--- a/LMCS-Paper/Paper.thy Sat May 12 21:05:59 2012 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2766 +0,0 @@
-(*<*)
-theory Paper
-imports "../Nominal/Nominal2"
- "~~/src/HOL/Library/LaTeXsugar"
-begin
-
-consts
- fv :: "'a \<Rightarrow> 'b"
- abs_set :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
- alpha_bn :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
- abs_set2 :: "'a \<Rightarrow> perm \<Rightarrow> 'b \<Rightarrow> 'c"
- equ2 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
- Abs_dist :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
- Abs_print :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
-
-definition
- "equal \<equiv> (op =)"
-
-fun alpha_set_ex where
- "alpha_set_ex (bs, x) R f (cs, y) = (\<exists>pi. alpha_set (bs, x) R f pi (cs, y))"
-
-fun alpha_res_ex where
- "alpha_res_ex (bs, x) R f pi (cs, y) = (\<exists>pi. alpha_res (bs, x) R f pi (cs, y))"
-
-fun alpha_lst_ex where
- "alpha_lst_ex (bs, x) R f pi (cs, y) = (\<exists>pi. alpha_lst (bs, x) R f pi (cs, y))"
-
-
-
-notation (latex output)
- swap ("'(_ _')" [1000, 1000] 1000) and
- fresh ("_ # _" [51, 51] 50) and
- fresh_star ("_ #\<^sup>* _" [51, 51] 50) and
- supp ("supp _" [78] 73) and
- uminus ("-_" [78] 73) and
- If ("if _ then _ else _" 10) and
- alpha_set_ex ("_ \<approx>\<^raw:\,\raisebox{-1pt}{\makebox[0mm][l]{$_{\textit{set}}$}}>\<^bsup>_, _\<^esup> _") and
- alpha_lst_ex ("_ \<approx>\<^raw:\,\raisebox{-1pt}{\makebox[0mm][l]{$_{\textit{list}}$}}>\<^bsup>_, _\<^esup> _") and
- alpha_res_ex ("_ \<approx>\<^raw:\,\raisebox{-1pt}{\makebox[0mm][l]{$_{\textit{set+}}$}}>\<^bsup>_, _\<^esup> _") and
- abs_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
- abs_set2 ("_ \<approx>\<^raw:\raisebox{-1pt}{\makebox[0mm][l]{$\,_{\textit{list}}$}}>\<^bsup>_\<^esup> _") and
- fv ("fa'(_')" [100] 100) and
- equal ("=") and
- alpha_abs_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
- alpha_abs_lst ("_ \<approx>\<^raw:{$\,_{\textit{abs\_list}}$}> _") and
- alpha_abs_res ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set+}}$}> _") and
- Abs_set ("[_]\<^bsub>set\<^esub>._" [20, 101] 999) and
- Abs_lst ("[_]\<^bsub>list\<^esub>._" [20, 101] 999) and
- Abs_dist ("[_]\<^bsub>#list\<^esub>._" [20, 101] 999) and
- Abs_res ("[_]\<^bsub>set+\<^esub>._") and
- Abs_print ("_\<^bsub>set\<^esub>._") and
- Cons ("_::_" [78,77] 73) and
- supp_set ("aux _" [1000] 10) and
- alpha_bn ("_ \<approx>bn _")
-
-consts alpha_trm ::'a
-consts fa_trm :: 'a
-consts fa_trm_al :: 'a
-consts alpha_trm2 ::'a
-consts fa_trm2 :: 'a
-consts fa_trm2_al :: 'a
-consts supp2 :: 'a
-consts ast :: 'a
-consts ast' :: 'a
-consts bn_al :: "'b \<Rightarrow> 'a"
-notation (latex output)
- alpha_trm ("\<approx>\<^bsub>trm\<^esub>") and
- fa_trm ("fa\<^bsub>trm\<^esub>") and
- fa_trm_al ("fa\<AL>\<^bsub>trm\<^esub>") and
- alpha_trm2 ("'(\<approx>\<^bsub>assn\<^esub>, \<approx>\<^bsub>trm\<^esub>')") and
- fa_trm2 ("'(fa\<^bsub>assn\<^esub>, fa\<^bsub>trm\<^esub>')") and
- fa_trm2_al ("'(fa\<AL>\<^bsub>assn\<^esub>, fa\<AL>\<^bsub>trm\<^esub>')") and
- ast ("'(as, t')") and
- ast' ("'(as', t\<PRIME> ')") and
- equ2 ("'(=, =')") and
- supp2 ("'(supp, supp')") and
- bn_al ("bn\<^sup>\<alpha> _" [100] 100)
-(*>*)
-
-
-section {* Introduction *}
-
-text {*
- So far, Nominal Isabelle provided a mechanism for constructing alpha-equated
- terms, for example lambda-terms
-
- \[
- @{text "t ::= x | t t | \<lambda>x. t"}
- \]\smallskip
-
- \noindent
- where free and bound variables have names. For such alpha-equated 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{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,
- respectively, of the form
-
- \begin{equation}\label{tysch}
- \begin{array}{l}
- @{text "T ::= x | T \<rightarrow> T"}\hspace{15mm}
- @{text "S ::= \<forall>{x\<^isub>1,\<dots>, x\<^isub>n}. T"}
- \end{array}
- \end{equation}\smallskip
-
- \noindent
- and the @{text "\<forall>"}-quantification 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, like @{text "\<forall>x\<^isub>1.\<forall>x\<^isub>2...\<forall>x\<^isub>n.T"}, this leads to a rather clumsy
- formalisation of W. For example, the usual definition for a
- type being an instance of a type-scheme requires in the iterated version
- the following auxiliary \emph{unbinding relation}:
-
- \[
- \infer{@{text T} \hookrightarrow ([], @{text T})}{}\qquad
- \infer{\forall @{text x.S} \hookrightarrow (@{text x}\!::\!@{text xs}, @{text T})}
- {@{text S} \hookrightarrow (@{text xs}, @{text T})}
- \]\smallskip
-
- \noindent
- Its purpose is to relate a type-scheme with a list of type-variables and a type. It is used to
- address the following problem:
- Given a type-scheme, say @{text S}, how does one get access to the bound type-variables
- and the type-part of @{text S}? The unbinding relation gives an answer to this problem, though
- in general it will only provide \emph{a} list of type-variables together with \emph{a} type that are
- ``alpha-equivalent'' to @{text S}. This is because unbinding is a relation; it cannot be a function
- for alpha-equated type-schemes. With the unbinding relation
- in place, we can define when a type @{text T} is an instance of a type-scheme @{text S} as follows:
-
- \[
- @{text "T \<prec> S \<equiv> \<exists>xs T' \<sigma>. S \<hookrightarrow> (xs, T') \<and> dom \<sigma> = set xs \<and> \<sigma>(T') = T"}
- \]\smallskip
-
- \noindent
- This means there exists a list of type-variables @{text xs} and a type @{text T'} to which
- the type-scheme @{text S} unbinds, and there exists a substitution @{text "\<sigma>"} whose domain is
- @{text xs} (seen as set) such that @{text "\<sigma>(T') = T"}.
- The problem with this definition is that we cannot follow the usual proofs
- that are by induction on the type-part of the type-scheme (since it is under
- an existential quantifier and only an alpha-variant). The implementation of
- type-schemes using iterations of single binders
- prevents us from directly ``unbinding'' the bound type-variables and the type-part.
- Clearly, a more dignified approach for formalising algorithm W is desirable.
- The purpose of this paper is to introduce general binders, which
- allow us to represent type-schemes so that they can bind multiple variables at once
- and as a result solve this problem more straightforwardly.
- The need of iterating single binders is also one reason
- why the existing Nominal Isabelle and similar theorem provers that only provide
- mechanisms for binding single variables have so far not fared very 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 the 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 in \eqref{ex1} below the first pair of type-schemes as alpha-equivalent,
- but assuming that @{text x}, @{text y} and @{text z} are distinct variables,
- the second pair should \emph{not} be alpha-equivalent:
-
- \begin{equation}\label{ex1}
- @{text "\<forall>{x, y}. x \<rightarrow> y \<approx>\<^isub>\<alpha> \<forall>{x, y}. y \<rightarrow> x"}\hspace{10mm}
- @{text "\<forall>{x, y}. x \<rightarrow> y \<notapprox>\<^isub>\<alpha> \<forall>{z}. z \<rightarrow> z"}
- \end{equation}\smallskip
-
- \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}\smallskip
-
- \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 for
- type-schemes by Leroy in \cite[Page 18--19]{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}\smallskip
-
- \noindent
- we might not care in which order the assignments @{text "x = 3"} and
- \mbox{@{text "y = 2"}} are given, but it would be often unusual (particularly
- in strict languages) to regard \eqref{one} as alpha-equivalent with
-
- \[
- @{text "\<LET> x = 3 \<AND> y = 2 \<AND> z = foo \<IN> x - y \<END>"}
- \]\smallskip
-
- \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 like
-
- \begin{equation}\label{two}
- @{text "\<LET> (x, y) = (3, 2) \<IN> x - y \<END>"}
- \end{equation}\smallskip
-
- \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
-
- \[
- @{text "\<LET> (y, x) = (3, 2) \<IN> x - y \<END>"}
- \]\smallskip
-
- \noindent
- As a result, we provide three general binding mechanisms each of which binds
- multiple variables at once, and let the user choose which one is intended
- when formalising a term-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
-
- \[
- @{text "\<LET> x\<^isub>1 = t\<^isub>1 \<AND> \<dots> \<AND> x\<^isub>n = t\<^isub>n \<IN> s \<END>"}
- \]\smallskip
-
- \noindent
- we care about the information that there are as many bound variables @{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
-
- \[
- @{text "\<LET> (\<lambda>x\<^isub>1\<dots>x\<^isub>n . s) [t\<^isub>1,\<dots>,t\<^isub>n]"}
- \]\smallskip
-
- \noindent
- where the notation @{text "\<lambda>_ . _"} indicates that the list of @{text
- "x\<^isub>i"} becomes bound in @{text s}. In this representation the term
- \mbox{@{text "\<LET> (\<lambda>x . s) [t\<^isub>1, t\<^isub>2]"}} is a perfectly
- legal instance, but the lengths of the two lists do not agree. 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 in
- very messy reasoning (see for example~\cite{BengtsonParow09}). To avoid
- this, we will allow type specifications for @{text "\<LET>"}s as follows
-
- \[
- \mbox{\begin{tabular}{r@ {\hspace{2mm}}r@ {\hspace{2mm}}ll}
- @{text trm} & @{text "::="} & @{text "\<dots>"} \\
- & @{text "|"} & @{text "\<LET> as::assn s::trm"}\hspace{2mm}
- \isacommand{binds} @{text "bn(as)"} \isacommand{in} @{text "s"}\\[1mm]
- @{text assn} & @{text "::="} & @{text "\<ANIL>"}\\
- & @{text "|"} & @{text "\<ACONS> name trm assn"}
- \end{tabular}}
- \]\smallskip
-
- \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 can be defined by recursion over @{text
- assn} as follows
-
- \[
- @{text "bn(\<ANIL>) ="}~@{term "{}"} \hspace{10mm}
- @{text "bn(\<ACONS> x t as) = {x} \<union> bn(as)"}
- \]\smallskip
-
- \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{binds} @{text "bn(as)"} \isacommand{in} @{text "s"}. This binding
- clause states that all the names the function @{text "bn(as)"} returns
- should be bound in @{text s}. This style of specifying terms and bindings
- is heavily inspired by the syntax of the Ott-tool \cite{ott-jfp}. Our work
- extends Ott in several aspects: one is that we support three binding
- modes---Ott has only one, namely the one where the order of binders matters.
- Another is that our reasoning infrastructure, like strong induction principles
- and the notion of free variables, is derived from first principles within
- the Isabelle/HOL theorem prover.
-
- However, we will not be able to cope with all specifications that are
- allowed by Ott. One reason is that Ott lets the user specify `empty' types
- like \mbox{@{text "t ::= t t | \<lambda>x. t"}} 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
- \cite{Berghofer99}. 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, for example, that the two type-schemes
-
- \[
- @{text "\<forall>{x}. x \<rightarrow> y = \<forall>{x, z}. x \<rightarrow> y"}
- \]\smallskip
-
- \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 with 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
- Isabelle/HOL is a rather non-trivial task. For every
- specification we will need to construct type(s) 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 Isabelle/HOL can be illustrated by the following picture:
-
- \begin{equation}\label{picture}
- \mbox{\begin{tikzpicture}[scale=1.1]
- %\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, 0.5) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]classes\end{tabular}};
- \draw (-2.4, 0.5) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};
- \draw (1.8, 0.48) node[right=-0.1mm]
- {\small\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ (sets of raw terms)\end{tabular}};
- \draw (0.9, -0.35) node {\footnotesize\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};
- \draw (-3.25, 0.55) node {\small\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] {\footnotesize{}isomorphism};
-
- \end{tikzpicture}}
- \end{equation}\smallskip
-
- \noindent
- We take as the starting point a definition of raw terms (defined as a
- datatype in Isabelle/HOL); then identify 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 (the
- non-emptiness requirement is always satisfied whenever the raw terms are
- definable as datatype in Isabelle/HOL and our relation for alpha-equivalence
- is 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 \cite{KaliszykUrban11}
- 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
- as follows
-
- \[
- \mbox{\begin{tabular}{l@ {\hspace{1mm}}r@ {\hspace{1mm}}l}
- @{text "fv(x)"} & @{text "\<equiv>"} & @{text "{x}"}\\
- @{text "fv(t\<^isub>1 t\<^isub>2)"} & @{text "\<equiv>"} & @{text "fv(t\<^isub>1) \<union> fv(t\<^isub>2)"}\\
- @{text "fv(\<lambda>x.t)"} & @{text "\<equiv>"} & @{text "fv(t) - {x}"}
- \end{tabular}}
- \]\smallskip
-
- \noindent
- then with the help of the quotient package we can obtain a function @{text "fv\<^sup>\<alpha>"}
- operating on quotients, that is alpha-equivalence classes of lambda-terms. This
- lifted function is characterised by the equations
-
- \[
- \mbox{\begin{tabular}{l@ {\hspace{1mm}}r@ {\hspace{1mm}}l}
- @{text "fv\<^sup>\<alpha>(x)"} & @{text "="} & @{text "{x}"}\\
- @{text "fv\<^sup>\<alpha>(t\<^isub>1 t\<^isub>2)"} & @{text "="} & @{text "fv\<^sup>\<alpha>(t\<^isub>1) \<union> fv\<^sup>\<alpha>(t\<^isub>2)"}\\
- @{text "fv\<^sup>\<alpha>(\<lambda>x.t)"} & @{text "="} & @{text "fv\<^sup>\<alpha>(t) - {x}"}
- \end{tabular}}
- \]\smallskip
-
- \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
- `raw' definitions and theorems are respectful w.r.t.~alpha-equivalence.
- For example, we will not be able to lift a bound-variable function. Although
- this function can be defined for raw terms, it does not respect
- alpha-equivalence and therefore cannot be lifted.
- To sum up, every lifting
- of theorems to the quotient level needs proofs of some respectfulness
- properties (see \cite{Homeier05}). In the paper we show that we are able to
- automate these proofs and as a result can automatically establish a reasoning
- infrastructure for alpha-equated terms.\smallskip
-
- The examples we have in mind where our reasoning infrastructure will be
- helpful include the term language of Core-Haskell (see
- Figure~\ref{corehas}). This term language involves patterns that have lists
- of type-, coercion- and term-variables, all of which are bound in @{text
- "\<CASE>"}-expressions. In these patterns we do not know in advance how many
- variables need to be bound. Another example is the algorithm W,
- which includes multiple binders in type-schemes.\medskip
-
- \noindent
- {\bf Contributions:} We provide three new definitions for when terms
- involving general binders are alpha-equivalent. These definitions are
- inspired by earlier work of Pitts \cite{Pitts04}. By means of automati\-cally-generated
- 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 automatically derive strong
- induction principles that have the variable convention already built in.
- For this we simplify the earlier automated proofs by using the proving tools
- from the function package~\cite{Krauss09} of Isabelle/HOL. The method
- behind our specification of general binders is taken from the Ott-tool, but
- we introduce crucial restrictions, and also extensions, so that our
- specifications make sense for reasoning about alpha-equated terms. The main
- improvement over Ott is that we introduce three binding modes (only one is
- present in Ott), provide formalised definitions for alpha-equivalence and
- for free variables of our terms, and also derive a reasoning infrastructure
- for our specifications from `first principles' inside a theorem prover.
-
-
- \begin{figure}[t]
- \begin{boxedminipage}{\linewidth}
- \begin{center}
- \begin{tabular}{@ {\hspace{8mm}}r@ {\hspace{2mm}}r@ {\hspace{2mm}}l}
- \multicolumn{3}{@ {}l}{Type Kinds}\\
- @{text "\<kappa>"} & @{text "::="} & @{text "\<star> | \<kappa>\<^isub>1 \<rightarrow> \<kappa>\<^isub>2"}\smallskip\\
- \multicolumn{3}{@ {}l}{Coercion Kinds}\\
- @{text "\<iota>"} & @{text "::="} & @{text "\<sigma>\<^isub>1 \<sim> \<sigma>\<^isub>2"}\smallskip\\
- \multicolumn{3}{@ {}l}{Types}\\
- @{text "\<sigma>"} & @{text "::="} & @{text "a | T | \<sigma>\<^isub>1 \<sigma>\<^isub>2 | S\<^isub>n"}$\;\overline{@{text "\<sigma>"}}$@{text "\<^sup>n"}
- @{text "| \<forall>a:\<kappa>. \<sigma> | \<iota> \<Rightarrow> \<sigma>"}\smallskip\\
- \multicolumn{3}{@ {}l}{Coercion Types}\\
- @{text "\<gamma>"} & @{text "::="} & @{text "c | C | \<gamma>\<^isub>1 \<gamma>\<^isub>2 | S\<^isub>n"}$\;\overline{@{text "\<gamma>"}}$@{text "\<^sup>n"}
- @{text "| \<forall>c:\<iota>. \<gamma> | \<iota> \<Rightarrow> \<gamma> | refl \<sigma> | sym \<gamma> | \<gamma>\<^isub>1 \<circ> \<gamma>\<^isub>2"}\\
- & @{text "|"} & @{text "\<gamma> @ \<sigma> | left \<gamma> | right \<gamma> | \<gamma>\<^isub>1 \<sim> \<gamma>\<^isub>2 | rightc \<gamma> | leftc \<gamma> | \<gamma>\<^isub>1 \<triangleright> \<gamma>\<^isub>2"}\smallskip\\
- \multicolumn{3}{@ {}l}{Terms}\\
- @{text "e"} & @{text "::="} & @{text "x | K | \<Lambda>a:\<kappa>. e | \<Lambda>c:\<iota>. e | e \<sigma> | e \<gamma> | \<lambda>x:\<sigma>. e | e\<^isub>1 e\<^isub>2"}\\
- & @{text "|"} & @{text "\<LET> x:\<sigma> = e\<^isub>1 \<IN> e\<^isub>2 | \<CASE> e\<^isub>1 \<OF>"}$\;\overline{@{text "p \<rightarrow> e\<^isub>2"}}$ @{text "| e \<triangleright> \<gamma>"}\smallskip\\
- \multicolumn{3}{@ {}l}{Patterns}\\
- @{text "p"} & @{text "::="} & @{text "K"}$\;\overline{@{text "a:\<kappa>"}}\;\overline{@{text "c:\<iota>"}}\;\overline{@{text "x:\<sigma>"}}$\smallskip\\
- \multicolumn{3}{@ {}l}{Constants}\\
- & @{text C} & coercion constants\\
- & @{text T} & value type constructors\\
- & @{text "S\<^isub>n"} & n-ary type functions (which need to be fully applied)\\
- & @{text K} & data constructors\smallskip\\
- \multicolumn{3}{@ {}l}{Variables}\\
- & @{text a} & type variables\\
- & @{text c} & coercion variables\\
- & @{text x} & term variables\\
- \end{tabular}
- \end{center}
- \end{boxedminipage}
- \caption{The System @{text "F\<^isub>C"}
- \cite{CoreHaskell}, also often referred to as \emph{Core-Haskell}. In this
- version of @{text "F\<^isub>C"} we made a modification by separating the
- grammars for type kinds and coercion kinds, as well as for types and coercion
- types. For this paper the interesting term-constructor is @{text "\<CASE>"},
- which binds multiple type-, coercion- and term-variables (the overlines stand for lists).\label{corehas}}
- \end{figure}
-*}
-
-section {* A Short Review of the Nominal Logic Work *}
-
-text {*
- At its core, Nominal Isabelle is an adaptation of the nominal logic work by
- Pitts \cite{Pitts03}. This adaptation for Isabelle/HOL is described in
- \cite{HuffmanUrban10} (including proofs). We shall briefly review this work
- to aid the description of what follows.
