Add Prepare explanation. Restructure
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JanUlrich
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unify.tex
419
unify.tex
@ -25,35 +25,10 @@ Each calculation step of the algorithm is expressed as a transformation rule con
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The input is shown above the line, the output below, and additional premises are displayed on the right side.
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If the wildcard environment $\wildcardEnv$ and the constraint set $C$ match the pattern declared as the input
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the transformation will reduce them into the specified output.
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The \rulename{Subst} rule for example takes a constraint set that has atleas one constraint of the form
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The \rulename{Subst} rule for example takes a constraint set that has atleast one constraint of the form
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$\ntv{a} \doteq \type{T}$ and replaces every occurence of $\ntv{a}$ by $\type{T}$
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in the wildcard environment $\wildcardEnv{}$ aswell as the remaining constraint set $C$.
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The \rulename{Upper} and \rulename{Lower} conversions (figure \ref{fig:wildcard-rules}) replace wildcards with their respective bounds when appearing in a subtype constraint.
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\rulename{Lower} has to check if the wildcard is part of the input wildcards $\Delta_{in}$.
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If that is the case the wildcard can be part of the type solution and stays in the constraint set.
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\textit{Note:} The subtype constraints in these rules are annotated with numbers $\lessdot_1$.
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All rule inputs containing subtype constraints $(\lessdot)$ are always meant for both,
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subtype constraints and capture constraints ($\lessdotCC$) aswell.
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If multiple constraints are stated in the input format they will be annotated with numbers which map them to the constraints used in the output of the rule.
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Constraints with the same number stay the same kind.
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So if the input to \rulename{Upper} is $\rwildcard{A} \lessdotCC \type{G}$ the output will be something like $\type{U} \lessdotCC \type{G}$.
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If the input is a normal subtype constraint the output has to be a normal subtype constraint too.
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%TODO: Rephrase
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%The reason being that capture constraints are treated like regular subtype constraints in these rules.
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%All transformations for subtype constraints work for capture constraints aswell.
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%For clarification Subtype constraints are marked with a number.
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But the \rulename{Adopt} rule for example takes multiple subtype constraints and also adds a new one.
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Here the numbered annotations are necessary.
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\textit{Example:} Having the constraints
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$\ntv{a} \lessdotCC \wtv{b}, \ntv{a} \lessdot \type{String}, \wtv{b} \lessdot \type{Object}$
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would lead to
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$\wtv{b} \lessdot \type{String}, \ntv{a} \lessdotCC \wtv{b}, \ntv{a} \lessdot \type{String}, \wtv{b} \lessdotCC \type{Object}$
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after applying \rulename{Adopt}.
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Note that the new generated constraint $\wtv{b} \lessdot \type{String}$ is a normal subtype constraint.
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%The type placeholders which are annotated as wildcard placeholders also stay wildcard placeholders.
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%The only rule that replaces wildcard type placeholders with regular placeholders is the \rulename{Normalize} rule.
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\begin{figure}
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\begin{center}
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\leavevmode
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@ -114,6 +89,167 @@ Note that the new generated constraint $\wtv{b} \lessdot \type{String}$ is a nor
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\caption{Wildcard reduce rules}\label{fig:wildcard-rules}
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\end{figure}
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The \rulename{Upper} and \rulename{Lower} conversions (figure \ref{fig:wildcard-rules}) replace wildcards with their respective bounds when appearing in a subtype constraint.
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\rulename{Lower} has to check if the wildcard is part of the input wildcards $\Delta_{in}$.
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If that is the case the wildcard can be part of the type solution and stays in the constraint set.
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\textit{Note:} The subtype constraints in these rules are annotated with numbers $\lessdot_1$.
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All rule inputs containing subtype constraints $(\lessdot)$ are always meant for both,
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subtype constraints and capture constraints ($\lessdotCC$).
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If multiple constraints are stated in the input format they will be annotated with numbers which map them to the constraints used in the output of the rule.
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Constraints with the same number stay the same kind.
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So if the input to \rulename{Upper} is $\rwildcard{A} \lessdotCC \type{G}$ the output will be something like $\type{U} \lessdotCC \type{G}$.
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If the input is a normal subtype constraint the output has to be a normal subtype constraint too.
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%TODO: Rephrase
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%The reason being that capture constraints are treated like regular subtype constraints in these rules.
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%All transformations for subtype constraints work for capture constraints aswell.
