theory Proof

 imports Main

 begin

 chapter {* Proofs \label{ch: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 typed according to

   the following three different modes of operation:

   \begin{description}

   \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{description}

   The proof mode indicator may be understood as an instruction to the

   writer, telling 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 (extensible)

   language emerging that way from the different types of proof

   commands.  The main ideas of the overall Isar framework are

   explained in \chref{ch:isar-framework}.

 *}

 section {* Proof structure *}

 subsection {* Example proofs *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "example_proof"} & : & @{text "local_theory \<rightarrow> proof(state)"} \\

   \end{matharray}

   \begin{description}

   \item @{command "example_proof"} opens an empty proof body.  This

   allows to experiment with Isar, without producing any persistent

   result.

   Structurally, this is like a vacous @{command "lemma"} statement

   followed by ``@{command "proof"}~@{text "-"}'', which means the

   example proof may be closed by a regular @{command "qed"}, or

   discontinued by @{command "oops"}.

   \end{description}

 *}

 subsection {* Blocks *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "next"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\

     @{command_def "{"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\

     @{command_def "}"} & : & @{text "proof(state) \<rightarrow> 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{description}

   \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{description}

 *}

 subsection {* Omitting proofs *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "oops"} & : & @{text "proof \<rightarrow> local_theory | 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 \chref{ch:document-prep}.

   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 {* Statements *}

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

 text {*

   \begin{matharray}{rcl}

     @{command_def "fix"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\

     @{command_def "assume"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\

     @{command_def "presume"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\

     @{command_def "def"} & : & @{text "proof(state) \<rightarrow> 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{description}

   \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{description}

   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.

 *}

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

 text {*

   \begin{matharray}{rcl}

     @{command_def "let"} & : & @{text "proof(state) \<rightarrow> 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{description}

   \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{description}

   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}).

 *}

 subsection {* Facts and forward chaining *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "note"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\

     @{command_def "then"} & : & @{text "proof(state) \<rightarrow> proof(chain)"} \\

     @{command_def "from"} & : & @{text "proof(state) \<rightarrow> proof(chain)"} \\

     @{command_def "with"} & : & @{text "proof(state) \<rightarrow> proof(chain)"} \\

     @{command_def "using"} & : & @{text "proof(prove) \<rightarrow> proof(prove)"} \\

     @{command_def "unfolding"} & : & @{text "proof(prove) \<rightarrow> 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{description}

   \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{description}

   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.

 *}

 subsection {* Goals \label{sec:goals} *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "lemma"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

     @{command_def "theorem"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

     @{command_def "corollary"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

     @{command_def "schematic_lemma"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

     @{command_def "schematic_theorem"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

     @{command_def "schematic_corollary"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\

     @{command_def "have"} & : & @{text "proof(state) | proof(chain) \<rightarrow> proof(prove)"} \\

     @{command_def "show"} & : & @{text "proof(state) | proof(chain) \<rightarrow> proof(prove)"} \\

     @{command_def "hence"} & : & @{text "proof(state) \<rightarrow> proof(prove)"} \\

     @{command_def "thus"} & : & @{text "proof(state) \<rightarrow> proof(prove)"} \\

     @{command_def "print_statement"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

   \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 (@{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' |

      'schematic\_lemma' | 'schematic\_theorem' | 'schematic\_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{description}

   \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 "schematic_lemma"}, @{command "schematic_theorem"},

   @{command "schematic_corollary"} are similar to @{command "lemma"},

   @{command "theorem"}, @{command "corollary"}, respectively but allow

   the statement to contain unbound schematic variables.

   Under normal circumstances, an Isar proof text needs to specify

   claims explicitly.  Schematic goals are more like goals in Prolog,

   where certain results are synthesized in the course of reasoning.

   With schematic statements, the inherent compositionality of Isar

   proofs is lost, which also impacts performance, because proof

   checking is forced into sequential mode.

   \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{description}

   Any goal statement causes some term abbreviations (such as

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

   \secref{sec:term-abbrev}.

   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}).

 *}

 section {* Refinement steps *}

 subsection {* Proof method expressions \label{sec:proof-meth} *}

 text {*

   Proof methods are either basic ones, or expressions composed of

   methods via ``@{verbatim ","}'' (sequential composition),

   ``@{verbatim "|"}'' (alternative choices), ``@{verbatim "?"}'' 

   (try), ``@{verbatim "+"}'' (repeat at least once), ``@{verbatim

   "["}@{text n}@{verbatim "]"}'' (restriction to first @{text n}

   sub-goals, with default @{text "n = 1"}).  In practice, proof

   methods are usually just a comma separated list of

   \railqtok{nameref}~\railnonterm{args} specifications.  Note that

   parentheses may be dropped for single method specifications (with no

   arguments).

