\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 usually 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 actual theory 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 [$\THEORY~A = B@1 + \cdots + B@n$] 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
$\isarkeyword{files}$.
\item [$\CONTEXT~B$] enters an existing theory context, basically in read-only
mode, so only a limited set of commands may be performed. 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, instead 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
another 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}
{\LaTeX} package to be included}
\end{descr}
Any markup command (except $\isarkeyword{text_raw}$) corresponds to a {\LaTeX}
macro with the name derived from \verb,\isamarkup, (e.g.\
\verb,\isamarkupchapter, for $\isarkeyword{chapter}$). The \railqtoken{text}
argument is passed to that macro unchanged, i.e.\ any {\LaTeX} commands may be
included here.
\medskip Further 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, as may be observed when printing terms or
theorems.
\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}
\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}
\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 default
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 must 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 has to refer 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}
\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)} \\
\isarcmd{\{\{} & : & \isartrans{proof(state)}{proof(state)} \\
\isarcmd{\}\}} & : & \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{typ}\indexisarcmd{term}\indexisarcmd{prop}\indexisarcmd{thm}
\begin{matharray}{rcl}
\isarcmd{typ} & : & \isarkeep{theory~|~proof} \\
\isarcmd{term} & : & \isarkeep{theory~|~proof} \\
\isarcmd{prop} & : & \isarkeep{theory~|~proof} \\
\isarcmd{thm} & : & \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}
'typ' type
;
'term' term
;
'prop' prop
;
'thm' thmrefs
;
\end{rail}
\begin{descr}
\item [$\isarkeyword{typ}~\tau$] reads and prints types of the meta-logic
according to the current theory or proof context.
\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 [$\isarkeyword{thm}~thms$] 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.
\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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