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\begin{isabellebody}%
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\def\isabellecontext{Proof}%
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%
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\isadelimtheory
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\isanewline
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\isanewline
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%
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\endisadelimtheory
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%
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\isatagtheory
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\isacommand{theory}\isamarkupfalse%
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\ Proof\isanewline
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\isakeyword{imports}\ Main\isanewline
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\isakeyword{begin}%
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isamarkupchapter{Proofs%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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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{description}
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\item \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} 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 \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} 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 \isa{{\isachardoublequote}proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} is intermediate between \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} and \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}}: existing facts (i.e.\
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the contents of the special ``\indexref{}{fact}{this}\hyperlink{fact.this}{\mbox{\isa{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{description}
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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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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Proof structure%
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}
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\isamarkuptrue%
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%
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\isamarkupsubsection{Blocks%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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\indexdef{}{command}{next}\hypertarget{command.next}{\hyperlink{command.next}{\mbox{\isa{\isacommand{next}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
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\indexdef{}{command}{\{}\hypertarget{command.braceleft}{\hyperlink{command.braceleft}{\mbox{\isa{\isacommand{{\isacharbraceleft}}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
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\indexdef{}{command}{\}}\hypertarget{command.braceright}{\hyperlink{command.braceright}{\mbox{\isa{\isacommand{{\isacharbraceright}}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
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\end{matharray}
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While Isar is inherently block-structured, opening and closing
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blocks is mostly handled rather casually, with little explicit
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user-intervention. Any local goal statement automatically opens
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\emph{two} internal blocks, which are closed again when concluding
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the sub-proof (by \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} etc.). Sections of different
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context within a sub-proof may be switched via \hyperlink{command.next}{\mbox{\isa{\isacommand{next}}}},
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which is just a single block-close followed by block-open again.
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The effect of \hyperlink{command.next}{\mbox{\isa{\isacommand{next}}}} is to reset the local proof context;
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there is no goal focus involved here!
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For slightly more advanced applications, there are explicit block
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parentheses as well. These typically achieve a stronger forward
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style of reasoning.
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\begin{description}
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\item \hyperlink{command.next}{\mbox{\isa{\isacommand{next}}}} switches to a fresh block within a
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sub-proof, resetting the local context to the initial one.
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\item \hyperlink{command.braceleft}{\mbox{\isa{\isacommand{{\isacharbraceleft}}}}} and \hyperlink{command.braceright}{\mbox{\isa{\isacommand{{\isacharbraceright}}}}} explicitly open and close
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blocks. Any current facts pass through ``\hyperlink{command.braceleft}{\mbox{\isa{\isacommand{{\isacharbraceleft}}}}}''
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unchanged, while ``\hyperlink{command.braceright}{\mbox{\isa{\isacommand{{\isacharbraceright}}}}}'' causes any result to be
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\emph{exported} into the enclosing context. Thus fixed variables
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are generalized, assumptions discharged, and local definitions
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unfolded (cf.\ \secref{sec:proof-context}). There is no difference
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of \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} and \hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}} in this mode of
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forward reasoning --- in contrast to plain backward reasoning with
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the result exported at \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} time.
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\end{description}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Omitting proofs%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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\indexdef{}{command}{oops}\hypertarget{command.oops}{\hyperlink{command.oops}{\mbox{\isa{\isacommand{oops}}}}} & : & \isa{{\isachardoublequote}proof\ {\isasymrightarrow}\ local{\isacharunderscore}theory\ {\isacharbar}\ theory{\isachardoublequote}} \\
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\end{matharray}
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The \hyperlink{command.oops}{\mbox{\isa{\isacommand{oops}}}} command discontinues the current proof
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attempt, while considering the partial proof text as properly
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processed. This is conceptually quite different from ``faking''
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actual proofs via \indexref{}{command}{sorry}\hyperlink{command.sorry}{\mbox{\isa{\isacommand{sorry}}}} (see
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\secref{sec:proof-steps}): \hyperlink{command.oops}{\mbox{\isa{\isacommand{oops}}}} does not observe the
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proof structure at all, but goes back right to the theory level.
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Furthermore, \hyperlink{command.oops}{\mbox{\isa{\isacommand{oops}}}} does not produce any result theorem
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--- there is no intended claim to be able to complete the proof
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anyhow.
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A typical application of \hyperlink{command.oops}{\mbox{\isa{\isacommand{oops}}}} is to explain Isar proofs
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\emph{within} the system itself, in conjunction with the document
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preparation tools of Isabelle described in \cite{isabelle-sys}.
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Thus partial or even wrong proof attempts can be discussed in a
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logically sound manner. Note that the Isabelle {\LaTeX} macros can
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be easily adapted to print something like ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' instead of
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the keyword ``\hyperlink{command.oops}{\mbox{\isa{\isacommand{oops}}}}''.
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\medskip The \hyperlink{command.oops}{\mbox{\isa{\isacommand{oops}}}} command is undo-able, unlike
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\indexref{}{command}{kill}\hyperlink{command.kill}{\mbox{\isa{\isacommand{kill}}}} (see \secref{sec:history}). The effect is to
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get back to the theory just before the opening of the proof.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Statements%
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}
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\isamarkuptrue%
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%
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\isamarkupsubsection{Context elements \label{sec:proof-context}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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\indexdef{}{command}{fix}\hypertarget{command.fix}{\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
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\indexdef{}{command}{assume}\hypertarget{command.assume}{\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
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\indexdef{}{command}{presume}\hypertarget{command.presume}{\hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
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\indexdef{}{command}{def}\hypertarget{command.def}{\hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
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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 ``\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{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 \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymphi}{\isacharbrackleft}x{\isacharbrackright}{\isachardoublequote}} exported from
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the context will be universally closed wrt.\ \isa{x} at the
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outermost level: \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymAnd}x{\isachardot}\ {\isasymphi}{\isacharbrackleft}x{\isacharbrackright}{\isachardoublequote}} (this is expressed in normal
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form using Isabelle's meta-variables).
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Similarly, introducing some assumption \isa{{\isasymchi}} 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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\isa{{\isachardoublequote}{\isasymchi}\ {\isasymturnstile}\ {\isasymphi}{\isachardoublequote}} exported from the context becomes conditional wrt.\
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the assumption: \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymchi}\ {\isasymLongrightarrow}\ {\isasymphi}{\isachardoublequote}}. 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 \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}
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insists on solving the subgoal by unification with some premise of
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the goal, \hyperlink{command.presume}{\mbox{\isa{\isacommand{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 ``\hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}}~\isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}}'', are achieved by combining ``\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{x}'' with
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another version of assumption that causes any hypothetical equation
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\isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}} to be eliminated by the reflexivity rule. Thus,
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exporting some result \isa{{\isachardoublequote}x\ {\isasymequiv}\ t\ {\isasymturnstile}\ {\isasymphi}{\isacharbrackleft}x{\isacharbrackright}{\isachardoublequote}} yields \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymphi}{\isacharbrackleft}t{\isacharbrackright}{\isachardoublequote}}.
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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{description}
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\item \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{x} introduces a local variable \isa{x} that is \emph{arbitrary, but fixed.}
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\item \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} and \hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} introduce a local fact \isa{{\isachardoublequote}{\isasymphi}\ {\isasymturnstile}\ {\isasymphi}{\isachardoublequote}} by
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assumption. Subsequent results applied to an enclosing goal (e.g.\
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by \indexref{}{command}{show}\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}) are handled as follows: \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} expects to be able to unify with existing premises in the
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goal, while \hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}} leaves \isa{{\isasymphi}} as new subgoals.
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Several lists of assumptions may be given (separated by
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\indexref{}{keyword}{and}\hyperlink{keyword.and}{\mbox{\isa{\isakeyword{and}}}}; the resulting list of current facts consists
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of all of these concatenated.
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\item \hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}}~\isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}} introduces a local
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(non-polymorphic) definition. In results exported from the context,
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\isa{x} is replaced by \isa{t}. Basically, ``\hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}}~\isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}}'' abbreviates ``\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{x}~\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}}'', with the resulting
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hypothetical equation solved by reflexivity.
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The default name for the definitional equation is \isa{x{\isacharunderscore}def}.
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Several simultaneous definitions may be given at the same time.
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\end{description}
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The special name \indexref{}{fact}{prems}\hyperlink{fact.prems}{\mbox{\isa{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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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Term abbreviations \label{sec:term-abbrev}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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\indexdef{}{command}{let}\hypertarget{command.let}{\hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
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\indexdef{}{keyword}{is}\hypertarget{keyword.is}{\hyperlink{keyword.is}{\mbox{\isa{\isakeyword{is}}}}} & : & syntax \\
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\end{matharray}
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Abbreviations may be either bound by explicit \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}~\isa{{\isachardoublequote}p\ {\isasymequiv}\ t{\isachardoublequote}} statements, or by annotating assumptions or
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goal statements with a list of patterns ``\isa{{\isachardoublequote}{\isacharparenleft}{\isasymIS}\ p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub n{\isacharparenright}{\isachardoublequote}}''. In both cases, higher-order matching is invoked to
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bind extra-logical term variables, which may be either named
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schematic variables of the form \isa{{\isacharquery}x}, or nameless dummies
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``\hyperlink{variable.underscore}{\mbox{\isa{{\isacharunderscore}}}}'' (underscore). Note that in the \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}
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form the patterns occur on the left-hand side, while the \hyperlink{keyword.is}{\mbox{\isa{\isakeyword{is}}}} patterns are in postfix position.
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Polymorphism of term bindings is handled in Hindley-Milner style,
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similar to ML. Type variables referring to local assumptions or
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open goal statements are \emph{fixed}, while those of finished
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results or bound by \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}} may occur in \emph{arbitrary}
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instances later. Even though actual polymorphism should be rarely
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used in practice, this mechanism is essential to achieve proper
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incremental type-inference, as the user proceeds to build up the
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Isar proof text from left to right.
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\medskip Term abbreviations are quite different from local
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definitions as introduced via \hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}} (see
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\secref{sec:proof-context}). The latter are visible within the
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logic as actual equations, while abbreviations disappear during the
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input process just after type checking. Also note that \hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}} does not support polymorphism.
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\begin{rail}
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'let' ((term + 'and') '=' term + 'and')
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;
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\end{rail}
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The syntax of \hyperlink{keyword.is}{\mbox{\isa{\isakeyword{is}}}} patterns follows \railnonterm{termpat}
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or \railnonterm{proppat} (see \secref{sec:term-decls}).
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\begin{description}
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\item \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}~\isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}\ {\isacharequal}\ t\isactrlsub {\isadigit{1}}\ {\isasymAND}\ {\isasymdots}\ p\isactrlsub n\ {\isacharequal}\ t\isactrlsub n{\isachardoublequote}} binds any
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text variables in patterns \isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ p\isactrlsub n{\isachardoublequote}} by simultaneous
|
|
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higher-order matching against terms \isa{{\isachardoublequote}t\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ t\isactrlsub n{\isachardoublequote}}.
|
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|
|
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\item \isa{{\isachardoublequote}{\isacharparenleft}{\isasymIS}\ p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub n{\isacharparenright}{\isachardoublequote}} resembles \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}, but
|
|
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matches \isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ p\isactrlsub n{\isachardoublequote}} against the preceding statement. Also
|
|
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note that \hyperlink{keyword.is}{\mbox{\isa{\isakeyword{is}}}} is not a separate command, but part of
|
|
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others (such as \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}, \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}} etc.).
|
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|
|
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\end{description}
|
|
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|
|
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Some \emph{implicit} term abbreviations\index{term abbreviations}
|
|
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for goals and facts are available as well. For any open goal,
|
|
282 |
\indexref{}{variable}{thesis}\hyperlink{variable.thesis}{\mbox{\isa{thesis}}} refers to its object-level statement,
|
|
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abstracted over any meta-level parameters (if present). Likewise,
|
|
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\indexref{}{variable}{this}\hyperlink{variable.this}{\mbox{\isa{this}}} is bound for fact statements resulting from
|
|
285 |
assumptions or finished goals. In case \hyperlink{variable.this}{\mbox{\isa{this}}} refers to
|
|
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an object-logic statement that is an application \isa{{\isachardoublequote}f\ t{\isachardoublequote}}, then
|
|
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\isa{t} is bound to the special text variable ``\hyperlink{variable.dots}{\mbox{\isa{{\isasymdots}}}}''
|
|
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(three dots). The canonical application of this convenience are
|
|
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calculational proofs (see \secref{sec:calculation}).%
|
|
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\end{isamarkuptext}%
|
|
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\isamarkuptrue%
|
|
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%
|
|
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\isamarkupsubsection{Facts and forward chaining%
|
|
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}
|
|
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\isamarkuptrue%
|
|
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%
|
|
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\begin{isamarkuptext}%
|
|
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\begin{matharray}{rcl}
|
|
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\indexdef{}{command}{note}\hypertarget{command.note}{\hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
|
|
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\indexdef{}{command}{then}\hypertarget{command.then}{\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} \\
|
|
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\indexdef{}{command}{from}\hypertarget{command.from}{\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} \\
|
|
302 |
\indexdef{}{command}{with}\hypertarget{command.with}{\hyperlink{command.with}{\mbox{\isa{\isacommand{with}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} \\
|
|
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\indexdef{}{command}{using}\hypertarget{command.using}{\hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
|
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\indexdef{}{command}{unfolding}\hypertarget{command.unfolding}{\hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
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|
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\end{matharray}
|
|
306 |
|
|
307 |
New facts are established either by assumption or proof of local
|
|
308 |
statements. Any fact will usually be involved in further proofs,
|
|
309 |
either as explicit arguments of proof methods, or when forward
|
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|
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chaining towards the next goal via \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} (and variants);
|
|
311 |
\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}} and \hyperlink{command.with}{\mbox{\isa{\isacommand{with}}}} are composite forms
|
|
312 |
involving \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}. The \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}} elements
|
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|
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augments the collection of used facts \emph{after} a goal has been
|
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|
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stated. Note that the special theorem name \indexref{}{fact}{this}\hyperlink{fact.this}{\mbox{\isa{this}}} refers
|
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|
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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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|
|
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\begin{rail}
|
|
319 |
'note' (thmdef? thmrefs + 'and')
|
|
320 |
;
|
|
321 |
('from' | 'with' | 'using' | 'unfolding') (thmrefs + 'and')
|
|
322 |
;
|
|
323 |
\end{rail}
|
|
324 |
|
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|
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\begin{description}
|
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|
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|
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|
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\item \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isachardoublequote}a\ {\isacharequal}\ b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}} recalls existing facts
|
|
328 |
\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub n{\isachardoublequote}}, binding the result as \isa{a}. Note that
|
|
329 |
attributes may be involved as well, both on the left and right hand
|
|
330 |
sides.
