:: Connectives and Subformulae of the First Order Language
:: by Grzegorz Bancerek
::
:: Received November 23, 1989
:: Copyright (c) 1990-2011 Association of Mizar Users

begin

theorem :: QC_LANG2:1

canceled;

theorem Th2: :: QC_LANG2:2

for p being Element of QC-WFF holds the_argument_of ('not' p) = p

proof end;

theorem Th3: :: QC_LANG2:3

for p, q, p1, q1 being Element of QC-WFF st p '&' q = p1 '&' q1 holds
( p = p1 & q = q1 )

proof end;

theorem Th4: :: QC_LANG2:4

for p being Element of QC-WFF st p is conjunctive holds
p = (the_left_argument_of p) '&' (the_right_argument_of p)

proof end;

theorem Th5: :: QC_LANG2:5

for p, q being Element of QC-WFF holds
( the_left_argument_of (p '&' q) = p & the_right_argument_of (p '&' q) = q )

proof end;

theorem Th6: :: QC_LANG2:6

for x, y being bound_QC-variable
for p, q being Element of QC-WFF st All (x,p) = All (y,q) holds
( x = y & p = q )

proof end;

theorem Th7: :: QC_LANG2:7

for p being Element of QC-WFF st p is universal holds
p = All ((bound_in p),(the_scope_of p))

proof end;

theorem Th8: :: QC_LANG2:8

for x being bound_QC-variable
for p being Element of QC-WFF holds
( bound_in (All (x,p)) = x & the_scope_of (All (x,p)) = p )

proof end;

definition

func FALSUM -> QC-formula equals :: QC_LANG2:def 1
'not' VERUM;
correctness
coherence
'not' VERUM is QC-formula;
;
let p, q be Element of QC-WFF ;
func p => q -> QC-formula equals :: QC_LANG2:def 2
'not' (p '&' ('not' q));
correctness
coherence
'not' (p '&' ('not' q)) is QC-formula;
;
func p 'or' q -> QC-formula equals :: QC_LANG2:def 3
'not' (('not' p) '&' ('not' q));
correctness
coherence
'not' (('not' p) '&' ('not' q)) is QC-formula;
;

end;

:: deftheorem defines FALSUM QC_LANG2:def 1 :
FALSUM = 'not' VERUM;

:: deftheorem defines => QC_LANG2:def 2 :
for p, q being Element of QC-WFF holds p => q = 'not' (p '&' ('not' q));

:: deftheorem defines 'or' QC_LANG2:def 3 :
for p, q being Element of QC-WFF holds p 'or' q = 'not' (('not' p) '&' ('not' q));

definition

let p, q be Element of QC-WFF ;
func p <=> q -> QC-formula equals :: QC_LANG2:def 4
(p => q) '&' (q => p);
correctness
coherence
(p => q) '&' (q => p) is QC-formula;
;

end;

:: deftheorem defines <=> QC_LANG2:def 4 :
for p, q being Element of QC-WFF holds p <=> q = (p => q) '&' (q => p);

definition

let x be bound_QC-variable;
let p be Element of QC-WFF ;
func Ex (x,p) -> QC-formula equals :: QC_LANG2:def 5
'not' (All (x,('not' p)));
correctness
coherence
'not' (All (x,('not' p))) is QC-formula;
;

end;

:: deftheorem defines Ex QC_LANG2:def 5 :
for x being bound_QC-variable
for p being Element of QC-WFF holds Ex (x,p) = 'not' (All (x,('not' p)));

theorem :: QC_LANG2:9

canceled;

theorem :: QC_LANG2:10

canceled;

theorem :: QC_LANG2:11

canceled;

theorem :: QC_LANG2:12

canceled;

theorem :: QC_LANG2:13

( FALSUM is negative & the_argument_of FALSUM = VERUM ) by Th2, QC_LANG1:def 18;

theorem :: QC_LANG2:14

for p, q being Element of QC-WFF holds p 'or' q = ('not' p) => q ;

theorem :: QC_LANG2:15

canceled;

theorem :: QC_LANG2:16

for p, q, p1, q1 being Element of QC-WFF st p 'or' q = p1 'or' q1 holds
( p = p1 & q = q1 )

