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(*<*)
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theory Paper
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imports "Quotient"
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"LaTeXsugar"
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"../Nominal/FSet"
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begin
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notation (latex output)
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rel_conj ("_ \<circ>\<circ>\<circ> _" [53, 53] 52) and
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pred_comp ("_ \<circ>\<circ> _" [1, 1] 30) and
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"op -->" (infix "\<longrightarrow>" 100) and
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"==>" (infix "\<Longrightarrow>" 100) and
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fun_map ("_ \<^raw:\mbox{\singlearr}> _" 51) and
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fun_rel ("_ \<^raw:\mbox{\doublearr}> _" 51) and
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list_eq (infix "\<approx>" 50) and (* Not sure if we want this notation...? *)
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fempty ("\<emptyset>") and
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funion ("_ \<union> _") and
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finsert ("{_} \<union> _") and
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Cons ("_::_") and
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concat ("flat") and
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fconcat ("\<Union>")
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1994
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ML {*
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fun nth_conj n (_, r) = nth (HOLogic.dest_conj r) n;
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fun style_lhs_rhs proj = Scan.succeed (fn ctxt => fn t =>
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let
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val concl =
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Object_Logic.drop_judgment (ProofContext.theory_of ctxt) (Logic.strip_imp_concl t)
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in
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case concl of (_ $ l $ r) => proj (l, r)
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| _ => error ("Binary operator expected in term: " ^ Syntax.string_of_term ctxt concl)
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end);
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*}
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setup {*
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Term_Style.setup "rhs1" (style_lhs_rhs (nth_conj 0)) #>
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Term_Style.setup "rhs2" (style_lhs_rhs (nth_conj 1)) #>
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Term_Style.setup "rhs3" (style_lhs_rhs (nth_conj 2))
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*}
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(*>*)
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section {* Introduction *}
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text {*
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\begin{flushright}
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{\em ``Not using a [quotient] package has its advantages: we do not have to\\
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collect all the theorems we shall ever want into one giant list;''}\\
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Larry Paulson \cite{Paulson06}
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\end{flushright}
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\noindent
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Isabelle is a popular generic theorem prover in which many logics can be
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implemented. The most widely used one, however, is Higher-Order Logic
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(HOL). This logic consists of a small number of axioms and inference rules
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over a simply-typed term-language. Safe reasoning in HOL is ensured by two
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very restricted mechanisms for extending the logic: one is the definition of
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new constants in terms of existing ones; the other is the introduction of
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new types by identifying non-empty subsets in existing types. It is well
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understood how to use both mechanisms for dealing with quotient
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constructions in HOL (see \cite{Homeier05,Paulson06}). For example the
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integers in Isabelle/HOL are constructed by a quotient construction over the
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type @{typ "nat \<times> nat"} and the equivalence relation
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
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@{text "(n\<^isub>1, n\<^isub>2) \<approx> (m\<^isub>1, m\<^isub>2) \<equiv> n\<^isub>1 + m\<^isub>2 = m\<^isub>1 + n\<^isub>2"}\hfill\numbered{natpairequiv}
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\end{isabelle}
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\noindent
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This constructions yields the new type @{typ int} and definitions for @{text
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"0"} and @{text "1"} of type @{typ int} can be given in terms of pairs of
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natural numbers (namely @{text "(0, 0)"} and @{text "(1, 0)"}). Operations
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such as @{text "add"} with type @{typ "int \<Rightarrow> int \<Rightarrow> int"} can be defined in
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terms of operations on pairs of natural numbers (namely @{text
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"add_pair (n\<^isub>1, m\<^isub>1) (n\<^isub>2,
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m\<^isub>2) \<equiv> (n\<^isub>1 + n\<^isub>2, m\<^isub>1 + m\<^isub>2)"}).
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Similarly one can construct the type of finite sets, written @{term "\<alpha> fset"},
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by quotienting the type @{text "\<alpha> list"} according to the equivalence relation
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
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@{text "xs \<approx> ys \<equiv> (\<forall>x. memb x xs \<longleftrightarrow> memb x ys)"}\hfill\numbered{listequiv}
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\end{isabelle}
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\noindent
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which states that two lists are equivalent if every element in one list is
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also member in the other. The empty finite set, written @{term "{||}"}, can
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then be defined as the empty list and the union of two finite sets, written
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@{text "\<union>"}, as list append.
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Quotients are important in a variety of areas, but they are really ubiquitous in
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the area of reasoning about programming language calculi. A simple example
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is the lambda-calculus, whose raw terms are defined as
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
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@{text "t ::= x | t t | \<lambda>x.t"}\hfill\numbered{lambda}
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\end{isabelle}
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\noindent
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The problem with this definition arises, for instance, when one attempts to
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prove formally the substitution lemma \cite{Barendregt81} by induction
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over the structure of terms. This can be fiendishly complicated (see
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\cite[Pages 94--104]{CurryFeys58} for some ``rough'' sketches of a proof
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about raw lambda-terms). In contrast, if we reason about
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$\alpha$-equated lambda-terms, that means terms quotient according to
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$\alpha$-equivalence, then the reasoning infrastructure provided,
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for example, by Nominal Isabelle \cite{UrbanKaliszyk11} makes the formal
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proof of the substitution lemma almost trivial.
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The difficulty is that in order to be able to reason about integers, finite
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sets or $\alpha$-equated lambda-terms one needs to establish a reasoning
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infrastructure by transferring, or \emph{lifting}, definitions and theorems
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from the raw type @{typ "nat \<times> nat"} to the quotient type @{typ int}
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(similarly for finite sets and $\alpha$-equated lambda-terms). This lifting
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usually requires a \emph{lot} of tedious reasoning effort \cite{Paulson06}.
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It is feasible to do this work manually, if one has only a few quotient
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constructions at hand. But if they have to be done over and over again, as in
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Nominal Isabelle, then manual reasoning is not an option.
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The purpose of a \emph{quotient package} is to ease the lifting of theorems
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and automate the reasoning as much as possible. In the
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context of HOL, there have been a few quotient packages already
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\cite{harrison-thesis,Slotosch97}. The most notable one is by Homeier
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\cite{Homeier05} implemented in HOL4. The fundamental construction these
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quotient packages perform can be illustrated by the following picture:
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\begin{center}
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\mbox{}\hspace{20mm}\begin{tikzpicture}
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%%\draw[step=2mm] (-4,-1) grid (4,1);
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\draw[very thick] (0.7,0.3) circle (4.85mm);
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\draw[rounded corners=1mm, very thick] ( 0.0,-0.9) rectangle ( 1.8, 0.9);
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\draw[rounded corners=1mm, very thick] (-1.95,0.8) rectangle (-2.9,-0.195);
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\draw (-2.0, 0.8) -- (0.7,0.8);
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\draw (-2.0,-0.195) -- (0.7,-0.195);
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\draw ( 0.7, 0.23) node {\begin{tabular}{@ {}c@ {}}equiv-\\[-1mm]clas.\end{tabular}};
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\draw (-2.45, 0.35) node {\begin{tabular}{@ {}c@ {}}new\\[-1mm]type\end{tabular}};
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\draw (1.8, 0.35) node[right=-0.1mm]
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{\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ (sets of raw elements)\end{tabular}};
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\draw (0.9, -0.55) node {\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}};
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\draw[->, very thick] (-1.8, 0.36) -- (-0.1,0.36);
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\draw[<-, very thick] (-1.8, 0.16) -- (-0.1,0.16);
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\draw (-0.95, 0.26) node[above=0.4mm] {@{text Rep}};
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\draw (-0.95, 0.26) node[below=0.4mm] {@{text Abs}};
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\end{tikzpicture}
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\end{center}
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\noindent
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The starting point is an existing type, to which we refer as the
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\emph{raw type} and over which an equivalence relation given by the user is
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defined. With this input the package introduces a new type, to which we
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refer as the \emph{quotient type}. This type comes with an
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\emph{abstraction} and a \emph{representation} function, written @{text Abs}
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and @{text Rep}.\footnote{Actually slightly more basic functions are given;
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the functions @{text Abs} and @{text Rep} need to be derived from them. We
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will show the details later. } They relate elements in the
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existing type to elements in the new type and vice versa, and can be uniquely
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identified by their quotient type. For example for the integer quotient construction
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the types of @{text Abs} and @{text Rep} are
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
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@{text "Abs :: nat \<times> nat \<Rightarrow> int"}\hspace{10mm}@{text "Rep :: int \<Rightarrow> nat \<times> nat"}
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\end{isabelle}
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+ − 171
\noindent
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We therefore often write @{text Abs_int} and @{text Rep_int} if the
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typing information is important.
