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(* $Id$ *)
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theory Proof
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imports Main
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begin
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chapter {* Proofs *}
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text {*
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Proof commands perform transitions of Isar/VM machine
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configurations, which are block-structured, consisting of a stack of
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nodes with three main components: logical proof context, current
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facts, and open goals. Isar/VM transitions are \emph{typed}
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according to the following three different modes of operation:
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\begin{descr}
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\item [@{text "proof(prove)"}] means that a new goal has just been
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stated that is now to be \emph{proven}; the next command may refine
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it by some proof method, and enter a sub-proof to establish the
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actual result.
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\item [@{text "proof(state)"}] is like a nested theory mode: the
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context may be augmented by \emph{stating} additional assumptions,
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intermediate results etc.
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\item [@{text "proof(chain)"}] is intermediate between @{text
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"proof(state)"} and @{text "proof(prove)"}: existing facts (i.e.\
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the contents of the special ``@{fact_ref this}'' register) have been
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just picked up in order to be used when refining the goal claimed
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next.
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\end{descr}
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The proof mode indicator may be read as a verb telling the writer
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what kind of operation may be performed next. The corresponding
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typings of proof commands restricts the shape of well-formed proof
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texts to particular command sequences. So dynamic arrangements of
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commands eventually turn out as static texts of a certain structure.
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\Appref{ap:refcard} gives a simplified grammar of the overall
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(extensible) language emerging that way.
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*}
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section {* Context elements \label{sec:proof-context} *}
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text {*
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\begin{matharray}{rcl}
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@{command_def "fix"} & : & \isartrans{proof(state)}{proof(state)} \\
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@{command_def "assume"} & : & \isartrans{proof(state)}{proof(state)} \\
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@{command_def "presume"} & : & \isartrans{proof(state)}{proof(state)} \\
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@{command_def "def"} & : & \isartrans{proof(state)}{proof(state)} \\
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\end{matharray}
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The logical proof context consists of fixed variables and
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assumptions. The former closely correspond to Skolem constants, or
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meta-level universal quantification as provided by the Isabelle/Pure
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logical framework. Introducing some \emph{arbitrary, but fixed}
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variable via ``@{command "fix"}~@{text x}'' results in a local value
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that may be used in the subsequent proof as any other variable or
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constant. Furthermore, any result @{text "\<turnstile> \<phi>[x]"} exported from
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the context will be universally closed wrt.\ @{text x} at the
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outermost level: @{text "\<turnstile> \<And>x. \<phi>[x]"} (this is expressed in normal
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form using Isabelle's meta-variables).
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Similarly, introducing some assumption @{text \<chi>} has two effects.
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On the one hand, a local theorem is created that may be used as a
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fact in subsequent proof steps. On the other hand, any result
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@{text "\<chi> \<turnstile> \<phi>"} exported from the context becomes conditional wrt.\
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the assumption: @{text "\<turnstile> \<chi> \<Longrightarrow> \<phi>"}. Thus, solving an enclosing goal
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using such a result would basically introduce a new subgoal stemming
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from the assumption. How this situation is handled depends on the
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version of assumption command used: while @{command "assume"}
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insists on solving the subgoal by unification with some premise of
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the goal, @{command "presume"} leaves the subgoal unchanged in order
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to be proved later by the user.
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Local definitions, introduced by ``@{command "def"}~@{text "x \<equiv>
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t"}'', are achieved by combining ``@{command "fix"}~@{text x}'' with
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another version of assumption that causes any hypothetical equation
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@{text "x \<equiv> t"} to be eliminated by the reflexivity rule. Thus,
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exporting some result @{text "x \<equiv> t \<turnstile> \<phi>[x]"} yields @{text "\<turnstile>
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\<phi>[t]"}.
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\begin{rail}
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'fix' (vars + 'and')
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;
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('assume' | 'presume') (props + 'and')
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;
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'def' (def + 'and')
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;
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def: thmdecl? \\ name ('==' | equiv) term termpat?
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;
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\end{rail}
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\begin{descr}
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\item [@{command "fix"}~@{text x}] introduces a local variable
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@{text x} that is \emph{arbitrary, but fixed.}
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\item [@{command "assume"}~@{text "a: \<phi>"} and @{command
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"presume"}~@{text "a: \<phi>"}] introduce a local fact @{text "\<phi> \<turnstile> \<phi>"} by
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assumption. Subsequent results applied to an enclosing goal (e.g.\
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by @{command_ref "show"}) are handled as follows: @{command
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"assume"} expects to be able to unify with existing premises in the
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goal, while @{command "presume"} leaves @{text \<phi>} as new subgoals.
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Several lists of assumptions may be given (separated by
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@{keyword_ref "and"}; the resulting list of current facts consists
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of all of these concatenated.
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\item [@{command "def"}~@{text "x \<equiv> t"}] introduces a local
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(non-polymorphic) definition. In results exported from the context,
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@{text x} is replaced by @{text t}. Basically, ``@{command
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"def"}~@{text "x \<equiv> t"}'' abbreviates ``@{command "fix"}~@{text
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x}~@{command "assume"}~@{text "x \<equiv> t"}'', with the resulting
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hypothetical equation solved by reflexivity.
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The default name for the definitional equation is @{text x_def}.
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Several simultaneous definitions may be given at the same time.
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\end{descr}
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The special name @{fact_ref prems} refers to all assumptions of the
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current context as a list of theorems. This feature should be used
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with great care! It is better avoided in final proof texts.
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*}
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section {* Facts and forward chaining *}
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text {*
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\begin{matharray}{rcl}
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@{command_def "note"} & : & \isartrans{proof(state)}{proof(state)} \\
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@{command_def "then"} & : & \isartrans{proof(state)}{proof(chain)} \\
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@{command_def "from"} & : & \isartrans{proof(state)}{proof(chain)} \\
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@{command_def "with"} & : & \isartrans{proof(state)}{proof(chain)} \\
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@{command_def "using"} & : & \isartrans{proof(prove)}{proof(prove)} \\
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@{command_def "unfolding"} & : & \isartrans{proof(prove)}{proof(prove)} \\
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\end{matharray}
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New facts are established either by assumption or proof of local
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statements. Any fact will usually be involved in further proofs,
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either as explicit arguments of proof methods, or when forward
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chaining towards the next goal via @{command "then"} (and variants);
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@{command "from"} and @{command "with"} are composite forms
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involving @{command "note"}. The @{command "using"} elements
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augments the collection of used facts \emph{after} a goal has been
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stated. Note that the special theorem name @{fact_ref this} refers
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to the most recently established facts, but only \emph{before}
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issuing a follow-up claim.
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\begin{rail}
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'note' (thmdef? thmrefs + 'and')
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;
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('from' | 'with' | 'using' | 'unfolding') (thmrefs + 'and')
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;
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\end{rail}
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\begin{descr}
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\item [@{command "note"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"}]
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recalls existing facts @{text "b\<^sub>1, \<dots>, b\<^sub>n"}, binding
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the result as @{text a}. Note that attributes may be involved as
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well, both on the left and right hand sides.
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\item [@{command "then"}] indicates forward chaining by the current
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facts in order to establish the goal to be claimed next. The
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initial proof method invoked to refine that will be offered the
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facts to do ``anything appropriate'' (see also
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\secref{sec:proof-steps}). For example, method @{method_ref rule}
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(see \secref{sec:pure-meth-att}) would typically do an elimination
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rather than an introduction. Automatic methods usually insert the
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facts into the goal state before operation. This provides a simple
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scheme to control relevance of facts in automated proof search.
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\item [@{command "from"}~@{text b}] abbreviates ``@{command
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"note"}~@{text b}~@{command "then"}''; thus @{command "then"} is
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equivalent to ``@{command "from"}~@{text this}''.
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\item [@{command "with"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"}]
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abbreviates ``@{command "from"}~@{text "b\<^sub>1 \<dots> b\<^sub>n \<AND>
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this"}''; thus the forward chaining is from earlier facts together
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with the current ones.
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\item [@{command "using"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"}] augments
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the facts being currently indicated for use by a subsequent
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refinement step (such as @{command_ref "apply"} or @{command_ref
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"proof"}).
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\item [@{command "unfolding"}~@{text "b\<^sub>1 \<dots> b\<^sub>n"}] is
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structurally similar to @{command "using"}, but unfolds definitional
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equations @{text "b\<^sub>1, \<dots> b\<^sub>n"} throughout the goal state
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and facts.
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\end{descr}
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Forward chaining with an empty list of theorems is the same as not
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chaining at all. Thus ``@{command "from"}~@{text nothing}'' has no
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effect apart from entering @{text "prove(chain)"} mode, since
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@{fact_ref nothing} is bound to the empty list of theorems.
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Basic proof methods (such as @{method_ref rule}) expect multiple
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facts to be given in their proper order, corresponding to a prefix
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of the premises of the rule involved. Note that positions may be
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easily skipped using something like @{command "from"}~@{text "_
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\<AND> a \<AND> b"}, for example. This involves the trivial rule
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@{text "PROP \<psi> \<Longrightarrow> PROP \<psi>"}, which is bound in Isabelle/Pure as
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``@{fact_ref "_"}'' (underscore).
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Automated methods (such as @{method simp} or @{method auto}) just
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insert any given facts before their usual operation. Depending on
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the kind of procedure involved, the order of facts is less
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significant here.
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*}
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section {* Goal statements \label{sec:goals} *}
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text {*
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\begin{matharray}{rcl}
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@{command_def "lemma"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
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@{command_def "theorem"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
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@{command_def "corollary"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
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@{command_def "have"} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\
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@{command_def "show"} & : & \isartrans{proof(state) ~|~ proof(chain)}{proof(prove)} \\
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@{command_def "hence"} & : & \isartrans{proof(state)}{proof(prove)} \\
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@{command_def "thus"} & : & \isartrans{proof(state)}{proof(prove)} \\
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@{command_def "print_statement"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
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\end{matharray}
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From a theory context, proof mode is entered by an initial goal
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command such as @{command "lemma"}, @{command "theorem"}, or
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@{command "corollary"}. Within a proof, new claims may be
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introduced locally as well; four variants are available here to
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indicate whether forward chaining of facts should be performed
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initially (via @{command_ref "then"}), and whether the final result
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is meant to solve some pending goal.
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Goals may consist of multiple statements, resulting in a list of
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facts eventually. A pending multi-goal is internally represented as
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a meta-level conjunction (printed as @{text "&&"}), which is usually
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split into the corresponding number of sub-goals prior to an initial
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method application, via @{command_ref "proof"}
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(\secref{sec:proof-steps}) or @{command_ref "apply"}
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(\secref{sec:tactic-commands}). The @{method_ref induct} method
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covered in \secref{sec:cases-induct} acts on multiple claims
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simultaneously.
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Claims at the theory level may be either in short or long form. A
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short goal merely consists of several simultaneous propositions
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(often just one). A long goal includes an explicit context
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specification for the subsequent conclusion, involving local
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parameters and assumptions. Here the role of each part of the
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statement is explicitly marked by separate keywords (see also
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\secref{sec:locale}); the local assumptions being introduced here
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are available as @{fact_ref assms} in the proof. Moreover, there
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are two kinds of conclusions: @{element_def "shows"} states several
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simultaneous propositions (essentially a big conjunction), while
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@{element_def "obtains"} claims several simultaneous simultaneous
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contexts of (essentially a big disjunction of eliminated parameters
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and assumptions, cf.\ \secref{sec:obtain}).
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\begin{rail}
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('lemma' | 'theorem' | 'corollary') target? (goal | longgoal)
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;
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('have' | 'show' | 'hence' | 'thus') goal
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;
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'print\_statement' modes? thmrefs
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;
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goal: (props + 'and')
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;
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longgoal: thmdecl? (contextelem *) conclusion
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;
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conclusion: 'shows' goal | 'obtains' (parname? case + '|')
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;
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case: (vars + 'and') 'where' (props + 'and')
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;
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\end{rail}
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\begin{descr}
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\item [@{command "lemma"}~@{text "a: \<phi>"}] enters proof mode with
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@{text \<phi>} as main goal, eventually resulting in some fact @{text "\<turnstile>
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\<phi>"} to be put back into the target context. An additional
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\railnonterm{context} specification may build up an initial proof
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context for the subsequent claim; this includes local definitions
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and syntax as well, see the definition of @{syntax contextelem} in
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\secref{sec:locale}.
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\item [@{command "theorem"}~@{text "a: \<phi>"} and @{command
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"corollary"}~@{text "a: \<phi>"}] are essentially the same as @{command
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"lemma"}~@{text "a: \<phi>"}, but the facts are internally marked as
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being of a different kind. This discrimination acts like a formal
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comment.
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\item [@{command "have"}~@{text "a: \<phi>"}] claims a local goal,
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eventually resulting in a fact within the current logical context.
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This operation is completely independent of any pending sub-goals of
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an enclosing goal statements, so @{command "have"} may be freely
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used for experimental exploration of potential results within a
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proof body.
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\item [@{command "show"}~@{text "a: \<phi>"}] is like @{command
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"have"}~@{text "a: \<phi>"} plus a second stage to refine some pending
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sub-goal for each one of the finished result, after having been
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exported into the corresponding context (at the head of the
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sub-proof of this @{command "show"} command).
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To accommodate interactive debugging, resulting rules are printed
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before being applied internally. Even more, interactive execution
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of @{command "show"} predicts potential failure and displays the
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resulting error as a warning beforehand. Watch out for the
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following message:
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%FIXME proper antiquitation
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\begin{ttbox}
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Problem! Local statement will fail to solve any pending goal
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\end{ttbox}
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\item [@{command "hence"}] abbreviates ``@{command "then"}~@{command
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"have"}'', i.e.\ claims a local goal to be proven by forward
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chaining the current facts. Note that @{command "hence"} is also
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equivalent to ``@{command "from"}~@{text this}~@{command "have"}''.
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\item [@{command "thus"}] abbreviates ``@{command "then"}~@{command
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"show"}''. Note that @{command "thus"} is also equivalent to
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``@{command "from"}~@{text this}~@{command "show"}''.
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\item [@{command "print_statement"}~@{text a}] prints facts from the
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current theory or proof context in long statement form, according to
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the syntax for @{command "lemma"} given above.
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\end{descr}
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Any goal statement causes some term abbreviations (such as
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@{variable_ref "?thesis"}) to be bound automatically, see also
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\secref{sec:term-abbrev}.
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The optional case names of @{element_ref "obtains"} have a twofold
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meaning: (1) during the of this claim they refer to the the local
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context introductions, (2) the resulting rule is annotated
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accordingly to support symbolic case splits when used with the
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@{method_ref cases} method (cf.\ \secref{sec:cases-induct}).
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\medskip
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\begin{warn}
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Isabelle/Isar suffers theory-level goal statements to contain
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\emph{unbound schematic variables}, although this does not conform
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to the aim of human-readable proof documents! The main problem
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with schematic goals is that the actual outcome is usually hard to
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predict, depending on the behavior of the proof methods applied
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during the course of reasoning. Note that most semi-automated
|
|
356 |
methods heavily depend on several kinds of implicit rule
|
|
357 |
declarations within the current theory context. As this would
|
|
358 |
also result in non-compositional checking of sub-proofs,
|
|
359 |
\emph{local goals} are not allowed to be schematic at all.
|
|
360 |
Nevertheless, schematic goals do have their use in Prolog-style
|
|
361 |
interactive synthesis of proven results, usually by stepwise
|
|
362 |
refinement via emulation of traditional Isabelle tactic scripts
|
|
363 |
(see also \secref{sec:tactic-commands}). In any case, users
|
|
364 |
should know what they are doing.
|
|
365 |
\end{warn}
|
|
366 |
*}
|
|
367 |
|
|
368 |
|
|
369 |
section {* Initial and terminal proof steps \label{sec:proof-steps} *}
|
|
370 |
|
|
371 |
text {*
|
|
372 |
\begin{matharray}{rcl}
|
|
373 |
@{command_def "proof"} & : & \isartrans{proof(prove)}{proof(state)} \\
|
|
374 |
@{command_def "qed"} & : & \isartrans{proof(state)}{proof(state) ~|~ theory} \\
|
|
375 |
@{command_def "by"} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
|
|
376 |
@{command_def ".."} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
|
|
377 |
@{command_def "."} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
|
|
378 |
@{command_def "sorry"} & : & \isartrans{proof(prove)}{proof(state) ~|~ theory} \\
|
|
379 |
\end{matharray}
|
|
380 |
|
|
381 |
Arbitrary goal refinement via tactics is considered harmful.
|
|
382 |
Structured proof composition in Isar admits proof methods to be
|
|
383 |
invoked in two places only.
|
|
384 |
|
|
385 |
\begin{enumerate}
|
|
386 |
|
|
387 |
\item An \emph{initial} refinement step @{command_ref
|
|
388 |
"proof"}~@{text "m\<^sub>1"} reduces a newly stated goal to a number
|
|
389 |
of sub-goals that are to be solved later. Facts are passed to
|
|
390 |
@{text "m\<^sub>1"} for forward chaining, if so indicated by @{text
|
|
391 |
"proof(chain)"} mode.
|
|
392 |
|
|
393 |
\item A \emph{terminal} conclusion step @{command_ref "qed"}~@{text
|
|
394 |
"m\<^sub>2"} is intended to solve remaining goals. No facts are
|
|
395 |
passed to @{text "m\<^sub>2"}.
|
|
396 |
|
|
397 |
\end{enumerate}
|
|
398 |
|
|
399 |
The only other (proper) way to affect pending goals in a proof body
|
|
400 |
is by @{command_ref "show"}, which involves an explicit statement of
|
|
401 |
what is to be solved eventually. Thus we avoid the fundamental
|
|
402 |
problem of unstructured tactic scripts that consist of numerous
|
|
403 |
consecutive goal transformations, with invisible effects.
|
|
404 |
|
|
405 |
\medskip As a general rule of thumb for good proof style, initial
|
|
406 |
proof methods should either solve the goal completely, or constitute
|
|
407 |
some well-understood reduction to new sub-goals. Arbitrary
|
|
408 |
automatic proof tools that are prone leave a large number of badly
|
|
409 |
structured sub-goals are no help in continuing the proof document in
|
|
410 |
an intelligible manner.
|
|
411 |
|
|
412 |
Unless given explicitly by the user, the default initial method is
|
|
413 |
``@{method_ref rule}'', which applies a single standard elimination
|
|
414 |
or introduction rule according to the topmost symbol involved.
|
|
415 |
There is no separate default terminal method. Any remaining goals
|
|
416 |
are always solved by assumption in the very last step.
|
|
417 |
|
|
418 |
\begin{rail}
|
|
419 |
'proof' method?
|
|
420 |
;
|
|
421 |
'qed' method?
|
|
422 |
;
|
|
423 |
'by' method method?
|
|
424 |
;
|
|
425 |
('.' | '..' | 'sorry')
|
|
426 |
;
|
|
427 |
\end{rail}
|
|
428 |
|
|
429 |
\begin{descr}
|
|
430 |
|
|
431 |
\item [@{command "proof"}~@{text "m\<^sub>1"}] refines the goal by
|
|
432 |
proof method @{text "m\<^sub>1"}; facts for forward chaining are
|
|
433 |
passed if so indicated by @{text "proof(chain)"} mode.
|
|
434 |
|
|
435 |
\item [@{command "qed"}~@{text "m\<^sub>2"}] refines any remaining
|
|
436 |
goals by proof method @{text "m\<^sub>2"} and concludes the
|
|
437 |
sub-proof by assumption. If the goal had been @{text "show"} (or
|
|
438 |
@{text "thus"}), some pending sub-goal is solved as well by the rule
|
|
439 |
resulting from the result \emph{exported} into the enclosing goal
|
|
440 |
context. Thus @{text "qed"} may fail for two reasons: either @{text
|
|
441 |
"m\<^sub>2"} fails, or the resulting rule does not fit to any
|
|
442 |
pending goal\footnote{This includes any additional ``strong''
|
|
443 |
assumptions as introduced by @{command "assume"}.} of the enclosing
|
|
444 |
context. Debugging such a situation might involve temporarily
|
|
445 |
changing @{command "show"} into @{command "have"}, or weakening the
|
|
446 |
local context by replacing occurrences of @{command "assume"} by
|
|
447 |
@{command "presume"}.
