--- a/doc-src/IsarRef/Thy/document/Proof.tex Fri May 09 23:35:57 2008 +0200
+++ b/doc-src/IsarRef/Thy/document/Proof.tex Sat May 10 00:14:00 2008 +0200
@@ -24,6 +24,997 @@
}
\isamarkuptrue%
%
+\begin{isamarkuptext}%
+Proof commands perform 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 different modes of operation:
+
+ \begin{descr}
+
+ \item [\isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}}] 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, and enter a sub-proof to establish the
+ actual result.
+
+ \item [\isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}}] is like a nested theory mode: the
+ context may be augmented by \emph{stating} additional assumptions,
+ intermediate results etc.
+
+ \item [\isa{{\isachardoublequote}proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}}] is intermediate between \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} and \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}}: existing facts (i.e.\
+ the contents of the special ``\indexref{}{fact}{this}\mbox{\isa{this}}'' register) have been
+ just picked up in order to be used when refining the goal claimed
+ next.
+
+ \end{descr}
+
+ The proof mode indicator may be read as a verb telling the writer
+ what kind of operation may be performed next. The corresponding
+ typings of proof commands restricts the shape of well-formed proof
+ texts to particular command sequences. So dynamic arrangements of
+ commands eventually turn out as static texts of a certain structure.
+ \Appref{ap:refcard} gives a simplified grammar of the overall
+ (extensible) language emerging that way.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Context elements \label{sec:proof-context}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{}{command}{fix}\mbox{\isa{\isacommand{fix}}} & : & \isartrans{proof(state)}{proof(state)} \\
+ \indexdef{}{command}{assume}\mbox{\isa{\isacommand{assume}}} & : & \isartrans{proof(state)}{proof(state)} \\
+ \indexdef{}{command}{presume}\mbox{\isa{\isacommand{presume}}} & : & \isartrans{proof(state)}{proof(state)} \\
+ \indexdef{}{command}{def}\mbox{\isa{\isacommand{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 ``\mbox{\isa{\isacommand{fix}}}~\isa{x}'' results in a local value
+ that may be used in the subsequent proof as any other variable or
+ constant. Furthermore, any result \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymphi}{\isacharbrackleft}x{\isacharbrackright}{\isachardoublequote}} exported from
+ the context will be universally closed wrt.\ \isa{x} at the
+ outermost level: \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ {\isasymphi}{\isacharbrackleft}x{\isacharbrackright}{\isachardoublequote}} (this is expressed in normal
+ form using Isabelle's meta-variables).
+
+ Similarly, introducing some assumption \isa{{\isasymchi}} 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
+ \isa{{\isachardoublequote}{\isasymchi}\ {\isasymturnstile}\ {\isasymphi}{\isachardoublequote}} exported from the context becomes conditional wrt.\
+ the assumption: \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymchi}\ {\isasymLongrightarrow}\ {\isasymphi}{\isachardoublequote}}. 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
+ version of assumption command used: while \mbox{\isa{\isacommand{assume}}}
+ insists on solving the subgoal by unification with some premise of
+ the goal, \mbox{\isa{\isacommand{presume}}} leaves the subgoal unchanged in order
+ to be proved later by the user.
+
+ Local definitions, introduced by ``\mbox{\isa{\isacommand{def}}}~\isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}}'', are achieved by combining ``\mbox{\isa{\isacommand{fix}}}~\isa{x}'' with
+ another version of assumption that causes any hypothetical equation
+ \isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}} to be eliminated by the reflexivity rule. Thus,
+ exporting some result \isa{{\isachardoublequote}x\ {\isasymequiv}\ t\ {\isasymturnstile}\ {\isasymphi}{\isacharbrackleft}x{\isacharbrackright}{\isachardoublequote}} yields \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymphi}{\isacharbrackleft}t{\isacharbrackright}{\isachardoublequote}}.
+
+ \begin{rail}
+ 'fix' (vars + 'and')
+ ;
+ ('assume' | 'presume') (props + 'and')
+ ;
+ 'def' (def + 'and')
+ ;
+ def: thmdecl? \\ name ('==' | equiv) term termpat?
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{fix}}}~\isa{x}] introduces a local variable
+ \isa{x} that is \emph{arbitrary, but fixed.}
+
+ \item [\mbox{\isa{\isacommand{assume}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} and \mbox{\isa{\isacommand{presume}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}}] introduce a local fact \isa{{\isachardoublequote}{\isasymphi}\ {\isasymturnstile}\ {\isasymphi}{\isachardoublequote}} by
+ assumption. Subsequent results applied to an enclosing goal (e.g.\
+ by \indexref{}{command}{show}\mbox{\isa{\isacommand{show}}}) are handled as follows: \mbox{\isa{\isacommand{assume}}} expects to be able to unify with existing premises in the
+ goal, while \mbox{\isa{\isacommand{presume}}} leaves \isa{{\isasymphi}} as new subgoals.
+
+ Several lists of assumptions may be given (separated by
+ \indexref{}{keyword}{and}\mbox{\isa{\isakeyword{and}}}; the resulting list of current facts consists
+ of all of these concatenated.
+
+ \item [\mbox{\isa{\isacommand{def}}}~\isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}}] introduces a local
+ (non-polymorphic) definition. In results exported from the context,
+ \isa{x} is replaced by \isa{t}. Basically, ``\mbox{\isa{\isacommand{def}}}~\isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}}'' abbreviates ``\mbox{\isa{\isacommand{fix}}}~\isa{x}~\mbox{\isa{\isacommand{assume}}}~\isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}}'', with the resulting
+ hypothetical equation solved by reflexivity.
+
+ The default name for the definitional equation is \isa{x{\isacharunderscore}def}.
+ Several simultaneous definitions may be given at the same time.
+
+ \end{descr}
+
+ The special name \indexref{}{fact}{prems}\mbox{\isa{prems}} refers to all assumptions of the
+ current context as a list of theorems. This feature should be used
+ with great care! It is better avoided in final proof texts.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Facts and forward chaining%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{}{command}{note}\mbox{\isa{\isacommand{note}}} & : & \isartrans{proof(state)}{proof(state)} \\
+ \indexdef{}{command}{then}\mbox{\isa{\isacommand{then}}} & : & \isartrans{proof(state)}{proof(chain)} \\
+ \indexdef{}{command}{from}\mbox{\isa{\isacommand{from}}} & : & \isartrans{proof(state)}{proof(chain)} \\
+ \indexdef{}{command}{with}\mbox{\isa{\isacommand{with}}} & : & \isartrans{proof(state)}{proof(chain)} \\
+ \indexdef{}{command}{using}\mbox{\isa{\isacommand{using}}} & : & \isartrans{proof(prove)}{proof(prove)} \\
+ \indexdef{}{command}{unfolding}\mbox{\isa{\isacommand{unfolding}}} & : & \isartrans{proof(prove)}{proof(prove)} \\
+ \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 \mbox{\isa{\isacommand{then}}} (and variants);
+ \mbox{\isa{\isacommand{from}}} and \mbox{\isa{\isacommand{with}}} are composite forms
+ involving \mbox{\isa{\isacommand{note}}}. The \mbox{\isa{\isacommand{using}}} elements
+ augments the collection of used facts \emph{after} a goal has been
+ stated. Note that the special theorem name \indexref{}{fact}{this}\mbox{\isa{this}} refers
+ to the most recently established facts, but only \emph{before}
+ issuing a follow-up claim.
