Qpaper / Clarify the typing system and composition of quotients issue.
(* How to change the notation for \<lbrakk> \<rbrakk> meta-level implications? *)(*<*)theory Paperimports "Quotient" "LaTeXsugar" "../Nominal/FSet"beginnotation (latex output) rel_conj ("_ \<circ>\<circ>\<circ> _" [53, 53] 52) and pred_comp ("_ \<circ>\<circ> _") and "op -->" (infix "\<longrightarrow>" 100) and "==>" (infix "\<Longrightarrow>" 100) and fun_map ("_ \<^raw:\mbox{\singlearr}> _" 51) and fun_rel ("_ \<^raw:\mbox{\doublearr}> _" 51) and list_eq (infix "\<approx>" 50) and (* Not sure if we want this notation...? *) fempty ("\<emptyset>") and funion ("_ \<union> _") and finsert ("{_} \<union> _") and Cons ("_::_") and concat ("flat") and fconcat ("\<Union>")ML {*fun nth_conj n (_, r) = nth (HOLogic.dest_conj r) n;fun style_lhs_rhs proj = Scan.succeed (fn ctxt => fn t => let val concl = Object_Logic.drop_judgment (ProofContext.theory_of ctxt) (Logic.strip_imp_concl t) in case concl of (_ $ l $ r) => proj (l, r) | _ => error ("Binary operator expected in term: " ^ Syntax.string_of_term ctxt concl) end);*}setup {* Term_Style.setup "rhs1" (style_lhs_rhs (nth_conj 0)) #> Term_Style.setup "rhs2" (style_lhs_rhs (nth_conj 1)) #> Term_Style.setup "rhs3" (style_lhs_rhs (nth_conj 2))*}(*>*)section {* Introduction *}text {* \begin{flushright} {\em ``Not using a [quotient] package has its advantages: we do not have to\\ collect all the theorems we shall ever want into one giant list;''}\\ Larry Paulson \cite{Paulson06} \end{flushright} \noindent Isabelle is a popular generic theorem prover in which many logics can be implemented. The most widely used one, however, is Higher-Order Logic (HOL). This logic consists of a small number of axioms and inference rules over a simply-typed term-language. Safe reasoning in HOL is ensured by two very restricted mechanisms for extending the logic: one is the definition of new constants in terms of existing ones; the other is the introduction of new types by identifying non-empty subsets in existing types. It is well understood how to use both mechanisms for dealing with quotient constructions in HOL (see \cite{Homeier05,Paulson06}). For example the integers in Isabelle/HOL are constructed by a quotient construction over the type @{typ "nat \<times> nat"} and the equivalence relation \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% @{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} \end{isabelle} \noindent This constructions yields the new type @{typ int} and definitions for @{text "0"} and @{text "1"} of type @{typ int} can be given in terms of pairs of natural numbers (namely @{text "(0, 0)"} and @{text "(1, 0)"}). Operations such as @{text "add"} with type @{typ "int \<Rightarrow> int \<Rightarrow> int"} can be defined in terms of operations on pairs of natural numbers (namely @{text "add_pair (n\<^isub>1, m\<^isub>1) (n\<^isub>2, m\<^isub>2) \<equiv> (n\<^isub>1 + n\<^isub>2, m\<^isub>1 + m\<^isub>2)"}). Similarly one can construct the type of finite sets, written @{term "\<alpha> fset"}, by quotienting the type @{text "\<alpha> list"} according to the equivalence relation \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% @{text "xs \<approx> ys \<equiv> (\<forall>x. memb x xs \<longleftrightarrow> memb x ys)"}\hfill\numbered{listequiv} \end{isabelle} \noindent which states that two lists are equivalent if every element in one list is also member in the other. The empty finite set, written @{term "{||}"}, can then be defined as the empty list and the union of two finite sets, written @{text "\<union>"}, as list append. Quotients are important in a variety of areas, but they are ubiquitous in the area of reasoning about programming language calculi. A simple example is the lambda-calculus, whose raw terms are defined as \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% @{text "t ::= x | t t | \<lambda>x.t"}\hfill\numbered{lambda} \end{isabelle} \noindent The problem with this definition arises, for instance, when one attempts to prove formally the substitution lemma \cite{Barendregt81} by induction over the structure of terms. This can be fiendishly complicated (see \cite[Pages 94--104]{CurryFeys58} for some ``rough'' sketches of a proof about raw lambda-terms). In contrast, if we reason about $\alpha$-equated lambda-terms, that means terms quotient according to $\alpha$-equivalence, then the reasoning infrastructure provided, for example, by Nominal Isabelle \cite{UrbanKaliszyk11} makes the formal proof of the substitution lemma almost trivial. The difficulty is that in order to be able to reason about integers, finite sets or $\alpha$-equated lambda-terms one needs to establish a reasoning infrastructure by transferring, or \emph{lifting}, definitions and theorems from the raw type @{typ "nat \<times> nat"} to the quotient type @{typ int} (similarly for finite sets and $\alpha$-equated lambda-terms). This lifting usually requires a \emph{lot} of tedious reasoning effort \cite{Paulson06}. It is feasible to do this work manually, if one has only a few quotient constructions at hand. But if they have to be done over and over again, as in Nominal Isabelle, then manual reasoning is not an option. The purpose of a \emph{quotient package} is to ease the lifting of theorems and automate the reasoning as much as possible. In the context of HOL, there have been a few quotient packages already \cite{harrison-thesis,Slotosch97}. The most notable one is by Homeier \cite{Homeier05} implemented in HOL4. The fundamental construction these quotient packages perform can be illustrated by the following picture: \begin{center} \mbox{}\hspace{20mm}\begin{tikzpicture} %%\draw[step=2mm] (-4,-1) grid (4,1); \draw[very thick] (0.7,0.3) circle (4.85mm); \draw[rounded corners=1mm, very thick] ( 0.0,-0.9) rectangle ( 1.8, 0.9); \draw[rounded corners=1mm, very thick] (-1.95,0.8) rectangle (-2.9,-0.195); \draw (-2.0, 0.8) -- (0.7,0.8); \draw (-2.0,-0.195) -- (0.7,-0.195); \draw ( 0.7, 0.23) node {\begin{tabular}{@ {}c@ {}}equiv-\\[-1mm]clas.\end{tabular}}; \draw (-2.45, 0.35) node {\begin{tabular}{@ {}c@ {}}new\\[-1mm]type\end{tabular}}; \draw (1.8, 0.35) node[right=-0.1mm] {\begin{tabular}{@ {}l@ {}}existing\\[-1mm] type\\ (sets of raw elements)\end{tabular}}; \draw (0.9, -0.55) node {\begin{tabular}{@ {}l@ {}}non-empty\\[-1mm]subset\end{tabular}}; \draw[->, very thick] (-1.8, 0.36) -- (-0.1,0.36); \draw[<-, very thick] (-1.8, 0.16) -- (-0.1,0.16); \draw (-0.95, 0.26) node[above=0.4mm] {@{text Rep}}; \draw (-0.95, 0.26) node[below=0.4mm] {@{text Abs}}; \end{tikzpicture} \end{center} \noindent The starting point is an existing type, to which we refer as the \emph{raw type}, over which an equivalence relation given by the user is defined. With this input the package introduces a new type, to which we refer as the \emph{quotient type}. This type comes with an \emph{abstraction} and a \emph{representation} function, written @{text Abs} and @{text Rep}.\footnote{Actually slightly more basic functions are given; the functions @{text Abs} and @{text Rep} need to be derived from them. We will show the details later. } These functions relate elements in the existing type to elements in the new type and vice versa; they can be uniquely identified by their quotient type. For example for the integer quotient construction the types of @{text Abs} and @{text Rep} are \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% @{text "Abs :: nat \<times> nat \<Rightarrow> int"}\hspace{10mm}@{text "Rep :: int \<Rightarrow> nat \<times> nat"} \end{isabelle} \noindent We therefore often write @{text Abs_int} and @{text Rep_int} if the typing information is important. Every abstraction and representation function stands for an isomorphism between the non-empty subset and elements in the new type. They are necessary for making definitions involving the new type. For example @{text "0"} and @{text "1"} of type @{typ int} can be defined as \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% @{text "0 \<equiv> Abs_int (0, 0)"}\hspace{10mm}@{text "1 \<equiv> Abs_int (1, 0)"} \end{isabelle} \noindent Slightly more complicated is the definition of @{text "add"} having type @{typ "int \<Rightarrow> int \<Rightarrow> int"}. Its definition is as follows \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% @{text "add n m \<equiv> Abs_int (add_pair (Rep_int n) (Rep_int m))"} \hfill\numbered{adddef} \end{isabelle} \noindent where we take the representation of the arguments @{text n} and @{text m}, add them according to the function @{text "add_pair"} and then take the abstraction of the result. This is all straightforward and the existing quotient packages can deal with such definitions. But what is surprising that none of them can deal with slightly more complicated definitions involving \emph{compositions} of quotients. Such compositions are needed for example in case of quotienting lists to yield finite sets and the operator that flattens lists of lists, defined as follows @{thm [display, indent=10] concat.simps(1) concat.simps(2)[no_vars]} \noindent We expect that the corresponding operator on finite sets, written @{term "fconcat"}, builds finite unions of finite sets: @{thm [display, indent=10] fconcat_empty[no_vars] fconcat_insert[no_vars]} \noindent The quotient package should automatically provide us with a definition for @{text "\<Union>"} in terms of @{text flat}, @{text Rep_fset} and @{text Abs_fset}. The problem is that the method used in the existing quotient packages of just taking the representation of the arguments and then taking the abstraction of the result is \emph{not} enough. The reason is that in case of @{text "\<Union>"} we obtain the incorrect definition @{text [display, indent=10] "\<Union> S \<equiv> Abs_fset (flat (Rep_fset S))"} \noindent where the right-hand side is not even typable! This problem can be remedied in the existing quotient packages by introducing an intermediate step and reasoning about flattening of lists of finite sets. However, this remedy is rather cumbersome and inelegant in light of our work, which can deal with such definitions directly. The solution is that we need to build aggregate representation and abstraction functions, which in case of @{text "\<Union>"} generate the following definition @{text [display, indent=10] "\<Union> S \<equiv> Abs_fset (flat ((map Rep_fset \<circ> Rep_fset) S))"} \noindent where @{term map} is the usual mapping function for lists. In this paper we will present a formal definition of our aggregate abstraction and representation functions (this definition was omitted in \cite{Homeier05}). They generate definitions, like the one above for @{text "\<Union>"}, according to the type of the raw constant and the type of the quotient constant. This means we also have to extend the notions of \emph{aggregate equivalence relation}, \emph{respectfulness} and \emph{preservation} from Homeier \cite{Homeier05}. In addition we are able to address the criticism by Paulson \cite{Paulson06} cited at the beginning of this section about having to collect theorems that are lifted from the raw level to the quotient level into one giant list. Our quotient package is the first one that is modular so that it allows to lift single theorems separately. This has the advantage for the user of being able to develop a formal theory interactively as a natural progression. A pleasing side-result of the modularity is that we are able to clearly specify what is involved in the lifting process (this was only hinted at in \cite{Homeier05} and implemented as a ``rough recipe'' in ML-code). The paper is organised as follows: Section \ref{sec:prelims} presents briefly some necessary preliminaries; Section \ref{sec:type} describes the definitions of quotient types and shows how definitions of constants can be made over quotient types. Section \ref{sec:resp} introduces the notions of respectfullness and preservation; Section \ref{sec:lift} describes the lifting of theorems, and Section \ref{sec:conc} concludes and compares our results to existing work.