(*<*)

theory Paper

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

begin

consts

  fv :: "'a \<Rightarrow> 'b"

  abs_set :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" 

definition

 "equal \<equiv> (op =)" 

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) and

  abs_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and

  fv ("fv'(_')" [100] 100) and

  equal ("=") and

  alpha_abs ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and 

  Abs ("[_]\<^raw:$\!$>\<^bsub>set\<^esub>._") and

  Abs_lst ("[_]\<^raw:$\!$>\<^bsub>list\<^esub>._") and

  Abs_res ("[_]\<^raw:$\!$>\<^bsub>res\<^esub>._") 

(*>*)

section {* Introduction *}

text {*

  So far, Nominal Isabelle provides a mechanism for constructing

  alpha-equated terms, for example

  \begin{center}

  @{text "t ::= x | t t | \<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}

  @{text "T ::= x | T \<rightarrow> T"}\hspace{5mm}

  @{text "S ::= \<forall>{x\<^isub>1,\<dots>, x\<^isub>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}

  @{text "\<forall>{x,y}. x \<rightarrow> y  \<approx>\<^isub>\<alpha>  \<forall>{y,x}. y \<rightarrow> x"} 

  \end{equation}

  \noindent

  but assuming that @{text x}, @{text y} and @{text z} are distinct variables,

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

  %

  \begin{equation}\label{ex2}

  @{text "\<forall>{x,y}. x \<rightarrow> y  \<notapprox>\<^isub>\<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}

  @{text "\<forall>{x}. x \<rightarrow> y  \<approx>\<^isub>\<alpha>  \<forall>{x,z}. x \<rightarrow> y"}

  \end{equation}

  \noindent

  where @{text 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}

  @{text "\<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}

  @{text "\<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 @{text "\<LET>"}s containing patterns

  %

  \begin{equation}\label{two}

  @{text "\<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}

  @{text "\<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 a problem that has been pointed out by Pottier \cite{Pottier06} and

  Cheney \cite{Cheney05}: in @{text "\<LET>"}-constructs of the form

  %

  \begin{center}

  @{text "\<LET> x\<^isub>1 = t\<^isub>1 \<AND> \<dots> \<AND> x\<^isub>n = t\<^isub>n \<IN> s \<END>"}

  \end{center}

  \noindent

  which bind all the @{text "x\<^isub>i"} in @{text s}, we might not care

  about the order in which the @{text "x\<^isub>i = t\<^isub>i"} are given,

  but we do care about the information that there are as many @{text

  "x\<^isub>i"} as there are @{text "t\<^isub>i"}. We lose this information if

  we represent the @{text "\<LET>"}-constructor by something like

  %

  \begin{center}

  @{text "\<LET> [x\<^isub>1,\<dots>,x\<^isub>n].s [t\<^isub>1,\<dots>,t\<^isub>n]"}

  \end{center}

  \noindent

  where the notation @{text "[_]._"} indicates that the @{text "x\<^isub>i"}

  become bound in @{text s}. In this representation the term 

  \mbox{@{text "\<LET> [x].s [t\<^isub>1,t\<^isub>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}

  @{text trm} & @{text "::="}  & @{text "\<dots>"}\\ 

              & @{text "|"}    & @{text "\<LET> a::assn s::trm"}\hspace{4mm} 

                                 \isacommand{bind} @{text "bn(a)"} \isacommand{in} @{text "s"}\\[1mm]

  @{text assn} & @{text "::="} & @{text "\<ANIL>"}\\

               & @{text "|"}   & @{text "\<ACONS> name trm assn"}

  \end{tabular}

  \end{center}

  \noindent

  where @{text assn} is an auxiliary type representing a list of assignments

  and @{text bn} an auxiliary function identifying the variables to be bound

  by the @{text "\<LET>"}. This function is defined by recursion over @{text

  assn} as follows

  \begin{center}

  @{text "bn(\<ANIL>) ="} @{term "{}"} \hspace{5mm} 

  @{text "bn(\<ACONS> x t as) = {x} \<union> bn(as)"} 

  \end{center}

  \noindent

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

  example @{text "s::trm"}, and a binding clause, in this case

  \isacommand{bind} @{text "bn(a)"} \isacommand{in} @{text "s"}, that states

  to bind in @{text s} all the names the function call @{text "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}

  @{text "t ::= t t | \<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}

  @{text "\<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}

  @{text "fv(x) = {x}"}\hspace{10mm}

  @{text "fv(t\<^isub>1 t\<^isub>2) = fv(t\<^isub>1) \<union> fv(t\<^isub>2)"}\\[1mm]

  @{text "fv(\<lambda>x.t) = fv(t) - {x}"}

  \end{center}

  \noindent

  then with not too great effort we obtain a function @{text "fv\<^sup>\<alpha>"}

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

  lifted function is characterised by the equations

  \begin{center}

  @{text "fv\<^sup>\<alpha>(x) = {x}"}\hspace{10mm}

  @{text "fv\<^sup>\<alpha>(t\<^isub>1 t\<^isub>2) = fv\<^sup>\<alpha>(t\<^isub>1) \<union> fv\<^sup>\<alpha>(t\<^isub>2)"}\\[1mm]

  @{text "fv\<^sup>\<alpha>(\<lambda>x.t) = fv\<^sup>\<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}.

*}

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 with deriving the reasoning infrastructure

  manageable, we will wherever possible state definitions and perform proofs

  on the user-level of Isabelle/HOL, as opposed to write custom ML-code that

  generates them anew for each specification. To that end, we will consider

  first 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 @{text "(as, x)"} and

  @{text "(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 @{text "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,

  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}{@{term "(as, x) \<approx>gen R fv p (bs, y)"} @{text "\<equiv>"}\hspace{30mm}}\\

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

  @{text "\<and>"} & @{term "(fv(x) - as) \<sharp>* p"}\\

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

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

  \end{array}

  \end{equation}

  \noindent

  Note that this relation is dependent on the permutation @{text

  "p"}. Alpha-equivalence between two pairs is then the relation where we

  existentially quantify over this @{text "p"}. Also note that the relation is

  dependent on a free-variable function @{text "fv"} and a relation @{text

  "R"}. The reason for this extra generality is that we will use

  $\approx_{\textit{set}}$ for both ``raw'' terms and alpha-equated terms. In

  the latter case, $R$ will be replaced by equality @{text "="} and for raw terms we

  will prove that @{text "fv"} is equal to the support of @{text

  x} and @{text 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}{@{term "(as, x) \<approx>lst R fv p (bs, y)"} @{text "\<equiv>"}\hspace{30mm}}\\[1mm]

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

  \wedge     & @{term "(fv(x) - set as) \<sharp>* p"}\\

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

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

  \end{array}