:: MSAFREE1 semantic presentation
theorem Th1: :: MSAFREE1:1
theorem Th2: :: MSAFREE1:2
theorem Th3: :: MSAFREE1:3
:: deftheorem Def1 defines Flatten MSAFREE1:def 1 :
theorem Th4: :: MSAFREE1:4
:: deftheorem Def2 defines disjoint_valued MSAFREE1:def 2 :
:: deftheorem Def3 defines SingleAlg MSAFREE1:def 3 :
Lemma7:
for b1 being non empty ManySortedSign holds
( SingleAlg b1 is non-empty & SingleAlg b1 is disjoint_valued )
theorem Th5: :: MSAFREE1:5
scheme :: MSAFREE1:sch 2
s2{ F
1()
-> non
empty non
void ManySortedSign , F
2()
-> V5 ManySortedSet of the
carrier of F
1(), F
3()
-> V5 ManySortedSet of the
carrier of F
1(), F
4(
set )
-> Element of
Union F
3(), F
5(
set ,
set ,
set )
-> Element of
Union F
3(), F
6()
-> ManySortedFunction of
FreeSort F
2(),F
3(), F
7()
-> ManySortedFunction of
FreeSort F
2(),F
3() } :
provided
E9:
for b
1 being
OperSymbol of F
1()
for b
2 being
Element of
Args b
1,
(FreeMSA F2())for b
3 being
FinSequence of
Union F
3() holds
( b
3 = (Flatten F6()) * b
2 implies
(F6() . (the_result_sort_of b1)) . ((Den b1,(FreeMSA F2())) . b2) = F
5(b
1,b
2,b
3) )
and
E10:
for b
1 being
SortSymbol of F
1()
for b
2 being
set holds
( b
2 in FreeGen b
1,F
2() implies
(F6() . b1) . b
2 = F
4(b
2) )
and
E11:
for b
1 being
OperSymbol of F
1()
for b
2 being
Element of
Args b
1,
(FreeMSA F2())for b
3 being
FinSequence of
Union F
3() holds
( b
3 = (Flatten F7()) * b
2 implies
(F7() . (the_result_sort_of b1)) . ((Den b1,(FreeMSA F2())) . b2) = F
5(b
1,b
2,b
3) )
and
E12:
for b
1 being
SortSymbol of F
1()
for b
2 being
set holds
( b
2 in FreeGen b
1,F
2() implies
(F7() . b1) . b
2 = F
4(b
2) )
scheme :: MSAFREE1:sch 3
s3{ F
1()
-> non
empty non
void ManySortedSign , F
2()
-> V5 ManySortedSet of the
carrier of F
1(), F
3()
-> non-empty MSAlgebra of F
1(), P
1[
set ,
set ,
set ], F
4()
-> ManySortedFunction of
(FreeMSA F2()),F
3(), F
5()
-> ManySortedFunction of
(FreeMSA F2()),F
3() } :
provided
E9:
F
4()
is_homomorphism FreeMSA F
2(),F
3()
and
E10:
for b
1 being
SortSymbol of F
1()
for b
2, b
3 being
set holds
( b
3 in FreeGen b
1,F
2() implies (
(F4() . b1) . b
3 = b
2 iff P
1[b
1,b
2,b
3] ) )
and
E11:
F
5()
is_homomorphism FreeMSA F
2(),F
3()
and
E12:
for b
1 being
SortSymbol of F
1()
for b
2, b
3 being
set holds
( b
3 in FreeGen b
1,F
2() implies (
(F5() . b1) . b
3 = b
2 iff P
1[b
1,b
2,b
3] ) )