:: QC_LANG3 semantic presentation
scheme :: QC_LANG3:sch 1
s1{ F
1()
-> non
empty set , F
2()
-> Function of
QC-WFF ,F
1(), F
3()
-> Function of
QC-WFF ,F
1(), F
4()
-> Element of F
1(), F
5(
set )
-> Element of F
1(), F
6(
set )
-> Element of F
1(), F
7(
set ,
set )
-> Element of F
1(), F
8(
set ,
set )
-> Element of F
1() } :
provided
scheme :: QC_LANG3:sch 2
s2{ F
1()
-> non
empty set , F
2()
-> Element of F
1(), F
3()
-> Element of
QC-WFF , F
4(
Element of
QC-WFF )
-> Element of F
1(), F
5(
Element of F
1())
-> Element of F
1(), F
6(
Element of F
1(),
Element of F
1())
-> Element of F
1(), F
7(
Element of
QC-WFF ,
Element of F
1())
-> Element of F
1() } :
( ex b
1 being
Element of F
1()ex b
2 being
Function of
QC-WFF ,F
1() st
( b
1 = b
2 . F
3() & ( for b
3 being
Element of
QC-WFF for b
4, b
5 being
Element of F
1() holds
( ( b
3 = VERUM implies b
2 . b
3 = F
2() ) & ( b
3 is
atomic implies b
2 . b
3 = F
4(b
3) ) & ( b
3 is
negative & b
4 = b
2 . (the_argument_of b3) implies b
2 . b
3 = F
5(b
4) ) & ( b
3 is
conjunctive & b
4 = b
2 . (the_left_argument_of b3) & b
5 = b
2 . (the_right_argument_of b3) implies b
2 . b
3 = F
6(b
4,b
5) ) & ( b
3 is
universal & b
4 = b
2 . (the_scope_of b3) implies b
2 . b
3 = F
7(b
3,b
4) ) ) ) ) & ( for b
1, b
2 being
Element of F
1() holds
( ex b
3 being
Function of
QC-WFF ,F
1() st
( b
1 = b
3 . F
3() & ( for b
4 being
Element of
QC-WFF for b
5, b
6 being
Element of F
1() holds
( ( b
4 = VERUM implies b
3 . b
4 = F
2() ) & ( b
4 is
atomic implies b
3 . b
4 = F
4(b
4) ) & ( b
4 is
negative & b
5 = b
3 . (the_argument_of b4) implies b
3 . b
4 = F
5(b
5) ) & ( b
4 is
conjunctive & b
5 = b
3 . (the_left_argument_of b4) & b
6 = b
3 . (the_right_argument_of b4) implies b
3 . b
4 = F
6(b
5,b
6) ) & ( b
4 is
universal & b
5 = b
3 . (the_scope_of b4) implies b
3 . b
4 = F
7(b
4,b
5) ) ) ) ) & ex b
3 being
Function of
QC-WFF ,F
1() st
( b
2 = b
3 . F
3() & ( for b
4 being
Element of
QC-WFF for b
5, b
6 being
Element of F
1() holds
( ( b
4 = VERUM implies b
3 . b
4 = F
2() ) & ( b
4 is
atomic implies b
3 . b
4 = F
4(b
4) ) & ( b
4 is
negative & b
5 = b
3 . (the_argument_of b4) implies b
3 . b
4 = F
5(b
5) ) & ( b
4 is
conjunctive & b
5 = b
3 . (the_left_argument_of b4) & b
6 = b
3 . (the_right_argument_of b4) implies b
3 . b
4 = F
6(b
5,b
6) ) & ( b
4 is
universal & b
5 = b
3 . (the_scope_of b4) implies b
3 . b
4 = F
7(b
4,b
5) ) ) ) ) implies b
1 = b
2 ) ) )
theorem Th1: :: QC_LANG3:1
canceled;
theorem Th2: :: QC_LANG3:2
canceled;
theorem Th3: :: QC_LANG3:3
:: deftheorem Def1 QC_LANG3:def 1 :
canceled;
:: deftheorem Def2 defines variables_in QC_LANG3:def 2 :
theorem Th4: :: QC_LANG3:4
