:: CQC_SIM1 semantic presentation
begin
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
let x, y be set ; ::_thesis: for f being Function holds Im ((f +* (x .--> y)),x) = {y}
let f be Function; ::_thesis: Im ((f +* (x .--> y)),x) = {y}
now__::_thesis:_for_u_being_set_holds_
(_(_u_in_(f_+*_(x_.-->_y))_.:_{x}_implies_u_=_y_)_&_(_u_=_y_implies_u_in_(f_+*_(x_.-->_y))_.:_{x}_)_)
let u be set ; ::_thesis: ( ( u in (f +* (x .--> y)) .: {x} implies u = y ) & ( u = y implies u in (f +* (x .--> y)) .: {x} ) )
thus ( u in (f +* (x .--> y)) .: {x} implies u = y ) ::_thesis: ( u = y implies u in (f +* (x .--> y)) .: {x} )
proof
assume u in (f +* (x .--> y)) .: {x} ; ::_thesis: u = y
then consider z being set such that
z in dom (f +* (x .--> y)) and
A1: z in {x} and
A2: u = (f +* (x .--> y)) . z by FUNCT_1:def_6;
z in dom (x .--> y) by A1, FUNCOP_1:13;
then u = (x .--> y) . z by A2, FUNCT_4:13;
hence u = y by A1, FUNCOP_1:7; ::_thesis: verum
end;
A3: x in {x} by TARSKI:def_1;
then A4: x in dom (x .--> y) by FUNCOP_1:13;
then A5: x in dom (f +* (x .--> y)) by FUNCT_4:12;
(x .--> y) . x = y by A3, FUNCOP_1:7;
then y = (f +* (x .--> y)) . x by A4, FUNCT_4:13;
hence ( u = y implies u in (f +* (x .--> y)) .: {x} ) by A3, A5, FUNCT_1:def_6; ::_thesis: verum
end;
hence Im ((f +* (x .--> y)),x) = {y} by TARSKI:def_1; ::_thesis: verum
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
let K, L be set ; ::_thesis: for x, y being set
for f being Function holds (f +* (L --> y)) .: K c= (f .: K) \/ {y}
let x, y be set ; ::_thesis: for f being Function holds (f +* (L --> y)) .: K c= (f .: K) \/ {y}
let f be Function; ::_thesis: (f +* (L --> y)) .: K c= (f .: K) \/ {y}
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in (f +* (L --> y)) .: K or z in (f .: K) \/ {y} )
assume z in (f +* (L --> y)) .: K ; ::_thesis: z in (f .: K) \/ {y}
then consider u being set such that
A1: u in dom (f +* (L --> y)) and
A2: u in K and
A3: z = (f +* (L --> y)) . u by FUNCT_1:def_6;
A4: dom (L --> y) = L by FUNCOP_1:13;
now__::_thesis:_(_(_u_in_L_&_z_in_{y}_)_or_(_not_u_in_L_&_z_in_f_.:_K_)_)
percases ( u in L or not u in L ) ;
caseA5: u in L ; ::_thesis: z in {y}
then z = (L --> y) . u by A3, A4, FUNCT_4:13;
then z = y by A5, FUNCOP_1:7;
hence z in {y} by TARSKI:def_1; ::_thesis: verum
end;
caseA6: not u in L ; ::_thesis: z in f .: K
then A7: z = f . u by A3, A4, FUNCT_4:11;
u in dom f by A1, A4, A6, FUNCT_4:12;
hence z in f .: K by A2, A7, FUNCT_1:def_6; ::_thesis: verum
end;
end;
end;
hence z in (f .: K) \/ {y} by XBOOLE_0:def_3; ::_thesis: verum
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
let x, y be set ; ::_thesis: for g being Function
for A being set holds (g +* (x .--> y)) .: (A \ {x}) = g .: (A \ {x})
let g be Function; ::_thesis: for A being set holds (g +* (x .--> y)) .: (A \ {x}) = g .: (A \ {x})
let A be set ; ::_thesis: (g +* (x .--> y)) .: (A \ {x}) = g .: (A \ {x})
thus (g +* (x .--> y)) .: (A \ {x}) c= g .: (A \ {x}) :: according to XBOOLE_0:def_10 ::_thesis: g .: (A \ {x}) c= (g +* (x .--> y)) .: (A \ {x})
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in (g +* (x .--> y)) .: (A \ {x}) or u in g .: (A \ {x}) )
A1: dom (x .--> y) = {x} by FUNCOP_1:13;
assume u in (g +* (x .--> y)) .: (A \ {x}) ; ::_thesis: u in g .: (A \ {x})
then consider z being set such that
A2: z in dom (g +* (x .--> y)) and
A3: z in A \ {x} and
A4: u = (g +* (x .--> y)) . z by FUNCT_1:def_6;
A5: not z in {x} by A3, XBOOLE_0:def_5;
then A6: z in dom g by A2, A1, FUNCT_4:12;
u = g . z by A4, A5, A1, FUNCT_4:11;
hence u in g .: (A \ {x}) by A3, A6, FUNCT_1:def_6; ::_thesis: verum
end;
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in g .: (A \ {x}) or u in (g +* (x .--> y)) .: (A \ {x}) )
assume u in g .: (A \ {x}) ; ::_thesis: u in (g +* (x .--> y)) .: (A \ {x})
then consider z being set such that
A7: z in dom g and
A8: z in A \ {x} and
A9: u = g . z by FUNCT_1:def_6;
not z in {x} by A8, XBOOLE_0:def_5;
then not z in dom (x .--> y) by FUNCOP_1:13;
then A10: u = (g +* (x .--> y)) . z by A9, FUNCT_4:11;
z in dom (g +* (x .--> y)) by A7, FUNCT_4:12;
hence u in (g +* (x .--> y)) .: (A \ {x}) by A8, A10, FUNCT_1:def_6; ::_thesis: verum
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
let x, y be set ; ::_thesis: 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}
let g be Function; ::_thesis: for A being set st not y in g .: (A \ {x}) holds
(g +* (x .--> y)) .: (A \ {x}) = ((g +* (x .--> y)) .: A) \ {y}
let A be set ; ::_thesis: ( not y in g .: (A \ {x}) implies (g +* (x .--> y)) .: (A \ {x}) = ((g +* (x .--> y)) .: A) \ {y} )
assume A1: not y in g .: (A \ {x}) ; ::_thesis: (g +* (x .--> y)) .: (A \ {x}) = ((g +* (x .--> y)) .: A) \ {y}
thus (g +* (x .--> y)) .: (A \ {x}) c= ((g +* (x .--> y)) .: A) \ {y} :: according to XBOOLE_0:def_10 ::_thesis: ((g +* (x .--> y)) .: A) \ {y} c= (g +* (x .--> y)) .: (A \ {x})
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in (g +* (x .--> y)) .: (A \ {x}) or u in ((g +* (x .--> y)) .: A) \ {y} )
A2: dom (x .--> y) = {x} by FUNCOP_1:13;
assume A3: u in (g +* (x .--> y)) .: (A \ {x}) ; ::_thesis: u in ((g +* (x .--> y)) .: A) \ {y}
then consider z being set such that
A4: z in dom (g +* (x .--> y)) and
A5: z in A \ {x} and
A6: u = (g +* (x .--> y)) . z by FUNCT_1:def_6;
A7: not z in {x} by A5, XBOOLE_0:def_5;
then A8: z in dom g by A4, A2, FUNCT_4:12;
u = g . z by A6, A7, A2, FUNCT_4:11;
then u <> y by A1, A5, A8, FUNCT_1:def_6;
then A9: not u in {y} by TARSKI:def_1;
(g +* (x .--> y)) .: (A \ {x}) c= (g +* (x .--> y)) .: A by RELAT_1:123;
hence u in ((g +* (x .--> y)) .: A) \ {y} by A3, A9, XBOOLE_0:def_5; ::_thesis: verum
end;
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in ((g +* (x .--> y)) .: A) \ {y} or u in (g +* (x .--> y)) .: (A \ {x}) )
assume A10: u in ((g +* (x .--> y)) .: A) \ {y} ; ::_thesis: u in (g +* (x .--> y)) .: (A \ {x})
then consider z being set such that
A11: z in dom (g +* (x .--> y)) and
A12: z in A and
A13: u = (g +* (x .--> y)) . z by FUNCT_1:def_6;
now__::_thesis:_not_z_in_{x}
assume A14: z in {x} ; ::_thesis: contradiction
then z in dom (x .--> y) by FUNCOP_1:13;
then u = (x .--> y) . z by A13, FUNCT_4:13;
then u = y by A14, FUNCOP_1:7;
then u in {y} by TARSKI:def_1;
hence contradiction by A10, XBOOLE_0:def_5; ::_thesis: verum
end;
then z in A \ {x} by A12, XBOOLE_0:def_5;
hence u in (g +* (x .--> y)) .: (A \ {x}) by A11, A13, FUNCT_1:def_6; ::_thesis: verum
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 Element of NAT ex P being QC-pred_symbol of k,A ex ll being CQC-variable_list of k,A st p = P ! ll
proof
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A st p is atomic holds
ex k being Element of NAT ex P being QC-pred_symbol of k,A ex ll being CQC-variable_list of k,A st p = P ! ll
let p be Element of CQC-WFF A; ::_thesis: ( p is atomic implies ex k being Element of NAT ex P being QC-pred_symbol of k,A ex ll being CQC-variable_list of k,A st p = P ! ll )
assume p is atomic ; ::_thesis: ex k being Element of NAT ex P being QC-pred_symbol of k,A ex ll being CQC-variable_list of k,A st p = P ! ll
then consider k being Element of NAT , P being QC-pred_symbol of k,A, ll being QC-variable_list of k,A such that
A1: p = P ! ll by QC_LANG1:def_18;
A2: { (ll . m) where m is Element of NAT : ( 1 <= m & m <= len ll & ll . m in fixed_QC-variables A ) } = {} by A1, CQC_LANG:7;
{ (ll . i) where i is Element of NAT : ( 1 <= i & i <= len ll & ll . i in free_QC-variables A ) } = {} by A1, CQC_LANG:7;
then ll is CQC-variable_list of k,A by A2, CQC_LANG:5;
hence ex k being Element of NAT ex P being QC-pred_symbol of k,A ex ll being CQC-variable_list of k,A st p = P ! ll by A1; ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: 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
let p be Element of CQC-WFF A; ::_thesis: ( p is negative implies ex q being Element of CQC-WFF A st p = 'not' q )
assume p is negative ; ::_thesis: ex q being Element of CQC-WFF A st p = 'not' q
then consider r being Element of QC-WFF A such that
A1: p = 'not' r by QC_LANG1:def_19;
r is Element of CQC-WFF A by A1, CQC_LANG:8;
hence ex q being Element of CQC-WFF A st p = 'not' q by A1; ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: 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
let p be Element of CQC-WFF A; ::_thesis: ( p is conjunctive implies ex q, r being Element of CQC-WFF A st p = q '&' r )
assume p is conjunctive ; ::_thesis: ex q, r being Element of CQC-WFF A st p = q '&' r
then consider q, r being Element of QC-WFF A such that
A1: p = q '&' r by QC_LANG1:def_20;
A2: r is Element of CQC-WFF A by A1, CQC_LANG:9;
q is Element of CQC-WFF A by A1, CQC_LANG:9;
hence ex q, r being Element of CQC-WFF A st p = q '&' r by A1, A2; ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: 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)
let p be Element of CQC-WFF A; ::_thesis: ( p is universal implies ex x being Element of bound_QC-variables A ex q being Element of CQC-WFF A st p = All (x,q) )
assume p is universal ; ::_thesis: ex x being Element of bound_QC-variables A ex q being Element of CQC-WFF A st p = All (x,q)
then consider x being bound_QC-variable of A, q being Element of QC-WFF A such that
A1: p = All (x,q) by QC_LANG1:def_21;
q is Element of CQC-WFF A by A1, CQC_LANG:13;
hence ex x being Element of bound_QC-variables A ex q being Element of CQC-WFF A st p = All (x,q) by A1; ::_thesis: verum
end;
theorem Th9: :: CQC_SIM1:9
for l being FinSequence holds rng l = { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l ) }
proof
let l be FinSequence; ::_thesis: rng l = { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l ) }
thus rng l c= { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l ) } :: according to XBOOLE_0:def_10 ::_thesis: { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l ) } c= rng l
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in rng l or a in { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l ) } )
assume a in rng l ; ::_thesis: a in { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l ) }
then consider x being set such that
A1: x in dom l and
A2: a = l . x by FUNCT_1:def_3;
reconsider k = x as Element of NAT by A1;
A3: k <= len l by A1, FINSEQ_3:25;
1 <= k by A1, FINSEQ_3:25;
hence a in { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l ) } by A2, A3; ::_thesis: verum
end;
thus { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l ) } c= rng l ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l ) } or a in rng l )
assume a in { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l ) } ; ::_thesis: a in rng l
then consider k being Element of NAT such that
A4: a = l . k and
A5: 1 <= k and
A6: k <= len l ;
k in dom l by A5, A6, FINSEQ_3:25;
hence a in rng l by A4, FUNCT_1:def_3; ::_thesis: verum
end;
end;
scheme :: CQC_SIM1:sch 1
QCFuncExN{ F1() -> QC-alphabet , F2() -> non empty set , F3() -> Element of F2(), F4( set ) -> Element of F2(), F5( set , set ) -> Element of F2(), F6( set , set , set ) -> Element of F2(), F7( set , set ) -> 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 . (the_argument_of p)),p) ) & ( p is conjunctive implies F . p = F6((F . (the_left_argument_of p)),(F . (the_right_argument_of p)),p) ) & ( p is universal implies F . p = F7((F . (the_scope_of p)),p) ) ) ) )
proof
defpred S1[ Function of (QC-WFF F1()),F2(), Nat] means for p being Element of QC-WFF F1() st len (@ p) <= $2 holds
( ( p = VERUM F1() implies $1 . p = F3() ) & ( p is atomic implies $1 . p = F4(p) ) & ( p is negative implies $1 . p = F5(($1 . (the_argument_of p)),p) ) & ( p is conjunctive implies $1 . p = F6(($1 . (the_left_argument_of p)),($1 . (the_right_argument_of p)),p) ) & ( p is universal implies $1 . p = F7(($1 . (the_scope_of p)),p) ) );
defpred S2[ Element of F2(), Function of (QC-WFF F1()),F2(), Element of QC-WFF F1()] means ( ( $3 = VERUM F1() implies $1 = F3() ) & ( $3 is atomic implies $1 = F4($3) ) & ( $3 is negative implies $1 = F5(($2 . (the_argument_of $3)),$3) ) & ( $3 is conjunctive implies $1 = F6(($2 . (the_left_argument_of $3)),($2 . (the_right_argument_of $3)),$3) ) & ( $3 is universal implies $1 = F7(($2 . (the_scope_of $3)),$3) ) );
defpred S3[ Element of NAT ] means ex F being Function of (QC-WFF F1()),F2() st S1[F,$1];
A1: for n being Element of NAT st S3[n] holds
S3[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S3[n] implies S3[n + 1] )
given F being Function of (QC-WFF F1()),F2() such that A2: S1[F,n] ; ::_thesis: S3[n + 1]
defpred S4[ Element of QC-WFF F1(), Element of F2()] means ( ( len (@ $1) <> n + 1 implies $2 = F . $1 ) & ( len (@ $1) = n + 1 implies S2[$2,F,$1] ) );
A3: for x being Element of QC-WFF F1() ex y being Element of F2() st S4[x,y]
proof
let p be Element of QC-WFF F1(); ::_thesis: ex y being Element of F2() st S4[p,y]
now__::_thesis:_(_(_len_(@_p)_<>_n_+_1_&_ex_y_being_Element_of_F2()_st_y_=_F_._p_)_or_(_len_(@_p)_=_n_+_1_&_p_=_VERUM_F1()_&_ex_y_being_Element_of_F2()_st_S2[y,F,p]_)_or_(_len_(@_p)_=_n_+_1_&_p_is_atomic_&_ex_y_being_Element_of_F2()_st_S2[y,F,p]_)_or_(_len_(@_p)_=_n_+_1_&_p_is_negative_&_ex_y_being_Element_of_F2()_st_S2[y,F,p]_)_or_(_len_(@_p)_=_n_+_1_&_p_is_conjunctive_&_ex_y_being_Element_of_F2()_st_S2[y,F,p]_)_or_(_len_(@_p)_=_n_+_1_&_p_is_universal_&_ex_y_being_Element_of_F2()_st_S2[y,F,p]_)_)
percases ( len (@ p) <> n + 1 or ( len (@ p) = n + 1 & p = VERUM F1() ) or ( len (@ p) = n + 1 & p is atomic ) or ( len (@ p) = n + 1 & p is negative ) or ( len (@ p) = n + 1 & p is conjunctive ) or ( len (@ p) = n + 1 & p is universal ) ) by QC_LANG1:9;
case len (@ p) <> n + 1 ; ::_thesis: ex y being Element of F2() st y = F . p
take y = F . p; ::_thesis: y = F . p
thus y = F . p ; ::_thesis: verum
end;
caseA4: ( len (@ p) = n + 1 & p = VERUM F1() ) ; ::_thesis: ex y being Element of F2() st S2[y,F,p]
take y = F3(); ::_thesis: S2[y,F,p]
thus S2[y,F,p] by A4, QC_LANG1:20; ::_thesis: verum
end;
caseA5: ( len (@ p) = n + 1 & p is atomic ) ; ::_thesis: ex y being Element of F2() st S2[y,F,p]
take y = F4(p); ::_thesis: S2[y,F,p]
thus S2[y,F,p] by A5, QC_LANG1:20; ::_thesis: verum
end;
caseA6: ( len (@ p) = n + 1 & p is negative ) ; ::_thesis: ex y being Element of F2() st S2[y,F,p]
take y = F5((F . (the_argument_of p)),p); ::_thesis: S2[y,F,p]
thus S2[y,F,p] by A6, QC_LANG1:20; ::_thesis: verum
end;
caseA7: ( len (@ p) = n + 1 & p is conjunctive ) ; ::_thesis: ex y being Element of F2() st S2[y,F,p]
take y = F6((F . (the_left_argument_of p)),(F . (the_right_argument_of p)),p); ::_thesis: S2[y,F,p]
thus S2[y,F,p] by A7, QC_LANG1:20; ::_thesis: verum
end;
caseA8: ( len (@ p) = n + 1 & p is universal ) ; ::_thesis: ex y being Element of F2() st S2[y,F,p]
take y = F7((F . (the_scope_of p)),p); ::_thesis: S2[y,F,p]
thus S2[y,F,p] by A8, QC_LANG1:20; ::_thesis: verum
end;
end;
end;
hence ex y being Element of F2() st S4[p,y] ; ::_thesis: verum
end;
consider G being Function of (QC-WFF F1()),F2() such that
A9: for p being Element of QC-WFF F1() holds S4[p,G . p] from FUNCT_2:sch_3(A3);
take H = G; ::_thesis: S1[H,n + 1]
thus S1[H,n + 1] ::_thesis: verum
proof
let p be Element of QC-WFF F1(); ::_thesis: ( len (@ p) <= n + 1 implies ( ( p = VERUM F1() implies H . p = F3() ) & ( p is atomic implies H . p = F4(p) ) & ( p is negative implies H . p = F5((H . (the_argument_of p)),p) ) & ( p is conjunctive implies H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p) ) & ( p is universal implies H . p = F7((H . (the_scope_of p)),p) ) ) )
assume A10: len (@ p) <= n + 1 ; ::_thesis: ( ( p = VERUM F1() implies H . p = F3() ) & ( p is atomic implies H . p = F4(p) ) & ( p is negative implies H . p = F5((H . (the_argument_of p)),p) ) & ( p is conjunctive implies H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p) ) & ( p is universal implies H . p = F7((H . (the_scope_of p)),p) ) )
thus ( p = VERUM F1() implies H . p = F3() ) ::_thesis: ( ( p is atomic implies H . p = F4(p) ) & ( p is negative implies H . p = F5((H . (the_argument_of p)),p) ) & ( p is conjunctive implies H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p) ) & ( p is universal implies H . p = F7((H . (the_scope_of p)),p) ) )
proof
now__::_thesis:_(_p_=_VERUM_F1()_implies_H_._p_=_F3()_)
percases ( len (@ p) <> n + 1 or len (@ p) = n + 1 ) ;
supposeA11: len (@ p) <> n + 1 ; ::_thesis: ( p = VERUM F1() implies H . p = F3() )
then A12: H . p = F . p by A9;
len (@ p) <= n by A10, A11, NAT_1:8;
hence ( p = VERUM F1() implies H . p = F3() ) by A2, A12; ::_thesis: verum
end;
suppose len (@ p) = n + 1 ; ::_thesis: ( p = VERUM F1() implies H . p = F3() )
hence ( p = VERUM F1() implies H . p = F3() ) by A9; ::_thesis: verum
end;
end;
end;
hence ( p = VERUM F1() implies H . p = F3() ) ; ::_thesis: verum
end;
thus ( p is atomic implies H . p = F4(p) ) ::_thesis: ( ( p is negative implies H . p = F5((H . (the_argument_of p)),p) ) & ( p is conjunctive implies H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p) ) & ( p is universal implies H . p = F7((H . (the_scope_of p)),p) ) )
proof
now__::_thesis:_(_p_is_atomic_implies_H_._p_=_F4(p)_)
percases ( len (@ p) <> n + 1 or len (@ p) = n + 1 ) ;
supposeA13: len (@ p) <> n + 1 ; ::_thesis: ( p is atomic implies H . p = F4(p) )
then A14: H . p = F . p by A9;
len (@ p) <= n by A10, A13, NAT_1:8;
hence ( p is atomic implies H . p = F4(p) ) by A2, A14; ::_thesis: verum
end;
suppose len (@ p) = n + 1 ; ::_thesis: ( p is atomic implies H . p = F4(p) )
hence ( p is atomic implies H . p = F4(p) ) by A9; ::_thesis: verum
end;
end;
end;
hence ( p is atomic implies H . p = F4(p) ) ; ::_thesis: verum
end;
thus ( p is negative implies H . p = F5((H . (the_argument_of p)),p) ) ::_thesis: ( ( p is conjunctive implies H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p) ) & ( p is universal implies H . p = F7((H . (the_scope_of p)),p) ) )
proof
assume A15: p is negative ; ::_thesis: H . p = F5((H . (the_argument_of p)),p)
then len (@ (the_argument_of p)) <> n + 1 by A10, QC_LANG1:14;
then A16: H . (the_argument_of p) = F . (the_argument_of p) by A9;
now__::_thesis:_H_._p_=_F5((H_._(the_argument_of_p)),p)
percases ( len (@ p) <> n + 1 or len (@ p) = n + 1 ) ;
supposeA17: len (@ p) <> n + 1 ; ::_thesis: H . p = F5((H . (the_argument_of p)),p)
then A18: H . p = F . p by A9;
len (@ p) <= n by A10, A17, NAT_1:8;
hence H . p = F5((H . (the_argument_of p)),p) by A2, A15, A16, A18; ::_thesis: verum
end;
suppose len (@ p) = n + 1 ; ::_thesis: H . p = F5((H . (the_argument_of p)),p)
hence H . p = F5((H . (the_argument_of p)),p) by A9, A15, A16; ::_thesis: verum
end;
end;
end;
hence H . p = F5((H . (the_argument_of p)),p) ; ::_thesis: verum
end;
thus ( p is conjunctive implies H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p) ) ::_thesis: ( p is universal implies H . p = F7((H . (the_scope_of p)),p) )
proof
assume A19: p is conjunctive ; ::_thesis: H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p)
then len (@ (the_right_argument_of p)) <> n + 1 by A10, QC_LANG1:15;
then A20: H . (the_right_argument_of p) = F . (the_right_argument_of p) by A9;
len (@ (the_left_argument_of p)) <> n + 1 by A10, A19, QC_LANG1:15;
then A21: H . (the_left_argument_of p) = F . (the_left_argument_of p) by A9;
now__::_thesis:_H_._p_=_F6((H_._(the_left_argument_of_p)),(H_._(the_right_argument_of_p)),p)
percases ( len (@ p) <> n + 1 or len (@ p) = n + 1 ) ;
supposeA22: len (@ p) <> n + 1 ; ::_thesis: H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p)
then A23: H . p = F . p by A9;
len (@ p) <= n by A10, A22, NAT_1:8;
hence H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p) by A2, A19, A21, A20, A23; ::_thesis: verum
end;
suppose len (@ p) = n + 1 ; ::_thesis: H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p)
hence H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p) by A9, A19, A21, A20; ::_thesis: verum
end;
end;
end;
hence H . p = F6((H . (the_left_argument_of p)),(H . (the_right_argument_of p)),p) ; ::_thesis: verum
end;
thus ( p is universal implies H . p = F7((H . (the_scope_of p)),p) ) ::_thesis: verum
proof
assume A24: p is universal ; ::_thesis: H . p = F7((H . (the_scope_of p)),p)
then len (@ (the_scope_of p)) <> n + 1 by A10, QC_LANG1:16;
then A25: H . (the_scope_of p) = F . (the_scope_of p) by A9;
