new rule VarA

language2.tex

@@ -631,6 +631,9 @@ takes as input $\Gamma$, $e$, and $t$, and returns an environment that
 extends $\Gamma$ with hypotheses on the occurrences of $e$ that are
 the most general that can be deduced by assuming that $e\in t$ succeeds.
 
+The notation $\tyof{e}{\Gamma}$ denotes the type that can be deduced for the occurence $e$ under the type environment $\Gamma$ in the algorithmic type system given in Section~\ref{sec:algorules}.
+That is, $\tyof{e}{\Gamma}=\ts$ if and only if $\Gamma\vdashA e:\ts$ is provable. 
+
 We start by defining the algorithm for each single occurrence, that is for the deduction of $\pvdash \Gamma e t \varpi:t'$. This is obtained by defining two mutually recursive functions $\constrf$ and $\env{}{}$:
 \newlength{\sk}
 \setlength{\sk}{-1.3pt}
@@ -677,7 +680,8 @@ We start by defining the algorithm for each single occurrence, that is for the d
 
 
 
-All the functions above are defined if and only if so all their subexpressions are (i.e., $\occ e{\varpi.i}$ must be defined).
+All the functions above are defined if and only if the initial path $\varpi$ is valid for $e$ (i.e. $\occ e{\varpi}$ is defined)
+and $e$ is well-typed (so that all $\tyof {\occ e{\varpi}} \Gamma$ encountered are defined).
 
 Each case of the definition of the $\constrf$ function corresponds to the
 application of a logical rule (\emph{cf.} Footnote~\ref{fo:rules}) in
@@ -700,9 +704,6 @@ remaining structural rule, \Rule{Psubs}, is accounted for by the use
 of the operators $\worra{}{}$ and $\boldsymbol{\pi}_i$ in
 the definition of $\constr{}{}$.
 
-
-The notation $\tyof{e}{\Gamma}$ denotes the type that can be deduced for the occurence $e$ under the type environment $\Gamma$ in the algorithmic type system given in Section~\ref{sec:algorules}.
-That is, $\tyof{e}{\Gamma}=\ts$ if and only if $\Gamma\vdashA e:\ts$ is provable. 
 \footnote{Note that the definition is well-founded.
 This can be seen by analyzing the rule \Rule{Case\Aa}: the definition of $\Refine {e,t} \Gamma$ and  $\Refine {e,\neg t} \Gamma$ use
 $\tyof{\occ e{\varpi}}\Gamma$, and this is defined for all $\varpi$ since the first premisses of \Rule{Case\Aa} states that
@@ -791,11 +792,16 @@ We now have all  notions needed for our typing algorithm, which is defined by th
   { }
   { \Gamma, (e:\Empty) \vdashA e': \Empty }
   { \begin{array}{c}\text{\tiny with priority over}\\[-1.8mm]\text{\tiny all the other rules}\end{array}}
+  \qquad
+  \Infer[Var\Aa]
+      { }
+      { \Gamma \vdashA x: \Gamma(x) }
+      { x\in\dom\Gamma}
   \\
   \Infer[Env\Aa]
       { \Gamma\setminus\{e\} \vdashA e : \ts }
       { \Gamma \vdashA e: \Gamma(e) \tsand \ts }
-      { e\in\dom\Gamma}
+      { e\in\dom\Gamma \text{ and } e \text{ not a variable}}
   \qquad
   \Infer[Const\Aa]
       { }