\chapter{Basic Isar Language Elements}\label{ch:pure-syntax}

Subsequently, we introduce the main part of the basic Isar theory and proof

commands as provided by Isabelle/Pure.  Chapter~\ref{ch:gen-tools} describes

further Isar elements provided by generic tools and packages (such as the

Simplifier) that are either part of Pure Isabelle or pre-loaded by most object

logics.  Chapter~\ref{ch:hol-tools} refers to actual object-logic specific

elements of Isabelle/HOL.

\medskip

Isar commands may be either \emph{proper} document constructors, or

\emph{improper commands}.  Some proof methods and attributes introduced later

are classified as improper as well.  Improper Isar language elements, which

are subsequently marked by $^*$, are often helpful when developing proof

documents, while their use is discouraged for the final outcome.  Typical

examples are diagnostic commands that print terms or theorems according to the

current context; other commands even emulate old-style tactical theorem

proving, which facilitates porting of legacy proof scripts.

\section{Theory commands}

\subsection{Defining theories}\label{sec:begin-thy}

\indexisarcmd{header}\indexisarcmd{theory}\indexisarcmd{end}\indexisarcmd{context}

\begin{matharray}{rcl}

  \isarcmd{header} & : & \isarkeep{toplevel} \\

  \isarcmd{theory} & : & \isartrans{\cdot}{theory} \\

  \isarcmd{context}^* & : & \isartrans{\cdot}{theory} \\

  \isarcmd{end} & : & \isartrans{theory}{\cdot} \\

\end{matharray}

Isabelle/Isar ``new-style'' theories are either defined via theory files or

interactively.  Both theory-level specifications and proofs are handled

uniformly --- occasionally definitional mechanisms even require some explicit

proof as well.  In contrast, ``old-style'' Isabelle theories support batch

processing only, with the proof scripts collected in separate ML files.

The first actual command of any theory has to be $\THEORY$, starting a new

theory based on the merge of existing ones.  Just preceding $\THEORY$, there

may be an optional $\isarkeyword{header}$ declaration, which is relevant to

document preparation only; it acts very much like a special pre-theory markup

command (cf.\ \S\ref{sec:markup-thy} and \S\ref{sec:markup-thy}).  The theory

context may be also changed by $\CONTEXT$ without creating a new theory.  In

both cases, $\END$ concludes the theory development; it has to be the very

last command in a theory file.

\begin{rail}

  'header' text

  ;

  'theory' name '=' (name + '+') filespecs? ':'

  ;

  'context' name

  ;

  'end'

  ;;

  filespecs: 'files' ((name | parname) +);

\end{rail}

\begin{descr}

\item [$\isarkeyword{header}~text$] provides plain text markup just preceding

  the formal begin of a theory.  In actual document preparation the

  corresponding {\LaTeX} macro \verb,\isamarkupheader, may be redefined to

  produce chapter or section headings.  See also \S\ref{sec:markup-thy} and

  \S\ref{sec:markup-prf} for further markup commands.

\item [$\THEORY~A = B@1 + \cdots + B@n\colon$] commences a new theory $A$

  based on existing ones $B@1 + \cdots + B@n$.  Isabelle's theory loader

  system ensures that any of the base theories are properly loaded (and fully

  up-to-date when $\THEORY$ is executed interactively).  The optional

  $\isarkeyword{files}$ specification declares additional dependencies on ML

  files.  Unless put in parentheses, any file will be loaded immediately via

  $\isarcmd{use}$ (see also \S\ref{sec:ML}).  The optional ML file

  \texttt{$A$.ML} that may be associated with any theory should \emph{not} be

  included in $\isarkeyword{files}$, though.

\item [$\CONTEXT~B$] enters an existing theory context, basically in read-only

  mode, so only a limited set of commands may be performed without destroying

  the theory.  Just as for $\THEORY$, the theory loader ensures that $B$ is

  loaded and up-to-date.

\item [$\END$] concludes the current theory definition or context switch.

Note that this command cannot be undone, but the whole theory definition has

to be retracted.

\end{descr}

\subsection{Theory markup commands}\label{sec:markup-thy}

\indexisarcmd{chapter}\indexisarcmd{section}\indexisarcmd{subsection}

\indexisarcmd{subsubsection}\indexisarcmd{text}\indexisarcmd{text-raw}

\begin{matharray}{rcl}

  \isarcmd{chapter} & : & \isartrans{theory}{theory} \\

  \isarcmd{section} & : & \isartrans{theory}{theory} \\

  \isarcmd{subsection} & : & \isartrans{theory}{theory} \\

  \isarcmd{subsubsection} & : & \isartrans{theory}{theory} \\

  \isarcmd{text} & : & \isartrans{theory}{theory} \\

  \isarcmd{text_raw} & : & \isartrans{theory}{theory} \\

\end{matharray}

Apart from formal comments (see \S\ref{sec:comments}), markup commands provide

a structured way to insert text into the document generated from a theory (see

\cite{isabelle-sys} for more information on Isabelle's document preparation

tools).

\railalias{textraw}{text\_raw}

\railterm{textraw}

\begin{rail}

  ('chapter' | 'section' | 'subsection' | 'subsubsection' | 'text' | textraw) text

  ;

\end{rail}

\begin{descr}

\item [$\isarkeyword{chapter}$, $\isarkeyword{section}$,

  $\isarkeyword{subsection}$, and $\isarkeyword{subsubsection}$] mark chapter

  and section headings.

