diff -r b7e524e7ee83 -r 95df71c3df2f Pearl-jv/Paper.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Pearl-jv/Paper.thy Thu Apr 08 09:12:13 2010 +0200 @@ -0,0 +1,1238 @@ +(*<*) +theory Paper +imports "../Nominal-General/Nominal2_Base" + "../Nominal-General/Nominal2_Atoms" + "../Nominal-General/Nominal2_Eqvt" + "../Nominal-General/Atoms" + "LaTeXsugar" +begin + +notation (latex output) + sort_of ("sort _" [1000] 100) and + Abs_perm ("_") and + Rep_perm ("_") and + swap ("'(_ _')" [1000, 1000] 1000) and + fresh ("_ # _" [51, 51] 50) and + Cons ("_::_" [78,77] 73) and + supp ("supp _" [78] 73) and + uminus ("-_" [78] 73) and + atom ("|_|") and + If ("if _ then _ else _" 10) and + Rep_name ("\_\") and + Abs_name ("\_\") and + Rep_var ("\_\") and + Abs_var ("\_\") and + sort_of_ty ("sort'_ty _") + +(* BH: uncomment if you really prefer the dot notation +syntax (latex output) + "_Collect" :: "pttrn => bool => 'a set" ("(1{_ . _})") +*) + +(* sort is used in Lists for sorting *) +hide const sort + +abbreviation + "sort \ sort_of" + +abbreviation + "sort_ty \ sort_of_ty" + +(*>*) + +section {* Introduction *} + +text {* + Nominal Isabelle is a definitional extension of the Isabelle/HOL theorem + prover providing a proving infrastructure for convenient reasoning about + programming languages. It has been used to formalise an equivalence checking + algorithm for LF \cite{UrbanCheneyBerghofer08}, + Typed Scheme~\cite{TobinHochstadtFelleisen08}, several calculi for concurrency + \cite{BengtsonParrow07} 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}. + + At its core Nominal Isabelle is based on the nominal logic work of Pitts et + al \cite{GabbayPitts02,Pitts03}. The most basic notion in this work is a + sort-respecting permutation operation defined over a countably infinite + collection of sorted atoms. The atoms are used for representing variables + that might be bound. Multiple sorts are necessary for being + able to represent different kinds of variables. For example, in the language + Mini-ML there are bound term variables and bound type variables; each kind + needs to be represented by a different sort of atoms. + + Unfortunately, the type system of Isabelle/HOL is not a good fit for the way + atoms and sorts are used in the original formulation of the nominal logic work. + Therefore it was decided in earlier versions of Nominal Isabelle to use a + separate type for each sort of atoms and let the type system enforce the + sort-respecting property of permutations. Inspired by the work on nominal + unification \cite{UrbanPittsGabbay04}, it seemed best at the time to also + implement permutations concretely as lists of pairs of atoms. Thus Nominal + Isabelle used the two-place permutation operation with the generic type + + @{text [display,indent=10] "_ \ _ :: (\ \ \) list \ \ \ \"} + + \noindent + where @{text "\"} stands for the type of atoms and @{text "\"} for the type + of the objects on which the permutation acts. For atoms of type @{text "\"} + the permutation operation is defined over the length of lists as follows + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}r@ {\hspace{2mm}}c@ {\hspace{2mm}}l} + @{text "[] \ c"} & @{text "="} & @{text c}\\ + @{text "(a b)::\ \ c"} & @{text "="} & + $\begin{cases} @{text a} & \textrm{if}~@{text "\ \ c = b"}\\ + @{text b} & \textrm{if}~@{text "\ \ c = a"}\\ + @{text "\ \ c"} & \textrm{otherwise}\end{cases}$ + \end{tabular}\hfill\numbered{atomperm} + \end{isabelle} + + \noindent + where we write @{text "(a b)"} for a swapping of atoms @{text "a"} and + @{text "b"}. For atoms of different type, the permutation operation + is defined as @{text "\ \ c \ c"}. + + With the list representation of permutations it is impossible to state an + ``ill-sorted'' permutation, since the type system excludes lists containing + atoms of different type. Another advantage of the list representation is that + the basic operations on permutations are already defined in the list library: + composition of two permutations (written @{text "_ @ _"}) is just list append, + and inversion of a permutation (written @{text "_\<^sup>-\<^sup>1"}) is just + list reversal. A disadvantage is that permutations do not have unique + representations as lists; we had to explicitly identify permutations according + to the relation + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{text "\\<^isub>1 \ \\<^isub>2 \ \a. \\<^isub>1 \ a = \\<^isub>2 \ a"} + \end{tabular}\hfill\numbered{permequ} + \end{isabelle} + + When lifting the permutation operation to other types, for example sets, + functions and so on, we needed to ensure that every definition is + well-behaved in the sense that it satisfies the following three + \emph{permutation properties}: + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}r@ {\hspace{4mm}}p{10cm}} + i) & @{text "[] \ x = x"}\\ + ii) & @{text "(\\<^isub>1 @ \\<^isub>2) \ x = \\<^isub>1 \ (\\<^isub>2 \ x)"}\\ + iii) & if @{text "\\<^isub>1 \ \\<^isub>2"} then @{text "\\<^isub>1 \ x = \\<^isub>2 \ x"} + \end{tabular}\hfill\numbered{permprops} + \end{isabelle} + + \noindent + From these properties we were able to derive most facts about permutations, and + the type classes of Isabelle/HOL allowed us to reason abstractly about these + three properties, and then let the type system automatically enforce these + properties for each type. + + The major problem with Isabelle/HOL's type classes, however, is that they + support operations with only a single type parameter and the permutation + operations @{text "_ \ _"} used above in the permutation properties + contain two! To work around this obstacle, Nominal Isabelle + required the user to + declare up-front the collection of \emph{all} atom types, say @{text + "\\<^isub>1,\,\\<^isub>n"}. From this collection it used custom ML-code to + generate @{text n} type classes corresponding to the permutation properties, + whereby in these type classes the permutation operation is restricted to + + @{text [display,indent=10] "_ \ _ :: (\\<^isub>i \ \\<^isub>i) list \ \ \ \"} + + \noindent + This operation has only a single type parameter @{text "\"} (the @{text "\\<^isub>i"} are the + atom types given by the user). + + While the representation of permutations-as-lists solved the + ``sort-respecting'' requirement and the declaration of all atom types + up-front solved the problem with Isabelle/HOL's type classes, this setup + caused several problems for formalising the nominal logic work: First, + Nominal Isabelle had to generate @{text "n\<^sup>2"} definitions for the + permutation operation over @{text "n"} types of atoms. Second, whenever we + need to generalise induction hypotheses by quantifying over permutations, we + have to build cumbersome quantifications