|
|
448 |
|
|
449 |
\item [@{command "by"}~@{text "m\<^sub>1 m\<^sub>2"}] is a
|
|
450 |
\emph{terminal proof}\index{proof!terminal}; it abbreviates
|
|
451 |
@{command "proof"}~@{text "m\<^sub>1"}~@{text "qed"}~@{text
|
|
452 |
"m\<^sub>2"}, but with backtracking across both methods. Debugging
|
|
453 |
an unsuccessful @{command "by"}~@{text "m\<^sub>1 m\<^sub>2"}
|
|
454 |
command can be done by expanding its definition; in many cases
|
|
455 |
@{command "proof"}~@{text "m\<^sub>1"} (or even @{text
|
|
456 |
"apply"}~@{text "m\<^sub>1"}) is already sufficient to see the
|
|
457 |
problem.
|
|
458 |
|
|
459 |
\item [``@{command ".."}''] is a \emph{default
|
|
460 |
proof}\index{proof!default}; it abbreviates @{command "by"}~@{text
|
|
461 |
"rule"}.
|
|
462 |
|
|
463 |
\item [``@{command "."}''] is a \emph{trivial
|
|
464 |
proof}\index{proof!trivial}; it abbreviates @{command "by"}~@{text
|
|
465 |
"this"}.
|
|
466 |
|
|
467 |
\item [@{command "sorry"}] is a \emph{fake proof}\index{proof!fake}
|
|
468 |
pretending to solve the pending claim without further ado. This
|
|
469 |
only works in interactive development, or if the @{ML
|
|
470 |
quick_and_dirty} flag is enabled (in ML). Facts emerging from fake
|
|
471 |
proofs are not the real thing. Internally, each theorem container
|
|
472 |
is tainted by an oracle invocation, which is indicated as ``@{text
|
|
473 |
"[!]"}'' in the printed result.
|
|
474 |
|
|
475 |
The most important application of @{command "sorry"} is to support
|
|
476 |
experimentation and top-down proof development.
|
|
477 |
|
|
478 |
\end{descr}
|
|
479 |
*}
|
|
480 |
|
|
481 |
|
|
482 |
section {* Fundamental methods and attributes \label{sec:pure-meth-att} *}
|
|
483 |
|
|
484 |
text {*
|
|
485 |
The following proof methods and attributes refer to basic logical
|
|
486 |
operations of Isar. Further methods and attributes are provided by
|
|
487 |
several generic and object-logic specific tools and packages (see
|
|
488 |
\chref{ch:gen-tools} and \chref{ch:hol}).
|
|
489 |
|
|
490 |
\begin{matharray}{rcl}
|
|
491 |
@{method_def "-"} & : & \isarmeth \\
|
|
492 |
@{method_def "fact"} & : & \isarmeth \\
|
|
493 |
@{method_def "assumption"} & : & \isarmeth \\
|
|
494 |
@{method_def "this"} & : & \isarmeth \\
|
|
495 |
@{method_def "rule"} & : & \isarmeth \\
|
|
496 |
@{method_def "iprover"} & : & \isarmeth \\[0.5ex]
|
26901
|
497 |
@{attribute_def (Pure) "intro"} & : & \isaratt \\
|
|
498 |
@{attribute_def (Pure) "elim"} & : & \isaratt \\
|
|
499 |
@{attribute_def (Pure) "dest"} & : & \isaratt \\
|
26870
|
500 |
@{attribute_def "rule"} & : & \isaratt \\[0.5ex]
|
|
501 |
@{attribute_def "OF"} & : & \isaratt \\
|
|
502 |
@{attribute_def "of"} & : & \isaratt \\
|
|
503 |
@{attribute_def "where"} & : & \isaratt \\
|
|
504 |
\end{matharray}
|
|
505 |
|
|
506 |
\begin{rail}
|
|
507 |
'fact' thmrefs?
|
|
508 |
;
|
|
509 |
'rule' thmrefs?
|
|
510 |
;
|
|
511 |
'iprover' ('!' ?) (rulemod *)
|
|
512 |
;
|
|
513 |
rulemod: ('intro' | 'elim' | 'dest') ((('!' | () | '?') nat?) | 'del') ':' thmrefs
|
|
514 |
;
|
|
515 |
('intro' | 'elim' | 'dest') ('!' | () | '?') nat?
|
|
516 |
;
|
|
517 |
'rule' 'del'
|
|
518 |
;
|
|
519 |
'OF' thmrefs
|
|
520 |
;
|
|
521 |
'of' insts ('concl' ':' insts)?
|
|
522 |
;
|
|
523 |
'where' ((name | var | typefree | typevar) '=' (type | term) * 'and')
|
|
524 |
;
|
|
525 |
\end{rail}
|
|
526 |
|
|
527 |
\begin{descr}
|
|
528 |
|
|
529 |
\item [``@{method "-"}'' (minus)] does nothing but insert the
|
|
530 |
forward chaining facts as premises into the goal. Note that command
|
|
531 |
@{command_ref "proof"} without any method actually performs a single
|
|
532 |
reduction step using the @{method_ref rule} method; thus a plain
|
|
533 |
\emph{do-nothing} proof step would be ``@{command "proof"}~@{text
|
|
534 |
"-"}'' rather than @{command "proof"} alone.
|
|
535 |
|
|
536 |
\item [@{method "fact"}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}] composes
|
|
537 |
some fact from @{text "a\<^sub>1, \<dots>, a\<^sub>n"} (or implicitly from
|
|
538 |
the current proof context) modulo unification of schematic type and
|
|
539 |
term variables. The rule structure is not taken into account, i.e.\
|
|
540 |
meta-level implication is considered atomic. This is the same
|
|
541 |
principle underlying literal facts (cf.\ \secref{sec:syn-att}):
|
|
542 |
``@{command "have"}~@{text "\<phi>"}~@{command "by"}~@{text fact}'' is
|
|
543 |
equivalent to ``@{command "note"}~@{verbatim "`"}@{text \<phi>}@{verbatim
|
|
544 |
"`"}'' provided that @{text "\<turnstile> \<phi>"} is an instance of some known
|
|
545 |
@{text "\<turnstile> \<phi>"} in the proof context.
|
|
546 |
|
|
547 |
\item [@{method assumption}] solves some goal by a single assumption
|
|
548 |
step. All given facts are guaranteed to participate in the
|
|
549 |
refinement; this means there may be only 0 or 1 in the first place.
|
|
550 |
Recall that @{command "qed"} (\secref{sec:proof-steps}) already
|
|
551 |
concludes any remaining sub-goals by assumption, so structured
|
|
552 |
proofs usually need not quote the @{method assumption} method at
|
|
553 |
all.
|
|
554 |
|
|
555 |
\item [@{method this}] applies all of the current facts directly as
|
|
556 |
rules. Recall that ``@{command "."}'' (dot) abbreviates ``@{command
|
|
557 |
"by"}~@{text this}''.
|
|
558 |
|
|
559 |
\item [@{method rule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}] applies some
|
|
560 |
rule given as argument in backward manner; facts are used to reduce
|
|
561 |
the rule before applying it to the goal. Thus @{method rule}
|
|
562 |
without facts is plain introduction, while with facts it becomes
|
|
563 |
elimination.
|
|
564 |
|
|
565 |
When no arguments are given, the @{method rule} method tries to pick
|
|
566 |
appropriate rules automatically, as declared in the current context
|
26901
|
567 |
using the @{attribute (Pure) intro}, @{attribute (Pure) elim},
|
|
568 |
@{attribute (Pure) dest} attributes (see below). This is the
|
|
569 |
default behavior of @{command "proof"} and ``@{command ".."}''
|
|
570 |
(double-dot) steps (see \secref{sec:proof-steps}).
|
26870
|
571 |
|
|
572 |
\item [@{method iprover}] performs intuitionistic proof search,
|
|
573 |
depending on specifically declared rules from the context, or given
|
|
574 |
as explicit arguments. Chained facts are inserted into the goal
|
|
575 |
before commencing proof search; ``@{method iprover}@{text "!"}''
|
|
576 |
means to include the current @{fact prems} as well.
|
|
577 |
|
26901
|
578 |
Rules need to be classified as @{attribute (Pure) intro},
|
|
579 |
@{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
|
|
580 |
``@{text "!"}'' indicator refers to ``safe'' rules, which may be
|
|
581 |
applied aggressively (without considering back-tracking later).
|
|
582 |
Rules declared with ``@{text "?"}'' are ignored in proof search (the
|
|
583 |
single-step @{method rule} method still observes these). An
|
|
584 |
explicit weight annotation may be given as well; otherwise the
|
|
585 |
number of rule premises will be taken into account here.
|
26870
|
586 |
|
26901
|
587 |
\item [@{attribute (Pure) intro}, @{attribute (Pure) elim}, and
|
|
588 |
@{attribute (Pure) dest}] declare introduction, elimination, and
|
|
589 |
destruct rules, to be used with the @{method rule} and @{method
|
|
590 |
iprover} methods. Note that the latter will ignore rules declared
|
|
591 |
with ``@{text "?"}'', while ``@{text "!"}'' are used most
|
|
592 |
aggressively.
|
26870
|
593 |
|
|
594 |
The classical reasoner (see \secref{sec:classical}) introduces its
|
|
595 |
own variants of these attributes; use qualified names to access the
|
26901
|
596 |
present versions of Isabelle/Pure, i.e.\ @{attribute (Pure)
|
|
597 |
"Pure.intro"}.
|
26870
|
598 |
|
|
599 |
\item [@{attribute rule}~@{text del}] undeclares introduction,
|
|
600 |
elimination, or destruct rules.