\item [$\TEXT$] specifies paragraphs of plain text, including references to

  formal entities.\footnote{The latter feature is not yet supported.

    Nevertheless, any source text of the form

    ``\texttt{\at\ttlbrace$\dots$\ttrbrace}'' should be considered as reserved

    for future use.}

\item [$\isarkeyword{text_raw}$] inserts {\LaTeX} source into the output,

  without additional markup.  Thus the full range of document manipulations

  becomes available.  A typical application would be to emit

  \verb,\begin{comment}, and \verb,\end{comment}, commands to exclude certain

  parts from the final document.\footnote{This requires the \texttt{comment}

    package to be included in {\LaTeX}.}

\end{descr}

Any markup command (except $\isarkeyword{text_raw}$) corresponds to a {\LaTeX}

macro with the name prefixed by \verb,\isamarkup, (e.g.\ 

\verb,\isamarkupchapter, for $\isarkeyword{chapter}$). The \railqtoken{text}

argument is passed to that macro unchanged, i.e.\ further {\LaTeX} commands

may be included here as well.

\medskip Additional markup commands are available for proofs (see

\S\ref{sec:markup-prf}).  Also note that the $\isarkeyword{header}$

declaration (see \S\ref{sec:begin-thy}) admits to insert document markup

elements just preceding the actual theory definition.

\subsection{Type classes and sorts}\label{sec:classes}

\indexisarcmd{classes}\indexisarcmd{classrel}\indexisarcmd{defaultsort}

\begin{matharray}{rcl}

  \isarcmd{classes} & : & \isartrans{theory}{theory} \\

  \isarcmd{classrel} & : & \isartrans{theory}{theory} \\

  \isarcmd{defaultsort} & : & \isartrans{theory}{theory} \\

\end{matharray}

\begin{rail}

  'classes' (classdecl comment? +)

  ;

  'classrel' nameref '<' nameref comment?

  ;

  'defaultsort' sort comment?

  ;

\end{rail}

\begin{descr}

\item [$\isarkeyword{classes}~c<\vec c$] declares class $c$ to be a subclass

  of existing classes $\vec c$.  Cyclic class structures are ruled out.

\item [$\isarkeyword{classrel}~c@1<c@2$] states a subclass relation between

  existing classes $c@1$ and $c@2$.  This is done axiomatically!  The

  $\isarkeyword{instance}$ command (see \S\ref{sec:axclass}) provides a way to

  introduce proven class relations.

\item [$\isarkeyword{defaultsort}~s$] makes sort $s$ the new default sort for

  any type variables given without sort constraints.  Usually, the default

  sort would be only changed when defining new logics.

\end{descr}

\subsection{Primitive types and type abbreviations}\label{sec:types-pure}

\indexisarcmd{typedecl}\indexisarcmd{types}\indexisarcmd{nonterminals}\indexisarcmd{arities}

\begin{matharray}{rcl}

  \isarcmd{types} & : & \isartrans{theory}{theory} \\

  \isarcmd{typedecl} & : & \isartrans{theory}{theory} \\

  \isarcmd{nonterminals} & : & \isartrans{theory}{theory} \\

  \isarcmd{arities} & : & \isartrans{theory}{theory} \\

\end{matharray}

\begin{rail}

  'types' (typespec '=' type infix? comment? +)

  ;

  'typedecl' typespec infix? comment?

  ;

  'nonterminals' (name +) comment?

  ;

  'arities' (nameref '::' arity comment? +)

  ;

\end{rail}

\begin{descr}

\item [$\TYPES~(\vec\alpha)t = \tau$] introduces \emph{type synonym}

  $(\vec\alpha)t$ for existing type $\tau$.  Unlike actual type definitions,

  as are available in Isabelle/HOL for example, type synonyms are just purely

  syntactic abbreviations without any logical significance.  Internally, type

  synonyms are fully expanded.

\item [$\isarkeyword{typedecl}~(\vec\alpha)t$] declares a new type constructor

  $t$, intended as an actual logical type.  Note that object-logics such as

  Isabelle/HOL override $\isarkeyword{typedecl}$ by their own version.

\item [$\isarkeyword{nonterminals}~\vec c$] declares $0$-ary type constructors

  $\vec c$ to act as purely syntactic types, i.e.\ nonterminal symbols of

  Isabelle's inner syntax of terms or types.

\item [$\isarkeyword{arities}~t::(\vec s)s$] augments Isabelle's order-sorted

  signature of types by new type constructor arities.  This is done

  axiomatically!  The $\isarkeyword{instance}$ command (see

  \S\ref{sec:axclass}) provides a way to introduce proven type arities.

\end{descr}

\subsection{Constants and simple definitions}\label{sec:consts}

\indexisarcmd{consts}\indexisarcmd{defs}\indexisarcmd{constdefs}\indexoutertoken{constdecl}

\begin{matharray}{rcl}

  \isarcmd{consts} & : & \isartrans{theory}{theory} \\

  \isarcmd{defs} & : & \isartrans{theory}{theory} \\

  \isarcmd{constdefs} & : & \isartrans{theory}{theory} \\

\end{matharray}

\begin{rail}

  'consts' (constdecl +)

  ;

  'defs' (axmdecl prop comment? +)

  ;

  'constdefs' (constdecl prop comment? +)

  ;

  constdecl: name '::' type mixfix? comment?