*}section {* Preliminaries and General Quotients\label{sec:prelims} *}text {* We describe here briefly the most basic notions of HOL we rely on, and the important definitions given by Homeier for quotients \cite{Homeier05}. At its core HOL is based on a simply-typed term language, where types are recorded in Church-style fashion (that means, we can infer the type of a term and its subterms without any additional information). The grammars for types and terms are as follows \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% \begin{tabular}{@ {}rl@ {\hspace{3mm}}l@ {}} @{text "\<sigma>, \<tau> ::="} & @{text "\<alpha> | (\<sigma>,\<dots>, \<sigma>) \<kappa>"} & (type variables and type constructors)\\ @{text "t, s ::="} & @{text "x\<^isup>\<sigma> | c\<^isup>\<sigma> | t t | \<lambda>x\<^isup>\<sigma>. t"} & (variables, constants, applications and abstractions)\\ \end{tabular} \end{isabelle} \noindent We often write just @{text \<kappa>} for @{text "() \<kappa>"}, and use @{text "\<alpha>s"} and @{text "\<sigma>s"} to stand for collections of type variables and types, respectively. The type of a term is often made explicit by writing @{text "t :: \<sigma>"}. HOL contains a type @{typ bool} for booleans and the function type, written @{text "\<sigma> \<Rightarrow> \<tau>"}. HOL also contains many primitive and defined constants; for example equality @{text "= :: \<sigma> \<Rightarrow> \<sigma> \<Rightarrow> bool"} and the identity function @{text "id :: \<sigma> => \<sigma>"} (the former being primitive and the latter being defined as @{text "\<lambda>x\<^sup>\<sigma>. x\<^sup>\<sigma>"}). An important point to note is that theorems in HOL can be seen as a subset of terms that are constructed specially (namely through axioms and prove rules). As a result we are able later on to define automatic proof procedures showing that one theorem implies another by decomposing the term underlying the first theorem. Like Homeier, our work relies on map-functions defined for every type constructor, like @{text map} for lists. Homeier describes others for products, sums, options and also the following map for function types @{thm [display, indent=10] fun_map_def[no_vars, THEN eq_reflection]} \noindent Using this map-function, we can give the following, equivalent, but more uniform, definition for @{text add} shown in \eqref{adddef}: @{text [display, indent=10] "add \<equiv> (Rep_int \<singlearr> Rep_int \<singlearr> Abs_int) add_pair"} \noindent We will sometimes refer to a map-function defined for a type-constructor @{text \<kappa>} as @{text "map_\<kappa>"}. It will also be necessary to have operators, referred to as @{text "rel_\<kappa>"}, which define equivalence relations in terms of constituent equivalence relations. For example given two equivalence relations @{text "R\<^isub>1"} and @{text "R\<^isub>2"}, we can define an equivalence relations over products as follows % @{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"} \noindent Similarly, Homeier defines the following operator for defining equivalence relations over function types: % @{thm [display, indent=10] fun_rel_def[of "R\<^isub>1" "R\<^isub>2", no_vars, THEN eq_reflection]} The central definition in Homeier's work \cite{Homeier05} relates equivalence relations, abstraction and representation functions: \begin{definition}[Quotient Types] Given a relation $R$, an abstraction function $Abs$ and a representation function $Rep$, the predicate @{term "Quotient R Abs Rep"} means \begin{enumerate} \item @{thm (rhs1) Quotient_def[of "R", no_vars]} \item @{thm (rhs2) Quotient_def[of "R", no_vars]} \item @{thm (rhs3) Quotient_def[of "R", no_vars]} \end{enumerate} \end{definition} \noindent The value of this definition is that validity of @{text Quotient} can be proved in terms of the validity of @{text "Quotient"} over the constituent types. For example Homeier proves the following property for higher-order quotient types: \begin{proposition}[Function Quotient]\label{funquot} @{thm[mode=IfThen] fun_quotient[where ?R1.0="R\<^isub>1" and ?R2.0="R\<^isub>2" and ?abs1.0="Abs\<^isub>1" and ?abs2.0="Abs\<^isub>2" and ?rep1.0="Rep\<^isub>1" and ?rep2.0="Rep\<^isub>2"]} \end{proposition} \noindent We will heavily rely on this part of Homeier's work including an extension to deal with compositions of equivalence relations defined as follows \begin{definition}[Composition of Relations] @{abbrev "rel_conj R\<^isub>1 R\<^isub>2"} where @{text "\<circ>\<circ>"} is the predicate composition defined by the rule % @{thm [mode=Rule, display, indent=10] pred_compI[of "R\<^isub>1" "x" "y" "R\<^isub>2" "z"]} \end{definition} \noindent Unfortunately a quotient type theorem, like Proposition \ref{funquot}, for the composition of any two quotients in not true (it is not even typable in the HOL type system). However, we can prove useful instances for compatible containers. We will show such example in Section \ref{sec:resp}.*}section {* Quotient Types and Quotient Definitions\label{sec:type} *}text {* The first step in a quotient construction is to take a name for the new type, say @{text "\<kappa>\<^isub>q"}, and an equivalence relation, say @{text R}, defined over a raw type, say @{text "\<sigma>"}. The type of the equivalence relation must be of type @{text "\<sigma> \<Rightarrow> \<sigma> \<Rightarrow> bool"}. The user-visible part of the declaration is therefore \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% \isacommand{quotient\_type}~~@{text "\<alpha>s \<kappa>\<^isub>q = \<sigma> / R"} \end{isabelle} \noindent and a proof that @{text "R"} is indeed an equivalence relation. Two concrete examples are \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% \begin{tabular}{@ {}l} \isacommand{quotient\_type}~~@{text "int = nat \<times> nat / \<approx>\<^bsub>nat \<times> nat\<^esub>"}\\ \isacommand{quotient\_type}~~@{text "\<alpha> fset = \<alpha> list / \<approx>\<^bsub>list\<^esub>"} \end{tabular} \end{isabelle} \noindent which introduce the type of integers and of finite sets using the equivalence relations @{text "\<approx>\<^bsub>nat \<times> nat\<^esub>"} and @{text "\<approx>\<^bsub>list\<^esub>"} defined earlier in \eqref{natpairequiv} and \eqref{listequiv}, respectively (the proofs about being equivalence relations is omitted). Given this data, we declare internally the quotient types as \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% \isacommand{typedef}~~@{text "\<alpha>s \<kappa>\<^isub>q = {c. \<exists>x. c = R x}"} \end{isabelle} \noindent where the right-hand side is the (non-empty) set of equivalence classes of @{text "R"}. The restriction in this declaration is that the type variables in the raw type @{text "\<sigma>"} must be included in the type variables @{text "\<alpha>s"} declared for @{text "\<kappa>\<^isub>q"}. HOL will provide us