-
- Two central notions in the nominal logic work are sorted atoms and
- sort-respecting permutations of atoms. We will use the letters @{text "a, b,
- c, \<dots>"} to stand for atoms and @{text "\<pi>, \<pi>\<^isub>1, \<dots>"} to stand for permutations,
- which in Nominal Isabelle have type @{typ perm}. The purpose of atoms is to
- represent variables, be they bound or free. The sorts of atoms can be used
- to represent different kinds of variables, such as the term-, coercion- and
- type-variables in Core-Haskell. It is assumed that there is an infinite
- supply of atoms for each sort. In the interest of brevity, we shall restrict
- ourselves in what follows to 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 "_ \<bullet> _ "} and having the type @{text "perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
- where the generic type @{text "\<beta>"} is the type of the object
- over which the permutation
- acts. In Nominal Isabelle, the identity permutation is written as @{term "0::perm"},
- the composition of two permutations @{term "\<pi>\<^isub>1"} and @{term "\<pi>\<^isub>2"} as \mbox{@{term "\<pi>\<^isub>1 + \<pi>\<^isub>2"}}
- (even if this operation is non-commutative),
- and the inverse permutation of @{term "\<pi>"} as @{text "- \<pi>"}. The permutation
- operation is defined over Isabelle/HOL's type-hierarchy \cite{HuffmanUrban10};
- for example permutations acting on atoms, products, lists, permutations, sets,
- functions and booleans are given by:
-
- \begin{equation}\label{permute}
- \mbox{\begin{tabular}{@ {}c@ {\hspace{10mm}}c@ {}}
- \begin{tabular}{@ {}l@ {}}
- @{text "\<pi> \<bullet> a \<equiv> \<pi> a"}\\
- @{thm permute_prod.simps[where p="\<pi>", no_vars, THEN eq_reflection]}\\[2mm]
- @{thm permute_list.simps(1)[where p="\<pi>", no_vars, THEN eq_reflection]}\\
- @{thm permute_list.simps(2)[where p="\<pi>", no_vars, THEN eq_reflection]}\\
- \end{tabular} &
- \begin{tabular}{@ {}l@ {}}
- @{thm permute_perm_def[where p="\<pi>" and q="\<pi>'", no_vars, THEN eq_reflection]}\\
- @{thm permute_set_def[where p="\<pi>", no_vars, THEN eq_reflection]}\\
- @{text "\<pi> \<bullet> f \<equiv> \<lambda>x. \<pi> \<bullet> (f (- \<pi> \<bullet> x))"}\\
- @{thm permute_bool_def[where p="\<pi>", no_vars, THEN eq_reflection]}
- \end{tabular}
- \end{tabular}}
- \end{equation}\smallskip
-
- \noindent
- Concrete permutations in Nominal Isabelle are built up from swappings,
- written as \mbox{@{text "(a b)"}}, which are permutations that behave
- as follows:
-
- \[
- @{text "(a b) = \<lambda>c. if a = c then b else if b = c then a else c"}
- \]\smallskip
-
- 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. Its definition depends only on the
- permutation operation and on the notion of equality defined for the type of
- @{text x}, namely:
-
- \begin{equation}\label{suppdef}
- @{thm supp_def[no_vars, THEN eq_reflection]}
- \end{equation}\smallskip
-
- \noindent
- There is also the derived notion for when an atom @{text a} is \emph{fresh}
- for an @{text x}, defined as
-
- \[
- @{thm fresh_def[no_vars]}
- \]\smallskip
-
- \noindent
- We 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}, leaves
- @{text x} unchanged, namely
-
- \begin{prop}\label{swapfreshfresh}
- If @{thm (prem 1) swap_fresh_fresh[no_vars]} and @{thm (prem 2) swap_fresh_fresh[no_vars]}
- then @{thm (concl) swap_fresh_fresh[no_vars]}.
- \end{prop}
-
- While often the support of an object can be relatively easily
- described, for example for atoms, products, lists, function applications,
- booleans and permutations as follows
-
- \begin{equation}\label{supps}\mbox{
- \begin{tabular}{c@ {\hspace{10mm}}c}
- \begin{tabular}{rcl}
- @{term "supp a"} & $=$ & @{term "{a}"}\\
- @{term "supp (x, y)"} & $=$ & @{term "supp x \<union> supp y"}\\
- @{term "supp []"} & $=$ & @{term "{}"}\\
- @{term "supp (x#xs)"} & $=$ & @{term "supp x \<union> supp xs"}\\
- \end{tabular}
- &
- \begin{tabular}{rcl}
- @{text "supp (f x)"} & @{text "\<subseteq>"} & @{term "supp f \<union> supp x"}\\
- @{term "supp b"} & $=$ & @{term "{}"}\\
- @{term "supp \<pi>"} & $=$ & @{term "{a. \<pi> \<bullet> a \<noteq> a}"}
- \end{tabular}
- \end{tabular}}
- \end{equation}\smallskip
-
- \noindent
- in some cases it can be difficult to characterise the support precisely, and
- only an approximation can be established (as for function applications
- above). Reasoning about such approximations can be simplified with the
- notion \emph{supports}, defined as follows:
-
- \begin{defi}
- A set @{text S} \emph{supports} @{text x}, if for all atoms @{text a} and @{text b}
- not in @{text S} we have @{term "(a \<rightleftharpoons> b) \<bullet> x = x"}.
- \end{defi}
-
- \noindent
- The main point of @{text supports} is that we can establish the following
- two properties.
-
- \begin{prop}\label{supportsprop}
- Given a set @{text "bs"} of atoms.\\
- {\it (i)} If @{thm (prem 1) supp_is_subset[where S="bs", no_vars]}
- and @{thm (prem 2) supp_is_subset[where S="bs", no_vars]} then
- @{thm (concl) supp_is_subset[where S="bs", no_vars]}.\\
- {\it (ii)} @{thm supp_supports[no_vars]}.
- \end{prop}
-
- Another important notion in the nominal logic work is \emph{equivariance}.
- For a function @{text f} to be equivariant
- it is required that every permutation leaves @{text f} unchanged, that is
-
- \begin{equation}\label{equivariancedef}
- @{term "\<forall>\<pi>. \<pi> \<bullet> f = f"}\;.
- \end{equation}\smallskip
-
- \noindent
- If a function is of type @{text "\<alpha> \<Rightarrow> \<beta>"}, say, this definition is equivalent to
- the fact that a permutation applied to the application
- @{text "f x"} can be moved to the argument @{text x}. That means for
- such functions, we have for all permutations @{text "\<pi>"}:
-
- \begin{equation}\label{equivariance}
- @{text "\<pi> \<bullet> f = f"} \;\;\;\;\textit{if and only if}\;\;\;\;
- @{text "\<forall>x. \<pi> \<bullet> (f x) = f (\<pi> \<bullet> x)"}\;.
- \end{equation}\smallskip
-
- \noindent
- There is
- also a similar property for relations, which are in HOL functions of type @{text "\<alpha> \<Rightarrow> \<beta> \<Rightarrow> bool"}.
- Suppose a relation @{text R}, then for all permutations @{text \<pi>}:
-
- \[
- @{text "\<pi> \<bullet> R = R"} \;\;\;\;\textit{if and only if}\;\;\;\;
- @{text "\<forall>x y."}~~@{text "x R y"} \;\textit{implies}\; @{text "(\<pi> \<bullet> x) R (\<pi> \<bullet> y)"}\;.
- \]\smallskip
-
- \noindent
- Note that from property \eqref{equivariancedef} and the definition of @{text supp}, we
- can easily deduce that for a function being equivariant is equivalent to having empty support.
-
- Using freshness, the nominal logic work provides us with general means for renaming
- binders.
-
- \noindent
- While in the older version of Nominal Isabelle, we used extensively
- Proposition~\ref{swapfreshfresh} to rename single binders, this property
- proved too unwieldy for dealing with multiple binders. For such binders the
- following generalisations turned out to be easier to use.
-
- \begin{prop}\label{supppermeq}
- @{thm[mode=IfThen] supp_perm_eq[where p="\<pi>", no_vars]}
- \end{prop}
-
- \begin{prop}\label{avoiding}
- For a finite set @{text as} and a finitely supported @{text x} with
- @{term "as \<sharp>* x"} and also a finitely supported @{text c}, there
- exists a permutation @{text "\<pi>"} such that @{term "(\<pi> \<bullet> as) \<sharp>* c"} and
- @{term "supp x \<sharp>* \<pi>"}.
- \end{prop}
-
- \noindent
- The idea behind the second property is that given a finite set @{text as}
- of binders (being bound, or fresh, in @{text x} is ensured by the
- assumption @{term "as \<sharp>* x"}), then there exists a permutation @{text "\<pi>"} such that
- the renamed binders @{term "\<pi> \<bullet> as"} avoid @{text c} (which can be arbitrarily chosen
- as long as it is finitely supported) and also @{text "\<pi>"} does not affect anything
- in the support of @{text x} (that is @{term "supp x \<sharp>* \<pi>"}). The last
- fact and Property~\ref{supppermeq} allow us to `rename' just the binders
- @{text as} in @{text x}, because @{term "\<pi> \<bullet> x = x"}.
-
- Note that @{term "supp x \<sharp>* \<pi>"}
- is equivalent with @{term "supp \<pi> \<sharp>* x"}, which means we could also formulate
- Propositions \ref{supppermeq} and \ref{avoiding} in the other `direction'; however the
- reasoning infrastructure of Nominal Isabelle is set up so that it provides more
- automation for the formulation given above.
-
- Most properties given in this section are described in detail in \cite{HuffmanUrban10}
- and all are formalised in Isabelle/HOL. In the next sections we will make
- use of these properties in order to define alpha-equivalence in
- the presence of multiple binders.
-*}
-
-
-section {* Abstractions\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 writing 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 of atoms @{text
- "as"} in the body @{text "x"}.
-
- The first question we have to answer is when two pairs @{text "(as, x)"} and
- @{text "(bs, y)"} are alpha-equivalent? (For the moment we are interested in
- the notion of alpha-equivalence that is \emph{not} preserved by adding
- vacuous binders.) To answer this question, we identify four conditions: {\it (i)}
- given a free-atom function @{text "fa"} of type \mbox{@{text "\<beta> \<Rightarrow> atom
- set"}}, then @{text "(as, x)"} and @{text "(bs, y)"} need to have the same set of free
- atoms; moreover there must be a permutation @{text \<pi>} such that {\it
- (ii)} @{text \<pi>} leaves the free atoms of @{text "(as, x)"} and @{text "(bs, y)"} unchanged, but
- {\it (iii)} `moves' their bound names so that we obtain modulo a relation,
- say \mbox{@{text "_ R _"}}, two equivalent terms. We also require that {\it (iv)}
- @{text \<pi>} makes the sets of abstracted atoms @{text as} and @{text bs} equal. The
- requirements {\it (i)} to {\it (iv)} can be stated formally as:
-
- \begin{defi}[Alpha-Equivalence for Set-Bindings]\label{alphaset}\mbox{}\\
- \begin{tabular}{@ {\hspace{10mm}}l@ {\hspace{5mm}}rl}
- @{term "alpha_set_ex (as, x) R fa (bs, y)"}\hspace{2mm}@{text "\<equiv>"} &
- \multicolumn{2}{@ {}l}{if there exists a @{text "\<pi>"} such that:}\\
- & \mbox{\it (i)} & @{term "fa(x) - as = fa(y) - bs"}\\
- & \mbox{\it (ii)} & @{term "(fa(x) - as) \<sharp>* \<pi>"}\\
- & \mbox{\it (iii)} & @{text "(\<pi> \<bullet> x) R y"} \\
- & \mbox{\it (iv)} & @{term "(\<pi> \<bullet> as) = bs"} \\
- \end{tabular}
- \end{defi}
-
- \noindent
- Note that the relation is
- dependent on a free-atom function @{text "fa"} and a relation @{text
- "R"}. The reason for this extra generality is that we will use
- $\approx_{\,\textit{set}}^{\textit{R}, \textit{fa}}$ for both raw terms and
- alpha-equated terms. In
- the latter case, @{text R} will be replaced by equality @{text "="} and we
- will prove that @{text "fa"} is equal to @{text "supp"}.
-
- Definition \ref{alphaset} does not make any distinction between the
- order of abstracted atoms. 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{defi}[Alpha-Equivalence for List-Bindings]\label{alphalist}\mbox{}\\
- \begin{tabular}{@ {\hspace{10mm}}l@ {\hspace{5mm}}rl}
- @{term "alpha_lst_ex (as, x) R fa (bs, y)"}\hspace{2mm}@{text "\<equiv>"} &
- \multicolumn{2}{@ {}l}{if there exists a @{text "\<pi>"} such that:}\\
- & \mbox{\it (i)} & @{term "fa(x) - (set as) = fa(y) - (set bs)"}\\
- & \mbox{\it (ii)} & @{term "(fa(x) - set as) \<sharp>* \<pi>"}\\
- & \mbox{\it (iii)} & @{text "(\<pi> \<bullet> x) R y"}\\
- & \mbox{\it (iv)} & @{term "(\<pi> \<bullet> as) = bs"}\\
- \end{tabular}
- \end{defi}
-
- \noindent
- where @{term set} is the function that coerces a list of atoms into a set of atoms.
- Now the last clause ensures that the order of the binders matters (since @{text as}
- and @{text bs} are lists of atoms).
-
- If we do not want to make any difference between the order of binders \emph{and}
- also allow vacuous binders, that means according to Pitts~\cite{Pitts04}
- \emph{restrict} atoms, then we keep sets of binders, but drop
- condition {\it (iv)} in Definition~\ref{alphaset}:
-
- \begin{defi}[Alpha-Equivalence for Set+-Bindings]\label{alphares}\mbox{}\\
- \begin{tabular}{@ {\hspace{10mm}}l@ {\hspace{5mm}}rl}
- @{term "alpha_res_ex (as, x) R fa (bs, y)"}\hspace{2mm}@{text "\<equiv>"} &
- \multicolumn{2}{@ {}l}{if there exists a @{text "\<pi>"} such that:}\\
- & \mbox{\it (i)} & @{term "fa(x) - as = fa(y) - bs"}\\
- & \mbox{\it (ii)} & @{term "(fa(x) - as) \<sharp>* \<pi>"}\\
- & \mbox{\it (iii)} & @{text "(\<pi> \<bullet> x) R y"}\\
- \end{tabular}
- \end{defi}
-
-
- It might be useful to consider first some examples how these definitions
- of alpha-equivalence pan out in practice. For this consider the case of
- abstracting a set of atoms over types (as in type-schemes). We set
- @{text R} to be the usual equality @{text "="} and for @{text "fa(T)"} we
- define
-
- \[
- @{text "fa(x) \<equiv> {x}"} \hspace{10mm} @{text "fa(T\<^isub>1 \<rightarrow> T\<^isub>2) \<equiv> fa(T\<^isub>1) \<union> fa(T\<^isub>2)"}
- \]\smallskip
-
- \noindent
- Now recall the examples shown in \eqref{ex1} and
- \eqref{ex3}. It can be easily checked that @{text "({x, y}, x \<rightarrow> y)"} and
- @{text "({x, y}, y \<rightarrow> x)"} are alpha-equivalent according to
- $\approx_{\,\textit{set}}$ and $\approx_{\,\textit{set+}}$ by taking @{text "\<pi>"} to
- be the swapping @{term "(x \<rightleftharpoons> y)"}. In case of @{text "x \<noteq> y"}, then @{text
- "([x, y], x \<rightarrow> y)"} $\not\approx_{\,\textit{list}}$ @{text "([y, x], x \<rightarrow> y)"}
- since there is no permutation that makes the lists @{text "[x, y]"} and
- @{text "[y, x]"} equal, and also leaves the type \mbox{@{text "x \<rightarrow> y"}}
- unchanged. Another example is @{text "({x}, x)"} $\approx_{\,\textit{set+}}$
- @{text "({x, y}, x)"} which holds by taking @{text "\<pi>"} to be the identity
- permutation. However, if @{text "x \<noteq> y"}, then @{text "({x}, x)"}
- $\not\approx_{\,\textit{set}}$ @{text "({x, y}, x)"} since there is no
- permutation that makes the sets @{text "{x}"} and @{text "{x, y}"} equal
- (similarly for $\approx_{\,\textit{list}}$). It can also relatively easily be
- shown that all three notions of alpha-equivalence coincide, if we only
- abstract a single atom. In this case they also agree with the alpha-equivalence
- used in older versions of Nominal Isabelle \cite{Urban08}.\footnote{We omit a
- proof of this fact since the details are hairy and not really important for the
- purpose of this paper.}
-
- In the rest of this section we are going to show that the alpha-equivalences
- really lead to abstractions where some atoms are bound (or more precisely
- removed from the support). For this we will consider three abstraction
- types that are quotients of the relations
-
- \begin{equation}
- \begin{array}{r}
- @{term "alpha_set_ex (as, x) equal supp (bs, y)"}\smallskip\\
- @{term "alpha_res_ex (as, x) equal supp (bs, y)"}\smallskip\\
- @{term "alpha_lst_ex (as, x) equal supp (bs, y)"}\\
- \end{array}
- \end{equation}\smallskip
-
- \noindent
- Note that in these relations we replaced the free-atom function @{text "fa"}
- with @{term "supp"} and the relation @{text R} with equality. We can show
- the following two properties:
-
- \begin{lem}\label{alphaeq}
- The relations $\approx_{\,\textit{set}}^{=, \textit{supp}}$,
- $\approx_{\,\textit{set+}}^{=, \textit{supp}}$
- and $\approx_{\,\textit{list}}^{=, \textit{supp}}$ are
- equivalence relations and equivariant.
