:: A Classical First Order Language
:: by Czes{\l}aw Byli\'nski
::
:: Received May 11, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies SUBSET_1, NUMBERS, QC_LANG1, FINSEQ_1, PARTFUN1, XXREAL_0,
FUNCT_1, RELAT_1, TARSKI, FUNCOP_1, QC_LANG3, XBOOLE_0, ZF_MODEL,
FINSEQ_2, ZF_LANG, CARD_1, REALSET1, XBOOLEAN, BVFUNC_2, MARGREL1,
CLASSES2, CQC_LANG, ZFMISC_1, NAT_1;
notations TARSKI, XBOOLE_0, ENUMSET1, SUBSET_1, CARD_1, ORDINAL1, NUMBERS,
RELAT_1, FUNCT_1, FUNCT_2, BINOP_1, PARTFUN1, FUNCOP_1, FUNCT_4,
FINSEQ_1, FINSEQ_2, QC_LANG1, QC_LANG2, QC_LANG3, XXREAL_0, NAT_1,
ZFMISC_1;
constructors ENUMSET1, PARTFUN1, BINOP_1, FUNCOP_1, FUNCT_4, XXREAL_0,
MEMBERED, QC_LANG2, QC_LANG3, FINSEQ_2, RELSET_1, NUMBERS;
registrations XBOOLE_0, RELSET_1, FUNCOP_1, MEMBERED, QC_LANG1, XXREAL_0,
FINSEQ_2;
requirements NUMERALS, SUBSET, BOOLE;
begin
reserve A for QC-alphabet;
reserve i,j,k for Nat;
theorem :: CQC_LANG:1
for x being set holds x in QC-variables(A) iff x in
fixed_QC-variables(A) or x in free_QC-variables(A) or
x in bound_QC-variables(A);
definition
let A;
mode Substitution of A is PartFunc of free_QC-variables(A),QC-variables(A);
end;
reserve f for Substitution of A;
definition
let A;
let l be FinSequence of QC-variables(A);
let f;
func Subst(l,f) -> FinSequence of QC-variables(A) means
:: CQC_LANG:def 1
len it = len l &
for k st 1 <= k & k <= len l holds (l.k in dom f implies it.k = f.(l.k)) & (not
l.k in dom f implies it.k = l.k);
end;
registration
let A,k;
let l be QC-variable_list of k, A;
let f;
cluster Subst(l,f) -> k-element;
end;
theorem :: CQC_LANG:2
for x being bound_QC-variable of A, a being free_QC-variable of A holds
a .--> x is Substitution of A;
definition
let A;
let a be free_QC-variable of A, x be bound_QC-variable of A;
redefine func a .--> x -> Substitution of A;
end;
theorem :: CQC_LANG:3
for x being bound_QC-variable of A, a being free_QC-variable of A,
l, ll being FinSequence of QC-variables(A)
holds
f = a .--> x & ll = Subst(l,f) & 1 <= k & k <= len l implies (l.
k = a implies ll.k = x) & (l.k <> a implies ll.k = l.k);
definition
let A;
func CQC-WFF(A) -> Subset of QC-WFF(A) equals
:: CQC_LANG:def 2
{s where s is QC-formula of A: Fixed s = {} & Free s = {} };
end;
registration
let A;
cluster CQC-WFF(A) -> non empty;
end;
theorem :: CQC_LANG:4
for p being Element of QC-WFF(A) holds
p is Element of CQC-WFF(A) iff Fixed p = {} & Free p = {};
registration
let A;
let k be Nat;
cluster bound_QC-variables(A)-valued for QC-variable_list of k, A;
end;
definition
let A; let k be Nat;
mode CQC-variable_list of k,
A is bound_QC-variables(A)-valued QC-variable_list of k,A;
end;
theorem :: CQC_LANG:5
for l being QC-variable_list of k, A holds l is CQC-variable_list
of k,A iff { l.i : 1 <= i & i <= len l & l.i in free_QC-variables(A) }
= {} & { l.j
: 1 <= j & j <= len l & l.j in fixed_QC-variables(A) } = {};
theorem :: CQC_LANG:6
VERUM(A) is Element of CQC-WFF(A);
theorem :: CQC_LANG:7
for P being QC-pred_symbol of k,A for l being QC-variable_list of
k,A holds P!l is Element of CQC-WFF(A) iff
{ l.i : 1 <= i & i <= len l & l.i in
free_QC-variables(A) } = {} & { l.j : 1 <= j & j <= len l & l.j in
fixed_QC-variables(A) } = {};
definition
let k,A;
let P be QC-pred_symbol of k,A;
