latex on intro

tesi/tesi_unicode.org

@@ -209,6 +209,72 @@ branches are related to the low level representation of values in
 Lambda code. For example, cons cells \texttt{x::xs} or tuples
 \texttt{(x,y)} are blocks with tag 0.
 
+We can define this parametrized grammar for decision trees
+A decision tree is defined as either a Leaf, a Failure terminal or
+an intermediate node with different children sharing the same accessor /a/
+and an optional fallback.
+
+The parametrized grammar $D(\pi, e)$ describes the common structure of
+source and decision trees. We use $\pi$ for the \emph{conditions} on
+each branch, and $a$ for our \emph{accessors}, which give a symbolic
+description of a sub-value of the scrutinee. Source conditions $\pi_S$
+are just datatype constructors; target conditions $\pi_T$ are
+arbitrary sets of low level OCaml values.
+\begin{comment}
+ possible immediate-integer or block-tag values.
+\end{comment}
+%
+\begin{mathpar}
+
+\begin{mathpar}
+  \begin{array}{l@{~}r@{~}r@{~}l}
+    \text{\emph{decision trees}} & D(\pi, e)
+    & \bnfeq & \Leaf {\cle(a)} \\
+    & & \bnfor & \Failure \\
+    & & \bnfor & \Switch a {\Fam {i \in I} {\pi_i, D_i}} \Dfb \\
+    & & \bnfor & \Guard {\cle(a)} {D_0} {D_1} \\
+    & & \bnfor & Unreachable 
+  \end{array}
+  \quad
+  \begin{array}{l@{~}r@{~}r@{~}l}
+    \text{\emph{environment}} & \sigma(v)
+    & \bnfeq & [x_1 \mapsto v_1, \dots, v_n \mapsto v_n] \\
+    \text{\emph{closed term}} & \cle(v)
+    & \bnfeq & (\sigma(v), e) \\
+    \text{\emph{accessors}} & a
+    & \bnfeq & \Root \;\bnfor\; a.n \quad (n \in \mathbb{N}) \\
+  \end{array}
+    \quad
+  \begin{array}{l}
+  \pi_S : \text{datatype constructors}
+  \\
+  \pi_T \subseteq        \{ \Int n \mid n \in \mathbb{Z} \}
+                  \uplus \{ \Tag n \mid n \in \mathbb{N} \}
+  \\[1em]
+  a(v_S), a(v_T), D_S(v_S), D_T(v_T) \quad (\text{\emph{omitted})}
+  \end{array}
+\end{mathpar}
+The tree $\Leaf \cle$ returns a leaf expression $e$ in a captured
+environment $\sigma$ mapping variables to accessors.
+%
+$\Failure$ expresses match failure, when no clause matches the input
+value.
+%
+$\Switch a {\Fam {i \in I} {\pi_i, D_i}} \Dfb$ has one subtree $D_i$
+for every head constructor that appears in the pattern matching
+clauses, and a fallback case for the constructors.
+%
+$\Guard \cle {D_0} {D_1}$ represents a \texttt{when}-guard on a closed
+expression $\cle$, expected to be of boolean type, with sub-trees
+$D_0$ for the \texttt{true} case and $D_1$ for the \texttt{false}
+case.
+
+We write $a(v)$ for the sub-value of the (source or target) value $v$
+that is reachable at the accessor $a$, and $D(v)$ for the result of
+running a value $v$ against a decision tree $D$; this results in
+a (source or target) matching run $R(v)$, just like running the value
+against a program.
+
 To check the equivalence of a source and a target decision tree,
 we proceed by case analysis.
 If we have two terminals, such as leaves in the previous example,
@@ -710,7 +776,7 @@ Execution follows the rules of Hoare logic:
 
 - Conditional : 
 \begin{mathpar}
-\infer { \{P \wedge b \} c_1 \{Q\}, \{P\wedge\not b\}c_2\{Q\} }
+\infer { \{P \wedge b \} c_1 \{Q\}, \{P\wedge\neg b\}c_2\{Q\} }
 { \{P\}\text{if b then $c_1$ else $c_2$}\{Q\} }
 \end{mathpar}
 
@@ -1299,58 +1365,6 @@ captured by the relation $\progrel {t_S} {t_T}$.
 # TODO eq_muovi
 
 \subsection{Grammar of the decision trees}
-A decision tree is defined as either a Leaf, a Failure terminal or
-an intermediate node with different children sharing the same accessor /a/
-and an optional fallback.
-
-The parametrized grammar $D(\pi, e)$ describes the common structure of
-source and decision trees. We use $\pi$ for the \emph{conditions} on
-each branch, and $a$ for our \emph{accessors}, which give a symbolic
-description of a sub-value of the scrutinee. Source conditions $\pi_S$
-are just datatype constructors; target conditions $\pi_T$ are
-arbitrary sets of possible immediate-integer or block-tag values.
-%
-\begin{mathpar}
-  \begin{array}{l@{~}r@{~}r@{~}l}
-    \text{\emph{decision trees}} & D(\pi, e)
-    & \bnfeq & \Leaf {\cle(a)} \\
-    & & \bnfor & \Failure \\
-    & & \bnfor & \Switch a {\Fam {i \in I} {\pi_i, D_i}} \Dfb \\
-    & & \bnfor & \Guard {\cle(a)} {D_0} {D_1} \\
-    & & \bnfor & Unreachable \\
-    \text{\emph{accessors}} & a
-    & \bnfeq & \Root \;\bnfor\; a.n \quad (n \in \mathbb{N}) \\
-  \end{array}
-  \quad
-  \begin{array}{l}
-  \pi_S : \text{datatype constructors}
-  \\
-  \pi_T \subseteq        \{ \Int n \mid n \in \mathbb{Z} \}
-                  \uplus \{ \Tag n \mid n \in \mathbb{N} \}
-  \\[1em]
-  a(v_S), a(v_T), D_S(v_S), D_T(v_T) \quad (\text{\emph{omitted})}
-  \end{array}
-\end{mathpar}
-The tree $\Leaf \cle$ returns a leaf expression $e$ in a captured
-environment $\sigma$ mapping variables to accessors.
-%
-$\Failure$ expresses match failure, when no clause matches the input
-value.
-%
-$\Switch a {\Fam {i \in I} {\pi_i, D_i}} \Dfb$ has one subtree $D_i$
-for every head constructor that appears in the pattern matching
-clauses, and a fallback case for the constructors.
-%
-$\Guard \cle {D_0} {D_1}$ represents a \texttt{when}-guard on a closed
-expression $\cle$, expected to be of boolean type, with sub-trees
-$D_0$ for the \texttt{true} case and $D_1$ for the \texttt{false}
-case.
-
-We write $a(v)$ for the sub-value of the (source or target) value $v$
-that is reachable at the accessor $a$, and $D(v)$ for the result of
-running a value $v$ against a decision tree $D$; this results in
-a (source or target) matching run $R(v)$, just like running the value
-against a program.
 
 \subsection{From source programs to decision trees}
 Let's consider some trivial examples: