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(*<*)
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theory Paper
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imports "../Nominal/Test" "LaTeXsugar"
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begin
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consts
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fv :: "'a \<Rightarrow> 'b"
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abs_set :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
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alpha_bn :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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definition
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"equal \<equiv> (op =)"
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notation (latex output)
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swap ("'(_ _')" [1000, 1000] 1000) and
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fresh ("_ # _" [51, 51] 50) and
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fresh_star ("_ #\<^sup>* _" [51, 51] 50) and
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supp ("supp _" [78] 73) and
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uminus ("-_" [78] 73) and
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If ("if _ then _ else _" 10) and
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alpha_gen ("_ \<approx>\<^raw:\raisebox{-1pt}{\makebox[0mm][l]{$\,_{\textit{set}}$}}>\<^bsup>_,_,_\<^esup> _") and
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alpha_lst ("_ \<approx>\<^raw:\raisebox{-1pt}{\makebox[0mm][l]{$\,_{\textit{list}}$}}>\<^bsup>_,_,_\<^esup> _") and
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alpha_res ("_ \<approx>\<^raw:\raisebox{-1pt}{\makebox[0mm][l]{$\,_{\textit{res}}$}}>\<^bsup>_,_,_\<^esup> _") and
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abs_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
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fv ("fv'(_')" [100] 100) and
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equal ("=") and
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alpha_abs ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
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Abs ("[_]\<^raw:$\!$>\<^bsub>set\<^esub>._" [20, 101] 999) and
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Abs_lst ("[_]\<^raw:$\!$>\<^bsub>list\<^esub>._") and
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Abs_res ("[_]\<^raw:$\!$>\<^bsub>res\<^esub>._") and
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Cons ("_::_" [78,77] 73) and
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supp_gen ("aux _" [1000] 10) and
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alpha_bn ("_ \<approx>bn _")
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(*>*)
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section {* Introduction *}
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text {*
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So far, Nominal Isabelle provides a mechanism for constructing
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alpha-equated terms, for example
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\begin{center}
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@{text "t ::= x | t t | \<lambda>x. t"}
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\end{center}
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\noindent
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where free and bound variables have names. For such alpha-equated terms, Nominal Isabelle
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derives automatically a reasoning infrastructure that has been used
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successfully in formalisations of an equivalence checking algorithm for LF
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\cite{UrbanCheneyBerghofer08}, Typed
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Scheme~\cite{TobinHochstadtFelleisen08}, several calculi for concurrency
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\cite{BengtsonParow09} and a strong normalisation result
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for cut-elimination in classical logic \cite{UrbanZhu08}. It has also been
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used by Pollack for formalisations in the locally-nameless approach to
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binding \cite{SatoPollack10}.
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However, Nominal Isabelle has fared less well in a formalisation of
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the algorithm W \cite{UrbanNipkow09}, where types and type-schemes are,
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respectively, of the form
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%
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\begin{equation}\label{tysch}
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\begin{array}{l}
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@{text "T ::= x | T \<rightarrow> T"}\hspace{5mm}
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@{text "S ::= \<forall>{x\<^isub>1,\<dots>, x\<^isub>n}. T"}
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\end{array}
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\end{equation}
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\noindent
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changeset
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and the @{text "\<forall>"}-quantification binds a finite (possibly empty) set of
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type-variables. While it is possible to implement this kind of more general
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binders by iterating single binders, this leads to a rather clumsy
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formalisation of W. The need of iterating single binders is also one reason
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why Nominal Isabelle and similar theorem provers that only provide
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mechanisms for binding single variables have not fared extremely well with the
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more advanced tasks in the POPLmark challenge \cite{challenge05}, because
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also there one would like to bind multiple variables at once.
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Binding multiple variables has interesting properties that cannot be captured
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easily by iterating single binders. For example in case of type-schemes we do not
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want to make a distinction about the order of the bound variables. Therefore
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we would like to regard the following two type-schemes as alpha-equivalent
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%
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\begin{equation}\label{ex1}
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@{text "\<forall>{x, y}. x \<rightarrow> y \<approx>\<^isub>\<alpha> \<forall>{y, x}. y \<rightarrow> x"}
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\end{equation}
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\noindent
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but assuming that @{text x}, @{text y} and @{text z} are distinct variables,
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the following two should \emph{not} be alpha-equivalent
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%
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\begin{equation}\label{ex2}
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@{text "\<forall>{x, y}. x \<rightarrow> y \<notapprox>\<^isub>\<alpha> \<forall>{z}. z \<rightarrow> z"}
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\end{equation}
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\noindent
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Moreover, we like to regard type-schemes as alpha-equivalent, if they differ
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only on \emph{vacuous} binders, such as
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%
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\begin{equation}\label{ex3}
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@{text "\<forall>{x}. x \<rightarrow> y \<approx>\<^isub>\<alpha> \<forall>{x, z}. x \<rightarrow> y"}
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\end{equation}
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\noindent
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where @{text z} does not occur freely in the type. In this paper we will
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give a general binding mechanism and associated notion of alpha-equivalence
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that can be used to faithfully represent this kind of binding in Nominal
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Isabelle. The difficulty of finding the right notion for alpha-equivalence
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can be appreciated in this case by considering that the definition given by
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Leroy in \cite{Leroy92} is incorrect (it omits a side-condition).
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However, the notion of alpha-equivalence that is preserved by vacuous
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binders is not always wanted. For example in terms like
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%
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\begin{equation}\label{one}
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@{text "\<LET> x = 3 \<AND> y = 2 \<IN> x - y \<END>"}
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\end{equation}
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\noindent
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we might not care in which order the assignments $x = 3$ and $y = 2$ are
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given, but it would be unusual to regard \eqref{one} as alpha-equivalent
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with
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%
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\begin{center}
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@{text "\<LET> x = 3 \<AND> y = 2 \<AND> z = loop \<IN> x - y \<END>"}
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\end{center}
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\noindent
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Therefore we will also provide a separate binding mechanism for cases in
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which the order of binders does not matter, but the ``cardinality'' of the
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binders has to agree.
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However, we found that this is still not sufficient for dealing with
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language constructs frequently occurring in programming language
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research. For example in @{text "\<LET>"}s containing patterns like
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%
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\begin{equation}\label{two}
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@{text "\<LET> (x, y) = (3, 2) \<IN> x - y \<END>"}
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\end{equation}
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\noindent
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we want to bind all variables from the pattern inside the body of the
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$\mathtt{let}$, but we also care about the order of these variables, since
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we do not want to regard \eqref{two} as alpha-equivalent with
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%
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\begin{center}
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@{text "\<LET> (y, x) = (3, 2) \<IN> x - y \<END>"}
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\end{center}
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\noindent
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As a result, we provide three general binding mechanisms each of which binds
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multiple variables at once, and let the user chose which one is intended
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when formalising a term-calculus.
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By providing these general binding mechanisms, however, we have to work
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around a problem that has been pointed out by Pottier \cite{Pottier06} and
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Cheney \cite{Cheney05}: in @{text "\<LET>"}-constructs of the form
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%
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\begin{center}
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@{text "\<LET> x\<^isub>1 = t\<^isub>1 \<AND> \<dots> \<AND> x\<^isub>n = t\<^isub>n \<IN> s \<END>"}
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\end{center}
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\noindent
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which bind all the @{text "x\<^isub>i"} in @{text s}, we might not care
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about the order in which the @{text "x\<^isub>i = t\<^isub>i"} are given,
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but we do care about the information that there are as many @{text
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"x\<^isub>i"} as there are @{text "t\<^isub>i"}. We lose this information if
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we represent the @{text "\<LET>"}-constructor by something like
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%
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\begin{center}
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@{text "\<LET> [x\<^isub>1,\<dots>,x\<^isub>n].s [t\<^isub>1,\<dots>,t\<^isub>n]"}
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\end{center}
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\noindent
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where the notation @{text "[_]._"} indicates that the list of @{text "x\<^isub>i"}
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becomes bound in @{text s}. In this representation the term
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\mbox{@{text "\<LET> [x].s [t\<^isub>1, t\<^isub>2]"}} is a perfectly legal
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instance, but the lengths of two lists do not agree. To exclude such terms,
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additional predicates about well-formed
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terms are needed in order to ensure that the two lists are of equal
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length. This can result into very messy reasoning (see for
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example~\cite{BengtsonParow09}). To avoid this, we will allow type
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specifications for $\mathtt{let}$s as follows
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%
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\begin{center}
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\begin{tabular}{r@ {\hspace{2mm}}r@ {\hspace{2mm}}l}
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@{text trm} & @{text "::="} & @{text "\<dots>"}\\
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& @{text "|"} & @{text "\<LET> as::assn s::trm"}\hspace{4mm}
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\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text "s"}\\[1mm]
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@{text assn} & @{text "::="} & @{text "\<ANIL>"}\\
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& @{text "|"} & @{text "\<ACONS> name trm assn"}
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\end{tabular}
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\end{center}
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\noindent
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where @{text assn} is an auxiliary type representing a list of assignments
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and @{text bn} an auxiliary function identifying the variables to be bound
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by the @{text "\<LET>"}. This function can be defined by recursion over @{text
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assn} as follows
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\begin{center}
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@{text "bn(\<ANIL>) ="} @{term "{}"} \hspace{5mm}
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@{text "bn(\<ACONS> x t as) = {x} \<union> bn(as)"}
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\end{center}
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\noindent
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The scope of the binding is indicated by labels given to the types, for
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example @{text "s::trm"}, and a binding clause, in this case
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\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text "s"}. This binding
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clause states to bind in @{text s} all the names the function call @{text
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"bn(as)"} returns. This style of specifying terms and bindings is heavily
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inspired by the syntax of the Ott-tool \cite{ott-jfp}.
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changeset
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However, we will not be able to cope with all specifications that are
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allowed by Ott. One reason is that Ott lets the user to specify ``empty''
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types like
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\begin{center}
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@{text "t ::= t t | \<lambda>x. t"}
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\end{center}
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\noindent
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where no clause for variables is given. Arguably, such specifications make
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some sense in the context of Coq's type theory (which Ott supports), but not
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at all in a HOL-based environment where every datatype must have a non-empty
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set-theoretic model \cite{Berghofer99}.
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Another reason is that we establish the reasoning infrastructure
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for alpha-\emph{equated} terms. In contrast, Ott produces a reasoning
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infrastructure in Isabelle/HOL for
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\emph{non}-alpha-equated, or ``raw'', terms. While our alpha-equated terms
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and the raw terms produced by Ott use names for bound variables,
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there is a key difference: working with alpha-equated terms means, for example,
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that the two type-schemes
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\begin{center}
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@{text "\<forall>{x}. x \<rightarrow> y = \<forall>{x, z}. x \<rightarrow> y"}
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\end{center}
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\noindent
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are not just alpha-equal, but actually \emph{equal}! As a result, we can
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only support specifications that make sense on the level of alpha-equated
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terms (offending specifications, which for example bind a variable according
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to a variable bound somewhere else, are not excluded by Ott, but we have
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to).
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Our insistence on reasoning with alpha-equated terms comes from the
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wealth of experience we gained with the older version of Nominal Isabelle:
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for non-trivial properties, reasoning about alpha-equated terms is much
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easier than reasoning with raw terms. The fundamental reason for this is
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that the HOL-logic underlying Nominal Isabelle allows us to replace
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``equals-by-equals''. In contrast, replacing
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``alpha-equals-by-alpha-equals'' in a representation based on raw terms
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requires a lot of extra reasoning work.
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Although in informal settings a reasoning infrastructure for alpha-equated
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terms is nearly always taken for granted, establishing it automatically in
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the Isabelle/HOL theorem prover is a rather non-trivial task. For every
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specification we will need to construct a type containing as elements the
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alpha-equated terms. To do so, we use the standard HOL-technique of defining
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a new type by identifying a non-empty subset of an existing type. The
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construction we perform in Isabelle/HOL can be illustrated by the following picture:
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\begin{center}
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\begin{tikzpicture}
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%\draw[step=2mm] (-4,-1) grid (4,1);
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\draw[very thick] (0.7,0.4) circle (4.25mm);
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\draw[rounded corners=1mm, very thick] ( 0.0,-0.8) rectangle ( 1.8, 0.9);
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\draw[rounded corners=1mm, very thick] (-1.95,0.85) rectangle (-2.85,-0.05);
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\draw (-2.0, 0.845) -- (0.7,0.845);
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\draw (-2.0,-0.045) -- (0.7,-0.045);
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\draw ( 0.7, 0.4) node {\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]clas.\end{tabular}};
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\draw (-2.4, 0.4) node {\begin{tabular}{@ {}c@ {}}$\alpha$-eq.\\[-1mm]terms\end{tabular}};
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\draw (1.8, 0.48) node[right=-0.1mm]
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{\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ (sets of raw terms)\end{tabular}};
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\draw (0.9, -0.35) node {\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};
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\draw (-3.25, 0.55) node {\begin{tabular}{@ {}l@ {}}new\\[-1mm]type\end{tabular}};
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\draw[<->, very thick] (-1.8, 0.3) -- (-0.1,0.3);
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\draw (-0.95, 0.3) node[above=0mm] {isomorphism};
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\end{tikzpicture}
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\end{center}
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\noindent
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We take as the starting point a definition of raw terms (defined as a
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datatype in Isabelle/HOL); identify then the alpha-equivalence classes in
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the type of sets of raw terms according to our alpha-equivalence relation
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and finally define the new type as these alpha-equivalence classes
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(non-emptiness is satisfied whenever the raw terms are definable as datatype
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in Isabelle/HOL and the property that our relation for alpha-equivalence is
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indeed an equivalence relation).