like + + @{text [display,indent=10] "\\\<^isub>1 \ \\\<^isub>n. \"} + + \noindent + where the @{text "\\<^isub>i"} are of type @{text "(\\<^isub>i \ \\<^isub>i) list"}. + The reason is that the permutation operation behaves differently for + every @{text "\\<^isub>i"}. Third, although the notion of support + + @{text [display,indent=10] "supp _ :: \ \ \ set"} + + \noindent + which we will define later, has a generic type @{text "\ set"}, it cannot be + used to express the support of an object over \emph{all} atoms. The reason + is again that support can behave differently for each @{text + "\\<^isub>i"}. This problem is annoying, because if we need to know in + a statement that an object, say @{text "x"}, is finitely supported we end up + with having to state premises of the form + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{text "finite ((supp x) :: \\<^isub>1 set) , \, finite ((supp x) :: \\<^isub>n set)"} + \end{tabular}\hfill\numbered{fssequence} + \end{isabelle} + + \noindent + Sometimes we can avoid such premises completely, if @{text x} is a member of a + \emph{finitely supported type}. However, keeping track of finitely supported + types requires another @{text n} type classes, and for technical reasons not + all types can be shown to be finitely supported. + + The real pain of having a separate type for each atom sort arises, however, + from another permutation property + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}r@ {\hspace{4mm}}p{10cm}} + iv) & @{text "\\<^isub>1 \ (\\<^isub>2 \ x) = (\\<^isub>1 \ \\<^isub>2) \ (\\<^isub>1 \ x)"} + \end{tabular} + \end{isabelle} + + \noindent + where permutation @{text "\\<^isub>1"} has type @{text "(\ \ \) list"}, + @{text "\\<^isub>2"} type @{text "(\' \ \') list"} and @{text x} type @{text + "\"}. This property is needed in order to derive facts about how + permutations of different types interact, which is not covered by the + permutation properties @{text "i"}-@{text "iii"} shown in + \eqref{permprops}. The problem is that this property involves three type + parameters. In order to use again Isabelle/HOL's type class mechanism with + only permitting a single type parameter, we have to instantiate the atom + types. Consequently we end up with an additional @{text "n\<^sup>2"} + slightly different type classes for this permutation property. + + While the problems and pain can be almost completely hidden from the user in + the existing implementation of Nominal Isabelle, the work is \emph{not} + pretty. It requires a large amount of custom ML-code and also forces the + user to declare up-front all atom-types that are ever going to be used in a + formalisation. In this paper we set out to solve the problems with multiple + type parameters in the permutation operation, and in this way can dispense + with the large amounts of custom ML-code for generating multiple variants + for some basic definitions. The result is that we can implement a pleasingly + simple formalisation of the nominal logic work.\smallskip + + \noindent + {\bf Contributions of the paper:} Our use of a single atom type for representing + atoms of different sorts and of functions for representing + permutations is not novel, but drastically reduces the number of type classes to just + two (permutation types and finitely supported types) that we need in order + reason abstractly about properties from the nominal logic work. The novel + technical contribution of this paper is a mechanism for dealing with + ``Church-style'' lambda-terms \cite{Church40} and HOL-based languages + \cite{PittsHOL4} where variables and variable binding depend on type + annotations. +*} + +section {* Sorted Atoms and Sort-Respecting Permutations *} + +text {* + In the nominal logic work of Pitts, binders and bound variables are + represented by \emph{atoms}. As stated above, we need to have different + \emph{sorts} of atoms to be able to bind different kinds of variables. A + basic requirement is that there must be a countably infinite number of atoms + of each sort. Unlike in our earlier work, where we identified each sort with + a separate type, we implement here atoms to be +*} + + datatype atom\ = Atom\ string nat + +text {* + \noindent + whereby the string argument specifies the sort of the atom.\footnote{A similar + design choice was made by Gunter et al \cite{GunterOsbornPopescu09} + for their variables.} (The use type + \emph{string} is merely for convenience; any countably infinite type would work + as well.) + We have an auxiliary function @{text sort} that is defined as @{thm + sort_of.simps[no_vars]}, and we clearly have for every finite set @{text X} of + atoms and every sort @{text s} the property: + + \begin{proposition}\label{choosefresh} + @{text "If finite X then there exists an atom a such that + sort a = s and a \ X"}. + \end{proposition} + + For implementing sort-respecting permutations, we use functions of type @{typ + "atom => atom"} that @{text "i)"} are bijective; @{text "ii)"} are the + identity on all atoms, except a finite number of them; and @{text "iii)"} map + each atom to one of the same sort. These properties can be conveniently stated + for a function @{text \} as follows: + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{r@ {\hspace{4mm}}l} + i) & @{term "bij \"}\\ + ii) & @{term "finite {a. \ a \ a}"}\\ + iii) & @{term "\a. sort (\ a) = sort a"} + \end{tabular}\hfill\numbered{permtype} + \end{isabelle} + + \noindent + Like all HOL-based theorem provers, Isabelle/HOL allows us to + introduce a new type @{typ perm} that includes just those functions + satisfying all three properties. For example the identity function, + written @{term id}, is included in @{typ perm}. Also function composition, + written \mbox{@{text "_ \ _"}}, and function inversion, given by Isabelle/HOL's + inverse operator and written \mbox{@{text "inv _"}}, preserve the properties + @{text "i"}-@{text "iii"}. + + However, a moment of thought is needed about how to construct non-trivial + permutations. In the nominal logic work it turned out to be most convenient + to work with swappings, written @{text "(a b)"}. In our setting the + type of swappings must be + + @{text [display,indent=10] "(_ _) :: atom \ atom \ perm"} + + \noindent + but since permutations are required to respect sorts, we must carefully + consider what happens if a user states a swapping of atoms with different + sorts. In earlier versions of Nominal Isabelle, we avoided this problem by + using different types for different sorts; the type system prevented users + from stating ill-sorted swappings. Here, however, definitions such + as\footnote{To increase legibility, we omit here and in what follows the + @{term Rep_perm} and @{term "Abs_perm"} wrappers that are needed in our + implementation since we defined permutation not to be the full function space, + but only those functions of type @{typ perm} satisfying properties @{text + i}-@{text "iii"}.