author | Christian Urban <urbanc@in.tum.de> |
Thu, 19 Apr 2018 13:58:22 +0100 | |
branch | Nominal2-Isabelle2016-1 |
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parent 3111 | 60c4c93b30d5 |
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(*<*) |
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theory Paper |
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imports "../Nominal/Nominal2" |
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"~~/src/HOL/Library/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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abs_set2 :: "'a \<Rightarrow> perm \<Rightarrow> 'b \<Rightarrow> 'c" |
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Abs_dist :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" |
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Abs_print :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" |
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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_set ("_ \<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]{$_{\textit{set+}}$}}>\<^bsup>_, _, _\<^esup> _") and |
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abs_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and |
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abs_set2 ("_ \<approx>\<^raw:\raisebox{-1pt}{\makebox[0mm][l]{$\,_{\textit{list}}$}}>\<^bsup>_\<^esup> _") and |
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fv ("fa'(_')" [100] 100) and |
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equal ("=") and |
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alpha_abs_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and |
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Abs_set ("[_]\<^bsub>set\<^esub>._" [20, 101] 999) and |
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Abs_lst ("[_]\<^bsub>list\<^esub>._") and |
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Abs_dist ("[_]\<^bsub>#list\<^esub>._") and |
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Abs_res ("[_]\<^bsub>set+\<^esub>._") and |
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Abs_print ("_\<^bsub>set\<^esub>._") and |
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Cons ("_::_" [78,77] 73) and |
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supp_set ("aux _" [1000] 10) and |
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alpha_bn ("_ \<approx>bn _") |
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consts alpha_trm ::'a |
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consts fa_trm :: 'a |
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consts alpha_trm2 ::'a |
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consts fa_trm2 :: 'a |
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consts ast :: 'a |
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consts ast' :: 'a |
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notation (latex output) |
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alpha_trm ("\<approx>\<^bsub>trm\<^esub>") and |
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fa_trm ("fa\<^bsub>trm\<^esub>") and |
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alpha_trm2 ("'(\<approx>\<^bsub>assn\<^esub>, \<approx>\<^bsub>trm\<^esub>')") and |
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fa_trm2 ("'(fa\<^bsub>assn\<^esub>, fa\<^bsub>trm\<^esub>')") and |
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ast ("'(as, t')") and |
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ast' ("'(as', t\<PRIME> ')") |
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(*>*) |
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section {* Introduction *} |
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text {* |
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So far, Nominal Isabelle provided a mechanism for constructing |
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$\alpha$-equated terms, for example lambda-terms, |
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@{text "t ::= x | t t | \<lambda>x. t"}, |
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where free and bound variables have names. For such $\alpha$-equated terms, |
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Nominal Isabelle derives automatically a reasoning infrastructure that has |
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been used successfully in formalisations of an equivalence checking |
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algorithm for LF \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 for cut-elimination |
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in classical logic \cite{UrbanZhu08}. It has also been used by Pollack for |
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formalisations in the locally-nameless approach to binding |
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\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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\begin{equation}\label{tysch} |
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\begin{array}{l} |
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@{text "T ::= x | T \<rightarrow> T"}\hspace{9mm} |
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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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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. |
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%The need of iterating single binders is also one reason |
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%why Nominal Isabelle |
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% and similar theorem provers that only provide |
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%mechanisms for binding single variables |
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%has 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 the 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 first pair of type-schemes as $\alpha$-equivalent, |
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but assuming that @{text x}, @{text y} and @{text z} are distinct variables, |
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the second pair should \emph{not} be $\alpha$-equivalent: |
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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"}\hspace{10mm} |
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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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% |
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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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\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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% |
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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. |
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%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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\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 @{text "x = 3"} and |
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\mbox{@{text "y = 2"}} are given, but it would be often unusual to regard |
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\eqref{one} as $\alpha$-equivalent with |
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% |
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\begin{center} |
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@{text "\<LET> x = 3 \<AND> y = 2 \<AND> z = foo \<IN> x - y \<END>"} |
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\end{center} |
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% |
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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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\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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% |
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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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% |
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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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in a formalisation. |
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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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\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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we care about the |
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information that there are as many bound variables @{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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\begin{center} |
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@{text "\<LET> (\<lambda>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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% |
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\noindent |
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where the notation @{text "\<lambda>_ . _"} indicates that the list of @{text |
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"x\<^isub>i"} becomes bound in @{text s}. In this representation the term |
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\mbox{@{text "\<LET> (\<lambda>x . s) [t\<^isub>1, t\<^isub>2]"}} is a perfectly legal |
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instance, but the lengths of the two lists do not agree. To exclude such |
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terms, additional predicates about well-formed terms are needed in order to |
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ensure that the two lists are of equal length. This can result in very messy |
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reasoning (see for example~\cite{BengtsonParow09}). To avoid this, we will |
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allow type specifications for @{text "\<LET>"}s as follows |
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\begin{center} |
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\begin{tabular}{r@ {\hspace{2mm}}r@ {\hspace{2mm}}cl} |
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@{text trm} & @{text "::="} & @{text "\<dots>"} |
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& @{text "|"} @{text "\<LET> as::assn s::trm"}\hspace{2mm} |
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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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% |
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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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% |
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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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% |
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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 that all the names the function @{text |
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"bn(as)"} returns should be bound in @{text s}. 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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|
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%Though, Ott |
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%has only one binding mode, namely the one where the order of |
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%binders matters. Consequently, type-schemes with binding sets |
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%of names cannot be modelled in Ott. |
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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 specify ``empty'' |
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types like @{text "t ::= t t | \<lambda>x. t"} |
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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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||
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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 with $\alpha$-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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|
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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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Isabelle/HOL is a rather non-trivial task. For every |
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specification we will need to construct type(s) containing as elements the |
2341 | 276 |
$\alpha$-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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% |
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\begin{center} |
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\begin{tikzpicture}[scale=0.89] |
1552 | 282 |
%\draw[step=2mm] (-4,-1) grid (4,1); |
283 |
||
284 |
\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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||
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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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||
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\draw ( 0.7, 0.4) node {\footnotesize\begin{tabular}{@ {}c@ {}}$\alpha$-\\[-1mm]clas.\end{tabular}}; |
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\draw (-2.4, 0.4) node {\footnotesize\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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{\footnotesize\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ (sets of raw terms)\end{tabular}}; |
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\draw (0.9, -0.35) node {\footnotesize\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}}; |
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\draw (-3.25, 0.55) node {\footnotesize\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] {\footnotesize{}isomorphism}; |
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\end{tikzpicture} |
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\end{center} |
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% |
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\noindent |
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We take as the starting point a definition of raw terms (defined as a |
2361 | 306 |
datatype in Isabelle/HOL); then identify the $\alpha$-equivalence classes in |
2341 | 307 |
the type of sets of raw terms according to our $\alpha$-equivalence relation, |
308 |
and finally define the new type as these $\alpha$-equivalence classes |
|
1657 | 309 |
(non-emptiness is satisfied whenever the raw terms are definable as datatype |
2516 | 310 |
in Isabelle/HOL and our relation for $\alpha$-equivalence is |
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an equivalence relation). |
|
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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 |
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%$\alpha$-equated terms. That is not the case for example for terms using the |
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%locally nameless representation of binders \cite{McKinnaPollack99}: in this |
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%representation there are ``junk'' terms that need to be excluded by |
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%reasoning about a well-formedness predicate. |
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|
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The problem with introducing a new type in Isabelle/HOL is that in order to |
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be useful, a reasoning infrastructure needs to be ``lifted'' from the |
|
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underlying subset to the new type. This is usually a tricky and arduous |
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task. To ease it, we re-implemented in Isabelle/HOL \cite{KaliszykUrban11} the quotient package |
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described by Homeier \cite{Homeier05} for the HOL4 system. This package |
325 |
allows us to lift definitions and theorems involving raw terms to |
|
2341 | 326 |
definitions and theorems involving $\alpha$-equated terms. For example if we |
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define the free-variable function over raw lambda-terms |
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|
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\begin{center} |
|
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@{text "fv(x) = {x}"}\hspace{8mm} |
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@{text "fv(t\<^isub>1 t\<^isub>2) = fv(t\<^isub>1) \<union> fv(t\<^isub>2)"}\hspace{8mm} |
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@{text "fv(\<lambda>x.t) = fv(t) - {x}"} |
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\end{center} |
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||
335 |
\noindent |
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then with the help of the quotient package we can obtain a function @{text "fv\<^sup>\<alpha>"} |
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operating on quotients, or $\alpha$-equivalence classes of lambda-terms. This |
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lifted function is characterised by the equations |
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|
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\begin{center} |
|
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@{text "fv\<^sup>\<alpha>(x) = {x}"}\hspace{8mm} |
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@{text "fv\<^sup>\<alpha>(t\<^isub>1 t\<^isub>2) = fv\<^sup>\<alpha>(t\<^isub>1) \<union> fv\<^sup>\<alpha>(t\<^isub>2)"}\hspace{8mm} |
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@{text "fv\<^sup>\<alpha>(\<lambda>x.t) = fv\<^sup>\<alpha>(t) - {x}"} |
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\end{center} |
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||
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\noindent |
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(Note that this means also the term-constructors for variables, applications |
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and lambda are lifted to the quotient level.) This construction, of course, |
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only works if $\alpha$-equivalence is indeed an equivalence relation, and the |
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``raw'' definitions and theorems are respectful w.r.t.~$\alpha$-equivalence. |
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%For example, we will not be able to lift a bound-variable function. Although |
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%this function can be defined for raw terms, it does not respect |
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%$\alpha$-equivalence and therefore cannot be lifted. |
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To sum up, every lifting |
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of theorems to the quotient level needs proofs of some respectfulness |
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properties (see \cite{Homeier05}). In the paper we show that we are able to |
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automate these proofs and as a result can automatically establish a reasoning |
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infrastructure for $\alpha$-equated terms.\smallskip |
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%The examples we have in mind where our reasoning infrastructure will be |
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%helpful includes the term language of Core-Haskell. This term language |
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%involves patterns that have lists of type-, coercion- and term-variables, |
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%all of which are bound in @{text "\<CASE>"}-expressions. In these |
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%patterns we do not know in advance how many variables need to |
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%be bound. Another example is the specification of SML, which includes |
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%includes bindings as in type-schemes.\medskip |
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\noindent |
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{\bf Contributions:} We provide three new definitions for when terms |
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involving general binders are $\alpha$-equivalent. These definitions are |
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inspired by earlier work of Pitts \cite{Pitts04}. By means of automatic |
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proofs, we establish a reasoning infrastructure for $\alpha$-equated |
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terms, including properties about support, freshness and equality |
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conditions for $\alpha$-equated terms. We are also able to derive strong |
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induction principles that have the variable convention already built in. |
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The method behind our specification of general binders is taken |
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from the Ott-tool, but we introduce crucial restrictions, and also extensions, so |
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that our specifications make sense for reasoning about $\alpha$-equated terms. |
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The main improvement over Ott is that we introduce three binding modes |
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(only one is present in Ott), provide formalised definitions for $\alpha$-equivalence and |
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for free variables of our terms, and also derive a reasoning infrastructure |
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for our specifications from ``first principles''. |
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|
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%\begin{figure} |
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%\begin{boxedminipage}{\linewidth} |
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%%\begin{center} |
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%\begin{tabular}{r@ {\hspace{1mm}}r@ {\hspace{2mm}}l} |
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%\multicolumn{3}{@ {}l}{Type Kinds}\\ |
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%@{text "\<kappa>"} & @{text "::="} & @{text "\<star> | \<kappa>\<^isub>1 \<rightarrow> \<kappa>\<^isub>2"}\smallskip\\ |
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%\multicolumn{3}{@ {}l}{Coercion Kinds}\\ |
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%@{text "\<iota>"} & @{text "::="} & @{text "\<sigma>\<^isub>1 \<sim> \<sigma>\<^isub>2"}\smallskip\\ |
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%\multicolumn{3}{@ {}l}{Types}\\ |
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%@{text "\<sigma>"} & @{text "::="} & @{text "a | T | \<sigma>\<^isub>1 \<sigma>\<^isub>2 | S\<^isub>n"}$\;\overline{@{text "\<sigma>"}}$@{text "\<^sup>n"} |
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%@{text "| \<forall>a:\<kappa>. \<sigma> | \<iota> \<Rightarrow> \<sigma>"}\smallskip\\ |
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%\multicolumn{3}{@ {}l}{Coercion Types}\\ |
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%@{text "\<gamma>"} & @{text "::="} & @{text "c | C | \<gamma>\<^isub>1 \<gamma>\<^isub>2 | S\<^isub>n"}$\;\overline{@{text "\<gamma>"}}$@{text "\<^sup>n"} |
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%@{text "| \<forall>c:\<iota>. \<gamma> | \<iota> \<Rightarrow> \<gamma> "}\\ |
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%& @{text "|"} & @{text "refl \<sigma> | sym \<gamma> | \<gamma>\<^isub>1 \<circ> \<gamma>\<^isub>2 | \<gamma> @ \<sigma> | left \<gamma> | right \<gamma>"}\\ |
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%& @{text "|"} & @{text "\<gamma>\<^isub>1 \<sim> \<gamma>\<^isub>2 | rightc \<gamma> | leftc \<gamma> | \<gamma>\<^isub>1 \<triangleright> \<gamma>\<^isub>2"}\smallskip\\ |
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%\multicolumn{3}{@ {}l}{Terms}\\ |
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%@{text "e"} & @{text "::="} & @{text "x | K | \<Lambda>a:\<kappa>. e | \<Lambda>c:\<iota>. e | e \<sigma> | e \<gamma>"}\\ |
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%& @{text "|"} & @{text "\<lambda>x:\<sigma>. e | e\<^isub>1 e\<^isub>2 | \<LET> x:\<sigma> = e\<^isub>1 \<IN> e\<^isub>2"}\\ |
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%& @{text "|"} & @{text "\<CASE> e\<^isub>1 \<OF>"}$\;\overline{@{text "p \<rightarrow> e\<^isub>2"}}$ @{text "| e \<triangleright> \<gamma>"}\smallskip\\ |
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%\multicolumn{3}{@ {}l}{Patterns}\\ |
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%@{text "p"} & @{text "::="} & @{text "K"}$\;\overline{@{text "a:\<kappa>"}}\;\overline{@{text "c:\<iota>"}}\;\overline{@{text "x:\<sigma>"}}$\smallskip\\ |
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%\multicolumn{3}{@ {}l}{Constants}\\ |
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%& @{text C} & coercion constants\\ |
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%& @{text T} & value type constructors\\ |
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%& @{text "S\<^isub>n"} & n-ary type functions (which need to be fully applied)\\ |
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%& @{text K} & data constructors\smallskip\\ |
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%\multicolumn{3}{@ {}l}{Variables}\\ |
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%& @{text a} & type variables\\ |
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%& @{text c} & coercion variables\\ |
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%& @{text x} & term variables\\ |
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%\end{tabular} |
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%\end{center} |
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%\end{boxedminipage} |
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%\caption{The System @{text "F\<^isub>C"} |
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%\cite{CoreHaskell}, also often referred to as \emph{Core-Haskell}. In this |
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%version of @{text "F\<^isub>C"} we made a modification by separating the |
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%grammars for type kinds and coercion kinds, as well as for types and coercion |
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%types. For this paper the interesting term-constructor is @{text "\<CASE>"}, |
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%which binds multiple type-, coercion- and term-variables.\label{corehas}} |
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%\end{figure} |
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*} |
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section {* A Short Review of the Nominal Logic Work *} |
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text {* |
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At its core, Nominal Isabelle is an adaption of the nominal logic work by |
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Pitts \cite{Pitts03}. This adaptation for Isabelle/HOL is described in |
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\cite{HuffmanUrban10} (including proofs). We shall briefly review this work |
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to aid the description of what follows. |
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Two central notions in the nominal logic work are sorted atoms and |
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sort-respecting permutations of atoms. We will use the letters @{text "a, |
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b, c, \<dots>"} to stand for atoms and @{text "p, q, \<dots>"} to stand for |
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permutations. The purpose of atoms is to represent variables, be they bound or free. |
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%The sorts of atoms can be used to represent different kinds of |
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%variables, such as the term-, coercion- and type-variables in Core-Haskell. |
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It is assumed that there is an infinite supply of atoms for each |
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sort. In the interest of brevity, we shall restrict ourselves |
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in what follows to only one sort of atoms. |
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Permutations are bijective functions from atoms to atoms that are |
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the identity everywhere except on a finite number of atoms. There is a |
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two-place permutation operation written |
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@{text "_ \<bullet> _ :: perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"} |
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where the generic type @{text "\<beta>"} is the type of the object |
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over which the permutation |
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acts. In Nominal Isabelle, the identity permutation is written as @{term "0::perm"}, |
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the composition of two permutations @{term p} and @{term q} as \mbox{@{term "p + q"}}, |
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and the inverse permutation of @{term p} as @{text "- p"}. The permutation |
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operation is defined over the type-hierarchy \cite{HuffmanUrban10}; |
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for example permutations acting on products, lists, sets, functions and booleans are |
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|
457 |
given by: |
1702 | 458 |
% |
2516 | 459 |
%\begin{equation}\label{permute} |
460 |
%\mbox{\begin{tabular}{@ {}c@ {\hspace{10mm}}c@ {}} |
|
461 |
%\begin{tabular}{@ {}l@ {}} |
|
462 |
%@{thm permute_prod.simps[no_vars, THEN eq_reflection]}\\[2mm] |
|
463 |
%@{thm permute_list.simps(1)[no_vars, THEN eq_reflection]}\\ |
|
464 |
%@{thm permute_list.simps(2)[no_vars, THEN eq_reflection]}\\ |
|
465 |
%\end{tabular} & |
|
466 |
%\begin{tabular}{@ {}l@ {}} |
|
467 |
%@{thm permute_set_eq[no_vars, THEN eq_reflection]}\\ |
|
468 |
%@{text "p \<bullet> f \<equiv> \<lambda>x. p \<bullet> (f (- p \<bullet> x))"}\\ |
|
469 |
%@{thm permute_bool_def[no_vars, THEN eq_reflection]} |
|
470 |
%\end{tabular} |
|
471 |
%\end{tabular}} |
|
472 |
%\end{equation} |
|
473 |
% |
|
474 |
\begin{center} |
|
475 |
\mbox{\begin{tabular}{@ {}c@ {\hspace{4mm}}c@ {\hspace{4mm}}c@ {}} |
|
1690 | 476 |
\begin{tabular}{@ {}l@ {}} |
2516 | 477 |
@{thm permute_prod.simps[no_vars, THEN eq_reflection]}\\ |
478 |
@{thm permute_bool_def[no_vars, THEN eq_reflection]} |
|
479 |
\end{tabular} & |
|
480 |
\begin{tabular}{@ {}l@ {}} |
|
1690 | 481 |
@{thm permute_list.simps(1)[no_vars, THEN eq_reflection]}\\ |
482 |
@{thm permute_list.simps(2)[no_vars, THEN eq_reflection]}\\ |
|
483 |
\end{tabular} & |
|
484 |
\begin{tabular}{@ {}l@ {}} |
|
485 |
@{thm permute_set_eq[no_vars, THEN eq_reflection]}\\ |
|
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|
486 |
@{text "p \<bullet> f \<equiv> \<lambda>x. p \<bullet> (f (- p \<bullet> x))"}\\ |
1690 | 487 |
\end{tabular} |
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|
488 |
\end{tabular}} |
2516 | 489 |
\end{center} |
1690 | 490 |
|
491 |
\noindent |
|
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|
492 |
Concrete permutations in Nominal Isabelle are built up from swappings, |
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|
493 |
written as \mbox{@{text "(a b)"}}, which are permutations that behave |
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|
494 |
as follows: |
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|
495 |
% |
1703 | 496 |
\begin{center} |
497 |
@{text "(a b) = \<lambda>c. if a = c then b else if b = c then a else c"} |
|
498 |
\end{center} |
|
499 |
||
1570 | 500 |
The most original aspect of the nominal logic work of Pitts is a general |
1703 | 501 |
definition for the notion of the ``set of free variables of an object @{text |
