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(* How to change the notation for \<lbrakk> \<rbrakk> meta-level implications? *)
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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 ("_ OOO _" [53, 53] 52) and
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"op -->" (infix "\<rightarrow>" 100) and
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"==>" (infix "\<Rightarrow>" 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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An area where quotients are ubiquitous is reasoning about programming language
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calculi. A simple example 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 when one attempts to
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prove formally, for example, 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 to 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}, 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. } These functions relate elements in the
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existing type to ones in the new type and vice versa; they can be uniquely
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identified by their 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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\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}
+ − 183
+ − 184
\noindent
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Slightly more complicated is the definition of @{text "add"} having type
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+ − 186
@{typ "int \<Rightarrow> int \<Rightarrow> int"}. Its definition is as follows
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@{text [display, indent=10] "add n m \<equiv> Abs_int (add_pair (Rep_int n) (Rep_int m))"}
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\noindent
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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
+ − 194
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
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in case of quotienting lists to yield finite sets and the operator that
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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 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 case in case
+ − 214
of @{text "\<Union>"} we obtain the incorrect definition
+ − 215
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@{text [display, indent=10] "\<Union> S \<equiv> Abs_fset (flat (Rep_fset S))"}
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\noindent
+ − 219
where the right-hand side is not even typable! This problem can be remedied in the
+ − 220
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
+ − 223
definitions directly. The solution is that we need to build aggregate
+ − 224
representation and abstraction functions, which in case of @{text "\<Union>"}
+ − 225
generate the following definition
+ − 226
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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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+ − 229
\noindent
2223
+ − 230
where @{term map} is the usual mapping function for lists. In this paper we
2224
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will present a formal definition of our aggregate abstraction and
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representation functions (this definition was omitted in \cite{Homeier05}).
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They generate definitions, like the one above for @{text "\<Union>"},
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according to the type of the raw constant and the type
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of the quotient constant. This means we also have to extend the notions
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of \emph{aggregate equivalence relation}, \emph{respectfulness} and \emph{preservation}
2231
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from Homeier \cite{Homeier05}.
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We are also able to address the criticism by Paulson \cite{Paulson06} cited
+ − 240
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
+ − 242
quotient package is the first one that is modular so that it allows to lift
+ − 243
single theorems separately. This has the advantage for the user to develop a
+ − 244
formal theory interactively an a natural progression. A pleasing result of
+ − 245
the modularity is also that we are able to clearly specify what needs to be
+ − 246
done in the lifting process (this was only hinted at in \cite{Homeier05} and
+ − 247
implemented as a ``rough recipe'' in ML-code).
+ − 248
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The paper is organised as follows: Section \ref{sec:prelims} presents briefly
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some necessary preliminaries; Section \ref{sec:type} describes the definitions
+ − 252
of quotient types and shows how definition of constants can be made over
+ − 253
quotient types. \ldots
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*}
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section {* Preliminaries and General Quotient\label{sec:prelims} *}
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text {*
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In this section we present the definitions of a quotient that follow
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closely those given by Homeier.
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\begin{definition}[Quotient]
+ − 264
A relation $R$ with an abstraction function $Abs$
+ − 265
and a representation function $Rep$ is a \emph{quotient}
+ − 266
if and only if:
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\begin{enumerate}
+ − 269
\item @{thm (rhs1) Quotient_def[of "R", no_vars]}
+ − 270
\item @{thm (rhs2) Quotient_def[of "R", no_vars]}
+ − 271
\item @{thm (rhs3) Quotient_def[of "R", no_vars]}
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\end{enumerate}
+ − 273
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\end{definition}
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\begin{definition}[Relation map and function map]\\
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@{thm fun_rel_def[of "R1" "R2", no_vars]}\\
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@{thm fun_map_def[no_vars]}
+ − 279
\end{definition}
+ − 280
+ − 281
The main theorems for building higher order quotients is:
+ − 282
\begin{lemma}[Function Quotient]
+ − 283
If @{thm (prem 1) fun_quotient[no_vars]} and @{thm (prem 2) fun_quotient[no_vars]}
+ − 284
then @{thm (concl) fun_quotient[no_vars]}
+ − 285
\end{lemma}
+ − 286
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Higher Order Logic
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Types:
+ − 291
\begin{eqnarray}\nonumber
+ − 292
@{text "\<sigma> ::="} & @{text "\<alpha>"} & \textrm{(type variable)} \\ \nonumber
+ − 293
@{text "|"} & @{text "(\<sigma>,\<dots>,\<sigma>)\<kappa>"} & \textrm{(type construction)}
+ − 294
\end{eqnarray}
+ − 295
+ − 296
Terms:
+ − 297
\begin{eqnarray}\nonumber
+ − 298
@{text "t ::="} & @{text "x\<^isup>\<sigma>"} & \textrm{(variable)} \\ \nonumber
+ − 299
@{text "|"} & @{text "c\<^isup>\<sigma>"} & \textrm{(constant)} \\ \nonumber
+ − 300
@{text "|"} & @{text "t t"} & \textrm{(application)} \\ \nonumber
+ − 301
@{text "|"} & @{text "\<lambda>x\<^isup>\<sigma>. t"} & \textrm{(abstraction)} \\ \nonumber
+ − 302
\end{eqnarray}
+ − 303
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{\it Say more about containers / maping functions }
+ − 305
2237
+ − 306
Such maps for most common types (list, pair, sum,
+ − 307
option, \ldots) are described in Homeier, and we assume that @{text "map"}
+ − 308
is the function that returns a map for a given type.
