A derivation of the declarative type system is said positive iff it does not contain any rule \Rule{Abs-}.

\end{definition}

\begin{definition}[Acceptable derivation]

A derivation of the declarative type system is said acceptable iff any application of \Rule{PAppL}

\begin{definition}[Rank-0 negated derivation]

A derivation of the declarative type system is said rank-0 negated iff any application of \Rule{PAppL}

has a positive derivation as first premise ($\pvdash\Gamma e t \varpi.1:t_1$).

\end{definition}

...

...

@@ -1302,7 +1302,7 @@

The rest follows.

\end{proof}

\begin{theorem}[Completeness of the algorithm for positive derivations]

\begin{theorem}[Completeness for positive derivations]

For every $\Gamma$, $e$, $t$ such that we have a positive derivation of $\Gamma\vdash e:t$,

there exists a global parameter $n_o$ with which $\tsrep{\tyof e \Gamma}\leq t$.

...

...

@@ -1434,14 +1434,38 @@

\end{description}

\end{proof}

\begin{theorem}[Completeness of the algorithm for acceptable derivations]

For any $\Gamma$, $e$, $t$ such that we have an acceptable derivation of $\Gamma\vdash e:t$, there exists a global parameter $n_o$

\begin{corollary}\label{app:completeness}

If we restrict our language so that the only functional type that can be used in a test is $\arrow\Empty\Any$

and such that there is no negated arrows in the domain of lambda-abstractions,

then the algorithmic type system becomes complete

(i.e. any expression that can be typed in the declarative type system is also typeable in the algorithmic one)

and type schemes can be removed from it (we can use regular types instead).

\end{corollary}

\begin{proof}

With such restrictions, the rule \Rule{Abs-} is not needed anymore.

Indeed, when typing an application $e_1 e_2$, the negative part of the type of $e_1$ (i.e. the $N_i$ part of its DNF)

is ignored by the operator $\apply{}{}$.

Moreover, as there is no negated arrows in the domain of lambda-abstractions,

the negative arrows of the type of $e_2$ can also be ignored.

Similarly, negative arrows can also be ignored when refining an application ($\worra{}{}$ also ignore the negative part

of the type of $e_1$).

Finally, as the only functional type that we can test is $\arrow\Empty\Any$, a functional type

cannot be refined to $\Empty$ due to its negative part, and thus we can ignore its negative part

(it makes no difference relatively to the rule \Rule{Efq}).

\end{proof}

\begin{theorem}[Completeness for rank-0 negated derivations]

For every $\Gamma$, $e$, $t$ such that we have a rank-0 negated derivation of $\Gamma\vdash e:t$, there exists a global parameter $n_o$

with which $\tyof e \Gamma\leq t$.

More precisely:

\begin{align*}

&\forall\Gamma, e, t.\ \Gamma\vdash e:t \text{ has an acceptable derivation }\Rightarrow\tyof e \Gamma\leq t\\

&\forall\Gamma, \Gamma', e, t.\ \Gamma\evdash e t \Gamma' \text{ has an acceptable derivation }\Rightarrow\Refine{e,t}\Gamma\leqA\Gamma' \text{ (for $n_o$ large enough)}

&\forall\Gamma, e, t.\ \Gamma\vdash e:t \text{ has a rank-0 negated derivation }\Rightarrow\tyof e \Gamma\leq t\\

&\forall\Gamma, \Gamma', e, t.\ \Gamma\evdash e t \Gamma' \text{ has a rank-0 negated derivation }\Rightarrow\Refine{e,t}\Gamma\leqA\Gamma' \text{ (for $n_o$ large enough)}