author | nipkow |
Tue, 20 Sep 2011 05:47:11 +0200 | |
changeset 45014 | 0e847655b2d8 |
parent 44164 | 238c5ea1e2ce |
child 45103 | a45121ffcfcb |
permissions | -rw-r--r-- |
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\begin{isabellebody}% |
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\def\isabellecontext{Proof}% |
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\isadelimtheory |
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\endisadelimtheory |
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\isatagtheory |
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\isacommand{theory}\isamarkupfalse% |
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\ Proof\isanewline |
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\isakeyword{imports}\ Base\ Main\isanewline |
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\isakeyword{begin}% |
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\endisatagtheory |
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{\isafoldtheory}% |
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\isadelimtheory |
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\endisadelimtheory |
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\isamarkupchapter{Proofs \label{ch:proofs}% |
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} |
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\isamarkuptrue% |
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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 typed according to |
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the following three different modes of operation: |
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\begin{description} |
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\item \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\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{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\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{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} is intermediate between \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\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 understood as an instruction to the |
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writer, telling what kind of operation may be performed next. The |
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corresponding typings of proof commands restricts the shape of |
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well-formed proof texts to particular command sequences. So dynamic |
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arrangements of commands eventually turn out as static texts of a |
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certain structure. |
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\Appref{ap:refcard} gives a simplified grammar of the (extensible) |
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language emerging that way from the different types of proof |
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commands. The main ideas of the overall Isar framework are |
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explained in \chref{ch:isar-framework}.% |
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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{Formal notepad% |
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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}{notepad}\hypertarget{command.notepad}{\hyperlink{command.notepad}{\mbox{\isa{\isacommand{notepad}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
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\end{matharray} |
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\begin{railoutput} |
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\rail@begin{1}{} |
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\rail@term{\hyperlink{command.notepad}{\mbox{\isa{\isacommand{notepad}}}}}[] |
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\rail@term{\isa{\isakeyword{begin}}}[] |
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\rail@end |
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\rail@begin{1}{} |
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\rail@term{\hyperlink{command.end}{\mbox{\isa{\isacommand{end}}}}}[] |
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\rail@end |
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\end{railoutput} |
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\begin{description} |
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\item \hyperlink{command.notepad}{\mbox{\isa{\isacommand{notepad}}}}~\hyperlink{keyword.begin}{\mbox{\isa{\isakeyword{begin}}}} opens a proof state |
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without any goal statement. This allows to experiment with Isar, |
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without producing any persistent result. |
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The notepad can be closed by \hyperlink{command.end}{\mbox{\isa{\isacommand{end}}}} or discontinued by |
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\hyperlink{command.oops}{\mbox{\isa{\isacommand{oops}}}}. |
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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{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{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
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\indexdef{}{command}{\{}\hypertarget{command.braceleft}{\hyperlink{command.braceleft}{\mbox{\isa{\isacommand{{\isaliteral{7B}{\isacharbraceleft}}}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
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\indexdef{}{command}{\}}\hypertarget{command.braceright}{\hyperlink{command.braceright}{\mbox{\isa{\isacommand{{\isaliteral{7D}{\isacharbraceright}}}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\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{{\isaliteral{7B}{\isacharbraceleft}}}}}} and \hyperlink{command.braceright}{\mbox{\isa{\isacommand{{\isaliteral{7D}{\isacharbraceright}}}}}} explicitly open and close |
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blocks. Any current facts pass through ``\hyperlink{command.braceleft}{\mbox{\isa{\isacommand{{\isaliteral{7B}{\isacharbraceleft}}}}}}'' |
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unchanged, while ``\hyperlink{command.braceright}{\mbox{\isa{\isacommand{{\isaliteral{7D}{\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{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{7C}{\isacharbar}}\ theory{\isaliteral{22}{\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 \chref{ch:document-prep}. |
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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{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\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{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
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\indexdef{}{command}{assume}\hypertarget{command.assume}{\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
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\indexdef{}{command}{presume}\hypertarget{command.presume}{\hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
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\indexdef{}{command}{def}\hypertarget{command.def}{\hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\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{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\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{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\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{{\isaliteral{5C3C6368693E}{\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{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6368693E}{\isasymchi}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} exported from the context becomes conditional wrt.\ |
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the assumption: \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C6368693E}{\isasymchi}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\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. |
216 |
||
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Local definitions, introduced by ``\hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\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{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\isachardoublequote}}} to be eliminated by the reflexivity rule. Thus, |
220 |
exporting some result \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{5B}{\isacharbrackleft}}x{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}} yields \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{5B}{\isacharbrackleft}}t{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}. |
|
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|
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\begin{railoutput} |
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\rail@begin{2}{} |
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\rail@term{\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}}[] |
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\rail@plus |
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\rail@nont{\hyperlink{syntax.vars}{\mbox{\isa{vars}}}}[] |
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\rail@nextplus{1} |
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\rail@cterm{\isa{\isakeyword{and}}}[] |
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\rail@endplus |
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\rail@end |
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\rail@begin{2}{} |
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\rail@bar |
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\rail@term{\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}}[] |
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\rail@nextbar{1} |
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\rail@term{\hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}}}[] |
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\rail@endbar |
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\rail@plus |
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\rail@nont{\hyperlink{syntax.props}{\mbox{\isa{props}}}}[] |
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\rail@nextplus{1} |
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\rail@cterm{\isa{\isakeyword{and}}}[] |
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\rail@endplus |
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\rail@end |
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\rail@begin{2}{} |
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\rail@term{\hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}}}[] |
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\rail@plus |
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\rail@nont{\isa{def}}[] |
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\rail@nextplus{1} |
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\rail@cterm{\isa{\isakeyword{and}}}[] |
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\rail@endplus |
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\rail@end |
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\rail@begin{5}{\isa{def}} |
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\rail@bar |
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\rail@nextbar{1} |
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\rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[] |
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\rail@endbar |
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\rail@cr{3} |
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\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[] |
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\rail@bar |
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\rail@term{\isa{{\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}}}[] |
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\rail@nextbar{4} |
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\rail@term{\isa{{\isaliteral{5C3C65717569763E}{\isasymequiv}}}}[] |
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\rail@endbar |
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\rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[] |
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\rail@bar |
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\rail@nextbar{4} |
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\rail@nont{\hyperlink{syntax.term-pat}{\mbox{\isa{term{\isaliteral{5F}{\isacharunderscore}}pat}}}}[] |
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\rail@endbar |
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\rail@end |
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\end{railoutput} |
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|
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|
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\begin{description} |
26870 | 273 |
|
28788 | 274 |
\item \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{x} introduces a local variable \isa{x} that is \emph{arbitrary, but fixed.} |
26870 | 275 |
|
40406 | 276 |
\item \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} and \hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} introduce a local fact \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7068693E}{\isasymphi}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} by |
26870 | 277 |
assumption. Subsequent results applied to an enclosing goal (e.g.\ |
26902 | 278 |
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 |
40406 | 279 |
goal, while \hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}} leaves \isa{{\isaliteral{5C3C7068693E}{\isasymphi}}} as new subgoals. |
26870 | 280 |
|
281 |
Several lists of assumptions may be given (separated by |
|
26902 | 282 |
\indexref{}{keyword}{and}\hyperlink{keyword.and}{\mbox{\isa{\isakeyword{and}}}}; the resulting list of current facts consists |
26870 | 283 |
of all of these concatenated. |
284 |
||
40406 | 285 |
\item \hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\isachardoublequote}}} introduces a local |
26870 | 286 |
(non-polymorphic) definition. In results exported from the context, |
40406 | 287 |
\isa{x} is replaced by \isa{t}. Basically, ``\hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\isachardoublequote}}}'' abbreviates ``\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{x}~\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\isachardoublequote}}}'', with the resulting |
26870 | 288 |
hypothetical equation solved by reflexivity. |
289 |
||
40406 | 290 |
The default name for the definitional equation is \isa{x{\isaliteral{5F}{\isacharunderscore}}def}. |
26870 | 291 |
Several simultaneous definitions may be given at the same time. |
292 |
||
28788 | 293 |
\end{description} |
26870 | 294 |
|
26902 | 295 |
The special name \indexref{}{fact}{prems}\hyperlink{fact.prems}{\mbox{\isa{prems}}} refers to all assumptions of the |
26870 | 296 |
current context as a list of theorems. This feature should be used |
297 |
with great care! It is better avoided in final proof texts.% |
|
298 |
\end{isamarkuptext}% |
|
299 |
\isamarkuptrue% |
|
300 |
% |
|
28788 | 301 |
\isamarkupsubsection{Term abbreviations \label{sec:term-abbrev}% |
26870 | 302 |
} |
303 |
\isamarkuptrue% |
|
304 |
% |
|
305 |
\begin{isamarkuptext}% |
|
306 |
\begin{matharray}{rcl} |
|
40406 | 307 |
\indexdef{}{command}{let}\hypertarget{command.let}{\hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
28788 | 308 |
\indexdef{}{keyword}{is}\hypertarget{keyword.is}{\hyperlink{keyword.is}{\mbox{\isa{\isakeyword{is}}}}} & : & syntax \\ |
309 |
\end{matharray} |
|
310 |
||
40406 | 311 |
Abbreviations may be either bound by explicit \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}p\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\isachardoublequote}}} statements, or by annotating assumptions or |
312 |
goal statements with a list of patterns ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C49533E}{\isasymIS}}\ p\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ p\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}''. In both cases, higher-order matching is invoked to |
|
28788 | 313 |
bind extra-logical term variables, which may be either named |
40406 | 314 |
schematic variables of the form \isa{{\isaliteral{3F}{\isacharquery}}x}, or nameless dummies |
315 |
``\hyperlink{variable.underscore}{\mbox{\isa{{\isaliteral{5F}{\isacharunderscore}}}}}'' (underscore). Note that in the \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}} |
|
28788 | 316 |
form the patterns occur on the left-hand side, while the \hyperlink{keyword.is}{\mbox{\isa{\isakeyword{is}}}} patterns are in postfix position. |
317 |
||
318 |
Polymorphism of term bindings is handled in Hindley-Milner style, |
|
319 |
similar to ML. Type variables referring to local assumptions or |
|
320 |
open goal statements are \emph{fixed}, while those of finished |
|
321 |
results or bound by \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}} may occur in \emph{arbitrary} |
|
322 |
instances later. Even though actual polymorphism should be rarely |
|
323 |
used in practice, this mechanism is essential to achieve proper |
|
324 |
incremental type-inference, as the user proceeds to build up the |
|
325 |
Isar proof text from left to right. |
|
326 |
||
327 |
\medskip Term abbreviations are quite different from local |
|
328 |
definitions as introduced via \hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}} (see |
|
329 |
\secref{sec:proof-context}). The latter are visible within the |
|
330 |
logic as actual equations, while abbreviations disappear during the |
|
331 |
input process just after type checking. Also note that \hyperlink{command.def}{\mbox{\isa{\isacommand{def}}}} does not support polymorphism. |
|
332 |
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\begin{railoutput} |
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\rail@begin{3}{} |
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\rail@term{\hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}}[] |
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\rail@plus |
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\rail@plus |
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\rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[] |
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\rail@nextplus{1} |
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\rail@cterm{\isa{\isakeyword{and}}}[] |
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\rail@endplus |
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\rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[] |
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\rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[] |
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\rail@nextplus{2} |
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\rail@cterm{\isa{\isakeyword{and}}}[] |
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\rail@endplus |
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\rail@end |
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\end{railoutput} |
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|
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The syntax of \hyperlink{keyword.is}{\mbox{\isa{\isakeyword{is}}}} patterns follows \hyperlink{syntax.term-pat}{\mbox{\isa{term{\isaliteral{5F}{\isacharunderscore}}pat}}} or |
352 |
\hyperlink{syntax.prop-pat}{\mbox{\isa{prop{\isaliteral{5F}{\isacharunderscore}}pat}}} (see \secref{sec:term-decls}). |
|
28788 | 353 |
|
354 |
\begin{description} |
|
355 |
||
40406 | 356 |
\item \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}p\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ p\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3D}{\isacharequal}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} binds any |
357 |
text variables in patterns \isa{{\isaliteral{22}{\isachardoublequote}}p\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ p\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} by simultaneous |
|
358 |
higher-order matching against terms \isa{{\isaliteral{22}{\isachardoublequote}}t\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}. |
|
28788 | 359 |
|
40406 | 360 |
\item \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C49533E}{\isasymIS}}\ p\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ p\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} resembles \hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}}, but |
361 |
matches \isa{{\isaliteral{22}{\isachardoublequote}}p\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ p\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} against the preceding statement. Also |
|
28788 | 362 |
note that \hyperlink{keyword.is}{\mbox{\isa{\isakeyword{is}}}} is not a separate command, but part of |
363 |
others (such as \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}, \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}} etc.). |
|
364 |
||
365 |
\end{description} |
|
366 |
||
367 |
Some \emph{implicit} term abbreviations\index{term abbreviations} |
|
368 |
for goals and facts are available as well. For any open goal, |
|
369 |
\indexref{}{variable}{thesis}\hyperlink{variable.thesis}{\mbox{\isa{thesis}}} refers to its object-level statement, |
|
370 |
abstracted over any meta-level parameters (if present). Likewise, |
|
371 |
\indexref{}{variable}{this}\hyperlink{variable.this}{\mbox{\isa{this}}} is bound for fact statements resulting from |
|
372 |
assumptions or finished goals. In case \hyperlink{variable.this}{\mbox{\isa{this}}} refers to |
|
40406 | 373 |
an object-logic statement that is an application \isa{{\isaliteral{22}{\isachardoublequote}}f\ t{\isaliteral{22}{\isachardoublequote}}}, then |
374 |
\isa{t} is bound to the special text variable ``\hyperlink{variable.dots}{\mbox{\isa{{\isaliteral{5C3C646F74733E}{\isasymdots}}}}}'' |
|
28788 | 375 |
(three dots). The canonical application of this convenience are |
376 |
calculational proofs (see \secref{sec:calculation}).% |
|
377 |
\end{isamarkuptext}% |
|
378 |
\isamarkuptrue% |
|
379 |
% |
|
380 |
\isamarkupsubsection{Facts and forward chaining% |
|
381 |
} |
|
382 |
\isamarkuptrue% |
|
383 |
% |
|
384 |
\begin{isamarkuptext}% |
|
385 |
\begin{matharray}{rcl} |
|
40406 | 386 |
\indexdef{}{command}{note}\hypertarget{command.note}{\hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
387 |
\indexdef{}{command}{then}\hypertarget{command.then}{\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
388 |
\indexdef{}{command}{from}\hypertarget{command.from}{\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
389 |
\indexdef{}{command}{with}\hypertarget{command.with}{\hyperlink{command.with}{\mbox{\isa{\isacommand{with}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
390 |
\indexdef{}{command}{using}\hypertarget{command.using}{\hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
391 |
\indexdef{}{command}{unfolding}\hypertarget{command.unfolding}{\hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
26870 | 392 |
\end{matharray} |
393 |
||
394 |
New facts are established either by assumption or proof of local |
|
395 |
statements. Any fact will usually be involved in further proofs, |
|
396 |
either as explicit arguments of proof methods, or when forward |
|
26902 | 397 |
chaining towards the next goal via \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} (and variants); |
398 |
\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}} and \hyperlink{command.with}{\mbox{\isa{\isacommand{with}}}} are composite forms |
|
399 |
involving \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}. The \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}} elements |
|
26870 | 400 |
augments the collection of used facts \emph{after} a goal has been |
26902 | 401 |
stated. Note that the special theorem name \indexref{}{fact}{this}\hyperlink{fact.this}{\mbox{\isa{this}}} refers |
26870 | 402 |
to the most recently established facts, but only \emph{before} |
403 |
issuing a follow-up claim. |
|
404 |
||
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\begin{railoutput} |
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\rail@begin{3}{} |
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\rail@term{\hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}}[] |
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\rail@plus |
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\rail@bar |
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\rail@nextbar{1} |
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\rail@nont{\hyperlink{syntax.thmdef}{\mbox{\isa{thmdef}}}}[] |
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\rail@endbar |
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\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[] |
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\rail@nextplus{2} |
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\rail@cterm{\isa{\isakeyword{and}}}[] |
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\rail@endplus |
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\rail@end |
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\rail@begin{4}{} |
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\rail@bar |
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\rail@term{\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}}[] |
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\rail@nextbar{1} |
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\rail@term{\hyperlink{command.with}{\mbox{\isa{\isacommand{with}}}}}[] |
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\rail@nextbar{2} |
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\rail@term{\hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}}[] |
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\rail@nextbar{3} |
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\rail@term{\hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}}}[] |
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\rail@endbar |
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\rail@plus |
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\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[] |
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\rail@nextplus{1} |
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\rail@cterm{\isa{\isakeyword{and}}}[] |
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\rail@endplus |
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\rail@end |
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\end{railoutput} |
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|
26870 | 436 |
|
28788 | 437 |
\begin{description} |
26870 | 438 |
|
40406 | 439 |
\item \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{3D}{\isacharequal}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} recalls existing facts |
440 |
\isa{{\isaliteral{22}{\isachardoublequote}}b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}, binding the result as \isa{a}. Note that |
|
28788 | 441 |
attributes may be involved as well, both on the left and right hand |
442 |
sides. |
|
26870 | 443 |
|
28788 | 444 |
\item \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}} indicates forward chaining by the current |
26870 | 445 |
facts in order to establish the goal to be claimed next. The |
446 |
initial proof method invoked to refine that will be offered the |
|
447 |
facts to do ``anything appropriate'' (see also |
|
42626 | 448 |
\secref{sec:proof-steps}). For example, method \hyperlink{method.Pure.rule}{\mbox{\isa{rule}}} |
26870 | 449 |
(see \secref{sec:pure-meth-att}) would typically do an elimination |
450 |
rather than an introduction. Automatic methods usually insert the |
|
451 |
facts into the goal state before operation. This provides a simple |
|
452 |
scheme to control relevance of facts in automated proof search. |
|
453 |
||
28788 | 454 |
\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 |
26902 | 455 |
equivalent to ``\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{this}''. |
26870 | 456 |
|
40406 | 457 |
\item \hyperlink{command.with}{\mbox{\isa{\isacommand{with}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} abbreviates ``\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ this{\isaliteral{22}{\isachardoublequote}}}''; thus the forward chaining |
28788 | 458 |
is from earlier facts together with the current ones. |
26870 | 459 |
|
40406 | 460 |
\item \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} augments the facts being |
28788 | 461 |
currently indicated for use by a subsequent refinement step (such as |
462 |
\indexref{}{command}{apply}\hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}} or \indexref{}{command}{proof}\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}). |
|
26870 | 463 |
|
40406 | 464 |
\item \hyperlink{command.unfolding}{\mbox{\isa{\isacommand{unfolding}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} is structurally |
28788 | 465 |
similar to \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}, but unfolds definitional equations |
40406 | 466 |
\isa{{\isaliteral{22}{\isachardoublequote}}b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} throughout the goal state and facts. |
26870 | 467 |
|
28788 | 468 |
\end{description} |
26870 | 469 |
|
470 |
Forward chaining with an empty list of theorems is the same as not |
|
26902 | 471 |
chaining at all. Thus ``\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{nothing}'' has no |
40406 | 472 |
effect apart from entering \isa{{\isaliteral{22}{\isachardoublequote}}prove{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} mode, since |
26902 | 473 |
\indexref{}{fact}{nothing}\hyperlink{fact.nothing}{\mbox{\isa{nothing}}} is bound to the empty list of theorems. |
26870 | 474 |
|
42626 | 475 |
Basic proof methods (such as \indexref{Pure}{method}{rule}\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}) expect multiple |
26870 | 476 |
facts to be given in their proper order, corresponding to a prefix |
477 |
of the premises of the rule involved. Note that positions may be |
|
40406 | 478 |
easily skipped using something like \hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ a\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ b{\isaliteral{22}{\isachardoublequote}}}, for example. This involves the trivial rule |
479 |
\isa{{\isaliteral{22}{\isachardoublequote}}PROP\ {\isaliteral{5C3C7073693E}{\isasympsi}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ PROP\ {\isaliteral{5C3C7073693E}{\isasympsi}}{\isaliteral{22}{\isachardoublequote}}}, which is bound in Isabelle/Pure as |
|
480 |
``\indexref{}{fact}{\_}\hyperlink{fact.underscore}{\mbox{\isa{{\isaliteral{5F}{\isacharunderscore}}}}}'' (underscore). |
|
26870 | 481 |
|
26902 | 482 |
Automated methods (such as \hyperlink{method.simp}{\mbox{\isa{simp}}} or \hyperlink{method.auto}{\mbox{\isa{auto}}}) just |
26870 | 483 |
insert any given facts before their usual operation. Depending on |
484 |
the kind of procedure involved, the order of facts is less |
|
485 |
significant here.% |
|
486 |
\end{isamarkuptext}% |
|
487 |
\isamarkuptrue% |
|
488 |
% |
|
28788 | 489 |
\isamarkupsubsection{Goals \label{sec:goals}% |
26870 | 490 |
} |
491 |
\isamarkuptrue% |
|
492 |
% |
|
493 |
\begin{isamarkuptext}% |
|
494 |
\begin{matharray}{rcl} |
|
40406 | 495 |
\indexdef{}{command}{lemma}\hypertarget{command.lemma}{\hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
496 |
\indexdef{}{command}{theorem}\hypertarget{command.theorem}{\hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
497 |
\indexdef{}{command}{corollary}\hypertarget{command.corollary}{\hyperlink{command.corollary}{\mbox{\isa{\isacommand{corollary}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
498 |
\indexdef{}{command}{schematic\_lemma}\hypertarget{command.schematic-lemma}{\hyperlink{command.schematic-lemma}{\mbox{\isa{\isacommand{schematic{\isaliteral{5F}{\isacharunderscore}}lemma}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
499 |
\indexdef{}{command}{schematic\_theorem}\hypertarget{command.schematic-theorem}{\hyperlink{command.schematic-theorem}{\mbox{\isa{\isacommand{schematic{\isaliteral{5F}{\isacharunderscore}}theorem}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
500 |
\indexdef{}{command}{schematic\_corollary}\hypertarget{command.schematic-corollary}{\hyperlink{command.schematic-corollary}{\mbox{\isa{\isacommand{schematic{\isaliteral{5F}{\isacharunderscore}}corollary}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
501 |
\indexdef{}{command}{have}\hypertarget{command.have}{\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
502 |
\indexdef{}{command}{show}\hypertarget{command.show}{\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
503 |
\indexdef{}{command}{hence}\hypertarget{command.hence}{\hyperlink{command.hence}{\mbox{\isa{\isacommand{hence}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
504 |
\indexdef{}{command}{thus}\hypertarget{command.thus}{\hyperlink{command.thus}{\mbox{\isa{\isacommand{thus}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
505 |
\indexdef{}{command}{print\_statement}\hypertarget{command.print-statement}{\hyperlink{command.print-statement}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}statement}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
26870 | 506 |
\end{matharray} |
507 |
||
508 |
From a theory context, proof mode is entered by an initial goal |
|
26902 | 509 |
command such as \hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}}, \hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}, or |
510 |
\hyperlink{command.corollary}{\mbox{\isa{\isacommand{corollary}}}}. Within a proof, new claims may be |
|
26870 | 511 |
introduced locally as well; four variants are available here to |
512 |
indicate whether forward chaining of facts should be performed |
|
26902 | 513 |
initially (via \indexref{}{command}{then}\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}), and whether the final result |
26870 | 514 |
is meant to solve some pending goal. |
515 |
||
516 |
Goals may consist of multiple statements, resulting in a list of |
|
517 |
facts eventually. A pending multi-goal is internally represented as |
|
40406 | 518 |
a meta-level conjunction (\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{26}{\isacharampersand}}{\isaliteral{26}{\isacharampersand}}{\isaliteral{26}{\isacharampersand}}{\isaliteral{22}{\isachardoublequote}}}), which is usually |
26870 | 519 |
split into the corresponding number of sub-goals prior to an initial |
26902 | 520 |
method application, via \indexref{}{command}{proof}\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} |
521 |
(\secref{sec:proof-steps}) or \indexref{}{command}{apply}\hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}} |
|
522 |
(\secref{sec:tactic-commands}). The \indexref{}{method}{induct}\hyperlink{method.induct}{\mbox{\isa{induct}}} method |
|
26870 | 523 |
covered in \secref{sec:cases-induct} acts on multiple claims |
524 |
simultaneously. |
|
525 |
||
526 |
Claims at the theory level may be either in short or long form. A |
|
527 |
short goal merely consists of several simultaneous propositions |
|
528 |
(often just one). A long goal includes an explicit context |
|
529 |
specification for the subsequent conclusion, involving local |
|
530 |
parameters and assumptions. Here the role of each part of the |
|
531 |
statement is explicitly marked by separate keywords (see also |
|
532 |
\secref{sec:locale}); the local assumptions being introduced here |
|
26902 | 533 |
are available as \indexref{}{fact}{assms}\hyperlink{fact.assms}{\mbox{\isa{assms}}} in the proof. Moreover, there |
534 |
are two kinds of conclusions: \indexdef{}{element}{shows}\hypertarget{element.shows}{\hyperlink{element.shows}{\mbox{\isa{\isakeyword{shows}}}}} states several |
|
26870 | 535 |
simultaneous propositions (essentially a big conjunction), while |
26902 | 536 |
\indexdef{}{element}{obtains}\hypertarget{element.obtains}{\hyperlink{element.obtains}{\mbox{\isa{\isakeyword{obtains}}}}} claims several simultaneous simultaneous |
26870 | 537 |
contexts of (essentially a big disjunction of eliminated parameters |
538 |
and assumptions, cf.\ \secref{sec:obtain}). |
|
539 |
||
42596
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|
540 |
\begin{railoutput} |
42662 | 541 |
\rail@begin{6}{} |
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modernized rail diagrams using @{rail} antiquotation;
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542 |
\rail@bar |
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modernized rail diagrams using @{rail} antiquotation;
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543 |
\rail@term{\hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
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parents:
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changeset
|
544 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
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parents:
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changeset
|
545 |
\rail@term{\hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
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parents:
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|
546 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
547 |
\rail@term{\hyperlink{command.corollary}{\mbox{\isa{\isacommand{corollary}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
548 |
\rail@nextbar{3} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
549 |
\rail@term{\hyperlink{command.schematic-lemma}{\mbox{\isa{\isacommand{schematic{\isaliteral{5F}{\isacharunderscore}}lemma}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
550 |
\rail@nextbar{4} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
551 |
\rail@term{\hyperlink{command.schematic-theorem}{\mbox{\isa{\isacommand{schematic{\isaliteral{5F}{\isacharunderscore}}theorem}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
552 |
\rail@nextbar{5} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
553 |
\rail@term{\hyperlink{command.schematic-corollary}{\mbox{\isa{\isacommand{schematic{\isaliteral{5F}{\isacharunderscore}}corollary}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
554 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
555 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
556 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
557 |
\rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
558 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
559 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
560 |
\rail@nont{\isa{goal}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
561 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
562 |
\rail@nont{\isa{longgoal}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
563 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
564 |
\rail@end |
42662 | 565 |
\rail@begin{4}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
566 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
567 |
\rail@term{\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
568 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
569 |
\rail@term{\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
570 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
571 |
\rail@term{\hyperlink{command.hence}{\mbox{\isa{\isacommand{hence}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
572 |
\rail@nextbar{3} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
573 |
\rail@term{\hyperlink{command.thus}{\mbox{\isa{\isacommand{thus}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
574 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
575 |
\rail@nont{\isa{goal}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
576 |
\rail@end |
42662 | 577 |
\rail@begin{2}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
578 |
\rail@term{\hyperlink{command.print-statement}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}statement}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
579 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
580 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
581 |
\rail@nont{\hyperlink{syntax.modes}{\mbox{\isa{modes}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
582 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
583 |
\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
584 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
585 |
\rail@begin{2}{\isa{goal}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
586 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
587 |
\rail@nont{\hyperlink{syntax.props}{\mbox{\isa{props}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
588 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
589 |
\rail@cterm{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
590 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
591 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
592 |
\rail@begin{2}{\isa{longgoal}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
593 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
594 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
595 |
\rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
596 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
597 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
598 |
\rail@nextplus{1} |
42617 | 599 |
\rail@cnont{\hyperlink{syntax.context-elem}{\mbox{\isa{context{\isaliteral{5F}{\isacharunderscore}}elem}}}}[] |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
600 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
601 |
\rail@nont{\isa{conclusion}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
602 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
603 |
\rail@begin{4}{\isa{conclusion}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
604 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
605 |
\rail@term{\isa{\isakeyword{shows}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
606 |
\rail@nont{\isa{goal}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
607 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
608 |
\rail@term{\isa{\isakeyword{obtains}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
609 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
610 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
611 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
612 |
\rail@nont{\hyperlink{syntax.parname}{\mbox{\isa{parname}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
613 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
614 |
\rail@nont{\isa{case}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
615 |
\rail@nextplus{3} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
616 |
\rail@cterm{\isa{{\isaliteral{7C}{\isacharbar}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
