( not 0 <= n or not (seq_a^ (a,b,c)) . n <= 0 )
assume n >= 0 ; ::_thesis: not (seq_a^ (a,b,c)) . n <= 0
(seq_a^ (a,b,c)) . n = a to_power ((b * n) + c) by Def1;
hence not (seq_a^ (a,b,c)) . n <= 0 by POWER:34; ::_thesis: verum
end;
end;
Lm3: for a, b, c being Real st a > 0 & c > 0 & c <> 1 holds
a to_power b = c to_power (b * (log (c,a)))
proof
let a, b, c be Real; ::_thesis: ( a > 0 & c > 0 & c <> 1 implies a to_power b = c to_power (b * (log (c,a))) )
assume that
A1: a > 0 and
A2: c > 0 and
A3: c <> 1 ; ::_thesis: a to_power b = c to_power (b * (log (c,a)))
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 a to_power b = c to_power (b * (log (c,a))) by A2, A3, A4, POWER:def_3; ::_thesis: verum
end;
theorem :: ASYMPT_1:3
for a, b being positive Real st a < b holds
not seq_a^ (b,1,0) in Big_Oh (seq_a^ (a,1,0))
proof
let a, b be positive Real; ::_thesis: ( a < b implies not seq_a^ (b,1,0) in Big_Oh (seq_a^ (a,1,0)) )
assume A1: a < b ; ::_thesis: not seq_a^ (b,1,0) in Big_Oh (seq_a^ (a,1,0))
set g = seq_a^ (a,1,0);
set f = seq_a^ (b,1,0);
hereby ::_thesis: verum
set d = (log (2,b)) - (log (2,a));
assume seq_a^ (b,1,0) in Big_Oh (seq_a^ (a,1,0)) ; ::_thesis: contradiction
then consider s being Element of Funcs (NAT,REAL) such that
A2: s = seq_a^ (b,1,0) and
A3: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( s . n <= c * ((seq_a^ (a,1,0)) . n) & s . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
A4: c > 0 and
A5: for n being Element of NAT st n >= N holds
( s . n <= c * ((seq_a^ (a,1,0)) . n) & s . n >= 0 ) by A3;
set N0 = [/((log (2,c)) / ((log (2,b)) - (log (2,a))))\];
set N1 = max (N,[/((log (2,c)) / ((log (2,b)) - (log (2,a))))\]);
A6: max (N,[/((log (2,c)) / ((log (2,b)) - (log (2,a))))\]) >= N by XXREAL_0:25;
A7: ( max (N,[/((log (2,c)) / ((log (2,b)) - (log (2,a))))\]) = N or max (N,[/((log (2,c)) / ((log (2,b)) - (log (2,a))))\]) = [/((log (2,c)) / ((log (2,b)) - (log (2,a))))\] ) by XXREAL_0:16;
A8: max (N,[/((log (2,c)) / ((log (2,b)) - (log (2,a))))\]) >= [/((log (2,c)) / ((log (2,b)) - (log (2,a))))\] by XXREAL_0:25;
reconsider N1 = max (N,[/((log (2,c)) / ((log (2,b)) - (log (2,a))))\]) as Element of NAT by A6, A7, INT_1:3;
set n = N1 + 1;
set e = 2 to_power ((N1 + 1) * (log (2,a)));
A9: 2 to_power ((N1 + 1) * (log (2,a))) > 0 by POWER:34;
A10: [/((log (2,c)) / ((log (2,b)) - (log (2,a))))\] >= (log (2,c)) / ((log (2,b)) - (log (2,a))) by INT_1:def_7;
log (2,b) > (log (2,a)) + 0 by A1, POWER:57;
then A11: (log (2,b)) - (log (2,a)) > 0 by XREAL_1:20;
A12: N1 + 1 > N1 + 0 by XREAL_1:8;
then N1 + 1 > [/((log (2,c)) / ((log (2,b)) - (log (2,a))))\] by A8, XXREAL_0:2;
then N1 + 1 > (log (2,c)) / ((log (2,b)) - (log (2,a))) by A10, XXREAL_0:2;
then (N1 + 1) * ((log (2,b)) - (log (2,a))) > ((log (2,c)) / ((log (2,b)) - (log (2,a)))) * ((log (2,b)) - (log (2,a))) by A11, XREAL_1:68;
then (N1 + 1) * ((log (2,b)) - (log (2,a))) > log (2,c) by A11, XCMPLX_1:87;
then 2 to_power ((N1 + 1) * ((log (2,b)) - (log (2,a)))) > 2 to_power (log (2,c)) by POWER:39;
then 2 to_power (((N1 + 1) * (log (2,b))) - ((N1 + 1) * (log (2,a)))) > c by A4, POWER:def_3;
then (2 to_power ((N1 + 1) * (log (2,b)))) / (2 to_power ((N1 + 1) * (log (2,a)))) > c by POWER:29;
then ((2 to_power ((N1 + 1) * (log (2,b)))) / (2 to_power ((N1 + 1) * (log (2,a))))) * (2 to_power ((N1 + 1) * (log (2,a)))) > c * (2 to_power ((N1 + 1) * (log (2,a)))) by A9, XREAL_1:68;
then 2 to_power ((N1 + 1) * (log (2,b))) > c * (2 to_power ((N1 + 1) * (log (2,a)))) by A9, XCMPLX_1:87;
then b to_power (N1 + 1) > c * (2 to_power ((N1 + 1) * (log (2,a)))) by Lm3;
then A13: b to_power (N1 + 1) > c * (a to_power (N1 + 1)) by Lm3;
N1 + 1 > N by A6, A12, XXREAL_0:2;
then (seq_a^ (b,1,0)) . (N1 + 1) <= c * ((seq_a^ (a,1,0)) . (N1 + 1)) by A2, A5;
then b to_power ((1 * (N1 + 1)) + 0) <= c * ((seq_a^ (a,1,0)) . (N1 + 1)) by Def1;
hence contradiction by A13, Def1; ::_thesis: verum
end;
end;
definition
func seq_logn -> Real_Sequence means :Def2: :: ASYMPT_1:def 2
( it . 0 = 0 & ( for n being Element of NAT st n > 0 holds
it . n = log (2,n) ) );
existence
ex b1 being Real_Sequence st
( b1 . 0 = 0 & ( for n being Element of NAT st n > 0 holds
b1 . n = log (2,n) ) )
proof
defpred S1[ Element of NAT , Real] means ( ( $1 = 0 implies $2 = 0 ) & ( $1 > 0 implies $2 = log (2,$1) ) );
A1: for x being Element of NAT ex y being Element of REAL st S1[x,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of REAL st S1[n,y]
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
end;
end;
consider h being Function of NAT,REAL such that
A2: for x being Element of NAT holds S1[x,h . x] from FUNCT_2:sch_3(A1);
take h ; ::_thesis: ( h . 0 = 0 & ( for n being Element of NAT st n > 0 holds
h . n = log (2,n) ) )
thus h . 0 = 0 by A2; ::_thesis: for n being Element of NAT st n > 0 holds
h . n = log (2,n)
let n be Element of NAT ; ::_thesis: ( n > 0 implies h . n = log (2,n) )
thus ( n > 0 implies h . n = log (2,n) ) by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being Real_Sequence st b1 . 0 = 0 & ( for n being Element of NAT st n > 0 holds
b1 . n = log (2,n) ) & b2 . 0 = 0 & ( for n being Element of NAT st n > 0 holds
b2 . n = log (2,n) ) holds
b1 = b2
proof
let j, k be Real_Sequence; ::_thesis: ( j . 0 = 0 & ( for n being Element of NAT st n > 0 holds
j . n = log (2,n) ) & k . 0 = 0 & ( for n being Element of NAT st n > 0 holds
k . n = log (2,n) ) implies j = k )
assume that
A3: j . 0 = 0 and
A4: for n being Element of NAT st n > 0 holds
j . n = log (2,n) and
A5: k . 0 = 0 and
A6: for n being Element of NAT st n > 0 holds
k . n = log (2,n) ; ::_thesis: j = k
now__::_thesis:_for_n_being_Element_of_NAT_holds_j_._n_=_k_._n
let n be Element of NAT ; ::_thesis: j . b1 = k . b1
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: j . b1 = k . b1
hence j . n = k . n by A3, A5; ::_thesis: verum
end;
supposeA7: n > 0 ; ::_thesis: j . b1 = k . b1
then j . n = log (2,n) by A4;
hence j . n = k . n by A6, A7; ::_thesis: verum
end;
end;
end;
hence j = k by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines seq_logn ASYMPT_1:def_2_:_
for b1 being Real_Sequence holds
( b1 = seq_logn iff ( b1 . 0 = 0 & ( for n being Element of NAT st n > 0 holds
b1 . n = log (2,n) ) ) );
definition
let a be Real;
func seq_n^ a -> Real_Sequence means :Def3: :: ASYMPT_1:def 3
( it . 0 = 0 & ( for n being Element of NAT st n > 0 holds
it . n = n to_power a ) );
existence
ex b1 being Real_Sequence st
( b1 . 0 = 0 & ( for n being Element of NAT st n > 0 holds
b1 . n = n to_power a ) )
proof
defpred S1[ Element of NAT , Real] means ( ( $1 = 0 implies $2 = 0 ) & ( $1 > 0 implies $2 = $1 to_power a ) );
A1: for x being Element of NAT ex y being Element of REAL st S1[x,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of REAL st S1[n,y]
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
end;
end;
consider h being Function of NAT,REAL such that
A2: for x being Element of NAT holds S1[x,h . x] from FUNCT_2:sch_3(A1);
take h ; ::_thesis: ( h . 0 = 0 & ( for n being Element of NAT st n > 0 holds
h . n = n to_power a ) )
thus h . 0 = 0 by A2; ::_thesis: for n being Element of NAT st n > 0 holds
h . n = n to_power a
let n be Element of NAT ; ::_thesis: ( n > 0 implies h . n = n to_power a )
thus ( n > 0 implies h . n = n to_power a ) by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being Real_Sequence st b1 . 0 = 0 & ( for n being Element of NAT st n > 0 holds
b1 . n = n to_power a ) & b2 . 0 = 0 & ( for n being Element of NAT st n > 0 holds
b2 . n = n to_power a ) holds
b1 = b2
proof
let j, k be Real_Sequence; ::_thesis: ( j . 0 = 0 & ( for n being Element of NAT st n > 0 holds
j . n = n to_power a ) & k . 0 = 0 & ( for n being Element of NAT st n > 0 holds
k . n = n to_power a ) implies j = k )
assume that
A3: j . 0 = 0 and
A4: for n being Element of NAT st n > 0 holds
j . n = n to_power a and
A5: k . 0 = 0 and
A6: for n being Element of NAT st n > 0 holds
k . n = n to_power a ; ::_thesis: j = k
now__::_thesis:_for_n_being_Element_of_NAT_holds_j_._n_=_k_._n
let n be Element of NAT ; ::_thesis: j . b1 = k . b1
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: j . b1 = k . b1
hence j . n = k . n by A3, A5; ::_thesis: verum
end;
supposeA7: n > 0 ; ::_thesis: j . b1 = k . b1
then j . n = n to_power a by A4;
hence j . n = k . n by A6, A7; ::_thesis: verum
end;
end;
end;
hence j = k by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines seq_n^ ASYMPT_1:def_3_:_
for a being Real
for b2 being Real_Sequence holds
( b2 = seq_n^ a iff ( b2 . 0 = 0 & ( for n being Element of NAT st n > 0 holds
b2 . n = n to_power a ) ) );
registration
cluster seq_logn -> eventually-positive ;
coherence
seq_logn is eventually-positive
proof
take 2 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 2 <= b1 or not seq_logn . b1 <= 0 )
set f = seq_logn ;
let n be Element of NAT ; ::_thesis: ( not 2 <= n or not seq_logn . n <= 0 )
assume A1: n >= 2 ; ::_thesis: not seq_logn . n <= 0
then A2: log (2,n) >= log (2,2) by PRE_FF:10;
seq_logn . n = log (2,n) by A1, Def2;
hence not seq_logn . n <= 0 by A2, POWER:52; ::_thesis: verum
end;
end;
registration
let a be Real;
cluster seq_n^ a -> eventually-positive ;
coherence
seq_n^ a is eventually-positive
proof
take 1 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 1 <= b1 or not (seq_n^ a) . b1 <= 0 )
set f = seq_n^ a;
let n be Element of NAT ; ::_thesis: ( not 1 <= n or not (seq_n^ a) . n <= 0 )
assume A1: n >= 1 ; ::_thesis: not (seq_n^ a) . n <= 0
then (seq_n^ a) . n = n to_power a by Def3;
hence not (seq_n^ a) . n <= 0 by A1, POWER:34; ::_thesis: verum
end;
end;
Lm4: for f, g being Real_Sequence
for n being Element of NAT holds (f /" g) . n = (f . n) / (g . n)
proof
let f, g be Real_Sequence; ::_thesis: for n being Element of NAT holds (f /" g) . n = (f . n) / (g . n)
let n be Element of NAT ; ::_thesis: (f /" g) . n = (f . n) / (g . n)
thus (f /" g) . n = (f . n) * ((g ") . n) by SEQ_1:8
.= (f . n) * ((g . n) ") by VALUED_1:10
.= (f . n) / (g . n) ; ::_thesis: verum
end;
Lm5: for f, g being eventually-nonnegative Real_Sequence holds
( ( f in Big_Oh g & g in Big_Oh f ) iff Big_Oh f = Big_Oh g )
proof
let f, g be eventually-nonnegative Real_Sequence; ::_thesis: ( ( f in Big_Oh g & g in Big_Oh f ) iff Big_Oh f = Big_Oh g )
hereby ::_thesis: ( Big_Oh f = Big_Oh g implies ( f in Big_Oh g & g in Big_Oh f ) )
assume that
A1: f in Big_Oh g and
A2: g in Big_Oh f ; ::_thesis: Big_Oh f = Big_Oh g
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; ::_thesis: verum
end;
thus ( Big_Oh f = Big_Oh g implies ( f in Big_Oh g & g in Big_Oh f ) ) by ASYMPT_0:10; ::_thesis: verum
end;
theorem Th4: :: ASYMPT_1:4
for f, g being eventually-nonnegative Real_Sequence holds
( Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g iff ( f in Big_Oh g & not f in Big_Omega g ) )
proof
let f, g be eventually-nonnegative Real_Sequence; ::_thesis: ( 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 ) )
hereby ::_thesis: ( f in Big_Oh g & not f in Big_Omega g implies ( Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g ) )
assume that
A1: Big_Oh f c= Big_Oh g and
A2: not Big_Oh f = Big_Oh g ; ::_thesis: ( f in Big_Oh g & not f in Big_Omega g )
A3: f in Big_Oh f by ASYMPT_0:10;
now__::_thesis:_not_f_in_Big_Omega_g
assume f in Big_Omega g ; ::_thesis: contradiction
then g in Big_Oh f by ASYMPT_0:19;
hence contradiction by A1, A2, A3, Lm5; ::_thesis: verum
end;
hence ( f in Big_Oh g & not f in Big_Omega g ) by A1, A3; ::_thesis: verum
end;
assume that
A4: f in Big_Oh g and
A5: not f in Big_Omega g ; ::_thesis: ( Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g )
now__::_thesis:_for_x_being_set_st_x_in_Big_Oh_f_holds_
x_in_Big_Oh_g
let x be set ; ::_thesis: ( x in Big_Oh f implies x in Big_Oh g )
assume x in Big_Oh f ; ::_thesis: x in Big_Oh g
then consider t being Element of Funcs (NAT,REAL) such that
A6: x = t and
A7: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
c > 0 and
A8: for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) by A7;
now__::_thesis:_ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
t_._n_>=_0
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
t . n >= 0
let n be Element of NAT ; ::_thesis: ( n >= N implies t . n >= 0 )
assume n >= N ; ::_thesis: t . n >= 0
hence t . n >= 0 by A8; ::_thesis: verum
end;
then A9: t is eventually-nonnegative by ASYMPT_0:def_2;
t in Big_Oh f by A7;
hence x in Big_Oh g by A4, A6, A9, ASYMPT_0:12; ::_thesis: verum
end;
hence Big_Oh f c= Big_Oh g by TARSKI:def_3; ::_thesis: not Big_Oh f = Big_Oh g
assume Big_Oh f = Big_Oh g ; ::_thesis: contradiction
then g in Big_Oh f by Lm5;
hence contradiction by A5, ASYMPT_0:19; ::_thesis: verum
end;
Lm6: for a, b, c being real number st 0 < a & a <= b & c >= 0 holds
a to_power c <= b to_power c
proof
let a, b, c be real number ; ::_thesis: ( 0 < a & a <= b & c >= 0 implies a to_power c <= b to_power c )
assume that
A1: 0 < a and
A2: a <= b and
A3: c >= 0 ; ::_thesis: a to_power c <= b to_power c
percases ( c = 0 or c > 0 ) by A3;
supposeA4: c = 0 ; ::_thesis: a to_power c <= b to_power c
then a to_power c = 1 by POWER:24;
hence a to_power c <= b to_power c by A4, POWER:24; ::_thesis: verum
end;
supposeA5: c > 0 ; ::_thesis: a to_power c <= b to_power c
percases ( a = b or a < b ) by A2, XXREAL_0:1;
suppose a = b ; ::_thesis: a to_power c <= b to_power c
hence a to_power c <= b to_power c ; ::_thesis: verum
end;
suppose a < b ; ::_thesis: a to_power c <= b to_power c
hence a to_power c <= b to_power c by A1, A5, POWER:37; ::_thesis: verum
end;
end;
end;
end;
end;
Lm7: for n being Nat st n >= 4 holds
(2 * n) + 3 < 2 to_power n
proof
defpred S1[ Nat] means (2 * $1) + 3 < 2 to_power $1;
A1: for k being Nat st k >= 4 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 4 & S1[k] implies S1[k + 1] )
assume that
A2: k >= 4 and
A3: (2 * k) + 3 < 2 to_power k ; ::_thesis: S1[k + 1]
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 S1[k + 1] by POWER:27; ::_thesis: verum
end;
A5: S1[4] by POWER:62;
for n being Nat st n >= 4 holds
S1[n] from NAT_1:sch_8(A5, A1);
hence for n being Nat st n >= 4 holds
(2 * n) + 3 < 2 to_power n ; ::_thesis: verum
end;
Lm8: for n being Element of NAT st n >= 6 holds
(n + 1) ^2 < 2 to_power n
proof
defpred S1[ Nat] means ($1 + 1) ^2 < 2 to_power $1;
A1: for k being Nat st k >= 6 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 6 & S1[k] implies S1[k + 1] )
assume that
A2: k >= 6 and
A3: (k + 1) ^2 < 2 to_power k ; ::_thesis: S1[k + 1]
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 S1[k + 1] by POWER:27; ::_thesis: verum
end;
A5: S1[6] by POWER:64;
for n being Nat st n >= 6 holds
S1[n] from NAT_1:sch_8(A5, A1);
hence for n being Element of NAT st n >= 6 holds
(n + 1) ^2 < 2 to_power n ; ::_thesis: verum
end;
Lm9: for c being Real st c > 6 holds
c ^2 < 2 to_power c
proof
A1: 5 = 6 - 1 ;
let c be Real; ::_thesis: ( c > 6 implies c ^2 < 2 to_power c )
assume A2: c > 6 ; ::_thesis: c ^2 < 2 to_power c
set i0 = [\c/];
set i1 = [/c\];
percases ( [\c/] = [/c\] or not [\c/] = [/c\] ) ;
suppose [\c/] = [/c\] ; ::_thesis: c ^2 < 2 to_power c
then c is Integer by INT_1:34;
then reconsider c = 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 c ^2 < 2 to_power c by A3, XXREAL_0:2; ::_thesis: verum
end;
suppose not [\c/] = [/c\] ; ::_thesis: c ^2 < 2 to_power c
then A4: [\c/] + 1 = [/c\] by INT_1:41;
then A5: [\c/] = [/c\] - 1 ;
A6: [/c\] >= c by INT_1:def_7;
then reconsider i1 = [/c\] as Element of NAT by A2, INT_1:3;
i1 > 6 by A2, A6, XXREAL_0:2;
then A7: [\c/] > 5 by A1, A5, XREAL_1:9;
then reconsider i0 = [\c/] 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 c ^2 < 2 to_power c by A8, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
Lm10: for e being positive Real
for f being Real_Sequence st ( for n being Element of NAT 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; ::_thesis: for f being Real_Sequence st ( for n being Element of NAT 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 )
let f be Real_Sequence; ::_thesis: ( ( for n being Element of NAT st n > 0 holds
f . n = log (2,(n to_power e)) ) implies ( f /" (seq_n^ e) is convergent & lim (f /" (seq_n^ e)) = 0 ) )
assume A1: for n being Element of NAT st n > 0 holds
f . n = log (2,(n to_power e)) ; ::_thesis: ( f /" (seq_n^ e) is convergent & lim (f /" (seq_n^ e)) = 0 )
set g = seq_n^ e;
set h = f /" (seq_n^ e);
A2: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
abs_(((f_/"_(seq_n^_e))_._n)_-_0)_<_p
let p be real number ; ::_thesis: ( p > 0 implies ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs (((f /" (seq_n^ e)) . n) - 0) < p )
reconsider p1 = p as Real by XREAL_0:def_1;
set i0 = [/((7 / p1) to_power (1 / e))\];
set i1 = [/((p1 to_power (- (2 / e))) + 1)\];
set N = max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2);
A3: max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) >= max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\]) by XXREAL_0:25;
A4: max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) is Integer
proof
percases ( max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) = max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\]) or max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) = 2 ) by XXREAL_0:16;
suppose max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) = max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\]) ; ::_thesis: max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) is Integer
hence max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) is Integer by XXREAL_0:16; ::_thesis: verum
end;
suppose max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) = 2 ; ::_thesis: max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) is Integer
hence max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) is Integer ; ::_thesis: verum
end;
end;
end;
A5: (p to_power (- (2 / e))) + 1 > (p to_power (- (2 / e))) + 0 by XREAL_1:8;
[/((p1 to_power (- (2 / e))) + 1)\] >= (p to_power (- (2 / e))) + 1 by INT_1:def_7;
then A6: [/((p1 to_power (- (2 / e))) + 1)\] > p to_power (- (2 / e)) by A5, XXREAL_0:2;
assume A7: p > 0 ; ::_thesis: ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs (((f /" (seq_n^ e)) . n) - 0) < p
then A8: p1 to_power 2 > 0 by POWER:34;
max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\]) >= [/((p1 to_power (- (2 / e))) + 1)\] by XXREAL_0:25;
then A9: max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) >= [/((p1 to_power (- (2 / e))) + 1)\] by A3, XXREAL_0:2;
A10: [/((7 / p1) to_power (1 / e))\] >= (7 / p) to_power (1 / e) by INT_1:def_7;
max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\]) >= [/((7 / p1) to_power (1 / e))\] by XXREAL_0:25;
then A11: max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) >= [/((7 / p1) to_power (1 / e))\] by A3, XXREAL_0:2;
A12: max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) >= 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;
reconsider N = max ((max ([/((7 / p1) to_power (1 / e))\],[/((p1 to_power (- (2 / e))) + 1)\])),2) as Element of NAT by A12, A4, INT_1:3;
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
abs (((f /" (seq_n^ e)) . n) - 0) < p
let n be Element of NAT ; ::_thesis: ( n >= N implies abs (((f /" (seq_n^ e)) . n) - 0) < p )
set c = p1 * (n to_power e);
assume A16: n >= N ; ::_thesis: abs (((f /" (seq_n^ e)) . n) - 0) < p
then n >= [/((7 / p1) to_power (1 / e))\] 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 >= [/((p1 to_power (- (2 / e))) + 1)\] 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 * (p1 * (n to_power e))) < 1 / 1 by XREAL_1:88;
2 to_power (p1 * (n to_power e)) > 0 by POWER:34;
then A19: (2 to_power (p1 * (n to_power e))) / ((p1 * (n to_power e)) * p) < (2 to_power (p1 * (n to_power e))) * 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 p1 * (n to_power e) < (2 to_power (p1 * (n to_power e))) / (p1 * (n to_power e)) by A17, XREAL_1:81;
then n to_power e < ((2 to_power (p1 * (n to_power e))) / (p1 * (n to_power e))) / p by A7, XREAL_1:81;
then n to_power e < (2 to_power (p1 * (n to_power e))) / ((p1 * (n to_power e)) * p) by XCMPLX_1:78;
then n to_power e < 2 to_power (p1 * (n to_power e)) by A19, XXREAL_0:2;
then log (2,(n to_power e)) < log (2,(2 to_power (p1 * (n to_power e)))) by A20, POWER:57;
then log (2,(n to_power e)) < (p1 * (n to_power e)) * (log (2,2)) by POWER:55;
then log (2,(n to_power e)) < (p1 * (n to_power e)) * 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;
(f /" (seq_n^ e)) . n = (f . n) / ((seq_n^ e) . n) by Lm4
.= (log (2,(n to_power e))) / ((seq_n^ e) . n) by A1, A12, A16
.= (log (2,(n to_power e))) / (n to_power e) by A12, A16, Def3 ;
hence abs (((f /" (seq_n^ e)) . n) - 0) < p by A20, A21, A22, ABSVALUE:def_1; ::_thesis: verum
end;
hence f /" (seq_n^ e) is convergent by SEQ_2:def_6; ::_thesis: lim (f /" (seq_n^ e)) = 0
hence lim (f /" (seq_n^ e)) = 0 by A2, SEQ_2:def_7; ::_thesis: verum
end;
Lm11: for e being Real st e > 0 holds
( seq_logn /" (seq_n^ e) is convergent & lim (seq_logn /" (seq_n^ e)) = 0 )
proof
set f = seq_logn ;
let e be Real; ::_thesis: ( e > 0 implies ( seq_logn /" (seq_n^ e) is convergent & lim (seq_logn /" (seq_n^ e)) = 0 ) )
assume e > 0 ; ::_thesis: ( seq_logn /" (seq_n^ e) is convergent & lim (seq_logn /" (seq_n^ e)) = 0 )
then reconsider e = e as positive Real ;
set g = seq_n^ e;
set h = seq_logn /" (seq_n^ e);
ex s being Real_Sequence st
( s . 0 = 0 & ( for n being Element of NAT st n > 0 holds
s . n = log (2,(n to_power e)) ) )
proof
defpred S1[ Element of NAT , Real] means ( ( $1 = 0 implies $2 = 0 ) & ( $1 > 0 implies $2 = log (2,($1 to_power e)) ) );
A1: for x being Element of NAT ex y being Element of REAL st S1[x,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of REAL st S1[n,y]
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
end;
end;
consider h being Function of NAT,REAL such that
A2: for x being Element of NAT holds S1[x,h . x] from FUNCT_2:sch_3(A1);
take h ; ::_thesis: ( h . 0 = 0 & ( for n being Element of NAT st n > 0 holds
h . n = log (2,(n to_power e)) ) )
thus h . 0 = 0 by A2; ::_thesis: for n being Element of NAT st n > 0 holds
h . n = log (2,(n to_power e))
let n be Element of NAT ; ::_thesis: ( n > 0 implies h . n = log (2,(n to_power e)) )
thus ( n > 0 implies h . n = log (2,(n to_power e)) ) by A2; ::_thesis: verum
end;
then consider p being Real_Sequence such that
A3: p . 0 = 0 and
A4: for n being Element of NAT st n > 0 holds
p . n = log (2,(n to_power e)) ;
set q = p /" (seq_n^ e);
A5: p /" (seq_n^ e) is convergent by A4, Lm10;
A6: 1 = e / e by XCMPLX_1:60
.= e * (1 / e) ;
A7: for n being Element of NAT holds (seq_logn /" (seq_n^ e)) . n = (1 / e) * ((p /" (seq_n^ e)) . n)
proof
let n be Element of NAT ; ::_thesis: (seq_logn /" (seq_n^ e)) . n = (1 / e) * ((p /" (seq_n^ e)) . n)
A8: (seq_logn /" (seq_n^ e)) . n = (seq_logn . n) / ((seq_n^ e) . n) by Lm4;
A9: (p /" (seq_n^ e)) . n = (p . n) / ((seq_n^ e) . n) by Lm4;
percases ( n = 0 or n > 0 ) ;
supposeA10: n = 0 ; ::_thesis: (seq_logn /" (seq_n^ e)) . n = (1 / e) * ((p /" (seq_n^ e)) . n)
then (seq_logn /" (seq_n^ e)) . n = 0 / ((seq_n^ e) . n) by A8, Def2
.= 0 * (1 / e) ;
hence (seq_logn /" (seq_n^ e)) . n = (1 / e) * ((p /" (seq_n^ e)) . n) by A3, A9, A10; ::_thesis: verum
end;
supposeA11: n > 0 ; ::_thesis: (seq_logn /" (seq_n^ e)) . n = (1 / e) * ((p /" (seq_n^ e)) . n)
then A12: n to_power e > 0 by POWER:34;
(seq_logn /" (seq_n^ e)) . n = (log (2,n)) / ((seq_n^ e) . n) by A8, A11, Def2
.= (log (2,(n to_power (e * (1 / e))))) / ((seq_n^ e) . n) by A6, POWER:25
.= (log (2,((n to_power e) to_power (1 / e)))) / ((seq_n^ e) . n) by A11, POWER:33
.= ((1 / e) * (log (2,(n to_power e)))) / ((seq_n^ e) . n) by A12, POWER:55
.= ((1 / e) * (log (2,(n to_power e)))) * (((seq_n^ e) . n) ")
.= (1 / e) * ((log (2,(n to_power e))) * (((seq_n^ e) . n) "))
.= (1 / e) * ((log (2,(n to_power e))) / ((seq_n^ e) . n)) ;
hence (seq_logn /" (seq_n^ e)) . n = (1 / e) * ((p /" (seq_n^ e)) . n) by A4, A9, A11; ::_thesis: verum
end;
end;
end;
then A13: seq_logn /" (seq_n^ e) = (1 / e) (#) (p /" (seq_n^ e)) by SEQ_1:9;
A14: lim (p /" (seq_n^ e)) = 0 by A4, Lm10;
lim (seq_logn /" (seq_n^ e)) = lim ((1 / e) (#) (p /" (seq_n^ e))) by A7, SEQ_1:9
.= (1 / e) * 0 by A5, A14, SEQ_2:8 ;
hence ( seq_logn /" (seq_n^ e) is convergent & lim (seq_logn /" (seq_n^ e)) = 0 ) by A5, A13, SEQ_2:7; ::_thesis: verum
end;
theorem Th5: :: ASYMPT_1:5
( 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 (seq_logn /" (seq_n^ (1 / 2))) = 0 by Lm11;
A2: seq_logn /" (seq_n^ (1 / 2)) is convergent by Lm11;
then not seq_n^ (1 / 2) in Big_Oh seq_logn by A1, ASYMPT_0:16;
then A3: not seq_logn in Big_Omega (seq_n^ (1 / 2)) by ASYMPT_0:19;
seq_logn in Big_Oh (seq_n^ (1 / 2)) by A2, A1, ASYMPT_0:16;
hence ( Big_Oh seq_logn c= Big_Oh (seq_n^ (1 / 2)) & not Big_Oh seq_logn = Big_Oh (seq_n^ (1 / 2)) ) by A3, Th4; ::_thesis: verum
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 ( seq_n^ (1 / 2) in Big_Omega seq_logn & not seq_logn in Big_Omega (seq_n^ (1 / 2)) ) by Th4, Th5, ASYMPT_0:19; ::_thesis: verum
end;
Lm12: for f being Real_Sequence
for N being Element of NAT st ( for n being Element of NAT st n <= N holds
f . n >= 0 ) holds
Sum (f,N) >= 0
proof
let f be Real_Sequence; ::_thesis: for N being Element of NAT st ( for n being Element of NAT st n <= N holds
f . n >= 0 ) holds
Sum (f,N) >= 0
defpred S1[ Element of NAT ] means ( ( for n being Element of NAT st n <= $1 holds
f . n >= 0 ) implies Sum (f,$1) >= 0 );
A1: for N being Element of NAT st S1[N] holds
S1[N + 1]
proof
let N be Element of NAT ; ::_thesis: ( S1[N] implies S1[N + 1] )
assume A2: ( ( for n being Element of NAT st n <= N holds
f . n >= 0 ) implies Sum (f,N) >= 0 ) ; ::_thesis: S1[N + 1]
assume A3: for n being Element of NAT st n <= N + 1 holds
f . n >= 0 ; ::_thesis: Sum (f,(N + 1)) >= 0
A4: now__::_thesis:_for_n_being_Element_of_NAT_st_n_<=_N_holds_
f_._n_>=_0
let n be Element of NAT ; ::_thesis: ( n <= N implies f . n >= 0 )
assume n <= N ; ::_thesis: f . n >= 0
then n + 0 <= N + 1 by XREAL_1:7;
hence f . n >= 0 by A3; ::_thesis: verum
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 Sum (f,(N + 1)) >= 0 by SERIES_1:def_5; ::_thesis: verum
end;
A5: S1[ 0 ]
proof
assume for n being Element of NAT st n <= 0 holds
f . n >= 0 ; ::_thesis: Sum (f,0) >= 0
then f . 0 >= 0 ;
then (Partial_Sums f) . 0 >= 0 by SERIES_1:def_1;
hence Sum (f,0) >= 0 by SERIES_1:def_5; ::_thesis: verum
end;
for N being Element of NAT holds S1[N] from NAT_1:sch_1(A5, A1);
hence for N being Element of NAT st ( for n being Element of NAT st n <= N holds
f . n >= 0 ) holds
Sum (f,N) >= 0 ; ::_thesis: verum
end;
Lm13: for f, g being Real_Sequence
for N being Element of NAT st ( for n being Element of NAT st n <= N holds
f . n <= g . n ) holds
Sum (f,N) <= Sum (g,N)
proof
let f, g be Real_Sequence; ::_thesis: for N being Element of NAT st ( for n being Element of NAT st n <= N holds
f . n <= g . n ) holds
Sum (f,N) <= Sum (g,N)
defpred S1[ Element of NAT ] means ( ( for n being Element of NAT st n <= $1 holds
f . n <= g . n ) implies Sum (f,$1) <= Sum (g,$1) );
A1: for N being Element of NAT st S1[N] holds
S1[N + 1]
proof
let N be Element of NAT ; ::_thesis: ( S1[N] implies S1[N + 1] )
assume A2: ( ( for n being Element of NAT st n <= N holds
f . n <= g . n ) implies Sum (f,N) <= Sum (g,N) ) ; ::_thesis: S1[N + 1]
assume A3: for n being Element of NAT st n <= N + 1 holds
f . n <= g . n ; ::_thesis: Sum (f,(N + 1)) <= Sum (g,(N + 1))
A4: now__::_thesis:_for_n_being_Element_of_NAT_st_n_<=_N_holds_
f_._n_<=_g_._n
let n be Element of NAT ; ::_thesis: ( n <= N implies f . n <= g . n )
assume n <= N ; ::_thesis: f . n <= g . n
then n + 0 <= N + 1 by XREAL_1:7;
hence f . n <= g . n by A3; ::_thesis: verum
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 Sum (f,(N + 1)) <= Sum (g,(N + 1)) by SERIES_1:def_5; ::_thesis: verum
end;
A5: S1[ 0 ]
proof
assume for n being Element of NAT st n <= 0 holds
f . n <= g . n ; ::_thesis: Sum (f,0) <= Sum (g,0)
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 Sum (f,0) <= Sum (g,0) by SERIES_1:def_5; ::_thesis: verum
end;
for N being Element of NAT holds S1[N] from NAT_1:sch_1(A5, A1);
hence for N being Element of NAT st ( for n being Element of NAT st n <= N holds
f . n <= g . n ) holds
Sum (f,N) <= Sum (g,N) ; ::_thesis: verum
end;
Lm14: for f being Real_Sequence
for b being Real st f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = b ) holds
for N being Element of NAT holds Sum (f,N) = b * N
proof
let f be Real_Sequence; ::_thesis: for b being Real st f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = b ) holds
for N being Element of NAT holds Sum (f,N) = b * N
let b be Real; ::_thesis: ( f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = b ) implies for N being Element of NAT holds Sum (f,N) = b * N )
defpred S1[ Element of NAT ] means Sum (f,$1) = b * $1;
assume that
A1: f . 0 = 0 and
A2: for n being Element of NAT st n > 0 holds
f . n = b ; ::_thesis: for N being Element of NAT holds Sum (f,N) = b * N
A3: for N being Element of NAT st S1[N] holds
S1[N + 1]
proof
let N be Element of NAT ; ::_thesis: ( S1[N] implies S1[N + 1] )
assume A4: Sum (f,N) = b * N ; ::_thesis: S1[N + 1]
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 S1[N + 1] ; ::_thesis: verum
end;
(Partial_Sums f) . 0 = 0 by A1, SERIES_1:def_1;
then A5: S1[ 0 ] by SERIES_1:def_5;
for N being Element of NAT holds S1[N] from NAT_1:sch_1(A5, A3);
hence for N being Element of NAT holds Sum (f,N) = b * N ; ::_thesis: verum
end;
Lm15: for f being Real_Sequence
for N, M being Nat holds (Sum (f,N,M)) + (f . (N + 1)) = Sum (f,(N + 1),M)
proof
let f be Real_Sequence; ::_thesis: for N, M being Nat holds (Sum (f,N,M)) + (f . (N + 1)) = Sum (f,(N + 1),M)
let N, M be Nat; ::_thesis: (Sum (f,N,M)) + (f . (N + 1)) = Sum (f,(N + 1),M)
A1: N in NAT by ORDINAL1:def_12;
(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 A1, 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 (Sum (f,N,M)) + (f . (N + 1)) = Sum (f,(N + 1),M) ; ::_thesis: verum
end;
Lm16: for f, g being Real_Sequence
for M, N being Element of NAT st N >= M + 1 & ( for n being Element of NAT st M + 1 <= n & n <= N holds
f . n <= g . n ) holds
Sum (f,N,M) <= Sum (g,N,M)
proof
let f, g be Real_Sequence; ::_thesis: for M, N being Element of NAT st N >= M + 1 & ( for n being Element of NAT st M + 1 <= n & n <= N holds
f . n <= g . n ) holds
Sum (f,N,M) <= Sum (g,N,M)
let M be Element of NAT ; ::_thesis: for N being Element of NAT st N >= M + 1 & ( for n being Element of NAT st M + 1 <= n & n <= N holds
f . n <= g . n ) holds
Sum (f,N,M) <= Sum (g,N,M)
defpred S1[ Nat] means ( ( for n being Element of NAT st M + 1 <= n & n <= $1 holds
f . n <= g . n ) implies Sum (f,$1,M) <= Sum (g,$1,M) );
A1: for N1 being Nat st N1 >= M + 1 & S1[N1] holds
S1[N1 + 1]
proof
let N1 be Nat; ::_thesis: ( N1 >= M + 1 & S1[N1] implies S1[N1 + 1] )
assume that
A2: N1 >= M + 1 and
A3: ( ( for n being Element of NAT st M + 1 <= n & n <= N1 holds
f . n <= g . n ) implies Sum (f,N1,M) <= Sum (g,N1,M) ) ; ::_thesis: S1[N1 + 1]
assume A4: for n being Element of NAT st M + 1 <= n & n <= N1 + 1 holds
f . n <= g . n ; ::_thesis: Sum (f,(N1 + 1),M) <= Sum (g,(N1 + 1),M)
A5: now__::_thesis:_for_n_being_Element_of_NAT_st_M_+_1_<=_n_&_n_<=_N1_holds_
f_._n_<=_g_._n
let n be Element of NAT ; ::_thesis: ( M + 1 <= n & n <= N1 implies f . n <= g . n )
assume that
A6: M + 1 <= n and
A7: n <= N1 ; ::_thesis: f . n <= g . n
n + 0 <= N1 + 1 by A7, XREAL_1:7;
hence f . n <= g . n by A4, A6; ::_thesis: verum
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 Sum (f,(N1 + 1),M) <= Sum (g,(N1 + 1),M) by Lm15; ::_thesis: verum
end;
A8: S1[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 being Element of NAT st M + 1 <= n & n <= M + 1 holds
f . n <= g . n ; ::_thesis: Sum (f,(M + 1),M) <= Sum (g,(M + 1),M)
hence Sum (f,(M + 1),M) <= Sum (g,(M + 1),M) by A10, A9; ::_thesis: verum
end;
for N being Nat st N >= M + 1 holds
S1[N] from NAT_1:sch_8(A8, A1);
hence for N being Element of NAT st N >= M + 1 & ( for n being Element of NAT st M + 1 <= n & n <= N holds
f . n <= g . n ) holds
Sum (f,N,M) <= Sum (g,N,M) ; ::_thesis: verum
end;