now__::_thesis:_H_._p_=_F7((H_._(the_scope_of_p)),p)
percases ( len (@ p) <> n + 1 or len (@ p) = n + 1 ) ;
supposeA26: len (@ p) <> n + 1 ; ::_thesis: H . p = F7((H . (the_scope_of p)),p)
then A27: H . p = F . p by A9;
len (@ p) <= n by A10, A26, NAT_1:8;
hence H . p = F7((H . (the_scope_of p)),p) by A2, A24, A25, A27; ::_thesis: verum
end;
suppose len (@ p) = n + 1 ; ::_thesis: H . p = F7((H . (the_scope_of p)),p)
hence H . p = F7((H . (the_scope_of p)),p) by A9, A24, A25; ::_thesis: verum
end;
end;
end;
hence H . p = F7((H . (the_scope_of p)),p) ; ::_thesis: verum
end;
end;
end;
defpred S4[ set , set ] means ex p being Element of QC-WFF F1() st
( p = $1 & ( for g being Function of (QC-WFF F1()),F2() st S1[g, len (@ p)] holds
$2 = g . p ) );
A28: S3[ 0 ]
proof
set F = the Function of (QC-WFF F1()),F2();
take the Function of (QC-WFF F1()),F2() ; ::_thesis: S1[ the Function of (QC-WFF F1()),F2(), 0 ]
thus S1[ the Function of (QC-WFF F1()),F2(), 0 ] by QC_LANG1:10; ::_thesis: verum
end;
A29: for n being Element of NAT holds S3[n] from NAT_1:sch_1(A28, A1);
A30: for x being set st x in QC-WFF F1() holds
ex y being set st S4[x,y]
proof
let x be set ; ::_thesis: ( x in QC-WFF F1() implies ex y being set st S4[x,y] )
assume x in QC-WFF F1() ; ::_thesis: ex y being set st S4[x,y]
then reconsider x9 = x as Element of QC-WFF F1() ;
consider F being Function of (QC-WFF F1()),F2() such that
A31: S1[F, len (@ x9)] by A29;
take F . x ; ::_thesis: S4[x,F . x]
take x9 ; ::_thesis: ( x9 = x & ( for g being Function of (QC-WFF F1()),F2() st S1[g, len (@ x9)] holds
F . x = g . x9 ) )
thus x = x9 ; ::_thesis: for g being Function of (QC-WFF F1()),F2() st S1[g, len (@ x9)] holds
F . x = g . x9
let G be Function of (QC-WFF F1()),F2(); ::_thesis: ( S1[G, len (@ x9)] implies F . x = G . x9 )
assume A32: S1[G, len (@ x9)] ; ::_thesis: F . x = G . x9
defpred S5[ Element of QC-WFF F1()] means ( len (@ $1) <= len (@ x9) implies F . $1 = G . $1 );
A33: now__::_thesis:_for_p_being_Element_of_QC-WFF_F1()_holds_
(_(_p_is_atomic_implies_S5[p]_)_&_S5[_VERUM_F1()]_&_(_p_is_negative_&_S5[_the_argument_of_p]_implies_S5[p]_)_&_(_p_is_conjunctive_&_S5[_the_left_argument_of_p]_&_S5[_the_right_argument_of_p]_implies_S5[p]_)_&_(_p_is_universal_&_S5[_the_scope_of_p]_implies_S5[p]_)_)
let p be Element of QC-WFF F1(); ::_thesis: ( ( p is atomic implies S5[p] ) & S5[ VERUM F1()] & ( p is negative & S5[ the_argument_of p] implies S5[p] ) & ( p is conjunctive & S5[ the_left_argument_of p] & S5[ the_right_argument_of p] implies S5[p] ) & ( p is universal & S5[ the_scope_of p] implies S5[p] ) )
thus ( p is atomic implies S5[p] ) ::_thesis: ( S5[ VERUM F1()] & ( p is negative & S5[ the_argument_of p] implies S5[p] ) & ( p is conjunctive & S5[ the_left_argument_of p] & S5[ the_right_argument_of p] implies S5[p] ) & ( p is universal & S5[ the_scope_of p] implies S5[p] ) )
proof
assume that
A34: p is atomic and
A35: len (@ p) <= len (@ x9) ; ::_thesis: F . p = G . p
thus F . p = F4(p) by A31, A34, A35
.= G . p by A32, A34, A35 ; ::_thesis: verum
end;
thus S5[ VERUM F1()] ::_thesis: ( ( p is negative & S5[ the_argument_of p] implies S5[p] ) & ( p is conjunctive & S5[ the_left_argument_of p] & S5[ the_right_argument_of p] implies S5[p] ) & ( p is universal & S5[ the_scope_of p] implies S5[p] ) )
proof
assume A36: len (@ (VERUM F1())) <= len (@ x9) ; ::_thesis: F . (VERUM F1()) = G . (VERUM F1())
hence F . (VERUM F1()) = F3() by A31
.= G . (VERUM F1()) by A32, A36 ;
::_thesis: verum
end;
thus ( p is negative & S5[ the_argument_of p] implies S5[p] ) ::_thesis: ( ( p is conjunctive & S5[ the_left_argument_of p] & S5[ the_right_argument_of p] implies S5[p] ) & ( p is universal & S5[ the_scope_of p] implies S5[p] ) )
proof
assume that
A37: p is negative and
A38: S5[ the_argument_of p] and
A39: len (@ p) <= len (@ x9) ; ::_thesis: F . p = G . p
len (@ (the_argument_of p)) < len (@ p) by A37, QC_LANG1:14;
hence F . p = F5((G . (the_argument_of p)),p) by A31, A37, A38, A39, XXREAL_0:2
.= G . p by A32, A37, A39 ;
::_thesis: verum
end;
thus ( p is conjunctive & S5[ the_left_argument_of p] & S5[ the_right_argument_of p] implies S5[p] ) ::_thesis: ( p is universal & S5[ the_scope_of p] implies S5[p] )
proof
assume that
A40: p is conjunctive and
A41: S5[ the_left_argument_of p] and
A42: S5[ the_right_argument_of p] and
A43: len (@ p) <= len (@ x9) ; ::_thesis: F . p = G . p
A44: len (@ (the_right_argument_of p)) < len (@ p) by A40, QC_LANG1:15;
len (@ (the_left_argument_of p)) < len (@ p) by A40, QC_LANG1:15;
hence F . p = F6((G . (the_left_argument_of p)),(G . (the_right_argument_of p)),p) by A31, A40, A41, A42, A43, A44, XXREAL_0:2
.= G . p by A32, A40, A43 ;
::_thesis: verum
end;
thus ( p is universal & S5[ the_scope_of p] implies S5[p] ) ::_thesis: verum
proof
assume that
A45: p is universal and
A46: S5[ the_scope_of p] and
A47: len (@ p) <= len (@ x9) ; ::_thesis: F . p = G . p
len (@ (the_scope_of p)) < len (@ p) by A45, QC_LANG1:16;
hence F . p = F7((G . (the_scope_of p)),p) by A31, A45, A46, A47, XXREAL_0:2
.= G . p by A32, A45, A47 ;
::_thesis: verum
end;
end;
for p being Element of QC-WFF F1() holds S5[p] from QC_LANG1:sch_2(A33);
hence F . x = G . x9 ; ::_thesis: verum
end;
consider F being Function such that
A48: dom F = QC-WFF F1() and
A49: for x being set st x in QC-WFF F1() holds
S4[x,F . x] from CLASSES1:sch_1(A30);
rng F c= F2()
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng F or y in F2() )
assume y in rng F ; ::_thesis: y in F2()
then consider x being set such that
A50: x in QC-WFF F1() and
A51: y = F . x by A48, FUNCT_1:def_3;
consider p being Element of QC-WFF F1() such that
p = x and
A52: for g being Function of (QC-WFF F1()),F2() st S1[g, len (@ p)] holds
y = g . p by A49, A50, A51;
consider G being Function of (QC-WFF F1()),F2() such that
A53: S1[G, len (@ p)] by A29;
y = G . p by A52, A53;
hence y in F2() ; ::_thesis: verum
end;
then reconsider F = F as Function of (QC-WFF F1()),F2() by A48, FUNCT_2:def_1, RELSET_1:4;
consider p1 being Element of QC-WFF F1() such that
A54: p1 = VERUM F1() and
A55: for g being Function of (QC-WFF F1()),F2() st S1[g, len (@ p1)] holds
F . (VERUM F1()) = g . p1 by A49;
take F ; ::_thesis: ( 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 . (the_argument_of p)),p) ) & ( p is conjunctive implies F . p = F6((F . (the_left_argument_of p)),(F . (the_right_argument_of p)),p) ) & ( p is universal implies F . p = F7((F . (the_scope_of p)),p) ) ) ) )
consider G being Function of (QC-WFF F1()),F2() such that
A56: S1[G, len (@ p1)] by A29;
F . (VERUM F1()) = G . (VERUM F1()) by A54, A55, A56;
hence F . (VERUM F1()) = F3() by A54, A56; ::_thesis: 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 . (the_argument_of p)),p) ) & ( p is conjunctive implies F . p = F6((F . (the_left_argument_of p)),(F . (the_right_argument_of p)),p) ) & ( p is universal implies F . p = F7((F . (the_scope_of p)),p) ) )
let p be Element of QC-WFF F1(); ::_thesis: ( ( p is atomic implies F . p = F4(p) ) & ( p is negative implies F . p = F5((F . (the_argument_of p)),p) ) & ( p is conjunctive implies F . p = F6((F . (the_left_argument_of p)),(F . (the_right_argument_of p)),p) ) & ( p is universal implies F . p = F7((F . (the_scope_of p)),p) ) )
consider p1 being Element of QC-WFF F1() such that
A57: p1 = p and
A58: for g being Function of (QC-WFF F1()),F2() st S1[g, len (@ p1)] holds
F . p = g . p1 by A49;
consider G being Function of (QC-WFF F1()),F2() such that
A59: S1[G, len (@ p1)] by A29;
set p9 = the_scope_of p;
A60: ex p1 being Element of QC-WFF F1() st
( p1 = the_scope_of p & ( for g being Function of (QC-WFF F1()),F2() st S1[g, len (@ p1)] holds
F . (the_scope_of p) = g . p1 ) ) by A49;
A61: F . p = G . p by A57, A58, A59;
hence ( p is atomic implies F . p = F4(p) ) by A57, A59; ::_thesis: ( ( p is negative implies F . p = F5((F . (the_argument_of p)),p) ) & ( p is conjunctive implies F . p = F6((F . (the_left_argument_of p)),(F . (the_right_argument_of p)),p) ) & ( p is universal implies F . p = F7((F . (the_scope_of p)),p) ) )
A62: for k being Element of NAT st k < len (@ p) holds
S1[G,k]
proof
let k be Element of NAT ; ::_thesis: ( k < len (@ p) implies S1[G,k] )
assume A63: k < len (@ p) ; ::_thesis: S1[G,k]
let p9 be Element of QC-WFF F1(); ::_thesis: ( len (@ p9) <= k implies ( ( p9 = VERUM F1() implies G . p9 = F3() ) & ( p9 is atomic implies G . p9 = F4(p9) ) & ( p9 is negative implies G . p9 = F5((G . (the_argument_of p9)),p9) ) & ( p9 is conjunctive implies G . p9 = F6((G . (the_left_argument_of p9)),(G . (the_right_argument_of p9)),p9) ) & ( p9 is universal implies G . p9 = F7((G . (the_scope_of p9)),p9) ) ) )
assume len (@ p9) <= k ; ::_thesis: ( ( p9 = VERUM F1() implies G . p9 = F3() ) & ( p9 is atomic implies G . p9 = F4(p9) ) & ( p9 is negative implies G . p9 = F5((G . (the_argument_of p9)),p9) ) & ( p9 is conjunctive implies G . p9 = F6((G . (the_left_argument_of p9)),(G . (the_right_argument_of p9)),p9) ) & ( p9 is universal implies G . p9 = F7((G . (the_scope_of p9)),p9) ) )
then len (@ p9) <= len (@ p) by A63, XXREAL_0:2;
hence ( ( p9 = VERUM F1() implies G . p9 = F3() ) & ( p9 is atomic implies G . p9 = F4(p9) ) & ( p9 is negative implies G . p9 = F5((G . (the_argument_of p9)),p9) ) & ( p9 is conjunctive implies G . p9 = F6((G . (the_left_argument_of p9)),(G . (the_right_argument_of p9)),p9) ) & ( p9 is universal implies G . p9 = F7((G . (the_scope_of p9)),p9) ) ) by A57, A59; ::_thesis: verum
end;
thus ( p is negative implies F . p = F5((F . (the_argument_of p)),p) ) ::_thesis: ( ( p is conjunctive implies F . p = F6((F . (the_left_argument_of p)),(F . (the_right_argument_of p)),p) ) & ( p is universal implies F . p = F7((F . (the_scope_of p)),p) ) )
proof
set p9 = the_argument_of p;
set k = len (@ (the_argument_of p));
A64: ex p1 being Element of QC-WFF F1() st
( p1 = the_argument_of p & ( for g being Function of (QC-WFF F1()),F2() st S1[g, len (@ p1)] holds
F . (the_argument_of p) = g . p1 ) ) by A49;
assume A65: p is negative ; ::_thesis: F . p = F5((F . (the_argument_of p)),p)
then len (@ (the_argument_of p)) < len (@ p) by QC_LANG1:14;
then S1[G, len (@ (the_argument_of p))] by A62;
then F . (the_argument_of p) = G . (the_argument_of p) by A64;
hence F . p = F5((F . (the_argument_of p)),p) by A57, A59, A61, A65; ::_thesis: verum
end;
thus ( p is conjunctive implies F . p = F6((F . (the_left_argument_of p)),(F . (the_right_argument_of p)),p) ) ::_thesis: ( p is universal implies F . p = F7((F . (the_scope_of p)),p) )
proof
set p99 = the_right_argument_of p;
set p9 = the_left_argument_of p;
set k9 = len (@ (the_left_argument_of p));
set k99 = len (@ (the_right_argument_of p));
A66: ex p2 being Element of QC-WFF F1() st
( p2 = the_right_argument_of p & ( for g being Function of (QC-WFF F1()),F2() st S1[g, len (@ p2)] holds
F . (the_right_argument_of p) = g . p2 ) ) by A49;
assume A67: p is conjunctive ; ::_thesis: F . p = F6((F . (the_left_argument_of p)),(F . (the_right_argument_of p)),p)
then len (@ (the_left_argument_of p)) < len (@ p) by QC_LANG1:15;
then A68: S1[G, len (@ (the_left_argument_of p))] by A62;
len (@ (the_right_argument_of p)) < len (@ p) by A67, QC_LANG1:15;
then S1[G, len (@ (the_right_argument_of p))] by A62;
then A69: F . (the_right_argument_of p) = G . (the_right_argument_of p) by A66;
ex p1 being Element of QC-WFF F1() st
( p1 = the_left_argument_of p & ( for g being Function of (QC-WFF F1()),F2() st S1[g, len (@ p1)] holds
F . (the_left_argument_of p) = g . p1 ) ) by A49;
then F . (the_left_argument_of p) = G . (the_left_argument_of p) by A68;
hence F . p = F6((F . (the_left_argument_of p)),(F . (the_right_argument_of p)),p) by A57, A59, A61, A67, A69; ::_thesis: verum
end;
set k = len (@ (the_scope_of p));
assume A70: p is universal ; ::_thesis: F . p = F7((F . (the_scope_of p)),p)
then len (@ (the_scope_of p)) < len (@ p) by QC_LANG1:16;
then S1[G, len (@ (the_scope_of p))] by A62;
then F . (the_scope_of p) = G . (the_scope_of p) by A60;
hence F . p = F7((F . (the_scope_of p)),p) by A57, A59, A61, A70; ::_thesis: verum
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( set , set , set ) -> Element of Funcs (F2(),F3()), F6( set , set ) -> Element of Funcs (F2(),F3()), F7( set , set , set , set ) -> Element of Funcs (F2(),F3()), F8( set , set , set ) -> Element of Funcs (F2(),F3()) } :
ex F being Function of (CQC-WFF F1()),(Funcs (F2(),F3())) st
( F . (VERUM F1()) = F4() & ( for k being Element of 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 . ('not' r) = F6((F . r),r) & F . (r '&' s) = F7((F . r),(F . s),r,s) & F . (All (x,r)) = F8(x,(F . r),r) ) ) )
proof
deffunc H1( set , Element of QC-WFF F1()) -> Element of Funcs (F2(),F3()) = F8((bound_in $2),$1,(the_scope_of $2));
deffunc H2( set , set , Element of QC-WFF F1()) -> Element of Funcs (F2(),F3()) = F7($1,$2,(the_left_argument_of $3),(the_right_argument_of $3));
deffunc H3( set , Element of QC-WFF F1()) -> Element of Funcs (F2(),F3()) = F6($1,(the_argument_of $2));
deffunc H4( Element of QC-WFF F1()) -> Element of Funcs (F2(),F3()) = F5((the_arity_of (the_pred_symbol_of $1)),(the_pred_symbol_of $1),(the_arguments_of $1));
consider F being Function of (QC-WFF F1()),(Funcs (F2(),F3())) such that
A1: ( F . (VERUM F1()) = F4() & ( for p being Element of QC-WFF F1() holds
( ( p is atomic implies F . p = H4(p) ) & ( p is negative implies F . p = H3(F . (the_argument_of p),p) ) & ( p is conjunctive implies F . p = H2(F . (the_left_argument_of p),F . (the_right_argument_of p),p) ) & ( p is universal implies F . p = H1(F . (the_scope_of p),p) ) ) ) ) from CQC_SIM1:sch_1();
reconsider G = F | (CQC-WFF F1()) as Function of (CQC-WFF F1()),(Funcs (F2(),F3())) by FUNCT_2:32;
take G ; ::_thesis: ( G . (VERUM F1()) = F4() & ( for k being Element of NAT
for l being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds G . (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
( G . ('not' r) = F6((G . r),r) & G . (r '&' s) = F7((G . r),(G . s),r,s) & G . (All (x,r)) = F8(x,(G . r),r) ) ) )
thus G . (VERUM F1()) = F4() by A1, FUNCT_1:49; ::_thesis: ( ( for k being Element of NAT
for l being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds G . (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
( G . ('not' r) = F6((G . r),r) & G . (r '&' s) = F7((G . r),(G . s),r,s) & G . (All (x,r)) = F8(x,(G . r),r) ) ) )
thus for k being Element of NAT
for l being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds G . (P ! l) = F5(k,P,l) ::_thesis: for r, s being Element of CQC-WFF F1()
for x being Element of bound_QC-variables F1() holds
( G . ('not' r) = F6((G . r),r) & G . (r '&' s) = F7((G . r),(G . s),r,s) & G . (All (x,r)) = F8(x,(G . r),r) )
proof
let k be Element of NAT ; ::_thesis: for l being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds G . (P ! l) = F5(k,P,l)
let l be CQC-variable_list of k,F1(); ::_thesis: for P being QC-pred_symbol of k,F1() holds G . (P ! l) = F5(k,P,l)
let P be QC-pred_symbol of k,F1(); ::_thesis: G . (P ! l) = F5(k,P,l)
A2: the_arity_of P = k by QC_LANG1:11;
A3: P ! l is atomic by QC_LANG1:def_18;
then A4: the_arguments_of (P ! l) = l by QC_LANG1:def_23;
A5: the_pred_symbol_of (P ! l) = P by A3, QC_LANG1:def_22;
thus G . (P ! l) = F . (P ! l) by FUNCT_1:49
.= F5(k,P,l) by A1, A3, A4, A5, A2 ; ::_thesis: verum
end;
let r, s be Element of CQC-WFF F1(); ::_thesis: for x being Element of bound_QC-variables F1() holds
( G . ('not' r) = F6((G . r),r) & G . (r '&' s) = F7((G . r),(G . s),r,s) & G . (All (x,r)) = F8(x,(G . r),r) )
let x be Element of bound_QC-variables F1(); ::_thesis: ( G . ('not' r) = F6((G . r),r) & G . (r '&' s) = F7((G . r),(G . s),r,s) & G . (All (x,r)) = F8(x,(G . r),r) )
set r9 = G . r;
set s9 = G . s;
A6: G . r = F . r by FUNCT_1:49;
A7: 'not' r is negative by QC_LANG1:def_19;
then A8: the_argument_of ('not' r) = r by QC_LANG1:def_24;
thus G . ('not' r) = F . ('not' r) by FUNCT_1:49
.= F6((G . r),r) by A1, A6, A7, A8 ; ::_thesis: ( G . (r '&' s) = F7((G . r),(G . s),r,s) & G . (All (x,r)) = F8(x,(G . r),r) )
A9: G . s = F . s by FUNCT_1:49;
A10: r '&' s is conjunctive by QC_LANG1:def_20;
then A11: the_left_argument_of (r '&' s) = r by QC_LANG1:def_25;
A12: the_right_argument_of (r '&' s) = s by A10, QC_LANG1:def_26;
thus G . (r '&' s) = F . (r '&' s) by FUNCT_1:49
.= F7((G . r),(G . s),r,s) by A1, A6, A9, A10, A11, A12 ; ::_thesis: G . (All (x,r)) = F8(x,(G . r),r)
A13: All (x,r) is universal by QC_LANG1:def_21;
then A14: bound_in (All (x,r)) = x by QC_LANG1:def_27;
A15: the_scope_of (All (x,r)) = r by A13, QC_LANG1:def_28;
thus G . (All (x,r)) = F . (All (x,r)) by FUNCT_1:49
.= F8(x,(G . r),r) by A1, A6, A13, A14, A15 ; ::_thesis: verum
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( set , set , set ) -> Function of F2(),F3(), F8( set , set ) -> Function of F2(),F3(), F9( set , set , set , set ) -> Function of F2(),F3(), F10( set , set , set ) -> Function of F2(),F3() } :
F4() = F5()
provided
A1: F4() . (VERUM F1()) = F6() and
A2: for k being Element of 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() . ('not' r) = 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 Element of 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() . ('not' r) = F8((F5() . r),r) & F5() . (r '&' s) = F9((F5() . r),(F5() . s),r,s) & F5() . (All (x,r)) = F10(x,(F5() . r),r) )
proof
defpred S1[ set ] means F4() . $1 = F5() . $1;
A7: for r, s being Element of CQC-WFF F1()
for x being bound_QC-variable of F1()
for k being Element of NAT
for ll being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds
( S1[ VERUM F1()] & S1[P ! ll] & ( S1[r] implies S1[ 'not' r] ) & ( S1[r] & S1[s] implies S1[r '&' s] ) & ( S1[r] implies S1[ All (x,r)] ) )
proof
let r, s be Element of CQC-WFF F1(); ::_thesis: for x being bound_QC-variable of F1()
for k being Element of NAT
for ll being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds
( S1[ VERUM F1()] & S1[P ! ll] & ( S1[r] implies S1[ 'not' r] ) & ( S1[r] & S1[s] implies S1[r '&' s] ) & ( S1[r] implies S1[ All (x,r)] ) )
let x be Element of bound_QC-variables F1(); ::_thesis: for k being Element of NAT
for ll being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds
( S1[ VERUM F1()] & S1[P ! ll] & ( S1[r] implies S1[ 'not' r] ) & ( S1[r] & S1[s] implies S1[r '&' s] ) & ( S1[r] implies S1[ All (x,r)] ) )
let k be Element of NAT ; ::_thesis: for ll being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds
( S1[ VERUM F1()] & S1[P ! ll] & ( S1[r] implies S1[ 'not' r] ) & ( S1[r] & S1[s] implies S1[r '&' s] ) & ( S1[r] implies S1[ All (x,r)] ) )
let ll be CQC-variable_list of k,F1(); ::_thesis: for P being QC-pred_symbol of k,F1() holds
( S1[ VERUM F1()] & S1[P ! ll] & ( S1[r] implies S1[ 'not' r] ) & ( S1[r] & S1[s] implies S1[r '&' s] ) & ( S1[r] implies S1[ All (x,r)] ) )
let P be QC-pred_symbol of k,F1(); ::_thesis: ( S1[ VERUM F1()] & S1[P ! ll] & ( S1[r] implies S1[ 'not' r] ) & ( S1[r] & S1[s] implies S1[r '&' s] ) & ( S1[r] implies S1[ All (x,r)] ) )
thus F4() . (VERUM F1()) = F5() . (VERUM F1()) by A1, A4; ::_thesis: ( S1[P ! ll] & ( S1[r] implies S1[ 'not' r] ) & ( S1[r] & S1[s] implies S1[r '&' s] ) & ( S1[r] implies S1[ All (x,r)] ) )
F4() . (P ! ll) = F7(k,P,ll) by A2;
hence F4() . (P ! ll) = F5() . (P ! ll) by A5; ::_thesis: ( ( S1[r] implies S1[ 'not' r] ) & ( S1[r] & S1[s] implies S1[r '&' s] ) & ( S1[r] implies S1[ All (x,r)] ) )
F4() . ('not' r) = F8((F4() . r),r) by A3;
hence ( F4() . r = F5() . r implies F4() . ('not' r) = F5() . ('not' r) ) by A6; ::_thesis: ( ( S1[r] & S1[s] implies S1[r '&' s] ) & ( S1[r] implies S1[ All (x,r)] ) )
F4() . (r '&' s) = F9((F4() . r),(F4() . s),r,s) by A3;
hence ( F4() . r = F5() . r & F4() . s = F5() . s implies F4() . (r '&' s) = F5() . (r '&' s) ) by A6; ::_thesis: ( S1[r] implies S1[ All (x,r)] )
F4() . (All (x,r)) = F10(x,(F4() . r),r) by A3;
hence ( S1[r] implies S1[ All (x,r)] ) by A6; ::_thesis: verum
end;
for r being Element of CQC-WFF F1() holds S1[r] from CQC_LANG:sch_1(A7);
hence F4() = F5() by FUNCT_2:63; ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds p is_subformula_of 'not' p
let p be Element of CQC-WFF A; ::_thesis: p is_subformula_of 'not' p
p is_proper_subformula_of 'not' p by QC_LANG2:66;
