:: Similarity of Formulae
:: by Agata Darmochwa{\l} and Andrzej Trybulec
::
:: Copyright (c) 1991-2021 Association of Mizar Users

definition
let A be QC-alphabet ;
let b be bound_QC-variable of A;
func x. b -> QC-symbol of A means :Def1: :: CQC_SIM1:def 1
x. it = b;
existence
ex b1 being QC-symbol of A st x. b1 = b
by QC_LANG3:30;
uniqueness
for b1, b2 being QC-symbol of A st x. b1 = b & x. b2 = b holds
b1 = b2
by XTUPLE_0:1;
end;

:: deftheorem Def1 defines x. CQC_SIM1:def 1 :
for A being QC-alphabet
for b being bound_QC-variable of A
for b3 being QC-symbol of A holds
( b3 = x. b iff x. b3 = b );

theorem Th1: :: CQC_SIM1:1
for x, y being set
for f being Function holds Im ((f +* (x .--> y)),x) = {y}
proof end;

theorem Th2: :: CQC_SIM1:2
for K, L, x, y being set
for f being Function holds (f +* (L --> y)) .: K c= (f .: K) \/ {y}
proof end;

theorem Th3: :: CQC_SIM1:3
for x, y being set
for g being Function
for A being set holds (g +* (x .--> y)) .: (A \ {x}) = g .: (A \ {x})
proof end;

theorem Th4: :: CQC_SIM1:4
for x, y being set
for g being Function
for A being set st not y in g .: (A \ {x}) holds
(g +* (x .--> y)) .: (A \ {x}) = ((g +* (x .--> y)) .: A) \ {y}
proof end;

theorem Th5: :: CQC_SIM1:5
for A being QC-alphabet
for p being Element of CQC-WFF A st p is atomic holds
ex k being Nat ex P being QC-pred_symbol of k,A ex ll being CQC-variable_list of k,A st p = P ! ll
proof end;

theorem :: CQC_SIM1:6
for A being QC-alphabet
for p being Element of CQC-WFF A st p is negative holds
ex q being Element of CQC-WFF A st p = 'not' q
proof end;

theorem :: CQC_SIM1:7
for A being QC-alphabet
for p being Element of CQC-WFF A st p is conjunctive holds
ex q, r being Element of CQC-WFF A st p = q '&' r
proof end;

theorem :: CQC_SIM1:8
for A being QC-alphabet
for p being Element of CQC-WFF A st p is universal holds
ex x being Element of bound_QC-variables A ex q being Element of CQC-WFF A st p = All (x,q)
proof end;

theorem Th9: :: CQC_SIM1:9
for l being FinSequence holds rng l = { (l . i) where i is Nat : ( 1 <= i & i <= len l ) }
proof end;

scheme :: CQC_SIM1:sch 1
QCFuncExN{ F1() -> QC-alphabet , F2() -> non empty set , F3() -> Element of F2(), F4( object ) -> Element of F2(), F5( object , object ) -> Element of F2(), F6( object , object , object ) -> Element of F2(), F7( object , object ) -> Element of F2() } :
ex F being Function of (QC-WFF F1()),F2() st
( F . (VERUM F1()) = F3() & ( for p being Element of QC-WFF F1() holds
( ( p is atomic implies F . p = F4(p) ) & ( p is negative implies F . p = F5((F . ()),p) ) & ( p is conjunctive implies F . p = F6((),(),p) ) & ( p is universal implies F . p = F7((F . ()),p) ) ) ) )
proof end;

scheme :: CQC_SIM1:sch 2
CQCF2FuncEx{ F1() -> QC-alphabet , F2() -> non empty set , F3() -> non empty set , F4() -> Element of Funcs (F2(),F3()), F5( object , object , object ) -> Element of Funcs (F2(),F3()), F6( object , object ) -> Element of Funcs (F2(),F3()), F7( object , object , object , object ) -> Element of Funcs (F2(),F3()), F8( object , object , object ) -> Element of Funcs (F2(),F3()) } :
ex F being Function of (CQC-WFF F1()),(Funcs (F2(),F3())) st
( F . (VERUM F1()) = F4() & ( for k being Nat
for l being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds F . (P ! l) = F5(k,P,l) ) & ( for r, s being Element of CQC-WFF F1()
for x being Element of bound_QC-variables F1() holds
( F . () = F6((F . r),r) & F . (r '&' s) = F7((F . r),(F . s),r,s) & F . (All (x,r)) = F8(x,(F . r),r) ) ) )
proof end;