+
+ \begin{rail}
+ 'note' (thmdef? thmrefs + 'and')
+ ;
+ ('from' | 'with' | 'using' | 'unfolding') (thmrefs + 'and')
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{note}}}~\isa{{\isachardoublequote}a\ {\isacharequal}\ b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}}]
+ recalls existing facts \isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub n{\isachardoublequote}}, binding
+ the result as \isa{a}. Note that attributes may be involved as
+ well, both on the left and right hand sides.
+
+ \item [\mbox{\isa{\isacommand{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'' (see also
+ \secref{sec:proof-steps}). For example, method \indexref{}{method}{rule}\mbox{\isa{rule}}
+ (see \secref{sec:pure-meth-att}) would typically do an elimination
+ rather than an introduction. Automatic methods usually insert the
+ facts into the goal state before operation. This provides a simple
+ scheme to control relevance of facts in automated proof search.
+
+ \item [\mbox{\isa{\isacommand{from}}}~\isa{b}] abbreviates ``\mbox{\isa{\isacommand{note}}}~\isa{b}~\mbox{\isa{\isacommand{then}}}''; thus \mbox{\isa{\isacommand{then}}} is
+ equivalent to ``\mbox{\isa{\isacommand{from}}}~\isa{this}''.
+
+ \item [\mbox{\isa{\isacommand{with}}}~\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}}]
+ abbreviates ``\mbox{\isa{\isacommand{from}}}~\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n\ {\isasymAND}\ this{\isachardoublequote}}''; thus the forward chaining is from earlier facts together
+ with the current ones.
+
+ \item [\mbox{\isa{\isacommand{using}}}~\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}}] augments
+ the facts being currently indicated for use by a subsequent
+ refinement step (such as \indexref{}{command}{apply}\mbox{\isa{\isacommand{apply}}} or \indexref{}{command}{proof}\mbox{\isa{\isacommand{proof}}}).
+
+ \item [\mbox{\isa{\isacommand{unfolding}}}~\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}}] is
+ structurally similar to \mbox{\isa{\isacommand{using}}}, but unfolds definitional
+ equations \isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}} throughout the goal state
+ and facts.
+
+ \end{descr}
+
+ Forward chaining with an empty list of theorems is the same as not
+ chaining at all. Thus ``\mbox{\isa{\isacommand{from}}}~\isa{nothing}'' has no
+ effect apart from entering \isa{{\isachardoublequote}prove{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} mode, since
+ \indexref{}{fact}{nothing}\mbox{\isa{nothing}} is bound to the empty list of theorems.
+
+ Basic proof methods (such as \indexref{}{method}{rule}\mbox{\isa{rule}}) 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 something like \mbox{\isa{\isacommand{from}}}~\isa{{\isachardoublequote}{\isacharunderscore}\ {\isasymAND}\ a\ {\isasymAND}\ b{\isachardoublequote}}, for example. This involves the trivial rule
+ \isa{{\isachardoublequote}PROP\ {\isasympsi}\ {\isasymLongrightarrow}\ PROP\ {\isasympsi}{\isachardoublequote}}, which is bound in Isabelle/Pure as
+ ``\indexref{}{fact}{\_}\mbox{\isa{{\isacharunderscore}}}'' (underscore).
+
+ Automated methods (such as \mbox{\isa{simp}} or \mbox{\isa{auto}}) just
+ insert any given facts before their usual operation. Depending on
+ the kind of procedure involved, the order of facts is less
+ significant here.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Goal statements \label{sec:goals}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{}{command}{lemma}\mbox{\isa{\isacommand{lemma}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
+ \indexdef{}{command}{theorem}\mbox{\isa{\isacommand{theorem}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
+ \indexdef{}{command}{corollary}\mbox{\isa{\isacommand{corollary}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
+ \indexdef{}{command}{have}\mbox{\isa{\isacommand{have}}} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\
+ \indexdef{}{command}{show}\mbox{\isa{\isacommand{show}}} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\
+ \indexdef{}{command}{hence}\mbox{\isa{\isacommand{hence}}} & : & \isartrans{proof(state)}{proof(prove)} \\
+ \indexdef{}{command}{thus}\mbox{\isa{\isacommand{thus}}} & : & \isartrans{proof(state)}{proof(prove)} \\
+ \indexdef{}{command}{print\_statement}\mbox{\isa{\isacommand{print{\isacharunderscore}statement}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
+ \end{matharray}
+
+ From a theory context, proof mode is entered by an initial goal
+ command such as \mbox{\isa{\isacommand{lemma}}}, \mbox{\isa{\isacommand{theorem}}}, or
+ \mbox{\isa{\isacommand{corollary}}}. Within a proof, new claims may be
+ introduced locally as well; four variants are available here to
+ indicate whether forward chaining of facts should be performed
+ initially (via \indexref{}{command}{then}\mbox{\isa{\isacommand{then}}}), and whether the final result
+ is meant to solve some pending goal.
+
+ Goals may consist of multiple statements, resulting in a list of
+ facts eventually. A pending multi-goal is internally represented as
+ a meta-level conjunction (printed as \isa{{\isachardoublequote}{\isacharampersand}{\isacharampersand}{\isachardoublequote}}), which is usually
+ split into the corresponding number of sub-goals prior to an initial
+ method application, via \indexref{}{command}{proof}\mbox{\isa{\isacommand{proof}}}
+ (\secref{sec:proof-steps}) or \indexref{}{command}{apply}\mbox{\isa{\isacommand{apply}}}
+ (\secref{sec:tactic-commands}). The \indexref{}{method}{induct}\mbox{\isa{induct}} method
+ covered in \secref{sec:cases-induct} acts on multiple claims
+ simultaneously.
+
+ Claims at the theory level may be either in short or long form. A
+ short goal merely consists of several simultaneous propositions
+ (often just one). A long goal includes an explicit context
+ specification for the subsequent conclusion, involving local
+ parameters and assumptions. Here the role of each part of the
+ statement is explicitly marked by separate keywords (see also
+ \secref{sec:locale}); the local assumptions being introduced here
+ are available as \indexref{}{fact}{assms}\mbox{\isa{assms}} in the proof. Moreover, there
+ are two kinds of conclusions: \indexdef{}{element}{shows}\mbox{\isa{\isakeyword{shows}}} states several
+ simultaneous propositions (essentially a big conjunction), while
+ \indexdef{}{element}{obtains}\mbox{\isa{\isakeyword{obtains}}} claims several simultaneous simultaneous
+ contexts of (essentially a big disjunction of eliminated parameters
+ and assumptions, cf.\ \secref{sec:obtain}).
+
+ \begin{rail}
+ ('lemma' | 'theorem' | 'corollary') target? (goal | longgoal)
+ ;
+ ('have' | 'show' | 'hence' | 'thus') goal
+ ;
+ 'print\_statement' modes? thmrefs
+ ;
+
+ goal: (props + 'and')
+ ;
+ longgoal: thmdecl? (contextelem *) conclusion
+ ;
+ conclusion: 'shows' goal | 'obtains' (parname? case + '|')
+ ;
+ case: (vars + 'and') 'where' (props + 'and')
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{lemma}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}}] enters proof mode with
+ \isa{{\isasymphi}} as main goal, eventually resulting in some fact \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymphi}{\isachardoublequote}} to be put back into the target context. An additional
+ \railnonterm{context} specification may build up an initial proof
+ context for the subsequent claim; this includes local definitions
+ and syntax as well, see the definition of \mbox{\isa{contextelem}} in
+ \secref{sec:locale}.
