src/Doc/Isar_Ref/Proof.thy
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     1 (*:maxLineLen=78:*)
     2 
     3 theory Proof
     4   imports Main Base
     5 begin
     6 
     7 chapter \<open>Proofs \label{ch:proofs}\<close>
     8 
     9 text \<open>
    10   Proof commands perform transitions of Isar/VM machine configurations, which
    11   are block-structured, consisting of a stack of nodes with three main
    12   components: logical proof context, current facts, and open goals. Isar/VM
    13   transitions are typed according to the following three different modes of
    14   operation:
    15 
    16     \<^descr> \<open>proof(prove)\<close> means that a new goal has just been stated that is now to
    17     be \<^emph>\<open>proven\<close>; the next command may refine it by some proof method, and
    18     enter a sub-proof to establish the actual result.
    19 
    20     \<^descr> \<open>proof(state)\<close> is like a nested theory mode: the context may be
    21     augmented by \<^emph>\<open>stating\<close> additional assumptions, intermediate results etc.
    22 
    23     \<^descr> \<open>proof(chain)\<close> is intermediate between \<open>proof(state)\<close> and
    24     \<open>proof(prove)\<close>: existing facts (i.e.\ the contents of the special
    25     @{fact_ref this} register) have been just picked up in order to be used
    26     when refining the goal claimed next.
    27 
    28   The proof mode indicator may be understood as an instruction to the writer,
    29   telling what kind of operation may be performed next. The corresponding
    30   typings of proof commands restricts the shape of well-formed proof texts to
    31   particular command sequences. So dynamic arrangements of commands eventually
    32   turn out as static texts of a certain structure.
    33 
    34   \Appref{ap:refcard} gives a simplified grammar of the (extensible) language
    35   emerging that way from the different types of proof commands. The main ideas
    36   of the overall Isar framework are explained in \chref{ch:isar-framework}.
    37 \<close>
    38 
    39 
    40 section \<open>Proof structure\<close>
    41 
    42 subsection \<open>Formal notepad\<close>
    43 
    44 text \<open>
    45   \begin{matharray}{rcl}
    46     @{command_def "notepad"} & : & \<open>local_theory \<rightarrow> proof(state)\<close> \\
    47   \end{matharray}
    48 
    49   @{rail \<open>
    50     @@{command notepad} @'begin'
    51     ;
    52     @@{command end}
    53   \<close>}
    54 
    55   \<^descr> @{command "notepad"}~@{keyword "begin"} opens a proof state without any
    56   goal statement. This allows to experiment with Isar, without producing any
    57   persistent result. The notepad is closed by @{command "end"}.
    58 \<close>
    59 
    60 
    61 subsection \<open>Blocks\<close>
    62 
    63 text \<open>
    64   \begin{matharray}{rcl}
    65     @{command_def "next"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
    66     @{command_def "{"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
    67     @{command_def "}"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
    68   \end{matharray}
    69 
    70   While Isar is inherently block-structured, opening and closing blocks is
    71   mostly handled rather casually, with little explicit user-intervention. Any
    72   local goal statement automatically opens \<^emph>\<open>two\<close> internal blocks, which are
    73   closed again when concluding the sub-proof (by @{command "qed"} etc.).
    74   Sections of different context within a sub-proof may be switched via
    75   @{command "next"}, which is just a single block-close followed by block-open
    76   again. The effect of @{command "next"} is to reset the local proof context;
    77   there is no goal focus involved here!
    78 
    79   For slightly more advanced applications, there are explicit block
    80   parentheses as well. These typically achieve a stronger forward style of
    81   reasoning.
    82 
    83   \<^descr> @{command "next"} switches to a fresh block within a sub-proof, resetting
    84   the local context to the initial one.
    85 
    86   \<^descr> @{command "{"} and @{command "}"} explicitly open and close blocks. Any
    87   current facts pass through ``@{command "{"}'' unchanged, while ``@{command
    88   "}"}'' causes any result to be \<^emph>\<open>exported\<close> into the enclosing context. Thus
    89   fixed variables are generalized, assumptions discharged, and local
    90   definitions unfolded (cf.\ \secref{sec:proof-context}). There is no
    91   difference of @{command "assume"} and @{command "presume"} in this mode of
    92   forward reasoning --- in contrast to plain backward reasoning with the
    93   result exported at @{command "show"} time.
    94 \<close>
    95 
    96 
    97 subsection \<open>Omitting proofs\<close>
    98 
    99 text \<open>
   100   \begin{matharray}{rcl}
   101     @{command_def "oops"} & : & \<open>proof \<rightarrow> local_theory | theory\<close> \\
   102   \end{matharray}
   103 
   104   The @{command "oops"} command discontinues the current proof attempt, while
   105   considering the partial proof text as properly processed. This is
   106   conceptually quite different from ``faking'' actual proofs via @{command_ref
   107   "sorry"} (see \secref{sec:proof-steps}): @{command "oops"} does not observe
   108   the proof structure at all, but goes back right to the theory level.
   109   Furthermore, @{command "oops"} does not produce any result theorem --- there
   110   is no intended claim to be able to complete the proof in any way.
   111 
   112   A typical application of @{command "oops"} is to explain Isar proofs
   113   \<^emph>\<open>within\<close> the system itself, in conjunction with the document preparation
   114   tools of Isabelle described in \chref{ch:document-prep}. Thus partial or
   115   even wrong proof attempts can be discussed in a logically sound manner. Note
   116   that the Isabelle {\LaTeX} macros can be easily adapted to print something
   117   like ``\<open>\<dots>\<close>'' instead of the keyword ``@{command "oops"}''.
   118 \<close>
   119 
   120 
   121 section \<open>Statements\<close>
   122 
   123 subsection \<open>Context elements \label{sec:proof-context}\<close>
   124 
   125 text \<open>
   126   \begin{matharray}{rcl}
   127     @{command_def "fix"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
   128     @{command_def "assume"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
   129     @{command_def "presume"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
   130     @{command_def "define"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
   131     @{command_def "def"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
   132   \end{matharray}
   133 
   134   The logical proof context consists of fixed variables and assumptions. The
   135   former closely correspond to Skolem constants, or meta-level universal
   136   quantification as provided by the Isabelle/Pure logical framework.
   137   Introducing some \<^emph>\<open>arbitrary, but fixed\<close> variable via ``@{command
   138   "fix"}~\<open>x\<close>'' results in a local value that may be used in the subsequent
   139   proof as any other variable or constant. Furthermore, any result \<open>\<turnstile> \<phi>[x]\<close>
   140   exported from the context will be universally closed wrt.\ \<open>x\<close> at the
   141   outermost level: \<open>\<turnstile> \<And>x. \<phi>[x]\<close> (this is expressed in normal form using
   142   Isabelle's meta-variables).
   143 
   144   Similarly, introducing some assumption \<open>\<chi>\<close> has two effects. On the one hand,
   145   a local theorem is created that may be used as a fact in subsequent proof
   146   steps. On the other hand, any result \<open>\<chi> \<turnstile> \<phi>\<close> exported from the context
   147   becomes conditional wrt.\ the assumption: \<open>\<turnstile> \<chi> \<Longrightarrow> \<phi>\<close>. Thus, solving an
   148   enclosing goal using such a result would basically introduce a new subgoal
   149   stemming from the assumption. How this situation is handled depends on the
   150   version of assumption command used: while @{command "assume"} insists on
   151   solving the subgoal by unification with some premise of the goal, @{command
   152   "presume"} leaves the subgoal unchanged in order to be proved later by the
   153   user.
   154 
   155   Local definitions, introduced by ``\<^theory_text>\<open>define x where x = t\<close>'', are achieved
   156   by combining ``@{command "fix"}~\<open>x\<close>'' with another version of assumption
   157   that causes any hypothetical equation \<open>x \<equiv> t\<close> to be eliminated by the
   158   reflexivity rule. Thus, exporting some result \<open>x \<equiv> t \<turnstile> \<phi>[x]\<close> yields \<open>\<turnstile>
   159   \<phi>[t]\<close>.
   160 
   161   @{rail \<open>
   162     @@{command fix} @{syntax vars}
   163     ;
   164     (@@{command assume} | @@{command presume}) concl prems @{syntax for_fixes}
   165     ;
   166     concl: (@{syntax props} + @'and')
   167     ;
   168     prems: (@'if' (@{syntax props'} + @'and'))?
   169     ;
   170     @@{command define} @{syntax vars} @'where'
   171       (@{syntax props} + @'and') @{syntax for_fixes}
   172     ;
   173     @@{command def} (def + @'and')
   174     ;
   175     def: @{syntax thmdecl}? \<newline>
   176       @{syntax name} ('==' | '\<equiv>') @{syntax term} @{syntax term_pat}?
   177   \<close>}
   178 
   179   \<^descr> @{command "fix"}~\<open>x\<close> introduces a local variable \<open>x\<close> that is \<^emph>\<open>arbitrary,
   180   but fixed\<close>.
   181 
   182   \<^descr> @{command "assume"}~\<open>a: \<phi>\<close> and @{command "presume"}~\<open>a: \<phi>\<close> introduce a
   183   local fact \<open>\<phi> \<turnstile> \<phi>\<close> by assumption. Subsequent results applied to an enclosing
   184   goal (e.g.\ by @{command_ref "show"}) are handled as follows: @{command
   185   "assume"} expects to be able to unify with existing premises in the goal,
   186   while @{command "presume"} leaves \<open>\<phi>\<close> as new subgoals.
   187 
   188   Several lists of assumptions may be given (separated by @{keyword_ref
   189   "and"}; the resulting list of current facts consists of all of these
   190   concatenated.
   191 
   192   A structured assumption like \<^theory_text>\<open>assume "B x" if "A x" for x\<close> is equivalent to
   193   \<^theory_text>\<open>assume "\<And>x. A x \<Longrightarrow> B x"\<close>, but vacuous quantification is avoided: a
   194   for-context only effects propositions according to actual use of variables.
   195 
   196   \<^descr> \<^theory_text>\<open>define x where "x = t"\<close> introduces a local (non-polymorphic) definition.
   197   In results that are exported from the context, \<open>x\<close> is replaced by \<open>t\<close>.
   198 
   199   Internally, equational assumptions are added to the context in Pure form,
   200   using \<open>x \<equiv> t\<close> instead of \<open>x = t\<close> or \<open>x \<longleftrightarrow> t\<close> from the object-logic. When
   201   exporting results from the context, \<open>x\<close> is generalized and the assumption
   202   discharged by reflexivity, causing the replacement by \<open>t\<close>.
   203 
   204   The default name for the definitional fact is \<open>x_def\<close>. Several simultaneous
   205   definitions may be given as well, with a collective default name.
