(*:maxLineLen=78:*)
theory Syntax
imports Base
begin
chapter \<open>Concrete syntax and type-checking\<close>
text \<open>Pure \<open>\<lambda>\<close>-calculus as introduced in \chref{ch:logic} is
an adequate foundation for logical languages --- in the tradition of
\<^emph>\<open>higher-order abstract syntax\<close> --- but end-users require
additional means for reading and printing of terms and types. This
important add-on outside the logical core is called \<^emph>\<open>inner
syntax\<close> in Isabelle jargon, as opposed to the \<^emph>\<open>outer syntax\<close> of
the theory and proof language @{cite "isabelle-isar-ref"}.
For example, according to @{cite church40} quantifiers are represented as
higher-order constants \<open>All :: ('a \<Rightarrow> bool) \<Rightarrow> bool\<close> such that \<open>All (\<lambda>x::'a. B x)\<close> faithfully represents the idea that is displayed in
Isabelle as \<open>\<forall>x::'a. B x\<close> via @{keyword "binder"} notation.
Moreover, type-inference in the style of Hindley-Milner @{cite hindleymilner}
(and extensions) enables users to write \<open>\<forall>x. B x\<close> concisely, when
the type \<open>'a\<close> is already clear from the
context.\<^footnote>\<open>Type-inference taken to the extreme can easily confuse
users. Beginners often stumble over unexpectedly general types inferred by
the system.\<close>
\<^medskip>
The main inner syntax operations are \<^emph>\<open>read\<close> for
parsing together with type-checking, and \<^emph>\<open>pretty\<close> for formatted
output. See also \secref{sec:read-print}.
Furthermore, the input and output syntax layers are sub-divided into
separate phases for \<^emph>\<open>concrete syntax\<close> versus \<^emph>\<open>abstract
syntax\<close>, see also \secref{sec:parse-unparse} and
\secref{sec:term-check}, respectively. This results in the
following decomposition of the main operations:
\<^item> \<open>read = parse; check\<close>
\<^item> \<open>pretty = uncheck; unparse\<close>
For example, some specification package might thus intercept syntax
processing at a well-defined stage after \<open>parse\<close>, to a augment the
resulting pre-term before full type-reconstruction is performed by \<open>check\<close>. Note that the formal status of bound variables, versus free
variables, versus constants must not be changed between these phases.
\<^medskip>
In general, \<open>check\<close> and \<open>uncheck\<close> operate
simultaneously on a list of terms. This is particular important for
type-checking, to reconstruct types for several terms of the same context
and scope. In contrast, \<open>parse\<close> and \<open>unparse\<close> operate separately
on single terms.
There are analogous operations to read and print types, with the same
sub-division into phases.
\<close>
section \<open>Reading and pretty printing \label{sec:read-print}\<close>
text \<open>
Read and print operations are roughly dual to each other, such that for the
user \<open>s' = pretty (read s)\<close> looks similar to the original source
text \<open>s\<close>, but the details depend on many side-conditions. There are
also explicit options to control the removal of type information in the
output. The default configuration routinely looses information, so \<open>t' = read (pretty t)\<close> might fail, or produce a differently typed term, or
a completely different term in the face of syntactic overloading.
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML Syntax.read_typs: "Proof.context -> string list -> typ list"} \\
@{index_ML Syntax.read_terms: "Proof.context -> string list -> term list"} \\
@{index_ML Syntax.read_props: "Proof.context -> string list -> term list"} \\[0.5ex]
@{index_ML Syntax.read_typ: "Proof.context -> string -> typ"} \\
@{index_ML Syntax.read_term: "Proof.context -> string -> term"} \\
@{index_ML Syntax.read_prop: "Proof.context -> string -> term"} \\[0.5ex]
@{index_ML Syntax.pretty_typ: "Proof.context -> typ -> Pretty.T"} \\
@{index_ML Syntax.pretty_term: "Proof.context -> term -> Pretty.T"} \\
@{index_ML Syntax.string_of_typ: "Proof.context -> typ -> string"} \\
@{index_ML Syntax.string_of_term: "Proof.context -> term -> string"} \\
\end{mldecls}
\<^descr> @{ML Syntax.read_typs}~\<open>ctxt strs\<close> parses and checks a
simultaneous list of source strings as types of the logic.