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|
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|
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The fact that we obtain an isomorphism between the new type and the
|
|
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non-empty subset shows that the new type is a faithful representation of
|
|
300 |
alpha-equated terms. That is not the case for example for terms using the
|
|
301 |
locally nameless representation of binders \cite{McKinnaPollack99}: in this
|
|
302 |
representation there are ``junk'' terms that need to be excluded by
|
|
303 |
reasoning about a well-formedness predicate.
|
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|
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|
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|
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The problem with introducing a new type in Isabelle/HOL is that in order to
|
|
306 |
be useful, a reasoning infrastructure needs to be ``lifted'' from the
|
|
307 |
underlying subset to the new type. This is usually a tricky and arduous
|
|
308 |
task. To ease it, we re-implemented in Isabelle/HOL the quotient package
|
|
309 |
described by Homeier \cite{Homeier05} for the HOL4 system. This package
|
|
310 |
allows us to lift definitions and theorems involving raw terms to
|
|
311 |
definitions and theorems involving alpha-equated terms. For example if we
|
|
312 |
define the free-variable function over raw lambda-terms
|
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|
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|
|
314 |
\begin{center}
|
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|
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@{text "fv(x) = {x}"}\hspace{10mm}
|
|
316 |
@{text "fv(t\<^isub>1 t\<^isub>2) = fv(t\<^isub>1) \<union> fv(t\<^isub>2)"}\\[1mm]
|
|
317 |
@{text "fv(\<lambda>x.t) = fv(t) - {x}"}
|
1577
|
318 |
\end{center}
|
|
319 |
|
|
320 |
\noindent
|
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|
321 |
then with the help of the quotient package we can obtain a function @{text "fv\<^sup>\<alpha>"}
|
1617
|
322 |
operating on quotients, or alpha-equivalence classes of lambda-terms. This
|
1628
|
323 |
lifted function is characterised by the equations
|
1577
|
324 |
|
|
325 |
\begin{center}
|
1657
|
326 |
@{text "fv\<^sup>\<alpha>(x) = {x}"}\hspace{10mm}
|
|
327 |
@{text "fv\<^sup>\<alpha>(t\<^isub>1 t\<^isub>2) = fv\<^sup>\<alpha>(t\<^isub>1) \<union> fv\<^sup>\<alpha>(t\<^isub>2)"}\\[1mm]
|
|
328 |
@{text "fv\<^sup>\<alpha>(\<lambda>x.t) = fv\<^sup>\<alpha>(t) - {x}"}
|
1577
|
329 |
\end{center}
|
|
330 |
|
|
331 |
\noindent
|
|
332 |
(Note that this means also the term-constructors for variables, applications
|
|
333 |
and lambda are lifted to the quotient level.) This construction, of course,
|
1694
|
334 |
only works if alpha-equivalence is indeed an equivalence relation, and the
|
|
335 |
lifted definitions and theorems are respectful w.r.t.~alpha-equivalence.
|
|
336 |
For example, we will not be able to lift a bound-variable function. Although
|
|
337 |
this function can be defined for raw terms, it does not respect
|
|
338 |
alpha-equivalence and therefore cannot be lifted. To sum up, every lifting
|
|
339 |
of theorems to the quotient level needs proofs of some respectfulness
|
|
340 |
properties (see \cite{Homeier05}). In the paper we show that we are able to
|
|
341 |
automate these proofs and therefore can establish a reasoning infrastructure
|
|
342 |
for alpha-equated terms.
|
1667
|
343 |
|
|
344 |
The examples we have in mind where our reasoning infrastructure will be
|
1694
|
345 |
helpful includes the term language of System @{text "F\<^isub>C"}, also
|
|
346 |
known as Core-Haskell (see Figure~\ref{corehas}). This term language
|
1711
|
347 |
involves patterns that have lists of type-, coercion- and term-variables,
|
1703
|
348 |
all of which are bound in @{text "\<CASE>"}-expressions. One
|
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changeset
|
349 |
difficulty is that we do not know in advance how many variables need to
|
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changeset
|
350 |
be bound. Another is that each bound variable comes with a kind or type
|
1694
|
351 |
annotation. Representing such binders with single binders and reasoning
|
|
352 |
about them in a theorem prover would be a major pain. \medskip
|
1506
|
353 |
|
1528
|
354 |
\noindent
|
|
355 |
{\bf Contributions:} We provide new definitions for when terms
|
|
356 |
involving multiple binders are alpha-equivalent. These definitions are
|
1607
|
357 |
inspired by earlier work of Pitts \cite{Pitts04}. By means of automatic
|
1528
|
358 |
proofs, we establish a reasoning infrastructure for alpha-equated
|
|
359 |
terms, including properties about support, freshness and equality
|
1607
|
360 |
conditions for alpha-equated terms. We are also able to derive, at the moment
|
|
361 |
only manually, strong induction principles that
|
|
362 |
have the variable convention already built in.
|
1667
|
363 |
|
|
364 |
\begin{figure}
|
1687
|
365 |
\begin{boxedminipage}{\linewidth}
|
|
366 |
\begin{center}
|
1699
|
367 |
\begin{tabular}{r@ {\hspace{1mm}}r@ {\hspace{2mm}}l}
|
1690
|
368 |
\multicolumn{3}{@ {}l}{Type Kinds}\\
|
1699
|
369 |
@{text "\<kappa>"} & @{text "::="} & @{text "\<star> | \<kappa>\<^isub>1 \<rightarrow> \<kappa>\<^isub>2"}\smallskip\\
|
1690
|
370 |
\multicolumn{3}{@ {}l}{Coercion Kinds}\\
|
1699
|
371 |
@{text "\<iota>"} & @{text "::="} & @{text "\<sigma>\<^isub>1 \<sim> \<sigma>\<^isub>2"}\smallskip\\
|
1690
|
372 |
\multicolumn{3}{@ {}l}{Types}\\
|
1694
|
373 |
@{text "\<sigma>"} & @{text "::="} & @{text "a | T | \<sigma>\<^isub>1 \<sigma>\<^isub>2 | S\<^isub>n"}$\;\overline{@{text "\<sigma>"}}$@{text "\<^sup>n"}
|
1699
|
374 |
@{text "| \<forall>a:\<kappa>. \<sigma> | \<iota> \<Rightarrow> \<sigma>"}\smallskip\\
|
1690
|
375 |
\multicolumn{3}{@ {}l}{Coercion Types}\\
|
1694
|
376 |
@{text "\<gamma>"} & @{text "::="} & @{text "c | C | \<gamma>\<^isub>1 \<gamma>\<^isub>2 | S\<^isub>n"}$\;\overline{@{text "\<gamma>"}}$@{text "\<^sup>n"}
|
1699
|
377 |
@{text "| \<forall>c:\<iota>. \<gamma> | \<iota> \<Rightarrow> \<gamma> "}\\
|
|
378 |
& @{text "|"} & @{text "refl \<sigma> | sym \<gamma> | \<gamma>\<^isub>1 \<circ> \<gamma>\<^isub>2 | \<gamma> @ \<sigma> | left \<gamma> | right \<gamma>"}\\
|
|
379 |
& @{text "|"} & @{text "\<gamma>\<^isub>1 \<sim> \<gamma>\<^isub>2 | rightc \<gamma> | leftc \<gamma> | \<gamma>\<^isub>1 \<triangleright> \<gamma>\<^isub>2"}\smallskip\\
|
1690
|
380 |
\multicolumn{3}{@ {}l}{Terms}\\
|
1699
|
381 |
@{text "e"} & @{text "::="} & @{text "x | K | \<Lambda>a:\<kappa>. e | \<Lambda>c:\<iota>. e | e \<sigma> | e \<gamma>"}\\
|
|
382 |
& @{text "|"} & @{text "\<lambda>x:\<sigma>. e | e\<^isub>1 e\<^isub>2 | \<LET> x:\<sigma> = e\<^isub>1 \<IN> e\<^isub>2"}\\
|
|
383 |
& @{text "|"} & @{text "\<CASE> e\<^isub>1 \<OF>"}$\;\overline{@{text "p \<rightarrow> e\<^isub>2"}}$ @{text "| e \<triangleright> \<gamma>"}\smallskip\\
|
1690
|
384 |
\multicolumn{3}{@ {}l}{Patterns}\\
|
1699
|
385 |
@{text "p"} & @{text "::="} & @{text "K"}$\;\overline{@{text "a:\<kappa>"}}\;\overline{@{text "c:\<iota>"}}\;\overline{@{text "x:\<sigma>"}}$\smallskip\\
|
1690
|
386 |
\multicolumn{3}{@ {}l}{Constants}\\
|
1699
|
387 |
& @{text C} & coercion constants\\
|
|
388 |
& @{text T} & value type constructors\\
|
|
389 |
& @{text "S\<^isub>n"} & n-ary type functions (which need to be fully applied)\\
|
|
390 |
& @{text K} & data constructors\smallskip\\
|
1690
|
391 |
\multicolumn{3}{@ {}l}{Variables}\\
|
1699
|
392 |
& @{text a} & type variables\\
|
|
393 |
& @{text c} & coercion variables\\
|
|
394 |
& @{text x} & term variables\\
|
1687
|
395 |
\end{tabular}
|
|
396 |
\end{center}
|
|
397 |
\end{boxedminipage}
|
1699
|
398 |
\caption{The term-language of System @{text "F\<^isub>C"}
|
|
399 |
\cite{CoreHaskell}, also often referred to as \emph{Core-Haskell}. In this
|
|
400 |
version of the term-language we made a modification by separating the
|
1711
|
401 |
grammars for type kinds and coercion kinds, as well as for types and coercion
|
1702
|
402 |
types. For this paper the interesting term-constructor is @{text "\<CASE>"},
|
|
403 |
which binds multiple type-, coercion- and term-variables.\label{corehas}}
|
1667
|
404 |
\end{figure}
|
1485
|
405 |
*}
|
|
406 |
|
1493
|
407 |
section {* A Short Review of the Nominal Logic Work *}
|
|
408 |
|
|
409 |
text {*
|
1556
|
410 |
At its core, Nominal Isabelle is an adaption of the nominal logic work by
|
|
411 |
Pitts \cite{Pitts03}. This adaptation for Isabelle/HOL is described in
|
1694
|
412 |
\cite{HuffmanUrban10} (including proofs). We shall briefly review this work
|
|
413 |
to aid the description of what follows.
|
|
414 |
|
1711
|
415 |
Two central notions in the nominal logic work are sorted atoms and
|
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changeset
|
416 |
sort-respecting permutations of atoms. We will use the letters @{text "a,
|
1711
|
417 |
b, c, \<dots>"} to stand for atoms and @{text "p, q, \<dots>"} to stand for
|
|
418 |
permutations. The sorts of atoms can be used to represent different kinds of
|
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changeset
|
419 |
variables, such as the term-, coercion- and type-variables in Core-Haskell.
|
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changeset
|
420 |
It is assumed that there is an infinite supply of atoms for each
|
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changeset
|
421 |
sort. However, in order to simplify the description, we shall restrict ourselves
|
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changeset
|
422 |
in what follows to only one sort of atoms.
|
1493
|
423 |
|
|
424 |
Permutations are bijective functions from atoms to atoms that are
|
|
425 |
the identity everywhere except on a finite number of atoms. There is a
|
|
426 |
two-place permutation operation written
|
1617
|
427 |
%
|
1703
|
428 |
\begin{center}
|
|
429 |
@{text "_ \<bullet> _ :: perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
|
|
430 |
\end{center}
|
1493
|
431 |
|
|
432 |
\noindent
|
1628
|
433 |
in which the generic type @{text "\<beta>"} stands for the type of the object
|
1694
|
434 |
over which the permutation
|
1617
|
435 |
acts. In Nominal Isabelle, the identity permutation is written as @{term "0::perm"},
|
1690
|
436 |
the composition of two permutations @{term p} and @{term q} as \mbox{@{term "p + q"}},
|
1570
|
437 |
and the inverse permutation of @{term p} as @{text "- p"}. The permutation
|
1703
|
438 |
operation is defined by induction over the type-hierarchy (see \cite{HuffmanUrban10});
|
1727
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changeset
|
439 |
for example permutations acting on products, lists, sets, functions and booleans is
|
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changeset
|
440 |
given by:
|
1702
|
441 |
%
|
1703
|
442 |
\begin{equation}\label{permute}
|
1694
|
443 |
\mbox{\begin{tabular}{@ {}cc@ {}}
|
1690
|
444 |
\begin{tabular}{@ {}l@ {}}
|
|
445 |
@{thm permute_prod.simps[no_vars, THEN eq_reflection]}\\[2mm]
|
|
446 |
@{thm permute_list.simps(1)[no_vars, THEN eq_reflection]}\\
|
|
447 |
@{thm permute_list.simps(2)[no_vars, THEN eq_reflection]}\\
|
|
448 |
\end{tabular} &
|
|
449 |
\begin{tabular}{@ {}l@ {}}
|
|
450 |
@{thm permute_set_eq[no_vars, THEN eq_reflection]}\\
|
1694
|
451 |
@{text "p \<bullet> f \<equiv> \<lambda>x. p \<bullet> (f (- p \<bullet> x))"}\\
|
1690
|
452 |
@{thm permute_bool_def[no_vars, THEN eq_reflection]}\\
|
|
453 |
\end{tabular}
|
1694
|
454 |
\end{tabular}}
|
|
455 |
\end{equation}
|
1690
|
456 |
|
|
457 |
\noindent
|
1730
|
458 |
Concrete permutations in Nominal Isabelle are built up from swappings,
|
|
459 |
written as \mbox{@{text "(a b)"}}, which are permutations that behave
|
|
460 |
as follows:
|
1617
|
461 |
%
|
1703
|
462 |
\begin{center}
|
|
463 |
@{text "(a b) = \<lambda>c. if a = c then b else if b = c then a else c"}
|
|
464 |
\end{center}
|
|
465 |
|
1570
|
466 |
The most original aspect of the nominal logic work of Pitts is a general
|
1703
|
467 |
definition for the notion of the ``set of free variables of an object @{text
|
1570
|
468 |
"x"}''. This notion, written @{term "supp x"}, is general in the sense that
|
1628
|
469 |
it applies not only to lambda-terms (alpha-equated or not), but also to lists,
|
1570
|
470 |
products, sets and even functions. The definition depends only on the
|
|
471 |
permutation operation and on the notion of equality defined for the type of
|
|
472 |
@{text x}, namely:
|
1617
|
473 |
%
|
1703
|
474 |
\begin{equation}\label{suppdef}
|
|
475 |
@{thm supp_def[no_vars, THEN eq_reflection]}
|
|
476 |
\end{equation}
|
1493
|
477 |
|
|
478 |
\noindent
|
|
479 |
There is also the derived notion for when an atom @{text a} is \emph{fresh}
|
|
480 |
for an @{text x}, defined as
|
1617
|
481 |
%
|
1703
|
482 |
\begin{center}
|
|
483 |
@{thm fresh_def[no_vars]}
|
|
484 |
\end{center}
|
1493
|
485 |
|
|
486 |
\noindent
|
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Christian Urban <urbanc@in.tum.de>
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changeset
|
487 |
We also use for sets of atoms the abbreviation
|
1703
|
488 |
@{thm (lhs) fresh_star_def[no_vars]}, defined as
|
1517
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
489 |
@{thm (rhs) fresh_star_def[no_vars]}.
|
1493
|
490 |
A striking consequence of these definitions is that we can prove
|
|
491 |
without knowing anything about the structure of @{term x} that
|
|
492 |
swapping two fresh atoms, say @{text a} and @{text b}, leave
|
1506
|
493 |
@{text x} unchanged.
