:: MSUALG_4 semantic presentation
:: deftheorem Def1 defines Relation-yielding MSUALG_4:def 1 :
:: deftheorem Def2 defines ManySortedRelation MSUALG_4:def 2 :
:: deftheorem Def3 defines MSEquivalence_Relation-like MSUALG_4:def 3 :
:: deftheorem Def4 MSUALG_4:def 4 :
canceled;
:: deftheorem Def5 defines MSEquivalence-like MSUALG_4:def 5 :
theorem Th1: :: MSUALG_4:1
:: deftheorem Def6 defines MSCongruence-like MSUALG_4:def 6 :
:: deftheorem Def7 defines Class MSUALG_4:def 7 :
:: deftheorem Def8 defines Class MSUALG_4:def 8 :
:: deftheorem Def9 defines # MSUALG_4:def 9 :
definition
let S be non
empty non
void ManySortedSign ;
let o be
OperSymbol of
S;
let A be
non-empty MSAlgebra of
S;
let R be
MSCongruence of
A;
func QuotRes c4,
c2 -> Function of
(the Sorts of a3 * the ResultSort of a1) . a2,
((Class a4) * the ResultSort of a1) . a2 means :
Def10:
:: MSUALG_4:def 10
for
x being
Element of the
Sorts of
A . (the_result_sort_of o) holds
it . x = Class R,
x;
existence
ex b1 being Function of (the Sorts of A * the ResultSort of S) . o,((Class R) * the ResultSort of S) . o st
for x being Element of the Sorts of A . (the_result_sort_of o) holds b1 . x = Class R,x
uniqueness
for b1, b2 being Function of (the Sorts of A * the ResultSort of S) . o,((Class R) * the ResultSort of S) . o st ( for x being Element of the Sorts of A . (the_result_sort_of o) holds b1 . x = Class R,x ) & ( for x being Element of the Sorts of A . (the_result_sort_of o) holds b2 . x = Class R,x ) holds
b1 = b2
func QuotArgs c4,
c2 -> Function of
((the Sorts of a3 # ) * the Arity of a1) . a2,
(((Class a4) # ) * the Arity of a1) . a2 means :: MSUALG_4:def 11
for
x being
Element of
Args o,
A holds
it . x = R # x;
existence
ex b1 being Function of ((the Sorts of A # ) * the Arity of S) . o,(((Class R) # ) * the Arity of S) . o st
for x being Element of Args o,A holds b1 . x = R # x
uniqueness
for b1, b2 being Function of ((the Sorts of A # ) * the Arity of S) . o,(((Class R) # ) * the Arity of S) . o st ( for x being Element of Args o,A holds b1 . x = R # x ) & ( for x being Element of Args o,A holds b2 . x = R # x ) holds
b1 = b2
end;
:: deftheorem Def10 defines QuotRes MSUALG_4:def 10 :
:: deftheorem Def11 defines QuotArgs MSUALG_4:def 11 :
definition
let S be non
empty non
void ManySortedSign ;
let A be
non-empty MSAlgebra of
S;
let R be
MSCongruence of
A;
func QuotRes c3 -> ManySortedFunction of the
Sorts of
a2 * the
ResultSort of
a1,
(Class a3) * the
ResultSort of
a1 means :: MSUALG_4:def 12
for
o being
OperSymbol of
S holds
it . o = QuotRes R,
o;
existence
ex b1 being ManySortedFunction of the Sorts of A * the ResultSort of S,(Class R) * the ResultSort of S st
for o being OperSymbol of S holds b1 . o = QuotRes R,o
uniqueness
for b1, b2 being ManySortedFunction of the Sorts of A * the ResultSort of S,(Class R) * the ResultSort of S st ( for o being OperSymbol of S holds b1 . o = QuotRes R,o ) & ( for o being OperSymbol of S holds b2 . o = QuotRes R,o ) holds
b1 = b2
func QuotArgs c3 -> ManySortedFunction of
(the Sorts of a2 # ) * the
Arity of
a1,
((Class a3) # ) * the
Arity of
a1 means :: MSUALG_4:def 13