scheme :: CQC_SIM1:sch 3
CQCF2FUniq{ F1() -> QC-alphabet , F2() -> non empty set , F3() -> non empty set , F4() -> Function of (CQC-WFF F1()),(Funcs (F2(),F3())), F5() -> Function of (CQC-WFF F1()),(Funcs (F2(),F3())), F6() -> Function of F2(),F3(), F7( object , object , object ) -> Function of F2(),F3(), F8( object , object ) -> Function of F2(),F3(), F9( object , object , object , object ) -> Function of F2(),F3(), F10( object , object , object ) -> Function of F2(),F3() } :
F4() = F5()
provided
A1: F4() . (VERUM F1()) = F6() and
A2: for k being Nat
for ll being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds F4() . (P ! ll) = F7(k,P,ll) and
A3: for r, s being Element of CQC-WFF F1()
for x being Element of bound_QC-variables F1() holds
( F4() . () = F8((F4() . r),r) & F4() . (r '&' s) = F9((F4() . r),(F4() . s),r,s) & F4() . (All (x,r)) = F10(x,(F4() . r),r) ) and
A4: F5() . (VERUM F1()) = F6() and
A5: for k being Nat
for ll being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds F5() . (P ! ll) = F7(k,P,ll) and
A6: for r, s being Element of CQC-WFF F1()
for x being Element of bound_QC-variables F1() holds
( F5() . () = F8((F5() . r),r) & F5() . (r '&' s) = F9((F5() . r),(F5() . s),r,s) & F5() . (All (x,r)) = F10(x,(F5() . r),r) )
proof end;

theorem Th10: :: CQC_SIM1:10
for A being QC-alphabet
for p being Element of CQC-WFF A holds p is_subformula_of 'not' p
proof end;

theorem Th11: :: CQC_SIM1:11
for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( p is_subformula_of p '&' q & q is_subformula_of p '&' q )
proof end;

theorem Th12: :: CQC_SIM1:12
for A being QC-alphabet
for p being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds p is_subformula_of All (x,p)
proof end;

theorem Th13: :: CQC_SIM1:13
for A being QC-alphabet
for k being Nat
for l being CQC-variable_list of k,A
for i being Nat st 1 <= i & i <= len l holds
l . i in bound_QC-variables A
proof end;

definition
let A be QC-alphabet ;
let D be non empty set ;
let f be Function of D,();
func NEGATIVE f -> Element of Funcs (D,()) means :Def2: :: CQC_SIM1:def 2
for a being Element of D
for p being Element of CQC-WFF A st p = f . a holds
it . a = 'not' p;
existence
ex b1 being Element of Funcs (D,()) st
for a being Element of D
for p being Element of CQC-WFF A st p = f . a holds
b1 . a = 'not' p
proof end;
uniqueness
for b1, b2 being Element of Funcs (D,()) st ( for a being Element of D
for p being Element of CQC-WFF A st p = f . a holds
b1 . a = 'not' p ) & ( for a being Element of D
for p being Element of CQC-WFF A st p = f . a holds
b2 . a = 'not' p ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines NEGATIVE CQC_SIM1:def 2 :
for A being QC-alphabet
for D being non empty set
for f being Function of D,()
for b4 being Element of Funcs (D,()) holds
( b4 = NEGATIVE f iff for a being Element of D
for p being Element of CQC-WFF A st p = f . a holds
b4 . a = 'not' p );

