--- a/Pearl-jv/Paper.thy Mon Jun 06 13:11:04 2011 +0100
+++ b/Pearl-jv/Paper.thy Tue Jun 07 08:52:59 2011 +0100
@@ -80,7 +80,7 @@
type to represent atoms of different sorts. The other is how to
present sort-respecting permutations. For them we use the standard
technique of HOL-formalisations of introducing an appropriate
- substype of functions from atoms to atoms.
+ subtype of functions from atoms to atoms.
The nominal logic work has been the starting point for a number of proving
infrastructures, most notable by Norrish \cite{norrish04} in HOL4, by
@@ -323,7 +323,7 @@
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
\begin{tabular}{@ {}l}
- i)~~@{thm add_assoc[where a="\<pi>\<^isub>1" and b="\<pi>\<^isub>2" and c="\<pi>\<^isub>3"]}\\
+ i)~~@{thm add_assoc[where a="\<pi>\<^isub>1" and b="\<pi>\<^isub>2" and c="\<pi>\<^isub>3"]}\smallskip\\
ii)~~@{thm monoid_add_class.add_0_left[where a="\<pi>::perm"]} \hspace{9mm}
iii)~~@{thm monoid_add_class.add_0_right[where a="\<pi>::perm"]} \hspace{9mm}
iv)~~@{thm group_add_class.left_minus[where a="\<pi>::perm"]}
@@ -357,7 +357,7 @@
\noindent
whereby @{text "\<beta>"} is a generic type for the object @{text
- x}.\footnote{We will use the standard notation @{text "((op \<bullet>) \<pi>)
+ x}.\footnote{We will write @{text "((op \<bullet>) \<pi>)
x"} for this operation in the few cases where we need to indicate
that it is a function applied with two arguments.} The definition
of this operation will be given by in terms of `induction' over this
@@ -506,7 +506,8 @@
\emph{equivariance} and the \emph{equivariance principle}. These
notions allows us to characterise how permutations act upon compound
statements in HOL by analysing how these statements are constructed.
- The notion of equivariance can defined as follows:
+ The notion of equivariance means that an object is invariant under
+ any permutations. This can be defined as follows:
\begin{definition}[Equivariance]\label{equivariance}
An object @{text "x"} of permutation type is \emph{equivariant} provided
@@ -518,8 +519,8 @@
@{text x} is a constant, but of course there is no way in
Isabelle/HOL to restrict this definition to just these cases.
- There are a number of equivalent formulations for the equivariance
- property. For example, assuming @{text f} is a function of permutation
+ There are a number of equivalent formulations for equivariance.
+ For example, assuming @{text f} is a function of permutation
type @{text "\<alpha> \<Rightarrow> \<beta>"}, then equivariance of @{text f} can also be stated as
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
@@ -529,7 +530,7 @@
\end{isabelle}
\noindent
- We will call this formulation of equivariance in \emph{fully applied form}.
+ We will say this formulation of equivariance is in \emph{fully applied form}.
To see that this formulation implies the definition, we just unfold
the definition of the permutation operation for functions and
simplify with the equation and the cancellation property shown in
@@ -602,8 +603,8 @@
legibility we leave the typing information implicit. We also assume
the usual notions for free and bound variables of a HOL-term.
Furthermore, HOL-terms are regarded as equal modulo alpha-, beta-
- and eta-equivalence. The equivariance principle can now be stated
- formally as follows:
+ and eta-equivalence. The equivariance principle can now
+ be stated formally as follows:
\begin{theorem}[Equivariance Principle]\label{eqvtprin}
Suppose a HOL-term @{text t} whose constants are all equivariant. For any
@@ -615,9 +616,20 @@
\noindent
The significance of this principle is that we can automatically establish
the equivariance of a constant for which equivariance is not yet
- known. For this we only have to make sure that the definiens of this
- constant is a HOL-term whose constants are all equivariant. For example
- the universal quantifier @{text "\<forall>"} is definied in HOL as
+ known. For this we only have to establish that the definiens of this
+ constant is a HOL-term whose constants are all equivariant.
+ This meshes well with how HOL is designed: except for a few axioms, every constant
+ is defined in terms of existing constants. For example an alternative way
+ to deduce that @{term True} is equivariant is to look at its
+ definition
+
+ \begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
+ @{thm True_def}
+ \end{isabelle}
+
+ \noindent
+ and observing that the only constant in the definiens, namely @{text "="}, is
+ equivariant. Similarly, the universal quantifier @{text "\<forall>"} is definied in HOL as
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
@{text "\<forall>x. P x \<equiv> "}~@{thm (rhs) All_def[no_vars]}
@@ -629,7 +641,11 @@
the equivariance principle gives us
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
- @{text "\<pi> \<bullet> (\<forall>x. P x) \<equiv> \<pi> \<bullet> (P = (\<lambda>x. True)) = ((\<pi> \<bullet> P) = (\<lambda>x. True)) \<equiv> \<forall>x. (\<pi> \<bullet> P) x"}
+ \begin{tabular}{r@ {\hspace{1mm}}c@ {\hspace{1mm}}l}
+ @{text "\<pi> \<bullet> (\<forall>x. P x)"} & @{text "\<equiv>"} & @{text "\<pi> \<bullet> (P = (\<lambda>x. True))"}\\
+ & @{text "="} & @{text "(\<pi> \<bullet> P) = (\<lambda>x. True)"}\\
+ & @{text "\<equiv>"} & @{text "\<forall>x. (\<pi> \<bullet> P) x"}
+ \end{tabular}
\end{isabelle}
\noindent
@@ -653,7 +669,8 @@
\noindent
with all constants on the right-hand side being equivariant. With this kind
- of reasoning we can build up a database of equivariant constants.
