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(* Title: Doc/Datatypes/Datatypes.thy |
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Author: Jasmin Blanchette, TU Muenchen |
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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/BNF/BNF_Decl" |
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"~~/src/HOL/BNF/More_BNFs" |
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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 usually all that is needed to port existing theories |
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to 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 theories @{theory BNF_Decl} and @{theory More_BNFs}, located in the |
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@{text "~~/src/HOL/BNF"} directory. |
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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{If the @{text quick_and_dirty} option is enabled, some of the |
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internal constructions and most of the internal proof obligations are skipped.} |
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The package's metatheory is described in a pair of papers |
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\cite{traytel-et-al-2012,blanchette-et-al-wit}. The central notion is that of a |
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\emph{bounded natural functor} (BNF)---a well-behaved type constructor for which |
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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_new}, \keyw{fun}, and \keyw{function}. |
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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:registering-bounded-natural-functors}, ``Registering |
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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 wrap_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:standard-ml-interface}, ``Standard ML Interface,'' |
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%describes the package's programmatic interface. |
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%\item Section \ref{sec:interoperability}, ``Interoperability,'' |
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%is concerned with the packages' interaction with other Isabelle packages and |
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%tools, such as the code generator and the counterexample generators. |
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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{lore{\color{white}NOSPAM}\kern-\wd\boxA{}nz.panny@\allowbreak |
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\allowbreak tum.\allowbreak de}} |
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\newcommand\authoremailiii{\texttt{pope{\color{white}NOSPAM}\kern-\wd\boxA{}scua@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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\newcommand\authoremailiv{\texttt{tray{\color{white}NOSPAM}\kern-\wd\boxA{}tel@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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The commands @{command datatype_new} and @{command primrec_new} are expected to |
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replace \keyw{datatype} and \keyw{primrec} in a future release. Authors of new |
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theories are encouraged to use the new commands, and maintainers of older |
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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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and \authoremailiv. |
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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 |
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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 | Suc 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_Suc odd_nat |
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and odd_nat = Odd_Suc 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) fn = Fn "'a \<Rightarrow> 'b" |
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datatype_new 'a also_wrong = W1 | W2 (*<*)'a |
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typ (*>*)"('a also_wrong, 'a) fn" |
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text {* |
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\noindent |
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This 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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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) fn"}, the type variable @{typ 'a} is dead and @{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:registering-bounded-natural-functors} explains how to register |
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arbitrary type constructors as BNFs. |
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Here is another example that fails: |
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*} |
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datatype_new 'a pow_list = PNil 'a (*<*)'a |
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datatype_new 'a pow_list' = PNil' 'a (*>*)| PCons "('a * 'a) pow_list" |
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text {* |
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\noindent |
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This one features a different flavor of nesting, where the recursive call in the |
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type specification occurs around (rather than inside) another type constructor. |
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*} |
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subsubsection {* Auxiliary Constants and Properties |
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\label{sssec:datatype-auxiliary-constants-and-properties} *} |
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text {* |
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The @{command datatype_new} command introduces various constants in addition to |
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the constructors. With each datatype are associated set functions, a map |
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function, a relator, discriminators, and selectors, all of which can be given |
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custom names. In the example below, the familiar names @{text null}, @{text hd}, |
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@{text tl}, @{text set}, @{text map}, and @{text list_all2}, override the |
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default names @{text is_Nil}, @{text un_Cons1}, @{text un_Cons2}, |
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@{text set_list}, @{text map_list}, and @{text rel_list}: |
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*} |
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(*<*) |
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no_translations |
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"[x, xs]" == "x # [xs]" |
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"[x]" == "x # []" |
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no_notation |
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Nil ("[]") and |
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Cons (infixr "#" 65) |
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hide_type list |
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hide_const Nil Cons hd tl set map list_all2 |
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context early begin |
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(*>*) |
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datatype_new (set: 'a) list (map: map rel: list_all2) = |
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null: Nil (defaults tl: Nil) |
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| Cons (hd: 'a) (tl: "'a list") |
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text {* |
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\noindent |
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\begin{tabular}{@ {}ll@ {}} |
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Constructors: & |
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@{text "Nil \<Colon> 'a list"} \\ |
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& |
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@{text "Cons \<Colon> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list"} \\ |
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Discriminator: & |
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@{text "null \<Colon> 'a list \<Rightarrow> bool"} \\ |
|
379 |
Selectors: & |
|
380 |
@{text "hd \<Colon> 'a list \<Rightarrow> 'a"} \\ |
|
381 |
& |
|
382 |
@{text "tl \<Colon> 'a list \<Rightarrow> 'a list"} \\ |
|
383 |
Set function: & |
|
384 |
@{text "set \<Colon> 'a list \<Rightarrow> 'a set"} \\ |
|
385 |
Map function: & |
|
386 |
@{text "map \<Colon> ('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list"} \\ |
|
387 |
Relator: & |
|
388 |
@{text "list_all2 \<Colon> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"} |
|
389 |
\end{tabular} |
|
390 |
||
391 |
The discriminator @{const null} and the selectors @{const hd} and @{const tl} |
|
392 |
are characterized as follows: |
|
52822 | 393 |
% |
53025 | 394 |
\[@{thm list.collapse(1)[of xs, no_vars]} |
395 |
\qquad @{thm list.collapse(2)[of xs, no_vars]}\] |
|
52822 | 396 |
% |
54187 | 397 |
For two-constructor datatypes, a single discriminator constant is sufficient. |
398 |
The discriminator associated with @{const Cons} is simply |
|
53491 | 399 |
@{term "\<lambda>xs. \<not> null xs"}. |
52822 | 400 |
|
53553 | 401 |
The @{text defaults} clause following the @{const Nil} constructor specifies a |
402 |
default value for selectors associated with other constructors. Here, it is used |
|
403 |
to ensure that the tail of the empty list is itself (instead of being left |
|
53535 | 404 |
unspecified). |
52822 | 405 |
|
53617 | 406 |
Because @{const Nil} is nullary, it is also possible to use |
53491 | 407 |
@{term "\<lambda>xs. xs = Nil"} as a discriminator. This is specified by |
53534 | 408 |
entering ``@{text "="}'' instead of the identifier @{const null}. Although this |
53535 | 409 |
may look appealing, the mixture of constructors and selectors in the |
53534 | 410 |
characteristic theorems can lead Isabelle's automation to switch between the |
411 |
constructor and the destructor view in surprising ways. |
|
52822 | 412 |
|
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|
413 |
The usual mixfix syntax annotations are available for both types and |
c7364dca96f2
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|
414 |
constructors. For example: |
52805 | 415 |
*} |
52794 | 416 |
|
53025 | 417 |
(*<*) |
418 |
end |
|
419 |
(*>*) |
|
53552 | 420 |
datatype_new ('a, 'b) prod (infixr "*" 20) = Pair 'a 'b |
421 |
||
422 |
text {* \blankline *} |
|
52822 | 423 |
|
52841 | 424 |
datatype_new (set: 'a) list (map: map rel: list_all2) = |
52822 | 425 |
null: Nil ("[]") |
52841 | 426 |
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65) |
427 |
||
428 |
text {* |
|
53535 | 429 |
\noindent |
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|
430 |
Incidentally, this is how the traditional syntax can be set up: |
52841 | 431 |
*} |
432 |
||
433 |
syntax "_list" :: "args \<Rightarrow> 'a list" ("[(_)]") |
|
434 |
||
53552 | 435 |
text {* \blankline *} |
436 |
||
52841 | 437 |
translations |
438 |
"[x, xs]" == "x # [xs]" |
|
439 |
"[x]" == "x # []" |
|
52822 | 440 |
|
52824 | 441 |
|
53617 | 442 |
subsection {* Command Syntax |
443 |
\label{ssec:datatype-command-syntax} *} |
|
444 |
||
445 |
||
53621 | 446 |
subsubsection {* \keyw{datatype\_new} |
447 |
\label{sssec:datatype-new} *} |
|
52794 | 448 |
|
52822 | 449 |
text {* |
53829 | 450 |
\begin{matharray}{rcl} |
451 |
@{command_def "datatype_new"} & : & @{text "local_theory \<rightarrow> local_theory"} |
|
452 |
\end{matharray} |
|
52822 | 453 |
|
52824 | 454 |
@{rail " |
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|
455 |
@@{command datatype_new} target? @{syntax dt_options}? \<newline> |
52824 | 456 |
(@{syntax dt_name} '=' (@{syntax ctor} + '|') + @'and') |
52828 | 457 |
; |
54626 | 458 |
@{syntax_def dt_options}: '(' (('no_discs_sels' | 'no_code' | 'rep_compat') + ',') ')' |
52824 | 459 |
"} |
460 |
||
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|
461 |
The syntactic entity \synt{target} can be used to specify a local |
53534 | 462 |
context---e.g., @{text "(in linorder)"}. It is documented in the Isar reference |
463 |
manual \cite{isabelle-isar-ref}. |
|
464 |
% |
|
54832 | 465 |
The optional target is potentially followed by datatype-specific options: |
52822 | 466 |
|
52824 | 467 |
\begin{itemize} |
468 |
\setlength{\itemsep}{0pt} |
|
469 |
||
470 |
\item |
|
53623 | 471 |
The @{text "no_discs_sels"} option indicates that no discriminators or selectors |
53543 | 472 |
should be generated. |
52822 | 473 |
|
52824 | 474 |
\item |
54626 | 475 |
The @{text "no_code"} option indicates that the datatype should not be |
476 |
registered for code generation. |
|
477 |
||
478 |
\item |
|
53644 | 479 |
The @{text "rep_compat"} option indicates that the generated names should |
480 |
contain optional (and normally not displayed) ``@{text "new."}'' components to |
|
481 |
prevent clashes with a later call to \keyw{rep\_datatype}. See |
|
52824 | 482 |
Section~\ref{ssec:datatype-compatibility-issues} for details. |
483 |
\end{itemize} |
|
52822 | 484 |
|
52827 | 485 |
The left-hand sides of the datatype equations specify the name of the type to |
53534 | 486 |
define, its type parameters, and additional information: |
52822 | 487 |
|
52824 | 488 |
@{rail " |
53534 | 489 |
@{syntax_def dt_name}: @{syntax tyargs}? name @{syntax map_rel}? mixfix? |
52824 | 490 |
; |
53534 | 491 |
@{syntax_def tyargs}: typefree | '(' ((name ':')? typefree + ',') ')' |
52824 | 492 |
; |
53534 | 493 |
@{syntax_def map_rel}: '(' ((('map' | 'rel') ':' name) +) ')' |
52824 | 494 |
"} |
52822 | 495 |
|
52827 | 496 |
\noindent |
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|
497 |
The syntactic entity \synt{name} denotes an identifier, \synt{typefree} |
53534 | 498 |
denotes fixed type variable (@{typ 'a}, @{typ 'b}, \ldots), and \synt{mixfix} |
499 |
denotes the usual parenthesized mixfix notation. They are documented in the Isar |
|
500 |
reference manual \cite{isabelle-isar-ref}. |
|
52822 | 501 |
|
52827 | 502 |
The optional names preceding the type variables allow to override the default |
54491 | 503 |
names of the set functions (@{text set1_t}, \ldots, @{text setM_t}). |
53647 | 504 |
