\chapter{Syntax primitives}

The rather generic framework of Isabelle/Isar syntax emerges from three main

syntactic categories: \emph{commands} of the top-level Isar engine (covering

theory and proof elements), \emph{methods} for general goal refinements

(analogous to traditional ``tactics''), and \emph{attributes} for operations

on facts (within a certain context).  Here we give a reference of basic

syntactic entities underlying Isabelle/Isar syntax in a bottom-up manner.

Concrete theory and proof language elements will be introduced later on.

\medskip

In order to get started with writing well-formed Isabelle/Isar documents, the

most important aspect to be noted is the difference of \emph{inner} versus

\emph{outer} syntax.  Inner syntax is that of Isabelle types and terms of the

logic, while outer syntax is that of Isabelle/Isar theory sources (including

proofs).  As a general rule, inner syntax entities may occur only as

\emph{atomic entities} within outer syntax.  For example, the string

\texttt{"x + y"} and identifier \texttt{z} are legal term specifications

within a theory, while \texttt{x + y} is not.

\begin{warn}

  Old-style Isabelle theories used to fake parts of the inner syntax of types,

  with rather complicated rules when quotes may be omitted.  Despite the minor

  drawback of requiring quotes more often, the syntax of Isabelle/Isar is

  somewhat simpler and more robust in that respect.

\end{warn}

Printed theory documents usually omit quotes to gain readability (this is a

matter of {\LaTeX} macro setup, say via \verb,\isabellestyle,, see also

\cite{isabelle-sys}).  Experienced users of Isabelle/Isar may easily

reconstruct the lost technical information, while mere readers need not care

about quotes at all.

\medskip

Isabelle/Isar input may contain any number of input termination characters

``\texttt{;}'' (semicolon) to separate commands explicitly.  This is

particularly useful in interactive shell sessions to make clear where the

current command is intended to end.  Otherwise, the interpreter loop will

continue to issue a secondary prompt ``\verb,#,'' until an end-of-command is

clearly recognized from the input syntax, e.g.\ encounter of the next command

keyword.

Advanced interfaces such as Proof~General \cite{proofgeneral} do not require

explicit semicolons, the amount of input text is determined automatically by

inspecting the present content of the Emacs text buffer.  In the printed

presentation of Isabelle/Isar documents semicolons are omitted altogether for

readability.

\begin{warn}

  Proof~General requires certain syntax classification tables in order to

  achieve properly synchronized interaction with the Isabelle/Isar process.

  These tables need to be consistent with the Isabelle version and particular

  logic image to be used in a running session (common object-logics may well

  change the outer syntax).  The standard setup should work correctly with any

  of the ``official'' logic images derived from Isabelle/HOL (including HOLCF

  etc.).  Users of alternative logics may need to tell Proof~General

  explicitly, e.g.\ by giving an option \verb,-k ZF, (in conjunction with

  \verb,-l ZF, to specify the default logic image).

\end{warn}

\section{Lexical matters}\label{sec:lex-syntax}

The Isabelle/Isar outer syntax provides token classes as presented below; most

of these coincide with the inner lexical syntax as presented in

\cite{isabelle-ref}.

\indexoutertoken{ident}\indexoutertoken{longident}\indexoutertoken{symident}

\indexoutertoken{nat}\indexoutertoken{var}\indexoutertoken{typefree}

\indexoutertoken{typevar}\indexoutertoken{string}\indexoutertoken{verbatim}

\begin{matharray}{rcl}

  ident & = & letter\,quasiletter^* \\

  longident & = & ident (\verb,.,ident)^+ \\

  symident & = & sym^+ ~|~ \verb,\<,ident\verb,>, \\

  nat & = & digit^+ \\

  var & = & ident ~|~ \verb,?,ident ~|~ \verb,?,ident\verb,.,nat \\

  typefree & = & \verb,',ident \\

  typevar & = & typefree ~|~ \verb,?,typefree ~|~ \verb,?,typefree\verb,.,nat \\

  string & = & \verb,", ~\dots~ \verb,", \\

  verbatim & = & \verb,{*, ~\dots~ \verb,*}, \\[1ex]

  letter & = & latin ~|~ \verb,\<,latin\verb,>, ~|~ \verb,\<,latin\,latin\verb,>, ~|~ greek ~|~ \\

