(*<*)
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{equation}\label{tysch}
\begin{array}{l}
T ::= x \mid T \rightarrow T \hspace{5mm} S ::= \forall \{x_1,\ldots, x_n\}. T
\end{array}
\end{equation}
\noindent
and the quantification $\forall$ binds 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 cannot be captured
easily by iterating single binders. For example 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{equation}\label{ex1}
\forall \{x, y\}. x \rightarrow y \;\approx_\alpha\; \forall \{y, x\}. y \rightarrow x
\end{equation}
\noindent
but the following two should \emph{not} be alpha-equivalent
%
\begin{equation}\label{ex2}
\forall \{x, y\}. x \rightarrow y \;\not\approx_\alpha\; \forall \{z\}. z \rightarrow z
\end{equation}
\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{equation}\label{ex3}
\forall \{x\}. x \rightarrow y \;\approx_\alpha\; \forall \{x, z\}. x \rightarrow y
\end{equation}
\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
always 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 regard \eqref{two} as alpha-equivalent 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 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 a perfectly legal instance. To exclude such terms an additional
predicate about well-formed terms is needed in order 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 type specifications
for $\mathtt{let}$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 is defined by recursion over $assn$ as follows
\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 to bind in $s$ all the names the
function $bn\,(a)$ returns. 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 Ott allows ``empty'' specifications
like
\begin{center}
$t ::= t\;t \mid \lambda x. t$
\end{center}
\noindent
where no clause for variables is given. Such specifications make some sense in
the context of Coq's type theory (which Ott supports), but not at al in a HOL-based
theorem prover where every datatype must have a non-empty set-theoretic model.
Another reason is that we establish the reasoning infrastructure
for alpha-\emph{equated} terms. In contrast, Ott produces 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 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, but we
have to). 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 for this 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. The construction we
perform in HOL is illustrated by the following picture:
\begin{center}
\begin{tikzpicture}
%\draw[step=2mm] (-4,-1) grid (4,1);
\draw[very thick] (0.7,0.4) circle (4.25mm);
\draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9);
\draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05);
\draw (-2.0, 0.845) -- (0.7,0.845);
\draw (-2.0,-0.045) -- (0.7,-0.045);
\draw ( 0.7, 0.4) node {\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]clas.\end{tabular}};
\draw (-2.4, 0.4) node {\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};
\draw (1.8, 0.48) node[right=-0.1mm]
{\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ (sets of raw terms)\end{tabular}};
\draw (0.9, -0.35) node {\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};
\draw (-3.25, 0.55) node {\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}};
\draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3);
\draw (-0.95, 0.3) node[above=0mm] {isomorphism};
\end{tikzpicture}
\end{center}
\noindent
We take as the starting point a definition of raw terms (defined as a
datatype in Isabelle/HOL); identify then the
alpha-equivalence classes in the type of sets of raw terms, according to our
alpha-equivalence relation and finally define the new type as these
alpha-equivalence classes (non-emptiness is satisfied whenever the raw terms are
definable as datatype in Isabelle/HOL and the fact that our relation for alpha is an
equivalence relation).
The fact that we obtain an isomorphism between between the new type and the non-empty
subset shows that the new type is a faithful representation of alpha-equated terms.
That is not the case for example in the representation of terms using the locally
nameless representation of binders \cite{McKinnaPollack99}: there are ``junk'' terms that need to
be excluded by reasoning about a well-formedness predicate.
The problem with introducing a new type in Isabelle/HOL is that in order to be useful,
a reasoning infrastructure needs to be ``lifted'' from the underlying subset to
the new type. This is usually a tricky and arduous task. To ease it
we re-implemented in Isabelle/HOL the quotient package described by Homeier
\cite{Homeier05}. This package
allows us to lift definitions and theorems involving raw terms
to definitions and theorems involving alpha-equated terms. For example
if we define the free-variable function over lambda terms
\begin{center}
$\fv(x) = \{x\}$\hspace{10mm}
$\fv(t_1\;t_2) = \fv(t_1) \cup \fv(t_2)$\\[1mm]
$\fv(\lambda x.t) = \fv(t) - \{x\}$
\end{center}
\noindent
then with not too great effort we obtain a function $\fv_\alpha$
operating on quotients, or alpha-equivalence classes of terms, as follows
\begin{center}
$\fv_\alpha(x) = \{x\}$\hspace{10mm}
$\fv_\alpha(t_1\;t_2) = \fv_\alpha(t_1) \cup \fv_\alpha(t_2)$\\[1mm]
$\fv_\alpha(\lambda x.t) = \fv_\alpha(t) - \{x\}$
\end{center}
\noindent
(Note that this means also the term-constructors for variables, applications
and lambda are lifted to the quotient level.) This construction, of course,
only works if alpha is an equivalence relation, and the definitions and theorems
are respectful w.r.t.~alpha-equivalence. Hence we will not be able to lift this
a bound-variable function to alpha-equated terms (since it does not respect
alpha-equivalence). To sum up, every lifting needs proofs of some respectfulness
properties. These proofs we are able automate and therefore establish a
useful reasoning infrastructure for alpha-equated lambda 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 re also able to derive, at the moment
only manually, 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 an adaption of the nominal logic work by
Pitts \cite{Pitts03}. This adaptation for Isabelle/HOL is described in
\cite{HuffmanUrban10}, which we review here briefly to aid the description
of what follows. Two central notions in the nominal logic work are sorted
atoms and sort-respecting permutations of atoms. The sorts can be used to
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 object 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 of @{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 is a general
definition for the notion of ``the set of free variables of an object @{text
"x"}''. This notion, written @{term "supp x"}, is general in the sense that
it applies not only to lambda-terms alpha-equated or not, 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 {* General Binders *}
text {*
In Nominal Isabelle the user is expected to write down a specification of a
term-calculus and a reasoning infrastructure is then automatically derived
from this specifcation (remember that Nominal Isabelle is a definitional
extension of Isabelle/HOL and cannot introduce new axioms).
