\documentclass[10pt, sigplan, nonacm]{acmart}

\usepackage[utf8]{inputenc}
\usepackage{formal-grammar}
\usepackage[dvipsnames]{xcolor}
\usepackage{mathpartir}
\usepackage{hyperref}
\usepackage{url}
\usepackage{stmaryrd}
\usepackage{minted}
\usemintedstyle{tango}

\DeclareUnicodeCharacter{2200}{$\forall$}
\DeclareUnicodeCharacter{03C3}{$\sigma$}
\DeclareUnicodeCharacter{03C4}{$\tau$}
\DeclareUnicodeCharacter{03BB}{$\lambda$}
\DeclareUnicodeCharacter{21A6}{$\mapsto$}
\DeclareUnicodeCharacter{039B}{$\Lambda$}
\DeclareUnicodeCharacter{03B1}{$\alpha$}
\DeclareUnicodeCharacter{03B2}{$\beta$}
\DeclareUnicodeCharacter{03B3}{$\gamma$}
\DeclareUnicodeCharacter{03B4}{$\delta$}
\DeclareUnicodeCharacter{0393}{$\Gamma$}

\newcommand{\metavariable}[1]{\textcolor{teal}{#1}}
\newcommand{\typeterminal}[1]{\textcolor{brown}{#1}}
\newcommand{\exprterminal}[1]{\textcolor{Sepia}{#1}}

\newcommand{\seltype}[0]{ {\textsc{\footnotesize{Typ}}} }
\newcommand{\selexpr}[0]{ {\textsc{\footnotesize{Exp}}} }
\newcommand{\selval}[0]{ {\textsc{\footnotesize{Val}}} }

\newcommand{\exprvars}[0]{ V }
\newcommand{\typevars}[0]{ \Upsilon }
\newcommand{\typenames}[0]{ \Sigma }

\newcommand{\typenonterm}[1]{ \nonterm{ T_\seltype \footnotesize{\textsf{$(\typenames, #1)$}}}}
\newcommand{\exprnonterm}[2]{ \nonterm{ T_\selexpr \footnotesize{\textsf{$(\typenames, #1, #2)$}}}}
\newcommand{\valnonterm}[2]{ \nonterm{ T_\selval \footnotesize{\textsf{$(\typenames, #1, #2)$}}}}


\newcommand{\todo}[1]{ {\textcolor{red}{\textbf{TODO:} #1}} }

\title{A functional core calculus with ladder-types}
\author{Michael Sippel}
\email{michael.sippel@mailbox.tu-dresden.de}

\makeatletter
\let\@authorsaddresses\@empty
\makeatother
\begin{document}


\begin{abstract}
This work explores the idea of \emph{representational polymorphism}
to treat the coexistence of multiple equivalent representational forms for a single abstract concept.


interchangeability
%Our goal is a type system to support the seamless integration of 
%which may arise by consequence of external interfaces or internal optimization.

For the study of its formalism, we extend the \emph{polymorphic lambda-calculus} by a new type-constructor,
called the \emph{ladder-type} in order to encode a \emph{represented-as} relationship into our type-terms.
Based on this extended type-structure, we first define a subtyping relation to capture
a notion of structural embedding of higher-level types into lower-level types
which is then relaxed into \emph{semantic subtyping},
where for a certain expected type, an equivalent representation implementing the same abstract type
is accepted as well. In that case, a coercion is inserted implicitly to transform the underlying datastructure
while keeping all semantical properties of the type intact.
We specify our typing-rules accordingly, give an algorithm that manifests all implicit coercions in a program
and prove its \emph{soundness}.

\end{abstract}

\maketitle
\tableofcontents


%\newpage
\section{Introduction}
While certain representational forms might be fixed already at the boundaries of an application,
internally, some other representations might be desired for reasons of simplicity and efficiency.
Further, differing complexity-profiles of certain representations might even have the potential to complement
each other and coexist in a single application.
Often however, implementations become heavily dependent on concrete data formats
and require technical knowledge of the low-level data structures.
Making use of multiple such representations additionally requires careful transformation of data.

\todo{serialization}
\todo{memory layout optimizations}
\todo{difference to traditional coercions (static cast)}
\todo{relation with inheritance based subtyping:  bottom-up vs top-down inheritance vs ladder-types}

\todo{related work: type specific languages}

In order to facilitate programming at "high-level", we introduce a type-system that is able to disambiguate
this multiplicity of representations and facilitate implicit coercions between them.
We claim this to aid in (1) forgetting details about representational details during program composition
and (2) keeping the system flexible enough to introduce representational optimizations at a later stage without
compromising semantic correctness.


\section{Core Calculus}
\subsection{Syntax}

The calculus is made up of a \emph{type-level} language, and an \emph{expression-level} language. The formal grammar for both is given in \ref{gr:core}.
In addition to \emph{function types} \(\tau_1 \rightarrow \tau_2\), \emph{universal types} \(\forall\alpha.\tau\) and type specialization \(<\tau_1 \tau_2>\),
which are already known from SystemF,
types can be of the form \(\tau_1 \sim \tau_2\) to denote a \emph{ladder type} to formalizes the notion of a type \(\tau_1\) being represented in terms of type \(\tau_2\).
Similar to SystemF, expressions can be \emph{variables}, \emph{type-abstractions}, \emph{}

Coq definitions of the abstract syntax can be found in \hyperref[coq:terms]{\texttt{terms.v}}.

\begin{figure}[h]
\label{gr:core}

\begin{grammar}

\firstcase{ T }{
	\metavariable{\sigma}
}{Base Type}

\otherform{
	\metavariable{\alpha}
}{Type Variable}

\otherform{
	\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \nonterm{T}
}{Universal Type}

\otherform{
	\typeterminal{<} \nonterm{T} \quad \nonterm{T} \typeterminal{>}
}{Specialized Type}

