nominal2: comparison Paper/Paper.thy

equal deleted inserted replaced

-:9db725590fe9
+:ddf409b2da2b
 \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, an additional
+would be a perfectly legal instance. To exclude such terms, additional
-predicate about well-formed terms is needed in order to ensure that the two
+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}
 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''
+``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.
 \end{center}
 \noindent
 then with not too great effort we obtain a function $\fv^\alpha$
 operating on quotients, or alpha-equivalence classes of lambda-terms. This
-function is characterised by the equations
+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]
 $\fv^\alpha(\lambda x.t) = \fv^\alpha(t) - \{x\}$
 \end{center}
 \noindent
 (Note that this means also the term-constructors for variables, applications
 and lambda are lifted to the quotient level.)  This construction, of course,
-only works if alpha-equivalence is an equivalence relation, and the lifted
+only works if alpha-equivalence is indeed an equivalence relation, and the lifted
 definitions and theorems are respectful w.r.t.~alpha-equivalence.  Accordingly, we
 will not be able to lift a bound-variable function to alpha-equated terms
 (since it does not respect alpha-equivalence). To sum up, every lifting of
 theorems to the quotient level needs proofs of some respectfulness
 properties. In the paper we show that we are able to automate these
 two-place permutation operation written
 %
 @{text[display,indent=5] "_ \<bullet> _  ::  perm \<Rightarrow> \<beta> \<Rightarrow> \<beta>"}
 \noindent
-with a generic type in which @{text "\<beta>"} stands for the type of the object
+in which the generic type @{text "\<beta>"} stands for the type of the object
 on which the permutation
 acts. In Nominal Isabelle, the identity permutation is written as @{term "0::perm"},
 the composition of two permutations @{term p} and @{term q} as \mbox{@{term "p + q"}}
 and the inverse permutation of @{term p} as @{text "- p"}. The permutation
 operation is defined for products, lists, sets, functions, booleans etc
-(see \cite{HuffmanUrban10}). In the nominal logic work, concrete
+(see \cite{HuffmanUrban10}). Concrete  permutations are build up from
-permutations are usually build up from swappings, written as @{text "(a b)"},
+swappings, written as @{text "(a b)"},
 which are permutations that behave as follows:
 %
 @{text[display,indent=5] "(a b) = \<lambda>c. if a = c then b else if b = c then a else c"}
 The most original aspect of the nominal logic work of Pitts is a general
 definition for the notion of ``the set of free variables of an object @{text
 "x"}''.  This notion, written @{term "supp x"}, is general in the sense that
-it applies not only to lambda-terms alpha-equated or not, but also to lists,
+it applies not only to lambda-terms (alpha-equated or not), but also to lists,
 products, sets and even functions. The definition depends only on the
 permutation operation and on the notion of equality defined for the type of
 @{text x}, namely:
 %
 @{thm[display,indent=5] supp_def[no_vars, THEN eq_reflection]}
 Our specifications for term-calculi are heavily
 inspired by the syntax of the Ott-tool \cite{ott-jfp}. A specification is
 a collection of (possibly mutual recursive) type declarations, say
 $ty^\alpha_1$, $ty^\alpha_2$, \ldots $ty^\alpha_n$, and an
 associated collection of binding function declarations, say
-$bn^\alpha_1$, \ldots $bn^\alpha_m$. The syntax for a specification is
+$bn^\alpha_1$, \ldots $bn^\alpha_m$. The syntax in Nominal Isabelle
-rougly as follows:
+for such specifications is rougly as follows:
+%
 \begin{equation}\label{scheme}
 \mbox{\begin{tabular}{@ {\hspace{-9mm}}p{1.8cm}l}
 type \mbox{declaration part} &
 $\begin{cases}
 \mbox{\begin{tabular}{l}
 \begin{center}
 $C^\alpha\;label_1\!::\!ty'_1\;\ldots label_l\!::\!ty'_l\;\;\textit{binding\_clauses}$
 \end{center}
 \noindent
-whereby some of the $ty'_k$ are contained in the set of $ty^\alpha_i$
+whereby some of the $ty'_k$ (or their type components) are contained in the
+set of $ty^\alpha_i$
 declared in \eqref{scheme}. In this case we will call
 the corresponding argument a \emph{recursive argument}.  The labels
 annotated on the types are optional and can be used in the (possibly empty)
 list of \emph{binding clauses}.  These clauses indicate the binders and the
 scope of the binders in a term-constructor.  They come in three \emph{modes}
 \isacommand{bind\_res}\; {\it binders}\; \isacommand{in}\; {\it label}\\
 \end{tabular}
 \end{center}
 \noindent
-The first mode is for binding lists of atoms (order matters); in the second sets
+The first mode is for binding lists of atoms (the order matters); the second is for sets
-of binders (order does not matter, but cardinality does) and in the last
+of binders (the order does not matter, but cardinality does) and the last is for
 sets of binders (with vacuous binders preserving alpha-equivalence).
