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(* Title: Doc/Datatypes/Datatypes.thy |
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Author: Jasmin Blanchette, TU Muenchen |
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Author: Martin Desharnais, Ecole de technologie superieure |
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Author: Lorenz Panny, TU Muenchen |
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Author: Andrei Popescu, TU Muenchen |
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Author: Dmitriy Traytel, TU Muenchen |
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Tutorial for (co)datatype definitions with the new package. |
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*) |
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theory Datatypes |
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imports |
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Setup |
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"~~/src/HOL/Library/BNF_Axiomatization" |
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"~~/src/HOL/Library/Cardinal_Notations" |
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"~~/src/HOL/Library/FSet" |
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"~~/src/HOL/Library/Simps_Case_Conv" |
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begin |
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section {* Introduction |
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\label{sec:introduction} *} |
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text {* |
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The 2013 edition of Isabelle introduced a new definitional package for freely |
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generated datatypes and codatatypes. The datatype support is similar to that |
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provided by the earlier package due to Berghofer and Wenzel |
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\cite{Berghofer-Wenzel:1999:TPHOL}, documented in the Isar reference manual |
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\cite{isabelle-isar-ref}; indeed, replacing the keyword \keyw{datatype} by |
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@{command datatype_new} is often all that is needed to port existing theories to |
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use the new package. |
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Perhaps the main advantage of the new package is that it supports recursion |
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through a large class of non-datatypes, such as finite sets: |
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*} |
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datatype_new 'a tree\<^sub>f\<^sub>s = Node\<^sub>f\<^sub>s (lbl\<^sub>f\<^sub>s: 'a) (sub\<^sub>f\<^sub>s: "'a tree\<^sub>f\<^sub>s fset") |
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text {* |
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\noindent |
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Another strong point is the support for local definitions: |
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*} |
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context linorder |
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begin |
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datatype_new flag = Less | Eq | Greater |
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end |
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text {* |
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\noindent |
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Furthermore, the package provides a lot of convenience, including automatically |
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generated discriminators, selectors, and relators as well as a wealth of |
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properties about them. |
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In addition to inductive datatypes, the new package supports coinductive |
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datatypes, or \emph{codatatypes}, which allow infinite values. For example, the |
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following command introduces the type of lazy lists, which comprises both finite |
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and infinite values: |
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*} |
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(*<*) |
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locale early |
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locale late |
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(*>*) |
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codatatype (*<*)(in early) (*>*)'a llist = LNil | LCons 'a "'a llist" |
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text {* |
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\noindent |
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Mixed inductive--coinductive recursion is possible via nesting. Compare the |
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following four Rose tree examples: |
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*} |
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datatype_new (*<*)(in early) (*>*)'a tree\<^sub>f\<^sub>f = Node\<^sub>f\<^sub>f 'a "'a tree\<^sub>f\<^sub>f list" |
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datatype_new (*<*)(in early) (*>*)'a tree\<^sub>f\<^sub>i = Node\<^sub>f\<^sub>i 'a "'a tree\<^sub>f\<^sub>i llist" |
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codatatype (*<*)(in early) (*>*)'a tree\<^sub>i\<^sub>f = Node\<^sub>i\<^sub>f 'a "'a tree\<^sub>i\<^sub>f list" |
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codatatype (*<*)(in early) (*>*)'a tree\<^sub>i\<^sub>i = Node\<^sub>i\<^sub>i 'a "'a tree\<^sub>i\<^sub>i llist" |
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text {* |
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The first two tree types allow only paths of finite length, whereas the last two |
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allow infinite paths. Orthogonally, the nodes in the first and third types have |
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finitely many direct subtrees, whereas those of the second and fourth may have |
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infinite branching. |
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The package is part of @{theory Main}. Additional functionality is provided by |
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the theory @{theory BNF_Axiomatization}, located in the directory |
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\verb|~~/src/HOL/Library|. |
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The package, like its predecessor, fully adheres to the LCF philosophy |
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\cite{mgordon79}: The characteristic theorems associated with the specified |
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(co)datatypes are derived rather than introduced axiomatically.% |
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\footnote{However, some of the |
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internal constructions and most of the internal proof obligations are skipped |
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if the @{text quick_and_dirty} option is enabled.} |
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The package is described in a number of papers |
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\cite{traytel-et-al-2012,blanchette-et-al-wit,blanchette-et-al-2014-impl,panny-et-al-2014}. |
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The central notion is that of a \emph{bounded natural functor} (BNF)---a |
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well-behaved type constructor for which nested (co)recursion is supported. |
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This tutorial is organized as follows: |
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\begin{itemize} |
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\setlength{\itemsep}{0pt} |
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\item Section \ref{sec:defining-datatypes}, ``Defining Datatypes,'' |
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describes how to specify datatypes using the @{command datatype_new} command. |
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\item Section \ref{sec:defining-recursive-functions}, ``Defining Recursive |
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Functions,'' describes how to specify recursive functions using |
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@{command primrec}. |
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\item Section \ref{sec:defining-codatatypes}, ``Defining Codatatypes,'' |
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describes how to specify codatatypes using the @{command codatatype} command. |
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\item Section \ref{sec:defining-corecursive-functions}, ``Defining Corecursive |
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Functions,'' describes how to specify corecursive functions using the |
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@{command primcorec} and @{command primcorecursive} commands. |
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\item Section \ref{sec:introducing-bounded-natural-functors}, ``Introducing |
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Bounded Natural Functors,'' explains how to use the @{command bnf} command |
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to register arbitrary type constructors as BNFs. |
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\item Section |
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\ref{sec:deriving-destructors-and-theorems-for-free-constructors}, |
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``Deriving Destructors and Theorems for Free Constructors,'' explains how to |
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use the command @{command free_constructors} to derive destructor constants |
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and theorems for freely generated types, as performed internally by @{command |
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datatype_new} and @{command codatatype}. |
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%\item Section \ref{sec:using-the-standard-ml-interface}, ``Using the Standard |
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ML Interface,'' %describes the package's programmatic interface. |
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\item Section \ref{sec:controlling-plugins}, ``Controlling Plugins,'' |
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is concerned with the package's interoperability with other Isabelle packages |
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and tools, such as the code generator, Transfer, Lifting, and Quickcheck. |
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%\item Section \ref{sec:known-bugs-and-limitations}, ``Known Bugs and |
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%Limitations,'' concludes with known open issues at the time of writing. |
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\end{itemize} |
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\newbox\boxA |
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\setbox\boxA=\hbox{\texttt{NOSPAM}} |
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\newcommand\authoremaili{\texttt{blan{\color{white}NOSPAM}\kern-\wd\boxA{}chette@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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\newcommand\authoremailii{\texttt{desh{\color{white}NOSPAM}\kern-\wd\boxA{}arna@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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\newcommand\authoremailiii{\texttt{lore{\color{white}NOSPAM}\kern-\wd\boxA{}nz.panny@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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\newcommand\authoremailiv{\texttt{pope{\color{white}NOSPAM}\kern-\wd\boxA{}scua@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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\newcommand\authoremailv{\texttt{tray{\color{white}NOSPAM}\kern-\wd\boxA{}tel@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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The command @{command datatype_new} is expected to replace \keyw{datatype} in a |
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future release. Authors of new theories are encouraged to use the new commands, |
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and maintainers of older theories may want to consider upgrading. |
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Comments and bug reports concerning either the tool or this tutorial should be |
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directed to the authors at \authoremaili, \authoremailii, \authoremailiii, |
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\authoremailiv, and \authoremailv. |
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*} |
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section {* Defining Datatypes |
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\label{sec:defining-datatypes} *} |
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text {* |
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Datatypes can be specified using the @{command datatype_new} command. |
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*} |
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subsection {* Introductory Examples |
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\label{ssec:datatype-introductory-examples} *} |
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text {* |
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Datatypes are illustrated through concrete examples featuring different flavors |
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of recursion. More examples can be found in the directory |
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\verb|~~/src/HOL/|\allowbreak\verb|BNF/Examples|. |
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*} |
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subsubsection {* Nonrecursive Types |
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\label{sssec:datatype-nonrecursive-types} *} |
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text {* |
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Datatypes are introduced by specifying the desired names and argument types for |
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their constructors. \emph{Enumeration} types are the simplest form of datatype. |
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All their constructors are nullary: |
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*} |
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datatype_new trool = Truue | Faalse | Perhaaps |
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text {* |
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\noindent |
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Here, @{const Truue}, @{const Faalse}, and @{const Perhaaps} have the type @{typ trool}. |
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Polymorphic types are possible, such as the following option type, modeled after |
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its homologue from the @{theory Option} theory: |
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*} |
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(*<*) |
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hide_const None Some map_option |
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hide_type option |
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(*>*) |
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datatype_new 'a option = None | Some 'a |
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text {* |
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\noindent |
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The constructors are @{text "None :: 'a option"} and |
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@{text "Some :: 'a \<Rightarrow> 'a option"}. |
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The next example has three type parameters: |
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*} |
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datatype_new ('a, 'b, 'c) triple = Triple 'a 'b 'c |
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text {* |
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\noindent |
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The constructor is |
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@{text "Triple :: 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) triple"}. |
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Unlike in Standard ML, curried constructors are supported. The uncurried variant |
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is also possible: |
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*} |
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datatype_new ('a, 'b, 'c) triple\<^sub>u = Triple\<^sub>u "'a * 'b * 'c" |
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text {* |
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\noindent |
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Occurrences of nonatomic types on the right-hand side of the equal sign must be |
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enclosed in double quotes, as is customary in Isabelle. |
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*} |
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subsubsection {* Simple Recursion |
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\label{sssec:datatype-simple-recursion} *} |
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text {* |
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Natural numbers are the simplest example of a recursive type: |
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*} |
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datatype_new nat = Zero | Succ nat |
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text {* |
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\noindent |
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Lists were shown in the introduction. Terminated lists are a variant that |
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stores a value of type @{typ 'b} at the very end: |
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*} |
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datatype_new (*<*)(in early) (*>*)('a, 'b) tlist = TNil 'b | TCons 'a "('a, 'b) tlist" |
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subsubsection {* Mutual Recursion |
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\label{sssec:datatype-mutual-recursion} *} |
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text {* |
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\emph{Mutually recursive} types are introduced simultaneously and may refer to |
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each other. The example below introduces a pair of types for even and odd |
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natural numbers: |
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*} |
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datatype_new even_nat = Even_Zero | Even_Succ odd_nat |
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and odd_nat = Odd_Succ even_nat |
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text {* |
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\noindent |
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Arithmetic expressions are defined via terms, terms via factors, and factors via |
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expressions: |
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*} |
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datatype_new ('a, 'b) exp = |
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Term "('a, 'b) trm" | Sum "('a, 'b) trm" "('a, 'b) exp" |
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and ('a, 'b) trm = |
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Factor "('a, 'b) fct" | Prod "('a, 'b) fct" "('a, 'b) trm" |
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and ('a, 'b) fct = |
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Const 'a | Var 'b | Expr "('a, 'b) exp" |
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subsubsection {* Nested Recursion |
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\label{sssec:datatype-nested-recursion} *} |
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text {* |
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\emph{Nested recursion} occurs when recursive occurrences of a type appear under |
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a type constructor. The introduction showed some examples of trees with nesting |