} + + @{text [display,indent=10] "(a b) \ \c. if a = c then b else (if b = c then a else c)"} + + \noindent + do not work in general, because the type system does not prevent @{text a} + and @{text b} from having different sorts---in which case the function would + violate property @{text iii}. We could make the definition of swappings + partial by adding the precondition @{term "sort a = sort b"}, + which would mean that in case @{text a} and @{text b} have different sorts, + the value of @{text "(a b)"} is unspecified. However, this looked like a + cumbersome solution, since sort-related side conditions would be required + everywhere, even to unfold the definition. It turned out to be more + convenient to actually allow the user to state ``ill-sorted'' swappings but + limit their ``damage'' by defaulting to the identity permutation in the + ill-sorted case: + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}rl} + @{text "(a b) \"} & @{text "if (sort a = sort b)"}\\ + & \hspace{3mm}@{text "then \c. if a = c then b else (if b = c then a else c)"}\\ + & \hspace{3mm}@{text "else id"} + \end{tabular}\hfill\numbered{swapdef} + \end{isabelle} + + \noindent + This function is bijective, the identity on all atoms except + @{text a} and @{text b}, and sort respecting. Therefore it is + a function in @{typ perm}. + + One advantage of using functions instead of lists as a representation for + permutations is that for example the swappings + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{thm swap_commute[no_vars]}\hspace{10mm} + @{text "(a a) = id"} + \end{tabular}\hfill\numbered{swapeqs} + \end{isabelle} + + \noindent + are \emph{equal}. We do not have to use the equivalence relation shown + in~\eqref{permequ} to identify them, as we would if they had been represented + as lists of pairs. Another advantage of the function representation is that + they form an (additive non-commutative) group provided we define + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{thm zero_perm_def[no_vars, THEN eq_reflection]} \hspace{4mm} + @{thm plus_perm_def[where p="\\<^isub>1" and q="\\<^isub>2", THEN eq_reflection]} \hspace{4mm} + @{thm uminus_perm_def[where p="\", THEN eq_reflection]} \hspace{4mm} + @{thm diff_def[where x="\\<^isub>1" and y="\\<^isub>2"]} + \end{tabular} + \end{isabelle} + + \noindent + and verify the simple properties + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{thm add_assoc[where a="\\<^isub>1" and b="\\<^isub>2" and c="\\<^isub>3"]} \hspace{3mm} + @{thm monoid_add_class.add_0_left[where a="\::perm"]} \hspace{3mm} + @{thm monoid_add_class.add_0_right[where a="\::perm"]} \hspace{3mm} + @{thm group_add_class.left_minus[where a="\::perm"]} + \end{tabular} + \end{isabelle} + + \noindent + Again this is in contrast to the list-of-pairs representation which does not + form a group. The technical importance of this fact is that for groups we + can rely on Isabelle/HOL's rich simplification infrastructure. This will + come in handy when we have to do calculations with permutations. However, + note that in this case Isabelle/HOL neglects well-entrenched mathematical + terminology that associates with an additive group a commutative + operation. Obviously, permutations are not commutative in general, because @{text + "p + q \ q + p"}. However, it is quite difficult to work around this + idiosyncrasy of Isabelle/HOL, unless we develop our own algebraic hierarchy + and infrastructure. But since the point of this paper is to implement the + nominal theory as smoothly as possible in Isabelle/HOL, we will follow its + characterisation of additive groups. + + By formalising permutations abstractly as functions, and using a single type + for all atoms, we can now restate the \emph{permutation properties} from + \eqref{permprops} as just the two equations + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}r@ {\hspace{4mm}}p{10cm}} + i) & @{thm permute_zero[no_vars]}\\ + ii) & @{thm permute_plus[where p="\\<^isub>1" and q="\\<^isub>2",no_vars]} + \end{tabular}\hfill\numbered{newpermprops} + \end{isabelle} + + \noindent + in which the permutation operations are of type @{text "perm \ \ \ \"} and so + have only a single type parameter. Consequently, these properties are + compatible with the one-parameter restriction of Isabelle/HOL's type classes. + There is no need to introduce a separate type class instantiated for each + sort, like in the old approach. + + The next notion allows us to establish generic lemmas involving the + permutation operation. + + \begin{definition} + A type @{text "\"} is a \emph{permutation type} if the permutation + properties in \eqref{newpermprops} are satisfied for every @{text "x"} of type + @{text "\"}. + \end{definition} + + \noindent + First, it follows from the laws governing + groups that a permutation and its inverse cancel each other. That is, for any + @{text "x"} of a permutation type: + + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{thm permute_minus_cancel(1)[where p="\", no_vars]}\hspace{10mm} + @{thm permute_minus_cancel(2)[where p="\", no_vars]} + \end{tabular}\hfill\numbered{cancel} + \end{isabelle} + + \noindent + Consequently, in a permutation type the permutation operation @{text "\ \ _"} is bijective, + which in turn implies the property + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{thm (lhs) permute_eq_iff[where p="\", no_vars]} + $\;$if and only if$\;$ + @{thm (rhs) permute_eq_iff[where p="\", no_vars]}. + \end{tabular}\hfill\numbered{permuteequ} + \end{isabelle} + + \noindent + In order to lift the permutation operation to other types, we can define for: + + \begin{isabelle} + \begin{tabular}{@ {}c@ {\hspace{-1mm}}c@ {}} + \begin{tabular}{@ {}r@ {\hspace{2mm}}l@ {}} + atoms: & @{thm permute_atom_def[where p="\",no_vars, THEN eq_reflection]}\\ + functions: & @{text "\ \ f \ \x. \ \ (f ((-\) \ x))"}\\ + permutations: & @{thm permute_perm_def[where p="\" and q="\'", THEN eq_reflection]}\\ + sets: & @{thm permute_set_eq[where p="\", no_vars, THEN eq_reflection]}\\ + booleans: & @{thm permute_bool_def[where p="\", no_vars, THEN eq_reflection]}\\ + \end{tabular} & + \begin{tabular}{@ {}r@ {\hspace{2mm}}l@ {}} + lists: & @{thm permute_list.simps(1)[where p="\", no_vars, THEN eq_reflection]}\\ + & @{thm permute_list.simps(2)[where p="\", no_vars, THEN eq_reflection]}\\[2mm] + products: & @{thm permute_prod.simps[where p="\", no_vars, THEN eq_reflection]}\\ + nats: & @{thm permute_nat_def[where p="\", no_vars, THEN eq_reflection]}\\ + \end{tabular} + \end{tabular} + \end{isabelle} + + \noindent + and then establish: + + \begin{theorem} + If @{text \}, @{text "\\<^isub>1"} and @{text "\\<^isub>2"} are permutation types, + then so are @{text "atom"}, @{text "\\<^isub>1 \ \\<^isub>2"}, + @{text perm}, @{term "\ set"}, @{term "\ list"}, @{term "\\<^isub>1 \ \\<^isub>2"}, + @{text bool} and @{text "nat"}. + \end{theorem} + + \begin{proof} + All statements are by unfolding the definitions of the permutation operations and simple + calculations involving addition and minus. With permutations for example we + have + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}[b]{@ {}rcl} + @{text "0 \ \'"} & @{text "\"} & @{text "0 + \' - 0 = \'"}\\ + @{text "(\\<^isub>1 + \\<^isub>2) \ \'"} & @{text "\"} & @{text "(\\<^isub>1 + \\<^isub>2) + \' - (\\<^isub>1 + \\<^isub>2)"}\\ + & @{text "="} & @{text "(\\<^isub>1 + \\<^isub>2) + \' - \\<^isub>2 - \\<^isub>1"}\\ + & @{text "="} & @{text "\\<^isub>1 + (\\<^isub>2 + \' - \\<^isub>2) - \\<^isub>1"} @{text "\"} @{text "\\<^isub>1 \ \\<^isub>2 \ \'"} + \end{tabular}\hfill\qed + \end{isabelle} + \end{proof} + + \noindent + The main point is that the above reasoning blends smoothly with the reasoning + infrastructure of Isabelle/HOL; no custom ML-code is necessary and a single + type class suffices. We can also show once and for all that the following + property---which caused so many headaches in our earlier setup---holds for any + permutation type. + + \begin{lemma}\label{permutecompose} + Given @{term x} is of permutation type, then + @{text "\\<^isub>1 \ (\\<^isub>2 \ x) = (\\<^isub>1 \ \\<^isub>2) \ (\\<^isub>1 \ x)"}. + \end{lemma} + + \begin{proof} The proof is as follows: + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}[b]{@ {}rcl@ {\hspace{8mm}}l} + @{text "\\<^isub>1 \ \\<^isub>2 \ x"} + & @{text "="} & @{text "\\<^isub>1 \ \\<^isub>2 \ (-\\<^isub>1) \ \\<^isub>1 \ x"} & by \eqref{cancel}\\ + & @{text "="} & @{text "(\\<^isub>1 + \\<^isub>2 - \\<^isub>1) \ \\<^isub>1 \ x"} & by {\rm(\ref{newpermprops}.@{text "ii"})}\\ + & @{text "\"} & @{text "(\\<^isub>1 \ \\<^isub>2) \ (\\<^isub>1 \ x)"}\\ + \end{tabular}\hfill\qed + \end{isabelle} + \end{proof} + +*} + +section {* Equivariance *} + +text {* + + One huge advantage of using bijective permutation functions (as opposed to + non-bijective renaming substitutions employed in traditional works syntax) is + the property of \emph{equivariance} + and the fact that most HOL-functions (this includes constants) whose argument + and result types are permutation types satisfy this property: + + \begin{definition}\label{equivariance} + A function @{text f} is \emph{equivariant} if @{term "\\. \ \ f = f"}. + \end{definition} + + \noindent + There are a number of equivalent formulations for the equivariance property. + For example, assuming @{text f} is of type @{text "\ \ \"}, then equivariance + can also be stated as + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{text "\\ x. \ \ (f x) = f (\ \ x)"} + \end{tabular}\hfill\numbered{altequivariance} + \end{isabelle} + + \noindent + To see that this formulation implies the definition, we just unfold the + definition of the permutation operation for functions and simplify with the equation + and the cancellation property shown in \eqref{cancel}. To see the other direction, we use + the fact + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{text "\ \ (f x) = (\ \ f) (\ \ x)"} + \end{tabular}\hfill\numbered{permutefunapp} + \end{isabelle} + + \noindent + which follows again directly + from the definition of the permutation operation for functions and the cancellation + property. Similarly for functions with more than one argument. + + Both formulations of equivariance have their advantages and disadvantages: + \eqref{altequivariance} is often easier to establish. For example we + can easily show that equality is equivariant + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{thm eq_eqvt[where p="\", no_vars]} + \end{tabular} + \end{isabelle} + + \noindent + using the permutation operation on booleans and property \eqref{permuteequ}. + Lemma~\ref{permutecompose} establishes that the permutation operation is + equivariant. It is also easy to see that the boolean operators, like + @{text "\"}, @{text "\"} and @{text "\"} are all equivariant. Furthermore + a simple calculation will show that our swapping functions are equivariant, that is + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{thm swap_eqvt[where p="\", no_vars]} + \end{tabular}\hfill\numbered{swapeqvt} + \end{isabelle} + + \noindent + for all @{text a}, @{text b} and @{text \}. These equivariance properties + are tremendously helpful later on when we have to push permutations inside + terms. +*} + + +section {* Support and Freshness *} + +text {* + The most original aspect of the nominal logic work of Pitts et al is a general + definition for ``the set of free variables of an object @{text "x"}''. This + definition is general in the sense that it applies not only to lambda-terms, + but also to lists, products, sets and even functions. The definition depends + only on the permutation operation and on the notion of equality defined for + the type of @{text x}, namely: + + @{thm [display,indent=10] supp_def[no_vars, THEN eq_reflection]} + + \noindent + (Note that due to the definition of swapping in \eqref{swapdef}, we do not + need to explicitly restrict @{text a} and @{text b} to have the same sort.) + There is also the derived notion for when an atom @{text a} is \emph{fresh} + for an @{text x}, defined as + + @{thm [display,indent=10] fresh_def[no_vars]} + + \noindent + 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. For the proof we use the following lemma + about swappings applied to an @{text x}: + + \begin{lemma}\label{swaptriple} + Assuming @{text x} is of permutation type, and @{text a}, @{text b} and @{text c} + have the same sort, then @{thm (prem 3) swap_rel_trans[no_vars]} and + @{thm (prem 4) swap_rel_trans[no_vars]} imply @{thm (concl) swap_rel_trans[no_vars]}. + \end{lemma} + + \begin{proof} + The cases where @{text "a = c"} and @{text "b = c"} are immediate. + For the remaining case it is, given our assumptions, easy to calculate + that the permutations + + @{thm [display,indent=10] (concl) swap_triple[no_vars]} + + \noindent + are equal. The lemma is then by application of the second permutation + property shown in \eqref{newpermprops}.