:: ASYMPT_1 semantic presentation
Lemma47:
for n being Element of NAT st n >= 2 holds
2 to_power n > n + 1
theorem Th1: :: ASYMPT_1:1
Lemma112:
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
theorem Th2: :: ASYMPT_1:2
:: deftheorem Def1 defines seq_a^ ASYMPT_1:def 1 :
Lemma118:
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))
theorem Th3: :: ASYMPT_1:3
:: deftheorem Def2 defines seq_logn ASYMPT_1:def 2 :
:: deftheorem Def3 defines seq_n^ ASYMPT_1:def 3 :
Lemma121:
for f, g being Real_Sequence
for n being Element of NAT holds (f /" g) . n = (f . n) / (g . n)
Lemma122:
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 )
theorem Th4: :: ASYMPT_1:4
Lemma124:
for a, b, c being real number st 0 < a & a <= b & c >= 0 holds
a to_power c <= b to_power c
Lemma125:
2 to_power 2 = 4
Lemma126:
2 to_power 3 = 8
Lemma127:
2 to_power 4 = 16
Lemma128:
2 to_power 5 = 32
Lemma129:
2 to_power 6 = 64
Lemma130:
for n being Element of NAT st n >= 4 holds
(2 * n) + 3 < 2 to_power n
Lemma131:
for n being Element of NAT st n >= 6 holds
(n + 1) ^2 < 2 to_power n
Lemma132:
for c being Real st c > 6 holds
c ^2 < 2 to_power c
Lemma135:
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 )
Lemma137:
for e being Real st e > 0 holds
( seq_logn /" (seq_n^ e) is convergent & lim (seq_logn /" (seq_n^ e)) = 0 )
theorem Th5: :: ASYMPT_1:5
theorem Th6: :: ASYMPT_1:6
Lemma139:
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
Lemma140:
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
Lemma141:
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
Lemma142:
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
Lemma143:
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
Lemma144:
for n being Element of NAT holds [/(n / 2)\] <= n
Lemma145:
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)
theorem Th7: :: ASYMPT_1:7
theorem Th8: :: ASYMPT_1:8
:: deftheorem Def4 defines seq_const ASYMPT_1:def 4 :
Lemma146:
for a, b, c being Real st a > 1 & b >= a & c >= 1 holds
log a,c >= log b,c
theorem Th9: :: ASYMPT_1:9
theorem Th10: :: ASYMPT_1:10
theorem Th11: :: ASYMPT_1:11
theorem Th12: :: ASYMPT_1:12
Lemma152:
for a being positive Real
for b, c being Real holds seq_a^ a,b,c is eventually-positive
;
theorem Th13: :: ASYMPT_1:13
:: deftheorem Def5 defines seq_n! ASYMPT_1:def 5 :
theorem Th14: :: ASYMPT_1:14
theorem Th15: :: ASYMPT_1:15
theorem Th16: :: ASYMPT_1:16
theorem Th17: :: ASYMPT_1:17
theorem Th18: :: ASYMPT_1:18
theorem Th19: :: ASYMPT_1:19
theorem Th20: :: ASYMPT_1:20
Lemma155:
for n being Element of NAT holds ((n ^2 ) - n) + 1 > 0
Lemma156:
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 )
Lemma157:
for n being Element of NAT st n >= 1 holds
((n ^2 ) - n) + 1 <= n ^2
Lemma158:
for n being Element of NAT st n >= 1 holds
n ^2 <= 2 * (((n ^2 ) - n) + 1)
Lemma159:
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)
theorem Th21: :: ASYMPT_1:21
theorem Th22: :: ASYMPT_1:22
theorem Th23: :: ASYMPT_1:23
theorem Th24: :: ASYMPT_1:24
theorem Th25: :: ASYMPT_1:25
Lemma162:
2 to_power 12 = 4096
Lemma163:
for n being Element of NAT st n >= 3 holds
n ^2 > (2 * n) + 1
Lemma164:
for n being Element of NAT st n >= 10 holds
2 to_power (n - 1) > (2 * n) ^2
Lemma165:
for n being Element of NAT st n >= 9 holds
(n + 1) to_power 6 < 2 * (n to_power 6)
Lemma170:
for n being Element of NAT st n >= 30 holds
2 to_power n > n to_power 6
Lemma171:
for x being Real st x > 9 holds
