Name_Space: proper configuration options long_names, short_names, unique_names instead of former unsynchronized references;
theory Inner_Syntax
imports Main
begin
chapter {* Inner syntax --- the term language \label{ch:inner-syntax} *}
section {* Printing logical entities *}
subsection {* Diagnostic commands *}
text {*
\begin{matharray}{rcl}
@{command_def "typ"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{command_def "term"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{command_def "prop"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{command_def "thm"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{command_def "prf"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{command_def "full_prf"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{command_def "pr"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
\end{matharray}
These diagnostic commands assist interactive development by printing
internal logical entities in a human-readable fashion.
\begin{rail}
'typ' modes? type
;
'term' modes? term
;
'prop' modes? prop
;
'thm' modes? thmrefs
;
( 'prf' | 'full_prf' ) modes? thmrefs?
;
'pr' modes? nat?
;
modes: '(' (name + ) ')'
;
\end{rail}
\begin{description}
\item @{command "typ"}~@{text \<tau>} reads and prints types of the
meta-logic according to the current theory or proof context.
\item @{command "term"}~@{text t} and @{command "prop"}~@{text \<phi>}
read, type-check and print terms or propositions according to the
current theory or proof context; the inferred type of @{text t} is
output as well. Note that these commands are also useful in
inspecting the current environment of term abbreviations.
\item @{command "thm"}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} retrieves
theorems from the current theory or proof context. Note that any
attributes included in the theorem specifications are applied to a
temporary context derived from the current theory or proof; the
result is discarded, i.e.\ attributes involved in @{text "a\<^sub>1,
\<dots>, a\<^sub>n"} do not have any permanent effect.
\item @{command "prf"} displays the (compact) proof term of the
current proof state (if present), or of the given theorems. Note
that this requires proof terms to be switched on for the current
object logic (see the ``Proof terms'' section of the Isabelle
reference manual for information on how to do this).
\item @{command "full_prf"} is like @{command "prf"}, but displays
the full proof term, i.e.\ also displays information omitted in the
compact proof term, which is denoted by ``@{text _}'' placeholders
there.
\item @{command "pr"}~@{text "goals"} prints the current proof state
(if present), including current facts and goals. The optional limit
arguments affect the number of goals to be displayed, which is
initially 10. Omitting limit value leaves the current setting
unchanged.
\end{description}
All of the diagnostic commands above admit a list of @{text modes}
to be specified, which is appended to the current print mode (see
also \cite{isabelle-ref}). Thus the output behavior may be modified
according particular print mode features. For example, @{command
"pr"}~@{text "(latex xsymbols)"} would print the current proof state
with mathematical symbols and special characters represented in
{\LaTeX} source, according to the Isabelle style
\cite{isabelle-sys}.
Note that antiquotations (cf.\ \secref{sec:antiq}) provide a more
systematic way to include formal items into the printed text
document.
*}
subsection {* Details of printed content *}
text {*
\begin{mldecls}
@{index_ML show_types: "bool Config.T"} & default @{ML false} \\
@{index_ML show_sorts: "bool Config.T"} & default @{ML false} \\
@{index_ML show_consts: "bool Config.T"} & default @{ML false} \\
@{index_ML show_abbrevs: "bool Config.T"} & default @{ML true} \\
@{index_ML show_brackets: "bool Config.T"} & default @{ML false} \\
@{index_ML Name_Space.long_names: "bool Config.T"} & default @{ML false} \\
@{index_ML Name_Space.short_names: "bool Config.T"} & default @{ML false} \\
@{index_ML Name_Space.unique_names: "bool Config.T"} & default @{ML true} \\
@{index_ML eta_contract: "bool Config.T"} & default @{ML true} \\
@{index_ML Goal_Display.goals_limit: "int Config.T"} & default @{ML 10} \\
@{index_ML Goal_Display.show_main_goal: "bool Config.T"} & default @{ML false} \\
@{index_ML show_hyps: "bool Config.T"} & default @{ML false} \\
@{index_ML show_tags: "bool Config.T"} & default @{ML false} \\
@{index_ML show_question_marks: "bool Config.T"} & default @{ML true} \\
\end{mldecls}
These global ML variables control the detail of information that is
displayed for types, terms, theorems, goals etc.
In interactive sessions, the user interface usually manages these
global parameters of the Isabelle process, even with some concept of
persistence. Nonetheless it is occasionally useful to manipulate ML
variables directly, e.g.\ using @{command "ML_val"} or @{command
"ML_command"}.
Batch-mode logic sessions may be configured by putting appropriate
ML text directly into the @{verbatim ROOT.ML} file.
\begin{description}
\item @{ML show_types} and @{ML show_sorts} control printing of type
constraints for term variables, and sort constraints for type
variables. By default, neither of these are shown in output. If
@{ML show_sorts} is set to @{ML true}, types are always shown as
well.
Note that displaying types and sorts may explain why a polymorphic
inference rule fails to resolve with some goal, or why a rewrite
rule does not apply as expected.
\item @{ML show_consts} controls printing of types of constants when
displaying a goal state.
