(* $Id$ *)

 theory Proof

 imports Main

 begin

 chapter {* Proofs *}

 text {*

   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 [@{text "proof(prove)"}] means that a new goal has just been

   stated that is now to be \emph{proven}; the next command may refine

   it by some proof method, and enter a sub-proof to establish the

   actual result.

   \item [@{text "proof(state)"}] is like a nested theory mode: the

   context may be augmented by \emph{stating} additional assumptions,

   intermediate results etc.

   \item [@{text "proof(chain)"}] is intermediate between @{text

   "proof(state)"} and @{text "proof(prove)"}: existing facts (i.e.\

   the contents of the special ``@{fact_ref 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.

 *}

 section {* Context elements \label{sec:proof-context} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "fix"} & : & \isartrans{proof(state)}{proof(state)} \\

     @{command_def "assume"} & : & \isartrans{proof(state)}{proof(state)} \\

     @{command_def "presume"} & : & \isartrans{proof(state)}{proof(state)} \\

     @{command_def "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 ``@{command "fix"}~@{text x}'' results in a local value

   that may be used in the subsequent proof as any other variable or

   constant.  Furthermore, any result @{text "\<turnstile> \<phi>[x]"} exported from

   the context will be universally closed wrt.\ @{text x} at the

   outermost level: @{text "\<turnstile> \<And>x. \<phi>[x]"} (this is expressed in normal

   form using Isabelle's meta-variables).

   Similarly, introducing some assumption @{text \<chi>} has two effects.

   On the one hand, a local theorem is created that may be used as a

   fact in subsequent proof steps.  On the other hand, any result

   @{text "\<chi> \<turnstile> \<phi>"} exported from the context becomes conditional wrt.\

   the assumption: @{text "\<turnstile> \<chi> \<Longrightarrow> \<phi>"}.  Thus, solving an enclosing goal

   using such a result would basically introduce a new subgoal stemming

   from the assumption.  How this situation is handled depends on the

   version of assumption command used: while @{command "assume"}

   insists on solving the subgoal by unification with some premise of

   the goal, @{command "presume"} leaves the subgoal unchanged in order

   to be proved later by the user.

   Local definitions, introduced by ``@{command "def"}~@{text "x \<equiv>

   t"}'', are achieved by combining ``@{command "fix"}~@{text x}'' with

   another version of assumption that causes any hypothetical equation

   @{text "x \<equiv> t"} to be eliminated by the reflexivity rule.  Thus,

   exporting some result @{text "x \<equiv> t \<turnstile> \<phi>[x]"} yields @{text "\<turnstile>

   \<phi>[t]"}.

   \begin{rail}

     'fix' (vars + 'and')

     ;

     ('assume' | 'presume') (props + 'and')

     ;

     'def' (def + 'and')

     ;

     def: thmdecl? \\ name ('==' | equiv) term termpat?

     ;

   \end{rail}

   \begin{descr}

   \item [@{command "fix"}~@{text x}] introduces a local variable

   @{text x} that is \emph{arbitrary, but fixed.}

   \item [@{command "assume"}~@{text "a: \<phi>"} and @{command

   "presume"}~@{text "a: \<phi>"}] introduce a local fact @{text "\<phi> \<turnstile> \<phi>"} by

   assumption.  Subsequent results applied to an enclosing goal (e.g.\

   by @{command_ref "show"}) are handled as follows: @{command

   "assume"} expects to be able to unify with existing premises in the

   goal, while @{command "presume"} leaves @{text \<phi>} as new subgoals.

   Several lists of assumptions may be given (separated by

   @{keyword_ref "and"}; the resulting list of current facts consists

   of all of these concatenated.

   \item [@{command "def"}~@{text "x \<equiv> t"}] introduces a local

   (non-polymorphic) definition.  In results exported from the context,

   @{text x} is replaced by @{text t}.  Basically, ``@{command

   "def"}~@{text "x \<equiv> t"}'' abbreviates ``@{command "fix"}~@{text

   x}~@{command "assume"}~@{text "x \<equiv> t"}'', with the resulting

   hypothetical equation solved by reflexivity.