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through lists. A more complex example, that reuses our @{type option} type, |
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follows: |
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*} |
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datatype_new 'a btree = |
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BNode 'a "'a btree option" "'a btree option" |
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text {* |
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\noindent |
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Not all nestings are admissible. For example, this command will fail: |
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*} |
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datatype_new 'a wrong = W1 | W2 (*<*)'a |
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typ (*>*)"'a wrong \<Rightarrow> 'a" |
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text {* |
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\noindent |
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The issue is that the function arrow @{text "\<Rightarrow>"} allows recursion |
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only through its right-hand side. This issue is inherited by polymorphic |
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datatypes defined in terms of~@{text "\<Rightarrow>"}: |
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*} |
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datatype_new ('a, 'b) fun_copy = Fun "'a \<Rightarrow> 'b" |
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datatype_new 'a also_wrong = W1 | W2 (*<*)'a |
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typ (*>*)"('a also_wrong, 'a) fun_copy" |
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text {* |
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\noindent |
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The following definition of @{typ 'a}-branching trees is legal: |
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*} |
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datatype_new 'a ftree = FTLeaf 'a | FTNode "'a \<Rightarrow> 'a ftree" |
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text {* |
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\noindent |
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And so is the definition of hereditarily finite sets: |
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*} |
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datatype_new hfset = HFSet "hfset fset" |
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text {* |
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\noindent |
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In general, type constructors @{text "('a\<^sub>1, \<dots>, 'a\<^sub>m) t"} |
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allow recursion on a subset of their type arguments @{text 'a\<^sub>1}, \ldots, |
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@{text 'a\<^sub>m}. These type arguments are called \emph{live}; the remaining |
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type arguments are called \emph{dead}. In @{typ "'a \<Rightarrow> 'b"} and |
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@{typ "('a, 'b) fun_copy"}, the type variable @{typ 'a} is dead and |
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@{typ 'b} is live. |
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Type constructors must be registered as BNFs to have live arguments. This is |
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done automatically for datatypes and codatatypes introduced by the @{command |
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datatype_new} and @{command codatatype} commands. |
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Section~\ref{sec:introducing-bounded-natural-functors} explains how to |
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register arbitrary type constructors as BNFs. |
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Here is another example that fails: |
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*} |
341 |
||
54187 | 342 |
datatype_new 'a pow_list = PNil 'a (*<*)'a |
343 |
datatype_new 'a pow_list' = PNil' 'a (*>*)| PCons "('a * 'a) pow_list" |
|
344 |
||
345 |
text {* |
|
346 |
\noindent |
|
55351 | 347 |
This attempted definition features a different flavor of nesting, where the |
348 |
recursive call in the type specification occurs around (rather than inside) |
|
349 |
another type constructor. |
|
54187 | 350 |
*} |
351 |
||
58245 | 352 |
|
54187 | 353 |
subsubsection {* Auxiliary Constants and Properties |
354 |
\label{sssec:datatype-auxiliary-constants-and-properties} *} |
|
52805 | 355 |
|
356 |
text {* |
|
53491 | 357 |
The @{command datatype_new} command introduces various constants in addition to |
53617 | 358 |
the constructors. With each datatype are associated set functions, a map |
359 |
function, a relator, discriminators, and selectors, all of which can be given |
|
54187 | 360 |
custom names. In the example below, the familiar names @{text null}, @{text hd}, |
361 |
@{text tl}, @{text set}, @{text map}, and @{text list_all2}, override the |
|
362 |
default names @{text is_Nil}, @{text un_Cons1}, @{text un_Cons2}, |
|
54491 | 363 |
@{text set_list}, @{text map_list}, and @{text rel_list}: |
52822 | 364 |
*} |
365 |
||
52841 | 366 |
(*<*) |
367 |
no_translations |
|
368 |
"[x, xs]" == "x # [xs]" |
|
369 |
"[x]" == "x # []" |
|
370 |
||
371 |
no_notation |
|
372 |
Nil ("[]") and |
|
373 |
Cons (infixr "#" 65) |
|
374 |
||
53543 | 375 |
hide_type list |
58245 | 376 |
hide_const Nil Cons case_list hd tl set map list_all2 rec_list size_list pred_list |
53025 | 377 |
|
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378 |
context early begin |
52841 | 379 |
(*>*) |
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|
380 |
datatype_new (set: 'a) list = |
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|
381 |
null: Nil |
53025 | 382 |
| Cons (hd: 'a) (tl: "'a list") |
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|
383 |
for |
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|
384 |
map: map |
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|
385 |
rel: list_all2 |
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|
386 |
where |
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|
387 |
"tl Nil = Nil" |
52822 | 388 |
|
389 |
text {* |
|
390 |
\noindent |
|
55353 | 391 |
The types of the constants that appear in the specification are listed below. |
55351 | 392 |
|
393 |
\medskip |
|
54187 | 394 |
|
395 |
\begin{tabular}{@ {}ll@ {}} |
|
396 |
Constructors: & |
|
397 |
@{text "Nil \<Colon> 'a list"} \\ |
|
398 |
& |
|
399 |
@{text "Cons \<Colon> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list"} \\ |
|
400 |
Discriminator: & |
|
401 |
@{text "null \<Colon> 'a list \<Rightarrow> bool"} \\ |
|
402 |
Selectors: & |
|
403 |
@{text "hd \<Colon> 'a list \<Rightarrow> 'a"} \\ |
|
404 |
& |
|
405 |
@{text "tl \<Colon> 'a list \<Rightarrow> 'a list"} \\ |
|
406 |
Set function: & |
|
407 |
@{text "set \<Colon> 'a list \<Rightarrow> 'a set"} \\ |
|
408 |
Map function: & |
|
409 |
@{text "map \<Colon> ('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list"} \\ |
|
410 |
Relator: & |
|
411 |
@{text "list_all2 \<Colon> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"} |
|
412 |
\end{tabular} |
|
413 |
||
55351 | 414 |
\medskip |
415 |
||
54187 | 416 |
The discriminator @{const null} and the selectors @{const hd} and @{const tl} |
55351 | 417 |
are characterized by the following conditional equations: |
52822 | 418 |
% |
53025 | 419 |
\[@{thm list.collapse(1)[of xs, no_vars]} |
420 |
\qquad @{thm list.collapse(2)[of xs, no_vars]}\] |
|
52822 | 421 |
% |
54187 | 422 |
For two-constructor datatypes, a single discriminator constant is sufficient. |
423 |
The discriminator associated with @{const Cons} is simply |
|
53491 | 424 |
@{term "\<lambda>xs. \<not> null xs"}. |
52822 | 425 |
|
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|
426 |
The \keyw{where} clause at the end of the command specifies a default value for |
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|
427 |
selectors applied to constructors on which they are not a priori specified. |
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|
428 |
Here, it is used to ensure that the tail of the empty list is itself (instead of |
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|
429 |
being left unspecified). |
52822 | 430 |
|
53617 | 431 |
Because @{const Nil} is nullary, it is also possible to use |
57091 | 432 |
@{term "\<lambda>xs. xs = Nil"} as a discriminator. This is the default behavior |
433 |
if we omit the identifier @{const null} and the associated colon. Some users |
|
434 |
argue against this, because the mixture of constructors and selectors in the |
|
435 |
characteristic theorems can lead Isabelle's automation to switch between the |
|
436 |
constructor and the destructor view in surprising ways. |
|
52822 | 437 |
|
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|
438 |
The usual mixfix syntax annotations are available for both types and |
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|
439 |
constructors. For example: |
52805 | 440 |
*} |
52794 | 441 |
|
53025 | 442 |
(*<*) |
443 |
end |
|
444 |
(*>*) |
|
53552 | 445 |
datatype_new ('a, 'b) prod (infixr "*" 20) = Pair 'a 'b |
446 |
||
447 |
text {* \blankline *} |
|
52822 | 448 |
|
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|
449 |
datatype_new (set: 'a) list = |
52822 | 450 |
null: Nil ("[]") |
52841 | 451 |
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65) |
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|
452 |
for |
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|
453 |
map: map |
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|
454 |
rel: list_all2 |
52841 | 455 |
|
456 |
text {* |
|
53535 | 457 |
\noindent |
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|
458 |
Incidentally, this is how the traditional syntax can be set up: |
52841 | 459 |
*} |
460 |
||
461 |
syntax "_list" :: "args \<Rightarrow> 'a list" ("[(_)]") |
|
462 |
||
53552 | 463 |
text {* \blankline *} |
464 |
||
52841 | 465 |
translations |
466 |
"[x, xs]" == "x # [xs]" |
|
467 |
"[x]" == "x # []" |
|
52822 | 468 |
|
52824 | 469 |
|
53617 | 470 |
subsection {* Command Syntax |
471 |
\label{ssec:datatype-command-syntax} *} |
|
472 |
||
57542 | 473 |
subsubsection {* \keyw{datatype_new} |
53621 | 474 |
\label{sssec:datatype-new} *} |
52794 | 475 |
|
52822 | 476 |
text {* |
53829 | 477 |
\begin{matharray}{rcl} |
478 |
@{command_def "datatype_new"} & : & @{text "local_theory \<rightarrow> local_theory"} |
|
479 |
\end{matharray} |
|
52822 | 480 |
|
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|
481 |
@{rail \<open> |
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|
482 |
@@{command datatype_new} target? @{syntax dt_options}? \<newline> |
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|
483 |
(@{syntax dt_name} '=' (@{syntax dt_ctor} + '|') \<newline> |
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|
484 |
@{syntax map_rel}? (@'where' (prop + '|'))? + @'and') |
52828 | 485 |
; |
58190 | 486 |
@{syntax_def plugins}: 'plugins' ('only' | 'del') ':' (name +) |
487 |
; |
|
58192 | 488 |
@{syntax_def dt_options}: '(' ((@{syntax plugins} | 'discs_sels') + ',') ')' |
57206
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|
489 |
; |
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|
490 |
@{syntax_def map_rel}: @'for' ((('map' | 'rel') ':' name) +) |
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|
491 |
\<close>} |
52824 | 492 |
|
55351 | 493 |
\medskip |
494 |
||
495 |
\noindent |
|
55460
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|
496 |
The @{command datatype_new} command introduces a set of mutually recursive |
3f4efd7d950d
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blanchet
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changeset
|
497 |
datatypes specified by their constructors. |
3f4efd7d950d
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diff
changeset
|
498 |
|
53863
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|
499 |
The syntactic entity \synt{target} can be used to specify a local |
57200
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|
500 |
context (e.g., @{text "(in linorder)"} \cite{isabelle-isar-ref}), |
aab87ffa60cc
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|
501 |
and \synt{prop} denotes a HOL proposition. |
55460
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|
502 |
|
58190 | 503 |
The optional target is optionally followed by a combination of the following |
56124 | 504 |
options: |
52822 | 505 |
|
52824 | 506 |
\begin{itemize} |
507 |
\setlength{\itemsep}{0pt} |
|
508 |
||
509 |
\item |
|
58190 | 510 |
The @{text "plugins"} option indicates which plugins should be enabled |
58235 | 511 |
(@{text only}) or disabled (@{text del}). Wildcards (``@{text "*"}'') are |
512 |
allowed (e.g., @{text "quickcheck_*"}). By default, all plugins are enabled. |
|
58190 | 513 |
|
514 |
\item |
|
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|
515 |
The @{text "discs_sels"} option indicates that discriminators and selectors |
589ec121ce1a
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|
516 |
should be generated. The option is implicitly enabled if names are specified for |
589ec121ce1a
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|
517 |
discriminators or selectors. |
52824 | 518 |
\end{itemize} |
52822 | 519 |
|
57200
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|
520 |
The optional \keyw{where} clause specifies default values for selectors. |
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|
521 |
Each proposition must be an equation of the form |
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|
522 |
@{text "un_D (C \<dots>) = \<dots>"}, where @{text C} is a constructor and |
aab87ffa60cc
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|
523 |
@{text un_D} is a selector. |
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|
524 |
|
52827 | 525 |
The left-hand sides of the datatype equations specify the name of the type to |
53534 | 526 |
define, its type parameters, and additional information: |
52822 | 527 |
|
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|
528 |
@{rail \<open> |
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|
529 |
@{syntax_def dt_name}: @{syntax tyargs}? name mixfix? |
52824 | 530 |
; |
57092 | 531 |
@{syntax_def tyargs}: typefree | '(' (('dead' | name ':')? typefree + ',') ')' |
55112
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|
532 |
\<close>} |
52822 | 533 |
|
55351 | 534 |
\medskip |
535 |
||
52827 | 536 |
\noindent |
55474 | 537 |
The syntactic entity \synt{name} denotes an identifier, \synt{mixfix} denotes |
538 |
the usual parenthesized mixfix notation, and \synt{typefree} denotes fixed type |
|
539 |
variable (@{typ 'a}, @{typ 'b}, \ldots) \cite{isabelle-isar-ref}. |
|
52822 | 540 |
|
52827 | 541 |
The optional names preceding the type variables allow to override the default |
57988 | 542 |
names of the set functions (@{text set\<^sub>1_t}, \ldots, @{text set\<^sub>m_t}). Type |
58190 | 543 |
arguments can be marked as dead by entering @{text dead} in front of the |
58220 | 544 |
type variable (e.g., @{text "(dead 'a)"}); otherwise, they are live or dead |
55705 | 545 |
(and a set function is generated or not) depending on where they occur in the |
546 |
right-hand sides of the definition. Declaring a type argument as dead can speed |
|
547 |
up the type definition but will prevent any later (co)recursion through that |
|
548 |
type argument. |
|
549 |
||
53647 | 550 |
Inside a mutually recursive specification, all defined datatypes must |
551 |
mention exactly the same type variables in the same order. |
|
52822 | 552 |
|
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|
553 |
@{rail \<open> |
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|
554 |
@{syntax_def dt_ctor}: (name ':')? name (@{syntax dt_ctor_arg} * ) mixfix? |
55112
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|
555 |
\<close>} |
52824 | 556 |
|
53535 | 557 |
\medskip |
558 |
||
52827 | 559 |
\noindent |
53863
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changeset
|
560 |
The main constituents of a constructor specification are the name of the |
52827 | 561 |
constructor and the list of its argument types. An optional discriminator name |
57091 | 562 |
can be supplied at the front to override the default, which is |
563 |
@{text "\<lambda>x. x = C\<^sub>j"} for nullary constructors and |
|
564 |
@{text t.is_C\<^sub>j} otherwise. |
|
52822 | 565 |
|
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|
566 |
@{rail \<open> |
55472
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|
567 |
@{syntax_def dt_ctor_arg}: type | '(' name ':' type ')' |
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|
568 |
\<close>} |
52827 | 569 |
|
53535 | 570 |
\medskip |
571 |
||
52827 | 572 |
\noindent |
55474 | 573 |
The syntactic entity \synt{type} denotes a HOL type \cite{isabelle-isar-ref}. |
574 |
||
52827 | 575 |
In addition to the type of a constructor argument, it is possible to specify a |
576 |
name for the corresponding selector to override the default name |
|
53554 | 577 |
(@{text un_C\<^sub>ji}). The same selector names can be reused for several |
578 |
constructors as long as they share the same type. |
|
52822 | 579 |
*} |
580 |
||
53617 | 581 |
|
57542 | 582 |
subsubsection {* \keyw{datatype_compat} |
56644 | 583 |
\label{sssec:datatype-compat} *} |
53617 | 584 |
|
585 |
text {* |
|
53829 | 586 |
\begin{matharray}{rcl} |
55531
601ca8efa000
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|
587 |
@{command_def "datatype_compat"} & : & @{text "local_theory \<rightarrow> local_theory"} |
53829 | 588 |
\end{matharray} |
589 |
||
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|
590 |
@{rail \<open> |
55531
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|
591 |
@@{command datatype_compat} (name +) |
55112
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|
592 |
\<close>} |
53829 | 593 |
|
55351 | 594 |
\medskip |
595 |
||
53829 | 596 |
\noindent |
55531
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changeset
|
597 |
The @{command datatype_compat} command registers new-style datatypes as |
58245 | 598 |
old-style datatypes and invokes the old-style plugins. For example: |
53621 | 599 |
*} |
600 |
||
55531
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|
601 |
datatype_compat even_nat odd_nat |
53621 | 602 |
|
603 |
text {* \blankline *} |
|
604 |
||
58121 | 605 |
ML {* Old_Datatype_Data.get_info @{theory} @{type_name even_nat} *} |
53621 | 606 |
|
607 |
text {* |
|
55474 | 608 |
The syntactic entity \synt{name} denotes an identifier \cite{isabelle-isar-ref}. |
609 |
||
58212 | 610 |
The command can be useful because the old datatype package provides some |
611 |
functionality that is not yet replicated in the new package. |
|
55474 | 612 |
|
613 |
A few remarks concern nested recursive datatypes: |
|
53748 | 614 |
|
615 |
\begin{itemize} |
|
53749
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|
616 |
\setlength{\itemsep}{0pt} |
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|
617 |
|
55871 | 618 |
\item The old-style, nested-as-mutual induction rule and recursor theorems are |
619 |
generated under their usual names but with ``@{text "compat_"}'' prefixed |
|
58215 | 620 |
(e.g., @{text compat_tree.induct}, @{text compat_tree.inducts}, and |
621 |
@{text compat_tree.rec}). |
|
53748 | 622 |
|
623 |
\item All types through which recursion takes place must be new-style datatypes |
|
624 |
or the function type. In principle, it should be possible to support old-style |
|
55474 | 625 |
datatypes as well, but this has not been implemented yet (and there is currently |
626 |
no way to register old-style datatypes as new-style datatypes). |
|
53748 | 627 |
\end{itemize} |
628 |
||
55531
601ca8efa000
renamed 'datatype_new_compat' to 'datatype_compat'
blanchet
parents:
55530
diff
changeset
|
629 |
An alternative to @{command datatype_compat} is to use the old package's |
57542 | 630 |
\keyw{rep_\allowbreak datatype} command. The associated proof obligations must then be |
53748 | 631 |
discharged manually. |
53617 | 632 |
*} |
633 |
||
634 |
||
635 |
subsection {* Generated Constants |
|
636 |
\label{ssec:datatype-generated-constants} *} |
|
637 |
||
638 |
text {* |
|
53623 | 639 |
Given a datatype @{text "('a\<^sub>1, \<dots>, 'a\<^sub>m) t"} |
53617 | 640 |
with $m > 0$ live type variables and $n$ constructors |
641 |
@{text "t.C\<^sub>1"}, \ldots, @{text "t.C\<^sub>n"}, the |
|
642 |
following auxiliary constants are introduced: |
|
643 |
||
55353 | 644 |
\medskip |
645 |
||
646 |
\begin{tabular}{@ {}ll@ {}} |
|
647 |
Case combinator: & |
|
648 |
@{text t.case_t} (rendered using the familiar @{text case}--@{text of} syntax) \\ |
|
649 |
Discriminators: & |
|
57988 | 650 |
@{text t.is_C\<^sub>1}$, \ldots, $@{text t.is_C\<^sub>n} \\ |
55353 | 651 |
Selectors: & |
652 |
