doc-src/TutorialI/Ifexpr/Ifexpr.thy

 The aim of this case study is twofold: it shows how to model boolean
 expressions and some algorithms for manipulating them, and it demonstrates
 the constructs introduced above.
 *}
 
-subsubsection{*How Can We Model Boolean Expressions?*}
+subsubsection{*Modelling Boolean Expressions*}
 
 text{*
 We want to represent boolean expressions built up from variables and
 constants by negation and conjunction. The following datatype serves exactly
 that purpose:

 @{term"Const False"}. Variables are represented by terms of the form
 @{term"Var n"}, where @{term"n"} is a natural number (type @{typ"nat"}).
 For example, the formula $P@0 \land \neg P@1$ is represented by the term
 @{term"And (Var 0) (Neg(Var 1))"}.
 
-\subsubsection{What is the Value of a Boolean Expression?}
+\subsubsection{The Value of a Boolean Expression}
 
 The value of a boolean expression depends on the value of its variables.
 Hence the function @{text"value"} takes an additional parameter, an
 \emph{environment} of type @{typ"nat => bool"}, which maps variables to their
 values:

 "valif (VIF x)    env = env x"
 "valif (IF b t e) env = (if valif b env then valif t env
                                         else valif e env)";
 
 text{*
-\subsubsection{Transformation Into and of If-Expressions}
+\subsubsection{Converting Boolean and If-Expressions}
 
-\REMARK{is this the title you wanted?}
 The type @{typ"boolex"} is close to the customary representation of logical
 formulae, whereas @{typ"ifex"} is designed for efficiency. It is easy to
 translate from @{typ"boolex"} into @{typ"ifex"}:
 *}