   The default name for the definitional equation is @{text x_def}.

   Several simultaneous definitions may be given at the same time.

   \end{descr}

   The special name @{fact_ref 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.

 *}

 section {* Facts and forward chaining *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "note"} & : & \isartrans{proof(state)}{proof(state)} \\

     @{command_def "then"} & : & \isartrans{proof(state)}{proof(chain)} \\

     @{command_def "from"} & : & \isartrans{proof(state)}{proof(chain)} \\

     @{command_def "with"} & : & \isartrans{proof(state)}{proof(chain)} \\

     @{command_def "using"} & : & \isartrans{proof(prove)}{proof(prove)} \\

     @{command_def "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 @{command "then"} (and variants);

   @{command "from"} and @{command "with"} are composite forms

   involving @{command "note"}.  The @{command "using"} elements

   augments the collection of used facts \emph{after} a goal has been

   stated.  Note that the special theorem name @{fact_ref 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 [@{command "note"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"}]

   recalls existing facts @{text "b\<^sub>1, \<dots>, b\<^sub>n"}, binding

   the result as @{text a}.  Note that attributes may be involved as

   well, both on the left and right hand sides.

   \item [@{command "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 @{method_ref 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 [@{command "from"}~@{text b}] abbreviates ``@{command

   "note"}~@{text b}~@{command "then"}''; thus @{command "then"} is

   equivalent to ``@{command "from"}~@{text this}''.

   \item [@{command "with"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"}]

   abbreviates ``@{command "from"}~@{text "b\<^sub>1 \<dots> b\<^sub>n \<AND>

   this"}''; thus the forward chaining is from earlier facts together

   with the current ones.

   \item [@{command "using"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"}] augments

   the facts being currently indicated for use by a subsequent

   refinement step (such as @{command_ref "apply"} or @{command_ref

   "proof"}).

   \item [@{command "unfolding"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"}] is

   structurally similar to @{command "using"}, but unfolds definitional

   equations @{text "b\<^sub>1, \<dots> b\<^sub>n"} 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 ``@{command "from"}~@{text nothing}'' has no

   effect apart from entering @{text "prove(chain)"} mode, since

   @{fact_ref nothing} is bound to the empty list of theorems.

   Basic proof methods (such as @{method_ref 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 @{command "from"}~@{text "_

   \<AND> a \<AND> b"}, for example.  This involves the trivial rule

   @{text "PROP \<psi> \<Longrightarrow> PROP \<psi>"}, which is bound in Isabelle/Pure as

   ``@{fact_ref "_"}'' (underscore).

   Automated methods (such as @{method simp} or @{method 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.

 *}

 section {* Goal statements \label{sec:goals} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "lemma"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\

     @{command_def "theorem"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\

     @{command_def "corollary"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\

     @{command_def "have"} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\

     @{command_def "show"} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\

     @{command_def "hence"} & : & \isartrans{proof(state)}{proof(prove)} \\

     @{command_def "thus"} & : & \isartrans{proof(state)}{proof(prove)} \\

     @{command_def "print_statement"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   \end{matharray}

   From a theory context, proof mode is entered by an initial goal

   command such as @{command "lemma"}, @{command "theorem"}, or

   @{command "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 @{command_ref "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 @{text "&&"}), which is usually

   split into the corresponding number of sub-goals prior to an initial

   method application, via @{command_ref "proof"}

   (\secref{sec:proof-steps}) or @{command_ref "apply"}

   (\secref{sec:tactic-commands}).  The @{method_ref 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 @{fact_ref assms} in the proof.  Moreover, there

   are two kinds of conclusions: @{element_def "shows"} states several

   simultaneous propositions (essentially a big conjunction), while

   @{element_def "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 [@{command "lemma"}~@{text "a: \<phi>"}] enters proof mode with

   @{text \<phi>} as main goal, eventually resulting in some fact @{text "\<turnstile>

   \<phi>"} 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 @{syntax contextelem} in

   \secref{sec:locale}.

