:: ASYMPT_1 semantic presentation
Lemma1:
for b1 being Nat holds
not ( b1 >= 2 & not 2 to_power b1 > b1 + 1 )
theorem Th1: :: ASYMPT_1:1
Lemma2:
for b1 being logbase Real
for b2 being Real_Sequence holds
( b1 > 1 & b2 . 0 = 0 & ( for b3 being Nat holds
( b3 > 0 implies b2 . b3 = log b1,b3 ) ) implies b2 is eventually-positive )
theorem Th2: :: ASYMPT_1:2
:: deftheorem Def1 defines seq_a^ ASYMPT_1:def 1 :
Lemma4:
for b1, b2, b3 being Real holds
( b1 > 0 & b3 > 0 & b3 <> 1 implies b1 to_power b2 = b3 to_power (b2 * (log b3,b1)) )
theorem Th3: :: ASYMPT_1:3
:: deftheorem Def2 defines seq_logn ASYMPT_1:def 2 :
:: deftheorem Def3 defines seq_n^ ASYMPT_1:def 3 :
Lemma7:
for b1, b2 being Real_Sequence
for b3 being Nat holds (b1 /" b2) . b3 = (b1 . b3) / (b2 . b3)
Lemma8:
for b1, b2 being eventually-nonnegative Real_Sequence holds
( ( b1 in Big_Oh b2 & b2 in Big_Oh b1 ) iff Big_Oh b1 = Big_Oh b2 )
theorem Th4: :: ASYMPT_1:4
Lemma10:
for b1, b2, b3 being real number holds
( 0 < b1 & b1 <= b2 & b3 >= 0 implies b1 to_power b3 <= b2 to_power b3 )
Lemma11:
2 to_power 2 = 4
Lemma12:
2 to_power 3 = 8
Lemma13:
2 to_power 4 = 16
Lemma14:
2 to_power 5 = 32
Lemma15:
2 to_power 6 = 64
Lemma16:
for b1 being Nat holds
not ( b1 >= 4 & not (2 * b1) + 3 < 2 to_power b1 )
Lemma17:
for b1 being Nat holds
not ( b1 >= 6 & not (b1 + 1) ^2 < 2 to_power b1 )
Lemma18:
for b1 being Real holds
not ( b1 > 6 & not b1 ^2 < 2 to_power b1 )
Lemma19:
for b1 being positive Real
for b2 being Real_Sequence holds
( b2 . 0 = 0 & ( for b3 being Nat holds
( b3 > 0 implies b2 . b3 = log 2,(b3 to_power b1) ) ) implies ( b2 /" (seq_n^ b1) is convergent & lim (b2 /" (seq_n^ b1)) = 0 ) )
Lemma20:
for b1 being Real holds
( b1 > 0 implies ( seq_logn /" (seq_n^ b1) is convergent & lim (seq_logn /" (seq_n^ b1)) = 0 ) )
theorem Th5: :: ASYMPT_1:5
theorem Th6: :: ASYMPT_1:6
Lemma22:
for b1 being Real_Sequence
for b2 being Nat holds
( ( for b3 being Nat holds
( b3 <= b2 implies b1 . b3 >= 0 ) ) implies Sum b1,b2 >= 0 )
Lemma23:
for b1, b2 being Real_Sequence
for b3 being Nat holds
( ( for b4 being Nat holds
( b4 <= b3 implies b1 . b4 <= b2 . b4 ) ) implies Sum b1,b3 <= Sum b2,b3 )
Lemma24:
for b1 being Real_Sequence
for b2 being Real holds
( b1 . 0 = 0 & ( for b3 being Nat holds
( b3 > 0 implies b1 . b3 = b2 ) ) implies for b3 being Nat holds Sum b1,b3 = b2 * b3 )
Lemma25:
for b1 being Real_Sequence
for b2, b3 being Nat holds (Sum b1,b2,b3) + (b1 . (b2 + 1)) = Sum b1,(b2 + 1),b3
Lemma26:
for b1, b2 being Real_Sequence
for b3, b4 being Nat holds
( b4 >= b3 + 1 & ( for b5 being Nat holds