hence p is_subformula_of 'not' p by QC_LANG2:def_21; ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds
( p is_subformula_of p '&' q & q is_subformula_of p '&' q )
let p, q be Element of CQC-WFF A; ::_thesis: ( p is_subformula_of p '&' q & q is_subformula_of p '&' q )
A1: q is_proper_subformula_of p '&' q by QC_LANG2:69;
p is_proper_subformula_of p '&' q by QC_LANG2:69;
hence ( p is_subformula_of p '&' q & q is_subformula_of p '&' q ) by A1, QC_LANG2:def_21; ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: 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)
let p be Element of CQC-WFF A; ::_thesis: for x being Element of bound_QC-variables A holds p is_subformula_of All (x,p)
let x be Element of bound_QC-variables A; ::_thesis: p is_subformula_of All (x,p)
p is_proper_subformula_of All (x,p) by QC_LANG2:71;
hence p is_subformula_of All (x,p) by QC_LANG2:def_21; ::_thesis: verum
end;
theorem Th13: :: CQC_SIM1:13
for A being QC-alphabet
for k being Element of NAT
for l being CQC-variable_list of k,A
for i being Element of NAT st 1 <= i & i <= len l holds
l . i in bound_QC-variables A
proof
let A be QC-alphabet ; ::_thesis: for k being Element of NAT
for l being CQC-variable_list of k,A
for i being Element of NAT st 1 <= i & i <= len l holds
l . i in bound_QC-variables A
let k be Element of NAT ; ::_thesis: for l being CQC-variable_list of k,A
for i being Element of NAT st 1 <= i & i <= len l holds
l . i in bound_QC-variables A
let l be CQC-variable_list of k,A; ::_thesis: for i being Element of NAT st 1 <= i & i <= len l holds
l . i in bound_QC-variables A
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len l implies l . i in bound_QC-variables A )
assume that
A1: 1 <= i and
A2: i <= len l ; ::_thesis: l . i in bound_QC-variables A
i in dom l by A1, A2, FINSEQ_3:25;
then A3: l . i in rng l by FUNCT_1:def_3;
rng l c= bound_QC-variables A by RELAT_1:def_19;
hence l . i in bound_QC-variables A by A3; ::_thesis: verum
end;
definition
let A be QC-alphabet ;
let D be non empty set ;
let f be Function of D,(CQC-WFF A);
func NEGATIVE f -> Element of Funcs (D,(CQC-WFF A)) 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,(CQC-WFF A)) 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
defpred S1[ set , set ] means for p being Element of CQC-WFF A st p = f . $1 holds
$2 = 'not' p;
A1: for e being Element of D ex u being Element of CQC-WFF A st S1[e,u]
proof
let e be Element of D; ::_thesis: ex u being Element of CQC-WFF A st S1[e,u]
reconsider p = f . e as Element of CQC-WFF A ;
take 'not' p ; ::_thesis: S1[e, 'not' p]
thus S1[e, 'not' p] ; ::_thesis: verum
end;
consider F being Function of D,(CQC-WFF A) such that
A2: for e being Element of D holds S1[e,F . e] from FUNCT_2:sch_3(A1);
F is Element of Funcs (D,(CQC-WFF A)) by FUNCT_2:8;
hence ex b1 being Element of Funcs (D,(CQC-WFF A)) st
for a being Element of D
for p being Element of CQC-WFF A st p = f . a holds
b1 . a = 'not' p by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of Funcs (D,(CQC-WFF A)) 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
let F, G be Element of Funcs (D,(CQC-WFF A)); ::_thesis: ( ( for a being Element of D
for p being Element of CQC-WFF A st p = f . a holds
F . a = 'not' p ) & ( for a being Element of D
for p being Element of CQC-WFF A st p = f . a holds
G . a = 'not' p ) implies F = G )
assume A3: for a being Element of D
for p being Element of CQC-WFF A st p = f . a holds
F . a = 'not' p ; ::_thesis: ( ex a being Element of D ex p being Element of CQC-WFF A st
( p = f . a & not G . a = 'not' p ) or F = G )
assume A4: for a being Element of D
for p being Element of CQC-WFF A st p = f . a holds
G . a = 'not' p ; ::_thesis: F = G
for a being Element of D holds F . a = G . a
proof
let a be Element of D; ::_thesis: F . a = G . a
consider p being Element of CQC-WFF A such that
A5: p = f . a ;
thus F . a = 'not' p by A3, A5
.= G . a by A4, A5 ; ::_thesis: verum
end;
hence F = G by FUNCT_2:63; ::_thesis: verum
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,(CQC-WFF A)
for b4 being Element of Funcs (D,(CQC-WFF A)) 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 [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A);
let n be Element of NAT ;
func CON (f,g,n) -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) means :Def3: :: CQC_SIM1:def 3
for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
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 ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) st
for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
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
defpred S1[ Element of QC-symbols A, set , set ] means for p, q being Element of CQC-WFF A st p = f . [$1,$2] & q = g . [($1 + n),$2] holds
$3 = p '&' q;
A1: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) ex u being Element of CQC-WFF A st S1[t,h,u]
proof
let t be QC-symbol of A; ::_thesis: for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) ex u being Element of CQC-WFF A st S1[t,h,u]
let h be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ex u being Element of CQC-WFF A st S1[t,h,u]
reconsider p = f . [t,h] as Element of CQC-WFF A ;
reconsider q = g . [(t + n),h] as Element of CQC-WFF A ;
take p '&' q ; ::_thesis: S1[t,h,p '&' q]
let p1, q1 be Element of CQC-WFF A; ::_thesis: ( p1 = f . [t,h] & q1 = g . [(t + n),h] implies p '&' q = p1 '&' q1 )
assume that
A2: p1 = f . [t,h] and
A3: q1 = g . [(t + n),h] ; ::_thesis: p '&' q = p1 '&' q1
thus p '&' q = p1 '&' q1 by A2, A3; ::_thesis: verum
end;
consider F being Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A) such that
A4: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds S1[t,h,F . (t,h)] from BINOP_1:sch_3(A1);
reconsider F = F as Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) by FUNCT_2:8;
take F ; ::_thesis: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
F . (t,h) = p '&' q
let t be QC-symbol of A; ::_thesis: for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
F . (t,h) = p '&' q
let h be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
F . (t,h) = p '&' q
let p, q be Element of CQC-WFF A; ::_thesis: ( p = f . (t,h) & q = g . ((t + n),h) implies F . (t,h) = p '&' q )
assume that
A5: p = f . (t,h) and
A6: q = g . ((t + n),h) ; ::_thesis: F . (t,h) = p '&' q
thus F . (t,h) = p '&' q by A4, A5, A6; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) st ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
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 ((bound_QC-variables A),(bound_QC-variables A))
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
let F, G be Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)); ::_thesis: ( ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
F . (t,h) = p '&' q ) & ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
G . (t,h) = p '&' q ) implies F = G )
assume A7: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
F . (t,h) = p '&' q ; ::_thesis: ( ex t being QC-symbol of A ex h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) ex p, q being Element of CQC-WFF A st
( p = f . (t,h) & q = g . ((t + n),h) & not G . (t,h) = p '&' q ) or F = G )
assume A8: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p, q being Element of CQC-WFF A st p = f . (t,h) & q = g . ((t + n),h) holds
G . (t,h) = p '&' q ; ::_thesis: F = G
for a being Element of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds F . a = G . a
proof
let a be Element of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]; ::_thesis: F . a = G . a
consider k being Element of QC-symbols A, h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) such that
A9: a = [k,h] by DOMAIN_1:1;
reconsider q = g . ((k + n),h) as Element of CQC-WFF A ;
reconsider p = f . (k,h) as Element of CQC-WFF A ;
F . (k,h) = p '&' q by A7
.= G . (k,h) by A8 ;
hence F . a = G . a by A9; ::_thesis: verum
end;
hence F = G by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines CON CQC_SIM1:def_3_:_
for A being QC-alphabet
for f, g being Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)
for n being Element of NAT
for b5 being Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) holds
( b5 = CON (f,g,n) iff for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
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 ((bound_QC-variables A),(bound_QC-variables A)) holds h +* (x .--> (x. t)) is Function of (bound_QC-variables A),(bound_QC-variables A)
proof
let A be QC-alphabet ; ::_thesis: for t being QC-symbol of A
for x being Element of bound_QC-variables A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds h +* (x .--> (x. t)) is Function of (bound_QC-variables A),(bound_QC-variables A)
let t be QC-symbol of A; ::_thesis: for x being Element of bound_QC-variables A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds h +* (x .--> (x. t)) is Function of (bound_QC-variables A),(bound_QC-variables A)
let x be Element of bound_QC-variables A; ::_thesis: for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds h +* (x .--> (x. t)) is Function of (bound_QC-variables A),(bound_QC-variables A)
let h be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: h +* (x .--> (x. t)) is Function of (bound_QC-variables A),(bound_QC-variables A)
A1: rng (h +* (x .--> (x. t))) c= (rng h) \/ (rng (x .--> (x. t))) by FUNCT_4:17;
A2: rng (x .--> (x. t)) c= bound_QC-variables A by RELAT_1:def_19;
rng h c= bound_QC-variables A by RELAT_1:def_19;
then A3: (rng h) \/ (rng (x .--> (x. t))) c= bound_QC-variables A by A2, XBOOLE_1:8;
dom (h +* (x .--> (x. t))) = (dom h) \/ (dom ({x} --> (x. t))) by FUNCT_4:def_1
.= (dom h) \/ {x} by FUNCOP_1:13
.= (bound_QC-variables A) \/ {x} by FUNCT_2:52
.= bound_QC-variables A by ZFMISC_1:40 ;
hence h +* (x .--> (x. t)) is Function of (bound_QC-variables A),(bound_QC-variables A) by A1, A3, FUNCT_2:2, XBOOLE_1:1; ::_thesis: verum
end;
definition
let A be QC-alphabet ;
let f be Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A);
let x be bound_QC-variable of A;
func UNIVERSAL (x,f) -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) means :Def4: :: CQC_SIM1:def 4
for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
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 ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) st
for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
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
defpred S1[ Element of QC-symbols A, Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)), set ] means for p being Element of CQC-WFF A st p = f . [($1 ++),($2 +* ({x} --> (x. $1)))] holds
$3 = All ((x. $1),p);
A1: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) ex u being Element of CQC-WFF A st S1[t,h,u]
proof
let t be QC-symbol of A; ::_thesis: for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) ex u being Element of CQC-WFF A st S1[t,h,u]
let h be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ex u being Element of CQC-WFF A st S1[t,h,u]
reconsider h2 = h +* (x .--> (x. t)) as Function of (bound_QC-variables A),(bound_QC-variables A) by Lm1;
reconsider h2 = h2 as Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) by FUNCT_2:8;
reconsider q = f . [(t ++),h2] as Element of CQC-WFF A ;
take All ((x. t),q) ; ::_thesis: S1[t,h, All ((x. t),q)]
thus S1[t,h, All ((x. t),q)] ; ::_thesis: verum
end;
consider F being Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A) such that
A2: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds S1[t,h,F . (t,h)] from BINOP_1:sch_3(A1);
reconsider F = F as Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) by FUNCT_2:8;
take F ; ::_thesis: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
F . (t,h) = All ((x. t),p)
let t be QC-symbol of A; ::_thesis: for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
F . (t,h) = All ((x. t),p)
let h be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
F . (t,h) = All ((x. t),p)
let p be Element of CQC-WFF A; ::_thesis: ( p = f . ((t ++),(h +* (x .--> (x. t)))) implies F . (t,h) = All ((x. t),p) )
assume p = f . ((t ++),(h +* (x .--> (x. t)))) ; ::_thesis: F . (t,h) = All ((x. t),p)
hence F . (t,h) = All ((x. t),p) by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) st ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
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 ((bound_QC-variables A),(bound_QC-variables A))
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
let F, G be Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)); ::_thesis: ( ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
F . (t,h) = All ((x. t),p) ) & ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
G . (t,h) = All ((x. t),p) ) implies F = G )
assume A3: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
F . (t,h) = All ((x. t),p) ; ::_thesis: ( ex t being QC-symbol of A ex h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) ex p being Element of CQC-WFF A st
( p = f . ((t ++),(h +* (x .--> (x. t)))) & not G . (t,h) = All ((x. t),p) ) or F = G )
assume A4: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for p being Element of CQC-WFF A st p = f . ((t ++),(h +* (x .--> (x. t)))) holds
G . (t,h) = All ((x. t),p) ; ::_thesis: F = G
for a being Element of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds F . a = G . a
proof
let a be Element of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]; ::_thesis: F . a = G . a
consider k being Element of QC-symbols A, h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) such that
A5: a = [k,h] by DOMAIN_1:1;
reconsider h2 = h +* (x .--> (x. k)) as Function of (bound_QC-variables A),(bound_QC-variables A) by Lm1;
reconsider h2 = h2 as Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) by FUNCT_2:8;
reconsider p = f . ((k ++),h2) as Element of CQC-WFF A ;
F . (k,h) = All ((x. k),p) by A3
.= G . (k,h) by A4 ;
hence F . a = G . a by A5; ::_thesis: verum
end;
hence F = G by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def4 defines UNIVERSAL CQC_SIM1:def_4_:_
for A being QC-alphabet
for f being Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)
for x being bound_QC-variable of A
for b4 being Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) holds
( b4 = UNIVERSAL (x,f) iff for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
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 Element of NAT
for f being CQC-variable_list of k,A holds f is FinSequence of bound_QC-variables A
proof
let A be QC-alphabet ; ::_thesis: for k being Element of NAT
for f being CQC-variable_list of k,A holds f is FinSequence of bound_QC-variables A
let k be Element of NAT ; ::_thesis: for f being CQC-variable_list of k,A holds f is FinSequence of bound_QC-variables A
let f be CQC-variable_list of k,A; ::_thesis: f is FinSequence of bound_QC-variables A
rng f c= bound_QC-variables A by RELAT_1:def_19;
hence f is FinSequence of bound_QC-variables A by FINSEQ_1:def_4; ::_thesis: verum
end;
definition
let A be QC-alphabet ;
let k be Element of NAT ;
let l be CQC-variable_list of k,A;
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A));
:: original: *
redefine funcf * l -> CQC-variable_list of k,A;
coherence
l * f is CQC-variable_list of k,A
proof
reconsider l9 = l as FinSequence of bound_QC-variables A by Lm2;
reconsider h = f * l9 as FinSequence of bound_QC-variables A by FINSEQ_2:32;
len h = len l9 by FINSEQ_2:33
.= k by CARD_1:def_7 ;
hence l * f is CQC-variable_list of k,A by CARD_1:def_7, FINSEQ_2:24; ::_thesis: verum
end;
end;
definition
let A be QC-alphabet ;
let k be Element of 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 ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) means :Def5: :: CQC_SIM1:def 5
for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds it . (t,h) = P ! (h * l);
existence
ex b1 being Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) st
for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds b1 . (t,h) = P ! (h * l)
proof
deffunc H1( set , Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))) -> Element of CQC-WFF A = P ! ($2 * l);
consider f being Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A) such that
A1: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds f . (t,h) = H1(t,h) from BINOP_1:sch_4();
f is Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) by FUNCT_2:8;
hence ex b1 being Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) st
for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds b1 . (t,h) = P ! (h * l) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) st ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds b1 . (t,h) = P ! (h * l) ) & ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds b2 . (t,h) = P ! (h * l) ) holds
b1 = b2
proof
let F, G be Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)); ::_thesis: ( ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds F . (t,h) = P ! (h * l) ) & ( for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds G . (t,h) = P ! (h * l) ) implies F = G )
assume A2: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds F . (t,h) = P ! (h * l) ; ::_thesis: ( ex t being QC-symbol of A ex h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st not G . (t,h) = P ! (h * l) or F = G )
assume A3: for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds G . (t,h) = P ! (h * l) ; ::_thesis: F = G
for a being Element of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds F . a = G . a
proof
let a be Element of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]; ::_thesis: F . a = G . a
consider k being Element of QC-symbols A, f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) such that
A4: a = [k,f] by DOMAIN_1:1;
F . (k,f) = P ! (f * l) by A2
.= G . (k,f) by A3 ;
hence F . a = G . a by A4; ::_thesis: verum
end;
hence F = G by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def5 defines ATOMIC CQC_SIM1:def_5_:_
for A being QC-alphabet
for k being Element of 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 ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) holds
( b5 = ATOMIC (P,l) iff for t being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) 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 (CQC-WFF A),NAT st
( it = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of 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 . ('not' r) = 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 (CQC-WFF A),NAT st
( b1 = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of 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 . ('not' r) = 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 (CQC-WFF A),NAT st
( b1 = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of 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 . ('not' r) = F . r & F . (r '&' s) = (F . r) + (F . s) & F . (All (x,r)) = (F . r) + 1 ) ) ) & ex F being Function of (CQC-WFF A),NAT st
( b2 = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of 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 . ('not' r) = F . r & F . (r '&' s) = (F . r) + (F . s) & F . (All (x,r)) = (F . r) + 1 ) ) ) holds
b1 = b2;
proof
thus ( ex d being Element of NAT ex F being Function of (CQC-WFF A),NAT st
( d = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = H1(k,P,l) & F . ('not' r) = H2(F . r) & F . (r '&' s) = H3(F . r,F . s) & F . (All (x,r)) = H4(x,F . r) ) ) ) & ( for d1, d2 being Element of NAT st ex F being Function of (CQC-WFF A),NAT st
( d1 = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = H1(k,P,l) & F . ('not' r) = H2(F . r) & F . (r '&' s) = H3(F . r,F . s) & F . (All (x,r)) = H4(x,F . r) ) ) ) & ex F being Function of (CQC-WFF A),NAT st
( d2 = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = H1(k,P,l) & F . ('not' r) = H2(F . r) & F . (r '&' s) = H3(F . r,F . s) & F . (All (x,r)) = H4(x,F . r) ) ) ) holds
d1 = d2 ) ) from CQC_LANG:sch_4(); ::_thesis: verum
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 (CQC-WFF A),NAT st
( b3 = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of 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 . ('not' r) = 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 (CQC-WFF A),(Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)));
let x be Element of CQC-WFF A;
:: original: .
