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(* Title: Doc/Datatypes/Datatypes.thy |
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Author: Jasmin Blanchette, TU Muenchen |
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Author: Martin Desharnais, Ecole de technologie superieure |
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Author: Lorenz Panny, TU Muenchen |
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Author: Andrei Popescu, TU Muenchen |
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Author: Dmitriy Traytel, TU Muenchen |
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Tutorial for (co)datatype definitions. |
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*) |
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theory Datatypes |
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imports |
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Setup |
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"~~/src/HOL/Library/BNF_Axiomatization" |
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"~~/src/HOL/Library/Cardinal_Notations" |
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"~~/src/HOL/Library/FSet" |
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"~~/src/HOL/Library/Simps_Case_Conv" |
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begin |
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section {* Introduction |
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\label{sec:introduction} *} |
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text {* |
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The 2013 edition of Isabelle introduced a definitional package for freely |
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generated datatypes and codatatypes. This package replaces the earlier |
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implementation due to Berghofer and Wenzel @{cite "Berghofer-Wenzel:1999:TPHOL"}. |
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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 '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 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 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 (*<*)(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 (*<*)(in early) (*>*)'a tree\<^sub>f\<^sub>i = Node\<^sub>f\<^sub>i 'a "'a tree\<^sub>f\<^sub>i llist" |
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codatatype (*<*)(in early) (*>*)'a tree\<^sub>i\<^sub>f = Node\<^sub>i\<^sub>f 'a "'a tree\<^sub>i\<^sub>f list" |
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codatatype (*<*)(in early) (*>*)'a tree\<^sub>i\<^sub>i = Node\<^sub>i\<^sub>i 'a "'a tree\<^sub>i\<^sub>i llist" |
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text {* |
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The first two tree types allow only paths of finite length, whereas the last two |
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allow infinite paths. Orthogonally, the nodes in the first and third types have |
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finitely many direct subtrees, whereas those of the second and fourth may have |
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infinite branching. |
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The package is part of @{theory Main}. Additional functionality is provided by |
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the theory @{theory BNF_Axiomatization}, located in the directory |
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\verb|~~/src/HOL/Library|. |
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The package, like its predecessor, fully adheres to the LCF philosophy |
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@{cite mgordon79}: The characteristic theorems associated with the specified |
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(co)datatypes are derived rather than introduced axiomatically.% |
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\footnote{However, some of the |
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internal constructions and most of the internal proof obligations are omitted |
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if the @{text quick_and_dirty} option is enabled.} |
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The package is described in a number of papers |
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@{cite "traytel-et-al-2012" and "blanchette-et-al-wit" and |
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"blanchette-et-al-2014-impl" and "panny-et-al-2014"}. |
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The central notion is that of a \emph{bounded natural functor} (BNF)---a |
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well-behaved type constructor for which nested (co)recursion is supported. |
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This tutorial is organized as follows: |
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\begin{itemize} |
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\setlength{\itemsep}{0pt} |
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\item Section \ref{sec:defining-datatypes}, ``Defining Datatypes,'' |
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describes how to specify datatypes using the @{command datatype} command. |
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\item Section \ref{sec:defining-primitively-recursive-functions}, ``Defining |
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Primitively Recursive Functions,'' describes how to specify functions |
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using @{command primrec}. (A separate tutorial @{cite "isabelle-function"} |
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describes the more general \keyw{fun} and \keyw{function} commands.) |
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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-primitively-corecursive-functions}, |
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``Defining Primitively Corecursive Functions,'' describes how to specify |
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functions using the @{command primcorec} and |
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@{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 free_constructors} to derive destructor constants |
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and theorems for freely generated types, as performed internally by |
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@{command datatype} and @{command codatatype}. |
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%\item Section \ref{sec:using-the-standard-ml-interface}, ``Using the Standard |
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%ML Interface,'' describes the package's programmatic interface. |
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\item Section \ref{sec:selecting-plugins}, ``Selecting Plugins,'' is concerned |
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with the package's interoperability with other Isabelle packages and tools, such |
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as the code generator, Transfer, Lifting, and Quickcheck. |
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\item Section \ref{sec:known-bugs-and-limitations}, ``Known Bugs and |
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Limitations,'' concludes with known open issues at the time of writing. |
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\end{itemize} |
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\newbox\boxA |
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\setbox\boxA=\hbox{\texttt{NOSPAM}} |
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\newcommand\authoremaili{\texttt{blan{\color{white}NOSPAM}\kern-\wd\boxA{}chette@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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\newcommand\authoremailii{\texttt{desh{\color{white}NOSPAM}\kern-\wd\boxA{}arna@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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\newcommand\authoremailiii{\texttt{lore{\color{white}NOSPAM}\kern-\wd\boxA{}nz.panny@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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\newcommand\authoremailiv{\texttt{pope{\color{white}NOSPAM}\kern-\wd\boxA{}scua@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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\newcommand\authoremailv{\texttt{tray{\color{white}NOSPAM}\kern-\wd\boxA{}tel@\allowbreak |
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in.\allowbreak tum.\allowbreak de}} |
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Comments and bug reports concerning either the package or this tutorial should |
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be directed to the authors at \authoremaili, \authoremailii, \authoremailiii, |
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\authoremailiv, and \authoremailv. |
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*} |
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section {* Defining Datatypes |
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\label{sec:defining-datatypes} *} |
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text {* |
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Datatypes can be specified using the @{command datatype} 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|Datatype_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 trool = Truue | Faalse | Perhaaps |
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text {* |
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\noindent |
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@{const Truue}, @{const Faalse}, and @{const Perhaaps} have the type @{typ trool}. |
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Polymorphic types are possible, such as the following option type, modeled after |
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its homologue from the @{theory Option} theory: |
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*} |
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(*<*) |
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hide_const None Some map_option |
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hide_type option |
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(*>*) |
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datatype 'a option = None | Some 'a |
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text {* |
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\noindent |
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The constructors are @{text "None \<Colon> 'a option"} and |
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@{text "Some \<Colon> 'a \<Rightarrow> 'a option"}. |
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The next example has three type parameters: |
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*} |
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datatype ('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 \<Colon> '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 ('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 nat = Zero | Succ nat |
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text {* |
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\noindent |
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Lists were shown in the introduction. Terminated lists are a variant that |
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stores a value of type @{typ 'b} at the very end: |
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*} |
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datatype (*<*)(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 even_nat = Even_Zero | Even_Succ odd_nat |
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and odd_nat = Odd_Succ even_nat |
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text {* |
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\noindent |
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Arithmetic expressions are defined via terms, terms via factors, and factors via |
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expressions: |
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*} |
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datatype ('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 '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 '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 ('a, 'b) fun_copy = Fun "'a \<Rightarrow> 'b" |
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datatype 'a also_wrong = W1 | W2 (*<*)'a |
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typ (*>*)"('a also_wrong, 'a) fun_copy" |
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text {* |
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\noindent |
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The following definition of @{typ 'a}-branching trees is legal: |
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*} |
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datatype 'a ftree = FTLeaf 'a | FTNode "'a \<Rightarrow> 'a ftree" |
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text {* |
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\noindent |
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And so is the definition of hereditarily finite sets: |
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*} |
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datatype hfset = HFSet "hfset fset" |
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text {* |
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\noindent |
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In general, type constructors @{text "('a\<^sub>1, \<dots>, 'a\<^sub>m) t"} |
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allow recursion on a subset of their type arguments @{text 'a\<^sub>1}, \ldots, |
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@{text 'a\<^sub>m}. These type arguments are called \emph{live}; the remaining |
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type arguments are called \emph{dead}. In @{typ "'a \<Rightarrow> 'b"} and |
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@{typ "('a, 'b) fun_copy"}, the type variable @{typ 'a} is dead and |
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@{typ 'b} is live. |
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Type constructors must be registered as BNFs to have live arguments. This is |
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done automatically for datatypes and codatatypes introduced by the |
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@{command datatype} and @{command codatatype} commands. |
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Section~\ref{sec:registering-bounded-natural-functors} explains how to |
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register arbitrary type constructors as BNFs. |
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Here is another example that fails: |
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*} |
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datatype 'a pow_list = PNil 'a (*<*)'a |
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datatype '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 attempted definition features a different flavor of nesting, where the |
342 |
recursive call in the type specification occurs around (rather than inside) |
|
343 |
another type constructor. |
|
54187 | 344 |
*} |
345 |
||
58245 | 346 |
|
59284 | 347 |
subsubsection {* Auxiliary Constants |
348 |
\label{sssec:datatype-auxiliary-constants} *} |
|
52805 | 349 |
|
350 |
text {* |
|
58310 | 351 |
The @{command datatype} command introduces various constants in addition to |
53617 | 352 |
the constructors. With each datatype are associated set functions, a map |
353 |
function, a relator, discriminators, and selectors, all of which can be given |
|
54187 | 354 |
custom names. In the example below, the familiar names @{text null}, @{text hd}, |
59284 | 355 |
@{text tl}, @{text set}, @{text map}, and @{text list_all2} override the |
54187 | 356 |
default names @{text is_Nil}, @{text un_Cons1}, @{text un_Cons2}, |
54491 | 357 |
@{text set_list}, @{text map_list}, and @{text rel_list}: |
52822 | 358 |
*} |
359 |
||
52841 | 360 |
(*<*) |
361 |
no_translations |
|
362 |
"[x, xs]" == "x # [xs]" |
|
363 |
"[x]" == "x # []" |
|
364 |
||
365 |
no_notation |
|
366 |
Nil ("[]") and |
|
367 |
Cons (infixr "#" 65) |
|
368 |
||
53543 | 369 |
hide_type list |
58245 | 370 |
hide_const Nil Cons case_list hd tl set map list_all2 rec_list size_list pred_list |
53025 | 371 |
|
59284 | 372 |
context early |
373 |
begin |
|
52841 | 374 |
(*>*) |
58310 | 375 |
datatype (set: 'a) list = |
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376 |
null: Nil |
53025 | 377 |
| Cons (hd: 'a) (tl: "'a list") |
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|
378 |
for |
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|
379 |
map: map |
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|
380 |
rel: list_all2 |
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|
381 |
where |
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|
382 |
"tl Nil = Nil" |
52822 | 383 |
|
384 |
text {* |
|
385 |
\noindent |
|
55353 | 386 |
The types of the constants that appear in the specification are listed below. |
55351 | 387 |
|
388 |
\medskip |
|
54187 | 389 |
|
390 |
\begin{tabular}{@ {}ll@ {}} |
|
391 |
Constructors: & |
|
392 |
@{text "Nil \<Colon> 'a list"} \\ |
|
393 |
& |
|
394 |
@{text "Cons \<Colon> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list"} \\ |
|
395 |
Discriminator: & |
|
396 |
@{text "null \<Colon> 'a list \<Rightarrow> bool"} \\ |
|
397 |
Selectors: & |
|
398 |
@{text "hd \<Colon> 'a list \<Rightarrow> 'a"} \\ |
|
399 |
& |
|
400 |
@{text "tl \<Colon> 'a list \<Rightarrow> 'a list"} \\ |
|
401 |
Set function: & |
|
402 |
@{text "set \<Colon> 'a list \<Rightarrow> 'a set"} \\ |
|
403 |
Map function: & |
|
404 |
@{text "map \<Colon> ('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list"} \\ |
|
405 |
Relator: & |
|
406 |
@{text "list_all2 \<Colon> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"} |
|
407 |
\end{tabular} |
|
408 |
||
55351 | 409 |
\medskip |
410 |
||
54187 | 411 |
The discriminator @{const null} and the selectors @{const hd} and @{const tl} |
55351 | 412 |
are characterized by the following conditional equations: |