Inside a mutually recursive specification, all defined datatypes must |
505 |
mention exactly the same type variables in the same order. |
|
52822 | 506 |
|
52824 | 507 |
@{rail " |
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|
508 |
@{syntax_def ctor}: (name ':')? name (@{syntax ctor_arg} * ) \<newline> |
53534 | 509 |
@{syntax dt_sel_defaults}? mixfix? |
52824 | 510 |
"} |
511 |
||
53535 | 512 |
\medskip |
513 |
||
52827 | 514 |
\noindent |
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|
515 |
The main constituents of a constructor specification are the name of the |
52827 | 516 |
constructor and the list of its argument types. An optional discriminator name |
53554 | 517 |
can be supplied at the front to override the default name |
518 |
(@{text t.is_C\<^sub>j}). |
|
52822 | 519 |
|
52824 | 520 |
@{rail " |
53534 | 521 |
@{syntax_def ctor_arg}: type | '(' name ':' type ')' |
52827 | 522 |
"} |
523 |
||
53535 | 524 |
\medskip |
525 |
||
52827 | 526 |
\noindent |
527 |
In addition to the type of a constructor argument, it is possible to specify a |
|
528 |
name for the corresponding selector to override the default name |
|
53554 | 529 |
(@{text un_C\<^sub>ji}). The same selector names can be reused for several |
530 |
constructors as long as they share the same type. |
|
52827 | 531 |
|
532 |
@{rail " |
|
53621 | 533 |
@{syntax_def dt_sel_defaults}: '(' 'defaults' (name ':' term +) ')' |
52824 | 534 |
"} |
52827 | 535 |
|
536 |
\noindent |
|
537 |
Given a constructor |
|
538 |
@{text "C \<Colon> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<Rightarrow> \<sigma>\<^sub>p \<Rightarrow> \<sigma>"}, |
|
539 |
default values can be specified for any selector |
|
540 |
@{text "un_D \<Colon> \<sigma> \<Rightarrow> \<tau>"} |
|
53535 | 541 |
associated with other constructors. The specified default value must be of type |
52828 | 542 |
@{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<Rightarrow> \<sigma>\<^sub>p \<Rightarrow> \<tau>"} |
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|
543 |
(i.e., it may depend on @{text C}'s arguments). |
52822 | 544 |
*} |
545 |
||
53617 | 546 |
|
53621 | 547 |
subsubsection {* \keyw{datatype\_new\_compat} |
548 |
\label{sssec:datatype-new-compat} *} |
|
53617 | 549 |
|
550 |
text {* |
|
53829 | 551 |
\begin{matharray}{rcl} |
552 |
@{command_def "datatype_new_compat"} & : & @{text "local_theory \<rightarrow> local_theory"} |
|
553 |
\end{matharray} |
|
554 |
||
555 |
@{rail " |
|
556 |
@@{command datatype_new_compat} names |
|
557 |
"} |
|
558 |
||
559 |
\noindent |
|
53621 | 560 |
The old datatype package provides some functionality that is not yet replicated |
561 |
in the new package: |
|
562 |
||
563 |
\begin{itemize} |
|
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|
564 |
\setlength{\itemsep}{0pt} |
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|
565 |
|
53621 | 566 |
\item It is integrated with \keyw{fun} and \keyw{function} |
567 |
\cite{isabelle-function}, Nitpick \cite{isabelle-nitpick}, Quickcheck, |
|
568 |
and other packages. |
|
569 |
||
570 |
\item It is extended by various add-ons, notably to produce instances of the |
|
571 |
@{const size} function. |
|
572 |
\end{itemize} |
|
573 |
||
574 |
\noindent |
|
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diff
changeset
|
575 |
New-style datatypes can in most cases be registered as old-style datatypes using |
53829 | 576 |
@{command datatype_new_compat}. The \textit{names} argument is a space-separated |
577 |
list of type names that are mutually recursive. For example: |
|
53621 | 578 |
*} |
579 |
||
53623 | 580 |
datatype_new_compat even_nat odd_nat |
53621 | 581 |
|
582 |
text {* \blankline *} |
|
583 |
||
53623 | 584 |
thm even_nat_odd_nat.size |
53621 | 585 |
|
586 |
text {* \blankline *} |
|
587 |
||
53623 | 588 |
ML {* Datatype_Data.get_info @{theory} @{type_name even_nat} *} |
53621 | 589 |
|
590 |
text {* |
|
53748 | 591 |
A few remarks concern nested recursive datatypes only: |
592 |
||
593 |
\begin{itemize} |
|
53749
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adapted primcorec documentation to reflect the three views
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changeset
|
594 |
\setlength{\itemsep}{0pt} |
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blanchet
parents:
53748
diff
changeset
|
595 |
|
53748 | 596 |
\item The old-style, nested-as-mutual induction rule, iterator theorems, and |
597 |
recursor theorems are generated under their usual names but with ``@{text |
|
598 |
"compat_"}'' prefixed (e.g., @{text compat_tree.induct}). |
|
599 |
||
600 |
\item All types through which recursion takes place must be new-style datatypes |
|
601 |
or the function type. In principle, it should be possible to support old-style |
|
602 |
datatypes as well, but the command does not support this yet (and there is |
|
603 |
currently no way to register old-style datatypes as new-style datatypes). |
|
54184 | 604 |
|
605 |
\item The recursor produced for types that recurse through functions has a |
|
606 |
different signature than with the old package. This makes it impossible to use |
|
607 |
the old \keyw{primrec} command. |
|
53748 | 608 |
\end{itemize} |
609 |
||
610 |
An alternative to @{command datatype_new_compat} is to use the old package's |
|
611 |
\keyw{rep\_datatype} command. The associated proof obligations must then be |
|
612 |
discharged manually. |
|
53617 | 613 |
*} |
614 |
||
615 |
||
616 |
subsection {* Generated Constants |
|
617 |
\label{ssec:datatype-generated-constants} *} |
|
618 |
||
619 |
text {* |
|
53623 | 620 |
Given a datatype @{text "('a\<^sub>1, \<dots>, 'a\<^sub>m) t"} |
53617 | 621 |
with $m > 0$ live type variables and $n$ constructors |
622 |
@{text "t.C\<^sub>1"}, \ldots, @{text "t.C\<^sub>n"}, the |
|
623 |
following auxiliary constants are introduced: |
|
624 |
||
625 |
\begin{itemize} |
|
626 |
\setlength{\itemsep}{0pt} |
|
627 |
||
54494 | 628 |
\item \relax{Case combinator}: @{text t.case_t} (rendered using the familiar |
53617 | 629 |
@{text case}--@{text of} syntax) |
630 |
||
631 |
\item \relax{Discriminators}: @{text "t.is_C\<^sub>1"}, \ldots, |
|
632 |
@{text "t.is_C\<^sub>n"} |
|
633 |
||
634 |
\item \relax{Selectors}: |
|
635 |
@{text t.un_C\<^sub>11}$, \ldots, @{text t.un_C\<^sub>1k\<^sub>1}, \\ |
|
636 |
\phantom{\relax{Selectors:}} \quad\vdots \\ |
|
637 |
\phantom{\relax{Selectors:}} @{text t.un_C\<^sub>n1}$, \ldots, @{text t.un_C\<^sub>nk\<^sub>n}. |
|
638 |
||
639 |
\item \relax{Set functions} (or \relax{natural transformations}): |
|
54491 | 640 |
@{text set1_t}, \ldots, @{text t.setm_t} |
641 |
||
642 |
\item \relax{Map function} (or \relax{functorial action}): @{text t.map_t} |
|
643 |
||
644 |
\item \relax{Relator}: @{text t.rel_t} |
|
645 |
||
54494 | 646 |
\item \relax{Iterator}: @{text t.fold_t} |
647 |
||
648 |
\item \relax{Recursor}: @{text t.rec_t} |
|
53617 | 649 |
|
650 |
\end{itemize} |
|
651 |
||
652 |
\noindent |
|
653 |
The case combinator, discriminators, and selectors are collectively called |
|
654 |
\emph{destructors}. The prefix ``@{text "t."}'' is an optional component of the |
|
54491 | 655 |
names and is normally hidden. |
53617 | 656 |
*} |
657 |
||
658 |
||
52840 | 659 |
subsection {* Generated Theorems |
660 |
\label{ssec:datatype-generated-theorems} *} |
|
52828 | 661 |
|
662 |
text {* |
|
53544 | 663 |
The characteristic theorems generated by @{command datatype_new} are grouped in |
53623 | 664 |
three broad categories: |
53535 | 665 |
|
53543 | 666 |
\begin{itemize} |
53749
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blanchet
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diff
changeset
|
667 |
\setlength{\itemsep}{0pt} |
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blanchet
parents:
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changeset
|
668 |
|
53543 | 669 |
\item The \emph{free constructor theorems} are properties about the constructors |
670 |
and destructors that can be derived for any freely generated type. Internally, |
|
53542 | 671 |
the derivation is performed by @{command wrap_free_constructors}. |
53535 | 672 |
|
53552 | 673 |
\item The \emph{functorial theorems} are properties of datatypes related to |
674 |
their BNF nature. |
|
675 |
||
676 |
\item The \emph{inductive theorems} are properties of datatypes related to |
|
53544 | 677 |
their inductive nature. |
53552 | 678 |
|
53543 | 679 |
\end{itemize} |
53535 | 680 |
|
681 |
\noindent |
|
53542 | 682 |
The full list of named theorems can be obtained as usual by entering the |
53543 | 683 |
command \keyw{print\_theorems} immediately after the datatype definition. |
53542 | 684 |
This list normally excludes low-level theorems that reveal internal |
53552 | 685 |
constructions. To make these accessible, add the line |
53542 | 686 |
*} |
53535 | 687 |
|
53542 | 688 |
declare [[bnf_note_all]] |
689 |
(*<*) |
|
690 |
declare [[bnf_note_all = false]] |
|
691 |
(*>*) |
|
53535 | 692 |
|
53552 | 693 |
text {* |
694 |
\noindent |
|
695 |
to the top of the theory file. |
|
696 |
*} |
|
53535 | 697 |
|
53621 | 698 |
subsubsection {* Free Constructor Theorems |
699 |
\label{sssec:free-constructor-theorems} *} |
|
53535 | 700 |
|
53543 | 701 |
(*<*) |
53837 | 702 |
consts nonnull :: 'a |
53543 | 703 |
(*>*) |
704 |
||
53535 | 705 |
text {* |
54621 | 706 |
The free constructor theorems are partitioned in three subgroups. The first |
707 |
subgroup of properties is concerned with the constructors. They are listed below |
|
708 |
for @{typ "'a list"}: |
|
53543 | 709 |
|
53552 | 710 |
\begin{indentblock} |
53543 | 711 |
\begin{description} |
53544 | 712 |
|
53642 | 713 |
\item[@{text "t."}\hthm{inject} @{text "[iff, induct_simp]"}\rm:] ~ \\ |
53544 | 714 |
@{thm list.inject[no_vars]} |
715 |
||
53642 | 716 |
\item[@{text "t."}\hthm{distinct} @{text "[simp, induct_simp]"}\rm:] ~ \\ |
53543 | 717 |
@{thm list.distinct(1)[no_vars]} \\ |
718 |
@{thm list.distinct(2)[no_vars]} |
|
719 |
||
53642 | 720 |
\item[@{text "t."}\hthm{exhaust} @{text "[cases t, case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
53543 | 721 |
@{thm list.exhaust[no_vars]} |
722 |
||
53642 | 723 |
\item[@{text "t."}\hthm{nchotomy}\rm:] ~ \\ |
53543 | 724 |
@{thm list.nchotomy[no_vars]} |
725 |
||
726 |
\end{description} |
|
53552 | 727 |
\end{indentblock} |
53543 | 728 |
|
729 |
\noindent |
|
53621 | 730 |
In addition, these nameless theorems are registered as safe elimination rules: |
731 |
||
732 |
\begin{indentblock} |
|
733 |
\begin{description} |
|
734 |
||
54386 | 735 |
\item[@{text "t."}\hthm{distinct {\upshape[}THEN notE}@{text ", elim!"}\hthm{\upshape]}\rm:] ~ \\ |
53621 | 736 |
@{thm list.distinct(1)[THEN notE, elim!, no_vars]} \\ |
737 |
@{thm list.distinct(2)[THEN notE, elim!, no_vars]} |
|
738 |
||
739 |
\end{description} |
|
740 |
\end{indentblock} |
|
741 |
||
742 |
\noindent |
|
53543 | 743 |
The next subgroup is concerned with the case combinator: |
744 |
||
53552 | 745 |
\begin{indentblock} |
53543 | 746 |
\begin{description} |
53544 | 747 |
|
53798 | 748 |
\item[@{text "t."}\hthm{case} @{text "[simp, code]"}\rm:] ~ \\ |
53543 | 749 |
@{thm list.case(1)[no_vars]} \\ |
750 |
@{thm list.case(2)[no_vars]} |
|
751 |
||
53642 | 752 |
\item[@{text "t."}\hthm{case\_cong}\rm:] ~ \\ |
53543 | 753 |
@{thm list.case_cong[no_vars]} |
754 |
||
53642 | 755 |
\item[@{text "t."}\hthm{weak\_case\_cong} @{text "[cong]"}\rm:] ~ \\ |
53543 | 756 |
@{thm list.weak_case_cong[no_vars]} |
757 |
||
53642 | 758 |
\item[@{text "t."}\hthm{split}\rm:] ~ \\ |
53543 | 759 |
@{thm list.split[no_vars]} |
760 |
||
53642 | 761 |
\item[@{text "t."}\hthm{split\_asm}\rm:] ~ \\ |
53543 | 762 |
@{thm list.split_asm[no_vars]} |
763 |
||
53544 | 764 |
\item[@{text "t."}\hthm{splits} = @{text "split split_asm"}] |
53543 | 765 |
|
766 |
\end{description} |
|
53552 | 767 |
\end{indentblock} |
53543 | 768 |
|
769 |
\noindent |
|
54621 | 770 |
The third subgroup revolves around discriminators and selectors: |
53543 | 771 |
|
53552 | 772 |
\begin{indentblock} |
53543 | 773 |
\begin{description} |
53544 | 774 |
|
53694 | 775 |
\item[@{text "t."}\hthm{disc} @{text "[simp]"}\rm:] ~ \\ |
776 |
@{thm list.disc(1)[no_vars]} \\ |
|
777 |
@{thm list.disc(2)[no_vars]} |
|
778 |
||
53703 | 779 |
\item[@{text "t."}\hthm{discI}\rm:] ~ \\ |
780 |
@{thm list.discI(1)[no_vars]} \\ |
|
781 |
@{thm list.discI(2)[no_vars]} |
|
782 |
||
53805 | 783 |
\item[@{text "t."}\hthm{sel} @{text "[simp, code]"}\rm:] ~ \\ |
53694 | 784 |
@{thm list.sel(1)[no_vars]} \\ |
785 |
@{thm list.sel(2)[no_vars]} |
|
53543 | 786 |
|
53642 | 787 |
\item[@{text "t."}\hthm{collapse} @{text "[simp]"}\rm:] ~ \\ |
53543 | 788 |
@{thm list.collapse(1)[no_vars]} \\ |
789 |
@{thm list.collapse(2)[no_vars]} |
|
790 |
||
53837 | 791 |
\item[@{text "t."}\hthm{disc\_exclude} @{text "[dest]"}\rm:] ~ \\ |
53543 | 792 |
These properties are missing for @{typ "'a list"} because there is only one |
793 |
proper discriminator. Had the datatype been introduced with a second |
|
53837 | 794 |
discriminator called @{const nonnull}, they would have read thusly: \\[\jot] |
795 |
@{prop "null list \<Longrightarrow> \<not> nonnull list"} \\ |
|
796 |
@{prop "nonnull list \<Longrightarrow> \<not> null list"} |
|
53543 | 797 |
|
53642 | 798 |
\item[@{text "t."}\hthm{disc\_exhaust} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
53543 | 799 |
@{thm list.disc_exhaust[no_vars]} |
800 |
||
53916 | 801 |
\item[@{text "t."}\hthm{sel\_exhaust} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
802 |
@{thm list.sel_exhaust[no_vars]} |
|
803 |
||
53642 | 804 |
\item[@{text "t."}\hthm{expand}\rm:] ~ \\ |
53543 | 805 |
@{thm list.expand[no_vars]} |
806 |
||
53917 | 807 |
\item[@{text "t."}\hthm{sel\_split}\rm:] ~ \\ |
808 |
@{thm list.sel_split[no_vars]} |
|
809 |
||
810 |
\item[@{text "t."}\hthm{sel\_split\_asm}\rm:] ~ \\ |
|
811 |
@{thm list.sel_split_asm[no_vars]} |
|
812 |
||
54491 | 813 |
\item[@{text "t."}\hthm{case\_eq\_if}\rm:] ~ \\ |
814 |
@{thm list.case_eq_if[no_vars]} |
|
53543 | 815 |
|
816 |
\end{description} |
|
53552 | 817 |
\end{indentblock} |
54152 | 818 |
|
819 |
\noindent |
|
820 |
In addition, equational versions of @{text t.disc} are registered with the @{text "[code]"} |
|
821 |
attribute. |
|
53552 | 822 |
*} |
823 |
||
824 |
||
53621 | 825 |
subsubsection {* Functorial Theorems |
826 |
\label{sssec:functorial-theorems} *} |
|
53552 | 827 |
|
828 |
text {* |
|
54621 | 829 |
The functorial theorems are partitioned in two subgroups. The first subgroup |
830 |
consists of properties involving the constructors and either a set function, the |
|
831 |
map function, or the relator: |
|
53552 | 832 |
|
833 |
\begin{indentblock} |
|
834 |
\begin{description} |
|
835 |
||
53798 | 836 |
\item[@{text "t."}\hthm{set} @{text "[simp, code]"}\rm:] ~ \\ |
53694 | 837 |
@{thm list.set(1)[no_vars]} \\ |
838 |
@{thm list.set(2)[no_vars]} |
|
53552 | 839 |
|
53798 | 840 |