   206 
   207   \<^medskip>
   208   It is also possible to abstract over local parameters as follows: \<^theory_text>\<open>define f
   209   :: "'a \<Rightarrow> 'b" where "f x = t" for x :: 'a\<close>.
   210 
   211   \<^descr> \<^theory_text>\<open>def x \<equiv> t\<close> introduces a local (non-polymorphic) definition. This is an
   212   old form of \<^theory_text>\<open>define x where "x = t"\<close>.
   213 \<close>
   214 
   215 
   216 subsection \<open>Term abbreviations \label{sec:term-abbrev}\<close>
   217 
   218 text \<open>
   219   \begin{matharray}{rcl}
   220     @{command_def "let"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
   221     @{keyword_def "is"} & : & syntax \\
   222   \end{matharray}
   223 
   224   Abbreviations may be either bound by explicit @{command "let"}~\<open>p \<equiv> t\<close>
   225   statements, or by annotating assumptions or goal statements with a list of
   226   patterns ``\<^theory_text>\<open>(is p\<^sub>1 \<dots> p\<^sub>n)\<close>''. In both cases, higher-order matching is
   227   invoked to bind extra-logical term variables, which may be either named
   228   schematic variables of the form \<open>?x\<close>, or nameless dummies ``@{variable _}''
   229   (underscore). Note that in the @{command "let"} form the patterns occur on
   230   the left-hand side, while the @{keyword "is"} patterns are in postfix
   231   position.
   232 
   233   Polymorphism of term bindings is handled in Hindley-Milner style, similar to
   234   ML. Type variables referring to local assumptions or open goal statements
   235   are \<^emph>\<open>fixed\<close>, while those of finished results or bound by @{command "let"}
   236   may occur in \<^emph>\<open>arbitrary\<close> instances later. Even though actual polymorphism
   237   should be rarely used in practice, this mechanism is essential to achieve
   238   proper incremental type-inference, as the user proceeds to build up the Isar
   239   proof text from left to right.
   240 
   241   \<^medskip>
   242   Term abbreviations are quite different from local definitions as introduced
   243   via @{command "define"} (see \secref{sec:proof-context}). The latter are
   244   visible within the logic as actual equations, while abbreviations disappear
   245   during the input process just after type checking. Also note that @{command
   246   "define"} does not support polymorphism.
   247 
   248   @{rail \<open>
   249     @@{command let} ((@{syntax term} + @'and') '=' @{syntax term} + @'and')
   250   \<close>}
   251 
   252   The syntax of @{keyword "is"} patterns follows @{syntax term_pat} or
   253   @{syntax prop_pat} (see \secref{sec:term-decls}).
   254 
   255     \<^descr> \<^theory_text>\<open>let p\<^sub>1 = t\<^sub>1 and \<dots> p\<^sub>n = t\<^sub>n\<close> binds any text variables in patterns
   256     \<open>p\<^sub>1, \<dots>, p\<^sub>n\<close> by simultaneous higher-order matching against terms \<open>t\<^sub>1, \<dots>,
   257     t\<^sub>n\<close>.
   258 
   259     \<^descr> \<^theory_text>\<open>(is p\<^sub>1 \<dots> p\<^sub>n)\<close> resembles @{command "let"}, but matches \<open>p\<^sub>1, \<dots>, p\<^sub>n\<close>
   260     against the preceding statement. Also note that @{keyword "is"} is not a
   261     separate command, but part of others (such as @{command "assume"},
   262     @{command "have"} etc.).
   263 
   264   Some \<^emph>\<open>implicit\<close> term abbreviations\index{term abbreviations} for goals and
   265   facts are available as well. For any open goal, @{variable_ref thesis}
   266   refers to its object-level statement, abstracted over any meta-level
   267   parameters (if present). Likewise, @{variable_ref this} is bound for fact
   268   statements resulting from assumptions or finished goals. In case @{variable
   269   this} refers to an object-logic statement that is an application \<open>f t\<close>, then
   270   \<open>t\<close> is bound to the special text variable ``@{variable "\<dots>"}'' (three dots).
   271   The canonical application of this convenience are calculational proofs (see
   272   \secref{sec:calculation}).
   273 \<close>
   274 
   275 
   276 subsection \<open>Facts and forward chaining \label{sec:proof-facts}\<close>
   277 
   278 text \<open>
   279   \begin{matharray}{rcl}
   280     @{command_def "note"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
   281     @{command_def "then"} & : & \<open>proof(state) \<rightarrow> proof(chain)\<close> \\
   282     @{command_def "from"} & : & \<open>proof(state) \<rightarrow> proof(chain)\<close> \\
   283     @{command_def "with"} & : & \<open>proof(state) \<rightarrow> proof(chain)\<close> \\
   284     @{command_def "using"} & : & \<open>proof(prove) \<rightarrow> proof(prove)\<close> \\
   285     @{command_def "unfolding"} & : & \<open>proof(prove) \<rightarrow> proof(prove)\<close> \\
   286     @{method_def "use"} & : & \<open>method\<close> \\
   287     @{fact_def "method_facts"} & : & \<open>fact\<close> \\
   288   \end{matharray}
   289 
   290   New facts are established either by assumption or proof of local statements.
   291   Any fact will usually be involved in further proofs, either as explicit
   292   arguments of proof methods, or when forward chaining towards the next goal
   293   via @{command "then"} (and variants); @{command "from"} and @{command
   294   "with"} are composite forms involving @{command "note"}. The @{command
   295   "using"} elements augments the collection of used facts \<^emph>\<open>after\<close> a goal has
   296   been stated. Note that the special theorem name @{fact_ref this} refers to
   297   the most recently established facts, but only \<^emph>\<open>before\<close> issuing a follow-up
   298   claim.
   299 
   300   @{rail \<open>
   301     @@{command note} (@{syntax thmdef}? @{syntax thms} + @'and')
   302     ;
   303     (@@{command from} | @@{command with} | @@{command using} | @@{command unfolding})
   304       (@{syntax thms} + @'and')
   305     ;
   306     @{method use} @{syntax thms} @'in' @{syntax method}
   307   \<close>}
   308 
   309   \<^descr> @{command "note"}~\<open>a = b\<^sub>1 \<dots> b\<^sub>n\<close> recalls existing facts \<open>b\<^sub>1, \<dots>, b\<^sub>n\<close>,
   310   binding the result as \<open>a\<close>. Note that attributes may be involved as well,
   311   both on the left and right hand sides.
   312 
   313   \<^descr> @{command "then"} indicates forward chaining by the current facts in order
   314   to establish the goal to be claimed next. The initial proof method invoked
   315   to refine that will be offered the facts to do ``anything appropriate'' (see
   316   also \secref{sec:proof-steps}). For example, method @{method (Pure) rule}
   317   (see \secref{sec:pure-meth-att}) would typically do an elimination rather
   318   than an introduction. Automatic methods usually insert the facts into the
   319   goal state before operation. This provides a simple scheme to control
   320   relevance of facts in automated proof search.
   321 
   322   \<^descr> @{command "from"}~\<open>b\<close> abbreviates ``@{command "note"}~\<open>b\<close>~@{command
   323   "then"}''; thus @{command "then"} is equivalent to ``@{command
   324   "from"}~\<open>this\<close>''.
   325 
   326   \<^descr> @{command "with"}~\<open>b\<^sub>1 \<dots> b\<^sub>n\<close> abbreviates ``@{command "from"}~\<open>b\<^sub>1 \<dots> b\<^sub>n
   327   \<AND> this\<close>''; thus the forward chaining is from earlier facts together
   328   with the current ones.
   329 
   330   \<^descr> @{command "using"}~\<open>b\<^sub>1 \<dots> b\<^sub>n\<close> augments the facts to be used by a
   331   subsequent refinement step (such as @{command_ref "apply"} or @{command_ref
   332   "proof"}).
   333 
   334   \<^descr> @{command "unfolding"}~\<open>b\<^sub>1 \<dots> b\<^sub>n\<close> is structurally similar to @{command
   335   "using"}, but unfolds definitional equations \<open>b\<^sub>1 \<dots> b\<^sub>n\<close> throughout the goal
   336   state and facts. See also the proof method @{method_ref unfold}.
   337 
   338   \<^descr> \<^theory_text>\<open>(use b\<^sub>1 \<dots> b\<^sub>n in method)\<close> uses the facts in the given method
   339   expression. The facts provided by the proof state (via @{command "using"}
   340   etc.) are ignored, but it is possible to refer to @{fact method_facts}
   341   explicitly.
   342 
   343   \<^descr> @{fact method_facts} is a dynamic fact that refers to the currently used
   344   facts of the goal state.
   345 
   346 
   347   Forward chaining with an empty list of theorems is the same as not chaining
   348   at all. Thus ``@{command "from"}~\<open>nothing\<close>'' has no effect apart from
   349   entering \<open>prove(chain)\<close> mode, since @{fact_ref nothing} is bound to the
   350   empty list of theorems.
   351 
   352   Basic proof methods (such as @{method_ref (Pure) rule}) expect multiple
   353   facts to be given in their proper order, corresponding to a prefix of the
   354   premises of the rule involved. Note that positions may be easily skipped
   355   using something like @{command "from"}~\<open>_ \<AND> a \<AND> b\<close>, for example.
   356   This involves the trivial rule \<open>PROP \<psi> \<Longrightarrow> PROP \<psi>\<close>, which is bound in
   357   Isabelle/Pure as ``@{fact_ref "_"}'' (underscore).
   358 
   359   Automated methods (such as @{method simp} or @{method auto}) just insert any
   360   given facts before their usual operation. Depending on the kind of procedure
   361   involved, the order of facts is less significant here.
   362 \<close>
   363 
   364 
   365 subsection \<open>Goals \label{sec:goals}\<close>
   366 
   367 text \<open>
   368   \begin{matharray}{rcl}
   369     @{command_def "lemma"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close> \\
   370     @{command_def "theorem"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close> \\
   371     @{command_def "corollary"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close> \\
   372     @{command_def "proposition"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close> \\
   373     @{command_def "schematic_goal"} & : & \<open>local_theory \<rightarrow> proof(prove)\<close> \\
   374     @{command_def "have"} & : & \<open>proof(state) | proof(chain) \<rightarrow> proof(prove)\<close> \\
   375     @{command_def "show"} & : & \<open>proof(state) | proof(chain) \<rightarrow> proof(prove)\<close> \\
   376     @{command_def "hence"} & : & \<open>proof(state) \<rightarrow> proof(prove)\<close> \\
   377     @{command_def "thus"} & : & \<open>proof(state) \<rightarrow> proof(prove)\<close> \\
   378     @{command_def "print_statement"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
   379   \end{matharray}
   380 
   381   From a theory context, proof mode is entered by an initial goal command such
   382   as @{command "lemma"}. Within a proof context, new claims may be introduced
   383   locally; there are variants to interact with the overall proof structure
   384   specifically, such as @{command have} or @{command show}.