Lm17: for n being Element of NAT holds [/(n / 2)\] <= n
proof
let n be Element of NAT ; ::_thesis: [/(n / 2)\] <= n
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: [/(n / 2)\] <= n
hence [/(n / 2)\] <= n by INT_1:30; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: [/(n / 2)\] <= n
then A1: n >= 0 + 1 by NAT_1:13;
percases ( n = 1 or n > 1 ) by A1, XXREAL_0:1;
supposeA2: n = 1 ; ::_thesis: [/(n / 2)\] <= n
now__::_thesis:_not_[/(1_/_2)\]_>_1
assume [/(1 / 2)\] > 1 ; ::_thesis: contradiction
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; ::_thesis: verum
end;
hence [/(n / 2)\] <= n by A2; ::_thesis: verum
end;
suppose n > 1 ; ::_thesis: [/(n / 2)\] <= n
then A4: n >= 1 + 1 by NAT_1:13;
A5: now__::_thesis:_not_(n_/_2)_+_1_>_n
assume (n / 2) + 1 > n ; ::_thesis: contradiction
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; ::_thesis: verum
end;
[/(n / 2)\] < (n / 2) + 1 by INT_1:def_7;
hence [/(n / 2)\] <= n by A5, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
end;
end;
Lm18: for f being Real_Sequence
for b being Real
for N being Element of NAT st f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = b ) holds
for M being Element of NAT holds Sum (f,N,M) = b * (N - M)
proof
let f be Real_Sequence; ::_thesis: for b being Real
for N being Element of NAT st f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = b ) holds
for M being Element of NAT holds Sum (f,N,M) = b * (N - M)
let b be Real; ::_thesis: for N being Element of NAT st f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = b ) holds
for M being Element of NAT holds Sum (f,N,M) = b * (N - M)
let N be Element of NAT ; ::_thesis: ( f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = b ) implies for M being Element of NAT holds Sum (f,N,M) = b * (N - M) )
assume that
A1: f . 0 = 0 and
A2: for n being Element of NAT st n > 0 holds
f . n = b ; ::_thesis: for M being Element of NAT holds Sum (f,N,M) = b * (N - M)
defpred S1[ Element of NAT ] means Sum (f,N,$1) = b * (N - $1);
A3: for M being Element of NAT st S1[M] holds
S1[M + 1]
proof
let M be Element of NAT ; ::_thesis: ( S1[M] implies S1[M + 1] )
assume A4: Sum (f,N,M) = b * (N - M) ; ::_thesis: S1[M + 1]
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 S1[M + 1] ; ::_thesis: verum
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: S1[ 0 ] ;
for M being Element of NAT holds S1[M] from NAT_1:sch_1(A5, A3);
hence for M being Element of NAT holds Sum (f,N,M) = b * (N - M) ; ::_thesis: verum
end;
theorem :: ASYMPT_1:7
for f being Real_Sequence
for k being Element of NAT st ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) holds
f in Big_Theta (seq_n^ (k + 1))
proof
let f be Real_Sequence; ::_thesis: for k being Element of NAT st ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) holds
f in Big_Theta (seq_n^ (k + 1))
let k be Element of NAT ; ::_thesis: ( ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) implies f in Big_Theta (seq_n^ (k + 1)) )
assume A1: for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ; ::_thesis: f in Big_Theta (seq_n^ (k + 1))
set g = seq_n^ (k + 1);
A2: f is Element of Funcs (NAT,REAL) by FUNCT_2:8;
A3: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_1_holds_
(_((2_to_power_(k_+_1))_")_*_((seq_n^_(k_+_1))_._n)_<=_f_._n_&_f_._n_>=_0_)
set p = seq_n^ k;
let n be Element of NAT ; ::_thesis: ( n >= 1 implies ( ((2 to_power (k + 1)) ") * ((seq_n^ (k + 1)) . n) <= f . n & f . n >= 0 ) )
set n1 = [/(n / 2)\];
ex s being Real_Sequence st
( s . 0 = 0 & ( for m being Element of NAT st m > 0 holds
s . m = (n / 2) to_power k ) )
proof
defpred S1[ Element of NAT , Real] means ( ( $1 = 0 implies $2 = 0 ) & ( $1 > 0 implies $2 = (n / 2) to_power k ) );
A4: for x being Element of NAT ex y being Element of REAL st S1[x,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of REAL st S1[n,y]
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
end;
end;
consider h being Function of NAT,REAL such that
A5: for x being Element of NAT holds S1[x,h . x] from FUNCT_2:sch_3(A4);
take h ; ::_thesis: ( h . 0 = 0 & ( for m being Element of NAT st m > 0 holds
h . m = (n / 2) to_power k ) )
thus h . 0 = 0 by A5; ::_thesis: for m being Element of NAT st m > 0 holds
h . m = (n / 2) to_power k
let n be Element of NAT ; ::_thesis: ( n > 0 implies h . n = (n / 2) to_power k )
thus ( n > 0 implies h . n = (n / 2) to_power k ) by A5; ::_thesis: verum
end;
then consider q being Real_Sequence such that
A6: q . 0 = 0 and
A7: for m being Element of NAT st m > 0 holds
q . m = (n / 2) to_power k ;
A8: [/(n / 2)\] >= n / 2 by INT_1:def_7;
then reconsider n1 = [/(n / 2)\] as Element of NAT by INT_1:3;
set n2 = n1 - 1;
assume A9: n >= 1 ; ::_thesis: ( ((2 to_power (k + 1)) ") * ((seq_n^ (k + 1)) . n) <= f . n & f . n >= 0 )
then A10: n * (2 ") > 0 * (2 ") by XREAL_1:68;
then A11: (n / 2) to_power k > 0 by POWER:34;
now__::_thesis:_not_n1_-_1_<_0
assume n1 - 1 < 0 ; ::_thesis: contradiction
then n1 - 1 <= - 1 by INT_1:8;
then (n1 - 1) + 1 <= (- 1) + 1 by XREAL_1:6;
hence contradiction by A10, INT_1:def_7; ::_thesis: verum
end;
then reconsider n2 = n1 - 1 as Element of NAT by INT_1:3;
A12: now__::_thesis:_not_n_-_n2_<_n_/_2
[/(n / 2)\] < (n / 2) + 1 by INT_1:def_7;
then n2 < n / 2 by XREAL_1:19;
then A13: (n / 2) + n2 < (n / 2) + (n / 2) by XREAL_1:6;
assume n - n2 < n / 2 ; ::_thesis: contradiction
hence contradiction by A13, XREAL_1:19; ::_thesis: verum
end;
Sum (q,n,n2) = (n - n2) * ((n / 2) to_power k) by A6, A7, Lm18;
then Sum (q,n,n2) >= (n / 2) * ((n / 2) to_power k) by A12, A11, 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 A10, POWER:27;
then A14: Sum (q,n,n2) >= (n to_power (k + 1)) / (2 to_power (k + 1)) by A9, POWER:31;
A15: f . n = Sum ((seq_n^ k),n) by A1;
A16: now__::_thesis:_for_m_being_Element_of_NAT_st_m_<=_n_holds_
(seq_n^_k)_._m_>=_0
let m be Element of NAT ; ::_thesis: ( m <= n implies (seq_n^ k) . b1 >= 0 )
assume m <= n ; ::_thesis: (seq_n^ k) . b1 >= 0
percases ( m = 0 or m > 0 ) ;
suppose m = 0 ; ::_thesis: (seq_n^ k) . b1 >= 0
hence (seq_n^ k) . m >= 0 by Def3; ::_thesis: verum
end;
suppose m > 0 ; ::_thesis: (seq_n^ k) . b1 >= 0
then (seq_n^ k) . m = m to_power k by Def3;
hence (seq_n^ k) . m >= 0 ; ::_thesis: verum
end;
end;
end;
now__::_thesis:_for_m_being_Element_of_NAT_st_m_<=_n2_holds_
(seq_n^_k)_._m_>=_0
let m be Element of NAT ; ::_thesis: ( m <= n2 implies (seq_n^ k) . m >= 0 )
n1 <= n1 + 1 by NAT_1:11;
then A17: n2 <= n1 by XREAL_1:20;
A18: n1 <= n by Lm17;
assume m <= n2 ; ::_thesis: (seq_n^ k) . m >= 0
then m <= n1 by A17, XXREAL_0:2;
then m <= n by A18, XXREAL_0:2;
hence (seq_n^ k) . m >= 0 by A16; ::_thesis: verum
end;
then Sum ((seq_n^ k),n2) >= 0 by Lm12;
then A19: (Sum ((seq_n^ k),n)) + (Sum ((seq_n^ k),n2)) >= (Sum ((seq_n^ k),n)) + 0 by XREAL_1:7;
A20: for N0 being Element of NAT st n1 <= N0 & N0 <= n holds
q . N0 <= (seq_n^ k) . N0
proof
let N0 be Element of NAT ; ::_thesis: ( n1 <= N0 & N0 <= n implies q . N0 <= (seq_n^ k) . N0 )
assume that
A21: n1 <= N0 and
N0 <= n ; ::_thesis: q . N0 <= (seq_n^ k) . N0
A22: N0 >= n / 2 by A8, A21, XXREAL_0:2;
A23: (seq_n^ k) . N0 = N0 to_power k by A10, A8, A21, Def3;
q . N0 = (n / 2) to_power k by A7, A10, A8, A21;
hence q . N0 <= (seq_n^ k) . N0 by A10, A23, A22, Lm6; ::_thesis: verum
end;
n >= n2 + 1 by Lm17;
then Sum ((seq_n^ k),n,n2) >= Sum (q,n,n2) by A20, Lm16;
then A24: Sum ((seq_n^ k),n,n2) >= (n to_power (k + 1)) * ((2 to_power (k + 1)) ") by A14, XXREAL_0:2;
Sum ((seq_n^ k),n,n2) = (Sum ((seq_n^ k),n)) - (Sum ((seq_n^ k),n2)) by SERIES_1:def_6;
then A25: Sum ((seq_n^ k),n) >= Sum ((seq_n^ k),n,n2) by A19, XREAL_1:20;
(seq_n^ (k + 1)) . n = n to_power (k + 1) by A9, Def3;
hence ((2 to_power (k + 1)) ") * ((seq_n^ (k + 1)) . n) <= f . n by A15, A25, A24, XXREAL_0:2; ::_thesis: f . n >= 0
Sum ((seq_n^ k),n) >= 0 by A16, Lm12;
hence f . n >= 0 by A1; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_1_holds_
(_f_._n_<=_1_*_((seq_n^_(k_+_1))_._n)_&_f_._n_>=_0_)
set p = seq_n^ k;
let n be Element of NAT ; ::_thesis: ( n >= 1 implies ( f . n <= 1 * ((seq_n^ (k + 1)) . n) & f . n >= 0 ) )
assume A26: n >= 1 ; ::_thesis: ( f . n <= 1 * ((seq_n^ (k + 1)) . n) & f . n >= 0 )
ex s being Real_Sequence st
( s . 0 = 0 & ( for m being Element of NAT st m > 0 holds
s . m = n to_power k ) )
proof
defpred S1[ Element of NAT , Real] means ( ( $1 = 0 implies $2 = 0 ) & ( $1 > 0 implies $2 = n to_power k ) );
A27: for x being Element of NAT ex y being Element of REAL st S1[x,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of REAL st S1[n,y]
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
end;
end;
consider h being Function of NAT,REAL such that
A28: for x being Element of NAT holds S1[x,h . x] from FUNCT_2:sch_3(A27);
take h ; ::_thesis: ( h . 0 = 0 & ( for m being Element of NAT st m > 0 holds
h . m = n to_power k ) )
thus h . 0 = 0 by A28; ::_thesis: for m being Element of NAT st m > 0 holds
h . m = n to_power k
let n be Element of NAT ; ::_thesis: ( n > 0 implies h . n = n to_power k )
thus ( n > 0 implies h . n = n to_power k ) by A28; ::_thesis: verum
end;
then consider q being Real_Sequence such that
A29: q . 0 = 0 and
A30: for m being Element of NAT st m > 0 holds
q . m = n to_power k ;
now__::_thesis:_for_m_being_Element_of_NAT_st_m_<=_n_holds_
(seq_n^_k)_._m_<=_q_._m
let m be Element of NAT ; ::_thesis: ( m <= n implies (seq_n^ k) . b1 <= q . b1 )
assume A31: m <= n ; ::_thesis: (seq_n^ k) . b1 <= q . b1
percases ( m = 0 or m > 0 ) ;
suppose m = 0 ; ::_thesis: (seq_n^ k) . b1 <= q . b1
hence (seq_n^ k) . m <= q . m by A29, Def3; ::_thesis: verum
end;
supposeA32: m > 0 ; ::_thesis: (seq_n^ k) . b1 <= q . b1
then A33: q . m = n to_power k by A30;
(seq_n^ k) . m = m to_power k by A32, Def3;
hence (seq_n^ k) . m <= q . m by A31, A32, A33, Lm6; ::_thesis: verum
end;
end;
end;
then A34: Sum ((seq_n^ k),n) <= Sum (q,n) by Lm13;
Sum (q,n) = (n to_power k) * n by A29, A30, Lm14
.= (n to_power k) * (n to_power 1) by POWER:25
.= n to_power (k + 1) by A26, POWER:27
.= (seq_n^ (k + 1)) . n by A26, Def3 ;
hence f . n <= 1 * ((seq_n^ (k + 1)) . n) by A1, A34; ::_thesis: f . n >= 0
A35: now__::_thesis:_for_m_being_Element_of_NAT_st_m_<=_n_holds_
(seq_n^_k)_._m_>=_0
let m be Element of NAT ; ::_thesis: ( m <= n implies (seq_n^ k) . b1 >= 0 )
assume m <= n ; ::_thesis: (seq_n^ k) . b1 >= 0
percases ( m = 0 or m > 0 ) ;
suppose m = 0 ; ::_thesis: (seq_n^ k) . b1 >= 0
hence (seq_n^ k) . m >= 0 by Def3; ::_thesis: verum
end;
suppose m > 0 ; ::_thesis: (seq_n^ k) . b1 >= 0
then (seq_n^ k) . m = m to_power k by Def3;
hence (seq_n^ k) . m >= 0 ; ::_thesis: verum
end;
end;
end;
f . n = Sum ((seq_n^ k),n) by A1;
hence f . n >= 0 by A35, Lm12; ::_thesis: verum
end;
then A36: f in Big_Oh (seq_n^ (k + 1)) by A2;
2 to_power (k + 1) > 0 by POWER:34;
then f in Big_Omega (seq_n^ (k + 1)) by A2, A3;
hence f in Big_Theta (seq_n^ (k + 1)) by A36, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: ASYMPT_1:8
for f being Real_Sequence st ( for n being Element of NAT st n > 0 holds
f . n = n to_power (log (2,n)) ) holds
ex s being eventually-positive Real_Sequence st
( s = f & not s is smooth )
proof
let f be Real_Sequence; ::_thesis: ( ( for n being Element of NAT st n > 0 holds
f . n = n to_power (log (2,n)) ) implies ex s being eventually-positive Real_Sequence st
( s = f & not s is smooth ) )
assume A1: for n being Element of NAT st n > 0 holds
f . n = n to_power (log (2,n)) ; ::_thesis: ex s being eventually-positive Real_Sequence st
( s = f & not s is smooth )
A2: f is eventually-positive
proof
take 1 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 1 <= b1 or not f . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 1 <= n or not f . n <= 0 )
assume A3: n >= 1 ; ::_thesis: not f . n <= 0
then f . n = n to_power (log (2,n)) by A1;
hence not f . n <= 0 by A3, POWER:34; ::_thesis: verum
end;
set g = f taken_every 2;
reconsider f = f as eventually-positive Real_Sequence by A2;
take f ; ::_thesis: ( f = f & not f is smooth )
now__::_thesis:_not_f_is_smooth
assume f is smooth ; ::_thesis: contradiction
then f is_smooth_wrt 2 by ASYMPT_0:def_17;
then f taken_every 2 in Big_Oh f by ASYMPT_0:def_16;
then consider t being Element of Funcs (NAT,REAL) such that
A4: t = f taken_every 2 and
A5: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
A6: c > 0 and
A7: for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) by A5;
A8: sqrt c > 0 by A6, SQUARE_1:25;
set N0 = [/((sqrt c) / (sqrt 2))\];
reconsider N2 = max (N,[/((sqrt c) / (sqrt 2))\]) as Integer by XXREAL_0:16;
set N1 = max (N2,2);
A9: max (N2,2) >= N2 by XXREAL_0:25;
N2 >= [/((sqrt c) / (sqrt 2))\] by XXREAL_0:25;
then A10: max (N2,2) >= [/((sqrt c) / (sqrt 2))\] by A9, XXREAL_0:2;
A11: max (N2,2) is Integer by XXREAL_0:16;
N2 >= N by XXREAL_0:25;
then A12: max (N2,2) >= N by A9, XXREAL_0:2;
max (N2,2) >= 2 by XXREAL_0:25;
then reconsider N1 = max (N2,2) as Element of NAT by A11, INT_1:3;
set n = N1 + 1;
A13: (N1 + 1) to_power (log (2,(N1 + 1))) > 0 by POWER:34;
A14: 2 * (N1 + 1) > 2 * 0 by XREAL_1:68;
A15: sqrt 2 <> 0 by SQUARE_1:25;
A16: sqrt 2 > 0 by SQUARE_1:25;
A17: [/((sqrt c) / (sqrt 2))\] >= (sqrt c) / (sqrt 2) by INT_1:def_7;
A18: N1 + 1 > N1 + 0 by XREAL_1:8;
then N1 + 1 > [/((sqrt c) / (sqrt 2))\] by A10, XXREAL_0:2;
then N1 + 1 > (sqrt c) / (sqrt 2) by A17, XXREAL_0:2;
then (N1 + 1) * (sqrt 2) > ((sqrt c) / (sqrt 2)) * (sqrt 2) by A16, XREAL_1:68;
then (N1 + 1) * (sqrt 2) > sqrt c by A15, XCMPLX_1:87;
then ((N1 + 1) * (sqrt 2)) ^2 > (sqrt c) ^2 by A8, SQUARE_1:16;
then ((N1 + 1) ^2) * ((sqrt 2) ^2) > c by A6, SQUARE_1:def_2;
then A19: 2 * ((N1 + 1) ^2) > c by SQUARE_1:def_2;
(2 * ((N1 + 1) ^2)) * ((N1 + 1) to_power (log (2,(N1 + 1)))) = ((2 * (N1 + 1)) * (N1 + 1)) * ((N1 + 1) to_power (log (2,(N1 + 1))))
.= ((2 * (N1 + 1)) * (2 to_power (log (2,(N1 + 1))))) * ((N1 + 1) to_power (log (2,(N1 + 1)))) by POWER:def_3
.= (2 * (N1 + 1)) * ((2 to_power (log (2,(N1 + 1)))) * ((N1 + 1) to_power (log (2,(N1 + 1)))))
.= (2 * (N1 + 1)) * ((2 * (N1 + 1)) to_power (log (2,(N1 + 1)))) by POWER:30
.= ((2 * (N1 + 1)) to_power 1) * ((2 * (N1 + 1)) to_power (log (2,(N1 + 1)))) by POWER:25
.= (2 * (N1 + 1)) to_power (1 + (log (2,(N1 + 1)))) by A14, POWER:27
.= (2 * (N1 + 1)) to_power ((log (2,2)) + (log (2,(N1 + 1)))) by POWER:52
.= (2 * (N1 + 1)) to_power (log (2,(2 * (N1 + 1)))) by POWER:53 ;
then (2 * (N1 + 1)) to_power (log (2,(2 * (N1 + 1)))) > c * ((N1 + 1) to_power (log (2,(N1 + 1)))) by A13, A19, XREAL_1:68;
then f . (2 * (N1 + 1)) > c * ((N1 + 1) to_power (log (2,(N1 + 1)))) by A1, A14;
then t . (N1 + 1) > c * ((N1 + 1) to_power (log (2,(N1 + 1)))) by A4, ASYMPT_0:def_15;
then A20: t . (N1 + 1) > c * (f . (N1 + 1)) by A1;
N1 + 1 > N by A12, A18, XXREAL_0:2;
hence contradiction by A7, A20; ::_thesis: verum
end;
hence ( f = f & not f is smooth ) ; ::_thesis: verum
end;
definition
let b be Real;
func seq_const b -> Real_Sequence equals :: ASYMPT_1:def 4
NAT --> b;
coherence
NAT --> b is Real_Sequence ;
end;
:: deftheorem defines seq_const ASYMPT_1:def_4_:_
for b being Real holds seq_const b = NAT --> b;
registration
cluster seq_const 1 -> eventually-nonnegative ;
coherence
seq_const 1 is eventually-nonnegative
proof
take 0 ; :: according to ASYMPT_0:def_2 ::_thesis: for b1 being Element of NAT holds
( not 0 <= b1 or 0 <= (seq_const 1) . b1 )
let n be Element of NAT ; ::_thesis: ( not 0 <= n or 0 <= (seq_const 1) . n )
assume n >= 0 ; ::_thesis: 0 <= (seq_const 1) . n
thus 0 <= (seq_const 1) . n ; ::_thesis: verum
end;
end;
Lm19: for a, b, c being Real st a > 1 & b >= a & c >= 1 holds
log (a,c) >= log (b,c)
proof
let a, b, c be Real; ::_thesis: ( a > 1 & b >= a & c >= 1 implies log (a,c) >= log (b,c) )
assume that
A1: a > 1 and
A2: b >= a and
A3: c >= 1 ; ::_thesis: log (a,c) >= log (b,c)
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 log (a,c) >= log (b,c) by A1, A2, A3, POWER:56; ::_thesis: verum
end;
theorem Th9: :: ASYMPT_1:9
for f being eventually-nonnegative Real_Sequence ex F being FUNCTION_DOMAIN of NAT , REAL st
( F = {(seq_n^ 1)} & ( f in F to_power (Big_Oh (seq_const 1)) implies ex N being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= f . n & f . n <= c * ((seq_n^ k) . n) ) ) ) ) & ( ex N being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= f . n & f . n <= c * ((seq_n^ k) . n) ) ) ) implies f in F to_power (Big_Oh (seq_const 1)) ) )
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; ::_thesis: ex F being FUNCTION_DOMAIN of NAT , REAL st
( F = {(seq_n^ 1)} & ( h in F to_power (Big_Oh (seq_const 1)) implies ex N being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= c * ((seq_n^ k) . n) ) ) ) ) & ( ex N being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= c * ((seq_n^ k) . n) ) ) ) implies h in F to_power (Big_Oh (seq_const 1)) ) )
take F ; ::_thesis: ( F = {(seq_n^ 1)} & ( h in F to_power (Big_Oh (seq_const 1)) implies ex N being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= c * ((seq_n^ k) . n) ) ) ) ) & ( ex N being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= c * ((seq_n^ k) . n) ) ) ) implies h in F to_power (Big_Oh (seq_const 1)) ) )
thus F = {(seq_n^ 1)} ; ::_thesis: ( h in F to_power (Big_Oh (seq_const 1)) iff ex N being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= c * ((seq_n^ k) . n) ) ) ) )
now__::_thesis:_(_(_h_in_F_to_power_(Big_Oh_(seq_const_1))_implies_ex_N_being_Element_of_NAT_ex_i,_k_being_Element_of_NAT_st_
(_i_>_0_&_(_for_n_being_Element_of_NAT_st_n_>=_N_holds_
(_1_<=_h_._n_&_h_._n_<=_i_*_((seq_n^_k)_._n)_)_)_)_)_&_(_ex_N0_being_Element_of_NAT_ex_c_being_Real_ex_k_being_Element_of_NAT_st_
(_c_>_0_&_(_for_n_being_Element_of_NAT_st_n_>=_N0_holds_
(_1_<=_h_._n_&_h_._n_<=_c_*_((seq_n^_k)_._n)_)_)_)_implies_h_in_F_to_power_(Big_Oh_(seq_const_1))_)_)
hereby ::_thesis: ( ex N0 being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N0 holds
( 1 <= h . n & h . n <= c * ((seq_n^ k) . n) ) ) ) implies h in F to_power (Big_Oh (seq_const 1)) )
reconsider i = 1 as Element of NAT ;
assume h in F to_power (Big_Oh (seq_const 1)) ; ::_thesis: ex N being Element of NAT ex i, k being Element of NAT st
( i > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= i * ((seq_n^ k) . n) ) ) )
then consider t being Element of Funcs (NAT,REAL) such that
A1: h = t and
A2: ex f, g being Element of Funcs (NAT,REAL) ex N being Element of NAT st
( f in F & g in Big_Oh (seq_const 1) & ( for n being Element of NAT st n >= N holds
t . n = (f . n) to_power (g . n) ) ) ;
consider f, g being Element of Funcs (NAT,REAL), N0 being Element of NAT such that
A3: f in F and
A4: g in Big_Oh (seq_const 1) and
A5: for n being Element of NAT st n >= N0 holds
t . n = (f . n) to_power (g . n) by A2;
consider g9 being Element of Funcs (NAT,REAL) such that
A6: g = g9 and
A7: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( g9 . n <= c * ((seq_const 1) . n) & g9 . n >= 0 ) ) ) by A4;
consider c being Real, N1 being Element of NAT such that
A8: c > 0 and
A9: for n being Element of NAT st n >= N1 holds
( g9 . n <= c * ((seq_const 1) . n) & g9 . n >= 0 ) by A7;
set k = [/c\];
A10: [/c\] > 0 by A8, INT_1:def_7;
set N = max (2,(max (N0,N1)));
A11: max (2,(max (N0,N1))) >= max (N0,N1) by XXREAL_0:25;
max (N0,N1) >= N0 by XXREAL_0:25;
then A12: max (2,(max (N0,N1))) >= N0 by A11, XXREAL_0:2;
A13: [/c\] >= c by INT_1:def_7;
reconsider k = [/c\] as Element of NAT by A10, INT_1:3;
take N = max (2,(max (N0,N1))); ::_thesis: ex i, k being Element of NAT st
( i > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= i * ((seq_n^ k) . n) ) ) )
take i = i; ::_thesis: ex k being Element of NAT st
( i > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= i * ((seq_n^ k) . n) ) ) )
take k = k; ::_thesis: ( i > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= i * ((seq_n^ k) . n) ) ) )
thus i > 0 ; ::_thesis: for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= i * ((seq_n^ k) . n) )
let n be Element of NAT ; ::_thesis: ( n >= N implies ( 1 <= h . n & h . n <= i * ((seq_n^ k) . n) ) )
assume A14: n >= N ; ::_thesis: ( 1 <= h . n & h . n <= i * ((seq_n^ k) . n) )
A15: N >= 2 by XXREAL_0:25;
then n >= 2 by A14, XXREAL_0:2;
then A16: n > 1 by XXREAL_0:2;
then A17: n to_power c <= n to_power k by A13, PRE_FF:8;
f = seq_n^ 1 by A3, TARSKI:def_1;
then f . n = n to_power 1 by A15, A14, Def3
.= n by POWER:25 ;
then A18: h . n = n to_power (g . n) by A1, A5, A12, A14, XXREAL_0:2;
max (N0,N1) >= N1 by XXREAL_0:25;
then N >= N1 by A11, XXREAL_0:2;
then A19: n >= N1 by A14, XXREAL_0:2;
then g9 . n >= 0 by A9;
then n to_power (g . n) >= n to_power 0 by A6, A16, PRE_FF:8;
hence 1 <= h . n by A18, POWER:24; ::_thesis: h . n <= i * ((seq_n^ k) . n)
A20: (seq_const 1) . n = 1 by FUNCOP_1:7;
g . n <= c * ((seq_const 1) . 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; ::_thesis: verum
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 being Element of NAT , c being Real, k being Element of NAT such that c > 0 and
A21: for n being Element of NAT st n >= N0 holds
( 1 <= h . n & h . n <= c * ((seq_n^ k) . n) ) ; ::_thesis: h in F to_power (Big_Oh (seq_const 1))
set N = max (N0,2);
defpred S1[ Element of NAT , Real] means ( ( $1 < max (N0,2) implies $2 = 1 ) & ( $1 >= max (N0,2) implies $2 = log ($1,(t . $1)) ) );
A22: max (N0,2) >= 2 by XXREAL_0:25;
then A23: max (N0,2) > 1 by XXREAL_0:2;
A24: for x being Element of NAT ex y being Element of REAL st S1[x,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of REAL st S1[n,y]
percases ( n < max (N0,2) or n >= max (N0,2) ) ;
suppose n < max (N0,2) ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
suppose n >= max (N0,2) ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
end;
end;
consider g being Function of NAT,REAL such that
A25: for x being Element of NAT holds S1[x,g . x] from FUNCT_2:sch_3(A24);
A26: max (N0,2) >= N0 by XXREAL_0:25;
A27: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_max_(N0,2)_holds_
(f_._n)_to_power_(g_._n)_=_t_._n
let n be Element of NAT ; ::_thesis: ( n >= max (N0,2) implies (f . n) to_power (g . n) = t . n )
assume A28: n >= max (N0,2) ; ::_thesis: (f . n) to_power (g . n) = t . n
then n >= N0 by A26, XXREAL_0:2;
then A29: t . n >= 1 by A21;
thus (f . n) to_power (g . n) = (n to_power 1) to_power (g . n) by A22, A28, Def3
.= n to_power (g . n) by POWER:25
.= n to_power (1 * (log (n,(t . n)))) by A25, A28
.= t . n by A23, A28, A29, POWER:def_3 ; ::_thesis: verum
end;
set c1 = max (c,2);
A30: max (N0,2) <> 1 by XXREAL_0:25;
set a = log ((max (N0,2)),(max (c,2)));
set b = k + (log ((max (N0,2)),(max (c,2))));
A31: max (c,2) >= 2 by XXREAL_0:25;
then A32: max (c,2) > 1 by XXREAL_0:2;
A33: f in F by TARSKI:def_1;
A34: g is Element of Funcs (NAT,REAL) by FUNCT_2:8;
A35: max (N0,2) > 0 by XXREAL_0:25;
now__::_thesis:_(_k_+_(log_((max_(N0,2)),(max_(c,2))))_>_0_&_(_for_n_being_Element_of_NAT_st_n_>=_max_(N0,2)_holds_
(_g_._n_<=_(k_+_(log_((max_(N0,2)),(max_(c,2)))))_*_((seq_const_1)_._n)_&_g_._n_>=_0_)_)_)
log ((max (N0,2)),1) = 0 by A35, A30, POWER:51;
then log ((max (N0,2)),(max (c,2))) > 0 by A23, A32, POWER:57;
hence k + (log ((max (N0,2)),(max (c,2)))) > 0 ; ::_thesis: for n being Element of NAT st n >= max (N0,2) holds
( g . n <= (k + (log ((max (N0,2)),(max (c,2))))) * ((seq_const 1) . n) & g . n >= 0 )
let n be Element of NAT ; ::_thesis: ( n >= max (N0,2) implies ( g . n <= (k + (log ((max (N0,2)),(max (c,2))))) * ((seq_const 1) . n) & g . n >= 0 ) )
A36: (seq_const 1) . n = 1 by FUNCOP_1:7;
assume A37: n >= max (N0,2) ; ::_thesis: ( g . n <= (k + (log ((max (N0,2)),(max (c,2))))) * ((seq_const 1) . n) & g . n >= 0 )
then A38: n <> 1 by A22, XXREAL_0:2;
A39: (seq_n^ k) . n = n to_power k by A22, A37, Def3;
then A40: c * ((seq_n^ k) . n) <= (max (c,2)) * ((seq_n^ k) . n) by XREAL_1:64, XXREAL_0:25;
(seq_n^ k) . n > 0 by A22, A37, A39, POWER:34;
then A41: log (n,((max (c,2)) * ((seq_n^ k) . n))) = (log (n,(max (c,2)))) + (log (n,(n to_power k))) by A22, A31, A37, A38, A39, POWER:53
.= (log (n,(max (c,2)))) + (k * (log (n,n))) by A22, A37, A38, POWER:55
.= (log (n,(max (c,2)))) + (k * 1) by A22, A37, A38, POWER:52 ;
log ((max (N0,2)),(max (c,2))) >= log (n,(max (c,2))) by A23, A32, A37, Lm19;
then A42: (log (n,(max (c,2)))) + k <= (log ((max (N0,2)),(max (c,2)))) + k by XREAL_1:6;
A43: n >= N0 by A26, A37, XXREAL_0:2;
then A44: 1 <= t . n by A21;
t . n = (f . n) to_power (g . n) by A27, A37
.= (n to_power 1) to_power (g . n) by A22, A37, Def3
.= n to_power (g . n) by POWER:25 ;
then A45: log (n,(t . n)) = (g . n) * (log (n,n)) by A22, A37, A38, POWER:55
.= (g . n) * 1 by A22, A37, A38, POWER:52 ;
n >= 2 by A22, A37, XXREAL_0:2;
then A46: n > 1 by XXREAL_0:2;
t . n <= c * ((seq_n^ k) . n) by A21, A43;
then t . n <= (max (c,2)) * ((seq_n^ k) . n) by A40, XXREAL_0:2;
then log (n,(t . n)) <= log (n,((max (c,2)) * ((seq_n^ k) . n))) by A46, A44, PRE_FF:10;
hence g . n <= (k + (log ((max (N0,2)),(max (c,2))))) * ((seq_const 1) . n) by A45, A41, A42, A36, XXREAL_0:2; ::_thesis: g . n >= 0
g . n = log (n,(t . n)) by A25, A37;
then g . n >= log (n,1) by A46, A44, PRE_FF:10;
hence g . n >= 0 by A22, A37, A38, POWER:51; ::_thesis: verum
end;
then g in Big_Oh (seq_const 1) by A34;
hence h in F to_power (Big_Oh (seq_const 1)) by A34, A27, A33; ::_thesis: verum
end;
hence ( h in F to_power (Big_Oh (seq_const 1)) iff ex N being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= h . n & h . n <= c * ((seq_n^ k) . n) ) ) ) ) ; ::_thesis: verum
end;
begin
theorem :: ASYMPT_1:10
for f being Real_Sequence st ( for n being Element of NAT holds f . n = ((3 * (10 to_power 6)) - ((18 * (10 to_power 3)) * n)) + (27 * (n ^2)) ) holds
f in Big_Oh (seq_n^ 2)
proof
set g = seq_n^ 2;
consider t1 being Element of NAT such that
A1: t1 = (10 * 10) * 10 ;
consider t2 being Element of NAT such that
A2: t2 = t1 * t1 ;
t1 = 10 * (10 ^2) by A1;
then t1 = 10 * (10 to_power 2) by POWER:46;
then t1 = (10 to_power 1) * (10 to_power 2) by POWER:25;
then A3: t1 = 10 to_power (1 + 2) by POWER:27;
then A4: t2 = 10 to_power (3 + 3) by A2, POWER:27
.= 10 to_power 6 ;
A5: 10 to_power 3 = 10 to_power (2 + 1)
.= (10 to_power 2) * (10 to_power 1) by POWER:27
.= (10 to_power 2) * 10 by POWER:25
.= (10 ^2) * 10 by POWER:46
.= 1000 ;
A6: for n being Element of NAT st n >= 400 holds
((18 * t1) * n) - (3 * t2) < 27 * (n ^2)
proof
defpred S1[ Nat] means ((18 * t1) * $1) - (3 * t2) < 27 * ($1 ^2);
A7: for k being Nat st k >= 400 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 400 & S1[k] implies S1[k + 1] )
assume that
A8: k >= 400 and
A9: ((18 * t1) * k) - (3 * t2) < 27 * (k ^2) ; ::_thesis: S1[k + 1]
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 S1[k + 1] by A11, XXREAL_0:2; ::_thesis: verum
end;
A12: S1[400] by A2, A3, A5;
for n being Nat st n >= 400 holds
S1[n] from NAT_1:sch_8(A12, A7);
hence for n being Element of NAT st n >= 400 holds
((18 * t1) * n) - (3 * t2) < 27 * (n ^2) ; ::_thesis: verum
end;
let f be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds f . n = ((3 * (10 to_power 6)) - ((18 * (10 to_power 3)) * n)) + (27 * (n ^2)) ) implies f in Big_Oh (seq_n^ 2) )
assume A13: for n being Element of NAT holds f . n = ((3 * (10 to_power 6)) - ((18 * (10 to_power 3)) * n)) + (27 * (n ^2)) ; ::_thesis: f in Big_Oh (seq_n^ 2)
A14: for n being Element of NAT st n >= 400 holds
f . n <= 27 * (n ^2)
proof
let n be Element of NAT ; ::_thesis: ( n >= 400 implies f . n <= 27 * (n ^2) )
assume A15: n >= 400 ; ::_thesis: f . n <= 27 * (n ^2)
now__::_thesis:_not_f_._n_>_27_*_(n_^2)
assume f . n > 27 * (n ^2) ; ::_thesis: contradiction
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; ::_thesis: verum
end;
hence f . n <= 27 * (n ^2) ; ::_thesis: verum
end;
A17: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_400_holds_
(_f_._n_<=_27_*_((seq_n^_2)_._n)_&_f_._n_>=_0_)
let n be Element of NAT ; ::_thesis: ( n >= 400 implies ( f . n <= 27 * ((seq_n^ 2) . n) & f . n >= 0 ) )
assume A18: n >= 400 ; ::_thesis: ( f . n <= 27 * ((seq_n^ 2) . n) & f . n >= 0 )
then f . n <= 27 * (n ^2) by A14;
then f . n <= 27 * (n to_power 2) by POWER:46;
hence f . n <= 27 * ((seq_n^ 2) . n) by A18, Def3; ::_thesis: f . n >= 0
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; ::_thesis: verum
end;
f is Element of Funcs (NAT,REAL) by FUNCT_2:8;
hence f in Big_Oh (seq_n^ 2) by A17; ::_thesis: verum
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__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_2_holds_
(_(seq_n^_2)_._n_<=_1_*_((seq_n^_3)_._n)_&_(seq_n^_2)_._n_>=_0_)
let n be Element of NAT ; ::_thesis: ( n >= 2 implies ( (seq_n^ 2) . n <= 1 * ((seq_n^ 3) . n) & (seq_n^ 2) . n >= 0 ) )
assume A2: n >= 2 ; ::_thesis: ( (seq_n^ 2) . n <= 1 * ((seq_n^ 3) . n) & (seq_n^ 2) . n >= 0 )
then A3: n > 1 by XXREAL_0:2;
A4: (seq_n^ 2) . n = n to_power 2 by A2, Def3;
(seq_n^ 3) . n = n to_power 3 by A2, Def3;
hence (seq_n^ 2) . n <= 1 * ((seq_n^ 3) . n) by A3, A4, POWER:39; ::_thesis: (seq_n^ 2) . n >= 0
thus (seq_n^ 2) . n >= 0 by A4; ::_thesis: verum
end;
seq_n^ 2 is Element of Funcs (NAT,REAL) by FUNCT_2:8;
hence seq_n^ 2 in Big_Oh (seq_n^ 3) by A1; ::_thesis: verum
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__::_thesis:_not_seq_n^_2_in_Big_Omega_(seq_n^_3)
assume seq_n^ 2 in Big_Omega (seq_n^ 3) ; ::_thesis: contradiction
then consider s being Element of Funcs (NAT,REAL) such that
A1: s = seq_n^ 2 and
A2: ex d being Real ex N being Element of NAT st
( d > 0 & ( for n being Element of NAT st n >= N holds
( d * ((seq_n^ 3) . n) <= s . n & s . n >= 0 ) ) ) ;
consider d being Real, N being Element of NAT such that
A3: d > 0 and
A4: for n being Element of NAT st n >= N holds
( d * ((seq_n^ 3) . n) <= s . n & s . n >= 0 ) by A2;
A5: N + 2 > 1 + 0 by XREAL_1:8;
ex n being Element of NAT st
( n >= N & d * ((seq_n^ 3) . n) > s . n )
proof
take n = max (N,[/((N + 2) / d)\]); ::_thesis: ( n is Element of REAL & n is Element of NAT & n >= N & d * ((seq_n^ 3) . n) > s . n )
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 = n as Element of NAT by A6, A7, INT_1:3;
A10: ((seq_n^ 2) . 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 * ((seq_n^ 3) . 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 * ((seq_n^ 3) . n)) * (n to_power (- 2)) > ((seq_n^ 2) . n) * (n to_power (- 2)) by A5, A10, A13, XXREAL_0:2;
hence ( n is Element of REAL & n is Element of NAT & n >= N & d * ((seq_n^ 3) . n) > s . n ) by A1, A11, XREAL_1:64, XXREAL_0:25; ::_thesis: verum
end;
hence contradiction by A4; ::_thesis: verum
end;
hence not seq_n^ 2 in Big_Omega (seq_n^ 3) ; ::_thesis: verum
end;
theorem :: ASYMPT_1:13
ex s being eventually-positive Real_Sequence st
( s = seq_a^ (2,1,1) & seq_a^ (2,1,0) in Big_Theta s )
proof
reconsider g = seq_a^ (2,1,1) as eventually-positive Real_Sequence ;
set f = seq_a^ (2,1,0);
take g ; ::_thesis: ( g = seq_a^ (2,1,1) & seq_a^ (2,1,0) in Big_Theta g )
thus g = seq_a^ (2,1,1) ; ::_thesis: seq_a^ (2,1,0) in Big_Theta g
A1: seq_a^ (2,1,0) is Element of Funcs (NAT,REAL) by FUNCT_2:8;
A2: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_2_holds_
(_(2_to_power_(-_1))_*_(g_._n)_<=_(seq_a^_(2,1,0))_._n_&_(seq_a^_(2,1,0))_._n_<=_1_*_(g_._n)_)
let n be Element of NAT ; ::_thesis: ( n >= 2 implies ( (2 to_power (- 1)) * (g . n) <= (seq_a^ (2,1,0)) . n & (seq_a^ (2,1,0)) . n <= 1 * (g . n) ) )
assume n >= 2 ; ::_thesis: ( (2 to_power (- 1)) * (g . n) <= (seq_a^ (2,1,0)) . n & (seq_a^ (2,1,0)) . n <= 1 * (g . n) )
A3: (seq_a^ (2,1,0)) . 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
.= (seq_a^ (2,1,0)) . n by A3 ;
hence (2 to_power (- 1)) * (g . n) <= (seq_a^ (2,1,0)) . n ; ::_thesis: (seq_a^ (2,1,0)) . n <= 1 * (g . n)
n + 0 <= n + 1 by XREAL_1:7;
hence (seq_a^ (2,1,0)) . n <= 1 * (g . n) by A3, A4, PRE_FF:8; ::_thesis: verum
end;
A5: 2 to_power (- 1) > 0 by POWER:34;
Big_Theta g = { s where s is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (g . n) <= s . n & s . n <= c * (g . n) ) ) ) } by ASYMPT_0:27;
hence seq_a^ (2,1,0) in Big_Theta g by A1, A5, A2; ::_thesis: verum
end;
definition
let a be Element of NAT ;
func seq_n! a -> Real_Sequence means :Def5: :: ASYMPT_1:def 5
for n being Element of NAT holds it . n = (n + a) ! ;
existence
ex b1 being Real_Sequence st
for n being Element of NAT holds b1 . n = (n + a) !