617 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
618 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
619 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
620 |
\rail@begin{2}{\isa{case}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
621 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
622 |
\rail@nont{\hyperlink{syntax.vars}{\mbox{\isa{vars}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
623 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
624 |
\rail@cterm{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
625 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
626 |
\rail@term{\isa{\isakeyword{where}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
627 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
628 |
\rail@nont{\hyperlink{syntax.props}{\mbox{\isa{props}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
629 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
630 |
\rail@cterm{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
631 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
632 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
633 |
\end{railoutput} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
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parents:
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changeset
|
634 |
|
26870 | 635 |
|
28788 | 636 |
\begin{description} |
26870 | 637 |
|
40406 | 638 |
\item \hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} enters proof mode with |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
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|
639 |
\isa{{\isaliteral{5C3C7068693E}{\isasymphi}}} as main goal, eventually resulting in some fact \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} to be put back into the target context. An additional \hyperlink{syntax.context}{\mbox{\isa{context}}} specification may build up an initial proof context for the |
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640 |
subsequent claim; this includes local definitions and syntax as |
42617 | 641 |
well, see the definition of \hyperlink{syntax.context-elem}{\mbox{\isa{context{\isaliteral{5F}{\isacharunderscore}}elem}}} in |
26870 | 642 |
\secref{sec:locale}. |
643 |
||
40406 | 644 |
\item \hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} and \hyperlink{command.corollary}{\mbox{\isa{\isacommand{corollary}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} are essentially the same as \hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}}, but the facts are internally marked as |
26870 | 645 |
being of a different kind. This discrimination acts like a formal |
646 |
comment. |
|
36321 | 647 |
|
40406 | 648 |
\item \hyperlink{command.schematic-lemma}{\mbox{\isa{\isacommand{schematic{\isaliteral{5F}{\isacharunderscore}}lemma}}}}, \hyperlink{command.schematic-theorem}{\mbox{\isa{\isacommand{schematic{\isaliteral{5F}{\isacharunderscore}}theorem}}}}, |
649 |
\hyperlink{command.schematic-corollary}{\mbox{\isa{\isacommand{schematic{\isaliteral{5F}{\isacharunderscore}}corollary}}}} are similar to \hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}}, |
|
36321 | 650 |
\hyperlink{command.theorem}{\mbox{\isa{\isacommand{theorem}}}}, \hyperlink{command.corollary}{\mbox{\isa{\isacommand{corollary}}}}, respectively but allow |
651 |
the statement to contain unbound schematic variables. |
|
652 |
||
653 |
Under normal circumstances, an Isar proof text needs to specify |
|
654 |
claims explicitly. Schematic goals are more like goals in Prolog, |
|
655 |
where certain results are synthesized in the course of reasoning. |
|
656 |
With schematic statements, the inherent compositionality of Isar |
|
657 |
proofs is lost, which also impacts performance, because proof |
|
658 |
checking is forced into sequential mode. |
|
26870 | 659 |
|
40406 | 660 |
\item \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} claims a local goal, |
26870 | 661 |
eventually resulting in a fact within the current logical context. |
662 |
This operation is completely independent of any pending sub-goals of |
|
26902 | 663 |
an enclosing goal statements, so \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}} may be freely |
26870 | 664 |
used for experimental exploration of potential results within a |
665 |
proof body. |
|
666 |
||
40406 | 667 |
\item \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} is like \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} plus a second stage to refine some pending |
26870 | 668 |
sub-goal for each one of the finished result, after having been |
669 |
exported into the corresponding context (at the head of the |
|
26902 | 670 |
sub-proof of this \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} command). |
26870 | 671 |
|
672 |
To accommodate interactive debugging, resulting rules are printed |
|
673 |
before being applied internally. Even more, interactive execution |
|
26902 | 674 |
of \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} predicts potential failure and displays the |
26870 | 675 |
resulting error as a warning beforehand. Watch out for the |
676 |
following message: |
|
677 |
||
678 |
%FIXME proper antiquitation |
|
679 |
\begin{ttbox} |
|
680 |
Problem! Local statement will fail to solve any pending goal |
|
681 |
\end{ttbox} |
|
682 |
||
28788 | 683 |
\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 | 684 |
chaining the current facts. Note that \hyperlink{command.hence}{\mbox{\isa{\isacommand{hence}}}} is also |
685 |
equivalent to ``\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{this}~\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}''. |
|
26870 | 686 |
|
28788 | 687 |
\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 | 688 |
``\hyperlink{command.from}{\mbox{\isa{\isacommand{from}}}}~\isa{this}~\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}''. |
26870 | 689 |
|
40406 | 690 |
\item \hyperlink{command.print-statement}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}statement}}}}~\isa{a} prints facts from the |
26870 | 691 |
current theory or proof context in long statement form, according to |
26902 | 692 |
the syntax for \hyperlink{command.lemma}{\mbox{\isa{\isacommand{lemma}}}} given above. |
26870 | 693 |
|
28788 | 694 |
\end{description} |
26870 | 695 |
|
696 |
Any goal statement causes some term abbreviations (such as |
|
40406 | 697 |
\indexref{}{variable}{?thesis}\hyperlink{variable.?thesis}{\mbox{\isa{{\isaliteral{3F}{\isacharquery}}thesis}}}) to be bound automatically, see also |
26961 | 698 |
\secref{sec:term-abbrev}. |
26870 | 699 |
|
26902 | 700 |
The optional case names of \indexref{}{element}{obtains}\hyperlink{element.obtains}{\mbox{\isa{\isakeyword{obtains}}}} have a twofold |
26870 | 701 |
meaning: (1) during the of this claim they refer to the the local |
702 |
context introductions, (2) the resulting rule is annotated |
|
703 |
accordingly to support symbolic case splits when used with the |
|
36321 | 704 |
\indexref{}{method}{cases}\hyperlink{method.cases}{\mbox{\isa{cases}}} method (cf.\ \secref{sec:cases-induct}).% |
26870 | 705 |
\end{isamarkuptext}% |
706 |
\isamarkuptrue% |
|
707 |
% |
|
28788 | 708 |
\isamarkupsection{Refinement steps% |
709 |
} |
|
710 |
\isamarkuptrue% |
|
711 |
% |
|
712 |
\isamarkupsubsection{Proof method expressions \label{sec:proof-meth}% |
|
713 |
} |
|
714 |
\isamarkuptrue% |
|
715 |
% |
|
716 |
\begin{isamarkuptext}% |
|
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717 |
Proof methods are either basic ones, or expressions composed |
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718 |
of methods via ``\verb|,|'' (sequential composition), |
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|
719 |
``\verb||\verb,|,\verb||'' (alternative choices), ``\verb|?|'' |
28788 | 720 |
(try), ``\verb|+|'' (repeat at least once), ``\verb|[|\isa{n}\verb|]|'' (restriction to first \isa{n} |
40406 | 721 |
sub-goals, with default \isa{{\isaliteral{22}{\isachardoublequote}}n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}). In practice, proof |
42596
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722 |
methods are usually just a comma separated list of \hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}~\hyperlink{syntax.args}{\mbox{\isa{args}}} specifications. Note that parentheses may |
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723 |
be dropped for single method specifications (with no arguments). |
28788 | 724 |
|
42596
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|
725 |
\begin{railoutput} |
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|
726 |
\rail@begin{5}{\indexdef{}{syntax}{method}\hypertarget{syntax.method}{\hyperlink{syntax.method}{\mbox{\isa{method}}}}} |
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|
727 |
\rail@bar |
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|
728 |
\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[] |
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|
729 |
\rail@nextbar{1} |
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|
730 |
\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[] |
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|
731 |
\rail@nont{\isa{methods}}[] |
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|
732 |
\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[] |
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|
733 |
\rail@endbar |
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|
734 |
\rail@bar |
6c621a9d612a
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|
735 |
\rail@nextbar{1} |
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|
736 |
\rail@term{\isa{{\isaliteral{3F}{\isacharquery}}}}[] |
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|
737 |
\rail@nextbar{2} |
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|
738 |
\rail@term{\isa{{\isaliteral{2B}{\isacharplus}}}}[] |
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|
739 |
\rail@nextbar{3} |
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|
740 |
\rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[] |
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|
741 |
\rail@bar |
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|
742 |
\rail@nextbar{4} |
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|
743 |
\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[] |
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|
744 |
\rail@endbar |
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|
745 |
\rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[] |
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|
746 |
\rail@endbar |
6c621a9d612a
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changeset
|
747 |
\rail@end |
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|
748 |
\rail@begin{4}{\isa{methods}} |
6c621a9d612a
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|
749 |
\rail@plus |
6c621a9d612a
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changeset
|
750 |
\rail@bar |
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parents:
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changeset
|
751 |
\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[] |
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|
752 |
\rail@nont{\hyperlink{syntax.args}{\mbox{\isa{args}}}}[] |
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changeset
|
753 |
\rail@nextbar{1} |
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changeset
|
754 |
\rail@nont{\hyperlink{syntax.method}{\mbox{\isa{method}}}}[] |
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|
755 |
\rail@endbar |
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changeset
|
756 |
\rail@nextplus{2} |
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|
757 |
\rail@bar |
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|
758 |
\rail@term{\isa{{\isaliteral{2C}{\isacharcomma}}}}[] |
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|
759 |
\rail@nextbar{3} |
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|
760 |
\rail@term{\isa{{\isaliteral{7C}{\isacharbar}}}}[] |
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|
761 |
\rail@endbar |
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|
762 |
\rail@endplus |
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|
763 |
\rail@end |
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|
764 |
\end{railoutput} |
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|
765 |
|
28788 | 766 |
|
767 |
Proper Isar proof methods do \emph{not} admit arbitrary goal |
|
768 |
addressing, but refer either to the first sub-goal or all sub-goals |
|
40406 | 769 |
uniformly. The goal restriction operator ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5B}{\isacharbrackleft}}n{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}'' |
28788 | 770 |
evaluates a method expression within a sandbox consisting of the |
771 |
first \isa{n} sub-goals (which need to exist). For example, the |
|
40406 | 772 |
method ``\isa{{\isaliteral{22}{\isachardoublequote}}simp{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{5B}{\isacharbrackleft}}{\isadigit{3}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}'' simplifies the first three |
773 |
sub-goals, while ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}rule\ foo{\isaliteral{2C}{\isacharcomma}}\ simp{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}'' simplifies all |
|
774 |
new goals that emerge from applying rule \isa{{\isaliteral{22}{\isachardoublequote}}foo{\isaliteral{22}{\isachardoublequote}}} to the |
|
28788 | 775 |
originally first one. |
776 |
||
777 |
Improper methods, notably tactic emulations, offer a separate |
|
778 |
low-level goal addressing scheme as explicit argument to the |
|
40406 | 779 |
individual tactic being involved. Here ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}'' refers to |
780 |
all goals, and ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5B}{\isacharbrackleft}}n{\isaliteral{2D}{\isacharminus}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}'' to all goals starting from \isa{{\isaliteral{22}{\isachardoublequote}}n{\isaliteral{22}{\isachardoublequote}}}. |
|
28788 | 781 |
|
42596
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|
782 |
\begin{railoutput} |
42705 | 783 |
\rail@begin{4}{\indexdef{}{syntax}{goal\_spec}\hypertarget{syntax.goal-spec}{\hyperlink{syntax.goal-spec}{\mbox{\isa{goal{\isaliteral{5F}{\isacharunderscore}}spec}}}}} |
42596
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|
784 |
\rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[] |
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|
785 |
\rail@bar |
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|
786 |
\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[] |
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|
787 |
\rail@term{\isa{{\isaliteral{2D}{\isacharminus}}}}[] |
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|
788 |
\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[] |
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|
789 |
\rail@nextbar{1} |
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|
790 |
\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[] |
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|
791 |
\rail@term{\isa{{\isaliteral{2D}{\isacharminus}}}}[] |
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|
792 |
\rail@nextbar{2} |
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|
793 |
\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[] |
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|
794 |
\rail@nextbar{3} |
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changeset
|
795 |
\rail@term{\isa{{\isaliteral{21}{\isacharbang}}}}[] |
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|
796 |
\rail@endbar |
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|
797 |
\rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[] |
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|
798 |
\rail@end |
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|
799 |
\end{railoutput}% |
28788 | 800 |
\end{isamarkuptext}% |
801 |
\isamarkuptrue% |
|
802 |
% |
|
803 |
\isamarkupsubsection{Initial and terminal proof steps \label{sec:proof-steps}% |
|
26870 | 804 |
} |
805 |
\isamarkuptrue% |
|
806 |
% |
|
807 |
\begin{isamarkuptext}% |
|
808 |
\begin{matharray}{rcl} |
|
40406 | 809 |
\indexdef{}{command}{proof}\hypertarget{command.proof}{\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
810 |
\indexdef{}{command}{qed}\hypertarget{command.qed}{\hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{7C}{\isacharbar}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\ |
|
811 |
\indexdef{}{command}{by}\hypertarget{command.by}{\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{7C}{\isacharbar}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\ |
|
812 |
\indexdef{}{command}{..}\hypertarget{command.ddot}{\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{7C}{\isacharbar}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\ |
|
813 |
\indexdef{}{command}{.}\hypertarget{command.dot}{\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{7C}{\isacharbar}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\ |
|
814 |
\indexdef{}{command}{sorry}\hypertarget{command.sorry}{\hyperlink{command.sorry}{\mbox{\isa{\isacommand{sorry}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{7C}{\isacharbar}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\ |
|
26870 | 815 |
\end{matharray} |
816 |
||
817 |
Arbitrary goal refinement via tactics is considered harmful. |
|
818 |
Structured proof composition in Isar admits proof methods to be |
|
819 |
invoked in two places only. |
|
820 |
||
821 |
\begin{enumerate} |
|
822 |
||
40406 | 823 |
\item An \emph{initial} refinement step \indexref{}{command}{proof}\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} reduces a newly stated goal to a number |
26870 | 824 |
of sub-goals that are to be solved later. Facts are passed to |
40406 | 825 |
\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} for forward chaining, if so indicated by \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} mode. |
26870 | 826 |
|
40406 | 827 |
\item A \emph{terminal} conclusion step \indexref{}{command}{qed}\hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}} is intended to solve remaining goals. No facts are |
828 |
passed to \isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}}. |
|
26870 | 829 |
|
830 |
\end{enumerate} |
|
831 |
||
832 |
The only other (proper) way to affect pending goals in a proof body |
|
26902 | 833 |
is by \indexref{}{command}{show}\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}, which involves an explicit statement of |
26870 | 834 |
what is to be solved eventually. Thus we avoid the fundamental |
835 |
problem of unstructured tactic scripts that consist of numerous |
|
836 |
consecutive goal transformations, with invisible effects. |
|
837 |
||
838 |
\medskip As a general rule of thumb for good proof style, initial |
|
839 |
proof methods should either solve the goal completely, or constitute |
|
840 |
some well-understood reduction to new sub-goals. Arbitrary |
|
841 |
automatic proof tools that are prone leave a large number of badly |
|
842 |
structured sub-goals are no help in continuing the proof document in |
|
843 |
an intelligible manner. |
|
844 |
||
845 |
Unless given explicitly by the user, the default initial method is |
|
42626 | 846 |
\indexref{Pure}{method}{rule}\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}} (or its classical variant \indexref{}{method}{rule}\hyperlink{method.rule}{\mbox{\isa{rule}}}), which applies a single standard elimination or introduction |
847 |
rule according to the topmost symbol involved. There is no separate |
|
848 |
default terminal method. Any remaining goals are always solved by |
|
849 |
assumption in the very last step. |
|
26870 | 850 |
|
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|
851 |
\begin{railoutput} |
42662 | 852 |
\rail@begin{2}{} |
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853 |
\rail@term{\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}}[] |
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854 |
\rail@bar |
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855 |
\rail@nextbar{1} |
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|
856 |
\rail@nont{\isa{method}}[] |
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|
857 |
\rail@endbar |
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|
858 |
\rail@end |
42662 | 859 |
\rail@begin{2}{} |
42596
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|
860 |
\rail@term{\hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}}[] |
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|
861 |
\rail@bar |
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|
862 |
\rail@nextbar{1} |
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|
863 |
\rail@nont{\isa{method}}[] |
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864 |
\rail@endbar |
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|
865 |
\rail@end |
42662 | 866 |
\rail@begin{2}{} |
42596
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|
867 |
\rail@term{\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}}[] |
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|
868 |
\rail@nont{\isa{method}}[] |
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|
869 |
\rail@bar |
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|
870 |
\rail@nextbar{1} |