canceled;
theorem Th5: :: QC_LANG3:5
canceled;
theorem Th6: :: QC_LANG3:6
deffunc H1( Element of QC-WFF ) -> Element of bool bound_QC-variables = still_not-bound_in a1;
deffunc H2( Element of QC-WFF ) -> Element of bool bound_QC-variables = still_not-bound_in (the_arguments_of a1);
deffunc H3( Subset of bound_QC-variables ) -> Subset of bound_QC-variables = a1;
deffunc H4( Subset of bound_QC-variables , Subset of bound_QC-variables ) -> Element of bool bound_QC-variables = a1 \/ a2;
deffunc H5( Element of QC-WFF , Subset of bound_QC-variables ) -> Element of bool bound_QC-variables = a2 \ {(bound_in a1)};
Lemma1:
for b1 being QC-formula
for b2 being Subset of bound_QC-variables holds
( b2 = H1(b1) iff ex b3 being Function of QC-WFF , bool bound_QC-variables st
( b2 = b3 . b1 & ( for b4 being Element of QC-WFF
for b5, b6 being Subset of bound_QC-variables holds
( ( b4 = VERUM implies b3 . b4 = {} bound_QC-variables ) & ( b4 is atomic implies b3 . b4 = H2(b4) ) & ( b4 is negative & b5 = b3 . (the_argument_of b4) implies b3 . b4 = H3(b5) ) & ( b4 is conjunctive & b5 = b3 . (the_left_argument_of b4) & b6 = b3 . (the_right_argument_of b4) implies b3 . b4 = H4(b5,b6) ) & ( b4 is universal & b5 = b3 . (the_scope_of b4) implies b3 . b4 = H5(b4,b5) ) ) ) ) )
theorem Th7: :: QC_LANG3:7
theorem Th8: :: QC_LANG3:8
theorem Th9: :: QC_LANG3:9
theorem Th10: :: QC_LANG3:10
theorem Th11: :: QC_LANG3:11
theorem Th12: :: QC_LANG3:12
theorem Th13: :: QC_LANG3:13
theorem Th14: :: QC_LANG3:14
theorem Th15: :: QC_LANG3:15
theorem Th16: :: QC_LANG3:16
theorem Th17: :: QC_LANG3:17
theorem Th18: :: QC_LANG3:18
theorem Th19: :: QC_LANG3:19
theorem Th20: :: QC_LANG3:20
theorem Th21: :: QC_LANG3:21
theorem Th22: :: QC_LANG3:22
theorem Th23: :: QC_LANG3:23
theorem Th24: :: QC_LANG3:24
theorem Th25: :: QC_LANG3:25
theorem Th26: :: QC_LANG3:26
theorem Th27: :: QC_LANG3:27
theorem Th28: :: QC_LANG3:28
theorem Th29: :: QC_LANG3:29
theorem Th30: :: QC_LANG3:30
theorem Th31: :: QC_LANG3:31
theorem Th32: :: QC_LANG3:32
theorem Th33: :: QC_LANG3:33
:: deftheorem Def3 defines x. QC_LANG3:def 3 :
theorem Th34: :: QC_LANG3:34
canceled;
theorem Th35: :: QC_LANG3:35
theorem Th36: :: QC_LANG3:36
:: deftheorem Def4 defines a. QC_LANG3:def 4 :
theorem Th37: :: QC_LANG3:37
canceled;
theorem Th38: :: QC_LANG3:38
theorem Th39: :: QC_LANG3:39
theorem Th40: :: QC_LANG3:40
theorem Th41: :: QC_LANG3:41
theorem Th42: :: QC_LANG3:42
definition
let c
1 be non
empty Subset of
QC-variables ;
let c
2 be
Element of
QC-WFF ;
func Vars c
2,c
1 -> Subset of a
1 means :
Def5:
:: QC_LANG3:def 5
ex b
1 being
Function of
QC-WFF ,
bool a
1 st
( a
3 = b
1 . a
2 & ( for b
