Article MSALIMIT, MML version 4.99.1005
:: MSALIMIT:funcreg 1
registration
let a1 be non empty set;
let a2 be non empty non void ManySortedSign;
let a3 be MSAlgebra-Family of a1,a2;
let a4 be Element of a1;
let a5 be Element of the OperSymbols of a2;
cluster ((OPER a3) . a4) . a5 -> Relation-like Function-like;
end;
:: MSALIMIT:funcreg 2
registration
let a1 be non empty set;
let a2 be non empty non void ManySortedSign;
let a3 be MSAlgebra-Family of a1,a2;
let a4 be Element of the carrier of a2;
cluster (SORTS a3) . a4 -> functional;
end;
:: MSALIMIT:modenot 1 => MSALIMIT:mode 1
definition
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
mode OrderedAlgFam of A1,A2 -> MSAlgebra-Family of the carrier of a1,a2 means
ex b1 being Function-yielding ManySortedSet of the InternalRel of a1 st
for b2, b3, b4 being Element of the carrier of a1
st b3 <= b2 & b4 <= b3
holds ex b5 being ManySortedFunction of the Sorts of it . b2,the Sorts of it . b3 st
ex b6 being ManySortedFunction of the Sorts of it . b3,the Sorts of it . b4 st
b5 = b1 .(b3,b2) & b6 = b1 .(b4,b3) & b1 .(b4,b2) = b6 ** b5 & b5 is_homomorphism it . b2,it . b3;
end;
:: MSALIMIT:dfs 1
definiens
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
let a3 be MSAlgebra-Family of the carrier of a1,a2;
To prove
a3 is OrderedAlgFam of a1,a2
it is sufficient to prove
thus ex b1 being Function-yielding ManySortedSet of the InternalRel of a1 st
for b2, b3, b4 being Element of the carrier of a1
st b3 <= b2 & b4 <= b3
holds ex b5 being ManySortedFunction of the Sorts of a3 . b2,the Sorts of a3 . b3 st
ex b6 being ManySortedFunction of the Sorts of a3 . b3,the Sorts of a3 . b4 st
b5 = b1 .(b3,b2) & b6 = b1 .(b4,b3) & b1 .(b4,b2) = b6 ** b5 & b5 is_homomorphism a3 . b2,a3 . b3;
:: MSALIMIT:def 1
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty non void ManySortedSign
for b3 being MSAlgebra-Family of the carrier of b1,b2 holds
b3 is OrderedAlgFam of b1,b2
iff
ex b4 being Function-yielding ManySortedSet of the InternalRel of b1 st
for b5, b6, b7 being Element of the carrier of b1
st b6 <= b5 & b7 <= b6
holds ex b8 being ManySortedFunction of the Sorts of b3 . b5,the Sorts of b3 . b6 st
ex b9 being ManySortedFunction of the Sorts of b3 . b6,the Sorts of b3 . b7 st
b8 = b4 .(b6,b5) & b9 = b4 .(b7,b6) & b4 .(b7,b5) = b9 ** b8 & b8 is_homomorphism b3 . b5,b3 . b6;
:: MSALIMIT:modenot 2 => MSALIMIT:mode 2
definition
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
let a3 be OrderedAlgFam of a1,a2;
mode Binding of A3 -> Function-yielding ManySortedSet of the InternalRel of a1 means
for b1, b2, b3 being Element of the carrier of a1
st b2 <= b1 & b3 <= b2
holds ex b4 being ManySortedFunction of the Sorts of a3 . b1,the Sorts of a3 . b2 st
ex b5 being ManySortedFunction of the Sorts of a3 . b2,the Sorts of a3 . b3 st
b4 = it .(b2,b1) & b5 = it .(b3,b2) & it .(b3,b1) = b5 ** b4 & b4 is_homomorphism a3 . b1,a3 . b2;
end;
:: MSALIMIT:dfs 2
definiens
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
let a3 be OrderedAlgFam of a1,a2;
let a4 be Function-yielding ManySortedSet of the InternalRel of a1;
To prove
a4 is Binding of a3
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1
st b2 <= b1 & b3 <= b2
holds ex b4 being ManySortedFunction of the Sorts of a3 . b1,the Sorts of a3 . b2 st
ex b5 being ManySortedFunction of the Sorts of a3 . b2,the Sorts of a3 . b3 st
b4 = a4 .(b2,b1) & b5 = a4 .(b3,b2) & a4 .(b3,b1) = b5 ** b4 & b4 is_homomorphism a3 . b1,a3 . b2;
:: MSALIMIT:def 2
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty non void ManySortedSign
for b3 being OrderedAlgFam of b1,b2
for b4 being Function-yielding ManySortedSet of the InternalRel of b1 holds
b4 is Binding of b3
iff
for b5, b6, b7 being Element of the carrier of b1
st b6 <= b5 & b7 <= b6
holds ex b8 being ManySortedFunction of the Sorts of b3 . b5,the Sorts of b3 . b6 st
ex b9 being ManySortedFunction of the Sorts of b3 . b6,the Sorts of b3 . b7 st
b8 = b4 .(b6,b5) & b9 = b4 .(b7,b6) & b4 .(b7,b5) = b9 ** b8 & b8 is_homomorphism b3 . b5,b3 . b6;
:: MSALIMIT:funcnot 1 => MSALIMIT:func 1
definition
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
let a3 be OrderedAlgFam of a1,a2;
let a4 be Binding of a3;
let a5, a6 be Element of the carrier of a1;
assume a6 <= a5;
func bind(A4,A5,A6) -> ManySortedFunction of the Sorts of a3 . a5,the Sorts of a3 . a6 equals
a4 .(a6,a5);
end;
:: MSALIMIT:def 3
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty non void ManySortedSign
for b3 being OrderedAlgFam of b1,b2
for b4 being Binding of b3
for b5, b6 being Element of the carrier of b1
st b6 <= b5
holds bind(b4,b5,b6) = b4 .(b6,b5);
:: MSALIMIT:th 1
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2, b3, b4 being Element of the carrier of b1
for b5 being non empty non void ManySortedSign
for b6 being OrderedAlgFam of b1,b5
for b7 being Binding of b6
st b3 <= b2 & b4 <= b3
holds (bind(b7,b3,b4)) ** bind(b7,b2,b3) = bind(b7,b2,b4);
:: MSALIMIT:attrnot 1 => MSALIMIT:attr 1
definition
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
let a3 be OrderedAlgFam of a1,a2;
let a4 be Binding of a3;
attr a4 is normalized means
for b1 being Element of the carrier of a1 holds
a4 .(b1,b1) = id the Sorts of a3 . b1;
end;
:: MSALIMIT:dfs 4
definiens
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
let a3 be OrderedAlgFam of a1,a2;
let a4 be Binding of a3;
To prove
a4 is normalized
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
a4 .(b1,b1) = id the Sorts of a3 . b1;
:: MSALIMIT:def 4
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty non void ManySortedSign
for b3 being OrderedAlgFam of b1,b2
for b4 being Binding of b3 holds
b4 is normalized(b1, b2, b3)
iff
for b5 being Element of the carrier of b1 holds
b4 .(b5,b5) = id the Sorts of b3 . b5;
:: MSALIMIT:th 2
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty non void ManySortedSign
for b3 being OrderedAlgFam of b1,b2
for b4 being Binding of b3
for b5, b6 being Element of the carrier of b1
st b6 <= b5
for b7 being ManySortedFunction of the Sorts of b3 . b5,the Sorts of b3 . b6
st b7 = bind(b4,b5,b6)
holds b7 is_homomorphism b3 . b5,b3 . b6;
:: MSALIMIT:funcnot 2 => MSALIMIT:func 2
definition
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
let a3 be OrderedAlgFam of a1,a2;
let a4 be Binding of a3;
func Normalized A4 -> Binding of a3 means
for b1, b2 being Element of the carrier of a1
st b2 <= b1
holds it .(b2,b1) = IFEQ(b2,b1,id the Sorts of a3 . b1,(bind(a4,b1,b2)) ** id the Sorts of a3 . b1);
end;
:: MSALIMIT:def 5
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty non void ManySortedSign
for b3 being OrderedAlgFam of b1,b2
for b4, b5 being Binding of b3 holds
b5 = Normalized b4
iff
for b6, b7 being Element of the carrier of b1
st b7 <= b6
holds b5 .(b7,b6) = IFEQ(b7,b6,id the Sorts of b3 . b6,(bind(b4,b6,b7)) ** id the Sorts of b3 . b6);
:: MSALIMIT:th 3
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty non void ManySortedSign
for b3 being OrderedAlgFam of b1,b2
for b4 being Binding of b3
for b5, b6 being Element of the carrier of b1
st b6 <= b5 & b5 <> b6
holds b4 .(b6,b5) = (Normalized b4) .(b6,b5);
:: MSALIMIT:funcreg 3
registration
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
let a3 be OrderedAlgFam of a1,a2;
let a4 be Binding of a3;
cluster Normalized a4 -> normalized;
end;
:: MSALIMIT:exreg 1
registration
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
let a3 be OrderedAlgFam of a1,a2;
cluster Relation-like Function-like Function-yielding normalized Binding of a3;
end;
:: MSALIMIT:th 4
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty non void ManySortedSign