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|
871 |
\rail@nont{\isa{method}}[] |
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|
872 |
\rail@endbar |
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|
873 |
\rail@end |
42662 | 874 |
\rail@begin{3}{} |
42596
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parents:
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|
875 |
\rail@bar |
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parents:
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|
876 |
\rail@term{\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}}}}}}[] |
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|
877 |
\rail@nextbar{1} |
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|
878 |
\rail@term{\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}}[] |
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|
879 |
\rail@nextbar{2} |
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|
880 |
\rail@term{\hyperlink{command.sorry}{\mbox{\isa{\isacommand{sorry}}}}}[] |
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parents:
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|
881 |
\rail@endbar |
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parents:
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|
882 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
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parents:
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|
883 |
\end{railoutput} |
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|
884 |
|
26870 | 885 |
|
28788 | 886 |
\begin{description} |
26870 | 887 |
|
40406 | 888 |
\item \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} refines the goal by proof |
889 |
method \isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}; facts for forward chaining are passed if so |
|
890 |
indicated by \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} mode. |
|
26870 | 891 |
|
40406 | 892 |
\item \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}} refines any remaining goals by |
893 |
proof method \isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}} and concludes the sub-proof by assumption. |
|
894 |
If the goal had been \isa{{\isaliteral{22}{\isachardoublequote}}show{\isaliteral{22}{\isachardoublequote}}} (or \isa{{\isaliteral{22}{\isachardoublequote}}thus{\isaliteral{22}{\isachardoublequote}}}), some |
|
28788 | 895 |
pending sub-goal is solved as well by the rule resulting from the |
40406 | 896 |
result \emph{exported} into the enclosing goal context. Thus \isa{{\isaliteral{22}{\isachardoublequote}}qed{\isaliteral{22}{\isachardoublequote}}} may fail for two reasons: either \isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}} fails, or the |
28788 | 897 |
resulting rule does not fit to any pending goal\footnote{This |
898 |
includes any additional ``strong'' assumptions as introduced by |
|
899 |
\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}.} of the enclosing context. Debugging such a |
|
900 |
situation might involve temporarily changing \hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}} into |
|
901 |
\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}, or weakening the local context by replacing |
|
902 |
occurrences of \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} by \hyperlink{command.presume}{\mbox{\isa{\isacommand{presume}}}}. |
|
26870 | 903 |
|
40406 | 904 |
\item \hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}} is a \emph{terminal |
905 |
proof}\index{proof!terminal}; it abbreviates \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}~\isa{{\isaliteral{22}{\isachardoublequote}}qed{\isaliteral{22}{\isachardoublequote}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}}, but with |
|
28788 | 906 |
backtracking across both methods. Debugging an unsuccessful |
40406 | 907 |
\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}} command can be done by expanding its |
908 |
definition; in many cases \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} (or even |
|
909 |
\isa{{\isaliteral{22}{\isachardoublequote}}apply{\isaliteral{22}{\isachardoublequote}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}) is already sufficient to see the |
|
26870 | 910 |
problem. |
911 |
||
40406 | 912 |
\item ``\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}'' is a \emph{default |
913 |
proof}\index{proof!default}; it abbreviates \hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}rule{\isaliteral{22}{\isachardoublequote}}}. |
|
26870 | 914 |
|
40406 | 915 |
\item ``\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}}}}}'' is a \emph{trivial |
916 |
proof}\index{proof!trivial}; it abbreviates \hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}this{\isaliteral{22}{\isachardoublequote}}}. |
|
26870 | 917 |
|
28788 | 918 |
\item \hyperlink{command.sorry}{\mbox{\isa{\isacommand{sorry}}}} is a \emph{fake proof}\index{proof!fake} |
26870 | 919 |
pretending to solve the pending claim without further ado. This |
920 |
only works in interactive development, or if the \verb|quick_and_dirty| flag is enabled (in ML). Facts emerging from fake |
|
921 |
proofs are not the real thing. Internally, each theorem container |
|
40406 | 922 |
is tainted by an oracle invocation, which is indicated as ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{21}{\isacharbang}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}}'' in the printed result. |
26870 | 923 |
|
26902 | 924 |
The most important application of \hyperlink{command.sorry}{\mbox{\isa{\isacommand{sorry}}}} is to support |
26870 | 925 |
experimentation and top-down proof development. |
926 |
||
28788 | 927 |
\end{description}% |
26870 | 928 |
\end{isamarkuptext}% |
929 |
\isamarkuptrue% |
|
930 |
% |
|
28788 | 931 |
\isamarkupsubsection{Fundamental methods and attributes \label{sec:pure-meth-att}% |
26870 | 932 |
} |
933 |
\isamarkuptrue% |
|
934 |
% |
|
935 |
\begin{isamarkuptext}% |
|
936 |
The following proof methods and attributes refer to basic logical |
|
937 |
operations of Isar. Further methods and attributes are provided by |
|
938 |
several generic and object-logic specific tools and packages (see |
|
939 |
\chref{ch:gen-tools} and \chref{ch:hol}). |
|
940 |
||
941 |
\begin{matharray}{rcl} |
|
40406 | 942 |
\indexdef{}{method}{-}\hypertarget{method.-}{\hyperlink{method.-}{\mbox{\isa{{\isaliteral{2D}{\isacharminus}}}}}} & : & \isa{method} \\ |
28788 | 943 |
\indexdef{}{method}{fact}\hypertarget{method.fact}{\hyperlink{method.fact}{\mbox{\isa{fact}}}} & : & \isa{method} \\ |
944 |
\indexdef{}{method}{assumption}\hypertarget{method.assumption}{\hyperlink{method.assumption}{\mbox{\isa{assumption}}}} & : & \isa{method} \\ |
|
945 |
\indexdef{}{method}{this}\hypertarget{method.this}{\hyperlink{method.this}{\mbox{\isa{this}}}} & : & \isa{method} \\ |
|
42626 | 946 |
\indexdef{Pure}{method}{rule}\hypertarget{method.Pure.rule}{\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}} & : & \isa{method} \\ |
28788 | 947 |
\indexdef{Pure}{attribute}{intro}\hypertarget{attribute.Pure.intro}{\hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}}} & : & \isa{attribute} \\ |
948 |
\indexdef{Pure}{attribute}{elim}\hypertarget{attribute.Pure.elim}{\hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}} & : & \isa{attribute} \\ |
|
949 |
\indexdef{Pure}{attribute}{dest}\hypertarget{attribute.Pure.dest}{\hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}} & : & \isa{attribute} \\ |
|
42626 | 950 |
\indexdef{Pure}{attribute}{rule}\hypertarget{attribute.Pure.rule}{\hyperlink{attribute.Pure.rule}{\mbox{\isa{rule}}}} & : & \isa{attribute} \\[0.5ex] |
28788 | 951 |
\indexdef{}{attribute}{OF}\hypertarget{attribute.OF}{\hyperlink{attribute.OF}{\mbox{\isa{OF}}}} & : & \isa{attribute} \\ |
952 |
\indexdef{}{attribute}{of}\hypertarget{attribute.of}{\hyperlink{attribute.of}{\mbox{\isa{of}}}} & : & \isa{attribute} \\ |
|
953 |
\indexdef{}{attribute}{where}\hypertarget{attribute.where}{\hyperlink{attribute.where}{\mbox{\isa{where}}}} & : & \isa{attribute} \\ |
|
26870 | 954 |
\end{matharray} |
955 |
||
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|
956 |
\begin{railoutput} |
42662 | 957 |
\rail@begin{2}{} |
42596
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|
958 |
\rail@term{\hyperlink{method.fact}{\mbox{\isa{fact}}}}[] |
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959 |
\rail@bar |
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|
960 |
\rail@nextbar{1} |
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|
961 |
\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[] |
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diff
changeset
|
962 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
963 |
\rail@end |
42662 | 964 |
\rail@begin{2}{} |
42626 | 965 |
\rail@term{\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}}[] |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
966 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
967 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
968 |
\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
969 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
970 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
971 |
\rail@begin{4}{\isa{rulemod}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
972 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
973 |
\rail@term{\isa{intro}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
974 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
975 |
\rail@term{\isa{elim}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
976 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
977 |
\rail@term{\isa{dest}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
978 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
979 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
980 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
981 |
\rail@term{\isa{{\isaliteral{21}{\isacharbang}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
982 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
983 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
984 |
\rail@term{\isa{{\isaliteral{3F}{\isacharquery}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
985 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
986 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
987 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
988 |
\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
989 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
990 |
\rail@nextbar{3} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
991 |
\rail@term{\isa{del}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
992 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
993 |
\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
994 |
\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
995 |
\rail@end |
42662 | 996 |
\rail@begin{3}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
997 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
998 |
\rail@term{\hyperlink{attribute.intro}{\mbox{\isa{intro}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
999 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1000 |
\rail@term{\hyperlink{attribute.elim}{\mbox{\isa{elim}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1001 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1002 |
\rail@term{\hyperlink{attribute.dest}{\mbox{\isa{dest}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1003 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1004 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1005 |
\rail@term{\isa{{\isaliteral{21}{\isacharbang}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1006 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1007 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1008 |
\rail@term{\isa{{\isaliteral{3F}{\isacharquery}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1009 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1010 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1011 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1012 |
\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1013 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1014 |
\rail@end |
42662 | 1015 |
\rail@begin{1}{} |
42626 | 1016 |
\rail@term{\hyperlink{attribute.Pure.rule}{\mbox{\isa{rule}}}}[] |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1017 |
\rail@term{\isa{del}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1018 |
\rail@end |
42662 | 1019 |
\rail@begin{1}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1020 |
\rail@term{\hyperlink{attribute.OF}{\mbox{\isa{OF}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1021 |
\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1022 |
\rail@end |
42662 | 1023 |
\rail@begin{2}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1024 |
\rail@term{\hyperlink{attribute.of}{\mbox{\isa{of}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1025 |
\rail@nont{\hyperlink{syntax.insts}{\mbox{\isa{insts}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1026 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1027 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1028 |
\rail@term{\isa{concl}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1029 |
\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1030 |
\rail@nont{\hyperlink{syntax.insts}{\mbox{\isa{insts}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1031 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1032 |
\rail@end |
42662 | 1033 |
\rail@begin{6}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1034 |
\rail@term{\hyperlink{attribute.where}{\mbox{\isa{where}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1035 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1036 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1037 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1038 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1039 |
\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1040 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1041 |
\rail@nont{\hyperlink{syntax.var}{\mbox{\isa{var}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1042 |
\rail@nextbar{3} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1043 |
\rail@nont{\hyperlink{syntax.typefree}{\mbox{\isa{typefree}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1044 |
\rail@nextbar{4} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1045 |
\rail@nont{\hyperlink{syntax.typevar}{\mbox{\isa{typevar}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1046 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1047 |
\rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1048 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1049 |
\rail@nont{\hyperlink{syntax.type}{\mbox{\isa{type}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1050 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1051 |
\rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1052 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1053 |
\rail@nextplus{5} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1054 |
\rail@cterm{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1055 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1056 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1057 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1058 |
\end{railoutput} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1059 |
|
26870 | 1060 |
|
28788 | 1061 |
\begin{description} |
26870 | 1062 |
|
40406 | 1063 |
\item ``\hyperlink{method.-}{\mbox{\isa{{\isaliteral{2D}{\isacharminus}}}}}'' (minus) does nothing but insert the forward |
28788 | 1064 |
chaining facts as premises into the goal. Note that command |
26902 | 1065 |
\indexref{}{command}{proof}\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} without any method actually performs a single |
42626 | 1066 |
reduction step using the \indexref{Pure}{method}{rule}\hyperlink{method.Pure.rule}{\mbox{\isa{rule}}} method; thus a plain |
40406 | 1067 |
\emph{do-nothing} proof step would be ``\hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2D}{\isacharminus}}{\isaliteral{22}{\isachardoublequote}}}'' rather than \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} alone. |
26870 | 1068 |
|
40406 | 1069 |
\item \hyperlink{method.fact}{\mbox{\isa{fact}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} composes some fact from |
1070 |
\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} (or implicitly from the current proof context) |
|
28788 | 1071 |
modulo unification of schematic type and term variables. The rule |
1072 |
structure is not taken into account, i.e.\ meta-level implication is |
|
1073 |
considered atomic. This is the same principle underlying literal |
|
40406 | 1074 |
facts (cf.\ \secref{sec:syn-att}): ``\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{fact}'' is equivalent to ``\hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\verb|`|\isa{{\isaliteral{5C3C7068693E}{\isasymphi}}}\verb|`|'' provided that |
1075 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} is an instance of some known \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} in the |
|
28788 | 1076 |
proof context. |
26870 | 1077 |
|
28788 | 1078 |
\item \hyperlink{method.assumption}{\mbox{\isa{assumption}}} solves some goal by a single assumption |
26870 | 1079 |
step. All given facts are guaranteed to participate in the |
1080 |
refinement; this means there may be only 0 or 1 in the first place. |
|
26902 | 1081 |
Recall that \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} (\secref{sec:proof-steps}) already |
26870 | 1082 |
concludes any remaining sub-goals by assumption, so structured |
26902 | 1083 |
proofs usually need not quote the \hyperlink{method.assumption}{\mbox{\isa{assumption}}} method at |
26870 | 1084 |
all. |
1085 |
||
28788 | 1086 |
\item \hyperlink{method.this}{\mbox{\isa{this}}} applies all of the current facts directly as |
40406 | 1087 |
rules. Recall that ``\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}}}}}'' (dot) abbreviates ``\hyperlink{command.by}{\mbox{\isa{\isacommand{by}}}}~\isa{this}''. |
26870 | 1088 |
|
42626 | 1089 |
\item \hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} applies some rule given as |
28788 | 1090 |
argument in backward manner; facts are used to reduce the rule |
42626 | 1091 |
before applying it to the goal. Thus \hyperlink{method.Pure.rule}{\mbox{\isa{rule}}} without facts |
28788 | 1092 |
is plain introduction, while with facts it becomes elimination. |
26870 | 1093 |
|
42626 | 1094 |
When no arguments are given, the \hyperlink{method.Pure.rule}{\mbox{\isa{rule}}} method tries to pick |
26870 | 1095 |
appropriate rules automatically, as declared in the current context |
26902 | 1096 |
using the \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}}, \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, |
1097 |
\hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}} attributes (see below). This is the |
|
40406 | 1098 |
default behavior of \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} and ``\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}'' |
26902 | 1099 |
(double-dot) steps (see \secref{sec:proof-steps}). |
26870 | 1100 |
|
28788 | 1101 |
\item \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}}, \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, and |
1102 |
\hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}} declare introduction, elimination, and |
|
42626 | 1103 |
destruct rules, to be used with method \hyperlink{method.Pure.rule}{\mbox{\isa{rule}}}, and similar |
30172 | 1104 |
tools. Note that the latter will ignore rules declared with |
40406 | 1105 |
``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}}'', while ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{21}{\isacharbang}}{\isaliteral{22}{\isachardoublequote}}}'' are used most aggressively. |
26870 | 1106 |
|
1107 |
The classical reasoner (see \secref{sec:classical}) introduces its |
|
1108 |
own variants of these attributes; use qualified names to access the |
|
40406 | 1109 |
present versions of Isabelle/Pure, i.e.\ \hyperlink{attribute.Pure.Pure.intro}{\mbox{\isa{Pure{\isaliteral{2E}{\isachardot}}intro}}}. |
26870 | 1110 |
|
42626 | 1111 |
\item \hyperlink{attribute.Pure.rule}{\mbox{\isa{rule}}}~\isa{del} undeclares introduction, |
26870 | 1112 |
elimination, or destruct rules. |
1113 |
||
40406 | 1114 |
\item \hyperlink{attribute.OF}{\mbox{\isa{OF}}}~\isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} applies some |
1115 |
theorem to all of the given rules \isa{{\isaliteral{22}{\isachardoublequote}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} |
|
30463 | 1116 |
(in parallel). This corresponds to the \verb|op MRS| operation in |
1117 |
ML, but note the reversed order. Positions may be effectively |
|
40406 | 1118 |
skipped by including ``\isa{{\isaliteral{5F}{\isacharunderscore}}}'' (underscore) as argument. |
26870 | 1119 |
|
40406 | 1120 |
\item \hyperlink{attribute.of}{\mbox{\isa{of}}}~\isa{{\isaliteral{22}{\isachardoublequote}}t\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} performs positional |
1121 |
instantiation of term variables. The terms \isa{{\isaliteral{22}{\isachardoublequote}}t\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} are |
|
28788 | 1122 |
substituted for any schematic variables occurring in a theorem from |
40406 | 1123 |
left to right; ``\isa{{\isaliteral{5F}{\isacharunderscore}}}'' (underscore) indicates to skip a |
1124 |
position. Arguments following a ``\isa{{\isaliteral{22}{\isachardoublequote}}concl{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}'' specification |
