QC_LANG2: Connectives and Subformulae of the First Order Language

:: Connectives and Subformulae of the First Order Language
:: by Grzegorz Bancerek
::
:: Received November 23, 1989
:: Copyright (c) 1990-2021 Association of Mizar Users

theorem Th1: :: QC_LANG2:1

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

proof end;

theorem Th2: :: QC_LANG2:2

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

proof end;

theorem Th3: :: QC_LANG2:3

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

proof end;

theorem Th4: :: QC_LANG2:4

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

proof end;

theorem Th5: :: QC_LANG2:5

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

proof end;

theorem Th6: :: QC_LANG2:6

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

proof end;

theorem Th7: :: QC_LANG2:7

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

proof end;

definition

let A be QC-alphabet ;

func FALSUM A -> QC-formula of A equals :: QC_LANG2:def 1
'not' (VERUM A);

correctness ;
let p, q be Element of QC-WFF A;

func p => q -> QC-formula of A equals :: QC_LANG2:def 2
'not' (p '&' ('not' q));

correctness ;

func p 'or' q -> QC-formula of A equals :: QC_LANG2:def 3
'not' (('not' p) '&' ('not' q));

correctness ;

end;

:: deftheorem defines FALSUM QC_LANG2:def 1 :

:: deftheorem defines => QC_LANG2:def 2 :

:: deftheorem defines 'or' QC_LANG2:def 3 :

definition

let A be QC-alphabet ;
let p, q be Element of QC-WFF A;

func p <=> q -> QC-formula of A equals :: QC_LANG2:def 4
(p => q) '&' (q => p);

correctness ;

end;

:: deftheorem defines <=> QC_LANG2:def 4 :

definition

let A be QC-alphabet ;
let x be bound_QC-variable of A;
let p be Element of QC-WFF A;

func Ex (x,p) -> QC-formula of A equals :: QC_LANG2:def 5
'not' (All (x,('not' p)));

correctness ;

end;

:: deftheorem defines Ex QC_LANG2:def 5 :

theorem :: QC_LANG2:8

for A being QC-alphabet holds
( FALSUM A is negative & the_argument_of (FALSUM A) = VERUM A ) by Th1;

theorem :: QC_LANG2:9

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

theorem :: QC_LANG2:10

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

proof end;

theorem Th11: :: QC_LANG2:11

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

proof end;

theorem :: QC_LANG2:12

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

proof end;

theorem Th13: :: QC_LANG2:13

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

proof end;

definition

let A be QC-alphabet ;
let x, y be bound_QC-variable of A;
let p be Element of QC-WFF A;

func All (x,y,p) -> QC-formula of A equals :: QC_LANG2:def 6
All (x,(All (y,p)));

correctness ;

func Ex (x,y,p) -> QC-formula of A equals :: QC_LANG2:def 7
Ex (x,(Ex (y,p)));

correctness ;

end;

:: deftheorem defines All QC_LANG2:def 6 :

:: deftheorem defines Ex QC_LANG2:def 7 :

theorem :: QC_LANG2:14

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

theorem Th15: :: QC_LANG2:15

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

proof end;

theorem :: QC_LANG2:16

for A being QC-alphabet
for x, y, z being bound_QC-variable of A
for p, q being Element of QC-WFF A st All (x,y,p) = All (z,q) holds
( x = z & All (y,p) = q ) by Th5;

theorem Th17: :: QC_LANG2:17

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

proof end;

theorem :: QC_LANG2:18

for A being QC-alphabet
for x, y, z being bound_QC-variable of A
for p, q being Element of QC-WFF A st Ex (x,y,p) = Ex (z,q) holds
( x = z & Ex (y,p) = q ) by Th13;

theorem :: QC_LANG2:19

for A being QC-alphabet
for x, y being bound_QC-variable of A
for p being Element of QC-WFF A 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 Th7;

definition

let A be QC-alphabet ;
let x, y, z be bound_QC-variable of A;
let p be Element of QC-WFF A;

func All (x,y,z,p) -> QC-formula of A equals :: QC_LANG2:def 8
All (x,(All (y,z,p)));

correctness ;

func Ex (x,y,z,p) -> QC-formula of A equals :: QC_LANG2:def 9
Ex (x,(Ex (y,z,p)));

correctness ;

end;

:: deftheorem defines All QC_LANG2:def 8 :

:: deftheorem defines Ex QC_LANG2:def 9 :

theorem :: QC_LANG2:20

for A being QC-alphabet
for x, y, z being bound_QC-variable of A
for p being Element of QC-WFF A 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_LANG2:21

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

proof end;

theorem :: QC_LANG2:22

for A being QC-alphabet
for x, y, z being bound_QC-variable of A
for p, q being Element of QC-WFF A
for t being bound_QC-variable of A st All (x,y,z,p) = All (t,q) holds
( x = t & All (y,z,p) = q ) by Th5;

theorem :: QC_LANG2:23

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

proof end;

theorem :: QC_LANG2:24

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

proof end;

theorem :: QC_LANG2:25

for A being QC-alphabet
for x, y, z being bound_QC-variable of A
for p, q being Element of QC-WFF A
for t being bound_QC-variable of A st Ex (x,y,z,p) = Ex (t,q) holds
( x = t & Ex (y,z,p) = q ) by Th13;

theorem :: QC_LANG2:26

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

proof end;

theorem :: QC_LANG2:27

for A being QC-alphabet
for x, y, z being bound_QC-variable of A
for p being Element of QC-WFF A 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 Th7;

definition

let A be QC-alphabet ;
let H be Element of QC-WFF A;

attr H is disjunctive means :: QC_LANG2:def 10
ex p, q being Element of QC-WFF A st H = p 'or' q;

attr H is conditional means :: QC_LANG2:def 11
ex p, q being Element of QC-WFF A st H = p => q;

attr H is biconditional means :: QC_LANG2:def 12
ex p, q being Element of QC-WFF A st H = p <=> q;

attr H is existential means :: QC_LANG2:def 13
ex x being bound_QC-variable of A ex p being Element of QC-WFF A st H = Ex (x,p);

end;

:: deftheorem defines disjunctive QC_LANG2:def 10 :

:: deftheorem defines conditional QC_LANG2:def 11 :

:: deftheorem defines biconditional QC_LANG2:def 12 :

:: deftheorem defines existential QC_LANG2:def 13 :

theorem :: QC_LANG2:28

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

definition

let A be QC-alphabet ;
let H be Element of QC-WFF A;

func the_left_disjunct_of H -> QC-formula of A equals :: QC_LANG2:def 14
the_argument_of (the_left_argument_of (the_argument_of H));

correctness ;

func the_right_disjunct_of H -> QC-formula of A equals :: QC_LANG2:def 15
the_argument_of (the_right_argument_of (the_argument_of H));

correctness ;

func the_antecedent_of H -> QC-formula of A equals :: QC_LANG2:def 16
the_left_argument_of (the_argument_of H);

correctness ;

end;

:: deftheorem defines the_left_disjunct_of QC_LANG2:def 14 :

:: deftheorem defines the_right_disjunct_of QC_LANG2:def 15 :

:: deftheorem defines the_antecedent_of QC_LANG2:def 16 :

notation

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

end;

definition

let A be QC-alphabet ;
let H be Element of QC-WFF A;

func the_left_side_of H -> QC-formula of A equals :: QC_LANG2:def 17
the_antecedent_of (the_left_argument_of H);

correctness ;

func the_right_side_of H -> QC-formula of A equals :: QC_LANG2:def 18
the_consequent_of (the_left_argument_of H);

correctness ;

end;

:: deftheorem defines the_left_side_of QC_LANG2:def 17 :

:: deftheorem defines the_right_side_of QC_LANG2:def 18 :

theorem Th29: :: QC_LANG2:29

for A being QC-alphabet
for F, G being Element of QC-WFF A 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 Th30: :: QC_LANG2:30

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

proof end;

theorem Th31: :: QC_LANG2:31

for A being QC-alphabet
for F, G being Element of QC-WFF A 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_LANG2:32

for A being QC-alphabet
for x being bound_QC-variable of A
for H being Element of QC-WFF A holds the_argument_of (Ex (x,H)) = All (x,('not' H)) by Th1;

theorem :: QC_LANG2:33

for A being QC-alphabet
for H being Element of QC-WFF A 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_LANG2:34

for A being QC-alphabet
for H being Element of QC-WFF A 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_LANG2:35

for A being QC-alphabet
for H being Element of QC-WFF A st H is biconditional holds
( H is conjunctive & the_left_argument_of H is conditional & the_right_argument_of H is conditional ) by Th4;

theorem :: QC_LANG2:36

for A being QC-alphabet
for H being Element of QC-WFF A 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_LANG2:37

