--- a/Paper/Paper.thy Fri Mar 26 16:20:39 2010 +0100
+++ b/Paper/Paper.thy Fri Mar 26 16:46:40 2010 +0100
@@ -3,15 +3,35 @@
imports "../Nominal/Test" "LaTeXsugar"
begin
+consts
+ fv :: "'a \<Rightarrow> 'b"
+ abs_set :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
+ Abs_lst :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
+ Abs_res :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"
+
+definition
+ "equal \<equiv> (op =)"
+
notation (latex output)
swap ("'(_ _')" [1000, 1000] 1000) and
fresh ("_ # _" [51, 51] 50) and
fresh_star ("_ #* _" [51, 51] 50) and
supp ("supp _" [78] 73) and
uminus ("-_" [78] 73) and
- If ("if _ then _ else _" 10)
+ If ("if _ then _ else _" 10) and
+ alpha_gen ("_ \<approx>\<^raw:\makebox[0mm][l]{$\,_{\textit{set}}$}>\<^bsup>_,_,_\<^esup> _") and
+ alpha_lst ("_ \<approx>\<^raw:\makebox[0mm][l]{$\,_{\textit{list}}$}>\<^bsup>_,_,_\<^esup> _") and
+ alpha_res ("_ \<approx>\<^raw:\makebox[0mm][l]{$\,_{\textit{res}}$}>\<^bsup>_,_,_\<^esup> _") and
+ abs_set ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
+ fv ("fv'(_')" [100] 100) and
+ equal ("=") and
+ alpha_abs ("_ \<approx>\<^raw:{$\,_{\textit{abs\_set}}$}> _") and
+ Abs ("[_]\<^raw:$\!$>\<^bsub>set\<^esub>._") and
+ Abs_lst ("[_]\<^raw:$\!$>\<^bsub>list\<^esub>._") and
+ Abs_res ("[_]\<^raw:$\!$>\<^bsub>res\<^esub>._")
(*>*)
+
section {* Introduction *}
text {*
@@ -19,12 +39,12 @@
alpha-equated terms, for example
\begin{center}
- $t ::= x \mid t\;t \mid \lambda x. t$
+ @{text "t ::= x | t t | \<lambda>x. t"}
\end{center}
\noindent
where free and bound variables have names. For such terms Nominal Isabelle
- derives automatically a reasoning infrastructure that has been used
+ derives automatically a reasoning infrastructure that has been used
successfully in formalisations of an equivalence checking algorithm for LF
\cite{UrbanCheneyBerghofer08}, Typed
Scheme~\cite{TobinHochstadtFelleisen08}, several calculi for concurrency
@@ -39,7 +59,8 @@
%
\begin{equation}\label{tysch}
\begin{array}{l}
- T ::= x \mid T \rightarrow T \hspace{5mm} S ::= \forall \{x_1,\ldots, x_n\}. T
+ @{text "T ::= x | T \<rightarrow> T"}\hspace{5mm}
+ @{text "S ::= \<forall>{x\<^isub>1,\<dots>, x\<^isub>n}. T"}
\end{array}
\end{equation}
@@ -59,38 +80,38 @@
we would like to regard the following two type-schemes as alpha-equivalent
%
\begin{equation}\label{ex1}
- \forall \{x, y\}. x \rightarrow y \;\approx_\alpha\; \forall \{y, x\}. y \rightarrow x
+ @{text "\<forall>{x,y}. x \<rightarrow> y \<approx>\<^isub>\<alpha> \<forall>{y,x}. y \<rightarrow> x"}
\end{equation}
\noindent
- but assuming that $x$, $y$ and $z$ are distinct variables,
+ but assuming that @{text x}, @{text y} and @{text z} are distinct variables,
the following two should \emph{not} be alpha-equivalent
%
\begin{equation}\label{ex2}
- \forall \{x, y\}. x \rightarrow y \;\not\approx_\alpha\; \forall \{z\}. z \rightarrow z
+ @{text "\<forall>{x,y}. x \<rightarrow> y \<notapprox>\<^isub>\<alpha> \<forall>{z}. z \<rightarrow> z"}
\end{equation}
\noindent
- Moreover, we like to regard type-schemes as
- alpha-equivalent, if they differ only on \emph{vacuous} binders, such as
+ Moreover, we like to regard type-schemes as alpha-equivalent, if they differ
+ only on \emph{vacuous} binders, such as
%
\begin{equation}\label{ex3}
- \forall \{x\}. x \rightarrow y \;\approx_\alpha\; \forall \{x, z\}. x \rightarrow y
+ @{text "\<forall>{x}. x \<rightarrow> y \<approx>\<^isub>\<alpha> \<forall>{x,z}. x \<rightarrow> y"}
\end{equation}
\noindent
- where $z$ does not occur freely in the type.