   \indexouternonterm{method}

   \begin{rail}

     method: (nameref | '(' methods ')') (() | '?' | '+' | '[' nat? ']')

     ;

     methods: (nameref args | method) + (',' | '|')

     ;

   \end{rail}

   Proper Isar proof methods do \emph{not} admit arbitrary goal

   addressing, but refer either to the first sub-goal or all sub-goals

   uniformly.  The goal restriction operator ``@{text "[n]"}''

   evaluates a method expression within a sandbox consisting of the

   first @{text n} sub-goals (which need to exist).  For example, the

   method ``@{text "simp_all[3]"}'' simplifies the first three

   sub-goals, while ``@{text "(rule foo, simp_all)[]"}'' simplifies all

   new goals that emerge from applying rule @{text "foo"} to the

   originally first one.

   Improper methods, notably tactic emulations, offer a separate

   low-level goal addressing scheme as explicit argument to the

   individual tactic being involved.  Here ``@{text "[!]"}'' refers to

   all goals, and ``@{text "[n-]"}'' to all goals starting from @{text

   "n"}.

   \indexouternonterm{goalspec}

   \begin{rail}

     goalspec: '[' (nat '-' nat | nat '-' | nat | '!' ) ']'

     ;

   \end{rail}

 *}

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

 text {*

   \begin{matharray}{rcl}

     @{command_def "proof"} & : & @{text "proof(prove) \<rightarrow> proof(state)"} \\

     @{command_def "qed"} & : & @{text "proof(state) \<rightarrow> proof(state) | local_theory | theory"} \\

     @{command_def "by"} & : & @{text "proof(prove) \<rightarrow> proof(state) | local_theory | theory"} \\

     @{command_def ".."} & : & @{text "proof(prove) \<rightarrow> proof(state) | local_theory | theory"} \\

     @{command_def "."} & : & @{text "proof(prove) \<rightarrow> proof(state) | local_theory | theory"} \\

     @{command_def "sorry"} & : & @{text "proof(prove) \<rightarrow> proof(state) | local_theory | 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{description}

   \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{description}

 *}

 subsection {* 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 "-"} & : & @{text method} \\

     @{method_def "fact"} & : & @{text method} \\

     @{method_def "assumption"} & : & @{text method} \\

     @{method_def "this"} & : & @{text method} \\

     @{method_def "rule"} & : & @{text method} \\

     @{attribute_def (Pure) "intro"} & : & @{text attribute} \\

     @{attribute_def (Pure) "elim"} & : & @{text attribute} \\

     @{attribute_def (Pure) "dest"} & : & @{text attribute} \\

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

     @{attribute_def "OF"} & : & @{text attribute} \\

     @{attribute_def "of"} & : & @{text attribute} \\

     @{attribute_def "where"} & : & @{text attribute} \\

   \end{matharray}

   \begin{rail}

     'fact' thmrefs?

     ;

     'rule' thmrefs?

     ;

     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{description}

   \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 (Pure) intro}, @{attribute (Pure) elim},

   @{attribute (Pure) dest} attributes (see below).  This is the

   default behavior of @{command "proof"} and ``@{command ".."}'' 

   (double-dot) steps (see \secref{sec:proof-steps}).

   \item @{attribute (Pure) intro}, @{attribute (Pure) elim}, and

   @{attribute (Pure) dest} declare introduction, elimination, and

   destruct rules, to be used with method @{method rule}, and similar

   tools.  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)

   "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{description}

 *}

 subsection {* 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>*"} & : & @{text "proof(prove) \<rightarrow> proof(prove)"} \\

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

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

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

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

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

   \end{matharray}

   \begin{rail}

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

     ;

     'defer' nat?

     ;

     'prefer' nat

     ;

   \end{rail}

   \begin{description}

   \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 @{text 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{description}

   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.

 *}

 subsection {* Defining proof methods *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "method_setup"} & : & @{text "theory \<rightarrow> theory"} \\

   \end{matharray}

   \begin{rail}

     'method\_setup' name '=' text text

     ;

   \end{rail}

   \begin{description}

   \item @{command "method_setup"}~@{text "name = text description"}

   defines a proof method in the current theory.  The given @{text

   "text"} has to be an ML expression of type

   @{ML_type "(Proof.context -> Proof.method) context_parser"}, cf.\

   basic parsers defined in structure @{ML_struct Args} and @{ML_struct

   Attrib}.  There are also combinators like @{ML METHOD} and @{ML

   SIMPLE_METHOD} to turn certain tactic forms into official proof

   methods; the primed versions refer to tactics with explicit goal

   addressing.

   Here are some example method definitions:

   \end{description}

 *}

     method_setup my_method1 = {*

       Scan.succeed (K (SIMPLE_METHOD' (fn i: int => no_tac)))

     *}  "my first method (without any arguments)"

     method_setup my_method2 = {*

       Scan.succeed (fn ctxt: Proof.context =>

         SIMPLE_METHOD' (fn i: int => no_tac))

     *}  "my second method (with context)"

     method_setup my_method3 = {*

       Attrib.thms >> (fn thms: thm list => fn ctxt: Proof.context =>

         SIMPLE_METHOD' (fn i: int => no_tac))

     *}  "my third method (with theorem arguments and context)"

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

 text {*

   \begin{matharray}{rcl}

     @{command_def "obtain"} & : & @{text "proof(state) | proof(chain) \<rightarrow> proof(prove)"} \\

     @{command_def "guess"}@{text "\<^sup>*"} & : & @{text "proof(state) | proof(chain) \<rightarrow> 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"}~@{method 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"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\

     @{command_def "finally"} & : & @{text "proof(state) \<rightarrow> proof(chain)"} \\

     @{command_def "moreover"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\

     @{command_def "ultimately"} & : & @{text "proof(state) \<rightarrow> proof(chain)"} \\

     @{command_def "print_trans_rules"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

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

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

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

   \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+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{description}

   \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"}@{text "?"} 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{description}

 *}

 section {* Proof by cases and induction \label{sec:cases-induct} *}

 subsection {* Rule contexts *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "case"} & : & @{text "proof(state) \<rightarrow> proof(state)"} \\

     @{command_def "print_cases"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{attribute_def case_names} & : & @{text attribute} \\

     @{attribute_def case_conclusion} & : & @{text attribute} \\

     @{attribute_def params} & : & @{text attribute} \\

     @{attribute_def consumes} & : & @{text attribute} \\

   \end{matharray}

   The puristic way to build up Isar proof contexts is by explicit

   language elements like @{command "fix"}, @{command "assume"},

   @{command "let"} (see \secref{sec:proof-context}).  This is adequate

   for plain natural deduction, but easily becomes unwieldy in concrete

   verification tasks, which typically involve big induction rules with

   several cases.

   The @{command "case"} command provides a shorthand to refer to a

   local context symbolically: certain proof methods provide an

   environment of named ``cases'' of the form @{text "c: x\<^sub>1, \<dots>,

   x\<^sub>m, \<phi>\<^sub>1, \<dots>, \<phi>\<^sub>n"}; the effect of ``@{command

   "case"}~@{text c}'' is then equivalent to ``@{command "fix"}~@{text

   "x\<^sub>1 \<dots> x\<^sub>m"}~@{command "assume"}~@{text "c: \<phi>\<^sub>1 \<dots>

   \<phi>\<^sub>n"}''.  Term bindings may be covered as well, notably

   @{variable ?case} for the main conclusion.

   By default, the ``terminology'' @{text "x\<^sub>1, \<dots>, x\<^sub>m"} of

   a case value is marked as hidden, i.e.\ there is no way to refer to

   such parameters in the subsequent proof text.  After all, original

   rule parameters stem from somewhere outside of the current proof

   text.  By using the explicit form ``@{command "case"}~@{text "(c

   y\<^sub>1 \<dots> y\<^sub>m)"}'' instead, the proof author is able to

   chose local names that fit nicely into the current context.

   \medskip It is important to note that proper use of @{command

   "case"} does not provide means to peek at the current goal state,

   which is not directly observable in Isar!  Nonetheless, goal

   refinement commands do provide named cases @{text "goal\<^sub>i"}

   for each subgoal @{text "i = 1, \<dots>, n"} of the resulting goal state.

   Using this extra feature requires great care, because some bits of

   the internal tactical machinery intrude the proof text.  In

   particular, parameter names stemming from the left-over of automated

   reasoning tools are usually quite unpredictable.