\otherform{
	\nonterm{T} \quad \typeterminal{\rightarrow} \quad \nonterm{T}
}{Function Type}

\otherform{
	\nonterm{T} \quad \typeterminal{\rightarrow_\text{morph}} \quad \nonterm{T}
}{Morphism Type}

\otherform{
	\nonterm{T} \quad \typeterminal{\sim} \quad \nonterm{T}
}{Ladder Type}

$$\\$$

\firstcase{ E
% T_\selexpr
}
	{ \metavariable{x}
} {Variable}

\otherform{
	\exprterminal{let} \quad \metavariable{x} \quad \exprterminal{=} \quad 
	\nonterm{ E }
	\quad \exprterminal{in} \quad
	\nonterm{ E }
}{Variable Binding}

\otherform{
	$$ \exprterminal{\Lambda} \metavariable{\alpha}
	\quad \exprterminal{\mapsto} \quad $$
	\nonterm{ E }
}{Type Abstraction}

\otherform{
	$$ \exprterminal{\lambda} \metavariable{x} $$
	\exprterminal{:} \nonterm{ T }
    \quad $$\exprterminal{\mapsto}$$ \quad
	\nonterm{ E }
}{Value Abstraction}

\otherform{
	$$ \exprterminal{\lambda} \metavariable{x} $$
	\exprterminal{:} \nonterm{ T }
    \quad $$\exprterminal{\mapsto_\text{morph}}$$ \quad
	\nonterm{ E }
}{Value Morphism}

\otherform{
	\nonterm{ E }
	\quad
	\nonterm{ T }
}{Type Application}

\otherform{
	\nonterm{ E }
	\quad
    \nonterm{ E }
}{Value Application}

\otherform{
    \nonterm{ E }
	\quad
	\exprterminal{as}
	\quad
    \nonterm{ T }
}{Ascription}

%\otherform{
%    \nonterm{ E }
%	\quad
%	\exprterminal{to}
%	\quad
%    \nonterm{ T }
%}{Transformation}
%\otherform{\exprterminal{(} \quad \nonterm{E} \quad \exprterminal{)}}{Parenthesis}


$$\\$$

\firstcase{V}{
	\nonterm{ V } \quad
	\exprterminal{as} \quad
	\nonterm{ T }
}{Ascribed Value}

\otherform{
	\exprterminal{\Lambda} \metavariable{\alpha} \quad
	\exprterminal{\mapsto} \quad
	\nonterm{ V }
}{Type-Abstraction Value}

\otherform{
	\exprterminal{\lambda} \metavariable{x}
	\exprterminal{:}
	\nonterm{ T } \quad
	\exprterminal{\mapsto} \quad
	\nonterm{ E }
}{Abstraction Value}

\end{grammar}



\caption{Syntax of the core calculus with colors for \metavariable{metavariables}, \typeterminal{type-level terminal symbols}, \exprterminal{expression-level terminal symbols}
where $\typenames, \typevars, \exprvars$ are mutually disjoint, countable sets of symbols to denote atomic type identifiers (\(\typenames\)), typevariables (\(\typevars\)), and expression variables (\(\exprvars\)).
By default, assume \(\metavariable{\sigma} \in \typenames\), \(\metavariable{\alpha} \in \typevars\) and \(\metavariable{x} \in \exprvars\).
For simplicity, we write \(\metavariable{e} \in \nonterm{E}\) to say that the term \metavariable{e} is contained in the language generated by the nonterminal \(\nonterm{E}\).
$$\\$$}

\end{figure}

\begin{example}
\label{ex:terms}
Let \(\Sigma = \{ \text{Digit}, \text{Char}, \text{Seq}, \text{UTF-8}, \mathbb{N}, PosInt \} \cup \{ 0, 1, 2, ... \}\).
The following terms are valid types over \(\Sigma\):

\begin{enumerate}
\item \typeterminal{<Seq Char>}\\ %\( \in \typenonterm{\emptyset}\)\\
"sequence of characters"
\item \typeterminal{<Seq <Digit 10> \(\sim\) Char>}\\ %\( \in \typenonterm{\emptyset}\)\\
"sequence of decimal digits, where each digit is represented as character"
\item \typeterminal{<Seq Item> \(\rightarrow \mathbb{N} \rightarrow\) Item}\\ %\( \in \typenonterm{\{Item\}}\)\\
"function that maps a sequence of items and a natural number to an item"\\
%Note: this type contains the free variable \typeterminal{Item}
\item \typeterminal{<Seq Char> \(\sim\) UTF-8 \(\rightarrow \mathbb{N} \rightarrow\) Char }\\ %\( \in \typenonterm{\emptyset}\)\\
"function that takes a sequence of chars, represented as UTF-8 string, and a natural number to return a character"
\item \typeterminal{\(\forall\) Radix . <Seq <Digit Radix> \(\sim\) Char>}\\ %\(\in \typenonterm{\emptyset} \)\\
"given the parameter \typeterminal{Radix}, a sequence of digits where each digit is represented as character"
%Note: this type-term is \emph{ground} (i.e. \(\typevars = \emptyset\)), since \typeterminal{Radix} is bound by \(\typeterminal{\forall}\)
\item \typeterminal{
	\(\forall\) SrcRadix.\\
	\(\forall\) DstRadix.\\
	\(\mathbb{N} \sim\) <PosInt SrcRadix> \(\sim\) <Seq <Digit SrcRadix> \(\sim\) Char>\\
	\(\rightarrow_{morph}\)\\
	\(\mathbb{N} \sim\) <PosInt DstRadix> \(\sim\) <Seq <Digit DstRadix> \(\sim\) Char>\\
}\\ %\(\in \typenonterm{\emptyset} \)\\
"morphism function that maps the \typeterminal{PosInt} representation of \(\typeterminal{\mathbb{N}}\) with radix \typeterminal{SrcRadix} to the \typeterminal{PosInt} representation of radix \typeterminal{DstRadix}"
\end{enumerate}
\end{example}


