diff --git a/tesi/.gitignore b/tesi/.gitignore
index bbd865c..03ea693 100644
--- a/tesi/.gitignore
+++ b/tesi/.gitignore
@@ -3,6 +3,7 @@ tesi.tex
 
 tesi_unicode.tex
 part4_unicode.tex
+part4.tex
                      
 
 ## Core latex/pdflatex auxiliary files:
diff --git a/tesi/gabriel/part4.pdf b/tesi/gabriel/part4.pdf
index c744289..4d09d10 100644
Binary files a/tesi/gabriel/part4.pdf and b/tesi/gabriel/part4.pdf differ
diff --git a/tesi/gabriel/part4.tex b/tesi/gabriel/part4.tex
index 05c321e..a0d244c 100644
--- a/tesi/gabriel/part4.tex
+++ b/tesi/gabriel/part4.tex
@@ -1,4 +1,4 @@
-% Created 2020-04-11 Sat 00:25
+% Created 2020-04-11 Sat 00:26
 % Intended LaTeX compiler: pdflatex
 \documentclass[11pt]{article}
 \usepackage[utf8]{inputenc}
@@ -64,10 +64,10 @@ xleftmargin=1em,
 \begin{document}
 
 \section{Translation validation of the Pattern Matching Compiler}
-\label{sec:org45bb046}
+\label{sec:org3d79c1b}
 
 \subsection{Source program}
-\label{sec:orgf4d7ca7}
+\label{sec:orgc2c278d}
 The algorithm takes as its input a source program and translates it
 into an algebraic data structure which type we call \emph{decision\_tree}.
 
@@ -412,7 +412,7 @@ TODO: fix \{0}
 
 \begin{enumerate}
 \item Proof:
-\label{sec:org9868f7d}
+\label{sec:org160a47b}
 Let \(m\) be a matrix with \[Group(m) = (k_r \to m_r)^k, m_{wild}\]
 Let \((v_i)^i\) be an input matrix with \(v_0 = k(v'_l)^l\) for some k.
 We proceed by case analysis:
@@ -472,7 +472,7 @@ We can also show that a$_i$ = (a\(_{\text{0.l}}\))$^l$ \sout{+} a\(_{\text{i$\in
 
 
 \subsection{Target translation}
-\label{sec:org5f1edb6}
+\label{sec:orgd2ea1d0}
 
 The target program of the following general form is parsed using a parser
 generated by Menhir, a LR(1) parser generator for the OCaml programming language.
@@ -560,7 +560,7 @@ $\forall$v$_T$, t$_T$(v$_T$) = $\llbracket$t$_T$$\rrbracket$$_T$(v$_T$)\\
 
 
 \subsection{Equivalence checking}
-\label{sec:org4d9383d}
+\label{sec:org33b978b}
 
 The equivalence checking algorithm takes as input a domain of
 possible values \emph{S} and a pair of source and target decision trees and