   Under normal circumstances, the text of cases emerge from standard

   elimination or induction rules, which in turn are derived from

   previous theory specifications in a canonical way (say from

   @{command "inductive"} definitions).

   \medskip Proper cases are only available if both the proof method

   and the rules involved support this.  By using appropriate

   attributes, case names, conclusions, and parameters may be also

   declared by hand.  Thus variant versions of rules that have been

   derived manually become ready to use in advanced case analysis

   later.

   \begin{rail}

     'case' (caseref | '(' caseref ((name | underscore) +) ')')

     ;

     caseref: nameref attributes?

     ;

     'case\_names' (name +)

     ;

     'case\_conclusion' name (name *)

     ;

     'params' ((name *) + 'and')

     ;

     'consumes' nat?

     ;

   \end{rail}

   \begin{description}

   \item @{command "case"}~@{text "(c x\<^sub>1 \<dots> x\<^sub>m)"} invokes a named local

   context @{text "c: x\<^sub>1, \<dots>, x\<^sub>m, \<phi>\<^sub>1, \<dots>, \<phi>\<^sub>m"}, as provided by an

   appropriate proof method (such as @{method_ref cases} and

   @{method_ref induct}).  The command ``@{command "case"}~@{text "(c

   x\<^sub>1 \<dots> x\<^sub>m)"}'' abbreviates ``@{command "fix"}~@{text "x\<^sub>1 \<dots>

   x\<^sub>m"}~@{command "assume"}~@{text "c: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}''.

   \item @{command "print_cases"} prints all local contexts of the

   current state, using Isar proof language notation.

   \item @{attribute case_names}~@{text "c\<^sub>1 \<dots> c\<^sub>k"} declares names for

   the local contexts of premises of a theorem; @{text "c\<^sub>1, \<dots>, c\<^sub>k"}

   refers to the \emph{suffix} of the list of premises.

   \item @{attribute case_conclusion}~@{text "c d\<^sub>1 \<dots> d\<^sub>k"} declares

   names for the conclusions of a named premise @{text c}; here @{text

   "d\<^sub>1, \<dots>, d\<^sub>k"} refers to the prefix of arguments of a logical formula

   built by nesting a binary connective (e.g.\ @{text "\<or>"}).

   Note that proof methods such as @{method induct} and @{method

   coinduct} already provide a default name for the conclusion as a

   whole.  The need to name subformulas only arises with cases that

   split into several sub-cases, as in common co-induction rules.

   \item @{attribute params}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots> q\<^sub>1 \<dots> q\<^sub>n"} renames

   the innermost parameters of premises @{text "1, \<dots>, n"} of some

   theorem.  An empty list of names may be given to skip positions,

   leaving the present parameters unchanged.

   Note that the default usage of case rules does \emph{not} directly

   expose parameters to the proof context.

   \item @{attribute consumes}~@{text n} declares the number of ``major

   premises'' of a rule, i.e.\ the number of facts to be consumed when

   it is applied by an appropriate proof method.  The default value of

   @{attribute consumes} is @{text "n = 1"}, which is appropriate for

   the usual kind of cases and induction rules for inductive sets (cf.\

   \secref{sec:hol-inductive}).  Rules without any @{attribute

   consumes} declaration given are treated as if @{attribute

   consumes}~@{text 0} had been specified.

   Note that explicit @{attribute consumes} declarations are only

   rarely needed; this is already taken care of automatically by the

   higher-level @{attribute cases}, @{attribute induct}, and

   @{attribute coinduct} declarations.

   \end{description}

 *}

 subsection {* Proof methods *}

 text {*

   \begin{matharray}{rcl}

     @{method_def cases} & : & @{text method} \\

     @{method_def induct} & : & @{text method} \\

     @{method_def coinduct} & : & @{text method} \\

   \end{matharray}

   The @{method cases}, @{method induct}, and @{method coinduct}

   methods provide a uniform interface to common proof techniques over

   datatypes, inductive predicates (or sets), recursive functions etc.

   The corresponding rules may be specified and instantiated in a

   casual manner.  Furthermore, these methods provide named local

   contexts that may be invoked via the @{command "case"} proof command

   within the subsequent proof text.  This accommodates compact proof

   texts even when reasoning about large specifications.

   The @{method induct} method also provides some additional

   infrastructure in order to be applicable to structure statements

   (either using explicit meta-level connectives, or including facts

   and parameters separately).  This avoids cumbersome encoding of

   ``strengthened'' inductive statements within the object-logic.

   \begin{rail}

     'cases' (insts * 'and') rule?