+ − 309
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{\it say something about our use of @{text "\<sigma>s"}}
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+ − 311
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+ − 312
*}
+ − 313
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+ − 314
section {* Quotient Types and Quotient Definitions\label{sec:type} *}
1978
+ − 315
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+ − 316
text {*
2247
+ − 317
The first step in a quotient construction is to take a name for the new
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+ − 318
type, say @{text "\<kappa>\<^isub>q"}, and an equivalence relation, say @{text R},
2247
+ − 319
defined over a raw type, say @{text "\<sigma>"}. The type of the equivalence
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relation must be of type @{text "\<sigma> \<Rightarrow> \<sigma> \<Rightarrow> bool"}. The user-visible part of
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+ − 321
the declaration is therefore
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+ − 322
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+ − 323
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 324
\isacommand{quotient\_type}~~@{text "\<alpha>s \<kappa>\<^isub>q = \<sigma> / R"}
+ − 325
\end{isabelle}
+ − 326
+ − 327
\noindent
2237
+ − 328
and a proof that @{text "R"} is indeed an equivalence relation. Two concrete
+ − 329
examples are
+ − 330
+ − 331
+ − 332
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 333
\begin{tabular}{@ {}l}
+ − 334
\isacommand{quotient\_type}~~@{text "int = nat \<times> nat / \<approx>\<^bsub>nat \<times> nat\<^esub>"}\\
+ − 335
\isacommand{quotient\_type}~~@{text "\<alpha> fset = \<alpha> list / \<approx>\<^bsub>list\<^esub>"}
+ − 336
\end{tabular}
+ − 337
\end{isabelle}
+ − 338
+ − 339
\noindent
+ − 340
which introduce the type of integers and of finite sets using the
+ − 341
equivalence relations @{text "\<approx>\<^bsub>nat \<times> nat\<^esub>"} and @{text
+ − 342
"\<approx>\<^bsub>list\<^esub>"} defined earlier in \eqref{natpairequiv} and
2247
+ − 343
\eqref{listequiv}, respectively (the proofs about being equivalence
+ − 344
relations is omitted). Given this data, we declare internally
2237
+ − 345
the quotient types as
2234
+ − 346
+ − 347
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 348
\isacommand{typedef}~~@{text "\<alpha>s \<kappa>\<^isub>q = {c. \<exists>x. c = R x}"}
+ − 349
\end{isabelle}
+ − 350
+ − 351
\noindent
2247
+ − 352
where the right-hand side is the (non-empty) set of equivalence classes of
2237
+ − 353
@{text "R"}. The restriction in this declaration is that the type variables
+ − 354
in the raw type @{text "\<sigma>"} must be included in the type variables @{text
2247
+ − 355
"\<alpha>s"} declared for @{text "\<kappa>\<^isub>q"}. HOL will provide us with the following
2237
+ − 356
abstraction and representation functions having the type
2182
+ − 357
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+ − 358
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 359
@{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"}
+ − 360
\end{isabelle}
+ − 361
2235
+ − 362
\noindent
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They relate the new quotient type and equivalence classes of the raw
2235
+ − 364
type. However, as Homeier \cite{Homeier05} noted, it is much more convenient
+ − 365
to work with the following derived abstraction and representation functions
+ − 366
2234
+ − 367
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 368
@{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)"}
+ − 369
\end{isabelle}
+ − 370
+ − 371
\noindent
2235
+ − 372
on the expense of having to use Hilbert's choice operator @{text "\<epsilon>"} in the
2237
+ − 373
definition of @{text "Rep_\<kappa>\<^isub>q"}. These derived notions relate the
+ − 374
quotient type and the raw type directly, as can be seen from their type,
+ − 375
namely @{text "\<sigma> \<Rightarrow> \<alpha>s \<kappa>\<^isub>q"} and @{text "\<alpha>s \<kappa>\<^isub>q \<Rightarrow> \<sigma>"},
+ − 376
respectively. Given that @{text "R"} is an equivalence relation, the
+ − 377
following property
+ − 378
2234
+ − 379
+ − 380
@{text [display, indent=10] "Quotient R Abs_\<kappa>\<^isub>q Rep_\<kappa>\<^isub>q"}
+ − 381
+ − 382
\noindent
2247
+ − 383
holds (for the proof see \cite{Homeier05}) for every quotient type defined
+ − 384
as above.
2182
+ − 385
2247
+ − 386
The next step in a quotient construction is to introduce definitions of new constants
+ − 387
involving the quotient type. These definitions need to be given in terms of concepts
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+ − 388
of the raw type (remember this is the only way how to extend HOL
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+ − 389
with new definitions). For the user visible is the declaration
2235
+ − 390
+ − 391
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
2237
+ − 392
\isacommand{quotient\_definition}~~@{text "c :: \<tau>"}~~\isacommand{is}~~@{text "t :: \<sigma>"}
2235
+ − 393
\end{isabelle}
+ − 394
2237
+ − 395
\noindent
+ − 396
where @{text t} is the definiens (its type @{text \<sigma>} can always be inferred)
+ − 397
and @{text "c"} is the name of definiendum, whose type @{text "\<tau>"} needs to be
+ − 398
given explicitly (the point is that @{text "\<tau>"} and @{text "\<sigma>"} can only differ
2238
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+ − 399
in places where a quotient and raw type are involved). Two concrete examples are
2188
+ − 400
2237
+ − 401
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 402
\begin{tabular}{@ {}l}
+ − 403
\isacommand{quotient\_definition}~~@{text "0 :: int"}~~\isacommand{is}~~@{text "(0::nat, 0::nat)"}\\
+ − 404
\isacommand{quotient\_definition}~~@{text "\<Union> :: (\<alpha> fset) fset \<Rightarrow> \<alpha> fset"}~~%
+ − 405
\isacommand{is}~~@{text "flat"}
+ − 406
\end{tabular}
+ − 407
\end{isabelle}
+ − 408
+ − 409
\noindent
+ − 410
The first one declares zero for integers and the second the operator for
2238
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+ − 411
building unions of finite sets.
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+ − 412
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+ − 413
The problem for us is that from such declarations we need to derive proper
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+ − 414
definitions using the @{text "Abs"} and @{text "Rep"} functions for the
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+ − 415
quotient types involved. The data we rely on is the given quotient type
2247
+ − 416
@{text "\<tau>"} and the raw type @{text "\<sigma>"}. They allow us to define \emph{aggregate
+ − 417
abstraction} and \emph{representation functions} using the functions @{text "ABS (\<sigma>,
+ − 418
\<tau>)"} and @{text "REP (\<sigma>, \<tau>)"} whose clauses we given below. The idea behind
+ − 419
these two functions is to recursively descend into the raw types @{text \<sigma>} and
+ − 420
quotient types @{text \<tau>}, and generate the appropriate
2238
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+ − 421
@{text "Abs"} and @{text "Rep"} in places where the types differ. Therefore
2247
+ − 422
we generate just the identity whenever the types are equal. All clauses
2238
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changeset
+ − 423
are as follows:
2182
+ − 424
2227
+ − 425
\begin{center}
2238
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+ − 426
\begin{tabular}{rcl}
2227
+ − 427
\multicolumn{3}{@ {\hspace{-4mm}}l}{equal types:}\\
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+ − 428
@{text "ABS (\<sigma>, \<sigma>)"} & $\dn$ & @{text "id :: \<sigma> \<Rightarrow> \<sigma>"}\\
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+ − 429
@{text "REP (\<sigma>, \<sigma>)"} & $\dn$ & @{text "id :: \<sigma> \<Rightarrow> \<sigma>"}\smallskip\\
2227
+ − 430
\multicolumn{3}{@ {\hspace{-4mm}}l}{function types:}\\
2233
+ − 431
@{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)"}\\
+ − 432
@{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
+ − 433
\multicolumn{3}{@ {\hspace{-4mm}}l}{equal type constructors:}\\
2232
+ − 434
@{text "ABS (\<sigma>s \<kappa>, \<tau>s \<kappa>)"} & $\dn$ & @{text "map_\<kappa> (ABS (\<sigma>s, \<tau>s))"}\\
+ − 435
@{text "REP (\<sigma>s \<kappa>, \<tau>s \<kappa>)"} & $\dn$ & @{text "map_\<kappa> (REP (\<sigma>s, \<tau>s))"}\smallskip\\
2227
+ − 436
\multicolumn{3}{@ {\hspace{-4mm}}l}{unequal type constructors:}\\
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+ − 437
@{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)))"}\\
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+ − 438
@{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
+ − 439
\end{tabular}\hfill\numbered{ABSREP}
2227
+ − 440
\end{center}
2234
+ − 441
%
2232
+ − 442
\noindent
2237
+ − 443
where in the last two clauses we have that the quotient type @{text "\<alpha>s
2238
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+ − 444
\<kappa>\<^isub>q"} is the quotient of the raw type @{text "\<rho>s \<kappa>"} (for example
2237
+ − 445
@{text "int"} and @{text "nat \<times> nat"}, or @{text "\<alpha> fset"} and @{text "\<alpha>
+ − 446
list"}). The quotient construction ensures that the type variables in @{text
2247
+ − 447
"\<rho>s"} must be among the @{text "\<alpha>s"}. The @{text "\<sigma>s'"} are given by the
2238
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+ − 448
matchers for the @{text "\<alpha>s"} when matching @{text "\<rho>s \<kappa>"} against
8ddf1330f2ed
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+ − 449
@{text "\<sigma>s \<kappa>"}. The
2237
+ − 450
function @{text "MAP"} calculates an \emph{aggregate map-function} for a raw
+ − 451
type as follows:
+ − 452
%
2227
+ − 453
\begin{center}
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+ − 454
\begin{tabular}{rcl}
2237
+ − 455
@{text "MAP' (\<alpha>)"} & $\dn$ & @{text "a\<^sup>\<alpha>"}\\
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+ − 456
@{text "MAP' (\<kappa>)"} & $\dn$ & @{text "id :: \<kappa> \<Rightarrow> \<kappa>"}\\
2232
+ − 457
@{text "MAP' (\<sigma>s \<kappa>)"} & $\dn$ & @{text "map_\<kappa> (MAP'(\<sigma>s))"}\smallskip\\
2233
+ − 458
@{text "MAP (\<sigma>)"} & $\dn$ & @{text "\<lambda>as. MAP'(\<sigma>)"}
2238
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+ − 459
\end{tabular}
2227
+ − 460
\end{center}
2237
+ − 461
%
2232
+ − 462
\noindent
2233
+ − 463
In this definition we abuse the fact that we can interpret type-variables @{text \<alpha>} as
2238
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+ − 464
term variables @{text a}. In the last clause we build an abstraction over all
2247
+ − 465
term-variables inside map-function generated by the auxiliary function
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+ − 466
@{text "MAP'"}.
2247
+ − 467
The need of aggregate map-functions can be seen in cases where we build quotients,
+ − 468
say @{text "(\<alpha>, \<beta>) \<kappa>\<^isub>q"}, out of compound raw types, say @{text "(\<alpha> list) \<times> \<beta>"}.
+ − 469
In this case @{text MAP} generates the
+ − 470
aggregate map-function:
2232
+ − 471
2233
+ − 472
@{text [display, indent=10] "\<lambda>a b. map_prod (map a) b"}
+ − 473
+ − 474
\noindent
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changeset
+ − 475
which we need to define the aggregate abstraction and representation functions.
8ddf1330f2ed
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+ − 476
2247
+ − 477
To see how these definitions pan out in practise, let us return to our
+ − 478
example about @{term "concat"} and @{term "fconcat"}, where we have the raw type
+ − 479
@{text "(\<alpha> list) list \<Rightarrow> \<alpha> list"} and the quotient type @{text "(\<alpha> fset) fset \<Rightarrow> \<alpha>
+ − 480
fset"}. Feeding them into @{text ABS} gives us (after some @{text "\<beta>"}-simplifications)
+ − 481
the abstraction function
2233
+ − 482
+ − 483
@{text [display, indent=10] "(map (map id \<circ> Rep_fset) \<circ> Rep_fset) \<singlearr> Abs_fset \<circ> map id"}
+ − 484
+ − 485
\noindent
2247
+ − 486
In our implementation we further
+ − 487
simplify this function by rewriting with the usual laws about @{text
2238
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changeset
+ − 488
"map"}s and @{text "id"}, namely @{term "map id = id"} and @{text "f \<circ> id =
8ddf1330f2ed
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diff
changeset
+ − 489
id \<circ> f = f"}. This gives us the abstraction function
2237
+ − 490
2233
+ − 491
@{text [display, indent=10] "(map Rep_fset \<circ> Rep_fset) \<singlearr> Abs_fset"}
+ − 492
+ − 493
\noindent
+ − 494
which we can use for defining @{term "fconcat"} as follows
+ − 495
+ − 496
@{text [display, indent=10] "\<Union> \<equiv> ((map Rep_fset \<circ> Rep_fset) \<singlearr> Abs_fset) flat"}
2232
+ − 497
2237
+ − 498
\noindent
2247
+ − 499
Note that by using the operator @{text "\<singlearr>"} and special clauses
+ − 500
for function types in \eqref{ABSREP}, we do not have to
2238
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diff
changeset
+ − 501
distinguish between arguments and results: the representation and abstraction
8ddf1330f2ed
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diff
changeset
+ − 502
functions are just inverses of each other, which we can combine using
8ddf1330f2ed
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diff
changeset
+ − 503
@{text "\<singlearr>"} to deal uniformly with arguments of functions and
2247
+ − 504
their result. Consequently, all definitions in the quotient package
2238
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changeset
+ − 505
are of the general form
2188
+ − 506
2237
+ − 507
@{text [display, indent=10] "c \<equiv> ABS (\<sigma>, \<tau>) t"}
2227
+ − 508
2237
+ − 509
\noindent
2238
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+ − 510
where @{text \<sigma>} is the type of the definiens @{text "t"} and @{text "\<tau>"} the
2247
+ − 511
type of the defined quotient constant @{text "c"}. This data can be easily
+ − 512
generated from the declaration given by the user.