- In this paper we will give a general binding mechanism and associated
- notion of alpha-equivalence that can be used to faithfully represent
- this kind of binding in Nominal Isabelle. The difficulty of finding the right notion
- for alpha-equivalence can be appreciated in this case by considering that the
- definition given by Leroy in \cite{Leroy92} is incorrect (it omits a side-condition).
+ where @{text z} does not occur freely in the type. In this paper we will
+ give a general binding mechanism and associated notion of alpha-equivalence
+ that can be used to faithfully represent this kind of binding in Nominal
+ Isabelle. The difficulty of finding the right notion for alpha-equivalence
+ can be appreciated in this case by considering that the definition given by
+ Leroy in \cite{Leroy92} is incorrect (it omits a side-condition).
- However, the notion of alpha-equivalence that is preserved by vacuous binders is not
- always wanted. For example in terms like
+ However, the notion of alpha-equivalence that is preserved by vacuous
+ binders is not always wanted. For example in terms like
%
\begin{equation}\label{one}
- \LET x = 3 \AND y = 2 \IN x\,-\,y \END
+ @{text "\<LET> x = 3 \<AND> y = 2 \<IN> x - y \<END>"}
\end{equation}
\noindent
@@ -99,7 +120,7 @@
with
%
\begin{center}
- $\LET x = 3 \AND y = 2 \AND z = loop \IN x\,-\,y \END$
+ @{text "\<LET> x = 3 \<AND> y = 2 \<AND> z = loop \<IN> x - y \<END>"}
\end{center}
\noindent
@@ -109,10 +130,10 @@
However, we found that this is still not sufficient for dealing with
language constructs frequently occurring in programming language
- research. For example in $\mathtt{let}$s containing patterns
+ research. For example in @{text "\<LET>"}s containing patterns
%
\begin{equation}\label{two}
- \LET (x, y) = (3, 2) \IN x\,-\,y \END
+ @{text "\<LET> (x, y) = (3, 2) \<IN> x - y \<END>"}
\end{equation}
\noindent
@@ -121,72 +142,79 @@
we do not want to regard \eqref{two} as alpha-equivalent with
%
\begin{center}
- $\LET (y, x) = (3, 2) \IN x\,- y\,\END$
+ @{text "\<LET> (y, x) = (3, 2) \<IN> x - y \<END>"}
\end{center}
\noindent
- As a result, we provide three general binding mechanisms each of which binds multiple
- variables at once, and let the user chose which one is intended when formalising a
- programming language calculus.
+ As a result, we provide three general binding mechanisms each of which binds
+ multiple variables at once, and let the user chose which one is intended
+ when formalising a programming language calculus.
- By providing these general binding mechanisms, however, we have to work around
- a problem that has been pointed out by Pottier \cite{Pottier06} and Cheney
- \cite{Cheney05}: in $\mathtt{let}$-constructs of the form
+ By providing these general binding mechanisms, however, we have to work
+ around a problem that has been pointed out by Pottier \cite{Pottier06} and
+ Cheney \cite{Cheney05}: in @{text "\<LET>"}-constructs of the form
%
\begin{center}
- $\LET x_1 = t_1 \AND \ldots \AND x_n = t_n \IN s \END$
+ @{text "\<LET> x\<^isub>1 = t\<^isub>1 \<AND> \<dots> \<AND> x\<^isub>n = t\<^isub>n \<IN> s \<END>"}
\end{center}
\noindent
- which bind all the $x_i$ in $s$, we might not care about the order in
- which the $x_i = t_i$ are given, but we do care about the information that there are
- as many $x_i$ as there are $t_i$. We lose this information if we represent the
- $\mathtt{let}$-constructor by something like
+ which bind all the @{text "x\<^isub>i"} in @{text s}, we might not care
+ about the order in which the @{text "x\<^isub>i = t\<^isub>i"} are given,
+ but we do care about the information that there are as many @{text
+ "x\<^isub>i"} as there are @{text "t\<^isub>i"}. We lose this information if
+ we represent the @{text "\<LET>"}-constructor by something like
%
\begin{center}
- $\LET [x_1,\ldots,x_n].s\;\; [t_1,\ldots,t_n]$
+ @{text "\<LET> [x\<^isub>1,\<dots>,x\<^isub>n].s [t\<^isub>1,\<dots>,t\<^isub>n]"}
\end{center}
\noindent
- where the notation $[\_\!\_].\_\!\_$ indicates that the $x_i$ become bound
- in $s$. In this representation the term \mbox{$\LET [x].s\;\;[t_1,t_2]$}
- would be a perfectly legal instance. To exclude such terms, additional