         &   & \verb,\<^isub>, ~|~ \verb,\<^isup>, \\

  quasiletter & = & letter ~|~ digit ~|~ \verb,_, ~|~ \verb,', \\

  latin & = & \verb,a, ~|~ \dots ~|~ \verb,z, ~|~ \verb,A, ~|~ \dots ~|~ \verb,Z, \\

  digit & = & \verb,0, ~|~ \dots ~|~ \verb,9, \\

  sym & = & \verb,!, ~|~ \verb,#, ~|~ \verb,$, ~|~ \verb,%, ~|~ \verb,&, ~|~  %$

   \verb,*, ~|~ \verb,+, ~|~ \verb,-, ~|~ \verb,/, ~|~ \verb,:, ~|~ \\

  & & \verb,<, ~|~ \verb,=, ~|~ \verb,>, ~|~ \verb,?, ~|~ \texttt{\at} ~|~

  \verb,^, ~|~ \verb,_, ~|~ \verb,`, ~|~ \verb,|, ~|~ \verb,~, \\

greek & = & \verb,\<alpha>, ~|~ \verb,\<beta>, ~|~ \verb,\<gamma>, ~|~ \verb,\<delta>, ~| \\

      &   & \verb,\<epsilon>, ~|~ \verb,\<zeta>, ~|~ \verb,\<eta>, ~|~ \verb,\<theta>, ~| \\

      &   & \verb,\<iota>, ~|~ \verb,\<kappa>, ~|~ \verb,\<mu>, ~|~ \verb,\<nu>, ~| \\

      &   & \verb,\<xi>, ~|~ \verb,\<pi>, ~|~ \verb,\<rho>, ~|~ \verb,\<sigma>, ~| \\

      &   & \verb,\<tau>, ~|~ \verb,\<upsilon>, ~|~ \verb,\<phi>, ~|~ \verb,\<psi>, ~| \\

      &   & \verb,\<omega>, ~|~ \verb,\<Gamma>, ~|~ \verb,\<Delta>, ~|~ \verb,\<Theta>, ~| \\

      &   & \verb,\<Lambda>, ~|~ \verb,\<Xi>, ~|~ \verb,\<Pi>, ~|~ \verb,\<Sigma>, ~| \\

      &   & \verb,\<Upsilon>, ~|~ \verb,\<Phi>, ~|~ \verb,\<Psi>, ~|~ \verb,\<Omega>, \\

\end{matharray}

The syntax of $string$ admits any characters, including newlines; ``\verb|"|''

(double-quote) and ``\verb|\|'' (backslash) need to be escaped by a backslash.

The body of $verbatim$ may consist of any text not containing ``\verb|*}|'';

this allows convenient inclusion of quotes without further escapes.  The greek

letters do \emph{not} include \verb,\<lambda>,, which is already used

differently in the meta-logic.

Common mathematical symbols such as $\forall$ are represented in Isabelle as

\verb,\<forall>,.  There are infinitely many legal symbols like this, although

proper presentation is left to front-end tools such as {\LaTeX} or

Proof~General with the X-Symbol package.  A list of standard Isabelle symbols

that work well with these tools is given in \cite[appendix~A]{isabelle-sys}.

Comments take the form \texttt{(*~\dots~*)} and may be nested, although

user-interface tools may prevent this.  Note that \texttt{(*~\dots~*)}

indicate source comments only, which are stripped after lexical analysis of

the input.  The Isar document syntax also provides formal comments that are

considered as part of the text (see \S\ref{sec:comments}).

\begin{warn}

  Proof~General does not handle nested comments properly; it is also unable to

  keep \verb,(*,\,/\,\verb,{*, and \verb,*),\,/\,\verb,*}, apart, despite

  their rather different meaning.  These are inherent problems of Emacs

  legacy.  Users should not be overly aggressive about nesting or alternating

  these delimiters.