1570 | 502 |
"x"}''. This notion, written @{term "supp x"}, is general in the sense that |
2341 | 503 |
it applies not only to lambda-terms ($\alpha$-equated or not), but also to lists, |
1570 | 504 |
products, sets and even functions. The definition depends only on the |
505 |
permutation operation and on the notion of equality defined for the type of |
|
506 |
@{text x}, namely: |
|
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|
507 |
% |
1703 | 508 |
\begin{equation}\label{suppdef} |
509 |
@{thm supp_def[no_vars, THEN eq_reflection]} |
|
510 |
\end{equation} |
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|
511 |
|
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|
512 |
\noindent |
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|
513 |
There is also the derived notion for when an atom @{text a} is \emph{fresh} |
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514 |
for an @{text x}, defined as @{thm fresh_def[no_vars]}. |
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|
515 |
We use for sets of atoms the abbreviation |
1703 | 516 |
@{thm (lhs) fresh_star_def[no_vars]}, defined as |
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|
517 |
@{thm (rhs) fresh_star_def[no_vars]}. |
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|
518 |
A striking consequence of these definitions is that we can prove |
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|
519 |
without knowing anything about the structure of @{term x} that |
2140 | 520 |
swapping two fresh atoms, say @{text a} and @{text b}, leaves |
2516 | 521 |
@{text x} unchanged, namely if @{text "a \<FRESH> x"} and @{text "b \<FRESH> x"} |
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|
522 |
then @{term "(a \<rightleftharpoons> b) \<bullet> x = x"}. |
2555 | 523 |
% |
2516 | 524 |
%\begin{myproperty}\label{swapfreshfresh} |
525 |
%@{thm[mode=IfThen] swap_fresh_fresh[no_vars]} |
|
526 |
%\end{myproperty} |
|
2555 | 527 |
% |
2511 | 528 |
%While often the support of an object can be relatively easily |
529 |
%described, for example for atoms, products, lists, function applications, |
|
530 |
%booleans and permutations as follows |
|
531 |
%% |
|
532 |
%\begin{center} |
|
533 |
%\begin{tabular}{c@ {\hspace{10mm}}c} |
|
534 |
%\begin{tabular}{rcl} |
|
535 |
%@{term "supp a"} & $=$ & @{term "{a}"}\\ |
|
536 |
%@{term "supp (x, y)"} & $=$ & @{term "supp x \<union> supp y"}\\ |
|
537 |
%@{term "supp []"} & $=$ & @{term "{}"}\\ |
|
538 |
%@{term "supp (x#xs)"} & $=$ & @{term "supp x \<union> supp xs"}\\ |
|
539 |
%\end{tabular} |
|
540 |
%& |
|
541 |
%\begin{tabular}{rcl} |
|
542 |
%@{text "supp (f x)"} & @{text "\<subseteq>"} & @{term "supp f \<union> supp x"}\\ |
|
543 |
%@{term "supp b"} & $=$ & @{term "{}"}\\ |
|
544 |
%@{term "supp p"} & $=$ & @{term "{a. p \<bullet> a \<noteq> a}"} |
|
545 |
%\end{tabular} |
|
546 |
%\end{tabular} |
|
547 |
%\end{center} |
|
1690 | 548 |
% |
2511 | 549 |
%\noindent |
550 |
%in some cases it can be difficult to characterise the support precisely, and |
|
551 |
%only an approximation can be established (as for functions above). |
|
2555 | 552 |
% |
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|
553 |
%Reasoning about |
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|
554 |
%such approximations can be simplified with the notion \emph{supports}, defined |
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|
555 |
%as follows: |
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|
556 |
% |
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|
557 |
%\begin{definition} |
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|
558 |
%A set @{text S} \emph{supports} @{text x} if for all atoms @{text a} and @{text b} |
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|
559 |
%not in @{text S} we have @{term "(a \<rightleftharpoons> b) \<bullet> x = x"}. |
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|
560 |
%\end{definition} |
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|
561 |
% |
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|
562 |
%\noindent |
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|
563 |
%The main point of @{text supports} is that we can establish the following |
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|
564 |
%two properties. |
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|
565 |
% |
2514
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|
566 |
%\begin{myproperty}\label{supportsprop} |
2508
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|
567 |
%Given a set @{text "as"} of atoms. |
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|
568 |
%{\it (i)} @{thm[mode=IfThen] supp_is_subset[where S="as", no_vars]} |
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|
569 |
%{\it (ii)} @{thm supp_supports[no_vars]}. |
2514
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|
570 |
%\end{myproperty} |
1711 | 571 |
% |
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|
572 |
%Another important notion in the nominal logic work is \emph{equivariance}. |
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|
573 |
%For a function @{text f}, say of type @{text "\<alpha> \<Rightarrow> \<beta>"}, to be equivariant |
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|
574 |
%it is required that every permutation leaves @{text f} unchanged, that is |
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|
575 |
%% |
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|
576 |
%\begin{equation}\label{equivariancedef} |
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|
577 |
%@{term "\<forall>p. p \<bullet> f = f"} |
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|
578 |
%\end{equation} |
1703 | 579 |
% |
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|
580 |
%\noindent or equivalently that a permutation applied to the application |
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|
581 |
%@{text "f x"} can be moved to the argument @{text x}. That means for equivariant |
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|
582 |
%functions @{text f}, we have for all permutations @{text p}: |
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|
583 |
%% |
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|
584 |
%\begin{equation}\label{equivariance} |
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|
585 |
%@{text "p \<bullet> f = f"} \;\;\;\textit{if and only if}\;\;\; |
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|
586 |
%@{text "p \<bullet> (f x) = f (p \<bullet> x)"} |
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|
587 |
%\end{equation} |
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|
588 |
% |
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|
589 |
%\noindent |
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|
590 |
%From property \eqref{equivariancedef} and the definition of @{text supp}, we |
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|
591 |
%can easily deduce that equivariant functions have empty support. There is |
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|
592 |
%also a similar notion for equivariant relations, say @{text R}, namely the property |
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|
593 |
%that |
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|
594 |
%% |
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|
595 |
%\begin{center} |
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|
596 |
%@{text "x R y"} \;\;\textit{implies}\;\; @{text "(p \<bullet> x) R (p \<bullet> y)"} |
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|
597 |
%\end{center} |
2511 | 598 |
% |
599 |
%Using freshness, the nominal logic work provides us with general means for renaming |
|
600 |
%binders. |
|
601 |
% |
|
2555 | 602 |
%\noindent |
2511 | 603 |
While in the older version of Nominal Isabelle, we used extensively |
604 |
%Property~\ref{swapfreshfresh} |
|
605 |
this property to rename single binders, it %%this property |
|
2341 | 606 |
proved too unwieldy for dealing with multiple binders. For such binders the |
1727
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|
607 |
following generalisations turned out to be easier to use. |
1711 | 608 |
|
2514
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|
609 |
\begin{myproperty}\label{supppermeq} |
1711 | 610 |
@{thm[mode=IfThen] supp_perm_eq[no_vars]} |
2514
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|
611 |
\end{myproperty} |
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|
612 |
|
2514
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|
613 |
\begin{myproperty}\label{avoiding} |
1716 | 614 |
For a finite set @{text as} and a finitely supported @{text x} with |
615 |
@{term "as \<sharp>* x"} and also a finitely supported @{text c}, there |
|
616 |
exists a permutation @{text p} such that @{term "(p \<bullet> as) \<sharp>* c"} and |
|
1711 | 617 |
@{term "supp x \<sharp>* p"}. |
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|
618 |
\end{myproperty} |
1711 | 619 |
|
620 |
\noindent |
|
1716 | 621 |
The idea behind the second property is that given a finite set @{text as} |
1727
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diff
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|
622 |
of binders (being bound, or fresh, in @{text x} is ensured by the |
1716 | 623 |
assumption @{term "as \<sharp>* x"}), then there exists a permutation @{text p} such that |
1727
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624 |
the renamed binders @{term "p \<bullet> as"} avoid @{text c} (which can be arbitrarily chosen |
1730
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diff
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|
625 |
as long as it is finitely supported) and also @{text "p"} does not affect anything |
1711 | 626 |
in the support of @{text x} (that is @{term "supp x \<sharp>* p"}). The last |
627 |
fact and Property~\ref{supppermeq} allow us to ``rename'' just the binders |
|
1727
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diff
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|
628 |
@{text as} in @{text x}, because @{term "p \<bullet> x = x"}. |
1711 | 629 |
|
2128 | 630 |
Most properties given in this section are described in detail in \cite{HuffmanUrban10} |
2512 | 631 |
and all are formalised in Isabelle/HOL. In the next sections we will make |
2341 | 632 |
extensive use of these properties in order to define $\alpha$-equivalence in |
1737
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633 |
the presence of multiple binders. |
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634 |
*} |
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635 |
|
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636 |
|
2345 | 637 |
section {* General Bindings\label{sec:binders} *} |
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638 |
|
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|
639 |
text {* |
1587 | 640 |
In Nominal Isabelle, the user is expected to write down a specification of a |
641 |
term-calculus and then a reasoning infrastructure is automatically derived |
|
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|
642 |
from this specification (remember that Nominal Isabelle is a definitional |
1587 | 643 |
extension of Isabelle/HOL, which does not introduce any new axioms). |
1579 | 644 |
|
1657 | 645 |
In order to keep our work with deriving the reasoning infrastructure |
646 |
manageable, we will wherever possible state definitions and perform proofs |
|
2516 | 647 |
on the ``user-level'' of Isabelle/HOL, as opposed to write custom ML-code. % that |
648 |
%generates them anew for each specification. |
|
649 |
To that end, we will consider |
|
1657 | 650 |
first pairs @{text "(as, x)"} of type @{text "(atom set) \<times> \<beta>"}. These pairs |
2128 | 651 |
are intended to represent the abstraction, or binding, of the set of atoms @{text |
1657 | 652 |
"as"} in the body @{text "x"}. |
1570 | 653 |
|
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|
654 |
The first question we have to answer is when two pairs @{text "(as, x)"} and |
2341 | 655 |
@{text "(bs, y)"} are $\alpha$-equivalent? (For the moment we are interested in |
656 |
the notion of $\alpha$-equivalence that is \emph{not} preserved by adding |
|
657 |
vacuous binders.) To answer this question, we identify four conditions: {\it (i)} |
|
2347 | 658 |
given a free-atom function @{text "fa"} of type \mbox{@{text "\<beta> \<Rightarrow> atom |
1657 | 659 |
set"}}, then @{text x} and @{text y} need to have the same set of free |
2348 | 660 |
atoms; moreover there must be a permutation @{text p} such that {\it |
2347 | 661 |
(ii)} @{text p} leaves the free atoms of @{text x} and @{text y} unchanged, but |
2341 | 662 |
{\it (iii)} ``moves'' their bound names so that we obtain modulo a relation, |
663 |
say \mbox{@{text "_ R _"}}, two equivalent terms. We also require that {\it (iv)} |
|
1662 | 664 |
@{text p} makes the sets of abstracted atoms @{text as} and @{text bs} equal. The |
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|
665 |
requirements {\it (i)} to {\it (iv)} can be stated formally as the conjunction of: |
1556 | 666 |
% |
1572 | 667 |
\begin{equation}\label{alphaset} |
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|
668 |
\begin{array}{@ {\hspace{10mm}}l@ {\hspace{5mm}}l@ {\hspace{10mm}}l@ {\hspace{5mm}}l} |
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|
669 |
\multicolumn{4}{l}{@{term "(as, x) \<approx>set R fa p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}}\\[1mm] |
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|
670 |
\mbox{\it (i)} & @{term "fa(x) - as = fa(y) - bs"} & |
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671 |
\mbox{\it (iii)} & @{text "(p \<bullet> x) R y"} \\ |
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|
672 |
\mbox{\it (ii)} & @{term "(fa(x) - as) \<sharp>* p"} & |
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|
673 |
\mbox{\it (iv)} & @{term "(p \<bullet> as) = bs"} \\ |
1572 | 674 |
\end{array} |
1556 | 675 |
\end{equation} |
2516 | 676 |
% |
1556 | 677 |
\noindent |
2175 | 678 |
Note that this relation depends on the permutation @{text |
2341 | 679 |
"p"}; $\alpha$-equivalence between two pairs is then the relation where we |
1657 | 680 |
existentially quantify over this @{text "p"}. Also note that the relation is |
2347 | 681 |
dependent on a free-atom function @{text "fa"} and a relation @{text |
1657 | 682 |
"R"}. The reason for this extra generality is that we will use |
2341 | 683 |
$\approx_{\,\textit{set}}$ for both ``raw'' terms and $\alpha$-equated terms. In |
684 |
the latter case, @{text R} will be replaced by equality @{text "="} and we |
|
2347 | 685 |
will prove that @{text "fa"} is equal to @{text "supp"}. |
1572 | 686 |
|
687 |
The definition in \eqref{alphaset} does not make any distinction between the |
|
2347 | 688 |
order of abstracted atoms. If we want this, then we can define $\alpha$-equivalence |
1579 | 689 |
for pairs of the form \mbox{@{text "(as, x)"}} with type @{text "(atom list) \<times> \<beta>"} |
690 |
as follows |
|
1572 | 691 |
% |
692 |
\begin{equation}\label{alphalist} |
|
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|
693 |
\begin{array}{@ {\hspace{10mm}}l@ {\hspace{5mm}}l@ {\hspace{10mm}}l@ {\hspace{5mm}}l} |
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|
694 |
\multicolumn{4}{l}{@{term "(as, x) \<approx>lst R fa p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}}\\[1mm] |
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|
695 |
\mbox{\it (i)} & @{term "fa(x) - (set as) = fa(y) - (set bs)"} & |
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|
696 |
\mbox{\it (iii)} & @{text "(p \<bullet> x) R y"}\\ |
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|
697 |
\mbox{\it (ii)} & @{term "(fa(x) - set as) \<sharp>* p"} & |
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|
698 |
\mbox{\it (iv)} & @{term "(p \<bullet> as) = bs"}\\ |
1572 | 699 |
\end{array} |
700 |
\end{equation} |
|
2516 | 701 |
% |
1572 | 702 |
\noindent |
2341 | 703 |
where @{term set} is the function that coerces a list of atoms into a set of atoms. |
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|
704 |
Now the last clause ensures that the order of the binders matters (since @{text as} |
9e09253c80cf
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diff
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|
705 |
and @{text bs} are lists of atoms). |
1556 | 706 |
|
1657 | 707 |
If we do not want to make any difference between the order of binders \emph{and} |
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|
708 |
also allow vacuous binders, that means \emph{restrict} names, then we keep sets of binders, but drop |
734341a79028
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diff
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|
709 |
condition {\it (iv)} in \eqref{alphaset}: |
1572 | 710 |
% |
1579 | 711 |
\begin{equation}\label{alphares} |
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|
712 |
\begin{array}{@ {\hspace{10mm}}l@ {\hspace{5mm}}l@ {\hspace{10mm}}l@ {\hspace{5mm}}l} |
2347 | 713 |
\multicolumn{2}{l}{@{term "(as, x) \<approx>res R fa p (bs, y)"}\hspace{2mm}@{text "\<equiv>"}}\\[1mm] |
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|
714 |
\mbox{\it (i)} & @{term "fa(x) - as = fa(y) - bs"} & |
6d9018d62b40
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|
715 |
\mbox{\it (iii)} & @{text "(p \<bullet> x) R y"}\\ |
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|
716 |
\mbox{\it (ii)} & @{term "(fa(x) - as) \<sharp>* p"}\\ |
1572 | 717 |
\end{array} |
718 |
\end{equation} |
|
1556 | 719 |
|
2516 | 720 |
It might be useful to consider first some examples how these definitions |
2341 | 721 |
of $\alpha$-equivalence pan out in practice. For this consider the case of |
2347 | 722 |
abstracting a set of atoms over types (as in type-schemes). We set |
723 |
@{text R} to be the usual equality @{text "="} and for @{text "fa(T)"} we |
|
2341 | 724 |
define |
2516 | 725 |
% |
1572 | 726 |
\begin{center} |
2347 | 727 |
@{text "fa(x) = {x}"} \hspace{5mm} @{text "fa(T\<^isub>1 \<rightarrow> T\<^isub>2) = fa(T\<^isub>1) \<union> fa(T\<^isub>2)"} |
1572 | 728 |
\end{center} |
729 |
||
730 |
\noindent |
|
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|
731 |
Now recall the examples shown in \eqref{ex1} and |
1687 | 732 |
\eqref{ex3}. It can be easily checked that @{text "({x, y}, x \<rightarrow> y)"} and |
2341 | 733 |
@{text "({y, x}, y \<rightarrow> x)"} are $\alpha$-equivalent according to |
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|
734 |
$\approx_{\,\textit{set}}$ and $\approx_{\,\textit{set+}}$ by taking @{text p} to |
2175 | 735 |
be the swapping @{term "(x \<rightleftharpoons> y)"}. In case of @{text "x \<noteq> y"}, then @{text |
2341 | 736 |
"([x, y], x \<rightarrow> y)"} $\not\approx_{\,\textit{list}}$ @{text "([y, x], x \<rightarrow> y)"} |
2175 | 737 |
since there is no permutation that makes the lists @{text "[x, y]"} and |
738 |
@{text "[y, x]"} equal, and also leaves the type \mbox{@{text "x \<rightarrow> y"}} |
|
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changeset
|
739 |
unchanged. Another example is @{text "({x}, x)"} $\approx_{\,\textit{set+}}$ |
2175 | 740 |
@{text "({x, y}, x)"} which holds by taking @{text p} to be the identity |
741 |
permutation. However, if @{text "x \<noteq> y"}, then @{text "({x}, x)"} |
|
2341 | 742 |
$\not\approx_{\,\textit{set}}$ @{text "({x, y}, x)"} since there is no |
2175 | 743 |
permutation that makes the sets @{text "{x}"} and @{text "{x, y}"} equal |
2341 | 744 |
(similarly for $\approx_{\,\textit{list}}$). It can also relatively easily be |
745 |
shown that all three notions of $\alpha$-equivalence coincide, if we only |
|
2175 | 746 |
abstract a single atom. |
1579 | 747 |
|
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diff
changeset
|
748 |
In the rest of this section we are going to introduce three abstraction |
cfd3a7368543
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diff
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|
749 |
types. For this we define |
1657 | 750 |
% |
751 |
\begin{equation} |
|
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diff
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|
752 |
@{term "abs_set (as, x) (bs, x) \<equiv> \<exists>p. alpha_set (as, x) equal supp p (bs, x)"} |
1657 | 753 |
\end{equation} |
754 |
||
1579 | 755 |
\noindent |
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|
756 |
(similarly for $\approx_{\,\textit{abs\_set+}}$ |
2341 | 757 |
and $\approx_{\,\textit{abs\_list}}$). We can show that these relations are equivalence |
2512 | 758 |
relations. %% and equivariant. |
1579 | 759 |
|
1739 | 760 |
\begin{lemma}\label{alphaeq} |
2341 | 761 |
The relations $\approx_{\,\textit{abs\_set}}$, $\approx_{\,\textit{abs\_list}}$ |
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|
762 |
and $\approx_{\,\textit{abs\_set+}}$ are equivalence relations. %, and if |
2512 | 763 |
%@{term "abs_set (as, x) (bs, y)"} then also |
764 |
%@{term "abs_set (p \<bullet> as, p \<bullet> x) (p \<bullet> bs, p \<bullet> y)"} (similarly for the other two relations). |
|
1657 | 765 |
\end{lemma} |
766 |
||
767 |
\begin{proof} |
|
768 |
Reflexivity is by taking @{text "p"} to be @{text "0"}. For symmetry we have |
|
769 |
a permutation @{text p} and for the proof obligation take @{term "-p"}. In case |
|
1662 | 770 |
of transitivity, we have two permutations @{text p} and @{text q}, and for the |
771 |
proof obligation use @{text "q + p"}. All conditions are then by simple |
|
1657 | 772 |
calculations. |
773 |
\end{proof} |
|
774 |
||
775 |
\noindent |
|
2343 | 776 |
This lemma allows us to use our quotient package for introducing |
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diff
changeset
|
777 |
new types @{text "\<beta> abs_set"}, @{text "\<beta> abs_set+"} and @{text "\<beta> abs_list"} |
2341 | 778 |
representing $\alpha$-equivalence classes of pairs of type |
2128 | 779 |
@{text "(atom set) \<times> \<beta>"} (in the first two cases) and of type @{text "(atom list) \<times> \<beta>"} |
780 |
(in the third case). |
|
2514
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changeset
|
781 |
The elements in these types will be, respectively, written as |
69780ae147f5
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|
782 |
% |
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|
783 |
%\begin{center} |
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changeset
|
784 |
@{term "Abs_set as x"}, %\hspace{5mm} |
69780ae147f5
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changeset
|
785 |
@{term "Abs_res as x"} and %\hspace{5mm} |
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changeset
|
786 |
@{term "Abs_lst as x"}, |
69780ae147f5
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|
787 |
%\end{center} |
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|
788 |
% |
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|
789 |
%\noindent |
1859 | 790 |
indicating that a set (or list) of atoms @{text as} is abstracted in @{text x}. We will |
1716 | 791 |
call the types \emph{abstraction types} and their elements |
1752
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|
792 |
\emph{abstractions}. The important property we need to derive is the support of |
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diff
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|
793 |
abstractions, namely: |
1662 | 794 |
|
2507 | 795 |
\begin{theorem}[Support of Abstractions]\label{suppabs} |
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796 |
Assuming @{text x} has finite support, then |
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|
797 |
|
1662 | 798 |
\begin{center} |
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|
799 |
\begin{tabular}{l} |
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800 |
@{thm (lhs) supp_Abs(1)[no_vars]} $\;\;=\;\;$ |
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|
801 |
@{thm (lhs) supp_Abs(2)[no_vars]} $\;\;=\;\;$ @{thm (rhs) supp_Abs(2)[no_vars]}, and\\ |
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|
802 |
@{thm (lhs) supp_Abs(3)[where bs="bs", no_vars]} $\;\;=\;\;$ |
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|
803 |
@{thm (rhs) supp_Abs(3)[where bs="bs", no_vars]} |
1687 | 804 |
\end{tabular} |
1662 | 805 |
\end{center} |
2507 | 806 |
\end{theorem} |
1662 | 807 |
|
808 |
\noindent |
|
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|
809 |
This theorem states that the bound names do not appear in the support. |
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|
810 |
For brevity we omit the proof and again refer the reader to |
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|
811 |
our formalisation in Isabelle/HOL. |
1662 | 812 |
|
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|
813 |
%\noindent |
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|
814 |
%Below we will show the first equation. The others |
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|
815 |
%follow by similar arguments. By definition of the abstraction type @{text "abs_set"} |
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|
816 |
%we have |
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|
817 |
%% |
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|
818 |
%\begin{equation}\label{abseqiff} |
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parents:
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diff
changeset
|
819 |
%@{thm (lhs) Abs_eq_iff(1)[where bs="as" and bs'="bs", no_vars]} \;\;\text{if and only if}\;\; |
60c4c93b30d5
made all papers work again
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changeset
|
820 |
%@{thm (rhs) Abs_eq_iff(1)[where bs="as" and bs'="bs", no_vars]} |
2508
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|
821 |
%\end{equation} |
6d9018d62b40
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changeset
|
822 |
% |
6d9018d62b40
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changeset
|
823 |
%\noindent |
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|
824 |
%and also |
6d9018d62b40
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changeset
|
825 |
% |
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|
826 |
%\begin{equation}\label{absperm} |
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|
827 |
%%@%{%thm %permute_Abs[no_vars]}% |
6d9018d62b40
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|
828 |
%\end{equation} |
1662 | 829 |
|
2508
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|
830 |
%\noindent |
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|
831 |
%The second fact derives from the definition of permutations acting on pairs |
6d9018d62b40
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|
832 |
%\eqref{permute} and $\alpha$-equivalence being equivariant |
6d9018d62b40
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|
833 |
%(see Lemma~\ref{alphaeq}). With these two facts at our disposal, we can show |
6d9018d62b40
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|
834 |
%the following lemma about swapping two atoms in an abstraction. |
6d9018d62b40
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changeset
|
835 |
% |
6d9018d62b40
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changeset
|
836 |
%\begin{lemma} |
6d9018d62b40
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|
837 |
%@{thm[mode=IfThen] Abs_swap1(1)[where bs="as", no_vars]} |
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|
838 |
%\end{lemma} |
6d9018d62b40
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|
839 |
% |
6d9018d62b40
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|
840 |
%\begin{proof} |
6d9018d62b40
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|
841 |
%This lemma is straightforward using \eqref{abseqiff} and observing that |
6d9018d62b40
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|
842 |
%the assumptions give us @{term "(a \<rightleftharpoons> b) \<bullet> (supp x - as) = (supp x - as)"}. |
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|
843 |
%Moreover @{text supp} and set difference are equivariant (see \cite{HuffmanUrban10}). |
6d9018d62b40
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|
844 |
%\end{proof} |
1687 | 845 |
% |
2508
6d9018d62b40
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changeset
|
846 |
%\noindent |
6d9018d62b40
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|
847 |
%Assuming that @{text "x"} has finite support, this lemma together |
6d9018d62b40
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|
848 |
%with \eqref{absperm} allows us to show |
6d9018d62b40
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parents:
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changeset
|
849 |
% |
6d9018d62b40
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changeset
|
850 |
%\begin{equation}\label{halfone} |
6d9018d62b40
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|
851 |
%@{thm Abs_supports(1)[no_vars]} |
6d9018d62b40
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|
852 |
%\end{equation} |
6d9018d62b40
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parents:
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changeset
|
853 |
% |
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changeset
|
854 |
%\noindent |
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changeset
|
855 |
%which by Property~\ref{supportsprop} gives us ``one half'' of |
6d9018d62b40
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|
856 |
%Theorem~\ref{suppabs}. The ``other half'' is a bit more involved. To establish |
6d9018d62b40
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|
857 |
%it, we use a trick from \cite{Pitts04} and first define an auxiliary |
6d9018d62b40
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|
858 |
%function @{text aux}, taking an abstraction as argument: |
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|
859 |
%@{thm supp_set.simps[THEN eq_reflection, no_vars]}. |
1687 | 860 |
% |
2508
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|
861 |
%Using the second equation in \eqref{equivariance}, we can show that |
6d9018d62b40
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|
862 |
%@{text "aux"} is equivariant (since @{term "p \<bullet> (supp x - as) = (supp (p \<bullet> x)) - (p \<bullet> as)"}) |
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|
863 |
%and therefore has empty support. |
6d9018d62b40
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|
864 |
%This in turn means |
6d9018d62b40
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|
865 |
% |
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|
866 |
%\begin{center} |
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|
867 |
%@{term "supp (supp_gen (Abs_set as x)) \<subseteq> supp (Abs_set as x)"} |
6d9018d62b40
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|
868 |
%\end{center} |
1703 | 869 |
% |
2508
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|
870 |
%\noindent |
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|
871 |
%using \eqref{suppfun}. Assuming @{term "supp x - as"} is a finite set, |
6d9018d62b40
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|
872 |
%we further obtain |
1703 | 873 |
% |
2508
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|
874 |
%\begin{equation}\label{halftwo} |
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|
875 |
%@{thm (concl) Abs_supp_subset1(1)[no_vars]} |
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|
876 |
%\end{equation} |
6d9018d62b40
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|
877 |
% |
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|
878 |
%\noindent |
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|
879 |
%since for finite sets of atoms, @{text "bs"}, we have |
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|
880 |
%@{thm (concl) supp_finite_atom_set[where S="bs", no_vars]}. |
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|
881 |
%Finally, taking \eqref{halfone} and \eqref{halftwo} together establishes |
6d9018d62b40
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|
882 |
%Theorem~\ref{suppabs}. |
1703 | 883 |
|
1737
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diff
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|
884 |
The method of first considering abstractions of the |
2471
894599a50af3
got rid of Nominal2_Supp (is now in Nomina2_Base)
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diff
changeset
|
885 |
form @{term "Abs_set as x"} etc is motivated by the fact that |
1956
028705c98304
more polishing on the paper
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diff
changeset
|
886 |
we can conveniently establish at the Isabelle/HOL level |
028705c98304
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diff
changeset
|
887 |
properties about them. It would be |
028705c98304
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diff
changeset
|
888 |
laborious to write custom ML-code that derives automatically such properties |
1730
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polished and removed tys from bn-functions.
