--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Pearl/Paper.thy Sat Apr 03 21:53:04 2010 +0200
@@ -0,0 +1,1232 @@
+(*<*)
+theory Paper
+imports "../Nominal/Nominal2_Base"
+ "../Nominal/Nominal2_Atoms"
+ "../Nominal/Nominal2_Eqvt"
+ "../Nominal/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 ("\<lfloor>_\<rfloor>") and
+ Abs_name ("\<lceil>_\<rceil>") and
+ Rep_var ("\<lfloor>_\<rfloor>") and
+ Abs_var ("\<lceil>_\<rceil>") 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 \<equiv> sort_of"
+
+abbreviation
+ "sort_ty \<equiv> 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,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}.
+
+ 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 list of pairs of atoms. Thus Nominal
+ Isabelle used the two-place permutation operation with the generic type
+
+ @{text [display,indent=10] "_ \<bullet> _ :: (\<alpha> \<times> \<alpha>) list \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
+
+ \noindent
+ where @{text "\<alpha>"} stands for the type of atoms and @{text "\<beta>"} for the type
+ of the objects on which the permutation acts. For atoms of type @{text "\<alpha>"}
+ the permutation operation is defined over the length of lists as follows
+
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ \begin{tabular}{@ {}r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
+ @{text "[] \<bullet> c"} & @{text "="} & @{text c}\\
+ @{text "(a b)::\<pi> \<bullet> c"} & @{text "="} &
+ $\begin{cases} @{text a} & \textrm{if}~@{text "\<pi> \<bullet> c = b"}\\
+ @{text b} & \textrm{if}~@{text "\<pi> \<bullet> c = a"}\\
+ @{text "\<pi> \<bullet> c"} & \textrm{otherwise}\end{cases}$
+ \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 "\<pi> \<bullet> c \<equiv> 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 "\<pi>\<^isub>1 \<sim> \<pi>\<^isub>2 \<equiv> \<forall>a. \<pi>\<^isub>1 \<bullet> a = \<pi>\<^isub>2 \<bullet> 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 "[] \<bullet> x = x"}\\
+ ii) & @{text "(\<pi>\<^isub>1 @ \<pi>\<^isub>2) \<bullet> x = \<pi>\<^isub>1 \<bullet> (\<pi>\<^isub>2 \<bullet> x)"}\\
+ iii) & if @{text "\<pi>\<^isub>1 \<sim> \<pi>\<^isub>2"} then @{text "\<pi>\<^isub>1 \<bullet> x = \<pi>\<^isub>2 \<bullet> 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 "_ \<bullet> _"} used above in the permutation properties
+ contain two! To work around this obstacle, Nominal Isabelle
+ required from the user to
+ declare up-front the collection of \emph{all} atom types, say @{text
+ "\<alpha>\<^isub>1,\<dots>,\<alpha>\<^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] "_ \<bullet> _ :: (\<alpha>\<^isub>i \<times> \<alpha>\<^isub>i) list \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
+
+ \noindent
+ This operation has only a single type parameter @{text "\<beta>"} (the @{text "\<alpha>\<^isub>i"} are the
+ atom types given by the user).
+
+ While the representation of permutations-as-list 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] "\<forall>\<pi>\<^isub>1 \<dots> \<forall>\<pi>\<^isub>n. \<dots>"}
+
+ \noindent
+ where the @{text "\<pi>\<^isub>i"} are of type @{text "(\<alpha>\<^isub>i \<times> \<alpha>\<^isub>i) list"}.
