
diff -r 1d70813ae674 -r 0368aef38e6a Paper/Paper.thy
--- a/Paper/Paper.thy	Mon Mar 22 10:21:14 2010 +0100
+++ b/Paper/Paper.thy	Mon Mar 22 11:55:29 2010 +0100
@@ -57,25 +57,25 @@
   by iterating single binders. For example in the case of type-schemes we do not
   like to make a distinction about the order of the bound variables. Therefore
   we would like to regard the following two type-schemes as alpha-equivalent
-
-  \begin{center}
-  $\forall \{x, y\}. x \rightarrow y  \;\approx_\alpha\; \forall \{y, x\}. y \rightarrow x$ 
-  \end{center}
+  %
+  \begin{equation}\label{ex1}
+  \forall \{x, y\}. x \rightarrow y  \;\approx_\alpha\; \forall \{y, x\}. y \rightarrow x 
+  \end{equation}
 
   \noindent
   but  the following two should \emph{not} be alpha-equivalent
-
-  \begin{center}
-  $\forall \{x, y\}. x \rightarrow y  \;\not\approx_\alpha\; \forall \{z\}. z \rightarrow z$ 
-  \end{center}
+  %
+  \begin{equation}\label{ex2}
+  \forall \{x, y\}. x \rightarrow y  \;\not\approx_\alpha\; \forall \{z\}. z \rightarrow z 
+  \end{equation}
 
   \noindent
   assuming that $x$, $y$ and $z$ are distinct. Moreover, we like to regard type-schemes as 
   alpha-equivalent, if they differ only on \emph{vacuous} binders, such as
-
-  \begin{center}
-  $\forall \{x\}. x \rightarrow y  \;\approx_\alpha\; \forall \{x, z\}. x \rightarrow y$ 
-  \end{center}
+  %
+  \begin{equation}\label{ex3}
+  \forall \{x\}. x \rightarrow y  \;\approx_\alpha\; \forall \{x, z\}. x \rightarrow y
+  \end{equation}
 
   \noindent
   where $z$ does not occur freely in the type.
@@ -268,7 +268,7 @@
   be excluded by reasoning about a well-formedness predicate.
 
   The problem with introducing a new type in Isabelle/HOL is that in order to be useful 
-  a resoning infrastructure needs to be ``lifted'' from the underlying subset to 
+  a reasoning infrastructure needs to be ``lifted'' from the underlying subset to 
   the new type. This is usually a tricky and arduous task. To ease it
   we reimplemented in Isabelle/HOL the quotient package described by Homeier 
   \cite{Homeier05}. Given that alpha is an equivalence relation, this package 
@@ -279,7 +279,7 @@
   variables of raw terms to alpha-equated terms (since this function respects 
   alpha-equivalence), but we will not be able to do this with a bound-variable
   function (since it does not respect alpha-equivalence). As a result, each
-  lifting needs some respectulness proofs which we automated.\medskip
+  lifting needs some respectfulness proofs which we automated.\medskip
 
   \noindent
   {\bf Contributions:}  We provide new definitions for when terms
@@ -363,8 +363,8 @@
 section {* General Binders *}
 
 text {*
-  In order to keep our work managable we give need to give definitions  
-  and perform proofs inside Isabelle whereever possible, as opposed to write
+  In order to keep our work manageable we give need to give definitions  
+  and perform proofs inside Isabelle wherever possible, as opposed to write
   custom ML-code that generates them  for each 
   instance of a term-calculus. To this end we will first consider pairs
 
@@ -378,45 +378,82 @@
   in the body $x$ (similarly to type-schemes given in \eqref{tysch}). 
 
   The first question we have to answer is when we should consider pairs such as
-  $(as, x)$ and $(bs, y)$ as alpha-equivelent? (At the moment we are interested in
+  $(as, x)$ and $(bs, y)$ as alpha-equivalent? (At the moment we are interested in
   the notion of alpha-equivalence that is \emph{not} preserved by adding 
-  vacuous binders.) For this we identify four conditions: i) given a free-variable function
-  of type \mbox{@{text "fv :: \<beta> \<Rightarrow> atom set"}}, then $x$ and $y$ 
-  need to have the same set of free variables; ii) there must be a permutation,
-  say $p$, that leaves the free variables $x$ and $y$ unchanged, but ``moves'' their bound names 
-  so that we obtain modulo a relation, say @{text "_ R _"}, 
-  two equal terms. We also require that $p$ makes the abstracted sets equal. These 
-  requirements can be stated formally as
-
-  \begin{center}
-  \begin{tabular}{rcl}
-  a
-  \end{tabular}
-  \end{center}
-
-
-  Assuming we are given a free-variable function, say 
-  \mbox{@{text "fv :: \<beta> \<Rightarrow> atom set"}}, then we expect for two alpha-equivelent
-  pairs that their sets of free variables aggree. That is
+  vacuous binders.) To answer this we identify four conditions: {\it i)} given 
+  a free-variable function of type \mbox{@{text "fv :: \<beta> \<Rightarrow> atom set"}}, then $x$ and $y$ 
+  need to have the same set of free variables; moreover there must be a permutation,
+  $p$ that {\it ii)} leaves the free variables $x$ and $y$ unchanged, 
+  but {\it iii)} ``moves'' their bound names so that we obtain modulo a relation, 
+  say \mbox{@{text "_ R _"}}, two equal terms. We also require {\it iv)} that $p$ makes 
+  the abstracted sets $as$ and $bs$ equal (since at the moment we do not want 
+  that the sets $as$ and $bs$ differ on vacuous binders). These requirements can 
+  be stated formally as follows
   %
-  \begin{equation}\label{four}
-  \mbox{@{text "(as, x) \<approx> (bs, y)"} \hspace{2mm}implies\hspace{2mm} @{text "fv(x) - as = fv(y) - bs"}} 
+  \begin{equation}\label{alphaset}
+  \begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
+  \multicolumn{2}{l}{(as, x) \approx_{set}^{\fv, R, p} (bs, y) \;\dn\hspace{30mm}\;}\\[1mm]
+             & @{text "fv(x) - as = fv(y) - bs"}\\
+  \wedge     & @{text "fv(x) - as #* p"}\\
+  \wedge     & @{text "(p \<bullet> x) R y"}\\
+  \wedge     & @{text "(p \<bullet> as) = bs"}\\ 
+  \end{array}
   \end{equation}
 
