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%% $Id$
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\chapter{Proof Management: The Subgoal Module}
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\index{proofs|(}
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\index{subgoal module|(}
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\index{reading!goals|see{proofs, starting}}
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The subgoal module stores the current proof state\index{proof state} and
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many previous states; commands can produce new states or return to previous
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ones.  The {\em state list\/} at level $n$ is a list of pairs
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\[ [(\psi@n,\Psi@n),\; (\psi@{n-1},\Psi@{n-1}),\; \ldots,\; (\psi@0,[])] \]
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where $\psi@n$ is the current proof state, $\psi@{n-1}$ is the previous
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one, \ldots, and $\psi@0$ is the initial proof state.  The $\Psi@i$ are
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sequences (lazy lists) of proof states, storing branch points where a
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tactic returned a list longer than one.  The state lists permit various
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forms of backtracking.
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Chopping elements from the state list reverts to previous proof states.
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Besides this, the \ttindex{undo} command keeps a list of state lists.  The
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module actually maintains a stack of state lists, to support several
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proofs at the same time.
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The subgoal module always contains some proof state.  At the start of the
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Isabelle session, this state consists of a dummy formula.
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\section{Basic commands}
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Most proofs begin with {\tt goal} or {\tt goalw} and require no other
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commands than {\tt by}, {\tt chop} and {\tt undo}.  They typically end
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with a call to {\tt qed}.
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\subsection{Starting a backward proof}
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\index{proofs!starting}
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\begin{ttbox} 
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goal        : theory -> string -> thm list 
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goalw       : theory -> thm list -> string -> thm list 
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goalw_cterm : thm list -> cterm -> thm list 
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premises    : unit -> thm list
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\end{ttbox}
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The {\tt goal} commands start a new proof by setting the goal.  They
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replace the current state list by a new one consisting of the initial proof
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state.  They also empty the \ttindex{undo} list; this command cannot be
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undone!
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They all return a list of meta-hypotheses taken from the main goal.  If
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this list is non-empty, bind its value to an \ML{} identifier by typing
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something like
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\begin{ttbox} 
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val prems = goal{\it theory\/ formula};
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\end{ttbox}\index{assumptions!of main goal}
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These assumptions serve as the premises when you are deriving a rule.
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They are also stored internally and can be retrieved later by the
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function {\tt premises}.  When the proof is finished, {\tt qed}
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compares the stored assumptions with the actual assumptions in the
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proof state.
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\index{definitions!unfolding}
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Some of the commands unfold definitions using meta-rewrite rules.  This
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expansion affects both the initial subgoal and the premises, which would
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otherwise require use of {\tt rewrite_goals_tac} and
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{\tt rewrite_rule}.
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\index{*"!"! symbol!in main goal}
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If the main goal has the form {\tt"!!{\it vars}.\ \ldots"}, with an
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outermost quantifier, then the list of premises will be empty.  Subgoal~1
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will contain the meta-quantified {\it vars\/} as parameters and the goal's
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premises as assumptions.  This avoids having to call
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\ttindex{cut_facts_tac} with the list of premises (\S\ref{cut_facts_tac}).
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\begin{ttdescription}
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\item[\ttindexbold{goal} {\it theory} {\it formula};] 
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begins a new proof, where {\it theory} is usually an \ML\ identifier
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and the {\it formula\/} is written as an \ML\ string.
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\item[\ttindexbold{goalw} {\it theory} {\it defs} {\it formula};] 
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is like {\tt goal} but also applies the list of {\it defs\/} as
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meta-rewrite rules to the first subgoal and the premises.
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\index{meta-rewriting}
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\item[\ttindexbold{goalw_cterm} {\it theory} {\it defs} {\it ct};] is
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  a version of {\tt goalw} for programming applications.  The main
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  goal is supplied as a cterm, not as a string.  Typically, the cterm
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  is created from a term~$t$ by \hbox{\tt cterm_of (sign_of thy) $t$}.
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  \index{*cterm_of}\index{*sign_of}
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\item[\ttindexbold{premises}()] 
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returns the list of meta-hypotheses associated with the current proof (in
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case you forgot to bind them to an \ML{} identifier).