   385 
   386   Goals may consist of multiple statements, resulting in a list of facts
   387   eventually. A pending multi-goal is internally represented as a meta-level
   388   conjunction (\<open>&&&\<close>), which is usually split into the corresponding number of
   389   sub-goals prior to an initial method application, via @{command_ref "proof"}
   390   (\secref{sec:proof-steps}) or @{command_ref "apply"}
   391   (\secref{sec:tactic-commands}). The @{method_ref induct} method covered in
   392   \secref{sec:cases-induct} acts on multiple claims simultaneously.
   393 
   394   Claims at the theory level may be either in short or long form. A short goal
   395   merely consists of several simultaneous propositions (often just one). A
   396   long goal includes an explicit context specification for the subsequent
   397   conclusion, involving local parameters and assumptions. Here the role of
   398   each part of the statement is explicitly marked by separate keywords (see
   399   also \secref{sec:locale}); the local assumptions being introduced here are
   400   available as @{fact_ref assms} in the proof. Moreover, there are two kinds
   401   of conclusions: @{element_def "shows"} states several simultaneous
   402   propositions (essentially a big conjunction), while @{element_def "obtains"}
   403   claims several simultaneous simultaneous contexts of (essentially a big
   404   disjunction of eliminated parameters and assumptions, cf.\
   405   \secref{sec:obtain}).
   406 
   407   @{rail \<open>
   408     (@@{command lemma} | @@{command theorem} | @@{command corollary} |
   409      @@{command proposition} | @@{command schematic_goal})
   410       (long_statement | short_statement)
   411     ;
   412     (@@{command have} | @@{command show} | @@{command hence} | @@{command thus})
   413       stmt cond_stmt @{syntax for_fixes}
   414     ;
   415     @@{command print_statement} @{syntax modes}? @{syntax thms}
   416     ;
   417 
   418     stmt: (@{syntax props} + @'and')
   419     ;
   420     cond_stmt: ((@'if' | @'when') stmt)?
   421     ;
   422     short_statement: stmt (@'if' stmt)? @{syntax for_fixes}
   423     ;
   424     long_statement: @{syntax thmdecl}? context conclusion
   425     ;
   426     context: (@{syntax_ref "includes"}?) (@{syntax context_elem} *)
   427     ;
   428     conclusion: @'shows' stmt | @'obtains' @{syntax obtain_clauses}
   429     ;
   430     @{syntax_def obtain_clauses}: (@{syntax par_name}? obtain_case + '|')
   431     ;
   432     @{syntax_def obtain_case}: @{syntax vars} @'where'
   433       (@{syntax thmdecl}? (@{syntax prop}+) + @'and')
   434   \<close>}
   435 
   436   \<^descr> @{command "lemma"}~\<open>a: \<phi>\<close> enters proof mode with \<open>\<phi>\<close> as main goal,
   437   eventually resulting in some fact \<open>\<turnstile> \<phi>\<close> to be put back into the target
   438   context.
   439 
   440   A @{syntax long_statement} may build up an initial proof context for the
   441   subsequent claim, potentially including local definitions and syntax; see
   442   also @{syntax "includes"} in \secref{sec:bundle} and @{syntax context_elem}
   443   in \secref{sec:locale}.
   444 
   445   A @{syntax short_statement} consists of propositions as conclusion, with an
   446   option context of premises and parameters, via \<^verbatim>\<open>if\<close>/\<^verbatim>\<open>for\<close> in postfix
   447   notation, corresponding to \<^verbatim>\<open>assumes\<close>/\<^verbatim>\<open>fixes\<close> in the long prefix notation.
   448 
   449   Local premises (if present) are called ``\<open>assms\<close>'' for @{syntax
   450   long_statement}, and ``\<open>that\<close>'' for @{syntax short_statement}.
   451 
   452   \<^descr> @{command "theorem"}, @{command "corollary"}, and @{command "proposition"}
   453   are the same as @{command "lemma"}. The different command names merely serve
   454   as a formal comment in the theory source.
   455 
   456   \<^descr> @{command "schematic_goal"} is similar to @{command "theorem"}, but allows
   457   the statement to contain unbound schematic variables.
   458 
   459   Under normal circumstances, an Isar proof text needs to specify claims
   460   explicitly. Schematic goals are more like goals in Prolog, where certain
   461   results are synthesized in the course of reasoning. With schematic
   462   statements, the inherent compositionality of Isar proofs is lost, which also
   463   impacts performance, because proof checking is forced into sequential mode.
   464 
   465   \<^descr> @{command "have"}~\<open>a: \<phi>\<close> claims a local goal, eventually resulting in a
   466   fact within the current logical context. This operation is completely
   467   independent of any pending sub-goals of an enclosing goal statements, so
   468   @{command "have"} may be freely used for experimental exploration of
   469   potential results within a proof body.
   470 
   471   \<^descr> @{command "show"}~\<open>a: \<phi>\<close> is like @{command "have"}~\<open>a: \<phi>\<close> plus a second
   472   stage to refine some pending sub-goal for each one of the finished result,
   473   after having been exported into the corresponding context (at the head of
   474   the sub-proof of this @{command "show"} command).
   475 
   476   To accommodate interactive debugging, resulting rules are printed before
   477   being applied internally. Even more, interactive execution of @{command
   478   "show"} predicts potential failure and displays the resulting error as a
   479   warning beforehand. Watch out for the following message:
   480   @{verbatim [display] \<open>Local statement fails to refine any pending goal\<close>}
   481 
   482   \<^descr> @{command "hence"} expands to ``@{command "then"}~@{command "have"}'' and
   483   @{command "thus"} expands to ``@{command "then"}~@{command "show"}''. These
   484   conflations are left-over from early history of Isar. The expanded syntax is
   485   more orthogonal and improves readability and maintainability of proofs.
   486 
   487   \<^descr> @{command "print_statement"}~\<open>a\<close> prints facts from the current theory or
   488   proof context in long statement form, according to the syntax for @{command
   489   "lemma"} given above.
   490 
   491 
   492   Any goal statement causes some term abbreviations (such as @{variable_ref
   493   "?thesis"}) to be bound automatically, see also \secref{sec:term-abbrev}.
   494 
   495   Structured goal statements involving @{keyword_ref "if"} or @{keyword_ref
   496   "when"} define the special fact @{fact_ref that} to refer to these
   497   assumptions in the proof body. The user may provide separate names according
   498   to the syntax of the statement.
   499 \<close>
   500 
   501 
   502 section \<open>Calculational reasoning \label{sec:calculation}\<close>
   503 
   504 text \<open>
   505   \begin{matharray}{rcl}
   506     @{command_def "also"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
   507     @{command_def "finally"} & : & \<open>proof(state) \<rightarrow> proof(chain)\<close> \\
   508     @{command_def "moreover"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
   509     @{command_def "ultimately"} & : & \<open>proof(state) \<rightarrow> proof(chain)\<close> \\
   510     @{command_def "print_trans_rules"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
   511     @{attribute trans} & : & \<open>attribute\<close> \\
   512     @{attribute sym} & : & \<open>attribute\<close> \\
   513     @{attribute symmetric} & : & \<open>attribute\<close> \\
   514   \end{matharray}
   515 
   516   Calculational proof is forward reasoning with implicit application of
   517   transitivity rules (such those of \<open>=\<close>, \<open>\<le>\<close>, \<open><\<close>). Isabelle/Isar maintains an
   518   auxiliary fact register @{fact_ref calculation} for accumulating results
   519   obtained by transitivity composed with the current result. Command @{command
   520   "also"} updates @{fact calculation} involving @{fact this}, while @{command
   521   "finally"} exhibits the final @{fact calculation} by forward chaining
   522   towards the next goal statement. Both commands require valid current facts,
   523   i.e.\ may occur only after commands that produce theorems such as @{command
   524   "assume"}, @{command "note"}, or some finished proof of @{command "have"},
   525   @{command "show"} etc. The @{command "moreover"} and @{command "ultimately"}
   526   commands are similar to @{command "also"} and @{command "finally"}, but only
   527   collect further results in @{fact calculation} without applying any rules
   528   yet.
   529 
   530   Also note that the implicit term abbreviation ``\<open>\<dots>\<close>'' has its canonical
   531   application with calculational proofs. It refers to the argument of the
   532   preceding statement. (The argument of a curried infix expression happens to
   533   be its right-hand side.)
   534 
   535   Isabelle/Isar calculations are implicitly subject to block structure in the
   536   sense that new threads of calculational reasoning are commenced for any new
   537   block (as opened by a local goal, for example). This means that, apart from
   538   being able to nest calculations, there is no separate \<^emph>\<open>begin-calculation\<close>
   539   command required.
   540 
   541   \<^medskip>
   542   The Isar calculation proof commands may be defined as follows:\<^footnote>\<open>We suppress
   543   internal bookkeeping such as proper handling of block-structure.\<close>
   544 
   545   \begin{matharray}{rcl}
   546     @{command "also"}\<open>\<^sub>0\<close> & \equiv & @{command "note"}~\<open>calculation = this\<close> \\
   547     @{command "also"}\<open>\<^sub>n\<^sub>+\<^sub>1\<close> & \equiv & @{command "note"}~\<open>calculation = trans [OF calculation this]\<close> \\[0.5ex]
   548     @{command "finally"} & \equiv & @{command "also"}~@{command "from"}~\<open>calculation\<close> \\[0.5ex]
   549     @{command "moreover"} & \equiv & @{command "note"}~\<open>calculation = calculation this\<close> \\
   550     @{command "ultimately"} & \equiv & @{command "moreover"}~@{command "from"}~\<open>calculation\<close> \\
   551   \end{matharray}
   552 
   553   @{rail \<open>
   554     (@@{command also} | @@{command finally}) ('(' @{syntax thms} ')')?
   555     ;
   556     @@{attribute trans} (() | 'add' | 'del')
   557   \<close>}
   558 
   559   \<^descr> @{command "also"}~\<open>(a\<^sub>1 \<dots> a\<^sub>n)\<close> maintains the auxiliary @{fact
   560   calculation} register as follows. The first occurrence of @{command "also"}
   561   in some calculational thread initializes @{fact calculation} by @{fact
   562   this}. Any subsequent @{command "also"} on the same level of block-structure
   563   updates @{fact calculation} by some transitivity rule applied to @{fact
   564   calculation} and @{fact this} (in that order). Transitivity rules are picked
   565   from the current context, unless alternative rules are given as explicit
   566   arguments.