definition
let A be QC-alphabet ;
let f, g be Function of [:(),(Funcs (,)):],();
let n be Nat;
func CON (f,g,n) -> Element of Funcs ([:(),(Funcs (,)):],()) means :Def3: :: CQC_SIM1:def 3
for t being QC-symbol of A
for h being Element of Funcs (,)
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
it . (t,h) = p '&' q;
existence
ex b1 being Element of Funcs ([:(),(Funcs (,)):],()) st
for t being QC-symbol of A
for h being Element of Funcs (,)
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
b1 . (t,h) = p '&' q
proof end;
uniqueness
for b1, b2 being Element of Funcs ([:(),(Funcs (,)):],()) st ( for t being QC-symbol of A
for h being Element of Funcs (,)
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
b1 . (t,h) = p '&' q ) & ( for t being QC-symbol of A
for h being Element of Funcs (,)
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
b2 . (t,h) = p '&' q ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines CON CQC_SIM1:def 3 :
for A being QC-alphabet
for f, g being Function of [:(),(Funcs (,)):],()
for n being Nat
for b5 being Element of Funcs ([:(),(Funcs (,)):],()) holds
( b5 = CON (f,g,n) iff for t being QC-symbol of A
for h being Element of Funcs (,)
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
b5 . (t,h) = p '&' q );

Lm1: for A being QC-alphabet
for t being QC-symbol of A
for x being Element of bound_QC-variables A
for h being Element of Funcs (,) holds h +* (x .--> (x. t)) is Function of ,

proof end;

definition
let A be QC-alphabet ;
let f be Function of [:(),(Funcs (,)):],();
let x be bound_QC-variable of A;
func UNIVERSAL (x,f) -> Element of Funcs ([:(),(Funcs (,)):],()) means :Def4: :: CQC_SIM1:def 4
for t being QC-symbol of A
for h being Element of Funcs (,)
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
it . (t,h) = All ((x. t),p);
existence
ex b1 being Element of Funcs ([:(),(Funcs (,)):],()) st
for t being QC-symbol of A
for h being Element of Funcs (,)
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
b1 . (t,h) = All ((x. t),p)
proof end;
uniqueness
for b1, b2 being Element of Funcs ([:(),(Funcs (,)):],()) st ( for t being QC-symbol of A
for h being Element of Funcs (,)
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
b1 . (t,h) = All ((x. t),p) ) & ( for t being QC-symbol of A
for h being Element of Funcs (,)
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
b2 . (t,h) = All ((x. t),p) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines UNIVERSAL CQC_SIM1:def 4 :
for A being QC-alphabet
for f being Function of [:(),(Funcs (,)):],()
for x being bound_QC-variable of A
for b4 being Element of Funcs ([:(),(Funcs (,)):],()) holds
( b4 = UNIVERSAL (x,f) iff for t being QC-symbol of A
for h being Element of Funcs (,)
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
b4 . (t,h) = All ((x. t),p) );

Lm2: for A being QC-alphabet
for k being Nat
for f being CQC-variable_list of k,A holds f is FinSequence of bound_QC-variables A

proof end;

definition
let A be QC-alphabet ;
let k be Nat;
let l be CQC-variable_list of k,A;
let f be Element of Funcs (,);
:: original: *
redefine func f * l -> CQC-variable_list of k,A;
coherence
l * f is CQC-variable_list of k,A
proof end;
end;

definition
let A be QC-alphabet ;
let k be Nat;
let P be QC-pred_symbol of k,A;
let l be CQC-variable_list of k,A;
func ATOMIC (P,l) -> Element of Funcs ([:(),(Funcs (,)):],()) means :Def5: :: CQC_SIM1:def 5
for t being QC-symbol of A
for h being Element of Funcs (,) holds it . (t,h) = P ! (h * l);
existence
ex b1 being Element of Funcs ([:(),(Funcs (,)):],()) st
for t being QC-symbol of A
for h being Element of Funcs (,) holds b1 . (t,h) = P ! (h * l)
proof end;
uniqueness
for b1, b2 being Element of Funcs ([:(),(Funcs (,)):],()) st ( for t being QC-symbol of A
for h being Element of Funcs (,) holds b1 . (t,h) = P ! (h * l) ) & ( for t being QC-symbol of A
for h being Element of Funcs (,) holds b2 . (t,h) = P ! (h * l) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines ATOMIC CQC_SIM1:def 5 :
for A being QC-alphabet
for k being Nat
for P being QC-pred_symbol of k,A
for l being CQC-variable_list of k,A
for b5 being Element of Funcs ([:(),(Funcs (,)):],()) holds
( b5 = ATOMIC (P,l) iff for t being QC-symbol of A
for h being Element of Funcs (,) holds b5 . (t,h) = P ! (h * l) );

