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Capture-avoiding substitution.

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Daniel Hillerström
2020-01-25 19:31:39 +00:00
parent 6d8c70a8c6
commit cecd23e853
3 changed files with 139 additions and 45 deletions
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12
macros.tex
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@@ -71,6 +71,7 @@
\newcommand{\typc}[3]{#1 \vdash #2 \eff #3} \newcommand{\typc}[3]{#1 \vdash #2 \eff #3}
\newcommand{\FTV}{\ensuremath{\mathrm{FTV}}} \newcommand{\FTV}{\ensuremath{\mathrm{FTV}}}
\newcommand{\FV}{\ensuremath{\mathrm{FV}}}
\newcommand{\reducesto}[0]{\ensuremath{\leadsto}} \newcommand{\reducesto}[0]{\ensuremath{\leadsto}}
\newcommand{\stepsto}[0]{\ensuremath{\longrightarrow}} \newcommand{\stepsto}[0]{\ensuremath{\longrightarrow}}
@@ -105,8 +106,8 @@
\newcommand{\CompCat}{\CatName{Comp}} \newcommand{\CompCat}{\CatName{Comp}}
\newcommand{\ValCat}{\CatName{Val}} \newcommand{\ValCat}{\CatName{Val}}
\newcommand{\VarCat}{\CatName{Var}} \newcommand{\VarCat}{\CatName{Var}}
\newcommand{\ValTypeCat}{\CatName{TVal}} \newcommand{\ValTypeCat}{\CatName{VType}}
\newcommand{\CompTypeCat}{\CatName{TComp}} \newcommand{\CompTypeCat}{\CatName{CType}}
\newcommand{\PresenceCat}{\CatName{Presence}} \newcommand{\PresenceCat}{\CatName{Presence}}
\newcommand{\TypeCat}{\CatName{Type}} \newcommand{\TypeCat}{\CatName{Type}}
\newcommand{\TyVarCat}{\CatName{TVar}} \newcommand{\TyVarCat}{\CatName{TVar}}
@@ -145,4 +146,9 @@
%% %%
%% Defined-as equality %% Defined-as equality
%% %%
\newcommand{\defas}[0]{\mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\text{def}}}}{=}}} \newcommand{\defas}[0]{\mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\text{def}}}}{=}}}
%%
%% Partiality
%%
\newcommand{\pto}[0]{\ensuremath{\rightharpoonup}}

10
thesis.bib
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@@ -759,3 +759,13 @@
publisher = {Indiana University}, publisher = {Indiana University},
address = {Indianapolis, IN, USA}, address = {Indianapolis, IN, USA},
} }
# The original lambda calculus reference
@InProceedings{Church32,
author = {Alonzo Church},
title = {A Set of Postulates for the Foundation of Logic},
year = {1932},
booktitle = {Annals of Mathematics},
pages = {346--366},
volume = {33}
}

162
thesis.tex
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@@ -50,6 +50,15 @@
\textquotedblleft={ ,150}, % left quotation mark, space from right \textquotedblleft={ ,150}, % left quotation mark, space from right
\textquotedblright={150, }} % right quotation mark, space from left \textquotedblright={150, }} % right quotation mark, space from left
%%
%% Theorem environments
%%
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
%% %%
%% Load macros. %% Load macros.
%% %%
@@ -500,11 +509,11 @@ $\TyVarCat$, to be generated by:
\mid \lambda x^A .\, M \mid \Lambda \alpha^K .\, M \mid \lambda x^A .\, M \mid \Lambda \alpha^K .\, M
\mid \Record{} \mid \Record{\ell = V;W} \mid (\ell~V)^R \\ \mid \Record{} \mid \Record{\ell = V;W} \mid (\ell~V)^R \\
& & &\\ & & &\\
\slab{Computations} &M,N \in \CompCat &::= & V\,W \mid V\,A\\ \slab{Computations} &M,N \in \CompCat &::= & V\,W \mid V\,T\\
& &\mid& \Let\; \Record{\ell=x;y} = V \; \In \; N\\ & &\mid& \Let\; \Record{\ell=x;y} = V \; \In \; N\\
& &\mid& \Case\; V \{\ell~x \mapsto M; y \mapsto N\} \mid \Absurd^C~V\\ & &\mid& \Case\; V \{\ell~x \mapsto M; y \mapsto N\} \mid \Absurd^C~V\\
& &\mid& \Return~V \mid \Let \; x \revto M \; \In \; N\\ & &\mid& \Return~V \mid \Let \; x \revto M \; \In \; N\\
\slab{Terms} &T \in \TermCat &::= & x \mid V \mid M \slab{Terms} &t \in \TermCat &::= & x \mid V \mid M
\end{syntax} \end{syntax}
\caption{Term syntax of \BCalc{}.} \caption{Term syntax of \BCalc{}.}
@@ -552,6 +561,37 @@ 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
$\FV : \TermCat \to \VarCat$ to compute the free variables of any
given term.
