(*<*)

theory Paper

imports "../Nominal/Test" "LaTeXsugar"

begin

notation (latex output)

  swap ("'(_ _')" [1000, 1000] 1000) and

  fresh ("_ # _" [51, 51] 50) and

  fresh_star ("_ #* _" [51, 51] 50) and

  supp ("supp _" [78] 73) and

  uminus ("-_" [78] 73) and

  If  ("if _ then _ else _" 10)

(*>*)

section {* Introduction *}

text {*

  So far, Nominal Isabelle provides a mechanism for constructing

  alpha-equated terms such as

  \begin{center}

  $t ::= x \mid t\;t \mid \lambda x. t$

  \end{center}

  \noindent

  where free and bound variables have names.  For such terms Nominal Isabelle

  derives automatically a reasoning infrastructure, which has been used

  successfully in formalisations of 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}.

  However, Nominal Isabelle has fared less well in a formalisation of

  the algorithm W \cite{UrbanNipkow09}, where types and type-schemes

  are of the form

  \begin{center}

  \begin{tabular}{l}

  $T ::= x \mid T \rightarrow T$ \hspace{5mm} $S ::= \forall \{x_1,\ldots, x_n\}. T$

  \end{tabular}

  \end{center}

  \noindent

  type-variables.  While it is possible to implement this kind of more general

  binders by iterating single binders, this leads to a rather clumsy

  formalisation of W. The need of iterating single binders is also one reason

  why Nominal Isabelle and similar theorem provers that only provide

  mechanisms for binding single variables have not fared extremely well with the

  more advanced tasks in the POPLmark challenge \cite{challenge05}, because

  also there one would like to bind multiple variables at once.

  Binding multiple variables has interesting properties that are not captured

  by iterating single binders. For example in the case of type-schemes we do not

  like to make a distinction about the order of the bound variables. Therefore

  we would like to regard the following two type-schemes as alpha-equivalent

  \begin{center}

  $\forall \{x, y\}. x \rightarrow y  \;\approx_\alpha\; \forall \{y, x\}. y \rightarrow x$ 

  \end{center}

  \noindent

  but  the following two should \emph{not} be alpha-equivalent

  \begin{center}

  $\forall \{x, y\}. x \rightarrow y  \;\not\approx_\alpha\; \forall \{z\}. z \rightarrow z$ 

  \end{center}

  \noindent

  assuming that $x$, $y$ and $z$ are distinct. Moreover, we like to regard type-schemes as 

  alpha-equivalent, if they differ only on \emph{vacuous} binders, such as

  \begin{center}

  $\forall \{x\}. x \rightarrow y  \;\approx_\alpha\; \forall \{x, z\}. x \rightarrow y$ 

  \end{center}

  \noindent

  where $z$ does not occur freely in the type.

  In this paper we will give a general binding mechanism and associated

  notion of alpha-equivalence that can be used to faithfully represent

  this kind of binding in Nominal Isabelle.  The difficulty of finding the right notion 

  for alpha-equivalence in this case can be appreciated by considering that the 

  definition given by Leroy in \cite{Leroy92} is incorrect (it omits a side-condition).

  However, the notion of alpha-equivalence that is preserved by vacuous binders is not

  \begin{equation}\label{one}

  \LET x = 3 \AND y = 2 \IN x\,-\,y \END

  \end{equation}

  \noindent

  we might not care in which order the assignments $x = 3$ and $y = 2$ are

  given, but it would be unusual to regard \eqref{one} as alpha-equivalent 

  with

  \begin{center}

  $\LET x = 3 \AND y = 2 \AND z = loop \IN x\,-\,y \END$

  \end{center}

  \noindent

  Therefore we will also provide a separate binding mechanism for cases in

  which the order of binders does not matter, but the ``cardinality'' of the

  binders has to agree.

  However, we found that this is still not sufficient for dealing with

  language constructs frequently occurring in programming language

  research. For example in $\mathtt{let}$s containing patterns

  \begin{equation}\label{two}

  \LET (x, y) = (3, 2) \IN x\,-\,y \END

  \end{equation}

  \noindent

  we want to bind all variables from the pattern inside the body of the

  $\mathtt{let}$, but we also care about the order of these variables, since

  \begin{center}

  $\LET (y, x) = (3, 2) \IN x\,- y\,\END$

  \end{center}

  \noindent

  As a result, we provide three general binding mechanisms each of which binds multiple

  programming language calculus.