|
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|
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|
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|
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\item \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} indicates forward chaining by the current
|
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|
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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
|
|
335 |
facts to do ``anything appropriate'' (see also
|
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|
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\secref{sec:proof-steps}). For example, method \indexref{}{method}{rule}\hyperlink{method.rule}{\mbox{\isa{rule}}}
|
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|
337 |
(see \secref{sec:pure-meth-att}) would typically do an elimination
|
|
338 |
rather than an introduction. Automatic methods usually insert the
|
|
339 |
facts into the goal state before operation. This provides a simple
|
|
340 |
scheme to control relevance of facts in automated proof search.
|
|
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|
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|
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\item \hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{b} abbreviates ``\hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{b}~\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}''; thus \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} is
|
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|
343 |
equivalent to ``\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{this}''.
|
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|
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|
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|
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\item \hyperlink{command.with}{\mbox{\isa{\isacommand{with}}}}~\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}} abbreviates ``\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n\ {\isasymAND}\ this{\isachardoublequote}}''; thus the forward chaining
|
|
346 |
is from earlier facts together with the current ones.
|
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|
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|
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|
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\item \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}} augments the facts being
|
|
349 |
currently indicated for use by a subsequent refinement step (such as
|
|
350 |
\indexref{}{command}{apply}\hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}} or \indexref{}{command}{proof}\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}).
|
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|
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|
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|
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\item \hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}}~\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}} is structurally
|
|
353 |
similar to \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}, but unfolds definitional equations
|
|
354 |
\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}\ b\isactrlsub n{\isachardoublequote}} throughout the goal state and facts.
|
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|
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|
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|
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\end{description}
|
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|
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|
|
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Forward chaining with an empty list of theorems is the same as not
|
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|
359 |
chaining at all. Thus ``\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{nothing}'' has no
|
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|
360 |
effect apart from entering \isa{{\isachardoublequote}prove{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} mode, since
|
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|
361 |
\indexref{}{fact}{nothing}\hyperlink{fact.nothing}{\mbox{\isa{nothing}}} is bound to the empty list of theorems.
|
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|
362 |
|
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|
363 |
Basic proof methods (such as \indexref{}{method}{rule}\hyperlink{method.rule}{\mbox{\isa{rule}}}) expect multiple
|
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|
364 |
facts to be given in their proper order, corresponding to a prefix
|
|
365 |
of the premises of the rule involved. Note that positions may be
|
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|
366 |
easily skipped using something like \hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{{\isachardoublequote}{\isacharunderscore}\ {\isasymAND}\ a\ {\isasymAND}\ b{\isachardoublequote}}, for example. This involves the trivial rule
|
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|
367 |
\isa{{\isachardoublequote}PROP\ {\isasympsi}\ {\isasymLongrightarrow}\ PROP\ {\isasympsi}{\isachardoublequote}}, which is bound in Isabelle/Pure as
|
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|
368 |
``\indexref{}{fact}{\_}\hyperlink{fact.underscore}{\mbox{\isa{{\isacharunderscore}}}}'' (underscore).
|
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|
369 |
|
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|
370 |
Automated methods (such as \hyperlink{method.simp}{\mbox{\isa{simp}}} or \hyperlink{method.auto}{\mbox{\isa{auto}}}) just
|
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|
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insert any given facts before their usual operation. Depending on
|
|
372 |
the kind of procedure involved, the order of facts is less
|
|
373 |
significant here.%
|
|
374 |
\end{isamarkuptext}%
|
|
375 |
\isamarkuptrue%
|
|
376 |
%
|
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|
377 |
\isamarkupsubsection{Goals \label{sec:goals}%
|
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|
378 |
}
|
|
379 |
\isamarkuptrue%
|
|
380 |
%
|
|
381 |
\begin{isamarkuptext}%
|
|
382 |
\begin{matharray}{rcl}
|
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|
383 |
\indexdef{}{command}{lemma}\hypertarget{command.lemma}{\hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
|
384 |
\indexdef{}{command}{theorem}\hypertarget{command.theorem}{\hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
|
385 |
\indexdef{}{command}{corollary}\hypertarget{command.corollary}{\hyperlink{command.corollary}{\mbox{\isa{\isacommand{corollary}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
|
386 |
\indexdef{}{command}{have}\hypertarget{command.have}{\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ proof{\isacharparenleft}chain{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
|
387 |
\indexdef{}{command}{show}\hypertarget{command.show}{\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ proof{\isacharparenleft}chain{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
|
388 |
\indexdef{}{command}{hence}\hypertarget{command.hence}{\hyperlink{command.hence}{\mbox{\isa{\isacommand{hence}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
|
389 |
\indexdef{}{command}{thus}\hypertarget{command.thus}{\hyperlink{command.thus}{\mbox{\isa{\isacommand{thus}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
|
390 |
\indexdef{}{command}{print\_statement}\hypertarget{command.print-statement}{\hyperlink{command.print-statement}{\mbox{\isa{\isacommand{print{\isacharunderscore}statement}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
|
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|
391 |
\end{matharray}
|
|
392 |
|
|
393 |
From a theory context, proof mode is entered by an initial goal
|
26902
|
394 |
command such as \hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}}, \hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}, or
|
|
395 |
\hyperlink{command.corollary}{\mbox{\isa{\isacommand{corollary}}}}. Within a proof, new claims may be
|
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|
396 |
introduced locally as well; four variants are available here to
|
|
397 |
indicate whether forward chaining of facts should be performed
|
26902
|
398 |
initially (via \indexref{}{command}{then}\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}), and whether the final result
|
26870
|
399 |
is meant to solve some pending goal.
|
|
400 |
|
|
401 |
Goals may consist of multiple statements, resulting in a list of
|
|
402 |
facts eventually. A pending multi-goal is internally represented as
|
|
403 |
a meta-level conjunction (printed as \isa{{\isachardoublequote}{\isacharampersand}{\isacharampersand}{\isachardoublequote}}), which is usually
|
|
404 |
split into the corresponding number of sub-goals prior to an initial
|
26902
|
405 |
method application, via \indexref{}{command}{proof}\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}
|
|
406 |
(\secref{sec:proof-steps}) or \indexref{}{command}{apply}\hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}
|
|
407 |
(\secref{sec:tactic-commands}). The \indexref{}{method}{induct}\hyperlink{method.induct}{\mbox{\isa{induct}}} method
|
26870
|
408 |
covered in \secref{sec:cases-induct} acts on multiple claims
|
|
409 |
simultaneously.
|
|
410 |
|
|
411 |
Claims at the theory level may be either in short or long form. A
|
|
412 |
short goal merely consists of several simultaneous propositions
|
|
413 |
(often just one). A long goal includes an explicit context
|
|
414 |
specification for the subsequent conclusion, involving local
|
|
415 |
parameters and assumptions. Here the role of each part of the
|
|
416 |
statement is explicitly marked by separate keywords (see also
|
|
417 |
\secref{sec:locale}); the local assumptions being introduced here
|
26902
|
418 |
are available as \indexref{}{fact}{assms}\hyperlink{fact.assms}{\mbox{\isa{assms}}} in the proof. Moreover, there
|
|
419 |
are two kinds of conclusions: \indexdef{}{element}{shows}\hypertarget{element.shows}{\hyperlink{element.shows}{\mbox{\isa{\isakeyword{shows}}}}} states several
|
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|
420 |
simultaneous propositions (essentially a big conjunction), while
|
26902
|
421 |
\indexdef{}{element}{obtains}\hypertarget{element.obtains}{\hyperlink{element.obtains}{\mbox{\isa{\isakeyword{obtains}}}}} claims several simultaneous simultaneous
|
26870
|
422 |
contexts of (essentially a big disjunction of eliminated parameters
|
|
423 |
and assumptions, cf.\ \secref{sec:obtain}).
|
|
424 |
|
|
425 |
\begin{rail}
|
|
426 |
('lemma' | 'theorem' | 'corollary') target? (goal | longgoal)
|
|
427 |
;
|
|
428 |
('have' | 'show' | 'hence' | 'thus') goal
|
|
429 |
;
|
|
430 |
'print\_statement' modes? thmrefs
|
|
431 |
;
|
|
432 |
|
|
433 |
goal: (props + 'and')
|
|
434 |
;
|
|
435 |
longgoal: thmdecl? (contextelem *) conclusion
|
|
436 |
;
|
|
437 |
conclusion: 'shows' goal | 'obtains' (parname? case + '|')
|
|
438 |
;
|
|
439 |
case: (vars + 'and') 'where' (props + 'and')
|
|
440 |
;
|
|
441 |
\end{rail}
|
|
442 |
|
28788
|
443 |
\begin{description}
|
26870
|
444 |
|
28788
|
445 |
\item \hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} enters proof mode with
|
26870
|
446 |
\isa{{\isasymphi}} as main goal, eventually resulting in some fact \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymphi}{\isachardoublequote}} to be put back into the target context. An additional
|
|
447 |
\railnonterm{context} specification may build up an initial proof
|
|
448 |
context for the subsequent claim; this includes local definitions
|
26902
|
449 |
and syntax as well, see the definition of \hyperlink{syntax.contextelem}{\mbox{\isa{contextelem}}} in
|
26870
|
450 |
\secref{sec:locale}.
|
|
451 |
|
28788
|
452 |
\item \hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} and \hyperlink{command.corollary}{\mbox{\isa{\isacommand{corollary}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} are essentially the same as \hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}}, but the facts are internally marked as
|
26870
|
453 |
being of a different kind. This discrimination acts like a formal
|
|
454 |
comment.
|
|
455 |
|
28788
|
456 |
\item \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} claims a local goal,
|
26870
|
457 |
eventually resulting in a fact within the current logical context.
|
|
458 |
This operation is completely independent of any pending sub-goals of
|
26902
|
459 |
an enclosing goal statements, so \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}} may be freely
|
26870
|
460 |
used for experimental exploration of potential results within a
|
|
461 |
proof body.
|
|
462 |
|
28788
|
463 |
\item \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} is like \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isachardoublequote}a{\isacharcolon}\ {\isasymphi}{\isachardoublequote}} plus a second stage to refine some pending
|
26870
|
464 |
sub-goal for each one of the finished result, after having been
|
|
465 |
exported into the corresponding context (at the head of the
|
26902
|
466 |
sub-proof of this \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} command).
|
26870
|
467 |
|
|
468 |
To accommodate interactive debugging, resulting rules are printed
|
|
469 |
before being applied internally. Even more, interactive execution
|
26902
|
470 |
of \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} predicts potential failure and displays the
|
26870
|
471 |
resulting error as a warning beforehand. Watch out for the
|
|
472 |
following message:
|
|
473 |
|
|
474 |
%FIXME proper antiquitation
|
|
475 |
\begin{ttbox}
|
|
476 |
Problem! Local statement will fail to solve any pending goal
|
|
477 |
\end{ttbox}
|
|
478 |
|
28788
|
479 |
\item \hyperlink{command.hence}{\mbox{\isa{\isacommand{hence}}}} abbreviates ``\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}~\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}'', i.e.\ claims a local goal to be proven by forward
|
26902
|
480 |
chaining the current facts. Note that \hyperlink{command.hence}{\mbox{\isa{\isacommand{hence}}}} is also
|
|
481 |
equivalent to ``\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{this}~\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}''.
|
26870
|
482 |
|
28788
|
483 |
\item \hyperlink{command.thus}{\mbox{\isa{\isacommand{thus}}}} abbreviates ``\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}~\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}''. Note that \hyperlink{command.thus}{\mbox{\isa{\isacommand{thus}}}} is also equivalent to
|
26902
|
484 |
``\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{this}~\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}''.
|
26870
|
485 |
|
28788
|
486 |
\item \hyperlink{command.print-statement}{\mbox{\isa{\isacommand{print{\isacharunderscore}statement}}}}~\isa{a} prints facts from the
|
26870
|
487 |
current theory or proof context in long statement form, according to
|
26902
|
488 |
the syntax for \hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}} given above.
|
26870
|
489 |
|
28788
|
490 |
\end{description}
|
26870
|
491 |
|
|
492 |
Any goal statement causes some term abbreviations (such as
|
26902
|
493 |
\indexref{}{variable}{?thesis}\hyperlink{variable.?thesis}{\mbox{\isa{{\isacharquery}thesis}}}) to be bound automatically, see also
|
26961
|
494 |
\secref{sec:term-abbrev}.