\item[@{text "t."}\hthm{map} @{text "[simp, code]"}\rm:] ~ \\ |
53552 | 841 |
@{thm list.map(1)[no_vars]} \\ |
842 |
@{thm list.map(2)[no_vars]} |
|
843 |
||
54146 | 844 |
\item[@{text "t."}\hthm{rel\_inject} @{text "[simp]"}\rm:] ~ \\ |
53552 | 845 |
@{thm list.rel_inject(1)[no_vars]} \\ |
846 |
@{thm list.rel_inject(2)[no_vars]} |
|
847 |
||
54146 | 848 |
\item[@{text "t."}\hthm{rel\_distinct} @{text "[simp]"}\rm:] ~ \\ |
53552 | 849 |
@{thm list.rel_distinct(1)[no_vars]} \\ |
850 |
@{thm list.rel_distinct(2)[no_vars]} |
|
851 |
||
852 |
\end{description} |
|
853 |
\end{indentblock} |
|
54146 | 854 |
|
855 |
\noindent |
|
856 |
In addition, equational versions of @{text t.rel_inject} and @{text |
|
857 |
rel_distinct} are registered with the @{text "[code]"} attribute. |
|
54621 | 858 |
|
859 |
The second subgroup consists of more abstract properties of the set functions, |
|
860 |
the map function, and the relator: |
|
861 |
||
862 |
\begin{indentblock} |
|
863 |
\begin{description} |
|
864 |
||
865 |
\item[@{text "t."}\hthm{map\_comp}\rm:] ~ \\ |
|
866 |
@{thm list.map_cong0[no_vars]} |
|
867 |
||
54624
36301c99ed26
revert making 'map_cong' a 'cong' -- it breaks too many proofs in the AFP
blanchet
parents:
54621
diff
changeset
|
868 |
\item[@{text "t."}\hthm{map\_cong} @{text "[fundef_cong]"}\rm:] ~ \\ |
54621 | 869 |
@{thm list.map_cong[no_vars]} |
870 |
||
871 |
\item[@{text "t."}\hthm{map\_id}\rm:] ~ \\ |
|
872 |
@{thm list.map_id[no_vars]} |
|
873 |
||
874 |
\item[@{text "t."}\hthm{rel\_compp}\rm:] ~ \\ |
|
875 |
@{thm list.rel_compp[no_vars]} |
|
876 |
||
877 |
\item[@{text "t."}\hthm{rel\_conversep}\rm:] ~ \\ |
|
878 |
@{thm list.rel_conversep[no_vars]} |
|
879 |
||
880 |
\item[@{text "t."}\hthm{rel\_eq}\rm:] ~ \\ |
|
881 |
@{thm list.rel_eq[no_vars]} |
|
882 |
||
883 |
\item[@{text "t."}\hthm{rel\_flip}\rm:] ~ \\ |
|
884 |
@{thm list.rel_flip[no_vars]} |
|
885 |
||
886 |
\item[@{text "t."}\hthm{rel\_mono}\rm:] ~ \\ |
|
887 |
@{thm list.rel_mono[no_vars]} |
|
888 |
||
889 |
\item[@{text "t."}\hthm{set\_map}\rm:] ~ \\ |
|
890 |
@{thm list.set_map[no_vars]} |
|
891 |
||
892 |
\end{description} |
|
893 |
\end{indentblock} |
|
53535 | 894 |
*} |
895 |
||
896 |
||
53621 | 897 |
subsubsection {* Inductive Theorems |
898 |
\label{sssec:inductive-theorems} *} |
|
53535 | 899 |
|
900 |
text {* |
|
53623 | 901 |
The inductive theorems are as follows: |
53544 | 902 |
|
53552 | 903 |
\begin{indentblock} |
53544 | 904 |
\begin{description} |
905 |
||
53642 | 906 |
\item[@{text "t."}\hthm{induct} @{text "[induct t, case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
53544 | 907 |
@{thm list.induct[no_vars]} |
908 |
||
53642 | 909 |
\item[@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
53544 | 910 |
Given $m > 1$ mutually recursive datatypes, this induction rule can be used to |
911 |
prove $m$ properties simultaneously. |
|
52828 | 912 |
|
53798 | 913 |
\item[@{text "t."}\hthm{fold} @{text "[simp, code]"}\rm:] ~ \\ |
53544 | 914 |
@{thm list.fold(1)[no_vars]} \\ |
915 |
@{thm list.fold(2)[no_vars]} |
|
916 |
||
53798 | 917 |
\item[@{text "t."}\hthm{rec} @{text "[simp, code]"}\rm:] ~ \\ |
53544 | 918 |
@{thm list.rec(1)[no_vars]} \\ |
919 |
@{thm list.rec(2)[no_vars]} |
|
920 |
||
921 |
\end{description} |
|
53552 | 922 |
\end{indentblock} |
53544 | 923 |
|
924 |
\noindent |
|
925 |
For convenience, @{command datatype_new} also provides the following collection: |
|
926 |
||
53552 | 927 |
\begin{indentblock} |
53544 | 928 |
\begin{description} |
929 |
||
930 |
\item[@{text "t."}\hthm{simps} = @{text t.inject} @{text t.distinct} @{text t.case} @{text t.rec} @{text t.fold} @{text t.map} @{text t.rel_inject}] ~ \\ |
|
53694 | 931 |
@{text t.rel_distinct} @{text t.set} |
53544 | 932 |
|
933 |
\end{description} |
|
53552 | 934 |
\end{indentblock} |
52828 | 935 |
*} |
936 |
||
52794 | 937 |
|
52827 | 938 |
subsection {* Compatibility Issues |
52824 | 939 |
\label{ssec:datatype-compatibility-issues} *} |
52794 | 940 |
|
52828 | 941 |
text {* |
53997 | 942 |
The command @{command datatype_new} has been designed to be highly compatible |
943 |
with the old \keyw{datatype}, to ease migration. There are nonetheless a few |
|
53647 | 944 |
incompatibilities that may arise when porting to the new package: |
945 |
||
946 |
\begin{itemize} |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
947 |
\setlength{\itemsep}{0pt} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
948 |
|
53647 | 949 |
\item \emph{The Standard ML interfaces are different.} Tools and extensions |
950 |
written to call the old ML interfaces will need to be adapted to the new |
|
951 |
interfaces. Little has been done so far in this direction. Whenever possible, it |
|
952 |
is recommended to use @{command datatype_new_compat} or \keyw{rep\_datatype} |
|
953 |
to register new-style datatypes as old-style datatypes. |
|
954 |
||
54537 | 955 |
\item \emph{The constants @{text t_case} and @{text t_rec} are now called |
956 |
@{text case_t} and @{text rec_t}.} |
|
957 |
||
958 |
\item \emph{The recursor @{text rec_t} has a different signature for nested |
|
54185 | 959 |
recursive datatypes.} In the old package, nested recursion through non-functions |
960 |
was internally reduced to mutual recursion. This reduction was visible in the |
|
961 |
type of the recursor, used by \keyw{primrec}. Recursion through functions was |
|
962 |
handled specially. In the new package, nested recursion (for functions and |
|
963 |
non-functions) is handled in a more modular fashion. The old-style recursor can |
|
964 |
be generated on demand using @{command primrec_new}, as explained in |
|
53647 | 965 |
Section~\ref{sssec:primrec-nested-as-mutual-recursion}, if the recursion is via |
966 |
new-style datatypes. |
|
967 |
||
54287 | 968 |
\item \emph{Accordingly, the induction rule is different for nested recursive |
969 |
datatypes.} Again, the old-style induction rule can be generated on demand using |
|
970 |
@{command primrec_new}, as explained in |
|
53647 | 971 |
Section~\ref{sssec:primrec-nested-as-mutual-recursion}, if the recursion is via |
972 |
new-style datatypes. |
|
52828 | 973 |
|
53863
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
974 |
\item \emph{The internal constructions are completely different.} Proof texts |
53647 | 975 |
that unfold the definition of constants introduced by \keyw{datatype} will be |
976 |
difficult to port. |
|
977 |
||
978 |
\item \emph{A few theorems have different names.} |
|
53997 | 979 |
The properties @{text t.cases} and @{text t.recs} have been renamed |
53647 | 980 |
@{text t.case} and @{text t.rec}. For non-mutually recursive datatypes, |
981 |
@{text t.inducts} is available as @{text t.induct}. |
|
982 |
For $m > 1$ mutually recursive datatypes, |
|
53997 | 983 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m.inducts(i)"} has been renamed |
53647 | 984 |
@{text "t\<^sub>i.induct"}. |
985 |
||
986 |
\item \emph{The @{text t.simps} collection has been extended.} |
|
987 |
Previously available theorems are available at the same index. |
|
988 |
||
989 |
\item \emph{Variables in generated properties have different names.} This is |
|
990 |
rarely an issue, except in proof texts that refer to variable names in the |
|
991 |
@{text "[where \<dots>]"} attribute. The solution is to use the more robust |
|
992 |
@{text "[of \<dots>]"} syntax. |
|
993 |
\end{itemize} |
|
994 |
||
995 |
In the other direction, there is currently no way to register old-style |
|
996 |
datatypes as new-style datatypes. If the goal is to define new-style datatypes |
|
997 |
with nested recursion through old-style datatypes, the old-style |
|
998 |
datatypes can be registered as a BNF |
|
999 |
(Section~\ref{sec:registering-bounded-natural-functors}). If the goal is |
|
1000 |
to derive discriminators and selectors, this can be achieved using @{command |
|
1001 |
wrap_free_constructors} |
|
1002 |
(Section~\ref{sec:deriving-destructors-and-theorems-for-free-constructors}). |
|
52828 | 1003 |
*} |
1004 |
||
52792 | 1005 |
|
52827 | 1006 |
section {* Defining Recursive Functions |
52805 | 1007 |
\label{sec:defining-recursive-functions} *} |
1008 |
||
1009 |
text {* |
|
54183 | 1010 |
Recursive functions over datatypes can be specified using the @{command |
1011 |
primrec_new} command, which supports primitive recursion, or using the more |
|
1012 |
general \keyw{fun} and \keyw{function} commands. Here, the focus is on @{command |
|
53644 | 1013 |
primrec_new}; the other two commands are described in a separate tutorial |
53646 | 1014 |
\cite{isabelle-function}. |
52828 | 1015 |
|
53621 | 1016 |
%%% TODO: partial_function |
52805 | 1017 |
*} |
52792 | 1018 |
|
52805 | 1019 |
|
53617 | 1020 |
subsection {* Introductory Examples |
1021 |
\label{ssec:primrec-introductory-examples} *} |
|
52828 | 1022 |
|
53646 | 1023 |
text {* |
1024 |
Primitive recursion is illustrated through concrete examples based on the |
|
1025 |
datatypes defined in Section~\ref{ssec:datatype-introductory-examples}. More |
|
1026 |
examples can be found in the directory \verb|~~/src/HOL/BNF/Examples|. |
|
1027 |
*} |
|
1028 |
||
53621 | 1029 |
|
1030 |
subsubsection {* Nonrecursive Types |
|
1031 |
\label{sssec:primrec-nonrecursive-types} *} |
|
52828 | 1032 |
|
52841 | 1033 |
text {* |
53621 | 1034 |
Primitive recursion removes one layer of constructors on the left-hand side in |
1035 |
each equation. For example: |
|
52841 | 1036 |
*} |
1037 |
||
1038 |
primrec_new bool_of_trool :: "trool \<Rightarrow> bool" where |
|
53621 | 1039 |
"bool_of_trool Faalse \<longleftrightarrow> False" | |
1040 |
"bool_of_trool Truue \<longleftrightarrow> True" |
|
52841 | 1041 |
|
53621 | 1042 |
text {* \blankline *} |
52841 | 1043 |
|
53025 | 1044 |
primrec_new the_list :: "'a option \<Rightarrow> 'a list" where |
1045 |
"the_list None = []" | |
|
1046 |
"the_list (Some a) = [a]" |
|
52841 | 1047 |
|
53621 | 1048 |
text {* \blankline *} |
1049 |
||
53025 | 1050 |
primrec_new the_default :: "'a \<Rightarrow> 'a option \<Rightarrow> 'a" where |
1051 |
"the_default d None = d" | |
|
1052 |
"the_default _ (Some a) = a" |
|
52843 | 1053 |
|
53621 | 1054 |
text {* \blankline *} |
1055 |
||
52841 | 1056 |
primrec_new mirrror :: "('a, 'b, 'c) triple \<Rightarrow> ('c, 'b, 'a) triple" where |
1057 |
"mirrror (Triple a b c) = Triple c b a" |
|
1058 |
||
53621 | 1059 |
text {* |
1060 |
\noindent |
|
1061 |
The equations can be specified in any order, and it is acceptable to leave out |
|
1062 |
some cases, which are then unspecified. Pattern matching on the left-hand side |
|
1063 |
is restricted to a single datatype, which must correspond to the same argument |
|
1064 |
in all equations. |
|
1065 |
*} |
|
52828 | 1066 |
|
53621 | 1067 |
|
1068 |
subsubsection {* Simple Recursion |
|
1069 |
\label{sssec:primrec-simple-recursion} *} |
|
52828 | 1070 |
|
52841 | 1071 |
text {* |
53621 | 1072 |
For simple recursive types, recursive calls on a constructor argument are |
1073 |
allowed on the right-hand side: |
|
52841 | 1074 |
*} |
1075 |
||
53330
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1076 |
primrec_new replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where |
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1077 |
"replicate Zero _ = []" | |
53644 | 1078 |
"replicate (Suc n) x = x # replicate n x" |
52841 | 1079 |
|
53621 | 1080 |
text {* \blankline *} |
52843 | 1081 |
|
53332 | 1082 |
primrec_new at :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" where |
53644 | 1083 |
"at (x # xs) j = |
52843 | 1084 |
(case j of |
53644 | 1085 |
Zero \<Rightarrow> x |
1086 |
| Suc j' \<Rightarrow> at xs j')" |
|
52843 | 1087 |
|
53621 | 1088 |
text {* \blankline *} |
1089 |
||
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1090 |
primrec_new (*<*)(in early) (*>*)tfold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) tlist \<Rightarrow> 'b" where |
53644 | 1091 |
"tfold _ (TNil y) = y" | |
1092 |
"tfold f (TCons x xs) = f x (tfold f xs)" |
|
52841 | 1093 |
|
53025 | 1094 |
text {* |
53621 | 1095 |
\noindent |
54402 | 1096 |
Pattern matching is only available for the argument on which the recursion takes |
1097 |
place. Fortunately, it is easy to generate pattern-maching equations using the |
|
1098 |
\keyw{simps\_of\_case} command provided by the theory |
|
1099 |
\verb|~~/src/HOL/Library/Simps_Case_Conv|. |
|
1100 |
*} |
|
1101 |
||
1102 |
simps_of_case at_simps: at.simps |
|
1103 |
||
1104 |
text {* |
|
1105 |
This generates the lemma collection @{thm [source] at_simps}: |
|
1106 |
% |
|
1107 |
\[@{thm at_simps(1)[no_vars]} |
|
1108 |
\qquad @{thm at_simps(2)[no_vars]}\] |
|
1109 |
% |
|
54184 | 1110 |
The next example is defined using \keyw{fun} to escape the syntactic |
1111 |
restrictions imposed on primitive recursive functions. The |
|
1112 |
@{command datatype_new_compat} command is needed to register new-style datatypes |
|
1113 |
for use with \keyw{fun} and \keyw{function} |
|
53621 | 1114 |
(Section~\ref{sssec:datatype-new-compat}): |
53025 | 1115 |
*} |
52828 | 1116 |
|
53621 | 1117 |
datatype_new_compat nat |
1118 |
||
1119 |
text {* \blankline *} |
|
1120 |
||
1121 |
fun at_least_two :: "nat \<Rightarrow> bool" where |
|
1122 |
"at_least_two (Suc (Suc _)) \<longleftrightarrow> True" | |
|
1123 |
"at_least_two _ \<longleftrightarrow> False" |
|
1124 |
||
1125 |
||
1126 |
subsubsection {* Mutual Recursion |
|
1127 |
\label{sssec:primrec-mutual-recursion} *} |
|
52828 | 1128 |
|
52841 | 1129 |
text {* |
53621 | 1130 |
The syntax for mutually recursive functions over mutually recursive datatypes |
1131 |
is straightforward: |
|
52841 | 1132 |
*} |
1133 |
||
1134 |
primrec_new |
|
53623 | 1135 |
nat_of_even_nat :: "even_nat \<Rightarrow> nat" and |
1136 |
nat_of_odd_nat :: "odd_nat \<Rightarrow> nat" |
|
52841 | 1137 |
where |
53623 | 1138 |
"nat_of_even_nat Even_Zero = Zero" | |
1139 |
"nat_of_even_nat (Even_Suc n) = Suc (nat_of_odd_nat n)" | |
|
1140 |
"nat_of_odd_nat (Odd_Suc n) = Suc (nat_of_even_nat n)" |
|
52841 | 1141 |
|
53752 | 1142 |
text {* \blankline *} |
1143 |
||
52841 | 1144 |
primrec_new |
53330
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1145 |
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
|
1146 |
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
|
1147 |
eval\<^sub>f :: "('a \<Rightarrow> int) \<Rightarrow> ('b \<Rightarrow> int) \<Rightarrow> ('a, 'b) fct \<Rightarrow> int" |
52841 | 1148 |
where |
1149 |
"eval\<^sub>e \<gamma> \<xi> (Term t) = eval\<^sub>t \<gamma> \<xi> t" | |
|
1150 |
"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
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parents:
53262
diff
changeset
|
1151 |
"eval\<^sub>t \<gamma> \<xi> (Factor f) = eval\<^sub>f \<gamma> \<xi> f" | |
52841 | 1152 |
"eval\<^sub>t \<gamma> \<xi> (Prod f t) = eval\<^sub>f \<gamma> \<xi> f + eval\<^sub>t \<gamma> \<xi> t" | |
1153 |
"eval\<^sub>f \<gamma> _ (Const a) = \<gamma> a" | |