- \end{lem}
-
- \begin{proof}
- Reflexivity is by taking @{text "\<pi>"} to be @{text "0"}. For symmetry we have
- a permutation @{text "\<pi>"} and for the proof obligation take @{term "-
- \<pi>"}. In case of transitivity, we have two permutations @{text "\<pi>\<^isub>1"}
- and @{text "\<pi>\<^isub>2"}, and for the proof obligation use @{text
- "\<pi>\<^isub>1 + \<pi>\<^isub>2"}. Equivariance means @{term "alpha_set_ex (\<pi> \<bullet> as,
- \<pi> \<bullet> x) equal supp (\<pi> \<bullet> bs, \<pi> \<bullet> y)"} holds provided \mbox{@{term
- "alpha_set_ex (as, x) equal supp(bs, y)"}} holds. From the assumption we
- have a permutation @{text "\<pi>'"} and for the proof obligation use @{text "\<pi> \<bullet>
- \<pi>'"}. To show equivariance, we need to `pull out' the permutations,
- which is possible since all operators, namely as @{text "#\<^sup>*, -, =, \<bullet>,
- set"} and @{text "supp"}, are equivariant (see
- \cite{HuffmanUrban10}). Finally, we apply the permutation operation on
- booleans.
- \end{proof}
-
- \noindent
- Recall the picture shown in \eqref{picture} about new types in HOL.
- The lemma above allows us to use our quotient package for introducing
- new types @{text "\<beta> abs\<^bsub>set\<^esub>"}, @{text "\<beta> abs\<^bsub>set+\<^esub>"} and @{text "\<beta> abs\<^bsub>list\<^esub>"}
- representing alpha-equivalence classes of pairs of type
- @{text "(atom set) \<times> \<beta>"} (in the first two cases) and of type @{text "(atom list) \<times> \<beta>"}
- (in the third case).
- The elements in these types will be, respectively, written as
-
- \[
- @{term "Abs_set as x"} \hspace{10mm}
- @{term "Abs_res as x"} \hspace{10mm}
- @{term "Abs_lst as x"}
- \]\smallskip
-
- \noindent
- indicating that a set (or list) of atoms @{text as} is abstracted in @{text x}. We will
- call the types \emph{abstraction types} and their elements
- \emph{abstractions}. The important property we need to derive is the support of
- abstractions, namely:
-
- \begin{thm}[Support of Abstractions]\label{suppabs}
- Assuming @{text x} has finite support, then
-
- \[
- \begin{array}{l@ {\;=\;}l}
- @{thm (lhs) supp_Abs(1)[no_vars]} & @{thm (rhs) supp_Abs(1)[no_vars]}\\
- @{thm (lhs) supp_Abs(2)[no_vars]} & @{thm (rhs) supp_Abs(2)[no_vars]}\\
- @{thm (lhs) supp_Abs(3)[where bs="as", no_vars]} &
- @{thm (rhs) supp_Abs(3)[where bs="as", no_vars]}\\
- \end{array}
- \]\smallskip
- \end{thm}
-
- \noindent
- In effect, this theorem states that the atoms @{text "as"} are bound in the
- abstraction. As stated earlier, this can be seen as a litmus test that our
- Definitions \ref{alphaset}, \ref{alphalist} and \ref{alphares} capture the
- idea of alpha-equivalence relations. Below we will give the proof for the
- first equation of Theorem \ref{suppabs}. The others follow by similar
- arguments. By definition of the abstraction type @{text
- "abs\<^bsub>set\<^esub>"} we have
-
- \begin{equation}\label{abseqiff}
- @{thm (lhs) Abs_eq_iff(1)[where bs="as" and bs'="bs", no_vars]} \;\;\;\text{if and only if}\;\;\;
- @{term "alpha_set_ex (as, x) equal supp (bs, y)"}
- \end{equation}\smallskip
-
- \noindent
- and also set
-
- \begin{equation}\label{absperm}
- @{thm permute_Abs(1)[where p="\<pi>", no_vars, THEN eq_reflection]}
- \end{equation}\smallskip
-
- \noindent
- With this at our disposal, we can show
- the following lemma about swapping two atoms in an abstraction.
-
- \begin{lem}
- If @{thm (prem 1) Abs_swap1(1)[where bs="as", no_vars]} and
- @{thm (prem 2) Abs_swap1(1)[where bs="as", no_vars]} then
- @{thm (concl) Abs_swap1(1)[where bs="as", no_vars]}
- \end{lem}
-
- \begin{proof}
- If @{term "a = b"} the lemma is immediate, since @{term "(a \<rightleftharpoons> b)"} is then
- the identity permutation.
- Also in the other case the lemma is straightforward using \eqref{abseqiff}
- and observing that the assumptions give us @{term "(a \<rightleftharpoons> b) \<bullet> (supp x - as) =
- (supp x - as)"}. We therefore can use the swapping @{term "(a \<rightleftharpoons> b)"} as
- the permutation for the proof obligation.
- \end{proof}
-
- \noindent
- This lemma together
- with \eqref{absperm} allows us to show
-
- \begin{equation}\label{halfone}
- @{thm Abs_supports(1)[no_vars]}
- \end{equation}\smallskip
-
- \noindent
- which by Property~\ref{supportsprop} gives us `one half' of
- Theorem~\ref{suppabs}. To establish the `other half', we
- use a trick from \cite{Pitts04} and first define an auxiliary
- function @{text aux}, taking an abstraction as argument
-
- \[
- @{thm supp_set.simps[THEN eq_reflection, no_vars]}
- \]\smallskip
-
- \noindent
- Using the second equation in \eqref{equivariance}, we can show that
- @{text "aux"} is equivariant (since @{term "\<pi> \<bullet> (supp x - as) = (supp (\<pi> \<bullet> x)) - (\<pi> \<bullet> as)"})
- and therefore has empty support.
- This in turn means
-
- \[
- @{term "supp (supp_set (Abs_set as x)) \<subseteq> supp (Abs_set as x)"}
- \]\smallskip
-
- \noindent
- using the fact about the support of function applications in \eqref{supps}. Assuming
- @{term "supp x - as"} is a finite set, we further obtain
-
- \begin{equation}\label{halftwo}
- @{thm (concl) Abs_supp_subset1(1)[no_vars]}
- \end{equation}\smallskip
-
- \noindent
- This is because for every finite set of atoms, say @{text "bs"}, we have
- @{thm (concl) supp_finite_atom_set[where S="bs", no_vars]}.\footnote{Note that this is not
- the case for infinite sets.}
- Finally, taking \eqref{halfone} and \eqref{halftwo} together establishes
- the first equation of Theorem~\ref{suppabs}. The others are similar.
-
- Recall the definition of support given in \eqref{suppdef}, and note the difference between
- the support of a raw pair and an abstraction
-
- \[
- @{term "supp (as, x) = supp as \<union> supp x"}\hspace{15mm}
- @{term "supp (Abs_set as x) = supp x - as"}
- \]\smallskip
-
- \noindent
- While the permutation operations behave in both cases the same (a permutation
- is just moved to the arguments), the notion of equality is different for pairs and
- abstractions. Therefore we have different supports. In case of abstractions,
- we have established in Theorem~\ref{suppabs} that bound atoms are removed from
- the support of the abstractions' bodies.
-
- The method of first considering abstractions of the form @{term "Abs_set as
- x"} etc is motivated by the fact that we can conveniently establish at the
- Isabelle/HOL level properties about them. It would be extremely laborious
- to write custom ML-code that derives automatically such properties for every
- term-constructor that binds some atoms. Also the generality of the
- definitions for alpha-equivalence will help us in the next sections.
-*}
-
-section {* Specifying General Bindings\label{sec:spec} *}
-
-text {*
- Our choice of syntax for specifications is influenced by the existing
- datatype package of Isabelle/HOL \cite{Berghofer99}
- and by the syntax of the
- Ott-tool \cite{ott-jfp}. For us a specification of a term-calculus is a
- collection of (possibly mutually recursive) type declarations, say @{text
- "ty\<AL>\<^isub>1, \<dots>, ty\<AL>\<^isub>n"}, and an associated collection of
- binding functions, say @{text "bn\<AL>\<^isub>1, \<dots>, bn\<AL>\<^isub>m"}. The
- syntax in Nominal Isabelle for such specifications is schematically as follows:
-
- \begin{equation}\label{scheme}
- \mbox{\begin{tabular}{@ {}p{2.5cm}l}
- type \mbox{declaration part} &
- $\begin{cases}
- \mbox{\begin{tabular}{l}
- \isacommand{nominal\_datatype} @{text "ty\<AL>\<^isub>1 = \<dots>"}\\
- \isacommand{and} @{text "ty\<AL>\<^isub>2 = \<dots>"}\\
- \raisebox{2mm}{$\ldots$}\\[-2mm]
- \isacommand{and} @{text "ty\<AL>\<^isub>n = \<dots>"}\\
- \end{tabular}}
- \end{cases}$\\[2mm]
- binding \mbox{function part} &
- $\begin{cases}
- \mbox{\begin{tabular}{l}
- \isacommand{binder} @{text "bn\<AL>\<^isub>1"} \isacommand{and} \ldots \isacommand{and} @{text "bn\<AL>\<^isub>m"}\\
- \isacommand{where}\\
- \raisebox{2mm}{$\ldots$}\\[-2mm]
- \end{tabular}}
- \end{cases}$\\
- \end{tabular}}
- \end{equation}\smallskip
-
- \noindent
- Every type declaration @{text ty}$^\alpha_{1..n}$ consists of a collection
- of term-constructors, each of which comes with a list of labelled types that
- stand for the types of the arguments of the term-constructor. For example a
- term-constructor @{text "C\<^sup>\<alpha>"} might be specified with
-
- \[
- @{text "C\<^sup>\<alpha> label\<^isub>1::ty"}\mbox{$'_1$} @{text "\<dots> label\<^isub>l::ty"}\mbox{$'_l\;\;\;\;\;$}
- @{text "binding_clauses"}
- \]\smallskip
-
- \noindent
- whereby some of the @{text ty}$'_{1..l}$ (or their components) can be
- contained in the collection of @{text ty}$^\alpha_{1..n}$ declared in
- \eqref{scheme}. In this case we will call the corresponding argument a
- \emph{recursive argument} of @{text "C\<^sup>\<alpha>"}. The types of such
- recursive arguments need to satisfy a `positivity' restriction, which
- ensures that the type has a set-theoretic semantics (see
- \cite{Berghofer99}). If the types are polymorphic, we require the
- type variables to stand for types that are finitely supported and over which
- a permutation operation is defined.
- The labels @{text "label"}$_{1..l}$ annotated on the types are optional. Their
- purpose is to be used in the (possibly empty) list of \emph{binding
- clauses}, which indicate the binders and their scope in a term-constructor.
- They come in three \emph{modes}:
-
-
- \[\mbox{
- \begin{tabular}{@ {}l@ {}}
- \isacommand{binds} {\it binders} \isacommand{in} {\it bodies}\\
- \isacommand{binds (set)} {\it binders} \isacommand{in} {\it bodies}\\
- \isacommand{binds (set+)} {\it binders} \isacommand{in} {\it bodies}
- \end{tabular}}
- \]\smallskip
-
- \noindent
- The first mode is for binding lists of atoms (the order of bound atoms
- matters); the second is for sets of binders (the order does not matter, but
- the cardinality does) and the last is for sets of binders (with vacuous
- binders preserving alpha-equivalence). As indicated, the labels in the
- `\isacommand{in}-part' of a binding clause will be called \emph{bodies};
- the `\isacommand{binds}-part' will be called \emph{binders}. In contrast to
- Ott, we allow multiple labels in binders and bodies. For example we allow
- binding clauses of the form:
-
- \[\mbox{
- \begin{tabular}{@ {}ll@ {}}
- @{text "Foo\<^isub>1 x::name y::name t::trm s::trm"} &
- \isacommand{binds} @{text "x y"} \isacommand{in} @{text "t s"}\\
- @{text "Foo\<^isub>2 x::name y::name t::trm s::trm"} &
- \isacommand{binds} @{text "x y"} \isacommand{in} @{text "t"},
- \isacommand{binds} @{text "x y"} \isacommand{in} @{text "s"}\\
- \end{tabular}}
- \]\smallskip
-
- \noindent
- Similarly for the other binding modes. Interestingly, in case of
- \isacommand{binds (set)} and \isacommand{binds (set+)} the binding clauses
- above will make a difference to the semantics of the specifications (the
- corresponding alpha-equivalence will differ). We will show this later with
- an example.
-
-
- There are also some restrictions we need to impose on our binding clauses in
- comparison to Ott. The main idea behind these restrictions is
- that we obtain a notion of alpha-equivalence where it is ensured
- that within a given scope an atom occurrence cannot be both bound and free
- at the same time. The first restriction is that a body can only occur in
- \emph{one} binding clause of a term constructor. So for example
-
- \[\mbox{
- @{text "Foo x::name y::name t::trm"}\hspace{3mm}
- \isacommand{binds} @{text "x"} \isacommand{in} @{text "t"},
- \isacommand{binds} @{text "y"} \isacommand{in} @{text "t"}}
- \]\smallskip
-
- \noindent
- is not allowed. This ensures that the bound atoms of a body cannot be free
- at the same time by specifying an alternative binder for the same body.
-
- For binders we distinguish between \emph{shallow} and \emph{deep} binders.
- Shallow binders are just labels. The restriction we need to impose on them
- is that in case of \isacommand{binds (set)} and \isacommand{binds (set+)} the
- labels must either refer to atom types or to sets of atom types; in case of
- \isacommand{binds} the labels must refer to atom types or to lists of atom
- types. Two examples for the use of shallow binders are the specification of
- lambda-terms, where a single name is bound, and type-schemes, where a finite
- set of names is bound:
-
- \[\mbox{
- \begin{tabular}{@ {}c@ {\hspace{8mm}}c@ {}}
- \begin{tabular}{@ {}l}
- \isacommand{nominal\_datatype} @{text lam} $=$\\
- \hspace{2mm}\phantom{$\mid$}~@{text "Var name"}\\
- \hspace{2mm}$\mid$~@{text "App lam lam"}\\
- \hspace{2mm}$\mid$~@{text "Lam x::name t::lam"}\hspace{3mm}%
- \isacommand{binds} @{text x} \isacommand{in} @{text t}\\
- \\
- \end{tabular} &
- \begin{tabular}{@ {}l@ {}}
- \isacommand{nominal\_datatype}~@{text ty} $=$\\
- \hspace{2mm}\phantom{$\mid$}~@{text "TVar name"}\\
- \hspace{2mm}$\mid$~@{text "TFun ty ty"}\\
- \isacommand{and}~@{text "tsc ="}\\
- \hspace{2mm}\phantom{$\mid$}~@{text "TAll xs::(name fset) T::ty"}\hspace{3mm}%
- \isacommand{binds (set+)} @{text xs} \isacommand{in} @{text T}\\
- \end{tabular}
- \end{tabular}}
- \]\smallskip
-
-
- \noindent
- In these specifications @{text "name"} refers to a (concrete) atom type, and @{text
- "fset"} to the type of finite sets. Note that for @{text Lam} it does not
- matter which binding mode we use. The reason is that we bind only a single
- @{text name}, in which case all three binding modes coincide. However, having
- \isacommand{binds (set)} or just \isacommand{binds}
- in the second case makes a difference to the semantics of the specification
- (which we will define in the next section).
-
- A \emph{deep} binder uses an auxiliary binding function that `picks' out
- the atoms in one argument of the term-constructor, which can be bound in
- other arguments and also in the same argument (we will call such binders
- \emph{recursive}, see below). The binding functions are
- expected to return either a set of atoms (for \isacommand{binds (set)} and
- \isacommand{binds (set+)}) or a list of atoms (for \isacommand{binds}). They need
- to be defined by recursion over the corresponding type; the equations
- must be given in the binding function part of the scheme shown in
- \eqref{scheme}. For example a term-calculus containing @{text "Let"}s with
- tuple patterns may be specified as:
-
- \begin{equation}\label{letpat}
- \mbox{%
- \begin{tabular}{l}
- \isacommand{nominal\_datatype} @{text trm} $=$\\
- \hspace{5mm}\phantom{$\mid$}~@{term "Var name"}\\
- \hspace{5mm}$\mid$~@{term "App trm trm"}\\
- \hspace{5mm}$\mid$~@{text "Lam x::name t::trm"}
- \;\;\isacommand{binds} @{text x} \isacommand{in} @{text t}\\
- \hspace{5mm}$\mid$~@{text "Let_pat p::pat trm t::trm"}
- \;\;\isacommand{binds} @{text "bn(p)"} \isacommand{in} @{text t}\\
- \isacommand{and} @{text pat} $=$\\
- \hspace{5mm}\phantom{$\mid$}~@{text "PVar name"}\\
- \hspace{5mm}$\mid$~@{text "PTup pat pat"}\\
- \isacommand{binder}~@{text "bn::pat \<Rightarrow> atom list"}\\
- \isacommand{where}~@{text "bn(PVar x) = [atom x]"}\\
- \hspace{5mm}$\mid$~@{text "bn(PTup p\<^isub>1 p\<^isub>2) = bn(p\<^isub>1) @ bn(p\<^isub>2)"}\smallskip\\
- \end{tabular}}
- \end{equation}\smallskip
-
- \noindent
- In this specification the function @{text "bn"} determines which atoms of
- the pattern @{text p} (fifth line) are bound in the argument @{text "t"}. Note that in the
- second-last @{text bn}-clause the function @{text "atom"} coerces a name
- into the generic atom type of Nominal Isabelle \cite{HuffmanUrban10}. This
- allows us to treat binders of different atom type uniformly.