let l be CQC-variable_list of k,A;
redefine func P!l -> Element of CQC-WFF(A);
end;
theorem :: CQC_LANG:8
for p being Element of QC-WFF(A) holds
'not' p is Element of CQC-WFF(A) iff p is Element of CQC-WFF(A);
theorem :: CQC_LANG:9
for p,q being Element of QC-WFF(A) holds
p '&' q is Element of CQC-WFF(A) iff p is Element of CQC-WFF(A) & q is
Element of CQC-WFF(A);
definition
let A;
redefine func VERUM(A) -> Element of CQC-WFF(A);
let r be Element of CQC-WFF(A);
redefine func 'not' r -> Element of CQC-WFF(A);
let s be Element of CQC-WFF(A);
redefine func r '&' s -> Element of CQC-WFF(A);
end;
theorem :: CQC_LANG:10
for r,s being Element of CQC-WFF(A) holds
r => s is Element of CQC-WFF(A);
theorem :: CQC_LANG:11
for r,s being Element of CQC-WFF(A) holds
r 'or' s is Element of CQC-WFF(A);
theorem :: CQC_LANG:12
for r,s being Element of CQC-WFF(A) holds
r <=> s is Element of CQC-WFF(A);
definition
let A;
let r,s be Element of CQC-WFF(A);
redefine func r => s -> Element of CQC-WFF(A);
redefine func r 'or' s -> Element of CQC-WFF(A);
redefine func r <=> s -> Element of CQC-WFF(A);
end;
theorem :: CQC_LANG:13
for x being bound_QC-variable of A, p being Element of QC-WFF(A) holds
All(x,p) is Element of CQC-WFF(A) iff p is Element of CQC-WFF(A);
definition
let A;
let x be bound_QC-variable of A,r be Element of CQC-WFF(A);
redefine func All(x,r) -> Element of CQC-WFF(A);
end;
theorem :: CQC_LANG:14
for x being bound_QC-variable of A,r being Element of CQC-WFF(A) holds
Ex(x,r) is Element of CQC-WFF(A);
definition
let A;
let x be bound_QC-variable of A,r be Element of CQC-WFF(A);
redefine func Ex(x,r) -> Element of CQC-WFF(A);
end;
scheme :: CQC_LANG:sch 1
CQCInd { A() -> QC-alphabet, P[set] }
:
for r being Element of CQC-WFF(A()) holds P[r]
provided
for r,s being Element of CQC-WFF(A())
for x being bound_QC-variable of A() for k
for l being CQC-variable_list of k, A() for P being
QC-pred_symbol of k,A() holds P[VERUM(A())] & P[P!l] &
(P[r] implies P['not' r]) & (P[r]
& P[s] implies P[r '&' s]) & (P[r] implies P[All(x, r)]);
scheme :: CQC_LANG:sch 2
CQCFuncEx { Al() -> QC-alphabet, D() -> non empty set,
V() -> (Element of D()), A(set,set,set) -> (Element of D()),
N(set) -> (Element of D()), C(set,set) -> (Element of D()),
Q(set,set) -> Element of D()} : ex F being Function of CQC-WFF(Al()),
D() st F.VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al()
for P being QC-pred_symbol of k,Al() holds F.(P!l) = A(k,P,l) &
F.('not' r) = N(F.r) & F.(r '&' s) = C(F.r,F.s)& F.All(x,r) = Q(x,F.r);
scheme :: CQC_LANG:sch 3
CQCFuncUniq { Al() -> QC-alphabet, D() -> non empty set,
F1() -> (Function of CQC-WFF(Al()), D()),
F2()-> (Function of CQC-WFF(Al()), D()),
V() -> (Element of D()), A(set,set,set) -> (Element of D()),
N(set) -> (Element of D()), C(set,set) -> (Element of D()),
Q(set,set) -> Element of D()} : F1() = F2()
provided
F1().VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al()
for P being QC-pred_symbol of k,Al() holds F1().(P!l) = A(k,P,l) &
F1().('not' r) = N(F1().r) & F1().(r '&' s) = C(F1().r,F1().s) &
F1().All(x,r) = Q(x,F1().r) and
F2().VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al()
for P being QC-pred_symbol of k,Al() holds F2().(P!l) =
A(k,P,l) & F2().('not' r) =
