(*<*)
theory Paper
imports "../Nominal/Test" "LaTeXsugar"
begin
consts
fv :: "'a \<Rightarrow> 'b"
abs_set :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
Abs_lst :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
Abs_res :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
definition
"equal \<equiv> (op =)"
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) and
alpha_gen ("_ \<approx>\<^raw:\makebox[0mm][l]{$\,_{\textit{set}}$}>\<^bsup>_,_,_\<^esup> _") and
alpha_lst ("_ \<approx>\<^raw:\makebox[0mm][l]{$\,_{\textit{list}}$}>\<^bsup>_,_,_\<^esup> _") and
alpha_res ("_ \<approx>\<^raw:\makebox[0mm][l]{$\,_{\textit{res}}$}>\<^bsup>_,_,_\<^esup> _") and
abs_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
fv ("fv'(_')" [100] 100) and
equal ("=") and
alpha_abs ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
Abs ("[_]\<^raw:$\!$>\<^bsub>set\<^esub>._") and
Abs_lst ("[_]\<^raw:$\!$>\<^bsub>list\<^esub>._") and
Abs_res ("[_]\<^raw:$\!$>\<^bsub>res\<^esub>._")
(*>*)
section {* Introduction *}
text {*
So far, Nominal Isabelle provides a mechanism for constructing
alpha-equated terms, for example
\begin{center}
@{text "t ::= x | t t | \<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}
@{text "T ::= x | T \<rightarrow> T"}\hspace{5mm}
@{text "S ::= \<forall>{x\<^isub>1,\<dots>, x\<^isub>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}
@{text "\<forall>{x,y}. x \<rightarrow> y \<approx>\<^isub>\<alpha> \<forall>{y,x}. y \<rightarrow> x"}
\end{equation}
\noindent
but assuming that @{text x}, @{text y} and @{text z} are distinct variables,
the following two should \emph{not} be alpha-equivalent
%
\begin{equation}\label{ex2}
@{text "\<forall>{x,y}. x \<rightarrow> y \<notapprox>\<^isub>\<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}
@{text "\<forall>{x}. x \<rightarrow> y \<approx>\<^isub>\<alpha> \<forall>{x,z}. x \<rightarrow> y"}
\end{equation}
\noindent
where @{text 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}
@{text "\<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}
@{text "\<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 @{text "\<LET>"}s containing patterns
%
\begin{equation}\label{two}
@{text "\<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}
@{text "\<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 \cite{Pottier06} and
Cheney \cite{Cheney05}: in @{text "\<LET>"}-constructs of the form
%
\begin{center}
@{text "\<LET> x\<^isub>1 = t\<^isub>1 \<AND> \<dots> \<AND> x\<^isub>n = t\<^isub>n \<IN> s \<END>"}
\end{center}
\noindent
which bind all the @{text "x\<^isub>i"} in @{text s}, we might not care
about the order in which the @{text "x\<^isub>i = t\<^isub>i"} are given,
but we do care about the information that there are as many @{text
"x\<^isub>i"} as there are @{text "t\<^isub>i"}. We lose this information if
we represent the @{text "\<LET>"}-constructor by something like
%
\begin{center}
@{text "\<LET> [x\<^isub>1,\<dots>,x\<^isub>n].s [t\<^isub>1,\<dots>,t\<^isub>n]"}
\end{center}
\noindent
where the notation @{text "[_]._"} indicates that the @{text "x\<^isub>i"}
become bound in @{text s}. In this representation the term
\mbox{@{text "\<LET> [x].s [t\<^isub>1,t\<^isub>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}
@{text trm} & @{text "::="} & @{text "\<dots>"}\\
& @{text "|"} & @{text "\<LET> a::assn s::trm"}\hspace{4mm}
\isacommand{bind} @{text "bn(a)"} \isacommand{in} @{text "s"}\\[1mm]
@{text assn} & @{text "::="} & @{text "\<ANIL>"}\\
