diff -r a6f3e1b08494 -r b6873d123f9b Quotient-Paper/Paper.thy
--- a/Quotient-Paper/Paper.thy	Sat May 12 21:05:59 2012 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1364 +0,0 @@
-(*<*)
-theory Paper
-imports "Quotient" 
-        "~~/src/HOL/Library/Quotient_Syntax"
-        "~~/src/HOL/Library/LaTeXsugar"
-        "~~/src/HOL/Quotient_Examples/FSet"
-begin
-
-(****
-
-** things to do for the next version
-*
-* - what are quot_thms?
-* - what do all preservation theorems look like,
-    in particular preservation for quotient
-    compositions
-  - explain how Quotient R Abs Rep is proved (j-version)
-  - give an example where precise specification helps (core Haskell in nominal?)
-
-  - Mention Andreas Lochbiler in Acknowledgements and 'desceding'.
-
-*)
-
-notation (latex output)
-  rel_conj ("_ \<circ>\<circ>\<circ> _" [53, 53] 52) and
-  pred_comp ("_ \<circ>\<circ> _" [1, 1] 30) and
-  implies (infix "\<longrightarrow>" 100) and
-  "==>" (infix "\<Longrightarrow>" 100) and
-  map_fun ("_ \<singlearr> _" 51) and
-  fun_rel ("_ \<doublearr> _" 51) and
-  list_eq (infix "\<approx>" 50) and (* Not sure if we want this notation...? *)
-  empty_fset ("\<emptyset>") and
-  union_fset ("_ \<union> _") and
-  insert_fset ("{_} \<union> _") and 
-  Cons ("_::_") and
-  concat ("flat") and
-  concat_fset ("\<Union>") and
-  Quotient ("Quot _ _ _")
-
-  
-
-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 (Proof_Context.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))
-*}
-
-lemma insert_preserve2:
-  shows "((rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op #) =
-         (id ---> rep_fset ---> abs_fset) op #"
-  by (simp add: fun_eq_iff abs_o_rep[OF Quotient_fset] map_id Quotient_abs_rep[OF Quotient_fset])
-
-lemma list_all2_symp:
-  assumes a: "equivp R"
-  and b: "list_all2 R xs ys"
-  shows "list_all2 R ys xs"
-using list_all2_lengthD[OF b] b
-apply(induct xs ys rule: list_induct2)
-apply(auto intro: equivp_symp[OF a])
-done
-
-lemma concat_rsp_unfolded:
-  "\<lbrakk>list_all2 list_eq a ba; list_eq ba bb; list_all2 list_eq bb b\<rbrakk> \<Longrightarrow> list_eq (concat a) (concat b)"
-proof -
-  fix a b ba bb
-  assume a: "list_all2 list_eq a ba"
-  assume b: "list_eq ba bb"
-  assume c: "list_all2 list_eq bb b"
-  have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" proof
-    fix x
-    show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" proof
-      assume d: "\<exists>xa\<in>set a. x \<in> set xa"
-      show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
-    next
-      assume e: "\<exists>xa\<in>set b. x \<in> set xa"
-      have a': "list_all2 list_eq ba a" by (rule list_all2_symp[OF list_eq_equivp, OF a])
-      have b': "list_eq bb ba" by (rule equivp_symp[OF list_eq_equivp, OF b])
-      have c': "list_all2 list_eq b bb" by (rule list_all2_symp[OF list_eq_equivp, OF c])
-      show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
-    qed
-  qed
-  then show "list_eq (concat a) (concat b)" by auto
-qed
-
-(*>*)
-
-
-section {* Introduction *}
-
-text {* 
-  \noindent
-  One might think quotients have been studied to death, but in the context of
-  theorem provers many questions concerning them are far from settled. In
-  this paper we address the question of how to establish a convenient reasoning 
-  infrastructure
-  for quotient constructions in the Isabelle/HOL 
-  theorem prover. Higher-Order Logic (HOL) 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. Previous work has shown 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 really 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}.  
-  In principle 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:
-
-%%% FIXME: Referee 1 says:
-%%% Diagram is unclear.  Firstly, isn't an existing type a "set (not sets) of raw elements"?
-%%% Secondly, isn't the _set of_ equivalence classes mapped to and from the new type?
-%%% Thirdly, what do the words "non-empty subset" refer to ?
-
-%%% Cezary: I like the diagram, maybe 'new type' could be outside, but otherwise
-%%% I wouldn't change it.
-
-  \begin{center}
-  \mbox{}\hspace{20mm}\begin{tikzpicture}[scale=0.9]
-  %%\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} and over which an equivalence relation is given by the user. 
-  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. } They relate elements in the
-  existing type to elements in the new type, % and vice versa, 
-  and 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 is
-  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
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{thm concat.simps(1)[THEN eq_reflection]}\hspace{10mm} 
-  @{thm concat.simps(2)[THEN eq_reflection, no_vars]}
-  \end{isabelle}
-
-  \noindent
-  where @{text "@"} is the usual 
-  list append. We expect that the corresponding 
-  operator on finite sets, written @{term "fconcat"},
-  builds finite unions of finite sets:
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{thm concat_empty_fset[THEN eq_reflection, no_vars]}\hspace{10mm} 
-  @{thm concat_insert_fset[THEN eq_reflection, no_vars]}
-  \end{isabelle}
-
-  \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
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "\<Union> S \<equiv> Abs_fset (flat (Rep_fset S))"}
-  \end{isabelle}
-
-  \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
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "\<Union> S \<equiv> Abs_fset (flat ((map_list Rep_fset \<circ> Rep_fset) S))"}
-  \end{isabelle}
-
-  \noindent
-  where @{term map_list} 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 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). A pleasing side-result
-  is that our procedure for lifting theorems is completely deterministic
-  following the structure of the theorem being lifted and the theorem
-  on the quotient level. Space constraints, unfortunately, allow us to only 
-  sketch this part of our work in Section 5 and we defer the reader to a longer
-  version for the details. However, we will give in Section 3 and 4 all 
-  definitions that specify the input and output data of our three-step 
-  lifting procedure. Appendix A gives an example how our quotient 
-  package works in practise.