|
1154 |
"eval\<^sub>f _ \<xi> (Var b) = \<xi> b" | |
|
1155 |
"eval\<^sub>f \<gamma> \<xi> (Expr e) = eval\<^sub>e \<gamma> \<xi> e" |
|
1156 |
||
53621 | 1157 |
text {* |
1158 |
\noindent |
|
53647 | 1159 |
Mutual recursion is possible within a single type, using \keyw{fun}: |
53621 | 1160 |
*} |
52828 | 1161 |
|
53621 | 1162 |
fun |
1163 |
even :: "nat \<Rightarrow> bool" and |
|
1164 |
odd :: "nat \<Rightarrow> bool" |
|
1165 |
where |
|
1166 |
"even Zero = True" | |
|
1167 |
"even (Suc n) = odd n" | |
|
1168 |
"odd Zero = False" | |
|
1169 |
"odd (Suc n) = even n" |
|
1170 |
||
1171 |
||
1172 |
subsubsection {* Nested Recursion |
|
1173 |
\label{sssec:primrec-nested-recursion} *} |
|
1174 |
||
1175 |
text {* |
|
1176 |
In a departure from the old datatype package, nested recursion is normally |
|
1177 |
handled via the map functions of the nesting type constructors. For example, |
|
1178 |
recursive calls are lifted to lists using @{const map}: |
|
1179 |
*} |
|
52828 | 1180 |
|
52843 | 1181 |
(*<*) |
53644 | 1182 |
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 | 1183 |
(*>*) |
53028 | 1184 |
primrec_new at\<^sub>f\<^sub>f :: "'a tree\<^sub>f\<^sub>f \<Rightarrow> nat list \<Rightarrow> 'a" where |
1185 |
"at\<^sub>f\<^sub>f (Node\<^sub>f\<^sub>f a ts) js = |
|
52843 | 1186 |
(case js of |
1187 |
[] \<Rightarrow> a |
|
53028 | 1188 |
| j # js' \<Rightarrow> at (map (\<lambda>t. at\<^sub>f\<^sub>f t js') ts) j)" |
52843 | 1189 |
|
53025 | 1190 |
text {* |
53647 | 1191 |
\noindent |
53621 | 1192 |
The next example features recursion through the @{text option} type. Although |
53623 | 1193 |
@{text option} is not a new-style datatype, it is registered as a BNF with the |
54491 | 1194 |
map function @{const map_option}: |
53025 | 1195 |
*} |
52843 | 1196 |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1197 |
primrec_new (*<*)(in early) (*>*)sum_btree :: "('a\<Colon>{zero,plus}) btree \<Rightarrow> 'a" where |
52843 | 1198 |
"sum_btree (BNode a lt rt) = |
54491 | 1199 |
a + the_default 0 (map_option sum_btree lt) + |
1200 |
the_default 0 (map_option sum_btree rt)" |
|
52843 | 1201 |
|
53136 | 1202 |
text {* |
53621 | 1203 |
\noindent |
1204 |
The same principle applies for arbitrary type constructors through which |
|
1205 |
recursion is possible. Notably, the map function for the function type |
|
1206 |
(@{text \<Rightarrow>}) is simply composition (@{text "op \<circ>"}): |
|
53136 | 1207 |
*} |
52828 | 1208 |
|
54182 | 1209 |
primrec_new (*<*)(in early) (*>*)relabel_ft :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where |
1210 |
"relabel_ft f (FTLeaf x) = FTLeaf (f x)" | |
|
1211 |
"relabel_ft f (FTNode g) = FTNode (relabel_ft f \<circ> g)" |
|
1212 |
||
1213 |
text {* |
|
1214 |
\noindent |
|
1215 |
For convenience, recursion through functions can also be expressed using |
|
1216 |
$\lambda$-abstractions and function application rather than through composition. |
|
1217 |
For example: |
|
1218 |
*} |
|
1219 |
||
1220 |
primrec_new relabel_ft :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where |
|
1221 |
"relabel_ft f (FTLeaf x) = FTLeaf (f x)" | |
|
1222 |
"relabel_ft f (FTNode g) = FTNode (\<lambda>x. relabel_ft f (g x))" |
|
52828 | 1223 |
|
54183 | 1224 |
text {* \blankline *} |
1225 |
||
1226 |
primrec_new subtree_ft :: "'a \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where |
|
1227 |
"subtree_ft x (FTNode g) = g x" |
|
1228 |
||
52843 | 1229 |
text {* |
53621 | 1230 |
\noindent |
54182 | 1231 |
For recursion through curried $n$-ary functions, $n$ applications of |
1232 |
@{term "op \<circ>"} are necessary. The examples below illustrate the case where |
|
1233 |
$n = 2$: |
|
53621 | 1234 |
*} |
1235 |
||
54182 | 1236 |
datatype_new 'a ftree2 = FTLeaf2 'a | FTNode2 "'a \<Rightarrow> 'a \<Rightarrow> 'a ftree2" |
1237 |
||
1238 |
text {* \blankline *} |
|
1239 |
||
1240 |
primrec_new (*<*)(in early) (*>*)relabel_ft2 :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree2 \<Rightarrow> 'a ftree2" where |
|
1241 |
"relabel_ft2 f (FTLeaf2 x) = FTLeaf2 (f x)" | |
|
1242 |
"relabel_ft2 f (FTNode2 g) = FTNode2 (op \<circ> (op \<circ> (relabel_ft2 f)) g)" |
|
1243 |
||
1244 |
text {* \blankline *} |
|
1245 |
||
1246 |
primrec_new relabel_ft2 :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree2 \<Rightarrow> 'a ftree2" where |
|
1247 |
"relabel_ft2 f (FTLeaf2 x) = FTLeaf2 (f x)" | |
|
1248 |
"relabel_ft2 f (FTNode2 g) = FTNode2 (\<lambda>x y. relabel_ft2 f (g x y))" |
|
54031 | 1249 |
|
54183 | 1250 |
text {* \blankline *} |
1251 |
||
1252 |
primrec_new subtree_ft2 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a ftree2 \<Rightarrow> 'a ftree2" where |
|
1253 |
"subtree_ft2 x y (FTNode2 g) = g x y" |
|
1254 |
||
53621 | 1255 |
|
1256 |
subsubsection {* Nested-as-Mutual Recursion |
|
53644 | 1257 |
\label{sssec:primrec-nested-as-mutual-recursion} *} |
53621 | 1258 |
|
53749
b37db925b663
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diff
changeset
|
1259 |
(*<*) |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1260 |
locale n2m begin |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1261 |
(*>*) |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1262 |
|
53621 | 1263 |
text {* |
1264 |
For compatibility with the old package, but also because it is sometimes |
|
1265 |
convenient in its own right, it is possible to treat nested recursive datatypes |
|
1266 |
as mutually recursive ones if the recursion takes place though new-style |
|
1267 |
datatypes. For example: |
|
52843 | 1268 |
*} |
1269 |
||
53331
20440c789759
prove theorem in the right context (that knows about local variables)
traytel
parents:
53330
diff
changeset
|
1270 |
primrec_new |
53647 | 1271 |
at\<^sub>f\<^sub>f :: "'a tree\<^sub>f\<^sub>f \<Rightarrow> nat list \<Rightarrow> 'a" and |
1272 |
ats\<^sub>f\<^sub>f :: "'a tree\<^sub>f\<^sub>f list \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> 'a" |
|
52843 | 1273 |
where |
53647 | 1274 |
"at\<^sub>f\<^sub>f (Node\<^sub>f\<^sub>f a ts) js = |
52843 | 1275 |
(case js of |
1276 |
[] \<Rightarrow> a |
|
53647 | 1277 |
| j # js' \<Rightarrow> ats\<^sub>f\<^sub>f ts j js')" | |
1278 |
"ats\<^sub>f\<^sub>f (t # ts) j = |
|
52843 | 1279 |
(case j of |
53647 | 1280 |
Zero \<Rightarrow> at\<^sub>f\<^sub>f t |
1281 |
| Suc j' \<Rightarrow> ats\<^sub>f\<^sub>f ts j')" |
|
52843 | 1282 |
|
53647 | 1283 |
text {* |
1284 |
\noindent |
|
54287 | 1285 |
Appropriate induction rules are generated as |
54031 | 1286 |
@{thm [source] at\<^sub>f\<^sub>f.induct}, |
1287 |
@{thm [source] ats\<^sub>f\<^sub>f.induct}, and |
|
54287 | 1288 |
@{thm [source] at\<^sub>f\<^sub>f_ats\<^sub>f\<^sub>f.induct}. The |
1289 |
induction rules and the underlying recursors are generated on a per-need basis |
|
1290 |
and are kept in a cache to speed up subsequent definitions. |
|
53647 | 1291 |
|
1292 |
Here is a second example: |
|
1293 |
*} |
|
53621 | 1294 |
|
53331
20440c789759
prove theorem in the right context (that knows about local variables)
traytel
parents:
53330
diff
changeset
|
1295 |
primrec_new |
53330
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1296 |
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
|
1297 |
sum_btree_option :: "'a btree option \<Rightarrow> 'a" |
52843 | 1298 |
where |
1299 |
"sum_btree (BNode a lt rt) = |
|
53025 | 1300 |
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
|
1301 |
"sum_btree_option None = 0" | |
53025 | 1302 |
"sum_btree_option (Some t) = sum_btree t" |
52843 | 1303 |
|
1304 |
text {* |
|
53621 | 1305 |
% * can pretend a nested type is mutually recursive (if purely inductive) |
1306 |
% * avoids the higher-order map |
|
1307 |
% * e.g. |
|
1308 |
||
53617 | 1309 |
% * this can always be avoided; |
1310 |
% * e.g. in our previous example, we first mapped the recursive |
|
1311 |
% calls, then we used a generic at function to retrieve the result |
|
1312 |
% |
|
1313 |
% * there's no hard-and-fast rule of when to use one or the other, |
|
1314 |
% just like there's no rule when to use fold and when to use |
|
1315 |
% primrec_new |
|
1316 |
% |
|
1317 |
% * higher-order approach, considering nesting as nesting, is more |
|
1318 |
% compositional -- e.g. we saw how we could reuse an existing polymorphic |
|
53647 | 1319 |
% at or the_default, whereas @{const ats\<^sub>f\<^sub>f} is much more specific |
53617 | 1320 |
% |
1321 |
% * but: |
|
1322 |
% * is perhaps less intuitive, because it requires higher-order thinking |
|
1323 |
% * may seem inefficient, and indeed with the code generator the |
|
1324 |
% mutually recursive version might be nicer |
|
1325 |
% * is somewhat indirect -- must apply a map first, then compute a result |
|
1326 |
% (cannot mix) |
|
53647 | 1327 |
% * the auxiliary functions like @{const ats\<^sub>f\<^sub>f} are sometimes useful in own right |
53617 | 1328 |
% |
1329 |
% * impact on automation unclear |
|
1330 |
% |
|
52843 | 1331 |
*} |
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1332 |
(*<*) |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1333 |
end |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1334 |
(*>*) |
52843 | 1335 |
|
52824 | 1336 |
|
53617 | 1337 |
subsection {* Command Syntax |
1338 |
\label{ssec:primrec-command-syntax} *} |
|
1339 |
||
1340 |
||
53621 | 1341 |
subsubsection {* \keyw{primrec\_new} |
1342 |
\label{sssec:primrec-new} *} |
|
52828 | 1343 |
|
1344 |
text {* |
|
53829 | 1345 |
\begin{matharray}{rcl} |
1346 |
@{command_def "primrec_new"} & : & @{text "local_theory \<rightarrow> local_theory"} |
|
1347 |
\end{matharray} |
|
52794 | 1348 |
|
52840 | 1349 |
@{rail " |
55029
61a6bf7d4b02
clarified @{rail} syntax: prefer explicit \<newline> symbol;
wenzelm
parents:
54958
diff
changeset
|
1350 |
@@{command primrec_new} target? fixes \<newline> @'where' (@{syntax pr_equation} + '|') |
52840 | 1351 |
; |
53829 | 1352 |
@{syntax_def pr_equation}: thmdecl? prop |
52840 | 1353 |
"} |
52828 | 1354 |
*} |
1355 |
||
52840 | 1356 |
|
53619 | 1357 |
(* |
52840 | 1358 |
subsection {* Generated Theorems |
1359 |
\label{ssec:primrec-generated-theorems} *} |
|
52824 | 1360 |
|
52828 | 1361 |
text {* |
53617 | 1362 |
% * synthesized nonrecursive definition |
1363 |
% * user specification is rederived from it, exactly as entered |
|
1364 |
% |
|
1365 |
% * induct |
|
1366 |
% * mutualized |
|
1367 |
% * without some needless induction hypotheses if not used |
|
1368 |
% * fold, rec |
|
1369 |
% * mutualized |
|
52828 | 1370 |
*} |
53619 | 1371 |
*) |
1372 |
||
52824 | 1373 |
|
52840 | 1374 |
subsection {* Recursive Default Values for Selectors |
53623 | 1375 |
\label{ssec:primrec-recursive-default-values-for-selectors} *} |
52827 | 1376 |
|
1377 |
text {* |
|
1378 |
A datatype selector @{text un_D} can have a default value for each constructor |
|
1379 |
on which it is not otherwise specified. Occasionally, it is useful to have the |
|
1380 |
default value be defined recursively. This produces a chicken-and-egg situation |
|
53621 | 1381 |
that may seem unsolvable, because the datatype is not introduced yet at the |
52827 | 1382 |
moment when the selectors are introduced. Of course, we can always define the |
1383 |
selectors manually afterward, but we then have to state and prove all the |
|
1384 |
characteristic theorems ourselves instead of letting the package do it. |
|
1385 |
||
1386 |
Fortunately, there is a fairly elegant workaround that relies on overloading and |
|
1387 |
that avoids the tedium of manual derivations: |
|
1388 |
||
1389 |
\begin{enumerate} |
|
1390 |
\setlength{\itemsep}{0pt} |
|
1391 |
||
1392 |
\item |
|
1393 |
Introduce a fully unspecified constant @{text "un_D\<^sub>0 \<Colon> 'a"} using |
|
1394 |
@{keyword consts}. |
|
1395 |
||
1396 |
\item |
|
53535 | 1397 |
Define the datatype, specifying @{text "un_D\<^sub>0"} as the selector's default |
1398 |
value. |
|
52827 | 1399 |
|
1400 |
\item |
|
53535 | 1401 |
Define the behavior of @{text "un_D\<^sub>0"} on values of the newly introduced |
1402 |
datatype using the \keyw{overloading} command. |
|
52827 | 1403 |
|
1404 |
\item |
|
1405 |
Derive the desired equation on @{text un_D} from the characteristic equations |
|
1406 |
for @{text "un_D\<^sub>0"}. |
|
1407 |
\end{enumerate} |
|
1408 |
||
53619 | 1409 |
\noindent |
52827 | 1410 |
The following example illustrates this procedure: |
1411 |
*} |
|
1412 |
||
1413 |
consts termi\<^sub>0 :: 'a |
|
1414 |
||
53619 | 1415 |
text {* \blankline *} |
1416 |
||
53491 | 1417 |
datatype_new ('a, 'b) tlist = |
52827 | 1418 |
TNil (termi: 'b) (defaults ttl: TNil) |
53491 | 1419 |
| TCons (thd: 'a) (ttl : "('a, 'b) tlist") (defaults termi: "\<lambda>_ xs. termi\<^sub>0 xs") |
52827 | 1420 |
|
53619 | 1421 |
text {* \blankline *} |
1422 |
||
52827 | 1423 |
overloading |
53491 | 1424 |
termi\<^sub>0 \<equiv> "termi\<^sub>0 \<Colon> ('a, 'b) tlist \<Rightarrow> 'b" |
52827 | 1425 |
begin |
53491 | 1426 |
primrec_new termi\<^sub>0 :: "('a, 'b) tlist \<Rightarrow> 'b" where |
53621 | 1427 |
"termi\<^sub>0 (TNil y) = y" | |
1428 |
"termi\<^sub>0 (TCons x xs) = termi\<^sub>0 xs" |
|
52827 | 1429 |
end |
1430 |
||
53619 | 1431 |
text {* \blankline *} |
1432 |
||
52827 | 1433 |
lemma terminal_TCons[simp]: "termi (TCons x xs) = termi xs" |
1434 |
by (cases xs) auto |
|
1435 |
||
1436 |
||
52828 | 1437 |
subsection {* Compatibility Issues |
53617 | 1438 |
\label{ssec:primrec-compatibility-issues} *} |
52828 | 1439 |
|
1440 |
text {* |
|
53997 | 1441 |
The command @{command primrec_new} has been designed to be highly compatible |
1442 |
with the old \keyw{primrec}, to ease migration. There is nonetheless at least |
|
1443 |
one incompatibility that may arise when porting to the new package: |
|
1444 |
||
1445 |
\begin{itemize} |
|
1446 |
\setlength{\itemsep}{0pt} |
|
1447 |
||
54185 | 1448 |
\item \emph{Some theorems have different names.} |
53997 | 1449 |
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
|
1450 |
@{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
|
1451 |
subcollections @{text "f\<^sub>i.simps"}. |
53997 | 1452 |
\end{itemize} |
52828 | 1453 |
*} |
52794 | 1454 |
|
1455 |
||
52827 | 1456 |
section {* Defining Codatatypes |
52805 | 1457 |
\label{sec:defining-codatatypes} *} |
1458 |
||
1459 |
text {* |
|
53829 | 1460 |
Codatatypes can be specified using the @{command codatatype} command. The |
53623 | 1461 |
command is first illustrated through concrete examples featuring different |
1462 |
flavors of corecursion. More examples can be found in the directory |
|
53997 | 1463 |
\verb|~~/src/HOL/|\allowbreak\verb|BNF/Examples|. The |
1464 |
\emph{Archive of Formal Proofs} also includes some useful codatatypes, notably |
|
1465 |
for lazy lists \cite{lochbihler-2010}. |
|
52805 | 1466 |
*} |
52792 | 1467 |
|
52824 | 1468 |
|
53617 | 1469 |
subsection {* Introductory Examples |
1470 |
\label{ssec:codatatype-introductory-examples} *} |
|
52794 | 1471 |
|
53623 | 1472 |
|
1473 |