\begin{definition}[Substitution in Types]
Given a type-variable assignment \(\psi_t = \{ \metavariable{\alpha_1} \mapsto \metavariable{\tau_1}, \quad \metavariable{\alpha_2} \mapsto \metavariable{\tau_2}, \quad \dots \}\),
the thereby induced substitution \(\overline{\psi_t}\) replaces all \emph{free} occurences of the variables \(\metavariable{\alpha_i}\) in a type-term \(\metavariable{\xi} \in \nonterm{T}\) recursively with the type-term given by \(\psi_t(\metavariable{\alpha_i})\)
, while occurences of bound variables are left untouched.
Further, we assume that for all \(\tau_i\), all variable names are disjunct with the free variables of the term to which the substitution is applied.
Coq definition is at \hyperref[coq:subst-type]{subst.v:\ref{coq:subst-type}}.

\[\overline{\psi_t} \metavariable{\xi} = \begin{cases}
\metavariable{\xi} \quad \text{if } \metavariable{\xi} \in \typenames\\
\metavariable{\xi} \quad \text{if }  \metavariable{\xi} \in \{\metavariable{\alpha_1}, \metavariable{\alpha_2}, \dots\} \text{ and } \metavariable{\xi} \notin \text{Dom}(\psi_t)\\
\metavariable{\tau} \quad \text{if }  \metavariable{\xi} \in \{\metavariable{\alpha_1}, \metavariable{\alpha_2}, \dots\} \text{ and } (\metavariable{\xi} \mapsto \metavariable{\tau}) \in \psi_t\\
\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \overline{\psi_t}\metavariable{\xi'} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{\forall}  \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \text{ and } \metavariable{\alpha} \notin \text{Dom}(\psi_t)\\
\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \overline{\psi_t \setminus \{\metavariable{\alpha} \mapsto \metavariable{\tau}\}}\metavariable{\xi'} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{\forall}  \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \text{ and } \{\metavariable{\alpha} \mapsto \metavariable{\tau}\} \in \psi_t\\
\typeterminal{<} (\overline{\psi_t} \metavariable{\xi_1}) \quad (\overline{\psi_t} \metavariable{\xi_2}) \typeterminal{>} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{<} \metavariable{\xi_1} \quad \metavariable{\xi_2} \typeterminal{>}\\
(\overline{\psi_t} \metavariable{\xi_1}) \typeterminal{\rightarrow} (\overline{\psi_t} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\rightarrow} \metavariable{\xi_2}\\
(\overline{\psi_t} \metavariable{\xi_1}) \typeterminal{\rightarrow_{morph}} (\overline{\psi_t} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\rightarrow_{morph}} \metavariable{\xi_2}\\
(\overline{\psi_t} \metavariable{\xi_1}) \typeterminal{\sim} (\overline{\psi_t} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\sim} \metavariable{\xi_2}\\
\end{cases}\]

\[\overline{\psi_t} \metavariable{e} = \begin{cases}
\metavariable{e} \quad \text{ if } \metavariable{e} \text{ is a variable}
\\

\exprterminal{\text{let } \metavariable{x} = }\overline{\psi_t}\metavariable{a} \exprterminal{\text{ in }} \overline{\psi_t}\metavariable{e'}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{\text{let } \metavariable{x} = \metavariable{a} \text{ in } \metavariable{e'}}
\\

\exprterminal{\Lambda \metavariable{\alpha} \mapsto} \overline{\psi_t}\metavariable{e'} 
\quad \text{ if \metavariable{e} is of the form } \exprterminal{\Lambda \metavariable{\alpha} \mapsto \metavariable{e'}}
\\

\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto} \overline{\psi_t}\metavariable{e'} 
\quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto \metavariable{e'}}
\\

\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto_\text{morph}} \overline{\psi_t}\metavariable{e'} 
\quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto_\text{morph} \metavariable{e'}}
\\

\overline{\psi_t} \metavariable{e'} \overline{\psi_t}\metavariable{\tau}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{( \metavariable{e'} \metavariable{\tau} )}
\\

\overline{\psi_t} \metavariable{e_1} \overline{\psi_t} \metavariable{e_2}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{(\metavariable{e_1} \metavariable{e_2})}
\\

\overline{\psi_t} \metavariable{e'} \exprterminal{\text{ as }} \overline{\psi_t}\metavariable{\tau}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{ \metavariable{e'} \text{ as } \metavariable{\tau} }

\end{cases}\]


\end{definition}





\begin{definition}[Substitution in Expressions]
Given an expression-variable assignment \(\psi_e = \{ \metavariable{x_1} \mapsto \metavariable{t_1}, \quad \metavariable{x_2} \mapsto \metavariable{t_2}, \quad \dots \}\),
the thereby induced substitution \(\overline{\psi_e}\) replaces all \emph{free} occurences of the expression variables \(\metavariable{x_i}\)
in an expression \(e \in \nonterm{E} \) with the \(\psi_e(\metavariable{x_i})\)

\[\overline{\psi_e} \metavariable{e} = \begin{cases}

\metavariable{e} \quad \text{if } \metavariable{e} \in \exprvars \text{ and } \metavariable{e} \notin \text{Dom}(\psi_e)\\
\metavariable{t} \quad \text{if } \metavariable{e} \in \exprvars \text{ and } (\metavariable{e}\mapsto\metavariable{t}) \in \psi_e\\

\exprterminal{\text{let } \metavariable{x} \text{ = }} \overline{\psi_e}\metavariable{a} \exprterminal{\text{ in }} \overline{\psi_e}\metavariable{e'}
\quad \text{if }
\\