@{text t.un_C\<^sub>11}$, \ldots, $@{text t.un_C\<^sub>1k\<^sub>1} \\ |
|
653 |
& \quad\vdots \\ |
|
654 |
& @{text t.un_C\<^sub>n1}$, \ldots, $@{text t.un_C\<^sub>nk\<^sub>n} \\ |
|
655 |
Set functions: & |
|
57988 | 656 |
@{text t.set\<^sub>1_t}, \ldots, @{text t.set\<^sub>m_t} \\ |
55353 | 657 |
Map function: & |
658 |
@{text t.map_t} \\ |
|
659 |
Relator: & |
|
660 |
@{text t.rel_t} \\ |
|
661 |
Recursor: & |
|
58190 | 662 |
@{text t.rec_t} |
55353 | 663 |
\end{tabular} |
664 |
||
665 |
\medskip |
|
53617 | 666 |
|
667 |
\noindent |
|
58245 | 668 |
In addition, some of the plugins introduce their own constants |
669 |
(Section~\ref{sec:plugins}). The case combinator, discriminators, and selectors |
|
670 |
are collectively called \emph{destructors}. The prefix ``@{text "t."}'' is an |
|
671 |
optional component of the names and is normally hidden. |
|
53617 | 672 |
*} |
673 |
||
674 |
||
52840 | 675 |
subsection {* Generated Theorems |
676 |
\label{ssec:datatype-generated-theorems} *} |
|
52828 | 677 |
|
678 |
text {* |
|
53544 | 679 |
The characteristic theorems generated by @{command datatype_new} are grouped in |
53623 | 680 |
three broad categories: |
53535 | 681 |
|
53543 | 682 |
\begin{itemize} |
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
683 |
\setlength{\itemsep}{0pt} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
684 |
|
56655 | 685 |
\item The \emph{free constructor theorems} |
686 |
(Section~\ref{sssec:free-constructor-theorems}) are properties of the |
|
687 |
constructors and destructors that can be derived for any freely generated type. |
|
688 |
Internally, the derivation is performed by @{command free_constructors}. |
|
689 |
||
690 |
\item The \emph{functorial theorems} (Section~\ref{sssec:functorial-theorems}) |
|
691 |
are properties of datatypes related to their BNF nature. |
|
692 |
||
693 |
\item The \emph{inductive theorems} (Section~\ref{sssec:inductive-theorems}) |
|
694 |
are properties of datatypes related to their inductive nature. |
|
53552 | 695 |
|
53543 | 696 |
\end{itemize} |
53535 | 697 |
|
698 |
\noindent |
|
53542 | 699 |
The full list of named theorems can be obtained as usual by entering the |
57542 | 700 |
command \keyw{print_theorems} immediately after the datatype definition. |
58245 | 701 |
This list includes theorems produced by plugins (Section~\ref{sec:plugins}), |
702 |
but normally excludes low-level theorems that reveal internal constructions. To |
|
703 |
make these accessible, add the line |
|
53542 | 704 |
*} |
53535 | 705 |
|
53542 | 706 |
declare [[bnf_note_all]] |
707 |
(*<*) |
|
708 |
declare [[bnf_note_all = false]] |
|
709 |
(*>*) |
|
53535 | 710 |
|
53552 | 711 |
text {* |
712 |
\noindent |
|
713 |
to the top of the theory file. |
|
714 |
*} |
|
53535 | 715 |
|
58245 | 716 |
|
53621 | 717 |
subsubsection {* Free Constructor Theorems |
718 |
\label{sssec:free-constructor-theorems} *} |
|
53535 | 719 |
|
53543 | 720 |
(*<*) |
53837 | 721 |
consts nonnull :: 'a |
53543 | 722 |
(*>*) |
723 |
||
53535 | 724 |
text {* |
54621 | 725 |
The free constructor theorems are partitioned in three subgroups. The first |
726 |
subgroup of properties is concerned with the constructors. They are listed below |
|
727 |
for @{typ "'a list"}: |
|
53543 | 728 |
|
53552 | 729 |
\begin{indentblock} |
53543 | 730 |
\begin{description} |
53544 | 731 |
|
53642 | 732 |
\item[@{text "t."}\hthm{inject} @{text "[iff, induct_simp]"}\rm:] ~ \\ |
53544 | 733 |
@{thm list.inject[no_vars]} |
734 |
||
53642 | 735 |
\item[@{text "t."}\hthm{distinct} @{text "[simp, induct_simp]"}\rm:] ~ \\ |
53543 | 736 |
@{thm list.distinct(1)[no_vars]} \\ |
737 |
@{thm list.distinct(2)[no_vars]} |
|
738 |
||
53642 | 739 |
\item[@{text "t."}\hthm{exhaust} @{text "[cases t, case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
53543 | 740 |
@{thm list.exhaust[no_vars]} |
741 |
||
53642 | 742 |
\item[@{text "t."}\hthm{nchotomy}\rm:] ~ \\ |
53543 | 743 |
@{thm list.nchotomy[no_vars]} |
744 |
||
745 |
\end{description} |
|
53552 | 746 |
\end{indentblock} |
53543 | 747 |
|
748 |
\noindent |
|
53621 | 749 |
In addition, these nameless theorems are registered as safe elimination rules: |
750 |
||
751 |
\begin{indentblock} |
|
752 |
\begin{description} |
|
753 |
||
54386 | 754 |
\item[@{text "t."}\hthm{distinct {\upshape[}THEN notE}@{text ", elim!"}\hthm{\upshape]}\rm:] ~ \\ |
53621 | 755 |
@{thm list.distinct(1)[THEN notE, elim!, no_vars]} \\ |
756 |
@{thm list.distinct(2)[THEN notE, elim!, no_vars]} |
|
757 |
||
758 |
\end{description} |
|
759 |
\end{indentblock} |
|
760 |
||
761 |
\noindent |
|
53543 | 762 |
The next subgroup is concerned with the case combinator: |
763 |
||
53552 | 764 |
\begin{indentblock} |
53543 | 765 |
\begin{description} |
53544 | 766 |
|
53798 | 767 |
\item[@{text "t."}\hthm{case} @{text "[simp, code]"}\rm:] ~ \\ |
53543 | 768 |
@{thm list.case(1)[no_vars]} \\ |
769 |
@{thm list.case(2)[no_vars]} |
|
770 |
||
57542 | 771 |
\item[@{text "t."}\hthm{case_cong} @{text "[fundef_cong]"}\rm:] ~ \\ |
53543 | 772 |
@{thm list.case_cong[no_vars]} |
773 |
||
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
774 |
\item[@{text "t."}\hthm{case_cong_weak} @{text "[cong]"}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
775 |
@{thm list.case_cong_weak[no_vars]} |
53543 | 776 |
|
53642 | 777 |
\item[@{text "t."}\hthm{split}\rm:] ~ \\ |
53543 | 778 |
@{thm list.split[no_vars]} |
779 |
||
57542 | 780 |
\item[@{text "t."}\hthm{split_asm}\rm:] ~ \\ |
53543 | 781 |
@{thm list.split_asm[no_vars]} |
782 |
||
53544 | 783 |
\item[@{text "t."}\hthm{splits} = @{text "split split_asm"}] |
53543 | 784 |
|
785 |
\end{description} |
|
53552 | 786 |
\end{indentblock} |
53543 | 787 |
|
788 |
\noindent |
|
54621 | 789 |
The third subgroup revolves around discriminators and selectors: |
53543 | 790 |
|
53552 | 791 |
\begin{indentblock} |
53543 | 792 |
\begin{description} |
53544 | 793 |
|
53694 | 794 |
\item[@{text "t."}\hthm{disc} @{text "[simp]"}\rm:] ~ \\ |
795 |
@{thm list.disc(1)[no_vars]} \\ |
|
796 |
@{thm list.disc(2)[no_vars]} |
|
797 |
||
53703 | 798 |
\item[@{text "t."}\hthm{discI}\rm:] ~ \\ |
799 |
@{thm list.discI(1)[no_vars]} \\ |
|
800 |
@{thm list.discI(2)[no_vars]} |
|
801 |
||
53805 | 802 |
\item[@{text "t."}\hthm{sel} @{text "[simp, code]"}\rm:] ~ \\ |
53694 | 803 |
@{thm list.sel(1)[no_vars]} \\ |
804 |
@{thm list.sel(2)[no_vars]} |
|
53543 | 805 |
|
53642 | 806 |
\item[@{text "t."}\hthm{collapse} @{text "[simp]"}\rm:] ~ \\ |
53543 | 807 |
@{thm list.collapse(1)[no_vars]} \\ |
58151
414deb2ef328
take out 'x = C' of the simplifier for unit types
blanchet
parents:
58121
diff
changeset
|
808 |
@{thm list.collapse(2)[no_vars]} \\ |
414deb2ef328
take out 'x = C' of the simplifier for unit types
blanchet
parents:
58121
diff
changeset
|
809 |
(The @{text "[simp]"} attribute is exceptionally omitted for datatypes equipped |
414deb2ef328
take out 'x = C' of the simplifier for unit types
blanchet
parents:
58121
diff
changeset
|
810 |
with a single nullary constructor, because a property of the form |
414deb2ef328
take out 'x = C' of the simplifier for unit types
blanchet
parents:
58121
diff
changeset
|
811 |
@{prop "x = C"} is not suitable as a simplification rule.) |
53543 | 812 |
|
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
813 |
\item[@{text "t."}\hthm{distinct_disc} @{text "[dest]"}\rm:] ~ \\ |
53543 | 814 |
These properties are missing for @{typ "'a list"} because there is only one |
815 |
proper discriminator. Had the datatype been introduced with a second |
|
53837 | 816 |
discriminator called @{const nonnull}, they would have read thusly: \\[\jot] |
817 |
@{prop "null list \<Longrightarrow> \<not> nonnull list"} \\ |
|
818 |
@{prop "nonnull list \<Longrightarrow> \<not> null list"} |
|
53543 | 819 |
|
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
820 |
\item[@{text "t."}\hthm{exhaust_disc} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
821 |
@{thm list.exhaust_disc[no_vars]} |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
822 |
|
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
823 |
\item[@{text "t."}\hthm{exhaust_sel} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
824 |
@{thm list.exhaust_sel[no_vars]} |
53916 | 825 |
|
53642 | 826 |
\item[@{text "t."}\hthm{expand}\rm:] ~ \\ |
53543 | 827 |
@{thm list.expand[no_vars]} |
828 |
||
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
829 |
\item[@{text "t."}\hthm{split_sel}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
830 |
@{thm list.split_sel[no_vars]} |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
831 |
|
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
832 |
\item[@{text "t."}\hthm{split_sel_asm}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
833 |
@{thm list.split_sel_asm[no_vars]} |
53917 | 834 |
|
57984
cbe9a16f8e11
added collection theorem for consistency and convenience
blanchet
parents:
57983
diff
changeset
|
835 |
\item[@{text "t."}\hthm{split_sels} = @{text "split_sel split_sel_asm"}] |
cbe9a16f8e11
added collection theorem for consistency and convenience
blanchet
parents:
57983
diff
changeset
|
836 |
|
57542 | 837 |
\item[@{text "t."}\hthm{case_eq_if}\rm:] ~ \\ |
54491 | 838 |
@{thm list.case_eq_if[no_vars]} |
53543 | 839 |
|
840 |
\end{description} |
|
53552 | 841 |
\end{indentblock} |
54152 | 842 |
|
843 |
\noindent |
|
58151
414deb2ef328
take out 'x = C' of the simplifier for unit types
blanchet
parents:
58121
diff
changeset
|
844 |
In addition, equational versions of @{text t.disc} are registered with the |
414deb2ef328
take out 'x = C' of the simplifier for unit types
blanchet
parents:
58121
diff
changeset
|
845 |
@{text "[code]"} attribute. |
53552 | 846 |
*} |
847 |
||
848 |
||
53621 | 849 |
subsubsection {* Functorial Theorems |
850 |
\label{sssec:functorial-theorems} *} |
|
53552 | 851 |
|
852 |
text {* |
|
54621 | 853 |
The functorial theorems are partitioned in two subgroups. The first subgroup |
56992 | 854 |
consists of properties involving the constructors or the destructors and either |
855 |
a set function, the map function, or the relator: |
|
53552 | 856 |
|
857 |
\begin{indentblock} |
|
858 |
\begin{description} |
|
859 |
||
58094 | 860 |
\item[@{text "t."}\hthm{case_transfer}\rm:] ~ \\ |
861 |
@{thm list.case_transfer[no_vars]} |
|
862 |
||
58000 | 863 |
\item[@{text "t."}\hthm{ctr_transfer}\rm:] ~ \\ |
864 |
@{thm list.ctr_transfer(1)[no_vars]} \\ |
|
865 |
@{thm list.ctr_transfer(2)[no_vars]} |
|
866 |
||
58096 | 867 |
\item[@{text "t."}\hthm{disc_transfer}\rm:] ~ \\ |
868 |
@{thm list.disc_transfer(1)[no_vars]} \\ |
|
869 |
@{thm list.disc_transfer(2)[no_vars]} |
|
870 |
||
53798 | 871 |
\item[@{text "t."}\hthm{set} @{text "[simp, code]"}\rm:] ~ \\ |
53694 | 872 |
@{thm list.set(1)[no_vars]} \\ |
873 |
@{thm list.set(2)[no_vars]} |
|
53552 | 874 |
|
57988 | 875 |
\item[@{text "t."}\hthm{set_cases} @{text "[consumes 1, cases set: set\<^sub>i_t]"}\rm:] ~ \\ |
57894 | 876 |
@{thm list.set_cases[no_vars]} |
877 |
||
57892 | 878 |
\item[@{text "t."}\hthm{set_intros}\rm:] ~ \\ |
879 |
@{thm list.set_intros(1)[no_vars]} \\ |
|
880 |
@{thm list.set_intros(2)[no_vars]} |
|
881 |
||
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
882 |
\item[@{text "t."}\hthm{set_sel}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
883 |
@{thm list.set_sel[no_vars]} |
57153 | 884 |
|
53798 | 885 |
\item[@{text "t."}\hthm{map} @{text "[simp, code]"}\rm:] ~ \\ |
53552 | 886 |
@{thm list.map(1)[no_vars]} \\ |
887 |
@{thm list.map(2)[no_vars]} |
|
888 |
||
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
889 |
\item[@{text "t."}\hthm{map_disc_iff} @{text "[simp]"}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
890 |
@{thm list.map_disc_iff[no_vars]} |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
891 |
|
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
892 |
\item[@{text "t."}\hthm{map_sel}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
893 |
@{thm list.map_sel[no_vars]} |
57047 | 894 |
|
57542 | 895 |
\item[@{text "t."}\hthm{rel_inject} @{text "[simp]"}\rm:] ~ \\ |
53552 | 896 |
@{thm list.rel_inject(1)[no_vars]} \\ |
897 |
@{thm list.rel_inject(2)[no_vars]} |
|
898 |
||
57542 | 899 |
\item[@{text "t."}\hthm{rel_distinct} @{text "[simp]"}\rm:] ~ \\ |
57526 | 900 |
@{thm list.rel_distinct(1)[no_vars]} \\ |
901 |
@{thm list.rel_distinct(2)[no_vars]} |
|
902 |
||
57542 | 903 |
\item[@{text "t."}\hthm{rel_intros}\rm:] ~ \\ |
57494 | 904 |
@{thm list.rel_intros(1)[no_vars]} \\ |
905 |
@{thm list.rel_intros(2)[no_vars]} |
|
906 |
||
57558
6bb3dd7f8097
took out 'rel_cases' for now because of failing tactic
blanchet
parents:
57542
diff
changeset
|
907 |
% FIXME (and add @ before antiquotation below) |
6bb3dd7f8097
took out 'rel_cases' for now because of failing tactic
blanchet
parents:
57542
diff
changeset
|
908 |
%\item[@{text "t."}\hthm{rel_cases} @{text "[consumes 1, case_names t\<^sub>1 \<dots> t\<^sub>m, cases pred]"}\rm:] ~ \\ |
6bb3dd7f8097
took out 'rel_cases' for now because of failing tactic
blanchet
parents:
57542
diff
changeset
|
909 |
%{thm list.rel_cases[no_vars]} |
53552 | 910 |
|
57564 | 911 |
\item[@{text "t."}\hthm{rel_sel}\rm:] ~ \\ |
912 |
@{thm list.rel_sel[no_vars]} |
|
913 |
||
53552 | 914 |
\end{description} |
915 |
\end{indentblock} |
|
54146 | 916 |
|
917 |
\noindent |
|
918 |
In addition, equational versions of @{text t.rel_inject} and @{text |
|
919 |
rel_distinct} are registered with the @{text "[code]"} attribute. |
|
54621 | 920 |
|
921 |
The second subgroup consists of more abstract properties of the set functions, |
|
922 |
the map function, and the relator: |
|
923 |
||
924 |
\begin{indentblock} |
|
925 |
\begin{description} |
|
926 |
||
57969 | 927 |
\item[@{text "t."}\hthm{inj_map}\rm:] ~ \\ |
928 |
@{thm list.inj_map[no_vars]} |
|
929 |
||
57971 | 930 |
\item[@{text "t."}\hthm{inj_map_strong}\rm:] ~ \\ |
931 |
@{thm list.inj_map_strong[no_vars]} |
|
932 |
||
57542 | 933 |
\item[@{text "t."}\hthm{set_map}\rm:] ~ \\ |
56992 | 934 |
@{thm list.set_map[no_vars]} |
935 |
||
58107 | 936 |
\item[@{text "t."}\hthm{set_transfer}\rm:] ~ \\ |
937 |
@{thm list.set_transfer[no_vars]} |
|
938 |
||
58245 | 939 |
\item[@{text "t."}\hthm{map_cong0}\rm:] ~ \\ |
54621 | 940 |
@{thm list.map_cong0[no_vars]} |
941 |
||
57542 | 942 |
\item[@{text "t."}\hthm{map_cong} @{text "[fundef_cong]"}\rm:] ~ \\ |
54621 | 943 |
@{thm list.map_cong[no_vars]} |
944 |
||
57982 | 945 |
\item[@{text "t."}\hthm{map_cong_simp}\rm:] ~ \\ |
946 |
@{thm list.map_cong_simp[no_vars]} |
|
947 |
||
57542 | 948 |
\item[@{text "t."}\hthm{map_id}\rm:] ~ \\ |
54621 | 949 |
@{thm list.map_id[no_vars]} |
950 |
||
57542 | 951 |
\item[@{text "t."}\hthm{map_id0}\rm:] ~ \\ |
56955 | 952 |
@{thm list.map_id0[no_vars]} |
953 |
||
57542 | 954 |
\item[@{text "t."}\hthm{map_ident}\rm:] ~ \\ |
56904 | 955 |
@{thm list.map_ident[no_vars]} |
956 |
||
58103 | 957 |
\item[@{text "t."}\hthm{map_transfer}\rm:] ~ \\ |
958 |
@{thm list.map_transfer[no_vars]} |
|
959 |
||
58244 | 960 |
\item[@{text "t."}\hthm{rel_compp} @{text "[relator_distr]"}\rm:] ~ \\ |
58245 | 961 |
@{thm list.rel_compp[no_vars]} \\ |
58244 | 962 |
(The @{text "[relator_distr]"} attribute is set by the @{text lifting} plugin, |
963 |
Section~\ref{ssec:lifting}.) |
|
54621 | 964 |
|
57542 | 965 |
\item[@{text "t."}\hthm{rel_conversep}\rm:] ~ \\ |
54621 | 966 |
@{thm list.rel_conversep[no_vars]} |
967 |
||
57542 | 968 |
\item[@{text "t."}\hthm{rel_eq}\rm:] ~ \\ |
54621 | 969 |
@{thm list.rel_eq[no_vars]} |
970 |
||
57542 | 971 |
\item[@{text "t."}\hthm{rel_flip}\rm:] ~ \\ |
54621 | 972 |
@{thm list.rel_flip[no_vars]} |
973 |
||
57933 | 974 |
\item[@{text "t."}\hthm{rel_map}\rm:] ~ \\ |
975 |
@{thm list.rel_map(1)[no_vars]} \\ |
|
976 |
@{thm list.rel_map(2)[no_vars]} |
|
977 |
||
58244 | 978 |
\item[@{text "t."}\hthm{rel_mono} @{text "[relator_mono]"}\rm:] ~ \\ |
58245 | 979 |
@{thm list.rel_mono[no_vars]} \\ |
58244 | 980 |
(The @{text "[relator_distr]"} attribute is set by the @{text lifting} plugin, |
981 |
Section~\ref{ssec:lifting}.) |
|
54621 | 982 |
|
58105 | 983 |
\item[@{text "t."}\hthm{rel_transfer}\rm:] ~ \\ |
984 |
@{thm list.rel_transfer[no_vars]} |
|
985 |
||
54621 | 986 |
\end{description} |
987 |
\end{indentblock} |
|
53535 | 988 |
*} |
989 |
||
990 |
||
53621 | 991 |
subsubsection {* Inductive Theorems |
992 |
\label{sssec:inductive-theorems} *} |
|
53535 | 993 |
|
994 |
text {* |
|
53623 | 995 |
The inductive theorems are as follows: |
53544 | 996 |
|
53552 | 997 |
\begin{indentblock} |
53544 | 998 |
\begin{description} |
999 |
||
57304 | 1000 |
\item[@{text "t."}\hthm{induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n, induct t]"}\rm:] ~ \\ |
53544 | 1001 |
@{thm list.induct[no_vars]} |
1002 |
||
57542 | 1003 |
\item[@{text "t."}\hthm{rel_induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n, induct pred]"}\rm:] ~ \\ |
57472 | 1004 |
@{thm list.rel_induct[no_vars]} |
1005 |
||
57701 | 1006 |
\item[\begin{tabular}{@ {}l@ {}} |
1007 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm: \\ |
|
1008 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{rel_induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm: \\ |
|
1009 |
\end{tabular}] ~ \\ |
|
57472 | 1010 |
Given $m > 1$ mutually recursive datatypes, this induction rule can be used to |
1011 |
prove $m$ properties simultaneously. |
|
1012 |
||
53798 | 1013 |
\item[@{text "t."}\hthm{rec} @{text "[simp, code]"}\rm:] ~ \\ |
53544 | 1014 |
@{thm list.rec(1)[no_vars]} \\ |
1015 |
@{thm list.rec(2)[no_vars]} |
|
1016 |
||
1017 |
\end{description} |
|
53552 | 1018 |
\end{indentblock} |
53544 | 1019 |
|
1020 |
\noindent |
|
1021 |
For convenience, @{command datatype_new} also provides the following collection: |
|
1022 |
||
53552 | 1023 |
\begin{indentblock} |
53544 | 1024 |
\begin{description} |
1025 |
||
55871 | 1026 |
\item[@{text "t."}\hthm{simps} = @{text t.inject} @{text t.distinct} @{text t.case} @{text t.rec} @{text t.map} @{text t.rel_inject}] ~ \\ |
53694 | 1027 |
@{text t.rel_distinct} @{text t.set} |
53544 | 1028 |
|
1029 |
\end{description} |
|
53552 | 1030 |
\end{indentblock} |
52828 | 1031 |
*} |
1032 |
||
52794 | 1033 |
|
52827 | 1034 |
subsection {* Compatibility Issues |
52824 | 1035 |
\label{ssec:datatype-compatibility-issues} *} |
52794 | 1036 |
|
52828 | 1037 |
text {* |
53997 | 1038 |
The command @{command datatype_new} has been designed to be highly compatible |
1039 |
with the old \keyw{datatype}, to ease migration. There are nonetheless a few |
|
53647 | 1040 |
incompatibilities that may arise when porting to the new package: |
1041 |
||
1042 |
\begin{itemize} |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1043 |
\setlength{\itemsep}{0pt} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1044 |
|
53647 | 1045 |
\item \emph{The Standard ML interfaces are different.} Tools and extensions |
1046 |
written to call the old ML interfaces will need to be adapted to the new |
|
58245 | 1047 |
interfaces. If possible, it is recommended to register new-style datatypes as |
1048 |
old-style datatypes using the @{command datatype_compat} command. |
|
54537 | 1049 |
|
1050 |
\item \emph{The recursor @{text rec_t} has a different signature for nested |
|
54185 | 1051 |
recursive datatypes.} In the old package, nested recursion through non-functions |
1052 |
was internally reduced to mutual recursion. This reduction was visible in the |
|
1053 |
type of the recursor, used by \keyw{primrec}. Recursion through functions was |
|
1054 |
handled specially. In the new package, nested recursion (for functions and |
|
1055 |
non-functions) is handled in a more modular fashion. The old-style recursor can |
|
56655 | 1056 |
be generated on demand using @{command primrec} if the recursion is via |
1057 |
new-style datatypes, as explained in |
|
1058 |
Section~\ref{sssec:primrec-nested-as-mutual-recursion}. |
|
53647 | 1059 |
|
54287 | 1060 |
\item \emph{Accordingly, the induction rule is different for nested recursive |
1061 |
datatypes.} Again, the old-style induction rule can be generated on demand using |
|
56655 | 1062 |
@{command primrec} if the recursion is via new-style datatypes, as explained in |
1063 |
Section~\ref{sssec:primrec-nested-as-mutual-recursion}. |
|
52828 | 1064 |
|
53863
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
1065 |
\item \emph{The internal constructions are completely different.} Proof texts |
53647 | 1066 |
that unfold the definition of constants introduced by \keyw{datatype} will be |
1067 |
difficult to port. |
|
1068 |
||
58207 | 1069 |
\item \emph{Some constants and theorems have different names.} |
55641
5b466efedd2c
renamed 'recs' and 'cases' theorems 'rec' and 'case' in old datatype package, for consistency with new package
blanchet
parents:
55531
diff
changeset
|
1070 |
For non-mutually recursive datatypes, |
5b466efedd2c
renamed 'recs' and 'cases' theorems 'rec' and 'case' in old datatype package, for consistency with new package
blanchet
parents:
55531
diff
changeset
|
1071 |
the alias @{text t.inducts} for @{text t.induct} is no longer generated. |
53647 | 1072 |
For $m > 1$ mutually recursive datatypes, |
58207 | 1073 |
@{text "rec_t\<^sub>1_\<dots>_t\<^sub>m_i"} has been renamed |
1074 |
@{text "rec_t\<^sub>i"} for each @{text "i \<in> {1, \<dots>, t}"}, |
|
53997 | 1075 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m.inducts(i)"} has been renamed |
