Add GenDelta for WTVS. Restructure step 2. Comment out old version
This commit is contained in:
Andreas Stadelmeier
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17559170d0
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f2002ea833
@ -378,6 +378,7 @@ $\Delta \vdash \sigma(C') \implies \Delta \vdash \sigma(C)$
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\begin{description}
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\item[AddDelta] $C$ is not changed
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\item[GenDelta] by definition, S-Var-Left, and S-Trans %The generated type variable is unique due to the solved form property. %and the solved form property (no $\tv{a}$ in $C$)
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\item[GenDelta'] same as GenDelta by setting $\sigma'(\wtv{b}) = \rwildcard{B}$
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\item[GenSigma] by definition and S-Refl.
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% holds for $\set{\tv{a} \doteq \type{G}}$ by definition.
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% Holds for $C$ by assumption and because $\tv{a} \notin C$ by solved form definition ($[\type{G}/\tv{a}]C = C$).
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269
unify.tex
269
unify.tex
@ -830,6 +830,20 @@ This builds a search tree over multiple possible solutions.
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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{SameW}
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& $
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \type{G} \lessdot \wtv{a}\\
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\hline
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\wildcardEnv \vdash
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C \cup \set{
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\wtv{a} \doteq \type{G}
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}
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\end{array}
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$
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\\\\
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\cdashline{1-2} \\
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\rulename{Same}
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& $
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\begin{array}[c]{l}
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@ -876,20 +890,20 @@ This builds a search tree over multiple possible solutions.
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\ol{X} \notin \wildcardEnv \cup \Delta,\, \Delta' = \Delta \cap \text{fv}([\ol{T}/\ol{X}]\ol{N})
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\end{array}
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$
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\\\\
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\hline \\
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\rulename{SameW}
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& $
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \type{G} \lessdot \wtv{a}\\
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\hline
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\wildcardEnv \vdash
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C \cup \set{
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\wtv{a} \doteq \type{G}
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}
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\end{array}
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$
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% \\\\
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% \hline \\
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% \rulename{SameW}
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% & $
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% \begin{array}[c]{l}
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% \wildcardEnv \vdash
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% C \cup \type{G} \lessdot \wtv{a}\\
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% \hline
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% \wildcardEnv \vdash
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% C \cup \set{
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% \wtv{a} \doteq \type{G}
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% }
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% \end{array}
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% $
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\\\\
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\cdashline{1-2} \\
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\rulename{\generalizeRule{}W}
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@ -953,6 +967,8 @@ This builds a search tree over multiple possible solutions.
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\label{fig:step2-rules}
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\end{figure}
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$\type{T} \lessdot \ntv{a}$ constraints have three and $\type{T} \lessdot \wtv{a}$ constraints have five possible transformations.
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\begin{figure}
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\begin{center}
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\fbox{
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@ -986,110 +1002,110 @@ This builds a search tree over multiple possible solutions.
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\label{fig:step2-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{Same}
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& $
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\begin{array}[c]{l}
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\wildcardEnv \vdash
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C \cup \type{G} \lessdot \tv{a}\\
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\hline
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\wildcardEnv \vdash
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C \cup \set{
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\tv{a} \doteq \type{G}
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}
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\end{array} \quad \begin{array}[c]{l}
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\text{fv}(\type{G}) \in \Delta_{in}
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\end{array}
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$
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\\\\
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\rulename{\generalizeRule}
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& $
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a}\\
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\hline
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\wildcardEnv \vdash C \cup \set{\wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a},
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\tv{a} \doteq \wctype{\overline{\wildcard{X}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{X}}},
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%\overline{\tv{l} \lessdot \tv{u}}, % not needed, due to subst and reduce rule which are used afterwards
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\overline{\tv{u} \lessdot \type{S}}
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}
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\end{array} \quad \begin{array}[c]{l}
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\texttt{class} \ \exptype{C}{\ol{X \triangleleft \type{S}}} \triangleleft \exptype{D}{\ol{N}} \\
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\text{fresh}\ \overline{\wildcard{X}{\tv{u}}{\tv{l}}}
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\end{array}
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$
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\\\\
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\rulename{Subst-X}
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& $
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\begin{array}[c]{l}