   567 
   568   \<^descr> @{command "finally"}~\<open>(a\<^sub>1 \<dots> a\<^sub>n)\<close> maintains @{fact calculation} in the
   569   same way as @{command "also"} and then concludes the current calculational
   570   thread. The final result is exhibited as fact for forward chaining towards
   571   the next goal. Basically, @{command "finally"} abbreviates @{command
   572   "also"}~@{command "from"}~@{fact calculation}. Typical idioms for concluding
   573   calculational proofs are ``@{command "finally"}~@{command
   574   "show"}~\<open>?thesis\<close>~@{command "."}'' and ``@{command "finally"}~@{command
   575   "have"}~\<open>\<phi>\<close>~@{command "."}''.
   576 
   577   \<^descr> @{command "moreover"} and @{command "ultimately"} are analogous to
   578   @{command "also"} and @{command "finally"}, but collect results only,
   579   without applying rules.
   580 
   581   \<^descr> @{command "print_trans_rules"} prints the list of transitivity rules (for
   582   calculational commands @{command "also"} and @{command "finally"}) and
   583   symmetry rules (for the @{attribute symmetric} operation and single step
   584   elimination patters) of the current context.
   585 
   586   \<^descr> @{attribute trans} declares theorems as transitivity rules.
   587 
   588   \<^descr> @{attribute sym} declares symmetry rules, as well as @{attribute
   589   "Pure.elim"}\<open>?\<close> rules.
   590 
   591   \<^descr> @{attribute symmetric} resolves a theorem with some rule declared as
   592   @{attribute sym} in the current context. For example, ``@{command
   593   "assume"}~\<open>[symmetric]: x = y\<close>'' produces a swapped fact derived from that
   594   assumption.
   595 
   596   In structured proof texts it is often more appropriate to use an explicit
   597   single-step elimination proof, such as ``@{command "assume"}~\<open>x =
   598   y\<close>~@{command "then"}~@{command "have"}~\<open>y = x\<close>~@{command ".."}''.
   599 \<close>
   600 
   601 
   602 section \<open>Refinement steps\<close>
   603 
   604 subsection \<open>Proof method expressions \label{sec:proof-meth}\<close>
   605 
   606 text \<open>
   607   Proof methods are either basic ones, or expressions composed of methods via
   608   ``\<^verbatim>\<open>,\<close>'' (sequential composition), ``\<^verbatim>\<open>;\<close>'' (structural composition),
   609   ``\<^verbatim>\<open>|\<close>'' (alternative choices), ``\<^verbatim>\<open>?\<close>'' (try), ``\<^verbatim>\<open>+\<close>'' (repeat at least
   610   once), ``\<^verbatim>\<open>[\<close>\<open>n\<close>\<^verbatim>\<open>]\<close>'' (restriction to first \<open>n\<close> subgoals). In practice,
   611   proof methods are usually just a comma separated list of @{syntax
   612   name}~@{syntax args} specifications. Note that parentheses may be dropped
   613   for single method specifications (with no arguments). The syntactic
   614   precedence of method combinators is \<^verbatim>\<open>|\<close> \<^verbatim>\<open>;\<close> \<^verbatim>\<open>,\<close> \<^verbatim>\<open>[]\<close> \<^verbatim>\<open>+\<close> \<^verbatim>\<open>?\<close> (from low
   615   to high).
   616 
   617   @{rail \<open>
   618     @{syntax_def method}:
   619       (@{syntax name} | '(' methods ')') (() | '?' | '+' | '[' @{syntax nat}? ']')
   620     ;
   621     methods: (@{syntax name} @{syntax args} | @{syntax method}) + (',' | ';' | '|')
   622   \<close>}
   623 
   624   Regular Isar proof methods do \<^emph>\<open>not\<close> admit direct goal addressing, but refer
   625   to the first subgoal or to all subgoals uniformly. Nonetheless, the
   626   subsequent mechanisms allow to imitate the effect of subgoal addressing that
   627   is known from ML tactics.
   628 
   629   \<^medskip>
   630   Goal \<^emph>\<open>restriction\<close> means the proof state is wrapped-up in a way that
   631   certain subgoals are exposed, and other subgoals are ``parked'' elsewhere.
   632   Thus a proof method has no other chance than to operate on the subgoals that
   633   are presently exposed.
   634 
   635   Structural composition ``\<open>m\<^sub>1\<close>\<^verbatim>\<open>;\<close>~\<open>m\<^sub>2\<close>'' means that method \<open>m\<^sub>1\<close> is
   636   applied with restriction to the first subgoal, then \<open>m\<^sub>2\<close> is applied
   637   consecutively with restriction to each subgoal that has newly emerged due to
   638   \<open>m\<^sub>1\<close>. This is analogous to the tactic combinator @{ML_op THEN_ALL_NEW} in
   639   Isabelle/ML, see also @{cite "isabelle-implementation"}. For example, \<open>(rule
   640   r; blast)\<close> applies rule \<open>r\<close> and then solves all new subgoals by \<open>blast\<close>.
   641 
   642   Moreover, the explicit goal restriction operator ``\<open>[n]\<close>'' exposes only the
   643   first \<open>n\<close> subgoals (which need to exist), with default \<open>n = 1\<close>. For example,
   644   the method expression ``\<open>simp_all[3]\<close>'' simplifies the first three subgoals,
   645   while ``\<open>(rule r, simp_all)[]\<close>'' simplifies all new goals that emerge from
   646   applying rule \<open>r\<close> to the originally first one.
   647 
   648   \<^medskip>
   649   Improper methods, notably tactic emulations, offer low-level goal addressing
   650   as explicit argument to the individual tactic being involved. Here ``\<open>[!]\<close>''
   651   refers to all goals, and ``\<open>[n-]\<close>'' to all goals starting from \<open>n\<close>.
   652 
   653   @{rail \<open>
   654     @{syntax_def goal_spec}:
   655       '[' (@{syntax nat} '-' @{syntax nat} | @{syntax nat} '-' | @{syntax nat} | '!' ) ']'
   656   \<close>}
   657 \<close>
   658 
   659 
   660 subsection \<open>Initial and terminal proof steps \label{sec:proof-steps}\<close>
   661 
   662 text \<open>
   663   \begin{matharray}{rcl}
   664     @{command_def "proof"} & : & \<open>proof(prove) \<rightarrow> proof(state)\<close> \\
   665     @{command_def "qed"} & : & \<open>proof(state) \<rightarrow> proof(state) | local_theory | theory\<close> \\
   666     @{command_def "by"} & : & \<open>proof(prove) \<rightarrow> proof(state) | local_theory | theory\<close> \\
   667     @{command_def ".."} & : & \<open>proof(prove) \<rightarrow> proof(state) | local_theory | theory\<close> \\
   668     @{command_def "."} & : & \<open>proof(prove) \<rightarrow> proof(state) | local_theory | theory\<close> \\
   669     @{command_def "sorry"} & : & \<open>proof(prove) \<rightarrow> proof(state) | local_theory | theory\<close> \\
   670     @{method_def standard} & : & \<open>method\<close> \\
   671   \end{matharray}
   672 
   673   Arbitrary goal refinement via tactics is considered harmful. Structured
   674   proof composition in Isar admits proof methods to be invoked in two places
   675   only.
   676 
   677     \<^enum> An \<^emph>\<open>initial\<close> refinement step @{command_ref "proof"}~\<open>m\<^sub>1\<close> reduces a
   678     newly stated goal to a number of sub-goals that are to be solved later.
   679     Facts are passed to \<open>m\<^sub>1\<close> for forward chaining, if so indicated by
   680     \<open>proof(chain)\<close> mode.
   681 
   682     \<^enum> A \<^emph>\<open>terminal\<close> conclusion step @{command_ref "qed"}~\<open>m\<^sub>2\<close> is intended to
   683     solve remaining goals. No facts are passed to \<open>m\<^sub>2\<close>.
   684 
   685   The only other (proper) way to affect pending goals in a proof body is by
   686   @{command_ref "show"}, which involves an explicit statement of what is to be
   687   solved eventually. Thus we avoid the fundamental problem of unstructured
   688   tactic scripts that consist of numerous consecutive goal transformations,
   689   with invisible effects.
   690 
   691   \<^medskip>
   692   As a general rule of thumb for good proof style, initial proof methods
   693   should either solve the goal completely, or constitute some well-understood
   694   reduction to new sub-goals. Arbitrary automatic proof tools that are prone
   695   leave a large number of badly structured sub-goals are no help in continuing
   696   the proof document in an intelligible manner.
   697 
   698   Unless given explicitly by the user, the default initial method is @{method
   699   standard}, which subsumes at least @{method_ref (Pure) rule} or its
   700   classical variant @{method_ref (HOL) rule}. These methods apply a single
   701   standard elimination or introduction rule according to the topmost logical
   702   connective involved. There is no separate default terminal method. Any
   703   remaining goals are always solved by assumption in the very last step.
   704 
   705   @{rail \<open>
   706     @@{command proof} method?
   707     ;
   708     @@{command qed} method?
   709     ;
   710     @@{command "by"} method method?
   711     ;
   712     (@@{command "."} | @@{command ".."} | @@{command sorry})
   713   \<close>}
   714 
   715   \<^descr> @{command "proof"}~\<open>m\<^sub>1\<close> refines the goal by proof method \<open>m\<^sub>1\<close>; facts for
   716   forward chaining are passed if so indicated by \<open>proof(chain)\<close> mode.
   717 
   718   \<^descr> @{command "qed"}~\<open>m\<^sub>2\<close> refines any remaining goals by proof method \<open>m\<^sub>2\<close>
   719   and concludes the sub-proof by assumption. If the goal had been \<open>show\<close>, some
   720   pending sub-goal is solved as well by the rule resulting from the result
   721   \<^emph>\<open>exported\<close> into the enclosing goal context. Thus \<open>qed\<close> may fail for two
   722   reasons: either \<open>m\<^sub>2\<close> fails, or the resulting rule does not fit to any
   723   pending goal\<^footnote>\<open>This includes any additional ``strong'' assumptions as
   724   introduced by @{command "assume"}.\<close> of the enclosing context. Debugging such
   725   a situation might involve temporarily changing @{command "show"} into
   726   @{command "have"}, or weakening the local context by replacing occurrences
   727   of @{command "assume"} by @{command "presume"}.
   728 
   729   \<^descr> @{command "by"}~\<open>m\<^sub>1 m\<^sub>2\<close> is a \<^emph>\<open>terminal proof\<close>\index{proof!terminal}; it
   730   abbreviates @{command "proof"}~\<open>m\<^sub>1\<close>~@{command "qed"}~\<open>m\<^sub>2\<close>, but with
   731   backtracking across both methods. Debugging an unsuccessful @{command
   732   "by"}~\<open>m\<^sub>1 m\<^sub>2\<close> command can be done by expanding its definition; in many
   733   cases @{command "proof"}~\<open>m\<^sub>1\<close> (or even \<open>apply\<close>~\<open>m\<^sub>1\<close>) is already sufficient
   734   to see the problem.