2 to_power x > (2 * x) ^2
Lemma172:
ex N being Element of NAT st
for n being Element of NAT st n >= N holds
(sqrt n) - (log 2,n) > 1
Lemma173:
(4 + 1) ! = 120
Lemma174:
for n being Element of NAT st n >= 10 holds
(2 to_power (2 * n)) / (n ! ) < 1 / (2 to_power (n - 9))
Lemma175:
for n being Element of NAT st n >= 3 holds
2 * (n - 2) >= n - 1
Lemma176:
5 to_power 5 = 3125
Lemma177:
4 to_power 4 = 256
Lemma178:
for a, b, d, e being Real holds (a / b) / (d / e) = (a / d) * (e / b)
Lemma179:
for x being real number st x >= 0 holds
sqrt x = x to_power (1 / 2)
Lemma180:
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
Lemma181:
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 non-decreasing
Lemma182:
for n being Element of NAT st n >= 1 holds
((n + 1) / n) to_power n <= ((n + 2) / (n + 1)) to_power (n + 1)
theorem Th26: :: ASYMPT_1:26
theorem Th27: :: ASYMPT_1:27
theorem Th28: :: ASYMPT_1:28
theorem Th29: :: ASYMPT_1:29
theorem Th30: :: ASYMPT_1:30
theorem Th31: :: ASYMPT_1:31
theorem Th32: :: ASYMPT_1:32
Lemma184:
for k, n being Element of NAT st k <= n holds
n choose k >= ((n + 1) choose k) / (n + 1)
theorem Th33: :: ASYMPT_1:33
Lemma194:
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
Lemma195:
for n being Element of NAT st n >= 4 holds
n * (log 2,n) >= 2 * n
theorem Th34: :: ASYMPT_1:34
theorem Th35: :: ASYMPT_1:35
definition
let n be
Element of
NAT ;
let a be
positive Real,
b be
positive Real;
defpred S1[
Element of
NAT ,
FinSequence of
REAL ,
set ]
means ex
n1 being
Element of
NAT st
(
n1 = [/(((a1 + 1) + 1) / 2)\] &
a3 = a2 ^ <*((4 * (a2 /. n1)) + (b * ((a1 + 1) + 1)))*> );
E49:
for
n being
Element of
NAT for
x,
y1,
y2 being
Element of
REAL * st
S1[
n,
x,
y1] &
S1[
n,
x,
y2] holds
y1 = y2
;
func Prob28 c1,
c2,
c3 -> Real means :
Def6:
:: ASYMPT_1:def 6
it = 0
if n = 0
otherwise ex
l being
Element of
NAT ex
prob28 being
Function of
NAT ,
REAL * st
(
l + 1
= n &
it = (prob28 . l) /. n &
prob28 . 0
= <*a*> & ( for
n being
Element of
NAT ex
n1 being
Element of
NAT st
(
n1 = [/(((n + 1) + 1) / 2)\] &
prob28 . (n + 1) = (prob28 . n) ^ <*((4 * ((prob28 . n) /. n1)) + (b * ((n + 1) + 1)))*> ) ) );
consistency
for b1 being Real holds verum
;
existence
( ( n = 0 implies ex b1 being Real st b1 = 0 ) & ( not n = 0 implies ex b1 being Real ex l being Element of NAT ex prob28 being Function of NAT ,REAL * st
( l + 1 = n & b1 = (prob28 . l) /. n & prob28 . 0 = <*a*> & ( for n being Element of NAT ex n1 being Element of NAT st
( n1 = [/(((n + 1) + 1) / 2)\] & prob28 . (n + 1) = (prob28 . n) ^ <*((4 * ((prob28 . n) /. n1)) + (b * ((n + 1) + 1)))*> ) ) ) ) )
uniqueness
for b1, b2 being Real holds
( ( n = 0 & b1 = 0 & b2 = 0 implies b1 = b2 ) & ( not n = 0 & ex l being Element of NAT ex prob28 being Function of NAT ,REAL * st
( l + 1 = n & b1 = (prob28 . l) /. n & prob28 . 0 = <*a*> & ( for n being Element of NAT ex n1 being Element of NAT st
( n1 = [/(((n + 1) + 1) / 2)\] & prob28 . (n + 1) = (prob28 . n) ^ <*((4 * ((prob28 . n) /. n1)) + (b * ((n + 1) + 1)))*> ) ) ) & ex l being Element of NAT ex prob28 being Function of NAT ,REAL * st
( l + 1 = n & b2 = (prob28 . l) /. n & prob28 . 0 = <*a*> & ( for n being Element of NAT ex n1 being Element of NAT st
( n1 = [/(((n + 1) + 1) / 2)\] & prob28 . (n + 1) = (prob28 . n) ^ <*((4 * ((prob28 . n) /. n1)) + (b * ((n + 1) + 1)))*> ) ) ) implies b1 = b2 ) )
end;
:: deftheorem Def6 defines Prob28 ASYMPT_1:def 6 :