proof
deffunc H1( Element of NAT ) -> Element of NAT = ($1 + a) ! ;
consider h being Function of NAT,REAL such that
A1: for n being Element of NAT holds h . n = H1(n) from FUNCT_2:sch_4();
take h ; ::_thesis: for n being Element of NAT holds h . n = (n + a) !
let n be Element of NAT ; ::_thesis: h . n = (n + a) !
thus h . n = (n + a) ! by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = (n + a) ! ) & ( for n being Element of NAT holds b2 . n = (n + a) ! ) holds
b1 = b2
proof
let j, k be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds j . n = (n + a) ! ) & ( for n being Element of NAT holds k . n = (n + a) ! ) implies j = k )
assume that
A2: for n being Element of NAT holds j . n = (n + a) ! and
A3: for n being Element of NAT holds k . n = (n + a) ! ; ::_thesis: j = k
now__::_thesis:_for_n_being_Element_of_NAT_holds_j_._n_=_k_._n
let n be Element of NAT ; ::_thesis: j . n = k . n
thus j . n = (n + a) ! by A2
.= k . n by A3 ; ::_thesis: verum
end;
hence j = k by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def5 defines seq_n! ASYMPT_1:def_5_:_
for a being Element of NAT
for b2 being Real_Sequence holds
( b2 = seq_n! a iff for n being Element of NAT holds b2 . n = (n + a) ! );
registration
let a be Element of NAT ;
cluster seq_n! a -> eventually-positive ;
coherence
seq_n! a is eventually-positive
proof
take 0 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 0 <= b1 or not (seq_n! a) . b1 <= 0 )
set f = seq_n! a;
let n be Element of NAT ; ::_thesis: ( not 0 <= n or not (seq_n! a) . n <= 0 )
assume n >= 0 ; ::_thesis: not (seq_n! a) . n <= 0
(seq_n! a) . n = (n + a) ! by Def5;
hence not (seq_n! a) . n <= 0 by NEWTON:17; ::_thesis: verum
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 (seq_n! 1) = { s where s is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * ((seq_n! 1) . n) <= s . n & s . n <= c * ((seq_n! 1) . n) ) ) ) } by ASYMPT_0:27;
now__::_thesis:_not_seq_n!_0_in_Big_Theta_(seq_n!_1)
assume seq_n! 0 in Big_Theta (seq_n! 1) ; ::_thesis: contradiction
then consider s being Element of Funcs (NAT,REAL) such that
A2: s = seq_n! 0 and
A3: ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * ((seq_n! 1) . n) <= s . n & s . n <= c * ((seq_n! 1) . n) ) ) ) by A1;
consider c, d being Real, N being Element of NAT such that
c > 0 and
A4: d > 0 and
A5: for n being Element of NAT st n >= N holds
( d * ((seq_n! 1) . n) <= s . n & s . n <= c * ((seq_n! 1) . n) ) by A3;
ex n being Element of NAT st
( n >= N & d * ((seq_n! 1) . n) > (seq_n! 0) . 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)\]); ::_thesis: ( n is Element of REAL & n is Element of NAT & n >= N & d * ((seq_n! 1) . n) > (seq_n! 0) . n )
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 = 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: ((seq_n! 0) . n) * ((n !) ") = ((n + 0) !) * ((n !) ") by Def5
.= 1 by A11, XCMPLX_0:def_7 ;
(d * ((seq_n! 1) . n)) * ((n !) ") = (d * ((n + 1) !)) * ((n !) ") by Def5
.= (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 * ((seq_n! 1) . n)) * ((n !) ") > 1 by A12, A8, XXREAL_0:2;
hence ( n is Element of REAL & n is Element of NAT & n >= N & d * ((seq_n! 1) . n) > (seq_n! 0) . n ) by A13, XREAL_1:64, XXREAL_0:25; ::_thesis: verum
end;
hence contradiction by A2, A5; ::_thesis: verum
end;
hence not seq_n! 0 in Big_Theta (seq_n! 1) ; ::_thesis: verum
end;
begin
Lm20: now__::_thesis:_for_a,_b,_c,_d_being_Real_st_0_<=_b_&_a_<=_b_&_0_<=_c_&_c_<=_d_holds_
a_*_c_<=_b_*_d
let a, b, c, d be Real; ::_thesis: ( 0 <= b & a <= b & 0 <= c & c <= d implies a * c <= b * d )
assume that
A1: 0 <= b and
A2: a <= b and
A3: 0 <= c and
A4: c <= d ; ::_thesis: a * c <= b * 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; ::_thesis: verum
end;
theorem :: ASYMPT_1:15
for f being Real_Sequence st f in Big_Oh (seq_n^ 1) holds
f (#) f in Big_Oh (seq_n^ 2)
proof
let f be Real_Sequence; ::_thesis: ( f in Big_Oh (seq_n^ 1) implies f (#) f in Big_Oh (seq_n^ 2) )
set h = seq_n^ 2;
set g = seq_n^ 1;
assume f in Big_Oh (seq_n^ 1) ; ::_thesis: f (#) f in Big_Oh (seq_n^ 2)
then consider t being Element of Funcs (NAT,REAL) such that
A1: t = f and
A2: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * ((seq_n^ 1) . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
A3: c > 0 and
A4: for n being Element of NAT st n >= N holds
( t . n <= c * ((seq_n^ 1) . n) & t . n >= 0 ) by A2;
set d = max (c,(c * c));
A5: 0 to_power 1 = 0 by POWER:def_2;
A6: now__::_thesis:_ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
(_(t_(#)_t)_._n_<=_(max_(c,(c_*_c)))_*_((seq_n^_2)_._n)_&_(t_(#)_t)_._n_>=_0_)
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
( (t (#) t) . n <= (max (c,(c * c))) * ((seq_n^ 2) . n) & (t (#) t) . n >= 0 )
let n be Element of NAT ; ::_thesis: ( n >= N implies ( (t (#) t) . n <= (max (c,(c * c))) * ((seq_n^ 2) . n) & (t (#) t) . n >= 0 ) )
assume A7: n >= N ; ::_thesis: ( (t (#) t) . n <= (max (c,(c * c))) * ((seq_n^ 2) . n) & (t (#) t) . n >= 0 )
then A8: t . n >= 0 by A4;
for n being Element of NAT holds (seq_n^ 1) . n <= (seq_n^ 2) . n
proof
let n be Element of NAT ; ::_thesis: (seq_n^ 1) . n <= (seq_n^ 2) . n
percases ( n = 0 or n > 0 ) ;
supposeA9: n = 0 ; ::_thesis: (seq_n^ 1) . n <= (seq_n^ 2) . n
then (seq_n^ 1) . n = 0 by Def3;
hence (seq_n^ 1) . n <= (seq_n^ 2) . n by A9, Def3; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: (seq_n^ 1) . n <= (seq_n^ 2) . n
then A10: n >= 0 + 1 by NAT_1:13;
thus (seq_n^ 1) . n <= (seq_n^ 2) . n ::_thesis: verum
proof
percases ( n = 1 or n > 1 ) by A10, XXREAL_0:1;
supposeA11: n = 1 ; ::_thesis: (seq_n^ 1) . n <= (seq_n^ 2) . n
A12: 1 to_power 2 = 1 by POWER:26;
1 to_power 1 = 1 by POWER:26;
then (seq_n^ 1) . n = 1 to_power 2 by A11, A12, Def3;
hence (seq_n^ 1) . n <= (seq_n^ 2) . n by A11, Def3; ::_thesis: verum
end;
supposeA13: n > 1 ; ::_thesis: (seq_n^ 1) . n <= (seq_n^ 2) . n
then n to_power 1 < n to_power 2 by POWER:39;
then (seq_n^ 1) . n < n to_power 2 by A13, Def3;
hence (seq_n^ 1) . n <= (seq_n^ 2) . n by A13, Def3; ::_thesis: verum
end;
end;
end;
end;
end;
end;
then A14: (seq_n^ 2) . n >= (seq_n^ 1) . n ;
(seq_n^ 1) . n >= 0
proof
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: (seq_n^ 1) . n >= 0
hence (seq_n^ 1) . n >= 0 by Def3; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: (seq_n^ 1) . n >= 0
then (seq_n^ 1) . n = n to_power 1 by Def3
.= n by POWER:25 ;
hence (seq_n^ 1) . n >= 0 ; ::_thesis: verum
end;
end;
end;
then A15: (c * c) * ((seq_n^ 2) . n) <= (max (c,(c * c))) * ((seq_n^ 2) . n) by A14, XREAL_1:64, XXREAL_0:25;
A16: (n to_power 1) * (n to_power 1) = n to_power (1 + 1)
proof
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: (n to_power 1) * (n to_power 1) = n to_power (1 + 1)
hence (n to_power 1) * (n to_power 1) = n to_power (1 + 1) by A5, POWER:def_2; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: (n to_power 1) * (n to_power 1) = n to_power (1 + 1)
hence (n to_power 1) * (n to_power 1) = n to_power (1 + 1) by POWER:27; ::_thesis: verum
end;
end;
end;
A17: ((seq_n^ 1) . n) * ((seq_n^ 1) . n) = (seq_n^ 2) . n
proof
percases ( n = 0 or n > 0 ) ;
supposeA18: n = 0 ; ::_thesis: ((seq_n^ 1) . n) * ((seq_n^ 1) . n) = (seq_n^ 2) . n
hence ((seq_n^ 1) . n) * ((seq_n^ 1) . n) = 0 * ((seq_n^ 1) . n) by Def3
.= (seq_n^ 2) . n by A18, Def3 ;
::_thesis: verum
end;
supposeA19: n > 0 ; ::_thesis: ((seq_n^ 1) . n) * ((seq_n^ 1) . n) = (seq_n^ 2) . n
hence ((seq_n^ 1) . n) * ((seq_n^ 1) . n) = (n to_power 1) * ((seq_n^ 1) . n) by Def3
.= n to_power (1 + 1) by A16, A19, Def3
.= (seq_n^ 2) . n by A19, Def3 ;
::_thesis: verum
end;
end;
end;
t . n <= c * ((seq_n^ 1) . n) by A4, A7;
then (t . n) * (t . n) <= (c * ((seq_n^ 1) . n)) * (c * ((seq_n^ 1) . n)) by A8, Lm20;
then (t . n) * (t . n) <= (max (c,(c * c))) * ((seq_n^ 2) . n) by A17, A15, XXREAL_0:2;
hence (t (#) t) . n <= (max (c,(c * c))) * ((seq_n^ 2) . n) by SEQ_1:8; ::_thesis: (t (#) t) . n >= 0
(t . n) * (t . n) >= (t . n) * 0 by A8;
hence (t (#) t) . n >= 0 by SEQ_1:8; ::_thesis: verum
end;
A20: t (#) t is Element of Funcs (NAT,REAL) by FUNCT_2:8;
max (c,(c * c)) > 0 by A3, XXREAL_0:25;
hence f (#) f in Big_Oh (seq_n^ 2) by A1, A20, A6; ::_thesis: verum
end;
begin
theorem :: ASYMPT_1:16
ex s being eventually-positive Real_Sequence st
( s = seq_a^ (2,1,0) & 2 (#) (seq_n^ 1) in Big_Oh (seq_n^ 1) & not seq_a^ (2,2,0) in Big_Oh s )
proof
reconsider q = seq_a^ (2,1,0) as eventually-positive Real_Sequence ;
set p = seq_a^ (2,2,0);
set g = seq_n^ 1;
set f = 2 (#) (seq_n^ 1);
take q ; ::_thesis: ( q = seq_a^ (2,1,0) & 2 (#) (seq_n^ 1) in Big_Oh (seq_n^ 1) & not seq_a^ (2,2,0) in Big_Oh q )
thus q = seq_a^ (2,1,0) ; ::_thesis: ( 2 (#) (seq_n^ 1) in Big_Oh (seq_n^ 1) & not seq_a^ (2,2,0) in Big_Oh q )
A1: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_0_holds_
(_(2_(#)_(seq_n^_1))_._n_<=_2_*_((seq_n^_1)_._n)_&_(2_(#)_(seq_n^_1))_._n_>=_0_)
let n be Element of NAT ; ::_thesis: ( n >= 0 implies ( (2 (#) (seq_n^ 1)) . n <= 2 * ((seq_n^ 1) . n) & (2 (#) (seq_n^ 1)) . n >= 0 ) )
assume n >= 0 ; ::_thesis: ( (2 (#) (seq_n^ 1)) . n <= 2 * ((seq_n^ 1) . n) & (2 (#) (seq_n^ 1)) . n >= 0 )
thus (2 (#) (seq_n^ 1)) . n <= 2 * ((seq_n^ 1) . n) by SEQ_1:9; ::_thesis: (2 (#) (seq_n^ 1)) . n >= 0
A2: (seq_n^ 1) . n = n
proof
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: (seq_n^ 1) . n = n
hence (seq_n^ 1) . n = n by Def3; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: (seq_n^ 1) . n = n
hence (seq_n^ 1) . n = n to_power 1 by Def3
.= n by POWER:25 ;
::_thesis: verum
end;
end;
end;
2 * n >= 2 * 0 ;
hence (2 (#) (seq_n^ 1)) . n >= 0 by A2, SEQ_1:9; ::_thesis: verum
end;
2 (#) (seq_n^ 1) is Element of Funcs (NAT,REAL) by FUNCT_2:8;
hence 2 (#) (seq_n^ 1) in Big_Oh (seq_n^ 1) by A1; ::_thesis: not seq_a^ (2,2,0) in Big_Oh q
now__::_thesis:_not_seq_a^_(2,2,0)_in_Big_Oh_q
assume seq_a^ (2,2,0) in Big_Oh q ; ::_thesis: contradiction
then consider t being Element of Funcs (NAT,REAL) such that
A3: t = seq_a^ (2,2,0) and
A4: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (q . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
A5: c > 0 and
A6: for n being Element of NAT st n >= N holds
( t . n <= c * (q . n) & t . n >= 0 ) by A4;
ex n being Element of NAT st
( n >= N & t . n > c * (q . n) )
proof
take n = max (N,[/((log (2,c)) + 1)\]); ::_thesis: ( n is Element of REAL & n is Element of NAT & n >= N & t . n > c * (q . n) )
A7: n >= N by XXREAL_0:25;
n is Integer by XXREAL_0:16;
then reconsider n = 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;
((seq_a^ (2,2,0)) . 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 ((seq_a^ (2,2,0)) . n) * (2 to_power (- n)) > (c * (q . n)) * (2 to_power (- n)) by A8, A12, XXREAL_0:2;
hence ( n is Element of REAL & n is Element of NAT & n >= N & t . n > c * (q . n) ) by A3, A9, XREAL_1:64, XXREAL_0:25; ::_thesis: verum
end;
hence contradiction by A6; ::_thesis: verum
end;
hence not seq_a^ (2,2,0) in Big_Oh q ; ::_thesis: verum
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 = (seq_n^ (log (2,3))) /" (seq_n^ (159 / 100));
assume A1: log (2,3) < 159 / 100 ; ::_thesis: ( 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)) )
then A2: (log (2,3)) - (log (2,3)) < (159 / 100) - (log (2,3)) by XREAL_1:9;
A3: ((159 / 100) - (log (2,3))) / 2 <> 0 by A1;
A4: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N1_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N1_holds_
abs_((((seq_n^_(log_(2,3)))_/"_(seq_n^_(159_/_100)))_._n)_-_0)_<_p
A5: ((159 / 100) - (log (2,3))) * (1 / 2) < ((159 / 100) - (log (2,3))) * 1 by A2, XREAL_1:68;
let p be real number ; ::_thesis: ( p > 0 implies ex N1 being Element of NAT st
for n being Element of NAT st n >= N1 holds
abs ((((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n) - 0) < p )
assume A6: p > 0 ; ::_thesis: ex N1 being Element of NAT st
for n being Element of NAT st n >= N1 holds
abs ((((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n) - 0) < p
reconsider p1 = p as Real by XREAL_0:def_1;
A7: (1 / p1) to_power (1 / (((159 / 100) - (log (2,3))) / 2)) > 0 by A6, POWER:34;
set N1 = max ([/((1 / p1) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\],2);
A8: max ([/((1 / p1) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\],2) >= [/((1 / p) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\] by XXREAL_0:25;
A9: max ([/((1 / p1) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\],2) is Integer by XXREAL_0:16;
A10: max ([/((1 / p1) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\],2) >= 2 by XXREAL_0:25;
then A11: max ([/((1 / p1) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\],2) > 1 by XXREAL_0:2;
reconsider N1 = max ([/((1 / p1) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\],2) as Element of NAT by A10, A9, INT_1:3;
take N1 = N1; ::_thesis: for n being Element of NAT st n >= N1 holds
abs ((((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n) - 0) < p
let n be Element of NAT ; ::_thesis: ( n >= N1 implies abs ((((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n) - 0) < p )
A12: ((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n = ((seq_n^ (log (2,3))) . n) / ((seq_n^ (159 / 100)) . n) by Lm4;
assume A13: n >= N1 ; ::_thesis: abs ((((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n) - 0) < p
then (seq_n^ (log (2,3))) . n = n to_power (log (2,3)) by A10, Def3;
then A14: ((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n = (n to_power (log (2,3))) / (n to_power (159 / 100)) by A10, A13, A12, Def3
.= n to_power ((log (2,3)) - (159 / 100)) by A10, A13, POWER:29
.= n to_power (- ((159 / 100) - (log (2,3)))) ;
[/((1 / p) to_power (1 / (((159 / 100) - (log (2,3))) / 2)))\] >= (1 / p) to_power (1 / (((159 / 100) - (log (2,3))) / 2)) by INT_1:def_7;
then N1 >= (1 / p) to_power (1 / (((159 / 100) - (log (2,3))) / 2)) by A8, XXREAL_0:2;
then n >= (1 / p) to_power (1 / (((159 / 100) - (log (2,3))) / 2)) by A13, XXREAL_0:2;
then n to_power (((159 / 100) - (log (2,3))) / 2) >= ((1 / p) to_power (1 / (((159 / 100) - (log (2,3))) / 2))) to_power (((159 / 100) - (log (2,3))) / 2) by A2, A7, Lm6;
then n to_power (((159 / 100) - (log (2,3))) / 2) >= (1 / p1) to_power ((1 / (((159 / 100) - (log (2,3))) / 2)) * (((159 / 100) - (log (2,3))) / 2)) by A6, POWER:33;
then n to_power (((159 / 100) - (log (2,3))) / 2) >= (1 / p) to_power 1 by A3, XCMPLX_1:87;
then n to_power (((159 / 100) - (log (2,3))) / 2) >= 1 / p1 by POWER:25;
then 1 / (n to_power (((159 / 100) - (log (2,3))) / 2)) <= 1 / (p ") by A6, XREAL_1:85;
then A15: n to_power (- (((159 / 100) - (log (2,3))) / 2)) <= p by A10, A13, POWER:28;
n > 1 by A11, A13, XXREAL_0:2;
then A16: n to_power (((159 / 100) - (log (2,3))) / 2) < n to_power ((159 / 100) - (log (2,3))) by A5, POWER:39;
n to_power (((159 / 100) - (log (2,3))) / 2) > 0 by A10, A13, POWER:34;
then 1 / (n to_power (((159 / 100) - (log (2,3))) / 2)) > 1 / (n to_power ((159 / 100) - (log (2,3)))) by A16, XREAL_1:88;
then n to_power (- (((159 / 100) - (log (2,3))) / 2)) > 1 / (n to_power ((159 / 100) - (log (2,3)))) by A10, A13, POWER:28;
then ((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n < n to_power (- (((159 / 100) - (log (2,3))) / 2)) by A10, A13, A14, POWER:28;
then A17: ((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n < p by A15, XXREAL_0:2;
((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n > 0 by A10, A13, A14, POWER:34;
hence abs ((((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) . n) - 0) < p by A17, ABSVALUE:def_1; ::_thesis: verum
end;
then A18: (seq_n^ (log (2,3))) /" (seq_n^ (159 / 100)) is convergent by SEQ_2:def_6;
then A19: lim ((seq_n^ (log (2,3))) /" (seq_n^ (159 / 100))) = 0 by A4, SEQ_2:def_7;
hence seq_n^ (log (2,3)) in Big_Oh (seq_n^ (159 / 100)) by A18, ASYMPT_0:16; ::_thesis: ( 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)) )
A20: not seq_n^ (159 / 100) in Big_Oh (seq_n^ (log (2,3))) by A18, A19, ASYMPT_0:16;
hence not seq_n^ (log (2,3)) in Big_Omega (seq_n^ (159 / 100)) by ASYMPT_0:19; ::_thesis: not seq_n^ (log (2,3)) in Big_Theta (seq_n^ (159 / 100))
not seq_n^ (log (2,3)) in Big_Omega (seq_n^ (159 / 100)) by A20, ASYMPT_0:19;
hence not seq_n^ (log (2,3)) in Big_Theta (seq_n^ (159 / 100)) by XBOOLE_0:def_4; ::_thesis: verum
end;
begin
theorem :: ASYMPT_1:18
for f, g being Real_Sequence st ( for n being Element of NAT holds f . n = n mod 2 ) & ( for n being Element of NAT holds g . n = (n + 1) mod 2 ) holds
ex s, s1 being eventually-nonnegative Real_Sequence st
( s = f & s1 = g & not s in Big_Oh s1 & not s1 in Big_Oh s )
proof
let f, g be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds f . n = n mod 2 ) & ( for n being Element of NAT holds g . n = (n + 1) mod 2 ) implies ex s, s1 being eventually-nonnegative Real_Sequence st
( s = f & s1 = g & not s in Big_Oh s1 & not s1 in Big_Oh s ) )
assume that
A1: for n being Element of NAT holds f . n = n mod 2 and
A2: for n being Element of NAT holds g . n = (n + 1) mod 2 ; ::_thesis: ex s, s1 being eventually-nonnegative Real_Sequence st
( s = f & s1 = g & not s in Big_Oh s1 & not s1 in Big_Oh s )
g is eventually-nonnegative
proof
take 0 ; :: according to ASYMPT_0:def_2 ::_thesis: for b1 being Element of NAT holds
( not 0 <= b1 or 0 <= g . b1 )
let n be Element of NAT ; ::_thesis: ( not 0 <= n or 0 <= g . n )
assume n >= 0 ; ::_thesis: 0 <= g . n
A3: g . n = (n + 1) mod 2 by A2;
percases ( g . n = 0 or g . n = 1 ) by A3, NAT_D:12;
suppose g . n = 0 ; ::_thesis: 0 <= g . n
hence 0 <= g . n ; ::_thesis: verum
end;
suppose g . n = 1 ; ::_thesis: 0 <= g . n
hence 0 <= g . n ; ::_thesis: verum
end;
end;
end;
then reconsider g = g as eventually-nonnegative Real_Sequence ;
f is eventually-nonnegative
proof
take 0 ; :: according to ASYMPT_0:def_2 ::_thesis: for b1 being Element of NAT holds
( not 0 <= b1 or 0 <= f . b1 )
let n be Element of NAT ; ::_thesis: ( not 0 <= n or 0 <= f . n )
assume n >= 0 ; ::_thesis: 0 <= f . n
A4: f . n = n mod 2 by A1;
percases ( f . n = 0 or f . n = 1 ) by A4, NAT_D:12;
suppose f . n = 0 ; ::_thesis: 0 <= f . n
hence 0 <= f . n ; ::_thesis: verum
end;
suppose f . n = 1 ; ::_thesis: 0 <= f . n
hence 0 <= f . n ; ::_thesis: verum
end;
end;
end;
then reconsider f = f as eventually-nonnegative Real_Sequence ;
A5: now__::_thesis:_not_g_in_Big_Oh_f
assume g in Big_Oh f ; ::_thesis: contradiction
then consider t being Element of Funcs (NAT,REAL) such that
A6: t = g and
A7: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
c > 0 and
A8: for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) by A7;
ex n being Element of NAT st
( n >= N & t . n > c * (f . n) )
proof
percases ( N mod 2 = 0 or N mod 2 = 1 ) by NAT_D:12;
supposeA9: N mod 2 = 0 ; ::_thesis: ex n being Element of NAT st
( n >= N & t . n > c * (f . n) )
then f . N = 0 by A1;
then A10: c * (f . N) = 0 ;
t . N = (N + 1) mod 2 by A2, A6
.= (0 + (1 mod 2)) mod 2 by A9, EULER_2:6
.= (0 + 1) mod 2 by NAT_D:14
.= 1 by NAT_D:14 ;
hence ex n being Element of NAT st
( n >= N & t . n > c * (f . n) ) by A10; ::_thesis: verum
end;
supposeA11: N mod 2 = 1 ; ::_thesis: ex n being Element of NAT st
( n >= N & t . n > c * (f . n) )
f . (N + 1) = (N + 1) mod 2 by A1
.= (1 + (1 mod 2)) mod 2 by A11, EULER_2:6
.= (1 + 1) mod 2 by NAT_D:14
.= 0 by NAT_D:25 ;
then A12: c * (f . (N + 1)) = 0 ;
A13: N + 1 >= N by NAT_1:13;
t . (N + 1) = ((N + 1) + 1) mod 2 by A2, A6
.= (N + (1 + 1)) mod 2
.= (1 + (2 mod 2)) mod 2 by A11, EULER_2:6
.= (1 + 0) mod 2 by NAT_D:25
.= 1 by NAT_D:14 ;
hence ex n being Element of NAT st
( n >= N & t . n > c * (f . n) ) by A13, A12; ::_thesis: verum
end;
end;
end;
hence contradiction by A8; ::_thesis: verum
end;
take f ; ::_thesis: ex s1 being eventually-nonnegative Real_Sequence st
( f = f & s1 = g & not f in Big_Oh s1 & not s1 in Big_Oh f )
take g ; ::_thesis: ( f = f & g = g & not f in Big_Oh g & not g in Big_Oh f )
now__::_thesis:_not_f_in_Big_Oh_g
assume f in Big_Oh g ; ::_thesis: contradiction
then consider t being Element of Funcs (NAT,REAL) such that
A14: t = f and
A15: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (g . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
c > 0 and
A16: for n being Element of NAT st n >= N holds
( t . n <= c * (g . n) & t . n >= 0 ) by A15;
ex n being Element of NAT st
( n >= N & t . n > c * (g . n) )
proof
percases ( N mod 2 = 0 or N mod 2 = 1 ) by NAT_D:12;
supposeA17: N mod 2 = 0 ; ::_thesis: ex n being Element of NAT st
( n >= N & t . n > c * (g . n) )
g . (N + 1) = ((N + 1) + 1) mod 2 by A2
.= (N + (1 + 1)) mod 2
.= (0 + (2 mod 2)) mod 2 by A17, EULER_2:6
.= (0 + 0) mod 2 by NAT_D:25
.= 0 by NAT_D:26 ;
then A18: c * (g . (N + 1)) = 0 ;
A19: N + 1 >= N by NAT_1:13;
t . (N + 1) = (N + 1) mod 2 by A1, A14
.= (0 + (1 mod 2)) mod 2 by A17, EULER_2:6
.= (0 + 1) mod 2 by NAT_D:14
.= 1 by NAT_D:14 ;
hence ex n being Element of NAT st
( n >= N & t . n > c * (g . n) ) by A19, A18; ::_thesis: verum
end;
supposeA20: N mod 2 = 1 ; ::_thesis: ex n being Element of NAT st
( n >= N & t . n > c * (g . n) )
g . N = (N + 1) mod 2 by A2
.= (1 + (1 mod 2)) mod 2 by A20, EULER_2:6
.= (1 + 1) mod 2 by NAT_D:14
.= 0 by NAT_D:25 ;
then A21: c * (g . N) = 0 ;
t . N = 1 by A1, A14, A20;
hence ex n being Element of NAT st
( n >= N & t . n > c * (g . n) ) by A21; ::_thesis: verum
end;
end;
end;
hence contradiction by A16; ::_thesis: verum
end;
hence ( f = f & g = g & not f in Big_Oh g & not g in Big_Oh f ) by A5; ::_thesis: verum
end;
begin
theorem :: ASYMPT_1:19
for f, g being eventually-nonnegative Real_Sequence holds
( Big_Oh f = Big_Oh g iff f in Big_Theta g )
proof
let f, g be eventually-nonnegative Real_Sequence; ::_thesis: ( Big_Oh f = Big_Oh g iff f in Big_Theta g )
hereby ::_thesis: ( f in Big_Theta g implies Big_Oh f = Big_Oh g )
assume A1: Big_Oh f = Big_Oh g ; ::_thesis: f in Big_Theta 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; ::_thesis: verum
end;
assume A3: f in Big_Theta g ; ::_thesis: Big_Oh f = Big_Oh g
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_Big_Oh_f_implies_x_in_Big_Oh_g_)_&_(_x_in_Big_Oh_g_implies_x_in_Big_Oh_f_)_)
let x be set ; ::_thesis: ( ( x in Big_Oh f implies x in Big_Oh g ) & ( x in Big_Oh g implies x in Big_Oh f ) )
hereby ::_thesis: ( x in Big_Oh g implies x in Big_Oh f )
assume x in Big_Oh f ; ::_thesis: x in Big_Oh g
then consider t being Element of Funcs (NAT,REAL) such that
A4: x = t and
A5: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
c > 0 and
A6: for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) by A5;
now__::_thesis:_ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
t_._n_>=_0
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
t . n >= 0
let n be Element of NAT ; ::_thesis: ( n >= N implies t . n >= 0 )
assume n >= N ; ::_thesis: t . n >= 0
hence t . n >= 0 by A6; ::_thesis: verum
end;
then A7: t is eventually-nonnegative by ASYMPT_0:def_2;
A8: 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, A7, A8, ASYMPT_0:12; ::_thesis: verum
end;
assume x in Big_Oh g ; ::_thesis: x in Big_Oh f
then consider t being Element of Funcs (NAT,REAL) such that
A9: x = t and
A10: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (g . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
c > 0 and
A11: for n being Element of NAT st n >= N holds
( t . n <= c * (g . n) & t . n >= 0 ) by A10;
now__::_thesis:_ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
t_._n_>=_0
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
t . n >= 0
let n be Element of NAT ; ::_thesis: ( n >= N implies t . n >= 0 )
assume n >= N ; ::_thesis: t . n >= 0
hence t . n >= 0 by A11; ::_thesis: verum
end;
then A12: t is eventually-nonnegative by ASYMPT_0:def_2;
f in Big_Omega g by A3, XBOOLE_0:def_4;
then A13: g in Big_Oh f by ASYMPT_0:19;
t in Big_Oh g by A10;
hence x in Big_Oh f by A9, A12, A13, ASYMPT_0:12; ::_thesis: verum
end;
hence Big_Oh f = Big_Oh g by TARSKI:1; ::_thesis: verum
end;
theorem :: ASYMPT_1:20
for f, g being eventually-nonnegative Real_Sequence holds
( f in Big_Theta g iff Big_Theta f = Big_Theta g )
proof
let f, g be eventually-nonnegative Real_Sequence; ::_thesis: ( f in Big_Theta g iff Big_Theta f = Big_Theta g )
A1: Big_Theta g = { s where s is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (g . n) <= s . n & s . n <= c * (g . n) ) ) ) } by ASYMPT_0:27;
consider N2 being Element of NAT such that
A2: for n being Element of NAT st n >= N2 holds
g . n >= 0 by ASYMPT_0:def_2;
consider N1 being Element of NAT such that
A3: for n being Element of NAT st n >= N1 holds
f . n >= 0 by ASYMPT_0:def_2;
A4: Big_Theta f = { s where s is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (f . n) <= s . n & s . n <= c * (f . n) ) ) ) } by ASYMPT_0:27;
hereby ::_thesis: ( Big_Theta f = Big_Theta g implies f in Big_Theta g )
assume A5: f in Big_Theta g ; ::_thesis: Big_Theta f = Big_Theta g
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_Big_Theta_f_implies_x_in_Big_Theta_g_)_&_(_x_in_Big_Theta_g_implies_x_in_Big_Theta_f_)_)
let x be set ; ::_thesis: ( ( x in Big_Theta f implies x in Big_Theta g ) & ( x in Big_Theta g implies x in Big_Theta f ) )
A6: g in Big_Theta f by A5, ASYMPT_0:29;
hereby ::_thesis: ( x in Big_Theta g implies x in Big_Theta f )
assume x in Big_Theta f ; ::_thesis: x in Big_Theta g
then consider s being Element of Funcs (NAT,REAL) such that
A7: s = x and
A8: ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (f . n) <= s . n & s . n <= c * (f . n) ) ) ) by A4;
consider c, d being Real, N3 being Element of NAT such that
c > 0 and
A9: d > 0 and
A10: for n being Element of NAT st n >= N3 holds
( d * (f . n) <= s . n & s . n <= c * (f . n) ) by A8;
set N = max (N1,N3);
A11: max (N1,N3) >= N3 by XXREAL_0:25;
A12: max (N1,N3) >= N1 by XXREAL_0:25;
now__::_thesis:_ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
s_._n_>=_0
take N = max (N1,N3); ::_thesis: for n being Element of NAT st n >= N holds
s . n >= 0
let n be Element of NAT ; ::_thesis: ( n >= N implies s . n >= 0 )
assume A13: n >= N ; ::_thesis: s . n >= 0
then n >= N1 by A12, XXREAL_0:2;
then f . n >= 0 by A3;
then A14: d * (f . n) >= d * 0 by A9;
n >= N3 by A11, A13, XXREAL_0:2;
hence s . n >= 0 by A10, A14; ::_thesis: verum
end;
then A15: s is eventually-nonnegative by ASYMPT_0:def_2;
s in Big_Theta f by A4, A8;
hence x in Big_Theta g by A5, A7, A15, ASYMPT_0:30; ::_thesis: verum
end;
assume x in Big_Theta g ; ::_thesis: x in Big_Theta f
then consider s being Element of Funcs (NAT,REAL) such that
A16: s = x and
A17: ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (g . n) <= s . n & s . n <= c * (g . n) ) ) ) by A1;
consider c, d being Real, N3 being Element of NAT such that
c > 0 and
A18: d > 0 and
A19: for n being Element of NAT st n >= N3 holds
( d * (g . n) <= s . n & s . n <= c * (g . n) ) by A17;
set N = max (N2,N3);
A20: max (N2,N3) >= N3 by XXREAL_0:25;
A21: max (N2,N3) >= N2 by XXREAL_0:25;
now__::_thesis:_ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
s_._n_>=_0
take N = max (N2,N3); ::_thesis: for n being Element of NAT st n >= N holds
s . n >= 0
let n be Element of NAT ; ::_thesis: ( n >= N implies s . n >= 0 )
assume A22: n >= N ; ::_thesis: s . n >= 0
then n >= N2 by A21, XXREAL_0:2;
then g . n >= 0 by A2;
then A23: d * (g . n) >= d * 0 by A18;
n >= N3 by A20, A22, XXREAL_0:2;
hence s . n >= 0 by A19, A23; ::_thesis: verum
end;
then A24: s is eventually-nonnegative by ASYMPT_0:def_2;
s in Big_Theta g by A1, A17;
hence x in Big_Theta f by A16, A24, A6, ASYMPT_0:30; ::_thesis: verum
end;
hence Big_Theta f = Big_Theta g by TARSKI:1; ::_thesis: verum
end;
assume Big_Theta f = Big_Theta g ; ::_thesis: f in Big_Theta g
hence f in Big_Theta g by ASYMPT_0:28; ::_thesis: verum
end;
begin
Lm21: for n being Element of NAT holds ((n ^2) - n) + 1 > 0
proof
defpred S1[ Element of NAT ] means (($1 ^2) - $1) + 1 > 0 ;
A1: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume ((k ^2) - k) + 1 > 0 ; ::_thesis: S1[k + 1]
then (((k ^2) - k) + 1) + (2 * k) > 0 + 0 ;
hence S1[k + 1] ; ::_thesis: verum
end;
A2: S1[ 0 ] ;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A2, A1);
hence for n being Element of NAT holds ((n ^2) - n) + 1 > 0 ; ::_thesis: verum
end;
Lm22: for f, g being Real_Sequence
for N being Element of NAT
for c being Real st f is convergent & lim f = c & ( for n being Element of NAT st n >= N holds
f . n = g . n ) holds
( g is convergent & lim g = c )
proof
let f, g be Real_Sequence; ::_thesis: for N being Element of NAT
for c being Real st f is convergent & lim f = c & ( for n being Element of NAT st n >= N holds
f . n = g . n ) holds
( g is convergent & lim g = c )
let N be Element of NAT ; ::_thesis: for c being Real st f is convergent & lim f = c & ( for n being Element of NAT st n >= N holds
f . n = g . n ) holds
( g is convergent & lim g = c )
let c be Real; ::_thesis: ( f is convergent & lim f = c & ( for n being Element of NAT st n >= N holds
f . n = g . n ) implies ( g is convergent & lim g = c ) )
assume that
A1: f is convergent and
A2: lim f = c and
A3: for n being Element of NAT st n >= N holds
f . n = g . n ; ::_thesis: ( g is convergent & lim g = c )
A4: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N1_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N1_holds_
abs_((g_._n)_-_c)_<_p
let p be real number ; ::_thesis: ( p > 0 implies ex N1 being Element of NAT st
for n being Element of NAT st n >= N1 holds
abs ((g . n) - c) < p )
assume p > 0 ; ::_thesis: ex N1 being Element of NAT st
for n being Element of NAT st n >= N1 holds
abs ((g . n) - c) < p
then consider M being Element of NAT such that
A5: for n being Element of NAT st n >= M holds
abs ((f . n) - c) < p by A1, A2, SEQ_2:def_7;