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\wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
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C \cup \rwildcard{X} \lessdot \tv{a}\\
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\hline
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\wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
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C \cup \set{
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\tv{a} \doteq \rwildcard{X}
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}
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\end{array} \quad \begin{array}[c]{l}
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\rwildcard{X} \in \Delta_{in}
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\end{array}
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$
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\\\\
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\rulename{Gen-X}
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& $
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\begin{array}[c]{l}
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\wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
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C \cup \rwildcard{X} \lessdot \tv{a}\\
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\hline
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\wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
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C \cup \set{
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\type{U} \lessdot \tv{a}
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}
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\end{array}
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$
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\\\\
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\rulename{Super}
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& $
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a}\\
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\hline
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\wildcardEnv \vdash C \cup \set{ \wctype{\Delta'}{D}{[\ol{T}/\ol{X}]\ol{N}} \lessdot \tv{a} }
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%\set{\wctype{\ol{\wtype{W}}}{D}{[\ol{X}/\ol{Y}]\ol{Z}} \lessdot \tv{a}}
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\end{array} \quad
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\begin{array}{l}
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\texttt{class} \ \exptype{C}{\ol{X}} \triangleleft \exptype{D}{\ol{N}} \\
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\ol{X} \notin \wildcardEnv \cup \Delta,\, \Delta' = \Delta \cap \text{fv}([\ol{T}/\ol{X}]\ol{N})
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\end{array}
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$
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\\\\
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\rulename{Settle}
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& $
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \type{N},
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\tv{a} \lessdot \tv{b}}
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\\
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\hline
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\wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \tv{b}, \tv{b} \lessdot \type{N} }
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\end{array}
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$
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\\\\
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\rulename{Raise}
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& $
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\begin{array}[c]{l}
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\wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \type{N},
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\tv{a} \lessdot \tv{b}}
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\\
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\hline
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\wildcardEnv \vdash C \cup \set{\tv{a} \lessdot \type{N}, \type{N} \lessdot \tv{b} }
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\end{array}
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$
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\\\\
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\end{tabular}}
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\end{center}
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\caption{Step 2 branching: Multiple rules can be applied to the same constraint}
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\label{fig:step2-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{Same}
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% & $
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% \begin{array}[c]{l}
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% \wildcardEnv \vdash
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% C \cup \type{G} \lessdot \tv{a}\\
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% \hline
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% \wildcardEnv \vdash
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% C \cup \set{
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% \tv{a} \doteq \type{G}
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% }
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% \end{array} \quad \begin{array}[c]{l}
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% \text{fv}(\type{G}) \in \Delta_{in}
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% \end{array}
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% $
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% \\\\
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% \rulename{\generalizeRule}
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% & $
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% \begin{array}[c]{l}
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% \wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a}\\
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% \hline
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% \wildcardEnv \vdash C \cup \set{\wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a},
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% \tv{a} \doteq \wctype{\overline{\wildcard{X}{\tv{u}}{\tv{l}}}}{C}{\overline{\rwildcard{X}}},
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% %\overline{\tv{l} \lessdot \tv{u}}, % not needed, due to subst and reduce rule which are used afterwards
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% \overline{\tv{u} \lessdot \type{S}}
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% }
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% \end{array} \quad \begin{array}[c]{l}
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% \texttt{class} \ \exptype{C}{\ol{X \triangleleft \type{S}}} \triangleleft \exptype{D}{\ol{N}} \\
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% \text{fresh}\ \overline{\wildcard{X}{\tv{u}}{\tv{l}}}
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% \end{array}
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% $
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% \\\\
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% \rulename{Subst-X}
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% & $
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% \begin{array}[c]{l}
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% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
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% C \cup \rwildcard{X} \lessdot \tv{a}\\
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% \hline
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% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
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% C \cup \set{
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% \tv{a} \doteq \rwildcard{X}
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% }