+
+ \item [\mbox{\isa{\isacommand{theorem}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} and \mbox{\isa{\isacommand{corollary}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}}] are essentially the same as \mbox{\isa{\isacommand{lemma}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}}, but the facts are internally marked as
+ being of a different kind. This discrimination acts like a formal
+ comment.
+
+ \item [\mbox{\isa{\isacommand{have}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}}] claims a local goal,
+ eventually resulting in a fact within the current logical context.
+ This operation is completely independent of any pending sub-goals of
+ an enclosing goal statements, so \mbox{\isa{\isacommand{have}}} may be freely
+ used for experimental exploration of potential results within a
+ proof body.
+
+ \item [\mbox{\isa{\isacommand{show}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}}] is like \mbox{\isa{\isacommand{have}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} plus a second stage to refine some pending
+ sub-goal for each one of the finished result, after having been
+ exported into the corresponding context (at the head of the
+ sub-proof of this \mbox{\isa{\isacommand{show}}} command).
+
+ To accommodate interactive debugging, resulting rules are printed
+ before being applied internally. Even more, interactive execution
+ of \mbox{\isa{\isacommand{show}}} predicts potential failure and displays the
+ resulting error as a warning beforehand. Watch out for the
+ following message:
+
+ %FIXME proper antiquitation
+ \begin{ttbox}
+ Problem! Local statement will fail to solve any pending goal
+ \end{ttbox}
+
+ \item [\mbox{\isa{\isacommand{hence}}}] abbreviates ``\mbox{\isa{\isacommand{then}}}~\mbox{\isa{\isacommand{have}}}'', i.e.\ claims a local goal to be proven by forward
+ chaining the current facts. Note that \mbox{\isa{\isacommand{hence}}} is also
+ equivalent to ``\mbox{\isa{\isacommand{from}}}~\isa{this}~\mbox{\isa{\isacommand{have}}}''.
+
+ \item [\mbox{\isa{\isacommand{thus}}}] abbreviates ``\mbox{\isa{\isacommand{then}}}~\mbox{\isa{\isacommand{show}}}''. Note that \mbox{\isa{\isacommand{thus}}} is also equivalent to
+ ``\mbox{\isa{\isacommand{from}}}~\isa{this}~\mbox{\isa{\isacommand{show}}}''.
+
+ \item [\mbox{\isa{\isacommand{print{\isacharunderscore}statement}}}~\isa{a}] prints facts from the
+ current theory or proof context in long statement form, according to
+ the syntax for \mbox{\isa{\isacommand{lemma}}} given above.
+
+ \end{descr}
+
+ Any goal statement causes some term abbreviations (such as
+ \indexref{}{variable}{?thesis}\mbox{\isa{{\isacharquery}thesis}}) to be bound automatically, see also
+ \secref{sec:term-abbrev}. Furthermore, the local context of a
+ (non-atomic) goal is provided via the \indexref{}{case}{rule\_context}\mbox{\isa{rule{\isacharunderscore}context}} case.
+
+ The optional case names of \indexref{}{element}{obtains}\mbox{\isa{\isakeyword{obtains}}} have a twofold
+ meaning: (1) during the of this claim they refer to the the local
+ context introductions, (2) the resulting rule is annotated
+ accordingly to support symbolic case splits when used with the
+ \indexref{}{method}{cases}\mbox{\isa{cases}} method (cf. \secref{sec:cases-induct}).
+
+ \medskip
+
+ \begin{warn}
+ Isabelle/Isar suffers theory-level goal statements to contain
+ \emph{unbound schematic variables}, although this does not conform
+ to the aim of human-readable proof documents! The main problem
+ with schematic goals is that the actual outcome is usually hard to
+ predict, depending on the behavior of the proof methods applied
+ during the course of reasoning. Note that most semi-automated
+ methods heavily depend on several kinds of implicit rule
+ declarations within the current theory context. As this would
+ also result in non-compositional checking of sub-proofs,
+ \emph{local goals} are not allowed to be schematic at all.
+ Nevertheless, schematic goals do have their use in Prolog-style
+ interactive synthesis of proven results, usually by stepwise
+ refinement via emulation of traditional Isabelle tactic scripts
+ (see also \secref{sec:tactic-commands}). In any case, users
+ should know what they are doing.
+ \end{warn}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Initial and terminal proof steps \label{sec:proof-steps}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{}{command}{proof}\mbox{\isa{\isacommand{proof}}} & : & \isartrans{proof(prove)}{proof(state)} \\
+ \indexdef{}{command}{qed}\mbox{\isa{\isacommand{qed}}} & : & \isartrans{proof(state)}{proof(state) ~|~ theory} \\
+ \indexdef{}{command}{by}\mbox{\isa{\isacommand{by}}} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
+ \indexdef{}{command}{..}\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
+ \indexdef{}{command}{.}\mbox{\isa{\isacommand{{\isachardot}}}} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
+ \indexdef{}{command}{sorry}\mbox{\isa{\isacommand{sorry}}} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
+ \end{matharray}
+
+ Arbitrary goal refinement via tactics is considered harmful.
+ Structured proof composition in Isar admits proof methods to be
+ invoked in two places only.
+
+ \begin{enumerate}
+
+ \item An \emph{initial} refinement step \indexref{}{command}{proof}\mbox{\isa{\isacommand{proof}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}} reduces a newly stated goal to a number
+ of sub-goals that are to be solved later. Facts are passed to
+ \isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}} for forward chaining, if so indicated by \isa{{\isachardoublequote}proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} mode.
+
+ \item A \emph{terminal} conclusion step \indexref{}{command}{qed}\mbox{\isa{\isacommand{qed}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}} is intended to solve remaining goals. No facts are
+ passed to \isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}}.
+
+ \end{enumerate}
+
+ The only other (proper) way to affect pending goals in a proof body
+ is by \indexref{}{command}{show}\mbox{\isa{\isacommand{show}}}, which involves an explicit statement of
+ what is to be solved eventually. Thus we avoid the fundamental
+ problem of unstructured tactic scripts that consist of numerous
+ consecutive goal transformations, with invisible effects.
+
+ \medskip As a general rule of thumb for good proof style, 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
+ an intelligible manner.
+
+ Unless given explicitly by the user, the default initial method is
+ ``\indexref{}{method}{rule}\mbox{\isa{rule}}'', which applies a single standard elimination
+ or introduction rule according to the topmost symbol involved.
+ There is no separate default terminal method. Any remaining goals
+ are always solved by assumption in the very last step.
+
+ \begin{rail}
+ 'proof' method?
+ ;
+ 'qed' method?
+ ;
+ 'by' method method?
+ ;
+ ('.' | '..' | 'sorry')
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{proof}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}}] refines the goal by
+ proof method \isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}}; facts for forward chaining are
+ passed if so indicated by \isa{{\isachardoublequote}proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} mode.
