Move stuff around to get rid of a big blank space

cbpv.mng

@@ -86,18 +86,7 @@ and the challenges posed by using Lean in \cref{sec:discussion}.
 
 \section{Syntax and Typing} \label{sec:syntax}
 
-\begin{align*}
+The grammar of the CBPV for this project is given below.
-  [[A]] &\Coloneqq [[Unit]] \mid [[A1 + A2]] \mid [[U B]] & \textit{value types} \\
-  [[B]] &\Coloneqq [[A → B]] \mid [[F A]] & \textit{computation types} \\
-  [[v]], [[w]] &\Coloneqq [[x]] \mid [[unit]] \mid [[inl v]] \mid [[inr v]] \mid [[thunk m]] & \textit{values} \\
-  [[m]], [[n]] &\Coloneqq [[force v]] \mid [[λx. m]] \mid [[m v]] \mid [[ret v]] & \textit{computations} \\
-  & \mid [[let x m n]] \\
-  & \mid [[case v | y.m | z.n]] \\
-  [[G]] &\Coloneqq [[•]] \mid [[G, x : A]] & \textit{typing contexts} \\
-  [[s]] &\Coloneqq [[•]] \mid [[s, x ↦ v]] & \textit{simultaneous substitutions}
-\end{align*}
-
-The grammar of the CBPV for this project is given above.
 Terms and their types are divided between \emph{values} and \emph{computations}.
 Values are variables, unit, injections into a sum type, or a thunked computation,
 while computations are forcing thunks,
@@ -115,6 +104,17 @@ with $[[•]]$ as the identity substitution.
 Applying a substitution is represented as $[[v{s}]]$ and $[[m{s}]]$,
 and implemented in terms of renamings,
 which are mappings from variables to other variables.
+%
+\begin{align*}
+  [[A]] &\Coloneqq [[Unit]] \mid [[A1 + A2]] \mid [[U B]] & \textit{value types} \\
+  [[B]] &\Coloneqq [[A → B]] \mid [[F A]] & \textit{computation types} \\
+  [[v]], [[w]] &\Coloneqq [[x]] \mid [[unit]] \mid [[inl v]] \mid [[inr v]] \mid [[thunk m]] & \textit{values} \\
+  [[m]], [[n]] &\Coloneqq [[force v]] \mid [[λx. m]] \mid [[m v]] \mid [[ret v]] & \textit{computations} \\
+  & \mid [[let x m n]] \\
+  & \mid [[case v | y.m | z.n]] \\
+  [[G]] &\Coloneqq [[•]] \mid [[G, x : A]] & \textit{typing contexts} \\
+  [[s]] &\Coloneqq [[•]] \mid [[s, x ↦ v]] & \textit{simultaneous substitutions}
+\end{align*}
 
 This is not the usual complete CBPV language,
 since it's missing both value tuples and computation tuples.

cbpv.tex

@@ -86,18 +86,7 @@ and the challenges posed by using Lean in \cref{sec:discussion}.
 