  ;

\end{rail}

\begin{descr}

\item [$\CONSTS~c::\sigma$] declares constant $c$ to have any instance of type

  scheme $\sigma$.  The optional mixfix annotations may attach concrete syntax

  to the constants declared.

\item [$\DEFS~name: eqn$] introduces $eqn$ as a definitional axiom for some

  existing constant.  See \cite[\S6]{isabelle-ref} for more details on the

  form of equations admitted as constant definitions.

\item [$\isarkeyword{constdefs}~c::\sigma~eqn$] combines declarations and

  definitions of constants, using canonical name $c_def$ for the definitional

  axiom.

\end{descr}

\subsection{Syntax and translations}\label{sec:syn-trans}

\indexisarcmd{syntax}\indexisarcmd{translations}

\begin{matharray}{rcl}

  \isarcmd{syntax} & : & \isartrans{theory}{theory} \\

  \isarcmd{translations} & : & \isartrans{theory}{theory} \\

\end{matharray}

\begin{rail}

  'syntax' ('(' name 'output'? ')')? (constdecl +)

  ;

  'translations' (transpat ('==' | '=>' | '<=') transpat comment? +)

  ;

  transpat: ('(' nameref ')')? string

  ;

\end{rail}

\begin{descr}

\item [$\isarkeyword{syntax}~(mode)~decls$] is similar to $\CONSTS~decls$,

  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 $mode$ argument refers to

  the print mode that the grammar rules belong; unless there is the

  \texttt{output} flag given, all productions are added both to the input and

  output grammar.

\item [$\isarkeyword{translations}~rules$] specifies syntactic translation

  rules (i.e.\ \emph{macros}): parse~/ print rules (\texttt{==}), parse rules

  (\texttt{=>}), or print rules (\texttt{<=}).  Translation patterns may be

  prefixed by the syntactic category to be used for parsing; the default is

  \texttt{logic}.

\end{descr}

\subsection{Axioms and theorems}

\indexisarcmd{axioms}\indexisarcmd{theorems}\indexisarcmd{lemmas}

\begin{matharray}{rcl}

  \isarcmd{axioms} & : & \isartrans{theory}{theory} \\

  \isarcmd{theorems} & : & \isartrans{theory}{theory} \\

  \isarcmd{lemmas} & : & \isartrans{theory}{theory} \\

\end{matharray}

\begin{rail}

  'axioms' (axmdecl prop comment? +)

  ;

  ('theorems' | 'lemmas') thmdef? thmrefs

  ;

\end{rail}

\begin{descr}

\item [$\isarkeyword{axioms}~a: \phi$] introduces arbitrary statements as

  axioms of the meta-logic.  In fact, axioms are ``axiomatic theorems'', and

  may be referred later just as any other theorem.

  Axioms are usually only introduced when declaring new logical systems.

  Everyday work is typically done the hard way, with proper definitions and

  actual theorems.

\item [$\isarkeyword{theorems}~a = \vec b$] stores lists of existing theorems.

  Typical applications would also involve attributes, to augment the

  Simplifier context, for example.

\item [$\isarkeyword{lemmas}$] is similar to $\isarkeyword{theorems}$, but

  tags the results as ``lemma''.

\end{descr}

\subsection{Name spaces}

\indexisarcmd{global}\indexisarcmd{local}

\begin{matharray}{rcl}

  \isarcmd{global} & : & \isartrans{theory}{theory} \\

  \isarcmd{local} & : & \isartrans{theory}{theory} \\

\end{matharray}

Isabelle organizes any kind of name declarations (of types, constants,

theorems etc.)  by hierarchically structured name spaces.  Normally the user

never has to control the behavior of name space entry by hand, yet the

following commands provide some way to do so.

\begin{descr}

\item [$\isarkeyword{global}$ and $\isarkeyword{local}$] change the current

  name declaration mode.  Initially, theories start in $\isarkeyword{local}$

  mode, causing all names to be automatically qualified by the theory name.

  Changing this to $\isarkeyword{global}$ causes all names to be declared

  without the theory prefix, until $\isarkeyword{local}$ is declared again.

\end{descr}

\subsection{Incorporating ML code}\label{sec:ML}

\indexisarcmd{use}\indexisarcmd{ML}\indexisarcmd{ML-setup}\indexisarcmd{setup}

\begin{matharray}{rcl}

  \isarcmd{use} & : & \isartrans{\cdot}{\cdot} \\

  \isarcmd{ML} & : & \isartrans{\cdot}{\cdot} \\

  \isarcmd{ML_setup} & : & \isartrans{theory}{theory} \\

  \isarcmd{setup} & : & \isartrans{theory}{theory} \\

\end{matharray}

\railalias{MLsetup}{ML\_setup}

\railterm{MLsetup}

\begin{rail}

  'use' name

  ;

  ('ML' | MLsetup | 'setup') text

  ;

\end{rail}

\begin{descr}

\item [$\isarkeyword{use}~file$] reads and executes ML commands from $file$.