-*}
-
-section {* Preliminaries and General\\ Quotients\label{sec:prelims} *}
-
-text {*
-  \noindent
-  We will give in this section a crude overview of HOL and describe the main
-  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 always infer the type of 
-  a term and its subterms without any additional information). The grammars
-  for types and terms are
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  \begin{tabular}{@ {}c@ {\hspace{10mm}}c@ {}}
-  @{text "\<sigma>, \<tau> ::= \<alpha> | (\<sigma>,\<dots>, \<sigma>) \<kappa>"} & 
-  @{text "t, s ::= x\<^isup>\<sigma> | c\<^isup>\<sigma> | t t | \<lambda>x\<^isup>\<sigma>. t"}\\
-  \end{tabular}
-  \end{isabelle}
-
-  \noindent
-  with types being either type variables or type constructors and terms
-  being variables, constants, applications or abstractions.
-  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 includes 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, a primitive constant is equality, with type @{text "= :: \<sigma> \<Rightarrow> \<sigma> \<Rightarrow>
-  bool"}, and the identity function with type @{text "id :: \<sigma> \<Rightarrow> \<sigma>"} is
-  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 proof
-  rules). As a result we are able to define automatic proof
-  procedures showing that one theorem implies another by decomposing the term
-  underlying the first theorem.
-
-  Like Homeier's, our work relies on map-functions defined for every type
-  constructor taking some arguments, for example @{text map_list} for lists. Homeier
-  describes in \cite{Homeier05} map-functions for products, sums, options and
-  also the following map for function types
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{thm map_fun_def[no_vars, THEN eq_reflection]}
-  \end{isabelle}
-
-  \noindent
-  Using this map-function, we can give the following, equivalent, but more 
-  uniform definition for @{text add} shown in \eqref{adddef}:
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "add \<equiv> (Rep_int \<singlearr> Rep_int \<singlearr> Abs_int) add_pair"}
-  \end{isabelle}
-
-  \noindent
-  Using extensionality and unfolding the definition of @{text "\<singlearr>"}, 
-  we can get back to \eqref{adddef}. 
-  In what follows we shall use the convention to write @{text "map_\<kappa>"} for a map-function 
-  of the type-constructor @{text \<kappa>}. 
-  %% a general type for map all types is difficult to give (algebraic types are
-  %% easy, but for example the function type is not algebraic
-  %For a type @{text \<kappa>} with arguments @{text "\<alpha>\<^isub>1\<^isub>\<dots>\<^isub>n"} the
-  %type of the function @{text "map_\<kappa>"} has to be @{text "\<alpha>\<^isub>1\<Rightarrow>\<dots>\<Rightarrow>\<alpha>\<^isub>n\<Rightarrow>\<alpha>\<^isub>1\<dots>\<alpha>\<^isub>n \<kappa>"}. 
-  %For example @{text "map_list"}
-  %has to have the type @{text "\<alpha>\<Rightarrow>\<alpha> list"}.
-  In our implementation we maintain
-  a database of these map-functions that can be dynamically extended.
-
-  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
-  
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "(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"}
-  \end{isabelle}
-
-  \noindent
-  Homeier gives also the following operator for defining equivalence 
-  relations over function types
-  %
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{thm fun_rel_def[of "R\<^isub>1" "R\<^isub>2", no_vars, THEN eq_reflection]}
-  \hfill\numbered{relfun}
-  \end{isabelle}
-  
-  \noindent
-  In the context of quotients, the following two notions from \cite{Homeier05} 
-  are needed later on.
-
-  \begin{definition}[Respects]\label{def:respects}
-  An element @{text "x"} respects a relation @{text "R"} provided @{text "R x x"}.
-  \end{definition}
-
-  \begin{definition}[Bounded $\forall$ and $\lambda$]\label{def:babs}
-  @{text "\<forall>x \<in> S. P x"} holds if for all @{text x}, @{text "x \<in> S"} implies @{text "P x"};
-  and @{text "(\<lambda>x \<in> S. f x) x = f x"} provided @{text "x \<in> S"}.
-  \end{definition}
-
-  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"}
-  holds if and only if
-  \begin{isabelle}\ \ \ \ \ %%%%
-  \begin{tabular}{rl}
-  (i) & \begin{isa}@{thm (rhs1) Quotient_def[of "R", no_vars]}\end{isa}\\
-  (ii) & \begin{isa}@{thm (rhs2) Quotient_def[of "R", no_vars]}\end{isa}\\
-  (iii) & \begin{isa}@{thm (rhs3) Quotient_def[of "R", no_vars]}\end{isa}\\
-  \end{tabular}
-  \end{isabelle}
-  \end{definition}
-
-  \noindent
-  The value of this definition lies in the fact that validity of @{term "Quotient R Abs Rep"} can 
-  often be proved in terms of the validity of @{term "Quot"} over the constituent 
-  types of @{text "R"}, @{text Abs} and @{text Rep}. 
-  For example Homeier proves the following property for higher-order quotient
-  types:
- 
-  \begin{proposition}\label{funquot}
-  \begin{isa}
-  @{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{isa}
-  \end{proposition}
-
-  \noindent
-  As a result, Homeier is able to build an automatic prover that can nearly
-  always discharge a proof obligation involving @{text "Quot"}. Our quotient
-  package makes heavy 
-  use of this part of Homeier's work including an extension 
-  for dealing with \emph{conjugations} of equivalence relations\footnote{That are 
-  symmetric by definition.} defined as follows:
-
-%%% FIXME Referee 2 claims that composition-of-relations means OO, and this is also
-%%% what wikipedia says. Any idea for a different name? Conjugation of Relations?
-
-  \begin{definition}%%[Composition of Relations]
-  @{abbrev "rel_conj R\<^isub>1 R\<^isub>2"} where @{text "\<circ>\<circ>"} is the predicate
-  composition defined by 
-  @{thm (concl) pred_compI[of "R\<^isub>1" "x" "y" "R\<^isub>2" "z"]}
-  holds if and only if there exists a @{text y} such that @{thm (prem 1) pred_compI[of "R\<^isub>1" "x" "y" "R\<^isub>2" "z"]} and
-  @{thm (prem 2) pred_compI[of "R\<^isub>1" "x" "y" "R\<^isub>2" "z"]}.