|
26870
|
495 |
|
26902
|
496 |
The optional case names of \indexref{}{element}{obtains}\hyperlink{element.obtains}{\mbox{\isa{\isakeyword{obtains}}}} have a twofold
|
26870
|
497 |
meaning: (1) during the of this claim they refer to the the local
|
|
498 |
context introductions, (2) the resulting rule is annotated
|
|
499 |
accordingly to support symbolic case splits when used with the
|
27124
|
500 |
\indexref{}{method}{cases}\hyperlink{method.cases}{\mbox{\isa{cases}}} method (cf.\ \secref{sec:cases-induct}).
|
26870
|
501 |
|
|
502 |
\medskip
|
|
503 |
|
|
504 |
\begin{warn}
|
|
505 |
Isabelle/Isar suffers theory-level goal statements to contain
|
|
506 |
\emph{unbound schematic variables}, although this does not conform
|
|
507 |
to the aim of human-readable proof documents! The main problem
|
|
508 |
with schematic goals is that the actual outcome is usually hard to
|
|
509 |
predict, depending on the behavior of the proof methods applied
|
|
510 |
during the course of reasoning. Note that most semi-automated
|
|
511 |
methods heavily depend on several kinds of implicit rule
|
|
512 |
declarations within the current theory context. As this would
|
|
513 |
also result in non-compositional checking of sub-proofs,
|
|
514 |
\emph{local goals} are not allowed to be schematic at all.
|
|
515 |
Nevertheless, schematic goals do have their use in Prolog-style
|
|
516 |
interactive synthesis of proven results, usually by stepwise
|
|
517 |
refinement via emulation of traditional Isabelle tactic scripts
|
|
518 |
(see also \secref{sec:tactic-commands}). In any case, users
|
|
519 |
should know what they are doing.
|
|
520 |
\end{warn}%
|
|
521 |
\end{isamarkuptext}%
|
|
522 |
\isamarkuptrue%
|
|
523 |
%
|
28788
|
524 |
\isamarkupsection{Refinement steps%
|
|
525 |
}
|
|
526 |
\isamarkuptrue%
|
|
527 |
%
|
|
528 |
\isamarkupsubsection{Proof method expressions \label{sec:proof-meth}%
|
|
529 |
}
|
|
530 |
\isamarkuptrue%
|
|
531 |
%
|
|
532 |
\begin{isamarkuptext}%
|
|
533 |
Proof methods are either basic ones, or expressions composed of
|
|
534 |
methods via ``\verb|,|'' (sequential composition),
|
|
535 |
``\verb||\verb,|,\verb||'' (alternative choices), ``\verb|?|''
|
|
536 |
(try), ``\verb|+|'' (repeat at least once), ``\verb|[|\isa{n}\verb|]|'' (restriction to first \isa{n}
|
|
537 |
sub-goals, with default \isa{{\isachardoublequote}n\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequote}}). In practice, proof
|
|
538 |
methods are usually just a comma separated list of
|
|
539 |
\railqtok{nameref}~\railnonterm{args} specifications. Note that
|
|
540 |
parentheses may be dropped for single method specifications (with no
|
|
541 |
arguments).
|
|
542 |
|
|
543 |
\indexouternonterm{method}
|
|
544 |
\begin{rail}
|
|
545 |
method: (nameref | '(' methods ')') (() | '?' | '+' | '[' nat? ']')
|
|
546 |
;
|
|
547 |
methods: (nameref args | method) + (',' | '|')
|
|
548 |
;
|
|
549 |
\end{rail}
|
|
550 |
|
|
551 |
Proper Isar proof methods do \emph{not} admit arbitrary goal
|
|
552 |
addressing, but refer either to the first sub-goal or all sub-goals
|
|
553 |
uniformly. The goal restriction operator ``\isa{{\isachardoublequote}{\isacharbrackleft}n{\isacharbrackright}{\isachardoublequote}}''
|
|
554 |
evaluates a method expression within a sandbox consisting of the
|
|
555 |
first \isa{n} sub-goals (which need to exist). For example, the
|
|
556 |
method ``\isa{{\isachardoublequote}simp{\isacharunderscore}all{\isacharbrackleft}{\isadigit{3}}{\isacharbrackright}{\isachardoublequote}}'' simplifies the first three
|
|
557 |
sub-goals, while ``\isa{{\isachardoublequote}{\isacharparenleft}rule\ foo{\isacharcomma}\ simp{\isacharunderscore}all{\isacharparenright}{\isacharbrackleft}{\isacharbrackright}{\isachardoublequote}}'' simplifies all
|
|
558 |
new goals that emerge from applying rule \isa{{\isachardoublequote}foo{\isachardoublequote}} to the
|
|
559 |
originally first one.
|
|
560 |
|
|
561 |
Improper methods, notably tactic emulations, offer a separate
|
|
562 |
low-level goal addressing scheme as explicit argument to the
|
|
563 |
individual tactic being involved. Here ``\isa{{\isachardoublequote}{\isacharbrackleft}{\isacharbang}{\isacharbrackright}{\isachardoublequote}}'' refers to
|
|
564 |
all goals, and ``\isa{{\isachardoublequote}{\isacharbrackleft}n{\isacharminus}{\isacharbrackright}{\isachardoublequote}}'' to all goals starting from \isa{{\isachardoublequote}n{\isachardoublequote}}.
|
|
565 |
|
|
566 |
\indexouternonterm{goalspec}
|
|
567 |
\begin{rail}
|
|
568 |
goalspec: '[' (nat '-' nat | nat '-' | nat | '!' ) ']'
|
|
569 |
;
|
|
570 |
\end{rail}%
|
|
571 |
\end{isamarkuptext}%
|
|
572 |
\isamarkuptrue%
|
|
573 |
%
|
|
574 |
\isamarkupsubsection{Initial and terminal proof steps \label{sec:proof-steps}%
|
26870
|
575 |
}
|
|
576 |
\isamarkuptrue%
|
|
577 |
%
|
|
578 |
\begin{isamarkuptext}%
|
|
579 |
\begin{matharray}{rcl}
|
28788
|
580 |
\indexdef{}{command}{proof}\hypertarget{command.proof}{\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
|
|
581 |
\indexdef{}{command}{qed}\hypertarget{command.qed}{\hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ local{\isacharunderscore}theory\ {\isacharbar}\ theory{\isachardoublequote}} \\
|
|
582 |
\indexdef{}{command}{by}\hypertarget{command.by}{\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ local{\isacharunderscore}theory\ {\isacharbar}\ theory{\isachardoublequote}} \\
|
|
583 |
\indexdef{}{command}{..}\hypertarget{command.ddot}{\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ local{\isacharunderscore}theory\ {\isacharbar}\ theory{\isachardoublequote}} \\
|
|
584 |
\indexdef{}{command}{.}\hypertarget{command.dot}{\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isachardot}}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ local{\isacharunderscore}theory\ {\isacharbar}\ theory{\isachardoublequote}} \\
|
|
585 |
\indexdef{}{command}{sorry}\hypertarget{command.sorry}{\hyperlink{command.sorry}{\mbox{\isa{\isacommand{sorry}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ local{\isacharunderscore}theory\ {\isacharbar}\ theory{\isachardoublequote}} \\
|
26870
|
586 |
\end{matharray}
|
|
587 |
|
|
588 |
Arbitrary goal refinement via tactics is considered harmful.
|
|
589 |
Structured proof composition in Isar admits proof methods to be
|
|
590 |
invoked in two places only.
|
|
591 |
|
|
592 |
\begin{enumerate}
|
|
593 |
|
26902
|
594 |
\item An \emph{initial} refinement step \indexref{}{command}{proof}\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}} reduces a newly stated goal to a number
|
26870
|
595 |
of sub-goals that are to be solved later. Facts are passed to
|
|
596 |
\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}} for forward chaining, if so indicated by \isa{{\isachardoublequote}proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} mode.
|
|
597 |
|
26902
|
598 |
\item A \emph{terminal} conclusion step \indexref{}{command}{qed}\hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}} is intended to solve remaining goals. No facts are
|
26870
|
599 |
passed to \isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}}.
|
|
600 |
|
|
601 |
\end{enumerate}
|
|
602 |
|
|
603 |
The only other (proper) way to affect pending goals in a proof body
|
26902
|
604 |
is by \indexref{}{command}{show}\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}, which involves an explicit statement of
|
26870
|
605 |
what is to be solved eventually. Thus we avoid the fundamental
|
|
606 |
problem of unstructured tactic scripts that consist of numerous
|
|
607 |
consecutive goal transformations, with invisible effects.
|
|
608 |
|
|
609 |
\medskip As a general rule of thumb for good proof style, initial
|
|
610 |
proof methods should either solve the goal completely, or constitute
|
|
611 |
some well-understood reduction to new sub-goals. Arbitrary
|
|
612 |
automatic proof tools that are prone leave a large number of badly
|
|
613 |
structured sub-goals are no help in continuing the proof document in
|
|
614 |
an intelligible manner.
|
|
615 |
|
|
616 |
Unless given explicitly by the user, the default initial method is
|
26902
|
617 |
``\indexref{}{method}{rule}\hyperlink{method.rule}{\mbox{\isa{rule}}}'', which applies a single standard elimination
|
26870
|
618 |
or introduction rule according to the topmost symbol involved.
|
|
619 |
There is no separate default terminal method. Any remaining goals
|
|
620 |
are always solved by assumption in the very last step.
|
|
621 |
|
|
622 |
\begin{rail}
|
|
623 |
'proof' method?
|
|
624 |
;
|
|
625 |
'qed' method?
|
|
626 |
;
|
|
627 |
'by' method method?
|
|
628 |
;
|
|
629 |
('.' | '..' | 'sorry')
|
|
630 |
;
|
|
631 |
\end{rail}
|
|
632 |
|
28788
|
633 |
\begin{description}
|
26870
|
634 |
|
28788
|
635 |
\item \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}} refines the goal by proof
|
|
636 |
method \isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}}; facts for forward chaining are passed if so
|
|
637 |
indicated by \isa{{\isachardoublequote}proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} mode.
|
26870
|
638 |
|
28788
|
639 |
\item \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}} refines any remaining goals by
|
|
640 |
proof method \isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}} and concludes the sub-proof by assumption.
|
|
641 |
If the goal had been \isa{{\isachardoublequote}show{\isachardoublequote}} (or \isa{{\isachardoublequote}thus{\isachardoublequote}}), some
|
|
642 |
pending sub-goal is solved as well by the rule resulting from the
|
|
643 |
result \emph{exported} into the enclosing goal context. Thus \isa{{\isachardoublequote}qed{\isachardoublequote}} may fail for two reasons: either \isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}} fails, or the
|
|
644 |
resulting rule does not fit to any pending goal\footnote{This
|
|
645 |
includes any additional ``strong'' assumptions as introduced by
|
|
646 |
\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}.} of the enclosing context. Debugging such a
|
|
647 |
situation might involve temporarily changing \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} into
|
|
648 |
\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}, or weakening the local context by replacing
|
|
649 |
occurrences of \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} by \hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}}.
|
26870
|
650 |
|
28788
|
651 |
\item \hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}\ m\isactrlsub {\isadigit{2}}{\isachardoublequote}} is a \emph{terminal
|
|
652 |
proof}\index{proof!terminal}; it abbreviates \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}}~\isa{{\isachardoublequote}qed{\isachardoublequote}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{2}}{\isachardoublequote}}, but with
|
|
653 |
backtracking across both methods. Debugging an unsuccessful
|
|
654 |
\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}\ m\isactrlsub {\isadigit{2}}{\isachardoublequote}} command can be done by expanding its
|
|
655 |
definition; in many cases \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}} (or even
|
|
656 |
\isa{{\isachardoublequote}apply{\isachardoublequote}}~\isa{{\isachardoublequote}m\isactrlsub {\isadigit{1}}{\isachardoublequote}}) is already sufficient to see the
|
26870
|
657 |
problem.
|
|
658 |
|
28788
|
659 |
\item ``\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}}}'' is a \emph{default
|
26902
|
660 |
proof}\index{proof!default}; it abbreviates \hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isachardoublequote}rule{\isachardoublequote}}.
|
26870
|
661 |
|
28788
|
662 |
\item ``\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isachardot}}}}}'' is a \emph{trivial
|
26902
|
663 |
proof}\index{proof!trivial}; it abbreviates \hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isachardoublequote}this{\isachardoublequote}}.
|
26870
|
664 |
|
28788
|
665 |
\item \hyperlink{command.sorry}{\mbox{\isa{\isacommand{sorry}}}} is a \emph{fake proof}\index{proof!fake}
|
26870
|
666 |
pretending to solve the pending claim without further ado. This
|
|
667 |
only works in interactive development, or if the \verb|quick_and_dirty| flag is enabled (in ML). Facts emerging from fake
|
|
668 |
proofs are not the real thing. Internally, each theorem container
|
|
669 |
is tainted by an oracle invocation, which is indicated as ``\isa{{\isachardoublequote}{\isacharbrackleft}{\isacharbang}{\isacharbrackright}{\isachardoublequote}}'' in the printed result.
|
|
670 |
|
26902
|
671 |
The most important application of \hyperlink{command.sorry}{\mbox{\isa{\isacommand{sorry}}}} is to support
|
26870
|
672 |
experimentation and top-down proof development.