proof end;

theorem Th17: :: QC_LANG2:17

for p, q, p1, q1 being Element of QC-WFF st p => q = p1 => q1 holds
( p = p1 & q = q1 )

proof end;

theorem :: QC_LANG2:18

for p, q, p1, q1 being Element of QC-WFF st p <=> q = p1 <=> q1 holds
( p = p1 & q = q1 )

proof end;

theorem Th19: :: QC_LANG2:19

for x, y being bound_QC-variable
for p, q being Element of QC-WFF st Ex (x,p) = Ex (y,q) holds
( x = y & p = q )

proof end;

definition

let x, y be bound_QC-variable;
let p be Element of QC-WFF ;
func All (x,y,p) -> QC-formula equals :: QC_LANG2:def 6
All (x,(All (y,p)));
correctness
coherence
All (x,(All (y,p))) is QC-formula;
;
func Ex (x,y,p) -> QC-formula equals :: QC_LANG2:def 7
Ex (x,(Ex (y,p)));
correctness
coherence
Ex (x,(Ex (y,p))) is QC-formula;
;

end;

:: deftheorem defines All QC_LANG2:def 6 :
for x, y being bound_QC-variable
for p being Element of QC-WFF holds All (x,y,p) = All (x,(All (y,p)));

:: deftheorem defines Ex QC_LANG2:def 7 :
for x, y being bound_QC-variable
for p being Element of QC-WFF holds Ex (x,y,p) = Ex (x,(Ex (y,p)));

theorem :: QC_LANG2:20

for x, y being bound_QC-variable
for p being Element of QC-WFF holds
( All (x,y,p) = All (x,(All (y,p))) & Ex (x,y,p) = Ex (x,(Ex (y,p))) ) ;

theorem Th21: :: QC_LANG2:21

for p1, p2 being Element of QC-WFF
for x1, x2, y1, y2 being bound_QC-variable st All (x1,y1,p1) = All (x2,y2,p2) holds
( x1 = x2 & y1 = y2 & p1 = p2 )

proof end;

theorem :: QC_LANG2:22

for x, y, z being bound_QC-variable
for p, q being Element of QC-WFF st All (x,y,p) = All (z,q) holds
( x = z & All (y,p) = q ) by Th6;

theorem Th23: :: QC_LANG2:23

for p1, p2 being Element of QC-WFF
for x1, x2, y1, y2 being bound_QC-variable st Ex (x1,y1,p1) = Ex (x2,y2,p2) holds
( x1 = x2 & y1 = y2 & p1 = p2 )

proof end;

theorem :: QC_LANG2:24

for x, y, z being bound_QC-variable
for p, q being Element of QC-WFF st Ex (x,y,p) = Ex (z,q) holds
( x = z & Ex (y,p) = q ) by Th19;

theorem :: QC_LANG2:25

for x, y being bound_QC-variable
for p being Element of QC-WFF holds
( All (x,y,p) is universal & bound_in (All (x,y,p)) = x & the_scope_of (All (x,y,p)) = All (y,p) ) by Th8, QC_LANG1:def 20;

definition

let x, y, z be bound_QC-variable;
let p be Element of QC-WFF ;
func All (x,y,z,p) -> QC-formula equals :: QC_LANG2:def 8
All (x,(All (y,z,p)));
correctness
coherence
All (x,(All (y,z,p))) is QC-formula;
;
func Ex (x,y,z,p) -> QC-formula equals :: QC_LANG2:def 9
Ex (x,(Ex (y,z,p)));
correctness
coherence
Ex (x,(Ex (y,z,p))) is QC-formula;
;

end;

:: deftheorem defines All QC_LANG2:def 8 :
for x, y, z being bound_QC-variable
for p being Element of QC-WFF holds All (x,y,z,p) = All (x,(All (y,z,p)));