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Every abstraction and representation function stands for an isomorphism
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between the non-empty subset and elements in the new type. They are
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necessary for making definitions involving the new type. For example @{text
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"0"} and @{text "1"} of type @{typ int} can be defined as
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
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@{text "0 \<equiv> Abs_int (0, 0)"}\hspace{10mm}@{text "1 \<equiv> Abs_int (1, 0)"}
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\end{isabelle}
+ − 184
+ − 185
\noindent
2224
+ − 186
Slightly more complicated is the definition of @{text "add"} having type
2222
+ − 187
@{typ "int \<Rightarrow> int \<Rightarrow> int"}. Its definition is as follows
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+ − 188
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 190
@{text "add n m \<equiv> Abs_int (add_pair (Rep_int n) (Rep_int m))"}
+ − 191
\hfill\numbered{adddef}
+ − 192
\end{isabelle}
+ − 193
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\noindent
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+ − 195
where we take the representation of the arguments @{text n} and @{text m},
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add them according to the function @{text "add_pair"} and then take the
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abstraction of the result. This is all straightforward and the existing
+ − 198
quotient packages can deal with such definitions. But what is surprising
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that none of them can deal with slightly more complicated definitions involving
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\emph{compositions} of quotients. Such compositions are needed for example
2247
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in case of quotienting lists to yield finite sets and the operator that
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+ − 202
flattens lists of lists, defined as follows
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@{thm [display, indent=10] concat.simps(1) concat.simps(2)[no_vars]}
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\noindent
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We expect that the corresponding operator on finite sets, written @{term "fconcat"},
2248
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builds finite unions of finite sets:
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@{thm [display, indent=10] fconcat_empty[no_vars] fconcat_insert[no_vars]}
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\noindent
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The quotient package should automatically provide us with a definition for @{text "\<Union>"} in
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terms of @{text flat}, @{text Rep_fset} and @{text Abs_fset}. The problem is
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that the method used in the existing quotient
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packages of just taking the representation of the arguments and then taking
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the abstraction of the result is \emph{not} enough. The reason is that in case
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of @{text "\<Union>"} we obtain the incorrect definition
+ − 219
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@{text [display, indent=10] "\<Union> S \<equiv> Abs_fset (flat (Rep_fset S))"}
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+ − 222
\noindent
+ − 223
where the right-hand side is not even typable! This problem can be remedied in the
+ − 224
existing quotient packages by introducing an intermediate step and reasoning
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about flattening of lists of finite sets. However, this remedy is rather
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cumbersome and inelegant in light of our work, which can deal with such
+ − 227
definitions directly. The solution is that we need to build aggregate
+ − 228
representation and abstraction functions, which in case of @{text "\<Union>"}
+ − 229
generate the following definition
+ − 230
2234
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@{text [display, indent=10] "\<Union> S \<equiv> Abs_fset (flat ((map Rep_fset \<circ> Rep_fset) S))"}
2221
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+ − 233
\noindent
2223
+ − 234
where @{term map} is the usual mapping function for lists. In this paper we
2224
+ − 235
will present a formal definition of our aggregate abstraction and
2223
+ − 236
representation functions (this definition was omitted in \cite{Homeier05}).
2224
+ − 237
They generate definitions, like the one above for @{text "\<Union>"},
2226
+ − 238
according to the type of the raw constant and the type
2224
+ − 239
of the quotient constant. This means we also have to extend the notions
2237
+ − 240
of \emph{aggregate equivalence relation}, \emph{respectfulness} and \emph{preservation}
2231
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from Homeier \cite{Homeier05}.
2223
+ − 242
2252
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In addition we are able to address the criticism by Paulson \cite{Paulson06} cited
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at the beginning of this section about having to collect theorems that are
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lifted from the raw level to the quotient level into one giant list. Our
+ − 246
quotient package is the first one that is modular so that it allows to lift
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single theorems separately. This has the advantage for the user of being able to develop a
+ − 248
formal theory interactively as a natural progression. A pleasing side-result of
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the modularity is that we are able to clearly specify what is involved
+ − 250
in the lifting process (this was only hinted at in \cite{Homeier05} and
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implemented as a ``rough recipe'' in ML-code).
+ − 252
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The paper is organised as follows: Section \ref{sec:prelims} presents briefly
2247
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some necessary preliminaries; Section \ref{sec:type} describes the definitions
2252
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of quotient types and shows how definitions of constants can be made over
+ − 257
quotient types. Section \ref{sec:resp} introduces the notions of respectfullness
2256
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and preservation; Section \ref{sec:lift} describes the lifting of theorems,
+ − 259
and Section \ref{sec:conc} concludes and compares our results to existing
+ − 260
work.
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*}
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section {* Preliminaries and General Quotients\label{sec:prelims} *}
1978
+ − 264
+ − 265
text {*
2269
+ − 266
We describe in this section briefly the most basic notions of HOL we rely on, and
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the important definitions given by Homeier for quotients \cite{Homeier05}.
+ − 268
+ − 269
At its core HOL is based on a simply-typed term language, where types are
2269
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recorded in Church-style fashion (that means, we can always infer the type of
2258
+ − 271
a term and its subterms without any additional information). The grammars
+ − 272
for types and terms are as follows
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+ − 274
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 275
\begin{tabular}{@ {}rl@ {\hspace{3mm}}l@ {}}
+ − 276
@{text "\<sigma>, \<tau> ::="} & @{text "\<alpha> | (\<sigma>,\<dots>, \<sigma>) \<kappa>"} & (type variables and type constructors)\\
+ − 277
@{text "t, s ::="} & @{text "x\<^isup>\<sigma> | c\<^isup>\<sigma> | t t | \<lambda>x\<^isup>\<sigma>. t"} &
+ − 278
(variables, constants, applications and abstractions)\\
+ − 279
\end{tabular}
+ − 280
\end{isabelle}
+ − 281
+ − 282
\noindent
+ − 283
We often write just @{text \<kappa>} for @{text "() \<kappa>"}, and use @{text "\<alpha>s"} and
2258
+ − 284
@{text "\<sigma>s"} to stand for collections of type variables and types,
+ − 285
respectively. The type of a term is often made explicit by writing @{text
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"t :: \<sigma>"}. HOL includes a type @{typ bool} for booleans and the function
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type, written @{text "\<sigma> \<Rightarrow> \<tau>"}. HOL also contains
2269
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many primitive and defined constants; this includes equality, with type @{text "= :: \<sigma> \<Rightarrow>
+ − 289
\<sigma> \<Rightarrow> bool"}, and the identity function, with type @{text "id :: \<sigma> => \<sigma>"} (the former
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being primitive and the latter being defined as @{text
+ − 291
"\<lambda>x\<^sup>\<sigma>. x\<^sup>\<sigma>"}).
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An important point to note is that theorems in HOL can be seen as a subset
+ − 294
of terms that are constructed specially (namely through axioms and prove
2269
+ − 295
rules). As a result we are able to define automatic proof
2258
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procedures showing that one theorem implies another by decomposing the term
+ − 297
underlying the first theorem.
+ − 298
+ − 299
Like Homeier, our work relies on map-functions defined for every type constructor,
2269
+ − 300
like @{text map} for lists. Homeier describes in \cite{Homeier05} map-functions
+ − 301
for products, sums,
2258
+ − 302
options and also the following map for function types
+ − 303
+ − 304
@{thm [display, indent=10] fun_map_def[no_vars, THEN eq_reflection]}
+ − 305
+ − 306
\noindent
+ − 307
Using this map-function, we can give the following, equivalent, but more
+ − 308
uniform, definition for @{text add} shown in \eqref{adddef}:
2256
+ − 309
2258
+ − 310
@{text [display, indent=10] "add \<equiv> (Rep_int \<singlearr> Rep_int \<singlearr> Abs_int) add_pair"}
2182
+ − 311
2258
+ − 312
\noindent
2269
+ − 313
Using extensionality and unfolding the definition, we can get back to \eqref{adddef}.
+ − 314
In what follows we shall use the terminology @{text "map_\<kappa>"} for a map-function
+ − 315
defined for the type-constructor @{text \<kappa>}. In our implementation we have
+ − 316
a database of map-functions that can be dynamically extended.
2258
+ − 317
+ − 318
It will also be necessary to have operators, referred to as @{text "rel_\<kappa>"},
+ − 319
which define equivalence relations in terms of constituent equivalence
+ − 320
relations. For example given two equivalence relations @{text "R\<^isub>1"}
+ − 321
and @{text "R\<^isub>2"}, we can define an equivalence relations over
+ − 322
products as follows
+ − 323
%
+ − 324
@{text [display, indent=10] "(R\<^isub>1 \<tripple> R\<^isub>2) (x\<^isub>1, x\<^isub>2) (y\<^isub>1, y\<^isub>2) \<equiv> R\<^isub>1 x\<^isub>1 y\<^isub>1 \<and> R\<^isub>2 x\<^isub>2 y\<^isub>2"}
1978
+ − 325
2258
+ − 326
\noindent
2269
+ − 327
Homeier gives also the following operator for defining equivalence
+ − 328
relations over function types
2258
+ − 329
%
+ − 330
@{thm [display, indent=10] fun_rel_def[of "R\<^isub>1" "R\<^isub>2", no_vars, THEN eq_reflection]}
+ − 331
+ − 332
The central definition in Homeier's work \cite{Homeier05} relates equivalence
+ − 333
relations, abstraction and representation functions:
+ − 334
+ − 335
\begin{definition}[Quotient Types]
+ − 336
Given a relation $R$, an abstraction function $Abs$
+ − 337
and a representation function $Rep$, the predicate @{term "Quotient R Abs Rep"}
+ − 338
means
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+ − 339
\begin{enumerate}
+ − 340
\item @{thm (rhs1) Quotient_def[of "R", no_vars]}
+ − 341
\item @{thm (rhs2) Quotient_def[of "R", no_vars]}
+ − 342
\item @{thm (rhs3) Quotient_def[of "R", no_vars]}
+ − 343
\end{enumerate}
+ − 344
\end{definition}
+ − 345
2258
+ − 346
\noindent
2269
+ − 347
The value of this definition is that validity of @{text "Quotient R Abs Rep"} can
+ − 348
often be proved in terms of the validity of @{text "Quotient"} over the constituent
+ − 349
types of @{text "R"}, @{text Abs} and @{text Rep}.