subsubsection {* Simple Corecursion |
|
1474 |
\label{sssec:codatatype-simple-corecursion} *} |
|
1475 |
||
52805 | 1476 |
text {* |
53863
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
1477 |
Noncorecursive codatatypes coincide with the corresponding datatypes, so they |
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
1478 |
are useless in practice. \emph{Corecursive codatatypes} have the same syntax |
53623 | 1479 |
as recursive datatypes, except for the command name. For example, here is the |
1480 |
definition of lazy lists: |
|
1481 |
*} |
|
1482 |
||
1483 |
codatatype (lset: 'a) llist (map: lmap rel: llist_all2) = |
|
1484 |
lnull: LNil (defaults ltl: LNil) |
|
1485 |
| LCons (lhd: 'a) (ltl: "'a llist") |
|
1486 |
||
1487 |
text {* |
|
1488 |
\noindent |
|
1489 |
Lazy lists can be infinite, such as @{text "LCons 0 (LCons 0 (\<dots>))"} and |
|
53647 | 1490 |
@{text "LCons 0 (LCons 1 (LCons 2 (\<dots>)))"}. Here is a related type, that of |
1491 |
infinite streams: |
|
1492 |
*} |
|
1493 |
||
1494 |
codatatype (sset: 'a) stream (map: smap rel: stream_all2) = |
|
1495 |
SCons (shd: 'a) (stl: "'a stream") |
|
1496 |
||
1497 |
text {* |
|
1498 |
\noindent |
|
1499 |
Another interesting type that can |
|
53623 | 1500 |
be defined as a codatatype is that of the extended natural numbers: |
1501 |
*} |
|
1502 |
||
53644 | 1503 |
codatatype enat = EZero | ESuc enat |
53623 | 1504 |
|
1505 |
text {* |
|
1506 |
\noindent |
|
1507 |
This type has exactly one infinite element, @{text "ESuc (ESuc (ESuc (\<dots>)))"}, |
|
1508 |
that represents $\infty$. In addition, it has finite values of the form |
|
1509 |
@{text "ESuc (\<dots> (ESuc EZero)\<dots>)"}. |
|
53675 | 1510 |
|
1511 |
Here is an example with many constructors: |
|
52805 | 1512 |
*} |
53623 | 1513 |
|
53675 | 1514 |
codatatype 'a process = |
1515 |
Fail |
|
1516 |
| Skip (cont: "'a process") |
|
1517 |
| Action (prefix: 'a) (cont: "'a process") |
|
1518 |
| Choice (left: "'a process") (right: "'a process") |
|
1519 |
||
53750 | 1520 |
text {* |
53829 | 1521 |
\noindent |
53750 | 1522 |
Notice that the @{const cont} selector is associated with both @{const Skip} |
54146 | 1523 |
and @{const Action}. |
53750 | 1524 |
*} |
1525 |
||
53623 | 1526 |
|
1527 |
subsubsection {* Mutual Corecursion |
|
1528 |
\label{sssec:codatatype-mutual-corecursion} *} |
|
1529 |
||
1530 |
text {* |
|
1531 |
\noindent |
|
1532 |
The example below introduces a pair of \emph{mutually corecursive} types: |
|
1533 |
*} |
|
1534 |
||
1535 |
codatatype even_enat = Even_EZero | Even_ESuc odd_enat |
|
1536 |
and odd_enat = Odd_ESuc even_enat |
|
1537 |
||
1538 |
||
1539 |
subsubsection {* Nested Corecursion |
|
1540 |
\label{sssec:codatatype-nested-corecursion} *} |
|
1541 |
||
1542 |
text {* |
|
1543 |
\noindent |
|
53675 | 1544 |
The next examples feature \emph{nested corecursion}: |
53623 | 1545 |
*} |
1546 |
||
53644 | 1547 |
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 | 1548 |
|
53752 | 1549 |
text {* \blankline *} |
1550 |
||
53644 | 1551 |
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 | 1552 |
|
53752 | 1553 |
text {* \blankline *} |
1554 |
||
53675 | 1555 |
codatatype 'a state_machine = |
53751 | 1556 |
State_Machine (accept: bool) (trans: "'a \<Rightarrow> 'a state_machine") |
53675 | 1557 |
|
52824 | 1558 |
|
53617 | 1559 |
subsection {* Command Syntax |
1560 |
\label{ssec:codatatype-command-syntax} *} |
|
52805 | 1561 |
|
53619 | 1562 |
|
53621 | 1563 |
subsubsection {* \keyw{codatatype} |
1564 |
\label{sssec:codatatype} *} |
|
53619 | 1565 |
|
52824 | 1566 |
text {* |
53829 | 1567 |
\begin{matharray}{rcl} |
1568 |
@{command_def "codatatype"} & : & @{text "local_theory \<rightarrow> local_theory"} |
|
1569 |
\end{matharray} |
|
1570 |
||
1571 |
@{rail " |
|
55029
61a6bf7d4b02
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|
1572 |
@@{command codatatype} target? \<newline> |
53829 | 1573 |
(@{syntax dt_name} '=' (@{syntax ctor} + '|') + @'and') |
1574 |
"} |
|
1575 |
||
1576 |
\noindent |
|
52827 | 1577 |
Definitions of codatatypes have almost exactly the same syntax as for datatypes |
53829 | 1578 |
(Section~\ref{ssec:datatype-command-syntax}). The @{text "no_discs_sels"} option |
1579 |
is not available, because destructors are a crucial notion for codatatypes. |
|
53623 | 1580 |
*} |
1581 |
||
1582 |
||
1583 |
subsection {* Generated Constants |
|
1584 |
\label{ssec:codatatype-generated-constants} *} |
|
1585 |
||
1586 |
text {* |
|
1587 |
Given a codatatype @{text "('a\<^sub>1, \<dots>, 'a\<^sub>m) t"} |
|
1588 |
with $m > 0$ live type variables and $n$ constructors @{text "t.C\<^sub>1"}, |
|
1589 |
\ldots, @{text "t.C\<^sub>n"}, the same auxiliary constants are generated as for |
|
1590 |
datatypes (Section~\ref{ssec:datatype-generated-constants}), except that the |
|
1591 |
iterator and the recursor are replaced by dual concepts: |
|
1592 |
||
1593 |
\begin{itemize} |
|
1594 |
\setlength{\itemsep}{0pt} |
|
1595 |
||
54494 | 1596 |
\item \relax{Coiterator}: @{text unfold_t} |
1597 |
||
1598 |
\item \relax{Corecursor}: @{text corec_t} |
|
53623 | 1599 |
|
1600 |
\end{itemize} |
|
1601 |
*} |
|
1602 |
||
1603 |
||
1604 |
subsection {* Generated Theorems |
|
1605 |
\label{ssec:codatatype-generated-theorems} *} |
|
1606 |
||
1607 |
text {* |
|
53829 | 1608 |
The characteristic theorems generated by @{command codatatype} are grouped in |
53623 | 1609 |
three broad categories: |
1610 |
||
1611 |
\begin{itemize} |
|
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|
1612 |
\setlength{\itemsep}{0pt} |
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|
1613 |
|
53623 | 1614 |
\item The \emph{free constructor theorems} are properties about the constructors |
1615 |
and destructors that can be derived for any freely generated type. |
|
1616 |
||
1617 |
\item The \emph{functorial theorems} are properties of datatypes related to |
|
1618 |
their BNF nature. |
|
1619 |
||
1620 |
\item The \emph{coinductive theorems} are properties of datatypes related to |
|
1621 |
their coinductive nature. |
|
1622 |
\end{itemize} |
|
1623 |
||
1624 |
\noindent |
|
1625 |
The first two categories are exactly as for datatypes and are described in |
|
53642 | 1626 |
Sections |
1627 |
\ref{sssec:free-constructor-theorems}~and~\ref{sssec:functorial-theorems}. |
|
52824 | 1628 |
*} |
1629 |
||
53617 | 1630 |
|
53623 | 1631 |
subsubsection {* Coinductive Theorems |
1632 |
\label{sssec:coinductive-theorems} *} |
|
1633 |
||
1634 |
text {* |
|
54031 | 1635 |
The coinductive theorems are listed below for @{typ "'a llist"}: |
53623 | 1636 |
|
1637 |
\begin{indentblock} |
|
1638 |
\begin{description} |
|
1639 |
||
53643 | 1640 |
\item[\begin{tabular}{@ {}l@ {}} |
1641 |
@{text "t."}\hthm{coinduct} @{text "[coinduct t, consumes m, case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
|
1642 |
\phantom{@{text "t."}\hthm{coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> D\<^sub>n]"}\rm: |
|
1643 |
\end{tabular}] ~ \\ |
|
53623 | 1644 |
@{thm llist.coinduct[no_vars]} |
53617 | 1645 |
|
53643 | 1646 |
\item[\begin{tabular}{@ {}l@ {}} |
1647 |
@{text "t."}\hthm{strong\_coinduct} @{text "[consumes m, case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
|
1648 |
\phantom{@{text "t."}\hthm{strong\_coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> D\<^sub>n]"}\rm: |
|
1649 |
\end{tabular}] ~ \\ |
|
1650 |
@{thm llist.strong_coinduct[no_vars]} |
|
53617 | 1651 |
|
53643 | 1652 |
\item[\begin{tabular}{@ {}l@ {}} |
1653 |
@{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]"} \\ |
|
1654 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{strong\_coinduct} @{text "[case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
|
1655 |
\phantom{@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{strong\_coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> D\<^sub>n]"}\rm: |
|
1656 |
\end{tabular}] ~ \\ |
|
1657 |
Given $m > 1$ mutually corecursive codatatypes, these coinduction rules can be |
|
1658 |
used to prove $m$ properties simultaneously. |
|
1659 |
||
54031 | 1660 |
\item[@{text "t."}\hthm{unfold}\rm:] ~ \\ |
53623 | 1661 |
@{thm llist.unfold(1)[no_vars]} \\ |
1662 |
@{thm llist.unfold(2)[no_vars]} |
|
1663 |
||
54031 | 1664 |
\item[@{text "t."}\hthm{corec}\rm:] ~ \\ |
53623 | 1665 |
@{thm llist.corec(1)[no_vars]} \\ |
1666 |
@{thm llist.corec(2)[no_vars]} |
|
1667 |
||
53703 | 1668 |
\item[@{text "t."}\hthm{disc\_unfold}\rm:] ~ \\ |
53643 | 1669 |
@{thm llist.disc_unfold(1)[no_vars]} \\ |
1670 |
@{thm llist.disc_unfold(2)[no_vars]} |
|
1671 |
||
53703 | 1672 |
\item[@{text "t."}\hthm{disc\_corec}\rm:] ~ \\ |
53643 | 1673 |
@{thm llist.disc_corec(1)[no_vars]} \\ |
1674 |
@{thm llist.disc_corec(2)[no_vars]} |
|
1675 |
||
1676 |
\item[@{text "t."}\hthm{disc\_unfold\_iff} @{text "[simp]"}\rm:] ~ \\ |
|
1677 |
@{thm llist.disc_unfold_iff(1)[no_vars]} \\ |
|
1678 |
@{thm llist.disc_unfold_iff(2)[no_vars]} |
|
1679 |
||
1680 |
\item[@{text "t."}\hthm{disc\_corec\_iff} @{text "[simp]"}\rm:] ~ \\ |
|
1681 |
@{thm llist.disc_corec_iff(1)[no_vars]} \\ |
|
1682 |
@{thm llist.disc_corec_iff(2)[no_vars]} |
|
1683 |
||
1684 |
\item[@{text "t."}\hthm{sel\_unfold} @{text "[simp]"}\rm:] ~ \\ |
|
1685 |
@{thm llist.sel_unfold(1)[no_vars]} \\ |
|
1686 |
@{thm llist.sel_unfold(2)[no_vars]} |
|
1687 |
||
1688 |
\item[@{text "t."}\hthm{sel\_corec} @{text "[simp]"}\rm:] ~ \\ |
|
1689 |
@{thm llist.sel_corec(1)[no_vars]} \\ |
|
1690 |
@{thm llist.sel_corec(2)[no_vars]} |
|
1691 |
||
53623 | 1692 |
\end{description} |
1693 |
\end{indentblock} |
|
1694 |
||
1695 |
\noindent |
|
53829 | 1696 |
For convenience, @{command codatatype} also provides the following collection: |
53623 | 1697 |
|
1698 |
\begin{indentblock} |
|
1699 |
\begin{description} |
|
1700 |
||
54031 | 1701 |
\item[@{text "t."}\hthm{simps} = @{text t.inject} @{text t.distinct} @{text t.case} @{text t.disc_corec} @{text t.disc_corec_iff}] ~ \\ |
1702 |
@{text t.sel_corec} @{text t.disc_unfold} @{text t.disc_unfold_iff} @{text t.sel_unfold} @{text t.map} \\ |
|
1703 |
@{text t.rel_inject} @{text t.rel_distinct} @{text t.set} |
|
53623 | 1704 |
|
1705 |
\end{description} |
|
1706 |
\end{indentblock} |
|
1707 |
*} |
|
52805 | 1708 |
|
1709 |
||
52827 | 1710 |
section {* Defining Corecursive Functions |
52805 | 1711 |
\label{sec:defining-corecursive-functions} *} |
1712 |
||
1713 |
text {* |
|
54183 | 1714 |
Corecursive functions can be specified using the @{command primcorec} and |
1715 |
\keyw{prim\-corec\-ursive} commands, which support primitive corecursion, or |
|
1716 |
using the more general \keyw{partial\_function} command. Here, the focus is on |
|
1717 |
the first two. More examples can be found in the directory |
|
53753
ae7f50e70c09
renamed "primcorec" to "primcorecursive", to open the door to a 'theory -> theory' command called "primcorec" (cf. "fun" vs. "function")
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53752
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|
1718 |
\verb|~~/src/HOL/BNF/Examples|. |
53644 | 1719 |
|
53749
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adapted primcorec documentation to reflect the three views
blanchet
parents:
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|
1720 |
Whereas recursive functions consume datatypes one constructor at a time, |
b37db925b663
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blanchet
parents:
53748
diff
changeset
|
1721 |
corecursive functions construct codatatypes one constructor at a time. |
53752 | 1722 |
Partly reflecting a lack of agreement among proponents of coalgebraic methods, |
1723 |
Isabelle supports three competing syntaxes for specifying a function $f$: |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
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diff
changeset
|
1724 |
|
b37db925b663
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changeset
|
1725 |
\begin{itemize} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
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diff
changeset
|
1726 |
\setlength{\itemsep}{0pt} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
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53748
diff
changeset
|
1727 |
|
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
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changeset
|
1728 |
\abovedisplayskip=.5\abovedisplayskip |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1729 |
\belowdisplayskip=.5\belowdisplayskip |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
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diff
changeset
|
1730 |
|
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
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changeset
|
1731 |
\item The \emph{destructor view} specifies $f$ by implications of the form |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1732 |
\[@{text "\<dots> \<Longrightarrow> is_C\<^sub>j (f x\<^sub>1 \<dots> x\<^sub>n)"}\] and |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1733 |
equations of the form |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1734 |
\[@{text "un_C\<^sub>ji (f x\<^sub>1 \<dots> x\<^sub>n) = \<dots>"}\] |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
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changeset
|
1735 |
This style is popular in the coalgebraic literature. |
b37db925b663
adapted primcorec documentation to reflect the three views
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changeset
|
1736 |
|
b37db925b663
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|
1737 |
\item The \emph{constructor view} specifies $f$ by equations of the form |
54183 | 1738 |
\[@{text "\<dots> \<Longrightarrow> f x\<^sub>1 \<dots> x\<^sub>n = C\<^sub>j \<dots>"}\] |
53752 | 1739 |
This style is often more concise than the previous one. |
53749
b37db925b663
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blanchet
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changeset
|
1740 |
|
b37db925b663
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|
1741 |
\item The \emph{code view} specifies $f$ by a single equation of the form |
b37db925b663
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changeset
|
1742 |
\[@{text "f x\<^sub>1 \<dots> x\<^sub>n = \<dots>"}\] |
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|
1743 |
with restrictions on the format of the right-hand side. Lazy functional |
b37db925b663
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blanchet
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53748
diff
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|
1744 |
programming languages such as Haskell support a generalized version of this |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1745 |
style. |
b37db925b663
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diff
changeset
|
1746 |
\end{itemize} |
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|
1747 |
|
53753