deffunc H1( set , set , set ) -> Element of NAT = 0 ;

deffunc H2( Element of NAT ) -> Element of NAT = $1; deffunc H3( Element of NAT , Element of NAT ) -> Element of NAT =$1 + $2; deffunc H4( set , Element of NAT ) -> Element of NAT =$2 + 1;

definition
let A be QC-alphabet ;
let p be Element of CQC-WFF A;
func QuantNbr p -> Element of NAT means :Def6: :: CQC_SIM1:def 6
ex F being Function of (),NAT st
( it = F . p & F . () = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = 0 & F . () = F . r & F . (r '&' s) = (F . r) + (F . s) & F . (All (x,r)) = (F . r) + 1 ) ) );
correctness
existence
ex b1 being Element of NAT ex F being Function of (),NAT st
( b1 = F . p & F . () = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = 0 & F . () = F . r & F . (r '&' s) = (F . r) + (F . s) & F . (All (x,r)) = (F . r) + 1 ) ) )
;
uniqueness
for b1, b2 being Element of NAT st ex F being Function of (),NAT st
( b1 = F . p & F . () = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = 0 & F . () = F . r & F . (r '&' s) = (F . r) + (F . s) & F . (All (x,r)) = (F . r) + 1 ) ) ) & ex F being Function of (),NAT st
( b2 = F . p & F . () = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = 0 & F . () = F . r & F . (r '&' s) = (F . r) + (F . s) & F . (All (x,r)) = (F . r) + 1 ) ) ) holds
b1 = b2
;
proof end;
end;

:: deftheorem Def6 defines QuantNbr CQC_SIM1:def 6 :
for A being QC-alphabet
for p being Element of CQC-WFF A
for b3 being Element of NAT holds
( b3 = QuantNbr p iff ex F being Function of (),NAT st
( b3 = F . p & F . () = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = 0 & F . () = F . r & F . (r '&' s) = (F . r) + (F . s) & F . (All (x,r)) = (F . r) + 1 ) ) ) );

definition
let A be QC-alphabet ;
let f be Function of (),(Funcs ([:(),(Funcs (,)):],()));
let x be Element of CQC-WFF A;
:: original: .
redefine func f . x -> Element of Funcs ([:(),(Funcs (,)):],());
coherence
f . x is Element of Funcs ([:(),(Funcs (,)):],())
proof end;
end;