|
|
494 |
|
1711
|
495 |
\begin{property}\label{swapfreshfresh}
|
1506
|
496 |
@{thm[mode=IfThen] swap_fresh_fresh[no_vars]}
|
1517
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corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
497 |
\end{property}
|
1506
|
498 |
|
1711
|
499 |
While often the support of an object can be relatively easily
|
1730
|
500 |
described, for example for atoms, products, lists, function applications,
|
|
501 |
booleans and permutations\\[-6mm]
|
1690
|
502 |
%
|
|
503 |
\begin{eqnarray}
|
1703
|
504 |
@{term "supp a"} & = & @{term "{a}"}\\
|
1690
|
505 |
@{term "supp (x, y)"} & = & @{term "supp x \<union> supp y"}\\
|
|
506 |
@{term "supp []"} & = & @{term "{}"}\\
|
1711
|
507 |
@{term "supp (x#xs)"} & = & @{term "supp x \<union> supp xs"}\\
|
1730
|
508 |
@{text "supp (f x)"} & @{text "\<subseteq>"} & @{term "supp f \<union> supp x"}\label{suppfun}\\
|
1703
|
509 |
@{term "supp b"} & = & @{term "{}"}\\
|
|
510 |
@{term "supp p"} & = & @{term "{a. p \<bullet> a \<noteq> a}"}
|
1690
|
511 |
\end{eqnarray}
|
|
512 |
|
|
513 |
\noindent
|
1730
|
514 |
in some cases it can be difficult to characterise the support precisely, and
|
|
515 |
only an approximation can be established (see \eqref{suppfun} above). Reasoning about
|
|
516 |
such approximations can be simplified with the notion \emph{supports}, defined
|
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changeset
|
517 |
as follows:
|
1693
|
518 |
|
|
519 |
\begin{defn}
|
|
520 |
A set @{text S} \emph{supports} @{text x} if for all atoms @{text a} and @{text b}
|
|
521 |
not in @{text S} we have @{term "(a \<rightleftharpoons> b) \<bullet> x = x"}.
|
|
522 |
\end{defn}
|
1690
|
523 |
|
1693
|
524 |
\noindent
|
1727
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diff
changeset
|
525 |
The main point of @{text supports} is that we can establish the following
|
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diff
changeset
|
526 |
two properties.
|
1693
|
527 |
|
1703
|
528 |
\begin{property}\label{supportsprop}
|
|
529 |
{\it i)} @{thm[mode=IfThen] supp_is_subset[no_vars]}\\
|
1693
|
530 |
{\it ii)} @{thm supp_supports[no_vars]}.
|
|
531 |
\end{property}
|
|
532 |
|
|
533 |
Another important notion in the nominal logic work is \emph{equivariance}.
|
1703
|
534 |
For a function @{text f}, say of type @{text "\<alpha> \<Rightarrow> \<beta>"}, to be equivariant
|
1727
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changeset
|
535 |
it is required that every permutation leaves @{text f} unchanged, that is
|
1711
|
536 |
%
|
|
537 |
\begin{equation}\label{equivariancedef}
|
|
538 |
@{term "\<forall>p. p \<bullet> f = f"}
|
|
539 |
\end{equation}
|
|
540 |
|
|
541 |
\noindent or equivalently that a permutation applied to the application
|
1730
|
542 |
@{text "f x"} can be moved to the argument @{text x}. That means for equivariant
|
|
543 |
functions @{text f} we have for all permutations @{text p}
|
1703
|
544 |
%
|
|
545 |
\begin{equation}\label{equivariance}
|
1711
|
546 |
@{text "p \<bullet> f = f"} \;\;\;\textit{if and only if}\;\;\;
|
|
547 |
@{text "p \<bullet> (f x) = f (p \<bullet> x)"}
|
1703
|
548 |
\end{equation}
|
1694
|
549 |
|
|
550 |
\noindent
|
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diff
changeset
|
551 |
From property \eqref{equivariancedef} and the definition of @{text supp}, we
|
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diff
changeset
|
552 |
can be easily deduce that an equivariant function has empty support.
|
1711
|
553 |
|
1727
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changeset
|
554 |
Finally, the nominal logic work provides us with convenient means to rename
|
1711
|
555 |
binders. While in the older version of Nominal Isabelle, we used extensively
|
|
556 |
Property~\ref{swapfreshfresh} for renaming single binders, this property
|
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diff
changeset
|
557 |
proved unwieldy for dealing with multiple binders. For such pinders the
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
558 |
following generalisations turned out to be easier to use.
|
1711
|
559 |
|
|
560 |
\begin{property}\label{supppermeq}
|
|
561 |
@{thm[mode=IfThen] supp_perm_eq[no_vars]}
|
|
562 |
\end{property}
|
1517
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
563 |
|
1711
|
564 |
\begin{property}
|
1716
|
565 |
For a finite set @{text as} and a finitely supported @{text x} with
|
|
566 |
@{term "as \<sharp>* x"} and also a finitely supported @{text c}, there
|
|
567 |
exists a permutation @{text p} such that @{term "(p \<bullet> as) \<sharp>* c"} and
|
1711
|
568 |
@{term "supp x \<sharp>* p"}.
|
|
569 |
\end{property}
|
|
570 |
|
|
571 |
\noindent
|
1716
|
572 |
The idea behind the second property is that given a finite set @{text as}
|
1727
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Christian Urban <urbanc@in.tum.de>
diff
changeset
|
573 |
of binders (being bound, or fresh, in @{text x} is ensured by the
|
1716
|
574 |
assumption @{term "as \<sharp>* x"}), then there exists a permutation @{text p} such that
|
1727
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Christian Urban <urbanc@in.tum.de>
diff
changeset
|
575 |
the renamed binders @{term "p \<bullet> as"} avoid @{text c} (which can be arbitrarily chosen
|
1730
|
576 |
as long as it is finitely supported) and also @{text "p"} does not affect anything
|
1711
|
577 |
in the support of @{text x} (that is @{term "supp x \<sharp>* p"}). The last
|
|
578 |
fact and Property~\ref{supppermeq} allow us to ``rename'' just the binders
|
1727
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Christian Urban <urbanc@in.tum.de>
diff
changeset
|
579 |
@{text as} in @{text x}, because @{term "p \<bullet> x = x"}.
|
1711
|
580 |
|
1730
|
581 |
All properties given in this section are formalised in Isabelle/HOL and
|
1727
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Christian Urban <urbanc@in.tum.de>
diff
changeset
|
582 |
most of proofs are described in \cite{HuffmanUrban10} to which we refer the
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
583 |
reader. In the next sections we will make extensively use of these
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
584 |
properties in order to define alpha-equivalence in the presence of multiple
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
585 |
binders.
|
1493
|
586 |
*}
|
|
587 |
|
1485
|
588 |
|
1620
|
589 |
section {* General Binders\label{sec:binders} *}
|
1485
|
590 |
|
1517
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
591 |
text {*
|
1587
|
592 |
In Nominal Isabelle, the user is expected to write down a specification of a
|
|
593 |
term-calculus and then a reasoning infrastructure is automatically derived
|
1617
|
594 |
from this specification (remember that Nominal Isabelle is a definitional
|
1587
|
595 |
extension of Isabelle/HOL, which does not introduce any new axioms).
|
1579
|
596 |
|
1657
|
597 |
In order to keep our work with deriving the reasoning infrastructure
|
|
598 |
manageable, we will wherever possible state definitions and perform proofs
|
|
599 |
on the user-level of Isabelle/HOL, as opposed to write custom ML-code that
|
|
600 |
generates them anew for each specification. To that end, we will consider
|
|
601 |
first pairs @{text "(as, x)"} of type @{text "(atom set) \<times> \<beta>"}. These pairs
|
|
602 |
are intended to represent the abstraction, or binding, of the set @{text
|
|
603 |
"as"} in the body @{text "x"}.
|
1570
|
604 |
|
1727
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diff
changeset
|
605 |
The first question we have to answer is when two pairs @{text "(as, x)"} and
|
1657
|
606 |
@{text "(bs, y)"} are alpha-equivalent? (At the moment we are interested in
|
|
607 |
the notion of alpha-equivalence that is \emph{not} preserved by adding
|
|
608 |
vacuous binders.) To answer this, we identify four conditions: {\it i)}
|
|
609 |
given a free-variable function @{text "fv"} of type \mbox{@{text "\<beta> \<Rightarrow> atom
|
|
610 |
set"}}, then @{text x} and @{text y} need to have the same set of free
|
|
611 |
variables; moreover there must be a permutation @{text p} such that {\it
|
1687
|
612 |
ii)} @{text p} leaves the free variables of @{text x} and @{text y} unchanged, but
|
1657
|
613 |
{\it iii)} ``moves'' their bound names so that we obtain modulo a relation,
|
1662
|
614 |
say \mbox{@{text "_ R _"}}, two equivalent terms. We also require {\it iv)} that
|
|
615 |
@{text p} makes the sets of abstracted atoms @{text as} and @{text bs} equal. The
|
1657
|
616 |
requirements {\it i)} to {\it iv)} can be stated formally as follows:
|
1556
|
617 |
%
|
1572
|
618 |
\begin{equation}\label{alphaset}
|
|
619 |
\begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
|
1687
|
620 |
\multicolumn{2}{l}{@{term "(as, x) \<approx>gen R fv p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}\hspace{30mm}}\\[1mm]
|
1657
|
621 |
& @{term "fv(x) - as = fv(y) - bs"}\\
|
|
622 |
@{text "\<and>"} & @{term "(fv(x) - as) \<sharp>* p"}\\
|
|
623 |
@{text "\<and>"} & @{text "(p \<bullet> x) R y"}\\
|
|
624 |
@{text "\<and>"} & @{term "(p \<bullet> as) = bs"}\\
|
1572
|
625 |
\end{array}
|
1556
|
626 |
\end{equation}
|
|
627 |
|
|
628 |
\noindent
|
1657
|
629 |
Note that this relation is dependent on the permutation @{text
|
|
630 |
"p"}. Alpha-equivalence between two pairs is then the relation where we
|
|
631 |
existentially quantify over this @{text "p"}. Also note that the relation is
|
|
632 |
dependent on a free-variable function @{text "fv"} and a relation @{text
|
|
633 |
"R"}. The reason for this extra generality is that we will use
|
|
634 |
$\approx_{\textit{set}}$ for both ``raw'' terms and alpha-equated terms. In
|
1716
|
635 |
the latter case, $R$ will be replaced by equality @{text "="} and we
|
1727
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Christian Urban <urbanc@in.tum.de>
diff
changeset
|
636 |
will prove that @{text "fv"} is equal to @{text "supp"}.
|
1572
|
637 |
|
|
638 |
The definition in \eqref{alphaset} does not make any distinction between the
|
1579
|
639 |
order of abstracted variables. If we want this, then we can define alpha-equivalence
|
|
640 |
for pairs of the form \mbox{@{text "(as, x)"}} with type @{text "(atom list) \<times> \<beta>"}
|
|
641 |
as follows
|
1572
|
642 |
%
|
|
643 |
\begin{equation}\label{alphalist}
|
|
644 |
\begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
|
1687
|
645 |
\multicolumn{2}{l}{@{term "(as, x) \<approx>lst R fv p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}\hspace{30mm}}\\[1mm]
|
1657
|
646 |
& @{term "fv(x) - (set as) = fv(y) - (set bs)"}\\
|
|
647 |
\wedge & @{term "(fv(x) - set as) \<sharp>* p"}\\
|
1572
|
648 |
\wedge & @{text "(p \<bullet> x) R y"}\\
|
1657
|
649 |
\wedge & @{term "(p \<bullet> as) = bs"}\\
|
1572
|
650 |
\end{array}
|
|
651 |
\end{equation}
|
|
652 |
|
|
653 |
\noindent
|
1657
|
654 |
where @{term set} is a function that coerces a list of atoms into a set of atoms.
|
|
655 |
Now the last clause ensures that the order of the binders matters.
|
1556
|
656 |
|
1657
|
657 |
If we do not want to make any difference between the order of binders \emph{and}
|
1579
|
658 |
also allow vacuous binders, then we keep sets of binders, but drop the fourth
|
|
659 |
condition in \eqref{alphaset}:
|
1572
|
660 |
%
|
1579
|
661 |
\begin{equation}\label{alphares}
|
1572
|
662 |
\begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
|
1687
|
663 |
\multicolumn{2}{l}{@{term "(as, x) \<approx>res R fv p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}\hspace{30mm}}\\[1mm]
|
1657
|
664 |
& @{term "fv(x) - as = fv(y) - bs"}\\
|
|
665 |
\wedge & @{term "(fv(x) - as) \<sharp>* p"}\\
|
1572
|
666 |
\wedge & @{text "(p \<bullet> x) R y"}\\
|
|
667 |
\end{array}
|
|
668 |
\end{equation}
|
1556
|
669 |
|
1662
|
670 |
It might be useful to consider some examples for how these definitions of alpha-equivalence
|
|
671 |
pan out in practise.
|
1579
|
672 |
For this consider the case of abstracting a set of variables over types (as in type-schemes).
|
1657
|
673 |
We set @{text R} to be the equality and for @{text "fv(T)"} we define
|
1572
|
674 |
|
|
675 |
\begin{center}
|
1657
|
676 |
@{text "fv(x) = {x}"} \hspace{5mm} @{text "fv(T\<^isub>1 \<rightarrow> T\<^isub>2) = fv(T\<^isub>1) \<union> fv(T\<^isub>2)"}
|
1572
|
677 |
\end{center}
|
|
678 |
|
|
679 |
\noindent
|
1657
|
680 |
Now recall the examples shown in \eqref{ex1}, \eqref{ex2} and
|
1687
|
681 |
\eqref{ex3}. It can be easily checked that @{text "({x, y}, x \<rightarrow> y)"} and
|
1727
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Christian Urban <urbanc@in.tum.de>
diff
changeset
|
682 |
@{text "({y, x}, y \<rightarrow> x)"} are alpha-equivalent according to $\approx_{\textit{set}}$ and
|
1657
|
683 |
$\approx_{\textit{res}}$ by taking @{text p} to be the swapping @{term "(x \<rightleftharpoons>
|
|
684 |
y)"}. In case of @{text "x \<noteq> y"}, then @{text "([x, y], x \<rightarrow> y)"}
|
1687
|
685 |
$\not\approx_{\textit{list}}$ @{text "([y, x], x \<rightarrow> y)"} since there is no permutation
|
1657
|
686 |
that makes the lists @{text "[x, y]"} and @{text "[y, x]"} equal, and also
|
|
687 |
leaves the type \mbox{@{text "x \<rightarrow> y"}} unchanged. Another example is
|
1687
|
688 |
@{text "({x}, x)"} $\approx_{\textit{res}}$ @{text "({x, y}, x)"} which holds by
|
1657
|
689 |
taking @{text p} to be the
|
|
690 |
identity permutation. However, if @{text "x \<noteq> y"}, then @{text "({x}, x)"}
|
1687
|
691 |
$\not\approx_{\textit{set}}$ @{text "({x, y}, x)"} since there is no permutation
|
1657
|
692 |
that makes the
|
1687
|
693 |
sets @{text "{x}"} and @{text "{x, y}"} equal (similarly for $\approx_{\textit{list}}$).
|
|
694 |
It can also relatively easily be shown that all tree notions of alpha-equivalence
|
|
695 |
coincide, if we only abstract a single atom.