2258
+ − 350
For example Homeier proves the following property for higher-order quotient
+ − 351
types:
+ − 352
2269
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\begin{proposition}\label{funquot}
2258
+ − 354
@{thm[mode=IfThen] fun_quotient[where ?R1.0="R\<^isub>1" and ?R2.0="R\<^isub>2"
+ − 355
and ?abs1.0="Abs\<^isub>1" and ?abs2.0="Abs\<^isub>2" and ?rep1.0="Rep\<^isub>1" and ?rep2.0="Rep\<^isub>2"]}
+ − 356
\end{proposition}
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2271
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\begin{definition}[Respects]\label{def:respects}
2268
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An element @{text "x"} respects a relation @{text "R"} if and only if @{text "R x x"}.
+ − 360
\end{definition}
+ − 361
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+ − 362
\noindent
2269
+ − 363
As a result, Homeier was able to build an automatic prover that can nearly
+ − 364
always discharge a proof obligation involving @{text "Quotient"}. Our quotient
+ − 365
package makes heavy
+ − 366
use of this part of Homeier's work including an extension
2258
+ − 367
to deal with compositions of equivalence relations defined as follows
2234
+ − 368
2258
+ − 369
\begin{definition}[Composition of Relations]
+ − 370
@{abbrev "rel_conj R\<^isub>1 R\<^isub>2"} where @{text "\<circ>\<circ>"} is the predicate
+ − 371
composition defined by the rule
+ − 372
%
+ − 373
@{thm [mode=Rule, display, indent=10] pred_compI[of "R\<^isub>1" "x" "y" "R\<^isub>2" "z"]}
+ − 374
\end{definition}
2237
+ − 375
2258
+ − 376
\noindent
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+ − 377
Unfortunately a quotient type theorem, like Proposition \ref{funquot}, for
dcffc2f132c9
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diff
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+ − 378
the composition of any two quotients in not true (it is not even typable in
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diff
changeset
+ − 379
the HOL type system). However, we can prove useful instances for compatible
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diff
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+ − 380
containers. We will show such example in Section \ref{sec:resp}.
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+ − 381
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+ − 382
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+ − 383
*}
+ − 384
2237
+ − 385
section {* Quotient Types and Quotient Definitions\label{sec:type} *}
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+ − 386
2234
+ − 387
text {*
2247
+ − 388
The first step in a quotient construction is to take a name for the new
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+ − 389
type, say @{text "\<kappa>\<^isub>q"}, and an equivalence relation, say @{text R},
2247
+ − 390
defined over a raw type, say @{text "\<sigma>"}. The type of the equivalence
2269
+ − 391
relation must be @{text "\<sigma> \<Rightarrow> \<sigma> \<Rightarrow> bool"}. The user-visible part of
+ − 392
the quotient type declaration is therefore
2234
+ − 393
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+ − 394
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
2269
+ − 395
\isacommand{quotient\_type}~~@{text "\<alpha>s \<kappa>\<^isub>q = \<sigma> / R"}\hfill\numbered{typedecl}
2235
+ − 396
\end{isabelle}
+ − 397
+ − 398
\noindent
2237
+ − 399
and a proof that @{text "R"} is indeed an equivalence relation. Two concrete
+ − 400
examples are
+ − 401
+ − 402
+ − 403
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 404
\begin{tabular}{@ {}l}
+ − 405
\isacommand{quotient\_type}~~@{text "int = nat \<times> nat / \<approx>\<^bsub>nat \<times> nat\<^esub>"}\\
+ − 406
\isacommand{quotient\_type}~~@{text "\<alpha> fset = \<alpha> list / \<approx>\<^bsub>list\<^esub>"}
+ − 407
\end{tabular}
+ − 408
\end{isabelle}
+ − 409
+ − 410
\noindent
+ − 411
which introduce the type of integers and of finite sets using the
+ − 412
equivalence relations @{text "\<approx>\<^bsub>nat \<times> nat\<^esub>"} and @{text
2269
+ − 413
"\<approx>\<^bsub>list\<^esub>"} defined in \eqref{natpairequiv} and
2247
+ − 414
\eqref{listequiv}, respectively (the proofs about being equivalence
2269
+ − 415
relations is omitted). Given this data, we declare for \eqref{typedecl} internally
2237
+ − 416
the quotient types as
2234
+ − 417
+ − 418
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 419
\isacommand{typedef}~~@{text "\<alpha>s \<kappa>\<^isub>q = {c. \<exists>x. c = R x}"}
+ − 420
\end{isabelle}
+ − 421
+ − 422
\noindent
2247
+ − 423
where the right-hand side is the (non-empty) set of equivalence classes of
2237
+ − 424
@{text "R"}. The restriction in this declaration is that the type variables
+ − 425
in the raw type @{text "\<sigma>"} must be included in the type variables @{text
2247
+ − 426
"\<alpha>s"} declared for @{text "\<kappa>\<^isub>q"}. HOL will provide us with the following
2269
+ − 427
abstraction and representation functions
2182
+ − 428
2234
+ − 429
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 430
@{text "abs_\<kappa>\<^isub>q :: \<sigma> set \<Rightarrow> \<alpha>s \<kappa>\<^isub>q"}\hspace{10mm}@{text "rep_\<kappa>\<^isub>q :: \<alpha>s \<kappa>\<^isub>q \<Rightarrow> \<sigma> set"}
+ − 431
\end{isabelle}
+ − 432
2235
+ − 433
\noindent
2269
+ − 434
As can be seen from the type, they relate the new quotient type and equivalence classes of the raw
2235
+ − 435
type. However, as Homeier \cite{Homeier05} noted, it is much more convenient
+ − 436
to work with the following derived abstraction and representation functions
+ − 437
2234
+ − 438
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 439
@{text "Abs_\<kappa>\<^isub>q x \<equiv> abs_\<kappa>\<^isub>q (R x)"}\hspace{10mm}@{text "Rep_\<kappa>\<^isub>q x \<equiv> \<epsilon> (rep_\<kappa>\<^isub>q x)"}
+ − 440
\end{isabelle}
+ − 441
+ − 442
\noindent
2235
+ − 443
on the expense of having to use Hilbert's choice operator @{text "\<epsilon>"} in the
2237
+ − 444
definition of @{text "Rep_\<kappa>\<^isub>q"}. These derived notions relate the
+ − 445
quotient type and the raw type directly, as can be seen from their type,
+ − 446
namely @{text "\<sigma> \<Rightarrow> \<alpha>s \<kappa>\<^isub>q"} and @{text "\<alpha>s \<kappa>\<^isub>q \<Rightarrow> \<sigma>"},
+ − 447
respectively. Given that @{text "R"} is an equivalence relation, the
+ − 448
following property
+ − 449
2258
+ − 450
\begin{proposition}
2252
+ − 451
@{text "Quotient R Abs_\<kappa>\<^isub>q Rep_\<kappa>\<^isub>q"}
2258
+ − 452
\end{proposition}
2234
+ − 453
+ − 454
\noindent
2252
+ − 455
holds for every quotient type defined
+ − 456
as above (for the proof see \cite{Homeier05}).
2182
+ − 457
2247
+ − 458
The next step in a quotient construction is to introduce definitions of new constants
+ − 459
involving the quotient type. These definitions need to be given in terms of concepts
2238
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 460
of the raw type (remember this is the only way how to extend HOL
2269
+ − 461
with new definitions). For the user the visible part of such definitions is the declaration
2235
+ − 462
+ − 463
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
2237
+ − 464
\isacommand{quotient\_definition}~~@{text "c :: \<tau>"}~~\isacommand{is}~~@{text "t :: \<sigma>"}
2235
+ − 465
\end{isabelle}
+ − 466
2237
+ − 467
\noindent
+ − 468
where @{text t} is the definiens (its type @{text \<sigma>} can always be inferred)
+ − 469
and @{text "c"} is the name of definiendum, whose type @{text "\<tau>"} needs to be
+ − 470
given explicitly (the point is that @{text "\<tau>"} and @{text "\<sigma>"} can only differ
2269
+ − 471
in places where a quotient and raw type is involved). Two concrete examples are
2188
+ − 472
2237
+ − 473
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 474
\begin{tabular}{@ {}l}
+ − 475
\isacommand{quotient\_definition}~~@{text "0 :: int"}~~\isacommand{is}~~@{text "(0::nat, 0::nat)"}\\
+ − 476
\isacommand{quotient\_definition}~~@{text "\<Union> :: (\<alpha> fset) fset \<Rightarrow> \<alpha> fset"}~~%
+ − 477
\isacommand{is}~~@{text "flat"}
+ − 478
\end{tabular}
+ − 479
\end{isabelle}
+ − 480
+ − 481
\noindent
+ − 482
The first one declares zero for integers and the second the operator for
2238
8ddf1330f2ed
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Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 483
building unions of finite sets.