set N1 = max (N,M);
A6: max (N,M) >= N by XXREAL_0:25;
take N1 = max (N,M); ::_thesis: for n being Element of NAT st n >= N1 holds
abs ((g . n) - c) < p
let n be Element of NAT ; ::_thesis: ( n >= N1 implies abs ((g . n) - c) < p )
assume A7: n >= N1 ; ::_thesis: abs ((g . n) - c) < p
N1 >= M by XXREAL_0:25;
then n >= M by A7, XXREAL_0:2;
then abs ((f . n) - c) < p by A5;
hence abs ((g . n) - c) < p by A3, A6, A7, XXREAL_0:2; ::_thesis: verum
end;
hence g is convergent by SEQ_2:def_6; ::_thesis: lim g = c
hence lim g = c by A4, SEQ_2:def_7; ::_thesis: verum
end;
Lm23: for n being Element of NAT st n >= 1 holds
((n ^2) - n) + 1 <= n ^2
proof
let n be Element of NAT ; ::_thesis: ( n >= 1 implies ((n ^2) - n) + 1 <= n ^2 )
assume A1: n >= 1 ; ::_thesis: ((n ^2) - n) + 1 <= n ^2
now__::_thesis:_not_((n_^2)_-_n)_+_1_>_n_^2
assume ((n ^2) - n) + 1 > n ^2 ; ::_thesis: contradiction
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; ::_thesis: verum
end;
hence ((n ^2) - n) + 1 <= n ^2 ; ::_thesis: verum
end;
Lm24: for n being Element of NAT st n >= 1 holds
n ^2 <= 2 * (((n ^2) - n) + 1)
proof
defpred S1[ Nat] means $1 ^2 <= 2 * ((($1 ^2) - $1) + 1);
A1: for k being Nat st k >= 1 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 1 & S1[k] implies S1[k + 1] )
assume that
A2: k >= 1 and
A3: k ^2 <= 2 * (((k ^2) - k) + 1) ; ::_thesis: S1[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 S1[k + 1] by A4, XXREAL_0:2; ::_thesis: verum
end;
A6: S1[1] ;
for n being Nat st n >= 1 holds
S1[n] from NAT_1:sch_8(A6, A1);
hence for n being Element of NAT st n >= 1 holds
n ^2 <= 2 * (((n ^2) - n) + 1) ; ::_thesis: verum
end;
Lm25: for e being Real st 0 < e & e < 1 holds
ex N being Element of NAT st
for n being Element of NAT 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; ::_thesis: ( 0 < e & e < 1 implies ex N being Element of NAT st
for n being Element of NAT st n >= N holds
(n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n)) )
assume that
A1: 0 < e and
A2: e < 1 ; ::_thesis: ex N being Element of NAT st
for n being Element of NAT st n >= N holds
(n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n))
set d = log (2,(1 + e));
set g = seq_n^ e;
set h = seq_logn /" (seq_n^ e);
A3: seq_logn /" (seq_n^ e) is convergent by A1, Lm11;
A4: lim (seq_logn /" (seq_n^ e)) = 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: log (2,(1 + e)) > 0 by POWER:51;
then (log (2,(1 + e))) * (1 / 16) > (log (2,(1 + e))) * 0 by XREAL_1:68;
then consider N being Element of NAT such that
A6: for n being Element of NAT st n >= N holds
abs (((seq_logn /" (seq_n^ e)) . n) - 0) < (log (2,(1 + e))) / 16 by A3, A4, SEQ_2:def_7;
ex N being Element of NAT st
for n being Element of NAT st n >= N holds
(n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n))
proof
set N1 = max (2,N);
A7: max (2,N) >= 2 by XXREAL_0:25;
A8: max (2,N) >= N by XXREAL_0:25;
now__::_thesis:_ex_N1_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N1_holds_
(n_*_(log_(2,(1_+_e))))_-_(8_*_(log_(2,n)))_>_8_*_(log_(2,n))
take N1 = max (2,N); ::_thesis: for n being Element of NAT st n >= N1 holds
(n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n))
let n be Element of NAT ; ::_thesis: ( n >= N1 implies (n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n)) )
assume A9: n >= N1 ; ::_thesis: (n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n))
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: (seq_logn /" (seq_n^ e)) . n = (seq_logn . n) / ((seq_n^ e) . n) by Lm4
.= (log (2,n)) / ((seq_n^ e) . 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 abs (((seq_logn /" (seq_n^ e)) . n) - 0) < (log (2,(1 + e))) / 16 by A6;
then (seq_logn /" (seq_n^ e)) . n < (log (2,(1 + e))) / 16 by A12, A13, A10, ABSVALUE:def_1;
then A15: (log (2,n)) / n < (log (2,(1 + e))) / 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 / ((log (2,(1 + e))) / 16) by A15, XREAL_1:88;
then n / (log (2,n)) > 1 / ((log (2,(1 + e))) / 16) by XCMPLX_1:57;
then n / (log (2,n)) > 16 / (log (2,(1 + e))) by XCMPLX_1:57;
then (log (2,(1 + e))) * (n / (log (2,n))) > (16 / (log (2,(1 + e)))) * (log (2,(1 + e))) by A5, XREAL_1:68;
then (log (2,(1 + e))) * (n / (log (2,n))) > 16 by A5, XCMPLX_1:87;
then ((log (2,(1 + e))) * (n / (log (2,n)))) * (log (2,n)) > 16 * (log (2,n)) by A12, XREAL_1:68;
then (log (2,(1 + e))) * ((n / (log (2,n))) * (log (2,n))) > 16 * (log (2,n)) ;
then (log (2,(1 + e))) * n > (8 + 8) * (log (2,n)) by A12, XCMPLX_1:87;
then ((log (2,(1 + e))) * n) - (8 * (log (2,n))) > ((8 * (log (2,n))) + (8 * (log (2,n)))) - (8 * (log (2,n))) by XREAL_1:9;
hence (n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n)) ; ::_thesis: verum
end;
hence ex N being Element of NAT st
for n being Element of NAT st n >= N holds
(n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n)) ; ::_thesis: verum
end;
hence ex N being Element of NAT st
for n being Element of NAT st n >= N holds
(n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n)) ; ::_thesis: verum
end;
theorem :: ASYMPT_1:21
for e being Real
for f being Real_Sequence st 0 < e & ( for n being Element of NAT st n > 0 holds
f . n = n * (log (2,n)) ) holds
ex s being eventually-positive Real_Sequence st
( s = f & Big_Oh s c= Big_Oh (seq_n^ (1 + e)) & not Big_Oh s = Big_Oh (seq_n^ (1 + e)) )
proof
set seq = seq_logn ;
let e be Real; ::_thesis: for f being Real_Sequence st 0 < e & ( for n being Element of NAT st n > 0 holds
f . n = n * (log (2,n)) ) holds
ex s being eventually-positive Real_Sequence st
( s = f & Big_Oh s c= Big_Oh (seq_n^ (1 + e)) & not Big_Oh s = Big_Oh (seq_n^ (1 + e)) )
let f be Real_Sequence; ::_thesis: ( 0 < e & ( for n being Element of NAT st n > 0 holds
f . n = n * (log (2,n)) ) implies ex s being eventually-positive Real_Sequence st
( s = f & Big_Oh s c= Big_Oh (seq_n^ (1 + e)) & not Big_Oh s = Big_Oh (seq_n^ (1 + e)) ) )
assume that
A1: 0 < e and
A2: for n being Element of NAT st n > 0 holds
f . n = n * (log (2,n)) ; ::_thesis: ex s being eventually-positive Real_Sequence st
( s = f & Big_Oh s c= Big_Oh (seq_n^ (1 + e)) & not Big_Oh s = Big_Oh (seq_n^ (1 + e)) )
set seq1 = seq_n^ e;
set p = seq_logn /" (seq_n^ e);
A3: lim (seq_logn /" (seq_n^ e)) = 0 by A1, Lm11;
f is eventually-positive
proof
take 2 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 2 <= b1 or not f . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 2 <= n or not f . n <= 0 )
assume A4: n >= 2 ; ::_thesis: not f . n <= 0
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 not f . n <= 0 by A2, A4; ::_thesis: verum
end;
then reconsider f = f as eventually-positive Real_Sequence ;
set g = seq_n^ (1 + e);
set h = f /" (seq_n^ (1 + e));
A5: for n being Element of NAT st n >= 1 holds
(f /" (seq_n^ (1 + e))) . n = (seq_logn /" (seq_n^ e)) . n
proof
let n be Element of NAT ; ::_thesis: ( n >= 1 implies (f /" (seq_n^ (1 + e))) . n = (seq_logn /" (seq_n^ e)) . n )
assume A6: n >= 1 ; ::_thesis: (f /" (seq_n^ (1 + e))) . n = (seq_logn /" (seq_n^ e)) . n
(f /" (seq_n^ (1 + e))) . n = (f . n) / ((seq_n^ (1 + e)) . n) by Lm4
.= (n * (log (2,n))) / ((seq_n^ (1 + e)) . n) by A2, A6
.= (n * (log (2,n))) / (n to_power (1 + e)) by A6, 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 A6, POWER:29
.= (log (2,n)) * (n to_power (1 + ((- 1) + (- e))))
.= (log (2,n)) * (1 / (n to_power e)) by A6, POWER:28
.= (log (2,n)) / (n to_power e)
.= (seq_logn . n) / (n to_power e) by A6, Def2
.= (seq_logn . n) / ((seq_n^ e) . n) by A6, Def3
.= (seq_logn /" (seq_n^ e)) . n by Lm4 ;
hence (f /" (seq_n^ (1 + e))) . n = (seq_logn /" (seq_n^ e)) . n ; ::_thesis: verum
end;
A7: seq_logn /" (seq_n^ e) is convergent by A1, Lm11;
then A8: lim (f /" (seq_n^ (1 + e))) = 0 by A3, A5, Lm22;
A9: f /" (seq_n^ (1 + e)) is convergent by A7, A3, A5, Lm22;
then not seq_n^ (1 + e) in Big_Oh f by A8, ASYMPT_0:16;
then A10: not f in Big_Omega (seq_n^ (1 + e)) by ASYMPT_0:19;
take f ; ::_thesis: ( f = f & Big_Oh f c= Big_Oh (seq_n^ (1 + e)) & not Big_Oh f = Big_Oh (seq_n^ (1 + e)) )
f in Big_Oh (seq_n^ (1 + e)) by A9, A8, ASYMPT_0:16;
hence ( f = f & Big_Oh f c= Big_Oh (seq_n^ (1 + e)) & not Big_Oh f = Big_Oh (seq_n^ (1 + e)) ) by A10, Th4; ::_thesis: verum
end;
theorem :: ASYMPT_1:22
for e being Real
for g being Real_Sequence st e < 1 & ( for n being Element of NAT st n > 1 holds
g . n = (n to_power 2) / (log (2,n)) ) holds
ex s being eventually-positive Real_Sequence st
( s = g & Big_Oh (seq_n^ (1 + e)) c= Big_Oh s & not Big_Oh (seq_n^ (1 + e)) = Big_Oh s )
proof
set seq = seq_logn ;
let e be Real; ::_thesis: for g being Real_Sequence st e < 1 & ( for n being Element of NAT st n > 1 holds
g . n = (n to_power 2) / (log (2,n)) ) holds
ex s being eventually-positive Real_Sequence st
( s = g & Big_Oh (seq_n^ (1 + e)) c= Big_Oh s & not Big_Oh (seq_n^ (1 + e)) = Big_Oh s )
let g be Real_Sequence; ::_thesis: ( e < 1 & ( for n being Element of NAT st n > 1 holds
g . n = (n to_power 2) / (log (2,n)) ) implies ex s being eventually-positive Real_Sequence st
( s = g & Big_Oh (seq_n^ (1 + e)) c= Big_Oh s & not Big_Oh (seq_n^ (1 + e)) = Big_Oh s ) )
assume that
A1: e < 1 and
A2: for n being Element of NAT st n > 1 holds
g . n = (n to_power 2) / (log (2,n)) ; ::_thesis: ex s being eventually-positive Real_Sequence st
( s = g & Big_Oh (seq_n^ (1 + e)) c= Big_Oh s & not Big_Oh (seq_n^ (1 + e)) = Big_Oh s )
set seq1 = seq_n^ (1 - e);
set p = seq_logn /" (seq_n^ (1 - e));
set f = seq_n^ (1 + e);
set h = (seq_n^ (1 + e)) /" g;
g is eventually-positive
proof
take 2 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 2 <= b1 or not g . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 2 <= n or not g . n <= 0 )
assume A3: n >= 2 ; ::_thesis: not g . n <= 0
then log (2,n) >= log (2,2) by PRE_FF:10;
then A4: log (2,n) >= 1 by POWER:52;
n > 1 by A3, XXREAL_0:2;
then A5: g . n = (n to_power 2) / (log (2,n)) by A2
.= (n to_power 2) * ((log (2,n)) ") ;
n to_power 2 > 0 by A3, POWER:34;
then (n to_power 2) * ((log (2,n)) ") > (n to_power 2) * 0 by A4, XREAL_1:68;
hence not g . n <= 0 by A5; ::_thesis: verum
end;
then reconsider g = g as eventually-positive Real_Sequence ;
A6: (1 + e) - 2 = e - 1 ;
A7: for n being Element of NAT st n >= 2 holds
((seq_n^ (1 + e)) /" g) . n = (seq_logn /" (seq_n^ (1 - e))) . n
proof
let n be Element of NAT ; ::_thesis: ( n >= 2 implies ((seq_n^ (1 + e)) /" g) . n = (seq_logn /" (seq_n^ (1 - e))) . n )
assume A8: n >= 2 ; ::_thesis: ((seq_n^ (1 + e)) /" g) . n = (seq_logn /" (seq_n^ (1 - e))) . n
then A9: n > 1 by XXREAL_0:2;
((seq_n^ (1 + e)) /" g) . n = ((seq_n^ (1 + e)) . n) / (g . n) by Lm4
.= (n to_power (1 + e)) / (g . n) by A8, Def3
.= (n to_power (1 + e)) / ((n to_power 2) / (log (2,n))) by A2, A9
.= (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 A6, A8, POWER:29
.= (log (2,n)) * (1 / (n to_power (1 - e))) by A8, POWER:28
.= (log (2,n)) / (n to_power (1 - e))
.= (seq_logn . n) / (n to_power (1 - e)) by A8, Def2
.= (seq_logn . n) / ((seq_n^ (1 - e)) . n) by A8, Def3
.= (seq_logn /" (seq_n^ (1 - e))) . n by Lm4 ;
hence ((seq_n^ (1 + e)) /" g) . n = (seq_logn /" (seq_n^ (1 - e))) . n ; ::_thesis: verum
end;
take g ; ::_thesis: ( g = g & Big_Oh (seq_n^ (1 + e)) c= Big_Oh g & not Big_Oh (seq_n^ (1 + e)) = Big_Oh g )
0 + e < 1 by A1;
then A10: 0 < 1 - e by XREAL_1:20;
then A11: seq_logn /" (seq_n^ (1 - e)) is convergent by Lm11;
A12: lim (seq_logn /" (seq_n^ (1 - e))) = 0 by A10, Lm11;
then A13: lim ((seq_n^ (1 + e)) /" g) = 0 by A11, A7, Lm22;
A14: (seq_n^ (1 + e)) /" g is convergent by A11, A12, A7, Lm22;
then not g in Big_Oh (seq_n^ (1 + e)) by A13, ASYMPT_0:16;
then A15: not seq_n^ (1 + e) in Big_Omega g by ASYMPT_0:19;
seq_n^ (1 + e) in Big_Oh g by A14, A13, ASYMPT_0:16;
hence ( g = g & Big_Oh (seq_n^ (1 + e)) c= Big_Oh g & not Big_Oh (seq_n^ (1 + e)) = Big_Oh g ) by A15, Th4; ::_thesis: verum
end;
theorem :: ASYMPT_1:23
for f being Real_Sequence st ( for n being Element of NAT st n > 1 holds
f . n = (n to_power 2) / (log (2,n)) ) holds
ex s being eventually-positive Real_Sequence st
( s = f & Big_Oh s c= Big_Oh (seq_n^ 8) & not Big_Oh s = Big_Oh (seq_n^ 8) )
proof
set g = seq_n^ 8;
let f be Real_Sequence; ::_thesis: ( ( for n being Element of NAT st n > 1 holds
f . n = (n to_power 2) / (log (2,n)) ) implies ex s being eventually-positive Real_Sequence st
( s = f & Big_Oh s c= Big_Oh (seq_n^ 8) & not Big_Oh s = Big_Oh (seq_n^ 8) ) )
assume A1: for n being Element of NAT st n > 1 holds
f . n = (n to_power 2) / (log (2,n)) ; ::_thesis: ex s being eventually-positive Real_Sequence st
( s = f & Big_Oh s c= Big_Oh (seq_n^ 8) & not Big_Oh s = Big_Oh (seq_n^ 8) )
A2: f is eventually-positive
proof
take 2 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 2 <= b1 or not f . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 2 <= n or not f . n <= 0 )
assume A3: n >= 2 ; ::_thesis: not f . n <= 0
then log (2,n) >= log (2,2) by PRE_FF:10;
then A4: log (2,n) >= 1 by POWER:52;
n > 1 by A3, XXREAL_0:2;
then A5: f . n = (n to_power 2) / (log (2,n)) by A1
.= (n to_power 2) * ((log (2,n)) ") ;
n to_power 2 > 0 by A3, POWER:34;
then (n to_power 2) * ((log (2,n)) ") > (n to_power 2) * 0 by A4, XREAL_1:68;
hence not f . n <= 0 by A5; ::_thesis: verum
end;
set h = f /" (seq_n^ 8);
reconsider f = f as eventually-positive Real_Sequence by A2;
A6: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
abs_(((f_/"_(seq_n^_8))_._n)_-_0)_<_p
A7: log (2,3) > log (2,2) by POWER:57;
let p be real number ; ::_thesis: ( p > 0 implies ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs (((f /" (seq_n^ 8)) . n) - 0) < p )
assume A8: p > 0 ; ::_thesis: ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs (((f /" (seq_n^ 8)) . n) - 0) < p
A9: [/(p to_power (- (1 / 6)))\] >= p to_power (- (1 / 6)) by INT_1:def_7;
reconsider p1 = p as Real by XREAL_0:def_1;
set N = max (3,[/(p1 to_power (- (1 / 6)))\]);
A10: max (3,[/(p1 to_power (- (1 / 6)))\]) >= 3 by XXREAL_0:25;
A11: max (3,[/(p1 to_power (- (1 / 6)))\]) is Integer by XXREAL_0:16;
A12: max (3,[/(p1 to_power (- (1 / 6)))\]) >= [/(p to_power (- (1 / 6)))\] by XXREAL_0:25;
reconsider N = max (3,[/(p1 to_power (- (1 / 6)))\]) as Element of NAT by A10, A11, INT_1:3;
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
abs (((f /" (seq_n^ 8)) . n) - 0) < p
let n be Element of NAT ; ::_thesis: ( n >= N implies abs (((f /" (seq_n^ 8)) . n) - 0) < p )
assume A13: n >= N ; ::_thesis: abs (((f /" (seq_n^ 8)) . n) - 0) < p
then A14: n >= 3 by A10, XXREAL_0:2;
then A15: n > 1 by XXREAL_0:2;
A16: (f /" (seq_n^ 8)) . n = (f . n) / ((seq_n^ 8) . n) by Lm4
.= ((n to_power 2) / (log (2,n))) / ((seq_n^ 8) . n) by A1, A15
.= ((n to_power 2) / (log (2,n))) / (n to_power 8) by A10, A13, Def3
.= ((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 A10, A13, POWER:29
.= ((log (2,n)) ") * (n to_power (- 6))
.= ((log (2,n)) ") * (1 / (n to_power 6)) by A10, A13, 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 A12, A13, XXREAL_0:2;
then A17: n >= p to_power (- (1 / 6)) by A9, XXREAL_0:2;
p1 to_power (- (1 / 6)) > 0 by A8, POWER:34;
then n to_power 6 >= (p to_power (- (1 / 6))) to_power 6 by A17, Lm6;
then A18: n to_power 6 >= p1 to_power ((- (1 / 6)) * 6) by A8, POWER:33;
p1 to_power (- 1) > 0 by A8, POWER:34;
then 1 / (n to_power 6) <= 1 / (p to_power (- 1)) by A18, XREAL_1:85;
then 1 / (n to_power 6) <= 1 / (1 / (p1 to_power 1)) by A8, POWER:28;
then A19: 1 / (n to_power 6) <= p by POWER:25;
log (2,n) >= log (2,3) by A14, PRE_FF:10;
then log (2,n) > log (2,2) by A7, XXREAL_0:2;
then A20: log (2,n) > 1 by POWER:52;
A21: n to_power 6 > 0 by A10, A13, POWER:34;
then (n to_power 6) * 1 < (n to_power 6) * (log (2,n)) by A20, XREAL_1:68;
then (f /" (seq_n^ 8)) . n < 1 / (n to_power 6) by A21, A16, XREAL_1:88;
then (f /" (seq_n^ 8)) . n < p by A19, XXREAL_0:2;
hence abs (((f /" (seq_n^ 8)) . n) - 0) < p by A20, A16, ABSVALUE:def_1; ::_thesis: verum
end;
then A22: f /" (seq_n^ 8) is convergent by SEQ_2:def_6;
then A23: lim (f /" (seq_n^ 8)) = 0 by A6, SEQ_2:def_7;
then not seq_n^ 8 in Big_Oh f by A22, ASYMPT_0:16;
then A24: not f in Big_Omega (seq_n^ 8) by ASYMPT_0:19;
take f ; ::_thesis: ( f = f & Big_Oh f c= Big_Oh (seq_n^ 8) & not Big_Oh f = Big_Oh (seq_n^ 8) )
f in Big_Oh (seq_n^ 8) by A22, A23, ASYMPT_0:16;
hence ( f = f & Big_Oh f c= Big_Oh (seq_n^ 8) & not Big_Oh f = Big_Oh (seq_n^ 8) ) by A24, Th4; ::_thesis: verum
end;
theorem :: ASYMPT_1:24
for g being Real_Sequence st ( for n being Element of NAT holds g . n = (((n ^2) - n) + 1) to_power 4 ) holds
ex s being eventually-positive Real_Sequence st
( s = g & Big_Oh (seq_n^ 8) = Big_Oh s )
proof
let g be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds g . n = (((n ^2) - n) + 1) to_power 4 ) implies ex s being eventually-positive Real_Sequence st
( s = g & Big_Oh (seq_n^ 8) = Big_Oh s ) )
assume A1: for n being Element of NAT holds g . n = (((n ^2) - n) + 1) to_power 4 ; ::_thesis: ex s being eventually-positive Real_Sequence st
( s = g & Big_Oh (seq_n^ 8) = Big_Oh s )
g is eventually-positive
proof
take 0 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 0 <= b1 or not g . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 0 <= n or not g . n <= 0 )
assume n >= 0 ; ::_thesis: not g . n <= 0
g . n = (((n ^2) - n) + 1) to_power 4 by A1;
hence not g . n <= 0 by Lm21, POWER:34; ::_thesis: verum
end;
then reconsider g = g as eventually-positive Real_Sequence ;
take g ; ::_thesis: ( g = g & Big_Oh (seq_n^ 8) = Big_Oh g )
set f = seq_n^ 8;
A2: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_1_holds_
(_g_._n_<=_1_*_((seq_n^_8)_._n)_&_g_._n_>=_0_)
let n be Element of NAT ; ::_thesis: ( n >= 1 implies ( g . n <= 1 * ((seq_n^ 8) . n) & g . n >= 0 ) )
A3: g . n = (((n ^2) - n) + 1) to_power 4 by A1;
assume A4: n >= 1 ; ::_thesis: ( g . n <= 1 * ((seq_n^ 8) . n) & g . n >= 0 )
then A5: ((n ^2) - n) + 1 <= n ^2 by Lm23;
(seq_n^ 8) . n = n to_power (2 * 4) by A4, Def3
.= (n to_power 2) to_power 4 by A4, POWER:33
.= (n ^2) to_power 4 by POWER:46 ;
hence g . n <= 1 * ((seq_n^ 8) . n) by A3, A5, Lm6, Lm21; ::_thesis: g . n >= 0
thus g . n >= 0 by A3, Lm21, POWER:34; ::_thesis: verum
end;
A6: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_1_holds_
(_(seq_n^_8)_._n_<=_16_*_(g_._n)_&_(seq_n^_8)_._n_>=_0_)
let n be Element of NAT ; ::_thesis: ( n >= 1 implies ( (seq_n^ 8) . n <= 16 * (g . n) & (seq_n^ 8) . n >= 0 ) )
A7: g . n = (((n ^2) - n) + 1) to_power 4 by A1;
A8: ((n ^2) - n) + 1 > 0 by Lm21;
assume A9: n >= 1 ; ::_thesis: ( (seq_n^ 8) . n <= 16 * (g . n) & (seq_n^ 8) . n >= 0 )
then A10: (seq_n^ 8) . n = n to_power (2 * 4) by Def3
.= (n to_power 2) to_power 4 by A9, POWER:33
.= (n ^2) to_power 4 by POWER:46 ;
A11: n * n > n * 0 by A9, XREAL_1:68;
n ^2 <= 2 * (((n ^2) - n) + 1) by A9, Lm24;
then (seq_n^ 8) . n <= (2 * (((n ^2) - n) + 1)) to_power 4 by A10, A11, Lm6;
hence (seq_n^ 8) . n <= 16 * (g . n) by A7, A8, POWER:30, POWER:62; ::_thesis: (seq_n^ 8) . n >= 0
thus (seq_n^ 8) . n >= 0 by A10, A11, POWER:34; ::_thesis: verum
end;
seq_n^ 8 is Element of Funcs (NAT,REAL) by FUNCT_2:8;
then A12: seq_n^ 8 in Big_Oh g by A6;
g is Element of Funcs (NAT,REAL) by FUNCT_2:8;
then g in Big_Oh (seq_n^ 8) by A2;
hence ( g = g & Big_Oh (seq_n^ 8) = Big_Oh g ) by A12, Lm5; ::_thesis: verum
end;
theorem :: ASYMPT_1:25
for e being Real st 0 < e & e < 1 holds
ex s being eventually-positive Real_Sequence st
( s = seq_a^ ((1 + e),1,0) & Big_Oh (seq_n^ 8) c= Big_Oh s & not Big_Oh (seq_n^ 8) = Big_Oh s )
proof
set f = seq_n^ 8;
let e be Real; ::_thesis: ( 0 < e & e < 1 implies ex s being eventually-positive Real_Sequence st
( s = seq_a^ ((1 + e),1,0) & Big_Oh (seq_n^ 8) c= Big_Oh s & not Big_Oh (seq_n^ 8) = Big_Oh s ) )
assume that
A1: 0 < e and
A2: e < 1 ; ::_thesis: ex s being eventually-positive Real_Sequence st
( s = seq_a^ ((1 + e),1,0) & Big_Oh (seq_n^ 8) c= Big_Oh s & not Big_Oh (seq_n^ 8) = Big_Oh s )
consider N being Element of NAT such that
A3: for n being Element of NAT 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 = (seq_n^ 8) /" (seq_a^ ((1 + e),1,0));
reconsider g = seq_a^ ((1 + e),1,0) as eventually-positive Real_Sequence by A1;
take g ; ::_thesis: ( g = seq_a^ ((1 + e),1,0) & Big_Oh (seq_n^ 8) c= Big_Oh g & not Big_Oh (seq_n^ 8) = Big_Oh g )
thus g = seq_a^ ((1 + e),1,0) ; ::_thesis: ( Big_Oh (seq_n^ 8) c= Big_Oh g & not Big_Oh (seq_n^ 8) = Big_Oh g )
A4: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N1_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N1_holds_
abs_((((seq_n^_8)_/"_(seq_a^_((1_+_e),1,0)))_._n)_-_0)_<_p
let p be real number ; ::_thesis: ( p > 0 implies ex N1 being Element of NAT st
for n being Element of NAT st n >= N1 holds
abs ((((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n) - 0) < p )
assume A5: p > 0 ; ::_thesis: ex N1 being Element of NAT st
for n being Element of NAT st n >= N1 holds
abs ((((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n) - 0) < p
reconsider p1 = p as Real by XREAL_0:def_1;
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: max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) >= N by XXREAL_0:25;
A8: max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) is Integer
proof
percases ( max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) = N or max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) = max ([/((1 / p) to_power (1 / 8))\],2) ) by XXREAL_0:16;
suppose max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) = N ; ::_thesis: max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) is Integer
hence max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) is Integer ; ::_thesis: verum
end;
suppose max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) = max ([/((1 / p) to_power (1 / 8))\],2) ; ::_thesis: max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) is Integer
hence max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) is Integer by XXREAL_0:16; ::_thesis: verum
end;
end;
end;
A9: max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) >= 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: max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) >= [/((1 / p) to_power (1 / 8))\] by A9, XXREAL_0:2;
reconsider N1 = max (N,(max ([/((1 / p1) to_power (1 / 8))\],2))) as Element of NAT by A7, A8, INT_1:3;
take N1 = N1; ::_thesis: for n being Element of NAT st n >= N1 holds
abs ((((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n) - 0) < p
let n be Element of NAT ; ::_thesis: ( n >= N1 implies abs ((((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n) - 0) < p )
assume A11: n >= N1 ; ::_thesis: abs ((((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n) - 0) < p
then n >= N by A7, XXREAL_0:2;
then (n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n)) by A3;
then A12: 2 to_power ((n * (log (2,(1 + e)))) - (8 * (log (2,n)))) > 2 to_power (8 * (log (2,n))) by POWER:39;
A13: max ([/((1 / p) to_power (1 / 8))\],2) >= 2 by XXREAL_0:25;
A14: g . n = (1 + e) to_power ((1 * n) + 0) by Def1;
((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n = ((seq_n^ 8) . n) / (g . n) by Lm4;
then A15: ((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n = (n to_power 8) / ((1 + e) to_power n) by A9, A13, A11, A14, Def3
.= (2 to_power (8 * (log (2,n)))) / ((1 + e) to_power n) by A9, A13, A11, 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 A11, 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, A13, A11, Lm3;
then A16: 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 A12, 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 ((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n < 2 to_power (- (8 * (log (2,n)))) by A15, POWER:28;
then A17: ((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n < p by A16, XXREAL_0:2;
((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n > 0 by A15, POWER:34;
hence abs ((((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) . n) - 0) < p by A17, ABSVALUE:def_1; ::_thesis: verum
end;
then A18: (seq_n^ 8) /" (seq_a^ ((1 + e),1,0)) is convergent by SEQ_2:def_6;
then A19: lim ((seq_n^ 8) /" (seq_a^ ((1 + e),1,0))) = 0 by A4, SEQ_2:def_7;
then not g in Big_Oh (seq_n^ 8) by A18, ASYMPT_0:16;
then A20: not seq_n^ 8 in Big_Omega g by ASYMPT_0:19;
seq_n^ 8 in Big_Oh g by A18, A19, ASYMPT_0:16;
hence ( Big_Oh (seq_n^ 8) c= Big_Oh g & not Big_Oh (seq_n^ 8) = Big_Oh g ) by A20, Th4; ::_thesis: verum
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 ; ::_thesis: verum
end;
Lm27: for n being Nat st n >= 3 holds
n ^2 > (2 * n) + 1
proof
defpred S1[ Nat] means $1 ^2 > (2 * $1) + 1;
A1: for n being Nat st n >= 3 & S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( n >= 3 & S1[n] implies S1[n + 1] )
assume that
A2: n >= 3 and
A3: n ^2 > (2 * n) + 1 ; ::_thesis: S1[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; ::_thesis: verum
end;
A5: S1[3] ;
for n being Nat st n >= 3 holds
S1[n] from NAT_1:sch_8(A5, A1);
hence for n being Nat st n >= 3 holds
n ^2 > (2 * n) + 1 ; ::_thesis: verum
end;
Lm28: for n being Element of NAT st n >= 10 holds
2 to_power (n - 1) > (2 * n) ^2
proof
defpred S1[ Nat] means 2 to_power ($1 - 1) > (2 * $1) ^2 ;
A1: for n being Nat st n >= 10 & S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( n >= 10 & S1[n] implies S1[n + 1] )
assume that
A2: n >= 10 and
A3: 2 to_power (n - 1) > (2 * n) ^2 ; ::_thesis: S1[n + 1]
A4: now__::_thesis:_not_((2_*_n)_^2)_*_2_<=_(2_*_2)_*_(((n_*_n)_+_(2_*_n))_+_1)
assume ((2 * n) ^2) * 2 <= (2 * 2) * (((n * n) + (2 * n)) + 1) ; ::_thesis: contradiction
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; ::_thesis: verum
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; ::_thesis: verum
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: S1[10] ;
for n being Nat st n >= 10 holds
S1[n] from NAT_1:sch_8(A6, A1);
hence for n being Element of NAT st n >= 10 holds
2 to_power (n - 1) > (2 * n) ^2 ; ::_thesis: verum
end;
Lm29: for n being Nat st n >= 9 holds
(n + 1) to_power 6 < 2 * (n to_power 6)
proof
defpred S1[ Nat] means ($1 + 1) to_power 6 < 2 * ($1 to_power 6);
A1: for n being Nat st n >= 9 & S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( n >= 9 & S1[n] implies S1[n + 1] )
assume that
A2: n >= 9 and
A3: (n + 1) to_power 6 < 2 * (n to_power 6) ; ::_thesis: S1[n + 1]
((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__::_thesis:_not_(n_+_2)_/_(n_+_1)_>=_(n_+_1)_/_n
assume (n + 2) / (n + 1) >= (n + 1) / n ; ::_thesis: contradiction
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 ; ::_thesis: verum
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 S1[n + 1] by A6, XCMPLX_1:87; ::_thesis: verum
end;
A7: S1[9]
proof
A8: 9 to_power 2 = 9 to_power (1 + 1)
.= (9 to_power 1) * (9 to_power 1) by POWER:27
.= 9 * (9 to_power 1) by POWER:25
.= 9 * 9 by POWER:25
.= 81 ;
2 * (9 to_power 4) = 2 * (9 to_power (2 + 2))
.= 2 * (81 * 81) by A8, POWER:27
.= 13122 ;
then A9: (13122 * 9) * 9 = (2 * ((9 to_power 4) * 9)) * 9
.= (2 * ((9 to_power 4) * (9 to_power 1))) * 9 by POWER:25
.= (2 * (9 to_power (4 + 1))) * 9 by POWER:27
.= 2 * ((9 to_power 5) * 9)
.= 2 * ((9 to_power 5) * (9 to_power 1)) by POWER:25
.= 2 * (9 to_power (5 + 1)) by POWER:27
.= 2 * (9 to_power 6) ;
consider t6 being Element of NAT such that
A10: t6 = ((((10 * 10) * 10) * 10) * 10) * 10 ;
A11: 10 to_power 3 = 10 to_power (2 + 1)
.= (10 to_power 2) * (10 to_power 1) by POWER:27
.= (10 to_power (1 + 1)) * 10 by POWER:25
.= ((10 to_power 1) * (10 to_power 1)) * 10 by POWER:27
.= (10 * (10 to_power 1)) * 10 by POWER:25
.= (10 * 10) * 10 by POWER:25 ;
10 to_power 6 = 10 to_power (3 + 3)
.= ((10 * 10) * 10) * ((10 * 10) * 10) by A11, POWER:27
.= t6 by A10 ;
hence S1[9] by A10, A9; ::_thesis: verum
end;
for n being Nat st n >= 9 holds
S1[n] from NAT_1:sch_8(A7, A1);
hence for n being Nat st n >= 9 holds
(n + 1) to_power 6 < 2 * (n to_power 6) ; ::_thesis: verum
end;
Lm30: for n being Element of NAT st n >= 30 holds
2 to_power n > n to_power 6
proof
defpred S1[ Nat] means 2 to_power $1 > $1 to_power 6;
A1: for n being Nat st n >= 30 & S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( n >= 30 & S1[n] implies S1[n + 1] )
assume that
A2: n >= 30 and
A3: 2 to_power n > n to_power 6 ; ::_thesis: S1[n + 1]
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 S1[n + 1] by A5, A4, XXREAL_0:2; ::_thesis: verum
end;
2 to_power 30 = 2 to_power (5 * 6)
.= 32 to_power 6 by POWER:33, POWER:63 ;
then A6: S1[30] by POWER:37;
for n being Nat st n >= 30 holds
S1[n] from NAT_1:sch_8(A6, A1);
hence for n being Element of NAT st n >= 30 holds
2 to_power n > n to_power 6 ; ::_thesis: verum
end;
Lm31: for x being Real st x > 9 holds
2 to_power x > (2 * x) ^2
proof
let x be Real; ::_thesis: ( x > 9 implies 2 to_power x > (2 * x) ^2 )
assume A1: x > 9 ; ::_thesis: 2 to_power x > (2 * x) ^2
set n = [/x\];
A2: [/x\] >= x by INT_1:def_7;
then reconsider n = [/x\] 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
percases ( x is Integer or not x is Integer ) ;
suppose x is Integer ; ::_thesis: [/x\] - [\x/] <= 1
then [\x/] = [/x\] by INT_1:34;
hence [/x\] - [\x/] <= 1 ; ::_thesis: verum
end;
suppose x is not Integer ; ::_thesis: [/x\] - [\x/] <= 1
then not [\x/] = [/x\] by INT_1:34;
then [\x/] + 1 = [/x\] by INT_1:41;
hence [/x\] - [\x/] <= 1 ; ::_thesis: verum
end;
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 2 to_power x > (2 * x) ^2 by A3, XXREAL_0:2; ::_thesis: verum
end;
Lm32: ex N being Element of NAT st
for n being Element of NAT st n >= N holds
(sqrt n) - (log (2,n)) > 1
proof
ex N being Element of NAT st
for n being Element of NAT st n >= N holds
n / 2 > (log (2,n)) * (log (2,n))
proof
reconsider N = 2 to_power 10 as Element of NAT ;
