--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ESOP-Paper/Paper.thy Tue Mar 29 23:52:14 2011 +0200
@@ -0,0 +1,2393 @@
+(*<*)
+theory Paper
+imports "../Nominal/Nominal2"
+ "~~/src/HOL/Library/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_set ("_ \<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]{$_{\textit{set+}}$}}>\<^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_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
+ Abs_set ("[_]\<^bsub>set\<^esub>._" [20, 101] 999) and
+ Abs_lst ("[_]\<^bsub>list\<^esub>._") and
+ Abs_dist ("[_]\<^bsub>#list\<^esub>._") and
+ Abs_res ("[_]\<^bsub>set+\<^esub>._") and
+ Abs_print ("_\<^bsub>set\<^esub>._") and
+ Cons ("_::_" [78,77] 73) and
+ supp_set ("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,
+ @{text "t ::= x | t t | \<lambda>x. t"},
+ 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{9mm}
+ @{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
+ %has 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 first pair of type-schemes as $\alpha$-equivalent,
+ but assuming that @{text x}, @{text y} and @{text z} are distinct variables,
+ the second pair should \emph{not} be $\alpha$-equivalent:
+ %
+ \begin{equation}\label{ex1}
+ @{text "\<forall>{x, y}. x \<rightarrow> y \<approx>\<^isub>\<alpha> \<forall>{y, x}. y \<rightarrow> x"}\hspace{10mm}
+ @{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 often unusual to regard
+ \eqref{one} as $\alpha$-equivalent with
+ %
+ \begin{center}
+ @{text "\<LET> x = 3 \<AND> y = 2 \<AND> z = foo \<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
+ in a formalisation.
+ %%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
+ we care about the
+ information that there are as many bound variables @{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}}cl}
+ @{text trm} & @{text "::="} & @{text "\<dots>"}
+ & @{text "|"} @{text "\<LET> as::assn s::trm"}\hspace{2mm}
+ \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}.
+
+ %Though, Ott
+ %has only one binding mode, namely the one where the order of
+ %binders matters. Consequently, type-schemes with binding sets
+ %of names cannot be modelled in Ott.
+
+ 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 @{text "t ::= t t | \<lambda>x. t"}
+ 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
+ Isabelle/HOL is a rather non-trivial task. For every
+ specification we will need to construct type(s) 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}[scale=0.89]
+ %\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 {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]clas.\end{tabular}};
+ \draw (-2.4, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};
+ \draw (1.8, 0.48) node[right=-0.1mm]
+ {\footnotesize\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ (sets of raw terms)\end{tabular}};
+ \draw (0.9, -0.35) node {\footnotesize\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};
+ \draw (-3.25, 0.55) node {\footnotesize\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] {\footnotesize{}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 our relation for $\alpha$-equivalence is
+ 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 \cite{KaliszykUrban11} 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{8mm}
+ @{text "fv(t\<^isub>1 t\<^isub>2) = fv(t\<^isub>1) \<union> fv(t\<^isub>2)"}\hspace{8mm}
+ @{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{8mm}
+ @{text "fv\<^sup>\<alpha>(t\<^isub>1 t\<^isub>2) = fv\<^sup>\<alpha>(t\<^isub>1) \<union> fv\<^sup>\<alpha>(t\<^isub>2)"}\hspace{8mm}
+ @{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.\smallskip
+
+ %The examples we have in mind where our reasoning infrastructure will be
+ %helpful includes the term language of Core-Haskell. This term language
+ %involves patterns that have lists of type-, coercion- and term-variables,
+ %all of which are bound in @{text "\<CASE>"}-expressions. In these
+ %patterns we do not know in advance how many variables need to
+ %be bound. Another example is the specification of SML, which includes
+ %includes bindings as in type-schemes.\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.
+ The main improvement over Ott is that we introduce three binding modes
+ (only one is present in Ott), provide formalised definitions for $\alpha$-equivalence and
+ for free variables of our terms, and also derive a reasoning infrastructure
+ for our specifications from ``first principles''.