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renamed "primcorec" to "primcorecursive", to open the door to a 'theory -> theory' command called "primcorec" (cf. "fun" vs. "function")
blanchet
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53752
diff
changeset
|
1748 |
All three styles are available as input syntax. Whichever syntax is chosen, |
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
|
1749 |
characteristic theorems for all three styles are generated. |
53749
b37db925b663
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changeset
|
1750 |
|
52828 | 1751 |
%%% TODO: partial_function? E.g. for defining tail recursive function on lazy |
1752 |
%%% lists (cf. terminal0 in TLList.thy) |
|
52805 | 1753 |
*} |
1754 |
||
52824 | 1755 |
|
53617 | 1756 |
subsection {* Introductory Examples |
1757 |
\label{ssec:primcorec-introductory-examples} *} |
|
52805 | 1758 |
|
53646 | 1759 |
text {* |
1760 |
Primitive corecursion is illustrated through concrete examples based on the |
|
1761 |
codatatypes defined in Section~\ref{ssec:codatatype-introductory-examples}. More |
|
53749
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|
1762 |
examples can be found in the directory \verb|~~/src/HOL/BNF/Examples|. The code |
b37db925b663
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blanchet
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53748
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|
1763 |
view is favored in the examples below. Sections |
b37db925b663
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diff
changeset
|
1764 |
\ref{ssec:primrec-constructor-view} and \ref{ssec:primrec-destructor-view} |
b37db925b663
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changeset
|
1765 |
present the same examples expressed using the constructor and destructor views. |
53646 | 1766 |
*} |
1767 |
||
53644 | 1768 |
subsubsection {* Simple Corecursion |
1769 |
\label{sssec:primcorec-simple-corecursion} *} |
|
1770 |
||
53646 | 1771 |
text {* |
53752 | 1772 |
Following the code view, corecursive calls are allowed on the right-hand side as |
1773 |
long as they occur under a constructor, which itself appears either directly to |
|
1774 |
the right of the equal sign or in a conditional expression: |
|
53646 | 1775 |
*} |
1776 |
||
53826 | 1777 |
primcorec literate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist" where |
54072 | 1778 |
"literate g x = LCons x (literate g (g x))" |
53647 | 1779 |
|
53677 | 1780 |
text {* \blankline *} |
1781 |
||
53826 | 1782 |
primcorec siterate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a stream" where |
54072 | 1783 |
"siterate g x = SCons x (siterate g (g x))" |
53644 | 1784 |
|
53646 | 1785 |
text {* |
1786 |
\noindent |
|
1787 |
The constructor ensures that progress is made---i.e., the function is |
|
53749
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|
1788 |
\emph{productive}. The above functions compute the infinite lazy list or stream |
54072 | 1789 |
@{text "[x, g x, g (g x), \<dots>]"}. Productivity guarantees that prefixes |
1790 |
@{text "[x, g x, g (g x), \<dots>, (g ^^ k) x]"} of arbitrary finite length |
|
53749
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adapted primcorec documentation to reflect the three views
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|
1791 |
@{text k} can be computed by unfolding the code equation a finite number of |
53863
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|
1792 |
times. |
53646 | 1793 |
|
53752 | 1794 |
Corecursive functions construct codatatype values, but nothing prevents them |
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|
1795 |
from also consuming such values. The following function drops every second |
53675 | 1796 |
element in a stream: |
1797 |
*} |
|
1798 |
||
53826 | 1799 |
primcorec every_snd :: "'a stream \<Rightarrow> 'a stream" where |
53675 | 1800 |
"every_snd s = SCons (shd s) (stl (stl s))" |
1801 |
||
1802 |
text {* |
|
53752 | 1803 |
\noindent |
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|
1804 |
Constructs such as @{text "let"}---@{text "in"}, @{text |
53646 | 1805 |
"if"}---@{text "then"}---@{text "else"}, and @{text "case"}---@{text "of"} may |
1806 |
appear around constructors that guard corecursive calls: |
|
1807 |
*} |
|
1808 |
||
54072 | 1809 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53644 | 1810 |
"lappend xs ys = |
1811 |
(case xs of |
|
1812 |
LNil \<Rightarrow> ys |
|
1813 |
| LCons x xs' \<Rightarrow> LCons x (lappend xs' ys))" |
|
1814 |
||
53646 | 1815 |
text {* |
53752 | 1816 |
\noindent |
54402 | 1817 |
Pattern matching is not supported by @{command primcorec}. Fortunately, it is |
1818 |
easy to generate pattern-maching equations using the \keyw{simps\_of\_case} |
|
1819 |
command provided by the theory \verb|~~/src/HOL/Library/Simps_Case_Conv|. |
|
1820 |
*} |
|
1821 |
||
1822 |
simps_of_case lappend_simps: lappend.code |
|
1823 |
||
1824 |
text {* |
|
1825 |
This generates the lemma collection @{thm [source] lappend_simps}: |
|
1826 |
% |
|
1827 |
\[@{thm lappend_simps(1)[no_vars]} |
|
1828 |
\qquad @{thm lappend_simps(2)[no_vars]}\] |
|
1829 |
% |
|
53646 | 1830 |
Corecursion is useful to specify not only functions but also infinite objects: |
1831 |
*} |
|
1832 |
||
53826 | 1833 |
primcorec infty :: enat where |
53644 | 1834 |
"infty = ESuc infty" |
1835 |
||
53646 | 1836 |
text {* |
53752 | 1837 |
\noindent |
1838 |
The example below constructs a pseudorandom process value. It takes a stream of |
|
53675 | 1839 |
actions (@{text s}), a pseudorandom function generator (@{text f}), and a |
1840 |
pseudorandom seed (@{text n}): |
|
1841 |
*} |
|
1842 |
||
54072 | 1843 |
primcorec |
53752 | 1844 |
random_process :: "'a stream \<Rightarrow> (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> 'a process" |
1845 |
where |
|
53675 | 1846 |
"random_process s f n = |
1847 |
(if n mod 4 = 0 then |
|
1848 |
Fail |
|
1849 |
else if n mod 4 = 1 then |
|
1850 |
Skip (random_process s f (f n)) |
|
1851 |
else if n mod 4 = 2 then |
|
1852 |
Action (shd s) (random_process (stl s) f (f n)) |
|
1853 |
else |
|
1854 |
Choice (random_process (every_snd s) (f \<circ> f) (f n)) |
|
1855 |
(random_process (every_snd (stl s)) (f \<circ> f) (f (f n))))" |
|
1856 |
||
1857 |
text {* |
|
1858 |
\noindent |
|
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|
1859 |
The main disadvantage of the code view is that the conditions are tested |
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1860 |
sequentially. This is visible in the generated theorems. The constructor and |
53752 | 1861 |
destructor views offer nonsequential alternatives. |
53675 | 1862 |
*} |
1863 |
||
53644 | 1864 |
|
1865 |
subsubsection {* Mutual Corecursion |
|
1866 |
\label{sssec:primcorec-mutual-corecursion} *} |
|
1867 |
||
53647 | 1868 |
text {* |
1869 |
The syntax for mutually corecursive functions over mutually corecursive |
|
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|
1870 |
datatypes is unsurprising: |
53647 | 1871 |
*} |
1872 |
||
53826 | 1873 |
primcorec |
53644 | 1874 |
even_infty :: even_enat and |
1875 |
odd_infty :: odd_enat |
|
1876 |
where |
|
1877 |
"even_infty = Even_ESuc odd_infty" | |
|
1878 |
"odd_infty = Odd_ESuc even_infty" |
|
1879 |
||
1880 |
||
1881 |
subsubsection {* Nested Corecursion |
|
1882 |
\label{sssec:primcorec-nested-corecursion} *} |
|
1883 |
||
53647 | 1884 |
text {* |
1885 |
The next pair of examples generalize the @{const literate} and @{const siterate} |
|
1886 |
functions (Section~\ref{sssec:primcorec-nested-corecursion}) to possibly |
|
1887 |
infinite trees in which subnodes are organized either as a lazy list (@{text |
|
54072 | 1888 |
tree\<^sub>i\<^sub>i}) or as a finite set (@{text tree\<^sub>i\<^sub>s}). They rely on the map functions of |
1889 |
the nesting type constructors to lift the corecursive calls: |
|
53647 | 1890 |
*} |
1891 |
||
53826 | 1892 |
primcorec iterate\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
54072 | 1893 |
"iterate\<^sub>i\<^sub>i g x = Node\<^sub>i\<^sub>i x (lmap (iterate\<^sub>i\<^sub>i g) (g x))" |
53644 | 1894 |
|
53677 | 1895 |
text {* \blankline *} |
1896 |
||
53826 | 1897 |
primcorec iterate\<^sub>i\<^sub>s :: "('a \<Rightarrow> 'a fset) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>s" where |
54072 | 1898 |
"iterate\<^sub>i\<^sub>s g x = Node\<^sub>i\<^sub>s x (fimage (iterate\<^sub>i\<^sub>s g) (g x))" |
53644 | 1899 |
|
52805 | 1900 |
text {* |
53752 | 1901 |
\noindent |
54072 | 1902 |
Both examples follow the usual format for constructor arguments associated |
1903 |
with nested recursive occurrences of the datatype. Consider |
|
1904 |
@{const iterate\<^sub>i\<^sub>i}. The term @{term "g x"} constructs an @{typ "'a llist"} |
|
1905 |
value, which is turned into an @{typ "'a tree\<^sub>i\<^sub>i llist"} value using |
|
1906 |
@{const lmap}. |
|
1907 |
||
1908 |
This format may sometimes feel artificial. The following function constructs |
|
1909 |
a tree with a single, infinite branch from a stream: |
|
1910 |
*} |
|
1911 |
||
1912 |
primcorec tree\<^sub>i\<^sub>i_of_stream :: "'a stream \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
|
1913 |
"tree\<^sub>i\<^sub>i_of_stream s = |
|
1914 |
Node\<^sub>i\<^sub>i (shd s) (lmap tree\<^sub>i\<^sub>i_of_stream (LCons (stl s) LNil))" |
|
1915 |
||
1916 |
text {* |
|
1917 |
\noindent |
|
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|
1918 |
A more natural syntax, also supported by Isabelle, is to move corecursive calls |
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|
1919 |
under constructors: |
54072 | 1920 |
*} |
1921 |
||
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|
1922 |
primcorec (*<*)(in late) (*>*)tree\<^sub>i\<^sub>i_of_stream :: "'a stream \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
54072 | 1923 |
"tree\<^sub>i\<^sub>i_of_stream s = Node\<^sub>i\<^sub>i (shd s) (LCons (tree\<^sub>i\<^sub>i_of_stream (stl s)) LNil)" |
1924 |
||
1925 |
text {* |
|
1926 |
The next example illustrates corecursion through functions, which is a bit |
|
1927 |
special. Deterministic finite automata (DFAs) are traditionally defined as |
|
1928 |
5-tuples @{text "(Q, \<Sigma>, \<delta>, q\<^sub>0, F)"}, where @{text Q} is a finite set of states, |
|
53675 | 1929 |
@{text \<Sigma>} is a finite alphabet, @{text \<delta>} is a transition function, @{text q\<^sub>0} |
1930 |
is an initial state, and @{text F} is a set of final states. The following |
|
1931 |
function translates a DFA into a @{type state_machine}: |
|
1932 |
*} |
|
1933 |
||
54071 | 1934 |
primcorec |
1935 |
(*<*)(in early) (*>*)sm_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> 'a state_machine" |
|
53752 | 1936 |
where |
54182 | 1937 |
"sm_of_dfa \<delta> F q = State_Machine (q \<in> F) (sm_of_dfa \<delta> F \<circ> \<delta> q)" |
53675 | 1938 |
|
53751 | 1939 |
text {* |
1940 |
\noindent |
|
1941 |
The map function for the function type (@{text \<Rightarrow>}) is composition |
|
54181 | 1942 |
(@{text "op \<circ>"}). For convenience, corecursion through functions can |
54182 | 1943 |
also be expressed using $\lambda$-abstractions and function application rather |
54031 | 1944 |
than through composition. For example: |
53751 | 1945 |
*} |
1946 |
||
53826 | 1947 |
primcorec |
53752 | 1948 |
sm_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> 'a state_machine" |
1949 |
where |
|
54182 | 1950 |
"sm_of_dfa \<delta> F q = State_Machine (q \<in> F) (\<lambda>a. sm_of_dfa \<delta> F (\<delta> q a))" |
53752 | 1951 |
|
1952 |
text {* \blankline *} |
|
1953 |
||
53826 | 1954 |
primcorec empty_sm :: "'a state_machine" where |
53752 | 1955 |
"empty_sm = State_Machine False (\<lambda>_. empty_sm)" |
53751 | 1956 |
|
53752 | 1957 |
text {* \blankline *} |
1958 |
||
53826 | 1959 |
primcorec not_sm :: "'a state_machine \<Rightarrow> 'a state_machine" where |
53752 | 1960 |
"not_sm M = State_Machine (\<not> accept M) (\<lambda>a. not_sm (trans M a))" |
53751 | 1961 |
|
53752 | 1962 |
text {* \blankline *} |
1963 |
||
53826 | 1964 |
primcorec |
53752 | 1965 |
or_sm :: "'a state_machine \<Rightarrow> 'a state_machine \<Rightarrow> 'a state_machine" |
1966 |
where |
|
54072 | 1967 |
"or_sm M N = State_Machine (accept M \<or> accept N) |
1968 |
(\<lambda>a. or_sm (trans M a) (trans N a))" |
|
53751 | 1969 |
|
54182 | 1970 |
text {* |
1971 |
\noindent |
|
1972 |
For recursion through curried $n$-ary functions, $n$ applications of |
|
1973 |
@{term "op \<circ>"} are necessary. The examples below illustrate the case where |
|
1974 |
$n = 2$: |
|
1975 |
*} |
|
1976 |
||
1977 |
codatatype ('a, 'b) state_machine2 = |
|
1978 |
State_Machine2 (accept2: bool) (trans2: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state_machine2") |
|
1979 |
||
1980 |
text {* \blankline *} |
|
1981 |
||
1982 |
primcorec |
|
1983 |
(*<*)(in early) (*>*)sm2_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> ('a, 'b) state_machine2" |
|
1984 |
where |
|
1985 |
"sm2_of_dfa \<delta> F q = State_Machine2 (q \<in> F) (op \<circ> (op \<circ> (sm2_of_dfa \<delta> F)) (\<delta> q))" |
|
1986 |
||
1987 |
text {* \blankline *} |
|
1988 |
||
1989 |
primcorec |
|
1990 |
sm2_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> ('a, 'b) state_machine2" |
|
1991 |
where |
|
1992 |
"sm2_of_dfa \<delta> F q = State_Machine2 (q \<in> F) (\<lambda>a b. sm2_of_dfa \<delta> F (\<delta> q a b))" |
|
1993 |
||
53644 | 1994 |
|
1995 |
subsubsection {* Nested-as-Mutual Corecursion |
|
1996 |
\label{sssec:primcorec-nested-as-mutual-corecursion} *} |
|
1997 |
||
53647 | 1998 |
text {* |
1999 |
Just as it is possible to recurse over nested recursive datatypes as if they |
|
2000 |
were mutually recursive |
|
2001 |
(Section~\ref{sssec:primrec-nested-as-mutual-recursion}), it is possible to |
|
53752 | 2002 |
pretend that nested codatatypes are mutually corecursive. For example: |
53647 | 2003 |
*} |
2004 |
||
54287 | 2005 |
(*<*) |
2006 |
context late |
|
2007 |
begin |
|
2008 |
(*>*) |
|
54072 | 2009 |
primcorec |
54287 | 2010 |
iterate\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>i" and |
53644 | 2011 |
iterates\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a llist \<Rightarrow> 'a tree\<^sub>i\<^sub>i llist" |
2012 |
where |
|
54072 | 2013 |
"iterate\<^sub>i\<^sub>i g x = Node\<^sub>i\<^sub>i x (iterates\<^sub>i\<^sub>i g (g x))" | |
2014 |
"iterates\<^sub>i\<^sub>i g xs = |
|
53644 | 2015 |
(case xs of |
2016 |
LNil \<Rightarrow> LNil |
|
54072 | 2017 |
| LCons x xs' \<Rightarrow> LCons (iterate\<^sub>i\<^sub>i g x) (iterates\<^sub>i\<^sub>i g xs'))" |
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|
2018 |
|
54287 | 2019 |
text {* |
2020 |
\noindent |
|
2021 |
Coinduction rules are generated as |
|
2022 |
@{thm [source] iterate\<^sub>i\<^sub>i.coinduct}, |
|
2023 |
@{thm [source] iterates\<^sub>i\<^sub>i.coinduct}, and |
|
2024 |
@{thm [source] iterate\<^sub>i\<^sub>i_iterates\<^sub>i\<^sub>i.coinduct} |
