Fix Related Work. Adapt Soundness proof
This commit is contained in:
Andreas Stadelmeier
parent
c86dc891f3
commit
0e157cf427
@ -63,14 +63,14 @@ indeed the transitiv closure of $\QMextends{\theta} \sub \theta'$ and $\theta' \
|
||||
algorithm for this type system, which he proved as sound and complete.
|
||||
|
||||
The problem of his type system is in the lossing reflexivity as shown in
|
||||
example \ref{intro-example1}. First approaches to solve this problem he gave in
|
||||
example \ref{lst:intro-example-typeless}. First approaches to solve this problem he gave in
|
||||
\cite{plue24_1}, where he fixes that no pairwise different nodes in the
|
||||
abstract syntax gets the same type variable and that no pairwise different type
|
||||
variables are equalized. In \cite{PH23} he showed how his type inference
|
||||
algorithm suffices theese properties.
|
||||
|
||||
In Pl\"umicke's work there is indeed a formal definition of the subtying
|
||||
ordering and a correctness proof of the type unification algorithms but no
|
||||
soundness proof of the type system, itself. Therefore we choose for our type
|
||||
inference algorithms with wildcars the approach of ???????
|
||||
% In Pl\"umicke's work there is indeed a formal definition of the subtying
|
||||
% ordering and a correctness proof of the type unification algorithms but no
|
||||
% soundness proof of the type system, itself. Therefore we choose for our type
|
||||
% inference algorithms with wildcars the approach of ???????
|
||||
|
||||
|
||||
182
soundness.tex
182
soundness.tex
@ -95,21 +95,21 @@
|
||||
\unify{}'s type solutions for a constraint set generated by $\typeExpr{}$ are correct.
|
||||
\begin{description}
|
||||
\item[if] $\typeExpr{}(\mtypeEnvironment{}, \texttt{e}, \tv{a}) = (\Delta', C)$
|
||||
and $(\Delta_u, \sigma) = \unify{}(\Delta', C)$ and let $\Delta= \Delta_u \cup \Delta'$
|
||||
$\Delta, \Delta' \vdash $
|
||||
and $(\Delta_u, \sigma) = \unify{}(\Delta', C)$ % and let $\Delta= \Delta_u \cup \Delta'$
|
||||
% $\Delta, \Delta' \vdash $
|
||||
% , with $C = \set{ \overline{ \type{S} \lessdot \type{T} } \cup \overline{ \type{S'} \lessdotCC \type{T'} } }$
|
||||
% and $\vdash \ol{L} : \mtypeEnvironment{}$
|
||||
% and $\Gamma \subseteq \mtypeEnvironment{}$
|
||||
% \item[given] a $(\Delta, \sigma)$ with $\Delta \vdash \overline{\sigma(\type{S}) <: \sigma(\type{T})}$
|
||||
% and there exists a $\Delta'$ with $\Delta, \Delta' \vdash \overline{\CC{}(\sigma(\type{S'})) <: \sigma(\type{T'})}$
|
||||
%\item[then] there is a completion $|\texttt{e}|$ with $\Delta|\Gamma \vdash |\texttt{e}| : \sigma(\tv{a})$
|
||||
\item[then] $\Delta|\Gamma \vdash \texttt{e} : \sigma(\tv{a})$
|
||||
\item[then] $\Delta,\Delta'|\Gamma \vdash \texttt{e} : \sigma(\tv{a})$
|
||||
\end{description}
|
||||
\end{lemma}
|
||||
Regular type placeholders represent type annotations.
|
||||
These are the only types a \wildFJ{} program needs to be correctly typed.
|
||||
The type placeholders flagged as wildcard placeholders are intermediate types
|
||||
used in let statements and as type parameters for generic method calls.
|
||||
% Regular type placeholders represent type annotations.
|
||||
% These are the only types a \wildFJ{} program needs to be correctly typed.
|
||||
% The type placeholders flagged as wildcard placeholders are intermediate types
|
||||
% used in let statements and as type parameters for generic method calls.
|
||||
|
||||
%Unify needs to return S aswell and guarantee that the \Delta' environment are the wildcards
|
||||
% only used inside the constraint the wildcard variable occurs
|
||||
@ -121,33 +121,44 @@ used in let statements and as type parameters for generic method calls.
|
||||
By structural induction over the expression $\texttt{e}$.
|
||||
\begin{description}
|
||||
\item[$\texttt{let}\ \texttt{x} = \texttt{t}_1 \ \texttt{in}\ \expr{t}_2$]
|
||||
$\Delta|\Gamma \vdash \expr{t}_1: \type{T}_1$ by assumption.
