(*<*)+ −
theory Paper+ −
imports "Quotient"+ −
"LaTeXsugar"+ −
"../Nominal/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?)+ −
+ −
- Quote from Peter:+ −
+ −
One might think quotient have been studied to death, but+ −
+ −
- 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+ −
"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 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}. + −
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}+ −
%%\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 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. } 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+ −
+ −
@{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_list Rep_fset \<circ> Rep_fset) S))"}+ −
+ −
\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 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. Homeier's and+ −
also our quotient package are modular so that they allow lifting+ −
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 respectfulness+ −
and preservation; Section \ref{sec:lift} describes the lifting of theorems;+ −
Section \ref{sec:examples} presents some examples+ −
and Section \ref{sec:conc} concludes and compares our results to existing + −
work.+ −
*}+ −
+ −
section {* Preliminaries and General Quotients\label{sec:prelims} *}+ −
+ −
text {*+ −
We 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 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 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+ −
+ −
@{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+ −
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>}. For a type @{text \<kappa>} with arguments @{text "\<alpha>\<^isub>1\<^isub>\<dots>\<^isub>n"} the+ −
type of @{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+ −
%+ −
@{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+ −
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 Quantification and Bounded Abstractions]\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) = 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{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 lies in the fact that validity of @{text "Quotient R Abs Rep"} can + −
often be proved in terms of the validity of @{text "Quotient"} 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}+ −
@{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+ −
As a result, Homeier is able to build an automatic prover that can nearly+ −
always discharge a proof obligation involving @{text "Quotient"}. Our quotient+ −
package makes heavy + −
use of this part of Homeier's work including an extension + −
for dealing with compositions of equivalence relations 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+ −
+ −
@{text [display, indent=2] "Quotient (rel_list R \<circ>\<circ>\<circ> \<approx>\<^bsub>list\<^esub>) (Abs_fset \<circ> map_list Abs) (map_list Rep \<circ> Rep_fset)"}+ −
+ −
\vspace{-.5mm}+ −
*}+ −
+ −
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 @{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. 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 is 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}+ −
@{text "Quotient R Abs_\<kappa>\<^isub>q Rep_\<kappa>\<^isub>q"}.+ −
\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}{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 with @{text "\<alpha>s+ −
\<kappa>\<^isub>q"} being the quotient of the raw type @{text "\<rho>s \<kappa>"}:}\\+ −
@{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 we rely 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"}). 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>"}. 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 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.+ −
+ −
@{text [display, indent=10] "\<lambda>a b. map_prod (map_list a) b"}+ −
+ −
\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+ −
+ −
@{text [display, indent=10] "(map_list (map_list id \<circ> Rep_fset) \<circ> Rep_fset) \<singlearr> Abs_fset \<circ> map_list id"}+ −
+ −
\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+ −
+ −
@{text [display, indent=10] "(map_list 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_list 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 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 {*+ −
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}\ \ \ \ \ \ \ \ \ \ %%%+ −
@{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+ −
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}{rcl}+ −
\multicolumn{3}{@ {\hspace{-4mm}}l}{equal types:}\\ + −
@{text "REL (\<sigma>, \<sigma>)"} & $\dn$ & @{text "= :: \<sigma> \<Rightarrow> \<sigma> \<Rightarrow> bool"}\smallskip\\+ −
\multicolumn{3}{@ {\hspace{-4mm}}l}{equal type constructors:}\\ + −
@{text "REL (\<sigma>s \<kappa>, \<tau>s \<kappa>)"} & $\dn$ & @{text "rel_\<kappa> (REL (\<sigma>s, \<tau>s))"}\smallskip\\+ −
\multicolumn{3}{@ {\hspace{-4mm}}l}{unequal type constructors with @{text "\<alpha>s+ −
\<kappa>\<^isub>q"} being the quotient of the raw type @{text "\<rho>s \<kappa>"}:}\smallskip\\+ −
@{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}:+ −
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+ −
+ −
@{text [display, indent=10] "REL (\<sigma>, \<tau>) c\<^isub>r c\<^isub>r"}+ −
+ −
\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+ −
+ −
@{text [display, indent=10] "(\<approx>\<^isub>\<alpha> \<doublearr> =) bn bn"}+ −
+ −
\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+ −
+ −