redefine funcf . x -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A));
coherence
f . x is Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A))
proof
thus f . x is Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) ; ::_thesis: verum
end;
end;
definition
let A be QC-alphabet ;
func SepFunc A -> Function of (CQC-WFF A),(Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A))) means :Def7: :: CQC_SIM1:def 7
( it . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) & ( for k being Element of 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 . ('not' r) = NEGATIVE (it . r) & it . (r '&' s) = CON ((it . r),(it . s),(QuantNbr r)) & it . (All (x,r)) = UNIVERSAL (x,(it . r)) ) ) );
existence
ex b1 being Function of (CQC-WFF A),(Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A))) st
( b1 . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) & ( for k being Element of 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 . ('not' r) = NEGATIVE (b1 . r) & b1 . (r '&' s) = CON ((b1 . r),(b1 . s),(QuantNbr r)) & b1 . (All (x,r)) = UNIVERSAL (x,(b1 . r)) ) ) )
proof
deffunc H5( Element of NAT , QC-pred_symbol of $1,A, CQC-variable_list of $1,A) -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) = ATOMIC ($2,$3);
set D = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):];
deffunc H6( Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A), set ) -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) = NEGATIVE $1;
deffunc H7( Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A), Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A), Element of CQC-WFF A, set ) -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) = CON ($1,$2,(QuantNbr $3));
deffunc H8( Element of bound_QC-variables A, Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A), set ) -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) = UNIVERSAL ($1,$2);
reconsider V = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) as Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A) ;
reconsider V = V as Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) by FUNCT_2:8;
consider F being Function of (CQC-WFF A),(Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A))) such that
A1: F . (VERUM A) = V and
A2: for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds F . (P ! l) = H5(k,P,l) and
A3: for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( F . ('not' r) = H6(F . r,r) & F . (r '&' s) = H7(F . r,F . s,r,s) & F . (All (x,r)) = H8(x,F . r,r) ) from CQC_SIM1:sch_2();
take F ; ::_thesis: ( F . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) & ( for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds F . (P ! l) = ATOMIC (P,l) ) & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( F . ('not' r) = NEGATIVE (F . r) & F . (r '&' s) = CON ((F . r),(F . s),(QuantNbr r)) & F . (All (x,r)) = UNIVERSAL (x,(F . r)) ) ) )
thus ( F . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) & ( for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds F . (P ! l) = ATOMIC (P,l) ) & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( F . ('not' r) = NEGATIVE (F . r) & F . (r '&' s) = CON ((F . r),(F . s),(QuantNbr r)) & F . (All (x,r)) = UNIVERSAL (x,(F . r)) ) ) ) by A1, A2, A3; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (CQC-WFF A),(Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A))) st b1 . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) & ( for k being Element of 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 . ('not' r) = NEGATIVE (b1 . r) & b1 . (r '&' s) = CON ((b1 . r),(b1 . s),(QuantNbr r)) & b1 . (All (x,r)) = UNIVERSAL (x,(b1 . r)) ) ) & b2 . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) & ( for k being Element of 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 . ('not' r) = NEGATIVE (b2 . r) & b2 . (r '&' s) = CON ((b2 . r),(b2 . s),(QuantNbr r)) & b2 . (All (x,r)) = UNIVERSAL (x,(b2 . r)) ) ) holds
b1 = b2
proof
deffunc H5( Element of NAT , QC-pred_symbol of $1,A, CQC-variable_list of $1,A) -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) = ATOMIC ($2,$3);
set D = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):];
deffunc H6( Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A), set ) -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) = NEGATIVE $1;
deffunc H7( Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A), Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A), Element of CQC-WFF A, set ) -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) = CON ($1,$2,(QuantNbr $3));
deffunc H8( Element of bound_QC-variables A, Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A), set ) -> Element of Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A)) = UNIVERSAL ($1,$2);
reconsider V = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) as Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A) ;
let F, G be Function of (CQC-WFF A),(Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A))); ::_thesis: ( F . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) & ( for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds F . (P ! l) = ATOMIC (P,l) ) & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( F . ('not' r) = NEGATIVE (F . r) & F . (r '&' s) = CON ((F . r),(F . s),(QuantNbr r)) & F . (All (x,r)) = UNIVERSAL (x,(F . r)) ) ) & G . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) & ( for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds G . (P ! l) = ATOMIC (P,l) ) & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( G . ('not' r) = NEGATIVE (G . r) & G . (r '&' s) = CON ((G . r),(G . s),(QuantNbr r)) & G . (All (x,r)) = UNIVERSAL (x,(G . r)) ) ) implies F = G )
assume that
A4: F . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) and
A5: for k being Element of NAT
for ll being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds F . (P ! ll) = H5(k,P,ll) and
A6: for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( F . ('not' r) = H6(F . r,r) & F . (r '&' s) = H7(F . r,F . s,r,s) & F . (All (x,r)) = H8(x,F . r,r) ) and
A7: G . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) and
A8: for k being Element of NAT
for ll being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds G . (P ! ll) = H5(k,P,ll) and
A9: for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds
( G . ('not' r) = H6(G . r,r) & G . (r '&' s) = H7(G . r,G . s,r,s) & G . (All (x,r)) = H8(x,G . r,r) ) ; ::_thesis: F = G
A10: G . (VERUM A) = V by A7;
A11: F . (VERUM A) = V by A4;
thus F = G from CQC_SIM1:sch_3(A11, A5, A6, A10, A8, A9); ::_thesis: verum
end;
end;
:: deftheorem Def7 defines SepFunc CQC_SIM1:def_7_:_
for A being QC-alphabet
for b2 being Function of (CQC-WFF A),(Funcs ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A))) holds
( b2 = SepFunc A iff ( b2 . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) & ( for k being Element of 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 . ('not' r) = NEGATIVE (b2 . r) & b2 . (r '&' s) = CON ((b2 . r),(b2 . s),(QuantNbr r)) & 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 ((bound_QC-variables A),(bound_QC-variables A));
func SepFunc (p,t,f) -> Element of CQC-WFF A equals :: CQC_SIM1:def 8
((SepFunc A) . p) . [t,f];
correctness
coherence
((SepFunc A) . 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 ((bound_QC-variables A),(bound_QC-variables A)) holds SepFunc (p,t,f) = ((SepFunc A) . p) . [t,f];
theorem :: CQC_SIM1:14
for A being QC-alphabet holds QuantNbr (VERUM A) = 0
proof
let A be QC-alphabet ; ::_thesis: QuantNbr (VERUM A) = 0
deffunc H5( Element of CQC-WFF A) -> Element of NAT = QuantNbr $1;
A1: for p being Element of CQC-WFF A
for d being Element of NAT holds
( d = H5(p) iff ex F being Function of (CQC-WFF A),NAT st
( d = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = H1(k,P,l) & F . ('not' r) = H2(F . r) & F . (r '&' s) = H3(F . r,F . s) & F . (All (x,r)) = H4(x,F . r) ) ) ) ) by Def6;
thus H5( VERUM A) = 0 from CQC_LANG:sch_5(A1); ::_thesis: verum
end;
theorem :: CQC_SIM1:15
for A being QC-alphabet
for k being Element of 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
let A be QC-alphabet ; ::_thesis: for k being Element of NAT
for ll being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds QuantNbr (P ! ll) = 0
let k be Element of NAT ; ::_thesis: for ll being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds QuantNbr (P ! ll) = 0
let ll be CQC-variable_list of k,A; ::_thesis: for P being QC-pred_symbol of k,A holds QuantNbr (P ! ll) = 0
let P be QC-pred_symbol of k,A; ::_thesis: QuantNbr (P ! ll) = 0
deffunc H5( Element of CQC-WFF A) -> Element of NAT = QuantNbr $1;
A1: for p being Element of CQC-WFF A
for d being Element of NAT holds
( d = H5(p) iff ex F being Function of (CQC-WFF A),NAT st
( d = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = H1(k,P,l) & F . ('not' r) = H2(F . r) & F . (r '&' s) = H3(F . r,F . s) & F . (All (x,r)) = H4(x,F . r) ) ) ) ) by Def6;
thus H5(P ! ll) = H1(k,P,ll) from CQC_LANG:sch_6(A1); ::_thesis: verum
end;
theorem :: CQC_SIM1:16
for A being QC-alphabet
for p being Element of CQC-WFF A holds QuantNbr ('not' p) = QuantNbr p
proof
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds QuantNbr ('not' p) = QuantNbr p
let p be Element of CQC-WFF A; ::_thesis: QuantNbr ('not' p) = QuantNbr p
deffunc H5( Element of CQC-WFF A) -> Element of NAT = QuantNbr $1;
A1: for p being Element of CQC-WFF A
for d being Element of NAT holds
( d = H5(p) iff ex F being Function of (CQC-WFF A),NAT st
( d = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = H1(k,P,l) & F . ('not' r) = H2(F . r) & F . (r '&' s) = H3(F . r,F . s) & F . (All (x,r)) = H4(x,F . r) ) ) ) ) by Def6;
thus H5( 'not' p) = H2(H5(p)) from CQC_LANG:sch_7(A1); ::_thesis: verum
end;
theorem :: CQC_SIM1:17
for A being QC-alphabet
for p, q being Element of CQC-WFF A holds QuantNbr (p '&' q) = (QuantNbr p) + (QuantNbr q)
proof
let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds QuantNbr (p '&' q) = (QuantNbr p) + (QuantNbr q)
let p, q be Element of CQC-WFF A; ::_thesis: QuantNbr (p '&' q) = (QuantNbr p) + (QuantNbr q)
deffunc H5( Element of CQC-WFF A) -> Element of NAT = QuantNbr $1;
A1: for p being Element of CQC-WFF A
for d being Element of NAT holds
( d = H5(p) iff ex F being Function of (CQC-WFF A),NAT st
( d = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = H1(k,P,l) & F . ('not' r) = H2(F . r) & F . (r '&' s) = H3(F . r,F . s) & F . (All (x,r)) = H4(x,F . r) ) ) ) ) by Def6;
thus H5(p '&' q) = H3(H5(p),H5(q)) from CQC_LANG:sch_8(A1); ::_thesis: verum
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)) = (QuantNbr p) + 1
proof
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A
for x being Element of bound_QC-variables A holds QuantNbr (All (x,p)) = (QuantNbr p) + 1
let p be Element of CQC-WFF A; ::_thesis: for x being Element of bound_QC-variables A holds QuantNbr (All (x,p)) = (QuantNbr p) + 1
let x be Element of bound_QC-variables A; ::_thesis: QuantNbr (All (x,p)) = (QuantNbr p) + 1
deffunc H5( Element of CQC-WFF A) -> Element of NAT = QuantNbr $1;
A1: for p being Element of CQC-WFF A
for d being Element of NAT holds
( d = H5(p) iff ex F being Function of (CQC-WFF A),NAT st
( d = F . p & F . (VERUM A) = 0 & ( for r, s being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for k being Element of NAT
for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds
( F . (P ! l) = H1(k,P,l) & F . ('not' r) = H2(F . r) & F . (r '&' s) = H3(F . r,F . s) & F . (All (x,r)) = H4(x,F . r) ) ) ) ) by Def6;
thus H5( All (x,p)) = H4(x,H5(p)) from CQC_LANG:sch_9(A1); ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: for p being Element of QC-WFF A holds still_not-bound_in p is finite
defpred S1[ Element of QC-WFF A] means still_not-bound_in $1 is finite ;
A1: for p being Element of QC-WFF A holds
( ( p is atomic implies S1[p] ) & S1[ VERUM A] & ( p is negative & S1[ the_argument_of p] implies S1[p] ) & ( p is conjunctive & S1[ the_left_argument_of p] & S1[ the_right_argument_of p] implies S1[p] ) & ( p is universal & S1[ the_scope_of p] implies S1[p] ) )
proof
let p be Element of QC-WFF A; ::_thesis: ( ( p is atomic implies S1[p] ) & S1[ VERUM A] & ( p is negative & S1[ the_argument_of p] implies S1[p] ) & ( p is conjunctive & S1[ the_left_argument_of p] & S1[ the_right_argument_of p] implies S1[p] ) & ( p is universal & S1[ the_scope_of p] implies S1[p] ) )
thus ( p is atomic implies still_not-bound_in p is finite ) ::_thesis: ( S1[ VERUM A] & ( p is negative & S1[ the_argument_of p] implies S1[p] ) & ( p is conjunctive & S1[ the_left_argument_of p] & S1[ the_right_argument_of p] implies S1[p] ) & ( p is universal & S1[ the_scope_of p] implies S1[p] ) )
proof
deffunc H5( set ) -> set = (the_arguments_of p) . $1;
defpred S2[ Element of NAT ] means ( 1 <= $1 & $1 <= len (the_arguments_of p) );
defpred S3[ Element of NAT ] means ( 1 <= $1 & $1 <= len (the_arguments_of p) & (the_arguments_of p) . $1 in bound_QC-variables A );
A2: for k being Element of NAT st S3[k] holds
S2[k] ;
A3: { H5(k) where k is Element of NAT : S3[k] } c= { H5(n) where n is Element of NAT : S2[n] } from FRAENKEL:sch_1(A2);
assume p is atomic ; ::_thesis: still_not-bound_in p is finite
then still_not-bound_in p = still_not-bound_in (the_arguments_of p) by QC_LANG3:4
.= variables_in ((the_arguments_of p),(bound_QC-variables A)) by QC_LANG3:2
.= { ((the_arguments_of p) . k) where k is Element of NAT : ( 1 <= k & k <= len (the_arguments_of p) & (the_arguments_of p) . k in bound_QC-variables A ) } ;
then still_not-bound_in p c= rng (the_arguments_of p) by A3, Th9;
hence still_not-bound_in p is finite ; ::_thesis: verum
end;
thus still_not-bound_in (VERUM A) is finite by QC_LANG3:3; ::_thesis: ( ( p is negative & S1[ the_argument_of p] implies S1[p] ) & ( p is conjunctive & S1[ the_left_argument_of p] & S1[ the_right_argument_of p] implies S1[p] ) & ( p is universal & S1[ the_scope_of p] implies S1[p] ) )
thus ( p is negative & still_not-bound_in (the_argument_of p) is finite implies still_not-bound_in p is finite ) by QC_LANG3:6; ::_thesis: ( ( p is conjunctive & S1[ the_left_argument_of p] & S1[ the_right_argument_of p] implies S1[p] ) & ( p is universal & S1[ the_scope_of p] implies S1[p] ) )
thus ( p is conjunctive & still_not-bound_in (the_left_argument_of p) is finite & still_not-bound_in (the_right_argument_of p) is finite implies still_not-bound_in p is finite ) ::_thesis: ( p is universal & S1[ the_scope_of p] implies S1[p] )
proof
assume that
A4: p is conjunctive and
A5: still_not-bound_in (the_left_argument_of p) is finite and
A6: still_not-bound_in (the_right_argument_of p) is finite ; ::_thesis: still_not-bound_in p is finite
still_not-bound_in p = (still_not-bound_in (the_left_argument_of p)) \/ (still_not-bound_in (the_right_argument_of p)) by A4, QC_LANG3:9;
hence still_not-bound_in p is finite by A5, A6; ::_thesis: verum
end;
assume that
A7: p is universal and
A8: still_not-bound_in (the_scope_of p) is finite ; ::_thesis: S1[p]
still_not-bound_in p = (still_not-bound_in (the_scope_of p)) \ {(bound_in p)} by A7, QC_LANG3:11;
hence S1[p] by A8; ::_thesis: verum
end;
thus for p being Element of QC-WFF A holds S1[p] from QC_LANG1:sch_2(A1); ::_thesis: verum
end;
scheme :: CQC_SIM1:sch 4
MaxFinDomElem{ F1() -> non empty set , F2() -> set , P1[ set , set ] } :
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
reconsider X = F2() as finite set by A1;
A4: X <> {} by A1;
defpred S1[ set , set ] means ( not $1 in X or ( $2 in X & P1[$1,$2] ) );
A5: for x, y, z being set st S1[x,y] & S1[y,z] holds
S1[x,z] by A1, A3;
A6: for x, y being set holds
( S1[x,y] or S1[y,x] ) by A1, A2;
consider x being set such that
A7: x in X and
A8: for y being set st y in X holds
S1[x,y] from CARD_2:sch_2(A4, A6, A5);
reconsider x = x as Element of F1() by A1, A7;
take x ; ::_thesis: ( x in F2() & ( for y being Element of F1() st y in F2() holds
P1[x,y] ) )
thus x in F2() by A7; ::_thesis: for y being Element of F1() st y in F2() holds
P1[x,y]
let y be Element of F1(); ::_thesis: ( y in F2() implies P1[x,y] )
assume y in F2() ; ::_thesis: P1[x,y]
hence P1[x,y] by A7, A8; ::_thesis: verum
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
id X is Function of X,X ;
hence id X is Element of Funcs (X,X) by FUNCT_2:9; ::_thesis: verum
end;
end;
definition
let A be QC-alphabet ;
let p be Element of CQC-WFF A;
func NBI p -> Subset of (QC-symbols A) 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 (QC-symbols A)
proof
defpred S1[ QC-symbol of A] means for u being QC-symbol of A st $1 <= u holds
not x. u in still_not-bound_in p;
{ t where t is QC-symbol of A : S1[t] } c= QC-symbols A from FRAENKEL:sch_10();
hence { 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 (QC-symbols A) ; ::_thesis: verum
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
set A2 = { 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 } ;
ex t being QC-symbol of A st t in { 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 }
proof
now__::_thesis:_ex_t_being_QC-symbol_of_A_st_t_in__{__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__}_
percases ( still_not-bound_in p = {} or still_not-bound_in p <> {} ) ;
suppose still_not-bound_in p = {} ; ::_thesis: ex t being QC-symbol of A st t in { 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 }
then for u being QC-symbol of A st 0 A <= u holds
not x. u in still_not-bound_in p ;
then 0 A in { 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 } ;
hence ex t being QC-symbol of A st t in { 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 } ; ::_thesis: verum
end;
supposeA1: still_not-bound_in p <> {} ; ::_thesis: ex t being QC-symbol of A st t in { 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 }
defpred S1[ QC-symbol of A] means x. A in still_not-bound_in p;
defpred S2[ set , set ] means for t being QC-symbol of A st t = p holds
x. t = A;
A2: { t where t is QC-symbol of A : S1[t] } c= QC-symbols A from FRAENKEL:sch_10();
A3: for e being set st e in still_not-bound_in p holds
ex b being set st
( b in QC-symbols A & S2[e,b] )
proof
let e be set ; ::_thesis: ( e in still_not-bound_in p implies ex b being set st
( b in QC-symbols A & S2[e,b] ) )
assume e in still_not-bound_in p ; ::_thesis: ex b being set st
( b in QC-symbols A & S2[e,b] )
then reconsider e = e as bound_QC-variable of A ;
consider t being QC-symbol of A such that
A4: x. t = e by QC_LANG3:30;
reconsider t = t as set ;
take t ; ::_thesis: ( t in QC-symbols A & S2[e,t] )
thus ( t in QC-symbols A & S2[e,t] ) by A4; ::_thesis: verum
end;
consider f being Function such that
A5: ( dom f = still_not-bound_in p & rng f c= QC-symbols A ) and
for e being set st e in still_not-bound_in p holds
S2[e,f . e] from FUNCT_1:sch_5(A3);
reconsider f = f as Function of (still_not-bound_in p),(QC-symbols A) by A5, FUNCT_2:def_1, RELSET_1:4;
set x = the Element of still_not-bound_in p;
reconsider x = the Element of still_not-bound_in p as bound_QC-variable of A by A1, TARSKI:def_3;
consider t being QC-symbol of A such that
A6: x. t = x by QC_LANG3:30;
A7: ex a being set st a in { z where z is QC-symbol of A : x. z in still_not-bound_in p }
proof
take t ; ::_thesis: t in { z where z is QC-symbol of A : x. z in still_not-bound_in p }
thus t in { z where z is QC-symbol of A : x. z in still_not-bound_in p } by A1, A6; ::_thesis: verum
end;
defpred S3[ QC-symbol of A, QC-symbol of A] means p <= A;
A8: for t, u being QC-symbol of A holds
( S3[t,u] or S3[u,t] ) by QC_LANG1:24;
A9: for t, u, v being QC-symbol of A st S3[t,u] & S3[u,v] holds
S3[t,v] by QC_LANG1:21;
A10: still_not-bound_in p is finite by Th19;
deffunc H5( bound_QC-variable of A) -> QC-symbol of A = x. A;
A11: { H5(b) where b is Element of bound_QC-variables A : b in still_not-bound_in p } is finite from FRAENKEL:sch_21(A10);
{ (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p } = { w where w is QC-symbol of A : x. w in still_not-bound_in p }
proof
set S1 = { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p } ;
set S2 = { w where w is QC-symbol of A : x. w in still_not-bound_in p } ;
for s being set st s in { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p } holds
s in { w where w is QC-symbol of A : x. w in still_not-bound_in p }
proof
let s be set ; ::_thesis: ( s in { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p } implies s in { w where w is QC-symbol of A : x. w in still_not-bound_in p } )
assume s in { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p } ; ::_thesis: s in { w where w is QC-symbol of A : x. w in still_not-bound_in p }
then consider b being Element of bound_QC-variables A such that
A12: ( s = x. b & b in still_not-bound_in p ) ;
reconsider s1 = s as QC-symbol of A by A12;
x. s1 = b by A12, Def1;
hence s in { w where w is QC-symbol of A : x. w in still_not-bound_in p } by A12; ::_thesis: verum
end;
hence { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p } c= { w where w is QC-symbol of A : x. w in still_not-bound_in p } by TARSKI:def_3; :: according to XBOOLE_0:def_10 ::_thesis: { w where w is QC-symbol of A : x. w in still_not-bound_in p } c= { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p }
for s being set st s in { w where w is QC-symbol of A : x. w in still_not-bound_in p } holds
s in { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p }
proof
let s be set ; ::_thesis: ( s in { w where w is QC-symbol of A : x. w in still_not-bound_in p } implies s in { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p } )
assume s in { w where w is QC-symbol of A : x. w in still_not-bound_in p } ; ::_thesis: s in { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p }
then consider w being QC-symbol of A such that
A13: ( s = w & x. w in still_not-bound_in p ) ;
x. (x. w) = w by Def1;
hence s in { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p } by A13; ::_thesis: verum
end;
hence { w where w is QC-symbol of A : x. w in still_not-bound_in p } c= { (x. b) where b is Element of bound_QC-variables A : b in still_not-bound_in p } by TARSKI:def_3; ::_thesis: verum