:: deftheorem defines Ex QC_LANG2:def 9 :
for x, y, z being bound_QC-variable
for p being Element of QC-WFF holds Ex (x,y,z,p) = Ex (x,(Ex (y,z,p)));

theorem :: QC_LANG2:26

for x, y, z being bound_QC-variable
for p being Element of QC-WFF holds
( All (x,y,z,p) = All (x,(All (y,z,p))) & Ex (x,y,z,p) = Ex (x,(Ex (y,z,p))) ) ;

theorem :: QC_LANG2:27

for p1, p2 being Element of QC-WFF
for x1, x2, y1, y2, z1, z2 being bound_QC-variable st All (x1,y1,z1,p1) = All (x2,y2,z2,p2) holds
( x1 = x2 & y1 = y2 & z1 = z2 & p1 = p2 )

proof end;

theorem :: QC_LANG2:28

for x, y, z being bound_QC-variable
for p, q being Element of QC-WFF
for t being bound_QC-variable st All (x,y,z,p) = All (t,q) holds
( x = t & All (y,z,p) = q ) by Th6;

theorem :: QC_LANG2:29

for x, y, z being bound_QC-variable
for p, q being Element of QC-WFF
for t, s being bound_QC-variable st All (x,y,z,p) = All (t,s,q) holds
( x = t & y = s & All (z,p) = q )

proof end;

theorem :: QC_LANG2:30

for p1, p2 being Element of QC-WFF
for x1, x2, y1, y2, z1, z2 being bound_QC-variable st Ex (x1,y1,z1,p1) = Ex (x2,y2,z2,p2) holds
( x1 = x2 & y1 = y2 & z1 = z2 & p1 = p2 )

proof end;

theorem :: QC_LANG2:31

for x, y, z being bound_QC-variable
for p, q being Element of QC-WFF
for t being bound_QC-variable st Ex (x,y,z,p) = Ex (t,q) holds
( x = t & Ex (y,z,p) = q ) by Th19;

theorem :: QC_LANG2:32

for x, y, z being bound_QC-variable
for p, q being Element of QC-WFF
for t, s being bound_QC-variable st Ex (x,y,z,p) = Ex (t,s,q) holds
( x = t & y = s & Ex (z,p) = q )

proof end;

theorem :: QC_LANG2:33

for x, y, z being bound_QC-variable
for p being Element of QC-WFF holds
( All (x,y,z,p) is universal & bound_in (All (x,y,z,p)) = x & the_scope_of (All (x,y,z,p)) = All (y,z,p) ) by Th8, QC_LANG1:def 20;

definition

let H be Element of QC-WFF ;
attr H is disjunctive means :: QC_LANG2:def 10
ex p, q being Element of QC-WFF st H = p 'or' q;
attr H is conditional means :Def11: :: QC_LANG2:def 11
ex p, q being Element of QC-WFF st H = p => q;
attr H is biconditional means :: QC_LANG2:def 12
ex p, q being Element of QC-WFF st H = p <=> q;
attr H is existential means :Def13: :: QC_LANG2:def 13
ex x being bound_QC-variable ex p being Element of QC-WFF st H = Ex (x,p);

end;

:: deftheorem defines disjunctive QC_LANG2:def 10 :
for H being Element of QC-WFF holds
( H is disjunctive iff ex p, q being Element of QC-WFF st H = p 'or' q );

:: deftheorem Def11 defines conditional QC_LANG2:def 11 :
for H being Element of QC-WFF holds
( H is conditional iff ex p, q being Element of QC-WFF st H = p => q );

:: deftheorem defines biconditional QC_LANG2:def 12 :
for H being Element of QC-WFF holds
( H is biconditional iff ex p, q being Element of QC-WFF st H = p <=> q );

:: deftheorem Def13 defines existential QC_LANG2:def 13 :
for H being Element of QC-WFF holds
( H is existential iff ex x being bound_QC-variable ex p being Element of QC-WFF st H = Ex (x,p) );

theorem :: QC_LANG2:34

canceled;

theorem :: QC_LANG2:35

canceled;

theorem :: QC_LANG2:36

canceled;

theorem :: QC_LANG2:37

canceled;

theorem :: QC_LANG2:38

for x, y, z being bound_QC-variable
for p being Element of QC-WFF holds
( Ex (x,y,p) is existential & Ex (x,y,z,p) is existential ) by Def13;

