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Giuseppe Castagna
occurrence-typing
Commits
31db3325
Commit
31db3325
authored
Feb 26, 2020
by
Mickael Laurent
Browse files
update completeness theorem in the proofs
parent
6237061c
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proofs.tex
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31db3325
...
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@@ -954,8 +954,8 @@ can be extended to type schemes (see also~\citep[\S4.4]{Cas15} for a detailed de
We present here a refinement of the algorithmic type system presented in
\ref
{
sec:algorules
}
that associates to an expression a type scheme instead of a regular type.
This allows to type expressions more precisely and thus to have a more powerful
completeness theorem in regards to the
declarative type system.
This allows to type expressions more precisely and thus to have a more powerful
(but still partial) completeness theorem in regards to the
declarative type system.
The results about this new type system will be used in
\ref
{
sec:proofs
_
algorithmic
_
without
_
ts
}
in order to obtain a soundness and completeness
theorem for the algorithmic type system presented in
\ref
{
sec:algorules
}
.
...
...
@@ -1756,11 +1756,9 @@ theorem for the algorithmic type system presented in \ref{sec:algorules}.
\end{proof}
\begin{theorem}
[Completeness of the algorithmic type system for positive expressions]
\label
{
completenessA
}
If we restrict the language to positive expressions
$
e
_
+
$
,
the algorithmic type system without type schemes is complete.
More precisely:
$
\forall
\Gamma
, e
_
+
, t.
\ \Gamma
\vdash
e
_
+
:t
\Rightarrow
\exists
t'.
\ \Gamma
\vdashA
e
_
+
: t'
$
For every type environment
$
\Gamma
$
and positive expression
$
e
_
+
$
, if
$
\Gamma\vdash
e
_
+
: t
$
, then there exist
$
n
_
o
$
and
$
t'
$
such that
$
\Gamma\vdashA
e
_
+
: t'
$
.
\end{theorem}
\begin{proof}
...
...
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