end;
then A14: ( { w where w is QC-symbol of A : x. w in still_not-bound_in p } is finite & { z where z is QC-symbol of A : x. z in still_not-bound_in p } <> {} & { v where v is QC-symbol of A : x. v in still_not-bound_in p } c= QC-symbols A ) by A11, A2, A7;
consider v being QC-symbol of A such that
v in { w where w is QC-symbol of A : x. w in still_not-bound_in p } and
A15: for t being QC-symbol of A st t in { z where z is QC-symbol of A : x. z in still_not-bound_in p } holds
S3[v,t] from CQC_SIM1:sch_4(A14, A8, A9);
now__::_thesis:_ex_n_being_Element_of_QC-symbols_A_st_
(_n_=_v_++_&_(_for_z_being_QC-symbol_of_A_st_v_++_<=_z_holds_
not_x._z_in_still_not-bound_in_p_)_)
take n = v ++ ; ::_thesis: ( n = v ++ & ( for z being QC-symbol of A st v ++ <= z holds
not x. z in still_not-bound_in p ) )
thus n = v ++ ; ::_thesis: for z being QC-symbol of A st v ++ <= z holds
not x. z in still_not-bound_in p
let z be QC-symbol of A; ::_thesis: ( v ++ <= z implies not x. z in still_not-bound_in p )
assume that
A16: v ++ <= z and
A17: x. z in still_not-bound_in p ; ::_thesis: contradiction
z in { w where w is QC-symbol of A : x. w in still_not-bound_in p } by A17;
then z <= v by A15;
then v ++ <= v by A16, QC_LANG1:21;
then not v < v ++ by QC_LANG1:25;
hence contradiction by QC_LANG1:27; ::_thesis: verum
end;
then v ++ in { 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 } ;
hence ex t being QC-symbol of A st t in { 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 } ; ::_thesis: verum
end;
end;
end;
hence ex t being QC-symbol of A st t in { 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 } ; ::_thesis: verum
end;
hence not NBI p is empty ; ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds
( index p = 0 A iff p is closed )
let p be Element of CQC-WFF A; ::_thesis: ( index p = 0 A iff p is closed )
thus ( index p = 0 A implies p is closed ) ::_thesis: ( p is closed implies index p = 0 A )
proof
assume index p = 0 A ; ::_thesis: p is closed
then 0 A in NBI p by QC_LANG1:def_35;
then consider t being QC-symbol of A such that
A1: ( t = 0 A & ( for u being QC-symbol of A st t <= u holds
not x. u in still_not-bound_in p ) ) ;
now__::_thesis:_not_still_not-bound_in_p_<>_{}
set a = the Element of still_not-bound_in p;
assume A2: still_not-bound_in p <> {} ; ::_thesis: contradiction
then reconsider a = the Element of still_not-bound_in p as bound_QC-variable of A by TARSKI:def_3;
consider u being QC-symbol of A such that
A3: x. u = a by QC_LANG3:30;
not t <= u by A1, A2, A3;
hence contradiction by A1, QC_LANG1:def_36; ::_thesis: verum
end;
hence p is closed by QC_LANG1:def_31; ::_thesis: verum
end;
assume p is closed ; ::_thesis: index p = 0 A
then for t being QC-symbol of A st 0 A <= t holds
not x. t in still_not-bound_in p by QC_LANG1:def_31;
then A4: 0 A in NBI p ;
0 A = min (NBI p)
proof
assume min (NBI p) <> 0 A ; ::_thesis: contradiction
then consider t being QC-symbol of A such that
A5: ( 0 A <> t & t = min (NBI p) ) ;
t <= 0 A by A4, A5, QC_LANG1:def_35;
then t < 0 A by A5, QC_LANG1:def_34;
then not 0 A <= t by QC_LANG1:25;
hence contradiction by QC_LANG1:def_36; ::_thesis: verum
end;
hence index p = 0 A ; ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: 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
let t be QC-symbol of A; ::_thesis: for p being Element of CQC-WFF A st x. t in still_not-bound_in p holds
t < index p
let p be Element of CQC-WFF A; ::_thesis: ( x. t in still_not-bound_in p implies t < index p )
assume A1: x. t in still_not-bound_in p ; ::_thesis: t < index p
now__::_thesis:_not_min_(NBI_p)_<=_t
min (NBI p) in NBI p by QC_LANG1:def_35;
then A2: ex u being QC-symbol of A st
( u = min (NBI p) & ( for t being QC-symbol of A st u <= t holds
not x. t in still_not-bound_in p ) ) ;
assume min (NBI p) <= t ; ::_thesis: contradiction
hence contradiction by A1, A2; ::_thesis: verum
end;
hence t < index p by QC_LANG1:25; ::_thesis: verum
end;
theorem Th22: :: CQC_SIM1:22
for A being QC-alphabet holds index (VERUM A) = 0 A
proof
let A be QC-alphabet ; ::_thesis: index (VERUM A) = 0 A
VERUM A is closed by QC_LANG3:20;
hence index (VERUM A) = 0 A by Th20; ::_thesis: verum
end;
theorem Th23: :: CQC_SIM1:23
for A being QC-alphabet
for p being Element of CQC-WFF A holds index ('not' p) = index p
proof
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds index ('not' p) = index p
let p be Element of CQC-WFF A; ::_thesis: index ('not' p) = index p
still_not-bound_in p = still_not-bound_in ('not' p) by QC_LANG3:7;
hence index ('not' p) = index p ; ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A holds
( index p <= index (p '&' q) & index q <= index (p '&' q) )
let p, q be Element of CQC-WFF A; ::_thesis: ( index p <= index (p '&' q) & index q <= index (p '&' q) )
A1: still_not-bound_in (p '&' q) = (still_not-bound_in p) \/ (still_not-bound_in q) by QC_LANG3:10;
A2: NBI (p '&' q) c= NBI q
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in NBI (p '&' q) or e in NBI q )
assume e in NBI (p '&' q) ; ::_thesis: e in NBI q
then consider t being QC-symbol of A such that
A3: t = e and
A4: for u being QC-symbol of A st t <= u holds
not x. u in still_not-bound_in (p '&' q) ;
now__::_thesis:_for_u_being_QC-symbol_of_A_st_t_<=_u_holds_
not_x._u_in_still_not-bound_in_q
let u be QC-symbol of A; ::_thesis: ( t <= u implies not x. u in still_not-bound_in q )
assume A5: t <= u ; ::_thesis: not x. u in still_not-bound_in q
still_not-bound_in q c= still_not-bound_in (p '&' q) by A1, XBOOLE_1:7;
hence not x. u in still_not-bound_in q by A4, A5; ::_thesis: verum
end;
hence e in NBI q by A3; ::_thesis: verum
end;
NBI (p '&' q) c= NBI p
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in NBI (p '&' q) or e in NBI p )
assume e in NBI (p '&' q) ; ::_thesis: e in NBI p
then consider t being QC-symbol of A such that
A6: t = e and
A7: for u being QC-symbol of A st t <= u holds
not x. u in still_not-bound_in (p '&' q) ;
now__::_thesis:_for_u_being_QC-symbol_of_A_st_t_<=_u_holds_
not_x._u_in_still_not-bound_in_p
let u be QC-symbol of A; ::_thesis: ( t <= u implies not x. u in still_not-bound_in p )
assume A8: t <= u ; ::_thesis: not x. u in still_not-bound_in p
still_not-bound_in p c= still_not-bound_in (p '&' q) by A1, XBOOLE_1:7;
hence not x. u in still_not-bound_in p by A7, A8; ::_thesis: verum
end;
hence e in NBI p by A6; ::_thesis: verum
end;
hence ( index p <= index (p '&' q) & index q <= index (p '&' q) ) by A2, QC_LANG1:28; ::_thesis: verum
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,(index p),(id (bound_QC-variables A)));
coherence
SepFunc (p,(index p),(id (bound_QC-variables A))) 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,(index p),(id (bound_QC-variables A)));
theorem :: CQC_SIM1:25
for A being QC-alphabet holds SepVar (VERUM A) = VERUM A
proof
let A be QC-alphabet ; ::_thesis: SepVar (VERUM A) = VERUM A
index (VERUM A) = 0 A by Th22;
hence SepVar (VERUM A) = ([:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A)) . [(0 A),(id (bound_QC-variables A))] by Def7
.= VERUM A by FUNCOP_1:7 ;
::_thesis: verum
end;
scheme :: CQC_SIM1:sch 5
CQCInd{ F1() -> QC-alphabet , P1[ set ] } :
for r being Element of CQC-WFF F1() holds P1[r]
provided
A1: P1[ VERUM F1()] and
A2: for k being Element of 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
A6: for r, s being Element of CQC-WFF F1()
for x being bound_QC-variable of F1()
for k being Element of NAT
for l being CQC-variable_list of k,F1()
for P being QC-pred_symbol of k,F1() holds
( P1[ VERUM F1()] & P1[P ! l] & ( P1[r] implies P1[ 'not' r] ) & ( P1[r] & P1[s] implies P1[r '&' s] ) & ( P1[r] implies P1[ All (x,r)] ) ) by A1, A2, A3, A4, A5;
thus for r being Element of CQC-WFF F1() holds P1[r] from CQC_LANG:sch_1(A6); ::_thesis: verum
end;
theorem Th26: :: CQC_SIM1:26
for A being QC-alphabet
for k being Element of 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
let A be QC-alphabet ; ::_thesis: for k being Element of 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
let k be Element of NAT ; ::_thesis: for ll being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds SepVar (P ! ll) = P ! ll
let ll be CQC-variable_list of k,A; ::_thesis: for P being QC-pred_symbol of k,A holds SepVar (P ! ll) = P ! ll
let P be QC-pred_symbol of k,A; ::_thesis: SepVar (P ! ll) = P ! ll
A1: dom ll = dom ll ;
rng ll c= bound_QC-variables A by RELAT_1:def_19;
then reconsider lf = ll as PartFunc of NAT,(bound_QC-variables A) by A1, RELSET_1:4;
A2: (id (bound_QC-variables A)) * lf = ll by PARTFUN1:7;
thus SepVar (P ! ll) = (ATOMIC (P,ll)) . ((index (P ! ll)),(id (bound_QC-variables A))) by Def7
.= P ! ll by A2, Def5 ; ::_thesis: verum
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
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A st p is atomic holds
SepVar p = p
let p be Element of CQC-WFF A; ::_thesis: ( p is atomic implies SepVar p = p )
assume p is atomic ; ::_thesis: SepVar p = p
then ex k being Element of NAT ex P being QC-pred_symbol of k,A ex ll being CQC-variable_list of k,A st p = P ! ll by Th5;
hence SepVar p = p by Th26; ::_thesis: verum
end;
theorem Th28: :: CQC_SIM1:28
for A being QC-alphabet
for p being Element of CQC-WFF A holds SepVar ('not' p) = 'not' (SepVar p)
proof
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds SepVar ('not' p) = 'not' (SepVar p)
let p be Element of CQC-WFF A; ::_thesis: SepVar ('not' p) = 'not' (SepVar p)
reconsider FP = (SepFunc A) . p as Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A) ;
thus SepVar ('not' p) = (NEGATIVE FP) . [(index ('not' p)),(id (bound_QC-variables A))] by Def7
.= (NEGATIVE FP) . [(index p),(id (bound_QC-variables A))] by Th23
.= 'not' (SepVar p) by Def2 ; ::_thesis: verum
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' (SepVar q)
proof
let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A st p is negative & q = the_argument_of p holds
SepVar p = 'not' (SepVar q)
let p, q be Element of CQC-WFF A; ::_thesis: ( p is negative & q = the_argument_of p implies SepVar p = 'not' (SepVar q) )
assume that
A1: p is negative and
A2: q = the_argument_of p ; ::_thesis: SepVar p = 'not' (SepVar q)
p = 'not' q by A1, A2, QC_LANG1:def_24;
hence SepVar p = 'not' (SepVar q) by Th28; ::_thesis: verum
end;
definition
let A be QC-alphabet ;
let p be Element of CQC-WFF A;
let X be Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):];
predX is_Sep-closed_on p means :Def12: :: CQC_SIM1:def 12
( [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in X & ( for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X ) );
end;
:: deftheorem Def12 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 [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
( X is_Sep-closed_on p iff ( [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in X & ( for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) 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 [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] means :Def13: :: CQC_SIM1:def 13
( it is_Sep-closed_on p & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
it c= D ) );
existence
ex b1 being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st
( b1 is_Sep-closed_on p & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
b1 c= D ) )
proof
set S = [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):];
set A2 = { X where X is Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] : X is_Sep-closed_on p } ;
{ X where X is Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] : X is_Sep-closed_on p } c= bool [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { X where X is Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] : X is_Sep-closed_on p } or a in bool [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] )
assume a in { X where X is Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] : X is_Sep-closed_on p } ; ::_thesis: a in bool [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]
then ex X being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st
( X = a & X is_Sep-closed_on p ) ;
hence a in bool [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ; ::_thesis: verum
end;
then reconsider A2 = { X where X is Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] : X is_Sep-closed_on p } as Subset-Family of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ;
take X = meet A2; ::_thesis: ( X is_Sep-closed_on p & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
X c= D ) )
set B = [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):];
[#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] is_Sep-closed_on p
proof
thus [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ; :: according to CQC_SIM1:def_12 ::_thesis: ( ( for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
[q,t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ) & ( for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
( [q,t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] & [r,(t + (QuantNbr q)),K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ) ) & ( 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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ) )
thus for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
[q,t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ; ::_thesis: ( ( for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
( [q,t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] & [r,(t + (QuantNbr q)),K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ) ) & ( 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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ) )
thus for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
( [q,t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] & [r,(t + (QuantNbr q)),K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ) ; ::_thesis: 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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]
let q be Element of CQC-WFF A; ::_thesis: for x being Element of bound_QC-variables A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]
let x be Element of bound_QC-variables A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]
let t be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(All (x,q)),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] implies [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] )
assume [(All (x,q)),t,K,f] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ; ::_thesis: [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]
A1: rng (f +* (x .--> (x. t))) c= (rng f) \/ (rng (x .--> (x. t))) by FUNCT_4:17;
A2: rng (x .--> (x. t)) = {(x. t)} by FUNCOP_1:8;
A3: (bound_QC-variables A) \/ {(x. t)} = bound_QC-variables A by ZFMISC_1:40;
rng f c= bound_QC-variables A by RELAT_1:def_19;
then (rng f) \/ (rng (x .--> (x. t))) c= bound_QC-variables A by A2, A3, XBOOLE_1:9;
then A4: rng (f +* (x .--> (x. t))) c= bound_QC-variables A by A1, XBOOLE_1:1;
dom (f +* (x .--> (x. t))) = (dom f) \/ (dom (x .--> (x. t))) by FUNCT_4:def_1
.= (bound_QC-variables A) \/ (dom (x .--> (x. t))) by FUNCT_2:def_1
.= (bound_QC-variables A) \/ {x} by FUNCOP_1:13
.= bound_QC-variables A by ZFMISC_1:40 ;
then f +* (x .--> (x. t)) is Function of (bound_QC-variables A),(bound_QC-variables A) by A4, FUNCT_2:def_1, RELSET_1:4;
then reconsider ff = f +* (x .--> (x. t)) as Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) by FUNCT_2:8;
[q,(t ++),(K \/ {.x.}),ff] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ;
hence [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ; ::_thesis: verum
end;
then A5: [#] [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] in A2 ;
for Y being set st Y in A2 holds
[p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in Y
proof
let Y be set ; ::_thesis: ( Y in A2 implies [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in Y )
assume Y in A2 ; ::_thesis: [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in Y
then ex X being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st
( X = Y & X is_Sep-closed_on p ) ;
hence [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in Y by Def12; ::_thesis: verum
end;
hence [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in X by A5, SETFAM_1:def_1; :: according to CQC_SIM1:def_12 ::_thesis: ( ( for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X ) & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
X c= D ) )
thus for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in X holds
[q,t,K,f] in X ::_thesis: ( ( for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X ) & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
X c= D ) )
proof
let q be Element of CQC-WFF A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in X holds
[q,t,K,f] in X
let t be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in X holds
[q,t,K,f] in X
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in X holds
[q,t,K,f] in X
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [('not' q),t,K,f] in X implies [q,t,K,f] in X )
assume A6: [('not' q),t,K,f] in X ; ::_thesis: [q,t,K,f] in X
for Y being set st Y in A2 holds
[q,t,K,f] in Y
proof
let Y be set ; ::_thesis: ( Y in A2 implies [q,t,K,f] in Y )
assume A7: Y in A2 ; ::_thesis: [q,t,K,f] in Y
then A8: ex X being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st
( X = Y & X is_Sep-closed_on p ) ;
[('not' q),t,K,f] in Y by A6, A7, SETFAM_1:def_1;
hence [q,t,K,f] in Y by A8, Def12; ::_thesis: verum
end;
hence [q,t,K,f] in X by A5, SETFAM_1:def_1; ::_thesis: verum
end;
thus for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X ) ::_thesis: ( ( 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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X ) & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
X c= D ) )
proof
let q, r be Element of CQC-WFF A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X )
let t be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X )
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X )
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(q '&' r),t,K,f] in X implies ( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X ) )
assume A9: [(q '&' r),t,K,f] in X ; ::_thesis: ( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X )
for Y being set st Y in A2 holds
[q,t,K,f] in Y
proof
let Y be set ; ::_thesis: ( Y in A2 implies [q,t,K,f] in Y )
assume A10: Y in A2 ; ::_thesis: [q,t,K,f] in Y
then A11: ex X being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st
( X = Y & X is_Sep-closed_on p ) ;
[(q '&' r),t,K,f] in Y by A9, A10, SETFAM_1:def_1;
hence [q,t,K,f] in Y by A11, Def12; ::_thesis: verum
end;
hence [q,t,K,f] in X by A5, SETFAM_1:def_1; ::_thesis: [r,(t + (QuantNbr q)),K,f] in X
for Y being set st Y in A2 holds
[r,(t + (QuantNbr q)),K,f] in Y
proof
let Y be set ; ::_thesis: ( Y in A2 implies [r,(t + (QuantNbr q)),K,f] in Y )
assume A12: Y in A2 ; ::_thesis: [r,(t + (QuantNbr q)),K,f] in Y
then A13: ex X being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st
( X = Y & X is_Sep-closed_on p ) ;
[(q '&' r),t,K,f] in Y by A9, A12, SETFAM_1:def_1;
hence [r,(t + (QuantNbr q)),K,f] in Y by A13, Def12; ::_thesis: verum
end;
hence [r,(t + (QuantNbr q)),K,f] in X by A5, SETFAM_1:def_1; ::_thesis: verum
end;
thus 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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X ::_thesis: for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
X c= D
proof
let q be Element of CQC-WFF A; ::_thesis: for x being Element of bound_QC-variables A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
let x be Element of bound_QC-variables A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
let t be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(All (x,q)),t,K,f] in X implies [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X )
assume A14: [(All (x,q)),t,K,f] in X ; ::_thesis: [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
for Y being set st Y in A2 holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in Y
proof
let Y be set ; ::_thesis: ( Y in A2 implies [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in Y )
assume A15: Y in A2 ; ::_thesis: [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in Y
then A16: ex X being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st
( X = Y & X is_Sep-closed_on p ) ;
[(All (x,q)),t,K,f] in Y by A14, A15, SETFAM_1:def_1;
hence [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in Y by A16, Def12; ::_thesis: verum
end;
hence [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X by A5, SETFAM_1:def_1; ::_thesis: verum
end;
let D be Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]; ::_thesis: ( D is_Sep-closed_on p implies X c= D )
assume D is_Sep-closed_on p ; ::_thesis: X c= D
then D in A2 ;
hence X c= D by SETFAM_1:3; ::_thesis: verum
end;
uniqueness
for b1, b2 being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st b1 is_Sep-closed_on p & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
b1 c= D ) & b2 is_Sep-closed_on p & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
b2 c= D ) holds
b1 = b2
proof
let D1, D2 be Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):]; ::_thesis: ( D1 is_Sep-closed_on p & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
D1 c= D ) & D2 is_Sep-closed_on p & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
D2 c= D ) implies D1 = D2 )
assume that
A17: D1 is_Sep-closed_on p and
A18: for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
D1 c= D and
A19: D2 is_Sep-closed_on p and
A20: for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] st D is_Sep-closed_on p holds
D2 c= D ; ::_thesis: D1 = D2
thus ( D1 c= D2 & D2 c= D1 ) by A17, A18, A19, A20; :: according to XBOOLE_0:def_10 ::_thesis: verum
end;
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 [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] holds
( b3 = SepQuadruples p iff ( b3 is_Sep-closed_on p & ( for D being Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] 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,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in SepQuadruples p