definition
let A be QC-alphabet ;
func SepFunc A -> Function of (),(Funcs ([:(),(Funcs (,)):],())) means :Def7: :: CQC_SIM1:def 7
( it . () = [:(),(Funcs (,)):] --> () & ( for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds it . (P ! l) = ATOMIC (P,l) ) & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( it . () = NEGATIVE (it . r) & it . (r '&' s) = CON ((it . r),(it . s),()) & it . (All (x,r)) = UNIVERSAL (x,(it . r)) ) ) );
existence
ex b1 being Function of (),(Funcs ([:(),(Funcs (,)):],())) st
( b1 . () = [:(),(Funcs (,)):] --> () & ( for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds b1 . (P ! l) = ATOMIC (P,l) ) & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( b1 . () = NEGATIVE (b1 . r) & b1 . (r '&' s) = CON ((b1 . r),(b1 . s),()) & b1 . (All (x,r)) = UNIVERSAL (x,(b1 . r)) ) ) )
proof end;
uniqueness
for b1, b2 being Function of (),(Funcs ([:(),(Funcs (,)):],())) st b1 . () = [:(),(Funcs (,)):] --> () & ( for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds b1 . (P ! l) = ATOMIC (P,l) ) & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( b1 . () = NEGATIVE (b1 . r) & b1 . (r '&' s) = CON ((b1 . r),(b1 . s),()) & b1 . (All (x,r)) = UNIVERSAL (x,(b1 . r)) ) ) & b2 . () = [:(),(Funcs (,)):] --> () & ( for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds b2 . (P ! l) = ATOMIC (P,l) ) & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( b2 . () = NEGATIVE (b2 . r) & b2 . (r '&' s) = CON ((b2 . r),(b2 . s),()) & b2 . (All (x,r)) = UNIVERSAL (x,(b2 . r)) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines SepFunc CQC_SIM1:def 7 :
for A being QC-alphabet
for b2 being Function of (),(Funcs ([:(),(Funcs (,)):],())) holds
( b2 = SepFunc A iff ( b2 . () = [:(),(Funcs (,)):] --> () & ( for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds b2 . (P ! l) = ATOMIC (P,l) ) & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( b2 . () = NEGATIVE (b2 . r) & b2 . (r '&' s) = CON ((b2 . r),(b2 . s),()) & b2 . (All (x,r)) = UNIVERSAL (x,(b2 . r)) ) ) ) );

definition
let A be QC-alphabet ;
let p be Element of CQC-WFF A;
let t be QC-symbol of A;
let f be Element of Funcs (,);
func SepFunc (p,t,f) -> Element of CQC-WFF A equals :: CQC_SIM1:def 8
(() . p) . [t,f];
correctness
coherence
(() . p) . [t,f] is Element of CQC-WFF A
;
;
end;

:: deftheorem defines SepFunc CQC_SIM1:def 8 :
for A being QC-alphabet
for p being Element of CQC-WFF A
for t being QC-symbol of A
for f being Element of Funcs (,) holds SepFunc (p,t,f) = (() . p) . [t,f];

theorem :: CQC_SIM1:14
for A being QC-alphabet holds QuantNbr () = 0
proof end;

theorem :: CQC_SIM1:15
for A being QC-alphabet
for k being Nat
for ll being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds QuantNbr (P ! ll) = 0
proof end;

theorem :: CQC_SIM1:16
for A being QC-alphabet
for p being Element of CQC-WFF A holds QuantNbr () = QuantNbr p
proof end;

theorem :: CQC_SIM1:17
for A being QC-alphabet
for p, q being Element of CQC-WFF A holds QuantNbr (p '&' q) = () + ()
proof end;

theorem :: CQC_SIM1:18
for A being QC-alphabet
for p being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds QuantNbr (All (x,p)) = () + 1
proof end;

theorem Th19: :: CQC_SIM1:19
for A being QC-alphabet
for p being Element of QC-WFF A holds still_not-bound_in p is finite
proof end;

scheme :: CQC_SIM1:sch 4
MaxFinDomElem{ F1() -> non empty set , F2() -> set , P1[ object , object ] } :
ex x being Element of F1() st
( x in F2() & ( for y being Element of F1() st y in F2() holds
P1[x,y] ) )
provided
A1: ( F2() is finite & F2() <> {} & F2() c= F1() ) and
A2: for x, y being Element of F1() holds
( P1[x,y] or P1[y,x] ) and
A3: for x, y, z being Element of F1() st P1[x,y] & P1[y,z] holds
P1[x,z]
proof end;

definition
let X be set ;
:: original: id
redefine func id X -> Element of Funcs (X,X);
coherence
id X is Element of Funcs (X,X)
proof end;
end;

definition
let A be QC-alphabet ;
let p be Element of CQC-WFF A;
func NBI p -> Subset of () equals :: CQC_SIM1:def 9
{ t where t is QC-symbol of A : for u being QC-symbol of A st t <= u holds
not x. u in still_not-bound_in p
}
;
coherence
{ t where t is QC-symbol of A : for u being QC-symbol of A st t <= u holds
not x. u in still_not-bound_in p
}
is Subset of ()
proof end;
end;