-  \end{definition}
-
-  \noindent
-  Unfortunately a general quotient theorem for @{text "\<circ>\<circ>\<circ>"}, analogous to the one
-  for @{text "\<singlearr>"} given in Proposition \ref{funquot}, would not be true
-  in general. It cannot even be stated inside HOL, because of restrictions on types.
-  However, we can prove specific instances of a
-  quotient theorem for composing particular quotient relations.
-  For example, to lift theorems involving @{term flat} the quotient theorem for 
-  composing @{text "\<approx>\<^bsub>list\<^esub>"} will be necessary: given @{term "Quotient R Abs Rep"} 
-  with @{text R} being an equivalence relation, then
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  \begin{tabular}{r@ {\hspace{1mm}}l}
-  @{text  "Quot"} & @{text "(rel_list R \<circ>\<circ>\<circ> \<approx>\<^bsub>list\<^esub>)"}\\
-                  & @{text "(Abs_fset \<circ> map_list Abs)"} @{text "(map_list Rep \<circ> Rep_fset)"}\\
-  \end{tabular}
-  \end{isabelle}
-*}
-
-section {* Quotient Types and Quotient\\ Definitions\label{sec:type} *}
-
-text {*
-  \noindent
-  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 @{text "\<sigma> \<Rightarrow> \<sigma> \<Rightarrow> bool"}. The user-visible part of
-  the quotient type declaration is therefore
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  \isacommand{quotient\_type}~~@{text "\<alpha>s \<kappa>\<^isub>q = \<sigma> / R"}\hfill\numbered{typedecl}
-  \end{isabelle}
-
-  \noindent
-  and a proof that @{text "R"} is indeed an equivalence relation. The @{text "\<alpha>s"}
-  indicate the arity of the new type and the type-variables of @{text "\<sigma>"} can only
-  be contained in @{text "\<alpha>s"}. 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 in \eqref{natpairequiv} and
-  \eqref{listequiv}, respectively (the proofs about being equivalence
-  relations are omitted).  Given this data, we define for declarations shown in
-  \eqref{typedecl} the quotient types internally 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 constraint 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 then provide us with the following
-  abstraction and representation functions 
-
-  \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 
-  As can be seen from the type, 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 holds  for every quotient type 
-  (for the proof see \cite{Homeier05}).
-
-  \begin{proposition}
-  \begin{isa}@{term "Quotient R Abs_\<kappa>\<^isub>q Rep_\<kappa>\<^isub>q"}.\end{isa}
-  \end{proposition}
-
-  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 the visible part of such definitions 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 is 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 (@{text "flat"} having the type 
-  @{text "(\<alpha> list) list \<Rightarrow> \<alpha> list"}). 
-
-  From such declarations given by the user, the quotient package needs to derive proper
-  definitions using @{text "Abs"} and @{text "Rep"}. 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 shall give below. The idea behind
-  these two functions is to simultaneously 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. On the ``way'' down,
-  however we might have to use map-functions to let @{text Abs} and @{text Rep} act
-  over the appropriate types. In what follows we use the short-hand notation 
-  @{text "ABS (\<sigma>s, \<tau>s)"} to mean @{text "ABS (\<sigma>\<^isub>1, \<tau>\<^isub>1)\<dots>ABS (\<sigma>\<^isub>n, \<tau>\<^isub>n)"}; similarly 
-  for @{text REP}.
-  %
-  \begin{center}
-  \hfill
-  \begin{tabular}{@ {\hspace{2mm}}l@ {}}
-  \multicolumn{1}{@ {}l}{equal types:}\\ 
-  @{text "ABS (\<sigma>, \<sigma>)"} $\dn$ @{text "id :: \<sigma> \<Rightarrow> \<sigma>"}\hspace{5mm}%\\
-  @{text "REP (\<sigma>, \<sigma>)"} $\dn$ @{text "id :: \<sigma> \<Rightarrow> \<sigma>"}\smallskip\\
-  \multicolumn{1}{@ {}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{1}{@ {}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{1}{@ {}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
-  In the last two clauses are subtle. We rely in them on the fact that the 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"}). This data is given by declarations shown in \eqref{typedecl}.
-  The quotient construction ensures that the type variables in @{text
-  "\<rho>s \<kappa>"} must be among the @{text "\<alpha>s"}. The @{text "\<sigma>s'"} are given by the
-  substitutions for the @{text "\<alpha>s"} when matching  @{text "\<sigma>s \<kappa>"} against
-  @{text "\<rho>s \<kappa>"}. This calculation determines what are the types in place
-  of the type variables @{text "\<alpha>s"} in the instance of 
-  quotient type @{text "\<alpha>s \<kappa>\<^isub>q"}---namely @{text "\<tau>s"}, and the corresponding 
-  types in place of the @{text "\<alpha>s"} in the raw type @{text "\<rho>s \<kappa>"}---namely @{text "\<sigma>s'"}. The
-  function @{text "MAP"} calculates an \emph{aggregate map-function} for a raw
-  type as follows:
-  %
-  \begin{center}
-  \begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
-  @{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 in the first clause we can interpret type-variables @{text \<alpha>} as 
-  term variables @{text a}. In the last clause we build an abstraction over all
-  term-variables of the map-function generated by the auxiliary function 
-  @{text "MAP'"}.
-  The need for 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:
-
-%%% FIXME: Reviewer 2 asks: last two lines defining ABS and REP for
-%%% unequal type constructors: How are the $\varrho$s defined? The
-%%% following paragraph mentions them, but this paragraph is unclear,
-%%% since it then mentions $\alpha$s, which do not seem to be defined
-%%% either. As a result, I do not understand the first two sentences
-%%% in this paragraph. I can imagine roughly what the following
-%%% sentence `The $\sigma$s' are given by the matchers for the
-%%% $\alpha$s$ when matching $\varrho$s $\kappa$ against $\sigma$s
-%%% $\kappa$.' means, but also think that it is too vague.