52822 | 413 |
% |
53025 | 414 |
\[@{thm list.collapse(1)[of xs, no_vars]} |
415 |
\qquad @{thm list.collapse(2)[of xs, no_vars]}\] |
|
52822 | 416 |
% |
54187 | 417 |
For two-constructor datatypes, a single discriminator constant is sufficient. |
418 |
The discriminator associated with @{const Cons} is simply |
|
53491 | 419 |
@{term "\<lambda>xs. \<not> null xs"}. |
52822 | 420 |
|
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421 |
The \keyw{where} clause at the end of the command specifies a default value for |
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|
422 |
selectors applied to constructors on which they are not a priori specified. |
59282 | 423 |
In the example, it is used to ensure that the tail of the empty list is itself |
424 |
(instead of being left unspecified). |
|
52822 | 425 |
|
53617 | 426 |
Because @{const Nil} is nullary, it is also possible to use |
57091 | 427 |
@{term "\<lambda>xs. xs = Nil"} as a discriminator. This is the default behavior |
428 |
if we omit the identifier @{const null} and the associated colon. Some users |
|
429 |
argue against this, because the mixture of constructors and selectors in the |
|
430 |
characteristic theorems can lead Isabelle's automation to switch between the |
|
431 |
constructor and the destructor view in surprising ways. |
|
52822 | 432 |
|
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|
433 |
The usual mixfix syntax annotations are available for both types and |
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|
434 |
constructors. For example: |
52805 | 435 |
*} |
52794 | 436 |
|
53025 | 437 |
(*<*) |
438 |
end |
|
439 |
(*>*) |
|
58310 | 440 |
datatype ('a, 'b) prod (infixr "*" 20) = Pair 'a 'b |
53552 | 441 |
|
442 |
text {* \blankline *} |
|
52822 | 443 |
|
58310 | 444 |
datatype (set: 'a) list = |
52822 | 445 |
null: Nil ("[]") |
52841 | 446 |
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65) |
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|
447 |
for |
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|
448 |
map: map |
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|
449 |
rel: list_all2 |
52841 | 450 |
|
451 |
text {* |
|
53535 | 452 |
\noindent |
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|
453 |
Incidentally, this is how the traditional syntax can be set up: |
52841 | 454 |
*} |
455 |
||
456 |
syntax "_list" :: "args \<Rightarrow> 'a list" ("[(_)]") |
|
457 |
||
53552 | 458 |
text {* \blankline *} |
459 |
||
52841 | 460 |
translations |
461 |
"[x, xs]" == "x # [xs]" |
|
462 |
"[x]" == "x # []" |
|
52822 | 463 |
|
52824 | 464 |
|
53617 | 465 |
subsection {* Command Syntax |
466 |
\label{ssec:datatype-command-syntax} *} |
|
467 |
||
58310 | 468 |
subsubsection {* \keyw{datatype} |
53621 | 469 |
\label{sssec:datatype-new} *} |
52794 | 470 |
|
52822 | 471 |
text {* |
53829 | 472 |
\begin{matharray}{rcl} |
58310 | 473 |
@{command_def "datatype"} & : & @{text "local_theory \<rightarrow> local_theory"} |
53829 | 474 |
\end{matharray} |
52822 | 475 |
|
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|
476 |
@{rail \<open> |
59282 | 477 |
@@{command datatype} target? @{syntax dt_options}? @{syntax dt_spec} |
52828 | 478 |
; |
59280 | 479 |
@{syntax_def dt_options}: '(' ((@{syntax plugins} | 'discs_sels') + ',') ')' |
58190 | 480 |
; |
59280 | 481 |
@{syntax_def plugins}: 'plugins' ('only' | 'del') ':' (name +) |
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|
482 |
; |
59282 | 483 |
@{syntax_def dt_spec}: (@{syntax dt_name} '=' (@{syntax dt_ctor} + '|') \<newline> |
484 |
@{syntax map_rel}? (@'where' (prop + '|'))? + @'and') |
|
485 |
; |
|
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|
486 |
@{syntax_def map_rel}: @'for' ((('map' | 'rel') ':' name) +) |
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|
487 |
\<close>} |
52824 | 488 |
|
55351 | 489 |
\medskip |
490 |
||
491 |
\noindent |
|
58310 | 492 |
The @{command datatype} command introduces a set of mutually recursive |
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|
493 |
datatypes specified by their constructors. |
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|
494 |
|
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|
495 |
The syntactic entity \synt{target} can be used to specify a local |
58620 | 496 |
context (e.g., @{text "(in linorder)"} @{cite "isabelle-isar-ref"}), |
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|
497 |
and \synt{prop} denotes a HOL proposition. |
55460
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|
498 |
|
58190 | 499 |
The optional target is optionally followed by a combination of the following |
56124 | 500 |
options: |
52822 | 501 |
|
52824 | 502 |
\begin{itemize} |
503 |
\setlength{\itemsep}{0pt} |
|
504 |
||
505 |
\item |
|
59280 | 506 |
The @{text plugins} option indicates which plugins should be enabled |
507 |
(@{text only}) or disabled (@{text del}). By default, all plugins are enabled. |
|
58190 | 508 |
|
509 |
\item |
|
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|
510 |
The @{text "discs_sels"} option indicates that discriminators and selectors |
589ec121ce1a
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|
511 |
should be generated. The option is implicitly enabled if names are specified for |
589ec121ce1a
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|
512 |
discriminators or selectors. |
52824 | 513 |
\end{itemize} |
52822 | 514 |
|
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|
515 |
The optional \keyw{where} clause specifies default values for selectors. |
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|
516 |
Each proposition must be an equation of the form |
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|
517 |
@{text "un_D (C \<dots>) = \<dots>"}, where @{text C} is a constructor and |
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|
518 |
@{text un_D} is a selector. |
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|
519 |
|
52827 | 520 |
The left-hand sides of the datatype equations specify the name of the type to |
53534 | 521 |
define, its type parameters, and additional information: |
52822 | 522 |
|
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|
523 |
@{rail \<open> |
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|
524 |
@{syntax_def dt_name}: @{syntax tyargs}? name mixfix? |
52824 | 525 |
; |
57092 | 526 |
@{syntax_def tyargs}: typefree | '(' (('dead' | name ':')? typefree + ',') ')' |
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|
527 |
\<close>} |
52822 | 528 |
|
55351 | 529 |
\medskip |
530 |
||
52827 | 531 |
\noindent |
55474 | 532 |
The syntactic entity \synt{name} denotes an identifier, \synt{mixfix} denotes |
533 |
the usual parenthesized mixfix notation, and \synt{typefree} denotes fixed type |
|
58620 | 534 |
variable (@{typ 'a}, @{typ 'b}, \ldots) @{cite "isabelle-isar-ref"}. |
52822 | 535 |
|
52827 | 536 |
The optional names preceding the type variables allow to override the default |
57988 | 537 |
names of the set functions (@{text set\<^sub>1_t}, \ldots, @{text set\<^sub>m_t}). Type |
58190 | 538 |
arguments can be marked as dead by entering @{text dead} in front of the |
58220 | 539 |
type variable (e.g., @{text "(dead 'a)"}); otherwise, they are live or dead |
55705 | 540 |
(and a set function is generated or not) depending on where they occur in the |
541 |
right-hand sides of the definition. Declaring a type argument as dead can speed |
|
542 |
up the type definition but will prevent any later (co)recursion through that |
|
543 |
type argument. |
|
544 |
||
53647 | 545 |
Inside a mutually recursive specification, all defined datatypes must |
546 |
mention exactly the same type variables in the same order. |
|
52822 | 547 |
|
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|
548 |
@{rail \<open> |
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|
549 |
@{syntax_def dt_ctor}: (name ':')? name (@{syntax dt_ctor_arg} * ) mixfix? |
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|
550 |
\<close>} |
52824 | 551 |
|
53535 | 552 |
\medskip |
553 |
||
52827 | 554 |
\noindent |
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|
555 |
The main constituents of a constructor specification are the name of the |
52827 | 556 |
constructor and the list of its argument types. An optional discriminator name |
59722 | 557 |
can be supplied at the front. If discriminators are enabled (cf.\ the |
558 |
@{text "discs_sels"} option) but no name is supplied, the default is |
|
57091 | 559 |
@{text "\<lambda>x. x = C\<^sub>j"} for nullary constructors and |
560 |
@{text t.is_C\<^sub>j} otherwise. |
|
52822 | 561 |
|
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|
562 |
@{rail \<open> |
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|
563 |
@{syntax_def dt_ctor_arg}: type | '(' name ':' type ')' |
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|
564 |
\<close>} |
52827 | 565 |
|
53535 | 566 |
\medskip |
567 |
||
52827 | 568 |
\noindent |
58620 | 569 |
The syntactic entity \synt{type} denotes a HOL type @{cite "isabelle-isar-ref"}. |
55474 | 570 |
|
52827 | 571 |
In addition to the type of a constructor argument, it is possible to specify a |
59722 | 572 |
name for the corresponding selector. The same selector name can be reused for |
573 |
arguments to several constructors as long as the arguments share the same type. |
|
574 |
If selectors are enabled (cf.\ the @{text "discs_sels"} option) but no name is |
|
575 |
supplied, the default name is @{text un_C\<^sub>ji}. |
|
52822 | 576 |
*} |
577 |
||
53617 | 578 |
|
57542 | 579 |
subsubsection {* \keyw{datatype_compat} |
56644 | 580 |
\label{sssec:datatype-compat} *} |
53617 | 581 |
|
582 |
text {* |
|
53829 | 583 |
\begin{matharray}{rcl} |
55531
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|
584 |
@{command_def "datatype_compat"} & : & @{text "local_theory \<rightarrow> local_theory"} |
53829 | 585 |
\end{matharray} |
586 |
||
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|
587 |
@{rail \<open> |
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|
588 |
@@{command datatype_compat} (name +) |
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|
589 |
\<close>} |
53829 | 590 |
|
55351 | 591 |
\medskip |
592 |
||
53829 | 593 |
\noindent |
55531
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|
594 |
The @{command datatype_compat} command registers new-style datatypes as |
58245 | 595 |
old-style datatypes and invokes the old-style plugins. For example: |
53621 | 596 |
*} |
597 |
||
55531
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|
598 |
datatype_compat even_nat odd_nat |
53621 | 599 |
|
600 |
text {* \blankline *} |
|
601 |
||
58121 | 602 |
ML {* Old_Datatype_Data.get_info @{theory} @{type_name even_nat} *} |
53621 | 603 |
|
604 |
text {* |
|
58620 | 605 |
The syntactic entity \synt{name} denotes an identifier @{cite "isabelle-isar-ref"}. |
55474 | 606 |
|
60135
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|
607 |
The command is sometimes useful when migrating from the old datatype package to |
852f7a49ec0c
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|
608 |
the new one. |
55474 | 609 |
|
610 |
A few remarks concern nested recursive datatypes: |
|
53748 | 611 |
|
612 |
\begin{itemize} |
|
53749
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|
613 |
\setlength{\itemsep}{0pt} |
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|
614 |
|
55871 | 615 |
\item The old-style, nested-as-mutual induction rule and recursor theorems are |
616 |
generated under their usual names but with ``@{text "compat_"}'' prefixed |
|
58215 | 617 |
(e.g., @{text compat_tree.induct}, @{text compat_tree.inducts}, and |
618 |
@{text compat_tree.rec}). |
|
53748 | 619 |
|
620 |
\item All types through which recursion takes place must be new-style datatypes |
|
59722 | 621 |
or the function type. |
53748 | 622 |
\end{itemize} |
53617 | 623 |
*} |
624 |
||
625 |
||
626 |
subsection {* Generated Constants |
|
627 |
\label{ssec:datatype-generated-constants} *} |
|
628 |
||
629 |
text {* |
|
59722 | 630 |
Given a datatype @{text "('a\<^sub>1, \<dots>, 'a\<^sub>m) t"} with $m$ live type variables |
59284 | 631 |
and $n$ constructors @{text "t.C\<^sub>1"}, \ldots, @{text "t.C\<^sub>n"}, the following |
632 |
auxiliary constants are introduced: |
|
53617 | 633 |
|
55353 | 634 |
\medskip |
635 |
||
636 |
\begin{tabular}{@ {}ll@ {}} |
|
637 |
Case combinator: & |
|
638 |
@{text t.case_t} (rendered using the familiar @{text case}--@{text of} syntax) \\ |
|
639 |
Discriminators: & |
|
57988 | 640 |
@{text t.is_C\<^sub>1}$, \ldots, $@{text t.is_C\<^sub>n} \\ |
55353 | 641 |
Selectors: & |
642 |
@{text t.un_C\<^sub>11}$, \ldots, $@{text t.un_C\<^sub>1k\<^sub>1} \\ |
|
643 |
& \quad\vdots \\ |
|
644 |
& @{text t.un_C\<^sub>n1}$, \ldots, $@{text t.un_C\<^sub>nk\<^sub>n} \\ |
|
645 |
Set functions: & |
|
57988 | 646 |
@{text t.set\<^sub>1_t}, \ldots, @{text t.set\<^sub>m_t} \\ |
55353 | 647 |
Map function: & |
648 |
@{text t.map_t} \\ |
|
649 |
Relator: & |
|
650 |
@{text t.rel_t} \\ |
|
651 |
Recursor: & |
|
58190 | 652 |
@{text t.rec_t} |
55353 | 653 |
\end{tabular} |
654 |
||
655 |
\medskip |
|
53617 | 656 |
|
657 |
\noindent |
|
59722 | 658 |
The discriminators and selectors are generated only if the @{text "discs_sels"} |
659 |
option is enabled or if names are specified for discriminators or selectors. |
|
660 |
The set functions, map function, and relator are generated only if $m > 0$. |
|
661 |
||
58245 | 662 |
In addition, some of the plugins introduce their own constants |
59282 | 663 |
(Section~\ref{sec:selecting-plugins}). The case combinator, discriminators, |
58508 | 664 |
and selectors are collectively called \emph{destructors}. The prefix |
665 |
``@{text "t."}'' is an optional component of the names and is normally hidden. |
|
53617 | 666 |
*} |
667 |
||
668 |
||
52840 | 669 |
subsection {* Generated Theorems |
670 |
\label{ssec:datatype-generated-theorems} *} |
|
52828 | 671 |
|
672 |
text {* |
|
58310 | 673 |
The characteristic theorems generated by @{command datatype} are grouped in |
53623 | 674 |
three broad categories: |
53535 | 675 |
|
53543 | 676 |
\begin{itemize} |
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
677 |
\setlength{\itemsep}{0pt} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
678 |
|
56655 | 679 |
\item The \emph{free constructor theorems} |
680 |
(Section~\ref{sssec:free-constructor-theorems}) are properties of the |
|
681 |
constructors and destructors that can be derived for any freely generated type. |
|
682 |
Internally, the derivation is performed by @{command free_constructors}. |
|
683 |
||
684 |
\item The \emph{functorial theorems} (Section~\ref{sssec:functorial-theorems}) |
|
685 |
are properties of datatypes related to their BNF nature. |
|
686 |
||
687 |
\item The \emph{inductive theorems} (Section~\ref{sssec:inductive-theorems}) |
|
688 |
are properties of datatypes related to their inductive nature. |
|
53552 | 689 |
|
53543 | 690 |
\end{itemize} |
53535 | 691 |
|
692 |
\noindent |
|
53542 | 693 |
The full list of named theorems can be obtained as usual by entering the |
57542 | 694 |
command \keyw{print_theorems} immediately after the datatype definition. |
58508 | 695 |
This list includes theorems produced by plugins |
59282 | 696 |
(Section~\ref{sec:selecting-plugins}), but normally excludes low-level |
58508 | 697 |
theorems that reveal internal constructions. To make these accessible, add |
698 |
the line |
|
53542 | 699 |
*} |
53535 | 700 |
|
53542 | 701 |
declare [[bnf_note_all]] |
702 |
(*<*) |
|
703 |
declare [[bnf_note_all = false]] |
|
704 |
(*>*) |
|
53535 | 705 |
|
53552 | 706 |
text {* |
707 |
\noindent |
|
708 |
to the top of the theory file. |
|
709 |
*} |
|
53535 | 710 |
|
58245 | 711 |
|
53621 | 712 |
subsubsection {* Free Constructor Theorems |
713 |
\label{sssec:free-constructor-theorems} *} |
|
53535 | 714 |
|
53543 | 715 |
(*<*) |
53837 | 716 |
consts nonnull :: 'a |
53543 | 717 |
(*>*) |
718 |
||
53535 | 719 |
text {* |
54621 | 720 |
The free constructor theorems are partitioned in three subgroups. The first |
721 |
subgroup of properties is concerned with the constructors. They are listed below |
|
722 |
for @{typ "'a list"}: |
|
53543 | 723 |
|
53552 | 724 |
\begin{indentblock} |
53543 | 725 |
\begin{description} |
53544 | 726 |
|
53642 | 727 |
\item[@{text "t."}\hthm{inject} @{text "[iff, induct_simp]"}\rm:] ~ \\ |
53544 | 728 |
@{thm list.inject[no_vars]} |
729 |
||
53642 | 730 |
\item[@{text "t."}\hthm{distinct} @{text "[simp, induct_simp]"}\rm:] ~ \\ |
53543 | 731 |
@{thm list.distinct(1)[no_vars]} \\ |
732 |
@{thm list.distinct(2)[no_vars]} |
|
733 |
||
53642 | 734 |
\item[@{text "t."}\hthm{exhaust} @{text "[cases t, case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
53543 | 735 |
@{thm list.exhaust[no_vars]} |
736 |
||
53642 | 737 |
\item[@{text "t."}\hthm{nchotomy}\rm:] ~ \\ |
53543 | 738 |
@{thm list.nchotomy[no_vars]} |
739 |
||
740 |
\end{description} |
|
53552 | 741 |
\end{indentblock} |
53543 | 742 |
|
743 |
\noindent |
|
53621 | 744 |
In addition, these nameless theorems are registered as safe elimination rules: |
745 |
||
746 |
\begin{indentblock} |
|
747 |
\begin{description} |
|
748 |
||
54386 | 749 |
\item[@{text "t."}\hthm{distinct {\upshape[}THEN notE}@{text ", elim!"}\hthm{\upshape]}\rm:] ~ \\ |
53621 | 750 |
@{thm list.distinct(1)[THEN notE, elim!, no_vars]} \\ |
751 |
@{thm list.distinct(2)[THEN notE, elim!, no_vars]} |
|
752 |
||
753 |
\end{description} |
|
754 |
\end{indentblock} |
|
755 |
||
756 |
\noindent |
|
53543 | 757 |
The next subgroup is concerned with the case combinator: |
758 |
||
53552 | 759 |
\begin{indentblock} |
53543 | 760 |
\begin{description} |
53544 | 761 |
|
53798 | 762 |
\item[@{text "t."}\hthm{case} @{text "[simp, code]"}\rm:] ~ \\ |
53543 | 763 |
@{thm list.case(1)[no_vars]} \\ |
58335
a5a3b576fcfb
generate 'code' attribute only if 'code' plugin is enabled
blanchet
parents:
58311
diff
changeset
|
764 |
@{thm list.case(2)[no_vars]} \\ |
59284 | 765 |
The @{text "[code]"} attribute is set by the @{text code} plugin |
766 |
(Section~\ref{ssec:code-generator}). |
|
53543 | 767 |
|
57542 | 768 |
\item[@{text "t."}\hthm{case_cong} @{text "[fundef_cong]"}\rm:] ~ \\ |
53543 | 769 |
@{thm list.case_cong[no_vars]} |
770 |
||
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
771 |
\item[@{text "t."}\hthm{case_cong_weak} @{text "[cong]"}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
772 |
@{thm list.case_cong_weak[no_vars]} |
53543 | 773 |
|
59268 | 774 |
\item[@{text "t."}\hthm{case_distrib}\rm:] ~ \\ |
775 |
@{thm list.case_distrib[no_vars]} |
|
776 |
||
53642 | 777 |