|
|
601 |
|
|
602 |
\item [@{attribute OF}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}] applies some
|
|
603 |
theorem to all of the given rules @{text "a\<^sub>1, \<dots>, a\<^sub>n"}
|
|
604 |
(in parallel). This corresponds to the @{ML "op MRS"} operation in
|
|
605 |
ML, but note the reversed order. Positions may be effectively
|
|
606 |
skipped by including ``@{text _}'' (underscore) as argument.
|
|
607 |
|
|
608 |
\item [@{attribute of}~@{text "t\<^sub>1 \<dots> t\<^sub>n"}] performs
|
|
609 |
positional instantiation of term variables. The terms @{text
|
|
610 |
"t\<^sub>1, \<dots>, t\<^sub>n"} are substituted for any schematic
|
26888
|
611 |
variables occurring in a theorem from left to right; ``@{text _}''
|
|
612 |
(underscore) indicates to skip a position. Arguments following a
|
|
613 |
``@{text "concl:"}'' specification refer to positions of the
|
|
614 |
conclusion of a rule.
|
26870
|
615 |
|
|
616 |
\item [@{attribute "where"}~@{text "x\<^sub>1 = t\<^sub>1 \<AND> \<dots>
|
|
617 |
x\<^sub>n = t\<^sub>n"}] performs named instantiation of schematic
|
|
618 |
type and term variables occurring in a theorem. Schematic variables
|
|
619 |
have to be specified on the left-hand side (e.g.\ @{text "?x1.3"}).
|
|
620 |
The question mark may be omitted if the variable name is a plain
|
|
621 |
identifier without index. As type instantiations are inferred from
|
|
622 |
term instantiations, explicit type instantiations are seldom
|
|
623 |
necessary.
|
|
624 |
|
|
625 |
\end{descr}
|
|
626 |
*}
|
|
627 |
|
|
628 |
|
|
629 |
section {* Term abbreviations \label{sec:term-abbrev} *}
|
|
630 |
|
|
631 |
text {*
|
|
632 |
\begin{matharray}{rcl}
|
|
633 |
@{command_def "let"} & : & \isartrans{proof(state)}{proof(state)} \\
|
|
634 |
@{keyword_def "is"} & : & syntax \\
|
|
635 |
\end{matharray}
|
|
636 |
|
|
637 |
Abbreviations may be either bound by explicit @{command
|
|
638 |
"let"}~@{text "p \<equiv> t"} statements, or by annotating assumptions or
|
|
639 |
goal statements with a list of patterns ``@{text "(\<IS> p\<^sub>1 \<dots>
|
|
640 |
p\<^sub>n)"}''. In both cases, higher-order matching is invoked to
|
|
641 |
bind extra-logical term variables, which may be either named
|
|
642 |
schematic variables of the form @{text ?x}, or nameless dummies
|
|
643 |
``@{variable _}'' (underscore). Note that in the @{command "let"}
|
|
644 |
form the patterns occur on the left-hand side, while the @{keyword
|
|
645 |
"is"} patterns are in postfix position.
|
|
646 |
|
|
647 |
Polymorphism of term bindings is handled in Hindley-Milner style,
|
|
648 |
similar to ML. Type variables referring to local assumptions or
|
|
649 |
open goal statements are \emph{fixed}, while those of finished
|
|
650 |
results or bound by @{command "let"} may occur in \emph{arbitrary}
|
|
651 |
instances later. Even though actual polymorphism should be rarely
|
|
652 |
used in practice, this mechanism is essential to achieve proper
|
|
653 |
incremental type-inference, as the user proceeds to build up the
|
|
654 |
Isar proof text from left to right.
|
|
655 |
|
|
656 |
\medskip Term abbreviations are quite different from local
|
|
657 |
definitions as introduced via @{command "def"} (see
|
|
658 |
\secref{sec:proof-context}). The latter are visible within the
|
|
659 |
logic as actual equations, while abbreviations disappear during the
|
|
660 |
input process just after type checking. Also note that @{command
|
|
661 |
"def"} does not support polymorphism.
|
|
662 |
|
|
663 |
\begin{rail}
|
|
664 |
'let' ((term + 'and') '=' term + 'and')
|
|
665 |
;
|
|
666 |
\end{rail}
|
|
667 |
|
|
668 |
The syntax of @{keyword "is"} patterns follows \railnonterm{termpat}
|
|
669 |
or \railnonterm{proppat} (see \secref{sec:term-decls}).
|
|
670 |
|
|
671 |
\begin{descr}
|
|
672 |
|
|
673 |
\item [@{command "let"}~@{text "p\<^sub>1 = t\<^sub>1 \<AND> \<dots>
|
|
674 |
p\<^sub>n = t\<^sub>n"}] binds any text variables in patterns @{text
|
|
675 |
"p\<^sub>1, \<dots>, p\<^sub>n"} by simultaneous higher-order matching
|
|
676 |
against terms @{text "t\<^sub>1, \<dots>, t\<^sub>n"}.
|
|
677 |
|
|
678 |
\item [@{text "(\<IS> p\<^sub>1 \<dots> p\<^sub>n)"}] resembles @{command
|
|
679 |
"let"}, but matches @{text "p\<^sub>1, \<dots>, p\<^sub>n"} against the
|
|
680 |
preceding statement. Also note that @{keyword "is"} is not a
|
|
681 |
separate command, but part of others (such as @{command "assume"},
|
|
682 |
@{command "have"} etc.).
|
|
683 |
|
|
684 |
\end{descr}
|
|
685 |
|
|
686 |
Some \emph{implicit} term abbreviations\index{term abbreviations}
|
|
687 |
for goals and facts are available as well. For any open goal,
|
|
688 |
@{variable_ref thesis} refers to its object-level statement,
|
|
689 |
abstracted over any meta-level parameters (if present). Likewise,
|
|
690 |
@{variable_ref this} is bound for fact statements resulting from
|
|
691 |
assumptions or finished goals. In case @{variable this} refers to
|
|
692 |
an object-logic statement that is an application @{text "f t"}, then
|
|
693 |
@{text t} is bound to the special text variable ``@{variable "\<dots>"}''
|
|
694 |
(three dots). The canonical application of this convenience are
|
|
695 |
calculational proofs (see \secref{sec:calculation}).
|
|
696 |
*}
|
|
697 |
|
|
698 |
|
|
699 |
section {* Block structure *}
|
|
700 |
|
|
701 |
text {*
|
|
702 |
\begin{matharray}{rcl}
|
|
703 |
@{command_def "next"} & : & \isartrans{proof(state)}{proof(state)} \\
|
|
704 |
@{command_def "{"} & : & \isartrans{proof(state)}{proof(state)} \\
|
|
705 |
@{command_def "}"} & : & \isartrans{proof(state)}{proof(state)} \\
|
|
706 |
\end{matharray}
|
|
707 |
|
|
708 |
While Isar is inherently block-structured, opening and closing
|
|
709 |
blocks is mostly handled rather casually, with little explicit
|
|
710 |
user-intervention. Any local goal statement automatically opens
|
|
711 |
\emph{two} internal blocks, which are closed again when concluding
|
|
712 |
the sub-proof (by @{command "qed"} etc.). Sections of different
|
|
713 |
context within a sub-proof may be switched via @{command "next"},
|
|
714 |
which is just a single block-close followed by block-open again.
|
|
715 |
The effect of @{command "next"} is to reset the local proof context;
|
|
716 |
there is no goal focus involved here!
|
|
717 |
|
|
718 |
For slightly more advanced applications, there are explicit block
|
|
719 |
parentheses as well. These typically achieve a stronger forward
|
|
720 |
style of reasoning.
|
|
721 |
|
|
722 |
\begin{descr}
|
|
723 |
|
|
724 |
\item [@{command "next"}] switches to a fresh block within a
|
|
725 |
sub-proof, resetting the local context to the initial one.
|
|
726 |
|
|
727 |
\item [@{command "{"} and @{command "}"}] explicitly open and close
|
|
728 |
blocks. Any current facts pass through ``@{command "{"}''
|
|
729 |
unchanged, while ``@{command "}"}'' causes any result to be
|
|
730 |
\emph{exported} into the enclosing context. Thus fixed variables
|
|
731 |
are generalized, assumptions discharged, and local definitions
|
|
732 |
unfolded (cf.\ \secref{sec:proof-context}). There is no difference
|
|
733 |
of @{command "assume"} and @{command "presume"} in this mode of
|
|
734 |
forward reasoning --- in contrast to plain backward reasoning with
|
|
735 |
the result exported at @{command "show"} time.
|
|
736 |
|
|
737 |
\end{descr}
|
|
738 |
*}
|
|
739 |
|
|
740 |
|
|
741 |
section {* Emulating tactic scripts \label{sec:tactic-commands} *}
|
|
742 |
|
|
743 |
text {*
|
|
744 |
The Isar provides separate commands to accommodate tactic-style
|
|
745 |
proof scripts within the same system. While being outside the
|
|
746 |
orthodox Isar proof language, these might come in handy for
|
|
747 |
interactive exploration and debugging, or even actual tactical proof
|
|
748 |
within new-style theories (to benefit from document preparation, for
|
|
749 |
example). See also \secref{sec:tactics} for actual tactics, that
|
|
750 |
have been encapsulated as proof methods. Proper proof methods may
|
|
751 |
be used in scripts, too.
|
|
752 |
|
|
753 |
\begin{matharray}{rcl}
|
|
754 |
@{command_def "apply"}@{text "\<^sup>*"} & : & \isartrans{proof(prove)}{proof(prove)} \\
|
|
755 |
@{command_def "apply_end"}@{text "\<^sup>*"} & : & \isartrans{proof(state)}{proof(state)} \\
|
|
756 |
@{command_def "done"}@{text "\<^sup>*"} & : & \isartrans{proof(prove)}{proof(state)} \\
|
|
757 |
@{command_def "defer"}@{text "\<^sup>*"} & : & \isartrans{proof}{proof} \\
|
|
758 |
@{command_def "prefer"}@{text "\<^sup>*"} & : & \isartrans{proof}{proof} \\
|
|
759 |
@{command_def "back"}@{text "\<^sup>*"} & : & \isartrans{proof}{proof} \\
|
|
760 |
\end{matharray}
|
|
761 |
|
|
762 |
\begin{rail}
|
|
763 |
( 'apply' | 'apply\_end' ) method
|
|
764 |
;
|
|
765 |
'defer' nat?
