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colored.tex
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\subsection{External Language with bidirectional checking}
\begin{figure}
$$
\begin{array}{lrll}
% \ltiE{}, \ltiF{} &::=& ... \alt
% \ltiufun{\ltivar{}}{\ltiE{}} \alt
% \ltiapp{\ltiF{}}{\ltiE{}}
% &\mbox{Terms} \\
% \ltiSyn{\ltiT{}}, \ltiSyn{\ltiS{}}, \ltiSyn{\ltiR{}} &::=&
% \ltiSyn{\ltitvar{}} \alt
% \ltiSyn{\ltiTop} \alt
% \ltiSyn{\ltiBot} \alt
% \ltiSyn{\ltiRec{\hastype{\ltivar{1}}{\ltiSyn{\ltiT{1}}}, ..., \hastype{\ltivar{n}}{\ltiSyn{\ltiT{n}}}}}
% \\&\alt&
% \ltiSyn{\ltiArrowMono{\ltiSyn{\ltiT{}}}{\ltiSyn{\ltiS{}}}} \alt
% \ltiSyn{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}
% &\mbox{Synthesized types} \\
% \ltiChk{\ltiT{}}, \ltiChk{\ltiS{}}, \ltiChk{\ltiR{}} &::=&
% \ltiChk{\ltitvar{}} \alt
% \ltiChk{\ltiTop} \alt
% \ltiChk{\ltiBot} \alt
% \ltiChk{\ltiRec{\hastype{\ltivar{1}}{\ltiChk{\ltiT{1}}}, ..., \hastype{\ltivar{n}}{\ltiChk{\ltiT{n}}}}}
% \\&\alt&
% \ltiChk{\ltiArrowMono{\ltiChk{\ltiT{}}}{\ltiChk{\ltiS{}}}} \alt
% \ltiChk{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}
% &\mbox{Inherited types} \\
% \ltiSynChk{\ltiT{}}, \ltiSynChk{\ltiS{}}, \ltiSynChk{\ltiR{}} &::=&
% \ltiSyn{\ltitvar{}} \alt
% \ltiSyn{\ltiTop} \alt
% \ltiSyn{\ltiBot} \alt
% \ltiSyn{\ltiRec{\hastype{\ltivar{1}}{\ltiSynChk{\ltiT{1}}},...,\hastype{\ltivar{n}}{\ltiSynChk{\ltiT{n}}}}}
% \\&\alt&
% \ltiSyn{\ltiArrowMono{\ltiSynChk{\ltiT{}}}{\ltiSynChk{\ltiS{}}}} \alt
% \ltiSyn{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}
% \\&\alt&
% \ltiChk{\ltitvar{}} \alt
% \ltiChk{\ltiTop} \alt
% \ltiChk{\ltiBot} \alt
% \ltiChk{\ltiRec{\hastype{\ltivar{1}}{\ltiSynChk{\ltiT{1}}},...,\hastype{\ltivar{n}}{\ltiSynChk{\ltiT{n}}}}}
% \\&\alt&
% \ltiChk{\ltiArrowMono{\ltiSynChk{\ltiT{}}}{\ltiSynChk{\ltiS{}}}} \alt
% \ltiChk{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}
% &\mbox{Colored types} \\
\ltiSynChk{\ltiT{}}, \ltiSynChk{\ltiS{}}, \ltiSynChk{\ltiR{}} &::=&
\ltiSynChk{\ltitvar{}} \alt
\ltiSynChk{\ltiTop} \alt
\ltiSynChk{\ltiBot} \alt
\ltiSynChk{\ltiRec{\hastype{\ltivar{1}}{\ltiSynChk{\ltiT{1}}},...,\hastype{\ltivar{n}}{\ltiSynChk{\ltiT{n}}}}}
\\&\alt&
\ltiSynChk{\ltiArrow{\ltiSynChk{\ltiT{}}}{\ova{\ltitvar{}}}{\ltiSynChk{\ltiS{}}}} \alt
\ltiSynChk{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}
&\mbox{Colored types} \\
\ltiEnv{} &::=& \hastype{\ltivar{}}{\ltiSyn{\ltiT{}}} \alt
\ltiEmptyEnv \alt
\ltiSyn{\ltitvar{}} \alt
\ltiEnvConcat{\ltiEnv{}}{\ltiEnvp{}}
&\mbox{Environments} \\
\ltiFuel{} &::=& \ltinat{}
&\mbox{Fuel} \\
\end{array}
$$
\caption{Colored Type System Syntax (External Language),
where $\ltinat{} \in \mathbb{N}$.