( b3 + 1 <= b5 & b5 <= b4 implies b1 . b5 <= b2 . b5 ) ) implies Sum b1,b4,b3 <= Sum b2,b4,b3 )
Lemma27:
for b1 being Nat holds [/(b1 / 2)\] <= b1
Lemma28:
for b1 being Real_Sequence
for b2 being Real
for b3 being Nat holds
( b1 . 0 = 0 & ( for b4 being Nat holds
( b4 > 0 implies b1 . b4 = b2 ) ) implies for b4 being Nat holds Sum b1,b3,b4 = b2 * (b3 - b4) )
theorem Th7: :: ASYMPT_1:7
theorem Th8: :: ASYMPT_1:8
:: deftheorem Def4 defines seq_const ASYMPT_1:def 4 :
Lemma29:
for b1, b2, b3 being Real holds
( b1 > 1 & b2 >= b1 & b3 >= 1 implies log b1,b3 >= log b2,b3 )
theorem Th9: :: ASYMPT_1:9
theorem Th10: :: ASYMPT_1:10
theorem Th11: :: ASYMPT_1:11
theorem Th12: :: ASYMPT_1:12
Lemma31:
for b1 being positive Real
for b2, b3 being Real holds seq_a^ b1,b2,b3 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
Lemma34:
for b1 being Nat holds
((b1 ^2 ) - b1) + 1 > 0
Lemma35:
for b1, b2 being Real_Sequence
for b3 being Nat
for b4 being Real holds
( b1 is convergent & lim b1 = b4 & ( for b5 being Nat holds
( b5 >= b3 implies b1 . b5 = b2 . b5 ) ) implies ( b2 is convergent & lim b2 = b4 ) )
Lemma36:
for b1 being Nat holds
( b1 >= 1 implies ((b1 ^2 ) - b1) + 1 <= b1 ^2 )
Lemma37:
for b1 being Nat holds
( b1 >= 1 implies b1 ^2 <= 2 * (((b1 ^2 ) - b1) + 1) )
Lemma38:
for b1 being Real holds
not ( 0 < b1 & b1 < 1 & ( for b2 being Nat holds
ex b3 being Nat st
( b3 >= b2 & not (b3 * (log 2,(1 + b1))) - (8 * (log 2,b3)) > 8 * (log 2,b3) ) ) )
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
Lemma39:
2 to_power 12 = 4096
Lemma40:
for b1 being Nat holds
not ( b1 >= 3 & not b1 ^2 > (2 * b1) + 1 )
Lemma41:
for b1 being Nat holds
not ( b1 >= 10 & not 2 to_power (b1 - 1) > (2 * b1) ^2 )
Lemma42:
for b1 being Nat holds
not ( b1 >= 9 & not (b1 + 1) to_power 6 < 2 * (b1 to_power 6) )
Lemma43:
for b1 being Nat holds
not ( b1 >= 30 & not 2 to_power b1 > b1 to_power 6 )
Lemma44:
for b1 being Real holds
not ( b1 > 9 & not 2 to_power b1 > (2 * b1) ^2 )
Lemma45:
ex b1 being Nat st
for b2 being Nat holds
not ( b2 >= b1 & not (sqrt b2) - (log 2,b2) > 1 )
Lemma46:
(4 + 1) ! = 120
Lemma47:
for b1 being Nat holds
not ( b1 >= 10 & not (2 to_power (2 * b1)) / (b1 ! ) < 1 / (2 to_power (b1 - 9)) )
Lemma48:
for b1 being Nat holds
( b1 >= 3 implies 2 * (b1 - 2) >= b1 - 1 )
Lemma49:
5 to_power 5 = 3125
Lemma50:
4 to_power 4 = 256
Lemma51:
for b1, b2, b3, b4 being Real holds (b1 / b2) / (b3 / b4) = (b1 / b3) * (b4 / b2)
Lemma52:
for b1 being real number holds
( b1 >= 0 implies sqrt b1 = b1 to_power (1 / 2) )
Lemma53:
ex b1 being Nat st
for b2 being Nat holds
not ( b2 >= b1 & not b2 - ((sqrt b2) * (log 2,b2)) > b2 / 2 )