|
|
766 |
;
|
|
767 |
'prefer' nat
|
|
768 |
;
|
|
769 |
\end{rail}
|
|
770 |
|
|
771 |
\begin{descr}
|
|
772 |
|
|
773 |
\item [@{command "apply"}~@{text m}] applies proof method @{text m}
|
|
774 |
in initial position, but unlike @{command "proof"} it retains
|
|
775 |
``@{text "proof(prove)"}'' mode. Thus consecutive method
|
|
776 |
applications may be given just as in tactic scripts.
|
|
777 |
|
|
778 |
Facts are passed to @{text m} as indicated by the goal's
|
|
779 |
forward-chain mode, and are \emph{consumed} afterwards. Thus any
|
|
780 |
further @{command "apply"} command would always work in a purely
|
|
781 |
backward manner.
|
|
782 |
|
|
783 |
\item [@{command "apply_end"}~@{text "m"}] applies proof method
|
|
784 |
@{text m} as if in terminal position. Basically, this simulates a
|
|
785 |
multi-step tactic script for @{command "qed"}, but may be given
|
|
786 |
anywhere within the proof body.
|
|
787 |
|
26894
|
788 |
No facts are passed to @{text m} here. Furthermore, the static
|
26870
|
789 |
context is that of the enclosing goal (as for actual @{command
|
|
790 |
"qed"}). Thus the proof method may not refer to any assumptions
|
|
791 |
introduced in the current body, for example.
|
|
792 |
|
|
793 |
\item [@{command "done"}] completes a proof script, provided that
|
|
794 |
the current goal state is solved completely. Note that actual
|
|
795 |
structured proof commands (e.g.\ ``@{command "."}'' or @{command
|
|
796 |
"sorry"}) may be used to conclude proof scripts as well.
|
|
797 |
|
|
798 |
\item [@{command "defer"}~@{text n} and @{command "prefer"}~@{text
|
|
799 |
n}] shuffle the list of pending goals: @{command "defer"} puts off
|
|
800 |
sub-goal @{text n} to the end of the list (@{text "n = 1"} by
|
|
801 |
default), while @{command "prefer"} brings sub-goal @{text n} to the
|
|
802 |
front.
|
|
803 |
|
|
804 |
\item [@{command "back"}] does back-tracking over the result
|
|
805 |
sequence of the latest proof command. Basically, any proof command
|
|
806 |
may return multiple results.
|
|
807 |
|
|
808 |
\end{descr}
|
|
809 |
|
|
810 |
Any proper Isar proof method may be used with tactic script commands
|
|
811 |
such as @{command "apply"}. A few additional emulations of actual
|
|
812 |
tactics are provided as well; these would be never used in actual
|
|
813 |
structured proofs, of course.
|
|
814 |
*}
|
|
815 |
|
|
816 |
|
|
817 |
section {* Omitting proofs *}
|
|
818 |
|
|
819 |
text {*
|
|
820 |
\begin{matharray}{rcl}
|
|
821 |
@{command_def "oops"} & : & \isartrans{proof}{theory} \\
|
|
822 |
\end{matharray}
|
|
823 |
|
|
824 |
The @{command "oops"} command discontinues the current proof
|
|
825 |
attempt, while considering the partial proof text as properly
|
|
826 |
processed. This is conceptually quite different from ``faking''
|
|
827 |
actual proofs via @{command_ref "sorry"} (see
|
|
828 |
\secref{sec:proof-steps}): @{command "oops"} does not observe the
|
|
829 |
proof structure at all, but goes back right to the theory level.
|
|
830 |
Furthermore, @{command "oops"} does not produce any result theorem
|
|
831 |
--- there is no intended claim to be able to complete the proof
|
|
832 |
anyhow.
|
|
833 |
|
|
834 |
A typical application of @{command "oops"} is to explain Isar proofs
|
|
835 |
\emph{within} the system itself, in conjunction with the document
|
|
836 |
preparation tools of Isabelle described in \cite{isabelle-sys}.
|
|
837 |
Thus partial or even wrong proof attempts can be discussed in a
|
|
838 |
logically sound manner. Note that the Isabelle {\LaTeX} macros can
|
|
839 |
be easily adapted to print something like ``@{text "\<dots>"}'' instead of
|
|
840 |
the keyword ``@{command "oops"}''.
|
|
841 |
|
|
842 |
\medskip The @{command "oops"} command is undo-able, unlike
|
|
843 |
@{command_ref "kill"} (see \secref{sec:history}). The effect is to
|
|
844 |
get back to the theory just before the opening of the proof.
|
|
845 |
*}
|
|
846 |
|
|
847 |
|
|
848 |
section {* Generalized elimination \label{sec:obtain} *}
|
|
849 |
|
|
850 |
text {*
|
|
851 |
\begin{matharray}{rcl}
|
|
852 |
@{command_def "obtain"} & : & \isartrans{proof(state)}{proof(prove)} \\
|
|
853 |
@{command_def "guess"}@{text "\<^sup>*"} & : & \isartrans{proof(state)}{proof(prove)} \\
|
|
854 |
\end{matharray}
|
|
855 |
|
|
856 |
Generalized elimination means that additional elements with certain
|
|
857 |
properties may be introduced in the current context, by virtue of a
|
|
858 |
locally proven ``soundness statement''. Technically speaking, the
|
|
859 |
@{command "obtain"} language element is like a declaration of
|
|
860 |
@{command "fix"} and @{command "assume"} (see also see
|
|
861 |
\secref{sec:proof-context}), together with a soundness proof of its
|
|
862 |
additional claim. According to the nature of existential reasoning,
|
|
863 |
assumptions get eliminated from any result exported from the context
|
|
864 |
later, provided that the corresponding parameters do \emph{not}
|
|
865 |
occur in the conclusion.
|
|
866 |
|
|
867 |
\begin{rail}
|
|
868 |
'obtain' parname? (vars + 'and') 'where' (props + 'and')
|
|
869 |
;
|
|
870 |
'guess' (vars + 'and')
|
|
871 |
;
|
|
872 |
\end{rail}
|
|
873 |
|
|
874 |
The derived Isar command @{command "obtain"} is defined as follows
|
|
875 |
(where @{text "b\<^sub>1, \<dots>, b\<^sub>k"} shall refer to (optional)
|
|
876 |
facts indicated for forward chaining).
|
|
877 |
\begin{matharray}{l}
|
|
878 |
@{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]
|
|
879 |
\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"} \\
|
|
880 |
\quad @{command "proof"}~@{text succeed} \\
|
|
881 |
\qquad @{command "fix"}~@{text thesis} \\
|
|
882 |
\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"} \\
|
|
883 |
\qquad @{command "then"}~@{command "show"}~@{text thesis} \\
|
|
884 |
\quad\qquad @{command "apply"}~@{text -} \\
|
|
885 |
\quad\qquad @{command "using"}~@{text "b\<^sub>1 \<dots> b\<^sub>k \<langle>proof\<rangle>"} \\
|
|
886 |
\quad @{command "qed"} \\
|
|
887 |
\quad @{command "fix"}~@{text "x\<^sub>1 \<dots> x\<^sub>m"}~@{command "assume"}@{text "\<^sup>* a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"} \\
|
|
888 |
\end{matharray}
|
|
889 |
|
|
890 |
Typically, the soundness proof is relatively straight-forward, often
|
|
891 |
just by canonical automated tools such as ``@{command "by"}~@{text
|
|
892 |
simp}'' or ``@{command "by"}~@{text blast}''. Accordingly, the
|
|
893 |
``@{text that}'' reduction above is declared as simplification and
|
|
894 |
introduction rule.
|
|
895 |
|
|
896 |
In a sense, @{command "obtain"} represents at the level of Isar
|
|
897 |
proofs what would be meta-logical existential quantifiers and
|
|
898 |
conjunctions. This concept has a broad range of useful
|
|
899 |
applications, ranging from plain elimination (or introduction) of
|
|
900 |
object-level existential and conjunctions, to elimination over
|
|
901 |
results of symbolic evaluation of recursive definitions, for
|
|
902 |
example. Also note that @{command "obtain"} without parameters acts
|
|
903 |
much like @{command "have"}, where the result is treated as a
|
|
904 |
genuine assumption.
|
|
905 |
|
|
906 |
An alternative name to be used instead of ``@{text that}'' above may
|
|
907 |
be given in parentheses.
|
|
908 |
|
|
909 |
\medskip The improper variant @{command "guess"} is similar to
|
|
910 |
@{command "obtain"}, but derives the obtained statement from the
|
|
911 |
course of reasoning! The proof starts with a fixed goal @{text
|
|
912 |
thesis}. The subsequent proof may refine this to anything of the
|
|
913 |
form like @{text "\<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots>
|
|
914 |
\<phi>\<^sub>n \<Longrightarrow> thesis"}, but must not introduce new subgoals. The
|
|
915 |
final goal state is then used as reduction rule for the obtain
|
|
916 |
scheme described above. Obtained parameters @{text "x\<^sub>1, \<dots>,
|
|
917 |
x\<^sub>m"} are marked as internal by default, which prevents the
|
|
918 |
proof context from being polluted by ad-hoc variables. The variable
|
|
919 |
names and type constraints given as arguments for @{command "guess"}
|
|
920 |
specify a prefix of obtained parameters explicitly in the text.
|
|
921 |
|
|
922 |
It is important to note that the facts introduced by @{command
|
|
923 |
"obtain"} and @{command "guess"} may not be polymorphic: any
|
|
924 |
type-variables occurring here are fixed in the present context!