|
|
673 |
|
28788
|
674 |
\end{description}%
|
26870
|
675 |
\end{isamarkuptext}%
|
|
676 |
\isamarkuptrue%
|
|
677 |
%
|
28788
|
678 |
\isamarkupsubsection{Fundamental methods and attributes \label{sec:pure-meth-att}%
|
26870
|
679 |
}
|
|
680 |
\isamarkuptrue%
|
|
681 |
%
|
|
682 |
\begin{isamarkuptext}%
|
|
683 |
The following proof methods and attributes refer to basic logical
|
|
684 |
operations of Isar. Further methods and attributes are provided by
|
|
685 |
several generic and object-logic specific tools and packages (see
|
|
686 |
\chref{ch:gen-tools} and \chref{ch:hol}).
|
|
687 |
|
|
688 |
\begin{matharray}{rcl}
|
28788
|
689 |
\indexdef{}{method}{-}\hypertarget{method.-}{\hyperlink{method.-}{\mbox{\isa{{\isacharminus}}}}} & : & \isa{method} \\
|
|
690 |
\indexdef{}{method}{fact}\hypertarget{method.fact}{\hyperlink{method.fact}{\mbox{\isa{fact}}}} & : & \isa{method} \\
|
|
691 |
\indexdef{}{method}{assumption}\hypertarget{method.assumption}{\hyperlink{method.assumption}{\mbox{\isa{assumption}}}} & : & \isa{method} \\
|
|
692 |
\indexdef{}{method}{this}\hypertarget{method.this}{\hyperlink{method.this}{\mbox{\isa{this}}}} & : & \isa{method} \\
|
|
693 |
\indexdef{}{method}{rule}\hypertarget{method.rule}{\hyperlink{method.rule}{\mbox{\isa{rule}}}} & : & \isa{method} \\
|
|
694 |
\indexdef{}{method}{iprover}\hypertarget{method.iprover}{\hyperlink{method.iprover}{\mbox{\isa{iprover}}}} & : & \isa{method} \\[0.5ex]
|
|
695 |
\indexdef{Pure}{attribute}{intro}\hypertarget{attribute.Pure.intro}{\hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}}} & : & \isa{attribute} \\
|
|
696 |
\indexdef{Pure}{attribute}{elim}\hypertarget{attribute.Pure.elim}{\hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}} & : & \isa{attribute} \\
|
|
697 |
\indexdef{Pure}{attribute}{dest}\hypertarget{attribute.Pure.dest}{\hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}} & : & \isa{attribute} \\
|
|
698 |
\indexdef{}{attribute}{rule}\hypertarget{attribute.rule}{\hyperlink{attribute.rule}{\mbox{\isa{rule}}}} & : & \isa{attribute} \\[0.5ex]
|
|
699 |
\indexdef{}{attribute}{OF}\hypertarget{attribute.OF}{\hyperlink{attribute.OF}{\mbox{\isa{OF}}}} & : & \isa{attribute} \\
|
|
700 |
\indexdef{}{attribute}{of}\hypertarget{attribute.of}{\hyperlink{attribute.of}{\mbox{\isa{of}}}} & : & \isa{attribute} \\
|
|
701 |
\indexdef{}{attribute}{where}\hypertarget{attribute.where}{\hyperlink{attribute.where}{\mbox{\isa{where}}}} & : & \isa{attribute} \\
|
26870
|
702 |
\end{matharray}
|
|
703 |
|
|
704 |
\begin{rail}
|
|
705 |
'fact' thmrefs?
|
|
706 |
;
|
|
707 |
'rule' thmrefs?
|
|
708 |
;
|
|
709 |
'iprover' ('!' ?) (rulemod *)
|
|
710 |
;
|
|
711 |
rulemod: ('intro' | 'elim' | 'dest') ((('!' | () | '?') nat?) | 'del') ':' thmrefs
|
|
712 |
;
|
|
713 |
('intro' | 'elim' | 'dest') ('!' | () | '?') nat?
|
|
714 |
;
|
|
715 |
'rule' 'del'
|
|
716 |
;
|
|
717 |
'OF' thmrefs
|
|
718 |
;
|
|
719 |
'of' insts ('concl' ':' insts)?
|
|
720 |
;
|
|
721 |
'where' ((name | var | typefree | typevar) '=' (type | term) * 'and')
|
|
722 |
;
|
|
723 |
\end{rail}
|
|
724 |
|
28788
|
725 |
\begin{description}
|
26870
|
726 |
|
28788
|
727 |
\item ``\hyperlink{method.-}{\mbox{\isa{{\isacharminus}}}}'' (minus) does nothing but insert the forward
|
|
728 |
chaining facts as premises into the goal. Note that command
|
26902
|
729 |
\indexref{}{command}{proof}\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} without any method actually performs a single
|
|
730 |
reduction step using the \indexref{}{method}{rule}\hyperlink{method.rule}{\mbox{\isa{rule}}} method; thus a plain
|
|
731 |
\emph{do-nothing} proof step would be ``\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' rather than \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} alone.
|
26870
|
732 |
|
28788
|
733 |
\item \hyperlink{method.fact}{\mbox{\isa{fact}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}} composes some fact from
|
|
734 |
\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ a\isactrlsub n{\isachardoublequote}} (or implicitly from the current proof context)
|
|
735 |
modulo unification of schematic type and term variables. The rule
|
|
736 |
structure is not taken into account, i.e.\ meta-level implication is
|
|
737 |
considered atomic. This is the same principle underlying literal
|
|
738 |
facts (cf.\ \secref{sec:syn-att}): ``\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isachardoublequote}{\isasymphi}{\isachardoublequote}}~\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{fact}'' is equivalent to ``\hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\verb|`|\isa{{\isasymphi}}\verb|`|'' provided that
|
|
739 |
\isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymphi}{\isachardoublequote}} is an instance of some known \isa{{\isachardoublequote}{\isasymturnstile}\ {\isasymphi}{\isachardoublequote}} in the
|
|
740 |
proof context.
|
26870
|
741 |
|
28788
|
742 |
\item \hyperlink{method.assumption}{\mbox{\isa{assumption}}} solves some goal by a single assumption
|
26870
|
743 |
step. All given facts are guaranteed to participate in the
|
|
744 |
refinement; this means there may be only 0 or 1 in the first place.
|
26902
|
745 |
Recall that \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} (\secref{sec:proof-steps}) already
|
26870
|
746 |
concludes any remaining sub-goals by assumption, so structured
|
26902
|
747 |
proofs usually need not quote the \hyperlink{method.assumption}{\mbox{\isa{assumption}}} method at
|
26870
|
748 |
all.
|
|
749 |
|
28788
|
750 |
\item \hyperlink{method.this}{\mbox{\isa{this}}} applies all of the current facts directly as
|
26902
|
751 |
rules. Recall that ``\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isachardot}}}}}'' (dot) abbreviates ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{this}''.
|
26870
|
752 |
|
28788
|
753 |
\item \hyperlink{method.rule}{\mbox{\isa{rule}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}} applies some rule given as
|
|
754 |
argument in backward manner; facts are used to reduce the rule
|
|
755 |
before applying it to the goal. Thus \hyperlink{method.rule}{\mbox{\isa{rule}}} without facts
|
|
756 |
is plain introduction, while with facts it becomes elimination.
|
26870
|
757 |
|
26902
|
758 |
When no arguments are given, the \hyperlink{method.rule}{\mbox{\isa{rule}}} method tries to pick
|
26870
|
759 |
appropriate rules automatically, as declared in the current context
|
26902
|
760 |
using the \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}}, \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}},
|
|
761 |
\hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}} attributes (see below). This is the
|
|
762 |
default behavior of \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} and ``\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}}}''
|
|
763 |
(double-dot) steps (see \secref{sec:proof-steps}).
|
26870
|
764 |
|
28788
|
765 |
\item \hyperlink{method.iprover}{\mbox{\isa{iprover}}} performs intuitionistic proof search,
|
26870
|
766 |
depending on specifically declared rules from the context, or given
|
|
767 |
as explicit arguments. Chained facts are inserted into the goal
|
28788
|
768 |
before commencing proof search; ``\hyperlink{method.iprover}{\mbox{\isa{iprover}}}\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}''
|
26902
|
769 |
means to include the current \hyperlink{fact.prems}{\mbox{\isa{prems}}} as well.
|
26870
|
770 |
|
26902
|
771 |
Rules need to be classified as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}},
|
|
772 |
\hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, or \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}; here the
|
|
773 |
``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' indicator refers to ``safe'' rules, which may be
|
|
774 |
applied aggressively (without considering back-tracking later).
|
|
775 |
Rules declared with ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' are ignored in proof search (the
|
|
776 |
single-step \hyperlink{method.rule}{\mbox{\isa{rule}}} method still observes these). An
|
|
777 |
explicit weight annotation may be given as well; otherwise the
|
|
778 |
number of rule premises will be taken into account here.
|
26870
|
779 |
|
28788
|
780 |
\item \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}}, \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, and
|
|
781 |
\hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}} declare introduction, elimination, and
|
26902
|
782 |
destruct rules, to be used with the \hyperlink{method.rule}{\mbox{\isa{rule}}} and \hyperlink{method.iprover}{\mbox{\isa{iprover}}} methods. Note that the latter will ignore rules declared
|
|
783 |
with ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'', while ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' are used most
|
|
784 |
aggressively.
|
26870
|
785 |
|
|
786 |
The classical reasoner (see \secref{sec:classical}) introduces its
|
|
787 |
own variants of these attributes; use qualified names to access the
|
26902
|
788 |
present versions of Isabelle/Pure, i.e.\ \hyperlink{attribute.Pure.Pure.intro}{\mbox{\isa{Pure{\isachardot}intro}}}.
|
26870
|
789 |
|
28788
|
790 |
\item \hyperlink{attribute.rule}{\mbox{\isa{rule}}}~\isa{del} undeclares introduction,
|
26870
|
791 |
elimination, or destruct rules.
|
|
792 |
|
28788
|
793 |
\item \hyperlink{attribute.OF}{\mbox{\isa{OF}}}~\isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isachardoublequote}} applies some theorem to all
|
|
794 |
of the given rules \isa{{\isachardoublequote}a\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ a\isactrlsub n{\isachardoublequote}} (in parallel). This
|
|
795 |
corresponds to the \verb|op MRS| operation in ML,
|
|
796 |
but note the reversed order. Positions may be effectively skipped
|
|
797 |
by including ``\isa{{\isacharunderscore}}'' (underscore) as argument.
|
26870
|
798 |
|
28788
|
799 |
\item \hyperlink{attribute.of}{\mbox{\isa{of}}}~\isa{{\isachardoublequote}t\isactrlsub {\isadigit{1}}\ {\isasymdots}\ t\isactrlsub n{\isachardoublequote}} performs positional
|
|
800 |
instantiation of term variables. The terms \isa{{\isachardoublequote}t\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ t\isactrlsub n{\isachardoublequote}} are
|
|
801 |
substituted for any schematic variables occurring in a theorem from
|
|
802 |
left to right; ``\isa{{\isacharunderscore}}'' (underscore) indicates to skip a
|
|
803 |
position. Arguments following a ``\isa{{\isachardoublequote}concl{\isacharcolon}{\isachardoublequote}}'' specification
|
|
804 |
refer to positions of the conclusion of a rule.
|
26870
|
805 |
|
28788
|
806 |
\item \hyperlink{attribute.where}{\mbox{\isa{where}}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isacharequal}\ t\isactrlsub {\isadigit{1}}\ {\isasymAND}\ {\isasymdots}\ x\isactrlsub n\ {\isacharequal}\ t\isactrlsub n{\isachardoublequote}}
|
|
807 |
performs named instantiation of schematic type and term variables
|
|
808 |
occurring in a theorem. Schematic variables have to be specified on
|
|
809 |
the left-hand side (e.g.\ \isa{{\isachardoublequote}{\isacharquery}x{\isadigit{1}}{\isachardot}{\isadigit{3}}{\isachardoublequote}}). The question mark may
|
|
810 |
be omitted if the variable name is a plain identifier without index.
|
|
811 |
As type instantiations are inferred from term instantiations,
|
|
812 |
explicit type instantiations are seldom necessary.
|
26870
|
813 |
|
28788
|
814 |
\end{description}%
|
26870
|
815 |
\end{isamarkuptext}%
|
|
816 |
\isamarkuptrue%
|
|
817 |
%
|
28788
|
818 |
\isamarkupsubsection{Emulating tactic scripts \label{sec:tactic-commands}%
|
26870
|
819 |
}
|
|
820 |
\isamarkuptrue%
|
|
821 |
%
|
|
822 |
\begin{isamarkuptext}%
|
|
823 |
The Isar provides separate commands to accommodate tactic-style
|
|
824 |
proof scripts within the same system. While being outside the
|
|
825 |
orthodox Isar proof language, these might come in handy for
|
|
826 |
interactive exploration and debugging, or even actual tactical proof
|
|
827 |
within new-style theories (to benefit from document preparation, for
|
|
828 |
example). See also \secref{sec:tactics} for actual tactics, that
|
|
829 |
have been encapsulated as proof methods. Proper proof methods may
|
|
830 |
be used in scripts, too.
