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@ -11,7 +11,7 @@
\usepackage[capitalize,nameinlink,noabbrev]{cleveref}
\usepackage{mathpartir,mathtools,stmaryrd,latexsym}
\setlist{nosep}
\setlist{noitemsep}
\hypersetup{
colorlinks=true,
urlcolor=blue,
@ -295,7 +295,7 @@ directly as a relation between a type and a term;
this presentation is given in \cref{fig:lr-alt}.
Unfortunately, this is not well defined,
since the logical relation appears in a negative position
(to the left of an implication) in the premise of \rref{LRPP-Arr}.
(to the left of an implication) in the premise of \rref{LR'-Arr}.
The alternate presentation can be interpreted as a function
match on the term and the type and returning the conjunction of its premises,
but I wanted to use a mutually defined inductive definition.
@ -324,7 +324,7 @@ determinism, which states that the interpretation indeed behaves like a function
from types to sets of terms;
and backward closure, which states that the interpretations of computation types
are backward closed under multi-step head reduction.
The last property is why \rref{LRPP-F-var} needs to include
The last property is why \rref{LR'-F-var} needs to include
computations that \emph{reduce to} strongly neutral terms or returns,
not merely ones that are such terms.
@ -691,7 +691,7 @@ traditional strong normalization of well typed terms holds as a corollary.
By induction on the derivation of $[[m ∈ SNe]]$.
\end{proof}
\begin{theorem}[Soundness (\textsf{SN})] \leavevmode
\begin{theorem}[Soundness (\textsf{SN})] \label{thm:soundness-sn} \leavevmode
\begin{enumerate}
\item If $[[m ∈ SNe]]$ then $[[m ∈ sn]]$.
\item If $[[v ∈ SN]]$ then $[[v ∈ sn]]$.
@ -716,4 +716,85 @@ traditional strong normalization of well typed terms holds as a corollary.
\section{Discussion} \label{sec:discussion}
Overall, the inductive characterization of strongly normal terms is convenient to work with.
Its congruence and inversion properties are exactly what are needed
to show \nameref{lem:closure} and \nameref{lem:adequacy} of the logical relation,
which are already built in by its inductive nature.
Proving soundness itself relies mostly on the properties of the logical relation,
along with a few more congruence properties of strong head reduction.
Knowing the general recipe for constructing the inductive definition
and for the structure of a proof by logical relations
makes it easy to adapt the proof of strong normalization to simply typed CBPV.
There are only a few pitfalls related to simply typed CBPV specifically:
\begin{itemize}[rightmargin=\leftmargin]
\item I originally included a corresponding notion of reduction for values
in the mutual inductive which reduced under subterms.
This is unnecessary not only because values have no $\beta$-reductions,
but also because subterms of strongly normal values need to be strongly normal anyway,
so there is no need to reduce subterms.
Having reduction for values therefore probably characterizes the same set of strongly normal terms,
but made proofs unnecessarily difficult to complete.
\item Similarly, I originally included premises asserting other subterms of head reduction
must be strongly normal even if they appear in the reduced term on the right-hand side.
This changes what terms may reduce,
but not what is included in the set of strongly normal terms,
since \rref{SN-Red} requires that the right-hand term is strongly normal anyway.
The soundness proof still went through,
but this design is hard to justify by first principles,
and adds proof burden to the lemmas for the traditional presentation
of strongly normal terms later on.
\item The lack of a notion of reduction for values creates an asymmetry
that initially led me to omit the fact that the computations
in the interpretation of $[[F A]]$ in \rref{LR-F} must \emph{reduce}
to strongly neutral terms instead of merely \emph{being} strongly neutral.
This makes \nameref{lem:closure} impossible to prove,
since strongly neutral terms are not backward closed.
\end{itemize}
The inductive characterization is easy to work with
because all the hard work seems to lie in showing it sound
with respect to the traditional presentation of strongly normal terms.
This proof took about as much time as the entire rest of the development.