Note that the output can be enormous, because polymorphic constants
often occur at several different type instances.
\item @{ML show_abbrevs} controls folding of constant abbreviations.
\item @{ML show_brackets} controls bracketing in pretty printed
output. If set to @{ML true}, all sub-expressions of the pretty
printing tree will be parenthesized, even if this produces malformed
term syntax! This crude way of showing the internal structure of
pretty printed entities may occasionally help to diagnose problems
with operator priorities, for example.
\item @{ML Name_Space.long_names}, @{ML Name_Space.short_names}, and
@{ML Name_Space.unique_names} control the way of printing fully
qualified internal names in external form. See also
\secref{sec:antiq} for the document antiquotation options of the
same names.
\item @{ML eta_contract} controls @{text "\<eta>"}-contracted printing of
terms.
The @{text \<eta>}-contraction law asserts @{prop "(\<lambda>x. f x) \<equiv> f"},
provided @{text x} is not free in @{text f}. It asserts
\emph{extensionality} of functions: @{prop "f \<equiv> g"} if @{prop "f x \<equiv>
g x"} for all @{text x}. Higher-order unification frequently puts
terms into a fully @{text \<eta>}-expanded form. For example, if @{text
F} has type @{text "(\<tau> \<Rightarrow> \<tau>) \<Rightarrow> \<tau>"} then its expanded form is @{term
"\<lambda>h. F (\<lambda>x. h x)"}.
Setting @{ML eta_contract} makes Isabelle perform @{text
\<eta>}-contractions before printing, so that @{term "\<lambda>h. F (\<lambda>x. h x)"}
appears simply as @{text F}.
Note that the distinction between a term and its @{text \<eta>}-expanded
form occasionally matters. While higher-order resolution and
rewriting operate modulo @{text "\<alpha>\<beta>\<eta>"}-conversion, some other tools
might look at terms more discretely.
\item @{ML Goal_Display.goals_limit} controls the maximum number of
subgoals to be shown in goal output.
\item @{ML Goal_Display.show_main_goal} controls whether the main
result to be proven should be displayed. This information might be
relevant for schematic goals, to inspect the current claim that has
been synthesized so far.
\item @{ML show_hyps} controls printing of implicit hypotheses of
local facts. Normally, only those hypotheses are displayed that are
\emph{not} covered by the assumptions of the current context: this
situation indicates a fault in some tool being used.
By setting @{ML show_hyps} to @{ML true}, output of \emph{all}
hypotheses can be enforced, which is occasionally useful for
diagnostic purposes.
\item @{ML show_tags} controls printing of extra annotations within
theorems, such as internal position information, or the case names
being attached by the attribute @{attribute case_names}.
Note that the @{attribute tagged} and @{attribute untagged}
attributes provide low-level access to the collection of tags
associated with a theorem.
\item @{ML show_question_marks} controls printing of question marks
for schematic variables, such as @{text ?x}. Only the leading
question mark is affected, the remaining text is unchanged
(including proper markup for schematic variables that might be
relevant for user interfaces).
\end{description}
*}
subsection {* Printing limits *}
text {*
\begin{mldecls}
@{index_ML Pretty.margin_default: "int Unsynchronized.ref"} \\
@{index_ML print_depth: "int -> unit"} \\
\end{mldecls}
These ML functions set limits for pretty printed text.
\begin{description}
\item @{ML Pretty.margin_default} indicates the global default for
the right margin of the built-in pretty printer, with initial value
76. Note that user-interfaces typically control margins
automatically when resizing windows, or even bypass the formatting
engine of Isabelle/ML altogether and do it within the front end via
Isabelle/Scala.
\item @{ML print_depth}~@{text n} limits the printing depth of the
ML toplevel pretty printer; the precise effect depends on the ML
compiler and run-time system. Typically @{text n} should be less
than 10. Bigger values such as 100--1000 are useful for debugging.
\end{description}
*}
section {* Mixfix annotations *}
text {* Mixfix annotations specify concrete \emph{inner syntax} of
Isabelle types and terms. Locally fixed parameters in toplevel
theorem statements, locale specifications etc.\ also admit mixfix
annotations.
\indexouternonterm{infix}\indexouternonterm{mixfix}\indexouternonterm{structmixfix}
\begin{rail}
infix: '(' ('infix' | 'infixl' | 'infixr') string nat ')'
;
mixfix: infix | '(' string prios? nat? ')' | '(' 'binder' string prios? nat ')'
;
structmixfix: mixfix | '(' 'structure' ')'
;
prios: '[' (nat + ',') ']'
;
\end{rail}
Here the \railtok{string} specifications refer to the actual mixfix
template, which may include literal text, spacing, blocks, and
arguments (denoted by ``@{text _}''); the special symbol
``@{verbatim "\<index>"}'' (printed as ``@{text "\<index>"}'') represents an index
argument that specifies an implicit structure reference (see also
\secref{sec:locale}). Infix and binder declarations provide common
abbreviations for particular mixfix declarations. So in practice,
mixfix templates mostly degenerate to literal text for concrete
syntax, such as ``@{verbatim "++"}'' for an infix symbol.