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%For clarification Subtype constraints are marked with a number.
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But the \rulename{Adopt} rule for example takes multiple subtype constraints and also adds a new one.
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Here the numbered annotations are necessary.
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\textit{Example:} Having the constraints
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$\ntv{a} \lessdotCC \wtv{b}, \ntv{a} \lessdot \type{String}, \wtv{b} \lessdot \type{Object}$
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would lead to
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$\wtv{b} \lessdot \type{String}, \ntv{a} \lessdotCC \wtv{b}, \ntv{a} \lessdot \type{String}, \wtv{b} \lessdotCC \type{Object}$
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after applying \rulename{Adopt}.
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Note that the new generated constraint $\wtv{b} \lessdot \type{String}$ is a normal subtype constraint.
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%The type placeholders which are annotated as wildcard placeholders also stay wildcard placeholders.
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%The only rule that replaces wildcard type placeholders with regular placeholders is the \rulename{Normalize} rule.
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\begin{figure}
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\begin{center}
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\leavevmode
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\fbox{
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\begin{tabular}[t]{l@{~}l}
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\rulename{Prepare} %The lessdotCC constraint only ensures that the left side looses its wildcardEnvironment.
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%It does not ensure that the left side doesn't contain free variables. If you want to ensure that you have to give the left side a normal placeholder
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \wctype{\Delta'}{C}{\ol{T}} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdotCC \wctype{\Delta'}{C}{\ol{T}} } \\
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\text{fv}(\type{N'}) \subseteq \Delta_{in} \\
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\text{wtv}(\type{N'}) = \emptyset
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Trim}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\Delta,\Delta'}{C}{\ol{S}} \lessdot \type{T} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\text{fv}(\ol{S}) \cap \Delta' = \emptyset
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Clear}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \cup \set{\wildcard{A}{\type{U}}{\type{L}}} \vdash
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C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\
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\hline
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\vspace*{-0.4cm}\\
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\subst{\type{U}}{\rwildcard{A}}\wildcardEnv \vdash
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[\type{U}/\rwildcard{A}]C \cup \, [\type{U}/\rwildcard{A}]\set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T}, \type{U} \doteq \type{L} } \\
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\Delta \neq \emptyset\\
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\rwildcard{A} \in \text{fv}(\type{T})
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Exclude}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\
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\hline
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\vspace*{-0.4cm}\\
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\subst{\tv{a}}{\wtv{a}}\wildcardEnv \vdash
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[\tv{a}/\wtv{a}]C \cup \, [\tv{a}/\wtv{a}]\set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\Delta \neq \emptyset\\
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\wtv{a} \in \text{fv}(\type{T}), \tv{a} \ \text{fresh}
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\end{array}
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\end{array}
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$
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\end{tabular}}
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\end{center}
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\caption{Dealing with wildcard types on the left side of a subtype constraint}
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\end{figure}
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An existential type can never be a subtype of a regular type:
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$\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \nless: \exptype{C}{\rwildcard{X}}$.
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This is a problem for constraints of the form $\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \lessdot \type{N}$.
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Take the constraint $\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \lessdot \exptype{C}{\wtv{a}}$ for example.
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After applying a reduction we get $\rwildcard{X} \doteq \wtv{a}$ which is not a valid solution.
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The problem is that we loose the information of the left side being an existential type with the reduction step.
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Therefore we have to assure beforehand that errors (like $\rwildcard{X} \doteq \wtv{a}$ in this example) cannot occur.
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But we alsod don't want to exclude reductions for these kind of constraints in general.
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The solution is to check if there are any free variables or wildcard placeholders on the right side of the constraint.
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If that is the case one of the rules \rulename{Trim}, \rulename{Clear}, or \rulename{Exclude} have to be applied.
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In our example this would be the \rulename{Exclude} rule replacing the wildcard placeholder with $\wtv{x}$ with a normal placeholder.