definition

let H be Element of QC-WFF ;
func the_left_disjunct_of H -> QC-formula equals :: QC_LANG2:def 14
the_argument_of (the_left_argument_of (the_argument_of H));
correctness
coherence
the_argument_of (the_left_argument_of (the_argument_of H)) is QC-formula;
;
func the_right_disjunct_of H -> QC-formula equals :: QC_LANG2:def 15
the_argument_of (the_right_argument_of (the_argument_of H));
correctness
coherence
the_argument_of (the_right_argument_of (the_argument_of H)) is QC-formula;
;
func the_antecedent_of H -> QC-formula equals :: QC_LANG2:def 16
the_left_argument_of (the_argument_of H);
correctness
coherence
the_left_argument_of (the_argument_of H) is QC-formula;
;

end;

:: deftheorem defines the_left_disjunct_of QC_LANG2:def 14 :
for H being Element of QC-WFF holds the_left_disjunct_of H = the_argument_of (the_left_argument_of (the_argument_of H));

:: deftheorem defines the_right_disjunct_of QC_LANG2:def 15 :
for H being Element of QC-WFF holds the_right_disjunct_of H = the_argument_of (the_right_argument_of (the_argument_of H));

:: deftheorem defines the_antecedent_of QC_LANG2:def 16 :
for H being Element of QC-WFF holds the_antecedent_of H = the_left_argument_of (the_argument_of H);

notation

let H be Element of QC-WFF ;
synonym the_consequent_of H for the_right_disjunct_of H;

end;

definition

let H be Element of QC-WFF ;
canceled;
func the_left_side_of H -> QC-formula equals :: QC_LANG2:def 18
the_antecedent_of (the_left_argument_of H);
correctness
coherence
the_antecedent_of (the_left_argument_of H) is QC-formula;
;
func the_right_side_of H -> QC-formula equals :: QC_LANG2:def 19
the_consequent_of (the_left_argument_of H);
correctness
coherence
the_consequent_of (the_left_argument_of H) is QC-formula;
;

end;

:: deftheorem QC_LANG2:def 17 :
canceled;

:: deftheorem defines the_left_side_of QC_LANG2:def 18 :
for H being Element of QC-WFF holds the_left_side_of H = the_antecedent_of (the_left_argument_of H);

:: deftheorem defines the_right_side_of QC_LANG2:def 19 :
for H being Element of QC-WFF holds the_right_side_of H = the_consequent_of (the_left_argument_of H);

theorem :: QC_LANG2:39

canceled;

theorem :: QC_LANG2:40

canceled;

theorem :: QC_LANG2:41

canceled;

theorem :: QC_LANG2:42

canceled;

theorem :: QC_LANG2:43

canceled;

theorem :: QC_LANG2:44

canceled;

theorem Th45: :: QC_LANG2:45

for F, G being Element of QC-WFF holds
( the_left_disjunct_of (F 'or' G) = F & the_right_disjunct_of (F 'or' G) = G & the_argument_of (F 'or' G) = ('not' F) '&' ('not' G) )

proof end;

theorem Th46: :: QC_LANG2:46

for F, G being Element of QC-WFF holds
( the_antecedent_of (F => G) = F & the_consequent_of (F => G) = G & the_argument_of (F => G) = F '&' ('not' G) )

proof end;

theorem Th47: :: QC_LANG2:47

for F, G being Element of QC-WFF holds
( the_left_side_of (F <=> G) = F & the_right_side_of (F <=> G) = G & the_left_argument_of (F <=> G) = F => G & the_right_argument_of (F <=> G) = G => F )

proof end;

theorem :: QC_LANG2:48

for x being bound_QC-variable
for H being Element of QC-WFF holds the_argument_of (Ex (x,H)) = All (x,('not' H)) by Th2;

theorem :: QC_LANG2:49

for H being Element of QC-WFF st H is disjunctive holds
( H is conditional & H is negative & the_argument_of H is conjunctive & the_left_argument_of (the_argument_of H) is negative & the_right_argument_of (the_argument_of H) is negative )

proof end;

theorem :: QC_LANG2:50

for H being Element of QC-WFF st H is conditional holds
( H is negative & the_argument_of H is conjunctive & the_right_argument_of (the_argument_of H) is negative )

proof end;

