:: SUBLEMMA semantic presentation
theorem Th1: :: SUBLEMMA:1
theorem Th2: :: SUBLEMMA:2
:: deftheorem Def1 defines . SUBLEMMA:def 1 :
:: deftheorem Def2 defines Val_S SUBLEMMA:def 2 :
theorem Th3: :: SUBLEMMA:3
:: deftheorem Def3 defines |= SUBLEMMA:def 3 :
theorem Th4: :: SUBLEMMA:4
theorem Th5: :: SUBLEMMA:5
theorem Th6: :: SUBLEMMA:6
theorem Th7: :: SUBLEMMA:7
theorem Th8: :: SUBLEMMA:8
canceled;
theorem Th9: :: SUBLEMMA:9
theorem Th10: :: SUBLEMMA:10
theorem Th11: :: SUBLEMMA:11
theorem Th12: :: SUBLEMMA:12
theorem Th13: :: SUBLEMMA:13
theorem Th14: :: SUBLEMMA:14
theorem Th15: :: SUBLEMMA:15
theorem Th16: :: SUBLEMMA:16
theorem Th17: :: SUBLEMMA:17
theorem Th18: :: SUBLEMMA:18
theorem Th19: :: SUBLEMMA:19
theorem Th20: :: SUBLEMMA:20
:: deftheorem Def4 defines CQCSub_& SUBLEMMA:def 4 :
theorem Th21: :: SUBLEMMA:21
theorem Th22: :: SUBLEMMA:22
theorem Th23: :: SUBLEMMA:23
theorem Th24: :: SUBLEMMA:24
theorem Th25: :: SUBLEMMA:25
theorem Th26: :: SUBLEMMA:26
for b
1 being non
empty set for b
2 being
interpretation of b
1for b
3, b
4 being
Element of
CQC-Sub-WFF holds
( b
3 `2 = b
4 `2 & ( for b
5 being
Element of
Valuations_in b
1 holds
( b
2,b
5 |= CQC_Sub b
3 iff b
2,b
5 . (Val_S b5,b3) |= b
3 ) ) & ( for b
5 being
Element of
Valuations_in b
1 holds
( b
2,b
5 |= CQC_Sub b
4 iff b
2,b
5 . (Val_S b5,b4) |= b
4 ) ) implies for b
5 being
Element of
Valuations_in b
1 holds
( b
2,b
5 |= CQC_Sub (CQCSub_& b3,b4) iff b
2,b
5 . (Val_S b5,(CQCSub_& b3,b4)) |= CQCSub_& b
3,b
4 ) )
theorem Th27: :: SUBLEMMA:27
:: deftheorem Def5 defines CQC-WFF-like SUBLEMMA:def 5 :
:: deftheorem Def6 defines CQCSub_All SUBLEMMA:def 6 :
theorem Th28: :: SUBLEMMA:28
:: deftheorem Def7 defines CQCSub_the_scope_of SUBLEMMA:def 7 :
:: deftheorem Def8 defines CQCQuant SUBLEMMA:def 8 :
theorem Th29: :: SUBLEMMA:29
theorem Th30: :: SUBLEMMA:30
theorem Th31: :: SUBLEMMA:31
for b
1 being
bound_QC-variablefor b
2 being
Element of
CQC-Sub-WFF for b
3 being
second_Q_comp of
[b2,b1] holds
(
[b2,b1] is
quantifiable implies (
CQCSub_the_scope_of (CQCSub_All [b2,b1],b3) = b
2 &
CQCQuant (CQCSub_All [b2,b1],b3),
(CQC_Sub (CQCSub_the_scope_of (CQCSub_All [b2,b1],b3))) = CQCQuant (CQCSub_All [b2,b1],b3),
(CQC_Sub b2) ) )
theorem Th32: :: SUBLEMMA:32
theorem Th33: :: SUBLEMMA:33
theorem Th34: :: SUBLEMMA:34
theorem Th35: :: SUBLEMMA:35
theorem Th36: :: SUBLEMMA:36
theorem Th37: :: SUBLEMMA:37
theorem Th38: :: SUBLEMMA:38
theorem Th39: :: SUBLEMMA:39
for b
1 being
bound_QC-variablefor b
2 being
Element of
CQC-Sub-WFF for b
3 being
second_Q_comp of
[b2,b1] holds
(
[b2,b1] is
quantifiable & b
1 in rng (RestrictSub b1,(All b1,(b2 `1 )),b3) implies ( not
S_Bound (@ (CQCSub_All [b2,b1],b3)) in rng (RestrictSub b1,(All b1,(b2 `1 )),b3) & not
S_Bound (@ (CQCSub_All [b2,b1],b3)) in Bound_Vars (b2 `1 ) ) )
theorem Th40: :: SUBLEMMA:40