-
- For deep binders we allow binding clauses such as
-
- \[\mbox{
- \begin{tabular}{ll}
- @{text "Bar p::pat t::trm"} &
- \isacommand{binds} @{text "bn(p)"} \isacommand{in} @{text "p t"} \\
- \end{tabular}}
- \]\smallskip
-
-
- \noindent
- where the argument of the deep binder also occurs in the body. We call such
- binders \emph{recursive}. To see the purpose of such recursive binders,
- compare `plain' @{text "Let"}s and @{text "Let_rec"}s in the following
- specification:
-
- \begin{equation}\label{letrecs}
- \mbox{%
- \begin{tabular}{@ {}l@ {}l}
- \isacommand{nominal\_datatype}~@{text "trm ="}\\
- \hspace{5mm}\phantom{$\mid$}~\ldots\\
- \hspace{5mm}$\mid$~@{text "Let as::assn t::trm"}
- & \hspace{-19mm}\isacommand{binds} @{text "bn(as)"} \isacommand{in} @{text t}\\
- \hspace{5mm}$\mid$~@{text "Let_rec as::assn t::trm"}
- & \hspace{-19mm}\isacommand{binds} @{text "bn(as)"} \isacommand{in} @{text "as t"}\\
- \isacommand{and} @{text "assn"} $=$\\
- \hspace{5mm}\phantom{$\mid$}~@{text "ANil"}\\
- \hspace{5mm}$\mid$~@{text "ACons name trm assn"}\\
- \isacommand{binder} @{text "bn::assn \<Rightarrow> atom list"}\\
- \isacommand{where}~@{text "bn(ANil) = []"}\\
- \hspace{5mm}$\mid$~@{text "bn(ACons a t as) = [atom a] @ bn(as)"}\\
- \end{tabular}}
- \end{equation}\smallskip
-
- \noindent
- The difference is that with @{text Let} we only want to bind the atoms @{text
- "bn(as)"} in the term @{text t}, but with @{text "Let_rec"} we also want to bind the atoms
- inside the assignment. This difference has consequences for the associated
- notions of free-atoms and alpha-equivalence.
-
- To make sure that atoms bound by deep binders cannot be free at the
- same time, we cannot have more than one binding function for a deep binder.
- Consequently we exclude specifications such as
-
- \[\mbox{
- \begin{tabular}{@ {}l@ {\hspace{2mm}}l@ {}}
- @{text "Baz\<^isub>1 p::pat t::trm"} &
- \isacommand{binds} @{text "bn\<^isub>1(p) bn\<^isub>2(p)"} \isacommand{in} @{text "p t"}\\
- @{text "Baz\<^isub>2 p::pat t\<^isub>1::trm t\<^isub>2::trm"} &
- \isacommand{binds} @{text "bn\<^isub>1(p)"} \isacommand{in} @{text "p t\<^isub>1"},
- \isacommand{binds} @{text "bn\<^isub>2(p)"} \isacommand{in} @{text "p t\<^isub>2"}\\
- \end{tabular}}
- \]\smallskip
-
- \noindent
- Otherwise it is possible that @{text "bn\<^isub>1"} and @{text "bn\<^isub>2"} pick
- out different atoms to become bound, respectively be free,
- in @{text "p"}.\footnote{Since the Ott-tool does not derive a reasoning
- infrastructure for
- alpha-equated terms with deep binders, it can permit such specifications.}
-
-
- We also need to restrict the form of the binding functions in order to
- ensure the @{text "bn"}-functions can be defined for alpha-equated
- terms. The main restriction is that we cannot return an atom in a binding
- function that is also bound in the corresponding term-constructor.
- Consider again the specification for @{text "trm"} and a contrived
- version for assignments @{text "assn"}:
-
- \begin{equation}\label{bnexp}
- \mbox{%
- \begin{tabular}{@ {}l@ {}}
- \isacommand{nominal\_datatype}~@{text "trm ="}~\ldots\\
- \isacommand{and} @{text "assn"} $=$\\
- \hspace{5mm}\phantom{$\mid$}~@{text "ANil'"}\\
- \hspace{5mm}$\mid$~@{text "ACons' x::name y::name t::trm assn"}
- \;\;\isacommand{binds} @{text "y"} \isacommand{in} @{text t}\\
- \isacommand{binder} @{text "bn::assn \<Rightarrow> atom list"}\\
- \isacommand{where}~@{text "bn(ANil') = []"}\\
- \hspace{5mm}$\mid$~@{text "bn(ACons' x y t as) = [atom x] @ bn(as)"}\\
- \end{tabular}}
- \end{equation}\smallskip
-
- \noindent
- In this example the term constructor @{text "ACons'"} has four arguments with
- a binding clause involving two of them. This constructor is also used in the definition
- of the binding function. The restriction we have to impose is that the
- binding function can only return free atoms, that is the ones that are \emph{not}
- mentioned in a binding clause. Therefore @{text "y"} cannot be used in the
- binding function @{text "bn"} (since it is bound in @{text "ACons'"} by the
- binding clause), but @{text x} can (since it is a free atom). This
- restriction is sufficient for lifting the binding function to alpha-equated
- terms. If we would permit @{text "bn"} to return @{text "y"},
- then it would not be respectful and therefore cannot be lifted to
- alpha-equated lambda-terms.
-
- In the version of Nominal Isabelle described here, we also adopted the
- restriction from the Ott-tool that binding functions can only return: the
- empty set or empty list (as in case @{text ANil'}), a singleton set or
- singleton list containing an atom (case @{text PVar} in \eqref{letpat}), or
- unions of atom sets or appended atom lists (case @{text ACons'}). This
- restriction will simplify some automatic definitions and proofs later on.
-
- To sum up this section, we introduced nominal datatype
- specifications, which are like standard datatype specifications in
- Isabelle/HOL but extended with binding clauses and specifications for binding
- functions. Each constructor argument in our specification can also
- have an optional label. These labels are used in the binding clauses
- of a constructor; there can be several binding clauses for each
- constructor, but bodies of binding clauses can only occur in a
- single one. Binding clauses come in three modes: \isacommand{binds},
- \isacommand{binds (set)} and \isacommand{binds (set+)}. Binders
- fall into two categories: shallow binders and deep binders. Shallow
- binders can occur in more than one binding clause and only have to
- respect the binding mode (i.e.~be of the right type). Deep binders
- can also occur in more than one binding clause, unless they are
- recursive in which case they can only occur once. Each of the deep
- binders can only have a single binding function. Binding functions
- are defined by recursion over a nominal datatype. They can
- return the empty set, singleton atoms and unions of sets of atoms
- (for binding modes \isacommand{binds (set)} and \isacommand{binds
- (set+)}), and the empty list, singleton atoms and appended lists of
- atoms (for mode \isacommand{bind}). However, they can only return
- atoms that are not mentioned in any binding clause.
-
- In order to
- simplify our definitions of free atoms and alpha-equivalence we define next, we
- shall assume specifications of term-calculi are implicitly
- \emph{completed}. By this we mean that for every argument of a
- term-constructor that is \emph{not} already part of a binding clause
- given by the user, we add implicitly a special \emph{empty} binding
- clause, written \isacommand{binds}~@{term
- "{}"}~\isacommand{in}~@{text "labels"}. In case of the lambda-terms,
- the completion produces
-
- \[\mbox{
- \begin{tabular}{@ {}l@ {\hspace{-1mm}}}
- \isacommand{nominal\_datatype} @{text lam} =\\
- \hspace{5mm}\phantom{$\mid$}~@{text "Var x::name"}
- \;\;\isacommand{binds}~@{term "{}"}~\isacommand{in}~@{text "x"}\\
- \hspace{5mm}$\mid$~@{text "App t\<^isub>1::lam t\<^isub>2::lam"}
- \;\;\isacommand{binds}~@{term "{}"}~\isacommand{in}~@{text "t\<^isub>1 t\<^isub>2"}\\
- \hspace{5mm}$\mid$~@{text "Lam x::name t::lam"}
- \;\;\isacommand{binds}~@{text x} \isacommand{in} @{text t}\\
- \end{tabular}}
- \]\smallskip
-
- \noindent
- The point of completion is that we can make definitions over the binding
- clauses and be sure to have captured all arguments of a term constructor.
-*}
-
-section {* Alpha-Equivalence and Free Atoms\label{sec:alpha} *}
-
-text {*
- Having dealt with all syntax matters, the problem now is how we can turn
- specifications into actual type definitions in Isabelle/HOL and then
- establish a reasoning infrastructure for them. As Pottier and Cheney pointed
- out \cite{Cheney05,Pottier06}, just re-arranging the arguments of
- term-constructors so that binders and their bodies are next to each other
- will result in inadequate representations in cases like \mbox{@{text "Let
- x\<^isub>1 = t\<^isub>1\<dots>x\<^isub>n = t\<^isub>n in s"}}. Therefore we will
- first extract `raw' datatype definitions from the specification and then
- define explicitly an alpha-equivalence relation over them. We subsequently
- construct the quotient of the datatypes according to our alpha-equivalence.
-
-
- The `raw' datatype definition can be obtained by stripping off the
- binding clauses and the labels from the types given by the user. We also have to invent
- new names for the types @{text "ty\<^sup>\<alpha>"} and the term-constructors @{text "C\<^sup>\<alpha>"}.
- In our implementation we just use the affix ``@{text "_raw"}''.
- But for the purpose of this paper, we use the superscript @{text "_\<^sup>\<alpha>"} to indicate
- that a notion is given for alpha-equivalence classes and leave it out
- for the corresponding notion given on the raw level. So for example
- we have @{text "ty\<^sup>\<alpha> / ty"} and @{text "C\<^sup>\<alpha> / C"}
- where @{term ty} is the type used in the quotient construction for
- @{text "ty\<^sup>\<alpha>"} and @{text "C"} is the term-constructor of the raw type @{text "ty"},
- respectively @{text "C\<^sup>\<alpha>"} is the corresponding term-constructor of @{text "ty\<^sup>\<alpha>"}.
-
- The resulting datatype definition is legal in Isabelle/HOL provided the datatypes are
- non-empty and the types in the constructors only occur in positive
- position (see \cite{Berghofer99} for an in-depth description of the datatype package
- in Isabelle/HOL).
- We subsequently define each of the user-specified binding
- functions @{term "bn"}$_{1..m}$ by recursion over the corresponding
- raw datatype. We also define permutation operations by
- recursion so that for each term constructor @{text "C"} we have that
-
- \begin{equation}\label{ceqvt}
- @{text "\<pi> \<bullet> (C z\<^isub>1 \<dots> z\<^isub>n) = C (\<pi> \<bullet> z\<^isub>1) \<dots> (\<pi> \<bullet> z\<^isub>n)"}
- \end{equation}\smallskip
-
- \noindent
- We will need this operation later when we define the notion of alpha-equivalence.
-
- The first non-trivial step we have to perform is the generation of
- \emph{free-atom functions} from the specifications.\footnote{Admittedly, the
- details of our definitions will be somewhat involved. However they are still
- conceptually simple in comparison with the `positional' approach taken in
- Ott \cite[Pages 88--95]{ott-jfp}, which uses the notions of \emph{occurrences} and
- \emph{partial equivalence relations} over sets of occurrences.} For the
- \emph{raw} types @{text "ty"}$_{1..n}$ we define the free-atom functions
-
- \begin{equation}\label{fvars}
- \mbox{@{text "fa_ty"}$_{1..n}$}
- \end{equation}\smallskip
-
- \noindent
- by recursion.
- We define these functions together with auxiliary free-atom functions for
- the binding functions. Given raw binding functions @{text "bn"}$_{1..m}$
- we define
-
- \[
- @{text "fa_bn"}\mbox{$_{1..m}$}.
- \]\smallskip
-
- \noindent
- The reason for this setup is that in a deep binder not all atoms have to be
- bound, as we saw in \eqref{letrecs} with the example of `plain' @{text Let}s. We need
- therefore functions that calculate those free atoms in deep binders.
-
- While the idea behind these free-atom functions is simple (they just
- collect all atoms that are not bound), because of our rather complicated
- binding mechanisms their definitions are somewhat involved. Given
- a raw term-constructor @{text "C"} of type @{text ty} and some associated
- binding clauses @{text "bc\<^isub>1\<dots>bc\<^isub>k"}, the result of @{text
- "fa_ty (C z\<^isub>1 \<dots> z\<^isub>n)"} will be the union @{text
- "fa(bc\<^isub>1) \<union> \<dots> \<union> fa(bc\<^isub>k)"} where we will define below what @{text "fa"} for a binding
- clause means. We only show the details for the mode \isacommand{binds (set)} (the other modes are similar).
- Suppose a binding clause @{text bc\<^isub>i} is of the form
-
- \[
- \mbox{\isacommand{binds (set)} @{text "b\<^isub>1\<dots>b\<^isub>p"} \isacommand{in} @{text "d\<^isub>1\<dots>d\<^isub>q"}}
- \]\smallskip
-
- \noindent
- in which the body-labels @{text "d"}$_{1..q}$ refer to types @{text
- ty}$_{1..q}$, and the binders @{text b}$_{1..p}$ either refer to labels of
- atom types (in case of shallow binders) or to binding functions taking a
- single label as argument (in case of deep binders). Assuming @{text "D"}
- stands for the set of free atoms of the bodies, @{text B} for the set of
- binding atoms in the binders and @{text "B'"} for the set of free atoms in
- non-recursive deep binders, then the free atoms of the binding clause @{text
- bc\<^isub>i} are
-
- \begin{equation}\label{fadef}
- \mbox{@{text "fa(bc\<^isub>i) \<equiv> (D - B) \<union> B'"}}.
- \end{equation}\smallskip
-
- \noindent
- The set @{text D} is formally defined as
-
- \[
- @{text "D \<equiv> fa_ty\<^isub>1 d\<^isub>1 \<union> ... \<union> fa_ty\<^isub>q d\<^isub>q"}
- \]\smallskip
-
- \noindent
- where in case @{text "d\<^isub>i"} refers to one of the raw types @{text "ty"}$_{1..n}$ from the
- specification, the function @{text "fa_ty\<^isub>i"} is the corresponding free-atom function
- we are defining by recursion; otherwise we set \mbox{@{text "fa_ty\<^isub>i \<equiv> supp"}}. The reason
- for the latter is that @{text "ty"}$_i$ is not a type that is part of the specification, and
- we assume @{text supp} is the generic function that characterises the free variables of
- a type (in fact in the next section we will show that the free-variable functions we
- define here, are equal to the support once lifted to alpha-equivalence classes).
-
- In order to formally define the set @{text B} we use the following auxiliary @{text "bn"}-functions
- for atom types to which shallow binders may refer\\[-4mm]
-
- \begin{equation}\label{bnaux}\mbox{
- \begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
- @{text "bn\<^bsub>atom\<^esup> a"} & @{text "\<equiv>"} & @{text "{atom a}"}\\
- @{text "bn\<^bsub>atom_set\<^esup> as"} & @{text "\<equiv>"} & @{text "atoms as"}\\
- @{text "bn\<^bsub>atom_list\<^esub> as"} & @{text "\<equiv>"} & @{text "atoms (set as)"}
- \end{tabular}}
- \end{equation}\smallskip
-
- \noindent
- Like the function @{text atom}, the function @{text "atoms"} coerces
- a set of atoms to a set of the generic atom type.
- It is defined as @{text "atoms as \<equiv> {atom a | a \<in> as}"}.
- The set @{text B} in \eqref{fadef} is then formally defined as
-
- \begin{equation}\label{bdef}
- @{text "B \<equiv> bn_ty\<^isub>1 b\<^isub>1 \<union> ... \<union> bn_ty\<^isub>p b\<^isub>p"}
- \end{equation}\smallskip
-
- \noindent
- where we use the auxiliary binding functions from \eqref{bnaux} for shallow
- binders (that means when @{text "ty"}$_i$ is of type @{text "atom"}, @{text "atom set"} or
- @{text "atom list"}).
-
- The set @{text "B'"} in \eqref{fadef} collects all free atoms in
- non-recursive deep binders. Let us assume these binders in the binding
- clause @{text "bc\<^isub>i"} are
-
- \[
- \mbox{@{text "bn\<^isub>1 l\<^isub>1, \<dots>, bn\<^isub>r l\<^isub>r"}}
- \]\smallskip
-
- \noindent
- with @{text "l"}$_{1..r}$ $\subseteq$ @{text "b"}$_{1..p}$ and
- none of the @{text "l"}$_{1..r}$ being among the bodies
- @{text "d"}$_{1..q}$. The set @{text "B'"} is defined as
-
- \begin{equation}\label{bprimedef}
- @{text "B' \<equiv> fa_bn\<^isub>1 l\<^isub>1 \<union> ... \<union> fa_bn\<^isub>r l\<^isub>r"}
- \end{equation}\smallskip
-
- \noindent
- This completes all clauses for the free-atom functions @{text "fa_ty"}$_{1..n}$.
-
- Note that for non-recursive deep binders, we have to add in \eqref{fadef}
- the set of atoms that are left unbound by the binding functions @{text
- "bn"}$_{1..m}$. We used for
- the definition of this set the functions @{text "fa_bn"}$_{1..m}$. The
- definition for those functions needs to be extracted from the clauses the
- user provided for @{text "bn"}$_{1..m}$ Assume the user specified a @{text
- bn}-clause of the form
-
- \[
- @{text "bn (C z\<^isub>1 \<dots> z\<^isub>s) = rhs"}
- \]\smallskip
-
- \noindent
- where the @{text "z"}$_{1..s}$ are of types @{text "ty"}$_{1..s}$. For
- each of the arguments we calculate the free atoms as follows:
-
- \[\mbox{
- \begin{tabular}{c@ {\hspace{2mm}}p{0.9\textwidth}}
- $\bullet$ & @{term "fa_ty\<^isub>i z\<^isub>i"} provided @{text "z\<^isub>i"} does not occur in @{text "rhs"}\\
- & (that means nothing is bound in @{text "z\<^isub>i"} by the binding function),\smallskip\\
- $\bullet$ & @{term "fa_bn\<^isub>i z\<^isub>i"} provided @{text "z\<^isub>i"} occurs in @{text "rhs"}
- with the recursive call @{text "bn\<^isub>i z\<^isub>i"}\\
- & (that means whatever is `left over' from the @{text "bn"}-function is free)\smallskip\\
- $\bullet$ & @{term "{}"} provided @{text "z\<^isub>i"} occurs in @{text "rhs"},
- but without a recursive call\\
- & (that means @{text "z\<^isub>i"} is supposed to become bound by the binding function)\\
- \end{tabular}}
- \]\smallskip
-
- \noindent
- For defining @{text "fa_bn (C z\<^isub>1 \<dots> z\<^isub>n)"} we just union up all these sets.