2 being
Element of
QC-WFF for b
3, b
4 being
Subset of a
1 holds
( ( b
2 = VERUM implies b
1 . b
2 = {} a
1 ) & ( b
2 is
atomic implies b
1 . b
2 = variables_in (the_arguments_of b2),a
1 ) & ( b
2 is
negative & b
3 = b
1 . (the_argument_of b2) implies b
1 . b
2 = b
3 ) & ( b
2 is
conjunctive & b
3 = b
1 . (the_left_argument_of b2) & b
4 = b
1 . (the_right_argument_of b2) implies b
1 . b
2 = b
3 \/ b
4 ) & ( b
2 is
universal & b
3 = b
1 . (the_scope_of b2) implies b
1 . b
2 = b
3 ) ) ) );
correctness
existence
ex b1 being Subset of c1ex b2 being Function of QC-WFF , bool c1 st
( b1 = b2 . c2 & ( for b3 being Element of QC-WFF
for b4, b5 being Subset of c1 holds
( ( b3 = VERUM implies b2 . b3 = {} c1 ) & ( b3 is atomic implies b2 . b3 = variables_in (the_arguments_of b3),c1 ) & ( b3 is negative & b4 = b2 . (the_argument_of b3) implies b2 . b3 = b4 ) & ( b3 is conjunctive & b4 = b2 . (the_left_argument_of b3) & b5 = b2 . (the_right_argument_of b3) implies b2 . b3 = b4 \/ b5 ) & ( b3 is universal & b4 = b2 . (the_scope_of b3) implies b2 . b3 = b4 ) ) ) );
uniqueness
for b1, b2 being Subset of c1 holds
( ex b3 being Function of QC-WFF , bool c1 st
( b1 = b3 . c2 & ( for b4 being Element of QC-WFF
for b5, b6 being Subset of c1 holds
( ( b4 = VERUM implies b3 . b4 = {} c1 ) & ( b4 is atomic implies b3 . b4 = variables_in (the_arguments_of b4),c1 ) & ( b4 is negative & b5 = b3 . (the_argument_of b4) implies b3 . b4 = b5 ) & ( b4 is conjunctive & b5 = b3 . (the_left_argument_of b4) & b6 = b3 . (the_right_argument_of b4) implies b3 . b4 = b5 \/ b6 ) & ( b4 is universal & b5 = b3 . (the_scope_of b4) implies b3 . b4 = b5 ) ) ) ) & ex b3 being Function of QC-WFF , bool c1 st
( b2 = b3 . c2 & ( for b4 being Element of QC-WFF
for b5, b6 being Subset of c1 holds
( ( b4 = VERUM implies b3 . b4 = {} c1 ) & ( b4 is atomic implies b3 . b4 = variables_in (the_arguments_of b4),c1 ) & ( b4 is negative & b5 = b3 . (the_argument_of b4) implies b3 . b4 = b5 ) & ( b4 is conjunctive & b5 = b3 . (the_left_argument_of b4) & b6 = b3 . (the_right_argument_of b4) implies b3 . b4 = b5 \/ b6 ) & ( b4 is universal & b5 = b3 . (the_scope_of b4) implies b3 . b4 = b5 ) ) ) ) implies b1 = b2 );
end;
:: deftheorem Def5 defines Vars QC_LANG3:def 5 :
E22:
now
let c
1 be non
empty Subset of
QC-variables ;
deffunc H
6(
Element of
QC-WFF )
-> Subset of c
1 =
Vars a
1,c
1;
deffunc H
7(
Element of
QC-WFF )
-> Subset of c
1 =
variables_in (the_arguments_of a1),c
1;
deffunc H
8(
Subset of c
1)
-> Subset of c
1 = a
1;
deffunc H
9(
Subset of c
1,
Subset of c
1)
-> Element of
bool c
1 = a
1 \/ a
2;
deffunc H
10(
Element of
QC-WFF ,
Subset of c
1)
