(*<*)theory Paperimports "../Nominal/Test" "LaTeXsugar"beginconsts fv :: "'a \<Rightarrow> 'b" abs_set :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" alpha_bn :: "'a \<Rightarrow> 'a \<Rightarrow> bool"definition "equal \<equiv> (op =)" notation (latex output) swap ("'(_ _')" [1000, 1000] 1000) and fresh ("_ # _" [51, 51] 50) and fresh_star ("_ #\<^sup>* _" [51, 51] 50) and supp ("supp _" [78] 73) and uminus ("-_" [78] 73) and If ("if _ then _ else _" 10) and alpha_gen ("_ \<approx>\<^raw:\raisebox{-1pt}{\makebox[0mm][l]{$\,_{\textit{set}}$}}>\<^bsup>_, _, _\<^esup> _") and alpha_lst ("_ \<approx>\<^raw:\raisebox{-1pt}{\makebox[0mm][l]{$\,_{\textit{list}}$}}>\<^bsup>_, _, _\<^esup> _") and alpha_res ("_ \<approx>\<^raw:\raisebox{-1pt}{\makebox[0mm][l]{$\,_{\textit{res}}$}}>\<^bsup>_, _, _\<^esup> _") and abs_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and fv ("fv'(_')" [100] 100) and equal ("=") and alpha_abs ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and Abs ("[_]\<^raw:$\!$>\<^bsub>set\<^esub>._" [20, 101] 999) and Abs_lst ("[_]\<^raw:$\!$>\<^bsub>list\<^esub>._") and Abs_res ("[_]\<^raw:$\!$>\<^bsub>res\<^esub>._") and Cons ("_::_" [78,77] 73) and supp_gen ("aux _" [1000] 10) and alpha_bn ("_ \<approx>bn _")(*>*)section {* Introduction *}text {* @{text "(1, (2, 3))"} So far, Nominal Isabelle provides a mechanism for constructing alpha-equated terms, for example \begin{center} @{text "t ::= x | t t | \<lambda>x. t"} \end{center} \noindent where free and bound variables have names. For such alpha-equated terms, Nominal Isabelle derives automatically a reasoning infrastructure that has been used successfully in formalisations of an equivalence checking algorithm for LF \cite{UrbanCheneyBerghofer08}, Typed Scheme~\cite{TobinHochstadtFelleisen08}, several calculi for concurrency \cite{BengtsonParow09} and a strong normalisation result for cut-elimination in classical logic \cite{UrbanZhu08}. It has also been used by Pollack for formalisations in the locally-nameless approach to binding \cite{SatoPollack10}. However, Nominal Isabelle has fared less well in a formalisation of the algorithm W \cite{UrbanNipkow09}, where types and type-schemes are, respectively, of the form % \begin{equation}\label{tysch} \begin{array}{l} @{text "T ::= x | T \<rightarrow> T"}\hspace{5mm} @{text "S ::= \<forall>{x\<^isub>1,\<dots>, x\<^isub>n}. T"} \end{array} \end{equation} \noindent and the @{text "\<forall>"}-quantification binds a finite (possibly empty) set of type-variables. While it is possible to implement this kind of more general binders by iterating single binders, this leads to a rather clumsy formalisation of W. The need of iterating single binders is also one reason why Nominal Isabelle and similar theorem provers that only provide mechanisms for binding single variables have not fared extremely well with the more advanced tasks in the POPLmark challenge \cite{challenge05}, because also there one would like to bind multiple variables at once. Binding multiple variables has interesting properties that cannot be captured easily by iterating single binders. For example in case of type-schemes we do not want to make a distinction about the order of the bound variables. Therefore we would like to regard the following two type-schemes as alpha-equivalent % \begin{equation}\label{ex1} @{text "\<forall>{x, y}. x \<rightarrow> y \<approx>\<^isub>\<alpha> \<forall>{y, x}. y \<rightarrow> x"} \end{equation} \noindent but assuming that @{text x}, @{text y} and @{text z} are distinct variables, the following two should \emph{not} be alpha-equivalent % \begin{equation}\label{ex2} @{text "\<forall>{x, y}. x \<rightarrow> y \<notapprox>\<^isub>\<alpha> \<forall>{z}. z \<rightarrow> z"} \end{equation} \noindent Moreover, we like to regard type-schemes as alpha-equivalent, if they differ only on \emph{vacuous} binders, such as % \begin{equation}\label{ex3} @{text "\<forall>{x}. x \<rightarrow> y \<approx>\<^isub>\<alpha> \<forall>{x, z}. x \<rightarrow> y"} \end{equation} \noindent where @{text z} does not occur freely in the type. In this paper we will give a general binding mechanism and associated notion of alpha-equivalence that can be used to faithfully represent this kind of binding in Nominal Isabelle. The difficulty of finding the right notion for alpha-equivalence can be appreciated in this case by considering that the definition given by Leroy in \cite{Leroy92} is incorrect (it omits a side-condition). However, the notion of alpha-equivalence that is preserved by vacuous binders is not always wanted. For example in terms like % \begin{equation}\label{one} @{text "\<LET> x = 3 \<AND> y = 2 \<IN> x - y \<END>"} \end{equation} \noindent we might not care in which order the assignments $x = 3$ and $y = 2$ are given, but it would be unusual to regard \eqref{one} as alpha-equivalent with % \begin{center} @{text "\<LET> x = 3 \<AND> y = 2 \<AND> z = loop \<IN> x - y \<END>"} \end{center} \noindent Therefore we will also provide a separate binding mechanism for cases in which the order of binders does not matter, but the ``cardinality'' of the binders has to agree. However, we found that this is still not sufficient for dealing with language constructs frequently occurring in programming language research. For example in @{text "\<LET>"}s containing patterns like % \begin{equation}\label{two} @{text "\<LET> (x, y) = (3, 2) \<IN> x - y \<END>"} \end{equation} \noindent we want to bind all variables from the pattern inside the body of the $\mathtt{let}$, but we also care about the order of these variables, since we do not want to regard \eqref{two} as alpha-equivalent with % \begin{center} @{text "\<LET> (y, x) = (3, 2) \<IN> x - y \<END>"} \end{center} \noindent As a result, we provide three general binding mechanisms each of which binds multiple variables at once, and let the user chose which one is intended when formalising a term-calculus. By providing these general binding mechanisms, however, we have to work around a problem that has been pointed out by Pottier \cite{Pottier06} and Cheney \cite{Cheney05}: in @{text "\<LET>"}-constructs of the form % \begin{center} @{text "\<LET> x\<^isub>1 = t\<^isub>1 \<AND> \<dots> \<AND> x\<^isub>n = t\<^isub>n \<IN> s \<END>"} \end{center} \noindent which bind all the @{text "x\<^isub>i"} in @{text s}, we might not care about the order in which the @{text "x\<^isub>i = t\<^isub>i"} are given, but we do care about the information that there are as many @{text "x\<^isub>i"} as there are @{text "t\<^isub>i"}. We lose this information if we represent the @{text "\<LET>"}-constructor by something like % \begin{center} @{text "\<LET> [x\<^isub>1,\<dots>,x\<^isub>n].s [t\<^isub>1,\<dots>,t\<^isub>n]"} \end{center} \noindent where the notation @{text "[_]._"} indicates that the list of @{text "x\<^isub>i"} becomes bound in @{text s}. In this representation the term \mbox{@{text "\<LET> [x].s [t\<^isub>1, t\<^isub>2]"}} is a perfectly legal instance, but the lengths of 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 into very messy reasoning (see for example~\cite{BengtsonParow09}). To avoid this, we will allow type specifications for $\mathtt{let}$s as follows % \begin{center} \begin{tabular}{r@ {\hspace{2mm}}r@ {\hspace{2mm}}l} @{text trm} & @{text "::="} & @{text "\<dots>"}\\ & @{text "|"} & @{text "\<LET> as::assn s::trm"}\hspace{4mm} \isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text "s"}\\[1mm] @{text assn} & @{text "::="} & @{text "\<ANIL>"}\\ & @{text "|"} & @{text "\<ACONS> name trm assn"} \end{tabular} \end{center} \noindent where @{text assn} is an auxiliary type representing a list of assignments and @{text bn} an auxiliary function identifying the variables to be bound by the @{text "\<LET>"}. This function can be defined by recursion over @{text assn} as follows \begin{center} @{text "bn(\<ANIL>) ="} @{term "{}"} \hspace{5mm} @{text "bn(\<ACONS> x t as) = {x} \<union> bn(as)"} \end{center} \noindent The scope of the binding is indicated by labels given to the types, for example @{text "s::trm"}, and a binding clause, in this case \isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text "s"}. This binding clause states to bind in @{text s} all the names the function call @{text "bn(as)"} returns. This style of specifying terms and bindings is heavily inspired by the syntax of the Ott-tool \cite{ott-jfp}. However, we will not be able to cope with all specifications that are allowed by Ott. One reason is that Ott lets the user to specify ``empty'' types like \begin{center} @{text "t ::= t t | \<lambda>x. t"} \end{center} \noindent where no clause for variables is given. Arguably, such specifications make some sense in the context of Coq's type theory (which Ott supports), but not at all in a HOL-based environment where every datatype must have a non-empty set-theoretic model \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 about alpha-equated terms is much easier than reasoning with raw terms. The fundamental reason for this is that the HOL-logic underlying Nominal Isabelle allows us to replace ``equals-by-equals''. In contrast, replacing ``alpha-equals-by-alpha-equals'' in a representation based on raw terms requires a lot of extra reasoning work. Although in informal settings a reasoning infrastructure for alpha-equated terms is nearly always taken for granted, establishing it automatically in the Isabelle/HOL theorem prover is a rather non-trivial task. For every specification we will need to construct a type containing as elements the alpha-equated terms. To do so, we use the standard HOL-technique of defining a new type by identifying a non-empty subset of an existing type. The construction we perform in Isabelle/HOL can be illustrated by the following picture: \begin{center} \begin{tikzpicture} %\draw[step=2mm] (-4,-1) grid (4,1); \draw[very thick] (0.7,0.4) circle (4.25mm); \draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9); \draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05); \draw (-2.0, 0.845) -- (0.7,0.845); \draw (-2.0,-0.045) -- (0.7,-0.045); \draw ( 0.7, 0.4) node {\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]clas.