|
|
831 |
|
|
832 |
\begin{matharray}{rcl}
|
28788
|
833 |
\indexdef{}{command}{apply}\hypertarget{command.apply}{\hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
|
834 |
\indexdef{}{command}{apply\_end}\hypertarget{command.apply-end}{\hyperlink{command.apply-end}{\mbox{\isa{\isacommand{apply{\isacharunderscore}end}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
|
|
835 |
\indexdef{}{command}{done}\hypertarget{command.done}{\hyperlink{command.done}{\mbox{\isa{\isacommand{done}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ local{\isacharunderscore}theory\ {\isacharbar}\ theory{\isachardoublequote}} \\
|
|
836 |
\indexdef{}{command}{defer}\hypertarget{command.defer}{\hyperlink{command.defer}{\mbox{\isa{\isacommand{defer}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof\ {\isasymrightarrow}\ proof{\isachardoublequote}} \\
|
|
837 |
\indexdef{}{command}{prefer}\hypertarget{command.prefer}{\hyperlink{command.prefer}{\mbox{\isa{\isacommand{prefer}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof\ {\isasymrightarrow}\ proof{\isachardoublequote}} \\
|
|
838 |
\indexdef{}{command}{back}\hypertarget{command.back}{\hyperlink{command.back}{\mbox{\isa{\isacommand{back}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof\ {\isasymrightarrow}\ proof{\isachardoublequote}} \\
|
26870
|
839 |
\end{matharray}
|
|
840 |
|
|
841 |
\begin{rail}
|
|
842 |
( 'apply' | 'apply\_end' ) method
|
|
843 |
;
|
|
844 |
'defer' nat?
|
|
845 |
;
|
|
846 |
'prefer' nat
|
|
847 |
;
|
|
848 |
\end{rail}
|
|
849 |
|
28788
|
850 |
\begin{description}
|
26870
|
851 |
|
28788
|
852 |
\item \hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}~\isa{m} applies proof method \isa{m} in
|
|
853 |
initial position, but unlike \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} it retains ``\isa{{\isachardoublequote}proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}}'' mode. Thus consecutive method applications may be
|
|
854 |
given just as in tactic scripts.
|
26870
|
855 |
|
|
856 |
Facts are passed to \isa{m} as indicated by the goal's
|
|
857 |
forward-chain mode, and are \emph{consumed} afterwards. Thus any
|
26902
|
858 |
further \hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}} command would always work in a purely
|
26870
|
859 |
backward manner.
|
|
860 |
|
28788
|
861 |
\item \hyperlink{command.apply-end}{\mbox{\isa{\isacommand{apply{\isacharunderscore}end}}}}~\isa{{\isachardoublequote}m{\isachardoublequote}} applies proof method \isa{m} as if in terminal position. Basically, this simulates a
|
26902
|
862 |
multi-step tactic script for \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}, but may be given
|
26870
|
863 |
anywhere within the proof body.
|
|
864 |
|
26895
|
865 |
No facts are passed to \isa{m} here. Furthermore, the static
|
26902
|
866 |
context is that of the enclosing goal (as for actual \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}). Thus the proof method may not refer to any assumptions
|
26870
|
867 |
introduced in the current body, for example.
|
|
868 |
|
28788
|
869 |
\item \hyperlink{command.done}{\mbox{\isa{\isacommand{done}}}} completes a proof script, provided that the
|
|
870 |
current goal state is solved completely. Note that actual
|
26902
|
871 |
structured proof commands (e.g.\ ``\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isachardot}}}}}'' or \hyperlink{command.sorry}{\mbox{\isa{\isacommand{sorry}}}}) may be used to conclude proof scripts as well.
|
26870
|
872 |
|
28788
|
873 |
\item \hyperlink{command.defer}{\mbox{\isa{\isacommand{defer}}}}~\isa{n} and \hyperlink{command.prefer}{\mbox{\isa{\isacommand{prefer}}}}~\isa{n}
|
|
874 |
shuffle the list of pending goals: \hyperlink{command.defer}{\mbox{\isa{\isacommand{defer}}}} puts off
|
26870
|
875 |
sub-goal \isa{n} to the end of the list (\isa{{\isachardoublequote}n\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequote}} by
|
26902
|
876 |
default), while \hyperlink{command.prefer}{\mbox{\isa{\isacommand{prefer}}}} brings sub-goal \isa{n} to the
|
26870
|
877 |
front.
|
|
878 |
|
28788
|
879 |
\item \hyperlink{command.back}{\mbox{\isa{\isacommand{back}}}} does back-tracking over the result sequence
|
|
880 |
of the latest proof command. Basically, any proof command may
|
|
881 |
return multiple results.
|
26870
|
882 |
|
28788
|
883 |
\end{description}
|
26870
|
884 |
|
|
885 |
Any proper Isar proof method may be used with tactic script commands
|
26902
|
886 |
such as \hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}. A few additional emulations of actual
|
26870
|
887 |
tactics are provided as well; these would be never used in actual
|
|
888 |
structured proofs, of course.%
|
|
889 |
\end{isamarkuptext}%
|
|
890 |
\isamarkuptrue%
|
|
891 |
%
|
28788
|
892 |
\isamarkupsubsection{Defining proof methods%
|
26870
|
893 |
}
|
|
894 |
\isamarkuptrue%
|
|
895 |
%
|
|
896 |
\begin{isamarkuptext}%
|
|
897 |
\begin{matharray}{rcl}
|
28788
|
898 |
\indexdef{}{command}{method\_setup}\hypertarget{command.method-setup}{\hyperlink{command.method-setup}{\mbox{\isa{\isacommand{method{\isacharunderscore}setup}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
|
26870
|
899 |
\end{matharray}
|
|
900 |
|
28788
|
901 |
\begin{rail}
|
|
902 |
'method\_setup' name '=' text text
|
|
903 |
;
|
|
904 |
\end{rail}
|
|
905 |
|
|
906 |
\begin{description}
|
|
907 |
|
|
908 |
\item \hyperlink{command.method-setup}{\mbox{\isa{\isacommand{method{\isacharunderscore}setup}}}}~\isa{{\isachardoublequote}name\ {\isacharequal}\ text\ description{\isachardoublequote}}
|
|
909 |
defines a proof method in the current theory. The given \isa{{\isachardoublequote}text{\isachardoublequote}} has to be an ML expression of type \verb|Args.src -> Proof.context -> Proof.method|. Parsing concrete
|
|
910 |
method syntax from \verb|Args.src| input can be quite tedious in
|
|
911 |
general. The following simple examples are for methods without any
|
|
912 |
explicit arguments, or a list of theorems, respectively.
|
26870
|
913 |
|
28788
|
914 |
%FIXME proper antiquotations
|
|
915 |
{\footnotesize
|
|
916 |
\begin{verbatim}
|
|
917 |
Method.no_args (Method.METHOD (fn facts => foobar_tac))
|
|
918 |
Method.thms_args (fn thms => Method.METHOD (fn facts => foobar_tac))
|
|
919 |
Method.ctxt_args (fn ctxt => Method.METHOD (fn facts => foobar_tac))
|
|
920 |
Method.thms_ctxt_args (fn thms => fn ctxt =>
|
|
921 |
Method.METHOD (fn facts => foobar_tac))
|
|
922 |
\end{verbatim}
|
|
923 |
}
|
26870
|
924 |
|
28788
|
925 |
Note that mere tactic emulations may ignore the \isa{facts}
|
|
926 |
parameter above. Proper proof methods would do something
|
|
927 |
appropriate with the list of current facts, though. Single-rule
|
|
928 |
methods usually do strict forward-chaining (e.g.\ by using \verb|Drule.multi_resolves|), while automatic ones just insert the facts
|
|
929 |
using \verb|Method.insert_tac| before applying the main tactic.
|
|
930 |
|
|
931 |
\end{description}%
|
26870
|
932 |
\end{isamarkuptext}%
|
|
933 |
\isamarkuptrue%
|
|
934 |
%
|
|
935 |
\isamarkupsection{Generalized elimination \label{sec:obtain}%
|
|
936 |
}
|
|
937 |
\isamarkuptrue%
|
|
938 |
%
|
|
939 |
\begin{isamarkuptext}%
|
|
940 |
\begin{matharray}{rcl}
|
28788
|
941 |
\indexdef{}{command}{obtain}\hypertarget{command.obtain}{\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ proof{\isacharparenleft}chain{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
|
942 |
\indexdef{}{command}{guess}\hypertarget{command.guess}{\hyperlink{command.guess}{\mbox{\isa{\isacommand{guess}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isacharbar}\ proof{\isacharparenleft}chain{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
|
26870
|
943 |
\end{matharray}
|
|
944 |
|
|
945 |
Generalized elimination means that additional elements with certain
|
|
946 |
properties may be introduced in the current context, by virtue of a
|
|
947 |
locally proven ``soundness statement''. Technically speaking, the
|
26902
|
948 |
\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} language element is like a declaration of
|
|
949 |
\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}} and \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} (see also see
|
26870
|
950 |
\secref{sec:proof-context}), together with a soundness proof of its
|
|
951 |
additional claim. According to the nature of existential reasoning,
|
|
952 |
assumptions get eliminated from any result exported from the context
|
|
953 |
later, provided that the corresponding parameters do \emph{not}
|
|
954 |
occur in the conclusion.
|
|
955 |
|
|
956 |
\begin{rail}
|
|
957 |
'obtain' parname? (vars + 'and') 'where' (props + 'and')
|
|
958 |
;
|
|
959 |
'guess' (vars + 'and')
|
|
960 |
;
|
|
961 |
\end{rail}
|
|
962 |
|
26902
|
963 |
The derived Isar command \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} is defined as follows
|
26870
|
964 |
(where \isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k{\isachardoublequote}} shall refer to (optional)
|
|
965 |
facts indicated for forward chaining).
|
|
966 |
\begin{matharray}{l}
|
26902
|
967 |
\isa{{\isachardoublequote}{\isasymlangle}using\ b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub k{\isasymrangle}{\isachardoublequote}}~~\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m\ {\isasymWHERE}\ a{\isacharcolon}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ {\isasymphi}\isactrlsub n\ \ {\isasymlangle}proof{\isasymrangle}\ {\isasymequiv}{\isachardoublequote}} \\[1ex]
|
|
968 |
\quad \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isachardoublequote}{\isasymAnd}thesis{\isachardot}\ {\isacharparenleft}{\isasymAnd}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardot}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymphi}\isactrlsub n\ {\isasymLongrightarrow}\ thesis{\isacharparenright}\ {\isasymLongrightarrow}\ thesis{\isachardoublequote}} \\
|
|
969 |
\quad \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{succeed} \\
|
|
970 |
\qquad \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{thesis} \\
|
|
971 |
\qquad \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isachardoublequote}that\ {\isacharbrackleft}Pure{\isachardot}intro{\isacharquery}{\isacharbrackright}{\isacharcolon}\ {\isasymAnd}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardot}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymphi}\isactrlsub n\ {\isasymLongrightarrow}\ thesis{\isachardoublequote}} \\
|
|
972 |
\qquad \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}~\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}~\isa{thesis} \\
|
|
973 |
\quad\qquad \hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}~\isa{{\isacharminus}} \\
|
|
974 |
\quad\qquad \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isasymdots}\ b\isactrlsub k\ \ {\isasymlangle}proof{\isasymrangle}{\isachardoublequote}} \\
|
|
975 |
\quad \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} \\
|
|
976 |
\quad \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardoublequote}}~\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}\ a{\isacharcolon}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ {\isasymphi}\isactrlsub n{\isachardoublequote}} \\
|
26870
|
977 |
\end{matharray}
|
|
978 |
|
|
979 |
Typically, the soundness proof is relatively straight-forward, often
|
26902
|
980 |
just by canonical automated tools such as ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{simp}'' or ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{blast}''. Accordingly, the
|
26870
|
981 |
``\isa{that}'' reduction above is declared as simplification and
|
|
982 |
introduction rule.
|
|
983 |
|
26902
|
984 |
In a sense, \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} represents at the level of Isar
|
26870
|
985 |
proofs what would be meta-logical existential quantifiers and
|
|
986 |
conjunctions. This concept has a broad range of useful
|
|
987 |
applications, ranging from plain elimination (or introduction) of
|
|
988 |
object-level existential and conjunctions, to elimination over
|
|
989 |
results of symbolic evaluation of recursive definitions, for
|
26902
|
990 |
example. Also note that \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} without parameters acts
|
|
991 |
much like \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}, where the result is treated as a
|
26870
|
992 |
genuine assumption.
|
|
993 |
|
|
994 |
An alternative name to be used instead of ``\isa{that}'' above may
|
|
995 |
be given in parentheses.
|
|
996 |
|
26902
|
997 |
\medskip The improper variant \hyperlink{command.guess}{\mbox{\isa{\isacommand{guess}}}} is similar to
|
|
998 |
\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}, but derives the obtained statement from the
|
26870
|
999 |
course of reasoning! The proof starts with a fixed goal \isa{thesis}. The subsequent proof may refine this to anything of the
|
|
1000 |
form like \isa{{\isachardoublequote}{\isasymAnd}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardot}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymphi}\isactrlsub n\ {\isasymLongrightarrow}\ thesis{\isachardoublequote}}, but must not introduce new subgoals. The
|
|
1001 |
final goal state is then used as reduction rule for the obtain
|
|
1002 |
scheme described above. Obtained parameters \isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlsub m{\isachardoublequote}} are marked as internal by default, which prevents the
|
|
1003 |
proof context from being polluted by ad-hoc variables. The variable
|
26902
|
1004 |
names and type constraints given as arguments for \hyperlink{command.guess}{\mbox{\isa{\isacommand{guess}}}}
|
26870
|
1005 |
specify a prefix of obtained parameters explicitly in the text.
