:: QMAX_1 semantic presentation
:: deftheorem Def1 defines Probabilities QMAX_1:def 1 :
:: deftheorem Def2 defines Obs QMAX_1:def 2 :
:: deftheorem Def3 defines Sts QMAX_1:def 3 :
:: deftheorem Def4 defines Meas QMAX_1:def 4 :
reconsider X = {0} as non empty set ;
consider P being Function of Borel_Sets , REAL such that
Lemma42:
for D being Subset of REAL st D in Borel_Sets holds
( ( 0 in D implies P . D = 1 ) & ( not 0 in D implies P . D = 0 ) )
by PROB_1:60;
Lemma43:
for A being Event of Borel_Sets holds
( ( 0 in A implies P . A = 1 ) & ( not 0 in A implies P . A = 0 ) )
Lemma44:
for A being Event of Borel_Sets holds 0 <= P . A
Lemma45:
P . REAL = 1
Lemma47:
for A, B being Event of Borel_Sets st A misses B holds
P . (A \/ B) = (P . A) + (P . B)
for ASeq being SetSequence of Borel_Sets st ASeq is non-increasing holds
( P * ASeq is convergent & lim (P * ASeq) = P . (Intersection ASeq) )
then reconsider P = P as Probability of Borel_Sets by , , , PROB_1:def 13;
reconsider f = {[[0,0],P]} as Function by GRFUNC_1:15;
Lemma59:
( dom f = {[0,0]} & rng f = {P} )
by RELAT_1:23;
then Lemma60:
dom f = [:X,X:]
by ZFMISC_1:35;
P in Probabilities Borel_Sets
by ;
then
rng f c= Probabilities Borel_Sets
by , ZFMISC_1:37;
then reconsider Y = f as Function of [:X,X:], Probabilities Borel_Sets by , FUNCT_2:def 1, RELSET_1:11;
E62:
now
thus
for
A1,
A2 being
Element of
Obs QM_Str(#
X,
X,
Y #) st ( for
s being
Element of
Sts QM_Str(#
X,
X,
Y #) holds
Meas A1,
s = Meas A2,
s ) holds
A1 = A2
thus
for
s1,
s2 being
Element of
Sts QM_Str(#
X,
X,
Y #) st ( for
A being
Element of
Obs QM_Str(#
X,
X,
Y #) holds
Meas A,
s1 = Meas A,
s2 ) holds
s1 = s2
thus
for
s1,
s2 being
Element of
Sts QM_Str(#
X,
X,
Y #)
for
t being
Real st 0
<= t &
t <= 1 holds
ex
s being
Element of
Sts QM_Str(#
X,
X,
Y #) st
for
A being
Element of
Obs QM_Str(#
X,
X,
Y #)
for
E being
Event of
Borel_Sets holds
(Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E))
proof
let s1 be
Element of
Sts QM_Str(#
X,
X,
Y #),
s2 be
Element of
Sts QM_Str(#
X,
X,
Y #);
E34:
(
s1 = 0 &
s2 = 0 )
by TARSKI:def 1;
let t be
Real;
assume
( 0
<= t &
t <= 1 )
;
take
s2
;
let A be
Element of
Obs QM_Str(#
X,
X,
Y #);
let E be
Event of
Borel_Sets ;
thus
(Meas A,s2) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E))
by ;
end;
end;
definition
let IT be
QM_Str ;
attr a1 is
Quantum_Mechanics-like means :
Def5:
:: QMAX_1:def 5
( ( for
A1,
A2 being
Element of
Obs X st ( for
s being
Element of
Sts X holds
Meas A1,
s = Meas A2,
s ) holds
A1 = A2 ) & ( for
s1,
s2 being
Element of
Sts X st ( for
A being
Element of
Obs X holds
Meas A,
s1 = Meas A,
s2 ) holds
s1 = s2 ) & ( for
s1,
s2 being
Element of
Sts X for
t being
Real st 0
<= t &
t <= 1 holds
ex
s being
Element of
Sts X st
for
A being
Element of
Obs X for
E being
Event of
Borel_Sets holds
(Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E)) ) );
end;
:: deftheorem Def5 defines Quantum_Mechanics-like QMAX_1:def 5 :
for
IT being
QM_Str holds
(
IT is
Quantum_Mechanics-like iff ( ( for
A1,
A2 being
Element of
Obs IT st ( for
s being
Element of
Sts IT holds
Meas A1,
s = Meas A2,
s ) holds
A1 = A2 ) & ( for
s1,
s2 being
Element of
Sts IT st ( for
A being
Element of
Obs IT holds
Meas A,
s1 = Meas A,
s2 ) holds
s1 = s2 ) & ( for
s1,
s2 being
Element of
Sts IT for
t being
Real st 0
<= t &
t <= 1 holds
ex
s being
Element of
Sts IT st
for
A being
Element of
Obs IT for
E being
Event of
