author | haftmann |
Thu, 17 Jun 2010 10:45:10 +0200 | |
changeset 37444 | 2e7e7ff21e25 |
parent 37422 | 6d19e4e6ebf5 |
child 37749 | c7e15d59c58d |
permissions | -rw-r--r-- |
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\begin{isabellebody}% |
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\def\isabellecontext{HOL{\isacharunderscore}Specific}% |
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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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\ HOL{\isacharunderscore}Specific\isanewline |
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\isakeyword{imports}\ Main\isanewline |
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\isakeyword{begin}% |
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\endisatagtheory |
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{\isafoldtheory}% |
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\isadelimtheory |
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\endisadelimtheory |
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\isamarkupchapter{Isabelle/HOL \label{ch:hol}% |
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} |
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\isamarkuptrue% |
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\isamarkupsection{Typedef axiomatization \label{sec:hol-typedef}% |
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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{HOL}{command}{typedef}\hypertarget{command.HOL.typedef}{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\ |
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\end{matharray} |
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\begin{rail} |
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'typedef' altname? abstype '=' repset |
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; |
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altname: '(' (name | 'open' | 'open' name) ')' |
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; |
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abstype: typespecsorts mixfix? |
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; |
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repset: term ('morphisms' name name)? |
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; |
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\end{rail} |
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\begin{description} |
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\item \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isacharequal}\ A{\isachardoublequote}} |
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axiomatizes a Gordon/HOL-style type definition in the background |
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theory of the current context, depending on a non-emptiness result |
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of the set \isa{A} (which needs to be proven interactively). |
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The raw type may not depend on parameters or assumptions of the |
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context --- this is logically impossible in Isabelle/HOL --- but the |
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non-emptiness property can be local, potentially resulting in |
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multiple interpretations in target contexts. Thus the established |
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bijection between the representing set \isa{A} and the new type |
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\isa{t} may semantically depend on local assumptions. |
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By default, \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}} defines both a type \isa{t} |
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and a set (term constant) of the same name, unless an alternative |
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base name is given in parentheses, or the ``\isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}}'' |
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declaration is used to suppress a separate constant definition |
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altogether. The injection from type to set is called \isa{Rep{\isacharunderscore}t}, |
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its inverse \isa{Abs{\isacharunderscore}t} --- this may be changed via an explicit |
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\hyperlink{keyword.HOL.morphisms}{\mbox{\isa{\isakeyword{morphisms}}}} declaration. |
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Theorems \isa{Rep{\isacharunderscore}t}, \isa{Rep{\isacharunderscore}t{\isacharunderscore}inverse}, and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inverse} provide the most basic characterization as a |
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corresponding injection/surjection pair (in both directions). Rules |
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\isa{Rep{\isacharunderscore}t{\isacharunderscore}inject} and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inject} provide a slightly |
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more convenient view on the injectivity part, suitable for automated |
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proof tools (e.g.\ in \hyperlink{attribute.simp}{\mbox{\isa{simp}}} or \hyperlink{attribute.iff}{\mbox{\isa{iff}}} |
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declarations). Rules \isa{Rep{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Rep{\isacharunderscore}t{\isacharunderscore}induct}, and |
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\isa{Abs{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Abs{\isacharunderscore}t{\isacharunderscore}induct} provide alternative views |
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on surjectivity; these are already declared as set or type rules for |
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the generic \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} methods. |
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An alternative name for the set definition (and other derived |
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entities) may be specified in parentheses; the default is to use |
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\isa{t} as indicated before. |
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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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\isamarkupsection{Adhoc tuples% |
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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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\hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{attribute} \\ |
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\end{matharray} |
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\begin{rail} |
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'split\_format' ((( name * ) + 'and') | ('(' 'complete' ')')) |
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; |
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\end{rail} |
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\begin{description} |
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\item \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}~\isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub m\ {\isasymAND}\ {\isasymdots}\ {\isasymAND}\ q\isactrlsub {\isadigit{1}}\ {\isasymdots}\ q\isactrlsub n{\isachardoublequote}} puts expressions of low-level tuple types into |
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canonical form as specified by the arguments given; the \isa{i}-th |
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collection of arguments refers to occurrences in premise \isa{i} |
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of the rule. The ``\isa{{\isachardoublequote}{\isacharparenleft}complete{\isacharparenright}{\isachardoublequote}}'' option causes \emph{all} |
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arguments in function applications to be represented canonically |
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according to their tuple type structure. |
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Note that these operations tend to invent funny names for new local |
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parameters to be introduced. |
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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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\isamarkupsection{Records \label{sec:hol-record}% |
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} |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
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In principle, records merely generalize the concept of tuples, where |
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components may be addressed by labels instead of just position. The |
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logical infrastructure of records in Isabelle/HOL is slightly more |
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advanced, though, supporting truly extensible record schemes. This |
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admits operations that are polymorphic with respect to record |
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extension, yielding ``object-oriented'' effects like (single) |
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inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more |
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details on object-oriented verification and record subtyping in HOL.% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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% |