& @{text "|"} & @{text "\<ACONS> name trm assn"}
\end{tabular}
\end{center}
\noindent
where @{text assn} is an auxiliary type representing a list of assignments
and @{text bn} an auxiliary function identifying the variables to be bound
by the @{text "\<LET>"}. This function is defined by recursion over @{text
assn} as follows
\begin{center}
@{text "bn(\<ANIL>) ="} @{term "{}"} \hspace{5mm}
@{text "bn(\<ACONS> x t as) = {x} \<union> bn(as)"}
\end{center}
\noindent
The scope of the binding is indicated by labels given to the types, for
example @{text "s::trm"}, and a binding clause, in this case
\isacommand{bind} @{text "bn(a)"} \isacommand{in} @{text "s"}, that states
to bind in @{text s} all the names the function call @{text "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}
@{text "t ::= t t | \<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}
@{text "\<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}
@{text "fv(x) = {x}"}\hspace{10mm}
@{text "fv(t\<^isub>1 t\<^isub>2) = fv(t\<^isub>1) \<union> fv(t\<^isub>2)"}\\[1mm]
@{text "fv(\<lambda>x.t) = fv(t) - {x}"}
\end{center}
\noindent
then with not too great effort we obtain a function @{text "fv\<^sup>\<alpha>"}
operating on quotients, or alpha-equivalence classes of lambda-terms. This
lifted function is characterised by the equations
\begin{center}
@{text "fv\<^sup>\<alpha>(x) = {x}"}\hspace{10mm}
@{text "fv\<^sup>\<alpha>(t\<^isub>1 t\<^isub>2) = fv\<^sup>\<alpha>(t\<^isub>1) \<union> fv\<^sup>\<alpha>(t\<^isub>2)"}\\[1mm]
@{text "fv\<^sup>\<alpha>(\<lambda>x.t) = fv\<^sup>\<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 with deriving the reasoning infrastructure
manageable, we will wherever possible state definitions and perform proofs
on the user-level of Isabelle/HOL, as opposed to write custom ML-code that
generates them anew for each specification. To that end, we will consider
first 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 @{text "(as, x)"} and
@{text "(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 @{text "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}{@{term "(as, x) \<approx>gen R fv p (bs, y)"} @{text "\<equiv>"}\hspace{30mm}}\\
& @{term "fv(x) - as = fv(y) - bs"}\\
@{text "\<and>"} & @{term "(fv(x) - as) \<sharp>* p"}\\
@{text "\<and>"} & @{text "(p \<bullet> x) R y"}\\
@{text "\<and>"} & @{term "(p \<bullet> as) = bs"}\\
\end{array}
\end{equation}
\noindent
Note that this relation is dependent on the permutation @{text
"p"}. Alpha-equivalence between two pairs is then the relation where we
existentially quantify over this @{text "p"}. Also note that the relation is
dependent on a free-variable function @{text "fv"} and a relation @{text
"R"}. The reason for this extra generality is that we will use
$\approx_{\textit{set}}$ for both ``raw'' terms and alpha-equated terms. In
the latter case, $R$ will be replaced by equality @{text "="} and for raw terms we
will prove that @{text "fv"} is equal to the support of @{text
x} and @{text 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}{@{term "(as, x) \<approx>lst R fv p (bs, y)"} @{text "\<equiv>"}\hspace{30mm}}\\[1mm]
& @{term "fv(x) - (set as) = fv(y) - (set bs)"}\\
\wedge & @{term "(fv(x) - set as) \<sharp>* p"}\\
\wedge & @{text "(p \<bullet> x) R y"}\\
\wedge & @{term "(p \<bullet> as) = bs"}\\
\end{array}
\end{equation}
\noindent
where @{term set} is a function that coerces a list of atoms into a set of atoms.
Now the last clause ensures that the order of the binders matters.