+
+
+ %\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. 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
+ @{text "_ \<bullet> _ :: perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
+ where the generic type @{text "\<beta>"} is 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 over the type-hierarchy \cite{HuffmanUrban10};
+ for example permutations acting on products, lists, sets, functions and booleans are
+ given by:
+ %
+ %\begin{equation}\label{permute}
+ %\mbox{\begin{tabular}{@ {}c@ {\hspace{10mm}}c@ {}}
+ %\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}
+ %
+ \begin{center}
+ \mbox{\begin{tabular}{@ {}c@ {\hspace{4mm}}c@ {\hspace{4mm}}c@ {}}
+ \begin{tabular}{@ {}l@ {}}
+ @{thm permute_prod.simps[no_vars, THEN eq_reflection]}\\
+ @{thm permute_bool_def[no_vars, THEN eq_reflection]}
+ \end{tabular} &
+ \begin{tabular}{@ {}l@ {}}
+ @{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))"}\\
+ \end{tabular}
+ \end{tabular}}
+ \end{center}
+
+ \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 @{thm fresh_def[no_vars]}.
+ 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, namely if @{text "a \<FRESH> x"} and @{text "b \<FRESH> x"}
+ then @{term "(a \<rightleftharpoons> b) \<bullet> x = x"}.
+ %
+ %\begin{myproperty}\label{swapfreshfresh}
+ %@{thm[mode=IfThen] swap_fresh_fresh[no_vars]}
+ %\end{myproperty}
+ %
+ %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{center}
+ %\begin{tabular}{c@ {\hspace{10mm}}c}
+ %\begin{tabular}{rcl}
+ %@{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"}\\
+ %\end{tabular}
+ %&
+ %\begin{tabular}{rcl}
+ %@{text "supp (f x)"} & @{text "\<subseteq>"} & @{term "supp f \<union> supp x"}\\
+ %@{term "supp b"} & $=$ & @{term "{}"}\\
+ %@{term "supp p"} & $=$ & @{term "{a. p \<bullet> a \<noteq> a}"}
+ %\end{tabular}
+ %\end{tabular}
+ %\end{center}
+ %
+ %\noindent
+ %in some cases it can be difficult to characterise the support precisely, and
+ %only an approximation can be established (as for functions above).
+ %
+ %Reasoning about
+ %such approximations can be simplified with the notion \emph{supports}, defined
+ %as follows:
+ %
+ %\begin{definition}
+ %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{definition}
+ %
+ %\noindent
+ %The main point of @{text supports} is that we can establish the following
+ %two properties.
+ %
+ %\begin{myproperty}\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{myproperty}
+ %
+ %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}
+ %
+ %Using freshness, the nominal logic work provides us with general means for renaming
+ %binders.
+ %
+ %\noindent
+ While in the older version of Nominal Isabelle, we used extensively
+ %Property~\ref{swapfreshfresh}
+ this property to rename single binders, it %%this property
+ proved too unwieldy for dealing with multiple binders. For such binders the
+ following generalisations turned out to be easier to use.
+
+ \begin{myproperty}\label{supppermeq}
+ @{thm[mode=IfThen] supp_perm_eq[no_vars]}
+ \end{myproperty}
+
+ \begin{myproperty}\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{myproperty}
+
+ \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 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 the conjunction of:
+ %
+ \begin{equation}\label{alphaset}
+ \begin{array}{@ {\hspace{10mm}}l@ {\hspace{5mm}}l@ {\hspace{10mm}}l@ {\hspace{5mm}}l}
+ \multicolumn{4}{l}{@{term "(as, x) \<approx>set R fa p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}}\\[1mm]
+ \mbox{\it (i)} & @{term "fa(x) - as = fa(y) - bs"} &
+ \mbox{\it (iii)} & @{text "(p \<bullet> x) R y"} \\
+ \mbox{\it (ii)} & @{term "(fa(x) - as) \<sharp>* p"} &
+ \mbox{\it (iv)} & @{term "(p \<bullet> as) = bs"} \\
+ \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}}l@ {\hspace{5mm}}l@ {\hspace{10mm}}l@ {\hspace{5mm}}l}
+ \multicolumn{4}{l}{@{term "(as, x) \<approx>lst R fa p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}}\\[1mm]
+ \mbox{\it (i)} & @{term "fa(x) - (set as) = fa(y) - (set bs)"} &
+ \mbox{\it (iii)} & @{text "(p \<bullet> x) R y"}\\
+ \mbox{\it (ii)} & @{term "(fa(x) - set as) \<sharp>* p"} &
+ \mbox{\it (iv)} & @{term "(p \<bullet> as) = bs"}\\
+ \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, that means \emph{restrict} names, then we keep sets of binders, but drop
+ condition {\it (iv)} in \eqref{alphaset}:
+ %
+ \begin{equation}\label{alphares}
+ \begin{array}{@ {\hspace{10mm}}l@ {\hspace{5mm}}l@ {\hspace{10mm}}l@ {\hspace{5mm}}l}
+ \multicolumn{2}{l}{@{term "(as, x) \<approx>res R fa p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}}\\[1mm]
+ \mbox{\it (i)} & @{term "fa(x) - as = fa(y) - bs"} &
+ \mbox{\it (iii)} & @{text "(p \<bullet> x) R y"}\\
+ \mbox{\it (ii)} & @{term "(fa(x) - as) \<sharp>* p"}\\
+ \end{array}
+ \end{equation}
+
+ It might be useful to consider first some examples 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} 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 $\approx_{\,\textit{set+}}$ 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)"} $\approx_{\,\textit{set+}}$
+ @{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_set (as, x) equal supp p (bs, x)"}
+ \end{equation}
+
+ \noindent
+ (similarly for $\approx_{\,\textit{abs\_set+}}$
+ 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 $\approx_{\,\textit{abs\_set+}}$ 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_set+"} 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_set as x"}, %\hspace{5mm}
+ @{term "Abs_res as x"} and %\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{theorem}[Support of Abstractions]\label{suppabs}
+ Assuming @{text x} has finite support, then
+
+ \begin{center}
+ \begin{tabular}{l}
+ @{thm (lhs) supp_Abs(1)[no_vars]} $\;\;=\;\;$
+ @{thm (lhs) supp_Abs(2)[no_vars]} $\;\;=\;\;$ @{thm (rhs) supp_Abs(2)[no_vars]}, and\\
+ @{thm (lhs) supp_Abs(3)[where bs="bs", no_vars]} $\;\;=\;\;$
+ @{thm (rhs) supp_Abs(3)[where bs="bs", no_vars]}
+ \end{tabular}
+ \end{center}
+ \end{theorem}
+
+ \noindent
+ This theorem states that the bound names do not appear in the support.
+ For brevity we omit the proof and again refer the reader to
+ our formalisation in Isabelle/HOL.
+
+ %\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:
+ %@{thm supp_set.simps[THEN eq_reflection, no_vars]}.
+ %
+ %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_set as x)) \<subseteq> supp (Abs_set 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) Abs_supp_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_set 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 sections.