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\end{ttdescription}
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\subsection{Applying a tactic}
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\index{tactics!commands for applying}
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\begin{ttbox} 
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by   : tactic -> unit
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byev : tactic list -> unit
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\end{ttbox}
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These commands extend the state list.  They apply a tactic to the current
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proof state.  If the tactic succeeds, it returns a non-empty sequence of
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next states.  The head of the sequence becomes the next state, while the
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tail is retained for backtracking (see~{\tt back}).
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\begin{ttdescription} \item[\ttindexbold{by} {\it tactic};] 
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applies the {\it tactic\/} to the proof state.
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\item[\ttindexbold{byev} {\it tactics};] 
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applies the list of {\it tactics}, one at a time.  It is useful for testing
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calls to {\tt prove_goal}, and abbreviates \hbox{\tt by (EVERY {\it
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tactics})}.
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\end{ttdescription}
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\noindent{\it Error indications:}\nobreak
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\begin{itemize}
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\item {\footnotesize\tt "by:\ tactic failed"} means that the tactic
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  returned an empty sequence when applied to the current proof state.
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\item {\footnotesize\tt "Warning:\ same as previous level"} means that the
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  new proof state is identical to the previous state.
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\item{\footnotesize\tt "Warning:\ signature of proof state has changed"}
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  means that some rule was applied whose theory is outside the theory of
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  the initial proof state.  This could signify a mistake such as expressing
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  the goal in intuitionistic logic and proving it using classical logic.
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\end{itemize}
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\subsection{Extracting and storing the proved theorem}
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\label{ExtractingAndStoringTheProvedTheorem}
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\index{theorems!extracting proved}
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\begin{ttbox} 
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qed       : string -> unit
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result    : unit -> thm
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uresult   : unit -> thm
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bind_thm  : string * thm -> unit
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store_thm : string * thm -> thm
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{qed} $name$] is the usual way of ending a proof.
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  It combines {\tt result} and {\tt bind_thm}: it gets the theorem using {\tt
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  result()} and stores it in Isabelle's theorem database.  See below for
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  details. 
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\item[\ttindexbold{result}()]\index{assumptions!of main goal}
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  returns the final theorem, after converting the free variables to
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  schematics.  It discharges the assumptions supplied to the matching 
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  {\tt goal} command.  
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  It raises an exception unless the proof state passes certain checks.  There
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  must be no assumptions other than those supplied to {\tt goal}.  There
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  must be no subgoals.  The theorem proved must be a (first-order) instance
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  of the original goal, as stated in the {\tt goal} command.  This allows
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  {\bf answer extraction} --- instantiation of variables --- but no other
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  changes to the main goal.  The theorem proved must have the same signature
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  as the initial proof state.
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  These checks are needed because an Isabelle tactic can return any proof
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  state at all.
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\item[\ttindexbold{uresult}()] is like {\tt result()} but omits the checks.
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  It is needed when the initial goal contains function unknowns, when
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  definitions are unfolded in the main goal (by calling
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  \ttindex{rewrite_tac}),\index{definitions!unfolding} or when
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  \ttindex{assume_ax} has been used.
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\item[\ttindexbold{bind_thm}($name$, $thm$)]\index{theorems!storing}
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  stores {\tt standard($thm$)} (see \S\ref{MiscellaneousForwardRules})
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  in Isabelle's theorem database and in the {\ML} variable $name$.  The
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  theorem can be retrieved from the database using {\tt get_thm}
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  (see \S\ref{BasicOperationsOnTheories}).
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\item[\ttindexbold{store_thm}($name$, $thm$)]\index{theorems!storing} stores
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  $thm$ in Isabelle's theorem database and returns that theorem.
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\end{ttdescription}
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\subsection{Retrieving theorems}
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\index{theorems!retrieving}
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The following functions retrieve theorems (together with their names) from
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the theorem database.  Hence they can only find theorems that have explicitly
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been stored in the database using \ttindex{qed}, \ttindex{bind_thm} or
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related functions.
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\begin{ttbox} 
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findI           :         int -> (string * thm) list
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findE           :  int -> int -> (string * thm) list
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findEs          :         int -> (string * thm) list
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thms_containing : string list -> (string * thm) list
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{findI} $i$]\index{assumptions!of main goal}
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  returns all ``introduction rules'' applicable to subgoal $i$ --- all
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  theorems whose conclusion matches (rather than unifies with) subgoal
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  $i$.  Useful in connection with {\tt resolve_tac}.