N(F2().r) & F2().(r '&' s) = C(F2().r,F2().s) & F2().All(x,r) = Q(x,F2().r);
scheme :: CQC_LANG:sch 4
CQCDefcorrectness { Al() -> QC-alphabet, D() -> non empty set,
p() -> (Element of CQC-WFF(Al())), V() -> (Element of D()),
A(set,set,set) -> (Element of D()), N(set) -> (Element of D()),
C(set,set) -> (Element of D()), Q(set,set) -> Element of D()}
:
(ex d being Element of D() st ex F being Function of CQC-WFF(Al()),
D() st d = F.p() & F.VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al() for P being
QC-pred_symbol of k,Al() holds F.(P!l) = A(k,P,l) & F.('not' r) = N(F.r) &
F.(r '&' s) = C(F.r,F.s) & F.All(x,r) = Q(x,F.r) ) &
for d1,d2 being Element of D() st (ex F being Function of CQC-WFF(Al()),
D() st d1 = F.p() & F.VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al()
for P being QC-pred_symbol of k,Al()
holds F.(P!l) = A(k,P,l) & F.('not' r) = N(F.r) &
F.(r '&' s) = C(F.r,F.s) & F.All(x,r) = Q(x,F.r) ) &
(ex F being Function of CQC-WFF(Al()), D() st d2 = F.p() &
F.VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al()
for P being QC-pred_symbol of k,Al() holds F.(P!l) = A(k,P,l) &
F.('not' r) = N(F.r) & F.(r '&' s) = C(F.r,F.s) &
F.All(x,r) = Q(x,F.r) ) holds d1 = d2;
scheme :: CQC_LANG:sch 5
CQCDefVERUM { Al() -> QC-alphabet, D() -> non empty set,
F(set) -> (Element of D()), V() -> (Element of D()),
A(set,set,set) -> (Element of D()), N(set) -> (Element of D()),
C(set,set) -> (Element of D()), Q(set,set) -> Element of D()} :
F(VERUM(Al())) = V()
provided
for p being (Element of CQC-WFF(Al())), d being Element of D() holds
d = F (p) iff ex F being Function of CQC-WFF(Al()), D() st
d = F.p & F.VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al()
for P being QC-pred_symbol of k,Al() holds F.(P!l) = A(k,P,l) &
F.('not' r) = N(F.r) & F.(r '&' s) = C(F.r,F.s) & F.All(x,r) = Q(x,F.r);
scheme :: CQC_LANG:sch 6
CQCDefatomic { Al() -> QC-alphabet, D() -> non empty set,
V() -> (Element of D()), F(set) -> (Element of D()),
A(set,set,set) -> (Element of D()), k() -> Nat, P()
-> (QC-pred_symbol of k(),Al()), l() -> (CQC-variable_list of k(), Al()),
N(set) -> (Element of D()), C(set,set) -> (Element of D()),
Q(set,set) -> Element of D()}
: F(P()!l()) = A(k(),P(),l())
provided
for p being (Element of CQC-WFF(Al())), d being Element of D() holds d = F
(p) iff ex F being Function of CQC-WFF(Al()), D()
st d = F.p & F.VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al()
for P being QC-pred_symbol of k,Al() holds
F.(P!l) = A(k,P,l) & F.('not' r) = N(F.r) & F.(r '&' s) = C(F.r,F.s) &
F.All(x,r) = Q(x,F.r);
scheme :: CQC_LANG:sch 7
CQCDefnegative { Al() -> QC-alphabet, D() -> non empty set,
F(set) -> (Element of D()), V() -> (Element of D()),
A(set,set,set) -> (Element of D()), N(set) -> (Element of D()),
r() -> (Element of CQC-WFF(Al())), C(set,set) -> (Element of D()),
Q(set,set) -> Element of D()}
:
F('not' r()) = N(F(r()))
provided
for p being (Element of CQC-WFF(Al())), d being Element of D() holds
d = F (p) iff ex F being Function of CQC-WFF(Al()), D() st
d = F.p & F.VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al()
for P being QC-pred_symbol of k,Al() holds
F.(P!l) = A(k,P,l) & F.('not' r) = N(F.r) & F.(r '&' s) = C(F.r,F.s) &
F.All(x,r) = Q(x,F.r);
scheme :: CQC_LANG:sch 8