- predicates about well-formed terms are needed in order to ensure that the two
- lists are of equal length. This can result into very messy reasoning (see
- for example~\cite{BengtsonParow09}). To avoid this, we will allow type specifications
- for $\mathtt{let}$s as follows
+ where the notation @{text "[_]._"} indicates that the @{text "x\<^isub>i"}
+ become bound in @{text s}. In this representation the term
+ \mbox{@{text "\<LET> [x].s [t\<^isub>1,t\<^isub>2]"}} would be a perfectly legal
+ instance. To exclude such terms, additional predicates about well-formed
+ terms are needed in order to ensure that the two lists are of equal
+ length. This can result into very messy reasoning (see for
+ example~\cite{BengtsonParow09}). To avoid this, we will allow type
+ specifications for $\mathtt{let}$s as follows
%
\begin{center}
\begin{tabular}{r@ {\hspace{2mm}}r@ {\hspace{2mm}}l}
- $trm$ & $::=$ & \ldots\\
- & $\mid$ & $\mathtt{let}\;a\!::\!assn\;\;s\!::\!trm\quad\mathtt{bind}\;bn\,(a) \IN s$\\[1mm]
- $assn$ & $::=$ & $\mathtt{anil}$\\
- & $\mid$ & $\mathtt{acons}\;\;name\;\;trm\;\;assn$
+ @{text trm} & @{text "::="} & @{text "\<dots>"}\\
+ & @{text "|"} & @{text "\<LET> a::assn s::trm"}\hspace{4mm}
+ \isacommand{bind} @{text "bn(a)"} \isacommand{in} @{text "s"}\\[1mm]
+ @{text assn} & @{text "::="} & @{text "\<ANIL>"}\\
+ & @{text "|"} & @{text "\<ACONS> name trm assn"}
\end{tabular}
\end{center}
\noindent
- where $assn$ is an auxiliary type representing a list of assignments
- and $bn$ an auxiliary function identifying the variables to be bound by
- the $\mathtt{let}$. This function is defined by recursion over $assn$ as follows
+ where @{text assn} is an auxiliary type representing a list of assignments
+ and @{text bn} an auxiliary function identifying the variables to be bound
+ by the @{text "\<LET>"}. This function is defined by recursion over @{text
+ assn} as follows
\begin{center}
- $bn\,(\mathtt{anil}) = \varnothing \qquad bn\,(\mathtt{acons}\;x\;t\;as) = \{x\} \cup bn\,(as)$
+ @{text "bn(\<ANIL>) ="} @{term "{}"} \hspace{5mm}
+ @{text "bn(\<ACONS> x t as) = {x} \<union> bn(as)"}
\end{center}
\noindent
The scope of the binding is indicated by labels given to the types, for
- example \mbox{$s\!::\!trm$}, and a binding clause, in this case
- $\mathtt{bind}\;bn\,(a) \IN s$, that states to bind in $s$ all the names the
- function call $bn\,(a)$ returns. This style of specifying terms and bindings is
- heavily inspired by the syntax of the Ott-tool \cite{ott-jfp}.
+ example @{text "s::trm"}, and a binding clause, in this case
+ \isacommand{bind} @{text "bn(a)"} \isacommand{in} @{text "s"}, that states
+ to bind in @{text s} all the names the function call @{text "bn(a)"} returns.
+ This style of specifying terms and bindings is heavily inspired by the
+ syntax of the Ott-tool \cite{ott-jfp}.
+
However, we will not be able to deal with all specifications that are
allowed by Ott. One reason is that Ott lets the user to specify ``empty''
types like
\begin{center}
- $t ::= t\;t \mid \lambda x. t$
+ @{text "t ::= t t | \<lambda>x. t"}
\end{center}
\noindent
@@ -204,32 +232,31 @@
two type-schemes (with $x$, $y$ and $z$ being distinct)
\begin{center}
- $\forall \{x\}. x \rightarrow y \;=\; \forall \{x, z\}. x \rightarrow y$
+ @{text "\<forall>{x}. x \<rightarrow> y = \<forall>{x,z}. x \<rightarrow> y"}
\end{center}
\noindent
- are not just alpha-equal, but actually \emph{equal}. As a
- result, we can only support specifications that make sense on the level of
- alpha-equated terms (offending specifications, which for example bind a variable
- according to a variable bound somewhere else, are not excluded by Ott, but we
- have to). Our
- insistence on reasoning with alpha-equated terms comes from the wealth of
- experience we gained with the older version of Nominal Isabelle: for
- non-trivial properties, reasoning about alpha-equated terms is much easier
- than reasoning with raw terms. The fundamental reason for this is that the
- HOL-logic underlying Nominal Isabelle allows us to replace
- ``equals-by-equals''. In contrast, replacing ``alpha-equals-by-alpha-equals''
- in a representation based on raw terms requires a lot of extra reasoning work.