+ The reason is that the permutation operation behaves differently for
+ every @{text "\<alpha>\<^isub>i"}. Third, although the notion of support
+
+ @{text [display,indent=10] "supp _ :: \<beta> \<Rightarrow> \<alpha> set"}
+
+ \noindent
+ which we will define later, has a generic type @{text "\<alpha> 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
+ "\<alpha>\<^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) :: \<alpha>\<^isub>1 set) , \<dots>, finite ((supp x) :: \<alpha>\<^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 "\<pi>\<^isub>1 \<bullet> (\<pi>\<^isub>2 \<bullet> x) = (\<pi>\<^isub>1 \<bullet> \<pi>\<^isub>2) \<bullet> (\<pi>\<^isub>1 \<bullet> x)"}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ where permutation @{text "\<pi>\<^isub>1"} has type @{text "(\<alpha> \<times> \<alpha>) list"},
+ @{text "\<pi>\<^isub>2"} type @{text "(\<alpha>' \<times> \<alpha>') list"} and @{text x} type @{text
+ "\<beta>"}. 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:} We use a single atom type for representing
+ atoms of different sorts and use functions for representing
+ permutations. This drastically reduces the number of type classes to just
+ two (permutation types and finitely supported types), which 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\<iota> = Atom\<iota> string nat
+
+text {*
+ \noindent
+ whereby the string argument specifies the sort of the atom. (We use type
+ \emph{string} 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 \<notin> 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 \<pi>} as follows:
+
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ \begin{tabular}{@ {}r@ {\hspace{4mm}}p{10cm}}
+ i) & @{term "bij \<pi>"}\\
+ ii) & @{term "finite {a. \<pi> a \<noteq> a}"}\\
+ iii) & @{term "\<forall>a. sort (\<pi> 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 "_ \<circ> _"}}, 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 \<Rightarrow> atom \<Rightarrow> 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) \<equiv> \<lambda>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) \<equiv>"} & @{text "if (sort a = sort b)"}\\
+ & \hspace{3mm}@{text "then \<lambda>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) 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="\<pi>\<^isub>1" and q="\<pi>\<^isub>2", THEN eq_reflection]} \hspace{4mm}
+ @{thm uminus_perm_def[where p="\<pi>", THEN eq_reflection]} \hspace{4mm}
+ @{thm diff_def[where x="\<pi>\<^isub>1" and y="\<pi>\<^isub>2"]}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ and verify the simple properties
+
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm add_assoc[where a="\<pi>\<^isub>1" and b="\<pi>\<^isub>2" and c="\<pi>\<^isub>3"]} \hspace{3mm}
+ @{thm monoid_add_class.add_0_left[where a="\<pi>::perm"]} \hspace{3mm}
+ @{thm monoid_add_class.add_0_right[where a="\<pi>::perm"]} \hspace{3mm}
+ @{thm group_add_class.left_minus[where a="\<pi>::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.
+
+ 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="\<pi>\<^isub>1" and q="\<pi>\<^isub>2",no_vars]}
+ \end{tabular}\hfill\numbered{newpermprops}
+ \end{isabelle}
+
+ \noindent
+ in which the permutation operations are of type @{text "perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"} 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.
+
+ We call type @{text "\<beta>"} a \emph{permutation type} if the permutation
+ properties in \eqref{newpermprops} are satisfied for every @{text "x"} of type
+ @{text "\<beta>"}. This notion allows us to establish generic lemmas, which are
+ applicable to any permutation type. 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="\<pi>", no_vars]}\hspace{10mm}
+ @{thm permute_minus_cancel(2)[where p="\<pi>", no_vars]}
+ \end{tabular}\hfill\numbered{cancel}
+ \end{isabelle}
+
+ \noindent
+ Consequently, in a permutation type the permutation operation @{text "\<pi> \<bullet> _"} is bijective,
+ which in turn implies the property
+
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm (lhs) permute_eq_iff[where p="\<pi>", no_vars]}
+ $\;$if and only if$\;$