+
+ \item [\mbox{\isa{\isacommand{qed}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}}] refines any remaining
+ goals by proof method \isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}} and concludes the
+ sub-proof by assumption. If the goal had been \isa{{\isachardoublequote}show{\isachardoublequote}} (or
+ \isa{{\isachardoublequote}thus{\isachardoublequote}}), some pending sub-goal is solved as well by the rule
+ resulting from the result \emph{exported} into the enclosing goal
+ context. Thus \isa{{\isachardoublequote}qed{\isachardoublequote}} may fail for two reasons: either \isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}} fails, or the resulting rule does not fit to any
+ pending goal\footnote{This includes any additional ``strong''
+ assumptions as introduced by \mbox{\isa{\isacommand{assume}}}.} of the enclosing
+ context. Debugging such a situation might involve temporarily
+ changing \mbox{\isa{\isacommand{show}}} into \mbox{\isa{\isacommand{have}}}, or weakening the
+ local context by replacing occurrences of \mbox{\isa{\isacommand{assume}}} by
+ \mbox{\isa{\isacommand{presume}}}.
+
+ \item [\mbox{\isa{\isacommand{by}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}\ m\isactrlsub {\isadigit{2}}{\isachardoublequote}}] is a
+ \emph{terminal proof}\index{proof!terminal}; it abbreviates
+ \mbox{\isa{\isacommand{proof}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}}~\isa{{\isachardoublequote}qed{\isachardoublequote}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}}, but with backtracking across both methods. Debugging
+ an unsuccessful \mbox{\isa{\isacommand{by}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}\ m\isactrlsub {\isadigit{2}}{\isachardoublequote}}
+ command can be done by expanding its definition; in many cases
+ \mbox{\isa{\isacommand{proof}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}} (or even \isa{{\isachardoublequote}apply{\isachardoublequote}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}}) is already sufficient to see the
+ problem.
+
+ \item [``\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}}''] is a \emph{default
+ proof}\index{proof!default}; it abbreviates \mbox{\isa{\isacommand{by}}}~\isa{{\isachardoublequote}rule{\isachardoublequote}}.
+
+ \item [``\mbox{\isa{\isacommand{{\isachardot}}}}''] is a \emph{trivial
+ proof}\index{proof!trivial}; it abbreviates \mbox{\isa{\isacommand{by}}}~\isa{{\isachardoublequote}this{\isachardoublequote}}.
+
+ \item [\mbox{\isa{\isacommand{sorry}}}] is a \emph{fake proof}\index{proof!fake}
+ pretending to solve the pending claim without further ado. This
+ only works in interactive development, or if the \verb|quick_and_dirty| flag is enabled (in ML). Facts emerging from fake
+ proofs are not the real thing. Internally, each theorem container
+ is tainted by an oracle invocation, which is indicated as ``\isa{{\isachardoublequote}{\isacharbrackleft}{\isacharbang}{\isacharbrackright}{\isachardoublequote}}'' in the printed result.
+
+ The most important application of \mbox{\isa{\isacommand{sorry}}} is to support
+ experimentation and top-down proof development.
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Fundamental methods and attributes \label{sec:pure-meth-att}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The following proof methods and attributes refer to basic logical
+ operations of Isar. Further methods and attributes are provided by
+ several generic and object-logic specific tools and packages (see
+ \chref{ch:gen-tools} and \chref{ch:hol}).
+
+ \begin{matharray}{rcl}
+ \indexdef{}{method}{-}\mbox{\isa{{\isacharminus}}} & : & \isarmeth \\
+ \indexdef{}{method}{fact}\mbox{\isa{fact}} & : & \isarmeth \\
+ \indexdef{}{method}{assumption}\mbox{\isa{assumption}} & : & \isarmeth \\
+ \indexdef{}{method}{this}\mbox{\isa{this}} & : & \isarmeth \\
+ \indexdef{}{method}{rule}\mbox{\isa{rule}} & : & \isarmeth \\
+ \indexdef{}{method}{iprover}\mbox{\isa{iprover}} & : & \isarmeth \\[0.5ex]
+ \indexdef{}{attribute}{intro}\mbox{\isa{intro}} & : & \isaratt \\
+ \indexdef{}{attribute}{elim}\mbox{\isa{elim}} & : & \isaratt \\
+ \indexdef{}{attribute}{dest}\mbox{\isa{dest}} & : & \isaratt \\
+ \indexdef{}{attribute}{rule}\mbox{\isa{rule}} & : & \isaratt \\[0.5ex]
+ \indexdef{}{attribute}{OF}\mbox{\isa{OF}} & : & \isaratt \\
+ \indexdef{}{attribute}{of}\mbox{\isa{of}} & : & \isaratt \\
+ \indexdef{}{attribute}{where}\mbox{\isa{where}} & : & \isaratt \\
+ \end{matharray}
+
+ \begin{rail}
+ 'fact' thmrefs?
+ ;
+ 'rule' thmrefs?
+ ;
+ 'iprover' ('!' ?) (rulemod *)
+ ;
+ rulemod: ('intro' | 'elim' | 'dest') ((('!' | () | '?') nat?) | 'del') ':' thmrefs
+ ;
+ ('intro' | 'elim' | 'dest') ('!' | () | '?') nat?
+ ;
+ 'rule' 'del'
+ ;
+ 'OF' thmrefs
+ ;
+ 'of' insts ('concl' ':' insts)?
+ ;
+ 'where' ((name | var | typefree | typevar) '=' (type | term) * 'and')
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [``\mbox{\isa{{\isacharminus}}}'' (minus)] does nothing but insert the
+ forward chaining facts as premises into the goal. Note that command
+ \indexref{}{command}{proof}\mbox{\isa{\isacommand{proof}}} without any method actually performs a single
+ reduction step using the \indexref{}{method}{rule}\mbox{\isa{rule}} method; thus a plain
+ \emph{do-nothing} proof step would be ``\mbox{\isa{\isacommand{proof}}}~\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' rather than \mbox{\isa{\isacommand{proof}}} alone.
+
+ \item [\mbox{\isa{fact}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}] composes
+ some fact from \isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ a\isactrlsub n{\isachardoublequote}} (or implicitly from
+ the current proof context) modulo unification of schematic type and
+ term variables. The rule structure is not taken into account, i.e.\
+ meta-level implication is considered atomic. This is the same
+ principle underlying literal facts (cf.\ \secref{sec:syn-att}):
+ ``\mbox{\isa{\isacommand{have}}}~\isa{{\isachardoublequote}{\isasymphi}{\isachardoublequote}}~\mbox{\isa{\isacommand{by}}}~\isa{fact}'' is
+ equivalent to ``\mbox{\isa{\isacommand{note}}}~\verb|`|\isa{{\isasymphi}}\verb|`|'' provided that \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymphi}{\isachardoublequote}} is an instance of some known
+ \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymphi}{\isachardoublequote}} in the proof context.
+
+ \item [\mbox{\isa{assumption}}] solves some goal by a single assumption
+ step. All given facts are guaranteed to participate in the
+ refinement; this means there may be only 0 or 1 in the first place.
+ Recall that \mbox{\isa{\isacommand{qed}}} (\secref{sec:proof-steps}) already
+ concludes any remaining sub-goals by assumption, so structured
+ proofs usually need not quote the \mbox{\isa{assumption}} method at
+ all.
+
+ \item [\mbox{\isa{this}}] applies all of the current facts directly as
+ rules. Recall that ``\mbox{\isa{\isacommand{{\isachardot}}}}'' (dot) abbreviates ``\mbox{\isa{\isacommand{by}}}~\isa{this}''.
+
+ \item [\mbox{\isa{rule}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}] applies some
+ rule given as argument in backward manner; facts are used to reduce
+ the rule before applying it to the goal. Thus \mbox{\isa{rule}}
+ without facts is plain introduction, while with facts it becomes
+ elimination.