+*}
+
+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}{@ {}p{2.5cm}l}
+ type \mbox{declaration part} &
+ $\begin{cases}
+ \mbox{\small\begin{tabular}{l}
+ \isacommand{nominal\_datatype} @{text "ty\<AL>\<^isub>1 = \<dots>"}\\
+ \isacommand{and} @{text "ty\<AL>\<^isub>2 = \<dots>"}\\
+ \raisebox{2mm}{$\ldots$}\\[-2mm]
+ \isacommand{and} @{text "ty\<AL>\<^isub>n = \<dots>"}\\
+ \end{tabular}}
+ \end{cases}$\\
+ binding \mbox{function part} &
+ $\begin{cases}
+ \mbox{\small\begin{tabular}{l}
+ \isacommand{binder} @{text "bn\<AL>\<^isub>1"} \isacommand{and} \ldots \isacommand{and} @{text "bn\<AL>\<^isub>m"}\\
+ \isacommand{where}\\
+ \raisebox{2mm}{$\ldots$}\\[-2mm]
+ \end{tabular}}
+ \end{cases}$\\
+ \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 (set+)} {\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 (set+)} 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 (set+)} 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}\small
+ \begin{tabular}{@ {}c@ {\hspace{7mm}}c@ {}}
+ \begin{tabular}{@ {}l}
+ \isacommand{nominal\_datatype} @{text lam} $=$\\
+ \hspace{2mm}\phantom{$\mid$}~@{text "Var name"}\\
+ \hspace{2mm}$\mid$~@{text "App lam lam"}\\
+ \hspace{2mm}$\mid$~@{text "Lam x::name t::lam"}~~\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"}~~%
+ \isacommand{bind (set+)} @{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 (set+)}) 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{\small%
+ \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} $=$
+ @{text PNil}
+ $\mid$~@{text "PVar name"}
+ $\mid$~@{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{\small%
+ \begin{tabular}{@ {}l@ {}}
+ \isacommand{nominal\_datatype}~@{text "trm ="}~\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 "assn"} $=$
+ @{text "ANil"}
+ $\mid$~@{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}\small
+ \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 with deep binders, 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 trm} (the only one that
+ is \emph{not} used in the definition of the binding function). This restriction
+ is sufficient for lifting the binding function to $\alpha$-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-terms, the completion produces
+
+ \begin{center}\small
+ \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
+ construct the quotient of 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 given for $\alpha$-equivalence classes and leave it out
+ for the corresponding notion given on the ``raw'' level. So for example
+ we have @{text "ty\<^sup>\<alpha> \<mapsto> ty"} and @{text "C\<^sup>\<alpha> \<mapsto> C"}
+ 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>"}$_{1..n}$
+ %\end{equation}
+ %
+ %\noindent
+ by 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>"}$_{1..m}$.
+ %\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\\[-2mm]
+ %
+ \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\\[-4mm]
+ %
+ %\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}
+ %
+ \begin{center}
+ @{text "bn\<^bsub>atom\<^esub> a \<equiv> {atom a}"}\hfill
+ @{text "bn\<^bsub>atom_set\<^esub> as \<equiv> atoms as"}\hfill
+ @{text "bn\<^bsub>atom_list\<^esub> as \<equiv> atoms (set as)"}
+ \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\\[-4mm]
+ %
+ \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}
+ \mbox{@{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\\[-5mm]
+ %
+ \begin{center}
+ @{text "B' \<equiv> fa_bn\<^isub>1 l\<^isub>1 \<union> ... \<union> fa_bn\<^isub>r l\<^isub>r"}\\[-9mm]
+ \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{0.9\textwidth}}
+ $\bullet$ & @{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}\small
+ \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
+ of 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"}$_{1..n}$.
+ %\end{center}
+ %
+ %\noindent
+ Like with the free-atom functions, we also need to
+ define auxiliary $\alpha$-equivalence relations
+ %
+ %\begin{center}
+ @{text "\<approx>bn\<^isub>"}$_{1..m}$