If we do not want to make any difference between the order of binders \emph{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}{@{term "(as, x) \<approx>res R fv p (bs, y)"} @{text "\<equiv>"}\hspace{30mm}}\\[1mm]
& @{term "fv(x) - as = fv(y) - bs"}\\
\wedge & @{term "(fv(x) - as) \<sharp>* 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 @{text R} to be the equality and for @{text "fv(T)"} we define
\begin{center}
@{text "fv(x) = {x}"} \hspace{5mm} @{text "fv(T\<^isub>1 \<rightarrow> T\<^isub>2) = fv(T\<^isub>1) \<union> fv(T\<^isub>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 $\approx_{\textit{set}}$ and
$\approx_{\textit{res}}$ by taking @{text p} to be the swapping @{term "(x \<rightleftharpoons>
y)"}. In case of @{text "x \<noteq> y"}, then @{text "([x, y], x \<rightarrow> y)"}
$\not\approx_{\textit{list}}$ @{text "([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 example is
@{text "({x}, x)"} $\approx_{\textit{res}}$ @{text "({x,y}, x)"} which holds by
taking @{text p} to be the
identity permutation. However, if @{text "x \<noteq> y"}, then @{text "({x}, x)"}
$\not\approx_{\textit{set}}$ @{text "({x,y}, x)"} since there is no permutation
that makes the
sets @{text "{x}"} and @{text "{x,y}"} equal (similarly for $\approx_{\textit{list}}$).
\end{exmple}
% looks too ugly
%\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}
In the rest of this section we are going to introduce a type- and term-constructor
for abstractions. For this we define
%
\begin{equation}
@{term "abs_set (as, x) (bs, x) \<equiv> \<exists>p. alpha_gen (as, x) equal supp p (bs, x)"}
\end{equation}
\noindent
Similarly for @{text "abs_list"} and @{text "abs_res"}. We can show that these
relations are equivalence relations and equivariant
(we only show the $\approx_{\textit{abs\_set}}$-case).
\begin{lemma}
$\approx_{\textit{abs\_set}}$ is an equivalence
relations, and if @{term "abs_set (as, x) (bs, x)"} then also
@{term "abs_set (p \<bullet> as, p \<bullet> x) (p \<bullet> bs, p \<bullet> x)"}.
\end{lemma}
\begin{proof}
Reflexivity is by taking @{text "p"} to be @{text "0"}. For symmetry we have
a permutation @{text p} and for the proof obligation take @{term "-p"}. In case
of transitivity we have two permutations @{text p} and @{text q}, and for the
proof obligation use @{text "q + p"}. All the conditions are then by simple
calculations.
\end{proof}
\noindent
The following lemma (and similar ones for $\approx_{\textit{abs\_list}}$ and
$\approx_{\textit{abs\_res}}$) will be crucial below:
\begin{lemma}
@{thm[mode=IfThen] alpha_abs_swap[no_vars]}
\end{lemma}
\begin{proof}
This lemma is straightforward by observing that the assumptions give us
@{term "(a \<rightleftharpoons> b) \<bullet> (supp x - bs) = (supp x - bs)"} and that @{text supp}
is equivariant.
\end{proof}
\noindent
We are also define the following
@{text "aux (as, x) \<equiv> supp x - as"}
\noindent
This allows us to use our quotient package and introduce new types
@{text "\<beta> abs_set"}, @{text "\<beta> abs_res"} and @{text "\<beta> abs_list"}
representing the alpha-equivalence classes. Elements in these types
we will, respectively, write as:
\begin{center}
@{term "Abs as x"} \hspace{5mm}
@{term "Abs_lst as x"} \hspace{5mm}
@{term "Abs_res as x"}
\end{center}
\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 existing
datatype package of Isabelle/HOL and by the syntax of the Ott-tool
\cite{ott-jfp}. A specification is a collection of (possibly mutual
recursive) type declarations, say @{text "ty"}$^\alpha_1$, \ldots,
@{text ty}$^\alpha_n$, and an associated collection
of binding functions, say @{text bn}$^\alpha_1$, \ldots, @{text
bn}$^\alpha_m$. The syntax in Nominal Isabelle for such specifications is
rougly as follows:
%
\begin{equation}\label{scheme}
\mbox{\begin{tabular}{@ {\hspace{-5mm}}p{1.8cm}l}
type \mbox{declaration part} &
$\begin{cases}
\mbox{\begin{tabular}{l}
\isacommand{nominal\_datatype} @{text ty}$^\alpha_1 = \ldots$\\
\isacommand{and} @{text ty}$^\alpha_2 = \ldots$\\
$\ldots$\\
\isacommand{and} @{text ty}$^\alpha_n = \ldots$\\