|
1579
|
696 |
|
1657
|
697 |
% looks too ugly
|
|
698 |
%\noindent
|
|
699 |
%Let $\star$ range over $\{set, res, list\}$. We prove next under which
|
|
700 |
%conditions the $\approx\hspace{0.05mm}_\star^{\fv, R, p}$ are equivalence
|
|
701 |
%relations and equivariant:
|
|
702 |
%
|
|
703 |
%\begin{lemma}
|
|
704 |
%{\it i)} Given the fact that $x\;R\;x$ holds, then
|
|
705 |
%$(as, x) \approx\hspace{0.05mm}^{\fv, R, 0}_\star (as, x)$. {\it ii)} Given
|
|
706 |
%that @{text "(p \<bullet> x) R y"} implies @{text "(-p \<bullet> y) R x"}, then
|
|
707 |
%$(as, x) \approx\hspace{0.05mm}^{\fv, R, p}_\star (bs, y)$ implies
|
|
708 |
%$(bs, y) \approx\hspace{0.05mm}^{\fv, R, - p}_\star (as, x)$. {\it iii)} Given
|
|
709 |
%that @{text "(p \<bullet> x) R y"} and @{text "(q \<bullet> y) R z"} implies
|
|
710 |
%@{text "((q + p) \<bullet> x) R z"}, then $(as, x) \approx\hspace{0.05mm}^{\fv, R, p}_\star (bs, y)$
|
|
711 |
%and $(bs, y) \approx\hspace{0.05mm}^{\fv, R, q}_\star (cs, z)$ implies
|
|
712 |
%$(as, x) \approx\hspace{0.05mm}^{\fv, R, q + p}_\star (cs, z)$. Given
|
|
713 |
%@{text "(q \<bullet> x) R y"} implies @{text "(p \<bullet> (q \<bullet> x)) R (p \<bullet> y)"} and
|
|
714 |
%@{text "p \<bullet> (fv x) = fv (p \<bullet> x)"} then @{text "p \<bullet> (fv y) = fv (p \<bullet> y)"}, then
|
|
715 |
%$(as, x) \approx\hspace{0.05mm}^{\fv, R, q}_\star (bs, y)$ implies
|
|
716 |
%$(p \;\isasymbullet\; as, p \;\isasymbullet\; x) \approx\hspace{0.05mm}^{\fv, R, q}_\star
|
|
717 |
%(p \;\isasymbullet\; bs, p \;\isasymbullet\; y)$.
|
|
718 |
%\end{lemma}
|
|
719 |
|
|
720 |
%\begin{proof}
|
|
721 |
%All properties are by unfolding the definitions and simple calculations.
|
|
722 |
%\end{proof}
|
|
723 |
|
|
724 |
|
1730
|
725 |
In the rest of this section we are going to introduce three abstraction
|
|
726 |
types. For this we define
|
1657
|
727 |
%
|
|
728 |
\begin{equation}
|
|
729 |
@{term "abs_set (as, x) (bs, x) \<equiv> \<exists>p. alpha_gen (as, x) equal supp p (bs, x)"}
|
|
730 |
\end{equation}
|
|
731 |
|
1579
|
732 |
\noindent
|
1687
|
733 |
(similarly for $\approx_{\textit{abs\_list}}$
|
|
734 |
and $\approx_{\textit{abs\_res}}$). We can show that these relations are equivalence
|
|
735 |
relations and equivariant.
|
1579
|
736 |
|
1687
|
737 |
\begin{lemma}\label{alphaeq} The relations
|
|
738 |
$\approx_{\textit{abs\_set}}$,
|
|
739 |
$\approx_{\textit{abs\_list}}$
|
|
740 |
and $\approx_{\textit{abs\_res}}$
|
|
741 |
are equivalence
|
1662
|
742 |
relations, and if @{term "abs_set (as, x) (bs, y)"} then also
|
1687
|
743 |
@{term "abs_set (p \<bullet> as, p \<bullet> x) (p \<bullet> bs, p \<bullet> y)"} (similarly for
|
|
744 |
the other two relations).
|
1657
|
745 |
\end{lemma}
|
|
746 |
|
|
747 |
\begin{proof}
|
|
748 |
Reflexivity is by taking @{text "p"} to be @{text "0"}. For symmetry we have
|
|
749 |
a permutation @{text p} and for the proof obligation take @{term "-p"}. In case
|
1662
|
750 |
of transitivity, we have two permutations @{text p} and @{text q}, and for the
|
|
751 |
proof obligation use @{text "q + p"}. All conditions are then by simple
|
1657
|
752 |
calculations.
|
|
753 |
\end{proof}
|
|
754 |
|
|
755 |
\noindent
|
1687
|
756 |
This lemma allows us to use our quotient package and introduce
|
1662
|
757 |
new types @{text "\<beta> abs_set"}, @{text "\<beta> abs_res"} and @{text "\<beta> abs_list"}
|
1687
|
758 |
representing alpha-equivalence classes of pairs. The elements in these types
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
759 |
will be, respectively, written as:
|
1657
|
760 |
|
|
761 |
\begin{center}
|
|
762 |
@{term "Abs as x"} \hspace{5mm}
|
|
763 |
@{term "Abs_lst as x"} \hspace{5mm}
|
|
764 |
@{term "Abs_res as x"}
|
|
765 |
\end{center}
|
|
766 |
|
1662
|
767 |
\noindent
|
1730
|
768 |
indicating that a set (or list) @{text as} is abstracted in @{text x}. We will
|
1716
|
769 |
call the types \emph{abstraction types} and their elements
|
1730
|
770 |
\emph{abstractions}. The important property we need to derive is what the
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
771 |
support of abstractions is, namely:
|
1662
|
772 |
|
1687
|
773 |
\begin{thm}[Support of Abstractions]\label{suppabs}
|
1703
|
774 |
Assuming @{text x} has finite support, then\\[-6mm]
|
1662
|
775 |
\begin{center}
|
1687
|
776 |
\begin{tabular}{l@ {\hspace{2mm}}c@ {\hspace{2mm}}l}
|
|
777 |
@{thm (lhs) supp_abs(1)[no_vars]} & $=$ & @{thm (rhs) supp_abs(1)[no_vars]}\\
|
|
778 |
@{thm (lhs) supp_abs(2)[no_vars]} & $=$ & @{thm (rhs) supp_abs(2)[no_vars]}\\
|
1716
|
779 |
@{thm (lhs) supp_abs(3)[where bs="as", no_vars]} & $=$ & @{thm (rhs) supp_abs(3)[where bs="as", no_vars]}
|
1687
|
780 |
\end{tabular}
|
1662
|
781 |
\end{center}
|
1687
|
782 |
\end{thm}
|
1662
|
783 |
|
|
784 |
\noindent
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
785 |
Below we will show the first equation. The others
|
1730
|
786 |
follow by similar arguments. By definition of the abstraction type @{text "abs_set"}
|
1687
|
787 |
we have
|
|
788 |
%
|
|
789 |
\begin{equation}\label{abseqiff}
|
1703
|
790 |
@{thm (lhs) abs_eq_iff(1)[where bs="as" and cs="bs", no_vars]} \;\;\text{if and only if}\;\;
|
1687
|
791 |
@{thm (rhs) abs_eq_iff(1)[where bs="as" and cs="bs", no_vars]}
|
|
792 |
\end{equation}
|
|
793 |
|
|
794 |
\noindent
|
1703
|
795 |
and also
|
|
796 |
%
|
|
797 |
\begin{equation}
|
|
798 |
@{thm permute_Abs[no_vars]}
|
|
799 |
\end{equation}
|
1662
|
800 |
|
1703
|
801 |
\noindent
|
1716
|
802 |
The second fact derives from the definition of permutations acting on pairs
|
|
803 |
(see \eqref{permute}) and alpha-equivalence being equivariant
|
|
804 |
(see Lemma~\ref{alphaeq}). With these two facts at our disposal, we can show
|
|
805 |
the following lemma about swapping two atoms.
|
1703
|
806 |
|
1662
|
807 |
\begin{lemma}
|
1716
|
808 |
@{thm[mode=IfThen] abs_swap1(1)[where bs="as", no_vars]}
|
1662
|
809 |
\end{lemma}
|
|
810 |
|
|
811 |
\begin{proof}
|
1730
|
812 |
This lemma is straightforward using \eqref{abseqiff} and observing that
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
813 |
the assumptions give us @{term "(a \<rightleftharpoons> b) \<bullet> (supp x - as) = (supp x - as)"}.
|
1730
|
814 |
Moreover @{text supp} and set difference are equivariant (see \cite{HuffmanUrban10}).
|
1662
|
815 |
\end{proof}
|
1587
|
816 |
|
1687
|
817 |
\noindent
|
1716
|
818 |
This lemma allows us to show
|
1687
|
819 |
%
|
|
820 |
\begin{equation}\label{halfone}
|
|
821 |
@{thm abs_supports(1)[no_vars]}
|
|
822 |
\end{equation}
|
|
823 |
|
|
824 |
\noindent
|
1716
|
825 |
which by Property~\ref{supportsprop} gives us ``one half'' of
|
|
826 |
Thm~\ref{suppabs}. The ``other half'' is a bit more involved. To establish
|
|
827 |
it, we use a trick from \cite{Pitts04} and first define an auxiliary
|
1727
fd2913415a73
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Christian Urban <urbanc@in.tum.de>
diff
changeset
|
828 |
function taking an abstraction as argument:
|
1687
|
829 |
%
|
|
830 |
\begin{center}
|
1703
|
831 |
@{thm supp_gen.simps[THEN eq_reflection, no_vars]}
|
1687
|
832 |
\end{center}
|
|
833 |
|
1703
|
834 |
\noindent
|
|
835 |
Using the second equation in \eqref{equivariance}, we can show that
|
1716
|
836 |
@{text "aux"} is equivariant (since @{term "p \<bullet> (supp x - as) =
|
|
837 |
(supp (p \<bullet> x)) - (p \<bullet> as)"}) and therefore has empty support.
|
|
838 |
This in turn means
|
1703
|
839 |
%
|
|
840 |
\begin{center}
|
1716
|
841 |
@{term "supp (supp_gen (Abs as x)) \<subseteq> supp (Abs as x)"}
|
1703
|
842 |
\end{center}
|
1687
|
843 |
|
|
844 |
\noindent
|
1716
|
845 |
using \eqref{suppfun}. Assuming @{term "supp x - as"} is a finite set
|
|
846 |
we further obtain
|
1703
|
847 |
%
|
|
848 |
\begin{equation}\label{halftwo}
|
|
849 |
@{thm (concl) supp_abs_subset1(1)[no_vars]}
|
|
850 |
\end{equation}
|
|
851 |
|
|
852 |
\noindent
|
1719
|
853 |
since for finite sets, @{text "S"}, we have @{thm (concl) supp_finite_atom_set[no_vars]}.
|
1703
|
854 |
|
1727
fd2913415a73
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Christian Urban <urbanc@in.tum.de>
diff
changeset
|
855 |
Finally, taking \eqref{halfone} and \eqref{halftwo} together establishes of
|
1730
|
856 |
Theorem~\ref{suppabs}. The method of first considering abstractions of the
|
|
857 |
form @{term "Abs as x"} etc is motivated by the fact that properties about them
|
|
858 |
can be conveninetly established at the Isabelle/HOL level. It would be
|
|
859 |
difficult to write custom ML-code that derives automatically such properties
|
|
860 |
for every term-constructor that binds some atoms. Also the generality of
|
|
861 |
the definitions for alpha-equivalence will also help us in the next section.
|
1517
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
862 |
*}
|
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
863 |
|
1717
|
864 |
section {* Alpha-Equivalence and Free Variables\label{sec:alpha} *}
|
1491
|
865 |
|
1520
|
866 |
text {*
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
867 |
Our choice of syntax for specifications is influenced by the existing
|
1719
|
868 |
datatype package of Isabelle/HOL \cite{Berghofer99} and by the syntax of the Ott-tool
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
869 |
\cite{ott-jfp}. For us a specification of a term-calculus is a collection of (possibly mutual
|
1637
|
870 |
recursive) type declarations, say @{text "ty"}$^\alpha_1$, \ldots,
|
|
871 |
@{text ty}$^\alpha_n$, and an associated collection
|
|
872 |
of binding functions, say @{text bn}$^\alpha_1$, \ldots, @{text
|
|
873 |
bn}$^\alpha_m$. The syntax in Nominal Isabelle for such specifications is
|
1693
|
874 |
roughly as follows:
|
1628
|
875 |
%
|
1619
|
876 |
\begin{equation}\label{scheme}
|
1636
|
877 |
\mbox{\begin{tabular}{@ {\hspace{-5mm}}p{1.8cm}l}
|
1617
|
878 |
type \mbox{declaration part} &
|
1611
|
879 |
$\begin{cases}
|
|
880 |
\mbox{\begin{tabular}{l}
|
1637
|
881 |
\isacommand{nominal\_datatype} @{text ty}$^\alpha_1 = \ldots$\\
|
|
882 |
\isacommand{and} @{text ty}$^\alpha_2 = \ldots$\\
|
1587
|
883 |
$\ldots$\\
|
1637
|
884 |
\isacommand{and} @{text ty}$^\alpha_n = \ldots$\\
|
1611
|
885 |
\end{tabular}}
|
|
886 |
\end{cases}$\\
|
1617
|
887 |
binding \mbox{function part} &
|
1611
|
888 |
$\begin{cases}
|
|
889 |
\mbox{\begin{tabular}{l}
|
1637
|
890 |
\isacommand{with} @{text bn}$^\alpha_1$ \isacommand{and} \ldots \isacommand{and} @{text bn}$^\alpha_m$\\
|
1611
|
891 |
\isacommand{where}\\
|
1587
|
892 |
$\ldots$\\
|
1611
|
893 |
\end{tabular}}
|
|
894 |
\end{cases}$\\
|
1619
|
895 |
\end{tabular}}
|
|
896 |
\end{equation}
|
1587
|
897 |
|
|
898 |
\noindent
|
1637
|
899 |
Every type declaration @{text ty}$^\alpha_{1..n}$ consists of a collection of
|
1611
|
900 |
term-constructors, each of which comes with a list of labelled
|
1620
|
901 |
types that stand for the types of the arguments of the term-constructor.