  By providing these general binding mechanisms, however, we have to work around 

  a problem that has been pointed out by Pottier in \cite{Pottier06}: in 

  $\mathtt{let}$-constructs of the form

  \begin{center}

  $\LET x_1 = t_1 \AND \ldots \AND x_n = t_n \IN s \END$

  \end{center}

  \noindent

  which bind all the $x_i$ in $s$, we might not care about the order in 

  which the $x_i = t_i$ are given, but we do care about the information that there are 

  as many $x_i$ as there are $t_i$. We lose this information if we represent the 

  $\mathtt{let}$-constructor by something like 

  \begin{center}

  $\LET [x_1,\ldots,x_n].s\;\; [t_1,\ldots,t_n]$

  \end{center}

  \noindent

  \begin{center}

  \begin{tabular}{r@ {\hspace{2mm}}r@ {\hspace{2mm}}l}

  $trm$ & $=$  & \ldots\\ 

        & $\mid$ & $\mathtt{let}\;a\!::\!assn\;\;s\!::\!trm\quad\mathtt{bind}\;bn\,(a) \IN s$\\[1mm]

  $assn$ & $=$  & $\mathtt{anil}$\\

         & $\mid$ & $\mathtt{acons}\;\;name\;\;trm\;\;assn$

  \end{tabular}

  \end{center}

  \noindent

  where $assn$ is an auxiliary type representing a list of assignments

  and $bn$ an auxiliary function identifying the variables to be bound by 

  \begin{center}

  $bn\,(\mathtt{anil}) = \varnothing \qquad bn\,(\mathtt{acons}\;x\;t\;as) = \{x\} \cup bn\,(as)$ 

  \end{center}

  \noindent

  The scope of the binding is indicated by labels given to the types, for

  example \mbox{$s\!::\!trm$}, and a binding clause, in this case

  $\mathtt{bind}\;bn\,(a) \IN s$, that states to bind all the names the function

  $bn$ returns in $s$.  This style of specifying terms and bindings is heavily

  inspired by the syntax of the Ott-tool \cite{ott-jfp}.

  However, we will not be able to deal with all specifications that are

  allowed by Ott. One reason is that we establish the reasoning infrastructure

  for alpha-\emph{equated} terms. In contrast, Ott produces for a subset of

  its specifications a reasoning infrastructure in Isabelle/HOL for

  \emph{non}-alpha-equated, or ``raw'', terms. While our alpha-equated terms

  and the raw terms produced by Ott use names for bound variables,

  there is a key difference: working with alpha-equated terms means that the

  two type-schemes with $x$, $y$ and $z$ being distinct

  \begin{center}

  $\forall \{x\}. x \rightarrow y  \;=\; \forall \{x, z\}. x \rightarrow y$ 

  \end{center}

  \noindent

  are not just alpha-equal, but actually equal. As a

  result, we can only support specifications that make sense on the level of

  alpha-equated terms (offending specifications, which for example bind a variable

  according to a variable bound somewhere else, are not excluded by Ott, but we 

  have to).  Our

  insistence on reasoning with alpha-equated terms comes from the wealth of

  experience we gained with the older version of Nominal Isabelle: for

  non-trivial properties, reasoning about alpha-equated terms is much easier

  than reasoning with raw terms. The fundamental reason for this is that the

  HOL-logic underlying Nominal Isabelle allows us to replace

  ``equals-by-equals''. In contrast replacing ``alpha-equals-by-alpha-equals''

  in a representation based on raw terms requires a lot of extra reasoning work.

  Although in informal settings a reasoning infrastructure for alpha-equated 

  terms (that have names for bound variables) is nearly always taken for granted, establishing 

  it automatically in the Isabelle/HOL theorem prover is a rather non-trivial task. 