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Afterwards \rulename{Prepare} can be used eventually leading to the erasure of the wildcard $\rwildcard{X}$ by equalizing its upper and lower bounds:
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\begin{displaymath}
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\prftree[r]{\rulename{Contract}}{
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\prftree[r]{\rulename{Reduce}}{
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\prftree[r]{\rulename{Equals}}{
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\prftree[r]{\rulename{Reduce}}{
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\prftree[r]{\rulename{Capture}}{
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\prftree[r]{\rulename{Prepare}}{
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\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\ntv{x}}}
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}
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{
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\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\exptype{List}{\rwildcard{X}}} \lessdotCC \exptype{List}{\exptype{List}{\ntv{x}}}
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}
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}
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{
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\wildcard{X}{\tv{u}}{\tv{l}} \vdash \exptype{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\ntv{x}}}
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}
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}
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{
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\wildcard{X}{\tv{u}}{\tv{l}} \vdash \exptype{List}{\rwildcard{X}} \doteq \exptype{List}{\ntv{x}}
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}}
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{
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\wildcard{X}{\tv{u}}{\tv{l}} \vdash \exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\ntv{x}}, \exptype{List}{\ntv{x}} \lessdot \exptype{List}{\rwildcard{X}}
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}
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}
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{
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\wildcard{X}{\tv{u}}{\tv{l}} \vdash \ntv{x} \doteq \rwildcard{X}
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}}{
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\text{equalize upper and lower bound of }\rwildcard{X}: \quad
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\ntv{x} \doteq \tv{u}, \tv{u} \doteq \tv{l}
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}
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\end{displaymath}
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Note that the \rulename{Prepare} rule is always applied together with the \rulename{Capture} and the \rulename{Reduce} rule:
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\rulename{Trim} removes unused wildcard declarations
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whereas \rulename{Clear} and \rulename{Exclude} remove wildcard placeholders or wildcards to
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allow the constraint to be processed by \rulename{Prepare}.
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\subsection{Adding Wildcards to the mix}
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\unify{} is able to create wildcard solutions even when the input set of constraints do not contain any wildcard variables.
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@ -899,95 +1035,6 @@ Prepare, Capture, Reduce, Trim, Clear, Exclude, Adapt
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\caption{Constraint reduce rules}\label{fig:reduce-rules}
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\end{figure}
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\begin{figure}
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\begin{center}
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\leavevmode
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\fbox{
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\begin{tabular}[t]{l@{~}l}
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\rulename{Prepare} %The lessdotCC constraint only ensures that the left side looses its wildcardEnvironment.
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%It does not ensure that the left side doesn't contain free variables. If you want to ensure that you have to give the left side a normal placeholder
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \wctype{\Delta'}{C}{\ol{T}} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdotCC \wctype{\Delta'}{C}{\ol{T}} } \\
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\text{fv}(\wctype{\Delta'}{C}{\ol{T}}) \subseteq \Delta_{in}
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Trim}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\Delta,\Delta'}{C}{\ol{S}} \lessdot \type{T} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\text{fv}(\ol{S}) \cap \Delta' = \emptyset
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Clear}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \cup \set{\wildcard{A}{\type{U}}{\type{L}}} \vdash
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C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\
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\hline
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\vspace*{-0.4cm}\\
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\subst{\type{U}}{\rwildcard{A}}\wildcardEnv \vdash
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[\type{U}/\rwildcard{A}]C \cup \, [\type{U}/\rwildcard{A}]\set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T}, \type{U} \doteq \type{L} } \\
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\Delta \neq \emptyset\\
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\rwildcard{A} \in \text{fv}(\type{T})
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Exclude}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\
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\hline
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\vspace*{-0.4cm}\\
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\subst{\tv{a}}{\wtv{a}}\wildcardEnv \vdash
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[\tv{a}/\wtv{a}]C \cup \, [\tv{a}/\wtv{a}]\set{ \wctype{\Delta}{C}{\ol{S}} \lessdot \type{T} } \\
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\Delta \neq \emptyset\\
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\wtv{a} \in \text{fv}(\type{T}), \tv{a} \ \text{fresh}
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\end{array}
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\end{array}
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$
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\end{tabular}}
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\end{center}
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\caption{Dealing with wildcard types on the left side of a subtype constraint}
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\end{figure}
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\begin{figure}
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If we find an illicit constraint assigning a type containing free variables to a type placeholder not flagged as a wildcard placeholder the algorithm fails.
