+\subsection{Bidirectional type checking}
+
+We start by describing bidirectional type checking since it calls for fairly
+different typing rules that what we have seen up to now. The idea is to have
+two kind of terms: terms for which a type can always be inferred, and terms
+that need to be checked against a type. A nice observation is that this
+duality runs through the semantics of the terms: data destructors (function
+application, record projections, primitive re cursors) \emph{infer} types,
+while data constructors (abstractions, record/data types data constructors)
+need to be checked. In the literature these terms are respectively known as
+
+To introduce the concept and notation, we will revisit the STLC in a
+bidirectional style. The presentation follows \cite{Loh2010}.
+
+% TODO do this --- is it even necessary
+
+% \subsubsection{Declarations and contexts}
+
+% A \mykant declaration can be one of 4 kinds:
+
+% \begin{description}
+% \item[Value] A declared variable, together with a type and a body.
+% \item[Postulate] An abstract variable, with a type but no body.
+% \item[Inductive data] A datatype, with a type constructor and various data
+% constructors---somewhat similar to what we find in Haskell. A primitive
+% recursor (or `destructor') will be generated automatically.
+% \item[Record] A record, which consists of one data constructor and various
+% fields, with no recursive occurrences. We will explain the need for records
+% later.
+% \end{description}
+
+% The syntax of
+
+\subsection{Base terms and types}
+
+Let us begin by describing the primitives available without the user defining
+any data types, and without equality. The syntax given here is the one of the
+core (`desugared') terms, and the way we handle variables and substitution is
+left unspecified, and explained in section \ref{sec:term-repr}, along with
+other implementation issues. We are also going to give an account of the
+implicit type hierarchy separately in section \ref{sec:term-hierarchy}, so as
+not to clutter derivation rules too much, and just treat types as
+impredicative for the time being.
+
+\mydesc{syntax}{ }{
+ $
+ \begin{array}{r@{\ }c@{\ }l}
+ \mytmsyn & ::= & \mynamesyn \mysynsep \mytyp \\
+ & | & \myfora{\myb{x}}{\mytmsyn}{\mytmsyn} \mysynsep
+ \myabss{\myb{x}}{\mytmsyn}{\mytmsyn} \mysynsep
+ (\myapp{\mytmsyn}{\mytmsyn}) \mysynsep
+ (\myann{\mytmsyn}{\mytmsyn}) \\
+ \mynamesyn & ::= & \myb{x} \mysynsep \myfun{f}
+ \end{array}
+ $
+}
+
+The syntax for our calculus includes just two basic constructs: abstractions
+and $\mytyp$s. Everything else will be provided by user-definable constructs.
+Since we let the user define values, we will need a context capable of
+carrying the body of variables along with their type. We also want to make
+sure not to have duplicate top names, so we enforce that.
+
+% \mytyc{D} \mysynsep \mytyc{D}.\mydc{c}
+% \mysynsep \mytyc{D}.\myfun{f} \mysynsep
+
