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In-Depth Information
where
h
q
=
h
q
E
=
is
−
1
T
{
r
q
(
x
)
,
⊥}
,
h
1
=
ret(
I
,D)
◦
in
ret(
I
,D)
◦
τ
J(h
q
)
,
h
2
=
is
−
1
Tret(
(τ
−
1
:
h
q
→
h
q
E
, that is,
J(h
q
)
)
OP
I
,D)
◦
in
Tret(
I
,D)
◦
and
β
1
=
T
e
(h
1
;
h
2
)
β
1
={
id
r
q
(
x
)
,q
⊥
}:{
⊥}
→{
⊥}
r
q
(
x
)
,
r
q
(
x
)
,
.
⊥
}:
h
q
E
→
h
q
, so that
β
2
=
β
1
are
the identity morphism for the instance database composed of the single computed
query-relation
Analogously, we obtain
β
2
={
id
,q
r
q
(
x
)
r
q
(
x
)
.
Let us show that the morphism
β
1
:
h
q
→
h
q
E
in the example above can be di-
rectly obtained from the commutative diagram (1) in
DB
, presented in the proof of
Theorem
8
:
Tf
in
))
OP
:
h
q
→
h
q
E
.
Corollary 16
It holds that β
1
=
(T
e
(f
in
;
Proof
From the commutative diagram (1) in
DB
in the proof of Theorem
8
, and
from Theorem
4
, we have that
T
e
(f
in
;
in
h
q
E
:
h
q
E
→
in
OP
=
h
q
◦
T(Tf
in
◦
◦
Tf
in
)
h
q
E
)
h
q
. Thus, its dual is equal to
=
in
OP
in
h
q
E
OP
Tf
in
)
OP
T
e
(f
in
;
h
q
◦
T(Tf
in
◦
h
q
E
)
◦
in
OP
h
q
E
)
OP
=
h
q
E
◦
T(Tf
in
◦
◦
in
h
q
h
q
E
◦
T
h
OP
in
◦
in
OP
q
E
◦
Tf
OP
=
in
h
q
in
OP
Th
OP
Tf
OP
in
h
q
:
h
q
→
h
q
E
.
=
h
q
E
◦
q
E
◦
in
◦
