\chapter{Basic Isar 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 as provided by generic tools and packages that are
either part of Pure Isabelle, or pre-loaded by most object logics (such as the
Simplifier). See chapter~\ref{ch:hol-tools} for actual object-logic specific
elements (for Isabelle/HOL).
\medskip
Isar commands may be either \emph{proper} document constructors, or
\emph{improper commands} (indicated by $^*$). Some proof methods and
attributes introduced later may be classified as improper as well. Improper
Isar language elements might be helpful when developing proof documents, while
their use is strongly discouraged for the final version. 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{theory}\indexisarcmd{end}\indexisarcmd{context}
\begin{matharray}{rcl}
\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 manual
proof. In contrast, ``old-style'' Isabelle theories support batch processing
only, with the proof scripts collected in separate ML files.
The first command of any theory has to be $\THEORY$, starting a new theory
based on the merge of existing ones. 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 of any proper
theory file.
\begin{rail}
'theory' name '=' (name + '+') filespecs? ':'
;
'context' name
;
'end'
;;
filespecs: 'files' ((name | parname) +);
\end{rail}
\begin{descr}
\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}).
\item [$\CONTEXT~B$] enters an existing theory context $B$, 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 theory definition
itself has to be retracted.
\end{descr}
\subsection{Formal comments}\label{sec:formal-cmt-thy}
\indexisarcmd{title}\indexisarcmd{chapter}\indexisarcmd{section}\indexisarcmd{subsection}
\indexisarcmd{subsubsection}\indexisarcmd{text}
\begin{matharray}{rcl}
\isarcmd{title} & : & \isartrans{theory}{theory} \\
\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} \\
\end{matharray}
There are several commands to include \emph{formal comments} in theory
specification (a few more are available for proofs, see
\S\ref{sec:formal-cmt-prf}). In contrast to source-level comments
\verb|(*|\dots\verb|*)|, which are stripped at the lexical level, any text
given as formal comment is meant to be part of the actual document.
Consequently, it would be included in the final printed version.
Apart from plain prose, formal comments may also refer to logical entities of
the theory context (types, terms, theorems etc.). Proper processing of the
text would then include some further consistency checks with the items
declared in the current theory, e.g.\ type-checking of included
terms.\footnote{The current version of Isabelle/Isar does not process formal
comments in any such way. This will be available as part of the automatic
theory and proof document preparation system (using (PDF){\LaTeX}) that is
planned for the near future.}
\begin{rail}
'title' text text? text?
;
('chapter' | 'section' | 'subsection' | 'subsubsection' | 'text') text
;
\end{rail}
\begin{descr}
\item [$\isarkeyword{title}~title~author~date$] specifies the document title
just as in typical {\LaTeX} documents.
\item [$\isarkeyword{chapter}~text$, $\isarkeyword{section}~text$,
$\isarkeyword{subsection}~text$, and $\isarkeyword{subsubsection}~text$]
mark chapter and section headings.
\item [$\TEXT~text$] specifies an actual body of prose text, including
references to formal entities.\footnote{The latter feature is not yet
exploited. Nevertheless, any text of the form \texttt{\at\{\dots\}}
should be considered as reserved for future use.}
\end{descr}
\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 [$\isarkeyword{classes}~c<\vec c ~\dots$] 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
introduce proven class relations.
\item [$\isarkeyword{defaultsort}~s$] makes sort $s$ the new default sort for
any type variables input 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 [$\TYPES~(\vec\alpha)t = \tau~\dots$] 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 some logics such as
Isabelle/HOL provide their own version of $\isarkeyword{typedecl}$.
\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 [$\isarkeyword{arities}~t::(\vec s)s~\dots$] 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 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' (thmdecl? prop comment? +)
;
'constdefs' (constdecl prop comment? +)
;
constdecl: name '::' type mixfix? comment?
;
\end{rail}
\begin{descr}
\item [$\CONSTS~c::\sigma~\dots$] declares constant $c$ to have any instance
of type scheme $\sigma$. The optional mixfix annotations may attach
concrete syntax constants.
\item [$\DEFS~name: eqn~\dots$] 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 [$\isarkeyword{constdefs}~c::\sigma~eqn~\dots$] combines constant
declarations and definitions, 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. 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 (also known as \emph{macros}): parse/print rules (\texttt{==}), parse
rules (\texttt{=>}), 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 [$\isarkeyword{axioms}~name: \phi~\dots$] introduces arbitrary
statements as logical axioms. In fact, axioms are ``axiomatic theorems'',
and may be referred just as any other theorem later.
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 [$\isarkeyword{theorems}~name = thms$] stores lists of existing theorems
as $name$. Typical applications would also involve attributes (to augment
the default simpset, 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 organises any kind of names (of types, constants, theorems etc.) by
hierarchically structured name spaces. Normally the user never has to control
the behaviour 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 as
base names only, until $\isarkeyword{local}$ is declared again.