\item[@{text "t."}\hthm{split}\rm:] ~ \\ |
53543 | 778 |
@{thm list.split[no_vars]} |
779 |
||
57542 | 780 |
\item[@{text "t."}\hthm{split_asm}\rm:] ~ \\ |
53543 | 781 |
@{thm list.split_asm[no_vars]} |
782 |
||
53544 | 783 |
\item[@{text "t."}\hthm{splits} = @{text "split split_asm"}] |
53543 | 784 |
|
785 |
\end{description} |
|
53552 | 786 |
\end{indentblock} |
53543 | 787 |
|
788 |
\noindent |
|
54621 | 789 |
The third subgroup revolves around discriminators and selectors: |
53543 | 790 |
|
53552 | 791 |
\begin{indentblock} |
53543 | 792 |
\begin{description} |
53544 | 793 |
|
53694 | 794 |
\item[@{text "t."}\hthm{disc} @{text "[simp]"}\rm:] ~ \\ |
795 |
@{thm list.disc(1)[no_vars]} \\ |
|
796 |
@{thm list.disc(2)[no_vars]} |
|
797 |
||
53703 | 798 |
\item[@{text "t."}\hthm{discI}\rm:] ~ \\ |
799 |
@{thm list.discI(1)[no_vars]} \\ |
|
800 |
@{thm list.discI(2)[no_vars]} |
|
801 |
||
53805 | 802 |
\item[@{text "t."}\hthm{sel} @{text "[simp, code]"}\rm:] ~ \\ |
53694 | 803 |
@{thm list.sel(1)[no_vars]} \\ |
58335
a5a3b576fcfb
generate 'code' attribute only if 'code' plugin is enabled
blanchet
parents:
58311
diff
changeset
|
804 |
@{thm list.sel(2)[no_vars]} \\ |
59284 | 805 |
The @{text "[code]"} attribute is set by the @{text code} plugin |
806 |
(Section~\ref{ssec:code-generator}). |
|
53543 | 807 |
|
53642 | 808 |
\item[@{text "t."}\hthm{collapse} @{text "[simp]"}\rm:] ~ \\ |
53543 | 809 |
@{thm list.collapse(1)[no_vars]} \\ |
58151
414deb2ef328
take out 'x = C' of the simplifier for unit types
blanchet
parents:
58121
diff
changeset
|
810 |
@{thm list.collapse(2)[no_vars]} \\ |
59284 | 811 |
The @{text "[simp]"} attribute is exceptionally omitted for datatypes equipped |
58151
414deb2ef328
take out 'x = C' of the simplifier for unit types
blanchet
parents:
58121
diff
changeset
|
812 |
with a single nullary constructor, because a property of the form |
59284 | 813 |
@{prop "x = C"} is not suitable as a simplification rule. |
53543 | 814 |
|
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
815 |
\item[@{text "t."}\hthm{distinct_disc} @{text "[dest]"}\rm:] ~ \\ |
53543 | 816 |
These properties are missing for @{typ "'a list"} because there is only one |
59284 | 817 |
proper discriminator. If the datatype had been introduced with a second |
53837 | 818 |
discriminator called @{const nonnull}, they would have read thusly: \\[\jot] |
819 |
@{prop "null list \<Longrightarrow> \<not> nonnull list"} \\ |
|
820 |
@{prop "nonnull list \<Longrightarrow> \<not> null list"} |
|
53543 | 821 |
|
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
822 |
\item[@{text "t."}\hthm{exhaust_disc} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
823 |
@{thm list.exhaust_disc[no_vars]} |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
824 |
|
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
825 |
\item[@{text "t."}\hthm{exhaust_sel} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
826 |
@{thm list.exhaust_sel[no_vars]} |
53916 | 827 |
|
53642 | 828 |
\item[@{text "t."}\hthm{expand}\rm:] ~ \\ |
53543 | 829 |
@{thm list.expand[no_vars]} |
830 |
||
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
831 |
\item[@{text "t."}\hthm{split_sel}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
832 |
@{thm list.split_sel[no_vars]} |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
833 |
|
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
834 |
\item[@{text "t."}\hthm{split_sel_asm}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
835 |
@{thm list.split_sel_asm[no_vars]} |
53917 | 836 |
|
57984
cbe9a16f8e11
added collection theorem for consistency and convenience
blanchet
parents:
57983
diff
changeset
|
837 |
\item[@{text "t."}\hthm{split_sels} = @{text "split_sel split_sel_asm"}] |
cbe9a16f8e11
added collection theorem for consistency and convenience
blanchet
parents:
57983
diff
changeset
|
838 |
|
57542 | 839 |
\item[@{text "t."}\hthm{case_eq_if}\rm:] ~ \\ |
54491 | 840 |
@{thm list.case_eq_if[no_vars]} |
53543 | 841 |
|
59273 | 842 |
\item[@{text "t."}\hthm{disc_eq_case}\rm:] ~ \\ |
843 |
@{thm list.disc_eq_case(1)[no_vars]} \\ |
|
844 |
@{thm list.disc_eq_case(2)[no_vars]} |
|
845 |
||
53543 | 846 |
\end{description} |
53552 | 847 |
\end{indentblock} |
54152 | 848 |
|
849 |
\noindent |
|
58151
414deb2ef328
take out 'x = C' of the simplifier for unit types
blanchet
parents:
58121
diff
changeset
|
850 |
In addition, equational versions of @{text t.disc} are registered with the |
59284 | 851 |
@{text "[code]"} attribute. The @{text "[code]"} attribute is set by the |
852 |
@{text code} plugin (Section~\ref{ssec:code-generator}). |
|
53552 | 853 |
*} |
854 |
||
855 |
||
53621 | 856 |
subsubsection {* Functorial Theorems |
857 |
\label{sssec:functorial-theorems} *} |
|
53552 | 858 |
|
859 |
text {* |
|
54621 | 860 |
The functorial theorems are partitioned in two subgroups. The first subgroup |
56992 | 861 |
consists of properties involving the constructors or the destructors and either |
862 |
a set function, the map function, or the relator: |
|
53552 | 863 |
|
864 |
\begin{indentblock} |
|
865 |
\begin{description} |
|
866 |
||
58915 | 867 |
\item[@{text "t."}\hthm{case_transfer} @{text "[transfer_rule]"}\rm:] ~ \\ |
868 |
@{thm list.case_transfer[no_vars]} \\ |
|
59284 | 869 |
The @{text "[transfer_rule]"} attribute is set by the @{text transfer} plugin |
59824 | 870 |
(Section~\ref{ssec:transfer}) for type constructors with no dead type arguments. |
58915 | 871 |
|
872 |
\item[@{text "t."}\hthm{sel_transfer} @{text "[transfer_rule]"}\rm:] ~ \\ |
|
58677 | 873 |
This property is missing for @{typ "'a list"} because there is no common |
58915 | 874 |
selector to all constructors. \\ |
59284 | 875 |
The @{text "[transfer_rule]"} attribute is set by the @{text transfer} plugin |
59824 | 876 |
(Section~\ref{ssec:transfer}) for type constructors with no dead type arguments. |
58915 | 877 |
|
878 |
\item[@{text "t."}\hthm{ctr_transfer} @{text "[transfer_rule]"}\rm:] ~ \\ |
|
58000 | 879 |
@{thm list.ctr_transfer(1)[no_vars]} \\ |
58915 | 880 |
@{thm list.ctr_transfer(2)[no_vars]} \\ |
59284 | 881 |
The @{text "[transfer_rule]"} attribute is set by the @{text transfer} plugin |
59824 | 882 |
(Section~\ref{ssec:transfer}) for type constructors with no dead type arguments. |
58915 | 883 |
|
884 |
\item[@{text "t."}\hthm{disc_transfer} @{text "[transfer_rule]"}\rm:] ~ \\ |
|
58096 | 885 |
@{thm list.disc_transfer(1)[no_vars]} \\ |
58915 | 886 |
@{thm list.disc_transfer(2)[no_vars]} \\ |
59284 | 887 |
The @{text "[transfer_rule]"} attribute is set by the @{text transfer} plugin |
59824 | 888 |
(Section~\ref{ssec:transfer}) for type constructors with no dead type arguments. |
58096 | 889 |
|
53798 | 890 |
\item[@{text "t."}\hthm{set} @{text "[simp, code]"}\rm:] ~ \\ |
53694 | 891 |
@{thm list.set(1)[no_vars]} \\ |
58335
a5a3b576fcfb
generate 'code' attribute only if 'code' plugin is enabled
blanchet
parents:
58311
diff
changeset
|
892 |
@{thm list.set(2)[no_vars]} \\ |
59284 | 893 |
The @{text "[code]"} attribute is set by the @{text code} plugin |
894 |
(Section~\ref{ssec:code-generator}). |
|
53552 | 895 |
|
57988 | 896 |
\item[@{text "t."}\hthm{set_cases} @{text "[consumes 1, cases set: set\<^sub>i_t]"}\rm:] ~ \\ |
57894 | 897 |
@{thm list.set_cases[no_vars]} |
898 |
||
57892 | 899 |
\item[@{text "t."}\hthm{set_intros}\rm:] ~ \\ |
900 |
@{thm list.set_intros(1)[no_vars]} \\ |
|
901 |
@{thm list.set_intros(2)[no_vars]} |
|
902 |
||
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
903 |
\item[@{text "t."}\hthm{set_sel}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
904 |
@{thm list.set_sel[no_vars]} |
57153 | 905 |
|
53798 | 906 |
\item[@{text "t."}\hthm{map} @{text "[simp, code]"}\rm:] ~ \\ |
53552 | 907 |
@{thm list.map(1)[no_vars]} \\ |
58335
a5a3b576fcfb
generate 'code' attribute only if 'code' plugin is enabled
blanchet
parents:
58311
diff
changeset
|
908 |
@{thm list.map(2)[no_vars]} \\ |
59284 | 909 |
The @{text "[code]"} attribute is set by the @{text code} plugin |
910 |
(Section~\ref{ssec:code-generator}). |
|
53552 | 911 |
|
57983
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
912 |
\item[@{text "t."}\hthm{map_disc_iff} @{text "[simp]"}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
913 |
@{thm list.map_disc_iff[no_vars]} |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
914 |
|
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
915 |
\item[@{text "t."}\hthm{map_sel}\rm:] ~ \\ |
6edc3529bb4e
reordered some (co)datatype property names for more consistency
blanchet
parents:
57982
diff
changeset
|
916 |
@{thm list.map_sel[no_vars]} |
57047 | 917 |
|
57542 | 918 |
\item[@{text "t."}\hthm{rel_inject} @{text "[simp]"}\rm:] ~ \\ |
53552 | 919 |
@{thm list.rel_inject(1)[no_vars]} \\ |
920 |
@{thm list.rel_inject(2)[no_vars]} |
|
921 |
||
57542 | 922 |
\item[@{text "t."}\hthm{rel_distinct} @{text "[simp]"}\rm:] ~ \\ |
57526 | 923 |
@{thm list.rel_distinct(1)[no_vars]} \\ |
924 |
@{thm list.rel_distinct(2)[no_vars]} |
|
925 |
||
57542 | 926 |
\item[@{text "t."}\hthm{rel_intros}\rm:] ~ \\ |
57494 | 927 |
@{thm list.rel_intros(1)[no_vars]} \\ |
928 |
@{thm list.rel_intros(2)[no_vars]} |
|
929 |
||
58474 | 930 |
\item[@{text "t."}\hthm{rel_cases} @{text "[consumes 1, case_names t\<^sub>1 \<dots> t\<^sub>m, cases pred]"}\rm:] ~ \\ |
931 |
@{thm list.rel_cases[no_vars]} |
|
53552 | 932 |
|
57564 | 933 |
\item[@{text "t."}\hthm{rel_sel}\rm:] ~ \\ |
934 |
@{thm list.rel_sel[no_vars]} |
|
935 |
||
53552 | 936 |
\end{description} |
937 |
\end{indentblock} |
|
54146 | 938 |
|
939 |
\noindent |
|
940 |
In addition, equational versions of @{text t.rel_inject} and @{text |
|
59284 | 941 |
rel_distinct} are registered with the @{text "[code]"} attribute. The |
942 |
@{text "[code]"} attribute is set by the @{text code} plugin |
|
943 |
(Section~\ref{ssec:code-generator}). |
|
54621 | 944 |
|
945 |
The second subgroup consists of more abstract properties of the set functions, |
|
946 |
the map function, and the relator: |
|
947 |
||
948 |
\begin{indentblock} |
|
949 |
\begin{description} |
|
950 |
||
57969 | 951 |
\item[@{text "t."}\hthm{inj_map}\rm:] ~ \\ |
952 |
@{thm list.inj_map[no_vars]} |
|
953 |
||
57971 | 954 |
\item[@{text "t."}\hthm{inj_map_strong}\rm:] ~ \\ |
955 |
@{thm list.inj_map_strong[no_vars]} |
|
956 |
||
57542 | 957 |
\item[@{text "t."}\hthm{set_map}\rm:] ~ \\ |
56992 | 958 |
@{thm list.set_map[no_vars]} |
959 |
||
58915 | 960 |
\item[@{text "t."}\hthm{set_transfer} @{text "[transfer_rule]"}\rm:] ~ \\ |
961 |
@{thm list.set_transfer[no_vars]} \\ |
|
59284 | 962 |
The @{text "[transfer_rule]"} attribute is set by the @{text transfer} plugin |
59824 | 963 |
(Section~\ref{ssec:transfer}) for type constructors with no dead type arguments. |
58107 | 964 |
|
58245 | 965 |
\item[@{text "t."}\hthm{map_cong0}\rm:] ~ \\ |
54621 | 966 |
@{thm list.map_cong0[no_vars]} |
967 |
||
57542 | 968 |
\item[@{text "t."}\hthm{map_cong} @{text "[fundef_cong]"}\rm:] ~ \\ |
54621 | 969 |
@{thm list.map_cong[no_vars]} |
970 |
||
57982 | 971 |
\item[@{text "t."}\hthm{map_cong_simp}\rm:] ~ \\ |
972 |
@{thm list.map_cong_simp[no_vars]} |
|
973 |
||
59793 | 974 |
\item[@{text "t."}\hthm{map_id0}\rm:] ~ \\ |
975 |
@{thm list.map_id0[no_vars]} |
|
976 |
||
57542 | 977 |
\item[@{text "t."}\hthm{map_id}\rm:] ~ \\ |
54621 | 978 |
@{thm list.map_id[no_vars]} |
979 |
||
57542 | 980 |
\item[@{text "t."}\hthm{map_ident}\rm:] ~ \\ |
56904 | 981 |
@{thm list.map_ident[no_vars]} |
982 |
||
58915 | 983 |
\item[@{text "t."}\hthm{map_transfer} @{text "[transfer_rule]"}\rm:] ~ \\ |
984 |
@{thm list.map_transfer[no_vars]} \\ |
|
59284 | 985 |
The @{text "[transfer_rule]"} attribute is set by the @{text transfer} plugin |
59824 | 986 |
(Section~\ref{ssec:transfer}) for type constructors with no dead type arguments. |
58103 | 987 |
|
58244 | 988 |
\item[@{text "t."}\hthm{rel_compp} @{text "[relator_distr]"}\rm:] ~ \\ |
58245 | 989 |
@{thm list.rel_compp[no_vars]} \\ |
59284 | 990 |
The @{text "[relator_distr]"} attribute is set by the @{text lifting} plugin |
991 |
(Section~\ref{ssec:lifting}). |
|
54621 | 992 |
|
57542 | 993 |
\item[@{text "t."}\hthm{rel_conversep}\rm:] ~ \\ |
54621 | 994 |
@{thm list.rel_conversep[no_vars]} |
995 |
||
57542 | 996 |
\item[@{text "t."}\hthm{rel_eq}\rm:] ~ \\ |
54621 | 997 |
@{thm list.rel_eq[no_vars]} |
998 |
||
57542 | 999 |
\item[@{text "t."}\hthm{rel_flip}\rm:] ~ \\ |
54621 | 1000 |
@{thm list.rel_flip[no_vars]} |
1001 |
||
57933 | 1002 |
\item[@{text "t."}\hthm{rel_map}\rm:] ~ \\ |
1003 |
@{thm list.rel_map(1)[no_vars]} \\ |
|
1004 |
@{thm list.rel_map(2)[no_vars]} |
|
1005 |
||
58344 | 1006 |
\item[@{text "t."}\hthm{rel_mono} @{text "[mono, relator_mono]"}\rm:] ~ \\ |
58245 | 1007 |
@{thm list.rel_mono[no_vars]} \\ |
59284 | 1008 |
The @{text "[relator_mono]"} attribute is set by the @{text lifting} plugin |
1009 |
(Section~\ref{ssec:lifting}). |
|
54621 | 1010 |
|
59793 | 1011 |
\item[@{text "t."}\hthm{rel_refl}\rm:] ~ \\ |
1012 |
@{thm list.rel_refl[no_vars]} |
|
1013 |
||
58915 | 1014 |
\item[@{text "t."}\hthm{rel_transfer} @{text "[transfer_rule]"}\rm:] ~ \\ |
1015 |
@{thm list.rel_transfer[no_vars]} \\ |
|
59284 | 1016 |
The @{text "[transfer_rule]"} attribute is set by the @{text transfer} plugin |
59824 | 1017 |
(Section~\ref{ssec:transfer}) for type constructors with no dead type arguments. |
58105 | 1018 |
|
54621 | 1019 |
\end{description} |
1020 |
\end{indentblock} |
|
53535 | 1021 |
*} |
1022 |
||
1023 |
||
53621 | 1024 |
subsubsection {* Inductive Theorems |
1025 |
\label{sssec:inductive-theorems} *} |
|
53535 | 1026 |
|
1027 |
text {* |
|
53623 | 1028 |
The inductive theorems are as follows: |
53544 | 1029 |
|
53552 | 1030 |
\begin{indentblock} |
53544 | 1031 |
\begin{description} |
1032 |
||
57304 | 1033 |
\item[@{text "t."}\hthm{induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n, induct t]"}\rm:] ~ \\ |
53544 | 1034 |
@{thm list.induct[no_vars]} |
1035 |
||
57542 | 1036 |
\item[@{text "t."}\hthm{rel_induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n, induct pred]"}\rm:] ~ \\ |
57472 | 1037 |
@{thm list.rel_induct[no_vars]} |
1038 |
||
57701 | 1039 |
\item[\begin{tabular}{@ {}l@ {}} |
1040 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm: \\ |
|
1041 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{rel_induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm: \\ |
|
1042 |
\end{tabular}] ~ \\ |
|
57472 | 1043 |
Given $m > 1$ mutually recursive datatypes, this induction rule can be used to |
1044 |
prove $m$ properties simultaneously. |
|
1045 |
||
53798 | 1046 |
\item[@{text "t."}\hthm{rec} @{text "[simp, code]"}\rm:] ~ \\ |
53544 | 1047 |
@{thm list.rec(1)[no_vars]} \\ |
58335
a5a3b576fcfb
generate 'code' attribute only if 'code' plugin is enabled
blanchet
parents:
58311
diff
changeset
|
1048 |
@{thm list.rec(2)[no_vars]} \\ |
59284 | 1049 |
The @{text "[code]"} attribute is set by the @{text code} plugin |
1050 |
(Section~\ref{ssec:code-generator}). |
|
53544 | 1051 |
|
58733 | 1052 |
\item[@{text "t."}\hthm{rec_o_map}\rm:] ~ \\ |
1053 |
@{thm list.rec_o_map[no_vars]} |
|
1054 |
||
58915 | 1055 |
\item[@{text "t."}\hthm{rec_transfer} @{text "[transfer_rule]"}\rm:] ~ \\ |
1056 |
@{thm list.rec_transfer[no_vars]} \\ |
|
59284 | 1057 |
The @{text "[transfer_rule]"} attribute is set by the @{text transfer} plugin |
59824 | 1058 |
(Section~\ref{ssec:transfer}) for type constructors with no dead type arguments. |
58447 | 1059 |
|
53544 | 1060 |
\end{description} |
53552 | 1061 |
\end{indentblock} |
53544 | 1062 |
|
1063 |
\noindent |
|
58310 | 1064 |
For convenience, @{command datatype} also provides the following collection: |
53544 | 1065 |
|
53552 | 1066 |
\begin{indentblock} |
53544 | 1067 |
\begin{description} |
1068 |
||
59284 | 1069 |
\item[@{text "t."}\hthm{simps}] = @{text t.inject} @{text t.distinct} @{text t.case} @{text t.rec} @{text t.map} @{text t.rel_inject} \\ |
53694 | 1070 |
@{text t.rel_distinct} @{text t.set} |
53544 | 1071 |
|
1072 |
\end{description} |
|
53552 | 1073 |
\end{indentblock} |
52828 | 1074 |
*} |
1075 |
||
52794 | 1076 |
|
52827 | 1077 |
subsection {* Compatibility Issues |
52824 | 1078 |
\label{ssec:datatype-compatibility-issues} *} |
52794 | 1079 |
|
52828 | 1080 |
text {* |
58310 | 1081 |
The command @{command datatype} has been designed to be highly compatible |
58305
57752a91eec4
renamed 'datatype' to 'old_datatype'; 'datatype' is now alias for 'datatype_new'
blanchet
parents:
58298
diff
changeset
|
1082 |
with the old command (which is now called \keyw{old_datatype}), to ease |
57752a91eec4
renamed 'datatype' to 'old_datatype'; 'datatype' is now alias for 'datatype_new'
blanchet
parents:
58298
diff
changeset
|
1083 |
migration. There are nonetheless a few incompatibilities that may arise when |
57752a91eec4
renamed 'datatype' to 'old_datatype'; 'datatype' is now alias for 'datatype_new'
blanchet
parents:
58298
diff
changeset
|
1084 |
porting: |
53647 | 1085 |
|
1086 |
\begin{itemize} |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1087 |
\setlength{\itemsep}{0pt} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1088 |
|
53647 | 1089 |
\item \emph{The Standard ML interfaces are different.} Tools and extensions |
1090 |
written to call the old ML interfaces will need to be adapted to the new |
|
60135
852f7a49ec0c
updated docs, esp. relating to 'datatype_compat'
blanchet
parents:
60134
diff
changeset
|
1091 |
interfaces. The @{text BNF_LFP_Compat} structure provides convenience functions |
852f7a49ec0c
updated docs, esp. relating to 'datatype_compat'
blanchet
parents:
60134
diff
changeset
|
1092 |
that simulate the old interfaces in terms of the new ones. |
54537 | 1093 |
|
1094 |
\item \emph{The recursor @{text rec_t} has a different signature for nested |
|
54185 | 1095 |
recursive datatypes.} In the old package, nested recursion through non-functions |
1096 |
was internally reduced to mutual recursion. This reduction was visible in the |
|
1097 |
type of the recursor, used by \keyw{primrec}. Recursion through functions was |
|
1098 |
handled specially. In the new package, nested recursion (for functions and |
|
1099 |
non-functions) is handled in a more modular fashion. The old-style recursor can |
|
56655 | 1100 |
be generated on demand using @{command primrec} if the recursion is via |
1101 |
new-style datatypes, as explained in |
|
1102 |
Section~\ref{sssec:primrec-nested-as-mutual-recursion}. |
|
53647 | 1103 |
|
54287 | 1104 |
\item \emph{Accordingly, the induction rule is different for nested recursive |
1105 |
datatypes.} Again, the old-style induction rule can be generated on demand using |
|
56655 | 1106 |
