(*<*)
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, for example
\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 that 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 case of type-schemes we do not
want 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 assuming that $x$, $y$ and $z$ are distinct variables,
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
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 can be appreciated in this case 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} and Cheney in
\cite{Cheney05}: 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, additional
predicates about well-formed terms are 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 call $bn\,(a)$ 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 lets the user to specify ``empty''
types like
\begin{center}
$t ::= t\;t \mid \lambda x. t$
\end{center}
\noindent
where no clause for variables is given. Arguably, such specifications make
some sense in the context of Coq's type theory (which Ott supports), but not
at all in a HOL-based environment 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 \emph{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 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 can be 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-equivalence is indeed an equivalence relation).
The fact that we obtain an isomorphism 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 for terms using the locally
nameless representation of binders \cite{McKinnaPollack99}: in this representation
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} for the HOL4 system. 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 raw 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 lambda-terms. This
lifted function is characterised by the equations
\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-equivalence is indeed an equivalence relation, and the lifted
definitions and theorems are respectful w.r.t.~alpha-equivalence. Accordingly, we
will not be able to lift a bound-variable function to alpha-equated terms
(since it does not respect alpha-equivalence). To sum up, every lifting of
theorems to the quotient level needs proofs of some respectfulness
properties. In the paper we show that we are able to automate these
proofs and therefore can establish a reasoning infrastructure for
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{Pitts04}. 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 are also able to derive, at the moment
only manually, strong induction principles that
have 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 one 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> _ :: perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
\noindent
in which the generic type @{text "\<beta>"} stands 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}). Concrete permutations are build up from
swappings, written as @{text "(a b)"},
which are permutations that behave as follows:
%
@{text[display,indent=5] "(a b) = \<lambda>c. if a = c then b else if b = c then a else c"}
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\label{sec:binders} *}
text {*
In Nominal Isabelle, the user is expected to write down a specification of a
term-calculus and then a reasoning infrastructure is automatically derived
from this specification (remember that Nominal Isabelle is a definitional
extension of Isabelle/HOL, which does not introduce any 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 anew for each specification. 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 @{text "as"}
in the body @{text "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 @{text x} and @{text y}
need to have the same set of free variables; moreover there must be a permutation
@{text p} such that {\it ii)} it leaves the free variables of @{text x} and @{text y} unchanged,
but {\it iii)} ``moves'' their bound names so that we obtain modulo a relation,
say \mbox{@{text "_ R _"}}, two equal terms. We also require {\it iv)} that @{text p} makes
the abstracted sets @{text as} and @{text 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 $y$.
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 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 also leaves the
type \mbox{@{text "x \<rightarrow> y"}} unchanged. Another examples is
$(\{x\}, x) \approx_{res} (\{x,y\}, x)$ which holds by taking $p$ to be the identity permutation.
However, if @{text "x \<noteq> y"}, then
$(\{x\}, x) \not\approx_{set} (\{x,y\}, x)$ 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}
\begin{lemma}
$supp ([as]set. x) = supp x - as$
\end{lemma}
*}
section {* Alpha-Equivalence and Free Variables *}
text {*
Our specifications for term-calculi are heavily
inspired by the syntax of the Ott-tool \cite{ott-jfp}. A specification is
a collection of (possibly mutual recursive) type declarations, say
$ty^\alpha_1$, $ty^\alpha_2$, \ldots $ty^\alpha_n$, and an
associated collection of binding function declarations, say
$bn^\alpha_1$, \ldots $bn^\alpha_m$. The syntax in Nominal Isabelle
for such specifications is rougly as follows:
%
\begin{equation}\label{scheme}
\mbox{\begin{tabular}{@ {\hspace{-9mm}}p{1.8cm}l}
type \mbox{declaration part} &
$\begin{cases}
\mbox{\begin{tabular}{l}
\isacommand{nominal\_datatype} $ty^\alpha_1 = \ldots$\\
\isacommand{and} $ty^\alpha_2 = \ldots$\\
$\ldots$\\
\isacommand{and} $ty^\alpha_n = \ldots$\\
\end{tabular}}
\end{cases}$\\
binding \mbox{function part} &
$\begin{cases}
\mbox{\begin{tabular}{l}
\isacommand{with} $bn^\alpha_1$ \isacommand{and} \ldots \isacommand{and} $bn^\alpha_m$
$\ldots$\\
\isacommand{where}\\
$\ldots$\\
\end{tabular}}
\end{cases}$\\
\end{tabular}}
\end{equation}
\noindent
Every type declaration $ty^\alpha_i$ consists of a collection of
term-constructors, each of which comes with a list of labelled
types that stand for the types of the arguments of the term-constructor.