|
1637
|
902 |
For example a term-constructor @{text "C\<^sup>\<alpha>"} might have
|
1611
|
903 |
|
|
904 |
\begin{center}
|
1637
|
905 |
@{text "C\<^sup>\<alpha> label\<^isub>1::ty"}$'_1$ @{text "\<dots> label\<^isub>l::ty"}$'_l\;\;$ @{text "binding_clauses"}
|
1611
|
906 |
\end{center}
|
1587
|
907 |
|
1611
|
908 |
\noindent
|
1730
|
909 |
whereby some of the @{text ty}$'_{1..l}$ (or their components) are contained
|
|
910 |
in the collection of @{text ty}$^\alpha_{1..n}$ declared in
|
|
911 |
\eqref{scheme}. In this case we will call the corresponding argument a
|
|
912 |
\emph{recursive argument} of @{text "C\<^sup>\<alpha>"}. There are ``positivity''
|
|
913 |
restrictions imposed in the type of such recursive arguments, which ensure
|
|
914 |
that the type has a set-theoretic semantics \cite{Berghofer99}. The labels
|
|
915 |
annotated on the types are optional. Their purpose is to be used in the
|
|
916 |
(possibly empty) list of \emph{binding clauses}, which indicate the binders
|
|
917 |
and their scope in a term-constructor. They come in three \emph{modes}:
|
1587
|
918 |
|
1611
|
919 |
\begin{center}
|
1617
|
920 |
\begin{tabular}{l}
|
|
921 |
\isacommand{bind}\; {\it binders}\; \isacommand{in}\; {\it label}\\
|
|
922 |
\isacommand{bind\_set}\; {\it binders}\; \isacommand{in}\; {\it label}\\
|
|
923 |
\isacommand{bind\_res}\; {\it binders}\; \isacommand{in}\; {\it label}\\
|
|
924 |
\end{tabular}
|
1611
|
925 |
\end{center}
|
|
926 |
|
|
927 |
\noindent
|
1730
|
928 |
The first mode is for binding lists of atoms (the order of binders matters);
|
|
929 |
the second is for sets of binders (the order does not matter, but the
|
|
930 |
cardinality does) and the last is for sets of binders (with vacuous binders
|
|
931 |
preserving alpha-equivalence). The ``\isacommand{in}-part'' of a binding
|
|
932 |
clause will be called the \emph{body} of the abstraction; the
|
|
933 |
``\isacommand{bind}-part'' will be the \emph{binder} of the binding clause.
|
1620
|
934 |
|
1719
|
935 |
In addition we distinguish between \emph{shallow} and \emph{deep}
|
1620
|
936 |
binders. Shallow binders are of the form \isacommand{bind}\; {\it label}\;
|
1637
|
937 |
\isacommand{in}\; {\it label'} (similar for the other two modes). The
|
1620
|
938 |
restriction we impose on shallow binders is that the {\it label} must either
|
|
939 |
refer to a type that is an atom type or to a type that is a finite set or
|
1637
|
940 |
list of an atom type. Two examples for the use of shallow binders are the
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
941 |
specification of lambda-terms, where a single name is bound, and
|
1637
|
942 |
type-schemes, where a finite set of names is bound:
|
1611
|
943 |
|
|
944 |
\begin{center}
|
1612
|
945 |
\begin{tabular}{@ {}cc@ {}}
|
|
946 |
\begin{tabular}{@ {}l@ {\hspace{-1mm}}}
|
1719
|
947 |
\isacommand{nominal\_datatype} @{text lam} =\\
|
|
948 |
\hspace{5mm}\phantom{$\mid$}~@{text "Var name"}\\
|
|
949 |
\hspace{5mm}$\mid$~@{text "App lam lam"}\\
|
|
950 |
\hspace{5mm}$\mid$~@{text "Lam x::name t::lam"}\\
|
|
951 |
\hspace{21mm}\isacommand{bind} @{text x} \isacommand{in} @{text t}\\
|
1611
|
952 |
\end{tabular} &
|
1612
|
953 |
\begin{tabular}{@ {}l@ {}}
|
1719
|
954 |
\isacommand{nominal\_datatype}~@{text ty} =\\
|
|
955 |
\hspace{5mm}\phantom{$\mid$}~@{text "TVar name"}\\
|
|
956 |
\hspace{5mm}$\mid$~@{text "TFun ty ty"}\\
|
|
957 |
\isacommand{and}~@{text "tsc = All xs::(name fset) T::ty"}\\
|
|
958 |
\hspace{24mm}\isacommand{bind\_res} @{text xs} \isacommand{in} @{text T}\\
|
1611
|
959 |
\end{tabular}
|
|
960 |
\end{tabular}
|
|
961 |
\end{center}
|
1587
|
962 |
|
1612
|
963 |
\noindent
|
1637
|
964 |
Note that in this specification \emph{name} refers to an atom type.
|
1628
|
965 |
If we have shallow binders that ``share'' a body, for instance $t$ in
|
1637
|
966 |
the following term-constructor
|
1620
|
967 |
|
|
968 |
\begin{center}
|
|
969 |
\begin{tabular}{ll}
|
1719
|
970 |
@{text "Foo x::name y::name t::lam"} &
|
1723
|
971 |
\isacommand{bind} @{text x} \isacommand{in} @{text t},\;
|
|
972 |
\isacommand{bind} @{text y} \isacommand{in} @{text t}
|
1620
|
973 |
\end{tabular}
|
|
974 |
\end{center}
|
|
975 |
|
|
976 |
\noindent
|
1628
|
977 |
then we have to make sure the modes of the binders agree. We cannot
|
1637
|
978 |
have, for instance, in the first binding clause the mode \isacommand{bind}
|
|
979 |
and in the second \isacommand{bind\_set}.
|
1620
|
980 |
|
|
981 |
A \emph{deep} binder uses an auxiliary binding function that ``picks'' out
|
1636
|
982 |
the atoms in one argument of the term-constructor, which can be bound in
|
1628
|
983 |
other arguments and also in the same argument (we will
|
1637
|
984 |
call such binders \emph{recursive}, see below).
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
985 |
The corresponding binding functions are expected to return either a set of atoms
|
1620
|
986 |
(for \isacommand{bind\_set} and \isacommand{bind\_res}) or a list of atoms
|
|
987 |
(for \isacommand{bind}). They can be defined by primitive recursion over the
|
|
988 |
corresponding type; the equations must be given in the binding function part of
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
989 |
the scheme shown in \eqref{scheme}. For example a term-calculus containing lets
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
990 |
with tuple patterns might be specified as:
|
1617
|
991 |
|
1619
|
992 |
\begin{center}
|
|
993 |
\begin{tabular}{l}
|
1719
|
994 |
\isacommand{nominal\_datatype} @{text trm} =\\
|
|
995 |
\hspace{5mm}\phantom{$\mid$}~@{term "Var name"}\\
|
|
996 |
\hspace{5mm}$\mid$~@{term "App trm trm"}\\
|
|
997 |
\hspace{5mm}$\mid$~@{text "Lam x::name t::trm"}
|
|
998 |
\;\;\isacommand{bind} @{text x} \isacommand{in} @{text t}\\
|
|
999 |
\hspace{5mm}$\mid$~@{text "Let p::pat trm t::trm"}
|
|
1000 |
\;\;\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t}\\
|
|
1001 |
\isacommand{and} @{text pat} =\\
|
|
1002 |
\hspace{5mm}\phantom{$\mid$}~@{text PNil}\\
|
|
1003 |
\hspace{5mm}$\mid$~@{text "PVar name"}\\
|
|
1004 |
\hspace{5mm}$\mid$~@{text "PTup pat pat"}\\
|
|
1005 |
\isacommand{with}~@{text "bn::pat \<Rightarrow> atom list"}\\
|
|
1006 |
\isacommand{where}~@{text "bn(PNil) = []"}\\
|
|
1007 |
\hspace{5mm}$\mid$~@{text "bn(PVar x) = [atom x]"}\\
|
|
1008 |
\hspace{5mm}$\mid$~@{text "bn(PTup p\<^isub>1 p\<^isub>2) = bn(p\<^isub>1) @ bn(p\<^isub>2)"}\\
|
1619
|
1009 |
\end{tabular}
|
|
1010 |
\end{center}
|
1617
|
1011 |
|
1619
|
1012 |
\noindent
|
1637
|
1013 |
In this specification the function @{text "bn"} determines which atoms of @{text p} are
|
1719
|
1014 |
bound in the argument @{text "t"}. Note that in the second last clause the function @{text "atom"}
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1015 |
coerces a name into the generic atom type of Nominal Isabelle \cite{HuffmanUrban10}. This allows
|
1637
|
1016 |
us to treat binders of different atom type uniformly.
|
|
1017 |
|
|
1018 |
As will shortly become clear, we cannot return an atom in a binding function
|
|
1019 |
that is also bound in the corresponding term-constructor. That means in the
|
1723
|
1020 |
example above that the term-constructors @{text PVar} and @{text PTup} must not have a
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1021 |
binding clause. In the version of Nominal Isabelle described here, we also adopted
|
1637
|
1022 |
the restriction from the Ott-tool that binding functions can only return:
|
1723
|
1023 |
the empty set or empty list (as in case @{text PNil}), a singleton set or singleton
|
|
1024 |
list containing an atom (case @{text PVar}), or unions of atom sets or appended atom
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1025 |
lists (case @{text PTup}). This restriction will simplify definitions and
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1026 |
proofs later on.
|
1719
|
1027 |
|
|
1028 |
The most drastic restriction we have to impose on deep binders is that
|
1637
|
1029 |
we cannot have ``overlapping'' deep binders. Consider for example the
|
|
1030 |
term-constructors:
|
1617
|
1031 |
|
1620
|
1032 |
\begin{center}
|
|
1033 |
\begin{tabular}{ll}
|
1719
|
1034 |
@{text "Foo p::pat q::pat t::trm"} &
|
|
1035 |
\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t},\;
|
|
1036 |
\isacommand{bind} @{text "bn(q)"} \isacommand{in} @{text t}\\
|
|
1037 |
@{text "Foo' x::name p::pat t::trm"} &
|
|
1038 |
\isacommand{bind} @{text x} \isacommand{in} @{text t},\;
|
|
1039 |
\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t}
|
1620
|
1040 |
|
|
1041 |
\end{tabular}
|
|
1042 |
\end{center}
|
|
1043 |
|
|
1044 |
\noindent
|
1730
|
1045 |
In the first case we might bind all atoms from the pattern @{text p} in @{text t}
|
1637
|
1046 |
and also all atoms from @{text q} in @{text t}. As a result we have no way
|
|
1047 |
to determine whether the binder came from the binding function @{text
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1048 |
"bn(p)"} or @{text "bn(q)"}. Similarly in the second case. The reason why
|
1693
|
1049 |
we must exclude such specifications is that they cannot be represent by
|
1637
|
1050 |
the general binders described in Section \ref{sec:binders}. However
|
|
1051 |
the following two term-constructors are allowed
|
1620
|
1052 |
|
|
1053 |
\begin{center}
|
|
1054 |
\begin{tabular}{ll}
|
1719
|
1055 |
@{text "Bar p::pat t::trm s::trm"} &
|
|
1056 |
\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t},\;
|
|
1057 |
\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text s}\\
|
|
1058 |
@{text "Bar' p::pat t::trm"} &
|
|
1059 |
\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text p},\;
|
|
1060 |
\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t}\\
|
1620
|
1061 |
\end{tabular}
|
|
1062 |
\end{center}
|
|
1063 |
|
|
1064 |
\noindent
|
1628
|
1065 |
since there is no overlap of binders.
|
1619
|
1066 |
|
1637
|
1067 |
Note that in the last example we wrote {\it\isacommand{bind}\;bn(p)\;\isacommand{in}\;p}.
|
1693
|
1068 |
Whenever such a binding clause is present, we will call the binder \emph{recursive}.
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1069 |
To see the purpose for such recursive binders, compare ``plain'' @{text "Let"}s and @{text "Let_rec"}s:
|
1725
|
1070 |
%
|
|
1071 |
\begin{equation}\label{letrecs}
|
|
1072 |
\mbox{%
|
1637
|
1073 |
\begin{tabular}{@ {}l@ {}}
|
1725
|
1074 |
\isacommand{nominal\_datatype}~@{text "trm ="}\\
|
1636
|
1075 |
\hspace{5mm}\phantom{$\mid$}\ldots\\
|
1725
|
1076 |
\hspace{5mm}$\mid$~@{text "Let as::assn t::trm"}
|
|
1077 |
\;\;\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text t}\\
|
|
1078 |
\hspace{5mm}$\mid$~@{text "Let_rec as::assn t::trm"}
|
|
1079 |
\;\;\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text t},
|
|
1080 |
\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text as}\\
|
1636
|
1081 |
\isacommand{and} {\it assn} =\\
|
1725
|
1082 |
\hspace{5mm}\phantom{$\mid$}~@{text "ANil"}\\
|
|
1083 |
\hspace{5mm}$\mid$~@{text "ACons name trm assn"}\\
|
|
1084 |
\isacommand{with} @{text "bn::assn \<Rightarrow> atom list"}\\
|
|
1085 |
\isacommand{where}~@{text "bn(ANil) = []"}\\
|
|
1086 |
\hspace{5mm}$\mid$~@{text "bn(ACons a t as) = [atom a] @ bn(as)"}\\
|
|
1087 |
\end{tabular}}
|
|
1088 |
\end{equation}
|
1636
|
1089 |
|
|
1090 |
\noindent
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1091 |
The difference is that with @{text Let} we only want to bind the atoms @{text
|
1730
|
1092 |
"bn(as)"} in the term @{text t}, but with @{text "Let_rec"} we also want to bind the atoms
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1093 |
inside the assignment. This difference has consequences for the free-variable
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1094 |
function and alpha-equivalence relation, which we are going to describe in the
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1095 |
rest of this section.
|
1637
|
1096 |
|
|
1097 |
Having dealt with all syntax matters, the problem now is how we can turn
|
|
1098 |
specifications into actual type definitions in Isabelle/HOL and then
|
|
1099 |
establish a reasoning infrastructure for them. Because of the problem
|
|
1100 |
Pottier and Cheney pointed out, we cannot in general re-arrange arguments of
|
|
1101 |
term-constructors so that binders and their bodies are next to each other, and
|
|
1102 |
then use the type constructors @{text "abs_set"}, @{text "abs_res"} and
|
|
1103 |
@{text "abs_list"} from Section \ref{sec:binders}. Therefore we will first
|
1719
|
1104 |
extract datatype definitions from the specification and then define
|
|
1105 |
independently an alpha-equivalence relation over them.
|
1637
|
1106 |
|
|
1107 |
|
1724
|
1108 |
The datatype definition can be obtained by stripping off the
|
1637
|
1109 |
binding clauses and the labels on the types. We also have to invent
|
|
1110 |
new names for the types @{text "ty\<^sup>\<alpha>"} and term-constructors @{text "C\<^sup>\<alpha>"}
|
1730
|
1111 |
given by user. In our implementation we just use the affix ``@{text "_raw"}''.
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1112 |
But for the purpose of this paper, we just use the superscript @{text "_\<^sup>\<alpha>"} to indicate
|
1724
|
1113 |
that a notion is defined over alpha-equivalence classes and leave it out
|
|
1114 |
for the corresponding notion defined on the ``raw'' level. So for example
|
|
1115 |
we have
|
|
1116 |
|
1636
|
1117 |
\begin{center}
|
1723
|
1118 |
@{text "ty\<^sup>\<alpha> \<mapsto> ty"} \hspace{2mm}and\hspace{2mm} @{text "C\<^sup>\<alpha> \<mapsto> C"}
|
1636
|
1119 |
\end{center}
|
|
1120 |
|
|
1121 |
\noindent
|
1730
|
1122 |
where @{term ty} is the type used in the quotient construction for
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1123 |
@{text "ty\<^sup>\<alpha>"} and @{text "C"} is the term-constructor on the ``raw'' type @{text "ty"}.
