:: QMAX_1 semantic presentation

definition
let X be non empty set ;
let S be SigmaField of X;
func Probabilities c2 -> set means :Def1: :: QMAX_1:def 1
for x being set holds
( x in it iff x is Probability of S );
existence
ex b1 being set st
for x being set holds
( x in b1 iff x is Probability of S )
proof end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff x is Probability of S ) ) & ( for x being set holds
( x in b2 iff x is Probability of S ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines Probabilities QMAX_1:def 1 :
for X being non empty set
for S being SigmaField of X
for b3 being set holds
( b3 = Probabilities S iff for x being set holds
( x in b3 iff x is Probability of S ) );

registration
let X be non empty set ;
let S be SigmaField of X;
cluster Probabilities a2 -> non empty ;
coherence
not Probabilities S is empty
proof end;
end;

definition
attr a1 is strict;
struct QM_Str -> ;
aggr QM_Str(# Observables, States, Quantum_Probability #) -> QM_Str ;
sel Observables c1 -> non empty set ;
sel States c1 -> non empty set ;
sel Quantum_Probability c1 -> Function of [:the Observables of a1,the States of a1:], Probabilities Borel_Sets ;
end;

definition
let Q be QM_Str ;
func Obs c1 -> set equals :: QMAX_1:def 2
the Observables of Q;
coherence
the Observables of Q is set
;
func Sts c1 -> set equals :: QMAX_1:def 3
the States of Q;
coherence
the States of Q is set
;
end;

:: deftheorem Def2 defines Obs QMAX_1:def 2 :
for Q being QM_Str holds Obs Q = the Observables of Q;

:: deftheorem Def3 defines Sts QMAX_1:def 3 :
for Q being QM_Str holds Sts Q = the States of Q;

registration
let Q be QM_Str ;
cluster Obs a1 -> non empty ;
coherence
not Obs Q is empty
;
cluster Sts a1 -> non empty ;
coherence
not Sts Q is empty
;
end;

definition
let Q be QM_Str ;
let A1 be Element of Obs Q;
let s be Element of Sts Q;
func Meas c2,c3 -> Probability of Borel_Sets equals :: QMAX_1:def 4
the Quantum_Probability of Q . [A1,s];
coherence
the Quantum_Probability of Q . [A1,s] is Probability of Borel_Sets
proof end;
end;

:: deftheorem Def4 defines Meas QMAX_1:def 4 :
for Q being QM_Str
for A1 being Element of Obs Q
for s being Element of Sts Q holds Meas A1,s = the Quantum_Probability of Q . [A1,s];

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 ) )
proof end;

Lemma44: for A being Event of Borel_Sets holds 0 <= P . A
proof end;

Lemma45: P . REAL = 1
proof end;

Lemma47: for A, B being Event of Borel_Sets st A misses B holds
P . (A \/ B) = (P . A) + (P . B)
proof end;

for ASeq being SetSequence of Borel_Sets st ASeq is non-increasing holds
( P * ASeq is convergent & lim (P * ASeq) = P . (Intersection ASeq) )
proof end;

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
proof
let A1 be Element of Obs QM_Str(# X,X,Y #), A2 be Element of Obs QM_Str(# X,X,Y #);
( A1 = 0 & A2 = 0 ) by TARSKI:def 1;
hence ( ( for s being Element of Sts QM_Str(# X,X,Y #) holds Meas A1,s = Meas A2,s ) implies A1 = A2 ) ;
end;
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
proof
let s1 be Element of Sts QM_Str(# X,X,Y #), s2 be Element of Sts QM_Str(# X,X,Y #);
( s1 = 0 & s2 = 0 ) by TARSKI:def 1;
hence ( ( for A being Element of Obs QM_Str(# X,X,Y #) holds Meas A,s1 = Meas A,s2 ) implies s1 = s2 ) ;
end;
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)) ) ) );

registration
cluster strict Quantum_Mechanics-like QM_Str ;
existence
ex b1 being QM_Str st
( b1 is strict & b1 is Quantum_Mechanics-like )
proof end;
end;

definition
mode Quantum_Mechanics is Quantum_Mechanics-like QM_Str ;
end;

definition
let X be set ;
attr a2 is strict;
struct POI_Str of c1 -> ;
aggr POI_Str(# Ordering, Involution #) -> POI_Str of a1;
sel Ordering c2 -> Relation of a1;
sel Involution c2 -> Function of a1,a1;
end;

definition
let X1 be set ;
let Inv be Function of X1,X1;
pred c2 is_an_involution_in c1 means :: QMAX_1:def 6
for x1 being Element of X holds P . (P . x1) = x1;
end;

