(*<*)
theory Paper
imports "../Nominal/Test" "LaTeXsugar"
begin
notation (latex output)
swap ("'(_ _')" [1000, 1000] 1000) and
fresh ("_ # _" [51, 51] 50) and
fresh_star ("_ #* _" [51, 51] 50) and
supp ("supp _" [78] 73) and
uminus ("-_" [78] 73) and
If ("if _ then _ else _" 10)
(*>*)
section {* Introduction *}
text {*
So far, Nominal Isabelle provides a mechanism for constructing
alpha-equated terms such as
\begin{center}
$t ::= x \mid t\;t \mid \lambda x. t$
\end{center}
\noindent
where free and bound variables have names. For such terms Nominal Isabelle
derives automatically a reasoning infrastructure, which has been used
successfully in formalisations of an equivalence checking algorithm for LF
\cite{UrbanCheneyBerghofer08}, Typed
Scheme~\cite{TobinHochstadtFelleisen08}, several calculi for concurrency
\cite{BengtsonParrow07,BengtsonParow09} and a strong normalisation result
for cut-elimination in classical logic \cite{UrbanZhu08}. It has also been
used by Pollack for formalisations in the locally-nameless approach to
binding \cite{SatoPollack10}.
However, Nominal Isabelle has fared less well in a formalisation of
the algorithm W \cite{UrbanNipkow09}, where types and type-schemes
are of the form
\begin{center}
\begin{tabular}{l}
$T ::= x \mid T \rightarrow T$ \hspace{5mm} $S ::= \forall \{x_1,\ldots, x_n\}. T$
\end{tabular}
\end{center}
\noindent
and the quantification binds at once a finite (possibly empty) set of
type-variables. While it is possible to implement this kind of more general
binders by iterating single binders, this leads to a rather clumsy
formalisation of W. The need of iterating single binders is also one reason
why Nominal Isabelle and similar theorem provers that only provide
mechanisms for binding single variables have not fared extremely well with the
more advanced tasks in the POPLmark challenge \cite{challenge05}, because
also there one would like to bind multiple variables at once.
Binding multiple variables has interesting properties that are not captured
by iterating single binders. First, in the case of type-schemes, we do not
like to make a distinction about the order of the bound variables. Therefore
we would like to regard the following two type-schemes as alpha-equivalent
\begin{center}
$\forall \{x, y\}. x \rightarrow y \;\approx_\alpha\; \forall \{y, x\}. y \rightarrow x$
\end{center}
\noindent
but the following two should \emph{not} be alpha-equivalent
\begin{center}
$\forall \{x, y\}. x \rightarrow y \;\not\approx_\alpha\; \forall \{z\}. z \rightarrow z$
\end{center}
\noindent
assuming that $x$, $y$ and $z$ are distinct. Moreover, we like to regard type-schemes as
alpha-equivalent, if they differ only on \emph{vacuous} binders, such as
\begin{center}
$\forall \{x\}. x \rightarrow y \;\approx_\alpha\; \forall \{x, z\}. x \rightarrow y$
\end{center}
\noindent
where $z$ does not occur freely in the type.
In this paper we will give a general binding mechanism and associated
notion of alpha-equivalence that can be used to faithfully represent
this kind of binding in Nominal Isabelle. The difficulty of finding the right notion
for alpha-equivalence in this case can be appreciated by considering that the
definition given by Leroy in \cite{Leroy92} is incorrect (it omits a side-condition).
However, the notion of alpha-equivalence that is preserved by vacuous binders is not
alway wanted. For example in terms like
\begin{equation}\label{one}
\LET x = 3 \AND y = 2 \IN x\,-\,y \END
\end{equation}
\noindent
we might not care in which order the assignments $x = 3$ and $y = 2$ are
given, but it would be unusual to regard \eqref{one} as alpha-equivalent
with
\begin{center}
$\LET x = 3 \AND y = 2 \AND z = loop \IN x\,-\,y \END$
\end{center}
\noindent
Therefore we will also provide a separate binding mechanism for cases in
which the order of binders does not matter, but the ``cardinality'' of the
binders has to agree.
However, we found that this is still not sufficient for dealing with
language constructs frequently occurring in programming language
research. For example in $\mathtt{let}$s containing patterns
\begin{equation}\label{two}
\LET (x, y) = (3, 2) \IN x\,-\,y \END
\end{equation}
\noindent
we want to bind all variables from the pattern inside the body of the
$\mathtt{let}$, but we also care about the order of these variables, since
we do not want to identify \eqref{two} with
\begin{center}
$\LET (y, x) = (3, 2) \IN x\,- y\,\END$
\end{center}
\noindent
As a result, we provide three general binding mechanisms each of which binds multiple
variables at once, and we let the user chose which one is intended when formalising a
programming language calculus.
