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fix type substitution

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Daniel Hillerström
2020-04-03 17:25:27 +01:00
parent 32f4e2a506
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thesis.tex
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@@ -593,6 +593,59 @@ to $\alpha$-conversion~\cite{Church32} of types.
\] \]
% %
\paragraph{Type substitution}
We define a type substitution map,
$\sigma : (\TyVarCat \times \TypeCat)^\ast$ as list of pairs mapping a
type variable to its replacement. The domain of a substitution map is
set generated by projecting the first component, i.e.
%
\[
\bl
\dom : (\TyVarCat \times \TypeCat)^\ast \to \TyVarCat\\
\dom(\sigma) \defas \{ \alpha \mid (\alpha,\_) \in \sigma \}
\el
\]
%
The application of a type substitution map on a type term, written
$T\sigma$ for some type $T$, is defined inductively on the type
structure as follows.
%
\[
\ba[t]{@{~}l@{~}c@{~}r}
\multicolumn{3}{c}{
\begin{eqs}
(A \eff E)\sigma &\defas& A\sigma \eff E\sigma\\
(A \to C)\sigma &\defas& A\sigma \to C\sigma\\
(\forall \alpha^K.C)\sigma &\defas& \forall \alpha^K.C\sigma \quad \text{if } \alpha \notin \dom(\sigma)\\
\alpha\sigma &\defas& \begin{cases}
A & \text{if } (\alpha,A) \in \sigma\\
\alpha & \text{otherwise}
\end{cases}
\end{eqs}}\\
\begin{eqs}
\Record{R}\sigma &\defas& \Record{R[B/\beta]}\\
{[R]}\sigma &\defas& [R\sigma]\\
\{R\}\sigma &\defas& \{R\sigma\}\\
\cdot\sigma &\defas& \cdot\\
\rho\sigma &\defas& \begin{cases}
R & \text{if } (\rho, R) \in \sigma\\
\rho & \text{otherwise}
\end{cases}\\
\end{eqs}
& ~~~~~~~~~~ &
\begin{eqs}
(\ell : P;R)\sigma &\defas& (\ell : P\sigma; R\sigma)\\
\theta\sigma &\defas& \begin{cases}
P & \text{if } (\theta,P) \in \sigma\\
\theta & \text{otherwise}
\end{cases}\\
\Abs\sigma &\defas& \Abs\\
\Pre{A}\sigma &\defas& \Pre{A\sigma}
\end{eqs}
\ea
\]
%
\paragraph{Types and their inhabitants} \paragraph{Types and their inhabitants}
We now have the basic vocabulary to construct types in $\BCalc$. For We now have the basic vocabulary to construct types in $\BCalc$. For
instance, the signature of the standard polymorphic identity function instance, the signature of the standard polymorphic identity function
@@ -680,6 +733,9 @@ above, an inhabitant of this type performs no effects of its own as
the (right-most) effect row is a singleton row containing a distinct the (right-most) effect row is a singleton row containing a distinct
effect variable $\varepsilon'$. effect variable $\varepsilon'$.
\paragraph{Syntactic sugar}
Detail the syntactic sugar\dots
\subsection{Terms} \subsection{Terms}
\label{sec:base-language-terms} \label{sec:base-language-terms}
% %
@@ -702,7 +758,7 @@ effect variable $\varepsilon'$.
\end{figure} \end{figure}
% %
The syntax for terms is given in The syntax for terms is given in
Figure~\ref{fig:base-language-term-syntax}. We assume countably Figure~\ref{fig:base-language-term-syntax}. We assume a countably
infinite set of names $\VarCat$ from which we draw fresh variable infinite set of names $\VarCat$ from which we draw fresh variable
names. We shall typically denote term variables by $x$, $y$, or $z$. names. We shall typically denote term variables by $x$, $y$, or $z$.
% %
@@ -711,11 +767,11 @@ The syntax partitions terms into values and computations.
Value terms comprise variables ($x$), lambda abstraction Value terms comprise variables ($x$), lambda abstraction
($\lambda x^A . \, M$), type abstraction ($\Lambda \alpha^K . \, M$), ($\lambda x^A . \, M$), type abstraction ($\Lambda \alpha^K . \, M$),
and the introduction forms for records and variants. Records are and the introduction forms for records and variants. Records are
introduced using the empty record $\Record{}$ and record extension introduced using the empty record $(\Record{})$ and record extension
$\Record{\ell = V; W}$, whilst variants are introduced using injection $(\Record{\ell = V; W})$, whilst variants are introduced using
$(\ell~V)^R$, which injects a field with label $\ell$ and value $V$ injection $((\ell~V)^R)$, which injects a field with label $\ell$ and
into a row whose type is $R$. We include the row type annotation in value $V$ into a row whose type is $R$. We include the row type
order to support bottom-up type reconstruction. annotation in to support bottom-up type reconstruction.