\end{tabular}}; \draw (-2.4, 0.4) node {\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}}; \draw (1.8, 0.48) node[right=-0.1mm] {\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ (sets of raw terms)\end{tabular}}; \draw (0.9, -0.35) node {\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}}; \draw (-3.25, 0.55) node {\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}}; \draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3); \draw (-0.95, 0.3) node[above=0mm] {isomorphism}; \end{tikzpicture} \end{center} \noindent We take as the starting point a definition of raw terms (defined as a datatype in Isabelle/HOL); identify then the alpha-equivalence classes in the type of sets of raw terms according to our alpha-equivalence relation and finally define the new type as these alpha-equivalence classes (non-emptiness is satisfied whenever the raw terms are definable as datatype in Isabelle/HOL and the property that our relation for alpha-equivalence is indeed an equivalence relation). The fact that we obtain an isomorphism between the new type and the non-empty subset shows that the new type is a faithful representation of alpha-equated terms. That is not the case for example for terms using the locally nameless representation of binders \cite{McKinnaPollack99}: in this representation there are ``junk'' terms that need to be excluded by reasoning about a well-formedness predicate. The problem with introducing a new type in Isabelle/HOL is that in order to be useful, a reasoning infrastructure needs to be ``lifted'' from the underlying subset to the new type. This is usually a tricky and arduous task. To ease it, we re-implemented in Isabelle/HOL the quotient package described by Homeier \cite{Homeier05} for the HOL4 system. This package allows us to lift definitions and theorems involving raw terms to definitions and theorems involving alpha-equated terms. For example if we define the free-variable function over raw lambda-terms \begin{center} @{text "fv(x) = {x}"}\hspace{10mm} @{text "fv(t\<^isub>1 t\<^isub>2) = fv(t\<^isub>1) \<union> fv(t\<^isub>2)"}\\[1mm] @{text "fv(\<lambda>x.t) = fv(t) - {x}"} \end{center} \noindent then with the help of the quotient package we can obtain a function @{text "fv\<^sup>\<alpha>"} operating on quotients, or alpha-equivalence classes of lambda-terms. This lifted function is characterised by the equations \begin{center} @{text "fv\<^sup>\<alpha>(x) = {x}"}\hspace{10mm} @{text "fv\<^sup>\<alpha>(t\<^isub>1 t\<^isub>2) = fv\<^sup>\<alpha>(t\<^isub>1) \<union> fv\<^sup>\<alpha>(t\<^isub>2)"}\\[1mm] @{text "fv\<^sup>\<alpha>(\<lambda>x.t) = fv\<^sup>\<alpha>(t) - {x}"} \end{center} \noindent (Note that this means also the term-constructors for variables, applications and lambda are lifted to the quotient level.) This construction, of course, only works if alpha-equivalence is indeed an equivalence relation, and the lifted 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 therefore can establish a reasoning infrastructure for alpha-equated terms. The examples we have in mind where our reasoning infrastructure will be helpful includes the term language of System @{text "F\<^isub>C"}, also known as Core-Haskell (see Figure~\ref{corehas}). This term language involves patterns that have lists of type-, coercion- and term-variables, all of which are bound in @{text "\<CASE>"}-expressions. One difficulty is that we do not know in advance how many variables need to be bound. Another is that each bound variable comes with a kind or type annotation. Representing such binders with single binders and reasoning about them in a theorem prover would be a major pain. \medskip \noindent {\bf Contributions:} We provide new definitions for when terms involving multiple binders are alpha-equivalent. These definitions are inspired by earlier work of Pitts \cite{Pitts04}. By means of automatic proofs, we establish a reasoning infrastructure for alpha-equated terms, including properties about support, freshness and equality conditions for alpha-equated terms. We are also able to derive, at the moment only manually, strong induction principles that have the variable convention already built in. \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 term-language of System @{text "F\<^isub>C"} \cite{CoreHaskell}, also often referred to as \emph{Core-Haskell}. In this version of the term-language 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 sorts of atoms can be used to represent different kinds of variables, such as the term-, coercion- and type-variables in Core-Haskell. It is assumed that there is an infinite supply of atoms for each sort. However, in order to simplify the description, we shall restrict ourselves in what follows to only one sort of atoms. Permutations are bijective functions from atoms to atoms that are the identity everywhere except on a finite number of atoms. There is a two-place permutation operation written % \begin{center} @{text "_ \<bullet> _ :: perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"} \end{center} \noindent in which the generic type @{text "\<beta>"} stands for the type of the object over which the permutation acts. In Nominal Isabelle, the identity permutation is written as @{term "0::perm"}, the composition of two permutations @{term p} and @{term q} as \mbox{@{term "p + q"}}, and the inverse permutation of @{term p} as @{text "- p"}. The permutation operation is defined by induction over the type-hierarchy (see \cite{HuffmanUrban10}); for example permutations acting on products, lists, sets, functions and booleans is given by: % \begin{equation}\label{permute} \mbox{\begin{tabular}{@ {}cc@ {}} \begin{tabular}{@ {}l@ {}} @{thm permute_prod.simps[no_vars, THEN eq_reflection]}\\[2mm] @{thm permute_list.simps(1)[no_vars, THEN eq_reflection]}\\ @{thm permute_list.simps(2)[no_vars, THEN eq_reflection]}\\ \end{tabular} & \begin{tabular}{@ {}l@ {}} @{thm permute_set_eq[no_vars, THEN eq_reflection]}\\ @{text "p \<bullet> f \<equiv> \<lambda>x. p \<bullet> (f (- p \<bullet> x))"}\\ @{thm permute_bool_def[no_vars, THEN eq_reflection]}\\ \end{tabular} \end{tabular}} \end{equation} \noindent Concrete permutations in Nominal Isabelle are built up from swappings, written as \mbox{@{text "(a b)"}}, which are permutations that behave as follows: % \begin{center} @{text "(a b) = \<lambda>c. if a = c then b else if b = c then a else c"} \end{center} The most original aspect of the nominal logic work of Pitts is a general definition for the notion of the ``set of free variables of an object @{text "x"}''. This notion, written @{term "supp x"}, is general in the sense that it applies not only to lambda-terms (alpha-equated or not), but also to lists, products, sets and even functions. The definition depends only on the permutation operation and on the notion of equality defined for the type of @{text x}, namely: % \begin{equation}\label{suppdef} @{thm supp_def[no_vars, THEN eq_reflection]} \end{equation} \noindent There is also the derived notion for when an atom @{text a} is \emph{fresh} for an @{text x}, defined as % \begin{center} @{thm fresh_def[no_vars]} \end{center} \noindent We also use for sets of atoms the abbreviation @{thm (lhs) fresh_star_def[no_vars]}, defined as @{thm (rhs) fresh_star_def[no_vars]}. A striking consequence of these definitions is that we can prove without knowing anything about the structure of @{term x} that swapping two fresh atoms, say @{text a} and @{text b}, leave @{text x} unchanged. \begin{property}\label{swapfreshfresh} @{thm[mode=IfThen] swap_fresh_fresh[no_vars]} \end{property} While often the support of an object can be relatively easily described, for example for atoms, products, lists, function applications, booleans and permutations\\[-6mm] % \begin{eqnarray} @{term "supp a"} & = & @{term "{a}"}\\ @{term "supp (x, y)"} & = & @{term "supp x \<union> supp y"}\\ @{term "supp []"} & = & @{term "{}"}\\ @{term "supp (x#xs)"} & = & @{term "supp x \<union> supp xs"}\\ @{text "supp (f x)"} & @{text "\<subseteq>"} & @{term "supp f \<union> supp x"}\label{suppfun}\\ @{term "supp