   \item [@{command "theorem"}~@{text "a: \<phi>"} and @{command

   "corollary"}~@{text "a: \<phi>"}] are essentially the same as @{command

   "lemma"}~@{text "a: \<phi>"}, but the facts are internally marked as

   being of a different kind.  This discrimination acts like a formal

   comment.

   \item [@{command "have"}~@{text "a: \<phi>"}] 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 @{command "have"} may be freely

   used for experimental exploration of potential results within a

   proof body.

   \item [@{command "show"}~@{text "a: \<phi>"}] is like @{command

   "have"}~@{text "a: \<phi>"} 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 @{command "show"} command).

   To accommodate interactive debugging, resulting rules are printed

   before being applied internally.  Even more, interactive execution

   of @{command "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 [@{command "hence"}] abbreviates ``@{command "then"}~@{command

   "have"}'', i.e.\ claims a local goal to be proven by forward

   chaining the current facts.  Note that @{command "hence"} is also

   equivalent to ``@{command "from"}~@{text this}~@{command "have"}''.

   \item [@{command "thus"}] abbreviates ``@{command "then"}~@{command

   "show"}''.  Note that @{command "thus"} is also equivalent to

   ``@{command "from"}~@{text this}~@{command "show"}''.

   \item [@{command "print_statement"}~@{text a}] prints facts from the

   current theory or proof context in long statement form, according to

   the syntax for @{command "lemma"} given above.

   \end{descr}

   Any goal statement causes some term abbreviations (such as

   @{variable_ref "?thesis"}) to be bound automatically, see also

   \secref{sec:term-abbrev}.  Furthermore, the local context of a

   (non-atomic) goal is provided via the @{case_ref rule_context} case.

   The optional case names of @{element_ref "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

   @{method_ref 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}

 *}

 section {* Initial and terminal proof steps \label{sec:proof-steps} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "proof"} & : & \isartrans{proof(prove)}{proof(state)} \\

     @{command_def "qed"} & : & \isartrans{proof(state)}{proof(state) ~|~ theory} \\

     @{command_def "by"} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

     @{command_def ".."} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

     @{command_def "."} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\

     @{command_def "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 @{command_ref

   "proof"}~@{text "m\<^sub>1"} reduces a newly stated goal to a number

   of sub-goals that are to be solved later.  Facts are passed to

   @{text "m\<^sub>1"} for forward chaining, if so indicated by @{text

   "proof(chain)"} mode.

   \item A \emph{terminal} conclusion step @{command_ref "qed"}~@{text

   "m\<^sub>2"} is intended to solve remaining goals.  No facts are

   passed to @{text "m\<^sub>2"}.

   \end{enumerate}

   The only other (proper) way to affect pending goals in a proof body

   is by @{command_ref "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

   ``@{method_ref 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 [@{command "proof"}~@{text "m\<^sub>1"}] refines the goal by

   proof method @{text "m\<^sub>1"}; facts for forward chaining are

   passed if so indicated by @{text "proof(chain)"} mode.

   \item [@{command "qed"}~@{text "m\<^sub>2"}] refines any remaining

   goals by proof method @{text "m\<^sub>2"} and concludes the

   sub-proof by assumption.  If the goal had been @{text "show"} (or

   @{text "thus"}), some pending sub-goal is solved as well by the rule

   resulting from the result \emph{exported} into the enclosing goal

   context.  Thus @{text "qed"} may fail for two reasons: either @{text

   "m\<^sub>2"} fails, or the resulting rule does not fit to any

   pending goal\footnote{This includes any additional ``strong''

   assumptions as introduced by @{command "assume"}.} of the enclosing

   context.  Debugging such a situation might involve temporarily

   changing @{command "show"} into @{command "have"}, or weakening the

   local context by replacing occurrences of @{command "assume"} by

   @{command "presume"}.