 In addition we distinguish between \emph{shallow} binders and \emph{deep}
 binders.  Shallow binders are of the form \isacommand{bind}\; {\it label}\;
-\isacommand{in}\; {\it another\_label} (similar the other two modes). The
+\isacommand{in}\; {\it another\_label} (similar for the other two modes). The
 restriction we impose on shallow binders is that the {\it label} must either
 refer to a type that is an atom type or to a type that is a finite set or
 list of an atom type. For example the specifications of lambda-terms, where
 a single name is bound, and type-schemes, where a finite set of names is
 bound, use shallow binders (the type \emph{name} is an atom type):
 \end{tabular}
 \end{tabular}
 \end{center}
 \noindent
-If we have shallow binders that ``share'' a body, for example
+If we have shallow binders that ``share'' a body, for instance $t$ in
+the term-constructor Foo$_0$
 \begin{center}
 \begin{tabular}{ll}
 \it {\rm Foo}$_0$ x::name y::name t::lam & \it
 \isacommand{bind}\;x\;\isacommand{in}\;t,\;
 \isacommand{bind}\;y\;\isacommand{in}\;t
 \end{tabular}
 \end{center}
 \noindent
-then we have to make sure the modes of the binders agree. For example we cannot
+then we have to make sure the modes of the binders agree. We cannot
 have in the first binding clause the mode \isacommand{bind} and in the second
 \isacommand{bind\_set}.
 A \emph{deep} binder uses an auxiliary binding function that ``picks'' out
-the atoms in one argument of the term-constructor that can be bound in one
+the atoms in one argument of the term-constructor that can be bound in
-or more of the other arguments and also can be bound in the same argument (we will
+other arguments and also in the same argument (we will
 call such binders \emph{recursive}).
 The binding functions are expected to return either a set of atoms
 (for \isacommand{bind\_set} and \isacommand{bind\_res}) or a list of atoms
 (for \isacommand{bind}). They can be defined by primitive recursion over the
 corresponding type; the equations must be given in the binding function part of
-the scheme shown in \eqref{scheme}.
+the scheme shown in \eqref{scheme}. For example for a calculus containing lets
+with tuple patterns, you might declare
-In the present version of Nominal Isabelle, we
-adopted the restrictions the Ott-tool imposes on the binding functions, namely a
-binding function can only return the empty set, a singleton set containing an
-atom or unions of atom sets. For example for lets with tuple patterns you might
-define
 \begin{center}
 \begin{tabular}{l}
 \isacommand{nominal\_datatype} {\it trm} =\\
 \hspace{5mm}\phantom{$\mid$} Var\;{\it name}\\
 \hspace{5mm}$\mid$ App\;{\it trm}\;{\it trm}\\
 \hspace{5mm}$\mid$ Lam\;{\it x::name}\;{\it t::trm}
 \;\;\isacommand{bind} {\it x} \isacommand{in} {\it t}\\
 \hspace{5mm}$\mid$ Let\;{\it p::pat}\;{\it trm}\; {\it t::trm}
-\;\;\isacommand{bind} {\it bn(p)} \isacommand{in} {\it t}\\
+\;\;\isacommand{bind\_set} {\it bn(p)} \isacommand{in} {\it t}\\
 \isacommand{and} {\it pat} =\\
 \hspace{5mm}\phantom{$\mid$} PNo\\
 \hspace{5mm}$\mid$ PVr\;{\it name}\\
 \hspace{5mm}$\mid$ PPr\;{\it pat}\;{\it pat}\\
 \isacommand{with} {\it bn::pat $\Rightarrow$ atom set}\\
-\isacommand{where} $bn(\textrm{PNo}) = \varnothing$\\
+\isacommand{where} $bn(\textrm{PNil}) = \varnothing$\\
-\hspace{5mm}$\mid$ $bn(\textrm{PVr}\;x) = \{atom\; x\}$\\
+\hspace{5mm}$\mid$ $bn(\textrm{PVar}\;x) = \{atom\; x\}$\\
-\hspace{5mm}$\mid$ $bn(\textrm{PPr}\;p_1\;p_2) = bn(p_1) \cup bn(p_2)$\\
+\hspace{5mm}$\mid$ $bn(\textrm{PPrd}\;p_1\;p_2) = bn(p_1) \cup bn(p_2)$\\
 \end{tabular}
 \end{center}
 \noindent
 In this specification the function $atom$ coerces a name into the generic
 atom type of Nominal Isabelle. This allows us to treat binders of different
-type uniformly. As will shortly become clear, we cannot return an atom
+atom type uniformly. As will shortly become clear, we cannot return an atom in a
-in a binding function that also is bound in the term-constructor.
