:: Equations in Many Sorted Algebras
:: by Artur Korni{\l}owicz
::
:: Received May 30, 1997
:: Copyright (c) 1997-2018 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 XBOOLE_0, FUNCT_1, RELAT_1, TARSKI, PBOOLE, MEMBER_1, MSUALG_3,
FUNCT_2, FUNCT_6, PZFMISC1, SUBSET_1, REALSET1, STRUCT_0, MSUALG_1,
MSUALG_2, UNIALG_2, PARTFUN1, PRALG_2, CARD_3, RLVECT_2, PRELAMB,
MSAFREE, FINSET_1, MSAFREE2, MSUALG_6, MARGREL1, NAT_1, GROUP_6,
CLOSURE2, ZFMISC_1, FINSEQ_1, CARD_1, PRALG_3, FUNCT_4, FUNCOP_1,
NUMBERS, ZF_LANG, MCART_1, ZF_MODEL, WELLORD1, MSUALG_4, EQUATION,
AFINSQ_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, SUBSET_1, ORDINAL1, NUMBERS,
RELAT_1, FUNCT_1, PBOOLE, RELSET_1, PARTFUN1, STRUCT_0, FUNCT_2, MCART_1,
FINSET_1, FINSEQ_1, FINSEQ_2, FUNCT_4, FUNCT_6, PZFMISC1, CARD_3,
AFINSQ_1, MSUALG_1, PRALG_2, FUNCOP_1, MSUALG_2, PRALG_3, MSUALG_3,
MSUALG_4, MSAFREE, MSAFREE2, CLOSURE2, MSUALG_6;
constructors FUNCT_4, PZFMISC1, MSUALG_3, MSAFREE2, CLOSURE2, MSUALG_6,
PRALG_3, RELSET_1, XTUPLE_0, NUMBERS, AFINSQ_1;
registrations XBOOLE_0, SUBSET_1, ORDINAL1, RELSET_1, FUNCT_2, FUNCOP_1,
FINSET_1, FINSEQ_1, STRUCT_0, MSUALG_1, MSUALG_2, RELAT_1, FUNCT_1,
PRALG_2, MSUALG_3, MSAFREE, MSUALG_4, EXTENS_1, CLOSURE2, MSUALG_6,
PRALG_3, MSUALG_9, MSSUBFAM, PBOOLE, AUTALG_1, AFINSQ_1;
requirements SUBSET, BOOLE, NUMERALS;
definitions TARSKI, PBOOLE, PRALG_2, MSUALG_2, MSUALG_3, FINSET_1, MSAFREE,
MSAFREE2, XBOOLE_0, PZFMISC1;
equalities PRALG_2, MSUALG_1;
expansions TARSKI, PBOOLE, MSUALG_2, MSUALG_3, MSAFREE, XBOOLE_0, PZFMISC1;
theorems MSUALG_1, EXTENS_1, MSAFREE2, MSSCYC_1, CARD_3, FUNCT_4, FUNCOP_1,
FUNCT_1, FUNCT_2, MCART_1, MSAFREE, FINSEQ_1, CLOSURE2, FINSET_1,
MSUALG_2, MSUALG_3, MSUALG_4, MSUALG_9, PBOOLE, PARTFUN1, MSUALG_6,
MSSUBFAM, INSTALG1, PRALG_2, PRALG_3, RELAT_1, TARSKI, ZFMISC_1,
RELSET_1, XBOOLE_0, XBOOLE_1, FUNCT_6, FINSEQ_2, AFINSQ_1, CARD_1;
schemes MSSUBFAM, PBOOLE, FUNCT_2, XBOOLE_0;
begin :: On the Functions and Many Sorted Functions
reserve I for set;
theorem Th1:
for A being set, B, C being non empty set for f being Function of
A, B, g being Function of B, C st rng (g * f) = C holds rng g = C
proof
let A be set, B, C be non empty set, f be Function of A, B, g be Function of
B, C such that
A1: rng (g * f) = C;
thus rng g c= C;
let q be object;
assume q in C;
then consider x being object such that
A2: x in dom (g * f) and
A3: q = (g * f).x by A1,FUNCT_1:def 3;
A4: dom f = A by FUNCT_2:def 1;
then
A5: f.x in rng f by A2,FUNCT_1:def 3;
dom (g * f) = A by FUNCT_2:def 1;
then
A6: rng f c= dom g by A4,FUNCT_1:15;
q = g.(f.x) by A2,A3,FUNCT_1:12;
hence thesis by A6,A5,FUNCT_1:def 3;
end;
theorem
for A being ManySortedSet of I, B, C being non-empty ManySortedSet of
I for f being ManySortedFunction of A, B, g being ManySortedFunction of B, C st
g ** f is "onto" holds g is "onto"
proof
let A be ManySortedSet of I, B, C be non-empty ManySortedSet of I, f be
ManySortedFunction of A, B, g be ManySortedFunction of B, C such that
A1: g ** f is "onto";
let i be set;
assume
A2: i in I;
then
A3: f.i is Function of A.i, B.i & g.i is Function of B.i, C.i by PBOOLE:def 15;
rng (g.i * f.i) = rng ((g ** f).i) by A2,MSUALG_3:2
.= C.i by A1,A2;
hence thesis by A2,A3,Th1;
end;
theorem Th3: :: see PRALG_2:5
for A, B being non empty set, C, y being set for f being Function
st f in Funcs(A,Funcs(B,C)) & y in B holds dom ((commute f).y) = A & rng ((
commute f).y) c= C
proof
let A, B be non empty set, C, y be set, f be Function such that
A1: f in Funcs(A,Funcs(B,C)) and
A2: y in B;
set cf = commute f;
cf in Funcs(B,Funcs(A,C)) by A1,FUNCT_6:55;
then
A3: ex g being Function st g = cf & dom g = B & rng g c= Funcs(A,C) by
FUNCT_2:def 2;
then cf.y in rng cf by A2,FUNCT_1:def 3;
then ex t being Function st t = ((commute f).y) & dom t = A & rng t c= C by
A3,FUNCT_2:def 2;
hence thesis;
end;
theorem
for A, B being ManySortedSet of I st A is_transformable_to B for f
being ManySortedFunction of I st doms f = A & rngs f c= B holds
f is ManySortedFunction of A, B
proof
let A, B be ManySortedSet of I such that
for i being set st i in I holds B.i = {} implies A.i = {};
let f be ManySortedFunction of I such that
A1: doms f = A and
A2: rngs f c= B;
let i be object;
assume
A3: i in I;
then reconsider J = I as non empty set;
reconsider s = i as Element of J by A3;
A4: dom (f.s) = A.s by A1,MSSUBFAM:14;
rng (f.s) = (rngs f).s by MSSUBFAM:13;
then rng (f.s) c= B.s by A2;
hence thesis by A4,FUNCT_2:2;
end;
theorem Th5:
for A, B being ManySortedSet of I, F being ManySortedFunction of
A, B for C, E being ManySortedSubset of A, D being ManySortedSubset of C st E =
D holds (F || C) || D = F || E
proof
let A, B be ManySortedSet of I, F be ManySortedFunction of A, B, C, E be
ManySortedSubset of A, D be ManySortedSubset of C such that
A1: E = D;
now
let i be object such that
A2: i in I;
D c= C by PBOOLE:def 18;
then
A4: D.i c= C.i by A2;
(F||C).i is Function of C.i, B.i by A2,PBOOLE:def 15;
then reconsider fc = (F.i) | (C.i) as Function of C.i, B.i by A2,
MSAFREE:def 1;
thus ((F || C) || D).i = (F || C).i | (D.i) by A2,MSAFREE:def 1
.= fc | (D.i) by A2,MSAFREE:def 1
.= F.i | (D.i) by A4,RELAT_1:74
.= (F || E).i by A1,A2,MSAFREE:def 1;
end;
hence thesis;
end;
theorem Th6:
for B being non-empty ManySortedSet of I, C being ManySortedSet of I
for A being ManySortedSubset of C, F being ManySortedFunction of A, B
ex G being ManySortedFunction of C, B st G || A = F
proof
let B be non-empty ManySortedSet of I, C be ManySortedSet of I, A be
ManySortedSubset of C, F be ManySortedFunction of A, B;
defpred P[object,object,object] means
ex f being Function of A.$3, B.$3 st f = F.$3 &
($2 in A.$3 implies $1 = f.$2);
A1: for i being object st i in I
for x being object st x in C.i ex y being
object st y in B.i & P[y,x,i]
proof
let i be object;
assume
A2: i in I;
then reconsider f = F.i as Function of A.i, B.i by PBOOLE:def 15;
let x be object such that
x in C.i;
per cases;
suppose
A3: x in A.i;
take f.x;
thus f.x in B.i by A2,A3,FUNCT_2:5;
take f;
thus thesis;
end;
suppose
A4: not x in A.i;
consider y being object such that
A5: y in B.i by A2,XBOOLE_0:def 1;
take y;
thus y in B.i by A5;
take f;
thus thesis by A4;
end;
end;
consider G being ManySortedFunction of C, B such that
A6: for i being object st i in I
ex g being Function of C.i, B.i st g = G.i &
for x being object st x in C.i holds P[g.x,x,i] from MSSUBFAM:sch 1(A1);
take G;
now
let i be object;
assume
A7: i in I;
then reconsider f = F.i as Function of A.i, B.i by PBOOLE:def 15;
A8: dom f = A.i by A7,FUNCT_2:def 1;
consider g being Function of C.i, B.i such that
A9: g = G.i and
A10: for x being object st x in C.i holds P[g.x,x,i] by A6,A7;
A c= C by PBOOLE:def 18;
then
A11: A.i c= C.i by A7;
A12: for x being object st x in A.i holds f.x = (g|(A.i)).x
proof
let x be object;
assume
A13: x in A.i;
then ex h being Function of A.i, B.i st h = F.i & (x in A.i implies g.x =
h.x) by A10,A11;
hence thesis by A13,FUNCT_1:49;
end;
dom g = C.i by A7,FUNCT_2:def 1;
then
A14: dom (g|(A.i)) = A.i by A11,RELAT_1:62;
thus (G || A).i = g|(A.i) by A7,A9,MSAFREE:def 1
.= F.i by A8,A14,A12,FUNCT_1:2;
end;
hence thesis;
end;
definition
let I be set, A be ManySortedSet of I, F be ManySortedFunction of I;
func F""A -> ManySortedSet of I means
:Def1:
for i being object st i in I holds it.i = (F.i)"(A.i);
existence
proof
deffunc F(object) = (F.$1)"(A.$1);
ex f being ManySortedSet of I st
for i being object st i in I holds f.i = F(i) from PBOOLE:sch 4;
hence thesis;
end;
uniqueness
proof
let X, Y be ManySortedSet of I such that
A1: for i being object st i in I holds X.i = (F.i)"(A.i) and
A2: for i being object st i in I holds Y.i = (F.i)"(A.i);
now
let i be object;
assume
A3: i in I;
hence X.i = (F.i)"(A.i) by A1
.= Y.i by A2,A3;
end;
hence X = Y;
end;
end;
theorem
for A, B, C being ManySortedSet of I, F being ManySortedFunction of A,
B holds F.:.:C is ManySortedSubset of B
proof
let A, B, C be ManySortedSet of I, F be ManySortedFunction of A, B;
let i be object;
assume
A1: i in I;
then reconsider J = I as non empty set;
reconsider j = i as Element of J by A1;
reconsider A1 = A, B1 = B as ManySortedSet of J;
reconsider F1 = F as ManySortedFunction of A1, B1;
(F1.j).:(C.j) c= B.j;
hence thesis by PBOOLE:def 20;
end;
theorem
for A, B, C being ManySortedSet of I, F being ManySortedFunction of A,
B holds F""C is ManySortedSubset of A
proof
let A, B, C be ManySortedSet of I, F be ManySortedFunction of A, B;
let i be object;
assume
A1: i in I;
then reconsider J = I as non empty set;
reconsider j = i as Element of J by A1;
reconsider A1 = A, B1 = B as ManySortedSet of J;
reconsider F1 = F as ManySortedFunction of A1, B1;