%
\[
\bl
\ba[t]{@{~}l@{~}c@{~}l}
\begin{eqs}
\FV(x) &\defas& \{x\}\\
\FV(\lambda x^A.M) &\defas& \FV(M) \setminus \{x\}\\
\FV(\Lambda \alpha^K.M) &\defas& \FV(M)\\[1.0ex]
\FV(V\,W) &\defas& \FV(V) \cup \FV(W)\\
\FV(\Return~V) &\defas& \FV(V)\\
\end{eqs}
& \qquad\qquad &
\begin{eqs}
\FV(\Record{}) &\defas& \emptyset\\
\FV(\Record{\ell = V; W}) &\defas& \FV(V) \cup \FV(W)\\
\FV((\ell~V)^R) &\defas& \FV(V)\\[1.0ex]
\FV(V\,T) &\defas& \FV(V)\\
\FV(\Absurd^C~V) &\defas& \FV(V)\\
\end{eqs}
\ea\\
\begin{eqs}
\FV(\Let\;x \revto M \;\In\;N) &\defas& \FV(M) \cup (\FV(N) \setminus \{x\})\\
\FV(\Let\;\Record{\ell=x;y} = V\;\In\;N) &\defas& \FV(V) \cup (\FV(N) \setminus \{x, y\})\\
\FV(\Case~V~\{\ell\,x \mapsto M; y \mapsto N\} &\defas& \FV(V) \cup (\FV(M) \setminus \{x\}) \cup (\FV(N) \setminus \{y\})
\end{eqs}
\el
\]
\subsection{Typing rules} \subsection{Typing rules}
\label{sec:base-language-type-rules} \label{sec:base-language-type-rules}
% %
@@ -660,13 +700,17 @@ of \emph{free type variables} ($\FTV$) to ensure that we do not
inadvertently capture a free type variable from the context. inadvertently capture a free type variable from the context.
% %
We define $\FTV$ by mutual induction over type environments, $\Gamma$, We define $\FTV$ by mutual induction over type environments, $\Gamma$,
and the type structure, $T$, as follows. and the type structure, $T$. Note that we always work up to
$\alpha$-conversion~\cite{Church32} of types.
% %
\[ \[
\ba[t]{@{~}l@{~~~~~~}c@{~}l} \ba[t]{@{~}l@{~~~~~~}c@{~}l}
\multicolumn{3}{c}{\begin{eqs}
\FTV &:& \TypeCat \to \TyVarCat
\end{eqs}}\\
\ba[t]{@{}l} \ba[t]{@{}l}
\begin{eqs} \begin{eqs}
\FTV &:& \ValTypeCat \to \TyVarCat\\ % \FTV &:& \ValTypeCat \to \TyVarCat\\
\FTV(\alpha) &\defas& \{\alpha\}\\ \FTV(\alpha) &\defas& \{\alpha\}\\
\FTV(\forall \alpha^K.C) &\defas& \FTV(C) \setminus \{\alpha\}\\ \FTV(\forall \alpha^K.C) &\defas& \FTV(C) \setminus \{\alpha\}\\
\FTV(A \to C) &\defas& \FTV(A) \cup \FTV(C)\\ \FTV(A \to C) &\defas& \FTV(A) \cup \FTV(C)\\
@@ -683,12 +727,12 @@ and the type structure, $T$, as follows.
% \FTV([R]) &\defas& \FTV(R)\\ % \FTV([R]) &\defas& \FTV(R)\\
% \FTV(\Record{R}) &\defas& \FTV(R)\\ % \FTV(\Record{R}) &\defas& \FTV(R)\\
% \FTV(\{R\}) &\defas& \FTV(R)\\ % \FTV(\{R\}) &\defas& \FTV(R)\\
\FTV &:& \RowCat \to \TyVarCat\\ % \FTV &:& \RowCat \to \TyVarCat\\
\FTV(\cdot) &\defas& \emptyset\\ \FTV(\cdot) &\defas& \emptyset\\
\FTV(\rho) &\defas& \{\rho\}\\ \FTV(\rho) &\defas& \{\rho\}\\
\FTV(l:P;R) &\defas& \FTV(P) \cup \FTV(R)\\[1.0ex] \FTV(l:P;R) &\defas& \FTV(P) \cup \FTV(R)\\
\FTV &:& \PresenceCat \to \TyVarCat\\ % \FTV &:& \PresenceCat \to \TyVarCat\\
\FTV(\theta) &\defas& \{\theta\}\\ \FTV(\theta) &\defas& \{\theta\}\\
\FTV(\Abs) &\defas& \emptyset\\ \FTV(\Abs) &\defas& \emptyset\\
\FTV(\Pre{A}) &\defas& \FTV(A)\\ \FTV(\Pre{A}) &\defas& \FTV(A)\\
@@ -747,16 +791,13 @@ defined as follows.
