rewording

language2.tex

@@ -619,50 +619,17 @@ 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}
+\setlength{\sk}{-1.5pt}
   \begin{eqnarray}
-    \constr\epsilon{\Gamma,e,t} & = & t\\[\sk]\label{uno}
-    \constr{\varpi.0}{\Gamma,e,t} & = & \neg(\arrow{\env {\Gamma,e,t}{(\varpi.1)}}{\neg \env {\Gamma,e,t} (\varpi)})\\[\sk]\label{due}
-    \constr{\varpi.1}{\Gamma,e,t} & = & \worra{\tsrep {\tyof{\occ e{\varpi.0}}\Gamma}}{\env {\Gamma,e,t} (\varpi)}\\[\sk]\label{tre}
-    \constr{\varpi.l}{\Gamma,e,t} & = & \bpl{\env {\Gamma,e,t} (\varpi)}\\[\sk]\label{quattro}
-    \constr{\varpi.r}{\Gamma,e,t} & = & \bpr{\env {\Gamma,e,t} (\varpi)}\\[\sk]\label{cinque}
-    \constr{\varpi.f}{\Gamma,e,t} & = & \pair{\env {\Gamma,e,t} (\varpi)}\Any\\[\sk]\label{sei}
-    \constr{\varpi.s}{\Gamma,e,t} & = & \pair\Any{\env {\Gamma,e,t} (\varpi)}\\[1.5mm]\label{sette}
+    \constr\epsilon{\Gamma,e,t} & = & t\label{uno}\\[\sk]
+    \constr{\varpi.0}{\Gamma,e,t} & = & \neg(\arrow{\env {\Gamma,e,t}{(\varpi.1)}}{\neg \env {\Gamma,e,t} (\varpi)})\label{due}\\[\sk]
+    \constr{\varpi.1}{\Gamma,e,t} & = & \worra{\tsrep {\tyof{\occ e{\varpi.0}}\Gamma}}{\env {\Gamma,e,t} (\varpi)}\label{tre}\\[\sk]
+    \constr{\varpi.l}{\Gamma,e,t} & = & \bpl{\env {\Gamma,e,t} (\varpi)}\label{quattro}\\[\sk]
+    \constr{\varpi.r}{\Gamma,e,t} & = & \bpr{\env {\Gamma,e,t} (\varpi)}\label{cinque}\\[\sk]
+    \constr{\varpi.f}{\Gamma,e,t} & = & \pair{\env {\Gamma,e,t} (\varpi)}\Any\label{sei}\\[\sk]
+    \constr{\varpi.s}{\Gamma,e,t} & = & \pair\Any{\env {\Gamma,e,t} (\varpi)}\label{sette}\\[1.5mm]
     \env {\Gamma,e,t} (\varpi) & = & \constr \varpi {\Gamma,e,t} \land \tsrep {\tyof {\occ e \varpi} \Gamma}\label{otto}
   \end{eqnarray}
-
-  
-
-
-%% \[
-%% \begin{array}{lcl}
-%%   \typep+\epsilon{\Gamma,e,t} & = & t\\
-%%   \typep-\epsilon{\Gamma,e,t} & = & \neg t\\
-%%   \typep{p}{\varpi.0}{\Gamma,e,t} & = & \neg(\arrow{\Gp p{\Gamma,e,t}{(\varpi.1)}}{\neg \Gp p {\Gamma,e,t} (\varpi)})\\
-%%   \typep{p}{\varpi.1}{\Gamma,e,t} & = & \worra{\tsrep {\tyof{\occ e{\varpi.0}}\Gamma}}{\Gp p {\Gamma,e,t} (\varpi)}\\
-%%   \typep{p}{\varpi.l}{\Gamma,e,t} & = & \bpl{\Gp p {\Gamma,e,t} (\varpi)}\\
-%%   \typep{p}{\varpi.r}{\Gamma,e,t} & = & \bpr{\Gp p {\Gamma,e,t} (\varpi)}\\
-%%   \typep{p}{\varpi.f}{\Gamma,e,t} & = & \pair{\Gp p {\Gamma,e,t} (\varpi)}\Any\\
-%%   \typep{p}{\varpi.s}{\Gamma,e,t} & = & \pair\Any{\Gp p {\Gamma,e,t} (\varpi)}\\ \\
-%%   \Gp p {\Gamma,e,t} (\varpi) & = & \typep p \varpi {\Gamma,e,t} \land \tsrep {\tyof {\occ e \varpi} \Gamma}\\
-%%   %\underbrace{\land \tyof {\occ e \varpi} {\Gamma\setminus\{\occ e \varpi\}}}_{\text{if $\occ e \varpi$ is not a variable}}}\\
-%% \end{array}
-%% \]
-
-%% \begin{align*}
-%%   &(\RefineStep p {e,t} (\Gamma))(e') = 
-%%     \left\{\begin{array}{ll}
-%%       %\tyof {e'} \Gamma \tsand
-%%       \bigwedge_{\{\varpi \alt \occ e \varpi \equiv e'\}}
-%%       \Gp p {\Gamma,e,t} (\varpi) & \text{if } \exists \varpi.\ \occ e \varpi \equiv e' \\
-%%       \Gamma(e') & \text{otherwise, if } e' \in \dom \Gamma\\
-%%       \text{undefined} & \text{otherwise}
-%%     \end{array}\right.\\
-%%   &\Refine p {e,t} \Gamma=\fixpoint_\Gamma (\RefineStep p {e,t})\qquad \text{(for the order $\leq$ on environments, defined in the proofs section)}
-%% \end{align*}
-
-
-
 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 (which implies that all $\tyof {\occ e{\varpi}} \Gamma$