paper: define morphism-path relation, redefine typing-relation \& translation

paper/main.tex

@@ -6,7 +6,7 @@
 \usepackage{mathpartir}
 \usepackage{hyperref}
 \usepackage{url}
-
+\usepackage{stmaryrd}
 \usepackage{minted}
 \usemintedstyle{tango}
 
@@ -21,6 +21,7 @@
 \DeclareUnicodeCharacter{03B3}{$\gamma$}
 \DeclareUnicodeCharacter{03B4}{$\delta$}
 \DeclareUnicodeCharacter{0393}{$\Gamma$}
+\DeclareUnicodeCharacter{211D}{$\mathbb{R}$}
 
 \newcommand{\metavariable}[1]{\textcolor{teal}{#1}}
 \newcommand{\typeterminal}[1]{\textcolor{brown}{#1}}
@@ -522,34 +523,66 @@ while preserving its semantics.
 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
+of the form \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)"
-\begin{itemize}
-\item \(\Gamma \vdash \metavariable{e} : \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is of type \(\metavariable{\tau}\)" and
-\item \(\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau} \quad\) read as "expression \(\metavariable{e}\) is compatible with type \(\metavariable{\tau}\)"
-\end{itemize}
 can be derived from the context \(\Gamma\) where \(\metavariable{e} \in \nonterm{E}\) and \(\metavariable{\tau} \in \nonterm{T}\).
 
+\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}
+
+	\inferrule[M-Sub]{
+		\metavariable{\tau} \leq \metavariable{\tau'}
+	}{
+		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
+	}
+
+	\inferrule[M-Single]{
+		(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
+	}{
+		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
+	}
+
+	\inferrule[M-Chain]{
+		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
+		\Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
+	}{
+		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
+	}
+	
+	\inferrule[M-MapSeq]{
+		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
+	}{
+		\Gamma \vdash
+		\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
+		\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
+	}
+\end{mathpar}
 
-\begin{definition}[Representational Well-Typedness]
-An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{representationally well-typed} if there exists a type \(\metavariable{\tau} \in \nonterm{T}\),
-such that \( \emptyset \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}.
 \end{definition}
 
-\begin{definition}[Compatible Well-Typedness]
+\begin{example}[Morphism Graph]
-An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{compatibly well-typed} if there exists a type \(\metavariable{\tau} \in \nonterm{T}\),
+Assume \(\Gamma := \{\\
-such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) by \ref{def:typerules} and \ref{def:semtyperules}.
+	\exprterminal{\text{degrees-to-turns}} : \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R} \rightarrow_\text{morph} \text{Angle}\sim\text{Turns}\sim\mathbb{R}},\\
-\end{definition}
+	\exprterminal{\text{turns-to-radians}} : \typeterminal{\text{Angle}\sim\text{Turns}\sim\mathbb{R} \rightarrow_\text{morph} \text{Angle}\sim\text{Radians}\sim\mathbb{R}},\\
+\}\).
+Then
+\begin{itemize}
+\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\mathbb{R}}\) (by \textsc{M-Sub})
+\item \(\Gamma \vdash \typeterminal{\text{Angle}\sim\text{Degrees}\sim\mathbb{R}} \leadsto \typeterminal{\text{Angle}\sim\text{Radians}\sim\mathbb{R}}\) (by \textsc{M-Chain})
+\item \(\Gamma \vdash \typeterminal{\langle\text{Seq }\text{Angle}\sim\text{Degrees}\sim\mathbb{R}\rangle} \leadsto \typeterminal{\langle\text{Seq }\text{Angle}\sim\text{Radians}\sim\mathbb{R}\rangle}\) (by \textsc{M-MapSeq})
+\end{itemize}
+
+\end{example}
 
 
-\begin{definition}["is" Typing Relation]
+\begin{definition}[Typing Relation]
 
 \label{def:typerules}
 \begin{mathpar}
 
 	\inferrule[T-Variable]{
-%		\metavariable{x} \in \exprvars\\
-%		\metavariable{\tau} \in \nonterm{T}\\
 		\metavariable{x}:\metavariable{\tau} \in \Gamma\\
 	}{
 		\Gamma \vdash \metavariable{x}:\metavariable{\tau}
@@ -563,46 +596,39 @@ such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) b
 	}
 
 	\inferrule[T-TypeAbs]{
-%		\metavariable{\tau} \in \nonterm{T} \\
-%		\metavariable{e} \in \nonterm{E} \\
 		\Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
 	}{
         \Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau}
 	}
 