   \item [@{command "by"}~@{text "m\<^sub>1 m\<^sub>2"}] is a

   \emph{terminal proof}\index{proof!terminal}; it abbreviates

   @{command "proof"}~@{text "m\<^sub>1"}~@{text "qed"}~@{text

   "m\<^sub>2"}, but with backtracking across both methods.  Debugging

   an unsuccessful @{command "by"}~@{text "m\<^sub>1 m\<^sub>2"}

   command can be done by expanding its definition; in many cases

   @{command "proof"}~@{text "m\<^sub>1"} (or even @{text

   "apply"}~@{text "m\<^sub>1"}) is already sufficient to see the

   problem.

   \item [``@{command ".."}''] is a \emph{default

   proof}\index{proof!default}; it abbreviates @{command "by"}~@{text

   "rule"}.

   \item [``@{command "."}''] is a \emph{trivial

   proof}\index{proof!trivial}; it abbreviates @{command "by"}~@{text

   "this"}.

   \item [@{command "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 @{ML

   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 ``@{text

   "[!]"}'' in the printed result.

   The most important application of @{command "sorry"} is to support

   experimentation and top-down proof development.

   \end{descr}

 *}

 section {* Fundamental methods and attributes \label{sec:pure-meth-att} *}

 text {*

   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}

     @{method_def "-"} & : & \isarmeth \\

     @{method_def "fact"} & : & \isarmeth \\

     @{method_def "assumption"} & : & \isarmeth \\

     @{method_def "this"} & : & \isarmeth \\

     @{method_def "rule"} & : & \isarmeth \\

     @{method_def "iprover"} & : & \isarmeth \\[0.5ex]

     @{attribute_def "intro"} & : & \isaratt \\

     @{attribute_def "elim"} & : & \isaratt \\

     @{attribute_def "dest"} & : & \isaratt \\

     @{attribute_def "rule"} & : & \isaratt \\[0.5ex]

     @{attribute_def "OF"} & : & \isaratt \\

     @{attribute_def "of"} & : & \isaratt \\

     @{attribute_def "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 [``@{method "-"}'' (minus)] does nothing but insert the

   forward chaining facts as premises into the goal.  Note that command

   @{command_ref "proof"} without any method actually performs a single

   reduction step using the @{method_ref rule} method; thus a plain

   \emph{do-nothing} proof step would be ``@{command "proof"}~@{text

   "-"}'' rather than @{command "proof"} alone.

   \item [@{method "fact"}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}] composes

   some fact from @{text "a\<^sub>1, \<dots>, a\<^sub>n"} (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}):

   ``@{command "have"}~@{text "\<phi>"}~@{command "by"}~@{text fact}'' is

   equivalent to ``@{command "note"}~@{verbatim "`"}@{text \<phi>}@{verbatim

   "`"}'' provided that @{text "\<turnstile> \<phi>"} is an instance of some known

   @{text "\<turnstile> \<phi>"} in the proof context.

   \item [@{method 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 @{command "qed"} (\secref{sec:proof-steps}) already

   concludes any remaining sub-goals by assumption, so structured

   proofs usually need not quote the @{method assumption} method at

   all.

   \item [@{method this}] applies all of the current facts directly as

   rules.  Recall that ``@{command "."}'' (dot) abbreviates ``@{command

   "by"}~@{text this}''.

   \item [@{method rule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}] applies some

   rule given as argument in backward manner; facts are used to reduce

   the rule before applying it to the goal.  Thus @{method rule}

   without facts is plain introduction, while with facts it becomes

   elimination.

   When no arguments are given, the @{method rule} method tries to pick

   appropriate rules automatically, as declared in the current context

   using the @{attribute intro}, @{attribute elim}, @{attribute dest}

   attributes (see below).  This is the default behavior of @{command

   "proof"} and ``@{command ".."}'' (double-dot) steps (see

   \secref{sec:proof-steps}).

   \item [@{method 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; ``@{method iprover}@{text "!"}'' 

   means to include the current @{fact prems} as well.