(F1.j)"(C.j) c= A.j;
hence thesis by Def1;
end;
theorem
for A, B being ManySortedSet of I, F being ManySortedFunction of A, B
st F is "onto" holds F.:.:A = B
proof
let A, B be ManySortedSet of I, F be ManySortedFunction of A, B such that
A1: F is "onto";
now
let i be object;
assume
A2: i in I;
then
A3: F.i is Function of A.i, B.i by PBOOLE:def 15;
per cases;
suppose
B.i = {} implies A.i = {};
thus (F.:.:A).i = (F.i).:(A.i) by A2,PBOOLE:def 20
.= rng (F.i) by A3,RELSET_1:22
.= B.i by A1,A2;
end;
suppose
A4: not (B.i = {} implies A.i = {});
then
A5: F.i = {} by A3;
thus (F.:.:A).i = (F.i).:(A.i) by A2,PBOOLE:def 20
.= B.i by A4,A5;
end;
end;
hence thesis;
end;
theorem
for A, B being ManySortedSet of I, F being ManySortedFunction of A, B
st A is_transformable_to B holds F""B = A
proof
let A, B be ManySortedSet of I, F be ManySortedFunction of A, B such that
A1: A is_transformable_to B;
now
let i be object;
assume
A2: i in I;
then
A3: F.i is Function of A.i, B.i by PBOOLE:def 15;
A4: B.i = {} implies A.i = {} by A1,A2;
thus (F""B).i = (F.i)"(B.i) by A2,Def1
.= A.i by A4,A3,FUNCT_2:40;
end;
hence thesis;
end;
theorem
for A being ManySortedSet of I, F being ManySortedFunction of I st A
c= rngs F holds F.:.:(F""A) = A
proof
let A be ManySortedSet of I, F be ManySortedFunction of I such that
A1: A c= rngs F;
now
let i be object;
assume
A2: i in I;
then A.i c= (rngs F).i by A1;
then
A3: A.i c= rng (F.i) by A2,MSSUBFAM:13;
thus (F.:.:(F""A)).i = (F.i).:((F""A).i) by A2,PBOOLE:def 20
.= (F.i).:((F.i)"(A.i)) by A2,Def1
.= A.i by A3,FUNCT_1:77;
end;
hence thesis;
end;
theorem
for f being ManySortedFunction of I for X being ManySortedSet of I
holds f.:.:X c= rngs f
proof
let f be ManySortedFunction of I;
let X be ManySortedSet of I;
let i be object;
assume
A1: i in I;
then (f.:.:X).i = (f.i).:(X.i) by PBOOLE:def 20;
then (f.:.:X).i c= rng (f.i) by RELAT_1:111;
hence thesis by A1,MSSUBFAM:13;
end;
theorem Th13:
for f being ManySortedFunction of I holds f.:.:(doms f) = rngs f
proof
let f be ManySortedFunction of I;
now
let i be object;
assume
A1: i in I;
hence (f.:.:(doms f)).i = (f.i).:((doms f).i) by PBOOLE:def 20
.= (f.i).:(dom (f.i)) by A1,MSSUBFAM:14
.= rng (f.i) by RELAT_1:113
.= (rngs f).i by A1,MSSUBFAM:13;
end;
hence thesis;
end;
theorem Th14:
for f being ManySortedFunction of I holds f""(rngs f) = doms f
proof
let f be ManySortedFunction of I;
now
let i be object;
assume
A1: i in I;
hence (f""(rngs f)).i = (f.i)"((rngs f).i) by Def1
.= (f.i)"(rng (f.i)) by A1,MSSUBFAM:13
.= dom (f.i) by RELAT_1:134
.= (doms f).i by A1,MSSUBFAM:14;
end;
hence thesis;
end;
theorem
for A being ManySortedSet of I holds (id A).:.:A = A
proof
let A be ManySortedSet of I;
A1: rngs (id A) = A by EXTENS_1:9;
doms (id A) = A by MSSUBFAM:17;
hence thesis by A1,Th13;
end;
theorem
for A being ManySortedSet of I holds (id A)""A = A
proof
let A be ManySortedSet of I;
A1: rngs (id A) = A by EXTENS_1:9;
doms (id A) = A by MSSUBFAM:17;
hence thesis by A1,Th14;
end;
begin :: On the Many Sorted Algebras
reserve S for non empty non void ManySortedSign,
U0, U1 for non-empty MSAlgebra over S;
theorem Th17:
for A being MSAlgebra over S holds A is MSSubAlgebra of the MSAlgebra of A
by MSUALG_2:5;
theorem Th18:
for U0 being MSAlgebra over S, A being MSSubAlgebra of U0 for o
being OperSymbol of S, x being set st x in Args(o,A) holds x in Args(o,U0)
proof
let U0 be MSAlgebra over S, A be MSSubAlgebra of U0, o be OperSymbol of S, x
be set such that
A1: x in Args(o,A);
reconsider B0 = the Sorts of A as MSSubset of U0 by MSUALG_2:def 9;
the MSAlgebra of U0 = the MSAlgebra of U0;
then U0 is MSSubAlgebra of U0 by MSUALG_2:5;
then reconsider B1 = the Sorts of U0 as MSSubset of U0 by MSUALG_2:def 9;
B0 c= B1 by PBOOLE:def 18;
then ((B0# * the Arity of S).o) c= ((B1# * the Arity of S).o) by MSUALG_2:2;
hence thesis by A1;
end;
theorem Th19:
for U0 being MSAlgebra over S, A being MSSubAlgebra of U0 for o
being OperSymbol of S, x being set st x in Args(o,A) holds Den(o,A).x = Den(o,
U0).x
proof
let U0 be MSAlgebra over S, A be MSSubAlgebra of U0, o be OperSymbol of S, x
be set such that
A1: x in Args(o,A);
reconsider B = the Sorts of A as MSSubset of U0 by MSUALG_2:def 9;
B is opers_closed by MSUALG_2:def 9;
then
A2: B is_closed_on o;
thus Den(o,A).x = (Opers(U0,B).o).x by MSUALG_2:def 9
.= (o/.B).x by MSUALG_2:def 8
.= ((Den(o,U0)) | ((B# * the Arity of S).o)).x by A2,MSUALG_2:def 7
.= Den(o,U0).x by A1,FUNCT_1:49;
end;
theorem Th20:
for F being MSAlgebra-Family of I, S, B being MSSubAlgebra of
product F for o being OperSymbol of S, x being object st x in Args(o,B)
holds Den(o,B).x is Function & Den(o,product F).x is Function
proof
let F be MSAlgebra-Family of I, S, B be MSSubAlgebra of product F, o be
OperSymbol of S, x be object;
set r = the_result_sort_of o;
assume
A1: x in Args(o,B);
then x in Args(o,product F) by Th18;
then Den(o,product F).x in product Carrier(F,r) by PRALG_3:19;
then Den(o,B).x in product Carrier(F,r) by A1,Th19;
then reconsider p = Den(o,B).x as Element of (SORTS F).r by PRALG_2:def 10;
A2: p is Function;
hence Den(o,B).x is Function;
thus thesis by A1,A2,Th19;
end;
definition
let S be non void non empty ManySortedSign, A be MSAlgebra over S, B be
MSSubAlgebra of A;
func SuperAlgebraSet B -> set means
:Def2:
for x being object holds x in it iff ex C
being strict MSSubAlgebra of A st x = C & B is MSSubAlgebra of C;
existence
proof
defpred P[object] means
ex C being strict MSSubAlgebra of A st $1 = C & B is
MSSubAlgebra of C;
consider IT being set such that
A1: for x being object holds x in IT iff x in MSSub A & P[x] from
XBOOLE_0:sch 1;
take IT;
let x be object;
thus x in IT implies ex C being strict MSSubAlgebra of A st x = C & B is
MSSubAlgebra of C by A1;
given C being strict MSSubAlgebra of A such that
A2: x = C and
A3: B is MSSubAlgebra of C;
x in MSSub A by A2,MSUALG_2:def 19;
hence thesis by A1,A2,A3;
end;
uniqueness
proof
defpred P[object] means
ex C being strict MSSubAlgebra of A st $1 = C & B is
MSSubAlgebra of C;
for X1,X2 being set st (for x being object holds x in X1 iff P[x]) &
(for
x being object holds x in X2 iff P[x]) holds X1 = X2 from XBOOLE_0:sch 3;
hence thesis;
end;
end;
registration
let S be non void non empty ManySortedSign, A be MSAlgebra over S, B be
MSSubAlgebra of A;
cluster SuperAlgebraSet B -> non empty;
coherence
proof
the MSAlgebra of B is MSSubAlgebra of B by MSUALG_2:37;
then
A1: the MSAlgebra of B is MSSubAlgebra of A by MSUALG_2:6;
B is MSSubAlgebra of the MSAlgebra of B by Th17;
hence thesis by A1,Def2;
end;
end;
registration
let S be non empty non void ManySortedSign;
cluster strict non-empty free for MSAlgebra over S;
existence
proof
set X = the non-empty ManySortedSet of the carrier of S;
take FreeMSA X;
thus thesis;
end;
end;
registration
let S be non empty non void ManySortedSign, A be non-empty MSAlgebra over S,
X be non-empty finite-yielding MSSubset of A;
cluster GenMSAlg X -> finitely-generated;
coherence
proof
per cases;
case
S is non void;
let S9 be non void non empty ManySortedSign such that
A1: S9 = S;
let H be MSAlgebra over S9;
assume
A2: H = GenMSAlg X;
then reconsider T = X as MSSubset of H by A1,MSUALG_2:def 17;
GenMSAlg T = H by A1,A2,EXTENS_1:18;
then reconsider T as GeneratorSet of H by A1,A2,MSAFREE:3;
take T;
thus thesis;
end;
case
S is void;
hence thesis;
end;
end;
end;
registration
let S be non empty non void ManySortedSign, A be non-empty MSAlgebra over S;
cluster strict non-empty finitely-generated for MSSubAlgebra of A;
existence
proof
set G = the non-empty finite-yielding ManySortedSubset of the Sorts of A;
take GenMSAlg G;
thus thesis;
end;
end;
registration
let S be non empty non void ManySortedSign, A be feasible MSAlgebra over S;
cluster feasible for MSSubAlgebra of A;
existence
proof
the MSAlgebra of A = the MSAlgebra of A;
then reconsider B = A as MSSubAlgebra of A by MSUALG_2:5;
take B;
thus thesis;
end;
end;
theorem Th21:
for A being MSAlgebra over S, C being MSSubAlgebra of A for D
being ManySortedSubset of the Sorts of A st D = the Sorts of C for h being
ManySortedFunction of A, U0 for g being ManySortedFunction of C, U0 st g = h ||
D for o being OperSymbol of S, x being Element of Args(o,A) for y being Element
of Args(o,C) st Args(o,C) <> {} & x = y holds h#x = g#y
proof
let A be MSAlgebra over S, C be MSSubAlgebra of A, D be ManySortedSubset of
the Sorts of A such that
A1: D = the Sorts of C;
let h be ManySortedFunction of A, U0, g be ManySortedFunction of C, U0 such
that
A2: g = h || D;
let o be OperSymbol of S, x be Element of Args(o,A);
let y be Element of Args(o,C) such that
A3: Args(o,C) <> {} and
A4: x = y;