In order to keep our work manageable, we will wherever possible state
definitions and perform proofs inside Isabelle, as opposed to write custom
ML-code that generates them for each instance of a term-calculus. To that
end, we will consider pairs @{text "(as, x)"} of type @{text "(atom set) \<times> \<beta>"}.
These pairs are intended to represent the abstraction, or binding, of the set $as$
in the body $x$.
The first question we have to answer is when the pairs $(as, x)$ and $(bs, y)$ are
alpha-equivalent? (At the moment we are interested in
the notion of alpha-equivalence that is \emph{not} preserved by adding
vacuous binders.) To answer this, we identify four conditions: {\it i)} given
a free-variable function $\fv$ of type \mbox{@{text "\<beta> \<Rightarrow> atom set"}}, then $x$ and $y$
need to have the same set of free variables; moreover there must be a permutation,
$p$ so that {\it ii)} it leaves the free variables $x$ and $y$ unchanged,
but {\it iii)} ``moves'' their bound names such that we obtain modulo a relation,
say \mbox{@{text "_ R _"}}, two equal terms. We also require {\it iv)} that $p$ makes
the abstracted sets $as$ and $bs$ equal. The requirements {\it i)} to {\it iv)} can
be stated formally as follows:
%
\begin{equation}\label{alphaset}
\begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
\multicolumn{2}{l}{(as, x) \approx\hspace{0.05mm}_{set}^{\fv, R, p} (bs, y) \;\dn\hspace{30mm}\;}\\[1mm]
& @{text "fv(x) - as = fv(y) - bs"}\\
\wedge & @{text "fv(x) - as #* p"}\\
\wedge & @{text "(p \<bullet> x) R y"}\\
\wedge & @{text "(p \<bullet> as) = bs"}\\
\end{array}
\end{equation}
\noindent
Note that this relation is dependent on $p$. Alpha-equivalence is then the relation where
we existentially quantify over this $p$.
Also note that the relation is dependent on a free-variable function $\fv$ and a relation
$R$. The reason for this extra generality is that we will use $\approx_{set}$ for both
``raw'' terms and alpha-equated terms. In the latter case, $R$ will be replaced by
equality $(op =)$ and we are going to prove that $\fv$ will be equal to the support
of $x$ and $y$. To have these parameters means, however, we can derive properties about
them generically.
The definition in \eqref{alphaset} does not make any distinction between the
order of abstracted variables. If we want this, then we can define alpha-equivalence
for pairs of the form \mbox{@{text "(as, x)"}} with type @{text "(atom list) \<times> \<beta>"}
as follows
%
\begin{equation}\label{alphalist}
\begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
\multicolumn{2}{l}{(as, x) \approx\hspace{0.05mm}_{list}^{\fv, R, p} (bs, y) \;\dn\hspace{30mm}\;}\\[1mm]
& @{text "fv(x) - (set as) = fv(y) - (set bs)"}\\
\wedge & @{text "fv(x) - (set as) #* p"}\\
\wedge & @{text "(p \<bullet> x) R y"}\\
\wedge & @{text "(p \<bullet> as) = bs"}\\
\end{array}
\end{equation}
\noindent
where $set$ is just the function that coerces a list of atoms into a set of atoms.
