:: INSTALG1 semantic presentation
theorem Th1: :: INSTALG1:1
canceled;
theorem Th2: :: INSTALG1:2
theorem Th3: :: INSTALG1:3
theorem Th4: :: INSTALG1:4
theorem Th5: :: INSTALG1:5
theorem Th6: :: INSTALG1:6
for b
1 being non
empty non
void ManySortedSign for b
2, b
3, b
4, b
5 being
MSAlgebra of b
1 holds
(
MSAlgebra(# the
Sorts of b
2,the
Charact of b
2 #)
= MSAlgebra(# the
Sorts of b
4,the
Charact of b
4 #) &
MSAlgebra(# the
Sorts of b
3,the
Charact of b
3 #)
= MSAlgebra(# the
Sorts of b
5,the
Charact of b
5 #) implies for b
6 being
ManySortedFunction of b
2,b
3for b
7 being
ManySortedFunction of b
4,b
5 holds
( b
6 = b
7 implies for b
8 being
OperSymbol of b
1 holds
(
Args b
8,b
2 <> {} &
Args b
8,b
3 <> {} implies for b
9 being
Element of
Args b
8,b
2for b
10 being
Element of
Args b
8,b
4 holds
( b
9 = b
10 implies b
6 # b
9 = b
7 # b
10 ) ) ) )
theorem Th7: :: INSTALG1:7
for b
1 being non
empty non
void ManySortedSign for b
2, b
3, b
4, b
5 being
MSAlgebra of b
1 holds
(
MSAlgebra(# the
Sorts of b
2,the
Charact of b
2 #)
= MSAlgebra(# the
Sorts of b
4,the
Charact of b
4 #) &
MSAlgebra(# the
Sorts of b
3,the
Charact of b
3 #)
= MSAlgebra(# the
Sorts of b
5,the
Charact of b
5 #) & the
Sorts of b
2 is_transformable_to the
Sorts of b
3 implies for b
6 being
ManySortedFunction of b
2,b
3 holds
not ( b
6 is_homomorphism b
2,b
3 & ( for b
7 being
ManySortedFunction of b
4,b
5 holds
not ( b
7 = b
6 & b
7 is_homomorphism b
4,b
5 ) ) ) )
:: deftheorem Def1 defines feasible INSTALG1:def 1 :
theorem Th8: :: INSTALG1:8
theorem Th9: :: INSTALG1:9
theorem Th10: :: INSTALG1:10
:: deftheorem Def2 defines Subsignature INSTALG1:def 2 :
theorem Th11: :: INSTALG1:11
theorem Th12: :: INSTALG1:12
theorem Th13: :: INSTALG1:13
theorem Th14: :: INSTALG1:14
theorem Th15: :: INSTALG1:15
theorem Th16: :: INSTALG1:16
theorem Th17: :: INSTALG1:17
theorem Th18: :: INSTALG1:18
theorem Th19: :: INSTALG1:19
theorem Th20: :: INSTALG1:20
theorem Th21: :: INSTALG1:21
:: deftheorem Def3 defines | INSTALG1:def 3 :
:: deftheorem Def4 defines | INSTALG1:def 4 :
theorem Th22: :: INSTALG1:22
for b
1, b
2 being non
empty ManySortedSign for b
3, b
4 being
MSAlgebra of b
2 holds
(
MSAlgebra(# the
Sorts of b
3,the
Charact of b
3 #)
= MSAlgebra(# the
Sorts of b
4,the
Charact of b
4 #) implies for b
5, b
6 being
Function holds
( b
5,b
6 form_morphism_between b
1,b
2 implies b
3 | b
1,b
5,b
6 = b
4 | b
1,b
5,b
6 ) )
theorem Th23: :: INSTALG1:23
theorem Th24: :: INSTALG1:24
theorem Th25: :: INSTALG1:25
for b
1, b
2 being non
empty non
void ManySortedSign for b
3, b
4 being
Function holds
( b
3,b
4 form_morphism_between b
1,b
2 implies for b
5 being
MSAlgebra of b
2for b
6 being
OperSymbol of b
1for b
7 being
OperSymbol of b
2 holds
( b
7 = b
4 . b
6 implies (
Args b
7,b
5 = Args b
6,
(b5 | b1,b3,b4) &
Result b
6,
(b5 | b1,b3,b4) = Result b
7,b
5 ) ) )
theorem Th26: :: INSTALG1:26
theorem Th27: :: INSTALG1:27
theorem Th28: :: INSTALG1:28
for b
1, b
2, b
3 being non
empty ManySortedSign for b
4, b
5 being
Function holds
( b
4,b
5 form_morphism_between b
1,b
2 implies for b
6, b
7 being
Function holds
( b
6,b
7 form_morphism_between b
2,b
3 implies for b
8 being
MSAlgebra of b
3 holds b
8 | b
1,
(b6 * b4),
(b7 * b5) = (b8 | b2,b6,b7) | b
1,b
4,b
5 ) )
theorem Th29: :: INSTALG1:29
theorem Th30: :: INSTALG1:30
for b
1, b
2 being non
empty ManySortedSign for b
3 being
Function of the
carrier of b
1,the
carrier of b
2for b
4 being
Function holds
( b
3,b
4 form_morphism_between b
1,b
2 implies for b
5, b
6 being
MSAlgebra of b
2for b
7 being
ManySortedFunction of the
Sorts of b
5,the
Sorts of b
6 holds
b
7 * b
3 is
ManySortedFunction of the
Sorts of
(b5 | b1,b3,b4),the
Sorts of
(b6 | b1,b3,b4) )
theorem Th31: :: INSTALG1:31
theorem Th32: :: INSTALG1:32
theorem Th33: :: INSTALG1:33
theorem Th34: :: INSTALG1:34
for b
1, b
2 being non
empty non
void ManySortedSign for b
3, b
4 being
Function holds
( b
3,b
4 form_morphism_between b
1,b
2 implies for b
5, b
6 being
MSAlgebra of b
2for b
7 being
ManySortedFunction of b
5,b
6for b
8 being
ManySortedFunction of
(b5 | b1,b3,b4),
(b6 | b1,b3,b4) holds
( b
8 = b
7 * b
3 implies for b
9 being
OperSymbol of b
1for b
10 being
OperSymbol of b
2 holds
( b
10 = b
4 . b
9 &
Args b
10,b
5 <> {} &
Args b
10,b
6 <> {} implies for b
11 being
Element of
Args b
10,b
5for b
12 being
Element of
Args b
9,
(b5 | b1,b3,b4) holds
( b
11 = b
12 implies b
8 # b
12 = b
7 # b
11 ) ) ) )
theorem Th35: :: INSTALG1:35
for b
1, b
2 being non
empty non
void ManySortedSign for b
3, b
4 being
MSAlgebra of b
1 holds
( the
Sorts of b
3 is_transformable_to the
Sorts of b
4 implies for b
5 being
ManySortedFunction of b
3,b
4 holds
( b
5 is_homomorphism b
3,b
4 implies for b
6 being
Function of the
carrier of b
2,the
carrier of b
1for b
7 being
Function holds
not ( b
6,b
7 form_morphism_between b
2,b
1 & ( for b
8 being
ManySortedFunction of
(b3 | b2,b6,b7),
(b4 | b2,b6,b7) holds
not ( b
8 = b
5 * b
6 & b
8 is_homomorphism b
3 | b
2,b
6,b
7,b
4 | b
2,b
6,b
7 ) ) ) ) )
theorem Th36: :: INSTALG1:36
theorem Th37: :: INSTALG1:37
for b
1, b
2 being non
empty non
void ManySortedSign for b
3 being
non-empty MSAlgebra of b
1for b
4 being
Function of the
carrier of b
2,the
carrier of b
1for b
5 being
Function holds
( b
4,b
5 form_morphism_between b
2,b
1 implies for b
6 being
non-empty MSAlgebra of b