+
+ When no arguments are given, the \mbox{\isa{rule}} method tries to pick
+ appropriate rules automatically, as declared in the current context
+ using the \mbox{\isa{intro}}, \mbox{\isa{elim}}, \mbox{\isa{dest}}
+ attributes (see below). This is the default behavior of \mbox{\isa{\isacommand{proof}}} and ``\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}}'' (double-dot) steps (see
+ \secref{sec:proof-steps}).
+
+ \item [\mbox{\isa{iprover}}] performs intuitionistic proof search,
+ depending on specifically declared rules from the context, or given
+ as explicit arguments. Chained facts are inserted into the goal
+ before commencing proof search; ``\mbox{\isa{iprover}}\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}''
+ means to include the current \mbox{\isa{prems}} as well.
+
+ Rules need to be classified as \mbox{\isa{intro}}, \mbox{\isa{elim}}, or \mbox{\isa{dest}}; here the ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' indicator
+ refers to ``safe'' rules, which may be applied aggressively (without
+ considering back-tracking later). Rules declared with ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' are ignored in proof search (the single-step \mbox{\isa{rule}}
+ method still observes these). An explicit weight annotation may be
+ given as well; otherwise the number of rule premises will be taken
+ into account here.
+
+ \item [\mbox{\isa{intro}}, \mbox{\isa{elim}}, and \mbox{\isa{dest}}]
+ declare introduction, elimination, and destruct rules, to be used
+ with the \mbox{\isa{rule}} and \mbox{\isa{iprover}} methods. Note that
+ the latter will ignore rules declared with ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'', while
+ ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' are used most aggressively.
+
+ The classical reasoner (see \secref{sec:classical}) introduces its
+ own variants of these attributes; use qualified names to access the
+ present versions of Isabelle/Pure, i.e.\ \mbox{\isa{Pure{\isachardot}intro}}.
+
+ \item [\mbox{\isa{rule}}~\isa{del}] undeclares introduction,
+ elimination, or destruct rules.
+
+ \item [\mbox{\isa{OF}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}}] applies some
+ theorem to all of the given rules \isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ a\isactrlsub n{\isachardoublequote}}
+ (in parallel). This corresponds to the \verb|"op MRS"| operation in
+ ML, but note the reversed order. Positions may be effectively
+ skipped by including ``\isa{{\isacharunderscore}}'' (underscore) as argument.
+
+ \item [\mbox{\isa{of}}~\isa{{\isachardoublequote}t\isactrlsub {\isadigit{1}}\ {\isasymdots}\ t\isactrlsub n{\isachardoublequote}}] performs
+ positional instantiation of term variables. The terms \isa{{\isachardoublequote}t\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ t\isactrlsub n{\isachardoublequote}} are substituted for any schematic
+ variables occurring in a theorem from left to right; ``\isa{{\isacharunderscore}}'' (underscore) indicates to skip a position. Arguments following
+ a ``\mbox{\isa{\isakeyword{concl}}}\isa{{\isachardoublequote}{\isacharcolon}{\isachardoublequote}}'' specification refer to positions
+ of the conclusion of a rule.
+
+ \item [\mbox{\isa{where}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isacharequal}\ t\isactrlsub {\isadigit{1}}\ {\isasymAND}\ {\isasymdots}\ x\isactrlsub n\ {\isacharequal}\ t\isactrlsub n{\isachardoublequote}}] performs named instantiation of schematic
+ type and term variables occurring in a theorem. Schematic variables
+ have to be specified on the left-hand side (e.g.\ \isa{{\isachardoublequote}{\isacharquery}x{\isadigit{1}}{\isachardot}{\isadigit{3}}{\isachardoublequote}}).
+ The question mark may be omitted if the variable name is a plain
+ identifier without index. As type instantiations are inferred from
+ term instantiations, explicit type instantiations are seldom
+ necessary.
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Term abbreviations \label{sec:term-abbrev}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{}{command}{let}\mbox{\isa{\isacommand{let}}} & : & \isartrans{proof(state)}{proof(state)} \\
+ \indexdef{}{keyword}{is}\mbox{\isa{\isakeyword{is}}} & : & syntax \\
+ \end{matharray}
+
+ Abbreviations may be either bound by explicit \mbox{\isa{\isacommand{let}}}~\isa{{\isachardoublequote}p\ {\isasymequiv}\ t{\isachardoublequote}} statements, or by annotating assumptions or
+ goal statements with a list of patterns ``\isa{{\isachardoublequote}{\isacharparenleft}{\isasymIS}\ p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub n{\isacharparenright}{\isachardoublequote}}''. In both cases, higher-order matching is invoked to
+ bind extra-logical term variables, which may be either named
+ schematic variables of the form \isa{{\isacharquery}x}, or nameless dummies
+ ``\mbox{\isa{{\isacharunderscore}}}'' (underscore). Note that in the \mbox{\isa{\isacommand{let}}}
+ form the patterns occur on the left-hand side, while the \mbox{\isa{\isakeyword{is}}} patterns are in postfix position.
+
+ Polymorphism of term bindings is handled in Hindley-Milner style,
+ similar to ML. Type variables referring to local assumptions or
+ open goal statements are \emph{fixed}, while those of finished
+ results or bound by \mbox{\isa{\isacommand{let}}} may occur in \emph{arbitrary}
+ instances later. Even though actual polymorphism should be rarely
+ used in practice, this mechanism is essential to achieve proper
+ incremental type-inference, as the user proceeds to build up the
+ Isar proof text from left to right.
+
+ \medskip Term abbreviations are quite different from local
+ definitions as introduced via \mbox{\isa{\isacommand{def}}} (see
+ \secref{sec:proof-context}). The latter are visible within the
+ logic as actual equations, while abbreviations disappear during the
+ input process just after type checking. Also note that \mbox{\isa{\isacommand{def}}} does not support polymorphism.
+
+ \begin{rail}
+ 'let' ((term + 'and') '=' term + 'and')
+ ;
+ \end{rail}
+
+ The syntax of \mbox{\isa{\isakeyword{is}}} patterns follows \railnonterm{termpat}
+ or \railnonterm{proppat} (see \secref{sec:term-decls}).
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{let}}}~\isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}\ {\isacharequal}\ t\isactrlsub {\isadigit{1}}\ {\isasymAND}\ {\isasymdots}\ p\isactrlsub n\ {\isacharequal}\ t\isactrlsub n{\isachardoublequote}}] binds any text variables in patterns \isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ p\isactrlsub n{\isachardoublequote}} by simultaneous higher-order matching
+ against terms \isa{{\isachardoublequote}t\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ t\isactrlsub n{\isachardoublequote}}.
+
+ \item [\isa{{\isachardoublequote}{\isacharparenleft}{\isasymIS}\ p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub n{\isacharparenright}{\isachardoublequote}}] resembles \mbox{\isa{\isacommand{let}}}, but matches \isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ p\isactrlsub n{\isachardoublequote}} against the
+ preceding statement. Also note that \mbox{\isa{\isakeyword{is}}} is not a
+ separate command, but part of others (such as \mbox{\isa{\isacommand{assume}}},
+ \mbox{\isa{\isacommand{have}}} etc.).