If we do not want to make any difference between the order of binders and
also allow vacuous binders, then we keep sets of binders, but drop the fourth
condition in \eqref{alphaset}:
%
\begin{equation}\label{alphares}
\begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
\multicolumn{2}{l}{(as, x) \approx\hspace{0.05mm}_{res}^{\fv, R, p} (bs, y) \;\dn\hspace{30mm}\;}\\[1mm]
& @{text "fv(x) - as = fv(y) - bs"}\\
\wedge & @{text "fv(x) - as #* p"}\\
\wedge & @{text "(p \<bullet> x) R y"}\\
\end{array}
\end{equation}
\begin{exmple}\rm
It might be useful to consider some examples for how these definitions pan out in practise.
For this consider the case of abstracting a set of variables over types (as in type-schemes).
We set $R$ to be the equality and for $\fv(T)$ we define
\begin{center}
$\fv(x) = \{x\} \qquad \fv(T_1 \rightarrow T_2) = \fv(T_1) \cup \fv(T_2)$
\end{center}
\noindent
Now recall the examples shown in \eqref{ex1}, \eqref{ex2} and \eqref{ex3}. It can be easily
checked that @{text "({x, y}, x \<rightarrow> y)"} and
@{text "({y, x}, y \<rightarrow> x)"} are equal according to $\approx_{set}$ and $\approx_{res}$ by taking $p$ to
be the swapping @{term "(x \<rightleftharpoons> y)"}. In case of @{text "x \<noteq> y"} then
$([x, y], x \rightarrow y) \not\approx_{list} ([y,x], x \rightarrow y)$ since there is no permutation that
makes the lists @{text "[x, y]"} and @{text "[y, x]"} equal, and in addition leaves the
type \mbox{@{text "x \<rightarrow> y"}} unchanged. Again if @{text "x \<noteq> y"}, we have that
$(\{x\}, x) \approx_{res} (\{x,y\}, x)$ by taking $p$ to be the identity permutation.
However $(\{x\}, x) \not\approx_{set} (\{x,y\}, x)$ since there is no permutation that makes
the sets $\{x\}$ and $\{x,y\}$ equal (similarly for $\approx_{list}$).
\end{exmple}
\noindent
Let $\star$ range over $\{set, res, list\}$. We prove next under which
conditions the $\approx\hspace{0.05mm}_\star^{\fv, R, p}$ are equivalence
relations and equivariant:
\begin{lemma}
{\it i)} Given the fact that $x\;R\;x$ holds, then
$(as, x) \approx\hspace{0.05mm}^{\fv, R, 0}_\star (as, x)$. {\it ii)} Given
that @{text "(p \<bullet> x) R y"} implies @{text "(-p \<bullet> y) R x"}, then
$(as, x) \approx\hspace{0.05mm}^{\fv, R, p}_\star (bs, y)$ implies
$(bs, y) \approx\hspace{0.05mm}^{\fv, R, - p}_\star (as, x)$. {\it iii)} Given
that @{text "(p \<bullet> x) R y"} and @{text "(q \<bullet> y) R z"} implies
@{text "((q + p) \<bullet> x) R z"}, then $(as, x) \approx\hspace{0.05mm}^{\fv, R, p}_\star (bs, y)$
and $(bs, y) \approx\hspace{0.05mm}^{\fv, R, q}_\star (cs, z)$ implies
$(as, x) \approx\hspace{0.05mm}^{\fv, R, q + p}_\star (cs, z)$. Given
@{text "(q \<bullet> x) R y"} implies @{text "(p \<bullet> (q \<bullet> x)) R (p \<bullet> y)"} and
@{text "p \<bullet> (fv x) = fv (p \<bullet> x)"} then @{text "p \<bullet> (fv y) = fv (p \<bullet> y)"}, then
$(as, x) \approx\hspace{0.05mm}^{\fv, R, q}_\star (bs, y)$ implies
$(p \;\isasymbullet\; as, p \;\isasymbullet\; x) \approx\hspace{0.05mm}^{\fv, R, q}_\star
(p \;\isasymbullet\; bs, p \;\isasymbullet\; y)$.
\end{lemma}
\begin{proof}
All properties are by unfolding the definitions and simple calculations.
\end{proof}
*}
section {* Alpha-Equivalence and Free Variables *}
text {*
Restrictions
\begin{itemize}
\item non-emptiness
\item positive datatype definitions
\item finitely supported abstractions
\item respectfulness of the bn-functions\bigskip
\item binders can only have a ``single scope''
\item all bindings must have the same mode
\end{itemize}
*}
section {* Examples *}
section {* Adequacy *}
section {* Related Work *}
text {*
Ott is better with list dot specifications; subgrammars
untyped;
*}
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 patiently explaining some of the finer points about the abstract
definitions and about the implementation of the Ott-tool.
Lookup: Merlin paper by James Cheney; Mark Shinwell PhD
Future work: distinct list abstraction
*}
(*<*)
end
(*>*)