  The current theory context (if present) is passed down to the ML session,

  but may not be modified.  Furthermore, the file name is checked with the

  $\isarkeyword{files}$ dependency declaration given in the theory header (see

  also \S\ref{sec:begin-thy}).

\item [$\isarkeyword{ML}~text$] executes ML commands from $text$.  The theory

  context is passed in the same way as for $\isarkeyword{use}$.

\item [$\isarkeyword{ML_setup}~text$] executes ML commands from $text$.  The

  theory context is passed down to the ML session, and fetched back

  afterwards.  Thus $text$ may actually change the theory as a side effect.

\item [$\isarkeyword{setup}~text$] changes the current theory context by

  applying setup functions from $text$, which refers to an ML expression of

  type $(theory \to theory)~list$.  The $\isarkeyword{setup}$ command is the

  canonical way to initialize object-logic specific tools and packages written

  in ML.

\end{descr}

%FIXME remove!?

%\subsection{Syntax translation functions}

%\indexisarcmd{parse-ast-translation}\indexisarcmd{parse-translation}

%\indexisarcmd{print-translation}\indexisarcmd{typed-print-translation}

%\indexisarcmd{print-ast-translation}\indexisarcmd{token-translation}

%\begin{matharray}{rcl}

%  \isarcmd{parse_ast_translation} & : & \isartrans{theory}{theory} \\

%  \isarcmd{parse_translation} & : & \isartrans{theory}{theory} \\

%  \isarcmd{print_translation} & : & \isartrans{theory}{theory} \\

%  \isarcmd{typed_print_translation} & : & \isartrans{theory}{theory} \\

%  \isarcmd{print_ast_translation} & : & \isartrans{theory}{theory} \\

%  \isarcmd{token_translation} & : & \isartrans{theory}{theory} \\

%\end{matharray}

%Syntax translation functions written in ML admit almost arbitrary

%manipulations of Isabelle's inner syntax.  Any of the above commands have a

%single \railqtoken{text} argument that refers to an ML expression of

%appropriate type.  See \cite[\S8]{isabelle-ref} for more information on syntax

%transformations.

\subsection{Oracles}

\indexisarcmd{oracle}

\begin{matharray}{rcl}

  \isarcmd{oracle} & : & \isartrans{theory}{theory} \\

\end{matharray}

Oracles provide an interface to external reasoning systems, without giving up

control completely --- each theorem carries a derivation object recording any

oracle invocation.  See \cite[\S6]{isabelle-ref} for more information.

\begin{rail}

  'oracle' name '=' text comment?

  ;

\end{rail}

\begin{descr}

\item [$\isarkeyword{oracle}~name=text$] declares oracle $name$ to be ML

  function $text$, which has to be of type $Sign\mathord.sg \times

  Object\mathord.T \to term$.

\end{descr}

\section{Proof commands}

Proof commands provide transitions of Isar/VM machine configurations, which

are block-structured, consisting of a stack of nodes with three main

components: logical proof context, current facts, and open goals.  Isar/VM

transitions are \emph{typed} according to the following three three different

modes of operation:

\begin{descr}

\item [$proof(prove)$] means that a new goal has just been stated that is now

  to be \emph{proven}; the next command may refine it by some proof method

  (read: tactic), and enter a sub-proof to establish the actual result.

\item [$proof(state)$] is like an internal theory mode: the context may be

  augmented by \emph{stating} additional assumptions, intermediate result etc.

\item [$proof(chain)$] is intermediate between $proof(state)$ and

  $proof(prove)$: existing facts (i.e.\ the contents of $this$) have been just

  picked up in order to be used when refining the goal claimed next.

\end{descr}

\subsection{Proof markup commands}\label{sec:markup-prf}

\indexisarcmd{sect}\indexisarcmd{subsect}\indexisarcmd{subsect}

\indexisarcmd{txt}\indexisarcmd{txt-raw}

\begin{matharray}{rcl}

  \isarcmd{sect} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{subsect} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{subsubsect} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{txt} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{txt_raw} & : & \isartrans{proof(state)}{proof(state)} \\

\end{matharray}

These markup commands for proof mode closely correspond to the ones of theory

mode (see \S\ref{sec:markup-thy}).  Note that $\isarkeyword{txt_raw}$ is

special in the same way as $\isarkeyword{text_raw}$.

\railalias{txtraw}{txt\_raw}

\railterm{txtraw}

\begin{rail}

  ('sect' | 'subsect' | 'subsubsect' | 'txt' | txtraw) text

  ;

\end{rail}

\subsection{Proof context}\label{sec:proof-context}

\indexisarcmd{fix}\indexisarcmd{assume}\indexisarcmd{presume}\indexisarcmd{def}

\begin{matharray}{rcl}

  \isarcmd{fix} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{assume} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{presume} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{def} & : & \isartrans{proof(state)}{proof(state)} \\

\end{matharray}

The logical proof context consists of fixed variables and assumptions.  The

former closely correspond to Skolem constants, or meta-level universal

quantification as provided by the Isabelle/Pure logical framework.

Introducing some \emph{arbitrary, but fixed} variable via $\FIX x$ results in

a local object that may be used in the subsequent proof as any other variable

or constant.  Furthermore, any result $\edrv \phi[x]$ exported from the

current context will be universally closed wrt.\ $x$ at the outermost level:

$\edrv \All x \phi$; this is expressed using Isabelle's meta-variables.