\<^descr> @{ML Syntax.read_terms}~\<open>ctxt strs\<close> parses and checks a
simultaneous list of source strings as terms of the logic.
Type-reconstruction puts all parsed terms into the same scope: types of
free variables ultimately need to coincide.
If particular type-constraints are required for some of the arguments, the
read operations needs to be split into its parse and check phases. Then it
is possible to use @{ML Type.constraint} on the intermediate pre-terms
(\secref{sec:term-check}).
\<^descr> @{ML Syntax.read_props}~\<open>ctxt strs\<close> parses and checks a
simultaneous list of source strings as terms of the logic, with an implicit
type-constraint for each argument to enforce type @{typ prop}; this also
affects the inner syntax for parsing. The remaining type-reconstruction
works as for @{ML Syntax.read_terms}.
\<^descr> @{ML Syntax.read_typ}, @{ML Syntax.read_term}, @{ML Syntax.read_prop}
are like the simultaneous versions, but operate on a single argument only.
This convenient shorthand is adequate in situations where a single item in
its own scope is processed. Do not use @{ML "map o Syntax.read_term"} where
@{ML Syntax.read_terms} is actually intended!
\<^descr> @{ML Syntax.pretty_typ}~\<open>ctxt T\<close> and @{ML
Syntax.pretty_term}~\<open>ctxt t\<close> uncheck and pretty-print the given type
or term, respectively. Although the uncheck phase acts on a simultaneous
list as well, this is rarely used in practice, so only the singleton case is
provided as combined pretty operation. There is no distinction of term vs.\
proposition.
\<^descr> @{ML Syntax.string_of_typ} and @{ML Syntax.string_of_term} are
convenient compositions of @{ML Syntax.pretty_typ} and @{ML
Syntax.pretty_term} with @{ML Pretty.string_of} for output. The result may
be concatenated with other strings, as long as there is no further
formatting and line-breaking involved.
@{ML Syntax.read_term}, @{ML Syntax.read_prop}, and @{ML
Syntax.string_of_term} are the most important operations in practice.
\<^medskip>
Note that the string values that are passed in and out are
annotated by the system, to carry further markup that is relevant for the
Prover IDE @{cite "isabelle-jedit"}. User code should neither compose its
own input strings, nor try to analyze the output strings. Conceptually this
is an abstract datatype, encoded as concrete string for historical reasons.
The standard way to provide the required position markup for input works via
the outer syntax parser wrapper @{ML Parse.inner_syntax}, which is already
part of @{ML Parse.typ}, @{ML Parse.term}, @{ML Parse.prop}. So a string
obtained from one of the latter may be directly passed to the corresponding
read operation: this yields PIDE markup of the input and precise positions
for warning and error messages.
\<close>
section \<open>Parsing and unparsing \label{sec:parse-unparse}\<close>
text \<open>
Parsing and unparsing converts between actual source text and a certain
\<^emph>\<open>pre-term\<close> format, where all bindings and scopes are already resolved
faithfully. Thus the names of free variables or constants are determined in
the sense of the logical context, but type information might be still
missing. Pre-terms support an explicit language of \<^emph>\<open>type constraints\<close>
that may be augmented by user code to guide the later \<^emph>\<open>check\<close> phase.
Actual parsing is based on traditional lexical analysis and Earley parsing
for arbitrary context-free grammars. The user can specify the grammar
declaratively via mixfix annotations. Moreover, there are \<^emph>\<open>syntax
translations\<close> that can be augmented by the user, either declaratively via
@{command translations} or programmatically via @{command
parse_translation}, @{command print_translation} @{cite
"isabelle-isar-ref"}. The final scope-resolution is performed by the system,
according to name spaces for types, term variables and constants determined
by the context.
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML Syntax.parse_typ: "Proof.context -> string -> typ"} \\
@{index_ML Syntax.parse_term: "Proof.context -> string -> term"} \\
@{index_ML Syntax.parse_prop: "Proof.context -> string -> term"} \\[0.5ex]
@{index_ML Syntax.unparse_typ: "Proof.context -> typ -> Pretty.T"} \\
@{index_ML Syntax.unparse_term: "Proof.context -> term -> Pretty.T"} \\
\end{mldecls}
\<^descr> @{ML Syntax.parse_typ}~\<open>ctxt str\<close> parses a source string as
pre-type that is ready to be used with subsequent check operations.
