:: MSALIMIT semantic presentation
definition
let c
1 be non
empty Poset;
let c
2 be non
empty non
void ManySortedSign ;
mode OrderedAlgFam of c
1,c
2 -> MSAlgebra-Family of the
carrier of a
1,a
2 means :
Def1:
:: MSALIMIT:def 1
ex b
1 being
ManySortedFunction of the
InternalRel of a
1 st
for b
2, b
3, b
4 being
Element of a
1 holds
not ( b
2 >= b
3 & b
3 >= b
4 & ( for b
5 being
ManySortedFunction of
(a3 . b2),
(a3 . b3)for b
6 being
ManySortedFunction of
(a3 . b3),
(a3 . b4) holds
not ( b
5 = b
1 . b
3,b
2 & b
6 = b
1 . b
4,b
3 & b
1 . b
4,b
2 = b
6 ** b
5 & b
5 is_homomorphism a
3 . b
2,a
3 . b
3 ) ) );
existence
ex b1 being MSAlgebra-Family of the carrier of c1,c2ex b2 being ManySortedFunction of the InternalRel of c1 st
for b3, b4, b5 being Element of c1 holds
not ( b3 >= b4 & b4 >= b5 & ( for b6 being ManySortedFunction of (b1 . b3),(b1 . b4)
for b7 being ManySortedFunction of (b1 . b4),(b1 . b5) holds
not ( b6 = b2 . b4,b3 & b7 = b2 . b5,b4 & b2 . b5,b3 = b7 ** b6 & b6 is_homomorphism b1 . b3,b1 . b4 ) ) )
end;
:: deftheorem Def1 defines OrderedAlgFam MSALIMIT:def 1 :
for b
1 being non
empty Posetfor b
2 being non
empty non
void ManySortedSign for b
3 being
MSAlgebra-Family of the
carrier of b
1,b
2 holds
( b
3 is
OrderedAlgFam of b
1,b
2 iff ex b
4 being
ManySortedFunction of the
InternalRel of b
1 st
for b
5, b
6, b
7 being
Element of b
1 holds
not ( b
5 >= b
6 & b
6 >= b
7 & ( for b
8 being
ManySortedFunction of
(b3 . b5),
(b3 . b6)for b
9 being
ManySortedFunction of
(b3 . b6),
(b3 . b7) holds
not ( b
8 = b
4 . b
6,b
5 & b
9 = b
4 . b
7,b
6 & b
4 . b
7,b
5 = b
9 ** b
8 & b
8 is_homomorphism b
3 . b
5,b
3 . b
6 ) ) ) );
definition
let c
1 be non
empty Poset;
let c
2 be non
empty non
void ManySortedSign ;
let c
3 be
OrderedAlgFam of c
1,c
2;
mode Binding of c
3 -> ManySortedFunction of the
InternalRel of a
1 means :
Def2:
:: MSALIMIT:def 2
for b
1, b
2, b
3 being
Element of a
1 holds
not ( b
1 >= b
2 & b
2 >= b
3 & ( for b
4 being
ManySortedFunction of
(a3 . b1),
(a3 . b2)for b
5 being
ManySortedFunction of
(a3 . b2),
(a3 . b3) holds
not ( b
4 = a
4 . b
2,b
1 & b
5 = a
4 . b
3,b
2 & a
4 . b
3,b
1 = b
5 ** b
4 & b
4 is_homomorphism a
3 . b
1,a
3 . b
2 ) ) );
existence
ex b1 being ManySortedFunction of the InternalRel of c1 st
for b2, b3, b4 being Element of c1 holds
not ( b2 >= b3 & b3 >= b4 & ( for b5 being ManySortedFunction of (c3 . b2),(c3 . b3)
for b6 being ManySortedFunction of (c3 . b3),(c3 . b4) holds
not ( b5 = b1 . b3,b2 & b6 = b1 . b4,b3 & b1 . b4,b2 = b6 ** b5 & b5 is_homomorphism c3 . b2,c3 . b3 ) ) )
by Def1;
end;
:: deftheorem Def2 defines Binding MSALIMIT:def 2 :
:: deftheorem Def3 defines bind MSALIMIT:def 3 :
theorem Th1: :: MSALIMIT:1
:: deftheorem Def4 defines normalized MSALIMIT:def 4 :
theorem Th2: :: MSALIMIT:2
definition
let c
1 be non
empty Poset;
let c
2 be non
empty non
void ManySortedSign ;
let c
3 be
OrderedAlgFam of c
1,c
2;
let c
4 be
Binding of c
3;
func Normalized c
4 -> Binding of a
3 means :
Def5:
:: MSALIMIT:def 5
for b
1, b
2 being
Element of a
1 holds
( b
1 >= b
2 implies a
5 . b
2,b
1 = IFEQ b
2,b
1,
(id the Sorts of (a3 . b1)),
((bind a4,b1,b2) ** (id the Sorts of (a3 . b1))) );
existence