+ are not just alpha-equal, but actually \emph{equal}. As a result, we can
+ only support specifications that make sense on the level of alpha-equated
+ terms (offending specifications, which for example bind a variable according
+ to a variable bound somewhere else, are not excluded by Ott, but we have
+ to). Our insistence on reasoning with alpha-equated terms comes from the
+ wealth of experience we gained with the older version of Nominal Isabelle:
+ for non-trivial properties, reasoning about alpha-equated terms is much
+ easier than reasoning with raw terms. The fundamental reason for this is
+ that the HOL-logic underlying Nominal Isabelle allows us to replace
+ ``equals-by-equals''. In contrast, replacing
+ ``alpha-equals-by-alpha-equals'' in a representation based on raw terms
+ requires a lot of extra reasoning work.
- Although in informal settings a reasoning infrastructure for alpha-equated
- terms is nearly always taken for granted, establishing
- it automatically in the Isabelle/HOL theorem prover is a rather non-trivial task.
- For every specification we will need to construct a type containing as
- elements the alpha-equated terms. To do so, we use
- the standard HOL-technique of defining a new type by
- identifying a non-empty subset of an existing type. The construction we
- perform in HOL can be illustrated by the following picture:
-
+ Although in informal settings a reasoning infrastructure for alpha-equated
+ terms is nearly always taken for granted, establishing it automatically in
+ the Isabelle/HOL theorem prover is a rather non-trivial task. For every
+ specification we will need to construct a type containing as elements the
+ alpha-equated terms. To do so, we use the standard HOL-technique of defining
+ a new type by identifying a non-empty subset of an existing type. The
+ construction we perform in HOL can be illustrated by the following picture:
+
\begin{center}
\begin{tikzpicture}
%\draw[step=2mm] (-4,-1) grid (4,1);
@@ -255,45 +282,45 @@
\end{center}
\noindent
- We take as the starting point a definition of raw terms (defined as a
- datatype in Isabelle/HOL); identify then the
- alpha-equivalence classes in the type of sets of raw terms according to our
- alpha-equivalence relation and finally define the new type as these
- alpha-equivalence classes (non-emptiness is satisfied whenever the raw terms are
- definable as datatype in Isabelle/HOL and the fact that our relation for
- alpha-equivalence is indeed an equivalence relation).
+ We take as the starting point a definition of raw terms (defined as a
+ datatype in Isabelle/HOL); identify then the alpha-equivalence classes in
+ the type of sets of raw terms according to our alpha-equivalence relation
+ and finally define the new type as these alpha-equivalence classes
+ (non-emptiness is satisfied whenever the raw terms are definable as datatype
+ in Isabelle/HOL and the fact that our relation for alpha-equivalence is
+ indeed an equivalence relation).
- The fact that we obtain an isomorphism between the new type and the non-empty
- subset shows that the new type is a faithful representation of alpha-equated terms.
- That is not the case for example for terms using the locally
- nameless representation of binders \cite{McKinnaPollack99}: in this representation
- there are ``junk'' terms that need to
- be excluded by reasoning about a well-formedness predicate.