+ − 513
To increase the confidence of making correct definitions, we can prove
+ − 514
that the terms involved are all typable.
2227
+ − 515
+ − 516
\begin{lemma}
+ − 517
If @{text "ABS (\<sigma>, \<tau>)"} returns some abstraction function @{text "Abs"}
+ − 518
and @{text "REP (\<sigma>, \<tau>)"} some representation function @{text "Rep"},
+ − 519
then @{text "Abs"} is of type @{text "\<sigma> \<Rightarrow> \<tau>"} and @{text "Rep"} of type
+ − 520
@{text "\<tau> \<Rightarrow> \<sigma>"}.
+ − 521
\end{lemma}
2233
+ − 522
2237
+ − 523
\begin{proof}
2247
+ − 524
By induction and analysing the definitions of @{text "ABS"}, @{text "REP"}
+ − 525
and @{text "MAP"}. The cases of equal types and function types are
+ − 526
straightforward (the latter follows from @{text "\<singlearr>"} having the
+ − 527
type @{text "(\<alpha> \<Rightarrow> \<beta>) \<Rightarrow> (\<gamma> \<Rightarrow> \<delta>) \<Rightarrow> (\<beta> \<Rightarrow> \<gamma>) \<Rightarrow> (\<alpha> \<Rightarrow> \<delta>)"}). In case of equal type
+ − 528
constructors we can observe that a map-function after applying the functions
+ − 529
@{text "ABS (\<sigma>s, \<tau>s)"} produces a term of type @{text "\<sigma>s \<kappa> \<Rightarrow> \<tau>s \<kappa>"}. The
+ − 530
interesting case is the one with unequal type constructors. Since we know
+ − 531
the quotient is between @{text "\<alpha>s \<kappa>\<^isub>q"} and @{text "\<rho>s \<kappa>"}, we have
+ − 532
that @{text "Abs_\<kappa>\<^isub>q"} is of type @{text "\<rho>s \<kappa> \<Rightarrow> \<alpha>s
+ − 533
\<kappa>\<^isub>q"}. This type can be more specialised to @{text "\<rho>s[\<tau>s] \<kappa> \<Rightarrow> \<tau>s
+ − 534
\<kappa>\<^isub>q"} where the type variables @{text "\<alpha>s"} are instantiated with the
+ − 535
@{text "\<tau>s"}. The complete type can be calculated by observing that @{text
+ − 536
"MAP (\<rho>s \<kappa>)"}, after applying the functions @{text "ABS (\<sigma>s', \<tau>s)"} to it,
+ − 537
returns a term of type @{text "\<rho>s[\<sigma>s'] \<kappa> \<Rightarrow> \<rho>s[\<tau>s] \<kappa>"}. This type is
+ − 538
equivalent to @{text "\<sigma>s \<kappa> \<Rightarrow> \<rho>s[\<tau>s] \<kappa>"}, which we just have to compose with
+ − 539
@{text "\<rho>s[\<tau>s] \<kappa> \<Rightarrow> \<tau>s \<kappa>\<^isub>q"} according to the type of @{text "\<circ>"}.\qed
2237
+ − 540
\end{proof}
+ − 541
+ − 542
\noindent
2238
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+ − 543
The reader should note that this lemma fails for the abstraction and representation
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+ − 544
functions used, for example, in Homeier's quotient package.
2188
+ − 545
*}
+ − 546
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changeset
+ − 547
section {* Respectfulness and Preservation *}
2188
+ − 548
+ − 549
text {*
2247
+ − 550
The main point of the quotient package is to automatically ``lift'' theorems
+ − 551
involving constants over the raw type to theorems involving constants over
+ − 552
the quotient type. Before we can describe this lift process, we need to impose
+ − 553
some restrictions. The reason is that even if definitions for all raw constants
+ − 554
can be given, \emph{not} all theorems can be actually be lifted. Most notably is
+ − 555
the bound variable function, that is the constant @{text bn}, defined for
+ − 556
raw lambda-terms as follows
2188
+ − 557
2247
+ − 558
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ − 559
@{text "bn (x) \<equiv> \<emptyset>"}\hspace{5mm}
+ − 560
@{text "bn (t\<^isub>1 t\<^isub>2) \<equiv> bn (t\<^isub>1) \<union> bn (t\<^isub>2)"}\hspace{5mm}
+ − 561
@{text "bn (\<lambda>x. t) \<equiv> {x} \<union> bn (t)"}
+ − 562
\end{isabelle}
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 563
2247
+ − 564
\noindent
+ − 565
This constant just does not respect @{text "\<alpha>"}-equivalence and as
+ − 566
consequently no theorem involving this constant can be lifted to @{text
+ − 567
"\<alpha>"}-equated lambda terms. Homeier formulates the restrictions in terms of
+ − 568
the properties of \emph{respectfullness} and \emph{preservation}. We have
+ − 569
to slighlty extend Homeier's definitions in order to deal with quotient
+ − 570
compositions.
+ − 571
+ − 572
To formally define what respectfulness is, we have to first define
+ − 573
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
+ − 574
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 575
@{text [display] "GIVE DEFINITION HERE"}
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 576
2188
+ − 577
class returned by this constant depends only on the equivalence
2207
+ − 578
classes of the arguments applied to the constant. To automatically
+ − 579
lift a theorem that talks about a raw constant, to a theorem about
+ − 580
the quotient type a respectfulness theorem is required.