with the following abstraction and representation functions having the type \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% @{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"} \end{isabelle} \noindent They relate the new quotient type and equivalence classes of the raw type. However, as Homeier \cite{Homeier05} noted, it is much more convenient to work with the following derived abstraction and representation functions \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% @{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)"} \end{isabelle} \noindent on the expense of having to use Hilbert's choice operator @{text "\<epsilon>"} in the definition of @{text "Rep_\<kappa>\<^isub>q"}. These derived notions relate the quotient type and the raw type directly, as can be seen from their type, namely @{text "\<sigma> \<Rightarrow> \<alpha>s \<kappa>\<^isub>q"} and @{text "\<alpha>s \<kappa>\<^isub>q \<Rightarrow> \<sigma>"}, respectively. Given that @{text "R"} is an equivalence relation, the following property \begin{proposition} @{text "Quotient R Abs_\<kappa>\<^isub>q Rep_\<kappa>\<^isub>q"} \end{proposition} \noindent holds for every quotient type defined as above (for the proof see \cite{Homeier05}). The next step in a quotient construction is to introduce definitions of new constants involving the quotient type. These definitions need to be given in terms of concepts of the raw type (remember this is the only way how to extend HOL with new definitions). For the user visible is the declaration \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% \isacommand{quotient\_definition}~~@{text "c :: \<tau>"}~~\isacommand{is}~~@{text "t :: \<sigma>"} \end{isabelle} \noindent where @{text t} is the definiens (its type @{text \<sigma>} can always be inferred) and @{text "c"} is the name of definiendum, whose type @{text "\<tau>"} needs to be given explicitly (the point is that @{text "\<tau>"} and @{text "\<sigma>"} can only differ in places where a quotient and raw type are involved). Two concrete examples are \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% \begin{tabular}{@ {}l} \isacommand{quotient\_definition}~~@{text "0 :: int"}~~\isacommand{is}~~@{text "(0::nat, 0::nat)"}\\ \isacommand{quotient\_definition}~~@{text "\<Union> :: (\<alpha> fset) fset \<Rightarrow> \<alpha> fset"}~~% \isacommand{is}~~@{text "flat"} \end{tabular} \end{isabelle} \noindent The first one declares zero for integers and the second the operator for building unions of finite sets. The problem for us is that from such declarations we need to derive proper definitions using the @{text "Abs"} and @{text "Rep"} functions for the quotient types involved. The data we rely on is the given quotient type @{text "\<tau>"} and the raw type @{text "\<sigma>"}. They allow us to define \emph{aggregate abstraction} and \emph{representation functions} using the functions @{text "ABS (\<sigma>, \<tau>)"} and @{text "REP (\<sigma>, \<tau>)"} whose clauses we give below. The idea behind these two functions is to recursively descend into the raw types @{text \<sigma>} and quotient types @{text \<tau>}, and generate the appropriate @{text "Abs"} and @{text "Rep"} in places where the types differ. Therefore we generate just the identity whenever the types are equal. All clauses are as follows: \begin{center} \hfill \begin{tabular}{rcl} \multicolumn{3}{@ {\hspace{-4mm}}l}{equal types:}\\ @{text "ABS (\<sigma>, \<sigma>)"} & $\dn$ & @{text "id :: \<sigma> \<Rightarrow> \<sigma>"}\\ @{text "REP (\<sigma>, \<sigma>)"} & $\dn$ & @{text "id :: \<sigma> \<Rightarrow> \<sigma>"}\smallskip\\ \multicolumn{3}{@ {\hspace{-4mm}}l}{function types:}\\ @{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)"}\\ @{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\\ \multicolumn{3}{@ {\hspace{-4mm}}l}{equal type constructors:}\\ @{text "ABS (\<sigma>s \<kappa>, \<tau>s \<kappa>)"} & $\dn$ & @{text "map_\<kappa> (ABS (\<sigma>s, \<tau>s))"}\\ @{text "REP (\<sigma>s \<kappa>, \<tau>s \<kappa>)"} & $\dn$ & @{text "map_\<kappa> (REP (\<sigma>s, \<tau>s))"}\smallskip\\ \multicolumn{3}{@ {\hspace{-4mm}}l}{unequal type constructors:}\\ @{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)))"}\\ @{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"} \end{tabular}\hfill\numbered{ABSREP} \end{center} % \noindent where in the last two clauses we have that the quotient type @{text "\<alpha>s \<kappa>\<^isub>q"} is the quotient of the raw type @{text "\<rho>s \<kappa>"} (for example @{text "int"} and @{text "nat \<times> nat"}, or @{text "\<alpha> fset"} and @{text "\<alpha> list"}). The quotient construction ensures that the type variables in @{text "\<rho>s"} must be among the @{text "\<alpha>s"}. The @{text "\<sigma>s'"} are given by the matchers for the @{text "\<alpha>s"} when matching @{text "\<rho>s \<kappa>"} against @{text "\<sigma>s \<kappa>"}. The function @{text "MAP"} calculates an \emph{aggregate map-function} for a raw type as follows: % \begin{center} \begin{tabular}{rcl} @{text "MAP' (\<alpha>)"} & $\dn$ & @{text "a\<^sup>\<alpha>"}\\ @{text "MAP' (\<kappa>)"} & $\dn$ & @{text "id :: \<kappa> \<Rightarrow> \<kappa>"}\\ @{text "MAP' (\<sigma>s \<kappa>)"} & $\dn$ & @{text "map_\<kappa> (MAP'(\<sigma>s))"}\smallskip\\ @{text "MAP (\<sigma>)"} & $\dn$ & @{text "\<lambda>as. MAP'(\<sigma>)"} \end{tabular} \end{center} % \noindent In this definition we rely on the fact that we can interpret type-variables @{text \<alpha>} as term variables @{text a}. In the last clause we build an abstraction over all term-variables inside map-function generated by the auxiliary function @{text "MAP'"}. The need of aggregate map-functions can be seen in cases where we build quotients, say @{text "(\<alpha>, \<beta>) \<kappa>\<^isub>q"}, out of compound raw types, say @{text "(\<alpha> list) \<times> \<beta>"}. In this case @{text MAP} generates the aggregate map-function: @{text [display, indent=10] "\<lambda>a b. map_prod (map a) b"} \noindent which we need to define the aggregate abstraction and representation functions. To see how these definitions pan out in practise, let us return to our example about @{term "concat"} and @{term "fconcat"}, where we have the raw type @{text "(\<alpha> list) list \<Rightarrow> \<alpha> list"} and the quotient