\end{warn}

\section{Common syntax entities}

Subsequently, we introduce several basic syntactic entities, such as names,

terms, and theorem specifications, which have been factored out of the actual

Isar language elements to be described later.

Note that some of the basic syntactic entities introduced below (e.g.\

\railqtok{name}) act much like tokens rather than plain nonterminals (e.g.\

\railnonterm{sort}), especially for the sake of error messages.  E.g.\ syntax

elements like $\CONSTS$ referring to \railqtok{name} or \railqtok{type} would

really report a missing name or type rather than any of the constituent

primitive tokens such as \railtok{ident} or \railtok{string}.

\subsection{Names}

Entity \railqtok{name} usually refers to any name of types, constants,

theorems etc.\ that are to be \emph{declared} or \emph{defined} (so qualified

identifiers are excluded here).  Quoted strings provide an escape for

non-identifier names or those ruled out by outer syntax keywords (e.g.\

\verb|"let"|).  Already existing objects are usually referenced by

\railqtok{nameref}.

\indexoutertoken{name}\indexoutertoken{parname}\indexoutertoken{nameref}

\indexoutertoken{int}

\begin{rail}

  name: ident | symident | string | nat

  ;

  parname: '(' name ')'

  ;

  nameref: name | longident

  ;

  int: nat | '-' nat

  ;

\end{rail}

\subsection{Comments}\label{sec:comments}

Large chunks of plain \railqtok{text} are usually given \railtok{verbatim},

i.e.\ enclosed in \verb|{*|~\dots~\verb|*}|.  For convenience, any of the

smaller text units conforming to \railqtok{nameref} are admitted as well.  A

marginal \railnonterm{comment} is of the form \texttt{--} \railqtok{text}.

Any number of these may occur within Isabelle/Isar commands.

\indexoutertoken{text}\indexouternonterm{comment}

\begin{rail}

  text: verbatim | nameref

  ;

  comment: '--' text

  ;

\end{rail}

\subsection{Type classes, sorts and arities}

Classes are specified by plain names.  Sorts have a very simple inner syntax,

which is either a single class name $c$ or a list $\{c@1, \dots, c@n\}$

referring to the intersection of these classes.  The syntax of type arities is

given directly at the outer level.

\railalias{subseteq}{\isasymsubseteq}

\railterm{subseteq}

\indexouternonterm{sort}\indexouternonterm{arity}

\indexouternonterm{classdecl}

\begin{rail}

  classdecl: name (('<' | subseteq) (nameref + ','))?

  ;

  sort: nameref

  ;

  arity: ('(' (sort + ',') ')')? sort

  ;

\end{rail}

\subsection{Types and terms}\label{sec:types-terms}

The actual inner Isabelle syntax, that of types and terms of the logic, is far

too sophisticated in order to be modelled explicitly at the outer theory

level.  Basically, any such entity has to be quoted to turn it into a single

token (the parsing and type-checking is performed internally later).  For

convenience, a slightly more liberal convention is adopted: quotes may be

omitted for any type or term that is already atomic at the outer level.  For

example, one may just write \texttt{x} instead of \texttt{"x"}.  Note that

symbolic identifiers (e.g.\ \texttt{++} or $\forall$) are available as well,

provided these have not been superseded by commands or other keywords already

(e.g.\ \texttt{=} or \texttt{+}).

\indexoutertoken{type}\indexoutertoken{term}\indexoutertoken{prop}

\begin{rail}

  type: nameref | typefree | typevar

  ;

  term: nameref | var

  ;

  prop: term

  ;

\end{rail}

Positional instantiations are indicated by giving a sequence of terms, or the

placeholder ``$\_$'' (underscore), which means to skip a position.