   Rules need to be classified as @{attribute intro}, @{attribute

   elim}, or @{attribute dest}; here the ``@{text "!"}'' indicator

   refers to ``safe'' rules, which may be applied aggressively (without

   considering back-tracking later).  Rules declared with ``@{text

   "?"}'' are ignored in proof search (the single-step @{method 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 [@{attribute intro}, @{attribute elim}, and @{attribute dest}]

   declare introduction, elimination, and destruct rules, to be used

   with the @{method rule} and @{method iprover} methods.  Note that

   the latter will ignore rules declared with ``@{text "?"}'', while

   ``@{text "!"}''  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.\ @{attribute "Pure.intro"}.

   \item [@{attribute rule}~@{text del}] undeclares introduction,

   elimination, or destruct rules.

   \item [@{attribute OF}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}] applies some

   theorem to all of the given rules @{text "a\<^sub>1, \<dots>, a\<^sub>n"}

   (in parallel).  This corresponds to the @{ML "op MRS"} operation in

   ML, but note the reversed order.  Positions may be effectively

   skipped by including ``@{text _}'' (underscore) as argument.

   \item [@{attribute of}~@{text "t\<^sub>1 \<dots> t\<^sub>n"}] performs

   positional instantiation of term variables.  The terms @{text

   "t\<^sub>1, \<dots>, t\<^sub>n"} are substituted for any schematic

   variables occurring in a theorem from left to right; ``@{text _}''

   (underscore) indicates to skip a position.  Arguments following a

   ``@{text "concl:"}'' specification refer to positions of the

   conclusion of a rule.

   \item [@{attribute "where"}~@{text "x\<^sub>1 = t\<^sub>1 \<AND> \<dots>

   x\<^sub>n = t\<^sub>n"}] 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.\ @{text "?x1.3"}).

   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}

 *}

 section {* Term abbreviations \label{sec:term-abbrev} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "let"} & : & \isartrans{proof(state)}{proof(state)} \\

     @{keyword_def "is"} & : & syntax \\

   \end{matharray}

   Abbreviations may be either bound by explicit @{command

   "let"}~@{text "p \<equiv> t"} statements, or by annotating assumptions or

   goal statements with a list of patterns ``@{text "(\<IS> p\<^sub>1 \<dots>

   p\<^sub>n)"}''.  In both cases, higher-order matching is invoked to

   bind extra-logical term variables, which may be either named

   schematic variables of the form @{text ?x}, or nameless dummies

   ``@{variable _}'' (underscore). Note that in the @{command "let"}

   form the patterns occur on the left-hand side, while the @{keyword

   "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 @{command "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 @{command "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 @{command

   "def"} does not support polymorphism.

   \begin{rail}

     'let' ((term + 'and') '=' term + 'and')

     ;  

   \end{rail}

   The syntax of @{keyword "is"} patterns follows \railnonterm{termpat}

   or \railnonterm{proppat} (see \secref{sec:term-decls}).

   \begin{descr}

   \item [@{command "let"}~@{text "p\<^sub>1 = t\<^sub>1 \<AND> \<dots>

   p\<^sub>n = t\<^sub>n"}] binds any text variables in patterns @{text

   "p\<^sub>1, \<dots>, p\<^sub>n"} by simultaneous higher-order matching

   against terms @{text "t\<^sub>1, \<dots>, t\<^sub>n"}.

   \item [@{text "(\<IS> p\<^sub>1 \<dots> p\<^sub>n)"}] resembles @{command

   "let"}, but matches @{text "p\<^sub>1, \<dots>, p\<^sub>n"} against the

   preceding statement.  Also note that @{keyword "is"} is not a

   separate command, but part of others (such as @{command "assume"},

   @{command "have"} etc.).

   \end{descr}

   Some \emph{implicit} term abbreviations\index{term abbreviations}

   for goals and facts are available as well.  For any open goal,

   @{variable_ref thesis} refers to its object-level statement,

   abstracted over any meta-level parameters (if present).  Likewise,

   @{variable_ref this} is bound for fact statements resulting from

   assumptions or finished goals.  In case @{variable this} refers to

   an object-logic statement that is an application @{text "f t"}, then

   @{text t} is bound to the special text variable ``@{variable "\<dots>"}''

   (three dots).  The canonical application of this convenience are

   calculational proofs (see \secref{sec:calculation}).