+\mydesc{context validity:}{\myvalid{\myctx}}{
+ \centering{
+ \begin{tabular}{ccc}
+ \AxiomC{\phantom{$\myjud{\mytya}{\mytyp_l}$}}
+ \UnaryInfC{$\myvalid{\myemptyctx}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myjud{\mytya}{\mytyp}$}
+ \AxiomC{$\mynamesyn \not\in \myctx$}
+ \BinaryInfC{$\myvalid{\myctx ; \mynamesyn : \mytya}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myjud{\mytmt}{\mytya}$}
+ \AxiomC{$\myfun{f} \not\in \myctx$}
+ \BinaryInfC{$\myvalid{\myctx ; \myfun{f} \mapsto \mytmt : \mytya}$}
+ \DisplayProof
+ \end{tabular}
+ }
+}
+
+Now we can present the reduction rules, which are unsurprising. We have the
+usual functional application ($\beta$-reduction), but also a rule to replace
+names with their bodies, if in the context ($\delta$-reduction), and one to
+discard type annotations. For this reason the new reduction rules are
+dependent on the context:
+
+\mydesc{reduction:}{\myctx \vdash \mytmsyn \myred \mytmsyn}{
+ \centering{
+ \begin{tabular}{ccc}
+ \AxiomC{\phantom{$\myb{x} \mapsto \mytmt : \mytya \in \myctx$}}
+ \UnaryInfC{$\myctx \vdash \myapp{(\myabs{\myb{x}}{\mytmm})}{\mytmn}
+ \myred \mysub{\mytmm}{\myb{x}}{\mytmn}$}
+ \DisplayProof
+ &
+ \AxiomC{$\myfun{f} \mapsto \mytmt : \mytya \in \myctx$}
+ \UnaryInfC{$\myctx \vdash \myfun{f} \myred \mytmt$}
+ \DisplayProof
+ &
+ \AxiomC{\phantom{$\myb{x} \mapsto \mytmt : \mytya \in \myctx$}}
+ \UnaryInfC{$\myctx \vdash \myann{\mytmm}{\mytya} \myred \mytmm$}
+ \DisplayProof
+ \end{tabular}
+ }
+}
+
+We want to define a \emph{weak head normal form} (WHNF) for our terms, to give
+a syntax directed presentation of type rules with no `conversion' rule. We
+will consider all \emph{canonical} forms (abstractions and data constructors)
+to be in weak head normal form... % TODO finish
+
+We can now give types to our terms. Using our definition of WHNF, I will use
+$\mytmm \mynf \mytmn$ to indicate that $\mytmm$'s normal form is $\mytmn$.
+This way, we can avoid the non syntax-directed conversion rule, giving a more
+algorithmic presentation of type checking.
+
+\mydesc{typing:}{\myctx \vdash \mytmsyn \Leftrightarrow \mytysyn}{
+ \centering{
+ \begin{tabular}{ccc}
+ \AxiomC{$\myb{x} : A \in \myctx$ or $\myb{x} \mapsto \mytmt : A \in \myctx$}
+ \UnaryInfC{$\myinf{\myb{x}}{A}$}
+ \DisplayProof
+ &
+ \AxiomC{$\mychk{\mytmt}{\mytya}$}
+ \UnaryInfC{$\myinf{\myann{\mytmt}{\mytya}}{\mytya}$}
+ \DisplayProof
+ \end{tabular}
+ \myderivsp
+
+ \AxiomC{$\myinf{\mytmm}{\mytya}$}
+ \AxiomC{$\myctx \vdash \mytya \mynf \myfora{\myb{x}}{\mytyb}{\myse{C}}$}
+ \AxiomC{$\mychk{\mytmn}{\mytyb}$}
+ \TrinaryInfC{$\myinf{\myapp{\mytmm}{\mytmn}}{\mysub{\myse{C}}{\myb{x}}{\mytmn}}$}
+ \DisplayProof
+
+ \myderivsp
+
+ \AxiomC{$\myctx \vdash \mytya \mynf \myfora{\myb{x}}{\mytyb}{\myse{C}}$}
+ \AxiomC{$\mychkk{\myctx; \myb{x}: \mytyb}{\mytmt}{\myse{C}}$}
+ \BinaryInfC{$\mychk{\myabs{\myb{x}}{\mytmt}}{\mytya}$}
+ \DisplayProof
+ }
+}
+
+\subsection{Elaboration}
+
+\mydesc{syntax}{ }{