-
- To see how these definitions work in practice, let us reconsider the
- term-constructors @{text "Let"} and @{text "Let_rec"} shown in
- \eqref{letrecs} together with the term-constructors for assignments @{text
- "ANil"} and @{text "ACons"}. Since there is a binding function defined for
- assignments, we have three free-atom functions, namely @{text
- "fa\<^bsub>trm\<^esub>"}, @{text "fa\<^bsub>assn\<^esub>"} and @{text
- "fa\<^bsub>bn\<^esub>"} as follows:
-
- \[\mbox{
- \begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}}
- @{text "fa\<^bsub>trm\<^esub> (Let as t)"} & @{text "\<equiv>"} & @{text "(fa\<^bsub>trm\<^esub> t - set (bn as)) \<union> fa\<^bsub>bn\<^esub> as"}\\
- @{text "fa\<^bsub>trm\<^esub> (Let_rec as t)"} & @{text "\<equiv>"} & @{text "(fa\<^bsub>assn\<^esub> as \<union> fa\<^bsub>trm\<^esub> t) - set (bn as)"}\smallskip\\
-
- @{text "fa\<^bsub>assn\<^esub> (ANil)"} & @{text "\<equiv>"} & @{term "{}"}\\
- @{text "fa\<^bsub>assn\<^esub> (ACons a t as)"} & @{text "\<equiv>"} & @{text "(supp a) \<union> (fa\<^bsub>trm\<^esub> t) \<union> (fa\<^bsub>assn\<^esub> as)"}\smallskip\\
-
- @{text "fa\<^bsub>bn\<^esub> (ANil)"} & @{text "\<equiv>"} & @{term "{}"}\\
- @{text "fa\<^bsub>bn\<^esub> (ACons a t as)"} & @{text "\<equiv>"} & @{text "(fa\<^bsub>trm\<^esub> t) \<union> (fa\<^bsub>bn\<^esub> as)"}
- \end{tabular}}
- \]\smallskip
-
-
- \noindent
- Recall that @{text ANil} and @{text "ACons"} have no binding clause in the
- specification. The corresponding free-atom function @{text
- "fa\<^bsub>assn\<^esub>"} therefore returns all free atoms of an assignment
- (in case of @{text "ACons"}, they are given in terms of @{text supp}, @{text
- "fa\<^bsub>trm\<^esub>"} and @{text "fa\<^bsub>assn\<^esub>"}). The binding
- only takes place in @{text Let} and @{text "Let_rec"}. In case of @{text
- "Let"}, the binding clause specifies that all atoms given by @{text "set (bn
- as)"} have to be bound in @{text t}. Therefore we have to subtract @{text
- "set (bn as)"} from @{text "fa\<^bsub>trm\<^esub> t"}. However, we also need
- to add all atoms that are free in @{text "as"}. This is in contrast with
- @{text "Let_rec"} where we have a recursive binder to bind all occurrences
- of the atoms in @{text "set (bn as)"} also inside @{text "as"}. Therefore we
- have to subtract @{text "set (bn as)"} from both @{text
- "fa\<^bsub>trm\<^esub> t"} and @{text "fa\<^bsub>assn\<^esub> as"}. Like the
- function @{text "bn"}, the function @{text "fa\<^bsub>bn\<^esub>"} traverses
- the list of assignments, but instead returns the free atoms, which means in
- this example the free atoms in the argument @{text "t"}.
-
-
- An interesting point in this example is that a `naked' assignment (@{text
- "ANil"} or @{text "ACons"}) does not bind any atoms, even if the binding
- function is specified over assignments. Only in the context of a @{text Let}
- or @{text "Let_rec"}, where the binding clauses are given, will some atoms
- actually become bound. This is a phenomenon that has also been pointed out
- in \cite{ott-jfp}. For us this observation is crucial, because we would not
- be able to lift the @{text "bn"}-functions to alpha-equated terms if they
- act on atoms that are bound. In that case, these functions would \emph{not}
- respect alpha-equivalence.
-
- Having the free-atom functions at our disposal, we can next define the
- alpha-equivalence relations for the raw types @{text
- "ty"}$_{1..n}$. We write them as
-
- \[
- \mbox{@{text "\<approx>ty"}$_{1..n}$}.
- \]\smallskip
-
- \noindent
- Like with the free-atom functions, we also need to
- define auxiliary alpha-equivalence relations
-
- \[
- \mbox{@{text "\<approx>bn\<^isub>"}$_{1..m}$}
- \]\smallskip
-
- \noindent
- for the binding functions @{text "bn"}$_{1..m}$,
- To simplify our definitions we will use the following abbreviations for
- \emph{compound equivalence relations} and \emph{compound free-atom functions} acting on tuples.
-
- \[\mbox{
- \begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
- @{text "(x\<^isub>1,\<dots>, x\<^isub>n) (R\<^isub>1,\<dots>, R\<^isub>n) (y\<^isub>1,\<dots>, y\<^isub>n)"} & @{text "\<equiv>"} &
- @{text "x\<^isub>1 R\<^isub>1 y\<^isub>1 \<and> \<dots> \<and> x\<^isub>n R\<^isub>n y\<^isub>n"}\\
- @{text "(fa\<^isub>1,\<dots>, fa\<^isub>n) (x\<^isub>1,\<dots>, x\<^isub>n)"} & @{text "\<equiv>"} & @{text "fa\<^isub>1 x\<^isub>1 \<union> \<dots> \<union> fa\<^isub>n x\<^isub>n"}\\
- \end{tabular}}
- \]\smallskip
-
-
- The alpha-equivalence relations are defined as inductive predicates
- having a single clause for each term-constructor. Assuming a
- term-constructor @{text C} is of type @{text ty} and has the binding clauses
- @{term "bc"}$_{1..k}$, then the alpha-equivalence clause has the form
-
- \begin{equation}\label{gform}
- \mbox{\infer{@{text "C z\<^isub>1 \<dots> z\<^isub>n \<approx>ty C z\<PRIME>\<^isub>1 \<dots> z\<PRIME>\<^isub>n"}}
- {@{text "prems(bc\<^isub>1) \<dots> prems(bc\<^isub>k)"}}}
- \end{equation}\smallskip
-
- \noindent
- The task below is to specify what the premises corresponding to a binding
- clause are. To understand better what the general pattern is, let us first
- treat the special instance where @{text "bc\<^isub>i"} is the empty binding clause
- of the form
-
- \[
- \mbox{\isacommand{binds (set)} @{term "{}"} \isacommand{in} @{text "d\<^isub>1\<dots>d\<^isub>q"}.}
- \]\smallskip
-
- \noindent
- In this binding clause no atom is bound and we only have to `alpha-relate'
- the bodies. For this we build first the tuples @{text "D \<equiv> (d\<^isub>1,\<dots>,
- d\<^isub>q)"} and @{text "D' \<equiv> (d\<PRIME>\<^isub>1,\<dots>, d\<PRIME>\<^isub>q)"}
- whereby the labels @{text "d"}$_{1..q}$ refer to some of the arguments @{text
- "z"}$_{1..n}$ and respectively @{text "d\<PRIME>"}$_{1..q}$ to some of the @{text
- "z\<PRIME>"}$_{1..n}$ in \eqref{gform}. In order to relate two such
- tuples we define the compound alpha-equivalence relation @{text "R"} as
- follows
-
- \begin{equation}\label{rempty}
- \mbox{@{text "R \<equiv> (R\<^isub>1,\<dots>, R\<^isub>q)"}}
- \end{equation}\smallskip
-
- \noindent
- with @{text "R\<^isub>i"} being @{text "\<approx>ty\<^isub>i"} if the corresponding
- labels @{text "d\<^isub>i"} and @{text "d\<PRIME>\<^isub>i"} refer to a
- recursive argument of @{text C} and have type @{text "ty\<^isub>i"}; otherwise
- we take @{text "R\<^isub>i"} to be the equality @{text "="}. Again the
- latter is because @{text "ty\<^isub>i"} is then not part of the specified types
- and alpha-equivalence of any previously defined type is supposed to coincide
- with equality. This lets us now define the premise for an empty binding
- clause succinctly as @{text "prems(bc\<^isub>i) \<equiv> D R D'"}, which can be
- unfolded to the series of premises
-
- \[
- @{text "d\<^isub>1 R\<^isub>1 d\<PRIME>\<^isub>1 \<dots> d\<^isub>q R\<^isub>q d\<PRIME>\<^isub>q"}.
- \]\smallskip
-
- \noindent
- We will use the unfolded version in the examples below.
-
- Now suppose the binding clause @{text "bc\<^isub>i"} is of the general form
-
- \begin{equation}\label{nonempty}
- \mbox{\isacommand{binds (set)} @{text "b\<^isub>1\<dots>b\<^isub>p"} \isacommand{in} @{text "d\<^isub>1\<dots>d\<^isub>q"}.}
- \end{equation}\smallskip
-
- \noindent
- In this case we define a premise @{text P} using the relation
- $\approx_{\,\textit{set}}^{\textit{R}, \textit{fa}}$ given in Section~\ref{sec:binders} (similarly
- $\approx_{\,\textit{set+}}^{\textit{R}, \textit{fa}}$ and
- $\approx_{\,\textit{list}}^{\textit{R}, \textit{fa}}$ for the other
- binding modes). As above, we first build the tuples @{text "D"} and
- @{text "D'"} for the bodies @{text "d"}$_{1..q}$, and the corresponding
- compound alpha-relation @{text "R"} (shown in \eqref{rempty}).
- For $\approx_{\,\textit{set}}^{\textit{R}, \textit{fa}}$ we also need
- a compound free-atom function for the bodies defined as
-
- \[
- \mbox{@{text "fa \<equiv> (fa_ty\<^isub>1,\<dots>, fa_ty\<^isub>q)"}}
- \]\smallskip
-
- \noindent
- with the assumption that the @{text "d"}$_{1..q}$ refer to arguments of types @{text "ty"}$_{1..q}$.
- The last ingredient we need are the sets of atoms bound in the bodies.
- For this we take
-
- \[
- @{text "B \<equiv> bn_ty\<^isub>1 b\<^isub>1 \<union> \<dots> \<union> bn_ty\<^isub>p b\<^isub>p"}\;.\\
- \]\smallskip
-
- \noindent
- Similarly for @{text "B'"} using the labels @{text "b\<PRIME>"}$_{1..p}$. This
- lets us formally define the premise @{text P} for a non-empty binding clause as:
-
- \[
- \mbox{@{term "P \<equiv> alpha_set_ex (B, D) R fa (B', D')"}}\;.
- \]\smallskip
-
- \noindent
- This premise accounts for alpha-equivalence of the bodies of the binding
- clause. However, in case the binders have non-recursive deep binders, this
- premise is not enough: we also have to `propagate' alpha-equivalence
- inside the structure of these binders. An example is @{text "Let"} where we
- have to make sure the right-hand sides of assignments are
- alpha-equivalent. For this we use relations @{text "\<approx>bn"}$_{1..m}$ (which we
- will define shortly). Let us assume the non-recursive deep binders
- in @{text "bc\<^isub>i"} are
-
- \[
- @{text "bn\<^isub>1 l\<^isub>1, \<dots>, bn\<^isub>r l\<^isub>r"}.
- \]\smallskip
-
- \noindent
- The tuple @{text L} consists then of all these binders @{text "(l\<^isub>1,\<dots>,l\<^isub>r)"}
- (similarly @{text "L'"}) and the compound equivalence relation @{text "R'"}
- is @{text "(\<approx>bn\<^isub>1,\<dots>,\<approx>bn\<^isub>r)"}. All premises for @{text "bc\<^isub>i"} are then given by
-
- \[
- @{text "prems(bc\<^isub>i) \<equiv> P \<and> L R' L'"}
- \]\smallskip
-
- \noindent
- The auxiliary alpha-equivalence relations @{text "\<approx>bn"}$_{1..m}$
- in @{text "R'"} are defined as follows: assuming a @{text bn}-clause is of the form
-
- \[
- @{text "bn (C z\<^isub>1 \<dots> z\<^isub>s) = rhs"}
- \]\smallskip
-
- \noindent
- where the @{text "z"}$_{1..s}$ are of types @{text "ty"}$_{1..s}$,
- then the corresponding alpha-equivalence clause for @{text "\<approx>bn"} has the form
-
- \[
- \mbox{\infer{@{text "C z\<^isub>1 \<dots> z\<^isub>s \<approx>bn C z\<PRIME>\<^isub>1 \<dots> z\<PRIME>\<^isub>s"}}
- {@{text "z\<^isub>1 R\<^isub>1 z\<PRIME>\<^isub>1 \<dots> z\<^isub>s R\<^isub>s z\<PRIME>\<^isub>s"}}}
- \]\smallskip
-
- \noindent
- In this clause the relations @{text "R"}$_{1..s}$ are given by
-
- \[\mbox{
- \begin{tabular}{c@ {\hspace{2mm}}p{0.9\textwidth}}
- $\bullet$ & @{text "z\<^isub>i \<approx>ty z\<PRIME>\<^isub>i"} provided @{text "z\<^isub>i"} does not occur in @{text rhs} and
- is a recursive argument of @{text C},\smallskip\\
- $\bullet$ & @{text "z\<^isub>i = z\<PRIME>\<^isub>i"} provided @{text "z\<^isub>i"} does not occur in @{text rhs}
- and is a non-recursive argument of @{text C},\smallskip\\
- $\bullet$ & @{text "z\<^isub>i \<approx>bn\<^isub>i z\<PRIME>\<^isub>i"} provided @{text "z\<^isub>i"} occurs in @{text rhs}
- with the recursive call @{text "bn\<^isub>i x\<^isub>i"} and\smallskip\\
- $\bullet$ & @{text True} provided @{text "z\<^isub>i"} occurs in @{text rhs} but without a
- recursive call.
- \end{tabular}}
- \]\smallskip
-
- \noindent
- This completes the definition of alpha-equivalence. As a sanity check, we can show
- that the premises of empty binding clauses are a special case of the clauses for
- non-empty ones (we just have to unfold the definition of
- $\approx_{\,\textit{set}}^{\textit{R}, \textit{fa}}$ and take @{text "0"}
- for the existentially quantified permutation).
-
- Again let us take a look at a concrete example for these definitions. For
- the specification shown in \eqref{letrecs}
- we have three relations $\approx_{\textit{trm}}$, $\approx_{\textit{assn}}$ and
- $\approx_{\textit{bn}}$ with the following rules:
-
- \begin{equation}\label{rawalpha}\mbox{
- \begin{tabular}{@ {}c @ {}}
- \infer{@{text "Let as t \<approx>\<^bsub>trm\<^esub> Let as' t'"}}
- {@{term "alpha_lst_ex (bn as, t) alpha_trm fa_trm (bn as', t')"} &
- \hspace{5mm}@{text "as \<approx>\<^bsub>bn\<^esub> as'"}}\\
- \\
- \makebox[0mm]{\infer{@{text "Let_rec as t \<approx>\<^bsub>trm\<^esub> Let_rec as' t'"}}
- {@{term "alpha_lst_ex (bn as, ast) alpha_trm2 fa_trm2 (bn as', ast')"}}}\\
- \\
-
- \begin{tabular}{@ {}c @ {}}
- \infer{@{text "ANil \<approx>\<^bsub>assn\<^esub> ANil"}}{}\hspace{9mm}
- \infer{@{text "ACons a t as \<approx>\<^bsub>assn\<^esub> ACons a' t' as"}}
- {@{text "a = a'"} & \hspace{5mm}@{text "t \<approx>\<^bsub>trm\<^esub> t'"} & \hspace{5mm}@{text "as \<approx>\<^bsub>assn\<^esub> as'"}}
- \end{tabular}\\
- \\
-
- \begin{tabular}{@ {}c @ {}}
- \infer{@{text "ANil \<approx>\<^bsub>bn\<^esub> ANil"}}{}\hspace{9mm}
- \infer{@{text "ACons a t as \<approx>\<^bsub>bn\<^esub> ACons a' t' as"}}
- {@{text "t \<approx>\<^bsub>trm\<^esub> t'"} & \hspace{5mm}@{text "as \<approx>\<^bsub>bn\<^esub> as'"}}
- \end{tabular}
- \end{tabular}}
- \end{equation}\smallskip
-
- \noindent
- Notice the difference between $\approx_{\textit{assn}}$ and
- $\approx_{\textit{bn}}$: the latter only `tracks' alpha-equivalence of
- the components in an assignment that are \emph{not} bound. This is needed in the
- clause for @{text "Let"} (which has
- a non-recursive binder).
- The underlying reason is that the terms inside an assignment are not meant
- to be `under' the binder. Such a premise is \emph{not} needed in @{text "Let_rec"},
- because there all components of an assignment are `under' the binder.
- Note also that in case of more than one body (that is in the @{text "Let_rec"}-case above)
- we need to parametrise the relation $\approx_{\textit{list}}$ with a compound
- equivalence relation and a compound free-atom function. This is because the
- corresponding binding clause specifies a binder with two bodies, namely
- @{text "as"} and @{text "t"}.
-*}
-
-section {* Establishing the Reasoning Infrastructure *}
-
-text {*
- Having made all necessary definitions for raw terms, we can start with
- establishing the reasoning infrastructure for the alpha-equated types @{text
- "ty\<AL>"}$_{1..n}$, that is the types the user originally specified. We
- give in this section and the next the proofs we need for establishing this
- infrastructure. One point of our work is that we have completely
- automated these proofs in Isabelle/HOL.
-
- First we establish that the free-variable functions, the binding functions and the
- alpha-equi\-va\-lences are equivariant.
-
- \begin{lem}\mbox{}\\
- @{text "(i)"} The functions @{text "fa_ty"}$_{1..n}$, @{text "fa_bn"}$_{1..m}$ and
- @{text "bn"}$_{1..m}$ are equivariant.\\
- @{text "(ii)"} The relations @{text "\<approx>ty"}$_{1..n}$ and
- @{text "\<approx>bn"}$_{1..m}$ are equivariant.
- \end{lem}
-
- \begin{proof}
- The function package of Isabelle/HOL allows us to prove the first part by
- mutual induction over the definitions of the functions.\footnote{We have
- that the free-atom functions are terminating. From this the function
- package derives an induction principle~\cite{Krauss09}.} The second is by a
- straightforward induction over the rules of @{text "\<approx>ty"}$_{1..n}$ and
- @{text "\<approx>bn"}$_{1..m}$ using the first part.
- \end{proof}
-
- \noindent
- Next we establish that the alpha-equivalence relations defined in the
- previous section are indeed equivalence relations.
-
- \begin{lem}\label{equiv}
- The relations @{text "\<approx>ty"}$_{1..n}$ and @{text "\<approx>bn"}$_{1..m}$ are
- equivalence relations.
- \end{lem}
-
- \begin{proof}
- The proofs are by induction. The non-trivial
- cases involve premises built up by $\approx_{\textit{set}}$,
- $\approx_{\textit{set+}}$ and $\approx_{\textit{list}}$. They
- can be dealt with as in Lemma~\ref{alphaeq}. However, the transitivity
- case needs in addition the fact that the relations are equivariant.
- \end{proof}
-
- \noindent
- We can feed the last lemma into our quotient package and obtain new types
- @{text "ty"}$^\alpha_{1..n}$ representing alpha-equated terms of types
- @{text "ty"}$_{1..n}$. We also obtain definitions for the term-constructors
- @{text "C"}$^\alpha_{1..k}$ from the raw term-constructors @{text
- "C"}$_{1..k}$, and similar definitions for the free-atom functions @{text
- "fa_ty"}$^\alpha_{1..n}$ and @{text "fa_bn"}$^\alpha_{1..m}$ as well as the
- binding functions @{text "bn"}$^\alpha_{1..m}$. However, these definitions
- are not really useful to the user, since they are given in terms of the
- isomorphisms we obtained by creating new types in Isabelle/HOL (recall the
- picture shown in the Introduction).
