paper: define morphism-path relation, redefine typing-relation \& translation
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paper/main.tex
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paper/main.tex
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@ -6,7 +6,7 @@
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\usepackage{mathpartir}
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\usepackage{hyperref}
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\usepackage{url}
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\usepackage{stmaryrd}
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\usepackage{minted}
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\usemintedstyle{tango}
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@ -21,6 +21,7 @@
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\DeclareUnicodeCharacter{03B3}{$\gamma$}
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\DeclareUnicodeCharacter{03B4}{$\delta$}
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\DeclareUnicodeCharacter{0393}{$\Gamma$}
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\DeclareUnicodeCharacter{211D}{$\mathbb{R}$}
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\newcommand{\metavariable}[1]{\textcolor{teal}{#1}}
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\newcommand{\typeterminal}[1]{\textcolor{brown}{#1}}
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@ -522,34 +523,66 @@ while preserving its semantics.
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As usual, the typing-context \(\Gamma = \{ \metavariable{x_1} : \metavariable{\tau_1} , \quad \metavariable{x_2} : \metavariable{\tau_2} , \quad \ldots \}\)
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is a finite mapping which assigns variables \(\metavariable{x_i} \in \exprvars\) to types \(\metavariable{\tau_i} \in \nonterm{T}\).
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Using the inference rules given in \ref{def:typerules}, further typing-judgements
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of the form
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\begin{itemize}
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\item \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)" and
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\item \(\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is compatible with type \(\metavariable{\tau}\)"
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\end{itemize}
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of the form \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)"
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can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \nonterm{E}\) and \(\metavariable{\tau} \in \nonterm{T}\).
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\begin{definition}[Morphism Paths]
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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 \(\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\).
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\label{def:pathrules}
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\begin{mathpar}
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\inferrule[M-Sub]{
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\metavariable{\tau} \leq \metavariable{\tau'}
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}
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\inferrule[M-Single]{
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(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}
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\inferrule[M-Chain]{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
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\Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
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}
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\inferrule[M-MapSeq]{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}{
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\Gamma \vdash
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\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
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\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
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}
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\end{mathpar}
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\begin{definition}[Representational Well-Typedness]
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An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{representationally well-typed} if there exists a type \(\metavariable{\tau} \in \nonterm{T}\),
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such that \( \emptyset \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}.
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\end{definition}
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\begin{definition}[Compatible Well-Typedness]
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An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{compatibly well-typed} if there exists a type \(\metavariable{\tau} \in \nonterm{T}\),
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such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) by \ref{def:typerules} and \ref{def:semtyperules}.
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\end{definition}
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\begin{example}[Morphism Graph]
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Assume \(\Gamma := \{\\
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\exprterminal{\text{degrees-to-turns}} : \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R} \rightarrow_\text{morph} \text{Angle}\sim\text{Turns}\sim\mathbb{R}},\\
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\exprterminal{\text{turns-to-radians}} : \typeterminal{\text{Angle}\sim\text{Turns}\sim\mathbb{R} \rightarrow_\text{morph} \text{Angle}\sim\text{Radians}\sim\mathbb{R}},\\
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\}\).
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Then
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\begin{itemize}
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\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\mathbb{R}}\) (by \textsc{M-Sub})
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\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})
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\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})
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\end{itemize}
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\end{example}
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\begin{definition}["is" Typing Relation]
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\begin{definition}[Typing Relation]
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\label{def:typerules}
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\begin{mathpar}
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\inferrule[T-Variable]{
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% \metavariable{x} \in \exprvars\\
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% \metavariable{\tau} \in \nonterm{T}\\
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\metavariable{x}:\metavariable{\tau} \in \Gamma\\
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}{
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\Gamma \vdash \metavariable{x}:\metavariable{\tau}
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@ -563,46 +596,39 @@ such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) b
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}
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\inferrule[T-TypeAbs]{
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% \metavariable{\tau} \in \nonterm{T} \\
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% \metavariable{e} \in \nonterm{E} \\
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\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
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}{
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\Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau}
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}
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\inferrule[T-TypeApp]{
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% \metavariable{\tau} \in \nonterm{T} \\
