(*<*)
theory Paper
imports "../Nominal/NewParser" "LaTeXsugar"
begin
consts
fv :: "'a \<Rightarrow> 'b"
abs_set :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
alpha_bn :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
abs_set2 :: "'a \<Rightarrow> perm \<Rightarrow> 'b \<Rightarrow> 'c"
Abs_dist :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
Abs_print :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
definition
"equal \<equiv> (op =)"
notation (latex output)
swap ("'(_ _')" [1000, 1000] 1000) and
fresh ("_ # _" [51, 51] 50) and
fresh_star ("_ #\<^sup>* _" [51, 51] 50) and
supp ("supp _" [78] 73) and
uminus ("-_" [78] 73) and
If ("if _ then _ else _" 10) and
alpha_gen ("_ \<approx>\<^raw:\,\raisebox{-1pt}{\makebox[0mm][l]{$_{\textit{set}}$}}>\<^bsup>_, _, _\<^esup> _") and
alpha_lst ("_ \<approx>\<^raw:\,\raisebox{-1pt}{\makebox[0mm][l]{$_{\textit{list}}$}}>\<^bsup>_, _, _\<^esup> _") and
alpha_res ("_ \<approx>\<^raw:\,\raisebox{-1pt}{\makebox[0mm][l]{$_{\textit{res}}$}}>\<^bsup>_, _, _\<^esup> _") and
abs_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
abs_set2 ("_ \<approx>\<^raw:\raisebox{-1pt}{\makebox[0mm][l]{$\,_{\textit{list}}$}}>\<^bsup>_\<^esup> _") and
fv ("fa'(_')" [100] 100) and
equal ("=") and
alpha_abs ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
Abs ("[_]\<^bsub>set\<^esub>._" [20, 101] 999) and
Abs_lst ("[_]\<^bsub>list\<^esub>._") and
Abs_dist ("[_]\<^bsub>#list\<^esub>._") and
Abs_res ("[_]\<^bsub>res\<^esub>._") and
Abs_print ("_\<^bsub>set\<^esub>._") and
Cons ("_::_" [78,77] 73) and
supp_gen ("aux _" [1000] 10) and
alpha_bn ("_ \<approx>bn _")
consts alpha_trm ::'a
consts fa_trm :: 'a
consts alpha_trm2 ::'a
consts fa_trm2 :: 'a
consts ast :: 'a
consts ast' :: 'a
notation (latex output)
alpha_trm ("\<approx>\<^bsub>trm\<^esub>") and
fa_trm ("fa\<^bsub>trm\<^esub>") and
alpha_trm2 ("'(\<approx>\<^bsub>assn\<^esub>, \<approx>\<^bsub>trm\<^esub>')") and
fa_trm2 ("'(fa\<^bsub>assn\<^esub>, fa\<^bsub>trm\<^esub>')") and
ast ("'(as, t')") and
ast' ("'(as', t\<PRIME> ')")
(*>*)
section {* Introduction *}
text {*
So far, Nominal Isabelle provided a mechanism for constructing
$\alpha$-equated terms, for example lambda-terms
\begin{center}
@{text "t ::= x | t t | \<lambda>x. t"}
\end{center}
\noindent
where free and bound variables have names. For such $\alpha$-equated 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{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,
respectively, 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 @{text "\<forall>"}-quantification 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
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}. x \<rightarrow> y"}
\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 @{text "x = 3"} and
\mbox{@{text "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 like
%
\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 term-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> (\<lambda>x\<^isub>1\<dots>x\<^isub>n . s) [t\<^isub>1,\<dots>,t\<^isub>n]"}
\end{center}
\noindent
where the notation @{text "\<lambda>_ . _"} indicates that the list of @{text
"x\<^isub>i"} becomes bound in @{text s}. In this representation the term
\mbox{@{text "\<LET> (\<lambda>x . s) [t\<^isub>1, t\<^isub>2]"}} is a perfectly legal
instance, but the lengths of the two lists do not agree. 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 in very messy
reasoning (see for example~\cite{BengtsonParow09}). To avoid this, we will
allow type specifications for @{text "\<LET>"}s as follows
%
\begin{center}
\begin{tabular}{r@ {\hspace{2mm}}r@ {\hspace{2mm}}l}
@{text trm} & @{text "::="} & @{text "\<dots>"}\\
& @{text "|"} & @{text "\<LET> as::assn s::trm"}\hspace{4mm}
\isacommand{bind} @{text "bn(as)"} \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 can be 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(as)"} \isacommand{in} @{text "s"}. This binding
clause states that all the names the function @{text
"bn(as)"} returns should be bound in @{text s}. This style of specifying terms and bindings is heavily
inspired by the syntax of the Ott-tool \cite{ott-jfp}.
However, we will not be able to cope with all specifications that are
allowed by Ott. One reason is that Ott lets the user 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 \cite{Berghofer99}.
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, for example,
that the two type-schemes
\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 with $\alpha$-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 Isabelle/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); then identify 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 property 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 the help of the quotient package we can 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
``raw'' definitions and theorems are respectful w.r.t.~$\alpha$-equivalence.
For example, we will not be able to lift a bound-variable function. Although
this function can be defined for raw terms, it does not respect
$\alpha$-equivalence and therefore cannot be lifted. To sum up, every lifting
of theorems to the quotient level needs proofs of some respectfulness
properties (see \cite{Homeier05}). In the paper we show that we are able to
automate these proofs and as a result can automatically establish a reasoning
infrastructure for $\alpha$-equated terms.
The examples we have in mind where our reasoning infrastructure will be
helpful includes the term language of System @{text "F\<^isub>C"}, also
known as Core-Haskell (see Figure~\ref{corehas}). This term language
involves patterns that have lists of type-, coercion- and term-variables,
all of which are bound in @{text "\<CASE>"}-expressions. One
feature is that we do not know in advance how many variables need to
be bound. Another is that each bound variable comes with a kind or type
annotation. Representing such binders with single binders and reasoning
about them in a theorem prover would be a major pain. \medskip
\noindent
{\bf Contributions:} We provide three new definitions for when terms
involving general 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 strong
induction principles that have the variable convention already built in.
The method behind our specification of general binders is taken
from the Ott-tool, but we introduce crucial restrictions, and also extensions, so
that our specifications make sense for reasoning about $\alpha$-equated terms.
\begin{figure}
\begin{boxedminipage}{\linewidth}
\begin{center}
\begin{tabular}{r@ {\hspace{1mm}}r@ {\hspace{2mm}}l}
\multicolumn{3}{@ {}l}{Type Kinds}\\
@{text "\<kappa>"} & @{text "::="} & @{text "\<star> | \<kappa>\<^isub>1 \<rightarrow> \<kappa>\<^isub>2"}\smallskip\\
\multicolumn{3}{@ {}l}{Coercion Kinds}\\
@{text "\<iota>"} & @{text "::="} & @{text "\<sigma>\<^isub>1 \<sim> \<sigma>\<^isub>2"}\smallskip\\
\multicolumn{3}{@ {}l}{Types}\\
@{text "\<sigma>"} & @{text "::="} & @{text "a | T | \<sigma>\<^isub>1 \<sigma>\<^isub>2 | S\<^isub>n"}$\;\overline{@{text "\<sigma>"}}$@{text "\<^sup>n"}
@{text "| \<forall>a:\<kappa>. \<sigma> | \<iota> \<Rightarrow> \<sigma>"}\smallskip\\
\multicolumn{3}{@ {}l}{Coercion Types}\\
@{text "\<gamma>"} & @{text "::="} & @{text "c | C | \<gamma>\<^isub>1 \<gamma>\<^isub>2 | S\<^isub>n"}$\;\overline{@{text "\<gamma>"}}$@{text "\<^sup>n"}
@{text "| \<forall>c:\<iota>. \<gamma> | \<iota> \<Rightarrow> \<gamma> "}\\
& @{text "|"} & @{text "refl \<sigma> | sym \<gamma> | \<gamma>\<^isub>1 \<circ> \<gamma>\<^isub>2 | \<gamma> @ \<sigma> | left \<gamma> | right \<gamma>"}\\
& @{text "|"} & @{text "\<gamma>\<^isub>1 \<sim> \<gamma>\<^isub>2 | rightc \<gamma> | leftc \<gamma> | \<gamma>\<^isub>1 \<triangleright> \<gamma>\<^isub>2"}\smallskip\\
\multicolumn{3}{@ {}l}{Terms}\\
@{text "e"} & @{text "::="} & @{text "x | K | \<Lambda>a:\<kappa>. e | \<Lambda>c:\<iota>. e | e \<sigma> | e \<gamma>"}\\
& @{text "|"} & @{text "\<lambda>x:\<sigma>. e | e\<^isub>1 e\<^isub>2 | \<LET> x:\<sigma> = e\<^isub>1 \<IN> e\<^isub>2"}\\
& @{text "|"} & @{text "\<CASE> e\<^isub>1 \<OF>"}$\;\overline{@{text "p \<rightarrow> e\<^isub>2"}}$ @{text "| e \<triangleright> \<gamma>"}\smallskip\\
\multicolumn{3}{@ {}l}{Patterns}\\
@{text "p"} & @{text "::="} & @{text "K"}$\;\overline{@{text "a:\<kappa>"}}\;\overline{@{text "c:\<iota>"}}\;\overline{@{text "x:\<sigma>"}}$\smallskip\\
\multicolumn{3}{@ {}l}{Constants}\\
& @{text C} & coercion constants\\
& @{text T} & value type constructors\\
& @{text "S\<^isub>n"} & n-ary type functions (which need to be fully applied)\\
& @{text K} & data constructors\smallskip\\
\multicolumn{3}{@ {}l}{Variables}\\
& @{text a} & type variables\\
& @{text c} & coercion variables\\
& @{text x} & term variables\\
\end{tabular}
\end{center}
\end{boxedminipage}
\caption{The System @{text "F\<^isub>C"}
\cite{CoreHaskell}, also often referred to as \emph{Core-Haskell}. In this
version of @{text "F\<^isub>C"} we made a modification by separating the
grammars for type kinds and coercion kinds, as well as for types and coercion
types. For this paper the interesting term-constructor is @{text "\<CASE>"},
which binds multiple type-, coercion- and term-variables.\label{corehas}}
\end{figure}
*}
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} (including proofs). We shall briefly review this work
to aid the description of what follows.
Two central notions in the nominal logic work are sorted atoms and
sort-respecting permutations of atoms. We will use the letters @{text "a,
b, c, \<dots>"} to stand for atoms and @{text "p, q, \<dots>"} to stand for
permutations. The purpose of atoms is to represent variables, be they bound or free.
The sorts of atoms can be used to represent different kinds of
variables, such as the term-, coercion- and type-variables in Core-Haskell.