-
- The first useful property for the user is the fact that distinct
- term-constructors are not equal, that is the property
-
- \begin{equation}\label{distinctalpha}
- \mbox{@{text "C"}$^\alpha$~@{text "x\<^isub>1 \<dots> x\<^isub>r"}~@{text "\<noteq>"}~%
- @{text "D"}$^\alpha$~@{text "y\<^isub>1 \<dots> y\<^isub>s"}}
- \end{equation}\smallskip
-
- \noindent
- whenever @{text "C"}$^\alpha$~@{text "\<noteq>"}~@{text "D"}$^\alpha$.
- In order to derive this property, we use the definition of alpha-equivalence
- and establish that
-
- \begin{equation}\label{distinctraw}
- \mbox{@{text "C x\<^isub>1 \<dots> x\<^isub>r"}\;$\not\approx$@{text ty}\;@{text "D y\<^isub>1 \<dots> y\<^isub>s"}}
- \end{equation}\smallskip
-
- \noindent
- holds for the corresponding raw term-constructors.
- In order to deduce \eqref{distinctalpha} from \eqref{distinctraw}, our quotient
- package needs to know that the raw term-constructors @{text "C"} and @{text "D"}
- are \emph{respectful} w.r.t.~the alpha-equivalence relations (see \cite{Homeier05}).
- Given, for example, @{text "C"} is of type @{text "ty"} with argument types
- @{text "ty"}$_{1..r}$, respectfulness amounts to showing that
-
- \[\mbox{
- @{text "C x\<^isub>1 \<dots> x\<^isub>r \<approx>ty C x\<PRIME>\<^isub>1 \<dots> x\<PRIME>\<^isub>r"}
- }\]\smallskip
-
- \noindent
- holds under the assumptions \mbox{@{text
- "x\<^isub>i \<approx>ty\<^isub>i x\<PRIME>\<^isub>i"}} whenever @{text "x\<^isub>i"}
- and @{text "x\<PRIME>\<^isub>i"} are recursive arguments of @{text C}, and
- @{text "x\<^isub>i = x\<PRIME>\<^isub>i"} whenever they are non-recursive arguments
- (similarly for @{text "D"}). For this we have to show
- by induction over the definitions of alpha-equivalences the following
- auxiliary implications
-
- \begin{equation}\label{fnresp}\mbox{
- \begin{tabular}{lll}
- @{text "x \<approx>ty\<^isub>i x'"} & implies & @{text "fa_ty\<^isub>i x = fa_ty\<^isub>i x'"}\\
- @{text "x \<approx>ty\<^isub>l x'"} & implies & @{text "fa_bn\<^isub>j x = fa_bn\<^isub>j x'"}\\
- @{text "x \<approx>ty\<^isub>l x'"} & implies & @{text "bn\<^isub>j x = bn\<^isub>j x'"}\\
- @{text "x \<approx>ty\<^isub>l x'"} & implies & @{text "x \<approx>bn\<^isub>j x'"}\\
- \end{tabular}
- }\end{equation}\smallskip
-
- \noindent
- whereby @{text "ty\<^isub>l"} is the type over which @{text "bn\<^isub>j"}
- is defined. Whereas the first, second and last implication are true by
- how we stated our definitions, the third \emph{only} holds because of our
- restriction imposed on the form of the binding functions---namely \emph{not}
- to return any bound atoms. In Ott, in contrast, the user may define @{text
- "bn"}$_{1..m}$ so that they return bound atoms and in this case the third
- implication is \emph{not} true. A result is that in general the lifting of the
- corresponding binding functions in Ott to alpha-equated terms is impossible.
- Having established respectfulness for the raw term-constructors, the
- quotient package is able to automatically deduce \eqref{distinctalpha} from
- \eqref{distinctraw}.
-
- Next we can lift the permutation operations defined in \eqref{ceqvt}. In
- order to make this lifting to go through, we have to show that the
- permutation operations are respectful. This amounts to showing that the
- alpha-equivalence relations are equivariant, which
- we already established in Lemma~\ref{equiv}. As a result we can add the
- equations
-
- \begin{equation}\label{calphaeqvt}
- @{text "\<pi> \<bullet> (C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r) = C\<^sup>\<alpha> (\<pi> \<bullet> x\<^isub>1) \<dots> (\<pi> \<bullet> x\<^isub>r)"}
- \end{equation}\smallskip
-
- \noindent
- to our infrastructure. In a similar fashion we can lift the defining equations
- of the free-atom functions @{text "fa_ty\<AL>"}$_{1..n}$ and
- @{text "fa_bn\<AL>"}$_{1..m}$ as well as of the binding functions @{text
- "bn\<AL>"}$_{1..m}$ and size functions @{text "size_ty\<AL>"}$_{1..n}$.
- The latter are defined automatically for the raw types @{text "ty"}$_{1..n}$
- by the datatype package of Isabelle/HOL.
-
- We also need to lift the properties that characterise when two raw terms of the form
-
- \[
- \mbox{@{text "C x\<^isub>1 \<dots> x\<^isub>r \<approx>ty C x\<PRIME>\<^isub>1 \<dots> x\<PRIME>\<^isub>r"}}
- \]\smallskip
-
- \noindent
- are alpha-equivalent. This gives us conditions when the corresponding
- alpha-equated terms are \emph{equal}, namely
-
- \[
- @{text "C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r = C\<^sup>\<alpha> x\<PRIME>\<^isub>1 \<dots> x\<PRIME>\<^isub>r"}.
- \]\smallskip
-
- \noindent
- We call these conditions \emph{quasi-injectivity}. They correspond to the
- premises in our alpha-equiva\-lence relations, except that the
- relations @{text "\<approx>ty"}$_{1..n}$ are all replaced by equality (and similarly
- the free-atom and binding functions are replaced by their lifted
- counterparts). Recall the alpha-equivalence rules for @{text "Let"} and
- @{text "Let_rec"} shown in \eqref{rawalpha}. For @{text "Let\<^sup>\<alpha>"} and
- @{text "Let_rec\<^sup>\<alpha>"} we have
-
- \begin{equation}\label{alphalift}\mbox{
- \begin{tabular}{@ {}c @ {}}
- \infer{@{text "Let\<^sup>\<alpha> as t = Let\<^sup>\<alpha> as' t'"}}
- {@{term "alpha_lst_ex (bn_al as, t) equal fa_trm_al (bn as', t')"} &
- \hspace{5mm}@{text "as \<approx>\<AL>\<^bsub>bn\<^esub> as'"}}\\
- \\
- \makebox[0mm]{\infer{@{text "Let_rec\<^sup>\<alpha> as t = Let_rec\<^sup>\<alpha> as' t'"}}
- {@{term "alpha_lst_ex (bn_al as, ast) equ2 fa_trm2_al (bn_al as', ast')"}}}\\
- \end{tabular}}
- \end{equation}\smallskip
-
- We can also add to our infrastructure cases lemmas and a (mutual)
- induction principle for the types @{text "ty\<AL>"}$_{1..n}$. The cases
- lemmas allow the user to deduce a property @{text "P"} by exhaustively
- analysing how an element of a type, say @{text "ty\<AL>"}$_i$, can be
- constructed (that means one case for each of the term-constructors in @{text
- "ty\<AL>"}$_i\,$). The lifted cases lemma for a type @{text
- "ty\<AL>"}$_i\,$ looks as follows
-
- \begin{equation}\label{cases}
- \infer{P}
- {\begin{array}{l}
- @{text "\<forall>x\<^isub>1\<dots>x\<^isub>k. y = C\<AL>\<^isub>1 x\<^isub>1 \<dots> x\<^isub>k \<Rightarrow> P"}\\
- \hspace{5mm}\vdots\\
- @{text "\<forall>x\<^isub>1\<dots>x\<^isub>l. y = C\<AL>\<^isub>m x\<^isub>1 \<dots> x\<^isub>l \<Rightarrow> P"}\\
- \end{array}}
- \end{equation}\smallskip
-
- \noindent
- where @{text "y"} is a variable of type @{text "ty\<AL>"}$_i$ and @{text "P"} is the
- property that is established by the case analysis. Similarly, we have a (mutual)
- induction principle for the types @{text "ty\<AL>"}$_{1..n}$, which is of the
- form
-
- \begin{equation}\label{induct}
- \infer{@{text "P\<^isub>1 y\<^isub>1 \<and> \<dots> \<and> P\<^isub>n y\<^isub>n "}}
- {\begin{array}{l}
- @{text "\<forall>x\<^isub>1\<dots>x\<^isub>k. P\<^isub>i x\<^isub>i \<and> \<dots> \<and> P\<^isub>j x\<^isub>j \<Rightarrow> P (C\<AL>\<^isub>1 x\<^isub>1 \<dots> x\<^isub>k)"}\\
- \hspace{5mm}\vdots\\
- @{text "\<forall>x\<^isub>1\<dots>x\<^isub>l. P\<^isub>r x\<^isub>r \<and> \<dots> \<and> P\<^isub>s x\<^isub>s \<Rightarrow> P (C\<AL>\<^isub>m x\<^isub>1 \<dots> x\<^isub>l)"}\\
- \end{array}}
- \end{equation}\smallskip
-
- \noindent
- whereby the @{text P}$_{1..n}$ are the properties established by the
- induction, and the @{text y}$_{1..n}$ are of type @{text
- "ty\<AL>"}$_{1..n}$. Note that for the term constructor @{text
- "C"}$^\alpha_1$ the induction principle has a hypothesis of the form
-
- \[
- \mbox{@{text "\<forall>x\<^isub>1\<dots>x\<^isub>k. P\<^isub>i x\<^isub>i \<and> \<dots> \<and> P\<^isub>j x\<^isub>j \<Rightarrow> P (C\<AL>\<^sub>1 x\<^isub>1 \<dots> x\<^isub>k)"}}
- \]\smallskip
-
- \noindent
- in which the @{text "x"}$_{i..j}$ @{text "\<subseteq>"} @{text "x"}$_{1..k}$ are the
- recursive arguments of this term constructor (similarly for the other
- term-constructors).
-
- Recall the lambda-calculus with @{text "Let"}-patterns shown in
- \eqref{letpat}. The cases lemmas and the induction principle shown in
- \eqref{cases} and \eqref{induct} boil down in that example to the following three inference
- rules:
-
- \begin{equation}\label{inductex}\mbox{
- \begin{tabular}{c}
- \multicolumn{1}{@ {\hspace{-5mm}}l}{cases lemmas:}\smallskip\\
- \infer{@{text "P\<^bsub>trm\<^esub>"}}
- {\begin{array}{@ {}l@ {}}
- @{text "\<forall>x. y = Var\<^sup>\<alpha> x \<Rightarrow> P\<^bsub>trm\<^esub>"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2. y = App\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub>"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2. y = Lam\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub>"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2 x\<^isub>3. y = Let_pat\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 x\<^isub>3 \<Rightarrow> P\<^bsub>trm\<^esub>"}
- \end{array}}\hspace{10mm}
-
- \infer{@{text "P\<^bsub>pat\<^esub>"}}
- {\begin{array}{@ {}l@ {}}
- @{text "\<forall>x. y = PVar\<^sup>\<alpha> x \<Rightarrow> P\<^bsub>pat\<^esub>"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2. y = PTup\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 \<Rightarrow> P\<^bsub>pat\<^esub>"}
- \end{array}}\medskip\\
-
- \multicolumn{1}{@ {\hspace{-5mm}}l}{induction principle:}\smallskip\\
-
- \infer{@{text "P\<^bsub>trm\<^esub> y\<^isub>1 \<and> P\<^bsub>pat\<^esub> y\<^isub>2"}}
- {\begin{array}{@ {}l@ {}}
- @{text "\<forall>x. P\<^bsub>trm\<^esub> (Var\<^sup>\<alpha> x)"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2. P\<^bsub>trm\<^esub> x\<^isub>1 \<and> P\<^bsub>trm\<^esub> x\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub> (App\<^sup>\<alpha> x\<^isub>1 x\<^isub>2)"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2. P\<^bsub>trm\<^esub> x\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub> (Lam\<^sup>\<alpha> x\<^isub>1 x\<^isub>2)"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2 x\<^isub>3. P\<^bsub>pat\<^esub> x\<^isub>1 \<and> P\<^bsub>trm\<^esub> x\<^isub>2 \<and> P\<^bsub>trm\<^esub> x\<^isub>3 \<Rightarrow> P\<^bsub>trm\<^esub> (Let_pat\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 x\<^isub>3)"}\\
- @{text "\<forall>x. P\<^bsub>pat\<^esub> (PVar\<^sup>\<alpha> x)"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2. P\<^bsub>pat\<^esub> x\<^isub>1 \<and> P\<^bsub>pat\<^esub> x\<^isub>2 \<Rightarrow> P\<^bsub>pat\<^esub> (PTup\<^sup>\<alpha> x\<^isub>1 x\<^isub>2)"}
- \end{array}}
- \end{tabular}}
- \end{equation}\smallskip
-
- By working now completely on the alpha-equated level, we
- can first show using \eqref{calphaeqvt} and Property~\ref{swapfreshfresh} that the support of each term
- constructor is included in the support of its arguments,
- namely
-
- \[
- @{text "(supp x\<^isub>1 \<union> \<dots> \<union> supp x\<^isub>r) supports (C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r)"}
- \]\smallskip
-
- \noindent
- This allows us to prove using the induction principle for @{text "ty\<AL>"}$_{1..n}$
- that every element of type @{text "ty\<AL>"}$_{1..n}$ is finitely supported
- (using Proposition~\ref{supportsprop}{\it (i)}).
- Similarly, we can establish by induction that the free-atom functions and binding
- functions are equivariant, namely
-
- \[\mbox{
- \begin{tabular}{rcl}
- @{text "\<pi> \<bullet> (fa_ty\<AL>\<^isub>i x)"} & $=$ & @{text "fa_ty\<AL>\<^isub>i (\<pi> \<bullet> x)"}\\
- @{text "\<pi> \<bullet> (fa_bn\<AL>\<^isub>j x)"} & $=$ & @{text "fa_bn\<AL>\<^isub>j (\<pi> \<bullet> x)"}\\
- @{text "\<pi> \<bullet> (bn\<AL>\<^isub>j x)"} & $=$ & @{text "bn\<AL>\<^isub>j (\<pi> \<bullet> x)"}\\
- \end{tabular}}
- \]\smallskip
-
-
- \noindent
- Lastly, we can show that the support of elements in @{text
- "ty\<AL>"}$_{1..n}$ is the same as the free-atom functions @{text
- "fa_ty\<AL>"}$_{1..n}$. This fact is important in the nominal setting where
- the general theory is formulated in terms of support and freshness, but also
- provides evidence that our notions of free-atoms and alpha-equivalence
- `match up' correctly.
-
- \begin{thm}\label{suppfa}
- For @{text "x"}$_{1..n}$ with type @{text "ty\<AL>"}$_{1..n}$, we have
- @{text "supp x\<^isub>i = fa_ty\<AL>\<^isub>i x\<^isub>i"}.
- \end{thm}
-
- \begin{proof}
- The proof is by induction on @{text "x"}$_{1..n}$. In each case
- we unfold the definition of @{text "supp"}, move the swapping inside the
- term-constructors and then use the quasi-injectivity lemmas in order to complete the
- proof. For the abstraction cases we use then the facts derived in Theorem~\ref{suppabs},
- for which we have to know that every body of an abstraction is finitely supported.
- This, we have proved earlier.
- \end{proof}
-
- \noindent
- Consequently, we can replace the free-atom functions by @{text "supp"} in
- our quasi-injection lemmas. In the examples shown in \eqref{alphalift}, for instance,
- we obtain for @{text "Let\<^sup>\<alpha>"} and @{text "Let_rec\<^sup>\<alpha>"}
-
- \[\mbox{
- \begin{tabular}{@ {}c @ {}}
- \infer{@{text "Let\<^sup>\<alpha> as t = Let\<^sup>\<alpha> as' t'"}}
- {@{term "alpha_lst_ex (bn_al as, t) equal supp (bn_al as', t')"} &
- \hspace{5mm}@{text "as \<approx>\<AL>\<^bsub>bn\<^esub> as'"}}\\
- \\
- \makebox[0mm]{\infer{@{text "Let_rec\<^sup>\<alpha> as t = Let_rec\<^sup>\<alpha> as' t'"}}
- {@{term "alpha_lst_ex (bn_al as, ast) equ2 supp2 (bn_al as', ast')"}}}\\
- \end{tabular}}
- \]\smallskip
-
- \noindent
- Taking into account that the compound equivalence relation @{term
- "equ2"} and the compound free-atom function @{term "supp2"} are by
- definition equal to @{term "equal"} and @{term "supp"}, respectively, the
- above rules simplify further to
-
- \[\mbox{
- \begin{tabular}{@ {}c @ {}}
- \infer{@{text "Let\<^sup>\<alpha> as t = Let\<^sup>\<alpha> as' t'"}}
- {@{term "Abs_lst (bn_al as) t = Abs_lst (bn_al as') t'"} &
- \hspace{5mm}@{text "as \<approx>\<AL>\<^bsub>bn\<^esub> as'"}}\\
- \\
- \makebox[0mm]{\infer{@{text "Let_rec\<^sup>\<alpha> as t = Let_rec\<^sup>\<alpha> as' t'"}}
- {@{term "Abs_lst (bn_al as) ast = Abs_lst (bn_al as') ast'"}}}\\
- \end{tabular}}
- \]\smallskip
-
- \noindent
- which means we can characterise equality between term-constructors (on the
- alpha-equated level) in terms of equality between the abstractions defined
- in Section~\ref{sec:binders}. From this we can deduce the support for @{text
- "Let\<^sup>\<alpha>"} and @{text "Let_rec\<^sup>\<alpha>"}, namely
-
-
- \[\mbox{
- \begin{tabular}{l@ {\hspace{2mm}}l@ {\hspace{2mm}}l}
- @{text "supp (Let\<^sup>\<alpha> as t)"} & @{text "="} & @{text "(supp t - set (bn\<^sup>\<alpha> as)) \<union> fa\<AL>\<^bsub>bn\<^esub> as"}\\
- @{text "supp (Let_rec\<^sup>\<alpha> as t)"} & @{text "="} & @{text "(supp t \<union> supp as) - set (bn\<^sup>\<alpha> as)"}\\
- \end{tabular}}
- \]\smallskip
-
- \noindent
- using the support of abstractions derived in Theorem~\ref{suppabs}.
-
- To sum up this section, we have established a reasoning infrastructure for the
- types @{text "ty\<AL>"}$_{1..n}$ by first lifting definitions from the
- `raw' level to the quotient level and then by proving facts about
- these lifted definitions. All necessary proofs are generated automatically
- by custom ML-code.