+ The fact that we obtain an isomorphism between the new type and the
+ non-empty subset shows that the new type is a faithful representation of
+ alpha-equated terms. That is not the case for example for terms using the
+ locally nameless representation of binders \cite{McKinnaPollack99}: in this
+ representation there are ``junk'' terms that need to 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 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 re-implemented in Isabelle/HOL the quotient package described by Homeier
- \cite{Homeier05} for the HOL4 system. This package
- allows us to lift definitions and theorems involving raw terms
- to definitions and theorems involving alpha-equated terms. For example
- if we define the free-variable function over raw lambda-terms
+ The problem with introducing a new type in Isabelle/HOL is that in order to
+ be useful, 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 re-implemented in Isabelle/HOL the quotient package
+ described by Homeier \cite{Homeier05} for the HOL4 system. This package
+ allows us to lift definitions and theorems involving raw terms to
+ definitions and theorems involving alpha-equated terms. For example if we
+ define the free-variable function over raw lambda-terms
\begin{center}
- $\fv(x) = \{x\}$\hspace{10mm}
- $\fv(t_1\;t_2) = \fv(t_1) \cup \fv(t_2)$\\[1mm]
- $\fv(\lambda x.t) = \fv(t) - \{x\}$
+ @{text "fv(x) = {x}"}\hspace{10mm}
+ @{text "fv(t\<^isub>1 t\<^isub>2) = fv(t\<^isub>1) \<union> fv(t\<^isub>2)"}\\[1mm]
+ @{text "fv(\<lambda>x.t) = fv(t) - {x}"}
\end{center}
\noindent
- then with not too great effort we obtain a function $\fv^\alpha$
+ then with not too great effort we obtain a function @{text "fv\<^sup>\<alpha>"}
operating on quotients, or alpha-equivalence classes of lambda-terms. This
lifted function is characterised by the equations
\begin{center}
- $\fv^\alpha(x) = \{x\}$\hspace{10mm}
- $\fv^\alpha(t_1\;t_2) = \fv^\alpha(t_1) \cup \fv^\alpha(t_2)$\\[1mm]
- $\fv^\alpha(\lambda x.t) = \fv^\alpha(t) - \{x\}$
+ @{text "fv\<^sup>\<alpha>(x) = {x}"}\hspace{10mm}
+ @{text "fv\<^sup>\<alpha>(t\<^isub>1 t\<^isub>2) = fv\<^sup>\<alpha>(t\<^isub>1) \<union> fv\<^sup>\<alpha>(t\<^isub>2)"}\\[1mm]
+ @{text "fv\<^sup>\<alpha>(\<lambda>x.t) = fv\<^sup>\<alpha>(t) - {x}"}
\end{center}
\noindent
@@ -400,44 +427,47 @@
from this specification (remember that Nominal Isabelle is a definitional
extension of Isabelle/HOL, which does not introduce any new axioms).
-
- In order to keep our work manageable, we will wherever possible state
- definitions and perform proofs inside Isabelle, as opposed to write custom
- ML-code that generates them anew for each specification. To that
- end, we will consider pairs @{text "(as, x)"} of type @{text "(atom set) \<times> \<beta>"}.
- These pairs are intended to represent the abstraction, or binding, of the set @{text "as"}
- in the body @{text "x"}.
+ In order to keep our work with deriving the reasoning infrastructure
+ manageable, we will wherever possible state definitions and perform proofs
+ on the user-level of Isabelle/HOL, as opposed to write custom ML-code that
+ generates them anew for each specification. To that end, we will consider
+ first pairs @{text "(as, x)"} of type @{text "(atom set) \<times> \<beta>"}. These pairs
+ are intended to represent the abstraction, or binding, of the set @{text
+ "as"} in the body @{text "x"}.
- The first question we have to answer is when the pairs $(as, x)$ and $(bs, y)$ are
- alpha-equivalent? (At the moment we are interested in
- the notion of alpha-equivalence that is \emph{not} preserved by adding
- vacuous binders.) To answer this, we identify four conditions: {\it i)} given
- a free-variable function $\fv$ of type \mbox{@{text "\<beta> \<Rightarrow> atom set"}}, then @{text x} and @{text y}
- need to have the same set of free variables; moreover there must be a permutation
- @{text p} such that {\it ii)} it leaves the free variables of @{text x} and @{text 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 @{text p} makes
- the abstracted sets @{text as} and @{text bs} equal. The requirements {\it i)} to {\it iv)} can
- be stated formally as follows:
+ The first question we have to answer is when the pairs @{text "(as, x)"} and
+ @{text "(bs, y)"} are alpha-equivalent? (At the moment we are interested in
+ the notion of alpha-equivalence that is \emph{not} preserved by adding
+ vacuous binders.) To answer this, we identify four conditions: {\it i)}
+ given a free-variable function @{text "fv"} of type \mbox{@{text "\<beta> \<Rightarrow> atom
+ set"}}, then @{text x} and @{text y} need to have the same set of free
+ variables; moreover there must be a permutation @{text p} such that {\it
+ ii)} it leaves the free variables of @{text x} and @{text 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
+ @{text p} makes the abstracted sets @{text as} and @{text bs} equal. The
+ requirements {\it i)} to {\it iv)} can be stated formally as follows:
%
\begin{equation}\label{alphaset}
\begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
- \multicolumn{2}{l}{(as, x) \approx\hspace{0.05mm}_{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"}\\
+ \multicolumn{2}{l}{@{term "(as, x) \<approx>gen R fv p (bs, y)"} @{text "\<equiv>"}\hspace{30mm}}\\
+ & @{term "fv(x) - as = fv(y) - bs"}\\
+ @{text "\<and>"} & @{term "(fv(x) - as) \<sharp>* p"}\\
+ @{text "\<and>"} & @{text "(p \<bullet> x) R y"}\\
+ @{text "\<and>"} & @{term "(p \<bullet> as) = bs"}\\
\end{array}
\end{equation}
\noindent
- Note that this relation is dependent on $p$. Alpha-equivalence is then the relation where
- we existentially quantify over this $p$.