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1124 |
|
1637
|
1125 |
The resulting datatype definition is legal in Isabelle/HOL provided the datatypes are
|
|
1126 |
non-empty and the types in the constructors only occur in positive
|
1724
|
1127 |
position (see \cite{Berghofer99} for an indepth description of the datatype package
|
1637
|
1128 |
in Isabelle/HOL). We then define the user-specified binding
|
1730
|
1129 |
functions, called @{term "bn"}, by primitive recursion over the corresponding
|
|
1130 |
raw datatype. We can also easily define permutation operations by
|
1724
|
1131 |
primitive recursion so that for each term constructor @{text "C ty\<^isub>1 \<dots> ty\<^isub>n"}
|
|
1132 |
we have that
|
1587
|
1133 |
|
1628
|
1134 |
\begin{center}
|
1724
|
1135 |
@{text "p \<bullet> (C x\<^isub>1 \<dots> x\<^isub>n) = C (p \<bullet> x\<^isub>1) \<dots> (p \<bullet> x\<^isub>n)"}
|
1628
|
1136 |
\end{center}
|
|
1137 |
|
1719
|
1138 |
% TODO: we did not define permutation types
|
|
1139 |
%\noindent
|
|
1140 |
%From this definition we can easily show that the raw datatypes are
|
|
1141 |
%all permutation types (Def ??) by a simple structural induction over
|
|
1142 |
%the @{text "ty"}s.
|
1637
|
1143 |
|
1693
|
1144 |
The first non-trivial step we have to perform is the generation free-variable
|
1723
|
1145 |
functions from the specifications. Given the raw types @{text "ty\<^isub>1 \<dots> ty\<^isub>n"}
|
|
1146 |
we need to define free-variable functions
|
1637
|
1147 |
|
|
1148 |
\begin{center}
|
1723
|
1149 |
@{text "fv_ty\<^isub>1 :: ty\<^isub>1 \<Rightarrow> atom set \<dots> fv_ty\<^isub>n :: ty\<^isub>n \<Rightarrow> atom set"}
|
1637
|
1150 |
\end{center}
|
|
1151 |
|
|
1152 |
\noindent
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1153 |
We define them together with auxiliary free-variable functions for
|
1724
|
1154 |
the binding functions. Given binding functions
|
1730
|
1155 |
@{text "bn\<^isub>1 \<dots> bn\<^isub>m"} we need to define
|
1724
|
1156 |
%
|
1637
|
1157 |
\begin{center}
|
1730
|
1158 |
@{text "fv_bn\<^isub>1 :: ty\<^isub>1 \<Rightarrow> atom set \<dots> fv_bn\<^isub>m :: ty\<^isub>m \<Rightarrow> atom set"}
|
1637
|
1159 |
\end{center}
|
1636
|
1160 |
|
1637
|
1161 |
\noindent
|
1724
|
1162 |
The reason for this setup is that in a deep binder not all atoms have to be
|
1730
|
1163 |
bound, as we shall see in an example below. We need therefore the function
|
|
1164 |
that returns us those unbound atoms.
|
|
1165 |
|
|
1166 |
While the idea behind these
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1167 |
free-variable functions is clear (they just collect all atoms that are not bound),
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1168 |
because of the rather complicated binding mechanisms their definitions are
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1169 |
somewhat involved.
|
1723
|
1170 |
Given a term-constructor @{text "C"} of type @{text ty} with argument types
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1171 |
\mbox{@{text "ty\<^isub>1 \<dots> ty\<^isub>n"}}, the function
|
1723
|
1172 |
@{text "fv_ty (C x\<^isub>1 \<dots> x\<^isub>n)"} will be the union of the values
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1173 |
calculated next for each argument.
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1174 |
First we deal with the case that @{text "x\<^isub>i"} is a binder. From the binding clauses,
|
1724
|
1175 |
we can determine whether the argument is a shallow or deep
|
1723
|
1176 |
binder, and in the latter case also whether it is a recursive or
|
1724
|
1177 |
non-recursive binder.
|
1628
|
1178 |
|
|
1179 |
\begin{center}
|
1724
|
1180 |
\begin{tabular}{c@ {\hspace{2mm}}p{7cm}}
|
1636
|
1181 |
$\bullet$ & @{term "{}"} provided @{text "x\<^isub>i"} is a shallow binder\\
|
1730
|
1182 |
$\bullet$ & @{text "fv_bn x\<^isub>i"} provided @{text "x\<^isub>i"} is a deep
|
|
1183 |
non-recursive binder with the auxiliary binding function @{text "bn"}\\
|
|
1184 |
$\bullet$ & @{text "fv_ty\<^isub>i x\<^isub>i - bn x\<^isub>i"} provided @{text "x\<^isub>i"} is
|
|
1185 |
a deep recursive binder with the auxiliary binding function @{text "bn"}
|
1628
|
1186 |
\end{tabular}
|
|
1187 |
\end{center}
|
|
1188 |
|
1636
|
1189 |
\noindent
|
1724
|
1190 |
The first clause states that shallow binders do not contribute to the
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1191 |
free variables; in the second clause, we have to collect all
|
1730
|
1192 |
variables that are left unbound by the binding function @{text "bn"}---this
|
|
1193 |
is done with function @{text "fv_bn"}; in the third clause, since the
|
1724
|
1194 |
binder is recursive, we need to bind all variables specified by
|
1730
|
1195 |
@{text "bn"}---therefore we subtract @{text "bn x\<^isub>i"} from the free
|
1724
|
1196 |
variables of @{text "x\<^isub>i"}.
|
|
1197 |
|
|
1198 |
In case the argument is \emph{not} a binder, we need to consider
|
|
1199 |
whether the @{text "x\<^isub>i"} is the body of one or more binding clauses.
|
|
1200 |
In this case we first calculate the set @{text "bnds"} as follows:
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1201 |
either the corresponding binders are all shallow or there is a single deep binder.
|
1724
|
1202 |
In the former case we take @{text bnds} to be the union of all shallow
|
|
1203 |
binders; in the latter case, we just take the set of atoms specified by the
|
|
1204 |
binding function. The value for @{text "x\<^isub>i"} is then given by:
|
1709
|
1205 |
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1206 |
\begin{equation}\label{deepbody}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1207 |
\mbox{%
|
1724
|
1208 |
\begin{tabular}{c@ {\hspace{2mm}}p{7cm}}
|
1636
|
1209 |
$\bullet$ & @{text "{atom x\<^isub>i} - bnds"} provided @{term "x\<^isub>i"} is an atom\\
|
|
1210 |
$\bullet$ & @{text "(atoms x\<^isub>i) - bnds"} provided @{term "x\<^isub>i"} is a set of atoms\\
|
1657
|
1211 |
$\bullet$ & @{text "(atoms (set x\<^isub>i)) - bnds"} provided @{term "x\<^isub>i"} is a list of atoms\\
|
1724
|
1212 |
$\bullet$ & @{text "(fv_ty\<^isub>i x\<^isub>i) - bnds"} provided @{term "ty\<^isub>i"} is one of the raw datatypes
|
|
1213 |
corresponding to the types specified by the user\\
|
1715
3d6df74fc934
Avoid mentioning other nominal datatypes as it makes things too complicated.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
1214 |
% $\bullet$ & @{text "(fv\<^isup>\<alpha> x\<^isub>i) - bnds"} provided @{term "x\<^isub>i"} is a defined nominal datatype
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1215 |
% with a free-variable function @{text "fv\<^isup>\<alpha>"}\\
|
1709
|
1216 |
$\bullet$ & @{term "{}"} otherwise
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1217 |
\end{tabular}}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1218 |
\end{equation}
|
1628
|
1219 |
|
1723
|
1220 |
\noindent
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1221 |
Like the coercion function @{text atom} used above, @{text "atoms as"} coerces
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1222 |
the set @{text as} to the generic atom type.
|
1724
|
1223 |
It is defined as @{text "atom as \<equiv> {atom a | a \<in> as}"}.
|
1637
|
1224 |
|
1724
|
1225 |
The last case we need to consider is when @{text "x\<^isub>i"} is neither
|
|
1226 |
a binder nor a body of an abstraction. In this case it is defined
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1227 |
as in \eqref{deepbody}, except that we do not need to subtract the
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1228 |
set @{text bnds}.
|
1724
|
1229 |
|
1730
|
1230 |
Next, we need to define a free-variable function @{text "fv_bn\<^isub>j"} for
|
|
1231 |
each binding function @{text "bn\<^isub>j"}. The idea behind this
|
1724
|
1232 |
function is to compute the set of free atoms that are not bound by
|
1730
|
1233 |
@{text "bn\<^isub>j"}. Because of the restrictions we imposed on the
|
1724
|
1234 |
form of binding functions, this can be done automatically by recursively
|
|
1235 |
building up the the set of free variables from the arguments that are
|
|
1236 |
not bound. Let us assume one clause of the binding function is
|
1730
|
1237 |
@{text "bn\<^isub>j (C x\<^isub>1 \<dots> x\<^isub>n) = rhs"}, then @{text "fv_bn\<^isub>j"} is the
|
|
1238 |
union of the values calculated for @{text "x\<^isub>i"} of type @{text "ty\<^isub>i"}
|
1724
|
1239 |
as follows:
|
1637
|
1240 |
|
|
1241 |
\begin{center}
|
1724
|
1242 |
\begin{tabular}{c@ {\hspace{2mm}}p{7cm}}
|
1730
|
1243 |
\multicolumn{2}{l}{@{text "x\<^isub>i"} occurs in @{text "rhs"}:}\\
|
|
1244 |
$\bullet$ & @{term "{}"} provided @{term "x\<^isub>i"} is a single atom,
|
1709
|
1245 |
atom list or atom set\\
|
1730
|
1246 |
$\bullet$ & @{text "fv_bn x\<^isub>i"} in case @{text "rhs"} contains the
|
|
1247 |
recursive call @{text "bn x\<^isub>i"}\\[1mm]
|
1724
|
1248 |
%
|
1730
|
1249 |
\multicolumn{2}{l}{@{text "x\<^isub>i"} does not occur in @{text "rhs"}:}\\
|
|
1250 |
$\bullet$ & @{text "atoms x\<^isub>i"} provided @{term "x\<^isub>i"} is a set of atoms\\
|
|
1251 |
$\bullet$ & @{term "atoms (set x\<^isub>i)"} provided @{term "x\<^isub>i"} is a list of atoms\\
|
|
1252 |
$\bullet$ & @{text "fv_ty\<^isub>i x\<^isub>i"} provided @{term "ty\<^isub>i"} is one of the raw
|
1724
|
1253 |
types corresponding to the types specified by the user\\
|
1730
|
1254 |
% $\bullet$ & @{text "fv_ty\<^isup>\<alpha> x\<^isub>i - bnds"} provided @{term "x\<^isub>i"} is not in @{text "rhs"}
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1255 |
% and is an existing nominal datatype with the free-variable function @{text "fv\<^isup>\<alpha>"}\\
|
1706
|
1256 |
$\bullet$ & @{term "{}"} otherwise
|
1637
|
1257 |
\end{tabular}
|
|
1258 |
\end{center}
|
|
1259 |
|
1725
|
1260 |
\noindent
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1261 |
To see how these definitions work, let us consider again the term-constructors
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1262 |
@{text "Let"} and @{text "Let_rec"} from example shown in \eqref{letrecs}.
|
1725
|
1263 |
For this specification we need to define three functions, namely
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1264 |
@{text "fv\<^bsub>trm\<^esub>"}, @{text "fv\<^bsub>assn\<^esub>"} and @{text "fv\<^bsub>bn\<^esub>"}. They are as follows:
|
1725
|
1265 |
%
|
|
1266 |
\begin{center}
|
|
1267 |
\begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}}
|
|
1268 |
@{text "fv\<^bsub>trm\<^esub> (Let as t)"} & @{text "="} & @{text "fv\<^bsub>bn\<^esub> as \<union> (fv\<^bsub>trm\<^esub> t - set (bn as))"}\\
|
|
1269 |
@{text "fv\<^bsub>trm\<^esub> (Let_rec as t)"} & @{text "="} &\\
|
|
1270 |
\multicolumn{3}{r}{@{text "(fv\<^bsub>assn\<^esub> as - set (bn as)) \<union> (fv\<^bsub>trm\<^esub> t - set (bn as))"}}\\[1mm]
|
|
1271 |
|
|
1272 |
@{text "fv\<^bsub>assn\<^esub> (ANil)"} & @{text "="} & @{text "[]"}\\
|
|
1273 |
@{text "fv\<^bsub>assn\<^esub> (ACons a t as)"} & @{text "="} & @{text "{atom a} \<union> (fv\<^bsub>trm\<^esub> t) \<union> (fv\<^bsub>assn\<^esub> as)"}\\[1mm]
|
|
1274 |
|
|
1275 |
@{text "fv\<^bsub>bn\<^esub> (ANil)"} & @{text "="} & @{text "[]"}\\
|
|
1276 |
@{text "fv\<^bsub>bn\<^esub> (ACons a t as)"} & @{text "="} & @{text "(fv\<^bsub>trm\<^esub> t) \<union> (fv\<^bsub>bn\<^esub> as)"}
|
|
1277 |
\end{tabular}
|
|
1278 |
\end{center}
|
|
1279 |
|
|
1280 |
\noindent
|
|
1281 |
Since there are no binding clauses for the term-constructors @{text ANil}
|
|
1282 |
and @{text "ACons"}, the corresponding free-variable function @{text
|
|
1283 |
"fv\<^bsub>assn\<^esub>"} returns all atoms occuring in an assignment. The
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1284 |
binding only takes place in @{text Let} and @{text "Let_rec"}. In the @{text
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1285 |
"Let"}-clause we want to bind all atoms given by @{text "set (bn as)"} in
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1286 |
@{text t}. Therefore we have to subtract @{text "set (bn as)"} from @{text
|
1725
|
1287 |
"fv\<^bsub>trm\<^esub> t"}. However, we also need to add all atoms that are
|
|
1288 |
free in @{text "as"}. This is what the purpose of the function @{text
|
|
1289 |
"fv\<^bsub>bn\<^esub>"} is. In contrast, in @{text "Let_rec"} we have a
|
1727
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Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1290 |
recursive binder where we want to also bind all occurences of the atoms
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1291 |
@{text "bn as"} inside @{text "as"}. Therefore we have to subtract @{text
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1292 |
"set (bn as)"} from @{text "fv\<^bsub>assn\<^esub> as"}, as well as from
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1293 |
@{text "fv\<^bsub>trm\<^esub> t"}. An interesting point in this example is
|
1725
|
1294 |
that an assignment ``alone'' does not have any bound variables. Only in the
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1295 |
context of a @{text Let} or @{text "Let_rec"} will some atoms become bound
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1296 |
(teh term-constructors that have binding clauses). This is a phenomenon
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1297 |
that has also been pointed out in \cite{ott-jfp}.
|
1725
|
1298 |
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1299 |
Next we define alpha-equivalence realtions for the types @{text "ty\<^isub>1, \<dots>, ty\<^isub>n"}. We call them
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1300 |
@{text "\<approx>ty\<^isub>1 \<dots> \<approx>ty\<^isub>n"}. Like with the free-variable functions,
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1301 |
we also need to define auxiliary alpha-equivalence relations for the binding functions.