\hfill\qed + \end{proof} + + \begin{theorem}\label{swapfreshfresh} + Let @{text x} be of permutation type. + @{thm [mode=IfThen] swap_fresh_fresh[no_vars]} + \end{theorem} + + \begin{proof} + If @{text a} and @{text b} have different sort, then the swapping is the identity. + If they have the same sort, we know by definition of support that both + @{term "finite {c. (a \ c) \ x \ x}"} and @{term "finite {c. (b \ c) \ x \ x}"} + hold. So the union of these sets is finite too, and we know by Proposition~\ref{choosefresh} + that there is an atom @{term c}, with the same sort as @{term a} and @{term b}, + that satisfies \mbox{@{term "(a \ c) \ x = x"}} and @{term "(b \ c) \ x = x"}. + Now the theorem follows from Lemma~\ref{swaptriple}.\hfill\qed + \end{proof} + + \noindent + Two important properties that need to be established for later calculations is + that @{text "supp"} and freshness are equivariant. For this we first show that: + + \begin{lemma}\label{half} + If @{term x} is a permutation type, then @{thm (lhs) fresh_permute_iff[where p="\",no_vars]} + if and only if @{thm (rhs) fresh_permute_iff[where p="\",no_vars]}. + \end{lemma} + + \begin{proof} + \begin{isabelle} + \begin{tabular}[t]{c@ {\hspace{2mm}}l@ {\hspace{5mm}}l} + & \multicolumn{2}{@ {}l}{@{thm (lhs) fresh_permute_iff[where p="\",no_vars]} @{text "\"} + @{term "finite {b. (\ \ a \ b) \ \ \ x \ \ \ x}"}}\\ + @{text "\"} + & @{term "finite {b. (\ \ a \ \ \ b) \ \ \ x \ \ \ x}"} + & since @{text "\ \ _"} is bijective\\ + @{text "\"} + & @{term "finite {b. \ \ (a \ b) \ x \ \ \ x}"} + & by \eqref{permutecompose} and \eqref{swapeqvt}\\ + @{text "\"} + & @{term "finite {b. (a \ b) \ x \ x}"} @{text "\"} + @{thm (rhs) fresh_permute_iff[where p="\",no_vars]} + & by \eqref{permuteequ}\\ + \end{tabular} + \end{isabelle}\hfill\qed + \end{proof} + + \noindent + Together with the definition of the permutation operation on booleans, + we can immediately infer equivariance of freshness: + + @{thm [display,indent=10] fresh_eqvt[where p="\",no_vars]} + + \noindent + Now equivariance of @{text "supp"}, namely + + @{thm [display,indent=10] supp_eqvt[where p="\",no_vars]} + + \noindent + is by noting that @{thm supp_conv_fresh[no_vars]} and that freshness and + the logical connectives are equivariant. + + While the abstract properties of support and freshness, particularly + Theorem~\ref{swapfreshfresh}, are useful for developing Nominal Isabelle, + one often has to calculate the support of some concrete object. This is + straightforward for example for booleans, nats, products and lists: + + \begin{center} + \begin{tabular}{@ {}c@ {\hspace{2mm}}c@ {}} + \begin{tabular}{@ {}r@ {\hspace{2mm}}l} + @{text "booleans"}: & @{term "supp b = {}"}\\ + @{text "nats"}: & @{term "supp n = {}"}\\ + @{text "products"}: & @{thm supp_Pair[no_vars]}\\ + \end{tabular} & + \begin{tabular}{r@ {\hspace{2mm}}l@ {}} + @{text "lists:"} & @{thm supp_Nil[no_vars]}\\ + & @{thm supp_Cons[no_vars]}\\ + \end{tabular} + \end{tabular} + \end{center} + + \noindent + But establishing the support of atoms and permutations in our setup here is a bit + trickier. To do so we will use the following notion about a \emph{supporting set}. + + \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 \ b) \ x = x"}. + \end{definition} + + \noindent + The main motivation for this notion is that we can characterise @{text "supp x"} + as the smallest finite set that supports @{text "x"}. For this we prove: + + \begin{lemma}\label{supports} Let @{text x} be of permutation type. + \begin{isabelle} + \begin{tabular}{r@ {\hspace{4mm}}p{10cm}} + i) & @{thm[mode=IfThen] supp_is_subset[no_vars]}\\ + ii) & @{thm[mode=IfThen] supp_supports[no_vars]}\\ + iii) & @{thm (concl) supp_is_least_supports[no_vars]} + provided @{thm (prem 1) supp_is_least_supports[no_vars]}, + @{thm (prem 2) supp_is_least_supports[no_vars]} + and @{text "S"} is the least such set, that means formally, + for all @{text "S'"}, if @{term "finite S'"} and + @{term "S' supports x"} then @{text "S \ S'"}. + \end{tabular} + \end{isabelle} + \end{lemma} + + \begin{proof} + For @{text "i)"} we derive a contradiction by assuming there is an atom @{text a} + with @{term "a \ supp x"} and @{text "a \ S"}. Using the second fact, the + assumption that @{term "S supports x"} gives us that @{text S} is a superset of + @{term "{b. (a \ b) \ x \ x}"}, which is finite by the assumption of @{text S} + being finite. But this means @{term "a \ supp x"}, contradicting our assumption. + Property @{text "ii)"} is by a direct application of + Theorem~\ref{swapfreshfresh}. For the last property, part @{text "i)"} proves + one ``half'' of the claimed equation. The other ``half'' is by property + @{text "ii)"} and the fact that @{term "supp x"} is finite by @{text "i)"}.\hfill\qed + \end{proof} + + \noindent + These are all relatively straightforward proofs adapted from the existing + nominal logic work. However for establishing the support of atoms and + permutations we found the following ``optimised'' variant of @{text "iii)"} + more useful: + + \begin{lemma}\label{optimised} Let @{text x} be of permutation type. + We have that @{thm (concl) finite_supp_unique[no_vars]} + provided @{thm (prem 1) finite_supp_unique[no_vars]}, + @{thm (prem 2) finite_supp_unique[no_vars]}, and for + all @{text "a \ S"} and all @{text "b \ S"}, with @{text a} + and @{text b} having the same sort, then \mbox{@{term "(a \ b) \ x \ x"}} + \end{lemma} + + \begin{proof} + By Lemma \ref{supports}@{text ".iii)"} we have to show that for every finite + set @{text S'} that supports @{text x}, \mbox{@{text "S \ S'"}} holds. We will + assume that there is an atom @{text "a"} that is element of @{text S}, but + not @{text "S'"} and derive a contradiction. Since both @{text S} and + @{text S'} are finite, we can by Proposition \ref{choosefresh} obtain an atom + @{text b}, which has the same sort as @{text "a"} and for which we know + @{text "b \ S"} and @{text "b \ S'"}. Since we assumed @{text "a \ S'"} and + we have that @{text "S' supports x"}, we have on one hand @{term "(a \ b) \ x + = x"}. On the other hand, the fact @{text "a \ S"} and @{text "b \ S"} imply + @{term "(a \ b) \ x \ x"} using the assumed implication. This gives us the + contradiction.