 \section{Syntax and Typing} \label{sec:syntax}
 
-\begin{align*}
+The grammar of the CBPV for this project is given below.
-  \ottnt{A} &\Coloneqq  \kw{Unit}  \mid  \ottnt{A_{{\mathrm{1}}}}  +  \ottnt{A_{{\mathrm{2}}}}  \mid  \kw{U} \gap  \ottnt{B}  & \textit{value types} \\
-  \ottnt{B} &\Coloneqq  \ottnt{A}  \rightarrow  \ottnt{B}  \mid  \kw{F} \gap  \ottnt{A}  & \textit{computation types} \\
-  \ottnt{v}, \ottnt{w} &\Coloneqq \ottmv{x} \mid  \kw{unit}  \mid  \kw{inl} \gap  \ottnt{v}  \mid  \kw{inr} \gap  \ottnt{v}  \mid  \kw{thunk} \gap  \ottnt{m}  & \textit{values} \\
-  \ottnt{m}, \ottnt{n} &\Coloneqq  \kw{force} \gap  \ottnt{v}  \mid  \lambda  \ottmv{x}  \mathpunct{.}  \ottnt{m}  \mid  \ottnt{m}  \gap  \ottnt{v}  \mid  \kw{return} \gap  \ottnt{v}  & \textit{computations} \\
-  & \mid  \kw{let} \gap  \ottmv{x}  \leftarrow  \ottnt{m}  \gap \kw{in} \gap  \ottnt{n}  \\
-  & \mid  \kw{case} \gap  \ottnt{v}  \gap \kw{of} \gap \lbrace \kw{inl} \gap  \ottmv{y}  \Rightarrow  \ottnt{m}  \mathrel{;} \kw{inr} \gap  \ottmv{z}  \Rightarrow  \ottnt{n}  \rbrace  \\
-  \Gamma &\Coloneqq  \cdot  \mid  \Gamma ,  \ottmv{x}  \mathbin{:}  \ottnt{A}  & \textit{typing contexts} \\
-  \sigma &\Coloneqq  \cdot  \mid  \sigma ,  \ottmv{x}  \mapsto  \ottnt{v}  & \textit{simultaneous substitutions}
-\end{align*}
-
-The grammar of the CBPV for this project is given above.
 Terms and their types are divided between \emph{values} and \emph{computations}.
 Values are variables, unit, injections into a sum type, or a thunked computation,
 while computations are forcing thunks,
@@ -115,6 +104,17 @@ with $ \cdot $ as the identity substitution.
 Applying a substitution is represented as $ \ottnt{v} [  \sigma  ] $ and $ \ottnt{m} [  \sigma  ] $,
 and implemented in terms of renamings,
 which are mappings from variables to other variables.
+%
+\begin{align*}
+  \ottnt{A} &\Coloneqq  \kw{Unit}  \mid  \ottnt{A_{{\mathrm{1}}}}  +  \ottnt{A_{{\mathrm{2}}}}  \mid  \kw{U} \gap  \ottnt{B}  & \textit{value types} \\
+  \ottnt{B} &\Coloneqq  \ottnt{A}  \rightarrow  \ottnt{B}  \mid  \kw{F} \gap  \ottnt{A}  & \textit{computation types} \\
+  \ottnt{v}, \ottnt{w} &\Coloneqq \ottmv{x} \mid  \kw{unit}  \mid  \kw{inl} \gap  \ottnt{v}  \mid  \kw{inr} \gap  \ottnt{v}  \mid  \kw{thunk} \gap  \ottnt{m}  & \textit{values} \\
+  \ottnt{m}, \ottnt{n} &\Coloneqq  \kw{force} \gap  \ottnt{v}  \mid  \lambda  \ottmv{x}  \mathpunct{.}  \ottnt{m}  \mid  \ottnt{m}  \gap  \ottnt{v}  \mid  \kw{return} \gap  \ottnt{v}  & \textit{computations} \\
+  & \mid  \kw{let} \gap  \ottmv{x}  \leftarrow  \ottnt{m}  \gap \kw{in} \gap  \ottnt{n}  \\
+  & \mid  \kw{case} \gap  \ottnt{v}  \gap \kw{of} \gap \lbrace \kw{inl} \gap  \ottmv{y}  \Rightarrow  \ottnt{m}  \mathrel{;} \kw{inr} \gap  \ottmv{z}  \Rightarrow  \ottnt{n}  \rbrace  \\
+  \Gamma &\Coloneqq  \cdot  \mid  \Gamma ,  \ottmv{x}  \mathbin{:}  \ottnt{A}  & \textit{typing contexts} \\
+  \sigma &\Coloneqq  \cdot  \mid  \sigma ,  \ottmv{x}  \mapsto  \ottnt{v}  & \textit{simultaneous substitutions}
+\end{align*}
 
 This is not the usual complete CBPV language,
 since it's missing both value tuples and computation tuples.