:: deftheorem defines NBI CQC_SIM1:def 9 :
for A being QC-alphabet
for p being Element of CQC-WFF A holds NBI p = { t where t is QC-symbol of A : for u being QC-symbol of A st t <= u holds
not x. u in still_not-bound_in p
}
;

registration
let A be QC-alphabet ;
let p be Element of CQC-WFF A;
cluster NBI p -> non empty ;
coherence
not NBI p is empty
proof end;
end;

definition
let A be QC-alphabet ;
let p be Element of CQC-WFF A;
func index p -> QC-symbol of A equals :: CQC_SIM1:def 10
min (NBI p);
coherence
min (NBI p) is QC-symbol of A
;
end;

:: deftheorem defines index CQC_SIM1:def 10 :
for A being QC-alphabet
for p being Element of CQC-WFF A holds index p = min (NBI p);

theorem Th20: :: CQC_SIM1:20
for A being QC-alphabet
for p being Element of CQC-WFF A holds
( index p = 0 A iff p is closed )
proof end;

theorem Th21: :: CQC_SIM1:21
for A being QC-alphabet
for t being QC-symbol of A
for p being Element of CQC-WFF A st x. t in still_not-bound_in p holds
t < index p
proof end;

theorem Th22: :: CQC_SIM1:22
for A being QC-alphabet holds index () = 0 A
proof end;

theorem Th23: :: CQC_SIM1:23
for A being QC-alphabet
for p being Element of CQC-WFF A holds index () = index p
proof end;

theorem :: CQC_SIM1:24
for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( index p <= index (p '&' q) & index q <= index (p '&' q) )
proof end;

definition
let A be QC-alphabet ;
let p be Element of CQC-WFF A;
func SepVar p -> Element of CQC-WFF A equals :: CQC_SIM1:def 11
SepFunc (p,(),());
coherence
SepFunc (p,(),()) is Element of CQC-WFF A
;
end;

:: deftheorem defines SepVar CQC_SIM1:def 11 :
for A being QC-alphabet
for p being Element of CQC-WFF A holds SepVar p = SepFunc (p,(),());

theorem :: CQC_SIM1:25
for A being QC-alphabet holds SepVar () = VERUM A
proof end;

scheme :: CQC_SIM1:sch 5
CQCInd{ F1() -> QC-alphabet , P1[ object ] } :
for r being Element of CQC-WFF F1() holds P1[r]
provided
A1: P1[ VERUM F1()] and
A2: for k being Nat
for l being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds P1[P ! l] and
A3: for r being Element of CQC-WFF F1() st P1[r] holds
P1[ 'not' r] and
A4: for r, s being Element of CQC-WFF F1() st P1[r] & P1[s] holds
P1[r '&' s] and
A5: for r being Element of CQC-WFF F1()
for x being bound_QC-variable of F1() st P1[r] holds
P1[ All (x,r)]
proof end;

theorem Th26: :: CQC_SIM1:26
for A being QC-alphabet
for k being Nat
for ll being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds SepVar (P ! ll) = P ! ll
proof end;

theorem :: CQC_SIM1:27
for A being QC-alphabet
for p being Element of CQC-WFF A st p is atomic holds
SepVar p = p
proof end;

theorem Th28: :: CQC_SIM1:28
for A being QC-alphabet
for p being Element of CQC-WFF A holds SepVar () = 'not' ()
proof end;

theorem :: CQC_SIM1:29
for A being QC-alphabet
for p, q being Element of CQC-WFF A st p is negative & q = the_argument_of p holds
SepVar p = 'not' ()
proof end;

definition
let A be QC-alphabet ;
let p be Element of CQC-WFF A;
let X be Subset of [:(),(),(),(Funcs (,)):];
pred X is_Sep-closed_on p means :: CQC_SIM1:def 12
( [p,(),(),()] in X & ( for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(),t,K,f] in X holds
[q,t,K,f] in X ) & ( for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + ()),K,f] in X ) ) & ( for q being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X ) );
end;