Not only does it require setting up an entire single-step reduction relation
and proving all of its tedious congruence, renaming, substitution, and commutativity lemmas,
the intermediate lemmas for \nameref{thm:soundness-sn} are long and tricky.
In particular, the congruence, head expansion, and backward closure lemmas
require double or even triple induction.
\Rref{sn-App-bwd,sn-Let-bwd} are especially tricky,
since their double inductions are on two auxiliary derivations
produced from one of the actual premises,
and that premise unintuitively needs to stick around over the course of the inductions.
Without guidance from the POPLMark Reloaded paper,
I would have been stuck on these two particular lemmas for much longer.
The purpose of all the work done in \cref{sec:sn-acc}
is to show that $\kw{sn}$ satisfies properties that
otherwise hold by definition for $\kw{SN}$ as a constructor.
If proving that the latter is sound with respect to the former
already requires all this work showing that the former behaves exactly like the latter,
then what is the point of using the inductive characterization?
All constructions of $\kw{SN}$ could be directly replaced
by the congruence and backward closure lemmas we have proven for $\kw{sn}$.
Using the inductive characterization is beneficial only if we don't care
about its soundness with respect to the traditional presentation,
where strong normalization is not the end goal.
For example, in the metatheory of dependent type theory,
we care about normalization because we want to know
that the type theory is consistent,
and that definitional equality is decidable
so that a type checker can actually be implemented.
For the latter purpose,
all we require is \emph{weak normalization}:
that there exists \emph{some} reduction strategy that reduces terms to normal form.
The shape of the inductive characterization makes it easy to show
that a leftmost-outermost reduction strategy does so (\texttt{LeftmostOutermost.lean}).
While weak normalization can be proven directly using the logical relation,
generalizing to the inductive strong normalization is actually easier,
in addition to being more modular and not tying the logical relation
to the particularities of the chosen reduction strategy.
\subsection{Lean for PL Metatheory}
\end{document}

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cbpv.tex
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@ -11,7 +11,7 @@
\usepackage[capitalize,nameinlink,noabbrev]{cleveref}
\usepackage{mathpartir,mathtools,stmaryrd,latexsym}
\setlist{nosep}
\setlist{noitemsep}
\hypersetup{
colorlinks=true,
urlcolor=blue,
@ -295,7 +295,7 @@ directly as a relation between a type and a term;
this presentation is given in \cref{fig:lr-alt}.
Unfortunately, this is not well defined,
since the logical relation appears in a negative position
(to the left of an implication) in the premise of \rref{LRPP-Arr}.
(to the left of an implication) in the premise of \rref{LR'-Arr}.
The alternate presentation can be interpreted as a function
match on the term and the type and returning the conjunction of its premises,
but I wanted to use a mutually defined inductive definition.
@ -324,7 +324,7 @@ determinism, which states that the interpretation indeed behaves like a function
from types to sets of terms;
and backward closure, which states that the interpretations of computation types
are backward closed under multi-step head reduction.
The last property is why \rref{LRPP-F-var} needs to include
The last property is why \rref{LR'-F-var} needs to include
computations that \emph{reduce to} strongly neutral terms or returns,
not merely ones that are such terms.
@ -691,7 +691,7 @@ traditional strong normalization of well typed terms holds as a corollary.
By induction on the derivation of $ \ottnt{m} \in \kw{SNe} $.
\end{proof}
\begin{theorem}[Soundness (\textsf{SN})] \leavevmode
\begin{theorem}[Soundness (\textsf{SN})] \label{thm:soundness-sn} \leavevmode
\begin{enumerate}
\item If $ \ottnt{m} \in \kw{SNe} $ then $ \ottnt{m} \in \kw{sn} $.
\item If $ \ottnt{v} \in \kw{SN} $ then $ \ottnt{v} \in \kw{sn} $.
@ -716,4 +716,85 @@ traditional strong normalization of well typed terms holds as a corollary.
\section{Discussion} \label{sec:discussion}
Overall, the inductive characterization of strongly normal terms is convenient to work with.