|
28788 | 1125 |
refer to positions of the conclusion of a rule. |
26870 | 1126 |
|
40406 | 1127 |
\item \hyperlink{attribute.where}{\mbox{\isa{where}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3D}{\isacharequal}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} |
28788 | 1128 |
performs named instantiation of schematic type and term variables |
1129 |
occurring in a theorem. Schematic variables have to be specified on |
|
40406 | 1130 |
the left-hand side (e.g.\ \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}x{\isadigit{1}}{\isaliteral{2E}{\isachardot}}{\isadigit{3}}{\isaliteral{22}{\isachardoublequote}}}). The question mark may |
28788 | 1131 |
be omitted if the variable name is a plain identifier without index. |
1132 |
As type instantiations are inferred from term instantiations, |
|
1133 |
explicit type instantiations are seldom necessary. |
|
26870 | 1134 |
|
28788 | 1135 |
\end{description}% |
26870 | 1136 |
\end{isamarkuptext}% |
1137 |
\isamarkuptrue% |
|
1138 |
% |
|
28788 | 1139 |
\isamarkupsubsection{Emulating tactic scripts \label{sec:tactic-commands}% |
26870 | 1140 |
} |
1141 |
\isamarkuptrue% |
|
1142 |
% |
|
1143 |
\begin{isamarkuptext}% |
|
1144 |
The Isar provides separate commands to accommodate tactic-style |
|
1145 |
proof scripts within the same system. While being outside the |
|
1146 |
orthodox Isar proof language, these might come in handy for |
|
1147 |
interactive exploration and debugging, or even actual tactical proof |
|
1148 |
within new-style theories (to benefit from document preparation, for |
|
1149 |
example). See also \secref{sec:tactics} for actual tactics, that |
|
1150 |
have been encapsulated as proof methods. Proper proof methods may |
|
1151 |
be used in scripts, too. |
|
1152 |
||
1153 |
\begin{matharray}{rcl} |
|
40406 | 1154 |
\indexdef{}{command}{apply}\hypertarget{command.apply}{\hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
1155 |
\indexdef{}{command}{apply\_end}\hypertarget{command.apply-end}{\hyperlink{command.apply-end}{\mbox{\isa{\isacommand{apply{\isaliteral{5F}{\isacharunderscore}}end}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
1156 |
\indexdef{}{command}{done}\hypertarget{command.done}{\hyperlink{command.done}{\mbox{\isa{\isacommand{done}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{7C}{\isacharbar}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\ |
|
1157 |
\indexdef{}{command}{defer}\hypertarget{command.defer}{\hyperlink{command.defer}{\mbox{\isa{\isacommand{defer}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{22}{\isachardoublequote}}} \\ |
|
1158 |
\indexdef{}{command}{prefer}\hypertarget{command.prefer}{\hyperlink{command.prefer}{\mbox{\isa{\isacommand{prefer}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{22}{\isachardoublequote}}} \\ |
|
1159 |
\indexdef{}{command}{back}\hypertarget{command.back}{\hyperlink{command.back}{\mbox{\isa{\isacommand{back}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{22}{\isachardoublequote}}} \\ |
|
26870 | 1160 |
\end{matharray} |
1161 |
||
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modernized rail diagrams using @{rail} antiquotation;
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parents:
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|
1162 |
\begin{railoutput} |
42662 | 1163 |
\rail@begin{2}{} |
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modernized rail diagrams using @{rail} antiquotation;
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|
1164 |
\rail@bar |
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modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
1165 |
\rail@term{\hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}}[] |
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|
1166 |
\rail@nextbar{1} |
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parents:
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diff
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|
1167 |
\rail@term{\hyperlink{command.apply-end}{\mbox{\isa{\isacommand{apply{\isaliteral{5F}{\isacharunderscore}}end}}}}}[] |
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|
1168 |
\rail@endbar |
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modernized rail diagrams using @{rail} antiquotation;
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parents:
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diff
changeset
|
1169 |
\rail@nont{\hyperlink{syntax.method}{\mbox{\isa{method}}}}[] |
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|
1170 |
\rail@end |
42662 | 1171 |
\rail@begin{2}{} |
42596
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modernized rail diagrams using @{rail} antiquotation;
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parents:
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diff
changeset
|
1172 |
\rail@term{\hyperlink{command.defer}{\mbox{\isa{\isacommand{defer}}}}}[] |
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modernized rail diagrams using @{rail} antiquotation;
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parents:
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|
1173 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1174 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1175 |
\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[] |
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modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1176 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1177 |
\rail@end |
42662 | 1178 |
\rail@begin{1}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1179 |
\rail@term{\hyperlink{command.prefer}{\mbox{\isa{\isacommand{prefer}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1180 |
\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[] |
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modernized rail diagrams using @{rail} antiquotation;
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parents:
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diff
changeset
|
1181 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1182 |
\end{railoutput} |
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modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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|
1183 |
|
26870 | 1184 |
|
28788 | 1185 |
\begin{description} |
26870 | 1186 |
|
28788 | 1187 |
\item \hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}~\isa{m} applies proof method \isa{m} in |
40406 | 1188 |
initial position, but unlike \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}} it retains ``\isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' mode. Thus consecutive method applications may be |
28788 | 1189 |
given just as in tactic scripts. |
26870 | 1190 |
|
1191 |
Facts are passed to \isa{m} as indicated by the goal's |
|
1192 |
forward-chain mode, and are \emph{consumed} afterwards. Thus any |
|
26902 | 1193 |
further \hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}} command would always work in a purely |
26870 | 1194 |
backward manner. |
1195 |
||
40406 | 1196 |
\item \hyperlink{command.apply-end}{\mbox{\isa{\isacommand{apply{\isaliteral{5F}{\isacharunderscore}}end}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}m{\isaliteral{22}{\isachardoublequote}}} applies proof method \isa{m} as if in terminal position. Basically, this simulates a |
26902 | 1197 |
multi-step tactic script for \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}}, but may be given |
26870 | 1198 |
anywhere within the proof body. |
1199 |
||
26895 | 1200 |
No facts are passed to \isa{m} here. Furthermore, the static |
26902 | 1201 |
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 | 1202 |
introduced in the current body, for example. |
1203 |
||
28788 | 1204 |
\item \hyperlink{command.done}{\mbox{\isa{\isacommand{done}}}} completes a proof script, provided that the |
1205 |
current goal state is solved completely. Note that actual |
|
40406 | 1206 |
structured proof commands (e.g.\ ``\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}}}}}'' or \hyperlink{command.sorry}{\mbox{\isa{\isacommand{sorry}}}}) may be used to conclude proof scripts as well. |
26870 | 1207 |
|
28788 | 1208 |
\item \hyperlink{command.defer}{\mbox{\isa{\isacommand{defer}}}}~\isa{n} and \hyperlink{command.prefer}{\mbox{\isa{\isacommand{prefer}}}}~\isa{n} |
1209 |
shuffle the list of pending goals: \hyperlink{command.defer}{\mbox{\isa{\isacommand{defer}}}} puts off |
|
40406 | 1210 |
sub-goal \isa{n} to the end of the list (\isa{{\isaliteral{22}{\isachardoublequote}}n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} by |
26902 | 1211 |
default), while \hyperlink{command.prefer}{\mbox{\isa{\isacommand{prefer}}}} brings sub-goal \isa{n} to the |
26870 | 1212 |
front. |
1213 |
||
28788 | 1214 |
\item \hyperlink{command.back}{\mbox{\isa{\isacommand{back}}}} does back-tracking over the result sequence |
1215 |
of the latest proof command. Basically, any proof command may |
|
1216 |
return multiple results. |
|
26870 | 1217 |
|
28788 | 1218 |
\end{description} |
26870 | 1219 |
|
1220 |
Any proper Isar proof method may be used with tactic script commands |
|
26902 | 1221 |
such as \hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}. A few additional emulations of actual |
26870 | 1222 |
tactics are provided as well; these would be never used in actual |
1223 |
structured proofs, of course.% |
|
1224 |
\end{isamarkuptext}% |
|
1225 |
\isamarkuptrue% |
|
1226 |
% |
|
28788 | 1227 |
\isamarkupsubsection{Defining proof methods% |
26870 | 1228 |
} |
1229 |
\isamarkuptrue% |
|
1230 |
% |
|
1231 |
\begin{isamarkuptext}% |
|
1232 |
\begin{matharray}{rcl} |
|
40406 | 1233 |
\indexdef{}{command}{method\_setup}\hypertarget{command.method-setup}{\hyperlink{command.method-setup}{\mbox{\isa{\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\ |
26870 | 1234 |
\end{matharray} |
1235 |
||
42596
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modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
1236 |
\begin{railoutput} |
42813
6c841fa92fa2
optional description for 'attribute_setup' and 'method_setup';
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parents:
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diff
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|
1237 |
\rail@begin{2}{} |
42596
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modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1238 |
\rail@term{\hyperlink{command.method-setup}{\mbox{\isa{\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}}}}}[] |
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wenzelm
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|
1239 |
\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
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|
1240 |
\rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[] |
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modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
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|
1241 |
\rail@nont{\hyperlink{syntax.text}{\mbox{\isa{text}}}}[] |
42813
6c841fa92fa2
optional description for 'attribute_setup' and 'method_setup';
wenzelm
parents:
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diff
changeset
|
1242 |
\rail@bar |
6c841fa92fa2
optional description for 'attribute_setup' and 'method_setup';
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parents:
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diff
changeset
|
1243 |
\rail@nextbar{1} |
42596
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modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
1244 |
\rail@nont{\hyperlink{syntax.text}{\mbox{\isa{text}}}}[] |
42813
6c841fa92fa2
optional description for 'attribute_setup' and 'method_setup';
wenzelm
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|
1245 |
\rail@endbar |
42596
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modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
1246 |
\rail@end |
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|
1247 |
\end{railoutput} |
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modernized rail diagrams using @{rail} antiquotation;
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changeset
|
1248 |
|
28788 | 1249 |
|
1250 |
\begin{description} |
|
1251 |
||
40406 | 1252 |
\item \hyperlink{command.method-setup}{\mbox{\isa{\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}name\ {\isaliteral{3D}{\isacharequal}}\ text\ description{\isaliteral{22}{\isachardoublequote}}} |
1253 |
defines a proof method in the current theory. The given \isa{{\isaliteral{22}{\isachardoublequote}}text{\isaliteral{22}{\isachardoublequote}}} has to be an ML expression of type |
|
30548 | 1254 |
\verb|(Proof.context -> Proof.method) context_parser|, cf.\ |
1255 |
basic parsers defined in structure \verb|Args| and \verb|Attrib|. There are also combinators like \verb|METHOD| and \verb|SIMPLE_METHOD| to turn certain tactic forms into official proof |
|
1256 |
methods; the primed versions refer to tactics with explicit goal |
|
1257 |
addressing. |
|
26870 | 1258 |
|
30548 | 1259 |
Here are some example method definitions: |
28788 | 1260 |
|
1261 |
\end{description}% |
|
26870 | 1262 |
\end{isamarkuptext}% |
1263 |
\isamarkuptrue% |
|
1264 |
% |
|
30548 | 1265 |
\isadelimML |
42704 | 1266 |
\ \ % |
30548 | 1267 |
\endisadelimML |
1268 |
% |
|
1269 |
\isatagML |
|
40406 | 1270 |
\isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}\isamarkupfalse% |
1271 |
\ my{\isaliteral{5F}{\isacharunderscore}}method{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B2A}{\isacharverbatimopen}}\isanewline |
|
42704 | 1272 |
\ \ \ \ Scan{\isaliteral{2E}{\isachardot}}succeed\ {\isaliteral{28}{\isacharparenleft}}K\ {\isaliteral{28}{\isacharparenleft}}SIMPLE{\isaliteral{5F}{\isacharunderscore}}METHOD{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}fn\ i{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ no{\isaliteral{5F}{\isacharunderscore}}tac{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline |
1273 |
\ \ {\isaliteral{2A7D}{\isacharverbatimclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}my\ first\ method\ {\isaliteral{28}{\isacharparenleft}}without\ any\ arguments{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
|
30548 | 1274 |
\isanewline |
42704 | 1275 |
\ \ \isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}\isamarkupfalse% |
40406 | 1276 |
\ my{\isaliteral{5F}{\isacharunderscore}}method{\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B2A}{\isacharverbatimopen}}\isanewline |
42704 | 1277 |
\ \ \ \ Scan{\isaliteral{2E}{\isachardot}}succeed\ {\isaliteral{28}{\isacharparenleft}}fn\ ctxt{\isaliteral{3A}{\isacharcolon}}\ Proof{\isaliteral{2E}{\isachardot}}context\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\isanewline |
1278 |
\ \ \ \ \ \ SIMPLE{\isaliteral{5F}{\isacharunderscore}}METHOD{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}fn\ i{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ no{\isaliteral{5F}{\isacharunderscore}}tac{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline |
|
1279 |
\ \ {\isaliteral{2A7D}{\isacharverbatimclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}my\ second\ method\ {\isaliteral{28}{\isacharparenleft}}with\ context{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
|
30548 | 1280 |
\isanewline |
42704 | 1281 |
\ \ \isacommand{method{\isaliteral{5F}{\isacharunderscore}}setup}\isamarkupfalse% |
40406 | 1282 |
\ my{\isaliteral{5F}{\isacharunderscore}}method{\isadigit{3}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{7B2A}{\isacharverbatimopen}}\isanewline |
42704 | 1283 |
\ \ \ \ Attrib{\isaliteral{2E}{\isachardot}}thms\ {\isaliteral{3E}{\isachargreater}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}fn\ thms{\isaliteral{3A}{\isacharcolon}}\ thm\ list\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ fn\ ctxt{\isaliteral{3A}{\isacharcolon}}\ Proof{\isaliteral{2E}{\isachardot}}context\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\isanewline |
1284 |
\ \ \ \ \ \ SIMPLE{\isaliteral{5F}{\isacharunderscore}}METHOD{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}fn\ i{\isaliteral{3A}{\isacharcolon}}\ int\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3E}{\isachargreater}}\ no{\isaliteral{5F}{\isacharunderscore}}tac{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline |
|
1285 |
\ \ {\isaliteral{2A7D}{\isacharverbatimclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}my\ third\ method\ {\isaliteral{28}{\isacharparenleft}}with\ theorem\ arguments\ and\ context{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% |
|
30548 | 1286 |
\endisatagML |
1287 |
{\isafoldML}% |
|
1288 |
% |
|
1289 |
\isadelimML |
|
1290 |
% |
|
1291 |
\endisadelimML |
|
1292 |
% |
|
26870 | 1293 |
\isamarkupsection{Generalized elimination \label{sec:obtain}% |
1294 |
} |
|
1295 |
\isamarkuptrue% |
|
1296 |
% |
|
1297 |
\begin{isamarkuptext}% |
|
1298 |
\begin{matharray}{rcl} |
|
40406 | 1299 |
\indexdef{}{command}{obtain}\hypertarget{command.obtain}{\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
1300 |
\indexdef{}{command}{guess}\hypertarget{command.guess}{\hyperlink{command.guess}{\mbox{\isa{\isacommand{guess}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{7C}{\isacharbar}}\ proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
26870 | 1301 |
\end{matharray} |
1302 |
||
1303 |
Generalized elimination means that additional elements with certain |
|
1304 |
properties may be introduced in the current context, by virtue of a |
|
1305 |
locally proven ``soundness statement''. Technically speaking, the |
|
26902 | 1306 |
\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} language element is like a declaration of |
1307 |
\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}} and \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}} (see also see |
|
26870 | 1308 |
\secref{sec:proof-context}), together with a soundness proof of its |
1309 |
additional claim. According to the nature of existential reasoning, |
|
1310 |
assumptions get eliminated from any result exported from the context |
|
1311 |
later, provided that the corresponding parameters do \emph{not} |
|
1312 |
occur in the conclusion. |
|
1313 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1314 |
\begin{railoutput} |
42662 | 1315 |
\rail@begin{2}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1316 |
\rail@term{\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1317 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1318 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1319 |
\rail@nont{\hyperlink{syntax.parname}{\mbox{\isa{parname}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1320 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1321 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1322 |
\rail@nont{\hyperlink{syntax.vars}{\mbox{\isa{vars}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1323 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1324 |
\rail@cterm{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1325 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1326 |
\rail@term{\isa{\isakeyword{where}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1327 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1328 |
\rail@nont{\hyperlink{syntax.props}{\mbox{\isa{props}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1329 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1330 |
\rail@cterm{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1331 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1332 |
\rail@end |
42662 | 1333 |
\rail@begin{2}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1334 |
\rail@term{\hyperlink{command.guess}{\mbox{\isa{\isacommand{guess}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1335 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1336 |
\rail@nont{\hyperlink{syntax.vars}{\mbox{\isa{vars}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1337 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1338 |
\rail@cterm{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1339 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1340 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1341 |
\end{railoutput} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1342 |
|
26870 | 1343 |
|
26902 | 1344 |
The derived Isar command \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} is defined as follows |
40406 | 1345 |
(where \isa{{\isaliteral{22}{\isachardoublequote}}b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub k{\isaliteral{22}{\isachardoublequote}}} shall refer to (optional) |
26870 | 1346 |
facts indicated for forward chaining). |
1347 |
\begin{matharray}{l} |
|
40406 | 1348 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616E676C653E}{\isasymlangle}}using\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub k{\isaliteral{5C3C72616E676C653E}{\isasymrangle}}{\isaliteral{22}{\isachardoublequote}}}~~\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{5C3C57484552453E}{\isasymWHERE}}\ a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ \ {\isaliteral{5C3C6C616E676C653E}{\isasymlangle}}proof{\isaliteral{5C3C72616E676C653E}{\isasymrangle}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}} \\[1ex] |
1349 |
\quad \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}thesis{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{22}{\isachardoublequote}}} \\ |
|
29731 | 1350 |