type @{text "(\<alpha> fset) fset \<Rightarrow> \<alpha> fset"}. Feeding them into @{text ABS} gives us (after some @{text "\<beta>"}-simplifications) the abstraction function @{text [display, indent=10] "(map (map id \<circ> Rep_fset) \<circ> Rep_fset) \<singlearr> Abs_fset \<circ> map id"} \noindent In our implementation we further simplify this function by rewriting with the usual laws about @{text "map"}s and @{text "id"}, namely @{term "map id = id"} and @{text "f \<circ> id = id \<circ> f = f"}. This gives us the abstraction function @{text [display, indent=10] "(map Rep_fset \<circ> Rep_fset) \<singlearr> Abs_fset"} \noindent which we can use for defining @{term "fconcat"} as follows @{text [display, indent=10] "\<Union> \<equiv> ((map Rep_fset \<circ> Rep_fset) \<singlearr> Abs_fset) flat"} \noindent Note that by using the operator @{text "\<singlearr>"} and special clauses for function types in \eqref{ABSREP}, we do not have to distinguish between arguments and results, but can deal with them uniformly. Consequently, all definitions in the quotient package are of the general form @{text [display, indent=10] "c \<equiv> ABS (\<sigma>, \<tau>) t"} \noindent where @{text \<sigma>} is the type of the definiens @{text "t"} and @{text "\<tau>"} the type of the defined quotient constant @{text "c"}. This data can be easily generated from the declaration given by the user. To increase the confidence in this way of making definitions, we can prove that the terms involved are all typable. \begin{lemma} If @{text "ABS (\<sigma>, \<tau>)"} returns some abstraction function @{text "Abs"} and @{text "REP (\<sigma>, \<tau>)"} some representation function @{text "Rep"}, then @{text "Abs"} is of type @{text "\<sigma> \<Rightarrow> \<tau>"} and @{text "Rep"} of type @{text "\<tau> \<Rightarrow> \<sigma>"}. \end{lemma} \begin{proof} By induction and analysing the definitions of @{text "ABS"}, @{text "REP"} and @{text "MAP"}. The cases of equal types and function types are straightforward (the latter follows from @{text "\<singlearr>"} having the type @{text "(\<alpha> \<Rightarrow> \<beta>) \<Rightarrow> (\<gamma> \<Rightarrow> \<delta>) \<Rightarrow> (\<beta> \<Rightarrow> \<gamma>) \<Rightarrow> (\<alpha> \<Rightarrow> \<delta>)"}). In case of equal type constructors we can observe that a map-function after applying the functions @{text "ABS (\<sigma>s, \<tau>s)"} produces a term of type @{text "\<sigma>s \<kappa> \<Rightarrow> \<tau>s \<kappa>"}. The interesting case is the one with unequal type constructors. Since we know the quotient is between @{text "\<alpha>s \<kappa>\<^isub>q"} and @{text "\<rho>s \<kappa>"}, we have that @{text "Abs_\<kappa>\<^isub>q"} is of type @{text "\<rho>s \<kappa> \<Rightarrow> \<alpha>s \<kappa>\<^isub>q"}. This type can be more specialised to @{text "\<rho>s[\<tau>s] \<kappa> \<Rightarrow> \<tau>s \<kappa>\<^isub>q"} where the type variables @{text "\<alpha>s"} are instantiated with the @{text "\<tau>s"}. The complete type can be calculated by observing that @{text "MAP (\<rho>s \<kappa>)"}, after applying the functions @{text "ABS (\<sigma>s', \<tau>s)"} to it, returns a term of type @{text "\<rho>s[\<sigma>s'] \<kappa> \<Rightarrow> \<rho>s[\<tau>s] \<kappa>"}. This type is equivalent to @{text "\<sigma>s \<kappa> \<Rightarrow> \<rho>s[\<tau>s] \<kappa>"}, which we just have to compose with @{text "\<rho>s[\<tau>s] \<kappa> \<Rightarrow> \<tau>s \<kappa>\<^isub>q"} according to the type of @{text "\<circ>"}.\qed \end{proof} \noindent The reader should note that this lemma fails for the abstraction and representation functions used, for example, in Homeier's quotient package.*}section {* Respectfulness and Preservation \label{sec:resp} *}text {* The main point of the quotient package is to automatically ``lift'' theorems involving constants over the raw type to theorems involving constants over the quotient type. Before we can describe this lift process, we need to impose some restrictions. The reason is that even if definitions for all raw constants can be given, \emph{not} all theorems can be actually be lifted. Most notably is the bound variable function, that is the constant @{text bn}, defined for raw lambda-terms as follows \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%% @{text "bn (x) \<equiv> \<emptyset>"}\hspace{4mm} @{text "bn (t\<^isub>1 t\<^isub>2) \<equiv> bn (t\<^isub>1) \<union> bn (t\<^isub>2)"}\hspace{4mm} @{text "bn (\<lambda>x. t) \<equiv> {x} \<union> bn (t)"} \end{isabelle} \noindent This constant just does not respect @{text "\<alpha>"}-equivalence and as consequently no theorem involving this constant can be lifted to @{text "\<alpha>"}-equated lambda terms. Homeier formulates the restrictions in terms of the properties of \emph{respectfullness} and \emph{preservation}. We have to slightly extend Homeier's definitions in order to deal with quotient compositions. To formally define what respectfulness is, we have to first define the notion of \emph{aggregate equivalence relations}. TBD \begin{itemize} \item @{text "REL(\<alpha>\<^isub>1, \<alpha>\<^isub>2)"} = @{text "op ="} \item @{text "REL(\<sigma>, \<sigma>)"} = @{text "op ="} \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))"} \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"} \end{itemize} class returned by this constant depends only on the equivalence classes of the arguments applied to the constant. To automatically lift a theorem that talks about a raw constant, to a theorem about the quotient type a respectfulness theorem is required. A respectfulness condition for a constant can be expressed in terms of an aggregate relation between the constant and itself, for example the respectfullness for @{text "append"} can be stated as: @{text [display, indent=10] "(\<approx>\<^bsub>list\<^esub> \<doublearr> \<approx>\<^bsub>list\<^esub> \<doublearr> \<approx>\<^bsub>list\<^esub>) append append"} \noindent Which after unfolding the definition of @{term "op ===>"} is equivalent to: @{thm [display, indent=10] append_rsp_unfolded[no_vars]} \noindent An aggregate relation is defined in terms of relation composition, so we define it first: The aggregate relation for an aggregate raw type and quotient type is defined as: Again, the last case is novel, so lets look at the example