--- a/Quotient-Paper/Paper.thy Sat Jun 12 06:35:27 2010 +0200
+++ b/Quotient-Paper/Paper.thy Sat Jun 12 11:32:36 2010 +0200
@@ -85,7 +85,7 @@
@{text "\<union>"}, as list append.
An area where quotients are ubiquitous is reasoning about programming language
- calculi. A simple example is the lambda-calculus, whose ``raw'' terms are defined as
+ calculi. A simple example is the lambda-calculus, whose raw terms are defined as
@{text [display, indent=10] "t ::= x | t t | \<lambda>x.t"}
@@ -94,7 +94,7 @@
prove formally, for example, 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
+ 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
@@ -103,7 +103,7 @@
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}
+ 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 to this work manually, if one has only a few quotient
@@ -143,13 +143,16 @@
\end{center}
\noindent
- The starting point is an existing type over which an equivalence relation
- given by the user is defined. With this input the package introduces a new
- type that comes with associated \emph{abstraction} and \emph{representation}
- functions, written @{text Abs} and @{text Rep}. These functions relate
- elements in the existing type to ones in the new type and vice versa; they
- can be uniquely identified by their type. For example for the integer
- quotient construction the types of @{text Abs} and @{text Rep} are
+ The starting point is an existing type, often referred to as the
+ \emph{raw level}, 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 level}. This type comes with an
+ \emph{abstraction} and a \emph{representation} function, written @{text Abs}
+ and @{text Rep}. These functions relate elements in the existing type to
+ ones in the new type and vice versa; they can be uniquely identified by
+ their 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"}
@@ -157,7 +160,7 @@
\noindent
However we often leave this typing information implicit for better
- readability. Every abstraction and representation function represents an
+ readability. 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
@@ -175,13 +178,13 @@
\noindent
where we take the representation of the arguments @{text n} and @{text m},
- add them according to @{text "add_pair"} and then take the
+ add them according to the function @{text "add_pair"} and then take the
abstraction of the result. This is all straightforward and the existing
quotient packages can deal with such definitions. But what is surprising
that none of them can deal with slightly more complicated definitions involving
\emph{compositions} of quotients. Such compositions are needed for example
- in case of quotienting lists and the operator that flattens lists of lists, defined
- as follows
+ in case of quotienting lists to obtain 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]}
@@ -195,7 +198,7 @@
The quotient package should provide us with a definition for @{text "\<Union>"} in
terms of @{text flat}, @{text Rep} and @{text Abs} (the latter two having
the type @{text "\<alpha> fset \<Rightarrow> \<alpha> list"} and @{text "\<alpha> list \<Rightarrow> \<alpha> fset"},
- respectively). The problem is that the method is used in the existing quotient
+ respectively). The problem is that the method used in the existing quotient
packages of just taking the representation of the arguments and then take
the abstraction of the result is \emph{not} enough. The reason is that case in case
of @{text "\<Union>"} we obtain the incorrect definition
@@ -205,7 +208,7 @@
\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 faltening of lists of finite sets. However, this remedy is rather
+ 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>"}
@@ -218,14 +221,14 @@
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
+ 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{respectfulness} and \emph{preservation} from Homeier
\cite{Homeier05}.
We will also 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. Our quotient package is modular so that it
+ level to the quotient level. Our quotient package id the first one that is modular so that it
allows to lift single theorems separately. This has the advantage for the
user to develop a formal theory interactively an a natural progression. A
pleasing result of the modularity is also that we are able to clearly