-[-/-] &:& \TypeCat \times \TypeCat \times \TyVarCat \to \TypeCat\\ -[-/-] &:& \TypeCat \times \TypeCat \times \TyVarCat \to \TypeCat\\
(A \eff E)[B/\beta] &\defas& A[B/\beta] \eff E[B/\beta]\\ (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]\\ (A \to C)[B/\beta] &\defas& A[B/\beta] \to C[B/\beta]\\
(\forall \alpha^K.C)[B/\beta] &\defas& \begin{cases} (\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)\\
\forall \alpha^K.C & \text{if } \alpha = \beta\\
\forall \alpha^K.C[B/\beta] & \text{otherwise}
\end{cases}\\
\alpha[B/\beta] &\defas& \begin{cases} \alpha[B/\beta] &\defas& \begin{cases}
B & \text{if } \alpha = \beta\\ B & \text{if } \alpha = \beta\\
\alpha & \text{otherwise} \alpha & \text{otherwise}
\end{cases}\\ \end{cases}\\
\Record{R}[B/\beta] &\defas& \Record{R[B/\beta]}\\ \Record{R}[B/\beta] &\defas& \Record{R[B/\beta]}\\
{[R]}[B/\beta] &\defas& [R[B/\beta]]\\ {[R]}[B/\beta] &\defas& [R[B/\beta]]\\
\{R\}[B/\beta] &\defas& \{R[B/\beta]\}\\ \{R\}[B/\beta] &\defas& \{R[B/\beta]\}\\
\cdot[B/\beta] &\defas& \cdot\\ \cdot[B/\beta] &\defas& \cdot\\
\rho[B/\beta] &\defas& \begin{cases} \rho[B/\beta] &\defas& \begin{cases}
@@ -848,8 +889,9 @@ rule admits a continuation.
% %
The semantics are based on a substitution model of computation. Thus, The semantics are based on a substitution model of computation. Thus,
before presenting the reduction rules, we define an adequacy before presenting the reduction rules, we define an adequate
substitution function. substitution function. As usual we work up to
$\alpha$-conversion~\cite{Church32} of terms in $\BCalc{}$.
% %
\paragraph{Term substitution} \paragraph{Term substitution}
We write $M[V/x]$ for the substitution of some value $V$ for some We write $M[V/x]$ for the substitution of some value $V$ for some
@@ -870,42 +912,62 @@ and we realise it by pattern matching on the first argument.
V & \text{if } x = y\\ V & \text{if } x = y\\
x & \text{otherwise } x & \text{otherwise }
\end{cases}\\ \end{cases}\\
(\lambda x^A.M)[V/y] &\defas& \begin{cases} (\lambda x^A.M)[V/y] &\defas& \lambda x^A.M[V/y] \quad \text{if } x \neq y \text{ and } x \notin \FV(V)\\
\lambda x^A.M & \text{if } x = y\\
\lambda x^A.M[V/y] & \text{otherwise}
\end{cases}\\
(\Lambda \alpha^K. M)[V/y] &\defas& \Lambda \alpha^K. M[V/y]\\ (\Lambda \alpha^K. M)[V/y] &\defas& \Lambda \alpha^K. M[V/y]\\
\Unit[V/y] &\defas& \Unit\\ \Unit[V/y] &\defas& \Unit\\
\Record{\ell = W; W'}[V/y] &\defas& \Record{\ell = W[V/y]; W'[V/y]}\\ \Record{\ell = W; W'}[V/y] &\defas& \Record{\ell = W[V/y]; W'[V/y]}\\
(\ell~W)^R[V/y] &\defas& (\ell~W[V/y])^R\\ (\ell~W)^R[V/y] &\defas& (\ell~W[V/y])^R\\
(W\,W')[V/y] &\defas& W[V/y]\,W'[V/y]\\ (W\,W')[V/y] &\defas& W[V/y]\,W'[V/y]\\
(W\,A)[V/y] &\defas& W[V/y]~A\\ (W\,T)[V/y] &\defas& W[V/y]~T\\
(\Let\;\Record{\ell = x; y} = W\;\In\;N)[V/y] &\defas& \Let\;\Record{\ell = x; y} = W[V/y] \;\In\;N[V/y]\\ (\ba[t]{@{}l}
(\Case\;(\ell~V)^R\{\ba[t]{@{}l} \ell~x \mapsto M\\ \Let\;\Record{\ell = x; y} = W\\
; y \mapsto N \})[V/z]\ea \In\;N)[V/z]
&\defas& \begin{cases} \ea &\defas&
\Case\;(\ell~V)^R\{\ell~x \mapsto M; y \mapsto N \} & \text{if } x = y = z\\ \ba[t]{@{~}l}
\Case\;(\ell~V)^R\{\ba[t]{@{}l} \ell~x \mapsto M\\ \Let\;\Record{\ell = x; y} = W[V/z]\\