8ddf1330f2ed
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diff
changeset
+ − 484
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
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diff
changeset
+ − 485
The problem for us is that from such declarations we need to derive proper
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 486
definitions using the @{text "Abs"} and @{text "Rep"} functions for the
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 487
quotient types involved. The data we rely on is the given quotient type
2247
+ − 488
@{text "\<tau>"} and the raw type @{text "\<sigma>"}. They allow us to define \emph{aggregate
+ − 489
abstraction} and \emph{representation functions} using the functions @{text "ABS (\<sigma>,
2252
+ − 490
\<tau>)"} and @{text "REP (\<sigma>, \<tau>)"} whose clauses we give below. The idea behind
2247
+ − 491
these two functions is to recursively descend into the raw types @{text \<sigma>} and
+ − 492
quotient types @{text \<tau>}, and generate the appropriate
2238
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diff
changeset
+ − 493
@{text "Abs"} and @{text "Rep"} in places where the types differ. Therefore
2269
+ − 494
we generate just the identity whenever the types are equal. On the ``way'' down,
+ − 495
however we might have to use map-functions to let @{text Abs} and @{text Rep} act
+ − 496
over the appropriate types. The clauses of @{text ABS} and @{text REP}
+ − 497
are as follows (where we use the short-hand notation @{text "ABS (\<sigma>s, \<tau>s)"} to mean
+ − 498
@{text "ABS (\<sigma>\<^isub>1, \<tau>\<^isub>1)\<dots>ABS (\<sigma>\<^isub>i, \<tau>\<^isub>i)"}; similarly for @{text REP}):
2182
+ − 499
2227
+ − 500
\begin{center}
2252
+ − 501
\hfill
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
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diff
changeset
+ − 502
\begin{tabular}{rcl}
2227
+ − 503
\multicolumn{3}{@ {\hspace{-4mm}}l}{equal types:}\\
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
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diff
changeset
+ − 504
@{text "ABS (\<sigma>, \<sigma>)"} & $\dn$ & @{text "id :: \<sigma> \<Rightarrow> \<sigma>"}\\
8ddf1330f2ed
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diff
changeset
+ − 505
@{text "REP (\<sigma>, \<sigma>)"} & $\dn$ & @{text "id :: \<sigma> \<Rightarrow> \<sigma>"}\smallskip\\
2227
+ − 506
\multicolumn{3}{@ {\hspace{-4mm}}l}{function types:}\\
2233
+ − 507
@{text "ABS (\<sigma>\<^isub>1 \<Rightarrow> \<sigma>\<^isub>2, \<tau>\<^isub>1 \<Rightarrow> \<tau>\<^isub>2)"} & $\dn$ & @{text "REP (\<sigma>\<^isub>1, \<tau>\<^isub>1) \<singlearr> ABS (\<sigma>\<^isub>2, \<tau>\<^isub>2)"}\\
+ − 508
@{text "REP (\<sigma>\<^isub>1 \<Rightarrow> \<sigma>\<^isub>2, \<tau>\<^isub>1 \<Rightarrow> \<tau>\<^isub>2)"} & $\dn$ & @{text "ABS (\<sigma>\<^isub>1, \<tau>\<^isub>1) \<singlearr> REP (\<sigma>\<^isub>2, \<tau>\<^isub>2)"}\smallskip\\
2227
+ − 509
\multicolumn{3}{@ {\hspace{-4mm}}l}{equal type constructors:}\\
2232
+ − 510
@{text "ABS (\<sigma>s \<kappa>, \<tau>s \<kappa>)"} & $\dn$ & @{text "map_\<kappa> (ABS (\<sigma>s, \<tau>s))"}\\
+ − 511
@{text "REP (\<sigma>s \<kappa>, \<tau>s \<kappa>)"} & $\dn$ & @{text "map_\<kappa> (REP (\<sigma>s, \<tau>s))"}\smallskip\\
2227
+ − 512
\multicolumn{3}{@ {\hspace{-4mm}}l}{unequal type constructors:}\\
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diff
changeset
+ − 513
@{text "ABS (\<sigma>s \<kappa>, \<tau>s \<kappa>\<^isub>q)"} & $\dn$ & @{text "Abs_\<kappa>\<^isub>q \<circ> (MAP(\<rho>s \<kappa>) (ABS (\<sigma>s', \<tau>s)))"}\\
8ddf1330f2ed
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diff
changeset
+ − 514
@{text "REP (\<sigma>s \<kappa>, \<tau>s \<kappa>\<^isub>q)"} & $\dn$ & @{text "(MAP(\<rho>s \<kappa>) (REP (\<sigma>s', \<tau>s))) \<circ> Rep_\<kappa>\<^isub>q"}
2247
+ − 515
\end{tabular}\hfill\numbered{ABSREP}
2227
+ − 516
\end{center}
2234
+ − 517
%
2232
+ − 518
\noindent
2269
+ − 519
where in the last two clauses we have that the type @{text "\<alpha>s
2238
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diff
changeset
+ − 520
\<kappa>\<^isub>q"} is the quotient of the raw type @{text "\<rho>s \<kappa>"} (for example
2237
+ − 521
@{text "int"} and @{text "nat \<times> nat"}, or @{text "\<alpha> fset"} and @{text "\<alpha>
+ − 522
list"}). The quotient construction ensures that the type variables in @{text
2247
+ − 523
"\<rho>s"} must be among the @{text "\<alpha>s"}. The @{text "\<sigma>s'"} are given by the
2238
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diff
changeset
+ − 524
matchers for the @{text "\<alpha>s"} when matching @{text "\<rho>s \<kappa>"} against
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
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diff
changeset
+ − 525
@{text "\<sigma>s \<kappa>"}. The
2237
+ − 526
function @{text "MAP"} calculates an \emph{aggregate map-function} for a raw
+ − 527
type as follows:
+ − 528
%
2227
+ − 529
\begin{center}
2238
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diff
changeset
+ − 530
\begin{tabular}{rcl}
2237
+ − 531
@{text "MAP' (\<alpha>)"} & $\dn$ & @{text "a\<^sup>\<alpha>"}\\
2238
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diff
changeset
+ − 532
@{text "MAP' (\<kappa>)"} & $\dn$ & @{text "id :: \<kappa> \<Rightarrow> \<kappa>"}\\
2232
+ − 533
@{text "MAP' (\<sigma>s \<kappa>)"} & $\dn$ & @{text "map_\<kappa> (MAP'(\<sigma>s))"}\smallskip\\
2233
+ − 534
@{text "MAP (\<sigma>)"} & $\dn$ & @{text "\<lambda>as. MAP'(\<sigma>)"}
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
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diff
changeset
+ − 535
\end{tabular}
2227
+ − 536
\end{center}
2237
+ − 537
%
2232
+ − 538
\noindent
2252
+ − 539
In this definition we rely on the fact that we can interpret type-variables @{text \<alpha>} as
2238
8ddf1330f2ed
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diff
changeset
+ − 540
term variables @{text a}. In the last clause we build an abstraction over all
2247
+ − 541
term-variables inside map-function generated by the auxiliary function
2238
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completed proof and started section about respectfulness and preservation
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diff
changeset
+ − 542
@{text "MAP'"}.
2247
+ − 543
The need of aggregate map-functions can be seen in cases where we build quotients,
+ − 544
say @{text "(\<alpha>, \<beta>) \<kappa>\<^isub>q"}, out of compound raw types, say @{text "(\<alpha> list) \<times> \<beta>"}.
+ − 545
In this case @{text MAP} generates the
+ − 546
aggregate map-function:
2232
+ − 547
2233
+ − 548
@{text [display, indent=10] "\<lambda>a b. map_prod (map a) b"}
+ − 549
+ − 550
\noindent
2269
+ − 551
which we need in order to define the aggregate abstraction and representation
+ − 552
functions.
2238
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diff
changeset
+ − 553
2247
+ − 554
To see how these definitions pan out in practise, let us return to our
+ − 555
example about @{term "concat"} and @{term "fconcat"}, where we have the raw type
+ − 556
@{text "(\<alpha> list) list \<Rightarrow> \<alpha> list"} and the quotient type @{text "(\<alpha> fset) fset \<Rightarrow> \<alpha>
+ − 557
fset"}. Feeding them into @{text ABS} gives us (after some @{text "\<beta>"}-simplifications)
+ − 558
the abstraction function
2233
+ − 559
+ − 560
@{text [display, indent=10] "(map (map id \<circ> Rep_fset) \<circ> Rep_fset) \<singlearr> Abs_fset \<circ> map id"}
+ − 561
+ − 562
\noindent
2247
+ − 563
In our implementation we further
+ − 564
simplify this function by rewriting with the usual laws about @{text
2238
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diff
changeset
+ − 565
"map"}s and @{text "id"}, namely @{term "map id = id"} and @{text "f \<circ> id =
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 566
id \<circ> f = f"}. This gives us the abstraction function
2237
+ − 567
2233
+ − 568
@{text [display, indent=10] "(map Rep_fset \<circ> Rep_fset) \<singlearr> Abs_fset"}
+ − 569
+ − 570
\noindent
+ − 571
which we can use for defining @{term "fconcat"} as follows
+ − 572
+ − 573
@{text [display, indent=10] "\<Union> \<equiv> ((map Rep_fset \<circ> Rep_fset) \<singlearr> Abs_fset) flat"}
2232
+ − 574
2237
+ − 575
\noindent
2247
+ − 576
Note that by using the operator @{text "\<singlearr>"} and special clauses
+ − 577
for function types in \eqref{ABSREP}, we do not have to
2252
+ − 578
distinguish between arguments and results, but can deal with them uniformly.