@{text [display, indent=10] "\<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)"}+ −
+ −
\noindent+ −
In contrast, if we lift a theorem about @{text "append"} to a theorem describing + −
the union of finite sets, then we need to discharge the proof obligation+ −
+ −
@{text [display, indent=10] "(\<approx>\<^bsub>list\<^esub> \<doublearr> \<approx>\<^bsub>list\<^esub> \<doublearr> \<approx>\<^bsub>list\<^esub>) append append"}+ −
+ −
\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+ −
+ −
@{text [display, indent=10] "Quotient 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"}+ −
+ −
\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 + −
+ −
@{text [display, indent=10] "(Rep ---> map_list Rep ---> map_list Abs) cons = cons"}+ −
+ −
\noindent+ −
under the assumption @{text "Quotient R Abs Rep"}. Interestingly, 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 the corresponding 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+ −
+ −
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.+ −
+ −
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.+ −
+ −
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 simultaneously recursing on + −
the structure of @{text "raw_thm"} and @{text "quot_thm"} as follows:+ −
+ −
\begin{center}+ −
\begin{tabular}{rcl}+ −
\multicolumn{3}{@ {}l}{abstractions:}\smallskip\\+ −
@{text "REG (\<lambda>x\<^sup>\<sigma>. t, \<lambda>x\<^sup>\<tau>. s)"} & $\dn$ & + −
$\begin{cases}+ −
@{text "\<lambda>x\<^sup>\<sigma>. REG (t, s)"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\+ −
@{text "\<lambda>x\<^sup>\<sigma> \<in> Respects (REL (\<sigma>, \<tau>)). REG (t, s)"}+ −
\end{cases}$\smallskip\\+ −
\\+ −
\multicolumn{3}{@ {}l}{universal quantifiers:}\\+ −
@{text "REG (\<forall>x\<^sup>\<sigma>. t, \<forall>x\<^sup>\<tau>. s)"} & $\dn$ & + −
$\begin{cases}+ −
@{text "\<forall>x\<^sup>\<sigma>. REG (t, s)"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\+ −
@{text "\<forall>x\<^sup>\<sigma> \<in> Respects (REL (\<sigma>, \<tau>)). REG (t, s)"}+ −
\end{cases}$\smallskip\\+ −
\multicolumn{3}{@ {}l}{equality:}\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{3}{@ {}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}{rcl}+ −
\multicolumn{3}{@ {\hspace{-4mm}}l}{abstractions:}\\+ −
@{text "INJ (\<lambda>x. t :: \<sigma>, \<lambda>x. s :: \<tau>) "} & $\dn$ & + −
$\begin{cases}+ −
@{text "\<lambda>x. INJ (t, s)"} \quad\mbox{provided @{text "\<sigma> = \<tau>"}}\\+ −
@{text "REP (\<sigma>, \<tau>) (ABS (\<sigma>, \<tau>) (\<lambda>x. INJ (t, s)))"}+ −
\end{cases}$\\+ −
@{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)))"}\smallskip\\+ −
\multicolumn{3}{@ {\hspace{-4mm}}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{3}{@ {\hspace{-4mm}}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 proof step, 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. In this case we have + −
rules of the form+ −
+ −
@{text [display, indent=10] "(\<forall>x. R x \<longrightarrow> (P x \<longrightarrow> Q x)) \<longrightarrow> (\<forall>x. P x \<longrightarrow> \<forall>x \<in> R. Q x)"}+ −
+ −
\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+ −
+ −
@{text [display, indent=10] "\<forall>x \<in> Respects R. P x = \<forall>x. P x"} + −
+ −
\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?+ −
+ −
@{thm [display, indent=10] (concl) ball_reg_eqv_range[of R\<^isub>1 R\<^isub>2, no_vars]}+ −
+ −
\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 proof step, 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 and is defined for a goal being a relation between these two terms.+ −
+ −
\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:+ −
+ −
@{term [display, indent=10] "R x y \<longrightarrow> R x (Rep (Abs y))"}+ −
+ −
Otherwise we introduce an appropriate relation between the subterms+ −
and continue with two subgoals using the lemma:+ −
+ −
@{text [display, indent=10] "(R\<^isub>1 \<doublearr> R\<^isub>2) f g \<longrightarrow> R\<^isub>1 x y \<longrightarrow> R\<^isub>2 (f x) (g y)"}+ −
\end{itemize}+ −
+ −
We defined the theorem @{text "inj_thm"} in such a way that+ −
establishing the equivalence @{text "inj_thm \<longleftrightarrow> quot_thm"} can be+ −
achieved by rewriting @{text "inj_thm"} with the preservation theorems and quotient+ −
definitions. First the definitions of all lifted constants+ −
are used to fold the @{term Rep} with the raw constants. 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:+ −
+ −
@{thm [display, indent=10] (concl) lambda_prs[of _ "Abs\<^isub>1" "Rep\<^isub>1" _ "Abs\<^isub>2" "Rep\<^isub>2", no_vars]}+ −
+ −
\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}:+ −
+ −
@{thm [display, indent=10] (concl) Quotient_rel_rep[no_vars]}+ −
+ −
\noindent+ −
Finally, we rewrite with the preservation theorems. This will result+ −
in two equal terms that can be solved by reflexivity.+ −
*}+ −
+ −
+ −
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...+ −
+ −
In this section 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> (nat \<times> nat)"}\\+ −
\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 [display, indent=10] "0 + x = x"}+ −
+ −
\noindent+ −
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}).+ −
*}+ −
+ −
section {* Conclusion and Related Work\label{sec:conc}*}+ −
+ −
text {*+ −
+ −
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+ −
%%+ −
\medskip+ −
+ −
\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.+ −
+ −
*}+ −
+ −
+ −
+ −
(*<*)+ −
end+ −
(*>*)+ −