for
o being
OperSymbol of
S holds
it . o = QuotArgs R,
o;
existence
ex b1 being ManySortedFunction of (the Sorts of A # ) * the Arity of S,((Class R) # ) * the Arity of S st
for o being OperSymbol of S holds b1 . o = QuotArgs R,o
uniqueness
for b1, b2 being ManySortedFunction of (the Sorts of A # ) * the Arity of S,((Class R) # ) * the Arity of S st ( for o being OperSymbol of S holds b1 . o = QuotArgs R,o ) & ( for o being OperSymbol of S holds b2 . o = QuotArgs R,o ) holds
b1 = b2
end;
:: deftheorem Def12 defines QuotRes MSUALG_4:def 12 :
:: deftheorem Def13 defines QuotArgs MSUALG_4:def 13 :
theorem Th2: :: MSUALG_4:2
definition
let S be non
empty non
void ManySortedSign ;
let o be
OperSymbol of
S;
let A be
non-empty MSAlgebra of
S;
let R be
MSCongruence of
A;
func QuotCharact c4,
c2 -> Function of
(((Class a4) # ) * the Arity of a1) . a2,
((Class a4) * the ResultSort of a1) . a2 means :
Def14:
:: MSUALG_4:def 14
for
a being
Element of
Args o,
A st
R # a in (((Class R) # ) * the Arity of S) . o holds
it . (R # a) = ((QuotRes R,o) * (Den o,A)) . a;
existence
ex b1 being Function of (((Class R) # ) * the Arity of S) . o,((Class R) * the ResultSort of S) . o st
for a being Element of Args o,A st R # a in (((Class R) # ) * the Arity of S) . o holds
b1 . (R # a) = ((QuotRes R,o) * (Den o,A)) . a
uniqueness
for b1, b2 being Function of (((Class R) # ) * the Arity of S) . o,((Class R) * the ResultSort of S) . o st ( for a being Element of Args o,A st R # a in (((Class R) # ) * the Arity of S) . o holds
b1 . (R # a) = ((QuotRes R,o) * (Den o,A)) . a ) & ( for a being Element of Args o,A st R # a in (((Class R) # ) * the Arity of S) . o holds
b2 . (R # a) = ((QuotRes R,o) * (Den o,A)) . a ) holds
b1 = b2
end;
:: deftheorem Def14 defines QuotCharact MSUALG_4:def 14 :
:: deftheorem Def15 defines QuotCharact MSUALG_4:def 15 :
:: deftheorem Def16 defines QuotMSAlg MSUALG_4:def 16 :
definition
let S be non
empty non
void ManySortedSign ;
let U1 be
non-empty MSAlgebra of
S;
let R be
MSCongruence of
U1;
let s be
SortSymbol of
S;
func MSNat_Hom c2,
c3,
c4 -> Function of the
Sorts of
a2 . a4,
(Class a3) . a4 means :
Def17:
:: MSUALG_4:def 17
for
x being
set st
x in the
Sorts of
U1 . s holds
it . x = Class (R . s),
x;
existence
ex b1 being Function of the Sorts of U1 . s,(Class R) . s st
for x being set st x in the Sorts of U1 . s holds
b1 . x = Class (R . s),x
uniqueness
for b1, b2 being Function of the Sorts of U1 . s,(Class R) . s st ( for x being set st x in the Sorts of U1 . s holds
b1 . x = Class (R . s),x ) & ( for x being set st x in the Sorts of U1 . s holds
b2 . x = Class (R . s),x ) holds
b1 = b2
end;
:: deftheorem Def17 defines MSNat_Hom MSUALG_4:def 17 :
definition
let S be non
empty non
void ManySortedSign ;
let U1 be
non-empty MSAlgebra of
S;
let R be
MSCongruence of
U1;
func MSNat_Hom c2,
c3 -> ManySortedFunction of
a2,
(QuotMSAlg a2,a3) means :
Def18:
:: MSUALG_4:def 18
for
s being
SortSymbol of
S holds
it . s = MSNat_Hom U1,
R,
s;
existence
ex b1 being ManySortedFunction of U1,(QuotMSAlg U1,R) st
for s being SortSymbol of S holds b1 . s = MSNat_Hom U1,R,s