+ − 579
Consequently, all definitions in the quotient package
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 580
are of the general form
2188
+ − 581
2237
+ − 582
@{text [display, indent=10] "c \<equiv> ABS (\<sigma>, \<tau>) t"}
2227
+ − 583
2237
+ − 584
\noindent
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 585
where @{text \<sigma>} is the type of the definiens @{text "t"} and @{text "\<tau>"} the
2247
+ − 586
type of the defined quotient constant @{text "c"}. This data can be easily
+ − 587
generated from the declaration given by the user.
2252
+ − 588
To increase the confidence in this way of making definitions, we can prove
2247
+ − 589
that the terms involved are all typable.
2227
+ − 590
+ − 591
\begin{lemma}
+ − 592
If @{text "ABS (\<sigma>, \<tau>)"} returns some abstraction function @{text "Abs"}
+ − 593
and @{text "REP (\<sigma>, \<tau>)"} some representation function @{text "Rep"},
+ − 594
then @{text "Abs"} is of type @{text "\<sigma> \<Rightarrow> \<tau>"} and @{text "Rep"} of type
+ − 595
@{text "\<tau> \<Rightarrow> \<sigma>"}.
+ − 596
\end{lemma}
2233
+ − 597
2237
+ − 598
\begin{proof}
2269
+ − 599
By mutual induction and analysing the definitions of @{text "ABS"}, @{text "REP"}
2247
+ − 600
and @{text "MAP"}. The cases of equal types and function types are
+ − 601
straightforward (the latter follows from @{text "\<singlearr>"} having the
+ − 602
type @{text "(\<alpha> \<Rightarrow> \<beta>) \<Rightarrow> (\<gamma> \<Rightarrow> \<delta>) \<Rightarrow> (\<beta> \<Rightarrow> \<gamma>) \<Rightarrow> (\<alpha> \<Rightarrow> \<delta>)"}). In case of equal type
+ − 603
constructors we can observe that a map-function after applying the functions
+ − 604
@{text "ABS (\<sigma>s, \<tau>s)"} produces a term of type @{text "\<sigma>s \<kappa> \<Rightarrow> \<tau>s \<kappa>"}. The
+ − 605
interesting case is the one with unequal type constructors. Since we know
+ − 606
the quotient is between @{text "\<alpha>s \<kappa>\<^isub>q"} and @{text "\<rho>s \<kappa>"}, we have
+ − 607
that @{text "Abs_\<kappa>\<^isub>q"} is of type @{text "\<rho>s \<kappa> \<Rightarrow> \<alpha>s
+ − 608
\<kappa>\<^isub>q"}. This type can be more specialised to @{text "\<rho>s[\<tau>s] \<kappa> \<Rightarrow> \<tau>s
+ − 609
\<kappa>\<^isub>q"} where the type variables @{text "\<alpha>s"} are instantiated with the
+ − 610
@{text "\<tau>s"}. The complete type can be calculated by observing that @{text
+ − 611
"MAP (\<rho>s \<kappa>)"}, after applying the functions @{text "ABS (\<sigma>s', \<tau>s)"} to it,
+ − 612
returns a term of type @{text "\<rho>s[\<sigma>s'] \<kappa> \<Rightarrow> \<rho>s[\<tau>s] \<kappa>"}. This type is
+ − 613
equivalent to @{text "\<sigma>s \<kappa> \<Rightarrow> \<rho>s[\<tau>s] \<kappa>"}, which we just have to compose with
+ − 614
@{text "\<rho>s[\<tau>s] \<kappa> \<Rightarrow> \<tau>s \<kappa>\<^isub>q"} according to the type of @{text "\<circ>"}.\qed
2237
+ − 615
\end{proof}
+ − 616
+ − 617
\noindent
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 618
The reader should note that this lemma fails for the abstraction and representation
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 619
functions used, for example, in Homeier's quotient package.
2188
+ − 620
*}
+ − 621
2252
+ − 622
section {* Respectfulness and Preservation \label{sec:resp} *}
2188
+ − 623
+ − 624
text {*
2247
+ − 625
The main point of the quotient package is to automatically ``lift'' theorems
+ − 626
involving constants over the raw type to theorems involving constants over
+ − 627
the quotient type. Before we can describe this lift process, we need to impose
+ − 628
some restrictions. The reason is that even if definitions for all raw constants
+ − 629
can be given, \emph{not} all theorems can be actually be lifted. Most notably is
+ − 630
the bound variable function, that is the constant @{text bn}, defined for
+ − 631
raw lambda-terms as follows
2188
+ − 632
2247
+ − 633
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
2252
+ − 634
@{text "bn (x) \<equiv> \<emptyset>"}\hspace{4mm}
+ − 635
@{text "bn (t\<^isub>1 t\<^isub>2) \<equiv> bn (t\<^isub>1) \<union> bn (t\<^isub>2)"}\hspace{4mm}
2247
+ − 636
@{text "bn (\<lambda>x. t) \<equiv> {x} \<union> bn (t)"}
+ − 637
\end{isabelle}
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 638
2247
+ − 639
\noindent
+ − 640
This constant just does not respect @{text "\<alpha>"}-equivalence and as
+ − 641
consequently no theorem involving this constant can be lifted to @{text
+ − 642
"\<alpha>"}-equated lambda terms. Homeier formulates the restrictions in terms of
+ − 643
the properties of \emph{respectfullness} and \emph{preservation}. We have
2258
+ − 644
to slightly extend Homeier's definitions in order to deal with quotient
2247
+ − 645
compositions.
+ − 646
+ − 647
To formally define what respectfulness is, we have to first define
+ − 648
the notion of \emph{aggregate equivalence relations}.
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 649
2258
+ − 650
TBD
+ − 651
+ − 652
\begin{itemize}
+ − 653
\item @{text "REL(\<alpha>\<^isub>1, \<alpha>\<^isub>2)"} = @{text "op ="}
+ − 654
\item @{text "REL(\<sigma>, \<sigma>)"} = @{text "op ="}
+ − 655
\item @{text "REL((\<sigma>\<^isub>1,\<dots>,\<sigma>\<^isub>n))\<kappa>, (\<tau>\<^isub>1,\<dots>,\<tau>\<^isub>n))\<kappa>)"} = @{text "(rel \<kappa>) (REL(\<sigma>\<^isub>1,\<tau>\<^isub>1)) \<dots> (REL(\<sigma>\<^isub>n,\<tau>\<^isub>n))"}
+ − 656
\item @{text "REL((\<sigma>\<^isub>1,\<dots>,\<sigma>\<^isub>n))\<kappa>\<^isub>1, (\<tau>\<^isub>1,\<dots>,\<tau>\<^isub>m))\<kappa>\<^isub>2)"} = @{text "(rel \<kappa>\<^isub>1) (REL(\<rho>\<^isub>1,\<nu>\<^isub>1) \<dots> (REL(\<rho>\<^isub>p,\<nu>\<^isub>p) OOO Eqv_\<kappa>\<^isub>2"} provided @{text "\<eta> \<kappa>\<^isub>2 = (\<alpha>\<^isub>1\<dots>\<alpha>\<^isub>p)\<kappa>\<^isub>1 \<and> \<exists>s. s(\<sigma>s\<kappa>\<^isub>1)=\<rho>s\<kappa>\<^isub>1 \<and> s(\<tau>s\<kappa>\<^isub>2)=\<nu>s\<kappa>\<^isub>2"}
+ − 657
\end{itemize}
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 658
2188
+ − 659
class returned by this constant depends only on the equivalence
2207
+ − 660
classes of the arguments applied to the constant. To automatically
+ − 661
lift a theorem that talks about a raw constant, to a theorem about
+ − 662
the quotient type a respectfulness theorem is required.
+ − 663
+ − 664
A respectfulness condition for a constant can be expressed in
+ − 665
terms of an aggregate relation between the constant and itself,
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 666
for example the respectfullness for @{text "append"}
2188
+ − 667
can be stated as:
+ − 668
2258
+ − 669
@{text [display, indent=10] "(\<approx>\<^bsub>list\<^esub> \<doublearr> \<approx>\<^bsub>list\<^esub> \<doublearr> \<approx>\<^bsub>list\<^esub>) append append"}
2182
+ − 670
2190
+ − 671
\noindent
2228
+ − 672
Which after unfolding the definition of @{term "op ===>"} is equivalent to:
2188
+ − 673
2228
+ − 674
@{thm [display, indent=10] append_rsp_unfolded[no_vars]}
2188
+ − 675
2228
+ − 676
\noindent An aggregate relation is defined in terms of relation
+ − 677
composition, so we define it first:
2188
+ − 678
2258
+ − 679
2188
+ − 680
2207
+ − 681
The aggregate relation for an aggregate raw type and quotient type
+ − 682
is defined as:
2188
+ − 683
+ − 684
2207
+ − 685
Again, the last case is novel, so lets look at the example of
+ − 686
respectfullness for @{term concat}. The statement according to
+ − 687
the definition above is:
2190
+ − 688
2228
+ − 689
@{thm [display, indent=10] concat_rsp[no_vars]}
2189
+ − 690
2190
+ − 691
\noindent
+ − 692
By unfolding the definition of relation composition and relation map
+ − 693
we can see the equivalent statement just using the primitive list
+ − 694
equivalence relation:
+ − 695
2228
+ − 696
@{thm [display, indent=10] concat_rsp_unfolded[of "a" "a'" "b'" "b", no_vars]}
2189
+ − 697
2190
+ − 698
The statement reads that, for any lists of lists @{term a} and @{term b}
+ − 699
if there exist intermediate lists of lists @{term "a'"} and @{term "b'"}
+ − 700
such that each element of @{term a} is in the relation with an appropriate
+ − 701
element of @{term a'}, @{term a'} is in relation with @{term b'} and each
+ − 702
element of @{term b'} is in relation with the appropriate element of
+ − 703
@{term b}.
2189
+ − 704
+ − 705
2228
+ − 706
Sometimes a non-lifted polymorphic constant is instantiated to a
+ − 707
type being lifted. For example take the @{term "op #"} which inserts
+ − 708
an element in a list of pairs of natural numbers. When the theorem
+ − 709
is lifted, the pairs of natural numbers are to become integers, but
+ − 710
the head constant is still supposed to be the head constant, just
+ − 711
with a different type. To be able to lift such theorems
+ − 712
automatically, additional theorems provided by the user are
+ − 713
necessary, we call these \emph{preservation} theorems following
+ − 714
Homeier's naming.
2196
+ − 715
+ − 716
To lift theorems that talk about insertion in lists of lifted types
+ − 717
we need to know that for any quotient type with the abstraction and
+ − 718
representation functions @{text "Abs"} and @{text Rep} we have:
+ − 719
2228
+ − 720
@{thm [display, indent=10] (concl) cons_prs[no_vars]}
2196
+ − 721
+ − 722
This is not enough to lift theorems that talk about quotient compositions.
+ − 723
For some constants (for example empty list) it is possible to show a
+ − 724
general compositional theorem, but for @{term "op #"} it is necessary
+ − 725
to show that it respects the particular quotient type:
+ − 726
2228
+ − 727
@{thm [display, indent=10] insert_preserve2[no_vars]}
2190
+ − 728
2258
+ − 729
{\it Composition of Quotient theorems}
2189
+ − 730
2191
+ − 731
Given two quotients, one of which quotients a container, and the
+ − 732
other quotients the type in the container, we can write the
2193
+ − 733
composition of those quotients. To compose two quotient theorems
2207
+ − 734
we compose the relations with relation composition as defined above
+ − 735
and the abstraction and relation functions are the ones of the sub
+ − 736
quotients composed with the usual function composition.
+ − 737
The @{term "Rep"} and @{term "Abs"} functions that we obtain agree
+ − 738
with the definition of aggregate Abs/Rep functions and the
2193
+ − 739
relation is the same as the one given by aggregate relations.
+ − 740
This becomes especially interesting
2191
+ − 741
when we compose the quotient with itself, as there is no simple
+ − 742
intermediate step.
+ − 743
2242
+ − 744
Lets take again the example of @{term flat}. To be able to lift
2207
+ − 745
theorems that talk about it we provide the composition quotient
2266
dcffc2f132c9
Qpaper / Clarify the typing system and composition of quotients issue.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 746
theorem which allows quotienting inside the container:
2254
+ − 747
2266
dcffc2f132c9
Qpaper / Clarify the typing system and composition of quotients issue.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 748
If @{term R} is an equivalence relation and @{term "Quotient R Abs Rep"}
dcffc2f132c9
Qpaper / Clarify the typing system and composition of quotients issue.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 749
then
dcffc2f132c9
Qpaper / Clarify the typing system and composition of quotients issue.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 750
dcffc2f132c9
Qpaper / Clarify the typing system and composition of quotients issue.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 751
@{text [display, indent=10] "Quotient (list_rel R \<circ>\<circ>\<circ> \<approx>\<^bsub>list\<^esub>) (abs_fset \<circ> map Abs) (map Rep o rep_fset)"}
2188
+ − 752
2254
+ − 753
\noindent
+ − 754
this theorem will then instantiate the quotients needed in the
+ − 755
injection and cleaning proofs allowing the lifting procedure to
+ − 756
proceed in an unchanged way.
+ − 757
2192
+ − 758
*}
+ − 759
2256
+ − 760
section {* Lifting of Theorems\label{sec:lift} *}
1978
+ − 761
2194
+ − 762
text {*
2271
+ − 763
+ − 764
The core of a quotient package lifts an original theorem to a lifted
+ − 765
version. We will perform this operation in three phases. In the
+ − 766
following we call these phases \emph{regularization},
+ − 767
\emph{injection} and \emph{cleaning} following the names used in
+ − 768
Homeier's HOL4 implementation.
+ − 769
+ − 770
Regularization is supposed to change the quantifications and abstractions
+ − 771
in the theorem to quantification over variables that respect the relation
+ − 772
(definition \ref{def:respects}). Injection is supposed to add @{term Rep}
+ − 773
and @{term Abs} of appropriate types in front of constants and variables
+ − 774
of the raw type so that they can be replaced by the ones that include the
+ − 775
quotient type. Cleaning rewrites the obtained injected theorem with
+ − 776
preservation rules obtaining the desired goal theorem.
2193
+ − 777
2271
+ − 778
Most quotient packages take only an original theorem involving raw
+ − 779
types and lift it. The procedure in our package takes both an
+ − 780
original theorem involving raw types and a statement of the theorem
+ − 781
that it is supposed to produce. To simplify the use of the quotient
+ − 782
package we additionally provide an automated statement translation
+ − 783
mechanism which can produce the latter automatically given a list of
+ − 784
quotient types. It is possible that a user wants to lift only some
+ − 785
occurrences of a raw type. In this case the user specifies the
+ − 786
complete lifted goal instead of using the automated mechanism.
2193
+ − 787
2271
+ − 788
In the following we will first define the statement of the
+ − 789
regularized theorem based on the original theorem and the goal
+ − 790
theorem. Then we define the statement of the injected theorem, based
+ − 791
on the regularized theorem and the goal. We then show the 3 proofs,
+ − 792
all three can be performed independently from each other.
2197
+ − 793
2251
+ − 794
We define the function @{text REG}, which takes the statements
2207
+ − 795
of the raw theorem and the lifted theorem (both as terms) and
+ − 796
returns the statement of the regularized version. The intuition
+ − 797
behind this function is that it replaces quantifiers and
+ − 798
abstractions involving raw types by bounded ones, and equalities
+ − 799
involving raw types are replaced by appropriate aggregate
2251
+ − 800
equivalence relations. It is defined as follows:
1994
+ − 801
2244
+ − 802
\begin{center}
2273
+ − 803
\begin{longtable}{rcl}
+ − 804
\multicolumn{3}{@ {}l}{abstractions:}\smallskip\\
+ − 805
@{text "REG (\<lambda>x\<^sup>\<sigma>. t, \<lambda>x\<^sup>\<tau>. s)"} & $\dn$ &
+ − 806
$\begin{cases}
+ − 807
@{text "\<lambda>x\<^sup>\<sigma>. REG (t, s)"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
+ − 808
@{text "\<lambda>x\<^sup>\<sigma> \<in> Respects (REL (\<sigma>, \<tau>)). REG (t, s)"}
+ − 809
\end{cases}$\smallskip\\
+ − 810
\multicolumn{3}{@ {}l}{universal quantifiers:}\\
+ − 811
@{text "REG (\<forall>x\<^sup>\<sigma>. t, \<forall>x\<^sup>\<tau>. s)"} & $\dn$ &
+ − 812
$\begin{cases}
+ − 813
@{text "\<forall>x\<^sup>\<sigma>. REG (t, s)"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
+ − 814
@{text "\<forall>x\<^sup>\<sigma> \<in> Respects (REL (\<sigma>, \<tau>)). REG (t, s)"}
+ − 815
\end{cases}$\smallskip\\
+ − 816
\multicolumn{3}{@ {}l}{equality:}\smallskip\\
+ − 817
@{text "REG (=\<^bsup>\<sigma>\<Rightarrow>\<sigma>\<Rightarrow>bool\<^esup>, =\<^bsup>\<tau>\<Rightarrow>\<tau>\<Rightarrow>bool\<^esup>)"} & $\dn$ &
+ − 818
$\begin{cases}
+ − 819
@{text "="} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
+ − 820
@{text "REL (\<sigma>, \<tau>)"}\\
+ − 821
\end{cases}$\\
+ − 822
\multicolumn{3}{@ {}l}{applications, variables and constants:}\\
2244
+ − 823
@{text "REG (t\<^isub>1 t\<^isub>2, s\<^isub>1 s\<^isub>2)"} & $\dn$ & @{text "REG (t\<^isub>1, s\<^isub>1) REG (t\<^isub>2, s\<^isub>2)"}\\
2273
+ − 824
@{text "REG (x\<^isub>1, x\<^isub>2)"} & $\dn$ & @{text "x\<^isub>1"}\\
+ − 825
@{text "REG (c\<^isub>1, c\<^isub>2)"} & $\dn$ & @{text "c\<^isub>1"}\\[-5mm]
+ − 826
\end{longtable}
2244
+ − 827
\end{center}
2273
+ − 828
%
+ − 829
\noindent
2230
+ − 830
In the above definition we omitted the cases for existential quantifiers
2207
+ − 831
and unique existential quantifiers, as they are very similar to the cases
+ − 832
for the universal quantifier.
+ − 833
Next we define the function @{text INJ} which takes the statement of
+ − 834
the regularized theorems and the statement of the lifted theorem both as
2230
+ − 835
terms and returns the statement of the injected theorem:
2198
+ − 836
2245
+ − 837
\begin{center}
+ − 838
\begin{tabular}{rcl}
2273
+ − 839
\multicolumn{3}{@ {\hspace{-4mm}}l}{abstractions:}\\
+ − 840
@{text "INJ (\<lambda>x. t :: \<sigma>, \<lambda>x. s :: \<tau>) "} & $\dn$ &
+ − 841
$\begin{cases}
+ − 842
@{text "\<lambda>x. INJ (t, s)"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
+ − 843
@{text "REP (\<sigma>, \<tau>) (ABS (\<sigma>, \<tau>) (\<lambda>x. INJ (t, s)))"}
+ − 844
\end{cases}$\\
+ − 845
@{text "INJ (\<lambda>x \<in> R. t :: \<sigma>, \<lambda>x. s :: \<tau>) "} & $\dn$
+ − 846
& @{text "REP (\<sigma>, \<tau>) (ABS (\<sigma>, \<tau>) (\<lambda>x \<in> R. INJ (t, s)))"}\smallskip\\
+ − 847
\multicolumn{3}{@ {\hspace{-4mm}}l}{universal quantifiers:}\\
+ − 848
@{text "INJ (\<forall> t, \<forall> s) "} & $\dn$ & @{text "\<forall> INJ (t, s)"}\\
+ − 849
@{text "INJ (\<forall> t \<in> R, \<forall> s) "} & $\dn$ & @{text "\<forall> INJ (t, s) \<in> R"}\smallskip\\
+ − 850
\multicolumn{3}{@ {\hspace{-4mm}}l}{applications, variables and constants:}\smallskip\\
2245
+ − 851
@{text "INJ (t\<^isub>1 t\<^isub>2, s\<^isub>1 s\<^isub>2) "} & $\dn$ & @{text " INJ (t\<^isub>1, s\<^isub>1) INJ (t\<^isub>2, s\<^isub>2)"}\\
2273
+ − 852
@{text "INJ (x\<^isub>1\<^sup>\<sigma>, x\<^isub>2\<^sup>\<tau>) "} & $\dn$ &
+ − 853
$\begin{cases}
+ − 854
@{text "x\<^isub>1"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
+ − 855
@{text "REP (\<sigma>, \<tau>) (ABS (\<sigma>, \<tau>) x\<^isub>1)"}\\
+ − 856
\end{cases}$\\
+ − 857
@{text "INJ (c\<^isub>1\<^sup>\<sigma>, c\<^isub>2\<^sup>\<tau>) "} & $\dn$ &
+ − 858
$\begin{cases}
+ − 859
@{text "c\<^isub>1"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
+ − 860
@{text "REP (\<sigma>, \<tau>) (ABS (\<sigma>, \<tau>) c\<^isub>1)"}\\
+ − 861
\end{cases}$\\
2245
+ − 862
\end{tabular}
+ − 863
\end{center}
2198
+ − 864
2271
+ − 865
\noindent where the cases for existential quantifiers and unique existential
+ − 866
quantifiers have been omitted for clarity; are similar to universal quantifier.
2208
+ − 867
2271
+ − 868
We can now define the subgoals that will imply the lifted theorem. Given
+ − 869
the statement of the original theorem @{term t} and the statement of the
+ − 870
goal @{term g} the regularization subgoal is @{term "t \<longrightarrow> REG(t, g)"},
+ − 871
the injection subgoal is @{term "REG(t, g) = INJ(REG(t, g), g)"} and the
+ − 872
cleaning subgoal is @{term "INJ(REG(t, g), g) = g"}. We will now describe
+ − 873
the three tactics provided for these three subgoals.
2208
+ − 874
+ − 875
The injection and cleaning subgoals are always solved if the appropriate
+ − 876
respectfulness and preservation theorems are given. It is not the case
+ − 877
with regularization; sometimes a theorem given by the user does not
+ − 878
imply a regularized version and a stronger one needs to be proved. This
2242
+ − 879
is outside of the scope of the quotient package, so such obligations are
2273
+ − 880
left to the user. Take a simple statement for integers @{text "0 \<noteq> 1"}.
+ − 881
It does not follow from the fact that @{text "(0, 0) \<noteq> (1, 0)"} because
2242
+ − 882
of regularization. The raw theorem only shows that particular items in the
+ − 883
equivalence classes are not equal. A more general statement saying that
+ − 884
the classes are not equal is necessary.
2261
+ − 885
2271
+ − 886
In the proof of the regularization subgoal we always start with an implication.
2209
+ − 887
Isabelle provides a set of \emph{mono} rules, that are used to split implications
2230
+ − 888
of similar statements into simpler implication subgoals. These are enhanced
2249
+ − 889
with special quotient theorem in the regularization proof. Below we only show
2209
+ − 890
the versions for the universal quantifier. For the existential quantifier
2242
+ − 891
and abstraction they are analogous.
2199
+ − 892
2209
+ − 893
First, bounded universal quantifiers can be removed on the right:
2199
+ − 894
2249
+ − 895
@{thm [display, indent=10] ball_reg_right_unfolded[no_vars]}
2206
+ − 896
2209
+ − 897
They can be removed anywhere if the relation is an equivalence relation:
+ − 898
2265
+ − 899
@{thm [display, indent=10] (concl) ball_reg_eqv[no_vars]}
2209
+ − 900
2273
+ − 901
And finally it can be removed anywhere if @{term R\<^isub>2} is an equivalence relation:
2231
+ − 902
2273
+ − 903
@{thm [display, indent=10] (concl) ball_reg_eqv_range[of R\<^isub>1 R\<^isub>2, no_vars]}
2209
+ − 904
2242
+ − 905
The last theorem is new in comparison with Homeier's package. There the
2231
+ − 906
injection procedure would be used to prove goals with such shape, and there
2242
+ − 907
the equivalence assumption would be used. We use the above theorem directly
+ − 908
also for composed relations where the range type is a type for which we know an
2231
+ − 909
equivalence theorem. This allows separating regularization from injection.
2206
+ − 910
2211
+ − 911
The injection proof starts with an equality between the regularized theorem
+ − 912
and the injected version. The proof again follows by the structure of the
2242
+ − 913
two terms, and is defined for a goal being a relation between these two terms.
2199
+ − 914
2211
+ − 915
\begin{itemize}
+ − 916
\item For two constants, an appropriate constant respectfullness assumption is used.
2242
+ − 917
\item For two variables, we use the assumptions proved in regularization.
2211
+ − 918
\item For two abstractions, they are eta-expanded and beta-reduced.
2271
+ − 919
\item For two applications, if the right side is an application of
+ − 920
@{term Rep} to an @{term Abs} and @{term "Quotient R Rep Abs"} we
+ − 921
can reduce the injected pair using the theorem:
+ − 922
+ − 923
@{term [display, indent=10] "R x y \<longrightarrow> R x (Rep (Abs y))"}
+ − 924
+ − 925
otherwise we introduce an appropriate relation between the subterms
+ − 926
and continue with two subgoals using the lemma:
+ − 927
2273
+ − 928
@{text [display, indent=10] "(R\<^isub>1 \<doublearr> R\<^isub>2) f g \<longrightarrow> R\<^isub>1 x y \<longrightarrow> R\<^isub>2 (f x) (g y)"}
2271
+ − 929
2211
+ − 930
\end{itemize}
2199
+ − 931
2271
+ − 932
The cleaning subgoal has been defined in such a way that
+ − 933
establishing the goal theorem now consists only on rewriting the
+ − 934
injected theorem with the preservation theorems and quotient
+ − 935
definitions. First for all lifted constants, their definitions
+ − 936
are used to fold the @{term Rep} with the raw constant. Next for
+ − 937
all lambda abstractions and quantifications the lambda and
+ − 938
quantifier preservation theorems are used to replace the
+ − 939
variables that include raw types with respects by quantification
+ − 940
over variables that include quotient types. We show here only
+ − 941
the lambda preservation theorem; assuming
2273
+ − 942
@{term "Quotient R\<^isub>1 Abs\<^isub>1 Rep\<^isub>1"} and @{term "Quotient R\<^isub>2 Abs\<^isub>2 Rep\<^isub>2"}
+ − 943
hold, we have:
2211
+ − 944
2271
+ − 945
@{thm [display, indent=10] (concl) lambda_prs[no_vars]}
2199
+ − 946
2243
+ − 947
\noindent
2271
+ − 948
holds. Next relations over lifted types are folded to equality.
+ − 949
The following theorem has been shown in Homeier~\cite{Homeier05}:
2211
+ − 950
2271
+ − 951
@{thm [display, indent=10] (concl) Quotient_rel_rep[no_vars]}
2199
+ − 952
2271
+ − 953
\noindent
+ − 954
Finally the user given preservation theorems, that allow using
+ − 955
higher level operations and containers of types being lifted.