   735 
   736   \<^descr> ``@{command ".."}'' is a \<^emph>\<open>standard proof\<close>\index{proof!standard}; it
   737   abbreviates @{command "by"}~\<open>standard\<close>.
   738 
   739   \<^descr> ``@{command "."}'' is a \<^emph>\<open>trivial proof\<close>\index{proof!trivial}; it
   740   abbreviates @{command "by"}~\<open>this\<close>.
   741 
   742   \<^descr> @{command "sorry"} is a \<^emph>\<open>fake proof\<close>\index{proof!fake} pretending to
   743   solve the pending claim without further ado. This only works in interactive
   744   development, or if the @{attribute quick_and_dirty} is enabled. Facts
   745   emerging from fake proofs are not the real thing. Internally, the derivation
   746   object is tainted by an oracle invocation, which may be inspected via the
   747   theorem status @{cite "isabelle-implementation"}.
   748 
   749   The most important application of @{command "sorry"} is to support
   750   experimentation and top-down proof development.
   751 
   752   \<^descr> @{method standard} refers to the default refinement step of some Isar
   753   language elements (notably @{command proof} and ``@{command ".."}''). It is
   754   \<^emph>\<open>dynamically scoped\<close>, so the behaviour depends on the application
   755   environment.
   756 
   757   In Isabelle/Pure, @{method standard} performs elementary introduction~/
   758   elimination steps (@{method_ref (Pure) rule}), introduction of type classes
   759   (@{method_ref intro_classes}) and locales (@{method_ref intro_locales}).
   760 
   761   In Isabelle/HOL, @{method standard} also takes classical rules into account
   762   (cf.\ \secref{sec:classical}).
   763 \<close>
   764 
   765 
   766 subsection \<open>Fundamental methods and attributes \label{sec:pure-meth-att}\<close>
   767 
   768 text \<open>
   769   The following proof methods and attributes refer to basic logical operations
   770   of Isar. Further methods and attributes are provided by several generic and
   771   object-logic specific tools and packages (see \chref{ch:gen-tools} and
   772   \partref{part:hol}).
   773 
   774   \begin{matharray}{rcl}
   775     @{command_def "print_rules"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\[0.5ex]
   776     @{method_def "-"} & : & \<open>method\<close> \\
   777     @{method_def "goal_cases"} & : & \<open>method\<close> \\
   778     @{method_def "fact"} & : & \<open>method\<close> \\
   779     @{method_def "assumption"} & : & \<open>method\<close> \\
   780     @{method_def "this"} & : & \<open>method\<close> \\
   781     @{method_def (Pure) "rule"} & : & \<open>method\<close> \\
   782     @{attribute_def (Pure) "intro"} & : & \<open>attribute\<close> \\
   783     @{attribute_def (Pure) "elim"} & : & \<open>attribute\<close> \\
   784     @{attribute_def (Pure) "dest"} & : & \<open>attribute\<close> \\
   785     @{attribute_def (Pure) "rule"} & : & \<open>attribute\<close> \\[0.5ex]
   786     @{attribute_def "OF"} & : & \<open>attribute\<close> \\
   787     @{attribute_def "of"} & : & \<open>attribute\<close> \\
   788     @{attribute_def "where"} & : & \<open>attribute\<close> \\
   789   \end{matharray}
   790 
   791   @{rail \<open>
   792     @@{method goal_cases} (@{syntax name}*)
   793     ;
   794     @@{method fact} @{syntax thms}?
   795     ;
   796     @@{method (Pure) rule} @{syntax thms}?
   797     ;
   798     rulemod: ('intro' | 'elim' | 'dest')
   799       ((('!' | () | '?') @{syntax nat}?) | 'del') ':' @{syntax thms}
   800     ;
   801     (@@{attribute intro} | @@{attribute elim} | @@{attribute dest})
   802       ('!' | () | '?') @{syntax nat}?
   803     ;
   804     @@{attribute (Pure) rule} 'del'
   805     ;
   806     @@{attribute OF} @{syntax thms}
   807     ;
   808     @@{attribute of} @{syntax insts} ('concl' ':' @{syntax insts})? @{syntax for_fixes}
   809     ;
   810     @@{attribute "where"} @{syntax named_insts} @{syntax for_fixes}
   811   \<close>}
   812 
   813   \<^descr> @{command "print_rules"} prints rules declared via attributes @{attribute
   814   (Pure) intro}, @{attribute (Pure) elim}, @{attribute (Pure) dest} of
   815   Isabelle/Pure.
   816 
   817   See also the analogous @{command "print_claset"} command for similar rule
   818   declarations of the classical reasoner (\secref{sec:classical}).
   819 
   820   \<^descr> ``@{method "-"}'' (minus) inserts the forward chaining facts as premises
   821   into the goal, and nothing else.
   822 
   823   Note that command @{command_ref "proof"} without any method actually
   824   performs a single reduction step using the @{method_ref (Pure) rule} method;
   825   thus a plain \<^emph>\<open>do-nothing\<close> proof step would be ``@{command "proof"}~\<open>-\<close>''
   826   rather than @{command "proof"} alone.
   827 
   828   \<^descr> @{method "goal_cases"}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> turns the current subgoals into cases
   829   within the context (see also \secref{sec:cases-induct}). The specified case
   830   names are used if present; otherwise cases are numbered starting from 1.
   831 
   832   Invoking cases in the subsequent proof body via the @{command_ref case}
   833   command will @{command fix} goal parameters, @{command assume} goal
   834   premises, and @{command let} variable @{variable_ref ?case} refer to the
   835   conclusion.
   836 
   837   \<^descr> @{method "fact"}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> composes some fact from \<open>a\<^sub>1, \<dots>, a\<^sub>n\<close> (or
   838   implicitly from the current proof context) modulo unification of schematic
   839   type and term variables. The rule structure is not taken into account, i.e.\
   840   meta-level implication is considered atomic. This is the same principle
   841   underlying literal facts (cf.\ \secref{sec:syn-att}): ``@{command
   842   "have"}~\<open>\<phi>\<close>~@{command "by"}~\<open>fact\<close>'' is equivalent to ``@{command
   843   "note"}~\<^verbatim>\<open>`\<close>\<open>\<phi>\<close>\<^verbatim>\<open>`\<close>'' provided that \<open>\<turnstile> \<phi>\<close> is an instance of some known \<open>\<turnstile> \<phi>\<close>
   844   in the proof context.
   845 
   846   \<^descr> @{method assumption} solves some goal by a single assumption step. All
   847   given facts are guaranteed to participate in the refinement; this means
   848   there may be only 0 or 1 in the first place. Recall that @{command "qed"}
   849   (\secref{sec:proof-steps}) already concludes any remaining sub-goals by
   850   assumption, so structured proofs usually need not quote the @{method
   851   assumption} method at all.
   852 
   853   \<^descr> @{method this} applies all of the current facts directly as rules. Recall
   854   that ``@{command "."}'' (dot) abbreviates ``@{command "by"}~\<open>this\<close>''.
   855 
   856   \<^descr> @{method (Pure) rule}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> applies some rule given as argument in
   857   backward manner; facts are used to reduce the rule before applying it to the
   858   goal. Thus @{method (Pure) rule} without facts is plain introduction, while
   859   with facts it becomes elimination.
   860 
   861   When no arguments are given, the @{method (Pure) rule} method tries to pick
   862   appropriate rules automatically, as declared in the current context using
   863   the @{attribute (Pure) intro}, @{attribute (Pure) elim}, @{attribute (Pure)
   864   dest} attributes (see below). This is included in the standard behaviour of
   865   @{command "proof"} and ``@{command ".."}'' (double-dot) steps (see
   866   \secref{sec:proof-steps}).
   867 
   868   \<^descr> @{attribute (Pure) intro}, @{attribute (Pure) elim}, and @{attribute
   869   (Pure) dest} declare introduction, elimination, and destruct rules, to be
   870   used with method @{method (Pure) rule}, and similar tools. Note that the
   871   latter will ignore rules declared with ``\<open>?\<close>'', while ``\<open>!\<close>'' are used most
   872   aggressively.
   873 
   874   The classical reasoner (see \secref{sec:classical}) introduces its own
   875   variants of these attributes; use qualified names to access the present
   876   versions of Isabelle/Pure, i.e.\ @{attribute (Pure) "Pure.intro"}.
   877 
   878   \<^descr> @{attribute (Pure) rule}~\<open>del\<close> undeclares introduction, elimination, or
   879   destruct rules.
   880 
   881   \<^descr> @{attribute OF}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> applies some theorem to all of the given rules
   882   \<open>a\<^sub>1, \<dots>, a\<^sub>n\<close> in canonical right-to-left order, which means that premises
   883   stemming from the \<open>a\<^sub>i\<close> emerge in parallel in the result, without
   884   interfering with each other. In many practical situations, the \<open>a\<^sub>i\<close> do not
   885   have premises themselves, so \<open>rule [OF a\<^sub>1 \<dots> a\<^sub>n]\<close> can be actually read as
   886   functional application (modulo unification).
   887 
   888   Argument positions may be effectively skipped by using ``\<open>_\<close>'' (underscore),
   889   which refers to the propositional identity rule in the Pure theory.
   890 
   891   \<^descr> @{attribute of}~\<open>t\<^sub>1 \<dots> t\<^sub>n\<close> performs positional instantiation of term
   892   variables. The terms \<open>t\<^sub>1, \<dots>, t\<^sub>n\<close> are substituted for any schematic
   893   variables occurring in a theorem from left to right; ``\<open>_\<close>'' (underscore)
   894   indicates to skip a position. Arguments following a ``\<open>concl:\<close>''
   895   specification refer to positions of the conclusion of a rule.
   896 
   897   An optional context of local variables \<open>\<FOR> x\<^sub>1 \<dots> x\<^sub>m\<close> may be specified:
   898   the instantiated theorem is exported, and these variables become schematic
   899   (usually with some shifting of indices).
   900 
   901   \<^descr> @{attribute "where"}~\<open>x\<^sub>1 = t\<^sub>1 \<AND> \<dots> x\<^sub>n = t\<^sub>n\<close> performs named
   902   instantiation of schematic type and term variables occurring in a theorem.
   903   Schematic variables have to be specified on the left-hand side (e.g.\
   904   \<open>?x1.3\<close>). The question mark may be omitted if the variable name is a plain
   905   identifier without index. As type instantiations are inferred from term
   906   instantiations, explicit type instantiations are seldom necessary.
   907 
   908   An optional context of local variables \<open>\<FOR> x\<^sub>1 \<dots> x\<^sub>m\<close> may be specified
   909   as for @{attribute "of"} above.
   910 \<close>
   911 
   912 
   913 subsection \<open>Defining proof methods\<close>
   914 
   915 text \<open>
   916   \begin{matharray}{rcl}
   917     @{command_def "method_setup"} & : & \<open>local_theory \<rightarrow> local_theory\<close> \\
   918   \end{matharray}
   919 
   920   @{rail \<open>
   921     @@{command method_setup} @{syntax name} '=' @{syntax text} @{syntax text}?
   922   \<close>}
   923 
   924   \<^descr> @{command "method_setup"}~\<open>name = text description\<close> defines a proof method
   925   in the current context. The given \<open>text\<close> has to be an ML expression of type
   926   @{ML_type "(Proof.context -> Proof.method) context_parser"}, cf.\ basic
   927   parsers defined in structure @{ML_structure Args} and @{ML_structure
   928   Attrib}. There are also combinators like @{ML METHOD} and @{ML
   929   SIMPLE_METHOD} to turn certain tactic forms into official proof methods; the
   930   primed versions refer to tactics with explicit goal addressing.
   931 
   932   Here are some example method definitions:
   933 \<close>
   934 
   935 (*<*)experiment begin(*>*)
   936   method_setup my_method1 =
   937     \<open>Scan.succeed (K (SIMPLE_METHOD' (fn i: int => no_tac)))\<close>
   938     "my first method (without any arguments)"
   939 
   940   method_setup my_method2 =
   941     \<open>Scan.succeed (fn ctxt: Proof.context =>
   942       SIMPLE_METHOD' (fn i: int => no_tac))\<close>
   943     "my second method (with context)"
   944 
   945   method_setup my_method3 =
   946     \<open>Attrib.thms >> (fn thms: thm list => fn ctxt: Proof.context =>
   947       SIMPLE_METHOD' (fn i: int => no_tac))\<close>
   948     "my third method (with theorem arguments and context)"
   949 (*<*)end(*>*)
   950 
   951 
   952 section \<open>Proof by cases and induction \label{sec:cases-induct}\<close>
   953 
   954 subsection \<open>Rule contexts\<close>
   955 
   956 text \<open>
   957   \begin{matharray}{rcl}
   958     @{command_def "case"} & : & \<open>proof(state) \<rightarrow> proof(state)\<close> \\
   959     @{command_def "print_cases"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
   960     @{attribute_def case_names} & : & \<open>attribute\<close> \\
   961     @{attribute_def case_conclusion} & : & \<open>attribute\<close> \\
   962     @{attribute_def params} & : & \<open>attribute\<close> \\
   963     @{attribute_def consumes} & : & \<open>attribute\<close> \\
   964   \end{matharray}
   965 
   966   The puristic way to build up Isar proof contexts is by explicit language
   967   elements like @{command "fix"}, @{command "assume"}, @{command "let"} (see
   968   \secref{sec:proof-context}). This is adequate for plain natural deduction,
   969   but easily becomes unwieldy in concrete verification tasks, which typically
   970   involve big induction rules with several cases.
   971 
   972   The @{command "case"} command provides a shorthand to refer to a local
   973   context symbolically: certain proof methods provide an environment of named
   974   ``cases'' of the form \<open>c: x\<^sub>1, \<dots>, x\<^sub>m, \<phi>\<^sub>1, \<dots>, \<phi>\<^sub>n\<close>; the effect of
   975   ``@{command "case"}~\<open>c\<close>'' is then equivalent to ``@{command "fix"}~\<open>x\<^sub>1 \<dots>
   976   x\<^sub>m\<close>~@{command "assume"}~\<open>c: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n\<close>''. Term bindings may be covered as
   977   well, notably @{variable ?case} for the main conclusion.
   978 
   979   By default, the ``terminology'' \<open>x\<^sub>1, \<dots>, x\<^sub>m\<close> of a case value is marked as
   980   hidden, i.e.\ there is no way to refer to such parameters in the subsequent
   981   proof text. After all, original rule parameters stem from somewhere outside
   982   of the current proof text. By using the explicit form ``@{command
   983   "case"}~\<open>(c y\<^sub>1 \<dots> y\<^sub>m)\<close>'' instead, the proof author is able to chose local
   984   names that fit nicely into the current context.
   985 
   986   \<^medskip>
   987   It is important to note that proper use of @{command "case"} does not
   988   provide means to peek at the current goal state, which is not directly
   989   observable in Isar! Nonetheless, goal refinement commands do provide named
   990   cases \<open>goal\<^sub>i\<close> for each subgoal \<open>i = 1, \<dots>, n\<close> of the resulting goal state.
   991   Using this extra feature requires great care, because some bits of the
   992   internal tactical machinery intrude the proof text. In particular, parameter
   993   names stemming from the left-over of automated reasoning tools are usually
   994   quite unpredictable.
   995 
   996   Under normal circumstances, the text of cases emerge from standard
   997   elimination or induction rules, which in turn are derived from previous
   998   theory specifications in a canonical way (say from @{command "inductive"}
   999   definitions).
  1000 
  1001   \<^medskip>
  1002   Proper cases are only available if both the proof method and the rules
  1003   involved support this. By using appropriate attributes, case names,
  1004   conclusions, and parameters may be also declared by hand. Thus variant
  1005   versions of rules that have been derived manually become ready to use in
  1006   advanced case analysis later.
  1007 
  1008   @{rail \<open>
  1009     @@{command case} @{syntax thmdecl}? (name | '(' name (('_' | @{syntax name}) *) ')')
  1010     ;
  1011     @@{attribute case_names} ((@{syntax name} ( '[' (('_' | @{syntax name}) *) ']' ) ? ) +)
  1012     ;
  1013     @@{attribute case_conclusion} @{syntax name} (@{syntax name} * )
  1014     ;
  1015     @@{attribute params} ((@{syntax name} * ) + @'and')
  1016     ;
  1017     @@{attribute consumes} @{syntax int}?
  1018   \<close>}
  1019 
  1020   \<^descr> @{command "case"}~\<open>a: (c x\<^sub>1 \<dots> x\<^sub>m)\<close> invokes a named local context \<open>c:
  1021   x\<^sub>1, \<dots>, x\<^sub>m, \<phi>\<^sub>1, \<dots>, \<phi>\<^sub>m\<close>, as provided by an appropriate proof method (such
  1022   as @{method_ref cases} and @{method_ref induct}). The command ``@{command
  1023   "case"}~\<open>a: (c x\<^sub>1 \<dots> x\<^sub>m)\<close>'' abbreviates ``@{command "fix"}~\<open>x\<^sub>1 \<dots>
  1024   x\<^sub>m\<close>~@{command "assume"}~\<open>a.c: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n\<close>''. Each local fact is qualified by
  1025   the prefix \<open>a\<close>, and all such facts are collectively bound to the name \<open>a\<close>.
  1026 
  1027   The fact name is specification \<open>a\<close> is optional, the default is to re-use
  1028   \<open>c\<close>. So @{command "case"}~\<open>(c x\<^sub>1 \<dots> x\<^sub>m)\<close> is the same as @{command
  1029   "case"}~\<open>c: (c x\<^sub>1 \<dots> x\<^sub>m)\<close>.
  1030 
  1031   \<^descr> @{command "print_cases"} prints all local contexts of the current state,
  1032   using Isar proof language notation.
  1033 
  1034   \<^descr> @{attribute case_names}~\<open>c\<^sub>1 \<dots> c\<^sub>k\<close> declares names for the local contexts
  1035   of premises of a theorem; \<open>c\<^sub>1, \<dots>, c\<^sub>k\<close> refers to the \<^emph>\<open>prefix\<close> of the list
  1036   of premises. Each of the cases \<open>c\<^sub>i\<close> can be of the form \<open>c[h\<^sub>1 \<dots> h\<^sub>n]\<close> where
  1037   the \<open>h\<^sub>1 \<dots> h\<^sub>n\<close> are the names of the hypotheses in case \<open>c\<^sub>i\<close> from left to
  1038   right.
  1039 
  1040   \<^descr> @{attribute case_conclusion}~\<open>c d\<^sub>1 \<dots> d\<^sub>k\<close> declares names for the
  1041   conclusions of a named premise \<open>c\<close>; here \<open>d\<^sub>1, \<dots>, d\<^sub>k\<close> refers to the prefix
  1042   of arguments of a logical formula built by nesting a binary connective
  1043   (e.g.\ \<open>\<or>\<close>).
  1044 
  1045   Note that proof methods such as @{method induct} and @{method coinduct}
  1046   already provide a default name for the conclusion as a whole. The need to
  1047   name subformulas only arises with cases that split into several sub-cases,
  1048   as in common co-induction rules.
  1049 
  1050   \<^descr> @{attribute params}~\<open>p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots> q\<^sub>1 \<dots> q\<^sub>n\<close> renames the innermost
  1051   parameters of premises \<open>1, \<dots>, n\<close> of some theorem. An empty list of names may
  1052   be given to skip positions, leaving the present parameters unchanged.
  1053 
  1054   Note that the default usage of case rules does \<^emph>\<open>not\<close> directly expose
  1055   parameters to the proof context.
  1056 
  1057   \<^descr> @{attribute consumes}~\<open>n\<close> declares the number of ``major premises'' of a
  1058   rule, i.e.\ the number of facts to be consumed when it is applied by an
  1059   appropriate proof method. The default value of @{attribute consumes} is \<open>n =
  1060   1\<close>, which is appropriate for the usual kind of cases and induction rules for
  1061   inductive sets (cf.\ \secref{sec:hol-inductive}). Rules without any
  1062   @{attribute consumes} declaration given are treated as if @{attribute
  1063   consumes}~\<open>0\<close> had been specified.
  1064 
  1065   A negative \<open>n\<close> is interpreted relatively to the total number of premises of
  1066   the rule in the target context. Thus its absolute value specifies the
  1067   remaining number of premises, after subtracting the prefix of major premises
  1068   as indicated above. This form of declaration has the technical advantage of
  1069   being stable under more morphisms, notably those that export the result from
  1070   a nested @{command_ref context} with additional assumptions.
  1071 
  1072   Note that explicit @{attribute consumes} declarations are only rarely
  1073   needed; this is already taken care of automatically by the higher-level
  1074   @{attribute cases}, @{attribute induct}, and @{attribute coinduct}
  1075   declarations.