\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto }  \overline{\psi_e} \metavariable{e'}
\quad \text{if \metavariable{e} is of the form }
\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto \metavariable{e'}}
\text{ and } \metavariable{x} \notin \text{Dom}(\psi_e)
\\

\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto }  \overline{\psi_e \setminus \{\metavariable{x} \mapsto \metavariable{t}\}} \metavariable{e'}
\quad \text{if \metavariable{e} is of the form }
\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto \metavariable{e'}}
\text{ and } (\metavariable{x}\mapsto\metavariable{t}) \in \psi_e
\\


\end{cases}\]

\end{definition}

\subsection{Typing}

\subsubsection{Equivalence of Type Terms}

%We want distributivity of ladders over type-specialization as well as over function/morphism types.
\begin{definition}[Distributivity in Types]

\begin{mathpar}
\typeterminal{< \metavariable{\sigma}\sim\metavariable{\sigma'} \quad \metavariable{\tau} >}
\rightarrow_\text{distribute}
\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau} > \sim < \metavariable{\sigma'} \quad \metavariable{\tau} > }

\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau}\sim\metavariable{\tau'} >}
\rightarrow_\text{distribute}
\typeterminal{< \metavariable{\sigma} \quad \metavariable{\tau} > \sim < \metavariable{\sigma} \quad \metavariable{\tau'} > }

\typeterminal{ \metavariable{\sigma}\sim\metavariable{\sigma'} \rightarrow \metavariable{\tau} }
\rightarrow_\text{distribute}
\typeterminal{ (\metavariable{\sigma} \rightarrow \metavariable{\tau} ) \sim ( \metavariable{\sigma'} \rightarrow \metavariable{\tau} ) }

\typeterminal{ \metavariable{\sigma} \rightarrow \metavariable{\tau}\sim\metavariable{\tau'} }
\rightarrow_\text{distribute}
\typeterminal{ (\metavariable{\sigma} \rightarrow \metavariable{\tau} ) \sim ( \metavariable{\sigma} \rightarrow \metavariable{\tau'} ) }

\typeterminal{ \metavariable{\sigma}\sim\metavariable{\sigma'} \rightarrow_\text{morph} \metavariable{\tau} }
\rightarrow_\text{distribute}
\typeterminal{ (\metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau} ) \sim ( \metavariable{\sigma'} \rightarrow_\text{morph} \metavariable{\tau} ) }

\typeterminal{ \metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau}\sim\metavariable{\tau'} }
\rightarrow_\text{distribute}
\typeterminal{ (\metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau} ) \sim ( \metavariable{\sigma} \rightarrow_\text{morph} \metavariable{\tau'} ) }

\end{mathpar}


Let \(\rightarrow_\text{condense}\) be the inverse to \(\rightarrow_\text{distribute}\).

See \hyperref[coq:type-dist]{equiv.v:\ref{coq:type-dist}}.
\end{definition}

\begin{definition}[Alpha Conversion in Types]
\begin{mathpar}
\typeterminal{\forall \metavariable{\alpha} . \metavariable{\tau}}
\rightarrow_{\alpha}
\typeterminal{\forall \metavariable{\alpha'} . } \{ \metavariable{\alpha} \mapsto \metavariable{\alpha'} \} \metavariable{\tau}

\end{mathpar}
\end{definition}

\begin{definition}[Equivalence Relation]
Transitive closure over \(\rightarrow_\text{distribute}\), \(\rightarrow_\text{condense}\) and \(\rightarrow_\alpha\).


\begin{mathpar}

\inferrule[E-Refl]{
	\metavariable{\tau} \in \nonterm{T}
}{
	\metavariable{\tau} \equiv \metavariable{\tau}
}\and
\inferrule[E-Trans]{
	\metavariable{\tau_1} \equiv \metavariable{\tau_2}\\
	\metavariable{\tau_2} \equiv \metavariable{\tau_3}
}{
	\metavariable{\tau_1} \equiv \metavariable{\tau_3}
}

\inferrule[E-Rename]{
	\metavariable{\tau_1} \rightarrow_\alpha \metavariable{\tau_2}
}{
	\metavariable{\tau_1} \equiv \metavariable{\tau_2}
}

\inferrule[E-Distribute]{
	\metavariable{\tau_1} \rightarrow_\text{distribute} \metavariable{\tau_2}
}{
	\metavariable{\tau_1} \equiv \metavariable{\tau_2}
}\and
\inferrule[E-Condense]{
	\metavariable{\tau_1} \rightarrow_\text{condense} \metavariable{\tau_2}
}{
	\metavariable{\tau_1} \equiv \metavariable{\tau_2}
}

\end{mathpar}

See \hyperref[coq:type-equiv]{equiv.v:\ref{coq:type-equiv}}.
\end{definition}


\begin{lemma}[Symmetry of \(\equiv\)]

\begin{mathpar}
\inferrule[E-Symm]{
	\metavariable{\tau_1} \equiv \metavariable{\tau_2}
}{
	\metavariable{\tau_2} \equiv \metavariable{\tau_1}
}
\end{mathpar}

\begin{proof}
\(\rightarrow_{distribute}\) is the inverse of \(\rightarrow_{condense}\) and \(\rightarrow_{\alpha}\) is symmetric by itself.
\end{proof}

\end{lemma}

\subsubsection{Normal Forms}

\begin{definition}[Ladder Normal Form]
LNF is reached by exhaustive application of \(\rightarrow_\text{distribute}\).
\end{definition}

\begin{definition}[Parameter Normal Form]
PNF reached by exhaustive application of \(\rightarrow_\text{condense}\).
\end{definition}

\subsubsection{Subtype Relations}

We define two relations: first the syntatic subtype relation \(\leq\) and second the semantic subtype relation \(\precsim\).