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diff
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|
889 |
for every term-constructor that binds some atoms. Also the generality of |
2510
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|
890 |
the definitions for $\alpha$-equivalence will help us in the next sections. |
1517
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
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diff
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|
891 |
*} |
62d6f7acc110
corrected the strong induction principle in the lambda-calculus case; gave a second (oartial) version that is more elegant
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diff
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|
892 |
|
2345 | 893 |
section {* Specifying General Bindings\label{sec:spec} *} |
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diff
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|
894 |
|
1520
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diff
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|
895 |
text {* |
1727
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started to polish alpha-equivalence section, but needs more work
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diff
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|
896 |
Our choice of syntax for specifications is influenced by the existing |
2514
69780ae147f5
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changeset
|
897 |
datatype package of Isabelle/HOL %\cite{Berghofer99} |
69780ae147f5
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parents:
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changeset
|
898 |
and by the syntax of the |
1765
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changeset
|
899 |
Ott-tool \cite{ott-jfp}. For us a specification of a term-calculus is a |
9a894c42e80e
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|
900 |
collection of (possibly mutual recursive) type declarations, say @{text |
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parents:
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|
901 |
"ty\<AL>\<^isub>1, \<dots>, ty\<AL>\<^isub>n"}, and an associated collection of |
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changeset
|
902 |
binding functions, say @{text "bn\<AL>\<^isub>1, \<dots>, bn\<AL>\<^isub>m"}. The |
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parents:
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|
903 |
syntax in Nominal Isabelle for such specifications is roughly as follows: |
1628 | 904 |
% |
1619 | 905 |
\begin{equation}\label{scheme} |
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|
906 |
\mbox{\begin{tabular}{@ {}p{2.5cm}l} |
1617
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diff
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|
907 |
type \mbox{declaration part} & |
1611 | 908 |
$\begin{cases} |
2511 | 909 |
\mbox{\small\begin{tabular}{l} |
1765
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changeset
|
910 |
\isacommand{nominal\_datatype} @{text "ty\<AL>\<^isub>1 = \<dots>"}\\ |
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|
911 |
\isacommand{and} @{text "ty\<AL>\<^isub>2 = \<dots>"}\\ |
2510
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|
912 |
\raisebox{2mm}{$\ldots$}\\[-2mm] |
1765
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diff
changeset
|
913 |
\isacommand{and} @{text "ty\<AL>\<^isub>n = \<dots>"}\\ |
1611 | 914 |
\end{tabular}} |
915 |
\end{cases}$\\ |
|
1617
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parents:
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diff
changeset
|
916 |
binding \mbox{function part} & |
1611 | 917 |
$\begin{cases} |
2511 | 918 |
\mbox{\small\begin{tabular}{l} |
1954
23480003f9c5
some changes to the paper
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diff
changeset
|
919 |
\isacommand{binder} @{text "bn\<AL>\<^isub>1"} \isacommand{and} \ldots \isacommand{and} @{text "bn\<AL>\<^isub>m"}\\ |
1611 | 920 |
\isacommand{where}\\ |
2510
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parents:
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diff
changeset
|
921 |
\raisebox{2mm}{$\ldots$}\\[-2mm] |
1611 | 922 |
\end{tabular}} |
923 |
\end{cases}$\\ |
|
1619 | 924 |
\end{tabular}} |
925 |
\end{equation} |
|
1587 | 926 |
|
927 |
\noindent |
|
1637 | 928 |
Every type declaration @{text ty}$^\alpha_{1..n}$ consists of a collection of |
2341 | 929 |
term-constructors, each of which comes with a list of labelled |
1620 | 930 |
types that stand for the types of the arguments of the term-constructor. |
1765
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parents:
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diff
changeset
|
931 |
For example a term-constructor @{text "C\<^sup>\<alpha>"} might be specified with |
1611 | 932 |
|
933 |
\begin{center} |
|
1637 | 934 |
@{text "C\<^sup>\<alpha> label\<^isub>1::ty"}$'_1$ @{text "\<dots> label\<^isub>l::ty"}$'_l\;\;$ @{text "binding_clauses"} |
1611 | 935 |
\end{center} |
1587 | 936 |
|
1611 | 937 |
\noindent |
2510
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parents:
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changeset
|
938 |
whereby some of the @{text ty}$'_{1..l}$ %%(or their components) |
734341a79028
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parents:
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changeset
|
939 |
can be contained |
1730
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1728
diff
changeset
|
940 |
in the collection of @{text ty}$^\alpha_{1..n}$ declared in |
1737
8b6a285ad480
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parents:
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diff
changeset
|
941 |
\eqref{scheme}. |
1765
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Christian Urban <urbanc@in.tum.de>
parents:
1764
diff
changeset
|
942 |
In this case we will call the corresponding argument a |
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more on the lifting section
Christian Urban <urbanc@in.tum.de>
parents:
1764
diff
changeset
|
943 |
\emph{recursive argument} of @{text "C\<^sup>\<alpha>"}. |
9a894c42e80e
more on the lifting section
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parents:
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diff
changeset
|
944 |
%The types of such recursive |
1737
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Christian Urban <urbanc@in.tum.de>
parents:
1736
diff
changeset
|
945 |
%arguments need to satisfy a ``positivity'' |
8b6a285ad480
polished everything up to TODO
Christian Urban <urbanc@in.tum.de>
parents:
1736
diff
changeset
|
946 |
%restriction, which ensures that the type has a set-theoretic semantics |
8b6a285ad480
polished everything up to TODO
Christian Urban <urbanc@in.tum.de>
parents:
1736
diff
changeset
|
947 |
%\cite{Berghofer99}. |
8b6a285ad480
polished everything up to TODO
Christian Urban <urbanc@in.tum.de>
parents:
1736
diff
changeset
|
948 |
The labels |
1730
cfd3a7368543
polished and removed tys from bn-functions.
Christian Urban <urbanc@in.tum.de>
parents:
1728
diff
changeset
|
949 |
annotated on the types are optional. Their purpose is to be used in the |
cfd3a7368543
polished and removed tys from bn-functions.
Christian Urban <urbanc@in.tum.de>
parents:
1728
diff
changeset
|
950 |
(possibly empty) list of \emph{binding clauses}, which indicate the binders |
cfd3a7368543
polished and removed tys from bn-functions.
Christian Urban <urbanc@in.tum.de>
parents:
1728
diff
changeset
|
951 |
and their scope in a term-constructor. They come in three \emph{modes}: |
2512 | 952 |
% |
1611 | 953 |
\begin{center} |
2512 | 954 |
\begin{tabular}{@ {}l@ {}} |
2637
3890483c674f
final version of the ESOP paper; used set+ instead of res as requested by one reviewer
Christian Urban <urbanc@in.tum.de>
parents:
2605
diff
changeset
|
955 |
\isacommand{bind} {\it binders} \isacommand{in} {\it bodies}\;\;\;\, |
3890483c674f
final version of the ESOP paper; used set+ instead of res as requested by one reviewer
Christian Urban <urbanc@in.tum.de>
parents:
2605
diff
changeset
|
956 |
\isacommand{bind (set)} {\it binders} \isacommand{in} {\it bodies}\;\;\;\, |
3890483c674f
final version of the ESOP paper; used set+ instead of res as requested by one reviewer
Christian Urban <urbanc@in.tum.de>
parents:
2605
diff
changeset
|
957 |
\isacommand{bind (set+)} {\it binders} \isacommand{in} {\it bodies} |
1617
99cee15cb5ff
more tuning in the paper
Christian Urban <urbanc@in.tum.de>
parents:
1613
diff
changeset
|
958 |
\end{tabular} |
1611 | 959 |
\end{center} |
2512 | 960 |
% |
1611 | 961 |
\noindent |
1730
cfd3a7368543
polished and removed tys from bn-functions.
Christian Urban <urbanc@in.tum.de>
parents:
1728
diff
changeset
|
962 |
The first mode is for binding lists of atoms (the order of binders matters); |
cfd3a7368543
polished and removed tys from bn-functions.
Christian Urban <urbanc@in.tum.de>
parents:
1728
diff
changeset
|
963 |
the second is for sets of binders (the order does not matter, but the |
cfd3a7368543
polished and removed tys from bn-functions.
Christian Urban <urbanc@in.tum.de>
parents:
1728
diff
changeset
|
964 |
cardinality does) and the last is for sets of binders (with vacuous binders |
2359 | 965 |
preserving $\alpha$-equivalence). As indicated, the labels in the ``\isacommand{in}-part'' of a binding |
2343 | 966 |
clause will be called \emph{bodies}; the |
2163
5dc48e1af733
added comments about pottiers work
Christian Urban <urbanc@in.tum.de>
parents:
2156
diff
changeset
|
967 |
``\isacommand{bind}-part'' will be called \emph{binders}. In contrast to |
2510
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
968 |
Ott, we allow multiple labels in binders and bodies. |
1956
028705c98304
more polishing on the paper
Christian Urban <urbanc@in.tum.de>
parents:
1954
diff
changeset
|
969 |
|
2510
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
970 |
%For example we allow |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
971 |
%binding clauses of the form: |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
972 |
% |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
973 |
%\begin{center} |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
974 |
%\begin{tabular}{@ {}ll@ {}} |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
975 |
%@{text "Foo\<^isub>1 x::name y::name t::trm s::trm"} & |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
976 |
% \isacommand{bind} @{text "x y"} \isacommand{in} @{text "t s"}\\ |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
977 |
%@{text "Foo\<^isub>2 x::name y::name t::trm s::trm"} & |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
978 |
% \isacommand{bind} @{text "x y"} \isacommand{in} @{text "t"}, |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
979 |
% \isacommand{bind} @{text "x y"} \isacommand{in} @{text "s"}\\ |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
980 |
%\end{tabular} |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
981 |
%\end{center} |
1956
028705c98304
more polishing on the paper
Christian Urban <urbanc@in.tum.de>
parents:
1954
diff
changeset
|
982 |
|
028705c98304
more polishing on the paper
Christian Urban <urbanc@in.tum.de>
parents:
1954
diff
changeset
|
983 |
\noindent |
2510
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
984 |
%Similarly for the other binding modes. |
2488
1c18f2cf3923
a few more words about Ott
Christian Urban <urbanc@in.tum.de>
parents:
2471
diff
changeset
|
985 |
%Interestingly, in case of \isacommand{bind (set)} |
2637
3890483c674f
final version of the ESOP paper; used set+ instead of res as requested by one reviewer
Christian Urban <urbanc@in.tum.de>
parents:
2605
diff
changeset
|
986 |
%and \isacommand{bind (set+)} the binding clauses above will make a difference to the semantics |
2343 | 987 |
%of the specifications (the corresponding $\alpha$-equivalence will differ). We will |
988 |
%show this later with an example. |
|
2140 | 989 |
|
2347 | 990 |
There are also some restrictions we need to impose on our binding clauses in comparison to |
991 |
the ones of Ott. The |
|
2343 | 992 |
main idea behind these restrictions is that we obtain a sensible notion of |
2347 | 993 |
$\alpha$-equivalence where it is ensured that within a given scope an |
2348 | 994 |
atom occurrence cannot be both bound and free at the same time. The first |
2344 | 995 |
restriction is that a body can only occur in |
2343 | 996 |
\emph{one} binding clause of a term constructor (this ensures that the bound |
2347 | 997 |
atoms of a body cannot be free at the same time by specifying an |
2514
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
998 |
alternative binder for the same body). |
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
999 |
|
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
1000 |
For binders we distinguish between |
2343 | 1001 |
\emph{shallow} and \emph{deep} binders. Shallow binders are just |
1002 |
labels. The restriction we need to impose on them is that in case of |
|
2637
3890483c674f
final version of the ESOP paper; used set+ instead of res as requested by one reviewer
Christian Urban <urbanc@in.tum.de>
parents:
2605
diff
changeset
|
1003 |
\isacommand{bind (set)} and \isacommand{bind (set+)} the labels must either |
2343 | 1004 |
refer to atom types or to sets of atom types; in case of \isacommand{bind} |
1005 |
the labels must refer to atom types or lists of atom types. Two examples for |
|
1006 |
the use of shallow binders are the specification of lambda-terms, where a |
|
1007 |
single name is bound, and type-schemes, where a finite set of names is |
|
1008 |
bound: |
|
1009 |
||
2511 | 1010 |
\begin{center}\small |
2604
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
1011 |
\begin{tabular}{@ {}c@ {\hspace{7mm}}c@ {}} |
2508
6d9018d62b40
first part of sqeezing everything into 20 pages (at the moment we have 26)
Christian Urban <urbanc@in.tum.de>
parents:
2507
diff
changeset
|
1012 |
\begin{tabular}{@ {}l} |
2341 | 1013 |
\isacommand{nominal\_datatype} @{text lam} $=$\\ |
2604
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
1014 |
\hspace{2mm}\phantom{$\mid$}~@{text "Var name"}\\ |
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
1015 |
\hspace{2mm}$\mid$~@{text "App lam lam"}\\ |
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
1016 |
\hspace{2mm}$\mid$~@{text "Lam x::name t::lam"}~~\isacommand{bind} @{text x} \isacommand{in} @{text t}\\ |
1611 | 1017 |
\end{tabular} & |
1612 | 1018 |
\begin{tabular}{@ {}l@ {}} |
2341 | 1019 |
\isacommand{nominal\_datatype}~@{text ty} $=$\\ |
1719
0c3c66f5c0e7
removed "raw" distinction
Christian Urban <urbanc@in.tum.de>
parents:
1717
diff
changeset
|
1020 |
\hspace{5mm}\phantom{$\mid$}~@{text "TVar name"}\\ |
0c3c66f5c0e7
removed "raw" distinction
Christian Urban <urbanc@in.tum.de>
parents:
1717
diff
changeset
|
1021 |
\hspace{5mm}$\mid$~@{text "TFun ty ty"}\\ |
2604
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
1022 |
\isacommand{and}~@{text "tsc = All xs::(name fset) T::ty"}~~% |
2637
3890483c674f
final version of the ESOP paper; used set+ instead of res as requested by one reviewer
Christian Urban <urbanc@in.tum.de>
parents:
2605
diff
changeset
|
1023 |
\isacommand{bind (set+)} @{text xs} \isacommand{in} @{text T}\\ |
1611 | 1024 |
\end{tabular} |
1025 |
\end{tabular} |
|
1026 |
\end{center} |
|
1587 | 1027 |
|
1612 | 1028 |
\noindent |
2341 | 1029 |
In these specifications @{text "name"} refers to an atom type, and @{text |
1030 |
"fset"} to the type of finite sets. |
|
2156 | 1031 |
Note that for @{text lam} it does not matter which binding mode we use. The |
1032 |
reason is that we bind only a single @{text name}. However, having |
|
2488
1c18f2cf3923
a few more words about Ott
Christian Urban <urbanc@in.tum.de>
parents:
2471
diff
changeset
|
1033 |
\isacommand{bind (set)} or \isacommand{bind} in the second case makes a |
2345 | 1034 |
difference to the semantics of the specification (which we will define in the next section). |
2156 | 1035 |
|
2128 | 1036 |
|
2134 | 1037 |
A \emph{deep} binder uses an auxiliary binding function that ``picks'' out |
2156 | 1038 |
the atoms in one argument of the term-constructor, which can be bound in |
1039 |
other arguments and also in the same argument (we will call such binders |
|
2341 | 1040 |
\emph{recursive}, see below). The binding functions are |
2488
1c18f2cf3923
a few more words about Ott
Christian Urban <urbanc@in.tum.de>
parents:
2471
diff
changeset
|
1041 |
expected to return either a set of atoms (for \isacommand{bind (set)} and |
2637
3890483c674f
final version of the ESOP paper; used set+ instead of res as requested by one reviewer
Christian Urban <urbanc@in.tum.de>
parents:
2605
diff
changeset
|
1042 |
\isacommand{bind (set+)}) or a list of atoms (for \isacommand{bind}). They can |
2343 | 1043 |
be defined by recursion over the corresponding type; the equations |
2156 | 1044 |
must be given in the binding function part of the scheme shown in |
1045 |
\eqref{scheme}. For example a term-calculus containing @{text "Let"}s with |
|
1046 |
tuple patterns might be specified as: |
|
1764
9f55d7927e5b
more on the strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
1763
diff
changeset
|
1047 |
% |
9f55d7927e5b
more on the strong induction section
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parents:
1763
diff
changeset
|
1048 |
\begin{equation}\label{letpat} |
2511 | 1049 |
\mbox{\small% |
1619 | 1050 |
\begin{tabular}{l} |
2512 | 1051 |
\isacommand{nominal\_datatype} @{text trm} $=$\\ |
1719
0c3c66f5c0e7
removed "raw" distinction
Christian Urban <urbanc@in.tum.de>
parents:
1717
diff
changeset
|
1052 |
\hspace{5mm}\phantom{$\mid$}~@{term "Var name"}\\ |
0c3c66f5c0e7
removed "raw" distinction
Christian Urban <urbanc@in.tum.de>
parents:
1717
diff
changeset
|
1053 |
\hspace{5mm}$\mid$~@{term "App trm trm"}\\ |
0c3c66f5c0e7
removed "raw" distinction
Christian Urban <urbanc@in.tum.de>
parents:
1717
diff
changeset
|
1054 |
\hspace{5mm}$\mid$~@{text "Lam x::name t::trm"} |
0c3c66f5c0e7
removed "raw" distinction
Christian Urban <urbanc@in.tum.de>
parents:
1717
diff
changeset
|
1055 |
\;\;\isacommand{bind} @{text x} \isacommand{in} @{text t}\\ |
0c3c66f5c0e7
removed "raw" distinction
Christian Urban <urbanc@in.tum.de>
parents:
1717
diff
changeset
|
1056 |
\hspace{5mm}$\mid$~@{text "Let p::pat trm t::trm"} |
0c3c66f5c0e7
removed "raw" distinction
Christian Urban <urbanc@in.tum.de>
parents:
1717
diff
changeset
|
1057 |
\;\;\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text t}\\ |
2512 | 1058 |
\isacommand{and} @{text pat} $=$ |
1059 |
@{text PNil} |
|
1060 |
$\mid$~@{text "PVar name"} |
|
1061 |
$\mid$~@{text "PTup pat pat"}\\ |
|
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1062 |
\isacommand{binder}~@{text "bn::pat \<Rightarrow> atom list"}\\ |
1719
0c3c66f5c0e7
removed "raw" distinction
Christian Urban <urbanc@in.tum.de>
parents:
1717
diff
changeset
|
1063 |
\isacommand{where}~@{text "bn(PNil) = []"}\\ |
0c3c66f5c0e7
removed "raw" distinction
Christian Urban <urbanc@in.tum.de>
parents:
1717
diff
changeset
|
1064 |
\hspace{5mm}$\mid$~@{text "bn(PVar x) = [atom x]"}\\ |
2341 | 1065 |
\hspace{5mm}$\mid$~@{text "bn(PTup p\<^isub>1 p\<^isub>2) = bn(p\<^isub>1) @ bn(p\<^isub>2)"}\smallskip\\ |
1764
9f55d7927e5b
more on the strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
1763
diff
changeset
|
1066 |
\end{tabular}} |
9f55d7927e5b
more on the strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
1763
diff
changeset
|
1067 |
\end{equation} |
2516 | 1068 |
% |
1619 | 1069 |
\noindent |
2140 | 1070 |
In this specification the function @{text "bn"} determines which atoms of |
2346 | 1071 |
the pattern @{text p} are bound in the argument @{text "t"}. Note that in the |
2140 | 1072 |
second-last @{text bn}-clause the function @{text "atom"} coerces a name |
1073 |
into the generic atom type of Nominal Isabelle \cite{HuffmanUrban10}. This |
|
1074 |
allows us to treat binders of different atom type uniformly. |
|
1637 | 1075 |
|
2140 | 1076 |
As said above, for deep binders we allow binding clauses such as |
1077 |
% |
|
2516 | 1078 |
%\begin{center} |
1079 |
%\begin{tabular}{ll} |
|
1080 |
@{text "Bar p::pat t::trm"} %%%& |
|
1081 |
\isacommand{bind} @{text "bn(p)"} \isacommand{in} @{text "p t"} %%\\ |
|
1082 |
%\end{tabular} |
|
1083 |
%\end{center} |
|
1084 |
% |
|
1085 |
%\noindent |
|
2344 | 1086 |
where the argument of the deep binder also occurs in the body. We call such |
2140 | 1087 |
binders \emph{recursive}. To see the purpose of such recursive binders, |
1088 |
compare ``plain'' @{text "Let"}s and @{text "Let_rec"}s in the following |
|
1089 |
specification: |
|
2343 | 1090 |
% |
1725
1801cc460fc9
polished and added an example for fvars
Christian Urban <urbanc@in.tum.de>
parents:
1724
diff
changeset
|
1091 |
\begin{equation}\label{letrecs} |
2511 | 1092 |
\mbox{\small% |
1637 | 1093 |
\begin{tabular}{@ {}l@ {}} |
2512 | 1094 |
\isacommand{nominal\_datatype}~@{text "trm ="}~\ldots\\ |
1725
1801cc460fc9
polished and added an example for fvars
Christian Urban <urbanc@in.tum.de>
parents:
1724
diff
changeset
|
1095 |
\hspace{5mm}$\mid$~@{text "Let as::assn t::trm"} |
1801cc460fc9
polished and added an example for fvars
Christian Urban <urbanc@in.tum.de>
parents:
1724
diff
changeset
|
1096 |
\;\;\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text t}\\ |
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1097 |
\hspace{5mm}$\mid$~@{text "Let_rec as::assn t::trm"} |
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1098 |
\;\;\isacommand{bind} @{text "bn(as)"} \isacommand{in} @{text "as t"}\\ |
2512 | 1099 |
\isacommand{and} @{text "assn"} $=$ |
1100 |
@{text "ANil"} |
|
1101 |
$\mid$~@{text "ACons name trm assn"}\\ |
|
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1102 |
\isacommand{binder} @{text "bn::assn \<Rightarrow> atom list"}\\ |
1725
1801cc460fc9
polished and added an example for fvars
Christian Urban <urbanc@in.tum.de>
parents:
1724
diff
changeset
|
1103 |
\isacommand{where}~@{text "bn(ANil) = []"}\\ |
1801cc460fc9
polished and added an example for fvars
Christian Urban <urbanc@in.tum.de>
parents:
1724
diff
changeset
|
1104 |
\hspace{5mm}$\mid$~@{text "bn(ACons a t as) = [atom a] @ bn(as)"}\\ |
1801cc460fc9
polished and added an example for fvars
Christian Urban <urbanc@in.tum.de>
parents:
1724
diff
changeset
|
1105 |
\end{tabular}} |
1801cc460fc9
polished and added an example for fvars
Christian Urban <urbanc@in.tum.de>
parents:
1724
diff
changeset
|
1106 |
\end{equation} |
2516 | 1107 |
% |
1636 | 1108 |
\noindent |
1727
fd2913415a73
started to polish alpha-equivalence section, but needs more work
Christian Urban <urbanc@in.tum.de>
parents:
1726
diff
changeset
|
1109 |
The difference is that with @{text Let} we only want to bind the atoms @{text |
1730
cfd3a7368543
polished and removed tys from bn-functions.