\end{tabular}}
\end{cases}$\\
binding \mbox{function part} &
$\begin{cases}
\mbox{\begin{tabular}{l}
\isacommand{with} @{text bn}$^\alpha_1$ \isacommand{and} \ldots \isacommand{and} @{text bn}$^\alpha_m$\\
\isacommand{where}\\
$\ldots$\\
\end{tabular}}
\end{cases}$\\
\end{tabular}}
\end{equation}
\noindent
Every type declaration @{text ty}$^\alpha_{1..n}$ 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 a term-constructor @{text "C\<^sup>\<alpha>"} might have
\begin{center}
@{text "C\<^sup>\<alpha> label\<^isub>1::ty"}$'_1$ @{text "\<dots> label\<^isub>l::ty"}$'_l\;\;$ @{text "binding_clauses"}
\end{center}
\noindent
whereby some of the @{text ty}$'_{1..l}$ (or their components) are contained in the collection
of @{text ty}$^\alpha_{1..n}$ 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 their scope of
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 of binders matters); the second is for sets
of binders (the order does not matter, but the 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 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. Two examples for the use of shallow binders are the
specification of lambda-terms, where a single name is bound, and of
type-schemes, where a finite set of names is bound:
\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
Note that in this specification \emph{name} refers to an atom type.
If we have shallow binders that ``share'' a body, for instance $t$ in
the following term-constructor
\begin{center}
\begin{tabular}{ll}
\it {\rm Foo} 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, for instance, 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, which can be bound in
other arguments and also in the same argument (we will
call such binders \emph{recursive}, see below).
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 specify
\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} {\it bn(p)} \isacommand{in} {\it t}\\
\isacommand{and} {\it pat} =\\
\hspace{5mm}\phantom{$\mid$} PNil\\
\hspace{5mm}$\mid$ PVar\;{\it name}\\
\hspace{5mm}$\mid$ PTup\;{\it pat}\;{\it pat}\\
\isacommand{with} {\it bn::pat $\Rightarrow$ atom list}\\
\isacommand{where} $\textit{bn}(\textrm{PNil}) = []$\\
\hspace{5mm}$\mid$ $\textit{bn}(\textrm{PVar}\;x) = [\textit{atom}\; x]$\\
\hspace{5mm}$\mid$ $\textit{bn}(\textrm{PTup}\;p_1\;p_2) = \textit{bn}(p_1)\; @\;\textit{bn}(p_2)$\\
\end{tabular}
\end{center}
\noindent
In this specification the function @{text "bn"} determines which atoms of @{text p} are
bound in the argument @{text "t"}. Note that the second last clause the function @{text "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 is also bound in the corresponding term-constructor. That means in the
example above that the term-constructors PVar and PTup must not have a
binding clause. In the present version of Nominal Isabelle, we also adopted
the restriction from the Ott-tool that binding functions can only return:
the empty set or empty list (as in case PNil), a singleton set or singleton
list containing an atom (case PVar), or unions of atom sets or appended atom
lists (case PTup). This restriction will simplify proofs later on.
The 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} p::pat q::pat t::trm & \it \isacommand{bind}\;bn(p)\;\isacommand{in}\;t,\;
\isacommand{bind}\;bn(q)\;\isacommand{in}\;t\\
\it {\rm Foo}$'$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 @{text p} in @{text t}
and also all atoms from @{text q} in @{text t}. As a result we have no way
to determine whether the binder came from the binding function @{text
"bn(p)"} or @{text "bn(q)"}. Similarly in the second case. There the binder
@{text "bn(p)"} overlaps with the shallow binder @{text x}. The reason why
we must exclude such specifiactions is that they cannot be represent by
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} p::pat t::trm s::trm & \it \isacommand{bind}\;bn(p)\;\isacommand{in}\;t,\;
\isacommand{bind}\;bn(p)\;\isacommand{in}\;s\\
\it {\rm Bar}$'$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.