\begin{definition}[Syntactic Subtype (\(\tau_1\leq\tau_2\))]
\begin{mathpar}
\inferrule[S-Refl]{	
	\metavariable{\tau} \equiv \metavariable{\tau'}
}{
	\metavariable{\tau} \leq \metavariable{\tau'}
}

\inferrule[S-Syntactic]{
	\metavariable{\sigma} \leq \metavariable{\tau}
}{
	\metavariable{\sigma'} \typeterminal{\sim} \metavariable{\sigma} \leq \metavariable{\tau}
}
\end{mathpar}
\end{definition}

\begin{definition}[Semantic Subtype (\(\tau_1\precsim\tau_2\))]
\begin{mathpar}

\inferrule[S-Refl]{	
	\metavariable{\tau} \equiv \metavariable{\tau'}
}{
	\metavariable{\tau} \precsim \metavariable{\tau'}
}

\inferrule[S-Syntactic]{
	\metavariable{\sigma} \precsim \metavariable{\tau}
}{
	\metavariable{\sigma'} \typeterminal{\sim} \metavariable{\sigma} \precsim \metavariable{\tau}
}

\inferrule[S-Semantic]{
	\metavariable{\sigma} \equiv \metavariable{\tau}
}{
	\metavariable{\sigma} \typeterminal{\sim} \metavariable{\sigma'} \precsim \metavariable{\tau} \typeterminal{\sim} \metavariable{\tau'}
}


\end{mathpar}
\end{definition}


\begin{example}[Syntactic \& Semantic Subtypes]$\\$

\begin{enumerate}
\item \typeterminal{ <Digit 10> \(\sim\) Char \( \quad \leq \quad \) Char }\\
.. is a \emph{syntactic subtype}, because the representation of \typeterminal{<Digit 10>} is \emph{embedded} into \typeterminal{Char}.\\

\item \typeterminal{ <Digit 10> \(\sim\) Char \( \quad \precsim \quad \) <Digit 10> \(\sim\) machine.UInt64}\\
.. is a \emph{semantic subtype}, because the \typeterminal{Char} based representation can be transformed into a representation based on \typeterminal{machine.UInt64},
while preserving its semantics.
\end{enumerate}
\end{example}

\subsubsection{Inference of Expression Types}

As usual, the typing-context \(\Gamma = \{ \metavariable{x_1} : \metavariable{\tau_1} , \quad \metavariable{x_2} : \metavariable{\tau_2} , \quad \ldots \}\)
is a finite mapping which assigns variables \(\metavariable{x_i} \in \exprvars\) to types \(\metavariable{\tau_i} \in \nonterm{T}\).
Using the inference rules given in \ref{def:typerules}, further typing-judgements
of the form
\begin{itemize}
\item \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)" and
\item \(\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is compatible with type \(\metavariable{\tau}\)"
\end{itemize}
can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \nonterm{E}\) and \(\metavariable{\tau} \in \nonterm{T}\).



\begin{definition}[Syntactic Well-Typedness]
An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{syntactically well-typed} if there exists a type \(\metavariable{\tau} \in \nonterm{T}\),
such that \( \emptyset \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}.
\end{definition}

\begin{definition}[Semantic Well-Typedness]
An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{semantically well-typed} if there exists a type \(\metavariable{\tau} \in \nonterm{T}\),
such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) by \ref{def:typerules} and \ref{def:semtyperules}.
\end{definition}


\begin{definition}[Syntactic Typing Relation]

\label{def:typerules}
\begin{mathpar}

	\inferrule[T-Variable]{
%		\metavariable{x} \in \exprvars\\
%		\metavariable{\tau} \in \nonterm{T}\\
		\metavariable{x}:\metavariable{\tau} \in \Gamma\\
	}{
		\Gamma \vdash \metavariable{x}:\metavariable{\tau}
	}\and

	\inferrule[T-LetBinding]{
		\Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\
		\Gamma , \metavariable{x}:\metavariable{\sigma} \vdash \metavariable{t} : \metavariable{\tau}
	}{
		\Gamma \vdash (\exprterminal{\text{let }}\metavariable{x}\exprterminal{\text{ = }}\metavariable{e}\exprterminal{\text{ in }} \metavariable{t}) : \metavariable{\tau}
	}

	\inferrule[T-TypeAbs]{
%		\metavariable{\tau} \in \nonterm{T} \\
%		\metavariable{e} \in \nonterm{E} \\
		\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
	}{
        \Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau}
	}

	\inferrule[T-TypeApp]{
%		\metavariable{\tau} \in \nonterm{T} \\
		\Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\
		\metavariable{\sigma} \in \nonterm{T}
	}{
		\Gamma \vdash ( \metavariable{e} \quad \metavariable{\sigma} ) : \{\metavariable{\alpha} \mapsto \metavariable{\sigma}\} \metavariable{\tau}
	}


	\inferrule[T-Abs]{
%		\metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
%		\metavariable{e} \in \nonterm{E} \\
		\Gamma,\metavariable{x}:\metavariable{\sigma} \vdash \metavariable{e} : \metavariable{\tau} \\
	}{
        \Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\sigma} \exprterminal{\mapsto} \metavariable{e}) : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
	}

	\inferrule[T-App]{
		\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau} \\
		\Gamma \vdash \metavariable{a} : \metavariable{\sigma} \\
	}{
		\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
	}\and

	\inferrule[T-MorphAbs]{
%		\metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
%		\metavariable{e} \in \nonterm{E} \\
		\Gamma,\metavariable{x}:\metavariable{\tau} \vdash \metavariable{e} : \metavariable{\tau'} \\
		\metavariable{\tau} \precsim \metavariable{\tau'}
	}{
        \Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\tau} \exprterminal{\mapsto_{morph}} \metavariable{e}) : \metavariable{\tau}\typeterminal{\rightarrow_{morph}}\metavariable{\tau'}
	}\and

	\inferrule[T-MorphFun]{
        \Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
	}{
        \Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
	}\and