\medskip In full generality, mixfix declarations work as follows.
Suppose a constant @{text "c :: \<tau>\<^sub>1 \<Rightarrow> \<dots> \<tau>\<^sub>n \<Rightarrow> \<tau>"} is
annotated by @{text "(mixfix [p\<^sub>1, \<dots>, p\<^sub>n] p)"}, where @{text
"mixfix"} is a string @{text "d\<^sub>0 _ d\<^sub>1 _ \<dots> _ d\<^sub>n"} consisting of
delimiters that surround argument positions as indicated by
underscores.
Altogether this determines a production for a context-free priority
grammar, where for each argument @{text "i"} the syntactic category
is determined by @{text "\<tau>\<^sub>i"} (with priority @{text "p\<^sub>i"}), and
the result category is determined from @{text "\<tau>"} (with
priority @{text "p"}). Priority specifications are optional, with
default 0 for arguments and 1000 for the result.
Since @{text "\<tau>"} may be again a function type, the constant
type scheme may have more argument positions than the mixfix
pattern. Printing a nested application @{text "c t\<^sub>1 \<dots> t\<^sub>m"} for
@{text "m > n"} works by attaching concrete notation only to the
innermost part, essentially by printing @{text "(c t\<^sub>1 \<dots> t\<^sub>n) \<dots> t\<^sub>m"}
instead. If a term has fewer arguments than specified in the mixfix
template, the concrete syntax is ignored.
\medskip A mixfix template may also contain additional directives
for pretty printing, notably spaces, blocks, and breaks. The
general template format is a sequence over any of the following
entities.
\begin{description}
\item @{text "d"} is a delimiter, namely a non-empty sequence of
characters other than the following special characters:
\smallskip
\begin{tabular}{ll}
@{verbatim "'"} & single quote \\
@{verbatim "_"} & underscore \\
@{text "\<index>"} & index symbol \\
@{verbatim "("} & open parenthesis \\
@{verbatim ")"} & close parenthesis \\
@{verbatim "/"} & slash \\
\end{tabular}
\medskip
\item @{verbatim "'"} escapes the special meaning of these
meta-characters, producing a literal version of the following
character, unless that is a blank.
A single quote followed by a blank separates delimiters, without
affecting printing, but input tokens may have additional white space
here.
\item @{verbatim "_"} is an argument position, which stands for a
certain syntactic category in the underlying grammar.
\item @{text "\<index>"} is an indexed argument position; this is the place
where implicit structure arguments can be attached.
\item @{text "s"} is a non-empty sequence of spaces for printing.
This and the following specifications do not affect parsing at all.
\item @{verbatim "("}@{text n} opens a pretty printing block. The
optional number specifies how much indentation to add when a line
break occurs within the block. If the parenthesis is not followed
by digits, the indentation defaults to 0. A block specified via
@{verbatim "(00"} is unbreakable.
\item @{verbatim ")"} closes a pretty printing block.
\item @{verbatim "//"} forces a line break.
\item @{verbatim "/"}@{text s} allows a line break. Here @{text s}
stands for the string of spaces (zero or more) right after the
slash. These spaces are printed if the break is \emph{not} taken.
\end{description}
For example, the template @{verbatim "(_ +/ _)"} specifies an infix
operator. There are two argument positions; the delimiter
@{verbatim "+"} is preceded by a space and followed by a space or
line break; the entire phrase is a pretty printing block.
The general idea of pretty printing with blocks and breaks is also
described in \cite{paulson-ml2}.
*}
section {* Explicit notation *}
text {*
\begin{matharray}{rcll}
@{command_def "type_notation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "no_type_notation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "notation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "no_notation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
@{command_def "write"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\
\end{matharray}
\begin{rail}
('type_notation' | 'no_type_notation') target? mode? \\ (nameref mixfix + 'and')
;
('notation' | 'no_notation') target? mode? \\ (nameref structmixfix + 'and')
;
'write' mode? (nameref structmixfix + 'and')
;
\end{rail}
\begin{description}
\item @{command "type_notation"}~@{text "c (mx)"} associates mixfix
syntax with an existing type constructor. The arity of the
constructor is retrieved from the context.
\item @{command "no_type_notation"} is similar to @{command
"type_notation"}, but removes the specified syntax annotation from
the present context.
\item @{command "notation"}~@{text "c (mx)"} associates mixfix
syntax with an existing constant or fixed variable. The type
declaration of the given entity is retrieved from the context.
\item @{command "no_notation"} is similar to @{command "notation"},
but removes the specified syntax annotation from the present
context.
\item @{command "write"} is similar to @{command "notation"}, but
works within an Isar proof body.
\end{description}
Note that the more primitive commands @{command "syntax"} and
@{command "no_syntax"} (\secref{sec:syn-trans}) provide raw access
to the syntax tables of a global theory.