+
+ \end{descr}
+
+ Some \emph{implicit} term abbreviations\index{term abbreviations}
+ for goals and facts are available as well. For any open goal,
+ \indexref{}{variable}{thesis}\mbox{\isa{thesis}} refers to its object-level statement,
+ abstracted over any meta-level parameters (if present). Likewise,
+ \indexref{}{variable}{this}\mbox{\isa{this}} is bound for fact statements resulting from
+ assumptions or finished goals. In case \mbox{\isa{this}} refers to
+ an object-logic statement that is an application \isa{{\isachardoublequote}f\ t{\isachardoublequote}}, then
+ \isa{t} is bound to the special text variable ``\mbox{\isa{{\isasymdots}}}''
+ (three dots). The canonical application of this convenience are
+ calculational proofs (see \secref{sec:calculation}).%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Block structure%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{}{command}{next}\mbox{\isa{\isacommand{next}}} & : & \isartrans{proof(state)}{proof(state)} \\
+ \indexdef{}{command}{\{}\mbox{\isa{\isacommand{{\isacharbraceleft}}}} & : & \isartrans{proof(state)}{proof(state)} \\
+ \indexdef{}{command}{\}}\mbox{\isa{\isacommand{{\isacharbraceright}}}} & : & \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} internal blocks, which are closed again when concluding
+ the sub-proof (by \mbox{\isa{\isacommand{qed}}} etc.). Sections of different
+ context within a sub-proof may be switched via \mbox{\isa{\isacommand{next}}},
+ which is just a single block-close followed by block-open again.
+ The effect of \mbox{\isa{\isacommand{next}}} is to reset the local proof context;
+ 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 [\mbox{\isa{\isacommand{next}}}] switches to a fresh block within a
+ sub-proof, resetting the local context to the initial one.
+
+ \item [\mbox{\isa{\isacommand{{\isacharbraceleft}}}} and \mbox{\isa{\isacommand{{\isacharbraceright}}}}] explicitly open and close
+ blocks. Any current facts pass through ``\mbox{\isa{\isacommand{{\isacharbraceleft}}}}''
+ unchanged, while ``\mbox{\isa{\isacommand{{\isacharbraceright}}}}'' causes any result to be
+ \emph{exported} into the enclosing context. Thus fixed variables
+ are generalized, assumptions discharged, and local definitions
+ unfolded (cf.\ \secref{sec:proof-context}). There is no difference
+ of \mbox{\isa{\isacommand{assume}}} and \mbox{\isa{\isacommand{presume}}} in this mode of
+ forward reasoning --- in contrast to plain backward reasoning with
+ the result exported at \mbox{\isa{\isacommand{show}}} time.
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Emulating tactic scripts \label{sec:tactic-commands}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The Isar provides separate commands to accommodate tactic-style
+ proof scripts within the same system. While being outside the
+ orthodox Isar proof language, these might come in handy for
+ interactive exploration and debugging, or even actual tactical proof
+ within new-style theories (to benefit from document preparation, for
+ example). See also \secref{sec:tactics} for actual tactics, that
+ have been encapsulated as proof methods. Proper proof methods may
+ be used in scripts, too.
+
+ \begin{matharray}{rcl}
+ \indexdef{}{command}{apply}\mbox{\isa{\isacommand{apply}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{proof(prove)}{proof(prove)} \\
+ \indexdef{}{command}{apply\_end}\mbox{\isa{\isacommand{apply{\isacharunderscore}end}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{proof(state)}{proof(state)} \\
+ \indexdef{}{command}{done}\mbox{\isa{\isacommand{done}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{proof(prove)}{proof(state)} \\
+ \indexdef{}{command}{defer}\mbox{\isa{\isacommand{defer}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{proof}{proof} \\
+ \indexdef{}{command}{prefer}\mbox{\isa{\isacommand{prefer}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{proof}{proof} \\
+ \indexdef{}{command}{back}\mbox{\isa{\isacommand{back}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{proof}{proof} \\
+ \end{matharray}
+
+ \begin{rail}
+ ( 'apply' | 'apply\_end' ) method
+ ;
+ 'defer' nat?
+ ;
+ 'prefer' nat
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{apply}}}~\isa{m}] applies proof method \isa{m}
+ in initial position, but unlike \mbox{\isa{\isacommand{proof}}} it retains
+ ``\isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}}'' mode. Thus consecutive method
+ applications may be given just as in tactic scripts.
+
+ Facts are passed to \isa{m} as indicated by the goal's
+ forward-chain mode, and are \emph{consumed} afterwards. Thus any
+ further \mbox{\isa{\isacommand{apply}}} command would always work in a purely
+ backward manner.
+
+ \item [\mbox{\isa{\isacommand{apply{\isacharunderscore}end}}}~\isa{{\isachardoublequote}m{\isachardoublequote}}] applies proof method
+ \isa{m} as if in terminal position. Basically, this simulates a
+ multi-step tactic script for \mbox{\isa{\isacommand{qed}}}, but may be given
+ anywhere within the proof body.
+
+ No facts are passed to \mbox{\isa{m}} here. Furthermore, the static
+ context is that of the enclosing goal (as for actual \mbox{\isa{\isacommand{qed}}}). Thus the proof method may not refer to any assumptions
+ introduced in the current body, for example.
+
+ \item [\mbox{\isa{\isacommand{done}}}] completes a proof script, provided that
+ the current goal state is solved completely. Note that actual
+ structured proof commands (e.g.\ ``\mbox{\isa{\isacommand{{\isachardot}}}}'' or \mbox{\isa{\isacommand{sorry}}}) may be used to conclude proof scripts as well.
+
+ \item [\mbox{\isa{\isacommand{defer}}}~\isa{n} and \mbox{\isa{\isacommand{prefer}}}~\isa{n}] shuffle the list of pending goals: \mbox{\isa{\isacommand{defer}}} puts off
+ sub-goal \isa{n} to the end of the list (\isa{{\isachardoublequote}n\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequote}} by
+ default), while \mbox{\isa{\isacommand{prefer}}} brings sub-goal \isa{n} to the
+ front.
+
+ \item [\mbox{\isa{\isacommand{back}}}] does back-tracking over the result
+ sequence of the latest proof command. Basically, any proof command
+ may return multiple results.
+
+ \end{descr}
+
+ Any proper Isar proof method may be used with tactic script commands
+ such as \mbox{\isa{\isacommand{apply}}}. A few additional emulations of actual
+ tactics are provided as well; these would be never used in actual
+ structured proofs, of course.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Omitting proofs%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{}{command}{oops}\mbox{\isa{\isacommand{oops}}} & : & \isartrans{proof}{theory} \\
+ \end{matharray}
+
+ The \mbox{\isa{\isacommand{oops}}} command discontinues the current proof
+ attempt, while considering the partial proof text as properly
+ processed. This is conceptually quite different from ``faking''
+ actual proofs via \indexref{}{command}{sorry}\mbox{\isa{\isacommand{sorry}}} (see
+ \secref{sec:proof-steps}): \mbox{\isa{\isacommand{oops}}} does not observe the
+ proof structure at all, but goes back right to the theory level.
+ Furthermore, \mbox{\isa{\isacommand{oops}}} does not produce any result theorem
+ --- there is no intended claim to be able to complete the proof
+ anyhow.
+
+ A typical application of \mbox{\isa{\isacommand{oops}}} is to explain Isar proofs
+ \emph{within} the system itself, in conjunction with the document
+ preparation tools of Isabelle described in \cite{isabelle-sys}.