     ;

     'induct' (definsts * 'and') \\ arbitrary? taking? rule?

     ;

     'coinduct' insts taking rule?

     ;

     rule: ('type' | 'pred' | 'set') ':' (nameref +) | 'rule' ':' (thmref +)

     ;

     definst: name ('==' | equiv) term | inst

     ;

     definsts: ( definst *)

     ;

     arbitrary: 'arbitrary' ':' ((term *) 'and' +)

     ;

     taking: 'taking' ':' insts

     ;

   \end{rail}

   \begin{description}

   \item @{method cases}~@{text "insts R"} applies method @{method

   rule} with an appropriate case distinction theorem, instantiated to

   the subjects @{text insts}.  Symbolic case names are bound according

   to the rule's local contexts.

   The rule is determined as follows, according to the facts and

   arguments passed to the @{method cases} method:

   \medskip

   \begin{tabular}{llll}

     facts           &                 & arguments   & rule \\\hline

                     & @{method cases} &             & classical case split \\

                     & @{method cases} & @{text t}   & datatype exhaustion (type of @{text t}) \\

     @{text "\<turnstile> A t"} & @{method cases} & @{text "\<dots>"} & inductive predicate/set elimination (of @{text A}) \\

     @{text "\<dots>"}     & @{method cases} & @{text "\<dots> rule: R"} & explicit rule @{text R} \\

   \end{tabular}

   \medskip

   Several instantiations may be given, referring to the \emph{suffix}

   of premises of the case rule; within each premise, the \emph{prefix}

   of variables is instantiated.  In most situations, only a single

   term needs to be specified; this refers to the first variable of the

   last premise (it is usually the same for all cases).

   \item @{method induct}~@{text "insts R"} is analogous to the

   @{method cases} method, but refers to induction rules, which are

   determined as follows:

   \medskip

   \begin{tabular}{llll}

     facts           &                  & arguments            & rule \\\hline

                     & @{method induct} & @{text "P x"}        & datatype induction (type of @{text x}) \\

     @{text "\<turnstile> A x"} & @{method induct} & @{text "\<dots>"}          & predicate/set induction (of @{text A}) \\

     @{text "\<dots>"}     & @{method induct} & @{text "\<dots> rule: R"} & explicit rule @{text R} \\

   \end{tabular}

   \medskip

   Several instantiations may be given, each referring to some part of

   a mutual inductive definition or datatype --- only related partial

   induction rules may be used together, though.  Any of the lists of

   terms @{text "P, x, \<dots>"} refers to the \emph{suffix} of variables

   present in the induction rule.  This enables the writer to specify

   only induction variables, or both predicates and variables, for

   example.

   Instantiations may be definitional: equations @{text "x \<equiv> t"}

   introduce local definitions, which are inserted into the claim and

   discharged after applying the induction rule.  Equalities reappear

   in the inductive cases, but have been transformed according to the

   induction principle being involved here.  In order to achieve

   practically useful induction hypotheses, some variables occurring in

   @{text t} need to be fixed (see below).

   The optional ``@{text "arbitrary: x\<^sub>1 \<dots> x\<^sub>m"}''

   specification generalizes variables @{text "x\<^sub>1, \<dots>,

   x\<^sub>m"} of the original goal before applying induction.  Thus

   induction hypotheses may become sufficiently general to get the

   proof through.  Together with definitional instantiations, one may

   effectively perform induction over expressions of a certain

   structure.

   The optional ``@{text "taking: t\<^sub>1 \<dots> t\<^sub>n"}''

   specification provides additional instantiations of a prefix of

   pending variables in the rule.  Such schematic induction rules

   rarely occur in practice, though.

   \item @{method coinduct}~@{text "inst R"} is analogous to the

   @{method induct} method, but refers to coinduction rules, which are

   determined as follows:

   \medskip

   \begin{tabular}{llll}

     goal          &                    & arguments & rule \\\hline

                   & @{method coinduct} & @{text x} & type coinduction (type of @{text x}) \\

     @{text "A x"} & @{method coinduct} & @{text "\<dots>"} & predicate/set coinduction (of @{text A}) \\

     @{text "\<dots>"}   & @{method coinduct} & @{text "\<dots> rule: R"} & explicit rule @{text R} \\

   \end{tabular}

   Coinduction is the dual of induction.  Induction essentially

   eliminates @{text "A x"} towards a generic result @{text "P x"},

   while coinduction introduces @{text "A x"} starting with @{text "B

   x"}, for a suitable ``bisimulation'' @{text B}.  The cases of a

   coinduct rule are typically named after the predicates or sets being

   covered, while the conclusions consist of several alternatives being

   named after the individual destructor patterns.

