paper: adapt soundness theorems

paper/main.tex

@@ -518,17 +518,23 @@ while preserving its semantics.
 \end{enumerate}
 \end{example}
 
-\subsubsection{Inference of Expression Types}
+\subsubsection{Typing Context}
 
 As usual, the typing-context \(\Gamma = \{ \metavariable{x_1} : \metavariable{\tau_1} , \quad \metavariable{x_2} : \metavariable{\tau_2} , \quad \ldots \}\)
 is a finite mapping which assigns variables \(\metavariable{x_i} \in \exprvars\) to types \(\metavariable{\tau_i} \in \nonterm{T}\).
-Using the inference rules given in \ref{def:typerules}, further typing-judgements
+
-of the form \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)"
+%Using the inference rules given in \ref{def:pathrules} \ref{def:typerules}, further typing-judgements
-can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \nonterm{E}\) and \(\metavariable{\tau} \in \nonterm{T}\).
+%of the form \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)"
+%can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \nonterm{E}\) and \(\metavariable{\tau} \in \nonterm{T}\).
+
+\subsubsection{Morphism Graph}
+Every typing context \(\Gamma\) implies a \emph{Morphism Graph}, a directed graph whose vertices are types
+and the edges represent a type-transformations, as defined by morphisms.
+A type \(\metavariable{\tau}\) can be implicitly coerced into a type \(\metavariable{\tau'}\),
+provided there is a path from \(\metavariable{\tau}\) to \(\metavariable{\tau'}\) in the \emph{Morphism-Graph} of \(\Gamma\),
+written as \(\Gamma  \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\).
 
 \begin{definition}[Morphism Paths]
-Given a typing context \(\Gamma\), any type \(\metavariable{\tau}\) can be transformed into \(\metavariable{\tau'}\), provided there is a path from \(\metavariable{\tau}\) to \(\metavariable{\tau'}\) in the \emph{Morphism-Graph} of \(\Gamma\), written as \(\Gamma  \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\).
-
 \label{def:pathrules}
 \begin{mathpar}
 
@@ -783,6 +789,20 @@ D_1
 \and
 
 
+\Big{\llbracket}
+	\inferrule[T-MorphAbs]{
+		D_1 :: \Gamma,\metavariable{x}:\metavariable{\tau} \vdash \metavariable{e} : \metavariable{\tau'} \\
+		\metavariable{\tau} \precsim \metavariable{\tau'}
+	}{
+        \Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\tau} \exprterminal{\mapsto_{morph}} \metavariable{e}) : \metavariable{\tau}\typeterminal{\rightarrow_\text{morph}}\metavariable{\tau'}
+	}
+\Big{\rrbracket} = 
+\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\tau}
+\exprterminal{\mapsto_\text{morph}} \Big{\llbracket}D_1\Big{\rrbracket}
+\and
+
+
+
 \Big{\llbracket}
 	\inferrule[T-App]{
 		D_1 :: \Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
@@ -905,16 +925,17 @@ Evaluation of an expression \(\metavariable{e} \in \nonterm{E}\) is defined by e
 		\rightarrow_\beta
 		\{ \metavariable{x} \mapsto \metavariable{a} \} \metavariable{e}
 	}
-	
+
 	\inferrule[E-AppLamAscribe]{
 	}{
-		\exprterminal{( \lambda \metavariable{x}:\metavariable{\sigma} \mapsto \metavariable{e} )}
+		\exprterminal{(( \lambda \metavariable{x}:\metavariable{\sigma} \mapsto \metavariable{e} )}
 		\exprterminal{\text{ as }}
 		\typeterminal{\metavariable{\tau}}
-		\metavariable{e}
+		\exprterminal{)}
+		\metavariable{a}
 		\rightarrow_\beta
-		\metavariable{v}
+		\exprterminal{( \lambda \metavariable{x}:\metavariable{\sigma} \mapsto \metavariable{e} )}
-		\metavariable{e}
+		\metavariable{a}
 	}
 \end{mathpar}
 \end{definition}
@@ -924,10 +945,13 @@ Evaluation of an expression \(\metavariable{e} \in \nonterm{E}\) is defined by e
 
 \begin{lemma}[Preservation]
 \label{lemma:preservation}
-Assume the expression \(\metavariable{e}\) is well typed, i.e. \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\)
+Assume the expression \(\metavariable{e}\) is well typed,
-for some type \(\metavariable{\tau}\).
+i.e. there is a type-derivation tree
-Then forall \(\metavariable{e'}\) with \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\)
+\(D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\)
-it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as well.
+for some type \(\metavariable{\tau}\) and context \(\Gamma\).
+
+Then forall \(\metavariable{e'}\) with \(\llbracket D \rrbracket \rightarrow_{\beta} \metavariable{e'}\)
+it holds that \(\Gamma \vdash \metavariable{e'} : \metavariable{\tau}\) as well.
 
 \begin{proof}
 \todo{}
@@ -936,38 +960,31 @@ it holds that \(\emptyset \vdash \metavariable{e'} : \metavariable{\tau}\) as we
 
 \begin{lemma}[Progress]
 \label{lemma:progress}
-If \(\emptyset \vdash \metavariable{e} : \metavariable{\tau}\),
+Assume the expression \(\metavariable{e}\) is well typed,
-then either \(\metavariable{e}\) is a value
+i.e. there is a type-derivation tree
-or there exists some \(\metavariable{e'}\) such that \(\metavariable{e} \rightarrow_{\beta} \metavariable{e'}\)
+\(D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\)
+for some type \(\metavariable{\tau}\) and context \(\Gamma\).
+
+Then either \(\metavariable{e}\) is a value
+or there exists some \(\metavariable{e'}\) such that \(\llbracket D \rrbracket \rightarrow_{\beta} \metavariable{e'}\)
 
 \begin{proof}
 \todo{}
 \end{proof}
 \end{lemma}
 
-\begin{theorem}[Syntactic Type Soundness]
+\begin{theorem}[Soundness]
-\label{theorem:syntactic-soundness}
-No syntactically well-typed expression is stuck.
-
-Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\metavariable{\tau}\).
-Then it never occurs that \(\metavariable{e} \rightarrow_{\beta}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.
-
-\begin{proof}
-Follows from \ref{lemma:progress} and \ref{lemma:preservation}.
-\end{proof}
-\end{theorem}
-
-\begin{theorem}[Semantic Type Soundness]
 \label{theorem:semantic-soundness}
-No semantically well-typed expression is stuck.
+No well-typed expression is stuck.
 
-Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\).
+Assume the typing derivation \(D :: \Gamma \vdash \metavariable{e}:\metavariable{\tau}\).
 Then it never occurs that \(\llbracket D \rrbracket \rightarrow_{\beta}^{*} \metavariable{e'}\) where \metavariable{e'} is in normal form but not a value.
 
 \begin{proof}
-Assume the typing derivation \(D :: \emptyset \vdash \metavariable{e}:\approx\metavariable{\tau}\).
+\todo{}
-By \ref{lemma:translation}, \(\emptyset \vdash \llbracket D \rrbracket : \metavariable{\tau}\)
+%Assume the typing derivation \(D :: \Gamma \vdash \metavariable{e}:\approx\metavariable{\tau}\).
-and thus it follows by \ref{theorem:syntactic-soundness} that \metavariable{e} is not stuck.
+%By \ref{lemma:translation}, \(\Gamma \vdash \llbracket D \rrbracket : \metavariable{\tau}\)
+%and thus it follows by \ref{theorem:syntactic-soundness} that \metavariable{e} is not stuck.
 \end{proof}
 \end{theorem}