Similarly, introducing some assumption $\chi$ has two effects.  On the one

hand, a local theorem is created that may be used as a fact in subsequent

proof steps.  On the other hand, any result $\chi \drv \phi$ exported from the

context becomes conditional wrt.\ the assumption: $\edrv \chi \Imp \phi$.

Thus, solving an enclosing goal using such a result would basically introduce

a new subgoal stemming from the assumption.  How this situation is handled

depends on the actual version of assumption command used: while $\ASSUMENAME$

insists on solving the subgoal by unification with some premise of the goal,

$\PRESUMENAME$ leaves the subgoal unchanged in order to be proved later by the

user.

Local definitions, introduced by $\DEF{}{x \equiv t}$, are achieved by

combining $\FIX x$ with another kind of assumption that causes any

hypothetical equation $x \equiv t$ to be eliminated by reflexivity.  Thus,

exporting some result $x \equiv t \drv \phi[x]$ yields $\edrv \phi[t]$.

\begin{rail}

  'fix' (vars + 'and') comment?

  ;

  ('assume' | 'presume') (assm comment? + 'and')

  ;

  'def' thmdecl? \\ var '==' term termpat? comment?

  ;

  var: name ('::' type)?

  ;

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

  ;

  assm: thmdecl? (prop proppat? +)

  ;

\end{rail}

\begin{descr}

\item [$\FIX{x}$] introduces a local \emph{arbitrary, but fixed} variable $x$.

\item [$\ASSUME{a}{\Phi}$ and $\PRESUME{a}{\Phi}$] introduce local theorems

  $\Phi$ by assumption.  Subsequent results applied to an enclosing goal

  (e.g.\ by $\SHOWNAME$) are handled as follows: $\ASSUMENAME$ expects to be

  able to unify with existing premises in the goal, while $\PRESUMENAME$

  leaves $\Phi$ as new subgoals.

  Several lists of assumptions may be given (separated by

  $\isarkeyword{and}$); the resulting list of current facts consists of all of

  these concatenated.

\item [$\DEF{a}{x \equiv t}$] introduces a local (non-polymorphic) definition.

  In results exported from the context, $x$ is replaced by $t$.  Basically,

  $\DEF{}{x \equiv t}$ abbreviates $\FIX{x}~\PRESUME{}{x \equiv t}$, with the

  resulting hypothetical equation solved by reflexivity.

  The default name for the definitional equation is $x_def$.

\end{descr}

The special name $prems$\indexisarthm{prems} refers to all assumptions of the

current context as a list of theorems.

\subsection{Facts and forward chaining}

\indexisarcmd{note}\indexisarcmd{then}\indexisarcmd{from}\indexisarcmd{with}

\begin{matharray}{rcl}

  \isarcmd{note} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarcmd{then} & : & \isartrans{proof(state)}{proof(chain)} \\

  \isarcmd{from} & : & \isartrans{proof(state)}{proof(chain)} \\

  \isarcmd{with} & : & \isartrans{proof(state)}{proof(chain)} \\

\end{matharray}

New facts are established either by assumption or proof of local statements.

Any fact will usually be involved in further proofs, either as explicit

arguments of proof methods or when forward chaining towards the next goal via

$\THEN$ (and variants).  Note that the special theorem name

$this$.\indexisarthm{this} refers to the most recently established facts.

\begin{rail}

  'note' thmdef? thmrefs comment?

  ;

  'then' comment?

  ;

  ('from' | 'with') thmrefs comment?

  ;

\end{rail}

\begin{descr}

\item [$\NOTE{a}{\vec b}$] recalls existing facts $\vec b$, binding the result

  as $a$.  Note that attributes may be involved as well, both on the left and

  right hand sides.

\item [$\THEN$] indicates forward chaining by the current facts in order to

  establish the goal to be claimed next.  The initial proof method invoked to

  refine that will be offered the facts to do ``anything appropriate'' (cf.\ 

  also \S\ref{sec:proof-steps}).  For example, method $rule$ (see

  \S\ref{sec:pure-meth}) would typically do an elimination rather than an

  introduction.  Automatic methods usually insert the facts into the goal

  state before operation.

\item [$\FROM{\vec b}$] abbreviates $\NOTE{}{\vec b}~\THEN$; thus $\THEN$ is

  equivalent to $\FROM{this}$.

\item [$\WITH{\vec b}$] abbreviates $\FROM{\vec b~facts}$; thus the forward

  chaining is from earlier facts together with the current ones.

