:: 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 c1 = {0} as non empty set ;
consider c2 being Function of Borel_Sets , REAL such that
Lemma2:
for b1 being Subset of REAL holds
( b1 in Borel_Sets implies ( ( 0 in b1 implies c2 . b1 = 1 ) & ( not 0 in b1 implies c2 . b1 = 0 ) ) )
by PROB_1:60;
Lemma3:
for b1 being Event of Borel_Sets holds
( ( 0 in b1 implies c2 . b1 = 1 ) & ( not 0 in b1 implies c2 . b1 = 0 ) )
Lemma4:
for b1 being Event of Borel_Sets holds 0 <= c2 . b1
Lemma5:
c2 . REAL = 1
Lemma6:
for b1, b2 being Event of Borel_Sets holds
( b1 misses b2 implies c2 . (b1 \/ b2) = (c2 . b1) + (c2 . b2) )
for b1 being SetSequence of Borel_Sets holds
( b1 is non-increasing implies ( c2 * b1 is convergent & lim (c2 * b1) = c2 . (Intersection b1) ) )
then reconsider c3 = c2 as Probability of Borel_Sets by Lemma4, Lemma5, Lemma6, PROB_1:def 13;
reconsider c4 = {[[0,0],c3]} as Function by GRFUNC_1:15;
Lemma7:
( dom c4 = {[0,0]} & rng c4 = {c3} )
by RELAT_1:23;
then Lemma8:
dom c4 = [:c1,c1:]
by ZFMISC_1:35;
c3 in Probabilities Borel_Sets
by Def1;
then
rng c4 c= Probabilities Borel_Sets
by Lemma7, ZFMISC_1:37;
then reconsider c5 = c4 as Function of [:c1,c1:], Probabilities Borel_Sets by Lemma8, FUNCT_2:def 1, RELSET_1:11;
E9:
now
thus
for b
1, b
2 being
Element of
Obs QM_Str(# c
1,c
1,c
5 #) holds
( ( for b
3 being
Element of
Sts QM_Str(# c
1,c
1,c
5 #) holds
Meas b
1,b
3 = Meas b
2,b
3 ) implies b
1 = b
2 )
thus
for b
1, b
2 being
Element of
Sts QM_Str(# c
1,c
1,c
5 #) holds
( ( for b
3 being
Element of
Obs QM_Str(# c
1,c
1,c
5 #) holds
Meas b
3,b
1 = Meas b
3,b
2 ) implies b
1 = b
2 )
thus
for b
1, b
2 being
Element of
Sts QM_Str(# c
1,c
1,c
5 #)
for b
3 being
Real holds
not ( 0
<= b
3 & b
3 <= 1 & ( for b
4 being
Element of
Sts QM_Str(# c
1,c
1,c
5 #) holds
not for b
5 being
Element of
Obs QM_Str(# c
1,c
1,c
5 #)
for b
6 being
Event of
Borel_Sets holds
(Meas b5,b4) . b
6 = (b3 * ((Meas b5,b1) . b6)) + ((1 - b3) * ((Meas b5,b2) . b6)) ) )
end;
definition
let c
6 be
QM_Str ;
attr a
1 is
Quantum_Mechanics-like means :
Def5:
:: QMAX_1:def 5
( ( for b
1, b
2 being
Element of
Obs a
1 holds
( ( for b
3 being
Element of
Sts a
1 holds
Meas b
1,b
3 = Meas b
2,b
3 ) implies b
1 = b
2 ) ) & ( for b
1, b
2 being
Element of
Sts a
1 holds
( ( for b
3 being
Element of
Obs a
1 holds
Meas b
3,b
1 = Meas b
3,b
2 ) implies b
1 = b
2 ) ) & ( for b
1, b
2 being
Element of
Sts a
1for b
3 being
Real holds
not ( 0
<= b
3 & b
3 <= 1 & ( for b
4 being
Element of
Sts a
1 holds
not for b
5 being
Element of
Obs a
1for b
6 being
Event of
Borel_Sets holds
(Meas b5,b4) . b
6 = (b3 * ((Meas b5,b1) . b6)) + ((1 - b3) * ((Meas b5,b2) . b6)) ) ) ) );
end;
:: deftheorem Def5 defines Quantum_Mechanics-like QMAX_1:def 5 :