b"} & = & @{term "{}"}\\ @{term "supp p"} & = & @{term "{a. p \<bullet> a \<noteq> a}"} \end{eqnarray} \noindent in some cases it can be difficult to characterise the support precisely, and only an approximation can be established (see \eqref{suppfun} above). Reasoning about such approximations can be simplified with the notion \emph{supports}, defined as follows: \begin{defn} A set @{text S} \emph{supports} @{text x} if for all atoms @{text a} and @{text b} not in @{text S} we have @{term "(a \<rightleftharpoons> b) \<bullet> x = x"}. \end{defn} \noindent The main point of @{text supports} is that we can establish the following two properties. \begin{property}\label{supportsprop} {\it i)} @{thm[mode=IfThen] supp_is_subset[no_vars]}\\ {\it ii)} @{thm supp_supports[no_vars]}. \end{property} Another important notion in the nominal logic work is \emph{equivariance}. For a function @{text f}, say of type @{text "\<alpha> \<Rightarrow> \<beta>"}, to be equivariant it is required that every permutation leaves @{text f} unchanged, that is % \begin{equation}\label{equivariancedef} @{term "\<forall>p. p \<bullet> f = f"} \end{equation} \noindent or equivalently that a permutation applied to the application @{text "f x"} can be moved to the argument @{text x}. That means for equivariant functions @{text f} we have for all permutations @{text p} % \begin{equation}\label{equivariance} @{text "p \<bullet> f = f"} \;\;\;\textit{if and only if}\;\;\; @{text "p \<bullet> (f x) = f (p \<bullet> x)"} \end{equation} \noindent From property \eqref{equivariancedef} and the definition of @{text supp}, we can be easily deduce that an equivariant function has empty support. Finally, the nominal logic work provides us with convenient means to rename binders. While in the older version of Nominal Isabelle, we used extensively Property~\ref{swapfreshfresh} for renaming single binders, this property proved unwieldy for dealing with multiple binders. For such pinders the following generalisations turned out to be easier to use. \begin{property}\label{supppermeq} @{thm[mode=IfThen] supp_perm_eq[no_vars]} \end{property} \begin{property} For a finite set @{text as} and a finitely supported @{text x} with @{term "as \<sharp>* x"} and also a finitely supported @{text c}, there exists a permutation @{text p} such that @{term "(p \<bullet> as) \<sharp>* c"} and @{term "supp x \<sharp>* p"}. \end{property} \noindent The idea behind the second property is that given a finite set @{text as} of binders (being bound, or fresh, in @{text x} is ensured by the assumption @{term "as \<sharp>* x"}), then there exists a permutation @{text p} such that the renamed binders @{term "p \<bullet> as"} avoid @{text c} (which can be arbitrarily chosen as long as it is finitely supported) and also @{text "p"} does not affect anything in the support of @{text x} (that is @{term "supp x \<sharp>* p"}). The last fact and Property~\ref{supppermeq} allow us to ``rename'' just the binders @{text as} in @{text x}, because @{term "p \<bullet> x = x"}. Most properties given in this section are described in \cite{HuffmanUrban10} and of course all are formalised in Isabelle/HOL. In the next sections we will make extensively use of these properties in order to define alpha-equivalence in the presence of multiple binders.*}section {* General Binders\label{sec:binders} *}text {* In Nominal Isabelle, the user is expected to write down a specification of a term-calculus and then a reasoning infrastructure is automatically derived from this specification (remember that Nominal Isabelle is a definitional extension of Isabelle/HOL, which does not introduce any new axioms). In order to keep our work with deriving the reasoning infrastructure manageable, we will wherever possible state definitions and perform proofs on the user-level of Isabelle/HOL, as opposed to write custom ML-code that generates them anew for each specification. To that end, we will consider first pairs @{text "(as, x)"} of type @{text "(atom set) \<times> \<beta>"}. These pairs are intended to represent the abstraction, or binding, of the set @{text "as"} in the body @{text "x"}. The first question we have to answer is when two pairs @{text "(as, x)"} and @{text "(bs, y)"} are alpha-equivalent? (At the moment we are interested in the notion of alpha-equivalence that is \emph{not} preserved by adding vacuous binders.) To answer this, we identify four conditions: {\it i)} given a free-variable function @{text "fv"} of type \mbox{@{text "\<beta> \<Rightarrow> atom set"}}, then @{text x} and @{text y} need to have the same set of free variables; moreover there must be a permutation @{text p} such that {\it ii)} @{text p} leaves the free variables of @{text x} and @{text y} unchanged, but {\it iii)} ``moves'' their bound names so that we obtain modulo a relation, say \mbox{@{text "_ R _"}}, two equivalent terms. We also require {\it iv)} that @{text p} makes the sets of abstracted atoms @{text as} and @{text bs} equal. The requirements {\it i)} to {\it iv)} can be stated formally as follows: % \begin{equation}\label{alphaset} \begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l} \multicolumn{2}{l}{@{term "(as, x) \<approx>gen R fv p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}\hspace{30mm}}\\[1mm] & @{term "fv(x) - as = fv(y) - bs"}\\ @{text "\<and>"} & @{term "(fv(x) - as) \<sharp>* p"}\\ @{text "\<and>"} & @{text "(p \<bullet> x) R y"}\\ @{text "\<and>"} & @{term "(p \<bullet> as) = bs"}\\ \end{array} \end{equation} \noindent Note that this relation is dependent on the permutation @{text "p"}. Alpha-equivalence between two pairs is then the relation where we existentially quantify over this @{text "p"}. Also note that the relation is dependent on a free-variable function @{text "fv"} and a relation @{text "R"}. The reason for this extra generality is that we will use $\approx_{\textit{set}}$ for both ``raw'' terms and alpha-equated terms. In the latter case, $R$ will be replaced by equality @{text "="} and we will prove that @{text "fv"} is equal to @{text "supp"}. The definition in \eqref{alphaset} does not make any distinction between the order of abstracted variables. If we want this, then we can define alpha-equivalence for pairs of the form \mbox{@{text "(as, x)"}} with type @{text "(atom list) \<times> \<beta>"} as follows % \begin{equation}\label{alphalist} \begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l} \multicolumn{2}{l}{@{term "(as, x) \<approx>lst R fv p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}\hspace{30mm}}\\[1mm] & @{term "fv(x) - (set as) = fv(y) - (set bs)"}\\ \wedge & @{term "(fv(x) - set as) \<sharp>* p"}\\ \wedge & @{text "(p \<bullet> x) R y"}\\ \wedge & @{term "(p \<bullet> as) = bs"}\\ \end{array} \end{equation} \noindent where @{term set} is a function that coerces a list of atoms into a set of atoms. Now the last clause ensures that the order of the binders matters. If we do not want to make any difference between the order of binders \emph{and} also allow vacuous binders, then we keep sets of binders, but drop the fourth condition in \eqref{alphaset}: % \begin{equation}\label{alphares} \begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l} \multicolumn{2}{l}{@{term "(as, x) \<approx>res R fv p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}\hspace{30mm}}\\[1mm] & @{term "fv(x) - as = fv(y) - bs"}\\ \wedge & @{term "(fv(x) - as) \<sharp>* p"}\\ \wedge & @{text "(p \<bullet> x) R y"}\\ \end{array} \end{equation} It might be useful to consider some examples for how these definitions of alpha-equivalence pan out in practise. For this consider the case of abstracting a set of variables over types (as in type-schemes). We set @{text R} to be the equality and for @{text "fv(T)"} we define \begin{center} @{text "fv(x) = {x}"} \hspace{5mm} @{text "fv(T\<^isub>1 \<rightarrow> T\<^isub>2) = fv(T\<^isub>1) \<union> fv(T\<^isub>2)"} \end{center} \noindent Now recall the examples shown in \eqref{ex1}, \eqref{ex2} and \eqref{ex3}. It can be easily checked that @{text "({x, y}, x \<rightarrow> y)"} and @{text "({y, x}, y \<rightarrow> x)"} are alpha-equivalent according to $\approx_{\textit{set}}$ and $\approx_{\textit{res}}$ by taking @{text p} to be the swapping @{term "(x \<rightleftharpoons> y)"}. In case of @{text "x \<noteq> y"}, then @{text "([x, y], x \<rightarrow> y)"} $\not\approx_{\textit{list}}$ @{text "([y, x], x \<rightarrow> y)"} since there is no permutation that makes the lists @{text "[x, y]"} and @{text "[y, x]"} equal, and also leaves the type \mbox{@{text "x \<rightarrow> y"}} unchanged. Another example is @{text "({x}, x)"} $\approx_{\textit{res}}$ @{text "({x, y}, x)"} which holds by taking @{text p} to be the identity permutation. However, if @{text "x \<noteq> y"}, then @{text "({x}, x)"} $\not\approx_{\textit{set}}$ @{text "({x, y}, x)"} since there is no permutation that makes the sets @{text "{x}"} and @{text "{x, y}"} equal (similarly for $\approx_{\textit{list}}$). It can also relatively easily be shown that all tree notions of alpha-equivalence coincide, if we only abstract a single atom. % looks too ugly %\noindent %Let $\star$ range over $\{set, res, list\}$. We prove next under which %conditions the $\approx\hspace{0.05mm}_\star^{\fv, R, p}$ are equivalence %relations and equivariant: % %\begin{lemma} %{\it i)} Given the fact that $x\;R\;x$ holds, then %$(as, x) \approx\hspace{0.05mm}^{\fv, R, 0}_\star (as, x)$. {\it ii)} Given %that @{text "(p \<bullet> x) R y"} implies @{text "(-p \<bullet> y) R x"}, then %$(as, x) \approx\hspace{0.05mm}^{\fv, R, p}_\star (bs, y)$ implies %$(bs, y) \approx\hspace{0.05mm}^{\fv, R, - p}_\star (as, x)$. {\it iii)} Given %that @{text "(p \<bullet> x) R y"} and @{text "(q \<bullet> y) R z"} implies %@{text "((q + p) \<bullet> x) R z"}, then $(as, x) \approx\hspace{0.05mm}^{\fv, R, p}_\star (bs, y)$ %and $(bs, y) \approx\hspace{0.05mm}^{\fv, R, q}_\star (cs, z)$ implies %$(as, x) \approx\hspace{0.05mm}^{\fv, R, q + p}_\star (cs, z)$. Given %@{text "(q \<bullet> x) R y"} implies @{text "(p \<bullet> (q \<bullet> x)) R (p \<bullet> y)"} and %@{text "p \<bullet> (fv x) = fv (p \<bullet> x)"} then @{text "p \<bullet> (fv y) = fv (p \<bullet> y)"}, then %$(as, x) \approx\hspace{0.05mm}^{\fv, R, q}_\star (bs, y)$ implies %$(p \;\isasymbullet\; as, p \;\isasymbullet\; x) \approx\hspace{0.05mm}^{\fv, R, q}_\star %(p \;\isasymbullet\; bs, p \;\isasymbullet\; y)$. %\end{lemma} %\begin{proof} %All properties are by unfolding the definitions and simple calculations. %\end{proof} In the rest of this section we are going to introduce three abstraction types. For this we define % \begin{equation} @{term "abs_set (as, x) (bs, x) \<equiv> \<exists>p. alpha_gen (as, x) equal supp p (bs, x)"} \end{equation} \noindent (similarly for $\approx_{\textit{abs\_list}}$ and $\approx_{\textit{abs\_res}}$). We can show that these relations are equivalence relations and equivariant. \begin{lemma}\label{alphaeq} The relations $\approx_{\textit{abs\_set}}$, $\approx_{\textit{abs\_list}}$ and $\approx_{\textit{abs\_res}}$ are equivalence relations, and if @{term "abs_set (as, x) (bs, y)"} then also @{term "abs_set (p \<bullet> as, p \<bullet> x) (p \<bullet> bs, p \<bullet> y)"} (similarly for the other two relations). \end{lemma} \begin{proof} Reflexivity is by taking @{text "p"} to be @{text "0"}. For symmetry we have a permutation @{text p} and for the proof obligation take @{term "-p"}. In case of transitivity, we have two permutations @{text p} and @{text q}, and for the proof obligation use @{text "q + p"}. All conditions are then by simple calculations. \end{proof} \noindent This lemma allows us to use our quotient package and introduce new types @{text "\<beta> abs_set"}, @{text "\<beta> abs_res"} and @{text "\<beta> abs_list"} representing alpha-equivalence classes of pairs. The elements in these types will be, respectively, written as: \begin{center} @{term "Abs as x"} \hspace{5mm} @{term "Abs_lst as x"} \hspace{5mm} @{term "Abs_res as x"} \end{center} \noindent indicating that a set (or list) @{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 the support of abstractions, namely: \begin{thm}[Support of Abstractions]\label{suppabs} Assuming @{text x} has finite support, then\\[-6mm] \begin{center} \begin{tabular}{l@ {\hspace{2mm}}c@ {\hspace{2mm}}l} @{thm (lhs) supp_abs(1)[no_vars]} & $=$ & @{thm (rhs) supp_abs(1)[no_vars]}\\ @{thm (lhs) supp_abs(2)[no_vars]} & $=$ & @{thm (rhs) supp_abs(2)[no_vars]}\\ @{thm (lhs) supp_abs(3)[where bs="as", no_vars]} & $=$ & @{thm (rhs) supp_abs(3)[where bs="as", no_vars]} \end{tabular} \end{center} \end{thm} \noindent Below we will show the first equation. The others follow by similar arguments. By definition of the abstraction type @{text "abs_set"} we have % \begin{equation}\label{abseqiff} @{thm (lhs) abs_eq_iff(1)[where bs="as" and cs="bs", no_vars]} \;\;\text{if and only if}\;\; @{thm (rhs) abs_eq_iff(1)[where bs="as" and cs="bs", no_vars]} \end{equation} \noindent and also % \begin{equation} @{thm permute_Abs[no_vars]} \end{equation} \noindent The second fact derives from the definition of permutations acting on pairs (see \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. \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 This lemma 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 Thm~\ref{suppabs}. The ``other half'' is a bit more involved. To establish it, we use a trick from \cite{Pitts04} and first define an auxiliary function @{text aux}, taking an abstraction as argument: % \begin{center} @{thm supp_gen.simps[THEN eq_reflection, no_vars]} \end{center} \noindent Using the second equation in \eqref{equivariance}, we can show that @{text "aux"} is equivariant (since @{term "p \<bullet> (supp x - as) = (supp (p \<bullet> x)) - (p \<bullet> as)"}) and therefore has empty support. This in turn means % \begin{center} @{term "supp (supp_gen (Abs as x)) \<subseteq> supp (Abs as x)"} \end{center} \noindent using \eqref{suppfun}. Assuming @{term "supp x - as"} is a finite set we further obtain % \begin{equation}\label{halftwo} @{thm (concl) supp_abs_subset1(1)[no_vars]} \end{equation} \noindent since for finite sets of atoms, @{text "bs"}, we have @{thm (concl) supp_finite_atom_set[where S="bs", no_vars]}. Finally, taking \eqref{halfone} and \eqref{halftwo} together establishes Theorem~\ref{suppabs}. The method of first considering abstractions of the form @{term "Abs as x"} etc is motivated by the fact that properties about them can be conveninetly established at the Isabelle/HOL level. It would be difficult to write custom ML-code that derives automatically such properties for every term-constructor that binds some atoms. Also the generality of the definitions for alpha-equivalence will help us in the next section.*}section {* Alpha-Equivalence and Free Variables\label{sec:alpha} *}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"}$^\alpha_1$, \ldots, @{text ty}$^\alpha_n$, and an associated collection of binding functions, say @{text bn}$^\alpha_1$, \ldots, @{text bn}$^\alpha_m$. The syntax in Nominal Isabelle for such specifications is roughly as follows: % \begin{equation}\label{scheme} \mbox{\begin{tabular}{@ {\hspace{-5mm}}p{1.8cm}l} type \mbox{declaration part} & $\begin{cases} \mbox{\begin{tabular}{l} \isacommand{nominal\_datatype} @{text ty}$^\alpha_1 = \ldots$\\ \isacommand{and} @{text ty}$^\alpha_2 = \ldots$\\ $\ldots$\\ \isacommand{and} @{text ty}$^\alpha_n = \ldots$\\ \end{tabular}} \end{cases}$\\ binding \mbox{function part} & $\begin{cases} \mbox{\begin{tabular}{l} \isacommand{with} @{text bn}$^\alpha_1$ \isacommand{and} \ldots \isacommand{and} @{text bn}$^\alpha_m$\\ \isacommand{where}\\ $\ldots$\\ \end{tabular}} \end{cases}$\\ \end{tabular}} \end{equation} \noindent Every type declaration @{text ty}$^\alpha_{1..n}$ consists of a collection of term-constructors, each of which comes with a list of labelled types that stand for the types of the arguments of the term-constructor. For example a term-constructor @{text "C\<^sup>\<alpha>"} might have \begin{center} @{text "C\<^sup>\<alpha> label\<^isub>1::ty"}$'_1$ @{text "\<dots> label\<^isub>l::ty"}$'_l\;\;$ @{text "binding_clauses"} \end{center} \noindent whereby some of the @{text ty}$'_{1..l}$ (or their components) might 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 label}\\ \isacommand{bind\_set}\; {\it binders}\; \isacommand{in}\; {\it label}\\ \isacommand{bind\_res}\; {\it binders}\; \isacommand{in}\; {\it label}\\ \end{tabular} \end{center} \noindent The first mode is for binding lists of atoms (the order of binders matters); the second is for sets of binders (the order does not matter, but the cardinality does) and the last is for sets of binders (with vacuous binders preserving alpha-equivalence). The ``\isacommand{in}-part'' of a binding clause will be called the \emph{body}; the ``\isacommand{bind}-part'' will be the \emph{binder}. In addition we distinguish between \emph{shallow} and \emph{deep} binders. Shallow binders are of the form \isacommand{bind}\; {\it label}\; \isacommand{in}\; {\it label'} (similar for the other two modes). The restriction we impose on shallow binders is that the {\it label} must either refer to a type that is an atom type or to a type that is a finite set or list of an atom type. Two examples for the use of shallow binders are the specification of lambda-terms, where a single name is bound, and type-schemes, where a finite set of names is bound: \begin{center} \begin{tabular}{@ {}cc@ {}} \begin{tabular}{@ {}l@ {\hspace{-1mm}}} \isacommand{nominal\_datatype} @{text lam} =\\ \hspace{5mm}\phantom{$\mid$}~@{text "Var name"}\\ \hspace{5mm}$\mid$~@{text "App lam lam"}\\ \hspace{5mm}$\mid$~@{text "Lam x::name t::lam"}\\ \hspace{21mm}\isacommand{bind} @{text x} \isacommand{in} @{text t}\\ \end{tabular} & \begin{tabular}{@ {}l@ {}} \isacommand{nominal\_datatype}~@{text ty} =\\ \hspace{5mm}\phantom{$\mid$}~@{text "TVar name"}\\ \hspace{5mm}$\mid$~@{text "TFun ty ty"}\\ \isacommand{and}~@{text "tsc = All xs::(name fset) T::ty"}\\ \hspace{24mm}\isacommand{bind\_res} @{text xs} \isacommand{in} @{text T}\\ \end{tabular} \end{tabular} \end{center} \noindent Note that in this specification \emph{name} refers to an atom type. If we have shallow