\quad \hyperlink{command.proof}{\mbox{\isa{\isacommand{proof}}}}~\hyperlink{method.succeed}{\mbox{\isa{succeed}}} \\ |
26902 | 1351 |
\qquad \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{thesis} \\ |
40406 | 1352 |
\qquad \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}that\ {\isaliteral{5B}{\isacharbrackleft}}Pure{\isaliteral{2E}{\isachardot}}intro{\isaliteral{3F}{\isacharquery}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{22}{\isachardoublequote}}} \\ |
26902 | 1353 |
\qquad \hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}~\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}~\isa{thesis} \\ |
40406 | 1354 |
\quad\qquad \hyperlink{command.apply}{\mbox{\isa{\isacommand{apply}}}}~\isa{{\isaliteral{2D}{\isacharminus}}} \\ |
1355 |
\quad\qquad \hyperlink{command.using}{\mbox{\isa{\isacommand{using}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub k\ \ {\isaliteral{5C3C6C616E676C653E}{\isasymlangle}}proof{\isaliteral{5C3C72616E676C653E}{\isasymrangle}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
26902 | 1356 |
\quad \hyperlink{command.qed}{\mbox{\isa{\isacommand{qed}}}} \\ |
40406 | 1357 |
\quad \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}\ a{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} \\ |
26870 | 1358 |
\end{matharray} |
1359 |
||
1360 |
Typically, the soundness proof is relatively straight-forward, often |
|
26902 | 1361 |
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 | 1362 |
``\isa{that}'' reduction above is declared as simplification and |
1363 |
introduction rule. |
|
1364 |
||
26902 | 1365 |
In a sense, \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} represents at the level of Isar |
26870 | 1366 |
proofs what would be meta-logical existential quantifiers and |
1367 |
conjunctions. This concept has a broad range of useful |
|
1368 |
applications, ranging from plain elimination (or introduction) of |
|
1369 |
object-level existential and conjunctions, to elimination over |
|
1370 |
results of symbolic evaluation of recursive definitions, for |
|
26902 | 1371 |
example. Also note that \hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}} without parameters acts |
1372 |
much like \hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}, where the result is treated as a |
|
26870 | 1373 |
genuine assumption. |
1374 |
||
1375 |
An alternative name to be used instead of ``\isa{that}'' above may |
|
1376 |
be given in parentheses. |
|
1377 |
||
26902 | 1378 |
\medskip The improper variant \hyperlink{command.guess}{\mbox{\isa{\isacommand{guess}}}} is similar to |
1379 |
\hyperlink{command.obtain}{\mbox{\isa{\isacommand{obtain}}}}, but derives the obtained statement from the |
|
26870 | 1380 |
course of reasoning! The proof starts with a fixed goal \isa{thesis}. The subsequent proof may refine this to anything of the |
40406 | 1381 |
form like \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ thesis{\isaliteral{22}{\isachardoublequote}}}, but must not introduce new subgoals. The |
26870 | 1382 |
final goal state is then used as reduction rule for the obtain |
40406 | 1383 |
scheme described above. Obtained parameters \isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}} are marked as internal by default, which prevents the |
26870 | 1384 |
proof context from being polluted by ad-hoc variables. The variable |
26902 | 1385 |
names and type constraints given as arguments for \hyperlink{command.guess}{\mbox{\isa{\isacommand{guess}}}} |
26870 | 1386 |
specify a prefix of obtained parameters explicitly in the text. |
1387 |
||
26902 | 1388 |
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 | 1389 |
type-variables occurring here are fixed in the present context!% |
1390 |
\end{isamarkuptext}% |
|
1391 |
\isamarkuptrue% |
|
1392 |
% |
|
1393 |
\isamarkupsection{Calculational reasoning \label{sec:calculation}% |
|
1394 |
} |
|
1395 |
\isamarkuptrue% |
|
1396 |
% |
|
1397 |
\begin{isamarkuptext}% |
|
1398 |
\begin{matharray}{rcl} |
|
40406 | 1399 |
\indexdef{}{command}{also}\hypertarget{command.also}{\hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
1400 |
\indexdef{}{command}{finally}\hypertarget{command.finally}{\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
1401 |
\indexdef{}{command}{moreover}\hypertarget{command.moreover}{\hyperlink{command.moreover}{\mbox{\isa{\isacommand{moreover}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
1402 |
\indexdef{}{command}{ultimately}\hypertarget{command.ultimately}{\hyperlink{command.ultimately}{\mbox{\isa{\isacommand{ultimately}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}chain{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
1403 |
\indexdef{}{command}{print\_trans\_rules}\hypertarget{command.print-trans-rules}{\hyperlink{command.print-trans-rules}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}trans{\isaliteral{5F}{\isacharunderscore}}rules}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
28788 | 1404 |
\hyperlink{attribute.trans}{\mbox{\isa{trans}}} & : & \isa{attribute} \\ |
1405 |
\hyperlink{attribute.sym}{\mbox{\isa{sym}}} & : & \isa{attribute} \\ |
|
1406 |
\hyperlink{attribute.symmetric}{\mbox{\isa{symmetric}}} & : & \isa{attribute} \\ |
|
26870 | 1407 |
\end{matharray} |
1408 |
||
1409 |
Calculational proof is forward reasoning with implicit application |
|
40406 | 1410 |
of transitivity rules (such those of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3D}{\isacharequal}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C653E}{\isasymle}}{\isaliteral{22}{\isachardoublequote}}}, |
1411 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3C}{\isacharless}}{\isaliteral{22}{\isachardoublequote}}}). Isabelle/Isar maintains an auxiliary fact register |
|
26902 | 1412 |
\indexref{}{fact}{calculation}\hyperlink{fact.calculation}{\mbox{\isa{calculation}}} for accumulating results obtained by |
1413 |
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 |
|
1414 |
\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}} exhibits the final \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} by |
|
26870 | 1415 |
forward chaining towards the next goal statement. Both commands |
1416 |
require valid current facts, i.e.\ may occur only after commands |
|
26902 | 1417 |
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}}}} |
1418 |
commands are similar to \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}} and \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}, |
|
1419 |
but only collect further results in \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} without |
|
26870 | 1420 |
applying any rules yet. |
1421 |
||
40406 | 1422 |
Also note that the implicit term abbreviation ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}'' has |
26870 | 1423 |
its canonical application with calculational proofs. It refers to |
1424 |
the argument of the preceding statement. (The argument of a curried |
|
1425 |
infix expression happens to be its right-hand side.) |
|
1426 |
||
1427 |
Isabelle/Isar calculations are implicitly subject to block structure |
|
1428 |
in the sense that new threads of calculational reasoning are |
|
1429 |
commenced for any new block (as opened by a local goal, for |
|
1430 |
example). This means that, apart from being able to nest |
|
1431 |
calculations, there is no separate \emph{begin-calculation} command |
|
1432 |
required. |
|
1433 |
||
1434 |
\medskip The Isar calculation proof commands may be defined as |
|
1435 |
follows:\footnote{We suppress internal bookkeeping such as proper |
|
1436 |
handling of block-structure.} |
|
1437 |
||
1438 |
\begin{matharray}{rcl} |
|
40406 | 1439 |
\hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{0}}{\isaliteral{22}{\isachardoublequote}}} & \equiv & \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}calculation\ {\isaliteral{3D}{\isacharequal}}\ this{\isaliteral{22}{\isachardoublequote}}} \\ |
1440 |
\hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} & \equiv & \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}calculation\ {\isaliteral{3D}{\isacharequal}}\ trans\ {\isaliteral{5B}{\isacharbrackleft}}OF\ calculation\ this{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}} \\[0.5ex] |
|
26902 | 1441 |
\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] |
40406 | 1442 |
\hyperlink{command.moreover}{\mbox{\isa{\isacommand{moreover}}}} & \equiv & \hyperlink{command.note}{\mbox{\isa{\isacommand{note}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}calculation\ {\isaliteral{3D}{\isacharequal}}\ calculation\ this{\isaliteral{22}{\isachardoublequote}}} \\ |
26902 | 1443 |
\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 | 1444 |
\end{matharray} |
1445 |
||
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1446 |
\begin{railoutput} |
42662 | 1447 |
\rail@begin{2}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1448 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1449 |
\rail@term{\hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1450 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1451 |
\rail@term{\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1452 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1453 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1454 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1455 |
\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1456 |
\rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1457 |
\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1458 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1459 |
\rail@end |
42662 | 1460 |
\rail@begin{3}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1461 |
\rail@term{\hyperlink{attribute.trans}{\mbox{\isa{trans}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1462 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1463 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1464 |
\rail@term{\isa{add}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1465 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1466 |
\rail@term{\isa{del}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1467 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1468 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1469 |
\end{railoutput} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1470 |
|
26870 | 1471 |
|
28788 | 1472 |
\begin{description} |
26870 | 1473 |
|
40406 | 1474 |
\item \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} maintains the auxiliary |
28788 | 1475 |
\hyperlink{fact.calculation}{\mbox{\isa{calculation}}} register as follows. The first occurrence of |
1476 |
\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 |
|
1477 |
the same level of block-structure updates \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} by |
|
1478 |
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 |
|
1479 |
current context, unless alternative rules are given as explicit |
|
1480 |
arguments. |
|
26870 | 1481 |
|
40406 | 1482 |
\item \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}a\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ a\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} maintaining \hyperlink{fact.calculation}{\mbox{\isa{calculation}}} in the same way as \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}}, and concludes the |
28788 | 1483 |
current calculational thread. The final result is exhibited as fact |
1484 |
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 |
|
40406 | 1485 |
calculational proofs are ``\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}~\hyperlink{command.show}{\mbox{\isa{\isacommand{show}}}}~\isa{{\isaliteral{3F}{\isacharquery}}thesis}~\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}}}}}'' and ``\hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}~\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{5C3C7068693E}{\isasymphi}}}~\hyperlink{command.dot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}}}}}''. |
26870 | 1486 |
|
28788 | 1487 |
\item \hyperlink{command.moreover}{\mbox{\isa{\isacommand{moreover}}}} and \hyperlink{command.ultimately}{\mbox{\isa{\isacommand{ultimately}}}} are |
26902 | 1488 |
analogous to \hyperlink{command.also}{\mbox{\isa{\isacommand{also}}}} and \hyperlink{command.finally}{\mbox{\isa{\isacommand{finally}}}}, but collect |
26870 | 1489 |
results only, without applying rules. |
1490 |
||
40406 | 1491 |
\item \hyperlink{command.print-trans-rules}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}trans{\isaliteral{5F}{\isacharunderscore}}rules}}}} prints the list of transitivity |
28788 | 1492 |
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}}} |
1493 |
operation and single step elimination patters) of the current |
|
1494 |
context. |
|
26870 | 1495 |
|
28788 | 1496 |
\item \hyperlink{attribute.trans}{\mbox{\isa{trans}}} declares theorems as transitivity rules. |
26870 | 1497 |
|
28788 | 1498 |
\item \hyperlink{attribute.sym}{\mbox{\isa{sym}}} declares symmetry rules, as well as |
40406 | 1499 |
\hyperlink{attribute.Pure.elim}{\mbox{\isa{Pure{\isaliteral{2E}{\isachardot}}elim}}}\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}} rules. |
26870 | 1500 |
|
28788 | 1501 |
\item \hyperlink{attribute.symmetric}{\mbox{\isa{symmetric}}} resolves a theorem with some rule |
26902 | 1502 |
declared as \hyperlink{attribute.sym}{\mbox{\isa{sym}}} in the current context. For example, |
40406 | 1503 |
``\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5B}{\isacharbrackleft}}symmetric{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequote}}}'' produces a |
26870 | 1504 |
swapped fact derived from that assumption. |
1505 |
||
1506 |
In structured proof texts it is often more appropriate to use an |
|
40406 | 1507 |
explicit single-step elimination proof, such as ``\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.then}{\mbox{\isa{\isacommand{then}}}}~\hyperlink{command.have}{\mbox{\isa{\isacommand{have}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}y\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.ddot}{\mbox{\isa{\isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}}}}''. |
26870 | 1508 |
|
28788 | 1509 |
\end{description}% |
26870 | 1510 |
\end{isamarkuptext}% |
1511 |
\isamarkuptrue% |
|
1512 |
% |
|
27042 | 1513 |
\isamarkupsection{Proof by cases and induction \label{sec:cases-induct}% |
1514 |
} |
|
1515 |
\isamarkuptrue% |
|
1516 |
% |
|
1517 |
\isamarkupsubsection{Rule contexts% |
|
1518 |
} |
|
1519 |
\isamarkuptrue% |
|
1520 |
% |
|
1521 |
\begin{isamarkuptext}% |
|
1522 |
\begin{matharray}{rcl} |
|
40406 | 1523 |
\indexdef{}{command}{case}\hypertarget{command.case}{\hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}state{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\ |
1524 |
\indexdef{}{command}{print\_cases}\hypertarget{command.print-cases}{\hyperlink{command.print-cases}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}cases}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\ |
|
1525 |
\indexdef{}{attribute}{case\_names}\hypertarget{attribute.case-names}{\hyperlink{attribute.case-names}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}names}}}} & : & \isa{attribute} \\ |
|
1526 |
\indexdef{}{attribute}{case\_conclusion}\hypertarget{attribute.case-conclusion}{\hyperlink{attribute.case-conclusion}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}conclusion}}}} & : & \isa{attribute} \\ |
|
28788 | 1527 |
\indexdef{}{attribute}{params}\hypertarget{attribute.params}{\hyperlink{attribute.params}{\mbox{\isa{params}}}} & : & \isa{attribute} \\ |
1528 |
\indexdef{}{attribute}{consumes}\hypertarget{attribute.consumes}{\hyperlink{attribute.consumes}{\mbox{\isa{consumes}}}} & : & \isa{attribute} \\ |
|
27042 | 1529 |
\end{matharray} |
1530 |
||
1531 |
The puristic way to build up Isar proof contexts is by explicit |
|
1532 |
language elements like \hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}, \hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}, |
|
1533 |
\hyperlink{command.let}{\mbox{\isa{\isacommand{let}}}} (see \secref{sec:proof-context}). This is adequate |
|
1534 |
for plain natural deduction, but easily becomes unwieldy in concrete |
|
1535 |
verification tasks, which typically involve big induction rules with |
|
1536 |
several cases. |
|
1537 |
||
1538 |
The \hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}} command provides a shorthand to refer to a |
|
1539 |
local context symbolically: certain proof methods provide an |
|
40406 | 1540 |
environment of named ``cases'' of the form \isa{{\isaliteral{22}{\isachardoublequote}}c{\isaliteral{3A}{\isacharcolon}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\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{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}''. Term bindings may be covered as well, notably |
1541 |
\hyperlink{variable.?case}{\mbox{\isa{{\isaliteral{3F}{\isacharquery}}case}}} for the main conclusion. |
|
27042 | 1542 |
|
40406 | 1543 |
By default, the ``terminology'' \isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}} of |
27042 | 1544 |
a case value is marked as hidden, i.e.\ there is no way to refer to |
1545 |
such parameters in the subsequent proof text. After all, original |
|
1546 |
rule parameters stem from somewhere outside of the current proof |
|
40406 | 1547 |
text. By using the explicit form ``\hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}c\ y\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ y\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' instead, the proof author is able to |
27042 | 1548 |
chose local names that fit nicely into the current context. |
1549 |
||
1550 |
\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, |
|
1551 |
which is not directly observable in Isar! Nonetheless, goal |
|
40406 | 1552 |
refinement commands do provide named cases \isa{{\isaliteral{22}{\isachardoublequote}}goal\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} |
1553 |
for each subgoal \isa{{\isaliteral{22}{\isachardoublequote}}i\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{22}{\isachardoublequote}}} of the resulting goal state. |
|
27042 | 1554 |
Using this extra feature requires great care, because some bits of |
1555 |
the internal tactical machinery intrude the proof text. In |
|
1556 |
particular, parameter names stemming from the left-over of automated |
|
1557 |
reasoning tools are usually quite unpredictable. |
|
1558 |
||
1559 |
Under normal circumstances, the text of cases emerge from standard |
|
1560 |
elimination or induction rules, which in turn are derived from |
|
1561 |
previous theory specifications in a canonical way (say from |
|
1562 |
\hyperlink{command.inductive}{\mbox{\isa{\isacommand{inductive}}}} definitions). |
|
1563 |
||
1564 |
\medskip Proper cases are only available if both the proof method |
|
1565 |
and the rules involved support this. By using appropriate |
|
1566 |
attributes, case names, conclusions, and parameters may be also |
|
1567 |
declared by hand. Thus variant versions of rules that have been |
|
1568 |
derived manually become ready to use in advanced case analysis |
|
1569 |
later. |
|
1570 |
||
42596
6c621a9d612a
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|
1571 |
\begin{railoutput} |
42662 | 1572 |
\rail@begin{4}{} |
42596
6c621a9d612a
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parents:
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changeset
|
1573 |
\rail@term{\hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1574 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
1575 |
\rail@nont{\isa{caseref}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1576 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1577 |
\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1578 |
\rail@nont{\isa{caseref}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1579 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1580 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1581 |
\rail@term{\isa{{\isaliteral{5F}{\isacharunderscore}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1582 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1583 |
\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1584 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1585 |
\rail@nextplus{3} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1586 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1587 |
\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1588 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1589 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1590 |
\rail@begin{2}{\isa{caseref}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1591 |
\rail@nont{\isa{nameref}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1592 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1593 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1594 |
\rail@nont{\isa{attributes}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1595 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1596 |
\rail@end |
44164
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1597 |
\rail@begin{5}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1598 |
\rail@term{\hyperlink{attribute.case-names}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}names}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1599 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1600 |
\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[] |
44164
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
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diff
changeset
|
1601 |
\rail@bar |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1602 |
