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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 {*

  \begin{center}

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

  \end{center}

  \noindent

  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}.

  \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

  at once.

  \begin{center}

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

  \end{center}

  \noindent

  \begin{center}

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

  \end{center}

  \noindent

  \begin{center}

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

  \end{center}

  \noindent

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

  \begin{center}

  \end{center}

  \noindent

  \begin{center}

  \end{center}

  \noindent

  \begin{center}

  \end{center}

  \noindent

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

  \begin{center}

  \end{center}

  \noindent

  \begin{center}

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

  \end{center}

  \noindent

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

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

  \begin{center}

  \end{center}

  \noindent

  in $s$. In this representation we need additional predicates to ensure 

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

  Contributions:  We provide definitions for when terms

  involving general bindings are alpha-equivelent.

  %\begin{center}

  %\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}

  %\end{center}

  quotient package \cite{Homeier05}

*}

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

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

  many discussions about Nominal Isabelle. We thank Peter Sewell for 

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

  also for explaining some of the finer points of the OTT-tool.

*}

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end

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