changed statement of completeness and soundness

language.tex

@@ -361,23 +361,22 @@ of rules.
         { \pvdash \Gamma e t \varpi:t_1 \\ t_1\leq t_2 }
         { \pvdash \Gamma e t \varpi:t_2 }
         { }
-        \qquad
+        \quad
     \Infer[PInter]
         { \pvdash \Gamma e t \varpi:t_1 \\ \pvdash \Gamma e t \varpi:t_2 }
         { \pvdash \Gamma e t \varpi:t_1\land t_2 }
         { }
-\vspace{-1.2mm}\\
+        \quad
     \Infer[PTypeof]
         { \Gamma \vdash \occ e \varpi:t' }
         { \pvdash \Gamma e t \varpi:t' }
         { }
-        \qquad
+\vspace{-1.2mm}\\
     \Infer[PEps]
         { }
         { \pvdash \Gamma e t \epsilon:t }
         { }
         \qquad
-\vspace{-1.2mm}\\
     \Infer[PAppR]
         { \pvdash \Gamma e t \varpi.0:\arrow{t_1}{t_2} \\ \pvdash \Gamma e t \varpi:t_2'}
         { \pvdash \Gamma e t \varpi.1:\neg t_1 }
@@ -854,7 +853,7 @@ for variables.
 
 The algorithmic system above is sound with respect to the deductive one of Section~\ref{sec:static}
 \begin{theorem}[Soundness]
-For every $\Gamma$, $e$, $t$, $n_o$, if $\tyof e \Gamma \leq t$, then $\Gamma \vdash e:t$.
+For every $\Gamma$, $e$, $t$, $n_o$, if $\Gamma\vdashA e: t$, then $\Gamma \vdash e:t$.
 \end{theorem}
 We were not able to prove full completeness, just a partial form of it. As
 anticipated, the problems are twofold: $(i)$ the recursive nature of
@@ -872,7 +871,7 @@ types, since, intuitively, it corresponds to typing a language in
 which  in the types used in dynamic tests, a negated arrow never occurs  on the
 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 ${\tyof e \Gamma} \leq t$.
+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 $\Gamma\vdashA e: t'$ and $t'\leq t$.
 \end{theorem}
 \noindent The use of type schemes and of possibly diverging  iterations
 yield a system that may seem overly complicated. But it is important

main.tex

@@ -10,7 +10,7 @@
 %\documentclass[acmsmall,screen]{acmart}\settopmatter{}
 \newif\ifwithcomments
 \withcommentstrue
-\withcommentsfalse
+%\withcommentsfalse
 \newif\iflongversion
 \longversiontrue
 \longversionfalse

practical.tex

@@ -163,7 +163,7 @@ since, \texttt{or\_} has type\\
    (\lnot\True\to\lnot\True\to\False)$}
 We consider \True, \Any and $\lnot\True$ as candidate types for
 \texttt{x} which, in turn allows us to deduce a precise type given in the table. Finally, thanks to this rule it is no longer necessary to use a type case to force refinement. As a consequence we can define the functions \texttt{and\_} and \texttt{xor\_} more naturally as:
-\begin{alltt}\color{darkblue}
+\begin{alltt}\color{darkblue}\morecompact
 let and_ = fun (x : Any) -> fun (y : Any) -> not_ (or_ (not_ x) (not_ y))
 let xor_ = fun (x : Any) -> fun (y : Any) -> and_ (or_ x y) (not_ (and_ x y))
 \end{alltt}