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\isamarkupsubsection{Basic concepts% |
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} |
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\isamarkuptrue% |
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\begin{isamarkuptext}% |
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Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records |
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at the level of terms and types. The notation is as follows: |
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\begin{center} |
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\begin{tabular}{l|l|l} |
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& record terms & record types \\ \hline |
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fixed & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isasymrparr}{\isachardoublequote}} \\ |
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schematic & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} & |
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\isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ M{\isasymrparr}{\isachardoublequote}} \\ |
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\end{tabular} |
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\end{center} |
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\noindent The ASCII representation of \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} is \isa{{\isachardoublequote}{\isacharparenleft}{\isacharbar}\ x\ {\isacharequal}\ a\ {\isacharbar}{\isacharparenright}{\isachardoublequote}}. |
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A fixed record \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} has field \isa{x} of value |
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\isa{a} and field \isa{y} of value \isa{b}. The corresponding |
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type is \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isasymrparr}{\isachardoublequote}}, assuming that \isa{{\isachardoublequote}a\ {\isacharcolon}{\isacharcolon}\ A{\isachardoublequote}} |
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and \isa{{\isachardoublequote}b\ {\isacharcolon}{\isacharcolon}\ B{\isachardoublequote}}. |
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A record scheme like \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} contains fields |
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\isa{x} and \isa{y} as before, but also possibly further fields |
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as indicated by the ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' notation (which is actually part |
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of the syntax). The improper field ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' of a record |
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scheme is called the \emph{more part}. Logically it is just a free |
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variable, which is occasionally referred to as ``row variable'' in |
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the literature. The more part of a record scheme may be |
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instantiated by zero or more further components. For example, the |
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previous scheme may get instantiated to \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ z\ {\isacharequal}\ c{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isacharprime}{\isasymrparr}{\isachardoublequote}}, where \isa{m{\isacharprime}} refers to a different more part. |
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Fixed records are special instances of record schemes, where |
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``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' is properly terminated by the \isa{{\isachardoublequote}{\isacharparenleft}{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ unit{\isachardoublequote}} |
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element. In fact, \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} is just an abbreviation |
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for \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenright}{\isasymrparr}{\isachardoublequote}}. |
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\medskip Two key observations make extensible records in a simply |
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typed language like HOL work out: |
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\begin{enumerate} |
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\item the more part is internalized, as a free term or type |
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variable, |
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\item field names are externalized, they cannot be accessed within |
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the logic as first-class values. |
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\end{enumerate} |
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\medskip In Isabelle/HOL record types have to be defined explicitly, |
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fixing their field names and types, and their (optional) parent |
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record. Afterwards, records may be formed using above syntax, while |
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obeying the canonical order of fields as given by their declaration. |
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The record package provides several standard operations like |
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selectors and updates. The common setup for various generic proof |
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tools enable succinct reasoning patterns. See also the Isabelle/HOL |
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tutorial \cite{isabelle-hol-book} for further instructions on using |
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records in practice.% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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% |
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\isamarkupsubsection{Record specifications% |
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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{HOL}{command}{record}\hypertarget{command.HOL.record}{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
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\end{matharray} |
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\begin{rail} |
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'record' typespecsorts '=' (type '+')? (constdecl +) |
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; |
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\end{rail} |
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\begin{description} |
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\item \hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t\ {\isacharequal}\ {\isasymtau}\ {\isacharplus}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}} defines extensible record type \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}}, |
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derived from the optional parent record \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} by adding new |
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field components \isa{{\isachardoublequote}c\isactrlsub i\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} etc. |
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The type variables of \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i{\isachardoublequote}} need to be |
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covered by the (distinct) parameters \isa{{\isachardoublequote}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isachardoublequote}}. Type constructor \isa{t} has to be new, while \isa{{\isasymtau}} needs to specify an instance of an existing record type. At |
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least one new field \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} has to be specified. |
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Basically, field names need to belong to a unique record. This is |
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not a real restriction in practice, since fields are qualified by |
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the record name internally. |
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The parent record specification \isa{{\isasymtau}} is optional; if omitted |
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\isa{t} becomes a root record. The hierarchy of all records |
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declared within a theory context forms a forest structure, i.e.\ a |
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set of trees starting with a root record each. There is no way to |
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merge multiple parent records! |
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For convenience, \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is made a |
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type abbreviation for the fixed record type \isa{{\isachardoublequote}{\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isasymrparr}{\isachardoublequote}}, likewise is \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharcomma}\ {\isasymzeta}{\isacharparenright}\ t{\isacharunderscore}scheme{\isachardoublequote}} made an abbreviation for |
