(* $Id$ *)

theory Outer_Syntax

imports Main

begin

chapter {* Outer syntax *}

text {*

  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).  Subsequently 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 (specifications and

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

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

  @{verbatim "\"x + y\""} and identifier @{verbatim z} are legal term

  specifications within a theory, while @{verbatim "x + y"} without

  quotes is not.

  Printed theory documents usually omit quotes to gain readability

  (this is a matter of {\LaTeX} macro setup, say via @{verbatim

  "\\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 ``@{verbatim ";"}'' (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 ``@{verbatim "#"}'' until an end-of-command is

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

  next command keyword.

  More 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 @{verbatim "-k ZF"}

    (in conjunction with @{verbatim "-l ZF"}, to specify the default

    logic image).  Note that option @{verbatim "-L"} does both

    of this at the same time.

  \end{warn}

*}

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

text {*

  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}.

  \begin{matharray}{rcl}

    @{syntax_def ident} & = & letter\,quasiletter^* \\

    @{syntax_def longident} & = & ident (\verb,.,ident)^+ \\

    @{syntax_def symident} & = & sym^+ ~|~ \verb,\,\verb,<,ident\verb,>, \\

    @{syntax_def nat} & = & digit^+ \\

    @{syntax_def var} & = & ident ~|~ \verb,?,ident ~|~ \verb,?,ident\verb,.,nat \\

    @{syntax_def typefree} & = & \verb,',ident \\

    @{syntax_def typevar} & = & typefree ~|~ \verb,?,typefree ~|~ \verb,?,typefree\verb,.,nat \\

    @{syntax_def string} & = & \verb,", ~\dots~ \verb,", \\

    @{syntax_def altstring} & = & \backquote ~\dots~ \backquote \\

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

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

    \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,\<chi>, ~|~ \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 @{syntax string} admits any characters, including

  newlines; ``@{verbatim "\""}'' (double-quote) and ``@{verbatim

  "\\"}'' (backslash) need to be escaped by a backslash; arbitrary

  character codes may be specified as ``@{verbatim "\\"}@{text ddd}'',

  with three decimal digits.  Alternative strings according to

  @{syntax altstring} are analogous, using single back-quotes instead.

  The body of @{syntax verbatim} may consist of any text not

  containing ``@{verbatim "*"}@{verbatim "}"}''; this allows

  convenient inclusion of quotes without further escapes.  The greek

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

  differently in the meta-logic.

  Common mathematical symbols such as @{text \<forall>} are represented in

  Isabelle as @{verbatim \<forall>}.  There are infinitely many Isabelle

  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}.

  Source comments take the form @{verbatim "(*"}~@{text

  "\<dots>"}~@{verbatim "*)"} and may be nested, although user-interface

  tools might prevent this.  Note that this form indicates source

  comments only, which are stripped after lexical analysis of the

  input.  The Isar syntax also provides proper \emph{document

  comments} that are considered as part of the text (see

  \secref{sec:comments}).

*}

section {* Common syntax entities *}

text {*

  We now introduce several basic syntactic entities, such as names,

  terms, and theorem specifications, which are factored out of the

  actual Isar language elements to be described later.

*}

subsection {* Names *}

text {*

  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.\ quoted @{verbatim "\"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} *}

text {*

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

  \railtok{verbatim}, i.e.\ enclosed in @{verbatim "{"}@{verbatim

  "*"}~@{text "\<dots>"}~@{verbatim "*"}@{verbatim "}"}.  For convenience,

  any of the smaller text units conforming to \railqtok{nameref} are

  admitted as well.  A marginal \railnonterm{comment} is of the form

  @{verbatim "--"} \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 *}

text {*

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

  inner syntax, which is either a single class name @{text c} or a

  list @{text "{c\<^sub>1, \<dots>, c\<^sub>n}"} referring to the

  intersection of these classes.  The syntax of type arities is given

  directly at the outer level.

  \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} *}

text {*

  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 @{verbatim x} instead of quoted @{verbatim "\"x\""}.

  Note that symbolic identifiers (e.g.\ @{verbatim "++"} or @{text

  "\<forall>"} are available as well, provided these have not been superseded

  by commands or other keywords already (such as @{verbatim "="} or

  @{verbatim "+"}).

  \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 ``@{text _}'' (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 *}

text {*

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

  types and terms.  Some commands such as @{command "types"} (see

  \secref{sec:types-pure}) admit infixes only, while @{command

  "consts"} (see \secref{sec:consts}) and @{command "syntax"} (see

  \secref{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 ``@{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, or ``@{verbatim "++"}@{text "\<index>"}'' for an infix of

  an implicit structure.