@{command primrec} if the recursion is via new-style datatypes, as explained in |
58298 | 1107 |
Section~\ref{sssec:primrec-nested-as-mutual-recursion}. For recursion through |
1108 |
functions, the old-style induction rule can be obtained by applying the |
|
1109 |
@{text "[unfolded all_mem_range]"} attribute on @{text t.induct}. |
|
52828 | 1110 |
|
58336 | 1111 |
\item \emph{The @{const size} function has a slightly different definition.} |
1112 |
The new function returns @{text 1} instead of @{text 0} for some nonrecursive |
|
1113 |
constructors. This departure from the old behavior made it possible to implement |
|
58737 | 1114 |
@{const size} in terms of the generic function @{text "t.size_t"}. |
58339 | 1115 |
Moreover, the new function considers nested occurrences of a value, in the nested |
1116 |
recursive case. The old behavior can be obtained by disabling the @{text size} |
|
59282 | 1117 |
plugin (Section~\ref{sec:selecting-plugins}) and instantiating the |
58339 | 1118 |
@{text size} type class manually. |
58336 | 1119 |
|
53863
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
1120 |
\item \emph{The internal constructions are completely different.} Proof texts |
58311 | 1121 |
that unfold the definition of constants introduced by \keyw{old_datatype} will |
58305
57752a91eec4
renamed 'datatype' to 'old_datatype'; 'datatype' is now alias for 'datatype_new'
blanchet
parents:
58298
diff
changeset
|
1122 |
be difficult to port. |
53647 | 1123 |
|
58207 | 1124 |
\item \emph{Some constants and theorems have different names.} |
55641
5b466efedd2c
renamed 'recs' and 'cases' theorems 'rec' and 'case' in old datatype package, for consistency with new package
blanchet
parents:
55531
diff
changeset
|
1125 |
For non-mutually recursive datatypes, |
5b466efedd2c
renamed 'recs' and 'cases' theorems 'rec' and 'case' in old datatype package, for consistency with new package
blanchet
parents:
55531
diff
changeset
|
1126 |
the alias @{text t.inducts} for @{text t.induct} is no longer generated. |
53647 | 1127 |
For $m > 1$ mutually recursive datatypes, |
58207 | 1128 |
@{text "rec_t\<^sub>1_\<dots>_t\<^sub>m_i"} has been renamed |
1129 |
@{text "rec_t\<^sub>i"} for each @{text "i \<in> {1, \<dots>, t}"}, |
|
53997 | 1130 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m.inducts(i)"} has been renamed |
58207 | 1131 |
@{text "t\<^sub>i.induct"} for each @{text "i \<in> {1, \<dots>, t}"}, and the |
1132 |
collection @{text "t\<^sub>1_\<dots>_t\<^sub>m.size"} (generated by the |
|
1133 |
@{text size} plugin, Section~\ref{ssec:size}) has been divided into |
|
1134 |
@{text "t\<^sub>1.size"}, \ldots, @{text "t\<^sub>m.size"}. |
|
53647 | 1135 |
|
1136 |
\item \emph{The @{text t.simps} collection has been extended.} |
|
58207 | 1137 |
Previously available theorems are available at the same index as before. |
53647 | 1138 |
|
1139 |
\item \emph{Variables in generated properties have different names.} This is |
|
1140 |
rarely an issue, except in proof texts that refer to variable names in the |
|
1141 |
@{text "[where \<dots>]"} attribute. The solution is to use the more robust |
|
1142 |
@{text "[of \<dots>]"} syntax. |
|
1143 |
\end{itemize} |
|
1144 |
||
1145 |
In the other direction, there is currently no way to register old-style |
|
1146 |
datatypes as new-style datatypes. If the goal is to define new-style datatypes |
|
1147 |
with nested recursion through old-style datatypes, the old-style |
|
1148 |
datatypes can be registered as a BNF |
|
59298 | 1149 |
(Section~\ref{sec:registering-bounded-natural-functors}). If the goal is |
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
1150 |
to derive discriminators and selectors, this can be achieved using |
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
1151 |
@{command free_constructors} |
53647 | 1152 |
(Section~\ref{sec:deriving-destructors-and-theorems-for-free-constructors}). |
52828 | 1153 |
*} |
1154 |
||
52792 | 1155 |
|
59861 | 1156 |
section {* Defining Primitively Recursive Functions |
1157 |
\label{sec:defining-primitively-recursive-functions} *} |
|
52805 | 1158 |
|
1159 |
text {* |
|
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1160 |
Recursive functions over datatypes can be specified using the @{command primrec} |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1161 |
command, which supports primitive recursion, or using the more general |
59861 | 1162 |
\keyw{fun}, \keyw{function}, and \keyw{partial_function} commands. In this |
1163 |
tutorial, the focus is on @{command primrec}; \keyw{fun} and \keyw{function} are |
|
1164 |
described in a separate tutorial @{cite "isabelle-function"}. |
|
52805 | 1165 |
*} |
52792 | 1166 |
|
52805 | 1167 |
|
53617 | 1168 |
subsection {* Introductory Examples |
1169 |
\label{ssec:primrec-introductory-examples} *} |
|
52828 | 1170 |
|
53646 | 1171 |
text {* |
1172 |
Primitive recursion is illustrated through concrete examples based on the |
|
1173 |
datatypes defined in Section~\ref{ssec:datatype-introductory-examples}. More |
|
58309
a09ec6daaa19
renamed 'BNF_Examples' to 'Datatype_Examples' (cf. 'datatypes.pdf')
blanchet
parents:
58305
diff
changeset
|
1174 |
examples can be found in the directory \verb|~~/src/HOL/Datatype_Examples|. |
53646 | 1175 |
*} |
1176 |
||
53621 | 1177 |
|
1178 |
subsubsection {* Nonrecursive Types |
|
1179 |
\label{sssec:primrec-nonrecursive-types} *} |
|
52828 | 1180 |
|
52841 | 1181 |
text {* |
53621 | 1182 |
Primitive recursion removes one layer of constructors on the left-hand side in |
1183 |
each equation. For example: |
|
52841 | 1184 |
*} |
1185 |
||
60136 | 1186 |
primrec (nonexhaustive) bool_of_trool :: "trool \<Rightarrow> bool" where |
53621 | 1187 |
"bool_of_trool Faalse \<longleftrightarrow> False" | |
1188 |
"bool_of_trool Truue \<longleftrightarrow> True" |
|
52841 | 1189 |
|
53621 | 1190 |
text {* \blankline *} |
52841 | 1191 |
|
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1192 |
primrec the_list :: "'a option \<Rightarrow> 'a list" where |
53025 | 1193 |
"the_list None = []" | |
1194 |
"the_list (Some a) = [a]" |
|
52841 | 1195 |
|
53621 | 1196 |
text {* \blankline *} |
1197 |
||
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1198 |
primrec the_default :: "'a \<Rightarrow> 'a option \<Rightarrow> 'a" where |
53025 | 1199 |
"the_default d None = d" | |
1200 |
"the_default _ (Some a) = a" |
|
52843 | 1201 |
|
53621 | 1202 |
text {* \blankline *} |
1203 |
||
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1204 |
primrec mirrror :: "('a, 'b, 'c) triple \<Rightarrow> ('c, 'b, 'a) triple" where |
52841 | 1205 |
"mirrror (Triple a b c) = Triple c b a" |
1206 |
||
53621 | 1207 |
text {* |
1208 |
\noindent |
|
1209 |
The equations can be specified in any order, and it is acceptable to leave out |
|
1210 |
some cases, which are then unspecified. Pattern matching on the left-hand side |
|
1211 |
is restricted to a single datatype, which must correspond to the same argument |
|
1212 |
in all equations. |
|
1213 |
*} |
|
52828 | 1214 |
|
53621 | 1215 |
|
1216 |
subsubsection {* Simple Recursion |
|
1217 |
\label{sssec:primrec-simple-recursion} *} |
|
52828 | 1218 |
|
52841 | 1219 |
text {* |
53621 | 1220 |
For simple recursive types, recursive calls on a constructor argument are |
1221 |
allowed on the right-hand side: |
|
52841 | 1222 |
*} |
1223 |
||
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1224 |
primrec replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where |
53330
77da8d3c46e0
fixed docs w.r.t. availability of "primrec_new" and friends
blanchet
parents:
53262
diff
changeset
|
1225 |
"replicate Zero _ = []" | |
58245 | 1226 |
"replicate (Succ n) x = x # replicate n x" |
52841 | 1227 |
|
53621 | 1228 |
text {* \blankline *} |
52843 | 1229 |
|
60136 | 1230 |
primrec (nonexhaustive) at :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" where |
53644 | 1231 |
"at (x # xs) j = |
52843 | 1232 |
(case j of |
53644 | 1233 |
Zero \<Rightarrow> x |
58245 | 1234 |
| Succ j' \<Rightarrow> at xs j')" |
52843 | 1235 |
|
53621 | 1236 |
text {* \blankline *} |
1237 |
||
59284 | 1238 |
primrec (*<*)(in early) (transfer) (*>*)tfold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) tlist \<Rightarrow> 'b" where |
53644 | 1239 |
"tfold _ (TNil y) = y" | |
1240 |
"tfold f (TCons x xs) = f x (tfold f xs)" |
|
52841 | 1241 |
|
53025 | 1242 |
text {* |
53621 | 1243 |
\noindent |
54402 | 1244 |
Pattern matching is only available for the argument on which the recursion takes |
1245 |
place. Fortunately, it is easy to generate pattern-maching equations using the |
|
57542 | 1246 |
\keyw{simps_of_case} command provided by the theory |
55290 | 1247 |
\verb|~~/src/HOL/|\allowbreak\verb|Library/|\allowbreak\verb|Simps_Case_Conv|. |
54402 | 1248 |
*} |
1249 |
||
1250 |
simps_of_case at_simps: at.simps |
|
1251 |
||
1252 |
text {* |
|
1253 |
This generates the lemma collection @{thm [source] at_simps}: |
|
1254 |
% |
|
1255 |
\[@{thm at_simps(1)[no_vars]} |
|
1256 |
\qquad @{thm at_simps(2)[no_vars]}\] |
|
1257 |
% |
|
54184 | 1258 |
The next example is defined using \keyw{fun} to escape the syntactic |
60135
852f7a49ec0c
updated docs, esp. relating to 'datatype_compat'
blanchet
parents:
60134
diff
changeset
|
1259 |
restrictions imposed on primitively recursive functions: |
53025 | 1260 |
*} |
52828 | 1261 |
|
53621 | 1262 |
fun at_least_two :: "nat \<Rightarrow> bool" where |
58245 | 1263 |
"at_least_two (Succ (Succ _)) \<longleftrightarrow> True" | |
53621 | 1264 |
"at_least_two _ \<longleftrightarrow> False" |
1265 |
||
1266 |
||
1267 |
subsubsection {* Mutual Recursion |
|
1268 |
\label{sssec:primrec-mutual-recursion} *} |
|
52828 | 1269 |
|
52841 | 1270 |
text {* |
53621 | 1271 |
The syntax for mutually recursive functions over mutually recursive datatypes |
1272 |
is straightforward: |
|
52841 | 1273 |
*} |
1274 |
||
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|
1275 |
primrec |
53623 | 1276 |
nat_of_even_nat :: "even_nat \<Rightarrow> nat" and |
1277 |
nat_of_odd_nat :: "odd_nat \<Rightarrow> nat" |
|
52841 | 1278 |
where |
53623 | 1279 |
"nat_of_even_nat Even_Zero = Zero" | |
58245 | 1280 |
"nat_of_even_nat (Even_Succ n) = Succ (nat_of_odd_nat n)" | |
1281 |
"nat_of_odd_nat (Odd_Succ n) = Succ (nat_of_even_nat n)" |
|
52841 | 1282 |
|
53752 | 1283 |
text {* \blankline *} |
1284 |
||
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diff
changeset
|
1285 |
primrec |
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diff
changeset
|
1286 |
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
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53262
diff
changeset
|
1287 |
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
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diff
changeset
|
1288 |
eval\<^sub>f :: "('a \<Rightarrow> int) \<Rightarrow> ('b \<Rightarrow> int) \<Rightarrow> ('a, 'b) fct \<Rightarrow> int" |
52841 | 1289 |
where |
1290 |
"eval\<^sub>e \<gamma> \<xi> (Term t) = eval\<^sub>t \<gamma> \<xi> t" | |
|
1291 |
"eval\<^sub>e \<gamma> \<xi> (Sum t e) = eval\<^sub>t \<gamma> \<xi> t + eval\<^sub>e \<gamma> \<xi> e" | |
|
53330
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diff
changeset
|
1292 |
"eval\<^sub>t \<gamma> \<xi> (Factor f) = eval\<^sub>f \<gamma> \<xi> f" | |
52841 | 1293 |
"eval\<^sub>t \<gamma> \<xi> (Prod f t) = eval\<^sub>f \<gamma> \<xi> f + eval\<^sub>t \<gamma> \<xi> t" | |
1294 |
"eval\<^sub>f \<gamma> _ (Const a) = \<gamma> a" | |
|
1295 |
"eval\<^sub>f _ \<xi> (Var b) = \<xi> b" | |
|
1296 |
"eval\<^sub>f \<gamma> \<xi> (Expr e) = eval\<^sub>e \<gamma> \<xi> e" |
|
1297 |
||
53621 | 1298 |
text {* |
1299 |
\noindent |
|
53647 | 1300 |
Mutual recursion is possible within a single type, using \keyw{fun}: |
53621 | 1301 |
*} |
52828 | 1302 |
|
53621 | 1303 |
fun |
1304 |
even :: "nat \<Rightarrow> bool" and |
|
1305 |
odd :: "nat \<Rightarrow> bool" |
|
1306 |
where |
|
1307 |
"even Zero = True" | |
|
58245 | 1308 |
"even (Succ n) = odd n" | |
53621 | 1309 |
"odd Zero = False" | |
58245 | 1310 |
"odd (Succ n) = even n" |
53621 | 1311 |
|
1312 |
||
1313 |
subsubsection {* Nested Recursion |
|
1314 |
\label{sssec:primrec-nested-recursion} *} |
|
1315 |
||
1316 |
text {* |
|
1317 |
In a departure from the old datatype package, nested recursion is normally |
|
1318 |
handled via the map functions of the nesting type constructors. For example, |
|
1319 |
recursive calls are lifted to lists using @{const map}: |
|
1320 |
*} |
|
52828 | 1321 |
|
52843 | 1322 |
(*<*) |
58310 | 1323 |
datatype '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 | 1324 |
(*>*) |
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|
1325 |
primrec at\<^sub>f\<^sub>f :: "'a tree\<^sub>f\<^sub>f \<Rightarrow> nat list \<Rightarrow> 'a" where |
53028 | 1326 |
"at\<^sub>f\<^sub>f (Node\<^sub>f\<^sub>f a ts) js = |
52843 | 1327 |
(case js of |
1328 |
[] \<Rightarrow> a |
|
53028 | 1329 |
| j # js' \<Rightarrow> at (map (\<lambda>t. at\<^sub>f\<^sub>f t js') ts) j)" |
52843 | 1330 |
|
53025 | 1331 |
text {* |
53647 | 1332 |
\noindent |
53621 | 1333 |
The next example features recursion through the @{text option} type. Although |
53623 | 1334 |
@{text option} is not a new-style datatype, it is registered as a BNF with the |
54491 | 1335 |
map function @{const map_option}: |
53025 | 1336 |
*} |
52843 | 1337 |
|
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changeset
|
1338 |
primrec (*<*)(in early) (*>*)sum_btree :: "('a\<Colon>{zero,plus}) btree \<Rightarrow> 'a" where |
52843 | 1339 |
"sum_btree (BNode a lt rt) = |
54491 | 1340 |
a + the_default 0 (map_option sum_btree lt) + |
1341 |
the_default 0 (map_option sum_btree rt)" |
|
52843 | 1342 |
|
53136 | 1343 |
text {* |
53621 | 1344 |
\noindent |
1345 |
The same principle applies for arbitrary type constructors through which |
|
1346 |
recursion is possible. Notably, the map function for the function type |
|
1347 |
(@{text \<Rightarrow>}) is simply composition (@{text "op \<circ>"}): |
|
53136 | 1348 |
*} |
52828 | 1349 |
|
55530
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diff
changeset
|
1350 |
primrec (*<*)(in early) (*>*)relabel_ft :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where |
54182 | 1351 |
"relabel_ft f (FTLeaf x) = FTLeaf (f x)" | |
1352 |
"relabel_ft f (FTNode g) = FTNode (relabel_ft f \<circ> g)" |
|
1353 |
||
1354 |
text {* |
|
1355 |
\noindent |
|
1356 |
For convenience, recursion through functions can also be expressed using |
|
1357 |
$\lambda$-abstractions and function application rather than through composition. |
|
1358 |
For example: |
|
1359 |
*} |
|
1360 |
||
55530
3dfb724db099
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diff
changeset
|
1361 |
primrec relabel_ft :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where |
54182 | 1362 |
"relabel_ft f (FTLeaf x) = FTLeaf (f x)" | |
1363 |
"relabel_ft f (FTNode g) = FTNode (\<lambda>x. relabel_ft f (g x))" |
|
52828 | 1364 |
|
54183 | 1365 |
text {* \blankline *} |
1366 |
||
60136 | 1367 |
primrec (nonexhaustive) subtree_ft :: "'a \<Rightarrow> 'a ftree \<Rightarrow> 'a ftree" where |
54183 | 1368 |
"subtree_ft x (FTNode g) = g x" |
1369 |
||
52843 | 1370 |
text {* |
53621 | 1371 |
\noindent |
54182 | 1372 |
For recursion through curried $n$-ary functions, $n$ applications of |
1373 |
@{term "op \<circ>"} are necessary. The examples below illustrate the case where |
|
1374 |
$n = 2$: |
|
53621 | 1375 |
*} |
1376 |
||
58310 | 1377 |
datatype 'a ftree2 = FTLeaf2 'a | FTNode2 "'a \<Rightarrow> 'a \<Rightarrow> 'a ftree2" |
54182 | 1378 |
|
1379 |
text {* \blankline *} |
|
1380 |
||
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diff
changeset
|
1381 |
primrec (*<*)(in early) (*>*)relabel_ft2 :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree2 \<Rightarrow> 'a ftree2" where |
54182 | 1382 |
"relabel_ft2 f (FTLeaf2 x) = FTLeaf2 (f x)" | |
1383 |
"relabel_ft2 f (FTNode2 g) = FTNode2 (op \<circ> (op \<circ> (relabel_ft2 f)) g)" |
|
1384 |
||
1385 |
text {* \blankline *} |
|
1386 |
||
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diff
changeset
|
1387 |
primrec relabel_ft2 :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a ftree2 \<Rightarrow> 'a ftree2" where |
54182 | 1388 |
"relabel_ft2 f (FTLeaf2 x) = FTLeaf2 (f x)" | |
1389 |
"relabel_ft2 f (FTNode2 g) = FTNode2 (\<lambda>x y. relabel_ft2 f (g x y))" |
|
54031 | 1390 |
|
54183 | 1391 |
text {* \blankline *} |
1392 |
||
60136 | 1393 |
primrec (nonexhaustive) subtree_ft2 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a ftree2 \<Rightarrow> 'a ftree2" where |
54183 | 1394 |
"subtree_ft2 x y (FTNode2 g) = g x y" |
1395 |
||
53621 | 1396 |
|
1397 |
subsubsection {* Nested-as-Mutual Recursion |
|
53644 | 1398 |
\label{sssec:primrec-nested-as-mutual-recursion} *} |
53621 | 1399 |
|
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|
1400 |
(*<*) |
59284 | 1401 |
locale n2m |
1402 |
begin |
|
53749
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adapted primcorec documentation to reflect the three views
blanchet
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diff
changeset
|
1403 |
(*>*) |
b37db925b663
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blanchet
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changeset
|
1404 |
|
53621 | 1405 |
text {* |
1406 |
For compatibility with the old package, but also because it is sometimes |
|
1407 |
convenient in its own right, it is possible to treat nested recursive datatypes |
|
1408 |
as mutually recursive ones if the recursion takes place though new-style |
|
1409 |
datatypes. For example: |
|
52843 | 1410 |
*} |
1411 |
||
60136 | 1412 |
primrec (nonexhaustive) |
53647 | 1413 |
at\<^sub>f\<^sub>f :: "'a tree\<^sub>f\<^sub>f \<Rightarrow> nat list \<Rightarrow> 'a" and |
1414 |
ats\<^sub>f\<^sub>f :: "'a tree\<^sub>f\<^sub>f list \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> 'a" |
|
52843 | 1415 |
where |
53647 | 1416 |
"at\<^sub>f\<^sub>f (Node\<^sub>f\<^sub>f a ts) js = |
52843 | 1417 |
(case js of |
1418 |
[] \<Rightarrow> a |
|
53647 | 1419 |
| j # js' \<Rightarrow> ats\<^sub>f\<^sub>f ts j js')" | |
1420 |
"ats\<^sub>f\<^sub>f (t # ts) j = |
|
52843 | 1421 |
(case j of |
53647 | 1422 |
Zero \<Rightarrow> at\<^sub>f\<^sub>f t |
58245 | 1423 |
| Succ j' \<Rightarrow> ats\<^sub>f\<^sub>f ts j')" |
52843 | 1424 |
|
53647 | 1425 |
text {* |
1426 |
\noindent |
|
54287 | 1427 |
Appropriate induction rules are generated as |
54031 | 1428 |
@{thm [source] at\<^sub>f\<^sub>f.induct}, |
1429 |
@{thm [source] ats\<^sub>f\<^sub>f.induct}, and |
|
54287 | 1430 |
@{thm [source] at\<^sub>f\<^sub>f_ats\<^sub>f\<^sub>f.induct}. The |
1431 |
induction rules and the underlying recursors are generated on a per-need basis |