for b
1 being
bound_QC-variablefor b
2 being
Element of
CQC-Sub-WFF for b
3 being
second_Q_comp of
[b2,b1] holds
not (
[b2,b1] is
quantifiable & not b
1 in rng (RestrictSub b1,(All b1,(b2 `1 )),b3) &
S_Bound (@ (CQCSub_All [b2,b1],b3)) in rng (RestrictSub b1,(All b1,(b2 `1 )),b3) )
theorem Th41: :: SUBLEMMA:41
theorem Th42: :: SUBLEMMA:42
theorem Th43: :: SUBLEMMA:43
theorem Th44: :: SUBLEMMA:44
theorem Th45: :: SUBLEMMA:45
theorem Th46: :: SUBLEMMA:46
theorem Th47: :: SUBLEMMA:47
theorem Th48: :: SUBLEMMA:48
:: deftheorem Def9 defines | SUBLEMMA:def 9 :
theorem Th49: :: SUBLEMMA:49
theorem Th50: :: SUBLEMMA:50
theorem Th51: :: SUBLEMMA:51
definition
let c
1 be
Element of
CQC-Sub-WFF ;
let c
2 be
bound_QC-variable;
let c
3 be
second_Q_comp of
[c1,c2];
let c
4 be non
empty set ;
let c
5 be
Element of
Valuations_in c
4;
func NEx_Val c
5,c
1,c
2,c
3 -> Val_Sub of a
4 equals :: SUBLEMMA:def 10
(@ (RestrictSub a2,(All a2,(a1 `1 )),a3)) * a
5;
coherence
(@ (RestrictSub c2,(All c2,(c1 `1 )),c3)) * c5 is Val_Sub of c4
;
end;
:: deftheorem Def10 defines NEx_Val SUBLEMMA:def 10 :
theorem Th52: :: SUBLEMMA:52
for b
1 being
bound_QC-variablefor b
2 being
Element of
CQC-Sub-WFF for b
3 being
second_Q_comp of
[b2,b1] holds
(
[b2,b1] is
quantifiable & b
1 in rng (RestrictSub b1,(All b1,(b2 `1 )),b3) implies
S_Bound (@ (CQCSub_All [b2,b1],b3)) = x. (upVar (RestrictSub b1,(All b1,(b2 `1 )),b3),(b2 `1 )) )
theorem Th53: :: SUBLEMMA:53
theorem Th54: :: SUBLEMMA:54
for b
1 being
bound_QC-variablefor b
2 being non
empty set for b
3 being
Element of
Valuations_in b
2for b
4 being
Element of
CQC-Sub-WFF for b
5 being
second_Q_comp of
[b4,b1] holds
(
[b4,b1] is
quantifiable implies for b
6 being
Element of b
2 holds
(
Val_S (b3 . ((S_Bound (@ (CQCSub_All [b4,b1],b5))) | b6)),b
4 = (NEx_Val (b3 . ((S_Bound (@ (CQCSub_All [b4,b1],b5))) | b6)),b4,b1,b5) +* (b1 | b6) &
dom (RestrictSub b1,(All b1,(b4 `1 )),b5) misses {b1} ) )
theorem Th55: :: SUBLEMMA:55
for b
1 being
bound_QC-variablefor b
2 being non
empty set for b
3 being
interpretation of b
2for b
4 being
Element of
Valuations_in b
2for b
5 being
Element of
CQC-Sub-WFF for b
6 being
second_Q_comp of
[b5,b1] holds
(
[b5,b1] is
quantifiable implies ( ( for b
7 being
Element of b
2 holds b
3,
(b4 . ((S_Bound (@ (CQCSub_All [b5,b1],b6))) | b7)) . (Val_S (b4 . ((S_Bound (@ (CQCSub_All [b5,b1],b6))) | b7)),b5) |= b
5 ) iff for b
7 being
Element of b
2 holds b
3,
(b4 . ((S_Bound (@ (CQCSub_All [b5,b1],b6))) | b7)) . ((NEx_Val (b4 . ((S_Bound (@ (CQCSub_All [b5,b1],b6))) | b7)),b5,b1,b6) +* (b1 | b7)) |= b
5 ) )
theorem Th56: :: SUBLEMMA:56
for b
1 being
bound_QC-variablefor b
2 being non
empty set for b
3 being
Element of
Valuations_in b
2for b
4 being
Element of
CQC-Sub-WFF for b
5 being
second_Q_comp of
[b4,b1] holds
(
[b4,b1] is
quantifiable implies for b
6 being