-*}
-
-
-section {* Strong Induction Principles *}
-
-text {*
- In the previous section we derived induction principles for alpha-equated
- terms (see \eqref{induct} for the general form and \eqref{inductex} for an
- example). This was done by lifting the corresponding inductions principles
- for `raw' terms. We already employed these induction principles for
- deriving several facts about alpha-equated terms, including the property that
- the free-atom functions and the notion of support coincide. Still, we
- call these induction principles \emph{weak}, because for a term-constructor,
- say \mbox{@{text "C\<^sup>\<alpha> x\<^isub>1\<dots>x\<^isub>r"}}, the induction
- hypothesis requires us to establish (under some assumptions) a property
- @{text "P (C\<^sup>\<alpha> x\<^isub>1\<dots>x\<^isub>r)"} for \emph{all} @{text
- "x"}$_{1..r}$. The problem with this is that in the presence of binders we cannot make
- any assumptions about the atoms that are bound---for example assuming the variable convention.
- One obvious way around this
- problem is to rename bound atoms. Unfortunately, this leads to very clunky proofs
- and makes formalisations grievous experiences (especially in the context of
- multiple bound atoms).
-
- For the older versions of Nominal Isabelle we described in \cite{Urban08} a
- method for automatically strengthening weak induction principles. These
- stronger induction principles allow the user to make additional assumptions
- about bound atoms. The advantage of these assumptions is that they make in
- most cases any renaming of bound atoms unnecessary. To explain how the
- strengthening works, we use as running example the lambda-calculus with
- @{text "Let"}-patterns shown in \eqref{letpat}. Its weak induction principle
- is given in \eqref{inductex}. The stronger induction principle is as
- follows:
-
- \begin{equation}\label{stronginduct}
- \mbox{
- \begin{tabular}{@ {}c@ {}}
- \infer{@{text "P\<^bsub>trm\<^esub> c y\<^isub>1 \<and> P\<^bsub>pat\<^esub> c y\<^isub>2"}}
- {\begin{array}{l}
- @{text "\<forall>x c. P\<^bsub>trm\<^esub> c (Var\<^sup>\<alpha> x)"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2 c. (\<forall>d. P\<^bsub>trm\<^esub> d x\<^isub>1) \<and> (\<forall>d. P\<^bsub>trm\<^esub> d x\<^isub>2) \<Rightarrow> P\<^bsub>trm\<^esub> c (App\<^sup>\<alpha> x\<^isub>1 x\<^isub>2)"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2 c. atom x\<^isub>1 # c \<and> (\<forall>d. P\<^bsub>trm\<^esub> d x\<^isub>2) \<Rightarrow> P\<^bsub>trm\<^esub> c (Lam\<^sup>\<alpha> x\<^isub>1 x\<^isub>2)"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2 x\<^isub>3 c. (set (bn\<^sup>\<alpha> x\<^isub>1)) #\<^sup>* c \<and>"}\\
- \hspace{10mm}@{text "(\<forall>d. P\<^bsub>pat\<^esub> d x\<^isub>1) \<and> (\<forall>d. P\<^bsub>trm\<^esub> d x\<^isub>2) \<and> (\<forall>d. P\<^bsub>trm\<^esub> d x\<^isub>3) \<Rightarrow> P\<^bsub>trm\<^esub> c (Let_pat\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 x\<^isub>3)"}\\
- @{text "\<forall>x c. P\<^bsub>pat\<^esub> c (PVar\<^sup>\<alpha> x)"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2 c. (\<forall>d. P\<^bsub>pat\<^esub> d x\<^isub>1) \<and> (\<forall>d. P\<^bsub>pat\<^esub> d x\<^isub>2) \<Rightarrow> P\<^bsub>pat\<^esub> c (PTup\<^sup>\<alpha> x\<^isub>1 x\<^isub>2)"}
- \end{array}}
- \end{tabular}}
- \end{equation}\smallskip
-
-
- \noindent
- Notice that instead of establishing two properties of the form @{text "
- P\<^bsub>trm\<^esub> y\<^isub>1 \<and> P\<^bsub>pat\<^esub> y\<^isub>2"}, as the
- weak one does, the stronger induction principle establishes the properties
- of the form @{text " P\<^bsub>trm\<^esub> c y\<^isub>1 \<and>
- P\<^bsub>pat\<^esub> c y\<^isub>2"} in which the additional parameter @{text
- c} is assumed to be of finite support. The purpose of @{text "c"} is to
- `control' which freshness assumptions the binders should satisfy in the
- @{text "Lam\<^sup>\<alpha>"} and @{text "Let_pat\<^sup>\<alpha>"} cases: for @{text
- "Lam\<^sup>\<alpha>"} we can assume the bound atom @{text "x\<^isub>1"} is fresh
- for @{text "c"} (third line); for @{text "Let_pat\<^sup>\<alpha>"} we can assume
- all bound atoms from an assignment are fresh for @{text "c"} (fourth
- line). In order to see how an instantiation for @{text "c"} in the
- conclusion `controls' the premises, one has to take into account that
- Isabelle/HOL is a typed logic. That means if @{text "c"} is instantiated
- with, for example, a pair, then this type-constraint will be propagated to
- the premises. The main point is that if @{text "c"} is instantiated
- appropriately, then the user can mimic the usual convenient `pencil-and-paper'
- reasoning employing the variable convention about bound and free variables
- being distinct \cite{Urban08}.
-
- In what follows we will show that the weak induction principle in
- \eqref{inductex} implies the strong one \eqref{stronginduct}. This fact was established for
- single binders in \cite{Urban08} by some quite involved, nevertheless
- automated, induction proof. In this paper we simplify the proof by
- leveraging the automated proving tools from the function package of
- Isabelle/HOL \cite{Krauss09}. The reasoning principle behind these tools
- is well-founded induction. To use them in our setting, we have to discharge
- two proof obligations: one is that we have well-founded measures (one for
- each type @{text "ty"}$^\alpha_{1..n}$) that decrease in every induction
- step and the other is that we have covered all cases in the induction
- principle. Once these two proof obligations are discharged, the reasoning
- infrastructure of the function package will automatically derive the
- stronger induction principle. This way of establishing the stronger induction
- principle is considerably simpler than the earlier work presented in \cite{Urban08}.
-
- As measures we can use the size functions @{text "size_ty"}$^\alpha_{1..n}$,
- which we lifted in the previous section and which are all well-founded. It
- is straightforward to establish that the sizes decrease in every
- induction step. What is left to show is that we covered all cases.
- To do so, we have to derive stronger cases lemmas, which look in our
- running example as follows:
-
- \[\mbox{
- \begin{tabular}{@ {}c@ {\hspace{4mm}}c@ {}}
- \infer{@{text "P\<^bsub>trm\<^esub>"}}
- {\begin{array}{@ {}l@ {}}
- @{text "\<forall>x. y = Var\<^sup>\<alpha> x \<Rightarrow> P\<^bsub>trm\<^esub>"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2. y = App\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub>"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2. atom x\<^isub>1 # c \<and> y = Lam\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub>"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2 x\<^isub>3. set (bn\<^sup>\<alpha> x\<^isub>1) #\<^sup>* c \<and> y = Let_pat\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 x\<^isub>3 \<Rightarrow> P\<^bsub>trm\<^esub>"}
- \end{array}} &
-
- \infer{@{text "P\<^bsub>pat\<^esub>"}}
- {\begin{array}{@ {}l@ {}}
- @{text "\<forall>x. y = PVar\<^sup>\<alpha> x \<Rightarrow> P\<^bsub>pat\<^esub>"}\\
- @{text "\<forall>x\<^isub>1 x\<^isub>2. y = PTup\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 \<Rightarrow> P\<^bsub>pat\<^esub>"}
- \end{array}}
- \end{tabular}}
- \]\smallskip
-
- \noindent
- They are stronger in the sense that they allow us to assume in the @{text
- "Lam\<^sup>\<alpha>"} and @{text "Let_pat\<^sup>\<alpha>"} cases that the bound atoms
- avoid, or are fresh for, a context @{text "c"} (which is assumed to be finitely supported).
-
- These stronger cases lemmas can be derived from the `weak' cases lemmas
- given in \eqref{inductex}. This is trivial in case of patterns (the one on
- the right-hand side) since the weak and strong cases lemma coincide (there
- is no binding in patterns). Interesting are only the cases for @{text
- "Lam\<^sup>\<alpha>"} and @{text "Let_pat\<^sup>\<alpha>"}, where we have some binders and
- therefore have an additional assumption about avoiding @{text "c"}. Let us
- first establish the case for @{text "Lam\<^sup>\<alpha>"}. By the weak cases lemma
- \eqref{inductex} we can assume that
-
- \begin{equation}\label{assm}
- @{text "y = Lam\<^sup>\<alpha> x\<^isub>1 x\<^isub>2"}
- \end{equation}\smallskip
-
- \noindent
- holds, and need to establish @{text "P\<^bsub>trm\<^esub>"}. The stronger cases lemma has the
- corresponding implication
-
- \begin{equation}\label{imp}
- @{text "\<forall>x\<^isub>1 x\<^isub>2. atom x\<^isub>1 # c \<and> y = Lam\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub>"}
- \end{equation}\smallskip
-
- \noindent
- which we must use in order to infer @{text "P\<^bsub>trm\<^esub>"}. Clearly, we cannot
- use this implication directly, because we have no information whether or not @{text
- "x\<^isub>1"} is fresh for @{text "c"}. However, we can use Properties
- \ref{supppermeq} and \ref{avoiding} to rename @{text "x\<^isub>1"}. We know
- by Theorem~\ref{suppfa} that @{text "{atom x\<^isub>1} #\<^sup>* Lam\<^sup>\<alpha>
- x\<^isub>1 x\<^isub>2"} (since its support is @{text "supp x\<^isub>2 -
- {atom x\<^isub>1}"}). Property \ref{avoiding} provides us then with a
- permutation @{text "\<pi>"}, such that @{text "{atom (\<pi> \<bullet> x\<^isub>1)} #\<^sup>*
- c"} and \mbox{@{text "supp (Lam\<^sup>\<alpha> x\<^isub>1 x\<^isub>2) #\<^sup>* \<pi>"}} hold.
- By using Property \ref{supppermeq}, we can infer from the latter that
-
- \[
- @{text "Lam\<^sup>\<alpha> (\<pi> \<bullet> x\<^isub>1) (\<pi> \<bullet> x\<^isub>2) = Lam\<^sup>\<alpha> x\<^isub>1 x\<^isub>2"}
- \]\smallskip
-
- \noindent
- holds. We can use this equation in the assumption \eqref{assm}, and hence
- use the implication \eqref{imp} with the renamed @{text "\<pi> \<bullet> x\<^isub>1"}
- and @{text "\<pi> \<bullet> x\<^isub>2"} for concluding this case.
-
- The @{text "Let_pat\<^sup>\<alpha>"}-case involving a deep binder is slightly more complicated.
- We have the assumption
-
- \begin{equation}\label{assmtwo}
- @{text "y = Let_pat\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 x\<^isub>3"}
- \end{equation}\smallskip
-
- \noindent
- and the implication from the stronger cases lemma
-
- \begin{equation}\label{impletpat}
- @{text "\<forall>x\<^isub>1 x\<^isub>2 x\<^isub>3. set (bn\<^sup>\<alpha> x\<^isub>1) #\<^sup>* c \<and> y = Let_pat\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 x\<^isub>3 \<Rightarrow> P\<^bsub>trm\<^esub>"}
- \end{equation}\smallskip
-
- \noindent
- The reason that this case is more complicated is that we cannot directly apply Property
- \ref{avoiding} for obtaining a renaming permutation. Property \ref{avoiding} requires
- that the binders are fresh for the term in which we want to perform the renaming. But
- this is not true in terms such as (using an informal notation)
-
- \[
- @{text "Let (x, y) := (x, y) in (x, y)"}
- \]\smallskip
-
- \noindent
- where @{text x} and @{text y} are bound in the term, but are also free
- in the right-hand side of the assignment. We can, however, obtain such a renaming permutation, say
- @{text "\<pi>"}, for the abstraction @{term "Abs_lst (bn_al x\<^isub>1)
- x\<^isub>3"}. As a result we have \mbox{@{term "set (bn_al (\<pi> \<bullet> x\<^isub>1))
- \<sharp>* c"}} and @{term "Abs_lst (bn_al (\<pi> \<bullet> x\<^isub>1)) (\<pi> \<bullet> x\<^isub>3) =
- Abs_lst (bn_al x\<^isub>1) x\<^isub>3"} (remember @{text "set"} and @{text
- "bn\<^sup>\<alpha>"} are equivariant). Now the quasi-injective property for @{text
- "Let_pat\<^sup>\<alpha>"} states that
-
- \[
- \infer{@{text "Let_pat\<^sup>\<alpha> p t\<^isub>1 t\<^isub>2 = Let_pat\<^sup>\<alpha> p\<PRIME> t\<PRIME>\<^isub>1 t\<PRIME>\<^isub>2"}}
- {@{text "[bn\<^sup>\<alpha> p]\<^bsub>list\<^esub>. t\<^isub>2 = [bn\<^sup>\<alpha> p']\<^bsub>list\<^esub>. t\<PRIME>\<^isub>2"}\;\; &
- @{text "p \<approx>\<AL>\<^bsub>bn\<^esub> p\<PRIME>"}\;\; & @{text "t\<^isub>1 = t\<PRIME>\<^isub>1"}}
- \]\smallskip
-
- \noindent
- Since all atoms in a pattern are bound by @{text "Let_pat\<^sup>\<alpha>"}, we can infer
- that @{text "(\<pi> \<bullet> x\<^isub>1) \<approx>\<AL>\<^bsub>bn\<^esub> x\<^isub>1"} holds for every @{text "\<pi>"}. Therefore we have that
-
- \[
- @{text "Let_pat\<^sup>\<alpha> (\<pi> \<bullet> x\<^isub>1) x\<^isub>2 (\<pi> \<bullet> x\<^isub>3) = Let_pat\<^sup>\<alpha> x\<^isub>1 x\<^isub>2 x\<^isub>3"}
- \]\smallskip
-
- \noindent
- Taking the left-hand side in the assumption shown in \eqref{assmtwo}, we can use
- the implication \eqref{impletpat} from the stronger cases lemma to infer @{text "P\<^bsub>trm\<^esub>"}, as needed.
-
- The remaining difficulty is when a deep binder contains some atoms that are
- bound and some that are free. An example is @{text "Let\<^sup>\<alpha>"} in
- \eqref{letrecs}. In such cases @{text "(\<pi> \<bullet> x\<^isub>1)
- \<approx>\<AL>\<^bsub>bn\<^esub> x\<^isub>1"} does not hold in general. The idea however is
- that @{text "\<pi>"} only renames atoms that become bound. In this way @{text "\<pi>"}
- does not affect @{text "\<approx>\<AL>\<^bsub>bn\<^esub>"} (which only tracks alpha-equivalence of terms that are not
- under the binder). However, the problem is that the
- permutation operation @{text "\<pi> \<bullet> x\<^isub>1"} applies to all atoms in @{text "x\<^isub>1"}. To avoid this
- we introduce an auxiliary permutation operations, written @{text "_
- \<bullet>\<^bsub>bn\<^esub> _"}, for deep binders that only permutes bound atoms (or
- more precisely the atoms specified by the @{text "bn"}-functions) and leaves
- the other atoms unchanged. Like the functions @{text "fa_bn"}$_{1..m}$, we
- can define these permutation operations over raw terms analysing how the functions @{text
- "bn"}$_{1..m}$ are defined. Assuming the user specified a clause
-
- \[
- @{text "bn (C x\<^isub>1 \<dots> x\<^isub>r) = rhs"}
- \]\smallskip
-
- \noindent
- we define @{text "\<pi> \<bullet>\<^bsub>bn\<^esub> (C x\<^isub>1 \<dots> x\<^isub>r) \<equiv> C y\<^isub>1 \<dots> y\<^isub>r"} with @{text "y\<^isub>i"} determined as follows:
-
- \[\mbox{
- \begin{tabular}{c@ {\hspace{2mm}}p{0.9\textwidth}}
- $\bullet$ & @{text "y\<^isub>i \<equiv> x\<^isub>i"} provided @{text "x\<^isub>i"} does not occur in @{text "rhs"}\\
- $\bullet$ & @{text "y\<^isub>i \<equiv> \<pi> \<bullet>\<^bsub>bn\<^esub> x\<^isub>i"} provided @{text "bn x\<^isub>i"} is in @{text "rhs"}\\
- $\bullet$ & @{text "y\<^isub>i \<equiv> \<pi> \<bullet> x\<^isub>i"} otherwise
- \end{tabular}}
- \]\smallskip
-
- \noindent
- Using again the quotient package we can lift the auxiliary permutation operations
- @{text "_ \<bullet>\<^bsub>bn\<^esub> _"}
- to alpha-equated terms. Moreover we can prove the following two properties:
-
- \begin{lem}\label{permutebn}
- Given a binding function @{text "bn\<^sup>\<alpha>"} and auxiliary equivalence @{text "\<approx>\<AL>\<^bsub>bn\<^esub>"}
- then for all @{text "\<pi>"}\smallskip\\
- {\it (i)} @{text "\<pi> \<bullet> (bn\<^sup>\<alpha> x) = bn\<^sup>\<alpha> (\<pi> \<bullet>\<AL>\<^bsub>bn\<^esub> x)"} and\\
- {\it (ii)} @{text "(\<pi> \<bullet>\<AL>\<^bsub>bn\<^esub> x) \<approx>\<AL>\<^bsub>bn\<^esub> x"}.
- \end{lem}
-
- \begin{proof}
- By induction on @{text x}. The properties follow by unfolding of the
- definitions.
- \end{proof}
-
- \noindent
- The first property states that a permutation applied to a binding function
- is equivalent to first permuting the binders and then calculating the bound
- atoms. The second states that @{text "_ \<bullet>\<AL>\<^bsub>bn\<^esub> _"} preserves
- @{text "\<approx>\<AL>\<^bsub>bn\<^esub>"}. The main point of the auxiliary
- permutation functions is that they allow us to rename just the bound atoms in a
- term, without changing anything else.
-
- Having the auxiliary permutation function in place, we can now solve all remaining cases.