For example for a term-constructor $C^\alpha$ we might have
\begin{center}
$C^\alpha\;label_1\!::\!ty'_1\;\ldots label_l\!::\!ty'_l\;\;\textit{binding\_clauses}$
\end{center}
\noindent
whereby some of the $ty'_k$ (or their type components) are contained in the
set of $ty^\alpha_i$
declared in \eqref{scheme}. In this case we will call
the corresponding argument a \emph{recursive argument}. The labels
annotated on the types are optional and can be used in the (possibly empty)
list of \emph{binding clauses}. These clauses indicate the binders and the
scope of the binders in a term-constructor. They come in three \emph{modes}
\begin{center}
\begin{tabular}{l}
\isacommand{bind}\; {\it binders}\; \isacommand{in}\; {\it label}\\
\isacommand{bind\_set}\; {\it binders}\; \isacommand{in}\; {\it label}\\
\isacommand{bind\_res}\; {\it binders}\; \isacommand{in}\; {\it label}\\
\end{tabular}
\end{center}
\noindent
The first mode is for binding lists of atoms (the order matters); the second is for sets
of binders (the order does not matter, but cardinality does) and the last is for
sets of binders (with vacuous binders preserving alpha-equivalence).
In addition we distinguish between \emph{shallow} binders and \emph{deep}
binders. Shallow binders are of the form \isacommand{bind}\; {\it label}\;
\isacommand{in}\; {\it another\_label} (similar for the other two modes). The
restriction we impose on shallow binders is that the {\it label} must either
refer to a type that is an atom type or to a type that is a finite set or
list of an atom type. For example the specifications of lambda-terms, where
a single name is bound, and type-schemes, where a finite set of names is
bound, use shallow binders (the type \emph{name} is an atom type):
\begin{center}
\begin{tabular}{@ {}cc@ {}}
\begin{tabular}{@ {}l@ {\hspace{-1mm}}}
\isacommand{nominal\_datatype} {\it lam} =\\
\hspace{5mm}\phantom{$\mid$} Var\;{\it name}\\
\hspace{5mm}$\mid$ App\;{\it lam}\;{\it lam}\\
\hspace{5mm}$\mid$ Lam\;{\it x::name}\;{\it t::lam}\\
\hspace{21mm}\isacommand{bind} {\it x} \isacommand{in} {\it t}\\
\end{tabular} &
\begin{tabular}{@ {}l@ {}}
\isacommand{nominal\_datatype} {\it ty} =\\
\hspace{5mm}\phantom{$\mid$} TVar\;{\it name}\\
\hspace{5mm}$\mid$ TFun\;{\it ty}\;{\it ty}\\
\isacommand{and} {\it tsc} = All\;{\it xs::(name fset)}\;{\it T::ty}\\
\hspace{24mm}\isacommand{bind\_res} {\it xs} \isacommand{in} {\it T}\\
\end{tabular}
\end{tabular}
\end{center}
\noindent
If we have shallow binders that ``share'' a body, for instance $t$ in
the term-constructor Foo$_0$
\begin{center}
\begin{tabular}{ll}
\it {\rm Foo}$_0$ x::name y::name t::lam & \it
\isacommand{bind}\;x\;\isacommand{in}\;t,\;
\isacommand{bind}\;y\;\isacommand{in}\;t
\end{tabular}
\end{center}
\noindent
then we have to make sure the modes of the binders agree. We cannot
have in the first binding clause the mode \isacommand{bind} and in the second
\isacommand{bind\_set}.
A \emph{deep} binder uses an auxiliary binding function that ``picks'' out
the atoms in one argument of the term-constructor that can be bound in
other arguments and also in the same argument (we will
call such binders \emph{recursive}).