Element of b
2 holds
NEx_Val (b3 . ((S_Bound (@ (CQCSub_All [b4,b1],b5))) | b6)),b
4,b
1,b
5 = NEx_Val b
3,b
4,b
1,b
5 )
theorem Th57: :: SUBLEMMA:57
for b
1 being
bound_QC-variablefor b
2 being non
empty set for b
3 being
interpretation of b
2for b
4 being
Element of
Valuations_in b
2for b
5 being
Element of
CQC-Sub-WFF for b
6 being
second_Q_comp of
[b5,b1] holds
(
[b5,b1] is
quantifiable implies ( ( for b
7 being
Element of b
2 holds b
3,
(b4 . ((S_Bound (@ (CQCSub_All [b5,b1],b6))) | b7)) . ((NEx_Val (b4 . ((S_Bound (@ (CQCSub_All [b5,b1],b6))) | b7)),b5,b1,b6) +* (b1 | b7)) |= b
5 ) iff for b
7 being
Element of b
2 holds b
3,
(b4 . ((S_Bound (@ (CQCSub_All [b5,b1],b6))) | b7)) . ((NEx_Val b4,b5,b1,b6) +* (b1 | b7)) |= b
5 ) )
theorem Th58: :: SUBLEMMA:58
theorem Th59: :: SUBLEMMA:59
theorem Th60: :: SUBLEMMA:60
theorem Th61: :: SUBLEMMA:61
theorem Th62: :: SUBLEMMA:62
theorem Th63: :: SUBLEMMA:63
for b
1, b
2 being
Element of
CQC-WFF for b
3 being non
empty set for b
4 being
interpretation of b
3 holds
( ( for b
5, b
6 being
Element of
Valuations_in b
3 holds
( b
5 | (still_not-bound_in b1) = b
6 | (still_not-bound_in b1) implies ( b
4,b
5 |= b
1 iff b
4,b
6 |= b
1 ) ) ) & ( for b
5, b
6 being
Element of
Valuations_in b
3 holds
( b
5 | (still_not-bound_in b2) = b
6 | (still_not-bound_in b2) implies ( b
4,b
5 |= b
2 iff b
4,b
6 |= b
2 ) ) ) implies for b
5, b
6 being
Element of
Valuations_in b
3 holds
( b
5 | (still_not-bound_in (b1 '&' b2)) = b
6 | (still_not-bound_in (b1 '&' b2)) implies ( b
4,b
5 |= b
1 '&' b
2 iff b
4,b
6 |= b
1 '&' b
2 ) ) )
theorem Th64: :: SUBLEMMA:64
theorem Th65: :: SUBLEMMA:65
theorem Th66: :: SUBLEMMA:66
theorem Th67: :: SUBLEMMA:67
theorem Th68: :: SUBLEMMA:68
for b
1 being
Element of
CQC-WFF for b
2 being
bound_QC-variablefor b
3 being non
empty set for b
4 being
interpretation of b
3 holds
( ( for b
5, b
6 being
Element of
Valuations_in b
3 holds
( b
5 | (still_not-bound_in b1) = b
6 | (still_not-bound_in b1) implies ( b
4,b
5 |= b
1 iff b
4,b
6 |= b
1 ) ) ) implies for b
5, b
6 being
Element of
Valuations_in b
3 holds
( b
5 | (still_not-bound_in (All b2,b1)) = b
6 | (still_not-bound_in (All b2,b1)) implies ( b
4,b
5 |= All b
2,b
1 iff b
4,b
6 |= All b
2,b
1 ) ) )
theorem Th69: :: SUBLEMMA:69
canceled;
theorem Th70: :: SUBLEMMA:70
theorem Th71: :: SUBLEMMA:71
for b
1 being
bound_QC-variablefor b
2 being non
empty set for b
3 being
Element of
Valuations_in b
2for b
4 being
Element of
CQC-Sub-WFF for b
5 being
second_Q_comp of
[b4,b1]for b
6 being
Element of b
2 holds
(
[b4,b1] is
quantifiable implies
((b3 . ((S_Bound (@ (CQCSub_All [b4,b1],b5))) | b6)) . ((NEx_Val b3,b4,b1,b5) +* (b1 | b6))) | (still_not-bound_in (b4 `1 )) = (b3 . ((NEx_Val b3,b4,b1,b5) +* (b1 | b6))) | (still_not-bound_in (b4 `1 )) )
theorem Th72: :: SUBLEMMA:72
for b
1 being