set hx = h#x, gy = g#y;
set ar = the_arity_of o;
A5: y in Args(o,A) by A3,Th18;
then reconsider xx = x, yy = y as Function by MSUALG_6:1;
A6: dom yy = dom ar by A3,MSUALG_3:6;
A7: dom xx = dom ar by A5,MSUALG_3:6;
A8: now
let n be object;
assume
A9: n in dom ar;
then reconsider n9=n as Nat;
reconsider hn = h.(ar.n) as Function of (the Sorts of A).(ar.n), (the
Sorts of U0).(ar.n) by A9,PARTFUN1:4,PBOOLE:def 15;
ar.n in the carrier of S by A9,PARTFUN1:4;
then
A10: g.(ar.n9) = hn | (D.(ar.n)) by A2,MSAFREE:def 1;
dom ((the Sorts of C)*(the_arity_of o)) = dom ar by PARTFUN1:def 2;
then xx.n in ((the Sorts of C) * ar).n by A3,A4,A9,MSUALG_3:6;
then
A11: xx.n in D.(ar.n) by A1,A9,FUNCT_1:13;
thus hx.n = hx.n9 .= (h.(ar/.n)).(xx.n) by A5,A7,A9,MSUALG_3:24
.= (h.(ar.n)).(xx.n) by A9,PARTFUN1:def 6
.= (g.(ar.n)).(xx.n) by A10,A11,FUNCT_1:49
.= (g.(ar/.n)).(yy.n) by A4,A9,PARTFUN1:def 6
.= gy.n9 by A3,A6,A9,MSUALG_3:24
.= gy.n;
end;
dom hx = dom ar & dom gy = dom ar by MSUALG_3:6;
hence thesis by A8,FUNCT_1:2;
end;
theorem Th22:
for A being feasible MSAlgebra over S, C being feasible
MSSubAlgebra of A for D being ManySortedSubset of the Sorts of A st D = the
Sorts of C for h being ManySortedFunction of A, U0 st h is_homomorphism A, U0
for g being ManySortedFunction of C, U0 st g = h || D holds g is_homomorphism C
, U0
proof
let A be feasible MSAlgebra over S, C be feasible MSSubAlgebra of A, D be
ManySortedSubset of the Sorts of A such that
A1: D = the Sorts of C;
let h be ManySortedFunction of A, U0 such that
A2: h is_homomorphism A, U0;
let g be ManySortedFunction of C, U0 such that
A3: g = h || D;
let o be OperSymbol of S such that
A4: Args(o,C) <> {};
let x be Element of Args(o,C);
reconsider y = x as Element of Args(o,A) by A4,Th18;
set r = the_result_sort_of o;
A5: x in Args(o,A) by A4,Th18;
Result(o,C) <> {} by A4,MSUALG_6:def 1;
then Den(o,C).x in Result(o,C) by A4,FUNCT_2:5;
then
A6: Den(o,C).x in (the Sorts of C).((the ResultSort of S).o) by FUNCT_2:15;
Den(o,A).x = Den(o,C).x by A4,Th19;
hence (g.(the_result_sort_of o)).(Den(o,C).x) = (h.r).(Den(o,A).x) by A1,A3
,A6,INSTALG1:39
.= Den(o,U0).(h#y) by A2,A5
.= Den(o,U0).(g#x) by A1,A3,A4,Th21;
end;
theorem
for B being strict non-empty MSAlgebra over S for G being GeneratorSet
of U0, H being non-empty GeneratorSet of B for f being ManySortedFunction of U0
, B st H c= f.:.:G & f is_homomorphism U0, B holds f is_epimorphism U0, B
proof
let B be strict non-empty MSAlgebra over S, G be GeneratorSet of U0, H be
non-empty GeneratorSet of B, f be ManySortedFunction of U0, B such that
A1: H c= f.:.:G and
A2: f is_homomorphism U0, B;
reconsider I = the Sorts of Image f as non-empty MSSubset of B by
MSUALG_2:def 9;
f.:.:G c= f.:.:the Sorts of U0 by EXTENS_1:2;
then H c= f.:.:the Sorts of U0 by A1,PBOOLE:13;
then H c= the Sorts of Image f by A2,MSUALG_3:def 12;
then H is ManySortedSubset of the Sorts of Image f by PBOOLE:def 18;
then
A3: GenMSAlg H is MSSubAlgebra of GenMSAlg I by EXTENS_1:17;
reconsider I1 = the Sorts of Image f as non-empty MSSubset of Image f by
PBOOLE:def 18;
I is GeneratorSet of Image f & GenMSAlg I = GenMSAlg I1 by EXTENS_1:18
,MSAFREE2:6;
then GenMSAlg I = Image f by MSAFREE:3;
then B is MSSubAlgebra of Image f by A3,MSAFREE:3;
then Image f = B by MSUALG_2:7;
hence thesis by A2,MSUALG_3:19;
end;
theorem Th24:
for W being strict free non-empty MSAlgebra over S for F being
ManySortedFunction of U0, U1 st F is_epimorphism U0, U1 for G being
ManySortedFunction of W, U1 st G is_homomorphism W, U1 holds ex H being
ManySortedFunction of W, U0 st H is_homomorphism W, U0 & G = F ** H
proof
let W be strict free non-empty MSAlgebra over S, F be ManySortedFunction of
U0, U1 such that
A1: F is_epimorphism U0, U1;
set sU0 = the Sorts of U0, sU1 = the Sorts of U1, I = the carrier of S;
let G be ManySortedFunction of W, U1 such that
A2: G is_homomorphism W, U1;
consider Gen be GeneratorSet of W such that
A3: Gen is free by MSAFREE:def 6;
defpred P[object,object,object] means
ex f be Function of sU0.$3, sU1.$3 st ex g be
Function of Gen.$3, sU1.$3 st f = F.$3 & g = (G || Gen).$3 & $1 in f"{g.$2};
A4: for i being object st i in I
for x be object st x in Gen.i ex y be object st y in
sU0.i & P[y,x,i]
proof
let i be object such that
A5: i in I;
reconsider g = (G || Gen).i as Function of Gen.i, sU1.i by A5,PBOOLE:def 15
;
reconsider f = F.i as Function of sU0.i, sU1.i by A5,PBOOLE:def 15;
let x be object;
assume x in Gen.i;
then
A6: g.x in sU1.i by A5,FUNCT_2:5;
F is "onto" by A1;
then rng (F.i) = sU1.i by A5;
then f"{g.x} <> {} by A6,FUNCT_2:41;
then consider y be object such that
A7: y in f"{g.x} by XBOOLE_0:def 1;
take y;
thus y in sU0.i by A7;
thus thesis by A7;
end;
consider H be ManySortedFunction of Gen, sU0 such that
A8: for i be object st i in I
ex h be Function of Gen.i, sU0.i st h = H.i
& for x be object st x in Gen.i holds P[h.x,x,i] from MSSUBFAM:sch 1(A4);
consider H1 be ManySortedFunction of W, U0 such that
A9: H1 is_homomorphism W, U0 and
A10: H1 || Gen = H by A3;
now
let i be object;
assume
A11: i in I;
then reconsider f9 = F.i as Function of sU0.i, sU1.i by PBOOLE:def 15;
reconsider g9 = (G || Gen).i as Function of Gen.i, sU1.i by A11,
PBOOLE:def 15;
consider h be Function of Gen.i, sU0.i such that
A12: h = H.i and
A13: for x be object st x in Gen.i holds ex f be Function of sU0.i, sU1.i
st ex g be Function of Gen.i, sU1.i st f = F.i & g = (G || Gen).i & h.x in f"{g
.x} by A8,A11;
A14: now
let x be object;
assume
A15: x in Gen.i;
then consider
f be Function of sU0.i, sU1.i, g be Function of Gen.i, sU1.i
such that
A16: f = F.i & g = (G || Gen).i and
A17: h.x in f"{g.x} by A13;
f.(h.x) in {g.x} by A11,A17,FUNCT_2:38;
then f.(h.x) = g.x by TARSKI:def 1;
hence (f9*h).x = g9.x by A15,A16,A17,FUNCT_2:15;
end;
Gen.i = dom g9 & Gen.i = dom (f9*h) by A11,FUNCT_2:def 1;
then g9 = f9 * h by A14,FUNCT_1:2;
hence (G || Gen).i = (F ** H).i by A11,A12,MSUALG_3:2;
end;
then G || Gen = F ** (H1 || Gen) by A10;
then
A18: G || Gen = (F ** H1) || Gen by EXTENS_1:4;
take H1;
thus H1 is_homomorphism W, U0 by A9;
F is_homomorphism U0, U1 by A1;
then F ** H1 is_homomorphism W, U1 by A9,MSUALG_3:10;
hence thesis by A2,A18,EXTENS_1:19;
end;
theorem Th25:
for I being non empty finite set, A being non-empty MSAlgebra
over S for F being MSAlgebra-Family of I, S st for i being Element of I ex C
being strict non-empty finitely-generated MSSubAlgebra of A st C = F.i ex B
being strict non-empty finitely-generated MSSubAlgebra of A st for i being
Element of I holds F.i is MSSubAlgebra of B
proof
let I be non empty finite set, A be non-empty MSAlgebra over S, F be
MSAlgebra-Family of I, S such that
A1: for i being Element of I ex C being strict non-empty
finitely-generated MSSubAlgebra of A st C = F.i;
set Z = { C where C is Element of MSSub A : ex i being (Element of I), D
being strict non-empty finitely-generated MSSubAlgebra of A st C = F.i & C = D
};
set J = the carrier of S;
set m = the Element of I;
consider W being strict non-empty finitely-generated MSSubAlgebra of A such
that
A2: W = F.m by A1;
W is Element of MSSub A by MSUALG_2:def 19;
then W in Z by A2;
then reconsider Z as non empty set;
defpred P[set,set] means ex Q being strict non-empty MSSubAlgebra of A, G
being GeneratorSet of Q st $1 = Q & $2 = G & G is non-empty finite-yielding;
A3: for a being Element of Z ex b being Element of Bool the Sorts of A st P[
a,b]
proof
let a be Element of Z;
a in Z;
then consider C being Element of MSSub A such that
A4: a = C and
A5: ex i being (Element of I), D being strict non-empty
finitely-generated MSSubAlgebra of A st C = F.i & C = D;
consider i being (Element of I), D being strict non-empty
finitely-generated MSSubAlgebra of A such that
C = F.i and
A6: C = D by A5;
consider G being GeneratorSet of D such that
A7: G is finite-yielding by MSAFREE2:def 10;
consider S being non-empty finite-yielding ManySortedSubset of the Sorts
of D such that
A8: G c= S by A7,MSUALG_9:7;
the Sorts of D is MSSubset of A by MSUALG_2:def 9;
then S c= the Sorts of D & the Sorts of D c= the Sorts of A by
PBOOLE:def 18;
then S c= the Sorts of A by PBOOLE:13;
then S is ManySortedSubset of the Sorts of A by PBOOLE:def 18;
then reconsider b = S as Element of Bool the Sorts of A by CLOSURE2:def 1;
reconsider S as GeneratorSet of D by A8,MSSCYC_1:21;
take b, D;
take S;
thus a = D by A4,A6;
thus thesis;
end;
consider f being Function of Z, Bool the Sorts of A such that
A9: for B being Element of Z holds P[B,f.B] from FUNCT_2:sch 3(A3);
deffunc F(object) =
union { a where a is Element of bool ((the Sorts of A).$1)
: ex i being (Element of I), K being ManySortedSet of J st K = f.(F.i) & a = K.