Christian Urban <urbanc@in.tum.de>
parents:
1728
diff
changeset
|
1110 |
"bn(as)"} in the term @{text t}, but with @{text "Let_rec"} we also want to bind the atoms |
2346 | 1111 |
inside the assignment. This difference has consequences for the associated |
2347 | 1112 |
notions of free-atoms and $\alpha$-equivalence. |
2341 | 1113 |
|
2347 | 1114 |
To make sure that atoms bound by deep binders cannot be free at the |
2346 | 1115 |
same time, we cannot have more than one binding function for a deep binder. |
2344 | 1116 |
Consequently we exclude specifications such as |
2516 | 1117 |
% |
2604
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
1118 |
\begin{center}\small |
2341 | 1119 |
\begin{tabular}{@ {}l@ {\hspace{2mm}}l@ {}} |
1120 |
@{text "Baz\<^isub>1 p::pat t::trm"} & |
|
2140 | 1121 |
\isacommand{bind} @{text "bn\<^isub>1(p) bn\<^isub>2(p)"} \isacommand{in} @{text t}\\ |
2341 | 1122 |
@{text "Baz\<^isub>2 p::pat t\<^isub>1::trm t\<^isub>2::trm"} & |
1123 |
\isacommand{bind} @{text "bn\<^isub>1(p)"} \isacommand{in} @{text "t\<^isub>1"}, |
|
1124 |
\isacommand{bind} @{text "bn\<^isub>2(p)"} \isacommand{in} @{text "t\<^isub>2"}\\ |
|
2140 | 1125 |
\end{tabular} |
1126 |
\end{center} |
|
1127 |
||
1128 |
\noindent |
|
2344 | 1129 |
Otherwise it is possible that @{text "bn\<^isub>1"} and @{text "bn\<^isub>2"} pick |
2348 | 1130 |
out different atoms to become bound, respectively be free, in @{text "p"}. |
2359 | 1131 |
(Since the Ott-tool does not derive a reasoning infrastructure for |
2514
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
1132 |
$\alpha$-equated terms with deep binders, it can permit such specifications.) |
2343 | 1133 |
|
2344 | 1134 |
We also need to restrict the form of the binding functions in order |
2345 | 1135 |
to ensure the @{text "bn"}-functions can be defined for $\alpha$-equated |
2346 | 1136 |
terms. The main restriction is that we cannot return an atom in a binding function that is also |
1137 |
bound in the corresponding term-constructor. That means in \eqref{letpat} |
|
1138 |
that the term-constructors @{text PVar} and @{text PTup} may |
|
1961 | 1139 |
not have a binding clause (all arguments are used to define @{text "bn"}). |
1140 |
In contrast, in case of \eqref{letrecs} the term-constructor @{text ACons} |
|
2605
213786e0bd45
a bit more tuning of the paper
Christian Urban <urbanc@in.tum.de>
parents:
2604
diff
changeset
|
1141 |
may have a binding clause involving the argument @{text trm} (the only one that |
2343 | 1142 |
is \emph{not} used in the definition of the binding function). This restriction |
2510
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1143 |
is sufficient for lifting the binding function to $\alpha$-equated terms. |
2341 | 1144 |
|
1145 |
In the version of |
|
1961 | 1146 |
Nominal Isabelle described here, we also adopted the restriction from the |
1147 |
Ott-tool that binding functions can only return: the empty set or empty list |
|
1148 |
(as in case @{text PNil}), a singleton set or singleton list containing an |
|
1149 |
atom (case @{text PVar}), or unions of atom sets or appended atom lists |
|
2341 | 1150 |
(case @{text PTup}). This restriction will simplify some automatic definitions and proofs |
1961 | 1151 |
later on. |
1152 |
||
2347 | 1153 |
In order to simplify our definitions of free atoms and $\alpha$-equivalence, |
2343 | 1154 |
we shall assume specifications |
2341 | 1155 |
of term-calculi are implicitly \emph{completed}. By this we mean that |
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1156 |
for every argument of a term-constructor that is \emph{not} |
2163
5dc48e1af733
added comments about pottiers work
Christian Urban <urbanc@in.tum.de>
parents:
2156
diff
changeset
|
1157 |
already part of a binding clause given by the user, we add implicitly a special \emph{empty} binding |
5dc48e1af733
added comments about pottiers work
Christian Urban <urbanc@in.tum.de>
parents:
2156
diff
changeset
|
1158 |
clause, written \isacommand{bind}~@{term "{}"}~\isacommand{in}~@{text "labels"}. In case |
2510
734341a79028
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Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1159 |
of the lambda-terms, the completion produces |
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1160 |
|
2511 | 1161 |
\begin{center}\small |
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1162 |
\begin{tabular}{@ {}l@ {\hspace{-1mm}}} |
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1163 |
\isacommand{nominal\_datatype} @{text lam} =\\ |
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1164 |
\hspace{5mm}\phantom{$\mid$}~@{text "Var x::name"} |
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1165 |
\;\;\isacommand{bind}~@{term "{}"}~\isacommand{in}~@{text "x"}\\ |
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1166 |
\hspace{5mm}$\mid$~@{text "App t\<^isub>1::lam t\<^isub>2::lam"} |
2163
5dc48e1af733
added comments about pottiers work
Christian Urban <urbanc@in.tum.de>
parents:
2156
diff
changeset
|
1167 |
\;\;\isacommand{bind}~@{term "{}"}~\isacommand{in}~@{text "t\<^isub>1 t\<^isub>2"}\\ |
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1168 |
\hspace{5mm}$\mid$~@{text "Lam x::name t::lam"} |
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1169 |
\;\;\isacommand{bind}~@{text x} \isacommand{in} @{text t}\\ |
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1170 |
\end{tabular} |
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1171 |
\end{center} |
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1172 |
|
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1173 |
\noindent |
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1174 |
The point of completion is that we can make definitions over the binding |
1961 | 1175 |
clauses and be sure to have captured all arguments of a term constructor. |
2342 | 1176 |
*} |
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1177 |
|
2347 | 1178 |
section {* Alpha-Equivalence and Free Atoms\label{sec:alpha} *} |
2342 | 1179 |
|
1180 |
text {* |
|
1637 | 1181 |
Having dealt with all syntax matters, the problem now is how we can turn |
1182 |
specifications into actual type definitions in Isabelle/HOL and then |
|
1926 | 1183 |
establish a reasoning infrastructure for them. As |
1956
028705c98304
more polishing on the paper
Christian Urban <urbanc@in.tum.de>
parents:
1954
diff
changeset
|
1184 |
Pottier and Cheney pointed out \cite{Pottier06,Cheney05}, just |
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1185 |
re-arranging the arguments of |
1956
028705c98304
more polishing on the paper
Christian Urban <urbanc@in.tum.de>
parents:
1954
diff
changeset
|
1186 |
term-constructors so that binders and their bodies are next to each other will |
2347 | 1187 |
result in inadequate representations in cases like @{text "Let x\<^isub>1 = t\<^isub>1\<dots>x\<^isub>n = t\<^isub>n in s"}. |
2343 | 1188 |
Therefore we will first |
2346 | 1189 |
extract ``raw'' datatype definitions from the specification and then define |
2343 | 1190 |
explicitly an $\alpha$-equivalence relation over them. We subsequently |
2605
213786e0bd45
a bit more tuning of the paper
Christian Urban <urbanc@in.tum.de>
parents:
2604
diff
changeset
|
1191 |
construct the quotient of the datatypes according to our $\alpha$-equivalence. |
1637 | 1192 |
|
2346 | 1193 |
The ``raw'' datatype definition can be obtained by stripping off the |
1771
3e71af53cedb
submitted version (just in time ;o)
Christian Urban <urbanc@in.tum.de>
parents:
1770
diff
changeset
|
1194 |
binding clauses and the labels from the types. We also have to invent |
1637 | 1195 |
new names for the types @{text "ty\<^sup>\<alpha>"} and term-constructors @{text "C\<^sup>\<alpha>"} |
1756 | 1196 |
given by the user. In our implementation we just use the affix ``@{text "_raw"}''. |
1771
3e71af53cedb
submitted version (just in time ;o)
Christian Urban <urbanc@in.tum.de>
parents:
1770
diff
changeset
|
1197 |
But for the purpose of this paper, we use the superscript @{text "_\<^sup>\<alpha>"} to indicate |
2510
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1198 |
that a notion is given for $\alpha$-equivalence classes and leave it out |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1199 |
for the corresponding notion given on the ``raw'' level. So for example |
2511 | 1200 |
we have @{text "ty\<^sup>\<alpha> \<mapsto> ty"} and @{text "C\<^sup>\<alpha> \<mapsto> C"} |
1730
cfd3a7368543
polished and removed tys from bn-functions.
Christian Urban <urbanc@in.tum.de>
parents:
1728
diff
changeset
|
1201 |
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>
parents:
1726
diff
changeset
|
1202 |
@{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>
parents:
1726
diff
changeset
|
1203 |
|
2514
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
1204 |
%The resulting datatype definition is legal in Isabelle/HOL provided the datatypes are |
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
1205 |
%non-empty and the types in the constructors only occur in positive |
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
1206 |
%position (see \cite{Berghofer99} for an in-depth description of the datatype package |
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
1207 |
%in Isabelle/HOL). |
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
1208 |
We subsequently define each of the user-specified binding |
2348 | 1209 |
functions @{term "bn"}$_{1..m}$ by recursion over the corresponding |
1730
cfd3a7368543
polished and removed tys from bn-functions.
Christian Urban <urbanc@in.tum.de>
parents:
1728
diff
changeset
|
1210 |
raw datatype. We can also easily define permutation operations by |
2345 | 1211 |
recursion so that for each term constructor @{text "C"} we have that |
1766
a2d5f9ea17ad
completed lifting section
Christian Urban <urbanc@in.tum.de>
parents:
1765
diff
changeset
|
1212 |
% |
a2d5f9ea17ad
completed lifting section
Christian Urban <urbanc@in.tum.de>
parents:
1765
diff
changeset
|
1213 |
\begin{equation}\label{ceqvt} |
1961 | 1214 |
@{text "p \<bullet> (C z\<^isub>1 \<dots> z\<^isub>n) = C (p \<bullet> z\<^isub>1) \<dots> (p \<bullet> z\<^isub>n)"} |
1766
a2d5f9ea17ad
completed lifting section
Christian Urban <urbanc@in.tum.de>
parents:
1765
diff
changeset
|
1215 |
\end{equation} |
2343 | 1216 |
|
2341 | 1217 |
The first non-trivial step we have to perform is the generation of |
2348 | 1218 |
free-atom functions from the specification. For the |
2347 | 1219 |
\emph{raw} types @{text "ty"}$_{1..n}$ we define the free-atom functions |
2343 | 1220 |
% |
2512 | 1221 |
%\begin{equation}\label{fvars} |
1222 |
@{text "fa_ty\<^isub>"}$_{1..n}$ |
|
1223 |
%\end{equation} |
|
1224 |
% |
|
1225 |
%\noindent |
|
2508
6d9018d62b40
first part of sqeezing everything into 20 pages (at the moment we have 26)
Christian Urban <urbanc@in.tum.de>
parents:
2507
diff
changeset
|
1226 |
by recursion. |
2347 | 1227 |
We define these functions together with auxiliary free-atom functions for |
2343 | 1228 |
the binding functions. Given raw binding functions @{text "bn"}$_{1..m}$ |
1229 |
we define |
|
1230 |
% |
|
2512 | 1231 |
%\begin{center} |
1232 |
@{text "fa_bn\<^isub>"}$_{1..m}$. |
|
1233 |
%\end{center} |
|
1234 |
% |
|
1235 |
%\noindent |
|
2341 | 1236 |
The reason for this setup is that in a deep binder not all atoms have to be |
1237 |
bound, as we saw in the example with ``plain'' @{text Let}s. We need therefore a function |
|
2348 | 1238 |
that calculates those free atoms in a deep binder. |
2343 | 1239 |
|
2347 | 1240 |
While the idea behind these free-atom functions is clear (they just |
2343 | 1241 |
collect all atoms that are not bound), because of our rather complicated |
2345 | 1242 |
binding mechanisms their definitions are somewhat involved. Given |
2346 | 1243 |
a term-constructor @{text "C"} of type @{text ty} and some associated |
2344 | 1244 |
binding clauses @{text "bc\<^isub>1\<dots>bc\<^isub>k"}, the result of @{text |
2347 | 1245 |
"fa_ty (C z\<^isub>1 \<dots> z\<^isub>n)"} will be the union @{text |
2348 | 1246 |
"fa(bc\<^isub>1) \<union> \<dots> \<union> fa(bc\<^isub>k)"} where we will define below what @{text "fa"} for a binding |
2488
1c18f2cf3923
a few more words about Ott
Christian Urban <urbanc@in.tum.de>
parents:
2471
diff
changeset
|
1247 |
clause means. We only show the details for the mode \isacommand{bind (set)} (the other modes are similar). |
2345 | 1248 |
Suppose the binding clause @{text bc\<^isub>i} is of the form |
2343 | 1249 |
% |
2510
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1250 |
%\begin{center} |
2488
1c18f2cf3923
a few more words about Ott
Christian Urban <urbanc@in.tum.de>
parents:
2471
diff
changeset
|
1251 |
\mbox{\isacommand{bind (set)} @{text "b\<^isub>1\<dots>b\<^isub>p"} \isacommand{in} @{text "d\<^isub>1\<dots>d\<^isub>q"}} |
2510
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1252 |
%\end{center} |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1253 |
% |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1254 |
%\noindent |
2344 | 1255 |
in which the body-labels @{text "d"}$_{1..q}$ refer to types @{text ty}$_{1..q}$, |
1256 |
and the binders @{text b}$_{1..p}$ |
|
2343 | 1257 |
either refer to labels of atom types (in case of shallow binders) or to binding |
2348 | 1258 |
functions taking a single label as argument (in case of deep binders). Assuming |
1259 |
@{text "D"} stands for the set of free atoms of the bodies, @{text B} for the |
|
1260 |
set of binding atoms in the binders and @{text "B'"} for the set of free atoms in |
|
2344 | 1261 |
non-recursive deep binders, |
2516 | 1262 |
then the free atoms of the binding clause @{text bc\<^isub>i} are\\[-2mm] |
2343 | 1263 |
% |
2348 | 1264 |
\begin{equation}\label{fadef} |
1265 |
\mbox{@{text "fa(bc\<^isub>i) \<equiv> (D - B) \<union> B'"}}. |
|
1266 |
\end{equation} |
|
2516 | 1267 |
% |
2343 | 1268 |
\noindent |
2348 | 1269 |
The set @{text D} is formally defined as |
2343 | 1270 |
% |
2516 | 1271 |
%\begin{center} |
2347 | 1272 |
@{text "D \<equiv> fa_ty\<^isub>1 d\<^isub>1 \<union> ... \<union> fa_ty\<^isub>q d\<^isub>q"} |
2516 | 1273 |
%\end{center} |
1274 |
% |
|
1275 |
%\noindent |
|
2348 | 1276 |
where in case @{text "d\<^isub>i"} refers to one of the raw types @{text "ty"}$_{1..n}$ from the |
1277 |
specification, the function @{text "fa_ty\<^isub>i"} is the corresponding free-atom function |
|
2512 | 1278 |
we are defining by recursion; |
1279 |
%(see \eqref{fvars}); |
|
1280 |
otherwise we set @{text "fa_ty\<^isub>i d\<^isub>i = supp d\<^isub>i"}. |
|
2348 | 1281 |
|
1282 |
In order to formally define the set @{text B} we use the following auxiliary @{text "bn"}-functions |
|
2516 | 1283 |
for atom types to which shallow binders may refer\\[-4mm] |
2344 | 1284 |
% |
2510
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1285 |
%\begin{center} |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1286 |
%\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l} |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1287 |
%@{text "bn_atom a"} & @{text "\<equiv>"} & @{text "{atom a}"}\\ |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1288 |
%@{text "bn_atom_set as"} & @{text "\<equiv>"} & @{text "atoms as"}\\ |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1289 |
%@{text "bn_atom_list as"} & @{text "\<equiv>"} & @{text "atoms (set as)"} |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1290 |
%\end{tabular} |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1291 |
%\end{center} |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1292 |
% |
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1293 |
\begin{center} |
2516 | 1294 |
@{text "bn\<^bsub>atom\<^esub> a \<equiv> {atom a}"}\hfill |
1295 |
@{text "bn\<^bsub>atom_set\<^esub> as \<equiv> atoms as"}\hfill |
|
1296 |
@{text "bn\<^bsub>atom_list\<^esub> as \<equiv> atoms (set as)"} |
|
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1297 |
\end{center} |
2516 | 1298 |
% |
1954
23480003f9c5
some changes to the paper
Christian Urban <urbanc@in.tum.de>
parents:
1926
diff
changeset
|
1299 |
\noindent |
2348 | 1300 |
Like the function @{text atom}, the function @{text "atoms"} coerces |
2512 | 1301 |
a set of atoms to a set of the generic atom type. |
1302 |
%It is defined as @{text "atoms as \<equiv> {atom a | a \<in> as}"}. |
|
2516 | 1303 |
The set @{text B} is then formally defined as\\[-4mm] |
2344 | 1304 |
% |
1305 |
\begin{center} |
|
1306 |
@{text "B \<equiv> bn_ty\<^isub>1 b\<^isub>1 \<union> ... \<union> bn_ty\<^isub>p b\<^isub>p"} |
|
1307 |
\end{center} |
|
2516 | 1308 |
% |
2344 | 1309 |
\noindent |
2348 | 1310 |
where we use the auxiliary binding functions for shallow binders. |
2344 | 1311 |
The set @{text "B'"} collects all free atoms in non-recursive deep |
2347 | 1312 |
binders. Let us assume these binders in @{text "bc\<^isub>i"} are |
1956
028705c98304
more polishing on the paper
Christian Urban <urbanc@in.tum.de>
parents:
1954
diff
changeset
|
1313 |
% |
2510
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1314 |
%\begin{center} |
734341a79028
down to 24 pages and a bit
Christian Urban <urbanc@in.tum.de>
parents:
2509
diff
changeset
|
1315 |
\mbox{@{text "bn\<^isub>1 l\<^isub>1, \<dots>, bn\<^isub>r l\<^isub>r"}} |
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1316 |
%\end{center} |
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|
1317 |
% |
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|
1318 |
%\noindent |
2349 | 1319 |
with @{text "l"}$_{1..r}$ $\subseteq$ @{text "b"}$_{1..p}$ and none of the |
1320 |
@{text "l"}$_{1..r}$ being among the bodies @{text |
|
2516 | 1321 |
"d"}$_{1..q}$. The set @{text "B'"} is defined as\\[-5mm] |
2344 | 1322 |
% |
1323 |
\begin{center} |
|
2516 | 1324 |
@{text "B' \<equiv> fa_bn\<^isub>1 l\<^isub>1 \<union> ... \<union> fa_bn\<^isub>r l\<^isub>r"}\\[-9mm] |
1325 |
\end{center} |
|
1326 |
% |
|
1636 | 1327 |
\noindent |
2348 | 1328 |
This completes the definition of the free-atom functions @{text "fa_ty"}$_{1..n}$. |
2344 | 1329 |
|
2348 | 1330 |
Note that for non-recursive deep binders, we have to add in \eqref{fadef} |
1331 |
the set of atoms that are left unbound by the binding functions @{text |
|
2349 | 1332 |
"bn"}$_{1..m}$. We used for the definition of |
2348 | 1333 |
this set the functions @{text "fa_bn"}$_{1..m}$, which are also defined by mutual |
1334 |
recursion. Assume the user specified a @{text bn}-clause of the form |
|
1956
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|
1335 |
% |
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1336 |
%\begin{center} |
2347 | 1337 |
@{text "bn (C z\<^isub>1 \<dots> z\<^isub>s) = rhs"} |
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|
1338 |
%\end{center} |
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|
1339 |
% |
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|
1340 |
%\noindent |
2347 | 1341 |
where the @{text "z"}$_{1..s}$ are of types @{text "ty"}$_{1..s}$. For each of |
2345 | 1342 |
the arguments we calculate the free atoms as follows: |
2344 | 1343 |
% |
1344 |
\begin{center} |
|
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|
1345 |
\begin{tabular}{c@ {\hspace{2mm}}p{0.9\textwidth}} |
2347 | 1346 |
$\bullet$ & @{term "fa_ty\<^isub>i z\<^isub>i"} provided @{text "z\<^isub>i"} does not occur in @{text "rhs"} |
2348 | 1347 |
(that means nothing is bound in @{text "z\<^isub>i"} by the binding function),\\ |
2347 | 1348 |
$\bullet$ & @{term "fa_bn\<^isub>i z\<^isub>i"} provided @{text "z\<^isub>i"} occurs in @{text "rhs"} |
2345 | 1349 |
with the recursive call @{text "bn\<^isub>i z\<^isub>i"}, and\\ |
2344 | 1350 |
$\bullet$ & @{term "{}"} provided @{text "z\<^isub>i"} occurs in @{text "rhs"}, |
2347 | 1351 |
but without a recursive call. |
2344 | 1352 |
\end{tabular} |
1353 |
\end{center} |
|
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|
1354 |
% |
2344 | 1355 |
\noindent |
2348 | 1356 |
For defining @{text "fa_bn (C z\<^isub>1 \<dots> z\<^isub>n)"} we just union up all these sets. |
2344 | 1357 |
|
2348 | 1358 |
To see how these definitions work in practice, let us reconsider the |
1359 |
term-constructors @{text "Let"} and @{text "Let_rec"} shown in |
|
1360 |
\eqref{letrecs} together with the term-constructors for assignments @{text |
|
1361 |
"ANil"} and @{text "ACons"}. Since there is a binding function defined for |
|
1362 |
assignments, we have three free-atom functions, namely @{text |
