Expression/Type relevant checking.
$$
\newcommand{\tyck}[0]{\vdash_\texttt{c}}
\newcommand{\Gtyck}[0]{\Gamma \tyck}
\newcommand{\infer}[0]{\vdash_\texttt{i}}
\newcommand{\Ginfer}[0]{\Gamma \infer}
\Ginfer a \Rightarrow o
\quad
\Gtyck a:m \Rightarrow o
\quad
\Gamma \vdash A <: B
$$
check | Check an abstract term against an expected type and produce a well-typed term.
$$
\newcommand{\tyck}[4]{#1 \vdash_\texttt{c} #2 : #3 \Rightarrow #4}
\newcommand{\Gtyck}[3]{\tyck{\Gamma}{#1}{#2}{#3}}
\newcommand{\eval}[1]{\llbracket #1 \rrbracket} % {\texttt{eval}(#1)}
\newcommand{\cheval}[3]{#1 \vdash_\texttt{e} #2 \Rightarrow #3}
\newcommand{\Gcheval}[2]{\cheval{\Gamma}{#1}{#2}}
\newcommand{\subt}[0]{<:}
\newcommand{\xx}[0]{\texttt{x}}
\newcommand{\ty}[0]{\tau}
\newcommand{\piTy}[1]{\Pi \langle #1 \rangle}
\newcommand{\sigTy}[1]{\Sigma \langle #1 \rangle}
\newcommand{\cA}[0]{\mathcal A}
\newcommand{\cB}[0]{\mathcal B}
\cfrac{
\Gtyck{a}{\cA}{\alpha} \quad
\Gtyck{b}{\cB[\xx := \alpha]}{\beta}
}{
\Gcheval{a, b}{\sigTy{\xx : \cA . \cB}}
} \quad
\cfrac{
\tyck{\Gamma, \xx : \cB}{a}{\cA}{\alpha}
}{
\Gtyck{
\lambda \xx. \alpha}{\piTy{\xx : \cB . \cA}
}{
\lambda \langle \xx . \alpha \rangle
}
}
$$
|
check_row_polymorphic_type | Extracted from check .
$$
\newcommand{\ty}[0]{\tau}
\newcommand{\tyck}[4]{#1 \vdash_\texttt{c} #2 : #3 \Rightarrow #4}
\newcommand{\Gtyck}[3]{\tyck{\Gamma}{#1}{#2}{#3}}
\newcommand{\cheval}[3]{#1 \vdash_\texttt{e} #2 \Rightarrow #3}
\newcommand{\Gcheval}[2]{\cheval{\Gamma}{#1}{#2}}
\newcommand{\variant}[1]{\textbf{Sum}\ \{ #1 \}}
\newcommand{\record}[1]{\textbf{Rec}\ \{ #1 \}}
\newcommand{\variantR}[1]{\mathbb{Sum}\ #1}
\newcommand{\recordR}[1]{\mathbb{Rec}\ #1}
\newcommand{\variantextS}[2]{\variant{#1, \ldots = #2}}
\newcommand{\recordextS}[2]{\record{#1, \ldots = #2}}
\newcommand{\cA}[0]{\mathcal A}
\cfrac{
n \notin ns \quad \Gtyck{A}{\ty}{\cA}
}{
\cfrac{
\Gcheval{B}{\recordR{n,ns}}
}{
\Gcheval{\recordextS{n : A}{B}}{\recordR{ns}}
} \quad
\cfrac{
\Gcheval{B}{\variantR{n,ns}}
}{
\Gcheval{\variantextS{n : A}{B}}{\variantR{ns}}
}
}
\\
\cfrac{
}{
\Gtyck{\record{}}{\recordR{ns}}{\record{}} \quad
\Gtyck{\variant{}}{\variantR{ns}}{\variant{}}
}
$$
|
check_variant_or_cons | |
infer | Infer type of an abstract term, if it's well-typed.
$$
\newcommand{\tyck}[4]{#1 \vdash_\texttt{c} #2 : #3 \Rightarrow #4}
\newcommand{\Gtyck}[3]{\tyck{\Gamma}{#1}{#2}{#3}}
\newcommand{\eval}[1]{\llbracket #1 \rrbracket} % {\texttt{eval}(#1)}
\newcommand{\infer}[3]{#1 \vdash_\texttt{i} #2 \Leftarrow #3}
\newcommand{\Ginfer}[2]{\infer{\Gamma}{#1}{#2}}
\newcommand{\subt}[0]{<:}
\newcommand{\xx}[0]{\texttt{x}}
\newcommand{\ty}[0]{\tau}
\newcommand{\piTy}[1]{\Pi \langle #1 \rangle}
\newcommand{\sigTy}[1]{\Sigma \langle #1 \rangle}
\newcommand{\cA}[0]{\mathcal A}
\newcommand{\cB}[0]{\mathcal B}
\cfrac{
\Gtyck{A}{\ty}{\cA}\quad
\tyck{\Gamma, (\xx : \cA)}{B}{\ty}{\cB}
}{
\Ginfer{\piTy{\xx : A . B}}{\ty} \quad
\Ginfer{\sigTy{\xx : A . B}}{\ty}
} \quad
\cfrac{}{\Ginfer{\ty}{\ty}}
\\ \space \\
\cfrac{
\Ginfer{a}{\sigTy{\xx : \cA . \cB}}
}{
\Ginfer{a.1}{\cA} \quad
\Ginfer{a.2}{\cB[\xx := \eval{a.1}]}
} \quad
\cfrac{
\Ginfer{a}{\piTy{\xx : \cB . \cA}} \quad
\Gtyck{b}{\cB}{\beta}
}{
\Ginfer{a\ b}{\cA [\xx := \eval{b}]}
}
$$
|
subtype | Check if subtype is a subtype of supertype .
$$
\newcommand{\Gvdash}[0]{\Gamma \vdash}
\newcommand{\tyck}[4]{#1 \vdash_\texttt{c} #2 : #3 \Rightarrow #4}
\newcommand{\Gtyck}[3]{\tyck{\Gamma}{#1}{#2}{#3}}
\newcommand{\subt}[0]{<:}
\newcommand{\xx}[0]{\texttt{x}}
\newcommand{\ty}[0]{\tau}
\newcommand{\piTy}[1]{\Pi \langle #1 \rangle}
\newcommand{\sigTy}[1]{\Sigma \langle #1 \rangle}
\newcommand{\variant}[1]{\textbf{Sum}\ \{ #1 \}}
\newcommand{\record}[1]{\textbf{Rec}\ \{ #1 \}}
\newcommand{\cA}[0]{\mathcal A}
\newcommand{\cB}[0]{\mathcal B}
\newcommand{\ctyLab}[0]{\gamma}
\newcommand{\clabVal}[0]{\delta}
\newcommand{\recordext}[2]{\record{#1 \mid #2}}
\newcommand{\recExt}[1]{\mid #1}
\newcommand{\variantext}[2]{\variant{#1 \mid #2}}
\newcommand{\variantR}[1]{\mathbb{Sum}\ #1}
\newcommand{\recordR}[1]{\mathbb{Rec}\ #1}
\cfrac{
\Gvdash \cA_0 \simeq \cA_1 \quad
\Gvdash \cB_0 \subt \cB_1
}{
\Gvdash \piTy{
\xx : \cA_0 . \cB_0
} \subt \piTy{\xx : \cA_1 . \cB_1} \quad
\Gvdash \sigTy{
\xx : \cA_0 . \cB_0
} \subt \sigTy{\xx : \cA_1 . \cB_1}
} \\ \space \\
\cfrac{
\Gvdash \cA_0 \subt \cA_1 \quad \Gvdash \cA_1 \subt \cA_2
}{
\Gvdash \cA_0 \subt \cA_2
} \quad
\cfrac{\Gvdash \cA \simeq \cB}{\Gvdash \cB \simeq \cA \quad \Gvdash \cA \subt \cB}
\\ \space \\
\cfrac{
n \notin ns
}{
\Gvdash \variantR{n,ns} \subt \variantR{ns} \quad
\Gvdash \recordR{ns} \subt \recordR{n,ns}
}
\\ \space \\
\cfrac{
\Gtyck{A}{\ty}{\cA} \quad
(n : \cA) \notin \ctyLab_0
}{
\cfrac{
\Gvdash \variant{\ctyLab_1} \subt \variant{\ctyLab_0}
}{
\Gvdash \variant{\ctyLab_1} \subt \variant{n : \cA, \ctyLab_0}
} \quad
\cfrac{
\Gvdash \record{\ctyLab_0} \subt \record{\ctyLab_1}
}{
\Gvdash \record{n : \cA, \ctyLab_0} \subt \record{\ctyLab_1}
}
}
\\
\cfrac{
}{
\Gvdash \recordR{ns} \subt \ty \quad
\Gvdash \variantR{ns} \subt \ty
}
$$
|