|
1432 |
and are kept in a cache to speed up subsequent definitions. |
|
53647 | 1433 |
|
1434 |
Here is a second example: |
|
1435 |
*} |
|
53621 | 1436 |
|
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
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parents:
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diff
changeset
|
1437 |
primrec |
53330
77da8d3c46e0
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changeset
|
1438 |
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
|
1439 |
sum_btree_option :: "'a btree option \<Rightarrow> 'a" |
52843 | 1440 |
where |
1441 |
"sum_btree (BNode a lt rt) = |
|
53025 | 1442 |
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
|
1443 |
"sum_btree_option None = 0" | |
53025 | 1444 |
"sum_btree_option (Some t) = sum_btree t" |
52843 | 1445 |
|
1446 |
text {* |
|
53621 | 1447 |
% * can pretend a nested type is mutually recursive (if purely inductive) |
1448 |
% * avoids the higher-order map |
|
1449 |
% * e.g. |
|
1450 |
||
53617 | 1451 |
% * this can always be avoided; |
1452 |
% * e.g. in our previous example, we first mapped the recursive |
|
1453 |
% calls, then we used a generic at function to retrieve the result |
|
1454 |
% |
|
1455 |
% * there's no hard-and-fast rule of when to use one or the other, |
|
1456 |
% just like there's no rule when to use fold and when to use |
|
55530
3dfb724db099
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blanchet
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diff
changeset
|
1457 |
% primrec |
53617 | 1458 |
% |
1459 |
% * higher-order approach, considering nesting as nesting, is more |
|
1460 |
% compositional -- e.g. we saw how we could reuse an existing polymorphic |
|
53647 | 1461 |
% at or the_default, whereas @{const ats\<^sub>f\<^sub>f} is much more specific |
53617 | 1462 |
% |
1463 |
% * but: |
|
1464 |
% * is perhaps less intuitive, because it requires higher-order thinking |
|
1465 |
% * may seem inefficient, and indeed with the code generator the |
|
1466 |
% mutually recursive version might be nicer |
|
1467 |
% * is somewhat indirect -- must apply a map first, then compute a result |
|
1468 |
% (cannot mix) |
|
53647 | 1469 |
% * the auxiliary functions like @{const ats\<^sub>f\<^sub>f} are sometimes useful in own right |
53617 | 1470 |
% |
1471 |
% * impact on automation unclear |
|
1472 |
% |
|
52843 | 1473 |
*} |
53749
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changeset
|
1474 |
(*<*) |
b37db925b663
adapted primcorec documentation to reflect the three views
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diff
changeset
|
1475 |
end |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
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diff
changeset
|
1476 |
(*>*) |
52843 | 1477 |
|
52824 | 1478 |
|
53617 | 1479 |
subsection {* Command Syntax |
1480 |
\label{ssec:primrec-command-syntax} *} |
|
1481 |
||
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changeset
|
1482 |
subsubsection {* \keyw{primrec} |
53621 | 1483 |
\label{sssec:primrec-new} *} |
52828 | 1484 |
|
1485 |
text {* |
|
53829 | 1486 |
\begin{matharray}{rcl} |
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
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diff
changeset
|
1487 |
@{command_def "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} |
53829 | 1488 |
\end{matharray} |
52794 | 1489 |
|
55112
b1a5d603fd12
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diff
changeset
|
1490 |
@{rail \<open> |
59277 | 1491 |
@@{command primrec} target? @{syntax pr_options}? fixes \<newline> |
56123
a27859b0ef7d
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changeset
|
1492 |
@'where' (@{syntax pr_equation} + '|') |
a27859b0ef7d
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changeset
|
1493 |
; |
59282 | 1494 |
@{syntax_def pr_options}: '(' ((@{syntax plugins} | 'nonexhaustive' | 'transfer') + ',') ')' |
52840 | 1495 |
; |
53829 | 1496 |
@{syntax_def pr_equation}: thmdecl? prop |
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
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55029
diff
changeset
|
1497 |
\<close>} |
55460
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diff
changeset
|
1498 |
|
3f4efd7d950d
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diff
changeset
|
1499 |
\medskip |
3f4efd7d950d
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55459
diff
changeset
|
1500 |
|
3f4efd7d950d
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diff
changeset
|
1501 |
\noindent |
55530
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changeset
|
1502 |
The @{command primrec} command introduces a set of mutually recursive functions |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
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diff
changeset
|
1503 |
over datatypes. |
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
1504 |
|
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
1505 |
The syntactic entity \synt{target} can be used to specify a local context, |
55474 | 1506 |
\synt{fixes} denotes a list of names with optional type signatures, |
1507 |
\synt{thmdecl} denotes an optional name for the formula that follows, and |
|
58620 | 1508 |
\synt{prop} denotes a HOL proposition @{cite "isabelle-isar-ref"}. |
55460
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changeset
|
1509 |
|
59280 | 1510 |
The optional target is optionally followed by a combination of the following |
1511 |
options: |
|
56123
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changeset
|
1512 |
|
a27859b0ef7d
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changeset
|
1513 |
\begin{itemize} |
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
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55945
diff
changeset
|
1514 |
\setlength{\itemsep}{0pt} |
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changeset
|
1515 |
|
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
blanchet
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55945
diff
changeset
|
1516 |
\item |
59282 | 1517 |
The @{text plugins} option indicates which plugins should be enabled |
1518 |
(@{text only}) or disabled (@{text del}). By default, all plugins are enabled. |
|
1519 |
||
1520 |
\item |
|
1521 |
The @{text nonexhaustive} option indicates that the functions are not |
|
56123
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55945
diff
changeset
|
1522 |
necessarily specified for all constructors. It can be used to suppress the |
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
blanchet
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55945
diff
changeset
|
1523 |
warning that is normally emitted when some constructors are missing. |
59277 | 1524 |
|
1525 |
\item |
|
59282 | 1526 |
The @{text transfer} option indicates that an unconditional transfer rule |
59278 | 1527 |
should be generated and proved @{text "by transfer_prover"}. The |
1528 |
@{text "[transfer_rule]"} attribute is set on the generated theorem. |
|
56123
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
blanchet
parents:
55945
diff
changeset
|
1529 |
\end{itemize} |
a27859b0ef7d
document the new 'nonexhaustive' option (cf. 52e8f110fec3)
blanchet
parents:
55945
diff
changeset
|
1530 |
|
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
1531 |
%%% TODO: elaborate on format of the equations |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
1532 |
%%% TODO: elaborate on mutual and nested-to-mutual |
52828 | 1533 |
*} |
1534 |
||
52840 | 1535 |
|
1536 |
subsection {* Generated Theorems |
|
1537 |
\label{ssec:primrec-generated-theorems} *} |
|
52824 | 1538 |
|
59284 | 1539 |
(*<*) |
1540 |
context early |
|
1541 |
begin |
|
1542 |
(*>*) |
|
1543 |
||
52828 | 1544 |
text {* |
59284 | 1545 |
The @{command primrec} command generates the following properties (listed |
1546 |
for @{const tfold}): |
|
1547 |
||
1548 |
\begin{indentblock} |
|
1549 |
\begin{description} |
|
1550 |
||
1551 |
\item[@{text "f."}\hthm{simps} @{text "[simp, code]"}\rm:] ~ \\ |
|
1552 |
@{thm tfold.simps(1)[no_vars]} \\ |
|
1553 |
@{thm tfold.simps(2)[no_vars]} \\ |
|
1554 |
The @{text "[code]"} attribute is set by the @{text code} plugin |
|
1555 |
(Section~\ref{ssec:code-generator}). |
|
1556 |
||
1557 |
\item[@{text "f."}\hthm{transfer} @{text "[transfer_rule]"}\rm:] ~ \\ |
|
1558 |
@{thm tfold.transfer[no_vars]} \\ |
|
1559 |
This theorem is generated by the @{text transfer} plugin |
|
1560 |
(Section~\ref{ssec:transfer}) for functions declared with the @{text transfer} |
|
1561 |
option enabled. |
|
1562 |
||
1563 |
\item[@{text "f."}\hthm{induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
|
1564 |
This induction rule is generated for nested-as-mutual recursive functions |
|
1565 |
(Section~\ref{sssec:primrec-nested-as-mutual-recursion}). |
|
1566 |
||
1567 |
\item[@{text "f\<^sub>1_\<dots>_f\<^sub>m."}\hthm{induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n]"}\rm:] ~ \\ |
|
1568 |
This induction rule is generated for nested-as-mutual recursive functions |
|
1569 |
(Section~\ref{sssec:primrec-nested-as-mutual-recursion}). Given $m > 1$ mutually |
|
1570 |
recursive functions, this rule can be used to prove $m$ properties |
|
1571 |
simultaneously. |
|
1572 |
||
1573 |
\end{description} |
|
1574 |
\end{indentblock} |
|
52828 | 1575 |
*} |
59284 | 1576 |
|
1577 |
(*<*) |
|
1578 |
end |
|
1579 |
(*>*) |
|
53619 | 1580 |
|
52824 | 1581 |
|
52840 | 1582 |
subsection {* Recursive Default Values for Selectors |
53623 | 1583 |
\label{ssec:primrec-recursive-default-values-for-selectors} *} |
52827 | 1584 |
|
1585 |
text {* |
|
1586 |
A datatype selector @{text un_D} can have a default value for each constructor |
|
1587 |
on which it is not otherwise specified. Occasionally, it is useful to have the |
|
55354 | 1588 |
default value be defined recursively. This leads to a chicken-and-egg |
1589 |
situation, because the datatype is not introduced yet at the moment when the |
|
1590 |
selectors are introduced. Of course, we can always define the selectors |
|
1591 |
manually afterward, but we then have to state and prove all the characteristic |
|
1592 |
theorems ourselves instead of letting the package do it. |
|
1593 |
||
1594 |
Fortunately, there is a workaround that relies on overloading to relieve us |
|
1595 |
from the tedium of manual derivations: |
|
52827 | 1596 |
|
1597 |
\begin{enumerate} |
|
1598 |
\setlength{\itemsep}{0pt} |
|
1599 |
||
1600 |
\item |
|
1601 |
Introduce a fully unspecified constant @{text "un_D\<^sub>0 \<Colon> 'a"} using |
|
58931 | 1602 |
@{command consts}. |
52827 | 1603 |
|
1604 |
\item |
|
53535 | 1605 |
Define the datatype, specifying @{text "un_D\<^sub>0"} as the selector's default |
1606 |
value. |
|
52827 | 1607 |
|
1608 |
\item |
|
53535 | 1609 |
Define the behavior of @{text "un_D\<^sub>0"} on values of the newly introduced |
1610 |
datatype using the \keyw{overloading} command. |
|
52827 | 1611 |
|
1612 |
\item |
|
1613 |
Derive the desired equation on @{text un_D} from the characteristic equations |
|
1614 |
for @{text "un_D\<^sub>0"}. |
|
1615 |
\end{enumerate} |
|
1616 |
||
53619 | 1617 |
\noindent |
52827 | 1618 |
The following example illustrates this procedure: |
1619 |
*} |
|
1620 |
||
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1621 |
(*<*) |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1622 |
hide_const termi |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1623 |
(*>*) |
52827 | 1624 |
consts termi\<^sub>0 :: 'a |
1625 |
||
53619 | 1626 |
text {* \blankline *} |
1627 |
||
58310 | 1628 |
datatype ('a, 'b) tlist = |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1629 |
TNil (termi: 'b) |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1630 |
| TCons (thd: 'a) (ttl: "('a, 'b) tlist") |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1631 |
where |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1632 |
"ttl (TNil y) = TNil y" |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1633 |
| "termi (TCons _ xs) = termi\<^sub>0 xs" |
52827 | 1634 |
|
53619 | 1635 |
text {* \blankline *} |
1636 |
||
52827 | 1637 |
overloading |
53491 | 1638 |
termi\<^sub>0 \<equiv> "termi\<^sub>0 \<Colon> ('a, 'b) tlist \<Rightarrow> 'b" |
52827 | 1639 |
begin |
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1640 |
primrec termi\<^sub>0 :: "('a, 'b) tlist \<Rightarrow> 'b" where |
53621 | 1641 |
"termi\<^sub>0 (TNil y) = y" | |
1642 |
"termi\<^sub>0 (TCons x xs) = termi\<^sub>0 xs" |
|
52827 | 1643 |
end |
1644 |
||
53619 | 1645 |
text {* \blankline *} |
1646 |
||
55354 | 1647 |
lemma termi_TCons[simp]: "termi (TCons x xs) = termi xs" |
52827 | 1648 |
by (cases xs) auto |
1649 |
||
1650 |
||
52828 | 1651 |
subsection {* Compatibility Issues |
53617 | 1652 |
\label{ssec:primrec-compatibility-issues} *} |
52828 | 1653 |
|
1654 |
text {* |
|
55530
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1655 |
The command @{command primrec}'s behavior on new-style datatypes has been |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1656 |
designed to be highly compatible with that for old-style datatypes, to ease |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1657 |
migration. There is nonetheless at least one incompatibility that may arise when |
3dfb724db099
renamed 'primrec_new' to 'primrec', overriding the old command (which it still uses as a fallback for old-style datatypes)
blanchet
parents:
55474
diff
changeset
|
1658 |
porting to the new package: |
53997 | 1659 |
|
1660 |
\begin{itemize} |
|
1661 |
\setlength{\itemsep}{0pt} |
|
1662 |
||
54185 | 1663 |
\item \emph{Some theorems have different names.} |
53997 | 1664 |
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
|
1665 |
@{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
|
1666 |
subcollections @{text "f\<^sub>i.simps"}. |
53997 | 1667 |
\end{itemize} |
52828 | 1668 |
*} |
52794 | 1669 |
|
1670 |
||
52827 | 1671 |
section {* Defining Codatatypes |
52805 | 1672 |
\label{sec:defining-codatatypes} *} |
1673 |
||
1674 |
text {* |
|
53829 | 1675 |
Codatatypes can be specified using the @{command codatatype} command. The |
53623 | 1676 |
command is first illustrated through concrete examples featuring different |
1677 |
flavors of corecursion. More examples can be found in the directory |
|
60266 | 1678 |
\verb|~~/src/HOL/|\allowbreak\verb|Datatype_Examples|. The |
53997 | 1679 |
\emph{Archive of Formal Proofs} also includes some useful codatatypes, notably |
58620 | 1680 |
for lazy lists @{cite "lochbihler-2010"}. |
52805 | 1681 |
*} |
52792 | 1682 |
|
52824 | 1683 |
|
53617 | 1684 |
subsection {* Introductory Examples |
1685 |
\label{ssec:codatatype-introductory-examples} *} |
|
52794 | 1686 |
|
53623 | 1687 |
subsubsection {* Simple Corecursion |
1688 |
\label{sssec:codatatype-simple-corecursion} *} |
|
1689 |
||
52805 | 1690 |
text {* |
57542 | 1691 |
Non-corecursive codatatypes coincide with the corresponding datatypes, so they |
56947 | 1692 |
are rarely used in practice. \emph{Corecursive codatatypes} have the same syntax |
53623 | 1693 |
as recursive datatypes, except for the command name. For example, here is the |
1694 |
definition of lazy lists: |
|
1695 |
*} |
|
1696 |
||
57206
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1697 |
codatatype (lset: 'a) llist = |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1698 |
lnull: LNil |
53623 | 1699 |
| LCons (lhd: 'a) (ltl: "'a llist") |
57206
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1700 |
for |
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1701 |
map: lmap |
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1702 |
rel: llist_all2 |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1703 |
where |
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
1704 |
"ltl LNil = LNil" |
53623 | 1705 |
|
1706 |
text {* |
|
1707 |
\noindent |
|
1708 |
Lazy lists can be infinite, such as @{text "LCons 0 (LCons 0 (\<dots>))"} and |
|
53647 | 1709 |
@{text "LCons 0 (LCons 1 (LCons 2 (\<dots>)))"}. Here is a related type, that of |
1710 |
infinite streams: |
|
1711 |
*} |
|
1712 |
||
57206
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1713 |
codatatype (sset: 'a) stream = |
53647 | 1714 |
SCons (shd: 'a) (stl: "'a stream") |
57206
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1715 |
for |
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1716 |
map: smap |
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
1717 |
rel: stream_all2 |
53647 | 1718 |
|
1719 |
text {* |
|
1720 |
\noindent |
|
1721 |
Another interesting type that can |
|
53623 | 1722 |
be defined as a codatatype is that of the extended natural numbers: |
1723 |
*} |
|
1724 |
||
58245 | 1725 |
codatatype enat = EZero | ESucc enat |
53623 | 1726 |
|
1727 |
text {* |
|
1728 |
\noindent |
|
58245 | 1729 |
This type has exactly one infinite element, @{text "ESucc (ESucc (ESucc (\<dots>)))"}, |
53623 | 1730 |
that represents $\infty$. In addition, it has finite values of the form |
58245 | 1731 |
@{text "ESucc (\<dots> (ESucc EZero)\<dots>)"}. |
53675 | 1732 |
|
1733 |
Here is an example with many constructors: |
|
52805 | 1734 |
*} |
53623 | 1735 |
|
53675 | 1736 |
codatatype 'a process = |
1737 |
Fail |
|
1738 |
| Skip (cont: "'a process") |
|
1739 |
| Action (prefix: 'a) (cont: "'a process") |
|
1740 |
| Choice (left: "'a process") (right: "'a process") |
|
1741 |
||
53750 | 1742 |
text {* |
53829 | 1743 |
\noindent |
53750 | 1744 |
Notice that the @{const cont} selector is associated with both @{const Skip} |
54146 | 1745 |
and @{const Action}. |
53750 | 1746 |
*} |
1747 |
||
53623 | 1748 |
|
1749 |
subsubsection {* Mutual Corecursion |
|
1750 |
\label{sssec:codatatype-mutual-corecursion} *} |
|
1751 |
||
1752 |
text {* |
|
1753 |
\noindent |
|
1754 |
The example below introduces a pair of \emph{mutually corecursive} types: |
|
1755 |
*} |
|
1756 |
||
58245 | 1757 |
codatatype even_enat = Even_EZero | Even_ESucc odd_enat |
1758 |
and odd_enat = Odd_ESucc even_enat |
|
53623 | 1759 |
|
1760 |
||
1761 |
subsubsection {* Nested Corecursion |
|
1762 |
\label{sssec:codatatype-nested-corecursion} *} |
|
1763 |
||
1764 |
text {* |
|
1765 |
\noindent |
|
53675 | 1766 |
The next examples feature \emph{nested corecursion}: |
53623 | 1767 |
*} |
1768 |
||
53644 | 1769 |
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 | 1770 |
|
53752 | 1771 |
text {* \blankline *} |
1772 |
||
53644 | 1773 |
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 | 1774 |
|
53752 | 1775 |
text {* \blankline *} |
1776 |
||
55350 | 1777 |
codatatype 'a sm = SM (accept: bool) (trans: "'a \<Rightarrow> 'a sm") |
53675 | 1778 |
|
52824 | 1779 |
|
53617 | 1780 |