+ %\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,\<dots>, x\<^isub>n) (R\<^isub>1,\<dots>, R\<^isub>n) (x\<PRIME>\<^isub>1,\<dots>, x\<PRIME>\<^isub>n)"} & @{text "\<equiv>"} &
+ @{text "x\<^isub>1 R\<^isub>1 x\<PRIME>\<^isub>1 \<and> \<dots> \<and> x\<^isub>n R\<^isub>n x\<PRIME>\<^isub>n"}\\
+ @{text "(fa\<^isub>1,\<dots>, fa\<^isub>n) (x\<^isub>1,\<dots>, x\<^isub>n)"} & @{text "\<equiv>"} & @{text "fa\<^isub>1 x\<^isub>1 \<union> \<dots> \<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
+ $\approx_{\,\textit{set+}}$ 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>set 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{0.9\textwidth}}
+ $\bullet$ & @{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}\small
+ \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}\small
+ \begin{tabular}{@ {}c @ {}}
+ \infer{@{text "ANil \<approx>\<^bsub>assn\<^esub> ANil"}}{}\hspace{9mm}
+ \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}\small
+ \begin{tabular}{@ {}c @ {}}
+ \infer{@{text "ANil \<approx>\<^bsub>bn\<^esub> ANil"}}{}\hspace{9mm}
+ \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
+ clause for @{text "Let"} (which 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 $\approx_{\textit{set}}$,
+ $\approx_{\textit{set+}}$ 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 implications %facts
+ %
+ \begin{equation}\label{fnresp}
+ \mbox{%
+ \begin{tabular}{ll@ {\hspace{7mm}}ll}
+ \mbox{\it (i)} & @{text "x \<approx>ty\<^isub>i x\<PRIME>"}~~@{text "\<Rightarrow>"}~~@{text "fa_ty\<^isub>i x = fa_ty\<^isub>i x\<PRIME>"} &
+ \mbox{\it (iii)} & @{text "x \<approx>ty\<^isub>j x\<PRIME>"}~~@{text "\<Rightarrow>"}~~@{text "bn\<^isub>j x = bn\<^isub>j x\<PRIME>"}\\
+
+ \mbox{\it (ii)} & @{text "x \<approx>ty\<^isub>j x\<PRIME>"}~~@{text "\<Rightarrow>"}~~@{text "fa_bn\<^isub>j x = fa_bn\<^isub>j x\<PRIME>"} &
+ \mbox{\it (iv)} & @{text "x \<approx>ty\<^isub>j x\<PRIME>"}~~@{text "\<Rightarrow>"}~~@{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 \cite{HuffmanUrban10}.
+ %, 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 cases lemma (explained in the next section)
+ and a structural induction principle
+ for the types @{text "ty\<AL>"}$_{1..n}$. The conclusion of the induction principle is
+ 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@ {\hspace{10mm}}rcl}
+ @{text "p \<bullet> (fa_ty\<AL>\<^isub>i x)"} & $=$ & @{text "fa_ty\<AL>\<^isub>i (p \<bullet> x)"} &
+ @{text "p \<bullet> (bn\<AL>\<^isub>j x)"} & $=$ & @{text "bn\<AL>\<^isub>j (p \<bullet> x)"}\\
+ @{text "p \<bullet> (fa_bn\<AL>\<^isub>j x)"} & $=$ & @{text "fa_bn\<AL>\<^isub>j (p \<bullet> x)"}\\
+ \end{tabular}
+ \end{center}
+ %
+ \noindent
+ These properties can be established using the induction principle for the types @{text "ty\<AL>"}$_{1..n}$.
+ %%in \eqref{weakinduct}.
+ Having these equivariant properties established, we can
+ show that the support of term-constructors @{text "C\<^sup>\<alpha>"} is included in
+ the support of its arguments, that means
+
+ \begin{center}
+ @{text "supp (C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r) \<subseteq> (supp x\<^isub>1 \<union> \<dots> \<union> supp x\<^isub>r)"}
+ \end{center}
+
+ \noindent
+ holds. This allows us to prove by induction that
+ every @{text x} of type @{text "ty\<AL>"}$_{1..n}$ is finitely supported.
+ %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{theorem}
+ For @{text "x"}$_{1..n}$ with type @{text "ty\<AL>"}$_{1..n}$, we have
+ @{text "supp x\<^isub>i = fa_ty\<AL>\<^isub>i x\<^isub>i"}.
+ \end{theorem}
+
+ \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 establish 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 derived induction principles for $\alpha$-equated terms.
+ We call such induction principles \emph{weak}, because for a
+ term-constructor \mbox{@{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 bound atoms 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 bound atoms.
+ %These additional assumptions amount to a formal
+ %version of the informal variable convention for binders.