+ − 581
+ − 582
A respectfulness condition for a constant can be expressed in
+ − 583
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
+ − 584
for example the respectfullness for @{text "append"}
2188
+ − 585
can be stated as:
+ − 586
2238
8ddf1330f2ed
completed proof and started section about respectfulness and preservation
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 587
@{text [display, indent=10] "(R \<doublearr> R \<doublearr> R) append append"}
2182
+ − 588
2190
+ − 589
\noindent
2228
+ − 590
Which after unfolding the definition of @{term "op ===>"} is equivalent to:
2188
+ − 591
2228
+ − 592
@{thm [display, indent=10] append_rsp_unfolded[no_vars]}
2188
+ − 593
2228
+ − 594
\noindent An aggregate relation is defined in terms of relation
+ − 595
composition, so we define it first:
2188
+ − 596
+ − 597
\begin{definition}[Composition of Relations]
2190
+ − 598
@{abbrev "rel_conj R1 R2"} where @{text OO} is the predicate
+ − 599
composition @{thm pred_compI[no_vars]}
2188
+ − 600
\end{definition}
+ − 601
2207
+ − 602
The aggregate relation for an aggregate raw type and quotient type
+ − 603
is defined as:
2188
+ − 604
+ − 605
\begin{itemize}
2207
+ − 606
\item @{text "REL(\<alpha>\<^isub>1, \<alpha>\<^isub>2)"} = @{text "op ="}
+ − 607
\item @{text "REL(\<sigma>, \<sigma>)"} = @{text "op ="}
+ − 608
\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))"}
+ − 609
\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"}
2189
+ − 610
2188
+ − 611
\end{itemize}
+ − 612
2207
+ − 613
Again, the last case is novel, so lets look at the example of
+ − 614
respectfullness for @{term concat}. The statement according to
+ − 615
the definition above is:
2190
+ − 616
2228
+ − 617
@{thm [display, indent=10] concat_rsp[no_vars]}
2189
+ − 618
2190
+ − 619
\noindent
+ − 620
By unfolding the definition of relation composition and relation map
+ − 621
we can see the equivalent statement just using the primitive list
+ − 622
equivalence relation:
+ − 623
2228
+ − 624
@{thm [display, indent=10] concat_rsp_unfolded[of "a" "a'" "b'" "b", no_vars]}
2189
+ − 625
2190
+ − 626
The statement reads that, for any lists of lists @{term a} and @{term b}
+ − 627
if there exist intermediate lists of lists @{term "a'"} and @{term "b'"}
+ − 628
such that each element of @{term a} is in the relation with an appropriate
+ − 629
element of @{term a'}, @{term a'} is in relation with @{term b'} and each
+ − 630
element of @{term b'} is in relation with the appropriate element of
+ − 631
@{term b}.
2189
+ − 632
+ − 633
*}
+ − 634
+ − 635
2190
+ − 636
text {*
2228
+ − 637
Sometimes a non-lifted polymorphic constant is instantiated to a
+ − 638
type being lifted. For example take the @{term "op #"} which inserts
+ − 639
an element in a list of pairs of natural numbers. When the theorem
+ − 640
is lifted, the pairs of natural numbers are to become integers, but
+ − 641
the head constant is still supposed to be the head constant, just
+ − 642
with a different type. To be able to lift such theorems
+ − 643
automatically, additional theorems provided by the user are
+ − 644
necessary, we call these \emph{preservation} theorems following
+ − 645
Homeier's naming.
2196
+ − 646
+ − 647
To lift theorems that talk about insertion in lists of lifted types
+ − 648
we need to know that for any quotient type with the abstraction and
+ − 649
representation functions @{text "Abs"} and @{text Rep} we have:
+ − 650
2228
+ − 651
@{thm [display, indent=10] (concl) cons_prs[no_vars]}
2196
+ − 652
+ − 653
This is not enough to lift theorems that talk about quotient compositions.
+ − 654
For some constants (for example empty list) it is possible to show a
+ − 655
general compositional theorem, but for @{term "op #"} it is necessary
+ − 656
to show that it respects the particular quotient type:
+ − 657
2228
+ − 658
@{thm [display, indent=10] insert_preserve2[no_vars]}
2190
+ − 659
*}
+ − 660
2191
+ − 661
subsection {* Composition of Quotient theorems *}
2189
+ − 662
2191
+ − 663
text {*
+ − 664
Given two quotients, one of which quotients a container, and the
+ − 665
other quotients the type in the container, we can write the
2193
+ − 666
composition of those quotients. To compose two quotient theorems
2207
+ − 667
we compose the relations with relation composition as defined above
+ − 668
and the abstraction and relation functions are the ones of the sub
+ − 669
quotients composed with the usual function composition.
+ − 670
The @{term "Rep"} and @{term "Abs"} functions that we obtain agree
+ − 671
with the definition of aggregate Abs/Rep functions and the
2193
+ − 672
relation is the same as the one given by aggregate relations.
+ − 673
This becomes especially interesting
2191
+ − 674
when we compose the quotient with itself, as there is no simple
+ − 675
intermediate step.
+ − 676
2242
+ − 677
Lets take again the example of @{term flat}. To be able to lift
2207
+ − 678
theorems that talk about it we provide the composition quotient
+ − 679
theorems, which then lets us perform the lifting procedure in an
+ − 680
unchanged way:
2188
+ − 681
2190
+ − 682
@{thm [display] quotient_compose_list[no_vars]}
2192
+ − 683
*}
+ − 684
2191
+ − 685
2227
+ − 686
section {* Lifting of Theorems *}
1978
+ − 687
2194
+ − 688
text {*
+ − 689
The core of the quotient package takes an original theorem that
+ − 690
talks about the raw types, and the statement of the theorem that
+ − 691
it is supposed to produce. This is different from other existing
2207
+ − 692
quotient packages, where only the raw theorems were necessary.
2194
+ − 693
We notice that in some cases only some occurrences of the raw
+ − 694
types need to be lifted. This is for example the case in the
+ − 695
new Nominal package, where a raw datatype that talks about
+ − 696
pairs of natural numbers or strings (being lists of characters)
+ − 697
should not be changed to a quotient datatype with constructors
+ − 698
taking integers or finite sets of characters. To simplify the
+ − 699
use of the quotient package we additionally provide an automated
+ − 700
statement translation mechanism that replaces occurrences of
+ − 701
types that match given quotients by appropriate lifted types.
+ − 702
+ − 703
Lifting the theorems is performed in three steps. In the following
+ − 704
we call these steps \emph{regularization}, \emph{injection} and
+ − 705
\emph{cleaning} following the names used in Homeier's HOL
2197
+ − 706
implementation.