 *}

 section {* Block structure *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "next"} & : & \isartrans{proof(state)}{proof(state)} \\

     @{command_def "{"} & : & \isartrans{proof(state)}{proof(state)} \\

     @{command_def "}"} & : & \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 @{command "qed"} etc.).  Sections of different

   context within a sub-proof may be switched via @{command "next"},

   which is just a single block-close followed by block-open again.

   The effect of @{command "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 [@{command "next"}] switches to a fresh block within a

   sub-proof, resetting the local context to the initial one.

   \item [@{command "{"} and @{command "}"}] explicitly open and close

   blocks.  Any current facts pass through ``@{command "{"}''

   unchanged, while ``@{command "}"}'' 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 @{command "assume"} and @{command "presume"} in this mode of

   forward reasoning --- in contrast to plain backward reasoning with

   the result exported at @{command "show"} time.

   \end{descr}

 *}

 section {* Emulating tactic scripts \label{sec:tactic-commands} *}

 text {*

   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}

     @{command_def "apply"}@{text "\<^sup>*"} & : & \isartrans{proof(prove)}{proof(prove)} \\

     @{command_def "apply_end"}@{text "\<^sup>*"} & : & \isartrans{proof(state)}{proof(state)} \\

     @{command_def "done"}@{text "\<^sup>*"} & : & \isartrans{proof(prove)}{proof(state)} \\

     @{command_def "defer"}@{text "\<^sup>*"} & : & \isartrans{proof}{proof} \\

     @{command_def "prefer"}@{text "\<^sup>*"} & : & \isartrans{proof}{proof} \\

     @{command_def "back"}@{text "\<^sup>*"} & : & \isartrans{proof}{proof} \\

   \end{matharray}

   \begin{rail}

     ( 'apply' | 'apply\_end' ) method

     ;

     'defer' nat?

     ;

     'prefer' nat

     ;

   \end{rail}

   \begin{descr}

   \item [@{command "apply"}~@{text m}] applies proof method @{text m}

   in initial position, but unlike @{command "proof"} it retains

   ``@{text "proof(prove)"}'' mode.  Thus consecutive method

   applications may be given just as in tactic scripts.

   Facts are passed to @{text m} as indicated by the goal's

   forward-chain mode, and are \emph{consumed} afterwards.  Thus any

   further @{command "apply"} command would always work in a purely

   backward manner.

   \item [@{command "apply_end"}~@{text "m"}] applies proof method

   @{text m} as if in terminal position.  Basically, this simulates a

   multi-step tactic script for @{command "qed"}, but may be given

   anywhere within the proof body.

   No facts are passed to @{method m} here.  Furthermore, the static

   context is that of the enclosing goal (as for actual @{command

   "qed"}).  Thus the proof method may not refer to any assumptions

   introduced in the current body, for example.

   \item [@{command "done"}] completes a proof script, provided that

   the current goal state is solved completely.  Note that actual

   structured proof commands (e.g.\ ``@{command "."}'' or @{command

   "sorry"}) may be used to conclude proof scripts as well.

   \item [@{command "defer"}~@{text n} and @{command "prefer"}~@{text

   n}] shuffle the list of pending goals: @{command "defer"} puts off

   sub-goal @{text n} to the end of the list (@{text "n = 1"} by

   default), while @{command "prefer"} brings sub-goal @{text n} to the

   front.

   \item [@{command "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 @{command "apply"}.  A few additional emulations of actual

   tactics are provided as well; these would be never used in actual

   structured proofs, of course.

 *}

 section {* Omitting proofs *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "oops"} & : & \isartrans{proof}{theory} \\

   \end{matharray}

   The @{command "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 @{command_ref "sorry"} (see

   \secref{sec:proof-steps}): @{command "oops"} does not observe the

   proof structure at all, but goes back right to the theory level.

   Furthermore, @{command "oops"} does not produce any result theorem

   --- there is no intended claim to be able to complete the proof

   anyhow.