theorem :: QC_LANG2:51

for H being Element of QC-WFF st H is biconditional holds
( H is conjunctive & the_left_argument_of H is conditional & the_right_argument_of H is conditional )

proof end;

theorem :: QC_LANG2:52

for H being Element of QC-WFF st H is existential holds
( H is negative & the_argument_of H is universal & the_scope_of (the_argument_of H) is negative )

proof end;

theorem :: QC_LANG2:53

for H being Element of QC-WFF st H is disjunctive holds
H = (the_left_disjunct_of H) 'or' (the_right_disjunct_of H)

proof end;

theorem :: QC_LANG2:54

for H being Element of QC-WFF st H is conditional holds
H = (the_antecedent_of H) => (the_consequent_of H)

proof end;

theorem :: QC_LANG2:55

for H being Element of QC-WFF st H is biconditional holds
H = (the_left_side_of H) <=> (the_right_side_of H)

proof end;

theorem :: QC_LANG2:56

for H being Element of QC-WFF st H is existential holds
H = Ex ((bound_in (the_argument_of H)),(the_argument_of (the_scope_of (the_argument_of H))))

proof end;

definition

let G, H be Element of QC-WFF ;
pred G is_immediate_constituent_of H means :Def20: :: QC_LANG2:def 20
( H = 'not' G or ex F being Element of QC-WFF st
( H = G '&' F or H = F '&' G ) or ex x being bound_QC-variable st H = All (x,G) );

end;

:: deftheorem Def20 defines is_immediate_constituent_of QC_LANG2:def 20 :
for G, H being Element of QC-WFF holds
( G is_immediate_constituent_of H iff ( H = 'not' G or ex F being Element of QC-WFF st
( H = G '&' F or H = F '&' G ) or ex x being bound_QC-variable st H = All (x,G) ) );

theorem :: QC_LANG2:57

canceled;

theorem Th58: :: QC_LANG2:58

for H being Element of QC-WFF holds not H is_immediate_constituent_of VERUM

proof end;

theorem Th59: :: QC_LANG2:59

for H being Element of QC-WFF
for k being Element of NAT
for P being QC-pred_symbol of k
for V being QC-variable_list of k holds not H is_immediate_constituent_of P ! V

proof end;

theorem Th60: :: QC_LANG2:60

for F, H being Element of QC-WFF holds
( F is_immediate_constituent_of 'not' H iff F = H )

proof end;

theorem :: QC_LANG2:61

for H being Element of QC-WFF holds
( H is_immediate_constituent_of FALSUM iff H = VERUM ) by Th60;

theorem Th62: :: QC_LANG2:62

for F, G, H being Element of QC-WFF holds
( F is_immediate_constituent_of G '&' H iff ( F = G or F = H ) )

proof end;

theorem Th63: :: QC_LANG2:63

for F, H being Element of QC-WFF
for x being bound_QC-variable holds
( F is_immediate_constituent_of All (x,H) iff F = H )

proof end;

theorem Th64: :: QC_LANG2:64

for H, F being Element of QC-WFF st H is atomic holds
not F is_immediate_constituent_of H

proof end;

theorem Th65: :: QC_LANG2:65

for H, F being Element of QC-WFF st H is negative holds
( F is_immediate_constituent_of H iff F = the_argument_of H )

proof end;

theorem Th66: :: QC_LANG2:66

for H, F being Element of QC-WFF st H is conjunctive holds
( F is_immediate_constituent_of H iff ( F = the_left_argument_of H or F = the_right_argument_of H ) )

proof end;

theorem Th67: :: QC_LANG2:67

for H, F being Element of QC-WFF st H is universal holds
( F is_immediate_constituent_of H iff F = the_scope_of H )

proof end;

definition

let G, H be Element of QC-WFF ;
pred G is_subformula_of H means :Def21: :: QC_LANG2:def 21
ex n being Element of NAT ex L being FinSequence st
( 1 <= n & len L = n & L . 1 = G & L . n = H & ( for k being Element of NAT st 1 <= k & k < n holds
ex G1, H1 being Element of QC-WFF st
( L . k = G1 & L . (k + 1) = H1 & G1 is_immediate_constituent_of H1 ) ) );
reflexivity
for G being Element of QC-WFF ex n being Element of NAT ex L being FinSequence st
( 1 <= n & len L = n & L . 1 = G & L . n = G & ( for k being Element of NAT st 1 <= k & k < n holds
ex G1, H1 being Element of QC-WFF st
( L . k = G1 & L . (k + 1) = H1 & G1 is_immediate_constituent_of H1 ) ) )

proof end;

end;