2193
+ − 707
2197
+ − 708
We first define the statement of the regularized theorem based
+ − 709
on the original theorem and the goal theorem. Then we define
+ − 710
the statement of the injected theorem, based on the regularized
2208
+ − 711
theorem and the goal. We then show the 3 proofs, as all three
2197
+ − 712
can be performed independently from each other.
2193
+ − 713
2194
+ − 714
*}
1994
+ − 715
2197
+ − 716
subsection {* Regularization and Injection statements *}
1994
+ − 717
+ − 718
text {*
2197
+ − 719
2207
+ − 720
We first define the function @{text REG}, which takes the statements
+ − 721
of the raw theorem and the lifted theorem (both as terms) and
+ − 722
returns the statement of the regularized version. The intuition
+ − 723
behind this function is that it replaces quantifiers and
+ − 724
abstractions involving raw types by bounded ones, and equalities
+ − 725
involving raw types are replaced by appropriate aggregate
+ − 726
relations. It is defined as follows:
1994
+ − 727
2244
+ − 728
\begin{center}
+ − 729
\begin{tabular}{rcl}
+ − 730
\multicolumn{3}{@ {\hspace{-4mm}}l}{abstractions (with same types and different types):}\\
+ − 731
@{text "REG (\<lambda>x : \<sigma>. t, \<lambda>x : \<sigma>. s)"} & $\dn$ & @{text "\<lambda>x : \<sigma>. REG (t, s)"}\\
+ − 732
@{text "REG (\<lambda>x : \<sigma>. t, \<lambda>x : \<tau>. s)"} & $\dn$ & @{text "\<lambda>x : \<sigma> \<in> Res (REL (\<sigma>, \<tau>)). REG (t, s)"}\\
+ − 733
\multicolumn{3}{@ {\hspace{-4mm}}l}{quantification (over same types and different types):}\\
+ − 734
@{text "REG (\<forall>x : \<sigma>. t, \<forall>x : \<sigma>. s)"} & $\dn$ & @{text "\<forall>x : \<sigma>. REG (t, s)"}\\
+ − 735
@{text "REG (\<forall>x : \<sigma>. t, \<forall>x : \<tau>. s)"} & $\dn$ & @{text "\<forall>x : \<sigma> \<in> Res (REL (\<sigma>, \<tau>)). REG (t, s)"}\\
2245
+ − 736
\multicolumn{3}{@ {\hspace{-4mm}}l}{equalities (with same types and different types):}\\
2244
+ − 737
@{text "REG ((op =) : \<sigma>, (op =) : \<sigma>)"} & $\dn$ & @{text "(op =) : \<sigma>"}\\
+ − 738
@{text "REG ((op =) : \<sigma>, (op =) : \<tau>)"} & $\dn$ & @{text "REL (\<sigma>, \<tau>) : \<sigma>"}\\
2245
+ − 739
\multicolumn{3}{@ {\hspace{-4mm}}l}{applications, variables, constants:}\\
2244
+ − 740
@{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)"}\\
+ − 741
@{text "REG (v\<^isub>1, v\<^isub>2)"} & $\dn$ & @{text "v\<^isub>1"}\\
+ − 742
@{text "REG (c\<^isub>1, c\<^isub>2)"} & $\dn$ & @{text "c\<^isub>1"}\\
+ − 743
\end{tabular}
+ − 744
\end{center}
1994
+ − 745
2230
+ − 746
In the above definition we omitted the cases for existential quantifiers
2207
+ − 747
and unique existential quantifiers, as they are very similar to the cases
+ − 748
for the universal quantifier.
2197
+ − 749
2207
+ − 750
Next we define the function @{text INJ} which takes the statement of
+ − 751
the regularized theorems and the statement of the lifted theorem both as
2230
+ − 752
terms and returns the statement of the injected theorem:
2198
+ − 753
2245
+ − 754
\begin{center}
+ − 755
\begin{tabular}{rcl}
+ − 756
\multicolumn{3}{@ {\hspace{-4mm}}l}{abstractions (with same types and different types):}\\
+ − 757
@{text "INJ ((\<lambda>x. t) : \<sigma>, (\<lambda>x. s) : \<sigma>) "} & $\dn$ & @{text "\<lambda>x. INJ (t, s)"}\\
+ − 758
@{text "INJ ((\<lambda>x. t) : \<sigma>, (\<lambda>x. s) : \<tau>) "} & $\dn$ & @{text "REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (\<lambda>x. (INJ (t, s))))"}\\
+ − 759
@{text "INJ ((\<lambda>x \<in> R. t) : \<sigma>, (\<lambda>x. s) : \<tau>) "} & $\dn$ & @{text "REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (\<lambda>x \<in> R. (INJ (t, s))))"}\\
+ − 760
\multicolumn{3}{@ {\hspace{-4mm}}l}{quantification (over same types and different types):}\\
+ − 761
@{text "INJ (\<forall> t, \<forall> s) "} & $\dn$ & @{text "\<forall> (INJ (t, s))"}\\
+ − 762
@{text "INJ (\<forall> t \<in> R, \<forall> s) "} & $\dn$ & @{text "\<forall> INJ (t, s) \<in> R"}\\
+ − 763
\multicolumn{3}{@ {\hspace{-4mm}}l}{applications, variables, constants:}\\
+ − 764
@{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)"}\\
+ − 765
@{text "INJ (v\<^isub>1 : \<sigma>, v\<^isub>2 : \<sigma>) "} & $\dn$ & @{text "v\<^isub>1"}\\
+ − 766
@{text "INJ (v\<^isub>1 : \<sigma>, v\<^isub>2 : \<tau>) "} & $\dn$ & @{text "REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (v\<^isub>1))"}\\
+ − 767
@{text "INJ (c\<^isub>1 : \<sigma>, c\<^isub>2 : \<sigma>) "} & $\dn$ & @{text "c\<^isub>1"}\\
+ − 768
@{text "INJ (c\<^isub>1 : \<sigma>, c\<^isub>2 : \<tau>) "} & $\dn$ & @{text "REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (c\<^isub>1))"}\\
+ − 769
\end{tabular}
+ − 770
\end{center}
2198
+ − 771
+ − 772
For existential quantifiers and unique existential quantifiers it is
2230
+ − 773
defined similarly to the universal one.
2198
+ − 774
2197
+ − 775
*}
+ − 776
2208
+ − 777
subsection {* Proof procedure *}
+ − 778
2242
+ − 779
(* In the below the type-guiding 'QuotTrue' assumption is removed. We need it
+ − 780
only for bound variables without types, while in the paper presentation
+ − 781
variables are typed *)
2197
+ − 782
+ − 783
text {*
2208
+ − 784
+ − 785
With the above definitions of @{text "REG"} and @{text "INJ"} we can show
+ − 786
how the proof is performed. The first step is always the application of
+ − 787
of the following lemma:
+ − 788
2231
+ − 789
@{term [display, indent=10] "[|A; A --> B; B = C; C = D|] ==> D"}
2208
+ − 790
+ − 791
With @{text A} instantiated to the original raw theorem,
+ − 792
@{text B} instantiated to @{text "REG(A)"},
+ − 793
@{text C} instantiated to @{text "INJ(REG(A))"},
+ − 794
and @{text D} instantiated to the statement of the lifted theorem.
+ − 795
The first assumption can be immediately discharged using the original
+ − 796
theorem and the three left subgoals are exactly the subgoals of regularization,
+ − 797
injection and cleaning. The three can be proved independently by the
+ − 798
framework and in case there are non-solved subgoals they can be left
+ − 799
to the user.
+ − 800
+ − 801
The injection and cleaning subgoals are always solved if the appropriate
+ − 802
respectfulness and preservation theorems are given. It is not the case
+ − 803
with regularization; sometimes a theorem given by the user does not
+ − 804
imply a regularized version and a stronger one needs to be proved. This
2242
+ − 805
is outside of the scope of the quotient package, so such obligations are
+ − 806
left to the user. Take a simple statement for integers @{term "0 \<noteq> 1"}.
+ − 807
It does not follow from the fact that @{term "\<not> (0, 0) = (1, 0)"} because
+ − 808
of regularization. The raw theorem only shows that particular items in the
+ − 809
equivalence classes are not equal. A more general statement saying that
+ − 810
the classes are not equal is necessary.
2208
+ − 811
*}
+ − 812
+ − 813
subsection {* Proving Regularization *}
+ − 814
+ − 815
text {*
1994
+ − 816
2209
+ − 817
Isabelle provides a set of \emph{mono} rules, that are used to split implications
2230
+ − 818
of similar statements into simpler implication subgoals. These are enhanced
2249
+ − 819
with special quotient theorem in the regularization proof. Below we only show
2209
+ − 820
the versions for the universal quantifier. For the existential quantifier
2242
+ − 821
and abstraction they are analogous.
2199
+ − 822
2209
+ − 823
First, bounded universal quantifiers can be removed on the right:
2199
+ − 824
2249
+ − 825
@{thm [display, indent=10] ball_reg_right_unfolded[no_vars]}
2206
+ − 826
2209
+ − 827
They can be removed anywhere if the relation is an equivalence relation:
+ − 828
2231
+ − 829
@{thm [display, indent=10] ball_reg_eqv[no_vars]}
2209
+ − 830
+ − 831
And finally it can be removed anywhere if @{term R2} is an equivalence relation, then:
2231
+ − 832
+ − 833
@{thm [display, indent=10] (concl) ball_reg_eqv_range[no_vars]}
2209
+ − 834
2242
+ − 835
The last theorem is new in comparison with Homeier's package. There the
2231
+ − 836
injection procedure would be used to prove goals with such shape, and there
2242
+ − 837
the equivalence assumption would be used. We use the above theorem directly
+ − 838
also for composed relations where the range type is a type for which we know an
2231
+ − 839
equivalence theorem. This allows separating regularization from injection.
2209
+ − 840
2206
+ − 841
*}
+ − 842
+ − 843
(*
2231
+ − 844
@{thm bex_reg_eqv_range[no_vars]}
2199
+ − 845
@{thm [display] bex_reg_left[no_vars]}
+ − 846
@{thm [display] bex1_bexeq_reg[no_vars]}
2206
+ − 847
@{thm [display] bex_reg_eqv[no_vars]}
2209
+ − 848
@{thm [display] babs_reg_eqv[no_vars]}
+ − 849
@{thm [display] babs_simp[no_vars]}
2206
+ − 850
*)
1994
+ − 851
+ − 852
subsection {* Injection *}
+ − 853
2199
+ − 854
text {*
2211
+ − 855
The injection proof starts with an equality between the regularized theorem
+ − 856
and the injected version. The proof again follows by the structure of the
2242
+ − 857
two terms, and is defined for a goal being a relation between these two terms.
2199
+ − 858
2211
+ − 859
\begin{itemize}
+ − 860
\item For two constants, an appropriate constant respectfullness assumption is used.
2242
+ − 861
\item For two variables, we use the assumptions proved in regularization.
2211
+ − 862
\item For two abstractions, they are eta-expanded and beta-reduced.
+ − 863
\end{itemize}
2199
+ − 864
2211
+ − 865
Otherwise the two terms are applications. There are two cases: If there is a REP/ABS
+ − 866
in the injected theorem we can use the theorem:
+ − 867
2243
+ − 868
@{thm [display, indent=10] rep_abs_rsp[no_vars]}
2199
+ − 869
2243
+ − 870
\noindent
2211
+ − 871
and continue the proof.
2199
+ − 872
2211
+ − 873
Otherwise we introduce an appropriate relation between the subterms and continue with
+ − 874
two subgoals using the lemma:
+ − 875
2243
+ − 876
@{thm [display, indent=10] apply_rsp[no_vars]}
2199
+ − 877
+ − 878
*}
+ − 879
1994
+ − 880
subsection {* Cleaning *}
+ − 881
2212
+ − 882
text {*
+ − 883
The @{text REG} and @{text INJ} functions have been defined in such a way
+ − 884
that establishing the goal theorem now consists only on rewriting the
+ − 885
injected theorem with the preservation theorems.
+ − 886
+ − 887
\begin{itemize}
+ − 888
\item First for lifted constants, their definitions are the preservation rules for
+ − 889
them.
+ − 890
\item For lambda abstractions lambda preservation establishes
+ − 891
the equality between the injected theorem and the goal. This allows both
+ − 892
abstraction and quantification over lifted types.
2246
+ − 893
@{thm [display] (concl) lambda_prs[no_vars]}
2212
+ − 894
\item Relations over lifted types are folded with:
2246
+ − 895
@{thm [display] (concl) Quotient_rel_rep[no_vars]}
2212
+ − 896
\item User given preservation theorems, that allow using higher level operations
+ − 897
and containers of types being lifted. An example may be
2246
+ − 898
@{thm [display] (concl) map_prs(1)[of R1 Abs1 Rep1 R2 Abs2 Rep2,no_vars]}
2212
+ − 899
\end{itemize}
+ − 900
2246
+ − 901
*}
1994
+ − 902
+ − 903
section {* Examples *}
+ − 904
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(* Mention why equivalence *)
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text {*
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In this section we will show, a complete interaction with the quotient package
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for defining the type of integers by quotienting pairs of natural numbers and
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lifting theorems to integers. Our quotient package is fully compatible with
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Isabelle type classes, but for clarity we will not use them in this example.
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In a larger formalization of integers using the type class mechanism would
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provide many algebraic properties ``for free''.
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A user of our quotient package first needs to define a relation on
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the raw type, by which the quotienting will be performed. We give
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the same integer relation as the one presented in the introduction:
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
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\isacommand{fun}~~@{text "int_rel"}~~\isacommand{where}~~@{text "(m \<Colon> nat, n) int_rel (p, q) = (m + q = n + p)"}
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\end{isabelle}
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\noindent
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Next the quotient type is defined. This leaves a proof obligation that the
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relation is an equivalence relation which is solved automatically using the
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definitions:
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
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\isacommand{quotient\_type}~~@{text "int"}~~\isacommand{=}~~@{text "(nat \<times> nat)"}~~\isacommand{/}~~@{text "int_rel"}
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\end{isabelle}
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\noindent
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The user can then specify the constants on the quotient type:
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
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\begin{tabular}{@ {}l}
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\isacommand{quotient\_definition}~~@{text "0 :: int"}~~\isacommand{is}~~@{text "(0::nat, 0::nat)"}\\
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\isacommand{fun}~~@{text "plus_raw"}~~\isacommand{where}~~@{text "plus_raw (m :: nat, n) (p, q) = (m + p, n + q)"}\\
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\isacommand{quotient\_definition}~~@{text "(op +) \<Colon> (int \<Rightarrow> int \<Rightarrow> int)"}~~\isacommand{is}~~@{text "plus_raw"}\\
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\end{tabular}
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\end{isabelle}
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\noindent
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Lets first take a simple theorem about addition on the raw level:
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
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\isacommand{lemma}~~@{text "plus_zero_raw: int_rel (plus_raw (0, 0) x) x"}
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\end{isabelle}
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\noindent
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When the user tries to lift a theorem about integer addition, the respectfulness
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proof obligation is left, so let us prove it first:
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
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\isacommand{lemma}~~@{text "[quot_respect]: (int_rel \<Longrightarrow> int_rel \<Longrightarrow> int_rel) plus_raw plus_raw"}
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\end{isabelle}
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\noindent
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Can be proved automatically by the system just by unfolding the definition
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of @{text "op \<Longrightarrow>"}.
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Now the user can either prove a lifted lemma explicitly:
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
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\isacommand{lemma}~~@{text "0 + (x :: int) = x"}~~\isacommand{by}~~@{text "lifting plus_zero_raw"}
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\end{isabelle}
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\noindent
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Or in this simple case use the automated translation mechanism:
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\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %
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\isacommand{thm}~~@{text "plus_zero_raw[quot_lifted]"}
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\end{isabelle}
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\noindent
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obtaining the same result.
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*}
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1978
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section {* Related Work *}
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text {*
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\begin{itemize}
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\item Peter Homeier's package~\cite{Homeier05} (and related work from there)
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\item John Harrison's one~\cite{harrison-thesis} is the first one to lift theorems
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but only first order.
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\item PVS~\cite{PVS:Interpretations}
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\item MetaPRL~\cite{Nogin02}
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% \item Manually defined quotients in Isabelle/HOL Library (Markus's Quotient\_Type,
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% Dixon's FSet, \ldots)
1978
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\item Oscar Slotosch defines quotient-type automatically but no
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lifting~\cite{Slotosch97}.
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\item PER. And how to avoid it.
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\item Necessity of Hilbert Choice op and Larry's quotients~\cite{Paulson06}
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\item Setoids in Coq and \cite{ChicliPS02}
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\end{itemize}
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*}
1975
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
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section {* Conclusion *}
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text {*
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The code of the quotient package described here is already included in the
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standard distribution of Isabelle.\footnote{Avaiable from
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\href{http://isabelle.in.tum.de/}{http://isabelle.in.tum.de/}.} It is
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heavily used in Nominal Isabelle, which provides a convenient reasoning
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infrastructure for programming language calculi involving binders. Earlier
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versions of Nominal Isabelle have been used successfully in formalisations
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of an equivalence checking algorithm for LF \cite{UrbanCheneyBerghofer08},
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Typed Scheme~\cite{TobinHochstadtFelleisen08}, several calculi for
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concurrency \cite{BengtsonParow09} and a strong normalisation result for
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cut-elimination in classical logic \cite{UrbanZhu08}.
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*}
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subsection {* Contributions *}
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text {*
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We present the detailed lifting procedure, which was not shown before.
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The quotient package presented in this paper has the following
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advantages over existing packages:
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\begin{itemize}
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\item We define quotient composition, function map composition and
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relation map composition. This lets lifting polymorphic types with
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subtypes quotiented as well. We extend the notions of
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respectfulness and preservation to cope with quotient
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composition.
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\item We allow lifting only some occurrences of quotiented
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types. Rsp/Prs extended. (used in nominal)
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\item The quotient package is very modular. Definitions can be added
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separately, rsp and prs can be proved separately, Quotients and maps
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can be defined separately and theorems can
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be lifted on a need basis. (useful with type-classes).
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\item Can be used both manually (attribute, separate tactics,
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rsp/prs databases) and programatically (automated definition of
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lifted constants, the rsp proof obligations and theorem statement
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translation according to given quotients).
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\end{itemize}
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*}
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1975
b1281a0051ae
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Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
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(*<*)
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added stub for quotient paper; call with isabelle make qpaper
Christian Urban <urbanc@in.tum.de>
parents:
diff
changeset
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end
1978
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(*>*)