*}
section {* The Pure syntax \label{sec:pure-syntax} *}
subsection {* Priority grammars \label{sec:priority-grammar} *}
text {* A context-free grammar consists of a set of \emph{terminal
symbols}, a set of \emph{nonterminal symbols} and a set of
\emph{productions}. Productions have the form @{text "A = \<gamma>"},
where @{text A} is a nonterminal and @{text \<gamma>} is a string of
terminals and nonterminals. One designated nonterminal is called
the \emph{root symbol}. The language defined by the grammar
consists of all strings of terminals that can be derived from the
root symbol by applying productions as rewrite rules.
The standard Isabelle parser for inner syntax uses a \emph{priority
grammar}. Each nonterminal is decorated by an integer priority:
@{text "A\<^sup>(\<^sup>p\<^sup>)"}. In a derivation, @{text "A\<^sup>(\<^sup>p\<^sup>)"} may be rewritten
using a production @{text "A\<^sup>(\<^sup>q\<^sup>) = \<gamma>"} only if @{text "p \<le> q"}. Any
priority grammar can be translated into a normal context-free
grammar by introducing new nonterminals and productions.
\medskip Formally, a set of context free productions @{text G}
induces a derivation relation @{text "\<longrightarrow>\<^sub>G"} as follows. Let @{text
\<alpha>} and @{text \<beta>} denote strings of terminal or nonterminal symbols.
Then @{text "\<alpha> A\<^sup>(\<^sup>p\<^sup>) \<beta> \<longrightarrow>\<^sub>G \<alpha> \<gamma> \<beta>"} holds if and only if @{text G}
contains some production @{text "A\<^sup>(\<^sup>q\<^sup>) = \<gamma>"} for @{text "p \<le> q"}.
\medskip The following grammar for arithmetic expressions
demonstrates how binding power and associativity of operators can be
enforced by priorities.
\begin{center}
\begin{tabular}{rclr}
@{text "A\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>)"} & @{text "="} & @{verbatim "("} @{text "A\<^sup>(\<^sup>0\<^sup>)"} @{verbatim ")"} \\
@{text "A\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>)"} & @{text "="} & @{verbatim 0} \\
@{text "A\<^sup>(\<^sup>0\<^sup>)"} & @{text "="} & @{text "A\<^sup>(\<^sup>0\<^sup>)"} @{verbatim "+"} @{text "A\<^sup>(\<^sup>1\<^sup>)"} \\
@{text "A\<^sup>(\<^sup>2\<^sup>)"} & @{text "="} & @{text "A\<^sup>(\<^sup>3\<^sup>)"} @{verbatim "*"} @{text "A\<^sup>(\<^sup>2\<^sup>)"} \\
@{text "A\<^sup>(\<^sup>3\<^sup>)"} & @{text "="} & @{verbatim "-"} @{text "A\<^sup>(\<^sup>3\<^sup>)"} \\
\end{tabular}
\end{center}
The choice of priorities determines that @{verbatim "-"} binds
tighter than @{verbatim "*"}, which binds tighter than @{verbatim
"+"}. Furthermore @{verbatim "+"} associates to the left and
@{verbatim "*"} to the right.
\medskip For clarity, grammars obey these conventions:
\begin{itemize}
\item All priorities must lie between 0 and 1000.
\item Priority 0 on the right-hand side and priority 1000 on the
left-hand side may be omitted.
\item The production @{text "A\<^sup>(\<^sup>p\<^sup>) = \<alpha>"} is written as @{text "A = \<alpha>
(p)"}, i.e.\ the priority of the left-hand side actually appears in
a column on the far right.
\item Alternatives are separated by @{text "|"}.
\item Repetition is indicated by dots @{text "(\<dots>)"} in an informal
but obvious way.
\end{itemize}
Using these conventions, the example grammar specification above
takes the form:
\begin{center}
\begin{tabular}{rclc}
@{text A} & @{text "="} & @{verbatim "("} @{text A} @{verbatim ")"} \\
& @{text "|"} & @{verbatim 0} & \qquad\qquad \\
& @{text "|"} & @{text A} @{verbatim "+"} @{text "A\<^sup>(\<^sup>1\<^sup>)"} & @{text "(0)"} \\
& @{text "|"} & @{text "A\<^sup>(\<^sup>3\<^sup>)"} @{verbatim "*"} @{text "A\<^sup>(\<^sup>2\<^sup>)"} & @{text "(2)"} \\
& @{text "|"} & @{verbatim "-"} @{text "A\<^sup>(\<^sup>3\<^sup>)"} & @{text "(3)"} \\
\end{tabular}
\end{center}
*}
subsection {* The Pure grammar *}
text {*
The priority grammar of the @{text "Pure"} theory is defined as follows:
%FIXME syntax for "index" (?)