It is assumed that there is an infinite supply of atoms for each
sort. However, in the interest of brevity, we shall restrict ourselves
in what follows to 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
%
\begin{center}
@{text "_ \<bullet> _ :: perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
\end{center}
\noindent
in which the generic type @{text "\<beta>"} stands for the type of the object
over 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 by induction over the type-hierarchy \cite{HuffmanUrban10};
for example permutations acting on products, lists, sets, functions and booleans is
given by:
%
\begin{equation}\label{permute}
\mbox{\begin{tabular}{@ {}cc@ {}}
\begin{tabular}{@ {}l@ {}}
@{thm permute_prod.simps[no_vars, THEN eq_reflection]}\\[2mm]
@{thm permute_list.simps(1)[no_vars, THEN eq_reflection]}\\
@{thm permute_list.simps(2)[no_vars, THEN eq_reflection]}\\
\end{tabular} &
\begin{tabular}{@ {}l@ {}}
@{thm permute_set_eq[no_vars, THEN eq_reflection]}\\
@{text "p \<bullet> f \<equiv> \<lambda>x. p \<bullet> (f (- p \<bullet> x))"}\\
@{thm permute_bool_def[no_vars, THEN eq_reflection]}\\
\end{tabular}
\end{tabular}}
\end{equation}
\noindent
Concrete permutations in Nominal Isabelle are built up from swappings,
written as \mbox{@{text "(a b)"}}, which are permutations that behave
as follows:
%
\begin{center}
@{text "(a b) = \<lambda>c. if a = c then b else if b = c then a else c"}
\end{center}
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:
%
\begin{equation}\label{suppdef}
@{thm supp_def[no_vars, THEN eq_reflection]}
\end{equation}
\noindent
There is also the derived notion for when an atom @{text a} is \emph{fresh}
for an @{text x}, defined as
%
\begin{center}
@{thm fresh_def[no_vars]}
\end{center}
\noindent
We 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}, leaves
@{text x} unchanged:
\begin{property}\label{swapfreshfresh}
@{thm[mode=IfThen] swap_fresh_fresh[no_vars]}
\end{property}
While often the support of an object can be relatively easily
described, for example for atoms, products, lists, function applications,
booleans and permutations as follows
%
\begin{eqnarray}
@{term "supp a"} & = & @{term "{a}"}\\
@{term "supp (x, y)"} & = & @{term "supp x \<union> supp y"}\\
@{term "supp []"} & = & @{term "{}"}\\
@{term "supp (x#xs)"} & = & @{term "supp x \<union> supp xs"}\\
@{text "supp (f x)"} & @{text "\<subseteq>"} & @{term "supp f \<union> supp x"}\label{suppfun}\\
@{term "supp b"} & = & @{term "{}"}\\
@{term "supp p"} & = & @{term "{a. p \<bullet> a \<noteq> a}"}
\end{eqnarray}
\noindent
in some cases it can be difficult to characterise the support precisely, and
only an approximation can be established (see \eqref{suppfun} above). Reasoning about
such approximations can be simplified with the notion \emph{supports}, defined
as follows:
\begin{defn}
A set @{text S} \emph{supports} @{text x} if for all atoms @{text a} and @{text b}
not in @{text S} we have @{term "(a \<rightleftharpoons> b) \<bullet> x = x"}.
\end{defn}
\noindent
The main point of @{text supports} is that we can establish the following
two properties.
\begin{property}\label{supportsprop}
Given a set @{text "as"} of atoms.
{\it (i)} @{thm[mode=IfThen] supp_is_subset[where S="as", no_vars]}
{\it (ii)} @{thm supp_supports[no_vars]}.
\end{property}
Another important notion in the nominal logic work is \emph{equivariance}.
For a function @{text f}, say of type @{text "\<alpha> \<Rightarrow> \<beta>"}, to be equivariant
it is required that every permutation leaves @{text f} unchanged, that is
%
\begin{equation}\label{equivariancedef}
@{term "\<forall>p. p \<bullet> f = f"}
\end{equation}
\noindent or equivalently that a permutation applied to the application
@{text "f x"} can be moved to the argument @{text x}. That means for equivariant
functions @{text f}, we have for all permutations @{text p}:
%
\begin{equation}\label{equivariance}
@{text "p \<bullet> f = f"} \;\;\;\textit{if and only if}\;\;\;
@{text "p \<bullet> (f x) = f (p \<bullet> x)"}
\end{equation}
\noindent
From property \eqref{equivariancedef} and the definition of @{text supp}, we
can easily deduce that equivariant functions have empty support. There is
also a similar notion for equivariant relations, say @{text R}, namely the property
that
%
\begin{center}
@{text "x R y"} \;\;\textit{implies}\;\; @{text "(p \<bullet> x) R (p \<bullet> y)"}
\end{center}
Finally, the nominal logic work provides us with general means for renaming
binders. While in the older version of Nominal Isabelle, we used extensively
Property~\ref{swapfreshfresh} to rename single binders, this property
proved too unwieldy for dealing with multiple binders. For such binders the
following generalisations turned out to be easier to use.
\begin{property}\label{supppermeq}
@{thm[mode=IfThen] supp_perm_eq[no_vars]}
\end{property}
\begin{property}\label{avoiding}
For a finite set @{text as} and a finitely supported @{text x} with
@{term "as \<sharp>* x"} and also a finitely supported @{text c}, there
exists a permutation @{text p} such that @{term "(p \<bullet> as) \<sharp>* c"} and
@{term "supp x \<sharp>* p"}.
\end{property}
\noindent
The idea behind the second property is that given a finite set @{text as}
of binders (being bound, or fresh, in @{text x} is ensured by the
assumption @{term "as \<sharp>* x"}), then there exists a permutation @{text p} such that
the renamed binders @{term "p \<bullet> as"} avoid @{text c} (which can be arbitrarily chosen
as long as it is finitely supported) and also @{text "p"} does not affect anything
in the support of @{text x} (that is @{term "supp x \<sharp>* p"}). The last
fact and Property~\ref{supppermeq} allow us to ``rename'' just the binders
@{text as} in @{text x}, because @{term "p \<bullet> x = x"}.
Most properties given in this section are described in detail in \cite{HuffmanUrban10}
and of course all are formalised in Isabelle/HOL. In the next sections we will make
extensive use of these properties in order to define $\alpha$-equivalence in
the presence of multiple binders.
*}
section {* General Bindings\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 of atoms @{text
"as"} in the body @{text "x"}.
The first question we have to answer is when two pairs @{text "(as, x)"} and
@{text "(bs, y)"} are $\alpha$-equivalent? (For the moment we are interested in
the notion of $\alpha$-equivalence that is \emph{not} preserved by adding
vacuous binders.) To answer this question, we identify four conditions: {\it (i)}
given a free-atom function @{text "fa"} of type \mbox{@{text "\<beta> \<Rightarrow> atom
set"}}, then @{text x} and @{text y} need to have the same set of free
atoms; moreover there must be a permutation @{text p} such that {\it
(ii)} @{text p} leaves the free atoms 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 equivalent terms. We also require that {\it (iv)}
@{text p} makes the sets of abstracted atoms @{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@ {\hspace{4mm}}r}
\multicolumn{3}{l}{@{term "(as, x) \<approx>gen R fa p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}}\\[1mm]
& @{term "fa(x) - as = fa(y) - bs"} & \mbox{\it (i)}\\
@{text "\<and>"} & @{term "(fa(x) - as) \<sharp>* p"} & \mbox{\it (ii)}\\
@{text "\<and>"} & @{text "(p \<bullet> x) R y"} & \mbox{\it (iii)}\\
@{text "\<and>"} & @{term "(p \<bullet> as) = bs"} & \mbox{\it (iv)}\\
\end{array}
\end{equation}
\noindent
Note that this relation depends 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-atom function @{text "fa"} 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, @{text R} will be replaced by equality @{text "="} and we
will prove that @{text "fa"} is equal to @{text "supp"}.
The definition in \eqref{alphaset} does not make any distinction between the
order of abstracted atoms. 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@ {\hspace{4mm}}r}
\multicolumn{2}{l}{@{term "(as, x) \<approx>lst R fa p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}}\\[1mm]
& @{term "fa(x) - (set as) = fa(y) - (set bs)"} & \mbox{\it (i)}\\
\wedge & @{term "(fa(x) - set as) \<sharp>* p"} & \mbox{\it (ii)}\\
\wedge & @{text "(p \<bullet> x) R y"} & \mbox{\it (iii)}\\
\wedge & @{term "(p \<bullet> as) = bs"} & \mbox{\it (iv)}\\
\end{array}
\end{equation}
\noindent
where @{term set} is the function that coerces a list of atoms into a set of atoms.
Now the last clause ensures that the order of the binders matters (since @{text as}
and @{text bs} are lists of atoms).
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@ {\hspace{4mm}}r}
\multicolumn{2}{l}{@{term "(as, x) \<approx>res R fa p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}}\\[1mm]
& @{term "fa(x) - as = fa(y) - bs"} & \mbox{\it (i)}\\
\wedge & @{term "(fa(x) - as) \<sharp>* p"} & \mbox{\it (ii)}\\
\wedge & @{text "(p \<bullet> x) R y"} & \mbox{\it (iii)}\\
\end{array}
\end{equation}
It might be useful to consider first some examples about how these definitions
of $\alpha$-equivalence pan out in practice. For this consider the case of
abstracting a set of atoms over types (as in type-schemes). We set
@{text R} to be the usual equality @{text "="} and for @{text "fa(T)"} we
define
\begin{center}
@{text "fa(x) = {x}"} \hspace{5mm} @{text "fa(T\<^isub>1 \<rightarrow> T\<^isub>2) = fa(T\<^isub>1) \<union> fa(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 $\alpha$-equivalent 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 $\approx_{\,\textit{list}}$). It can also relatively easily be
shown that all three notions of $\alpha$-equivalence coincide, if we only
abstract a single atom.
In the rest of this section we are going to introduce three abstraction
types. 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 $\approx_{\,\textit{abs\_res}}$
and $\approx_{\,\textit{abs\_list}}$). We can show that these relations are equivalence
relations and equivariant.
\begin{lemma}\label{alphaeq}
The relations $\approx_{\,\textit{abs\_set}}$, $\approx_{\,\textit{abs\_list}}$
and $\approx_{\,\textit{abs\_res}}$ are equivalence relations, and if @{term
"abs_set (as, x) (bs, y)"} then also @{term "abs_set (p \<bullet> as, p \<bullet> x) (p \<bullet>
bs, p \<bullet> y)"} (similarly for the other two relations).
\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 conditions are then by simple
calculations.
\end{proof}
\noindent
This lemma allows us to use our quotient package for introducing
new types @{text "\<beta> abs_set"}, @{text "\<beta> abs_res"} and @{text "\<beta> abs_list"}
representing $\alpha$-equivalence classes of pairs of type
@{text "(atom set) \<times> \<beta>"} (in the first two cases) and of type @{text "(atom list) \<times> \<beta>"}
(in the third case).
The elements in these types will be, respectively, written as:
\begin{center}
@{term "Abs as x"} \hspace{5mm}
@{term "Abs_res as x"} \hspace{5mm}
@{term "Abs_lst as x"}
\end{center}
\noindent
indicating that a set (or list) of atoms @{text as} is abstracted in @{text x}. We will
call the types \emph{abstraction types} and their elements
\emph{abstractions}. The important property we need to derive is the support of
abstractions, namely:
\begin{thm}[Support of Abstractions]\label{suppabs}
Assuming @{text x} has finite support, then\\[-6mm]
\begin{center}
\begin{tabular}{l@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
@{thm (lhs) supp_abs(1)[no_vars]} & $=$ & @{thm (rhs) supp_abs(1)[no_vars]}\\
@{thm (lhs) supp_abs(2)[no_vars]} & $=$ & @{thm (rhs) supp_abs(2)[no_vars]}\\
@{thm (lhs) supp_abs(3)[where bs="as", no_vars]} & $=$ & @{thm (rhs) supp_abs(3)[where bs="as", no_vars]}
\end{tabular}
\end{center}
\end{thm}
\noindent
Below we will show the first equation. The others
follow by similar arguments. By definition of the abstraction type @{text "abs_set"}
we have
%
\begin{equation}\label{abseqiff}
@{thm (lhs) abs_eq_iff(1)[where bs="as" and cs="bs", no_vars]} \;\;\text{if and only if}\;\;
@{thm (rhs) abs_eq_iff(1)[where bs="as" and cs="bs", no_vars]}
\end{equation}
\noindent
and also
%
\begin{equation}\label{absperm}
@{thm permute_Abs[no_vars]}
\end{equation}
\noindent
The second fact derives from the definition of permutations acting on pairs
\eqref{permute} and $\alpha$-equivalence being equivariant
(see Lemma~\ref{alphaeq}). With these two facts at our disposal, we can show
the following lemma about swapping two atoms in an abstraction.
\begin{lemma}
@{thm[mode=IfThen] abs_swap1(1)[where bs="as", no_vars]}
\end{lemma}
\begin{proof}
This lemma is straightforward using \eqref{abseqiff} and observing that
the assumptions give us @{term "(a \<rightleftharpoons> b) \<bullet> (supp x - as) = (supp x - as)"}.
Moreover @{text supp} and set difference are equivariant (see \cite{HuffmanUrban10}).