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@ -1275,152 +1322,10 @@ $
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\label{fig:generation-rules}
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\end{figure}
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\begin{figure}
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\begin{center}
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\fbox{
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\begin{tabular}[t]{l@{~}l}
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\rulename{Equals} %TODO
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& $
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \, \set{ \wcNtype{\Delta}{N} \doteq \wcNtype{\Delta'}{N'} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv \vdash C \cup \,
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\set{
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\wcNtype{\Delta}{N} \lessdot \wcNtype{\Delta'}{N'}, \wcNtype{\Delta'}{N'} \lessdot \wcNtype{\Delta}{N}
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}
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\end{array} %\quad |\Delta| = |\Delta'|
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% \quad \text{fv}(\type{N}) = \text{fv}(\type{N'}) = \emptyset
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$
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\\\\
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\ruleReduceWC{}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \exptype{C}{\ol{S}} \lessdot
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\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}} } \\
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\hline
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\vspace*{-0.4cm}\\
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\wildcardEnv
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\vdash C \cup \, \set{
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\ol{\type{S}} \doteq [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{\type{T}},
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\ol{\wtv{a}} \lessdot [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{U}, [\ol{\wtv{a}}/\overline{\rwildcard{A}}]\ol{L} \lessdot \ol{\wtv{a}} }
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\end{array}
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%\quad \ol{Y} = \textit{fresh}(\ol{X})
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\quad \begin{array}[c]{l}
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\ol{\wtv{a}} \ \text{fresh}\\
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%\text{fv}(\exptype{C}{\ol{S}}) \subseteq \text{dom}(\overline{\wildcard{B}{\type{U'}}{\type{L'}}})
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%\text{dom}(\overline{\wildcard{A}{\type{U}}{\type{L}}}) \subseteq \text{fv}(\exptype{C}{\ol{T}}) \\
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%\text{fv}(\wctype{\overline{\wildcard{A}{\type{U}}{\type{L}}}}{C}{\ol{T}}) = \emptyset
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\end{array}
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\end{array}
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$
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\\\\
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\rulename{Capture}
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&
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$
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\begin{array}[c]{@{}ll}
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \, \set{ \wctype{\overline{\wildcard{B}{\type{U}}{\type{L}}}}{C}{\ol{S}} \lessdotCC \type{T} } \\
|
||||
\hline
|
||||
\vspace*{-0.4cm}\\
|
||||
\wildcardEnv \cup \overline{\wildcard{C}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{U}}{[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}]\type{L}}}
|
||||
\vdash C \cup \, \set{
|
||||
[\ol{\rwildcard{C}}/\ol{\rwildcard{B}}] \exptype{C}{\ol{S}} \lessdot \type{T} }
|
||||
\end{array}
|
||||
%\quad \ol{Y} = \textit{fresh}(\ol{X})
|
||||
\quad \begin{array}[c]{l}
|
||||
\ol{\rwildcard{C}} \ \text{fresh}\\
|
||||
%\text{fv}(\type{T}) \neq \emptyset
|
||||
\end{array}
|
||||
\end{array}
|
||||
$
|
||||
\\\\
|
||||
\rulename{Prepare} %The lessdotCC constraint only ensures that the left side looses its wildcardEnvironment.
|
||||
%It does not ensure that the left side doesn't contain free variables. If you want to ensure that you have to give the left side a normal placeholder
|
||||
&
|
||||
$
|
||||
\begin{array}[c]{@{}ll}
|
||||
\begin{array}[c]{l}
|
||||
\wildcardEnv \vdash
|
||||
C \cup \, \set{ \type{N} \lessdot \type{N'} } \\
|
||||
\hline
|
||||
\vspace*{-0.4cm}\\
|
||||
\wildcardEnv \vdash
|
||||
C \cup \, \set{ \type{N} \lessdotCC \type{N'} } \\
|
||||
\end{array}
|
||||
%\quad \ol{Y} = \textit{fresh}(\ol{X})
|
||||
\quad \begin{array}[c]{l}
|
||||
\text{fv}(\type{N'}) \subseteq \Delta_{in} \\
|
||||
\text{wtv}(\type{N'}) = \emptyset
|
||||
\end{array}
|
||||
\end{array}
|
||||
$
|
||||
\end{tabular}}
|
||||
\end{center}
|
||||
\caption{Type Reduction rules}\label{fig:reductionRules}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Explanations}
|
||||
|
||||
%The type reduction is done by the rules in figure \ref{fig:reductionRules}
|
||||
|
||||
|
||||
\begin{description}
|
||||
\item[Prepare]
|
||||
The \rulename{Prepare} transformation is always applied together with the \rulename{Reduce} transformation.
|
||||
\unify{} cannot reduce constraints without checking a few prerequisites.
