:: ENTROPY1 semantic presentation
theorem th477: :: ENTROPY1:1
theorem th300: :: ENTROPY1:2
theorem th300a: :: ENTROPY1:3
theorem th290b: :: ENTROPY1:4
for
a,
b being
Real st
a > 1 &
b > 1 holds
log a,
b > 0
theorem th290: :: ENTROPY1:5
theorem th291: :: ENTROPY1:6
theorem th306: :: ENTROPY1:7
theorem th306a: :: ENTROPY1:8
theorem th419: :: ENTROPY1:9
:: deftheorem Def109 defines nonnegative ENTROPY1:def 1 :
theorem th293a: :: ENTROPY1:10
:: deftheorem Def16 defines has_onlyone_value_in ENTROPY1:def 2 :
theorem :: ENTROPY1:11
theorem th232: :: ENTROPY1:12
theorem th233: :: ENTROPY1:13
theorem th390: :: ENTROPY1:14
theorem th391a: :: ENTROPY1:15
theorem th270a: :: ENTROPY1:16
theorem :: ENTROPY1:17
theorem th350: :: ENTROPY1:18
theorem th294: :: ENTROPY1:19
theorem th339: :: ENTROPY1:20
theorem th330: :: ENTROPY1:21
theorem th331: :: ENTROPY1:22
theorem th332: :: ENTROPY1:23
:: deftheorem Def15 defines diagonal ENTROPY1:def 3 :
theorem th231: :: ENTROPY1:24
:: deftheorem Def17 defines Vec2DiagMx ENTROPY1:def 4 :
theorem th234: :: ENTROPY1:25
theorem th235: :: ENTROPY1:26
theorem th236: :: ENTROPY1:27
theorem th237: :: ENTROPY1:28
theorem th409: :: ENTROPY1:29
theorem th409a: :: ENTROPY1:30
theorem th412: :: ENTROPY1:31
theorem th413: :: ENTROPY1:32
theorem th413a: :: ENTROPY1:33
theorem th414: :: ENTROPY1:34
theorem th415: :: ENTROPY1:35
theorem th416: :: ENTROPY1:36
theorem th420: :: ENTROPY1:37
theorem th421: :: ENTROPY1:38
:: deftheorem Def210 defines Mx2FinS ENTROPY1:def 5 :
theorem th410: :: ENTROPY1:39
theorem th417: :: ENTROPY1:40
theorem th418: :: ENTROPY1:41
theorem th422: :: ENTROPY1:42
theorem th423: :: ENTROPY1:43
theorem th447: :: ENTROPY1:44
theorem th450: :: ENTROPY1:45
theorem th450a: :: ENTROPY1:46
:: deftheorem Def299 defines FinSeq_log ENTROPY1:def 6 :
:: deftheorem defines Infor_FinSeq_of ENTROPY1:def 7 :
theorem th196: :: ENTROPY1:47
theorem th401: :: ENTROPY1:48
theorem :: ENTROPY1:49
theorem th292: :: ENTROPY1:50
theorem th293: :: ENTROPY1:51
theorem th405: :: ENTROPY1:52
definition
let MR be
Matrix of
REAL ;
assume A1:
MR is
m-nonnegative
;
func Infor_FinSeq_of MR -> Matrix of
REAL means :
Def198:
:: 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 Def198 defines Infor_FinSeq_of ENTROPY1:def 8 :
theorem th295: :: ENTROPY1:53
theorem th296: :: ENTROPY1:54
:: deftheorem defines Entropy ENTROPY1:def 9 :
theorem :: ENTROPY1:55
theorem :: ENTROPY1:56
theorem th310a: :: ENTROPY1:57
theorem :: ENTROPY1:58
theorem th451: :: ENTROPY1:59
theorem th452: :: ENTROPY1:60
:: deftheorem defines Entropy_of_Joint_Prob ENTROPY1:def 10 :
theorem :: ENTROPY1:61
:: deftheorem Def10 defines Entropy_of_Cond_Prob ENTROPY1:def 11 :
theorem th444: :: ENTROPY1:62
theorem th445: :: ENTROPY1:63
theorem th454: :: ENTROPY1:64
theorem :: ENTROPY1:65