$1 };
consider SOR being ManySortedSet of J such that
A10: for j being object st j in J holds SOR.j = F(j) from PBOOLE:sch 4;
SOR is ManySortedSubset of the Sorts of A
proof
let j be object such that
A11: j in J;
let q be object;
set UU = { a where a is Element of bool ((the Sorts of A).j) : ex i being
(Element of I), K being ManySortedSet of J st K = f.(F.i) & a = K.j };
assume q in SOR.j;
then q in union UU by A10,A11;
then consider w being set such that
A12: q in w and
A13: w in UU by TARSKI:def 4;
consider a being Element of bool ((the Sorts of A).j) such that
A14: w = a and
ex i being (Element of I), K being ManySortedSet of J st K = f.(F.i) &
a = K.j by A13;
thus thesis by A12,A14;
end;
then reconsider SOR as MSSubset of A;
SOR is non-empty
proof
set i = the Element of I;
consider C being strict non-empty finitely-generated MSSubAlgebra of A
such that
A15: C = F.i by A1;
C is Element of MSSub A by MSUALG_2:def 19;
then
A16: F.i in Z by A15;
then
A17: ex Q being strict non-empty MSSubAlgebra of A st ex G being
GeneratorSet of Q st F.i = Q & f.(F.i) = G & G is non-empty finite-yielding by
A9;
then reconsider
G1 = f.(F.i) as finite-yielding Element of Bool the Sorts of A
by A16,FUNCT_2:5;
let j be object such that
A18: j in J;
consider q be object such that
A19: q in G1.j by A18,A17,XBOOLE_0:def 1;
set UU = { a where a is Element of bool ((the Sorts of A).j) : ex i being
(Element of I), K being ManySortedSet of J st K = f.(F.i) & a = K.j };
G1 c= the Sorts of A by PBOOLE:def 18;
then G1.j c= (the Sorts of A).j by A18;
then G1.j in UU;
then q in union UU by A19,TARSKI:def 4;
hence thesis by A10,A18;
end;
then reconsider SOR as non-empty MSSubset of A;
A20: now
set i = the Element of I;
let j be set such that
A21: j in J;
set UU = { a where a is Element of bool ((the Sorts of A).j) : ex i being
(Element of I), K being ManySortedSet of J st K = f.(F.i) & a = K.j };
consider C being strict non-empty finitely-generated MSSubAlgebra of A
such that
A22: C = F.i by A1;
C is Element of MSSub A by MSUALG_2:def 19;
then
A23: F.i in Z by A22;
then ex Q being strict non-empty MSSubAlgebra of A st ex G being
GeneratorSet of Q st F.i = Q & f.(F.i) = G & G is non-empty finite-yielding by
A9;
then reconsider
G1 = f.(F.i) as finite-yielding Element of Bool the Sorts of A
by A23,FUNCT_2:5;
G1 c= the Sorts of A by PBOOLE:def 18;
then G1.j c= (the Sorts of A).j by A21;
then G1.j in UU;
hence UU is non empty;
end;
SOR is finite-yielding
proof
let j be object such that
A24: j in J;
set UU = { a where a is Element of bool ((the Sorts of A).j) : ex i being
(Element of I), K being ManySortedSet of J st K = f.(F.i) & a = K.j };
reconsider VV = UU as non empty set by A20,A24;
A25: now
defpred P[set,set] means ex L being ManySortedSet of J st f.(F.$1) = L &
$2 = L.j;
consider r being FinSequence such that
A26: rng r = I by FINSEQ_1:52;
A27: for a being Element of I ex b being Element of VV st P[a,b]
proof
let a be Element of I;
consider W being strict non-empty finitely-generated MSSubAlgebra of A
such that
A28: W = F.a by A1;
W is Element of MSSub A by MSUALG_2:def 19;
then
A29: W in Z by A28;
then ex Q being strict non-empty MSSubAlgebra of A st ex G being
GeneratorSet of Q st W = Q & f.W = G & G is non-empty finite-yielding by A9;
then reconsider
G1 = f.(F.a) as finite-yielding Element of Bool the Sorts
of A by A28,A29,FUNCT_2:5;
G1 c= the Sorts of A by PBOOLE:def 18;
then G1.j c= (the Sorts of A).j by A24;
then G1.j in UU;
then reconsider b = G1.j as Element of VV;
take b, G1;
thus thesis;
end;
consider h being Function of I, VV such that
A30: for i being Element of I holds P[i,h.i] from FUNCT_2:sch 3(A27);
A31: rng r = dom h by A26,FUNCT_2:def 1;
then reconsider p = h*r as FinSequence by FINSEQ_1:16;
A32: VV c= rng p
proof
let q be object;
assume q in VV;
then consider a being Element of bool ((the Sorts of A).j) such that
A33: q = a and
A34: ex i being (Element of I), K being ManySortedSet of J st K =
f.(F.i) & a = K.j;
consider i being (Element of I), K being ManySortedSet of J such that
A35: K = f.(F.i) & a = K.j by A34;
A36: rng p = rng h & dom h = I by A31,FUNCT_2:def 1,RELAT_1:28;
ex L being ManySortedSet of J st f.(F.i) = L & h.i = L.j by A30;
hence thesis by A33,A35,A36,FUNCT_1:def 3;
end;
take p;
rng p c= rng h
by FUNCT_1:14;
then rng p c= VV by XBOOLE_1:1;
hence rng p = VV by A32;
end;
for x being set st x in UU holds x is finite
proof
let x be set;
assume x in UU;
then consider a being Element of bool ((the Sorts of A).j) such that
A37: x = a and
A38: ex w being (Element of I), K being ManySortedSet of J st K = f.
(F.w) & a = K.j;
consider w being (Element of I), K being ManySortedSet of J such that
A39: K = f.(F.w) & a = K.j by A38;
A40: ex E being strict non-empty finitely-generated MSSubAlgebra of A st
E = F.w by A1;
then F.w is Element of MSSub A by MSUALG_2:def 19;
then F.w in Z by A40;
then P[F.w,f.(F.w)] by A9;
hence thesis by A37,A39;
end;
then union UU is finite by A25,FINSET_1:7;
hence thesis by A10,A24;
end;
then reconsider SOR as non-empty finite-yielding MSSubset of A;
take GenMSAlg SOR;
let i be Element of I;
consider C being strict non-empty finitely-generated MSSubAlgebra of A such
that
A41: C = F.i by A1;
C is Element of MSSub A by MSUALG_2:def 19;
then F.i in Z by A41;
then consider
Q being strict non-empty MSSubAlgebra of A, G being GeneratorSet of
Q such that
A42: F.i = Q and
A43: f.(F.i) = G and
A44: G is non-empty finite-yielding by A9;
the Sorts of Q is MSSubset of A by MSUALG_2:def 9;
then G c= the Sorts of Q & the Sorts of Q c= the Sorts of A by PBOOLE:def 18;
then
A45: G c= the Sorts of A by PBOOLE:13;
then reconsider G1 = G as non-empty MSSubset of A by A44,PBOOLE:def 18;
A46: G1 is ManySortedSubset of SOR
proof
let j be object such that
A47: j in J;
set UU = { a where a is Element of bool ((the Sorts of A).j) : ex i being
(Element of I), K being ManySortedSet of J st K = f.(F.i) & a = K.j };
G1.j c= (the Sorts of A).j by A45,A47;
then
A48: G1.j in UU by A43;
let q be object;
assume q in G1.j;
then q in union UU by A48,TARSKI:def 4;
hence thesis by A10,A47;
end;
C = GenMSAlg G by A41,A42,MSAFREE:3
.= GenMSAlg G1 by EXTENS_1:18;
hence thesis by A41,A46,EXTENS_1:17;
end;
theorem Th26:
for A, B being strict non-empty finitely-generated MSSubAlgebra
of U0 ex M being strict non-empty finitely-generated MSSubAlgebra of U0 st A is
MSSubAlgebra of M & B is MSSubAlgebra of M
proof
let A, B be strict non-empty finitely-generated MSSubAlgebra of U0;
len <%A,B%> = 2 by AFINSQ_1:38;
then dom <%A,B%> = {0,1} by CARD_1:50;
then reconsider F = <%A,B%> as ManySortedSet of {0,1}
by RELAT_1:def 18, PARTFUN1:def 2;
A1: F.1 = B;
A2: F.0 = A;
F is MSAlgebra-Family of {0,1}, S
proof
let i be object;
assume i in {0,1};
hence thesis by A1,TARSKI:def 2,A2;
end;
then reconsider F as MSAlgebra-Family of {0,1}, S;
for i being Element of {0,1} ex C being strict non-empty
finitely-generated MSSubAlgebra of U0 st C = F.i
proof
let i be Element of {0,1};
per cases by TARSKI:def 2;
suppose
A3: i = 0;
take A;
thus thesis by A3;
end;
suppose
A4: i = 1;
take B;
thus thesis by A4;
end;
end;
then consider
M being strict non-empty finitely-generated MSSubAlgebra of U0 such
that
A5: for i being Element of {0,1} holds F.i is MSSubAlgebra of M by Th25;
take M;
0 in {0,1} by TARSKI:def 2;
hence A is MSSubAlgebra of M by A5,A2;
1 in {0,1} by TARSKI:def 2;
hence thesis by A5,A1;
end;
theorem
for SG being non empty non void ManySortedSign for AG being non-empty
MSAlgebra over SG for C being set st C = { A where A is Element of MSSub AG :
ex R being strict non-empty finitely-generated MSSubAlgebra of AG st R = A }
for F being MSAlgebra-Family of C, SG st
for c being object st c in C holds c = F.