  For every specification we will need to construct a type containing as 

  elements the alpha-equated terms. To do so, we use 

  the standard HOL-technique of defining a new type by  

  identifying a non-empty subset of an existing type.   The construction we 

  perform in HOL is illustrated by the following picture:

  \begin{center}

  \begin{tikzpicture}

  %\draw[step=2mm] (-4,-1) grid (4,1);

  \draw[very thick] (0.7,0.4) circle (4.25mm);

  \draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9);

  \draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05);

  \draw (-2.0, 0.845) --  (0.7,0.845);

  \draw (-2.0,-0.045)  -- (0.7,-0.045);

  \draw ( 0.7, 0.4) node {\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]clas.\end{tabular}};

  \draw (-2.4, 0.4) node {\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};

  \draw (1.8, 0.48) node[right=-0.1mm]

    {\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ (sets of raw terms)\end{tabular}};

  \draw (0.9, -0.35) node {\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};

  \draw (-3.25, 0.55) node {\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}};

  \draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3);

  \draw (-0.95, 0.3) node[above=0mm] {isomorphism};

  \end{tikzpicture}

  \end{center}

  \noindent

  We take as the starting point a definition of raw terms (being defined as a 

  datatype in Isabelle/HOL); identify then the 

  alpha-equivalence classes in the type of sets of raw terms, according to our 

  alpha-equivalence relation and finally define the new type as these 

  alpha-equivalence classes (non-emptiness is satisfied whenever the raw terms are 

  definable as datatype in Isabelle/HOL and the fact that our relation for alpha is an 

  equivalence relation).\marginpar{\footnotesize Does Ott allow definitions of ``strange'' types?}

  The fact that we obtain an isomorphism between between the new type and the non-empty 

  subset shows that the new type is a faithful representation of alpha-equated terms. 

  That is different for example in the representation of terms using the locally 

  nameless representation of binders: there are non-well-formed terms that need to

  be excluded by reasoning about a well-formedness predicate.

  The problem with introducing a new type is that in order to be useful 

  a resoning infrastructure needs to be ``lifted'' from the underlying type

  and subset to the new type. This is usually a tricky task. To ease this task 

  we reimplemented in Isabelle/HOL the quotient package described by Homeier 

  \cite{Homeier05}. Gieven that alpha is an equivalence relation, this package 

  allows us to automatically lift definitions and theorems involving raw terms

  to definitions and theorems involving alpha-equated terms. This of course

  only works if the definitions and theorems are respectful w.r.t.~alpha-equivalence.

  Hence we will be able to lift, for instance, the function for free

  variables of raw terms to alpha-equated terms (since this function respects 

  alpha-equivalence), but we will not be able to do this with a bound-variable

  function (since it does not respect alpha-equivalence). As a result, each

  lifting needs some respectulness proofs which we automated.\medskip

  \noindent

  {\bf Contributions:}  We provide new definitions for when terms

  involving multiple binders are alpha-equivalent. These definitions are

  inspired by earlier work of Pitts \cite{}. By means of automatic

  proofs, we establish a reasoning infrastructure for alpha-equated

  terms, including properties about support, freshness and equality

  conditions for alpha-equated terms. We re also able to derive, at the moment 

  only manually, for these terms a strong induction principle that 

  has the variable convention already built in.

*}

section {* A Short Review of the Nominal Logic Work *}

text {*

  At its core, Nominal Isabelle is an adaption of the nominal logic work by

  Pitts \cite{Pitts03}. This adaptation for Isabelle/HOL is described in

  \cite{HuffmanUrban10}, which we review here briefly to aid the description

  of what follows. Two central notions in the nominal logic work are sorted

  atoms and permutations of atoms. The sorted atoms represent different kinds

  of variables, such as term- and type-variables in Core-Haskell, and it is

  assumed that there is an infinite supply of atoms for each sort. However, in

  order to simplify the description, we shall assume in what follows that

  there is only a single sort of atoms.

  Permutations are bijective functions from atoms to atoms that are 

  the identity everywhere except on a finite number of atoms. There is a 

  two-place permutation operation written

  @{text[display,indent=5] "_ \<bullet> _  ::  (\<alpha> \<times> \<alpha>) list \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}

  \noindent 

  with a generic type in which @{text "\<alpha>"} stands for the type of atoms 

  and @{text "\<beta>"} for the type of the objects on which the permutation 

  acts. In Nominal Isabelle the identity permutation is written as @{term "0::perm"},

  the composition of two permutations @{term p} and @{term q} as \mbox{@{term "p + q"}} 

  and the inverse permutation @{term p} as @{text "- p"}. The permutation

  operation is defined for products, lists, sets, functions, booleans etc 

  (see \cite{HuffmanUrban10}).