2 holds
( b
6 = b
3 | b
2,b
4,b
5 implies for b
7, b
8 being
SortSymbol of b
2for b
9 being
Function holds
( b
9 is_e.translation_of b
6,b
7,b
8 implies b
9 is_e.translation_of b
3,b
4 . b
7,b
4 . b
8 ) ) )
Lemma26:
for b1, b2 being non empty non void ManySortedSign
for b3 being non-empty MSAlgebra of b1
for b4 being Function of the carrier of b2,the carrier of b1
for b5 being Function holds
( b4,b5 form_morphism_between b2,b1 implies for b6 being non-empty MSAlgebra of b2 holds
( b6 = b3 | b2,b4,b5 implies for b7, b8 being SortSymbol of b2 holds
( TranslationRel b2 reduces b7,b8 implies ( TranslationRel b1 reduces b4 . b7,b4 . b8 & ( for b9 being Translation of b6,b7,b8 holds
b9 is Translation of b3,b4 . b7,b4 . b8 ) ) ) ) )
theorem Th38: :: INSTALG1:38
theorem Th39: :: INSTALG1:39
for b
1, b
2 being non
empty non
void ManySortedSign for b
3 being
non-empty MSAlgebra of b
1for b
4 being
Function of the
carrier of b
2,the
carrier of b
1for b
5 being
Function holds
( b
4,b
5 form_morphism_between b
2,b
1 implies for b
6 being
non-empty MSAlgebra of b
2 holds
( b
6 = b
3 | b
2,b
4,b
5 implies for b
7, b
8 being
SortSymbol of b
2 holds
(
TranslationRel b
2 reduces b
7,b
8 implies for b
9 being
Translation of b
6,b
7,b
8 holds
b
9 is
Translation of b
3,b
4 . b
7,b
4 . b
8 ) ) )
by Lemma26;
theorem Th40: :: INSTALG1:40
definition
let c
1, c
2 be non
empty non
void ManySortedSign ;
let c
3 be
V5 ManySortedSet of the
carrier of c
2;
let c
4 be
Function of the
carrier of c
1,the
carrier of c
2;
let c
5 be
Function;
assume E28:
c
4,c
5 form_morphism_between c
1,c
2
;
func hom c
4,c
5,c
3,c
1,c
2 -> ManySortedFunction of
(FreeMSA (a3 * a4)),
((FreeMSA a3) | a1,a4,a5) means :
Def5:
:: INSTALG1:def 5
( a
6 is_homomorphism FreeMSA (a3 * a4),
(FreeMSA a3) | a
1,a
4,a
5 & ( for b
1 being
SortSymbol of a
1for b
2 being
Element of
(a3 * a4) . b
1 holds
(a6 . b1) . (root-tree [b2,b1]) = root-tree [b2,(a4 . b1)] ) );
existence
ex b1 being ManySortedFunction of (FreeMSA (c3 * c4)),((FreeMSA c3) | c1,c4,c5) st
( b1 is_homomorphism FreeMSA (c3 * c4),(FreeMSA c3) | c1,c4,c5 & ( for b2 being SortSymbol of c1
for b3 being Element of (c3 * c4) . b2 holds (b1 . b2) . (root-tree [b3,b2]) = root-tree [b3,(c4 . b2)] ) )
uniqueness
for b1, b2 being ManySortedFunction of (FreeMSA (c3 * c4)),((FreeMSA c3) | c1,c4,c5) holds
( b1 is_homomorphism FreeMSA (c3 * c4),(FreeMSA c3) | c1,c4,c5 & ( for b3 being SortSymbol of c1
for b4 being Element of (c3 * c4) . b3 holds (b1 . b3) . (root-tree [b4,b3]) = root-tree [b4,(c4 . b3)] ) & b2 is_homomorphism FreeMSA (c3 * c4),(FreeMSA c3) | c1,c4,c5 & ( for b3 being SortSymbol of c1
