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-\chapter[Proof handling]{Proof handling\index{Proof editing}
-\label{Proof-handling}}
-
-In \Coq's proof editing mode all top-level commands documented in 
-Chapter~\ref{Vernacular-commands} remain available
-and the user has access to specialized commands dealing with proof
-development pragmas documented in this section. He can also use some
-other specialized commands called {\em tactics}.  They are the very
-tools allowing the user to deal with logical reasoning. They are
-documented in Chapter~\ref{Tactics}.\\ 
-When switching in editing proof mode, the prompt
-\index{Prompt} 
-{\tt Coq <} is changed into {\tt {\ident} <} where {\ident} is the
-declared name of the theorem currently edited.
-
-At each stage of a proof development, one has a list of goals to
-prove. Initially, the list consists only in the theorem itself. After
-having applied some tactics, the list of goals contains the subgoals
-generated by the tactics.
-
-To each subgoal is associated  a number of
-hypotheses called the {\em \index*{local context}} of the goal.
-Initially, the local context contains the local variables and
-hypotheses of the current section (see Section~\ref{Variable}) and the
-local variables and hypotheses of the theorem statement.  It is
-enriched by the use of certain tactics (see e.g. {\tt intro} in
-Section~\ref{intro}).
-
-When a proof is completed, the message {\tt Proof completed} is
-displayed. One can then register this proof as a defined constant in the
-environment. Because there exists a correspondence between proofs and
-terms of $\lambda$-calculus, known as the {\em Curry-Howard
-isomorphism} \cite{How80,Bar91,Gir89,Hue89}, \Coq~ stores proofs as
-terms of {\sc Cic}. Those terms are called {\em proof
-  terms}\index{Proof term}.
-
-It is possible to edit several proofs in parallel: see Section
-\ref{Resume}.
-
-\ErrMsg When one attempts to use a proof editing command out of the
-proof editing mode, \Coq~ raises the error message : \errindex{No focused
-  proof}.
-
-\section{Switching on/off the proof editing mode}
-
-The proof editing mode is entered by asserting a statement, which
-typically is the assertion of a theorem:
-
-\begin{quote}
-{\tt Theorem {\ident} \zeroone{\binders} : {\form}.\comindex{Theorem}
-\label{Theorem}}
-\end{quote}
-
-The list of assertion commands is given in
-Section~\ref{Assertions}. The command {\tt Goal} can also be used.
-
-\subsection[Goal {\form}.]{\tt Goal {\form}.\comindex{Goal}\label{Goal}}
-
-This is intended for quick assertion of statements, without knowing in
-advance which name to give to the assertion, typically for quick
-testing of the provability of a statement. If the proof of the
-statement is eventually completed and validated, the statement is then
-bound to the name {\tt Unnamed\_thm} (or a variant of this name not
-already used for another statement).
-
-\subsection[\tt Qed.]{\tt Qed.\comindex{Qed}\label{Qed}}
-This command is available in interactive editing proof mode when the
-proof is completed.  Then {\tt Qed} extracts a proof term from the
-proof script, switches back to {\Coq} top-level and attaches the
-extracted proof term to the declared name of the original goal. This
-name is added to the environment as an {\tt Opaque} constant.
-
-\begin{ErrMsgs}
-\item \errindex{Attempt to save an incomplete proof}
-%\item \ident\ \errindex{already exists}\\ 
-%  The implicit name is already defined. You have then to provide
-%  explicitly a new name (see variant 3 below).
-\item Sometimes an error occurs when building the proof term,
-because tactics do not enforce completely the term construction
-constraints.
-
-The user should also be aware of the fact that since the proof term is
-completely rechecked at this point, one may have to wait a while when
-the proof is large. In some exceptional cases one may even incur a
-memory overflow.
-\end{ErrMsgs}
-
-\begin{Variants}
-
-\item {\tt Defined.}
-\comindex{Defined} 
-\label{Defined} 
-
-  Defines the proved term as a transparent constant.
-
-\item {\tt Save.}
-\comindex{Save}
-
-  This is a deprecated equivalent to {\tt Qed}.
-
-\item {\tt Save {\ident}.}
-  
-  Forces the name of the original goal to be {\ident}.  This command
-  (and the following ones) can only be used if the original goal has
-  been opened using the {\tt Goal} command.
-
-\item {\tt Save Theorem {\ident}.} \\
- {\tt Save Lemma {\ident}.} \\
- {\tt Save Remark {\ident}.}\\
- {\tt Save Fact {\ident}.}
- {\tt Save Corollary {\ident}.}
- {\tt Save Proposition {\ident}.}
-
-  Are equivalent to {\tt Save {\ident}.} 
-\end{Variants}
-
-\subsection[\tt Admitted.]{\tt Admitted.\comindex{Admitted}\label{Admitted}}
-This command is available in interactive editing proof mode to give up
-the current proof and declare the initial goal as an axiom.