- Also note that the relation is dependent on a free-variable function $\fv$ and a relation
- $R$. The reason for this extra generality is that we will use $\approx_{set}$ for both
- ``raw'' terms and alpha-equated terms. In the latter case, $R$ will be replaced by
- equality $(op =)$ and we are going to prove that $\fv$ will be equal to the support
- of $x$ and $y$.
+ Note that this relation is dependent on the permutation @{text
+ "p"}. Alpha-equivalence between two pairs is then the relation where we
+ existentially quantify over this @{text "p"}. Also note that the relation is
+ dependent on a free-variable function @{text "fv"} and a relation @{text
+ "R"}. The reason for this extra generality is that we will use
+ $\approx_{\textit{set}}$ for both ``raw'' terms and alpha-equated terms. In
+ the latter case, $R$ will be replaced by equality @{text "="} and for raw terms we
+ will prove that @{text "fv"} is equal to the support of @{text
+ x} and @{text y}.
The definition in \eqref{alphaset} does not make any distinction between the
order of abstracted variables. If we want this, then we can define alpha-equivalence
@@ -446,26 +476,27 @@
%
\begin{equation}\label{alphalist}
\begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
- \multicolumn{2}{l}{(as, x) \approx\hspace{0.05mm}_{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"}\\
+ \multicolumn{2}{l}{@{term "(as, x) \<approx>lst R fv p (bs, y)"} @{text "\<equiv>"}\hspace{30mm}}\\[1mm]
+ & @{term "fv(x) - (set as) = fv(y) - (set bs)"}\\
+ \wedge & @{term "(fv(x) - set as) \<sharp>* p"}\\
\wedge & @{text "(p \<bullet> x) R y"}\\
- \wedge & @{text "(p \<bullet> as) = bs"}\\
+ \wedge & @{term "(p \<bullet> as) = bs"}\\
\end{array}
\end{equation}
\noindent
- where $set$ is the function that coerces a list of atoms into a set of atoms.
+ where @{term set} is a function that coerces a list of atoms into a set of atoms.
+ Now the last clause ensures that the order of the binders matters.
- If we do not want to make any difference between the order of binders and
+ If we do not want to make any difference between the order of binders \emph{and}
also allow vacuous binders, then we keep sets of binders, but drop the fourth
condition in \eqref{alphaset}:
%
\begin{equation}\label{alphares}
\begin{array}{@ {\hspace{10mm}}r@ {\hspace{2mm}}l}
- \multicolumn{2}{l}{(as, x) \approx\hspace{0.05mm}_{res}^{\fv, R, p} (bs, y) \;\dn\hspace{30mm}\;}\\[1mm]
- & @{text "fv(x) - as = fv(y) - bs"}\\
- \wedge & @{text "(fv(x) - as) #* p"}\\
+ \multicolumn{2}{l}{@{term "(as, x) \<approx>res R fv p (bs, y)"} @{text "\<equiv>"}\hspace{30mm}}\\[1mm]
+ & @{term "fv(x) - as = fv(y) - bs"}\\
+ \wedge & @{term "(fv(x) - as) \<sharp>* p"}\\
\wedge & @{text "(p \<bullet> x) R y"}\\
\end{array}
\end{equation}
@@ -473,52 +504,116 @@
\begin{exmple}\rm
It might be useful to consider some examples for how these definitions pan out in practise.
For this consider the case of abstracting a set of variables over types (as in type-schemes).
- We set $R$ to be the equality and for $\fv(T)$ we define
+ We set @{text R} to be the equality and for @{text "fv(T)"} we define
\begin{center}
- $\fv(x) = \{x\} \qquad \fv(T_1 \rightarrow T_2) = \fv(T_1) \cup \fv(T_2)$
+ @{text "fv(x) = {x}"} \hspace{5mm} @{text "fv(T\<^isub>1 \<rightarrow> T\<^isub>2) = fv(T\<^isub>1) \<union> fv(T\<^isub>2)"}
\end{center}
\noindent
- Now recall the examples shown 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 $\approx_{res}$ by taking $p$ to
- be the swapping @{term "(x \<rightleftharpoons> y)"}. In case of @{text "x \<noteq> y"}, then
- $([x, y], x \rightarrow y) \not\approx_{list} ([y,x], x \rightarrow y)$ since there is no permutation that
- makes the lists @{text "[x, y]"} and @{text "[y, x]"} equal, and also leaves the
- type \mbox{@{text "x \<rightarrow> y"}} unchanged. Another examples is
- $(\{x\}, x) \approx_{res} (\{x,y\}, x)$ which holds by taking $p$ to be the identity permutation.