|
|
1006 |
|
26902
|
1007 |
It is important to note that the facts introduced by \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} and \hyperlink{command.guess}{\mbox{\isa{\isacommand{guess}}}} may not be polymorphic: any
|
26870
|
1008 |
type-variables occurring here are fixed in the present context!%
|
|
1009 |
\end{isamarkuptext}%
|
|
1010 |
\isamarkuptrue%
|
|
1011 |
%
|
|
1012 |
\isamarkupsection{Calculational reasoning \label{sec:calculation}%
|
|
1013 |
}
|
|
1014 |
\isamarkuptrue%
|
|
1015 |
%
|
|
1016 |
\begin{isamarkuptext}%
|
|
1017 |
\begin{matharray}{rcl}
|
28788
|
1018 |
\indexdef{}{command}{also}\hypertarget{command.also}{\hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
|
|
1019 |
\indexdef{}{command}{finally}\hypertarget{command.finally}{\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} \\
|
|
1020 |
\indexdef{}{command}{moreover}\hypertarget{command.moreover}{\hyperlink{command.moreover}{\mbox{\isa{\isacommand{moreover}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
|
|
1021 |
\indexdef{}{command}{ultimately}\hypertarget{command.ultimately}{\hyperlink{command.ultimately}{\mbox{\isa{\isacommand{ultimately}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}chain{\isacharparenright}{\isachardoublequote}} \\
|
|
1022 |
\indexdef{}{command}{print\_trans\_rules}\hypertarget{command.print-trans-rules}{\hyperlink{command.print-trans-rules}{\mbox{\isa{\isacommand{print{\isacharunderscore}trans{\isacharunderscore}rules}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
|
|
1023 |
\hyperlink{attribute.trans}{\mbox{\isa{trans}}} & : & \isa{attribute} \\
|
|
1024 |
\hyperlink{attribute.sym}{\mbox{\isa{sym}}} & : & \isa{attribute} \\
|
|
1025 |
\hyperlink{attribute.symmetric}{\mbox{\isa{symmetric}}} & : & \isa{attribute} \\
|
26870
|
1026 |
\end{matharray}
|
|
1027 |
|
|
1028 |
Calculational proof is forward reasoning with implicit application
|
|
1029 |
of transitivity rules (such those of \isa{{\isachardoublequote}{\isacharequal}{\isachardoublequote}}, \isa{{\isachardoublequote}{\isasymle}{\isachardoublequote}},
|
|
1030 |
\isa{{\isachardoublequote}{\isacharless}{\isachardoublequote}}). Isabelle/Isar maintains an auxiliary fact register
|
26902
|
1031 |
\indexref{}{fact}{calculation}\hyperlink{fact.calculation}{\mbox{\isa{calculation}}} for accumulating results obtained by
|
|
1032 |
transitivity composed with the current result. Command \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}} updates \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} involving \hyperlink{fact.this}{\mbox{\isa{this}}}, while
|
|
1033 |
\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}} exhibits the final \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} by
|
26870
|
1034 |
forward chaining towards the next goal statement. Both commands
|
|
1035 |
require valid current facts, i.e.\ may occur only after commands
|
26902
|
1036 |
that produce theorems such as \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}, \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}, or some finished proof of \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}, \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} etc. The \hyperlink{command.moreover}{\mbox{\isa{\isacommand{moreover}}}} and \hyperlink{command.ultimately}{\mbox{\isa{\isacommand{ultimately}}}}
|
|
1037 |
commands are similar to \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}} and \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}},
|
|
1038 |
but only collect further results in \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} without
|
26870
|
1039 |
applying any rules yet.
|
|
1040 |
|
|
1041 |
Also note that the implicit term abbreviation ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' has
|
|
1042 |
its canonical application with calculational proofs. It refers to
|
|
1043 |
the argument of the preceding statement. (The argument of a curried
|
|
1044 |
infix expression happens to be its right-hand side.)
|
|
1045 |
|
|
1046 |
Isabelle/Isar calculations are implicitly subject to block structure
|
|
1047 |
in the sense that new threads of calculational reasoning are
|
|
1048 |
commenced for any new block (as opened by a local goal, for
|
|
1049 |
example). This means that, apart from being able to nest
|
|
1050 |
calculations, there is no separate \emph{begin-calculation} command
|
|
1051 |
required.
|
|
1052 |
|
|
1053 |
\medskip The Isar calculation proof commands may be defined as
|
|
1054 |
follows:\footnote{We suppress internal bookkeeping such as proper
|
|
1055 |
handling of block-structure.}
|
|
1056 |
|
|
1057 |
\begin{matharray}{rcl}
|
26902
|
1058 |
\hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}\isa{{\isachardoublequote}\isactrlsub {\isadigit{0}}{\isachardoublequote}} & \equiv & \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isachardoublequote}calculation\ {\isacharequal}\ this{\isachardoublequote}} \\
|
|
1059 |
\hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}\isa{{\isachardoublequote}\isactrlsub n\isactrlsub {\isacharplus}\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \equiv & \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isachardoublequote}calculation\ {\isacharequal}\ trans\ {\isacharbrackleft}OF\ calculation\ this{\isacharbrackright}{\isachardoublequote}} \\[0.5ex]
|
|
1060 |
\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}} & \equiv & \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}~\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{calculation} \\[0.5ex]
|
|
1061 |
\hyperlink{command.moreover}{\mbox{\isa{\isacommand{moreover}}}} & \equiv & \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isachardoublequote}calculation\ {\isacharequal}\ calculation\ this{\isachardoublequote}} \\
|
|
1062 |
\hyperlink{command.ultimately}{\mbox{\isa{\isacommand{ultimately}}}} & \equiv & \hyperlink{command.moreover}{\mbox{\isa{\isacommand{moreover}}}}~\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{calculation} \\
|
26870
|
1063 |
\end{matharray}
|
|
1064 |
|
|
1065 |
\begin{rail}
|
|
1066 |
('also' | 'finally') ('(' thmrefs ')')?
|
|
1067 |
;
|
|
1068 |
'trans' (() | 'add' | 'del')
|
|
1069 |
;
|
|
1070 |
\end{rail}
|
|
1071 |
|
28788
|
1072 |
\begin{description}
|
26870
|
1073 |
|
28788
|
1074 |
\item \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}~\isa{{\isachardoublequote}{\isacharparenleft}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isacharparenright}{\isachardoublequote}} maintains the auxiliary
|
|
1075 |
\hyperlink{fact.calculation}{\mbox{\isa{calculation}}} register as follows. The first occurrence of
|
|
1076 |
\hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}} in some calculational thread initializes \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} by \hyperlink{fact.this}{\mbox{\isa{this}}}. Any subsequent \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}} on
|
|
1077 |
the same level of block-structure updates \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} by
|
|
1078 |
some transitivity rule applied to \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} and \hyperlink{fact.this}{\mbox{\isa{this}}} (in that order). Transitivity rules are picked from the
|
|
1079 |
current context, unless alternative rules are given as explicit
|
|
1080 |
arguments.
|
26870
|
1081 |
|
28788
|
1082 |
\item \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}~\isa{{\isachardoublequote}{\isacharparenleft}a\isactrlsub {\isadigit{1}}\ {\isasymdots}\ a\isactrlsub n{\isacharparenright}{\isachardoublequote}} maintaining \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} in the same way as \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}, and concludes the
|
|
1083 |
current calculational thread. The final result is exhibited as fact
|
|
1084 |
for forward chaining towards the next goal. Basically, \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}} just abbreviates \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}~\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\hyperlink{fact.calculation}{\mbox{\isa{calculation}}}. Typical idioms for concluding
|
|
1085 |
calculational proofs are ``\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}~\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}~\isa{{\isacharquery}thesis}~\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isachardot}}}}}'' and ``\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}~\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isasymphi}}~\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isachardot}}}}}''.
|
26870
|
1086 |
|
28788
|
1087 |
\item \hyperlink{command.moreover}{\mbox{\isa{\isacommand{moreover}}}} and \hyperlink{command.ultimately}{\mbox{\isa{\isacommand{ultimately}}}} are
|
26902
|
1088 |
analogous to \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}} and \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}, but collect
|
26870
|
1089 |
results only, without applying rules.
|
|
1090 |
|
28788
|
1091 |
\item \hyperlink{command.print-trans-rules}{\mbox{\isa{\isacommand{print{\isacharunderscore}trans{\isacharunderscore}rules}}}} prints the list of transitivity
|
|
1092 |
rules (for calculational commands \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}} and \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}) and symmetry rules (for the \hyperlink{attribute.symmetric}{\mbox{\isa{symmetric}}}
|
|
1093 |
operation and single step elimination patters) of the current
|
|
1094 |
context.
|
26870
|
1095 |
|
28788
|
1096 |
\item \hyperlink{attribute.trans}{\mbox{\isa{trans}}} declares theorems as transitivity rules.
|
26870
|
1097 |
|
28788
|
1098 |
\item \hyperlink{attribute.sym}{\mbox{\isa{sym}}} declares symmetry rules, as well as
|
26902
|
1099 |
\hyperlink{attribute.Pure.elim}{\mbox{\isa{Pure{\isachardot}elim}}}\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}} rules.
|
26870
|
1100 |
|
28788
|
1101 |
\item \hyperlink{attribute.symmetric}{\mbox{\isa{symmetric}}} resolves a theorem with some rule
|
26902
|
1102 |
declared as \hyperlink{attribute.sym}{\mbox{\isa{sym}}} in the current context. For example,
|
|
1103 |
``\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isachardoublequote}{\isacharbrackleft}symmetric{\isacharbrackright}{\isacharcolon}\ x\ {\isacharequal}\ y{\isachardoublequote}}'' produces a
|
26870
|
1104 |
swapped fact derived from that assumption.
|
|
1105 |
|
|
1106 |
In structured proof texts it is often more appropriate to use an
|
26902
|
1107 |
explicit single-step elimination proof, such as ``\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isachardoublequote}x\ {\isacharequal}\ y{\isachardoublequote}}~\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}~\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isachardoublequote}y\ {\isacharequal}\ x{\isachardoublequote}}~\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isachardot}{\isachardot}}}}}''.
|
26870
|
1108 |
|
28788
|
1109 |
\end{description}%
|
26870
|
1110 |
\end{isamarkuptext}%
|
|
1111 |
\isamarkuptrue%
|
|
1112 |
%
|
27042
|
1113 |
\isamarkupsection{Proof by cases and induction \label{sec:cases-induct}%
|
|
1114 |
}
|
|
1115 |
\isamarkuptrue%
|
|
1116 |
%
|
|
1117 |
\isamarkupsubsection{Rule contexts%
|
|
1118 |
}
|
|
1119 |
\isamarkuptrue%
|
|
1120 |
%
|
|
1121 |
\begin{isamarkuptext}%
|
|
1122 |
\begin{matharray}{rcl}
|
28788
|
1123 |
\indexdef{}{command}{case}\hypertarget{command.case}{\hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}} & : & \isa{{\isachardoublequote}proof{\isacharparenleft}state{\isacharparenright}\ {\isasymrightarrow}\ proof{\isacharparenleft}state{\isacharparenright}{\isachardoublequote}} \\
|
|
1124 |
\indexdef{}{command}{print\_cases}\hypertarget{command.print-cases}{\hyperlink{command.print-cases}{\mbox{\isa{\isacommand{print{\isacharunderscore}cases}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
|
|
1125 |
\indexdef{}{attribute}{case\_names}\hypertarget{attribute.case-names}{\hyperlink{attribute.case-names}{\mbox{\isa{case{\isacharunderscore}names}}}} & : & \isa{attribute} \\
|
|
1126 |
\indexdef{}{attribute}{case\_conclusion}\hypertarget{attribute.case-conclusion}{\hyperlink{attribute.case-conclusion}{\mbox{\isa{case{\isacharunderscore}conclusion}}}} & : & \isa{attribute} \\
|
|
1127 |
\indexdef{}{attribute}{params}\hypertarget{attribute.params}{\hyperlink{attribute.params}{\mbox{\isa{params}}}} & : & \isa{attribute} \\
|
|
1128 |
\indexdef{}{attribute}{consumes}\hypertarget{attribute.consumes}{\hyperlink{attribute.consumes}{\mbox{\isa{consumes}}}} & : & \isa{attribute} \\
|
27042
|
1129 |
\end{matharray}
|
|
1130 |
|
|
1131 |
The puristic way to build up Isar proof contexts is by explicit
|
|
1132 |
language elements like \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}, \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}},
|
|
1133 |
\hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}} (see \secref{sec:proof-context}). This is adequate
|
|
1134 |
for plain natural deduction, but easily becomes unwieldy in concrete
|
|
1135 |
verification tasks, which typically involve big induction rules with
|
|
1136 |
several cases.
|
|
1137 |
|
|
1138 |
The \hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}} command provides a shorthand to refer to a
|
|
1139 |
local context symbolically: certain proof methods provide an
|
|
1140 |
environment of named ``cases'' of the form \isa{{\isachardoublequote}c{\isacharcolon}\ x\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlsub m{\isacharcomma}\ {\isasymphi}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymphi}\isactrlsub n{\isachardoublequote}}; the effect of ``\hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}~\isa{c}'' is then equivalent to ``\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardoublequote}}~\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isachardoublequote}c{\isacharcolon}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ {\isasymphi}\isactrlsub n{\isachardoublequote}}''. Term bindings may be covered as well, notably
|
|
1141 |
\hyperlink{variable.?case}{\mbox{\isa{{\isacharquery}case}}} for the main conclusion.
|
|
1142 |
|
|
1143 |
By default, the ``terminology'' \isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlsub m{\isachardoublequote}} of
|
|
1144 |
a case value is marked as hidden, i.e.\ there is no way to refer to
|
|
1145 |
such parameters in the subsequent proof text. After all, original
|
|
1146 |
rule parameters stem from somewhere outside of the current proof
|
|
1147 |
text. By using the explicit form ``\hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}~\isa{{\isachardoublequote}{\isacharparenleft}c\ y\isactrlsub {\isadigit{1}}\ {\isasymdots}\ y\isactrlsub m{\isacharparenright}{\isachardoublequote}}'' instead, the proof author is able to
|
|
1148 |
chose local names that fit nicely into the current context.