subsection {* Command Syntax |
1781 |
\label{ssec:codatatype-command-syntax} *} |
|
52805 | 1782 |
|
53621 | 1783 |
subsubsection {* \keyw{codatatype} |
1784 |
\label{sssec:codatatype} *} |
|
53619 | 1785 |
|
52824 | 1786 |
text {* |
53829 | 1787 |
\begin{matharray}{rcl} |
1788 |
@{command_def "codatatype"} & : & @{text "local_theory \<rightarrow> local_theory"} |
|
1789 |
\end{matharray} |
|
1790 |
||
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
1791 |
@{rail \<open> |
59282 | 1792 |
@@{command codatatype} target? @{syntax dt_options}? @{syntax dt_spec} |
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
1793 |
\<close>} |
53829 | 1794 |
|
55351 | 1795 |
\medskip |
1796 |
||
53829 | 1797 |
\noindent |
52827 | 1798 |
Definitions of codatatypes have almost exactly the same syntax as for datatypes |
57094
589ec121ce1a
don't generate discriminators and selectors for 'datatype_new' unless the user asked for it
blanchet
parents:
57092
diff
changeset
|
1799 |
(Section~\ref{ssec:datatype-command-syntax}). The @{text "discs_sels"} option |
589ec121ce1a
don't generate discriminators and selectors for 'datatype_new' unless the user asked for it
blanchet
parents:
57092
diff
changeset
|
1800 |
is superfluous because discriminators and selectors are always generated for |
589ec121ce1a
don't generate discriminators and selectors for 'datatype_new' unless the user asked for it
blanchet
parents:
57092
diff
changeset
|
1801 |
codatatypes. |
53623 | 1802 |
*} |
1803 |
||
1804 |
||
1805 |
subsection {* Generated Constants |
|
1806 |
\label{ssec:codatatype-generated-constants} *} |
|
1807 |
||
1808 |
text {* |
|
1809 |
Given a codatatype @{text "('a\<^sub>1, \<dots>, 'a\<^sub>m) t"} |
|
1810 |
with $m > 0$ live type variables and $n$ constructors @{text "t.C\<^sub>1"}, |
|
1811 |
\ldots, @{text "t.C\<^sub>n"}, the same auxiliary constants are generated as for |
|
1812 |
datatypes (Section~\ref{ssec:datatype-generated-constants}), except that the |
|
58190 | 1813 |
recursor is replaced by a dual concept: |
53623 | 1814 |
|
55354 | 1815 |
\medskip |
1816 |
||
1817 |
\begin{tabular}{@ {}ll@ {}} |
|
1818 |
Corecursor: & |
|
56655 | 1819 |
@{text t.corec_t} |
55354 | 1820 |
\end{tabular} |
53623 | 1821 |
*} |
1822 |
||
1823 |
||
1824 |
subsection {* Generated Theorems |
|
1825 |
\label{ssec:codatatype-generated-theorems} *} |
|
1826 |
||
1827 |
text {* |
|
53829 | 1828 |
The characteristic theorems generated by @{command codatatype} are grouped in |
53623 | 1829 |
three broad categories: |
1830 |
||
1831 |
\begin{itemize} |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1832 |
\setlength{\itemsep}{0pt} |
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
1833 |
|
56655 | 1834 |
\item The \emph{free constructor theorems} |
1835 |
(Section~\ref{sssec:free-constructor-theorems}) are properties of the |
|
1836 |
constructors and destructors that can be derived for any freely generated type. |
|
1837 |
||
1838 |
\item The \emph{functorial theorems} |
|
1839 |
(Section~\ref{sssec:functorial-theorems}) are properties of datatypes related to |
|
53623 | 1840 |
their BNF nature. |
1841 |
||
56655 | 1842 |
\item The \emph{coinductive theorems} (Section~\ref{sssec:coinductive-theorems}) |
1843 |
are properties of datatypes related to their coinductive nature. |
|
53623 | 1844 |
\end{itemize} |
1845 |
||
1846 |
\noindent |
|
56655 | 1847 |
The first two categories are exactly as for datatypes. |
52824 | 1848 |
*} |
1849 |
||
53617 | 1850 |
|
53623 | 1851 |
subsubsection {* Coinductive Theorems |
1852 |
\label{sssec:coinductive-theorems} *} |
|
1853 |
||
1854 |
text {* |
|
54031 | 1855 |
The coinductive theorems are listed below for @{typ "'a llist"}: |
53623 | 1856 |
|
1857 |
\begin{indentblock} |
|
1858 |
\begin{description} |
|
1859 |
||
53643 | 1860 |
\item[\begin{tabular}{@ {}l@ {}} |
57304 | 1861 |
@{text "t."}\hthm{coinduct} @{text "[consumes m, case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
1862 |
\phantom{@{text "t."}\hthm{coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> |
|
1863 |
D\<^sub>n, coinduct t]"}\rm: |
|
53643 | 1864 |
\end{tabular}] ~ \\ |
53623 | 1865 |
@{thm llist.coinduct[no_vars]} |
53617 | 1866 |
|
53643 | 1867 |
\item[\begin{tabular}{@ {}l@ {}} |
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|
1868 |
@{text "t."}\hthm{coinduct_strong} @{text "[consumes m, case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
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|
1869 |
\phantom{@{text "t."}\hthm{coinduct_strong} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> D\<^sub>n]"}\rm: |
53643 | 1870 |
\end{tabular}] ~ \\ |
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|
1871 |
@{thm llist.coinduct_strong[no_vars]} |
53617 | 1872 |
|
53643 | 1873 |
\item[\begin{tabular}{@ {}l@ {}} |
57304 | 1874 |
@{text "t."}\hthm{rel_coinduct} @{text "[consumes m, case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
1875 |
\phantom{@{text "t."}\hthm{rel_coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> |
|
1876 |
D\<^sub>n, coinduct pred]"}\rm: |
|
1877 |
\end{tabular}] ~ \\ |
|
1878 |
@{thm llist.rel_coinduct[no_vars]} |
|
1879 |
||
1880 |
\item[\begin{tabular}{@ {}l@ {}} |
|
53643 | 1881 |
@{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]"} \\ |
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|
1882 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{coinduct_strong} @{text "[case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
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|
1883 |
\phantom{@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{coinduct_strong} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> D\<^sub>n]"}\rm: \\ |
57542 | 1884 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{rel_coinduct} @{text "[case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
1885 |
\phantom{@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{rel_coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> D\<^sub>n]"}\rm: \\ |
|
53643 | 1886 |
\end{tabular}] ~ \\ |
1887 |
Given $m > 1$ mutually corecursive codatatypes, these coinduction rules can be |
|
1888 |
used to prove $m$ properties simultaneously. |
|
1889 |
||
57701 | 1890 |
\item[\begin{tabular}{@ {}l@ {}} |
1891 |
@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{set_induct} @{text "[case_names C\<^sub>1 \<dots> C\<^sub>n,"} \\ |
|
1892 |
\phantom{@{text "t\<^sub>1_\<dots>_t\<^sub>m."}\hthm{set_induct} @{text "["}}@{text "induct set: set\<^sub>j_t\<^sub>1, \<dots>, induct set: set\<^sub>j_t\<^sub>m]"}\rm: \\ |
|
1893 |
\end{tabular}] ~ \\ |
|
1894 |
@{thm llist.set_induct[no_vars]} \\ |
|
1895 |
If $m = 1$, the attribute @{text "[consumes 1]"} is generated as well. |
|
1896 |
||
54031 | 1897 |
\item[@{text "t."}\hthm{corec}\rm:] ~ \\ |
53623 | 1898 |
@{thm llist.corec(1)[no_vars]} \\ |
1899 |
@{thm llist.corec(2)[no_vars]} |
|
1900 |
||
57542 | 1901 |
\item[@{text "t."}\hthm{corec_code} @{text "[code]"}\rm:] ~ \\ |
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|
1902 |
@{thm llist.corec_code[no_vars]} \\ |
59284 | 1903 |
The @{text "[code]"} attribute is set by the @{text code} plugin |
1904 |
(Section~\ref{ssec:code-generator}). |
|
57490 | 1905 |
|
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|
1906 |
\item[@{text "t."}\hthm{corec_disc}\rm:] ~ \\ |
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|
1907 |
@{thm llist.corec_disc(1)[no_vars]} \\ |
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|
1908 |
@{thm llist.corec_disc(2)[no_vars]} |
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|
1909 |
|
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|
1910 |
\item[@{text "t."}\hthm{corec_disc_iff} @{text "[simp]"}\rm:] ~ \\ |
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|
1911 |
@{thm llist.corec_disc_iff(1)[no_vars]} \\ |
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|
1912 |
@{thm llist.corec_disc_iff(2)[no_vars]} |
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|
1913 |
|
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|
1914 |
\item[@{text "t."}\hthm{corec_sel} @{text "[simp]"}\rm:] ~ \\ |
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|
1915 |
@{thm llist.corec_sel(1)[no_vars]} \\ |
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|
1916 |
@{thm llist.corec_sel(2)[no_vars]} |
53643 | 1917 |
|
58735 | 1918 |
\item[@{text "t."}\hthm{map_o_corec}\rm:] ~ \\ |
1919 |
@{thm llist.map_o_corec[no_vars]} |
|
1920 |
||
58915 | 1921 |
\item[@{text "t."}\hthm{corec_transfer} @{text "[transfer_rule]"}\rm:] ~ \\ |
1922 |
@{thm llist.corec_transfer[no_vars]} \\ |
|
59284 | 1923 |
The @{text "[transfer_rule]"} attribute is set by the @{text transfer} plugin |
59824 | 1924 |
(Section~\ref{ssec:transfer}) for type constructors with no dead type arguments. |
58449 | 1925 |
|
53623 | 1926 |
\end{description} |
1927 |
\end{indentblock} |
|
1928 |
||
1929 |
\noindent |
|
53829 | 1930 |
For convenience, @{command codatatype} also provides the following collection: |
53623 | 1931 |
|
1932 |
\begin{indentblock} |
|
1933 |
\begin{description} |
|
1934 |
||
59284 | 1935 |
\item[@{text "t."}\hthm{simps}] = @{text t.inject} @{text t.distinct} @{text t.case} @{text t.corec_disc_iff} @{text t.corec_sel} \\ |
55896 | 1936 |
@{text t.map} @{text t.rel_inject} @{text t.rel_distinct} @{text t.set} |
53623 | 1937 |
|
1938 |
\end{description} |
|
1939 |
\end{indentblock} |
|
1940 |
*} |
|
52805 | 1941 |
|
1942 |
||
59861 | 1943 |
section {* Defining Primitively Corecursive Functions |
1944 |
\label{sec:defining-primitively-corecursive-functions} *} |
|
52805 | 1945 |
|
1946 |
text {* |
|
54183 | 1947 |
Corecursive functions can be specified using the @{command primcorec} and |
1948 |
\keyw{prim\-corec\-ursive} commands, which support primitive corecursion, or |
|
59282 | 1949 |
using the more general \keyw{partial_function} command. In this tutorial, the |
1950 |
focus is on the first two. More examples can be found in the directory |
|
1951 |
\verb|~~/src/HOL/Datatype_|\allowbreak\verb|Examples|. |
|
53644 | 1952 |
|
53749
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|
1953 |
Whereas recursive functions consume datatypes one constructor at a time, |
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|
1954 |
corecursive functions construct codatatypes one constructor at a time. |
53752 | 1955 |
Partly reflecting a lack of agreement among proponents of coalgebraic methods, |
1956 |
Isabelle supports three competing syntaxes for specifying a function $f$: |
|
53749
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changeset
|
1957 |
|
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|
1958 |
\begin{itemize} |
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diff
changeset
|
1959 |
\setlength{\itemsep}{0pt} |
b37db925b663
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changeset
|
1960 |
|
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|
1961 |
\abovedisplayskip=.5\abovedisplayskip |
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changeset
|
1962 |
\belowdisplayskip=.5\belowdisplayskip |
b37db925b663
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changeset
|
1963 |
|
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|
1964 |
\item The \emph{destructor view} specifies $f$ by implications of the form |
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diff
changeset
|
1965 |
\[@{text "\<dots> \<Longrightarrow> is_C\<^sub>j (f x\<^sub>1 \<dots> x\<^sub>n)"}\] and |
b37db925b663
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changeset
|
1966 |
equations of the form |
b37db925b663
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blanchet
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diff
changeset
|
1967 |
\[@{text "un_C\<^sub>ji (f x\<^sub>1 \<dots> x\<^sub>n) = \<dots>"}\] |
b37db925b663
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|
1968 |
This style is popular in the coalgebraic literature. |
b37db925b663
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diff
changeset
|
1969 |
|
b37db925b663
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|
1970 |
\item The \emph{constructor view} specifies $f$ by equations of the form |
54183 | 1971 |
\[@{text "\<dots> \<Longrightarrow> f x\<^sub>1 \<dots> x\<^sub>n = C\<^sub>j \<dots>"}\] |
53752 | 1972 |
This style is often more concise than the previous one. |
53749
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changeset
|
1973 |
|
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|
1974 |
\item The \emph{code view} specifies $f$ by a single equation of the form |
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changeset
|
1975 |
\[@{text "f x\<^sub>1 \<dots> x\<^sub>n = \<dots>"}\] |
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|
1976 |
with restrictions on the format of the right-hand side. Lazy functional |
b37db925b663
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|
1977 |
programming languages such as Haskell support a generalized version of this |
b37db925b663
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diff
changeset
|
1978 |
style. |
b37db925b663
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|
1979 |
\end{itemize} |
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|
1980 |
|
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
parents:
53752
diff
changeset
|
1981 |
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
|
1982 |
characteristic theorems for all three styles are generated. |
53749
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changeset
|
1983 |
|
52828 | 1984 |
%%% TODO: partial_function? E.g. for defining tail recursive function on lazy |
1985 |
%%% lists (cf. terminal0 in TLList.thy) |
|
52805 | 1986 |
*} |
1987 |
||
52824 | 1988 |
|
53617 | 1989 |
subsection {* Introductory Examples |
1990 |
\label{ssec:primcorec-introductory-examples} *} |
|
52805 | 1991 |
|
53646 | 1992 |
text {* |
1993 |
Primitive corecursion is illustrated through concrete examples based on the |
|
1994 |
codatatypes defined in Section~\ref{ssec:codatatype-introductory-examples}. More |
|
58309
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renamed 'BNF_Examples' to 'Datatype_Examples' (cf. 'datatypes.pdf')
blanchet
parents:
58305
diff
changeset
|
1995 |
examples can be found in the directory \verb|~~/src/HOL/Datatype_Examples|. The |
a09ec6daaa19
renamed 'BNF_Examples' to 'Datatype_Examples' (cf. 'datatypes.pdf')
blanchet
parents:
58305
diff
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|
1996 |
code view is favored in the examples below. Sections |
53749
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|
1997 |
\ref{ssec:primrec-constructor-view} and \ref{ssec:primrec-destructor-view} |
b37db925b663
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|
1998 |
present the same examples expressed using the constructor and destructor views. |
53646 | 1999 |
*} |
2000 |
||
58245 | 2001 |
|
53644 | 2002 |
subsubsection {* Simple Corecursion |
2003 |
\label{sssec:primcorec-simple-corecursion} *} |
|
2004 |
||
53646 | 2005 |
text {* |
53752 | 2006 |
Following the code view, corecursive calls are allowed on the right-hand side as |
2007 |
long as they occur under a constructor, which itself appears either directly to |
|
2008 |
the right of the equal sign or in a conditional expression: |
|
53646 | 2009 |
*} |
2010 |
||
59284 | 2011 |
primcorec (*<*)(transfer) (*>*)literate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist" where |
54072 | 2012 |
"literate g x = LCons x (literate g (g x))" |
53647 | 2013 |
|
53677 | 2014 |
text {* \blankline *} |
2015 |
||
53826 | 2016 |
primcorec siterate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a stream" where |
54072 | 2017 |
"siterate g x = SCons x (siterate g (g x))" |
53644 | 2018 |
|
53646 | 2019 |
text {* |
2020 |
\noindent |
|
2021 |
The constructor ensures that progress is made---i.e., the function is |
|
53749
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|
2022 |
\emph{productive}. The above functions compute the infinite lazy list or stream |
54072 | 2023 |
@{text "[x, g x, g (g x), \<dots>]"}. Productivity guarantees that prefixes |
2024 |
@{text "[x, g x, g (g x), \<dots>, (g ^^ k) x]"} of arbitrary finite length |
|
53749
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changeset
|
2025 |
@{text k} can be computed by unfolding the code equation a finite number of |
53863
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blanchet
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diff
changeset
|
2026 |
times. |
53646 | 2027 |
|
53752 | 2028 |
Corecursive functions construct codatatype values, but nothing prevents them |
53863
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blanchet
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diff
changeset
|
2029 |
from also consuming such values. The following function drops every second |
53675 | 2030 |
element in a stream: |
2031 |
*} |
|
2032 |
||
53826 | 2033 |
primcorec every_snd :: "'a stream \<Rightarrow> 'a stream" where |
53675 | 2034 |
"every_snd s = SCons (shd s) (stl (stl s))" |
2035 |
||
2036 |
text {* |
|
53752 | 2037 |
\noindent |
56124 | 2038 |
Constructs such as @{text "let"}--@{text "in"}, @{text |
2039 |
"if"}--@{text "then"}--@{text "else"}, and @{text "case"}--@{text "of"} may |
|
53646 | 2040 |
appear around constructors that guard corecursive calls: |
2041 |
*} |
|
2042 |
||
54072 | 2043 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53644 | 2044 |
"lappend xs ys = |
2045 |
(case xs of |
|
2046 |
LNil \<Rightarrow> ys |
|
2047 |
| LCons x xs' \<Rightarrow> LCons x (lappend xs' ys))" |
|
2048 |
||
53646 | 2049 |
text {* |
53752 | 2050 |
\noindent |
54402 | 2051 |
Pattern matching is not supported by @{command primcorec}. Fortunately, it is |
57542 | 2052 |
easy to generate pattern-maching equations using the \keyw{simps_of_case} |
54402 | 2053 |
command provided by the theory \verb|~~/src/HOL/Library/Simps_Case_Conv|. |
2054 |
*} |
|
2055 |
||
2056 |
simps_of_case lappend_simps: lappend.code |
|
2057 |
||
2058 |
text {* |
|
2059 |
This generates the lemma collection @{thm [source] lappend_simps}: |
|
2060 |
% |
|
55355 | 2061 |
\begin{gather*% |
2062 |
} |
|
2063 |
@{thm lappend_simps(1)[no_vars]} \\ |
|
2064 |
@{thm lappend_simps(2)[no_vars]} |
|
2065 |
\end{gather*% |
|
2066 |
} |
|
54402 | 2067 |
% |
53646 | 2068 |
Corecursion is useful to specify not only functions but also infinite objects: |
2069 |
*} |
|
2070 |
||
53826 | 2071 |
primcorec infty :: enat where |
58245 | 2072 |
"infty = ESucc infty" |
53644 | 2073 |
|
53646 | 2074 |
text {* |
53752 | 2075 |
\noindent |
2076 |
The example below constructs a pseudorandom process value. It takes a stream of |