|
1363 |
"fa\<^bsub>trm\<^esub>"}, @{text "fa\<^bsub>assn\<^esub>"} and @{text |
|
1364 |
"fa\<^bsub>bn\<^esub>"} as follows: |
|
1725
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changeset
|
1365 |
% |
2512 | 1366 |
\begin{center}\small |
1725
1801cc460fc9
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diff
changeset
|
1367 |
\begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}} |
2347 | 1368 |
@{text "fa\<^bsub>trm\<^esub> (Let as t)"} & @{text "="} & @{text "(fa\<^bsub>trm\<^esub> t - set (bn as)) \<union> fa\<^bsub>bn\<^esub> as"}\\ |
1369 |
@{text "fa\<^bsub>trm\<^esub> (Let_rec as t)"} & @{text "="} & @{text "(fa\<^bsub>assn\<^esub> as \<union> fa\<^bsub>trm\<^esub> t) - set (bn as)"}\\[1mm] |
|
1725
1801cc460fc9
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diff
changeset
|
1370 |
|
2347 | 1371 |
@{text "fa\<^bsub>assn\<^esub> (ANil)"} & @{text "="} & @{term "{}"}\\ |
1372 |
@{text "fa\<^bsub>assn\<^esub> (ACons a t as)"} & @{text "="} & @{text "(supp a) \<union> (fa\<^bsub>trm\<^esub> t) \<union> (fa\<^bsub>assn\<^esub> as)"}\\[1mm] |
|
1725
1801cc460fc9
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diff
changeset
|
1373 |
|
2347 | 1374 |
@{text "fa\<^bsub>bn\<^esub> (ANil)"} & @{text "="} & @{term "{}"}\\ |
1375 |
@{text "fa\<^bsub>bn\<^esub> (ACons a t as)"} & @{text "="} & @{text "(fa\<^bsub>trm\<^esub> t) \<union> (fa\<^bsub>bn\<^esub> as)"} |
|
1725
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|
1376 |
\end{tabular} |
1801cc460fc9
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diff
changeset
|
1377 |
\end{center} |
1801cc460fc9
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diff
changeset
|
1378 |
|
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diff
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|
1379 |
\noindent |
2349 | 1380 |
Recall that @{text ANil} and @{text "ACons"} have no |
2347 | 1381 |
binding clause in the specification. The corresponding free-atom |
2348 | 1382 |
function @{text "fa\<^bsub>assn\<^esub>"} therefore returns all free atoms |
2516 | 1383 |
of an assignment (in case of @{text "ACons"}, they are given in |
2360
99134763d03e
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diff
changeset
|
1384 |
terms of @{text supp}, @{text "fa\<^bsub>trm\<^esub>"} and @{text "fa\<^bsub>assn\<^esub>"}). |
2348 | 1385 |
The binding only takes place in @{text Let} and |
1386 |
@{text "Let_rec"}. In case of @{text "Let"}, the binding clause specifies |
|
2345 | 1387 |
that all atoms given by @{text "set (bn as)"} have to be bound in @{text |
1388 |
t}. Therefore we have to subtract @{text "set (bn as)"} from @{text |
|
2347 | 1389 |
"fa\<^bsub>trm\<^esub> t"}. However, we also need to add all atoms that are |
2348 | 1390 |
free in @{text "as"}. This is |
1391 |
in contrast with @{text "Let_rec"} where we have a recursive |
|
1392 |
binder to bind all occurrences of the atoms in @{text |
|
2345 | 1393 |
"set (bn as)"} also inside @{text "as"}. Therefore we have to subtract |
2348 | 1394 |
@{text "set (bn as)"} from both @{text "fa\<^bsub>trm\<^esub> t"} and @{text "fa\<^bsub>assn\<^esub> as"}. |
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|
1395 |
%Like the function @{text "bn"}, the function @{text "fa\<^bsub>bn\<^esub>"} traverses the |
734341a79028
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|
1396 |
%list of assignments, but instead returns the free atoms, which means in this |
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|
1397 |
%example the free atoms in the argument @{text "t"}. |
2348 | 1398 |
|
2345 | 1399 |
An interesting point in this |
2348 | 1400 |
example is that a ``naked'' assignment (@{text "ANil"} or @{text "ACons"}) does not bind any |
1401 |
atoms, even if the binding function is specified over assignments. |
|
1402 |
Only in the context of a @{text Let} or @{text "Let_rec"}, where the binding clauses are given, will |
|
1403 |
some atoms actually become bound. This is a phenomenon that has also been pointed |
|
2345 | 1404 |
out in \cite{ott-jfp}. For us this observation is crucial, because we would |
2348 | 1405 |
not be able to lift the @{text "bn"}-functions to $\alpha$-equated terms if they act on |
1406 |
atoms that are bound. In that case, these functions would \emph{not} respect |
|
2345 | 1407 |
$\alpha$-equivalence. |
1408 |
||
2348 | 1409 |
Next we define the $\alpha$-equivalence relations for the raw types @{text |
1410 |
"ty"}$_{1..n}$ from the specification. We write them as |
|
1733
6988077666dc
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|
1411 |
% |
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|
1412 |
%\begin{center} |
2512 | 1413 |
@{text "\<approx>ty"}$_{1..n}$. |
2510
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|
1414 |
%\end{center} |
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|
1415 |
% |
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|
1416 |
%\noindent |
2347 | 1417 |
Like with the free-atom functions, we also need to |
2345 | 1418 |
define auxiliary $\alpha$-equivalence relations |
1419 |
% |
|
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|
1420 |
%\begin{center} |
2512 | 1421 |
@{text "\<approx>bn\<^isub>"}$_{1..m}$ |
2510
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|
1422 |
%\end{center} |
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|
1423 |
% |
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|
1424 |
%\noindent |
2345 | 1425 |
for the binding functions @{text "bn"}$_{1..m}$, |
1426 |
To simplify our definitions we will use the following abbreviations for |
|
2580
6b3e8602edcf
implemented concrete suggestion of 3rd reviewer
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2555
diff
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|
1427 |
\emph{compound equivalence relations} and \emph{compound free-atom functions} acting on tuples. |
1733
6988077666dc
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Christian Urban <urbanc@in.tum.de>
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1732
diff
changeset
|
1428 |
% |
6988077666dc
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diff
changeset
|
1429 |
\begin{center} |
1737
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1736
diff
changeset
|
1430 |
\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l} |
2512 | 1431 |
@{text "(x\<^isub>1,\<dots>, x\<^isub>n) (R\<^isub>1,\<dots>, R\<^isub>n) (x\<PRIME>\<^isub>1,\<dots>, x\<PRIME>\<^isub>n)"} & @{text "\<equiv>"} & |
1432 |
@{text "x\<^isub>1 R\<^isub>1 x\<PRIME>\<^isub>1 \<and> \<dots> \<and> x\<^isub>n R\<^isub>n x\<PRIME>\<^isub>n"}\\ |
|
1433 |
@{text "(fa\<^isub>1,\<dots>, fa\<^isub>n) (x\<^isub>1,\<dots>, x\<^isub>n)"} & @{text "\<equiv>"} & @{text "fa\<^isub>1 x\<^isub>1 \<union> \<dots> \<union> fa\<^isub>n x\<^isub>n"}\\ |
|
1733
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1732
diff
changeset
|
1434 |
\end{tabular} |
6988077666dc
abbreviations for \<otimes> and \<oplus>
Christian Urban <urbanc@in.tum.de>
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1732
diff
changeset
|
1435 |
\end{center} |
6988077666dc
abbreviations for \<otimes> and \<oplus>
Christian Urban <urbanc@in.tum.de>
parents:
1732
diff
changeset
|
1436 |
|
1727
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diff
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|
1437 |
|
2348 | 1438 |
The $\alpha$-equivalence relations are defined as inductive predicates |
1439 |
having a single clause for each term-constructor. Assuming a |
|
1440 |
term-constructor @{text C} is of type @{text ty} and has the binding clauses |
|
1441 |
@{term "bc"}$_{1..k}$, then the $\alpha$-equivalence clause has the form |
|
1737
8b6a285ad480
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1736
diff
changeset
|
1442 |
% |
8b6a285ad480
polished everything up to TODO
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diff
changeset
|
1443 |
\begin{center} |
2345 | 1444 |
\mbox{\infer{@{text "C z\<^isub>1 \<dots> z\<^isub>n \<approx>ty C z\<PRIME>\<^isub>1 \<dots> z\<PRIME>\<^isub>n"}} |
2348 | 1445 |
{@{text "prems(bc\<^isub>1) \<dots> prems(bc\<^isub>k)"}}} |
1446 |
\end{center} |
|
1447 |
||
1448 |
\noindent |
|
1449 |
The task below is to specify what the premises of a binding clause are. As a |
|
1450 |
special instance, we first treat the case where @{text "bc\<^isub>i"} is the |
|
1451 |
empty binding clause of the form |
|
1452 |
% |
|
1453 |
\begin{center} |
|
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diff
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|
1454 |
\mbox{\isacommand{bind (set)} @{term "{}"} \isacommand{in} @{text "d\<^isub>1\<dots>d\<^isub>q"}.} |
2348 | 1455 |
\end{center} |
1456 |
||
1457 |
\noindent |
|
1458 |
In this binding clause no atom is bound and we only have to $\alpha$-relate the bodies. For this |
|
1459 |
we build first the tuples @{text "D \<equiv> (d\<^isub>1,\<dots>, d\<^isub>q)"} and @{text "D' \<equiv> (d\<PRIME>\<^isub>1,\<dots>, d\<PRIME>\<^isub>q)"} |
|
1460 |
whereby the labels @{text "d"}$_{1..q}$ refer to arguments @{text "z"}$_{1..n}$ and |
|
1461 |
respectively @{text "d\<PRIME>"}$_{1..q}$ to @{text "z\<PRIME>"}$_{1..n}$. In order to relate |
|
1462 |
two such tuples we define the compound $\alpha$-equivalence relation @{text "R"} as follows |
|
1463 |
% |
|
1464 |
\begin{equation}\label{rempty} |
|
1465 |
\mbox{@{text "R \<equiv> (R\<^isub>1,\<dots>, R\<^isub>q)"}} |
|
1466 |
\end{equation} |
|
1467 |
||
1468 |
\noindent |
|
1469 |
with @{text "R\<^isub>i"} being @{text "\<approx>ty\<^isub>i"} if the corresponding labels @{text "d\<^isub>i"} and |
|
1470 |
@{text "d\<PRIME>\<^isub>i"} refer |
|
1471 |
to a recursive argument of @{text C} with type @{text "ty\<^isub>i"}; otherwise |
|
1472 |
we take @{text "R\<^isub>i"} to be the equality @{text "="}. This lets us define |
|
1473 |
the premise for an empty binding clause succinctly as @{text "prems(bc\<^isub>i) \<equiv> D R D'"}, |
|
1474 |
which can be unfolded to the series of premises |
|
1475 |
% |
|
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|
1476 |
%\begin{center} |
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|
1477 |
@{text "d\<^isub>1 R\<^isub>1 d\<PRIME>\<^isub>1 \<dots> d\<^isub>q R\<^isub>q d\<PRIME>\<^isub>q"}. |
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|
1478 |
%\end{center} |
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|
1479 |
% |
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|
1480 |
%\noindent |
2348 | 1481 |
We will use the unfolded version in the examples below. |
1482 |
||
1483 |
Now suppose the binding clause @{text "bc\<^isub>i"} is of the general form |
|
1484 |
% |
|
1485 |
\begin{equation}\label{nonempty} |
|
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diff
changeset
|
1486 |
\mbox{\isacommand{bind (set)} @{text "b\<^isub>1\<dots>b\<^isub>p"} \isacommand{in} @{text "d\<^isub>1\<dots>d\<^isub>q"}.} |
2348 | 1487 |
\end{equation} |
1488 |
||
1489 |
\noindent |
|
1490 |
In this case we define a premise @{text P} using the relation |
|
1491 |
$\approx_{\,\textit{set}}$ given in Section~\ref{sec:binders} (similarly |
|
2637
3890483c674f
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Christian Urban <urbanc@in.tum.de>
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2605
diff
changeset
|
1492 |
$\approx_{\,\textit{set+}}$ and $\approx_{\,\textit{list}}$ for the other |
2348 | 1493 |
binding modes). This premise defines $\alpha$-equivalence of two abstractions |
2349 | 1494 |
involving multiple binders. As above, we first build the tuples @{text "D"} and |
2348 | 1495 |
@{text "D'"} for the bodies @{text "d"}$_{1..q}$, and the corresponding |
1496 |
compound $\alpha$-relation @{text "R"} (shown in \eqref{rempty}). |
|
2381 | 1497 |
For $\approx_{\,\textit{set}}$ we also need |
2348 | 1498 |
a compound free-atom function for the bodies defined as |
1499 |
% |
|
1500 |
\begin{center} |
|
1501 |
\mbox{@{text "fa \<equiv> (fa_ty\<^isub>1,\<dots>, fa_ty\<^isub>q)"}} |
|
1502 |
\end{center} |
|
1503 |
||
1504 |
\noindent |
|
2381 | 1505 |
with the assumption that the @{text "d"}$_{1..q}$ refer to arguments of types @{text "ty"}$_{1..q}$. |
2348 | 1506 |
The last ingredient we need are the sets of atoms bound in the bodies. |
1507 |
For this we take |
|
1508 |
||
1509 |
\begin{center} |
|
1510 |
@{text "B \<equiv> bn_ty\<^isub>1 b\<^isub>1 \<union> \<dots> \<union> bn_ty\<^isub>p b\<^isub>p"}\;.\\ |
|
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|
1511 |
\end{center} |
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|
1512 |
|
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|
1513 |
\noindent |
2348 | 1514 |
Similarly for @{text "B'"} using the labels @{text "b\<PRIME>"}$_{1..p}$. This |
1515 |
lets us formally define the premise @{text P} for a non-empty binding clause as: |
|
1516 |
% |
|
2349 | 1517 |
\begin{center} |
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894599a50af3
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|
1518 |
\mbox{@{term "P \<equiv> \<exists>p. (B, D) \<approx>set R fa p (B', D')"}}\;. |
2349 | 1519 |
\end{center} |
2348 | 1520 |
|
1521 |
\noindent |
|
1522 |
This premise accounts for $\alpha$-equivalence of the bodies of the binding |
|
2359 | 1523 |
clause. |
2349 | 1524 |
However, in case the binders have non-recursive deep binders, this premise |
1525 |
is not enough: |
|
2348 | 1526 |
we also have to ``propagate'' $\alpha$-equivalence inside the structure of |
1527 |
these binders. An example is @{text "Let"} where we have to make sure the |
|
2359 | 1528 |
right-hand sides of assignments are $\alpha$-equivalent. For this we use |
2348 | 1529 |
relations @{text "\<approx>bn"}$_{1..m}$ (which we will formally define shortly). |
1530 |
Let us assume the non-recursive deep binders in @{text "bc\<^isub>i"} are |
|
1531 |
% |
|
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|
1532 |
%\begin{center} |
2348 | 1533 |
@{text "bn\<^isub>1 l\<^isub>1, \<dots>, bn\<^isub>r l\<^isub>r"}. |
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|
1534 |
%\end{center} |
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|
1535 |
% |
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|
1536 |
%\noindent |
2359 | 1537 |
The tuple @{text L} is then @{text "(l\<^isub>1,\<dots>,l\<^isub>r)"} (similarly @{text "L'"}) |
1538 |
and the compound equivalence relation @{text "R'"} is @{text "(\<approx>bn\<^isub>1,\<dots>,\<approx>bn\<^isub>r)"}. |
|
2349 | 1539 |
All premises for @{text "bc\<^isub>i"} are then given by |
2348 | 1540 |
% |
1541 |
\begin{center} |
|
2359 | 1542 |
@{text "prems(bc\<^isub>i) \<equiv> P \<and> L R' L'"} |
2348 | 1543 |
\end{center} |
1544 |
||
2349 | 1545 |
\noindent |
2359 | 1546 |
The auxiliary $\alpha$-equivalence relations @{text "\<approx>bn"}$_{1..m}$ |
1547 |
in @{text "R'"} are defined as follows: assuming a @{text bn}-clause is of the form |
|
2348 | 1548 |
% |
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|
1549 |
%\begin{center} |
2348 | 1550 |
@{text "bn (C z\<^isub>1 \<dots> z\<^isub>s) = rhs"} |
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|
1551 |
%\end{center} |
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|
1552 |
% |
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|
1553 |
%\noindent |
2348 | 1554 |
where the @{text "z"}$_{1..s}$ are of types @{text "ty"}$_{1..s}$, |
2359 | 1555 |
then the corresponding $\alpha$-equivalence clause for @{text "\<approx>bn"} has the form |
2348 | 1556 |
% |
1557 |
\begin{center} |
|
1558 |
\mbox{\infer{@{text "C z\<^isub>1 \<dots> z\<^isub>s \<approx>bn C z\<PRIME>\<^isub>1 \<dots> z\<PRIME>\<^isub>s"}} |
|
1559 |
{@{text "z\<^isub>1 R\<^isub>1 z\<PRIME>\<^isub>1 \<dots> z\<^isub>s R\<^isub>s z\<PRIME>\<^isub>s"}}} |
|
1560 |
\end{center} |
|
1561 |
||
1562 |
\noindent |
|
2349 | 1563 |
In this clause the relations @{text "R"}$_{1..s}$ are given by |
2348 | 1564 |
|
1565 |
\begin{center} |
|
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|
1566 |
\begin{tabular}{c@ {\hspace{2mm}}p{0.9\textwidth}} |
2348 | 1567 |
$\bullet$ & @{text "z\<^isub>i \<approx>ty z\<PRIME>\<^isub>i"} provided @{text "z\<^isub>i"} does not occur in @{text rhs} and |
1568 |
is a recursive argument of @{text C},\\ |
|
1569 |
$\bullet$ & @{text "z\<^isub>i = z\<PRIME>\<^isub>i"} provided @{text "z\<^isub>i"} does not occur in @{text rhs} |
|
1570 |
and is a non-recursive argument of @{text C},\\ |
|
1571 |
$\bullet$ & @{text "z\<^isub>i \<approx>bn\<^isub>i z\<PRIME>\<^isub>i"} provided @{text "z\<^isub>i"} occurs in @{text rhs} |
|
1572 |
with the recursive call @{text "bn\<^isub>i x\<^isub>i"} and\\ |
|
1573 |
$\bullet$ & @{text True} provided @{text "z\<^isub>i"} occurs in @{text rhs} but without a |
|
1574 |
recursive call. |
|
1575 |
\end{tabular} |
|
1576 |
\end{center} |
|
1577 |
||
1578 |
\noindent |
|
1579 |
This completes the definition of $\alpha$-equivalence. As a sanity check, we can show |
|
1580 |
that the premises of empty binding clauses are a special case of the clauses for |
|
1581 |
non-empty ones (we just have to unfold the definition of $\approx_{\,\textit{set}}$ and take @{text "0"} |
|
2349 | 1582 |
for the existentially quantified permutation). |
2359 | 1583 |
|
1584 |
Again let us take a look at a concrete example for these definitions. For \eqref{letrecs} |
|
2348 | 1585 |
we have three relations $\approx_{\textit{trm}}$, $\approx_{\textit{assn}}$ and |
1586 |
$\approx_{\textit{bn}}$ with the following clauses: |
|
1739 | 1587 |
|
2512 | 1588 |
\begin{center}\small |
1739 | 1589 |
\begin{tabular}{@ {}c @ {}} |
2350 | 1590 |
\infer{@{text "Let as t \<approx>\<^bsub>trm\<^esub> Let as' t'"}} |
1591 |
{@{term "\<exists>p. (bn as, t) \<approx>lst alpha_trm fa_trm p (bn as', t')"} & @{text "as \<approx>\<^bsub>bn\<^esub> as'"}}\smallskip\\ |
|
2359 | 1592 |
\makebox[0mm]{\infer{@{text "Let_rec as t \<approx>\<^bsub>trm\<^esub> Let_rec as' t'"}} |
1593 |
{@{term "\<exists>p. (bn as, ast) \<approx>lst alpha_trm2 fa_trm2 p (bn as', ast')"}}} |
|
1739 | 1594 |
\end{tabular} |
1595 |
\end{center} |
|
1596 |
||
2512 | 1597 |
\begin{center}\small |
1739 | 1598 |
\begin{tabular}{@ {}c @ {}} |
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|
1599 |
\infer{@{text "ANil \<approx>\<^bsub>assn\<^esub> ANil"}}{}\hspace{9mm} |
1739 | 1600 |
\infer{@{text "ACons a t as \<approx>\<^bsub>assn\<^esub> ACons a' t' as"}} |
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|
1601 |
{@{text "a = a'"} & @{text "t \<approx>\<^bsub>trm\<^esub> t'"} & @{text "as \<approx>\<^bsub>assn\<^esub> as'"}} |
1739 | 1602 |
\end{tabular} |
1603 |
\end{center} |
|
1604 |
||
2512 | 1605 |
\begin{center}\small |
1739 | 1606 |
\begin{tabular}{@ {}c @ {}} |
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|
1607 |
\infer{@{text "ANil \<approx>\<^bsub>bn\<^esub> ANil"}}{}\hspace{9mm} |
1739 | 1608 |
\infer{@{text "ACons a t as \<approx>\<^bsub>bn\<^esub> ACons a' t' as"}} |
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diff
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|
1609 |
{@{text "t \<approx>\<^bsub>trm\<^esub> t'"} & @{text "as \<approx>\<^bsub>bn\<^esub> as'"}} |
1739 | 1610 |
\end{tabular} |
1611 |
\end{center} |
|
1612 |
||
1613 |
\noindent |
|
1614 |
Note the difference between $\approx_{\textit{assn}}$ and |
|
2341 | 1615 |
$\approx_{\textit{bn}}$: the latter only ``tracks'' $\alpha$-equivalence of |
2359 | 1616 |
the components in an assignment that are \emph{not} bound. This is needed in the |
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6b3e8602edcf
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changeset
|
1617 |
clause for @{text "Let"} (which has |
2510
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|
1618 |
a non-recursive binder). |
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|
1619 |
%The underlying reason is that the terms inside an assignment are not meant |
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diff
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|
1620 |
%to be ``under'' the binder. Such a premise is \emph{not} needed in @{text "Let_rec"}, |
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|
1621 |
%because there all components of an assignment are ``under'' the binder. |
1587 | 1622 |
*} |
1623 |
||
1739 | 1624 |
section {* Establishing the Reasoning Infrastructure *} |
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|
1625 |
|
a3ef7fba983f
Beginning of section 5.