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "\<lambda>a b. map_prod (map_list a) b"}
-  \end{isabelle}
-
-  \noindent
-  which is essential in order to define the corresponding 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 these types into @{text ABS} gives us (after some @{text "\<beta>"}-simplifications)
-  the abstraction function
-
-  \begin{isabelle}\ \ \ %%%
-  \begin{tabular}{l}
-  @{text "(map_list (map_list id \<circ> Rep_fset) \<circ> Rep_fset) \<singlearr>"}\\
-  \mbox{}\hspace{4.5cm}@{text " Abs_fset \<circ> map_list id"}
-  \end{tabular}
-  \end{isabelle}
-
-  \noindent
-  In our implementation we further
-  simplify this function by rewriting with the usual laws about @{text
-  "map"}s and @{text "id"}, for example @{term "map_list id = id"} and @{text "f \<circ> id =
-  id \<circ> f = f"}. This gives us the simpler abstraction function
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "(map_list Rep_fset \<circ> Rep_fset) \<singlearr> Abs_fset"}
-  \end{isabelle}
-
-  \noindent
-  which we can use for defining @{term "fconcat"} as follows
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "\<Union> \<equiv> ((map_list Rep_fset \<circ> Rep_fset) \<singlearr> Abs_fset) flat"}
-  \end{isabelle}
-
-  \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
-  
-  \begin{isabelle}\ \ \ \ \ %%%
-  \mbox{@{text "c \<equiv> ABS (\<sigma>, \<tau>) t"}}
-  \end{isabelle}
-  
-  \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 mutual induction and analysing the definitions of @{text "ABS"} and @{text "REP"}. 
-  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}
-*}
-
-section {* Respectfulness and\\ Preservation \label{sec:resp} *}
-
-text {*
-  \noindent
-  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 lifting process, we need to impose 
-  two restrictions in form of proof obligations that arise during the
-  lifting. The reason is that even if definitions for all raw constants 
-  can be given, \emph{not} all theorems can be lifted to the quotient type. Most 
-  notable is the bound variable function, that is the constant @{text bn}, 
-  defined 
-  for raw lambda-terms as follows
-
-  \begin{isabelle}
-  \begin{center}
-  @{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)"}\smallskip\\
-  @{text "bn (\<lambda>x. t) \<equiv> {x} \<union> bn (t)"}
-  \end{center}
-  \end{isabelle}
-
-  \noindent
-  We can generate a definition for this constant using @{text ABS} and @{text REP}.
-  But this constant does \emph{not} respect @{text "\<alpha>"}-equivalence and 
-  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{respectfulness} and \emph{preservation}. We have
-  to slightly extend Homeier's definitions in order to deal with quotient
-  compositions. 
-
-%%% FIXME: Reviewer 3 asks why are the definitions that follow enough to deal
-%%% with quotient composition.
-
-  To formally define what respectfulness is, we have to first define 
-  the notion of \emph{aggregate equivalence relations} using the function @{text "REL(\<sigma>, \<tau>)"}
-  The idea behind this function is to simultaneously descend into the raw types
-  @{text \<sigma>} and quotient types @{text \<tau>}, and generate the appropriate
-  quotient equivalence relations in places where the types differ and equalities
-  elsewhere.
-
-  \begin{center}
-  \hfill
-  \begin{tabular}{l}
-  \multicolumn{1}{@ {}l}{equal types:} 
-  @{text "REL (\<sigma>, \<sigma>)"} $\dn$ @{text "= :: \<sigma> \<Rightarrow> \<sigma> \<Rightarrow> bool"}\smallskip\\
-   \multicolumn{1}{@ {}l}{equal type constructors:}\\ 
-  @{text "REL (\<sigma>s \<kappa>, \<tau>s \<kappa>)"} $\dn$ @{text "rel_\<kappa> (REL (\<sigma>s, \<tau>s))"}\smallskip\\
-  \multicolumn{1}{@ {}l}{unequal type constructors:}\\
-  @{text "REL (\<sigma>s \<kappa>, \<tau>s \<kappa>\<^isub>q)"} $\dn$ @{text "rel_\<kappa>\<^isub>q (REL (\<sigma>s', \<tau>s))"}\\
-  \end{tabular}\hfill\numbered{REL}
-  \end{center}
-
-  \noindent
-  The @{text "\<sigma>s'"} in the last clause are calculated as in \eqref{ABSREP}:
-  again we know that type @{text "\<alpha>s \<kappa>\<^isub>q"} is the quotient of the raw type 
-  @{text "\<rho>s \<kappa>"}. The @{text "\<sigma>s'"} are the substitutions for @{text "\<alpha>s"} obtained by matching 
-  @{text "\<rho>s \<kappa>"} and @{text "\<sigma>s \<kappa>"}.
-
-  Let us return to the lifting procedure of theorems. Assume we have a theorem
-  that contains the raw constant @{text "c\<^isub>r :: \<sigma>"} and which we want to
-  lift to a theorem where @{text "c\<^isub>r"} is replaced by the corresponding
-  constant @{text "c\<^isub>q :: \<tau>"} defined over a quotient type. In this situation 
-  we generate the following proof obligation
-  
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "REL (\<sigma>, \<tau>) c\<^isub>r c\<^isub>r"}.
-  \end{isabelle}
-  
-  \noindent
-  Homeier calls these proof obligations \emph{respectfulness
-  theorems}. However, unlike his quotient package, we might have several
-  respectfulness theorems for one constant---he has at most one.
-  The reason is that because of our quotient compositions, the types
-  @{text \<sigma>} and @{text \<tau>} are not completely determined by @{text "c\<^bsub>r\<^esub>"}.
-  And for every instantiation of the types, a corresponding
-  respectfulness theorem is necessary.