 	\inferrule[T-TypeApp]{
-%		\metavariable{\tau} \in \nonterm{T} \\
 		\Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\
 		\metavariable{\sigma} \in \nonterm{T}
 	}{
 		\Gamma \vdash ( \metavariable{e} \quad \metavariable{\sigma} ) : \{\metavariable{\alpha} \mapsto \metavariable{\sigma}\} \metavariable{\tau}
 	}
 
-
 	\inferrule[T-Abs]{
-%		\metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
-%		\metavariable{e} \in \nonterm{E} \\
 		\Gamma,\metavariable{x}:\metavariable{\sigma} \vdash \metavariable{e} : \metavariable{\tau} \\
 	}{
         \Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\sigma} \exprterminal{\mapsto} \metavariable{e}) : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
 	}
 
-	\inferrule[T-App]{
-		\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau} \\
-		\Gamma \vdash \metavariable{a} : \metavariable{\sigma} \\
-	}{
-		\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
-	}\and
-
 	\inferrule[T-MorphAbs]{
-%		\metavariable{\sigma}, \metavariable{\tau} \in \nonterm{T} \\
-%		\metavariable{e} \in \nonterm{E} \\
 		\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_{morph}}\metavariable{\tau'}
 	}\and
 
+	\inferrule[T-App]{
+		\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
+		\Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\
+		\Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
+	}{
+		\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
+	}\and
+
 	\inferrule[T-MorphFun]{
         \Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
 	}{
@@ -626,95 +652,191 @@ such that \( \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau} \) b
 \end{mathpar}
 \end{definition}
 
-\begin{definition}["compatible" Typing Relation]
-\label{def:semtyperules}
-\begin{mathpar}
-	\inferrule[TCompat-NativeRepr]{
-		\Gamma\vdash \metavariable{e} : \metavariable{\tau}
-	}{
-		\Gamma\vdash \metavariable{e} :\approx \metavariable{\tau}
-	}
 
-	\inferrule[TCompat-Let]{
+\begin{definition}[Well-Typedness]
-		\Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\
+An expression \(\metavariable{e} \in \nonterm{E}\) is \textbf{well-typed} if there exist \(\Gamma\) and \(\metavariable{\tau}\)
-		\Gamma , \metavariable{x}:\metavariable{\sigma} \vdash \metavariable{t} :\approx \metavariable{\tau}
+such that \( \Gamma \vdash \metavariable{e} : \metavariable{\tau} \) by \ref{def:typerules}.
-	}{
-		\Gamma \vdash (\exprterminal{\text{let }}\metavariable{x}\exprterminal{\text{ = }}\metavariable{e}\exprterminal{\text{ in }} \metavariable{t}) :\approx \metavariable{\tau}
-	}
-
-
-	\inferrule[TCompat-Morph]{
-		\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau}\\
-%		\metavariable{\tau} \precsim \metavariable{\tau'}\\
-		%\exists \metavariable{h} \text{ s.t. }
-		\metavariable{h}:\typeterminal{\metavariable{\tau}\rightarrow_\text{morph}\metavariable{\tau'}} \in \Gamma
-	}{
-		\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
-	}
-
-	\inferrule[TCompat-App]{
-		\Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
-		\Gamma \vdash \metavariable{a} :\approx \metavariable{\sigma}
-	}{
-		\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} :\approx \metavariable{\tau}
-	}
-\end{mathpar}
 \end{definition}
 
 \subsection{Coercion Semantics}
+%We define the translation function \(\llbracket . \rrbracket\) which translates morphism-paths into
+%expressions that define a transformation function, and also translates type-derivations into expressions with expanded type coercions.
 
-We define the translation function \(\llbracket . \rrbracket\) which completes a \emph{semantically well-typed} expression
-by inserting all required coercions based on the typing derivation of the expression.
-The result shall be a \emph{syntactically well-typed} expression.
 
-We write \(C :: \sigma \precsim \tau\) to mean "C is a subtyping derivation tree whose conclusion is \(\sigma \precsim \tau\)".
+%which completes a \emph{semantically well-typed} expression
+%by inserting all required coercions based on the typing derivation of the expression.
+%The result shall be a \emph{syntactically well-typed} expression.
+
+We write \(C :: \tau \leadsto \tau'\) to mean "C is a morphism-path derivation tree whose conclusion is \(\tau \leadsto \tau'\)".
 Similarly, we write \(D :: \Gamma \vdash e : \tau\) to mean "D is a typing derivation whose conclusion is \(\Gamma \vdash e : \tau\)"
 
-\begin{definition}[Translation]
+\begin{definition}[Morphism Translation]
+%Translates a morphism-path derivation into an expression that defines a coercion function
+
 \begin{mathpar}
 