uniqueness
for b1, b2 being ManySortedFunction of U1,(QuotMSAlg U1,R) st ( for s being SortSymbol of S holds b1 . s = MSNat_Hom U1,R,s ) & ( for s being SortSymbol of S holds b2 . s = MSNat_Hom U1,R,s ) holds
b1 = b2
end;
:: deftheorem Def18 defines MSNat_Hom MSUALG_4:def 18 :
theorem Th3: :: MSUALG_4:3
definition
let S be non
empty non
void ManySortedSign ;
let U1 be
non-empty MSAlgebra of
S,
U2 be
non-empty MSAlgebra of
S;
let F be
ManySortedFunction of
U1,
U2;
let s be
SortSymbol of
S;
func MSCng c4,
c5 -> Equivalence_Relation of the
Sorts of
a2 . a5 means :
Def19:
:: MSUALG_4:def 19
for
x,
y being
Element of the
Sorts of
U1 . s holds
(
[x,y] in it iff
(F . s) . x = (F . s) . y );
existence
ex b1 being Equivalence_Relation of the Sorts of U1 . s st
for x, y being Element of the Sorts of U1 . s holds
( [x,y] in b1 iff (F . s) . x = (F . s) . y )
uniqueness
for b1, b2 being Equivalence_Relation of the Sorts of U1 . s st ( for x, y being Element of the Sorts of U1 . s holds
( [x,y] in b1 iff (F . s) . x = (F . s) . y ) ) & ( for x, y being Element of the Sorts of U1 . s holds
( [x,y] in b2 iff (F . s) . x = (F . s) . y ) ) holds
b1 = b2
end;
:: deftheorem Def19 defines MSCng MSUALG_4:def 19 :
:: deftheorem Def20 defines MSCng MSUALG_4:def 20 :
definition
let S be non
empty non
void ManySortedSign ;
let U1 be
non-empty MSAlgebra of
S,
U2 be
non-empty MSAlgebra of
S;
let F be
ManySortedFunction of
U1,
U2;
let s be
SortSymbol of
S;
assume E28:
F is_homomorphism U1,
U2
;
func MSHomQuot c4,
c5 -> Function of the
Sorts of
(QuotMSAlg a2,(MSCng a4)) . a5,the
Sorts of
a3 . a5 means :
Def21:
:: MSUALG_4:def 21
for
x being
Element of the
Sorts of
U1 . s holds
it . (Class (MSCng F,s),x) = (F . s) . x;
existence
ex b1 being Function of the Sorts of (QuotMSAlg U1,(MSCng F)) . s,the Sorts of U2 . s st
for x being Element of the Sorts of U1 . s holds b1 . (Class (MSCng F,s),x) = (F . s) . x
uniqueness
for b1, b2 being Function of the Sorts of (QuotMSAlg U1,(MSCng F)) . s,the Sorts of U2 . s st ( for x being Element of the Sorts of U1 . s holds b1 . (Class (MSCng F,s),x) = (F . s) . x ) & ( for x being Element of the Sorts of U1 . s holds b2 . (Class (MSCng F,s),x) = (F . s) . x ) holds
b1 = b2
end;
:: deftheorem Def21 defines MSHomQuot MSUALG_4:def 21 :
definition
let S be non
empty non
void ManySortedSign ;
let U1 be
non-empty MSAlgebra of
S,
U2 be
non-empty MSAlgebra of
S;
let F be
ManySortedFunction of
U1,
U2;
func MSHomQuot c4 -> ManySortedFunction of
(QuotMSAlg a2,(MSCng a4)),
a3 means :
Def22:
:: MSUALG_4:def 22
for
s being
SortSymbol of
S holds
it . s = MSHomQuot F,
s;
existence
ex b1 being ManySortedFunction of (QuotMSAlg U1,(MSCng F)),U2 st
for s being SortSymbol of S holds b1 . s = MSHomQuot F,s
uniqueness
for b1, b2 being ManySortedFunction of (QuotMSAlg U1,(MSCng F)),U2 st ( for s being SortSymbol of S holds b1 . s = MSHomQuot F,s ) & ( for s being SortSymbol of S holds b2 . s = MSHomQuot F,s ) holds
b1 = b2
end;
:: deftheorem Def22 defines MSHomQuot MSUALG_4:def 22 :
theorem Th4: :: MSUALG_4:4
theorem Th5: :: MSUALG_4:5
theorem Th6: :: MSUALG_4:6