\end{proof}
\noindent
Assuming that @{text "x"} has finite support, this lemma together
with \eqref{absperm} allows us to show
%
\begin{equation}\label{halfone}
@{thm abs_supports(1)[no_vars]}
\end{equation}
\noindent
which by Property~\ref{supportsprop} gives us ``one half'' of
Theorem~\ref{suppabs}. The ``other half'' is a bit more involved. To establish
it, we use a trick from \cite{Pitts04} and first define an auxiliary
function @{text aux}, taking an abstraction as argument:
%
\begin{center}
@{thm supp_gen.simps[THEN eq_reflection, no_vars]}
\end{center}
\noindent
Using the second equation in \eqref{equivariance}, we can show that
@{text "aux"} is equivariant (since @{term "p \<bullet> (supp x - as) =
(supp (p \<bullet> x)) - (p \<bullet> as)"}) and therefore has empty support.
This in turn means
%
\begin{center}
@{term "supp (supp_gen (Abs as x)) \<subseteq> supp (Abs as x)"}
\end{center}
\noindent
using \eqref{suppfun}. Assuming @{term "supp x - as"} is a finite set,
we further obtain
%
\begin{equation}\label{halftwo}
@{thm (concl) supp_abs_subset1(1)[no_vars]}
\end{equation}
\noindent
since for finite sets of atoms, @{text "bs"}, we have
@{thm (concl) supp_finite_atom_set[where S="bs", no_vars]}.
Finally, taking \eqref{halfone} and \eqref{halftwo} together establishes
Theorem~\ref{suppabs}.
The method of first considering abstractions of the
form @{term "Abs as x"} etc is motivated by the fact that
we can conveniently establish at the Isabelle/HOL level
properties about them. It would be
laborious to write custom ML-code that derives automatically such properties
for every term-constructor that binds some atoms. Also the generality of
the definitions for $\alpha$-equivalence will help us in the next section.
*}
section {* Specifying General Bindings\label{sec:spec} *}
text {*
Our choice of syntax for specifications is influenced by the existing
datatype package of Isabelle/HOL \cite{Berghofer99} and by the syntax of the
Ott-tool \cite{ott-jfp}. For us a specification of a term-calculus is a
collection of (possibly mutual recursive) type declarations, say @{text
"ty\<AL>\<^isub>1, \<dots>, ty\<AL>\<^isub>n"}, and an associated collection of
binding functions, say @{text "bn\<AL>\<^isub>1, \<dots>, bn\<AL>\<^isub>m"}. The
syntax in Nominal Isabelle for such specifications is roughly 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\<AL>\<^isub>1 = \<dots>"}\\
\isacommand{and} @{text "ty\<AL>\<^isub>2 = \<dots>"}\\
$\ldots$\\
\isacommand{and} @{text "ty\<AL>\<^isub>n = \<dots>"}\\
\end{tabular}}
\end{cases}$\\
binding \mbox{function part} &
$\begin{cases}
\mbox{\begin{tabular}{l}
\isacommand{binder} @{text "bn\<AL>\<^isub>1"} \isacommand{and} \ldots \isacommand{and} @{text "bn\<AL>\<^isub>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 be specified with
\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) can be 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} of @{text "C\<^sup>\<alpha>"}.
%The types of such recursive
%arguments need to satisfy a ``positivity''
%restriction, which ensures that the type has a set-theoretic semantics
%\cite{Berghofer99}.
The labels
annotated on the types are optional. Their purpose is to be used in the
(possibly empty) list of \emph{binding clauses}, which indicate the binders
and their scope in a term-constructor. They come in three \emph{modes}:
\begin{center}
\begin{tabular}{l}
\isacommand{bind}\; {\it binders}\; \isacommand{in}\; {\it bodies}\\
\isacommand{bind\_set}\; {\it binders}\; \isacommand{in}\; {\it bodies}\\
\isacommand{bind\_res}\; {\it binders}\; \isacommand{in}\; {\it bodies}\\
\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). As indicated, the labels in the ``\isacommand{in}-part'' of a binding
clause will be called \emph{bodies}; the
``\isacommand{bind}-part'' will be called \emph{binders}. In contrast to
Ott, we allow multiple labels in binders and bodies. For example we allow
binding clauses of the form:
\begin{center}
\begin{tabular}{@ {}ll@ {}}
@{text "Foo\<^isub>1 x::name y::name t::trm s::trm"} &
\isacommand{bind} @{text "x y"} \isacommand{in} @{text "t s"}\\
@{text "Foo\<^isub>2 x::name y::name t::trm s::trm"} &
\isacommand{bind} @{text "x y"} \isacommand{in} @{text "t"},
\isacommand{bind} @{text "x y"} \isacommand{in} @{text "s"}\\
\end{tabular}
\end{center}
\noindent
Similarly for the other binding modes.
%Interestingly, in case of \isacommand{bind\_set}
%and \isacommand{bind\_res} the binding clauses above will make a difference to the semantics
%of the specifications (the corresponding $\alpha$-equivalence will differ). We will
%show this later with an example.
There are also some restrictions we need to impose on our binding clauses in comparison to
the ones of Ott. The
main idea behind these restrictions is that we obtain a sensible notion of
$\alpha$-equivalence where it is ensured that within a given scope an
atom occurrence cannot be both bound and free at the same time. The first
restriction is that a body can only occur in
\emph{one} binding clause of a term constructor (this ensures that the bound
atoms of a body cannot be free at the same time by specifying an
alternative binder for the same body). For binders we distinguish between
\emph{shallow} and \emph{deep} binders. Shallow binders are just
labels. The restriction we need to impose on them is that in case of
\isacommand{bind\_set} and \isacommand{bind\_res} the labels must either
refer to atom types or to sets of atom types; in case of \isacommand{bind}
the labels must refer to atom types or lists of atom types. Two examples for
the use of shallow binders are the specification of lambda-terms, where a
single name is bound, and type-schemes, where a finite set of names is
bound:
\begin{center}
\begin{tabular}{@ {}cc@ {}}
\begin{tabular}{@ {}l@ {\hspace{-1mm}}}
\isacommand{nominal\_datatype} @{text lam} $=$\\
\hspace{5mm}\phantom{$\mid$}~@{text "Var name"}\\
\hspace{5mm}$\mid$~@{text "App lam lam"}\\
\hspace{5mm}$\mid$~@{text "Lam x::name t::lam"}\\
\hspace{21mm}\isacommand{bind} @{text x} \isacommand{in} @{text t}\\
\end{tabular} &
\begin{tabular}{@ {}l@ {}}
\isacommand{nominal\_datatype}~@{text ty} $=$\\
\hspace{5mm}\phantom{$\mid$}~@{text "TVar name"}\\
\hspace{5mm}$\mid$~@{text "TFun ty ty"}\\
\isacommand{and}~@{text "tsc = All xs::(name fset) T::ty"}\\
\hspace{24mm}\isacommand{bind\_res} @{text xs} \isacommand{in} @{text T}\\
\end{tabular}
\end{tabular}
\end{center}
\noindent
In these specifications @{text "name"} refers to an atom type, and @{text
"fset"} to the type of finite sets.
Note that for @{text lam} it does not matter which binding mode we use. The
reason is that we bind only a single @{text name}. However, having
\isacommand{bind\_set} or \isacommand{bind} in the second case makes a
difference to the semantics of the specification (which we will define in the next section).
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 recursion over the corresponding type; the equations
must be given in the binding function part of the scheme shown in
\eqref{scheme}. For example a term-calculus containing @{text "Let"}s with
tuple patterns might be specified as:
%
\begin{equation}\label{letpat}
\mbox{%
\begin{tabular}{l}
\isacommand{nominal\_datatype} @{text trm} =\\
\hspace{5mm}\phantom{$\mid$}~@{term "Var name"}\\
\hspace{5mm}$\mid$~@{term "App trm trm"}\\
\hspace{5mm}$\mid$~@{text "Lam x::name t::trm"}
\;\;\isacommand{bind} @{text x} \isacommand{in} @{text t}\\
\hspace{5mm}$\mid$~@{text "Let p::pat trm t::trm"}
\;\;\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t}\\
\isacommand{and} @{text pat} =\\
\hspace{5mm}\phantom{$\mid$}~@{text PNil}\\
\hspace{5mm}$\mid$~@{text "PVar name"}\\
\hspace{5mm}$\mid$~@{text "PTup pat pat"}\\
\isacommand{binder}~@{text "bn::pat \<Rightarrow> atom list"}\\
\isacommand{where}~@{text "bn(PNil) = []"}\\
\hspace{5mm}$\mid$~@{text "bn(PVar x) = [atom x]"}\\
\hspace{5mm}$\mid$~@{text "bn(PTup p\<^isub>1 p\<^isub>2) = bn(p\<^isub>1) @ bn(p\<^isub>2)"}\smallskip\\
\end{tabular}}
\end{equation}
\noindent
In this specification the function @{text "bn"} determines which atoms of
the pattern @{text p} are bound in the argument @{text "t"}. Note that in the
second-last @{text bn}-clause the function @{text "atom"} coerces a name
into the generic atom type of Nominal Isabelle \cite{HuffmanUrban10}. This
allows us to treat binders of different atom type uniformly.
As said above, for deep binders we allow binding clauses such as
%
\begin{center}
\begin{tabular}{ll}
@{text "Bar p::pat t::trm"} &
\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text "p t"}\\
\end{tabular}
\end{center}
\noindent
where the argument of the deep binder also occurs in the body. We call such
binders \emph{recursive}. To see the purpose of such recursive binders,
compare ``plain'' @{text "Let"}s and @{text "Let_rec"}s in the following
specification:
%
\begin{equation}\label{letrecs}
\mbox{%
\begin{tabular}{@ {}l@ {}}
\isacommand{nominal\_datatype}~@{text "trm ="}\\
\hspace{5mm}\phantom{$\mid$}\ldots\\
\hspace{5mm}$\mid$~@{text "Let as::assn t::trm"}
\;\;\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text t}\\
\hspace{5mm}$\mid$~@{text "Let_rec as::assn t::trm"}
\;\;\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text "as t"}\\
\isacommand{and} @{text "ass"} =\\
\hspace{5mm}\phantom{$\mid$}~@{text "ANil"}\\
\hspace{5mm}$\mid$~@{text "ACons name trm assn"}\\
\isacommand{binder} @{text "bn::assn \<Rightarrow> atom list"}\\
\isacommand{where}~@{text "bn(ANil) = []"}\\
\hspace{5mm}$\mid$~@{text "bn(ACons a t as) = [atom a] @ bn(as)"}\\
\end{tabular}}
\end{equation}
\noindent
The difference is that with @{text Let} we only want to bind the atoms @{text
"bn(as)"} in the term @{text t}, but with @{text "Let_rec"} we also want to bind the atoms
inside the assignment. This difference has consequences for the associated
notions of free-atoms and $\alpha$-equivalence.
To make sure that atoms bound by deep binders cannot be free at the
same time, we cannot have more than one binding function for a deep binder.
Consequently we exclude specifications such as
\begin{center}
\begin{tabular}{@ {}l@ {\hspace{2mm}}l@ {}}
@{text "Baz\<^isub>1 p::pat t::trm"} &
\isacommand{bind} @{text "bn\<^isub>1(p) bn\<^isub>2(p)"} \isacommand{in} @{text t}\\
@{text "Baz\<^isub>2 p::pat t\<^isub>1::trm t\<^isub>2::trm"} &
\isacommand{bind} @{text "bn\<^isub>1(p)"} \isacommand{in} @{text "t\<^isub>1"},
\isacommand{bind} @{text "bn\<^isub>2(p)"} \isacommand{in} @{text "t\<^isub>2"}\\
\end{tabular}
\end{center}
\noindent
Otherwise it is possible that @{text "bn\<^isub>1"} and @{text "bn\<^isub>2"} pick
out different atoms to become bound, respectively be free, in @{text "p"}.
(Since the Ott-tool does not derive a reasoning infrastructure for
$\alpha$-equated terms, it can permit such specifications.)
We also need to restrict the form of the binding functions in order
to ensure the @{text "bn"}-functions can be defined for $\alpha$-equated
terms. The main restriction is that we cannot return an atom in a binding function that is also
bound in the corresponding term-constructor. That means in \eqref{letpat}
that the term-constructors @{text PVar} and @{text PTup} may
not have a binding clause (all arguments are used to define @{text "bn"}).
In contrast, in case of \eqref{letrecs} the term-constructor @{text ACons}
may have a binding clause involving the argument @{text t} (the only one that
is \emph{not} used in the definition of the binding function). This restriction
is sufficient for having the binding function over $\alpha$-equated terms.
In the version of
Nominal Isabelle described here, we also adopted the restriction from the
Ott-tool that binding functions can only return: the empty set or empty list
(as in case @{text PNil}), a singleton set or singleton list containing an
atom (case @{text PVar}), or unions of atom sets or appended atom lists
(case @{text PTup}). This restriction will simplify some automatic definitions and proofs
later on.
In order to simplify our definitions of free atoms and $\alpha$-equivalence,
we shall assume specifications
of term-calculi are implicitly \emph{completed}. By this we mean that
for every argument of a term-constructor that is \emph{not}
already part of a binding clause given by the user, we add implicitly a special \emph{empty} binding
clause, written \isacommand{bind}~@{term "{}"}~\isacommand{in}~@{text "labels"}. In case
of the lambda-calculus, the completion produces
\begin{center}
\begin{tabular}{@ {}l@ {\hspace{-1mm}}}
\isacommand{nominal\_datatype} @{text lam} =\\
\hspace{5mm}\phantom{$\mid$}~@{text "Var x::name"}
\;\;\isacommand{bind}~@{term "{}"}~\isacommand{in}~@{text "x"}\\
\hspace{5mm}$\mid$~@{text "App t\<^isub>1::lam t\<^isub>2::lam"}
\;\;\isacommand{bind}~@{term "{}"}~\isacommand{in}~@{text "t\<^isub>1 t\<^isub>2"}\\
\hspace{5mm}$\mid$~@{text "Lam x::name t::lam"}
\;\;\isacommand{bind}~@{text x} \isacommand{in} @{text t}\\
\end{tabular}
\end{center}
\noindent
The point of completion is that we can make definitions over the binding
clauses and be sure to have captured all arguments of a term constructor.
*}
section {* Alpha-Equivalence and Free Atoms\label{sec:alpha} *}
text {*
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. As
Pottier and Cheney pointed out \cite{Pottier06,Cheney05}, just
re-arranging the arguments of
term-constructors so that binders and their bodies are next to each other will
result in inadequate representations in cases like @{text "Let x\<^isub>1 = t\<^isub>1\<dots>x\<^isub>n = t\<^isub>n in s"}.
Therefore we will first
extract ``raw'' datatype definitions from the specification and then define
explicitly an $\alpha$-equivalence relation over them. We subsequently
quotient the datatypes according to our $\alpha$-equivalence.
The ``raw'' datatype definition can be obtained by stripping off the
binding clauses and the labels from 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 the user. In our implementation we just use the affix ``@{text "_raw"}''.
But for the purpose of this paper, we use the superscript @{text "_\<^sup>\<alpha>"} to indicate
that a notion is defined over $\alpha$-equivalence classes and leave it out
for the corresponding notion defined on the ``raw'' level. So for example
we have
\begin{center}
@{text "ty\<^sup>\<alpha> \<mapsto> ty"} \hspace{2mm}and\hspace{2mm} @{text "C\<^sup>\<alpha> \<mapsto> C"}
\end{center}
\noindent
where @{term ty} is the type used in the quotient construction for
@{text "ty\<^sup>\<alpha>"} and @{text "C"} is the term-constructor on the ``raw'' type @{text "ty"}.
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{Berghofer99} for an in-depth description of the datatype package
in Isabelle/HOL). We subsequently define each of the user-specified binding
functions @{term "bn"}$_{1..m}$ by recursion over the corresponding
raw datatype. We can also easily define permutation operations by
recursion so that for each term constructor @{text "C"} we have that
%
\begin{equation}\label{ceqvt}
@{text "p \<bullet> (C z\<^isub>1 \<dots> z\<^isub>n) = C (p \<bullet> z\<^isub>1) \<dots> (p \<bullet> z\<^isub>n)"}
\end{equation}
The first non-trivial step we have to perform is the generation of
free-atom functions from the specification. For the
\emph{raw} types @{text "ty"}$_{1..n}$ we define the free-atom functions
%
\begin{equation}\label{fvars}
@{text "fa_ty\<^isub>1, \<dots>, fa_ty\<^isub>n"}
\end{equation}
\noindent
by mutual recursion.
We define these functions together with auxiliary free-atom functions for
the binding functions. Given raw binding functions @{text "bn"}$_{1..m}$
we define
%
\begin{center}
@{text "fa_bn\<^isub>1, \<dots>, fa_bn\<^isub>m"}
\end{center}
\noindent
The reason for this setup is that in a deep binder not all atoms have to be
bound, as we saw in the example with ``plain'' @{text Let}s. We need therefore a function
that calculates those free atoms in a deep binder.
While the idea behind these free-atom functions is clear (they just
collect all atoms that are not bound), because of our rather complicated
binding mechanisms their definitions are somewhat involved. Given
a term-constructor @{text "C"} of type @{text ty} and some associated
binding clauses @{text "bc\<^isub>1\<dots>bc\<^isub>k"}, the result of @{text
"fa_ty (C z\<^isub>1 \<dots> z\<^isub>n)"} will be the union @{text
"fa(bc\<^isub>1) \<union> \<dots> \<union> fa(bc\<^isub>k)"} where we will define below what @{text "fa"} for a binding
clause means. We only show the details for the mode \isacommand{bind\_set} (the other modes are similar).
Suppose the binding clause @{text bc\<^isub>i} is of the form
%
\begin{center}
\mbox{\isacommand{bind\_set} @{text "b\<^isub>1\<dots>b\<^isub>p"} \isacommand{in} @{text "d\<^isub>1\<dots>d\<^isub>q"}}
\end{center}
\noindent
in which the body-labels @{text "d"}$_{1..q}$ refer to types @{text ty}$_{1..q}$,
and the binders @{text b}$_{1..p}$
either refer to labels of atom types (in case of shallow binders) or to binding
functions taking a single label as argument (in case of deep binders). Assuming
@{text "D"} stands for the set of free atoms of the bodies, @{text B} for the
set of binding atoms in the binders and @{text "B'"} for the set of free atoms in
non-recursive deep binders,
then the free atoms of the binding clause @{text bc\<^isub>i} are
%
\begin{equation}\label{fadef}
\mbox{@{text "fa(bc\<^isub>i) \<equiv> (D - B) \<union> B'"}}.
\end{equation}
\noindent
The set @{text D} is formally defined as
%
\begin{center}
@{text "D \<equiv> fa_ty\<^isub>1 d\<^isub>1 \<union> ... \<union> fa_ty\<^isub>q d\<^isub>q"}
\end{center}
\noindent
where in case @{text "d\<^isub>i"} refers to one of the raw types @{text "ty"}$_{1..n}$ from the
specification, the function @{text "fa_ty\<^isub>i"} is the corresponding free-atom function
we are defining by recursion
(see \eqref{fvars}); otherwise we set @{text "fa_ty\<^isub>i d\<^isub>i = supp d\<^isub>i"}.
In order to formally define the set @{text B} we use the following auxiliary @{text "bn"}-functions
for atom types to which shallow binders may refer
%
\begin{center}
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
@{text "bn_atom a"} & @{text "\<equiv>"} & @{text "{atom a}"}\\
@{text "bn_atom_set as"} & @{text "\<equiv>"} & @{text "atoms as"}\\
@{text "bn_atom_list as"} & @{text "\<equiv>"} & @{text "atoms (set as)"}
\end{tabular}
\end{center}
\noindent
Like the function @{text atom}, the function @{text "atoms"} coerces
a set of atoms to a set of the generic atom type. It is defined as
@{text "atoms as \<equiv> {atom a | a \<in> as}"}.
The set @{text B} is then formally defined as
%
\begin{center}
@{text "B \<equiv> bn_ty\<^isub>1 b\<^isub>1 \<union> ... \<union> bn_ty\<^isub>p b\<^isub>p"}
\end{center}
\noindent
where we use the auxiliary binding functions for shallow binders.
The set @{text "B'"} collects all free atoms in non-recursive deep
binders. Let us assume these binders in @{text "bc\<^isub>i"} are
%
\begin{center}
@{text "bn\<^isub>1 l\<^isub>1, \<dots>, bn\<^isub>r l\<^isub>r"}
\end{center}
\noindent
with @{text "l"}$_{1..r}$ $\subseteq$ @{text "b"}$_{1..p}$ and none of the
@{text "l"}$_{1..r}$ being among the bodies @{text
"d"}$_{1..q}$. The set @{text "B'"} is defined as
%
\begin{center}
@{text "B' \<equiv> fa_bn\<^isub>1 l\<^isub>1 \<union> ... \<union> fa_bn\<^isub>r l\<^isub>r"}
\end{center}
\noindent
This completes the definition of the free-atom functions @{text "fa_ty"}$_{1..n}$.
Note that for non-recursive deep binders, we have to add in \eqref{fadef}
the set of atoms that are left unbound by the binding functions @{text
"bn"}$_{1..m}$. We used for the definition of
this set the functions @{text "fa_bn"}$_{1..m}$, which are also defined by mutual
recursion. Assume the user specified a @{text bn}-clause of the form
%
\begin{center}
@{text "bn (C z\<^isub>1 \<dots> z\<^isub>s) = rhs"}
\end{center}
\noindent
where the @{text "z"}$_{1..s}$ are of types @{text "ty"}$_{1..s}$. For each of
the arguments we calculate the free atoms as follows:
%
\begin{center}
\begin{tabular}{c@ {\hspace{2mm}}p{7cm}}
$\bullet$ & @{term "fa_ty\<^isub>i z\<^isub>i"} provided @{text "z\<^isub>i"} does not occur in @{text "rhs"}
(that means nothing is bound in @{text "z\<^isub>i"} by the binding function),\\
$\bullet$ & @{term "fa_bn\<^isub>i z\<^isub>i"} provided @{text "z\<^isub>i"} occurs in @{text "rhs"}
with the recursive call @{text "bn\<^isub>i z\<^isub>i"}, and\\
$\bullet$ & @{term "{}"} provided @{text "z\<^isub>i"} occurs in @{text "rhs"},
but without a recursive call.
\end{tabular}
\end{center}
\noindent
For defining @{text "fa_bn (C z\<^isub>1 \<dots> z\<^isub>n)"} we just union up all these sets.
To see how these definitions work in practice, let us reconsider the
term-constructors @{text "Let"} and @{text "Let_rec"} shown in
\eqref{letrecs} together with the term-constructors for assignments @{text
"ANil"} and @{text "ACons"}. Since there is a binding function defined for
assignments, we have three free-atom functions, namely @{text
"fa\<^bsub>trm\<^esub>"}, @{text "fa\<^bsub>assn\<^esub>"} and @{text
"fa\<^bsub>bn\<^esub>"} as follows:
%
\begin{center}
\begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}}
@{text "fa\<^bsub>trm\<^esub> (Let as t)"} & @{text "="} & @{text "(fa\<^bsub>trm\<^esub> t - set (bn as)) \<union> fa\<^bsub>bn\<^esub> as"}\\
@{text "fa\<^bsub>trm\<^esub> (Let_rec as t)"} & @{text "="} & @{text "(fa\<^bsub>assn\<^esub> as \<union> fa\<^bsub>trm\<^esub> t) - set (bn as)"}\\[1mm]
@{text "fa\<^bsub>assn\<^esub> (ANil)"} & @{text "="} & @{term "{}"}\\
@{text "fa\<^bsub>assn\<^esub> (ACons a t as)"} & @{text "="} & @{text "(supp a) \<union> (fa\<^bsub>trm\<^esub> t) \<union> (fa\<^bsub>assn\<^esub> as)"}\\[1mm]
@{text "fa\<^bsub>bn\<^esub> (ANil)"} & @{text "="} & @{term "{}"}\\
@{text "fa\<^bsub>bn\<^esub> (ACons a t as)"} & @{text "="} & @{text "(fa\<^bsub>trm\<^esub> t) \<union> (fa\<^bsub>bn\<^esub> as)"}
\end{tabular}
\end{center}
\noindent
Recall that @{text ANil} and @{text "ACons"} have no
binding clause in the specification. The corresponding free-atom
function @{text "fa\<^bsub>assn\<^esub>"} therefore returns all free atoms
occurring in an assignment (in case of @{text "ACons"}, they are given in
terms of @{text supp}, @{text "fa\<^bsub>trm\<^esub>"} and @{text "fa\<^bsub>assn\<^esub>"}).