Note that in the last example we wrote {\it\isacommand{bind}\;bn(p)\;\isacommand{in}\;p}.
Whenver such a binding clause is present, we will call the binder \emph{recursive}.
To see the purpose for this, consider ``plain'' Lets and Let\_recs:
\begin{center}
\begin{tabular}{@ {}l@ {}}
\isacommand{nominal\_datatype} {\it trm} =\\
\hspace{5mm}\phantom{$\mid$}\ldots\\
\hspace{5mm}$\mid$ Let\;{\it a::assn}\; {\it t::trm}
\;\;\isacommand{bind} {\it bn(a)} \isacommand{in} {\it t}\\
\hspace{5mm}$\mid$ Let\_rec\;{\it a::assn}\; {\it t::trm}
\;\;\isacommand{bind} {\it bn(a)} \isacommand{in} {\it t},
\isacommand{bind} {\it bn(a)} \isacommand{in} {\it a}\\
\isacommand{and} {\it assn} =\\
\hspace{5mm}\phantom{$\mid$} ANil\\
\hspace{5mm}$\mid$ ACons\;{\it name}\;{\it trm}\;{\it assn}\\
\isacommand{with} {\it bn::assn $\Rightarrow$ atom list}\\
\isacommand{where} $bn(\textrm{ANil}) = []$\\
\hspace{5mm}$\mid$ $bn(\textrm{ACons}\;x\;t\;a) = [atom\; x]\; @\; bn(a)$\\
\end{tabular}
\end{center}
\noindent
The difference is that with Let we only want to bind the atoms @{text
"bn(a)"} in the term @{text t}, but with Let\_rec we also want to bind the atoms
inside the assignment. This requires recursive binders and also has
consequences for the free variable function and alpha-equivalence relation,
which we are going to explain in the rest of this section.
Having dealt with all syntax matters, the problem now 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 bodies are next to each other, and
then use the type constructors @{text "abs_set"}, @{text "abs_res"} and
@{text "abs_list"} from Section \ref{sec:binders}. 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 off the
binding clauses and the labels on the types. We also have to invent
new names for the types @{text "ty\<^sup>\<alpha>"} and term-constructors @{text "C\<^sup>\<alpha>"}
given by user. In our implementation we just use an affix like
\begin{center}
@{text "ty\<^sup>\<alpha> \<mapsto> ty_raw"} \hspace{7mm} @{text "C\<^sup>\<alpha> \<mapsto> C_raw"}
\end{center}
\noindent
The resulting datatype definition is legal in Isabelle/HOL provided the datatypes are
non-empty and the types in the constructors only occur in positive
position (see \cite{} for an indepth explanataion of the datatype package
in Isabelle/HOL). We then define the user-specified binding
functions by primitive recursion over the raw datatypes. We can also
easily define a permutation operation by primitive recursion so that for each
term constructor @{text "C_raw ty\<^isub>1 \<dots> ty\<^isub>n"} we have that
\begin{center}
@{text "p \<bullet> (C_raw x\<^isub>1 \<dots> x\<^isub>n) \<equiv> C_raw (p \<bullet> x\<^isub>1) \<dots> (p \<bullet> x\<^isub>n)"}
\end{center}
\noindent
From this definition we can easily show that the raw datatypes are
all permutation types (Def ??) by a simple structural induction over
the @{text "ty_raw"}s.
The first non-trivial step we have to perform is the generatation free-variable
functions from the specifications. Given types @{text "ty\<^isub>1, \<dots>, ty\<^isub>n"}
we need to define the free-variable functions
\begin{center}
@{text "fv_ty\<^isub>1 :: ty\<^isub>1 \<Rightarrow> atom set \<dots> fv_ty\<^isub>n :: ty\<^isub>n \<Rightarrow> atom set"}
\end{center}
\noindent
and given binding functions @{text "bn_ty\<^isub>1, \<dots>, bn_ty\<^isub>m"} we also need to define
the free-variable functions
\begin{center}
@{text "fv_bn_ty\<^isub>1 :: ty\<^isub>1 \<Rightarrow> atom set \<dots> fv_bn_ty\<^isub>m :: ty\<^isub>m \<Rightarrow> atom set"}
\end{center}
\noindent
The basic idea behind these free-variable functions is to collect all atoms
that are not bound in a term constructor, but because of the rather
complicated binding mechanisms the details are somewhat involved.