now__::_thesis:_ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
n_/_2_>_(log_(2,n))_*_(log_(2,n))
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
n / 2 > (log (2,n)) * (log (2,n))
let n be Element of NAT ; ::_thesis: ( n >= N implies n / 2 > (log (2,n)) * (log (2,n)) )
set x = log (2,n);
A1: 2 to_power 9 > 0 by POWER:34;
assume A2: n >= N ; ::_thesis: n / 2 > (log (2,n)) * (log (2,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 log (2,n) > 9 * (log (2,2)) by POWER:55;
then A9: log (2,n) > 9 * 1 by POWER:52;
then A10: 2 * (log (2,n)) > 0 * (log (2,n)) by XREAL_1:68;
then (2 * (log (2,n))) * (2 * (log (2,n))) > 0 * (2 * (log (2,n))) by XREAL_1:68;
then log (2,(2 to_power (log (2,n)))) > log (2,((2 * (log (2,n))) ^2)) by A9, Lm31, POWER:57;
then (log (2,n)) * (log (2,2)) > log (2,((2 * (log (2,n))) ^2)) by POWER:55;
then (log (2,n)) * 1 > log (2,((2 * (log (2,n))) ^2)) by POWER:52;
then log (2,n) > log (2,((2 * (log (2,n))) to_power 2)) by POWER:46;
then log (2,n) > 2 * (log (2,(2 * (log (2,n))))) 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; ::_thesis: verum
end;
hence ex N being Element of NAT st
for n being Element of NAT st n >= N holds
n / 2 > (log (2,n)) * (log (2,n)) ; ::_thesis: verum
end;
then consider N3 being Element of NAT such that
A11: for n being Element of NAT st n >= N3 holds
n / 2 > (log (2,n)) * (log (2,n)) ;
now__::_thesis:_ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
n_/_3_>_2_*_(log_(2,n))
take N = 30; ::_thesis: for n being Element of NAT st n >= N holds
n / 3 > 2 * (log (2,n))
let n be Element of NAT ; ::_thesis: ( n >= N implies n / 3 > 2 * (log (2,n)) )
assume A12: n >= N ; ::_thesis: n / 3 > 2 * (log (2,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; ::_thesis: verum
end;
then consider N2 being Element of NAT such that
A14: for n being Element of NAT st n >= N2 holds
n / 3 > 2 * (log (2,n)) ;
now__::_thesis:_ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
n_/_6_>_1
take N = 7; ::_thesis: for n being Element of NAT st n >= N holds
n / 6 > 1
let n be Element of NAT ; ::_thesis: ( n >= N implies n / 6 > 1 )
assume n >= N ; ::_thesis: n / 6 > 1
then n > 6 by XXREAL_0:2;
then n / 6 > 6 / 6 by XREAL_1:74;
hence n / 6 > 1 ; ::_thesis: verum
end;
then consider N1 being Element of NAT such that
A15: for n being Element of NAT st n >= N1 holds
n / 6 > 1 ;
set N = max ((max (N1,2)),(max (N2,N3)));
A16: max ((max (N1,2)),(max (N2,N3))) >= max (N1,2) by XXREAL_0:25;
max (N1,2) >= 2 by XXREAL_0:25;
then A17: max ((max (N1,2)),(max (N2,N3))) >= 2 by A16, XXREAL_0:2;
A18: max ((max (N1,2)),(max (N2,N3))) >= max (N2,N3) by XXREAL_0:25;
max (N2,N3) >= N3 by XXREAL_0:25;
then A19: max ((max (N1,2)),(max (N2,N3))) >= N3 by A18, XXREAL_0:2;
max (N2,N3) >= N2 by XXREAL_0:25;
then A20: max ((max (N1,2)),(max (N2,N3))) >= N2 by A18, XXREAL_0:2;
max (N1,2) >= N1 by XXREAL_0:25;
then A21: max ((max (N1,2)),(max (N2,N3))) >= N1 by A16, XXREAL_0:2;
now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_max_((max_(N1,2)),(max_(N2,N3)))_holds_
(sqrt_n)_-_(log_(2,n))_>_1
let n be Element of NAT ; ::_thesis: ( n >= max ((max (N1,2)),(max (N2,N3))) implies (sqrt n) - (log (2,n)) > 1 )
A22: (1 + (2 * (log (2,n)))) + ((log (2,n)) * (log (2,n))) = (1 + (log (2,n))) ^2 ;
assume A23: n >= max ((max (N1,2)),(max (N2,N3))) ; ::_thesis: (sqrt n) - (log (2,n)) > 1
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; ::_thesis: verum
end;
hence ex N being Element of NAT st
for n being Element of NAT st n >= N holds
(sqrt n) - (log (2,n)) > 1 ; ::_thesis: verum
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 5 ! = 120 ; ::_thesis: verum
end;
Lm34: for n being Element of NAT st n >= 10 holds
(2 to_power (2 * n)) / (n !) < 1 / (2 to_power (n - 9))
proof
defpred S1[ 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 being Nat st k >= 10 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 10 & S1[k] implies S1[k + 1] )
assume that
A4: k >= 10 and
A5: (2 to_power (2 * k)) / (k !) < 1 / (2 to_power (k - 9)) ; ::_thesis: S1[k + 1]
A6: 2 to_power 1 > 0 by POWER:34;
A7: now__::_thesis:_not_(2_to_power_2)_/_(k_+_1)_>=_1_/_(2_to_power_1)
assume (2 to_power 2) / (k + 1) >= 1 / (2 to_power 1) ; ::_thesis: contradiction
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; ::_thesis: verum
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)) ; ::_thesis: verum
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: S1[10] by A1, POWER:25;
for n being Nat st n >= 10 holds
S1[n] from NAT_1:sch_8(A10, A3);
hence for n being Element of NAT st n >= 10 holds
(2 to_power (2 * n)) / (n !) < 1 / (2 to_power (n - 9)) ; ::_thesis: verum
end;
Lm35: for n being Element of NAT st n >= 3 holds
2 * (n - 2) >= n - 1
proof
defpred S1[ Nat] means 2 * ($1 - 2) >= $1 - 1;
A1: for n being Nat st n >= 3 & S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( n >= 3 & S1[n] implies S1[n + 1] )
assume that
n >= 3 and
A2: 2 * (n - 2) >= n - 1 ; ::_thesis: S1[n + 1]
(2 * (n - 2)) + 2 >= (n + (- 1)) + 1 by A2, XREAL_1:7;
hence 2 * ((n + 1) - 2) >= (n + 1) - 1 ; ::_thesis: verum
end;
A3: S1[3] ;
for n being Nat st n >= 3 holds
S1[n] from NAT_1:sch_8(A3, A1);
hence for n being Element of NAT st n >= 3 holds
2 * (n - 2) >= n - 1 ; ::_thesis: verum
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 5 to_power 5 = 3125 ; ::_thesis: verum
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 4 to_power 4 = 256 ; ::_thesis: verum
end;
Lm38: for a, b, d, e being Real holds (a / b) / (d / e) = (a / d) * (e / b)
proof
let a, b, d, e be Real; ::_thesis: (a / b) / (d / e) = (a / d) * (e / b)
thus (a / b) / (d / e) = (a * e) / (b * d) by XCMPLX_1:84
.= (a / d) * (e / b) by XCMPLX_1:76 ; ::_thesis: verum
end;
Lm39: for x being real number st x >= 0 holds
sqrt x = x to_power (1 / 2)
proof
let x be real number ; ::_thesis: ( x >= 0 implies sqrt x = x to_power (1 / 2) )
assume A1: x >= 0 ; ::_thesis: sqrt x = x to_power (1 / 2)
percases ( x = 0 or x > 0 ) by A1;
suppose x = 0 ; ::_thesis: sqrt x = x to_power (1 / 2)
hence sqrt x = x to_power (1 / 2) by POWER:def_2, SQUARE_1:17; ::_thesis: verum
end;
supposeA2: x > 0 ; ::_thesis: sqrt x = x to_power (1 / 2)
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 sqrt x = x to_power (1 / 2) by A3, SQUARE_1:22; ::_thesis: verum
end;
end;
end;
Lm40: ex N being Element of NAT st
for n being Element of NAT 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_logn /" (seq_n^ (1 / 2));
A1: lim (seq_logn /" (seq_n^ (1 / 2))) = 0 by Lm11;
seq_logn /" (seq_n^ (1 / 2)) is convergent by Lm11;
then consider N being Element of NAT such that
A2: for n being Element of NAT st n >= N holds
abs (((seq_logn /" (seq_n^ (1 / 2))) . n) - 0) < 1 / 2 by A1, SEQ_2:def_7;
set N1 = max (2,N);
A3: max (2,N) >= 2 by XXREAL_0:25;
A4: max (2,N) >= N by XXREAL_0:25;
now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_max_(2,N)_holds_
n_/_2_<_n_-_((sqrt_n)_*_(log_(2,n)))
let n be Element of NAT ; ::_thesis: ( n >= max (2,N) implies n / 2 < n - ((sqrt n) * (log (2,n))) )
assume A5: n >= max (2,N) ; ::_thesis: n / 2 < n - ((sqrt n) * (log (2,n)))
then A6: sqrt n > 0 by A3, SQUARE_1:25;
n >= N by A4, A5, XXREAL_0:2;
then abs (((seq_logn /" (seq_n^ (1 / 2))) . n) - 0) < 1 / 2 by A2;
then A7: (seq_logn /" (seq_n^ (1 / 2))) . n < 1 / 2 by ABSVALUE:def_1;
A8: sqrt n <> 0 by A3, A5, SQUARE_1:25;
(seq_logn /" (seq_n^ (1 / 2))) . n = (seq_logn . n) / ((seq_n^ (1 / 2)) . n) by Lm4
.= (log (2,n)) / ((seq_n^ (1 / 2)) . 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; ::_thesis: verum
end;
hence ex N being Element of NAT st
for n being Element of NAT st n >= N holds
n - ((sqrt n) * (log (2,n))) > n / 2 ; ::_thesis: verum
end;
Lm41: for s being Real_Sequence st ( for n being Element of NAT holds s . n = (1 + (1 / (n + 1))) to_power (n + 1) ) holds
s is V41()
proof
let s be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds s . n = (1 + (1 / (n + 1))) to_power (n + 1) ) implies s is V41() )
assume A1: for n being Element of NAT holds s . n = (1 + (1 / (n + 1))) to_power (n + 1) ; ::_thesis: s is V41()
now__::_thesis:_for_n_being_Element_of_NAT_holds_s_._(n_+_1)_>=_s_._n
let n be Element of NAT ; ::_thesis: s . (n + 1) >= s . n
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; ::_thesis: verum
end;
hence s is V41() by SEQM_3:def_8; ::_thesis: verum
end;
Lm42: for n being Element of NAT st n >= 1 holds
((n + 1) / n) to_power n <= ((n + 2) / (n + 1)) to_power (n + 1)
proof
deffunc H1( Element of NAT ) -> Element of REAL = (1 + (1 / ($1 + 1))) to_power ($1 + 1);
let n be Element of NAT ; ::_thesis: ( n >= 1 implies ((n + 1) / n) to_power n <= ((n + 2) / (n + 1)) to_power (n + 1) )
consider seq being Real_Sequence such that
A1: for n being Element of NAT holds seq . n = H1(n) from SEQ_1:sch_1();
assume A2: n >= 1 ; ::_thesis: ((n + 1) / n) to_power n <= ((n + 2) / (n + 1)) to_power (n + 1)
then n >= 0 + 1 ;
then n - 1 >= 0 by XREAL_1:19;
then reconsider m = n - 1 as Element of NAT by INT_1:3;
seq is V41() by A1, Lm41;
then seq . m <= seq . (m + 1) by SEQM_3:def_8;
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 ((n + 1) / n) to_power n <= ((n + 2) / (n + 1)) to_power (n + 1) ; ::_thesis: verum
end;
theorem :: ASYMPT_1:26
for f, g being Real_Sequence st ( for n being Element of NAT st n > 0 holds
f . n = n to_power (log (2,n)) ) & ( for n being Element of NAT st n > 0 holds
g . n = n to_power (sqrt n) ) holds
ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = g & Big_Oh s c= Big_Oh s1 & not Big_Oh s = Big_Oh s1 )
proof
let f, g be Real_Sequence; ::_thesis: ( ( for n being Element of NAT st n > 0 holds
f . n = n to_power (log (2,n)) ) & ( for n being Element of NAT st n > 0 holds
g . n = n to_power (sqrt n) ) implies ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = g & Big_Oh s c= Big_Oh s1 & not Big_Oh s = Big_Oh s1 ) )
assume that
A1: for n being Element of NAT st n > 0 holds
f . n = n to_power (log (2,n)) and
A2: for n being Element of NAT st n > 0 holds
g . n = n to_power (sqrt n) ; ::_thesis: ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = g & Big_Oh s c= Big_Oh s1 & not Big_Oh s = Big_Oh s1 )
set h = f /" g;
g is eventually-positive
proof
take 1 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 1 <= b1 or not g . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 1 <= n or not g . n <= 0 )
assume A3: n >= 1 ; ::_thesis: not g . n <= 0
then g . n = n to_power (sqrt n) by A2;
hence not g . n <= 0 by A3, POWER:34; ::_thesis: verum
end;
then reconsider g = g as eventually-positive Real_Sequence ;
f is eventually-positive
proof
take 1 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 1 <= b1 or not f . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 1 <= n or not f . n <= 0 )
assume A4: n >= 1 ; ::_thesis: not f . n <= 0
then f . n = n to_power (log (2,n)) by A1;
hence not f . n <= 0 by A4, POWER:34; ::_thesis: verum
end;
then reconsider f = f as eventually-positive Real_Sequence ;
take f ; ::_thesis: ex s1 being eventually-positive Real_Sequence st
( f = f & s1 = g & Big_Oh f c= Big_Oh s1 & not Big_Oh f = Big_Oh s1 )
take g ; ::_thesis: ( f = f & g = g & Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g )
consider N being Element of NAT such that
A5: for n being Element of NAT st n >= N holds
(sqrt n) - (log (2,n)) > 1 by Lm32;
A6: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N1_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N1_holds_
abs_(((f_/"_g)_._n)_-_0)_<_p
let p be real number ; ::_thesis: ( p > 0 implies ex N1 being Element of NAT st
for n being Element of NAT st n >= N1 holds
abs (((f /" g) . n) - 0) < p )
assume A7: p > 0 ; ::_thesis: ex N1 being Element of NAT st
for n being Element of NAT st n >= N1 holds
abs (((f /" g) . n) - 0) < p
set N1 = max (N,(max ([/(1 / p)\],2)));
A8: max (N,(max ([/(1 / p)\],2))) >= N by XXREAL_0:25;
A9: max (N,(max ([/(1 / p)\],2))) is Integer
proof
percases ( max (N,(max ([/(1 / p)\],2))) = N or max (N,(max ([/(1 / p)\],2))) = max ([/(1 / p)\],2) ) by XXREAL_0:16;
suppose max (N,(max ([/(1 / p)\],2))) = N ; ::_thesis: max (N,(max ([/(1 / p)\],2))) is Integer
hence max (N,(max ([/(1 / p)\],2))) is Integer ; ::_thesis: verum
end;
suppose max (N,(max ([/(1 / p)\],2))) = max ([/(1 / p)\],2) ; ::_thesis: max (N,(max ([/(1 / p)\],2))) is Integer
hence max (N,(max ([/(1 / p)\],2))) is Integer by XXREAL_0:16; ::_thesis: verum
end;
end;
end;
A10: max (N,(max ([/(1 / p)\],2))) >= max ([/(1 / p)\],2) by XXREAL_0:25;
max ([/(1 / p)\],2) >= [/(1 / p)\] by XXREAL_0:25;
then A11: max (N,(max ([/(1 / p)\],2))) >= [/(1 / p)\] by A10, XXREAL_0:2;
A12: max ([/(1 / p)\],2) >= 2 by XXREAL_0:25;
then max (N,(max ([/(1 / p)\],2))) >= 2 by A10, XXREAL_0:2;
then A13: max (N,(max ([/(1 / p)\],2))) > 1 by XXREAL_0:2;
reconsider N1 = max (N,(max ([/(1 / p)\],2))) as Element of NAT by A8, A9, INT_1:3;
take N1 = N1; ::_thesis: for n being Element of NAT st n >= N1 holds
abs (((f /" g) . n) - 0) < p
let n be Element of NAT ; ::_thesis: ( n >= N1 implies abs (((f /" g) . n) - 0) < p )
A14: (f /" g) . n = (f . n) / (g . n) by Lm4;
assume A15: n >= N1 ; ::_thesis: abs (((f /" g) . n) - 0) < p
then f . n = n to_power (log (2,n)) by A1, A10, A12;
then A16: (f /" g) . n = (n to_power (log (2,n))) / (n to_power (sqrt n)) by A2, A10, A12, A15, A14
.= n to_power ((log (2,n)) - (sqrt n)) by A10, A12, A15, POWER:29
.= n to_power (- ((sqrt n) - (log (2,n)))) ;
then A17: (f /" g) . n > 0 by A10, A12, A15, POWER:34;
n >= N by A8, A15, XXREAL_0:2;
then (sqrt n) - (log (2,n)) > 1 by A5;
then A18: (- 1) * ((sqrt n) - (log (2,n))) < (- 1) * 1 by XREAL_1:69;
n > 1 by A13, A15, XXREAL_0:2;
then A19: n to_power (- ((sqrt n) - (log (2,n)))) < n to_power (- 1) by A18, POWER:39;
[/(1 / p)\] >= 1 / p by INT_1:def_7;
then N1 >= 1 / p by A11, XXREAL_0:2;
then n >= 1 / p by A15, XXREAL_0:2;
then A20: 1 / n <= 1 / (1 / p) by A7, XREAL_1:85;
n to_power (- 1) = 1 / (n to_power 1) by A10, A12, A15, POWER:28
.= 1 / n by POWER:25 ;
then (f /" g) . n < p by A16, A19, A20, XXREAL_0:2;
hence abs (((f /" g) . n) - 0) < p by A17, ABSVALUE:def_1; ::_thesis: verum
end;
then A21: f /" g is convergent by SEQ_2:def_6;
then A22: lim (f /" g) = 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 ( f = f & g = g & Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g ) by A23, Th4; ::_thesis: verum
end;
theorem :: ASYMPT_1:27
for f being Real_Sequence st ( for n being Element of NAT st n > 0 holds
f . n = n to_power (sqrt n) ) holds
ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = seq_a^ (2,1,0) & Big_Oh s c= Big_Oh s1 & not Big_Oh s = Big_Oh s1 )
proof
set g = seq_a^ (2,1,0);
let f be Real_Sequence; ::_thesis: ( ( for n being Element of NAT st n > 0 holds
f . n = n to_power (sqrt n) ) implies ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = seq_a^ (2,1,0) & Big_Oh s c= Big_Oh s1 & not Big_Oh s = Big_Oh s1 ) )
assume A1: for n being Element of NAT st n > 0 holds
f . n = n to_power (sqrt n) ; ::_thesis: ex s, s1 being eventually-positive Real_Sequence st
( s = f & s1 = seq_a^ (2,1,0) & Big_Oh s c= Big_Oh s1 & not Big_Oh s = Big_Oh s1 )
A2: f is eventually-positive
proof
take 1 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 1 <= b1 or not f . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 1 <= n or not f . n <= 0 )
assume A3: n >= 1 ; ::_thesis: not f . n <= 0
then f . n = n to_power (sqrt n) by A1;
hence not f . n <= 0 by A3, POWER:34; ::_thesis: verum
end;
set h = f /" (seq_a^ (2,1,0));
reconsider f = f as eventually-positive Real_Sequence by A2;
reconsider g = seq_a^ (2,1,0) as eventually-positive Real_Sequence ;
take f ; ::_thesis: ex s1 being eventually-positive Real_Sequence st
( f = f & s1 = seq_a^ (2,1,0) & Big_Oh f c= Big_Oh s1 & not Big_Oh f = Big_Oh s1 )
take g ; ::_thesis: ( f = f & g = seq_a^ (2,1,0) & Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g )
consider N being Element of NAT such that
A4: for n being Element of NAT st n >= N holds
n - ((sqrt n) * (log (2,n))) > n / 2 by Lm40;
A5: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N1_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N1_holds_
abs_(((f_/"_(seq_a^_(2,1,0)))_._n)_-_0)_<_p
let p be real number ; ::_thesis: ( p > 0 implies ex N1 being Element of NAT st
for n being Element of NAT st n >= N1 holds
abs (((f /" (seq_a^ (2,1,0))) . n) - 0) < p )
assume A6: p > 0 ; ::_thesis: ex N1 being Element of NAT st
for n being Element of NAT st n >= N1 holds
abs (((f /" (seq_a^ (2,1,0))) . n) - 0) < p
set N1 = max (N,(max ((2 * [/(log (2,(1 / p)))\]),2)));
A7: max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) >= N by XXREAL_0:25;
A8: max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) is Integer
proof
percases ( max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) = N or max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) = max ((2 * [/(log (2,(1 / p)))\]),2) ) by XXREAL_0:16;
suppose max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) = N ; ::_thesis: max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) is Integer
hence max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) is Integer ; ::_thesis: verum
end;
suppose max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) = max ((2 * [/(log (2,(1 / p)))\]),2) ; ::_thesis: max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) is Integer
hence max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) is Integer by XXREAL_0:16; ::_thesis: verum
end;
end;
end;
A9: max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) >= 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 A10: max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) >= 2 * [/(log (2,(1 / p)))\] by A9, XXREAL_0:2;
reconsider N1 = max (N,(max ((2 * [/(log (2,(1 / p)))\]),2))) as Element of NAT by A7, A8, INT_1:3;
take N1 = N1; ::_thesis: for n being Element of NAT st n >= N1 holds
abs (((f /" (seq_a^ (2,1,0))) . n) - 0) < p
let n be Element of NAT ; ::_thesis: ( n >= N1 implies abs (((f /" (seq_a^ (2,1,0))) . n) - 0) < p )
A11: (f /" (seq_a^ (2,1,0))) . n = (f . n) / (g . n) by Lm4;
A12: [/(log (2,(1 / p)))\] >= log (2,(1 / p)) by INT_1:def_7;
assume A13: n >= N1 ; ::_thesis: abs (((f /" (seq_a^ (2,1,0))) . n) - 0) < p
then n >= 2 * [/(log (2,(1 / p)))\] by A10, XXREAL_0:2;
then n / 2 >= [/(log (2,(1 / p)))\] by XREAL_1:77;
then n / 2 >= log (2,(1 / p)) by A12, 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 A14: 2 to_power (- (n / 2)) <= 1 / (1 / p) by A6, POWER:def_3;
A15: g . n = 2 to_power ((1 * n) + 0) by Def1
.= 2 to_power n ;
A16: max ((2 * [/(log (2,(1 / p)))\]),2) >= 2 by XXREAL_0:25;
then f . n = n to_power (sqrt n) by A1, A9, A13
.= 2 to_power ((sqrt n) * (log (2,n))) by A9, A16, A13, Lm3 ;
then A17: (f /" (seq_a^ (2,1,0))) . n = 2 to_power (((sqrt n) * (log (2,n))) - n) by A11, A15, POWER:29
.= 2 to_power (- (n - ((sqrt n) * (log (2,n))))) ;
then A18: (f /" (seq_a^ (2,1,0))) . n > 0 by POWER:34;
n >= N by A7, A13, XXREAL_0:2;
then n - ((sqrt n) * (log (2,n))) > n / 2 by A4;
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 (f /" (seq_a^ (2,1,0))) . n < p by A17, A14, XXREAL_0:2;
hence abs (((f /" (seq_a^ (2,1,0))) . n) - 0) < p by A18, ABSVALUE:def_1; ::_thesis: verum
end;
then A19: f /" (seq_a^ (2,1,0)) is convergent by SEQ_2:def_6;
then A20: lim (f /" (seq_a^ (2,1,0))) = 0 by A5, 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 ( f = f & g = seq_a^ (2,1,0) & Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g ) by A21, Th4; ::_thesis: verum
end;
theorem :: ASYMPT_1:28
ex s, s1 being eventually-positive Real_Sequence st
( s = seq_a^ (2,1,0) & s1 = seq_a^ (2,1,1) & Big_Oh s = Big_Oh s1 )
proof
set g = seq_a^ (2,1,1);
set f = seq_a^ (2,1,0);
set h = (seq_a^ (2,1,0)) /" (seq_a^ (2,1,1));
reconsider f = seq_a^ (2,1,0) as eventually-positive Real_Sequence ;
reconsider g = seq_a^ (2,1,1) as eventually-positive Real_Sequence ;
take f ; ::_thesis: ex s1 being eventually-positive Real_Sequence st
( f = seq_a^ (2,1,0) & s1 = seq_a^ (2,1,1) & Big_Oh f = Big_Oh s1 )
take g ; ::_thesis: ( f = seq_a^ (2,1,0) & g = seq_a^ (2,1,1) & Big_Oh f = Big_Oh g )
thus ( f = seq_a^ (2,1,0) & g = seq_a^ (2,1,1) ) ; ::_thesis: Big_Oh f = Big_Oh g
A1: now__::_thesis:_for_n_being_Element_of_NAT_holds_((seq_a^_(2,1,0))_/"_(seq_a^_(2,1,1)))_._n_=_2_"
let n be Element of NAT ; ::_thesis: ((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) . n = 2 "
A2: g . n = 2 to_power ((1 * n) + 1) by Def1;
f . n = 2 to_power ((1 * n) + 0) by Def1;
then ((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) . 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 ((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) . n = 2 " ; ::_thesis: verum
end;
A3: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
abs_((((seq_a^_(2,1,0))_/"_(seq_a^_(2,1,1)))_._n)_-_(2_"))_<_p
let p be real number ; ::_thesis: ( p > 0 implies ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs ((((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) . n) - (2 ")) < p )
assume A4: p > 0 ; ::_thesis: ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs ((((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) . n) - (2 ")) < p
take N = 0 ; ::_thesis: for n being Element of NAT st n >= N holds
abs ((((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) . n) - (2 ")) < p
let n be Element of NAT ; ::_thesis: ( n >= N implies abs ((((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) . n) - (2 ")) < p )
assume n >= N ; ::_thesis: abs ((((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) . n) - (2 ")) < p
abs ((((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) . n) - (2 ")) = abs ((2 ") - (2 ")) by A1
.= 0 by ABSVALUE:2 ;
hence abs ((((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) . n) - (2 ")) < p by A4; ::_thesis: verum
end;
then A5: (seq_a^ (2,1,0)) /" (seq_a^ (2,1,1)) is convergent by SEQ_2:def_6;
then lim ((seq_a^ (2,1,0)) /" (seq_a^ (2,1,1))) > 0 by A3, SEQ_2:def_7;
hence Big_Oh f = Big_Oh g by A5, ASYMPT_0:15; ::_thesis: verum
end;
theorem :: ASYMPT_1:29
ex s, s1 being eventually-positive Real_Sequence st
( s = seq_a^ (2,1,0) & s1 = seq_a^ (2,2,0) & Big_Oh s c= Big_Oh s1 & not Big_Oh s = Big_Oh s1 )
proof
reconsider g = seq_a^ (2,2,0) as eventually-positive Real_Sequence ;
reconsider f = seq_a^ (2,1,0) as eventually-positive Real_Sequence ;
take f ; ::_thesis: ex s1 being eventually-positive Real_Sequence st
( f = seq_a^ (2,1,0) & s1 = seq_a^ (2,2,0) & Big_Oh f c= Big_Oh s1 & not Big_Oh f = Big_Oh s1 )
take g ; ::_thesis: ( f = seq_a^ (2,1,0) & g = seq_a^ (2,2,0) & Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g )
thus ( f = seq_a^ (2,1,0) & g = seq_a^ (2,2,0) ) ; ::_thesis: ( Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g )
set h = f /" g;
A1: for n being Element of NAT holds (f /" g) . n = 2 to_power (- n)
proof
let n be Element of NAT ; ::_thesis: (f /" g) . n = 2 to_power (- n)
(f /" g) . 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 (f /" g) . n = 2 to_power (- n) ; ::_thesis: verum
end;
A2: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
abs_(((f_/"_g)_._n)_-_0)_<_p
let p be real number ; ::_thesis: ( p > 0 implies ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs (((f /" g) . n) - 0) < p )
set N = max (1,([/(log (2,(1 / p)))\] + 1));
A3: max (1,([/(log (2,(1 / p)))\] + 1)) >= 1 by XXREAL_0:25;
A4: max (1,([/(log (2,(1 / p)))\] + 1)) 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;
reconsider N = max (1,([/(log (2,(1 / p)))\] + 1)) as Element of NAT by A3, A4, INT_1:3;
assume A7: p > 0 ; ::_thesis: ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs (((f /" g) . n) - 0) < p
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
abs (((f /" g) . n) - 0) < p
let n be Element of NAT ; ::_thesis: ( n >= N implies abs (((f /" g) . n) - 0) < p )
2 to_power N >= 2 to_power ([/(log (2,(1 / p)))\] + 1) by PRE_FF:8, XXREAL_0:25;
then A8: 2 to_power N > 2 to_power (log (2,(1 / p))) by A6, XXREAL_0:2;
assume n >= N ; ::_thesis: abs (((f /" g) . n) - 0) < p
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 A8, 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 A9: 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 A9, XREAL_1:68;
then A10: 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 A10, XCMPLX_0:def_7;
then A11: p > 1 / (2 to_power n) ;
A12: 2 to_power (- n) > 0 by POWER:34;
abs (((f /" g) . n) - 0) = abs (2 to_power (- n)) by A1;
then abs (((f /" g) . n) - 0) = 2 to_power (- n) by A12, ABSVALUE:def_1;
hence abs (((f /" g) . n) - 0) < p by A11, POWER:28; ::_thesis: verum
end;
then A13: f /" g is convergent by SEQ_2:def_6;
then A14: lim (f /" g) = 0 by A2, SEQ_2:def_7;
then not g in Big_Oh f by A13, ASYMPT_0:16;
then A15: not f in Big_Omega g by ASYMPT_0:19;
f in Big_Oh g by A13, A14, ASYMPT_0:16;
hence ( Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g ) by A15, Th4; ::_thesis: verum
end;
theorem :: ASYMPT_1:30
ex s being eventually-positive Real_Sequence st
( s = seq_a^ (2,2,0) & Big_Oh s c= Big_Oh (seq_n! 0) & not Big_Oh s = Big_Oh (seq_n! 0) )
proof
reconsider f = seq_a^ (2,2,0) as eventually-positive Real_Sequence ;
set g = seq_n! 0;
take f ; ::_thesis: ( f = seq_a^ (2,2,0) & Big_Oh f c= Big_Oh (seq_n! 0) & not Big_Oh f = Big_Oh (seq_n! 0) )
thus f = seq_a^ (2,2,0) ; ::_thesis: ( Big_Oh f c= Big_Oh (seq_n! 0) & not Big_Oh f = Big_Oh (seq_n! 0) )
set h = f /" (seq_n! 0);
A1: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
abs_(((f_/"_(seq_n!_0))_._n)_-_0)_<_p
let p be real number ; ::_thesis: ( p > 0 implies ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs (((f /" (seq_n! 0)) . n) - 0) < p )
assume A2: p > 0 ; ::_thesis: ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs (((f /" (seq_n! 0)) . n) - 0) < p
set N = max (10,[/(9 + (log (2,(1 / p))))\]);
A3: max (10,[/(9 + (log (2,(1 / p))))\]) >= 10 by XXREAL_0:25;
A4: max (10,[/(9 + (log (2,(1 / p))))\]) is Integer by XXREAL_0:16;
A5: max (10,[/(9 + (log (2,(1 / p))))\]) >= [/(9 + (log (2,(1 / p))))\] by XXREAL_0:25;
reconsider N = max (10,[/(9 + (log (2,(1 / p))))\]) as Element of NAT by A3, A4, INT_1:3;
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
abs (((f /" (seq_n! 0)) . n) - 0) < p
let n be Element of NAT ; ::_thesis: ( n >= N implies abs (((f /" (seq_n! 0)) . n) - 0) < p )
A6: [/(9 + (log (2,(1 / p))))\] >= 9 + (log (2,(1 / p))) by INT_1:def_7;
assume A7: n >= N ; ::_thesis: abs (((f /" (seq_n! 0)) . n) - 0) < p
then n >= [/(9 + (log (2,(1 / p))))\] by A5, XXREAL_0:2;
then n >= 9 + (log (2,(1 / p))) by A6, 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 A8: 1 / (2 to_power (n - 9)) <= 1 / (1 / p) by A2, XREAL_1:85;
A9: (f /" (seq_n! 0)) . n = (f . n) / ((seq_n! 0) . n) by Lm4
.= (2 to_power ((2 * n) + 0)) / ((seq_n! 0) . n) by Def1
.= (2 to_power ((2 * n) + 0)) / ((n + 0) !) by Def5
.= (2 to_power (2 * n)) / (n !) ;
n >= 10 by A3, A7, XXREAL_0:2;
then (f /" (seq_n! 0)) . n < 1 / (2 to_power (n - 9)) by A9, Lm34;
then (f /" (seq_n! 0)) . n < p by A8, XXREAL_0:2;
hence abs (((f /" (seq_n! 0)) . n) - 0) < p by A9, ABSVALUE:def_1; ::_thesis: verum
end;
then A10: f /" (seq_n! 0) is convergent by SEQ_2:def_6;
then A11: lim (f /" (seq_n! 0)) = 0 by A1, SEQ_2:def_7;
then not seq_n! 0 in Big_Oh f by A10, ASYMPT_0:16;
then A12: not f in Big_Omega (seq_n! 0) by ASYMPT_0:19;
f in Big_Oh (seq_n! 0) by A10, A11, ASYMPT_0:16;
hence ( Big_Oh f c= Big_Oh (seq_n! 0) & not Big_Oh f = Big_Oh (seq_n! 0) ) by A12, Th4; ::_thesis: verum
end;
theorem :: ASYMPT_1:31
( Big_Oh (seq_n! 0) c= Big_Oh (seq_n! 1) & not Big_Oh (seq_n! 0) = Big_Oh (seq_n! 1) )
proof
set g = seq_n! 1;
set f = seq_n! 0;
set h = (seq_n! 0) /" (seq_n! 1);
A1: for n being Element of NAT holds ((seq_n! 0) /" (seq_n! 1)) . n = 1 / (n + 1)
proof
let n be Element of NAT ; ::_thesis: ((seq_n! 0) /" (seq_n! 1)) . n = 1 / (n + 1)
A2: n ! <> 0 by NEWTON:17;
((seq_n! 0) /" (seq_n! 1)) . n = ((seq_n! 0) . n) / ((seq_n! 1) . n) by Lm4
.= ((n + 0) !) / ((seq_n! 1) . n) by Def5
.= (n !) / ((n + 1) !) by Def5
.= ((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 ((seq_n! 0) /" (seq_n! 1)) . n = 1 / (n + 1) ; ::_thesis: verum
end;
A3: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
abs_((((seq_n!_0)_/"_(seq_n!_1))_._n)_-_0)_<_p
let p be real number ; ::_thesis: ( p > 0 implies ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs ((((seq_n! 0) /" (seq_n! 1)) . n) - 0) < p )
assume A4: p > 0 ; ::_thesis: ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs ((((seq_n! 0) /" (seq_n! 1)) . n) - 0) < p
set N = max (1,[/(1 / p)\]);
A5: max (1,[/(1 / p)\]) >= 1 by XXREAL_0:25;
A6: max (1,[/(1 / p)\]) >= [/(1 / p)\] by XXREAL_0:25;
max (1,[/(1 / p)\]) is Integer by XXREAL_0:16;
then reconsider N = max (1,[/(1 / p)\]) 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;
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
abs ((((seq_n! 0) /" (seq_n! 1)) . n) - 0) < p
let n be Element of NAT ; ::_thesis: ( n >= N implies abs ((((seq_n! 0) /" (seq_n! 1)) . n) - 0) < p )
assume n >= N ; ::_thesis: abs ((((seq_n! 0) /" (seq_n! 1)) . n) - 0) < p
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 A8: ((seq_n! 0) /" (seq_n! 1)) . n < p by A1;
A9: 0 < 1 / (n + 1) ;
- p < - 0 by A4, XREAL_1:24;
then A10: - p < ((seq_n! 0) /" (seq_n! 1)) . n by A1, A9;
abs (((seq_n! 0) /" (seq_n! 1)) . n) < p
proof
percases ( ((seq_n! 0) /" (seq_n! 1)) . n >= 0 or ((seq_n! 0) /" (seq_n! 1)) . n < 0 ) ;
suppose ((seq_n! 0) /" (seq_n! 1)) . n >= 0 ; ::_thesis: abs (((seq_n! 0) /" (seq_n! 1)) . n) < p
hence abs (((seq_n! 0) /" (seq_n! 1)) . n) < p by A8, ABSVALUE:def_1; ::_thesis: verum
end;
supposeA11: ((seq_n! 0) /" (seq_n! 1)) . n < 0 ; ::_thesis: abs (((seq_n! 0) /" (seq_n! 1)) . n) < p