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\item[\ttindexbold{findE} $n$ $i$] returns all ``elimination rules''
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  applicable to premise $n$ of subgoal $i$ --- all those theorems whose
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  first premise matches premise $n$ of subgoal $i$.  Useful in connection with
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  {\tt eresolve_tac} and {\tt dresolve_tac}.
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\item[\ttindexbold{findEs} $i$] returns all ``elimination rules'' applicable
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  to subgoal $i$ --- all those theorems whose first premise matches some
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  premise of subgoal $i$.  Useful in connection with {\tt eresolve_tac} and
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  {\tt dresolve_tac}.
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\item[\ttindexbold{thms_containing} $strings$] returns all theorems that
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  contain all of the constants in $strings$.  Note that a few basic constants
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  like \verb$==>$ are ignored.
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\end{ttdescription}
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\subsection{Undoing and backtracking}
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\begin{ttbox} 
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chop    : unit -> unit
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choplev : int -> unit
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back    : unit -> unit
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undo    : unit -> unit
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{chop}();] 
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deletes the top level of the state list, cancelling the last \ttindex{by}
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command.  It provides a limited undo facility, and the {\tt undo} command
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can cancel it.
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\item[\ttindexbold{choplev} {\it n};] 
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truncates the state list to level~{\it n}, if $n\geq0$.  A negative value
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of~$n$ refers to the $n$th previous level: 
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\hbox{\verb|choplev ~1|} has the same effect as {\tt chop}.
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\item[\ttindexbold{back}();]
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searches the state list for a non-empty branch point, starting from the top
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level.  The first one found becomes the current proof state --- the most
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recent alternative branch is taken.  This is a form of interactive
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backtracking.
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\item[\ttindexbold{undo}();] 
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cancels the most recent change to the proof state by the commands \ttindex{by},
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{\tt chop}, {\tt choplev}, and~{\tt back}.  It {\bf cannot}
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cancel {\tt goal} or {\tt undo} itself.  It can be repeated to
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cancel a series of commands.
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\end{ttdescription}
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\goodbreak
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\noindent{\it Error indications for {\tt back}:}\par\nobreak
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\begin{itemize}
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\item{\footnotesize\tt"Warning:\ same as previous choice at this level"}
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  means {\tt back} found a non-empty branch point, but that it contained
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  the same proof state as the current one.
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\item{\footnotesize\tt "Warning:\ signature of proof state has changed"}
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  means the signature of the alternative proof state differs from that of
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  the current state.
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\item {\footnotesize\tt "back:\ no alternatives"} means {\tt back} could
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  find no alternative proof state.
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\end{itemize}
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\subsection{Printing the proof state}\label{sec:goals-printing}
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\index{proof state!printing of}
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\begin{ttbox} 
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pr    : unit -> unit
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prlev : int -> unit
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prlim : int -> unit
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goals_limit: int ref \hfill{\bf initially 10}
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\end{ttbox}
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See also the printing control options described 
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in~\S\ref{sec:printing-control}. 
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\begin{ttdescription}
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\item[\ttindexbold{pr}();] 
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prints the current proof state.
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\item[\ttindexbold{prlev} {\it n};] 
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prints the proof state at level {\it n}, if $n\geq0$.  A negative value
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of~$n$ refers to the $n$th previous level.  This command allows
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you to review earlier stages of the proof.
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\item[\ttindexbold{prlim} {\it k};] 
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prints the current proof state, limiting the number of subgoals to~$k$.  It
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updates {\tt goals_limit} (see below) and is helpful when there are many
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subgoals. 
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\item[\ttindexbold{goals_limit} := {\it k};] 
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specifies~$k$ as the maximum number of subgoals to print.
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\end{ttdescription}
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\subsection{Timing}
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\index{timing statistics}\index{proofs!timing}
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\begin{ttbox} 
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proof_timing: bool ref \hfill{\bf initially false}
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{proof_timing} := true;] 
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makes the \ttindex{by} and \ttindex{prove_goal} commands display how much
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processor time was spent.  This information is compiler-dependent.