58207 | 1076 |
@{text "t\<^sub>i.induct"} for each @{text "i \<in> {1, \<dots>, t}"}, and the |
1077 |
collection @{text "t\<^sub>1_\<dots>_t\<^sub>m.size"} (generated by the |
|
1078 |
@{text size} plugin, Section~\ref{ssec:size}) has been divided into |
|
1079 |
@{text "t\<^sub>1.size"}, \ldots, @{text "t\<^sub>m.size"}. |
|
53647 | 1080 |
|
1081 |
\item \emph{The @{text t.simps} collection has been extended.} |
|
58207 | 1082 |
Previously available theorems are available at the same index as before. |
53647 | 1083 |
|
1084 |
\item \emph{Variables in generated properties have different names.} This is |
|
1085 |
rarely an issue, except in proof texts that refer to variable names in the |
|
1086 |
@{text "[where \<dots>]"} attribute. The solution is to use the more robust |
|
1087 |
@{text "[of \<dots>]"} syntax. |
|
1088 |
\end{itemize} |
|
1089 |
||
1090 |
In the other direction, there is currently no way to register old-style |
|
1091 |
datatypes as new-style datatypes. If the goal is to define new-style datatypes |
|
1092 |
with nested recursion through old-style datatypes, the old-style |
|
1093 |
datatypes can be registered as a BNF |
|
55351 | 1094 |
(Section~\ref{sec:introducing-bounded-natural-functors}). If the goal is |
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
1095 |
to derive discriminators and selectors, this can be achieved using |
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
1096 |
@{command free_constructors} |
53647 | 1097 |
(Section~\ref{sec:deriving-destructors-and-theorems-for-free-constructors}). |
52828 | 1098 |
*} |
1099 |
||
52792 | 1100 |
|
52827 | 1101 |
section {* Defining Recursive Functions |
52805 | 1102 |
\label{sec:defining-recursive-functions} *} |
1103 |
||
1104 |
text {* |
|
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1105 |
Recursive functions over datatypes can be specified using the @{command primrec} |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1106 |
command, which supports primitive recursion, or using the more general |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1107 |
\keyw{fun} and \keyw{function} commands. Here, the focus is on |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1108 |
@{command primrec}; the other two commands are described in a separate |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1109 |
tutorial \cite{isabelle-function}. |
52828 | 1110 |
|
53621 | 1111 |
%%% TODO: partial_function |
52805 | 1112 |
*} |
52792 | 1113 |
|
52805 | 1114 |
|
53617 | 1115 |
subsection {* Introductory Examples |
1116 |
\label{ssec:primrec-introductory-examples} *} |
|
52828 | 1117 |
|
53646 | 1118 |
text {* |
1119 |
Primitive recursion is illustrated through concrete examples based on the |
|
1120 |
datatypes defined in Section~\ref{ssec:datatype-introductory-examples}. More |
|
55075 | 1121 |
examples can be found in the directory \verb|~~/src/HOL/BNF_Examples|. |
53646 | 1122 |
*} |
1123 |
||
53621 | 1124 |
|
1125 |
subsubsection {* Nonrecursive Types |
|
1126 |
\label{sssec:primrec-nonrecursive-types} *} |
|
52828 | 1127 |
|
52841 | 1128 |
text {* |
53621 | 1129 |
Primitive recursion removes one layer of constructors on the left-hand side in |
1130 |
each equation. For example: |
|
52841 | 1131 |
*} |
1132 |
||
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1133 |
primrec bool_of_trool :: "trool \<Rightarrow> bool" where |
53621 | 1134 |
"bool_of_trool Faalse \<longleftrightarrow> False" | |
1135 |
"bool_of_trool Truue \<longleftrightarrow> True" |
|
52841 | 1136 |
|
53621 | 1137 |
text {* \blankline *} |
52841 | 1138 |
|
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1139 |
primrec the_list :: "'a option \<Rightarrow> 'a list" where |
53025 | 1140 |
"the_list None = []" | |
1141 |
"the_list (Some a) = [a]" |
|
52841 | 1142 |
|
53621 | 1143 |
text {* \blankline *} |
1144 |
||
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1145 |
primrec the_default :: "'a \<Rightarrow> 'a option \<Rightarrow> 'a" where |
53025 | 1146 |
"the_default d None = d" | |
1147 |
"the_default _ (Some a) = a" |
|
52843 | 1148 |
|
53621 | 1149 |
text {* \blankline *} |
1150 |
||
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1151 |
primrec mirrror :: "('a, 'b, 'c) triple \<Rightarrow> ('c, 'b, 'a) triple" where |
52841 | 1152 |
"mirrror (Triple a b c) = Triple c b a" |
1153 |
||
53621 | 1154 |
text {* |
1155 |
\noindent |
|
1156 |
The equations can be specified in any order, and it is acceptable to leave out |
|
1157 |
some cases, which are then unspecified. Pattern matching on the left-hand side |
|
1158 |
is restricted to a single datatype, which must correspond to the same argument |
|
1159 |
in all equations. |
|
1160 |
*} |
|
52828 | 1161 |
|
53621 | 1162 |
|
1163 |
subsubsection {* Simple Recursion |
|
1164 |
\label{sssec:primrec-simple-recursion} *} |
|
52828 | 1165 |
|
52841 | 1166 |
text {* |
53621 | 1167 |
For simple recursive types, recursive calls on a constructor argument are |
1168 |
allowed on the right-hand side: |
|
52841 | 1169 |
*} |
1170 |
||
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1171 |
primrec replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where |
53330
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1172 |
"replicate Zero _ = []" | |
58245 | 1173 |
"replicate (Succ n) x = x # replicate n x" |
52841 | 1174 |
|
53621 | 1175 |
text {* \blankline *} |
52843 | 1176 |
|
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1177 |
primrec at :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" where |
53644 | 1178 |
"at (x # xs) j = |
52843 | 1179 |
(case j of |
53644 | 1180 |
Zero \<Rightarrow> x |
58245 | 1181 |
| Succ j' \<Rightarrow> at xs j')" |
52843 | 1182 |
|
53621 | 1183 |
text {* \blankline *} |
1184 |
||
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1185 |
primrec (*<*)(in early) (*>*)tfold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) tlist \<Rightarrow> 'b" where |
53644 | 1186 |
"tfold _ (TNil y) = y" | |
1187 |
"tfold f (TCons x xs) = f x (tfold f xs)" |
|
52841 | 1188 |
|
53025 | 1189 |
text {* |
53621 | 1190 |
\noindent |
54402 | 1191 |
Pattern matching is only available for the argument on which the recursion takes |
1192 |
place. Fortunately, it is easy to generate pattern-maching equations using the |
|
57542 | 1193 |
\keyw{simps_of_case} command provided by the theory |
55290 | 1194 |
\verb|~~/src/HOL/|\allowbreak\verb|Library/|\allowbreak\verb|Simps_Case_Conv|. |
54402 | 1195 |
*} |
1196 |
||
1197 |
simps_of_case at_simps: at.simps |
|
1198 |
||
1199 |
text {* |
|
1200 |
This generates the lemma collection @{thm [source] at_simps}: |
|
1201 |
% |
|
1202 |
\[@{thm at_simps(1)[no_vars]} |
|
1203 |
\qquad @{thm at_simps(2)[no_vars]}\] |
|
1204 |
% |
|
54184 | 1205 |
The next example is defined using \keyw{fun} to escape the syntactic |
55254 | 1206 |
restrictions imposed on primitively recursive functions. The |
55531
601ca8efa000
renamed 'datatype_new_compat' to 'datatype_compat'
blanchet
parents:
55530
diff
changeset
|
1207 |
@{command datatype_compat} command is needed to register new-style datatypes |
54184 | 1208 |
for use with \keyw{fun} and \keyw{function} |
56644 | 1209 |
(Section~\ref{sssec:datatype-compat}): |
53025 | 1210 |
*} |
52828 | 1211 |
|
55531
601ca8efa000
renamed 'datatype_new_compat' to 'datatype_compat'
blanchet
parents:
55530
diff
changeset
|
1212 |
datatype_compat nat |
53621 | 1213 |
|
1214 |
text {* \blankline *} |
|
1215 |
||
1216 |
fun at_least_two :: "nat \<Rightarrow> bool" where |
|
58245 | 1217 |
"at_least_two (Succ (Succ _)) \<longleftrightarrow> True" | |
53621 | 1218 |
"at_least_two _ \<longleftrightarrow> False" |
1219 |
||
1220 |
||
1221 |
subsubsection {* Mutual Recursion |
|
1222 |
\label{sssec:primrec-mutual-recursion} *} |
|
52828 | 1223 |
|
52841 | 1224 |
text {* |
53621 | 1225 |
The syntax for mutually recursive functions over mutually recursive datatypes |
1226 |
is straightforward: |
|
52841 | 1227 |
*} |
1228 |
||
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1229 |
primrec |
53623 | 1230 |
nat_of_even_nat :: "even_nat \<Rightarrow> nat" and |
1231 |
nat_of_odd_nat :: "odd_nat \<Rightarrow> nat" |
|
52841 | 1232 |
where |
53623 | 1233 |
"nat_of_even_nat Even_Zero = Zero" | |
58245 | 1234 |
"nat_of_even_nat (Even_Succ n) = Succ (nat_of_odd_nat n)" | |
1235 |
"nat_of_odd_nat (Odd_Succ n) = Succ (nat_of_even_nat n)" |
|
52841 | 1236 |
|
53752 | 1237 |
text {* \blankline *} |
1238 |
||
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1239 |
primrec |
53330
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1240 |
eval\<^sub>e :: "('a \<Rightarrow> int) \<Rightarrow> ('b \<Rightarrow> int) \<Rightarrow> ('a, 'b) exp \<Rightarrow> int" and |
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1241 |
eval\<^sub>t :: "('a \<Rightarrow> int) \<Rightarrow> ('b \<Rightarrow> int) \<Rightarrow> ('a, 'b) trm \<Rightarrow> int" and |
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1242 |
eval\<^sub>f :: "('a \<Rightarrow> int) \<Rightarrow> ('b \<Rightarrow> int) \<Rightarrow> ('a, 'b) fct \<Rightarrow> int" |
52841 | 1243 |
where |
1244 |
"eval\<^sub>e \<gamma> \<xi> (Term t) = eval\<^sub>t \<gamma> \<xi> t" | |
|
1245 |
"eval\<^sub>e \<gamma> \<xi> (Sum t e) = eval\<^sub>t \<gamma> \<xi> t + eval\<^sub>e \<gamma> \<xi> e" | |
|
53330
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1246 |
"eval\<^sub>t \<gamma> \<xi> (Factor f) = eval\<^sub>f \<gamma> \<xi> f" | |
52841 | 1247 |
"eval\<^sub>t \<gamma> \<xi> (Prod f t) = eval\<^sub>f \<gamma> \<xi> f + eval\<^sub>t \<gamma> \<xi> t" | |
1248 |
"eval\<^sub>f \<gamma> _ (Const a) = \<gamma> a" | |
|
1249 |
"eval\<^sub>f _ \<xi> (Var b) = \<xi> b" | |
|
1250 |
"eval\<^sub>f \<gamma> \<xi> (Expr e) = eval\<^sub>e \<gamma> \<xi> e" |
|
1251 |
||
53621 | 1252 |
text {* |
1253 |
\noindent |
|
53647 | 1254 |
Mutual recursion is possible within a single type, using \keyw{fun}: |
53621 | 1255 |
*} |
52828 | 1256 |
|
53621 | 1257 |
fun |
1258 |
even :: "nat \<Rightarrow> bool" and |
|
1259 |
odd :: "nat \<Rightarrow> bool" |
|
1260 |
where |
|
1261 |
"even Zero = True" | |
|
58245 | 1262 |
"even (Succ n) = odd n" | |
53621 | 1263 |
"odd Zero = False" | |
58245 | 1264 |
"odd (Succ n) = even n" |
53621 | 1265 |
|
1266 |
||
1267 |
subsubsection {* Nested Recursion |
|
1268 |
\label{sssec:primrec-nested-recursion} *} |
|
1269 |
||
1270 |
text {* |
|
1271 |
In a departure from the old datatype package, nested recursion is normally |
|
1272 |
handled via the map functions of the nesting type constructors. For example, |
|
1273 |
recursive calls are lifted to lists using @{const map}: |
|
1274 |
*} |
|
52828 | 1275 |
|
52843 | 1276 |
(*<*) |
53644 | 1277 |
datatype_new 'a tree\<^sub>f\<^sub>f = Node\<^sub>f\<^sub>f (lbl\<^sub>f\<^sub>f: 'a) (sub\<^sub>f\<^sub>f: "'a tree\<^sub>f\<^sub>f list") |
52843 | 1278 |
(*>*) |
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changeset
|
1279 |
primrec at\<^sub>f\<^sub>f :: "'a tree\<^sub>f\<^sub>f \<Rightarrow> nat list \<Rightarrow> 'a" where |
53028 | 1280 |
"at\<^sub>f\<^sub>f (Node\<^sub>f\<^sub>f a ts) js = |
52843 | 1281 |
(case js of |
1282 |
[] \<Rightarrow> a |
|
53028 | 1283 |
| j # js' \<Rightarrow> at (map (\<lambda>t. at\<^sub>f\<^sub>f t js') ts) j)" |
52843 | 1284 |
|
53025 | 1285 |
text {* |
53647 | 1286 |
\noindent |
53621 | 1287 |
The next example features recursion through the @{text option} type. Although |
53623 | 1288 |
@{text option} is not a new-style datatype, it is registered as a BNF with the |
54491 | 1289 |
map function @{const map_option}: |
53025 | 1290 |
*} |
52843 | 1291 |
|
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changeset
|
1292 |
primrec (*<*)(in early) (*>*)sum_btree :: "('a\<Colon>{zero,plus}) btree \<Rightarrow> 'a" where |
52843 | 1293 |
"sum_btree (BNode a lt rt) = |
54491 | 1294 |
a + the_default 0 (map_option sum_btree lt) + |
1295 |
the_default 0 (map_option sum_btree rt)" |
|
52843 | 1296 |
|
53136 | 1297 |
text {* |
53621 | 1298 |
\noindent |
1299 |
The same principle applies for arbitrary type constructors through which |
|
1300 |
recursion is possible. Notably, the map function for the function type |
|
1301 |
(@{text \<Rightarrow>}) is simply composition (@{text "op \<circ>"}): |
|
53136 | 1302 |
*} |
52828 | 1303 |
|
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changeset
|
1304 |
primrec (*<*)(in early) (*>*)relabel_ft :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where |
54182 | 1305 |
"relabel_ft f (FTLeaf x) = FTLeaf (f x)" | |
1306 |
"relabel_ft f (FTNode g) = FTNode (relabel_ft f \<circ> g)" |
|
1307 |
||
1308 |
text {* |
|
1309 |
\noindent |
|
1310 |
For convenience, recursion through functions can also be expressed using |
|
1311 |
$\lambda$-abstractions and function application rather than through composition. |
|
1312 |
For example: |
|
1313 |
*} |
|
1314 |
||
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changeset
|
1315 |
primrec relabel_ft :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where |
54182 | 1316 |
"relabel_ft f (FTLeaf x) = FTLeaf (f x)" | |
1317 |
"relabel_ft f (FTNode g) = FTNode (\<lambda>x. relabel_ft f (g x))" |
|
52828 | 1318 |
|
54183 | 1319 |
text {* \blankline *} |
1320 |
||
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changeset
|
1321 |
primrec subtree_ft :: "'a \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where |
54183 | 1322 |
"subtree_ft x (FTNode g) = g x" |
1323 |
||
52843 | 1324 |
text {* |
53621 | 1325 |
\noindent |
54182 | 1326 |
For recursion through curried $n$-ary functions, $n$ applications of |
1327 |
@{term "op \<circ>"} are necessary. The examples below illustrate the case where |
|
1328 |
$n = 2$: |
|
53621 | 1329 |
*} |
1330 |
||
54182 | 1331 |
datatype_new 'a ftree2 = FTLeaf2 'a | FTNode2 "'a \<Rightarrow> 'a \<Rightarrow> 'a ftree2" |
1332 |
||
1333 |
text {* \blankline *} |
|
1334 |
||
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changeset
|
1335 |
primrec (*<*)(in early) (*>*)relabel_ft2 :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree2 \<Rightarrow> 'a ftree2" where |
54182 | 1336 |
"relabel_ft2 f (FTLeaf2 x) = FTLeaf2 (f x)" | |
1337 |
"relabel_ft2 f (FTNode2 g) = FTNode2 (op \<circ> (op \<circ> (relabel_ft2 f)) g)" |
|
1338 |
||
1339 |
text {* \blankline *} |
|
1340 |
||
55530
3dfb724db099
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diff
changeset
|
1341 |
primrec relabel_ft2 :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree2 \<Rightarrow> 'a ftree2" where |
54182 | 1342 |
"relabel_ft2 f (FTLeaf2 x) = FTLeaf2 (f x)" | |
1343 |
"relabel_ft2 f (FTNode2 g) = FTNode2 (\<lambda>x y. relabel_ft2 f (g x y))" |
|
54031 | 1344 |
|
54183 | 1345 |
text {* \blankline *} |
1346 |
||
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diff
changeset
|
1347 |
primrec subtree_ft2 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a ftree2 \<Rightarrow> 'a ftree2" where |
54183 | 1348 |
"subtree_ft2 x y (FTNode2 g) = g x y" |
1349 |
||
53621 | 1350 |
|
1351 |
subsubsection {* Nested-as-Mutual Recursion |
|
53644 | 1352 |
\label{sssec:primrec-nested-as-mutual-recursion} *} |
53621 | 1353 |
|
53749
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changeset
|
1354 |
(*<*) |
b37db925b663
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diff
changeset
|
1355 |
locale n2m begin |
b37db925b663
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changeset
|
1356 |
(*>*) |
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blanchet
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changeset
|
1357 |
|
53621 | 1358 |
text {* |
1359 |
For compatibility with the old package, but also because it is sometimes |
|
1360 |
convenient in its own right, it is possible to treat nested recursive datatypes |
|
1361 |
as mutually recursive ones if the recursion takes place though new-style |
|
1362 |
datatypes. For example: |
|
52843 | 1363 |
*} |
1364 |
||
55530
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changeset
|
1365 |
primrec |
53647 | 1366 |
at\<^sub>f\<^sub>f :: "'a tree\<^sub>f\<^sub>f \<Rightarrow> nat list \<Rightarrow> 'a" and |
1367 |
ats\<^sub>f\<^sub>f :: "'a tree\<^sub>f\<^sub>f list \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> 'a" |
|
52843 | 1368 |
where |
53647 | 1369 |
"at\<^sub>f\<^sub>f (Node\<^sub>f\<^sub>f a ts) js = |
52843 | 1370 |
(case js of |
1371 |
[] \<Rightarrow> a |
|
53647 | 1372 |
| j # js' \<Rightarrow> ats\<^sub>f\<^sub>f ts j js')" | |
1373 |
"ats\<^sub>f\<^sub>f (t # ts) j = |
|
52843 | 1374 |
(case j of |
53647 | 1375 |
Zero \<Rightarrow> at\<^sub>f\<^sub>f t |
58245 | 1376 |
| Succ j' \<Rightarrow> ats\<^sub>f\<^sub>f ts j')" |
52843 | 1377 |
|
53647 | 1378 |
text {* |
1379 |
\noindent |
|
54287 | 1380 |
Appropriate induction rules are generated as |
54031 | 1381 |
@{thm [source] at\<^sub>f\<^sub>f.induct}, |
1382 |
@{thm [source] ats\<^sub>f\<^sub>f.induct}, and |
|
54287 | 1383 |
@{thm [source] at\<^sub>f\<^sub>f_ats\<^sub>f\<^sub>f.induct}. The |
1384 |
induction rules and the underlying recursors are generated on a per-need basis |
|
1385 |
and are kept in a cache to speed up subsequent definitions. |
|
53647 | 1386 |
|
1387 |
Here is a second example: |
|
1388 |
*} |
|
53621 | 1389 |
|
55530
3dfb724db099
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parents:
55474
diff
changeset
|
1390 |
primrec |
53330
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1391 |
sum_btree :: "('a\<Colon>{zero,plus}) btree \<Rightarrow> 'a" and |
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1392 |
sum_btree_option :: "'a btree option \<Rightarrow> 'a" |
52843 | 1393 |
where |
1394 |
"sum_btree (BNode a lt rt) = |
|
53025 | 1395 |
a + sum_btree_option lt + sum_btree_option rt" | |
53330
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1396 |
"sum_btree_option None = 0" | |
53025 | 1397 |
"sum_btree_option (Some t) = sum_btree t" |
52843 | 1398 |
|
1399 |
text {* |
|
53621 | 1400 |
% * can pretend a nested type is mutually recursive (if purely inductive) |
1401 |
% * avoids the higher-order map |
|
1402 |
% * e.g. |
|
1403 |
||
53617 | 1404 |
% * this can always be avoided; |
1405 |
% * e.g. in our previous example, we first mapped the recursive |
|
1406 |
% calls, then we used a generic at function to retrieve the result |
|
1407 |
% |
|
1408 |
% * there's no hard-and-fast rule of when to use one or the other, |
|
1409 |
% just like there's no rule when to use fold and when to use |
|
55530
3dfb724db099
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blanchet
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diff
changeset
|
1410 |
% primrec |
53617 | 1411 |
% |
1412 |
% * higher-order approach, considering nesting as nesting, is more |
|
1413 |
% compositional -- e.g. we saw how we could reuse an existing polymorphic |
|
53647 | 1414 |
% at or the_default, whereas @{const ats\<^sub>f\<^sub>f} is much more specific |
53617 | 1415 |
% |
1416 |
% * but: |
|
1417 |
% * is perhaps less intuitive, because it requires higher-order thinking |
|
1418 |
% * may seem inefficient, and indeed with the code generator the |
|
1419 |
% mutually recursive version might be nicer |
|
1420 |
% * is somewhat indirect -- must apply a map first, then compute a result |
|
1421 |
% (cannot mix) |
|
53647 | 1422 |
% * the auxiliary functions like @{const ats\<^sub>f\<^sub>f} are sometimes useful in own right |
53617 | 1423 |
% |
1424 |
% * impact on automation unclear |
|
1425 |
% |
|
52843 | 1426 |
*} |
53749
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diff
changeset
|
1427 |
(*<*) |
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diff
changeset
|
1428 |
end |
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blanchet
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diff
changeset
|
1429 |
(*>*) |
52843 | 1430 |
|
52824 | 1431 |
|
53617 | 1432 |
subsection {* Command Syntax |
1433 |
\label{ssec:primrec-command-syntax} *} |
|
1434 |
||
56123
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changeset
|
1435 |
subsubsection {* \keyw{primrec} |
53621 | 1436 |
\label{sssec:primrec-new} *} |
52828 | 1437 |
|
1438 |
text {* |
|
53829 | 1439 |
\begin{matharray}{rcl} |
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
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diff
changeset
|
1440 |
@{command_def "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} |
53829 | 1441 |
\end{matharray} |
52794 | 1442 |
|
55112
b1a5d603fd12
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wenzelm
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55029
diff
changeset
|
1443 |
@{rail \<open> |
56123
a27859b0ef7d
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55945
diff
changeset
|
1444 |
@@{command primrec} target? @{syntax pr_option}? fixes \<newline> |
a27859b0ef7d
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55945
diff
changeset
|
1445 |
@'where' (@{syntax pr_equation} + '|') |
a27859b0ef7d
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blanchet
parents:
55945
diff
changeset
|
1446 |
; |
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
blanchet
parents:
55945
diff
changeset
|
1447 |
@{syntax_def pr_option}: '(' 'nonexhaustive' ')' |
52840 | 1448 |
; |
53829 | 1449 |
@{syntax_def pr_equation}: thmdecl? prop |
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
1450 |
\<close>} |
55460