\rail@nextbar{1} |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1603 |
\rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[] |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1604 |
\rail@plus |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1605 |
\rail@bar |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1606 |
\rail@term{\isa{{\isaliteral{5F}{\isacharunderscore}}}}[] |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1607 |
\rail@nextbar{2} |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1608 |
\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[] |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1609 |
\rail@endbar |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1610 |
\rail@nextplus{3} |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1611 |
\rail@endplus |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1612 |
\rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[] |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1613 |
\rail@endbar |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1614 |
\rail@nextplus{4} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1615 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1616 |
\rail@end |
42662 | 1617 |
\rail@begin{2}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1618 |
\rail@term{\hyperlink{attribute.case-conclusion}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}conclusion}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
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|
1619 |
\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1620 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1621 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1622 |
\rail@cnont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1623 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1624 |
\rail@end |
42662 | 1625 |
\rail@begin{3}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1626 |
\rail@term{\hyperlink{attribute.params}{\mbox{\isa{params}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1627 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1628 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1629 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1630 |
\rail@cnont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1631 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1632 |
\rail@nextplus{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1633 |
\rail@cterm{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1634 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1635 |
\rail@end |
42662 | 1636 |
\rail@begin{2}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1637 |
\rail@term{\hyperlink{attribute.consumes}{\mbox{\isa{consumes}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1638 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1639 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1640 |
\rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1641 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1642 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1643 |
\end{railoutput} |
27042 | 1644 |
|
1645 |
||
28788 | 1646 |
\begin{description} |
27042 | 1647 |
|
40406 | 1648 |
\item \hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}c\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} invokes a named local |
1649 |
context \isa{{\isaliteral{22}{\isachardoublequote}}c{\isaliteral{3A}{\isacharcolon}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}}, as provided by an |
|
28788 | 1650 |
appropriate proof method (such as \indexref{}{method}{cases}\hyperlink{method.cases}{\mbox{\isa{cases}}} and |
40406 | 1651 |
\indexref{}{method}{induct}\hyperlink{method.induct}{\mbox{\isa{induct}}}). The command ``\hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}c\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' abbreviates ``\hyperlink{command.fix}{\mbox{\isa{\isacommand{fix}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}}~\hyperlink{command.assume}{\mbox{\isa{\isacommand{assume}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}''. |
27042 | 1652 |
|
40406 | 1653 |
\item \hyperlink{command.print-cases}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}cases}}}} prints all local contexts of the |
27042 | 1654 |
current state, using Isar proof language notation. |
1655 |
||
40406 | 1656 |
\item \hyperlink{attribute.case-names}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}names}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub k{\isaliteral{22}{\isachardoublequote}}} declares names for |
1657 |
the local contexts of premises of a theorem; \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub k{\isaliteral{22}{\isachardoublequote}}} |
|
44164
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1658 |
refers to the \emph{prefix} of the list of premises. Each of the |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1659 |
cases \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E697375623E}{}\isactrlisub i{\isaliteral{22}{\isachardoublequote}}} can be of the form \isa{{\isaliteral{22}{\isachardoublequote}}c{\isaliteral{5B}{\isacharbrackleft}}h\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ h\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequote}}} where |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1660 |
the \isa{{\isaliteral{22}{\isachardoublequote}}h\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ h\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}} are the names of the hypotheses in case \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E697375623E}{}\isactrlisub i{\isaliteral{22}{\isachardoublequote}}} |
238c5ea1e2ce
documented extended version of case_names attribute
nipkow
parents:
43633
diff
changeset
|
1661 |
from left to right. |
27042 | 1662 |
|
40406 | 1663 |
\item \hyperlink{attribute.case-conclusion}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}conclusion}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c\ d\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ d\isaliteral{5C3C5E7375623E}{}\isactrlsub k{\isaliteral{22}{\isachardoublequote}}} declares |
1664 |
names for the conclusions of a named premise \isa{c}; here \isa{{\isaliteral{22}{\isachardoublequote}}d\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ d\isaliteral{5C3C5E7375623E}{}\isactrlsub k{\isaliteral{22}{\isachardoublequote}}} refers to the prefix of arguments of a logical formula |
|
1665 |
built by nesting a binary connective (e.g.\ \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6F723E}{\isasymor}}{\isaliteral{22}{\isachardoublequote}}}). |
|
27042 | 1666 |
|
1667 |
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 |
|
1668 |
whole. The need to name subformulas only arises with cases that |
|
1669 |
split into several sub-cases, as in common co-induction rules. |
|
1670 |
||
40406 | 1671 |
\item \hyperlink{attribute.params}{\mbox{\isa{params}}}~\isa{{\isaliteral{22}{\isachardoublequote}}p\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ p\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{5C3C414E443E}{\isasymAND}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ q\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} renames |
1672 |
the innermost parameters of premises \isa{{\isaliteral{22}{\isachardoublequote}}{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{22}{\isachardoublequote}}} of some |
|
28788 | 1673 |
theorem. An empty list of names may be given to skip positions, |
1674 |
leaving the present parameters unchanged. |
|
27042 | 1675 |
|
1676 |
Note that the default usage of case rules does \emph{not} directly |
|
1677 |
expose parameters to the proof context. |
|
1678 |
||
28788 | 1679 |
\item \hyperlink{attribute.consumes}{\mbox{\isa{consumes}}}~\isa{n} declares the number of ``major |
1680 |
premises'' of a rule, i.e.\ the number of facts to be consumed when |
|
1681 |
it is applied by an appropriate proof method. The default value of |
|
40406 | 1682 |
\hyperlink{attribute.consumes}{\mbox{\isa{consumes}}} is \isa{{\isaliteral{22}{\isachardoublequote}}n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}, which is appropriate for |
28788 | 1683 |
the usual kind of cases and induction rules for inductive sets (cf.\ |
1684 |
\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 | 1685 |
|
1686 |
Note that explicit \hyperlink{attribute.consumes}{\mbox{\isa{consumes}}} declarations are only |
|
1687 |
rarely needed; this is already taken care of automatically by the |
|
1688 |
higher-level \hyperlink{attribute.cases}{\mbox{\isa{cases}}}, \hyperlink{attribute.induct}{\mbox{\isa{induct}}}, and |
|
1689 |
\hyperlink{attribute.coinduct}{\mbox{\isa{coinduct}}} declarations. |
|
1690 |
||
28788 | 1691 |
\end{description}% |
27042 | 1692 |
\end{isamarkuptext}% |
1693 |
\isamarkuptrue% |
|
1694 |
% |
|
1695 |
\isamarkupsubsection{Proof methods% |
|
1696 |
} |
|
1697 |
\isamarkuptrue% |
|
1698 |
% |
|
1699 |
\begin{isamarkuptext}% |
|
1700 |
\begin{matharray}{rcl} |
|
28788 | 1701 |
\indexdef{}{method}{cases}\hypertarget{method.cases}{\hyperlink{method.cases}{\mbox{\isa{cases}}}} & : & \isa{method} \\ |
1702 |
\indexdef{}{method}{induct}\hypertarget{method.induct}{\hyperlink{method.induct}{\mbox{\isa{induct}}}} & : & \isa{method} \\ |
|
45014
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1703 |
\indexdef{}{method}{induction}\hypertarget{method.induction}{\hyperlink{method.induction}{\mbox{\isa{induction}}}} & : & \isa{method} \\ |
28788 | 1704 |
\indexdef{}{method}{coinduct}\hypertarget{method.coinduct}{\hyperlink{method.coinduct}{\mbox{\isa{coinduct}}}} & : & \isa{method} \\ |
27042 | 1705 |
\end{matharray} |
1706 |
||
45014
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1707 |
The \hyperlink{method.cases}{\mbox{\isa{cases}}}, \hyperlink{method.induct}{\mbox{\isa{induct}}}, \hyperlink{method.induction}{\mbox{\isa{induction}}}, |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1708 |
and \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} |
27042 | 1709 |
methods provide a uniform interface to common proof techniques over |
1710 |
datatypes, inductive predicates (or sets), recursive functions etc. |
|
1711 |
The corresponding rules may be specified and instantiated in a |
|
1712 |
casual manner. Furthermore, these methods provide named local |
|
1713 |
contexts that may be invoked via the \hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}} proof command |
|
1714 |
within the subsequent proof text. This accommodates compact proof |
|
1715 |
texts even when reasoning about large specifications. |
|
1716 |
||
1717 |
The \hyperlink{method.induct}{\mbox{\isa{induct}}} method also provides some additional |
|
1718 |
infrastructure in order to be applicable to structure statements |
|
1719 |
(either using explicit meta-level connectives, or including facts |
|
1720 |
and parameters separately). This avoids cumbersome encoding of |
|
1721 |
``strengthened'' inductive statements within the object-logic. |
|
1722 |
||
45014
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1723 |
Method \hyperlink{method.induction}{\mbox{\isa{induction}}} differs from \hyperlink{method.induct}{\mbox{\isa{induct}}} only in |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1724 |
the names of the facts in the local context invoked by the \hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}} |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1725 |
command. |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1726 |
|
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1727 |
\begin{railoutput} |
42704 | 1728 |
\rail@begin{6}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1729 |
\rail@term{\hyperlink{method.cases}{\mbox{\isa{cases}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1730 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1731 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1732 |
\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1733 |
\rail@term{\isa{no{\isaliteral{5F}{\isacharunderscore}}simp}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1734 |
\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1735 |
\rail@endbar |
42704 | 1736 |
\rail@cr{3} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1737 |
\rail@bar |
42704 | 1738 |
\rail@nextbar{4} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1739 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1740 |
\rail@nont{\hyperlink{syntax.insts}{\mbox{\isa{insts}}}}[] |
42704 | 1741 |
\rail@nextplus{5} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1742 |
\rail@cterm{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1743 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1744 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1745 |
\rail@bar |
42704 | 1746 |
\rail@nextbar{4} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1747 |
\rail@nont{\isa{rule}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1748 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1749 |
\rail@end |
42662 | 1750 |
\rail@begin{6}{} |
45014
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1751 |
\rail@bar |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1752 |
\rail@term{\hyperlink{method.induct}{\mbox{\isa{induct}}}}[] |
45014
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1753 |
\rail@nextbar{1} |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1754 |
\rail@term{\hyperlink{method.induction}{\mbox{\isa{induction}}}}[] |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1755 |
\rail@endbar |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1756 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1757 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1758 |
\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1759 |
\rail@term{\isa{no{\isaliteral{5F}{\isacharunderscore}}simp}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1760 |
\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1761 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1762 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1763 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1764 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1765 |
\rail@nont{\isa{definsts}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1766 |
\rail@nextplus{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1767 |
\rail@cterm{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1768 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1769 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1770 |
\rail@cr{4} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1771 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1772 |
\rail@nextbar{5} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1773 |
\rail@nont{\isa{arbitrary}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1774 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1775 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1776 |
\rail@nextbar{5} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1777 |
\rail@nont{\isa{taking}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1778 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1779 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1780 |
\rail@nextbar{5} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1781 |
\rail@nont{\isa{rule}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1782 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1783 |
\rail@end |
42662 | 1784 |
\rail@begin{2}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1785 |
\rail@term{\hyperlink{method.coinduct}{\mbox{\isa{coinduct}}}}[] |
42617 | 1786 |
\rail@nont{\hyperlink{syntax.insts}{\mbox{\isa{insts}}}}[] |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1787 |
\rail@nont{\isa{taking}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1788 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1789 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1790 |
\rail@nont{\isa{rule}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1791 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1792 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1793 |
\rail@begin{5}{\isa{rule}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1794 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1795 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1796 |
\rail@term{\isa{type}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1797 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1798 |
\rail@term{\isa{pred}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1799 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1800 |
\rail@term{\isa{set}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1801 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1802 |
\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1803 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1804 |
\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1805 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
1806 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1807 |
\rail@nextbar{3} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1808 |
\rail@term{\isa{rule}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1809 |
\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1810 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1811 |
\rail@nont{\hyperlink{syntax.thmref}{\mbox{\isa{thmref}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1812 |
\rail@nextplus{4} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1813 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1814 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1815 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1816 |
\rail@begin{4}{\isa{definst}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1817 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1818 |
\rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1819 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1820 |
\rail@term{\isa{{\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1821 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1822 |
\rail@term{\isa{{\isaliteral{5C3C65717569763E}{\isasymequiv}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1823 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1824 |
\rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1825 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1826 |
\rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1827 |
\rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1828 |
\rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1829 |
\rail@nextbar{3} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1830 |
\rail@nont{\hyperlink{syntax.inst}{\mbox{\isa{inst}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1831 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1832 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1833 |
\rail@begin{2}{\isa{definsts}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1834 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1835 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1836 |
\rail@cnont{\isa{definst}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1837 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1838 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
1839 |
\rail@begin{3}{\isa{arbitrary}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
1840 |
\rail@term{\isa{arbitrary}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
1841 |
\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1842 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1843 |
\rail@plus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1844 |
\rail@nextplus{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1845 |
\rail@cnont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1846 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1847 |
\rail@term{\isa{\isakeyword{and}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1848 |
\rail@nextplus{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1849 |
\rail@endplus |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1850 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1851 |
\rail@begin{1}{\isa{taking}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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diff
changeset
|
1852 |
\rail@term{\isa{taking}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1853 |
\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[] |
42617 | 1854 |
\rail@nont{\hyperlink{syntax.insts}{\mbox{\isa{insts}}}}[] |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
1855 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
1856 |
\end{railoutput} |
27042 | 1857 |
|
1858 |
||
28788 | 1859 |
\begin{description} |
27042 | 1860 |
|
40406 | 1861 |
\item \hyperlink{method.cases}{\mbox{\isa{cases}}}~\isa{{\isaliteral{22}{\isachardoublequote}}insts\ R{\isaliteral{22}{\isachardoublequote}}} applies method \hyperlink{method.rule}{\mbox{\isa{rule}}} with an appropriate case distinction theorem, instantiated to |
27042 | 1862 |
the subjects \isa{insts}. Symbolic case names are bound according |
1863 |
to the rule's local contexts. |
|
1864 |
||
1865 |
The rule is determined as follows, according to the facts and |
|
1866 |
arguments passed to the \hyperlink{method.cases}{\mbox{\isa{cases}}} method: |
|
1867 |
||
1868 |
\medskip |
|
1869 |
\begin{tabular}{llll} |
|
1870 |
facts & & arguments & rule \\\hline |
|
1871 |
& \hyperlink{method.cases}{\mbox{\isa{cases}}} & & classical case split \\ |
|
1872 |
& \hyperlink{method.cases}{\mbox{\isa{cases}}} & \isa{t} & datatype exhaustion (type of \isa{t}) \\ |
|