-
-  Before lifting a theorem, we require the user to discharge
-  respectfulness proof obligations. In case of @{text bn}
-  this obligation is %%as follows
-   
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text  "(\<approx>\<^isub>\<alpha> \<doublearr> =) bn bn"}
-  \end{isabelle}
-  
-  \noindent
-  and the point is that the user cannot discharge it: because it is not true. To see this,
-  we can just unfold the definition of @{text "\<doublearr>"} \eqref{relfun} 
-  using extensionality to obtain the false statement
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "\<forall>t\<^isub>1 t\<^isub>2. if t\<^isub>1 \<approx>\<^isub>\<alpha> t\<^isub>2 then bn(t\<^isub>1) = bn(t\<^isub>2)"}
-  \end{isabelle}
-
-  \noindent
-  In contrast, lifting a theorem about @{text "append"} to a theorem describing 
-  the union of finite sets will mean to discharge the proof obligation
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "(\<approx>\<^bsub>list\<^esub> \<doublearr> \<approx>\<^bsub>list\<^esub> \<doublearr> \<approx>\<^bsub>list\<^esub>) append append"}
-  \end{isabelle}
-
-  \noindent
-  To do so, we have to establish
-  
-  \begin{isabelle}\ \ \ \ \ %%%
-  if @{text "xs \<approx>\<^bsub>list\<^esub> ys"} and  @{text "us \<approx>\<^bsub>list\<^esub> vs"}
-  then @{text "xs @ us \<approx>\<^bsub>list\<^esub> ys @ vs"}
-  \end{isabelle}
-
-  \noindent
-  which is straightforward given the definition shown in \eqref{listequiv}.
-
-  The second restriction we have to impose arises from non-lifted polymorphic
-  constants, which are instantiated to a type being quotient. For example,
-  take the @{term "cons"}-constructor to add a pair of natural numbers to a
-  list, whereby we assume the pair of natural numbers turns into an integer in
-  the quotient construction. The point is that we still want to use @{text
-  cons} for adding integers to lists---just with a different type. To be able
-  to lift such theorems, we need a \emph{preservation property} for @{text
-  cons}. Assuming we have a polymorphic raw constant @{text "c\<^isub>r :: \<sigma>"}
-  and a corresponding quotient constant @{text "c\<^isub>q :: \<tau>"}, then a
-  preservation property is as follows
-
-%%% FIXME: Reviewer 2 asks: You say what a preservation theorem is,
-%%% but not which preservation theorems you assume. Do you generate a
-%%% proof obligation for a preservation theorem for each raw constant
-%%% and its corresponding lifted constant?
-
-%%% Cezary: I think this would be a nice thing to do but we have not
-%%% done it, the theorems need to be 'guessed' from the remaining obligations
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "Quot R\<^bsub>\<alpha>s\<^esub> Abs\<^bsub>\<alpha>s\<^esub> Rep\<^bsub>\<alpha>s\<^esub> implies  ABS (\<sigma>, \<tau>) c\<^isub>r = c\<^isub>r"}
-  \end{isabelle}
-
-  \noindent
-  where the @{text "\<alpha>s"} stand for the type variables in the type of @{text "c\<^isub>r"}.
-  In case of @{text cons} (which has type @{text "\<alpha> \<Rightarrow> \<alpha> list \<Rightarrow> \<alpha> list"}) we have 
-
-  \begin{isabelle}\ \ \ \ \ %%%
-  @{text "(Rep \<singlearr> map_list Rep \<singlearr> map_list Abs) cons = cons"}
-  \end{isabelle}
-
-  \noindent
-  under the assumption @{term "Quotient R Abs Rep"}. The point is that if we have
-  an instance of @{text cons} where the type variable @{text \<alpha>} is instantiated
-  with @{text "nat \<times> nat"} and we also quotient this type to yield integers,
-  then we need to show this preservation property.
-
-  %%%@ {thm [display, indent=10] insert_preserve2[no_vars]}
-
-  %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_list Abs) (map_list 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 {*
-
-%%% FIXME Reviewer 3 asks: Section 5 shows the technicalities of
-%%% lifting theorems. But there is no clarification about the
-%%% correctness. A reader would also be interested in seeing some
-%%% discussions about the generality and limitation of the approach
-%%% proposed there
-
-  \noindent
-  The main benefit of a quotient package is to lift automatically theorems over raw
-  types to theorems over quotient types. We will perform this lifting in
-  three phases, called \emph{regularization},
-  \emph{injection} and \emph{cleaning} according to procedures in Homeier's ML-code.
-  Space restrictions, unfortunately, prevent us from giving anything but a sketch of these three
-  phases. However, we will precisely define the input and output data of these phases
-  (this was omitted in \cite{Homeier05}).
-
-  The purpose of regularization is to change the quantifiers and abstractions
-  in a ``raw'' theorem to quantifiers over variables that respect their respective relations
-  (Definition \ref{def:respects} states what respects means). The purpose of injection is to add @{term Rep}
-  and @{term Abs} of appropriate types in front of constants and variables
-  of the raw type so that they can be replaced by the corresponding constants from the
-  quotient type. The purpose of cleaning is to bring the theorem derived in the
-  first two phases into the form the user has specified. Abstractly, our
-  package establishes the following three proof steps:
-
-%%% FIXME: Reviewer 1 complains that the reader needs to guess the
-%%% meaning of reg_thm and inj_thm, as well as the arguments of REG
-%%% which are given above. I wouldn't change it.
-
-  \begin{center}
-  \begin{tabular}{l@ {\hspace{4mm}}l}
-  1.) Regularization & @{text "raw_thm \<longrightarrow> reg_thm"}\\
-  2.) Injection & @{text "reg_thm \<longleftrightarrow> inj_thm"}\\
-  3.) Cleaning & @{text "inj_thm \<longleftrightarrow> quot_thm"}\\
-  \end{tabular}
-  \end{center}
-
-  \noindent
-  which means, stringed together, the raw theorem implies the quotient theorem.
-  In contrast to other quotient packages, our package requires that the user specifies 
-  both, the @{text "raw_thm"} (as theorem) and the \emph{term} of the @{text "quot_thm"}.\footnote{Though we
-  also provide a fully automated mode, where the @{text "quot_thm"} is guessed
-  from the form of @{text "raw_thm"}.} As a result, the user has fine control
-  over which parts of a raw theorem should be lifted. 
-
-  The second and third proof step performed in package will always succeed if the appropriate
-  respectfulness and preservation theorems are given. In contrast, the first
-  proof step can fail: a theorem given by the user does not always
-  imply a regularized version and a stronger one needs to be proved. An example
-  for this kind of failure is the simple statement for integers @{text "0 \<noteq> 1"}.