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\isa{{\isachardoublequote}{\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}}. |
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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{Record operations% |
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} |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
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Any record definition of the form presented above produces certain |
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standard operations. Selectors and updates are provided for any |
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field, including the improper one ``\isa{more}''. There are also |
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cumulative record constructor functions. To simplify the |
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presentation below, we assume for now that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is a root record with fields \isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}}. |
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\medskip \textbf{Selectors} and \textbf{updates} are available for |
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any field (including ``\isa{more}''): |
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\begin{matharray}{lll} |
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\isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} \\ |
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\isa{{\isachardoublequote}c\isactrlsub i{\isacharunderscore}update{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}} \\ |
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\end{matharray} |
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There is special syntax for application of updates: \isa{{\isachardoublequote}r{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} abbreviates term \isa{{\isachardoublequote}x{\isacharunderscore}update\ a\ r{\isachardoublequote}}. Further notation for |
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repeated updates is also available: \isa{{\isachardoublequote}r{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isasymrparr}{\isasymlparr}y\ {\isacharcolon}{\isacharequal}\ b{\isasymrparr}{\isasymlparr}z\ {\isacharcolon}{\isacharequal}\ c{\isasymrparr}{\isachardoublequote}} may be written \isa{{\isachardoublequote}r{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isacharcomma}\ y\ {\isacharcolon}{\isacharequal}\ b{\isacharcomma}\ z\ {\isacharcolon}{\isacharequal}\ c{\isasymrparr}{\isachardoublequote}}. Note that |
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because of postfix notation the order of fields shown here is |
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reverse than in the actual term. Since repeated updates are just |
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function applications, fields may be freely permuted in \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isacharcomma}\ y\ {\isacharcolon}{\isacharequal}\ b{\isacharcomma}\ z\ {\isacharcolon}{\isacharequal}\ c{\isasymrparr}{\isachardoublequote}}, as far as logical equality is concerned. |
|
260 |
Thus commutativity of independent updates can be proven within the |
|
261 |
logic for any two fields, but not as a general theorem. |
|
262 |
||
263 |
\medskip The \textbf{make} operation provides a cumulative record |
|
264 |
constructor function: |
|
265 |
||
266 |
\begin{matharray}{lll} |
|
26852 | 267 |
\isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\ |
26849 | 268 |
\end{matharray} |
269 |
||
270 |
\medskip We now reconsider the case of non-root records, which are |
|
271 |
derived of some parent. In general, the latter may depend on |
|
272 |
another parent as well, resulting in a list of \emph{ancestor |
|
273 |
records}. Appending the lists of fields of all ancestors results in |
|
274 |
a certain field prefix. The record package automatically takes care |
|
275 |
of this by lifting operations over this context of ancestor fields. |
|
276 |
Assuming that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} has ancestor |
|
277 |
fields \isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isachardoublequote}}, |
|
278 |
the above record operations will get the following types: |
|
279 |
||
26852 | 280 |
\medskip |
281 |
\begin{tabular}{lll} |
|
282 |
\isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} \\ |
|
283 |
\isa{{\isachardoublequote}c\isactrlsub i{\isacharunderscore}update{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}} \\ |
|
284 |
\isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymrho}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymrho}\isactrlsub k\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\ |
|
285 |
\end{tabular} |
|
286 |
\medskip |
|
26849 | 287 |
|
26852 | 288 |
\noindent Some further operations address the extension aspect of a |
26849 | 289 |
derived record scheme specifically: \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} produces a |
290 |
record fragment consisting of exactly the new fields introduced here |
|
291 |
(the result may serve as a more part elsewhere); \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}} |
|
292 |
takes a fixed record and adds a given more part; \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} restricts a record scheme to a fixed record. |
|
293 |
||
26852 | 294 |
\medskip |
295 |
\begin{tabular}{lll} |
|
296 |
\isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\ |
|
297 |
\isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymzeta}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}} \\ |
|
298 |
\isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\ |
|
299 |
\end{tabular} |
|
300 |
\medskip |
|
26849 | 301 |
|
302 |
\noindent Note that \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} and \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} coincide |
|
303 |
for root records.% |
|
304 |
\end{isamarkuptext}% |
|
305 |
\isamarkuptrue% |
|
306 |
% |
|
307 |
\isamarkupsubsection{Derived rules and proof tools% |
|
308 |
} |
|
309 |
\isamarkuptrue% |
|
310 |
% |
|
311 |
\begin{isamarkuptext}% |
|
312 |
The record package proves several results internally, declaring |
|
313 |
these facts to appropriate proof tools. This enables users to |
|
314 |
reason about record structures quite conveniently. Assume that |
|
315 |
\isa{t} is a record type as specified above. |
|
316 |
||
317 |
\begin{enumerate} |
|
318 |
||
319 |
\item Standard conversions for selectors or updates applied to |
|
320 |
record constructor terms are made part of the default Simplifier |
|
321 |
context; thus proofs by reduction of basic operations merely require |
|
26902 | 322 |
the \hyperlink{method.simp}{\mbox{\isa{simp}}} method without further arguments. These rules |
26849 | 323 |
are available as \isa{{\isachardoublequote}t{\isachardot}simps{\isachardoublequote}}, too. |
324 |
||
325 |
\item Selectors applied to updated records are automatically reduced |
|
326 |
by an internal simplification procedure, which is also part of the |
|
327 |
standard Simplifier setup. |
|
328 |
||
329 |
\item Inject equations of a form analogous to \isa{{\isachardoublequote}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x{\isacharprime}{\isacharcomma}\ y{\isacharprime}{\isacharparenright}\ {\isasymequiv}\ x\ {\isacharequal}\ x{\isacharprime}\ {\isasymand}\ y\ {\isacharequal}\ y{\isacharprime}{\isachardoublequote}} are declared to the Simplifier and Classical |
|
26902 | 330 |
Reasoner as \hyperlink{attribute.iff}{\mbox{\isa{iff}}} rules. These rules are available as |
26849 | 331 |
\isa{{\isachardoublequote}t{\isachardot}iffs{\isachardoublequote}}. |
332 |
||
333 |
\item The introduction rule for record equality analogous to \isa{{\isachardoublequote}x\ r\ {\isacharequal}\ x\ r{\isacharprime}\ {\isasymLongrightarrow}\ y\ r\ {\isacharequal}\ y\ r{\isacharprime}\ {\isasymdots}\ {\isasymLongrightarrow}\ r\ {\isacharequal}\ r{\isacharprime}{\isachardoublequote}} is declared to the Simplifier, |
|
26902 | 334 |
and as the basic rule context as ``\hyperlink{attribute.intro}{\mbox{\isa{intro}}}\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}''. |
26849 | 335 |
The rule is called \isa{{\isachardoublequote}t{\isachardot}equality{\isachardoublequote}}. |
336 |
||
337 |
\item Representations of arbitrary record expressions as canonical |
|
26902 | 338 |
constructor terms are provided both in \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} format (cf.\ the generic proof methods of the same name, |
26849 | 339 |
\secref{sec:cases-induct}). Several variations are available, for |
340 |
fixed records, record schemes, more parts etc. |
|
341 |
||
342 |
The generic proof methods are sufficiently smart to pick the most |
|
343 |
sensible rule according to the type of the indicated record |
|
344 |
expression: users just need to apply something like ``\isa{{\isachardoublequote}{\isacharparenleft}cases\ r{\isacharparenright}{\isachardoublequote}}'' to a certain proof problem. |
|
345 |
||
346 |
\item The derived record operations \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}}, \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}}, \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}, \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} are \emph{not} |
|
347 |
treated automatically, but usually need to be expanded by hand, |
|
348 |
using the collective fact \isa{{\isachardoublequote}t{\isachardot}defs{\isachardoublequote}}. |
|
349 |
||
350 |
\end{enumerate}% |
|
351 |
\end{isamarkuptext}% |
|
352 |
\isamarkuptrue% |
|
353 |
% |
|
354 |
\isamarkupsection{Datatypes \label{sec:hol-datatype}% |
|
355 |
} |
|
356 |
\isamarkuptrue% |
|
357 |
% |
|
358 |
\begin{isamarkuptext}% |
|
359 |
\begin{matharray}{rcl} |
|
28788 | 360 |
\indexdef{HOL}{command}{datatype}\hypertarget{command.HOL.datatype}{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
