(*<*)

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, for example

  \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 that  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{equation}\label{tysch}

  \begin{array}{l}

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

  \end{array}

  \end{equation}

  \noindent

  and the quantification $\forall$ binds a finite (possibly empty) set of

  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 cannot be captured

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

  want 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{equation}\label{ex1}

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

  \end{equation}

  \noindent

  but assuming that $x$, $y$ and $z$ are distinct variables,

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

  %

  \begin{equation}\label{ex2}

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

  \end{equation}

  \noindent

  Moreover, we like to regard type-schemes as 

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

  %

  \begin{equation}\label{ex3}

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

  \end{equation}

  \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 can be appreciated in this case 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

  always wanted. For example in terms like

  %

  \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

  we do not want to regard \eqref{two} as alpha-equivalent with

  %

  \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

  variables at once, and let the user chose which one is intended when formalising a

  programming language calculus.

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

  %

  \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

  where the notation $[\_\!\_].\_\!\_$ indicates that the $x_i$ become bound

  in $s$. In this representation the term \mbox{$\LET [x].s\;\;[t_1,t_2]$}

  would be a perfectly legal instance. To exclude such terms, additional

  predicates about well-formed terms are needed in order to ensure that the two

  lists are of equal length. This can result into very messy reasoning (see

  for example~\cite{BengtsonParow09}). To avoid this, we will allow type specifications

  for $\mathtt{let}$s as follows

  %

  \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 

  the $\mathtt{let}$. This function is defined by recursion over $assn$ as follows

  \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 in $s$ all the names the

  function call $bn\,(a)$ returns.  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 Ott lets the user to specify ``empty'' 

  types like

  \begin{center}

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

  \end{center}

  \noindent

  where no clause for variables is given. Arguably, such specifications make

  some sense in the context of Coq's type theory (which Ott supports), but not

  at all in a HOL-based environment where every datatype must have a non-empty

  set-theoretic model.

  Another reason is that we establish the reasoning infrastructure

  for alpha-\emph{equated} terms. In contrast, Ott produces  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 \emph{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 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 can be 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 (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-equivalence is indeed an equivalence relation).

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

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

  That is not the case for example for terms using the locally 

  nameless representation of binders \cite{McKinnaPollack99}: in this representation 

  there are ``junk'' terms that need to

  be excluded by reasoning about a well-formedness predicate.

  The problem with introducing a new type in Isabelle/HOL is that in order to be useful, 

  a reasoning infrastructure needs to be ``lifted'' from the underlying subset to 

  the new type. This is usually a tricky and arduous task. To ease it,

  we re-implemented in Isabelle/HOL the quotient package described by Homeier 

  \cite{Homeier05} for the HOL4 system. This package 

  allows us to  lift definitions and theorems involving raw terms

  to definitions and theorems involving alpha-equated terms. For example

  if we define the free-variable function over raw lambda-terms

  \begin{center}

  $\fv(x) = \{x\}$\hspace{10mm}

  $\fv(t_1\;t_2) = \fv(t_1) \cup \fv(t_2)$\\[1mm]

  $\fv(\lambda x.t) = \fv(t) - \{x\}$

  \end{center}

  \noindent

  then with not too great effort we obtain a function $\fv^\alpha$

  operating on quotients, or alpha-equivalence classes of lambda-terms. This

  lifted function is characterised by the equations

  \begin{center}

  $\fv^\alpha(x) = \{x\}$\hspace{10mm}

  $\fv^\alpha(t_1\;t_2) = \fv^\alpha(t_1) \cup \fv^\alpha(t_2)$\\[1mm]

  $\fv^\alpha(\lambda x.t) = \fv^\alpha(t) - \{x\}$

  \end{center}

  \noindent

  (Note that this means also the term-constructors for variables, applications

  and lambda are lifted to the quotient level.)  This construction, of course,

  only works if alpha-equivalence is indeed an equivalence relation, and the lifted

  definitions and theorems are respectful w.r.t.~alpha-equivalence.  Accordingly, we

  will not be able to lift a bound-variable function to alpha-equated terms

  (since it does not respect alpha-equivalence). To sum up, every lifting of

  theorems to the quotient level needs proofs of some respectfulness

  properties. In the paper we show that we are able to automate these

  proofs and therefore can establish a reasoning infrastructure for

  alpha-equated terms.\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{Pitts04}. 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 are also able to derive, at the moment 

  only manually, strong induction principles that 

  have 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 sort-respecting permutations of atoms. The sorts can be used to

  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 one 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> _  ::  perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}

  \noindent 

  in which the generic type @{text "\<beta>"} stands for the type of the object 

  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 of @{term p} as @{text "- p"}. The permutation

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

  (see \cite{HuffmanUrban10}). Concrete  permutations are build up from 

  swappings, written as @{text "(a b)"},

  which are permutations that behave as follows:

  %

  @{text[display,indent=5] "(a b) = \<lambda>c. if a = c then b else if b = c then a else c"}

  The most original aspect of the nominal logic work of Pitts is a general

  definition for the notion of ``the set of free variables of an object @{text

  "x"}''.  This notion, written @{term "supp x"}, is general in the sense that

  it applies not only to lambda-terms (alpha-equated or not), 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\label{sec:binders} *}

text {*

  In Nominal Isabelle, the user is expected to write down a specification of a

  term-calculus and then a reasoning infrastructure is automatically derived

  from this specification (remember that Nominal Isabelle is a definitional

  extension of Isabelle/HOL, which does not introduce any new axioms).