-\Big{\llbracket} \inferrule[T-SemanticSubtype]{
+\Big{\llbracket}
-	D_1 :: \Gamma \vdash \metavariable{h}:\metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}\\
+	\inferrule[M-Sub]{
-    D_2 :: \Gamma \vdash \metavariable{e}:\metavariable{\tau}\\
+		\metavariable{\tau} \leq \metavariable{\tau'}
-%	C :: \metavariable{\tau} \precsim \metavariable{\tau'}
 	}{
-	\Gamma \vdash \metavariable{e} :\approx \metavariable{\tau'}
+		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
-}\Big{\rrbracket} = \exprterminal{(}
+	}
-\llbracket D_1 \rrbracket \llbracket D_2 \rrbracket
+\Big{\rrbracket} = \exprterminal{\lambda x:\metavariable{\tau} \mapsto x}
-%\metavariable{h} \llbracket D_2 \rrbracket
+\and
-\exprterminal{)}
-
-\Big{\llbracket} \inferrule[T-CoercedApp]{
-	D_1 :: \Gamma \vdash \metavariable{f}:\metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
-    D_2 :: \Gamma \vdash \metavariable{a}:\approx\metavariable{\sigma}
-}{
-	\Gamma \vdash \exprterminal{(\metavariable{f} \text{ } \metavariable{a})} : \metavariable{\tau}
-}\Big{\rrbracket} = \exprterminal{(}
-%\exprterminal{(}\metavariable{f} \llbracket D_2 \rrbracket \exprterminal{)}
-\llbracket D_1 \rrbracket \llbracket D_2 \rrbracket
-\exprterminal{)}
 
 \Big{\llbracket}
-\inferrule[\emph{Otherwise}]{}{
+	\inferrule[M-Single]{
-D :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}
+		(\metavariable{h} : \metavariable{\tau} \typeterminal{\rightarrow_\text{morph}} \metavariable{\tau'}) \in \Gamma
+	}{
+		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
 	}
-\Big{\rrbracket} = \metavariable{e}
+\Big{\rrbracket} = \metavariable{h}
+\and
+
+
+\Big{\llbracket}
+	\inferrule[M-Chain]{
+		C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}\\
+		C_2 :: \Gamma \vdash \metavariable{\tau'} \leadsto \metavariable{\tau''}
+	}{
+		\Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau''}
+	}
+\Big{\rrbracket} = \exprterminal{\lambda \text{x}:\metavariable{\tau} \mapsto}
+		\Big{\llbracket} C_2 \Big{\rrbracket}
+		\exprterminal{(}\Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{x})}
+\and
+
+
+\Big{\llbracket}
+	\inferrule[M-MapSeq]{
+		C_1 :: \Gamma \vdash \metavariable{\tau} \leadsto \metavariable{\tau'}
+	}{
+		\Gamma \vdash
+		\typeterminal{\langle\text{Seq } \metavariable{\tau}\rangle} \leadsto
+		\typeterminal{\langle\text{Seq } \metavariable{\tau'}\rangle}
+	}
+\Big{\rrbracket} = \exprterminal{\lambda \text{xs}:\typeterminal{\langle\text{Seq }\metavariable{\tau}\rangle} \mapsto}
+		\exprterminal{( \text{map}} \Big{\llbracket} C_1 \Big{\rrbracket} \exprterminal{\text{xs})}
 
 \end{mathpar}
 \end{definition}
 
-\begin{lemma}[Elimination of \(:\approx\)]
+\begin{definition}[Expression Translation]
-\label{lemma:translation}
+%Translates a type-derivation tree into a fully expanded expression
 