-  One might hope that it can be proved by lifting @{text "(0, 0) \<noteq> (1, 0)"},  
-  but this raw theorem only shows that two particular elements in the
-  equivalence classes are not equal. In order to obtain @{text "0 \<noteq> 1"}, a  
-  more general statement stipulating that the equivalence classes are not 
-  equal is necessary.  This kind of failure is beyond the scope where the 
-  quotient package can help: the user has to provide a raw theorem that
-  can be regularized automatically, or has to provide an explicit proof
-  for the first proof step. Homeier gives more details about this issue
-  in the long version of \cite{Homeier05}.
-
-  In the following we will first define the statement of the
-  regularized theorem based on @{text "raw_thm"} and
-  @{text "quot_thm"}. Then we define the statement of the injected theorem, based
-  on @{text "reg_thm"} and @{text "quot_thm"}. We then show the three proof steps,
-  which can all be performed independently from each other.
-
-  We first define the function @{text REG}, which takes the terms of the 
-  @{text "raw_thm"} and @{text "quot_thm"} as input and returns
-  @{text "reg_thm"}. The idea
-  behind this function is that it replaces quantifiers and
-  abstractions involving raw types by bounded ones, and equalities
-  involving raw types by appropriate aggregate
-  equivalence relations. It is defined by simultaneous recursion on 
-  the structure of  the terms of @{text "raw_thm"} and @{text "quot_thm"} as follows:
-  %
-  \begin{center}
-  \begin{tabular}{@ {}l@ {}}
-  \multicolumn{1}{@ {}l@ {}}{abstractions:}\\%%\smallskip\\
-  @{text "REG (\<lambda>x\<^sup>\<sigma>. t, \<lambda>x\<^sup>\<tau>. s)"} $\dn$ 
-  $\begin{cases}
-  @{text "\<lambda>x\<^sup>\<sigma>. REG (t, s)"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
-  @{text "\<lambda>x\<^sup>\<sigma> \<in> Resp (REL (\<sigma>, \<tau>)). REG (t, s)"}
-  \end{cases}$\\%%\smallskip\\
-  \multicolumn{1}{@ {}l@ {}}{universal quantifiers:}\\
-  @{text "REG (\<forall>x\<^sup>\<sigma>. t, \<forall>x\<^sup>\<tau>. s)"} $\dn$
-  $\begin{cases}
-  @{text "\<forall>x\<^sup>\<sigma>. REG (t, s)"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
-  @{text "\<forall>x\<^sup>\<sigma> \<in> Resp (REL (\<sigma>, \<tau>)). REG (t, s)"}
-  \end{cases}$\\%%\smallskip\\
-  \multicolumn{1}{@ {}l@ {}}{equality:  \hspace{3mm}%%}\smallskip\\
-  %% REL of two equal types is the equality so we do not need a separate case
-  @{text "REG (=\<^bsup>\<sigma>\<Rightarrow>\<sigma>\<Rightarrow>bool\<^esup>, =\<^bsup>\<tau>\<Rightarrow>\<tau>\<Rightarrow>bool\<^esup>)"} $\dn$ @{text "REL (\<sigma>, \<tau>)"}}\smallskip\\
-  \multicolumn{1}{@ {}l@ {}}{applications, variables and 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 (x\<^isub>1, x\<^isub>2)"} $\dn$ @{text "x\<^isub>1"}\\
-  @{text "REG (c\<^isub>1, c\<^isub>2)"} $\dn$ @{text "c\<^isub>1"}\\
-  \end{tabular}
-  \end{center}
-  %
-  \noindent
-  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 as argument
-  @{text "reg_thm"} and @{text "quot_thm"} (both as
-  terms) and returns @{text "inj_thm"}:
-
-  \begin{center}
-  \begin{tabular}{l}
-  \multicolumn{1}{@ {}l}{abstractions:}\\
-  @{text "INJ (\<lambda>x. t :: \<sigma>, \<lambda>x. s :: \<tau>) "} $\dn$\\ 
-  \hspace{18mm}$\begin{cases}
-  @{text "\<lambda>x. INJ (t, s)"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
-  @{text "REP (\<sigma>, \<tau>) (ABS (\<sigma>, \<tau>) (\<lambda>x. INJ (t, s)))"}
-  \end{cases}$\smallskip\\
-  @{text "INJ (\<lambda>x \<in> R. t :: \<sigma>, \<lambda>x. s :: \<tau>) "} $\dn$\\ 
-  \hspace{18mm}@{text "REP (\<sigma>, \<tau>) (ABS (\<sigma>, \<tau>) (\<lambda>x \<in> R. INJ (t, s)))"}\smallskip\\
-  \multicolumn{1}{@ {}l}{universal quantifiers:}\\
-  @{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"}\smallskip\\
-  \multicolumn{1}{@ {}l}{applications, variables and constants:}\smallskip\\
-  @{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 (x\<^isub>1\<^sup>\<sigma>, x\<^isub>2\<^sup>\<tau>) "} $\dn$
-  $\begin{cases}
-  @{text "x\<^isub>1"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
-  @{text "REP (\<sigma>, \<tau>) (ABS (\<sigma>, \<tau>) x\<^isub>1)"}\\
-  \end{cases}$\\
-  @{text "INJ (c\<^isub>1\<^sup>\<sigma>, c\<^isub>2\<^sup>\<tau>) "} $\dn$ 
-  $\begin{cases}
-  @{text "c\<^isub>1"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\
-  @{text "REP (\<sigma>, \<tau>) (ABS (\<sigma>, \<tau>) c\<^isub>1)"}\\
-  \end{cases}$\\
-  \end{tabular}
-  \end{center}
-
-  \noindent 
-  In this definition we again omitted the cases for existential and unique existential
-  quantifiers. 
-
-%%% FIXME: Reviewer2 citing following sentence: You mention earlier
-%%% that this implication may fail to be true. Does that meant that
-%%% the `first proof step' is a heuristic that proves the implication
-%%% raw_thm \implies reg_thm in some instances, but fails in others?
-%%% You should clarify under which circumstances the implication is
-%%% being proved here.
-%%% Cezary: It would be nice to cite Homeiers discussions in the
-%%% Quotient Package manual from HOL (the longer paper), do you agree?