|
2025 |
and analogously for @{text strong_coinduct}. These rules and the |
|
2026 |
underlying corecursors are generated on a per-need basis and are kept in a cache |
|
2027 |
to speed up subsequent definitions. |
|
2028 |
*} |
|
2029 |
||
2030 |
(*<*) |
|
2031 |
end |
|
2032 |
(*>*) |
|
2033 |
||
53749
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|
2034 |
|
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|
2035 |
subsubsection {* Constructor View |
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|
2036 |
\label{ssec:primrec-constructor-view} *} |
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|
2037 |
|
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|
2038 |
(*<*) |
54182 | 2039 |
locale ctr_view |
53749
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53748
diff
changeset
|
2040 |
begin |
b37db925b663
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|
2041 |
(*>*) |
b37db925b663
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changeset
|
2042 |
|
b37db925b663
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|
2043 |
text {* |
53750 | 2044 |
The constructor view is similar to the code view, but there is one separate |
2045 |
conditional equation per constructor rather than a single unconditional |
|
2046 |
equation. Examples that rely on a single constructor, such as @{const literate} |
|
2047 |
and @{const siterate}, are identical in both styles. |
|
2048 |
||
2049 |
Here is an example where there is a difference: |
|
53749
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|
2050 |
*} |
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|
2051 |
|
53826 | 2052 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53749
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blanchet
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changeset
|
2053 |
"lnull xs \<Longrightarrow> lnull ys \<Longrightarrow> lappend xs ys = LNil" | |
b37db925b663
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blanchet
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53748
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changeset
|
2054 |
"_ \<Longrightarrow> lappend xs ys = LCons (lhd (if lnull xs then ys else xs)) |
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blanchet
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53748
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changeset
|
2055 |
(if xs = LNil then ltl ys else lappend (ltl xs) ys)" |
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blanchet
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changeset
|
2056 |
|
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|
2057 |
text {* |
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blanchet
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|
2058 |
\noindent |
53752 | 2059 |
With the constructor view, we must distinguish between the @{const LNil} and |
2060 |
the @{const LCons} case. The condition for @{const LCons} is |
|
2061 |
left implicit, as the negation of that for @{const LNil}. |
|
53750 | 2062 |
|
2063 |
For this example, the constructor view is slighlty more involved than the |
|
2064 |
code equation. Recall the code view version presented in |
|
53749
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|
2065 |
Section~\ref{sssec:primcorec-simple-corecursion}. |
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|
2066 |
% TODO: \[{thm code_view.lappend.code}\] |
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changeset
|
2067 |
The constructor view requires us to analyze the second argument (@{term ys}). |
53752 | 2068 |
The code equation generated from the constructor view also suffers from this. |
53749
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blanchet
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|
2069 |
% TODO: \[{thm lappend.code}\] |
53750 | 2070 |
|
53752 | 2071 |
In contrast, the next example is arguably more naturally expressed in the |
2072 |
constructor view: |
|
53749
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|
2073 |
*} |
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|
2074 |
|
53831
80423b9080cf
support "of" syntax to disambiguate selector equations
panny
parents:
53829
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changeset
|
2075 |
primcorec |
53752 | 2076 |
random_process :: "'a stream \<Rightarrow> (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> 'a process" |
2077 |
where |
|
53749
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blanchet
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changeset
|
2078 |
"n mod 4 = 0 \<Longrightarrow> random_process s f n = Fail" | |
53752 | 2079 |
"n mod 4 = 1 \<Longrightarrow> |
2080 |
random_process s f n = Skip (random_process s f (f n))" | |
|
2081 |
"n mod 4 = 2 \<Longrightarrow> |
|
2082 |
random_process s f n = Action (shd s) (random_process (stl s) f (f n))" | |
|
2083 |
"n mod 4 = 3 \<Longrightarrow> |
|
2084 |
random_process s f n = Choice (random_process (every_snd s) f (f n)) |
|
53826 | 2085 |
(random_process (every_snd (stl s)) f (f n))" |
2086 |
(*<*) |
|
53644 | 2087 |
end |
2088 |
(*>*) |
|
52805 | 2089 |
|
53750 | 2090 |
text {* |
53752 | 2091 |
\noindent |
53750 | 2092 |
Since there is no sequentiality, we can apply the equation for @{const Choice} |
53752 | 2093 |
without having first to discharge @{term "n mod (4\<Colon>int) \<noteq> 0"}, |
2094 |
@{term "n mod (4\<Colon>int) \<noteq> 1"}, and |
|
2095 |
@{term "n mod (4\<Colon>int) \<noteq> 2"}. |
|
53750 | 2096 |
The price to pay for this elegance is that we must discharge exclusivity proof |
2097 |
obligations, one for each pair of conditions |
|
53752 | 2098 |
@{term "(n mod (4\<Colon>int) = i, n mod (4\<Colon>int) = j)"} |
2099 |
with @{term "i < j"}. If we prefer not to discharge any obligations, we can |
|
2100 |
enable the @{text "sequential"} option. This pushes the problem to the users of |
|
2101 |
the generated properties. |
|
53750 | 2102 |
%Here are more examples to conclude: |
2103 |
*} |
|
2104 |
||
52824 | 2105 |
|
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|
2106 |
subsubsection {* Destructor View |
53752 | 2107 |
\label{ssec:primrec-destructor-view} *} |
2108 |
||
2109 |
(*<*) |
|
54182 | 2110 |
locale dtr_view |
53752 | 2111 |
begin |
2112 |
(*>*) |
|
53749
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changeset
|
2113 |
|
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|
2114 |
text {* |
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changeset
|
2115 |
The destructor view is in many respects dual to the constructor view. Conditions |
b37db925b663
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changeset
|
2116 |
determine which constructor to choose, and these conditions are interpreted |
b37db925b663
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|
2117 |
sequentially or not depending on the @{text "sequential"} option. |
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changeset
|
2118 |
Consider the following examples: |
b37db925b663
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diff
changeset
|
2119 |
*} |
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parents:
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changeset
|
2120 |
|
53826 | 2121 |
primcorec literate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist" where |
53749
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|
2122 |
"\<not> lnull (literate _ x)" | |
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changeset
|
2123 |
"lhd (literate _ x) = x" | |
54072 | 2124 |
"ltl (literate g x) = literate g (g x)" |
53749
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changeset
|
2125 |
|
53752 | 2126 |
text {* \blankline *} |
2127 |
||
53826 | 2128 |
primcorec siterate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a stream" where |
53749
b37db925b663
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changeset
|
2129 |
"shd (siterate _ x) = x" | |
54072 | 2130 |
"stl (siterate g x) = siterate g (g x)" |
53749
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diff
changeset
|
2131 |
|
53752 | 2132 |
text {* \blankline *} |
2133 |
||
53826 | 2134 |
primcorec every_snd :: "'a stream \<Rightarrow> 'a stream" where |
53749
b37db925b663
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|
2135 |
"shd (every_snd s) = shd s" | |
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|
2136 |
"stl (every_snd s) = stl (stl s)" |
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changeset
|
2137 |
|
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|
2138 |
text {* |
b37db925b663
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diff
changeset
|
2139 |
\noindent |
b37db925b663
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parents:
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diff
changeset
|
2140 |
The first formula in the @{const literate} specification indicates which |
b37db925b663
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blanchet
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diff
changeset
|
2141 |
constructor to choose. For @{const siterate} and @{const every_snd}, no such |
b37db925b663
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diff
changeset
|
2142 |
formula is necessary, since the type has only one constructor. The last two |
b37db925b663
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diff
changeset
|
2143 |
formulas are equations specifying the value of the result for the relevant |
b37db925b663
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diff
changeset
|
2144 |
selectors. Corecursive calls appear directly to the right of the equal sign. |
b37db925b663
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diff
changeset
|
2145 |
Their arguments are unrestricted. |
53750 | 2146 |
|
2147 |
The next example shows how to specify functions that rely on more than one |
|
2148 |
constructor: |
|
2149 |
*} |
|
2150 |
||
53826 | 2151 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53750 | 2152 |
"lnull xs \<Longrightarrow> lnull ys \<Longrightarrow> lnull (lappend xs ys)" | |
2153 |
"lhd (lappend xs ys) = lhd (if lnull xs then ys else xs)" | |
|
2154 |
"ltl (lappend xs ys) = (if xs = LNil then ltl ys else lappend (ltl xs) ys)" |
|
2155 |
||
2156 |
text {* |
|
2157 |
\noindent |
|
2158 |
For a codatatype with $n$ constructors, it is sufficient to specify $n - 1$ |
|
2159 |
discriminator formulas. The command will then assume that the remaining |
|
2160 |
constructor should be taken otherwise. This can be made explicit by adding |
|
2161 |
*} |
|
2162 |
||
2163 |
(*<*) |
|
2164 |
end |
|
2165 |
||
54182 | 2166 |
locale dtr_view2 |
2167 |
begin |
|
2168 |
||
53826 | 2169 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53750 | 2170 |
"lnull xs \<Longrightarrow> lnull ys \<Longrightarrow> lnull (lappend xs ys)" | |
2171 |
(*>*) |
|
53752 | 2172 |
"_ \<Longrightarrow> \<not> lnull (lappend xs ys)" |
2173 |
(*<*) | |
|
53750 | 2174 |
"lhd (lappend xs ys) = lhd (if lnull xs then ys else xs)" | |
2175 |
"ltl (lappend xs ys) = (if xs = LNil then ltl ys else lappend (ltl xs) ys)" |
|
2176 |
(*>*) |
|
2177 |
||
2178 |
text {* |
|
2179 |
\noindent |
|
53752 | 2180 |
to the specification. The generated selector theorems are conditional. |
2181 |
||
2182 |
The next example illustrates how to cope with selectors defined for several |
|
53750 | 2183 |
constructors: |
53749
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|
2184 |
*} |
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changeset
|
2185 |
|
53831
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support "of" syntax to disambiguate selector equations
panny
parents:
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diff
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|
2186 |
primcorec |
53752 | 2187 |
random_process :: "'a stream \<Rightarrow> (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> 'a process" |
2188 |
where |
|
53749
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parents:
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diff
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|
2189 |
"n mod 4 = 0 \<Longrightarrow> is_Fail (random_process s f n)" | |
b37db925b663
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diff
changeset
|
2190 |
"n mod 4 = 1 \<Longrightarrow> is_Skip (random_process s f n)" | |
b37db925b663
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diff
changeset
|
2191 |
"n mod 4 = 2 \<Longrightarrow> is_Action (random_process s f n)" | |
b37db925b663
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diff
changeset
|
2192 |
"n mod 4 = 3 \<Longrightarrow> is_Choice (random_process s f n)" | |
53831
80423b9080cf
support "of" syntax to disambiguate selector equations
panny
parents:
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diff
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|
2193 |
"cont (random_process s f n) = random_process s f (f n)" of Skip | |
53749
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|
2194 |
"prefix (random_process s f n) = shd s" | |
53831
80423b9080cf
support "of" syntax to disambiguate selector equations
panny
parents:
53829
diff
changeset
|
2195 |
"cont (random_process s f n) = random_process (stl s) f (f n)" of Action | |
53749
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diff
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|
2196 |
"left (random_process s f n) = random_process (every_snd s) f (f n)" | |
53831
80423b9080cf
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parents:
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diff
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|
2197 |
"right (random_process s f n) = random_process (every_snd (stl s)) f (f n)" |
53749
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changeset
|
2198 |
|
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|
2199 |
text {* |
53750 | 2200 |
\noindent |
2201 |
Using the @{text "of"} keyword, different equations are specified for @{const |
|
2202 |
cont} depending on which constructor is selected. |
|
2203 |
||
2204 |
Here are more examples to conclude: |
|
53749
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diff
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|
2205 |
*} |
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|
2206 |
|
53826 | 2207 |
primcorec |
53749
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|
2208 |
even_infty :: even_enat and |
b37db925b663
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diff
changeset
|
2209 |
odd_infty :: odd_enat |
b37db925b663
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diff
changeset
|
2210 |
where |
b37db925b663
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diff
changeset
|
2211 |
"\<not> is_Even_EZero even_infty" | |
b37db925b663
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diff
changeset
|
2212 |
"un_Even_ESuc even_infty = odd_infty" | |
b37db925b663