40406 | 1873 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A\ t{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{method.cases}{\mbox{\isa{cases}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & inductive predicate/set elimination (of \isa{A}) \\ |
1874 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{method.cases}{\mbox{\isa{cases}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ rule{\isaliteral{3A}{\isacharcolon}}\ R{\isaliteral{22}{\isachardoublequote}}} & explicit rule \isa{R} \\ |
|
27042 | 1875 |
\end{tabular} |
1876 |
\medskip |
|
1877 |
||
1878 |
Several instantiations may be given, referring to the \emph{suffix} |
|
1879 |
of premises of the case rule; within each premise, the \emph{prefix} |
|
1880 |
of variables is instantiated. In most situations, only a single |
|
1881 |
term needs to be specified; this refers to the first variable of the |
|
40406 | 1882 |
last premise (it is usually the same for all cases). The \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}no{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} option can be used to disable pre-simplification of |
37364
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
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changeset
|
1883 |
cases (see the description of \hyperlink{method.induct}{\mbox{\isa{induct}}} below for details). |
27042 | 1884 |
|
45014
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1885 |
\item \hyperlink{method.induct}{\mbox{\isa{induct}}}~\isa{{\isaliteral{22}{\isachardoublequote}}insts\ R{\isaliteral{22}{\isachardoublequote}}} and |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1886 |
\hyperlink{method.induction}{\mbox{\isa{induction}}}~\isa{{\isaliteral{22}{\isachardoublequote}}insts\ R{\isaliteral{22}{\isachardoublequote}}} are analogous to the |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
1887 |
\hyperlink{method.cases}{\mbox{\isa{cases}}} method, but refer to induction rules, which are |
27042 | 1888 |
determined as follows: |
1889 |
||
1890 |
\medskip |
|
1891 |
\begin{tabular}{llll} |
|
1892 |
facts & & arguments & rule \\\hline |
|
40406 | 1893 |
& \hyperlink{method.induct}{\mbox{\isa{induct}}} & \isa{{\isaliteral{22}{\isachardoublequote}}P\ x{\isaliteral{22}{\isachardoublequote}}} & datatype induction (type of \isa{x}) \\ |
1894 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ A\ x{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{method.induct}{\mbox{\isa{induct}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & predicate/set induction (of \isa{A}) \\ |
|
1895 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{method.induct}{\mbox{\isa{induct}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ rule{\isaliteral{3A}{\isacharcolon}}\ R{\isaliteral{22}{\isachardoublequote}}} & explicit rule \isa{R} \\ |
|
27042 | 1896 |
\end{tabular} |
1897 |
\medskip |
|
1898 |
||
1899 |
Several instantiations may be given, each referring to some part of |
|
1900 |
a mutual inductive definition or datatype --- only related partial |
|
1901 |
induction rules may be used together, though. Any of the lists of |
|
40406 | 1902 |
terms \isa{{\isaliteral{22}{\isachardoublequote}}P{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} refers to the \emph{suffix} of variables |
27042 | 1903 |
present in the induction rule. This enables the writer to specify |
1904 |
only induction variables, or both predicates and variables, for |
|
1905 |
example. |
|
37364
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
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changeset
|
1906 |
|
40406 | 1907 |
Instantiations may be definitional: equations \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\isachardoublequote}}} |
27042 | 1908 |
introduce local definitions, which are inserted into the claim and |
1909 |
discharged after applying the induction rule. Equalities reappear |
|
1910 |
in the inductive cases, but have been transformed according to the |
|
1911 |
induction principle being involved here. In order to achieve |
|
1912 |
practically useful induction hypotheses, some variables occurring in |
|
37364
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
37361
diff
changeset
|
1913 |
\isa{t} need to be fixed (see below). Instantiations of the form |
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
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changeset
|
1914 |
\isa{t}, where \isa{t} is not a variable, are taken as a |
40406 | 1915 |
shorthand for \mbox{\isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ t{\isaliteral{22}{\isachardoublequote}}}}, where \isa{x} is a fresh |
37364
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
37361
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changeset
|
1916 |
variable. If this is not intended, \isa{t} has to be enclosed in |
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
37361
diff
changeset
|
1917 |
parentheses. By default, the equalities generated by definitional |
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
37361
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changeset
|
1918 |
instantiations are pre-simplified using a specific set of rules, |
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
37361
diff
changeset
|
1919 |
usually consisting of distinctness and injectivity theorems for |
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
37361
diff
changeset
|
1920 |
datatypes. This pre-simplification may cause some of the parameters |
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
37361
diff
changeset
|
1921 |
of an inductive case to disappear, or may even completely delete |
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
37361
diff
changeset
|
1922 |
some of the inductive cases, if one of the equalities occurring in |
40406 | 1923 |
their premises can be simplified to \isa{False}. The \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}no{\isaliteral{5F}{\isacharunderscore}}simp{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} option can be used to disable pre-simplification. |
37364
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
37361
diff
changeset
|
1924 |
Additional rules to be used in pre-simplification can be declared |
40406 | 1925 |
using the \indexdef{}{attribute}{induct\_simp}\hypertarget{attribute.induct-simp}{\hyperlink{attribute.induct-simp}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}simp}}}} attribute. |
37364
dfca6c4cd1e8
more uniform treatment of options and attributes, preferring formal markup over old-style LaTeX macros;
wenzelm
parents:
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diff
changeset
|
1926 |
|
40406 | 1927 |
The optional ``\isa{{\isaliteral{22}{\isachardoublequote}}arbitrary{\isaliteral{3A}{\isacharcolon}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}}'' |
43633 | 1928 |
specification generalizes variables \isa{{\isaliteral{22}{\isachardoublequote}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}} of the original goal before applying induction. One can |
1929 |
separate variables by ``\isa{{\isaliteral{22}{\isachardoublequote}}and{\isaliteral{22}{\isachardoublequote}}}'' to generalize them in other |
|
1930 |
goals then the first. Thus induction hypotheses may become |
|
1931 |
sufficiently general to get the proof through. Together with |
|
1932 |
definitional instantiations, one may effectively perform induction |
|
1933 |
over expressions of a certain structure. |
|
27042 | 1934 |
|
40406 | 1935 |
The optional ``\isa{{\isaliteral{22}{\isachardoublequote}}taking{\isaliteral{3A}{\isacharcolon}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}'' |
27042 | 1936 |
specification provides additional instantiations of a prefix of |
1937 |
pending variables in the rule. Such schematic induction rules |
|
1938 |
rarely occur in practice, though. |
|
1939 |
||
40406 | 1940 |
\item \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}}~\isa{{\isaliteral{22}{\isachardoublequote}}inst\ R{\isaliteral{22}{\isachardoublequote}}} is analogous to the |
27042 | 1941 |
\hyperlink{method.induct}{\mbox{\isa{induct}}} method, but refers to coinduction rules, which are |
1942 |
determined as follows: |
|
1943 |
||
1944 |
\medskip |
|
1945 |
\begin{tabular}{llll} |
|
1946 |
goal & & arguments & rule \\\hline |
|
1947 |
& \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} & \isa{x} & type coinduction (type of \isa{x}) \\ |
|
40406 | 1948 |
\isa{{\isaliteral{22}{\isachardoublequote}}A\ x{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & predicate/set coinduction (of \isa{A}) \\ |
1949 |
\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} & \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ rule{\isaliteral{3A}{\isacharcolon}}\ R{\isaliteral{22}{\isachardoublequote}}} & explicit rule \isa{R} \\ |
|
27042 | 1950 |
\end{tabular} |
1951 |
||
1952 |
Coinduction is the dual of induction. Induction essentially |
|
40406 | 1953 |
eliminates \isa{{\isaliteral{22}{\isachardoublequote}}A\ x{\isaliteral{22}{\isachardoublequote}}} towards a generic result \isa{{\isaliteral{22}{\isachardoublequote}}P\ x{\isaliteral{22}{\isachardoublequote}}}, |
1954 |
while coinduction introduces \isa{{\isaliteral{22}{\isachardoublequote}}A\ x{\isaliteral{22}{\isachardoublequote}}} starting with \isa{{\isaliteral{22}{\isachardoublequote}}B\ x{\isaliteral{22}{\isachardoublequote}}}, for a suitable ``bisimulation'' \isa{B}. The cases of a |
|
27042 | 1955 |
coinduct rule are typically named after the predicates or sets being |
1956 |
covered, while the conclusions consist of several alternatives being |
|
1957 |
named after the individual destructor patterns. |
|
1958 |
||
1959 |
The given instantiation refers to the \emph{suffix} of variables |
|
1960 |
occurring in the rule's major premise, or conclusion if unavailable. |
|
40406 | 1961 |
An additional ``\isa{{\isaliteral{22}{\isachardoublequote}}taking{\isaliteral{3A}{\isacharcolon}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}'' |
27042 | 1962 |
specification may be required in order to specify the bisimulation |
1963 |
to be used in the coinduction step. |
|
1964 |
||
28788 | 1965 |
\end{description} |
27042 | 1966 |
|
1967 |
Above methods produce named local contexts, as determined by the |
|
1968 |
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 |
|
1969 |
from the goal specification itself. Any persisting unresolved |
|
1970 |
schematic variables of the resulting rule will render the the |
|
40406 | 1971 |
corresponding case invalid. The term binding \hyperlink{variable.?case}{\mbox{\isa{{\isaliteral{3F}{\isacharquery}}case}}} for |
27042 | 1972 |
the conclusion will be provided with each case, provided that term |
1973 |
is fully specified. |
|
1974 |
||
40406 | 1975 |
The \hyperlink{command.print-cases}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}cases}}}} command prints all named cases present |
27042 | 1976 |
in the current proof state. |
1977 |
||
1978 |
\medskip Despite the additional infrastructure, both \hyperlink{method.cases}{\mbox{\isa{cases}}} |
|
1979 |
and \hyperlink{method.coinduct}{\mbox{\isa{coinduct}}} merely apply a certain rule, after |
|
1980 |
instantiation, while conforming due to the usual way of monotonic |
|
40406 | 1981 |
natural deduction: the context of a structured statement \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7068693E}{\isasymphi}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} |
27042 | 1982 |
reappears unchanged after the case split. |
1983 |
||
1984 |
The \hyperlink{method.induct}{\mbox{\isa{induct}}} method is fundamentally different in this |
|
1985 |
respect: the meta-level structure is passed through the |
|
1986 |
``recursive'' course involved in the induction. Thus the original |
|
1987 |
statement is basically replaced by separate copies, corresponding to |
|
1988 |
the induction hypotheses and conclusion; the original goal context |
|
1989 |
is no longer available. Thus local assumptions, fixed parameters |
|
1990 |
and definitions effectively participate in the inductive rephrasing |
|
1991 |
of the original statement. |
|
1992 |
||
45014
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New proof method "induction" that gives induction hypotheses the name IH.
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|
1993 |
In \hyperlink{method.induct}{\mbox{\isa{induct}}} proofs, local assumptions introduced by cases are split |
27042 | 1994 |
into two different kinds: \isa{hyps} stemming from the rule and |
1995 |
\isa{prems} from the goal statement. This is reflected in the |
|
40406 | 1996 |
extracted cases accordingly, so invoking ``\hyperlink{command.case}{\mbox{\isa{\isacommand{case}}}}~\isa{c}'' will provide separate facts \isa{c{\isaliteral{2E}{\isachardot}}hyps} and \isa{c{\isaliteral{2E}{\isachardot}}prems}, |
27042 | 1997 |
as well as fact \isa{c} to hold the all-inclusive list. |
1998 |
||
45014
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New proof method "induction" that gives induction hypotheses the name IH.
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changeset
|
1999 |
In \hyperlink{method.induction}{\mbox{\isa{induction}}} proofs, local assumptions introduced by cases are |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
2000 |
split into three different kinds: \isa{IH}, the induction hypotheses, |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
2001 |
\isa{hyps}, the remaining hypotheses stemming from the rule, and |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
2002 |
\isa{prems}, the assumptions from the goal statement. The names are |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
2003 |
\isa{c{\isaliteral{2E}{\isachardot}}IH}, \isa{c{\isaliteral{2E}{\isachardot}}hyps} and \isa{c{\isaliteral{2E}{\isachardot}}prems}, as above. |
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
parents:
44164
diff
changeset
|
2004 |
|
0e847655b2d8
New proof method "induction" that gives induction hypotheses the name IH.
nipkow
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44164
diff
changeset
|
2005 |
|
27042 | 2006 |
\medskip Facts presented to either method are consumed according to |
2007 |
the number of ``major premises'' of the rule involved, which is |
|
2008 |
usually 0 for plain cases and induction rules of datatypes etc.\ and |
|
2009 |
1 for rules of inductive predicates or sets and the like. The |
|
2010 |
remaining facts are inserted into the goal verbatim before the |
|
2011 |
actual \isa{cases}, \isa{induct}, or \isa{coinduct} rule is |
|
2012 |
applied.% |
|
2013 |
\end{isamarkuptext}% |
|
2014 |
\isamarkuptrue% |
|
2015 |
% |
|
2016 |
\isamarkupsubsection{Declaring rules% |
|
2017 |
} |
|
2018 |
\isamarkuptrue% |
|
2019 |
% |
|
2020 |
\begin{isamarkuptext}% |
|
2021 |
\begin{matharray}{rcl} |
|
40406 | 2022 |
\indexdef{}{command}{print\_induct\_rules}\hypertarget{command.print-induct-rules}{\hyperlink{command.print-induct-rules}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}induct{\isaliteral{5F}{\isacharunderscore}}rules}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\ |
28788 | 2023 |
\indexdef{}{attribute}{cases}\hypertarget{attribute.cases}{\hyperlink{attribute.cases}{\mbox{\isa{cases}}}} & : & \isa{attribute} \\ |
2024 |
\indexdef{}{attribute}{induct}\hypertarget{attribute.induct}{\hyperlink{attribute.induct}{\mbox{\isa{induct}}}} & : & \isa{attribute} \\ |
|
2025 |
\indexdef{}{attribute}{coinduct}\hypertarget{attribute.coinduct}{\hyperlink{attribute.coinduct}{\mbox{\isa{coinduct}}}} & : & \isa{attribute} \\ |
|
27042 | 2026 |
\end{matharray} |
2027 |
||
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changeset
|
2028 |
\begin{railoutput} |
42662 | 2029 |
\rail@begin{1}{} |
42596
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modernized rail diagrams using @{rail} antiquotation;
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changeset
|
2030 |
\rail@term{\hyperlink{attribute.cases}{\mbox{\isa{cases}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
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changeset
|
2031 |
\rail@nont{\isa{spec}}[] |
6c621a9d612a
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changeset
|
2032 |
\rail@end |
42662 | 2033 |
\rail@begin{1}{} |
42596
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modernized rail diagrams using @{rail} antiquotation;
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changeset
|
2034 |
\rail@term{\hyperlink{attribute.induct}{\mbox{\isa{induct}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
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changeset
|
2035 |
\rail@nont{\isa{spec}}[] |
6c621a9d612a
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changeset
|
2036 |
\rail@end |
42662 | 2037 |
\rail@begin{1}{} |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
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changeset
|
2038 |
\rail@term{\hyperlink{attribute.coinduct}{\mbox{\isa{coinduct}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
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diff
changeset
|
2039 |
\rail@nont{\isa{spec}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
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changeset
|
2040 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
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changeset
|
2041 |
\rail@begin{4}{\isa{spec}} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
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changeset
|
2042 |
\rail@bar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
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changeset
|
2043 |
\rail@bar |
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modernized rail diagrams using @{rail} antiquotation;
wenzelm
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changeset
|
2044 |
\rail@term{\isa{type}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
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changeset
|
2045 |
\rail@nextbar{1} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
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changeset
|
2046 |
\rail@term{\isa{pred}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
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diff
changeset
|
2047 |
\rail@nextbar{2} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
2048 |
\rail@term{\isa{set}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
2049 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
2050 |
\rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
2051 |
\rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
2052 |
\rail@nextbar{3} |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
2053 |
\rail@term{\isa{del}}[] |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
2054 |
\rail@endbar |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
40965
diff
changeset
|
2055 |
\rail@end |
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
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40965
diff
changeset
|
2056 |
\end{railoutput} |
27042 | 2057 |
|
2058 |
||
28788 | 2059 |
\begin{description} |
27042 | 2060 |
|
40406 | 2061 |
\item \hyperlink{command.print-induct-rules}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}induct{\isaliteral{5F}{\isacharunderscore}}rules}}}} prints cases and induct rules |
28788 | 2062 |
for predicates (or sets) and types of the current context. |
42596
6c621a9d612a
modernized rail diagrams using @{rail} antiquotation;
wenzelm
parents:
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changeset
|
2063 |
|
28788 | 2064 |
\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 | 2065 |
(co)inductive predicates (or sets) and types, using the |
2066 |
corresponding methods of the same name. Certain definitional |
|
2067 |
packages of object-logics usually declare emerging cases and |
|
2068 |
induction rules as expected, so users rarely need to intervene. |
|
2069 |
||
40406 | 2070 |
Rules may be deleted via the \isa{{\isaliteral{22}{\isachardoublequote}}del{\isaliteral{22}{\isachardoublequote}}} specification, which |
2071 |
covers all of the \isa{{\isaliteral{22}{\isachardoublequote}}type{\isaliteral{22}{\isachardoublequote}}}/\isa{{\isaliteral{22}{\isachardoublequote}}pred{\isaliteral{22}{\isachardoublequote}}}/\isa{{\isaliteral{22}{\isachardoublequote}}set{\isaliteral{22}{\isachardoublequote}}} |
|
27142 | 2072 |
sub-categories simultaneously. For example, \hyperlink{attribute.cases}{\mbox{\isa{cases}}}~\isa{del} removes any \hyperlink{attribute.cases}{\mbox{\isa{cases}}} rules declared for |
2073 |
some type, predicate, or set. |
|
27042 | 2074 |
|
40406 | 2075 |
Manual rule declarations usually refer to the \hyperlink{attribute.case-names}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}names}}} and \hyperlink{attribute.params}{\mbox{\isa{params}}} attributes to adjust names of |
27042 | 2076 |
cases and parameters of a rule; the \hyperlink{attribute.consumes}{\mbox{\isa{consumes}}} |
2077 |
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. |
|
2078 |
||
28788 | 2079 |
\end{description}% |
27042 | 2080 |
\end{isamarkuptext}% |
2081 |
\isamarkuptrue% |
|
2082 |
% |
|
26869 | 2083 |
\isadelimtheory |
2084 |
% |
|
2085 |
\endisadelimtheory |
|
2086 |
% |
|
2087 |
\isatagtheory |
|
2088 |
\isacommand{end}\isamarkupfalse% |
|
2089 |
% |
|
2090 |
\endisatagtheory |
|
2091 |
{\isafoldtheory}% |
|
2092 |
% |
|
2093 |
\isadelimtheory |
|
2094 |
% |
|
2095 |
\endisadelimtheory |
|
2096 |
\isanewline |
|
2097 |
\end{isabellebody}% |
|
2098 |
%%% Local Variables: |
|
2099 |
%%% mode: latex |
|
2100 |
%%% TeX-master: "root" |
|
2101 |
%%% End: |