By providing these general binding mechanisms, however, we have to work around
a problem that has been pointed out by Pottier in \cite{Pottier06}: in
$\mathtt{let}$-constructs of the form
\begin{center}
$\LET x_1 = t_1 \AND \ldots \AND x_n = t_n \IN s \END$
\end{center}
\noindent
which bind all the $x_i$ in $s$, we might not care about the order in
which the $x_i = t_i$ are given, but we do care about the information that there are
as many $x_i$ as there are $t_i$. We lose this information if we represent the
$\mathtt{let}$-constructor by something like
\begin{center}
$\LET [x_1,\ldots,x_n].s\;\; [t_1,\ldots,t_n]$
\end{center}
\noindent
where the notation $[\_\!\_].\_\!\_$ indicates that the $x_i$ become
bound in $s$. In this representation the term \mbox{$\LET [x].s\;\;[t_1,t_2]$}
would be perfectly legal instance, and so one needs additional predicates about terms
to ensure that the two lists are of equal length. This can result into very
messy reasoning (see for example~\cite{BengtsonParow09}).
To avoid this, we will allow for example to specify $\mathtt{let}$s
as follows
\begin{center}
\begin{tabular}{r@ {\hspace{2mm}}r@ {\hspace{2mm}}l}
$trm$ & $::=$ & \ldots\\
& $\mid$ & $\mathtt{let}\;a\!::\!assn\;\;s\!::\!trm\quad\mathtt{bind}\;bn\,(a) \IN s$\\[1mm]
$assn$ & $::=$ & $\mathtt{anil}$\\
& $\mid$ & $\mathtt{acons}\;\;name\;\;trm\;\;assn$
\end{tabular}
\end{center}
\noindent
where $assn$ is an auxiliary type representing a list of assignments
and $bn$ an auxiliary function identifying the variables to be bound by
the $\mathtt{let}$. This function can be defined as
\begin{center}
$bn\,(\mathtt{anil}) = \varnothing \qquad bn\,(\mathtt{acons}\;x\;t\;as) = \{x\} \cup bn\,(as)$
\end{center}
\noindent
The scope of the binding is indicated by labels given to the types, for
example \mbox{$s\!::\!trm$}, and a binding clause, in this case
$\mathtt{bind}\;bn\,(a) \IN s$, that states bind all the names the function
$bn$ returns in $s$. This style of specifying terms and bindings is heavily
inspired by the syntax of the Ott-tool \cite{ott-jfp}.
However, we will not be able to deal with all specifications that are
allowed by Ott. One reason is that we establish the reasoning infrastructure
for alpha-\emph{equated} terms. In contrast, Ott produces for a subset of
its specifications a reasoning infrastructure in Isabelle/HOL for
\emph{non}-alpha-equated, or ``raw'', terms. While our alpha-equated terms
and the raw terms produced by Ott use names for the bound variables,
there is a key difference: working with alpha-equated terms means that the
two type-schemes with $x$, $y$ and $z$ being distinct
\begin{center}
$\forall \{x\}. x \rightarrow y \;=\; \forall \{x, z\}. x \rightarrow y$
\end{center}
\noindent
are not just alpha-equal, but actually equal. As a
result, we can only support specifications that make sense on the level of
alpha-equated terms (offending specifications, which for example bind a variable
according to a variable bound somewhere else, are not excluded by Ott). Our
insistence on reasoning with alpha-equated terms comes from the wealth of
experience we gained with the older version of Nominal Isabelle: for
non-trivial properties, reasoning about alpha-equated terms is much easier
than reasoning with raw terms. The fundamental reason is that the
HOL-logic underlying Nominal Isabelle allows us to replace
``equals-by-equals''. In contrast replacing ``alpha-equals-by-alpha-equals''
in a representation based on raw terms requires a lot of extra reasoning work.
Although in informal settings a reasoning infrastructure for alpha-equated
terms (that have names for bound variables) is nearly always taken for granted, establishing
it automatically in the Isabelle/HOL theorem prover is a rather non-trivial task.