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

proof end;

theorem :: QC_LANG2:38

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

proof end;

theorem :: QC_LANG2:39

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

proof end;

theorem :: QC_LANG2:40

for A being QC-alphabet
for H being Element of QC-WFF A 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;

::
:: Immediate constituents of QC-formulae
::

definition

let A be QC-alphabet ;
let G, H be Element of QC-WFF A;

pred G is_immediate_constituent_of H means :: QC_LANG2:def 19
( H = 'not' G or ex F being Element of QC-WFF A st
( H = G '&' F or H = F '&' G ) or ex x being bound_QC-variable of A st H = All (x,G) );

end;

:: deftheorem defines is_immediate_constituent_of QC_LANG2:def 19 :

theorem Th41: :: QC_LANG2:41

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

proof end;

theorem Th42: :: QC_LANG2:42

for A being QC-alphabet
for H being Element of QC-WFF A
for k being Nat
for P being QC-pred_symbol of k,A
for V being QC-variable_list of k,A holds not H is_immediate_constituent_of P ! V

proof end;

theorem Th43: :: QC_LANG2:43

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

proof end;

theorem :: QC_LANG2:44

for A being QC-alphabet
for H being Element of QC-WFF A holds
( H is_immediate_constituent_of FALSUM A iff H = VERUM A ) by Th43;

theorem Th45: :: QC_LANG2:45

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

proof end;

theorem Th46: :: QC_LANG2:46

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

proof end;

theorem Th47: :: QC_LANG2:47

for A being QC-alphabet
for F, H being Element of QC-WFF A st H is atomic holds
not F is_immediate_constituent_of H by Th42;

theorem Th48: :: QC_LANG2:48

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

proof end;

theorem Th49: :: QC_LANG2:49

for A being QC-alphabet
for F, H being Element of QC-WFF A 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 Th50: :: QC_LANG2:50

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

proof end;

definition

let A be QC-alphabet ;
let G, H be Element of QC-WFF A;

pred G is_subformula_of H means :: QC_LANG2:def 20
ex n being Nat ex L being FinSequence st
( 1 <= n & len L = n & L . 1 = G & L . n = H & ( for k being Nat st 1 <= k & k < n holds
ex G1, H1 being Element of QC-WFF A st
( L . k = G1 & L . (k + 1) = H1 & G1 is_immediate_constituent_of H1 ) ) );

reflexivity

proof end;

end;

:: deftheorem defines is_subformula_of QC_LANG2:def 20 :

definition

let A be QC-alphabet ;
let H, F be Element of QC-WFF A;

pred H is_proper_subformula_of F means :Def21: :: QC_LANG2:def 21
( H is_subformula_of F & H <> F );

irreflexivity ;

end;

:: deftheorem Def21 defines is_proper_subformula_of QC_LANG2:def 21 :

theorem Th51: :: QC_LANG2:51

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

proof end;

theorem Th52: :: QC_LANG2:52

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

proof end;

theorem Th53: :: QC_LANG2:53

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

proof end;

theorem Th54: :: QC_LANG2:54

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

proof end;

theorem Th55: :: QC_LANG2:55

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

proof end;

theorem Th56: :: QC_LANG2:56

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

proof end;

theorem Th57: :: QC_LANG2:57

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

proof end;

theorem Th58: :: QC_LANG2:58

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

proof end;

theorem Th59: :: QC_LANG2:59

for A being QC-alphabet
for G, H being Element of QC-WFF A holds
( not G is_proper_subformula_of H or not H is_subformula_of G ) by Th58;

theorem :: QC_LANG2:60

for A being QC-alphabet
for G, H being Element of QC-WFF A holds
( not G is_proper_subformula_of H or not H is_proper_subformula_of G ) by Th59;

theorem Th61: :: QC_LANG2:61

for A being QC-alphabet
for G, H being Element of QC-WFF A holds
( not G is_subformula_of H or not H is_immediate_constituent_of G ) by Th53, Th59;