  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=5] supp_def[no_vars, THEN eq_reflection]}

  \noindent

  There is also the derived notion for when an atom @{text a} is \emph{fresh}

  for an @{text x}, defined as

  @{thm[display,indent=5] fresh_def[no_vars]}

  \noindent

  We also use for sets of atoms the abbreviation 

  @{thm (lhs) fresh_star_def[no_vars]} defined as 

  @{thm (rhs) fresh_star_def[no_vars]}.

  A striking consequence of these definitions is that we can prove

  without knowing anything about the structure of @{term x} that

  swapping two fresh atoms, say @{text a} and @{text b}, leave 

  @{text x} unchanged. 

  \begin{property}

  @{thm[mode=IfThen] swap_fresh_fresh[no_vars]}

  \end{property}

  \noindent

  For a proof see \cite{HuffmanUrban10}.

  \begin{property}

  @{thm[mode=IfThen] at_set_avoiding[no_vars]}

  \end{property}

*}

section {* General Binders *}

text {*

  In order to keep our work managable we like to state general definitions  

  and perform proofs inside Isabelle as much as possible, as opposed to write

  custom ML-code that generates appropriate definitions and proofs for each 

  instance of a term-calculus. For this, we like to consider pairs

  \begin{equation}\label{three}

  \mbox{@{text "(as, x) :: atom set \<times> \<beta>"}}

  \end{equation}

  \noindent

  consisting of a set of atoms and an object of generic type. The pairs

  are intended to be used for representing binding such as found in 

  type-schemes

  \begin{center}

  $\forall \{x_1,\ldots,x_n\}. T$

  \end{center}

  \noindent

  where the atoms $x_1,\ldots,x_n$ is intended to be in \eqref{three} the 

  set @{text as} of atoms we want to bind, and $T$, an object-level type, is 

  one instance for the generic $x$.

  The first question we have to answer is when we should consider the pairs 

  in \eqref{three} as alpha-equivelent? (At the moment we are interested in

  the notion of alpha-equivalence that is \emph{not} preserved by adding 

  vacuous binders.) Assuming we are given a free-variable function, say 

  \mbox{@{text "fv :: \<beta> \<Rightarrow> atom set"}}, then we expect for two alpha-equivelent

  pairs that their sets of free variables aggree. That is

  %

  \begin{equation}\label{four}

  \mbox{@{text "(as, x) \<approx> (bs, y)"} \hspace{2mm}implies\hspace{2mm} @{text "fv(x) - as = fv(y) - bs"}} 

  \end{equation}

  \noindent

  Next we expect that there is a permutation, say $p$, that leaves the 

  free variables unchanged, but ``moves'' the bound names in $x$ so that

  we obtain $y$ modulo a relation, say @{text "_ R _"}, that characterises when two

  elments of type $\beta$ are equivalent. We also expect that $p$ 

  makes the binders equal. We can formulate these requirements as: there

  exists a $p$ such that $i)$  @{term "(fv(x) - as) \<sharp>* p"}, $ii)$ @{text "(p \<bullet> x) R y"} and

  $iii)$ @{text "(p \<bullet> as) = bs"}. 

  We take now \eqref{four} and the three 

  General notion of alpha-equivalence (depends on a free-variable

  function and a relation).

*}

section {* Alpha-Equivalence and Free Variables *}

text {*

  Restrictions

  \begin{itemize}

  \item non-emptyness

  \item positive datatype definitions

  \item finitely supported abstractions

  \item respectfulness of the bn-functions\bigskip

  \item binders can only have a ``single scope''

  \end{itemize}

*}

section {* Examples *}

section {* Adequacy *}

section {* Related Work *}

section {* Conclusion *}

text {*

  Complication when the single scopedness restriction is lifted (two 

  overlapping permutations)

*}

text {*

  TODO: function definitions:

  \medskip

  \noindent

  {\bf Acknowledgements:} We are very grateful to Andrew Pitts for  

  many discussions about Nominal Isabelle. We thank Peter Sewell for 

  making the informal notes \cite{SewellBestiary} available to us and 

  also for patiently explaining some of the finer points about the abstract 

  definitions and about the implementation of the Ott-tool.

*}

(*<*)

end

(*>*)