-> Subset of c
1 = a
2;
E23:
for b
1 being
Element of
QC-WFF for b
2 being
Subset of c
1 holds
( b
2 = H
6(b
1) iff ex b
3 being
Function of
QC-WFF ,
bool c
1 st
( b
2 = b
3 . b
1 & ( for b
4 being
Element of
QC-WFF for b
5, b
6 being
Subset of c
1 holds
( ( b
4 = VERUM implies b
3 . b
4 = {} c
1 ) & ( b
4 is
atomic implies b
3 . b
4 = H
7(b
4) ) & ( b
4 is
negative & b
5 = b
3 . (the_argument_of b4) implies b
3 . b
4 = H
8(b
5) ) & ( b
4 is
conjunctive & b
5 = b
3 . (the_left_argument_of b4) & b
6 = b
3 . (the_right_argument_of b4) implies b
3 . b
4 = H
9(b
5,b
6) ) & ( b
4 is
universal & b
5 = b
3 . (the_scope_of b4) implies b
3 . b
4 = H
10(b
4,b
5) ) ) ) ) )
by Def5;
thus H
6(
VERUM ) =
{} c
1
from QC_LANG3:sch 3(E23)
.=
{}
;
thus
for b
1 being
Element of
QC-WFF holds
( b
1 is
atomic implies
Vars b
1,c
1 = variables_in (the_arguments_of b1),c
1 )
thus
for b
1 being
Element of
QC-WFF holds
( b
1 is
negative implies
Vars b
1,c
1 = Vars (the_argument_of b1),c
1 )
thus
for b
1 being
Element of
QC-WFF holds
( b
1 is
conjunctive implies
Vars b
1,c
1 = (Vars (the_left_argument_of b1),c1) \/ (Vars (the_right_argument_of b1),c1) )
thus
for b
1 being
Element of
QC-WFF holds
( b
1 is
universal implies
Vars b
1,c
1 = Vars (the_scope_of b1),c
1 )
end;
theorem Th43: :: QC_LANG3:43
canceled;
theorem Th44: :: QC_LANG3:44
canceled;
theorem Th45: :: QC_LANG3:45
canceled;
theorem Th46: :: QC_LANG3:46
theorem Th47: :: QC_LANG3:47
theorem Th48: :: QC_LANG3:48
theorem Th49: :: QC_LANG3:49
theorem Th50: :: QC_LANG3:50
theorem Th51: :: QC_LANG3:51
theorem Th52: :: QC_LANG3:52
theorem Th53: :: QC_LANG3:53
theorem Th54: :: QC_LANG3:54
theorem Th55: :: QC_LANG3:55
theorem Th56: :: QC_LANG3:56
theorem Th57: :: QC_LANG3:57
theorem Th58: :: QC_LANG3:58
theorem Th59: :: QC_LANG3:59
theorem Th60: :: QC_LANG3:60
theorem Th61: :: QC_LANG3:61
theorem Th62: :: QC_LANG3:62
theorem Th63: :: QC_LANG3:63
:: deftheorem Def6 defines Free QC_LANG3:def 6 :
theorem Th64: :: QC_LANG3:64
canceled;
theorem Th65: :: QC_LANG3:65
theorem Th66: :: QC_LANG3:66
theorem Th67: :: QC_LANG3:67
theorem Th68: :: QC_LANG3:68
theorem Th69: :: QC_LANG3:69
theorem Th70: :: QC_LANG3:70
theorem Th71: :: QC_LANG3:71
theorem Th72: :: QC_LANG3:72
theorem Th73: :: QC_LANG3:73
theorem Th74: :: QC_LANG3:74
:: deftheorem Def7 defines Fixed QC_LANG3:def 7 :
theorem Th75: :: QC_LANG3:75
canceled;
theorem Th76: :: QC_LANG3:76
theorem Th77: :: QC_LANG3:77
theorem Th78: :: QC_LANG3:78
theorem Th79: :: QC_LANG3:79
theorem Th80: :: QC_LANG3:80
theorem Th81: :: QC_LANG3:81
theorem Th82: :: QC_LANG3:82
theorem Th83: :: QC_LANG3:83
theorem Th84: :: QC_LANG3:84
theorem Th85: :: QC_LANG3:85