+binding function that also is bound in the term-constructor. In the present
+version of Nominal Isabelle, we adopted the restriction the Ott-tool imposes
-Like with shallow binders, deep binders with shared body need to have the
+on the binding functions, namely a binding function can only return the
-same binding mode. A more serious restriction we have to impose on deep binders
+empty set (case PNil), a singleton set containing an atom (case PVar) or
-is that we cannot have ``overlapping'' binders. Consider for example the
+unions of atom sets (case PPrd). Moreover, as with shallow binders, deep
-term-constructors:
+binders with shared body need to have the same binding mode. Finally, the
+most drastic restriction we have to impose on deep binders is that we cannot
+have ``overlapping'' deep binders. Consider for example the term-constructors:
 \begin{center}
 \begin{tabular}{ll}
 \it {\rm Foo}$_1$ p::pat q::pat t::trm & \it \isacommand{bind}\;bn(p)\;\isacommand{in}\;t,\;
 \isacommand{bind}\;bn(q)\;\isacommand{in}\;t\\
 \end{tabular}
 \end{center}
 \noindent
 In the first case we bind all atoms from the pattern $p$ in $t$ and also all atoms
-from $q$ in $t$. Therefore the binders overlap in $t$. Similarly in the second case:
+from $q$ in $t$. As a result we have no way to determine whether the binder came from the
-the binder $bn(p)$ overlap with the shallow binder $x$. We must exclude such specifiactions,
+binding function in $p$ or $q$. Similarly in the second case:
+the binder $bn(p)$ overlaps with the shallow binder $x$. We must exclude such specifiactions,
 as we will not be able to represent them using the general binders described in
 Section \ref{sec:binders}. However the following two term-constructors are allowed:
 \begin{center}
 \begin{tabular}{ll}
 \isacommand{bind}\;bn(p)\;\isacommand{in}\;t\\
 \end{tabular}
 \end{center}
 \noindent
-as no overlapping of binders occurs.
+since there is no overlap of binders.
+Now the question is how we can turn specifications into actual type
-Because of the problem Pottier pointed out in \cite{Pottier06}, the general
+definitions in Isabelle/HOL and then establish a reasoning infrastructure
-binders from the previous section cannot be used directly to represent w
+for them? Because of the problem Pottier and Cheney pointed out, we cannot
-be used directly
+in general re-arrange arguments of term-constructors so that binders and
+their scopes next to each other, an then use the type constructors
+@{text "abs_set"}, @{text "abs_res"} and @{text "abs_list"}. Therefore
+we will first extract datatype definitions from the specification and
+then define an alpha-equiavlence relation over them.
+The datatype definition can be obtained by just stripping of the
+binding clauses and the labels on the types. We also have to invent
+new names for the types, $ty^\alpha$ and term-constructors $C^\alpha$
+given by user. We just use an affix:
+\begin{center}
+$ty^\alpha \mapsto ty\_raw  \qquad C^\alpha \mapsto C\_raw$
+\end{center}
+\noindent
+This definition can be made, provided the usual conditions hold: the
+types must be non-empty and the types in the term-constructors need to
+be, what is called, positive position (see \cite{}). We then take the binding
+functions and define them by primitive recursion over the raw datatypes.
+binding function.
+The first non-trivial step is to read off from the specification free-variable
+functions. There are two kinds: free-variable functions corresponding to types,
+written $\fv\_ty$, and  free-variable functions corresponding to binding functions,
+written $\fv\_bn$. They have to be defined at the same time since there can
+be interdependencies. Given a term-constructor $C ty_1 \ldots ty_n$ and some binding
+clauses, the function $\fv (C x_1 \ldots x_n)$ will be the union of the values
+generated for each argument, say $x_i$, as follows:
+\begin{center}
+\begin{tabular}{cp{8cm}}
+$\bullet$ & if it is a shallow binder, then $\varnothing$\\
+$\bullet$ & if it is a deep binder, then $\fv_bn x_i$\\
+$\bullet$ & if
+\end{tabular}
+\end{center}
 *}
 text {*

changeset 1628	ddf409b2da2b
parent 1620	17a2c6fddc0c
child 1629	a0ca7d9f6781