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paper: define morphism-path relation, redefine typing-relation \& translation

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Michael Sippel 2024-09-05 11:20:21 +02:00
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@ -6,7 +6,7 @@
\usepackage{mathpartir}
\usepackage{hyperref}
\usepackage{url}
\usepackage{stmaryrd}
\usepackage{minted}
\usemintedstyle{tango}
@ -21,6 +21,7 @@
\DeclareUnicodeCharacter{03B3}{$\gamma$}
\DeclareUnicodeCharacter{03B4}{$\delta$}
\DeclareUnicodeCharacter{0393}{$\Gamma$}
\DeclareUnicodeCharacter{211D}{$\mathbb{R}$}
\newcommand{\metavariable}[1]{\textcolor{teal}{#1}}
\newcommand{\typeterminal}[1]{\textcolor{brown}{#1}}
@ -522,34 +523,66 @@ while preserving its semantics.
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}
of the form \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)"
can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \nonterm{E}\) and \(\metavariable{\tau} \in \nonterm{T}\).
\begin{definition}[Morphism Paths]
Given a typing context \(\Gamma\), any type \(\metavariable{\tau}\) can be transformed into \(\metavariable{\tau'}\), provided there is a path from \(\metavariable{\tau}\) to \(\metavariable{\tau'}\) in the \emph{Morphism-Graph} of \(\Gamma\), written as \(\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\).
\label{def:pathrules}
\begin{mathpar}
\inferrule[M-Sub]{
\metavariable{\tau} \leq \metavariable{\tau'}
}{
\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
}
\inferrule[M-Single]{
(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
}{
\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
}
\inferrule[M-Chain]{
\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
\Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
}{
\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
}
\inferrule[M-MapSeq]{
\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
}{
\Gamma \vdash
\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
}
\end{mathpar}
\begin{definition}[Representational Well-Typedness]
An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{representationally 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}[Compatible Well-Typedness]
An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{compatibly 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{example}[Morphism Graph]
Assume \(\Gamma := \{\\
\exprterminal{\text{degrees-to-turns}} : \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R} \rightarrow_\text{morph} \text{Angle}\sim\text{Turns}\sim\mathbb{R}},\\
\exprterminal{\text{turns-to-radians}} : \typeterminal{\text{Angle}\sim\text{Turns}\sim\mathbb{R} \rightarrow_\text{morph} \text{Angle}\sim\text{Radians}\sim\mathbb{R}},\\
\}\).
Then
\begin{itemize}
\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\mathbb{R}}\) (by \textsc{M-Sub})
\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\text{Angle}\sim\text{Radians}\sim\mathbb{R}}\) (by \textsc{M-Chain})
\item \(\Gamma \vdash \typeterminal{\langle\text{Seq }\text{Angle}\sim\text{Degrees}\sim\mathbb{R}\rangle} \leadsto \typeterminal{\langle\text{Seq }\text{Angle}\sim\text{Radians}\sim\mathbb{R}\rangle}\) (by \textsc{M-MapSeq})
\end{itemize}
\end{example}
\begin{definition}["is" Typing Relation]
\begin{definition}[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}
@ -563,46 +596,39 @@ such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) b
}
\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-App]{
\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
\Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\
\Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
}{
\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
}\and
\inferrule[T-MorphFun]{
\Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
}{
@ -626,95 +652,191 @@ such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) b
\end{mathpar}
\end{definition}
\begin{definition}["compatible" Typing Relation]
\label{def:semtyperules}
\begin{mathpar}
\inferrule[TCompat-NativeRepr]{
\Gamma\vdash \metavariable{e} : \metavariable{\tau}
}{
\Gamma\vdash \metavariable{e} :\approx \metavariable{\tau}
}
\inferrule[TCompat-Let]{
\Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\
\Gamma , \metavariable{x}:\metavariable{\sigma} \vdash \metavariable{t} :\approx \metavariable{\tau}
}{
\Gamma \vdash (\exprterminal{\text{let }}\metavariable{x}\exprterminal{\text{ = }}\metavariable{e}\exprterminal{\text{ in }} \metavariable{t}) :\approx \metavariable{\tau}
}
\inferrule[TCompat-Morph]{
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau}\\
% \metavariable{\tau} \precsim \metavariable{\tau'}\\
%\exists \metavariable{h} \text{ s.t. }
\metavariable{h}:\typeterminal{\metavariable{\tau}\rightarrow_\text{morph}\metavariable{\tau'}} \in \Gamma
}{
\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
}
\inferrule[TCompat-App]{
\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
\Gamma \vdash \metavariable{a} :\approx \metavariable{\sigma}
}{
\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} :\approx \metavariable{\tau}
}
\end{mathpar}
\begin{definition}[Well-Typedness]
An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{well-typed} if there exist \(\Gamma\) and \(\metavariable{\tau}\)
such that \( \Gamma \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}.