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diff
changeset
|
2213 |
"un_Odd_ESuc odd_infty = even_infty" |
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diff
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|
2214 |
|
53752 | 2215 |
text {* \blankline *} |
2216 |
||
53826 | 2217 |
primcorec iterate\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
54072 | 2218 |
"lbl\<^sub>i\<^sub>i (iterate\<^sub>i\<^sub>i g x) = x" | |
2219 |
"sub\<^sub>i\<^sub>i (iterate\<^sub>i\<^sub>i g x) = lmap (iterate\<^sub>i\<^sub>i g) (g x)" |
|
53749
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diff
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|
2220 |
(*<*) |
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|
2221 |
end |
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|
2222 |
(*>*) |
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|
2223 |
|
53750 | 2224 |
|
53617 | 2225 |
subsection {* Command Syntax |
2226 |
\label{ssec:primcorec-command-syntax} *} |
|
2227 |
||
2228 |
||
53826 | 2229 |
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
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53752
diff
changeset
|
2230 |
\label{sssec:primcorecursive-and-primcorec} *} |
52840 | 2231 |
|
2232 |
text {* |
|
53829 | 2233 |
\begin{matharray}{rcl} |
2234 |
@{command_def "primcorec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
|
2235 |
@{command_def "primcorecursive"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
|
2236 |
\end{matharray} |
|
52840 | 2237 |
|
2238 |
@{rail " |
|
55029
61a6bf7d4b02
clarified @{rail} syntax: prefer explicit \<newline> symbol;
wenzelm
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diff
changeset
|
2239 |
(@@{command primcorec} | @@{command primcorecursive}) target? \<newline> |
61a6bf7d4b02
clarified @{rail} syntax: prefer explicit \<newline> symbol;
wenzelm
parents:
54958
diff
changeset
|
2240 |
@{syntax pcr_option}? fixes @'where' |
53749
b37db925b663
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diff
changeset
|
2241 |
(@{syntax pcr_formula} + '|') |
b37db925b663
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parents:
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diff
changeset
|
2242 |
; |
53828 | 2243 |
@{syntax_def pcr_option}: '(' ('sequential' | 'exhaustive') ')' |
52840 | 2244 |
; |
53749
b37db925b663
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blanchet
parents:
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diff
changeset
|
2245 |
@{syntax_def pcr_formula}: thmdecl? prop (@'of' (term * ))? |
52840 | 2246 |
"} |
53749
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diff
changeset
|
2247 |
|
54832 | 2248 |
The optional target is potentially followed by a corecursion-specific option: |
53749
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diff
changeset
|
2249 |
|
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diff
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|
2250 |
\begin{itemize} |
b37db925b663
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parents:
53748
diff
changeset
|
2251 |
\setlength{\itemsep}{0pt} |
b37db925b663
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parents:
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diff
changeset
|
2252 |
|
b37db925b663
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diff
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|
2253 |
\item |
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diff
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|
2254 |
The @{text "sequential"} option indicates that the conditions in specifications |
b37db925b663
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parents:
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diff
changeset
|
2255 |
expressed using the constructor or destructor view are to be interpreted |
b37db925b663
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blanchet
parents:
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diff
changeset
|
2256 |
sequentially. |
53826 | 2257 |
|
2258 |
\item |
|
2259 |
The @{text "exhaustive"} option indicates that the conditions in specifications |
|
2260 |
expressed using the constructor or destructor view cover all possible cases. |
|
53749
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diff
changeset
|
2261 |
\end{itemize} |
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|
2262 |
|
53826 | 2263 |
\noindent |
2264 |
The @{command primcorec} command is an abbreviation for @{command primcorecursive} with |
|
2265 |
@{text "by auto?"} to discharge any emerging proof obligations. |
|
52840 | 2266 |
*} |
52794 | 2267 |
|
52824 | 2268 |
|
53619 | 2269 |
(* |
52840 | 2270 |
subsection {* Generated Theorems |
2271 |
\label{ssec:primcorec-generated-theorems} *} |
|
53619 | 2272 |
*) |
52794 | 2273 |
|
2274 |
||
53623 | 2275 |
(* |
2276 |
subsection {* Recursive Default Values for Selectors |
|
2277 |
\label{ssec:primcorec-recursive-default-values-for-selectors} *} |
|
2278 |
||
2279 |
text {* |
|
2280 |
partial_function to the rescue |
|
2281 |
*} |
|
2282 |
*) |
|
2283 |
||
2284 |
||
52827 | 2285 |
section {* Registering Bounded Natural Functors |
52805 | 2286 |
\label{sec:registering-bounded-natural-functors} *} |
52792 | 2287 |
|
52805 | 2288 |
text {* |
53647 | 2289 |
The (co)datatype package can be set up to allow nested recursion through |
2290 |
arbitrary type constructors, as long as they adhere to the BNF requirements and |
|
2291 |
are registered as BNFs. |
|
52805 | 2292 |
*} |
2293 |
||
52824 | 2294 |
|
53619 | 2295 |
(* |
53617 | 2296 |
subsection {* Introductory Example |
2297 |
\label{ssec:bnf-introductory-example} *} |
|
52805 | 2298 |
|
2299 |
text {* |
|
2300 |
More examples in \verb|~~/src/HOL/BNF/Basic_BNFs.thy| and |
|
2301 |
\verb|~~/src/HOL/BNF/More_BNFs.thy|. |
|
52806 | 2302 |
|
53617 | 2303 |
%Mention distinction between live and dead type arguments; |
2304 |
% * and existence of map, set for those |
|
2305 |
%mention =>. |
|
52805 | 2306 |
*} |
53619 | 2307 |
*) |
52794 | 2308 |
|
52824 | 2309 |
|
53617 | 2310 |
subsection {* Command Syntax |
2311 |
\label{ssec:bnf-command-syntax} *} |
|
2312 |
||
2313 |
||
53621 | 2314 |
subsubsection {* \keyw{bnf} |
2315 |
\label{sssec:bnf} *} |
|
52794 | 2316 |
|
53028 | 2317 |
text {* |
53829 | 2318 |
\begin{matharray}{rcl} |
2319 |
@{command_def "bnf"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
|
2320 |
\end{matharray} |
|
2321 |
||
53028 | 2322 |
@{rail " |
55029
61a6bf7d4b02
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wenzelm
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diff
changeset
|
2323 |
@@{command bnf} target? (name ':')? typ \<newline> |
61a6bf7d4b02
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wenzelm
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diff
changeset
|
2324 |
'map:' term ('sets:' (term +))? 'bd:' term \<newline> |
54421 | 2325 |
('wits:' (term +))? ('rel:' term)? |
53028 | 2326 |
"} |
2327 |
*} |
|
52805 | 2328 |
|
53617 | 2329 |
|
54187 | 2330 |
subsubsection {* \keyw{bnf\_decl} |
2331 |
\label{sssec:bnf-decl} *} |
|
2332 |
||
2333 |
text {* |
|
2334 |
%%% TODO: use command_def once the command is available |
|
2335 |
\begin{matharray}{rcl} |
|
2336 |
@{text "bnf_decl"} & : & @{text "local_theory \<rightarrow> local_theory"} |
|
2337 |
\end{matharray} |
|
2338 |
||
2339 |
@{rail " |
|
54602 | 2340 |
@@{command bnf_decl} target? @{syntax dt_name} |
2341 |
; |
|
2342 |
@{syntax_def dt_name}: @{syntax tyargs}? name @{syntax map_rel}? mixfix? |
|
2343 |
; |
|
2344 |
@{syntax_def tyargs}: typefree | '(' (((name | '-') ':')? typefree + ',') ')' |
|
2345 |
; |
|
2346 |
@{syntax_def map_rel}: '(' ((('map' | 'rel') ':' name) +) ')' |
|
54187 | 2347 |
"} |
54602 | 2348 |
|
2349 |
Declares a fresh type and fresh constants (map, set, relator, cardinal bound) |
|
2350 |
and asserts the bnf properties for these constants as axioms. Additionally, |
|
2351 |
type arguments may be marked as dead (by using @{syntax "-"} instead of a name for the |
|
2352 |
set function)---this is the only difference of @{syntax dt_name} compared to |
|
2353 |
the syntax used by the @{command datatype_new}/@{command codatatype} commands. |
|
2354 |
||
2355 |
The axioms are sound, since one there exists a bnf of any given arity. |
|
54187 | 2356 |
*} |
2357 |
||
2358 |
||
53621 | 2359 |
subsubsection {* \keyw{print\_bnfs} |
2360 |
\label{sssec:print-bnfs} *} |
|
53617 | 2361 |
|
2362 |
text {* |
|
53829 | 2363 |
\begin{matharray}{rcl} |
2364 |
@{command_def "print_bnfs"} & : & @{text "local_theory \<rightarrow>"} |
|
2365 |
\end{matharray} |
|
2366 |
||
53647 | 2367 |
@{rail " |
53829 | 2368 |
@@{command print_bnfs} |
53647 | 2369 |
"} |
53617 | 2370 |
*} |
2371 |
||
2372 |
||
2373 |
section {* Deriving Destructors and Theorems for Free Constructors |
|
2374 |
\label{sec:deriving-destructors-and-theorems-for-free-constructors} *} |
|
52794 | 2375 |
|
52805 | 2376 |
text {* |
53623 | 2377 |
The derivation of convenience theorems for types equipped with free constructors, |
53829 | 2378 |
as performed internally by @{command datatype_new} and @{command codatatype}, |
53623 | 2379 |
is available as a stand-alone command called @{command wrap_free_constructors}. |
52794 | 2380 |
|
53617 | 2381 |
% * need for this is rare but may arise if you want e.g. to add destructors to |
2382 |
% a type not introduced by ... |
|
2383 |
% |
|
2384 |
% * also useful for compatibility with old package, e.g. add destructors to |
|
2385 |
% old \keyw{datatype} |
|
2386 |
% |
|
2387 |
% * @{command wrap_free_constructors} |
|
54626 | 2388 |
% * @{text "no_discs_sels"}, @{text "no_code"}, @{text "rep_compat"} |
53617 | 2389 |
% * hack to have both co and nonco view via locale (cf. ext nats) |
54616 | 2390 |
% * code generator |
2391 |
% * eq, refl, simps |
|
52805 | 2392 |
*} |
52792 | 2393 |
|
52824 | 2394 |
|
53619 | 2395 |
(* |
53617 | 2396 |
subsection {* Introductory Example |
2397 |
\label{ssec:ctors-introductory-example} *} |
|
53619 | 2398 |
*) |
52794 | 2399 |
|
52824 | 2400 |
|
53617 | 2401 |
subsection {* Command Syntax |
2402 |
\label{ssec:ctors-command-syntax} *} |
|
2403 |
||
2404 |
||
53621 | 2405 |
subsubsection {* \keyw{wrap\_free\_constructors} |
53675 | 2406 |
\label{sssec:wrap-free-constructors} *} |
52828 | 2407 |
|
53018 | 2408 |
text {* |
53829 | 2409 |
\begin{matharray}{rcl} |
2410 |
@{command_def "wrap_free_constructors"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
|
2411 |
\end{matharray} |
|
53018 | 2412 |
|
2413 |
@{rail " |
|
55029
61a6bf7d4b02
clarified @{rail} syntax: prefer explicit \<newline> symbol;
wenzelm
parents:
54958
diff
changeset
|
2414 |
@@{command wrap_free_constructors} target? @{syntax dt_options} \<newline> |
53863
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
2415 |
term_list name @{syntax wfc_discs_sels}? |
53018 | 2416 |
; |
53863
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
2417 |
@{syntax_def wfc_discs_sels}: name_list (name_list_list name_term_list_list? )? |
53018 | 2418 |
; |
53534 | 2419 |
@{syntax_def name_term}: (name ':' term) |
54421 | 2420 |
; |
2421 |
X_list: '[' (X + ',') ']' |
|
53018 | 2422 |
"} |
2423 |
||
54626 | 2424 |
% options: no_discs_sels no_code rep_compat |
53028 | 2425 |
|
53829 | 2426 |
\noindent |
53542 | 2427 |
Section~\ref{ssec:datatype-generated-theorems} lists the generated theorems. |
53018 | 2428 |
*} |
52828 | 2429 |
|
52794 | 2430 |
|
53617 | 2431 |
(* |
52827 | 2432 |
section {* Standard ML Interface |
52805 | 2433 |
\label{sec:standard-ml-interface} *} |
52792 | 2434 |
|
52805 | 2435 |
text {* |
53623 | 2436 |
The package's programmatic interface. |
52805 | 2437 |
*} |
53617 | 2438 |
*) |
52794 | 2439 |
|
2440 |
||
53617 | 2441 |
(* |
52827 | 2442 |
section {* Interoperability |
52805 | 2443 |
\label{sec:interoperability} *} |
52794 | 2444 |
|
52805 | 2445 |
text {* |
53623 | 2446 |
The package's interaction with other Isabelle packages and tools, such as the |
2447 |
code generator and the counterexample generators. |
|
52805 | 2448 |
*} |
52794 | 2449 |
|
52824 | 2450 |
|
52828 | 2451 |
subsection {* Transfer and Lifting |
2452 |
\label{ssec:transfer-and-lifting} *} |
|
52794 | 2453 |
|
52824 | 2454 |
|
52828 | 2455 |
subsection {* Code Generator |
2456 |
\label{ssec:code-generator} *} |
|
52794 | 2457 |
|
52824 | 2458 |
|
52828 | 2459 |
subsection {* Quickcheck |
2460 |
\label{ssec:quickcheck} *} |
|
52794 | 2461 |
|
52824 | 2462 |
|
52828 | 2463 |
subsection {* Nitpick |
2464 |
\label{ssec:nitpick} *} |
|
52794 | 2465 |
|
52824 | 2466 |
|
52828 | 2467 |
subsection {* Nominal Isabelle |
2468 |
\label{ssec:nominal-isabelle} *} |
|
53617 | 2469 |
*) |
52794 | 2470 |
|
52805 | 2471 |
|
53617 | 2472 |
(* |
52827 | 2473 |
section {* Known Bugs and Limitations |
52805 | 2474 |
\label{sec:known-bugs-and-limitations} *} |
2475 |
||
2476 |
text {* |
|
53623 | 2477 |
Known open issues of the package. |
52805 | 2478 |
*} |
52794 | 2479 |
|
2480 |
text {* |
|
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
|
2481 |
%* primcorecursive and primcorec is unfinished |
53617 | 2482 |
% |
2483 |
%* slow n-ary mutual (co)datatype, avoid as much as possible (e.g. using nesting) |
|
2484 |
% |
|
2485 |
%* issues with HOL-Proofs? |
|
2486 |
% |
|
2487 |
%* partial documentation |
|
2488 |
% |
|
2489 |
%* no way to register "sum" and "prod" as (co)datatypes to enable N2M reduction for them |
|
2490 |
% (for @{command datatype_new_compat} and prim(co)rec) |
|
2491 |
% |
|
53619 | 2492 |
% * a fortiori, no way to register same type as both data- and codatatype |
53617 | 2493 |
% |
2494 |
%* no recursion through unused arguments (unlike with the old package) |
|
2495 |
% |
|
2496 |
%* in a locale, cannot use locally fixed types (because of limitation in typedef)? |
|
53619 | 2497 |
% |
2498 |
% *names of variables suboptimal |
|
52822 | 2499 |
*} |
53675 | 2500 |
*) |
52822 | 2501 |
|
2502 |
||
2503 |
text {* |
|
53863
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
2504 |
\section*{Acknowledgment} |
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
2505 |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
2506 |
Tobias Nipkow and Makarius Wenzel encouraged us to implement the new |
53617 | 2507 |
(co)datatype package. Andreas Lochbihler provided lots of comments on earlier |
2508 |
versions of the package, especially for the coinductive part. Brian Huffman |
|
55073 | 2509 |
suggested major simplifications to the internal constructions, many of which has |
53617 | 2510 |
yet to be implemented. Florian Haftmann and Christian Urban provided general |
53675 | 2511 |
advice on Isabelle and package writing. Stefan Milius and Lutz Schr\"oder |
54146 | 2512 |
found an elegant proof to eliminate one of the BNF assumptions. Andreas |
2513 |
Lochbihler and Christian Sternagel suggested many textual improvements to this |
|
2514 |
tutorial. |
|
52794 | 2515 |
*} |
53617 | 2516 |
|
52792 | 2517 |
end |