proof
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in SepQuadruples p
let p be Element of CQC-WFF A; ::_thesis: [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in SepQuadruples p
SepQuadruples p is_Sep-closed_on p by Def13;
hence [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in SepQuadruples p by Def12; ::_thesis: verum
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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in SepQuadruples p holds
[q,t,K,f] in SepQuadruples p
proof
let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in SepQuadruples p holds
[q,t,K,f] in SepQuadruples p
let p be Element of CQC-WFF A; ::_thesis: for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in SepQuadruples p holds
[q,t,K,f] in SepQuadruples p
SepQuadruples p is_Sep-closed_on p by Def13;
hence for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in SepQuadruples p holds
[q,t,K,f] in SepQuadruples p by Def12; ::_thesis: verum
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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in SepQuadruples p holds
( [q,t,K,f] in SepQuadruples p & [r,(t + (QuantNbr q)),K,f] in SepQuadruples p )
proof
let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in SepQuadruples p holds
( [q,t,K,f] in SepQuadruples p & [r,(t + (QuantNbr q)),K,f] in SepQuadruples p )
let p be Element of CQC-WFF A; ::_thesis: for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in SepQuadruples p holds
( [q,t,K,f] in SepQuadruples p & [r,(t + (QuantNbr q)),K,f] in SepQuadruples p )
SepQuadruples p is_Sep-closed_on p by Def13;
hence for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in SepQuadruples p holds
( [q,t,K,f] in SepQuadruples p & [r,(t + (QuantNbr q)),K,f] in SepQuadruples p ) by Def12; ::_thesis: verum
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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in SepQuadruples p
proof
let A be QC-alphabet ; ::_thesis: 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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in SepQuadruples p
let p be Element of CQC-WFF A; ::_thesis: 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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in SepQuadruples p
SepQuadruples p is_Sep-closed_on p by Def13;
hence 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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in SepQuadruples p by Def12; ::_thesis: verum
end;
theorem Th34: :: CQC_SIM1:34
for A being QC-alphabet
for t being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) holds
( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] or [('not' q),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 + (QuantNbr r) & [(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 ((bound_QC-variables A),(bound_QC-variables A)) 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
let A be QC-alphabet ; ::_thesis: for t being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) holds
( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] or [('not' q),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 + (QuantNbr r) & [(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 ((bound_QC-variables A),(bound_QC-variables A)) 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 ) ) )
let t be QC-symbol of A; ::_thesis: for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) holds
( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] or [('not' q),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 + (QuantNbr r) & [(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 ((bound_QC-variables A),(bound_QC-variables A)) 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 ) ) )
let q, p be Element of CQC-WFF A; ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) holds
( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] or [('not' q),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 + (QuantNbr r) & [(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 ((bound_QC-variables A),(bound_QC-variables A)) 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 ) ) )
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: for K being Finite_Subset of (bound_QC-variables A) holds
( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] or [('not' q),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 + (QuantNbr r) & [(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 ((bound_QC-variables A),(bound_QC-variables A)) 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 ) ) )
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: ( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] or [('not' q),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 + (QuantNbr r) & [(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 ((bound_QC-variables A),(bound_QC-variables A)) 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 ) ) )
assume that
A1: [q,t,K,f] in SepQuadruples p and
A2: [q,t,K,f] <> [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] and
A3: not [('not' q),t,K,f] in SepQuadruples p and
A4: for r being Element of CQC-WFF A holds not [(q '&' r),t,K,f] in SepQuadruples p and
A5: for r being Element of CQC-WFF A
for u being QC-symbol of A holds
( not t = u + (QuantNbr r) or not [(r '&' q),u,K,f] in SepQuadruples p ) and
A6: for x being Element of bound_QC-variables A
for u being QC-symbol of A
for h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) holds
( not u ++ = t or not h +* ({x} --> (x. u)) = f or ( not [(All (x,q)),u,K,h] in SepQuadruples p & not [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p ) ) ; ::_thesis: contradiction
reconsider Y = (SepQuadruples p) \ {[q,t,K,f]} as Subset of [:(CQC-WFF A),(QC-symbols A),(Fin (bound_QC-variables A)),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] ;
A7: SepQuadruples p is_Sep-closed_on p by Def13;
A8: for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in Y holds
[q,t,K,f] in Y
proof
let s be Element of CQC-WFF A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' s),t,K,f] in Y holds
[s,t,K,f] in Y
let u be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' s),u,K,f] in Y holds
[s,u,K,f] in Y
let L be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' s),u,L,f] in Y holds
[s,u,L,f] in Y
let h be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [('not' s),u,L,h] in Y implies [s,u,L,h] in Y )
assume A9: [('not' s),u,L,h] in Y ; ::_thesis: [s,u,L,h] in Y
then ( s <> q or u <> t or L <> K or f <> h ) by A3, XBOOLE_0:def_5;
then A10: [s,u,L,h] <> [q,t,K,f] by XTUPLE_0:5;
[('not' s),u,L,h] in SepQuadruples p by A9, XBOOLE_0:def_5;
then [s,u,L,h] in SepQuadruples p by A7, Def12;
hence [s,u,L,h] in Y by A10, ZFMISC_1:56; ::_thesis: verum
end;
A11: for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in Y holds
( [q,t,K,f] in Y & [r,(t + (QuantNbr q)),K,f] in Y )
proof
let s, r be Element of CQC-WFF A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(s '&' r),t,K,f] in Y holds
( [s,t,K,f] in Y & [r,(t + (QuantNbr s)),K,f] in Y )
let u be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(s '&' r),u,K,f] in Y holds
( [s,u,K,f] in Y & [r,(u + (QuantNbr s)),K,f] in Y )
let L be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(s '&' r),u,L,f] in Y holds
( [s,u,L,f] in Y & [r,(u + (QuantNbr s)),L,f] in Y )
let h be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(s '&' r),u,L,h] in Y implies ( [s,u,L,h] in Y & [r,(u + (QuantNbr s)),L,h] in Y ) )
assume [(s '&' r),u,L,h] in Y ; ::_thesis: ( [s,u,L,h] in Y & [r,(u + (QuantNbr s)),L,h] in Y )
then A12: [(s '&' r),u,L,h] in SepQuadruples p by XBOOLE_0:def_5;
then ( s <> q or u <> t or L <> K or f <> h ) by A4;
then A13: [s,u,L,h] <> [q,t,K,f] by XTUPLE_0:5;
[s,u,L,h] in SepQuadruples p by A7, A12, Def12;
hence [s,u,L,h] in Y by A13, ZFMISC_1:56; ::_thesis: [r,(u + (QuantNbr s)),L,h] in Y
( r <> q or L <> K or f <> h or u + (QuantNbr s) <> t ) by A5, A12;
then A14: [r,(u + (QuantNbr s)),L,h] <> [q,t,K,f] by XTUPLE_0:5;
[r,(u + (QuantNbr s)),L,h] in SepQuadruples p by A7, A12, Def12;
hence [r,(u + (QuantNbr s)),L,h] in Y by A14, ZFMISC_1:56; ::_thesis: verum
end;
A15: Y c= SepQuadruples p by XBOOLE_1:36;
A16: 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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in Y holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in Y
proof
let s be Element of CQC-WFF A; ::_thesis: for x being Element of bound_QC-variables A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,s)),t,K,f] in Y holds
[s,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in Y
let x be Element of bound_QC-variables A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,s)),t,K,f] in Y holds
[s,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in Y
let u be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,s)),u,K,f] in Y holds
[s,(u ++),(K \/ {x}),(f +* (x .--> (x. u)))] in Y
let L be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,s)),u,L,f] in Y holds
[s,(u ++),(L \/ {x}),(f +* (x .--> (x. u)))] in Y
let h be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(All (x,s)),u,L,h] in Y implies [s,(u ++),(L \/ {x}),(h +* (x .--> (x. u)))] in Y )
assume A17: [(All (x,s)),u,L,h] in Y ; ::_thesis: [s,(u ++),(L \/ {x}),(h +* (x .--> (x. u)))] in Y
now__::_thesis:_(_not_[(All_(x,q)),u,K,h]_in_SepQuadruples_p_&_not_[(All_(x,q)),u,(K_\_{x}),h]_in_SepQuadruples_p_&_s_=_q_implies_not_L_\/_{x}_=_K_)
assume that
A18: not [(All (x,q)),u,K,h] in SepQuadruples p and
A19: not [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p ; ::_thesis: ( s = q implies not L \/ {x} = K )
A20: ( s <> q or ( L <> K & L <> K \ {x} ) ) by A17, A18, A19, XBOOLE_0:def_5;
assume A21: s = q ; ::_thesis: not L \/ {x} = K
assume A23: L \/ {x} = K ; ::_thesis: contradiction
then K \ {x} = L \ {x} by XBOOLE_1:40;
hence contradiction by A20, A21, A23, ZFMISC_1:40, ZFMISC_1:57; ::_thesis: verum
end;
then ( s <> q or u ++ <> t or L \/ {x} <> K or f <> h +* ({x} --> (x. u)) ) by A6;
then A24: [s,(u ++),(L \/ {x}),(h +* (x .--> (x. u)))] <> [q,t,K,f] by XTUPLE_0:5;
[(All (x,s)),u,L,h] in SepQuadruples p by A17, XBOOLE_0:def_5;
then [s,(u ++),(L \/ {x}),(h +* (x .--> (x. u)))] in SepQuadruples p by A7, Def12;
hence [s,(u ++),(L \/ {x}),(h +* (x .--> (x. u)))] in Y by A24, ZFMISC_1:56; ::_thesis: verum
end;
[p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in SepQuadruples p by A7, Def12;
then [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in Y by A2, ZFMISC_1:56;
then Y is_Sep-closed_on p by A8, A11, A16, Def12;
then SepQuadruples p c= Y by Def13;
then Y = SepQuadruples p by A15, XBOOLE_0:def_10;
hence contradiction by A1, ZFMISC_1:57; ::_thesis: verum
end;
scheme :: CQC_SIM1:sch 6
Sepregression{ F1() -> QC-alphabet , F2() -> Element of CQC-WFF F1(), P1[ set , set , set , set ] } :
for q being Element of CQC-WFF F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [q,t,K,f] in SepQuadruples F2() holds
P1[q,t,K,f]
provided
A1: P1[F2(), index F2(), {}. (bound_QC-variables F1()), id (bound_QC-variables F1())] and
A2: for q being Element of CQC-WFF F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [('not' q),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 Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(q '&' r),t,K,f] in SepQuadruples F2() & P1[q '&' r,t,K,f] holds
( P1[q,t,K,f] & P1[r,t + (QuantNbr q),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 Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) 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
set Y = { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } ;
reconsider X = (SepQuadruples F2()) /\ { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } as Subset of [:(CQC-WFF F1()),(QC-symbols F1()),(Fin (bound_QC-variables F1())),(Funcs ((bound_QC-variables F1()),(bound_QC-variables F1()))):] ;
A5: SepQuadruples F2() is_Sep-closed_on F2() by Def13;
X is_Sep-closed_on F2()
proof
A6: [F2(),(index F2()),({}. (bound_QC-variables F1())),(id (bound_QC-variables F1()))] in { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } by A1;
[F2(),(index F2()),({}. (bound_QC-variables F1())),(id (bound_QC-variables F1()))] in SepQuadruples F2() by Th30;
hence [F2(),(index F2()),({}. (bound_QC-variables F1())),(id (bound_QC-variables F1()))] in X by A6, XBOOLE_0:def_4; :: according to CQC_SIM1:def_12 ::_thesis: ( ( for q being Element of CQC-WFF F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [('not' q),t,K,f] in X holds
[q,t,K,f] in X ) & ( for q, r being Element of CQC-WFF F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X ) ) & ( for q being Element of CQC-WFF F1()
for x being Element of bound_QC-variables F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X ) )
thus for q being Element of CQC-WFF F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [('not' q),t,K,f] in X holds
[q,t,K,f] in X ::_thesis: ( ( for q, r being Element of CQC-WFF F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X ) ) & ( for q being Element of CQC-WFF F1()
for x being Element of bound_QC-variables F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X ) )
proof
let q be Element of CQC-WFF F1(); ::_thesis: for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [('not' q),t,K,f] in X holds
[q,t,K,f] in X
let t be QC-symbol of F1(); ::_thesis: for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [('not' q),t,K,f] in X holds
[q,t,K,f] in X
let K be Finite_Subset of (bound_QC-variables F1()); ::_thesis: for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [('not' q),t,K,f] in X holds
[q,t,K,f] in X
let f be Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())); ::_thesis: ( [('not' q),t,K,f] in X implies [q,t,K,f] in X )
assume A7: [('not' q),t,K,f] in X ; ::_thesis: [q,t,K,f] in X
then A8: [('not' q),t,K,f] in SepQuadruples F2() by XBOOLE_0:def_4;
[('not' q),t,K,f] in { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } by A7, XBOOLE_0:def_4;
then consider p being Element of CQC-WFF F1(), L being Finite_Subset of (bound_QC-variables F1()), u being QC-symbol of F1(), h being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) such that
A9: [('not' q),t,K,f] = [p,u,L,h] and
A10: P1[p,u,L,h] ;
A11: t = u by A9, XTUPLE_0:5;
A12: f = h by A9, XTUPLE_0:5;
A13: K = L by A9, XTUPLE_0:5;
'not' q = p by A9, XTUPLE_0:5;
then P1[q,t,K,f] by A2, A8, A10, A11, A13, A12;
then A14: [q,t,K,f] in { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } ;
[q,t,K,f] in SepQuadruples F2() by A5, A8, Def12;
hence [q,t,K,f] in X by A14, XBOOLE_0:def_4; ::_thesis: verum
end;
thus for q, r being Element of CQC-WFF F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X ) ::_thesis: for q being Element of CQC-WFF F1()
for x being Element of bound_QC-variables F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
proof
let q, r be Element of CQC-WFF F1(); ::_thesis: for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X )
let t be QC-symbol of F1(); ::_thesis: for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X )
let K be Finite_Subset of (bound_QC-variables F1()); ::_thesis: for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(q '&' r),t,K,f] in X holds
( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X )
let f be Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())); ::_thesis: ( [(q '&' r),t,K,f] in X implies ( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X ) )
assume A15: [(q '&' r),t,K,f] in X ; ::_thesis: ( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X )
then A16: [(q '&' r),t,K,f] in SepQuadruples F2() by XBOOLE_0:def_4;
then A17: [r,(t + (QuantNbr q)),K,f] in SepQuadruples F2() by A5, Def12;
[(q '&' r),t,K,f] in { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } by A15, XBOOLE_0:def_4;
then consider p being Element of CQC-WFF F1(), L being Finite_Subset of (bound_QC-variables F1()), u being QC-symbol of F1(), h being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) such that
A18: [(q '&' r),t,K,f] = [p,u,L,h] and
A19: P1[p,u,L,h] ;
A20: t = u by A18, XTUPLE_0:5;
A21: f = h by A18, XTUPLE_0:5;
A22: K = L by A18, XTUPLE_0:5;
A23: q '&' r = p by A18, XTUPLE_0:5;
then P1[q,t,K,f] by A3, A16, A19, A20, A22, A21;
then A24: [q,t,K,f] in { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } ;
P1[r,t + (QuantNbr q),K,f] by A3, A16, A19, A23, A20, A22, A21;
then A25: [r,(t + (QuantNbr q)),K,f] in { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } ;
[q,t,K,f] in SepQuadruples F2() by A5, A16, Def12;
hence ( [q,t,K,f] in X & [r,(t + (QuantNbr q)),K,f] in X ) by A24, A25, A17, XBOOLE_0:def_4; ::_thesis: verum
end;
let q be Element of CQC-WFF F1(); ::_thesis: for x being Element of bound_QC-variables F1()
for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
let x be bound_QC-variable of F1(); ::_thesis: for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
let t be QC-symbol of F1(); ::_thesis: for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
let K be Finite_Subset of (bound_QC-variables F1()); ::_thesis: for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [(All (x,q)),t,K,f] in X holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
let f be Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())); ::_thesis: ( [(All (x,q)),t,K,f] in X implies [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X )
assume A26: [(All (x,q)),t,K,f] in X ; ::_thesis: [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X
then A27: [(All (x,q)),t,K,f] in SepQuadruples F2() by XBOOLE_0:def_4;
f +* (x .--> (x. t)) is Function of (bound_QC-variables F1()),(bound_QC-variables F1()) by Lm1;
then reconsider g = f +* (x .--> (x. t)) as Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) by FUNCT_2:8;
[(All (x,q)),t,K,f] in { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } by A26, XBOOLE_0:def_4;
then consider p being Element of CQC-WFF F1(), L being Finite_Subset of (bound_QC-variables F1()), u being QC-symbol of F1(), h being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) such that
A28: [(All (x,q)),t,K,f] = [p,u,L,h] and
A29: P1[p,u,L,h] ;
A30: t = u by A28, XTUPLE_0:5;
A31: f = h by A28, XTUPLE_0:5;
A32: K = L by A28, XTUPLE_0:5;
All (x,q) = p by A28, XTUPLE_0:5;
then P1[q,t ++ ,K \/ {x},g] by A4, A27, A29, A30, A32, A31;
then A33: [q,(t ++),(K \/ {.x.}),(f +* (x .--> (x. t)))] in { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } ;
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in SepQuadruples F2() by A5, A27, Def12;
hence [q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in X by A33, XBOOLE_0:def_4; ::_thesis: verum
end;
then A34: SepQuadruples F2() c= X by Def13;
let q be Element of CQC-WFF F1(); ::_thesis: for t being QC-symbol of F1()
for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [q,t,K,f] in SepQuadruples F2() holds
P1[q,t,K,f]
let t be QC-symbol of F1(); ::_thesis: for K being Finite_Subset of (bound_QC-variables F1())
for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [q,t,K,f] in SepQuadruples F2() holds
P1[q,t,K,f]
let K be Finite_Subset of (bound_QC-variables F1()); ::_thesis: for f being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) st [q,t,K,f] in SepQuadruples F2() holds
P1[q,t,K,f]
let f be Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())); ::_thesis: ( [q,t,K,f] in SepQuadruples F2() implies P1[q,t,K,f] )
assume [q,t,K,f] in SepQuadruples F2() ; ::_thesis: P1[q,t,K,f]
then [q,t,K,f] in { [s,v,M,g] where s is Element of CQC-WFF F1(), M is Finite_Subset of (bound_QC-variables F1()), v is QC-symbol of F1(), g is Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) : P1[s,v,M,g] } by A34, XBOOLE_0:def_4;
then consider p being Element of CQC-WFF F1(), L being Finite_Subset of (bound_QC-variables F1()), u being QC-symbol of F1(), h being Element of Funcs ((bound_QC-variables F1()),(bound_QC-variables F1())) such that
A35: [q,t,K,f] = [p,u,L,h] and
A36: P1[p,u,L,h] ;
A37: t = u by A35, XTUPLE_0:5;
A38: K = L by A35, XTUPLE_0:5;
q = p by A35, XTUPLE_0:5;
hence P1[q,t,K,f] by A35, A36, A37, A38, XTUPLE_0:5; ::_thesis: verum
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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,t,K,f] in SepQuadruples p holds
q is_subformula_of p
proof
let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,t,K,f] in SepQuadruples p holds
q is_subformula_of p
let p be Element of CQC-WFF A; ::_thesis: for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,t,K,f] in SepQuadruples p holds
q is_subformula_of p
defpred S1[ Element of CQC-WFF A, set , set , set ] means $1 is_subformula_of p;
A1: now__::_thesis:_for_q_being_Element_of_CQC-WFF_A
for_t_being_QC-symbol_of_A
for_K_being_Finite_Subset_of_(bound_QC-variables_A)
for_f_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_st_[('not'_q),t,K,f]_in_SepQuadruples_p_&_S1[_'not'_q,t,K,f]_holds_
S1[q,t,K,f]
let q be Element of CQC-WFF A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in SepQuadruples p & S1[ 'not' q,t,K,f] holds
S1[q,t,K,f]
let t be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in SepQuadruples p & S1[ 'not' q,t,K,f] holds
S1[q,t,K,f]
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in SepQuadruples p & S1[ 'not' q,t,K,f] holds
S1[q,t,K,f]
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [('not' q),t,K,f] in SepQuadruples p & S1[ 'not' q,t,K,f] implies S1[q,t,K,f] )
assume [('not' q),t,K,f] in SepQuadruples p ; ::_thesis: ( S1[ 'not' q,t,K,f] implies S1[q,t,K,f] )
q is_subformula_of 'not' q by Th10;
hence ( S1[ 'not' q,t,K,f] implies S1[q,t,K,f] ) by QC_LANG2:57; ::_thesis: verum
end;
A2: now__::_thesis:_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_Finite_Subset_of_(bound_QC-variables_A)
for_f_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_st_[(All_(x,q)),t,K,f]_in_SepQuadruples_p_&_S1[_All_(x,q),t,K,f]_holds_
S1[q,t_++_,K_\/_{x},f_+*_(x_.-->_(x._t))]
let q be Element of CQC-WFF A; ::_thesis: for x being Element of bound_QC-variables A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p & S1[ All (x,q),t,K,f] holds
