let m be Nat; for S being Language
for U being non empty set holds dom (ExIterator ((S,U) -TruthEval m)) = [:(U -InterpretersOf S),(m -ExFormulasOf S):]
let S be Language; for U being non empty set holds dom (ExIterator ((S,U) -TruthEval m)) = [:(U -InterpretersOf S),(m -ExFormulasOf S):]
let U be non empty set ; dom (ExIterator ((S,U) -TruthEval m)) = [:(U -InterpretersOf S),(m -ExFormulasOf S):]
set mm = m;
set II = U -InterpretersOf S;
set SS = AllSymbolsOf S;
set N = TheNorSymbOf S;
set Fm = (S,U) -TruthEval m;
set Phim = S -formulasOfMaxDepth m;
set IT = ExIterator ((S,U) -TruthEval m);
set L = LettersOf S;
deffunc H1( set , FinSequence) -> set = <*$1*> ^ $2;
defpred S1[] means verum;
set A = { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } ;
set LH = dom (ExIterator ((S,U) -TruthEval m));
set RH = [:(U -InterpretersOf S), { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } :];
(S,U) -TruthEval m is Element of Funcs ([:(U -InterpretersOf S),(S -formulasOfMaxDepth m):],BOOLEAN)
by Th8;
then reconsider Fmm = (S,U) -TruthEval m as Function of [:(U -InterpretersOf S),(S -formulasOfMaxDepth m):],BOOLEAN ;
A1:
dom Fmm = [:(U -InterpretersOf S),(S -formulasOfMaxDepth m):]
by FUNCT_2:def 1;
reconsider ITT = ExIterator ((S,U) -TruthEval m) as PartFunc of [:(U -InterpretersOf S),(((AllSymbolsOf S) *) \ {{}}):],BOOLEAN ;
A2:
[:(U -InterpretersOf S),(((AllSymbolsOf S) *) \ {{}}):] = { [x,y] where x is Element of U -InterpretersOf S, y is Element of ((AllSymbolsOf S) *) \ {{}} : verum }
by DOMAIN_1:19;
now for z being object st z in dom (ExIterator ((S,U) -TruthEval m)) holds
z in [:(U -InterpretersOf S), { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } :]let z be
object ;
( z in dom (ExIterator ((S,U) -TruthEval m)) implies z in [:(U -InterpretersOf S), { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } :] )assume A3:
z in dom (ExIterator ((S,U) -TruthEval m))
;
z in [:(U -InterpretersOf S), { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } :]then
z in [:(U -InterpretersOf S),(((AllSymbolsOf S) *) \ {{}}):]
;
then consider x being
Element of
U -InterpretersOf S,
y being
Element of
((AllSymbolsOf S) *) \ {{}} such that A4:
z = [x,y]
by A2;
consider v being
literal Element of
S,
w being
string of
S such that A5:
(
[x,w] in dom ((S,U) -TruthEval m) &
y = <*v*> ^ w )
by Def16, A3, A4;
reconsider phi =
w as
Element of
S -formulasOfMaxDepth m by A1, A5, ZFMISC_1:87;
reconsider vv =
v as
Element of
LettersOf S by FOMODEL1:def 14;
y = <*vv*> ^ phi
by A5;
then
y in { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] }
;
hence
z in [:(U -InterpretersOf S), { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } :]
by ZFMISC_1:def 2, A4;
verum end;
then A6:
dom (ExIterator ((S,U) -TruthEval m)) c= [:(U -InterpretersOf S), { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } :]
;
now for z being object st z in [:(U -InterpretersOf S), { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } :] holds
z in dom (ExIterator ((S,U) -TruthEval m))let z be
object ;
( z in [:(U -InterpretersOf S), { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } :] implies z in dom (ExIterator ((S,U) -TruthEval m)) )assume
z in [:(U -InterpretersOf S), { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } :]
;
z in dom (ExIterator ((S,U) -TruthEval m))then consider xx,
yy being
object such that A7:
(
xx in U -InterpretersOf S &
yy in { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } &
z = [xx,yy] )
by ZFMISC_1:def 2;
reconsider x =
xx as
Element of
U -InterpretersOf S by A7;
consider vv being
Element of
LettersOf S,
phi being
Element of
S -formulasOfMaxDepth m such that A8:
yy = H1(
vv,
phi)
by A7;
reconsider v =
vv as
literal Element of
S ;
reconsider w =
phi as
string of
S ;
(
[x,phi] in dom ((S,U) -TruthEval m) &
yy = <*v*> ^ w )
by A8, A1;
hence
z in dom (ExIterator ((S,U) -TruthEval m))
by A7, Def16;
verum end;
then
[:(U -InterpretersOf S), { H1(v,phi) where v is Element of LettersOf S, phi is Element of S -formulasOfMaxDepth m : S1[] } :] c= dom (ExIterator ((S,U) -TruthEval m))
;
hence
dom (ExIterator ((S,U) -TruthEval m)) = [:(U -InterpretersOf S),(m -ExFormulasOf S):]
by A6; verum