\indexoutertoken{inst}\indexoutertoken{insts}

\begin{rail}

  inst: underscore | term

  ;

  insts: (inst *)

  ;

\end{rail}

Type declarations and definitions usually refer to \railnonterm{typespec} on

the left-hand side.  This models basic type constructor application at the

outer syntax level.  Note that only plain postfix notation is available here,

but no infixes.

\indexouternonterm{typespec}

\begin{rail}

  typespec: (() | typefree | '(' ( typefree + ',' ) ')') name

  ;

\end{rail}

\subsection{Mixfix annotations}

Mixfix annotations specify concrete \emph{inner} syntax of Isabelle types and

terms.  Some commands such as $\TYPES$ (see \S\ref{sec:types-pure}) admit

infixes only, while $\CONSTS$ (see \S\ref{sec:consts}) and

$\isarkeyword{syntax}$ (see \S\ref{sec:syn-trans}) support the full range of

general mixfixes and binders.

\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

(see also \cite{isabelle-ref}), which may include literal text, spacing,

blocks, and arguments (denoted by ``$_$''); the special symbol \verb,\<index>,

(printed as ``\i'') represents an index argument that specifies an implicit

structure reference (see also \S\ref{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 ``\verb,++,'' for an infix symbol, or ``\verb,++,\i''

for an infix of an implicit structure.

\subsection{Proof methods}\label{sec:syn-meth}

Proof methods are either basic ones, or expressions composed of methods via

``\texttt{,}'' (sequential composition), ``\texttt{|}'' (alternative choices),

``\texttt{?}'' (try), ``\texttt{+}'' (repeat at least once).  In practice,

proof methods are usually just a comma separated list of

\railqtok{nameref}~\railnonterm{args} specifications.  Note that parentheses

may be dropped for single method specifications (with no arguments).

\indexouternonterm{method}

\begin{rail}

  method: (nameref | '(' methods ')') (() | '?' | '+')

  ;

  methods: (nameref args | method) + (',' | '|')

  ;

\end{rail}

Proper use of Isar proof methods does \emph{not} involve goal addressing.

Nevertheless, specifying goal ranges may occasionally come in handy in

emulating tactic scripts.  Note that $[n-]$ refers to all goals, starting from

$n$.  All goals may be specified by $[!]$, which is the same as $[1-]$.

\indexouternonterm{goalspec}

\begin{rail}

  goalspec: '[' (nat '-' nat | nat '-' | nat | '!' ) ']'

  ;

\end{rail}

\subsection{Attributes and theorems}\label{sec:syn-att}

Attributes (and proof methods, see \S\ref{sec:syn-meth}) have their own

``semi-inner'' syntax, in the sense that input conforming to

\railnonterm{args} below is parsed by the attribute a second time.  The

attribute argument specifications may be any sequence of atomic entities

(identifiers, strings etc.), or properly bracketed argument lists.  Below

\railqtok{atom} refers to any atomic entity, including any \railtok{keyword}

conforming to \railtok{symident}.

\indexoutertoken{atom}\indexouternonterm{args}\indexouternonterm{attributes}

\begin{rail}

  atom: nameref | typefree | typevar | var | nat | keyword

  ;

  arg: atom | '(' args ')' | '[' args ']'

  ;

  args: arg *

  ;

  attributes: '[' (nameref args * ',') ']'

  ;

\end{rail}

Theorem specifications come in several flavors: \railnonterm{axmdecl} and

\railnonterm{thmdecl} usually refer to axioms, assumptions or results of goal

statements, while \railnonterm{thmdef} collects lists of existing theorems.

Existing theorems are given by \railnonterm{thmref} and \railnonterm{thmrefs},

the former requires an actual singleton result.  Any of these theorem

specifications may include lists of attributes both on the left and right hand

sides; attributes are applied to any immediately preceding fact.  If names are

omitted, the theorems are not stored within the theorem database of the theory

or proof context; any given attributes are still applied, though.