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blanchet
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diff
changeset
|
1451 |
|
3f4efd7d950d
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blanchet
parents:
55459
diff
changeset
|
1452 |
\medskip |
3f4efd7d950d
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blanchet
parents:
55459
diff
changeset
|
1453 |
|
3f4efd7d950d
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blanchet
parents:
55459
diff
changeset
|
1454 |
\noindent |
55530
3dfb724db099
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55474
diff
changeset
|
1455 |
The @{command primrec} command introduces a set of mutually recursive functions |
3dfb724db099
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55474
diff
changeset
|
1456 |
over datatypes. |
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
1457 |
|
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
1458 |
The syntactic entity \synt{target} can be used to specify a local context, |
55474 | 1459 |
\synt{fixes} denotes a list of names with optional type signatures, |
1460 |
\synt{thmdecl} denotes an optional name for the formula that follows, and |
|
1461 |
\synt{prop} denotes a HOL proposition \cite{isabelle-isar-ref}. |
|
55460
3f4efd7d950d
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blanchet
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diff
changeset
|
1462 |
|
56124 | 1463 |
The optional target is optionally followed by the following option: |
56123
a27859b0ef7d
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55945
diff
changeset
|
1464 |
|
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
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parents:
55945
diff
changeset
|
1465 |
\begin{itemize} |
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
blanchet
parents:
55945
diff
changeset
|
1466 |
\setlength{\itemsep}{0pt} |
a27859b0ef7d
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55945
diff
changeset
|
1467 |
|
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
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55945
diff
changeset
|
1468 |
\item |
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
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55945
diff
changeset
|
1469 |
The @{text "nonexhaustive"} option indicates that the functions are not |
a27859b0ef7d
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blanchet
parents:
55945
diff
changeset
|
1470 |
necessarily specified for all constructors. It can be used to suppress the |
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
blanchet
parents:
55945
diff
changeset
|
1471 |
warning that is normally emitted when some constructors are missing. |
a27859b0ef7d
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55945
diff
changeset
|
1472 |
\end{itemize} |
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55945
diff
changeset
|
1473 |
|
55460
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55459
diff
changeset
|
1474 |
%%% TODO: elaborate on format of the equations |
3f4efd7d950d
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parents:
55459
diff
changeset
|
1475 |
%%% TODO: elaborate on mutual and nested-to-mutual |
52828 | 1476 |
*} |
1477 |
||
52840 | 1478 |
|
53619 | 1479 |
(* |
52840 | 1480 |
subsection {* Generated Theorems |
1481 |
\label{ssec:primrec-generated-theorems} *} |
|
52824 | 1482 |
|
52828 | 1483 |
text {* |
53617 | 1484 |
% * synthesized nonrecursive definition |
1485 |
% * user specification is rederived from it, exactly as entered |
|
1486 |
% |
|
1487 |
% * induct |
|
1488 |
% * mutualized |
|
1489 |
% * without some needless induction hypotheses if not used |
|
55871 | 1490 |
% * rec |
53617 | 1491 |
% * mutualized |
52828 | 1492 |
*} |
53619 | 1493 |
*) |
1494 |
||
52824 | 1495 |
|
52840 | 1496 |
subsection {* Recursive Default Values for Selectors |
53623 | 1497 |
\label{ssec:primrec-recursive-default-values-for-selectors} *} |
52827 | 1498 |
|
1499 |
text {* |
|
1500 |
A datatype selector @{text un_D} can have a default value for each constructor |
|
1501 |
on which it is not otherwise specified. Occasionally, it is useful to have the |
|
55354 | 1502 |
default value be defined recursively. This leads to a chicken-and-egg |
1503 |
situation, because the datatype is not introduced yet at the moment when the |
|
1504 |
selectors are introduced. Of course, we can always define the selectors |
|
1505 |
manually afterward, but we then have to state and prove all the characteristic |
|
1506 |
theorems ourselves instead of letting the package do it. |
|
1507 |
||
1508 |
Fortunately, there is a workaround that relies on overloading to relieve us |
|
1509 |
from the tedium of manual derivations: |
|
52827 | 1510 |
|
1511 |
\begin{enumerate} |
|
1512 |
\setlength{\itemsep}{0pt} |
|
1513 |
||
1514 |
\item |
|
1515 |
Introduce a fully unspecified constant @{text "un_D\<^sub>0 \<Colon> 'a"} using |
|
1516 |
@{keyword consts}. |
|
1517 |
||
1518 |
\item |
|
53535 | 1519 |
Define the datatype, specifying @{text "un_D\<^sub>0"} as the selector's default |
1520 |
value. |
|
52827 | 1521 |
|
1522 |
\item |
|
53535 | 1523 |
Define the behavior of @{text "un_D\<^sub>0"} on values of the newly introduced |
1524 |
datatype using the \keyw{overloading} command. |
|
52827 | 1525 |
|
1526 |
\item |
|
1527 |
Derive the desired equation on @{text un_D} from the characteristic equations |
|
1528 |
for @{text "un_D\<^sub>0"}. |
|
1529 |
\end{enumerate} |
|
1530 |
||
53619 | 1531 |
\noindent |
52827 | 1532 |
The following example illustrates this procedure: |
1533 |
*} |
|
1534 |
||
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1535 |
(*<*) |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1536 |
hide_const termi |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1537 |
(*>*) |
52827 | 1538 |
consts termi\<^sub>0 :: 'a |
1539 |
||
53619 | 1540 |
text {* \blankline *} |
1541 |
||
53491 | 1542 |
datatype_new ('a, 'b) tlist = |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1543 |
TNil (termi: 'b) |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1544 |
| TCons (thd: 'a) (ttl: "('a, 'b) tlist") |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1545 |
where |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1546 |
"ttl (TNil y) = TNil y" |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1547 |
| "termi (TCons _ xs) = termi\<^sub>0 xs" |
52827 | 1548 |
|
53619 | 1549 |
text {* \blankline *} |
1550 |
||
52827 | 1551 |
overloading |
53491 | 1552 |
termi\<^sub>0 \<equiv> "termi\<^sub>0 \<Colon> ('a, 'b) tlist \<Rightarrow> 'b" |
52827 | 1553 |
begin |
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1554 |
primrec termi\<^sub>0 :: "('a, 'b) tlist \<Rightarrow> 'b" where |
53621 | 1555 |
"termi\<^sub>0 (TNil y) = y" | |
1556 |
"termi\<^sub>0 (TCons x xs) = termi\<^sub>0 xs" |
|
52827 | 1557 |
end |
1558 |
||
53619 | 1559 |
text {* \blankline *} |
1560 |
||
55354 | 1561 |
lemma termi_TCons[simp]: "termi (TCons x xs) = termi xs" |
52827 | 1562 |
by (cases xs) auto |
1563 |
||
1564 |
||
52828 | 1565 |
subsection {* Compatibility Issues |
53617 | 1566 |
\label{ssec:primrec-compatibility-issues} *} |
52828 | 1567 |
|
1568 |
text {* |
|
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1569 |
The command @{command primrec}'s behavior on new-style datatypes has been |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1570 |
designed to be highly compatible with that for old-style datatypes, to ease |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1571 |
migration. There is nonetheless at least one incompatibility that may arise when |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1572 |
porting to the new package: |
53997 | 1573 |
|
1574 |
\begin{itemize} |
|
1575 |
\setlength{\itemsep}{0pt} |
|
1576 |
||
54185 | 1577 |
\item \emph{Some theorems have different names.} |
53997 | 1578 |
For $m > 1$ mutually recursive functions, |
54023
cede3c1d2417
minor doc fix (there is no guarantee that the equations for a given f_i are contiguous in the collection)
blanchet
parents:
54014
diff
changeset
|
1579 |
@{text "f\<^sub>1_\<dots>_f\<^sub>m.simps"} has been broken down into separate |
cede3c1d2417
minor doc fix (there is no guarantee that the equations for a given f_i are contiguous in the collection)
blanchet
parents:
54014
diff
changeset
|
1580 |
subcollections @{text "f\<^sub>i.simps"}. |
53997 | 1581 |
\end{itemize} |
52828 | 1582 |
*} |
52794 | 1583 |
|
1584 |
||
52827 | 1585 |
section {* Defining Codatatypes |
52805 | 1586 |
\label{sec:defining-codatatypes} *} |
1587 |
||
1588 |
text {* |
|
53829 | 1589 |
Codatatypes can be specified using the @{command codatatype} command. The |
53623 | 1590 |
command is first illustrated through concrete examples featuring different |
1591 |
flavors of corecursion. More examples can be found in the directory |
|
53997 | 1592 |
\verb|~~/src/HOL/|\allowbreak\verb|BNF/Examples|. The |
1593 |
\emph{Archive of Formal Proofs} also includes some useful codatatypes, notably |
|
1594 |
for lazy lists \cite{lochbihler-2010}. |
|
52805 | 1595 |
*} |
52792 | 1596 |
|
52824 | 1597 |
|
53617 | 1598 |
subsection {* Introductory Examples |
1599 |
\label{ssec:codatatype-introductory-examples} *} |
|
52794 | 1600 |
|
53623 | 1601 |
subsubsection {* Simple Corecursion |
1602 |
\label{sssec:codatatype-simple-corecursion} *} |
|
1603 |
||
52805 | 1604 |
text {* |
57542 | 1605 |
Non-corecursive codatatypes coincide with the corresponding datatypes, so they |
56947 | 1606 |
are rarely used in practice. \emph{Corecursive codatatypes} have the same syntax |
53623 | 1607 |
as recursive datatypes, except for the command name. For example, here is the |
1608 |
definition of lazy lists: |
|
1609 |
*} |
|
1610 |
||
57206
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1611 |
codatatype (lset: 'a) llist = |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1612 |
lnull: LNil |
53623 | 1613 |
| LCons (lhd: 'a) (ltl: "'a llist") |
57206
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1614 |
for |
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1615 |
map: lmap |
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1616 |
rel: llist_all2 |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1617 |
where |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1618 |
"ltl LNil = LNil" |
53623 | 1619 |
|
1620 |
text {* |
|
1621 |
\noindent |
|
1622 |
Lazy lists can be infinite, such as @{text "LCons 0 (LCons 0 (\<dots>))"} and |
|
53647 | 1623 |
@{text "LCons 0 (LCons 1 (LCons 2 (\<dots>)))"}. Here is a related type, that of |
1624 |
infinite streams: |
|
1625 |
*} |
|
1626 |
||
57206
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1627 |
codatatype (sset: 'a) stream = |
53647 | 1628 |
SCons (shd: 'a) (stl: "'a stream") |
57206
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1629 |
for |
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1630 |
map: smap |
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1631 |
rel: stream_all2 |
53647 | 1632 |
|
1633 |
text {* |
|
1634 |
\noindent |
|
1635 |
Another interesting type that can |
|
53623 | 1636 |
be defined as a codatatype is that of the extended natural numbers: |
1637 |
*} |
|
1638 |
||
58245 | 1639 |
codatatype enat = EZero | ESucc enat |
53623 | 1640 |
|
1641 |
text {* |
|
1642 |
\noindent |
|
58245 | 1643 |
This type has exactly one infinite element, @{text "ESucc (ESucc (ESucc (\<dots>)))"}, |
53623 | 1644 |
that represents $\infty$. In addition, it has finite values of the form |
58245 | 1645 |
@{text "ESucc (\<dots> (ESucc EZero)\<dots>)"}. |
53675 | 1646 |
|
1647 |
Here is an example with many constructors: |
|
52805 | 1648 |
*} |
53623 | 1649 |
|
53675 | 1650 |
codatatype 'a process = |
1651 |
Fail |
|
1652 |
| Skip (cont: "'a process") |
|
1653 |
| Action (prefix: 'a) (cont: "'a process") |
|
1654 |
| Choice (left: "'a process") (right: "'a process") |
|
1655 |
||
53750 | 1656 |
text {* |
53829 | 1657 |
\noindent |
53750 | 1658 |
Notice that the @{const cont} selector is associated with both @{const Skip} |
54146 | 1659 |
and @{const Action}. |
53750 | 1660 |
*} |
1661 |
||
53623 | 1662 |
|
1663 |
subsubsection {* Mutual Corecursion |
|
1664 |
\label{sssec:codatatype-mutual-corecursion} *} |
|
1665 |
||
1666 |
text {* |
|
1667 |
\noindent |
|
1668 |
The example below introduces a pair of \emph{mutually corecursive} types: |
|
1669 |
*} |
|
1670 |
||
58245 | 1671 |
codatatype even_enat = Even_EZero | Even_ESucc odd_enat |
1672 |
and odd_enat = Odd_ESucc even_enat |
|
53623 | 1673 |
|
1674 |
||
1675 |
subsubsection {* Nested Corecursion |
|
1676 |
\label{sssec:codatatype-nested-corecursion} *} |
|
1677 |
||
1678 |
text {* |
|
1679 |
\noindent |
|
53675 | 1680 |
The next examples feature \emph{nested corecursion}: |
53623 | 1681 |
*} |
1682 |
||
53644 | 1683 |
codatatype 'a tree\<^sub>i\<^sub>i = Node\<^sub>i\<^sub>i (lbl\<^sub>i\<^sub>i: 'a) (sub\<^sub>i\<^sub>i: "'a tree\<^sub>i\<^sub>i llist") |
53675 | 1684 |
|
53752 | 1685 |
text {* \blankline *} |
1686 |
||
53644 | 1687 |
codatatype 'a tree\<^sub>i\<^sub>s = Node\<^sub>i\<^sub>s (lbl\<^sub>i\<^sub>s: 'a) (sub\<^sub>i\<^sub>s: "'a tree\<^sub>i\<^sub>s fset") |
52805 | 1688 |
|
53752 | 1689 |
text {* \blankline *} |
1690 |
||
55350 | 1691 |
codatatype 'a sm = SM (accept: bool) (trans: "'a \<Rightarrow> 'a sm") |
53675 | 1692 |
|
52824 | 1693 |
|
53617 | 1694 |
subsection {* Command Syntax |
1695 |
\label{ssec:codatatype-command-syntax} *} |
|
52805 | 1696 |
|
53621 | 1697 |
subsubsection {* \keyw{codatatype} |
1698 |
\label{sssec:codatatype} *} |
|
53619 | 1699 |
|
52824 | 1700 |
text {* |
53829 | 1701 |
\begin{matharray}{rcl} |
1702 |
@{command_def "codatatype"} & : & @{text "local_theory \<rightarrow> local_theory"} |
|
1703 |
\end{matharray} |
|
1704 |
||
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
1705 |
@{rail \<open> |
55029
61a6bf7d4b02
clarified @{rail} syntax: prefer explicit \<newline> symbol;
wenzelm
parents:
54958
diff
changeset
|
1706 |
@@{command codatatype} target? \<newline> |
55472
990651bfc65b
updated docs to reflect the new 'free_constructors' syntax
blanchet
parents:
55468
diff
changeset
|
1707 |
(@{syntax dt_name} '=' (@{syntax dt_ctor} + '|') + @'and') |
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
1708 |
\<close>} |
53829 | 1709 |
|
55351 | 1710 |
\medskip |
1711 |
||
53829 | 1712 |
\noindent |
52827 | 1713 |
Definitions of codatatypes have almost exactly the same syntax as for datatypes |
57094
589ec121ce1a
don't generate discriminators and selectors for 'datatype_new' unless the user asked for it
blanchet
parents:
57092
diff
changeset
|
1714 |
(Section~\ref{ssec:datatype-command-syntax}). The @{text "discs_sels"} option |
589ec121ce1a
don't generate discriminators and selectors for 'datatype_new' unless the user asked for it
blanchet
parents:
57092
diff
changeset
|
1715 |
is superfluous because discriminators and selectors are always generated for |
589ec121ce1a
don't generate discriminators and selectors for 'datatype_new' unless the user asked for it
blanchet
parents:
57092
diff
changeset
|
1716 |
codatatypes. |
53623 | 1717 |
*} |
1718 |
||
1719 |
||
1720 |
subsection {* Generated Constants |
|
1721 |
\label{ssec:codatatype-generated-constants} *} |
|
1722 |
||
1723 |
text {* |
|
1724 |
Given a codatatype @{text "('a\<^sub>1, \<dots>, 'a\<^sub>m) t"} |
|
1725 |
with $m > 0$ live type variables and $n$ constructors @{text "t.C\<^sub>1"}, |
|
1726 |
\ldots, @{text "t.C\<^sub>n"}, the same auxiliary constants are generated as for |
|
1727 |
datatypes (Section~\ref{ssec:datatype-generated-constants}), except that the |
|
58190 | 1728 |
recursor is replaced by a dual concept: |
53623 | 1729 |
|
55354 | 1730 |
\medskip |
1731 |
||
1732 |
\begin{tabular}{@ {}ll@ {}} |
|
1733 |
Corecursor: & |
|
56655 | 1734 |
@{text t.corec_t} |
55354 | 1735 |
\end{tabular} |
53623 | 1736 |
*} |
1737 |
||
1738 |
||
1739 |
subsection {* Generated Theorems |
|
1740 |
\label{ssec:codatatype-generated-theorems} *} |
|
1741 |
||
1742 |
text {* |
|
53829 | 1743 |
The characteristic theorems generated by @{command codatatype} are grouped in |
53623 | 1744 |
three broad categories: |
1745 |
||
1746 |
\begin{itemize} |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1747 |
\setlength{\itemsep}{0pt} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1748 |
|
56655 | 1749 |
\item The \emph{free constructor theorems} |
1750 |
(Section~\ref{sssec:free-constructor-theorems}) are properties of the |
|
1751 |
constructors and destructors that can be derived for any freely generated type. |
|
1752 |
||
1753 |
\item The \emph{functorial theorems} |
|
1754 |
(Section~\ref{sssec:functorial-theorems}) are properties of datatypes related to |
|
53623 | 1755 |
their BNF nature. |
1756 |
||
56655 | 1757 |
\item The \emph{coinductive theorems} (Section~\ref{sssec:coinductive-theorems}) |
1758 |
are properties of datatypes related to their coinductive nature. |
|
53623 | 1759 |
\end{itemize} |
1760 |
||
1761 |
\noindent |
|
56655 | 1762 |
The first two categories are exactly as for datatypes. |
52824 | 1763 |
*} |
1764 |
||
53617 | 1765 |
|
53623 | 1766 |
subsubsection {* Coinductive Theorems |
1767 |
\label{sssec:coinductive-theorems} *} |
|
1768 |
||
1769 |
text {* |
|
54031 | 1770 |
The coinductive theorems are listed below for @{typ "'a llist"}: |
53623 | 1771 |
|
1772 |
\begin{indentblock} |
|
1773 |
\begin{description} |
|
1774 |
||
53643 | 1775 |
\item[\begin{tabular}{@ {}l@ {}} |
57304 | 1776 |
@{text "t."}\hthm{coinduct} @{text "[consumes m, case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
1777 |
\phantom{@{text "t."}\hthm{coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> |
|
1778 |
D\<^sub>n, coinduct t]"}\rm: |
|
53643 | 1779 |
\end{tabular}] ~ \\ |
53623 | 1780 |
@{thm llist.coinduct[no_vars]} |
53617 | 1781 |
|
53643 | 1782 |
\item[\begin{tabular}{@ {}l@ {}} |
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
1783 |
@{text "t."}\hthm{coinduct_strong} @{text "[consumes m, case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
1784 |
\phantom{@{text "t."}\hthm{coinduct_strong} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> D\<^sub>n]"}\rm: |
53643 | 1785 |
\end{tabular}] ~ \\ |
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
1786 |
@{thm llist.coinduct_strong[no_vars]} |
53617 | 1787 |
|
53643 | 1788 |
\item[\begin{tabular}{@ {}l@ {}} |
57304 | 1789 |
@{text "t."}\hthm{rel_coinduct} @{text "[consumes m, case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
1790 |
\phantom{@{text "t."}\hthm{rel_coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> |
|
1791 |
D\<^sub>n, coinduct pred]"}\rm: |
|
1792 |
\end{tabular}] ~ \\ |
|
1793 |
@{thm llist.rel_coinduct[no_vars]} |
|
1794 |
||
1795 |
\item[\begin{tabular}{@ {}l@ {}} |
|
53643 | 1796 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{coinduct} @{text "[case_names t\<^sub>1 \<dots> t\<^sub>m, case_conclusion D\<^sub>1 \<dots> D\<^sub>n]"} \\ |
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
1797 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{coinduct_strong} @{text "[case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
1798 |
\phantom{@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{coinduct_strong} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> D\<^sub>n]"}\rm: \\ |
57542 | 1799 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{rel_coinduct} @{text "[case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
1800 |
\phantom{@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{rel_coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> D\<^sub>n]"}\rm: \\ |
|
53643 | 1801 |
\end{tabular}] ~ \\ |
1802 |
Given $m > 1$ mutually corecursive codatatypes, these coinduction rules can be |
|
1803 |
used to prove $m$ properties simultaneously. |
|
1804 |
||
57701 | 1805 |
\item[\begin{tabular}{@ {}l@ {}} |
1806 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{set_induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n,"} \\ |
|
1807 |
\phantom{@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{set_induct} @{text "["}}@{text "induct set: set\<^sub>j_t\<^sub>1, \<dots>, induct set: set\<^sub>j_t\<^sub>m]"}\rm: \\ |
|
1808 |
\end{tabular}] ~ \\ |
|
1809 |
@{thm llist.set_induct[no_vars]} \\ |
|
1810 |
If $m = 1$, the attribute @{text "[consumes 1]"} is generated as well. |
|
1811 |
||
54031 | 1812 |
\item[@{text "t."}\hthm{corec}\rm:] ~ \\ |
53623 | 1813 |
@{thm llist.corec(1)[no_vars]} \\ |
1814 |
@{thm llist.corec(2)[no_vars]} |
|
1815 |
||
57542 | 1816 |
\item[@{text "t."}\hthm{corec_code} @{text "[code]"}\rm:] ~ \\ |
57490 | 1817 |
@{thm llist.corec_code[no_vars]} |
1818 |
||
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1819 |
\item[@{text "t."}\hthm{corec_disc}\rm:] ~ \\ |
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1820 |
@{thm llist.corec_disc(1)[no_vars]} \\ |
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|
1821 |
@{thm llist.corec_disc(2)[no_vars]} |
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|
1822 |
|
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|
1823 |
\item[@{text "t."}\hthm{corec_disc_iff} @{text "[simp]"}\rm:] ~ \\ |
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|
1824 |
@{thm llist.corec_disc_iff(1)[no_vars]} \\ |
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|
1825 |