+ $
+ \begin{array}{r@{\ }c@{\ }l}
+ \mydeclsyn & ::= & \myval{\myb{x}}{\mytmsyn}{\mytmsyn} \\
+ & | & \mypost{\myb{x}}{\mytmsyn} \\
+ & | & \myadt{\mytyc{D}}{\mytelesyn}{}{\mydc{c} : \mytelesyn\ |\ \cdots } \\
+ & | & \myreco{\mytyc{D}}{\mytelesyn}{}{\myfun{f} : \mytmsyn,\ \cdots } \\
+
+ \mytelesyn & ::= & \myemptytele \mysynsep \mytelesyn \mycc (\myb{x} {:} \mytmsyn)
+ \end{array}
+ $
+}
+
+\subsubsection{Values and postulated variables}
+
+\mydesc{context elaboration:}{\myelab{\mydeclsyn}{\myctx}}{
+ \centering{
+ \begin{tabular}{cc}
+ \AxiomC{$\myjud{\mytmt}{\mytya}$}
+ \AxiomC{$\myfun{f} \not\in \myctx$}
+ \BinaryInfC{
+ $\myctx \myelabt \myval{\myfun{f}}{\mytya}{\mytmt} \ \ \myelabf\ \ \myctx; \myfun{f} \mapsto \mytmt : \mytya$
+ }
+ \DisplayProof
+ &
+ \AxiomC{$\myjud{\mytya}{\mytyp}$}
+ \AxiomC{$\myfun{f} \not\in \myctx$}
+ \BinaryInfC{
+ $
+ \myctx \myelabt \mypost{\myfun{f}}{\mytya}
+ \ \ \myelabf\ \ \myctx; \myfun{f} : \mytya
+ $
+ }
+ \DisplayProof
+ \end{tabular}
+}
+}
+
+\subsubsection{User defined types}
+
+\mydesc{syntax}{ }{
+ $
+ \begin{array}{l}
+ \mynamesyn ::= \cdots \mysynsep \mytyc{D} \mysynsep \mytyc{D}.\mydc{c} \mysynsep \mytyc{D}.\myfun{f}
+ \end{array}
+ $
+}
+
+\subsubsection{Data types}
+
+\begin{figure}[t]
+ \mydesc{syntax elaboration:}{\myelab{\mydeclsyn}{\mytmsyn ::= \cdots}}{
+ \centering{
+ $
+ \begin{array}{r@{\ }c@{\ }l}
+ \myctx & \myelabt & \myadt{\mytyc{D}}{\mytele}{}{\cdots\ |\ \mydc{c}_n : \myvec{(\myb{x} {:} \mytya)} \ |\ \cdots } \\
+ & \myelabf &
+
+ \begin{array}{r@{\ }c@{\ }l}
+ \mytmsyn & ::= & \cdots \mysynsep \myapp{\mytyc{D}}{\myvec{\mytmsyn}} \mysynsep
+ \mytyc{D}.\mydc{c}_n \myappsp \myvec{\mytmsyn} \mysynsep \cdots \mysynsep \mytyc{D}.\myfun{elim} \myappsp \mytmsyn \\
+ \end{array}
+ \end{array}
+ $
+ }
+ }
+
+ \mydesc{context elaboration:}{\myelab{\mydeclsyn}{\myctx}}{
+ \centering{
+ \AxiomC{$\myinf{\mytele \myarr \mytyp}{\mytyp}$}
+ \AxiomC{$\mytyc{D} \not\in \myctx$}
+ \noLine
+ \BinaryInfC{$\myinff{\myctx;\ \mytyc{D} : \mytele \myarr \mytyp}{\mytele \mycc \mytele_i \myarr \myapp{\mytyc{D}}{\mytelee}}{\mytyp}\ \ \ (1 \leq i \leq n)$}
+ \noLine
+ \UnaryInfC{For each $(\myb{x} {:} \mytya)$ in each $\mytele_i$, if $\mytyc{D} \in \mytya$, then $\mytya = \myapp{\mytyc{D}}{\vec{\mytmt}}$.}
+ \UnaryInfC{$
+ \begin{array}{r@{\ }c@{\ }l}
+ \myctx & \myelabt & \myadt{\mytyc{D}}{\mytele}{}{ \mydc{c} : \mytele_1 \ |\ \cdots \ |\ \mydc{c}_n : \mytele_n } \\
+ & & \vspace{-0.2cm} \\
+ & \myelabf & \myctx;\ \mytyc{D} : \mytele \mycc \mytyp;\ \mytyc{D}.\mydc{c}_1 : \mytele \mycc \mytele_1 \myarr \myapp{\mytyc{D}}{\mytelee};\ \cdots;\ \mytyc{D}.\mydc{c}_n : \mytele \mycc \mytele_n \myarr \myapp{\mytyc{D}}{\mytelee}; \\
+ & &
+ \begin{array}{@{}r@{\ }l l}
+ \mytyc{D}.\myfun{elim} : & \mytele \myarr (\myb{x} {:} \myapp{\mytyc{D}}{\mytelee}) \myarr & \textbf{target} \\
+ & (\myb{P} {:} \myapp{\mytyc{D}}{\mytelee} \myarr \mytyp) \myarr & \textbf{motive} \\
+ & \left.