  In order to keep our work manageable, we will wherever possible state

  definitions and perform proofs inside Isabelle, as opposed to write custom

  ML-code that generates them anew for each specification. To that

  end, we will consider pairs @{text "(as, x)"} of type @{text "(atom set) \<times> \<beta>"}.

  These pairs are intended to represent the abstraction, or binding, of the set @{text "as"} 

  in the body @{text "x"}.

  The first question we have to answer is when the pairs $(as, x)$ and $(bs, y)$ are

  alpha-equivalent? (At the moment we are interested in

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

  vacuous binders.) To answer this, we identify four conditions: {\it i)} given 

  a free-variable function $\fv$ of type \mbox{@{text "\<beta> \<Rightarrow> atom set"}}, then @{text x} and @{text y} 

  need to have the same set of free variables; moreover there must be a permutation

  @{text p}  such that {\it ii)} it leaves the free variables of @{text x} and @{text y} unchanged, 

  but {\it iii)} ``moves'' their bound names so that we obtain modulo a relation, 

  say \mbox{@{text "_ R _"}}, two equal terms. We also require {\it iv)} that @{text p} makes 

  the abstracted sets @{text as} and @{text bs} equal. The requirements {\it i)} to {\it iv)} can 

  be stated formally as follows:

  %

  \begin{equation}\label{alphaset}

  \begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}

  \multicolumn{2}{l}{(as, x) \approx\hspace{0.05mm}_{set}^{\fv, R, p} (bs, y) \;\dn\hspace{30mm}\;}\\[1mm]

             & @{text "fv(x) - as = fv(y) - bs"}\\

  \wedge     & @{text "(fv(x) - as) #* p"}\\

  \wedge     & @{text "(p \<bullet> x) R y"}\\

  \wedge     & @{text "(p \<bullet> as) = bs"}\\ 

  \end{array}

  \end{equation}

  \noindent

  Note that this relation is dependent on $p$. Alpha-equivalence is then the relation where 

  we existentially quantify over this $p$. 

  Also note that the relation is dependent on a free-variable function $\fv$ and a relation 

  $R$. The reason for this extra generality is that we will use $\approx_{set}$ for both 

  ``raw'' terms and alpha-equated terms. In the latter case, $R$ will be replaced by 

  equality $(op =)$ and we are going to prove that $\fv$ will be equal to the support 

  of $x$ and $y$. 

  The definition in \eqref{alphaset} does not make any distinction between the

  order of abstracted variables. If we want this, then we can define alpha-equivalence 

  for pairs of the form \mbox{@{text "(as, x)"}} with type @{text "(atom list) \<times> \<beta>"} 

  as follows

  %

  \begin{equation}\label{alphalist}

  \begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}

  \multicolumn{2}{l}{(as, x) \approx\hspace{0.05mm}_{list}^{\fv, R, p} (bs, y) \;\dn\hspace{30mm}\;}\\[1mm]

             & @{text "fv(x) - (set as) = fv(y) - (set bs)"}\\

  \wedge     & @{text "(fv(x) - set as) #* p"}\\

  \wedge     & @{text "(p \<bullet> x) R y"}\\

  \wedge     & @{text "(p \<bullet> as) = bs"}\\ 

  \end{array}

  \end{equation}

  \noindent

  where $set$ is the function that coerces a list of atoms into a set of atoms.

  If we do not want to make any difference between the order of binders and

  also allow vacuous binders, then we keep sets of binders, but drop the fourth 

  condition in \eqref{alphaset}:

  %

  \begin{equation}\label{alphares}

  \begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}

  \multicolumn{2}{l}{(as, x) \approx\hspace{0.05mm}_{res}^{\fv, R, p} (bs, y) \;\dn\hspace{30mm}\;}\\[1mm]

             & @{text "fv(x) - as = fv(y) - bs"}\\

  \wedge     & @{text "(fv(x) - as) #* p"}\\

  \wedge     & @{text "(p \<bullet> x) R y"}\\

  \end{array}

  \end{equation}

  \begin{exmple}\rm

  It might be useful to consider some examples for how these definitions pan out in practise.

  For this consider the case of abstracting a set of variables over types (as in type-schemes). 

  We set $R$ to be the equality and for $\fv(T)$ we define

  \begin{center}

  $\fv(x) = \{x\}  \qquad \fv(T_1 \rightarrow T_2) = \fv(T_1) \cup \fv(T_2)$

  \end{center}