|
|
925 |
*}
|
|
926 |
|
|
927 |
|
|
928 |
section {* Calculational reasoning \label{sec:calculation} *}
|
|
929 |
|
|
930 |
text {*
|
|
931 |
\begin{matharray}{rcl}
|
|
932 |
@{command_def "also"} & : & \isartrans{proof(state)}{proof(state)} \\
|
|
933 |
@{command_def "finally"} & : & \isartrans{proof(state)}{proof(chain)} \\
|
|
934 |
@{command_def "moreover"} & : & \isartrans{proof(state)}{proof(state)} \\
|
|
935 |
@{command_def "ultimately"} & : & \isartrans{proof(state)}{proof(chain)} \\
|
|
936 |
@{command_def "print_trans_rules"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
|
|
937 |
@{attribute trans} & : & \isaratt \\
|
|
938 |
@{attribute sym} & : & \isaratt \\
|
|
939 |
@{attribute symmetric} & : & \isaratt \\
|
|
940 |
\end{matharray}
|
|
941 |
|
|
942 |
Calculational proof is forward reasoning with implicit application
|
|
943 |
of transitivity rules (such those of @{text "="}, @{text "\<le>"},
|
|
944 |
@{text "<"}). Isabelle/Isar maintains an auxiliary fact register
|
|
945 |
@{fact_ref calculation} for accumulating results obtained by
|
|
946 |
transitivity composed with the current result. Command @{command
|
|
947 |
"also"} updates @{fact calculation} involving @{fact this}, while
|
|
948 |
@{command "finally"} exhibits the final @{fact calculation} by
|
|
949 |
forward chaining towards the next goal statement. Both commands
|
|
950 |
require valid current facts, i.e.\ may occur only after commands
|
|
951 |
that produce theorems such as @{command "assume"}, @{command
|
|
952 |
"note"}, or some finished proof of @{command "have"}, @{command
|
|
953 |
"show"} etc. The @{command "moreover"} and @{command "ultimately"}
|
|
954 |
commands are similar to @{command "also"} and @{command "finally"},
|
|
955 |
but only collect further results in @{fact calculation} without
|
|
956 |
applying any rules yet.
|
|
957 |
|
|
958 |
Also note that the implicit term abbreviation ``@{text "\<dots>"}'' has
|
|
959 |
its canonical application with calculational proofs. It refers to
|
|
960 |
the argument of the preceding statement. (The argument of a curried
|
|
961 |
infix expression happens to be its right-hand side.)
|
|
962 |
|
|
963 |
Isabelle/Isar calculations are implicitly subject to block structure
|
|
964 |
in the sense that new threads of calculational reasoning are
|
|
965 |
commenced for any new block (as opened by a local goal, for
|
|
966 |
example). This means that, apart from being able to nest
|
|
967 |
calculations, there is no separate \emph{begin-calculation} command
|
|
968 |
required.
|
|
969 |
|
|
970 |
\medskip The Isar calculation proof commands may be defined as
|
|
971 |
follows:\footnote{We suppress internal bookkeeping such as proper
|
|
972 |
handling of block-structure.}
|
|
973 |
|
|
974 |
\begin{matharray}{rcl}
|
|
975 |
@{command "also"}@{text "\<^sub>0"} & \equiv & @{command "note"}~@{text "calculation = this"} \\
|
|
976 |
@{command "also"}@{text "\<^sub>n\<^sub>+\<^sub>1"} & \equiv & @{command "note"}~@{text "calculation = trans [OF calculation this]"} \\[0.5ex]
|
|
977 |
@{command "finally"} & \equiv & @{command "also"}~@{command "from"}~@{text calculation} \\[0.5ex]
|
|
978 |
@{command "moreover"} & \equiv & @{command "note"}~@{text "calculation = calculation this"} \\
|
|
979 |
@{command "ultimately"} & \equiv & @{command "moreover"}~@{command "from"}~@{text calculation} \\
|
|
980 |
\end{matharray}
|
|
981 |
|
|
982 |
\begin{rail}
|
|
983 |
('also' | 'finally') ('(' thmrefs ')')?
|
|
984 |
;
|
|
985 |
'trans' (() | 'add' | 'del')
|
|
986 |
;
|
|
987 |
\end{rail}
|
|
988 |
|
|
989 |
\begin{descr}
|
|
990 |
|
|
991 |
\item [@{command "also"}~@{text "(a\<^sub>1 \<dots> a\<^sub>n)"}]
|
|
992 |
maintains the auxiliary @{fact calculation} register as follows.
|
|
993 |
The first occurrence of @{command "also"} in some calculational
|
|
994 |
thread initializes @{fact calculation} by @{fact this}. Any
|
|
995 |
subsequent @{command "also"} on the same level of block-structure
|
|
996 |
updates @{fact calculation} by some transitivity rule applied to
|
|
997 |
@{fact calculation} and @{fact this} (in that order). Transitivity
|
|
998 |
rules are picked from the current context, unless alternative rules
|
|
999 |
are given as explicit arguments.
|
|
1000 |
|
|
1001 |
\item [@{command "finally"}~@{text "(a\<^sub>1 \<dots> a\<^sub>n)"}]
|
|
1002 |
maintaining @{fact calculation} in the same way as @{command
|
|
1003 |
"also"}, and concludes the current calculational thread. The final
|
|
1004 |
result is exhibited as fact for forward chaining towards the next
|
|
1005 |
goal. Basically, @{command "finally"} just abbreviates @{command
|
|
1006 |
"also"}~@{command "from"}~@{fact calculation}. Typical idioms for
|
|
1007 |
concluding calculational proofs are ``@{command "finally"}~@{command
|
|
1008 |
"show"}~@{text ?thesis}~@{command "."}'' and ``@{command
|
|
1009 |
"finally"}~@{command "have"}~@{text \<phi>}~@{command "."}''.
|
|
1010 |
|
|
1011 |
\item [@{command "moreover"} and @{command "ultimately"}] are
|
|
1012 |
analogous to @{command "also"} and @{command "finally"}, but collect
|
|
1013 |
results only, without applying rules.
|
|
1014 |
|
|
1015 |
\item [@{command "print_trans_rules"}] prints the list of
|
|
1016 |
transitivity rules (for calculational commands @{command "also"} and
|
|
1017 |
@{command "finally"}) and symmetry rules (for the @{attribute
|
|
1018 |
symmetric} operation and single step elimination patters) of the
|
|
1019 |
current context.
|
|
1020 |
|
|
1021 |
\item [@{attribute trans}] declares theorems as transitivity rules.
|
|
1022 |
|
|
1023 |
\item [@{attribute sym}] declares symmetry rules, as well as
|
26894
|
1024 |
@{attribute "Pure.elim"}@{text "?"} rules.
|
26870
|
1025 |
|
|
1026 |
\item [@{attribute symmetric}] resolves a theorem with some rule
|
|
1027 |
declared as @{attribute sym} in the current context. For example,
|
|
1028 |
``@{command "assume"}~@{text "[symmetric]: x = y"}'' produces a
|
|
1029 |
swapped fact derived from that assumption.
|
|
1030 |
|
|
1031 |
In structured proof texts it is often more appropriate to use an
|
|
1032 |
explicit single-step elimination proof, such as ``@{command
|
|
1033 |
"assume"}~@{text "x = y"}~@{command "then"}~@{command "have"}~@{text
|
|
1034 |
"y = x"}~@{command ".."}''.
|
|
1035 |
|
|
1036 |
\end{descr}
|
|
1037 |
*}
|
|
1038 |
|
27040
|
1039 |
section {* Proof by cases and induction \label{sec:cases-induct} *}
|
|
1040 |
|
|
1041 |
subsection {* Rule contexts *}
|
|
1042 |
|
|
1043 |
text {*
|
|
1044 |
\begin{matharray}{rcl}
|
|
1045 |
@{command_def "case"} & : & \isartrans{proof(state)}{proof(state)} \\
|
|
1046 |
@{command_def "print_cases"}@{text "\<^sup>*"} & : & \isarkeep{proof} \\
|
|
1047 |
@{attribute_def case_names} & : & \isaratt \\
|
|
1048 |
@{attribute_def case_conclusion} & : & \isaratt \\
|
|
1049 |
@{attribute_def params} & : & \isaratt \\
|
|
1050 |
@{attribute_def consumes} & : & \isaratt \\
|
|
1051 |
\end{matharray}
|
|
1052 |
|
|
1053 |
The puristic way to build up Isar proof contexts is by explicit
|
|
1054 |
language elements like @{command "fix"}, @{command "assume"},
|
|
1055 |
@{command "let"} (see \secref{sec:proof-context}). This is adequate
|
|
1056 |
for plain natural deduction, but easily becomes unwieldy in concrete
|
|
1057 |
verification tasks, which typically involve big induction rules with
|
|
1058 |
several cases.
|
|
1059 |
|
|
1060 |
The @{command "case"} command provides a shorthand to refer to a
|
|
1061 |
local context symbolically: certain proof methods provide an
|
|
1062 |
environment of named ``cases'' of the form @{text "c: x\<^sub>1, \<dots>,
|
|
1063 |
x\<^sub>m, \<phi>\<^sub>1, \<dots>, \<phi>\<^sub>n"}; the effect of ``@{command
|
|
1064 |
"case"}~@{text c}'' is then equivalent to ``@{command "fix"}~@{text
|
|
1065 |
"x\<^sub>1 \<dots> x\<^sub>m"}~@{command "assume"}~@{text "c: \<phi>\<^sub>1 \<dots>
|
|
1066 |
\<phi>\<^sub>n"}''. Term bindings may be covered as well, notably
|
|
1067 |
@{variable ?case} for the main conclusion.
|
|
1068 |
|
|
1069 |
By default, the ``terminology'' @{text "x\<^sub>1, \<dots>, x\<^sub>m"} of
|
|
1070 |
a case value is marked as hidden, i.e.\ there is no way to refer to
|
|
1071 |
such parameters in the subsequent proof text. After all, original
|
|
1072 |
rule parameters stem from somewhere outside of the current proof
|
|
1073 |
text. By using the explicit form ``@{command "case"}~@{text "(c
|
|
1074 |
y\<^sub>1 \<dots> y\<^sub>m)"}'' instead, the proof author is able to
|
|
1075 |
chose local names that fit nicely into the current context.
