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paper:work on syntax definition & substitutions

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Michael Sippel 2024-08-08 13:17:34 +02:00
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commit 35e0a9b62b
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1 changed files with 82 additions and 47 deletions
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paper/main.tex
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@ -116,6 +116,13 @@ $$\\$$
{ \metavariable{x} { \metavariable{x}
} {Variable} } {Variable}
\otherform{
\exprterminal{let} \quad \metavariable{x} \quad \exprterminal{=} \quad
\nonterm{ E }
\quad \exprterminal{in} \quad
\nonterm{ E }
}{Variable Binding}
\otherform{ \otherform{
$$ \exprterminal{\Lambda} \metavariable{\alpha} $$ \exprterminal{\Lambda} \metavariable{\alpha}
\quad \exprterminal{\mapsto} \quad $$ \quad \exprterminal{\mapsto} \quad $$
@ -136,13 +143,6 @@ $$\\$$
\nonterm{ E } \nonterm{ E }
}{Value Morphism} }{Value Morphism}
\otherform{
\exprterminal{let} \quad \metavariable{x} \quad \exprterminal{=} \quad
\nonterm{ E }
\quad \exprterminal{in} \quad
\nonterm{ E }
}{Variable Binding}
\otherform{ \otherform{
\nonterm{ E } \nonterm{ E }
\quad \quad
@ -161,23 +161,25 @@ $$\\$$
\exprterminal{as} \exprterminal{as}
\quad \quad
\nonterm{ T } \nonterm{ T }
}{Up-Cast} }{Ascription}
\otherform{ %\otherform{
\nonterm{ E } % \nonterm{ E }
\quad % \quad
\exprterminal{to} % \exprterminal{to}
\quad % \quad
\nonterm{ T } % \nonterm{ T }
}{Transformation} %}{Transformation}
\otherform{\exprterminal{(} \quad \nonterm{E} \quad \exprterminal{)}}{Parenthesis} %\otherform{\exprterminal{(} \quad \nonterm{E} \quad \exprterminal{)}}{Parenthesis}
$$\\$$ $$\\$$
\firstcase{V}{ \firstcase{V}{
\exprterminal{\epsilon} \nonterm{ V } \quad
}{Empty Value} \exprterminal{as} \quad
\nonterm{ T }
}{Ascribed Value}
\otherform{ \otherform{
\exprterminal{\Lambda} \metavariable{\alpha} \quad \exprterminal{\Lambda} \metavariable{\alpha} \quad
@ -193,19 +195,14 @@ $$\\$$
\nonterm{ E } \nonterm{ E }
}{Abstraction Value} }{Abstraction Value}
\otherform{
\nonterm{ V } \quad
\exprterminal{as} \quad
\nonterm{ T }
}{Cast Value}
\end{grammar} \end{grammar}
\caption{Syntax of the core calculus with colors for \metavariable{metavariables}, \typeterminal{type-level terminal symbols}, \exprterminal{expression-level terminal symbols} \caption{Syntax of the core calculus with colors for \metavariable{metavariables}, \typeterminal{type-level terminal symbols}, \exprterminal{expression-level terminal symbols}
where $\typenames, \typevars, \exprvars$ are mutually disjoint, countable sets of symbols to denote atomic type identifiers (\(\typenames\)), typevariables (\(\typevars\)), and expression variables (\(\exprvars\)). where $\typenames, \typevars, \exprvars$ are mutually disjoint, countable sets of symbols to denote atomic type identifiers (\(\typenames\)), typevariables (\(\typevars\)), and expression variables (\(\exprvars\)).
By default, assume \(\metavariable{\sigma} \in \typenames\), \(\metavariable{\alpha} \in \typevars\) and \(\metavariable{x} \in \exprvars\) By default, assume \(\metavariable{\sigma} \in \typenames\), \(\metavariable{\alpha} \in \typevars\) and \(\metavariable{x} \in \exprvars\).