bound_QC-variablefor b
2 being non
empty set for b
3 being
interpretation of b
2for b
4 being
Element of
Valuations_in b
2for b
5 being
Element of
CQC-Sub-WFF for b
6 being
second_Q_comp of
[b5,b1] holds
(
[b5,b1] is
quantifiable implies ( ( for b
7 being
Element of b
2 holds b
3,
(b4 . ((S_Bound (@ (CQCSub_All [b5,b1],b6))) | b7)) . ((NEx_Val b4,b5,b1,b6) +* (b1 | b7)) |= b
5 ) iff for b
7 being
Element of b
2 holds b
3,b
4 . ((NEx_Val b4,b5,b1,b6) +* (b1 | b7)) |= b
5 ) )
theorem Th73: :: SUBLEMMA:73
theorem Th74: :: SUBLEMMA:74
theorem Th75: :: SUBLEMMA:75
for b
1 being
bound_QC-variablefor b
2 being non
empty set for b
3 being
interpretation of b
2for b
4 being
Element of
Valuations_in b
2for b
5 being
Element of
CQC-Sub-WFF for b
6 being
second_Q_comp of
[b5,b1] holds
(
[b5,b1] is
quantifiable implies ( ( for b
7 being
Element of b
2 holds b
3,b
4 . ((NEx_Val b4,b5,b1,b6) +* (b1 | b7)) |= b
5 ) iff for b
7 being
Element of b
2 holds b
3,
(b4 . (NEx_Val b4,b5,b1,b6)) . (b1 | b7) |= b
5 ) )
theorem Th76: :: SUBLEMMA:76
theorem Th77: :: SUBLEMMA:77
canceled;
theorem Th78: :: SUBLEMMA:78
theorem Th79: :: SUBLEMMA:79
theorem Th80: :: SUBLEMMA:80
theorem Th81: :: SUBLEMMA:81
for b
1, b
2 being
Element of
CQC-WFF for b
3 being non
empty set for b
4 being
interpretation of b
3 holds
( ( for b
5 being
Element of
Valuations_in b
3for b
6, b
7, b
8 being
Val_Sub of b
3 holds
( ( for b
9 being
bound_QC-variable holds
not ( b
9 in dom b
7 & b
9 in still_not-bound_in b
1 ) ) & ( for b
9 being
bound_QC-variable holds
( b
9 in dom b
8 implies b
8 . b
9 = b
5 . b
9 ) ) &
dom b
6 misses dom b
8 implies ( b
4,b
5 . b
6 |= b
1 iff b
4,b
5 . ((b6 +* b7) +* b8) |= b
1 ) ) ) & ( for b
5 being
Element of
Valuations_in b
3for b
6, b
7, b
8 being
Val_Sub of b
3 holds
( ( for b
9 being
bound_QC-variable holds
not ( b
9 in dom b
7 & b
9 in still_not-bound_in b
2 ) ) & ( for b
9 being
bound_QC-variable holds
( b
9 in dom b
8 implies b
8 . b
9 = b
5 . b
9 ) ) &
dom b
6 misses dom b
8 implies ( b
4,b
5 . b
6 |= b
2 iff b
4,b
5 . ((b6 +* b7) +* b8) |= b
2 ) ) ) implies for b
5 being
Element of
Valuations_in b
3for b
6, b
7, b
8 being
Val_Sub of b
3 holds
( ( for b
9 being
bound_QC-variable holds
not ( b
9 in dom b
7 & b
9 in still_not-bound_in (b1 '&' b2) ) ) & ( for b
9 being
bound_QC-variable holds
( b
9 in dom b
8 implies b
8 . b
9 = b
5 . b
9 ) ) &
dom b
6 misses dom b
8 implies ( b
4,b
5 . b
6 |= b
1 '&' b
2 iff b
4,b
5 . ((b6 +* b7) +* b8) |= b
1 '&' b
2 ) ) )
theorem Th82: :: SUBLEMMA:82
theorem Th83: :: SUBLEMMA:83
theorem Th84: :: SUBLEMMA:84
for b
1 being
Element of
CQC-WFF for b
2 being
bound_QC-variablefor b
3 being non
empty set for b
4 being
interpretation of b
3 holds
( ( for b
5 being
Element of
Valuations_in b
3for b
6, b
7, b
8 being
Val_Sub of b
3 holds
( ( for b
9 being
bound_QC-variable holds