|
|
1149 |
|
|
1150 |
\medskip It is important to note that proper use of \hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}} does not provide means to peek at the current goal state,
|
|
1151 |
which is not directly observable in Isar! Nonetheless, goal
|
|
1152 |
refinement commands do provide named cases \isa{{\isachardoublequote}goal\isactrlsub i{\isachardoublequote}}
|
|
1153 |
for each subgoal \isa{{\isachardoublequote}i\ {\isacharequal}\ {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ n{\isachardoublequote}} of the resulting goal state.
|
|
1154 |
Using this extra feature requires great care, because some bits of
|
|
1155 |
the internal tactical machinery intrude the proof text. In
|
|
1156 |
particular, parameter names stemming from the left-over of automated
|
|
1157 |
reasoning tools are usually quite unpredictable.
|
|
1158 |
|
|
1159 |
Under normal circumstances, the text of cases emerge from standard
|
|
1160 |
elimination or induction rules, which in turn are derived from
|
|
1161 |
previous theory specifications in a canonical way (say from
|
|
1162 |
\hyperlink{command.inductive}{\mbox{\isa{\isacommand{inductive}}}} definitions).
|
|
1163 |
|
|
1164 |
\medskip Proper cases are only available if both the proof method
|
|
1165 |
and the rules involved support this. By using appropriate
|
|
1166 |
attributes, case names, conclusions, and parameters may be also
|
|
1167 |
declared by hand. Thus variant versions of rules that have been
|
|
1168 |
derived manually become ready to use in advanced case analysis
|
|
1169 |
later.
|
|
1170 |
|
|
1171 |
\begin{rail}
|
|
1172 |
'case' (caseref | '(' caseref ((name | underscore) +) ')')
|
|
1173 |
;
|
|
1174 |
caseref: nameref attributes?
|
|
1175 |
;
|
|
1176 |
|
|
1177 |
'case\_names' (name +)
|
|
1178 |
;
|
|
1179 |
'case\_conclusion' name (name *)
|
|
1180 |
;
|
|
1181 |
'params' ((name *) + 'and')
|
|
1182 |
;
|
|
1183 |
'consumes' nat?
|
|
1184 |
;
|
|
1185 |
\end{rail}
|
|
1186 |
|
28788
|
1187 |
\begin{description}
|
27042
|
1188 |
|
28788
|
1189 |
\item \hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}~\isa{{\isachardoublequote}{\isacharparenleft}c\ x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isacharparenright}{\isachardoublequote}} invokes a named local
|
|
1190 |
context \isa{{\isachardoublequote}c{\isacharcolon}\ x\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlsub m{\isacharcomma}\ {\isasymphi}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymphi}\isactrlsub m{\isachardoublequote}}, as provided by an
|
|
1191 |
appropriate proof method (such as \indexref{}{method}{cases}\hyperlink{method.cases}{\mbox{\isa{cases}}} and
|
|
1192 |
\indexref{}{method}{induct}\hyperlink{method.induct}{\mbox{\isa{induct}}}). The command ``\hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}~\isa{{\isachardoublequote}{\isacharparenleft}c\ x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isacharparenright}{\isachardoublequote}}'' abbreviates ``\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardoublequote}}~\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isachardoublequote}c{\isacharcolon}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ {\isasymphi}\isactrlsub n{\isachardoublequote}}''.
|
27042
|
1193 |
|
28788
|
1194 |
\item \hyperlink{command.print-cases}{\mbox{\isa{\isacommand{print{\isacharunderscore}cases}}}} prints all local contexts of the
|
27042
|
1195 |
current state, using Isar proof language notation.
|
|
1196 |
|
28788
|
1197 |
\item \hyperlink{attribute.case-names}{\mbox{\isa{case{\isacharunderscore}names}}}~\isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}\ {\isasymdots}\ c\isactrlsub k{\isachardoublequote}} declares names for
|
|
1198 |
the local contexts of premises of a theorem; \isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub k{\isachardoublequote}}
|
|
1199 |
refers to the \emph{suffix} of the list of premises.
|
27042
|
1200 |
|
28788
|
1201 |
\item \hyperlink{attribute.case-conclusion}{\mbox{\isa{case{\isacharunderscore}conclusion}}}~\isa{{\isachardoublequote}c\ d\isactrlsub {\isadigit{1}}\ {\isasymdots}\ d\isactrlsub k{\isachardoublequote}} declares
|
|
1202 |
names for the conclusions of a named premise \isa{c}; here \isa{{\isachardoublequote}d\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ d\isactrlsub k{\isachardoublequote}} refers to the prefix of arguments of a logical formula
|
|
1203 |
built by nesting a binary connective (e.g.\ \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}).
|
27042
|
1204 |
|
|
1205 |
Note that proof methods such as \hyperlink{method.induct}{\mbox{\isa{induct}}} and \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} already provide a default name for the conclusion as a
|
|
1206 |
whole. The need to name subformulas only arises with cases that
|
|
1207 |
split into several sub-cases, as in common co-induction rules.
|
|
1208 |
|
28788
|
1209 |
\item \hyperlink{attribute.params}{\mbox{\isa{params}}}~\isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub m\ {\isasymAND}\ {\isasymdots}\ q\isactrlsub {\isadigit{1}}\ {\isasymdots}\ q\isactrlsub n{\isachardoublequote}} renames
|
|
1210 |
the innermost parameters of premises \isa{{\isachardoublequote}{\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ n{\isachardoublequote}} of some
|
|
1211 |
theorem. An empty list of names may be given to skip positions,
|
|
1212 |
leaving the present parameters unchanged.
|
27042
|
1213 |
|
|
1214 |
Note that the default usage of case rules does \emph{not} directly
|
|
1215 |
expose parameters to the proof context.
|
|
1216 |
|
28788
|
1217 |
\item \hyperlink{attribute.consumes}{\mbox{\isa{consumes}}}~\isa{n} declares the number of ``major
|
|
1218 |
premises'' of a rule, i.e.\ the number of facts to be consumed when
|
|
1219 |
it is applied by an appropriate proof method. The default value of
|
|
1220 |
\hyperlink{attribute.consumes}{\mbox{\isa{consumes}}} is \isa{{\isachardoublequote}n\ {\isacharequal}\ {\isadigit{1}}{\isachardoublequote}}, which is appropriate for
|
|
1221 |
the usual kind of cases and induction rules for inductive sets (cf.\
|
|
1222 |
\secref{sec:hol-inductive}). Rules without any \hyperlink{attribute.consumes}{\mbox{\isa{consumes}}} declaration given are treated as if \hyperlink{attribute.consumes}{\mbox{\isa{consumes}}}~\isa{{\isadigit{0}}} had been specified.
|
27042
|
1223 |
|
|
1224 |
Note that explicit \hyperlink{attribute.consumes}{\mbox{\isa{consumes}}} declarations are only
|
|
1225 |
rarely needed; this is already taken care of automatically by the
|
|
1226 |
higher-level \hyperlink{attribute.cases}{\mbox{\isa{cases}}}, \hyperlink{attribute.induct}{\mbox{\isa{induct}}}, and
|
|
1227 |
\hyperlink{attribute.coinduct}{\mbox{\isa{coinduct}}} declarations.
|
|
1228 |
|
28788
|
1229 |
\end{description}%
|
27042
|
1230 |
\end{isamarkuptext}%
|
|
1231 |
\isamarkuptrue%
|
|
1232 |
%
|
|
1233 |
\isamarkupsubsection{Proof methods%
|
|
1234 |
}
|
|
1235 |
\isamarkuptrue%
|
|
1236 |
%
|
|
1237 |
\begin{isamarkuptext}%
|
|
1238 |
\begin{matharray}{rcl}
|
28788
|
1239 |
\indexdef{}{method}{cases}\hypertarget{method.cases}{\hyperlink{method.cases}{\mbox{\isa{cases}}}} & : & \isa{method} \\
|
|
1240 |
\indexdef{}{method}{induct}\hypertarget{method.induct}{\hyperlink{method.induct}{\mbox{\isa{induct}}}} & : & \isa{method} \\
|
|
1241 |
\indexdef{}{method}{coinduct}\hypertarget{method.coinduct}{\hyperlink{method.coinduct}{\mbox{\isa{coinduct}}}} & : & \isa{method} \\
|
27042
|
1242 |
\end{matharray}
|
|
1243 |
|
|
1244 |
The \hyperlink{method.cases}{\mbox{\isa{cases}}}, \hyperlink{method.induct}{\mbox{\isa{induct}}}, and \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}}
|
|
1245 |
methods provide a uniform interface to common proof techniques over
|
|
1246 |
datatypes, inductive predicates (or sets), recursive functions etc.
|
|
1247 |
The corresponding rules may be specified and instantiated in a
|
|
1248 |
casual manner. Furthermore, these methods provide named local
|
|
1249 |
contexts that may be invoked via the \hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}} proof command
|
|
1250 |
within the subsequent proof text. This accommodates compact proof
|
|
1251 |
texts even when reasoning about large specifications.
|
|
1252 |
|
|
1253 |
The \hyperlink{method.induct}{\mbox{\isa{induct}}} method also provides some additional
|
|
1254 |
infrastructure in order to be applicable to structure statements
|
|
1255 |
(either using explicit meta-level connectives, or including facts
|
|
1256 |
and parameters separately). This avoids cumbersome encoding of
|
|
1257 |
``strengthened'' inductive statements within the object-logic.
|
|
1258 |
|
|
1259 |
\begin{rail}
|
|
1260 |
'cases' (insts * 'and') rule?
|
|
1261 |
;
|
|
1262 |
'induct' (definsts * 'and') \\ arbitrary? taking? rule?
|
|
1263 |
;
|
|
1264 |
'coinduct' insts taking rule?
|
|
1265 |
;
|
|
1266 |
|
|
1267 |
rule: ('type' | 'pred' | 'set') ':' (nameref +) | 'rule' ':' (thmref +)
|
|
1268 |
;
|
|
1269 |
definst: name ('==' | equiv) term | inst
|
|
1270 |
;
|
|
1271 |
definsts: ( definst *)
|
|
1272 |
;
|
|
1273 |
arbitrary: 'arbitrary' ':' ((term *) 'and' +)
|
|
1274 |
;
|
|
1275 |
taking: 'taking' ':' insts
|
|
1276 |
;
|
|
1277 |
\end{rail}
|
|
1278 |
|
28788
|
1279 |
\begin{description}
|
27042
|
1280 |
|
28788
|
1281 |
\item \hyperlink{method.cases}{\mbox{\isa{cases}}}~\isa{{\isachardoublequote}insts\ R{\isachardoublequote}} applies method \hyperlink{method.rule}{\mbox{\isa{rule}}} with an appropriate case distinction theorem, instantiated to
|
27042
|
1282 |
the subjects \isa{insts}. Symbolic case names are bound according
|
|
1283 |
to the rule's local contexts.
|
|
1284 |
|
|
1285 |
The rule is determined as follows, according to the facts and
|
|
1286 |
arguments passed to the \hyperlink{method.cases}{\mbox{\isa{cases}}} method:
|
|
1287 |
|
|
1288 |
\medskip
|
|
1289 |
\begin{tabular}{llll}
|
|
1290 |
facts & & arguments & rule \\\hline
|
|
1291 |
& \hyperlink{method.cases}{\mbox{\isa{cases}}} & & classical case split \\
|
|
1292 |
& \hyperlink{method.cases}{\mbox{\isa{cases}}} & \isa{t} & datatype exhaustion (type of \isa{t}) \\
|
|
1293 |
\isa{{\isachardoublequote}{\isasymturnstile}\ A\ t{\isachardoublequote}} & \hyperlink{method.cases}{\mbox{\isa{cases}}} & \isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}} & inductive predicate/set elimination (of \isa{A}) \\
|
|
1294 |
\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}} & \hyperlink{method.cases}{\mbox{\isa{cases}}} & \isa{{\isachardoublequote}{\isasymdots}\ rule{\isacharcolon}\ R{\isachardoublequote}} & explicit rule \isa{R} \\
|
|
1295 |
\end{tabular}
|
|
1296 |
\medskip
|
|
1297 |
|
|
1298 |
Several instantiations may be given, referring to the \emph{suffix}
|
|
1299 |
of premises of the case rule; within each premise, the \emph{prefix}
|
|
1300 |
of variables is instantiated. In most situations, only a single
|
|
1301 |
term needs to be specified; this refers to the first variable of the
|
|
1302 |
last premise (it is usually the same for all cases).