:: deftheorem defines is_Sep-closed_on CQC_SIM1:def 12 :
for A being QC-alphabet
for p being Element of CQC-WFF A
for X being Subset of [:(),(),(),(Funcs (,)):] holds
( X is_Sep-closed_on p iff ( [p,(),(),()] in X & ( for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(),t,K,f] in X holds
[q,t,K,f] in X ) & ( for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + ()),K,f] in X ) ) & ( for q being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X ) ) );

definition
let A be QC-alphabet ;
let p be Element of CQC-WFF A;
func SepQuadruples p -> Subset of [:(),(),(),(Funcs (,)):] means :Def13: :: CQC_SIM1:def 13
( it is_Sep-closed_on p & ( for D being Subset of [:(),(),(),(Funcs (,)):] st D is_Sep-closed_on p holds
it c= D ) );
existence
ex b1 being Subset of [:(),(),(),(Funcs (,)):] st
( b1 is_Sep-closed_on p & ( for D being Subset of [:(),(),(),(Funcs (,)):] st D is_Sep-closed_on p holds
b1 c= D ) )
proof end;
uniqueness
for b1, b2 being Subset of [:(),(),(),(Funcs (,)):] st b1 is_Sep-closed_on p & ( for D being Subset of [:(),(),(),(Funcs (,)):] st D is_Sep-closed_on p holds
b1 c= D ) & b2 is_Sep-closed_on p & ( for D being Subset of [:(),(),(),(Funcs (,)):] st D is_Sep-closed_on p holds
b2 c= D ) holds
b1 = b2
;
end;

:: deftheorem Def13 defines SepQuadruples CQC_SIM1:def 13 :
for A being QC-alphabet
for p being Element of CQC-WFF A
for b3 being Subset of [:(),(),(),(Funcs (,)):] holds
( b3 = SepQuadruples p iff ( b3 is_Sep-closed_on p & ( for D being Subset of [:(),(),(),(Funcs (,)):] st D is_Sep-closed_on p holds
b3 c= D ) ) );

theorem Th30: :: CQC_SIM1:30
for A being QC-alphabet
for p being Element of CQC-WFF A holds [p,(),(),()] in SepQuadruples p
proof end;

theorem Th31: :: CQC_SIM1:31
for A being QC-alphabet
for p, q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(),t,K,f] in SepQuadruples p holds
proof end;

theorem Th32: :: CQC_SIM1:32
for A being QC-alphabet
for p, q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(q '&' r),t,K,f] in SepQuadruples p holds
( [q,t,K,f] in SepQuadruples p & [r,(t + ()),K,f] in SepQuadruples p )
proof end;

theorem Th33: :: CQC_SIM1:33
for A being QC-alphabet
for p, q being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(All (x,q)),t,K,f] in SepQuadruples p holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in SepQuadruples p
proof end;

theorem Th34: :: CQC_SIM1:34
for A being QC-alphabet
for t being QC-symbol of A
for p, q being Element of CQC-WFF A
for f being Element of Funcs (,)
for K being Element of Fin holds
( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(),(),()] or [(),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A st [(q '&' r),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A ex u being QC-symbol of A st
( t = u + () & [(r '&' q),u,K,f] in SepQuadruples p ) or ex x being Element of bound_QC-variables A ex u being QC-symbol of A ex h being Element of Funcs (,) st
( u ++ = t & h +* ({x} --> (x. u)) = f & ( [(All (x,q)),u,K,h] in SepQuadruples p or [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p ) ) )
proof end;