+ @{thm (rhs) permute_eq_iff[where p="\<pi>", 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="\<pi>",no_vars, THEN eq_reflection]}\\
+ functions: & @{text "\<pi> \<bullet> f \<equiv> \<lambda>x. \<pi> \<bullet> (f ((-\<pi>) \<bullet> x))"}\\
+ permutations: & @{thm permute_perm_def[where p="\<pi>" and q="\<pi>'", THEN eq_reflection]}\\
+ sets: & @{thm permute_set_eq[where p="\<pi>", no_vars, THEN eq_reflection]}\\
+ booleans: & @{thm permute_bool_def[where p="\<pi>", no_vars, THEN eq_reflection]}\\
+ \end{tabular} &
+ \begin{tabular}{@ {}r@ {\hspace{2mm}}l@ {}}
+ lists: & @{thm permute_list.simps(1)[where p="\<pi>", no_vars, THEN eq_reflection]}\\
+ & @{thm permute_list.simps(2)[where p="\<pi>", no_vars, THEN eq_reflection]}\\[2mm]
+ products: & @{thm permute_prod.simps[where p="\<pi>", no_vars, THEN eq_reflection]}\\
+ nats: & @{thm permute_nat_def[where p="\<pi>", no_vars, THEN eq_reflection]}\\
+ \end{tabular}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ and then establish:
+
+ \begin{theorem}
+ If @{text \<beta>}, @{text "\<beta>\<^isub>1"} and @{text "\<beta>\<^isub>2"} are permutation types,
+ then so are: @{text "atom"}, @{text "\<beta>\<^isub>1 \<Rightarrow> \<beta>\<^isub>2"},
+ @{text perm}, @{term "\<beta> set"}, @{term "\<beta> list"}, @{term "\<beta>\<^isub>1 \<times> \<beta>\<^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 \<bullet> \<pi>'"} & @{text "\<equiv>"} & @{text "0 + \<pi>' - 0 = \<pi>'"}\\
+ @{text "(\<pi>\<^isub>1 + \<pi>\<^isub>2) \<bullet> \<pi>'"} & @{text "\<equiv>"} & @{text "(\<pi>\<^isub>1 + \<pi>\<^isub>2) + \<pi>' - (\<pi>\<^isub>1 + \<pi>\<^isub>2)"}\\
+ & @{text "="} & @{text "(\<pi>\<^isub>1 + \<pi>\<^isub>2) + \<pi>' - \<pi>\<^isub>2 - \<pi>\<^isub>1"}\\
+ & @{text "="} & @{text "\<pi>\<^isub>1 + (\<pi>\<^isub>2 + \<pi>' - \<pi>\<^isub>2) - \<pi>\<^isub>1"}\\
+ & @{text "\<equiv>"} & @{text "\<pi>\<^isub>1 \<bullet> \<pi>\<^isub>2 \<bullet> \<pi>'"}
+ \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 "\<pi>\<^isub>1 \<bullet> (\<pi>\<^isub>2 \<bullet> x) = (\<pi>\<^isub>1 \<bullet> \<pi>\<^isub>2) \<bullet> (\<pi>\<^isub>1 \<bullet> x)"}.
+ \end{lemma}
+
+ \begin{proof} The proof is as follows:
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ \begin{tabular}[b]{@ {}rcl@ {\hspace{8mm}}l}
+ @{text "\<pi>\<^isub>1 \<bullet> \<pi>\<^isub>2 \<bullet> x"}
+ & @{text "="} & @{text "\<pi>\<^isub>1 \<bullet> \<pi>\<^isub>2 \<bullet> (-\<pi>\<^isub>1) \<bullet> \<pi>\<^isub>1 \<bullet> x"} & by \eqref{cancel}\\
+ & @{text "="} & @{text "(\<pi>\<^isub>1 + \<pi>\<^isub>2 - \<pi>\<^isub>1) \<bullet> \<pi>\<^isub>1 \<bullet> x"} & by {\rm(\ref{newpermprops}.@{text "ii"})}\\
+ & @{text "\<equiv>"} & @{text "(\<pi>\<^isub>1 \<bullet> \<pi>\<^isub>2) \<bullet> (\<pi>\<^isub>1 \<bullet> 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) 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 "\<forall>\<pi>. \<pi> \<bullet> 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 "\<alpha> \<Rightarrow> \<beta>"}, then equivariance
+ can also be stated as
+
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{text "\<forall>\<pi> x. \<pi> \<bullet> (f x) = f (\<pi> \<bullet> 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 "\<pi> \<bullet> (f x) = (\<pi> \<bullet> f) (\<pi> \<bullet> 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="\<pi>", 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 "\<and>"}, @{text "\<or>"} and @{text "\<longrightarrow>"} 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="\<pi>", no_vars]}
+ \end{tabular}\hfill\numbered{swapeqvt}
+ \end{isabelle}
+
+ \noindent
+ for all @{text a}, @{text b} and @{text \<pi>}. 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 \<rightleftharpoons> c) \<bullet> x \<noteq> x}"} and @{term "finite {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> 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 \<rightleftharpoons> c) \<bullet> x = x"}} and @{term "(b \<rightleftharpoons> c) \<bullet> 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="\<pi>",no_vars]}
+ if and only if @{thm (rhs) fresh_permute_iff[where p="\<pi>",no_vars]}.
+ \end{lemma}
+
+ \begin{proof}
+ \begin{isabelle}
+ \begin{tabular}[t]{c@ {\hspace{2mm}}l@ {\hspace{5mm}}l}
+ & @{thm (lhs) fresh_permute_iff[where p="\<pi>",no_vars]}
+ & \\
+ @{text "\<Leftrightarrow>"}
+ & @{term "finite {b. (\<pi> \<bullet> a \<rightleftharpoons> b) \<bullet> \<pi> \<bullet> x \<noteq> \<pi> \<bullet> x}"}
+ & by definition\\
+ @{text "\<Leftrightarrow>"}
+ & @{term "finite {b. (\<pi> \<bullet> a \<rightleftharpoons> \<pi> \<bullet> b) \<bullet> \<pi> \<bullet> x \<noteq> \<pi> \<bullet> x}"}
+ & since @{text "\<pi> \<bullet> _"} is bijective\\
+ @{text "\<Leftrightarrow>"}
+ & @{term "finite {b. \<pi> \<bullet> (a \<rightleftharpoons> b) \<bullet> x \<noteq> \<pi> \<bullet> x}"}
+ & by \eqref{permutecompose} and \eqref{swapeqvt}\\
+ @{text "\<Leftrightarrow>"}
+ & @{term "finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}"}
+ & by \eqref{permuteequ}\\
+ @{text "\<Leftrightarrow>"}
+ & @{thm (rhs) fresh_permute_iff[where p="\<pi>",no_vars]}
+ & by definition\\
+ \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="\<pi>",no_vars]}
+
+ \noindent
+ Now equivariance of @{text "supp"}, namely
+
+ @{thm [display,indent=10] supp_eqvt[where p="\<pi>",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 \<rightleftharpoons> b) \<bullet> 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:\\[2mm]
+
+ & \multicolumn{1}{c}{for all @{text "S'"}, if @{term "finite S'"} and
+ @{term "S' supports x"} then @{text "S \<subseteq> 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 \<in> supp x"} and @{text "a \<notin> S"}. Using the second fact, the
+ assumption that @{term "S supports x"} gives us that @{text S} is a superset of
+ @{term "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}"}, which is finite by the assumption of @{text S}
+ being finite. But this means @{term "a \<notin> 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.
+ Then @{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 \<in> S"} and all @{text "b \<notin> S"}, with @{text a}
+ and @{text b} having the same sort, then @{term "(a \<rightleftharpoons> b) \<bullet> x \<noteq> 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 \<subseteq> 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 \<notin> S"} and @{text "b \<notin> S'"}. Since we assumed @{text "a \<notin> S'"} and
+ we have that @{text "S' supports x"}, we have on one hand @{term "(a \<rightleftharpoons> b) \<bullet> x
+ = x"}. On the other hand, the fact @{text "a \<in> S"} and @{text "b \<notin> S"} imply
+ @{term "(a \<rightleftharpoons> b) \<bullet> x \<noteq> 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 "\<forall>a \<in> {c} b \<notin> {c}. sort a = sort b \<longrightarrow> (a b) \<bullet> c \<noteq> 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="\<pi>", 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 \<pi> \<bullet> a \<noteq> a, \<pi> \<bullet> b = b and sort a = sort b, then (a b) \<bullet> \<pi> \<noteq> \<pi>"}.
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ For this we observe that
+
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ \begin{tabular}{@ {}rcl}
+ @{thm (lhs) perm_swap_eq[where p="\<pi>", no_vars]} &
+ if and only if &
+ @{thm (rhs) perm_swap_eq[where p="\<pi>", 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 "(p \<bullet> a \<rightleftharpoons> b)"} and @{term "(a \<rightleftharpoons> 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 [display,indent=10] "finite (supp x)"}
+
+ \noindent
+ 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) \<equiv> i"}
+
+ \noindent
+ This function's support is the set of \emph{all} atoms. To establish this we show @{term "\<not> a \<sharp> nat_of"}.
+ This is equivalent to assuming the set @{term "{b. (a \<rightleftharpoons> b) \<bullet> nat_of \<noteq> nat_of}"} is finite
+ and deriving a contradiction. From the assumption we also know that
+ @{term "{a} \<union> {b. (a \<rightleftharpoons> b) \<bullet> nat_of \<noteq> nat_of}"} is finite. Then we can use
+ Proposition~\ref{choosefresh} to choose an atom @{text c} such that
+ @{term "c \<noteq> a"}, @{term "sort_of c = sort_of a"} and @{term "(a \<rightleftharpoons> c) \<bullet> 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 \<rightleftharpoons> c) \<bullet> (nat_of a)"} & by def.~of permutations on nats\\
+ & @{text "="} & @{term "((a \<rightleftharpoons> c) \<bullet> nat_of) ((a \<rightleftharpoons> c) \<bullet> a)"} & by \eqref{permutefunapp}\\
+ & @{text "="} & @{term "nat_of c"} & by assumptions on @{text c}\\
+ \end{tabular}\hfill\qed
+ \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 \<noteq> a"} about how we chose @{text c}. Similar examples for
+ constructions that have infinite support are given in~\cite{Cheney06}.
+*}
+
+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 \<rightleftharpoons> b) \<bullet> (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\<iota> 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 "\<lfloor>_\<rfloor> :: name \<Rightarrow> atom"} &
+ @{text "\<lceil>_\<rceil> :: atom \<Rightarrow> name"}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ With the definition @{thm permute_name_def [where p="\<pi>", 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.
+
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ \begin{tabular}{@ {}r@ {\hspace{4mm}}p{10cm}}
+ i) & if @{thm (lhs) atom_eq_iff [no_vars]} then @{thm (rhs) atom_eq_iff [no_vars]}\\
+ ii) & @{thm atom_eqvt[where p="\<pi>", 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 "\<alpha> \<Rightarrow> \<alpha>
+ \<Rightarrow> perm"}, where @{text "\<alpha>"} 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 "\<leftrightarrow>"}-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 "\<leftrightarrow>"} 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 \<leftrightarrow> b) \<bullet> (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 ("_ \<rightarrow> _")
+(*>*)
+
+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 "\<lambda>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 "\<lambda>x\<^isub>\<alpha>. x\<^isub>\<alpha> x\<^isub>\<beta>"}
+ \end{tabular}\hfill\numbered{church}
+ \end{isabelle}
+
+ \noindent
+ both variables and binders include typing information indicated by @{text
+ \<alpha>} and @{text \<beta>}. In this setting, we treat @{text
+ "x\<^isub>\<alpha>"} and @{text "x\<^isub>\<beta>"} as distinct variables
+ (assuming @{term "\<alpha>\<noteq>\<beta>"}) so that the variable @{text
+ "x\<^isub>\<alpha>"} is bound by the lambda-abstraction, but not @{text
+ "x\<^isub>\<beta>"}.
+
+ To illustrate how we can deal with this phenomenon, let us represent object
+ types like @{text \<alpha>} and @{text \<beta>} by the datatype
+
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ \isacommand{datatype}~~@{text "ty = TVar string | ty \<rightarrow> 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 \<leftrightarrow> y)"}, applied to the pair
+
+ @{text [display,indent=10] "((x, \<alpha>), (x, \<beta>))"}
+
+ \noindent
+ will always permute \emph{both} occurrences of @{text x}, even if the types
+ @{text "\<alpha>"} and @{text "\<beta>"} 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\<iota>\<iota> = Atom\<iota>\<iota> 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 \<iota>atom\<iota>\<iota>\<iota> = \<iota>Atom\<iota>\<iota> '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\<iota>\<iota>\<iota>\<iota> = Atom\<iota>\<iota>\<iota>\<iota> 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 (\<tau>\<^isub>1 \<rightarrow> \<tau>\<^isub>2) = Sort ''Fun'' [sort_ty \<tau>\<^isub>1, sort_ty \<tau>\<^isub>2]"}
+ \end{tabular}\hfill\numbered{sortty}
+ \end{isabelle}
+
+ \noindent
+ which can easily be shown to be injective.
+
+ Having settled on what the 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 "\<lfloor>_\<rfloor> :: var \<Rightarrow> atom"}} and
+ @{text "\<lceil>_\<rceil> :: atom \<Rightarrow> var"}, respectively.
+
+ We can define the permutation operation for @{text var} as @{thm
+ permute_var_def[where p="\<pi>", 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="\<alpha>", 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 \<alpha> \<leftrightarrow> Var y \<alpha>)"} and to prove the following
+ lemma
+*}
+
+ lemma
+ assumes asm: "\<alpha> \<noteq> \<beta>"
+ shows "(Var x \<alpha> \<leftrightarrow> Var y \<alpha>) \<bullet> (Var x \<alpha>, Var x \<beta>) = (Var y \<alpha>, Var x \<beta>)"
+ using asm by simp
+
+text {*
+ \noindent
+ As we expect, the atom @{term "Var x \<beta>"} 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
+
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ \isacommand{atom\_decl}~~@{text "var (ty)"}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ 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 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 \<pi>}, then we only have to state @{term "\<forall>\<pi>. P (\<pi> \<bullet> 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 permutation as functions so that the associated
+ permutation operation has only a single type parameter. From this we derive
+ most benefits 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.
+
+ 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 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.
+
+ Finally, our implementation of sorted atoms turned out powerful enough to
+ use it for representing variables that carry 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. We are not aware of any other approach proposed for
+ language formalisations 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}.\medskip
+
+ \noindent
+ {\bf Acknowledgements:} We are very grateful to Jesper Bengtson, Stefan
+ Berghofer and Cezary Kaliszyk for their comments on earlier versions
+ of this paper.
+
+*}
+
+
+(*<*)
+end
+(*>*)
\ No newline at end of file