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Thus, by applying the algorithm in the proof of Proposition
23
, we transform
this binary tree-term into the following path-term
r
t
RA
WHERE
C
ρ
nr(r)
&
S
WHERE
C
REDUCE
S
TO
nr(r),
⊗
¬
t
RA
)
∈
T
RA
X
is the unique leaf,
C
ρ
where
(r
⊗
is the renamed version of
C
w.r.t. Cartesian product renaming
ρ
and
S
=
name
1
,...,name
n
.
t
RA
)
Thus, this update of
r
is represented by the arrow
o
U
:
⊗
→
⊗
(r
o
U
(r
t
RA
)
in
RA
, equal to the following composition of atomic arrows:
_ REDUCE
TO
nr(r)
name
1
,...,name
n
¬
C)
◦
_
nr(r)
&
name
1
,...,name
n
◦
_ WHERE
C
ρ
◦
(
_ WHERE
,
α
1
(r)
=
o
U
(r
⊗
t
RA
)
such that the new extension of
r
∈
A
, which is gener-
ally different from
α(r)
.
Consequently, by the functor
Eval
DB
of Proposition
24
, we ob-
tain the morphism
Eval(o
U
)
:
B
→
α
1
(
A
)
, where
B
={
r
⊗
t
RA
#
,
⊥}
:
RA
→
is an
r
}
)
and
r
is a relational symbol
instance database in
DB
of a schema
B
=
(
{
,
∅
t
ρ
of the leaf term
r
RA
.
Let us show that there is a morphism
h
r
:
⊗
t
ρ
α
∗
(
A
)
→
B
such that
r
⊗
RA
is the result of an inter-schema mapping
M
AB
:
A
→
B
and the R-algebra
α
is
t
ρ
extended to
r
∈
B
by
α(r
)
RA
#
.
In fact, based on the logical equivalence for the algebraic operation of Carte-
sian product in Sect.
5.1
, and on the recursive definition of
t
RA
=
=
r
⊗
t
(n)
(with
t
(
0
)
=
r
), we can define a mapping
M
AB
={
Ψ
}:
A
→
B
such that
Ψ
is the
following SOtgd:
x
1
r(
x
0
)
r(
x
1
)
⇒
r
x
0
&
e
1
(
x
1
),...,e
n
(
x
1
)
∀
x
0
∀
∧
where for
i
are the tuples of variables and
e
i
are the
functional symbols (also with arity zero for the constants), and
ar(r
)
=
0
,
1,
x
i
=
x
i,
1
,...,x
i,n
=
2
n
.
Consequently,
h
r
=
α
∗
MakeOperads(
M
AB
)
=
α
∗
{
v
1
·
q
A,
1
,
1
r
∅
}
:
α
∗
(
A
)
→
α
∗
(
B
)
is a morphism in
DB
, where the R-algebra
α
is a mapping-interpretation
(Definition
11
) which satisfies this schema mapping
M
AB
. That is,
q
A,
1
∈
O(r, r, r
q
)
is an operation of the expression
(
_
)
1
(
x
0
)
(
_
)
2
(
x
1
)
⇒
(
_
)
x
0
&
e
1
(
x
1
),...,e
n
(
x
n
)
,
∧
O(r
q
,r
)
, and
α(v
1
)
is the identity function for the relational table
α(r
)
v
1
∈
=
t
ρ
α(r
q
)
RA
#
.
The information flux of this morphism in
DB
is equal to
=
r
⊗
Flux
α, MakeOperads(
M
AB
)
.
h
r
=