c ex PS being strict non-empty MSSubAlgebra of product F, BA being
ManySortedFunction of PS, AG st BA is_epimorphism PS, AG
proof
let SG be non empty non void ManySortedSign, AG be non-empty MSAlgebra over
SG, C be set such that
A1: C = { A where A is Element of MSSub AG : ex R being strict non-empty
finitely-generated MSSubAlgebra of AG st R = A };
A2: now
let A be strict non-empty finitely-generated MSSubAlgebra of AG;
A in MSSub AG by MSUALG_2:def 19;
hence A in C by A1;
end;
then reconsider CC = C as non empty set;
set T = the strict non-empty finitely-generated MSSubAlgebra of AG;
set I = the carrier of SG;
let F be MSAlgebra-Family of C, SG such that
A3: for c being object st c in C holds c = F.c;
reconsider FF = F as MSAlgebra-Family of CC, SG;
set pr = product FF;
defpred T[object,object] means
ex t being SortSymbol of SG, f being Element of (
SORTS FF).t st ex A being strict non-empty finitely-generated MSSubAlgebra of
AG st t = $1 & f = $2 & for B being strict non-empty finitely-generated
MSSubAlgebra of AG st A is MSSubAlgebra of B holds f.A = f.B;
consider SOR being ManySortedSet of I such that
A4: for i being object st i in I for e being object holds e in SOR.i iff e in
(SORTS FF).i & T[i,e] from PBOOLE:sch 2;
SOR is MSSubset of pr
proof
let i be object such that
A5: i in I;
let e be object;
assume e in SOR.i;
hence thesis by A4,A5;
end;
then reconsider SOR as MSSubset of pr;
SOR is opers_closed
proof
let o be OperSymbol of SG;
set r = the_result_sort_of o;
set ar = the_arity_of o;
let q be object such that
A6: q in rng ( (Den(o,pr)) | ((SOR# * the Arity of SG).o) );
reconsider q1 = q as Element of (SORTS FF).r by A6,PRALG_2:3;
consider g being object such that
A7: g in dom ((Den(o,pr)) | ((SOR# * the Arity of SG).o)) and
A8: q = ((Den(o,pr)) | ((SOR# * the Arity of SG).o)).g by A6,FUNCT_1:def 3;
A9: q = Den(o,pr).g by A7,A8,FUNCT_1:47;
A10: g in (SOR# * the Arity of SG).o by A7;
g in dom (Den(o,pr)) by A7,RELAT_1:57;
then reconsider g as Element of Args(o,pr);
T[r,q]
proof
take r;
take q1;
per cases;
suppose
A11: the_arity_of o = {};
set A = the strict non-empty finitely-generated MSSubAlgebra of AG;
Args(o,A) = {{}} by A11,PRALG_2:4;
then
A12: {} in Args(o,A) by TARSKI:def 1;
take A;
thus r = r;
thus q1 = q;
let B be strict non-empty finitely-generated MSSubAlgebra of AG such
that
A is MSSubAlgebra of B;
Args(o,B) = {{}} by A11,PRALG_2:4;
then
A13: {} in Args(o,B) by TARSKI:def 1;
B in MSSub AG by MSUALG_2:def 19;
then
A14: B in CC by A1;
A in MSSub AG by MSUALG_2:def 19;
then A in CC by A1;
then reconsider iA = A, iB = B as Element of CC by A14;
A15: iA = FF.iA by A3;
A16: iB = FF.iB by A3;
Args(o,pr) = {{}} by A11,PRALG_2:4;
then
A17: g = {} by TARSKI:def 1;
hence q1.A = const(o,pr).iA by A9,PRALG_3:def 1
.= const(o,FF.iA) by A11,PRALG_3:9
.= Den(o,FF.iA).{} by PRALG_3:def 1
.= Den(o,AG).{} by A15,A12,Th19
.= Den(o,FF.iB).{} by A16,A13,Th19
.= const(o,FF.iB) by PRALG_3:def 1
.= const(o,pr).iB by A11,PRALG_3:9
.= q1.B by A9,A17,PRALG_3:def 1;
end;
suppose
A18: the_arity_of o <> {};
then reconsider domar = dom ar as non empty finite set;
defpred P[set,set] means ex gn being Function, A being strict
non-empty finitely-generated MSSubAlgebra of AG st gn = g.$1 & (for B being
strict non-empty finitely-generated MSSubAlgebra of AG st A is MSSubAlgebra of
B holds gn.A = gn.B) & $2 = A;
g in SOR#.((the Arity of SG).o) by A10,FUNCT_2:15;
then
A19: g in product(SOR*ar) by FINSEQ_2:def 5;
then
A20: dom (SOR*ar) = dom g by CARD_3:9
.= domar by MSUALG_3:6;
A21: for a being Element of domar ex b being Element of MSSub AG st P[ a,b]
proof
let n be Element of domar;
g.n in (SOR*ar).n by A19,A20,CARD_3:9;
then ar.n in I & g.n in SOR.(ar.n) by FUNCT_1:13,PARTFUN1:4;
then consider
s being SortSymbol of SG, f being (Element of (SORTS FF).s)
, A being strict non-empty finitely-generated MSSubAlgebra of AG such that
s = ar.n and
A22: f = g.n and
A23: for B being strict non-empty finitely-generated
MSSubAlgebra of AG st A is MSSubAlgebra of B holds f.A = f.B by A4;
reconsider b = A as Element of MSSub AG by MSUALG_2:def 19;
take b, f;
take A;
thus f = g.n by A22;
thus for B being strict non-empty finitely-generated MSSubAlgebra of
AG st A is MSSubAlgebra of B holds f.A = f.B by A23;
thus thesis;
end;
consider KK being Function of domar, MSSub AG such that
A24: for n being Element of domar holds P[n,KK.n] from FUNCT_2:sch
3(A21);
reconsider KK as ManySortedSet of domar;
KK is MSAlgebra-Family of domar, SG
proof
let n be object;
assume n in domar;
then ex gn being Function st ex A being strict non-empty
finitely-generated MSSubAlgebra of AG st gn = g.n & (for B being strict
non-empty finitely-generated MSSubAlgebra of AG st A is MSSubAlgebra of B holds
gn.A = gn.B) & KK.n = A by A24;
hence thesis;
end;
then reconsider KK as MSAlgebra-Family of domar, SG;
for n being Element of domar ex C being strict non-empty
finitely-generated MSSubAlgebra of AG st C = KK.n
proof
let n be Element of domar;
ex gn being Function, A being strict non-empty
finitely-generated MSSubAlgebra of AG st gn = g.n & (for B being strict
non-empty finitely-generated MSSubAlgebra of AG st A is MSSubAlgebra of B holds
gn.A = gn.B) & KK.n = A by A24;
hence thesis;
end;
then consider
K being strict non-empty finitely-generated MSSubAlgebra of
AG such that
A25: for n being Element of domar holds KK.n is MSSubAlgebra of K by Th25;
take K;
thus r = r;
thus q1 = q;
let B be strict non-empty finitely-generated MSSubAlgebra of AG such
that
A26: K is MSSubAlgebra of B;
K in MSSub AG by MSUALG_2:def 19;
then
A27: K in CC by A1;
B in MSSub AG by MSUALG_2:def 19;
then B in CC by A1;
then reconsider iB = B, iK = K as Element of CC by A27;
A28: g in Funcs (dom ar, Funcs (CC, union the set of all
(the Sorts of FF.i).s
where i is (Element of CC), s is (Element of (the carrier of SG))
)) by PRALG_3:14;
then
A29: dom ((commute g).iB) = domar by Th3;
A30: dom ((commute g).iK) = domar by A28,Th3;
A31: for n being object st n in dom ((commute g).iK) holds ((commute g).