\indexouternonterm{thmdecl}\indexouternonterm{axmdecl}

\indexouternonterm{thmdef}\indexouternonterm{thmrefs}

\begin{rail}

  axmdecl: name attributes? ':'

  ;

  thmdecl: thmbind ':'

  ;

  thmdef: thmbind '='

  ;

  thmref: nameref attributes?

  ;

  thmrefs: thmref +

  ;

  thmbind: name attributes | name | attributes

  ;

\end{rail}

\subsection{Term patterns and declarations}\label{sec:term-decls}

Wherever explicit propositions (or term fragments) occur in a proof text,

casual binding of schematic term variables may be given specified via patterns

of the form ``$\ISS{p@1\;\dots}{p@n}$''.  There are separate versions

available for \railqtok{term}s and \railqtok{prop}s.  The latter provides a

$\CONCLNAME$ part with patterns referring the (atomic) conclusion of a rule.

\indexouternonterm{termpat}\indexouternonterm{proppat}

\begin{rail}

  termpat: '(' ('is' term +) ')'

  ;

  proppat: '(' (('is' prop +) | 'concl' ('is' prop +) | ('is' prop +) 'concl' ('is' prop +)) ')'

  ;

\end{rail}

Declarations of local variables $x :: \tau$ and logical propositions $a :

\phi$ represent different views on the same principle of introducing a local

scope.  In practice, one may usually omit the typing of $vars$ (due to

type-inference), and the naming of propositions (due to implicit references of

current facts).  In any case, Isar proof elements usually admit to introduce

multiple such items simultaneously.

\indexouternonterm{vars}\indexouternonterm{props}

\begin{rail}

  vars: (name+) ('::' type)?

  ;

  props: thmdecl? (prop proppat? +)

  ;

\end{rail}

The treatment of multiple declarations corresponds to the complementary focus

of $vars$ versus $props$: in ``$x@1~\dots~x@n :: \tau$'' the typing refers to

all variables, while in $a\colon \phi@1~\dots~\phi@n$ the naming refers to all

propositions collectively.  Isar language elements that refer to $vars$ or

$props$ typically admit separate typings or namings via another level of

iteration, with explicit $\AND$ separators; e.g.\ see $\FIXNAME$ and

$\ASSUMENAME$ in \S\ref{sec:proof-context}.

\subsection{Antiquotations}\label{sec:antiq}

\begin{matharray}{rcl}

  thm & : & \isarantiq \\

  prop & : & \isarantiq \\

  term & : & \isarantiq \\

  typ & : & \isarantiq \\

  text & : & \isarantiq \\

  goals & : & \isarantiq \\

  subgoals & : & \isarantiq \\

  prf & : & \isarantiq \\

  full_prf & : & \isarantiq \\

\end{matharray}

The text body of formal comments (see also \S\ref{sec:comments}) may contain

antiquotations of logical entities, such as theorems, terms and types, which

are to be presented in the final output produced by the Isabelle document

preparation system (see also \S\ref{sec:document-prep}).

Thus embedding of

``\texttt{{\at}{\ttlbrace}term~[show_types]~"f(x)~=~a~+~x"{\ttrbrace}}''

within a text block would cause

\isa{(f{\isasymColon}'a~{\isasymRightarrow}~'a)~(x{\isasymColon}'a)~=~(a{\isasymColon}'a)~+~x}

to appear in the final {\LaTeX} document.  Also note that theorem

antiquotations may involve attributes as well.  For example,

\texttt{{\at}{\ttlbrace}thm~sym~[no_vars]{\ttrbrace}} would print the

statement where all schematic variables have been replaced by fixed ones,

which are easier to read.

