Added file with strong calculus and definition of distinction

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+\documentclass[12pt]{article}\r
+\usepackage{blindtext}\r
+\usepackage{enumerate}\r
+\usepackage{hyperref, bookmark}\r
+\usepackage{amsmath, amsfonts, amssymb, amsthm}\r
+\usepackage{xcolor}\r
+\r
+\title{Strong Separation}\r
+% \author{---}\r
+\date{\vspace{-2em}\today{}}\r
+\r
+\input{macros}\r
+\renewcommand{\Lam}[2]{\lambda#1.\, \{\!\!\{#2\}\!\!\}}\r
+\r
+\begin{document}\r
+\r
+\maketitle\r
+\r
+\section*{The Calculus}\r
+\subparagraph{Syntax}\r
+\[\begin{array}{lll}\r
+  \tm, \tmtwo & \ddef & \var \mid \tm\,\tmtwo \mid \Lam\var{\tm\Comma\vec\tm} \\\r
+  n & \ddef & \Lam\var{n\Comma\vec n} \mid \var\,\vec n \\\r
+  \\\r
+  C & \ddef & \Box \mid C\,\tm \mid \tm\,C \\\r
+  P & \ddef & \vec\tm \Comma \Box \Comma \vec\tm \mid \vec\tm \Comma C[\Lam\var P] \Comma \vec\tm  \\\r
+\end{array}\]\r
+\r
+\subparagraph{Reduction rules}\r
+\[\begin{array}{lll}\r
+  P[C[(\Lam\var{\tm\Comma\vec\tm})\,\tmtwo]] & \Red{}{\var\in \tm,\vec\tm} &\r
+   P[C[\tm\Subst\var\tmtwo]\Comma \vec\tm\Subst\var\tmtwo]\\\r
+   P[C[(\Lam\var{\tm\Comma\vec\tm})\,\tmtwo]] & \Red{}{\var\not\in\tm,\vec\tm} &\r
+    P[C[\tm] \Comma \vec\tm\Comma\tmtwo]\\\r
+\end{array}\]\r
+\r
+\subparagraph{Properties}\r
+\begin{itemize}\r
+  \item Every term is normalizing iff it is strongly normalizing.\r
+  \item Ogni strategia e' perpetua!\r
+  \item \ldots\r
+\end{itemize}\r
+\r
+\clearpage\r
+\section*{Separation}\r
+\newcommand{\HeadOf}[1]{\operatorname{head}(#1)}\r
+\newcommand{\FstOf}[1]{\operatorname{fst}(#1)}\r
+\newcommand{\DegOf}[1]{\operatorname{deg}(#1)}\r
+\newcommand{\SubtmOf}[2]{#1\preceq #2}\r
+\begin{itemize}\r
+  % \item \textbf{$\boldsymbol\sigma$-separation.}\r
+  %  \textcolor{red}{come definirlo? con le variabili? con i termini?\r
+  %   problematico in cbv}\r
+  %  Two terms are $\sigma$-separable iff there exists a substitution\r
+  %   $\sigma$ such that \textcolor{red}{???}\r
+  % \item \textbf{Semi-$\boldsymbol\sigma$-separation.}\r
+  %  Two terms are semi-$\sigma$-separable iff there exists a substitution\r
+  %   $\sigma$ such that -- in short -- it makes one diverge and the other one converge.\r
+  % \item \textbf{Our subproblem:} Semi-$\sigma$-separating two (usual) $\boldsymbol\lambda$-terms\r
+   % (in deep normal form)\r
+   \item \textbf{Subterm:} $\SubtmOf\tm\tmtwo$ means that $\tm$ is an ($\eta/\Omega$)-subterm of $\tmtwo$.\r
+   \item \textbf{Distinction:} Let $T_x \defeq \{t \preceq T \mid \HeadOf{t} = \HeadOf{D} \}$.\r
+  Now: $C_x$ is $D$--\emph{distinct} iff it is empty, or there exists path $\pi$ s.t.:\r
+  \begin{itemize}\r
+    \item \emph{effective} for $D$, cioe' $\FstOf{\pi} \leq \DegOf{D}$;\r
+    \item $\forall t\in D_x$, $t_\pi \neq \Omega$;\r
+    \item $\exists t\in C_x$ s.t. $t \not\sim_\pi D$;\r
+    \item $\{t\in C_x \mid t \sim_\pi D\}$ is $D$--distinct.\r
+  \end{itemize}\r
+\end{itemize}\r
+\r
+\end{document}\r