Its congruence and inversion properties are exactly what are needed
to show \nameref{lem:closure} and \nameref{lem:adequacy} of the logical relation,
which are already built in by its inductive nature.
Proving soundness itself relies mostly on the properties of the logical relation,
along with a few more congruence properties of strong head reduction.
Knowing the general recipe for constructing the inductive definition
and for the structure of a proof by logical relations
makes it easy to adapt the proof of strong normalization to simply typed CBPV.
There are only a few pitfalls related to simply typed CBPV specifically:
\begin{itemize}[rightmargin=\leftmargin]
\item I originally included a corresponding notion of reduction for values
in the mutual inductive which reduced under subterms.
This is unnecessary not only because values have no $\beta$-reductions,
but also because subterms of strongly normal values need to be strongly normal anyway,
so there is no need to reduce subterms.
Having reduction for values therefore probably characterizes the same set of strongly normal terms,
but made proofs unnecessarily difficult to complete.
\item Similarly, I originally included premises asserting other subterms of head reduction
must be strongly normal even if they appear in the reduced term on the right-hand side.
This changes what terms may reduce,
but not what is included in the set of strongly normal terms,
since \rref{SN-Red} requires that the right-hand term is strongly normal anyway.
The soundness proof still went through,
but this design is hard to justify by first principles,
and adds proof burden to the lemmas for the traditional presentation
of strongly normal terms later on.
\item The lack of a notion of reduction for values creates an asymmetry
that initially led me to omit the fact that the computations
in the interpretation of $ \kw{F} \gap \ottnt{A} $ in \rref{LR-F} must \emph{reduce}
to strongly neutral terms instead of merely \emph{being} strongly neutral.
This makes \nameref{lem:closure} impossible to prove,
since strongly neutral terms are not backward closed.
\end{itemize}
The inductive characterization is easy to work with
because all the hard work seems to lie in showing it sound
with respect to the traditional presentation of strongly normal terms.
This proof took about as much time as the entire rest of the development.
Not only does it require setting up an entire single-step reduction relation
and proving all of its tedious congruence, renaming, substitution, and commutativity lemmas,
the intermediate lemmas for \nameref{thm:soundness-sn} are long and tricky.
In particular, the congruence, head expansion, and backward closure lemmas
require double or even triple induction.
\Rref{sn-App-bwd,sn-Let-bwd} are especially tricky,
since their double inductions are on two auxiliary derivations
produced from one of the actual premises,
and that premise unintuitively needs to stick around over the course of the inductions.
Without guidance from the POPLMark Reloaded paper,
I would have been stuck on these two particular lemmas for much longer.
The purpose of all the work done in \cref{sec:sn-acc}
is to show that $\kw{sn}$ satisfies properties that
otherwise hold by definition for $\kw{SN}$ as a constructor.
If proving that the latter is sound with respect to the former
already requires all this work showing that the former behaves exactly like the latter,
then what is the point of using the inductive characterization?
All constructions of $\kw{SN}$ could be directly replaced
by the congruence and backward closure lemmas we have proven for $\kw{sn}$.
Using the inductive characterization is beneficial only if we don't care
about its soundness with respect to the traditional presentation,
where strong normalization is not the end goal.
For example, in the metatheory of dependent type theory,
we care about normalization because we want to know
that the type theory is consistent,
and that definitional equality is decidable
so that a type checker can actually be implemented.
For the latter purpose,
all we require is \emph{weak normalization}:
that there exists \emph{some} reduction strategy that reduces terms to normal form.
The shape of the inductive characterization makes it easy to show
that a leftmost-outermost reduction strategy does so (\texttt{LeftmostOutermost.lean}).
While weak normalization can be proven directly using the logical relation,
generalizing to the inductive strong normalization is actually easier,
in addition to being more modular and not tying the logical relation
to the particularities of the chosen reduction strategy.
\subsection{Lean for PL Metatheory}
\end{document}