The binding functions are expected to return either a set of atoms
(for \isacommand{bind\_set} and \isacommand{bind\_res}) or a list of atoms
(for \isacommand{bind}). They can be defined by primitive recursion over the
corresponding type; the equations must be given in the binding function part of
the scheme shown in \eqref{scheme}. For example for a calculus containing lets
with tuple patterns, you might declare
\begin{center}
\begin{tabular}{l}
\isacommand{nominal\_datatype} {\it trm} =\\
\hspace{5mm}\phantom{$\mid$} Var\;{\it name}\\
\hspace{5mm}$\mid$ App\;{\it trm}\;{\it trm}\\
\hspace{5mm}$\mid$ Lam\;{\it x::name}\;{\it t::trm}
\;\;\isacommand{bind} {\it x} \isacommand{in} {\it t}\\
\hspace{5mm}$\mid$ Let\;{\it p::pat}\;{\it trm}\; {\it t::trm}
\;\;\isacommand{bind\_set} {\it bn(p)} \isacommand{in} {\it t}\\
\isacommand{and} {\it pat} =\\
\hspace{5mm}\phantom{$\mid$} PNo\\
\hspace{5mm}$\mid$ PVr\;{\it name}\\
\hspace{5mm}$\mid$ PPr\;{\it pat}\;{\it pat}\\
\isacommand{with} {\it bn::pat $\Rightarrow$ atom set}\\
\isacommand{where} $bn(\textrm{PNil}) = \varnothing$\\
\hspace{5mm}$\mid$ $bn(\textrm{PVar}\;x) = \{atom\; x\}$\\
\hspace{5mm}$\mid$ $bn(\textrm{PPrd}\;p_1\;p_2) = bn(p_1) \cup bn(p_2)$\\
\end{tabular}
\end{center}
\noindent
In this specification the function $atom$ coerces a name into the generic
atom type of Nominal Isabelle. This allows us to treat binders of different
atom type uniformly. As will shortly become clear, we cannot return an atom in a
binding function that also is bound in the term-constructor. In the present
version of Nominal Isabelle, we adopted the restriction the Ott-tool imposes
on the binding functions, namely a binding function can only return the
empty set (case PNil), a singleton set containing an atom (case PVar) or
unions of atom sets (case PPrd). Moreover, as with shallow binders, deep
binders with shared body need to have the same binding mode. Finally, the
most drastic restriction we have to impose on deep binders is that we cannot
have ``overlapping'' deep binders. Consider for example the term-constructors:
\begin{center}
\begin{tabular}{ll}
\it {\rm Foo}$_1$ p::pat q::pat t::trm & \it \isacommand{bind}\;bn(p)\;\isacommand{in}\;t,\;
\isacommand{bind}\;bn(q)\;\isacommand{in}\;t\\
\it {\rm Foo}$_2$ x::name p::pat t::trm & \it \it \isacommand{bind}\;x\;\isacommand{in}\;t,\;
\isacommand{bind}\;bn(p)\;\isacommand{in}\;t
\end{tabular}
\end{center}
\noindent
In the first case we bind all atoms from the pattern $p$ in $t$ and also all atoms
from $q$ in $t$. As a result we have no way to determine whether the binder came from the
binding function in $p$ or $q$. Similarly in the second case:
the binder $bn(p)$ overlaps with the shallow binder $x$. We must exclude such specifiactions,
as we will not be able to represent them using the general binders described in
Section \ref{sec:binders}. However the following two term-constructors are allowed:
\begin{center}
\begin{tabular}{ll}
\it {\rm Bar}$_1$ p::pat t::trm s::trm & \it \isacommand{bind}\;bn(p)\;\isacommand{in}\;t,\;
\isacommand{bind}\;bn(p)\;\isacommand{in}\;s\\
\it {\rm Bar}$_2$ p::pat t::trm & \it \isacommand{bind}\;bn(p)\;\isacommand{in}\;p,\;
\isacommand{bind}\;bn(p)\;\isacommand{in}\;t\\
\end{tabular}
\end{center}
\noindent
since there is no overlap of binders.
Now the question is how we can turn specifications into actual type
definitions in Isabelle/HOL and then establish a reasoning infrastructure
for them? Because of the problem Pottier and Cheney pointed out, we cannot
in general re-arrange arguments of term-constructors so that binders and
their scopes next to each other, an then use the type constructors
@{text "abs_set"}, @{text "abs_res"} and @{text "abs_list"}. Therefore
we will first extract datatype definitions from the specification and
then define an alpha-equiavlence relation over them.
The datatype definition can be obtained by just stripping of the
binding clauses and the labels on the types. We also have to invent
new names for the types, $ty^\alpha$ and term-constructors $C^\alpha$
given by user. We just use an affix:
\begin{center}
$ty^\alpha \mapsto ty\_raw \qquad C^\alpha \mapsto C\_raw$
\end{center}
\noindent
This definition can be made, provided the usual conditions hold: the
types must be non-empty and the types in the term-constructors need to
be, what is called, positive position (see \cite{}). We then take the binding
functions and define them by primitive recursion over the raw datatypes.
binding function.
The first non-trivial step is to read off from the specification free-variable
functions. There are two kinds: free-variable functions corresponding to types,
written $\fv\_ty$, and free-variable functions corresponding to binding functions,
written $\fv\_bn$. They have to be defined at the same time since there can
be interdependencies. Given a term-constructor $C ty_1 \ldots ty_n$ and some binding
clauses, the function $\fv (C x_1 \ldots x_n)$ will be the union of the values
generated for each argument, say $x_i$, as follows:
\begin{center}
\begin{tabular}{cp{8cm}}
$\bullet$ & if it is a shallow binder, then $\varnothing$\\
$\bullet$ & if it is a deep binder, then $\fv_bn x_i$\\
$\bullet$ & if
\end{tabular}
\end{center}
*}
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
(*>*)