Lemma54:
for b1 being Real_Sequence holds
( ( for b2 being Nat holds b1 . b2 = (1 + (1 / (b2 + 1))) to_power (b2 + 1) ) implies b1 is non-decreasing )
Lemma55:
for b1 being Nat holds
( b1 >= 1 implies ((b1 + 1) / b1) to_power b1 <= ((b1 + 2) / (b1 + 1)) to_power (b1 + 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
Lemma56:
for b1, b2 being Nat holds
( b1 <= b2 implies b2 choose b1 >= ((b2 + 1) choose b1) / (b2 + 1) )
theorem Th33: :: ASYMPT_1:33
Lemma57:
for b1 being Real_Sequence holds
( ( for b2 being Nat holds b1 . b2 = log 2,(b2 ! ) ) implies for b2 being Nat holds b1 . b2 = Sum seq_logn ,b2 )
Lemma58:
for b1 being Nat holds
( b1 >= 4 implies b1 * (log 2,b1) >= 2 * b1 )
theorem Th34: :: ASYMPT_1:34
theorem Th35: :: ASYMPT_1:35
definition
let c
1 be
Nat;
let c
2, c
3 be
positive Real;
defpred S
1[
Nat,
FinSequence of
REAL ,
set ] means ex b
1 being
Nat st
( b
1 = [/(((a1 + 1) + 1) / 2)\] & a
3 = a
2 ^ <*((4 * (a2 /. b1)) + (c3 * ((a1 + 1) + 1)))*> );
E60:
for b
1 being
Natfor b
2, b
3, b
4 being
Element of
REAL * holds
( S
1[b
1,b
2,b
3] & S
1[b
1,b
2,b
4] implies b
3 = b
4 )
;
func Prob28 c
1,c
2,c
3 -> Real means :
Def6:
:: ASYMPT_1:def 6
a
4 = 0
if a
1 = 0
otherwise ex b
1 being
Natex b
2 being
Function of
NAT ,
REAL * st
( b
1 + 1
= a
1 & a
4 = (b2 . b1) /. a
1 & b
2 . 0
= <*a2*> & ( for b
3 being
Nat holds
ex b
4 being
Nat st
( b
4 = [/(((b3 + 1) + 1) / 2)\] & b
2 . (b3 + 1) = (b2 . b3) ^ <*((4 * ((b2 . b3) /. b4)) + (a3 * ((b3 + 1) + 1)))*> ) ) );
consistency
for b1 being Real holds
verum
;
existence
( not ( c1 = 0 & ( for b1 being Real holds
not b1 = 0 ) ) & not ( not c1 = 0 & ( for b1 being Real
for b2 being Nat
for b3 being Function of NAT ,REAL * holds
not ( b2 + 1 = c1 & b1 = (b3 . b2) /. c1 & b3 . 0 = <*c2*> & ( for b4 being Nat holds
ex b5 being Nat st
( b5 = [/(((b4 + 1) + 1) / 2)\] & b3 . (b4 + 1) = (b3 . b4) ^ <*((4 * ((b3 . b4) /. b5)) + (c3 * ((b4 + 1) + 1)))*> ) ) ) ) ) )
uniqueness
for b1, b2 being Real holds
( ( c1 = 0 & b1 = 0 & b2 = 0 implies b1 = b2 ) & ( not c1 = 0 & ex b3 being Natex b4 being Function of NAT ,REAL * st
( b3 + 1 = c1 & b1 = (b4 . b3) /. c1 & b4 . 0 = <*c2*> & ( for b5 being Nat holds
ex b6 being Nat st
( b6 = [/(((b5 + 1) + 1) / 2)\] & b4 . (b5 + 1) = (b4 . b5) ^ <*((4 * ((b4 . b5) /. b6)) + (c3 * ((b5 + 1) + 1)))*> ) ) ) & ex b3 being Natex b4 being Function of NAT ,REAL * st
( b3 + 1 = c1 & b2 = (b4 . b3) /. c1 & b4 . 0 = <*c2*> & ( for b5 being Nat holds
ex b6 being Nat st
( b6 = [/(((b5 + 1) + 1) / 2)\] & b4 . (b5 + 1) = (b4 . b5) ^ <*((4 * ((b4 . b5) /. b6)) + (c3 * ((b5 + 1) + 1)))*> ) ) ) implies b1 = b2 ) )