|
||||
The body of the let statement will only use the free variables provided by $\Delta$ and $\Delta'$ (see lemma \ref{lemma:wildcardsStayInScope}).
|
||||
$\text{dom}(\Delta') \subseteq \text{fv}(\type{N})$ by lemma \ref{lemma:wildcardWellFormedness}.
|
||||
$\Delta \vdash \type{T}, \wcNtype{\Delta'}{N} \ \ok$ by lemma \ref{lemma:unifyWellFormedness}.
|
||||
$\Delta|\Gamma \vdash \expr{t}_1: \sigma(\tv{e}_1)$ by assumption.
|
||||
$\Delta \vdash \sigma(\tv{e}_1) <: \wcNtype{\Delta'}{N}$ by constraint $\tv{e}_1 \lessdot \tv{x}$ and lemma \ref{lemma:unifySoundness}.
|
||||
The body of the let statement will only use the free variables provided by $\Delta$ and $\Delta'$.
|
||||
We can prove this by lemma \ref{lemma:tvsNoFV} and the fact that the wildcard placeholders used generated the body of the let statement are not used outside of it.
|
||||
Therefore we can say $\Delta,\Delta' | \Gamma, \expr{x}:\type{N} \vdash \expr{t}_2 : \sigma(\tv{e}_2)$ by assumption
|
||||
and $\Delta, \Delta' \vdash \sigma(\tv{e}_2) <: \sigma(\tv{a})$ by lemma \ref{lemma:unifySoundness}
|
||||
given the constraint $\tv{e}_2 \lessdot \tv{a}$.
|
||||
\item[$\expr{v}.\texttt{f}$]
|
||||
We are alowwed to use capture conversion for $\expr{v}$ here.
|
||||
$\Delta \vdash \expr{v} : \sigma(\tv{a})$ by assumption.
|
||||
$\Delta \vdash \sigma(\tv{a}) <: \sigma(\exptype{C}{\ol{\wtv{a}}})$ and
|
||||
$\Delta \vdash \type{U}_i <: \sigma(\tv{a})$,
|
||||
because of the constraints $[\overline{\wtv{a}}/\ol{X}]\type{T} \lessdot \tv{a}$, $\tv{r} \lessdotCC \exptype{C}{\ol{\wtv{a}}}$ and lemma \ref{lemma:unifySoundness}.
|
||||
$\textit{fields}(\sigma(\exptype{C}{\overline{\wtv{a}}})) = \sigma([\overline{\wtv{a}}/\ol{X}]\type{T})$.
|
||||
% \item[$\texttt{let}\ \texttt{x} = \texttt{e} \ \texttt{in} \ \texttt{x}.\texttt{f}$]
|
||||
% $\Delta|\Gamma \vdash \expr{e}: \type{T}$ by assumption.
|
||||
% $\text{dom}(\Delta') \subseteq \text{fv}(\type{N})$ by lemma \ref{lemma:wildcardWellFormedness}.
|
||||
% $\Delta, \Delta' | \Gamma, \expr{x} : \type{T} \vdash \texttt{x}.\texttt{f}$
|
||||
% \item[$\texttt{e}.\texttt{f}$] Let $\sigma(\tv{r}) = \wcNtype{\Delta_c}{N}$,
|
||||
% then $\Delta|\Gamma \vdash \texttt{e} : \wcNtype{\Delta_c}{N}$ by assumption.
|
||||
% $\Delta', \Delta, \Delta_c \vdash \type{N} <: \sigma(\exptype{C}{\overline{\wtv{a}}})$ by premise.
|
||||
% %Let $\sigma(\tv{r}) = \wcNtype{\Delta'}{N}$.
|
||||
% %Let $\sigma([\ol{\wtv{a}}/\ol{X}]\type{T}) = \wcNtype{\Delta_t}{N_t}$.
|
||||
|
||||
\item[$\texttt{let}\ \texttt{x} = \texttt{e} \ \texttt{in} \ \texttt{x}.\texttt{f}$]
|
||||
$\Delta|\Gamma \vdash \expr{e}: \type{T}$ by assumption.
|
||||
$\text{dom}(\Delta') \subseteq \text{fv}(\type{N})$ by lemma \ref{lemma:wildcardWellFormedness}.
