(*<*)+ −
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, this leads to a rather clumsy+ −
formalisation of W. + −
+ −
{\bf add example about W}+ −
+ −
The need of iterating single binders is also one reason+ −
why Nominal Isabelle and similar theorem provers that only provide+ −
mechanisms for binding single variables have not fared extremely well with+ −
the more advanced tasks in the POPLmark challenge \cite{challenge05},+ −
because also there one would like to bind multiple variables at once.+ −
+ −
Binding multiple variables has interesting properties that cannot be captured+ −
easily by iterating single binders. For example in 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>{y, x}. 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 chose 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 automatic+ −
proofs, we establish a reasoning infrastructure for alpha-equated terms,+ −
including properties about support, freshness and equality conditions for+ −
alpha-equated terms. We are also able to 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 adaption 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> _ :: 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"}}, + −
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_eq[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 "as"} of atoms.\\+ −
{\it (i)} If @{thm (prem 1) supp_is_subset[where S="as", no_vars]}+ −
and @{thm (prem 2) supp_is_subset[where S="as", no_vars]} then + −
@{thm (concl) supp_is_subset[where S="as", 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}, say of type @{text "\<alpha> \<Rightarrow> \<beta>"}, 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 or equivalently that a permutation applied to the application+ −
@{text "f x"} can be moved to the argument @{text x}. That means for equivariant+ −
functions @{text f}, 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+ −
From property \eqref{equivariancedef} and the definition of @{text supp}, we + −
can easily deduce that equivariant functions have empty support. There is+ −
also a similar notion for equivariant relations, say @{text R}, namely the property+ −
that+ −
+ −
\[+ −
@{text "x R y"} \;\;\textit{implies}\;\; @{text "(\<pi> \<bullet> x) R (\<pi> \<bullet> y)"}+ −
\]\smallskip+ −
+ −
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 + −
Property~\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 x} and @{text 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 x} and @{text 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 "({y, x}, 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 relation 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+ −
+ −
\begin{equation}\label{absperm}+ −
@{thm permute_Abs(1)[where p="\<pi>", no_vars]}+ −
\end{equation}\smallskip+ −
+ −
\noindent+ −
The second fact derives from the definition of permutations acting on pairs + −
\eqref{permute} and alpha-equivalence being equivariant + −
(see Lemma~\ref{alphaeq}). With these two facts 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+ −
Assuming that @{text "x"} has finite support, 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}. The `other half' is a bit more involved. To establish + −
it, 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]}.+ −
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 mutual 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}). The labels 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 ="}~\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+ −
+ −
\[\mbox{+ −
\begin{tabular}{@ {}l@ {\hspace{2mm}}l@ {}}+ −
@{text "Baz\<^isub>1 p::pat t::trm"} & + −
\isacommand{binds} @{text "bn\<^isub>1(p) bn\<^isub>2(p)"} \isacommand{in} @{text t}\\+ −
@{text "Baz\<^isub>2 p::pat t\<^isub>1::trm t\<^isub>2::trm"} & + −
\isacommand{binds} @{text "bn\<^isub>1(p)"} \isacommand{in} @{text "t\<^isub>1"},+ −
\isacommand{binds} @{text "bn\<^isub>2(p)"} \isacommand{in} @{text "t\<^isub>2"}\\+ −
\end{tabular}}+ −
\]\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"}.+ −
(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.+ −
+ −
In order to simplify our definitions of free atoms and alpha-equivalence, + −
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 the 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+ −
+ −
\[+ −
\mbox{\infer{@{text "C z\<^isub>1 \<dots> z\<^isub>n \<approx>ty C z\<PRIME>\<^isub>1 \<dots> z\<PRIME>\<^isub>n"}}+ −
{@{text "prems(bc\<^isub>1) \<dots> prems(bc\<^isub>k)"}}} + −
\]\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 @{text+ −
"z\<PRIME>"}$_{1..n}$ of the term-constructor. 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 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 as \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 the 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 constructors @{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 `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 are 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 being 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' variables. This can be done in one step with our+ −
general binders described in this paper, but needs a cumbersome+ −
iteration with single binders. The resulting formal reasoning turned out to+ −
be rather unpleasant. + −
+ −
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' types to types of+ −
alpha-equivalence classes. For this we have established (automatically) that every+ −
term-constructor and function defined for `raw' types + −
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. 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.+ −
+ −
\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 imposed in this paper that we would like to lift in+ −
future work. One is the exclusion of nested datatype definitions. Nested+ −
datatype definitions 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}). To+ −
achieve this, we need more clever implementation than we have + −
at the moment. However, really lifting this restriction will involve major+ −
work on the underlying datatype package of Isabelle/HOL.+ −
+ −
A more interesting line of 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 about + −
an earlier version of this paper.+ −
*}+ −
+ −
+ −
(*<*)+ −
end+ −
(*>*)+ −