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diff
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|
1626 |
text {* |
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diff
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|
1627 |
Having made all necessary definitions for raw terms, we can start |
2361 | 1628 |
with establishing the reasoning infrastructure for the $\alpha$-equated types |
1629 |
@{text "ty\<AL>"}$_{1..n}$, that is the types the user originally specified. We sketch |
|
1630 |
in this section the proofs we need for establishing this infrastructure. One |
|
1631 |
main point of our work is that we have completely automated these proofs in Isabelle/HOL. |
|
1632 |
||
1633 |
First we establish that the |
|
2359 | 1634 |
$\alpha$-equivalence relations defined in the previous section are |
1766
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diff
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|
1635 |
equivalence relations. |
1717
a3ef7fba983f
Beginning of section 5.
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|
1636 |
|
1766
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diff
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|
1637 |
\begin{lemma}\label{equiv} |
2361 | 1638 |
Given the raw types @{text "ty"}$_{1..n}$ and binding functions |
1639 |
@{text "bn"}$_{1..m}$, the relations @{text "\<approx>ty"}$_{1..n}$ and |
|
2512 | 1640 |
@{text "\<approx>bn"}$_{1..m}$ are equivalence relations.%% and equivariant. |
1717
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diff
changeset
|
1641 |
\end{lemma} |
1739 | 1642 |
|
1643 |
\begin{proof} |
|
1752
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diff
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|
1644 |
The proof is by mutual induction over the definitions. The non-trivial |
2176
5054f170024e
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diff
changeset
|
1645 |
cases involve premises built up by $\approx_{\textit{set}}$, |
2637
3890483c674f
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diff
changeset
|
1646 |
$\approx_{\textit{set+}}$ and $\approx_{\textit{list}}$. They |
1752
9e09253c80cf
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diff
changeset
|
1647 |
can be dealt with as in Lemma~\ref{alphaeq}. |
1739 | 1648 |
\end{proof} |
1718 | 1649 |
|
1739 | 1650 |
\noindent |
1651 |
We can feed this lemma into our quotient package and obtain new types @{text |
|
2359 | 1652 |
"ty"}$^\alpha_{1..n}$ representing $\alpha$-equated terms of types @{text "ty"}$_{1..n}$. |
1653 |
We also obtain definitions for the term-constructors @{text |
|
2361 | 1654 |
"C"}$^\alpha_{1..k}$ from the raw term-constructors @{text |
1655 |
"C"}$_{1..k}$, and similar definitions for the free-atom functions @{text |
|
1656 |
"fa_ty"}$^\alpha_{1..n}$ and @{text "fa_bn"}$^\alpha_{1..m}$ as well as the binding functions @{text |
|
1657 |
"bn"}$^\alpha_{1..m}$. However, these definitions are not really useful to the |
|
1775 | 1658 |
user, since they are given in terms of the isomorphisms we obtained by |
1754 | 1659 |
creating new types in Isabelle/HOL (recall the picture shown in the |
1739 | 1660 |
Introduction). |
1661 |
||
2361 | 1662 |
The first useful property for the user is the fact that distinct |
1663 |
term-constructors are not |
|
1664 |
equal, that is |
|
1760
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1758
diff
changeset
|
1665 |
% |
0bb0f6e662a4
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diff
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|
1666 |
\begin{equation}\label{distinctalpha} |
2361 | 1667 |
\mbox{@{text "C"}$^\alpha$~@{text "x\<^isub>1 \<dots> x\<^isub>r"}~@{text "\<noteq>"}~% |
1668 |
@{text "D"}$^\alpha$~@{text "y\<^isub>1 \<dots> y\<^isub>s"}} |
|
1760
0bb0f6e662a4
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1758
diff
changeset
|
1669 |
\end{equation} |
0bb0f6e662a4
updated related work section
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1758
diff
changeset
|
1670 |
|
0bb0f6e662a4
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1758
diff
changeset
|
1671 |
\noindent |
2361 | 1672 |
whenever @{text "C"}$^\alpha$~@{text "\<noteq>"}~@{text "D"}$^\alpha$. |
1673 |
In order to derive this fact, we use the definition of $\alpha$-equivalence |
|
1674 |
and establish that |
|
1765
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diff
changeset
|
1675 |
% |
9a894c42e80e
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diff
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|
1676 |
\begin{equation}\label{distinctraw} |
2361 | 1677 |
\mbox{@{text "C x\<^isub>1 \<dots> x\<^isub>r"}\;$\not\approx$@{text ty}\;@{text "D y\<^isub>1 \<dots> y\<^isub>s"}} |
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diff
changeset
|
1678 |
\end{equation} |
1760
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diff
changeset
|
1679 |
|
0bb0f6e662a4
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1758
diff
changeset
|
1680 |
\noindent |
2361 | 1681 |
holds for the corresponding raw term-constructors. |
1682 |
In order to deduce \eqref{distinctalpha} from \eqref{distinctraw}, our quotient |
|
1683 |
package needs to know that the raw term-constructors @{text "C"} and @{text "D"} |
|
2341 | 1684 |
are \emph{respectful} w.r.t.~the $\alpha$-equivalence relations (see \cite{Homeier05}). |
2361 | 1685 |
Assuming, for example, @{text "C"} is of type @{text "ty"} with argument types |
1686 |
@{text "ty"}$_{1..r}$, respectfulness amounts to showing that |
|
2359 | 1687 |
% |
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changeset
|
1688 |
\begin{center} |
2361 | 1689 |
@{text "C x\<^isub>1 \<dots> x\<^isub>r \<approx>ty C x\<PRIME>\<^isub>1 \<dots> x\<PRIME>\<^isub>r"} |
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changeset
|
1690 |
\end{center} |
9a894c42e80e
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diff
changeset
|
1691 |
|
9a894c42e80e
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diff
changeset
|
1692 |
\noindent |
2361 | 1693 |
holds under the assumptions that we have \mbox{@{text |
1694 |
"x\<^isub>i \<approx>ty\<^isub>i x\<PRIME>\<^isub>i"}} whenever @{text "x\<^isub>i"} |
|
1695 |
and @{text "x\<PRIME>\<^isub>i"} are recursive arguments of @{text C} and |
|
1696 |
@{text "x\<^isub>i = x\<PRIME>\<^isub>i"} whenever they are non-recursive arguments. We can prove this |
|
1697 |
implication by applying the corresponding rule in our $\alpha$-equivalence |
|
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changeset
|
1698 |
definition and by establishing the following auxiliary implications %facts |
1766
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diff
changeset
|
1699 |
% |
a2d5f9ea17ad
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changeset
|
1700 |
\begin{equation}\label{fnresp} |
a2d5f9ea17ad
completed lifting section
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parents:
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diff
changeset
|
1701 |
\mbox{% |
2514
69780ae147f5
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diff
changeset
|
1702 |
\begin{tabular}{ll@ {\hspace{7mm}}ll} |
69780ae147f5
down to 21 pages and changed strong induction section
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2513
diff
changeset
|
1703 |
\mbox{\it (i)} & @{text "x \<approx>ty\<^isub>i x\<PRIME>"}~~@{text "\<Rightarrow>"}~~@{text "fa_ty\<^isub>i x = fa_ty\<^isub>i x\<PRIME>"} & |
69780ae147f5
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diff
changeset
|
1704 |
\mbox{\it (iii)} & @{text "x \<approx>ty\<^isub>j x\<PRIME>"}~~@{text "\<Rightarrow>"}~~@{text "bn\<^isub>j x = bn\<^isub>j x\<PRIME>"}\\ |
69780ae147f5
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diff
changeset
|
1705 |
|
69780ae147f5
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diff
changeset
|
1706 |
\mbox{\it (ii)} & @{text "x \<approx>ty\<^isub>j x\<PRIME>"}~~@{text "\<Rightarrow>"}~~@{text "fa_bn\<^isub>j x = fa_bn\<^isub>j x\<PRIME>"} & |
69780ae147f5
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diff
changeset
|
1707 |
\mbox{\it (iv)} & @{text "x \<approx>ty\<^isub>j x\<PRIME>"}~~@{text "\<Rightarrow>"}~~@{text "x \<approx>bn\<^isub>j x\<PRIME>"}\\ |
1766
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diff
changeset
|
1708 |
\end{tabular}} |
a2d5f9ea17ad
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diff
changeset
|
1709 |
\end{equation} |
1717
a3ef7fba983f
Beginning of section 5.
Cezary Kaliszyk <kaliszyk@in.tum.de>
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1716
diff
changeset
|
1710 |
|
1766
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diff
changeset
|
1711 |
\noindent |
2361 | 1712 |
They can be established by induction on @{text "\<approx>ty"}$_{1..n}$. Whereas the first, |
1713 |
second and last implication are true by how we stated our definitions, the |
|
1714 |
third \emph{only} holds because of our restriction |
|
1715 |
imposed on the form of the binding functions---namely \emph{not} returning |
|
1716 |
any bound atoms. In Ott, in contrast, the user may |
|
1717 |
define @{text "bn"}$_{1..m}$ so that they return bound |
|
1718 |
atoms and in this case the third implication is \emph{not} true. A |
|
2880 | 1719 |
result is that the lifting of the corresponding binding functions in Ott to $\alpha$-equated |
2361 | 1720 |
terms is impossible. |
1766
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|
1721 |
|
2361 | 1722 |
Having established respectfulness for the raw term-constructors, the |
1723 |
quotient package is able to automatically deduce \eqref{distinctalpha} from |
|
1724 |
\eqref{distinctraw}. Having the facts \eqref{fnresp} at our disposal, we can |
|
1725 |
also lift properties that characterise when two raw terms of the form |
|
1726 |
% |
|
1718 | 1727 |
\begin{center} |
2361 | 1728 |
@{text "C x\<^isub>1 \<dots> x\<^isub>r \<approx>ty C x\<PRIME>\<^isub>1 \<dots> x\<PRIME>\<^isub>r"} |
1766
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|
1729 |
\end{center} |
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|
1730 |
|
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|
1731 |
\noindent |
2341 | 1732 |
are $\alpha$-equivalent. This gives us conditions when the corresponding |
1733 |
$\alpha$-equated terms are \emph{equal}, namely |
|
2361 | 1734 |
% |
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|
1735 |
%\begin{center} |
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|
1736 |
@{text "C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r = C\<^sup>\<alpha> x\<PRIME>\<^isub>1 \<dots> x\<PRIME>\<^isub>r"}. |
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|
1737 |
%\end{center} |
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|
1738 |
% |
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|
1739 |
%\noindent |
2361 | 1740 |
We call these conditions as \emph{quasi-injectivity}. They correspond to |
1741 |
the premises in our $\alpha$-equivalence relations. |
|
1742 |
||
1743 |
Next we can lift the permutation |
|
1744 |
operations defined in \eqref{ceqvt}. In order to make this |
|
1745 |
lifting to go through, we have to show that the permutation operations are respectful. |
|
1746 |
This amounts to showing that the |
|
2512 | 1747 |
$\alpha$-equivalence relations are equivariant \cite{HuffmanUrban10}. |
1748 |
%, which we already established |
|
1749 |
%in Lemma~\ref{equiv}. |
|
1750 |
As a result we can add the equations |
|
2361 | 1751 |
% |
1752 |
\begin{equation}\label{calphaeqvt} |
|
1753 |
@{text "p \<bullet> (C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r) = C\<^sup>\<alpha> (p \<bullet> x\<^isub>1) \<dots> (p \<bullet> x\<^isub>r)"} |
|
1754 |
\end{equation} |
|
1755 |
||
1756 |
\noindent |
|
1757 |
to our infrastructure. In a similar fashion we can lift the defining equations |
|
1758 |
of the free-atom functions @{text "fn_ty\<AL>"}$_{1..n}$ and |
|
1759 |
@{text "fa_bn\<AL>"}$_{1..m}$ as well as of the binding functions @{text |
|
2363 | 1760 |
"bn\<AL>"}$_{1..m}$ and the size functions @{text "size_ty\<AL>"}$_{1..n}$. |
2361 | 1761 |
The latter are defined automatically for the raw types @{text "ty"}$_{1..n}$ |
1762 |
by the datatype package of Isabelle/HOL. |
|
1766
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|
1763 |
|
2512 | 1764 |
Finally we can add to our infrastructure a cases lemma (explained in the next section) |
1765 |
and a structural induction principle |
|
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|
1766 |
for the types @{text "ty\<AL>"}$_{1..n}$. The conclusion of the induction principle is |
2512 | 1767 |
of the form |
2361 | 1768 |
% |
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|
1769 |
%\begin{equation}\label{weakinduct} |
2361 | 1770 |
\mbox{@{text "P\<^isub>1 x\<^isub>1 \<and> \<dots> \<and> P\<^isub>n x\<^isub>n "}} |
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|
1771 |
%\end{equation} |
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|
1772 |
% |
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|
1773 |
%\noindent |
2361 | 1774 |
whereby the @{text P}$_{1..n}$ are predicates and the @{text x}$_{1..n}$ |
1775 |
have types @{text "ty\<AL>"}$_{1..n}$. This induction principle has for each |
|
1776 |
term constructor @{text "C"}$^\alpha$ a premise of the form |
|
1777 |
% |
|
1778 |
\begin{equation}\label{weakprem} |
|
1779 |
\mbox{@{text "\<forall>x\<^isub>1\<dots>x\<^isub>r. P\<^isub>i x\<^isub>i \<and> \<dots> \<and> P\<^isub>j x\<^isub>j \<Rightarrow> P (C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r)"}} |
|
1780 |
\end{equation} |
|
1781 |
||
1782 |
\noindent |
|
1783 |
in which the @{text "x"}$_{i..j}$ @{text "\<subseteq>"} @{text "x"}$_{1..r}$ are |
|
1784 |
the recursive arguments of @{text "C\<AL>"}. |
|
1785 |
||
1786 |
By working now completely on the $\alpha$-equated level, we |
|
1787 |
can first show that the free-atom functions and binding functions are |
|
1788 |
equivariant, namely |
|
1789 |
% |
|
1766
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|
1790 |
\begin{center} |
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|
1791 |
\begin{tabular}{rcl@ {\hspace{10mm}}rcl} |
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|
1792 |
@{text "p \<bullet> (fa_ty\<AL>\<^isub>i x)"} & $=$ & @{text "fa_ty\<AL>\<^isub>i (p \<bullet> x)"} & |
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|
1793 |
@{text "p \<bullet> (bn\<AL>\<^isub>j x)"} & $=$ & @{text "bn\<AL>\<^isub>j (p \<bullet> x)"}\\ |
2361 | 1794 |
@{text "p \<bullet> (fa_bn\<AL>\<^isub>j x)"} & $=$ & @{text "fa_bn\<AL>\<^isub>j (p \<bullet> x)"}\\ |
1718 | 1795 |
\end{tabular} |
1796 |
\end{center} |
|
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|
1797 |
% |
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|
1798 |
\noindent |
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|
1799 |
These properties can be established using the induction principle for the types @{text "ty\<AL>"}$_{1..n}$. |
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|
1800 |
%%in \eqref{weakinduct}. |
2361 | 1801 |
Having these equivariant properties established, we can |
2514
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|
1802 |
show that the support of term-constructors @{text "C\<^sup>\<alpha>"} is included in |
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|
1803 |
the support of its arguments, that means |
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|
1804 |
|
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|
1805 |
\begin{center} |
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|
1806 |
@{text "supp (C\<^sup>\<alpha> x\<^isub>1 \<dots> x\<^isub>r) \<subseteq> (supp x\<^isub>1 \<union> \<dots> \<union> supp x\<^isub>r)"} |
1766
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|
1807 |
\end{center} |
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|
1808 |
|
a2d5f9ea17ad
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changeset
|
1809 |
\noindent |
2514
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|
1810 |
holds. This allows us to prove by induction that |
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|
1811 |
every @{text x} of type @{text "ty\<AL>"}$_{1..n}$ is finitely supported. |
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|
1812 |
%This can be again shown by induction |
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|
1813 |
%over @{text "ty\<AL>"}$_{1..n}$. |
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|
1814 |
Lastly, we can show that the support of |
2361 | 1815 |
elements in @{text "ty\<AL>"}$_{1..n}$ is the same as @{text "fa_ty\<AL>"}$_{1..n}$. |
2363 | 1816 |
This fact is important in a nominal setting, but also provides evidence |
1817 |
that our notions of free-atoms and $\alpha$-equivalence are correct. |
|
1766
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|
1818 |
|
2514
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|
1819 |
\begin{theorem} |
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|
1820 |
For @{text "x"}$_{1..n}$ with type @{text "ty\<AL>"}$_{1..n}$, we have |
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changeset
|
1821 |
@{text "supp x\<^isub>i = fa_ty\<AL>\<^isub>i x\<^isub>i"}. |
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|
1822 |
\end{theorem} |
1766
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diff
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|
1823 |
|
1722 | 1824 |
\begin{proof} |
2361 | 1825 |
The proof is by induction. In each case |
1766
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diff
changeset
|
1826 |
we unfold the definition of @{text "supp"}, move the swapping inside the |
2361 | 1827 |
term-constructors and then use the quasi-injectivity lemmas in order to complete the |
1770
26e44bcddb5b
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1769
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|
1828 |
proof. For the abstraction cases we use the facts derived in Theorem~\ref{suppabs}. |
1722 | 1829 |
\end{proof} |
1721 | 1830 |
|
1766
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|
1831 |
\noindent |
2580
6b3e8602edcf
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|
1832 |
To sum up this section, we can establish automatically a reasoning infrastructure |
2361 | 1833 |
for the types @{text "ty\<AL>"}$_{1..n}$ |
1766
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diff
changeset
|
1834 |
by first lifting definitions from the raw level to the quotient level and |
2361 | 1835 |
then by establishing facts about these lifted definitions. All necessary proofs |
2512 | 1836 |
are generated automatically by custom ML-code. |
1837 |
||
1838 |
%This code can deal with |
|
1839 |
%specifications such as the one shown in Figure~\ref{nominalcorehas} for Core-Haskell. |
|
1728 | 1840 |
|
2508
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|
1841 |
%\begin{figure}[t!] |
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1842 |
%\begin{boxedminipage}{\linewidth} |
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|
1843 |
%\small |
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|
1844 |
%\begin{tabular}{l} |
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1845 |
%\isacommand{atom\_decl}~@{text "var cvar tvar"}\\[1mm] |
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1846 |
%\isacommand{nominal\_datatype}~@{text "tkind ="}\\ |
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|
1847 |
%\phantom{$|$}~@{text "KStar"}~$|$~@{text "KFun tkind tkind"}\\ |
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|
1848 |
%\isacommand{and}~@{text "ckind ="}\\ |
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|
1849 |
%\phantom{$|$}~@{text "CKSim ty ty"}\\ |
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|
1850 |
%\isacommand{and}~@{text "ty ="}\\ |
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|
1851 |
%\phantom{$|$}~@{text "TVar tvar"}~$|$~@{text "T string"}~$|$~@{text "TApp ty ty"}\\ |
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|
1852 |
%$|$~@{text "TFun string ty_list"}~% |
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|
1853 |
%$|$~@{text "TAll tv::tvar tkind ty::ty"} \isacommand{bind}~@{text "tv"}~\isacommand{in}~@{text ty}\\ |
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1854 |
%$|$~@{text "TArr ckind ty"}\\ |
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1855 |
%\isacommand{and}~@{text "ty_lst ="}\\ |
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1856 |
%\phantom{$|$}~@{text "TNil"}~$|$~@{text "TCons ty ty_lst"}\\ |
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|
1857 |
%\isacommand{and}~@{text "cty ="}\\ |
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1858 |
%\phantom{$|$}~@{text "CVar cvar"}~% |
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1859 |
%$|$~@{text "C string"}~$|$~@{text "CApp cty cty"}~$|$~@{text "CFun string co_lst"}\\ |
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|
1860 |
%$|$~@{text "CAll cv::cvar ckind cty::cty"} \isacommand{bind}~@{text "cv"}~\isacommand{in}~@{text cty}\\ |
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|
1861 |
%$|$~@{text "CArr ckind cty"}~$|$~@{text "CRefl ty"}~$|$~@{text "CSym cty"}~$|$~@{text "CCirc cty cty"}\\ |
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|
1862 |
%$|$~@{text "CAt cty ty"}~$|$~@{text "CLeft cty"}~$|$~@{text "CRight cty"}~$|$~@{text "CSim cty cty"}\\ |
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1863 |
%$|$~@{text "CRightc cty"}~$|$~@{text "CLeftc cty"}~$|$~@{text "Coerce cty cty"}\\ |
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1864 |
%\isacommand{and}~@{text "co_lst ="}\\ |
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1865 |
%\phantom{$|$}@{text "CNil"}~$|$~@{text "CCons cty co_lst"}\\ |
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|
1866 |
%\isacommand{and}~@{text "trm ="}\\ |
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1867 |
%\phantom{$|$}~@{text "Var var"}~$|$~@{text "K string"}\\ |
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1868 |
%$|$~@{text "LAM_ty tv::tvar tkind t::trm"} \isacommand{bind}~@{text "tv"}~\isacommand{in}~@{text t}\\ |
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1869 |
%$|$~@{text "LAM_cty cv::cvar ckind t::trm"} \isacommand{bind}~@{text "cv"}~\isacommand{in}~@{text t}\\ |
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1870 |
%$|$~@{text "App_ty trm ty"}~$|$~@{text "App_cty trm cty"}~$|$~@{text "App trm trm"}\\ |
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1871 |
%$|$~@{text "Lam v::var ty t::trm"} \isacommand{bind}~@{text "v"}~\isacommand{in}~@{text t}\\ |
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1872 |
%$|$~@{text "Let x::var ty trm t::trm"} \isacommand{bind}~@{text x}~\isacommand{in}~@{text t}\\ |
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1873 |
%$|$~@{text "Case trm assoc_lst"}~$|$~@{text "Cast trm co"}\\ |
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1874 |
%\isacommand{and}~@{text "assoc_lst ="}\\ |
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1875 |
%\phantom{$|$}~@{text ANil}~% |
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1876 |
%$|$~@{text "ACons p::pat t::trm assoc_lst"} \isacommand{bind}~@{text "bv p"}~\isacommand{in}~@{text t}\\ |
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1877 |
%\isacommand{and}~@{text "pat ="}\\ |
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1878 |
%\phantom{$|$}~@{text "Kpat string tvtk_lst tvck_lst vt_lst"}\\ |
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1879 |
%\isacommand{and}~@{text "vt_lst ="}\\ |
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1880 |
%\phantom{$|$}~@{text VTNil}~$|$~@{text "VTCons var ty vt_lst"}\\ |
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1881 |
%\isacommand{and}~@{text "tvtk_lst ="}\\ |
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1882 |
%\phantom{$|$}~@{text TVTKNil}~$|$~@{text "TVTKCons tvar tkind tvtk_lst"}\\ |
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1883 |
%\isacommand{and}~@{text "tvck_lst ="}\\ |
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1884 |
%\phantom{$|$}~@{text TVCKNil}~$|$ @{text "TVCKCons cvar ckind tvck_lst"}\\ |
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1885 |
%\isacommand{binder}\\ |
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1886 |
%@{text "bv :: pat \<Rightarrow> atom list"}~\isacommand{and}~% |
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1887 |
%@{text "bv1 :: vt_lst \<Rightarrow> atom list"}~\isacommand{and}\\ |
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1888 |
%@{text "bv2 :: tvtk_lst \<Rightarrow> atom list"}~\isacommand{and}~% |
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1889 |
%@{text "bv3 :: tvck_lst \<Rightarrow> atom list"}\\ |
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1890 |
%\isacommand{where}\\ |
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1891 |
%\phantom{$|$}~@{text "bv (K s tvts tvcs vs) = (bv3 tvts) @ (bv2 tvcs) @ (bv1 vs)"}\\ |
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1892 |
%$|$~@{text "bv1 VTNil = []"}\\ |
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1893 |
%$|$~@{text "bv1 (VTCons x ty tl) = (atom x)::(bv1 tl)"}\\ |
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1894 |
%$|$~@{text "bv2 TVTKNil = []"}\\ |
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1895 |
%$|$~@{text "bv2 (TVTKCons a ty tl) = (atom a)::(bv2 tl)"}\\ |
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1896 |
%$|$~@{text "bv3 TVCKNil = []"}\\ |
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1897 |
%$|$~@{text "bv3 (TVCKCons c cty tl) = (atom c)::(bv3 tl)"}\\ |
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1898 |
%\end{tabular} |
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1899 |
%\end{boxedminipage} |
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1900 |
%\caption{The nominal datatype declaration for Core-Haskell. For the moment we |
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1901 |
%do not support nested types; therefore we explicitly have to unfold the |
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1902 |
%lists @{text "co_lst"}, @{text "assoc_lst"} and so on. This will be improved |
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1903 |
%in a future version of Nominal Isabelle. Apart from that, the |
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1904 |
%declaration follows closely the original in Figure~\ref{corehas}. The |
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1905 |
%point of our work is that having made such a declaration in Nominal Isabelle, |
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1906 |
%one obtains automatically a reasoning infrastructure for Core-Haskell. |
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1907 |
%\label{nominalcorehas}} |
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1908 |
%\end{figure} |
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1909 |
*} |
1728 | 1910 |
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1587 | 1911 |
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1912 |
section {* Strong Induction Principles *} |
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1913 |
|
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1914 |
text {* |
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1915 |
In the previous section we derived induction principles for $\alpha$-equated terms. |
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1916 |
We call such induction principles \emph{weak}, because for a |
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1917 |
term-constructor \mbox{@{text "C\<^sup>\<alpha> x\<^isub>1\<dots>x\<^isub>r"}} |
2361 | 1918 |
the induction hypothesis requires us to establish the implications \eqref{weakprem}. |
2363 | 1919 |
The problem with these implications is that in general they are difficult to establish. |
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1920 |
The reason is that we cannot make any assumption about the bound atoms that might be in @{text "C\<^sup>\<alpha>"}. |
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1921 |
%%(for example we cannot assume the variable convention for them). |
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1922 |
|
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1923 |
In \cite{UrbanTasson05} we introduced a method for automatically |
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1924 |
strengthening weak induction principles for terms containing single |
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1925 |
binders. These stronger induction principles allow the user to make additional |
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1926 |
assumptions about bound atoms. |
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1927 |
%These additional assumptions amount to a formal |
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1928 |
%version of the informal variable convention for binders. |
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1929 |
To sketch how this strengthening extends to the case of multiple binders, we use as |
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1930 |
running example the term-constructors @{text "Lam"} and @{text "Let"} |
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1931 |
from example \eqref{letpat}. Instead of establishing @{text " P\<^bsub>trm\<^esub> t \<and> P\<^bsub>pat\<^esub> p"}, |
2517 | 1932 |
the stronger induction principle for \eqref{letpat} establishes properties @{text " P\<^bsub>trm\<^esub> c t \<and> P\<^bsub>pat\<^esub> c p"} |
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1933 |
where the additional parameter @{text c} controls |
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1934 |
which freshness assumptions the binders should satisfy. For the two term constructors |
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1935 |
this means that the user has to establish in inductions the implications |
2363 | 1936 |
% |
1937 |
\begin{center} |
|
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1938 |
\begin{tabular}{l} |
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1939 |
@{text "\<forall>a t c. {atom a} \<FRESH>\<^sup>* c \<and> (\<forall>d. P\<^bsub>trm\<^esub> d t) \<Rightarrow> P\<^bsub>trm\<^esub> c (Lam a t)"}\\ |
2517 | 1940 |
@{text "\<forall>p t c. (set (bn p)) \<FRESH>\<^sup>* c \<and> (\<forall>d. P\<^bsub>pat\<^esub> d p) \<and> (\<forall>d. P\<^bsub>trm\<^esub> d t) \<and> \<Rightarrow> P\<^bsub>trm\<^esub> c (Let p t)"}\\%[-0mm] |
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1941 |
\end{tabular} |
2363 | 1942 |
\end{center} |
1943 |
||
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1944 |
In \cite{UrbanTasson05} we showed how the weaker induction principles imply |
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1945 |
the stronger ones. This was done by some quite complicated, nevertheless automated, |
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1946 |
induction proof. In this paper we simplify this work by leveraging the automated proof |
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1947 |
methods from the function package of Isabelle/HOL. |
2517 | 1948 |
The reasoning principle these methods employ is well-founded induction. |
1949 |
To use them in our setting, we have to discharge |
|
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1950 |
two proof obligations: one is that we have |
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1951 |
well-founded measures (for each type @{text "ty"}$^\alpha_{1..n}$) that decrease in |
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1952 |
every induction step and the other is that we have covered all cases. |
2517 | 1953 |
As measures we use the size functions |
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1954 |
@{text "size_ty"}$^\alpha_{1..n}$, which we lifted in the previous section and which are |
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1955 |
all well-founded. %It is straightforward to establish that these measures decrease |
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1956 |
%in every induction step. |
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1957 |
|
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1958 |
What is left to show is that we covered all cases. To do so, we use |
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1959 |
a \emph{cases lemma} derived for each type. For the terms in \eqref{letpat} |
2517 | 1960 |
this lemma is of the form |
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1961 |
% |
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1962 |
\begin{equation}\label{weakcases} |
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1963 |
\infer{@{text "P\<^bsub>trm\<^esub>"}} |
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1964 |
{\begin{array}{l@ {\hspace{9mm}}l} |
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1965 |
@{text "\<forall>x. t = Var x \<Rightarrow> P\<^bsub>trm\<^esub>"} & @{text "\<forall>a t'. t = Lam a t' \<Rightarrow> P\<^bsub>trm\<^esub>"}\\ |
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1966 |
@{text "\<forall>t\<^isub>1 t\<^isub>2. t = App t\<^isub>1 t\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub>"} & @{text "\<forall>p t'. t = Let p t' \<Rightarrow> P\<^bsub>trm\<^esub>"}\\ |
2517 | 1967 |
\end{array}}\\[-1mm] |
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1968 |
\end{equation} |
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1969 |
% |
2517 | 1970 |
where we have a premise for each term-constructor. |
1971 |
The idea behind such cases lemmas is that we can conclude with a property @{text "P\<^bsub>trm\<^esub>"}, |
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1972 |
provided we can show that this property holds if we substitute for @{text "t"} all |
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1973 |
possible term-constructors. |
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1974 |
|
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1975 |
The only remaining difficulty is that in order to derive the stronger induction |
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1976 |
principles conveniently, the cases lemma in \eqref{weakcases} is too weak. For this note that |
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1977 |
in order to apply this lemma, we have to establish @{text "P\<^bsub>trm\<^esub>"} for \emph{all} @{text Lam}- and |
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1978 |
\emph{all} @{text Let}-terms. |
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1979 |
What we need instead is a cases lemma where we only have to consider terms that have |
2517 | 1980 |
binders that are fresh w.r.t.~a context @{text "c"}. This gives the implications |
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1981 |
% |
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1982 |
\begin{center} |
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1983 |
\begin{tabular}{l} |
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|
1984 |
@{text "\<forall>a t'. t = Lam a t' \<and> {atom a} \<FRESH>\<^sup>* c \<Rightarrow> P\<^bsub>trm\<^esub>"}\\ |
2517 | 1985 |
@{text "\<forall>p t'. t = Let p t' \<and> (set (bn p)) \<FRESH>\<^sup>* c \<Rightarrow> P\<^bsub>trm\<^esub>"}\\%[-2mm] |
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1986 |
\end{tabular} |
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1987 |
\end{center} |
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|
1988 |
% |
2517 | 1989 |
\noindent |
1990 |
which however can be relatively easily be derived from the implications in \eqref{weakcases} |
|
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1991 |
by a renaming using Properties \ref{supppermeq} and \ref{avoiding}. In the first case we know |
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1992 |
that @{text "{atom a} \<FRESH>\<^sup>* Lam a t"}. Property \eqref{avoiding} provides us therefore with |
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|
1993 |
a permutation @{text q}, such that @{text "{atom (q \<bullet> a)} \<FRESH>\<^sup>* c"} and |
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|
1994 |
@{text "supp (Lam a t) \<FRESH>\<^sup>* q"} hold. |
2517 | 1995 |
By using Property \ref{supppermeq}, we can infer from the latter |
1996 |
that @{text "Lam (q \<bullet> a) (q \<bullet> t) = Lam a t"} |
|
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1997 |
and we are done with this case. |
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1998 |
|
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1999 |
The @{text Let}-case involving a (non-recursive) deep binder is a bit more complicated. |
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2000 |
The reason is that the we cannot apply Property \ref{avoiding} to the whole term @{text "Let p t"}, |
2929 | 2001 |
because @{text p} might contain names bound by @{text bn}, but also some that are |
2517 | 2002 |
free. To solve this problem we have to introduce a permutation function that only |
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2003 |
permutes names bound by @{text bn} and leaves the other names unchanged. We do this again |
2517 | 2004 |
by lifting. For a |
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2005 |
clause @{text "bn (C x\<^isub>1 \<dots> x\<^isub>r) = rhs"}, we define |
2363 | 2006 |
% |
2007 |
\begin{center} |
|
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2008 |
@{text "p \<bullet>\<^bsub>bn\<^esub> (C x\<^isub>1 \<dots> x\<^isub>r) \<equiv> C y\<^isub>1 \<dots> y\<^isub>r"} with |
2516 | 2009 |
$\begin{cases} |
2010 |
\text{@{text "y\<^isub>i \<equiv> x\<^isub>i"} provided @{text "x\<^isub>i"} does not occur in @{text "rhs"}}\\ |
|
2011 |
\text{@{text "y\<^isub>i \<equiv> p \<bullet>\<^bsub>bn'\<^esub> x\<^isub>i"} provided @{text "bn' x\<^isub>i"} is in @{text "rhs"}}\\ |
|
2012 |
\text{@{text "y\<^isub>i \<equiv> p \<bullet> x\<^isub>i"} otherwise} |
|
2013 |
\end{cases}$ |
|
2363 | 2014 |
\end{center} |
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2015 |
% |
2516 | 2016 |
%\noindent |
2017 |
%with @{text "y\<^isub>i"} determined as follows: |
|
2018 |
% |
|
2019 |
%\begin{center} |
|
2020 |
%\begin{tabular}{c@ {\hspace{2mm}}p{0.9\textwidth}} |
|
2021 |
%$\bullet$ & @{text "y\<^isub>i \<equiv> x\<^isub>i"} provided @{text "x\<^isub>i"} does not occur in @{text "rhs"}\\ |
|
2022 |
%$\bullet$ & @{text "y\<^isub>i \<equiv> p \<bullet>\<^bsub>bn'\<^esub> x\<^isub>i"} provided @{text "bn' x\<^isub>i"} is in @{text "rhs"}\\ |
|
2023 |
%$\bullet$ & @{text "y\<^isub>i \<equiv> p \<bullet> x\<^isub>i"} otherwise |
|
2024 |
%\end{tabular} |
|
2025 |
%\end{center} |
|
2026 |
% |
|
2363 | 2027 |
\noindent |
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2028 |
Now Properties \ref{supppermeq} and \ref{avoiding} give us a permutation @{text q} such that |
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|
2029 |
@{text "(set (bn (q \<bullet>\<^bsub>bn\<^esub> p)) \<FRESH>\<^sup>* c"} holds and such that @{text "[q \<bullet>\<^bsub>bn\<^esub> p]\<^bsub>list\<^esub>.(q \<bullet> t)"} |
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|
2030 |
is equal to @{text "[p]\<^bsub>list\<^esub>. t"}. We can also show that @{text "(q \<bullet>\<^bsub>bn\<^esub> p) \<approx>\<^bsub>bn\<^esub> p"}. |
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|
2031 |
These facts establish that @{text "Let (q \<bullet>\<^bsub>bn\<^esub> p) (p \<bullet> t) = Let p t"}, as we need. This |
2517 | 2032 |
completes the non-trivial cases in \eqref{letpat} for strengthening the corresponding induction |
2033 |
principle. |
|
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|
2034 |
|
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2035 |
|
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2036 |
|
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2037 |
%A natural question is |
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2038 |
%whether we can also strengthen the weak induction principles involving |
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2039 |
%the general binders presented here. We will indeed be able to so, but for this we need an |
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2040 |
%additional notion for permuting deep binders. |
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2041 |
|
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2042 |
%Given a binding function @{text "bn"} we define an auxiliary permutation |
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2043 |
%operation @{text "_ \<bullet>\<^bsub>bn\<^esub> _"} which permutes only bound arguments in a deep binder. |
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2044 |
%Assuming a clause of @{text bn} is given as |
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2045 |
% |
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2046 |
%\begin{center} |
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|
2047 |
%@{text "bn (C x\<^isub>1 \<dots> x\<^isub>r) = rhs"}, |
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2048 |
%\end{center} |
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2049 |
|
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2050 |
%\noindent |
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2051 |
%then we define |
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|
2052 |
% |
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|
2053 |
%\begin{center} |
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|
2054 |
%@{text "p \<bullet>\<^bsub>bn\<^esub> (C x\<^isub>1 \<dots> x\<^isub>r) \<equiv> C y\<^isub>1 \<dots> y\<^isub>r"} |
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2055 |
%\end{center} |
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|
2056 |
|
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2057 |
%\noindent |
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|
2058 |
%with @{text "y\<^isub>i"} determined as follows: |
1770
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|
2059 |
% |
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|
2060 |
%\begin{center} |
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|
2061 |
%\begin{tabular}{c@ {\hspace{2mm}}p{7cm}} |
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|
2062 |
%$\bullet$ & @{text "y\<^isub>i \<equiv> x\<^isub>i"} provided @{text "x\<^isub>i"} does not occur in @{text "rhs"}\\ |
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2063 |
%$\bullet$ & @{text "y\<^isub>i \<equiv> p \<bullet>\<^bsub>bn'\<^esub> x\<^isub>i"} provided @{text "bn' x\<^isub>i"} is in @{text "rhs"}\\ |
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2064 |
%$\bullet$ & @{text "y\<^isub>i \<equiv> p \<bullet> x\<^isub>i"} otherwise |
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|
2065 |
%\end{tabular} |
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|
2066 |
%\end{center} |
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|
2067 |
|
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2068 |
%\noindent |
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|
2069 |
%Using again the quotient package we can lift the @{text "_ \<bullet>\<^bsub>bn\<^esub> _"} function to |
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2070 |
%$\alpha$-equated terms. We can then prove the following two facts |
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|
2071 |
|
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2072 |
%\begin{lemma}\label{permutebn} |
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2073 |
%Given a binding function @{text "bn\<^sup>\<alpha>"} then for all @{text p} |
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|
2074 |
%{\it (i)} @{text "p \<bullet> (bn\<^sup>\<alpha> x) = bn\<^sup>\<alpha> (p \<bullet>\<AL>\<^bsub>bn\<^esub> x)"} and {\it (ii)} |
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|
2075 |
% @{text "fa_bn\<^isup>\<alpha> x = fa_bn\<^isup>\<alpha> (p \<bullet>\<AL>\<^bsub>bn\<^esub> x)"}. |
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2076 |
%\end{lemma} |
1769 | 2077 |
|
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|
2078 |
%\begin{proof} |
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2079 |
%By induction on @{text x}. The equations follow by simple unfolding |
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2080 |
%of the definitions. |
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2081 |
%\end{proof} |
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|
2082 |
|
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2083 |
%\noindent |
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2084 |
%The first property states that a permutation applied to a binding function is |
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2085 |
%equivalent to first permuting the binders and then calculating the bound |
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2086 |
%atoms. The second amounts to the fact that permuting the binders has no |
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2087 |
%effect on the free-atom function. The main point of this permutation |
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2088 |
%function, however, is that if we have a permutation that is fresh |
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2089 |
%for the support of an object @{text x}, then we can use this permutation |
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2090 |
%to rename the binders in @{text x}, without ``changing'' @{text x}. In case of the |
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2091 |
%@{text "Let"} term-constructor from the example shown |
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2092 |
%in \eqref{letpat} this means for a permutation @{text "r"} |
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2093 |
%% |
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2094 |
%\begin{equation}\label{renaming} |
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2095 |
%\begin{array}{l} |
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2096 |
%\mbox{if @{term "supp (Abs_lst (bn p) t\<^isub>2) \<sharp>* r"}}\\ |
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2097 |
%\qquad\mbox{then @{text "Let p t\<^isub>1 t\<^isub>2 = Let (r \<bullet>\<AL>\<^bsub>bn_pat\<^esub> p) t\<^isub>1 (r \<bullet> t\<^isub>2)"}} |
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2098 |
%\end{array} |
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2099 |
%\end{equation} |
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2100 |
|
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2101 |
%\noindent |
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2102 |
%This fact will be crucial when establishing the strong induction principles below. |
2363 | 2103 |
|
2104 |
||
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2105 |
%In our running example about @{text "Let"}, the strong induction |
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2106 |
%principle means that instead |
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2107 |
%of establishing the implication |
2361 | 2108 |
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2109 |
%\begin{center} |
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2110 |
%@{text "\<forall>p t\<^isub>1 t\<^isub>2. P\<^bsub>pat\<^esub> p \<and> P\<^bsub>trm\<^esub> t\<^isub>1 \<and> P\<^bsub>trm\<^esub> t\<^isub>2 \<Rightarrow> P\<^bsub>trm\<^esub> (Let p t\<^isub>1 t\<^isub>2)"} |
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2111 |
%\end{center} |
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2112 |
% |
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2113 |
%\noindent |
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2114 |
%it is sufficient to establish the following implication |
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2115 |
% |
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2116 |
%\begin{equation}\label{strong} |
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2117 |
%\mbox{\begin{tabular}{l} |
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|
2118 |
%@{text "\<forall>p t\<^isub>1 t\<^isub>2 c."}\\ |
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|
2119 |
%\hspace{5mm}@{text "set (bn p) #\<^sup>* c \<and>"}\\ |
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2120 |
%\hspace{5mm}@{text "(\<forall>d. P\<^bsub>pat\<^esub> d p) \<and> (\<forall>d. P\<^bsub>trm\<^esub> d t\<^isub>1) \<and> (\<forall>d. P\<^bsub>trm\<^esub> d t\<^isub>2)"}\\ |
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2121 |
%\hspace{15mm}@{text "\<Rightarrow> P\<^bsub>trm\<^esub> c (Let p t\<^isub>1 t\<^isub>2)"} |
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2122 |
%\end{tabular}} |
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2123 |
%\end{equation} |
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2124 |
% |
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2125 |
%\noindent |
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2126 |
%While this implication contains an additional argument, namely @{text c}, and |
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2127 |
%also additional universal quantifications, it is usually easier to establish. |
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2128 |
%The reason is that we have the freshness |
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2129 |
%assumption @{text "set (bn\<^sup>\<alpha> p) #\<^sup>* c"}, whereby @{text c} can be arbitrarily |
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2130 |
%chosen by the user as long as it has finite support. |
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2131 |
% |
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2132 |
%Let us now show how we derive the strong induction principles from the |
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2133 |
%weak ones. In case of the @{text "Let"}-example we derive by the weak |
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|
2134 |
%induction the following two properties |
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|
2135 |
% |
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|
2136 |
%\begin{equation}\label{hyps} |
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|
2137 |
%@{text "\<forall>q c. P\<^bsub>trm\<^esub> c (q \<bullet> t)"} \hspace{4mm} |
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|
2138 |
%@{text "\<forall>q\<^isub>1 q\<^isub>2 c. P\<^bsub>pat\<^esub> (q\<^isub>1 \<bullet>\<AL>\<^bsub>bn\<^esub> (q\<^isub>2 \<bullet> p))"} |
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|
2139 |
%\end{equation} |
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2140 |
% |
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|
2141 |
%\noindent |
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|
2142 |
%For the @{text Let} term-constructor we therefore have to establish @{text "P\<^bsub>trm\<^esub> c (q \<bullet> Let p t\<^isub>1 t\<^isub>2)"} |
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|
2143 |
%assuming \eqref{hyps} as induction hypotheses (the first for @{text t\<^isub>1} and @{text t\<^isub>2}). |
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|
2144 |
%By Property~\ref{avoiding} we |
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|
2145 |
%obtain a permutation @{text "r"} such that |
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2146 |
% |
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|
2147 |
%\begin{equation}\label{rprops} |