:: deftheorem Def21 defines is_subformula_of QC_LANG2:def 21 :
for G, H being Element of QC-WFF holds
( G is_subformula_of H iff ex n being Element of NAT ex L being FinSequence st
( 1 <= n & len L = n & L . 1 = G & L . n = H & ( for k being Element of NAT st 1 <= k & k < n holds
ex G1, H1 being Element of QC-WFF st
( L . k = G1 & L . (k + 1) = H1 & G1 is_immediate_constituent_of H1 ) ) ) );

definition

let H, F be Element of QC-WFF ;
pred H is_proper_subformula_of F means :Def22: :: QC_LANG2:def 22
( H is_subformula_of F & H <> F );
irreflexivity
for H being Element of QC-WFF holds
( not H is_subformula_of H or not H <> H ) ;

end;

:: deftheorem Def22 defines is_proper_subformula_of QC_LANG2:def 22 :
for H, F being Element of QC-WFF holds
( H is_proper_subformula_of F iff ( H is_subformula_of F & H <> F ) );

theorem :: QC_LANG2:68

canceled;

theorem :: QC_LANG2:69

canceled;

theorem :: QC_LANG2:70

canceled;

theorem Th71: :: QC_LANG2:71

for H, F being Element of QC-WFF st H is_immediate_constituent_of F holds
len (@ H) < len (@ F)

proof end;

theorem Th72: :: QC_LANG2:72

for H, F being Element of QC-WFF st H is_immediate_constituent_of F holds
H is_subformula_of F

proof end;

theorem Th73: :: QC_LANG2:73

for H, F being Element of QC-WFF st H is_immediate_constituent_of F holds
H is_proper_subformula_of F

proof end;

theorem Th74: :: QC_LANG2:74

for H, F being Element of QC-WFF st H is_proper_subformula_of F holds
len (@ H) < len (@ F)

proof end;

theorem Th75: :: QC_LANG2:75

for H, F being Element of QC-WFF st H is_proper_subformula_of F holds
ex G being Element of QC-WFF st G is_immediate_constituent_of F

proof end;

theorem Th76: :: QC_LANG2:76

for F, G, H being Element of QC-WFF st F is_proper_subformula_of G & G is_proper_subformula_of H holds
F is_proper_subformula_of H

proof end;

theorem Th77: :: QC_LANG2:77

for F, G, H being Element of QC-WFF st F is_subformula_of G & G is_subformula_of H holds
F is_subformula_of H

proof end;

theorem Th78: :: QC_LANG2:78

for G, H being Element of QC-WFF st G is_subformula_of H & H is_subformula_of G holds
G = H

proof end;

theorem Th79: :: QC_LANG2:79

for G, H being Element of QC-WFF holds
( not G is_proper_subformula_of H or not H is_subformula_of G )

proof end;

theorem :: QC_LANG2:80

for G, H being Element of QC-WFF holds
( not G is_proper_subformula_of H or not H is_proper_subformula_of G )

proof end;

theorem Th81: :: QC_LANG2:81

for G, H being Element of QC-WFF holds
( not G is_subformula_of H or not H is_immediate_constituent_of G ) by Th73, Th79;

theorem Th82: :: QC_LANG2:82

for G, H being Element of QC-WFF holds
( not G is_proper_subformula_of H or not H is_immediate_constituent_of G )

proof end;

theorem Th83: :: QC_LANG2:83

for F, G, H being Element of QC-WFF st ( ( F is_proper_subformula_of G & G is_subformula_of H ) or ( F is_subformula_of G & G is_proper_subformula_of H ) or ( F is_subformula_of G & G is_immediate_constituent_of H ) or ( F is_immediate_constituent_of G & G is_subformula_of H ) or ( F is_proper_subformula_of G & G is_immediate_constituent_of H ) or ( F is_immediate_constituent_of G & G is_proper_subformula_of H ) ) holds
F is_proper_subformula_of H