|
53675 | 2077 |
actions (@{text s}), a pseudorandom function generator (@{text f}), and a |
2078 |
pseudorandom seed (@{text n}): |
|
2079 |
*} |
|
2080 |
||
59861 | 2081 |
(*<*) |
2082 |
context early |
|
2083 |
begin |
|
2084 |
(*>*) |
|
2085 |
primcorec |
|
53752 | 2086 |
random_process :: "'a stream \<Rightarrow> (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> 'a process" |
2087 |
where |
|
53675 | 2088 |
"random_process s f n = |
2089 |
(if n mod 4 = 0 then |
|
2090 |
Fail |
|
2091 |
else if n mod 4 = 1 then |
|
2092 |
Skip (random_process s f (f n)) |
|
2093 |
else if n mod 4 = 2 then |
|
2094 |
Action (shd s) (random_process (stl s) f (f n)) |
|
2095 |
else |
|
2096 |
Choice (random_process (every_snd s) (f \<circ> f) (f n)) |
|
2097 |
(random_process (every_snd (stl s)) (f \<circ> f) (f (f n))))" |
|
59861 | 2098 |
(*<*) |
2099 |
end |
|
2100 |
(*>*) |
|
53675 | 2101 |
|
2102 |
text {* |
|
2103 |
\noindent |
|
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|
2104 |
The main disadvantage of the code view is that the conditions are tested |
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|
2105 |
sequentially. This is visible in the generated theorems. The constructor and |
53752 | 2106 |
destructor views offer nonsequential alternatives. |
53675 | 2107 |
*} |
2108 |
||
53644 | 2109 |
|
2110 |
subsubsection {* Mutual Corecursion |
|
2111 |
\label{sssec:primcorec-mutual-corecursion} *} |
|
2112 |
||
53647 | 2113 |
text {* |
2114 |
The syntax for mutually corecursive functions over mutually corecursive |
|
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|
2115 |
datatypes is unsurprising: |
53647 | 2116 |
*} |
2117 |
||
53826 | 2118 |
primcorec |
53644 | 2119 |
even_infty :: even_enat and |
2120 |
odd_infty :: odd_enat |
|
2121 |
where |
|
58245 | 2122 |
"even_infty = Even_ESucc odd_infty" | |
2123 |
"odd_infty = Odd_ESucc even_infty" |
|
53644 | 2124 |
|
2125 |
||
2126 |
subsubsection {* Nested Corecursion |
|
2127 |
\label{sssec:primcorec-nested-corecursion} *} |
|
2128 |
||
53647 | 2129 |
text {* |
2130 |
The next pair of examples generalize the @{const literate} and @{const siterate} |
|
2131 |
functions (Section~\ref{sssec:primcorec-nested-corecursion}) to possibly |
|
2132 |
infinite trees in which subnodes are organized either as a lazy list (@{text |
|
54072 | 2133 |
tree\<^sub>i\<^sub>i}) or as a finite set (@{text tree\<^sub>i\<^sub>s}). They rely on the map functions of |
2134 |
the nesting type constructors to lift the corecursive calls: |
|
53647 | 2135 |
*} |
2136 |
||
53826 | 2137 |
primcorec iterate\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
54072 | 2138 |
"iterate\<^sub>i\<^sub>i g x = Node\<^sub>i\<^sub>i x (lmap (iterate\<^sub>i\<^sub>i g) (g x))" |
53644 | 2139 |
|
53677 | 2140 |
text {* \blankline *} |
2141 |
||
53826 | 2142 |
primcorec iterate\<^sub>i\<^sub>s :: "('a \<Rightarrow> 'a fset) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>s" where |
54072 | 2143 |
"iterate\<^sub>i\<^sub>s g x = Node\<^sub>i\<^sub>s x (fimage (iterate\<^sub>i\<^sub>s g) (g x))" |
53644 | 2144 |
|
52805 | 2145 |
text {* |
53752 | 2146 |
\noindent |
54072 | 2147 |
Both examples follow the usual format for constructor arguments associated |
2148 |
with nested recursive occurrences of the datatype. Consider |
|
2149 |
@{const iterate\<^sub>i\<^sub>i}. The term @{term "g x"} constructs an @{typ "'a llist"} |
|
2150 |
value, which is turned into an @{typ "'a tree\<^sub>i\<^sub>i llist"} value using |
|
2151 |
@{const lmap}. |
|
2152 |
||
2153 |
This format may sometimes feel artificial. The following function constructs |
|
2154 |
a tree with a single, infinite branch from a stream: |
|
2155 |
*} |
|
2156 |
||
2157 |
primcorec tree\<^sub>i\<^sub>i_of_stream :: "'a stream \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
|
2158 |
"tree\<^sub>i\<^sub>i_of_stream s = |
|
2159 |
Node\<^sub>i\<^sub>i (shd s) (lmap tree\<^sub>i\<^sub>i_of_stream (LCons (stl s) LNil))" |
|
2160 |
||
2161 |
text {* |
|
2162 |
\noindent |
|
54955
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|
2163 |
A more natural syntax, also supported by Isabelle, is to move corecursive calls |
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|
2164 |
under constructors: |
54072 | 2165 |
*} |
2166 |
||
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|
2167 |
primcorec (*<*)(in late) (*>*)tree\<^sub>i\<^sub>i_of_stream :: "'a stream \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
55350 | 2168 |
"tree\<^sub>i\<^sub>i_of_stream s = |
2169 |
Node\<^sub>i\<^sub>i (shd s) (LCons (tree\<^sub>i\<^sub>i_of_stream (stl s)) LNil)" |
|
54072 | 2170 |
|
2171 |
text {* |
|
2172 |
The next example illustrates corecursion through functions, which is a bit |
|
2173 |
special. Deterministic finite automata (DFAs) are traditionally defined as |
|
2174 |
5-tuples @{text "(Q, \<Sigma>, \<delta>, q\<^sub>0, F)"}, where @{text Q} is a finite set of states, |
|
53675 | 2175 |
@{text \<Sigma>} is a finite alphabet, @{text \<delta>} is a transition function, @{text q\<^sub>0} |
2176 |
is an initial state, and @{text F} is a set of final states. The following |
|
55350 | 2177 |
function translates a DFA into a state machine: |
53675 | 2178 |
*} |
2179 |
||
55350 | 2180 |
primcorec (*<*)(in early) (*>*)sm_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> 'a sm" where |
2181 |
"sm_of_dfa \<delta> F q = SM (q \<in> F) (sm_of_dfa \<delta> F \<circ> \<delta> q)" |
|
53675 | 2182 |
|
53751 | 2183 |
text {* |
2184 |
\noindent |
|
2185 |
The map function for the function type (@{text \<Rightarrow>}) is composition |
|
54181 | 2186 |
(@{text "op \<circ>"}). For convenience, corecursion through functions can |
54182 | 2187 |
also be expressed using $\lambda$-abstractions and function application rather |
54031 | 2188 |
than through composition. For example: |
53751 | 2189 |
*} |
2190 |
||
55350 | 2191 |
primcorec sm_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> 'a sm" where |
2192 |
"sm_of_dfa \<delta> F q = SM (q \<in> F) (\<lambda>a. sm_of_dfa \<delta> F (\<delta> q a))" |
|
53752 | 2193 |
|
2194 |
text {* \blankline *} |
|
2195 |
||
55350 | 2196 |
primcorec empty_sm :: "'a sm" where |
2197 |
"empty_sm = SM False (\<lambda>_. empty_sm)" |
|
53751 | 2198 |
|
53752 | 2199 |
text {* \blankline *} |
2200 |
||
55350 | 2201 |
primcorec not_sm :: "'a sm \<Rightarrow> 'a sm" where |
2202 |
"not_sm M = SM (\<not> accept M) (\<lambda>a. not_sm (trans M a))" |
|
53751 | 2203 |
|
53752 | 2204 |
text {* \blankline *} |
2205 |
||
55350 | 2206 |
primcorec or_sm :: "'a sm \<Rightarrow> 'a sm \<Rightarrow> 'a sm" where |
2207 |
"or_sm M N = |
|
2208 |
SM (accept M \<or> accept N) (\<lambda>a. or_sm (trans M a) (trans N a))" |
|
53751 | 2209 |
|
54182 | 2210 |
text {* |
2211 |
\noindent |
|
2212 |
For recursion through curried $n$-ary functions, $n$ applications of |
|
2213 |
@{term "op \<circ>"} are necessary. The examples below illustrate the case where |
|
2214 |
$n = 2$: |
|
2215 |
*} |
|
2216 |
||
55350 | 2217 |
codatatype ('a, 'b) sm2 = |
2218 |
SM2 (accept2: bool) (trans2: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) sm2") |
|
54182 | 2219 |
|
2220 |
text {* \blankline *} |
|
2221 |
||
2222 |
primcorec |
|
55350 | 2223 |
(*<*)(in early) (*>*)sm2_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> ('a, 'b) sm2" |
54182 | 2224 |
where |
55350 | 2225 |
"sm2_of_dfa \<delta> F q = SM2 (q \<in> F) (op \<circ> (op \<circ> (sm2_of_dfa \<delta> F)) (\<delta> q))" |
54182 | 2226 |
|
2227 |
text {* \blankline *} |
|
2228 |
||
2229 |
primcorec |
|
55350 | 2230 |
sm2_of_dfa :: "('q \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'q) \<Rightarrow> 'q set \<Rightarrow> 'q \<Rightarrow> ('a, 'b) sm2" |
54182 | 2231 |
where |
55350 | 2232 |
"sm2_of_dfa \<delta> F q = SM2 (q \<in> F) (\<lambda>a b. sm2_of_dfa \<delta> F (\<delta> q a b))" |
54182 | 2233 |
|
53644 | 2234 |
|
2235 |
subsubsection {* Nested-as-Mutual Corecursion |
|
2236 |
\label{sssec:primcorec-nested-as-mutual-corecursion} *} |
|
2237 |
||
53647 | 2238 |
text {* |
2239 |
Just as it is possible to recurse over nested recursive datatypes as if they |
|
2240 |
were mutually recursive |
|
2241 |
(Section~\ref{sssec:primrec-nested-as-mutual-recursion}), it is possible to |
|
53752 | 2242 |
pretend that nested codatatypes are mutually corecursive. For example: |
53647 | 2243 |
*} |
2244 |
||
54287 | 2245 |
(*<*) |
2246 |
context late |
|
2247 |
begin |
|
2248 |
(*>*) |
|
54072 | 2249 |
primcorec |
54287 | 2250 |
iterate\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>i" and |
53644 | 2251 |
iterates\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a llist \<Rightarrow> 'a tree\<^sub>i\<^sub>i llist" |
2252 |
where |
|
54072 | 2253 |
"iterate\<^sub>i\<^sub>i g x = Node\<^sub>i\<^sub>i x (iterates\<^sub>i\<^sub>i g (g x))" | |
2254 |
"iterates\<^sub>i\<^sub>i g xs = |
|
53644 | 2255 |
(case xs of |
2256 |
LNil \<Rightarrow> LNil |
|
54072 | 2257 |
| LCons x xs' \<Rightarrow> LCons (iterate\<^sub>i\<^sub>i g x) (iterates\<^sub>i\<^sub>i g xs'))" |
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|
2258 |
|
54287 | 2259 |
text {* |
2260 |
\noindent |
|
2261 |
Coinduction rules are generated as |
|
2262 |
@{thm [source] iterate\<^sub>i\<^sub>i.coinduct}, |
|
2263 |
@{thm [source] iterates\<^sub>i\<^sub>i.coinduct}, and |
|
2264 |
@{thm [source] iterate\<^sub>i\<^sub>i_iterates\<^sub>i\<^sub>i.coinduct} |
|
57983
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|
2265 |
and analogously for @{text coinduct_strong}. These rules and the |
54287 | 2266 |
underlying corecursors are generated on a per-need basis and are kept in a cache |
2267 |
to speed up subsequent definitions. |
|
2268 |
*} |
|
2269 |
||
2270 |
(*<*) |
|
2271 |
end |
|
2272 |
(*>*) |
|
2273 |
||
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|
2274 |
|
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|
2275 |
subsubsection {* Constructor View |
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|
2276 |
\label{ssec:primrec-constructor-view} *} |
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|
2277 |
|
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|
2278 |
(*<*) |
54182 | 2279 |
locale ctr_view |
53749
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|
2280 |
begin |
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|
2281 |
(*>*) |
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|
2282 |
|
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|
2283 |
text {* |
53750 | 2284 |
The constructor view is similar to the code view, but there is one separate |
2285 |
conditional equation per constructor rather than a single unconditional |
|
2286 |
equation. Examples that rely on a single constructor, such as @{const literate} |
|
2287 |
and @{const siterate}, are identical in both styles. |
|
2288 |
||
2289 |
Here is an example where there is a difference: |
|
53749
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|
2290 |
*} |
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|
2291 |
|
53826 | 2292 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53749
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|
2293 |
"lnull xs \<Longrightarrow> lnull ys \<Longrightarrow> lappend xs ys = LNil" | |
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|
2294 |
"_ \<Longrightarrow> lappend xs ys = LCons (lhd (if lnull xs then ys else xs)) |
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|
2295 |
(if xs = LNil then ltl ys else lappend (ltl xs) ys)" |
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|
2296 |
|
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|
2297 |
text {* |
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|
2298 |
\noindent |
53752 | 2299 |
With the constructor view, we must distinguish between the @{const LNil} and |
2300 |
the @{const LCons} case. The condition for @{const LCons} is |
|
2301 |
left implicit, as the negation of that for @{const LNil}. |
|
53750 | 2302 |
|
59861 | 2303 |
For this example, the constructor view is slightly more involved than the |
53750 | 2304 |
code equation. Recall the code view version presented in |
53749
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|
2305 |
Section~\ref{sssec:primcorec-simple-corecursion}. |
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|
2306 |
% TODO: \[{thm code_view.lappend.code}\] |
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|
2307 |
The constructor view requires us to analyze the second argument (@{term ys}). |
53752 | 2308 |
The code equation generated from the constructor view also suffers from this. |
53749
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|
2309 |
% TODO: \[{thm lappend.code}\] |
53750 | 2310 |
|
53752 | 2311 |
In contrast, the next example is arguably more naturally expressed in the |
2312 |
constructor view: |
|
53749
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|
2313 |
*} |
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|
2314 |
|
53831
80423b9080cf
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53829
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|
2315 |
primcorec |
53752 | 2316 |
random_process :: "'a stream \<Rightarrow> (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> 'a process" |
2317 |
where |
|
53749
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|
2318 |
"n mod 4 = 0 \<Longrightarrow> random_process s f n = Fail" | |
53752 | 2319 |
"n mod 4 = 1 \<Longrightarrow> |
2320 |
random_process s f n = Skip (random_process s f (f n))" | |
|
2321 |
"n mod 4 = 2 \<Longrightarrow> |
|
2322 |
random_process s f n = Action (shd s) (random_process (stl s) f (f n))" | |
|
2323 |
"n mod 4 = 3 \<Longrightarrow> |
|
2324 |
random_process s f n = Choice (random_process (every_snd s) f (f n)) |
|
53826 | 2325 |
(random_process (every_snd (stl s)) f (f n))" |
2326 |
(*<*) |
|
53644 | 2327 |
end |
2328 |
(*>*) |
|
52805 | 2329 |
|
53750 | 2330 |
text {* |
53752 | 2331 |
\noindent |
53750 | 2332 |
Since there is no sequentiality, we can apply the equation for @{const Choice} |
53752 | 2333 |
without having first to discharge @{term "n mod (4\<Colon>int) \<noteq> 0"}, |
2334 |
@{term "n mod (4\<Colon>int) \<noteq> 1"}, and |
|
2335 |
@{term "n mod (4\<Colon>int) \<noteq> 2"}. |
|
59284 | 2336 |
The price to pay for this elegance is that we must discharge exclusiveness proof |
53750 | 2337 |
obligations, one for each pair of conditions |
53752 | 2338 |
@{term "(n mod (4\<Colon>int) = i, n mod (4\<Colon>int) = j)"} |
2339 |
with @{term "i < j"}. If we prefer not to discharge any obligations, we can |
|
2340 |
enable the @{text "sequential"} option. This pushes the problem to the users of |
|
2341 |
the generated properties. |
|
53750 | 2342 |
%Here are more examples to conclude: |
2343 |
*} |
|
2344 |
||
52824 | 2345 |
|
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|
2346 |
subsubsection {* Destructor View |
53752 | 2347 |
\label{ssec:primrec-destructor-view} *} |
2348 |
||
2349 |
(*<*) |
|
54182 | 2350 |
locale dtr_view |
53752 | 2351 |
begin |
2352 |
(*>*) |
|
53749
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|
2353 |
|
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|
2354 |
text {* |
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|
2355 |
The destructor view is in many respects dual to the constructor view. Conditions |
b37db925b663
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|
2356 |
determine which constructor to choose, and these conditions are interpreted |
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|
2357 |
sequentially or not depending on the @{text "sequential"} option. |
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|
2358 |
Consider the following examples: |
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|
2359 |
*} |
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|
2360 |
|
53826 | 2361 |
primcorec literate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a llist" where |
53749
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|
2362 |
"\<not> lnull (literate _ x)" | |
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|
2363 |
"lhd (literate _ x) = x" | |
54072 | 2364 |
"ltl (literate g x) = literate g (g x)" |
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|
2365 |
|
53752 | 2366 |
text {* \blankline *} |
2367 |
||
53826 | 2368 |
primcorec siterate :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a stream" where |
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|
2369 |
"shd (siterate _ x) = x" | |
54072 | 2370 |
"stl (siterate g x) = siterate g (g x)" |
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|
2371 |
|
53752 | 2372 |
text {* \blankline *} |
2373 |
||
53826 | 2374 |
primcorec every_snd :: "'a stream \<Rightarrow> 'a stream" where |
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|
2375 |
"shd (every_snd s) = shd s" | |
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|
2376 |
"stl (every_snd s) = stl (stl s)" |
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changeset
|
2377 |
|
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|
2378 |
text {* |
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changeset
|
2379 |
\noindent |
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|
2380 |
The first formula in the @{const literate} specification indicates which |
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diff
changeset
|
2381 |
constructor to choose. For @{const siterate} and @{const every_snd}, no such |
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changeset
|
2382 |
formula is necessary, since the type has only one constructor. The last two |
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|
2383 |
formulas are equations specifying the value of the result for the relevant |
b37db925b663
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changeset
|
2384 |
selectors. Corecursive calls appear directly to the right of the equal sign. |
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|
2385 |
Their arguments are unrestricted. |
53750 | 2386 |
|
2387 |
The next example shows how to specify functions that rely on more than one |
|
2388 |
constructor: |
|
2389 |
*} |
|
2390 |
||
53826 | 2391 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53750 | 2392 |
"lnull xs \<Longrightarrow> lnull ys \<Longrightarrow> lnull (lappend xs ys)" | |
2393 |
"lhd (lappend xs ys) = lhd (if lnull xs then ys else xs)" | |
|
2394 |
"ltl (lappend xs ys) = (if xs = LNil then ltl ys else lappend (ltl xs) ys)" |
|
2395 |
||
2396 |
text {* |
|
2397 |
\noindent |
|
2398 |
For a codatatype with $n$ constructors, it is sufficient to specify $n - 1$ |
|
2399 |
discriminator formulas. The command will then assume that the remaining |