- For the @{text "Let\<^sup>\<alpha>"} term-constructor, for example, we can by Property \ref{avoiding}
- obtain a @{text "\<pi>"} such that
-
- \[
- @{text "(\<pi> \<bullet> (set (bn\<^sup>\<alpha> x\<^isub>1)) #\<^sup>* c"} \hspace{10mm}
- @{text "\<pi> \<bullet> [bn\<^sup>\<alpha> x\<^isub>1]\<^bsub>list\<^esub>. x\<^isub>2 = [bn\<^sup>\<alpha> x\<^isub>1]\<^bsub>list\<^esub>. x\<^isub>2"}
- \]\smallskip
-
- \noindent
- hold. Using the first part of Lemma \ref{permutebn}, we can simplify this
- to @{text "set (bn\<^sup>\<alpha> (\<pi> \<bullet>\<AL>\<^bsub>bn\<^esub> x\<^isub>1)) #\<^sup>* c"} and
- \mbox{@{text "[bn\<^sup>\<alpha> (\<pi> \<bullet>\<AL>\<^bsub>bn\<^esub> x\<^isub>1)]\<^bsub>list\<^esub>. (\<pi> \<bullet> x\<^isub>2) = [bn\<^sup>\<alpha> x\<^isub>1]\<^bsub>list\<^esub>. x\<^isub>2"}}. Since
- @{text "(\<pi> \<bullet>\<AL>\<^bsub>bn\<^esub> x\<^isub>1) \<approx>\<AL>\<^bsub>bn\<^esub> x\<^isub>1"} holds by the second part,
- we can infer that
-
- \[
- @{text "Let\<^sup>\<alpha> (\<pi> \<bullet>\<AL>\<^bsub>bn\<^esub> x\<^isub>1) (\<pi> \<bullet> x\<^isub>2) = Let\<^sup>\<alpha> x\<^isub>1 x\<^isub>2"}
- \]\smallskip
-
- \noindent
- holds. This allows us to use the implication from the strong cases
- lemma, and we are done.
-
- Consequently, we can discharge all proof-obligations about having `covered all
- cases'. This completes the proof establishing that the weak induction principles imply
- the strong induction principles. These strong induction principles have already proved
- being very useful in practice, particularly for proving properties about
- capture-avoiding substitution \cite{Urban08}.
-*}
-
-
-section {* Related Work\label{related} *}
-
-text {*
- To our knowledge the earliest usage of general binders in a theorem prover
- is described by Nara\-schew\-ski and Nipkow \cite{NaraschewskiNipkow99} with a
- formalisation of the algorithm W. This formalisation implements binding in
- type-schemes using a de-Bruijn indices representation. Since type-schemes in
- W contain only a single place where variables are bound, different indices
- do not refer to different binders (as in the usual de-Bruijn
- representation), but to different bound variables. A similar idea has been
- recently explored for general binders by Chargu\'eraud \cite{chargueraud09}
- in the locally nameless approach to
- binding. There, de-Bruijn indices consist of two
- numbers, one referring to the place where a variable is bound, and the other
- to which variable is bound. The reasoning infrastructure for both
- representations of bindings comes for free in theorem provers like
- Isabelle/HOL and Coq, since the corresponding term-calculi can be implemented
- as `normal' datatypes. However, in both approaches it seems difficult to
- achieve our fine-grained control over the `semantics' of bindings
- (i.e.~whether the order of binders should matter, or vacuous binders should
- be taken into account). To do so, one would require additional predicates
- that filter out unwanted terms. Our guess is that such predicates result in
- rather intricate formal reasoning. We are not aware of any formalisation of
- a non-trivial language that uses Chargu\'eraud's idea.
-
- Another technique for representing binding is higher-order abstract syntax
- (HOAS), which for example is implemented in the Twelf system \cite{pfenningsystem}.
- This representation technique supports very elegantly many aspects of
- \emph{single} binding, and impressive work by Lee et al~\cite{LeeCraryHarper07}
- has been done that uses HOAS for mechanising the metatheory of SML. We
- are, however, not aware how multiple binders of SML are represented in this
- work. Judging from the submitted Twelf-solution for the POPLmark challenge,
- HOAS cannot easily deal with binding constructs where the number of bound
- variables is not fixed. For example, in the second part of this challenge,
- @{text "Let"}s involve patterns that bind multiple variables at once. In
- such situations, HOAS seems to have to resort to the
- iterated-single-binders-approach with all the unwanted consequences when
- reasoning about the resulting terms.
-
-
- Two formalisations involving general binders have been
- performed in older
- versions of Nominal Isabelle (one about Psi-calculi and one about algorithm W
- \cite{BengtsonParow09,UrbanNipkow09}). Both
- use the approach based on iterated single binders. Our experience with
- the latter formalisation has been disappointing. The major pain arose from
- the need to `unbind' bound variables and the resulting formal reasoning turned out to
- be rather unpleasant. In contrast, the unbinding can be
- done in one step with our
- general binders described in this paper.
-
- The most closely related work to the one presented here is the Ott-tool by
- Sewell et al \cite{ott-jfp} and the C$\alpha$ml language by Pottier
- \cite{Pottier06}. Ott is a nifty front-end for creating \LaTeX{} documents
- from specifications of term-calculi involving general binders. For a subset
- of the specifications Ott can also generate theorem prover code using a `raw'
- representation of terms, and in Coq also a locally nameless
- representation. The developers of this tool have also put forward (on paper)
- a definition for alpha-equivalence and free variables for terms that can be
- specified in Ott. This definition is rather different from ours, not using
- any nominal techniques. To our knowledge there is no concrete mathematical
- result concerning this notion of alpha-equivalence and free variables. We
- have proved that our definitions lead to alpha-equated terms, whose support
- is as expected (that means bound atoms are removed from the support). We
- also showed that our specifications lift from `raw' terms to
- alpha-equivalence classes. For this we have established (automatically) that every
- term-constructor and function defined for `raw' terms
- is respectful w.r.t.~alpha-equivalence.
-
- Although we were heavily inspired by the syntax of Ott, its definition of
- alpha-equi\-valence is unsuitable for our extension of Nominal
- Isabelle. First, it is far too complicated to be a basis for automated
- proofs implemented on the ML-level of Isabelle/HOL. Second, it covers cases
- of binders depending on other binders, which just do not make sense for our
- alpha-equated terms (the corresponding @{text "fa"}-functions would not lift).
- Third, it allows empty types that have no meaning in a
- HOL-based theorem prover. We also had to generalise slightly Ott's binding
- clauses. In Ott one specifies binding clauses with a single body; we allow
- more than one. We have to do this, because this makes a difference for our
- notion of alpha-equivalence in case of \isacommand{binds (set)} and
- \isacommand{binds (set+)}. Consider the examples
-
- \[\mbox{
- \begin{tabular}{@ {}l@ {\hspace{2mm}}l@ {}}
- @{text "Foo\<^isub>1 xs::name fset t::trm s::trm"} &
- \isacommand{binds (set)} @{text "xs"} \isacommand{in} @{text "t s"}\\
- @{text "Foo\<^isub>2 xs::name fset t::trm s::trm"} &
- \isacommand{binds (set)} @{text "xs"} \isacommand{in} @{text "t"},
- \isacommand{binds (set)} @{text "xs"} \isacommand{in} @{text "s"}\\
- \end{tabular}}
- \]\smallskip
-
- \noindent
- In the first term-constructor we have a single body that happens to be
- `spread' over two arguments; in the second term-constructor we have two
- independent bodies in which the same variables are bound. As a result we
- have\footnote{Assuming @{term "a \<noteq> b"}, there is no permutation that can
- make @{text "(a, b)"} equal with both @{text "(a, b)"} and @{text "(b, a)"}, but
- there are two permutations so that we can make @{text "(a, b)"} and @{text
- "(a, b)"} equal with one permutation, and @{text "(a, b)"} and @{text "(b,
- a)"} with the other.}
-
-
- \[\mbox{
- \begin{tabular}{r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l}
- @{text "Foo\<^isub>1 {a, b} (a, b) (a, b)"} & $\not=$ &
- @{text "Foo\<^isub>1 {a, b} (a, b) (b, a)"}
- \end{tabular}}
- \]\smallskip
-
- \noindent
- but
-
- \[\mbox{
- \begin{tabular}{r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l}
- @{text "Foo\<^isub>2 {a, b} (a, b) (a, b)"} & $=$ &
- @{text "Foo\<^isub>2 {a, b} (a, b) (b, a)"}\\
- \end{tabular}}
- \]\smallskip
-
- \noindent
- and therefore need the extra generality to be able to distinguish
- between both specifications. Because of how we set up our
- definitions, we also had to impose some restrictions (like a single
- binding function for a deep binder) that are not present in Ott. Our
- expectation is that we can still cover many interesting term-calculi
- from programming language research, for example the Core-Haskell
- language from the Introduction. With the work presented in this
- paper we can define it formally as shown in
- Figure~\ref{nominalcorehas} and then Nominal Isabelle derives
- automatically a corresponding reasoning infrastructure. However we
- have found out that telescopes seem to not easily be representable
- in our framework. The reason is that we need to be able to lift our
- @{text bn}-functions to alpha-equated lambda-terms and therefore
- need to restrict what these @{text bn}-functions can return.
- Telescopes can be represented in the framework described in
- \cite{WeirichYorgeySheard11} using an extension of the usual
- locally-nameless representation.
-
- \begin{figure}[p]
- \begin{boxedminipage}{\linewidth}
- \small
- \begin{tabular}{l}
- \isacommand{atom\_decl}~@{text "var cvar tvar"}\\[1mm]
- \isacommand{nominal\_datatype}~@{text "tkind ="}~@{text "KStar"}~$|$~@{text "KFun tkind tkind"}\\
- \isacommand{and}~@{text "ckind ="}~@{text "CKSim ty ty"}\\
- \isacommand{and}~@{text "ty ="}~@{text "TVar tvar"}~$|$~@{text "T string"}~$|$~@{text "TApp ty ty"}\\
- $|$~@{text "TFun string ty_list"}~%
- $|$~@{text "TAll tv::tvar tkind ty::ty"}\hspace{3mm}\isacommand{binds}~@{text "tv"}~\isacommand{in}~@{text ty}\\
- $|$~@{text "TArr ckind ty"}\\
- \isacommand{and}~@{text "ty_lst ="}~@{text "TNil"}~$|$~@{text "TCons ty ty_lst"}\\
- \isacommand{and}~@{text "cty ="}~@{text "CVar cvar"}~%
- $|$~@{text "C string"}~$|$~@{text "CApp cty cty"}~$|$~@{text "CFun string co_lst"}\\
- $|$~@{text "CAll cv::cvar ckind cty::cty"}\hspace{3mm}\isacommand{binds}~@{text "cv"}~\isacommand{in}~@{text cty}\\
- $|$~@{text "CArr ckind cty"}~$|$~@{text "CRefl ty"}~$|$~@{text "CSym cty"}~$|$~@{text "CCirc cty cty"}\\
- $|$~@{text "CAt cty ty"}~$|$~@{text "CLeft cty"}~$|$~@{text "CRight cty"}~$|$~@{text "CSim cty cty"}\\
- $|$~@{text "CRightc cty"}~$|$~@{text "CLeftc cty"}~$|$~@{text "Coerce cty cty"}\\
- \isacommand{and}~@{text "co_lst ="}~@{text "CNil"}~$|$~@{text "CCons cty co_lst"}\\
- \isacommand{and}~@{text "trm ="}~@{text "Var var"}~$|$~@{text "K string"}\\
- $|$~@{text "LAM_ty tv::tvar tkind t::trm"}\hspace{3mm}\isacommand{binds}~@{text "tv"}~\isacommand{in}~@{text t}\\
- $|$~@{text "LAM_cty cv::cvar ckind t::trm"}\hspace{3mm}\isacommand{binds}~@{text "cv"}~\isacommand{in}~@{text t}\\
- $|$~@{text "App_ty trm ty"}~$|$~@{text "App_cty trm cty"}~$|$~@{text "App trm trm"}\\
- $|$~@{text "Lam v::var ty t::trm"}\hspace{3mm}\isacommand{binds}~@{text "v"}~\isacommand{in}~@{text t}\\
- $|$~@{text "Let x::var ty trm t::trm"}\hspace{3mm}\isacommand{binds}~@{text x}~\isacommand{in}~@{text t}\\
- $|$~@{text "Case trm assoc_lst"}~$|$~@{text "Cast trm co"}\\
- \isacommand{and}~@{text "assoc_lst ="}~@{text ANil}~%
- $|$~@{text "ACons p::pat t::trm assoc_lst"}\hspace{3mm}\isacommand{binds}~@{text "bv p"}~\isacommand{in}~@{text t}\\
- \isacommand{and}~@{text "pat ="}~@{text "Kpat string tvtk_lst tvck_lst vt_lst"}\\
- \isacommand{and}~@{text "vt_lst ="}~@{text VTNil}~$|$~@{text "VTCons var ty vt_lst"}\\
- \isacommand{and}~@{text "tvtk_lst ="}~@{text TVTKNil}~$|$~@{text "TVTKCons tvar tkind tvtk_lst"}\\
- \isacommand{and}~@{text "tvck_lst ="}~@{text TVCKNil}~$|$ @{text "TVCKCons cvar ckind tvck_lst"}\\
- \isacommand{binder}\\
- \;@{text "bv :: pat \<Rightarrow> atom list"}~\isacommand{and}\\
- \;@{text "bv\<^isub>1 :: vt_lst \<Rightarrow> atom list"}~\isacommand{and}\\
- \;@{text "bv\<^isub>2 :: tvtk_lst \<Rightarrow> atom list"}~\isacommand{and}\\
- \;@{text "bv\<^isub>3 :: tvck_lst \<Rightarrow> atom list"}\\
- \isacommand{where}\\
- \phantom{$|$}~@{text "bv (K s tvts tvcs vs) = (bv\<^isub>3 tvts) @ (bv\<^isub>2 tvcs) @ (bv\<^isub>1 vs)"}\\
- $|$~@{text "bv\<^isub>1 VTNil = []"}\\
- $|$~@{text "bv\<^isub>1 (VTCons x ty tl) = (atom x)::(bv\<^isub>1 tl)"}\\
- $|$~@{text "bv\<^isub>2 TVTKNil = []"}\\
- $|$~@{text "bv\<^isub>2 (TVTKCons a ty tl) = (atom a)::(bv\<^isub>2 tl)"}\\
- $|$~@{text "bv\<^isub>3 TVCKNil = []"}\\
- $|$~@{text "bv\<^isub>3 (TVCKCons c cty tl) = (atom c)::(bv\<^isub>3 tl)"}\\
- \end{tabular}
- \end{boxedminipage}
- \caption{A definition for Core-Haskell in Nominal Isabelle. For the moment we
- do not support nested types; therefore we explicitly have to unfold the
- lists @{text "co_lst"}, @{text "assoc_lst"} and so on. Apart from that limitation, the
- definition follows closely the original shown in Figure~\ref{corehas}. The
- point of our work is that having made such a definition in Nominal Isabelle,
- one obtains automatically a reasoning infrastructure for Core-Haskell.
- \label{nominalcorehas}}
- \end{figure}
- \afterpage{\clearpage}
-
- Pottier presents a programming language, called C$\alpha$ml, for
- representing terms with general binders inside OCaml \cite{Pottier06}. This
- language is implemented as a front-end that can be translated to OCaml with
- the help of a library. He presents a type-system in which the scope of
- general binders can be specified using special markers, written @{text
- "inner"} and @{text "outer"}. It seems our and his specifications can be
- inter-translated as long as ours use the binding mode \isacommand{binds}
- only. However, we have not proved this. Pottier gives a definition for
- alpha-equivalence, which also uses a permutation operation (like ours).
- Still, this definition is rather different from ours and he only proves that
- it defines an equivalence relation. A complete reasoning infrastructure is
- well beyond the purposes of his language. Similar work for Haskell with
- similar results was reported by Cheney \cite{Cheney05a} and more recently
- by Weirich et al \cite{WeirichYorgeySheard11}.
-
- In a slightly different domain (programming with dependent types),
- Altenkirch et al \cite{Altenkirch10} present a calculus with a notion of
- alpha-equivalence related to our binding mode \isacommand{binds (set+)}.
- Their definition is similar to the one by Pottier, except that it has a more
- operational flavour and calculates a partial (renaming) map. In this way,
- the definition can deal with vacuous binders. However, to our best
- knowledge, no concrete mathematical result concerning this definition of
- alpha-equivalence has been proved.
-*}
-
-section {* Conclusion *}
-
-text {*
-
- We have presented an extension of Nominal Isabelle for dealing with general
- binders, that is where term-constructors have multiple bound atoms. For this
- extension we introduced new definitions of alpha-equivalence and automated
- all necessary proofs in Isabelle/HOL. To specify general binders we used
- the syntax from Ott, but extended it in some places and restricted
- it in others so that the definitions make sense in the context of alpha-equated
- terms. We also introduced two binding modes (set and set+) that do not exist
- in Ott. We have tried out the extension with calculi such as Core-Haskell,
- type-schemes and approximately a dozen of other typical examples from
- programming language research~\cite{SewellBestiary}. The code will
- eventually become part of the Isabelle distribution.\footnote{It
- can be downloaded already from \href{http://isabelle.in.tum.de/nominal/download}
- {http://isabelle.in.tum.de/nominal/download}.}
-
- We have left out a discussion about how functions can be defined over
- alpha-equated terms involving general binders. In earlier versions of
- Nominal Isabelle this turned out to be a thorny issue. We hope to do better
- this time by using the function package \cite{Krauss09} that has recently
- been implemented in Isabelle/HOL and also by restricting function
- definitions to equivariant functions (for them we can provide more
- automation).
-
- There are some restrictions we had
- to impose in this paper that can be lifted using
- a recent reimplementation \cite{Traytel12} of the datatype package for Isabelle/HOL, which
- however is not yet part of the stable distribution.
- This reimplementation allows nested
- datatype definitions and would allow one to specify, for instance, the function kinds
- in Core-Haskell as @{text "TFun string (ty list)"} instead of the unfolded
- version @{text "TFun string ty_list"} (see Figure~\ref{nominalcorehas}). We can
- also use it to represent the @{text "Let"}-terms from the Introduction where
- the order of @{text "let"}-assignments does not matter. This means we can represent @{text "Let"}s
- such that the following two terms are equal
-
- \[
- @{text "Let x\<^isub>1 = t\<^isub>1 and x\<^isub>2 = t\<^isub>2 in s"} \;\;=\;\;
- @{text "Let x\<^isub>2 = t\<^isub>2 and x\<^isub>1 = t\<^isub>1 in s"}
- \]\smallskip
-
- \noindent
- For this we have to represent the @{text "Let"}-assignments as finite sets
- of pair and a binding function that picks out the left components to be bound in @{text s}.
-
- One line of future investigation is whether we can go beyond the
- simple-minded form of binding functions that we adopted from Ott. At the moment, binding
- functions can only return the empty set, a singleton atom set or unions
- of atom sets (similarly for lists). It remains to be seen whether
- properties like
-
- \[
- \mbox{@{text "fa_ty x = bn x \<union> fa_bn x"}}
- \]\smallskip
-
- \noindent
- allow us to support more interesting binding functions.
-
- We have also not yet played with other binding modes. For example we can
- imagine that there is need for a binding mode where instead of usual lists,
- we abstract lists of distinct elements (the corresponding type @{text
- "dlist"} already exists in the library of Isabelle/HOL). We expect the
- presented work can be extended to accommodate such binding modes.\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 of the Ott-tool. Stephanie Weirich
- suggested to separate the subgrammars of kinds and types in our Core-Haskell
- example. Ramana Kumar and Andrei Popescu helped us with comments for
- an earlier version of this paper.
-*}
-
-
-(*<*)
-end
-(*>*)