for b
1 being
QM_Str holds
( b
1 is
Quantum_Mechanics-like iff ( ( for b
2, b
3 being
Element of
Obs b
1 holds
( ( for b
4 being
Element of
Sts b
1 holds
Meas b
2,b
4 = Meas b
3,b
4 ) implies b
2 = b
3 ) ) & ( for b
2, b
3 being
Element of
Sts b
1 holds
( ( for b
4 being
Element of
Obs b
1 holds
Meas b
4,b
2 = Meas b
4,b
3 ) implies b
2 = b
3 ) ) & ( for b
2, b
3 being
Element of
Sts b
1for b
4 being
Real holds
not ( 0
<= b
4 & b
4 <= 1 & ( for b
5 being
Element of
Sts b
1 holds
not for b
6 being
Element of
Obs b
1for b
7 being
Event of
Borel_Sets holds
(Meas b6,b5) . b
7 = (b4 * ((Meas b6,b2) . b7)) + ((1 - b4) * ((Meas b6,b3) . b7)) ) ) ) ) );
:: 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 c
6 be
Quantum_Mechanics;
func OrdRel c
1 -> Relation of
Class (PropRel a1) means :
Def13:
:: QMAX_1:def 13
for b
1, b
2 being
Subset of
(Prop a1) holds
(
[b1,b2] in a
2 iff ( b
1 in Class (PropRel a1) & b
2 in Class (PropRel a1) & ( for b
3, b
4 being
Element of
Prop a
1 holds
( b
3 in b
1 & b
4 in b
2 implies b
3 |- b
4 ) ) ) );
existence
ex b1 being Relation of Class (PropRel c6) st
for b2, b3 being Subset of (Prop c6) holds
( [b2,b3] in b1 iff ( b2 in Class (PropRel c6) & b3 in Class (PropRel c6) & ( for b4, b5 being Element of Prop c6 holds
( b4 in b2 & b5 in b3 implies b4 |- b5 ) ) ) )
uniqueness
for b1, b2 being Relation of Class (PropRel c6) holds
( ( for b3, b4 being Subset of (Prop c6) holds
( [b3,b4] in b1 iff ( b3 in Class (PropRel c6) & b4 in Class (PropRel c6) & ( for b5, b6 being Element of Prop c6 holds
( b5 in b3 & b6 in b4 implies b5 |- b6 ) ) ) ) ) & ( for b3, b4 being Subset of (Prop c6) holds
( [b3,b4] in b2 iff ( b3 in Class (PropRel c6) & b4 in Class (PropRel c6) & ( for b5, b6 being Element of Prop c6 holds
( b5 in b3 & b6 in b4 implies b5 |- b6 ) ) ) ) ) implies 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 c
6 be
Quantum_Mechanics;
func InvRel c
1 -> Function of
Class (PropRel a1),
Class (PropRel a1) means :
Def14:
:: QMAX_1:def 14
for b
1 being
Element of
Prop a
1 holds a
2 . (Class (PropRel a1),b1) = Class (PropRel a1),
('not' b1);
existence
ex b1 being Function of Class (PropRel c6), Class (PropRel c6) st
for b2 being Element of Prop c6 holds b1 . (Class (PropRel c6),b2) = Class (PropRel c6),('not' b2)
uniqueness
for b1, b2 being Function of Class (PropRel c6), Class (PropRel c6) holds
( ( for b3 being Element of Prop c6 holds b1 . (Class (PropRel c6),b3) = Class (PropRel c6),('not' b3) ) & ( for b3 being Element of Prop c6 holds b2 . (Class (PropRel c6),b3) = Class (PropRel c6),('not' b3) ) implies b1 = b2 )
end;
:: deftheorem Def14 defines InvRel QMAX_1:def 14 :
theorem Th35: :: QMAX_1:35
canceled;
theorem Th36: :: QMAX_1:36