binders that ``share'' a body, for instance $t$ in the following term-constructor \begin{center} \begin{tabular}{ll} @{text "Foo x::name y::name t::lam"} & \isacommand{bind} @{text x} \isacommand{in} @{text t},\; \isacommand{bind} @{text y} \isacommand{in} @{text t} \end{tabular} \end{center} \noindent then we have to make sure the modes of the binders agree. We cannot have, for instance, in the first binding clause the mode \isacommand{bind} and in the second \isacommand{bind\_set}. A \emph{deep} binder uses an auxiliary binding function that ``picks'' out the atoms in one argument of the term-constructor, which can be bound in other arguments and also in the same argument (we will call such binders \emph{recursive}, see below). The corresponding binding functions are expected to return either a set of atoms (for \isacommand{bind\_set} and \isacommand{bind\_res}) or a list of atoms (for \isacommand{bind}). They can be defined by primitive recursion over the corresponding type; the equations must be given in the binding function part of the scheme shown in \eqref{scheme}. For example a term-calculus containing @{text "Let"}s with tuple patterns might be specified as: \begin{center} \begin{tabular}{l} \isacommand{nominal\_datatype} @{text trm} =\\ \hspace{5mm}\phantom{$\mid$}~@{term "Var name"}\\ \hspace{5mm}$\mid$~@{term "App trm trm"}\\ \hspace{5mm}$\mid$~@{text "Lam x::name t::trm"} \;\;\isacommand{bind} @{text x} \isacommand{in} @{text t}\\ \hspace{5mm}$\mid$~@{text "Let p::pat trm t::trm"} \;\;\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t}\\ \isacommand{and} @{text pat} =\\ \hspace{5mm}\phantom{$\mid$}~@{text PNil}\\ \hspace{5mm}$\mid$~@{text "PVar name"}\\ \hspace{5mm}$\mid$~@{text "PTup pat pat"}\\ \isacommand{with}~@{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)"}\\ \end{tabular} \end{center} \noindent In this specification the function @{text "bn"} determines which atoms of @{text p} are bound in the argument @{text "t"}. Note that in the second last 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 will shortly become clear, we cannot return an atom in a binding function that is also bound in the corresponding term-constructor. That means in the example above that the term-constructors @{text PVar} and @{text PTup} must not have a binding clause. 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 definitions and proofs later on. The most drastic restriction we have to impose on deep binders is that we cannot have ``overlapping'' deep binders. Consider for example the term-constructors: \begin{center} \begin{tabular}{ll} @{text "Foo p::pat q::pat t::trm"} & \isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t},\; \isacommand{bind} @{text "bn(q)"} \isacommand{in} @{text t}\\ @{text "Foo' x::name p::pat t::trm"} & \isacommand{bind} @{text x} \isacommand{in} @{text t},\; \isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t} \end{tabular} \end{center} \noindent In the first case we might bind all atoms from the pattern @{text p} in @{text t} and also all atoms from @{text q} in @{text t}. As a result we have no way to determine whether the binder came from the binding function @{text "bn(p)"} or @{text "bn(q)"}. Similarly in the second case. The reason why we must exclude such specifications is that they cannot be represent by the general binders described in Section \ref{sec:binders}. However the following two term-constructors are allowed \begin{center} \begin{tabular}{ll} @{text "Bar p::pat t::trm s::trm"} & \isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t},\; \isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text s}\\ @{text "Bar' p::pat t::trm"} & \isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text p},\; \isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t}\\ \end{tabular} \end{center} \noindent since there is no overlap of binders. Note that in the last example we wrote {\it\isacommand{bind}\;bn(p)\;\isacommand{in}\;p}. Whenever such a binding clause is present, we will call the binder \emph{recursive}. To see the purpose of such recursive binders, compare ``plain'' @{text "Let"}s and @{text "Let_rec"}s in the following specification: % \begin{equation}\label{letrecs} \mbox{% \begin{tabular}{@ {}l@ {}} \isacommand{nominal\_datatype}~@{text "trm ="}\\ \hspace{5mm}\phantom{$\mid$}\ldots\\ \hspace{5mm}$\mid$~@{text "Let as::assn t::trm"} \;\;\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text t}\\ \hspace{5mm}$\mid$~@{text "Let_rec as::assn t::trm"} \;\;\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text t}, \isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text as}\\ \isacommand{and} {\it assn} =\\ \hspace{5mm}\phantom{$\mid$}~@{text "ANil"}\\ \hspace{5mm}$\mid$~@{text "ACons name trm assn"}\\ \isacommand{with} @{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 free-variable function and alpha-equivalence relation, which we are going to describe in the rest of this section. Having dealt with all syntax matters, the problem now is how we can turn specifications into actual type definitions in Isabelle/HOL and then establish a reasoning infrastructure for them. Because of the problem Pottier and Cheney pointed out, we cannot in general re-arrange arguments of term-constructors so that binders and their bodies are next to each other, and then use the type constructors @{text "abs_set"}, @{text "abs_res"} and @{text "abs_list"} from Section \ref{sec:binders}. Therefore we will first extract datatype definitions from the specification and then define independently an alpha-equivalence relation over them. The datatype definition can be obtained by stripping off the binding clauses and the labels on the types. We also have to invent new names for the types @{text "ty\<^sup>\<alpha>"} and term-constructors @{text "C\<^sup>\<alpha>"} given by user. In our implementation we just use the affix ``@{text "_raw"}''. But for the purpose of this paper, we just use the superscript @{text "_\<^sup>\<alpha>"} to indicate that a notion is defined over alpha-equivalence classes and leave it out for the corresponding notion defined on the ``raw'' level. So for example we have \begin{center} @{text "ty\<^sup>\<alpha> \<mapsto> ty"} \hspace{2mm}and\hspace{2mm} @{text "C\<^sup>\<alpha> \<mapsto> C"} \end{center} \noindent where @{term ty} is the type used in the quotient construction for @{text "ty\<^sup>\<alpha>"} and @{text "C"} is the term-constructor on the ``raw'' type @{text "ty"}. The resulting datatype definition is legal in Isabelle/HOL provided the datatypes are non-empty and the types in the constructors only occur in positive position (see \cite{Berghofer99} for an indepth description of the datatype package in Isabelle/HOL). We then define the user-specified binding functions, called @{term "bn"}, by primitive recursion over the corresponding raw datatype. We can also easily define permutation operations by primitive recursion so that for each term constructor @{text "C ty\<^isub>1 \<dots> ty\<^isub>n"} we have that \begin{center} @{text "p \<bullet> (C x\<^isub>1 \<dots> x\<^isub>n) = C (p \<bullet> x\<^isub>1) \<dots> (p \<bullet> x\<^isub>n)"} \end{center} % TODO: we did not define permutation types %\noindent %From this definition we can easily show that the raw datatypes are %all permutation types (Def ??) by a simple structural induction over %the @{text "ty"}s. The first non-trivial step we have to perform is the generation free-variable functions from the specifications. Given the raw types @{text "ty\<^isub>1 \<dots> ty\<^isub>n"} we need to define free-variable functions \begin{center} @{text "fv_ty\<^isub>1 :: ty\<^isub>1 \<Rightarrow> atom set \<dots> fv_ty\<^isub>n :: ty\<^isub>n \<Rightarrow> atom set"} \end{center} \noindent We define them together with auxiliary free-variable functions for the binding functions. Given binding functions @{text "bn\<^isub>1 \<dots> bn\<^isub>m"} we need to define % \begin{center} @{text "fv_bn\<^isub>1 :: ty\<^isub>1 \<Rightarrow> atom set \<dots> fv_bn\<^isub>m :: ty\<^isub>m \<Rightarrow> atom set"} \end{center} \noindent The reason for this setup is that in a deep binder not all atoms have to be bound, as we shall see in an example below. We need therefore the function that calculates those unbound atoms. While the idea behind these free-variable functions is clear (they just collect all atoms that are not bound), because of the rather complicated binding mechanisms their definitions are somewhat involved. Given a term-constructor @{text "C"} of type @{text ty} with argument types \mbox{@{text "ty\<^isub>1 \<dots> ty\<^isub>n"}}, the function @{text "fv_ty (C x\<^isub>1 \<dots> x\<^isub>n)"} will be the union of the values calculated below for each argument. First we deal with the case that @{text "x\<^isub>i"} is a binder. From the binding clauses, we can determine whether the argument is a shallow or deep binder, and in the latter case also whether it is a recursive or non-recursive binder. \begin{center} \begin{tabular}{c@ {\hspace{2mm}}p{7cm}} $\bullet$ & @{term "{}"} provided @{text "x\<^isub>i"} is a shallow binder\\ $\bullet$ & @{text "fv_bn x\<^isub>i"} provided @{text "x\<^isub>i"} is a deep non-recursive binder with the auxiliary binding function @{text "bn"}\\ $\bullet$ & @{text "fv_ty\<^isub>i x\<^isub>i - bn x\<^isub>i"} provided @{text "x\<^isub>i"} is a deep recursive binder with the auxiliary binding function @{text "bn"} \end{tabular} \end{center} \noindent The first clause states that shallow binders do not contribute to the free variables; in the second clause, we have to collect all variables that are left unbound by the binding function @{text "bn"}---this is done with function @{text "fv_bn"}; in the third clause, since the binder is recursive, we need to bind all variables specified by @{text "bn"}---therefore we subtract @{text "bn x\<^isub>i"} from the free variables of @{text "x\<^isub>i"}. In case the argument is \emph{not} a binder, we need to consider whether the @{text "x\<^isub>i"} is the body of one or more binding clauses. In this case we first calculate the set @{text "bnds"} as follows: either the corresponding binders are all shallow or there is a single deep binder. In the former case we take @{text bnds} to be the union of all shallow binders; in the latter case, we just take the set of atoms specified by the corrsponding binding function. The value for @{text "x\<^isub>i"} is then given by: % \begin{equation}\label{deepbody} \mbox{% \begin{tabular}{c@ {\hspace{2mm}}p{7cm}} $\bullet$ & @{text "{atom x\<^isub>i} - bnds"} provided @{term "x\<^isub>i"} is an atom\\ $\bullet$ & @{text "(atoms x\<^isub>i) - bnds"} provided @{term "x\<^isub>i"} is a set of atoms\\ $\bullet$ & @{text "(atoms (set x\<^isub>i)) - bnds"} provided @{term "x\<^isub>i"} is a list of atoms\\ $\bullet$ & @{text "(fv_ty\<^isub>i x\<^isub>i) - bnds"} provided @{term "ty\<^isub>i"} is one of the raw datatypes corresponding to the types specified by the user\\% $\bullet$ & @{text "(fv\<^isup>\<alpha> x\<^isub>i) - bnds"} provided @{term "x\<^isub>i"} is a defined nominal datatype% with a free-variable function @{text "fv\<^isup>\<alpha>"}\\ $\bullet$ & @{term "{}"} otherwise \end{tabular}} \end{equation} \noindent Like the coercion function @{text atom} used earlier, @{text "atoms as"} coerces the set @{text as} to the generic atom type. It is defined as @{text "atoms as \<equiv> {atom a | a \<in> as}"}. The last case we need to consider is when @{text "x\<^isub>i"} is neither a binder nor a body of an abstraction. In this case it is defined as in \eqref{deepbody}, except that we do not need to subtract the set @{text bnds}. Next, we need to define a free-variable function @{text "fv_bn\<^isub>j"} for each binding function @{text "bn\<^isub>j"}. The idea behind these functions is to compute the set of free atoms that are not bound by @{text "bn\<^isub>j"}. Because of the restrictions we imposed on the form of binding functions, this can be done automatically by recursively building up the the set of free variables from the arguments that are not bound. Let us assume one clause of a binding function is @{text "bn\<^isub>j (C x\<^isub>1 \<dots> x\<^isub>n) = rhs"}, then @{text "fv_bn\<^isub>j"} is the union of the values calculated for @{text "x\<^isub>i"} of type @{text "ty\<^isub>i"} as follows: \begin{center} \begin{tabular}{c@ {\hspace{2mm}}p{7cm}} \multicolumn{2}{l}{@{text "x\<^isub>i"} occurs in @{text "rhs"}:}\\ $\bullet$ & @{term "{}"} provided @{term "x\<^isub>i"} is a single atom, atom list or atom set\\ $\bullet$ & @{text "fv_bn x\<^isub>i"} in case @{text "rhs"} contains the recursive call @{text "bn x\<^isub>i"}\medskip\\ \end{tabular} \end{center} \begin{center} \begin{tabular}{c@ {\hspace{2mm}}p{7cm}} \multicolumn{2}{l}{@{text "x\<^isub>i"} does not occur in @{text "rhs"}:}\\ $\bullet$ & @{text "atom x\<^isub>i"} provided @{term "x\<^isub>i"} is an atom\\ $\bullet$ & @{text "atoms x\<^isub>i"} provided @{term "x\<^isub>i"} is a set of atoms\\ $\bullet$ & @{term "atoms (set x\<^isub>i)"} provided @{term "x\<^isub>i"} is a list of atoms\\ $\bullet$ & @{text "fv_ty\<^isub>i x\<^isub>i"} provided @{term "ty\<^isub>i"} is one of the raw types corresponding to the types specified by the user\\% $\bullet$ & @{text "fv_ty\<^isup>\<alpha> x\<^isub>i - bnds"} provided @{term "x\<^isub>i"} is not in @{text "rhs"}% and is an existing nominal datatype with the free-variable function @{text "fv\<^isup>\<alpha>"}\\ $\bullet$ & @{term "{}"} otherwise \end{tabular} \end{center} \noindent To see how these definitions work in practise, let us reconsider the term-constructors @{text "Let"} and @{text "Let_rec"} from the example shown in \eqref{letrecs}. For this specification we need to define three free-variable functions, namely @{text "fv\<^bsub>trm\<^esub>"}, @{text "fv\<^bsub>assn\<^esub>"} and @{text "fv\<^bsub>bn\<^esub>"}. They are as follows: % \begin{center} \begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}} @{text "fv\<^bsub>trm\<^esub> (Let as t)"} & @{text "="} & @{text "fv\<^bsub>bn\<^esub> as \<union> (fv\<^bsub>trm\<^esub> t - set (bn as))"}\\ @{text "fv\<^bsub>trm\<^esub> (Let_rec as t)"} & @{text "="} &\\ \multicolumn{3}{r}{@{text "(fv\<^bsub>assn\<^esub> as - set (bn as)) \<union> (fv\<^bsub>trm\<^esub> t - set (bn as))"}}\\[1mm] @{text "fv\<^bsub>assn\<^esub> (ANil)"} & @{text "="} & @{text "[]"}\\ @{text "fv\<^bsub>assn\<^esub> (ACons a t as)"} & @{text "="} & @{text "{atom a} \<union> (fv\<^bsub>trm\<^esub> t) \<union> (fv\<^bsub>assn\<^esub> as)"}\\[1mm] @{text "fv\<^bsub>bn\<^esub> (ANil)"} & @{text "="} & @{text "[]"}\\ @{text "fv\<^bsub>bn\<^esub> (ACons a t as)"} & @{text "="} & @{text "(fv\<^bsub>trm\<^esub> t) \<union> (fv\<^bsub>bn\<^esub> as)"} \end{tabular} \end{center} \noindent Since there are no binding clauses for the term-constructors @{text ANil} and @{text "ACons"}, the corresponding free-variable function @{text "fv\<^bsub>assn\<^esub>"} returns all atoms occuring in an assignment. The binding only takes place in @{text Let} and @{text "Let_rec"}. In the @{text "Let"}-clause we want to bind all atoms given by @{text "set (bn as)"} in @{text t}. Therefore we have to subtract @{text "set (bn as)"} from @{text "fv\<^bsub>trm\<^esub> t"}. However, we also need to add all atoms that are free in @{text "as"}. This is what the purpose of the function @{text "fv\<^bsub>bn\<^esub>"} is. In contrast, in @{text "Let_rec"} we have a recursive binder where we want to also bind all occurences of the atoms @{text "bn as"} inside @{text "as"}. Therefore we have to subtract @{text "set (bn as)"} from @{text "fv\<^bsub>assn\<^esub> as"}, as well as from @{text "fv\<^bsub>trm\<^esub> t"}. An interesting point in this example is that an assignment ``alone'' does not have any bound variables. Only in the context of a @{text Let} or @{text "Let_rec"} will some atoms become bound. This is a phenomenon that has also been pointed out in \cite{ott-jfp}. We can also see that given a @{text "bn"}-function for a type @{text "ty"}, we have that % \begin{equation}\label{bnprop} @{text "fv_ty x = bn x \<union> fv_bn x"}. \end{equation} Next we define alpha-equivalence for the types @{text "ty\<^isub>1, \<dots>, ty\<^isub>n"}. We call them @{text "\<approx>ty\<^isub>1, \<dots>, \<approx>ty\<^isub>n"}. Like with the free-variable functions, we also need to define auxiliary alpha-equivalence relations for the binding functions. Say we have @{text "bn\<^isub>1, \<dots>, bn\<^isub>m"}, then we also define @{text "\<approx>bn\<^isub>1, \<dots>, \<approx>b\<^isub>m"}. To simplify our definitions we will use the following abbreviations for relations and free-variable acting on products. % \begin{center} \begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l} @{text "(x\<^isub>1, y\<^isub>1) (R\<^isub>1 \<otimes> R\<^isub>2) (x\<^isub>2, y\<^isub>2)"} & @{text "\<equiv>"} & @{text "x\<^isub>1 R\<^isub>1 y\<^isub>1 \<and> x\<^isub>2 R\<^isub>2 y\<^isub>2"}\\ @{text "(fv\<^isub>1 \<oplus> fv\<^isub>2) (x, y)"} & @{text "\<equiv>"} & @{text "fv\<^isub>1 x \<union> fv\<^isub>2 y"}\\ \end{tabular} \end{center} The relations for alpha-equivalence are inductively defined predicates, whose clauses have conclusions of the form % \begin{center} @{text "C x\<^isub>1 \<dots> x\<^isub>n \<approx>ty C y\<^isub>1 \<dots> y\<^isub>n"} \end{center} \noindent For what follows, let us assume @{text C} is of type @{text ty} and its arguments are given by @{text "C ty\<^isub>1 \<dots> ty\<^isub>n"}). The task now is to specify what the premises of these clauses are. For this we consider the pairs \mbox{@{text "(x\<^isub>i, y\<^isub>i)"}}, but instead of considering them in turn, it will be easier to analyse these pairs according to ``clusters'' of the binding clauses. Therefore we distinguish the following cases:*}(*<*)consts alpha_ty ::'aconsts alpha_trm ::'aconsts fv_trm :: 'aconsts alpha_trm2 ::'aconsts fv_trm2 :: 'anotation (latex output) alpha_ty ("\<approx>ty") and alpha_trm ("\<approx>\<^bsub>trm\<^esub>") and fv_trm ("fv\<^bsub>trm\<^esub>") and alpha_trm2 ("\<approx>\<^bsub>assn\<^esub> \<otimes> \<approx>\<^bsub>trm\<^esub>") and fv_trm2 ("fv\<^bsub>assn\<^esub> \<oplus> fv\<^bsub>trm\<^esub>") (*>*)text {* \begin{itemize} \item The @{text "(x\<^isub>i, y\<^isub>i)"} are