|
||||
$\Delta, \Delta' | \Gamma, \expr{x} : \type{T} \vdash \texttt{x}.\texttt{f}$
|
||||
\item[$\texttt{e}.\texttt{f}$] Let $\sigma(\tv{r}) = \wcNtype{\Delta_c}{N}$,
|
||||
then $\Delta|\Gamma \vdash \texttt{e} : \wcNtype{\Delta_c}{N}$ by assumption.
|
||||
$\Delta', \Delta, \Delta_c \vdash \type{N} <: \sigma(\exptype{C}{\overline{\wtv{a}}})$ by premise.
|
||||
%Let $\sigma(\tv{r}) = \wcNtype{\Delta'}{N}$.
|
||||
%Let $\sigma([\ol{\wtv{a}}/\ol{X}]\type{T}) = \wcNtype{\Delta_t}{N_t}$.
|
||||
% The completion of $|\texttt{e}.\texttt{f}|$ is $\texttt{let}\ \texttt{x} = \texttt{e} : \wcNtype{\Delta_c}{N}\ \texttt{in} \ \texttt{x}.\texttt{f}$
|
||||
|
||||
The completion of $|\texttt{e}.\texttt{f}|$ is $\texttt{let}\ \texttt{x} = \texttt{e} : \wcNtype{\Delta_c}{N}\ \texttt{in} \ \texttt{x}.\texttt{f}$
|
||||
% We now show
|
||||
% $\Delta|\Gamma \vdash \texttt{let}\ \texttt{x} = \texttt{e} : \wcNtype{\Delta_c}{N}\ \texttt{in} \ \texttt{x}.\texttt{f} : \sigma(a)$
|
||||
% by the T-Field rule.
|
||||
% $\Delta \vdash \wcNtype{\Delta_c}{N} <: \wcNtype{\Delta_c}{N}$ by S-Refl.
|
||||
% $\Delta, \Delta_c \vdash \type{U}_i <: \sigma(\tv{a})$,
|
||||
% because of the constraint $[\overline{\wtv{a}}/\ol{X}]\type{T} \lessdot \tv{a}$ and lemma \ref{lemma:unifySoundness}.
|
||||
% $\textit{fields}(\sigma(\exptype{C}{\overline{\wtv{a}}})) = \sigma([\overline{\wtv{a}}/\ol{X}]\type{T})$
|
||||
% and $\text{fv}(\type{U}_i) \subseteq \text{fv}(\type{N})$ by definition of $\textit{fields}$.
|
||||
|
||||
We now show
|
||||
$\Delta|\Gamma \vdash \texttt{let}\ \texttt{x} = \texttt{e} : \wcNtype{\Delta_c}{N}\ \texttt{in} \ \texttt{x}.\texttt{f} : \sigma(a)$
|
||||
by the T-Field rule.
|
||||
$\Delta \vdash \wcNtype{\Delta_c}{N} <: \wcNtype{\Delta_c}{N}$ by S-Refl.
|
||||
$\Delta, \Delta_c \vdash \type{U}_i <: \sigma(\tv{a})$,
|
||||
because of the constraint $[\overline{\wtv{a}}/\ol{X}]\type{T} \lessdot \tv{a}$ and lemma \ref{lemma:unifySoundness}.
|
||||
$\textit{fields}(\sigma(\exptype{C}{\overline{\wtv{a}}})) = \sigma([\overline{\wtv{a}}/\ol{X}]\type{T})$
|
||||
and $\text{fv}(\type{U}_i) \subseteq \text{fv}(\type{N})$ by definition of $\textit{fields}$.
|
||||
|
||||
$\text{dom}(\Delta_c) \subseteq \text{fv}{\type{N}}$ by lemma \ref{lemma:tvsNoFV}.
|
||||
% $\text{dom}(\Delta_c) \subseteq \text{fv}{\type{N}}$ by lemma \ref{lemma:tvsNoFV}.
|
||||
|
||||
% X.List<X> <. List<a?>
|
||||
% $\sigma(\ol{\tv{r}}) = \overline{\wcNtype{\Delta}{N}}$,
|
||||
@ -183,55 +194,65 @@ By structural induction over the expression $\texttt{e}$.
|
||||
% %Easy, because unify only generates substitutions for normal type placeholders which are OK
|
||||
|
||||
\item[$\texttt{e}.\texttt{m}(\ol{e})$]
|
||||
Lets have a look at the case where the receiver and parameter types are all named types.