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\Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\
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\metavariable{\sigma} \in \nonterm{T}
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}{
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\Gamma \vdash ( \metavariable{e} \quad \metavariable{\sigma} ) : \{\metavariable{\alpha} \mapsto \metavariable{\sigma}\} \metavariable{\tau}
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}
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\inferrule[T-Abs]{
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% \metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
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% \metavariable{e} \in \nonterm{E} \\
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\Gamma,\metavariable{x}:\metavariable{\sigma} \vdash \metavariable{e} : \metavariable{\tau} \\
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}{
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\Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\sigma} \exprterminal{\mapsto} \metavariable{e}) : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
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}
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\inferrule[T-App]{
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\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau} \\
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\Gamma \vdash \metavariable{a} : \metavariable{\sigma} \\
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}{
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\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
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}\and
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\inferrule[T-MorphAbs]{
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% \metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
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% \metavariable{e} \in \nonterm{E} \\
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\Gamma,\metavariable{x}:\metavariable{\tau} \vdash \metavariable{e} : \metavariable{\tau'} \\
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\metavariable{\tau} \precsim \metavariable{\tau'}
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}{
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\Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\tau} \exprterminal{\mapsto_{morph}} \metavariable{e}) : \metavariable{\tau}\typeterminal{\rightarrow_{morph}}\metavariable{\tau'}
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}\and
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\inferrule[T-App]{
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\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
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\Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\
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\Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
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}{
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\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
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}\and
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\inferrule[T-MorphFun]{
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\Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
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}{
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@ -626,95 +652,191 @@ such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) b
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\end{mathpar}
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\end{definition}
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\begin{definition}["compatible" Typing Relation]
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\label{def:semtyperules}
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\begin{mathpar}
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\inferrule[TCompat-NativeRepr]{
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\Gamma\vdash \metavariable{e} : \metavariable{\tau}
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}{
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\Gamma\vdash \metavariable{e} :\approx \metavariable{\tau}
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}
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\inferrule[TCompat-Let]{
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\Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\
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\Gamma , \metavariable{x}:\metavariable{\sigma} \vdash \metavariable{t} :\approx \metavariable{\tau}
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}{
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\Gamma \vdash (\exprterminal{\text{let }}\metavariable{x}\exprterminal{\text{ = }}\metavariable{e}\exprterminal{\text{ in }} \metavariable{t}) :\approx \metavariable{\tau}
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}
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\inferrule[TCompat-Morph]{
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\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau}\\
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% \metavariable{\tau} \precsim \metavariable{\tau'}\\
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%\exists \metavariable{h} \text{ s.t. }
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\metavariable{h}:\typeterminal{\metavariable{\tau}\rightarrow_\text{morph}\metavariable{\tau'}} \in \Gamma
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}{
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\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
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}
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\inferrule[TCompat-App]{
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\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
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\Gamma \vdash \metavariable{a} :\approx \metavariable{\sigma}
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}{
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\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} :\approx \metavariable{\tau}
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}
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\end{mathpar}
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\begin{definition}[Well-Typedness]
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An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{well-typed} if there exist \(\Gamma\) and \(\metavariable{\tau}\)
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such that \( \Gamma \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}.
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\end{definition}
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\subsection{Coercion Semantics}
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%We define the translation function \(\llbracket . \rrbracket\) which translates morphism-paths into
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%expressions that define a transformation function, and also translates type-derivations into expressions with expanded type coercions.
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We define the translation function \(\llbracket . \rrbracket\) which completes a \emph{semantically well-typed} expression
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by inserting all required coercions based on the typing derivation of the expression.
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The result shall be a \emph{syntactically well-typed} expression.
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We write \(C :: \sigma \precsim \tau\) to mean "C is a subtyping derivation tree whose conclusion is \(\sigma \precsim \tau\)".
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%which completes a \emph{semantically well-typed} expression
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%by inserting all required coercions based on the typing derivation of the expression.
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%The result shall be a \emph{syntactically well-typed} expression.
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We write \(C :: \tau \leadsto \tau'\) to mean "C is a morphism-path derivation tree whose conclusion is \(\tau \leadsto \tau'\)".