  1076 \<close>
  1077 
  1078 
  1079 subsection \<open>Proof methods\<close>
  1080 
  1081 text \<open>
  1082   \begin{matharray}{rcl}
  1083     @{method_def cases} & : & \<open>method\<close> \\
  1084     @{method_def induct} & : & \<open>method\<close> \\
  1085     @{method_def induction} & : & \<open>method\<close> \\
  1086     @{method_def coinduct} & : & \<open>method\<close> \\
  1087   \end{matharray}
  1088 
  1089   The @{method cases}, @{method induct}, @{method induction}, and @{method
  1090   coinduct} methods provide a uniform interface to common proof techniques
  1091   over datatypes, inductive predicates (or sets), recursive functions etc. The
  1092   corresponding rules may be specified and instantiated in a casual manner.
  1093   Furthermore, these methods provide named local contexts that may be invoked
  1094   via the @{command "case"} proof command within the subsequent proof text.
  1095   This accommodates compact proof texts even when reasoning about large
  1096   specifications.
  1097 
  1098   The @{method induct} method also provides some additional infrastructure in
  1099   order to be applicable to structure statements (either using explicit
  1100   meta-level connectives, or including facts and parameters separately). This
  1101   avoids cumbersome encoding of ``strengthened'' inductive statements within
  1102   the object-logic.
  1103 
  1104   Method @{method induction} differs from @{method induct} only in the names
  1105   of the facts in the local context invoked by the @{command "case"} command.
  1106 
  1107   @{rail \<open>
  1108     @@{method cases} ('(' 'no_simp' ')')? \<newline>
  1109       (@{syntax insts} * @'and') rule?
  1110     ;
  1111     (@@{method induct} | @@{method induction})
  1112       ('(' 'no_simp' ')')? (definsts * @'and') \<newline> arbitrary? taking? rule?
  1113     ;
  1114     @@{method coinduct} @{syntax insts} taking rule?
  1115     ;
  1116 
  1117     rule: ('type' | 'pred' | 'set') ':' (@{syntax name} +) | 'rule' ':' (@{syntax thm} +)
  1118     ;
  1119     definst: @{syntax name} ('==' | '\<equiv>') @{syntax term} | '(' @{syntax term} ')' | @{syntax inst}
  1120     ;
  1121     definsts: ( definst * )
  1122     ;
  1123     arbitrary: 'arbitrary' ':' ((@{syntax term} * ) @'and' +)
  1124     ;
  1125     taking: 'taking' ':' @{syntax insts}
  1126   \<close>}
  1127 
  1128   \<^descr> @{method cases}~\<open>insts R\<close> applies method @{method rule} with an
  1129   appropriate case distinction theorem, instantiated to the subjects \<open>insts\<close>.
  1130   Symbolic case names are bound according to the rule's local contexts.
  1131 
  1132   The rule is determined as follows, according to the facts and arguments
  1133   passed to the @{method cases} method:
  1134 
  1135   \<^medskip>
  1136   \begin{tabular}{llll}
  1137     facts           &                 & arguments   & rule \\\hline
  1138     \<open>\<turnstile> R\<close>   & @{method cases} &             & implicit rule \<open>R\<close> \\
  1139                     & @{method cases} &             & classical case split \\
  1140                     & @{method cases} & \<open>t\<close>   & datatype exhaustion (type of \<open>t\<close>) \\
  1141     \<open>\<turnstile> A t\<close> & @{method cases} & \<open>\<dots>\<close> & inductive predicate/set elimination (of \<open>A\<close>) \\
  1142     \<open>\<dots>\<close>     & @{method cases} & \<open>\<dots> rule: R\<close> & explicit rule \<open>R\<close> \\
  1143   \end{tabular}
  1144   \<^medskip>
  1145 
  1146   Several instantiations may be given, referring to the \<^emph>\<open>suffix\<close> of premises
  1147   of the case rule; within each premise, the \<^emph>\<open>prefix\<close> of variables is
  1148   instantiated. In most situations, only a single term needs to be specified;
  1149   this refers to the first variable of the last premise (it is usually the
  1150   same for all cases). The \<open>(no_simp)\<close> option can be used to disable
  1151   pre-simplification of cases (see the description of @{method induct} below
  1152   for details).
  1153 
  1154   \<^descr> @{method induct}~\<open>insts R\<close> and @{method induction}~\<open>insts R\<close> are analogous
  1155   to the @{method cases} method, but refer to induction rules, which are
  1156   determined as follows:
  1157 
  1158   \<^medskip>
  1159   \begin{tabular}{llll}
  1160     facts           &                  & arguments            & rule \\\hline
  1161                     & @{method induct} & \<open>P x\<close>        & datatype induction (type of \<open>x\<close>) \\
  1162     \<open>\<turnstile> A x\<close> & @{method induct} & \<open>\<dots>\<close>          & predicate/set induction (of \<open>A\<close>) \\
  1163     \<open>\<dots>\<close>     & @{method induct} & \<open>\<dots> rule: R\<close> & explicit rule \<open>R\<close> \\
  1164   \end{tabular}
  1165   \<^medskip>
  1166 
  1167   Several instantiations may be given, each referring to some part of a mutual
  1168   inductive definition or datatype --- only related partial induction rules
  1169   may be used together, though. Any of the lists of terms \<open>P, x, \<dots>\<close> refers to
  1170   the \<^emph>\<open>suffix\<close> of variables present in the induction rule. This enables the
  1171   writer to specify only induction variables, or both predicates and
  1172   variables, for example.
  1173 
  1174   Instantiations may be definitional: equations \<open>x \<equiv> t\<close> introduce local
  1175   definitions, which are inserted into the claim and discharged after applying
  1176   the induction rule. Equalities reappear in the inductive cases, but have
  1177   been transformed according to the induction principle being involved here.
  1178   In order to achieve practically useful induction hypotheses, some variables
  1179   occurring in \<open>t\<close> need to be fixed (see below). Instantiations of the form
  1180   \<open>t\<close>, where \<open>t\<close> is not a variable, are taken as a shorthand for \<open>x \<equiv> t\<close>,
  1181   where \<open>x\<close> is a fresh variable. If this is not intended, \<open>t\<close> has to be
  1182   enclosed in parentheses. By default, the equalities generated by
  1183   definitional instantiations are pre-simplified using a specific set of
  1184   rules, usually consisting of distinctness and injectivity theorems for
  1185   datatypes. This pre-simplification may cause some of the parameters of an
  1186   inductive case to disappear, or may even completely delete some of the
  1187   inductive cases, if one of the equalities occurring in their premises can be
  1188   simplified to \<open>False\<close>. The \<open>(no_simp)\<close> option can be used to disable
  1189   pre-simplification. Additional rules to be used in pre-simplification can be
  1190   declared using the @{attribute_def induct_simp} attribute.
  1191 
  1192   The optional ``\<open>arbitrary: x\<^sub>1 \<dots> x\<^sub>m\<close>'' specification generalizes variables
  1193   \<open>x\<^sub>1, \<dots>, x\<^sub>m\<close> of the original goal before applying induction. One can
  1194   separate variables by ``\<open>and\<close>'' to generalize them in other goals then the
  1195   first. Thus induction hypotheses may become sufficiently general to get the
  1196   proof through. Together with definitional instantiations, one may
  1197   effectively perform induction over expressions of a certain structure.
  1198 
  1199   The optional ``\<open>taking: t\<^sub>1 \<dots> t\<^sub>n\<close>'' specification provides additional
  1200   instantiations of a prefix of pending variables in the rule. Such schematic
  1201   induction rules rarely occur in practice, though.
  1202 
  1203   \<^descr> @{method coinduct}~\<open>inst R\<close> is analogous to the @{method induct} method,
  1204   but refers to coinduction rules, which are determined as follows:
  1205 
  1206   \<^medskip>
  1207   \begin{tabular}{llll}
  1208     goal          &                    & arguments & rule \\\hline
  1209                   & @{method coinduct} & \<open>x\<close> & type coinduction (type of \<open>x\<close>) \\
  1210     \<open>A x\<close> & @{method coinduct} & \<open>\<dots>\<close> & predicate/set coinduction (of \<open>A\<close>) \\
  1211     \<open>\<dots>\<close>   & @{method coinduct} & \<open>\<dots> rule: R\<close> & explicit rule \<open>R\<close> \\
  1212   \end{tabular}
  1213   \<^medskip>
  1214 
  1215   Coinduction is the dual of induction. Induction essentially eliminates \<open>A x\<close>
  1216   towards a generic result \<open>P x\<close>, while coinduction introduces \<open>A x\<close> starting
  1217   with \<open>B x\<close>, for a suitable ``bisimulation'' \<open>B\<close>. The cases of a coinduct
  1218   rule are typically named after the predicates or sets being covered, while
  1219   the conclusions consist of several alternatives being named after the
  1220   individual destructor patterns.
  1221 
  1222   The given instantiation refers to the \<^emph>\<open>suffix\<close> of variables occurring in
  1223   the rule's major premise, or conclusion if unavailable. An additional
  1224   ``\<open>taking: t\<^sub>1 \<dots> t\<^sub>n\<close>'' specification may be required in order to specify
  1225   the bisimulation to be used in the coinduction step.
  1226 
  1227 
  1228   Above methods produce named local contexts, as determined by the
  1229   instantiated rule as given in the text. Beyond that, the @{method induct}
  1230   and @{method coinduct} methods guess further instantiations from the goal
  1231   specification itself. Any persisting unresolved schematic variables of the
  1232   resulting rule will render the the corresponding case invalid. The term
  1233   binding @{variable ?case} for the conclusion will be provided with each
  1234   case, provided that term is fully specified.
  1235 
  1236   The @{command "print_cases"} command prints all named cases present in the
  1237   current proof state.
  1238 
  1239   \<^medskip>
  1240   Despite the additional infrastructure, both @{method cases} and @{method
  1241   coinduct} merely apply a certain rule, after instantiation, while conforming
  1242   due to the usual way of monotonic natural deduction: the context of a
  1243   structured statement \<open>\<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots> \<phi>\<^sub>n \<Longrightarrow> \<dots>\<close> reappears unchanged after
  1244   the case split.
  1245 
  1246   The @{method induct} method is fundamentally different in this respect: the
  1247   meta-level structure is passed through the ``recursive'' course involved in
  1248   the induction. Thus the original statement is basically replaced by separate
  1249   copies, corresponding to the induction hypotheses and conclusion; the
  1250   original goal context is no longer available. Thus local assumptions, fixed
  1251   parameters and definitions effectively participate in the inductive
  1252   rephrasing of the original statement.
  1253 
  1254   In @{method induct} proofs, local assumptions introduced by cases are split
  1255   into two different kinds: \<open>hyps\<close> stemming from the rule and \<open>prems\<close> from the
  1256   goal statement. This is reflected in the extracted cases accordingly, so
  1257   invoking ``@{command "case"}~\<open>c\<close>'' will provide separate facts \<open>c.hyps\<close> and
  1258   \<open>c.prems\<close>, as well as fact \<open>c\<close> to hold the all-inclusive list.