\end{definition}
\subsection{Coercion Semantics}
%We define the translation function \(\llbracket . \rrbracket\) which translates morphism-paths into
%expressions that define a transformation function, and also translates type-derivations into expressions with expanded type coercions.
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 \(\sigma \precsim \tau\)".
%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 \(C :: \tau \leadsto \tau'\) to mean "C is a morphism-path derivation tree whose conclusion is \(\tau \leadsto \tau'\)".
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{definition}[Morphism Translation]
%Translates a morphism-path derivation into an expression that defines a coercion function
\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[M-Sub]{
\metavariable{\tau} \leq \metavariable{\tau'}
}{
\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
}
\Big{\rrbracket} = \exprterminal{\lambda x:\metavariable{\tau} \mapsto x}
\and
\Big{\llbracket}
\inferrule[\emph{Otherwise}]{}{
D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}
}
\Big{\rrbracket} = \metavariable{e}
\inferrule[M-Single]{
(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
}{
\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
}
\Big{\rrbracket} = \metavariable{h}
\and
\Big{\llbracket}
\inferrule[M-Chain]{
C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
C_2 :: \Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
}{
\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
}
\Big{\rrbracket} = \exprterminal{\lambda \text{x}:\metavariable{\tau} \mapsto}
\Big{\llbracket} C_2 \Big{\rrbracket}
\exprterminal{(}\Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{x})}
\and
\Big{\llbracket}
\inferrule[M-MapSeq]{
C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
}{
\Gamma \vdash
\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
}
\Big{\rrbracket} = \exprterminal{\lambda \text{xs}:\typeterminal{\langle\text{Seq }\metavariable{\tau}\rangle} \mapsto}
\exprterminal{( \text{map}} \Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{xs})}
\end{mathpar}
\end{definition}
\begin{lemma}[Elimination of \(:\approx\)]
\label{lemma:translation}
\begin{definition}[Expression Translation]
%Translates a type-derivation tree into a fully expanded expression
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{mathpar}
\Big{\llbracket} \inferrule[T-Variable]{
\metavariable{x}:\metavariable{\tau} \in \Gamma
}{
\Gamma \vdash \metavariable{x}:\metavariable{\tau}
}\Big{\rrbracket} = \metavariable{x}
\and
\Big{\llbracket} \inferrule[T-LetBinding]{
D_1 ::\Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\
D_2 :: \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}
}\Big{\rrbracket} = \exprterminal{\text{let }\metavariable{x} = }
\Big{\llbracket} D_1 \Big{\rrbracket}
\exprterminal{\text{ in }}
\Big{\llbracket} D_2 \Big{\rrbracket}
\and
\Big{\llbracket}
\inferrule[T-TypeAbs]{
D_1 :: \Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
}{
\Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau}
}
\Big{\rrbracket} = \exprterminal{\Lambda \metavariable{\alpha} \mapsto} \Big{\llbracket} D_1 \Big{\rrbracket}
\and
\Big{\llbracket}
\inferrule[T-TypeApp]{
D_1 :: \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}
}
\Big{\rrbracket} =
\exprterminal{(}
\Big{\llbracket}
D_1
\Big{\rrbracket}
\metavariable{\sigma}
\exprterminal{)}
\and
\Big{\llbracket}
\inferrule[T-Abs]{
D_1 :: \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}
}
\Big{\rrbracket} =
\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\sigma}
\exprterminal{\mapsto} \Big{\llbracket}D_1\Big{\rrbracket}
\and
\Big{\llbracket}
\inferrule[T-App]{
D_1 :: \Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
D_2 :: \Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\\\
C :: \Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
}{
\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
}
\Big{\rrbracket} =
\Big{\llbracket}D_1\Big{\rrbracket}
\exprterminal{(}
\Big{\llbracket}C\Big{\rrbracket}
\Big{\llbracket}D_2\Big{\rrbracket}
\exprterminal{)}
\and
\Big{\llbracket}
\inferrule[T-MorphFun]{
D_1 :: \Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
}{
\Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
}
\Big{\rrbracket} = \Big{\llbracket} D_1 \Big{\rrbracket}
\and
\Big{\llbracket}
\inferrule[T-Ascension]{
D_1 :: \Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
\metavariable{\tau'} \leq \metavariable{\tau}
}{
\Gamma \vdash (\metavariable{e} \exprterminal{\text{ as }} \metavariable{\tau'}) : \metavariable{\tau'}
}
\Big{\rrbracket} =
\Big{\llbracket}D_1\Big{\rrbracket} \exprterminal{\text{ as }} \metavariable{\tau'}
\and
\Big{\llbracket}
\inferrule[T-Descension]{
D_1 :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
\metavariable{\tau} \leq \metavariable{\tau'}
}{
\Gamma \vdash \metavariable{e} : \metavariable{\tau'}
}
\Big{\rrbracket} =
\Big{\llbracket}
D_1
\Big{\rrbracket}
\end{mathpar}
\end{definition}
\begin{proof}
\todo{}
\end{proof}
\end{lemma}
\subsection{Evaluation}