   A typical application of @{command "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 ``@{text "\<dots>"}'' instead of

   the keyword ``@{command "oops"}''.

   \medskip The @{command "oops"} command is undo-able, unlike

   @{command_ref "kill"} (see \secref{sec:history}).  The effect is to

   get back to the theory just before the opening of the proof.

 *}

 section {* Generalized elimination \label{sec:obtain} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "obtain"} & : & \isartrans{proof(state)}{proof(prove)} \\

     @{command_def "guess"}@{text "\<^sup>*"} & : & \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

   @{command "obtain"} language element is like a declaration of

   @{command "fix"} and @{command "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 @{command "obtain"} is defined as follows

   (where @{text "b\<^sub>1, \<dots>, b\<^sub>k"} shall refer to (optional)

   facts indicated for forward chaining).

   \begin{matharray}{l}

     @{text "\<langle>using b\<^sub>1 \<dots> b\<^sub>k\<rangle>"}~~@{command "obtain"}~@{text "x\<^sub>1 \<dots> x\<^sub>m \<WHERE> a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n  \<langle>proof\<rangle> \<equiv>"} \\[1ex]

     \quad @{command "have"}~@{text "\<And>thesis. (\<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots> \<phi>\<^sub>n \<Longrightarrow> thesis) \<Longrightarrow> thesis"} \\

     \quad @{command "proof"}~@{text succeed} \\

     \qquad @{command "fix"}~@{text thesis} \\

     \qquad @{command "assume"}~@{text "that [Pure.intro?]: \<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots> \<phi>\<^sub>n \<Longrightarrow> thesis"} \\

     \qquad @{command "then"}~@{command "show"}~@{text thesis} \\

     \quad\qquad @{command "apply"}~@{text -} \\

     \quad\qquad @{command "using"}~@{text "b\<^sub>1 \<dots> b\<^sub>k  \<langle>proof\<rangle>"} \\

     \quad @{command "qed"} \\

     \quad @{command "fix"}~@{text "x\<^sub>1 \<dots> x\<^sub>m"}~@{command "assume"}@{text "\<^sup>* a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"} \\

   \end{matharray}

   Typically, the soundness proof is relatively straight-forward, often

   just by canonical automated tools such as ``@{command "by"}~@{text

   simp}'' or ``@{command "by"}~@{text blast}''.  Accordingly, the

   ``@{text that}'' reduction above is declared as simplification and

   introduction rule.

   In a sense, @{command "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 @{command "obtain"} without parameters acts

   much like @{command "have"}, where the result is treated as a

   genuine assumption.

   An alternative name to be used instead of ``@{text that}'' above may

   be given in parentheses.

   \medskip The improper variant @{command "guess"} is similar to

   @{command "obtain"}, but derives the obtained statement from the

   course of reasoning!  The proof starts with a fixed goal @{text

   thesis}.  The subsequent proof may refine this to anything of the

   form like @{text "\<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots>

   \<phi>\<^sub>n \<Longrightarrow> thesis"}, but must not introduce new subgoals.  The

   final goal state is then used as reduction rule for the obtain

   scheme described above.  Obtained parameters @{text "x\<^sub>1, \<dots>,

   x\<^sub>m"} 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 @{command "guess"}

   specify a prefix of obtained parameters explicitly in the text.

   It is important to note that the facts introduced by @{command

   "obtain"} and @{command "guess"} may not be polymorphic: any

   type-variables occurring here are fixed in the present context!