@{thm llist.corec_disc_iff(2)[no_vars]} |
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|
1826 |
|
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1827 |
\item[@{text "t."}\hthm{corec_sel} @{text "[simp]"}\rm:] ~ \\ |
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|
1828 |
@{thm llist.corec_sel(1)[no_vars]} \\ |
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|
1829 |
@{thm llist.corec_sel(2)[no_vars]} |
53643 | 1830 |
|
53623 | 1831 |
\end{description} |
1832 |
\end{indentblock} |
|
1833 |
||
1834 |
\noindent |
|
53829 | 1835 |
For convenience, @{command codatatype} also provides the following collection: |
53623 | 1836 |
|
1837 |
\begin{indentblock} |
|
1838 |
\begin{description} |
|
1839 |
||
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1840 |
\item[@{text "t."}\hthm{simps} = @{text t.inject} @{text t.distinct} @{text t.case} @{text t.corec_disc_iff}] @{text t.corec_sel} ~ \\ |
55896 | 1841 |
@{text t.map} @{text t.rel_inject} @{text t.rel_distinct} @{text t.set} |
53623 | 1842 |
|
1843 |
\end{description} |
|
1844 |
\end{indentblock} |
|
1845 |
*} |
|
52805 | 1846 |
|
1847 |
||
52827 | 1848 |
section {* Defining Corecursive Functions |
52805 | 1849 |
\label{sec:defining-corecursive-functions} *} |
1850 |
||
1851 |
text {* |
|
54183 | 1852 |
Corecursive functions can be specified using the @{command primcorec} and |
1853 |
\keyw{prim\-corec\-ursive} commands, which support primitive corecursion, or |
|
57542 | 1854 |
using the more general \keyw{partial_function} command. Here, the focus is on |
54183 | 1855 |
the first two. More examples can be found in the directory |
55075 | 1856 |
\verb|~~/src/HOL/BNF_Examples|. |
53644 | 1857 |
|
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|
1858 |
Whereas recursive functions consume datatypes one constructor at a time, |
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|
1859 |
corecursive functions construct codatatypes one constructor at a time. |
53752 | 1860 |
Partly reflecting a lack of agreement among proponents of coalgebraic methods, |
1861 |
Isabelle supports three competing syntaxes for specifying a function $f$: |
|
53749
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|
1862 |
|
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|
1863 |
\begin{itemize} |
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|
1864 |
\setlength{\itemsep}{0pt} |
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|
1865 |
|
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|
1866 |
\abovedisplayskip=.5\abovedisplayskip |
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|
1867 |
\belowdisplayskip=.5\belowdisplayskip |
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|
1868 |
|
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|
1869 |
\item The \emph{destructor view} specifies $f$ by implications of the form |
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|
1870 |
\[@{text "\<dots> \<Longrightarrow> is_C\<^sub>j (f x\<^sub>1 \<dots> x\<^sub>n)"}\] and |
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|
1871 |
equations of the form |
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|
1872 |
\[@{text "un_C\<^sub>ji (f x\<^sub>1 \<dots> x\<^sub>n) = \<dots>"}\] |
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|
1873 |
This style is popular in the coalgebraic literature. |
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|
1874 |
|
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|
1875 |
\item The \emph{constructor view} specifies $f$ by equations of the form |
54183 | 1876 |
\[@{text "\<dots> \<Longrightarrow> f x\<^sub>1 \<dots> x\<^sub>n = C\<^sub>j \<dots>"}\] |
53752 | 1877 |
This style is often more concise than the previous one. |
53749
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|
1878 |
|
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|
1879 |
\item The \emph{code view} specifies $f$ by a single equation of the form |
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|
1880 |
\[@{text "f x\<^sub>1 \<dots> x\<^sub>n = \<dots>"}\] |
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|
1881 |
with restrictions on the format of the right-hand side. Lazy functional |
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|
1882 |
programming languages such as Haskell support a generalized version of this |
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|
1883 |
style. |
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|
1884 |
\end{itemize} |
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|
1885 |
|
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|
1886 |
All three styles are available as input syntax. Whichever syntax is chosen, |
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|
1887 |
characteristic theorems for all three styles are generated. |
53749
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|
1888 |
|
52828 | 1889 |
%%% TODO: partial_function? E.g. for defining tail recursive function on lazy |
1890 |
%%% lists (cf. terminal0 in TLList.thy) |
|
52805 | 1891 |
*} |
1892 |
||
52824 | 1893 |
|
53617 | 1894 |
subsection {* Introductory Examples |
1895 |
\label{ssec:primcorec-introductory-examples} *} |
|
52805 | 1896 |
|
53646 | 1897 |
text {* |
1898 |
Primitive corecursion is illustrated through concrete examples based on the |
|
1899 |
codatatypes defined in Section~\ref{ssec:codatatype-introductory-examples}. More |
|
55075 | 1900 |
examples can be found in the directory \verb|~~/src/HOL/BNF_Examples|. The code |
53749
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|
1901 |
view is favored in the examples below. Sections |
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|
1902 |
\ref{ssec:primrec-constructor-view} and \ref{ssec:primrec-destructor-view} |
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|
1903 |
present the same examples expressed using the constructor and destructor views. |
53646 | 1904 |
*} |
1905 |
||
58245 | 1906 |
|
53644 | 1907 |
subsubsection {* Simple Corecursion |
1908 |
\label{sssec:primcorec-simple-corecursion} *} |
|
1909 |
||
53646 | 1910 |
text {* |
53752 | 1911 |
Following the code view, corecursive calls are allowed on the right-hand side as |
1912 |
long as they occur under a constructor, which itself appears either directly to |
|
1913 |
the right of the equal sign or in a conditional expression: |
|
53646 | 1914 |
*} |
1915 |
||
53826 | 1916 |
primcorec literate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist" where |
54072 | 1917 |
"literate g x = LCons x (literate g (g x))" |
53647 | 1918 |
|
53677 | 1919 |
text {* \blankline *} |
1920 |
||
53826 | 1921 |
primcorec siterate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a stream" where |
54072 | 1922 |
"siterate g x = SCons x (siterate g (g x))" |
53644 | 1923 |
|
53646 | 1924 |
text {* |
1925 |
\noindent |
|
1926 |
The constructor ensures that progress is made---i.e., the function is |
|
53749
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|
1927 |
\emph{productive}. The above functions compute the infinite lazy list or stream |
54072 | 1928 |
@{text "[x, g x, g (g x), \<dots>]"}. Productivity guarantees that prefixes |
1929 |
@{text "[x, g x, g (g x), \<dots>, (g ^^ k) x]"} of arbitrary finite length |
|
53749
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|
1930 |
@{text k} can be computed by unfolding the code equation a finite number of |
53863
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|
1931 |
times. |
53646 | 1932 |
|
53752 | 1933 |
Corecursive functions construct codatatype values, but nothing prevents them |
53863
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|
1934 |
from also consuming such values. The following function drops every second |
53675 | 1935 |
element in a stream: |
1936 |
*} |
|
1937 |
||
53826 | 1938 |
primcorec every_snd :: "'a stream \<Rightarrow> 'a stream" where |
53675 | 1939 |
"every_snd s = SCons (shd s) (stl (stl s))" |
1940 |
||
1941 |
text {* |
|
53752 | 1942 |
\noindent |
56124 | 1943 |
Constructs such as @{text "let"}--@{text "in"}, @{text |
1944 |
"if"}--@{text "then"}--@{text "else"}, and @{text "case"}--@{text "of"} may |
|
53646 | 1945 |
appear around constructors that guard corecursive calls: |
1946 |
*} |
|
1947 |
||
54072 | 1948 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53644 | 1949 |
"lappend xs ys = |
1950 |
(case xs of |
|
1951 |
LNil \<Rightarrow> ys |
|
1952 |
| LCons x xs' \<Rightarrow> LCons x (lappend xs' ys))" |
|
1953 |
||
53646 | 1954 |
text {* |
53752 | 1955 |
\noindent |
54402 | 1956 |
Pattern matching is not supported by @{command primcorec}. Fortunately, it is |
57542 | 1957 |
easy to generate pattern-maching equations using the \keyw{simps_of_case} |
54402 | 1958 |
command provided by the theory \verb|~~/src/HOL/Library/Simps_Case_Conv|. |
1959 |
*} |
|
1960 |
||
1961 |
simps_of_case lappend_simps: lappend.code |
|
1962 |
||
1963 |
text {* |
|
1964 |
This generates the lemma collection @{thm [source] lappend_simps}: |
|
1965 |
% |
|
55355 | 1966 |
\begin{gather*% |
1967 |
} |
|
1968 |
@{thm lappend_simps(1)[no_vars]} \\ |
|
1969 |
@{thm lappend_simps(2)[no_vars]} |
|
1970 |
\end{gather*% |
|
1971 |
} |
|
54402 | 1972 |
% |
53646 | 1973 |
Corecursion is useful to specify not only functions but also infinite objects: |
1974 |
*} |
|
1975 |
||
53826 | 1976 |
primcorec infty :: enat where |
58245 | 1977 |
"infty = ESucc infty" |
53644 | 1978 |
|
53646 | 1979 |
text {* |
53752 | 1980 |
\noindent |
1981 |
The example below constructs a pseudorandom process value. It takes a stream of |
|
53675 | 1982 |
actions (@{text s}), a pseudorandom function generator (@{text f}), and a |
1983 |
pseudorandom seed (@{text n}): |
|
1984 |
*} |
|
1985 |
||
54072 | 1986 |
primcorec |
53752 | 1987 |
random_process :: "'a stream \<Rightarrow> (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> 'a process" |
1988 |
where |
|
53675 | 1989 |
"random_process s f n = |
1990 |
(if n mod 4 = 0 then |
|
1991 |
Fail |
|
1992 |
else if n mod 4 = 1 then |
|
1993 |
Skip (random_process s f (f n)) |
|
1994 |
else if n mod 4 = 2 then |
|
1995 |
Action (shd s) (random_process (stl s) f (f n)) |
|
1996 |
else |
|
1997 |
Choice (random_process (every_snd s) (f \<circ> f) (f n)) |
|
1998 |
(random_process (every_snd (stl s)) (f \<circ> f) (f (f n))))" |
|
1999 |
||
2000 |
text {* |
|
2001 |
\noindent |
|
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|
2002 |
The main disadvantage of the code view is that the conditions are tested |
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|
2003 |
sequentially. This is visible in the generated theorems. The constructor and |
53752 | 2004 |
destructor views offer nonsequential alternatives. |
53675 | 2005 |
*} |
2006 |
||
53644 | 2007 |
|
2008 |
subsubsection {* Mutual Corecursion |
|
2009 |
\label{sssec:primcorec-mutual-corecursion} *} |
|
2010 |
||
53647 | 2011 |
text {* |
2012 |
The syntax for mutually corecursive functions over mutually corecursive |
|
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|
2013 |
datatypes is unsurprising: |
53647 | 2014 |
*} |
2015 |
||
53826 | 2016 |
primcorec |
53644 | 2017 |
even_infty :: even_enat and |
2018 |
odd_infty :: odd_enat |
|
2019 |
where |
|
58245 | 2020 |
"even_infty = Even_ESucc odd_infty" | |
2021 |
"odd_infty = Odd_ESucc even_infty" |
|
53644 | 2022 |
|
2023 |
||
2024 |
subsubsection {* Nested Corecursion |
|
2025 |
\label{sssec:primcorec-nested-corecursion} *} |
|
2026 |
||
53647 | 2027 |
text {* |
2028 |
The next pair of examples generalize the @{const literate} and @{const siterate} |
|
2029 |
functions (Section~\ref{sssec:primcorec-nested-corecursion}) to possibly |
|
2030 |
infinite trees in which subnodes are organized either as a lazy list (@{text |
|
54072 | 2031 |
tree\<^sub>i\<^sub>i}) or as a finite set (@{text tree\<^sub>i\<^sub>s}). They rely on the map functions of |
2032 |
the nesting type constructors to lift the corecursive calls: |
|
53647 | 2033 |
*} |
2034 |
||
53826 | 2035 |
primcorec iterate\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
54072 | 2036 |
"iterate\<^sub>i\<^sub>i g x = Node\<^sub>i\<^sub>i x (lmap (iterate\<^sub>i\<^sub>i g) (g x))" |
53644 | 2037 |
|
53677 | 2038 |
text {* \blankline *} |
2039 |
||
53826 | 2040 |
primcorec iterate\<^sub>i\<^sub>s :: "('a \<Rightarrow> 'a fset) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>s" where |
54072 | 2041 |
"iterate\<^sub>i\<^sub>s g x = Node\<^sub>i\<^sub>s x (fimage (iterate\<^sub>i\<^sub>s g) (g x))" |
53644 | 2042 |
|
52805 | 2043 |
text {* |
53752 | 2044 |
\noindent |
54072 | 2045 |
Both examples follow the usual format for constructor arguments associated |
2046 |
with nested recursive occurrences of the datatype. Consider |
|
2047 |
@{const iterate\<^sub>i\<^sub>i}. The term @{term "g x"} constructs an @{typ "'a llist"} |
|
2048 |
value, which is turned into an @{typ "'a tree\<^sub>i\<^sub>i llist"} value using |
|
2049 |
@{const lmap}. |
|
2050 |
||
2051 |
This format may sometimes feel artificial. The following function constructs |
|
2052 |
a tree with a single, infinite branch from a stream: |
|
2053 |
*} |
|
2054 |
||
2055 |
primcorec tree\<^sub>i\<^sub>i_of_stream :: "'a stream \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
|
2056 |
"tree\<^sub>i\<^sub>i_of_stream s = |
|
2057 |
Node\<^sub>i\<^sub>i (shd s) (lmap tree\<^sub>i\<^sub>i_of_stream (LCons (stl s) LNil))" |
|
2058 |
||
2059 |
text {* |
|
2060 |
\noindent |
|
54955
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|
2061 |
A more natural syntax, also supported by Isabelle, is to move corecursive calls |
cf8d429dc24e
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|
2062 |
under constructors: |
54072 | 2063 |
*} |
2064 |
||
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|
2065 |
primcorec (*<*)(in late) (*>*)tree\<^sub>i\<^sub>i_of_stream :: "'a stream \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
55350 | 2066 |
"tree\<^sub>i\<^sub>i_of_stream s = |
2067 |
Node\<^sub>i\<^sub>i (shd s) (LCons (tree\<^sub>i\<^sub>i_of_stream (stl s)) LNil)" |
|
54072 | 2068 |
|
2069 |
text {* |
|
2070 |
The next example illustrates corecursion through functions, which is a bit |
|
2071 |
special. Deterministic finite automata (DFAs) are traditionally defined as |
|
2072 |
5-tuples @{text "(Q, \<Sigma>, \<delta>, q\<^sub>0, F)"}, where @{text Q} is a finite set of states, |
|
53675 | 2073 |
@{text \<Sigma>} is a finite alphabet, @{text \<delta>} is a transition function, @{text q\<^sub>0} |
2074 |
is an initial state, and @{text F} is a set of final states. The following |
|
55350 | 2075 |
function translates a DFA into a state machine: |
53675 | 2076 |
*} |
2077 |
||
55350 | 2078 |
primcorec (*<*)(in early) (*>*)sm_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> 'a sm" where |
2079 |
"sm_of_dfa \<delta> F q = SM (q \<in> F) (sm_of_dfa \<delta> F \<circ> \<delta> q)" |
|
53675 | 2080 |
|
53751 | 2081 |
text {* |
2082 |
\noindent |
|
2083 |
The map function for the function type (@{text \<Rightarrow>}) is composition |
|
54181 | 2084 |
(@{text "op \<circ>"}). For convenience, corecursion through functions can |
54182 | 2085 |
also be expressed using $\lambda$-abstractions and function application rather |
54031 | 2086 |
than through composition. For example: |
53751 | 2087 |
*} |
2088 |
||
55350 | 2089 |
primcorec sm_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> 'a sm" where |
2090 |
"sm_of_dfa \<delta> F q = SM (q \<in> F) (\<lambda>a. sm_of_dfa \<delta> F (\<delta> q a))" |
|
53752 | 2091 |
|
2092 |
text {* \blankline *} |
|
2093 |
||
55350 | 2094 |
primcorec empty_sm :: "'a sm" where |
2095 |
"empty_sm = SM False (\<lambda>_. empty_sm)" |
|
53751 | 2096 |
|
53752 | 2097 |
text {* \blankline *} |
2098 |
||
55350 | 2099 |
primcorec not_sm :: "'a sm \<Rightarrow> 'a sm" where |
2100 |
"not_sm M = SM (\<not> accept M) (\<lambda>a. not_sm (trans M a))" |
|
53751 | 2101 |
|
53752 | 2102 |
text {* \blankline *} |
2103 |
||
55350 | 2104 |
primcorec or_sm :: "'a sm \<Rightarrow> 'a sm \<Rightarrow> 'a sm" where |
2105 |
"or_sm M N = |
|
2106 |
SM (accept M \<or> accept N) (\<lambda>a. or_sm (trans M a) (trans N a))" |
|
53751 | 2107 |
|
54182 | 2108 |
text {* |
2109 |
\noindent |
|
2110 |
For recursion through curried $n$-ary functions, $n$ applications of |
|
2111 |
@{term "op \<circ>"} are necessary. The examples below illustrate the case where |
|
2112 |
$n = 2$: |
|
2113 |
*} |
|
2114 |
||
55350 | 2115 |
codatatype ('a, 'b) sm2 = |
2116 |
SM2 (accept2: bool) (trans2: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) sm2") |
|
54182 | 2117 |
|
2118 |
text {* \blankline *} |
|
2119 |
||
2120 |
primcorec |
|
55350 | 2121 |
(*<*)(in early) (*>*)sm2_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> ('a, 'b) sm2" |
54182 | 2122 |
where |
55350 | 2123 |
"sm2_of_dfa \<delta> F q = SM2 (q \<in> F) (op \<circ> (op \<circ> (sm2_of_dfa \<delta> F)) (\<delta> q))" |
54182 | 2124 |
|
2125 |
text {* \blankline *} |
|
2126 |
||
2127 |
primcorec |
|
55350 | 2128 |
sm2_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> ('a, 'b) sm2" |
54182 | 2129 |
where |
55350 | 2130 |
"sm2_of_dfa \<delta> F q = SM2 (q \<in> F) (\<lambda>a b. sm2_of_dfa \<delta> F (\<delta> q a b))" |
54182 | 2131 |
|
53644 | 2132 |
|
2133 |
subsubsection {* Nested-as-Mutual Corecursion |
|
2134 |
\label{sssec:primcorec-nested-as-mutual-corecursion} *} |
|
2135 |
||
53647 | 2136 |
text {* |
2137 |
Just as it is possible to recurse over nested recursive datatypes as if they |
|
2138 |
were mutually recursive |
|
2139 |
(Section~\ref{sssec:primrec-nested-as-mutual-recursion}), it is possible to |
|
53752 | 2140 |
pretend that nested codatatypes are mutually corecursive. For example: |
53647 | 2141 |
*} |
2142 |
||
54287 | 2143 |
(*<*) |
2144 |
context late |
|
2145 |
begin |
|
2146 |
(*>*) |
|
54072 | 2147 |
primcorec |
54287 | 2148 |
iterate\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>i" and |
53644 | 2149 |
iterates\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a llist \<Rightarrow> 'a tree\<^sub>i\<^sub>i llist" |
2150 |
where |
|
54072 | 2151 |
"iterate\<^sub>i\<^sub>i g x = Node\<^sub>i\<^sub>i x (iterates\<^sub>i\<^sub>i g (g x))" | |
2152 |
"iterates\<^sub>i\<^sub>i g xs = |
|
53644 | 2153 |
(case xs of |
2154 |
LNil \<Rightarrow> LNil |
|
54072 | 2155 |
| LCons x xs' \<Rightarrow> LCons (iterate\<^sub>i\<^sub>i g x) (iterates\<^sub>i\<^sub>i g xs'))" |
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|
2156 |
|
54287 | 2157 |
text {* |
2158 |
\noindent |
|
2159 |
Coinduction rules are generated as |
|
2160 |
@{thm [source] iterate\<^sub>i\<^sub>i.coinduct}, |
|
2161 |
@{thm [source] iterates\<^sub>i\<^sub>i.coinduct}, and |
|
2162 |
@{thm [source] iterate\<^sub>i\<^sub>i_iterates\<^sub>i\<^sub>i.coinduct} |
|
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|
2163 |
and analogously for @{text coinduct_strong}. These rules and the |
54287 | 2164 |
underlying corecursors are generated on a per-need basis and are kept in a cache |
2165 |
to speed up subsequent definitions. |
|
2166 |
*} |
|
2167 |
||
2168 |
(*<*) |
|
2169 |
end |
|
2170 |
(*>*) |
|
2171 |
||
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|
2172 |
|
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|
2173 |
subsubsection {* Constructor View |
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|
2174 |
\label{ssec:primrec-constructor-view} *} |
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|
2175 |
|
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|
2176 |
(*<*) |
54182 | 2177 |
locale ctr_view |
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|
2178 |
begin |
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|
2179 |
(*>*) |
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|
2180 |
|
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|
2181 |
text {* |
53750 | 2182 |
The constructor view is similar to the code view, but there is one separate |
2183 |
conditional equation per constructor rather than a single unconditional |
|
2184 |
equation. Examples that rely on a single constructor, such as @{const literate} |
|
2185 |
and @{const siterate}, are identical in both styles. |
|
2186 |
||
2187 |
Here is an example where there is a difference: |
|
53749
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|
2188 |
*} |
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|
2189 |
|
53826 | 2190 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
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|
2191 |
"lnull xs \<Longrightarrow> lnull ys \<Longrightarrow> lappend xs ys = LNil" | |
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|
2192 |
"_ \<Longrightarrow> lappend xs ys = LCons (lhd (if lnull xs then ys else xs)) |
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|
2193 |
(if xs = LNil then ltl ys else lappend (ltl xs) ys)" |
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|
2194 |
|
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|
2195 |
text {* |
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|
2196 |
\noindent |
53752 | 2197 |
With the constructor view, we must distinguish between the @{const LNil} and |
2198 |
the @{const LCons} case. The condition for @{const LCons} is |
|
2199 |
left implicit, as the negation of that for @{const LNil}. |
|
53750 | 2200 |
|
2201 |