-
-  In the first phase, establishing @{text "raw_thm \<longrightarrow> reg_thm"}, we always 
-  start with an implication. Isabelle provides \emph{mono} rules that can split up 
-  the implications into simpler implicational subgoals. This succeeds for every
-  monotone connective, except in places where the function @{text REG} replaced,
-  for instance, a quantifier by a bounded quantifier. To decompose them, we have 
-  to prove that the relations involved are aggregate equivalence relations.
-
-  
-  %In this case we have 
-  %rules of the form
-  %
-  % \begin{isabelle}\ \ \ \ \ %%%
-  %@{text "(\<forall>x. R x \<longrightarrow> (P x \<longrightarrow> Q x)) \<longrightarrow> (\<forall>x. P x \<longrightarrow> \<forall>x \<in> R. Q x)"}
-  %\end{isabelle}
-
-  %\noindent
-  %They decompose a bounded quantifier on the right-hand side. We can decompose a
-  %bounded quantifier anywhere if R is an equivalence relation or
-  %if it is a relation over function types with the range being an equivalence
-  %relation. If @{text R} is an equivalence relation we can prove that
-
-  %\begin{isabelle}\ \ \ \ \ %%%
-  %@{text "\<forall>x \<in> Resp R. P x = \<forall>x. P x"}    
-  %\end{isabelle}
-
-  %\noindent
-  %If @{term R\<^isub>2} is an equivalence relation, we can prove that for any predicate @{term P}
-
-%%% FIXME Reviewer 1 claims the theorem is obviously false so maybe we
-%%% should include a proof sketch?
-
-  %\begin{isabelle}\ \ \ \ \ %%%
-  %@{thm (concl) ball_reg_eqv_range[of R\<^isub>1 R\<^isub>2, no_vars]}
-  %\end{isabelle}
-
-  %\noindent
-  %The last theorem is new in comparison with Homeier's package. There the
-  %injection procedure would be used to prove such goals and 
-  %the assumption about the equivalence relation would be used. We use the above theorem directly,
-  %because this allows us to completely separate the first and the second
-  %proof step into two independent ``units''.
-
-  The second phase, establishing @{text "reg_thm \<longleftrightarrow> inj_thm"},  starts with an equality
-  between the terms of the regularized theorem and the injected theorem.
-  The proof again follows the structure of the
-  two underlying terms taking respectfulness theorems into account.
-
-  %\begin{itemize}
-  %\item For two constants an appropriate respectfulness theorem is applied.
-  %\item For two variables, we use the assumptions proved in the regularization step.
-  %\item For two abstractions, we @{text "\<eta>"}-expand and @{text "\<beta>"}-reduce them.
-  %\item For two applications, we check that the right-hand side is an application of
-  %  @{term Rep} to an @{term Abs} and @{term "Quotient R Rep Abs"} holds. If yes then we
-  %  can apply the theorem:
-
-  %\begin{isabelle}\ \ \ \ \ %%%
-  %  @{term "R x y \<longrightarrow> R x (Rep (Abs y))"}
-  %\end{isabelle}
-
-  %  Otherwise we introduce an appropriate relation between the subterms
-  %  and continue with two subgoals using the lemma:
-
-  %\begin{isabelle}\ \ \ \ \ %%%
-  %  @{text "(R\<^isub>1 \<doublearr> R\<^isub>2) f g \<longrightarrow> R\<^isub>1 x y \<longrightarrow> R\<^isub>2 (f x) (g y)"}
-  %\end{isabelle}
-  %\end{itemize}
-
-  We defined the theorem @{text "inj_thm"} in such a way that
-  establishing in the third phase the equivalence @{text "inj_thm \<longleftrightarrow> quot_thm"} can be
-  achieved by rewriting @{text "inj_thm"} with the preservation theorems and quotient
-  definitions. This step also requires that the definitions of all lifted constants
-  are used to fold the @{term Rep} with the raw constants. We will give more details
-  about our lifting procedure in a longer version of this paper.
-
-  %Next for
-  %all abstractions and quantifiers the lambda and
-  %quantifier preservation theorems are used to replace the
-  %variables that include raw types with respects by quantifiers
-  %over variables that include quotient types. We show here only
-  %the lambda preservation theorem. Given
-  %@{term "Quotient R\<^isub>1 Abs\<^isub>1 Rep\<^isub>1"} and @{term "Quotient R\<^isub>2 Abs\<^isub>2 Rep\<^isub>2"}, we have:
-
-  %\begin{isabelle}\ \ \ \ \ %%%
-  %@{thm (concl) lambda_prs[of _ "Abs\<^isub>1" "Rep\<^isub>1" _ "Abs\<^isub>2" "Rep\<^isub>2", no_vars]}
-  %\end{isabelle}
-
-  %\noindent
-  %Next, relations over lifted types can be rewritten to equalities
-  %over lifted type. Rewriting is performed with the following theorem,
-  %which has been shown by Homeier~\cite{Homeier05}:
-
-  %\begin{isabelle}\ \ \ \ \ %%%
-  %@{thm (concl) Quotient_rel_rep[no_vars]}
-  %\end{isabelle}
-
-  
-  %Finally, we rewrite with the preservation theorems. This will result
-  %in two equal terms that can be solved by reflexivity.
-*}
-
-section {* Conclusion and Related Work\label{sec:conc}*}
-
-text {*
-
-  \noindent
-  The code of the quotient package and the examples described here are already
-  included in the standard distribution of Isabelle.
-  \footnote{Available from \href{http://isabelle.in.tum.de/}{http://isabelle.in.tum.de/}.}
-  The package is
-  heavily used in the new version of Nominal Isabelle, which provides a
-  convenient reasoning infrastructure for programming language calculi
-  involving general binders.  To achieve this, it builds types representing
-  @{text \<alpha>}-equivalent terms.  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}.