 *}

 section {* Calculational reasoning \label{sec:calculation} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "also"} & : & \isartrans{proof(state)}{proof(state)} \\

     @{command_def "finally"} & : & \isartrans{proof(state)}{proof(chain)} \\

     @{command_def "moreover"} & : & \isartrans{proof(state)}{proof(state)} \\

     @{command_def "ultimately"} & : & \isartrans{proof(state)}{proof(chain)} \\

     @{command_def "print_trans_rules"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

     @{attribute trans} & : & \isaratt \\

     @{attribute sym} & : & \isaratt \\

     @{attribute symmetric} & : & \isaratt \\

   \end{matharray}

   Calculational proof is forward reasoning with implicit application

   of transitivity rules (such those of @{text "="}, @{text "\<le>"},

   @{text "<"}).  Isabelle/Isar maintains an auxiliary fact register

   @{fact_ref calculation} for accumulating results obtained by

   transitivity composed with the current result.  Command @{command

   "also"} updates @{fact calculation} involving @{fact this}, while

   @{command "finally"} exhibits the final @{fact 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 @{command "assume"}, @{command

   "note"}, or some finished proof of @{command "have"}, @{command

   "show"} etc.  The @{command "moreover"} and @{command "ultimately"}

   commands are similar to @{command "also"} and @{command "finally"},

   but only collect further results in @{fact calculation} without

   applying any rules yet.

   Also note that the implicit term abbreviation ``@{text "\<dots>"}'' 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}

     @{command "also"}@{text "\<^sub>0"} & \equiv & @{command "note"}~@{text "calculation = this"} \\

     @{command "also"}@{text "\<^sub>n\<^sub>+\<^sub>1"} & \equiv & @{command "note"}~@{text "calculation = trans [OF calculation this]"} \\[0.5ex]

     @{command "finally"} & \equiv & @{command "also"}~@{command "from"}~@{text calculation} \\[0.5ex]

     @{command "moreover"} & \equiv & @{command "note"}~@{text "calculation = calculation this"} \\

     @{command "ultimately"} & \equiv & @{command "moreover"}~@{command "from"}~@{text calculation} \\

   \end{matharray}

   \begin{rail}

     ('also' | 'finally') ('(' thmrefs ')')?

     ;

     'trans' (() | 'add' | 'del')

     ;

   \end{rail}

   \begin{descr}

   \item [@{command "also"}~@{text "(a\<^sub>1 \<dots> a\<^sub>n)"}]

   maintains the auxiliary @{fact calculation} register as follows.

   The first occurrence of @{command "also"} in some calculational

   thread initializes @{fact calculation} by @{fact this}. Any

   subsequent @{command "also"} on the same level of block-structure

   updates @{fact calculation} by some transitivity rule applied to

   @{fact calculation} and @{fact this} (in that order).  Transitivity

   rules are picked from the current context, unless alternative rules

   are given as explicit arguments.

   \item [@{command "finally"}~@{text "(a\<^sub>1 \<dots> a\<^sub>n)"}]

   maintaining @{fact calculation} in the same way as @{command

   "also"}, and concludes the current calculational thread.  The final

   result is exhibited as fact for forward chaining towards the next

   goal. Basically, @{command "finally"} just abbreviates @{command

   "also"}~@{command "from"}~@{fact calculation}.  Typical idioms for

   concluding calculational proofs are ``@{command "finally"}~@{command

   "show"}~@{text ?thesis}~@{command "."}'' and ``@{command

   "finally"}~@{command "have"}~@{text \<phi>}~@{command "."}''.

   \item [@{command "moreover"} and @{command "ultimately"}] are

   analogous to @{command "also"} and @{command "finally"}, but collect

   results only, without applying rules.

   \item [@{command "print_trans_rules"}] prints the list of

   transitivity rules (for calculational commands @{command "also"} and

   @{command "finally"}) and symmetry rules (for the @{attribute

   symmetric} operation and single step elimination patters) of the

   current context.

   \item [@{attribute trans}] declares theorems as transitivity rules.

   \item [@{attribute sym}] declares symmetry rules, as well as

   @{attribute "Pure.elim?"} rules.

   \item [@{attribute symmetric}] resolves a theorem with some rule

   declared as @{attribute sym} in the current context.  For example,

   ``@{command "assume"}~@{text "[symmetric]: x = y"}'' 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 ``@{command

   "assume"}~@{text "x = y"}~@{command "then"}~@{command "have"}~@{text

   "y = x"}~@{command ".."}''.

   \end{descr}

 *}

 end