\<^descr> @{ML Syntax.parse_term}~\<open>ctxt str\<close> parses a source string as
pre-term that is ready to be used with subsequent check operations.
\<^descr> @{ML Syntax.parse_prop}~\<open>ctxt str\<close> parses a source string as
pre-term that is ready to be used with subsequent check operations. The
inner syntax category is @{typ prop} and a suitable type-constraint is
included to ensure that this information is observed in subsequent type
reconstruction.
\<^descr> @{ML Syntax.unparse_typ}~\<open>ctxt T\<close> unparses a type after
uncheck operations, to turn it into a pretty tree.
\<^descr> @{ML Syntax.unparse_term}~\<open>ctxt T\<close> unparses a term after
uncheck operations, to turn it into a pretty tree. There is no distinction
for propositions here.
These operations always operate on a single item; use the combinator @{ML
map} to apply them to a list.
\<close>
section \<open>Checking and unchecking \label{sec:term-check}\<close>
text \<open>These operations define the transition from pre-terms and
fully-annotated terms in the sense of the logical core
(\chref{ch:logic}).
The \<^emph>\<open>check\<close> phase is meant to subsume a variety of mechanisms
in the manner of ``type-inference'' or ``type-reconstruction'' or
``type-improvement'', not just type-checking in the narrow sense.
The \<^emph>\<open>uncheck\<close> phase is roughly dual, it prunes type-information
before pretty printing.
A typical add-on for the check/uncheck syntax layer is the @{command
abbreviation} mechanism @{cite "isabelle-isar-ref"}. Here the user specifies
syntactic definitions that are managed by the system as polymorphic \<open>let\<close> bindings. These are expanded during the \<open>check\<close> phase, and
contracted during the \<open>uncheck\<close> phase, without affecting the
type-assignment of the given terms.
\<^medskip>
The precise meaning of type checking depends on the context ---
additional check/uncheck modules might be defined in user space.
For example, the @{command class} command defines a context where
\<open>check\<close> treats certain type instances of overloaded
constants according to the ``dictionary construction'' of its
logical foundation. This involves ``type improvement''
(specialization of slightly too general types) and replacement by
certain locale parameters. See also @{cite "Haftmann-Wenzel:2009"}.
\<close>
text %mlref \<open>
\begin{mldecls}
@{index_ML Syntax.check_typs: "Proof.context -> typ list -> typ list"} \\
@{index_ML Syntax.check_terms: "Proof.context -> term list -> term list"} \\
@{index_ML Syntax.check_props: "Proof.context -> term list -> term list"} \\[0.5ex]
@{index_ML Syntax.uncheck_typs: "Proof.context -> typ list -> typ list"} \\
@{index_ML Syntax.uncheck_terms: "Proof.context -> term list -> term list"} \\
\end{mldecls}
\<^descr> @{ML Syntax.check_typs}~\<open>ctxt Ts\<close> checks a simultaneous list
of pre-types as types of the logic. Typically, this involves normalization
of type synonyms.
\<^descr> @{ML Syntax.check_terms}~\<open>ctxt ts\<close> checks a simultaneous list
of pre-terms as terms of the logic. Typically, this involves type-inference
and normalization term abbreviations. The types within the given terms are
treated in the same way as for @{ML Syntax.check_typs}.
Applications sometimes need to check several types and terms together. The
standard approach uses @{ML Logic.mk_type} to embed the language of types
into that of terms; all arguments are appended into one list of terms that
is checked; afterwards the type arguments are recovered with @{ML
Logic.dest_type}.
\<^descr> @{ML Syntax.check_props}~\<open>ctxt ts\<close> checks a simultaneous list
of pre-terms as terms of the logic, such that all terms are constrained by
type @{typ prop}. The remaining check operation works as @{ML
Syntax.check_terms} above.
\<^descr> @{ML Syntax.uncheck_typs}~\<open>ctxt Ts\<close> unchecks a simultaneous
list of types of the logic, in preparation of pretty printing.
\<^descr> @{ML Syntax.uncheck_terms}~\<open>ctxt ts\<close> unchecks a simultaneous
list of terms of the logic, in preparation of pretty printing. There is no
distinction for propositions here.
These operations always operate simultaneously on a list; use the combinator
@{ML singleton} to apply them to a single item.
\<close>
end