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1302 |
Say we have @{text "bn_ty\<^isub>1 \<dots> bn_ty\<^isub>m"}, we also define @{text "\<approx>bn_ty\<^isub>1 \<dots> \<approx>bn_ty\<^isub>n"}.
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1303 |
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1304 |
The relations are inductively defined predicates, whose clauses have
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1305 |
conclusions of the form @{text "C x\<^isub>1 \<dots> x\<^isub>n \<approx>ty C y\<^isub>1 \<dots> y\<^isub>n"} (let us assume
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1306 |
@{text C} is of type @{text ty} and its arguments are specified as @{text "C ty\<^isub>1 \<dots> ty\<^isub>n"}).
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1307 |
The task is to specify what the premises of these clauses are. For this we
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1308 |
consider the pairs @{text "(x\<^isub>i, y\<^isub>i)"} which necesarily must have the same type, say
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1309 |
@{text "ty\<^isub>i"}. For each of these pairs we calculate a premise as follows.
|
1705
|
1310 |
|
|
1311 |
\begin{center}
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1312 |
\begin{tabular}{c@ {\hspace{2mm}}p{7cm}}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1313 |
\multicolumn{2}{l}{@{text "x\<^isub>i"} is a binder:}\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1314 |
$\bullet$ & @{text "True"} provided @{text "x\<^isub>i"} is a shallow binder\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1315 |
$\bullet$ & @{text "x\<^isub>i \<approx>bn_ty\<^isub>i y\<^isub>i"} provided @{text "x\<^isub>i"} is a deep
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1316 |
non-recursive binder\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1317 |
$\bullet$ & @{text "True"} provided @{text "x\<^isub>i"} is a deep
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1318 |
recursive binder\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1319 |
\end{tabular}
|
1705
|
1320 |
\end{center}
|
|
1321 |
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1322 |
TODO BELOW
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1323 |
|
1705
|
1324 |
\begin{center}
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1325 |
\begin{tabular}{c@ {\hspace{2mm}}p{7cm}}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1326 |
\multicolumn{2}{l}{@{text "x\<^isub>i"} is a body where the binding clause has mode \isacommand{bind}:}\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1327 |
$\bullet$ & @{text "\<exists>p. (bnds_x\<^isub>i, x\<^isub>i) \<approx>lst (\<approx>ty\<^isub>i) fv_ty\<^isub>i p (bnds_y\<^isub>i, y\<^isub>j)"}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1328 |
provided @{text "x\<^isub>i"} has only shallow binders; in this case @{text "bnds_x\<^isub>i"} is the
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1329 |
union of all these shallow binders (similarly for @{text "bnds_y\<^isub>i"}\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1330 |
$\bullet$ & @{text "\<exists>p. (bn_ty\<^isub>j x\<^isub>j, x\<^isub>i) \<approx>lst (\<approx>ty\<^isub>i) fv_ty\<^isub>i p (bn_ty y\<^isub>j, y\<^isub>i)"}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1331 |
provided @{text "x\<^isub>i"} is a body with a deep non-recursive binder @{text x\<^isub>j}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1332 |
(similarly @{text "y\<^isub>j"} is the deep non-recursive binder for @{text "y\<^isub>i"})\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1333 |
$\bullet$ & @{text "\<exists>p (bn_ty\<^isub>i x\<^isub>i, (x\<^isub>j, x\<^isub>n)) \<approx>lst R fvs \<pi> (bn\<^isub>m y\<^isub>j, (y\<^isub>j, y\<^isub>n))"}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1334 |
provided @{text "x\<^isub>j"} is a deep recursive binder with the auxiliary binding
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1335 |
function @{text "bn\<^isub>m"} and permutation @{text "\<pi>"}, @{term "fvs"} is a compound
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1336 |
free variable function returning the union of appropriate @{term "fv_ty\<^isub>x"} and
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1337 |
@{term "R"} is the composition of equivalence relations @{text "\<approx>"} and @{text "\<approx>\<^isub>n"}\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1338 |
$\bullet$ & @{text "x\<^isub>j"} has a deep recursive binding\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1339 |
$\bullet$ & @{text "({x\<^isub>n}, x\<^isub>j) \<approx>gen R fv_ty \<pi> ({y\<^isub>n}, y\<^isub>j)"} provided @{text "x\<^isub>j"} has
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1340 |
a shallow binder @{text "x\<^isub>n"} with permutation @{text "\<pi>"}, @{term "R"} is the
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1341 |
alpha-equivalence for @{term "x\<^isub>j"}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1342 |
and @{term "fv_ty"} is the free-variable function for @{term "x\<^isub>j"}\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1343 |
$\bullet$ & @{text "(bn\<^isub>m x\<^isub>n, x\<^isub>j) \<approx>gen R fv_ty \<pi> (bn\<^isub>m y\<^isub>n, y\<^isub>j)"} provided @{text "x\<^isub>j"}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1344 |
has a deep non-recursive binder @{text "bn\<^isub>m x\<^isub>n"} with permutation @{text "\<pi>"}, @{term "R"} is the
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1345 |
alpha-equivalence for @{term "x\<^isub>j"}
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1346 |
and @{term "fv_ty"} is the free-variable function for @{term "x\<^isub>j"}\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1347 |
$\bullet$ & @{text "x\<^isub>j \<approx>\<^isub>j y\<^isub>j"} provided @{term "x\<^isub>j"} is one of the types being
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1348 |
defined\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1349 |
$\bullet$ & @{text "x\<^isub>j = y\<^isub>j"} otherwise\\
|
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1350 |
\end{tabular}
|
1705
|
1351 |
\end{center}
|
|
1352 |
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1353 |
, of a type @{text ty}, two instances
|
1719
|
1354 |
of this constructor are alpha-equivalent @{text "C x\<^isub>1 \<dots> x\<^isub>n \<approx> C y\<^isub>1 \<dots> y\<^isub>n"} if there
|
1707
|
1355 |
exist permutations @{text "\<pi>\<^isub>1 \<dots> \<pi>\<^isub>p"} (one for each bound argument) such that
|
1710
|
1356 |
the conjunction of equivalences defined below for each argument pair @{text "x\<^isub>j"}, @{text "y\<^isub>j"} holds.
|
1706
|
1357 |
For an argument pair @{text "x\<^isub>j"}, @{text "y\<^isub>j"} this holds if:
|
|
1358 |
|
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1359 |
|
1705
|
1360 |
|
1708
|
1361 |
The alpha-equivalence relations for binding functions are similar to the alpha-equivalences
|
|
1362 |
for their respective types, the difference is that they ommit checking the arguments that
|
|
1363 |
are bound. We assumed that there are no bindings in the type on which the binding function
|
|
1364 |
is defined so, there are no permutations involved. For a binding function clause
|
1719
|
1365 |
@{text "bn (C x\<^isub>1 \<dots> x\<^isub>n) = rhs"}, two instances of the constructor are equivalent
|
|
1366 |
@{text "C x\<^isub>1 \<dots> x\<^isub>n \<approx> C y\<^isub>1 \<dots> y\<^isub>n"} if:
|
1708
|
1367 |
\begin{center}
|
|
1368 |
\begin{tabular}{cp{7cm}}
|
1715
3d6df74fc934
Avoid mentioning other nominal datatypes as it makes things too complicated.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
1369 |
$\bullet$ & @{text "x\<^isub>j"} is not of a type being defined and occurs in @{text "rhs"}\\
|
3d6df74fc934
Avoid mentioning other nominal datatypes as it makes things too complicated.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
1370 |
$\bullet$ & @{text "x\<^isub>j = y\<^isub>j"} provided @{text "x\<^isub>j"} is not of a type being defined
|
3d6df74fc934
Avoid mentioning other nominal datatypes as it makes things too complicated.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
1371 |
and does not occur in @{text "rhs"}\\
|
3d6df74fc934
Avoid mentioning other nominal datatypes as it makes things too complicated.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
1372 |
$\bullet$ & @{text "x\<^isub>j \<approx>bn\<^isub>m y\<^isub>j"} provided @{text "x\<^isub>j"} is of a type being defined
|
3d6df74fc934
Avoid mentioning other nominal datatypes as it makes things too complicated.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
|
1373 |
occuring in @{text "rhs"} under the binding function @{text "bn\<^isub>m"}\\
|
1708
|
1374 |
$\bullet$ & @{text "x\<^isub>j \<approx> y\<^isub>j"} otherwise\\
|
|
1375 |
\end{tabular}
|
|
1376 |
\end{center}
|
|
1377 |
|
1587
|
1378 |
*}
|
|
1379 |
|
1717
|
1380 |
section {* The Lifting of Definitions and Properties *}
|
|
1381 |
|
|
1382 |
text {*
|
1721
|
1383 |
To define the quotient types we first need to show that the defined
|
|
1384 |
relations are equivalence relations.
|
1717
|
1385 |
|
1718
|
1386 |
\begin{lemma} The relations @{text "\<approx>\<^isub>1 \<dots> \<approx>\<^isub>1"} and @{text "\<approx>bn\<^isub>1 \<dots> \<approx>bn\<^isub>m"}
|
|
1387 |
defined as above are equivalence relations and are equivariant.
|
1717
|
1388 |
\end{lemma}
|
1718
|
1389 |
\begin{proof} Reflexivity by induction on the raw datatype. Symmetry,
|
|
1390 |
transitivity and equivariance by induction on the alpha equivalence
|
|
1391 |
relation. Using lemma \ref{alphaeq}, the conditions follow by simple
|
|
1392 |
calculations. \end{proof}
|
|
1393 |
|
|
1394 |
\noindent We then define the quotient types @{text "ty\<^isub>1\<^isup>\<alpha> \<dots> ty\<^isub>n\<^isup>\<alpha>"}. To lift
|
|
1395 |
the raw definitions to the quotient type, we need to prove that they
|
|
1396 |
\emph{respect} the relation. We follow the definition of respectfullness given
|
|
1397 |
by Homeier~\cite{Homeier05}. The intuition behind a respectfullness condition
|
|
1398 |
is that when a function (or constructor) is given arguments that are
|
|
1399 |
alpha-equivalent the results are also alpha equivalent. For arguments that are
|
|
1400 |
not of any of the relations taken into account, equivalence is replaced by
|
|
1401 |
equality. In particular the respectfullness condition for a @{text "bn"}
|
|
1402 |
function means that for alpha equivalent raw terms it returns the same bound
|
|
1403 |
names. Thanks to the restrictions on the binding functions introduced in
|
|
1404 |
Section~\ref{sec:alpha} we can show that are respectful.
|
1717
|
1405 |
|
1718
|
1406 |
\begin{lemma} The functions @{text "bn\<^isub>1 \<dots> bn\<^isub>m"}, @{text "fv_ty\<^isub>1 \<dots> fv_ty\<^isub>n"},
|
|
1407 |
the raw constructors, the raw permutations and @{text "\<approx>bn\<^isub>1 \<dots> \<approx>bn\<^isub>m"} are
|
|
1408 |
respectful w.r.t. the relations @{text "\<approx>\<^isub>1 \<dots> \<approx>\<^isub>n"}.
|
|
1409 |
\end{lemma}
|
|
1410 |
\begin{proof} Respectfullness of permutations is a direct consequence of
|
|
1411 |
equivariance. All other properties by induction on the alpha-equivalence
|
|
1412 |
relation. For @{text "bn"} the thesis follows by simple calculations thanks
|
|
1413 |
to the restrictions on the binding functions. For @{text "fv"} functions it
|
|
1414 |
follows using respectfullness of @{text "bn"}. For type constructors it is a
|
|
1415 |
simple calculation thanks to the way alpha-equivalence was defined. For @{text
|
|
1416 |
"alpha_bn"} after a second induction on the second relation by simple
|
|
1417 |
calculations. \end{proof}
|
1717
|
1418 |
|
1718
|
1419 |
With these respectfullness properties we can use the quotient package
|
|
1420 |
to define the above constants on the quotient level. We can then automatically
|
|
1421 |
lift the theorems that talk about the raw constants to theorems on the quotient
|
|
1422 |
level. The following lifted properties are proved:
|
1717
|
1423 |
|
1718
|
1424 |
\begin{center}
|
|
1425 |
\begin{tabular}{cp{7cm}}
|
1721
|
1426 |
%skipped permute_zero and permute_add, since we do not have a permutation
|
|
1427 |
%definition
|
1718
|
1428 |
$\bullet$ & permutation defining equations \\
|
|
1429 |
$\bullet$ & @{term "bn"} defining equations \\
|
|
1430 |
$\bullet$ & @{term "fv_ty"} and @{term "fv_bn"} defining equations \\
|
1721
|
1431 |
$\bullet$ & induction. The induction principle that we obtain by lifting
|
|
1432 |
is the weak induction principle, just on the term structure \\
|
|
1433 |
$\bullet$ & quasi-injectivity. This means the equations that specify
|
|
1434 |
when two constructors are equal and comes from lifting the alpha
|
|
1435 |
equivalence defining relations\\
|
1718
|
1436 |
$\bullet$ & distinctness\\
|
1721
|
1437 |
%may be skipped
|
1718
|
1438 |
$\bullet$ & equivariance of @{term "fv"} and @{term "bn"} functions\\
|
|
1439 |
\end{tabular}
|
|
1440 |
\end{center}
|
1717
|
1441 |
|
1721
|
1442 |
Notice that until now we have not said anything about the support of the
|
|
1443 |
defined type. This is because we could not use the general definition of
|
|
1444 |
support in lifted theorems, since it does not preserve the relation.
|
|
1445 |
Indeed, take the term @{text "\<lambda>x. x"}. The support of the term is empty @{term "{}"},
|
|
1446 |
since the @{term "x"} is bound. On the raw level, before the binding is
|
1722
|
1447 |
introduced the term has the support equal to @{text "{x}"}.
|
1721
|
1448 |
|
1722
|
1449 |
To show the support equations for the lifted types we want to use the
|
1728
|
1450 |
Theorem \ref{suppabs}, so we start with showing that they have a finite
|
1722
|
1451 |
support.
|
1721
|
1452 |
|
1722
|
1453 |
\begin{lemma} The types @{text "ty\<^isup>\<alpha>\<^isub>1 \<dots> ty\<^isup>\<alpha>\<^isub>n"} have finite support.