-
-\subsection[\tt Proof {\term}.]{\tt Proof {\term}.\comindex{Proof}
-\label{BeginProof}}
-This command applies in proof editing mode. It is equivalent to {\tt
-  exact {\term}; Save.} That is, you have to give the full proof in
-one gulp, as a proof term (see Section~\ref{exact}).
-
-\variant {\tt Proof.}
-  
-  Is a noop which is useful to delimit the sequence of tactic commands
-  which start a proof, after a {\tt Theorem} command.  It is a good
-  practice to use {\tt Proof.} as an opening parenthesis, closed in
-  the script with a closing {\tt Qed.}
-
-\SeeAlso {\tt Proof with {\tac}.} in Section~\ref{ProofWith}.
-
-\subsection[\tt Abort.]{\tt Abort.\comindex{Abort}}
-
-This command cancels the current proof development, switching back to
-the previous proof development, or to the \Coq\ toplevel if no other
-proof was edited.
-
-\begin{ErrMsgs}
-\item \errindex{No focused proof (No proof-editing in progress)}
-\end{ErrMsgs}
-
-\begin{Variants}
-
-\item {\tt Abort {\ident}.}
-
-  Aborts the editing of the proof named {\ident}.
-
-\item {\tt Abort All.}
-
-  Aborts all current goals, switching back to the \Coq\ toplevel.
-
-\end{Variants}
-
-%%%%
-\subsection[\tt Suspend.]{\tt Suspend.\comindex{Suspend}}
-
-This command applies in proof editing mode. It switches back to the
-\Coq\ toplevel, but without canceling the current proofs.
-
-%%%%
-\subsection[\tt Resume.]{\tt Resume.\comindex{Resume}\label{Resume}}
-
-This commands switches back to the editing of the last edited proof.
-
-\begin{ErrMsgs}
-\item \errindex{No proof-editing in progress}
-\end{ErrMsgs}
-
-\begin{Variants}
-
-\item {\tt Resume {\ident}.}
-
-  Restarts the editing of the proof named {\ident}. This can be used
-  to navigate between currently edited proofs.
-
-\end{Variants}
-
-\begin{ErrMsgs}
-\item \errindex{No such proof}
-\end{ErrMsgs}
-
-
-%%%%
-\subsection[\tt Existential {\num} := {\term}.]{\tt Existential  {\num} := {\term}.\comindex{Existential}
-\label{Existential}}
-
-This command allows to instantiate an existential variable. {\tt \num}
-is an index in the list of uninstantiated existential variables
-displayed by {\tt Show Existentials.} (described in Section~\ref{Show})
-
-This command is intended to be used to instantiate existential
-variables when the proof is completed but some uninstantiated
-existential variables remain. To instantiate existential variables
-during proof edition, you should use the tactic {\tt instantiate}.
-
-\SeeAlso {\tt instantiate (\num:= \term).} in Section~\ref{instantiate}.
-
-
-%%%%%%%%
-\section{Navigation in the proof tree}
-%%%%%%%%
-
-\subsection[\tt Undo.]{\tt Undo.\comindex{Undo}}
-
-This command cancels the effect of the last tactic command.  Thus, it
-backtracks one step.
-
-\begin{ErrMsgs}
-\item \errindex{No focused proof (No proof-editing in progress)}
-\item \errindex{Undo stack would be exhausted}
-\end{ErrMsgs}
-
-\begin{Variants}
-
-\item {\tt Undo {\num}.}
-
-  Repeats {\tt Undo} {\num} times.
-
-\end{Variants}
-
-\subsection[\tt Set Undo {\num}.]{\tt Set Undo {\num}.\comindex{Set Undo}}
-
-This command changes the maximum number of {\tt Undo}'s that will be
-possible when doing a proof. It only affects proofs started after
-this command, such that if you want to change the current undo limit
-inside a proof, you should first restart this proof.
-
-\subsection[\tt Unset Undo.]{\tt Unset Undo.\comindex{Unset Undo}}
-
-This command resets the default number of possible {\tt Undo} commands
-(which is currently 12).
-
-\subsection[\tt Restart.]{\tt Restart.\comindex{Restart}}
-This command restores the proof editing process to the original goal.
-
-\begin{ErrMsgs}
-\item \errindex{No focused proof to restart}
-\end{ErrMsgs}
-
-\subsection[\tt Focus.]{\tt Focus.\comindex{Focus}}
-This focuses the attention on the first subgoal to prove and the printing
-of the other subgoals is suspended until the focused subgoal is
-solved or unfocused. This is useful when there are many current
-subgoals which clutter your screen.
-
-\begin{Variant}
-\item {\tt Focus {\num}.}\\ 
-This focuses the attention on the $\num^{th}$ subgoal to prove.
-
-\end{Variant}
-
-\subsection[\tt Unfocus.]{\tt Unfocus.\comindex{Unfocus}}
-Turns off the focus mode.
-
-
-\section{Requesting information}
-
-\subsection[\tt Show.]{\tt Show.\comindex{Show}\label{Show}}
-This command displays the current goals.