of respectfullness for @{term concat}. The statement according to the definition above is: @{thm [display, indent=10] concat_rsp[no_vars]} \noindent By unfolding the definition of relation composition and relation map we can see the equivalent statement just using the primitive list equivalence relation: @{thm [display, indent=10] concat_rsp_unfolded[of "a" "a'" "b'" "b", no_vars]} The statement reads that, for any lists of lists @{term a} and @{term b} if there exist intermediate lists of lists @{term "a'"} and @{term "b'"} such that each element of @{term a} is in the relation with an appropriate element of @{term a'}, @{term a'} is in relation with @{term b'} and each element of @{term b'} is in relation with the appropriate element of @{term b}. Sometimes a non-lifted polymorphic constant is instantiated to a type being lifted. For example take the @{term "op #"} which inserts an element in a list of pairs of natural numbers. When the theorem is lifted, the pairs of natural numbers are to become integers, but the head constant is still supposed to be the head constant, just with a different type. To be able to lift such theorems automatically, additional theorems provided by the user are necessary, we call these \emph{preservation} theorems following Homeier's naming. To lift theorems that talk about insertion in lists of lifted types we need to know that for any quotient type with the abstraction and representation functions @{text "Abs"} and @{text Rep} we have: @{thm [display, indent=10] (concl) cons_prs[no_vars]} This is not enough to lift theorems that talk about quotient compositions. For some constants (for example empty list) it is possible to show a general compositional theorem, but for @{term "op #"} it is necessary to show that it respects the particular quotient type: @{thm [display, indent=10] insert_preserve2[no_vars]} {\it Composition of Quotient theorems} Given two quotients, one of which quotients a container, and the other quotients the type in the container, we can write the composition of those quotients. To compose two quotient theorems we compose the relations with relation composition as defined above and the abstraction and relation functions are the ones of the sub quotients composed with the usual function composition. The @{term "Rep"} and @{term "Abs"} functions that we obtain agree with the definition of aggregate Abs/Rep functions and the relation is the same as the one given by aggregate relations. This becomes especially interesting when we compose the quotient with itself, as there is no simple intermediate step. Lets take again the example of @{term flat}. To be able to lift theorems that talk about it we provide the composition quotient theorem which allows quotienting inside the container: If @{term R} is an equivalence relation and @{term "Quotient R Abs Rep"} then @{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)"} \noindent this theorem will then instantiate the quotients needed in the injection and cleaning proofs allowing the lifting procedure to proceed in an unchanged way.*}section {* Lifting of Theorems\label{sec:lift} *}text {* The core of our quotient package takes an original theorem involving raw types and a statement of the theorem that it is supposed to produce. This is different from existing quotient packages, where only the raw theorems are necessary. To simplify the use of the quotient package we additionally provide an automated statement translation mechanism which can produce the latter automatically given a list of quotient types. It is possible that a user wants to lift only some occurrences of a raw type. In this case the user specifies the complete lifted goal instead of using the automated mechanism. Lifting the theorems is performed in three steps. In the following we call these steps \emph{regularization}, \emph{injection} and \emph{cleaning} following the names used in Homeier's HOL4 implementation. We first define the statement of the regularized theorem based on the original theorem and the goal theorem. Then we define the statement of the injected theorem, based on the regularized theorem and the goal. We then show the 3 proofs, as all three can be performed independently from each other.*}text {* \textit{Regularization and Injection statements} *}text {* We define the function @{text REG}, which takes the statements of the raw theorem and the lifted theorem (both as terms) and returns the statement of the regularized version. The intuition behind this function is that it replaces quantifiers and abstractions involving raw types by bounded ones, and equalities involving raw types are replaced by appropriate aggregate equivalence relations. It is defined as follows: \begin{center} \begin{tabular}{rcl} \multicolumn{3}{@ {\hspace{-4mm}}l}{abstractions (with same types and different types):}\\ @{text "REG (\<lambda>x : \<sigma>. t, \<lambda>x : \<sigma>. s)"} & $\dn$ & @{text "\<lambda>x : \<sigma>. REG (t, s)"}\\ @{text "REG (\<lambda>x : \<sigma>. t, \<lambda>x : \<tau>. s)"} & $\dn$ & @{text "\<lambda>x : \<sigma> \<in> Res (REL (\<sigma>, \<tau>)). REG (t, s)"}\\ \multicolumn{3}{@ {\hspace{-4mm}}l}{quantification (over same types and different types):}\\ @{text "REG (\<forall>x : \<sigma>. t, \<forall>x : \<sigma>. s)"} & $\dn$ & @{text "\<forall>x : \<sigma>. REG (t, s)"}\\ @{text "REG (\<forall>x : \<sigma>. t, \<forall>x : \<tau>. s)"} & $\dn$ & @{text "\<forall>x : \<sigma> \<in> Res (REL (\<sigma>, \<tau>)). REG (t, s)"}\\ \multicolumn{3}{@ {\hspace{-4mm}}l}{equalities (with same types and different types):}\\ @{text "REG ((op =) : \<sigma>, (op =) : \<sigma>)"} & $\dn$ & @{text "(op =) : \<sigma>"}\\ @{text "REG ((op =) : \<sigma>, (op =) : \<tau>)"} & $\dn$ & @{text "REL (\<sigma>, \<tau>) : \<sigma>"}\\ \multicolumn{3}{@ {\hspace{-4mm}}l}{applications, variables, constants:}\\ @{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)"}\\ @{text "REG (v\<^isub>1, v\<^isub>2)"} & $\dn$ & @{text "v\<^isub>1"}\\ @{text "REG (c\<^isub>1, c\<^isub>2)"} & $\dn$ & @{text "c\<^isub>1"}\\ \end{tabular} \end{center} In the above definition we omitted the cases for existential quantifiers and unique existential quantifiers, as they are very similar to the cases for the universal quantifier. Next we define the function @{text INJ} which takes the statement of the regularized theorems and the statement of the lifted theorem both as terms and returns the statement of the injected theorem: \begin{center} \begin{tabular}{rcl} \multicolumn{3}{@ {\hspace{-4mm}}l}{abstractions (with same types and different types):}\\ @{text "INJ ((\<lambda>x. t) : \<sigma>, (\<lambda>x. s) : \<sigma>) "} & $\dn$ & @{text "\<lambda>x. INJ (t, s)"}\\ @{text "INJ ((\<lambda>x. t) : \<sigma>, (\<lambda>x. s) : \<tau>) "} & $\dn$ & @{text "REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (\<lambda>x. (INJ (t, s))))"}\\ @{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))))"}\\ \multicolumn{3}{@ {\hspace{-4mm}}l}{quantification (over same types and different types):}\\ @{text "INJ (\<forall> t, \<forall> s) "} & $\dn$ & @{text "\<forall> (INJ (t, s))"}\\ @{text "INJ (\<forall> t \<in> R, \<forall> s) "} & $\dn$ & @{text "\<forall> INJ (t, s) \<in> R"}\\ \multicolumn{3}{@ {\hspace{-4mm}}l}{applications, variables, constants:}\\ @{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)"}\\ @{text "INJ (v\<^isub>1 : \<sigma>, v\<^isub>2 : \<sigma>) "} & $\dn$ & @{text "v\<^isub>1"}\\ @{text "INJ (v\<^isub>1 : \<sigma>, v\<^isub>2 : \<tau>) "} & $\dn$ & @{text "REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (v\<^isub>1))"}\\ @{text "INJ (c\<^isub>1 : \<sigma>, c\<^isub>2 : \<sigma>) "} & $\dn$ & @{text "c\<^isub>1"}\\ @{text "INJ (c\<^isub>1 : \<sigma>, c\<^isub>2 : \<tau>) "} & $\dn$ & @{text "REP(\<sigma>,\<tau>) (ABS (\<sigma>,\<tau>) (c\<^isub>1))"}\\ \end{tabular} \end{center} For existential quantifiers and unique existential quantifiers it is defined similarly to the universal one.*}text {*\textit{Proof Procedure}*}(* In the below the type-guiding 'QuotTrue' assumption is removed. We need it only for bound variables without types, while in the paper presentation variables are typed *)text {* When lifting a theorem we first apply the following rule @{term [display, indent=10] "[|A; A --> B; B = C; C = D|] ==> D"} \noindent with @{text A} instantiated to the original raw theorem, @{text B} instantiated to @{text "REG(A)"}, @{text C} instantiated to @{text "INJ(REG(A))"}, and @{text D} instantiated to the statement of the lifted theorem. The first assumption can be immediately discharged using the original theorem and the three left subgoals are exactly the subgoals of regularization, injection and cleaning. The three can be proved independently by the framework and in case there are non-solved subgoals they can be left to the user. The injection and cleaning subgoals are always solved if the appropriate respectfulness and preservation theorems are given. It is not the case with regularization; sometimes a theorem given by the user does not imply a regularized version and a stronger one needs to be proved. This is outside of the scope of the quotient package, so such obligations are left to the user. Take a simple statement for integers @{term "0 \<noteq> 1"}. It does not follow from the fact that @{term "\<not> (0, 0) = (1, 0)"} because of regularization. The raw theorem only shows that particular items in the equivalence classes are not equal. A more general statement saying that the classes are not equal is necessary.*}text {* \textit{Proving Regularization} *}text {* Isabelle provides a set of \emph{mono} rules, that are used to split implications of similar statements into simpler implication subgoals. These are enhanced with special quotient theorem in the regularization proof. Below we only show the versions for the universal quantifier. For the existential quantifier and abstraction they are analogous. First, bounded universal quantifiers can be removed on the right: @{thm [display, indent=10] ball_reg_right_unfolded[no_vars]} They can be removed anywhere if the relation is an equivalence relation: @{thm [display, indent=10] (concl) ball_reg_eqv[no_vars]} And finally it can be removed anywhere if @{term R2} is an equivalence relation: @{thm [display, indent=10] (concl) ball_reg_eqv_range[no_vars]} The last theorem is new in comparison with Homeier's package. There the injection procedure would be used to prove goals with such shape, and there the equivalence assumption would be used. We use the above theorem directly also for composed relations where the range type is a type for which we know an equivalence theorem. This allows separating regularization from injection.*}text {* \textit{Proving Rep/Abs Injection} *}(* @{thm bex_reg_eqv_range[no_vars]} @{thm [display] bex_reg_left[no_vars]} @{thm [display] bex1_bexeq_reg[no_vars]} @{thm [display] bex_reg_eqv[no_vars]} @{thm [display] babs_reg_eqv[no_vars]} @{thm [display] babs_simp[no_vars]}*)text {* The injection proof starts with an equality between the regularized theorem and the injected version. The proof again follows by the structure of the two terms, and is defined for a goal being a relation between these two terms. \begin{itemize} \item For two constants, an appropriate constant respectfullness assumption is used. \item For two variables, we use the assumptions proved in regularization. \item For two abstractions, they are eta-expanded and beta-reduced. \end{itemize} Otherwise the two terms are applications. There are two cases: If there is a REP/ABS in the injected theorem we can use the theorem: @{thm [display, indent=10] rep_abs_rsp[no_vars]} \noindent and continue the proof. Otherwise we introduce an appropriate relation between the subterms and continue with two subgoals using the lemma: @{thm [display, indent=10] apply_rsp[no_vars]}*}text {* \textit{Cleaning} *}text {* The @{text REG} and @{text INJ} functions have been defined in such a way that establishing the goal theorem now consists only on rewriting the injected theorem with the preservation theorems. \begin{itemize} \item First for lifted constants, their definitions are the preservation rules for them. \item For lambda abstractions lambda preservation establishes the equality between the injected theorem and the goal. This allows both abstraction and quantification over lifted types. @{thm [display] (concl) lambda_prs[no_vars]} \item Relations over lifted types are folded with: @{thm [display] (concl) Quotient_rel_rep[no_vars]} \item User given preservation theorems, that allow using