For every specification we will need to construct a type containing as
elements the alpha-equated terms. To do so we use
the standard HOL-technique of defining a new type by
identifying a non-empty subset of an existing type. In our
case we take as the starting point the type of sets of raw
terms (the latter being defined as a datatype); identify the
alpha-equivalence classes according to our alpha-equivalence relation and
then define the new type as these alpha-equivalence classes. The construction we
can perform in HOL is illustrated by the following picture:
\begin{center}
figure
%\begin{pspicture}(0.5,0.0)(8,2.5)
%%\showgrid
%\psframe[linewidth=0.4mm,framearc=0.2](5,0.0)(7.7,2.5)
%\pscircle[linewidth=0.3mm,dimen=middle](6,1.5){0.6}
%\psframe[linewidth=0.4mm,framearc=0.2,dimen=middle](1.1,2.1)(2.3,0.9)
%\pcline[linewidth=0.4mm]{->}(2.6,1.5)(4.8,1.5)
%\pcline[linewidth=0.2mm](2.2,2.1)(6,2.1)
%\pcline[linewidth=0.2mm](2.2,0.9)(6,0.9)
%\rput(7.3,2.2){$\mathtt{phi}$}
%\rput(6,1.5){$\lama$}
%\rput[l](7.6,2.05){\begin{tabular}{l}existing\\[-1.6mm]type\end{tabular}}
%\rput[r](1.2,1.5){\begin{tabular}{l}new\\[-1.6mm]type\end{tabular}}
%\rput(6.1,0.5){\begin{tabular}{l}non-empty\\[-1.6mm]subset\end{tabular}}
%\rput[c](1.7,1.5){$\lama$}
%\rput(3.7,1.75){isomorphism}
%\end{pspicture}
\end{center}
\noindent
To ``lift'' the reasoning from the underlying type to the new type
is usually a tricky task. To ease this task we reimplemented in Isabelle/HOL
the quotient package described by Homeier \cite{Homeier05}. This
re-implementation will automate the proofs we require for our
reasoning infrastructure over alpha-equated terms.\medskip
\noindent
{\bf Contributions:} We provide new definitions for when terms
involving multiple binders are alpha-equivalent. These definitions are
inspired by earlier work of Pitts \cite{}. By means of automatic
proofs, we establish a reasoning infrastructure for alpha-equated
terms, including properties about support, freshness and equality
conditions for alpha-equated terms. We will also derive for these
terms a strong induction principle that has the variable convention
already built in.
*}
section {* A Short Review of the Nominal Logic Work *}
text {*
At its core, Nominal Isabelle is based on the nominal logic work by Pitts
\cite{Pitts03}. The implementation of this work are described in
\cite{HuffmanUrban10}, which we review here briefly to aid the description
of what follows in the next sections. Two central notions in the nominal
logic work are sorted atoms and permutations of atoms. The sorted atoms
represent different kinds of variables, such as term- and type-variables in
Core-Haskell, and it is assumed that there is an infinite supply of atoms
for each sort. However, in order to simplify the description, we
shall assume in what follows that there is only a single sort of atoms.
Permutations are bijective functions from atoms to atoms that are
the identity everywhere except on a finite number of atoms. There is a
two-place permutation operation written
@{text[display,indent=5] "_ \<bullet> _ :: (\<alpha> \<times> \<alpha>) list \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
\noindent
with a generic type in which @{text "\<alpha>"} stands for the type of atoms
and @{text "\<beta>"} for the type of the objects on which the permutation
acts. In Nominal Isabelle the identity permutation is written as @{term "0::perm"},
the composition of two permutations @{term p} and @{term q} as \mbox{@{term "p + q"}}
and the inverse permutation @{term p} as @{text "- p"}. The permutation
operation is defined for products, lists, sets, functions, booleans etc
(see \cite{HuffmanUrban10}).
The most original aspect of the nominal logic work of Pitts et al is a general
definition for ``the set of free variables of an object @{text "x"}''. This
definition is general in the sense that it applies not only to lambda-terms,
but also to lists, products, sets and even functions. The definition depends
only on the permutation operation and on the notion of equality defined for
the type of @{text x}, namely:
@{thm[display,indent=5] supp_def[no_vars, THEN eq_reflection]}
\noindent
There is also the derived notion for when an atom @{text a} is \emph{fresh}
for an @{text x}, defined as
@{thm[display,indent=5] fresh_def[no_vars]}
\noindent
We also use for sets of atoms the abbreviation
@{thm (lhs) fresh_star_def[no_vars]} defined as
@{thm (rhs) fresh_star_def[no_vars]}.
A striking consequence of these definitions is that we can prove
without knowing anything about the structure of @{term x} that
swapping two fresh atoms, say @{text a} and @{text b}, leave
@{text x} unchanged.
\begin{property}
@{thm[mode=IfThen] swap_fresh_fresh[no_vars]}
\end{property}
\noindent
For a proof see \cite{HuffmanUrban10}.
\begin{property}
@{thm[mode=IfThen] at_set_avoiding[no_vars]}
\end{property}
*}
section {* Abstractions *}
text {*
General notion of alpha-equivalence (depends on a free-variable
function and a relation).
*}
section {* Alpha-Equivalence and Free Variables *}
text {*
Restrictions
\begin{itemize}
\item non-emptyness
\item positive datatype definitions
\item finitely supported abstractions
\item respectfulness of the bn-functions\bigskip
\item binders can only have a ``single scope''
\end{itemize}
*}
section {* Examples *}
section {* Adequacy *}
section {* Related Work *}
section {* Conclusion *}
text {*
Complication when the single scopedness restriction is lifted (two
overlapping permutations)
*}
text {*
TODO: function definitions:
\medskip
\noindent
{\bf Acknowledgements:} We are very grateful to Andrew Pitts for
many discussions about Nominal Isabelle. We thank Peter Sewell for
making the informal notes \cite{SewellBestiary} available to us and
also for explaining some of the finer points about the abstract
definitions and about the implementation of the Ott-tool.
*}
(*<*)
end
(*>*)