+ To sketch how this strengthening extends to the case of multiple binders, we use as
+ running example the term-constructors @{text "Lam"} and @{text "Let"}
+ from example \eqref{letpat}. Instead of establishing @{text " P\<^bsub>trm\<^esub> t \<and> P\<^bsub>pat\<^esub> p"},
+ the stronger induction principle for \eqref{letpat} establishes properties @{text " P\<^bsub>trm\<^esub> c t \<and> P\<^bsub>pat\<^esub> c p"}
+ where the additional parameter @{text c} controls
+ which freshness assumptions the binders should satisfy. For the two term constructors
+ this means that the user has to establish in inductions the implications
+ %
+ \begin{center}
+ \begin{tabular}{l}
+ @{text "\<forall>a t c. {atom a} \<FRESH>\<^sup>* c \<and> (\<forall>d. P\<^bsub>trm\<^esub> d t) \<Rightarrow> P\<^bsub>trm\<^esub> c (Lam a t)"}\\
+ @{text "\<forall>p t c. (set (bn p)) \<FRESH>\<^sup>* c \<and> (\<forall>d. P\<^bsub>pat\<^esub> d p) \<and> (\<forall>d. P\<^bsub>trm\<^esub> d t) \<and> \<Rightarrow> P\<^bsub>trm\<^esub> c (Let p t)"}\\%[-0mm]
+ \end{tabular}
+ \end{center}
+
+ In \cite{UrbanTasson05} we showed how the weaker induction principles imply
+ the stronger ones. This was done by some quite complicated, nevertheless automated,
+ induction proof. In this paper we simplify this work by leveraging the automated proof
+ methods from the function package of Isabelle/HOL.
+ The reasoning principle these methods employ is well-founded induction.
+ To use them in our setting, we have to discharge
+ two proof obligations: one is that we have
+ well-founded measures (for each type @{text "ty"}$^\alpha_{1..n}$) that decrease in
+ every induction step and the other is that we have covered all cases.
+ As measures we use the size functions
+ @{text "size_ty"}$^\alpha_{1..n}$, which we lifted in the previous section and which are
+ all well-founded. %It is straightforward to establish that these measures decrease
+ %in every induction step.
+
+ What is left to show is that we covered all cases. To do so, we use
+ a \emph{cases lemma} derived for each type. For the terms in \eqref{letpat}
+ this lemma is of the form
+ %
+ \begin{equation}\label{weakcases}
+ \infer{@{text "P\<^bsub>trm\<^esub>"}}
+ {\begin{array}{l@ {\hspace{9mm}}l}
+ @{text "\<forall>x. t = Var x \<Rightarrow> P\<^bsub>trm\<^esub>"} & @{text "\<forall>a t'. t = Lam a t' \<Rightarrow> P\<^bsub>trm\<^esub>"}\\
+ @{text "\<forall>t\<^isub>1 t\<^isub>2. t = App t\<^isub>1 t\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub>"} & @{text "\<forall>p t'. t = Let p t' \<Rightarrow> P\<^bsub>trm\<^esub>"}\\
+ \end{array}}\\[-1mm]
+ \end{equation}
+ %
+ where we have a premise for each term-constructor.
+ The idea behind such cases lemmas is that we can conclude with a property @{text "P\<^bsub>trm\<^esub>"},
+ provided we can show that this property holds if we substitute for @{text "t"} all
+ possible term-constructors.
+
+ The only remaining difficulty is that in order to derive the stronger induction
+ principles conveniently, the cases lemma in \eqref{weakcases} is too weak. For this note that
+ in order to apply this lemma, we have to establish @{text "P\<^bsub>trm\<^esub>"} for \emph{all} @{text Lam}- and
+ \emph{all} @{text Let}-terms.
+ What we need instead is a cases lemma where we only have to consider terms that have
+ binders that are fresh w.r.t.~a context @{text "c"}. This gives the implications
+ %
+ \begin{center}
+ \begin{tabular}{l}
+ @{text "\<forall>a t'. t = Lam a t' \<and> {atom a} \<FRESH>\<^sup>* c \<Rightarrow> P\<^bsub>trm\<^esub>"}\\
+ @{text "\<forall>p t'. t = Let p t' \<and> (set (bn p)) \<FRESH>\<^sup>* c \<Rightarrow> P\<^bsub>trm\<^esub>"}\\%[-2mm]
+ \end{tabular}
+ \end{center}
+ %
+ \noindent
+ which however can be relatively easily be derived from the implications in \eqref{weakcases}
+ by a renaming using Properties \ref{supppermeq} and \ref{avoiding}. In the first case we know
+ that @{text "{atom a} \<FRESH>\<^sup>* Lam a t"}. Property \eqref{avoiding} provides us therefore with
+ a permutation @{text q}, such that @{text "{atom (q \<bullet> a)} \<FRESH>\<^sup>* c"} and
+ @{text "supp (Lam a t) \<FRESH>\<^sup>* q"} hold.
+ By using Property \ref{supppermeq}, we can infer from the latter
+ that @{text "Lam (q \<bullet> a) (q \<bullet> t) = Lam a t"}
+ and we are done with this case.
+
+ The @{text Let}-case involving a (non-recursive) deep binder is a bit more complicated.
+ The reason is that the we cannot apply Property \ref{avoiding} to the whole term @{text "Let p t"},
+ because @{text p} might contain names that are bound (by @{text bn}) and so are
+ free. To solve this problem we have to introduce a permutation function that only
+ permutes names bound by @{text bn} and leaves the other names unchanged. We do this again
+ by lifting. For a
+ clause @{text "bn (C x\<^isub>1 \<dots> x\<^isub>r) = rhs"}, 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"} with
+ $\begin{cases}
+ \text{@{text "y\<^isub>i \<equiv> x\<^isub>i"} provided @{text "x\<^isub>i"} does not occur in @{text "rhs"}}\\
+ \text{@{text "y\<^isub>i \<equiv> p \<bullet>\<^bsub>bn'\<^esub> x\<^isub>i"} provided @{text "bn' x\<^isub>i"} is in @{text "rhs"}}\\
+ \text{@{text "y\<^isub>i \<equiv> p \<bullet> x\<^isub>i"} otherwise}
+ \end{cases}$
+ \end{center}
+ %
+ %\noindent
+ %with @{text "y\<^isub>i"} determined as follows:
+ %
+ %\begin{center}
+ %\begin{tabular}{c@ {\hspace{2mm}}p{0.9\textwidth}}
+ %$\bullet$ & @{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
+ Now Properties \ref{supppermeq} and \ref{avoiding} give us a permutation @{text q} such that
+ @{text "(set (bn (q \<bullet>\<^bsub>bn\<^esub> p)) \<FRESH>\<^sup>* c"} holds and such that @{text "[q \<bullet>\<^bsub>bn\<^esub> p]\<^bsub>list\<^esub>.(q \<bullet> t)"}
+ is equal to @{text "[p]\<^bsub>list\<^esub>. t"}. We can also show that @{text "(q \<bullet>\<^bsub>bn\<^esub> p) \<approx>\<^bsub>bn\<^esub> p"}.
+ These facts establish that @{text "Let (q \<bullet>\<^bsub>bn\<^esub> p) (p \<bullet> t) = Let p t"}, as we need. This
+ completes the non-trivial cases in \eqref{letpat} for strengthening the corresponding induction
+ principle.
+
+
+
+ %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\label{related} *}
+
+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 technique for representing 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
+ seems to 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 no concrete mathematical result concerning this
+ notion of $\alpha$-equivalence. Also the definition for the
+ notion of free variables
+ is work in progress.
+
+ Although we were heavily inspired by the syntax of Ott,
+ its definition of $\alpha$-equi\-valence 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 (set+)}.
+ %
+ %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.
+ Similar work for Haskell with similar results was reported by Cheney \cite{Cheney05a}.
+
+ 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 (set+)}.
+ 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.\\[-7mm]
+*}
+
+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 also introduced two binding modes (set and set+) that do not
+ exist in Ott.
+ We have tried out the extension with calculi such as Core-Haskell, type-schemes
+ and approximately a dozen of other typical examples from programming
+ language research~\cite{SewellBestiary}.
+ %The 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 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 them we can
+ 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.
+ %
+ \smallskip
+ \noindent
+ {\bf Acknowledgements:} %We are very grateful to Andrew Pitts for
+ %many discussions about Nominal Isabelle.
+ We thank Peter Sewell for
+ making the informal notes \cite{SewellBestiary} available to us and
+ also for patiently explaining some of the finer points of the Ott-tool.\\[-7mm]
+ %Stephanie Weirich suggested to separate the subgrammars
+ %of kinds and types in our Core-Haskell example. \\[-6mm]
+*}
+
+
+(*<*)
+end
+(*>*)