scheme :: CQC_SIM1:sch 6
Sepregression{ F1() -> QC-alphabet , F2() -> Element of CQC-WFF F1(), P1[ object , object , object , object ] } :
for q being Element of CQC-WFF F1()
for t being QC-symbol of F1()
for K being Element of Fin ()
for f being Element of Funcs ((),()) st [q,t,K,f] in SepQuadruples F2() holds
P1[q,t,K,f]
provided
A1: P1[F2(), index F2(), {}. (), id ()] and
A2: for q being Element of CQC-WFF F1()
for t being QC-symbol of F1()
for K being Element of Fin ()
for f being Element of Funcs ((),()) st [(),t,K,f] in SepQuadruples F2() & P1[ 'not' q,t,K,f] holds
P1[q,t,K,f] and
A3: for q, r being Element of CQC-WFF F1()
for t being QC-symbol of F1()
for K being Element of Fin ()
for f being Element of Funcs ((),()) st [(q '&' r),t,K,f] in SepQuadruples F2() & P1[q '&' r,t,K,f] holds
( P1[q,t,K,f] & P1[r,t + (),K,f] ) and
A4: for q being Element of CQC-WFF F1()
for x being bound_QC-variable of F1()
for t being QC-symbol of F1()
for K being Element of Fin ()
for f being Element of Funcs ((),()) st [(All (x,q)),t,K,f] in SepQuadruples F2() & P1[ All (x,q),t,K,f] holds
P1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
proof end;

theorem Th35: :: CQC_SIM1:35
for A being QC-alphabet
for p, q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [q,t,K,f] in SepQuadruples p holds
q is_subformula_of p
proof end;

theorem :: CQC_SIM1:36
for A being QC-alphabet holds SepQuadruples () = {[(),(0 A),(),()]}
proof end;

theorem :: CQC_SIM1:37
for A being QC-alphabet
for k being Nat
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds SepQuadruples (P ! l) = {[(P ! l),(index (P ! l)),(),()]}
proof end;

theorem Th38: :: CQC_SIM1:38
for A being QC-alphabet
for p, q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [q,t,K,f] in SepQuadruples p holds
still_not-bound_in q c= \/ K
proof end;

theorem Th39: :: CQC_SIM1:39
for A being QC-alphabet
for t, u being QC-symbol of A
for p, q being Element of CQC-WFF A
for f being Element of Funcs (,)
for K being Element of Fin st [q,t,K,f] in SepQuadruples p & x. u in f .: K holds
u < t
proof end;

theorem :: CQC_SIM1:40
for A being QC-alphabet
for t being QC-symbol of A
for p, q being Element of CQC-WFF A
for f being Element of Funcs (,)
for K being Element of Fin st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: K
proof end;

theorem Th41: :: CQC_SIM1:41
for A being QC-alphabet
for t, u being QC-symbol of A
for p, q being Element of CQC-WFF A
for f being Element of Funcs (,)
for K being Element of Fin st [q,t,K,f] in SepQuadruples p & x. u in f .: holds
u < t
proof end;

theorem Th42: :: CQC_SIM1:42
for A being QC-alphabet
for t, u being QC-symbol of A
for p, q being Element of CQC-WFF A
for f being Element of Funcs (,)
for K being Element of Fin st [q,t,K,f] in SepQuadruples p & x. u in f .: holds
u < t
proof end;

theorem Th43: :: CQC_SIM1:43
for A being QC-alphabet
for t being QC-symbol of A
for p, q being Element of CQC-WFF A
for f being Element of Funcs (,)
for K being Element of Fin st [q,t,K,f] in SepQuadruples p holds
not x. t in f .:
proof end;

theorem Th44: :: CQC_SIM1:44
for A being QC-alphabet
for p being Element of CQC-WFF A holds still_not-bound_in p = still_not-bound_in ()
proof end;

theorem :: CQC_SIM1:45
for A being QC-alphabet
for p being Element of CQC-WFF A holds index p = index ()
proof end;

definition
let A be QC-alphabet ;
let p, q be Element of CQC-WFF A;
reflexivity
for p being Element of CQC-WFF A holds SepVar p = SepVar p
;
symmetry
for p, q being Element of CQC-WFF A st SepVar p = SepVar q holds
SepVar q = SepVar p
;
end;

:: deftheorem defines are_similar CQC_SIM1:def 14 :
for A being QC-alphabet
for p, q being Element of CQC-WFF A holds
( p,q are_similar iff SepVar p = SepVar q );

theorem :: CQC_SIM1:46
for A being QC-alphabet
for p, q, r being Element of CQC-WFF A st p,q are_similar & q,r are_similar holds
p,r are_similar ;