  1259 
  1260   In @{method induction} proofs, local assumptions introduced by cases are
  1261   split into three different kinds: \<open>IH\<close>, the induction hypotheses, \<open>hyps\<close>,
  1262   the remaining hypotheses stemming from the rule, and \<open>prems\<close>, the
  1263   assumptions from the goal statement. The names are \<open>c.IH\<close>, \<open>c.hyps\<close> and
  1264   \<open>c.prems\<close>, as above.
  1265 
  1266   \<^medskip>
  1267   Facts presented to either method are consumed according to the number of
  1268   ``major premises'' of the rule involved, which is usually 0 for plain cases
  1269   and induction rules of datatypes etc.\ and 1 for rules of inductive
  1270   predicates or sets and the like. The remaining facts are inserted into the
  1271   goal verbatim before the actual \<open>cases\<close>, \<open>induct\<close>, or \<open>coinduct\<close> rule is
  1272   applied.
  1273 \<close>
  1274 
  1275 
  1276 subsection \<open>Declaring rules\<close>
  1277 
  1278 text \<open>
  1279   \begin{matharray}{rcl}
  1280     @{command_def "print_induct_rules"}\<open>\<^sup>*\<close> & : & \<open>context \<rightarrow>\<close> \\
  1281     @{attribute_def cases} & : & \<open>attribute\<close> \\
  1282     @{attribute_def induct} & : & \<open>attribute\<close> \\
  1283     @{attribute_def coinduct} & : & \<open>attribute\<close> \\
  1284   \end{matharray}
  1285 
  1286   @{rail \<open>
  1287     @@{attribute cases} spec
  1288     ;
  1289     @@{attribute induct} spec
  1290     ;
  1291     @@{attribute coinduct} spec
  1292     ;
  1293 
  1294     spec: (('type' | 'pred' | 'set') ':' @{syntax name}) | 'del'
  1295   \<close>}
  1296 
  1297   \<^descr> @{command "print_induct_rules"} prints cases and induct rules for
  1298   predicates (or sets) and types of the current context.
  1299 
  1300   \<^descr> @{attribute cases}, @{attribute induct}, and @{attribute coinduct} (as
  1301   attributes) declare rules for reasoning about (co)inductive predicates (or
  1302   sets) and types, using the corresponding methods of the same name. Certain
  1303   definitional packages of object-logics usually declare emerging cases and
  1304   induction rules as expected, so users rarely need to intervene.
  1305 
  1306   Rules may be deleted via the \<open>del\<close> specification, which covers all of the
  1307   \<open>type\<close>/\<open>pred\<close>/\<open>set\<close> sub-categories simultaneously. For example, @{attribute
  1308   cases}~\<open>del\<close> removes any @{attribute cases} rules declared for some type,
  1309   predicate, or set.
  1310 
  1311   Manual rule declarations usually refer to the @{attribute case_names} and
  1312   @{attribute params} attributes to adjust names of cases and parameters of a
  1313   rule; the @{attribute consumes} declaration is taken care of automatically:
  1314   @{attribute consumes}~\<open>0\<close> is specified for ``type'' rules and @{attribute
  1315   consumes}~\<open>1\<close> for ``predicate'' / ``set'' rules.
  1316 \<close>
  1317 
  1318 
  1319 section \<open>Generalized elimination and case splitting \label{sec:obtain}\<close>
  1320 
  1321 text \<open>
  1322   \begin{matharray}{rcl}
  1323     @{command_def "consider"} & : & \<open>proof(state) | proof(chain) \<rightarrow> proof(prove)\<close> \\
  1324     @{command_def "obtain"} & : & \<open>proof(state) | proof(chain) \<rightarrow> proof(prove)\<close> \\
  1325     @{command_def "guess"}\<open>\<^sup>*\<close> & : & \<open>proof(state) | proof(chain) \<rightarrow> proof(prove)\<close> \\
  1326   \end{matharray}
  1327 
  1328   Generalized elimination means that hypothetical parameters and premises may
  1329   be introduced in the current context, potentially with a split into cases.
  1330   This works by virtue of a locally proven rule that establishes the soundness
  1331   of this temporary context extension. As representative examples, one may
  1332   think of standard rules from Isabelle/HOL like this:
  1333 
  1334   \<^medskip>
  1335   \begin{tabular}{ll}
  1336   \<open>\<exists>x. B x \<Longrightarrow> (\<And>x. B x \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close> \\
  1337   \<open>A \<and> B \<Longrightarrow> (A \<Longrightarrow> B \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close> \\
  1338   \<open>A \<or> B \<Longrightarrow> (A \<Longrightarrow> thesis) \<Longrightarrow> (B \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close> \\
  1339   \end{tabular}
  1340   \<^medskip>
  1341 
  1342   In general, these particular rules and connectives need to get involved at
  1343   all: this concept works directly in Isabelle/Pure via Isar commands defined
  1344   below. In particular, the logic of elimination and case splitting is
  1345   delegated to an Isar proof, which often involves automated tools.
  1346 
  1347   @{rail \<open>
  1348     @@{command consider} @{syntax obtain_clauses}
  1349     ;
  1350     @@{command obtain} @{syntax par_name}? @{syntax vars} \<newline>
  1351       @'where' concl prems @{syntax for_fixes}
  1352     ;
  1353     concl: (@{syntax props} + @'and')
  1354     ;
  1355     prems: (@'if' (@{syntax props'} + @'and'))?
  1356     ;
  1357     @@{command guess} @{syntax vars}
  1358   \<close>}
  1359 
  1360   \<^descr> @{command consider}~\<open>(a) \<^vec>x \<WHERE> \<^vec>A \<^vec>x | (b)
  1361   \<^vec>y \<WHERE> \<^vec>B \<^vec>y | \<dots>\<close> states a rule for case splitting
  1362   into separate subgoals, such that each case involves new parameters and
  1363   premises. After the proof is finished, the resulting rule may be used
  1364   directly with the @{method cases} proof method (\secref{sec:cases-induct}),
  1365   in order to perform actual case-splitting of the proof text via @{command
  1366   case} and @{command next} as usual.
  1367 
  1368   Optional names in round parentheses refer to case names: in the proof of the
  1369   rule this is a fact name, in the resulting rule it is used as annotation
  1370   with the @{attribute_ref case_names} attribute.
  1371 
  1372   \<^medskip>
  1373   Formally, the command @{command consider} is defined as derived Isar
  1374   language element as follows:
  1375 
  1376   \begin{matharray}{l}
  1377     @{command "consider"}~\<open>(a) \<^vec>x \<WHERE> \<^vec>A \<^vec>x | (b) \<^vec>y \<WHERE> \<^vec>B \<^vec>y | \<dots> \<equiv>\<close> \\[1ex]
  1378     \quad @{command "have"}~\<open>[case_names a b \<dots>]: thesis\<close> \\
  1379     \qquad \<open>\<IF> a [Pure.intro?]: \<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis\<close> \\
  1380     \qquad \<open>\<AND> b [Pure.intro?]: \<And>\<^vec>y. \<^vec>B \<^vec>y \<Longrightarrow> thesis\<close> \\
  1381     \qquad \<open>\<AND> \<dots>\<close> \\
  1382     \qquad \<open>\<FOR> thesis\<close> \\
  1383     \qquad @{command "apply"}~\<open>(insert a b \<dots>)\<close> \\
  1384   \end{matharray}
  1385 
  1386   See also \secref{sec:goals} for @{keyword "obtains"} in toplevel goal
  1387   statements, as well as @{command print_statement} to print existing rules in
  1388   a similar format.
  1389 
  1390   \<^descr> @{command obtain}~\<open>\<^vec>x \<WHERE> \<^vec>A \<^vec>x\<close> states a
  1391   generalized elimination rule with exactly one case. After the proof is
  1392   finished, it is activated for the subsequent proof text: the context is
  1393   augmented via @{command fix}~\<open>\<^vec>x\<close> @{command assume}~\<open>\<^vec>A
  1394   \<^vec>x\<close>, with special provisions to export later results by discharging
  1395   these assumptions again.
  1396 
  1397   Note that according to the parameter scopes within the elimination rule,
  1398   results \<^emph>\<open>must not\<close> refer to hypothetical parameters; otherwise the export
  1399   will fail! This restriction conforms to the usual manner of existential
  1400   reasoning in Natural Deduction.
  1401 
  1402   \<^medskip>
  1403   Formally, the command @{command obtain} is defined as derived Isar language
  1404   element as follows, using an instrumented variant of @{command assume}:
  1405 
  1406   \begin{matharray}{l}
  1407     @{command "obtain"}~\<open>\<^vec>x \<WHERE> a: \<^vec>A \<^vec>x  \<langle>proof\<rangle> \<equiv>\<close> \\[1ex]
  1408     \quad @{command "have"}~\<open>thesis\<close> \\
  1409     \qquad \<open>\<IF> that [Pure.intro?]: \<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis\<close> \\
  1410     \qquad \<open>\<FOR> thesis\<close> \\
  1411     \qquad @{command "apply"}~\<open>(insert that)\<close> \\
  1412     \qquad \<open>\<langle>proof\<rangle>\<close> \\
  1413     \quad @{command "fix"}~\<open>\<^vec>x\<close>~@{command "assume"}\<open>\<^sup>* a: \<^vec>A \<^vec>x\<close> \\
  1414   \end{matharray}
  1415 
  1416   \<^descr> @{command guess} is similar to @{command obtain}, but it derives the
  1417   obtained context elements from the course of tactical reasoning in the
  1418   proof. Thus it can considerably obscure the proof: it is classified as
  1419   \<^emph>\<open>improper\<close>.
  1420 
  1421   A proof with @{command guess} starts with a fixed goal \<open>thesis\<close>. The
  1422   subsequent refinement steps may turn this to anything of the form
  1423   \<open>\<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis\<close>, but without splitting into new
  1424   subgoals. The final goal state is then used as reduction rule for the obtain
  1425   pattern described above. Obtained parameters \<open>\<^vec>x\<close> are marked as
  1426   internal by default, and thus inaccessible in the proof text. The variable
  1427   names and type constraints given as arguments for @{command "guess"} specify
  1428   a prefix of accessible parameters.
  1429 
  1430 
  1431   In the proof of @{command consider} and @{command obtain} the local premises
  1432   are always bound to the fact name @{fact_ref that}, according to structured
  1433   Isar statements involving @{keyword_ref "if"} (\secref{sec:goals}).
  1434 
  1435   Facts that are established by @{command "obtain"} and @{command "guess"} may
  1436   not be polymorphic: any type-variables occurring here are fixed in the
  1437   present context. This is a natural consequence of the role of @{command fix}
  1438   and @{command assume} in these constructs.
  1439 \<close>
  1440 
  1441 end