proof end;

theorem :: QC_LANG2:84

for F being Element of QC-WFF holds not F is_proper_subformula_of VERUM by Th58, Th75;

theorem Th85: :: QC_LANG2:85

for F being Element of QC-WFF
for k being Element of NAT
for P being QC-pred_symbol of k
for V being QC-variable_list of k holds not F is_proper_subformula_of P ! V

proof end;

theorem Th86: :: QC_LANG2:86

for F, H being Element of QC-WFF holds
( F is_subformula_of H iff F is_proper_subformula_of 'not' H )

proof end;

theorem :: QC_LANG2:87

for F, H being Element of QC-WFF st 'not' F is_subformula_of H holds
F is_proper_subformula_of H

proof end;

theorem :: QC_LANG2:88

for F being Element of QC-WFF holds
( F is_proper_subformula_of FALSUM iff F is_subformula_of VERUM ) by Th86;

theorem Th89: :: QC_LANG2:89

for F, G, H being Element of QC-WFF holds
( ( F is_subformula_of G or F is_subformula_of H ) iff F is_proper_subformula_of G '&' H )

proof end;

theorem :: QC_LANG2:90

for F, G, H being Element of QC-WFF st F '&' G is_subformula_of H holds
( F is_proper_subformula_of H & G is_proper_subformula_of H )

proof end;

theorem Th91: :: QC_LANG2:91

for F, H being Element of QC-WFF
for x being bound_QC-variable holds
( F is_subformula_of H iff F is_proper_subformula_of All (x,H) )

proof end;

theorem :: QC_LANG2:92

for H, F being Element of QC-WFF
for x being bound_QC-variable st All (x,H) is_subformula_of F holds
H is_proper_subformula_of F

proof end;

theorem :: QC_LANG2:93

for F, G being Element of QC-WFF holds
( F '&' ('not' G) is_proper_subformula_of F => G & F is_proper_subformula_of F => G & 'not' G is_proper_subformula_of F => G & G is_proper_subformula_of F => G )

proof end;

theorem :: QC_LANG2:94

for F, G being Element of QC-WFF holds
( ('not' F) '&' ('not' G) is_proper_subformula_of F 'or' G & 'not' F is_proper_subformula_of F 'or' G & 'not' G is_proper_subformula_of F 'or' G & F is_proper_subformula_of F 'or' G & G is_proper_subformula_of F 'or' G )

proof end;

theorem :: QC_LANG2:95

for H, F being Element of QC-WFF st H is atomic holds
not F is_proper_subformula_of H

proof end;

theorem :: QC_LANG2:96

for H being Element of QC-WFF st H is negative holds
the_argument_of H is_proper_subformula_of H

proof end;

theorem :: QC_LANG2:97

for H being Element of QC-WFF st H is conjunctive holds
( the_left_argument_of H is_proper_subformula_of H & the_right_argument_of H is_proper_subformula_of H )

proof end;

theorem :: QC_LANG2:98

for H being Element of QC-WFF st H is universal holds
the_scope_of H is_proper_subformula_of H

proof end;

theorem Th99: :: QC_LANG2:99

for H being Element of QC-WFF holds
( H is_subformula_of VERUM iff H = VERUM )

proof end;

theorem Th100: :: QC_LANG2:100

for H being Element of QC-WFF
for k being Element of NAT
for P being QC-pred_symbol of k
for V being QC-variable_list of k holds
( H is_subformula_of P ! V iff H = P ! V )

proof end;

theorem Th101: :: QC_LANG2:101

for H being Element of QC-WFF holds
( H is_subformula_of FALSUM iff ( H = FALSUM or H = VERUM ) )

proof end;

definition

let H be Element of QC-WFF ;
func Subformulae H -> set means :Def23: :: QC_LANG2:def 23
for a being set holds
( a in it iff ex F being Element of QC-WFF st
( F = a & F is_subformula_of H ) );
existence
ex b₁ being set st
for a being set holds
( a in b₁ iff ex F being Element of QC-WFF st
( F = a & F is_subformula_of H ) )