+ of reasoning we can build up a database of equivariant constants, which will
+ be handy for more complex calculations later on.
Before we proceed, let us give a justification for the equivariance principle.
This justification cannot be given directly inside Isabelle/HOL since we cannot
@@ -670,14 +687,15 @@
permutation inside the term @{text t}. We have implemented this as a
conversion tactic on the ML-level of Isabelle/HOL. In what follows,
we will show that this tactic produces only finitely many equations
- and also show that is correct (in the sense of pushing a permutation
+ and also show that it is correct (in the sense of pushing a permutation
@{text "\<pi>"} inside a term and the only remaining instances of @{text
- "\<pi>"} are in front of the term's free variables). The tactic applies
- four `oriented' equations. We will first give a naive version of
- this tactic, which however in some cornercases produces incorrect
+ "\<pi>"} are in front of the term's free variables).
+
+ The tactic applies four `oriented' equations.
+ We will first give a naive version of
+ our tactic, which however in some corner cases produces incorrect
results or does not terminate. We then give a modification in order
to obtain the desired properties.
-
Consider the following for oriented equations
\begin{isabelle}\ \ \ \ \ \ \ \ \ \ %%%
@@ -697,10 +715,9 @@
and the fact that HOL-terms are equal modulo beta-equivalence.
The third is a consequence of \eqref{cancel} and the fourth from
Definition~\ref{equivariance}. Unfortunately, we have to be careful with
- the rules {\it i)} and {\it iv}) since they can lead to a loop whenever
- \mbox{@{text "t\<^isub>1 t\<^isub>2"}} is of the form @{text "((op \<bullet>) \<pi>') t"}.\footnote{Note we
- deviate here from our usual convention of writing the permutation operation infix,
- instead as an application.} Recall that we established in Lemma \ref{permutecompose} that the
+ the rules {\it i)} and {\it iv}) since they can lead to loops whenever
+ \mbox{@{text "t\<^isub>1 t\<^isub>2"}} is of the form @{text "((op \<bullet>) \<pi>') t"}.
+ Recall that we established in Lemma \ref{permutecompose} that the
constant @{text "(op \<bullet>)"} is equivariant and consider the infinite
reduction sequence
@@ -716,7 +733,7 @@
\end{isabelle}
\noindent
- where the last term is again an instance of rewrite rule {\it i}), but bigger.
+ where the last term is again an instance of rewrite rule {\it i}), but larger.
To avoid this loop we will apply the rewrite rule
using an `outside to inside' strategy. This strategy is sufficient
since we are only interested of rewriting terms of the form @{term
@@ -726,8 +743,8 @@
iii)} can `overlap'. For this note that the term @{term "\<pi>
\<bullet>(\<lambda>x. x)"} reduces by {\it ii)} to @{term "\<lambda>x. \<pi> \<bullet> (- \<pi>) \<bullet> x"}, to
which we can apply rule {\it iii)} in order to obtain @{term
- "\<lambda>x. x"}, as is desired---since there is no free variable in the original
- term. the permutation should completely vanish. However, the
+ "\<lambda>x. x"}, as is desired: since there is no free variable in the original
+ term, the permutation should completely vanish. However, the
subterm @{text "(- \<pi>) \<bullet> x"} is also an application. Consequently,
the term @{term "\<lambda>x. \<pi> \<bullet> (- \<pi>) \<bullet>x"} can also reduce to @{text "\<lambda>x. (- (\<pi>
\<bullet> \<pi>)) \<bullet> (\<pi> \<bullet> x)"} using {\it i)}. Given our strategy, we cannot
@@ -1222,11 +1239,11 @@
representation for permutations (for example @{term "(a \<rightleftharpoons> b)"} and
@{term "(b \<rightleftharpoons> a)"} are equal permutations), this representation does
not come automatically with an induction principle. Such an
- induction principle is however handy for generalising
- Lemma~\ref{swapfreshfresh} from swappings to permutations
+ induction principle is however useful for generalising
+ Lemma~\ref{swapfreshfresh} from swappings to permutations, namely
\begin{lemma}
- @{thm [mode=IfThen] perm_supp_eq[no_vars]}
+ @{thm [mode=IfThen] perm_supp_eq[where p="\<pi>", no_vars]}
\end{lemma}
\noindent
@@ -1238,7 +1255,7 @@
Using a the property from \cite{???}
\begin{lemma}\label{smallersupp}
- @{thm [mode=IfThen] smaller_supp[no_vars]}
+ @{thm [mode=IfThen] smaller_supp[where p="\<pi>", no_vars]}
\end{lemma}
*}