\end{descr}

Basic proof methods (such as $rule$, see \S\ref{sec:pure-meth}) expect

multiple facts to be given in their proper order, corresponding to a prefix of

the premises of the rule involved.  Note that positions may be easily skipped

using a form like $\FROM{\text{\texttt{_}}~a~b}$, for example.  This involves

the trivial rule $\PROP\psi \Imp \PROP\psi$, which is bound in Isabelle/Pure

as ``\texttt{_}'' (underscore).\indexisarthm{_@\texttt{_}}

\subsection{Goal statements}

\indexisarcmd{theorem}\indexisarcmd{lemma}

\indexisarcmd{have}\indexisarcmd{show}\indexisarcmd{hence}\indexisarcmd{thus}

\begin{matharray}{rcl}

  \isarcmd{theorem} & : & \isartrans{theory}{proof(prove)} \\

  \isarcmd{lemma} & : & \isartrans{theory}{proof(prove)} \\

  \isarcmd{have} & : & \isartrans{proof(state)}{proof(prove)} \\

  \isarcmd{show} & : & \isartrans{proof(state)}{proof(prove)} \\

  \isarcmd{hence} & : & \isartrans{proof(state)}{proof(prove)} \\

  \isarcmd{thus} & : & \isartrans{proof(state)}{proof(prove)} \\

\end{matharray}

Proof mode is entered from theory mode by initial goal commands $\THEOREMNAME$

and $\LEMMANAME$.  New local goals may be claimed within proof mode as well.

Four variants are available, indicating whether the result is meant to solve

some pending goal and whether forward chaining is employed.

\begin{rail}

  ('theorem' | 'lemma') goal

  ;

  ('have' | 'show' | 'hence' | 'thus') goal

  ;

  goal: thmdecl? proppat comment?

  ;

\end{rail}

\begin{descr}

\item [$\THEOREM{a}{\phi}$] enters proof mode with $\phi$ as main goal,

  eventually resulting in some theorem $\turn \phi$ put back into the theory.

\item [$\LEMMANAME$] is similar to $\THEOREMNAME$, but tags the result as

  ``lemma''.

\item [$\HAVE{a}{\phi}$] claims a local goal, eventually resulting in a

  theorem with the current assumption context as hypotheses.

\item [$\SHOW{a}{\phi}$] is similar to $\HAVE{a}{\phi}$, but solves some

  pending goal with the result \emph{exported} into the corresponding context

  (cf.\ \S\ref{sec:proof-context}).

\item [$\HENCENAME$] abbreviates $\THEN~\HAVENAME$, i.e.\ claims a local goal

  to be proven by forward chaining the current facts.  Note that $\HENCENAME$

  is also equivalent to $\FROM{this}~\HAVENAME$.

\item [$\THUSNAME$] abbreviates $\THEN~\SHOWNAME$.  Note that $\THUSNAME$ is

  also equivalent to $\FROM{this}~\SHOWNAME$.

\end{descr}

\subsection{Initial and terminal proof steps}\label{sec:proof-steps}

\indexisarcmd{proof}\indexisarcmd{qed}\indexisarcmd{by}

\indexisarcmd{.}\indexisarcmd{..}\indexisarcmd{sorry}

\begin{matharray}{rcl}

  \isarcmd{proof} & : & \isartrans{proof(prove)}{proof(state)} \\

  \isarcmd{qed} & : & \isartrans{proof(state)}{proof(state) ~|~ theory} \\

  \isarcmd{by} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

  \isarcmd{.\,.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

  \isarcmd{.} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

  \isarcmd{sorry} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

\end{matharray}

Arbitrary goal refinement via tactics is considered harmful.  Consequently the

Isar framework admits proof methods to be invoked in two places only.

\begin{enumerate}

\item An \emph{initial} refinement step $\PROOF{m@1}$ reduces a newly stated

  goal to a number of sub-goals that are to be solved later.  Facts are passed

  to $m@1$ for forward chaining, if so indicated by $proof(chain)$ mode.

\item A \emph{terminal} conclusion step $\QED{m@2}$ solves any remaining goals

  completely.  No facts are passed to $m@2$.

\end{enumerate}

The only other proper way to affect pending goals is by $\SHOWNAME$ (or

$\THUSNAME$), which involves an explicit statement of what is to be solved.

\medskip

Also note that initial proof methods should either solve the goal completely,

or constitute some well-understood reduction to new sub-goals.  Arbitrary

automatic proof tools that are prone leave a large number of badly structured

sub-goals are no help in continuing the proof document in any intelligible

way.  A much better technique would be to $\SHOWNAME$ some non-trivial

reduction as an explicit rule, which is solved completely by some automated

method, and then applied to some pending goal.

\medskip

Unless given explicitly by the user, the default initial method is

``$default$'', which is usually set up to apply a single standard elimination

or introduction rule according to the topmost symbol involved.  There is no

separate default terminal method; in any case the final step is to solve all

remaining goals by assumption, though.

\begin{rail}

  'proof' interest? meth? comment?

  ;

  'qed' meth? comment?

  ;

  'by' meth meth? comment?

  ;

  ('.' | '..' | 'sorry') comment?

  ;

  meth: method interest?

  ;

\end{rail}

\begin{descr}

\item [$\PROOF{m@1}$] refines the goal by proof method $m@1$; facts for

  forward chaining are passed if so indicated by $proof(chain)$ mode.

\item [$\QED{m@2}$] refines any remaining goals by proof method $m@2$ and

  concludes the sub-proof by assumption.  If the goal had been $\SHOWNAME$ (or

  $\THUSNAME$), some pending sub-goal is solved as well by the rule resulting

  from the result \emph{exported} into the enclosing goal context.  Thus

  $\QEDNAME$ may fail for two reasons: either $m@2$ fails, or the resulting

  rule does not fit to any pending goal\footnote{This includes any additional

    ``strong'' assumptions as introduced by $\ASSUMENAME$.} of the enclosing

  context.  Debugging such a situation might involve temporarily changing

  $\SHOWNAME$ into $\HAVENAME$, or weakening the local context by replacing

  some occurrences of $\ASSUMENAME$ by $\PRESUMENAME$.