+ Thus partial or even wrong proof attempts can be discussed in a
+ logically sound manner. Note that the Isabelle {\LaTeX} macros can
+ be easily adapted to print something like ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' instead of
+ the keyword ``\mbox{\isa{\isacommand{oops}}}''.
+
+ \medskip The \mbox{\isa{\isacommand{oops}}} command is undo-able, unlike
+ \indexref{}{command}{kill}\mbox{\isa{\isacommand{kill}}} (see \secref{sec:history}). The effect is to
+ get back to the theory just before the opening of the proof.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Generalized elimination \label{sec:obtain}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{}{command}{obtain}\mbox{\isa{\isacommand{obtain}}} & : & \isartrans{proof(state)}{proof(prove)} \\
+ \indexdef{}{command}{guess}\mbox{\isa{\isacommand{guess}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{proof(state)}{proof(prove)} \\
+ \end{matharray}
+
+ Generalized elimination means that additional elements with certain
+ properties may be introduced in the current context, by virtue of a
+ locally proven ``soundness statement''. Technically speaking, the
+ \mbox{\isa{\isacommand{obtain}}} language element is like a declaration of
+ \mbox{\isa{\isacommand{fix}}} and \mbox{\isa{\isacommand{assume}}} (see also see
+ \secref{sec:proof-context}), together with a soundness proof of its
+ additional claim. According to the nature of existential reasoning,
+ assumptions get eliminated from any result exported from the context
+ later, provided that the corresponding parameters do \emph{not}
+ occur in the conclusion.
+
+ \begin{rail}
+ 'obtain' parname? (vars + 'and') 'where' (props + 'and')
+ ;
+ 'guess' (vars + 'and')
+ ;
+ \end{rail}
+
+ The derived Isar command \mbox{\isa{\isacommand{obtain}}} is defined as follows
+ (where \isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k{\isachardoublequote}} shall refer to (optional)
+ facts indicated for forward chaining).
+ \begin{matharray}{l}
+ \isa{{\isachardoublequote}{\isasymlangle}using\ b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub k{\isasymrangle}{\isachardoublequote}}~~\mbox{\isa{\isacommand{obtain}}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m\ {\isasymWHERE}\ a{\isacharcolon}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ {\isasymphi}\isactrlsub n\ \ {\isasymlangle}proof{\isasymrangle}\ {\isasymequiv}{\isachardoublequote}} \\[1ex]
+ \quad \mbox{\isa{\isacommand{have}}}~\isa{{\isachardoublequote}{\isasymAnd}thesis{\isachardot}\ {\isacharparenleft}{\isasymAnd}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardot}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymphi}\isactrlsub n\ {\isasymLongrightarrow}\ thesis{\isacharparenright}\ {\isasymLongrightarrow}\ thesis{\isachardoublequote}} \\
+ \quad \mbox{\isa{\isacommand{proof}}}~\isa{succeed} \\
+ \qquad \mbox{\isa{\isacommand{fix}}}~\isa{thesis} \\
+ \qquad \mbox{\isa{\isacommand{assume}}}~\isa{{\isachardoublequote}that\ {\isacharbrackleft}Pure{\isachardot}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isasymAnd}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardot}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymphi}\isactrlsub n\ {\isasymLongrightarrow}\ thesis{\isachardoublequote}} \\
+ \qquad \mbox{\isa{\isacommand{then}}}~\mbox{\isa{\isacommand{show}}}~\isa{thesis} \\
+ \quad\qquad \mbox{\isa{\isacommand{apply}}}~\isa{{\isacharminus}} \\
+ \quad\qquad \mbox{\isa{\isacommand{using}}}~\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub k\ \ {\isasymlangle}proof{\isasymrangle}{\isachardoublequote}} \\
+ \quad \mbox{\isa{\isacommand{qed}}} \\
+ \quad \mbox{\isa{\isacommand{fix}}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardoublequote}}~\mbox{\isa{\isacommand{assume}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}\ a{\isacharcolon}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ {\isasymphi}\isactrlsub n{\isachardoublequote}} \\
+ \end{matharray}
+
+ Typically, the soundness proof is relatively straight-forward, often
+ just by canonical automated tools such as ``\mbox{\isa{\isacommand{by}}}~\isa{simp}'' or ``\mbox{\isa{\isacommand{by}}}~\isa{blast}''. Accordingly, the
+ ``\isa{that}'' reduction above is declared as simplification and
+ introduction rule.
+
+ In a sense, \mbox{\isa{\isacommand{obtain}}} represents at the level of Isar
+ proofs what would be meta-logical existential quantifiers and
+ conjunctions. This concept has a broad range of useful
+ applications, ranging from plain elimination (or introduction) of
+ object-level existential and conjunctions, to elimination over
+ results of symbolic evaluation of recursive definitions, for
+ example. Also note that \mbox{\isa{\isacommand{obtain}}} without parameters acts
+ much like \mbox{\isa{\isacommand{have}}}, where the result is treated as a
+ genuine assumption.
+
+ An alternative name to be used instead of ``\isa{that}'' above may
+ be given in parentheses.
+
+ \medskip The improper variant \mbox{\isa{\isacommand{guess}}} is similar to
+ \mbox{\isa{\isacommand{obtain}}}, but derives the obtained statement from the
+ course of reasoning! The proof starts with a fixed goal \isa{thesis}. The subsequent proof may refine this to anything of the
+ form like \isa{{\isachardoublequote}{\isasymAnd}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardot}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymphi}\isactrlsub n\ {\isasymLongrightarrow}\ thesis{\isachardoublequote}}, but must not introduce new subgoals. The
+ final goal state is then used as reduction rule for the obtain
+ scheme described above. Obtained parameters \isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlsub m{\isachardoublequote}} are marked as internal by default, which prevents the
+ proof context from being polluted by ad-hoc variables. The variable
+ names and type constraints given as arguments for \mbox{\isa{\isacommand{guess}}}
+ specify a prefix of obtained parameters explicitly in the text.
+
+ It is important to note that the facts introduced by \mbox{\isa{\isacommand{obtain}}} and \mbox{\isa{\isacommand{guess}}} may not be polymorphic: any
+ type-variables occurring here are fixed in the present context!%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Calculational reasoning \label{sec:calculation}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{}{command}{also}\mbox{\isa{\isacommand{also}}} & : & \isartrans{proof(state)}{proof(state)} \\
+ \indexdef{}{command}{finally}\mbox{\isa{\isacommand{finally}}} & : & \isartrans{proof(state)}{proof(chain)} \\
+ \indexdef{}{command}{moreover}\mbox{\isa{\isacommand{moreover}}} & : & \isartrans{proof(state)}{proof(state)} \\
+ \indexdef{}{command}{ultimately}\mbox{\isa{\isacommand{ultimately}}} & : & \isartrans{proof(state)}{proof(chain)} \\
+ \indexdef{}{command}{print\_trans\_rules}\mbox{\isa{\isacommand{print{\isacharunderscore}trans{\isacharunderscore}rules}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
+ \mbox{\isa{trans}} & : & \isaratt \\
+ \mbox{\isa{sym}} & : & \isaratt \\
+ \mbox{\isa{symmetric}} & : & \isaratt \\
+ \end{matharray}
+
+ Calculational proof is forward reasoning with implicit application
+ of transitivity rules (such those of \isa{{\isachardoublequote}{\isacharequal}{\isachardoublequote}}, \isa{{\isachardoublequote}{\isasymle}{\isachardoublequote}},
+ \isa{{\isachardoublequote}{\isacharless}{\isachardoublequote}}). Isabelle/Isar maintains an auxiliary fact register
+ \indexref{}{fact}{calculation}\mbox{\isa{calculation}} for accumulating results obtained by
+ transitivity composed with the current result. Command \mbox{\isa{\isacommand{also}}} updates \mbox{\isa{calculation}} involving \mbox{\isa{this}}, while
+ \mbox{\isa{\isacommand{finally}}} exhibits the final \mbox{\isa{calculation}} by
+ forward chaining towards the next goal statement. Both commands
+ require valid current facts, i.e.\ may occur only after commands
+ that produce theorems such as \mbox{\isa{\isacommand{assume}}}, \mbox{\isa{\isacommand{note}}}, or some finished proof of \mbox{\isa{\isacommand{have}}}, \mbox{\isa{\isacommand{show}}} etc. The \mbox{\isa{\isacommand{moreover}}} and \mbox{\isa{\isacommand{ultimately}}}
+ commands are similar to \mbox{\isa{\isacommand{also}}} and \mbox{\isa{\isacommand{finally}}},
+ but only collect further results in \mbox{\isa{calculation}} without
+ applying any rules yet.
