:: CQC_LANG semantic presentation
:: deftheorem Def1 defines IFEQ CQC_LANG:def 1 :
for b
1, b
2, b
3, b
4 being
set holds
( ( b
1 = b
2 implies
IFEQ b
1,b
2,b
3,b
4 = b
3 ) & ( not b
1 = b
2 implies
IFEQ b
1,b
2,b
3,b
4 = b
4 ) );
:: deftheorem Def2 defines .--> CQC_LANG:def 2 :
theorem Th1: :: CQC_LANG:1
canceled;
theorem Th2: :: CQC_LANG:2
canceled;
theorem Th3: :: CQC_LANG:3
canceled;
theorem Th4: :: CQC_LANG:4
canceled;
theorem Th5: :: CQC_LANG:5
theorem Th6: :: CQC_LANG:6
for b
1, b
2 being
set holds
(b1 .--> b2) . b
1 = b
2
Lemma4:
for b1 being bound_QC-variable holds
not b1 in fixed_QC-variables
theorem Th7: :: CQC_LANG:7
:: deftheorem Def3 defines Subst CQC_LANG:def 3 :
theorem Th8: :: CQC_LANG:8
canceled;
theorem Th9: :: CQC_LANG:9
canceled;
theorem Th10: :: CQC_LANG:10
theorem Th11: :: CQC_LANG:11
:: deftheorem Def4 defines CQC-WFF CQC_LANG:def 4 :
theorem Th12: :: CQC_LANG:12
canceled;
theorem Th13: :: CQC_LANG:13
:: deftheorem Def5 defines CQC-variable_list-like CQC_LANG:def 5 :
theorem Th14: :: CQC_LANG:14
canceled;
theorem Th15: :: CQC_LANG:15
theorem Th16: :: CQC_LANG:16
theorem Th17: :: CQC_LANG:17
theorem Th18: :: CQC_LANG:18
theorem Th19: :: CQC_LANG:19
theorem Th20: :: CQC_LANG:20
theorem Th21: :: CQC_LANG:21
theorem Th22: :: CQC_LANG:22
theorem Th23: :: CQC_LANG:23
theorem Th24: :: CQC_LANG:24
scheme :: CQC_LANG:sch 3
s3{ F
1()
-> non
empty set , F
2()
-> Function of
CQC-WFF ,F
1(), F
3()
-> Function of
CQC-WFF ,F
1(), F
4()
-> Element of F
1(), F
5(
set ,
set ,
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
E20:
( F
2()
. VERUM = F
4() & ( for b
1, b
2 being
Element of
CQC-WFF for b
3 being
bound_QC-variablefor b
4 being
Natfor b
5 being
CQC-variable_list of b
4for b
6 being
QC-pred_symbol of b
4 holds
( F
2()
. (b6 ! b5) = F
5(b
4,b
6,b
5) & F
2()
. ('not' b1) = F
6(
(F2() . b1)) & F
2()
. (b1 '&' b2) = F
7(
(F2() . b1),
(F2() . b2)) & F
2()
. (All b3,b1) = F
8(b
3,
(F2() . b1)) ) ) )
and
E21:
( F
3()
. VERUM = F
4() & ( for b
1, b
2 being
Element of
CQC-WFF for b
3 being
bound_QC-variablefor b
4 being
Natfor b
5 being
CQC-variable_list of b
4for b
6 being
QC-pred_symbol of b
4 holds
( F
3()
. (b6 ! b5) = F
5(b
4,b
6,b
5) & F
3()
. ('not' b1) = F
6(
(F3() . b1)) & F
3()
. (b1 '&' b2) = F
7(
(F3() . b1),
(F3() . b2)) & F
3()
. (All b3,b1) = F
8(b
3,
(F3() . b1)) ) ) )
scheme :: CQC_LANG:sch 4
s4{ F
1()
-> non
empty set , F
2()
-> Element of
CQC-WFF , F
3()
-> Element of F
1(), F
4(
set ,
set ,
set )
-> Element of F
1(), F
5(
set )
-> Element of F
1(), F
6(
set ,
set )
-> Element of F
1(), F
7(
set ,
set )
-> Element of F
1() } :
( ex b
1 being
Element of F
1()ex b
2 being
Function of
CQC-WFF ,F
1() st
( b
1 = b
2 . F
2() & b
2 . VERUM = F
3() & ( for b
3, b
4 being
Element of
CQC-WFF for b
5 being
bound_QC-variablefor b
6 being
Natfor b
7 being
CQC-variable_list of b
6for b
8 being
QC-pred_symbol of b
6 holds
( b
2 . (b8 ! b7) = F
4(b
6,b
8,b
7) & b
2 . ('not' b3) = F
5(
(b2 . b3)) & b
2 . (b3 '&' b4) = F
6(
(b2 . b3),
(b2 . b4)) & b
2 . (All b5,b3) = F
7(b
5,
(b2 . b3)) ) ) ) & ( for b
1, b
2 being
Element of F
1() holds
( ex b
3 being
Function of
CQC-WFF ,F
1() st
( b
1 = b
3 . F
2() & b
3 . VERUM = F
3() & ( for b
4, b
5 being
Element of
CQC-WFF for b
6 being
bound_QC-variablefor b
7 being
Natfor b
8 being
CQC-variable_list of b
7for b
9 being
QC-pred_symbol of b
7 holds
( b
3 . (b9 ! b8) = F
4(b
7,b
9,b
8) & b
3 . ('not' b4) = F
5(
(b3 . b4)) & b
3 . (b4 '&' b5) = F
6(
(b3 . b4),
(b3 . b5)) & b
3 . (All b6,b4) = F
7(b
6,
(b3 . b4)) ) ) ) & ex b
3 being
Function of
CQC-WFF ,F
1() st
( b
2 = b
3 . F
2() & b
3 . VERUM = F
3() & ( for b
4, b
5 being
Element of
CQC-WFF for b
6 being
bound_QC-variablefor b
7 being
Natfor b
8 being
CQC-variable_list of b
7for b
9 being
QC-pred_symbol of b
7 holds
( b
3 . (b9 ! b8) = F
4(b
7,b
9,b
8) & b
3 . ('not' b4) = F
5(
(b3 . b4)) & b
3 . (b4 '&' b5) = F
6(
(b3 . b4),
(b3 . b5)) & b
3 . (All b6,b4) = F
7(b
6,
(b3 . b4)) ) ) ) implies b
1 = b
2 ) ) )
scheme :: CQC_LANG:sch 5
s5{ F
1()
-> non
empty set , F
2(
set )
-> Element of F
1(), F
3()
-> Element of F
1(), F
4(
set ,
set ,
set )
-> Element of F
1(), F
5(
set )
-> Element of F
1(), F
6(
set ,
set )
-> Element of F
1(), F
7(
set ,
set )
-> Element of F
1() } :
provided
scheme :: CQC_LANG:sch 6
s6{ F
1()
-> non
empty set , F
2()
-> Element of F
1(), F
3(
set )
-> Element of F
1(), F
4(
set ,
set ,
set )
-> Element of F
1(), F
5()
-> Nat, F
6()
-> QC-pred_symbol of F
5(), F
7()
-> CQC-variable_list of F
5(), F
8(
set )
-> Element of F
1(), F
9(
set ,
set )
-> Element of F
1(), F
10(
set ,
set )
-> Element of F
1() } :
F
3(
(F6() ! F7()))
= F
4(F
5(),F
6(),F
7())
provided
scheme :: CQC_LANG:sch 7
s7{ F
1()
-> non
empty set , F
2(
set )
-> Element of F
1(), F
3()
-> Element of F
1(), F
4(
set ,
set ,
set )
-> Element of F
1(), F
5(
set )
-> Element of F
1(), F
6()
-> Element of
CQC-WFF , F
7(
set ,
set )
-> Element of F
1(), F
8(
set ,
set )
-> Element of F
1() } :
F
2(
('not' F6()))
= F
5(F
2(F
6()))
provided
scheme :: CQC_LANG:sch 8
s8{ F
1()
-> non
empty set , F
2(
set )
-> Element of F
1(), F
3()
-> Element of F
1(), F
4(
set ,
set ,
set )
-> Element of F
1(), F
5(
set )
-> Element of F
1(), F
6(
set ,
set )
-> Element of F
1(), F
7()
-> Element of
CQC-WFF , F
8()
-> Element of
CQC-WFF , F
9(
set ,
set )
-> Element of F
1() } :
F
2(
(F7() '&' F8()))
= F
6(F
2(F
7()),F
2(F
8()))
provided
scheme :: CQC_LANG:sch 9
s9{ F
1()
-> non
empty set , F
2(
set )
-> Element of F
1(), F
3()
-> Element of F
1(), F
4(
set ,
set ,
set )
-> Element of F
1(), F
5(
set )
-> Element of F
1(), F
6(
set ,
set )
-> Element of F
1(), F
7(
set ,
set )
-> Element of F
1(), F
8()
-> bound_QC-variable, F
9()
-> Element of
CQC-WFF } :
F
2(
(All F8(),F9()))
= F
7(F
8(),F
2(F
9()))
provided
Lemma20:
for b1 being bound_QC-variable
for b2, b3 being Function of QC-WFF , QC-WFF holds
( ( for b4 being Element of QC-WFF holds
( b2 . VERUM = VERUM & ( b4 is atomic implies b2 . b4 = (the_pred_symbol_of b4) ! (Subst (the_arguments_of b4),((a. 0) .--> b1)) ) & ( b4 is negative implies b2 . b4 = 'not' (b2 . (the_argument_of b4)) ) & ( b4 is conjunctive implies b2 . b4 = (b2 . (the_left_argument_of b4)) '&' (b2 . (the_right_argument_of b4)) ) & ( b4 is universal implies b2 . b4 = IFEQ (bound_in b4),b1,b4,(All (bound_in b4),(b2 . (the_scope_of b4))) ) ) ) & ( for b4 being Element of QC-WFF holds
( b3 . VERUM = VERUM & ( b4 is atomic implies b3 . b4 = (the_pred_symbol_of b4) ! (Subst (the_arguments_of b4),((a. 0) .--> b1)) ) & ( b4 is negative implies b3 . b4 = 'not' (b3 . (the_argument_of b4)) ) & ( b4 is conjunctive implies b3 . b4 = (b3 . (the_left_argument_of b4)) '&' (b3 . (the_right_argument_of b4)) ) & ( b4 is universal implies b3 . b4 = IFEQ (bound_in b4),b1,b4,(All (bound_in b4),(b3 . (the_scope_of b4))) ) ) ) implies b2 = b3 )
definition
let c
1 be
Element of
QC-WFF ;
let c
2 be
bound_QC-variable;
func c
1 . c
2 -> Element of
QC-WFF means :
Def6:
:: CQC_LANG:def 6
ex b
1 being
Function of
QC-WFF ,
QC-WFF st
( a
3 = b
1 . a
1 & ( for b
2 being
Element of
QC-WFF holds
( b
1 . VERUM = VERUM & ( b
2 is
atomic implies b
1 . b
2 = (the_pred_symbol_of b2) ! (Subst (the_arguments_of b2),((a. 0) .--> a2)) ) & ( b
2 is
negative implies b
1 . b
2 = 'not' (b1 . (the_argument_of b2)) ) & ( b
2 is
conjunctive implies b
1 . b
2 = (b1 . (the_left_argument_of b2)) '&' (b1 . (the_right_argument_of b2)) ) & ( b
2 is
universal implies b
1 . b
2 = IFEQ (bound_in b2),a
2,b
2,
(All (bound_in b2),(b1 . (the_scope_of b2))) ) ) ) );
existence
ex b1 being Element of QC-WFF ex b2 being Function of QC-WFF , QC-WFF st
( b1 = b2 . c1 & ( for b3 being Element of QC-WFF holds