\indexisarant{thm}\indexisarant{prop}\indexisarant{term}

\indexisarant{typ}\indexisarant{text}\indexisarant{goals}\indexisarant{subgoals}

\begin{rail}

  atsign lbrace antiquotation rbrace

  ;

  antiquotation:

    'thm' options thmrefs |

    'prop' options prop |

    'term' options term |

    'typ' options type |

    'text' options name |

    'goals' options |

    'subgoals' options |

    'prf' options thmrefs |

    'full\_prf' options thmrefs

  ;

  options: '[' (option * ',') ']'

  ;

  option: name | name '=' name

  ;

\end{rail}

Note that the syntax of antiquotations may \emph{not} include source comments

\texttt{(*~\dots~*)} or verbatim text \verb|{*|~\dots~\verb|*}|.

\begin{descr}

\item [$\at\{thm~\vec a\}$] prints theorems $\vec a$. Note that attribute

  specifications may be included as well (see also \S\ref{sec:syn-att}); the

  $no_vars$ operation (see \S\ref{sec:misc-meth-att}) would be particularly

  useful to suppress printing of schematic variables.

\item [$\at\{prop~\phi\}$] prints a well-typed proposition $\phi$.

\item [$\at\{term~t\}$] prints a well-typed term $t$.

\item [$\at\{typ~\tau\}$] prints a well-formed type $\tau$.

\item [$\at\{text~s\}$] prints uninterpreted source text $s$.  This is

  particularly useful to print portions of text according to the Isabelle

  {\LaTeX} output style, without demanding well-formedness (e.g.\ small pieces

  of terms that should not be parsed or type-checked yet).

\item [$\at\{goals\}$] prints the current \emph{dynamic} goal state.  This is

  mainly for support of tactic-emulation scripts within Isar --- presentation

  of goal states does not conform to actual human-readable proof documents.

  Please do not include goal states into document output unless you really

  know what you are doing!

\item [$\at\{subgoals\}$] behaves almost like $goals$, except that it does not

  print the main goal.

\item [$\at\{prf~\vec a\}$] prints the (compact) proof terms corresponding to

  the theorems $\vec a$. 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 [$\at\{full_prf~\vec a\}$] is like $\at\{prf~\vec a\}$, but displays

  the full proof terms, i.e.\ also displays information omitted in

  the compact proof term, which is denoted by ``$_$'' placeholders there.

\end{descr}

\medskip

The following options are available to tune the output.  Note that most of

these coincide with ML flags of the same names (see also \cite{isabelle-ref}).

\begin{descr}

\item[$show_types = bool$ and $show_sorts = bool$] control printing of

  explicit type and sort constraints.

\item[$show_structs = bool$] controls printing of implicit structures.

\item[$long_names = bool$] forces names of types and constants etc.\ to be

  printed in their fully qualified internal form.

\item[$eta_contract = bool$] prints terms in $\eta$-contracted form.

\item[$display = bool$] indicates if the text is to be output as multi-line

  ``display material'', rather than a small piece of text without line breaks

  (which is the default).

\item[$breaks = bool$] controls line breaks in non-display material.

\item[$quotes = bool$] indicates if the output should be enclosed in double

  quotes.

\item[$mode = name$] adds $name$ to the print mode to be used for presentation

  (see also \cite{isabelle-ref}).  Note that the standard setup for {\LaTeX}

  output is already present by default, including the modes ``$latex$'',

  ``$xsymbols$'', ``$symbols$''.

\item[$margin = nat$ and $indent = nat$] change the margin or indentation for

  pretty printing of display material.

\item[$source = bool$] prints the source text of the antiquotation arguments,

  rather than the actual value.  Note that this does not affect

  well-formedness checks of $thm$, $term$, etc. (only the $text$ antiquotation

  admits arbitrary output).

\item[$goals_limit = nat$] determines the maximum number of goals to be

  printed.

\item[$locale = name$] specifies an alternative context used for evaluating

  and printing the subsequent argument.

\end{descr}

For boolean flags, ``$name = true$'' may be abbreviated as ``$name$''.  All of

the above flags are disabled by default, unless changed from ML.

\medskip Note that antiquotations do not only spare the author from tedious

typing of logical entities, but also achieve some degree of

consistency-checking of informal explanations with formal developments:

well-formedness of terms and types with respect to the current theory or proof

context is ensured here.

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