end;
:: deftheorem Def6 defines Prob28 ASYMPT_1:def 6 :
definition
let c
1, c
2 be
positive Real;
func seq_prob28 c
1,c
2 -> Real_Sequence means :
Def7:
:: ASYMPT_1:def 7
for b
1 being
Nat holds a
3 . b
1 = Prob28 b
1,a
1,a
2;
existence
ex b1 being Real_Sequence st
for b2 being Nat holds b1 . b2 = Prob28 b2,c1,c2
uniqueness
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds b1 . b3 = Prob28 b3,c1,c2 ) & ( for b3 being Nat holds b2 . b3 = Prob28 b3,c1,c2 ) implies b1 = b2 )
end;
:: deftheorem Def7 defines seq_prob28 ASYMPT_1:def 7 :
Lemma62:
for b1 being Nat holds
not ( b1 >= 2 & not [/(b1 / 2)\] < b1 )
Lemma63:
for b1, b2 being positive Real holds
( Prob28 0,b1,b2 = 0 & Prob28 1,b1,b2 = b1 & ( for b3 being Nat holds
not ( b3 >= 2 & ( for b4 being Nat holds
not ( b4 = [/(b3 / 2)\] & Prob28 b3,b1,b2 = (4 * (Prob28 b4,b1,b2)) + (b2 * b3) ) ) ) ) )
theorem Th36: :: ASYMPT_1:36
:: deftheorem Def8 defines POWEROF2SET ASYMPT_1:def 8 :
Lemma64:
for b1 being Nat holds
not ( b1 >= 2 & not b1 ^2 > b1 + 1 )
Lemma65:
for b1 being Nat holds
not ( b1 >= 1 & not (2 to_power (b1 + 1)) - (2 to_power b1) > 1 )
Lemma66:
for b1 being Nat holds
not ( b1 >= 2 & (2 to_power b1) - 1 in POWEROF2SET )
theorem Th37: :: ASYMPT_1:37
theorem Th38: :: ASYMPT_1:38
theorem Th39: :: ASYMPT_1:39
Lemma67:
for b1 being Nat holds
not ( b1 >= 2 & not b1 ! > 1 )
Lemma68:
for b1, b2 being Nat holds
( b2 <= b1 implies b2 ! <= b1 ! )
Lemma69:
for b1 being Nat holds
not ( b1 >= 1 & ( for b2 being Nat holds
not ( b2 ! <= b1 & b1 < (b2 + 1) ! & ( for b3 being Nat holds
( b3 ! <= b1 & b1 < (b3 + 1) ! implies b3 = b2 ) ) ) ) )
:: deftheorem Def9 defines Step1 ASYMPT_1:def 9 :
for b
1, b
2 being
Nat holds
( ( b
1 <> 0 implies ( b
2 = Step1 b
1 iff ex b
3 being
Nat st
( b
3 ! <= b
1 & b
1 < (b3 + 1) ! & b
2 = b
3 ! ) ) ) & ( not b
1 <> 0 implies ( b
2 = Step1 b
1 iff b
2 = 0 ) ) );
Lemma71:
for b1 being Nat holds
not ( b1 >= 3 & not b1 ! > b1 )
theorem Th40: :: ASYMPT_1:40
Lemma72:
(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
for b
1 being
Nat holds
not ( b
1 >= 1 & ( for b
2 being
Nat holds
not ( b
2 ! <= b
1 & b
1 < (b2 + 1) ! & ( for b
3 being
Nat holds
( b
3 ! <= b
1 & b
1 < (b3 + 1) ! implies b
3 = b
2 ) ) ) ) )
by Lemma69;
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
for b
1 being
Real holds
not ( 0
< b
1 & b
1 < 1 & ( for b
2 being
Nat holds
ex b
3 being
Nat st
( b
3 >= b
2 & not
(b3 * (log 2,(1 + b1))) - (8 * (log 2,b3)) > 8
* (log 2,b3) ) ) )
by Lemma38;
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