*}

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

text {*

  Proof methods are either basic ones, or expressions composed of

  methods via ``@{verbatim ","}'' (sequential composition),

  ``@{verbatim "|"}'' (alternative choices), ``@{verbatim "?"}'' 

  (try), ``@{verbatim "+"}'' (repeat at least once), ``@{verbatim

  "["}@{text n}@{verbatim "]"}'' (restriction to first @{text n}

  sub-goals, with default @{text "n = 1"}).  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 ')') (() | '?' | '+' | '[' nat? ']')

    ;

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

    ;

  \end{rail}

  Proper Isar proof methods do \emph{not} admit arbitrary goal

  addressing, but refer either to the first sub-goal or all sub-goals

  uniformly.  The goal restriction operator ``@{text "[n]"}''

  evaluates a method expression within a sandbox consisting of the

  first @{text n} sub-goals (which need to exist).  For example, the

  method ``@{text "simp_all[3]"}'' simplifies the first three

  sub-goals, while ``@{text "(rule foo, simp_all)[]"}'' simplifies all

  new goals that emerge from applying rule @{text "foo"} to the

  originally first one.

  Improper methods, notably tactic emulations, offer a separate

  low-level goal addressing scheme as explicit argument to the

  individual tactic being involved.  Here ``@{text "[!]"}'' refers to

  all goals, and ``@{text "[n-]"}'' to all goals starting from @{text

  "n"}.

  \indexouternonterm{goalspec}

  \begin{rail}

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

    ;

  \end{rail}

*}

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

text {*

  Attributes (and proof methods, see \secref{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.

  There are three forms of theorem references:

  \begin{enumerate}

  \item named facts @{text "a"},

  \item selections from named facts @{text "a(i)"} or @{text "a(j - k)"},

  \item literal fact propositions using @{syntax_ref altstring} syntax

  @{verbatim "`"}@{text "\<phi>"}@{verbatim "`"} (see also method

  @{method_ref fact} in \secref{sec:pure-meth-att}).

  \end{enumerate}

  Any kind of theorem specification 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, but any given attributes are applied nonetheless.

  An extra pair of brackets around attributes (like ``@{text

  "[[simproc a]]"}'') abbreviates a theorem reference involving an

  internal dummy fact, which will be ignored later on.  So only the

  effect of the attribute on the background context will persist.

  This form of in-place declarations is particularly useful with

  commands like @{command "declare"} and @{command "using"}.

  \indexouternonterm{axmdecl}\indexouternonterm{thmdecl}

  \indexouternonterm{thmdef}\indexouternonterm{thmref}

  \indexouternonterm{thmrefs}\indexouternonterm{selection}

  \begin{rail}

    axmdecl: name attributes? ':'

    ;

    thmdecl: thmbind ':'

    ;

    thmdef: thmbind '='

    ;

    thmref: (nameref selection? | altstring) attributes? | '[' attributes ']'

    ;

    thmrefs: thmref +

    ;

    thmbind: name attributes | name | attributes

    ;

    selection: '(' ((nat | nat '-' nat?) + ',') ')'

    ;

  \end{rail}

*}

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

text {*

  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 ``@{text "(\<IS> p\<^sub>1 \<dots>

  p\<^sub>n)"}''.  This works both for \railqtok{term} and \railqtok{prop}.

  \indexouternonterm{termpat}\indexouternonterm{proppat}

  \begin{rail}

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

    ;

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

    ;

  \end{rail}

  \medskip Declarations of local variables @{text "x :: \<tau>"} and

  logical propositions @{text "a : \<phi>"} represent different views on

  the same principle of introducing a local scope.  In practice, one

  may usually omit the typing of \railnonterm{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 \railnonterm{vars} versus

  \railnonterm{props}.  In ``@{text "x\<^sub>1 \<dots> x\<^sub>n :: \<tau>"}''

  the typing refers to all variables, while in @{text "a: \<phi>\<^sub>1 \<dots>

  \<phi>\<^sub>n"} the naming refers to all propositions collectively.

  Isar language elements that refer to \railnonterm{vars} or

  \railnonterm{props} typically admit separate typings or namings via

  another level of iteration, with explicit @{keyword_ref "and"}

  separators; e.g.\ see @{command "fix"} and @{command "assume"} in

  \secref{sec:proof-context}.

*}

end