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Fix Type Substitution

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Andreas Stadelmeier 2024-03-27 14:12:50 +01:00
parent cec613b875
commit 495e37b370
1 changed files with 36 additions and 31 deletions
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67
unify.tex
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@ -579,34 +579,6 @@ $
\end{array} \end{array}
\end{array} \end{array}
$ $
\\\\
\rulename{Crunch}
& $\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash C \cup \, \set{ \tv{a} \doteq \wctype{\Delta', \set{\overline{\wildcard{X}{\type{U}}{\type{L}}}}}{C}{\ol{S}} } \\
\hline
\vspace*{-0.4cm}\\
\wildcardEnv \vdash
C \cup \set{ \tv{a} \doteq \wctype{\Delta'}{C}{[\ol{U}/\ol{X}]\ol{S}}}
\end{array}
&\begin{array}[c]{l}
\ol{U} = \ol{L}
\end{array}
\end{array}$
\\\\
\rulename{Crunch}
& $\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash C \cup \, \set{ \tv{a} \lessdot \wctype{\Delta', \set{\overline{\wildcard{X}{\type{U}}{\type{L}}}}}{C}{\ol{S}} } \\
\hline
\vspace*{-0.4cm}\\
\wildcardEnv \vdash
C \cup \set{ \tv{a} \lessdot \wctype{\Delta'}{C}{[\ol{U}/\ol{X}]\ol{S}}}
\end{array}
&\begin{array}[c]{l}
\ol{U} = \ol{L}
\end{array}
\end{array}$
\end{tabular}} \end{tabular}}
\end{center} \end{center}
\caption{Constraint reduce rules}\label{fig:reduce-rules} \caption{Constraint reduce rules}\label{fig:reduce-rules}
@ -728,10 +700,9 @@ To only rename the respective wildcards the reduce rule renames wildcards up to
\begin{align*} \begin{align*}
[\type{A}/\type{B}]\type{B} &= \type{A} \\ [\type{A}/\type{B}]\type{B} &= \type{A} \\
[\type{A}/\type{B}]\type{C} &= \type{C} & \text{if}\ \type{B} \neq \type{C}\\ [\type{A}/\type{B}]\type{C} &= \type{C} & \text{if}\ \type{B} \neq \type{C}\\
[\type{A}/\type{B}]\wcNtype{\Delta}{N} &= \wcNtype{\Delta}{[\type{A}/\type{B}]N} & \text{if}\ \type{B} \notin \Delta \\ [\type{A}/\type{B}]\wcNtype{\overline{\wildcard{X}{\type{U}}{\type{L}}}}{N} &= \wcNtype{\overline{\wildcard{X}{[\type{A}/\type{B}]\type{U}}{[\type{A}/\type{B}]\type{L}}}}{[\type{A}/\type{B}]N} & \text{if}\ \type{B} \notin \overline{\rwildcard{X}} \\
[\type{A}/\type{B}]\wcNtype{\Delta}{N} &= \wcNtype{\Delta}{N} & \text{if}\ \type{B} \in \Delta \\ [\type{A}/\type{B}]\wcNtype{\overline{\wildcard{X}{\type{U}}{\type{L}}}}{N} &= \wcNtype{\overline{\wildcard{X}{\type{U}}{\type{L}}}}{N} & \text{if}\ \type{B} \in \overline{\rwildcard{X}} \\
\end{align*} \end{align*}
\item[Free Variables:] \item[Free Variables:]
The $\text{fv}$ function assumes every wildcard type variable to be a free variable aswell. The $\text{fv}$ function assumes every wildcard type variable to be a free variable aswell.
@ -1195,3 +1166,37 @@ Every constraint that is not in solved form and is not able to be processed by a
The new constraint generated by the adopt rule may be eliminated by the match rule. The new constraint generated by the adopt rule may be eliminated by the match rule.
The adopt rule still needs to be applied only once per constraint. The adopt rule still needs to be applied only once per constraint.
Two additional rules can be added to remove clutter in the output.
Wildcards with same upper and lower bounds can be replaced with their upper bound.
\begin{tabular}[t]{l@{~}l}
\\\\
\rulename{Crunch}
& $\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash C \cup \, \set{ \tv{a} \doteq \wctype{\Delta', \set{\overline{\wildcard{X}{\type{U}}{\type{L}}}}}{C}{\ol{S}} } \\
\hline
\vspace*{-0.4cm}\\
\wildcardEnv \vdash
C \cup \set{ \tv{a} \doteq \wctype{\Delta'}{C}{[\ol{U}/\ol{X}]\ol{S}}}
\end{array}
&\begin{array}[c]{l}
\ol{U} = \ol{L}
\end{array}
\end{array}$
\\\\
\rulename{Crunch}
& $\begin{array}[c]{@{}ll}
\begin{array}[c]{l}
\wildcardEnv \vdash C \cup \, \set{ \tv{a} \lessdot \wctype{\Delta', \set{\overline{\wildcard{X}{\type{U}}{\type{L}}}}}{C}{\ol{S}} } \\
\hline
\vspace*{-0.4cm}\\
\wildcardEnv \vdash
C \cup \set{ \tv{a} \lessdot \wctype{\Delta'}{C}{[\ol{U}/\ol{X}]\ol{S}}}
\end{array}
&\begin{array}[c]{l}
\ol{U} = \ol{L}
\end{array}
\end{array}$
\end{tabular}