A type constructor's color \ltiSynChk{} stands for exactly one of either a blue up-tick \ltiSyn{} (``synthesized'')
or a red down-tick \ltiChk{} (``inherited'').
The types \ltiSyn{\ltiT{}} and \ltiChk{\ltiT{}} contain only synthesized and inherited constructors, respectively.
Both are structurally equivalent to \ltiSynChk{\ltiT{}}, differing only in color.
If the same constructor appears more than once in a rule without annotation,
they are assumed to have the same color, and similarly for two occurrences of \ltiSynChk{\ltiT{}}.
}
\end{figure}
\begin{figure}
\begin{mathpar}
\infer[]
{}
{\lticsubtype{\ltiT{}}{\ltiT{}}}
\infer[]
{
\lticfsubtype{\ltiFuel{}}{\ltiT{1}}{\ltiT{2}}{\ltiFuelp{}}
\\
\lticfsubtype{\ltiFuelp{}}{\ltiT{2}}{\ltiT{3}}{\ltiFuelpp{}}
}
{\lticfsubtype{\ltiFuel{}}{\ltiT{1}}{\ltiT{3}}{\ltiFuelpp{}}
}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltitvar{}}}{\ltiChk{\ltitvar{}}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiBot}}{\ltiChk{\ltiBot}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiTop}}{\ltiChk{\ltiTop}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiBot}}{\ltiChk{\ltiTop}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiRec{\hastype{\ltivar{1}}{\ltiSynChk{\ltiT{1}}},..., \hastype{\ltivar{n}}{\ltiSynChk{\ltiT{n}}},
\hastype{\ltivar{n+1}}{\ltiChk{\ltiTop}},..., \hastype{\ltivar{m}}{\ltiChk{\ltiTop}}
}}}
{\ltiChk{\ltiRec{\hastype{\ltivar{1}}{\ltiSynChk{\ltiT{1}}}, ..., \hastype{\ltivar{n}}{\ltiSynChk{\ltiT{n}}}}}}
}
\infer[]
{}
{
\lticsubtype{\ltiSyn{\ltiArrowMono{\ltiSynChk{\ltiT{}}}{\ltiSynChk{\ltiS{}}}}}
{\ltiChk{\ltiArrowMono{\ltiSynChk{\ltiT{}}}{\ltiSynChk{\ltiS{}}}}}
}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiBot}}{\ltiChk{\ltitvar{}}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiBot}}
{\ltiChk{\ltiRec{\hastype{\ltivar{1}}{\ltiSyn{\ltiBot}}, ..., \hastype{\ltivar{n}}{\ltiSyn{\ltiBot}}}}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiBot}}
{\ltiChk{\ltiArrowMono{\ltiSyn{\ltiTop}}{\ltiSyn{\ltiBot}}}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltitvar{}}}
{\ltiChk{\ltiTop{}}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiRec{\hastype{\ltivar{1}}{\ltiChk{\ltiTop}}, ..., \hastype{\ltivar{n}}{\ltiChk{\ltiTop}}}}}
{\ltiChk{\ltiTop{}}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiArrowMono{\ltiChk{\ltiBot}}{\ltiChk{\ltiTop}}}}
{\ltiChk{\ltiTop}}
}
\infer[]
{
\lticfsubtype{\ltiFuel{1}}{\ltiT{1}}{\ltiTp{1}}{\ltiFuel{2}}
\\
...
\\
\lticfsubtype{\ltiFuel{n}}{\ltiT{n}}{\ltiTp{n}}{\ltiFuel{n+1}}
}
{
\lticfsubtype{\ltiFuel{1}}
{\ltiRec{\hastype{\ltivar{1}}{\ltiT{1}},...,\hastype{\ltivar{n}}{\ltiT{n}}}}
{\ltiRec{\hastype{\ltivar{1}}{\ltiTp{1}},...,\hastype{\ltivar{n}}{\ltiTp{n}}}}
{\ltiFuel{n+1}}
}
\infer[SC-Arrow]
{
\lticfsubtypeinv{\ltiFuel{}}{\ltiT{1}}{\ltiTp{1}}{\ltiFuelp{}}
\\
\lticfsubtype{\ltiFuelp{}}{\ltiT{2}}{\ltiTp{2}}{\ltiFuelpp{}}
}
{
\lticfsubtype{\ltiFuel{}}
{\ltiArrow{\ltiTp{1}}{\ova{\ltitvar{}}}{\ltiT{2}}}
{\ltiArrow{\ltiT{1}}{\ova{\ltitvar{}}}{\ltiTp{2}}}
{\ltiFuelpp{}}
}
% \infer[SC-ArrowMonoR]
% {
% \ltiagreeexcepttvars{\ltiSp{}}{\ova{\ltitvar{}}}{\ltiChk{\ltiS{}}}
% \\
% \lticsubtype{\ltiSp{}}
% {\ltireplace{\ova{\ltiR{}}}
% {\ova{\ltitvar{}}}
% {\ltiChk{\ltiS{}}}}
% \\
% \lticsubtype{\ltireplace{\ova{\ltiR{}}}
% {\ova{\ltitvar{}}}
% {\ltiSyn{\ltiT{}}}}
% {\ltiTp{}}
% \\\\
% \forall \ova{\ltiRp{}}, \ltiTpp{}.
