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? (',' 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, prems"} prints the current

   proof state (if present), including the proof context, current facts

   and goals.  The optional limit arguments affect the number of goals

   and premises to be displayed, which is initially 10 for both.

   Omitting limit values 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 Unsynchronized.ref"} & default @{ML false} \\

     @{index_ML show_sorts: "bool Unsynchronized.ref"} & default @{ML false} \\

     @{index_ML show_consts: "bool Unsynchronized.ref"} & default @{ML false} \\

     @{index_ML long_names: "bool Unsynchronized.ref"} & default @{ML false} \\

     @{index_ML short_names: "bool Unsynchronized.ref"} & default @{ML false} \\

     @{index_ML unique_names: "bool Unsynchronized.ref"} & default @{ML true} \\

     @{index_ML show_brackets: "bool Unsynchronized.ref"} & default @{ML false} \\

     @{index_ML eta_contract: "bool Unsynchronized.ref"} & default @{ML true} \\

     @{index_ML goals_limit: "int Unsynchronized.ref"} & default @{ML 10} \\

     @{index_ML Proof.show_main_goal: "bool Unsynchronized.ref"} & default @{ML false} \\

     @{index_ML show_hyps: "bool Unsynchronized.ref"} & default @{ML false} \\

     @{index_ML show_tags: "bool Unsynchronized.ref"} & default @{ML false} \\

     @{index_ML show_question_marks: "bool Unsynchronized.ref"} & 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 long_names}, @{ML short_names}, and @{ML 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 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 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 goals_limit} controls the maximum number of subgoals to

   be shown in goal output.

   \item @{ML Proof.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.setdepth: "int -> unit"} \\

     @{index_ML Pretty.setmargin: "int -> unit"} \\

     @{index_ML print_depth: "int -> unit"} \\

   \end{mldecls}

   These ML functions set limits for pretty printed text.

   \begin{description}

   \item @{ML Pretty.setdepth}~@{text d} tells the pretty printer to

   limit the printing depth to @{text d}.  This affects the display of

   types, terms, theorems etc.  The default value is 0, which permits

   printing to an arbitrary depth.  Other useful values for @{text d}

   are 10 and 20.

   \item @{ML Pretty.setmargin}~@{text m} tells the pretty printer to

   assume a right margin (page width) of @{text m}.  The initial margin

   is 76, but user interfaces might adapt the margin automatically when

   resizing windows.

   \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.  Some commands such as @{command

   "typedecl"} admit infixes only, while @{command "definition"} etc.\

   support the full range of general mixfixes and binders.  Fixed

   parameters in toplevel theorem statements, locale specifications

   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 term notation *}

 text {*

   \begin{matharray}{rcll}

     @{command_def "notation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

     @{command_def "no_notation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\

   \end{matharray}

   \begin{rail}

     ('notation' | 'no\_notation') target? mode? \\ (nameref structmixfix + 'and')

     ;

   \end{rail}

   \begin{description}

   \item @{command "notation"}~@{text "c (mx)"} associates mixfix

   syntax with an existing constant or fixed variable.  This is a

   robust interface to the underlying @{command "syntax"} primitive

   (\secref{sec:syn-trans}).  Type declaration and internal syntactic

   representation 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.

   \end{description}

 *}

 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>)"} @{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 "nonterminals"} & : & @{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}

     'nonterminals' (name +)

     ;

     ('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 "nonterminals"}~@{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 * (bool -> 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 -> bool -> 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