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% \end{array} \quad \begin{array}[c]{l}
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% \rwildcard{X} \in \Delta_{in}
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% \end{array}
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% $
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% \\\\
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% \rulename{Gen-X}
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% & $
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% \begin{array}[c]{l}
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% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
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% C \cup \rwildcard{X} \lessdot \tv{a}\\
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% \hline
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% \wildcardEnv \cup \set{\wildcard{X}{\type{U}}{\type{L}}} \vdash
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% C \cup \set{
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% \type{U} \lessdot \tv{a}
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% }
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% \end{array}
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% $
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% \\\\
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% \rulename{Super}
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% & $
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% \begin{array}[c]{l}
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% \wildcardEnv \vdash C \cup \wctype{\Delta}{C}{\ol{T}} \lessdot \tv{a}\\
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% \hline
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% \wildcardEnv \vdash C \cup \set{ \wctype{\Delta'}{D}{[\ol{T}/\ol{X}]\ol{N}} \lessdot \tv{a} }
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% %\set{\wctype{\ol{\wtype{W}}}{D}{[\ol{X}/\ol{Y}]\ol{Z}} \lessdot \tv{a}}
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% \end{array} \quad
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% \begin{array}{l}
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% \texttt{class} \ \exptype{C}{\ol{X}} \triangleleft \exptype{D}{\ol{N}} \\
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% \ol{X} \notin \wildcardEnv \cup \Delta,\, \Delta' = \Delta \cap \text{fv}([\ol{T}/\ol{X}]\ol{N})
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% \end{array}
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% $
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% \\\\
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% \rulename{Settle}
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% & $
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% \begin{array}[c]{l}
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% \wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \type{N},
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% \tv{a} \lessdot \tv{b}}
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% \\
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% \hline
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% \wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \tv{b}, \tv{b} \lessdot \type{N} }
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% \end{array}
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% $
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% \\\\
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% \rulename{Raise}
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% & $
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% \begin{array}[c]{l}
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% \wildcardEnv \vdash C \cup \set{ \tv{a} \lessdot \type{N},
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% \tv{a} \lessdot \tv{b}}
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% \\
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% \hline
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% \wildcardEnv \vdash C \cup \set{\tv{a} \lessdot \type{N}, \type{N} \lessdot \tv{b} }
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% \end{array}
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% $
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% \\\\
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% \end{tabular}}
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% \end{center}
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% \caption{Step 2 branching: Multiple rules can be applied to the same constraint}
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% \label{fig:step2-rules}
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% \end{figure}
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%For every $\type{T} \lessdot \tv{a}$ constraint, the unify algorithm has to consider every possible supertype of $\type{T}$.
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@ -1330,6 +1346,8 @@ Otherwise the generation rules \rulename{GenSigma} and \rulename{GenDelta} will
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% \quad \rulename{AddDelta}
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% \end{displaymath}
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\dashbox{The \text{tph} function returns all type placeholders inside a given type.}
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\begin{figure}
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\begin{center}
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\fbox{
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@ -1347,6 +1365,19 @@ Otherwise the generation rules \rulename{GenSigma} and \rulename{GenDelta} will
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\end{array}
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$
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\\\\
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\rulename{GenDelta'}
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& $
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\deduction{
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\wildcardEnv \vdash C \cup \set{\wtv{b} \lessdot \type{T} } \implies \Delta, \sigma
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}{
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\wildcardEnv \vdash [\type{B}/\ntv{b}]C \implies \Delta \cup \set{\wildcard{B}{\type{T}}{\bot}}, \sigma
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} \quad
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\begin{array}{l}
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\tph(\type{T}) = \emptyset, \text{fv}(\type{T}) \subseteq \Delta \cup \Delta_{in} \\
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\rwildcard{B} \ \text{fresh}, \Delta, \Delta_{in} \vdash \type{T} \ \ok
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\end{array}
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$
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\\\\
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\rulename{GenSigma}
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& $
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\deduction{
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@ -1358,7 +1389,7 @@ Otherwise the generation rules \rulename{GenSigma} and \rulename{GenDelta} will
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} \quad
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\begin{array}{l}
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\tph(\type{T}) = \emptyset \\ %,\, \text{fv}(\type{T}) \subseteq \Delta \\ % T ok implies that
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\ntv{a} \notin \text{dom}(\sigma),\, \Delta, \Delta_{in} \vdash \type{T} \ \ok
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\ntv{a} \notin \text{dom}(\sigma),\, \Delta, \Delta_{in} \vdash \type{T} \ \ok % TODO: Is it possible to imply well-formedness as long as input is well-formed?
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\end{array}
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$
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\\\\
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