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Similarly, we write \(D :: \Gamma \vdash e : \tau\) to mean "D is a typing derivation whose conclusion is \(\Gamma \vdash e : \tau\)"
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\begin{definition}[Translation]
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\begin{definition}[Morphism Translation]
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%Translates a morphism-path derivation into an expression that defines a coercion function
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\begin{mathpar}
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\Big{\llbracket} \inferrule[T-SemanticSubtype]{
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D_1 :: \Gamma \vdash \metavariable{h}:\metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}\\
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D_2 :: \Gamma \vdash \metavariable{e}:\metavariable{\tau}\\
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% C :: \metavariable{\tau} \precsim \metavariable{\tau'}
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\Big{\llbracket}
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\inferrule[M-Sub]{
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\metavariable{\tau} \leq \metavariable{\tau'}
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}{
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\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
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}\Big{\rrbracket} = \exprterminal{(}
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\llbracket D_1 \rrbracket \llbracket D_2 \rrbracket
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%\metavariable{h} \llbracket D_2 \rrbracket
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\exprterminal{)}
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\Big{\llbracket} \inferrule[T-CoercedApp]{
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D_1 :: \Gamma \vdash \metavariable{f}:\metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
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D_2 :: \Gamma \vdash \metavariable{a}:\approx\metavariable{\sigma}
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}{
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\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} : \metavariable{\tau}
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}\Big{\rrbracket} = \exprterminal{(}
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%\exprterminal{(}\metavariable{f} \llbracket D_2 \rrbracket \exprterminal{)}
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\llbracket D_1 \rrbracket \llbracket D_2 \rrbracket
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\exprterminal{)}
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}
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\Big{\rrbracket} = \exprterminal{\lambda x:\metavariable{\tau} \mapsto x}
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\and
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\Big{\llbracket}
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\inferrule[\emph{Otherwise}]{}{
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D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}
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\inferrule[M-Single]{
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(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}
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\Big{\rrbracket} = \metavariable{e}
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\Big{\rrbracket} = \metavariable{h}
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\and
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\Big{\llbracket}
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\inferrule[M-Chain]{
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C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
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C_2 :: \Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
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}{
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\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
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}
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\Big{\rrbracket} = \exprterminal{\lambda \text{x}:\metavariable{\tau} \mapsto}
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\Big{\llbracket} C_2 \Big{\rrbracket}
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\exprterminal{(}\Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{x})}
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\and
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\Big{\llbracket}
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\inferrule[M-MapSeq]{
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C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
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}{
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\Gamma \vdash
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\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
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\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
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}
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\Big{\rrbracket} = \exprterminal{\lambda \text{xs}:\typeterminal{\langle\text{Seq }\metavariable{\tau}\rangle} \mapsto}
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\exprterminal{( \text{map}} \Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{xs})}
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\end{mathpar}
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\end{definition}
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\begin{lemma}[Elimination of \(:\approx\)]
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\label{lemma:translation}
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\begin{definition}[Expression Translation]
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%Translates a type-derivation tree into a fully expanded expression
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For all \emph{semantically well-typed} expressions \metavariable{e} with the typing derivation \(D :: \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau}\),
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the translation \(\llbracket D \rrbracket = \metavariable{e'}\), yields a \emph{syntactically well-typed} expression \metavariable{e'} with
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\(\emptyset \vdash \metavariable{e'} : \metavariable{\tau} \)
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\begin{mathpar}
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\Big{\llbracket} \inferrule[T-Variable]{
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\metavariable{x}:\metavariable{\tau} \in \Gamma
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}{
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\Gamma \vdash \metavariable{x}:\metavariable{\tau}
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}\Big{\rrbracket} = \metavariable{x}
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\and
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\Big{\llbracket} \inferrule[T-LetBinding]{
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D_1 ::\Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\
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D_2 :: \Gamma , \metavariable{x}:\metavariable{\sigma} \vdash \metavariable{t} : \metavariable{\tau}
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}{
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\Gamma \vdash (\exprterminal{\text{let }}\metavariable{x}\exprterminal{\text{ = }}\metavariable{e}\exprterminal{\text{ in }} \metavariable{t}) : \metavariable{\tau}
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}\Big{\rrbracket} = \exprterminal{\text{let }\metavariable{x} = }
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\Big{\llbracket} D_1 \Big{\rrbracket}
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\exprterminal{\text{ in }}
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\Big{\llbracket} D_2 \Big{\rrbracket}
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\and
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\Big{\llbracket}
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\inferrule[T-TypeAbs]{
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D_1 :: \Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
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}{
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\Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau}
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}
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\Big{\rrbracket} = \exprterminal{\Lambda \metavariable{\alpha} \mapsto} \Big{\llbracket} D_1 \Big{\rrbracket}
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\and
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\Big{\llbracket}
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\inferrule[T-TypeApp]{
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D_1 :: \Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\
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\metavariable{\sigma} \in \nonterm{T}
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}{
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\Gamma \vdash ( \metavariable{e} \quad \metavariable{\sigma} ) : \{\metavariable{\alpha} \mapsto \metavariable{\sigma}\} \metavariable{\tau}
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}
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\Big{\rrbracket} =
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\exprterminal{(}
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\Big{\llbracket}
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D_1
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\Big{\rrbracket}
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\metavariable{\sigma}
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\exprterminal{)}
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\and
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\Big{\llbracket}
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\inferrule[T-Abs]{
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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}
|
||||
|
|
|
|||
Loading…
Reference in a new issue