For simplicity, we write \(\metavariable{e} \in \nonterm{E}\) to say that the term \metavariable{e} is contained in the language generated by the nonterminal \(\nonterm{E}\).
$$\\$$} $$\\$$}
\end{figure} \end{figure}
@ -242,9 +239,9 @@ The following terms are valid types over \(\Sigma\):
\begin{definition}[Substitution in Types] \begin{definition}[Substitution in Types]
Given a type-variable assignment \(\psi_t = \{ \metavariable{\alpha_1} \mapsto \metavariable{\tau_1}, \quad \metavariable{\alpha_2} \mapsto \metavariable{\tau_2}, \quad \dots \}\), Given a type-variable assignment \(\psi_t = \{ \metavariable{\alpha_1} \mapsto \metavariable{\tau_1}, \quad \metavariable{\alpha_2} \mapsto \metavariable{\tau_2}, \quad \dots \}\),
the thereby induced, substitution \(\overline{\psi_t}\) replaces all \emph{free} occurences of the variables \(\metavariable{\alpha_i}\) in a type-term \(\metavariable{\xi} \in \typenonterm{\{\metavariable{\alpha_1}, \quad \metavariable{\alpha_2}, \quad \dots\}}\) recursively with the type-term given by \(\psi_t(\metavariable{\alpha_i})\). the thereby induced substitution \(\overline{\psi_t}\) replaces all \emph{free} occurences of the variables \(\metavariable{\alpha_i}\) in a type-term \(\metavariable{\xi} \in \nonterm{T}\) recursively with the type-term given by \(\psi_t(\metavariable{\alpha_i})\)
Occourences of bound variables \(\metavariable{\alpha_i}\) are , while occurences of bound variables are left untouched.
Further, we assume that for all \(\tau_i\), all variable names are disjunct with the free variables of the term to which the substitution is applied.
Coq definition is at \hyperref[coq:subst-type]{subst.v:\ref{coq:subst-type}}. Coq definition is at \hyperref[coq:subst-type]{subst.v:\ref{coq:subst-type}}.
\[\overline{\psi_t} \metavariable{\xi} = \begin{cases} \[\overline{\psi_t} \metavariable{\xi} = \begin{cases}
@ -261,9 +258,39 @@ Coq definition is at \hyperref[coq:subst-type]{subst.v:\ref{coq:subst-type}}.
\[\overline{\psi_t} \metavariable{e} = \begin{cases} \[\overline{\psi_t} \metavariable{e} = \begin{cases}
\metavariable{e} \quad \text{ if } \metavariable{e} \text{ is a variable} \metavariable{e} \quad \text{ if } \metavariable{e} \text{ is a variable}
\exprterminal{\Lambda \metavariable{\alpha} \mapsto} \overline{\psi_t}\metavariable{e'} \quad \text{ if \metavariable{e} is of the form } \exprterminal{\Lambda \metavariable{\alpha} \mapsto \metavariable{e'}}\\ \\
\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto} \overline{\psi_t}\metavariable{e'} \quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto \metavariable{e'}}\\
\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto_\text{morph}} \overline{\psi_t}\metavariable{e'} \quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto_\text{morph} \metavariable{e'}}\\ \exprterminal{\text{let } \metavariable{x} = }\overline{\psi_t}\metavariable{a} \exprterminal{\text{ in }} \overline{\psi_t}\metavariable{e'}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{\text{let } \metavariable{x} = \metavariable{a} \text{ in } \metavariable{e'}}
\\
\exprterminal{\Lambda \metavariable{\alpha} \mapsto} \overline{\psi_t}\metavariable{e'}
\quad \text{ if \metavariable{e} is of the form } \exprterminal{\Lambda \metavariable{\alpha} \mapsto \metavariable{e'}}
\\
\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto} \overline{\psi_t}\metavariable{e'}
\quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto \metavariable{e'}}
\\
\exprterminal{\lambda \metavariable{x} : }\overline{\psi_t}\metavariable{\tau} \exprterminal{\mapsto_\text{morph}} \overline{\psi_t}\metavariable{e'}
\quad \text{ if \metavariable{e} is of the form } \exprterminal{\lambda \metavariable{x} : \metavariable{\tau} \mapsto_\text{morph} \metavariable{e'}}
\\
\overline{\psi_t} \metavariable{e'} \overline{\psi_t}\metavariable{\tau}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{( \metavariable{e'} \metavariable{\tau} )}
\\
\overline{\psi_t} \metavariable{e_1} \overline{\psi_t} \metavariable{e_2}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{(\metavariable{e_1} \metavariable{e_2})}
\\
\overline{\psi_t} \metavariable{e'} \exprterminal{\text{ as }} \overline{\psi_t}\metavariable{\tau}
\quad \text{ if \metavariable{e} is of the form }
\exprterminal{ \metavariable{e'} \text{ as } \metavariable{\tau} }
\end{cases}\] \end{cases}\]
@ -276,23 +303,31 @@ Coq definition is at \hyperref[coq:subst-type]{subst.v:\ref{coq:subst-type}}.