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\end{ttdescription}
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\section{Shortcuts for applying tactics}
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\index{shortcuts!for {\tt by} commands}
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These commands call \ttindex{by} with common tactics.  Their chief purpose
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is to minimise typing, although the scanning shortcuts are useful in their
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own right.  Chapter~\ref{tactics} explains the tactics themselves.
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\subsection{Refining a given subgoal}
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\begin{ttbox} 
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ba  :             int -> unit
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br  : thm      -> int -> unit
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be  : thm      -> int -> unit
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bd  : thm      -> int -> unit
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brs : thm list -> int -> unit
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bes : thm list -> int -> unit
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bds : thm list -> int -> unit
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{ba} {\it i};] 
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performs \hbox{\tt by (assume_tac {\it i});}
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\item[\ttindexbold{br} {\it thm} {\it i};] 
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performs \hbox{\tt by (resolve_tac [{\it thm}] {\it i});}
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\item[\ttindexbold{be} {\it thm} {\it i};] 
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performs \hbox{\tt by (eresolve_tac [{\it thm}] {\it i});}
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\item[\ttindexbold{bd} {\it thm} {\it i};] 
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performs \hbox{\tt by (dresolve_tac [{\it thm}] {\it i});}
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\item[\ttindexbold{brs} {\it thms} {\it i};] 
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performs \hbox{\tt by (resolve_tac {\it thms} {\it i});}
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\item[\ttindexbold{bes} {\it thms} {\it i};] 
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performs \hbox{\tt by (eresolve_tac {\it thms} {\it i});}
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\item[\ttindexbold{bds} {\it thms} {\it i};] 
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performs \hbox{\tt by (dresolve_tac {\it thms} {\it i});}
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\end{ttdescription}
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\subsection{Scanning shortcuts}
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These shortcuts scan for a suitable subgoal (starting from subgoal~1).
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They refine the first subgoal for which the tactic succeeds.  Thus, they
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require less typing than {\tt br}, etc.  They display the selected
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subgoal's number; please watch this, for it may not be what you expect!
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\begin{ttbox} 
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fa  : unit     -> unit
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fr  : thm      -> unit
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fe  : thm      -> unit
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fd  : thm      -> unit
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frs : thm list -> unit
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fes : thm list -> unit
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fds : thm list -> unit
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{fa}();] 
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solves some subgoal by assumption.
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\item[\ttindexbold{fr} {\it thm};] 
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refines some subgoal using \hbox{\tt resolve_tac [{\it thm}]}
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\item[\ttindexbold{fe} {\it thm};] 
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refines some subgoal using \hbox{\tt eresolve_tac [{\it thm}]}
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\item[\ttindexbold{fd} {\it thm};] 
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refines some subgoal using \hbox{\tt dresolve_tac [{\it thm}]}
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\item[\ttindexbold{frs} {\it thms};] 
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refines some subgoal using \hbox{\tt resolve_tac {\it thms}}
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\item[\ttindexbold{fes} {\it thms};] 
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refines some subgoal using \hbox{\tt eresolve_tac {\it thms}} 
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\item[\ttindexbold{fds} {\it thms};] 
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refines some subgoal using \hbox{\tt dresolve_tac {\it thms}} 
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\end{ttdescription}
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\subsection{Other shortcuts}
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\begin{ttbox} 
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bw  : thm -> unit
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bws : thm list -> unit
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ren : string -> int -> unit
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{bw} {\it def};] 
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performs \hbox{\tt by (rewrite_goals_tac [{\it def}]);}
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It unfolds definitions in the subgoals (but not the main goal), by
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meta-rewriting with the given definition.
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\index{meta-rewriting}
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\item[\ttindexbold{bws}] 
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is like {\tt bw} but takes a list of definitions.
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\item[\ttindexbold{ren} {\it names} {\it i};] 
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performs \hbox{\tt by (rename_tac {\it names} {\it i});} it renames
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parameters in subgoal~$i$.  (Ignore the message {\footnotesize\tt Warning:\
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  same as previous level}.)
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\index{parameters!renaming}
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\end{ttdescription}
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\section{Executing batch proofs}
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\index{batch execution}%
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To save space below, let type \texttt{tacfun} abbreviate \texttt{thm list ->
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  tactic list}, which is the type of a tactical proof.