	\inferrule[T-Ascension]{
		\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
		\metavariable{\tau'} \leq \metavariable{\tau}
	}{
		\Gamma \vdash (\metavariable{e} \exprterminal{\text{ as }} \metavariable{\tau'}) : \metavariable{\tau'}
	}\and

	\inferrule[T-Descension]{
		\Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
		\metavariable{\tau} \leq \metavariable{\tau'}
	}{
		\Gamma \vdash \metavariable{e} : \metavariable{\tau'}
	}

\end{mathpar}
\end{definition}

\begin{definition}[Semantic Typing Relation]
\label{def:semtyperules}
\begin{mathpar}
	\inferrule[T-NativeRepr]{
		\Gamma\vdash \metavariable{e} : \metavariable{\tau}
	}{
		\Gamma\vdash \metavariable{e} :\approx \metavariable{\tau}
	}

	\inferrule[T-CoercedRepr]{
		\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau}\\
%		\metavariable{\tau} \precsim \metavariable{\tau'}\\
		%\exists \metavariable{h} \text{ s.t. }
		\Gamma \vdash \metavariable{h}: \typeterminal{\metavariable{\tau}\rightarrow_\text{morph}\metavariable{\tau'}}
	}{
		\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
	}


	\inferrule[T-CompatibleApp]{
		\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \rightarrow \metavariable{\tau}\\
		\Gamma \vdash \metavariable{a} :\approx \metavariable{\sigma}
	}{
		\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} : \metavariable{\tau}
	}
\end{mathpar}
\end{definition}

\subsection{Coercion Semantics}

We define the translation function \(\llbracket . \rrbracket\) which completes a \emph{semantically well-typed} expression
by inserting all required coercions based on the typing derivation of the expression.
The result shall be a \emph{syntactically well-typed} expression.

We write \(C :: \sigma \precsim \tau\) to mean "C is a subtyping derivation tree whose conclusion is \(\sigma \precsim \tau\)".
Similarly, we write \(D :: \Gamma \vdash e : \tau\) to mean "D is a typing derivation whose conclusion is \(\Gamma \vdash e : \tau\)"

\begin{definition}[Translation]
\begin{mathpar}

\Big{\llbracket} \inferrule[T-SemanticSubtype]{
	D_1 :: \Gamma \vdash \metavariable{h}:\metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}\\
    D_2 :: \Gamma \vdash \metavariable{e}:\metavariable{\tau}\\
%	C :: \metavariable{\tau} \precsim \metavariable{\tau'}
}{
	\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
}\Big{\rrbracket} = \exprterminal{(}
\llbracket D_1 \rrbracket \llbracket D_2 \rrbracket
%\metavariable{h} \llbracket D_2 \rrbracket
\exprterminal{)}

\Big{\llbracket} \inferrule[T-CoercedApp]{
	D_1 :: \Gamma \vdash \metavariable{f}:\metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
    D_2 :: \Gamma \vdash \metavariable{a}:\approx\metavariable{\sigma}
}{
	\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} : \metavariable{\tau}
}\Big{\rrbracket} = \exprterminal{(}
%\exprterminal{(}\metavariable{f} \llbracket D_2 \rrbracket \exprterminal{)}
\llbracket D_1 \rrbracket \llbracket D_2 \rrbracket
\exprterminal{)}

\Big{\llbracket}
\inferrule[\emph{Otherwise}]{}{
D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}
}
\Big{\rrbracket} = \metavariable{e}

\end{mathpar}
\end{definition}

\begin{lemma}[Elimination of \(:\approx\)]
\label{lemma:translation}

For all \emph{semantically well-typed} expressions \metavariable{e} with the typing derivation \(D :: \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau}\),
the translation \(\llbracket D \rrbracket = \metavariable{e'}\), yields a \emph{syntactically well-typed} expression \metavariable{e'} with
\(\emptyset \vdash \metavariable{e'} : \metavariable{\tau} \)

\begin{proof}
\todo{}
\end{proof}

\end{lemma}


\subsection{Evaluation}

Evaluation of an expression \(\metavariable{e} \in \nonterm{E}\) is defined by exhaustive application of the rewrite rule \(\rightarrow_\beta\) as in \ref{def:evalrules}.

\begin{definition}[Inference Rules for Evaluation]
\label{def:evalrules}

\begin{mathpar}

	\inferrule[E-App1]{
		\metavariable{e_1} \rightarrow_\beta \metavariable{e_1'}
	}{
		\metavariable{e_1} \metavariable{e_2}
		\rightarrow_\beta
		\metavariable{e_1'} \metavariable{e_2}
	}\and

	\inferrule[E-App2]{
		\metavariable{e_2} \rightarrow_\beta \metavariable{e_2'}
	}{
		\metavariable{v_1} \metavariable{e_2}
		\rightarrow_\beta
		\metavariable{v_1} \metavariable{e_2'}
	}\and

	\inferrule[E-TypApp]{
%		\metavariable{\tau} \in \typenonterm{\emptyset}\\
		\metavariable{e} \rightarrow_\beta \metavariable{e'}
	}{
		\metavariable{e}
		\metavariable{\tau}
		\rightarrow_\beta
		\metavariable{e'}
		\metavariable{\tau}
	}

	\newline

	\inferrule[E-TypAppLam]{
	}{
		\exprterminal{(\Lambda} \metavariable{\alpha}
		\exprterminal{\mapsto} \metavariable{e}
		\exprterminal{)}
		\metavariable{\tau}
		\rightarrow_\beta
		\{ \metavariable{\alpha} \mapsto \metavariable{\tau} \} \metavariable{e}
	}\and
	\inferrule[E-AppLam]{
	}{
		\exprterminal{(\lambda} \metavariable{x}
		\exprterminal{:} \metavariable{\sigma}
		\exprterminal{\mapsto} \metavariable{e}
		\exprterminal{)}
		\metavariable{a}
		\rightarrow_\beta
		\{ \metavariable{x} \mapsto \metavariable{a} \} \metavariable{e}
	}\and