A12: (- 1) * (- p) > (- 1) * (((seq_n! 0) /" (seq_n! 1)) . n) by A10, XREAL_1:69;
abs (((seq_n! 0) /" (seq_n! 1)) . n) = - (((seq_n! 0) /" (seq_n! 1)) . n) by A11, ABSVALUE:def_1;
hence abs (((seq_n! 0) /" (seq_n! 1)) . n) < p by A12; ::_thesis: verum
end;
end;
end;
hence abs ((((seq_n! 0) /" (seq_n! 1)) . n) - 0) < p ; ::_thesis: verum
end;
then A13: (seq_n! 0) /" (seq_n! 1) is convergent by SEQ_2:def_6;
then A14: lim ((seq_n! 0) /" (seq_n! 1)) = 0 by A3, SEQ_2:def_7;
then not seq_n! 1 in Big_Oh (seq_n! 0) by A13, ASYMPT_0:16;
then A15: not seq_n! 0 in Big_Omega (seq_n! 1) by ASYMPT_0:19;
seq_n! 0 in Big_Oh (seq_n! 1) by A13, A14, ASYMPT_0:16;
hence ( Big_Oh (seq_n! 0) c= Big_Oh (seq_n! 1) & not Big_Oh (seq_n! 0) = Big_Oh (seq_n! 1) ) by A15, Th4; ::_thesis: verum
end;
theorem :: ASYMPT_1:32
for g being Real_Sequence st ( for n being Element of NAT st n > 0 holds
g . n = n to_power n ) holds
ex s being eventually-positive Real_Sequence st
( s = g & Big_Oh (seq_n! 1) c= Big_Oh s & not Big_Oh (seq_n! 1) = Big_Oh s )
proof
set f = seq_n! 1;
let g be Real_Sequence; ::_thesis: ( ( for n being Element of NAT st n > 0 holds
g . n = n to_power n ) implies ex s being eventually-positive Real_Sequence st
( s = g & Big_Oh (seq_n! 1) c= Big_Oh s & not Big_Oh (seq_n! 1) = Big_Oh s ) )
assume A1: for n being Element of NAT st n > 0 holds
g . n = n to_power n ; ::_thesis: ex s being eventually-positive Real_Sequence st
( s = g & Big_Oh (seq_n! 1) c= Big_Oh s & not Big_Oh (seq_n! 1) = Big_Oh s )
A2: g is eventually-positive
proof
take 1 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 1 <= b1 or not g . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 1 <= n or not g . n <= 0 )
assume A3: n >= 1 ; ::_thesis: not g . n <= 0
then g . n = n to_power n by A1;
hence not g . n <= 0 by A3, POWER:34; ::_thesis: verum
end;
set h = (seq_n! 1) /" g;
reconsider g = g as eventually-positive Real_Sequence by A2;
deffunc H1( Element of NAT ) -> Element of REAL = (((seq_n! 1) /" g) . $1) / (((seq_n! 1) /" g) . ($1 + 1));
consider p being Real_Sequence such that
A4: for n being Element of NAT holds p . n = H1(n) from SEQ_1:sch_1();
defpred S1[ Nat] means p . $1 > 2;
A5: for n being Element of NAT st n > 0 holds
p . n = ((n + 1) / (n + 2)) * (((n + 1) / n) to_power n)
proof
let n be Element of NAT ; ::_thesis: ( n > 0 implies p . n = ((n + 1) / (n + 2)) * (((n + 1) / n) to_power n) )
assume A6: n > 0 ; ::_thesis: p . n = ((n + 1) / (n + 2)) * (((n + 1) / n) to_power n)
A7: (n + 1) ! > 0 by NEWTON:17;
p . n = (((seq_n! 1) /" g) . n) / (((seq_n! 1) /" g) . (n + 1)) by A4
.= (((seq_n! 1) . n) / (g . n)) / (((seq_n! 1) /" g) . (n + 1)) by Lm4
.= (((n + 1) !) / (g . n)) / (((seq_n! 1) /" g) . (n + 1)) by Def5
.= (((n + 1) !) / (n to_power n)) / (((seq_n! 1) /" g) . (n + 1)) by A1, A6
.= (((n + 1) !) / (n to_power n)) / (((seq_n! 1) . (n + 1)) / (g . (n + 1))) by Lm4
.= (((n + 1) !) / (n to_power n)) / ((((n + 1) + 1) !) / (g . (n + 1))) by Def5
.= (((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 A7, 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 A6, POWER:31
.= ((n + 1) * (1 / (n + 2))) * (((n + 1) / n) to_power n)
.= ((n + 1) / (n + 2)) * (((n + 1) / n) to_power n) ;
hence p . n = ((n + 1) / (n + 2)) * (((n + 1) / n) to_power n) ; ::_thesis: verum
end;
A8: for k being Nat st k >= 4 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 4 & S1[k] implies S1[k + 1] )
assume that
A9: k >= 4 and
A10: p . k > 2 ; ::_thesis: S1[k + 1]
(k + 2) * ((k + 1) ") > 0 * ((k + 1) ") by XREAL_1:68;
then A11: ((k + 2) / (k + 1)) to_power (k + 1) > 0 by POWER:34;
(k + 1) * (k ") > 0 * (k ") by A9, XREAL_1:68;
then ((k + 1) / k) to_power k > 0 by POWER:34;
then A12: (((k + 1) / k) to_power k) * ((((k + 2) / (k + 1)) to_power (k + 1)) ") > 0 * ((((k + 2) / (k + 1)) to_power (k + 1)) ") by A11, XREAL_1:68;
A13: k in NAT by ORDINAL1:def_12;
A14: now__::_thesis:_not_(k_+_1)_*_(k_+_3)_>=_(k_+_2)_*_(k_+_2)
assume (k + 1) * (k + 3) >= (k + 2) * (k + 2) ; ::_thesis: contradiction
then ((k * k) + (4 * k)) + 3 >= ((k * k) + (2 * (2 * k))) + (2 ^2) ;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
then (k + 1) * (k + 3) < 1 * ((k + 2) * (k + 2)) ;
then A15: ((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 A16: ((k + 1) * (k + 3)) * (((k + 2) * (k + 2)) ") > 0 * (((k + 2) * (k + 2)) ") by A14, XREAL_1:68;
k >= 1 by A9, XXREAL_0:2;
then ((k + 1) / k) to_power k <= 1 * (((k + 2) / (k + 1)) to_power (k + 1)) by A13, Lm42;
then (((k + 1) / k) to_power k) / (((k + 2) / (k + 1)) to_power (k + 1)) <= 1 by A11, XREAL_1:79;
then A17: (((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 A12, A16, A15, XREAL_1:98;
(k + 2) * ((k + 3) ") > 0 * ((k + 3) ") by XREAL_1:68;
then A18: ((k + 2) / (k + 3)) * (((k + 2) / (k + 1)) to_power (k + 1)) > ((k + 2) / (k + 3)) * 0 by A11, XREAL_1:68;
A19: p . (k + 1) = (((k + 1) + 1) / ((k + 1) + 2)) * (((k + (1 + 1)) / (k + 1)) to_power (k + 1)) by A5
.= ((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 A5, A9, A13
.= (((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 A19, A18, A17, XREAL_1:68;
then p . (k + 1) > p . k by A19, A18, XCMPLX_1:87;
hence S1[k + 1] by A10, XXREAL_0:2; ::_thesis: verum
end;
defpred S2[ Nat] means ((seq_n! 1) /" g) . $1 < 1 / ($1 - 2);
take g ; ::_thesis: ( g = g & Big_Oh (seq_n! 1) c= Big_Oh g & not Big_Oh (seq_n! 1) = Big_Oh g )
A20: for n being Element of NAT st n >= 1 holds
((seq_n! 1) /" g) . n > 0
proof
let n be Element of NAT ; ::_thesis: ( n >= 1 implies ((seq_n! 1) /" g) . n > 0 )
A21: (n + 1) ! > 0 by NEWTON:17;
assume A22: n >= 1 ; ::_thesis: ((seq_n! 1) /" g) . n > 0
then n to_power n > 0 by POWER:34;
then A23: ((n + 1) !) * (1 / (n to_power n)) > ((n + 1) !) * 0 by A21, XREAL_1:68;
((seq_n! 1) /" g) . n = ((seq_n! 1) . n) / (g . n) by Lm4
.= ((n + 1) !) / (g . n) by Def5
.= ((n + 1) !) / (n to_power n) by A1, A22 ;
hence ((seq_n! 1) /" g) . n > 0 by A23; ::_thesis: verum
end;
p . 4 = ((4 + 1) / (4 + 2)) * (((4 + 1) / 4) to_power 4) by A5
.= (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 A24: S1[4] ;
A25: for n being Nat st n >= 4 holds
S1[n] from NAT_1:sch_8(A24, A8);
A26: 3 = 4 - 1 ;
A27: for k being Nat st k >= 4 & S2[k] holds
S2[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 4 & S2[k] implies S2[k + 1] )
assume that
A28: k >= 4 and
A29: ((seq_n! 1) /" g) . k < 1 / (k - 2) ; ::_thesis: S2[k + 1]
A30: k in NAT by ORDINAL1:def_12;
A31: ((seq_n! 1) /" g) . (k + 1) > 0 by A20, NAT_1:11;
p . k > 2 by A25, A28;
then (((seq_n! 1) /" g) . k) / (((seq_n! 1) /" g) . (k + 1)) > 2 by A4, A30;
then ((((seq_n! 1) /" g) . k) / (((seq_n! 1) /" g) . (k + 1))) * (((seq_n! 1) /" g) . (k + 1)) > 2 * (((seq_n! 1) /" g) . (k + 1)) by A31, XREAL_1:68;
then ((seq_n! 1) /" g) . k > 2 * (((seq_n! 1) /" g) . (k + 1)) by A31, XCMPLX_1:87;
then A32: (((seq_n! 1) /" g) . k) / 2 > ((seq_n! 1) /" g) . (k + 1) by XREAL_1:81;
A33: k - 1 >= 3 by A26, A28, XREAL_1:9;
k >= 3 by A28, XXREAL_0:2;
then 2 * (k - 2) >= k - 1 by A30, Lm35;
then A34: 1 / (2 * (k - 2)) <= 1 / (k - 1) by A33, XREAL_1:85;
(((seq_n! 1) /" g) . k) * (1 / 2) < (1 / (k - 2)) * (1 / 2) by A29, XREAL_1:68;
then (((seq_n! 1) /" g) . k) / 2 < 1 / (2 * (k - 2)) by XCMPLX_1:102;
then (((seq_n! 1) /" g) . k) / 2 < 1 / (k - 1) by A34, XXREAL_0:2;
hence S2[k + 1] by A32, XXREAL_0:2; ::_thesis: verum
end;
((seq_n! 1) /" g) . 4 = ((seq_n! 1) . 4) / (g . 4) by Lm4
.= ((4 + 1) !) / (g . 4) by Def5
.= 120 / 256 by A1, Lm33, Lm37 ;
then A35: S2[4] ;
A36: for n being Nat st n >= 4 holds
S2[n] from NAT_1:sch_8(A35, A27);
A37: now__::_thesis:_for_p_being_real_number_st_p_>_0_holds_
ex_N_being_Element_of_NAT_st_
for_n_being_Element_of_NAT_st_n_>=_N_holds_
abs_((((seq_n!_1)_/"_g)_._n)_-_0)_<_p
let p be real number ; ::_thesis: ( p > 0 implies ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs ((((seq_n! 1) /" g) . n) - 0) < p )
set N = [/((1 / p) + 4)\];
A38: [/((1 / p) + 4)\] >= (1 / p) + 4 by INT_1:def_7;
assume A39: p > 0 ; ::_thesis: ex N being Element of NAT st
for n being Element of NAT st n >= N holds
abs ((((seq_n! 1) /" g) . n) - 0) < p
then A40: 4 + (1 / p) > 4 by XREAL_1:29;
then A41: [/((1 / p) + 4)\] >= 4 by A38, XXREAL_0:2;
reconsider N = [/((1 / p) + 4)\] as Element of NAT by A38, A40, INT_1:3;
take N = N; ::_thesis: for n being Element of NAT st n >= N holds
abs ((((seq_n! 1) /" g) . n) - 0) < p
let n be Element of NAT ; ::_thesis: ( n >= N implies abs ((((seq_n! 1) /" g) . n) - 0) < p )
assume A42: n >= N ; ::_thesis: abs ((((seq_n! 1) /" g) . n) - 0) < p
then A43: n >= 4 by A41, XXREAL_0:2;
then n >= 1 by XXREAL_0:2;
then A44: ((seq_n! 1) /" g) . n > 0 by A20;
A45: (1 / p) + 2 > 1 / p by XREAL_1:29;
n >= ((1 / p) + 2) + 2 by A38, A42, XXREAL_0:2;
then n - 2 >= (1 / p) + 2 by XREAL_1:19;
then n - 2 > 1 / p by A45, XXREAL_0:2;
then A46: 1 / (n - 2) < 1 / (1 / p) by A39, XREAL_1:88;
((seq_n! 1) /" g) . n < 1 / (n - 2) by A36, A43;
then ((seq_n! 1) /" g) . n < p by A46, XXREAL_0:2;
hence abs ((((seq_n! 1) /" g) . n) - 0) < p by A44, ABSVALUE:def_1; ::_thesis: verum
end;
then A47: (seq_n! 1) /" g is convergent by SEQ_2:def_6;
then A48: lim ((seq_n! 1) /" g) = 0 by A37, SEQ_2:def_7;
then not g in Big_Oh (seq_n! 1) by A47, ASYMPT_0:16;
then A49: not seq_n! 1 in Big_Omega g by ASYMPT_0:19;
seq_n! 1 in Big_Oh g by A47, A48, ASYMPT_0:16;
hence ( g = g & Big_Oh (seq_n! 1) c= Big_Oh g & not Big_Oh (seq_n! 1) = Big_Oh g ) by A49, Th4; ::_thesis: verum
end;
begin
Lm43: for k, n being Element of NAT st k <= n holds
n choose k >= ((n + 1) choose k) / (n + 1)
proof
let k, n be Element of NAT ; ::_thesis: ( k <= n implies n choose k >= ((n + 1) choose k) / (n + 1) )
set n1 = n + 1;
assume A1: k <= n ; ::_thesis: n choose k >= ((n + 1) choose k) / (n + 1)
then reconsider l = n - k as Element of NAT by INT_1:5;
set l1 = l + 1;
A2: l + 1 = (n + 1) - k ;
0 + 1 <= l + 1 by XREAL_1:6;
then 1 / 1 >= 1 / (l + 1) by XREAL_1:85;
then A3: (n choose k) * (1 / 1) >= (n choose k) * (1 / (l + 1)) by XREAL_1:64;
n + 0 <= n + 1 by XREAL_1:6;
then k <= n + 1 by A1, XXREAL_0:2;
then ((n + 1) choose k) / (n + 1) = (((n + 1) !) / ((k !) * ((l + 1) !))) / (n + 1) by A2, NEWTON:def_3
.= (((n + 1) * (n !)) / ((k !) * ((l + 1) !))) / ((n + 1) * 1) by NEWTON:15
.= (((n + 1) * (n !)) * (((k !) * ((l + 1) !)) ")) / ((n + 1) * 1)
.= ((n + 1) * ((n !) * (((k !) * ((l + 1) !)) "))) / ((n + 1) * 1)
.= ((n + 1) * ((n !) / ((k !) * ((l + 1) !)))) / ((n + 1) * 1)
.= ((n + 1) / (n + 1)) * (((n !) / ((k !) * ((l + 1) !))) / 1)
.= 1 * (((n !) / ((k !) * ((l + 1) !))) / 1) by XCMPLX_1:60
.= (n !) / ((k !) * ((l !) * (l + 1))) by NEWTON:15
.= ((n !) * 1) / (((k !) * (l !)) * (l + 1))
.= ((n !) / ((k !) * (l !))) * (1 / (l + 1)) by XCMPLX_1:76
.= (n choose k) * (1 / (l + 1)) by A1, NEWTON:def_3 ;
hence n choose k >= ((n + 1) choose k) / (n + 1) by A3; ::_thesis: verum
end;
theorem :: ASYMPT_1:33
for n being Element of NAT st n >= 1 holds
for f being Real_Sequence
for k being Element of NAT st ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) holds
f . n >= (n to_power (k + 1)) / (k + 1)
proof
defpred S1[ Nat] means for f being Real_Sequence
for k being Element of NAT st ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) holds
f . $1 >= ($1 to_power (k + 1)) / (k + 1);
A1: for n being Nat st n >= 1 & S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( n >= 1 & S1[n] implies S1[n + 1] )
assume that
n >= 1 and
A2: for f being Real_Sequence
for k being Element of NAT st ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) holds
f . n >= (n to_power (k + 1)) / (k + 1) ; ::_thesis: S1[n + 1]
reconsider n = n as Element of NAT by ORDINAL1:def_12;
let f be Real_Sequence; ::_thesis: for k being Element of NAT st ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) holds
f . (n + 1) >= ((n + 1) to_power (k + 1)) / (k + 1)
let k be Element of NAT ; ::_thesis: ( ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) implies f . (n + 1) >= ((n + 1) to_power (k + 1)) / (k + 1) )
assume A3: for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ; ::_thesis: f . (n + 1) >= ((n + 1) to_power (k + 1)) / (k + 1)
set R3 = (n,1) In_Power (k + 1);
len ((n,1) In_Power (k + 1)) = (k + 1) + 1 by NEWTON:def_4
.= k + 2 ;
then reconsider R3 = (n,1) In_Power (k + 1) as Element of (k + 2) -tuples_on REAL by FINSEQ_2:92;
set R2 = ((k + 1) ") * ((n,1) In_Power (k + 1));
len (((k + 1) ") * ((n,1) In_Power (k + 1))) = len ((n,1) In_Power (k + 1)) by NEWTON:2
.= (k + 1) + 1 by NEWTON:def_4
.= k + 2 ;
then reconsider R2 = ((k + 1) ") * ((n,1) In_Power (k + 1)) as Element of (k + 2) -tuples_on REAL by FINSEQ_2:92;
set R1 = <*((n to_power (k + 1)) / (k + 1))*> ^ ((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 ((seq_n^ k),n) >= (n to_power (k + 1)) / (k + 1) by A3;
then A5: (Partial_Sums (seq_n^ k)) . n >= (n to_power (k + 1)) / (k + 1) by SERIES_1:def_5;
(seq_n^ k) . (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)) + ((seq_n^ k) . (n + 1)) = Sum (<*((n to_power (k + 1)) / (k + 1))*> ^ ((n,1) In_Power k)) by RVSUM_1:76;
len ((n,1) In_Power k) = k + 1 by NEWTON:def_4;
then A7: len (<*((n to_power (k + 1)) / (k + 1))*> ^ ((n,1) In_Power k)) = (k + 1) + 1 by A4, FINSEQ_1:22
.= k + 2 ;
then reconsider R1 = <*((n to_power (k + 1)) / (k + 1))*> ^ ((n,1) In_Power k) as Element of (k + 2) -tuples_on REAL by FINSEQ_2:92;
A8: for i being Nat st i in Seg (k + 2) holds
R2 . i <= R1 . i
proof
set k1 = (k + 1) " ;
let i be Nat; ::_thesis: ( i in Seg (k + 2) implies R2 . i <= R1 . i )
assume A9: i in Seg (k + 2) ; ::_thesis: R2 . i <= R1 . i
A10: 1 <= i by A9, FINSEQ_1:1;
set r2 = R2 . i;
set r1 = R1 . i;
A11: i <= k + 2 by A9, FINSEQ_1:1;
percases ( i = 1 or i > 1 ) by A10, XXREAL_0:1;
supposeA12: i = 1 ; ::_thesis: R2 . i <= R1 . i
n |^ (k + 1) = R3 . 1 by NEWTON:28;
then R2 . i = ((k + 1) ") * (n |^ (k + 1)) by A12, RVSUM_1:45
.= (n to_power (k + 1)) / (k + 1) ;
hence R2 . i <= R1 . i by A12, FINSEQ_1:41; ::_thesis: verum
end;
supposeA13: i > 1 ; ::_thesis: R2 . i <= R1 . i
set i0 = i - 1;
set m = (i - 1) - 1;
A14: i - 1 > 1 - 1 by A13, XREAL_1:9;
then reconsider i0 = i - 1 as Element of NAT by INT_1:3;
set l = k - ((i - 1) - 1);
A15: i0 >= 0 + 1 by A14, INT_1:7;
then (i - 1) - 1 >= 0 by XREAL_1:19;
then reconsider m = (i - 1) - 1 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 k - ((i - 1) - 1) >= 0 by XREAL_1:19;
then reconsider l = k - ((i - 1) - 1) as Element of NAT by INT_1:3;
A19: (k + 1) - i0 = l ;
then A20: i0 + 0 <= k + 1 by XREAL_1:19;
reconsider i3 = (k + 1) - i0 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 . i = ((k + 1) ") * ((((k + 1) choose i0) * (n |^ i3)) * (1 |^ i0)) by RVSUM_1:45
.= ((k + 1) ") * ((((k + 1) choose l) * (n |^ l)) * (1 |^ i0)) by A20, NEWTON:20
.= ((k + 1) ") * ((((k + 1) choose l) * (n |^ l)) * 1) by NEWTON:10
.= (((k + 1) ") * ((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 . i = ((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 R2 . i <= R1 . i by A21, A22, XREAL_1:64; ::_thesis: verum
end;
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 ((seq_n^ k),(n + 1)) by A3
.= (Partial_Sums (seq_n^ k)) . (n + 1) by SERIES_1:def_5
.= ((Partial_Sums (seq_n^ k)) . n) + ((seq_n^ k) . (n + 1)) by SERIES_1:def_1 ;
then f . (n + 1) >= ((n to_power (k + 1)) / (k + 1)) + ((seq_n^ k) . (n + 1)) by A5, XREAL_1:6;
hence f . (n + 1) >= ((n + 1) to_power (k + 1)) / (k + 1) by A6, A23, XXREAL_0:2; ::_thesis: verum
end;
A24: S1[1]
proof
let f be Real_Sequence; ::_thesis: for k being Element of NAT st ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) holds
f . 1 >= (1 to_power (k + 1)) / (k + 1)
let k be Element of NAT ; ::_thesis: ( ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) implies f . 1 >= (1 to_power (k + 1)) / (k + 1) )
assume A25: for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ; ::_thesis: f . 1 >= (1 to_power (k + 1)) / (k + 1)
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 ((seq_n^ k),1) by A25
.= (Partial_Sums (seq_n^ k)) . (0 + 1) by SERIES_1:def_5
.= ((Partial_Sums (seq_n^ k)) . 0) + ((seq_n^ k) . 1) by SERIES_1:def_1
.= ((seq_n^ k) . 1) + ((seq_n^ k) . 0) by SERIES_1:def_1
.= (1 to_power k) + ((seq_n^ k) . 0) by Def3
.= 1 + ((seq_n^ k) . 0) by POWER:26
.= 1 + 0 by Def3
.= 1 / 1 ;
hence f . 1 >= (1 to_power (k + 1)) / (k + 1) by A26, A27, XREAL_1:85; ::_thesis: verum
end;
for n being Nat st n >= 1 holds
S1[n] from NAT_1:sch_8(A24, A1);
hence for n being Element of NAT st n >= 1 holds
for f being Real_Sequence
for k being Element of NAT st ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) holds
f . n >= (n to_power (k + 1)) / (k + 1) ; ::_thesis: verum
end;
begin
Lm44: for f being Real_Sequence st ( for n being Element of NAT holds f . n = log (2,(n !)) ) holds
for n being Element of NAT holds f . n = Sum (seq_logn,n)
proof
set g = seq_logn ;
let f be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds f . n = log (2,(n !)) ) implies for n being Element of NAT holds f . n = Sum (seq_logn,n) )
assume A1: for n being Element of NAT holds f . n = log (2,(n !)) ; ::_thesis: for n being Element of NAT holds f . n = Sum (seq_logn,n)
defpred S1[ Element of NAT ] means f . $1 = Sum (seq_logn,$1);
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: f . k = Sum (seq_logn,k) ; ::_thesis: S1[k + 1]
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 (seq_logn,k)) by A1, A3
.= (seq_logn . (k + 1)) + (Sum (seq_logn,k)) by Def2
.= (seq_logn . (k + 1)) + ((Partial_Sums seq_logn) . k) by SERIES_1:def_5
.= (Partial_Sums seq_logn) . (k + 1) by SERIES_1:def_1
.= Sum (seq_logn,(k + 1)) by SERIES_1:def_5 ;
hence S1[k + 1] ; ::_thesis: verum
end;
A5: Sum (seq_logn,0) = (Partial_Sums seq_logn) . 0 by SERIES_1:def_5
.= seq_logn . 0 by SERIES_1:def_1
.= 0 by Def2 ;
f . 0 = log (2,1) by A1, NEWTON:12
.= 0 by POWER:51 ;
then A6: S1[ 0 ] by A5;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A6, A2);
hence for n being Element of NAT holds f . n = Sum (seq_logn,n) ; ::_thesis: verum
end;
Lm45: for n being Element of NAT st n >= 4 holds
n * (log (2,n)) >= 2 * n
proof
let n be Element of NAT ; ::_thesis: ( n >= 4 implies n * (log (2,n)) >= 2 * n )
assume n >= 4 ; ::_thesis: n * (log (2,n)) >= 2 * n
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 n * (log (2,n)) >= 2 * n by XREAL_1:64; ::_thesis: verum
end;
theorem :: ASYMPT_1:34
for f, g being Real_Sequence st ( for n being Element of NAT st n > 0 holds
g . n = n * (log (2,n)) ) & ( for n being Element of NAT holds f . n = log (2,(n !)) ) holds
ex s being eventually-nonnegative Real_Sequence st
( s = g & f in Big_Theta s )
proof
set h = seq_logn ;
let f, g be Real_Sequence; ::_thesis: ( ( for n being Element of NAT st n > 0 holds
g . n = n * (log (2,n)) ) & ( for n being Element of NAT holds f . n = log (2,(n !)) ) implies ex s being eventually-nonnegative Real_Sequence st
( s = g & f in Big_Theta s ) )
assume that
A1: for n being Element of NAT st n > 0 holds
g . n = n * (log (2,n)) and
A2: for n being Element of NAT holds f . n = log (2,(n !)) ; ::_thesis: ex s being eventually-nonnegative Real_Sequence st
( s = g & f in Big_Theta s )
g is eventually-positive
proof
take 2 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 2 <= b1 or not g . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 2 <= n or not g . n <= 0 )
assume A3: n >= 2 ; ::_thesis: not g . n <= 0
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 A3, XREAL_1:68;
hence not g . n <= 0 by A1, A3; ::_thesis: verum
end;
then reconsider g = g as eventually-positive Real_Sequence ;
A4: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_4_holds_
(_(1_/_4)_*_(g_._n)_<=_f_._n_&_f_._n_<=_1_*_(g_._n)_)
let n be Element of NAT ; ::_thesis: ( n >= 4 implies ( (1 / 4) * (g . n) <= f . n & f . n <= 1 * (g . n) ) )
set n1 = [/(n / 2)\];
assume A5: n >= 4 ; ::_thesis: ( (1 / 4) * (g . n) <= f . n & f . n <= 1 * (g . n) )
then A6: (n / 2) * (log (2,(n / 2))) = (n / 2) * ((log (2,n)) - (log (2,2))) by POWER:54
.= (n / 2) * ((log (2,n)) - 1) by POWER:52
.= ((n * (log (2,n))) / 2) - (n / 2) ;
ex s being Real_Sequence st
( s . 0 = 0 & ( for m being Element of NAT st m > 0 holds
s . m = log (2,(n / 2)) ) )
proof
defpred S1[ Element of NAT , Real] means ( ( $1 = 0 implies $2 = 0 ) & ( $1 > 0 implies $2 = log (2,(n / 2)) ) );
A7: for x being Element of NAT ex y being Element of REAL st S1[x,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of REAL st S1[n,y]
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
end;
end;
consider h being Function of NAT,REAL such that
A8: for x being Element of NAT holds S1[x,h . x] from FUNCT_2:sch_3(A7);
take h ; ::_thesis: ( h . 0 = 0 & ( for m being Element of NAT st m > 0 holds
h . m = log (2,(n / 2)) ) )
thus h . 0 = 0 by A8; ::_thesis: for m being Element of NAT st m > 0 holds
h . m = log (2,(n / 2))
let n be Element of NAT ; ::_thesis: ( n > 0 implies h . n = log (2,(n / 2)) )
thus ( n > 0 implies h . n = log (2,(n / 2)) ) by A8; ::_thesis: verum
end;
then consider p being Real_Sequence such that
A9: p . 0 = 0 and
A10: for m being Element of NAT st m > 0 holds
p . m = log (2,(n / 2)) ;
A11: [/(n / 2)\] >= n / 2 by INT_1:def_7;
then reconsider n1 = [/(n / 2)\] as Element of NAT by INT_1:3;
set n2 = n1 - 1;
A12: n * (2 ") > 0 * (2 ") by A5, XREAL_1:68;
A13: now__::_thesis:_not_n1_-_1_<_0
assume n1 - 1 < 0 ; ::_thesis: contradiction
then n1 - 1 <= - 1 by INT_1:8;
then (n1 - 1) + 1 <= (- 1) + 1 by XREAL_1:6;
hence contradiction by A12, INT_1:def_7; ::_thesis: verum
end;
(n * (log (2,n))) * (4 ") >= (2 * n) * (4 ") by A5, 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 A14: ((n * (log (2,n))) / 2) - (n / 2) >= (n * (log (2,n))) / 4 by XREAL_1:19;
2 * 2 <= n by A5;
then 2 <= n / 2 by XREAL_1:77;
then log (2,2) <= log (2,(n / 2)) by PRE_FF:10;
then A15: 1 <= log (2,(n / 2)) by POWER:52;
reconsider n2 = n1 - 1 as Element of NAT by A13, INT_1:3;
A16: for k being Element of NAT st n2 + 1 <= k & k <= n holds
p . k <= seq_logn . k
proof
let k be Element of NAT ; ::_thesis: ( n2 + 1 <= k & k <= n implies p . k <= seq_logn . k )
assume that
A17: n2 + 1 <= k and
k <= n ; ::_thesis: p . k <= seq_logn . k
n / 2 <= k by A11, A17, XXREAL_0:2;
then log (2,(n / 2)) <= log (2,k) by A12, PRE_FF:10;
then p . k <= log (2,k) by A10, A17;
hence p . k <= seq_logn . k by A17, Def2; ::_thesis: verum
end;
n >= n1 by Lm17;
then A18: Sum (seq_logn,n,n2) >= Sum (p,n,n2) by A16, Lm16;
A19: now__::_thesis:_not_n_-_n2_<_n_/_2
[/(n / 2)\] < (n / 2) + 1 by INT_1:def_7;
then n2 < n / 2 by XREAL_1:19;
then A20: (n / 2) + n2 < (n / 2) + (n / 2) by XREAL_1:6;
assume n - n2 < n / 2 ; ::_thesis: contradiction
hence contradiction by A20, XREAL_1:19; ::_thesis: verum
end;
for k being Element of NAT st k <= n2 holds
seq_logn . k >= 0
proof
let k be Element of NAT ; ::_thesis: ( k <= n2 implies seq_logn . k >= 0 )
assume k <= n2 ; ::_thesis: seq_logn . k >= 0
percases ( k = 0 or k > 0 ) ;
suppose k = 0 ; ::_thesis: seq_logn . k >= 0
hence seq_logn . k >= 0 by Def2; ::_thesis: verum
end;
supposeA21: k > 0 ; ::_thesis: seq_logn . 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 seq_logn . k >= 0 by A21, Def2; ::_thesis: verum
end;
end;
end;
then Sum (seq_logn,n2) >= 0 by Lm12;
then (Sum (seq_logn,n)) + (Sum (seq_logn,n2)) >= (Sum (seq_logn,n)) + 0 by XREAL_1:6;
then Sum (seq_logn,n) >= (Sum (seq_logn,n)) - (Sum (seq_logn,n2)) by XREAL_1:20;
then A22: Sum (seq_logn,n) >= Sum (seq_logn,n,n2) by SERIES_1:def_6;
Sum (p,n,n2) = (n - n2) * (log (2,(n / 2))) by A9, A10, Lm18;
then A23: Sum (p,n,n2) >= (n / 2) * (log (2,(n / 2))) by A19, A15, XREAL_1:64;
(n * (log (2,n))) / 4 = (g . n) / 4 by A1, A5
.= (1 / 4) * (g . n) ;
then Sum (p,n,n2) >= (1 / 4) * (g . n) by A23, A6, A14, XXREAL_0:2;
then Sum (seq_logn,n,n2) >= (1 / 4) * (g . n) by A18, XXREAL_0:2;
then Sum (seq_logn,n) >= (1 / 4) * (g . n) by A22, XXREAL_0:2;
hence (1 / 4) * (g . n) <= f . n by A2, Lm44; ::_thesis: f . n <= 1 * (g . n)
ex s being Real_Sequence st
( s . 0 = 0 & ( for m being Element of NAT st m > 0 holds
s . m = log (2,n) ) )
proof
defpred S1[ Element of NAT , Real] means ( ( $1 = 0 implies $2 = 0 ) & ( $1 > 0 implies $2 = log (2,n) ) );
A24: for x being Element of NAT ex y being Element of REAL st S1[x,y]
proof
let n be Element of NAT ; ::_thesis: ex y being Element of REAL st S1[n,y]
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: ex y being Element of REAL st S1[n,y]
hence ex y being Element of REAL st S1[n,y] ; ::_thesis: verum
end;
end;
end;
consider h being Function of NAT,REAL such that
A25: for x being Element of NAT holds S1[x,h . x] from FUNCT_2:sch_3(A24);
take h ; ::_thesis: ( h . 0 = 0 & ( for m being Element of NAT st m > 0 holds
h . m = log (2,n) ) )
thus h . 0 = 0 by A25; ::_thesis: for m being Element of NAT st m > 0 holds
h . m = log (2,n)
let n be Element of NAT ; ::_thesis: ( n > 0 implies h . n = log (2,n) )
thus ( n > 0 implies h . n = log (2,n) ) by A25; ::_thesis: verum
end;
then consider q being Real_Sequence such that
A26: q . 0 = 0 and
A27: for m being Element of NAT st m > 0 holds
q . m = log (2,n) ;
A28: Sum (q,n) = n * (log (2,n)) by A26, A27, Lm14;
for k being Element of NAT st k <= n holds
seq_logn . k <= q . k
proof
let k be Element of NAT ; ::_thesis: ( k <= n implies seq_logn . k <= q . k )
assume A29: k <= n ; ::_thesis: seq_logn . k <= q . k
percases ( k = 0 or k > 0 ) ;
suppose k = 0 ; ::_thesis: seq_logn . k <= q . k
hence seq_logn . k <= q . k by A26, Def2; ::_thesis: verum
end;
supposeA30: k > 0 ; ::_thesis: seq_logn . k <= q . k
then log (2,k) <= log (2,n) by A29, PRE_FF:10;
then seq_logn . k <= log (2,n) by A30, Def2;
hence seq_logn . k <= q . k by A27, A30; ::_thesis: verum
end;
end;
end;
then A31: Sum (seq_logn,n) <= Sum (q,n) by Lm13;
log (2,(n !)) = f . n by A2
.= Sum (seq_logn,n) by A2, Lm44 ;
then log (2,(n !)) <= 1 * (g . n) by A1, A5, A31, A28;
hence f . n <= 1 * (g . n) by A2; ::_thesis: verum
end;
take g ; ::_thesis: ( g = g & f in Big_Theta g )
A32: f is Element of Funcs (NAT,REAL) by FUNCT_2:8;
Big_Theta g = { s where s is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (g . n) <= s . n & s . n <= c * (g . n) ) ) ) } by ASYMPT_0:27;
hence ( g = g & f in Big_Theta g ) by A32, A4; ::_thesis: verum
end;
begin
theorem :: ASYMPT_1:35
for f being eventually-nonnegative eventually-nondecreasing Real_Sequence
for t being Real_Sequence st ( for n being Element of NAT 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-nonnegative eventually-nondecreasing Real_Sequence; ::_thesis: for t being Real_Sequence st ( for n being Element of NAT holds
( ( n mod 2 = 0 implies t . n = 1 ) & ( n mod 2 = 1 implies t . n = n ) ) ) holds
not t in Big_Theta f
let t be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds
( ( n mod 2 = 0 implies t . n = 1 ) & ( n mod 2 = 1 implies t . n = n ) ) ) implies not t in Big_Theta f )
assume A1: for n being Element of NAT holds
( ( n mod 2 = 0 implies t . n = 1 ) & ( n mod 2 = 1 implies t . n = n ) ) ; ::_thesis: not t in Big_Theta f
A2: Big_Theta f = { s where s is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (f . n) <= s . n & s . n <= c * (f . n) ) ) ) } by ASYMPT_0:27;
hereby ::_thesis: verum
consider N0 being Element of NAT such that
A3: for n being Element of NAT st n >= N0 holds
f . n <= f . (n + 1) by ASYMPT_0:def_6;
assume t in Big_Theta f ; ::_thesis: contradiction
then consider s being Element of Funcs (NAT,REAL) such that
A4: s = t and
A5: ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (f . n) <= s . n & s . n <= c * (f . n) ) ) ) by A2;
consider c, d being Real, N being Element of NAT such that
A6: c > 0 and
A7: d > 0 and
A8: for n being Element of NAT 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: max (([/(c / d)\] + 1),(max (N,N0))) >= [/(c / d)\] + 1 by XXREAL_0:25;
A10: max (([/(c / d)\] + 1),(max (N,N0))) is Integer by XXREAL_0:16;
A11: max (([/(c / d)\] + 1),(max (N,N0))) >= max (N,N0) by XXREAL_0:25;
max (N,N0) >= N0 by XXREAL_0:25;
then A12: max (([/(c / d)\] + 1),(max (N,N0))) >= N0 by A11, XXREAL_0:2;
max (N,N0) >= N by XXREAL_0:25;
then A13: max (([/(c / d)\] + 1),(max (N,N0))) >= N by A11, XXREAL_0:2;
reconsider N1 = max (([/(c / d)\] + 1),(max (N,N0))) as Element of NAT by A11, A10, INT_1:3;
thus contradiction ::_thesis: verum
proof
percases ( N1 mod 2 = 1 or N1 mod 2 = 0 ) by NAT_D:12;
supposeA14: N1 mod 2 = 1 ; ::_thesis: contradiction
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 contradiction by A7, A18, XCMPLX_1:106; ::_thesis: verum
end;
supposeA20: N1 mod 2 = 0 ; ::_thesis: contradiction
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 contradiction by A7, A26, XCMPLX_1:106; ::_thesis: verum
end;
end;
end;
end;
end;
begin
Lm46: for n being Element of NAT st n >= 2 holds
[/(n / 2)\] < n
proof
let n be Element of NAT ; ::_thesis: ( n >= 2 implies [/(n / 2)\] < n )
assume A1: n >= 2 ; ::_thesis: [/(n / 2)\] < n
A2: now__::_thesis:_not_(n_/_2)_+_1_>_n
assume (n / 2) + 1 > n ; ::_thesis: contradiction
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; ::_thesis: verum
end;
[/(n / 2)\] < (n / 2) + 1 by INT_1:def_7;
hence [/(n / 2)\] < n by A2, XXREAL_0:2; ::_thesis: verum
end;
begin
definition
func POWEROF2SET -> non empty Subset of NAT equals :: ASYMPT_1:def 6
{ (2 to_power n) where n is Element of NAT : verum } ;
coherence
{ (2 to_power n) where n is Element of NAT : verum } is non empty Subset of NAT
proof
set IT = { (2 to_power n) where n is Element of NAT : verum } ;
A1: now__::_thesis:_for_x_being_set_st_x_in__{__(2_to_power_n)_where_n_is_Element_of_NAT_:_verum__}__holds_
x_in_NAT
let x be set ; ::_thesis: ( x in { (2 to_power n) where n is Element of NAT : verum } implies x in NAT )