higher level operations and containers of types being lifted. An example may be @{thm [display] (concl) map_prs(1)[of R1 Abs1 Rep1 R2 Abs2 Rep2,no_vars]} \end{itemize} *}section {* Examples *}(* Mention why equivalence *)text {* In this section we will show, a complete interaction with the quotient package for defining the type of integers by quotienting pairs of natural numbers and lifting theorems to integers. Our quotient package is fully compatible with Isabelle type classes, but for clarity we will not use them in this example. In a larger formalization of integers using the type class mechanism would provide many algebraic properties ``for free''. A user of our quotient package first needs to define a relation on the raw type, by which the quotienting will be performed. We give the same integer relation as the one presented in the introduction: \begin{isabelle}\ \ \ \ \ \ \ \ \ \ % \isacommand{fun}~~@{text "int_rel"}~~\isacommand{where}~~@{text "(m \<Colon> nat, n) int_rel (p, q) = (m + q = n + p)"} \end{isabelle} \noindent Next the quotient type is defined. This leaves a proof obligation that the relation is an equivalence relation which is solved automatically using the definitions: \begin{isabelle}\ \ \ \ \ \ \ \ \ \ % \isacommand{quotient\_type}~~@{text "int"}~~\isacommand{=}~~@{text "(nat \<times> nat)"}~~\isacommand{/}~~@{text "int_rel"} \end{isabelle} \noindent The user can then specify the constants on the quotient type: \begin{isabelle}\ \ \ \ \ \ \ \ \ \ % \begin{tabular}{@ {}l} \isacommand{quotient\_definition}~~@{text "0 :: int"}~~\isacommand{is}~~@{text "(0::nat, 0::nat)"}\\ \isacommand{fun}~~@{text "plus_raw"}~~\isacommand{where}~~@{text "plus_raw (m :: nat, n) (p, q) = (m + p, n + q)"}\\ \isacommand{quotient\_definition}~~@{text "(op +) \<Colon> (int \<Rightarrow> int \<Rightarrow> int)"}~~\isacommand{is}~~@{text "plus_raw"}\\ \end{tabular} \end{isabelle} \noindent Lets first take a simple theorem about addition on the raw level: \begin{isabelle}\ \ \ \ \ \ \ \ \ \ % \isacommand{lemma}~~@{text "plus_zero_raw: int_rel (plus_raw (0, 0) x) x"} \end{isabelle} \noindent When the user tries to lift a theorem about integer addition, the respectfulness proof obligation is left, so let us prove it first: \begin{isabelle}\ \ \ \ \ \ \ \ \ \ % \isacommand{lemma}~~@{text "[quot_respect]: (int_rel \<Longrightarrow> int_rel \<Longrightarrow> int_rel) plus_raw plus_raw"} \end{isabelle} \noindent Can be proved automatically by the system just by unfolding the definition of @{text "op \<Longrightarrow>"}. Now the user can either prove a lifted lemma explicitly: \begin{isabelle}\ \ \ \ \ \ \ \ \ \ % \isacommand{lemma}~~@{text "0 + (x :: int) = x"}~~\isacommand{by}~~@{text "lifting plus_zero_raw"} \end{isabelle} \noindent Or in this simple case use the automated translation mechanism: \begin{isabelle}\ \ \ \ \ \ \ \ \ \ % \isacommand{thm}~~@{text "plus_zero_raw[quot_lifted]"} \end{isabelle} \noindent obtaining the same result.*}section {* Conclusion and Related Work\label{sec:conc}*}text {* Oscar Slotosch~\cite{Slotosch97} implemented a mechanism that automatically defines quotient types for Isabelle/HOL. It did not include theorem lifting. John Harrison's quotient package~\cite{harrison-thesis} is the first one to lift theorems, however only first order. There is work on quotient types in non-HOL based systems and logical frameworks, namely theory interpretations in PVS~\cite{PVS:Interpretations}, new types in MetaPRL~\cite{Nogin02}, or the use of setoids in Coq, with some higher order issues~\cite{ChicliPS02}. Larry Paulson shows a construction of quotients that does not require the Hilbert Choice operator, again only first order~\cite{Paulson06}. The closest to our package is the package for HOL4 by Peter Homeier~\cite{Homeier05}, which is the first one to support lifting of higher order theorems. In comparison with this package we explore the notion of composition of quotients, which allows lifting constants like @{term "concat"} and theorems about it. The HOL4 package requires a big lists of constants, theorems to lift, respectfullness conditions as input. Our package is modularized, so that single definitions, single theorems or single respectfullness conditions etc can be added, which allows a natural use of the quotient package together with type-classes and locales. The code of the quotient package described here is already included in the standard distribution of Isabelle.\footnote{Available from \href{http://isabelle.in.tum.de/}{http://isabelle.in.tum.de/}.} It is heavily used in Nominal Isabelle, which provides a convenient reasoning infrastructure for programming language calculi involving binders. Earlier versions of Nominal Isabelle have been used successfully in formalisations of an equivalence checking algorithm for LF \cite{UrbanCheneyBerghofer08}, Typed Scheme~\cite{TobinHochstadtFelleisen08}, several calculi for concurrency \cite{BengtsonParow09} and a strong normalisation result for cut-elimination in classical logic \cite{UrbanZhu08}.*}subsection {* Contributions *}text {* We present the detailed lifting procedure, which was not shown before. The quotient package presented in this paper has the following advantages over existing packages: \begin{itemize} \item We define quotient composition, function map composition and relation map composition. This lets lifting polymorphic types with subtypes quotiented as well. We extend the notions of respectfulness and preservation to cope with quotient composition. \item We allow lifting only some occurrences of quotiented types. Rsp/Prs extended. (used in nominal) \item The quotient package is very modular. Definitions can be added separately, rsp and prs can be proved separately, Quotients and maps can be defined separately and theorems can be lifted on a need basis. (useful with type-classes). \item Can be used both manually (attribute, separate tactics, rsp/prs databases) and programatically (automated definition of lifted constants, the rsp proof obligations and theorem statement translation according to given quotients). \end{itemize} \medskip \noindent {\bf Acknowledgements:} We would like to thank Peter Homeier for the discussions about the HOL4 quotient package and explaining us its implementation details.*}(*<*)end(*>*)