Fix small typos

Reference Manual/kernel.tex

@@ -165,7 +165,7 @@ \subsection{Syntax}
 
 \subsection{Substitution}
 \label{subsec:substitution}
-On top of basic building blocks of terms and formulas, there is one important type of operations: substitution of schematic symbols, which has to be implemented in a capture-avoiding way. We start with the subcase of variable substitution:
+On top of basic building blocks of terms and formulas, there is one important type of operation: substitution of schematic symbols, which has to be implemented in a capture-avoiding way. We start with the subcase of variable substitution:
 \begin{definition}[Capture-avoiding Substitution of variables]
   Given a base term $t$, a variable $x$ and another term $r$, the substitution of $x$ by $r$ inside $t$ is denoted by $ t[x := r] $ and is computed by replacing all occurences of $x$ by $r$.
 
@@ -752,7 +752,7 @@ \subsection{Definitions}
     FunctionDefinition(f, y, lambda(Seq(x1,...,x2), φ)) 
   \end{lstlisting}
   corresponds to \\
-  \hspace*{1.3em}``For any $\vec{x}$, let $f^n(\vec{x}) \textnormal{ be the unique } y \textnormal{ such that } \phi$ holds."
+  \hspace*{1.3em}``For any $\vec{x}$, let $f^n(\vec{x}) \textnormal{ be the unique } y \textnormal{ such that } \varphi$ holds."
 
   \caption{Definitions in LISA.}
   \label{fig:definitions}
@@ -1078,7 +1078,7 @@ \subsection{How to write helpers}
   val f: SchematicFunctionLabel("f", 2)
 \end{lstlisting}
 even with the above \lstinline|apply| trick, \lstinline|f(x, y)| would not compile, since \lstinline|f| can apply to \lstinline|Term| arguments, but not to \lstinline|TermLabel|. Hence we first need to apply \lstinline|x| and \lstinline|y| to an empty list of argument, such as in \lstinline|f(x(), y())|.
-This can be done automatically with implicit conversions. Implicit conversion is the mechanism allowing to cast an object of a type to an other in a canonical way. It is define with the \lstinline|given| keyword:
+This can be done automatically with implicit conversions. Implicit conversion is the mechanism allowing to cast an object of a type to an other in a canonical way. It is defined with the \lstinline|given| keyword:
 %
 \begin{lstlisting}[language=Scala]
   given Conversion[TermLabel, Term] = (t: Term) => Term(t, Seq())

Reference Manual/quickguide.tex

@@ -186,7 +186,7 @@ \section{Writing theory files}
     have (Y) by Tactic2(s1)
   \end{lstlisting}
 \end{minipage}
-\lstinline|thenHave| allows us to not give a name to every step when we're doing linear reasoning. Finally, in lines 5 and 8, we see that tactic can refer not only to steps of the current proof, but also to previously proven theorems and axioms, such as \lstinline|emptySetAxiom|. The \lstinline|of| keyword indicates the the axiom (or step) is instantiated in a particular way. For example:
+\lstinline|thenHave| allows us to not give a name to every step when we're doing linear reasoning. Finally, in lines 5 and 8, we see that tactic can refer not only to steps of the current proof, but also to previously proven theorems and axioms, such as \lstinline|emptySetAxiom|. The \lstinline|of| keyword indicates the axiom (or step) is instantiated in a particular way. For example:
 \noindent\begin{minipage}{\linewidth}\vspace{1em}
   \begin{lstlisting}[language=lisa, frame=single]
     emptySetAxiom             // ==  !(x ∈ ∅)