S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let x be Element of bound_QC-variables A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p & S1[ All (x,q),t,K,f] holds
S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let t be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p & S1[ All (x,q),t,K,f] holds
S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p & S1[ All (x,q),t,K,f] holds
S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(All (x,q)),t,K,f] in SepQuadruples p & S1[ All (x,q),t,K,f] implies S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))] )
assume [(All (x,q)),t,K,f] in SepQuadruples p ; ::_thesis: ( S1[ All (x,q),t,K,f] implies S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))] )
q is_subformula_of All (x,q) by Th12;
hence ( S1[ All (x,q),t,K,f] implies S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))] ) by QC_LANG2:57; ::_thesis: verum
end;
A3: now__::_thesis:_for_q,_r_being_Element_of_CQC-WFF_A
for_t_being_QC-symbol_of_A
for_K_being_Finite_Subset_of_(bound_QC-variables_A)
for_f_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_st_[(q_'&'_r),t,K,f]_in_SepQuadruples_p_&_S1[q_'&'_r,t,K,f]_holds_
(_S1[q,t,K,f]_&_S1[r,t_+_(QuantNbr_q),K,f]_)
let q, r be Element of CQC-WFF A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in SepQuadruples p & S1[q '&' r,t,K,f] holds
( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] )
let t be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in SepQuadruples p & S1[q '&' r,t,K,f] holds
( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] )
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in SepQuadruples p & S1[q '&' r,t,K,f] holds
( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] )
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(q '&' r),t,K,f] in SepQuadruples p & S1[q '&' r,t,K,f] implies ( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] ) )
assume [(q '&' r),t,K,f] in SepQuadruples p ; ::_thesis: ( S1[q '&' r,t,K,f] implies ( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] ) )
A4: r is_subformula_of q '&' r by Th11;
q is_subformula_of q '&' r by Th11;
hence ( S1[q '&' r,t,K,f] implies ( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] ) ) by A4, QC_LANG2:57; ::_thesis: verum
end;
A5: S1[p, index p, {}. (bound_QC-variables A), id (bound_QC-variables A)] ;
thus for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,t,K,f] in SepQuadruples p holds
S1[q,t,K,f] from CQC_SIM1:sch_6(A5, A1, A3, A2); ::_thesis: verum
end;
theorem :: CQC_SIM1:36
for A being QC-alphabet holds SepQuadruples (VERUM A) = {[(VERUM A),(0 A),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]}
proof
let A be QC-alphabet ; ::_thesis: SepQuadruples (VERUM A) = {[(VERUM A),(0 A),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]}
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_SepQuadruples_(VERUM_A)_implies_x_=_[(VERUM_A),(0_A),({}._(bound_QC-variables_A)),(id_(bound_QC-variables_A))]_)_&_(_x_=_[(VERUM_A),(0_A),({}._(bound_QC-variables_A)),(id_(bound_QC-variables_A))]_implies_x_in_SepQuadruples_(VERUM_A)_)_)
let x be set ; ::_thesis: ( ( x in SepQuadruples (VERUM A) implies x = [(VERUM A),(0 A),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] ) & ( x = [(VERUM A),(0 A),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] implies x in SepQuadruples (VERUM A) ) )
thus ( x in SepQuadruples (VERUM A) implies x = [(VERUM A),(0 A),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] ) ::_thesis: ( x = [(VERUM A),(0 A),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] implies x in SepQuadruples (VERUM A) )
proof
assume A1: x in SepQuadruples (VERUM A) ; ::_thesis: x = [(VERUM A),(0 A),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]
then consider q being Element of CQC-WFF A, t being QC-symbol of A, K being Finite_Subset of (bound_QC-variables A), f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) such that
A2: x = [q,t,K,f] by DOMAIN_1:10;
A3: now__::_thesis:_for_x_being_Element_of_bound_QC-variables_A
for_v_being_QC-symbol_of_A
for_h_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_holds_
(_not_v_++_=_t_or_not_h_+*_({x}_-->_(x._v))_=_f_or_(_not_[(All_(x,q)),v,K,h]_in_SepQuadruples_(VERUM_A)_&_not_[(All_(x,q)),v,(K_\_{.x.}),h]_in_SepQuadruples_(VERUM_A)_)_)
given x being Element of bound_QC-variables A, v being QC-symbol of A, h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) such that v ++ = t and
h +* ({x} --> (x. v)) = f and
A4: ( [(All (x,q)),v,K,h] in SepQuadruples (VERUM A) or [(All (x,q)),v,(K \ {.x.}),h] in SepQuadruples (VERUM A) ) ; ::_thesis: contradiction
All (x,q) is_subformula_of VERUM A by A4, Th35;
then All (x,q) = VERUM A by QC_LANG2:79;
then VERUM A is universal by QC_LANG1:def_21;
hence contradiction by QC_LANG1:20; ::_thesis: verum
end;
A5: now__::_thesis:_for_r_being_Element_of_CQC-WFF_A
for_v_being_QC-symbol_of_A_holds_
(_not_t_=_v_+_(QuantNbr_r)_or_not_[(r_'&'_q),v,K,f]_in_SepQuadruples_(VERUM_A)_)
given r being Element of CQC-WFF A, v being QC-symbol of A such that t = v + (QuantNbr r) and
A6: [(r '&' q),v,K,f] in SepQuadruples (VERUM A) ; ::_thesis: contradiction
r '&' q is_subformula_of VERUM A by A6, Th35;
then r '&' q = VERUM A by QC_LANG2:79;
then VERUM A is conjunctive by QC_LANG1:def_20;
hence contradiction by QC_LANG1:20; ::_thesis: verum
end;
A7: now__::_thesis:_for_r_being_Element_of_CQC-WFF_A_holds_not_[(q_'&'_r),t,K,f]_in_SepQuadruples_(VERUM_A)
given r being Element of CQC-WFF A such that A8: [(q '&' r),t,K,f] in SepQuadruples (VERUM A) ; ::_thesis: contradiction
q '&' r is_subformula_of VERUM A by A8, Th35;
then q '&' r = VERUM A by QC_LANG2:79;
then VERUM A is conjunctive by QC_LANG1:def_20;
hence contradiction by QC_LANG1:20; ::_thesis: verum
end;
A9: now__::_thesis:_not_[('not'_q),t,K,f]_in_SepQuadruples_(VERUM_A)
assume [('not' q),t,K,f] in SepQuadruples (VERUM A) ; ::_thesis: contradiction
then 'not' q is_subformula_of VERUM A by Th35;
then 'not' q = VERUM A by QC_LANG2:79;
then VERUM A is negative by QC_LANG1:def_19;
hence contradiction by QC_LANG1:20; ::_thesis: verum
end;
A: index (VERUM A) = 0 A by Th22;
set p = VERUM A;
( [q,t,K,f] = [(VERUM A),(index (VERUM A)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] or [('not' q),t,K,f] in SepQuadruples (VERUM A) or ex r being Element of CQC-WFF A st [(q '&' r),t,K,f] in SepQuadruples (VERUM A) or ex r being Element of CQC-WFF A ex u being QC-symbol of A st
( t = u + (QuantNbr r) & [(r '&' q),u,K,f] in SepQuadruples (VERUM A) ) or ex x being Element of bound_QC-variables A ex u being QC-symbol of A ex h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st
( u ++ = t & h +* ({x} --> (x. u)) = f & ( [(All (x,q)),u,K,h] in SepQuadruples (VERUM A) or [(All (x,q)),u,(K \ {x}),h] in SepQuadruples (VERUM A) ) ) ) by A1, A2, Th34;
hence x = [(VERUM A),(0 A),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] by A2, A7, A5, A3, A9, A; ::_thesis: verum
end;
index (VERUM A) = 0 A by Th22;
hence ( x = [(VERUM A),(0 A),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] implies x in SepQuadruples (VERUM A) ) by Th30; ::_thesis: verum
end;
hence SepQuadruples (VERUM A) = {[(VERUM A),(0 A),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]} by TARSKI:def_1; ::_thesis: verum
end;
theorem :: CQC_SIM1:37
for A being QC-alphabet
for k being Element of 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)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]}
proof
let A be QC-alphabet ; ::_thesis: for k being Element of 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)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]}
let k be Element of NAT ; ::_thesis: 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)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]}
let l be CQC-variable_list of k,A; ::_thesis: for P being QC-pred_symbol of k,A holds SepQuadruples (P ! l) = {[(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]}
let P be QC-pred_symbol of k,A; ::_thesis: SepQuadruples (P ! l) = {[(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]}
A1: P ! l is atomic by QC_LANG1:def_18;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_SepQuadruples_(P_!_l)_implies_x_=_[(P_!_l),(index_(P_!_l)),({}._(bound_QC-variables_A)),(id_(bound_QC-variables_A))]_)_&_(_x_=_[(P_!_l),(index_(P_!_l)),({}._(bound_QC-variables_A)),(id_(bound_QC-variables_A))]_implies_x_in_SepQuadruples_(P_!_l)_)_)
let x be set ; ::_thesis: ( ( x in SepQuadruples (P ! l) implies x = [(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] ) & ( x = [(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] implies x in SepQuadruples (P ! l) ) )
thus ( x in SepQuadruples (P ! l) implies x = [(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] ) ::_thesis: ( x = [(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] implies x in SepQuadruples (P ! l) )
proof
assume A2: x in SepQuadruples (P ! l) ; ::_thesis: x = [(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]
then consider q being Element of CQC-WFF A, t being QC-symbol of A, K being Finite_Subset of (bound_QC-variables A), f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) such that
A3: x = [q,t,K,f] by DOMAIN_1:10;
A4: now__::_thesis:_for_x_being_Element_of_bound_QC-variables_A
for_u_being_QC-symbol_of_A
for_h_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_holds_
(_not_u_++_=_t_or_not_h_+*_({x}_-->_(x._u))_=_f_or_(_not_[(All_(x,q)),u,K,h]_in_SepQuadruples_(P_!_l)_&_not_[(All_(x,q)),u,(K_\_{.x.}),h]_in_SepQuadruples_(P_!_l)_)_)
given x being Element of bound_QC-variables A, u being QC-symbol of A, h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) such that u ++ = t and
h +* ({x} --> (x. u)) = f and
A5: ( [(All (x,q)),u,K,h] in SepQuadruples (P ! l) or [(All (x,q)),u,(K \ {.x.}),h] in SepQuadruples (P ! l) ) ; ::_thesis: contradiction
All (x,q) is_subformula_of P ! l by A5, Th35;
then All (x,q) = P ! l by QC_LANG2:80;
then P ! l is universal by QC_LANG1:def_21;
hence contradiction by A1, QC_LANG1:20; ::_thesis: verum
end;
A6: now__::_thesis:_for_r_being_Element_of_CQC-WFF_A
for_u_being_QC-symbol_of_A_holds_
(_not_t_=_u_+_(QuantNbr_r)_or_not_[(r_'&'_q),u,K,f]_in_SepQuadruples_(P_!_l)_)
given r being Element of CQC-WFF A, u being QC-symbol of A such that t = u + (QuantNbr r) and
A7: [(r '&' q),u,K,f] in SepQuadruples (P ! l) ; ::_thesis: contradiction
r '&' q is_subformula_of P ! l by A7, Th35;
then r '&' q = P ! l by QC_LANG2:80;
then P ! l is conjunctive by QC_LANG1:def_20;
hence contradiction by A1, QC_LANG1:20; ::_thesis: verum
end;
A8: now__::_thesis:_for_r_being_Element_of_CQC-WFF_A_holds_not_[(q_'&'_r),t,K,f]_in_SepQuadruples_(P_!_l)
given r being Element of CQC-WFF A such that A9: [(q '&' r),t,K,f] in SepQuadruples (P ! l) ; ::_thesis: contradiction
q '&' r is_subformula_of P ! l by A9, Th35;
then q '&' r = P ! l by QC_LANG2:80;
then P ! l is conjunctive by QC_LANG1:def_20;
hence contradiction by A1, QC_LANG1:20; ::_thesis: verum
end;
A10: now__::_thesis:_not_[('not'_q),t,K,f]_in_SepQuadruples_(P_!_l)
assume [('not' q),t,K,f] in SepQuadruples (P ! l) ; ::_thesis: contradiction
then 'not' q is_subformula_of P ! l by Th35;
then 'not' q = P ! l by QC_LANG2:80;
then P ! l is negative by QC_LANG1:def_19;
hence contradiction by A1, QC_LANG1:20; ::_thesis: verum
end;
set p = P ! l;
( [q,t,K,f] = [(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] or [('not' q),t,K,f] in SepQuadruples (P ! l) or ex r being Element of CQC-WFF A st [(q '&' r),t,K,f] in SepQuadruples (P ! l) or ex r being Element of CQC-WFF A ex u being QC-symbol of A st
( t = u + (QuantNbr r) & [(r '&' q),u,K,f] in SepQuadruples (P ! l) ) or ex x being Element of bound_QC-variables A ex u being QC-symbol of A ex h being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st
( u ++ = t & h +* ({x} --> (x. u)) = f & ( [(All (x,q)),u,K,h] in SepQuadruples (P ! l) or [(All (x,q)),u,(K \ {x}),h] in SepQuadruples (P ! l) ) ) ) by A2, Th34, A3;
hence x = [(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] by A3, A8, A6, A4, A10; ::_thesis: verum
end;
thus ( x = [(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] implies x in SepQuadruples (P ! l) ) by Th30; ::_thesis: verum
end;
hence SepQuadruples (P ! l) = {[(P ! l),(index (P ! l)),({}. (bound_QC-variables A)),(id (bound_QC-variables A))]} by TARSKI:def_1; ::_thesis: verum
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 Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,t,K,f] in SepQuadruples p holds
still_not-bound_in q c= (still_not-bound_in p) \/ K
proof
let A be QC-alphabet ; ::_thesis: for p, q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,t,K,f] in SepQuadruples p holds
still_not-bound_in q c= (still_not-bound_in p) \/ K
let p be Element of CQC-WFF A; ::_thesis: for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,t,K,f] in SepQuadruples p holds
still_not-bound_in q c= (still_not-bound_in p) \/ K
deffunc H5( QC-formula of A) -> Element of bool (bound_QC-variables A) = still_not-bound_in $1;
defpred S1[ QC-formula of A, set , set , set ] means H5($1) c= H5(p) \/ $3;
A1: for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),t,K,f] in SepQuadruples p & S1[ 'not' q,t,K,f] holds
S1[q,t,K,f] by QC_LANG3:7;
A2: now__::_thesis:_for_q,_r_being_Element_of_CQC-WFF_A
for_t_being_QC-symbol_of_A
for_K_being_Finite_Subset_of_(bound_QC-variables_A)
for_f_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_st_[(q_'&'_r),t,K,f]_in_SepQuadruples_p_&_S1[q_'&'_r,t,K,f]_holds_
(_S1[q,t,K,f]_&_S1[r,t_+_(QuantNbr_q),K,f]_)
let q, r be Element of CQC-WFF A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in SepQuadruples p & S1[q '&' r,t,K,f] holds
( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] )
let t be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in SepQuadruples p & S1[q '&' r,t,K,f] holds
( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] )
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),t,K,f] in SepQuadruples p & S1[q '&' r,t,K,f] holds
( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] )
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(q '&' r),t,K,f] in SepQuadruples p & S1[q '&' r,t,K,f] implies ( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] ) )
assume that
[(q '&' r),t,K,f] in SepQuadruples p and
A3: S1[q '&' r,t,K,f] ; ::_thesis: ( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] )
A4: still_not-bound_in (q '&' r) = (still_not-bound_in q) \/ (still_not-bound_in r) by QC_LANG3:10;
then A5: still_not-bound_in r c= still_not-bound_in (q '&' r) by XBOOLE_1:7;
still_not-bound_in q c= still_not-bound_in (q '&' r) by A4, XBOOLE_1:7;
hence ( S1[q,t,K,f] & S1[r,t + (QuantNbr q),K,f] ) by A3, A5, XBOOLE_1:1; ::_thesis: verum
end;
A6: now__::_thesis:_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_Finite_Subset_of_(bound_QC-variables_A)
for_f_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_st_[(All_(x,q)),t,K,f]_in_SepQuadruples_p_&_S1[_All_(x,q),t,K,f]_holds_
S1[q,t_++_,K_\/_{x},f_+*_(x_.-->_(x._t))]
let q be Element of CQC-WFF A; ::_thesis: for x being Element of bound_QC-variables A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p & S1[ All (x,q),t,K,f] holds
S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let x be Element of bound_QC-variables A; ::_thesis: for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p & S1[ All (x,q),t,K,f] holds
S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let t be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p & S1[ All (x,q),t,K,f] holds
S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),t,K,f] in SepQuadruples p & S1[ All (x,q),t,K,f] holds
S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(All (x,q)),t,K,f] in SepQuadruples p & S1[ All (x,q),t,K,f] implies S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))] )
assume that
[(All (x,q)),t,K,f] in SepQuadruples p and
A7: S1[ All (x,q),t,K,f] ; ::_thesis: S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
still_not-bound_in (All (x,q)) = (still_not-bound_in q) \ {x} by QC_LANG3:12;
then still_not-bound_in q c= ((still_not-bound_in p) \/ K) \/ {x} by A7, XBOOLE_1:44;
hence S1[q,t ++ ,K \/ {x},f +* (x .--> (x. t))] by XBOOLE_1:4; ::_thesis: verum
end;
A8: S1[p, index p, {}. (bound_QC-variables A), id (bound_QC-variables A)] ;
thus for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,t,K,f] in SepQuadruples p holds
S1[q,t,K,f] from CQC_SIM1:sch_6(A8, A1, A2, A6); ::_thesis: verum
end;
theorem Th39: :: CQC_SIM1:39
for A being QC-alphabet
for t, u being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: K holds
u < t
proof
let A be QC-alphabet ; ::_thesis: for t, u being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: K holds
u < t
let t, u be QC-symbol of A; ::_thesis: for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: K holds
u < t
let q, p be Element of CQC-WFF A; ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: K holds
u < t
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: K holds
u < t
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: ( [q,t,K,f] in SepQuadruples p & x. u in f .: K implies u < t )
defpred S1[ Element of CQC-WFF A, QC-symbol of A, Finite_Subset of (bound_QC-variables A), Function] means for u being QC-symbol of A st x. u in $4 .: $3 holds
u < $2;
A1: for q being Element of CQC-WFF A
for v being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),v,K,f] in SepQuadruples p & S1[ 'not' q,v,K,f] holds
S1[q,v,K,f] ;
A2: now__::_thesis:_for_q,_r_being_Element_of_CQC-WFF_A
for_v_being_QC-symbol_of_A
for_K_being_Finite_Subset_of_(bound_QC-variables_A)
for_f_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_st_[(q_'&'_r),v,K,f]_in_SepQuadruples_p_&_S1[q_'&'_r,v,K,f]_holds_
(_S1[q,v,K,f]_&_S1[r,v_+_(QuantNbr_q),K,f]_)
let q, r be Element of CQC-WFF A; ::_thesis: for v being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),v,K,f] in SepQuadruples p & S1[q '&' r,v,K,f] holds
( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] )
let v be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),v,K,f] in SepQuadruples p & S1[q '&' r,v,K,f] holds
( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] )
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),v,K,f] in SepQuadruples p & S1[q '&' r,v,K,f] holds
( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] )
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(q '&' r),v,K,f] in SepQuadruples p & S1[q '&' r,v,K,f] implies ( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] ) )
assume [(q '&' r),v,K,f] in SepQuadruples p ; ::_thesis: ( S1[q '&' r,v,K,f] implies ( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] ) )
assume A3: S1[q '&' r,v,K,f] ; ::_thesis: ( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] )
hence S1[q,v,K,f] ; ::_thesis: S1[r,v + (QuantNbr q),K,f]
thus S1[r,v + (QuantNbr q),K,f] ::_thesis: verum
proof
let u be QC-symbol of A; ::_thesis: ( x. u in f .: K implies u < v + (QuantNbr q) )
A4: v <= v + (QuantNbr q) by QC_LANG1:31;
assume x. u in f .: K ; ::_thesis: u < v + (QuantNbr q)
hence u < v + (QuantNbr q) by A3, A4, QC_LANG1:30; ::_thesis: verum
end;
end;
A5: now__::_thesis:_for_q_being_Element_of_CQC-WFF_A
for_x_being_Element_of_bound_QC-variables_A
for_v_being_QC-symbol_of_A
for_K_being_Finite_Subset_of_(bound_QC-variables_A)
for_f_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_st_[(All_(x,q)),v,K,f]_in_SepQuadruples_p_&_S1[_All_(x,q),v,K,f]_holds_
S1[q,v_++_,K_\/_{.x.},f_+*_(x_.-->_(x._v))]
let q be Element of CQC-WFF A; ::_thesis: for x being Element of bound_QC-variables A
for v being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),v,K,f] in SepQuadruples p & S1[ All (x,q),v,K,f] holds
S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]
let x be Element of bound_QC-variables A; ::_thesis: for v being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),v,K,f] in SepQuadruples p & S1[ All (x,q),v,K,f] holds
S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]
let v be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),v,K,f] in SepQuadruples p & S1[ All (x,q),v,K,f] holds
S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),v,K,f] in SepQuadruples p & S1[ All (x,q),v,K,f] holds
S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(All (x,q)),v,K,f] in SepQuadruples p & S1[ All (x,q),v,K,f] implies S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))] )
assume [(All (x,q)),v,K,f] in SepQuadruples p ; ::_thesis: ( S1[ All (x,q),v,K,f] implies S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))] )
assume A6: S1[ All (x,q),v,K,f] ; ::_thesis: S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]
thus S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))] ::_thesis: verum
proof
let u be QC-symbol of A; ::_thesis: ( x. u in (f +* (x .--> (x. v))) .: (K \/ {.x.}) implies u < v ++ )
assume x. u in (f +* (x .--> (x. v))) .: (K \/ {x}) ; ::_thesis: u < v ++
then x. u in ((f +* (x .--> (x. v))) .: K) \/ ((f +* (x .--> (x. v))) .: {x}) by RELAT_1:120;
then A7: ( x. u in (f +* (x .--> (x. v))) .: K or x. u in Im ((f +* (x .--> (x. v))),x) ) by XBOOLE_0:def_3;
(f +* (x .--> (x. v))) .: K c= (f .: K) \/ {(x. v)} by Th2;
then ( x. u in f .: K or x. u in {(x. v)} ) by A7, Th1, XBOOLE_0:def_3;
then ( u < v or x. u = x. v ) by A6, TARSKI:def_1;
then ( u < v or u = v ) by XTUPLE_0:1;
then ( u <= v & v < v ++ ) by QC_LANG1:22, QC_LANG1:27, QC_LANG1:def_34;
hence u < v ++ by QC_LANG1:29; ::_thesis: verum
end;
end;
A8: S1[p, index p, {}. (bound_QC-variables A), id (bound_QC-variables A)] ;
for q being Element of CQC-WFF A
for v being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,v,K,f] in SepQuadruples p holds
S1[q,v,K,f] from CQC_SIM1:sch_6(A8, A1, A2, A5);
hence ( [q,t,K,f] in SepQuadruples p & x. u in f .: K implies u < t ) ; ::_thesis: verum
end;
theorem :: CQC_SIM1:40
for A being QC-alphabet
for t being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: K
proof
let A be QC-alphabet ; ::_thesis: for t being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: K