%FIXME "op" versions of ==> etc. (?)
\begin{center}
\begin{supertabular}{rclr}
@{syntax_def (inner) any} & = & @{text "prop | logic"} \\\\
@{syntax_def (inner) prop} & = & @{verbatim "("} @{text prop} @{verbatim ")"} \\
& @{text "|"} & @{text "prop\<^sup>(\<^sup>4\<^sup>)"} @{verbatim "::"} @{text type} & @{text "(3)"} \\
& @{text "|"} & @{text "any\<^sup>(\<^sup>3\<^sup>)"} @{verbatim "=="} @{text "any\<^sup>(\<^sup>2\<^sup>)"} & @{text "(2)"} \\
& @{text "|"} & @{text "any\<^sup>(\<^sup>3\<^sup>)"} @{text "\<equiv>"} @{text "any\<^sup>(\<^sup>2\<^sup>)"} & @{text "(2)"} \\
& @{text "|"} & @{text "prop\<^sup>(\<^sup>3\<^sup>)"} @{verbatim "&&&"} @{text "prop\<^sup>(\<^sup>2\<^sup>)"} & @{text "(2)"} \\
& @{text "|"} & @{text "prop\<^sup>(\<^sup>2\<^sup>)"} @{verbatim "==>"} @{text "prop\<^sup>(\<^sup>1\<^sup>)"} & @{text "(1)"} \\
& @{text "|"} & @{text "prop\<^sup>(\<^sup>2\<^sup>)"} @{text "\<Longrightarrow>"} @{text "prop\<^sup>(\<^sup>1\<^sup>)"} & @{text "(1)"} \\
& @{text "|"} & @{verbatim "[|"} @{text prop} @{verbatim ";"} @{text "\<dots>"} @{verbatim ";"} @{text prop} @{verbatim "|]"} @{verbatim "==>"} @{text "prop\<^sup>(\<^sup>1\<^sup>)"} & @{text "(1)"} \\
& @{text "|"} & @{text "\<lbrakk>"} @{text prop} @{verbatim ";"} @{text "\<dots>"} @{verbatim ";"} @{text prop} @{text "\<rbrakk>"} @{text "\<Longrightarrow>"} @{text "prop\<^sup>(\<^sup>1\<^sup>)"} & @{text "(1)"} \\
& @{text "|"} & @{verbatim "!!"} @{text idts} @{verbatim "."} @{text prop} & @{text "(0)"} \\
& @{text "|"} & @{text "\<And>"} @{text idts} @{verbatim "."} @{text prop} & @{text "(0)"} \\
& @{text "|"} & @{verbatim OFCLASS} @{verbatim "("} @{text type} @{verbatim ","} @{text logic} @{verbatim ")"} \\
& @{text "|"} & @{verbatim SORT_CONSTRAINT} @{verbatim "("} @{text type} @{verbatim ")"} \\
& @{text "|"} & @{verbatim TERM} @{text logic} \\
& @{text "|"} & @{verbatim PROP} @{text aprop} \\\\
@{syntax_def (inner) aprop} & = & @{verbatim "("} @{text aprop} @{verbatim ")"} \\
& @{text "|"} & @{text "id | longid | var | "}@{verbatim "_"}@{text " | "}@{verbatim "..."} \\
& @{text "|"} & @{verbatim CONST} @{text "id | "}@{verbatim CONST} @{text "longid"} \\
& @{text "|"} & @{text "logic\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) any\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) \<dots> any\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>)"} & @{text "(999)"} \\\\
@{syntax_def (inner) logic} & = & @{verbatim "("} @{text logic} @{verbatim ")"} \\
& @{text "|"} & @{text "logic\<^sup>(\<^sup>4\<^sup>)"} @{verbatim "::"} @{text type} & @{text "(3)"} \\
& @{text "|"} & @{text "id | longid | var | "}@{verbatim "_"}@{text " | "}@{verbatim "..."} \\
& @{text "|"} & @{verbatim CONST} @{text "id | "}@{verbatim CONST} @{text "longid"} \\
& @{text "|"} & @{text "logic\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) any\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) \<dots> any\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>)"} & @{text "(999)"} \\
& @{text "|"} & @{verbatim "%"} @{text pttrns} @{verbatim "."} @{text "any\<^sup>(\<^sup>3\<^sup>)"} & @{text "(3)"} \\
& @{text "|"} & @{text \<lambda>} @{text pttrns} @{verbatim "."} @{text "any\<^sup>(\<^sup>3\<^sup>)"} & @{text "(3)"} \\
& @{text "|"} & @{verbatim TYPE} @{verbatim "("} @{text type} @{verbatim ")"} \\\\
@{syntax_def (inner) idt} & = & @{verbatim "("} @{text idt} @{verbatim ")"}@{text " | id | "}@{verbatim "_"} \\
& @{text "|"} & @{text id} @{verbatim "::"} @{text type} & @{text "(0)"} \\
& @{text "|"} & @{verbatim "_"} @{verbatim "::"} @{text type} & @{text "(0)"} \\\\
@{syntax_def (inner) idts} & = & @{text "idt | idt\<^sup>(\<^sup>1\<^sup>) idts"} & @{text "(0)"} \\\\
@{syntax_def (inner) pttrn} & = & @{text idt} \\\\
@{syntax_def (inner) pttrns} & = & @{text "pttrn | pttrn\<^sup>(\<^sup>1\<^sup>) pttrns"} & @{text "(0)"} \\\\
@{syntax_def (inner) type} & = & @{verbatim "("} @{text type} @{verbatim ")"} \\
& @{text "|"} & @{text "tid | tvar | "}@{verbatim "_"} \\
& @{text "|"} & @{text "tid"} @{verbatim "::"} @{text "sort | tvar "}@{verbatim "::"} @{text "sort | "}@{verbatim "_"} @{verbatim "::"} @{text "sort"} \\
& @{text "|"} & @{text "id | type\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) id | "}@{verbatim "("} @{text type} @{verbatim ","} @{text "\<dots>"} @{verbatim ","} @{text type} @{verbatim ")"} @{text id} \\