- However, if @{text "x \<noteq> y"}, then
- $(\{x\}, x) \not\approx_{set} (\{x,y\}, x)$ since there is no permutation that makes
- the sets $\{x\}$ and $\{x,y\}$ equal (similarly for $\approx_{list}$).
+ Now recall the examples shown 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 $\approx_{\textit{set}}$ and
+ $\approx_{\textit{res}}$ by taking @{text p} to be the swapping @{term "(x \<rightleftharpoons>
+ y)"}. In case of @{text "x \<noteq> y"}, then @{text "([x, y], x \<rightarrow> y)"}
+ $\not\approx_{\textit{list}}$ @{text "([y,x], x \<rightarrow> y)"} since there is no permutation
+ that makes the lists @{text "[x, y]"} and @{text "[y, x]"} equal, and also
+ leaves the type \mbox{@{text "x \<rightarrow> y"}} unchanged. Another example is
+ @{text "({x}, x)"} $\approx_{\textit{res}}$ @{text "({x,y}, x)"} which holds by
+ taking @{text p} to be the
+ identity permutation. However, if @{text "x \<noteq> y"}, then @{text "({x}, x)"}
+ $\not\approx_{\textit{set}}$ @{text "({x,y}, x)"} since there is no permutation
+ that makes the
+ sets @{text "{x}"} and @{text "{x,y}"} equal (similarly for $\approx_{\textit{list}}$).
\end{exmple}
+ % looks too ugly
+ %\noindent
+ %Let $\star$ range over $\{set, res, list\}$. We prove next under which
+ %conditions the $\approx\hspace{0.05mm}_\star^{\fv, R, p}$ are equivalence
+ %relations and equivariant:
+ %
+ %\begin{lemma}
+ %{\it i)} Given the fact that $x\;R\;x$ holds, then
+ %$(as, x) \approx\hspace{0.05mm}^{\fv, R, 0}_\star (as, x)$. {\it ii)} Given
+ %that @{text "(p \<bullet> x) R y"} implies @{text "(-p \<bullet> y) R x"}, then
+ %$(as, x) \approx\hspace{0.05mm}^{\fv, R, p}_\star (bs, y)$ implies
+ %$(bs, y) \approx\hspace{0.05mm}^{\fv, R, - p}_\star (as, x)$. {\it iii)} Given
+ %that @{text "(p \<bullet> x) R y"} and @{text "(q \<bullet> y) R z"} implies
+ %@{text "((q + p) \<bullet> x) R z"}, then $(as, x) \approx\hspace{0.05mm}^{\fv, R, p}_\star (bs, y)$
+ %and $(bs, y) \approx\hspace{0.05mm}^{\fv, R, q}_\star (cs, z)$ implies
+ %$(as, x) \approx\hspace{0.05mm}^{\fv, R, q + p}_\star (cs, z)$. Given
+ %@{text "(q \<bullet> x) R y"} implies @{text "(p \<bullet> (q \<bullet> x)) R (p \<bullet> y)"} and
+ %@{text "p \<bullet> (fv x) = fv (p \<bullet> x)"} then @{text "p \<bullet> (fv y) = fv (p \<bullet> y)"}, then
+ %$(as, x) \approx\hspace{0.05mm}^{\fv, R, q}_\star (bs, y)$ implies
+ %$(p \;\isasymbullet\; as, p \;\isasymbullet\; x) \approx\hspace{0.05mm}^{\fv, R, q}_\star
+ %(p \;\isasymbullet\; bs, p \;\isasymbullet\; y)$.
+ %\end{lemma}
+
+ %\begin{proof}
+ %All properties are by unfolding the definitions and simple calculations.
+ %\end{proof}
+
+
+ In the rest of this section we are going to introduce a type- and term-constructor
+ for abstractions. For this we define
+ %
+ \begin{equation}
+ @{term "abs_set (as, x) (bs, x) \<equiv> \<exists>p. alpha_gen (as, x) equal supp p (bs, x)"}
+ \end{equation}
+
\noindent
- Let $\star$ range over $\{set, res, list\}$. We prove next under which
- conditions the $\approx\hspace{0.05mm}_\star^{\fv, R, p}$ are equivalence
- relations and equivariant:
+ Similarly for @{text "abs_list"} and @{text "abs_res"}. We can show that these
+ relations are equivalence relations and equivariant
+ (we only show the $\approx_{\textit{abs\_set}}$-case).
