(*<*)
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
automatically alpha-equated terms such as
\begin{center}
$t ::= x \mid t\;t \mid \lambda x. t$
\end{center}
\noindent
For such terms it derives automatically a reasoning
infrastructure, which has been used 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 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 abstracts over a finite (possibly empty) set of type variables.
While it is possible to formalise such abstractions by iterating single bindings,
it leads to a very clumsy formalisation of W. This need of iterating single binders
for representing multiple binders is also the reason why Nominal Isabelle and other
theorem provers have not fared extremely well with the more advanced tasks
in the POPLmark challenge \cite{challenge05}, because also there one would be able
to aviod clumsy reasoning if there were a mechanisms for abstracting several variables
at once.
To see this, let us point out some interesting properties of binders abstracting multiple
variables. 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
In this paper we will give a general abstraction mechanism and associated
notion of alpha-equivalence that can be used to faithfully represent
type-schemes 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{center}
$\LET x = 3 \AND y = 2 \IN x\,\backslash\,y \END$
\end{center}
\noindent
we might not care in which order the assignments $x = 3$ and $y = 2$ are
given, but it would be unusual to regard the above term as alpha-equivalent
with
\begin{center}
$\LET x = 3 \AND y = 2 \AND z = loop \IN x\,\backslash\,y \END$
\end{center}
\noindent
Therefore we will also provide a separate abstraction mechanism for cases
in which the order of binders does not matter, but the ``cardinality'' of the
binders has to be the same.
However, we found that this is still not sufficient for covering language
constructs frequently occuring in programming language research. For example
in $\mathtt{let}$s involving patterns
\begin{center}
$\LET (x, y) = (3, 2) \IN x\,\backslash\,y \END$
\end{center}
\noindent
we want to bind all variables from the pattern (there might be an arbitrary
number of them) inside the body of the $\mathtt{let}$, but we also care about
the order of these variables, since we do not want to identify the above term
with
\begin{center}
$\LET (y, x) = (3, 2) \IN x\,\backslash y\,\END$
\end{center}
\noindent
As a result, we provide three general abstraction mechanisms for multiple binders
and allow the user to chose which one is intended when formalising a
programming language calculus.
By providing these general abstraction 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 as something like
\begin{center}
$\LET [x_1,\ldots,x_n].s\;\; [t_1,\ldots,t_n]$
\end{center}
\noindent
where the $[\_\!\_].\_\!\_$ indicates that a list of variables becomes bound
in $s$. In this representation we need additional predicates to ensure
that the two lists are of equal length. This can result into very
unintelligible reasoning (see for example~\cite{BengtsonParow09}).
Contributions: We provide definitions for when terms
involving general bindings are alpha-equivelent.
%\begin{center}
%\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}
quotient package \cite{Homeier05}
*}
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 of our work, we
shall assume in this paper 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 the
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 of the OTT-tool.
*}
(*<*)
end
(*>*)