The binding only takes place in @{text Let} and
@{text "Let_rec"}. In case of @{text "Let"}, the binding clause specifies
that all atoms given by @{text "set (bn as)"} have to be bound in @{text
t}. Therefore we have to subtract @{text "set (bn as)"} from @{text
"fa\<^bsub>trm\<^esub> t"}. However, we also need to add all atoms that are
free in @{text "as"}. This is
in contrast with @{text "Let_rec"} where we have a recursive
binder to bind all occurrences of the atoms in @{text
"set (bn as)"} also inside @{text "as"}. Therefore we have to subtract
@{text "set (bn as)"} from both @{text "fa\<^bsub>trm\<^esub> t"} and @{text "fa\<^bsub>assn\<^esub> as"}.
Like the function @{text "bn"}, the function @{text "fa\<^bsub>bn\<^esub>"} traverses the
list of assignments, but instead returns the free atoms, which means in this
example the free atoms in the argument @{text "t"}.
An interesting point in this
example is that a ``naked'' assignment (@{text "ANil"} or @{text "ACons"}) does not bind any
atoms, even if the binding function is specified over assignments.
Only in the context of a @{text Let} or @{text "Let_rec"}, where the binding clauses are given, will
some atoms actually become bound. This is a phenomenon that has also been pointed
out in \cite{ott-jfp}. For us this observation is crucial, because we would
not be able to lift the @{text "bn"}-functions to $\alpha$-equated terms if they act on
atoms that are bound. In that case, these functions would \emph{not} respect
$\alpha$-equivalence.
Next we define the $\alpha$-equivalence relations for the raw types @{text
"ty"}$_{1..n}$ from the specification. We write them as
%
\begin{center}
@{text "\<approx>ty\<^isub>1, \<dots>, \<approx>ty\<^isub>n"}.
\end{center}
\noindent
Like with the free-atom functions, we also need to
define auxiliary $\alpha$-equivalence relations
%
\begin{center}
@{text "\<approx>bn\<^isub>1, \<dots>, \<approx>bn\<^isub>m"}
\end{center}
\noindent
for the binding functions @{text "bn"}$_{1..m}$,
To simplify our definitions we will use the following abbreviations for
\emph{compound equivalence relations} and \emph{compound free-atom functions} acting on tuples
%
\begin{center}
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
@{text "(x\<^isub>1,.., x\<^isub>n) (R\<^isub>1,.., R\<^isub>n) (x\<PRIME>\<^isub>1,.., x\<PRIME>\<^isub>n)"} & @{text "\<equiv>"} & \\
\multicolumn{3}{r}{@{text "x\<^isub>1 R\<^isub>1 x\<PRIME>\<^isub>1 \<and> .. \<and> x\<^isub>n R\<^isub>n x\<PRIME>\<^isub>n"}}\\
@{text "(fa\<^isub>1,.., fa\<^isub>n) (x\<^isub>1,.., x\<^isub>n)"} & @{text "\<equiv>"} & @{text "fa\<^isub>1 x\<^isub>1 \<union> .. \<union> fa\<^isub>n x\<^isub>n"}\\
\end{tabular}
\end{center}
The $\alpha$-equivalence relations are defined as inductive predicates
having a single clause for each term-constructor. Assuming a
term-constructor @{text C} is of type @{text ty} and has the binding clauses
@{term "bc"}$_{1..k}$, then the $\alpha$-equivalence clause has the form
%
\begin{center}
\mbox{\infer{@{text "C z\<^isub>1 \<dots> z\<^isub>n \<approx>ty C z\<PRIME>\<^isub>1 \<dots> z\<PRIME>\<^isub>n"}}
{@{text "prems(bc\<^isub>1) \<dots> prems(bc\<^isub>k)"}}}
\end{center}
\noindent
The task below is to specify what the premises of a binding clause are. As a
special instance, we first treat the case where @{text "bc\<^isub>i"} is the
empty binding clause of the form
%
\begin{center}
\mbox{\isacommand{bind\_set} @{term "{}"} \isacommand{in} @{text "d\<^isub>1\<dots>d\<^isub>q"}.}
\end{center}
\noindent
In this binding clause no atom is bound and we only have to $\alpha$-relate the bodies. For this
we build first the tuples @{text "D \<equiv> (d\<^isub>1,\<dots>, d\<^isub>q)"} and @{text "D' \<equiv> (d\<PRIME>\<^isub>1,\<dots>, d\<PRIME>\<^isub>q)"}
whereby the labels @{text "d"}$_{1..q}$ refer to arguments @{text "z"}$_{1..n}$ and
respectively @{text "d\<PRIME>"}$_{1..q}$ to @{text "z\<PRIME>"}$_{1..n}$. In order to relate
two such tuples we define the compound $\alpha$-equivalence relation @{text "R"} as follows
%
\begin{equation}\label{rempty}
\mbox{@{text "R \<equiv> (R\<^isub>1,\<dots>, R\<^isub>q)"}}
\end{equation}
\noindent
with @{text "R\<^isub>i"} being @{text "\<approx>ty\<^isub>i"} if the corresponding labels @{text "d\<^isub>i"} and
@{text "d\<PRIME>\<^isub>i"} refer
to a recursive argument of @{text C} with type @{text "ty\<^isub>i"}; otherwise
we take @{text "R\<^isub>i"} to be the equality @{text "="}. This lets us define
the premise for an empty binding clause succinctly as @{text "prems(bc\<^isub>i) \<equiv> D R D'"},
which can be unfolded to the series of premises
%
\begin{center}
@{text "d\<^isub>1 R\<^isub>1 d\<PRIME>\<^isub>1 \<dots> d\<^isub>q R\<^isub>q d\<PRIME>\<^isub>q"}
\end{center}
\noindent
We will use the unfolded version in the examples below.
Now suppose the binding clause @{text "bc\<^isub>i"} is of the general form
%
\begin{equation}\label{nonempty}
\mbox{\isacommand{bind\_set} @{text "b\<^isub>1\<dots>b\<^isub>p"} \isacommand{in} @{text "d\<^isub>1\<dots>d\<^isub>q"}.}
\end{equation}
\noindent
In this case we define a premise @{text P} using the relation
$\approx_{\,\textit{set}}$ given in Section~\ref{sec:binders} (similarly
$\approx_{\,\textit{res}}$ and $\approx_{\,\textit{list}}$ for the other
binding modes). This premise defines $\alpha$-equivalence of two abstractions
involving multiple binders. As above, we first build the tuples @{text "D"} and
@{text "D'"} for the bodies @{text "d"}$_{1..q}$, and the corresponding
compound $\alpha$-relation @{text "R"} (shown in \eqref{rempty}).
For $\approx_{\,\textit{set}}$ we also need
a compound free-atom function for the bodies defined as
%
\begin{center}
\mbox{@{text "fa \<equiv> (fa_ty\<^isub>1,\<dots>, fa_ty\<^isub>q)"}}
\end{center}
\noindent
with the assumption that the @{text "d"}$_{1..q}$ refer to arguments of types @{text "ty"}$_{1..q}$.
The last ingredient we need are the sets of atoms bound in the bodies.
For this we take
\begin{center}
@{text "B \<equiv> bn_ty\<^isub>1 b\<^isub>1 \<union> \<dots> \<union> bn_ty\<^isub>p b\<^isub>p"}\;.\\
\end{center}
\noindent
Similarly for @{text "B'"} using the labels @{text "b\<PRIME>"}$_{1..p}$. This
lets us formally define the premise @{text P} for a non-empty binding clause as:
%
\begin{center}
\mbox{@{term "P \<equiv> \<exists>p. (B, D) \<approx>gen R fa p (B', D')"}}\;.
\end{center}
\noindent
This premise accounts for $\alpha$-equivalence of the bodies of the binding
clause.
However, in case the binders have non-recursive deep binders, this premise
is not enough:
we also have to ``propagate'' $\alpha$-equivalence inside the structure of
these binders. An example is @{text "Let"} where we have to make sure the
right-hand sides of assignments are $\alpha$-equivalent. For this we use
relations @{text "\<approx>bn"}$_{1..m}$ (which we will formally define shortly).
Let us assume the non-recursive deep binders in @{text "bc\<^isub>i"} are
%
\begin{center}
@{text "bn\<^isub>1 l\<^isub>1, \<dots>, bn\<^isub>r l\<^isub>r"}.
\end{center}
\noindent
The tuple @{text L} is then @{text "(l\<^isub>1,\<dots>,l\<^isub>r)"} (similarly @{text "L'"})
and the compound equivalence relation @{text "R'"} is @{text "(\<approx>bn\<^isub>1,\<dots>,\<approx>bn\<^isub>r)"}.
All premises for @{text "bc\<^isub>i"} are then given by
%
\begin{center}
@{text "prems(bc\<^isub>i) \<equiv> P \<and> L R' L'"}
\end{center}
\noindent
The auxiliary $\alpha$-equivalence relations @{text "\<approx>bn"}$_{1..m}$
in @{text "R'"} are defined as follows: assuming a @{text bn}-clause is of the form
%
\begin{center}
@{text "bn (C z\<^isub>1 \<dots> z\<^isub>s) = rhs"}
\end{center}
\noindent
where the @{text "z"}$_{1..s}$ are of types @{text "ty"}$_{1..s}$,
then the corresponding $\alpha$-equivalence clause for @{text "\<approx>bn"} has the form
%
\begin{center}
\mbox{\infer{@{text "C z\<^isub>1 \<dots> z\<^isub>s \<approx>bn C z\<PRIME>\<^isub>1 \<dots> z\<PRIME>\<^isub>s"}}
{@{text "z\<^isub>1 R\<^isub>1 z\<PRIME>\<^isub>1 \<dots> z\<^isub>s R\<^isub>s z\<PRIME>\<^isub>s"}}}
\end{center}
\noindent
In this clause the relations @{text "R"}$_{1..s}$ are given by
\begin{center}
\begin{tabular}{c@ {\hspace{2mm}}p{7cm}}
$\bullet$ & @{text "z\<^isub>i \<approx>ty z\<PRIME>\<^isub>i"} provided @{text "z\<^isub>i"} does not occur in @{text rhs} and
is a recursive argument of @{text C},\\
$\bullet$ & @{text "z\<^isub>i = z\<PRIME>\<^isub>i"} provided @{text "z\<^isub>i"} does not occur in @{text rhs}
and is a non-recursive argument of @{text C},\\
$\bullet$ & @{text "z\<^isub>i \<approx>bn\<^isub>i z\<PRIME>\<^isub>i"} provided @{text "z\<^isub>i"} occurs in @{text rhs}
with the recursive call @{text "bn\<^isub>i x\<^isub>i"} and\\
$\bullet$ & @{text True} provided @{text "z\<^isub>i"} occurs in @{text rhs} but without a
recursive call.
\end{tabular}
\end{center}
\noindent
This completes the definition of $\alpha$-equivalence. As a sanity check, we can show
that the premises of empty binding clauses are a special case of the clauses for
non-empty ones (we just have to unfold the definition of $\approx_{\,\textit{set}}$ and take @{text "0"}
for the existentially quantified permutation).