; y \mapsto N[V/z] \}\ea & \text{if } x = z \text{ and } y \neq z\\ \In\;N[V/z]
\Case\;(\ell~V)^R\{ \ba[t]{@{}l}\ell~x \mapsto M[V/z]\\ \ea \quad
; y \mapsto N \}\ea & \text{if } x \neq z \text{ and } y = z\\ \ba[t]{@{}l}
\Case\;(\ell~V)^R\{ \ba[t]{@{}l} \ell~x \mapsto M[V/z]\\ \text{if } x \neq z, y \neq z,\\
; y \mapsto N[V/z] \}\ea & \text{otherwise} \text{and } x,y\notin \FV(V)
\end{cases}\\ \ea\\
(\Let\;x \revto M \;\In\;N)[V/y] &\defas& \begin{cases} (\Case\;(\ell~W)^R\{
\Let\;x \revto M[V/y]\;\In\;N & \text{if } x = y\\ \ba[t]{@{}l}
\Let\;x \revto M[V/y]\;\In\;N[V/y] & \text{otherwise} \ell~x \mapsto M\\
\end{cases} ; y \mapsto N \})[V/z]
\end{eqs} \ea
\] &\defas&
\Case\;(\ell~W[V/z])^R\{
\ba[t]{@{}l}
\ell~x \mapsto M[V/z]\\
; y \mapsto N[V/z] \}
\ea\quad
\ba[t]{@{}l}
\text{if } x \neq z, y \neq z,\\
\text{and } x, y \notin \FV(V)
\ea\\
(\Let\;x \revto M \;\In\;N)[V/y] &\defas& \Let\;x \revto M[V/y] \;\In\;N[V/y] \quad\text{if } x \neq y \text{ and } x \notin \FV(V)
\end{eqs}
\]
%
% \begin{cases}
% \Let\;x \revto M[V/y]\;\In\;N & \text{if } x = y\\
% \Let\;x \revto M[V/y]\;\In\;N[V/y] & \text{otherwise}
% \end{cases}
% \end{eqs}
% %
We write $t[t_0/x_0,\dots,t_n/x_n]$ to mean the simultaneous
substitution of $t_0$ for $x_0$ up to $t_n$ for $x_n$ in $t$.
%
\paragraph{Reduction semantics} \paragraph{Reduction semantics}
Figure~\ref{fig:base-language-small-step} depicts the reduction The reduction relation $\reducesto : \CompCat \pto \CompCat$ is defined
rules. The application rules \semlab{App} and \semlab{TyApp} on computation terms. Figure~\ref{fig:base-language-small-step}
eliminates a lambda and type abstraction, respectively, by depicts the reduction rules. The application rules \semlab{App} and
substituting the argument for the parameter in their body computation \semlab{TyApp} eliminates a lambda and type abstraction, respectively,
$M$. by substituting the argument for the parameter in their body
computation $M$.
% %
Record splitting is handled by the \semlab{Split} rule: splitting on Record splitting is handled by the \semlab{Split} rule: splitting on
some label $\ell$ binds the payload $V$ to $x$ and the remainder $W$ some label $\ell$ binds the payload $V$ to $x$ and the remainder $W$
@@ -958,6 +1020,22 @@ Thus far we have defined the syntax, static semantics, and dynamic
semantics of \BCalc{}. In this section, we finish the definition of semantics of \BCalc{}. In this section, we finish the definition of
\BCalc{} by stating and proving some standard metatheoretic properties \BCalc{} by stating and proving some standard metatheoretic properties
about the language. about the language.
%
We begin by showing that type substitutions preserve typability.
%
\begin{lemma}[Preservation of typing under type substitution]
Let $\sigma$ be any type substitution and $V$ be a value and $M$ a
computation such that $\typ{\Delta;\Gamma}{V : A}$ and
$\typ{\Delta;\Gamma}{M : C}$, then
$\typ{\Delta;\sigma~\Gamma}{\sigma~V : \sigma~A}$ and
$\typ{\Delta;\sigma~\Gamma}{\sigma~M : \sigma~C}$.
\end{lemma}
%
\begin{proof}
By induction on the typing derivations.
\end{proof}
%
\section{Primitive effect: general recursion} \section{Primitive effect: general recursion}
\label{sec:base-language-recursion} \label{sec:base-language-recursion}