+ \begin{array}{@{}l}
+ (\mytele_1 \mycc \myhyps(\myb{P}, \mytele_1) \myarr \myapp{\myb{P}}{(\myapp{\mytyc{D}.\mydc{c}_1}{\mytelee_1})}) \myarr \\
+ \myind{3} \vdots \\
+ (\mytele_n \mycc \myhyps(\myb{P}, \mytele_n) \myarr \myapp{\myb{P}}{(\myapp{\mytyc{D}.\mydc{c}_n}{\mytelee_n})}) \myarr
+ \end{array} \right \}
+ & \textbf{methods} \\
+ & \myapp{\myb{P}}{\myb{x}} &
+ \end{array} \\
+ \\
+ \multicolumn{3}{l}{
+ \begin{array}{@{}l l@{\ } l@{} r c l}
+ \textbf{where} & \myhyps(\myb{P}, & \myemptytele &) & \mymetagoes & \myemptytele \\
+ & \myhyps(\myb{P}, & (\myb{r} {:} \myapp{\mytyc{D}}{\vec{\mytmt}}) \mycc \mytele &) & \mymetagoes & (\myb{r'} {:} \myapp{\myb{P}}{\myb{r}}) \mycc \myhyps(\myb{P}, \mytele) \\
+ & \myhyps(\myb{P}, & (\myb{x} {:} \mytya) \mycc \mytele & ) & \mymetagoes & \myhyps(\myb{P}, \mytele)
+ \end{array}
+ }
+ \end{array}
+ $}
+ \DisplayProof
+ }
+ }
+
+ \mydesc{reduction elaboration:}{\myctx \vdash \mytmsyn \myred \mytmsyn}{
+ \centering{
+ \AxiomC{$\mytyc{D} : \mytele \myarr \mytyp \in \myctx$}
+ \AxiomC{$\mytyc{D}.\mydc{c}_i : \mytele;\mytele_i \myarr \myapp{\mytyc{D}}{\mytelee} \in \myctx$}
+ \BinaryInfC{$
+ \begin{array}{c}
+ \myctx \vdash \myapp{\myapp{\myapp{\mytyc{D}.\myfun{elim}}{(\myapp{\mytyc{D}.\mydc{c}_i}{\vec{\myse{t}}})}}{\myse{P}}}{\vec{\myse{m}}} \myred \myapp{\myapp{\myse{m}_i}{\vec{\mytmt}}}{\myrecs(\myse{P}, \vec{m}, \mytele_i)} \\ \\
+ \begin{array}{@{}l l@{\ } l@{} r c l}
+ \textbf{where} & \myrecs(\myse{P}, \vec{m}, & \myemptytele &) & \mymetagoes & \myemptytele \\
+ & \myrecs(\myse{P}, \vec{m}, & (\myb{r} {:} \myapp{\mytyc{D}}{\vec{t}}); \mytele & ) & \mymetagoes & (\mytyc{D}.\myfun{elim} \myappsp \myb{r} \myappsp \myse{P} \myappsp \vec{m}); \myrecs(\myse{P}, \vec{m}, \mytele) \\
+ & \myrecs(\myse{P}, \vec{m}, & (\myb{x} {:} \mytya); \mytele &) & \mymetagoes & \myrecs(\myse{P}, \vec{m}, \mytele)
+ \end{array}
+ \end{array}
+ $}
+ \DisplayProof
+ }
+ }
+
+ \caption{Elaborations for data types.}
+ \label{fig:elab-adt}
+\end{figure}
+
+
+\subsubsection{Records}
+
+\begin{figure}[t]
+\mydesc{syntax elaboration:}{\myelab{\mydeclsyn}{\mytmsyn ::= \cdots}}{
+ \centering{