not ( b
9 in dom b
7 & b
9 in still_not-bound_in b
1 ) ) & ( for b
9 being
bound_QC-variable holds
( b
9 in dom b
8 implies b
8 . b
9 = b
5 . b
9 ) ) &
dom b
6 misses dom b
8 implies ( b
4,b
5 . b
6 |= b
1 iff b
4,b
5 . ((b6 +* b7) +* b8) |= b
1 ) ) ) implies for b
5 being
Element of
Valuations_in b
3for b
6, b
7, b
8 being
Val_Sub of b
3 holds
( ( for b
9 being
bound_QC-variable holds
not ( b
9 in dom b
7 & b
9 in still_not-bound_in (All b2,b1) ) ) & ( for b
9 being
bound_QC-variable holds
( b
9 in dom b
8 implies b
8 . b
9 = b
5 . b
9 ) ) &
dom b
6 misses dom b
8 implies ( b
4,b
5 . b
6 |= All b
2,b
1 iff b
4,b
5 . ((b6 +* b7) +* b8) |= All b
2,b
1 ) ) )
theorem Th85: :: SUBLEMMA:85
:: deftheorem Def11 defines RSub1 SUBLEMMA:def 11 :
:: deftheorem Def12 defines RSub2 SUBLEMMA:def 12 :
theorem Th86: :: SUBLEMMA:86
theorem Th87: :: SUBLEMMA:87
theorem Th88: :: SUBLEMMA:88
theorem Th89: :: SUBLEMMA:89
for b
1 being
bound_QC-variablefor b
2 being
Element of
CQC-Sub-WFF for b
3 being
second_Q_comp of
[b2,b1] holds
(
[b2,b1] is
quantifiable implies
@ ((CQCSub_All [b2,b1],b3) `2 ) = ((@ (RestrictSub b1,(All b1,(b2 `1 )),b3)) +* ((@ b3) | (RSub1 (All b1,(b2 `1 ))))) +* ((@ b3) | (RSub2 (All b1,(b2 `1 )),b3)) )
theorem Th90: :: SUBLEMMA:90
for b
1 being
bound_QC-variablefor b
2 being non
empty set for b
3 being
Element of
Valuations_in b
2for b
4 being
Element of
CQC-Sub-WFF for b
5 being
second_Q_comp of
[b4,b1] holds
not (
[b4,b1] is
quantifiable & ( for b
6, b
7 being
Val_Sub of b
2 holds
not ( ( for b
8 being
bound_QC-variable holds
not ( b
8 in dom b
6 & b
8 in still_not-bound_in (All b1,(b4 `1 )) ) ) & ( for b
8 being
bound_QC-variable holds
( b
8 in dom b
7 implies b
7 . b
8 = b
3 . b
8 ) ) &
dom (NEx_Val b3,b4,b1,b5) misses dom b
7 & b
3 . (Val_S b3,(CQCSub_All [b4,b1],b5)) = b
3 . (((NEx_Val b3,b4,b1,b5) +* b6) +* b7) ) ) )
theorem Th91: :: SUBLEMMA:91
for b
1 being
bound_QC-variablefor b
2 being non
empty set for b
3 being
interpretation of b
2for b
4 being
Element of
CQC-Sub-WFF for b
5 being
second_Q_comp of
[b4,b1] holds
(
[b4,b1] is
quantifiable implies for b
6 being
Element of
Valuations_in b
2 holds
( b
3,b
6 . (NEx_Val b6,b4,b1,b5) |= All b
1,
(b4 `1 ) iff b
3,b
6 . (Val_S b6,(CQCSub_All [b4,b1],b5)) |= CQCSub_All [b4,b1],b
5 ) )
theorem Th92: :: SUBLEMMA:92
for b
1 being
bound_QC-variablefor b
2 being non
empty set for b
3 being
interpretation of b
2for b
4 being
Element of
CQC-Sub-WFF for b
5 being
second_Q_comp of
[b4,b1] holds
(
[b4,b1] is
quantifiable & ( for b
6 being
Element of
Valuations_in b
2 holds
( b
3,b
6 |= CQC_Sub b
4 iff b
3,b
6 . (Val_S b6,b4) |= b
4 ) ) implies for b
6 being
Element of
Valuations_in b
2 holds
( b
3,b
6 |= CQC_Sub (CQCSub_All [b4,b1],b5) iff b
3,b
6 . (Val_S b6,(CQCSub_All [b4,b1],b5)) |= CQCSub_All [b4,b1],b
5 ) )
theorem Th93: :: SUBLEMMA:93