	\inferrule[E-AppLet]{
	}{
		\exprterminal{\text{let }}\metavariable{x}
		\exprterminal{\text{ = }}\metavariable{a}
		\exprterminal{\text{ in }}\metavariable{e}
		\rightarrow_\beta
		\{ \metavariable{x} \mapsto \metavariable{a} \} \metavariable{e}
	}\and
	\inferrule[E-Ascribe]{
	}{
		\metavariable{e}
		\exprterminal{\text{ as }}
		\metavariable{\tau}
		\rightarrow_\beta
		\metavariable{e}
	}
\end{mathpar}
\end{definition}


\subsection{Soundness}

\begin{lemma}[Preservation]
\label{lemma:preservation}
Assume the expression \(\metavariable{e}\) is well typed, i.e. \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\)
for some type \(\metavariable{\tau}\).
Then forall \(\metavariable{e'}\) with \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\)
it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as well.

\begin{proof}
\todo{}
\end{proof}
\end{lemma}

\begin{lemma}[Progress]
\label{lemma:progress}
If \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\),
then either \(\metavariable{e}\) is a value
or there exists some \(\metavariable{e'}\) such that \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\)

\begin{proof}
\todo{}
\end{proof}
\end{lemma}

\begin{theorem}[Syntactic Type Soundness]
\label{theorem:syntactic-soundness}
No syntactically well-typed expression is stuck.

Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\metavariable{\tau}\).
Then it never occurs that \(\metavariable{e} \rightarrow_{\beta}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.

\begin{proof}
Follows from \ref{lemma:progress} and \ref{lemma:preservation}.
\end{proof}
\end{theorem}

\begin{theorem}[Semantic Type Soundness]
\label{theorem:semantic-soundness}
No semantically well-typed expression is stuck.

Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\).
Then it never occurs that \(\llbracket D \rrbracket \rightarrow_{\beta}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.

\begin{proof}
Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\).
By \ref{lemma:translation}, \(\emptyset \vdash \llbracket D \rrbracket : \metavariable{\tau}\)
and thus it follows by \ref{theorem:syntactic-soundness} that \metavariable{e} is not stuck.
\end{proof}
\end{theorem}



\newpage
\section{Boehm-Berarducci Encoding}


\newcommand{\typvar}[1]{\textcolor{Emerald}{#1}}
\newcommand{\expvar}[1]{\textcolor{MidnightBlue}{#1}}


\subsection{Booleans}

\begin{definition}
Let \(\Sigma = \{ \text{Bool}, \text{BB-Bool} \}\).

\begin{itemize}
\item let {\textcolor{magenta}{bb-true}} = \exprterminal{\((\Lambda\typvar{\alpha}\mapsto\lambda\expvar{t}:\typvar{\alpha}\mapsto\lambda\expvar{f}:\typvar{\alpha}\mapsto\expvar{t})\) as} \typeterminal{Bool \(\sim\) BB-Bool}
\item let {\textcolor{magenta}{bb-false}} = \exprterminal{\((\Lambda\typvar{\alpha}\mapsto\lambda\expvar{t}:\typvar{\alpha}\mapsto\lambda\expvar{f}:\typvar{\alpha}\mapsto\expvar{f})\) as} \typeterminal{Bool \(\sim\) BB-Bool}
\end{itemize}

\end{definition}

\begin{lemma}
{\textcolor{magenta}{bb-true}} and {\textcolor{magenta}{bb-false}} are \emph{values}.

\begin{proof}
\todo{}
\end{proof}
\end{lemma}

\subsection{Conditionals}

\begin{itemize}
\item let {\textcolor{magenta}{bb-if}} = \exprterminal{
	\(\Lambda\typvar{\alpha}\)\\
	\(\mapsto\lambda\) \expvar{cond} : \typeterminal{ Bool \(\sim\) BB-Bool \(\sim \forall\typvar{\beta}.(\typvar{\beta}\rightarrow\typvar{\beta}\rightarrow\typvar{\beta})\)}\\
	\(\mapsto\lambda\) \expvar{then} : \(\typvar{\alpha}\)\\
	\(\mapsto\lambda\) \expvar{else} : \(\typvar{\alpha}\)\\
	\(\mapsto\) \expvar{cond} \typvar{\(\alpha\)} \expvar{then} \expvar{else}
}
\end{itemize}

\subsection{Natural Numbers}
\begin{definition}

Let \(\Sigma = \{ \mathbb{N}, \text{BB-Nat}\}\).

\begin{itemize}
\item let {\textcolor{magenta}{bb-zero}} = \(\exprterminal{( \Lambda \typvar{\alpha} \mapsto \lambda \expvar{s} : \typeterminal{\typvar{\alpha} \rightarrow \typvar{\alpha}} \mapsto \lambda \expvar{z} : \typvar{\alpha} \mapsto \expvar{z} ) \text{ as } }\)
\typeterminal{\(\mathbb{N} \sim \text{BB-Nat} \sim \forall \typvar{\alpha} . (\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha} \)}


\item let {\textcolor{magenta}{bb-one}} = \(\exprterminal{(\Lambda \typvar{\alpha} \mapsto \lambda \expvar{s} : \typeterminal{\typvar{\alpha} \rightarrow \typvar{\alpha}} \mapsto \lambda \expvar{z} : \typvar{\alpha} \mapsto \expvar{s} \expvar{z} ) \text{ as }}\)
\typeterminal{\(\mathbb{N} \sim \text{BB-Nat} \sim \forall \typvar{\alpha} . (\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha} \)}


