:: Some Properties of {F}ibonacci Numbers
:: by Magdalena Jastrz\c{e}bska and Adam Grabowski
::
:: Received May 10, 2004
:: Copyright (c) 2004 Association of Mizar Users
theorem :: FIB_NUM2:1
theorem Th2: :: FIB_NUM2:2
theorem Th3: :: FIB_NUM2:3
theorem Th4: :: FIB_NUM2:4
theorem Th5: :: FIB_NUM2:5
theorem Th6: :: FIB_NUM2:6
theorem Th7: :: FIB_NUM2:7
theorem Th8: :: FIB_NUM2:8
theorem Th9: :: FIB_NUM2:9
theorem Th10: :: FIB_NUM2:10
theorem Th11: :: FIB_NUM2:11
theorem Th12: :: FIB_NUM2:12
theorem Th13: :: FIB_NUM2:13
theorem Th14: :: FIB_NUM2:14
theorem :: FIB_NUM2:15
:: deftheorem defines Prefix FIB_NUM2:def 1 :
theorem Th16: :: FIB_NUM2:16
theorem Th17: :: FIB_NUM2:17
theorem Th18: :: FIB_NUM2:18
theorem Th19: :: FIB_NUM2:19
theorem Th20: :: FIB_NUM2:20
theorem Th21: :: FIB_NUM2:21
theorem Th22: :: FIB_NUM2:22
theorem Th23: :: FIB_NUM2:23
theorem Th24: :: FIB_NUM2:24
theorem Th25: :: FIB_NUM2:25
theorem Th26: :: FIB_NUM2:26
theorem Th27: :: FIB_NUM2:27
theorem Th28: :: FIB_NUM2:28
theorem Th29: :: FIB_NUM2:29
Lm1:
for k being Nat holds Fib ((2 * (k + 2)) + 1) = (Fib ((2 * k) + 3)) + (Fib ((2 * k) + 4))
theorem Th30: :: FIB_NUM2:30
theorem Th31: :: FIB_NUM2:31
theorem Th32: :: FIB_NUM2:32
theorem Th33: :: FIB_NUM2:33
theorem :: FIB_NUM2:34
theorem Th35: :: FIB_NUM2:35
theorem Th36: :: FIB_NUM2:36
theorem Th37: :: FIB_NUM2:37
theorem Th38: :: FIB_NUM2:38
theorem Th39: :: FIB_NUM2:39
theorem :: FIB_NUM2:40
theorem :: FIB_NUM2:41
theorem Th42: :: FIB_NUM2:42
theorem Th43: :: FIB_NUM2:43
theorem Th44: :: FIB_NUM2:44
theorem Th45: :: FIB_NUM2:45
theorem Th46: :: FIB_NUM2:46
theorem :: FIB_NUM2:47
theorem Th48: :: FIB_NUM2:48
theorem Th49: :: FIB_NUM2:49
for
k being
Nat holds
(
Fib k = 1 iff (
k = 1 or
k = 2 ) )
theorem Th50: :: FIB_NUM2:50
theorem Th51: :: FIB_NUM2:51
theorem :: FIB_NUM2:52
:: deftheorem Def2 defines FIB FIB_NUM2:def 2 :
:: deftheorem defines EvenNAT FIB_NUM2:def 3 :
:: deftheorem defines OddNAT FIB_NUM2:def 4 :
theorem Th53: :: FIB_NUM2:53
theorem Th54: :: FIB_NUM2:54
:: deftheorem defines EvenFibs FIB_NUM2:def 5 :
:: deftheorem defines OddFibs FIB_NUM2:def 6 :
theorem Th55: :: FIB_NUM2:55
theorem :: FIB_NUM2:56
theorem Th57: :: FIB_NUM2:57
theorem :: FIB_NUM2:58
theorem Th59: :: FIB_NUM2:59
theorem Th60: :: FIB_NUM2:60
theorem Th61: :: FIB_NUM2:61
theorem Th62: :: FIB_NUM2:62
theorem Th63: :: FIB_NUM2:63
theorem Th64: :: FIB_NUM2:64
theorem Th65: :: FIB_NUM2:65
theorem Th66: :: FIB_NUM2:66
theorem :: FIB_NUM2:67
theorem :: FIB_NUM2:68
theorem Th69: :: FIB_NUM2:69
theorem Th70: :: FIB_NUM2:70
theorem :: FIB_NUM2:71
theorem :: FIB_NUM2:72