|
|
1303 |
|
28788
|
1304 |
\item \hyperlink{method.induct}{\mbox{\isa{induct}}}~\isa{{\isachardoublequote}insts\ R{\isachardoublequote}} is analogous to the
|
27042
|
1305 |
\hyperlink{method.cases}{\mbox{\isa{cases}}} method, but refers to induction rules, which are
|
|
1306 |
determined as follows:
|
|
1307 |
|
|
1308 |
\medskip
|
|
1309 |
\begin{tabular}{llll}
|
|
1310 |
facts & & arguments & rule \\\hline
|
|
1311 |
& \hyperlink{method.induct}{\mbox{\isa{induct}}} & \isa{{\isachardoublequote}P\ x{\isachardoublequote}} & datatype induction (type of \isa{x}) \\
|
|
1312 |
\isa{{\isachardoublequote}{\isasymturnstile}\ A\ x{\isachardoublequote}} & \hyperlink{method.induct}{\mbox{\isa{induct}}} & \isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}} & predicate/set induction (of \isa{A}) \\
|
|
1313 |
\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}} & \hyperlink{method.induct}{\mbox{\isa{induct}}} & \isa{{\isachardoublequote}{\isasymdots}\ rule{\isacharcolon}\ R{\isachardoublequote}} & explicit rule \isa{R} \\
|
|
1314 |
\end{tabular}
|
|
1315 |
\medskip
|
|
1316 |
|
|
1317 |
Several instantiations may be given, each referring to some part of
|
|
1318 |
a mutual inductive definition or datatype --- only related partial
|
|
1319 |
induction rules may be used together, though. Any of the lists of
|
|
1320 |
terms \isa{{\isachardoublequote}P{\isacharcomma}\ x{\isacharcomma}\ {\isasymdots}{\isachardoublequote}} refers to the \emph{suffix} of variables
|
|
1321 |
present in the induction rule. This enables the writer to specify
|
|
1322 |
only induction variables, or both predicates and variables, for
|
|
1323 |
example.
|
|
1324 |
|
|
1325 |
Instantiations may be definitional: equations \isa{{\isachardoublequote}x\ {\isasymequiv}\ t{\isachardoublequote}}
|
|
1326 |
introduce local definitions, which are inserted into the claim and
|
|
1327 |
discharged after applying the induction rule. Equalities reappear
|
|
1328 |
in the inductive cases, but have been transformed according to the
|
|
1329 |
induction principle being involved here. In order to achieve
|
|
1330 |
practically useful induction hypotheses, some variables occurring in
|
|
1331 |
\isa{t} need to be fixed (see below).
|
|
1332 |
|
|
1333 |
The optional ``\isa{{\isachardoublequote}arbitrary{\isacharcolon}\ x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardoublequote}}''
|
|
1334 |
specification generalizes variables \isa{{\isachardoublequote}x\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ x\isactrlsub m{\isachardoublequote}} of the original goal before applying induction. Thus
|
|
1335 |
induction hypotheses may become sufficiently general to get the
|
|
1336 |
proof through. Together with definitional instantiations, one may
|
|
1337 |
effectively perform induction over expressions of a certain
|
|
1338 |
structure.
|
|
1339 |
|
|
1340 |
The optional ``\isa{{\isachardoublequote}taking{\isacharcolon}\ t\isactrlsub {\isadigit{1}}\ {\isasymdots}\ t\isactrlsub n{\isachardoublequote}}''
|
|
1341 |
specification provides additional instantiations of a prefix of
|
|
1342 |
pending variables in the rule. Such schematic induction rules
|
|
1343 |
rarely occur in practice, though.
|
|
1344 |
|
28788
|
1345 |
\item \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}}~\isa{{\isachardoublequote}inst\ R{\isachardoublequote}} is analogous to the
|
27042
|
1346 |
\hyperlink{method.induct}{\mbox{\isa{induct}}} method, but refers to coinduction rules, which are
|
|
1347 |
determined as follows:
|
|
1348 |
|
|
1349 |
\medskip
|
|
1350 |
\begin{tabular}{llll}
|
|
1351 |
goal & & arguments & rule \\\hline
|
|
1352 |
& \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} & \isa{x} & type coinduction (type of \isa{x}) \\
|
|
1353 |
\isa{{\isachardoublequote}A\ x{\isachardoublequote}} & \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} & \isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}} & predicate/set coinduction (of \isa{A}) \\
|
|
1354 |
\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}} & \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} & \isa{{\isachardoublequote}{\isasymdots}\ rule{\isacharcolon}\ R{\isachardoublequote}} & explicit rule \isa{R} \\
|
|
1355 |
\end{tabular}
|
|
1356 |
|
|
1357 |
Coinduction is the dual of induction. Induction essentially
|
|
1358 |
eliminates \isa{{\isachardoublequote}A\ x{\isachardoublequote}} towards a generic result \isa{{\isachardoublequote}P\ x{\isachardoublequote}},
|
|
1359 |
while coinduction introduces \isa{{\isachardoublequote}A\ x{\isachardoublequote}} starting with \isa{{\isachardoublequote}B\ x{\isachardoublequote}}, for a suitable ``bisimulation'' \isa{B}. The cases of a
|
|
1360 |
coinduct rule are typically named after the predicates or sets being
|
|
1361 |
covered, while the conclusions consist of several alternatives being
|
|
1362 |
named after the individual destructor patterns.
|
|
1363 |
|
|
1364 |
The given instantiation refers to the \emph{suffix} of variables
|
|
1365 |
occurring in the rule's major premise, or conclusion if unavailable.
|
|
1366 |
An additional ``\isa{{\isachardoublequote}taking{\isacharcolon}\ t\isactrlsub {\isadigit{1}}\ {\isasymdots}\ t\isactrlsub n{\isachardoublequote}}''
|
|
1367 |
specification may be required in order to specify the bisimulation
|
|
1368 |
to be used in the coinduction step.
|
|
1369 |
|
28788
|
1370 |
\end{description}
|
27042
|
1371 |
|
|
1372 |
Above methods produce named local contexts, as determined by the
|
|
1373 |
instantiated rule as given in the text. Beyond that, the \hyperlink{method.induct}{\mbox{\isa{induct}}} and \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} methods guess further instantiations
|
|
1374 |
from the goal specification itself. Any persisting unresolved
|
|
1375 |
schematic variables of the resulting rule will render the the
|
|
1376 |
corresponding case invalid. The term binding \hyperlink{variable.?case}{\mbox{\isa{{\isacharquery}case}}} for
|
|
1377 |
the conclusion will be provided with each case, provided that term
|
|
1378 |
is fully specified.
|
|
1379 |
|
|
1380 |
The \hyperlink{command.print-cases}{\mbox{\isa{\isacommand{print{\isacharunderscore}cases}}}} command prints all named cases present
|
|
1381 |
in the current proof state.
|
|
1382 |
|
|
1383 |
\medskip Despite the additional infrastructure, both \hyperlink{method.cases}{\mbox{\isa{cases}}}
|
|
1384 |
and \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} merely apply a certain rule, after
|
|
1385 |
instantiation, while conforming due to the usual way of monotonic
|
|
1386 |
natural deduction: the context of a structured statement \isa{{\isachardoublequote}{\isasymAnd}x\isactrlsub {\isadigit{1}}\ {\isasymdots}\ x\isactrlsub m{\isachardot}\ {\isasymphi}\isactrlsub {\isadigit{1}}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymphi}\isactrlsub n\ {\isasymLongrightarrow}\ {\isasymdots}{\isachardoublequote}}
|
|
1387 |
reappears unchanged after the case split.
|
|
1388 |
|
|
1389 |
The \hyperlink{method.induct}{\mbox{\isa{induct}}} method is fundamentally different in this
|
|
1390 |
respect: the meta-level structure is passed through the
|
|
1391 |
``recursive'' course involved in the induction. Thus the original
|
|
1392 |
statement is basically replaced by separate copies, corresponding to
|
|
1393 |
the induction hypotheses and conclusion; the original goal context
|
|
1394 |
is no longer available. Thus local assumptions, fixed parameters
|
|
1395 |
and definitions effectively participate in the inductive rephrasing
|
|
1396 |
of the original statement.
|
|
1397 |
|
|
1398 |
In induction proofs, local assumptions introduced by cases are split
|
|
1399 |
into two different kinds: \isa{hyps} stemming from the rule and
|
|
1400 |
\isa{prems} from the goal statement. This is reflected in the
|
|
1401 |
extracted cases accordingly, so invoking ``\hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}~\isa{c}'' will provide separate facts \isa{c{\isachardot}hyps} and \isa{c{\isachardot}prems},
|
|
1402 |
as well as fact \isa{c} to hold the all-inclusive list.
|
|
1403 |
|
|
1404 |
\medskip Facts presented to either method are consumed according to
|
|
1405 |
the number of ``major premises'' of the rule involved, which is
|
|
1406 |
usually 0 for plain cases and induction rules of datatypes etc.\ and
|
|
1407 |
1 for rules of inductive predicates or sets and the like. The
|
|
1408 |
remaining facts are inserted into the goal verbatim before the
|
|
1409 |
actual \isa{cases}, \isa{induct}, or \isa{coinduct} rule is
|
|
1410 |
applied.%
|
|
1411 |
\end{isamarkuptext}%
|
|
1412 |
\isamarkuptrue%
|
|
1413 |
%
|
|
1414 |
\isamarkupsubsection{Declaring rules%
|
|
1415 |
}
|
|
1416 |
\isamarkuptrue%
|
|
1417 |
%
|
|
1418 |
\begin{isamarkuptext}%
|
|
1419 |
\begin{matharray}{rcl}
|
28788
|
1420 |
\indexdef{}{command}{print\_induct\_rules}\hypertarget{command.print-induct-rules}{\hyperlink{command.print-induct-rules}{\mbox{\isa{\isacommand{print{\isacharunderscore}induct{\isacharunderscore}rules}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
|
|
1421 |
\indexdef{}{attribute}{cases}\hypertarget{attribute.cases}{\hyperlink{attribute.cases}{\mbox{\isa{cases}}}} & : & \isa{attribute} \\
|
|
1422 |
\indexdef{}{attribute}{induct}\hypertarget{attribute.induct}{\hyperlink{attribute.induct}{\mbox{\isa{induct}}}} & : & \isa{attribute} \\
|
|
1423 |
\indexdef{}{attribute}{coinduct}\hypertarget{attribute.coinduct}{\hyperlink{attribute.coinduct}{\mbox{\isa{coinduct}}}} & : & \isa{attribute} \\
|
27042
|
1424 |
\end{matharray}
|
|
1425 |
|
|
1426 |
\begin{rail}
|
|
1427 |
'cases' spec
|
|
1428 |
;
|
|
1429 |
'induct' spec
|
|
1430 |
;
|
|
1431 |
'coinduct' spec
|
|
1432 |
;
|
|
1433 |
|
27142
|
1434 |
spec: (('type' | 'pred' | 'set') ':' nameref) | 'del'
|
27042
|
1435 |
;
|
|
1436 |
\end{rail}
|
|
1437 |
|
28788
|
1438 |
\begin{description}
|
27042
|
1439 |
|
28788
|
1440 |
\item \hyperlink{command.print-induct-rules}{\mbox{\isa{\isacommand{print{\isacharunderscore}induct{\isacharunderscore}rules}}}} prints cases and induct rules
|
|
1441 |
for predicates (or sets) and types of the current context.
|
27042
|
1442 |
|
28788
|
1443 |
\item \hyperlink{attribute.cases}{\mbox{\isa{cases}}}, \hyperlink{attribute.induct}{\mbox{\isa{induct}}}, and \hyperlink{attribute.coinduct}{\mbox{\isa{coinduct}}} (as attributes) declare rules for reasoning about
|
27142
|
1444 |
(co)inductive predicates (or sets) and types, using the
|
|
1445 |
corresponding methods of the same name. Certain definitional
|
|
1446 |
packages of object-logics usually declare emerging cases and
|
|
1447 |
induction rules as expected, so users rarely need to intervene.
|
|
1448 |
|
|
1449 |
Rules may be deleted via the \isa{{\isachardoublequote}del{\isachardoublequote}} specification, which
|
|
1450 |
covers all of the \isa{{\isachardoublequote}type{\isachardoublequote}}/\isa{{\isachardoublequote}pred{\isachardoublequote}}/\isa{{\isachardoublequote}set{\isachardoublequote}}
|
|
1451 |
sub-categories simultaneously. For example, \hyperlink{attribute.cases}{\mbox{\isa{cases}}}~\isa{del} removes any \hyperlink{attribute.cases}{\mbox{\isa{cases}}} rules declared for
|
|
1452 |
some type, predicate, or set.
|
27042
|
1453 |
|
|
1454 |
Manual rule declarations usually refer to the \hyperlink{attribute.case-names}{\mbox{\isa{case{\isacharunderscore}names}}} and \hyperlink{attribute.params}{\mbox{\isa{params}}} attributes to adjust names of
|
|
1455 |
cases and parameters of a rule; the \hyperlink{attribute.consumes}{\mbox{\isa{consumes}}}
|
|
1456 |
declaration is taken care of automatically: \hyperlink{attribute.consumes}{\mbox{\isa{consumes}}}~\isa{{\isadigit{0}}} is specified for ``type'' rules and \hyperlink{attribute.consumes}{\mbox{\isa{consumes}}}~\isa{{\isadigit{1}}} for ``predicate'' / ``set'' rules.
|
|
1457 |
|
28788
|
1458 |
\end{description}%
|
27042
|
1459 |
\end{isamarkuptext}%
|
|
1460 |
\isamarkuptrue%
|
|
1461 |
%
|
26869
|
1462 |
\isadelimtheory
|
|
1463 |
%
|
|
1464 |
\endisadelimtheory
|
|
1465 |
%
|
|
1466 |
\isatagtheory
|
|
1467 |
\isacommand{end}\isamarkupfalse%
|
|
1468 |
%
|
|
1469 |
\endisatagtheory
|
|
1470 |
{\isafoldtheory}%
|
|
1471 |
%
|
|
1472 |
\isadelimtheory
|
|
1473 |
%
|
|
1474 |
\endisadelimtheory
|
|
1475 |
\isanewline
|
|
1476 |
\end{isabellebody}%
|
|
1477 |
%%% Local Variables:
|
|
1478 |
%%% mode: latex
|
|
1479 |
%%% TeX-master: "root"
|
|
1480 |
%%% End:
|