\item let {\textcolor{magenta}{bb-two}} = \(\exprterminal{(\Lambda \typvar{\alpha} \mapsto \lambda \expvar{s} : \typeterminal{\typvar{\alpha} \rightarrow \typvar{\alpha}} \mapsto \lambda \expvar{z} : \typvar{\alpha} \mapsto \expvar{s} (\expvar{s} \expvar{z}) ) \text{ as }}\)
\typeterminal{\(\mathbb{N} \sim \text{BB-Nat} \sim \forall \typvar{\alpha} . (\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha} \)}

\end{itemize}

\end{definition}


\begin{definition}
The successor function is given by:

let {\textcolor{magenta}{bb-succ}} =
\newline\(\exprterminal{\quad \lambda \expvar{n} :
\typeterminal{ \mathbb{N}
\sim \text{BB-Nat}
\sim \forall \typvar{\alpha} . ((\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha})}\\
}\)
\newline
\( \exprterminal{\quad \quad \mapsto ( \Lambda \typvar{\alpha} \mapsto \lambda \expvar{s} : \typeterminal{(\typvar{\alpha} \rightarrow \typvar{\alpha})} \mapsto \lambda \expvar{z} : \typvar{\alpha} }\)
\( \exprterminal{\mapsto \expvar{s} ( \expvar{n} \typvar{\alpha} \expvar{s} \expvar{z} )) \text{ as } }\)
\typeterminal{
\(\mathbb{N} \sim \text{BB-Nat} \sim \forall \typvar{\alpha} . (\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha}\)
}
\end{definition}

\begin{example}
\(\emptyset \vdash ( \textcolor{magenta}{\text{bb-succ} \quad \text{bb-one}} ) : \typeterminal{ \mathbb{N} \sim \text{BB-Nat} \sim \forall \typvar{\alpha} . (\typvar{\alpha} \rightarrow \typvar{\alpha}) \rightarrow \typvar{\alpha} \rightarrow \typvar{\alpha}) } \)
\end{example}

\begin{example}
\textcolor{magenta}{bb-succ bb-one} \(\rightarrow_\beta^{*}\) \textcolor{magenta}{bb-two}.
\end{example}

\begin{example}
\exprterminal{(}\textcolor{magenta}{bb-succ bb-false}\exprterminal{)} is stuck. \(\nexists\metavariable{v} . \) \exprterminal{(}\textcolor{magenta}{bb-succ bb-false}\exprterminal{)} \( \rightarrow_\text{eval} \metavariable{v}\) where \(\metavariable{v}\) is a \emph{value}.
\end{example}

\begin{example}
\exprterminal{(}\textcolor{magenta}{bb-succ bb-false}\exprterminal{)} is \emph{not} well-typed. \(\nexists\metavariable{\tau} . \emptyset \vdash \) \textcolor{magenta}{bb-succ bb-false} : \(\metavariable{\tau}\).
\end{example}

\subsection{Pairs}

\begin{definition}
pairs

\begin{itemize}
\item let {\textcolor{magenta}{bb-pair}} = \(\exprterminal{\Lambda \typvar{\alpha} \mapsto \Lambda \typvar{\beta} \mapsto \lambda \expvar{a}:\typvar{\alpha} \mapsto \lambda \expvar{b} : \typvar{\beta}}\)\\
\(\exprterminal{
	\mapsto (\Lambda\typvar{\gamma} \mapsto \lambda \expvar{s} : \typeterminal{\typvar{\alpha} \rightarrow \typvar{\beta} \rightarrow \typvar{\gamma}} \mapsto \expvar{s} \expvar{a} \expvar{b})}\)
	\exprterminal{ as } \typeterminal{\((\typvar{\alpha}\times\typvar{\beta})\) \(\sim\) <BB-Pair \(\typvar{\alpha}\text{ }\typvar{\beta}\)> \(\sim\) \(\forall \typvar{\gamma} . ((\typvar{\alpha} \rightarrow \typvar{\beta} \rightarrow \typvar{\gamma}) \rightarrow \typvar{\gamma})\) }\\

\item let {\textcolor{magenta}{bb-pair-first}} = \(\exprterminal{
	\Lambda \typvar{\alpha} \mapsto \Lambda \typvar{\beta}\\
	\mapsto \lambda \expvar{p} : \typeterminal{
		(\typvar{\alpha}\times\typvar{\beta}) \sim\text{ <BB-Pair \(\typvar{\alpha}\text{ }\typvar{\beta}\)> }\sim \forall \typvar{\gamma} . ((\typvar{\alpha} \rightarrow \typvar{\beta} \rightarrow \typvar{\gamma}) \rightarrow \typvar{\gamma} )
	}}\)\\
	\(\exprterminal{
	\mapsto \expvar{p} \typvar{\alpha} (\lambda \expvar{a}:\typvar{\alpha} \mapsto \lambda \expvar{b}:\typvar{\beta} \mapsto \expvar{a})
}\)\\

\item let {\textcolor{magenta}{bb-pair-second}} = \(\exprterminal{
	\Lambda \typvar{\alpha} \mapsto \Lambda \typvar{\beta}
	\mapsto \lambda \expvar{p} : \typeterminal{
		(\typvar{\alpha}\times\typvar{\beta}) \sim\text{<BB-Pair \(\typvar{\alpha}\text{ }\typvar{\beta}\)> }\sim \forall \typvar{\gamma} . ((\typvar{\alpha} \rightarrow \typvar{\beta} \rightarrow \typvar{\gamma}) \rightarrow \typvar{\gamma} )
	}}\)\\
	\(\exprterminal{
	\mapsto \expvar{p} \typvar{\beta} (\lambda \expvar{a}:\typvar{\alpha} \mapsto \lambda \expvar{b}:\typvar{\beta} \mapsto \expvar{b})
}\)\\

\end{itemize}
\end{definition}

\subsection{Lists}


\subsection{Algebraic Datatypes}
\todo{}

\input{appendix}

\end{document}