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|
2148 |
%@{term "(r \<bullet> set (bn (q \<bullet> p))) \<sharp>* c "}\hspace{4mm} |
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|
2149 |
%@{term "supp (Abs_lst (bn (q \<bullet> p)) (q \<bullet> t\<^isub>2)) \<sharp>* r"} |
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|
2150 |
%\end{equation} |
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2151 |
% |
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2152 |
%\noindent |
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|
2153 |
%hold. The latter fact and \eqref{renaming} give us |
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2154 |
%% |
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2155 |
%\begin{center} |
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|
2156 |
%\begin{tabular}{l} |
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|
2157 |
%@{text "Let (q \<bullet> p) (q \<bullet> t\<^isub>1) (q \<bullet> t\<^isub>2) ="} \\ |
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|
2158 |
%\hspace{15mm}@{text "Let (r \<bullet>\<AL>\<^bsub>bn\<^esub> (q \<bullet> p)) (q \<bullet> t\<^isub>1) (r \<bullet> (q \<bullet> t\<^isub>2))"} |
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|
2159 |
%\end{tabular} |
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|
2160 |
%\end{center} |
2361 | 2161 |
% |
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2162 |
%\noindent |
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|
2163 |
%So instead of proving @{text "P\<^bsub>trm\<^esub> c (q \<bullet> Let p t\<^isub>1 t\<^isub>2)"}, we can equally |
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|
2164 |
%establish @{text "P\<^bsub>trm\<^esub> c (Let (r \<bullet>\<AL>\<^bsub>bn\<^esub> (q \<bullet> p)) (q \<bullet> t\<^isub>1) (r \<bullet> (q \<bullet> t\<^isub>2)))"}. |
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|
2165 |
%To do so, we will use the implication \eqref{strong} of the strong induction |
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2166 |
%principle, which requires us to discharge |
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|
2167 |
%the following four proof obligations: |
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|
2168 |
%% |
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|
2169 |
%\begin{center} |
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|
2170 |
%\begin{tabular}{rl} |
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|
2171 |
%{\it (i)} & @{text "set (bn (r \<bullet>\<AL>\<^bsub>bn\<^esub> (q \<bullet> p))) #\<^sup>* c"}\\ |
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|
2172 |
%{\it (ii)} & @{text "\<forall>d. P\<^bsub>pat\<^esub> d (r \<bullet>\<AL>\<^bsub>bn\<^esub> (q \<bullet> p))"}\\ |
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|
2173 |
%{\it (iii)} & @{text "\<forall>d. P\<^bsub>trm\<^esub> d (q \<bullet> t\<^isub>1)"}\\ |
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|
2174 |
%{\it (iv)} & @{text "\<forall>d. P\<^bsub>trm\<^esub> d (r \<bullet> (q \<bullet> t\<^isub>2))"}\\ |
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|
2175 |
%\end{tabular} |
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|
2176 |
%\end{center} |
2361 | 2177 |
% |
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2178 |
%\noindent |
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|
2179 |
%The first follows from \eqref{rprops} and Lemma~\ref{permutebn}.{\it (i)}; the |
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|
2180 |
%others from the induction hypotheses in \eqref{hyps} (in the fourth case |
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|
2181 |
%we have to use the fact that @{term "(r \<bullet> (q \<bullet> t\<^isub>2)) = (r + q) \<bullet> t\<^isub>2"}). |
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2182 |
% |
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|
2183 |
%Taking now the identity permutation @{text 0} for the permutations in \eqref{hyps}, |
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2184 |
%we can establish our original goals, namely @{text "P\<^bsub>trm\<^esub> c t"} and \mbox{@{text "P\<^bsub>pat\<^esub> c p"}}. |
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2185 |
%This completes the proof showing that the weak induction principles imply |
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2186 |
%the strong induction principles. |
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2187 |
*} |
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2188 |
|
1702 | 2189 |
|
2742 | 2190 |
section {* Related Work\label{related} *} |
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2191 |
|
1570 | 2192 |
text {* |
2342 | 2193 |
To our knowledge the earliest usage of general binders in a theorem prover |
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2194 |
is described in \cite{NaraschewskiNipkow99} about a formalisation of the |
2361 | 2195 |
algorithm W. This formalisation implements binding in type-schemes using a |
2196 |
de-Bruijn indices representation. Since type-schemes in W contain only a single |
|
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2197 |
place where variables are bound, different indices do not refer to different binders (as in the usual |
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2198 |
de-Bruijn representation), but to different bound variables. A similar idea |
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2199 |
has been recently explored for general binders in the locally nameless |
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2200 |
approach to binding \cite{chargueraud09}. There, de-Bruijn indices consist |
2580
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2201 |
of two numbers, one referring to the place where a variable is bound, and the |
1760
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2202 |
other to which variable is bound. The reasoning infrastructure for both |
2163
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2203 |
representations of bindings comes for free in theorem provers like Isabelle/HOL or |
2342 | 2204 |
Coq, since the corresponding term-calculi can be implemented as ``normal'' |
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2205 |
datatypes. However, in both approaches it seems difficult to achieve our |
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2206 |
fine-grained control over the ``semantics'' of bindings (i.e.~whether the |
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2207 |
order of binders should matter, or vacuous binders should be taken into |
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2208 |
account). %To do so, one would require additional predicates that filter out |
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2209 |
%unwanted terms. Our guess is that such predicates result in rather |
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2210 |
%intricate formal reasoning. |
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2211 |
|
2516 | 2212 |
Another technique for representing binding is higher-order abstract syntax |
2213 |
(HOAS). %, which for example is implemented in the Twelf system. |
|
2214 |
This %%representation |
|
1760
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2215 |
technique supports very elegantly many aspects of \emph{single} binding, and |
2342 | 2216 |
impressive work has been done that uses HOAS for mechanising the metatheory |
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2217 |
of SML~\cite{LeeCraryHarper07}. We are, however, not aware how multiple |
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diff
changeset
|
2218 |
binders of SML are represented in this work. Judging from the submitted |
9f55d7927e5b
more on the strong induction section
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parents:
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diff
changeset
|
2219 |
Twelf-solution for the POPLmark challenge, HOAS cannot easily deal with |
2517 | 2220 |
binding constructs where the number of bound variables is not fixed. %For example |
2516 | 2221 |
In the second part of this challenge, @{text "Let"}s involve |
1764
9f55d7927e5b
more on the strong induction section
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parents:
1763
diff
changeset
|
2222 |
patterns that bind multiple variables at once. In such situations, HOAS |
2516 | 2223 |
seems to have to resort to the iterated-single-binders-approach with |
1764
9f55d7927e5b
more on the strong induction section
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parents:
1763
diff
changeset
|
2224 |
all the unwanted consequences when reasoning about the resulting terms. |
1740
2afee29cf81c
completed related work section
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parents:
1739
diff
changeset
|
2225 |
|
2512 | 2226 |
%Two formalisations involving general binders have been |
2227 |
%performed in older |
|
2228 |
%versions of Nominal Isabelle (one about Psi-calculi and one about algorithm W |
|
2229 |
%\cite{BengtsonParow09,UrbanNipkow09}). Both |
|
2230 |
%use the approach based on iterated single binders. Our experience with |
|
2231 |
%the latter formalisation has been disappointing. The major pain arose from |
|
2232 |
%the need to ``unbind'' variables. This can be done in one step with our |
|
2233 |
%general binders described in this paper, but needs a cumbersome |
|
2234 |
%iteration with single binders. The resulting formal reasoning turned out to |
|
2235 |
%be rather unpleasant. The hope is that the extension presented in this paper |
|
2236 |
%is a substantial improvement. |
|
1726
2eafd8ed4bbf
started with a related work section
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diff
changeset
|
2237 |
|
2163
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changeset
|
2238 |
The most closely related work to the one presented here is the Ott-tool |
5dc48e1af733
added comments about pottiers work
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parents:
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diff
changeset
|
2239 |
\cite{ott-jfp} and the C$\alpha$ml language \cite{Pottier06}. Ott is a nifty |
5dc48e1af733
added comments about pottiers work
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parents:
2156
diff
changeset
|
2240 |
front-end for creating \LaTeX{} documents from specifications of |
2343 | 2241 |
term-calculi involving general binders. For a subset of the specifications |
2163
5dc48e1af733
added comments about pottiers work
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parents:
2156
diff
changeset
|
2242 |
Ott can also generate theorem prover code using a raw representation of |
5dc48e1af733
added comments about pottiers work
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parents:
2156
diff
changeset
|
2243 |
terms, and in Coq also a locally nameless representation. The developers of |
5dc48e1af733
added comments about pottiers work
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parents:
2156
diff
changeset
|
2244 |
this tool have also put forward (on paper) a definition for |
2341 | 2245 |
$\alpha$-equivalence of terms that can be specified in Ott. This definition is |
2163
5dc48e1af733
added comments about pottiers work
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2156
diff
changeset
|
2246 |
rather different from ours, not using any nominal techniques. To our |
2604
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
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parents:
2580
diff
changeset
|
2247 |
knowledge there is no concrete mathematical result concerning this |
2580
6b3e8602edcf
implemented concrete suggestion of 3rd reviewer
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parents:
2555
diff
changeset
|
2248 |
notion of $\alpha$-equivalence. Also the definition for the |
6b3e8602edcf
implemented concrete suggestion of 3rd reviewer
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parents:
2555
diff
changeset
|
2249 |
notion of free variables |
6b3e8602edcf
implemented concrete suggestion of 3rd reviewer
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parents:
2555
diff
changeset
|
2250 |
is work in progress. |
2163
5dc48e1af733
added comments about pottiers work
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parents:
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diff
changeset
|
2251 |
|
2516 | 2252 |
Although we were heavily inspired by the syntax of Ott, |
2604
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
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parents:
2580
diff
changeset
|
2253 |
its definition of $\alpha$-equi\-valence is unsuitable for our extension of |
1760
0bb0f6e662a4
updated related work section
Christian Urban <urbanc@in.tum.de>
parents:
1758
diff
changeset
|
2254 |
Nominal Isabelle. First, it is far too complicated to be a basis for |
0bb0f6e662a4
updated related work section
Christian Urban <urbanc@in.tum.de>
parents:
1758
diff
changeset
|
2255 |
automated proofs implemented on the ML-level of Isabelle/HOL. Second, it |
0bb0f6e662a4
updated related work section
Christian Urban <urbanc@in.tum.de>
parents:
1758
diff
changeset
|
2256 |
covers cases of binders depending on other binders, which just do not make |
2341 | 2257 |
sense for our $\alpha$-equated terms. Third, it allows empty types that have no |
2363 | 2258 |
meaning in a HOL-based theorem prover. We also had to generalise slightly Ott's |
2163
5dc48e1af733
added comments about pottiers work
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parents:
2156
diff
changeset
|
2259 |
binding clauses. In Ott you specify binding clauses with a single body; we |
5dc48e1af733
added comments about pottiers work
Christian Urban <urbanc@in.tum.de>
parents:
2156
diff
changeset
|
2260 |
allow more than one. We have to do this, because this makes a difference |
2488
1c18f2cf3923
a few more words about Ott
Christian Urban <urbanc@in.tum.de>
parents:
2471
diff
changeset
|
2261 |
for our notion of $\alpha$-equivalence in case of \isacommand{bind (set)} and |
2637
3890483c674f
final version of the ESOP paper; used set+ instead of res as requested by one reviewer
Christian Urban <urbanc@in.tum.de>
parents:
2605
diff
changeset
|
2262 |
\isacommand{bind (set+)}. |
2513 | 2263 |
% |
2512 | 2264 |
%Consider the examples |
2265 |
% |
|
2266 |
%\begin{center} |
|
2267 |
%\begin{tabular}{@ {}l@ {\hspace{2mm}}l@ {}} |
|
2268 |
%@{text "Foo\<^isub>1 xs::name fset t::trm s::trm"} & |
|
2269 |
% \isacommand{bind (set)} @{text "xs"} \isacommand{in} @{text "t s"}\\ |
|
2270 |
%@{text "Foo\<^isub>2 xs::name fset t::trm s::trm"} & |
|
2271 |
% \isacommand{bind (set)} @{text "xs"} \isacommand{in} @{text "t"}, |
|
2272 |
% \isacommand{bind (set)} @{text "xs"} \isacommand{in} @{text "s"}\\ |
|
2273 |
%\end{tabular} |
|
2274 |
%\end{center} |
|
2275 |
% |
|
2276 |
%\noindent |
|
2277 |
%In the first term-constructor we have a single |
|
2278 |
%body that happens to be ``spread'' over two arguments; in the second term-constructor we have |
|
2279 |
%two independent bodies in which the same variables are bound. As a result we |
|
2280 |
%have |
|
2281 |
% |
|
2282 |
%\begin{center} |
|
2283 |
%\begin{tabular}{r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l} |
|
2284 |
%@{text "Foo\<^isub>1 {a, b} (a, b) (a, b)"} & $\not=$ & |
|
2285 |
%@{text "Foo\<^isub>1 {a, b} (a, b) (b, a)"}\\ |
|
2286 |
%@{text "Foo\<^isub>2 {a, b} (a, b) (a, b)"} & $=$ & |
|
2287 |
%@{text "Foo\<^isub>2 {a, b} (a, b) (b, a)"}\\ |
|
2288 |
%\end{tabular} |
|
2289 |
%\end{center} |
|
2513 | 2290 |
% |
2291 |
%\noindent |
|
2512 | 2292 |
%and therefore need the extra generality to be able to distinguish between |
2293 |
%both specifications. |
|
2362 | 2294 |
Because of how we set up our definitions, we also had to impose some restrictions |
2514
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
2295 |
(like a single binding function for a deep binder) that are not present in Ott. |
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
2296 |
%Our |
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
2297 |
%expectation is that we can still cover many interesting term-calculi from |
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
2298 |
%programming language research, for example Core-Haskell. |
2163
5dc48e1af733
added comments about pottiers work
Christian Urban <urbanc@in.tum.de>
parents:
2156
diff
changeset
|
2299 |
|
5dc48e1af733
added comments about pottiers work
Christian Urban <urbanc@in.tum.de>
parents:
2156
diff
changeset
|
2300 |
Pottier presents in \cite{Pottier06} a language, called C$\alpha$ml, for |
5dc48e1af733
added comments about pottiers work
Christian Urban <urbanc@in.tum.de>
parents:
2156
diff
changeset
|
2301 |
representing terms with general binders inside OCaml. This language is |
2362 | 2302 |
implemented as a front-end that can be translated to OCaml with the help of |
2163
5dc48e1af733
added comments about pottiers work
Christian Urban <urbanc@in.tum.de>
parents:
2156
diff
changeset
|
2303 |
a library. He presents a type-system in which the scope of general binders |
2362 | 2304 |
can be specified using special markers, written @{text "inner"} and |
2305 |
@{text "outer"}. It seems our and his specifications can be |
|
2306 |
inter-translated as long as ours use the binding mode |
|
2307 |
\isacommand{bind} only. |
|
2308 |
However, we have not proved this. Pottier gives a definition for |
|
2309 |
$\alpha$-equivalence, which also uses a permutation operation (like ours). |
|
2310 |
Still, this definition is rather different from ours and he only proves that |
|
2311 |
it defines an equivalence relation. A complete |
|
2163
5dc48e1af733
added comments about pottiers work
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parents:
2156
diff
changeset
|
2312 |
reasoning infrastructure is well beyond the purposes of his language. |
2514
69780ae147f5
down to 21 pages and changed strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
2513
diff
changeset
|
2313 |
Similar work for Haskell with similar results was reported by Cheney \cite{Cheney05a}. |
2362 | 2314 |
|
2218
502eaa199726
added to the popl-paper a pointer to work by Altenkirch
Christian Urban <urbanc@in.tum.de>
parents:
2203
diff
changeset
|
2315 |
In a slightly different domain (programming with dependent types), the |
502eaa199726
added to the popl-paper a pointer to work by Altenkirch
Christian Urban <urbanc@in.tum.de>
parents:
2203
diff
changeset
|
2316 |
paper \cite{Altenkirch10} presents a calculus with a notion of |
2637
3890483c674f
final version of the ESOP paper; used set+ instead of res as requested by one reviewer
Christian Urban <urbanc@in.tum.de>
parents:
2605
diff
changeset
|
2317 |
$\alpha$-equivalence related to our binding mode \isacommand{bind (set+)}. |
2363 | 2318 |
The definition in \cite{Altenkirch10} is similar to the one by Pottier, except that it |
2218
502eaa199726
added to the popl-paper a pointer to work by Altenkirch
Christian Urban <urbanc@in.tum.de>
parents:
2203
diff
changeset
|
2319 |
has a more operational flavour and calculates a partial (renaming) map. |
2363 | 2320 |
In this way, the definition can deal with vacuous binders. However, to our |
2321 |
best knowledge, no concrete mathematical result concerning this |
|
2516 | 2322 |
definition of $\alpha$-equivalence has been proved.\\[-7mm] |
1739 | 2323 |
*} |
2324 |
||
1493
52f68b524fd2
slightly more of the paper
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parents:
1491
diff
changeset
|
2325 |
section {* Conclusion *} |
1485
c004e7448dca
temporarily disabled tests in Nominal/ROOT
Christian Urban <urbanc@in.tum.de>
parents:
1484
diff
changeset
|
2326 |
|
c004e7448dca
temporarily disabled tests in Nominal/ROOT
Christian Urban <urbanc@in.tum.de>
parents:
1484
diff
changeset
|
2327 |
text {* |
2362 | 2328 |
We have presented an extension of Nominal Isabelle for dealing with |
2329 |
general binders, that is term-constructors having multiple bound |
|
2381 | 2330 |
variables. For this extension we introduced new definitions of |
2362 | 2331 |
$\alpha$-equivalence and automated all necessary proofs in Isabelle/HOL. |
2364 | 2332 |
To specify general binders we used the specifications from Ott, but extended them |
2333 |
in some places and restricted |
|
2512 | 2334 |
them in others so that they make sense in the context of $\alpha$-equated terms. |
2637
3890483c674f
final version of the ESOP paper; used set+ instead of res as requested by one reviewer
Christian Urban <urbanc@in.tum.de>
parents:
2605
diff
changeset
|
2335 |
We also introduced two binding modes (set and set+) that do not |
2488
1c18f2cf3923
a few more words about Ott
Christian Urban <urbanc@in.tum.de>
parents:
2471
diff
changeset
|
2336 |
exist in Ott. |
2517 | 2337 |
We have tried out the extension with calculi such as Core-Haskell, type-schemes |
2338 |
and approximately a dozen of other typical examples from programming |
|
2604
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
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parents:
2580
diff
changeset
|
2339 |
language research~\cite{SewellBestiary}. |
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
2340 |
%The code |
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
2341 |
%will eventually become part of the next Isabelle distribution.\footnote{For the moment |
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
2342 |
%it can be downloaded from the Mercurial repository linked at |
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
2343 |
%\href{http://isabelle.in.tum.de/nominal/download} |
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
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parents:
2580
diff
changeset
|
2344 |
%{http://isabelle.in.tum.de/nominal/download}.} |
1741 | 2345 |
|
1764
9f55d7927e5b
more on the strong induction section
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parents:
1763
diff
changeset
|
2346 |
We have left out a discussion about how functions can be defined over |
2362 | 2347 |
$\alpha$-equated terms involving general binders. In earlier versions of Nominal |
2604
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
2348 |
Isabelle this turned out to be a thorny issue. We |
1764
9f55d7927e5b
more on the strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
1763
diff
changeset
|
2349 |
hope to do better this time by using the function package that has recently |
9f55d7927e5b
more on the strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
1763
diff
changeset
|
2350 |
been implemented in Isabelle/HOL and also by restricting function |
2604
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
2351 |
definitions to equivariant functions (for them we can |
1764
9f55d7927e5b
more on the strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
1763
diff
changeset
|
2352 |
provide more automation). |
1741 | 2353 |
|
2512 | 2354 |
%There are some restrictions we imposed in this paper that we would like to lift in |
2355 |
%future work. One is the exclusion of nested datatype definitions. Nested |
|
2356 |
%datatype definitions allow one to specify, for instance, the function kinds |
|
2357 |
%in Core-Haskell as @{text "TFun string (ty list)"} instead of the unfolded |
|
2358 |
%version @{text "TFun string ty_list"} (see Figure~\ref{nominalcorehas}). To |
|
2359 |
%achieve this, we need a slightly more clever implementation than we have at the moment. |
|
1764
9f55d7927e5b
more on the strong induction section
Christian Urban <urbanc@in.tum.de>
parents:
1763
diff
changeset
|
2360 |
|
2512 | 2361 |
%A more interesting line of investigation is whether we can go beyond the |
2362 |
%simple-minded form of binding functions that we adopted from Ott. At the moment, binding |
|
2363 |
%functions can only return the empty set, a singleton atom set or unions |
|
2364 |
%of atom sets (similarly for lists). It remains to be seen whether |
|
2365 |
%properties like |
|
2366 |
%% |
|
2367 |
%\begin{center} |
|
2368 |
%@{text "fa_ty x = bn x \<union> fa_bn x"}. |
|
2369 |
%\end{center} |
|
2361 | 2370 |
% |
2512 | 2371 |
%\noindent |
2372 |
%allow us to support more interesting binding functions. |
|
2373 |
% |
|
2374 |
%We have also not yet played with other binding modes. For example we can |
|
2375 |
%imagine that there is need for a binding mode |
|
2376 |
%where instead of lists, we abstract lists of distinct elements. |
|
2377 |
%Once we feel confident about such binding modes, our implementation |
|
2378 |
%can be easily extended to accommodate them. |
|
2516 | 2379 |
% |
2380 |
\smallskip |
|
2362 | 2381 |
\noindent |
2511 | 2382 |
{\bf Acknowledgements:} %We are very grateful to Andrew Pitts for |
2383 |
%many discussions about Nominal Isabelle. |
|
2384 |
We thank Peter Sewell for |
|
2362 | 2385 |
making the informal notes \cite{SewellBestiary} available to us and |
2604
431cf4e6a7e2
brought the paper to 20 pages plus one page appendix
Christian Urban <urbanc@in.tum.de>
parents:
2580
diff
changeset
|
2386 |
also for patiently explaining some of the finer points of the Ott-tool.\\[-7mm] |
2511 | 2387 |
%Stephanie Weirich suggested to separate the subgrammars |
2516 | 2388 |
%of kinds and types in our Core-Haskell example. \\[-6mm] |
2519
3e9b4ce0aeca
added apendix to paper detailing one proof
Christian Urban <urbanc@in.tum.de>
parents:
2518
diff
changeset
|
2389 |
*} |
2341 | 2390 |
|
754
b85875d65b10
added a paper for possible notes
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
2391 |
|
b85875d65b10
added a paper for possible notes
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
2392 |
(*<*) |
b85875d65b10
added a paper for possible notes
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
|
2393 |
end |
1704 | 2394 |
(*>*) |