+
+ Also note that the implicit term abbreviation ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' has
+ its canonical application with calculational proofs. It refers to
+ the argument of the preceding statement. (The argument of a curried
+ infix expression happens to be its right-hand side.)
+
+ Isabelle/Isar calculations are implicitly subject to block structure
+ in the sense that new threads of calculational reasoning are
+ commenced for any new block (as opened by a local goal, for
+ example). This means that, apart from being able to nest
+ calculations, there is no separate \emph{begin-calculation} command
+ required.
+
+ \medskip The Isar calculation proof commands may be defined as
+ follows:\footnote{We suppress internal bookkeeping such as proper
+ handling of block-structure.}
+
+ \begin{matharray}{rcl}
+ \mbox{\isa{\isacommand{also}}}\isa{{\isachardoublequote}\isactrlsub {\isadigit{0}}{\isachardoublequote}} & \equiv & \mbox{\isa{\isacommand{note}}}~\isa{{\isachardoublequote}calculation\ {\isacharequal}\ this{\isachardoublequote}} \\
+ \mbox{\isa{\isacommand{also}}}\isa{{\isachardoublequote}\isactrlsub n\isactrlsub {\isacharplus}\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \equiv & \mbox{\isa{\isacommand{note}}}~\isa{{\isachardoublequote}calculation\ {\isacharequal}\ trans\ {\isacharbrackleft}OF\ calculation\ this{\isacharbrackright}{\isachardoublequote}} \\[0.5ex]
+ \mbox{\isa{\isacommand{finally}}} & \equiv & \mbox{\isa{\isacommand{also}}}~\mbox{\isa{\isacommand{from}}}~\isa{calculation} \\[0.5ex]
+ \mbox{\isa{\isacommand{moreover}}} & \equiv & \mbox{\isa{\isacommand{note}}}~\isa{{\isachardoublequote}calculation\ {\isacharequal}\ calculation\ this{\isachardoublequote}} \\
+ \mbox{\isa{\isacommand{ultimately}}} & \equiv & \mbox{\isa{\isacommand{moreover}}}~\mbox{\isa{\isacommand{from}}}~\isa{calculation} \\
+ \end{matharray}
+
+ \begin{rail}
+ ('also' | 'finally') ('(' thmrefs ')')?
+ ;
+ 'trans' (() | 'add' | 'del')
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{also}}}~\isa{{\isachardoublequote}{\isacharparenleft}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isacharparenright}{\isachardoublequote}}]
+ maintains the auxiliary \mbox{\isa{calculation}} register as follows.
+ The first occurrence of \mbox{\isa{\isacommand{also}}} in some calculational
+ thread initializes \mbox{\isa{calculation}} by \mbox{\isa{this}}. Any
+ subsequent \mbox{\isa{\isacommand{also}}} on the same level of block-structure
+ updates \mbox{\isa{calculation}} by some transitivity rule applied to
+ \mbox{\isa{calculation}} and \mbox{\isa{this}} (in that order). Transitivity
+ rules are picked from the current context, unless alternative rules
+ are given as explicit arguments.
+
+ \item [\mbox{\isa{\isacommand{finally}}}~\isa{{\isachardoublequote}{\isacharparenleft}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isacharparenright}{\isachardoublequote}}]
+ maintaining \mbox{\isa{calculation}} in the same way as \mbox{\isa{\isacommand{also}}}, and concludes the current calculational thread. The final
+ result is exhibited as fact for forward chaining towards the next
+ goal. Basically, \mbox{\isa{\isacommand{finally}}} just abbreviates \mbox{\isa{\isacommand{also}}}~\mbox{\isa{\isacommand{from}}}~\mbox{\isa{calculation}}. Typical idioms for
+ concluding calculational proofs are ``\mbox{\isa{\isacommand{finally}}}~\mbox{\isa{\isacommand{show}}}~\isa{{\isacharquery}thesis}~\mbox{\isa{\isacommand{{\isachardot}}}}'' and ``\mbox{\isa{\isacommand{finally}}}~\mbox{\isa{\isacommand{have}}}~\isa{{\isasymphi}}~\mbox{\isa{\isacommand{{\isachardot}}}}''.
+
+ \item [\mbox{\isa{\isacommand{moreover}}} and \mbox{\isa{\isacommand{ultimately}}}] are
+ analogous to \mbox{\isa{\isacommand{also}}} and \mbox{\isa{\isacommand{finally}}}, but collect
+ results only, without applying rules.
+
+ \item [\mbox{\isa{\isacommand{print{\isacharunderscore}trans{\isacharunderscore}rules}}}] prints the list of
+ transitivity rules (for calculational commands \mbox{\isa{\isacommand{also}}} and
+ \mbox{\isa{\isacommand{finally}}}) and symmetry rules (for the \mbox{\isa{symmetric}} operation and single step elimination patters) of the
+ current context.
+
+ \item [\mbox{\isa{trans}}] declares theorems as transitivity rules.
+
+ \item [\mbox{\isa{sym}}] declares symmetry rules, as well as
+ \mbox{\isa{Pure{\isachardot}elim{\isacharquery}}} rules.
+
+ \item [\mbox{\isa{symmetric}}] resolves a theorem with some rule
+ declared as \mbox{\isa{sym}} in the current context. For example,
+ ``\mbox{\isa{\isacommand{assume}}}~\isa{{\isachardoublequote}{\isacharbrackleft}symmetric{\isacharbrackright}{\isacharcolon}\ x\ {\isacharequal}\ y{\isachardoublequote}}'' produces a
+ swapped fact derived from that assumption.
+
+ In structured proof texts it is often more appropriate to use an
+ explicit single-step elimination proof, such as ``\mbox{\isa{\isacommand{assume}}}~\isa{{\isachardoublequote}x\ {\isacharequal}\ y{\isachardoublequote}}~\mbox{\isa{\isacommand{then}}}~\mbox{\isa{\isacommand{have}}}~\isa{{\isachardoublequote}y\ {\isacharequal}\ x{\isachardoublequote}}~\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}}''.
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
\isadelimtheory
%
\endisadelimtheory