|
|
1076 |
|
|
1077 |
\medskip It is important to note that proper use of @{command
|
|
1078 |
"case"} does not provide means to peek at the current goal state,
|
|
1079 |
which is not directly observable in Isar! Nonetheless, goal
|
|
1080 |
refinement commands do provide named cases @{text "goal\<^sub>i"}
|
|
1081 |
for each subgoal @{text "i = 1, \<dots>, n"} of the resulting goal state.
|
|
1082 |
Using this extra feature requires great care, because some bits of
|
|
1083 |
the internal tactical machinery intrude the proof text. In
|
|
1084 |
particular, parameter names stemming from the left-over of automated
|
|
1085 |
reasoning tools are usually quite unpredictable.
|
|
1086 |
|
|
1087 |
Under normal circumstances, the text of cases emerge from standard
|
|
1088 |
elimination or induction rules, which in turn are derived from
|
|
1089 |
previous theory specifications in a canonical way (say from
|
|
1090 |
@{command "inductive"} definitions).
|
|
1091 |
|
|
1092 |
\medskip Proper cases are only available if both the proof method
|
|
1093 |
and the rules involved support this. By using appropriate
|
|
1094 |
attributes, case names, conclusions, and parameters may be also
|
|
1095 |
declared by hand. Thus variant versions of rules that have been
|
|
1096 |
derived manually become ready to use in advanced case analysis
|
|
1097 |
later.
|
|
1098 |
|
|
1099 |
\begin{rail}
|
|
1100 |
'case' (caseref | '(' caseref ((name | underscore) +) ')')
|
|
1101 |
;
|
|
1102 |
caseref: nameref attributes?
|
|
1103 |
;
|
|
1104 |
|
|
1105 |
'case\_names' (name +)
|
|
1106 |
;
|
|
1107 |
'case\_conclusion' name (name *)
|
|
1108 |
;
|
|
1109 |
'params' ((name *) + 'and')
|
|
1110 |
;
|
|
1111 |
'consumes' nat?
|
|
1112 |
;
|
|
1113 |
\end{rail}
|
|
1114 |
|
|
1115 |
\begin{descr}
|
|
1116 |
|
|
1117 |
\item [@{command "case"}~@{text "(c x\<^sub>1 \<dots> x\<^sub>m)"}]
|
|
1118 |
invokes a named local context @{text "c: x\<^sub>1, \<dots>, x\<^sub>m,
|
|
1119 |
\<phi>\<^sub>1, \<dots>, \<phi>\<^sub>m"}, as provided by an appropriate
|
|
1120 |
proof method (such as @{method_ref cases} and @{method_ref induct}).
|
|
1121 |
The command ``@{command "case"}~@{text "(c x\<^sub>1 \<dots>
|
|
1122 |
x\<^sub>m)"}'' abbreviates ``@{command "fix"}~@{text "x\<^sub>1 \<dots>
|
|
1123 |
x\<^sub>m"}~@{command "assume"}~@{text "c: \<phi>\<^sub>1 \<dots>
|
|
1124 |
\<phi>\<^sub>n"}''.
|
|
1125 |
|
|
1126 |
\item [@{command "print_cases"}] prints all local contexts of the
|
|
1127 |
current state, using Isar proof language notation.
|
|
1128 |
|
|
1129 |
\item [@{attribute case_names}~@{text "c\<^sub>1 \<dots> c\<^sub>k"}]
|
|
1130 |
declares names for the local contexts of premises of a theorem;
|
|
1131 |
@{text "c\<^sub>1, \<dots>, c\<^sub>k"} refers to the \emph{suffix} of the
|
|
1132 |
list of premises.
|
|
1133 |
|
|
1134 |
\item [@{attribute case_conclusion}~@{text "c d\<^sub>1 \<dots>
|
|
1135 |
d\<^sub>k"}] declares names for the conclusions of a named premise
|
|
1136 |
@{text c}; here @{text "d\<^sub>1, \<dots>, d\<^sub>k"} refers to the
|
|
1137 |
prefix of arguments of a logical formula built by nesting a binary
|
|
1138 |
connective (e.g.\ @{text "\<or>"}).
|
|
1139 |
|
|
1140 |
Note that proof methods such as @{method induct} and @{method
|
|
1141 |
coinduct} already provide a default name for the conclusion as a
|
|
1142 |
whole. The need to name subformulas only arises with cases that
|
|
1143 |
split into several sub-cases, as in common co-induction rules.
|
|
1144 |
|
|
1145 |
\item [@{attribute params}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>
|
|
1146 |
q\<^sub>1 \<dots> q\<^sub>n"}] renames the innermost parameters of
|
|
1147 |
premises @{text "1, \<dots>, n"} of some theorem. An empty list of names
|
|
1148 |
may be given to skip positions, leaving the present parameters
|
|
1149 |
unchanged.
|
|
1150 |
|
|
1151 |
Note that the default usage of case rules does \emph{not} directly
|
|
1152 |
expose parameters to the proof context.
|
|
1153 |
|
|
1154 |
\item [@{attribute consumes}~@{text n}] declares the number of
|
|
1155 |
``major premises'' of a rule, i.e.\ the number of facts to be
|
|
1156 |
consumed when it is applied by an appropriate proof method. The
|
|
1157 |
default value of @{attribute consumes} is @{text "n = 1"}, which is
|
|
1158 |
appropriate for the usual kind of cases and induction rules for
|
|
1159 |
inductive sets (cf.\ \secref{sec:hol-inductive}). Rules without any
|
|
1160 |
@{attribute consumes} declaration given are treated as if
|
|
1161 |
@{attribute consumes}~@{text 0} had been specified.
|
|
1162 |
|
|
1163 |
Note that explicit @{attribute consumes} declarations are only
|
|
1164 |
rarely needed; this is already taken care of automatically by the
|
|
1165 |
higher-level @{attribute cases}, @{attribute induct}, and
|
|
1166 |
@{attribute coinduct} declarations.
|
|
1167 |
|
|
1168 |
\end{descr}
|
|
1169 |
*}
|
|
1170 |
|
|
1171 |
|
|
1172 |
subsection {* Proof methods *}
|
|
1173 |
|
|
1174 |
text {*
|
|
1175 |
\begin{matharray}{rcl}
|
|
1176 |
@{method_def cases} & : & \isarmeth \\
|
|
1177 |
@{method_def induct} & : & \isarmeth \\
|
|
1178 |
@{method_def coinduct} & : & \isarmeth \\
|
|
1179 |
\end{matharray}
|
|
1180 |
|
|
1181 |
The @{method cases}, @{method induct}, and @{method coinduct}
|
|
1182 |
methods provide a uniform interface to common proof techniques over
|
|
1183 |
datatypes, inductive predicates (or sets), recursive functions etc.
|
|
1184 |
The corresponding rules may be specified and instantiated in a
|
|
1185 |
casual manner. Furthermore, these methods provide named local
|
|
1186 |
contexts that may be invoked via the @{command "case"} proof command
|
|
1187 |
within the subsequent proof text. This accommodates compact proof
|
|
1188 |
texts even when reasoning about large specifications.
|
|
1189 |
|
|
1190 |
The @{method induct} method also provides some additional
|
|
1191 |
infrastructure in order to be applicable to structure statements
|
|
1192 |
(either using explicit meta-level connectives, or including facts
|
|
1193 |
and parameters separately). This avoids cumbersome encoding of
|
|
1194 |
``strengthened'' inductive statements within the object-logic.
|
|
1195 |
|
|
1196 |
\begin{rail}
|
|
1197 |
'cases' (insts * 'and') rule?
|
|
1198 |
;
|
|
1199 |
'induct' (definsts * 'and') \\ arbitrary? taking? rule?
|
|
1200 |
;
|
|
1201 |
'coinduct' insts taking rule?
|
|
1202 |
;
|
|
1203 |
|
|
1204 |
rule: ('type' | 'pred' | 'set') ':' (nameref +) | 'rule' ':' (thmref +)
|
|
1205 |
;
|
|
1206 |
definst: name ('==' | equiv) term | inst
|
|
1207 |
;
|
|
1208 |
definsts: ( definst *)
|
|
1209 |
;
|
|
1210 |
arbitrary: 'arbitrary' ':' ((term *) 'and' +)
|
|
1211 |
;
|
|
1212 |
taking: 'taking' ':' insts
|
|
1213 |
;
|
|
1214 |
\end{rail}
|
|
1215 |
|
|
1216 |
\begin{descr}
|
|
1217 |
|
|
1218 |
\item [@{method cases}~@{text "insts R"}] applies method @{method
|
|
1219 |
rule} with an appropriate case distinction theorem, instantiated to
|
|
1220 |
the subjects @{text insts}. Symbolic case names are bound according
|
|
1221 |
to the rule's local contexts.
|
|
1222 |
|
|
1223 |
The rule is determined as follows, according to the facts and
|
|
1224 |
arguments passed to the @{method cases} method:
|
|
1225 |
|
|
1226 |
\medskip
|
|
1227 |
\begin{tabular}{llll}
|
|
1228 |
facts & & arguments & rule \\\hline
|
|
1229 |
& @{method cases} & & classical case split \\
|
|
1230 |
& @{method cases} & @{text t} & datatype exhaustion (type of @{text t}) \\
|
|
1231 |
@{text "\<turnstile> A t"} & @{method cases} & @{text "\<dots>"} & inductive predicate/set elimination (of @{text A}) \\
|
|
1232 |
@{text "\<dots>"} & @{method cases} & @{text "\<dots> rule: R"} & explicit rule @{text R} \\
|
|
1233 |
\end{tabular}
|
|
1234 |
\medskip
|
|
1235 |
|
|
1236 |
Several instantiations may be given, referring to the \emph{suffix}
|
|
1237 |
of premises of the case rule; within each premise, the \emph{prefix}
|
|
1238 |
of variables is instantiated. In most situations, only a single
|
|
1239 |
term needs to be specified; this refers to the first variable of the
|
|
1240 |
last premise (it is usually the same for all cases).