iB).n = ((commute g).iK).n
proof
let x be object;
assume
A32: x in dom ((commute g).iK);
then consider gn being Function, A being strict non-empty
finitely-generated MSSubAlgebra of AG such that
A33: gn = g.x and
A34: ( for B being strict non-empty finitely-generated
MSSubAlgebra of AG st A is MSSubAlgebra of B holds gn.A = gn.B)& KK.x = A by
A24,A30;
A35: KK.x is MSSubAlgebra of K by A25,A30,A32;
thus ((commute g).iB).x = gn.iB by A30,A32,A33,PRALG_3:21
.= gn.A by A26,A34,A35,MSUALG_2:6
.= gn.iK by A25,A30,A32,A34
.= ((commute g).iK).x by A30,A32,A33,PRALG_3:21;
end;
A36: iK = FF.iK by A3;
A37: (commute g).iK is Element of Args(o,FF.iK) by A18,PRALG_3:20;
A38: (commute g).iB is Element of Args(o,FF.iB) by A18,PRALG_3:20;
A39: iB = FF.iB by A3;
thus q1.K = Den(o,FF.iK).((commute g).iK) by A9,A18,PRALG_3:22
.= Den(o,AG).((commute g).iK) by A36,A37,Th19
.= Den(o,AG).((commute g).iB) by A28,A29,A31,Th3,FUNCT_1:2
.= Den(o,FF.iB).((commute g).iB) by A39,A38,Th19
.= q1.B by A9,A18,PRALG_3:22;
end;
end;
then q in SOR.r by A4;
hence thesis by FUNCT_2:15;
end;
then
A40: pr|SOR = MSAlgebra (#SOR, Opers(pr,SOR)#) by MSUALG_2:def 15;
defpred Z[object,object,object] means
ex s being SortSymbol of SG, f being Element of
(SORTS FF).s st ex A being strict non-empty finitely-generated MSSubAlgebra of
AG st s = $3 & f = $2 & (for B being strict non-empty finitely-generated
MSSubAlgebra of AG st A is MSSubAlgebra of B holds f.A = f.B) & $1 = f.A;
SOR is non-empty
proof
defpred P[object] means
ex C being strict non-empty MSSubAlgebra of AG st $1
= C & C is finitely-generated;
let i be object;
consider ST1 being set such that
A41: for x being object holds x in ST1 iff x in SuperAlgebraSet T & P[x]
from XBOOLE_0:sch 1;
A42: ST1 c= CC
proof
let q be object;
assume q in ST1;
then
A43: ex C being strict non-empty MSSubAlgebra of AG st q = C & C is
finitely-generated by A41;
then q in MSSub AG by MSUALG_2:def 19;
hence thesis by A1,A43;
end;
defpred A[object,object] means
ex K being MSSubAlgebra of AG st ex t being set
st $1 = K & t in (the Sorts of K).i & $2 = t;
assume
A44: i in I;
then consider x being object such that
A45: x in (the Sorts of T).i by XBOOLE_0:def 1;
reconsider s = i as SortSymbol of SG by A44;
A46: for c being object st c in CC ex j being object st A[c,j]
proof
let c be object;
assume c in CC;
then consider C being Element of MSSub AG such that
A47: c = C and
A48: ex R being strict non-empty finitely-generated MSSubAlgebra of
AG st R = C by A1;
consider R being strict non-empty finitely-generated MSSubAlgebra of AG
such that
A49: R = C by A48;
consider t being object such that
A50: t in (the Sorts of R).i by A44,XBOOLE_0:def 1;
take t, R;
reconsider t as set by TARSKI:1;
take t;
thus c = R by A47,A49;
thus thesis by A50;
end;
consider f being ManySortedSet of CC such that
A51: for c being object st c in CC holds A[c,f.c] from PBOOLE:sch 3(A46);
set g = f +* (ST1 --> x);
A52: dom (ST1 --> x) = ST1 by FUNCOP_1:13;
A53: for a being object st a in dom (Carrier(FF,s)) holds g.a in (Carrier(FF,
s)).a
proof
let a be object;
assume a in dom (Carrier(FF,s));
then
A54: a in CC;
then
A55: ex U0 being MSAlgebra over SG st U0 = FF.a & (Carrier(FF,s)).a = (
the Sorts of U0).s by PRALG_2:def 9;
consider K being MSSubAlgebra of AG, t being set such that
A56: a = K & t in (the Sorts of K).i and
A57: f.a = t by A51,A54;
per cases;
suppose
A58: a in ST1;
then a in dom (ST1 --> x) by FUNCOP_1:13;
then g.a = (ST1 --> x).a by FUNCT_4:13;
then
A59: g.a = x by A58,FUNCOP_1:7;
a in SuperAlgebraSet T by A41,A58;
then consider M being strict MSSubAlgebra of AG such that
A60: a = M and
A61: T is MSSubAlgebra of M by Def2;
the Sorts of T is ManySortedSubset of the Sorts of M by A61,
MSUALG_2:def 9;
then the Sorts of T c= the Sorts of M by PBOOLE:def 18;
then (the Sorts of T).i c= (the Sorts of M).i by A44;
then x in (the Sorts of M).i by A45;
hence thesis by A3,A54,A55,A59,A60;
end;
suppose
not a in ST1;
then g.a = t by A52,A57,FUNCT_4:11;
hence thesis by A3,A54,A55,A56;
end;
end;
dom g = dom f \/ dom (ST1 --> x) by FUNCT_4:def 1
.= CC \/ dom (ST1 --> x) by PARTFUN1:def 2
.= CC \/ ST1 by FUNCOP_1:13
.= CC by A42,XBOOLE_1:12
.= dom (Carrier(FF,s)) by PARTFUN1:def 2;
then
A62: g in product Carrier(FF,s) by A53,CARD_3:9;
T[i,g]
proof
reconsider g1 = g as Element of (SORTS FF).s by A62,PRALG_2:def 10;
take s;
take g1;
take T;
thus s = i;
thus g1 = g;
let B be strict non-empty finitely-generated MSSubAlgebra of AG;
assume T is MSSubAlgebra of B;
then B in SuperAlgebraSet T by Def2;
then
A63: B in ST1 by A41;
T is MSSubAlgebra of T by MSUALG_2:5;
then T in SuperAlgebraSet T by Def2;
then
A64: T in ST1 by A41;
hence g1.T = (ST1 --> x).T by A52,FUNCT_4:13
.= x by A64,FUNCOP_1:7
.= (ST1 --> x).B by A63,FUNCOP_1:7
.= g1.B by A52,A63,FUNCT_4:13;
end;
hence thesis by A4;
end;
then reconsider
PS = pr|SOR as strict non-empty MSSubAlgebra of product F by A40,
MSUALG_1:def 3;
A65: now
let s be SortSymbol of SG, f be (Element of (SORTS FF).s), A be strict
non-empty finitely-generated MSSubAlgebra of AG;
A66: dom Carrier(FF,s) = CC by PARTFUN1:def 2;
A in MSSub AG by MSUALG_2:def 19;
then
A67: A in dom Carrier(FF,s) by A1,A66;
then consider U0 being MSAlgebra over SG such that
A68: U0 = FF.A and
A69: Carrier(FF,s).A = (the Sorts of U0).s by PRALG_2:def 9;
(SORTS FF).s = product Carrier(FF,s) by PRALG_2:def 10;
then
A70: f.A in Carrier(FF,s).A by A67,CARD_3:9;
FF.A = A by A3,A67;
then the Sorts of U0 is ManySortedSubset of the Sorts of AG by A68,
MSUALG_2:def 9;
then the Sorts of U0 c= the Sorts of AG by PBOOLE:def 18;
then (the Sorts of U0).s c= (the Sorts of AG).s;
hence f.A in (the Sorts of AG).s by A69,A70;
end;
A71: now
let i be object such that
A72: i in I;
let x be object;
assume x in (the Sorts of PS).i;
then consider
s being SortSymbol of SG, f being (Element of (SORTS FF).s), A
being strict non-empty finitely-generated MSSubAlgebra of AG such that
A73: s = i and
A74: f = x & for B being strict non-empty finitely-generated
MSSubAlgebra of AG st A is MSSubAlgebra of B holds f.A = f.B by A4,A40,A72;
reconsider y = f.A as object;
take y;
thus y in (the Sorts of AG).i by A65,A73;
thus Z[y,x,i] by A73,A74;
end;
consider BA being ManySortedFunction of PS, AG such that
A75: for i being object st i in I
ex g being Function of (the Sorts
of PS).i, (the Sorts of AG).i st g = BA.i &
for x being object st x in (the Sorts
of PS).i holds Z[g.x,x,i] from MSSUBFAM:sch 1(A71);
take PS;
take BA;
thus BA is_homomorphism PS, AG
proof
let o be OperSymbol of SG such that
Args(o,PS) <> {};
let k be Element of Args(o,PS);
set r = the_result_sort_of o, ar = the_arity_of o;
consider g being Function of (the Sorts of PS).r, (the Sorts of AG).r such
that
A76: g = BA.r and
A77: for k being object st k in (the Sorts of PS).r holds Z[g.k,k,r] by A75;
consider s being SortSymbol of SG, f being (Element of (SORTS FF).s), A
being strict non-empty finitely-generated MSSubAlgebra of AG such that
s = r and
A78: f = Den(o,PS).k and
A79: for B being strict non-empty finitely-generated MSSubAlgebra of
AG st A is MSSubAlgebra of B holds f.A = f.B and
A80: g.(Den(o,PS).k) = f.A by A77,MSUALG_9:18;
per cases;
suppose
A81: the_arity_of o = {};
A in MSSub AG by MSUALG_2:def 19;
then A in CC by A1;
then reconsider iA = A as Element of CC;
reconsider ff1 = Den(o,pr).k as Function by Th20;
A82: Den(o,pr).{} = const(o,pr) by PRALG_3:def 1;
Args(o,A) = {{}} by A81,PRALG_2:4;
then
A83: {} in Args(o,A) by TARSKI:def 1;
A84: Args(o,PS) = {{}} by A81,PRALG_2:4;
then
A85: k = {} by TARSKI:def 1;
thus BA.(the_result_sort_of o).(Den(o,PS).k) = ff1.A by A76,A78,A80,Th19
.= const(o,pr).iA by A84,A82,TARSKI:def 1
.= const(o,FF.iA) by A81,PRALG_3:9
.= Den(o,FF.iA).{} by PRALG_3:def 1
.= Den(o,A).{} by A3
.= Den(o,AG).{} by A83,Th19
.= Den(o,AG).(BA#k) by A81,A85,PRALG_3:11;
end;
suppose
A86: the_arity_of o <> {};
then reconsider domar = dom ar as non empty finite set;
defpred P[set,set] means ex kn being Function st ex A being strict
non-empty finitely-generated MSSubAlgebra of AG st kn = k.$1 & (for B being
strict non-empty finitely-generated MSSubAlgebra of AG st A is MSSubAlgebra of
B holds kn.A = kn.B) & BA.(ar.$1).kn = kn.A & $2 = A;
A87: for n being Element of domar ex b being Element of MSSub AG st P[n ,b]
proof
let n be Element of domar;
consider g being Function of (the Sorts of PS).(ar.n), (the Sorts of
AG).(ar.n) such that
A88: g = BA.(ar.n) and
A89: for x being object st x in (the Sorts of PS).(ar.n) holds Z[g.x
,x,ar. n] by A75,PARTFUN1:4;
k.n in (the Sorts of PS).(ar/.n) by MSUALG_6:2;
then k.n in (the Sorts of PS).(ar.n) by PARTFUN1:def 6;
then consider
s being SortSymbol of SG, f being (Element of (SORTS FF).s),
A being strict non-empty finitely-generated MSSubAlgebra of AG such that
s = ar.n and
A90: f = k.n and
A91: for B being strict non-empty finitely-generated MSSubAlgebra