definition
let a be
positive Real,
b be
positive Real;
func seq_prob28 c1,
c2 -> Real_Sequence means :
Def7:
:: ASYMPT_1:def 7
for
n being
Element of
NAT holds
it . n = Prob28 n,
a,
b;
existence
ex b1 being Real_Sequence st
for n being Element of NAT holds b1 . n = Prob28 n,a,b
uniqueness
for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = Prob28 n,a,b ) & ( for n being Element of NAT holds b2 . n = Prob28 n,a,b ) holds
b1 = b2
end;
:: deftheorem Def7 defines seq_prob28 ASYMPT_1:def 7 :
Lemma206:
for n being Element of NAT st n >= 2 holds
[/(n / 2)\] < n
Lemma207:
for a, b being positive Real holds
( Prob28 0,a,b = 0 & Prob28 1,a,b = a & ( for n being Element of NAT st n >= 2 holds
ex n1 being Element of NAT st
( n1 = [/(n / 2)\] & Prob28 n,a,b = (4 * (Prob28 n1,a,b)) + (b * n) ) ) )
theorem Th36: :: ASYMPT_1:36
:: deftheorem Def8 defines POWEROF2SET ASYMPT_1:def 8 :
Lemma210:
for n being Element of NAT st n >= 2 holds
n ^2 > n + 1
Lemma211:
for n being Element of NAT st n >= 1 holds
(2 to_power (n + 1)) - (2 to_power n) > 1
Lemma212:
for n being Element of NAT st n >= 2 holds
not (2 to_power n) - 1 in POWEROF2SET
theorem Th37: :: ASYMPT_1:37
theorem Th38: :: ASYMPT_1:38
theorem Th39: :: ASYMPT_1:39
Lemma216:
for n being Element of NAT st n >= 2 holds
n ! > 1
Lemma217:
for n1, n being Element of NAT st n <= n1 holds
n ! <= n1 !
Lemma218:
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 ) )
:: deftheorem Def9 defines Step1 ASYMPT_1:def 9 :
Lemma220:
for n being Element of NAT st n >= 3 holds
n ! > n
theorem Th40: :: ASYMPT_1:40
Lemma222:
(seq_n^ 1) - (seq_const 1) is eventually-positive
theorem Th41: :: ASYMPT_1:41
theorem Th42: :: ASYMPT_1:42
theorem Th43: :: ASYMPT_1:43
theorem Th44: :: ASYMPT_1:44
theorem Th45: :: ASYMPT_1:45
theorem Th46: :: ASYMPT_1:46
theorem Th47: :: ASYMPT_1:47
theorem Th48: :: ASYMPT_1:48
theorem Th49: :: ASYMPT_1:49
theorem Th50: :: ASYMPT_1:50
theorem Th51: :: ASYMPT_1:51
theorem Th52: :: ASYMPT_1:52
theorem Th53: :: ASYMPT_1:53
theorem Th54: :: ASYMPT_1:54
theorem Th55: :: ASYMPT_1:55
theorem Th56: :: ASYMPT_1:56
theorem Th57: :: ASYMPT_1:57
theorem Th58: :: ASYMPT_1:58
theorem Th59: :: ASYMPT_1:59
theorem Th60: :: ASYMPT_1:60
theorem Th61: :: ASYMPT_1:61
theorem Th62: :: ASYMPT_1:62
theorem Th63: :: ASYMPT_1:63
theorem Th64: :: ASYMPT_1:64
theorem Th65: :: ASYMPT_1:65
theorem Th66: :: ASYMPT_1:66
theorem Th67: :: ASYMPT_1:67
theorem Th68: :: ASYMPT_1:68
theorem Th69: :: ASYMPT_1:69
theorem Th70: :: ASYMPT_1:70
theorem Th71: :: ASYMPT_1:71
theorem Th72: :: ASYMPT_1:72
theorem Th73: :: ASYMPT_1:73
theorem Th74: :: ASYMPT_1:74
theorem Th75: :: ASYMPT_1:75
theorem Th76: :: ASYMPT_1:76
theorem Th77: :: ASYMPT_1:77
theorem Th78: :: ASYMPT_1:78
theorem Th79: :: ASYMPT_1:79
theorem Th80: :: ASYMPT_1:80
theorem Th81: :: ASYMPT_1:81
theorem Th82: :: ASYMPT_1:82
theorem Th83: :: ASYMPT_1:83
theorem Th84: :: ASYMPT_1:84
theorem Th85: :: ASYMPT_1:85
theorem Th86: :: ASYMPT_1:86
theorem Th87: :: ASYMPT_1:87
theorem Th88: :: ASYMPT_1:88
theorem Th89: :: ASYMPT_1:89
theorem Th90: :: ASYMPT_1:90
theorem Th91: :: ASYMPT_1:91
theorem Th92: :: ASYMPT_1:92
theorem Th93: :: ASYMPT_1:93
theorem Th94: :: ASYMPT_1:94
theorem Th95: :: ASYMPT_1:95
theorem Th96: :: ASYMPT_1:96
theorem Th97: :: ASYMPT_1:97
theorem Th98: :: ASYMPT_1:98
theorem Th99: :: ASYMPT_1:99
theorem Th100: :: ASYMPT_1:100
theorem Th101: :: ASYMPT_1:101