Again let us take a look at a concrete example for these definitions. For \eqref{letrecs}
we have three relations $\approx_{\textit{trm}}$, $\approx_{\textit{assn}}$ and
$\approx_{\textit{bn}}$ with the following clauses:
\begin{center}
\begin{tabular}{@ {}c @ {}}
\infer{@{text "Let as t \<approx>\<^bsub>trm\<^esub> Let as' t'"}}
{@{term "\<exists>p. (bn as, t) \<approx>lst alpha_trm fa_trm p (bn as', t')"} & @{text "as \<approx>\<^bsub>bn\<^esub> as'"}}\smallskip\\
\makebox[0mm]{\infer{@{text "Let_rec as t \<approx>\<^bsub>trm\<^esub> Let_rec as' t'"}}
{@{term "\<exists>p. (bn as, ast) \<approx>lst alpha_trm2 fa_trm2 p (bn as', ast')"}}}
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{@ {}c @ {}}
\infer{@{text "ANil \<approx>\<^bsub>assn\<^esub> ANil"}}{}\smallskip\\
\infer{@{text "ACons a t as \<approx>\<^bsub>assn\<^esub> ACons a' t' as"}}
{@{text "a = a'"} & @{text "t \<approx>\<^bsub>trm\<^esub> t'"} & @{text "as \<approx>\<^bsub>assn\<^esub> as'"}}
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{@ {}c @ {}}
\infer{@{text "ANil \<approx>\<^bsub>bn\<^esub> ANil"}}{}\smallskip\\
\infer{@{text "ACons a t as \<approx>\<^bsub>bn\<^esub> ACons a' t' as"}}
{@{text "t \<approx>\<^bsub>trm\<^esub> t'"} & @{text "as \<approx>\<^bsub>bn\<^esub> as'"}}
\end{tabular}
\end{center}
\noindent
Note the difference between $\approx_{\textit{assn}}$ and
$\approx_{\textit{bn}}$: the latter only ``tracks'' $\alpha$-equivalence of
the components in an assignment that are \emph{not} bound. This is needed in the
in the clause for @{text "Let"} (which is has
a non-recursive binder). The underlying reason is that the terms inside an assignment are not meant
to be ``under'' the binder. Such a premise is \emph{not} needed in @{text "Let_rec"},
because there all components of an assignment are ``under'' the binder.
*}
section {* Establishing the Reasoning Infrastructure *}
text {*
Having made all necessary definitions for raw terms, we can start
with establishing the reasoning infrastructure for the $\alpha$-equated types
@{text "ty\<AL>"}$_{1..n}$, that is the types the user originally specified. We sketch
in this section the proofs we need for establishing this infrastructure. One
main point of our work is that we have completely automated these proofs in Isabelle/HOL.
First we establish that the
$\alpha$-equivalence relations defined in the previous section are
equivalence relations.
\begin{lemma}\label{equiv}
Given the raw types @{text "ty"}$_{1..n}$ and binding functions
@{text "bn"}$_{1..m}$, the relations @{text "\<approx>ty"}$_{1..n}$ and
@{text "\<approx>bn"}$_{1..m}$ are equivalence relations and equivariant.
\end{lemma}
\begin{proof}
The proof is by mutual induction over the definitions. The non-trivial
cases involve premises built up by $\approx_{\textit{set}}$,
$\approx_{\textit{res}}$ and $\approx_{\textit{list}}$. They
can be dealt with as in Lemma~\ref{alphaeq}.
\end{proof}
\noindent
We can feed this lemma into our quotient package and obtain new types @{text
"ty"}$^\alpha_{1..n}$ representing $\alpha$-equated terms of types @{text "ty"}$_{1..n}$.
We also obtain definitions for the term-constructors @{text
"C"}$^\alpha_{1..k}$ from the raw term-constructors @{text
"C"}$_{1..k}$, and similar definitions for the free-atom functions @{text
"fa_ty"}$^\alpha_{1..n}$ and @{text "fa_bn"}$^\alpha_{1..m}$ as well as the binding functions @{text
"bn"}$^\alpha_{1..m}$. However, these definitions are not really useful to the
user, since they are given in terms of the isomorphisms we obtained by
creating new types in Isabelle/HOL (recall the picture shown in the
Introduction).
The first useful property for the user is the fact that distinct
term-constructors are not
equal, that is
%
\begin{equation}\label{distinctalpha}
\mbox{@{text "C"}$^\alpha$~@{text "x\<^isub>1 \<dots> x\<^isub>r"}~@{text "\<noteq>"}~%
@{text "D"}$^\alpha$~@{text "y\<^isub>1 \<dots> y\<^isub>s"}}
\end{equation}
\noindent
whenever @{text "C"}$^\alpha$~@{text "\<noteq>"}~@{text "D"}$^\alpha$.
In order to derive this fact, we use the definition of $\alpha$-equivalence
and establish that
%
\begin{equation}\label{distinctraw}
\mbox{@{text "C x\<^isub>1 \<dots> x\<^isub>r"}\;$\not\approx$@{text ty}\;@{text "D y\<^isub>1 \<dots> y\<^isub>s"}}
\end{equation}
\noindent
holds for the corresponding raw term-constructors.
In order to deduce \eqref{distinctalpha} from \eqref{distinctraw}, our quotient
package needs to know that the raw term-constructors @{text "C"} and @{text "D"}
are \emph{respectful} w.r.t.~the $\alpha$-equivalence relations (see \cite{Homeier05}).
Assuming, for example, @{text "C"} is of type @{text "ty"} with argument types
@{text "ty"}$_{1..r}$, respectfulness amounts to showing that
%
\begin{center}
@{text "C x\<^isub>1 \<dots> x\<^isub>r \<approx>ty C x\<PRIME>\<^isub>1 \<dots> x\<PRIME>\<^isub>r"}
\end{center}
\noindent
holds under the assumptions that we have \mbox{@{text
"x\<^isub>i \<approx>ty\<^isub>i x\<PRIME>\<^isub>i"}} whenever @{text "x\<^isub>i"}
and @{text "x\<PRIME>\<^isub>i"} are recursive arguments of @{text C} and
@{text "x\<^isub>i = x\<PRIME>\<^isub>i"} whenever they are non-recursive arguments. We can prove this
implication by applying the corresponding rule in our $\alpha$-equivalence
definition and by establishing the following auxiliary facts
%
\begin{equation}\label{fnresp}
\mbox{%
\begin{tabular}{l}
@{text "x \<approx>ty\<^isub>i x\<PRIME>"}~~implies~@{text "fa_ty\<^isub>i x = fa_ty\<^isub>i x\<PRIME>"}\\
@{text "x \<approx>ty\<^isub>j x\<PRIME>"}~~implies~@{text "fa_bn\<^isub>j x = fa_bn\<^isub>j x\<PRIME>"}\\
@{text "x \<approx>ty\<^isub>j x\<PRIME>"}~~implies~@{text "bn\<^isub>j x = bn\<^isub>j x\<PRIME>"}\\
@{text "x \<approx>ty\<^isub>j x\<PRIME>"}~~implies~@{text "x \<approx>bn\<^isub>j x\<PRIME>"}\\
\end{tabular}}
\end{equation}
\noindent
They can be established by induction on @{text "\<approx>ty"}$_{1..n}$. Whereas the first,
second and last implication are true by how we stated our definitions, the
third \emph{only} holds because of our restriction
imposed on the form of the binding functions---namely \emph{not} returning
any bound atoms. In Ott, in contrast, the user may
define @{text "bn"}$_{1..m}$ so that they return bound
atoms and in this case the third implication is \emph{not} true. A
result is that the lifing of the corresponding binding functions in Ott to $\alpha$-equated
terms is impossible.
Having established respectfulness for the raw term-constructors, the
quotient package is able to automatically deduce \eqref{distinctalpha} from
\eqref{distinctraw}. Having the facts \eqref{fnresp} at our disposal, we can
also lift properties that characterise when two raw terms of the form
%
\begin{center}
@{text "C x\<^isub>1 \<dots> x\<^isub>r \<approx>ty C x\<PRIME>\<^isub>1 \<dots> x\<PRIME>\<^isub>r"}
\end{center}
\noindent
are $\alpha$-equivalent. This gives us conditions when the corresponding
$\alpha$-equated terms are \emph{equal}, namely
%
\begin{center}
@{text "C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r = C\<^sup>\<alpha> x\<PRIME>\<^isub>1 \<dots> x\<PRIME>\<^isub>r"}
\end{center}
\noindent
We call these conditions as \emph{quasi-injectivity}. They correspond to
the premises in our $\alpha$-equivalence relations.
Next we can lift the permutation
operations defined in \eqref{ceqvt}. In order to make this
lifting to go through, we have to show that the permutation operations are respectful.
This amounts to showing that the
$\alpha$-equivalence relations are equivariant, which we already established
in Lemma~\ref{equiv}. As a result we can add the equations
%
\begin{equation}\label{calphaeqvt}
@{text "p \<bullet> (C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r) = C\<^sup>\<alpha> (p \<bullet> x\<^isub>1) \<dots> (p \<bullet> x\<^isub>r)"}
\end{equation}
\noindent
to our infrastructure. In a similar fashion we can lift the defining equations
of the free-atom functions @{text "fn_ty\<AL>"}$_{1..n}$ and
@{text "fa_bn\<AL>"}$_{1..m}$ as well as of the binding functions @{text
"bn\<AL>"}$_{1..m}$ and the size functions @{text "size_ty\<AL>"}$_{1..n}$.
The latter are defined automatically for the raw types @{text "ty"}$_{1..n}$
by the datatype package of Isabelle/HOL.
Finally we can add to our infrastructure a structural induction principle
for the types @{text "ty\<AL>"}$_{i..n}$ whose
conclusion of the form
%
\begin{equation}\label{weakinduct}
\mbox{@{text "P\<^isub>1 x\<^isub>1 \<and> \<dots> \<and> P\<^isub>n x\<^isub>n "}}
\end{equation}
\noindent
whereby the @{text P}$_{1..n}$ are predicates and the @{text x}$_{1..n}$
have types @{text "ty\<AL>"}$_{1..n}$. This induction principle has for each
term constructor @{text "C"}$^\alpha$ a premise of the form
%
\begin{equation}\label{weakprem}
\mbox{@{text "\<forall>x\<^isub>1\<dots>x\<^isub>r. P\<^isub>i x\<^isub>i \<and> \<dots> \<and> P\<^isub>j x\<^isub>j \<Rightarrow> P (C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r)"}}
\end{equation}
\noindent
in which the @{text "x"}$_{i..j}$ @{text "\<subseteq>"} @{text "x"}$_{1..r}$ are
the recursive arguments of @{text "C\<AL>"}.
By working now completely on the $\alpha$-equated level, we
can first show that the free-atom functions and binding functions are
equivariant, namely
%
\begin{center}
\begin{tabular}{rcl}
@{text "p \<bullet> (fa_ty\<AL>\<^isub>i x)"} & $=$ & @{text "fa_ty\<AL>\<^isub>i (p \<bullet> x)"}\\
@{text "p \<bullet> (fa_bn\<AL>\<^isub>j x)"} & $=$ & @{text "fa_bn\<AL>\<^isub>j (p \<bullet> x)"}\\
@{text "p \<bullet> (bn\<AL>\<^isub>j x)"} & $=$ & @{text "bn\<AL>\<^isub>j (p \<bullet> x)"}
\end{tabular}
\end{center}
\noindent
These properties can be established using the induction principle
in \eqref{weakinduct}.
Having these equivariant properties established, we can
show for every term-constructor @{text "C\<^sup>\<alpha>"} that
\begin{center}
@{text "(supp x\<^isub>1 \<union> \<dots> \<union> supp x\<^isub>r) supports (C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r)"}
\end{center}
\noindent
holds. This together with Property~\ref{supportsprop} allows us to prove
that every @{text x} of type @{text "ty\<AL>"}$_{1..n}$ is finitely supported,
namely @{text "finite (supp x)"}. This can be again shown by induction
over @{text "ty\<AL>"}$_{1..n}$. Lastly, we can show that the support of
elements in @{text "ty\<AL>"}$_{1..n}$ is the same as @{text "fa_ty\<AL>"}$_{1..n}$.
This fact is important in a nominal setting, but also provides evidence
that our notions of free-atoms and $\alpha$-equivalence are correct.
\begin{lemma}
For every @{text "x"} of type @{text "ty\<AL>"}$_{1..n}$, we have
@{text "supp x = fa_ty\<AL>\<^isub>i x"}.
\end{lemma}
\begin{proof}
The proof is by induction. In each case
we unfold the definition of @{text "supp"}, move the swapping inside the
term-constructors and then use the quasi-injectivity lemmas in order to complete the
proof. For the abstraction cases we use the facts derived in Theorem~\ref{suppabs}.
\end{proof}
\noindent
To sum up this section, we can established automatically a reasoning infrastructure
for the types @{text "ty\<AL>"}$_{1..n}$
by first lifting definitions from the raw level to the quotient level and
then by establishing facts about these lifted definitions. All necessary proofs
are generated automatically by custom ML-code. This code can deal with
specifications such as the one shown in Figure~\ref{nominalcorehas} for Core-Haskell.
\begin{figure}[t!]
\begin{boxedminipage}{\linewidth}
\small
\begin{tabular}{l}
\isacommand{atom\_decl}~@{text "var cvar tvar"}\\[1mm]
\isacommand{nominal\_datatype}~@{text "tkind ="}\\
\phantom{$|$}~@{text "KStar"}~$|$~@{text "KFun tkind tkind"}\\
\isacommand{and}~@{text "ckind ="}\\
\phantom{$|$}~@{text "CKSim ty ty"}\\
\isacommand{and}~@{text "ty ="}\\
\phantom{$|$}~@{text "TVar tvar"}~$|$~@{text "T string"}~$|$~@{text "TApp ty ty"}\\
$|$~@{text "TFun string ty_list"}~%
$|$~@{text "TAll tv::tvar tkind ty::ty"} \isacommand{bind}~@{text "tv"}~\isacommand{in}~@{text ty}\\
$|$~@{text "TArr ckind ty"}\\
\isacommand{and}~@{text "ty_lst ="}\\
\phantom{$|$}~@{text "TNil"}~$|$~@{text "TCons ty ty_lst"}\\
\isacommand{and}~@{text "cty ="}\\
\phantom{$|$}~@{text "CVar cvar"}~%
$|$~@{text "C string"}~$|$~@{text "CApp cty cty"}~$|$~@{text "CFun string co_lst"}\\
$|$~@{text "CAll cv::cvar ckind cty::cty"} \isacommand{bind}~@{text "cv"}~\isacommand{in}~@{text cty}\\
$|$~@{text "CArr ckind cty"}~$|$~@{text "CRefl ty"}~$|$~@{text "CSym cty"}~$|$~@{text "CCirc cty cty"}\\
$|$~@{text "CAt cty ty"}~$|$~@{text "CLeft cty"}~$|$~@{text "CRight cty"}~$|$~@{text "CSim cty cty"}\\
$|$~@{text "CRightc cty"}~$|$~@{text "CLeftc cty"}~$|$~@{text "Coerce cty cty"}\\
\isacommand{and}~@{text "co_lst ="}\\
\phantom{$|$}@{text "CNil"}~$|$~@{text "CCons cty co_lst"}\\
\isacommand{and}~@{text "trm ="}\\
\phantom{$|$}~@{text "Var var"}~$|$~@{text "K string"}\\
$|$~@{text "LAM_ty tv::tvar tkind t::trm"} \isacommand{bind}~@{text "tv"}~\isacommand{in}~@{text t}\\
$|$~@{text "LAM_cty cv::cvar ckind t::trm"} \isacommand{bind}~@{text "cv"}~\isacommand{in}~@{text t}\\
$|$~@{text "App_ty trm ty"}~$|$~@{text "App_cty trm cty"}~$|$~@{text "App trm trm"}\\
$|$~@{text "Lam v::var ty t::trm"} \isacommand{bind}~@{text "v"}~\isacommand{in}~@{text t}\\
$|$~@{text "Let x::var ty trm t::trm"} \isacommand{bind}~@{text x}~\isacommand{in}~@{text t}\\
$|$~@{text "Case trm assoc_lst"}~$|$~@{text "Cast trm co"}\\
\isacommand{and}~@{text "assoc_lst ="}\\
\phantom{$|$}~@{text ANil}~%
$|$~@{text "ACons p::pat t::trm assoc_lst"} \isacommand{bind}~@{text "bv p"}~\isacommand{in}~@{text t}\\
\isacommand{and}~@{text "pat ="}\\
\phantom{$|$}~@{text "Kpat string tvtk_lst tvck_lst vt_lst"}\\
\isacommand{and}~@{text "vt_lst ="}\\
\phantom{$|$}~@{text VTNil}~$|$~@{text "VTCons var ty vt_lst"}\\
\isacommand{and}~@{text "tvtk_lst ="}\\
\phantom{$|$}~@{text TVTKNil}~$|$~@{text "TVTKCons tvar tkind tvtk_lst"}\\
\isacommand{and}~@{text "tvck_lst ="}\\
\phantom{$|$}~@{text TVCKNil}~$|$ @{text "TVCKCons cvar ckind tvck_lst"}\\
\isacommand{binder}\\
@{text "bv :: pat \<Rightarrow> atom list"}~\isacommand{and}~%
@{text "bv1 :: vt_lst \<Rightarrow> atom list"}~\isacommand{and}\\
@{text "bv2 :: tvtk_lst \<Rightarrow> atom list"}~\isacommand{and}~%
@{text "bv3 :: tvck_lst \<Rightarrow> atom list"}\\
\isacommand{where}\\
\phantom{$|$}~@{text "bv (K s tvts tvcs vs) = (bv3 tvts) @ (bv2 tvcs) @ (bv1 vs)"}\\
$|$~@{text "bv1 VTNil = []"}\\
$|$~@{text "bv1 (VTCons x ty tl) = (atom x)::(bv1 tl)"}\\
$|$~@{text "bv2 TVTKNil = []"}\\
$|$~@{text "bv2 (TVTKCons a ty tl) = (atom a)::(bv2 tl)"}\\
$|$~@{text "bv3 TVCKNil = []"}\\
$|$~@{text "bv3 (TVCKCons c cty tl) = (atom c)::(bv3 tl)"}\\
\end{tabular}
\end{boxedminipage}
\caption{The nominal datatype declaration for Core-Haskell. For the moment we
do not support nested types; therefore we explicitly have to unfold the
lists @{text "co_lst"}, @{text "assoc_lst"} and so on. This will be improved
in a future version of Nominal Isabelle. Apart from that, the
declaration follows closely the original in Figure~\ref{corehas}. The
point of our work is that having made such a declaration in Nominal Isabelle,
one obtains automatically a reasoning infrastructure for Core-Haskell.
\label{nominalcorehas}}
\end{figure}
*}
section {* Strong Induction Principles *}
text {*
In the previous section we were able to provide induction principles that
allow us to perform structural inductions over $\alpha$-equated terms.
We call such induction principles \emph{weak}, because in case of the
term-constructor @{text "C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r"},
the induction hypothesis requires us to establish the implications \eqref{weakprem}.
The problem with these implications is that in general they are difficult to establish.
The reason is that we cannot make any assumption about the binders that might be in @{text "C\<^sup>\<alpha>"}
(for example we cannot assume the variable convention for them).
In \cite{UrbanTasson05} we introduced a method for automatically
strengthening weak induction principles for terms containing single
binders. These stronger induction principles allow the user to make additional
assumptions about binders.
These additional assumptions amount to a formal
version of the informal variable convention for binders. A natural question is
whether we can also strengthen the weak induction principles involving
the general binders presented here. We will indeed be able to so, but for this we need an
additional notion for permuting deep binders.
Given a binding function @{text "bn"} we define an auxiliary permutation
operation @{text "_ \<bullet>\<^bsub>bn\<^esub> _"} which permutes only bound arguments in a deep binder.
Assuming a clause of @{text bn} is given as
%
\begin{center}
@{text "bn (C x\<^isub>1 \<dots> x\<^isub>r) = rhs"},
\end{center}
\noindent
then we define
%
\begin{center}
@{text "p \<bullet>\<^bsub>bn\<^esub> (C x\<^isub>1 \<dots> x\<^isub>r) \<equiv> C y\<^isub>1 \<dots> y\<^isub>r"}
\end{center}
\noindent
with @{text "y\<^isub>i"} determined as follows:
%
\begin{center}
\begin{tabular}{c@ {\hspace{2mm}}p{7cm}}
$\bullet$ & @{text "y\<^isub>i \<equiv> x\<^isub>i"} provided @{text "x\<^isub>i"} does not occur in @{text "rhs"}\\
$\bullet$ & @{text "y\<^isub>i \<equiv> p \<bullet>\<^bsub>bn'\<^esub> x\<^isub>i"} provided @{text "bn' x\<^isub>i"} is in @{text "rhs"}\\
$\bullet$ & @{text "y\<^isub>i \<equiv> p \<bullet> x\<^isub>i"} otherwise
\end{tabular}
\end{center}
\noindent
Using again the quotient package we can lift the @{text "_ \<bullet>\<^bsub>bn\<^esub> _"} function to
$\alpha$-equated terms. We can then prove the following two facts
\begin{lemma}\label{permutebn}
Given a binding function @{text "bn\<^sup>\<alpha>"} then for all @{text p}
{\it (i)} @{text "p \<bullet> (bn\<^sup>\<alpha> x) = bn\<^sup>\<alpha> (p \<bullet>\<AL>\<^bsub>bn\<^esub> x)"} and {\it (ii)}
@{text "fa_bn\<^isup>\<alpha> x = fa_bn\<^isup>\<alpha> (p \<bullet>\<AL>\<^bsub>bn\<^esub> x)"}.
\end{lemma}
\begin{proof}
By induction on @{text x}. The equations follow by simple unfolding
of the definitions.
\end{proof}
\noindent
The first property states that a permutation applied to a binding function is
equivalent to first permuting the binders and then calculating the bound
atoms. The second amounts to the fact that permuting the binders has no
effect on the free-atom function. The main point of this permutation
function, however, is that if we have a permutation that is fresh
for the support of an object @{text x}, then we can use this permutation
to rename the binders in @{text x}, without ``changing'' @{text x}. In case of the
@{text "Let"} term-constructor from the example shown
in \eqref{letpat} this means for a permutation @{text "r"}
%
\begin{equation}\label{renaming}
\begin{array}{l}
\mbox{if @{term "supp (Abs_lst (bn p) t\<^isub>2) \<sharp>* r"}}\\
\qquad\mbox{then @{text "Let p t\<^isub>1 t\<^isub>2 = Let (r \<bullet>\<AL>\<^bsub>bn_pat\<^esub> p) t\<^isub>1 (r \<bullet> t\<^isub>2)"}}
\end{array}
\end{equation}
\noindent
This fact will be crucial when establishing the strong induction principles below.
In our running example about @{text "Let"}, the strong induction
principle means that instead
of establishing the implication
%
\begin{center}
@{text "\<forall>p t\<^isub>1 t\<^isub>2. P\<^bsub>pat\<^esub> p \<and> P\<^bsub>trm\<^esub> t\<^isub>1 \<and> P\<^bsub>trm\<^esub> t\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub> (Let p t\<^isub>1 t\<^isub>2)"}
\end{center}
\noindent
it is sufficient to establish the following implication
%
\begin{equation}\label{strong}
\mbox{\begin{tabular}{l}
@{text "\<forall>p t\<^isub>1 t\<^isub>2 c."}\\
\hspace{5mm}@{text "set (bn p) #\<^sup>* c \<and>"}\\
\hspace{5mm}@{text "(\<forall>d. P\<^bsub>pat\<^esub> d p) \<and> (\<forall>d. P\<^bsub>trm\<^esub> d t\<^isub>1) \<and> (\<forall>d. P\<^bsub>trm\<^esub> d t\<^isub>2)"}\\
\hspace{15mm}@{text "\<Rightarrow> P\<^bsub>trm\<^esub> c (Let p t\<^isub>1 t\<^isub>2)"}
\end{tabular}}
\end{equation}
\noindent
While this implication contains an additional argument, namely @{text c}, and
also additional universal quantifications, it is usually easier to establish.
The reason is that we have the freshness
assumption @{text "set (bn\<^sup>\<alpha> p) #\<^sup>* c"}, whereby @{text c} can be arbitrarily
chosen by the user as long as it has finite support.
Let us now show how we derive the strong induction principles from the
weak ones. In case of the @{text "Let"}-example we derive by the weak
induction the following two properties
%
\begin{equation}\label{hyps}
@{text "\<forall>q c. P\<^bsub>trm\<^esub> c (q \<bullet> t)"} \hspace{4mm}
@{text "\<forall>q\<^isub>1 q\<^isub>2 c. P\<^bsub>pat\<^esub> (q\<^isub>1 \<bullet>\<AL>\<^bsub>bn\<^esub> (q\<^isub>2 \<bullet> p))"}
\end{equation}
\noindent
For the @{text Let} term-constructor we therefore have to establish @{text "P\<^bsub>trm\<^esub> c (q \<bullet> Let p t\<^isub>1 t\<^isub>2)"}
assuming \eqref{hyps} as induction hypotheses (the first for @{text t\<^isub>1} and @{text t\<^isub>2}).