+ $
+ \begin{array}{r@{\ }c@{\ }l}
+ \myctx & \myelabt & \myadt{\mytyc{D}}{\mytele}{}{\cdots\ |\ \mydc{c}_n : \myvec{(\myb{x} {:} \mytya)} \ |\ \cdots } \\
+ & \myelabf &
+
+ \begin{array}{r@{\ }c@{\ }l}
+ \mytmsyn & ::= & \cdots \mysynsep \myapp{\mytyc{D}}{\myvec{\mytmsyn}} \mysynsep
+ \mytyc{D}.\mydc{c}_n \myappsp \myvec{\mytmsyn} \mysynsep \cdots \mysynsep \mytyc{D}.\myfun{elim} \myappsp \mytmsyn \\
+ \end{array}
+ \end{array}
+ $
+ }
+}
+
+
+\mydesc{context elaboration:}{\myelab{\mydeclsyn}{\myctx}}{
+ \centering{
+ \AxiomC{$\myinf{\mytele \myarr \mytyp}{\mytyp}$}
+ \AxiomC{$\mytyc{D} \not\in \myctx$}
+ \noLine
+ \BinaryInfC{$\myinff{\myctx; \mytele; (\myb{f}_j : \myse{F}_j)_{j=1}^{i - 1}}{F_i}{\mytyp} \myind{3} (1 \le i \le n)$}
+ \UnaryInfC{$
+ \begin{array}{r@{\ }c@{\ }l}
+ \myctx & \myelabt & \myreco{\mytyc{D}}{\mytele}{}{ \myfun{f}_1 : \myse{F}_1, \cdots, \myfun{f}_n : \myse{F}_n } \\
+ & & \vspace{-0.2cm} \\
+ & \myelabf & \myctx;\ \mytyc{D} : \mytele \myarr \mytyp;\\
+ & & \mytyc{D}.\myfun{f}_1 : \mytele \myarr \myapp{\mytyc{D}}{\mytelee} \myarr \myse{F}_1;\ \cdots;\ \mytyc{D}.\myfun{f}_n : \mytele \myarr (\myb{x} {:} \myapp{\mytyc{D}}{\mytelee}) \myarr \mysub{\myse{F}_n}{\myb{f}_i}{\myapp{\myfun{f}_i}{\myb{x}}}_{i = 1}^{n-1}; \\
+ & & \mytyc{D}.\mydc{constr} : \mytele \myarr \myse{F}_1 \myarr \cdots \myarr \myse{F}_n \myarr \myapp{\mytyc{D}}{\mytelee};
+ \end{array}
+ $}
+ \DisplayProof
+ }
+}
+
+ \mydesc{reduction elaboration:}{\myctx \vdash \mytmsyn \myred \mytmsyn}{
+ \centering{
+ \AxiomC{$\mytyc{D} \in \myctx$}
+ \UnaryInfC{$\myctx \vdash \myapp{\mytyc{D}.\myfun{f}_i}{(\mytyc{D}.\mydc{constr} \myappsp \vec{t})} \myred t_i$}
+ \DisplayProof
+ }
+ }
+
+ \caption{Elaborations for records.}
+ \label{fig:elab-adt}
+\end{figure}
+
+
+\subsection{Type hierarchy}
+\label{sec:term-hierarchy}
+
+\subsection{Defined and postulated variables}
+
+As mentioned, in \mykant\ we can defined top level variables, with or without
+a body. We call the variables
+
+\subsection{Observational equality}
+
+\section{\mykant : The practice}
+\label{sec:kant-practice}
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