for b4 being Element of (c3 * c4) . b3 holds (b2 . b3) . (root-tree [b4,b3]) = root-tree [b4,(c4 . b3)] ) implies b1 = b2 )
end;
:: deftheorem Def5 defines hom INSTALG1:def 5 :
for b
1, b
2 being non
empty non
void ManySortedSign for b
3 being
V5 ManySortedSet of the
carrier of b
2for b
4 being
Function of the
carrier of b
1,the
carrier of b
2for b
5 being
Function holds
( b
4,b
5 form_morphism_between b
1,b
2 implies for b
6 being
ManySortedFunction of
(FreeMSA (b3 * b4)),
((FreeMSA b3) | b1,b4,b5) holds
( b
6 = hom b
4,b
5,b
3,b
1,b
2 iff ( b
6 is_homomorphism FreeMSA (b3 * b4),
(FreeMSA b3) | b
1,b
4,b
5 & ( for b
7 being
SortSymbol of b
1for b
8 being
Element of
(b3 * b4) . b
7 holds
(b6 . b7) . (root-tree [b8,b7]) = root-tree [b8,(b4 . b7)] ) ) ) );
theorem Th41: :: INSTALG1:41
for b
1, b
2 being non
empty non
void ManySortedSign for b
3 being
V5 ManySortedSet of the
carrier of b
2for b
4 being
Function of the
carrier of b
1,the
carrier of b
2for b
5 being
Function holds
( b
4,b
5 form_morphism_between b
1,b
2 implies for b
6 being
OperSymbol of b
1for b
7 being
Element of
Args b
6,
(FreeMSA (b3 * b4))for b
8 being
FinSequence holds
( b
8 = (hom b4,b5,b3,b1,b2) # b
7 implies
((hom b4,b5,b3,b1,b2) . (the_result_sort_of b6)) . ([b6,the carrier of b1] -tree b7) = [(b5 . b6),the carrier of b2] -tree b
8 ) )
theorem Th42: :: INSTALG1:42
for b
1, b
2 being non
empty non
void ManySortedSign for b
3 being
V5 ManySortedSet of the
carrier of b
2for b
4 being
Function of the
carrier of b
1,the
carrier of b
2for b
5 being
Function holds
( b
4,b
5 form_morphism_between b
1,b
2 implies for b
6 being
Term of b
1,
(b3 * b4) holds
(
((hom b4,b5,b3,b1,b2) . (the_sort_of b6)) . b
6 is
CompoundTerm of b
2,b
3 iff b
6 is
CompoundTerm of b
1,b
3 * b
4 ) )
theorem Th43: :: INSTALG1:43
for b
1, b
2 being non
empty non
void ManySortedSign for b
3 being
V5 ManySortedSet of the
carrier of b
2for b
4 being
Function of the
carrier of b
1,the
carrier of b
2for b
5 being
one-to-one Function holds
( b
4,b
5 form_morphism_between b
1,b
2 implies
hom b
4,b
5,b
3,b
1,b
2 is_monomorphism FreeMSA (b3 * b4),
(FreeMSA b3) | b
1,b
4,b
5 )
theorem Th44: :: INSTALG1:44
theorem Th45: :: INSTALG1:45
for b
1, b
2, b
3 being non
empty non
void ManySortedSign for b
4 being
V5 ManySortedSet of the
carrier of b
3for b
5 being
Function of the
carrier of b
1,the
carrier of b
2for b
6 being
Function holds
( b
5,b
6 form_morphism_between b
1,b
2 implies for b
7 being
Function of the
carrier of b
2,the
carrier of b
3for b
8 being
Function holds
( b
7,b
8 form_morphism_between b
2,b
3 implies
hom (b7 * b5),
(b8 * b6),b
4,b
1,b
3 = ((hom b7,b8,b4,b2,b3) * b5) ** (hom b5,b6,(b4 * b7),b1,b2) ) )