   The given instantiation refers to the \emph{suffix} of variables

   occurring in the rule's major premise, or conclusion if unavailable.

   An additional ``@{text "taking: t\<^sub>1 \<dots> t\<^sub>n"}''

   specification may be required in order to specify the bisimulation

   to be used in the coinduction step.

   \end{description}

   Above methods produce named local contexts, as determined by the

   instantiated rule as given in the text.  Beyond that, the @{method

   induct} and @{method coinduct} methods guess further instantiations

   from the goal specification itself.  Any persisting unresolved

   schematic variables of the resulting rule will render the the

   corresponding case invalid.  The term binding @{variable ?case} for

   the conclusion will be provided with each case, provided that term

   is fully specified.

   The @{command "print_cases"} command prints all named cases present

   in the current proof state.

   \medskip Despite the additional infrastructure, both @{method cases}

   and @{method coinduct} merely apply a certain rule, after

   instantiation, while conforming due to the usual way of monotonic

   natural deduction: the context of a structured statement @{text

   "\<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots> \<phi>\<^sub>n \<Longrightarrow> \<dots>"}

   reappears unchanged after the case split.

   The @{method induct} method is fundamentally different in this

   respect: the meta-level structure is passed through the

   ``recursive'' course involved in the induction.  Thus the original

   statement is basically replaced by separate copies, corresponding to

   the induction hypotheses and conclusion; the original goal context

   is no longer available.  Thus local assumptions, fixed parameters

   and definitions effectively participate in the inductive rephrasing

   of the original statement.

   In induction proofs, local assumptions introduced by cases are split

   into two different kinds: @{text hyps} stemming from the rule and

   @{text prems} from the goal statement.  This is reflected in the

   extracted cases accordingly, so invoking ``@{command "case"}~@{text

   c}'' will provide separate facts @{text c.hyps} and @{text c.prems},

   as well as fact @{text c} to hold the all-inclusive list.

   \medskip Facts presented to either method are consumed according to

   the number of ``major premises'' of the rule involved, which is

   usually 0 for plain cases and induction rules of datatypes etc.\ and

   1 for rules of inductive predicates or sets and the like.  The

   remaining facts are inserted into the goal verbatim before the

   actual @{text cases}, @{text induct}, or @{text coinduct} rule is

   applied.

 *}

 subsection {* Declaring rules *}

 text {*

   \begin{matharray}{rcl}

     @{command_def "print_induct_rules"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\

     @{attribute_def cases} & : & @{text attribute} \\

     @{attribute_def induct} & : & @{text attribute} \\

     @{attribute_def coinduct} & : & @{text attribute} \\

   \end{matharray}

   \begin{rail}

     'cases' spec

     ;

     'induct' spec

     ;

     'coinduct' spec

     ;

     spec: (('type' | 'pred' | 'set') ':' nameref) | 'del'

     ;

   \end{rail}

   \begin{description}

   \item @{command "print_induct_rules"} prints cases and induct rules

   for predicates (or sets) and types of the current context.

   \item @{attribute cases}, @{attribute induct}, and @{attribute

   coinduct} (as attributes) declare rules for reasoning about

   (co)inductive predicates (or sets) and types, using the

   corresponding methods of the same name.  Certain definitional

   packages of object-logics usually declare emerging cases and

   induction rules as expected, so users rarely need to intervene.

   Rules may be deleted via the @{text "del"} specification, which

   covers all of the @{text "type"}/@{text "pred"}/@{text "set"}

   sub-categories simultaneously.  For example, @{attribute

   cases}~@{text del} removes any @{attribute cases} rules declared for

   some type, predicate, or set.

   Manual rule declarations usually refer to the @{attribute

   case_names} and @{attribute params} attributes to adjust names of

   cases and parameters of a rule; the @{attribute consumes}

   declaration is taken care of automatically: @{attribute

   consumes}~@{text 0} is specified for ``type'' rules and @{attribute

   consumes}~@{text 1} for ``predicate'' / ``set'' rules.

   \end{description}

 *}

 end