\hfill\qed + \end{proof} + + \noindent + Using this lemma we only have to show the following three proof-obligations + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}r@ {\hspace{4mm}}l} + i) & @{term "{c} supports c"}\\ + ii) & @{term "finite {c}"}\\ + iii) & @{text "\a \ {c} b \ {c}. sort a = sort b \ (a b) \ c \ c"} + \end{tabular} + \end{isabelle} + + \noindent + in order to establish that @{thm supp_atom[where a="c", no_vars]} holds. In + Isabelle/HOL these proof-obligations can be discharged by easy + simplifications. Similar proof-obligations arise for the support of + permutations, which is + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{thm supp_perm[where p="\", no_vars]} + \end{tabular} + \end{isabelle} + + \noindent + The only proof-obligation that is + interesting is the one where we have to show that + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{text "If \ \ a \ a, \ \ b = b and sort a = sort b, then (a b) \ \ \ \"}. + \end{tabular} + \end{isabelle} + + \noindent + For this we observe that + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}rcl} + @{thm (lhs) perm_swap_eq[where p="\", no_vars]} & + if and only if & + @{thm (rhs) perm_swap_eq[where p="\", no_vars]} + \end{tabular} + \end{isabelle} + + \noindent + holds by a simple calculation using the group properties of permutations. + The proof-obligation can then be discharged by analysing the inequality + between the permutations @{term "(\ \ a \ b)"} and @{term "(a \ b)"}. + + The main point about support is that whenever an object @{text x} has finite + support, then Proposition~\ref{choosefresh} allows us to choose for @{text x} a + fresh atom with arbitrary sort. This is an important operation in Nominal + Isabelle in situations where, for example, a bound variable needs to be + renamed. To allow such a choice, we only have to assume \emph{one} premise + of the form @{text "finite (supp x)"} + for each @{text x}. Compare that with the sequence of premises in our earlier + version of Nominal Isabelle (see~\eqref{fssequence}). For more convenience we + can define a type class for types where every element has finite support, and + prove that the types @{term "atom"}, @{term "perm"}, lists, products and + booleans are instances of this type class. Then \emph{no} premise is needed, + as the type system of Isabelle/HOL can figure out automatically when an object + is finitely supported. + + Unfortunately, this does not work for sets or Isabelle/HOL's function type. + There are functions and sets definable in Isabelle/HOL for which the finite + support property does not hold. A simple example of a function with + infinite support is the function that returns the natural number of an atom + + @{text [display, indent=10] "nat_of (Atom s i) \ i"} + + \noindent + This function's support is the set of \emph{all} atoms. To establish this we show @{term "\ a \ nat_of"}. + This is equivalent to assuming the set @{term "{b. (a \ b) \ nat_of \ nat_of}"} is finite + and deriving a contradiction. From the assumption we also know that + @{term "{a} \ {b. (a \ b) \ nat_of \ nat_of}"} is finite. Then we can use + Proposition~\ref{choosefresh} to choose an atom @{text c} such that + @{term "c \ a"}, @{term "sort_of c = sort_of a"} and @{term "(a \ c) \ nat_of = nat_of"}. + Now we can reason as follows: + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}[b]{@ {}rcl@ {\hspace{5mm}}l} + @{text "nat_of a"} & @{text "="} & @{text "(a \ c) \ (nat_of a)"} & by def.~of permutations on nats\\ + & @{text "="} & @{term "((a \ c) \ nat_of) ((a \ c) \ a)"} & by \eqref{permutefunapp}\\ + & @{text "="} & @{term "nat_of c"} & by assumptions on @{text c}\\ + \end{tabular} + \end{isabelle} + + + \noindent + But this means we have that @{term "nat_of a = nat_of c"} and @{term "sort_of a = sort_of c"}. + This implies that atoms @{term a} and @{term c} must be equal, which clashes with our + assumption @{term "c \ a"} about how we chose @{text c}. + Cheney \cite{Cheney06} gives similar examples for constructions that have infinite support. +*} + +section {* Induction Principles *} + + + + +section {* Concrete Atom Types *} + +text {* + + So far, we have presented a system that uses only a single multi-sorted atom + type. This design gives us the flexibility to define operations and prove + theorems that are generic with respect to atom sorts. For example, as + illustrated above the @{term supp} function returns a set that includes the + free atoms of \emph{all} sorts together; the flexibility offered by the new + atom type makes this possible. + + However, the single multi-sorted atom type does not make an ideal interface + for end-users of Nominal Isabelle. If sorts are not distinguished by + Isabelle's type system, users must reason about atom sorts manually. That + means subgoals involving sorts must be discharged explicitly within proof + scripts, instead of being inferred by Isabelle/HOL's type checker. In other + cases, lemmas might require additional side conditions about sorts to be true. + For example, swapping @{text a} and @{text b} in the pair \mbox{@{term "(a, + b)"}} will only produce the expected result if we state the lemma in + Isabelle/HOL as: +*} + + lemma + fixes a b :: "atom" + assumes asm: "sort a = sort b" + shows "(a \ b) \ (a, b) = (b, a)" + using asm by simp + +text {* + \noindent + Fortunately, it is possible to regain most of the type-checking automation + that is lost by moving to a single atom type. We accomplish this by defining + \emph{subtypes} of the generic atom type that only include atoms of a single + specific sort. We call such subtypes \emph{concrete atom types}. + + The following Isabelle/HOL command defines a concrete atom type called + \emph{name}, which consists of atoms whose sort equals the string @{term + "''name''"}. + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \isacommand{typedef}\ \ @{typ name} = @{term "{a. sort\ a = ''name''}"} + \end{isabelle} + + \noindent + This command automatically generates injective functions that map from the + concrete atom type into the generic atom type and back, called + representation and abstraction functions, respectively. We will write these + functions as follows: + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l@ {\hspace{10mm}}l} + @{text "\_\ :: name \ atom"} & + @{text "\_\ :: atom \ name"} + \end{tabular} + \end{isabelle} + + \noindent + With the definition @{thm permute_name_def [where p="\", THEN + eq_reflection, no_vars]}, it is straightforward to verify that the type + @{typ name} is a permutation type. + + In order to reason uniformly about arbitrary concrete atom types, we define a + type class that characterises type @{typ name} and other similarly-defined + types. The definition of the concrete atom type class is as follows: First, + every concrete atom type must be a permutation type. In addition, the class + defines an overloaded function that maps from the concrete type into the + generic atom type, which we will write @{text "|_|"}. For each class + instance, this function must be injective and equivariant, and its outputs + must all have the same sort, that is + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{r@ {\hspace{3mm}}l} + i) if @{thm (lhs) atom_eq_iff [no_vars]} then @{thm (rhs) atom_eq_iff [no_vars]}\\ + ii) @{thm atom_eqvt[where p="\", no_vars]}\\ + iii) @{thm sort_of_atom_eq [no_vars]} + \end{tabular}\hfill\numbered{atomprops} + \end{isabelle} + + \noindent + With the definition @{thm atom_name_def [THEN eq_reflection, no_vars]} we can + show that @{typ name} satisfies all the above requirements of a concrete atom + type. + + The whole point of defining the concrete atom type class was to let users + avoid explicit reasoning about sorts. This benefit is realised by defining a + special swapping operation of type @{text "\ \ \ + \ perm"}, where @{text "\"} is a concrete atom type: + + @{thm [display,indent=10] flip_def [THEN eq_reflection, no_vars]} + + \noindent + As a consequence of its type, the @{text "\"}-swapping + operation works just like the generic swapping operation, but it does not + require any sort-checking side conditions---the sort-correctness is ensured by + the types! For @{text "\"} we can establish the following + simplification rule: + + @{thm [display,indent=10] permute_flip_at[no_vars]} + + \noindent + If we now want to swap the \emph{concrete} atoms @{text a} and @{text b} + in the pair @{term "(a, b)"} we can establish the lemma as follows: +*} + + lemma + fixes a b :: "name" + shows "(a \ b) \ (a, b) = (b, a)" + by simp + +text {* + \noindent + There is no need to state an explicit premise involving sorts. + + We can automate the process of creating concrete atom types, so that users + can define a new one simply by issuing the command + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + \isacommand{atom\_decl}~~@{text "name"} + \end{tabular} + \end{isabelle} + + \noindent + This command can be implemented using less than 100 lines of custom ML-code. + In comparison, the old version of Nominal Isabelle included more than 1000 + lines of ML-code for creating concrete atom types, and for defining various + type classes and instantiating generic lemmas for them. In addition to + simplifying the ML-code, the setup here also offers user-visible improvements: + Now concrete atoms can be declared at any point of a formalisation, and + theories that separately declare different atom types can be merged + together---it is no longer required to collect all atom declarations in one + place. +*} + + +section {* Multi-Sorted Concrete Atoms *} + +(*<*) +datatype ty = TVar string | Fun ty ty ("_ \ _") +(*>*) + +text {* + The formalisation presented so far allows us to streamline proofs and reduce + the amount of custom ML-code in the existing implementation of Nominal + Isabelle. In this section we describe a mechanism that extends the + capabilities of Nominal Isabelle. This mechanism is about variables with + additional information, for example typing constraints. + While we leave a detailed treatment of binders and binding of variables for a + later paper, we will have a look here at how such variables can be + represented by concrete atoms. + + In the previous section we considered concrete atoms that can be used in + simple binders like \emph{@{text "\x. x"}}. Such concrete atoms do + not carry any information beyond their identities---comparing for equality + is really the only way to analyse ordinary concrete atoms. + However, in ``Church-style'' lambda-terms \cite{Church40} and in the terms + underlying HOL-systems \cite{PittsHOL4} binders and bound variables have a + more complicated structure. For example in the ``Church-style'' lambda-term + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{text "\x\<^isub>\. x\<^isub>\ x\<^isub>\"} + \end{tabular}\hfill\numbered{church} + \end{isabelle} + + \noindent + both variables and binders include typing information indicated by @{text \} + and @{text \}. In this setting, we treat @{text "x\<^isub>\"} and @{text + "x\<^isub>\"} as distinct variables (assuming @{term "\\\"}) so that the + variable @{text "x\<^isub>\"} is bound by the lambda-abstraction, but not + @{text "x\<^isub>\"}. + + To illustrate how we can deal with this phenomenon, let us represent object + types like @{text \} and @{text \} by the datatype + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + \isacommand{datatype}~~@{text "ty = TVar string | ty \ ty"} + \end{tabular} + \end{isabelle} + + \noindent + If we attempt to encode a variable naively as a pair of a @{text name} and a @{text ty}, we have the + problem that a swapping, say @{term "(x \ y)"}, applied to the pair @{text "((x, \), (x, \))"} + will always permute \emph{both} occurrences of @{text x}, even if the types + @{text "\"} and @{text "\"} are different. This is unwanted, because it will + eventually mean that both occurrences of @{text x} will become bound by a + corresponding binder. + + Another attempt might be to define variables as an instance of the concrete + atom type class, where a @{text ty} is somehow encoded within each variable. + Remember we defined atoms as the datatype: +*} + + datatype atom\\ = Atom\\ string nat + +text {* + \noindent + Considering our method of defining concrete atom types, the usage of a string + for the sort of atoms seems a natural choice. However, none of the results so + far depend on this choice and we are free to change it. + One possibility is to encode types or any other information by making the sort + argument parametric as follows: +*} + + datatype 'a \atom\\\ = \Atom\\ 'a nat + +text {* + \noindent + The problem with this possibility is that we are then back in the old + situation where our permutation operation is parametric in two types and + this would require to work around Isabelle/HOL's restriction on type + classes. Fortunately, encoding the types in a separate parameter is not + necessary for what we want to achieve, as we only have to know when two + types are equal or not. The solution is to use a different sort for each + object type. Then we can use the fact that permutations respect \emph{sorts} to + ensure that permutations also respect \emph{object types}. In order to do + this, we must define an injective function @{text "sort_ty"} mapping from + object types to sorts. For defining functions like @{text "sort_ty"}, it is + more convenient to use a tree datatype for sorts. Therefore we define +*} + + datatype sort = Sort string "(sort list)" + datatype atom\\\\ = Atom\\\\ sort nat + +text {* + \noindent + With this definition, + the sorts we considered so far can be encoded just as \mbox{@{text "Sort s []"}}. + The point, however, is that we can now define the function @{text sort_ty} simply as + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \begin{tabular}{@ {}l} + @{text "sort_ty (TVar s) = Sort ''TVar'' [Sort s []]"}\\ + @{text "sort_ty (\\<^isub>1 \ \\<^isub>2) = Sort ''Fun'' [sort_ty \\<^isub>1, sort_ty \\<^isub>2]"} + \end{tabular}\hfill\numbered{sortty} + \end{isabelle} + + \noindent + which can easily be shown to be injective. + + Having settled on what the sorts should be for ``Church-like'' atoms, we have to + give a subtype definition for concrete atoms. Previously we identified a subtype consisting + of atoms of only one specified sort. This must be generalised to all sorts the + function @{text "sort_ty"} might produce, i.e.~the + range of @{text "sort_ty"}. Therefore we define + + \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% + \isacommand{typedef}\ \ @{text var} = @{term "{a. sort a : range sort_ty}"} + \end{isabelle} + + \noindent + This command gives us again injective representation and abstraction + functions. We will write them also as \mbox{@{text "\_\ :: var \ atom"}} and + @{text "\_\ :: atom \ var"}, respectively. + + We can define the permutation operation for @{text var} as @{thm + permute_var_def[where p="\", THEN eq_reflection, no_vars]} and the + injective function to type @{typ atom} as @{thm atom_var_def[THEN + eq_reflection, no_vars]}. Finally, we can define a constructor function that + makes a @{text var} from a variable name and an object type: + + @{thm [display,indent=10] Var_def[where t="\", THEN eq_reflection, no_vars]} + + \noindent + With these definitions we can verify all the properties for concrete atom + types except Property \ref{atomprops}@{text ".iii)"}, which requires every + atom to have the same sort. This last property is clearly not true for type + @{text "var"}. + This fact is slightly unfortunate since this + property allowed us to use the type-checker in order to shield the user from + all sort-constraints. But this failure is expected here, because we cannot + burden the type-system of Isabelle/HOL with the task of deciding when two + object types are equal. This means we sometimes need to explicitly state sort + constraints or explicitly discharge them, but as we will see in the lemma + below this seems a natural price to pay in these circumstances. + + To sum up this section, the encoding of type-information into atoms allows us + to form the swapping @{term "(Var x \ \ Var y \)"} and to prove the following + lemma +*} + + lemma + assumes asm: "\ \ \" + shows "(Var x \ \ Var y \) \ (Var x \, Var x \) = (Var y \, Var x \)" + using asm by simp + +text {* + \noindent + As we expect, the atom @{term "Var x \"} is left unchanged by the + swapping. With this we can faithfully represent bindings in languages + involving ``Church-style'' terms and bindings as shown in \eqref{church}. We + expect that the creation of such atoms can be easily automated so that the + user just needs to specify \isacommand{atom\_decl}~~@{text "var (ty)"} + where the argument, or arguments, are datatypes for which we can automatically + define an injective function like @{text "sort_ty"} (see \eqref{sortty}). + Our hope is that with this approach Benzmueller and Paulson, the authors of + \cite{PaulsonBenzmueller}, can make headway with formalising their results + about simple type theory. + Because of its limitations, they did not attempt this with the old version + of Nominal Isabelle. We also hope we can make progress with formalisations of + HOL-based languages. +*} + + +section {* Conclusion *} + +text {* + This proof pearl describes a new formalisation of the nominal logic work by + Pitts et al. With the definitions we presented here, the formal reasoning blends + smoothly with the infrastructure of the Isabelle/HOL theorem prover. + Therefore the formalisation will be the underlying theory for a + new version of Nominal Isabelle. + + The main difference of this paper with respect to existing work on Nominal + Isabelle is the representation of atoms and permutations. First, we used a + single type for sorted atoms. This design choice means for a term @{term t}, + say, that its support is completely characterised by @{term "supp t"}, even + if the term contains different kinds of atoms. Also, whenever we have to + generalise an induction so that a property @{text P} is not just established + for all @{text t}, but for all @{text t} \emph{and} under all permutations + @{text \}, then we only have to state @{term "\\. P (\ \ t)"}. The reason is + that permutations can now consist of multiple swapping each of which can + swap different kinds of atoms. This simplifies considerably the reasoning + involved in building Nominal Isabelle. + + Second, we represented permutations as functions so that the associated + permutation operation has only a single type parameter. This is very convenient + because the abstract reasoning about permutations fits cleanly + with Isabelle/HOL's type classes. No custom ML-code is required to work + around rough edges. Moreover, by establishing that our permutations-as-functions + representation satisfy the group properties, we were able to use extensively + Isabelle/HOL's reasoning infrastructure for groups. This often reduced proofs + to simple calculations over @{text "+"}, @{text "-"} and @{text "0"}. + An interesting point is that we defined the swapping operation so that a + swapping of two atoms with different sorts is \emph{not} excluded, like + in our older work on Nominal Isabelle, but there is no ``effect'' of such + a swapping (it is defined as the identity). This is a crucial insight + in order to make the approach based on a single type of sorted atoms to work. + But of course it is analogous to the well-known trick of defining division by + zero to return zero. + + We noticed only one disadvantage of the permutations-as-functions: Over + lists we can easily perform inductions. For permutation made up from + functions, we have to manually derive an appropriate induction principle. We + can establish such a principle, but we have no real experience yet whether ours + is the most useful principle: such an induction principle was not needed in + any of the reasoning we ported from the old Nominal Isabelle, except + when showing that if @{term "\a \ supp x. a \ p"} implies @{term "p \ x = x"}. + + Finally, our implementation of sorted atoms turned out powerful enough to + use it for representing variables that carry on additional information, for + example typing annotations. This information is encoded into the sorts. With + this we can represent conveniently binding in ``Church-style'' lambda-terms + and HOL-based languages. While dealing with such additional information in + dependent type-theories, such as LF or Coq, is straightforward, we are not + aware of any other approach in a non-dependent HOL-setting that can deal + conveniently with such binders. + + 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}.\smallskip + + \noindent + {\bf Acknowledgements:} We are very grateful to Jesper Bengtson, Stefan + Berghofer and Cezary Kaliszyk for their comments on earlier versions + of this paper. We are also grateful to the anonymous referee who helped us to + put the work into the right context. +*} + + +(*<*) +end +(*>*) \ No newline at end of file