( b2 . VERUM = VERUM & ( b3 is atomic implies b2 . b3 = (the_pred_symbol_of b3) ! (Subst (the_arguments_of b3),((a. 0) .--> c2)) ) & ( b3 is negative implies b2 . b3 = 'not' (b2 . (the_argument_of b3)) ) & ( b3 is conjunctive implies b2 . b3 = (b2 . (the_left_argument_of b3)) '&' (b2 . (the_right_argument_of b3)) ) & ( b3 is universal implies b2 . b3 = IFEQ (bound_in b3),c2,b3,(All (bound_in b3),(b2 . (the_scope_of b3))) ) ) ) )
uniqueness
for b1, b2 being Element of QC-WFF holds
( ex b3 being Function of QC-WFF , QC-WFF st
( b1 = b3 . c1 & ( for b4 being Element of QC-WFF holds
( b3 . VERUM = VERUM & ( b4 is atomic implies b3 . b4 = (the_pred_symbol_of b4) ! (Subst (the_arguments_of b4),((a. 0) .--> c2)) ) & ( b4 is negative implies b3 . b4 = 'not' (b3 . (the_argument_of b4)) ) & ( b4 is conjunctive implies b3 . b4 = (b3 . (the_left_argument_of b4)) '&' (b3 . (the_right_argument_of b4)) ) & ( b4 is universal implies b3 . b4 = IFEQ (bound_in b4),c2,b4,(All (bound_in b4),(b3 . (the_scope_of b4))) ) ) ) ) & ex b3 being Function of QC-WFF , QC-WFF st
( b2 = b3 . c1 & ( for b4 being Element of QC-WFF holds
( b3 . VERUM = VERUM & ( b4 is atomic implies b3 . b4 = (the_pred_symbol_of b4) ! (Subst (the_arguments_of b4),((a. 0) .--> c2)) ) & ( b4 is negative implies b3 . b4 = 'not' (b3 . (the_argument_of b4)) ) & ( b4 is conjunctive implies b3 . b4 = (b3 . (the_left_argument_of b4)) '&' (b3 . (the_right_argument_of b4)) ) & ( b4 is universal implies b3 . b4 = IFEQ (bound_in b4),c2,b4,(All (bound_in b4),(b3 . (the_scope_of b4))) ) ) ) ) implies b1 = b2 )
by Lemma20;
end;
:: deftheorem Def6 defines . CQC_LANG:def 6 :
theorem Th25: :: CQC_LANG:25
canceled;
theorem Th26: :: CQC_LANG:26
canceled;
theorem Th27: :: CQC_LANG:27
canceled;
theorem Th28: :: CQC_LANG:28
theorem Th29: :: CQC_LANG:29
theorem Th30: :: CQC_LANG:30
theorem Th31: :: CQC_LANG:31
theorem Th32: :: CQC_LANG:32
theorem Th33: :: CQC_LANG:33
theorem Th34: :: CQC_LANG:34
Lemma29:
for b1 being bound_QC-variable
for b2 being Element of QC-WFF holds
( b2 is universal implies b2 . b1 = IFEQ (bound_in b2),b1,b2,(All (bound_in b2),((the_scope_of b2) . b1)) )
theorem Th35: :: CQC_LANG:35
theorem Th36: :: CQC_LANG:36
theorem Th37: :: CQC_LANG:37
theorem Th38: :: CQC_LANG:38
theorem Th39: :: CQC_LANG:39
theorem Th40: :: CQC_LANG:40
theorem Th41: :: CQC_LANG:41
theorem Th42: :: CQC_LANG:42
theorem Th43: :: CQC_LANG:43
for b
1, b
2, b
3 being
set holds
(b1,b2 :-> b3) . b
1,b
2 = b
3
theorem Th44: :: CQC_LANG:44
for b
1, b
2, b
3 being
set holds b
1,b
1 --> b
2,b
3 = b
1 .--> b
3
theorem Th45: :: CQC_LANG:45
theorem Th46: :: CQC_LANG:46
theorem Th47: :: CQC_LANG:47
for b
1 being
Functionfor b
2, b
3, b
4, b
5 being
set holds
( b
2 <> b
3 implies (
(b1 +* (b2,b3 --> b4,b5)) . b
2 = b
4 &
(b1 +* (b2,b3 --> b4,b5)) . b
3 = b
5 ) )