let t be QC-symbol of A; ::_thesis: for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: K
let q, p be Element of CQC-WFF A; ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: K
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: K
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: ( [q,t,K,f] in SepQuadruples p implies not x. t in f .: K )
assume A1: [q,t,K,f] in SepQuadruples p ; ::_thesis: not x. t in f .: K
assume x. t in f .: K ; ::_thesis: contradiction
then ( t < t & t <= t ) by A1, Th39, QC_LANG1:22;
hence contradiction by QC_LANG1:25; ::_thesis: verum
end;
theorem Th41: :: CQC_SIM1:41
for A being QC-alphabet
for t, u being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in p) holds
u < t
proof
let A be QC-alphabet ; ::_thesis: for t, u being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in p) holds
u < t
let t, u be QC-symbol of A; ::_thesis: for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in p) holds
u < t
let q, p be Element of CQC-WFF A; ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in p) holds
u < t
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in p) holds
u < t
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: ( [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in p) implies u < t )
defpred S1[ Element of CQC-WFF A, QC-symbol of A, Finite_Subset of (bound_QC-variables A), Function] means for u being QC-symbol of A st x. u in $4 .: (still_not-bound_in p) holds
u < $2;
A1: now__::_thesis:_for_q,_r_being_Element_of_CQC-WFF_A
for_v_being_QC-symbol_of_A
for_K_being_Finite_Subset_of_(bound_QC-variables_A)
for_f_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_st_[(q_'&'_r),v,K,f]_in_SepQuadruples_p_&_S1[q_'&'_r,v,K,f]_holds_
(_S1[q,v,K,f]_&_S1[r,v_+_(QuantNbr_q),K,f]_)
let q, r be Element of CQC-WFF A; ::_thesis: for v being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),v,K,f] in SepQuadruples p & S1[q '&' r,v,K,f] holds
( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] )
let v be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),v,K,f] in SepQuadruples p & S1[q '&' r,v,K,f] holds
( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] )
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),v,K,f] in SepQuadruples p & S1[q '&' r,v,K,f] holds
( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] )
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(q '&' r),v,K,f] in SepQuadruples p & S1[q '&' r,v,K,f] implies ( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] ) )
assume [(q '&' r),v,K,f] in SepQuadruples p ; ::_thesis: ( S1[q '&' r,v,K,f] implies ( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] ) )
assume A2: S1[q '&' r,v,K,f] ; ::_thesis: ( S1[q,v,K,f] & S1[r,v + (QuantNbr q),K,f] )
hence S1[q,v,K,f] ; ::_thesis: S1[r,v + (QuantNbr q),K,f]
thus S1[r,v + (QuantNbr q),K,f] ::_thesis: verum
proof
let u be QC-symbol of A; ::_thesis: ( x. u in f .: (still_not-bound_in p) implies u < v + (QuantNbr q) )
A3: v <= v + (QuantNbr q) by QC_LANG1:31;
assume x. u in f .: (still_not-bound_in p) ; ::_thesis: u < v + (QuantNbr q)
hence u < v + (QuantNbr q) by A2, A3, QC_LANG1:30; ::_thesis: verum
end;
end;
A4: S1[p, index p, {}. (bound_QC-variables A), id (bound_QC-variables A)]
proof
let u be QC-symbol of A; ::_thesis: ( x. u in (id (bound_QC-variables A)) .: (still_not-bound_in p) implies u < index p )
assume A5: x. u in (id (bound_QC-variables A)) .: (still_not-bound_in p) ; ::_thesis: u < index p
(id (bound_QC-variables A)) .: (still_not-bound_in p) = still_not-bound_in p by FUNCT_1:92;
hence u < index p by A5, Th21; ::_thesis: verum
end;
A6: now__::_thesis:_for_q_being_Element_of_CQC-WFF_A
for_x_being_Element_of_bound_QC-variables_A
for_v_being_QC-symbol_of_A
for_K_being_Finite_Subset_of_(bound_QC-variables_A)
for_f_being_Element_of_Funcs_((bound_QC-variables_A),(bound_QC-variables_A))_st_[(All_(x,q)),v,K,f]_in_SepQuadruples_p_&_S1[_All_(x,q),v,K,f]_holds_
S1[q,v_++_,K_\/_{.x.},f_+*_(x_.-->_(x._v))]
let q be Element of CQC-WFF A; ::_thesis: for x being Element of bound_QC-variables A
for v being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),v,K,f] in SepQuadruples p & S1[ All (x,q),v,K,f] holds
S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]
let x be Element of bound_QC-variables A; ::_thesis: for v being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),v,K,f] in SepQuadruples p & S1[ All (x,q),v,K,f] holds
S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]
let v be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),v,K,f] in SepQuadruples p & S1[ All (x,q),v,K,f] holds
S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),v,K,f] in SepQuadruples p & S1[ All (x,q),v,K,f] holds
S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(All (x,q)),v,K,f] in SepQuadruples p & S1[ All (x,q),v,K,f] implies S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))] )
assume [(All (x,q)),v,K,f] in SepQuadruples p ; ::_thesis: ( S1[ All (x,q),v,K,f] implies S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))] )
assume A7: S1[ All (x,q),v,K,f] ; ::_thesis: S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]
thus S1[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))] ::_thesis: verum
proof
let u be QC-symbol of A; ::_thesis: ( x. u in (f +* (x .--> (x. v))) .: (still_not-bound_in p) implies u < v ++ )
assume A8: x. u in (f +* (x .--> (x. v))) .: (still_not-bound_in p) ; ::_thesis: u < v ++
(f +* (x .--> (x. v))) .: (still_not-bound_in p) c= (f .: (still_not-bound_in p)) \/ {(x. v)} by Th2;
then ( x. u in f .: (still_not-bound_in p) or x. u in {(x. v)} ) by A8, XBOOLE_0:def_3;
then ( u < v or x. u = x. v ) by A7, TARSKI:def_1;
then ( u < v or u = v ) by XTUPLE_0:1;
then ( u <= v & v < v ++ ) by QC_LANG1:22, QC_LANG1:27, QC_LANG1:def_34;
hence u < v ++ by QC_LANG1:29; ::_thesis: verum
end;
end;
A9: for q being Element of CQC-WFF A
for v being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),v,K,f] in SepQuadruples p & S1[ 'not' q,v,K,f] holds
S1[q,v,K,f] ;
for q being Element of CQC-WFF A
for v being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,v,K,f] in SepQuadruples p holds
S1[q,v,K,f] from CQC_SIM1:sch_6(A4, A9, A1, A6);
hence ( [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in p) implies u < t ) ; ::_thesis: verum
end;
theorem Th42: :: CQC_SIM1:42
for A being QC-alphabet
for t, u being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in q) holds
u < t
proof
let A be QC-alphabet ; ::_thesis: for t, u being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in q) holds
u < t
let t, u be QC-symbol of A; ::_thesis: for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in q) holds
u < t
let q, p be Element of CQC-WFF A; ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in q) holds
u < t
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in q) holds
u < t
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: ( [q,t,K,f] in SepQuadruples p & x. u in f .: (still_not-bound_in q) implies u < t )
assume that
A1: [q,t,K,f] in SepQuadruples p and
A2: x. u in f .: (still_not-bound_in q) ; ::_thesis: u < t
f .: (still_not-bound_in q) c= f .: ((still_not-bound_in p) \/ K) by A1, Th38, RELAT_1:123;
then x. u in f .: ((still_not-bound_in p) \/ K) by A2;
then x. u in (f .: (still_not-bound_in p)) \/ (f .: K) by RELAT_1:120;
then ( x. u in f .: (still_not-bound_in p) or x. u in f .: K ) by XBOOLE_0:def_3;
hence u < t by A1, Th39, Th41; ::_thesis: verum
end;
theorem Th43: :: CQC_SIM1:43
for A being QC-alphabet
for t being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: (still_not-bound_in q)
proof
let A be QC-alphabet ; ::_thesis: for t being QC-symbol of A
for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: (still_not-bound_in q)
let t be QC-symbol of A; ::_thesis: for q, p being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: (still_not-bound_in q)
let q, p be Element of CQC-WFF A; ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: (still_not-bound_in q)
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: for K being Finite_Subset of (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p holds
not x. t in f .: (still_not-bound_in q)
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: ( [q,t,K,f] in SepQuadruples p implies not x. t in f .: (still_not-bound_in q) )
assume A1: [q,t,K,f] in SepQuadruples p ; ::_thesis: not x. t in f .: (still_not-bound_in q)
assume x. t in f .: (still_not-bound_in q) ; ::_thesis: contradiction
then ( t < t & t <= t ) by A1, Th42, QC_LANG1:22;
hence contradiction by QC_LANG1:25; ::_thesis: verum
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 (SepVar p)
proof
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds still_not-bound_in p = still_not-bound_in (SepVar p)
let p be Element of CQC-WFF A; ::_thesis: still_not-bound_in p = still_not-bound_in (SepVar p)
defpred S1[ Element of CQC-WFF A] means for t being QC-symbol of A
for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [$1,t,K,f] in SepQuadruples p holds
f .: (still_not-bound_in $1) = still_not-bound_in (((SepFunc A) . $1) . [t,f]);
A1: [p,(index p),({}. (bound_QC-variables A)),(id (bound_QC-variables A))] in SepQuadruples p by Th30;
A2: now__::_thesis:_for_r_being_Element_of_CQC-WFF_A_st_S1[r]_holds_
S1[_'not'_r]
let r be Element of CQC-WFF A; ::_thesis: ( S1[r] implies S1[ 'not' r] )
reconsider g = (SepFunc A) . r as Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A) ;
assume A3: S1[r] ; ::_thesis: S1[ 'not' r]
A4: (SepFunc A) . ('not' r) = NEGATIVE g by Def7;
thus S1[ 'not' r] ::_thesis: verum
proof
let u be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' r),u,K,f] in SepQuadruples p holds
f .: (still_not-bound_in ('not' r)) = still_not-bound_in (((SepFunc A) . ('not' r)) . [u,f])
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' r),u,K,f] in SepQuadruples p holds
f .: (still_not-bound_in ('not' r)) = still_not-bound_in (((SepFunc A) . ('not' r)) . [u,f])
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [('not' r),u,K,f] in SepQuadruples p implies f .: (still_not-bound_in ('not' r)) = still_not-bound_in (((SepFunc A) . ('not' r)) . [u,f]) )
assume [('not' r),u,K,f] in SepQuadruples p ; ::_thesis: f .: (still_not-bound_in ('not' r)) = still_not-bound_in (((SepFunc A) . ('not' r)) . [u,f])
then A5: [r,u,K,f] in SepQuadruples p by Th31;
set uf = [u,f];
reconsider r9 = g . [u,f] as Element of CQC-WFF A ;
A6: still_not-bound_in r9 = still_not-bound_in ('not' r9) by QC_LANG3:7;
A7: still_not-bound_in r = still_not-bound_in ('not' r) by QC_LANG3:7;
(NEGATIVE g) . [u,f] = 'not' r9 by Def2;
hence f .: (still_not-bound_in ('not' r)) = still_not-bound_in (((SepFunc A) . ('not' r)) . [u,f]) by A4, A3, A7, A6, A5; ::_thesis: verum
end;
end;
A8: now__::_thesis:_for_k_being_Element_of_NAT_
for_l_being_CQC-variable_list_of_k,A
for_P_being_QC-pred_symbol_of_k,A_holds_S1[P_!_l]
let k be Element of NAT ; ::_thesis: for l being CQC-variable_list of k,A
for P being QC-pred_symbol of k,A holds S1[P ! l]
let l be CQC-variable_list of k,A; ::_thesis: for P being QC-pred_symbol of k,A holds S1[P ! l]
let P be QC-pred_symbol of k,A; ::_thesis: S1[P ! l]
thus S1[P ! l] ::_thesis: verum
proof
let u be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(P ! l),u,K,f] in SepQuadruples p holds
f .: (still_not-bound_in (P ! l)) = still_not-bound_in (((SepFunc A) . (P ! l)) . [u,f])
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(P ! l),u,K,f] in SepQuadruples p holds
f .: (still_not-bound_in (P ! l)) = still_not-bound_in (((SepFunc A) . (P ! l)) . [u,f])
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(P ! l),u,K,f] in SepQuadruples p implies f .: (still_not-bound_in (P ! l)) = still_not-bound_in (((SepFunc A) . (P ! l)) . [u,f]) )
assume [(P ! l),u,K,f] in SepQuadruples p ; ::_thesis: f .: (still_not-bound_in (P ! l)) = still_not-bound_in (((SepFunc A) . (P ! l)) . [u,f])
set fl = f * l;
A9: f .: { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in bound_QC-variables A ) } = { ((f * l) . j) where j is Element of NAT : ( 1 <= j & j <= len (f * l) & (f * l) . j in bound_QC-variables A ) }
proof
A10: len (f * l) = k by CARD_1:def_7
.= len l by CARD_1:def_7 ;
thus f .: { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in bound_QC-variables A ) } c= { ((f * l) . j) where j is Element of NAT : ( 1 <= j & j <= len (f * l) & (f * l) . j in bound_QC-variables A ) } :: according to XBOOLE_0:def_10 ::_thesis: { ((f * l) . j) where j is Element of NAT : ( 1 <= j & j <= len (f * l) & (f * l) . j in bound_QC-variables A ) } c= f .: { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in bound_QC-variables A ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f .: { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in bound_QC-variables A ) } or x in { ((f * l) . j) where j is Element of NAT : ( 1 <= j & j <= len (f * l) & (f * l) . j in bound_QC-variables A ) } )
assume x in f .: { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in bound_QC-variables A ) } ; ::_thesis: x in { ((f * l) . j) where j is Element of NAT : ( 1 <= j & j <= len (f * l) & (f * l) . j in bound_QC-variables A ) }
then consider y being set such that
A11: ( y in dom f & y in { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in bound_QC-variables A ) } & x = f . y ) by FUNCT_1:def_6;
consider i being Element of NAT such that
A12: y = l . i and
A13: 1 <= i and
A14: i <= len l and
l . i in bound_QC-variables A by A11;
i in dom l by A13, A14, FINSEQ_3:25;
then A15: f . (l . i) = (f * l) . i by FUNCT_1:13;
(f * l) . i in bound_QC-variables A by A10, A13, A14, Th13;
hence x in { ((f * l) . j) where j is Element of NAT : ( 1 <= j & j <= len (f * l) & (f * l) . j in bound_QC-variables A ) } by A10, A11, A12, A13, A14, A15; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { ((f * l) . j) where j is Element of NAT : ( 1 <= j & j <= len (f * l) & (f * l) . j in bound_QC-variables A ) } or x in f .: { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in bound_QC-variables A ) } )
assume x in { ((f * l) . i) where i is Element of NAT : ( 1 <= i & i <= len (f * l) & (f * l) . i in bound_QC-variables A ) } ; ::_thesis: x in f .: { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in bound_QC-variables A ) }
then consider i being Element of NAT such that
A16: x = (f * l) . i and
A17: 1 <= i and
A18: i <= len (f * l) and
(f * l) . i in bound_QC-variables A ;
i in dom l by A10, A17, A18, FINSEQ_3:25;
then A19: (f * l) . i = f . (l . i) by FUNCT_1:13;
A20: l . i in bound_QC-variables A by A10, A17, A18, Th13;
then A21: l . i in dom f by FUNCT_2:def_1;
l . i in { (l . j) where j is Element of NAT : ( 1 <= j & j <= len l & l . j in bound_QC-variables A ) } by A10, A17, A18, A20;
hence x in f .: { (l . i) where i is Element of NAT : ( 1 <= i & i <= len l & l . i in bound_QC-variables A ) } by A16, A21, A19, FUNCT_1:def_6; ::_thesis: verum
end;
A22: f .: (still_not-bound_in (P ! l)) = f .: (still_not-bound_in l) by QC_LANG3:5
.= f .: (variables_in (l,(bound_QC-variables A))) by QC_LANG3:2
.= variables_in ((f * l),(bound_QC-variables A)) by A9
.= still_not-bound_in (f * l) by QC_LANG3:2
.= still_not-bound_in (P ! (f * l)) by QC_LANG3:5 ;
(ATOMIC (P,l)) . (u,f) = P ! (f * l) by Def5;
hence f .: (still_not-bound_in (P ! l)) = still_not-bound_in (((SepFunc A) . (P ! l)) . [u,f]) by A22, Def7; ::_thesis: verum
end;
end;
A23: now__::_thesis:_for_r_being_Element_of_CQC-WFF_A
for_x_being_Element_of_bound_QC-variables_A_st_S1[r]_holds_
S1[_All_(x,r)]
let r be Element of CQC-WFF A; ::_thesis: for x being Element of bound_QC-variables A st S1[r] holds
S1[ All (x,r)]
let x be Element of bound_QC-variables A; ::_thesis: ( S1[r] implies S1[ All (x,r)] )
assume A24: S1[r] ; ::_thesis: S1[ All (x,r)]
thus S1[ All (x,r)] ::_thesis: verum
proof
reconsider g = (SepFunc A) . r as Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A) ;
let u be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,r)),u,K,f] in SepQuadruples p holds
f .: (still_not-bound_in (All (x,r))) = still_not-bound_in (((SepFunc A) . (All (x,r))) . [u,f])
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,r)),u,K,f] in SepQuadruples p holds
f .: (still_not-bound_in (All (x,r))) = still_not-bound_in (((SepFunc A) . (All (x,r))) . [u,f])
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(All (x,r)),u,K,f] in SepQuadruples p implies f .: (still_not-bound_in (All (x,r))) = still_not-bound_in (((SepFunc A) . (All (x,r))) . [u,f]) )
assume A25: [(All (x,r)),u,K,f] in SepQuadruples p ; ::_thesis: f .: (still_not-bound_in (All (x,r))) = still_not-bound_in (((SepFunc A) . (All (x,r))) . [u,f])
A26: [r,(u ++),(K \/ {.x.}),(f +* (x .--> (x. u)))] in SepQuadruples p by A25, Th33;
f +* (x .--> (x. u)) is Function of (bound_QC-variables A),(bound_QC-variables A) by Lm1;
then reconsider fu = f +* (x .--> (x. u)) as Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) by FUNCT_2:8;
reconsider r99 = g . ((u ++),fu) as Element of CQC-WFF A ;
A27: (UNIVERSAL (x,g)) . (u,f) = All ((x. u),r99) by Def4;
A28: still_not-bound_in (All (x,r)) = (still_not-bound_in r) \ {x} by QC_LANG3:12;
then A29: not x. u in f .: ((still_not-bound_in r) \ {x}) by A25, Th43;
thus f .: (still_not-bound_in (All (x,r))) = fu .: ((still_not-bound_in r) \ {x}) by A28, Th3
.= (fu .: (still_not-bound_in r)) \ {(x. u)} by A29, Th4
.= (still_not-bound_in r99) \ {(x. u)} by A24, A26
.= still_not-bound_in (All ((x. u),r99)) by QC_LANG3:12
.= still_not-bound_in (((SepFunc A) . (All (x,r))) . [u,f]) by A27, Def7 ; ::_thesis: verum
end;
end;
A30: now__::_thesis:_for_r,_s_being_Element_of_CQC-WFF_A_st_S1[r]_&_S1[s]_holds_
S1[r_'&'_s]
let r, s be Element of CQC-WFF A; ::_thesis: ( S1[r] & S1[s] implies S1[r '&' s] )
assume that
A31: S1[r] and
A32: S1[s] ; ::_thesis: S1[r '&' s]
thus S1[r '&' s] ::_thesis: verum
proof
reconsider g = (SepFunc A) . r, h = (SepFunc A) . s as Function of [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):],(CQC-WFF A) ;
let u be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(r '&' s),u,K,f] in SepQuadruples p holds
f .: (still_not-bound_in (r '&' s)) = still_not-bound_in (((SepFunc A) . (r '&' s)) . [u,f])
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(r '&' s),u,K,f] in SepQuadruples p holds
f .: (still_not-bound_in (r '&' s)) = still_not-bound_in (((SepFunc A) . (r '&' s)) . [u,f])
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(r '&' s),u,K,f] in SepQuadruples p implies f .: (still_not-bound_in (r '&' s)) = still_not-bound_in (((SepFunc A) . (r '&' s)) . [u,f]) )
assume A33: [(r '&' s),u,K,f] in SepQuadruples p ; ::_thesis: f .: (still_not-bound_in (r '&' s)) = still_not-bound_in (((SepFunc A) . (r '&' s)) . [u,f])
reconsider r9 = g . (u,f), s9 = h . ((u + (QuantNbr r)),f) as Element of CQC-WFF A ;
A34: (CON (g,h,(QuantNbr r))) . (u,f) = r9 '&' s9 by Def3;
[r,u,K,f] in SepQuadruples p by A33, Th32;
then A35: f .: (still_not-bound_in r) = still_not-bound_in r9 by A31;
[s,(u + (QuantNbr r)),K,f] in SepQuadruples p by A33, Th32;
then A36: f .: (still_not-bound_in s) = still_not-bound_in s9 by A32;
thus f .: (still_not-bound_in (r '&' s)) = f .: ((still_not-bound_in r) \/ (still_not-bound_in s)) by QC_LANG3:10
.= (still_not-bound_in r9) \/ (still_not-bound_in s9) by A35, A36, RELAT_1:120
.= still_not-bound_in (r9 '&' s9) by QC_LANG3:10
.= still_not-bound_in (((SepFunc A) . (r '&' s)) . [u,f]) by A34, Def7 ; ::_thesis: verum
end;
end;
A37: (SepFunc A) . (VERUM A) = [:(QC-symbols A),(Funcs ((bound_QC-variables A),(bound_QC-variables A))):] --> (VERUM A) by Def7;
A38: S1[ VERUM A]
proof
let v be QC-symbol of A; ::_thesis: for K being Finite_Subset of (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(VERUM A),v,K,f] in SepQuadruples p holds
f .: (still_not-bound_in (VERUM A)) = still_not-bound_in (((SepFunc A) . (VERUM A)) . [v,f])
let K be Finite_Subset of (bound_QC-variables A); ::_thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(VERUM A),v,K,f] in SepQuadruples p holds
f .: (still_not-bound_in (VERUM A)) = still_not-bound_in (((SepFunc A) . (VERUM A)) . [v,f])
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); ::_thesis: ( [(VERUM A),v,K,f] in SepQuadruples p implies f .: (still_not-bound_in (VERUM A)) = still_not-bound_in (((SepFunc A) . (VERUM A)) . [v,f]) )
assume [(VERUM A),v,K,f] in SepQuadruples p ; ::_thesis: f .: (still_not-bound_in (VERUM A)) = still_not-bound_in (((SepFunc A) . (VERUM A)) . [v,f])
A39: still_not-bound_in (VERUM A) = {} by QC_LANG3:3;
then f .: (still_not-bound_in (VERUM A)) = {} ;
hence f .: (still_not-bound_in (VERUM A)) = still_not-bound_in (((SepFunc A) . (VERUM A)) . [v,f]) by A39, A37, FUNCOP_1:7; ::_thesis: verum
end;
A40: for q being Element of CQC-WFF A holds S1[q] from CQC_SIM1:sch_5(A38, A8, A2, A30, A23);
thus still_not-bound_in p = (id (bound_QC-variables A)) .: (still_not-bound_in p) by FUNCT_1:92
.= still_not-bound_in (SepVar p) by A40, A1 ; ::_thesis: verum
end;
theorem :: CQC_SIM1:45
for A being QC-alphabet
for p being Element of CQC-WFF A holds index p = index (SepVar p)
proof
let A be QC-alphabet ; ::_thesis: for p being Element of CQC-WFF A holds index p = index (SepVar p)
let p be Element of CQC-WFF A; ::_thesis: index p = index (SepVar p)
still_not-bound_in p = still_not-bound_in (SepVar p) by Th44;
hence index p = index (SepVar p) ; ::_thesis: verum
end;
definition
let A be QC-alphabet ;
let p, q be Element of CQC-WFF A;
predp,q are_similar means :Def14: :: CQC_SIM1:def 14
SepVar p = SepVar q;
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 Def14 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
proof
let A be QC-alphabet ; ::_thesis: for p, q, r being Element of CQC-WFF A st p,q are_similar & q,r are_similar holds
p,r are_similar
let p, q, r be Element of CQC-WFF A; ::_thesis: ( p,q are_similar & q,r are_similar implies p,r are_similar )
assume that
A1: p,q are_similar and
A2: q,r are_similar ; ::_thesis: p,r are_similar
A3: SepVar q = SepVar r by A2, Def14;
SepVar p = SepVar q by A1, Def14;
hence p,r are_similar by A3, Def14; ::_thesis: verum
end;