-For all \emph{semantically well-typed} expressions \metavariable{e} with the typing derivation \(D :: \emptyset \vdash \metavariable{e} :\approx \metavariable{\tau}\),
+\begin{mathpar}
-the translation \(\llbracket D \rrbracket = \metavariable{e'}\), yields a \emph{syntactically well-typed} expression \metavariable{e'} with
+\Big{\llbracket} \inferrule[T-Variable]{
-\(\emptyset \vdash \metavariable{e'} : \metavariable{\tau} \)
+	\metavariable{x}:\metavariable{\tau} \in \Gamma
+}{
+	\Gamma \vdash \metavariable{x}:\metavariable{\tau}
+}\Big{\rrbracket} = \metavariable{x}
+\and
+
+\Big{\llbracket} \inferrule[T-LetBinding]{
+	D_1 ::\Gamma \vdash \metavariable{e} : \metavariable{\sigma} \\
+	D_2 :: \Gamma , \metavariable{x}:\metavariable{\sigma} \vdash \metavariable{t} : \metavariable{\tau}
+}{
+	\Gamma \vdash (\exprterminal{\text{let }}\metavariable{x}\exprterminal{\text{ = }}\metavariable{e}\exprterminal{\text{ in }} \metavariable{t}) : \metavariable{\tau}
+}\Big{\rrbracket} = \exprterminal{\text{let }\metavariable{x} = }
+\Big{\llbracket} D_1 \Big{\rrbracket}
+\exprterminal{\text{ in }}
+\Big{\llbracket} D_2 \Big{\rrbracket}
+\and
+
+
+\Big{\llbracket} 
+	\inferrule[T-TypeAbs]{
+		D_1 :: \Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
+	}{
+        \Gamma \vdash (\exprterminal{\Lambda} \metavariable{\alpha} \exprterminal{\mapsto} \metavariable{e}) : \typeterminal{\forall}\metavariable{\alpha}\typeterminal{.}\metavariable{\tau}
+	}
+\Big{\rrbracket} = \exprterminal{\Lambda \metavariable{\alpha} \mapsto} \Big{\llbracket} D_1 \Big{\rrbracket}
+\and
+
+
+\Big{\llbracket}
+	\inferrule[T-TypeApp]{
+		D_1 :: \Gamma \vdash \metavariable{e} : \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\tau} \\
+		\metavariable{\sigma} \in \nonterm{T}
+	}{
+		\Gamma \vdash ( \metavariable{e} \quad \metavariable{\sigma} ) : \{\metavariable{\alpha} \mapsto \metavariable{\sigma}\} \metavariable{\tau}
+	}
+\Big{\rrbracket} =
+\exprterminal{(}
+\Big{\llbracket}
+D_1
+\Big{\rrbracket}
+\metavariable{\sigma}
+\exprterminal{)}
+\and
+
+
+\Big{\llbracket}
+	\inferrule[T-Abs]{
+		D_1 :: \Gamma,\metavariable{x}:\metavariable{\sigma} \vdash \metavariable{e} : \metavariable{\tau} \\
+	}{
+        \Gamma \vdash (\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\sigma} \exprterminal{\mapsto} \metavariable{e}) : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
+	}
+\Big{\rrbracket} = 
+\exprterminal{\lambda} \metavariable{x} \exprterminal{:} \metavariable{\sigma}
+\exprterminal{\mapsto} \Big{\llbracket}D_1\Big{\rrbracket}
+\and
+
+
+\Big{\llbracket}
+	\inferrule[T-App]{
+		D_1 :: \Gamma \vdash \metavariable{f} : \metavariable{\sigma} \typeterminal{\rightarrow} \metavariable{\tau}\\
+		D_2 :: \Gamma \vdash \metavariable{a} : \metavariable{\sigma'}\\\\
+		C :: \Gamma \vdash \metavariable{\sigma'} \leadsto \metavariable{\sigma}
+	}{
+		\Gamma \vdash (\metavariable{f} \quad \metavariable{a}) : \metavariable{\tau}
+	}
+\Big{\rrbracket} =
+	\Big{\llbracket}D_1\Big{\rrbracket}
+	\exprterminal{(}
+	\Big{\llbracket}C\Big{\rrbracket}
+	\Big{\llbracket}D_2\Big{\rrbracket}
+	\exprterminal{)}
+\and
+
+\Big{\llbracket}
+	\inferrule[T-MorphFun]{
+        D_1 :: \Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow_{morph}}\metavariable{\tau}
+	}{
+        \Gamma \vdash \metavariable{f} : \metavariable{\sigma}\typeterminal{\rightarrow}\metavariable{\tau}
+	}
+\Big{\rrbracket} = \Big{\llbracket} D_1 \Big{\rrbracket}
+\and
+
+\Big{\llbracket}
+	\inferrule[T-Ascension]{
+		D_1 :: \Gamma \vdash \metavariable{e} : \metavariable{\tau} \\
+		\metavariable{\tau'} \leq \metavariable{\tau}
+	}{
+		\Gamma \vdash (\metavariable{e} \exprterminal{\text{ as }} \metavariable{\tau'}) : \metavariable{\tau'}
+	}
+\Big{\rrbracket} =
+	\Big{\llbracket}D_1\Big{\rrbracket} \exprterminal{\text{ as }} \metavariable{\tau'}
+\and
+
+
+\Big{\llbracket}
+	\inferrule[T-Descension]{
+		D_1 :: \Gamma \vdash \metavariable{e} : \metavariable{\tau}\\
+		\metavariable{\tau} \leq \metavariable{\tau'}
+	}{
+		\Gamma \vdash \metavariable{e} : \metavariable{\tau'}
+	}
+\Big{\rrbracket} =
+	\Big{\llbracket}
+		D_1
+	\Big{\rrbracket}
+
+
+\end{mathpar}
+\end{definition}
 
-\begin{proof}
-\todo{}
-\end{proof}
 
-\end{lemma}
 
 
 \subsection{Evaluation}