of AG st A is MSSubAlgebra of B holds f.A = f.B and
A92: g.(k.n) = f.A by A89;
reconsider b = A as Element of MSSub AG by MSUALG_2:def 19;
take b, f;
take A;
thus f = k.n by A90;
thus for B being strict non-empty finitely-generated MSSubAlgebra of
AG st A is MSSubAlgebra of B holds f.A = f.B by A91;
thus BA.(ar.n).f = f.A by A88,A90,A92;
thus thesis;
end;
consider KK being Function of domar, MSSub AG such that
A93: for n being Element of domar holds P[n,KK.n] from FUNCT_2:sch
3 (A87);
reconsider KK as ManySortedSet of domar;
KK is MSAlgebra-Family of domar, SG
proof
let n be object;
assume n in domar;
then ex kn being Function st ex A being strict non-empty
finitely-generated MSSubAlgebra of AG st kn = k.n & (for B being strict
non-empty finitely-generated MSSubAlgebra of AG st A is MSSubAlgebra of B holds
kn.A = kn.B) & BA.(ar.n).kn = kn.A & KK.n = A by A93;
hence thesis;
end;
then reconsider KK as MSAlgebra-Family of domar, SG;
for n being Element of domar ex C being strict non-empty
finitely-generated MSSubAlgebra of AG st C = KK.n
proof
let n be Element of domar;
ex kn being Function st ex A being strict non-empty
finitely-generated MSSubAlgebra of AG st kn = k.n & (for B being strict
non-empty finitely-generated MSSubAlgebra of AG st A is MSSubAlgebra of B holds
kn.A = kn.B) & BA.(ar.n).kn = kn.A & KK.n = A by A93;
hence thesis;
end;
then consider
K being strict non-empty finitely-generated MSSubAlgebra of AG
such that
A94: for n being Element of domar holds KK.n is MSSubAlgebra of K by Th25;
consider MAX being strict non-empty finitely-generated MSSubAlgebra of
AG such that
A95: A is MSSubAlgebra of MAX and
A96: K is MSSubAlgebra of MAX by Th26;
MAX in MSSub AG by MSUALG_2:def 19;
then MAX in CC by A1;
then reconsider iA = MAX as Element of CC;
A97: k in Args(o,pr) by Th18;
then
A98: (commute k).iA is Element of Args(o,FF.iA) by A86,PRALG_3:20;
then (commute k).iA in Args(o,FF.iA);
then
A99: (commute k).iA in Args(o,MAX) by A3;
A100: k in Funcs (dom ar, Funcs (CC, union the set of all
(the Sorts of FF.i).m where
i is (Element of CC), m is (Element of (the carrier of SG))
)) by A97,PRALG_3:14;
then
A101: dom ((commute k).iA) = domar by Th3;
A102: for n being object st n in dom ((commute k).iA) holds (BA#k).n = ((
commute k).iA).n
proof
set fs = ((commute k).iA);
reconsider fs as Element of Args(o,MAX) by A3,A98;
let n be object;
assume
A103: n in dom ((commute k).iA);
then reconsider arn = ar.n as SortSymbol of SG by A101,PARTFUN1:4;
n in Seg len fs by A103,FINSEQ_1:def 3;
then reconsider n9=n as Nat;
consider kn being Function, Ki being strict non-empty
finitely-generated MSSubAlgebra of AG such that
A104: kn = k.n and
A105: for B being strict non-empty finitely-generated MSSubAlgebra
of AG st Ki is MSSubAlgebra of B holds kn.Ki = kn.B and
A106: BA.arn.kn = kn.Ki and
A107: KK.n = Ki by A93,A101,A103;
A108: Ki is MSSubAlgebra of K by A94,A101,A103,A107;
dom k = domar by A100,FUNCT_2:92;
hence (BA#k).n = (BA.(ar/.n9)).(k.n) by A101,A103,MSUALG_3:def 6
.= kn.(KK.n) by A101,A103,A104,A106,A107,PARTFUN1:def 6
.= kn.iA by A96,A105,A107,A108,MSUALG_2:6
.= ((commute k).iA).n by A97,A101,A103,A104,PRALG_3:21;
end;
reconsider ff1 = Den(o,pr).k as Function by Th20;
A109: dom (BA#k) = domar by MSUALG_6:2;
thus (BA.(the_result_sort_of o)).(Den(o,PS).k) = f.MAX by A76,A79,A80,A95
.= ff1.MAX by A78,Th19
.= Den(o,FF.iA).((commute k).iA) by A86,A97,PRALG_3:22
.= Den(o,MAX).((commute k).iA) by A3
.= Den(o,AG).((commute k).iA) by A99,Th19
.= Den(o,AG).(BA#k) by A100,A109,A102,Th3,FUNCT_1:2;
end;
end;
let i be set;
assume i in I;
then reconsider s = i as SortSymbol of SG;
consider ff being object such that
A110: ff in (the Sorts of PS).s by XBOOLE_0:def 1;
rng (BA.s) c= (the Sorts of AG).s;
hence rng (BA.i) c= (the Sorts of AG).i;
let y be object such that
A111: y in (the Sorts of AG).i;
A112: (SORTS FF).s = product Carrier(FF,s) by PRALG_2:def 10;
then ff in product Carrier(FF,s) by A4,A40,A110;
then consider aa being Function such that
ff = aa and
A113: dom aa = dom Carrier(FF,s) and
A114: for x being object st x in dom Carrier(FF,s) holds aa.x in Carrier(FF
,s).x by CARD_3:def 5;
defpred KK[object] means
ex Axx being MSSubAlgebra of AG st Axx = $1 & y in (
the Sorts of Axx).s;
consider WW being set such that
A115: for a being object holds a in WW iff a in CC & KK[a] from XBOOLE_0:
sch 1;
set XX = aa +* (WW --> y);
A116: dom (WW --> y) = WW by FUNCOP_1:13;
A117: for xx being Element of CC st KK[xx] holds XX.xx = y
proof
let xx be Element of CC;
assume KK[xx];
then
A118: xx in WW by A115;
hence XX.xx = (WW --> y).xx by A116,FUNCT_4:13
.= y by A118,FUNCOP_1:7;
end;
A119: dom Carrier(FF,s) = CC by PARTFUN1:def 2;
A120: for x being object st x in dom Carrier(FF,s)
holds XX.x in (Carrier(FF,s)).x
proof
let x be object;
assume
A121: x in dom Carrier(FF,s);
then consider C being Element of MSSub AG such that
A122: x = C and
A123: ex R being strict non-empty finitely-generated MSSubAlgebra of
AG st R = C by A1,A119;
consider R being strict non-empty finitely-generated MSSubAlgebra of AG
such that
A124: R = C by A123;
A125: R in CC by A1,A124;
then
A126: (ex U0 being MSAlgebra over SG st U0 = FF.R & Carrier(FF, s).R = (
the Sorts of U0).s )& FF.R = R by A3,PRALG_2:def 9;
per cases;
suppose
KK[x];
hence thesis by A117,A122,A124,A125,A126;
end;
suppose
not KK[x];
then not x in WW by A115;
then XX.x = aa.x by A116,FUNCT_4:11;
hence thesis by A114,A121;
end;
end;
A127: WW c= CC
by A115;
set L = the non-empty finite-yielding MSSubset of AG;
A128: dom (BA.s) = (the Sorts of PS).s by FUNCT_2:def 1;
set SY = {s} --> {y};
dom (L +* SY) = dom L \/ dom SY by FUNCT_4:def 1
.= I \/ dom SY by PARTFUN1:def 2
.= I \/ {s} by FUNCOP_1:13
.= I by ZFMISC_1:40;
then reconsider Y = L +* SY as ManySortedSet of I by PARTFUN1:def 2
,RELAT_1:def 18;
A129: s in {s} by TARSKI:def 1;
dom SY = {s} by FUNCOP_1:13;
then
A130: Y.s = SY.s by A129,FUNCT_4:13
.= {y} by A129,FUNCOP_1:7;
A131: now
let k be set;
assume that
k in I and
A132: k <> s;
not k in dom SY by A132,TARSKI:def 1;
hence Y.k = L.k by FUNCT_4:11;
end;
Y is ManySortedSubset of the Sorts of AG
proof
let j be object such that
A133: j in I;
let x be object such that
A134: x in Y.j;
per cases;
suppose
A135: j <> s;
L c= the Sorts of AG by PBOOLE:def 18;
then
A136: L.j c= (the Sorts of AG).j by A133;
x in L.j by A131,A133,A134,A135;
hence thesis by A136;
end;
suppose
j = s;
hence thesis by A111,A130,A134,TARSKI:def 1;
end;
end;
then reconsider Y as MSSubset of AG;
Y is non-empty
proof
let j be object such that
A137: j in I;
per cases;
suppose
j <> s;
then Y.j = L.j by A131,A137;
hence thesis by A137;
end;
suppose
j = s;
hence thesis by A130;
end;
end;
then reconsider Y as non-empty MSSubset of AG;
Y is finite-yielding
proof
let j be object such that
A138: j in I;
per cases;
suppose
j <> s;
then Y.j = L.j by A131,A138;
hence thesis;
end;
suppose
j = s;
hence thesis by A130;
end;
end;
then reconsider Y as non-empty finite-yielding MSSubset of AG;
Y is MSSubset of GenMSAlg Y by MSUALG_2:def 17;
then Y c= the Sorts of GenMSAlg Y by PBOOLE:def 18;
then
A139: Y.s c= (the Sorts of GenMSAlg Y).s;
dom XX = dom aa \/ dom (WW --> y) by FUNCT_4:def 1
.= CC \/ dom (WW --> y) by A113,PARTFUN1:def 2
.= CC \/ WW by FUNCOP_1:13
.= CC by A127,XBOOLE_1:12;
then reconsider XX as Element of (SORTS FF).s by A112,A119,A120,CARD_3:9;
consider h being Function of (the Sorts of PS).s, (the Sorts of AG).s such
that
A140: h = BA.s and
A141: for x being object st x in (the Sorts of PS).s holds Z[h.x,x,s] by A75;
A142: y in Y.s by A130,TARSKI:def 1;
then
A143: y in (the Sorts of GenMSAlg Y).s by A139;
T[s,XX]
proof
take s;
take XX;
take TT = GenMSAlg Y;
thus s = s;
thus XX = XX;
let B be strict non-empty finitely-generated MSSubAlgebra of AG;
assume TT is MSSubAlgebra of B;
then the Sorts of TT is ManySortedSubset of the Sorts of B by
MSUALG_2:def 9;
then the Sorts of TT c= the Sorts of B by PBOOLE:def 18;
then
A144: (the Sorts of TT).s c= (the Sorts of B).s;
A145: B in CC by A2;
TT in CC by A2;
hence XX.TT = y by A139,A142,A117
.= XX.B by A143,A117,A145,A144;
end;
then
A146: XX in SOR.s by A4;
then Z[h.XX,XX,s] by A40,A141;
then consider t being SortSymbol of SG, f being (Element of (SORTS FF).s), A
being strict non-empty finitely-generated MSSubAlgebra of AG such that
s = t and
A147: ( f = XX & for B being strict non-empty finitely-generated
MSSubAlgebra of AG st A is MSSubAlgebra of B holds f.A = f.B )& h.XX = f.A;
consider MAX being strict non-empty finitely-generated MSSubAlgebra of AG
such that
A148: A is MSSubAlgebra of MAX and
A149: GenMSAlg Y is MSSubAlgebra of MAX by Th26;
A150: MAX in CC by A2;
the Sorts of GenMSAlg Y is ManySortedSubset of the Sorts of MAX by A149,
MSUALG_2:def 9;
then the Sorts of GenMSAlg Y c= the Sorts of MAX by PBOOLE:def 18;
then
A151: (the Sorts of GenMSAlg Y).s c= (the Sorts of MAX).s;