assume x in { (2 to_power n) where n is Element of NAT : verum } ; ::_thesis: x in NAT
then ex n being Element of NAT st 2 to_power n = x ;
hence x in NAT ; ::_thesis: verum
end;
2 to_power 1 in { (2 to_power n) where n is Element of NAT : verum } ;
hence { (2 to_power n) where n is Element of NAT : verum } is non empty Subset of NAT by A1, TARSKI:def_3; ::_thesis: verum
end;
end;
:: deftheorem defines POWEROF2SET ASYMPT_1:def_6_:_
POWEROF2SET = { (2 to_power n) where n is Element of NAT : verum } ;
Lm47: for n being Element of NAT st n >= 2 holds
n ^2 > n + 1
proof
defpred S1[ Nat] means $1 ^2 > $1 + 1;
A1: for k being Nat st k >= 2 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 2 & S1[k] implies S1[k + 1] )
assume that
A2: k >= 2 and
A3: k ^2 > k + 1 ; ::_thesis: S1[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; ::_thesis: verum
end;
A5: S1[2] ;
for n being Nat st n >= 2 holds
S1[n] from NAT_1:sch_8(A5, A1);
hence for n being Element of NAT st n >= 2 holds
n ^2 > n + 1 ; ::_thesis: verum
end;
Lm48: for n being Element of NAT st n >= 1 holds
(2 to_power (n + 1)) - (2 to_power n) > 1
proof
let n be Element of NAT ; ::_thesis: ( n >= 1 implies (2 to_power (n + 1)) - (2 to_power n) > 1 )
assume n >= 1 ; ::_thesis: (2 to_power (n + 1)) - (2 to_power 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 (2 to_power (n + 1)) - (2 to_power n) > 1 by POWER:27; ::_thesis: verum
end;
Lm49: for n being Element of NAT st n >= 2 holds
not (2 to_power n) - 1 in POWEROF2SET
proof
A1: 1 = 2 - 1 ;
let n be Element of NAT ; ::_thesis: ( n >= 2 implies not (2 to_power n) - 1 in POWEROF2SET )
assume n >= 2 ; ::_thesis: not (2 to_power n) - 1 in POWEROF2SET
then A2: n - 1 >= 1 by A1, XREAL_1:9;
then n - 1 is Element of NAT by INT_1:3;
then (2 to_power ((n + (- 1)) + 1)) - (2 to_power (n - 1)) > 1 by A2, Lm48;
then 2 to_power n > 1 + (2 to_power (n - 1)) by XREAL_1:20;
then A3: (2 to_power n) - 1 > 2 to_power (n - 1) by XREAL_1:20;
assume (2 to_power n) - 1 in POWEROF2SET ; ::_thesis: contradiction
then consider m being Element of NAT such that
A4: 2 to_power m = (2 to_power n) - 1 ;
now__::_thesis:_not_m_>=_n
assume m >= n ; ::_thesis: contradiction
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; ::_thesis: verum
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; ::_thesis: verum
end;
theorem :: ASYMPT_1:36
for f being Real_Sequence st ( for n being Element of NAT 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 = (seq_n^ 1) taken_every 2;
set q = seq_logn /" (seq_n^ 1);
A1: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_1_holds_
(_((seq_n^_1)_taken_every_2)_._n_<=_2_*_((seq_n^_1)_._n)_&_((seq_n^_1)_taken_every_2)_._n_>=_0_)
let n be Element of NAT ; ::_thesis: ( n >= 1 implies ( ((seq_n^ 1) taken_every 2) . n <= 2 * ((seq_n^ 1) . n) & ((seq_n^ 1) taken_every 2) . n >= 0 ) )
assume A2: n >= 1 ; ::_thesis: ( ((seq_n^ 1) taken_every 2) . n <= 2 * ((seq_n^ 1) . n) & ((seq_n^ 1) taken_every 2) . n >= 0 )
then A3: 2 * n > 2 * 0 by XREAL_1:68;
A4: ((seq_n^ 1) taken_every 2) . n = (seq_n^ 1) . (2 * n) by ASYMPT_0:def_15
.= (2 * n) to_power 1 by A3, Def3
.= 2 * n by POWER:25 ;
(seq_n^ 1) . n = n to_power 1 by A2, Def3
.= n by POWER:25 ;
hence ((seq_n^ 1) taken_every 2) . n <= 2 * ((seq_n^ 1) . n) by A4; ::_thesis: ((seq_n^ 1) taken_every 2) . n >= 0
thus ((seq_n^ 1) taken_every 2) . n >= 0 by A4; ::_thesis: verum
end;
let f be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds
( ( n in POWEROF2SET implies f . n = n ) & ( not n in POWEROF2SET implies f . n = 2 to_power n ) ) ) implies ( 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 ) )
assume A5: for n being Element of NAT holds
( ( n in POWEROF2SET implies f . n = n ) & ( not n in POWEROF2SET implies f . n = 2 to_power n ) ) ; ::_thesis: ( 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 )
A6: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_1_&_n_in_POWEROF2SET_holds_
(_1_*_((seq_n^_1)_._n)_<=_f_._n_&_f_._n_<=_1_*_((seq_n^_1)_._n)_)
let n be Element of NAT ; ::_thesis: ( n >= 1 & n in POWEROF2SET implies ( 1 * ((seq_n^ 1) . n) <= f . n & f . n <= 1 * ((seq_n^ 1) . n) ) )
assume that
A7: n >= 1 and
A8: n in POWEROF2SET ; ::_thesis: ( 1 * ((seq_n^ 1) . n) <= f . n & f . n <= 1 * ((seq_n^ 1) . n) )
A9: (seq_n^ 1) . n = n to_power 1 by A7, Def3
.= n by POWER:25 ;
hence 1 * ((seq_n^ 1) . n) <= f . n by A5, A8; ::_thesis: f . n <= 1 * ((seq_n^ 1) . n)
thus f . n <= 1 * ((seq_n^ 1) . n) by A5, A8, A9; ::_thesis: verum
end;
f is Element of Funcs (NAT,REAL) by FUNCT_2:8;
hence f in Big_Theta ((seq_n^ 1),POWEROF2SET) by A6; ::_thesis: ( not f in Big_Theta (seq_n^ 1) & seq_n^ 1 is smooth & not f is eventually-nondecreasing )
A10: Big_Theta (seq_n^ 1) = { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * ((seq_n^ 1) . n) <= t . n & t . n <= c * ((seq_n^ 1) . n) ) ) ) } by ASYMPT_0:27;
hereby ::_thesis: ( seq_n^ 1 is smooth & not f is eventually-nondecreasing )
A11: lim (seq_logn /" (seq_n^ 1)) = 0 by Lm11;
seq_logn /" (seq_n^ 1) is convergent by Lm11;
then consider N0 being Element of NAT such that
A12: for m being Element of NAT st m >= N0 holds
abs (((seq_logn /" (seq_n^ 1)) . m) - 0) < 1 / 2 by A11, SEQ_2:def_7;
assume f in Big_Theta (seq_n^ 1) ; ::_thesis: contradiction
then consider t being Element of Funcs (NAT,REAL) such that
A13: t = f and
A14: ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * ((seq_n^ 1) . n) <= t . n & t . n <= c * ((seq_n^ 1) . n) ) ) ) by A10;
consider c, d being Real, N being Element of NAT such that
A15: c > 0 and
d > 0 and
A16: for n being Element of NAT st n >= N holds
( d * ((seq_n^ 1) . n) <= t . n & t . n <= c * ((seq_n^ 1) . n) ) by A14;
set N2 = max ((max (N0,N)),(max ([/c\],2)));
A17: max ((max (N0,N)),(max ([/c\],2))) >= max (N0,N) by XXREAL_0:25;
A18: max ((max (N0,N)),(max ([/c\],2))) is Integer
proof
percases ( max ((max (N0,N)),(max ([/c\],2))) = max (N0,N) or max ((max (N0,N)),(max ([/c\],2))) = max ([/c\],2) ) by XXREAL_0:16;
suppose max ((max (N0,N)),(max ([/c\],2))) = max (N0,N) ; ::_thesis: max ((max (N0,N)),(max ([/c\],2))) is Integer
hence max ((max (N0,N)),(max ([/c\],2))) is Integer ; ::_thesis: verum
end;
suppose max ((max (N0,N)),(max ([/c\],2))) = max ([/c\],2) ; ::_thesis: max ((max (N0,N)),(max ([/c\],2))) is Integer
hence max ((max (N0,N)),(max ([/c\],2))) is Integer by XXREAL_0:16; ::_thesis: verum
end;
end;
end;
max (N0,N) >= N0 by XXREAL_0:25;
then A19: max ((max (N0,N)),(max ([/c\],2))) >= N0 by A17, XXREAL_0:2;
A20: max ((max (N0,N)),(max ([/c\],2))) >= max ([/c\],2) by XXREAL_0:25;
max ([/c\],2) >= [/c\] by XXREAL_0:25;
then A21: max ((max (N0,N)),(max ([/c\],2))) >= [/c\] by A20, XXREAL_0:2;
A22: max ([/c\],2) >= 2 by XXREAL_0:25;
then A23: max ((max (N0,N)),(max ([/c\],2))) >= 2 by A20, XXREAL_0:2;
max (N0,N) >= N by XXREAL_0:25;
then A24: max ((max (N0,N)),(max ([/c\],2))) >= N by A17, XXREAL_0:2;
A25: [/c\] >= c by INT_1:def_7;
reconsider N2 = max ((max (N0,N)),(max ([/c\],2))) 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 (2 to_power N2) - 1 >= 0 by XREAL_1:19;
then reconsider N3 = (2 to_power N2) - 1 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: (seq_n^ 1) . 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: abs (((seq_logn /" (seq_n^ 1)) . N3) - 0) < 1 / 2 by A12;
(seq_logn /" (seq_n^ 1)) . N3 = (seq_logn . N3) / ((seq_n^ 1) . N3) by Lm4
.= (log (2,N3)) / ((seq_n^ 1) . 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; ::_thesis: verum
end;
now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_0_holds_
(seq_n^_1)_._n_<=_(seq_n^_1)_._(n_+_1)
let n be Element of NAT ; ::_thesis: ( n >= 0 implies (seq_n^ 1) . n <= (seq_n^ 1) . (n + 1) )
assume n >= 0 ; ::_thesis: (seq_n^ 1) . n <= (seq_n^ 1) . (n + 1)
A33: n + 0 <= n + 1 by XREAL_1:6;
A34: (seq_n^ 1) . n = n
proof
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: (seq_n^ 1) . n = n
hence (seq_n^ 1) . n = n by Def3; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: (seq_n^ 1) . n = n
hence (seq_n^ 1) . n = n to_power 1 by Def3
.= n by POWER:25 ;
::_thesis: verum
end;
end;
end;
(seq_n^ 1) . (n + 1) = (n + 1) to_power 1 by Def3
.= n + 1 by POWER:25 ;
hence (seq_n^ 1) . n <= (seq_n^ 1) . (n + 1) by A34, A33; ::_thesis: verum
end;
then A35: seq_n^ 1 is eventually-nondecreasing by ASYMPT_0:def_6;
(seq_n^ 1) taken_every 2 is Element of Funcs (NAT,REAL) by FUNCT_2:8;
then (seq_n^ 1) taken_every 2 in Big_Oh (seq_n^ 1) by A1;
then seq_n^ 1 is_smooth_wrt 2 by A35, ASYMPT_0:def_16;
hence seq_n^ 1 is smooth by ASYMPT_0:37; ::_thesis: not f is eventually-nondecreasing
A36: 3 = 4 - 1 ;
hereby ::_thesis: verum
assume f is eventually-nondecreasing ; ::_thesis: contradiction
then consider N being Element of NAT such that
A37: for n being Element of NAT st n >= N holds
f . n <= f . (n + 1) by ASYMPT_0:def_6;
set N1 = (2 to_power (N + 2)) - 1;
A38: 2 to_power 2 = 2 ^2 by POWER:46
.= 4 ;
A39: N + 2 >= 0 + 2 by XREAL_1:6;
then 2 to_power (N + 2) >= 2 to_power 2 by PRE_FF:8;
then (2 to_power (N + 2)) - 1 >= 3 by A36, A38, XREAL_1:9;
then reconsider N1 = (2 to_power (N + 2)) - 1 as Element of NAT by INT_1:3;
2 to_power (N + 2) > (N + 2) + 1 by A39, Lm1;
then A40: N1 > N + 2 by XREAL_1:20;
N + 2 >= N + 0 by XREAL_1:6;
then A41: N1 >= N by A40, XXREAL_0:2;
N1 + 1 in POWEROF2SET ;
then A42: f . (N1 + 1) = N1 + 1 by A5;
not N1 in POWEROF2SET by A39, Lm49;
then f . N1 = 2 to_power N1 by A5;
then f . N1 > f . (N1 + 1) by A42, A40, POWER:39;
hence contradiction by A37, A41; ::_thesis: verum
end;
end;
theorem :: ASYMPT_1:37
for f, g being Real_Sequence st ( for n being Element of NAT st n > 0 holds
f . n = n to_power (2 to_power [\(log (2,n))/]) ) & ( for n being Element of NAT st n > 0 holds
g . n = n to_power n ) holds
ex s being eventually-positive Real_Sequence st
( s = g & f in Big_Theta (s,POWEROF2SET) & not f in Big_Theta s & f is eventually-nondecreasing & s is eventually-nondecreasing & not s is_smooth_wrt 2 )
proof
set X = POWEROF2SET ;
let f, g be Real_Sequence; ::_thesis: ( ( for n being Element of NAT st n > 0 holds
f . n = n to_power (2 to_power [\(log (2,n))/]) ) & ( for n being Element of NAT st n > 0 holds
g . n = n to_power n ) implies ex s being eventually-positive Real_Sequence st
( s = g & f in Big_Theta (s,POWEROF2SET) & not f in Big_Theta s & f is eventually-nondecreasing & s is eventually-nondecreasing & not s is_smooth_wrt 2 ) )
assume that
A1: for n being Element of NAT st n > 0 holds
f . n = n to_power (2 to_power [\(log (2,n))/]) and
A2: for n being Element of NAT st n > 0 holds
g . n = n to_power n ; ::_thesis: ex s being eventually-positive Real_Sequence st
( s = g & f in Big_Theta (s,POWEROF2SET) & not f in Big_Theta s & f is eventually-nondecreasing & s is eventually-nondecreasing & not s is_smooth_wrt 2 )
A3: g is eventually-positive
proof
take 1 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 1 <= b1 or not g . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 1 <= n or not g . n <= 0 )
assume A4: n >= 1 ; ::_thesis: not g . n <= 0
then g . n = n to_power n by A2;
hence not g . n <= 0 by A4, POWER:34; ::_thesis: verum
end;
set h = g taken_every 2;
reconsider g = g as eventually-positive Real_Sequence by A3;
A5: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_1_&_n_in_POWEROF2SET_holds_
(_1_*_(g_._n)_<=_f_._n_&_f_._n_<=_1_*_(g_._n)_)
let n be Element of NAT ; ::_thesis: ( n >= 1 & n in POWEROF2SET implies ( 1 * (g . n) <= f . n & f . n <= 1 * (g . n) ) )
assume that
A6: n >= 1 and
A7: n in POWEROF2SET ; ::_thesis: ( 1 * (g . n) <= f . n & f . n <= 1 * (g . n) )
consider n1 being Element of NAT such that
A8: n = 2 to_power n1 by A7;
A9: f . n = n to_power (2 to_power [\(log (2,n))/]) by A1, A6;
log (2,n) = n1 * (log (2,2)) by A8, POWER:55
.= n1 * 1 by POWER:52 ;
then A10: f . n = n to_power n by A9, A8, INT_1:25;
hence 1 * (g . n) <= f . n by A2, A6; ::_thesis: f . n <= 1 * (g . n)
thus f . n <= 1 * (g . n) by A2, A6, A10; ::_thesis: verum
end;
A11: Big_Theta g = { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (g . n) <= t . n & t . n <= c * (g . n) ) ) ) } by ASYMPT_0:27;
A12: now__::_thesis:_not_f_in_Big_Theta_g
assume f in Big_Theta g ; ::_thesis: contradiction
then consider t being Element of Funcs (NAT,REAL) such that
A13: t = f and
A14: ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (g . n) <= t . n & t . n <= c * (g . n) ) ) ) by A11;
consider c, d being Real, N being Element of NAT such that
c > 0 and
A15: d > 0 and
A16: for n being Element of NAT st n >= N holds
( d * (g . n) <= t . n & t . n <= c * (g . n) ) by A14;
set N1 = max ([/(1 / d)\],(max (N,2)));
A17: max ([/(1 / d)\],(max (N,2))) >= [/(1 / d)\] by XXREAL_0:25;
A18: max ([/(1 / d)\],(max (N,2))) is Integer by XXREAL_0:16;
A19: max ([/(1 / d)\],(max (N,2))) >= max (N,2) by XXREAL_0:25;
max (N,2) >= N by XXREAL_0:25;
then A20: max ([/(1 / d)\],(max (N,2))) >= N by A19, XXREAL_0:2;
max (N,2) >= 2 by XXREAL_0:25;
then A21: max ([/(1 / d)\],(max (N,2))) >= 2 by A19, XXREAL_0:2;
reconsider N1 = max ([/(1 / d)\],(max (N,2))) as Element of NAT by A19, A18, INT_1:3;
reconsider N2 = 2 to_power N1 as Element of NAT ;
A22: N2 > N1 + 1 by A21, Lm1;
N1 > 1 by A21, 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 A23: (N1 + 1) * 1 > log (2,(N2 + 1)) by POWER:52;
A24: now__::_thesis:_not_[\(log_(2,(N2_+_1)))/]_>_N1
assume [\(log (2,(N2 + 1)))/] > N1 ; ::_thesis: contradiction
then A25: [\(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 A23, A25, XXREAL_0:2; ::_thesis: verum
end;
A26: g . (N2 + 1) = (N2 + 1) to_power (N2 + 1) by A2;
then A27: g . (N2 + 1) > 0 by POWER:34;
N1 + 1 > N1 + 0 by XREAL_1:8;
then A28: N2 > N1 by A22, XXREAL_0:2;
A29: N2 + 1 > N2 + 0 by XREAL_1:8;
then N2 + 1 > N1 by A28, XXREAL_0:2;
then N2 + 1 > N by A20, XXREAL_0:2;
then A30: d * (g . (N2 + 1)) <= t . (N2 + 1) by A16;
[/(1 / d)\] >= 1 / d by INT_1:def_7;
then N1 >= 1 / d by A17, XXREAL_0:2;
then N2 >= 1 / d by A28, XXREAL_0:2;
then A31: 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 A22, A29, POWER:57;
then [\(log (2,(N2 + 1)))/] >= [\N1/] by PRE_FF:9;
then A32: [\(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, A13;
then (g . (N2 + 1)) / (t . (N2 + 1)) = ((N2 + 1) to_power (N2 + 1)) / ((N2 + 1) to_power N2) by A26, A32, A24, 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 A15, A31, 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 A27, XREAL_1:68;
hence contradiction by A30, A27, XCMPLX_1:87; ::_thesis: verum
end;
A33: now__::_thesis:_not_g_is_smooth_wrt_2
assume g is_smooth_wrt 2 ; ::_thesis: contradiction
then g taken_every 2 in Big_Oh g by ASYMPT_0:def_16;
then consider t being Element of Funcs (NAT,REAL) such that
A34: t = g taken_every 2 and
A35: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (g . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
c > 0 and
A36: for n being Element of NAT st n >= N holds
( t . n <= c * (g . n) & t . n >= 0 ) by A35;
set N0 = max ([/c\],(max (N,2)));
A37: max ([/c\],(max (N,2))) >= [/c\] by XXREAL_0:25;
A38: max ([/c\],(max (N,2))) is Integer by XXREAL_0:16;
A39: max ([/c\],(max (N,2))) >= max (N,2) by XXREAL_0:25;
max (N,2) >= N by XXREAL_0:25;
then A40: max ([/c\],(max (N,2))) >= N by A39, XXREAL_0:2;
A41: max (N,2) >= 2 by XXREAL_0:25;
then A42: 2 * (max ([/c\],(max (N,2)))) > 1 * (max ([/c\],(max (N,2)))) by A39, XREAL_1:68;
A43: max ([/c\],(max (N,2))) >= 2 by A39, A41, XXREAL_0:2;
then A44: max ([/c\],(max (N,2))) > 1 by XXREAL_0:2;
reconsider N0 = max ([/c\],(max (N,2))) as Element of NAT by A39, A38, INT_1:3;
[/c\] >= c by INT_1:def_7;
then A45: N0 >= c by A37, XXREAL_0:2;
N0 >= 1 by A43, XXREAL_0:2;
then N0 + N0 >= N0 + 1 by XREAL_1:6;
then A46: N0 to_power (2 * N0) >= N0 to_power (N0 + 1) by A44, PRE_FF:8;
N0 to_power (N0 + 1) = (N0 to_power N0) * (N0 to_power 1) by A39, A41, POWER:27
.= (N0 to_power N0) * N0 by POWER:25 ;
then A47: N0 to_power (N0 + 1) >= c * (N0 to_power N0) by A45, XREAL_1:64;
(g taken_every 2) . N0 = g . (2 * N0) by ASYMPT_0:def_15
.= (2 * N0) to_power (2 * N0) by A2, A42 ;
then (g taken_every 2) . N0 > N0 to_power (2 * N0) by A39, A41, A42, POWER:37;
then (g taken_every 2) . N0 > N0 to_power (N0 + 1) by A46, XXREAL_0:2;
then (g taken_every 2) . N0 > c * (N0 to_power N0) by A47, XXREAL_0:2;
then (g taken_every 2) . N0 > c * (g . N0) by A2, A39, A41;
hence contradiction by A34, A36, A40; ::_thesis: verum
end;
A48: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_2_holds_
f_._n_<=_f_._(n_+_1)
let n be Element of NAT ; ::_thesis: ( n >= 2 implies f . n <= f . (n + 1) )
A49: f . (n + 1) = (n + 1) to_power (2 to_power [\(log (2,(n + 1)))/]) by A1;
assume A50: n >= 2 ; ::_thesis: f . n <= f . (n + 1)
then A51: f . n = n to_power (2 to_power [\(log (2,n))/]) by A1;
A52: n + 1 > n + 0 by XREAL_1:8;
then log (2,n) <= log (2,(n + 1)) by A50, POWER:57;
then [\(log (2,n))/] <= [\(log (2,(n + 1)))/] by PRE_FF:9;
then A53: 2 to_power [\(log (2,n))/] <= 2 to_power [\(log (2,(n + 1)))/] by PRE_FF:8;
n + 1 > 0 + 1 by A50, XREAL_1:8;
then A54: (n + 1) to_power (2 to_power [\(log (2,n))/]) <= (n + 1) to_power (2 to_power [\(log (2,(n + 1)))/]) by A53, PRE_FF:8;
log (2,n) >= log (2,2) by A50, 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 A50, A52, POWER:37;
hence f . n <= f . (n + 1) by A51, A49, A54, XXREAL_0:2; ::_thesis: verum
end;
A55: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_1_holds_
g_._n_<=_g_._(n_+_1)
let n be Element of NAT ; ::_thesis: ( n >= 1 implies g . n <= g . (n + 1) )
assume A56: n >= 1 ; ::_thesis: g . n <= g . (n + 1)
A57: n + 1 > n + 0 by XREAL_1:8;
then A58: n to_power n < (n + 1) to_power n by A56, POWER:37;
n + 1 >= 1 + 1 by A56, XREAL_1:6;
then n + 1 > 1 by XXREAL_0:2;
then A59: (n + 1) to_power n < (n + 1) to_power (n + 1) by A57, POWER:39;
A60: g . (n + 1) = (n + 1) to_power (n + 1) by A2;
g . n = n to_power n by A2, A56;
hence g . n <= g . (n + 1) by A60, A59, A58, XXREAL_0:2; ::_thesis: verum
end;
take g ; ::_thesis: ( g = g & f in Big_Theta (g,POWEROF2SET) & not f in Big_Theta g & f is eventually-nondecreasing & g is eventually-nondecreasing & not g is_smooth_wrt 2 )
f is Element of Funcs (NAT,REAL) by FUNCT_2:8;
hence ( g = g & f in Big_Theta (g,POWEROF2SET) & not f in Big_Theta g & f is eventually-nondecreasing & g is eventually-nondecreasing & not g is_smooth_wrt 2 ) by A5, A12, A48, A55, A33, ASYMPT_0:def_6; ::_thesis: verum
end;
theorem :: ASYMPT_1:38
for g being Real_Sequence st ( for n being Element of NAT holds
( ( n in POWEROF2SET implies g . n = n ) & ( not n in POWEROF2SET implies g . n = n to_power 2 ) ) ) holds
ex s being eventually-positive Real_Sequence st
( s = g & seq_n^ 1 in Big_Theta (s,POWEROF2SET) & not seq_n^ 1 in Big_Theta s & s taken_every 2 in Big_Oh s & seq_n^ 1 is eventually-nondecreasing & not s is eventually-nondecreasing )
proof
let g be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds
( ( n in POWEROF2SET implies g . n = n ) & ( not n in POWEROF2SET implies g . n = n to_power 2 ) ) ) implies ex s being eventually-positive Real_Sequence st
( s = g & seq_n^ 1 in Big_Theta (s,POWEROF2SET) & not seq_n^ 1 in Big_Theta s & s taken_every 2 in Big_Oh s & seq_n^ 1 is eventually-nondecreasing & not s is eventually-nondecreasing ) )
assume A1: for n being Element of NAT holds
( ( n in POWEROF2SET implies g . n = n ) & ( not n in POWEROF2SET implies g . n = n to_power 2 ) ) ; ::_thesis: ex s being eventually-positive Real_Sequence st
( s = g & seq_n^ 1 in Big_Theta (s,POWEROF2SET) & not seq_n^ 1 in Big_Theta s & s taken_every 2 in Big_Oh s & seq_n^ 1 is eventually-nondecreasing & not s is eventually-nondecreasing )
A2: g is eventually-positive
proof
take 1 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 1 <= b1 or not g . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 1 <= n or not g . n <= 0 )
assume A3: n >= 1 ; ::_thesis: not g . n <= 0
thus g . n > 0 ::_thesis: verum
proof
percases ( n in POWEROF2SET or not n in POWEROF2SET ) ;
suppose n in POWEROF2SET ; ::_thesis: g . n > 0
hence g . n > 0 by A1, A3; ::_thesis: verum
end;
suppose not n in POWEROF2SET ; ::_thesis: g . n > 0
then g . n = n to_power 2 by A1;
hence g . n > 0 by A3, POWER:34; ::_thesis: verum
end;
end;
end;
end;
set h = g taken_every 2;
reconsider g = g as eventually-positive Real_Sequence by A2;
take g ; ::_thesis: ( g = g & seq_n^ 1 in Big_Theta (g,POWEROF2SET) & not seq_n^ 1 in Big_Theta g & g taken_every 2 in Big_Oh g & seq_n^ 1 is eventually-nondecreasing & not g is eventually-nondecreasing )
set X = POWEROF2SET ;
set f = seq_n^ 1;
A4: g taken_every 2 is Element of Funcs (NAT,REAL) by FUNCT_2:8;
A5: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_0_holds_
(_(g_taken_every_2)_._n_<=_4_*_(g_._n)_&_(g_taken_every_2)_._n_>=_0_)
let n be Element of NAT ; ::_thesis: ( n >= 0 implies ( (g taken_every 2) . n <= 4 * (g . n) & (g taken_every 2) . n >= 0 ) )
assume n >= 0 ; ::_thesis: ( (g taken_every 2) . n <= 4 * (g . n) & (g taken_every 2) . n >= 0 )
A6: (g taken_every 2) . n = g . (2 * n) by ASYMPT_0:def_15;
thus (g taken_every 2) . n <= 4 * (g . n) ::_thesis: (g taken_every 2) . n >= 0
proof
percases ( n in POWEROF2SET or not n in POWEROF2SET ) ;
supposeA7: n in POWEROF2SET ; ::_thesis: (g taken_every 2) . n <= 4 * (g . n)
then consider m being Element of NAT such that
A8: n = 2 to_power m ;
2 * n = (2 to_power 1) * (2 to_power m) by A8, POWER:25
.= 2 to_power (m + 1) by POWER:27 ;
then 2 * n in POWEROF2SET ;
then A9: g . (2 * n) = 2 * n by A1;
g . n = n by A1, A7;
hence (g taken_every 2) . n <= 4 * (g . n) by A6, A9, XREAL_1:64; ::_thesis: verum
end;
supposeA10: not n in POWEROF2SET ; ::_thesis: (g taken_every 2) . n <= 4 * (g . n)
now__::_thesis:_not_2_*_n_in_POWEROF2SET
assume 2 * n in POWEROF2SET ; ::_thesis: contradiction
then consider m being Element of NAT such that
A11: 2 * n = 2 to_power m ;
thus contradiction ::_thesis: verum
proof
percases ( m = 0 or m > 0 ) ;
supposeA12: m = 0 ; ::_thesis: contradiction
A13: now__::_thesis:_1_/_2_is_not_Element_of_NAT
assume 1 / 2 is Element of NAT ; ::_thesis: contradiction
then 0 + 1 <= 1 / 2 by NAT_1:13;
hence contradiction ; ::_thesis: verum
end;
(n * 2) * (2 ") = 1 * (2 ") by A11, A12, POWER:24;
hence contradiction by A13; ::_thesis: verum
end;
suppose m > 0 ; ::_thesis: contradiction
then m >= 0 + 1 by NAT_1:13;
then m - 1 >= 0 by XREAL_1:19;
then A14: m - 1 is Element of NAT by INT_1:3;
2 * n = 2 to_power ((m + (- 1)) + 1) by A11
.= (2 to_power (m - 1)) * (2 to_power 1) by POWER:27
.= (2 to_power (m - 1)) * 2 by POWER:25 ;
hence contradiction by A10, A14; ::_thesis: verum
end;
end;
end;
end;
then A15: 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, A10
.= n ^2 by POWER:46 ;
hence (g taken_every 2) . n <= 4 * (g . n) by A15, ASYMPT_0:def_15; ::_thesis: verum
end;
end;
end;
thus (g taken_every 2) . n >= 0 ::_thesis: verum
proof
percases ( 2 * n in POWEROF2SET or not 2 * n in POWEROF2SET ) ;
suppose 2 * n in POWEROF2SET ; ::_thesis: (g taken_every 2) . n >= 0
hence (g taken_every 2) . n >= 0 by A1, A6; ::_thesis: verum
end;
suppose not 2 * n in POWEROF2SET ; ::_thesis: (g taken_every 2) . n >= 0
then g . (2 * n) = (2 * n) to_power 2 by A1;
hence (g taken_every 2) . n >= 0 by ASYMPT_0:def_15; ::_thesis: verum
end;
end;
end;
end;
A16: Big_Theta g = { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (g . n) <= t . n & t . n <= c * (g . n) ) ) ) } by ASYMPT_0:27;
A17: now__::_thesis:_not_seq_n^_1_in_Big_Theta_g
assume seq_n^ 1 in Big_Theta g ; ::_thesis: contradiction
then consider t being Element of Funcs (NAT,REAL) such that
A18: t = seq_n^ 1 and
A19: ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (g . n) <= t . n & t . n <= c * (g . n) ) ) ) by A16;
consider c, d being Real, N being Element of NAT such that
c > 0 and
A20: d > 0 and
A21: for n being Element of NAT st n >= N holds
( d * (g . n) <= t . n & t . n <= c * (g . n) ) by A19;
set N0 = max ((max (N,2)),[/(1 / d)\]);
A22: max ((max (N,2)),[/(1 / d)\]) >= max (N,2) by XXREAL_0:25;
max (N,2) >= N by XXREAL_0:25;
then A23: max ((max (N,2)),[/(1 / d)\]) >= N by A22, XXREAL_0:2;
A24: max ((max (N,2)),[/(1 / d)\]) >= [/(1 / d)\] by XXREAL_0:25;
A25: max (N,2) >= 2 by XXREAL_0:25;
then A26: max ((max (N,2)),[/(1 / d)\]) >= 2 by A22, XXREAL_0:2;
max ((max (N,2)),[/(1 / d)\]) is Integer by XXREAL_0:16;
then reconsider N0 = max ((max (N,2)),[/(1 / d)\]) as Element of NAT by A22, 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 (2 to_power N0) - 1 >= 0 by XREAL_1:19;
then reconsider N1 = (2 to_power N0) - 1 as Element of NAT by INT_1:3;
A27: [/(1 / d)\] >= 1 / d by INT_1:def_7;
not N1 in POWEROF2SET by A22, A25, Lm49, XXREAL_0:2;
then A28: g . N1 = N1 to_power 2 by A1
.= N1 ^2 by POWER:46 ;
2 to_power N0 > N0 + 1 by A26, Lm1;
then A29: N1 > N0 by XREAL_1:20;
then A30: N1 >= N by A23, XXREAL_0:2;
N1 > [/(1 / d)\] by A24, A29, XXREAL_0:2;
then N1 > 1 / d by A27, XXREAL_0:2;
then N1 * N1 > N1 * (1 / d) by A29, XREAL_1:68;
then d * (N1 ^2) > (N1 * (1 / d)) * d by A20, XREAL_1:68;
then d * (N1 ^2) > N1 * ((1 / d) * d) ;
then A31: d * (N1 ^2) > N1 * 1 by A20, XCMPLX_1:87;
t . N1 = N1 to_power 1 by A18, A29, Def3
.= N1 by POWER:25 ;
hence contradiction by A21, A30, A31, A28; ::_thesis: verum
end;
A32: 3 = 4 - 1 ;
A33: now__::_thesis:_not_g_is_eventually-nondecreasing
assume g is eventually-nondecreasing ; ::_thesis: contradiction
then consider N being Element of NAT such that
A34: for n being Element of NAT st n >= N holds
g . n <= g . (n + 1) by ASYMPT_0:def_6;
set N0 = max (N,1);
set N1 = (2 to_power (2 * (max (N,1)))) - 1;
A35: max (N,1) >= N by XXREAL_0:25;
2 to_power (2 * (max (N,1))) >= 2 to_power 0 by PRE_FF:8;
then 2 to_power (2 * (max (N,1))) >= 1 by POWER:24;
then (2 to_power (2 * (max (N,1)))) - 1 >= 1 - 1 by XREAL_1:9;
then reconsider N1 = (2 to_power (2 * (max (N,1)))) - 1 as Element of NAT by INT_1:3;
A36: 2 * (max (N,1)) >= 2 * 1 by XREAL_1:64, XXREAL_0:25;
then 2 to_power (2 * (max (N,1))) > (2 * (max (N,1))) + 1 by Lm1;
then A37: N1 > 2 * (max (N,1)) by XREAL_1:20;
2 to_power (2 * (max (N,1))) >= 2 to_power 2 by A36, PRE_FF:8;
then 2 to_power (2 * (max (N,1))) >= 2 ^2 by POWER:46;
then N1 >= 3 by A32, XREAL_1:9;
then N1 >= 2 by XXREAL_0:2;
then A38: N1 ^2 > N1 + 1 by Lm47;
2 * (max (N,1)) >= 2 * 1 by XREAL_1:64, XXREAL_0:25;
then not N1 in POWEROF2SET by Lm49;
then A39: g . N1 = N1 to_power 2 by A1;
2 * (max (N,1)) >= 1 * (max (N,1)) by XREAL_1:64;
then N1 >= max (N,1) by A37, XXREAL_0:2;
then A40: N1 >= N by A35, XXREAL_0:2;
N1 + 1 in POWEROF2SET ;
then g . (N1 + 1) = N1 + 1 by A1;
then g . N1 > g . (N1 + 1) by A39, A38, POWER:46;
hence contradiction by A34, A40; ::_thesis: verum
end;
A41: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_0_holds_
(seq_n^_1)_._n_<=_(seq_n^_1)_._(n_+_1)
let n be Element of NAT ; ::_thesis: ( n >= 0 implies (seq_n^ 1) . n <= (seq_n^ 1) . (n + 1) )
assume n >= 0 ; ::_thesis: (seq_n^ 1) . n <= (seq_n^ 1) . (n + 1)
A42: n + 0 <= n + 1 by XREAL_1:6;
A43: (seq_n^ 1) . n = n
proof
percases ( n = 0 or n > 0 ) ;
suppose n = 0 ; ::_thesis: (seq_n^ 1) . n = n
hence (seq_n^ 1) . n = n by Def3; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: (seq_n^ 1) . n = n
hence (seq_n^ 1) . n = n to_power 1 by Def3
.= n by POWER:25 ;
::_thesis: verum
end;
end;
end;
(seq_n^ 1) . (n + 1) = (n + 1) to_power 1 by Def3
.= n + 1 by POWER:25 ;
hence (seq_n^ 1) . n <= (seq_n^ 1) . (n + 1) by A43, A42; ::_thesis: verum
end;
A44: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_1_&_n_in_POWEROF2SET_holds_
(_1_*_(g_._n)_<=_(seq_n^_1)_._n_&_(seq_n^_1)_._n_<=_1_*_(g_._n)_)
let n be Element of NAT ; ::_thesis: ( n >= 1 & n in POWEROF2SET implies ( 1 * (g . n) <= (seq_n^ 1) . n & (seq_n^ 1) . n <= 1 * (g . n) ) )
assume that
A45: n >= 1 and
A46: n in POWEROF2SET ; ::_thesis: ( 1 * (g . n) <= (seq_n^ 1) . n & (seq_n^ 1) . n <= 1 * (g . n) )
A47: (seq_n^ 1) . n = n to_power 1 by A45, Def3
.= n by POWER:25 ;
hence 1 * (g . n) <= (seq_n^ 1) . n by A1, A46; ::_thesis: (seq_n^ 1) . n <= 1 * (g . n)
thus (seq_n^ 1) . n <= 1 * (g . n) by A1, A46, A47; ::_thesis: verum
end;
seq_n^ 1 is Element of Funcs (NAT,REAL) by FUNCT_2:8;
hence ( g = g & seq_n^ 1 in Big_Theta (g,POWEROF2SET) & not seq_n^ 1 in Big_Theta g & g taken_every 2 in Big_Oh g & seq_n^ 1 is eventually-nondecreasing & not g is eventually-nondecreasing ) by A44, A17, A4, A5, A41, A33, ASYMPT_0:def_6; ::_thesis: verum
end;
begin
Lm50: for n being Element of NAT st n >= 2 holds
n ! > 1
proof
defpred S1[ Nat] means $1 ! > 1;
A1: for k being Nat st k >= 2 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 2 & S1[k] implies S1[k + 1] )
assume that
A2: k >= 2 and
A3: k ! > 1 ; ::_thesis: S1[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 S1[k + 1] by NEWTON:15; ::_thesis: verum
end;
A5: S1[2] by NEWTON:14;
for n being Nat st n >= 2 holds
S1[n] from NAT_1:sch_8(A5, A1);
hence for n being Element of NAT st n >= 2 holds
n ! > 1 ; ::_thesis: verum
end;
Lm51: for n1, n being Element of NAT st n <= n1 holds
n ! <= n1 !
proof
defpred S1[ Element of NAT ] means for n being Element of NAT st n <= $1 holds
n ! <= $1 ! ;
A1: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A2: for n being Element of NAT st n <= k holds
n ! <= k ! ; ::_thesis: S1[k + 1]
let n be Element of NAT ; ::_thesis: ( n <= k + 1 implies n ! <= (k + 1) ! )
assume A3: n <= k + 1 ; ::_thesis: n ! <= (k + 1) !
percases ( n <= k or n = k + 1 ) by A3, NAT_1:8;
supposeA4: n <= k ; ::_thesis: n ! <= (k + 1) !
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 n ! <= (k + 1) ! by A5, XXREAL_0:2; ::_thesis: verum
end;
suppose n = k + 1 ; ::_thesis: n ! <= (k + 1) !