:: deftheorem Def6 defines is_an_involution_in QMAX_1:def 6 :
for X1 being set
for Inv being Function of X1,X1 holds
( Inv is_an_involution_in X1 iff for x1 being Element of X1 holds Inv . (Inv . x1) = x1 );

definition
let X1 be set ;
let W be POI_Str of X1;
pred c2 is_a_Quantuum_Logic_on c1 means :Def7: :: QMAX_1:def 7
ex Ord being Relation of X ex Inv being Function of X,X st
( P = POI_Str(# Ord,Inv #) & Ord partially_orders X & Inv is_an_involution_in X & ( for x, y being Element of X st [x,y] in Ord holds
[(Inv . y),(Inv . x)] in Ord ) );
end;

:: deftheorem Def7 defines is_a_Quantuum_Logic_on QMAX_1:def 7 :
for X1 being set
for W being POI_Str of X1 holds
( W is_a_Quantuum_Logic_on X1 iff ex Ord being Relation of X1 ex Inv being Function of X1,X1 st
( W = POI_Str(# Ord,Inv #) & Ord partially_orders X1 & Inv is_an_involution_in X1 & ( for x, y being Element of X1 st [x,y] in Ord holds
[(Inv . y),(Inv . x)] in Ord ) ) );

definition
let Q be Quantum_Mechanics;
func Prop c1 -> set equals :: QMAX_1:def 8
[:(Obs X),Borel_Sets :];
coherence
[:(Obs Q),Borel_Sets :] is set
;
end;

:: deftheorem Def8 defines Prop QMAX_1:def 8 :
for Q being Quantum_Mechanics holds Prop Q = [:(Obs Q),Borel_Sets :];

registration
let Q be Quantum_Mechanics;
cluster Prop a1 -> non empty ;
coherence
not Prop Q is empty
;
end;

definition
let Q be Quantum_Mechanics;
let p be Element of Prop Q;
redefine func `1 as c2 `1 -> Element of Obs a1;
coherence
p `1 is Element of Obs Q
by MCART_1:10;
redefine func `2 as c2 `2 -> Event of Borel_Sets ;
coherence
p `2 is Event of Borel_Sets
proof end;
end;

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
for Q being Quantum_Mechanics
for p being Element of Prop Q holds p = [(p `1 ),(p `2 )] by MCART_1:23;

theorem Th15: :: QMAX_1:15
canceled;

theorem Th16: :: QMAX_1:16
for Q being Quantum_Mechanics
for s being Element of Sts Q
for p being Element of Prop Q
for E being Event of Borel_Sets st E = (p `2 ) ` holds
(Meas (p `1 ),s) . (p `2 ) = 1 - ((Meas (p `1 ),s) . E)
proof end;

definition
let Q be Quantum_Mechanics;
let p be Element of Prop Q;
func 'not' c2 -> Element of Prop a1 equals :: QMAX_1:def 9
[(P `1 ),((P `2 ) ` )];
coherence
[(p `1 ),((p `2 ) ` )] is Element of Prop Q
proof end;
end;

:: deftheorem Def9 defines 'not' QMAX_1:def 9 :
for Q being Quantum_Mechanics
for p being Element of Prop Q holds 'not' p = [(p `1 ),((p `2 ) ` )];

definition
let Q be Quantum_Mechanics;
let p be Element of Prop Q;
let q be Element of Prop Q;
pred c2 |- c3 means :Def10: :: QMAX_1:def 10
for s being Element of Sts X holds (Meas (P `1 ),s) . (P `2 ) <= (Meas (P `1 ),s) . (P `2 );
end;

:: deftheorem Def10 defines |- QMAX_1:def 10 :
for Q being Quantum_Mechanics
for p, q being Element of Prop Q holds
( p |- q iff for s being Element of Sts Q holds (Meas (p `1 ),s) . (p `2 ) <= (Meas (q `1 ),s) . (q `2 ) );

definition
let Q be Quantum_Mechanics;
let p be Element of Prop Q;
let q be Element of Prop Q;
pred c2 <==> c3 means :Def11: :: QMAX_1:def 11
( P |- P & P |- P );
end;