For this example, the constructor view is slighlty more involved than the |
|
2202 |
code equation. Recall the code view version presented in |
|
53749
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|
2203 |
Section~\ref{sssec:primcorec-simple-corecursion}. |
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|
2204 |
% TODO: \[{thm code_view.lappend.code}\] |
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|
2205 |
The constructor view requires us to analyze the second argument (@{term ys}). |
53752 | 2206 |
The code equation generated from the constructor view also suffers from this. |
53749
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|
2207 |
% TODO: \[{thm lappend.code}\] |
53750 | 2208 |
|
53752 | 2209 |
In contrast, the next example is arguably more naturally expressed in the |
2210 |
constructor view: |
|
53749
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|
2211 |
*} |
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|
2212 |
|
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|
2213 |
primcorec |
53752 | 2214 |
random_process :: "'a stream \<Rightarrow> (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> 'a process" |
2215 |
where |
|
53749
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|
2216 |
"n mod 4 = 0 \<Longrightarrow> random_process s f n = Fail" | |
53752 | 2217 |
"n mod 4 = 1 \<Longrightarrow> |
2218 |
random_process s f n = Skip (random_process s f (f n))" | |
|
2219 |
"n mod 4 = 2 \<Longrightarrow> |
|
2220 |
random_process s f n = Action (shd s) (random_process (stl s) f (f n))" | |
|
2221 |
"n mod 4 = 3 \<Longrightarrow> |
|
2222 |
random_process s f n = Choice (random_process (every_snd s) f (f n)) |
|
53826 | 2223 |
(random_process (every_snd (stl s)) f (f n))" |
2224 |
(*<*) |
|
53644 | 2225 |
end |
2226 |
(*>*) |
|
52805 | 2227 |
|
53750 | 2228 |
text {* |
53752 | 2229 |
\noindent |
53750 | 2230 |
Since there is no sequentiality, we can apply the equation for @{const Choice} |
53752 | 2231 |
without having first to discharge @{term "n mod (4\<Colon>int) \<noteq> 0"}, |
2232 |
@{term "n mod (4\<Colon>int) \<noteq> 1"}, and |
|
2233 |
@{term "n mod (4\<Colon>int) \<noteq> 2"}. |
|
53750 | 2234 |
The price to pay for this elegance is that we must discharge exclusivity proof |
2235 |
obligations, one for each pair of conditions |
|
53752 | 2236 |
@{term "(n mod (4\<Colon>int) = i, n mod (4\<Colon>int) = j)"} |
2237 |
with @{term "i < j"}. If we prefer not to discharge any obligations, we can |
|
2238 |
enable the @{text "sequential"} option. This pushes the problem to the users of |
|
2239 |
the generated properties. |
|
53750 | 2240 |
%Here are more examples to conclude: |
2241 |
*} |
|
2242 |
||
52824 | 2243 |
|
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|
2244 |
subsubsection {* Destructor View |
53752 | 2245 |
\label{ssec:primrec-destructor-view} *} |
2246 |
||
2247 |
(*<*) |
|
54182 | 2248 |
locale dtr_view |
53752 | 2249 |
begin |
2250 |
(*>*) |
|
53749
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|
2251 |
|
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|
2252 |
text {* |
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|
2253 |
The destructor view is in many respects dual to the constructor view. Conditions |
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|
2254 |
determine which constructor to choose, and these conditions are interpreted |
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|
2255 |
sequentially or not depending on the @{text "sequential"} option. |
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|
2256 |
Consider the following examples: |
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|
2257 |
*} |
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|
2258 |
|
53826 | 2259 |
primcorec literate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist" where |
53749
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|
2260 |
"\<not> lnull (literate _ x)" | |
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|
2261 |
"lhd (literate _ x) = x" | |
54072 | 2262 |
"ltl (literate g x) = literate g (g x)" |
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|
2263 |
|
53752 | 2264 |
text {* \blankline *} |
2265 |
||
53826 | 2266 |
primcorec siterate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a stream" where |
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|
2267 |
"shd (siterate _ x) = x" | |
54072 | 2268 |
"stl (siterate g x) = siterate g (g x)" |
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|
2269 |
|
53752 | 2270 |
text {* \blankline *} |
2271 |
||
53826 | 2272 |
primcorec every_snd :: "'a stream \<Rightarrow> 'a stream" where |
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|
2273 |
"shd (every_snd s) = shd s" | |
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|
2274 |
"stl (every_snd s) = stl (stl s)" |
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|
2275 |
|
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|
2276 |
text {* |
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|
2277 |
\noindent |
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|
2278 |
The first formula in the @{const literate} specification indicates which |
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|
2279 |
constructor to choose. For @{const siterate} and @{const every_snd}, no such |
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|
2280 |
formula is necessary, since the type has only one constructor. The last two |
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|
2281 |
formulas are equations specifying the value of the result for the relevant |
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|
2282 |
selectors. Corecursive calls appear directly to the right of the equal sign. |
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|
2283 |
Their arguments are unrestricted. |
53750 | 2284 |
|
2285 |
The next example shows how to specify functions that rely on more than one |
|
2286 |
constructor: |
|
2287 |
*} |
|
2288 |
||
53826 | 2289 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53750 | 2290 |
"lnull xs \<Longrightarrow> lnull ys \<Longrightarrow> lnull (lappend xs ys)" | |
2291 |
"lhd (lappend xs ys) = lhd (if lnull xs then ys else xs)" | |
|
2292 |
"ltl (lappend xs ys) = (if xs = LNil then ltl ys else lappend (ltl xs) ys)" |
|
2293 |
||
2294 |
text {* |
|
2295 |
\noindent |
|
2296 |
For a codatatype with $n$ constructors, it is sufficient to specify $n - 1$ |
|
2297 |
discriminator formulas. The command will then assume that the remaining |
|
2298 |
constructor should be taken otherwise. This can be made explicit by adding |
|
2299 |
*} |
|
2300 |
||
2301 |
(*<*) |
|
2302 |
end |
|
2303 |
||
54182 | 2304 |
locale dtr_view2 |
2305 |
begin |
|
2306 |
||
53826 | 2307 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53750 | 2308 |
"lnull xs \<Longrightarrow> lnull ys \<Longrightarrow> lnull (lappend xs ys)" | |
55350 | 2309 |
"lhd (lappend xs ys) = lhd (if lnull xs then ys else xs)" | |
2310 |
"ltl (lappend xs ys) = (if xs = LNil then ltl ys else lappend (ltl xs) ys)" | |
|
53750 | 2311 |
(*>*) |
53752 | 2312 |
"_ \<Longrightarrow> \<not> lnull (lappend xs ys)" |
53750 | 2313 |
|
2314 |
text {* |
|
2315 |
\noindent |
|
53752 | 2316 |
to the specification. The generated selector theorems are conditional. |
2317 |
||
2318 |
The next example illustrates how to cope with selectors defined for several |
|
53750 | 2319 |
constructors: |
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|
2320 |
*} |
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|
2321 |
|
53831
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|
2322 |
primcorec |
53752 | 2323 |
random_process :: "'a stream \<Rightarrow> (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> 'a process" |
2324 |
where |
|
57091 | 2325 |
"n mod 4 = 0 \<Longrightarrow> random_process s f n = Fail" | |
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|
2326 |
"n mod 4 = 1 \<Longrightarrow> is_Skip (random_process s f n)" | |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2327 |
"n mod 4 = 2 \<Longrightarrow> is_Action (random_process s f n)" | |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2328 |
"n mod 4 = 3 \<Longrightarrow> is_Choice (random_process s f n)" | |
53831
80423b9080cf
support "of" syntax to disambiguate selector equations
panny
parents:
53829
diff
changeset
|
2329 |
"cont (random_process s f n) = random_process s f (f n)" of Skip | |
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2330 |
"prefix (random_process s f n) = shd s" | |
53831
80423b9080cf
support "of" syntax to disambiguate selector equations
panny
parents:
53829
diff
changeset
|
2331 |
"cont (random_process s f n) = random_process (stl s) f (f n)" of Action | |
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2332 |
"left (random_process s f n) = random_process (every_snd s) f (f n)" | |
53831
80423b9080cf
support "of" syntax to disambiguate selector equations
panny
parents:
53829
diff
changeset
|
2333 |
"right (random_process s f n) = random_process (every_snd (stl s)) f (f n)" |
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2334 |
|
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2335 |
text {* |
53750 | 2336 |
\noindent |
2337 |
Using the @{text "of"} keyword, different equations are specified for @{const |
|
2338 |
cont} depending on which constructor is selected. |
|
2339 |
||
2340 |
Here are more examples to conclude: |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2341 |
*} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2342 |
|
53826 | 2343 |
primcorec |
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2344 |
even_infty :: even_enat and |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2345 |
odd_infty :: odd_enat |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2346 |
where |
57091 | 2347 |
"even_infty \<noteq> Even_EZero" | |
58245 | 2348 |
"un_Even_ESucc even_infty = odd_infty" | |
2349 |
"un_Odd_ESucc odd_infty = even_infty" |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2350 |
|
53752 | 2351 |
text {* \blankline *} |
2352 |
||
53826 | 2353 |
primcorec iterate\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
54072 | 2354 |
"lbl\<^sub>i\<^sub>i (iterate\<^sub>i\<^sub>i g x) = x" | |
2355 |
"sub\<^sub>i\<^sub>i (iterate\<^sub>i\<^sub>i g x) = lmap (iterate\<^sub>i\<^sub>i g) (g x)" |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2356 |
(*<*) |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2357 |
end |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2358 |
(*>*) |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2359 |
|
53750 | 2360 |
|
53617 | 2361 |
subsection {* Command Syntax |
2362 |
\label{ssec:primcorec-command-syntax} *} |
|
2363 |
||
53826 | 2364 |
subsubsection {* \keyw{primcorec} and \keyw{primcorecursive} |
53753
ae7f50e70c09
renamed "primcorec" to "primcorecursive", to open the door to a 'theory -> theory' command called "primcorec" (cf. "fun" vs. "function")
blanchet
parents:
53752
diff
changeset
|
2365 |
\label{sssec:primcorecursive-and-primcorec} *} |
52840 | 2366 |
|
2367 |
text {* |
|
53829 | 2368 |
\begin{matharray}{rcl} |
2369 |
@{command_def "primcorec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
|
2370 |
@{command_def "primcorecursive"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
|
2371 |
\end{matharray} |
|
52840 | 2372 |
|
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2373 |
@{rail \<open> |
55029
61a6bf7d4b02
clarified @{rail} syntax: prefer explicit \<newline> symbol;
wenzelm
parents:
54958
diff
changeset
|
2374 |
(@@{command primcorec} | @@{command primcorecursive}) target? \<newline> |
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2375 |
@{syntax pcr_option}? fixes @'where' (@{syntax pcr_formula} + '|') |
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2376 |
; |
53828 | 2377 |
@{syntax_def pcr_option}: '(' ('sequential' | 'exhaustive') ')' |
52840 | 2378 |
; |
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2379 |
@{syntax_def pcr_formula}: thmdecl? prop (@'of' (term * ))? |
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2380 |
\<close>} |
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2381 |
|
55351 | 2382 |
\medskip |
2383 |
||
2384 |
\noindent |
|
55474 | 2385 |
The @{command primcorec} and @{command primcorecursive} commands introduce a set |
2386 |
of mutually corecursive functions over codatatypes. |
|
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2387 |
|
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2388 |
The syntactic entity \synt{target} can be used to specify a local context, |
55474 | 2389 |
\synt{fixes} denotes a list of names with optional type signatures, |
2390 |
\synt{thmdecl} denotes an optional name for the formula that follows, and |
|
2391 |
\synt{prop} denotes a HOL proposition \cite{isabelle-isar-ref}. |
|
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2392 |
|
56124 | 2393 |
The optional target is optionally followed by one or both of the following |
2394 |
options: |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2395 |
|
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2396 |
\begin{itemize} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2397 |
\setlength{\itemsep}{0pt} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2398 |
|
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2399 |
\item |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2400 |
The @{text "sequential"} option indicates that the conditions in specifications |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2401 |
expressed using the constructor or destructor view are to be interpreted |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2402 |
sequentially. |
53826 | 2403 |
|
2404 |
\item |
|
2405 |
The @{text "exhaustive"} option indicates that the conditions in specifications |
|
2406 |
expressed using the constructor or destructor view cover all possible cases. |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2407 |
\end{itemize} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2408 |
|
55474 | 2409 |
The @{command primcorec} command is an abbreviation for @{command |
2410 |
primcorecursive} with @{text "by auto?"} to discharge any emerging proof |
|
2411 |
obligations. |
|
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2412 |
|
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2413 |
%%% TODO: elaborate on format of the propositions |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2414 |
%%% TODO: elaborate on mutual and nested-to-mutual |
52840 | 2415 |
*} |
52794 | 2416 |
|
52824 | 2417 |
|
53619 | 2418 |
(* |
52840 | 2419 |
subsection {* Generated Theorems |
2420 |
\label{ssec:primcorec-generated-theorems} *} |
|
53619 | 2421 |
*) |
52794 | 2422 |
|
2423 |
||
53623 | 2424 |
(* |
2425 |
subsection {* Recursive Default Values for Selectors |
|
2426 |
\label{ssec:primcorec-recursive-default-values-for-selectors} *} |
|
2427 |
||
2428 |
text {* |
|
2429 |
partial_function to the rescue |
|
2430 |
*} |
|
2431 |
*) |
|
2432 |
||
2433 |
||
55351 | 2434 |
section {* Introducing Bounded Natural Functors |
2435 |
\label{sec:introducing-bounded-natural-functors} *} |
|
52792 | 2436 |
|
52805 | 2437 |
text {* |
53647 | 2438 |
The (co)datatype package can be set up to allow nested recursion through |
55350 | 2439 |
arbitrary type constructors, as long as they adhere to the BNF requirements |
2440 |
and are registered as BNFs. It is also possible to declare a BNF abstractly |
|
2441 |
without specifying its internal structure. |
|
52805 | 2442 |
*} |
2443 |
||
52824 | 2444 |
|
55350 | 2445 |
subsection {* Bounded Natural Functors |
2446 |
\label{ssec:bounded-natural-functors} *} |
|
2447 |
||
2448 |
text {* |
|
2449 |
Bounded natural functors (BNFs) are a semantic criterion for where |
|
2450 |
(co)re\-cur\-sion may appear on the right-hand side of an equation |
|
2451 |
\cite{traytel-et-al-2012,blanchette-et-al-wit}. |
|
2452 |
||
2453 |
An $n$-ary BNF is a type constructor equipped with a map function |
|
2454 |
(functorial action), $n$ set functions (natural transformations), |
|
2455 |
and an infinite cardinal bound that satisfy certain properties. |
|
2456 |
For example, @{typ "'a llist"} is a unary BNF. |
|
2457 |
Its relator @{text "llist_all2 \<Colon> |
|
2458 |
('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> |
|
2459 |
'a llist \<Rightarrow> 'b llist \<Rightarrow> bool"} |
|
2460 |
extends binary predicates over elements to binary predicates over parallel |
|
2461 |
lazy lists. The cardinal bound limits the number of elements returned by the |
|
2462 |
set function; it may not depend on the cardinality of @{typ 'a}. |
|
2463 |
||
2464 |
The type constructors introduced by @{command datatype_new} and |
|
2465 |
@{command codatatype} are automatically registered as BNFs. In addition, a |
|
2466 |
number of old-style datatypes and non-free types are preregistered. |
|
2467 |
||
2468 |
Given an $n$-ary BNF, the $n$ type variables associated with set functions, |
|
2469 |
and on which the map function acts, are \emph{live}; any other variables are |
|
2470 |
\emph{dead}. Nested (co)recursion can only take place through live variables. |
|
2471 |
*} |
|
2472 |
||
2473 |
||
2474 |
subsection {* Introductory Examples |
|
2475 |
\label{ssec:bnf-introductory-examples} *} |
|
52805 | 2476 |
|
2477 |
text {* |
|
55350 | 2478 |
The example below shows how to register a type as a BNF using the @{command bnf} |
2479 |
command. Some of the proof obligations are best viewed with the theory |
|
2480 |
@{theory Cardinal_Notations}, located in \verb|~~/src/HOL/Library|, |
|
2481 |
imported. |
|
2482 |
||
2483 |
The type is simply a copy of the function space @{typ "'d \<Rightarrow> 'a"}, where @{typ 'a} |
|
2484 |
is live and @{typ 'd} is dead. We introduce it together with its map function, |
|
2485 |
set function, and relator. |
|
52805 | 2486 |
*} |
55350 | 2487 |
|
2488 |
typedef ('d, 'a) fn = "UNIV \<Colon> ('d \<Rightarrow> 'a) set" |
|
2489 |
by simp |
|
2490 |
||
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2491 |
text {* \blankline *} |
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2492 |
|
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2493 |
setup_lifting type_definition_fn |
55350 | 2494 |
|
2495 |
text {* \blankline *} |
|
2496 |
||
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2497 |
lift_definition map_fn :: "('a \<Rightarrow> 'b) \<Rightarrow> ('d, 'a) fn \<Rightarrow> ('d, 'b) fn" is "op \<circ>" . |
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2498 |
lift_definition set_fn :: "('d, 'a) fn \<Rightarrow> 'a set" is range . |
55350 | 2499 |
|
2500 |
text {* \blankline *} |
|
2501 |
||
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2502 |
lift_definition |
55350 | 2503 |
rel_fn :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('d, 'a) fn \<Rightarrow> ('d, 'b) fn \<Rightarrow> bool" |
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2504 |
is |
55945 | 2505 |
"rel_fun (op =)" . |
55350 | 2506 |
|
2507 |
text {* \blankline *} |
|
2508 |
||
2509 |
bnf "('d, 'a) fn" |
|
2510 |
map: map_fn |
|
2511 |
sets: set_fn |
|
2512 |
bd: "natLeq +c |UNIV :: 'd set|" |
|
2513 |
rel: rel_fn |
|
2514 |
proof - |
|
2515 |
show "map_fn id = id" |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2516 |
by transfer auto |
55350 | 2517 |
next |
2518 |
fix F G show "map_fn (G \<circ> F) = map_fn G \<circ> map_fn F" |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2519 |
by transfer (auto simp add: comp_def) |
55350 | 2520 |
next |
2521 |
fix F f g |
|
2522 |
assume "\<And>x. x \<in> set_fn F \<Longrightarrow> f x = g x" |
|
2523 |
thus "map_fn f F = map_fn g F" |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2524 |
by transfer auto |
55350 | 2525 |
next |
2526 |
fix f show "set_fn \<circ> map_fn f = op ` f \<circ> set_fn" |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2527 |
by transfer (auto simp add: comp_def) |
55350 | 2528 |
next |
2529 |
show "card_order (natLeq +c |UNIV \<Colon> 'd set| )" |
|
2530 |
apply (rule card_order_csum) |
|
2531 |
apply (rule natLeq_card_order) |
|
2532 |
by (rule card_of_card_order_on) |
|
2533 |
next |
|
2534 |
show "cinfinite (natLeq +c |UNIV \<Colon> 'd set| )" |
|
2535 |
apply (rule cinfinite_csum) |
|
2536 |
apply (rule disjI1) |
|
2537 |
by (rule natLeq_cinfinite) |
|
2538 |
next |
|
2539 |
fix F :: "('d, 'a) fn" |
|
2540 |
have "|set_fn F| \<le>o |UNIV \<Colon> 'd set|" (is "_ \<le>o ?U") |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2541 |
by transfer (rule card_of_image) |
55350 | 2542 |
also have "?U \<le>o natLeq +c ?U" |
2543 |
by (rule ordLeq_csum2) (rule card_of_Card_order) |
|
2544 |
finally show "|set_fn F| \<le>o natLeq +c |UNIV \<Colon> 'd set|" . |
|
2545 |
next |
|
2546 |
fix R S |
|
2547 |
show "rel_fn R OO rel_fn S \<le> rel_fn (R OO S)" |
|
55945 | 2548 |
by (rule, transfer) (auto simp add: rel_fun_def) |
55350 | 2549 |
next |
2550 |
fix R |
|
2551 |
show "rel_fn R = |
|
57398 | 2552 |
(BNF_Def.Grp {x. set_fn x \<subseteq> Collect (split R)} (map_fn fst))\<inverse>\<inverse> OO |
2553 |
BNF_Def.Grp {x. set_fn x \<subseteq> Collect (split R)} (map_fn snd)" |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2554 |
unfolding Grp_def fun_eq_iff relcompp.simps conversep.simps |
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2555 |