theorem :: QC_LANG2:62

for A being QC-alphabet
for G, H being Element of QC-WFF A holds
( not G is_proper_subformula_of H or not H is_immediate_constituent_of G ) by Th53, Th59;

theorem Th63: :: QC_LANG2:63

for A being QC-alphabet
for F, G, H being Element of QC-WFF A 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 Th64: :: QC_LANG2:64

for A being QC-alphabet
for F being Element of QC-WFF A holds not F is_proper_subformula_of VERUM A

proof end;

theorem Th65: :: QC_LANG2:65

for A being QC-alphabet
for F being Element of QC-WFF A
for k being Nat
for P being QC-pred_symbol of k,A
for V being QC-variable_list of k,A holds not F is_proper_subformula_of P ! V

proof end;

theorem Th66: :: QC_LANG2:66

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

proof end;

theorem :: QC_LANG2:67

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

proof end;

theorem :: QC_LANG2:68

for A being QC-alphabet
for F being Element of QC-WFF A holds
( F is_proper_subformula_of FALSUM A iff F is_subformula_of VERUM A ) by Th66;

theorem Th69: :: QC_LANG2:69

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

proof end;

theorem :: QC_LANG2:70

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

proof end;

theorem Th71: :: QC_LANG2:71

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

proof end;

theorem :: QC_LANG2:72

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

proof end;

theorem :: QC_LANG2:73

for A being QC-alphabet
for F, G being Element of QC-WFF A 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_LANG2:74

for A being QC-alphabet
for F, G being Element of QC-WFF A 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_LANG2:75

for A being QC-alphabet
for F, H being Element of QC-WFF A st H is atomic holds
not F is_proper_subformula_of H by Th65;

theorem :: QC_LANG2:76

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

proof end;

theorem :: QC_LANG2:77

for A being QC-alphabet
for H being Element of QC-WFF A 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_LANG2:78

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

proof end;

theorem Th79: :: QC_LANG2:79

for A being QC-alphabet
for H being Element of QC-WFF A holds
( H is_subformula_of VERUM A iff H = VERUM A ) by Def21, Th64;

theorem Th80: :: QC_LANG2:80

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

proof end;

theorem Th81: :: QC_LANG2:81

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

proof end;

::
:: Set of subformulae of QC-formulae
::

definition

let A be QC-alphabet ;
let H be Element of QC-WFF A;

func Subformulae H -> set means :Def22: :: QC_LANG2:def 22
for a being object holds
( a in it iff ex F being Element of QC-WFF A st
( F = a & F is_subformula_of H ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def22 defines Subformulae QC_LANG2:def 22 :

theorem Th82: :: QC_LANG2:82

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

proof end;

theorem Th83: :: QC_LANG2:83

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

proof end;

theorem :: QC_LANG2:84

for A being QC-alphabet
for G, H being Element of QC-WFF A st G in Subformulae H holds
Subformulae G c= Subformulae H by Th82, Th83;

theorem Th85: :: QC_LANG2:85

for A being QC-alphabet holds Subformulae (VERUM A) = {(VERUM A)}

proof end;

theorem Th86: :: QC_LANG2:86

for A being QC-alphabet
for k being Nat
for P being QC-pred_symbol of k,A
for V being QC-variable_list of k,A holds Subformulae (P ! V) = {(P ! V)}

proof end;

theorem :: QC_LANG2:87

for A being QC-alphabet holds Subformulae (FALSUM A) = {(VERUM A),(FALSUM A)}

proof end;

theorem Th88: :: QC_LANG2:88

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

proof end;

theorem Th89: :: QC_LANG2:89

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

proof end;

theorem Th90: :: QC_LANG2:90

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

proof end;

theorem Th91: :: QC_LANG2:91

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

proof end;

theorem :: QC_LANG2:92

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

proof end;

theorem :: QC_LANG2:93

for A being QC-alphabet
for F, G being Element of QC-WFF A 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_LANG2:94

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

proof end;

theorem :: QC_LANG2:95

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

proof end;

theorem :: QC_LANG2:96

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

proof end;

theorem :: QC_LANG2:97

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

proof end;

theorem :: QC_LANG2:98

for A being QC-alphabet
for F, G, H being Element of QC-WFF A 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;