& @{text "|"} & @{text "longid | type\<^sup>(\<^sup>1\<^sup>0\<^sup>0\<^sup>0\<^sup>) longid"} \\
& @{text "|"} & @{verbatim "("} @{text type} @{verbatim ","} @{text "\<dots>"} @{verbatim ","} @{text type} @{verbatim ")"} @{text longid} \\
& @{text "|"} & @{text "type\<^sup>(\<^sup>1\<^sup>)"} @{verbatim "=>"} @{text type} & @{text "(0)"} \\
& @{text "|"} & @{text "type\<^sup>(\<^sup>1\<^sup>)"} @{text "\<Rightarrow>"} @{text type} & @{text "(0)"} \\
& @{text "|"} & @{verbatim "["} @{text type} @{verbatim ","} @{text "\<dots>"} @{verbatim ","} @{text type} @{verbatim "]"} @{verbatim "=>"} @{text type} & @{text "(0)"} \\
& @{text "|"} & @{verbatim "["} @{text type} @{verbatim ","} @{text "\<dots>"} @{verbatim ","} @{text type} @{verbatim "]"} @{text "\<Rightarrow>"} @{text type} & @{text "(0)"} \\\\
@{syntax_def (inner) sort} & = & @{text "id | longid | "}@{verbatim "{}"} \\
& @{text "|"} & @{verbatim "{"} @{text "(id | longid)"} @{verbatim ","} @{text "\<dots>"} @{verbatim ","} @{text "(id | longid)"} @{verbatim "}"} \\
\end{supertabular}
\end{center}
\medskip Here literal terminals are printed @{verbatim "verbatim"};
see also \secref{sec:inner-lex} for further token categories of the
inner syntax. The meaning of the nonterminals defined by the above
grammar is as follows:
\begin{description}
\item @{syntax_ref (inner) any} denotes any term.
\item @{syntax_ref (inner) prop} denotes meta-level propositions,
which are terms of type @{typ prop}. The syntax of such formulae of
the meta-logic is carefully distinguished from usual conventions for
object-logics. In particular, plain @{text "\<lambda>"}-term notation is
\emph{not} recognized as @{syntax (inner) prop}.
\item @{syntax_ref (inner) aprop} denotes atomic propositions, which
are embedded into regular @{syntax (inner) prop} by means of an
explicit @{verbatim PROP} token.
Terms of type @{typ prop} with non-constant head, e.g.\ a plain
variable, are printed in this form. Constants that yield type @{typ
prop} are expected to provide their own concrete syntax; otherwise
the printed version will appear like @{syntax (inner) logic} and
cannot be parsed again as @{syntax (inner) prop}.
\item @{syntax_ref (inner) logic} denotes arbitrary terms of a
logical type, excluding type @{typ prop}. This is the main
syntactic category of object-logic entities, covering plain @{text
\<lambda>}-term notation (variables, abstraction, application), plus
anything defined by the user.
When specifying notation for logical entities, all logical types
(excluding @{typ prop}) are \emph{collapsed} to this single category
of @{syntax (inner) logic}.
\item @{syntax_ref (inner) idt} denotes identifiers, possibly
constrained by types.
\item @{syntax_ref (inner) idts} denotes a sequence of @{syntax_ref
(inner) idt}. This is the most basic category for variables in
iterated binders, such as @{text "\<lambda>"} or @{text "\<And>"}.
\item @{syntax_ref (inner) pttrn} and @{syntax_ref (inner) pttrns}
denote patterns for abstraction, cases bindings etc. In Pure, these
categories start as a merely copy of @{syntax (inner) idt} and
@{syntax (inner) idts}, respectively. Object-logics may add
additional productions for binding forms.
\item @{syntax_ref (inner) type} denotes types of the meta-logic.
\item @{syntax_ref (inner) sort} denotes meta-level sorts.
\end{description}
Here are some further explanations of certain syntax features.
\begin{itemize}
\item In @{syntax (inner) idts}, note that @{text "x :: nat y"} is
parsed as @{text "x :: (nat y)"}, treating @{text y} like a type
constructor applied to @{text nat}. To avoid this interpretation,
write @{text "(x :: nat) y"} with explicit parentheses.
\item Similarly, @{text "x :: nat y :: nat"} is parsed as @{text "x ::
(nat y :: nat)"}. The correct form is @{text "(x :: nat) (y ::
nat)"}, or @{text "(x :: nat) y :: nat"} if @{text y} is last in the
sequence of identifiers.
\item Type constraints for terms bind very weakly. For example,
@{text "x < y :: nat"} is normally parsed as @{text "(x < y) ::
nat"}, unless @{text "<"} has a very low priority, in which case the
input is likely to be ambiguous. The correct form is @{text "x < (y
:: nat)"}.