proof end;

uniqueness
for b₁, b₂ being set st ( for a being set holds
( a in b₁ iff ex F being Element of QC-WFF st
( F = a & F is_subformula_of H ) ) ) & ( for a being set holds
( a in b₂ iff ex F being Element of QC-WFF st
( F = a & F is_subformula_of H ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def23 defines Subformulae QC_LANG2:def 23 :
for H being Element of QC-WFF
for b₂ being set holds
( b₂ = Subformulae H iff for a being set holds
( a in b₂ iff ex F being Element of QC-WFF st
( F = a & F is_subformula_of H ) ) );

theorem :: QC_LANG2:102

canceled;

theorem Th103: :: QC_LANG2:103

for G, H being Element of QC-WFF st G in Subformulae H holds
G is_subformula_of H

proof end;

theorem Th104: :: QC_LANG2:104

for F, H being Element of QC-WFF st F is_subformula_of H holds
Subformulae F c= Subformulae H

proof end;

theorem :: QC_LANG2:105

for G, H being Element of QC-WFF st G in Subformulae H holds
Subformulae G c= Subformulae H by Th103, Th104;

theorem :: QC_LANG2:106

canceled;

theorem Th107: :: QC_LANG2:107

Subformulae VERUM = {VERUM}

proof end;

theorem Th108: :: QC_LANG2:108

for k being Element of NAT
for P being QC-pred_symbol of k
for V being QC-variable_list of k holds Subformulae (P ! V) = {(P ! V)}

proof end;

theorem :: QC_LANG2:109

Subformulae FALSUM = {VERUM,FALSUM}

proof end;

theorem Th110: :: QC_LANG2:110

for H being Element of QC-WFF holds Subformulae ('not' H) = (Subformulae H) \/ {('not' H)}

proof end;

theorem Th111: :: QC_LANG2:111

for H, F being Element of QC-WFF holds Subformulae (H '&' F) = ((Subformulae H) \/ (Subformulae F)) \/ {(H '&' F)}

proof end;

theorem Th112: :: QC_LANG2:112

for H being Element of QC-WFF
for x being bound_QC-variable holds Subformulae (All (x,H)) = (Subformulae H) \/ {(All (x,H))}

proof end;

theorem Th113: :: QC_LANG2:113

for F, G being Element of QC-WFF holds Subformulae (F => G) = ((Subformulae F) \/ (Subformulae G)) \/ {('not' G),(F '&' ('not' G)),(F => G)}

proof end;

theorem :: QC_LANG2:114

for F, G being Element of QC-WFF holds Subformulae (F 'or' G) = ((Subformulae F) \/ (Subformulae G)) \/ {('not' G),('not' F),(('not' F) '&' ('not' G)),(F 'or' G)}

proof end;

theorem :: QC_LANG2:115

for F, G being Element of QC-WFF holds Subformulae (F <=> G) = ((Subformulae F) \/ (Subformulae G)) \/ {('not' G),(F '&' ('not' G)),(F => G),('not' F),(G '&' ('not' F)),(G => F),(F <=> G)}

proof end;

theorem :: QC_LANG2:116

for H being Element of QC-WFF holds
( ( H = VERUM or H is atomic ) iff Subformulae H = {H} )

proof end;

theorem :: QC_LANG2:117

for H being Element of QC-WFF st H is negative holds
Subformulae H = (Subformulae (the_argument_of H)) \/ {H}

proof end;

theorem :: QC_LANG2:118

for H being Element of QC-WFF st H is conjunctive holds
Subformulae H = ((Subformulae (the_left_argument_of H)) \/ (Subformulae (the_right_argument_of H))) \/ {H}

proof end;

theorem :: QC_LANG2:119

for H being Element of QC-WFF st H is universal holds
Subformulae H = (Subformulae (the_scope_of H)) \/ {H}

proof end;

theorem :: QC_LANG2:120

for H, G, F being Element of QC-WFF st ( H is_immediate_constituent_of G or H is_proper_subformula_of G or H is_subformula_of G ) & G in Subformulae F holds
H in Subformulae F

proof end;