Given a term-constructor @{text "C_raw ty\<^isub>1 \<dots> ty\<^isub>n"}, of type @{text ty} together with
some binding clauses, the function @{text "fv_ty (C_raw x\<^isub>1 \<dots> x\<^isub>n)"} will be
the union of the values defined below for each argument, say @{text "x\<^isub>i"} with type @{text "ty\<^isub>i"}.
From the binding clause of this term constructor, we can determine whether the
argument @{text "x\<^isub>i"} is a shallow or deep binder, and in the latter case also
whether it is a recursive or non-recursive of a binder. In these cases the value is:
\begin{center}
\begin{tabular}{cp{7cm}}
$\bullet$ & @{term "{}"} provided @{text "x\<^isub>i"} is a shallow binder\\
$\bullet$ & @{text "fv_bn_ty\<^isub>i x\<^isub>i"} provided @{text "x\<^isub>i"} is a deep non-recursive binder\\
$\bullet$ & @{text "fv_ty\<^isub>i x\<^isub>i - bn_ty\<^isub>i x\<^isub>i"} provided @{text "x\<^isub>i"} is a deep recursive binder\\
\end{tabular}
\end{center}
\noindent
In case the argument @{text "x\<^isub>i"} is not a binder, it might be a body of
one or more abstractions. There are two cases: either the
corresponding binders are all shallow or there is a single deep binder.
In the former case we build the union of all shallow binders; in the
later case we just take set or list of atoms the specified binding
function returns. Let @{text "bnds"} be an abbreviation of the bound
atoms. Then the value is given by:
\begin{center}
\begin{tabular}{cp{7cm}}
$\bullet$ & @{text "{atom x\<^isub>i} - bnds"} provided @{term "x\<^isub>i"} is an atom\\
$\bullet$ & @{text "(atoms x\<^isub>i) - bnds"} provided @{term "x\<^isub>i"} is a set of atoms\\
$\bullet$ & @{text "(atoms (set x\<^isub>i)) - bnds"} provided @{term "x\<^isub>i"} is a list of atoms\\
$\bullet$ & @{text "(fv_ty\<^isub>i x\<^isub>i) - bnds"} provided @{term "ty\<^isub>i"} is a nominal datatype\\
$\bullet$ & @{term "{}"} otherwise
\end{tabular}
\end{center}
\noindent
If the argument is neither a binder, nor a body of an abstraction, then the
value is defined as above, except that @{text "bnds"} is empty. i.e.~no atoms
are abstracted.
TODO
Given a clause of a binding function of the form
\begin{center}
@{text "bn_ty (C_raw x\<^isub>1 \<dots> x\<^isub>n) = rhs"}
\end{center}
\noindent
then the corresponding free-variable function @{text "fv_bn_ty\<^isub>i"} is the
union of the values calculated for the @{text "x\<^isub>j"} as follows:
\begin{center}
\begin{tabular}{cp{7cm}}
$\bullet$ & @{text "{}"} provided @{term "x\<^isub>j"} occurs in @{text "rhs"} and is an atom\\
$\bullet$ & @{text "fv_bn_ty x\<^isub>j"} provided @{term "x\<^isub>j"} occurs in @{text "rhs"}
with the recursive call @{text "bn_ty x\<^isub>j"}\\
$\bullet$ & @{text "(atoms x\<^isub>i) - bnds"} provided @{term "x\<^isub>i"} is a set of atoms\\
$\bullet$ & @{text "(atoml x\<^isub>i) - bnds"} provided @{term "x\<^isub>i"} is a list of atoms\\
$\bullet$ & @{text "(fv_ty\<^isub>i x\<^isub>i) - bnds"} provided @{term "ty\<^isub>i"} is a nominal datatype\\
$\bullet$ & @{term "{}"} otherwise
\end{tabular}
\end{center}
*}
section {* The Lifting of Definitions and Properties *}
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
(*>*)