\end{descr}
\subsection{Incorporating ML code}\label{sec:ML}
\indexisarcmd{use}\indexisarcmd{ML}\indexisarcmd{setup}
\begin{matharray}{rcl}
\isarcmd{use} & : & \isartrans{\cdot}{\cdot} \\
\isarcmd{ML} & : & \isartrans{\cdot}{\cdot} \\
\isarcmd{setup} & : & \isartrans{theory}{theory} \\
\end{matharray}
\begin{rail}
'use' name
;
'ML' text
;
'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.
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 [$\isarkeyword{ML}~text$] reads and executes ML commands from $text$.
The theory context is passed just as for $\isarkeyword{use}$.
\item [$\isarkeyword{setup}~text$] changes the current theory context by
applying setup functions $text$, which has to be an ML expression of type
$(theory \to theory)~list$. The $\isarkeyword{setup}$ command is the usual
way to initialise 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 \emph{proof context}, local \emph{facts}, and open
\emph{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
($\approx$ tactic), and enter a sub-proof to establish the final 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 an intermediate mode between $proof(state)$ and
$proof(prove)$: existing facts have been just picked up in order to use them
when refining the goal to be claimed next.
\end{descr}
\subsection{Formal comments}\label{sec:formal-cmt-prf}
\indexisarcmd{sect}\indexisarcmd{subsect}\indexisarcmd{subsect}\indexisarcmd{txt}
\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)} \\
\end{matharray}
These formal comments in proof mode closely correspond to the ones of theory
mode (see \S\ref{sec:formal-cmt-thy}).
\begin{rail}
('sect' | 'subsect' | 'subsubsect' | 'txt') 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 entity that may be used in the subsequent proof as any other variable
or constant. Furthermore, any result $\phi[x]$ exported from the current
context will be universally closed wrt.\ $x$ at the outermost level (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 $\phi$ exported from the context
becomes conditional wrt.\ the assumption. Thus, solving an enclosing goal
using this 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 $\ASSUMENAME$ solves the subgoal by unifying
with some premise of the goal, $\PRESUMENAME$ leaves the subgoal unchanged to
be proved later by the user.
Local definitions, introduced by $\DEF{a}{x \equiv t}$, are achieved by
combining $\FIX x$ with another version of assumption that causes any
hypothetical equation $x = t$ to be eliminated by reflexivity. Thus,
exporting some result $\phi[x]$ simply yields $\phi[t]$.
\begin{rail}
'fix' (var +) comment?
;
('assume' | 'presume') (assm comment? + 'and')
;
'def' thmdecl? \\ var '==' term termpat? comment?
;
var: 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
$\Phi$. Subsequent results used to solve some enclosing goal (e.g.\ via
$\SHOWNAME$) are handled as follows: $\ASSUMENAME$ expects to be able to
unify with existing premises in the goal, while $\PRESUMENAME$ leaves $\Phi$
as new subgoals. Note that several lists of assumptions may be given
(separated by \railterm{and}); the resulting list of current facts consists
of all of these.
\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 $\FIX{x}~\PRESUME{}{x \equiv t}$ (the
resulting hypothetical equation is solved by reflexivity, though).
\end{descr}
The internal register $prems$\indexisarreg{prems} refers to all current
assumptions 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
(via $\HAVENAME$ or $\SHOWNAME$). Any facts will usually be involved in
proofs of further results: either as explicit arguments of proof methods or
when forward chaining towards the next goal via $\THEN$ (and variants).
\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
solve that will be offered these facts to do ``anything appropriate'' (see
also \S\ref{sec:proof-steps}). For example, method $rule$ (see
\S\ref{sec:pure-meth}) would do an elimination rather than an introduction.
\item [$\FROM{\vec b}$] abbreviates $\NOTE{}{\vec b}~\THEN$; also note that
$\THEN$ is equivalent to $\FROM{facts}$.
\item [$\WITH{\vec b}$] abbreviates $\FROM{\vec b~facts}$; thus the forward
chaining is from earlier facts together with the current ones.
\end{descr}
Note that the internal register of \emph{current facts} may be referred as
theorem list $facts$.\indexisarreg{facts}
\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: 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 [$\THEOREM{name}{\phi}$] enters proof mode with $\phi$ as main goal,
eventually resulting in some theorem $\turn \phi$, which will be stored in
the theory.
\item [$\LEMMANAME$] is similar to $\THEOREMNAME$, but tags the result as
``lemma''.
\item [$\HAVE{name}{\phi}$] claims a local goal, eventually resulting in a
theorem with the current assumption context as hypotheses.
\item [$\SHOW{name}{\phi}$] is similar to $\HAVE{name}{\phi}$, but solves some
pending goal with the result \emph{exported} into the corresponding context.
\item [$\HENCE{name}{\phi}$] abbreviates $\THEN~\HAVE{name}{\phi}$, i.e.\
claims a local goal to be proven by forward chaining the current facts.