\begin{lemma}
- {\it i)} Given the fact that $x\;R\;x$ holds, then
- $(as, x) \approx\hspace{0.05mm}^{\fv, R, 0}_\star (as, x)$. {\it ii)} Given
- that @{text "(p \<bullet> x) R y"} implies @{text "(-p \<bullet> y) R x"}, then
- $(as, x) \approx\hspace{0.05mm}^{\fv, R, p}_\star (bs, y)$ implies
- $(bs, y) \approx\hspace{0.05mm}^{\fv, R, - p}_\star (as, x)$. {\it iii)} Given
- that @{text "(p \<bullet> x) R y"} and @{text "(q \<bullet> y) R z"} implies
- @{text "((q + p) \<bullet> x) R z"}, then $(as, x) \approx\hspace{0.05mm}^{\fv, R, p}_\star (bs, y)$
- and $(bs, y) \approx\hspace{0.05mm}^{\fv, R, q}_\star (cs, z)$ implies
- $(as, x) \approx\hspace{0.05mm}^{\fv, R, q + p}_\star (cs, z)$. Given
- @{text "(q \<bullet> x) R y"} implies @{text "(p \<bullet> (q \<bullet> x)) R (p \<bullet> y)"} and
- @{text "p \<bullet> (fv x) = fv (p \<bullet> x)"} then @{text "p \<bullet> (fv y) = fv (p \<bullet> y)"}, then
- $(as, x) \approx\hspace{0.05mm}^{\fv, R, q}_\star (bs, y)$ implies
- $(p \;\isasymbullet\; as, p \;\isasymbullet\; x) \approx\hspace{0.05mm}^{\fv, R, q}_\star
- (p \;\isasymbullet\; bs, p \;\isasymbullet\; y)$.
+ $\approx_{\textit{abs\_set}}$ is an equivalence
+ relations, and if @{term "abs_set (as, x) (bs, x)"} then also
+ @{term "abs_set (p \<bullet> as, p \<bullet> x) (p \<bullet> bs, p \<bullet> x)"}.
+ \end{lemma}
+
+ \begin{proof}
+ Reflexivity is by taking @{text "p"} to be @{text "0"}. For symmetry we have
+ a permutation @{text p} and for the proof obligation take @{term "-p"}. In case
+ of transitivity we have two permutations @{text p} and @{text q}, and for the
+ proof obligation use @{text "q + p"}. All the conditions are then by simple
+ calculations.
+ \end{proof}
+
+ \noindent
+ The following lemma (and similar ones for $\approx_{\textit{abs\_list}}$ and
+ $\approx_{\textit{abs\_res}}$) will be crucial below:
+
+ \begin{lemma}
+ @{thm[mode=IfThen] alpha_abs_swap[no_vars]}
\end{lemma}
-
+
\begin{proof}
- All properties are by unfolding the definitions and simple calculations.
+ This lemma is straightforward by observing that the assumptions give us
+ @{term "(a \<rightleftharpoons> b) \<bullet> (supp x - bs) = (supp x - bs)"} and that @{text supp}
+ is equivariant.
\end{proof}
+ \noindent
+ We are also define the following
+
+ @{text "aux (as, x) \<equiv> supp x - as"}
+
+
+
+ \noindent
+ This allows us to use our quotient package and introduce new types
+ @{text "\<beta> abs_set"}, @{text "\<beta> abs_res"} and @{text "\<beta> abs_list"}
+ representing the alpha-equivalence classes. Elements in these types
+ we will, respectively, write as:
+
+ \begin{center}
+ @{term "Abs as x"} \hspace{5mm}
+ @{term "Abs_lst as x"} \hspace{5mm}
+ @{term "Abs_res as x"}
+ \end{center}
+
\begin{lemma}
$supp ([as]set. x) = supp x - as$
@@ -834,7 +929,7 @@
\begin{tabular}{cp{7cm}}
$\bullet$ & @{text "{atom x\<^isub>i} - bnds"} provided @{term "x\<^isub>i"} is an atom\\
$\bullet$ & @{text "(atoms x\<^isub>i) - bnds"} provided @{term "x\<^isub>i"} is a set of atoms\\
- $\bullet$ & @{text "(atoml x\<^isub>i) - bnds"} provided @{term "x\<^isub>i"} is a list of atoms\\
+ $\bullet$ & @{text "(atoms (set x\<^isub>i)) - bnds"} provided @{term "x\<^isub>i"} is a list of atoms\\
$\bullet$ & @{text "(fv_ty\<^isub>i x\<^isub>i) - bnds"} provided @{term "ty\<^isub>i"} is a nominal datatype\\
$\bullet$ & @{term "{}"} otherwise
\end{tabular}