\begin{definition}[Substitution in Expressions] \begin{definition}[Substitution in Expressions]
\todo{complete} \todo{complete}
Given an expression-variable assignment \(\psi_e = \{ \metavariable{x_1} \mapsto \metavariable{t_1}, \quad \metavariable{x_2} \mapsto \metavariable{t_2}, \quad \dots \}\), Given an expression-variable assignment \(\psi_e = \{ \metavariable{x_1} \mapsto \metavariable{t_1}, \quad \metavariable{x_2} \mapsto \metavariable{t_2}, \quad \dots \}\),
the thereby induced, lexically scoped substitution \(\overline{\psi_e}\) replaces all \emph{free} occurences of the expression variables \(\metavariable{x_i}\) the thereby induced substitution \(\overline{\psi_e}\) replaces all \emph{free} occurences of the expression variables \(\metavariable{x_i}\)
in an expression \( in an expression \(e \in \nonterm{E} \) with the \(\psi_e(\metavariable{x_i})\)
e \in
\exprnonterm \[\overline{\psi_e} \metavariable{e} = \begin{cases}
{\typevars}
{\{\metavariable{x_1}, \quad \metavariable{x_2}, \quad \dots\}} \metavariable{e} \quad \text{if } \metavariable{e} \in \exprvars \text{ and } \metavariable{e} \notin \text{Dom}(\psi_e)\\
\). Lexical scoping is implemented by simply not substituting any bound occourences of variables \(\metavariable{\alpha_i}\). This allows to skip \(\alpha\)-conversion as done in classical \(\lambda\)-calculus. \metavariable{t} \quad \text{if } \metavariable{e} \in \exprvars \text{ and } (\metavariable{e}\mapsto\metavariable{t}) \in \psi_e\\
\exprterminal{\text{let } \metavariable{x} \text{ = }} \overline{\psi_e}\metavariable{a} \exprterminal{\text{ in }} \overline{\psi_e}\metavariable{e'}
\quad \text{if }
\\
\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto } \overline{\psi_e} \metavariable{e'}
\quad \text{if \metavariable{e} is of the form }
\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto \metavariable{e'}}
\text{ and } \metavariable{x} \notin \text{Dom}(\psi_e)
\\
\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto } \overline{\psi_e \setminus \{\metavariable{x} \mapsto \metavariable{t}\}} \metavariable{e'}
\quad \text{if \metavariable{e} is of the form }
\exprterminal{\lambda\metavariable{x}:\metavariable{\tau} \mapsto \metavariable{e'}}
\text{ and } (\metavariable{x}\mapsto\metavariable{t}) \in \psi_e
\\
\[\overline{\psi} \metavariable{e} = \begin{cases}
\metavariable{e} \quad \text{if } \metavariable{\xi} \in \typenames\\
\metavariable{\tau} \quad \text{if } (\metavariable{\xi} \mapsto \metavariable{\tau}) \in \psi\\
\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \overline{\psi}\metavariable{\xi'} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \text{ and } \metavariable{\alpha} \notin \text{Dom}(\psi)\\
\typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{\forall} \metavariable{\alpha} \typeterminal{.} \metavariable{\xi'} \text{ and } \metavariable{\alpha} \in \text{Dom}(\psi)\\
\typeterminal{<} (\overline{\psi} \metavariable{\xi_1}) \quad (\overline{\psi} \metavariable{\xi_2}) \typeterminal{>} \quad \text{if } \metavariable{\xi} \text{ is of the form } \typeterminal{<} \metavariable{\xi_1} \quad \metavariable{\xi_2} \typeterminal{>}\\
(\overline{\psi} \metavariable{\xi_1}) \typeterminal{\rightarrow} (\overline{\psi} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\rightarrow} \metavariable{\xi_2}\\
(\overline{\psi} \metavariable{\xi_1}) \typeterminal{\rightarrow_{morph}} (\overline{\psi} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\rightarrow_{morph}} \metavariable{\xi_2}\\
(\overline{\psi} \metavariable{\xi_1}) \typeterminal{\sim} (\overline{\psi} \metavariable{\xi_2}) \quad \text{if } \metavariable{\xi} \text{ is of the form } \metavariable{\xi_1} \typeterminal{\sim} \metavariable{\xi_2}\\
\end{cases}\] \end{cases}\]
\end{definition} \end{definition}