\item [$\BYY{m@1}{m@2}$] is a \emph{terminal proof}\index{proof!terminal}; it

  abbreviates $\PROOF{m@1}~\QED{m@2}$, with automatic backtracking across both

  methods.  Debugging an unsuccessful $\BYY{m@1}{m@2}$ commands might be done

  by expanding its definition; in many cases $\PROOF{m@1}$ is already

  sufficient to see what is going wrong.

\item [``$\DDOT$''] is a \emph{default proof}\index{proof!default}; it

  abbreviates $\BY{default}$.

\item [``$\DOT$''] is a \emph{trivial proof}\index{proof!trivial}; it

  abbreviates $\BY{assumption}$.

\item [$\isarkeyword{sorry}$] is a \emph{fake proof}\index{proof!fake};

  provided that \texttt{quick_and_dirty} is enabled, $\isarkeyword{sorry}$

  pretends to solve the goal without further ado.  Of course, the result is a

  fake theorem only, involving some oracle in its internal derivation object

  (this is indicated as ``$[!]$'' in the printed result).  The main

  application of $\isarkeyword{sorry}$ is to support experimentation and

  top-down proof development.

\end{descr}

\subsection{Improper proof steps}

The following commands emulate unstructured tactic scripts to some extent.

While these are anathema for writing proper Isar proof documents, they might

come in handy for interactive exploration and debugging.

\indexisarcmd{apply}\indexisarcmd{then-apply}\indexisarcmd{back}

\begin{matharray}{rcl}

  \isarcmd{apply}^* & : & \isartrans{proof}{proof} \\

  \isarcmd{then_apply}^* & : & \isartrans{proof}{proof} \\

  \isarcmd{back}^* & : & \isartrans{proof}{proof} \\

\end{matharray}

\railalias{thenapply}{then\_apply}

\railterm{thenapply}

\begin{rail}

  'apply' method

  ;

  thenapply method

  ;

  'back'

  ;

\end{rail}

\begin{descr}

\item [$\isarkeyword{apply}~(m)$] applies proof method $m$ in the plain old

  tactic sense.  Facts for forward chaining are reset.

\item [$\isarkeyword{then_apply}~(m)$] is similar to $\isarkeyword{apply}$,

  but keeps the goal's facts.

\item [$\isarkeyword{back}$] does back-tracking over the result sequence of

  the latest proof command.\footnote{Unlike the ML function \texttt{back}

    \cite{isabelle-ref}, the Isar command does not search upwards for further

    branch points.} Basically, any proof command may return multiple results.

\end{descr}

\subsection{Term abbreviations}\label{sec:term-abbrev}

\indexisarcmd{let}

\begin{matharray}{rcl}

  \isarcmd{let} & : & \isartrans{proof(state)}{proof(state)} \\

  \isarkeyword{is} & : & syntax \\

\end{matharray}

Abbreviations may be either bound by explicit $\LET{p \equiv t}$ statements,

or by annotating assumptions or goal statements ($\ASSUMENAME$, $\SHOWNAME$

etc.) with a list of patterns $\ISS{p@1 \dots}{p@n}$.  In both cases,

higher-order matching is invoked to bind extra-logical term variables, which

may be either named schematic variables of the form $\Var{x}$, or nameless

dummies ``\texttt{_}'' (underscore).\indexisarvar{_@\texttt{_}} Note that in

the $\LETNAME$ form the patterns occur on the left-hand side, while the

$\ISNAME$ patterns are in postfix position.

Term abbreviations are quite different from actual local definitions as

introduced via $\DEFNAME$ (see \S\ref{sec:proof-context}).  The latter are

visible within the logic as actual equations, while abbreviations disappear

during the input process just after type checking.

\begin{rail}

  'let' ((term + 'as') '=' term comment? + 'and')

  ;  

\end{rail}

The syntax of $\ISNAME$ patterns follows \railnonterm{termpat} or

\railnonterm{proppat} (see \S\ref{sec:term-pats}).

\begin{descr}

\item [$\LET{\vec p = \vec t}$] binds any text variables in patters $\vec p$

  by simultaneous higher-order matching against terms $\vec t$.

\item [$\IS{\vec p}$] resembles $\LETNAME$, but matches $\vec p$ against the

  preceding statement.  Also note that $\ISNAME$ is not a separate command,

  but part of others (such as $\ASSUMENAME$, $\HAVENAME$ etc.).

\end{descr}

A few \emph{automatic} term abbreviations\index{automatic abbreviation} for

goals and facts are available as well.  For any open goal,

$\Var{thesis_prop}$\indexisarvar{thesis-prop} refers to the full proposition

(which may be a rule), $\Var{thesis_concl}$\indexisarvar{thesis-concl} to its

(atomic) conclusion, and $\Var{thesis}$\indexisarvar{thesis} to its

object-logical statement.  The latter two abstract over any meta-level

parameters bound by $\Forall$.