apply transfer |
55945 | 2556 |
unfolding rel_fun_def subset_iff image_iff |
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2557 |
by auto (force, metis pair_collapse) |
55350 | 2558 |
qed |
2559 |
||
2560 |
text {* \blankline *} |
|
2561 |
||
2562 |
print_theorems |
|
2563 |
print_bnfs |
|
2564 |
||
2565 |
text {* |
|
2566 |
\noindent |
|
57542 | 2567 |
Using \keyw{print_theorems} and @{keyword print_bnfs}, we can contemplate and |
55350 | 2568 |
show the world what we have achieved. |
2569 |
||
2570 |
This particular example does not need any nonemptiness witness, because the |
|
2571 |
one generated by default is good enough, but in general this would be |
|
2572 |
necessary. See \verb|~~/src/HOL/Basic_BNFs.thy|, |
|
2573 |
\verb|~~/src/HOL/Library/FSet.thy|, and \verb|~~/src/HOL/Library/Multiset.thy| |
|
2574 |
for further examples of BNF registration, some of which feature custom |
|
2575 |
witnesses. |
|
2576 |
||
57542 | 2577 |
The next example declares a BNF axiomatically. This can be convenient for |
2578 |
reasoning abstractly about an arbitrary BNF. The @{command bnf_axiomatization} |
|
2579 |
command below introduces a type @{text "('a, 'b, 'c) F"}, three set constants, |
|
2580 |
a map function, a relator, and a nonemptiness witness that depends only on |
|
55350 | 2581 |
@{typ 'a}. (The type @{text "'a \<Rightarrow> ('a, 'b, 'c) F"} of |
2582 |
the witness can be read as an implication: If we have a witness for @{typ 'a}, |
|
2583 |
we can construct a witness for @{text "('a, 'b, 'c) F"}.) The BNF |
|
2584 |
properties are postulated as axioms. |
|
2585 |
*} |
|
2586 |
||
57542 | 2587 |
bnf_axiomatization (setA: 'a, setB: 'b, setC: 'c) F |
2588 |
[wits: "'a \<Rightarrow> ('a, 'b, 'c) F"] |
|
55350 | 2589 |
|
2590 |
text {* \blankline *} |
|
2591 |
||
2592 |
print_theorems |
|
2593 |
print_bnfs |
|
52794 | 2594 |
|
52824 | 2595 |
|
53617 | 2596 |
subsection {* Command Syntax |
2597 |
\label{ssec:bnf-command-syntax} *} |
|
2598 |
||
53621 | 2599 |
subsubsection {* \keyw{bnf} |
2600 |
\label{sssec:bnf} *} |
|
52794 | 2601 |
|
53028 | 2602 |
text {* |
53829 | 2603 |
\begin{matharray}{rcl} |
2604 |
@{command_def "bnf"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
|
2605 |
\end{matharray} |
|
2606 |
||
55112
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diff
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|
2607 |
@{rail \<open> |
55474 | 2608 |
@@{command bnf} target? (name ':')? type \<newline> |
55029
61a6bf7d4b02
clarified @{rail} syntax: prefer explicit \<newline> symbol;
wenzelm
parents:
54958
diff
changeset
|
2609 |
'map:' term ('sets:' (term +))? 'bd:' term \<newline> |
58190 | 2610 |
('wits:' (term +))? ('rel:' term)? \<newline> |
2611 |
@{syntax plugins}? |
|
55112
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changeset
|
2612 |
\<close>} |
55460
3f4efd7d950d
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parents:
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diff
changeset
|
2613 |
|
3f4efd7d950d
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parents:
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diff
changeset
|
2614 |
\medskip |
3f4efd7d950d
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parents:
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diff
changeset
|
2615 |
|
3f4efd7d950d
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parents:
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diff
changeset
|
2616 |
\noindent |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2617 |
The @{command bnf} command registers an existing type as a bounded natural |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2618 |
functor (BNF). The type must be equipped with an appropriate map function |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2619 |
(functorial action). In addition, custom set functions, relators, and |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2620 |
nonemptiness witnesses can be specified; otherwise, default versions are used. |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2621 |
|
55474 | 2622 |
The syntactic entity \synt{target} can be used to specify a local context, |
2623 |
\synt{type} denotes a HOL type, and \synt{term} denotes a HOL term |
|
2624 |
\cite{isabelle-isar-ref}. |
|
2625 |
||
2626 |
%%% TODO: elaborate on proof obligations |
|
53028 | 2627 |
*} |
52805 | 2628 |
|
53617 | 2629 |
|
57542 | 2630 |
subsubsection {* \keyw{bnf_axiomatization} |
56948 | 2631 |
\label{sssec:bnf-axiomatization} *} |
54187 | 2632 |
|
2633 |
text {* |
|
2634 |
\begin{matharray}{rcl} |
|
56942 | 2635 |
@{command_def "bnf_axiomatization"} & : & @{text "local_theory \<rightarrow> local_theory"} |
54187 | 2636 |
\end{matharray} |
2637 |
||
55112
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diff
changeset
|
2638 |
@{rail \<open> |
58190 | 2639 |
@@{command bnf_axiomatization} target? @{syntax plugins}? \<newline> |
2640 |
@{syntax tyargs}? name @{syntax wit_types}? \<newline> |
|
2641 |
mixfix? @{syntax map_rel}? |
|
54602 | 2642 |
; |
55350 | 2643 |
@{syntax_def wit_types}: '[' 'wits' ':' types ']' |
55112
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parents:
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changeset
|
2644 |
\<close>} |
54602 | 2645 |
|
55351 | 2646 |
\medskip |
2647 |
||
55350 | 2648 |
\noindent |
56942 | 2649 |
The @{command bnf_axiomatization} command declares a new type and associated constants |
55350 | 2650 |
(map, set, relator, and cardinal bound) and asserts the BNF properties for |
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2651 |
these constants as axioms. |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2652 |
|
55474 | 2653 |
The syntactic entity \synt{target} can be used to specify a local context, |
2654 |
\synt{name} denotes an identifier, \synt{typefree} denotes fixed type variable |
|
2655 |
(@{typ 'a}, @{typ 'b}, \ldots), and \synt{mixfix} denotes the usual |
|
2656 |
parenthesized mixfix notation \cite{isabelle-isar-ref}. |
|
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2657 |
|
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2658 |
Type arguments are live by default; they can be marked as dead by entering |
58220 | 2659 |
@{text dead} in front of the type variable (e.g., @{text "(dead 'a)"}) |
57092 | 2660 |
instead of an identifier for the corresponding set function. Witnesses can be |
2661 |
specified by their types. Otherwise, the syntax of @{command bnf_axiomatization} |
|
2662 |
is identical to the left-hand side of a @{command datatype_new} or |
|
2663 |
@{command codatatype} definition. |
|
55350 | 2664 |
|
2665 |
The command is useful to reason abstractly about BNFs. The axioms are safe |
|
57575 | 2666 |
because there exist BNFs of arbitrary large arities. Applications must import |
56942 | 2667 |
the theory @{theory BNF_Axiomatization}, located in the directory |
57542 | 2668 |
\verb|~~/src/|\allowbreak\verb|HOL/Library|, to use this functionality. |
54187 | 2669 |
*} |
2670 |
||
2671 |
||
57542 | 2672 |
subsubsection {* \keyw{print_bnfs} |
53621 | 2673 |
\label{sssec:print-bnfs} *} |
53617 | 2674 |
|
2675 |
text {* |
|
53829 | 2676 |
\begin{matharray}{rcl} |
2677 |
@{command_def "print_bnfs"} & : & @{text "local_theory \<rightarrow>"} |
|
2678 |
\end{matharray} |
|
2679 |
||
55112
b1a5d603fd12
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wenzelm
parents:
55029
diff
changeset
|
2680 |
@{rail \<open> |
53829 | 2681 |
@@{command print_bnfs} |
55112
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wenzelm
parents:
55029
diff
changeset
|
2682 |
\<close>} |
53617 | 2683 |
*} |
2684 |
||
2685 |
||
2686 |
section {* Deriving Destructors and Theorems for Free Constructors |
|
2687 |
\label{sec:deriving-destructors-and-theorems-for-free-constructors} *} |
|
52794 | 2688 |
|
52805 | 2689 |
text {* |
53623 | 2690 |
The derivation of convenience theorems for types equipped with free constructors, |
53829 | 2691 |
as performed internally by @{command datatype_new} and @{command codatatype}, |
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
2692 |
is available as a stand-alone command called @{command free_constructors}. |
52794 | 2693 |
|
53617 | 2694 |
% * need for this is rare but may arise if you want e.g. to add destructors to |
2695 |
% a type not introduced by ... |
|
2696 |
% |
|
2697 |
% * also useful for compatibility with old package, e.g. add destructors to |
|
2698 |
% old \keyw{datatype} |
|
2699 |
% |
|
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
2700 |
% * @{command free_constructors} |
58192 | 2701 |
% * @{text plugins}, @{text discs_sels} |
53617 | 2702 |
% * hack to have both co and nonco view via locale (cf. ext nats) |
52805 | 2703 |
*} |
52792 | 2704 |
|
52824 | 2705 |
|
53619 | 2706 |
(* |
53617 | 2707 |
subsection {* Introductory Example |
2708 |
\label{ssec:ctors-introductory-example} *} |
|
53619 | 2709 |
*) |
52794 | 2710 |
|
52824 | 2711 |
|
53617 | 2712 |
subsection {* Command Syntax |
2713 |
\label{ssec:ctors-command-syntax} *} |
|
2714 |
||
57542 | 2715 |
subsubsection {* \keyw{free_constructors} |
55472
990651bfc65b
updated docs to reflect the new 'free_constructors' syntax
blanchet
parents:
55468
diff
changeset
|
2716 |
\label{sssec:free-constructors} *} |
52828 | 2717 |
|
53018 | 2718 |
text {* |
53829 | 2719 |
\begin{matharray}{rcl} |
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
2720 |
@{command_def "free_constructors"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
53829 | 2721 |
\end{matharray} |
53018 | 2722 |
|
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2723 |
@{rail \<open> |
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
2724 |
@@{command free_constructors} target? @{syntax dt_options} \<newline> |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
2725 |
name 'for' (@{syntax fc_ctor} + '|') \<newline> |
57206
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
2726 |
(@'where' (prop + '|'))? |
53018 | 2727 |
; |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
2728 |
@{syntax_def fc_ctor}: (name ':')? term (name * ) |
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2729 |
\<close>} |
53018 | 2730 |
|
55351 | 2731 |
\medskip |
2732 |
||
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2733 |
\noindent |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2734 |
The @{command free_constructors} command generates destructor constants for |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2735 |
freely constructed types as well as properties about constructors and |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2736 |
destructors. It also registers the constants and theorems in a data structure |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2737 |
that is queried by various tools (e.g., \keyw{function}). |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2738 |
|
55474 | 2739 |
The syntactic entity \synt{target} can be used to specify a local context, |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
2740 |
\synt{name} denotes an identifier, \synt{prop} denotes a HOL proposition, and |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
2741 |
\synt{term} denotes a HOL term \cite{isabelle-isar-ref}. |
55474 | 2742 |
|
2743 |
The syntax resembles that of @{command datatype_new} and @{command codatatype} |
|
2744 |
definitions (Sections |
|
2745 |
\ref{ssec:datatype-command-syntax}~and~\ref{ssec:codatatype-command-syntax}). |
|
2746 |
A constructor is specified by an optional name for the discriminator, the |
|
2747 |
constructor itself (as a term), and a list of optional names for the selectors. |
|
53028 | 2748 |
|
53542 | 2749 |
Section~\ref{ssec:datatype-generated-theorems} lists the generated theorems. |
57542 | 2750 |
For bootstrapping reasons, the generally useful @{text "[fundef_cong]"} |
2751 |
attribute is not set on the generated @{text case_cong} theorem. It can be |
|
2752 |
added manually using \keyw{declare}. |
|
53018 | 2753 |
*} |
52828 | 2754 |
|
52794 | 2755 |
|
53617 | 2756 |
(* |
58245 | 2757 |
section {* Using the Standard ML Interface |
2758 |
\label{sec:using-the-standard-ml-interface} *} |
|
52792 | 2759 |
|
52805 | 2760 |
text {* |
53623 | 2761 |
The package's programmatic interface. |
52805 | 2762 |
*} |
53617 | 2763 |
*) |
52794 | 2764 |
|
2765 |
||
58245 | 2766 |
section {* Controlling Plugins |
2767 |
\label{sec:controlling-plugins} *} |
|
58190 | 2768 |
|
2769 |
text {* |
|
2770 |
Plugins extend the (co)datatype package to interoperate with other Isabelle |
|
58192 | 2771 |
packages and tools, such as the code generator, Transfer, Lifting, and |
2772 |
Quickcheck. They can be enabled or disabled individually using the |
|
58245 | 2773 |
@{syntax plugins} option to the commands @{command datatype_new}, |
2774 |
@{command codatatype}, @{command free_constructors}, @{command bnf}, and |
|
2775 |
@{command bnf_axiomatization}. |
|
58192 | 2776 |
For example: |
58190 | 2777 |
*} |
2778 |
||
58245 | 2779 |
datatype_new (plugins del: code "quickcheck_*") color = Red | Black |
58192 | 2780 |
|
2781 |
||
2782 |
subsection {* Code Generator |
|
2783 |
\label{ssec:code-generator} *} |
|
2784 |
||
2785 |
text {* |
|
58245 | 2786 |
The \hthm{code} plugin registers (co)datatypes and other freely generated types |
58244 | 2787 |
for code generation. No distinction is made between datatypes and codatatypes. |
2788 |
This means that for target languages with a strict evaluation strategy (e.g., |
|
2789 |
Standard ML), programs that attempt to produce infinite codatatype values will |
|
2790 |
not terminate. |
|
2791 |
||
2792 |
The plugin derives the following properties: |
|
2793 |
||
2794 |
\begin{indentblock} |
|
2795 |
\begin{description} |
|
2796 |
||
2797 |
\item[@{text "t."}\hthm{eq.refl} @{text "[code nbe]"}\rm:] ~ \\ |
|
2798 |
@{thm list.eq.refl[no_vars]} |
|
2799 |
||
2800 |
\item[@{text "t."}\hthm{eq.simps} @{text "[code]"}\rm:] ~ \\ |
|
2801 |
@{thm list.eq.simps(1)[no_vars]} \\ |
|
2802 |
@{thm list.eq.simps(2)[no_vars]} \\ |
|
2803 |
@{thm list.eq.simps(3)[no_vars]} \\ |
|
2804 |
@{thm list.eq.simps(4)[no_vars]} \\ |
|
2805 |
@{thm list.eq.simps(5)[no_vars]} \\ |
|
2806 |
@{thm list.eq.simps(6)[no_vars]} |
|
2807 |
||
2808 |
\end{description} |
|
2809 |
\end{indentblock} |
|
58192 | 2810 |
*} |
58190 | 2811 |
|
2812 |
||
2813 |
subsection {* Size |
|
2814 |
\label{ssec:size} *} |
|
52794 | 2815 |
|
52805 | 2816 |
text {* |
58245 | 2817 |
For each datatype, the \hthm{size} plugin generates a parameterized size |
58190 | 2818 |
function @{text "t.size_t"} as well as a specific instance |
2819 |
@{text "size \<Colon> t \<Rightarrow> bool"} belonging to the @{text size} type |
|
2820 |
class. The \keyw{fun} command relies on @{const size} to prove termination of |
|
2821 |
recursive functions on datatypes. |
|
2822 |
||
2823 |
The plugin derives the following properties: |
|
2824 |
||
2825 |
\begin{indentblock} |
|
2826 |
\begin{description} |
|
2827 |
||
2828 |
\item[@{text "t."}\hthm{size} @{text "[simp, code]"}\rm:] ~ \\ |
|
2829 |
@{thm list.size(1)[no_vars]} \\ |
|
2830 |
@{thm list.size(2)[no_vars]} \\ |
|
2831 |
@{thm list.size(3)[no_vars]} \\ |
|
2832 |
@{thm list.size(4)[no_vars]} |
|
2833 |
||
2834 |
\item[@{text "t."}\hthm{rec_o_map}\rm:] ~ \\ |
|
2835 |
@{thm list.rec_o_map[no_vars]} |
|
2836 |
||
2837 |
\item[@{text "t."}\hthm{size_o_map}\rm:] ~ \\ |
|
2838 |
@{thm list.size_o_map[no_vars]} |
|
2839 |
(This property is not generated for all datatypes.) |
|
2840 |
||
2841 |
\end{description} |
|
2842 |
\end{indentblock} |
|
52805 | 2843 |
*} |
52794 | 2844 |
|
52824 | 2845 |
|
58190 | 2846 |
subsection {* Transfer |
2847 |
\label{ssec:transfer} *} |
|
2848 |
||
2849 |
text {* |
|
58192 | 2850 |
For each (co)datatype with live type arguments and each manually registered BNF, |
58245 | 2851 |
the \hthm{transfer} plugin generates a predicator @{text "t.pred_t"} and |
58244 | 2852 |
properties that guide the Transfer tool. |
2853 |
||
2854 |
The plugin derives the following properties: |
|
2855 |
||
2856 |
\begin{indentblock} |
|
2857 |
\begin{description} |
|
2858 |
||
2859 |
\item[@{text "t."}\hthm{Domainp_rel} @{text "[relator_domain]"}\rm:] ~ \\ |
|
2860 |
@{thm list.Domainp_rel[no_vars]} |
|
2861 |
||
2862 |
\item[@{text "t."}\hthm{pred_inject} @{text "[simp]"}\rm:] ~ \\ |
|
2863 |
@{thm list.pred_inject(1)[no_vars]} \\ |
|
2864 |
@{thm list.pred_inject(2)[no_vars]} |
|
2865 |
||
2866 |
\item[@{text "t."}\hthm{rel_eq_onp}\rm:] ~ \\ |
|
2867 |
@{thm list.rel_eq_onp[no_vars]} |
|
2868 |
||
2869 |
\item[@{text "t."}\hthm{left_total_rel} @{text "[transfer]"}\rm:] ~ \\ |
|
2870 |
@{thm list.left_total_rel[no_vars]} |
|
2871 |
||
2872 |
\item[@{text "t."}\hthm{left_unique_rel} @{text "[transfer]"}\rm:] ~ \\ |
|
2873 |
@{thm list.left_unique_rel[no_vars]} |
|
2874 |
||
2875 |
\item[@{text "t."}\hthm{right_total_rel} @{text "[transfer]"}\rm:] ~ \\ |
|
2876 |
@{thm list.right_total_rel[no_vars]} |
|
2877 |
||
2878 |
\item[@{text "t."}\hthm{right_unique_rel} @{text "[transfer]"}\rm:] ~ \\ |
|
2879 |
@{thm list.right_unique_rel[no_vars]} |
|
2880 |
||
2881 |
\item[@{text "t."}\hthm{bi_total_rel} @{text "[transfer]"}\rm:] ~ \\ |
|
2882 |
@{thm list.bi_total_rel[no_vars]} |
|
2883 |
||
2884 |
\item[@{text "t."}\hthm{bi_unique_rel} @{text "[transfer]"}\rm:] ~ \\ |
|
2885 |
@{thm list.bi_unique_rel[no_vars]} |
|
2886 |
||
2887 |
\end{description} |
|
2888 |
\end{indentblock} |
|
58190 | 2889 |
*} |
2890 |
||
2891 |
||
2892 |
subsection {* Lifting |
|
2893 |
\label{ssec:lifting} *} |
|
2894 |
||
2895 |
text {* |
|
58192 | 2896 |
For each (co)datatype with live type arguments and each manually registered BNF, |
58245 | 2897 |
the \hthm{lifting} plugin generates properties and attributes that guide the |
58244 | 2898 |
Lifting tool. |
2899 |
||
2900 |
The plugin derives the following property: |
|
2901 |
||
2902 |
\begin{indentblock} |
|
2903 |
\begin{description} |
|
2904 |
||
2905 |
\item[@{text "t."}\hthm{Quotient} @{text "[quot_map]"}\rm:] ~ \\ |
|
2906 |
@{thm list.Quotient[no_vars]} |
|
2907 |
||
2908 |
\end{description} |
|
2909 |
\end{indentblock} |
|
2910 |
||
58245 | 2911 |
In addition, the plugin sets the @{text "[relator_eq_onp]"} attribute on a |
58244 | 2912 |
variant of the @{text t.rel_eq_onp} property generated by the @{text quotient} |
2913 |
plugin, the @{text "[relator_mono]"} attribute on @{text t.rel_mono}, and the |
|
2914 |
@{text "[relator_distr]"} attribute on @{text t.rel_compp}. |
|
58190 | 2915 |
*} |
52794 | 2916 |
|
52824 | 2917 |
|
52828 | 2918 |
subsection {* Quickcheck |
2919 |
\label{ssec:quickcheck} *} |
|
52794 | 2920 |
|
58190 | 2921 |
text {* |
58245 | 2922 |
The integration of datatypes with Quickcheck is accomplished by a number of |
2923 |
plugins that instantiate specific type classes. The plugins are listed below: |
|
2924 |
||
2925 |
\begin{indentblock} |
|
2926 |
\hthm{quickcheck_random} \\ |
|
2927 |
\hthm{quickcheck_exhaustive} \\ |
|
2928 |
\hthm{quickcheck_bounded_forall} \\ |
|
2929 |
\hthm{quickcheck_full_exhaustive} \\ |
|
2930 |
\hthm{quickcheck_narrowing} |
|
2931 |
\end{indentblock} |
|
58190 | 2932 |
*} |
52794 | 2933 |
|
52805 | 2934 |
|
53617 | 2935 |
(* |
52827 | 2936 |
section {* Known Bugs and Limitations |
52805 | 2937 |
\label{sec:known-bugs-and-limitations} *} |
2938 |
||
2939 |
text {* |
|
53623 | 2940 |
Known open issues of the package. |
52805 | 2941 |
*} |
52794 | 2942 |
|
2943 |
text {* |
|
53617 | 2944 |
%* slow n-ary mutual (co)datatype, avoid as much as possible (e.g. using nesting) |
2945 |
% |
|
2946 |
%* no way to register "sum" and "prod" as (co)datatypes to enable N2M reduction for them |
|
55531
601ca8efa000
renamed 'datatype_new_compat' to 'datatype_compat'
blanchet
parents:
55530
diff
changeset
|
2947 |
% (for @{command datatype_compat} and prim(co)rec) |
53617 | 2948 |
% |
53619 | 2949 |
% * a fortiori, no way to register same type as both data- and codatatype |
53617 | 2950 |
% |
2951 |
%* no recursion through unused arguments (unlike with the old package) |
|
2952 |
% |
|
2953 |
%* in a locale, cannot use locally fixed types (because of limitation in typedef)? |
|
53619 | 2954 |
% |
2955 |
% *names of variables suboptimal |
|
52822 | 2956 |
*} |
53675 | 2957 |
*) |
52822 | 2958 |
|
2959 |
||
2960 |
text {* |
|
53863
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
2961 |
\section*{Acknowledgment} |
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
2962 |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2963 |
Tobias Nipkow and Makarius Wenzel encouraged us to implement the new |
53617 | 2964 |
(co)datatype package. Andreas Lochbihler provided lots of comments on earlier |
56655 | 2965 |
versions of the package, especially on the coinductive part. Brian Huffman |
58245 | 2966 |
suggested major simplifications to the internal constructions. Ond\v{r}ej |
2967 |
Kun\v{c}ar implemented the @{text transfer} and @{text lifting} plugins. |
|
2968 |
Florian Haftmann and Christian Urban provided general advice on Isabelle and |
|
2969 |
package writing. Stefan Milius and Lutz Schr\"oder found an elegant proof that |
|
2970 |
eliminated one of the BNF proof obligations. Andreas Lochbihler and Christian |
|
2971 |
Sternagel suggested many textual improvements to this tutorial. |
|
52794 | 2972 |
*} |
53617 | 2973 |
|
52792 | 2974 |
end |