361 |
\indexdef{HOL}{command}{rep\_datatype}\hypertarget{command.HOL.rep-datatype}{\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\ |
|
26849 | 362 |
\end{matharray} |
363 |
||
364 |
\begin{rail} |
|
365 |
'datatype' (dtspec + 'and') |
|
366 |
; |
|
27452 | 367 |
'rep\_datatype' ('(' (name +) ')')? (term +) |
26849 | 368 |
; |
369 |
||
35351
7425aece4ee3
allow general mixfix syntax for type constructors;
wenzelm
parents:
34172
diff
changeset
|
370 |
dtspec: parname? typespec mixfix? '=' (cons + '|') |
26849 | 371 |
; |
31913 | 372 |
cons: name ( type * ) mixfix? |
26849 | 373 |
\end{rail} |
374 |
||
28788 | 375 |
\begin{description} |
26849 | 376 |
|
28788 | 377 |
\item \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} defines inductive datatypes in |
26849 | 378 |
HOL. |
379 |
||
28788 | 380 |
\item \hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}} represents existing types as |
26849 | 381 |
inductive ones, generating the standard infrastructure of derived |
382 |
concepts (primitive recursion etc.). |
|
383 |
||
28788 | 384 |
\end{description} |
26849 | 385 |
|
386 |
The induction and exhaustion theorems generated provide case names |
|
387 |
according to the constructors involved, while parameters are named |
|
388 |
after the types (see also \secref{sec:cases-induct}). |
|
389 |
||
390 |
See \cite{isabelle-HOL} for more details on datatypes, but beware of |
|
391 |
the old-style theory syntax being used there! Apart from proper |
|
392 |
proof methods for case-analysis and induction, there are also |
|
26907 | 393 |
emulations of ML tactics \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} available, see \secref{sec:hol-induct-tac}; these admit |
26849 | 394 |
to refer directly to the internal structure of subgoals (including |
395 |
internally bound parameters).% |
|
396 |
\end{isamarkuptext}% |
|
397 |
\isamarkuptrue% |
|
398 |
% |
|
399 |
\isamarkupsection{Recursive functions \label{sec:recursion}% |
|
400 |
} |
|
401 |
\isamarkuptrue% |
|
402 |
% |
|
403 |
\begin{isamarkuptext}% |
|
404 |
\begin{matharray}{rcl} |
|
28788 | 405 |
\indexdef{HOL}{command}{primrec}\hypertarget{command.HOL.primrec}{\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\ |
406 |
\indexdef{HOL}{command}{fun}\hypertarget{command.HOL.fun}{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\ |
|
407 |
\indexdef{HOL}{command}{function}\hypertarget{command.HOL.function}{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\ |
|
408 |
\indexdef{HOL}{command}{termination}\hypertarget{command.HOL.termination}{\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\ |
|
26849 | 409 |
\end{matharray} |
410 |
||
411 |
\begin{rail} |
|
412 |
'primrec' target? fixes 'where' equations |
|
413 |
; |
|
414 |
equations: (thmdecl? prop + '|') |
|
415 |
; |
|
26987 | 416 |
('fun' | 'function') target? functionopts? fixes 'where' clauses |
26849 | 417 |
; |
418 |
clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|') |
|
419 |
; |
|
26987 | 420 |
functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')' |
26849 | 421 |
; |
422 |
'termination' ( term )? |
|
423 |
\end{rail} |
|
424 |
||
28788 | 425 |
\begin{description} |
26849 | 426 |
|
28788 | 427 |
\item \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} defines primitive recursive |
26849 | 428 |
functions over datatypes, see also \cite{isabelle-HOL}. |
429 |
||
28788 | 430 |
\item \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} defines functions by general |
26849 | 431 |
wellfounded recursion. A detailed description with examples can be |
432 |
found in \cite{isabelle-function}. The function is specified by a |
|
433 |
set of (possibly conditional) recursive equations with arbitrary |
|
434 |
pattern matching. The command generates proof obligations for the |
|
435 |
completeness and the compatibility of patterns. |
|
436 |
||
437 |
The defined function is considered partial, and the resulting |
|
438 |
simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule |
|
439 |
(named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain |
|
26902 | 440 |
predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}} |
26849 | 441 |
command can then be used to establish that the function is total. |
442 |
||
28788 | 443 |
\item \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} is a shorthand notation for ``\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}~\isa{{\isachardoublequote}{\isacharparenleft}sequential{\isacharparenright}{\isachardoublequote}}, followed by automated |
444 |
proof attempts regarding pattern matching and termination. See |
|
445 |
\cite{isabelle-function} for further details. |
|
26849 | 446 |
|
28788 | 447 |
\item \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f} commences a |
26849 | 448 |
termination proof for the previously defined function \isa{f}. If |
449 |
this is omitted, the command refers to the most recent function |
|
450 |
definition. After the proof is closed, the recursive equations and |
|
451 |
the induction principle is established. |
|
452 |
||
28788 | 453 |
\end{description} |
26849 | 454 |
|
27452 | 455 |
Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} |
456 |
command accommodate |
|
26849 | 457 |
reasoning by induction (cf.\ \secref{sec:cases-induct}): rule \isa{{\isachardoublequote}c{\isachardot}induct{\isachardoublequote}} (where \isa{c} is the name of the function definition) |
458 |
refers to a specific induction rule, with parameters named according |
|
33857 | 459 |
to the user-specified equations. Cases are numbered (starting from 1). |
460 |
||
461 |
For \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}, the induction principle coincides |
|
27452 | 462 |
with structural recursion on the datatype the recursion is carried |
463 |
out. |
|
26849 | 464 |
|
465 |
The equations provided by these packages may be referred later as |
|
466 |
theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective) |
|
467 |
name of the functions defined. Individual equations may be named |
|
468 |
explicitly as well. |
|
469 |
||
26902 | 470 |
The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following |
26849 | 471 |
options. |
472 |
||
28788 | 473 |
\begin{description} |
26849 | 474 |
|
28788 | 475 |
\item \isa{sequential} enables a preprocessor which disambiguates |
476 |
overlapping patterns by making them mutually disjoint. Earlier |
|
477 |
equations take precedence over later ones. This allows to give the |
|
478 |
specification in a format very similar to functional programming. |
|
479 |
Note that the resulting simplification and induction rules |
|
480 |
correspond to the transformed specification, not the one given |
|
26849 | 481 |
originally. This usually means that each equation given by the user |
36139 | 482 |
may result in several theorems. Also note that this automatic |
26849 | 483 |
transformation only works for ML-style datatype patterns. |
484 |
||
28788 | 485 |
\item \isa{domintros} enables the automated generation of |
26849 | 486 |
introduction rules for the domain predicate. While mostly not |
487 |
needed, they can be helpful in some proofs about partial functions. |
|
488 |
||
28788 | 489 |
\item \isa{tailrec} generates the unconstrained recursive |
26849 | 490 |
equations even without a termination proof, provided that the |
491 |
function is tail-recursive. This currently only works |
|
492 |
||
28788 | 493 |
\item \isa{{\isachardoublequote}default\ d{\isachardoublequote}} allows to specify a default value for a |
26849 | 494 |
(partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}} |
495 |
whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}. |
|
496 |
||
28788 | 497 |
\end{description}% |
26849 | 498 |
\end{isamarkuptext}% |
499 |
\isamarkuptrue% |
|
500 |
% |
|
501 |
\isamarkupsubsection{Proof methods related to recursive definitions% |
|
502 |
} |
|
503 |
\isamarkuptrue% |
|
504 |
% |
|
505 |
\begin{isamarkuptext}% |
|
506 |
\begin{matharray}{rcl} |
|
28788 | 507 |
\indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isa{method} \\ |
508 |
\indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isa{method} \\ |
|
509 |
\indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isa{method} \\ |
|
33858 | 510 |
\indexdef{HOL}{method}{size\_change}\hypertarget{method.HOL.size-change}{\hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isacharunderscore}change}}}} & : & \isa{method} \\ |
26849 | 511 |
\end{matharray} |
512 |
||
513 |
\begin{rail} |
|
514 |
'relation' term |
|
515 |
; |
|
31913 | 516 |
'lexicographic\_order' ( clasimpmod * ) |
26849 | 517 |
; |
33858 | 518 |
'size\_change' ( orders ( clasimpmod * ) ) |
519 |
; |
|
520 |
orders: ( 'max' | 'min' | 'ms' ) * |
|
26849 | 521 |
\end{rail} |
522 |
||
28788 | 523 |
\begin{description} |
26849 | 524 |
|
28788 | 525 |
\item \hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}} is a specialized method to |
26849 | 526 |
solve goals regarding the completeness of pattern matching, as |
26902 | 527 |
required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\ |
26849 | 528 |
\cite{isabelle-function}). |
529 |
||
28788 | 530 |
\item \hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R} introduces a termination |
26849 | 531 |
proof using the relation \isa{R}. The resulting proof state will |
532 |
contain goals expressing that \isa{R} is wellfounded, and that the |
|
533 |
arguments of recursive calls decrease with respect to \isa{R}. |
|
534 |
Usually, this method is used as the initial proof step of manual |
|
535 |
termination proofs. |
|
536 |
||
28788 | 537 |
\item \hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}} attempts a fully |
26849 | 538 |
automated termination proof by searching for a lexicographic |
539 |
combination of size measures on the arguments of the function. The |
|
26902 | 540 |