|
|
1241 |
|
|
1242 |
\item [@{method induct}~@{text "insts R"}] is analogous to the
|
|
1243 |
@{method cases} method, but refers to induction rules, which are
|
|
1244 |
determined as follows:
|
|
1245 |
|
|
1246 |
\medskip
|
|
1247 |
\begin{tabular}{llll}
|
|
1248 |
facts & & arguments & rule \\\hline
|
|
1249 |
& @{method induct} & @{text "P x"} & datatype induction (type of @{text x}) \\
|
|
1250 |
@{text "\<turnstile> A x"} & @{method induct} & @{text "\<dots>"} & predicate/set induction (of @{text A}) \\
|
|
1251 |
@{text "\<dots>"} & @{method induct} & @{text "\<dots> rule: R"} & explicit rule @{text R} \\
|
|
1252 |
\end{tabular}
|
|
1253 |
\medskip
|
|
1254 |
|
|
1255 |
Several instantiations may be given, each referring to some part of
|
|
1256 |
a mutual inductive definition or datatype --- only related partial
|
|
1257 |
induction rules may be used together, though. Any of the lists of
|
|
1258 |
terms @{text "P, x, \<dots>"} refers to the \emph{suffix} of variables
|
|
1259 |
present in the induction rule. This enables the writer to specify
|
|
1260 |
only induction variables, or both predicates and variables, for
|
|
1261 |
example.
|
|
1262 |
|
|
1263 |
Instantiations may be definitional: equations @{text "x \<equiv> t"}
|
|
1264 |
introduce local definitions, which are inserted into the claim and
|
|
1265 |
discharged after applying the induction rule. Equalities reappear
|
|
1266 |
in the inductive cases, but have been transformed according to the
|
|
1267 |
induction principle being involved here. In order to achieve
|
|
1268 |
practically useful induction hypotheses, some variables occurring in
|
|
1269 |
@{text t} need to be fixed (see below).
|
|
1270 |
|
|
1271 |
The optional ``@{text "arbitrary: x\<^sub>1 \<dots> x\<^sub>m"}''
|
|
1272 |
specification generalizes variables @{text "x\<^sub>1, \<dots>,
|
|
1273 |
x\<^sub>m"} of the original goal before applying induction. Thus
|
|
1274 |
induction hypotheses may become sufficiently general to get the
|
|
1275 |
proof through. Together with definitional instantiations, one may
|
|
1276 |
effectively perform induction over expressions of a certain
|
|
1277 |
structure.
|
|
1278 |
|
|
1279 |
The optional ``@{text "taking: t\<^sub>1 \<dots> t\<^sub>n"}''
|
|
1280 |
specification provides additional instantiations of a prefix of
|
|
1281 |
pending variables in the rule. Such schematic induction rules
|
|
1282 |
rarely occur in practice, though.
|
|
1283 |
|
|
1284 |
\item [@{method coinduct}~@{text "inst R"}] is analogous to the
|
|
1285 |
@{method induct} method, but refers to coinduction rules, which are
|
|
1286 |
determined as follows:
|
|
1287 |
|
|
1288 |
\medskip
|
|
1289 |
\begin{tabular}{llll}
|
|
1290 |
goal & & arguments & rule \\\hline
|
|
1291 |
& @{method coinduct} & @{text x} & type coinduction (type of @{text x}) \\
|
|
1292 |
@{text "A x"} & @{method coinduct} & @{text "\<dots>"} & predicate/set coinduction (of @{text A}) \\
|
|
1293 |
@{text "\<dots>"} & @{method coinduct} & @{text "\<dots> rule: R"} & explicit rule @{text R} \\
|
|
1294 |
\end{tabular}
|
|
1295 |
|
|
1296 |
Coinduction is the dual of induction. Induction essentially
|
|
1297 |
eliminates @{text "A x"} towards a generic result @{text "P x"},
|
|
1298 |
while coinduction introduces @{text "A x"} starting with @{text "B
|
|
1299 |
x"}, for a suitable ``bisimulation'' @{text B}. The cases of a
|
|
1300 |
coinduct rule are typically named after the predicates or sets being
|
|
1301 |
covered, while the conclusions consist of several alternatives being
|
|
1302 |
named after the individual destructor patterns.
|
|
1303 |
|
|
1304 |
The given instantiation refers to the \emph{suffix} of variables
|
|
1305 |
occurring in the rule's major premise, or conclusion if unavailable.
|
|
1306 |
An additional ``@{text "taking: t\<^sub>1 \<dots> t\<^sub>n"}''
|
|
1307 |
specification may be required in order to specify the bisimulation
|
|
1308 |
to be used in the coinduction step.
|
|
1309 |
|
|
1310 |
\end{descr}
|
|
1311 |
|
|
1312 |
Above methods produce named local contexts, as determined by the
|
|
1313 |
instantiated rule as given in the text. Beyond that, the @{method
|
|
1314 |
induct} and @{method coinduct} methods guess further instantiations
|
|
1315 |
from the goal specification itself. Any persisting unresolved
|
|
1316 |
schematic variables of the resulting rule will render the the
|
|
1317 |
corresponding case invalid. The term binding @{variable ?case} for
|
|
1318 |
the conclusion will be provided with each case, provided that term
|
|
1319 |
is fully specified.
|
|
1320 |
|
|
1321 |
The @{command "print_cases"} command prints all named cases present
|
|
1322 |
in the current proof state.
|
|
1323 |
|
|
1324 |
\medskip Despite the additional infrastructure, both @{method cases}
|
|
1325 |
and @{method coinduct} merely apply a certain rule, after
|
|
1326 |
instantiation, while conforming due to the usual way of monotonic
|
|
1327 |
natural deduction: the context of a structured statement @{text
|
|
1328 |
"\<And>x\<^sub>1 \<dots> x\<^sub>m. \<phi>\<^sub>1 \<Longrightarrow> \<dots> \<phi>\<^sub>n \<Longrightarrow> \<dots>"}
|
|
1329 |
reappears unchanged after the case split.
|
|
1330 |
|
|
1331 |
The @{method induct} method is fundamentally different in this
|
|
1332 |
respect: the meta-level structure is passed through the
|
|
1333 |
``recursive'' course involved in the induction. Thus the original
|
|
1334 |
statement is basically replaced by separate copies, corresponding to
|
|
1335 |
the induction hypotheses and conclusion; the original goal context
|
|
1336 |
is no longer available. Thus local assumptions, fixed parameters
|
|
1337 |
and definitions effectively participate in the inductive rephrasing
|
|
1338 |
of the original statement.
|
|
1339 |
|
|
1340 |
In induction proofs, local assumptions introduced by cases are split
|
|
1341 |
into two different kinds: @{text hyps} stemming from the rule and
|
|
1342 |
@{text prems} from the goal statement. This is reflected in the
|
|
1343 |
extracted cases accordingly, so invoking ``@{command "case"}~@{text
|
|
1344 |
c}'' will provide separate facts @{text c.hyps} and @{text c.prems},
|
|
1345 |
as well as fact @{text c} to hold the all-inclusive list.
|
|
1346 |
|
|
1347 |
\medskip Facts presented to either method are consumed according to
|
|
1348 |
the number of ``major premises'' of the rule involved, which is
|
|
1349 |
usually 0 for plain cases and induction rules of datatypes etc.\ and
|
|
1350 |
1 for rules of inductive predicates or sets and the like. The
|
|
1351 |
remaining facts are inserted into the goal verbatim before the
|
|
1352 |
actual @{text cases}, @{text induct}, or @{text coinduct} rule is
|
|
1353 |
applied.
|
|
1354 |
*}
|
|
1355 |
|
|
1356 |
|
|
1357 |
subsection {* Declaring rules *}
|
|
1358 |
|
|
1359 |
text {*
|
|
1360 |
\begin{matharray}{rcl}
|
|
1361 |
@{command_def "print_induct_rules"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
|
|
1362 |
@{attribute_def cases} & : & \isaratt \\
|
|
1363 |
@{attribute_def induct} & : & \isaratt \\
|
|
1364 |
@{attribute_def coinduct} & : & \isaratt \\
|
|
1365 |
\end{matharray}
|
|
1366 |
|
|
1367 |
\begin{rail}
|
|
1368 |
'cases' spec
|
|
1369 |
;
|
|
1370 |
'induct' spec
|
|
1371 |
;
|
|
1372 |
'coinduct' spec
|
|
1373 |
;
|
|
1374 |
|
|
1375 |
spec: ('type' | 'pred' | 'set') ':' nameref
|
|
1376 |
;
|
|
1377 |
\end{rail}
|
|
1378 |
|
|
1379 |
\begin{descr}
|
|
1380 |
|
|
1381 |
\item [@{command "print_induct_rules"}] prints cases and induct
|
|
1382 |
rules for predicates (or sets) and types of the current context.
|
|
1383 |
|
|
1384 |
\item [@{attribute cases}, @{attribute induct}, and @{attribute
|
|
1385 |
coinduct}] (as attributes) augment the corresponding context of
|
|
1386 |
rules for reasoning about (co)inductive predicates (or sets) and
|
|
1387 |
types, using the corresponding methods of the same name. Certain
|
|
1388 |
definitional packages of object-logics usually declare emerging
|
|
1389 |
cases and induction rules as expected, so users rarely need to
|
|
1390 |
intervene.
|
|
1391 |
|
|
1392 |
Manual rule declarations usually refer to the @{attribute
|
|
1393 |
case_names} and @{attribute params} attributes to adjust names of
|
|
1394 |
cases and parameters of a rule; the @{attribute consumes}
|
|
1395 |
declaration is taken care of automatically: @{attribute
|
|
1396 |
consumes}~@{text 0} is specified for ``type'' rules and @{attribute
|
|
1397 |
consumes}~@{text 1} for ``predicate'' / ``set'' rules.
|
|
1398 |
|
|
1399 |
\end{descr}
|
|
1400 |
*}
|
|
1401 |
|
26869
|
1402 |
end
|