### update completeness theorem in the proofs

parent 6237061c
 ... @@ -954,8 +954,8 @@ can be extended to type schemes (see also~\citep[\S4.4]{Cas15} for a detailed de ... @@ -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} 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. 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 This allows to type expressions more precisely and thus to have a more powerful declarative type system. (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 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}. 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}. ... @@ -1756,11 +1756,9 @@ theorem for the algorithmic type system presented in \ref{sec:algorules}. \end{proof} \end{proof} \begin{theorem}[Completeness of the algorithmic type system for positive expressions]\label{completenessA} \begin{theorem}[Completeness of the algorithmic type system for positive expressions]\label{completenessA} If we restrict the language to positive expressions $e_+$, For every type environment $\Gamma$ and positive expression $e_+$, if the algorithmic type system without type schemes is complete. $\Gamma\vdash e_+: t$, then there exist $n_o$ and $t'$ such that $\Gamma\vdashA e_+: t'$. More precisely: $\forall \Gamma, e_+, t.\ \Gamma \vdash e_+:t \Rightarrow \exists t'.\ \Gamma \vdashA e_+ : t'$ \end{theorem} \end{theorem} \begin{proof} \begin{proof} ... ...
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