:: deftheorem Def11 defines <==> QMAX_1:def 11 :
for Q being Quantum_Mechanics
for p, q being Element of Prop Q holds
( p <==> q iff ( p |- q & q |- p ) );

theorem Th17: :: QMAX_1:17
canceled;

theorem Th18: :: QMAX_1:18
canceled;

theorem Th19: :: QMAX_1:19
canceled;

theorem Th20: :: QMAX_1:20
for Q being Quantum_Mechanics
for p, q being Element of Prop Q holds
( p <==> q iff for s being Element of Sts Q holds (Meas (p `1 ),s) . (p `2 ) = (Meas (q `1 ),s) . (q `2 ) )
proof end;

theorem Th21: :: QMAX_1:21
for Q being Quantum_Mechanics
for p being Element of Prop Q holds p |- p
proof end;

theorem Th22: :: QMAX_1:22
for Q being Quantum_Mechanics
for p, q, r being Element of Prop Q st p |- q & q |- r holds
p |- r
proof end;

theorem Th23: :: QMAX_1:23
for Q being Quantum_Mechanics
for p being Element of Prop Q holds p <==> p
proof end;

theorem Th24: :: QMAX_1:24
for Q being Quantum_Mechanics
for p, q being Element of Prop Q st p <==> q holds
q <==> p
proof end;

theorem Th25: :: QMAX_1:25
for Q being Quantum_Mechanics
for p, q, r being Element of Prop Q st p <==> q & q <==> r holds
p <==> r
proof end;

theorem Th26: :: QMAX_1:26
for Q being Quantum_Mechanics
for p being Element of Prop Q holds
( ('not' p) `1 = p `1 & ('not' p) `2 = (p `2 ) ` ) by MCART_1:7;

theorem Th27: :: QMAX_1:27
for Q being Quantum_Mechanics
for p being Element of Prop Q holds 'not' ('not' p) = p
proof end;

theorem Th28: :: QMAX_1:28
for Q being Quantum_Mechanics
for p, q being Element of Prop Q st p |- q holds
'not' q |- 'not' p
proof end;

definition
let Q be Quantum_Mechanics;
func PropRel c1 -> Equivalence_Relation of Prop a1 means :Def12: :: QMAX_1:def 12
for p, q being Element of Prop X holds
( [p,q] in it iff p <==> q );
existence
ex b1 being Equivalence_Relation of Prop Q st
for p, q being Element of Prop Q holds
( [p,q] in b1 iff p <==> q )
proof end;
uniqueness
for b1, b2 being Equivalence_Relation of Prop Q st ( for p, q being Element of Prop Q holds
( [p,q] in b1 iff p <==> q ) ) & ( for p, q being Element of Prop Q holds
( [p,q] in b2 iff p <==> q ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines PropRel QMAX_1:def 12 :
for Q being Quantum_Mechanics
for b2 being Equivalence_Relation of Prop Q holds
( b2 = PropRel Q iff for p, q being Element of Prop Q holds
( [p,q] in b2 iff p <==> q ) );

theorem Th29: :: QMAX_1:29
canceled;

theorem Th30: :: QMAX_1:30
for Q being Quantum_Mechanics
for B, C being Subset of (Prop Q) st B in Class (PropRel Q) & C in Class (PropRel Q) holds
for a, b, c, d being Element of Prop Q st a in B & b in B & c in C & d in C & a |- c holds
b |- d
proof end;

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 ) ) )
proof end;
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
proof end;
end;

:: deftheorem Def13 defines OrdRel QMAX_1:def 13 :
for Q being Quantum_Mechanics
for b2 being Relation of Class (PropRel Q) holds
( b2 = OrdRel Q iff 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 ) ) ) );

theorem Th31: :: QMAX_1:31
canceled;

theorem Th32: :: QMAX_1:32
for Q being Quantum_Mechanics
for p, q being Element of Prop Q holds
( p |- q iff [(Class (PropRel Q),p),(Class (PropRel Q),q)] in OrdRel Q )
proof end;

theorem Th33: :: QMAX_1:33
for Q being Quantum_Mechanics
for B, C being Subset of (Prop Q) st B in Class (PropRel Q) & C in Class (PropRel Q) holds
for p1, q1 being Element of Prop Q st p1 in B & q1 in B & 'not' p1 in C holds
'not' q1 in C
proof end;

theorem Th34: :: QMAX_1:34
for Q being Quantum_Mechanics
for B, C being Subset of (Prop Q) st B in Class (PropRel Q) & C in Class (PropRel Q) holds
for p, q being Element of Prop Q st 'not' p in C & 'not' q in C & p in B holds
q in B
proof end;

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)
proof end;
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
proof end;
end;

:: deftheorem Def14 defines InvRel QMAX_1:def 14 :
for Q being Quantum_Mechanics
for b2 being Function of Class (PropRel Q), Class (PropRel Q) holds
( b2 = InvRel Q iff for p being Element of Prop Q holds b2 . (Class (PropRel Q),p) = Class (PropRel Q),('not' p) );

theorem Th35: :: QMAX_1:35
canceled;

theorem Th36: :: QMAX_1:36
for Q being Quantum_Mechanics holds POI_Str(# (OrdRel Q),(InvRel Q) #) is_a_Quantuum_Logic_on Class (PropRel Q)
proof end;