QCDefconjunctive { Al() -> QC-alphabet, D() -> non empty set,
F(set) -> (Element of D()), V() -> (Element of D()),
A(set,set,set) -> (Element of D()), N(set) -> (Element of D()),
C(set,set) -> (Element of D()), r() -> (Element of CQC-WFF(Al())),
s() -> (Element of CQC-WFF(Al())), Q(set,set) -> Element of D()}
:
F(r() '&' s()) = C(F(r()), F(s()))
provided
for p being (Element of CQC-WFF(Al())), d being Element of D() holds d = F
(p) iff ex F being Function of CQC-WFF(Al()), D() st d = F.p &
F.VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al()
for P being QC-pred_symbol of k,Al() holds
F.(P!l) = A(k,P,l) & F.('not' r) = N(F.r) & F.(r '&' s) = C(F.r,F.s) &
F.All(x,r) = Q(x,F.r);
scheme :: CQC_LANG:sch 9
QCDefuniversal { Al() -> QC-alphabet, D() -> non empty set,
F(set) -> (Element of D()), V() -> (Element of D()),
A(set,set,set) -> (Element of D()), N(set) -> (Element of D()),
C(set,set) -> (Element of D()), Q(set,set) -> (Element of D()),
x() -> bound_QC-variable of Al(), r() -> Element of CQC-WFF(Al())}
:
F(All(x(),r())) = Q(x(),F(r()))
provided
for p being (Element of CQC-WFF(Al())), d being Element of D() holds d = F
(p) iff ex F being Function of CQC-WFF(Al()), D() st
d = F.p & F.VERUM(Al()) = V() &
for r,s being Element of CQC-WFF(Al())
for x being bound_QC-variable of Al()
for k
for l being CQC-variable_list of k, Al()
for P being QC-pred_symbol of k,Al() holds F.(P!l) = A(k,P,l) &
F.('not' r) = N(F.r) & F.(r '&' s) = C(F.r,F.s) & F.All(x,r) = Q(x,F.r);
reserve x,y for bound_QC-variable of A;
reserve a for free_QC-variable of A;
reserve p,q for Element of QC-WFF(A);
reserve l,l1,l2,ll for FinSequence of QC-variables(A);
reserve r,s for Element of CQC-WFF(A);
definition
let A,p,x;
func p.x -> Element of QC-WFF(A) means
:: CQC_LANG:def 3
ex F being Function of QC-WFF(A),
QC-WFF(A) st it = F.p & for q holds F.VERUM(A) = VERUM(A) &
(q is atomic implies F.q =
(the_pred_symbol_of q)!Subst(the_arguments_of q,(A)a.0.-->x)) &
(q is negative implies F.q = 'not' (F.the_argument_of q) ) &
(q is conjunctive implies F.q = (F.the_left_argument_of q) '&'
(F.the_right_argument_of q)) & (q is universal implies F.q =
IFEQ(bound_in q,x,q,All(bound_in q,F.the_scope_of q)));
end;
theorem :: CQC_LANG:15
VERUM(A).x = VERUM(A);
theorem :: CQC_LANG:16
p is atomic implies p.x = (the_pred_symbol_of p)!Subst(
the_arguments_of p,(A)a.0.-->x);
theorem :: CQC_LANG:17
for k being Nat
for P being QC-pred_symbol of k,A for l being QC-variable_list of
k,A holds (P!l).x = P!Subst(l,(A)a.0.-->x);
theorem :: CQC_LANG:18
p is negative implies p.x = 'not'((the_argument_of p).x);
theorem :: CQC_LANG:19
('not' p).x = 'not'(p.x);
theorem :: CQC_LANG:20
p is conjunctive implies p.x = ((the_left_argument_of p).x) '&'
((the_right_argument_of p).x);
theorem :: CQC_LANG:21
(p '&' q).x = (p.x) '&' (q.x);
theorem :: CQC_LANG:22
p is universal & bound_in p = x implies p.x = p;
theorem :: CQC_LANG:23
p is universal & bound_in p <> x implies p.x = All(bound_in p,(
the_scope_of p).x);
theorem :: CQC_LANG:24
(All(x,p)).x = All(x,p);
theorem :: CQC_LANG:25
x<>y implies (All(x,p)).y = All(x,p.y);
theorem :: CQC_LANG:26
Free p = {} implies p.x = p;
theorem :: CQC_LANG:27
r.x = r;
theorem :: CQC_LANG:28
Fixed(p.x) = Fixed p;