@@ -873,15 +873,14 @@ left-hand side of another negated arrow.

\begin{theorem}[Partial Completeness]

For every $\Gamma$, $e$, $t$, if $\Gamma\vdash e:t$ is derivable by a rank-0 negated derivation, then there exists $n_o$ such that ${\tyof e \Gamma}\leq t$.

\end{theorem}

The use of type schemes and the possible non convergence of iterations

The use of type schemes and of possibly non convergencing iterations

yield a system that may seem overly complicated. But it is important

to stress that this systems is defined only to study the declarative

type inference system of Section~\ref{sec:static} and in particular to prod

how close we can get to a complete algorithm to it. But for the

how close we can get to a complete algorithm for it. But for the

practical application type schemes are not needed, since they are

necessary only when type cases may specify types with negative arrows

and this in practice never happens: see Footnote

\ref{foo:typecase}. This is why for our implementation we use the

and this in practice never happens (see Footnote~\ref{foo:typecase} and Corollary~\ref{app:completeness}). This is why for our implementation we use the

CDuce library in which type schemes are absent and functions are typed

only by intersections of positive arrows. We present the implementation in Section~\ref{sec:practical}