for b3 being OrderedAlgFam of b1,b2
for b4 being normalized Binding of b3
for b5, b6 being Element of the carrier of b1
st b6 <= b5
holds (Normalized b4) .(b6,b5) = b4 .(b6,b5);
:: MSALIMIT:funcnot 3 => MSALIMIT:func 3
definition
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty non void ManySortedSign;
let a3 be OrderedAlgFam of a1,a2;
let a4 be Binding of a3;
func InvLim A4 -> strict MSSubAlgebra of product a3 means
for b1 being Element of the carrier of a2
for b2 being Element of (SORTS a3) . b1 holds
b2 in (the Sorts of it) . b1
iff
for b3, b4 being Element of the carrier of a1
st b4 <= b3
holds ((bind(a4,b3,b4)) . b1) . (b2 . b3) = b2 . b4;
end;
:: MSALIMIT:def 6
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty non void ManySortedSign
for b3 being OrderedAlgFam of b1,b2
for b4 being Binding of b3
for b5 being strict MSSubAlgebra of product b3 holds
b5 = InvLim b4
iff
for b6 being Element of the carrier of b2
for b7 being Element of (SORTS b3) . b6 holds
b7 in (the Sorts of b5) . b6
iff
for b8, b9 being Element of the carrier of b1
st b9 <= b8
holds ((bind(b4,b8,b9)) . b6) . (b7 . b8) = b7 . b9;
:: MSALIMIT:th 5
theorem
for b1 being non empty reflexive transitive antisymmetric discrete RelStr
for b2 being non empty non void ManySortedSign
for b3 being OrderedAlgFam of b1,b2
for b4 being normalized Binding of b3 holds
InvLim b4 = product b3;
:: MSALIMIT:attrnot 2 => MSALIMIT:attr 2
definition
let a1 be set;
attr a1 is MSS-membered means
for b1 being set
st b1 in a1
holds b1 is non empty strict non void ManySortedSign;
end;
:: MSALIMIT:dfs 7
definiens
let a1 be set;
To prove
a1 is MSS-membered
it is sufficient to prove
thus for b1 being set
st b1 in a1
holds b1 is non empty strict non void ManySortedSign;
:: MSALIMIT:def 7
theorem
for b1 being set holds
b1 is MSS-membered
iff
for b2 being set
st b2 in b1
holds b2 is non empty strict non void ManySortedSign;
:: MSALIMIT:exreg 2
registration
cluster non empty MSS-membered set;
end;
:: MSALIMIT:funcnot 4 => MSALIMIT:func 4
definition
func TrivialMSSign -> strict ManySortedSign means
it is empty & it is void;
end;
:: MSALIMIT:def 8
theorem
for b1 being strict ManySortedSign holds
b1 = TrivialMSSign
iff
b1 is empty & b1 is void;
:: MSALIMIT:funcreg 4
registration
cluster TrivialMSSign -> empty strict void;
end;
:: MSALIMIT:exreg 3
registration
cluster empty strict void ManySortedSign;
end;
:: MSALIMIT:th 6
theorem
for b1 being void ManySortedSign holds
id the carrier of b1,id the OperSymbols of b1 form_morphism_between b1,b1;
:: MSALIMIT:funcnot 5 => MSALIMIT:func 5
definition
let a1 be non empty set;
func MSS_set A1 -> set means
for b1 being set holds
b1 in it
iff
ex b2 being non empty strict non void ManySortedSign st
b1 = b2 & the carrier of b2 c= a1 & the OperSymbols of b2 c= a1;
end;
:: MSALIMIT:def 9
theorem
for b1 being non empty set
for b2 being set holds
b2 = MSS_set b1
iff
for b3 being set holds
b3 in b2
iff
ex b4 being non empty strict non void ManySortedSign st
b3 = b4 & the carrier of b4 c= b1 & the OperSymbols of b4 c= b1;
:: MSALIMIT:funcreg 5
registration
let a1 be non empty set;
cluster MSS_set a1 -> non empty MSS-membered;
end;
:: MSALIMIT:modenot 3 => MSALIMIT:mode 3
definition
let a1 be non empty MSS-membered set;
redefine mode Element of a1 -> non empty strict non void ManySortedSign;
end;
:: MSALIMIT:funcnot 6 => MSALIMIT:func 6
definition
let a1, a2 be ManySortedSign;
func MSS_morph(A1,A2) -> set means
for b1 being set holds
b1 in it
iff
ex b2, b3 being Relation-like Function-like set st
b1 = [b2,b3] & b2,b3 form_morphism_between a1,a2;
end;
:: MSALIMIT:def 10
theorem
for b1, b2 being ManySortedSign
for b3 being set holds
b3 = MSS_morph(b1,b2)
iff
for b4 being set holds
b4 in b3
iff
ex b5, b6 being Relation-like Function-like set st
b4 = [b5,b6] & b5,b6 form_morphism_between b1,b2;