All elimination forms are computation terms. Abstraction and type All elimination forms are computation terms. Abstraction and type
abstraction are eliminated using application ($V\,W$) and type abstraction are eliminated using application ($V\,W$) and type
@@ -741,8 +797,12 @@ kind information (term abstraction, type abstraction, injection,
operations, and empty cases). However, we shall omit these annotations operations, and empty cases). However, we shall omit these annotations
whenever they are clear from context. whenever they are clear from context.
\paragraph{Free variables} We define the function \paragraph{Free variables} A given term is said to be \emph{closed} if
$\FV : \TermCat \to \VarCat$ to compute the free variables of any every applied occurrence of a variable is preceded by some
corresponding binding occurrence. Any applied occurrence of a variable
that is not preceded by a binding occurrence is said be \emph{free
variable}. We define the function $\FV : \TermCat \to \VarCat$
inductively on the term structure to compute the free variables of any
given term. given term.
% %
\[ \[
@@ -771,6 +831,12 @@ given term.
\end{eqs} \end{eqs}
\el \el
\] \]
%
The function computes the set of free variables bottom-up. Most cases
are homomorphic on the syntax constructors. The interesting cases are
those constructs which feature term binders: lambda abstraction, let
bindings, pair destructing, and case splitting. In each of those cases
we subtract the relevant binder(s) from the set of free variables.
\subsection{Typing rules} \subsection{Typing rules}
\label{sec:base-language-type-rules} \label{sec:base-language-type-rules}
@@ -891,38 +957,7 @@ domain type of the abstractor agree.
The type application rule \tylab{PolyApp} tells us that a type The type application rule \tylab{PolyApp} tells us that a type
application $V\,A$ is well-typed whenever the abstractor term $V$ has application $V\,A$ is well-typed whenever the abstractor term $V$ has
the polymorphic type $\forall \alpha^K.C$ and the type $A$ has kind the polymorphic type $\forall \alpha^K.C$ and the type $A$ has kind
$K$. This rule makes use of type substitution. We write $C[A/\alpha]$ $K$. This rule makes use of type substitution.
to mean substitute some type $A$ for some type variable $\alpha$ in
some type $C$. We define type substitution as a ternary function
defined as follows.
%
\[
\begin{eqs}
-[-/-] &:& \TypeCat \times \TypeCat \times \TyVarCat \to \TypeCat\\
(A \eff E)[B/\beta] &\defas& A[B/\beta] \eff E[B/\beta]\\
(A \to C)[B/\beta] &\defas& A[B/\beta] \to C[B/\beta]\\
(\forall \alpha^K.C)[B/\beta] &\defas& \forall \alpha^K.C[B/\beta] \quad \text{if } \alpha \neq \beta \text{ and } \alpha \notin \FTV(B)\\
\alpha[B/\beta] &\defas& \begin{cases}
B & \text{if } \alpha = \beta\\
\alpha & \text{otherwise}
\end{cases}\\
\Record{R}[B/\beta] &\defas& \Record{R[B/\beta]}\\
{[R]}[B/\beta] &\defas& [R[B/\beta]]\\
\{R\}[B/\beta] &\defas& \{R[B/\beta]\}\\
\cdot[B/\beta] &\defas& \cdot\\
\rho[B/\beta] &\defas& \begin{cases}
B & \text{if } \rho = \beta\\
\rho & \text{otherwise}
\end{cases}\\
(\ell : P;R)[B/\beta] &\defas& (\ell : P[B/\beta]; R[B/\beta])\\
\theta[B/\beta] &\defas& \begin{cases}
B & \text{if } \rho = \beta\\
\theta & \text{otherwise}
\end{cases}\\
\Abs[B/\beta] &\defas& \Abs\\
\Pre{A}[B/\beta] &\defas& \Pre{A[B/\beta]}
\end{eqs}
\]
% %
The \tylab{Split} rule handles typing of record destructing. When The \tylab{Split} rule handles typing of record destructing. When
splitting a record term $V$ on some label $\ell$ binding it to $x$ and splitting a record term $V$ on some label $\ell$ binding it to $x$ and