(*<*)

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

  automatically alpha-equated terms such as

  \begin{center}

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

  \end{center}

  \noindent

  For such terms it derives automatically a reasoning

  infrastructure, which has been used 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 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

  and the quantification abstracts over a finite (possibly empty) set of type variables.

  While it is possible to formalise such abstractions by iterating single bindings, 

  it leads to a very clumsy formalisation of W. This need of iterating single binders 

  for representing multiple binders is also the reason why Nominal Isabelle and other 

  theorem provers have not fared extremely well with the more advanced tasks 

  in the POPLmark challenge \cite{challenge05}, because also there one would be able 

  to aviod clumsy reasoning if there were a mechanisms for abstracting several variables 

  at once.

  To see this, let us point out some interesting properties of binders abstracting multiple 

  variables. First, 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

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

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

  type-schemes 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

  alway wanted. For example in terms like

  \begin{center}

  $\LET x = 3 \AND y = 2 \IN x\,\backslash\,y \END$

  \end{center}

  \noindent

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

  given, but it would be unusual to regard the above term as alpha-equivalent 

  with

  \begin{center}

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

  \end{center}

  \noindent

  Therefore we will also provide a separate abstraction mechanism for cases 

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

  binders has to be the same.

  However, we found that this is still not sufficient for covering language 

  constructs frequently occuring in programming language research. For example 

  in $\mathtt{let}$s involving patterns 

  \begin{center}

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

  \end{center}

  \noindent

  we want to bind all variables from the pattern (there might be an arbitrary

  number of them) inside the body of the $\mathtt{let}$, but we also care about 

  the order of these variables, since we do not want to identify the above term 

  with

  \begin{center}

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

  \end{center}

  \noindent

  As a result, we provide three general abstraction mechanisms for multiple binders

  and allow the user to chose which one is intended when formalising a

  programming language calculus.

  By providing these general abstraction 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 as something like 

  \begin{center}

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

  \end{center}

  \noindent

  that the two lists are of equal length. This can result into very 

  unintelligible reasoning (see for example~\cite{BengtsonParow09}). 

  Contributions:  We provide definitions for when terms

  involving general bindings are alpha-equivelent.

  %\begin{pspicture}(0.5,0.0)(8,2.5)

  %%\showgrid

  %\psframe[linewidth=0.4mm,framearc=0.2](5,0.0)(7.7,2.5)

  %\pscircle[linewidth=0.3mm,dimen=middle](6,1.5){0.6}

  %\psframe[linewidth=0.4mm,framearc=0.2,dimen=middle](1.1,2.1)(2.3,0.9)

  %\pcline[linewidth=0.4mm]{->}(2.6,1.5)(4.8,1.5)

  %\pcline[linewidth=0.2mm](2.2,2.1)(6,2.1)

  %\pcline[linewidth=0.2mm](2.2,0.9)(6,0.9)

  %\rput(7.3,2.2){$\mathtt{phi}$}

  %\rput(6,1.5){$\lama$}

  %\rput[l](7.6,2.05){\begin{tabular}{l}existing\\[-1.6mm]type\end{tabular}}

  %\rput[r](1.2,1.5){\begin{tabular}{l}new\\[-1.6mm]type\end{tabular}}

  %\rput(6.1,0.5){\begin{tabular}{l}non-empty\\[-1.6mm]subset\end{tabular}}

  %\rput[c](1.7,1.5){$\lama$}

  %\rput(3.7,1.75){isomorphism}

  %\end{pspicture}

*}

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

text {*

  At its core, Nominal Isabelle is based on the nominal logic work by Pitts

  \cite{Pitts03}. The implementation of this work are described in

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

  of what follows in the next sections. 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 of our work, we

  shall assume in this paper 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 {* Abstractions *}

text {*

  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

  many discussions about Nominal Isabelle. We thank Peter Sewell for 

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

*}

(*<*)

end

(*>*)