By Property~\ref{avoiding} we
obtain a permutation @{text "r"} such that
%
\begin{equation}\label{rprops}
@{term "(r \<bullet> set (bn (q \<bullet> p))) \<sharp>* c "}\hspace{4mm}
@{term "supp (Abs_lst (bn (q \<bullet> p)) (q \<bullet> t\<^isub>2)) \<sharp>* r"}
\end{equation}
\noindent
hold. The latter fact and \eqref{renaming} give us
%
\begin{center}
\begin{tabular}{l}
@{text "Let (q \<bullet> p) (q \<bullet> t\<^isub>1) (q \<bullet> t\<^isub>2) ="} \\
\hspace{15mm}@{text "Let (r \<bullet>\<AL>\<^bsub>bn\<^esub> (q \<bullet> p)) (q \<bullet> t\<^isub>1) (r \<bullet> (q \<bullet> t\<^isub>2))"}
\end{tabular}
\end{center}
\noindent
So instead of proving @{text "P\<^bsub>trm\<^esub> c (q \<bullet> Let p t\<^isub>1 t\<^isub>2)"}, we can equally
establish @{text "P\<^bsub>trm\<^esub> c (Let (r \<bullet>\<AL>\<^bsub>bn\<^esub> (q \<bullet> p)) (q \<bullet> t\<^isub>1) (r \<bullet> (q \<bullet> t\<^isub>2)))"}.
To do so, we will use the implication \eqref{strong} of the strong induction
principle, which requires us to discharge
the following four proof obligations:
%
\begin{center}
\begin{tabular}{rl}
{\it (i)} & @{text "set (bn (r \<bullet>\<AL>\<^bsub>bn\<^esub> (q \<bullet> p))) #\<^sup>* c"}\\
{\it (ii)} & @{text "\<forall>d. P\<^bsub>pat\<^esub> d (r \<bullet>\<AL>\<^bsub>bn\<^esub> (q \<bullet> p))"}\\
{\it (iii)} & @{text "\<forall>d. P\<^bsub>trm\<^esub> d (q \<bullet> t\<^isub>1)"}\\
{\it (iv)} & @{text "\<forall>d. P\<^bsub>trm\<^esub> d (r \<bullet> (q \<bullet> t\<^isub>2))"}\\
\end{tabular}
\end{center}
\noindent
The first follows from \eqref{rprops} and Lemma~\ref{permutebn}.{\it (i)}; the
others from the induction hypotheses in \eqref{hyps} (in the fourth case
we have to use the fact that @{term "(r \<bullet> (q \<bullet> t\<^isub>2)) = (r + q) \<bullet> t\<^isub>2"}).
Taking now the identity permutation @{text 0} for the permutations in \eqref{hyps},
we can establish our original goals, namely @{text "P\<^bsub>trm\<^esub> c t"} and \mbox{@{text "P\<^bsub>pat\<^esub> c p"}}.
This completes the proof showing that the weak induction principles imply
the strong induction principles.
*}
section {* Related Work *}
text {*
To our knowledge the earliest usage of general binders in a theorem prover
is described in \cite{NaraschewskiNipkow99} about a formalisation of the
algorithm W. This formalisation implements binding in type-schemes using a
de-Bruijn indices representation. Since type-schemes in W contain only a single
place where variables are bound, different indices do not refer to different binders (as in the usual
de-Bruijn representation), but to different bound variables. A similar idea
has been recently explored for general binders in the locally nameless
approach to binding \cite{chargueraud09}. There, de-Bruijn indices consist
of two numbers, one referring to the place where a variable is bound and the
other to which variable is bound. The reasoning infrastructure for both
representations of bindings comes for free in theorem provers like Isabelle/HOL or
Coq, since the corresponding term-calculi can be implemented as ``normal''
datatypes. However, in both approaches it seems difficult to achieve our
fine-grained control over the ``semantics'' of bindings (i.e.~whether the
order of binders should matter, or vacuous binders should be taken into
account). To do so, one would require additional predicates that filter out
unwanted terms. Our guess is that such predicates result in rather
intricate formal reasoning.
Another representation technique for binding is higher-order abstract syntax
(HOAS), which for example is implemented in the Twelf system. This representation
technique supports very elegantly many aspects of \emph{single} binding, and
impressive work has been done that uses HOAS for mechanising the metatheory
of SML~\cite{LeeCraryHarper07}. We are, however, not aware how multiple
binders of SML are represented in this work. Judging from the submitted
Twelf-solution for the POPLmark challenge, HOAS cannot easily deal with
binding constructs where the number of bound variables is not fixed. For
example in the second part of this challenge, @{text "Let"}s involve
patterns that bind multiple variables at once. In such situations, HOAS
representations have to resort to the iterated-single-binders-approach with
all the unwanted consequences when reasoning about the resulting terms.
Two formalisations involving general binders have been
performed in older
versions of Nominal Isabelle (one about Psi-calculi and one about algorithm W
\cite{BengtsonParow09, UrbanNipkow09}). Both
use the approach based on iterated single binders. Our experience with
the latter formalisation has been disappointing. The major pain arose from
the need to ``unbind'' variables. This can be done in one step with our
general binders described in this paper, but needs a cumbersome
iteration with single binders. The resulting formal reasoning turned out to
be rather unpleasant. The hope is that the extension presented in this paper
is a substantial improvement.
The most closely related work to the one presented here is the Ott-tool
\cite{ott-jfp} and the C$\alpha$ml language \cite{Pottier06}. Ott is a nifty
front-end for creating \LaTeX{} documents from specifications of
term-calculi involving general binders. For a subset of the specifications
Ott can also generate theorem prover code using a raw representation of
terms, and in Coq also a locally nameless representation. The developers of
this tool have also put forward (on paper) a definition for
$\alpha$-equivalence of terms that can be specified in Ott. This definition is
rather different from ours, not using any nominal techniques. To our
knowledge there is also no concrete mathematical result concerning this
notion of $\alpha$-equivalence. A definition for the notion of free variables
is work in progress in Ott.
Although we were heavily inspired by the syntax in Ott,
its definition of $\alpha$-equivalence is unsuitable for our extension of
Nominal Isabelle. First, it is far too complicated to be a basis for
automated proofs implemented on the ML-level of Isabelle/HOL. Second, it
covers cases of binders depending on other binders, which just do not make
sense for our $\alpha$-equated terms. Third, it allows empty types that have no
meaning in a HOL-based theorem prover. We also had to generalise slightly Ott's
binding clauses. In Ott you specify binding clauses with a single body; we
allow more than one. We have to do this, because this makes a difference
for our notion of $\alpha$-equivalence in case of \isacommand{bind\_set} and
\isacommand{bind\_res}. Consider the examples
\begin{center}
\begin{tabular}{@ {}l@ {\hspace{2mm}}l@ {}}
@{text "Foo\<^isub>1 xs::name fset t::trm s::trm"} &
\isacommand{bind\_set} @{text "xs"} \isacommand{in} @{text "t s"}\\
@{text "Foo\<^isub>2 xs::name fset t::trm s::trm"} &
\isacommand{bind\_set} @{text "xs"} \isacommand{in} @{text "t"},
\isacommand{bind\_set} @{text "xs"} \isacommand{in} @{text "s"}\\
\end{tabular}
\end{center}
\noindent
In the first term-constructor we have a single
body that happens to be ``spread'' over two arguments; in the second term-constructor we have
two independent bodies in which the same variables are bound. As a result we
have
\begin{center}
\begin{tabular}{r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l}
@{text "Foo\<^isub>1 {a, b} (a, b) (a, b)"} & $\not=$ &
@{text "Foo\<^isub>1 {a, b} (a, b) (b, a)"}\\
@{text "Foo\<^isub>2 {a, b} (a, b) (a, b)"} & $=$ &
@{text "Foo\<^isub>2 {a, b} (a, b) (b, a)"}\\
\end{tabular}
\end{center}
\noindent
and therefore need the extra generality to be able to distinguish between
both specifications.
Because of how we set up our definitions, we also had to impose some restrictions
(like a single binding function for a deep binder) that are not present in Ott. Our
expectation is that we can still cover many interesting term-calculi from
programming language research, for example Core-Haskell.
Pottier presents in \cite{Pottier06} a language, called C$\alpha$ml, for
representing terms with general binders inside OCaml. This language is
implemented as a front-end that can be translated to OCaml with the help of
a library. He presents a type-system in which the scope of general binders
can be specified using special markers, written @{text "inner"} and
@{text "outer"}. It seems our and his specifications can be
inter-translated as long as ours use the binding mode
\isacommand{bind} only.
However, we have not proved this. Pottier gives a definition for
$\alpha$-equivalence, which also uses a permutation operation (like ours).
Still, this definition is rather different from ours and he only proves that
it defines an equivalence relation. A complete
reasoning infrastructure is well beyond the purposes of his language.
In a slightly different domain (programming with dependent types), the
paper \cite{Altenkirch10} presents a calculus with a notion of
$\alpha$-equivalence related to our binding mode \isacommand{bind\_res}.
The definition in \cite{Altenkirch10} is similar to the one by Pottier, except that it
has a more operational flavour and calculates a partial (renaming) map.
In this way, the definition can deal with vacuous binders. However, to our
best knowledge, no concrete mathematical result concerning this
definition of $\alpha$-equivalence has been proved.
*}
section {* Conclusion *}
text {*
We have presented an extension of Nominal Isabelle for dealing with
general binders, that is term-constructors having multiple bound
variables. For this extension we introduced new definitions of
$\alpha$-equivalence and automated all necessary proofs in Isabelle/HOL.
To specify general binders we used the specifications from Ott, but extended them
in some places and restricted
them in others so that they make sense in the context of $\alpha$-equated terms.
We have tried out the extension with terms from Core-Haskell, type-schemes
and the lambda-calculus, and our code
will eventually become part of the next Isabelle distribution.\footnote{For the moment
it can be downloaded from the Mercurial repository linked at
\href{http://isabelle.in.tum.de/nominal/download}
{http://isabelle.in.tum.de/nominal/download}.}
We have left out a discussion about how functions can be defined over
$\alpha$-equated terms involving general binders. In earlier versions of Nominal
Isabelle \cite{UrbanBerghofer06} this turned out to be a thorny issue. We
hope to do better this time by using the function package that has recently
been implemented in Isabelle/HOL and also by restricting function
definitions to equivariant functions (for such functions it is possible to
provide more automation).
There are some restrictions we imposed in this paper that we would like to lift in
future work. One is the exclusion of nested datatype definitions. Nested
datatype definitions allow one to specify, for instance, the function kinds
in Core-Haskell as @{text "TFun string (ty list)"} instead of the unfolded
version @{text "TFun string ty_list"} (see Figure~\ref{nominalcorehas}). To
achieve this, we need a slightly more clever implementation than we have at the moment.
A more interesting line of investigation is whether we can go beyond the
simple-minded form of binding functions that we adopted from Ott. At the moment, binding
functions can only return the empty set, a singleton atom set or unions
of atom sets (similarly for lists). It remains to be seen whether
properties like
%
\begin{center}
@{text "fa_ty x = bn x \<union> fa_bn x"}.
\end{center}
\noindent
allow us to support more interesting binding functions.
We have also not yet played with other binding modes. For example we can
imagine that there is need for a binding mode
where instead of lists, we abstract lists of distinct elements.
Once we feel confident about such binding modes, our implementation
can be easily extended to accommodate them.
\medskip
\noindent
{\bf Acknowledgements:} We are very grateful to Andrew Pitts for
many discussions about Nominal Isabelle. We also thank Peter Sewell for
making the informal notes \cite{SewellBestiary} available to us and
also for patiently explaining some of the finer points of the work on the Ott-tool.
Stephanie Weirich suggested to separate the subgrammars
of kinds and types in our Core-Haskell example.
*}
(*<*)
end
(*>*)