-
-\begin{Variants}
-\item {\tt Show {\num}.}\\ 
-  Displays only the {\num}-th subgoal.\\ 
-\begin{ErrMsgs}
-\item \errindex{No such goal}
-\item \errindex{No focused proof}
-\end{ErrMsgs}
-
-\item {\tt Show Implicits.}\comindex{Show Implicits}\\
-  Displays the current goals, printing the implicit arguments of
-  constants.
-
-\item {\tt Show Implicits {\num}.}\\
-  Same as above, only displaying the {\num}-th subgoal.
-
-\item {\tt Show Script.}\comindex{Show Script}\\
-  Displays the whole list of tactics applied from the beginning
-  of the current proof. 
-  This tactics script may contain some holes (subgoals not yet proved).
-  They are printed under the form \verb!<Your Tactic Text here>!.
-
-\item {\tt Show Tree.}\comindex{Show Tree}\\
-This command can be seen as a more structured way of
-displaying the state of the proof than that 
-provided by {\tt Show Script}. Instead of just giving
-the list of tactics that have been applied, it 
-shows the derivation tree constructed by then. 
-Each node of the tree contains the conclusion
-of the corresponding sub-derivation (i.e. a
-goal with its corresponding local context) and 
-the tactic that has generated all the 
-sub-derivations. The leaves of this tree are
-the goals which still remain to be proved.
-
-%\item {\tt Show Node}\comindex{Show Node}\\
-%        Not yet documented
-
-\item {\tt Show Proof.}\comindex{Show Proof}\\
-It displays the proof term generated by the 
-tactics that have been applied. 
-If the proof is not completed, this term contain holes,
-which correspond to the sub-terms which are still to be 
-constructed. These holes appear as a question mark indexed 
-by an integer, and applied to the list of variables in 
-the context, since it may depend on them. 
-The types obtained by abstracting away the context from the
-type of each hole-placer are also printed.
-
-\item {\tt Show Conjectures.}\comindex{Show Conjectures}\\
-It prints the list of the names of all the theorems that 
-are currently being proved.
-As it is possible to start proving a previous lemma during
-the proof of a theorem, this list may contain several 
-names. 
-
-\item{\tt Show Intro.}\comindex{Show Intro}\\
-If the current goal begins by at least one product, this command
-prints the name of the first product, as it would be generated by 
-an anonymous {\tt Intro}. The aim of this command is to ease the
-writing of more robust scripts. For example, with an appropriate 
-Proof General macro, it is possible to transform any anonymous {\tt
-  Intro} into a qualified one such as {\tt Intro y13}.
-In the case of a non-product goal, it prints nothing. 
-
-\item{\tt Show Intros.}\comindex{Show Intros}\\
-This command is similar to the previous one, it simulates the naming 
-process of an {\tt Intros}.
-
-\item{\tt Show Existentials}\comindex{Show Existentials}\\ It displays
-the set of all uninstantiated existential variables in the current proof tree, 
-along with the type and the context of each variable.
-
-\end{Variants}
-
-
-\subsection[\tt Guarded.]{\tt Guarded.\comindex{Guarded}\label{Guarded}}
-
-Some tactics (e.g. refine \ref{refine}) allow to build proofs using
-fixpoint or co-fixpoint constructions. Due to the incremental nature
-of interactive proof construction, the check of the termination (or
-guardedness) of the recursive calls in the fixpoint or cofixpoint
-constructions is postponed to the time of the completion of the proof.
-
-The command \verb!Guarded! allows to verify if the guard condition for
-fixpoint and cofixpoint is violated at some time of the construction
-of the proof without having to wait the completion of the proof."
-
-
-\section{Controlling the effect of proof editing commands}
-
-\subsection[\tt Set Hyps Limit {\num}.]{\tt Set Hyps Limit {\num}.\comindex{Set Hyps Limit}}
-This command sets the maximum number of hypotheses displayed in
-goals after the application of a tactic. 
-All the hypotheses remains usable in the proof development.
-
-
-\subsection[\tt Unset Hyps Limit.]{\tt Unset Hyps Limit.\comindex{Unset Hyps Limit}}
-This command goes back to the default mode which is to print all
-available hypotheses.
-
-% $Id: RefMan-pro.tex 13091 2010-06-08 13:56:19Z herbelin $
-
-
-\subsection[\tt Set Automatic Introduction.]{\tt Set Automatic Introduction.\comindex{Set Automatic Introduction}\comindex{Unset Automatic Introduction}\label{Set Automatic Introduction}}
-
-The option {\tt Automatic Introduction} controls the way binders are
-handled in assertion commands such as {\tt Theorem {\ident}
-  \zeroone{\binders} : {\form}}. When the option is set, which is the
-default, {\binders} are automatically put in the local context of the
-goal to prove.
-
-The option can be unset by issuing {\tt Unset Automatic Introduction}.
-When the option is unset, {\binders} are discharged on the statement
-to be proved and a tactic such as {\tt intro} (see
-Section~\ref{intro}) has to be used to move the assumptions to the
-local context.
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: "Reference-Manual"
-%%% End: