:: Definition and Some Properties of Information Entropy
:: by Bo Zhang and Yatsuka Nakamura
::
:: Received July 9, 2007
:: Copyright (c) 2007 Association of Mizar Users
theorem Th1: :: ENTROPY1:1
theorem Th2: :: ENTROPY1:2
for
r being
Real st
r > 0 holds
(
ln . r <= r - 1 & (
r = 1 implies
ln . r = r - 1 ) & (
ln . r = r - 1 implies
r = 1 ) & (
r <> 1 implies
ln . r < r - 1 ) & (
ln . r < r - 1 implies
r <> 1 ) )
theorem Th3: :: ENTROPY1:3
theorem Th4: :: ENTROPY1:4
theorem Th5: :: ENTROPY1:5
theorem Th6: :: ENTROPY1:6
theorem Th7: :: ENTROPY1:7
theorem Th8: :: ENTROPY1:8
theorem Th9: :: ENTROPY1:9
:: deftheorem Def1 defines nonnegative ENTROPY1:def 1 :
theorem Th10: :: ENTROPY1:10
:: deftheorem Def2 defines has_onlyone_value_in ENTROPY1:def 2 :
theorem :: ENTROPY1:11
theorem Th12: :: ENTROPY1:12
theorem Th13: :: ENTROPY1:13
theorem Th14: :: ENTROPY1:14
theorem Th15: :: ENTROPY1:15
theorem Th16: :: ENTROPY1:16
theorem :: ENTROPY1:17
theorem Th18: :: ENTROPY1:18
theorem Th19: :: ENTROPY1:19
theorem Th20: :: ENTROPY1:20
theorem Th21: :: ENTROPY1:21
theorem Th22: :: ENTROPY1:22
theorem Th23: :: ENTROPY1:23
:: deftheorem Def3 defines diagonal ENTROPY1:def 3 :
theorem Th24: :: ENTROPY1:24
:: deftheorem Def4 defines Vec2DiagMx ENTROPY1:def 4 :
theorem Th25: :: ENTROPY1:25
theorem Th26: :: ENTROPY1:26
theorem Th27: :: ENTROPY1:27
theorem Th28: :: ENTROPY1:28
theorem Th29: :: ENTROPY1:29
theorem Th30: :: ENTROPY1:30
theorem Th31: :: ENTROPY1:31
theorem Th32: :: ENTROPY1:32
theorem Th33: :: ENTROPY1:33
theorem Th34: :: ENTROPY1:34
theorem Th35: :: ENTROPY1:35
theorem Th36: :: ENTROPY1:36
theorem Th37: :: ENTROPY1:37
theorem Th38: :: ENTROPY1:38
:: deftheorem Def5 defines Mx2FinS ENTROPY1:def 5 :
theorem Th39: :: ENTROPY1:39
theorem Th40: :: ENTROPY1:40
theorem Th41: :: ENTROPY1:41
theorem Th42: :: ENTROPY1:42
theorem Th43: :: ENTROPY1:43
theorem Th44: :: ENTROPY1:44
theorem Th45: :: ENTROPY1:45
theorem :: ENTROPY1:46
:: deftheorem Def6 defines FinSeq_log ENTROPY1:def 6 :
:: deftheorem defines Infor_FinSeq_of ENTROPY1:def 7 :
theorem Th47: :: ENTROPY1:47
theorem Th48: :: ENTROPY1:48
theorem :: ENTROPY1:49
theorem Th50: :: ENTROPY1:50
theorem Th51: :: ENTROPY1:51
theorem Th52: :: ENTROPY1:52
definition
let MR be
Matrix of
REAL ;
assume A1:
MR is
m-nonnegative
;
func Infor_FinSeq_of MR -> Matrix of
REAL means :
Def8:
:: ENTROPY1:def 8
(
len it = len MR &
width it = width MR & ( for
k being
Element of
NAT st
k in dom it holds
it . k = mlt (Line MR,k),
(FinSeq_log 2,(Line MR,k)) ) );
existence
ex b1 being Matrix of REAL st
( len b1 = len MR & width b1 = width MR & ( for k being Element of NAT st k in dom b1 holds
b1 . k = mlt (Line MR,k),(FinSeq_log 2,(Line MR,k)) ) )
uniqueness
for b1, b2 being Matrix of REAL st len b1 = len MR & width b1 = width MR & ( for k being Element of NAT st k in dom b1 holds
b1 . k = mlt (Line MR,k),(FinSeq_log 2,(Line MR,k)) ) & len b2 = len MR & width b2 = width MR & ( for k being Element of NAT st k in dom b2 holds
b2 . k = mlt (Line MR,k),(FinSeq_log 2,(Line MR,k)) ) holds
b1 = b2
end;
:: deftheorem Def8 defines Infor_FinSeq_of ENTROPY1:def 8 :
theorem Th53: :: ENTROPY1:53
theorem Th54: :: ENTROPY1:54
:: deftheorem defines Entropy ENTROPY1:def 9 :
theorem :: ENTROPY1:55
theorem :: ENTROPY1:56
theorem Th57: :: ENTROPY1:57
theorem :: ENTROPY1:58
theorem Th59: :: ENTROPY1:59
theorem Th60: :: ENTROPY1:60
:: deftheorem defines Entropy_of_Joint_Prob ENTROPY1:def 10 :
theorem :: ENTROPY1:61
:: deftheorem Def11 defines Entropy_of_Cond_Prob ENTROPY1:def 11 :
theorem Th62: :: ENTROPY1:62
theorem Th63: :: ENTROPY1:63
theorem Th64: :: ENTROPY1:64
theorem :: ENTROPY1:65