Article MSUHOM_1, MML version 4.99.1005
:: MSUHOM_1:th 1
theorem
for b1, b2 being Relation-like Function-like set
for b3 being set
st proj2 b1 c= b3
holds b1 * (b2 | b3) = b1 * b2;
:: MSUHOM_1:th 2
theorem
for b1 being set
for b2 being Element of bool b1 holds
b2 * c= b1 *;
:: MSUHOM_1:th 3
theorem
for b1 being Relation-like Function-like set
for b2 being set
st b1 is Function-yielding
holds b1 | b2 is Function-yielding;
:: MSUHOM_1:th 4
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being ManySortedSet of b1 holds
(b3 | b2) # = b3 # | (b2 *);
:: MSUHOM_1:funcnot 1 => MSUHOM_1:func 1
definition
let a1 be non empty set;
let a2 be natural set;
let a3 be Element of a1;
redefine func a2 |-> a3 -> FinSequence of a1;
end;
:: MSUHOM_1:prednot 1 => MSUHOM_1:pred 1
definition
let a1, a2 be non empty ManySortedSign;
pred A1 <= A2 means
the carrier of a1 c= the carrier of a2 &
the OperSymbols of a1 c= the OperSymbols of a2 &
(the Arity of a2) | the OperSymbols of a1 = the Arity of a1 &
(the ResultSort of a2) | the OperSymbols of a1 = the ResultSort of a1;
reflexivity;
:: for a1 being non empty ManySortedSign holds
:: a1 <= a1;
end;
:: MSUHOM_1:dfs 1
definiens
let a1, a2 be non empty ManySortedSign;
To prove
a1 <= a2
it is sufficient to prove
thus the carrier of a1 c= the carrier of a2 &
the OperSymbols of a1 c= the OperSymbols of a2 &
(the Arity of a2) | the OperSymbols of a1 = the Arity of a1 &
(the ResultSort of a2) | the OperSymbols of a1 = the ResultSort of a1;
:: MSUHOM_1:def 1
theorem
for b1, b2 being non empty ManySortedSign holds
b1 <= b2
iff
the carrier of b1 c= the carrier of b2 &
the OperSymbols of b1 c= the OperSymbols of b2 &
(the Arity of b2) | the OperSymbols of b1 = the Arity of b1 &
(the ResultSort of b2) | the OperSymbols of b1 = the ResultSort of b1;
:: MSUHOM_1:th 5
theorem
for b1, b2, b3 being non empty ManySortedSign
st b1 <= b2 & b2 <= b3
holds b1 <= b3;
:: MSUHOM_1:th 6
theorem
for b1, b2 being non empty strict ManySortedSign
st b1 <= b2 & b2 <= b1
holds b1 = b2;
:: MSUHOM_1:th 7
theorem
for b1 being natural set
for b2 being non empty set
for b3 being Relation-like Function-like set
for b4 being Element of b2
for b5 being natural set
st 1 <= b5 & b5 <= b1
holds (b4 .--> b3) . ((b1 |-> b4) /. b5) = b3;
:: MSUHOM_1:th 8
theorem
for b1 being set
for b2 being Element of bool b1
for b3, b4 being ManySortedSet of b1
for b5 being ManySortedFunction of b3,b4
for b6, b7 being ManySortedSet of b2
st b6 = b3 | b2 & b7 = b4 | b2
holds b5 | b2 is ManySortedFunction of b6,b7;
:: MSUHOM_1:funcnot 2 => MSUHOM_1:func 2
definition
let a1, a2 be non empty strict non void ManySortedSign;
let a3 be strict non-empty MSAlgebra over a2;
assume a1 <= a2;
func A3 Over A1 -> strict non-empty MSAlgebra over a1 means
the Sorts of it = (the Sorts of a3) | the carrier of a1 &
the Charact of it = (the Charact of a3) | the OperSymbols of a1;
end;
:: MSUHOM_1:def 2
theorem
for b1, b2 being non empty strict non void ManySortedSign
for b3 being strict non-empty MSAlgebra over b2
st b1 <= b2
for b4 being strict non-empty MSAlgebra over b1 holds
b4 = b3 Over b1
iff
the Sorts of b4 = (the Sorts of b3) | the carrier of b1 &
the Charact of b4 = (the Charact of b3) | the OperSymbols of b1;
:: MSUHOM_1:th 9
theorem
for b1 being non empty strict non void ManySortedSign
for b2 being strict non-empty MSAlgebra over b1 holds
b2 = b2 Over b1;
:: MSUHOM_1:th 10
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