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\begin{ttbox}
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prove_goal :           theory ->             string -> tacfun -> thm
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qed_goal   : string -> theory ->             string -> tacfun -> unit
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prove_goalw:           theory -> thm list -> string -> tacfun -> thm
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qed_goalw  : string -> theory -> thm list -> string -> tacfun -> unit
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prove_goalw_cterm:               thm list -> cterm  -> tacfun -> thm
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\end{ttbox}
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These batch functions create an initial proof state, then apply a tactic to
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it, yielding a sequence of final proof states.  The head of the sequence is
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returned, provided it is an instance of the theorem originally proposed.
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The forms {\tt prove_goal}, {\tt prove_goalw} and {\tt prove_goalw_cterm}
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are analogous to {\tt goal}, {\tt goalw} and {\tt goalw_cterm}.
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The tactic is specified by a function from theorem lists to tactic lists.
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The function is applied to the list of meta-assumptions taken from
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the main goal.  The resulting tactics are applied in sequence (using {\tt
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  EVERY}).  For example, a proof consisting of the commands
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\begin{ttbox} 
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val prems = goal {\it theory} {\it formula};
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by \(tac@1\);  \ldots  by \(tac@n\);
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qed "my_thm";
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\end{ttbox}
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can be transformed to an expression as follows:
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\begin{ttbox} 
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qed_goal "my_thm" {\it theory} {\it formula}
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 (fn prems=> [ \(tac@1\), \ldots, \(tac@n\) ]);
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\end{ttbox}
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The methods perform identical processing of the initial {\it formula} and
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the final proof state.  But {\tt prove_goal} executes the tactic as a
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atomic operation, bypassing the subgoal module; the current interactive
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proof is unaffected.
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%
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\begin{ttdescription}
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\item[\ttindexbold{prove_goal} {\it theory} {\it formula} {\it tacsf};] 
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executes a proof of the {\it formula\/} in the given {\it theory}, using
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the given tactic function.
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\item[\ttindexbold{qed_goal} $name$ $theory$ $formula$ $tacsf$;]
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acts like {\tt prove_goal} but also stores the resulting theorem in
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Isabelle's theorem database and in the {\ML} variable $name$ (see
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\S\ref{ExtractingAndStoringTheProvedTheorem}).
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\item[\ttindexbold{prove_goalw} {\it theory} {\it defs} {\it formula} 
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      {\it tacsf};]\index{meta-rewriting}
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is like {\tt prove_goal} but also applies the list of {\it defs\/} as
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meta-rewrite rules to the first subgoal and the premises.
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\item[\ttindexbold{qed_goalw} $name$ $theory$ $defs$ $formula$ $tacsf$;]
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is analogous to {\tt qed_goal}.
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\item[\ttindexbold{prove_goalw_cterm} {\it theory} {\it defs} {\it ct}
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      {\it tacsf};] 
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is a version of {\tt prove_goalw} for programming applications.  The main
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goal is supplied as a cterm, not as a string.  Typically, the cterm is
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created from a term~$t$ as follows:
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\begin{ttbox}
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cterm_of (sign_of thy) \(t\)
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\end{ttbox}
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\index{*cterm_of}\index{*sign_of}
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\end{ttdescription}
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\section{Managing multiple proofs}
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\index{proofs!managing multiple}
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You may save the current state of the subgoal module and resume work on it
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later.  This serves two purposes.  
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\begin{enumerate}
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\item At some point, you may be uncertain of the next step, and
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wish to experiment.
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\item During a proof, you may see that a lemma should be proved first.
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\end{enumerate}
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Each saved proof state consists of a list of levels; \ttindex{chop} behaves
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independently for each of the saved proofs.  In addition, each saved state
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carries a separate \ttindex{undo} list.
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\subsection{The stack of proof states}
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\index{proofs!stacking}
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\begin{ttbox} 
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push_proof   : unit -> unit
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pop_proof    : unit -> thm list
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rotate_proof : unit -> thm list
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\end{ttbox}
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The subgoal module maintains a stack of proof states.  Most subgoal
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commands affect only the top of the stack.  The \ttindex{goal} command {\em
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replaces\/} the top of the stack; the only command that pushes a proof on the
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stack is {\tt push_proof}.