% (
% {\lticsubtype{\ltiSp{}}
% {\ltireplace{\ova{\ltiRp{}}}
% {\ova{\ltitvar{}}}
% {\ltiChk{\ltiS{}}}}}
% \ \
% \wedge
% \ \
% {\lticsubtype{\ltireplace{\ova{\ltiRp{}}}
% {\ova{\ltitvar{}}}
% {\ltiSyn{\ltiT{}}}}
% {\ltiTpp{}}}
% \ \
% \wedge
% \ \
% {\lticoincideinherited{\ltiTpp{}}{\ltiTp{}}}
% \ \
% \Rightarrow
% \ \
% {\lticsubtype{\ltireplace{\ova{\ltiR{}}}
% {\ova{\ltitvar{}}}
% {\ltiSyn{\ltiT{}}}}
% {\ltireplace{\ova{\ltiRp{}}}
% {\ova{\ltitvar{}}}
% {\ltiChk{\ltiT{}}}}}
% )
% }
% {
% \lticsubtype{\ltiChk{\ltiArrow{\ltiSyn{\ltiS{}}}{\ova{\ltitvar{}}}{\ltiSyn{\ltiT{}}}}}
% {\ltiArrowMono{\ltiSp{}}{\ltiTp{}}}
% }
\\
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}}
{\ltiChk{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiBot}}
{\ltiChk{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}}}
\infer[]
{}
{\lticsubtype{\ltiSyn{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}}
{\ltiChk{\ltiTop}}
}
\infer[SC-Closure]
{
\lticfjudgement{\ltiFuel{}}
{\ltiEnv{},\ova{\ltitvar{}}, \hastype{\ltivar{}}{\ltiSyn{\ltiT{}}}}
{\ltiE{}}
{\ltiSynChk{\ltiS{}}}
{\ltiFuelp{}}
}
{
\lticfsubtype{\ltiFuel{}}
{\ltiSyn{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}}
{\ltiChk{\ltiArrow{\ltiSyn{\ltiT{}}}{\ova{\ltitvar{}}}{\ltiSynChk{\ltiS{}}}}}
{\ltiFuelp{}}
}
\end{mathpar}
\caption{Subtyping for colored types
\lticfsubtype{\ltiFuel{}}{\ltiT{}}{\ltiS{}}{\ltiFuelp{}},
with
\lticfsubtype{\ltiFuel{}}{\ltiT{}}{\ltiS{}}{\ltiFuel{}}
abbreviated
to \lticsubtype{\ltiT{}}{\ltiS{}}.
The upside-down \lticsubtypesymbol relation
is defined like \lticsubtypesymbol, except with inverted colors.
% TODO can't have this command in a caption (rotatebox incompatibility I think)
%The relation
%\ltiinvertedleq
%is defined similarly, except with inverted colors.
}
\end{figure}
\begin{figure}
\begin{mathpar}
\infer[(C-Var)]
{
\ltiEnvLookup{\ltiEnv{}}{\ltivar{}} = {\ltiSyn{\ltiT{}}}
}
{
\lticfjudgement{\ltiFuel{}}{\ltiEnv{}}{\ltivar{}}{\ltiSyn{\ltiT{}}}{\ltiFuel{}}
}
\infer[(C-Sub)]
{\lticfjudgement{\ltiFuel{1}}
{\ltiEnv{}}
{\ltiE{}}
{\ltiT{}}
{\ltiFuel{2}}
\\
\lticfsubtype{\ltiFuel{2}}{\ltiT{}}{\ltiTp{}}{\ltiFuel{3}}
}
{\lticfjudgement{\ltiFuel{1}}
{\ltiEnv{}}
{\ltiE{}}
{\ltiTp{}}
{\ltiFuel{3}}
}
\infer[(C-Abs)]
{\lticfjudgement{\ltiFuel{}}
{\ltiEnv{}, \ltiSyn{\ova{\ltitvar{}}}, \hastype{\ltivar{}}{\ltiSyn{\ltiT{}}}}
{\ltiE{}}
{\ltiS{}}
{\ltiFuelp{}}
\\
\ova{\ltitvar{}} \not\in \ltitv{\ltiE{}}
}
{\lticfjudgement{\ltiFuel{}}
{\ltiEnv{}}
{\ltiufun{\ltivar{}}{\ltiE{}}}
{\ltiChk{\ltiArrow{\ltiT{}}{\ova{\ltitvar{}}}{\ltiSynChk{\ltiS{}}}}}
{\ltiFuelp{}}
}
\infer [(C-UAbs)]
{\lticfjudgement{\ltiFuel{}}
{\ltiEnv{}, \hastype{\ltivar{}}{\ltiSyn{\ltiBot{}}}}
{\ltiE{}}
{\ltiS{}}
{\ltiFuelp{}}
}
{\lticfjudgement{\ltiFuel{}}
{\ltiEnv{}}
{\ltiufun{\ltivar{}}{\ltiE{}}}
{\ltiSyn{\ltiClosure{\ltiEnv{}}{\ltiufun{\ltivar{}}{\ltiE{}}}}}
{\ltiFuelp{}}
}
\infer [($\text{C-Abs}_{tp}$)]
{
\lticfjudgement{\ltiFuel{}}
{\ltiEnv{}, \ltiSyn{\ova{\ltitvar{}}}, \hastype{\ltivar{}}{\ltiSyn{\ltiT{}}}}
{\ltiE{}}
{\ltiS{}}
{\ltiFuelp{}}
}
{
\lticfjudgement{\ltiFuel{}}
{\ltiEnv{}}
{\ltifun{\ova{\ltitvar{}}}{\ltivar{}}{\ltiT{}}{\ltiE{}}}
{\ltiSyn{\ltiArrow{\ltiT{}}{\ova{\ltitvar{}}}{\ltiSynChk{\ltiS{}}}}}
{\ltiFuelp{}}
}
\infer [(C-Sel)]
{
\lticfjudgement{\ltiFuel{}}
{\ltiEnv{}}
{\ltiE{}}
{\ltiChk{\ltiRec{\hastype{\ltivar{}}{\ltiSynChk{\ltiT{}}}}}}
{\ltiFuelp{}}
}
{
\lticfjudgement{\ltiFuel{}}
{\ltiEnv{}}
{\ltisel{\ltiE{}}{\ltivar{}}}
{\ltiT{}}
{\ltiFuelp{}}
}
\infer [(C-Rec)]
{
\lticfjudgement{\ltiFuel{i}}
{\ltiEnv{}}
{\ltiE{i}}
{\ltiSynChk{\ltiT{i}}}
{\ltiFuel{i+1}}
}
{
\lticfjudgement{\ltiFuel{1}}
{\ltiEnv{}}
{\ltiRec{\ova{\ltivar{} = \ltiE{}}^n}}
{\ltiSyn{\ltiRec{\ova{\hastype{\ltivar{}}{\ltiSynChk{\ltiT{}}}}^n}}}
{\ltiFuel{n+1}}
}
\infer [($\text{C-App}_{tp}$)]
{
\lticfjudgement{\ltiFuel{1}}
{\ltiEnv{}}
{\ltiF{}}
{\ltiChk{\ltiArrow{\ltiSyn{\ltiS{}}}{\ova{\ltitvar{}}}{\ltiSyn{\ltiT{}}}}}
{\ltiFuel{2}}
\\
\lticfjudgement{\ltiFuel{2}}
{\ltiEnv{}}
{\ltiE{}}
{\ltireplace{\ova{\ltiR{}}}
{\ova{\ltitvar{}}}
{\ltiChk{\ltiS{}}}}
{\ltiFuel{3}}
}
{
\lticfjudgement{\ltiFuel{1}}
{\ltiEnv{}}
{\ltiappinst{\ltiF{}}{\ova{\ltiR{}}}{\ltiE{}}}
{\ltireplace{\ova{\ltiR{}}}
{\ova{\ltitvar{}}}
{\ltiChk{\ltiT{}}}}
{\ltiFuel{3}}
}
\infer [(C-App)]
{
\lticjudgement{\ltiEnv{}}
{\ltiF{}}
{\ltiChk{\ltiArrow{\ltiSyn{\ltiS{}}}{\ova{\ltitvar{}}}{\ltiSyn{\ltiT{}}}}}
\\
\lticjudgement{\ltiEnv{}}
{\ltiE{}}
{\ltiSp{}}
\\
\ltiagreeexcepttvars{\ltiSp{}}{\ova{\ltitvar{}}}{\ltiChk{\ltiS{}}}
\\
\lticsubtype{\ltiSp{}}
{\ltireplace{\ova{\ltiR{}}}
{\ova{\ltitvar{}}}
{\ltiChk{\ltiS{}}}}
\\
\lticsubtype{\ltireplace{\ova{\ltiR{}}}
{\ova{\ltitvar{}}}
{\ltiSyn{\ltiT{}}}}
{\ltiTp{}}
\\\\
\forall \ova{\ltiRp{}}, \ltiTpp{}.
(
{\lticsubtype{\ltiSp{}}
{\ltireplace{\ova{\ltiRp{}}}
{\ova{\ltitvar{}}}
{\ltiChk{\ltiS{}}}}}
\ \
\wedge
\ \
{\lticsubtype{\ltireplace{\ova{\ltiRp{}}}
{\ova{\ltitvar{}}}
{\ltiSyn{\ltiT{}}}}
{\ltiTpp{}}}
\ \
\wedge
\ \
{\lticoincideinherited{\ltiTpp{}}{\ltiTp{}}}
\ \
\Rightarrow
\ \
{\lticsubtype{\ltireplace{\ova{\ltiR{}}}
{\ova{\ltitvar{}}}
{\ltiSyn{\ltiT{}}}}
{\ltireplace{\ova{\ltiRp{}}}
{\ova{\ltitvar{}}}
{\ltiChk{\ltiT{}}}}}
)
}
{
\lticjudgement{\ltiEnv{}}
{\ltiapp{\ltiF{}}{\ltiE{}}}
{\ltiTp{}}
}
% \infer [(C-App)]
% {
% \lticfjudgement{\ltiFuel{1}}
% {\ltiEnv{}}
% {\ltiF{}}
% {\ltiR{}}
% {\ltiFuel{2}}
% \\\\
% \lticfjudgement{\ltiFuel{2}}
% {\ltiEnv{}}
% {\ltiE{}}
% {\ltiSynChk{\ltiS{}}}
% {\ltiFuel{3}}
% \\\\
% \lticfsubtype{\ltiFuel{3}}{\ltiR{}}
% {\ltiChk{\ltiArrowMono{\ltiSynChk{\ltiS{}}}{\ltiSynChk{\ltiT{}}}}}
% {\ltiFuel{4}}
% }
% {
% \lticfjudgement{\ltiFuel{1}}
% {\ltiEnv{}}
% {\ltiapp{\ltiF{}}{\ltiE{}}}
% {\ltiSynChk{\ltiT{}}}
% {\ltiFuel{4}}
% }
\infer[C-AppClosure]
{
\lticfjudgement{\ltiFuel{1}}
{\ltiEnv{}}
{\ltiF{}}
{\ltiChk{\ltiClosure{\ltiEnvp{}}{\ltiufun{\ltivar{}}{\ltiEp{}}}}}
{\ltiFuel{2}}
\\
\lticfjudgement{\ltiFuel{2}}
{\ltiEnv{}}
{\ltiE{}}
{\ltiSynChk{\ltiS{}}}
{\ltiFuel{3}}
\\
\ltiFuel{3} > 0
\\
\lticfjudgement{\ltiFuel{3} - 1}
{\ltiEnvp{},\hastype{\ltivar{}}{\ltiSyn{\ltiS{}}}}
{\ltiEp{}}
{\ltiSynChk{\ltiT{}}}
{\ltiFuel{4}}
}
{
\lticfjudgement{\ltiFuel{1}}
{\ltiEnv{}}
{\ltiapp{\ltiF{}}{\ltiE{}}}
{\ltiSynChk{\ltiT{}}}
{\ltiFuel{4}}
}
\end{mathpar}
\caption{Polymorphic Colored Type System rules
\lticfjudgement{\ltiFuel{}}
{\ltiEnv{}}
{\ltiE{}}
{\ltiT{}}
{\ltiFuel{}},
abbreviated \lticjudgement{\ltiEnv{}}
{\ltiE{}}
{\ltiT{}}
when fuel is propagated left-to-right through premises,
as in \textsc{C-App}.
}
\end{figure}
% FIXME these 2 paragraphs used to apply to the internal language, now just the colored type system
%The last three rules
%of \figref{symbolic:figure:internal-language-type-system}
%shows our extensions to the type system.
%\textsc{I-UAbs} assigns unannotated functions a symbolic closure type.
%The rules \textsc{I-UApp} and \textsc{S-Closure} together handle
%checking symbolic closures.
%In \textsc{I-UApp}, the argument type \ltiS{} (which itself might contain
%symbolic closures via \textsc{I-UAbs}) is
%propagated to
%\textsc{S-Closure}
%where it is used as the parameter type to check the body of the symbolic
%closure.
%Moving the checking of a symbolic closure to
%\textsc{S-Closure},
%(instead of \textsc{I-UApp})
%allows symbolic closures to participate with other rules.
%In particular, checking \clj{(f (fn [x] (inc x)))},
%where \clj{f : [[Int -> Int] -> Int]},
%uses the \textsc{I-App} application rule,
%but manages to propagate \clj{x : Int}
%via \textsc{S-Closure}.
%
%Note that \textsc{S-Closure} uses the symbolic closure's definition type environment
%for type checking. The definition environment is (somewhat analogously to runtime closures)
%``captured'' by \textsc{I-UAbs},
%and carefully propagated in select rules.
%For example, in \textsc{S-UApp}
%the definition environment \ltiEnvp{} is passed to subtyping,
%rather than the application environment \ltiEnv{}.
%This arrangement correctly type checks \figref{symbolic:example:closure-red}.
\subsection{Type Inference Algorithm}
\begin{figure}
$$
\begin{array}{lrll}
\ltiC{} &::=& {\ltiCSet{\ltiCEntry{\ltiT{}}{\ltitvar{1}}{\ltiT{}},...,\ltiCEntry{\ltiT{}}{\ltitvar{n}}{\ltiT{}}}}
&\mbox{Constraint sets} \\
\end{array}
$$
\caption{Type Inference Syntax}
\end{figure}
\begin{figure}
\begin{mathpar}
\infer [(W-Var)]
{}
{
\ltijudgement{\ltiP{}}{\ltiEnv{}}{\ltivar{}}
{\ltimatchsuper{\ltiEnvLookup{\ltiEnv{}}{\ltivar{}}}{\ltiP{}}}
}
\infer [(W-Abs)]
{
\ltijudgement{\ltiP{}}
{\ltiEnv{},\hastype{\ltivar{}}{\ltiT{}}}
{\ltiE{}}
{\ltiS{}}
}
{
\ltijudgement{\ltiArrowMono{\ltiT{}}{\ltiP{}}}
{\ltiEnv{}}
{\ltiufun{\ltivar{}}{\ltiE{}}}
{\ltiArrowMono{\ltiT{}}{\ltiS{}}}
}
\infer [($\text{W-App}_{\ltiBot}$)]
{
\ltijudgement{\ltiWild{}}
{\ltiEnv{}}
{\ltiF{}}
{\ltiBot}
\\
\ltijudgement{\ltiTop}
{\ltiEnv{}}
{\ltiE{}}
{\ltiS{}}
}
{
\ltijudgement{\ltiP{}}{\ltiEnv{}}
{\ltiapp{\ltiF{}}{\ltiE{}}}
{\ltimatchsuper{\ltiBot}
{\ltiP{}}}
}
\infer [(W-Sel)]
{
\ltijudgement{\ltiRec{\hastype{\ltivar{}}{\ltiP{}}}}{\ltiEnv{}}
{\ltiE{}}
{\ltiRec{\hastype{\ltivar{}}{\ltiT{}}}}
}
{
\ltijudgement{\ltiP{}}{\ltiEnv{}}
{\ltisel{\ltiE{}}{\ltivar{}}}
{\ltiT{}}
}
\infer [($\text{W-Rec}_{\ltiWild{}}$)]
{
\ltijudgement{\ltiWild{}}
{\ltiEnv{}}
{\ltiE{1}}
{\ltiT{1}}
\\
...
\\
\ltijudgement{\ltiWild{}}
{\ltiEnv{}}
{\ltiE{n}}
{\ltiT{n}}
}
{
\ltijudgement{\ltiWild{}}
{\ltiEnv{}}
{\ltiRec{\ltivar{1} = \ltiE{1}, ..., \ltivar{n} = \ltiE{n}}}
{\ltiRec{\hastype{\ltivar{1}}{\ltiT{1}}, ..., \hastype{\ltivar{n}}{\ltiT{n}}}}
}
\infer [($\text{W-Rec}_{\ltiTop}$)]
{
\ltijudgement{\ltiTop}
{\ltiEnv{}}
{\ltiE{1}}
{\ltiT{1}}
\\
...
\\
\ltijudgement{\ltiTop}
{\ltiEnv{}}
{\ltiE{n}}
{\ltiT{n}}
}
{
\ltijudgement{\ltiTop}
{\ltiEnv{}}
{\ltiRec{\ltivar{1} = \ltiE{1}, ..., \ltivar{n} = \ltiE{n}}}
{\ltiRec{\hastype{\ltivar{1}}{\ltiT{1}}, ..., \hastype{\ltivar{n}}{\ltiT{n}}}}
}
\infer [(W-Rec)]
{
\ltijudgement{\ltiP{1}}
{\ltiEnv{}}
{\ltiE{1}}
{\ltiT{1}}
\\
...
\\
\ltijudgement{\ltiP{m}}
{\ltiEnv{}}
{\ltiE{m}}
{\ltiT{m}}
\\\\
\ltijudgement{\ltiTop}
{\ltiEnv{}}
{\ltiE{m+1}}
{\ltiT{m+1}}
\\
...
\\
\ltijudgement{\ltiTop}
{\ltiEnv{}}
{\ltiE{n}}
{\ltiT{n}}
}
{
\ltijudgement{\ltiRec{\hastype{\ltivar{1}}{\ltiP{1}}, ..., \hastype{\ltivar{m}}{\ltiP{m}}}}
{\ltiEnv{}}
{\ltiRec{\ltivar{1} = \ltiE{1}, ..., \ltivar{n} = \ltiE{n}}}
{\ltiRec{\hastype{\ltivar{1}}{\ltiT{1}}, ..., \hastype{\ltivar{m}}{\ltiT{m}}}}
}
\end{mathpar}
\caption{Type Inference algorithm rules
$\ltijudgement{\ltiP{}}
{\ltiEnv{}}
{\ltiE{}}
{\ltiT{}}$
}
\end{figure}
\section{Colored local type inference}
\begin{figure}
$$
\begin{array}{lrll}
\ltiE{}, \ltiF{} &::=& ... \alt
\ltifun{\ova{\ltitvar{}}}{\ltivar{}}{\ltiT{}}{\ltiE{}} \alt
\ltiappinst{\ltiF{}}{\ova{\ltiT{}}}{\ltiE{}}
&\mbox{Terms} \\
\ltiT{}, \ltiS{}, \ltiR{} &::=& ... \alt
\ltiArrow{\ltiT{}}{\ova{\ltitvar{}}}{\ltiS{}} \alt
&\mbox{Types} \\
\end{array}
$$
\caption{Internal Language syntax extensions for polymorphic functions}
\label{symbolic:figure:internal-language}
\end{figure}
\begin{figure}
$$
\begin{array}{lrll}
\end{array}
$$
\caption{Syntax Extensions for type argument inference.}
\end{figure}
\begin{figure}
\begin{mathpar}
\end{mathpar}
\caption{Polymorphic Colored Type System rules
$\lticjudgement{\ltiEnv{}}
{\ltiE{}}
{\ltiT{}}$
}
\end{figure}
\begin{figure}
\begin{mathpar}
\infer [($\text{W-Abs}_{tp,?}$)]
{
\ltijudgement{\ltiWild{}}
{\ltiEnv{},\overline{\ltitvar{}},\hastype{\ltivar{}}{\ltiT{}}}
{\ltiE{}}
{\ltiS{}}
}
{
\ltijudgement{\ltiWild{}}{\ltiEnv{}}
{\ltifun{\overline{\ltitvar{}}}{\ltivar{}}{\ltiT{}}{\ltiE{}}}
{\ltiArrow{\ltiT{}}{\overline{\ltitvar{}}}{\ltiS{}}}
}
\infer [($\text{W-Abs}_{tp}$)]
{
\ltijudgement{\ltiPp{}}{\ltiEnv{},\overline{\ltitvar{}},\hastype{\ltivar{}}{\ltiT{}}}
{\ltiE{}}
{\ltiS{}}
}
{
\ltijudgement{\ltiArrow{\ltiP{}}{\overline{\ltitvar{}}}{\ltiPp{}}}
{\ltiEnv{}}
{\ltifun{\overline{\ltitvar{}}}{\ltivar{}}{\ltiT{}}{\ltiE{}}}
{\ltimatchsuper{\ltiArrow{\ltiT{}}{\overline{\ltitvar{}}}{\ltiS{}}}
{\ltiArrow{\ltiP{}}{\overline{\ltitvar{}}}{\ltiPp{}}}}
}
\infer [($\text{W-Abs}_{tp,\ltiTop}$)]
{
\ltijudgement{\ltiTop}{\ltiEnv{},\overline{\ltitvar{}},\hastype{\ltivar{}}{\ltiT{}}}
{\ltiE{}}
{\ltiS{}}
}
{
\ltijudgement{\ltiTop}
{\ltiEnv{}}
{\ltifun{\overline{\ltitvar{}}}{\ltivar{}}{\ltiT{}}{\ltiE{}}}
{\ltiTop}
}
\infer [(W-AbsPoly)]
{
\ltijudgement{\ltiP{}}
{\ltiEnv{},\overline{\ltitvar{}},\hastype{\ltivar{}}{\ltiT{}}}
{\ltiE{}}
{\ltiS{}}
}
{
\ltijudgement{\ltiArrow{\ltiT{}}{\overline{\ltitvar{}}}{\ltiP{}}}
{\ltiEnv{}}
{\ltiufun{\ltivar{}}{\ltiE{}}}
{\ltiArrow{\ltiT{}}{\overline{\ltitvar{}}}{\ltiS{}}}
}
\infer [($\text{W-App}_{tp}$)]
{
\ltijudgement{\ltiWild{}}
{\ltiEnv{}}
{\ltiF{}}
{\ltiArrow{\ltiS{}}{\overline{\ltitvar{}}}{\ltiT{}}}
\\
\ltijudgement{\ltireplace{\overline{\ltiR{}}}{\overline{\ltitvar{}}}{\ltiS{}}}
{\ltiEnv{}}
{\ltiE{}}
{\ltireplace{\overline{\ltiR{}}}{\overline{\ltitvar{}}}{\ltiS{}}}
}
{
\ltijudgement{\ltiP{}}{\ltiEnv{}}
{\ltiappinst{\ltiF{}}{\overline{\ltiR{}}}{\ltiE{}}}
{\ltimatchsuper{\ltireplace{\overline{\ltiR{}}}{\overline{\ltitvar{}}}{\ltiT{}}}
{\ltiP{}}}
}
\infer [($\text{W-App}_{tp,\ltiBot}$)]
{
\ltijudgement{\ltiWild{}}
{\ltiEnv{}}
{\ltiF{}}
{\ltiBot}
\\
\ltijudgement{\ltiTop}
{\ltiEnv{}}
{\ltiE{}}
{\ltiS{}}
}
{
\ltijudgement{\ltiP{}}{\ltiEnv{}}
{\ltiappinst{\ltiF{}}{\overline{\ltiR{}}}{\ltiE{}}}
{\ltimatchsuper{\ltiBot}
{\ltiP{}}}
}
\end{mathpar}
\caption{Extensions to Type Inference algorithm for polymorphic functions.
}
\end{figure}
\begin{figure}
\begin{mathpar}
\infer [(W-App)]
{
\ltijudgement{\ltiWild{}}{\ltiEnv{}}{\ltiF{}}
{\ltiArrow{\ltiS{}}{\overline{\ltitvar{}}}{\ltiT{}}}
\\
\ltijudgement{\ltireplace{\ltiWild{}}{\overline{\ltitvar{}}}{\ltiS{}}}
{\ltiEnv{}}
{\ltiE{}}
{\ltiSp{}}
\\\\
\ltigenconstraint{}{\overline{\ltitvar{}}}{\ltiSp{}}{\ltiS{}}{\ltiC{1}}
\\
\ltigenconstraint{}{\overline{\ltitvar{}}}{\ltiT{}}{\ltimatchsub{\ltiTop}{\ltiP{}}}{\ltiC{2}}
}
{
\ltijudgement{\ltiP{}}{\ltiEnv{}}
{\ltiapp{\ltiF{}}{\ltiE{}}}
{\ltiSubst{\ltiCUnion{\ltiC{1}}{\ltiC{2}}}{\ltiT{}}{\ltimatchsuper{\ltiT{}}{\ltiP{}}}}
}
\end{mathpar}
\caption{
Extension to type inference algorithm for type argument reconstruction.
}
\end{figure}