ex b1 being Binding of c3 st
for b2, b3 being Element of c1 holds
( b2 >= b3 implies b1 . b3,b2 = IFEQ b3,b2,(id the Sorts of (c3 . b2)),((bind c4,b2,b3) ** (id the Sorts of (c3 . b2))) )
uniqueness
for b1, b2 being Binding of c3 holds
( ( for b3, b4 being Element of c1 holds
( b3 >= b4 implies b1 . b4,b3 = IFEQ b4,b3,(id the Sorts of (c3 . b3)),((bind c4,b3,b4) ** (id the Sorts of (c3 . b3))) ) ) & ( for b3, b4 being Element of c1 holds
( b3 >= b4 implies b2 . b4,b3 = IFEQ b4,b3,(id the Sorts of (c3 . b3)),((bind c4,b3,b4) ** (id the Sorts of (c3 . b3))) ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines Normalized MSALIMIT:def 5 :
theorem Th3: :: MSALIMIT:3
theorem Th4: :: MSALIMIT:4
definition
let c
1 be non
empty Poset;
let c
2 be non
empty non
void ManySortedSign ;
let c
3 be
OrderedAlgFam of c
1,c
2;
let c
4 be
Binding of c
3;
func InvLim c
4 -> strict MSSubAlgebra of
product a
3 means :
Def6:
:: MSALIMIT:def 6
for b
1 being
SortSymbol of a
2for b
2 being
Element of
(SORTS a3) . b
1 holds
( b
2 in the
Sorts of a
5 . b
1 iff for b
3, b
4 being
Element of a
1 holds
( b
3 >= b
4 implies
((bind a4,b3,b4) . b1) . (b2 . b3) = b
2 . b
4 ) );
existence
ex b1 being strict MSSubAlgebra of product c3 st
for b2 being SortSymbol of c2
for b3 being Element of (SORTS c3) . b2 holds
( b3 in the Sorts of b1 . b2 iff for b4, b5 being Element of c1 holds
( b4 >= b5 implies ((bind c4,b4,b5) . b2) . (b3 . b4) = b3 . b5 ) )
uniqueness
for b1, b2 being strict MSSubAlgebra of product c3 holds
( ( for b3 being SortSymbol of c2
for b4 being Element of (SORTS c3) . b3 holds
( b4 in the Sorts of b1 . b3 iff for b5, b6 being Element of c1 holds
( b5 >= b6 implies ((bind c4,b5,b6) . b3) . (b4 . b5) = b4 . b6 ) ) ) & ( for b3 being SortSymbol of c2
for b4 being Element of (SORTS c3) . b3 holds
( b4 in the Sorts of b2 . b3 iff for b5, b6 being Element of c1 holds
( b5 >= b6 implies ((bind c4,b5,b6) . b3) . (b4 . b5) = b4 . b6 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines InvLim MSALIMIT:def 6 :
theorem Th5: :: MSALIMIT:5
:: deftheorem Def7 defines MSS-membered MSALIMIT:def 7 :
:: deftheorem Def8 defines TrivialMSSign MSALIMIT:def 8 :
Lemma12:
for b1 being empty void ManySortedSign holds id the carrier of b1, id the OperSymbols of b1 form_morphism_between b1,b1
Lemma13:
for b1 being non empty void ManySortedSign holds id the carrier of b1, id the OperSymbols of b1 form_morphism_between b1,b1
theorem Th6: :: MSALIMIT:6
:: deftheorem Def9 defines MSS_set MSALIMIT:def 9 :
definition
let c
1, c
2 be
ManySortedSign ;
func MSS_morph c
1,c
2 -> set means :: MSALIMIT:def 10
for b
1 being
set holds
( b
1 in a
3 iff ex b
2, b
3 being
Function st
( b
1 = [b2,b3] & b
2,b
3 form_morphism_between a
1,a
2 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ex b3, b4 being Function st
( b2 = [b3,b4] & b3,b4 form_morphism_between c1,c2 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5 being Function st
( b3 = [b4,b5] & b4,b5 form_morphism_between c1,c2 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5 being Function st
( b3 = [b4,b5] & b4,b5 form_morphism_between c1,c2 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def10 defines MSS_morph MSALIMIT:def 10 :