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