|
||||
So $\sigma(\ol{\tv{r}}) = \ol{T} = \ol{\wcNtype{\Delta_u}{N}}$ and $\sigma(\tv{r}) = \type{T}_r = \wcNtype{\Delta_u}{N}$
|
||||
and a $\Delta'$, $\overline{\Delta}$ where $\Delta' \subseteq \Delta_u$, $\overline{\Delta \subseteq \Delta_u}$
|
||||
and $\text{dom}(\Delta') = (\text{fv}(\type{N})/\Delta)$, $\overline{\text{dom}(\Delta) = (\text{fv}(\type{N})/\Delta)}$ in
|
||||
Proof is analog to field access, except the $\Delta \vdash \ol{S}\ \ok$ premise.
|
||||
We know that $\unify{}(\Delta, [\overline{\wtv{b}}/\ol{Y}]\set{
|
||||
\ol{\tv{e}} \lessdotCC \ol{T}, \type{T} \lessdot \tv{a},
|
||||
\ol{Y} \lessdot \ol{N} } \cup C) = (\Delta', \sigma)$
|
||||
and if we assume $\sigma(\ol{\tv{e}}) = \ol{\wcNtype{\Delta}{N'}}$
|
||||
then the call to $\unify{}(\Delta \cup \overline{\Delta}, [\overline{\ntv{b}}/\ol{Y}]\set{
|
||||
\ol{N'} \lessdot \ol{T}, \type{T} \lessdot \tv{a},
|
||||
\ol{Y} \lessdot \ol{N} } \cup C)$ succeeds with $\overline{\ntv{b}}$ being normal placeholders this time,
|
||||
which proofs $\Delta \vdash \ol{S}\ \ok$ via lemma \ref{lemma:unifyWellFormedness}.
|
||||
%TODO: why does this call succeed?
|
||||
% Lets have a look at the case where the receiver and parameter types are all named types.
|
||||
% So $\sigma(\ol{\tv{r}}) = \ol{T} = \ol{\wcNtype{\Delta_u}{N}}$ and $\sigma(\tv{r}) = \type{T}_r = \wcNtype{\Delta_u}{N}$
|
||||
% and a $\Delta'$, $\overline{\Delta}$ where $\Delta' \subseteq \Delta_u$, $\overline{\Delta \subseteq \Delta_u}$
|
||||
% and $\text{dom}(\Delta') = (\text{fv}(\type{N})/\Delta)$, $\overline{\text{dom}(\Delta) = (\text{fv}(\type{N})/\Delta)}$ in
|
||||
|
||||
$\texttt{let}\ \texttt{x} : \wcNtype{\Delta'}{N} = \texttt{e} \ \texttt{in}\
|
||||
\texttt{let}\ \ol{x} : \overline{\wcNtype{\Delta'}{N}} = \ol{e} \ \texttt{in}\ \texttt{x}.\texttt{m}(\ol{x})$
|
||||
% $\texttt{let}\ \texttt{x} : \wcNtype{\Delta'}{N} = \texttt{e} \ \texttt{in}\
|
||||
% \texttt{let}\ \ol{x} : \overline{\wcNtype{\Delta'}{N}} = \ol{e} \ \texttt{in}\ \texttt{x}.\texttt{m}(\ol{x})$
|
||||
|
||||
%TODO: show that this type exists \wcNtype{\Delta'}{N} (a \Delta' which only contains free variables of the type N)
|
||||
% and which is a supertype of T_r (so no free variables in T_r)
|
||||
% Solution: Make this a lemma and guarantee that Unify does only create types Delta.T with \Delta subset fv(T)
|
||||
% This is possible because free variables (except those in \Delta_in) are never used in wildcard environments
|
||||
% %TODO: show that this type exists \wcNtype{\Delta'}{N} (a \Delta' which only contains free variables of the type N)
|
||||
% % and which is a supertype of T_r (so no free variables in T_r)
|
||||
% % Solution: Make this a lemma and guarantee that Unify does only create types Delta.T with \Delta subset fv(T)
|
||||
% % This is possible because free variables (except those in \Delta_in) are never used in wildcard environments
|
||||
|
||||
%By lemma \ref{lemma:unifyWeakening}
|
||||
We know that
|
||||
$\unify{}(\Delta, [\ol{\tv{a}}/\ol{X}][\ol{\tv{b}}/\ol{Y}]\set{ \overline{\wcNtype{\Delta}{N}} \lessdotCC \ol{T}, \wcNtype{\Delta'}{N} \lessdotCC \exptype{C}{\ol{X}}})$
|
||||
has a solution.