Fact statements resulting from assumptions or finished goals are bound as

$\Var{this_prop}$\indexisarvar{this-prop},

$\Var{this_concl}$\indexisarvar{this-concl}, and

$\Var{this}$\indexisarvar{this}, similar to $\Var{thesis}$ above.  In case

$\Var{this}$ refers to an object-logic statement that is an application

$f(t)$, then $t$ is bound to the special text variable

``$\dots$''\indexisarvar{\dots} (three dots).  The canonical application of

this feature are calculational proofs (see \S\ref{sec:calculation}).

\subsection{Block structure}

\indexisarcmd{next}\indexisarcmd{\{\{}\indexisarcmd{\}\}}

\begin{matharray}{rcl}

  \isarcmd{next} & : & \isartrans{proof(state)}{proof(state)} \\

  \BG & : & \isartrans{proof(state)}{proof(state)} \\

  \EN & : & \isartrans{proof(state)}{proof(state)} \\

\end{matharray}

While Isar is inherently block-structured, opening and closing blocks is

mostly handled rather casually, with little explicit user-intervention.  Any

local goal statement automatically opens \emph{two} blocks, which are closed

again when concluding the sub-proof (by $\QEDNAME$ etc.).  Sections of

different context within a sub-proof may be switched via $\isarkeyword{next}$,

which is just a single block-close followed by block-open again.  Thus the

effect of $\isarkeyword{next}$ is a local reset the proof

context.\footnote{There is no goal focus involved here!}

For slightly more advanced applications, there are explicit block parentheses

as well.  These typically achieve a stronger forward style of reasoning.

\begin{descr}

\item [$\isarkeyword{next}$] switches to a fresh block within a sub-proof,

  resetting the local context to the initial one.

\item [$\isarkeyword{\{\{}$ and $\isarkeyword{\}\}}$] explicitly open and

  close blocks.  Any current facts pass through ``$\isarkeyword{\{\{}$''

  unchanged, while ``$\isarkeyword{\}\}}$'' causes any result to be

  \emph{exported} into the enclosing context.  Thus fixed variables are

  generalized, assumptions discharged, and local definitions unfolded (cf.\ 

  \S\ref{sec:proof-context}).  There is no difference of $\ASSUMENAME$ and

  $\PRESUMENAME$ in this mode of forward reasoning --- in contrast to plain

  backward reasoning with the result exported at $\SHOWNAME$ time.

\end{descr}

\section{Other commands}

\subsection{Diagnostics}

\indexisarcmd{thm}\indexisarcmd{term}\indexisarcmd{prop}\indexisarcmd{typ}

\begin{matharray}{rcl}

  \isarcmd{thm} & : & \isarkeep{theory~|~proof} \\

  \isarcmd{term} & : & \isarkeep{theory~|~proof} \\

  \isarcmd{prop} & : & \isarkeep{theory~|~proof} \\

  \isarcmd{typ} & : & \isarkeep{theory~|~proof} \\

\end{matharray}

These commands are not part of the actual Isabelle/Isar syntax, but assist

interactive development.  Also note that $undo$ does not apply here, since the

theory or proof configuration is not changed.

\begin{rail}

  'thm' thmrefs

  ;

  'term' term

  ;

  'prop' prop

  ;

  'typ' type

  ;

\end{rail}

\begin{descr}

\item [$\isarkeyword{thm}~thms$] retrieves lists of 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

  $thms$ do not have any permanent effect.

\item [$\isarkeyword{term}~t$, $\isarkeyword{prop}~\phi$] read, type-checks

  and print terms or propositions according to the current theory or proof

  context; the inferred type of $t$ is output as well.  Note that these

  commands are also useful in inspecting the current environment of term

  abbreviations.

\item [$\isarkeyword{typ}~\tau$] reads and prints types of the meta-logic

  according to the current theory or proof context.

\end{descr}

\subsection{System operations}

\indexisarcmd{cd}\indexisarcmd{pwd}\indexisarcmd{use-thy}\indexisarcmd{use-thy-only}

\indexisarcmd{update-thy}\indexisarcmd{update-thy-only}

\begin{matharray}{rcl}

  \isarcmd{cd} & : & \isarkeep{\cdot} \\

  \isarcmd{pwd} & : & \isarkeep{\cdot} \\

  \isarcmd{use_thy} & : & \isarkeep{\cdot} \\

  \isarcmd{use_thy_only} & : & \isarkeep{\cdot} \\

  \isarcmd{update_thy} & : & \isarkeep{\cdot} \\

  \isarcmd{update_thy_only} & : & \isarkeep{\cdot} \\

\end{matharray}

\begin{descr}

\item [$\isarkeyword{cd}~name$] changes the current directory of the Isabelle

  process.

\item [$\isarkeyword{pwd}~$] prints the current working directory.

\item [$\isarkeyword{use_thy}$, $\isarkeyword{use_thy_only}$,

  $\isarkeyword{update_thy}$, and $\isarkeyword{update_thy_only}$] load some

  theory given as $name$ argument.  These commands are basically the same as

  the corresponding ML functions\footnote{For historic reasons, the original

    ML versions also change the theory context to that of the theory loaded.}

  (see also \cite[\S1,\S6]{isabelle-ref}).  Note that both the ML and Isar

  versions may load new- and old-style theories alike.

\end{descr}

Note that these system commands are scarcely used when working with the

Proof~General interface, since loading of theories is done fully

transparently.

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