Borel_Sets holds
(Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E)) ) ) );
:: deftheorem Def6 defines is_an_involution_in QMAX_1:def 6 :
:: deftheorem Def7 defines is_a_Quantuum_Logic_on QMAX_1:def 7 :
:: deftheorem Def8 defines Prop QMAX_1:def 8 :
theorem Th1: :: QMAX_1:1
canceled;
theorem Th2: :: QMAX_1:2
canceled;
theorem Th3: :: QMAX_1:3
canceled;
theorem Th4: :: QMAX_1:4
canceled;
theorem Th5: :: QMAX_1:5
canceled;
theorem Th6: :: QMAX_1:6
canceled;
theorem Th7: :: QMAX_1:7
canceled;
theorem Th8: :: QMAX_1:8
canceled;
theorem Th9: :: QMAX_1:9
canceled;
theorem Th10: :: QMAX_1:10
canceled;
theorem Th11: :: QMAX_1:11
canceled;
theorem Th12: :: QMAX_1:12
canceled;
theorem Th13: :: QMAX_1:13
canceled;
theorem Th14: :: QMAX_1:14
theorem Th15: :: QMAX_1:15
canceled;
theorem Th16: :: QMAX_1:16
:: deftheorem Def9 defines 'not' QMAX_1:def 9 :
:: deftheorem Def10 defines |- QMAX_1:def 10 :
:: deftheorem Def11 defines <==> QMAX_1:def 11 :
theorem Th17: :: QMAX_1:17
canceled;
theorem Th18: :: QMAX_1:18
canceled;
theorem Th19: :: QMAX_1:19
canceled;
theorem Th20: :: QMAX_1:20
theorem Th21: :: QMAX_1:21
theorem Th22: :: QMAX_1:22
theorem Th23: :: QMAX_1:23
theorem Th24: :: QMAX_1:24
theorem Th25: :: QMAX_1:25
theorem Th26: :: QMAX_1:26
theorem Th27: :: QMAX_1:27
theorem Th28: :: QMAX_1:28
:: deftheorem Def12 defines PropRel QMAX_1:def 12 :
theorem Th29: :: QMAX_1:29
canceled;
theorem Th30: :: QMAX_1:30
definition
let Q be
Quantum_Mechanics;
func OrdRel c1 -> Relation of
Class (PropRel a1) means :
Def13:
:: QMAX_1:def 13
for
B,
C being
Subset of
(Prop X) holds
(
[B,C] in it iff (
B in Class (PropRel X) &
C in Class (PropRel X) & ( for
p,
q being
Element of
Prop X st
p in B &
q in C holds
p |- q ) ) );
existence
ex b1 being Relation of Class (PropRel Q) st
for B, C being Subset of (Prop Q) holds
( [B,C] in b1 iff ( B in Class (PropRel Q) & C in Class (PropRel Q) & ( for p, q being Element of Prop Q st p in B & q in C holds
p |- q ) ) )
uniqueness
for b1, b2 being Relation of Class (PropRel Q) st ( for B, C being Subset of (Prop Q) holds
( [B,C] in b1 iff ( B in Class (PropRel Q) & C in Class (PropRel Q) & ( for p, q being Element of Prop Q st p in B & q in C holds
p |- q ) ) ) ) & ( for B, C being Subset of (Prop Q) holds
( [B,C] in b2 iff ( B in Class (PropRel Q) & C in Class (PropRel Q) & ( for p, q being Element of Prop Q st p in B & q in C holds
p |- q ) ) ) ) holds
b1 = b2
end;
:: deftheorem Def13 defines OrdRel QMAX_1:def 13 :
theorem Th31: :: QMAX_1:31
canceled;
theorem Th32: :: QMAX_1:32
theorem Th33: :: QMAX_1:33
theorem Th34: :: QMAX_1:34
definition
let Q be
Quantum_Mechanics;
func InvRel c1 -> Function of
Class (PropRel a1),
Class (PropRel a1) means :
Def14:
:: QMAX_1:def 14
for
p being
Element of
Prop X holds
it . (Class (PropRel X),p) = Class (PropRel X),
('not' p);
existence
ex b1 being Function of Class (PropRel Q), Class (PropRel Q) st
for p being Element of Prop Q holds b1 . (Class (PropRel Q),p) = Class (PropRel Q),('not' p)
uniqueness
for b1, b2 being Function of Class (PropRel Q), Class (PropRel Q) st ( for p being Element of Prop Q holds b1 . (Class (PropRel Q),p) = Class (PropRel Q),('not' p) ) & ( for p being Element of Prop Q holds b2 . (Class (PropRel Q),p) = Class (PropRel Q),('not' p) ) holds
b1 = b2
end;
:: deftheorem Def14 defines InvRel QMAX_1:def 14 :
theorem Th35: :: QMAX_1:35
canceled;
theorem Th36: :: QMAX_1:36