st b1,b2 are_similar
holds MSSign b1 = MSSign b2;
:: MSUHOM_1:funcnot 3 => MSUHOM_1:func 3
definition
let a1, a2 be non empty partial quasi_total non-empty UAStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
assume MSSign a1 = MSSign a2;
func MSAlg A3 -> ManySortedFunction of the Sorts of MSAlg a1,the Sorts of (MSAlg a2) Over MSSign a1 equals
{0} --> a3;
end;
:: MSUHOM_1:def 3
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st MSSign b1 = MSSign b2
holds MSAlg b3 = {0} --> b3;
:: MSUHOM_1:th 11
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1,b2 are_similar
for b4 being Element of the OperSymbols of MSSign b1 holds
(MSAlg b3) . the_result_sort_of b4 = b3;
:: MSUHOM_1:th 12
theorem
for b1 being non empty partial quasi_total non-empty UAStr
for b2 being Element of the OperSymbols of MSSign b1 holds
Den(b2,MSAlg b1) = (the charact of b1) . b2;
:: MSUHOM_1:th 13
theorem
for b1 being non empty partial quasi_total non-empty UAStr
for b2 being Element of the OperSymbols of MSSign b1 holds
Den(b2,MSAlg b1) is Element of Operations b1;
:: MSUHOM_1:th 14
theorem
for b1 being non empty partial quasi_total non-empty UAStr
for b2 being Element of the OperSymbols of MSSign b1
for b3 being Element of Args(b2,MSAlg b1) holds
b3 is FinSequence of the carrier of b1;
:: MSUHOM_1:th 15
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1,b2 are_similar
for b4 being Element of the OperSymbols of MSSign b1
for b5 being Element of Args(b4,MSAlg b1) holds
(MSAlg b3) # b5 = b5 * b3;
:: MSUHOM_1:th 16
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is_homomorphism b1,b2
holds MSAlg b3 is_homomorphism MSAlg b1,(MSAlg b2) Over MSSign b1;
:: MSUHOM_1:th 17
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1,b2 are_similar
holds MSAlg b3 is ManySortedSet of {0};
:: MSUHOM_1:th 18
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is_epimorphism b1,b2
holds MSAlg b3 is_epimorphism MSAlg b1,(MSAlg b2) Over MSSign b1;
:: MSUHOM_1:th 19
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is_monomorphism b1,b2
holds MSAlg b3 is_monomorphism MSAlg b1,(MSAlg b2) Over MSSign b1;
:: MSUHOM_1:th 20
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is_isomorphism b1,b2
holds MSAlg b3 is_isomorphism MSAlg b1,(MSAlg b2) Over MSSign b1;
:: MSUHOM_1:th 21
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1,b2 are_similar &
MSAlg b3 is_homomorphism MSAlg b1,(MSAlg b2) Over MSSign b1
holds b3 is_homomorphism b1,b2;
:: MSUHOM_1:th 22
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1,b2 are_similar &
MSAlg b3 is_epimorphism MSAlg b1,(MSAlg b2) Over MSSign b1
holds b3 is_epimorphism b1,b2;
:: MSUHOM_1:th 23
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1,b2 are_similar &
MSAlg b3 is_monomorphism MSAlg b1,(MSAlg b2) Over MSSign b1
holds b3 is_monomorphism b1,b2;
:: MSUHOM_1:th 24
theorem
for b1, b2 being non empty partial quasi_total non-empty UAStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b1,b2 are_similar &
MSAlg b3 is_isomorphism MSAlg b1,(MSAlg b2) Over MSSign b1
holds b3 is_isomorphism b1,b2;
:: MSUHOM_1:th 25
theorem
for b1 being non empty partial quasi_total non-empty UAStr holds
MSAlg id the carrier of b1 = id the Sorts of MSAlg b1;
:: MSUHOM_1:th 26
theorem
for b1, b2, b3 being non empty partial quasi_total non-empty UAStr
st b1,b2 are_similar & b2,b3 are_similar
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3 holds
(MSAlg b5) ** MSAlg b4 = MSAlg (b5 * b4);