hence n ! <= (k + 1) ! ; ::_thesis: verum
end;
end;
end;
A6: S1[ 0 ] ;
for n1 being Element of NAT holds S1[n1] from NAT_1:sch_1(A6, A1);
hence for n1, n being Element of NAT st n <= n1 holds
n ! <= n1 ! ; ::_thesis: verum
end;
Lm52: for k being Element of NAT st k >= 1 holds
ex n being Element of NAT st
( n ! <= k & k < (n + 1) ! & ( for m being Element of NAT st m ! <= k & k < (m + 1) ! holds
m = n ) )
proof
defpred S1[ Nat] means ex n being Element of NAT st
( n ! <= $1 & $1 < (n + 1) ! & ( for m being Element of NAT st m ! <= $1 & $1 < (m + 1) ! holds
m = n ) );
A1: for k being Nat st k >= 1 & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( k >= 1 & S1[k] implies S1[k + 1] )
assume that
k >= 1 and
A2: ex n being Element of NAT st
( n ! <= k & k < (n + 1) ! & ( for m being Element of NAT st m ! <= k & k < (m + 1) ! holds
m = n ) ) ; ::_thesis: S1[k + 1]
consider n being Element of NAT such that
A3: n ! <= k and
A4: k < (n + 1) ! and
for m being Element of NAT st m ! <= k & k < (m + 1) ! holds
m = n by A2;
A5: k + 1 <= (n + 1) ! by A4, INT_1:7;
percases ( k + 1 < (n + 1) ! or k + 1 = (n + 1) ! ) by A5, XXREAL_0:1;
supposeA6: k + 1 < (n + 1) ! ; ::_thesis: S1[k + 1]
take n ; ::_thesis: ( n ! <= k + 1 & k + 1 < (n + 1) ! & ( for m being Element of NAT st m ! <= k + 1 & k + 1 < (m + 1) ! holds
m = n ) )
k + 0 <= k + 1 by XREAL_1:6;
hence n ! <= k + 1 by A3, XXREAL_0:2; ::_thesis: ( k + 1 < (n + 1) ! & ( for m being Element of NAT st m ! <= k + 1 & k + 1 < (m + 1) ! holds
m = n ) )
thus k + 1 < (n + 1) ! by A6; ::_thesis: for m being Element of NAT st m ! <= k + 1 & k + 1 < (m + 1) ! holds
m = n
let m be Element of NAT ; ::_thesis: ( m ! <= k + 1 & k + 1 < (m + 1) ! implies m = n )
assume that
A7: m ! <= k + 1 and
A8: k + 1 < (m + 1) ! ; ::_thesis: m = n
now__::_thesis:_not_m_<>_n
assume A9: m <> n ; ::_thesis: contradiction
thus contradiction ::_thesis: verum
proof
percases ( m > n or m < n ) by A9, XXREAL_0:1;
suppose m > n ; ::_thesis: contradiction
then m >= n + 1 by NAT_1:13;
then m ! >= (n + 1) ! by Lm51;
hence contradiction by A6, A7, XXREAL_0:2; ::_thesis: verum
end;
suppose m < n ; ::_thesis: contradiction
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 contradiction by A8, A10, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
end;
hence m = n ; ::_thesis: verum
end;
supposeA11: k + 1 = (n + 1) ! ; ::_thesis: S1[k + 1]
take N = n + 1; ::_thesis: ( N ! <= k + 1 & k + 1 < (N + 1) ! & ( for m being Element of NAT st m ! <= k + 1 & k + 1 < (m + 1) ! holds
m = N ) )
thus N ! <= k + 1 by A11; ::_thesis: ( k + 1 < (N + 1) ! & ( for m being Element of NAT st m ! <= k + 1 & k + 1 < (m + 1) ! holds
m = N ) )
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; ::_thesis: for m being Element of NAT st m ! <= k + 1 & k + 1 < (m + 1) ! holds
m = N
let m be Element of NAT ; ::_thesis: ( m ! <= k + 1 & k + 1 < (m + 1) ! implies m = N )
assume that
A13: m ! <= k + 1 and
A14: k + 1 < (m + 1) ! ; ::_thesis: m = N
now__::_thesis:_not_m_<>_N
assume A15: m <> N ; ::_thesis: contradiction
thus contradiction ::_thesis: verum
proof
percases ( m > N or m < N ) by A15, XXREAL_0:1;
suppose m > N ; ::_thesis: contradiction
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 contradiction by A11, A16, NEWTON:15; ::_thesis: verum
end;
suppose m < N ; ::_thesis: contradiction
then m + 1 <= N by NAT_1:13;
hence contradiction by A11, A14, Lm51; ::_thesis: verum
end;
end;
end;
end;
hence m = N ; ::_thesis: verum
end;
end;
end;
A18: S1[1]
proof
take 1 ; ::_thesis: ( 1 ! <= 1 & 1 < (1 + 1) ! & ( for m being Element of NAT st m ! <= 1 & 1 < (m + 1) ! holds
m = 1 ) )
thus ( 1 ! <= 1 & 1 < (1 + 1) ! ) by NEWTON:13, NEWTON:14; ::_thesis: for m being Element of NAT st m ! <= 1 & 1 < (m + 1) ! holds
m = 1
let m be Element of NAT ; ::_thesis: ( m ! <= 1 & 1 < (m + 1) ! implies m = 1 )
assume that
A19: m ! <= 1 and
A20: 1 < (m + 1) ! ; ::_thesis: m = 1
A21: now__::_thesis:_not_m_>_1
assume m > 1 ; ::_thesis: contradiction
then m >= 1 + 1 by NAT_1:13;
hence contradiction by A19, Lm50; ::_thesis: verum
end;
m <> 0 by A20, NEWTON:13;
then m >= 0 + 1 by NAT_1:13;
hence m = 1 by A21, XXREAL_0:1; ::_thesis: verum
end;
for k being Nat st k >= 1 holds
S1[k] from NAT_1:sch_8(A18, A1);
hence for k being Element of NAT st k >= 1 holds
ex n being Element of NAT st
( n ! <= k & k < (n + 1) ! & ( for m being Element of NAT st m ! <= k & k < (m + 1) ! holds
m = n ) ) ; ::_thesis: verum
end;
definition
let x be Element of NAT ;
func Step1 x -> Element of NAT means :Def7: :: ASYMPT_1:def 7
ex n being Element of NAT st
( n ! <= x & x < (n + 1) ! & it = n ! ) if x <> 0
otherwise it = 0 ;
consistency
for b1 being Element of NAT holds verum ;
existence
( ( x <> 0 implies ex b1, n being Element of NAT st
( n ! <= x & x < (n + 1) ! & b1 = n ! ) ) & ( not x <> 0 implies ex b1 being Element of NAT st b1 = 0 ) )
proof
hereby ::_thesis: ( not x <> 0 implies ex b1 being Element of NAT st b1 = 0 )
assume x <> 0 ; ::_thesis: ex k1, m being Element of NAT st
( m ! <= x & x < (m + 1) ! & k1 = m ! )
then x >= 0 + 1 by NAT_1:13;
then consider k being Element of NAT such that
A1: k ! <= x and
A2: x < (k + 1) ! and
for m being Element of NAT st m ! <= x & x < (m + 1) ! holds
m = k by Lm52;
consider k1 being Real such that
A3: k1 = k ! ;
reconsider k1 = k1 as Element of NAT by A3;
take k1 = k1; ::_thesis: ex m being Element of NAT st
( m ! <= x & x < (m + 1) ! & k1 = m ! )
thus ex m being Element of NAT st
( m ! <= x & x < (m + 1) ! & k1 = m ! ) by A1, A2, A3; ::_thesis: verum
end;
thus ( not x <> 0 implies ex b1 being Element of NAT st b1 = 0 ) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of NAT holds
( ( x <> 0 & ex n being Element of NAT st
( n ! <= x & x < (n + 1) ! & b1 = n ! ) & ex n being Element of NAT st
( n ! <= x & x < (n + 1) ! & b2 = n ! ) implies b1 = b2 ) & ( not x <> 0 & b1 = 0 & b2 = 0 implies b1 = b2 ) )
proof
let n1, n2 be Element of NAT ; ::_thesis: ( ( x <> 0 & ex n being Element of NAT st
( n ! <= x & x < (n + 1) ! & n1 = n ! ) & ex n being Element of NAT st
( n ! <= x & x < (n + 1) ! & n2 = n ! ) implies n1 = n2 ) & ( not x <> 0 & n1 = 0 & n2 = 0 implies n1 = n2 ) )
now__::_thesis:_(_ex_n_being_Element_of_NAT_st_
(_n_!_<=_x_&_x_<_(n_+_1)_!_&_n1_=_n_!_)_&_ex_n_being_Element_of_NAT_st_
(_n_!_<=_x_&_x_<_(n_+_1)_!_&_n2_=_n_!_)_implies_n1_=_n2_)
assume that
A4: ex n being Element of NAT st
( n ! <= x & x < (n + 1) ! & n1 = n ! ) and
A5: ex n being Element of NAT st
( n ! <= x & x < (n + 1) ! & n2 = n ! ) ; ::_thesis: n1 = n2
consider n being Element of NAT such that
A6: n ! <= x and
A7: x < (n + 1) ! and
A8: n1 = n ! by A4;
consider m being Element of NAT such that
A9: m ! <= x and
A10: x < (m + 1) ! and
A11: n2 = m ! by A5;
now__::_thesis:_not_m_<>_n
assume A12: m <> n ; ::_thesis: contradiction
thus contradiction ::_thesis: verum
proof
percases ( m > n or m < n ) by A12, XXREAL_0:1;
suppose m > n ; ::_thesis: contradiction
then m >= n + 1 by INT_1:7;
then m ! >= (n + 1) ! by Lm51;
hence contradiction by A7, A9, XXREAL_0:2; ::_thesis: verum
end;
suppose m < n ; ::_thesis: contradiction
then m + 1 <= n by INT_1:7;
then (m + 1) ! <= n ! by Lm51;
hence contradiction by A6, A10, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
end;
hence n1 = n2 by A8, A11; ::_thesis: verum
end;
hence ( ( x <> 0 & ex n being Element of NAT st
( n ! <= x & x < (n + 1) ! & n1 = n ! ) & ex n being Element of NAT st
( n ! <= x & x < (n + 1) ! & n2 = n ! ) implies n1 = n2 ) & ( not x <> 0 & n1 = 0 & n2 = 0 implies n1 = n2 ) ) ; ::_thesis: verum
end;
end;
:: deftheorem Def7 defines Step1 ASYMPT_1:def_7_:_
for x, b2 being Element of NAT holds
( ( x <> 0 implies ( b2 = Step1 x iff ex n being Element of NAT st
( n ! <= x & x < (n + 1) ! & b2 = n ! ) ) ) & ( not x <> 0 implies ( b2 = Step1 x iff b2 = 0 ) ) );
Lm53: for n being Element of NAT st n >= 3 holds
n ! > n
proof
let n be Element of NAT ; ::_thesis: ( n >= 3 implies n ! > n )
assume A1: n >= 3 ; ::_thesis: n ! > n
set n1 = n - 1;
2 = 3 - 1 ;
then A2: n - 1 >= 2 by A1, XREAL_1:9;
then reconsider n1 = n - 1 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 n ! > n by A3, NEWTON:15; ::_thesis: verum
end;
theorem :: ASYMPT_1:39
for f being Real_Sequence st ( for n being Element of NAT holds f . n = Step1 n ) holds
ex s being eventually-positive Real_Sequence st
( s = f & f is eventually-nondecreasing & ( for n being Element of NAT holds f . n <= (seq_n^ 1) . n ) & not s is smooth )
proof
set g = seq_n^ 1;
let f be Real_Sequence; ::_thesis: ( ( for n being Element of NAT holds f . n = Step1 n ) implies ex s being eventually-positive Real_Sequence st
( s = f & f is eventually-nondecreasing & ( for n being Element of NAT holds f . n <= (seq_n^ 1) . n ) & not s is smooth ) )
assume A1: for n being Element of NAT holds f . n = Step1 n ; ::_thesis: ex s being eventually-positive Real_Sequence st
( s = f & f is eventually-nondecreasing & ( for n being Element of NAT holds f . n <= (seq_n^ 1) . n ) & not s is smooth )
f is eventually-positive
proof
take 1 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 1 <= b1 or not f . b1 <= 0 )
let n be Element of NAT ; ::_thesis: ( not 1 <= n or not f . n <= 0 )
assume n >= 1 ; ::_thesis: not f . n <= 0
then A2: ex m being Element of NAT st
( m ! <= n & n < (m + 1) ! & Step1 n = m ! ) by Def7;
f . n = Step1 n by A1;
hence not f . n <= 0 by A2, NEWTON:17; ::_thesis: verum
end;
then reconsider f = f as eventually-positive Real_Sequence ;
now__::_thesis:_for_k_being_Element_of_NAT_holds_f_._k_<=_f_._(k_+_1)
let k be Element of NAT ; ::_thesis: f . k <= f . (k + 1)
thus f . k <= f . (k + 1) ::_thesis: verum
proof
percases ( k = 0 or k > 0 ) ;
supposeA3: k = 0 ; ::_thesis: f . k <= f . (k + 1)
A4: f . (0 + 1) = Step1 1 by A1;
f . 0 = Step1 0 by A1
.= 0 by Def7 ;
hence f . k <= f . (k + 1) by A3, A4; ::_thesis: verum
end;
suppose k > 0 ; ::_thesis: f . k <= f . (k + 1)
then consider n1 being Element of NAT such that
A5: n1 ! <= k and
A6: k < (n1 + 1) ! and
A7: Step1 k = n1 ! by Def7;
A8: k + 1 <= (n1 + 1) ! by A6, INT_1:7;
A9: k <= k + 1 by NAT_1:11;
A10: f . k = n1 ! by A1, A7;
percases ( k + 1 < (n1 + 1) ! or k + 1 = (n1 + 1) ! ) by A8, XXREAL_0:1;
supposeA11: k + 1 < (n1 + 1) ! ; ::_thesis: f . k <= f . (k + 1)
n1 ! <= k + 1 by A9, A5, XXREAL_0:2;
then Step1 (k + 1) = n1 ! by A11, Def7;
hence f . k <= f . (k + 1) by A1, A10; ::_thesis: verum
end;
supposeA12: k + 1 = (n1 + 1) ! ; ::_thesis: f . k <= f . (k + 1)
A13: (n1 + 1) ! > 0 by NEWTON:17;
n1 + 2 > 0 + 1 by XREAL_1:8;
then 1 * ((n1 + 1) !) < (n1 + 2) * ((n1 + 1) !) by A13, XREAL_1:68;
then A14: k + 1 < ((n1 + 1) + 1) ! by A12, NEWTON:15;
f . (k + 1) = Step1 (k + 1) by A1
.= (n1 + 1) ! by A12, A14, Def7 ;
hence f . k <= f . (k + 1) by A10, Lm51, NAT_1:11; ::_thesis: verum
end;
end;
end;
end;
end;
end;
then A15: for k being Element of NAT st k >= 0 holds
f . k <= f . (k + 1) ;
A16: 1 = 2 - 1 ;
A17: now__::_thesis:_not_f_is_smooth
set h = f taken_every 2;
assume f is smooth ; ::_thesis: contradiction
then f is_smooth_wrt 2 by ASYMPT_0:def_17;
then f taken_every 2 in Big_Oh f by ASYMPT_0:def_16;
then consider t being Element of Funcs (NAT,REAL) such that
A18: t = f taken_every 2 and
A19: ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) ;
consider c being Real, N being Element of NAT such that
c > 0 and
A20: for n being Element of NAT st n >= N holds
( t . n <= c * (f . n) & t . n >= 0 ) by A19;
set n2 = max ((max (N,3)),([/c\] + 1));
A21: max ((max (N,3)),([/c\] + 1)) >= max (N,3) by XXREAL_0:25;
max (N,3) >= N by XXREAL_0:25;
then A22: max ((max (N,3)),([/c\] + 1)) >= N by A21, XXREAL_0:2;
A23: max ((max (N,3)),([/c\] + 1)) >= [/c\] + 1 by XXREAL_0:25;
A24: max ((max (N,3)),([/c\] + 1)) is Integer by XXREAL_0:16;
A25: max (N,3) >= 3 by XXREAL_0:25;
then A26: max ((max (N,3)),([/c\] + 1)) >= 3 by A21, XXREAL_0:2;
reconsider n2 = max ((max (N,3)),([/c\] + 1)) 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: (n2 !) - 1 >= 1 by A16, 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: n2 - 1 >= 1 by A16, A27, XREAL_1:9;
reconsider n1 = (n2 !) - 1 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 = n2 - 1 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, Def7
.= n2 * (n3 !) by NEWTON:15 ;
f . n1 = Step1 n1 by A1
.= n3 ! by A29, A36, A39, A40, Def7 ;
hence contradiction by A33, A32, A45, A42, A43, XREAL_1:68; ::_thesis: verum
end;
take f ; ::_thesis: ( f = f & f is eventually-nondecreasing & ( for n being Element of NAT holds f . n <= (seq_n^ 1) . n ) & not f is smooth )
now__::_thesis:_for_n_being_Element_of_NAT_holds_f_._n_<=_(seq_n^_1)_._n
let n be Element of NAT ; ::_thesis: f . n <= (seq_n^ 1) . n
thus f . n <= (seq_n^ 1) . n ::_thesis: verum
proof
percases ( n = 0 or n > 0 ) ;
supposeA46: n = 0 ; ::_thesis: f . n <= (seq_n^ 1) . n
f . 0 = Step1 0 by A1
.= 0 by Def7 ;
hence f . n <= (seq_n^ 1) . n by A46, Def3; ::_thesis: verum
end;
supposeA47: n > 0 ; ::_thesis: f . n <= (seq_n^ 1) . n
then A48: (seq_n^ 1) . n = n to_power 1 by Def3
.= n by POWER:25 ;
ex n1 being Element of NAT st
( n1 ! <= n & n < (n1 + 1) ! & Step1 n = n1 ! ) by A47, Def7;
hence f . n <= (seq_n^ 1) . n by A1, A48; ::_thesis: verum
end;
end;
end;
end;
hence ( f = f & f is eventually-nondecreasing & ( for n being Element of NAT holds f . n <= (seq_n^ 1) . n ) & not f is smooth ) by A15, A17, ASYMPT_0:def_6; ::_thesis: verum
end;
begin
Lm54: (seq_n^ 1) - (seq_const 1) is eventually-positive
proof
take 2 ; :: according to ASYMPT_0:def_4 ::_thesis: for b1 being Element of NAT holds
( not 2 <= b1 or not ((seq_n^ 1) - (seq_const 1)) . b1 <= 0 )
set g = seq_const 1;
set f = seq_n^ 1;
let n be Element of NAT ; ::_thesis: ( not 2 <= n or not ((seq_n^ 1) - (seq_const 1)) . n <= 0 )
A1: (seq_const 1) . n = 1 by FUNCOP_1:7;
assume A2: n >= 2 ; ::_thesis: not ((seq_n^ 1) - (seq_const 1)) . n <= 0
then A3: n > 1 + 0 by XXREAL_0:2;
A4: (seq_n^ 1) . n = n to_power 1 by A2, Def3
.= n by POWER:25 ;
((seq_n^ 1) - (seq_const 1)) . n = ((seq_n^ 1) . n) + ((- (seq_const 1)) . n) by SEQ_1:7
.= n + (- 1) by A4, A1, SEQ_1:10
.= n - 1 ;
hence not ((seq_n^ 1) - (seq_const 1)) . n <= 0 by A3, XREAL_1:20; ::_thesis: verum
end;
theorem :: ASYMPT_1:40
for F being eventually-nonnegative Real_Sequence st F = (seq_n^ 1) - (seq_const 1) holds
(Big_Theta F) + (Big_Theta (seq_n^ 1)) = Big_Theta (seq_n^ 1)
proof
set q = seq_const 1;
set p = seq_n^ 1;
set f = (seq_n^ 1) - (seq_const 1);
set g = seq_n^ 1;
A1: Big_Theta (seq_n^ 1) = { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * ((seq_n^ 1) . n) <= t . n & t . n <= c * ((seq_n^ 1) . n) ) ) ) } by ASYMPT_0:27;
let F be eventually-nonnegative Real_Sequence; ::_thesis: ( F = (seq_n^ 1) - (seq_const 1) implies (Big_Theta F) + (Big_Theta (seq_n^ 1)) = Big_Theta (seq_n^ 1) )
assume F = (seq_n^ 1) - (seq_const 1) ; ::_thesis: (Big_Theta F) + (Big_Theta (seq_n^ 1)) = Big_Theta (seq_n^ 1)
then A2: Big_Theta F = { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (((seq_n^ 1) - (seq_const 1)) . n) <= t . n & t . n <= c * (((seq_n^ 1) - (seq_const 1)) . n) ) ) ) } by ASYMPT_0:27;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_(Big_Theta_F)_+_(Big_Theta_(seq_n^_1))_implies_x_in_Big_Theta_(seq_n^_1)_)_&_(_x_in_Big_Theta_(seq_n^_1)_implies_x_in_(Big_Theta_F)_+_(Big_Theta_(seq_n^_1))_)_)
let x be set ; ::_thesis: ( ( x in (Big_Theta F) + (Big_Theta (seq_n^ 1)) implies x in Big_Theta (seq_n^ 1) ) & ( x in Big_Theta (seq_n^ 1) implies x in (Big_Theta F) + (Big_Theta (seq_n^ 1)) ) )
hereby ::_thesis: ( x in Big_Theta (seq_n^ 1) implies x in (Big_Theta F) + (Big_Theta (seq_n^ 1)) )
assume x in (Big_Theta F) + (Big_Theta (seq_n^ 1)) ; ::_thesis: x in Big_Theta (seq_n^ 1)
then consider t being Element of Funcs (NAT,REAL) such that
A3: t = x and
A4: ex f9, g9 being Element of Funcs (NAT,REAL) st
( f9 in Big_Theta F & g9 in Big_Theta (seq_n^ 1) & ( for n being Element of NAT holds t . n = (f9 . n) + (g9 . n) ) ) ;
consider f9, g9 being Element of Funcs (NAT,REAL) such that
A5: f9 in Big_Theta F and
A6: g9 in Big_Theta (seq_n^ 1) and
A7: for n being Element of NAT holds t . n = (f9 . n) + (g9 . n) by A4;
ex r being Element of Funcs (NAT,REAL) st
( r = f9 & ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * (((seq_n^ 1) - (seq_const 1)) . n) <= r . n & r . n <= c * (((seq_n^ 1) - (seq_const 1)) . n) ) ) ) ) by A2, A5;
then consider c1, d1 being Real, N1 being Element of NAT such that
A8: c1 > 0 and
A9: d1 > 0 and
A10: for n being Element of NAT st n >= N1 holds
( d1 * (((seq_n^ 1) - (seq_const 1)) . n) <= f9 . n & f9 . n <= c1 * (((seq_n^ 1) - (seq_const 1)) . n) ) ;
ex s being Element of Funcs (NAT,REAL) st
( s = g9 & ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * ((seq_n^ 1) . n) <= s . n & s . n <= c * ((seq_n^ 1) . n) ) ) ) ) by A1, A6;
then consider c2, d2 being Real, N2 being Element of NAT such that
A11: c2 > 0 and
A12: d2 > 0 and
A13: for n being Element of NAT st n >= N2 holds
( d2 * ((seq_n^ 1) . n) <= g9 . n & g9 . n <= c2 * ((seq_n^ 1) . n) ) ;
set d = d2;
set c = c1 + c2;
set N = max (1,(max (N1,N2)));
A14: max (1,(max (N1,N2))) >= 1 by XXREAL_0:25;
A15: max (1,(max (N1,N2))) >= max (N1,N2) by XXREAL_0:25;
max (N1,N2) >= N2 by XXREAL_0:25;
then A16: max (1,(max (N1,N2))) >= N2 by A15, XXREAL_0:2;
max (N1,N2) >= N1 by XXREAL_0:25;
then A17: max (1,(max (N1,N2))) >= N1 by A15, XXREAL_0:2;
now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_max_(1,(max_(N1,N2)))_holds_
(_d2_*_((seq_n^_1)_._n)_<=_t_._n_&_t_._n_<=_(c1_+_c2)_*_((seq_n^_1)_._n)_)
let n be Element of NAT ; ::_thesis: ( n >= max (1,(max (N1,N2))) implies ( d2 * ((seq_n^ 1) . n) <= t . n & t . n <= (c1 + c2) * ((seq_n^ 1) . n) ) )
A18: (seq_const 1) . n = 1 by FUNCOP_1:7;
assume A19: n >= max (1,(max (N1,N2))) ; ::_thesis: ( d2 * ((seq_n^ 1) . n) <= t . n & t . n <= (c1 + c2) * ((seq_n^ 1) . n) )
then A20: (seq_n^ 1) . 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: ((seq_n^ 1) - (seq_const 1)) . 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 * ((seq_n^ 1) . n) <= g9 . n by A13;
then A24: (d1 * (((seq_n^ 1) - (seq_const 1)) . n)) + (d2 * ((seq_n^ 1) . n)) <= (d1 * (((seq_n^ 1) - (seq_const 1)) . n)) + (g9 . n) by XREAL_1:6;
g9 . n <= c2 * ((seq_n^ 1) . n) by A13, A23;
then A25: (c1 * (((seq_n^ 1) - (seq_const 1)) . n)) + (g9 . n) <= (c1 * (((seq_n^ 1) - (seq_const 1)) . n)) + (c2 * ((seq_n^ 1) . n)) by XREAL_1:6;
A26: n >= N1 by A17, A19, XXREAL_0:2;
then f9 . n <= c1 * (((seq_n^ 1) - (seq_const 1)) . n) by A10;
then (f9 . n) + (g9 . n) <= (c1 * (((seq_n^ 1) - (seq_const 1)) . n)) + (g9 . n) by XREAL_1:6;
then A27: (f9 . n) + (g9 . n) <= (c1 * (((seq_n^ 1) - (seq_const 1)) . n)) + (c2 * ((seq_n^ 1) . n)) by A25, XXREAL_0:2;
d1 * (((seq_n^ 1) - (seq_const 1)) . n) <= f9 . n by A10, A26;
then (d1 * (((seq_n^ 1) - (seq_const 1)) . n)) + (g9 . n) <= (f9 . n) + (g9 . n) by XREAL_1:6;
then (d1 * (((seq_n^ 1) - (seq_const 1)) . n)) + (d2 * ((seq_n^ 1) . 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 d2 * ((seq_n^ 1) . n) <= t . n by A7, A20; ::_thesis: t . n <= (c1 + c2) * ((seq_n^ 1) . n)
(- 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 <= (c1 + c2) * ((seq_n^ 1) . n) by A7, A20; ::_thesis: verum
end;
hence x in Big_Theta (seq_n^ 1) by A1, A3, A8, A11, A12; ::_thesis: verum
end;
assume x in Big_Theta (seq_n^ 1) ; ::_thesis: x in (Big_Theta F) + (Big_Theta (seq_n^ 1))
then consider t being Element of Funcs (NAT,REAL) such that
A28: t = x and
A29: ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds
( d * ((seq_n^ 1) . n) <= t . n & t . n <= c * ((seq_n^ 1) . n) ) ) ) by A1;
consider c, d being Real, N being Element of NAT such that
A30: c > 0 and
A31: d > 0 and
A32: for n being Element of NAT st n >= N holds
( d * ((seq_n^ 1) . n) <= t . n & t . n <= c * ((seq_n^ 1) . n) ) by A29;
set f9 = (2 ") (#) t;
set g9 = (2 ") (#) t;
A33: (2 ") (#) t is Element of Funcs (NAT,REAL) by FUNCT_2:8;
A34: for n being Element of NAT holds t . n = (((2 ") (#) t) . n) + (((2 ") (#) t) . n)
proof
let n be Element of NAT ; ::_thesis: t . n = (((2 ") (#) t) . n) + (((2 ") (#) t) . n)
((2 ") (#) t) . n = (2 ") * (t . n) by SEQ_1:9;
hence t . n = (((2 ") (#) t) . n) + (((2 ") (#) t) . n) ; ::_thesis: verum
end;
A35: (2 ") * d > (2 ") * 0 by A31, XREAL_1:68;
set N0 = max (N,2);
A36: max (N,2) >= N by XXREAL_0:25;
A37: max (N,2) >= 2 by XXREAL_0:25;
reconsider N0 = max (N,2) as Element of NAT ;
A38: now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_N0_holds_
(_((2_")_*_d)_*_((seq_n^_1)_._n)_<=_((2_")_(#)_t)_._n_&_((2_")_(#)_t)_._n_<=_((2_")_*_c)_*_((seq_n^_1)_._n)_)
let n be Element of NAT ; ::_thesis: ( n >= N0 implies ( ((2 ") * d) * ((seq_n^ 1) . n) <= ((2 ") (#) t) . n & ((2 ") (#) t) . n <= ((2 ") * c) * ((seq_n^ 1) . n) ) )
assume n >= N0 ; ::_thesis: ( ((2 ") * d) * ((seq_n^ 1) . n) <= ((2 ") (#) t) . n & ((2 ") (#) t) . n <= ((2 ") * c) * ((seq_n^ 1) . n) )
then A39: n >= N by A36, XXREAL_0:2;
then A40: (2 ") * (t . n) <= (2 ") * (c * ((seq_n^ 1) . n)) by A32, XREAL_1:64;
(2 ") * (d * ((seq_n^ 1) . n)) <= (2 ") * (t . n) by A32, A39, XREAL_1:64;
hence ( ((2 ") * d) * ((seq_n^ 1) . n) <= ((2 ") (#) t) . n & ((2 ") (#) t) . n <= ((2 ") * c) * ((seq_n^ 1) . n) ) by A40, SEQ_1:9; ::_thesis: verum
end;
(2 ") * c > (2 ") * 0 by A30, XREAL_1:68;
then A41: (2 ") (#) t in Big_Theta (seq_n^ 1) by A1, A35, A33, A38;
now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_N0_holds_
(_((2_")_*_d)_*_(((seq_n^_1)_-_(seq_const_1))_._n)_<=_((2_")_(#)_t)_._n_&_((2_")_(#)_t)_._n_<=_c_*_(((seq_n^_1)_-_(seq_const_1))_._n)_)
let n be Element of NAT ; ::_thesis: ( n >= N0 implies ( ((2 ") * d) * (((seq_n^ 1) - (seq_const 1)) . n) <= ((2 ") (#) t) . n & ((2 ") (#) t) . n <= c * (((seq_n^ 1) - (seq_const 1)) . n) ) )
A42: (seq_const 1) . n = 1 by FUNCOP_1:7;
assume A43: n >= N0 ; ::_thesis: ( ((2 ") * d) * (((seq_n^ 1) - (seq_const 1)) . n) <= ((2 ") (#) t) . n & ((2 ") (#) t) . n <= c * (((seq_n^ 1) - (seq_const 1)) . n) )
then A44: (seq_n^ 1) . 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 * ((seq_n^ 1) . n)) <= (2 ") * (t . n) by A32, XREAL_1:64;
A48: ((seq_n^ 1) - (seq_const 1)) . n = ((seq_n^ 1) . n) + ((- (seq_const 1)) . n) by SEQ_1:7
.= ((seq_n^ 1) . n) + (- 1) by A42, SEQ_1:10
.= ((seq_n^ 1) . n) - 1
.= (n to_power 1) - 1 by A37, A43, Def3
.= n - 1 by POWER:25 ;
then ((seq_n^ 1) - (seq_const 1)) . n <= (seq_n^ 1) . n by A44, XREAL_1:13;
then ((2 ") * d) * (((seq_n^ 1) - (seq_const 1)) . n) <= ((2 ") * d) * ((seq_n^ 1) . n) by A31, XREAL_1:64;
then ((2 ") * d) * (((seq_n^ 1) - (seq_const 1)) . n) <= (2 ") * (t . n) by A47, XXREAL_0:2;
hence ((2 ") * d) * (((seq_n^ 1) - (seq_const 1)) . n) <= ((2 ") (#) t) . n by SEQ_1:9; ::_thesis: ((2 ") (#) t) . n <= c * (((seq_n^ 1) - (seq_const 1)) . n)
(2 ") * (t . n) <= (2 ") * (c * ((seq_n^ 1) . n)) by A32, A46, XREAL_1:64;
then (2 ") * (t . n) <= c * (((seq_n^ 1) - (seq_const 1)) . n) by A48, A44, A45, XXREAL_0:2;
hence ((2 ") (#) t) . n <= c * (((seq_n^ 1) - (seq_const 1)) . n) by SEQ_1:9; ::_thesis: verum
end;
then (2 ") (#) t in Big_Theta F by A2, A30, A35, A33;
hence x in (Big_Theta F) + (Big_Theta (seq_n^ 1)) by A28, A33, A41, A34; ::_thesis: verum
end;
hence (Big_Theta F) + (Big_Theta (seq_n^ 1)) = Big_Theta (seq_n^ 1) by TARSKI:1; ::_thesis: verum
end;
begin
theorem :: ASYMPT_1:41
ex F being FUNCTION_DOMAIN of NAT , REAL st
( F = {(seq_n^ 1)} & ( for n being Element of NAT 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 ; ::_thesis: ( F = {(seq_n^ 1)} & ( for n being Element of NAT holds (seq_n^ (- 1)) . n <= (seq_n^ 1) . n ) & not seq_n^ (- 1) in F to_power (Big_Oh (seq_const 1)) )
thus F = {(seq_n^ 1)} ; ::_thesis: ( ( for n being Element of NAT holds (seq_n^ (- 1)) . n <= (seq_n^ 1) . n ) & not seq_n^ (- 1) in F to_power (Big_Oh (seq_const 1)) )
A1: now__::_thesis:_for_n_being_Element_of_NAT_holds_(seq_n^_(-_1))_._n_<=_(seq_n^_1)_._n
let n be Element of NAT ; ::_thesis: (seq_n^ (- 1)) . b1 <= (seq_n^ 1) . b1
percases ( n = 0 or n > 0 ) ;
supposeA2: n = 0 ; ::_thesis: (seq_n^ (- 1)) . b1 <= (seq_n^ 1) . b1
then (seq_n^ (- 1)) . n = 0 by Def3;
hence (seq_n^ (- 1)) . n <= (seq_n^ 1) . n by A2, Def3; ::_thesis: verum
end;
supposeA3: n > 0 ; ::_thesis: (seq_n^ (- 1)) . b1 <= (seq_n^ 1) . b1
then A4: n >= 0 + 1 by INT_1:7;
A5: n to_power (- 1) <= n to_power 1
proof
percases ( n = 1 or n > 1 ) by A4, XXREAL_0:1;
supposeA6: n = 1 ; ::_thesis: n to_power (- 1) <= n to_power 1
then n to_power (- 1) = 1 by POWER:26;
hence n to_power (- 1) <= n to_power 1 by A6, POWER:26; ::_thesis: verum
end;
suppose n > 1 ; ::_thesis: n to_power (- 1) <= n to_power 1
hence n to_power (- 1) <= n to_power 1 by PRE_FF:8; ::_thesis: verum
end;
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; ::_thesis: verum
end;
end;
end;
now__::_thesis:_not_seq_n^_(-_1)_in_F_to_power_(Big_Oh_(seq_const_1))
assume A7: seq_n^ (- 1) in F to_power (Big_Oh (seq_const 1)) ; ::_thesis: contradiction
ex H being FUNCTION_DOMAIN of NAT , REAL st
( H = F & ( seq_n^ (- 1) in H to_power (Big_Oh (seq_const 1)) implies ex N being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= (seq_n^ (- 1)) . n & (seq_n^ (- 1)) . n <= c * ((seq_n^ k) . n) ) ) ) ) & ( ex N being Element of NAT ex c being Real ex k being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N holds
( 1 <= (seq_n^ (- 1)) . n & (seq_n^ (- 1)) . n <= c * ((seq_n^ k) . n) ) ) ) implies seq_n^ (- 1) in H to_power (Big_Oh (seq_const 1)) ) ) by Th9;
then consider N0 being Element of NAT , c being Real, k being Element of NAT such that
c > 0 and
A8: for n being Element of NAT st n >= N0 holds
( 1 <= (seq_n^ (- 1)) . n & (seq_n^ (- 1)) . n <= c * ((seq_n^ k) . n) ) by A7;
set N = max (N0,2);
A9: max (N0,2) >= 2 by XXREAL_0:25;
A10: max (N0,2) >= N0 by XXREAL_0:25;
now__::_thesis:_for_n_being_Element_of_NAT_holds_not_n_>=_max_(N0,2)
let n be Element of NAT ; ::_thesis: not n >= max (N0,2)
assume A11: n >= max (N0,2) ; ::_thesis: contradiction
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: (seq_n^ (- 1)) . n >= 1 by A8;
(seq_n^ (- 1)) . n = n to_power (- 1) by A9, A11, Def3;
hence contradiction by A13, A12, POWER:36; ::_thesis: verum
end;
hence contradiction ; ::_thesis: verum
end;
hence ( ( for n being Element of NAT holds (seq_n^ (- 1)) . n <= (seq_n^ 1) . n ) & not seq_n^ (- 1) in F to_power (Big_Oh (seq_const 1)) ) by A1; ::_thesis: verum
end;
begin
theorem :: ASYMPT_1:42
for c being non negative Real
for x, f being eventually-nonnegative Real_Sequence st ex e being Real ex N being Element of NAT st
( e > 0 & ( for n being Element of NAT st n >= N holds
f . n >= e ) ) & x in Big_Oh (c + f) holds
x in Big_Oh f
proof
let c be non negative Real; ::_thesis: for x, f being eventually-nonnegative Real_Sequence st ex e being Real ex N being Element of NAT st
( e > 0 & ( for n being Element of NAT st n >= N holds
f . n >= e ) ) & x in Big_Oh (c + f) holds
x in Big_Oh f
let x, f be eventually-nonnegative Real_Sequence; ::_thesis: ( ex e being Real ex N being Element of NAT st
( e > 0 & ( for n being Element of NAT st n >= N holds
f . n >= e ) ) & x in Big_Oh (c + f) implies x in Big_Oh f )
given e being Real, N0 being Element of NAT such that A1: e > 0 and
A2: for n being Element of NAT st n >= N0 holds
f . n >= e ; ::_thesis: ( not x in Big_Oh (c + f) or x in Big_Oh f )
assume x in Big_Oh (c + f) ; ::_thesis: x in Big_Oh f
then consider t being Element of Funcs (NAT,REAL) such that
A3: x = t and
A4: ex d being Real ex N being Element of NAT st
( d > 0 & ( for n being Element of NAT st n >= N holds
( t . n <= d * ((c + f) . n) & t . n >= 0 ) ) ) ;
consider d being Real, N1 being Element of NAT such that
A5: d > 0 and
A6: for n being Element of NAT 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: max ((2 * d),(((2 * d) * c) / e)) > 0 by XXREAL_0:25;
set N = max (N0,N1);
A8: max (N0,N1) >= N1 by XXREAL_0:25;
A9: max (N0,N1) >= N0 by XXREAL_0:25;
now__::_thesis:_for_n_being_Element_of_NAT_st_n_>=_max_(N0,N1)_holds_
(_t_._n_<=_(max_((2_*_d),(((2_*_d)_*_c)_/_e)))_*_(f_._n)_&_t_._n_>=_0_)
let n be Element of NAT ; ::_thesis: ( n >= max (N0,N1) implies ( t . n <= (max ((2 * d),(((2 * d) * c) / e))) * (f . n) & t . n >= 0 ) )
assume A10: n >= max (N0,N1) ; ::_thesis: ( t . n <= (max ((2 * d),(((2 * d) * c) / e))) * (f . n) & t . n >= 0 )
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 <= (max ((2 * d),(((2 * d) * c) / e))) * (f . n) ::_thesis: t . n >= 0
proof
percases ( c >= f . n or c < f . n ) ;
suppose c >= f . n ; ::_thesis: t . n <= (max ((2 * d),(((2 * d) * c) / e))) * (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;
(max ((2 * d),(((2 * d) * c) / e))) * e >= (((2 * d) * c) / e) * e by A1, XREAL_1:64, XXREAL_0:25;
then A15: t . n <= (max ((2 * d),(((2 * d) * c) / e))) * e by A14, XXREAL_0:2;
(max ((2 * d),(((2 * d) * c) / e))) * (f . n) >= (max ((2 * d),(((2 * d) * c) / e))) * e by A2, A7, A13, XREAL_1:64;
hence t . n <= (max ((2 * d),(((2 * d) * c) / e))) * (f . n) by A15, XXREAL_0:2; ::_thesis: verum
end;
suppose c < f . n ; ::_thesis: t . n <= (max ((2 * d),(((2 * d) * c) / e))) * (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 (max ((2 * d),(((2 * d) * c) / e))) * (f . n) >= (2 * d) * (f . n) by XREAL_1:64, XXREAL_0:25;
hence t . n <= (max ((2 * d),(((2 * d) * c) / e))) * (f . n) by A16, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
thus t . n >= 0 by A6, A11; ::_thesis: verum
end;
hence x in Big_Oh f by A3, A7; ::_thesis: verum
end;
begin
theorem :: ASYMPT_1:43
2 to_power 12 = 4096 by Lm26;
theorem :: ASYMPT_1:44
for n being Element of NAT st n >= 3 holds
n ^2 > (2 * n) + 1 by Lm27;
theorem :: ASYMPT_1:45
for n being Element of NAT st n >= 10 holds
2 to_power (n - 1) > (2 * n) ^2 by Lm28;
theorem :: ASYMPT_1:46
for n being Element of NAT st n >= 9 holds
(n + 1) to_power 6 < 2 * (n to_power 6) by Lm29;
theorem :: ASYMPT_1:47
for n being Element of NAT st n >= 30 holds
2 to_power n > n to_power 6 by Lm30;
theorem :: ASYMPT_1:48
for x being Real st x > 9 holds
2 to_power x > (2 * x) ^2 by Lm31;
theorem :: ASYMPT_1:49
ex N being Element of NAT st
for n being Element of NAT st n >= N holds
(sqrt n) - (log (2,n)) > 1 by Lm32;
theorem :: ASYMPT_1:50
for a, b, c being Real st a > 0 & c > 0 & c <> 1 holds
a to_power b = c to_power (b * (log (c,a))) by Lm3;
theorem :: 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 being Element of NAT holds ((n ^2) - n) + 1 > 0 by Lm21;
theorem :: ASYMPT_1:55
for n being Element of NAT st n >= 2 holds
n ! > 1 by Lm50;
theorem :: ASYMPT_1:56
for n1, n being Element of NAT st n <= n1 holds
n ! <= n1 ! by Lm51;
theorem :: ASYMPT_1:57
for k being Element of NAT st k >= 1 holds
ex n being Element of NAT st
( n ! <= k & k < (n + 1) ! & ( for m being Element of NAT st m ! <= k & k < (m + 1) ! holds
m = n ) ) by Lm52;
theorem :: ASYMPT_1:58
for n being Element of NAT st n >= 2 holds
[/(n / 2)\] < n by Lm46;
theorem :: ASYMPT_1:59
for n being Element of 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 being Element of NAT st n >= 2 holds
2 to_power n > n + 1 by Lm1;
theorem :: ASYMPT_1:62
for a being logbase Real
for f being Real_Sequence st a > 1 & f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = log (a,n) ) holds
f is eventually-positive by Lm2;
theorem :: ASYMPT_1:63
for f, g being eventually-nonnegative Real_Sequence holds
( ( f in Big_Oh g & g in Big_Oh f ) iff Big_Oh f = Big_Oh g ) by Lm5;
theorem :: ASYMPT_1:64
for a, b, c being Real st 0 < a & a <= b & c >= 0 holds
a to_power c <= b to_power c by Lm6;
theorem :: ASYMPT_1:65
for n being Element of NAT st n >= 4 holds
(2 * n) + 3 < 2 to_power n by Lm7;
theorem :: ASYMPT_1:66
for n being Element of NAT st n >= 6 holds
(n + 1) ^2 < 2 to_power n by Lm8;
theorem :: ASYMPT_1:67
for c being Real st c > 6 holds
c ^2 < 2 to_power c by Lm9;
theorem :: ASYMPT_1:68
for e being positive Real
for f being Real_Sequence st f . 0 = 0 & ( for n being Element of NAT 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 being 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 being Real_Sequence
for N being Element of NAT st ( for n being Element of NAT st n <= N holds
f . n >= 0 ) holds
Sum (f,N) >= 0 by Lm12;
theorem :: ASYMPT_1:71
for f, g being Real_Sequence
for N being Element of NAT st ( for n being Element of NAT st n <= N holds
f . n <= g . n ) holds
Sum (f,N) <= Sum (g,N) by Lm13;
theorem :: ASYMPT_1:72
for f being Real_Sequence
for b being Real st f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = b ) holds
for N being Element of NAT holds Sum (f,N) = b * N by Lm14;
theorem :: ASYMPT_1:73
for f being Real_Sequence
for N, M being 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 being Real_Sequence
for M, N being Element of NAT st N >= M + 1 & ( for n being Element of NAT st M + 1 <= n & n <= N holds
f . n <= g . n ) holds
Sum (f,N,M) <= Sum (g,N,M) by Lm16;
theorem :: ASYMPT_1:75
for n being Element of NAT holds [/(n / 2)\] <= n by Lm17;
theorem :: ASYMPT_1:76
for f being Real_Sequence
for b being Real
for N being Element of NAT st f . 0 = 0 & ( for n being Element of NAT st n > 0 holds
f . n = b ) holds
for M being Element of NAT holds Sum (f,N,M) = b * (N - M) by Lm18;
theorem :: ASYMPT_1:77
for f, g being Real_Sequence
for N being Element of NAT
for c being Real st f is convergent & lim f = c & ( for n being Element of NAT st n >= N holds
f . n = g . n ) holds
( g is convergent & lim g = c ) by Lm22;
theorem :: ASYMPT_1:78
for n being Element of NAT st n >= 1 holds
((n ^2) - n) + 1 <= n ^2 by Lm23;
theorem :: ASYMPT_1:79
for n being Element of NAT st n >= 1 holds
n ^2 <= 2 * (((n ^2) - n) + 1) by Lm24;
theorem :: ASYMPT_1:80
for e being Real st 0 < e & e < 1 holds
ex N being Element of NAT st
for n being Element of NAT 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 being Element of NAT st n >= 10 holds
(2 to_power (2 * n)) / (n !) < 1 / (2 to_power (n - 9)) by Lm34;
theorem :: ASYMPT_1:82
for n being Element of NAT st n >= 3 holds
2 * (n - 2) >= n - 1 by Lm35;
theorem :: ASYMPT_1:83
for c being real number st c >= 0 holds
c to_power (1 / 2) = sqrt c by Lm39;
theorem :: ASYMPT_1:84
ex N being Element of NAT st
for n being Element of NAT st n >= N holds
n - ((sqrt n) * (log (2,n))) > n / 2 by Lm40;
theorem :: ASYMPT_1:85
for s being Real_Sequence st ( for n being Element of NAT holds s . n = (1 + (1 / (n + 1))) to_power (n + 1) ) holds
s is V41() by Lm41;
theorem :: ASYMPT_1:86
for n being Element of NAT 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 being Element of NAT st k <= n holds
n choose k >= ((n + 1) choose k) / (n + 1) by Lm43;
theorem :: ASYMPT_1:88
for f being Real_Sequence st ( for n being Element of NAT holds f . n = log (2,(n !)) ) holds
for n being Element of NAT holds f . n = Sum (seq_logn,n) by Lm44;
theorem :: ASYMPT_1:89
for n being Element of NAT st n >= 4 holds
n * (log (2,n)) >= 2 * n by Lm45;
theorem :: ASYMPT_1:90
for n being Element of NAT st n >= 2 holds
n ^2 > n + 1 by Lm47;
theorem :: ASYMPT_1:91
for n being Element of NAT st n >= 1 holds
(2 to_power (n + 1)) - (2 to_power n) > 1 by Lm48;
theorem :: ASYMPT_1:92
for n being Element of NAT st n >= 2 holds
not (2 to_power n) - 1 in POWEROF2SET by Lm49;
theorem :: ASYMPT_1:93
for n, k being Element of NAT st k >= 1 & n ! <= k & k < (n + 1) ! holds
Step1 k = n ! by Def7;
theorem :: ASYMPT_1:94
for a, b, c being Real st a > 1 & b >= a & c >= 1 holds
log (a,c) >= log (b,c) by Lm19;