-
-
-  There is a wide range of existing literature for dealing with quotients
-  in theorem provers.  Slotosch~\cite{Slotosch97} implemented a mechanism that
-  automatically defines quotient types for Isabelle/HOL. But he did not
-  include theorem lifting.  Harrison's quotient package~\cite{harrison-thesis}
-  is the first one that is able to automatically lift theorems, however only
-  first-order theorems (that is theorems where abstractions, quantifiers and
-  variables do not involve functions that include the quotient type). There is
-  also some work on quotient types in non-HOL based systems and logical
-  frameworks, including theory interpretations in
-  PVS~\cite{PVS:Interpretations}, new types in MetaPRL~\cite{Nogin02}, and
-  setoids in Coq \cite{ChicliPS02}.  Paulson showed a construction of
-  quotients that does not require the Hilbert Choice operator, but also only
-  first-order theorems can be lifted~\cite{Paulson06}.  The most related work
-  to our package is the package for HOL4 by Homeier~\cite{Homeier05}.  He
-  introduced most of the abstract notions about quotients and also deals with
-  lifting of higher-order theorems. However, he cannot deal with quotient
-  compositions (needed for lifting theorems about @{text flat}). Also, a
-  number of his definitions, like @{text ABS}, @{text REP} and @{text INJ} etc
-  only exist in \cite{Homeier05} as ML-code, not included in the paper.
-  Like Homeier's, our quotient package can deal with partial equivalence
-  relations, but for lack of space we do not describe the mechanisms
-  needed for this kind of quotient constructions.
-
-%%% FIXME Reviewer 3 would like to know more about the lifting in Coq and PVS,
-%%% and some comparison. I don't think we have the space for any additions...
-
-  One feature of our quotient package is that when lifting theorems, the user
-  can precisely specify what the lifted theorem should look like. This feature
-  is necessary, for example, when lifting an induction principle for two
-  lists.  Assuming this principle has as the conclusion a predicate of the
-  form @{text "P xs ys"}, then we can precisely specify whether we want to
-  quotient @{text "xs"} or @{text "ys"}, or both. We found this feature very
-  useful in the new version of Nominal Isabelle, where such a choice is
-  required to generate a reasoning infrastructure for alpha-equated terms.
-%%
-%% give an example for this
-%%
-  \smallskip
-
-  \noindent
-  {\bf Acknowledgements:} We would like to thank Peter Homeier for the many
-  discussions about his HOL4 quotient package and explaining to us
-  some of its finer points in the implementation. Without his patient
-  help, this work would have been impossible.
-*}
-
-text_raw {*
-  %%\bibliographystyle{abbrv}
-  \bibliography{root}
-
-  \appendix
-
-*}
-
-section {* Examples \label{sec:examples} *}
-
-text {*
-
-%%% FIXME Reviewer 1 would like an example of regularized and injected
-%%% statements. He asks for the examples twice, but I would still ignore
-%%% it due to lack of space...
-
-  \noindent
-  In this appendix we will show a sequence of declarations for defining the 
-  type of integers by quotienting pairs of natural numbers, and
-  lifting one theorem. 
-
-  A user of our quotient package first needs to define a relation on
-  the raw type with which the quotienting will be performed. We give
-  the same integer relation as the one presented in \eqref{natpairequiv}:
-
-  \begin{isabelle}\ \ \ \ \ %
-  \begin{tabular}{@ {}l}
-  \isacommand{fun}~~@{text "int_rel :: (nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"}\\
-  \isacommand{where}~~@{text "int_rel (m, n) (p, q) = (m + q = n + p)"}
-  \end{tabular}
-  \end{isabelle}
-
-  \noindent
-  Next the quotient type must be defined. This generates a proof obligation that the
-  relation is an equivalence relation, which is solved automatically using the
-  definition of equivalence and extensionality:
-
-  \begin{isabelle}\ \ \ \ \ %
-  \begin{tabular}{@ {}l}
-  \isacommand{quotient\_type}~~@{text "int"}~~\isacommand{=}~~@{text "(nat \<times> nat)"}~~\isacommand{/}~~@{text "int_rel"}\\
-  \hspace{5mm}@{text "by (auto simp add: equivp_def expand_fun_eq)"}
-  \end{tabular}
-  \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)"}\\[3mm]
-  \isacommand{fun}~~@{text "add_pair"}\\
-  \isacommand{where}~~%
-  @{text "add_pair (m, n) (p, q) \<equiv> (m + p :: nat, n + q :: nat)"}\\
-  \isacommand{quotient\_definition}~~@{text "+ :: int \<Rightarrow> int \<Rightarrow> int"}~~%
-  \isacommand{is}~~@{text "add_pair"}\\
-  \end{tabular}
-  \end{isabelle}
-
-  \noindent
-  The following theorem about addition on the raw level can be proved.
-
-  \begin{isabelle}\ \ \ \ \ %
-  \isacommand{lemma}~~@{text "add_pair_zero: int_rel (add_pair (0, 0) x) x"}
-  \end{isabelle}
-
-  \noindent
-  If the user lifts this theorem, the quotient package performs all the lifting
-  automatically leaving the respectfulness proof for the constant @{text "add_pair"}
-  as the only remaining proof obligation. This property needs to be proved by the user:
-
-  \begin{isabelle}\ \ \ \ \ %
-  \begin{tabular}{@ {}l}
-  \isacommand{lemma}~~@{text "[quot_respect]:"}\\ 
-  @{text "(int_rel \<doublearr> int_rel \<doublearr> int_rel) add_pair add_pair"}
-  \end{tabular}
-  \end{isabelle}
-
-  \noindent
-  It can be discharged automatically by Isabelle when hinting to unfold the definition
-  of @{text "\<doublearr>"}.
-  After this, the user can prove the lifted lemma as follows:
-
-  \begin{isabelle}\ \ \ \ \ %
-  \isacommand{lemma}~~@{text "0 + (x :: int) = x"}~~\isacommand{by}~~@{text "lifting add_pair_zero"}
-  \end{isabelle}
-
-  \noindent
-  or by using the completely automated mode stating just:
-
-  \begin{isabelle}\ \ \ \ \ %
-  \isacommand{thm}~~@{text "add_pair_zero[quot_lifted]"}
-  \end{isabelle}
-
-  \noindent
-  Both methods give the same result, namely @{text "0 + x = x"}
-  where @{text x} is of type integer.
-  Although seemingly simple, arriving at this result without the help of a quotient
-  package requires a substantial reasoning effort (see \cite{Paulson06}).
-*}
-
-
-
-(*<*)
-end
-(*>*)