+ − 956
We show the preservation theorem for @{term map}. Again assuming
2273
+ − 957
that @{term "Quotient R\<^isub>1 Abs\<^isub>1 Rep\<^isub>1"} and @{term "Quotient R\<^isub>2 Abs\<^isub>2 Rep\<^isub>2"} hold,
2271
+ − 958
we have:
2212
+ − 959
2273
+ − 960
@{thm [display, indent=10] (concl) map_prs(1)[of R\<^isub>1 Abs\<^isub>1 Rep\<^isub>1 R\<^isub>2 Abs\<^isub>2 Rep\<^isub>2]}
2212
+ − 961
2246
+ − 962
*}
1994
+ − 963
+ − 964
section {* Examples *}
+ − 965
2210
+ − 966
(* Mention why equivalence *)
2206
+ − 967
2210
+ − 968
text {*
+ − 969
2239
+ − 970
In this section we will show, a complete interaction with the quotient package
2240
+ − 971
for defining the type of integers by quotienting pairs of natural numbers and
+ − 972
lifting theorems to integers. Our quotient package is fully compatible with
+ − 973
Isabelle type classes, but for clarity we will not use them in this example.
+ − 974
In a larger formalization of integers using the type class mechanism would
+ − 975
provide many algebraic properties ``for free''.
2210
+ − 976
2240
+ − 977
A user of our quotient package first needs to define a relation on
+ − 978
the raw type, by which the quotienting will be performed. We give
+ − 979
the same integer relation as the one presented in the introduction:
+ − 980
+ − 981
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
2273
+ − 982
\isacommand{fun}~~@{text "int_rel"}~~\isacommand{where}~~%
+ − 983
@{text "(m :: nat, n) int_rel (p, q) = (m + q = n + p)"}
2239
+ − 984
\end{isabelle}
2210
+ − 985
2239
+ − 986
\noindent
+ − 987
Next the quotient type is defined. This leaves a proof obligation that the
+ − 988
relation is an equivalence relation which is solved automatically using the
+ − 989
definitions:
2210
+ − 990
2240
+ − 991
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
2241
+ − 992
\isacommand{quotient\_type}~~@{text "int"}~~\isacommand{=}~~@{text "(nat \<times> nat)"}~~\isacommand{/}~~@{text "int_rel"}
2239
+ − 993
\end{isabelle}
2210
+ − 994
2239
+ − 995
\noindent
2210
+ − 996
The user can then specify the constants on the quotient type:
+ − 997
2240
+ − 998
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
+ − 999
\begin{tabular}{@ {}l}
2273
+ − 1000
\isacommand{quotient\_definition}~~@{text "0 :: int"}~~\isacommand{is}~~@{text "(0 :: nat, 0 :: nat)"}\\
+ − 1001
\isacommand{fun}~~@{text "plus_raw"}~~\isacommand{where}~~%
+ − 1002
@{text "plus_raw (m, n) (p, q) = (m + p :: nat, n + q :: nat)"}\\
+ − 1003
\isacommand{quotient\_definition}~~@{text "+ :: int \<Rightarrow> int \<Rightarrow> int"}~~%
+ − 1004
\isacommand{is}~~@{text "plus_raw"}\\
2240
+ − 1005
\end{tabular}
+ − 1006
\end{isabelle}
2210
+ − 1007
2240
+ − 1008
\noindent
2210
+ − 1009
Lets first take a simple theorem about addition on the raw level:
+ − 1010
2240
+ − 1011
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
2241
+ − 1012
\isacommand{lemma}~~@{text "plus_zero_raw: int_rel (plus_raw (0, 0) x) x"}
2240
+ − 1013
\end{isabelle}
2210
+ − 1014
2240
+ − 1015
\noindent
2210
+ − 1016
When the user tries to lift a theorem about integer addition, the respectfulness
+ − 1017
proof obligation is left, so let us prove it first:
+ − 1018
2240
+ − 1019
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
2273
+ − 1020
\isacommand{lemma}~~@{text "[quot_respect]:
+ − 1021
(int_rel \<doublearr> int_rel \<doublearr> int_rel) plus_raw plus_raw"}
2240
+ − 1022
\end{isabelle}
+ − 1023
+ − 1024
\noindent
2210
+ − 1025
Can be proved automatically by the system just by unfolding the definition
2273
+ − 1026
of @{text "\<doublearr>"}.
2230
+ − 1027
Now the user can either prove a lifted lemma explicitly:
2210
+ − 1028
2240
+ − 1029
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
+ − 1030
\isacommand{lemma}~~@{text "0 + (x :: int) = x"}~~\isacommand{by}~~@{text "lifting plus_zero_raw"}
+ − 1031
\end{isabelle}
2210
+ − 1032
2240
+ − 1033
\noindent
2210
+ − 1034
Or in this simple case use the automated translation mechanism:
+ − 1035
2240
+ − 1036
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
+ − 1037
\isacommand{thm}~~@{text "plus_zero_raw[quot_lifted]"}
+ − 1038
\end{isabelle}
2210
+ − 1039
2240
+ − 1040
\noindent
2210
+ − 1041
obtaining the same result.
+ − 1042
*}
2206
+ − 1043
2256
+ − 1044
section {* Conclusion and Related Work\label{sec:conc}*}
1978
+ − 1045
+ − 1046
text {*
2243
+ − 1047
2267
+ − 1048
The code of the quotient package and the examples described here are
+ − 1049
already included in the
2254
+ − 1050
standard distribution of Isabelle.\footnote{Available from
2237
+ − 1051
\href{http://isabelle.in.tum.de/}{http://isabelle.in.tum.de/}.} It is
+ − 1052
heavily used in Nominal Isabelle, which provides a convenient reasoning
+ − 1053
infrastructure for programming language calculi involving binders. Earlier
+ − 1054
versions of Nominal Isabelle have been used successfully in formalisations
+ − 1055
of an equivalence checking algorithm for LF \cite{UrbanCheneyBerghofer08},
+ − 1056
Typed Scheme~\cite{TobinHochstadtFelleisen08}, several calculi for
+ − 1057
concurrency \cite{BengtsonParow09} and a strong normalisation result for
+ − 1058
cut-elimination in classical logic \cite{UrbanZhu08}.
+ − 1059
2273
+ − 1060
Slotosch~\cite{Slotosch97} implemented a mechanism that automatically
2267
+ − 1061
defines quotient types for Isabelle/HOL. It did not include theorem lifting.
2273
+ − 1062
Harrison's quotient package~\cite{harrison-thesis} is the first one to
2267
+ − 1063
lift theorems, however only first order. There is work on quotient types in
+ − 1064
non-HOL based systems and logical frameworks, namely theory interpretations
+ − 1065
in PVS~\cite{PVS:Interpretations}, new types in MetaPRL~\cite{Nogin02}, or
+ − 1066
the use of setoids in Coq, with some higher order issues~\cite{ChicliPS02}.
2273
+ − 1067
Paulson shows a construction of quotients that does not require the
2267
+ − 1068
Hilbert Choice operator, again only first order~\cite{Paulson06}.
2273
+ − 1069
The closest to our package is the package for HOL4 by Homeier~\cite{Homeier05},
2267
+ − 1070
which is the first one to support lifting of higher order theorems.
2224
+ − 1071
+ − 1072
2267
+ − 1073
Our quotient package for the first time explore the notion of
+ − 1074
composition of quotients, which allows lifting constants like @{term
+ − 1075
"concat"} and theorems about it. We defined the composition of
+ − 1076
relations and showed examples of compositions of quotients which
+ − 1077
allows lifting polymorphic types with subtypes quotiented as well.
+ − 1078
We extended the notions of respectfullness and preservation;
+ − 1079
with quotient compositions there is more than one condition needed
+ − 1080
for a constant.
2224
+ − 1081
2267
+ − 1082
Our package is modularized, so that single definitions, single
+ − 1083
theorems or single respectfullness conditions etc can be added,
+ − 1084
which allows the use of the quotient package together with
+ − 1085
type-classes and locales. This has the advantage over packages
+ − 1086
requiring big lists as input for the user of being able to develop
+ − 1087
a theory progressively.
2224
+ − 1088
2267
+ − 1089
We allow lifting only some occurrences of quotiented types, which
2273
+ − 1090
is useful in Nominal Isabelle. The package can be used automatically with
2267
+ − 1091
an attribute, manually with separate tactics for parts of the lifting
+ − 1092
procedure, and programatically. Automated definitions of constants
+ − 1093
and respectfulness proof obligations are used in Nominal. Finally
+ − 1094
we streamlined and showed the detailed lifting procedure, which
+ − 1095
has not been presented before.
2263
+ − 1096
+ − 1097
\medskip
+ − 1098
\noindent
+ − 1099
{\bf Acknowledgements:} We would like to thank Peter Homeier for the
2273
+ − 1100
discussions about his HOL4 quotient package and explaining to us
+ − 1101
some its finer points in the implementation.
2263
+ − 1102
2224
+ − 1103
*}
+ − 1104
+ − 1105
2227
+ − 1106
1975
b1281a0051ae
added stub for quotient paper; call with isabelle make qpaper
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 1107
(*<*)
b1281a0051ae
added stub for quotient paper; call with isabelle make qpaper
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
+ − 1108
end
1978
+ − 1109
(*>*)