   \noindent
-  Next we expect that there is a permutation, say $p$, that leaves the 
-  free variables unchanged, but ``moves'' the bound names in $x$ so that
-  we obtain $y$ modulo a relation, say @{text "_ R _"}, that characterises when two
-  elments of type $\beta$ are equivalent. We also expect that $p$ 
-  makes the binders equal. We can formulate these requirements as: there
-  exists a $p$ such that $i)$  @{term "(fv(x) - as) \<sharp>* p"}, $ii)$ @{text "(p \<bullet> x) R y"} and
-  $iii)$ @{text "(p \<bullet> as) = bs"}. 
+  Alpha equivalence between such pairs is then the relation with the additional 
+  existential quantification over $p$. Note that this relation is additionally 
+  dependent on the free-variable function $\fv$ and the relation $R$. The reason 
+  for this generality is that we want to use $\approx_{set}$ for both ``raw'' terms
+  and alpha-equated terms. In the latter case, $R$ can be replaced by equality $(op =)$ and
+  we are going to prove that $\fv$ will be equal to the support of $x$ and $y$.
+
+  The definition in \eqref{alphaset} does not make any distinction between the
+  order of abstracted variables. If we do want this then we can define alpha-equivalence 
+  for pairs of the form \mbox{@{text "(as, x) :: (atom list) \<times> \<beta>"}} by
+  %
+  \begin{equation}\label{alphalist}
+  \begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
+  \multicolumn{2}{l}{(as, x) \approx_{list}^{\fv, R, p} (bs, y) \;\dn\hspace{30mm}\;}\\[1mm]
+             & @{text "fv(x) - (set as) = fv(y) - (set bs)"}\\
+  \wedge     & @{text "fv(x) - (set as) #* p"}\\
+  \wedge     & @{text "(p \<bullet> x) R y"}\\
+  \wedge     & @{text "(p \<bullet> as) = bs"}\\ 
+  \end{array}
+  \end{equation}
+  
+  \noindent
+  where $set$ is just the function that coerces a list of atoms into a set of atoms.
 
-  We take now \eqref{four} and the three 
- 
+  If we do not want to make any difference between the order of binders and
+  allow vacuous binders, then we just need to drop the fourth condition in \eqref{alphaset}
+  and define
+  %
+  \begin{equation}\label{alphaset}
+  \begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
+  \multicolumn{2}{l}{(as, x) \approx_{res}^{\fv, R, p} (bs, y) \;\dn\hspace{30mm}\;}\\[1mm]
+             & @{text "fv(x) - as = fv(y) - bs"}\\
+  \wedge     & @{text "fv(x) - as #* p"}\\
+  \wedge     & @{text "(p \<bullet> x) R y"}\\
+  \end{array}
+  \end{equation}
 
-  General notion of alpha-equivalence (depends on a free-variable
-  function and a relation).
+  To get a feeling how these definitions pan out in practise consider the case of 
+  abstracting names over types (like in type-schemes). For this we set $R$ to be 
+  the equality and for $\fv(T)$ we define
+
+  \begin{center}
+  $\fv(x) = \{x\}  \qquad \fv(T_1 \rightarrow T_2) = \fv(T_1) \cup \fv(T_2)$
+  \end{center}
+
+  \noindent
+  Now reacall the examples in \eqref{ex1}, \eqref{ex2} and \eqref{ex3}: it can be easily 
+  checked that @{text "({x,y}, x \<rightarrow> y)"} and
+  @{text "({y,x}, y \<rightarrow> x)"} are equal according to $\approx_{set}$ and $\approx_{ref}$ by taking $p$ to
+  be the swapping @{text "(x \<rightleftharpoons> y)"}; but assuming @{text "x \<noteq> y"} then for instance 
+  $([x,y], x \rightarrow y) \not\approx_{list} ([y,x], x \rightarrow y)$ since there is no permutation that 
+  makes the lists @{text "[x,y]"} and @{text "[y,x]"} equal, but leaves the type \mbox{@{text "x \<rightarrow> y"}}
+  unchanged.
 *}
 
 section {* Alpha-Equivalence and Free Variables *}
@@ -425,7 +462,7 @@
   Restrictions
 
   \begin{itemize}
-  \item non-emptyness
+  \item non-emptiness
   \item positive datatype definitions
   \item finitely supported abstractions
   \item respectfulness of the bn-functions\bigskip