h.XX = XX.MAX by A147,A148
.= y by A143,A117,A151,A150;
hence thesis by A40,A128,A140,A146,FUNCT_1:def 3;
end;
theorem
for U0 being feasible free MSAlgebra over S, A being free GeneratorSet
of U0 for Z being MSSubset of U0 st Z c= A & GenMSAlg Z is feasible holds
GenMSAlg Z is free
proof
let U0 be feasible free MSAlgebra over S, A be free GeneratorSet of U0, Z be
MSSubset of U0 such that
A1: Z c= A and
A2: GenMSAlg Z is feasible;
reconsider Z1 = Z as MSSubset of GenMSAlg Z by MSUALG_2:def 17;
the Sorts of GenMSAlg Z1 = the Sorts of GenMSAlg Z by EXTENS_1:18;
then reconsider z = Z as GeneratorSet of GenMSAlg Z by MSAFREE:def 4;
reconsider z1 = z as ManySortedSubset of A by A1,PBOOLE:def 18;
take z;
z is free
proof
reconsider D = the Sorts of GenMSAlg Z as MSSubset of U0 by MSUALG_2:def 9;
let U1 be non-empty MSAlgebra over S, g be ManySortedFunction of z, the
Sorts of U1;
consider G being ManySortedFunction of A, the Sorts of U1 such that
A3: G||z1 = g by Th6;
consider h being ManySortedFunction of U0, U1 such that
A4: h is_homomorphism U0, U1 and
A5: h || A = G by MSAFREE:def 5;
reconsider H = h || D as ManySortedFunction of GenMSAlg Z, U1;
take H;
thus H is_homomorphism GenMSAlg Z, U1 by A2,A4,Th22;
thus g = h || Z by A3,A5,Th5
.= H || z by Th5;
end;
hence thesis;
end;
begin :: Equations in Many Sorted Algebras
definition
let S be non empty non void ManySortedSign;
func TermAlg S -> MSAlgebra over S equals
FreeMSA ((the carrier of S) -->
NAT);
correctness;
end;
registration
let S be non empty non void ManySortedSign;
cluster TermAlg S -> strict non-empty free;
coherence;
end;
definition
let S be non empty non void ManySortedSign;
func Equations S -> ManySortedSet of the carrier of S equals
[|the Sorts of
TermAlg S, the Sorts of TermAlg S|];
correctness;
end;
registration
let S be non empty non void ManySortedSign;
cluster Equations S -> non-empty;
coherence;
end;
definition
let S be non empty non void ManySortedSign;
mode EqualSet of S is ManySortedSubset of Equations S;
end;
reserve s for SortSymbol of S;
reserve e for Element of (Equations S).s;
reserve E for EqualSet of S;
notation
let S be non empty non void ManySortedSign, s be SortSymbol of S, x, y be
Element of (the Sorts of TermAlg S).s;
synonym x '=' y for [x,y];
end;
definition
let S be non empty non void ManySortedSign, s be SortSymbol of S, x, y be
Element of (the Sorts of TermAlg S).s;
redefine func x '=' y -> Element of (Equations S).s;
coherence
proof
[x,y] in [:(the Sorts of TermAlg S).s, (the Sorts of TermAlg S).s:] by
ZFMISC_1:87;
hence thesis by PBOOLE:def 16;
end;
end;
theorem Th29:
e`1 in (the Sorts of TermAlg S).s
proof
set T = the Sorts of TermAlg S;
e is Element of [:T.s, T.s:] by PBOOLE:def 16;
hence thesis by MCART_1:10;
end;
theorem Th30:
e`2 in (the Sorts of TermAlg S).s
proof
set T = the Sorts of TermAlg S;
e is Element of [:T.s, T.s:] by PBOOLE:def 16;
hence thesis by MCART_1:10;
end;
definition
let S be non empty non void ManySortedSign, A be MSAlgebra over S, s be
SortSymbol of S, e be Element of (Equations S).s;
pred A |= e means
for h being ManySortedFunction of TermAlg S, A st
h is_homomorphism TermAlg S, A holds h.s.(e`1) = h.s.(e`2);
end;
definition
let S be non empty non void ManySortedSign, A be MSAlgebra over S, E be
EqualSet of S;
pred A |= E means
for s being SortSymbol of S for e being Element of
(Equations S).s st e in E.s holds A |= e;
end;
theorem Th31:
for U2 being strict non-empty MSSubAlgebra of U0 st U0 |= e holds U2 |= e
proof
let U2 be strict non-empty MSSubAlgebra of U0 such that
A1: U0 |= e;
let h be ManySortedFunction of TermAlg S, U2 such that
A2: h is_homomorphism TermAlg S, U2;
A3: the Sorts of TermAlg S is_transformable_to the Sorts of U2;
the Sorts of U2 is MSSubset of U0 by MSUALG_2:def 9;
then reconsider f1 = h as ManySortedFunction of TermAlg S, U0 by A3,
EXTENS_1:5;
f1 is_homomorphism TermAlg S, U0 by A2,MSUALG_9:15;
hence thesis by A1;
end;
theorem
for U2 being strict non-empty MSSubAlgebra of U0 st U0 |= E holds U2 |= E
by Th31;
theorem Th33:
U0, U1 are_isomorphic & U0 |= e implies U1 |= e
proof
assume that
A1: U0, U1 are_isomorphic and
A2: U0 |= e;
consider F be ManySortedFunction of U0, U1 such that
A3: F is_isomorphism U0, U1 by A1;
consider G be ManySortedFunction of U1, U0 such that
A4: G = F"";
F is "1-1" & F is "onto" by A3,MSUALG_3:13;
then
A5: (G.s) = (F.s)" by A4,MSUALG_3:def 4;
F is "onto" by A3,MSUALG_3:13;
then
A6: rng (F.s) = (the Sorts of U1).s;
let h1 be ManySortedFunction of TermAlg S, U1 such that
A7: h1 is_homomorphism TermAlg S, U1;
set F1 = G ** h1;
G is_isomorphism U1, U0 by A3,A4,MSUALG_3:14;
then G is_homomorphism U1, U0 by MSUALG_3:13;
then
A8: F1 is_homomorphism TermAlg S, U0 by A7,MSUALG_3:10;
F is "1-1" by A3,MSUALG_3:13;
then
A9: (F.s) is one-to-one by MSUALG_3:1;
(F1.s) = (G.s) * (h1.s) by MSUALG_3:2;
then
A10: (F.s) * (F1.s) = (F.s) * (G.s) * (h1.s) by RELAT_1:36
.= id ((the Sorts of U1).s) * (h1.s) by A5,A6,A9,FUNCT_2:29
.= h1.s by FUNCT_2:17;
A11: dom (F1.s) = (the Sorts of TermAlg S).s by FUNCT_2:def 1;
hence h1.s.(e`1) = (F.s).((F1.s).(e`1)) by A10,Th29,FUNCT_1:13
.= (F.s).((F1.s).(e`2)) by A2,A8
.= h1.s.(e`2) by A10,A11,Th30,FUNCT_1:13;
end;
theorem
U0, U1 are_isomorphic & U0 |= E implies U1 |= E
by Th33;
theorem Th35:
for R being MSCongruence of U0 st U0 |= e holds QuotMSAlg (U0,R) |= e
proof
let R be MSCongruence of U0 such that
A1: U0 |= e;
set n = (MSNat_Hom(U0,R)).s;
let h be ManySortedFunction of TermAlg S, QuotMSAlg (U0,R);
assume h is_homomorphism TermAlg S, QuotMSAlg (U0,R);
then consider h0 be ManySortedFunction of TermAlg S, U0 such that
A2: h0 is_homomorphism TermAlg S, U0 and
A3: h = (MSNat_Hom(U0,R)) ** h0 by Th24,MSUALG_4:3;
A4: dom (h0.s) = (the Sorts of TermAlg S).s by FUNCT_2:def 1;
thus h.s.(e`1) = (n*(h0.s)).(e`1) by A3,MSUALG_3:2
.= n.((h0.s).(e`1)) by A4,Th29,FUNCT_1:13
.= n.((h0.s).(e`2)) by A1,A2
.= (n*(h0.s)).(e`2) by A4,Th30,FUNCT_1:13
.= h.s.(e`2) by A3,MSUALG_3:2;
end;
theorem
for R being MSCongruence of U0 st U0 |= E holds QuotMSAlg (U0,R) |= E
by Th35;
Lm1: for I being non empty set, A being MSAlgebra-Family of I,S st for i being
Element of I holds A.i |= e holds product A |= e
proof
reconsider e2 = e`2 as Element of (the Sorts of TermAlg S).s by Th30;
reconsider e1 = e`1 as Element of (the Sorts of TermAlg S).s by Th29;
let I be non empty set, A be MSAlgebra-Family of I,S such that
A1: for i be Element of I holds A.i |= e;
let h be ManySortedFunction of TermAlg S, product A such that
A2: h is_homomorphism TermAlg S, product A;
A3: dom (h.s) = (the Sorts of TermAlg S).s by FUNCT_2:def 1;
A4: now
let i be Element of I;
set Z = proj(A,i) ** h;
A5: A.i |= e by A1;
proj(A,i) is_homomorphism product A, A.i by PRALG_3:24;
then
A6: proj(A,i) ** h is_homomorphism TermAlg S, A.i by A2,MSUALG_3:10;
thus (proj (Carrier(A,s), i)).((h.s).(e1)) = (proj(A,i).s).((h.s).(e1)) by
PRALG_3:def 2
.= ((proj(A,i).s)*(h.s)).(e1) by A3,FUNCT_1:13
.= Z.s.e1 by MSUALG_3:2
.= Z.s.e2 by A5,A6
.= ((proj(A,i).s)*(h.s)).(e2) by MSUALG_3:2
.= (proj(A,i).s).((h.s).(e2)) by A3,FUNCT_1:13
.= (proj (Carrier(A,s), i)).((h.s).(e2)) by PRALG_3:def 2;
end;
(the Sorts of product A).s = product Carrier(A,s) by PRALG_2:def 10;
hence thesis by A4,MSUALG_9:14;
end;
theorem Th37:
for F being MSAlgebra-Family of I, S st (for i being set st i in
I ex A being MSAlgebra over S st A = F.i & A |= e) holds product F |= e
proof
let F be MSAlgebra-Family of I, S such that
A1: for i being set st i in I ex A being MSAlgebra over S st A = F.i & A |= e;
per cases;
suppose
I = {};
then reconsider J = I as empty set;
reconsider FJ = F as MSAlgebra-Family of J, S;
thus product F |= e
proof
let h be ManySortedFunction of TermAlg S, product F such that
h is_homomorphism TermAlg S, product F;
A2: (the Sorts of product FJ).s = product Carrier(FJ,s) by PRALG_2:def 10
.= {{}} by CARD_3:10;
A3: h.s.(e`2) in (the Sorts of product FJ).s by Th30,FUNCT_2:5;
h.s.(e`1) in (the Sorts of product FJ).s by Th29,FUNCT_2:5;
hence h.s.(e`1) = {} by A2,TARSKI:def 1
.= h.s.(e`2) by A2,A3,TARSKI:def 1;
end;
end;
suppose
I <> {};
then reconsider J = I as non empty set;
reconsider F1 = F as MSAlgebra-Family of J, S;
now
let i be Element of J;
ex A being MSAlgebra over S st A = F1.i & A |= e by A1;
hence F1.i |= e;
end;
hence thesis by Lm1;
end;
end;
theorem
for F being MSAlgebra-Family of I, S st (for i being set st i in I ex
A being MSAlgebra over S st A = F.i & A |= E) holds product F |= E
proof
let F be MSAlgebra-Family of I, S such that
A1: for i being set st i in I ex A being MSAlgebra over S st A = F.i & A |= E;
let s be SortSymbol of S, e be Element of (Equations S).s such that
A2: e in E.s;
now
let i be set;
assume i in I;
then consider A being MSAlgebra over S such that
A3: A = F.i & A |= E by A1;
take A;
thus A = F.i & A |= e by A2,A3;
end;
hence thesis by Th37;
end;