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To save some point of the proof, call {\tt push_proof}.  You may now
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state a lemma using {\tt goal}, or simply continue to apply tactics.
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Later, you can return to the saved point by calling {\tt pop_proof} or 
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{\tt rotate_proof}. 
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To view the entire stack, call {\tt rotate_proof} repeatedly; as it rotates
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the stack, it prints the new top element.
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\begin{ttdescription}
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\item[\ttindexbold{push_proof}();]  
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duplicates the top element of the stack, pushing a copy of the current
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proof state on to the stack.
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\item[\ttindexbold{pop_proof}();]  
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discards the top element of the stack.  It returns the list of
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assumptions associated with the new proof;  you should bind these to an
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\ML\ identifier.  They can also be obtained by calling \ttindex{premises}.
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\item[\ttindexbold{rotate_proof}();]
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\index{assumptions!of main goal}
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rotates the stack, moving the top element to the bottom.  It returns the
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list of assumptions associated with the new proof.
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\end{ttdescription}
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\subsection{Saving and restoring proof states}
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\index{proofs!saving and restoring}
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\begin{ttbox} 
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save_proof    : unit -> proof
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restore_proof : proof -> thm list
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\end{ttbox}
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States of the subgoal module may be saved as \ML\ values of
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type~\mltydx{proof}, and later restored.
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\begin{ttdescription}
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\item[\ttindexbold{save_proof}();]  
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returns the current state, which is on top of the stack.
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\item[\ttindexbold{restore_proof} {\it prf};]\index{assumptions!of main goal}
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  replaces the top of the stack by~{\it prf}.  It returns the list of
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  assumptions associated with the new proof.
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\end{ttdescription}
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\section{Debugging and inspecting}
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\index{tactics!debugging}
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These functions can be useful when you are debugging a tactic.  They refer
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to the current proof state stored in the subgoal module.  A tactic
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should never call them; it should operate on the proof state supplied as its
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argument.
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\subsection{Reading and printing terms}
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\index{terms!reading of}\index{terms!printing of}\index{types!printing of}
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\begin{ttbox} 
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read    : string -> term
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prin    : term -> unit
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printyp : typ -> unit
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\end{ttbox}
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These read and print terms (or types) using the syntax associated with the
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proof state.
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\begin{ttdescription}
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\item[\ttindexbold{read} {\it string}]  
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reads the {\it string} as a term, without type checking.
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\item[\ttindexbold{prin} {\it t};]  
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prints the term~$t$ at the terminal.
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\item[\ttindexbold{printyp} {\it T};]  
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prints the type~$T$ at the terminal.
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\end{ttdescription}
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\subsection{Inspecting the proof state}
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\index{proofs!inspecting the state}
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\begin{ttbox} 
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topthm  : unit -> thm
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getgoal : int -> term
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gethyps : int -> thm list
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{topthm}()]  
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returns the proof state as an Isabelle theorem.  This is what \ttindex{by}
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would supply to a tactic at this point.  It omits the post-processing of
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\ttindex{result} and \ttindex{uresult}.
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\item[\ttindexbold{getgoal} {\it i}]  
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returns subgoal~$i$ of the proof state, as a term.  You may print
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this using {\tt prin}, though you may have to examine the internal
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data structure in order to locate the problem!
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\item[\ttindexbold{gethyps} {\it i}]
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  returns the hypotheses of subgoal~$i$ as meta-level assumptions.  In
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  these theorems, the subgoal's parameters become free variables.  This
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  command is supplied for debugging uses of \ttindex{METAHYPS}.
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\end{ttdescription}
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\subsection{Filtering lists of rules}
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\begin{ttbox} 
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filter_goal: (term*term->bool) -> thm list -> int -> thm list
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\end{ttbox}
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\begin{ttdescription}
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\item[\ttindexbold{filter_goal} {\it could} {\it ths} {\it i}] 
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applies \hbox{\tt filter_thms {\it could}} to subgoal~$i$ of the proof
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state and returns the list of theorems that survive the filtering. 
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\end{ttdescription}
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\index{subgoal module|)}
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\index{proofs|)}