|
||||
Also $\overline{\wcNtype{\Delta}{N}} \lessdot \ol{T}$ and $\wcNtype{\Delta'}{N} \lessdotCC \exptype{C}{\ol{X}}$ will be converted to
|
||||
$\ol{N} \lessdot \ol{T}$ and $\type{N} \lessdot \exptype{C}{\ol{X}}$ by the \rulename{Capture} rule.
|
||||
Which then results in the same as calling $\unify{}((\Delta, \Delta', \overline{\Delta}), [\ol{\tv{a}}/\ol{X}][\ol{\tv{b}}/\ol{Y}]\set{\ol{N} \lessdot \ol{T}, \type{N} \lessdot \exptype{C}{\ol{X}}})$,
|
||||
which shows $\Delta, \Delta', \overline{\Delta} \vdash \overline{N} <: [\ol{S}/\ol{X}]\overline{U}$.
|
||||
% %By lemma \ref{lemma:unifyWeakening}
|
||||
% We know that
|
||||
% $\unify{}(\Delta, [\ol{\tv{a}}/\ol{X}][\ol{\tv{b}}/\ol{Y}]\set{ \overline{\wcNtype{\Delta}{N}} \lessdotCC \ol{T}, \wcNtype{\Delta'}{N} \lessdotCC \exptype{C}{\ol{X}}})$
|
||||
% has a solution.
|
||||
% Also $\overline{\wcNtype{\Delta}{N}} \lessdot \ol{T}$ and $\wcNtype{\Delta'}{N} \lessdotCC \exptype{C}{\ol{X}}$ will be converted to
|
||||
% $\ol{N} \lessdot \ol{T}$ and $\type{N} \lessdot \exptype{C}{\ol{X}}$ by the \rulename{Capture} rule.
|
||||
% Which then results in the same as calling $\unify{}((\Delta, \Delta', \overline{\Delta}), [\ol{\tv{a}}/\ol{X}][\ol{\tv{b}}/\ol{Y}]\set{\ol{N} \lessdot \ol{T}, \type{N} \lessdot \exptype{C}{\ol{X}}})$,
|
||||
% which shows $\Delta, \Delta', \overline{\Delta} \vdash \overline{N} <: [\ol{S}/\ol{X}]\overline{U}$.
|
||||
|
||||
This implies $\text{dom}(\Delta') \subseteq \text{fv}(\type{N})$ and $\text{dom}(\Delta') \subseteq \text{fv}(\type{N})$
|
||||
$\Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} \type{T}$,
|
||||
$\Delta, \Delta', \overline{\Delta} \vdash \ol{N} <: [\ol{S}/\ol{X}]\ol{U}$,
|
||||
and $\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}$
|
||||
are guaranteed by the generated constraints and lemma \ref{lemma:unifySoundness}.% and \ref{lemma:unifyCC}.
|
||||
$\Delta, \Delta' \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}$ due to $\ol{T} = \ol{\wcNtype{\Delta}{N}}$ and S-Refl. $\Delta, \Delta' \vdash \type{T}_r <: \wcNtype{\Delta'}{N}$ accordingly.
|
||||
$\Delta|\Gamma \vdash \texttt{t}_r : \type{T}_r$ and $\Delta|\Gamma \vdash \ol{t} : \ol{T}_r$ by assumption.
|
||||
$\Delta, \Delta' \vdash \ol{\wcNtype{\Delta}{N}} \ \ok$ by lemma \ref{lemma:unifyWellFormedness},
|
||||
therefore $\Delta, \Delta', \overline{\Delta} \vdash \ol{N} \ \ok$, which implies $\Delta, \Delta', \overline{\Delta} \vdash \ol{U} \ \ok$
|
||||
%by lemma \ref{lemma:wfHereditary}
|
||||
and $\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok $ by premise of WF-Class.
|
||||
The same goes for $\wcNtype{\Delta'}{N}$.
|
||||
% This implies $\text{dom}(\Delta') \subseteq \text{fv}(\type{N})$ and $\text{dom}(\Delta') \subseteq \text{fv}(\type{N})$
|
||||
% $\Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} \type{T}$,
|
||||
% $\Delta, \Delta', \overline{\Delta} \vdash \ol{N} <: [\ol{S}/\ol{X}]\ol{U}$,
|
||||
% and $\Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'}$
|
||||
% are guaranteed by the generated constraints and lemma \ref{lemma:unifySoundness}.% and \ref{lemma:unifyCC}.
|
||||
% $\Delta, \Delta' \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}$ due to $\ol{T} = \ol{\wcNtype{\Delta}{N}}$ and S-Refl. $\Delta, \Delta' \vdash \type{T}_r <: \wcNtype{\Delta'}{N}$ accordingly.
|
||||
% $\Delta|\Gamma \vdash \texttt{t}_r : \type{T}_r$ and $\Delta|\Gamma \vdash \ol{t} : \ol{T}_r$ by assumption.
|
||||
% $\Delta, \Delta' \vdash \ol{\wcNtype{\Delta}{N}} \ \ok$ by lemma \ref{lemma:unifyWellFormedness},
|
||||
% therefore $\Delta, \Delta', \overline{\Delta} \vdash \ol{N} \ \ok$, which implies $\Delta, \Delta', \overline{\Delta} \vdash \ol{U} \ \ok$
|
||||
% %by lemma \ref{lemma:wfHereditary}
|
||||
% and $\Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok $ by premise of WF-Class.
|
||||
% The same goes for $\wcNtype{\Delta'}{N}$.
|
||||
|
||||
% \begin{gather}
|
||||
% \label{sp:0}
|
||||
% \Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} \type{T} \\
|
||||
% \label{sp:1}
|
||||
% \Delta, \Delta', \overline{\Delta} \vdash \ol{N} <: [\ol{S}/\ol{X}]\ol{U} \\
|
||||
% \label{sp:2}
|
||||
% \Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'} \\
|
||||
% \label{sp:3}
|
||||
% \Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \\
|
||||
% \label{sp:4}
|
||||
% \Delta, \Delta' \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}
|
||||
% \end{gather}
|
||||
% % \begin{gather}
|
||||
% % \label{sp:0}
|
||||
% % \Delta, \Delta', \overline{\Delta} \vdash [\ol{S}/\ol{X}]\type{U} \type{T} \\
|
||||
% % \label{sp:1}
|
||||
% % \Delta, \Delta', \overline{\Delta} \vdash \ol{N} <: [\ol{S}/\ol{X}]\ol{U} \\
|
||||
% % \label{sp:2}
|
||||
% % \Delta, \Delta', \overline{\Delta} \vdash \ol{S} <: [\ol{S}/\ol{X}]\ol{U'} \\
|
||||
% % \label{sp:3}
|
||||
% % \Delta, \Delta', \overline{\Delta} \vdash \ol{S} \ \ok \\
|
||||
% % \label{sp:4}
|
||||
% % \Delta, \Delta' \vdash \ol{T} <: \ol{\wcNtype{\Delta}{N}}
|
||||
% % \end{gather}
|
||||
|
||||
Method calls generate multiple constraints that share the same wildcard placeholders ($\ol{\wtv{a}}$, $\ol{\wtv{b}}$).
|
||||
% Method calls generate multiple constraints that share the same wildcard placeholders ($\ol{\wtv{a}}$, $\ol{\wtv{b}}$).
|
||||
|
||||
\end{description}
|
||||
|
||||
@ -333,15 +354,14 @@ Trivial. \unify{} fails when a constraint $\tv{a} \doteq \rwildcard{X}$ arises.
|
||||
% Delta |- N <. T
|
||||
% \end{lemma}
|
||||
|
||||
\begin{lemma}\label{lemma:wildcardsStayInScope}
|
||||
Free variables can only hop to another constraint if they share the same wildcard placeholder.
|
||||
\begin{description}
|
||||
\item[If] $(\sigma, \Delta) = \unify{} (\Delta', C \cup \set{\wcNtype{\Delta'}{N} \lessdot \tv{a}})$
|
||||
\item[and] $\tv{a}$ being a type placeholders used in $C$
|
||||
\item[then] $\text{fv}(\sigma(\tv{a})) \subseteq \Delta, \Delta'$
|
||||
\end{description}
|
||||
If $\wcNtype{\Delta'}{N} \lessdot \tv{a}$
|
||||
\end{lemma}
|
||||
% \begin{lemma}\label{lemma:wildcardsStayInScope}
|
||||
% Free variables can only hop to another constraint if they share the same wildcard placeholder.
|
||||
% \begin{description}
|
||||
% \item[If] $(\sigma, \Delta) = \unify{} (\Delta', C \cup \set{\wcNtype{\Delta'}{N} \lessdot \tv{a}})$
|
||||
% \item[and] $\tv{a}$ being a type placeholders used in $C$
|
||||
% \item[then] $\text{fv}(\sigma(\tv{a})) \subseteq \Delta, \Delta'$
|
||||
% \end{description}
|
||||
% \end{lemma}
|
||||
|
||||
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user