|
||||
%Take the constraint $\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \lessdot \exptype{C}{\wtv{a}}$ for example.
|
||||
%If we apply a reduction here we get $\rwildcard{X} \doteq \wtv{a}$.
|
||||
%The resulting $\sigma(\wtv{a}) = \rwildcard{X}$ seems like a correct substitution,
|
||||
%but by S-Exists $\wctype{\rwildcard{X}}{C}{\rwildcard{X}} \nless: \exptype{C}{\rwildcard{X}}$.
|
||||
%Reason: Free variables on the right side of a subtype relations are not allowed to show up as bound variables on the left side.
|
||||
%$\rwildcard{X}$ in this case.
|
||||
%Therefore the \rulename{Reduce} rule only reduces constraints where the left side does not declare any wildcards.
|
||||
%But if the right side neither contains wildcard type placeholders nor free variables the constraint can be reduced anyways.
|
||||
%The \rulename{Prepare} rule then converts this constraint to a capture constraint.
|
||||
%Afterwards the \rulename{Capture} rule removes the wildcard declarations on the left side an the constraint can be reduced.
|
||||
%We loose information during the unification process.
|
||||
When reducing the constraint
|
||||
$\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\wtv{x}}}$
|
||||
it turns into $\exptype{List}{\rwildcard{X}} \doteq \exptype{List}{\wtv{x}}$
|
||||
and now it seems that $\wtv{x} \doteq \rwildcard{X}$ is a correct solution.
|
||||
This is indeed wrong because $\wctype{\rwildcard{X}}{List}{\exptype{List}{\rwildcard{X}}} \nless: \exptype{List}{\exptype{List}{\rwildcard{X}}}$.
|
||||
A correct solution is to remove the wildcard $\rwildcard{X}$ if possible.
|
||||
% X.List<X> <. List<x> // incorrect
|
||||
% X.List<List<X>> <. List<List<x>> // incorrect
|
||||
Therefore the \rulename{Prepare} rule checks if there are any free variables or wildcard placeholders on the right side of the constraint.
|
||||
If that is the case one of the rules \rulename{Trim}, \rulename{Clear}, or \rulename{Exclude} have to be applied.
|
||||
In our example this would be the \rulename{Exclude} rule replacing the wildcard placeholder with $\wtv{x}$ with a normal placeholder.
|
||||
Afterwards \rulename{Prepare} can be used eventually leading to the erasure of the wildcard $\rwildcard{X}$ by equalizing its upper and lower bounds:
|
||||
\begin{displaymath}
|
||||
\prftree[r]{\rulename{Contract}}{
|
||||
\prftree[r]{\rulename{Reduce}}{
|
||||
\prftree[r]{\rulename{Equals}}{
|
||||
\prftree[r]{\rulename{Prepare}}{
|
||||
\wctype{\wildcard{X}{\tv{u}}{\tv{l}}}{List}{\exptype{List}{\rwildcard{X}}} \lessdot \exptype{List}{\exptype{List}{\ntv{x}}}
|
||||
}
|
||||
{
|
||||
\wildcard{X}{\tv{u}}{\tv{l}} \vdash \exptype{List}{\rwildcard{X}} \doteq \exptype{List}{\ntv{x}}
|
||||
}}
|
||||
{
|
||||
\exptype{List}{\rwildcard{X}} \lessdot \exptype{List}{\ntv{x}}, \exptype{List}{\ntv{x}} \lessdot \exptype{List}{\rwildcard{X}}
|
||||
}
|
||||
}
|
||||
{
|
||||
\wildcard{X}{\tv{u}}{\tv{l}} \vdash \ntv{x} \doteq \rwildcard{X}
|
||||
}}{
|
||||
\text{equalize upper and lower bound of }\rwildcard{X}: \quad
|
||||
\ntv{x} \doteq \tv{u}, \tv{u} \doteq \tv{l}
|
||||
}
|
||||
\end{displaymath}
|
||||
|
||||
\end{description}
|
||||
|
||||
\subsection{Examples}
|
||||
\textit{Example} of the type reduction rules in figure \ref{fig:reductionRules} with the input
|
||||
$\wctype{\rwildcard{X}}{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \rwildcard{X}} \lessdot \exptype{Pair}{\wctype{\rwildcard{Y}}{List}{\rwildcard{Y}}, \wtv{a}}$
|
||||
|
||||
Loading…
Reference in New Issue
Block a user