|
|
1454 |
\end{lemma}
|
|
1455 |
\begin{proof}
|
|
1456 |
By induction on the lifted types. For each constructor its support is
|
|
1457 |
supported by the union of the supports of all arguments. By induction
|
|
1458 |
hypothesis we know that each of the recursive arguments has finite
|
|
1459 |
support. We also know that atoms and finite atom sets and lists that
|
|
1460 |
occur in the constructors have finite support. A union of finite
|
|
1461 |
sets is finite thus the support of the constructor is finite.
|
|
1462 |
\end{proof}
|
1721
|
1463 |
|
1728
|
1464 |
% Very vague...
|
1722
|
1465 |
\begin{lemma} For each lifted type @{text "ty\<^isup>\<alpha>\<^isub>i"}, for every @{text "x"}
|
|
1466 |
of this type:
|
|
1467 |
\begin{center}
|
|
1468 |
@{term "supp x = fv_ty\<^isup>\<alpha>\<^isub>i x"}.
|
|
1469 |
\end{center}
|
|
1470 |
\end{lemma}
|
|
1471 |
\begin{proof}
|
1728
|
1472 |
We will show this by induction together with equations that characterize
|
|
1473 |
@{term "fv_bn\<^isup>\<alpha>\<^isub>"} in terms of @{term "alpha_bn\<^isup>\<alpha>"}. For each of @{text "fv_bn\<^isup>\<alpha>"}
|
|
1474 |
functions this equaton is:
|
|
1475 |
\begin{center}
|
|
1476 |
@{term "{a. infinite {b. \<not> alpha_bn\<^isup>\<alpha> ((a \<rightleftharpoons> b) \<bullet> x) x}} = fv_bn\<^isup>\<alpha> x"}
|
|
1477 |
\end{center}
|
|
1478 |
|
|
1479 |
In the induction we need to show these equations together with the goal
|
|
1480 |
for the appropriate constructors. We first transform the right hand sides.
|
|
1481 |
The free variable functions are applied to theirs respective constructors
|
|
1482 |
so we can apply the lifted free variable defining equations to obtain
|
|
1483 |
free variable functions applied to subterms minus binders. Using the
|
|
1484 |
induction hypothesis we can replace free variable functions applied to
|
|
1485 |
subterms by support. Using Theorem \ref{suppabs} we replace the differences
|
|
1486 |
by supports of appropriate abstractions.
|
|
1487 |
|
|
1488 |
Unfolding the definition of supports on both sides of the equations we
|
|
1489 |
obtain by simple calculations the equalities.
|
1722
|
1490 |
\end{proof}
|
1728
|
1491 |
|
1729
|
1492 |
%%% Without defining permute_bn, we cannot even write the substitution
|
|
1493 |
%%% of bindings in term constructors...
|
|
1494 |
|
|
1495 |
% With the above equations we can substitute free variables for support in
|
|
1496 |
% the lifted free variable equations, which gives us the support equations
|
|
1497 |
% for the term constructors. With this we can show that for each binding in
|
|
1498 |
% a constructors the bindings can be renamed.
|
1728
|
1499 |
|
1717
|
1500 |
*}
|
1587
|
1501 |
|
|
1502 |
text {*
|
1722
|
1503 |
%%% FIXME: The restricions should have already been described in previous sections?
|
1520
|
1504 |
Restrictions
|
|
1505 |
|
|
1506 |
\begin{itemize}
|
1572
|
1507 |
\item non-emptiness
|
1520
|
1508 |
\item positive datatype definitions
|
|
1509 |
\item finitely supported abstractions
|
|
1510 |
\item respectfulness of the bn-functions\bigskip
|
|
1511 |
\item binders can only have a ``single scope''
|
1577
|
1512 |
\item all bindings must have the same mode
|
1520
|
1513 |
\end{itemize}
|
|
1514 |
*}
|
|
1515 |
|
1493
|
1516 |
section {* Examples *}
|
1485
|
1517 |
|
1702
|
1518 |
text {*
|
|
1519 |
|
|
1520 |
\begin{figure}
|
|
1521 |
\begin{boxedminipage}{\linewidth}
|
|
1522 |
\small
|
|
1523 |
\begin{tabular}{l}
|
|
1524 |
\isacommand{atom\_decl}~@{text "var"}\\
|
|
1525 |
\isacommand{atom\_decl}~@{text "cvar"}\\
|
|
1526 |
\isacommand{atom\_decl}~@{text "tvar"}\\[1mm]
|
|
1527 |
\isacommand{nominal\_datatype}~@{text "tkind ="}\\
|
|
1528 |
\phantom{$|$}~@{text "KStar"}~$|$~@{text "KFun tkind tkind"}\\
|
|
1529 |
\isacommand{and}~@{text "ckind ="}\\
|
|
1530 |
\phantom{$|$}~@{text "CKSim ty ty"}\\
|
|
1531 |
\isacommand{and}~@{text "ty ="}\\
|
|
1532 |
\phantom{$|$}~@{text "TVar tvar"}~$|$~@{text "T string"}~$|$~@{text "TApp ty ty"}\\
|
|
1533 |
$|$~@{text "TFun string ty_list"}~%
|
|
1534 |
$|$~@{text "TAll tv::tvar tkind ty::ty"} \isacommand{bind}~@{text "tv"}~\isacommand{in}~@{text ty}\\
|
|
1535 |
$|$~@{text "TArr ckind ty"}\\
|
|
1536 |
\isacommand{and}~@{text "ty_lst ="}\\
|
|
1537 |
\phantom{$|$}~@{text "TNil"}~$|$~@{text "TCons ty ty_lst"}\\
|
|
1538 |
\isacommand{and}~@{text "cty ="}\\
|
|
1539 |
\phantom{$|$}~@{text "CVar cvar"}~%
|
|
1540 |
$|$~@{text "C string"}~$|$~@{text "CApp cty cty"}~$|$~@{text "CFun string co_lst"}\\
|
|
1541 |
$|$~@{text "CAll cv::cvar ckind cty::cty"} \isacommand{bind}~@{text "cv"}~\isacommand{in}~@{text cty}\\
|
|
1542 |
$|$~@{text "CArr ckind cty"}~$|$~@{text "CRefl ty"}~$|$~@{text "CSym cty"}~$|$~@{text "CCirc cty cty"}\\
|
|
1543 |
$|$~@{text "CAt cty ty"}~$|$~@{text "CLeft cty"}~$|$~@{text "CRight cty"}~$|$~@{text "CSim cty cty"}\\
|
|
1544 |
$|$~@{text "CRightc cty"}~$|$~@{text "CLeftc cty"}~$|$~@{text "Coerce cty cty"}\\
|
|
1545 |
\isacommand{and}~@{text "co_lst ="}\\
|
|
1546 |
\phantom{$|$}@{text "CNil"}~$|$~@{text "CCons cty co_lst"}\\
|
|
1547 |
\isacommand{and}~@{text "trm ="}\\
|
|
1548 |
\phantom{$|$}~@{text "Var var"}~$|$~@{text "K string"}\\
|
|
1549 |
$|$~@{text "LAM_ty tv::tvar tkind t::trm"} \isacommand{bind}~@{text "tv"}~\isacommand{in}~@{text t}\\
|
|
1550 |
$|$~@{text "LAM_cty cv::cvar ckind t::trm"} \isacommand{bind}~@{text "cv"}~\isacommand{in}~@{text t}\\
|
|
1551 |
$|$~@{text "App_ty trm ty"}~$|$~@{text "App_cty trm cty"}~$|$~@{text "App trm trm"}\\
|
|
1552 |
$|$~@{text "Lam v::var ty t::trm"} \isacommand{bind}~@{text "v"}~\isacommand{in}~@{text t}\\
|
|
1553 |
$|$~@{text "Let x::var ty trm t::trm"} \isacommand{bind}~{text x}~\isacommand{in}~{text t}\\
|
|
1554 |
$|$~@{text "Case trm assoc_lst"}~$|$~@{text "Cast trm co"}\\
|
|
1555 |
\isacommand{and}~@{text "assoc_lst ="}\\
|
|
1556 |
\phantom{$|$}~@{text ANil}~%
|
|
1557 |
$|$~@{text "ACons p::pat t::trm assoc_lst"} \isacommand{bind}~@{text "bv p"}~\isacommand{in}~@{text t}\\
|
|
1558 |
\isacommand{and}~@{text "pat ="}\\
|
|
1559 |
\phantom{$|$}~@{text "Kpat string tvtk_lst tvck_lst vt_lst"}\\
|
|
1560 |
\isacommand{and}~@{text "vt_lst ="}\\
|
|
1561 |
\phantom{$|$}~@{text VTNil}~$|$~@{text "VTCons var ty vt_lst"}\\
|
|
1562 |
\isacommand{and}~@{text "tvtk_lst ="}\\
|
|
1563 |
\phantom{$|$}~@{text TVTKNil}~$|$~@{text "TVTKCons tvar tkind tvtk_lst"}\\
|
|
1564 |
\isacommand{and}~@{text "tvck_lst ="}\\
|
|
1565 |
\phantom{$|$}~@{text TVCKNil}~$|$ @{text "TVCKCons cvar ckind tvck_lst"}\\
|
|
1566 |
\isacommand{binder}\\
|
|
1567 |
@{text "bv :: pat \<Rightarrow> atom list"}~\isacommand{and}~%
|
|
1568 |
@{text "bv1 :: vt_lst \<Rightarrow> atom list"}~\isacommand{and}\\
|
|
1569 |
@{text "bv2 :: tvtk_lst \<Rightarrow> atom list"}~\isacommand{and}~%
|
|
1570 |
@{text "bv3 :: tvck_lst \<Rightarrow> atom list"}\\
|
|
1571 |
\isacommand{where}\\
|
|
1572 |
\phantom{$|$}~@{text "bv (K s tvts tvcs vs) = (bv3 tvts) @ (bv2 tvcs) @ (bv1 vs)"}\\
|
|
1573 |
$|$~@{text "bv1 VTNil = []"}\\
|
|
1574 |
$|$~@{text "bv1 (VTCons x ty tl) = (atom x)::(bv1 tl)"}\\
|
|
1575 |
$|$~@{text "bv2 TVTKNil = []"}\\
|
|
1576 |
$|$~@{text "bv2 (TVTKCons a ty tl) = (atom a)::(bv2 tl)"}\\
|
|
1577 |
$|$~@{text "bv3 TVCKNil = []"}\\
|
|
1578 |
$|$~@{text "bv3 (TVCKCons c cty tl) = (atom c)::(bv3 tl)"}\\
|
|
1579 |
\end{tabular}
|
|
1580 |
\end{boxedminipage}
|
|
1581 |
\caption{\label{nominalcorehas}}
|
|
1582 |
\end{figure}
|
|
1583 |
*}
|
|
1584 |
|
|
1585 |
|
|
1586 |
|
|
1587 |
|
1517
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1588 |
section {* Adequacy *}
|
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1589 |
|
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1590 |
section {* Related Work *}
|
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
Christian Urban <urbanc@in.tum.de>
diff
changeset
|
1591 |
|
1570
|
1592 |
text {*
|
1730
|
1593 |
To our knowledge the earliest usage of general binders in a theorem prover setting is
|
1726
|
1594 |
in the paper \cite{NaraschewskiNipkow99}, which describes a formalisation of
|
|
1595 |
the algorithm W. This formalisation implements binding in type schemes using a
|
|
1596 |
a de-Bruijn indices representation. Also recently an extension for general binders
|
|
1597 |
has been proposed for the locally nameless approach to binding \cite{chargueraud09}. .
|
|
1598 |
But we have not yet seen it to be employed in a non-trivial formal verification.
|
|
1599 |
In both approaches, it seems difficult to achieve our fine-grained control over the
|
|
1600 |
``semantics'' of bindings (whether the order should matter, or vacous binders
|
|
1601 |
should be taken into account). To do so, it is necessary to introduce predicates
|
|
1602 |
that filter out some unwanted terms. This very likely results in intricate
|
|
1603 |
formal reasoning.
|
|
1604 |
|
|
1605 |
Higher-Order Abstract Syntax (HOAS) approaches to representing binders are
|
|
1606 |
nicely supported in the Twelf theorem prover and work is in progress to use
|
|
1607 |
HOAS in a mechanisation of the metatheory of SML
|
|
1608 |
\cite{LeeCraryHarper07}. HOAS supports elegantly reasoning about
|
|
1609 |
term-calculi with single binders. We are not aware how more complicated
|
|
1610 |
binders from SML are represented in HOAS, but we know that HOAS cannot
|
|
1611 |
easily deal with binding constructs where the number of bound variables is
|
|
1612 |
not fixed. An example is the second part of the POPLmark challenge where
|
|
1613 |
@{text "Let"}s involving patterns need to be formalised. In such situations
|
|
1614 |
HOAS needs to use essentially has to represent multiple binders with
|
|
1615 |
iterated single binders.
|
1570
|
1616 |
|
1726
|
1617 |
An attempt of representing general binders in the old version of Isabelle
|
|
1618 |
based also on iterating single binders is described in \cite{BengtsonParow09}.
|
|
1619 |
The reasoning there turned out to be quite complex.
|
|
1620 |
|
|
1621 |
Ott is better with list dot specifications; subgrammars, is untyped;
|
1570
|
1622 |
|
|
1623 |
*}
|
|
1624 |
|
|
1625 |
|
1493
|
1626 |
section {* Conclusion *}
|
1485
|
1627 |
|
|
1628 |
text {*
|
1520
|
1629 |
Complication when the single scopedness restriction is lifted (two
|
|
1630 |
overlapping permutations)
|
1662
|
1631 |
|
1726
|
1632 |
Future work: distinct list abstraction
|
|
1633 |
|
|
1634 |
TODO: function definitions:
|
|
1635 |
|
1662
|
1636 |
|
|
1637 |
The formalisation presented here will eventually become part of the
|
|
1638 |
Isabelle distribution, but for the moment it can be downloaded from
|
|
1639 |
the Mercurial repository linked at
|
|
1640 |
\href{http://isabelle.in.tum.de/nominal/download}
|
|
1641 |
{http://isabelle.in.tum.de/nominal/download}.\medskip
|
1520
|
1642 |
*}
|
|
1643 |
|
|
1644 |
text {*
|
1493
|
1645 |
\noindent
|
1528
|
1646 |
{\bf Acknowledgements:} We are very grateful to Andrew Pitts for
|
1506
|
1647 |
many discussions about Nominal Isabelle. We thank Peter Sewell for
|
|
1648 |
making the informal notes \cite{SewellBestiary} available to us and
|
1556
|
1649 |
also for patiently explaining some of the finer points about the abstract
|
1702
|
1650 |
definitions and about the implementation of the Ott-tool. We
|
|
1651 |
also thank Stephanie Weirich for suggesting to separate the subgrammars
|
|
1652 |
of kinds and types in our Core-Haskell example.
|
1485
|
1653 |
|
1726
|
1654 |
|
1577
|
1655 |
|
|
1656 |
|
754
|
1657 |
*}
|
|
1658 |
|
1484
|
1659 |
|
|
1660 |
|
754
|
1661 |
(*<*)
|
|
1662 |
end
|
1704
|
1663 |
(*>*)
|