method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method, |
26849 | 541 |
which it uses internally to prove local descents. The same context |
26902 | 542 |
modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see |
26849 | 543 |
\secref{sec:clasimp}. |
544 |
||
545 |
In case of failure, extensive information is printed, which can help |
|
546 |
to analyse the situation (cf.\ \cite{isabelle-function}). |
|
547 |
||
33858 | 548 |
\item \hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isacharunderscore}change}}} also works on termination goals, |
549 |
using a variation of the size-change principle, together with a |
|
550 |
graph decomposition technique (see \cite{krauss_phd} for details). |
|
551 |
Three kinds of orders are used internally: \isa{max}, \isa{min}, |
|
552 |
and \isa{ms} (multiset), which is only available when the theory |
|
553 |
\isa{Multiset} is loaded. When no order kinds are given, they are |
|
554 |
tried in order. The search for a termination proof uses SAT solving |
|
555 |
internally. |
|
556 |
||
557 |
For local descent proofs, the same context modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see \secref{sec:clasimp}. |
|
558 |
||
28788 | 559 |
\end{description}% |
26849 | 560 |
\end{isamarkuptext}% |
561 |
\isamarkuptrue% |
|
562 |
% |
|
563 |
\isamarkupsubsection{Old-style recursive function definitions (TFL)% |
|
564 |
} |
|
565 |
\isamarkuptrue% |
|
566 |
% |
|
567 |
\begin{isamarkuptext}% |
|
26907 | 568 |
The old TFL commands \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} and \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}} for defining recursive are mostly obsolete; \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} or \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} should be used instead. |
26849 | 569 |
|
570 |
\begin{matharray}{rcl} |
|
28788 | 571 |
\indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isacharparenright}{\isachardoublequote}} \\ |
572 |
\indexdef{HOL}{command}{recdef\_tc}\hypertarget{command.HOL.recdef-tc}{\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\ |
|
26849 | 573 |
\end{matharray} |
574 |
||
575 |
\begin{rail} |
|
576 |
'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints? |
|
577 |
; |
|
578 |
recdeftc thmdecl? tc |
|
579 |
; |
|
31913 | 580 |
hints: '(' 'hints' ( recdefmod * ) ')' |
26849 | 581 |
; |
582 |
recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod |
|
583 |
; |
|
584 |
tc: nameref ('(' nat ')')? |
|
585 |
; |
|
586 |
\end{rail} |
|
587 |
||
28788 | 588 |
\begin{description} |
26849 | 589 |
|
28788 | 590 |
\item \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} defines general well-founded |
26849 | 591 |
recursive functions (using the TFL package), see also |
592 |
\cite{isabelle-HOL}. The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells |
|
593 |
TFL to recover from failed proof attempts, returning unfinished |
|
594 |
results. The \isa{recdef{\isacharunderscore}simp}, \isa{recdef{\isacharunderscore}cong}, and \isa{recdef{\isacharunderscore}wf} hints refer to auxiliary rules to be used in the internal |
|
26902 | 595 |
automated proof process of TFL. Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}} |
26849 | 596 |
declarations (cf.\ \secref{sec:clasimp}) may be given to tune the |
597 |
context of the Simplifier (cf.\ \secref{sec:simplifier}) and |
|
598 |
Classical reasoner (cf.\ \secref{sec:classical}). |
|
599 |
||
28788 | 600 |
\item \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}} recommences the |
26849 | 601 |
proof for leftover termination condition number \isa{i} (default |
26902 | 602 |
1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of |
26849 | 603 |
constant \isa{c}. |
604 |
||
26902 | 605 |
Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish |
26849 | 606 |
its internal proofs without manual intervention. |
607 |
||
28788 | 608 |
\end{description} |
26849 | 609 |
|
26902 | 610 |
\medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared |
26849 | 611 |
globally, using the following attributes. |
612 |
||
613 |
\begin{matharray}{rcl} |
|
28788 | 614 |
\indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isa{attribute} \\ |
615 |
\indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isa{attribute} \\ |
|
616 |
\indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isa{attribute} \\ |
|
26849 | 617 |
\end{matharray} |
618 |
||
619 |
\begin{rail} |
|
620 |
('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') |
|
621 |
; |
|
622 |
\end{rail}% |
|
623 |
\end{isamarkuptext}% |
|
624 |
\isamarkuptrue% |
|
625 |
% |
|
626 |
\isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}% |
|
627 |
} |
|
628 |
\isamarkuptrue% |
|
629 |
% |
|
630 |
\begin{isamarkuptext}% |
|
631 |
An \textbf{inductive definition} specifies the least predicate (or |
|
632 |
set) \isa{R} closed under given rules: applying a rule to elements |
|
633 |
of \isa{R} yields a result within \isa{R}. For example, a |
|
634 |
structural operational semantics is an inductive definition of an |
|
635 |
evaluation relation. |
|
636 |
||
637 |
Dually, a \textbf{coinductive definition} specifies the greatest |
|
638 |
predicate~/ set \isa{R} that is consistent with given rules: every |
|
639 |
element of \isa{R} can be seen as arising by applying a rule to |
|
640 |
elements of \isa{R}. An important example is using bisimulation |
|
641 |
relations to formalise equivalence of processes and infinite data |
|
642 |
structures. |
|
643 |
||
644 |
\medskip The HOL package is related to the ZF one, which is |
|
645 |
described in a separate paper,\footnote{It appeared in CADE |
|
646 |
\cite{paulson-CADE}; a longer version is distributed with Isabelle.} |
|
647 |
which you should refer to in case of difficulties. The package is |
|
648 |
simpler than that of ZF thanks to implicit type-checking in HOL. |
|
649 |
The types of the (co)inductive predicates (or sets) determine the |
|
650 |
domain of the fixedpoint definition, and the package does not have |
|
651 |
to use inference rules for type-checking. |
|
652 |
||
653 |
\begin{matharray}{rcl} |
|
28788 | 654 |
\indexdef{HOL}{command}{inductive}\hypertarget{command.HOL.inductive}{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\ |
655 |
\indexdef{HOL}{command}{inductive\_set}\hypertarget{command.HOL.inductive-set}{\hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\ |
|
656 |
\indexdef{HOL}{command}{coinductive}\hypertarget{command.HOL.coinductive}{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\ |
|
657 |
\indexdef{HOL}{command}{coinductive\_set}\hypertarget{command.HOL.coinductive-set}{\hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\ |
|
658 |
\indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\ |
|
26849 | 659 |
\end{matharray} |
660 |
||
661 |
\begin{rail} |
|
662 |
('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\ |
|
663 |
('where' clauses)? ('monos' thmrefs)? |
|
664 |
; |
|
665 |
clauses: (thmdecl? prop + '|') |
|
666 |
; |
|
667 |
'mono' (() | 'add' | 'del') |
|
668 |
; |
|
669 |
\end{rail} |
|
670 |
||
28788 | 671 |
\begin{description} |
26849 | 672 |
|
28788 | 673 |
\item \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}} define (co)inductive predicates from the |
26902 | 674 |
introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part. The |
675 |
optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the |
|
26849 | 676 |
(co)inductive predicates that remain fixed throughout the |
26902 | 677 |
definition. The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains |
26849 | 678 |
\emph{monotonicity theorems}, which are required for each operator |
679 |
applied to a recursive set in the introduction rules. There |
|
680 |
\emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, |
|
681 |
for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule! |
|
682 |
||
28788 | 683 |
\item \hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}}} and \hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}} are wrappers for to the previous commands, |
26849 | 684 |
allowing the definition of (co)inductive sets. |
685 |
||
28788 | 686 |
\item \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} declares monotonicity rules. These |
26902 | 687 |
rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}. |
26849 | 688 |
|
28788 | 689 |
\end{description}% |
26849 | 690 |
\end{isamarkuptext}% |
691 |
\isamarkuptrue% |
|
692 |
% |
|
693 |
\isamarkupsubsection{Derived rules% |
|
694 |
} |
|
695 |
\isamarkuptrue% |
|
696 |
% |
|
697 |
\begin{isamarkuptext}% |
|
698 |
Each (co)inductive definition \isa{R} adds definitions to the |
|
699 |
theory and also proves some theorems: |
|
700 |
||
701 |
\begin{description} |
|
702 |
||
28788 | 703 |
\item \isa{R{\isachardot}intros} is the list of introduction rules as proven |
26849 | 704 |
theorems, for the recursive predicates (or sets). The rules are |
705 |
also available individually, using the names given them in the |
|
706 |
theory file; |
|
707 |
||
28788 | 708 |
\item \isa{R{\isachardot}cases} is the case analysis (or elimination) rule; |
26849 | 709 |
|
28788 | 710 |
\item \isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct} is the (co)induction |
26849 | 711 |
rule. |
712 |
||
713 |
\end{description} |
|
714 |
||
715 |
When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are |
|
716 |
defined simultaneously, the list of introduction rules is called |
|
717 |
\isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are |
|
718 |
called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list |
|
719 |
of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.% |
|
720 |
\end{isamarkuptext}% |
|
721 |
\isamarkuptrue% |
|
722 |
% |
|
723 |
\isamarkupsubsection{Monotonicity theorems% |
|
724 |
} |
|
725 |
\isamarkuptrue% |
|
726 |
% |
|
727 |
\begin{isamarkuptext}% |
|
728 |
Each theory contains a default set of theorems that are used in |
|
729 |
monotonicity proofs. New rules can be added to this set via the |
|
26902 | 730 |
\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute. The HOL theory \isa{Inductive} |
26849 | 731 |
shows how this is done. In general, the following monotonicity |
732 |
theorems may be added: |
|
733 |
||
734 |
\begin{itemize} |
|
735 |
||
736 |
\item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving |
|
737 |
monotonicity of inductive definitions whose introduction rules have |
|
738 |
premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}. |
|
739 |
||
740 |
\item Monotonicity theorems for logical operators, which are of the |
|
741 |
general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}. For example, in |
|
742 |
the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is |
|
743 |
\[ |
|
744 |
\infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}} |
|
745 |
\] |
|
746 |
||
747 |
\item De Morgan style equations for reasoning about the ``polarity'' |
|
748 |
of expressions, e.g. |
|
749 |
\[ |
|
750 |
\isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad |
|
751 |
\isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}} |
|
752 |
\] |
|
753 |
||
754 |
\item Equations for reducing complex operators to more primitive |
|
755 |
ones whose monotonicity can easily be proved, e.g. |
|
756 |
\[ |
|
757 |
\isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad |
|
758 |
\isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}} |
|
759 |
\] |
|
760 |
||
761 |
\end{itemize} |
|
762 |
||
763 |
%FIXME: Example of an inductive definition% |
|
764 |
\end{isamarkuptext}% |
|
765 |
\isamarkuptrue% |
|
766 |
% |
|
767 |
\isamarkupsection{Arithmetic proof support% |
|
768 |
} |
|
769 |
\isamarkuptrue% |
|
770 |
% |
|
771 |
\begin{isamarkuptext}% |
|
772 |
\begin{matharray}{rcl} |
|
28788 | 773 |
\indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\ |
30863 | 774 |
\indexdef{HOL}{attribute}{arith}\hypertarget{attribute.HOL.arith}{\hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{attribute} \\ |
28788 | 775 |
\indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isa{attribute} \\ |
26849 | 776 |
\end{matharray} |
777 |
||
26902 | 778 |
The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems |
26849 | 779 |
(on types \isa{nat}, \isa{int}, \isa{real}). Any current |
780 |
facts are inserted into the goal before running the procedure. |
|
781 |
||
30863 | 782 |
The \hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}} attribute declares facts that are |
783 |
always supplied to the arithmetic provers implicitly. |
|
26849 | 784 |
|
30863 | 785 |
The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split |
30865 | 786 |
rules to be expanded before \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} is invoked. |
30863 | 787 |
|
788 |
Note that a simpler (but faster) arithmetic prover is |
|
789 |
already invoked by the Simplifier.% |
|
26849 | 790 |
\end{isamarkuptext}% |
791 |
\isamarkuptrue% |
|
792 |
% |
|
30172 | 793 |
\isamarkupsection{Intuitionistic proof search% |
794 |
} |
|
795 |
\isamarkuptrue% |
|
796 |
% |
|
797 |
\begin{isamarkuptext}% |
|
798 |
\begin{matharray}{rcl} |
|
799 |
\indexdef{HOL}{method}{iprover}\hypertarget{method.HOL.iprover}{\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}} & : & \isa{method} \\ |
|
800 |
\end{matharray} |
|
801 |
||
802 |
\begin{rail} |
|
35613 | 803 |
'iprover' ( rulemod * ) |
30172 | 804 |
; |
805 |
\end{rail} |
|
806 |
||
807 |
The \hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}} method performs intuitionistic proof |
|
808 |
search, depending on specifically declared rules from the context, |
|
809 |
or given as explicit arguments. Chained facts are inserted into the |
|
35613 | 810 |
goal before commencing proof search. |
811 |
||
30172 | 812 |
Rules need to be classified as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}}, |
813 |
\hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, or \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}; here the |
|
814 |
``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' indicator refers to ``safe'' rules, which may be |
|
815 |
applied aggressively (without considering back-tracking later). |
|
816 |
Rules declared with ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' are ignored in proof search (the |
|
817 |
single-step \hyperlink{method.rule}{\mbox{\isa{rule}}} method still observes these). An |
|
818 |
explicit weight annotation may be given as well; otherwise the |
|
819 |
number of rule premises will be taken into account here.% |
|
820 |
\end{isamarkuptext}% |
|
821 |
\isamarkuptrue% |
|
822 |
% |
|
823 |
\isamarkupsection{Coherent Logic% |
|
824 |
} |
|
825 |
\isamarkuptrue% |
|
826 |
% |
|
827 |
\begin{isamarkuptext}% |
|
828 |
\begin{matharray}{rcl} |
|
829 |
\indexdef{HOL}{method}{coherent}\hypertarget{method.HOL.coherent}{\hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}}} & : & \isa{method} \\ |
|
830 |
\end{matharray} |
|
831 |
||
832 |
\begin{rail} |
|
833 |
'coherent' thmrefs? |
|
834 |
; |
|
835 |
\end{rail} |
|
836 |
||
837 |
The \hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}} method solves problems of |
|
838 |
\emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers |
|
839 |
applications in confluence theory, lattice theory and projective |
|
840 |
geometry. See \hyperlink{file.~~/src/HOL/ex/Coherent.thy}{\mbox{\isa{\isatt{{\isachartilde}{\isachartilde}{\isacharslash}src{\isacharslash}HOL{\isacharslash}ex{\isacharslash}Coherent{\isachardot}thy}}}} for some |
|
841 |
examples.% |
|
842 |
\end{isamarkuptext}% |
|
843 |
\isamarkuptrue% |
|
844 |
% |
|
31913 | 845 |
\isamarkupsection{Checking and refuting propositions% |
846 |
} |
|
847 |
\isamarkuptrue% |
|
848 |
% |
|
849 |
\begin{isamarkuptext}% |
|
850 |
Identifying incorrect propositions usually involves evaluation of |
|
851 |
particular assignments and systematic counter example search. This |
|
852 |
is supported by the following commands. |
|
853 |
||
854 |
\begin{matharray}{rcl} |
|
855 |
\indexdef{HOL}{command}{value}\hypertarget{command.HOL.value}{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\ |
|
856 |
\indexdef{HOL}{command}{quickcheck}\hypertarget{command.HOL.quickcheck}{\hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof\ {\isasymrightarrow}{\isachardoublequote}} \\ |
|
857 |
\indexdef{HOL}{command}{quickcheck\_params}\hypertarget{command.HOL.quickcheck-params}{\hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isacharunderscore}params}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} |
|
858 |
\end{matharray} |
|
859 |
||
860 |
\begin{rail} |
|
861 |
'value' ( ( '[' name ']' ) ? ) modes? term |
|
862 |
; |
|
863 |
||
864 |
'quickcheck' ( ( '[' args ']' ) ? ) nat? |
|
865 |
; |
|
866 |
||
867 |
'quickcheck_params' ( ( '[' args ']' ) ? ) |
|
868 |
; |
|
869 |
||
870 |
modes: '(' (name + ) ')' |
|
871 |
; |
|
872 |
||
873 |
args: ( name '=' value + ',' ) |
|
874 |
; |
|
875 |
\end{rail} |
|
876 |
||
877 |
\begin{description} |
|
878 |
||
879 |
\item \hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t} evaluates and prints a |
|
880 |
term; optionally \isa{modes} can be specified, which are |
|
881 |
appended to the current print mode (see also \cite{isabelle-ref}). |
|
882 |
Internally, the evaluation is performed by registered evaluators, |
|
883 |
which are invoked sequentially until a result is returned. |
|
884 |
Alternatively a specific evaluator can be selected using square |
|
37444 | 885 |
brackets; typical evaluators use the current set of code equations |
886 |
to normalize and include \isa{simp} for fully symbolic evaluation |
|
887 |
using the simplifier, \isa{nbe} for \emph{normalization by evaluation} |
|
888 |
and \emph{code} for code generation in SML. |
|
31913 | 889 |
|
890 |
\item \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}} tests the current goal for |
|
891 |
counter examples using a series of arbitrary assignments for its |
|
892 |
free variables; by default the first subgoal is tested, an other |
|
893 |
can be selected explicitly using an optional goal index. |
|
894 |
A number of configuration options are supported for |
|
895 |
\hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}, notably: |
|
896 |
||
897 |
\begin{description} |
|
898 |
||
899 |
\item[size] specifies the maximum size of the search space for |
|
900 |
assignment values. |
|
901 |
||
902 |
\item[iterations] sets how many sets of assignments are |
|
903 |
generated for each particular size. |
|
904 |
||
35351
7425aece4ee3
allow general mixfix syntax for type constructors;
wenzelm
parents:
34172
diff
changeset
|
905 |
\item[no\_assms] specifies whether assumptions in |
7425aece4ee3
allow general mixfix syntax for type constructors;
wenzelm
parents:
34172
diff
changeset
|
906 |
structured proofs should be ignored. |
7425aece4ee3
allow general mixfix syntax for type constructors;
wenzelm
parents:
34172
diff
changeset
|
907 |
|
31913 | 908 |
\end{description} |
909 |
||
910 |
These option can be given within square brackets. |
|
911 |
||
912 |
\item \hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isacharunderscore}params}}}} changes quickcheck |
|
913 |
configuration options persitently. |
|
914 |
||
915 |
\end{description}% |
|
916 |
\end{isamarkuptext}% |
|
917 |
\isamarkuptrue% |
|
918 |
% |
|
28788 | 919 |
\isamarkupsection{Unstructured case analysis and induction \label{sec:hol-induct-tac}% |
26849 | 920 |
} |
921 |
\isamarkuptrue% |
|
922 |
% |
|
923 |
\begin{isamarkuptext}% |
|
27124 | 924 |
The following tools of Isabelle/HOL support cases analysis and |
925 |
induction in unstructured tactic scripts; see also |
|
926 |
\secref{sec:cases-induct} for proper Isar versions of similar ideas. |
|
26849 | 927 |
|
928 |
\begin{matharray}{rcl} |
|
28788 | 929 |
\indexdef{HOL}{method}{case\_tac}\hypertarget{method.HOL.case-tac}{\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\ |
930 |
\indexdef{HOL}{method}{induct\_tac}\hypertarget{method.HOL.induct-tac}{\hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\ |
|
931 |
\indexdef{HOL}{method}{ind\_cases}\hypertarget{method.HOL.ind-cases}{\hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\ |
|
932 |
\indexdef{HOL}{command}{inductive\_cases}\hypertarget{command.HOL.inductive-cases}{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\ |
|
26849 | 933 |
\end{matharray} |
934 |
||
935 |
\begin{rail} |
|
936 |
'case\_tac' goalspec? term rule? |
|
937 |
; |
|
938 |
'induct\_tac' goalspec? (insts * 'and') rule? |
|
939 |
; |
|
940 |
'ind\_cases' (prop +) ('for' (name +)) ? |
|
941 |
; |
|
942 |
'inductive\_cases' (thmdecl? (prop +) + 'and') |
|
943 |
; |
|
944 |
||
945 |
rule: ('rule' ':' thmref) |
|
946 |
; |
|
947 |
\end{rail} |
|
948 |
||
28788 | 949 |
\begin{description} |
26849 | 950 |
|
28788 | 951 |
\item \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} admit |
952 |
to reason about inductive types. Rules are selected according to |
|
953 |
the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}} |
|
954 |
attributes, cf.\ \secref{sec:cases-induct}. The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this. |
|
27124 | 955 |
|
956 |
These unstructured tactics feature both goal addressing and dynamic |
|
26849 | 957 |
instantiation. Note that named rule cases are \emph{not} provided |
27124 | 958 |
as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof |
959 |
methods (see \secref{sec:cases-induct}). Unlike the \hyperlink{method.induct}{\mbox{\isa{induct}}} method, \hyperlink{method.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} does not handle structured rule |
|
960 |
statements, only the compact object-logic conclusion of the subgoal |
|
961 |
being addressed. |
|
26849 | 962 |
|
28788 | 963 |
\item \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} and \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provide an interface to the internal \verb|mk_cases| operation. Rules are simplified in an unrestricted |
26861 | 964 |
forward manner. |
26849 | 965 |
|
26907 | 966 |
While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the |
967 |
result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provides case split theorems at the theory level |
|
968 |
for later use. The \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} argument of the \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} method allows to specify a list of variables that should |
|
26849 | 969 |
be generalized before applying the resulting rule. |
970 |
||
28788 | 971 |
\end{description}% |
26849 | 972 |
\end{isamarkuptext}% |
973 |
\isamarkuptrue% |
|
974 |
% |
|
975 |
\isamarkupsection{Executable code% |
|
976 |
} |
|
977 |
\isamarkuptrue% |
|
978 |
% |
|
979 |
\begin{isamarkuptext}% |
|
980 |
Isabelle/Pure provides two generic frameworks to support code |
|
981 |
generation from executable specifications. Isabelle/HOL |
|
982 |
instantiates these mechanisms in a way that is amenable to end-user |
|
983 |
applications. |
|
984 |
||
37422 | 985 |
\medskip One framework generates code from functional programs |
986 |
(including overloading using type classes) to SML \cite{SML}, OCaml |
|
987 |
\cite{OCaml} and Haskell \cite{haskell-revised-report}. |
|
988 |
Conceptually, code generation is split up in three steps: |
|
989 |
\emph{selection} of code theorems, \emph{translation} into an |
|
990 |
abstract executable view and \emph{serialization} to a specific |
|
991 |
\emph{target language}. Inductive specifications can be executed |
|
992 |
using the predicate compiler which operates within HOL. |
|
993 |
See \cite{isabelle-codegen} for an introduction. |
|
994 |
||
995 |
\begin{matharray}{rcl} |
|
996 |
\indexdef{HOL}{command}{export\_code}\hypertarget{command.HOL.export-code}{\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\ |
|
997 |
\indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\ |
|
998 |
\indexdef{HOL}{command}{code\_abort}\hypertarget{command.HOL.code-abort}{\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
999 |
\indexdef{HOL}{command}{code\_datatype}\hypertarget{command.HOL.code-datatype}{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1000 |
\indexdef{HOL}{command}{print\_codesetup}\hypertarget{command.HOL.print-codesetup}{\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\ |
|
1001 |
\indexdef{HOL}{attribute}{code\_inline}\hypertarget{attribute.HOL.code-inline}{\hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isacharunderscore}inline}}}} & : & \isa{attribute} \\ |
|
1002 |
\indexdef{HOL}{attribute}{code\_post}\hypertarget{attribute.HOL.code-post}{\hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isacharunderscore}post}}}} & : & \isa{attribute} \\ |
|
1003 |
\indexdef{HOL}{command}{print\_codeproc}\hypertarget{command.HOL.print-codeproc}{\hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isacharunderscore}codeproc}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\ |
|
1004 |
\indexdef{HOL}{command}{code\_thms}\hypertarget{command.HOL.code-thms}{\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\ |
|
1005 |
\indexdef{HOL}{command}{code\_deps}\hypertarget{command.HOL.code-deps}{\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\ |
|
1006 |
\indexdef{HOL}{command}{code\_const}\hypertarget{command.HOL.code-const}{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1007 |
\indexdef{HOL}{command}{code\_type}\hypertarget{command.HOL.code-type}{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1008 |
\indexdef{HOL}{command}{code\_class}\hypertarget{command.HOL.code-class}{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1009 |
\indexdef{HOL}{command}{code\_instance}\hypertarget{command.HOL.code-instance}{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1010 |
\indexdef{HOL}{command}{code\_reserved}\hypertarget{command.HOL.code-reserved}{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1011 |
\indexdef{HOL}{command}{code\_monad}\hypertarget{command.HOL.code-monad}{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1012 |
\indexdef{HOL}{command}{code\_include}\hypertarget{command.HOL.code-include}{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1013 |
\indexdef{HOL}{command}{code\_modulename}\hypertarget{command.HOL.code-modulename}{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1014 |
\end{matharray} |
|
1015 |
||
1016 |
\begin{rail} |
|
1017 |
'export\_code' ( constexpr + ) \\ |
|
1018 |
( ( 'in' target ( 'module\_name' string ) ? \\ |
|
1019 |
( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ? |
|
1020 |
; |
|
1021 |
||
1022 |
const: term |
|
1023 |
; |
|
1024 |
||
1025 |
constexpr: ( const | 'name.*' | '*' ) |
|
1026 |
; |
|
1027 |
||
1028 |
typeconstructor: nameref |
|
1029 |
; |
|
1030 |
||
1031 |
class: nameref |
|
1032 |
; |
|
1033 |
||
1034 |
target: 'OCaml' | 'SML' | 'Haskell' |
|
1035 |
; |
|
1036 |
||
1037 |
'code' ( 'del' ) ? |
|
1038 |
; |
|
1039 |
||
1040 |
'code\_abort' ( const + ) |
|
1041 |
; |
|
1042 |
||
1043 |
'code\_datatype' ( const + ) |
|
1044 |
; |
|
1045 |
||
1046 |
'code_inline' ( 'del' ) ? |
|
1047 |
; |
|
1048 |
||
1049 |
'code_post' ( 'del' ) ? |
|
1050 |
; |
|
1051 |
||
1052 |
'code\_thms' ( constexpr + ) ? |
|
1053 |
; |
|
1054 |
||
1055 |
'code\_deps' ( constexpr + ) ? |
|
1056 |
; |
|
1057 |
||
1058 |
'code\_const' (const + 'and') \\ |
|
1059 |
( ( '(' target ( syntax ? + 'and' ) ')' ) + ) |
|
1060 |
; |
|
1061 |
||
1062 |
'code\_type' (typeconstructor + 'and') \\ |
|
1063 |
( ( '(' target ( syntax ? + 'and' ) ')' ) + ) |
|
1064 |
; |
|
1065 |
||
1066 |
'code\_class' (class + 'and') \\ |
|
1067 |
( ( '(' target \\ ( string ? + 'and' ) ')' ) + ) |
|
1068 |
; |
|
1069 |
||
1070 |
'code\_instance' (( typeconstructor '::' class ) + 'and') \\ |
|
1071 |
( ( '(' target ( '-' ? + 'and' ) ')' ) + ) |
|
1072 |
; |
|
1073 |
||
1074 |
'code\_reserved' target ( string + ) |
|
1075 |
; |
|
1076 |
||
1077 |
'code\_monad' const const target |
|
1078 |
; |
|
1079 |
||
1080 |
'code\_include' target ( string ( string | '-') ) |
|
1081 |
; |
|
1082 |
||
1083 |
'code\_modulename' target ( ( string string ) + ) |
|
1084 |
; |
|
1085 |
||
1086 |
syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string |
|
1087 |
; |
|
1088 |
||
1089 |
\end{rail} |
|
1090 |
||
1091 |
\begin{description} |
|
1092 |
||
1093 |
\item \hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} generates code for a given list |
|
1094 |
of constants in the specified target language(s). If no serialization |
|
1095 |
instruction is given, only abstract code is generated internally. |
|
1096 |
||
1097 |
Constants may be specified by giving them literally, referring to |
|
1098 |
all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently |
|
1099 |
available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}. |
|
1100 |
||
1101 |
By default, for each involved theory one corresponding name space |
|
1102 |
module is generated. Alternativly, a module name may be specified |
|
1103 |
after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is |
|
1104 |
placed in this module. |
|
1105 |
||
1106 |
For \emph{SML} and \emph{OCaml}, the file specification refers to a |
|
1107 |
single file; for \emph{Haskell}, it refers to a whole directory, |
|
1108 |
where code is generated in multiple files reflecting the module |
|
1109 |
hierarchy. The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard |
|
1110 |
output. For \emph{SML}, omitting the file specification compiles |
|
1111 |
code internally in the context of the current ML session. |
|
1112 |
||
1113 |
Serializers take an optional list of arguments in parentheses. For |
|
1114 |
\emph{SML} and \emph{OCaml}, ``\isa{no{\isacharunderscore}signatures}`` omits |
|
1115 |
explicit module signatures. |
|
1116 |
||
1117 |
For \emph{Haskell} a module name prefix may be given using the ``\isa{{\isachardoublequote}root{\isacharcolon}{\isachardoublequote}}'' argument; ``\isa{string{\isacharunderscore}classes}'' adds a ``\verb|deriving (Read, Show)|'' clause to each appropriate datatype |
|
1118 |
declaration. |
|
1119 |
||
1120 |
\item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}} explicitly selects (or with option |
|
1121 |
``\isa{{\isachardoublequote}del{\isachardoublequote}}'' deselects) a code equation for code |
|
1122 |
generation. Usually packages introducing code equations provide |
|
1123 |
a reasonable default setup for selection. |
|
1124 |
||
1125 |
\item \hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}} declares constants which are not |
|
1126 |
required to have a definition by means of code equations; if |
|
1127 |
needed these are implemented by program abort instead. |
|
1128 |
||
1129 |
\item \hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}} specifies a constructor set |
|
1130 |
for a logical type. |
|
1131 |
||
1132 |
\item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} gives an overview on |
|
1133 |
selected code equations and code generator datatypes. |
|
1134 |
||
1135 |
\item \hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isacharunderscore}inline}}} declares (or with |
|
1136 |
option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) inlining theorems which are |
|
1137 |
applied as rewrite rules to any code equation during |
|
1138 |
preprocessing. |
|
1139 |
||
1140 |
\item \hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isacharunderscore}post}}} declares (or with |
|
1141 |
option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) theorems which are |
|
1142 |
applied as rewrite rules to any result of an evaluation. |
|
1143 |
||
1144 |
\item \hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isacharunderscore}codeproc}}}} prints the setup |
|
1145 |
of the code generator preprocessor. |
|
1146 |
||
1147 |
\item \hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} prints a list of theorems |
|
1148 |
representing the corresponding program containing all given |
|
1149 |
constants after preprocessing. |
|
1150 |
||
1151 |
\item \hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} visualizes dependencies of |
|
1152 |
theorems representing the corresponding program containing all given |
|
1153 |
constants after preprocessing. |
|
1154 |
||
1155 |
\item \hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}} associates a list of constants |
|
1156 |
with target-specific serializations; omitting a serialization |
|
1157 |
deletes an existing serialization. |
|
1158 |
||
1159 |
\item \hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}} associates a list of type |
|
1160 |
constructors with target-specific serializations; omitting a |
|
1161 |
serialization deletes an existing serialization. |
|
1162 |
||
1163 |
\item \hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}} associates a list of classes |
|
1164 |
with target-specific class names; omitting a serialization deletes |
|
1165 |
an existing serialization. This applies only to \emph{Haskell}. |
|
1166 |
||
1167 |
\item \hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}} declares a list of type |
|
1168 |
constructor / class instance relations as ``already present'' for a |
|
1169 |
given target. Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing |
|
1170 |
``already present'' declaration. This applies only to |
|
1171 |
\emph{Haskell}. |
|
1172 |
||
1173 |
\item \hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}} declares a list of names as |
|
1174 |
reserved for a given target, preventing it to be shadowed by any |
|
1175 |
generated code. |
|
1176 |
||
1177 |
\item \hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}} provides an auxiliary mechanism |
|
1178 |
to generate monadic code for Haskell. |
|
1179 |
||
1180 |
\item \hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}} adds arbitrary named content |
|
1181 |
(``include'') to generated code. A ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' as last argument |
|
1182 |
will remove an already added ``include''. |
|
1183 |
||
1184 |
\item \hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}} declares aliasings from one |
|
1185 |
module name onto another. |
|
1186 |
||
1187 |
\end{description} |
|
1188 |
||
1189 |
The other framework generates code from both functional and relational |
|
26849 | 1190 |
programs to SML. See \cite{isabelle-HOL} for further information |
1191 |
(this actually covers the new-style theory format as well). |
|
1192 |
||
1193 |
\begin{matharray}{rcl} |
|
28788 | 1194 |
\indexdef{HOL}{command}{code\_module}\hypertarget{command.HOL.code-module}{\hyperlink{command.HOL.code-module}{\mbox{\isa{\isacommand{code{\isacharunderscore}module}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
1195 |
\indexdef{HOL}{command}{code\_library}\hypertarget{command.HOL.code-library}{\hyperlink{command.HOL.code-library}{\mbox{\isa{\isacommand{code{\isacharunderscore}library}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1196 |
\indexdef{HOL}{command}{consts\_code}\hypertarget{command.HOL.consts-code}{\hyperlink{command.HOL.consts-code}{\mbox{\isa{\isacommand{consts{\isacharunderscore}code}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1197 |
\indexdef{HOL}{command}{types\_code}\hypertarget{command.HOL.types-code}{\hyperlink{command.HOL.types-code}{\mbox{\isa{\isacommand{types{\isacharunderscore}code}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\ |
|
1198 |
\indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\ |
|
26849 | 1199 |
\end{matharray} |
1200 |
||
1201 |
\begin{rail} |
|
1202 |
( 'code\_module' | 'code\_library' ) modespec ? name ? \\ |
|
1203 |
( 'file' name ) ? ( 'imports' ( name + ) ) ? \\ |
|
1204 |
'contains' ( ( name '=' term ) + | term + ) |
|
1205 |
; |
|
1206 |
||
1207 |
modespec: '(' ( name * ) ')' |
|
1208 |
; |
|
1209 |
||
1210 |
'consts\_code' (codespec +) |
|
1211 |
; |
|
1212 |
||
1213 |
codespec: const template attachment ? |
|
1214 |
; |
|
1215 |
||
1216 |
'types\_code' (tycodespec +) |
|
1217 |
; |
|
1218 |
||
1219 |
tycodespec: name template attachment ? |
|
1220 |
; |
|
1221 |
||
1222 |
const: term |
|
1223 |
; |
|
1224 |
||
1225 |
template: '(' string ')' |
|
1226 |
; |
|
1227 |
||
1228 |
attachment: 'attach' modespec ? verblbrace text verbrbrace |
|
1229 |
; |
|
1230 |
||
1231 |
'code' (name)? |
|
1232 |
; |
|
37422 | 1233 |
\end{rail}% |
26849 | 1234 |
\end{isamarkuptext}% |
1235 |
\isamarkuptrue% |
|
1236 |
% |
|
27047 | 1237 |
\isamarkupsection{Definition by specification \label{sec:hol-specification}% |
1238 |
} |
|
1239 |
\isamarkuptrue% |
|
1240 |
% |
|
1241 |
\begin{isamarkuptext}% |
|
1242 |
\begin{matharray}{rcl} |
|
28788 | 1243 |
\indexdef{HOL}{command}{specification}\hypertarget{command.HOL.specification}{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\ |
1244 |
\indexdef{HOL}{command}{ax\_specification}\hypertarget{command.HOL.ax-specification}{\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\ |
|
27047 | 1245 |
\end{matharray} |
1246 |
||
1247 |
\begin{rail} |
|
1248 |
('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +) |
|
1249 |
; |
|
1250 |
decl: ((name ':')? term '(' 'overloaded' ')'?) |
|
1251 |
\end{rail} |
|
1252 |
||
28788 | 1253 |
\begin{description} |
27047 | 1254 |
|
28788 | 1255 |
\item \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up a |
27047 | 1256 |
goal stating the existence of terms with the properties specified to |
1257 |
hold for the constants given in \isa{decls}. After finishing the |
|
1258 |
proof, the theory will be augmented with definitions for the given |
|
1259 |
constants, as well as with theorems stating the properties for these |
|
1260 |
constants. |
|
1261 |
||
28788 | 1262 |
\item \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up |
1263 |
a goal stating the existence of terms with the properties specified |
|
1264 |
to hold for the constants given in \isa{decls}. After finishing |
|
1265 |
the proof, the theory will be augmented with axioms expressing the |
|
1266 |
properties given in the first place. |
|
27047 | 1267 |
|
28788 | 1268 |
\item \isa{decl} declares a constant to be defined by the |
27047 | 1269 |
specification given. The definition for the constant \isa{c} is |
1270 |
bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in |
|
1271 |
the declaration. Overloaded constants should be declared as such. |
|
1272 |
||
28788 | 1273 |
\end{description} |
27047 | 1274 |
|
1275 |
Whether to use \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} or \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} is to some extent a matter of style. \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} introduces no new axioms, and so by |
|
1276 |
construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} does introduce axioms, but only after the |
|
1277 |
user has explicitly proven it to be safe. A practical issue must be |
|
1278 |
considered, though: After introducing two constants with the same |
|
1279 |
properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove |
|
1280 |
that the two constants are, in fact, equal. If this might be a |
|
1281 |
problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.% |
|
1282 |
\end{isamarkuptext}% |
|
1283 |
\isamarkuptrue% |
|
1284 |
% |
|
26849 | 1285 |
\isadelimtheory |
1286 |
% |
|
1287 |
\endisadelimtheory |
|
1288 |
% |
|
1289 |
\isatagtheory |
|
26840 | 1290 |
\isacommand{end}\isamarkupfalse% |
1291 |
% |
|
1292 |
\endisatagtheory |
|
1293 |
{\isafoldtheory}% |
|
1294 |
% |
|
1295 |
\isadelimtheory |
|
1296 |
% |
|
1297 |
\endisadelimtheory |
|
26849 | 1298 |
\isanewline |
26840 | 1299 |
\end{isabellebody}% |
1300 |
%%% Local Variables: |
|
1301 |
%%% mode: latex |
|
1302 |
%%% TeX-master: "root" |
|
1303 |
%%% End: |