bodies of shallow binders with type @{text "ty"}. We assume the \mbox{@{text "(u\<^isub>1, v\<^isub>1),\<dots>,(u\<^isub>m, v\<^isub>m)"}} are the corresponding binders. For the binding mode \isacommand{bind\_set} we generate the premise \begin{center} @{term "\<exists>p. (u\<^isub>1 \<union> \<xi> \<union> u\<^isub>m, x\<^isub>i) \<approx>gen alpha_ty fv_ty\<^isub>i p (v\<^isub>1 \<union> \<xi> \<union> v\<^isub>m, y\<^isub>i)"} \end{center} For the binding mode \isacommand{bind} we use $\approx_{\textit{list}}$, and for binding mode \isacommand{bind\_res} we use $\approx_{\textit{res}}$. \item The @{text "(x\<^isub>i, y\<^isub>i)"} are deep non-recursive binders with type @{text "ty"} and @{text bn} being the auxiliary binding function. We assume @{text "(u\<^isub>1, v\<^isub>1),\<dots>,(u\<^isub>m, v\<^isub>m)"} are the corresponding bodies with types @{text "ty\<^isub>1,\<dots>, ty\<^isub>m"}. For the binding mode \isacommand{bind\_set} we generate two premises % \begin{center} @{text "x\<^isub>i \<approx>bn y\<^isub>i"}\hfill @{term "\<exists>p. (bn x\<^isub>i, (u\<^isub>1,\<xi>,u\<^isub>m)) \<approx>gen R fv p (bn y\<^isub>i, (v\<^isub>1,\<xi>,v\<^isub>m))"} \end{center} \noindent where @{text R} is @{text "\<approx>ty\<^isub>1 \<otimes> ... \<otimes> \<approx>ty\<^isub>m"} and @{text fv} is @{text "fv_ty\<^isub>1 \<oplus> ... \<oplus> fv_ty\<^isub>m"}. Similarly for the other binding modes. \item The @{text "(x\<^isub>i, y\<^isub>i)"} are deep recursive binders with type @{text "ty"} and with @{text bn} being the auxiliary binding function. We assume @{text "(u\<^isub>1, v\<^isub>1),\<dots>,(u\<^isub>m, v\<^isub>m)"} are the corresponding bodies with types @{text "ty\<^isub>1,\<dots>, ty\<^isub>m"}. For the binding mode \isacommand{bind\_set} we generate the premise % \begin{center} @{term "\<exists>p. (bn x\<^isub>i, (x\<^isub>i, u\<^isub>1,\<xi>,u\<^isub>m)) \<approx>gen R fv p (bn y\<^isub>i, (y\<^isub>i, v\<^isub>1,\<xi>,v\<^isub>m))"} \end{center} \noindent where @{text R} is @{text "\<approx>ty \<otimes> \<approx>ty\<^isub>1 \<otimes> ... \<otimes> \<approx>ty\<^isub>m"} and @{text fv} is @{text "fv_ty \<oplus> fv_ty\<^isub>1 \<oplus> ... \<oplus> fv_ty\<^isub>m"}. Similarly for the other modes. \end{itemize} \noindent From these definition it is clear why we can only support one binding mode per binder and body, as we cannot mix the relations $\approx_{\textit{set}}$, $\approx_{\textit{list}}$ and $\approx_{\textit{res}}$. It is also clear why we had to impose the restriction of excluding overlapping binders, as these would need to be translated to separate abstractions. The only cases that are not covered by the rules above are the cases where @{text "(x\<^isub>i, y\<^isub>i)"} is neither a binder nor a body. Then we just generate @{text "x\<^isub>i \<approx>ty y\<^isub>i"} provided the type of @{text "x\<^isub>i"} and @{text "y\<^isub>i"} is @{text ty} and the arguments are recursive arguments of the term-constructor. If they are non-recursive arguments then we generate just @{text "x\<^isub>i = y\<^isub>i"}. {\bf TODO (I do not understand the definition below yet).} The alpha-equivalence relations for binding functions are similar to the alpha-equivalences for their respective types, the difference is that they ommit checking the arguments that are bound. We assumed that there are no bindings in the type on which the binding function is defined so, there are no permutations involved. For a binding function clause @{text "bn (C x\<^isub>1 \<dots> x\<^isub>n) = rhs"}, two instances of the constructor are equivalent @{text "C x\<^isub>1 \<dots> x\<^isub>n \<approx> C y\<^isub>1 \<dots> y\<^isub>n"} if: \begin{center} \begin{tabular}{cp{7cm}} $\bullet$ & @{text "x\<^isub>j"} is not of a type being defined and occurs in @{text "rhs"}\\ $\bullet$ & @{text "x\<^isub>j = y\<^isub>j"} provided @{text "x\<^isub>j"} is not of a type being defined and does not occur in @{text "rhs"}\\ $\bullet$ & @{text "x\<^isub>j \<approx>bn\<^isub>m y\<^isub>j"} provided @{text "x\<^isub>j"} is of a type being defined occuring in @{text "rhs"} under the binding function @{text "bn\<^isub>m"}\\ $\bullet$ & @{text "x\<^isub>j \<approx> y\<^isub>j"} otherwise\\ \end{tabular} \end{center} Again lets take a look at an example for these definitions. For \eqref{letrecs} we have three relations, namely $\approx_{\textit{trm}}$, $\approx_{\textit{assn}}$ and $\approx_{\textit{bn}}$, with the clauses as follows: \begin{center} \begin{tabular}{@ {}c @ {}} \infer{@{text "Let as t \<approx>\<^bsub>trm\<^esub> Let as' t'"}} {@{text "as \<approx>\<^bsub>bn\<^esub> as'"} & @{term "\<exists>p. (bn as, t) \<approx>lst alpha_trm fv_trm p (bn as', t')"}}\smallskip\\ \infer{@{text "Let_rec as t \<approx>\<^bsub>trm\<^esub> Let_rec as' t'"}} {@{term "\<exists>p. (bn as, (as, t)) \<approx>lst alpha_trm2 fv_trm2 p (bn as', (as', t'))"}} \end{tabular} \end{center} \begin{center} \begin{tabular}{@ {}c @ {}} \infer{@{text "ANil \<approx>\<^bsub>assn\<^esub> ANil"}}{}\\ \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'"}}\medskip\\ \end{tabular} \end{center} \begin{center} \begin{tabular}{@ {}c @ {}} \infer{@{text "ANil \<approx>\<^bsub>bn\<^esub> ANil"}}{}\\ \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'"}}\medskip\\ \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. Therefore we have a $\approx_{\textit{bn}}$-premise in the clause for @{text "Let"} (which is a non-recursive binder), since the terms inside an assignment are not meant to be under the binder. Such a premise is not needed in @{text "Let_rec"}, because there everything from the assignment is under the binder. *}section {* Establishing the Reasoning Infrastructure *}text {* Having made all these definition for raw terms, we can start to introduce the resoning infrastructure for the specified types. For this we first have to show that the alpha-equivalence relation defined in the previous sections are indeed equivalence relations. \begin{lemma} Given the raw types @{text "ty\<^isub>1, \<dots>, ty\<^isub>n"} and binding functions @{text "bn\<^isub>1, \<dots>, bn\<^isub>m"}, the relations @{text "\<approx>ty\<^isub>1, \<dots>, \<approx>ty\<^isub>n"} and @{text "\<approx>bn\<^isub>1 \<dots> \<approx>bn\<^isub>m"} are equivalence relations and equivariant. \end{lemma} \begin{proof} The proof is by induction over the definitions. The non-trivial cases involves premises build up by $\approx_{\textit{set}}$, $\approx_{\textit{res}}$ and $\approx_{\textit{list}}$. They can be dealt with like 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"}$^\alpha_{1..m}$ from the raw term-constructors @{text "C"}$_{1..m}$. Similarly for the free-variable functions @{text "fv_ty"}$^\alpha_{1..n}$ and the binding functions @{text "bn"}$^\alpha_{1..k}$. However, these definitions are of not much use for the user, since the are formulated in terms of the isomorphism we obtained by cerating a new type in Isabelle/HOL (remember the picture shown in the Introduction). {\bf TODO below.} then define the quotient types @{text "ty\<^isub>1\<^isup>\<alpha> \<dots> ty\<^isub>n\<^isup>\<alpha>"}. To lift the raw definitions to the quotient type, we need to prove that they \emph{respect} the relation. We follow the definition of respectfullness given by Homeier~\cite{Homeier05}. The intuition behind a respectfullness condition is that when a function (or constructor) is given arguments that are alpha-equivalent the results are also alpha equivalent. For arguments that are not of any of the relations taken into account, equivalence is replaced by equality. In particular the respectfullness condition for a @{text "bn"} function means that for alpha equivalent raw terms it returns the same bound names. Thanks to the restrictions on the binding functions introduced in Section~\ref{sec:alpha} we can show that are respectful. \begin{lemma} The functions @{text "bn\<^isub>1 \<dots> bn\<^isub>m"}, @{text "fv_ty\<^isub>1 \<dots> fv_ty\<^isub>n"}, the raw constructors, the raw permutations and @{text "\<approx>bn\<^isub>1 \<dots> \<approx>bn\<^isub>m"} are respectful w.r.t. the relations @{text "\<approx>\<^isub>1 \<dots> \<approx>\<^isub>n"}. \end{lemma} \begin{proof} Respectfullness of permutations is a direct consequence of equivariance. All other properties by induction on the alpha-equivalence relation. For @{text "bn"} the thesis follows by simple calculations thanks to the restrictions on the binding functions. For @{text "fv"} functions it follows using respectfullness of @{text "bn"}. For type constructors it is a simple calculation thanks to the way alpha-equivalence was defined. For @{text "alpha_bn"} after a second induction on the second relation by simple calculations. \end{proof} With these respectfullness properties we can use the quotient package to define the above constants on the quotient level. We can then automatically lift the theorems that talk about the raw constants to theorems on the quotient level. The following lifted properties are proved: \begin{center} \begin{tabular}{cp{7cm}}%skipped permute_zero and permute_add, since we do not have a permutation%definition $\bullet$ & permutation defining equations \\ $\bullet$ & @{term "bn"} defining equations \\ $\bullet$ & @{term "fv_ty"} and @{term "fv_bn"} defining equations \\ $\bullet$ & induction. The induction principle that we obtain by lifting is the weak induction principle, just on the term structure \\ $\bullet$ & quasi-injectivity. This means the equations that specify when two constructors are equal and comes from lifting the alpha equivalence defining relations\\ $\bullet$ & distinctness\\%may be skipped $\bullet$ & equivariance of @{term "fv"} and @{term "bn"} functions\\ \end{tabular} \end{center} Until now we have not said anything about the support of the defined type. This is because we could not use the general definition of support in lifted theorems, since it does not preserve the relation. Indeed, take the term @{text "\<lambda>x. x"}. The support of the term is empty @{term "{}"}, since the @{term "x"} is bound. On the raw level, before the binding is introduced the term has the support equal to @{text "{x}"}. To show the support equations for the lifted types we want to use the Theorem \ref{suppabs}, so we start with showing that they have a finite support. \begin{lemma} The types @{text "ty\<^isup>\<alpha>\<^isub>1 \<dots> ty\<^isup>\<alpha>\<^isub>n"} have finite support. \end{lemma} \begin{proof} By induction on the lifted types. For each constructor its support is supported by the union of the supports of all arguments. By induction hypothesis we know that each of the recursive arguments has finite support. We also know that atoms and finite atom sets and lists that occur in the constructors have finite support. A union of finite sets is finite thus the support of the constructor is finite. \end{proof}% Very vague... \begin{lemma} For each lifted type @{text "ty\<^isup>\<alpha>\<^isub>i"}, for every @{text "x"} of this type: \begin{center} @{term "supp x = fv_ty\<^isup>\<alpha>\<^isub>i x"}. \end{center} \end{lemma} \begin{proof} We will show this by induction together with equations that characterize @{term "fv_bn\<^isup>\<alpha>\<^isub>"} in terms of @{term "alpha_bn\<^isup>\<alpha>"}. For each of @{text "fv_bn\<^isup>\<alpha>"} functions this equaton is: \begin{center} @{term "{a. infinite {b. \<not> alpha_bn\<^isup>\<alpha> ((a \<rightleftharpoons> b) \<bullet> x) x}} = fv_bn\<^isup>\<alpha> x"} \end{center} In the induction we need to show these equations together with the goal for the appropriate constructors. We first transform the right hand sides. The free variable functions are applied to theirs respective constructors so we can apply the lifted free variable defining equations to obtain free variable functions applied to subterms minus binders. Using the induction hypothesis we can replace free variable functions applied to subterms by support. Using Theorem \ref{suppabs} we replace the differences by supports of appropriate abstractions. Unfolding the definition of supports on both sides of the equations we obtain by simple calculations the equalities. \end{proof} With the above equations we can substitute free variables for support in the lifted free variable equations, which gives us the support equations for the term constructors. With this we can show that for each binding in a constructors the bindings can be renamed. To rename a shallow binder or a deep recursive binder a permutation is sufficient. This is not the case for a deep non-recursive bindings. Take the term @{text "Let (ACons x (Vr x) ANil) (Vr x)"}, representing the language construct @{text "let x = x in x"}. To rename the binder the permutation cannot be applied to the whole assignment since it would rename the free @{term "x"} as well. To avoid this we introduce a new construction operation that applies a permutation under a binding function. For each binding function @{text "bn\<^isub>j :: ty\<^isub>i \<Rightarrow> \<dots>"} we define a permutation operation @{text "\<bullet>bn\<^isub>j\<^isub> :: perm \<Rightarrow> ty\<^isub>i \<Rightarrow> ty\<^isub>i"}. This operation permutes only the arguments that are bound by the binding function while also descending in the recursive subcalls. For each term constructor @{text "C x\<^isub>1 \<dots> x\<^isub>n"} the @{text "\<bullet>bn\<^isub>j"} operation applied to a permutation @{text "\<pi>"} and to this constructor equals the constructor applied to the values for each argument. Provided @{text "bn\<^isub>j (C x\<^isub>1 \<dots> x\<^isub>n) = rhs"}, the value for an argument @{text "x\<^isub>j"} is: \begin{center} \begin{tabular}{c@ {\hspace{2mm}}p{7cm}} $\bullet$ & @{text "x\<^isub>i"} provided @{text "x\<^isub>i"} is not in @{text "rhs"}\\ $\bullet$ & @{text "\<pi> \<bullet> x\<^isub>i"} provided @{text "x\<^isub>i"} is in @{text "rhs"} without a binding function\\ $\bullet$ & @{text "\<pi> \<bullet>bn\<^isub>k x\<^isub>i"} provided @{text "bn\<^isub>k x\<^isub>i"} is @{text "rhs"}\\ \end{tabular} \end{center} The definition of @{text "\<bullet>bn"} is respectful (simple induction) so we can lift it and obtain @{text "\<bullet>bn\<^isup>\<alpha>"}. A property that shows that this defnition is correct is:%% We could get this from ExLet/perm_bn lemma. \begin{lemma} For every bn function @{text "bn\<^isup>\<alpha>\<^isub>i"}, \begin{center} @{text "\<pi> \<bullet> bn\<^isup>\<alpha>\<^isub>i l = bn\<^isup>\<alpha>\<^isub>i(p \<bullet>bn\<^isup>\<alpha>\<^isub>i l)"} \end{center} \end{lemma} \begin{proof} By induction on the lifted type it follows from the definitions of permutations on the lifted type and the lifted defining eqautions of @{text "\<bullet>bn"}. \end{proof}*}text {* \begin{figure} \begin{boxedminipage}{\linewidth} \small \begin{tabular}{l} \isacommand{atom\_decl}~@{text "var"}\\ \isacommand{atom\_decl}~@{text "cvar"}\\ \isacommand{atom\_decl}~@{text "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. At 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 one obtains automatically a reasoning infrastructure for Core-Haskell.\label{nominalcorehas}} \end{figure}*}(* @{thm "ACons_subst[no_vars]"}*)section {* Strong Induction Principles *}section {* Related Work *}text {* To our knowledge the earliest usage of general binders in a theorem prover setting is in the paper \cite{NaraschewskiNipkow99}, which describes a formalisation of the algorithm W. This formalisation implements binding in type schemes using a a de-Bruijn indices representation. Also recently an extension for general binders has been proposed for the locally nameless approach to binding \cite{chargueraud09}. . But we have not yet seen it to be employed in a non-trivial formal verification. In both approaches, it seems difficult to achieve our fine-grained control over the ``semantics'' of bindings (whether the order should matter, or vacous binders should be taken into account). To do so, it is necessary to introduce predicates that filter out some unwanted terms. This very likely results in intricate formal reasoning. Higher-Order Abstract Syntax (HOAS) approaches to representing binders are nicely supported in the Twelf theorem prover and work is in progress to use HOAS in a mechanisation of the metatheory of SML \cite{LeeCraryHarper07}. HOAS supports elegantly reasoning about term-calculi with single binders. We are not aware how more complicated binders from SML are represented in HOAS, but we know that HOAS cannot easily deal with binding constructs where the number of bound variables is not fixed. An example is the second part of the POPLmark challenge where @{text "Let"}s involving patterns need to be formalised. In such situations HOAS needs to use essentially has to represent multiple binders with iterated single binders. An attempt of representing general binders in the old version of Isabelle based also on iterating single binders is described in \cite{BengtsonParow09}. The reasoning there turned out to be quite complex. Ott is better with list dot specifications; subgrammars, is untyped; *}text {*%%% FIXME: The restricions should have already been described in previous sections? Restrictions \begin{itemize} \item non-emptiness \item positive datatype definitions \item finitely supported abstractions \item respectfulness of the bn-functions\bigskip \item binders can only have a ``single scope'' \item all bindings must have the same mode \end{itemize}*}section {* Conclusion *}text {* Complication when the single scopedness restriction is lifted (two overlapping permutations) Future work: distinct list abstraction TODO: function definitions: The formalisation presented here will eventually become part of the Isabelle distribution, but 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}.\medskip \noindent {\bf Acknowledgements:} We are very grateful to Andrew Pitts for many discussions about Nominal Isabelle. We thank Peter Sewell for making the informal notes \cite{SewellBestiary} available to us and also for patiently explaining some of the finer points about the abstract definitions and about the implementation of the Ott-tool. We also thank Stephanie Weirich for suggesting to separate the subgrammars of kinds and types in our Core-Haskell example. *}(*<*)end(*>*)