\item Constraints may be either written with two literal colons
``@{verbatim "::"}'' or the double-colon symbol @{verbatim "\<Colon>"},
which actually looks exactly the same in some {\LaTeX} styles.
\item Dummy variables (written as underscore) may occur in different
roles.
\begin{description}
\item A type ``@{text "_"}'' or ``@{text "_ :: sort"}'' acts like an
anonymous inference parameter, which is filled-in according to the
most general type produced by the type-checking phase.
\item A bound ``@{text "_"}'' refers to a vacuous abstraction, where
the body does not refer to the binding introduced here. As in the
term @{term "\<lambda>x _. x"}, which is @{text "\<alpha>"}-equivalent to @{text
"\<lambda>x y. x"}.
\item A free ``@{text "_"}'' refers to an implicit outer binding.
Higher definitional packages usually allow forms like @{text "f x _
= x"}.
\item A schematic ``@{text "_"}'' (within a term pattern, see
\secref{sec:term-decls}) refers to an anonymous variable that is
implicitly abstracted over its context of locally bound variables.
For example, this allows pattern matching of @{text "{x. f x = g
x}"} against @{text "{x. _ = _}"}, or even @{text "{_. _ = _}"} by
using both bound and schematic dummies.
\end{description}
\item The three literal dots ``@{verbatim "..."}'' may be also
written as ellipsis symbol @{verbatim "\<dots>"}. In both cases this
refers to a special schematic variable, which is bound in the
context. This special term abbreviation works nicely with
calculational reasoning (\secref{sec:calculation}).
\end{itemize}
*}
section {* Lexical matters \label{sec:inner-lex} *}
text {* The inner lexical syntax vaguely resembles the outer one
(\secref{sec:outer-lex}), but some details are different. There are
two main categories of inner syntax tokens:
\begin{enumerate}
\item \emph{delimiters} --- the literal tokens occurring in
productions of the given priority grammar (cf.\
\secref{sec:priority-grammar});
\item \emph{named tokens} --- various categories of identifiers etc.
\end{enumerate}
Delimiters override named tokens and may thus render certain
identifiers inaccessible. Sometimes the logical context admits
alternative ways to refer to the same entity, potentially via
qualified names.
\medskip The categories for named tokens are defined once and for
all as follows, reusing some categories of the outer token syntax
(\secref{sec:outer-lex}).
\begin{center}
\begin{supertabular}{rcl}
@{syntax_def (inner) id} & = & @{syntax_ref ident} \\
@{syntax_def (inner) longid} & = & @{syntax_ref longident} \\
@{syntax_def (inner) var} & = & @{syntax_ref var} \\
@{syntax_def (inner) tid} & = & @{syntax_ref typefree} \\
@{syntax_def (inner) tvar} & = & @{syntax_ref typevar} \\
@{syntax_def (inner) num} & = & @{syntax_ref nat}@{text " | "}@{verbatim "-"}@{syntax_ref nat} \\
@{syntax_def (inner) float_token} & = & @{syntax_ref nat}@{verbatim "."}@{syntax_ref nat}@{text " | "}@{verbatim "-"}@{syntax_ref nat}@{verbatim "."}@{syntax_ref nat} \\
@{syntax_def (inner) xnum} & = & @{verbatim "#"}@{syntax_ref nat}@{text " | "}@{verbatim "#-"}@{syntax_ref nat} \\
@{syntax_def (inner) xstr} & = & @{verbatim "''"} @{text "\<dots>"} @{verbatim "''"} \\
\end{supertabular}
\end{center}
The token categories @{syntax (inner) num}, @{syntax (inner)
float_token}, @{syntax (inner) xnum}, and @{syntax (inner) xstr} are
not used in Pure. Object-logics may implement numerals and string
constants by adding appropriate syntax declarations, together with
some translation functions (e.g.\ see Isabelle/HOL).
The derived categories @{syntax_def (inner) num_const} and
@{syntax_def (inner) float_const} provide robust access to @{syntax
(inner) num}, and @{syntax (inner) float_token}, respectively: the
syntax tree holds a syntactic constant instead of a free variable.
*}
section {* Syntax and translations \label{sec:syn-trans} *}
text {*
\begin{matharray}{rcl}
@{command_def "nonterminal"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "syntax"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "no_syntax"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "translations"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "no_translations"} & : & @{text "theory \<rightarrow> theory"} \\
\end{matharray}
\begin{rail}
'nonterminal' (name + 'and')
;
('syntax' | 'no_syntax') mode? (constdecl +)
;
('translations' | 'no_translations') (transpat ('==' | '=>' | '<=' | rightleftharpoons | rightharpoonup | leftharpoondown) transpat +)
;
mode: ('(' ( name | 'output' | name 'output' ) ')')
;
transpat: ('(' nameref ')')? string
;
\end{rail}
\begin{description}
\item @{command "nonterminal"}~@{text c} declares a type
constructor @{text c} (without arguments) to act as purely syntactic
type: a nonterminal symbol of the inner syntax.