\item [$\THUS{name}{\phi}$] abbreviates $\THEN~\SHOW{name}{\phi}$.
\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 refinements 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
intermediate 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
pending goals completely. No facts are passed to $m@2$.
\end{enumerate}
The only other proper way to affect pending goals is by $\SHOWNAME$, which
involves an explicit statement of what is solved.
\medskip
Also note that initial proof methods should either solve the goal completely,
or constitute some well-understood deterministic 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. The default
terminal method is ``$finish$''; it solves all goals by assumption.
\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 pending goal by proof method $m@1$; facts
for forward chaining are passed if so indicated by $proof(chain)$.
\item [$\QED{m@2}$] refines any remaining goals by proof method $m@1$ and
concludes the sub-proof. If the goal had been $\SHOWNAME$, some pending
sub-goal is solved as well by the rule resulting from the result exported to
the enclosing goal context. Thus $\QEDNAME$ may fail for two reasons:
either $m@2$ fails to solve all remaining goals completely, or the resulting
rule does not resolve with any enclosing goal. Debugging such a situation
might involve temporarily changing $\SHOWNAME$ into $\HAVENAME$, or
weakening the local context by replacing $\ASSUMENAME$ by $\PRESUMENAME$.
\item [$\BYY{m@1}{m@2}$] is a \emph{terminal proof}; 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 simply
expanding the abbreviation by hand; note that $\PROOF{m@1}$ is usually
sufficient to see what is going wrong.
\item [$\DDOT$] is a \emph{default proof}; it abbreviates $\BY{default}$.
\item [$\DOT$] is a \emph{trivial proof}, it abbreviates $\BY{-}$, where
method ``$-$'' does nothing except inserting any facts into the proof state.
\item [$\isarkeyword{sorry}$] is a \emph{fake proof}; provided that
\texttt{quick_and_dirty} is enabled, $\isarkeyword{sorry}$ pretends to solve
the goal without much 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 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 exploring 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 ignored.
\item [$\isarkeyword{then_apply}~m$] is similar to $\isarkeyword{apply}$, but
observes the goal's facts.
\item [$\isarkeyword{back}$] does back-tracking over the result sequence of
the last proof command. 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 $\IS{p@1 \dots p@n}$. In both cases,
higher-order matching is applied to bind extra-logical text
variables\index{text variables}, which may be either of the form $\VVar{x}$
(token class \railtoken{textvar}, see \S\ref{sec:lex-syntax}) or nameless
dummies ``\verb,_,'' (underscore).\index{dummy variables} Note that in the
$\LETNAME$ form the patterns occur on the left-hand side, while the $\ISNAME$
patterns are in postfix position.
Note that 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}
Furthermore, a few automatic term abbreviations\index{automatic abbreviation}
for goals and facts are available. For any open goal, $\VVar{thesis}$ refers
to its object-logic statement, $\VVar{thesis_prop}$ to the full proposition
(which may be a rule), and $\VVar{thesis_concl}$ to its (atomic) conclusion.
Facts (i.e.\ assumptions and finished goals) that have an application $f(x)$
as object-logic statement get $x$ bound to the special text variable
``$\dots$'' (three dots). The canonical application of this feature are
calculational proofs, see \S\ref{sec:calculation}.
\subsection{Block structure}
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 are typically switched via
$\isarkeyword{next}$, which is just a single block-close followed by
block-open again. Thus the effect of $\isarkeyword{next}$ is to reset the
proof context to that of the head of the sub-proof. Note that there is no
goal focus involved here!
For slightly more advanced applications, there are explicit block parentheses
as well. These typically achieve a strong forward style of reasoning.
\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}
\begin{descr}
\item [$\isarkeyword{next}$] switches to a fresh block within a sub-proof,
resetting the context to the initial one.
\item [$\isarkeyword{\{\{}$ and $\isarkeyword{\}\}}$] explicitly open and
close blocks. Any current facts pass through $\isarkeyword{\{\{}$
unchanged, while $\isarkeyword{\}\}}$ causes them to be \emph{exported} into
the enclosing context. Thus fixed variables are generalised, assumptions
discharged, and local definitions eliminated.
\end{descr}
\section{Other commands}
The following commands are not part of the actual proper or improper
Isabelle/Isar syntax, but assist interactive development, for example. Also
note that $undo$ does not apply here, since the theory or proof configuration
is not changed.
\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}
\begin{rail}
'typ' type
;
'term' term
;
'prop' prop
;
'thm' thmrefs
;
\end{rail}
\begin{descr}
\item [$\isarkeyword{typ}~\tau$, $\isarkeyword{term}~t$,
$\isarkeyword{prop}~\phi$] read and print types / terms / propositions
according to the current theory or proof context.
\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.
\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 exactly the same as the
corresponding ML functions (see also \cite[\S1 and \S6]{isabelle-ref}).
Note that both the ML and Isar versions of these commands may load new- and
old-style theories alike.
\end{descr}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "isar-ref"
%%% End: