--- a/Pearl/Paper.thy Tue Feb 19 05:38:46 2013 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1231 +0,0 @@
-(*<*)
-theory Paper
-imports "../Nominal/Nominal2_Base"
- "../Nominal/Atoms"
- "~~/src/HOL/Library/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} 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] "_ \<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}\\
- \end{tabular}\hspace{12mm}
- \begin{tabular}{@ {}r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
- @{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 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-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] "\<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:} Using a single atom type to represent
- atoms of different sorts and representing permutations as functions are not
- new ideas. The main contribution of this paper is to show an example of how
- to make better theorem proving tools by choosing the right level of
- abstraction for the underlying theory---our design choices take advantage of
- Isabelle's type system, type classes, and reasoning infrastructure.
- The novel
- technical contribution 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.\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 \<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}\ \ \ \ \ \ \ \ \ \ %%%
- i)~~~@{term "bij \<pi>"}\hspace{5mm}
- ii)~~~@{term "finite {a. \<pi> a \<noteq> a}"}\hspace{5mm}
- iii)~~~@{term "\<forall>a. sort (\<pi> a) = sort a"}\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 a (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="\<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 minus_perm_def[where ?p1.0="\<pi>\<^isub>1" and ?p2.0="\<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 we can rely on
- Isabelle/HOL's existing simplification infrastructure for groups, which will
- come in handy when we have to do calculations with permutations.
- Note that Isabelle/HOL defies standard conventions of mathematical notation
- by using additive syntax even for non-commutative groups. Obviously,
- composition of permutations is not commutative in general---@{text
- "\<pi>\<^sub>1 + \<pi>\<^sub>2 \<noteq> \<pi>\<^sub>2 + \<pi>\<^sub>1"}. But since the point of this paper is to implement the
- nominal theory as smoothly as possible in Isabelle/HOL, we tolerate
- the non-standard notation in order to reuse the existing libraries.
-
- 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.
-
- The next notion allows us to establish generic lemmas involving the
- permutation operation.
-
- \begin{definition}
- A type @{text "\<beta>"} is a \emph{permutation type} if the permutation
- properties in \eqref{newpermprops} are satisfied for every @{text "x"} of type
- @{text "\<beta>"}.
- \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="\<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 { *
-
- An \emph{equivariant} function or predicate is one that is invariant under
- the swapping of atoms. Having a notion of equivariance with nice logical
- properties is a major advantage of bijective permutations over traditional
- renaming substitutions \cite[\S2]{Pitts03}. Equivariance can be defined
- uniformly for all permutation types, and it is satisfied by most HOL
- functions and constants.
-
- \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}
- & \multicolumn{2}{@ {}l}{@{thm (lhs) fresh_permute_iff[where p="\<pi>",no_vars]} @{text "\<equiv>"}
- @{term "finite {b. (\<pi> \<bullet> a \<rightleftharpoons> b) \<bullet> \<pi> \<bullet> x \<noteq> \<pi> \<bullet> x}"}}\\
- @{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}"} @{text "\<equiv>"}
- @{thm (rhs) fresh_permute_iff[where p="\<pi>",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="\<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,
- 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.
- 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 \<in> S"} and all @{text "b \<notin> S"}, with @{text a}
- and @{text b} having the same sort, \mbox{@{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 "(\<pi> \<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 "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) \<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}
- \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}.
- Cheney \cite{Cheney06} gives similar examples for constructions that have infinite support.
-*}
-
-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, that is
-
- \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
- i) \hspace{1mm}if @{thm (lhs) atom_eq_iff [no_vars]} then @{thm (rhs) atom_eq_iff [no_vars]}\hspace{4mm}
- ii) \hspace{1mm}@{thm atom_eqvt[where p="\<pi>", no_vars]}\hspace{4mm}
- iii) \hspace{1mm}@{thm sort_of_atom_eq [no_vars]}
- \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 "((x, \<alpha>), (x, \<beta>))"}
- 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 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 \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 can make
- headway with formalising their results
- about simple type theory \cite{PaulsonBenzmueller}.
- 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 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 permutations 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 "\<forall>a \<in> supp x. a \<sharp> p"} implies @{term "p \<bullet> 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