\item @{command "syntax"}~@{text "(mode) decls"} is similar to
@{command "consts"}~@{text decls}, except that the actual logical
signature extension is omitted. Thus the context free grammar of
Isabelle's inner syntax may be augmented in arbitrary ways,
independently of the logic. The @{text mode} argument refers to the
print mode that the grammar rules belong; unless the @{keyword_ref
"output"} indicator is given, all productions are added both to the
input and output grammar.
\item @{command "no_syntax"}~@{text "(mode) decls"} removes grammar
declarations (and translations) resulting from @{text decls}, which
are interpreted in the same manner as for @{command "syntax"} above.
\item @{command "translations"}~@{text rules} specifies syntactic
translation rules (i.e.\ macros): parse~/ print rules (@{text "\<rightleftharpoons>"}),
parse rules (@{text "\<rightharpoonup>"}), or print rules (@{text "\<leftharpoondown>"}).
Translation patterns may be prefixed by the syntactic category to be
used for parsing; the default is @{text logic}.
\item @{command "no_translations"}~@{text rules} removes syntactic
translation rules, which are interpreted in the same manner as for
@{command "translations"} above.
\end{description}
*}
section {* Syntax translation functions \label{sec:tr-funs} *}
text {*
\begin{matharray}{rcl}
@{command_def "parse_ast_translation"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "parse_translation"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "print_translation"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "typed_print_translation"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def "print_ast_translation"} & : & @{text "theory \<rightarrow> theory"} \\
\end{matharray}
\begin{rail}
( 'parse_ast_translation' | 'parse_translation' | 'print_translation' |
'typed_print_translation' | 'print_ast_translation' ) ('(advanced)')? text
;
\end{rail}
Syntax translation functions written in ML admit almost arbitrary
manipulations of Isabelle's inner syntax. Any of the above commands
have a single \railqtok{text} argument that refers to an ML
expression of appropriate type, which are as follows by default:
%FIXME proper antiquotations
\begin{ttbox}
val parse_ast_translation : (string * (ast list -> ast)) list
val parse_translation : (string * (term list -> term)) list
val print_translation : (string * (term list -> term)) list
val typed_print_translation : (string * (typ -> term list -> term)) list
val print_ast_translation : (string * (ast list -> ast)) list
\end{ttbox}
If the @{text "(advanced)"} option is given, the corresponding
translation functions may depend on the current theory or proof
context. This allows to implement advanced syntax mechanisms, as
translations functions may refer to specific theory declarations or
auxiliary proof data.
See also \cite{isabelle-ref} for more information on the general
concept of syntax transformations in Isabelle.
%FIXME proper antiquotations
\begin{ttbox}
val parse_ast_translation:
(string * (Proof.context -> ast list -> ast)) list
val parse_translation:
(string * (Proof.context -> term list -> term)) list
val print_translation:
(string * (Proof.context -> term list -> term)) list
val typed_print_translation:
(string * (Proof.context -> typ -> term list -> term)) list
val print_ast_translation:
(string * (Proof.context -> ast list -> ast)) list
\end{ttbox}
*}
section {* Inspecting the syntax *}
text {*
\begin{matharray}{rcl}
@{command_def "print_syntax"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
\end{matharray}
\begin{description}
\item @{command "print_syntax"} prints the inner syntax of the
current context. The output can be quite large; the most important
sections are explained below.
\begin{description}
\item @{text "lexicon"} lists the delimiters of the inner token
language; see \secref{sec:inner-lex}.
\item @{text "prods"} lists the productions of the underlying
priority grammar; see \secref{sec:priority-grammar}.
The nonterminal @{text "A\<^sup>(\<^sup>p\<^sup>)"} is rendered in plain text as @{text
"A[p]"}; delimiters are quoted. Many productions have an extra
@{text "\<dots> => name"}. These names later become the heads of parse
trees; they also guide the pretty printer.
Productions without such parse tree names are called \emph{copy
productions}. Their right-hand side must have exactly one
nonterminal symbol (or named token). The parser does not create a
new parse tree node for copy productions, but simply returns the
parse tree of the right-hand symbol.
If the right-hand side of a copy production consists of a single
nonterminal without any delimiters, then it is called a \emph{chain
production}. Chain productions act as abbreviations: conceptually,
they are removed from the grammar by adding new productions.
Priority information attached to chain productions is ignored; only
the dummy value @{text "-1"} is displayed.
\item @{text "print modes"} lists the alternative print modes
provided by this grammar; see \secref{sec:print-modes}.
\item @{text "parse_rules"} and @{text "print_rules"} relate to
syntax translations (macros); see \secref{sec:syn-trans}.
\item @{text "parse_ast_translation"} and @{text
"print_ast_translation"} list sets of constants that invoke
translation functions for abstract syntax trees, which are only
required in very special situations; see \secref{sec:tr-funs}.
\item @{text "parse_translation"} and @{text "print_translation"}
list the sets of constants that invoke regular translation
functions; see \secref{sec:tr-funs}.
\end{description}
\end{description}
*}
end