|
2400 |
constructor should be taken otherwise. This can be made explicit by adding |
|
2401 |
*} |
|
2402 |
||
2403 |
(*<*) |
|
2404 |
end |
|
2405 |
||
54182 | 2406 |
locale dtr_view2 |
2407 |
begin |
|
2408 |
||
53826 | 2409 |
primcorec lappend :: "'a llist \<Rightarrow> 'a llist \<Rightarrow> 'a llist" where |
53750 | 2410 |
"lnull xs \<Longrightarrow> lnull ys \<Longrightarrow> lnull (lappend xs ys)" | |
55350 | 2411 |
"lhd (lappend xs ys) = lhd (if lnull xs then ys else xs)" | |
2412 |
"ltl (lappend xs ys) = (if xs = LNil then ltl ys else lappend (ltl xs) ys)" | |
|
53750 | 2413 |
(*>*) |
53752 | 2414 |
"_ \<Longrightarrow> \<not> lnull (lappend xs ys)" |
53750 | 2415 |
|
2416 |
text {* |
|
2417 |
\noindent |
|
53752 | 2418 |
to the specification. The generated selector theorems are conditional. |
2419 |
||
2420 |
The next example illustrates how to cope with selectors defined for several |
|
53750 | 2421 |
constructors: |
53749
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changeset
|
2422 |
*} |
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changeset
|
2423 |
|
53831
80423b9080cf
support "of" syntax to disambiguate selector equations
panny
parents:
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diff
changeset
|
2424 |
primcorec |
53752 | 2425 |
random_process :: "'a stream \<Rightarrow> (int \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> 'a process" |
2426 |
where |
|
57091 | 2427 |
"n mod 4 = 0 \<Longrightarrow> random_process s f n = Fail" | |
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|
2428 |
"n mod 4 = 1 \<Longrightarrow> is_Skip (random_process s f n)" | |
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diff
changeset
|
2429 |
"n mod 4 = 2 \<Longrightarrow> is_Action (random_process s f n)" | |
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diff
changeset
|
2430 |
"n mod 4 = 3 \<Longrightarrow> is_Choice (random_process s f n)" | |
53831
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parents:
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diff
changeset
|
2431 |
"cont (random_process s f n) = random_process s f (f n)" of Skip | |
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diff
changeset
|
2432 |
"prefix (random_process s f n) = shd s" | |
53831
80423b9080cf
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parents:
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diff
changeset
|
2433 |
"cont (random_process s f n) = random_process (stl s) f (f n)" of Action | |
53749
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|
2434 |
"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
changeset
|
2435 |
"right (random_process s f n) = random_process (every_snd (stl s)) f (f n)" |
53749
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diff
changeset
|
2436 |
|
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|
2437 |
text {* |
53750 | 2438 |
\noindent |
2439 |
Using the @{text "of"} keyword, different equations are specified for @{const |
|
2440 |
cont} depending on which constructor is selected. |
|
2441 |
||
2442 |
Here are more examples to conclude: |
|
53749
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changeset
|
2443 |
*} |
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changeset
|
2444 |
|
53826 | 2445 |
primcorec |
53749
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diff
changeset
|
2446 |
even_infty :: even_enat and |
b37db925b663
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parents:
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diff
changeset
|
2447 |
odd_infty :: odd_enat |
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parents:
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diff
changeset
|
2448 |
where |
57091 | 2449 |
"even_infty \<noteq> Even_EZero" | |
58245 | 2450 |
"un_Even_ESucc even_infty = odd_infty" | |
2451 |
"un_Odd_ESucc odd_infty = even_infty" |
|
53749
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changeset
|
2452 |
|
53752 | 2453 |
text {* \blankline *} |
2454 |
||
53826 | 2455 |
primcorec iterate\<^sub>i\<^sub>i :: "('a \<Rightarrow> 'a llist) \<Rightarrow> 'a \<Rightarrow> 'a tree\<^sub>i\<^sub>i" where |
54072 | 2456 |
"lbl\<^sub>i\<^sub>i (iterate\<^sub>i\<^sub>i g x) = x" | |
2457 |
"sub\<^sub>i\<^sub>i (iterate\<^sub>i\<^sub>i g x) = lmap (iterate\<^sub>i\<^sub>i g) (g x)" |
|
53749
b37db925b663
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diff
changeset
|
2458 |
(*<*) |
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diff
changeset
|
2459 |
end |
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parents:
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diff
changeset
|
2460 |
(*>*) |
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changeset
|
2461 |
|
53750 | 2462 |
|
53617 | 2463 |
subsection {* Command Syntax |
2464 |
\label{ssec:primcorec-command-syntax} *} |
|
2465 |
||
53826 | 2466 |
subsubsection {* \keyw{primcorec} and \keyw{primcorecursive} |
53753
ae7f50e70c09
renamed "primcorec" to "primcorecursive", to open the door to a 'theory -> theory' command called "primcorec" (cf. "fun" vs. "function")
blanchet
parents:
53752
diff
changeset
|
2467 |
\label{sssec:primcorecursive-and-primcorec} *} |
52840 | 2468 |
|
2469 |
text {* |
|
53829 | 2470 |
\begin{matharray}{rcl} |
2471 |
@{command_def "primcorec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\ |
|
2472 |
@{command_def "primcorecursive"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
|
2473 |
\end{matharray} |
|
52840 | 2474 |
|
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2475 |
@{rail \<open> |
55029
61a6bf7d4b02
clarified @{rail} syntax: prefer explicit \<newline> symbol;
wenzelm
parents:
54958
diff
changeset
|
2476 |
(@@{command primcorec} | @@{command primcorecursive}) target? \<newline> |
59277 | 2477 |
@{syntax pcr_options}? fixes @'where' (@{syntax pcr_formula} + '|') |
53749
b37db925b663
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parents:
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diff
changeset
|
2478 |
; |
59282 | 2479 |
@{syntax_def pcr_options}: '(' ((@{syntax plugins} | 'sequential' | 'exhaustive' | 'transfer') + ',') ')' |
52840 | 2480 |
; |
53749
b37db925b663
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parents:
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diff
changeset
|
2481 |
@{syntax_def pcr_formula}: thmdecl? prop (@'of' (term * ))? |
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2482 |
\<close>} |
53749
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diff
changeset
|
2483 |
|
55351 | 2484 |
\medskip |
2485 |
||
2486 |
\noindent |
|
55474 | 2487 |
The @{command primcorec} and @{command primcorecursive} commands introduce a set |
2488 |
of mutually corecursive functions over codatatypes. |
|
55460
3f4efd7d950d
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diff
changeset
|
2489 |
|
3f4efd7d950d
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diff
changeset
|
2490 |
The syntactic entity \synt{target} can be used to specify a local context, |
55474 | 2491 |
\synt{fixes} denotes a list of names with optional type signatures, |
2492 |
\synt{thmdecl} denotes an optional name for the formula that follows, and |
|
58620 | 2493 |
\synt{prop} denotes a HOL proposition @{cite "isabelle-isar-ref"}. |
55460
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changeset
|
2494 |
|
59280 | 2495 |
The optional target is optionally followed by a combination of the following |
56124 | 2496 |
options: |
53749
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diff
changeset
|
2497 |
|
b37db925b663
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parents:
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changeset
|
2498 |
\begin{itemize} |
b37db925b663
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parents:
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diff
changeset
|
2499 |
\setlength{\itemsep}{0pt} |
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parents:
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changeset
|
2500 |
|
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changeset
|
2501 |
\item |
59282 | 2502 |
The @{text plugins} option indicates which plugins should be enabled |
2503 |
(@{text only}) or disabled (@{text del}). By default, all plugins are enabled. |
|
2504 |
||
2505 |
\item |
|
2506 |
The @{text sequential} option indicates that the conditions in specifications |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
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diff
changeset
|
2507 |
expressed using the constructor or destructor view are to be interpreted |
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blanchet
parents:
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diff
changeset
|
2508 |
sequentially. |
53826 | 2509 |
|
2510 |
\item |
|
59282 | 2511 |
The @{text exhaustive} option indicates that the conditions in specifications |
53826 | 2512 |
expressed using the constructor or destructor view cover all possible cases. |
59284 | 2513 |
This generally gives rise to an additional proof obligation. |
59277 | 2514 |
|
2515 |
\item |
|
59282 | 2516 |
The @{text transfer} option indicates that an unconditional transfer rule |
59278 | 2517 |
should be generated and proved @{text "by transfer_prover"}. The |
2518 |
@{text "[transfer_rule]"} attribute is set on the generated theorem. |
|
53749
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parents:
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diff
changeset
|
2519 |
\end{itemize} |
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parents:
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diff
changeset
|
2520 |
|
55474 | 2521 |
The @{command primcorec} command is an abbreviation for @{command |
2522 |
primcorecursive} with @{text "by auto?"} to discharge any emerging proof |
|
2523 |
obligations. |
|
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2524 |
|
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2525 |
%%% TODO: elaborate on format of the propositions |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2526 |
%%% TODO: elaborate on mutual and nested-to-mutual |
52840 | 2527 |
*} |
52794 | 2528 |
|
52824 | 2529 |
|
52840 | 2530 |
subsection {* Generated Theorems |
2531 |
\label{ssec:primcorec-generated-theorems} *} |
|
59284 | 2532 |
|
2533 |
text {* |
|
2534 |
The @{command primcorec} and @{command primcorecursive} commands generate the |
|
2535 |
following properties (listed for @{const literate}): |
|
2536 |
||
2537 |
\begin{indentblock} |
|
2538 |
\begin{description} |
|
2539 |
||
2540 |
\item[@{text "f."}\hthm{code} @{text "[code]"}\rm:] ~ \\ |
|
2541 |
@{thm literate.code[no_vars]} \\ |
|
2542 |
The @{text "[code]"} attribute is set by the @{text code} plugin |
|
2543 |
(Section~\ref{ssec:code-generator}). |
|
2544 |
||
2545 |
\item[@{text "f."}\hthm{ctr}\rm:] ~ \\ |
|
2546 |
@{thm literate.ctr[no_vars]} |
|
2547 |
||
2548 |
\item[@{text "f."}\hthm{disc} @{text "[simp, code]"}\rm:] ~ \\ |
|
2549 |
@{thm literate.disc[no_vars]} \\ |
|
2550 |
The @{text "[code]"} attribute is set by the @{text code} plugin |
|
2551 |
(Section~\ref{ssec:code-generator}). The @{text "[simp]"} attribute is set only |
|
2552 |
for functions for which @{text f.disc_iff} is not available. |
|
2553 |
||
2554 |
\item[@{text "f."}\hthm{disc_iff} @{text "[simp]"}\rm:] ~ \\ |
|
2555 |
@{thm literate.disc_iff[no_vars]} \\ |
|
2556 |
This property is generated only for functions declared with the |
|
2557 |
@{text exhaustive} option or whose conditions are trivially exhaustive. |
|
2558 |
||
2559 |
\item[@{text "f."}\hthm{sel} @{text "[simp, code]"}\rm:] ~ \\ |
|
2560 |
@{thm literate.disc[no_vars]} \\ |
|
2561 |
The @{text "[code]"} attribute is set by the @{text code} plugin |
|
2562 |
(Section~\ref{ssec:code-generator}). |
|
2563 |
||
2564 |
\item[@{text "f."}\hthm{exclude}\rm:] ~ \\ |
|
2565 |
These properties are missing for @{const literate} because no exclusiveness |
|
2566 |
proof obligations arose. In general, the properties correspond to the |
|
2567 |
discharged proof obligations. |
|
2568 |
||
2569 |
\item[@{text "f."}\hthm{exhaust}\rm:] ~ \\ |
|
2570 |
This property is missing for @{const literate} because no exhaustiveness |
|
2571 |
proof obligation arose. In general, the property correspond to the discharged |
|
2572 |
proof obligation. |
|
2573 |
||
2574 |
\item[\begin{tabular}{@ {}l@ {}} |
|
2575 |
@{text "f."}\hthm{coinduct} @{text "[consumes m, case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
|
2576 |
\phantom{@{text "f."}\hthm{coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> |
|
2577 |
D\<^sub>n]"}\rm: |
|
2578 |
\end{tabular}] ~ \\ |
|
2579 |
This coinduction rule is generated for nested-as-mutual corecursive functions |
|
2580 |
(Section~\ref{sssec:primcorec-nested-as-mutual-corecursion}). |
|
2581 |
||
2582 |
\item[\begin{tabular}{@ {}l@ {}} |
|
2583 |
@{text "f."}\hthm{coinduct_strong} @{text "[consumes m, case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
|
2584 |
\phantom{@{text "f."}\hthm{coinduct_strong} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> |
|
2585 |
D\<^sub>n]"}\rm: |
|
2586 |
\end{tabular}] ~ \\ |
|
2587 |
This coinduction rule is generated for nested-as-mutual corecursive functions |
|
2588 |
(Section~\ref{sssec:primcorec-nested-as-mutual-corecursion}). |
|
2589 |
||
2590 |
\item[\begin{tabular}{@ {}l@ {}} |
|
2591 |
@{text "f\<^sub>1_\<dots>_f\<^sub>m."}\hthm{coinduct} @{text "[case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
|
2592 |
\phantom{@{text "f."}\hthm{coinduct} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> |
|
2593 |
D\<^sub>n]"}\rm: |
|
2594 |
\end{tabular}] ~ \\ |
|
2595 |
This coinduction rule is generated for nested-as-mutual corecursive functions |
|
2596 |
(Section~\ref{sssec:primcorec-nested-as-mutual-corecursion}). Given $m > 1$ |
|
2597 |
mutually corecursive functions, this rule can be used to prove $m$ properties |
|
2598 |
simultaneously. |
|
2599 |
||
2600 |
\item[\begin{tabular}{@ {}l@ {}} |
|
2601 |
@{text "f\<^sub>1_\<dots>_f\<^sub>m."}\hthm{coinduct_strong} @{text "[case_names t\<^sub>1 \<dots> t\<^sub>m,"} \\ |
|
2602 |
\phantom{@{text "f."}\hthm{coinduct_strong} @{text "["}}@{text "case_conclusion D\<^sub>1 \<dots> |
|
2603 |
D\<^sub>n]"}\rm: |
|
2604 |
\end{tabular}] ~ \\ |
|
2605 |
This coinduction rule is generated for nested-as-mutual corecursive functions |
|
2606 |
(Section~\ref{sssec:primcorec-nested-as-mutual-corecursion}). Given $m > 1$ |
|
2607 |
mutually corecursive functions, this rule can be used to prove $m$ properties |
|
2608 |
simultaneously. |
|
2609 |
||
2610 |
\end{description} |
|
2611 |
\end{indentblock} |
|
2612 |
||
2613 |
\noindent |
|
2614 |
For convenience, @{command primcorec} and @{command primcorecursive} also |
|
2615 |
provide the following collection: |
|
2616 |
||
2617 |
\begin{indentblock} |
|
2618 |
\begin{description} |
|
2619 |
||
2620 |
\item[@{text "f."}\hthm{simps}] = @{text f.disc_iff} (or @{text f.disc}) @{text t.sel} |
|
2621 |
||
2622 |
\end{description} |
|
2623 |
\end{indentblock} |
|
2624 |
*} |
|
52794 | 2625 |
|
2626 |
||
53623 | 2627 |
(* |
2628 |
subsection {* Recursive Default Values for Selectors |
|
2629 |
\label{ssec:primcorec-recursive-default-values-for-selectors} *} |
|
2630 |
||
2631 |
text {* |
|
2632 |
partial_function to the rescue |
|
2633 |
*} |
|
2634 |
*) |
|
2635 |
||
2636 |
||
59298 | 2637 |
section {* Registering Bounded Natural Functors |
2638 |
\label{sec:registering-bounded-natural-functors} *} |
|
52792 | 2639 |
|
52805 | 2640 |
text {* |
53647 | 2641 |
The (co)datatype package can be set up to allow nested recursion through |
55350 | 2642 |
arbitrary type constructors, as long as they adhere to the BNF requirements |
2643 |
and are registered as BNFs. It is also possible to declare a BNF abstractly |
|
2644 |
without specifying its internal structure. |
|
52805 | 2645 |
*} |
2646 |
||
52824 | 2647 |
|
55350 | 2648 |
subsection {* Bounded Natural Functors |
2649 |
\label{ssec:bounded-natural-functors} *} |
|
2650 |
||
2651 |
text {* |
|
2652 |
Bounded natural functors (BNFs) are a semantic criterion for where |
|
2653 |
(co)re\-cur\-sion may appear on the right-hand side of an equation |
|
58620 | 2654 |
@{cite "traytel-et-al-2012" and "blanchette-et-al-wit"}. |
55350 | 2655 |
|
2656 |
An $n$-ary BNF is a type constructor equipped with a map function |
|
2657 |
(functorial action), $n$ set functions (natural transformations), |
|
2658 |
and an infinite cardinal bound that satisfy certain properties. |
|
2659 |
For example, @{typ "'a llist"} is a unary BNF. |
|
2660 |
Its relator @{text "llist_all2 \<Colon> |
|
2661 |
('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> |
|
2662 |
'a llist \<Rightarrow> 'b llist \<Rightarrow> bool"} |
|
2663 |
extends binary predicates over elements to binary predicates over parallel |
|
2664 |
lazy lists. The cardinal bound limits the number of elements returned by the |
|
2665 |
set function; it may not depend on the cardinality of @{typ 'a}. |
|
2666 |
||
58310 | 2667 |
The type constructors introduced by @{command datatype} and |
55350 | 2668 |
@{command codatatype} are automatically registered as BNFs. In addition, a |
2669 |
number of old-style datatypes and non-free types are preregistered. |
|
2670 |
||
2671 |
Given an $n$-ary BNF, the $n$ type variables associated with set functions, |
|
2672 |
and on which the map function acts, are \emph{live}; any other variables are |
|
2673 |
\emph{dead}. Nested (co)recursion can only take place through live variables. |
|
2674 |
*} |
|
2675 |
||
2676 |
||
2677 |
subsection {* Introductory Examples |
|
2678 |
\label{ssec:bnf-introductory-examples} *} |
|
52805 | 2679 |
|
2680 |
text {* |
|
55350 | 2681 |
The example below shows how to register a type as a BNF using the @{command bnf} |
2682 |
command. Some of the proof obligations are best viewed with the theory |
|
2683 |
@{theory Cardinal_Notations}, located in \verb|~~/src/HOL/Library|, |
|
2684 |
imported. |
|
2685 |
||
2686 |
The type is simply a copy of the function space @{typ "'d \<Rightarrow> 'a"}, where @{typ 'a} |
|
2687 |
is live and @{typ 'd} is dead. We introduce it together with its map function, |
|
2688 |
set function, and relator. |
|
52805 | 2689 |
*} |
55350 | 2690 |
|
2691 |
typedef ('d, 'a) fn = "UNIV \<Colon> ('d \<Rightarrow> 'a) set" |
|
2692 |
by simp |
|
2693 |
||
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2694 |
text {* \blankline *} |
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2695 |
|
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2696 |
setup_lifting type_definition_fn |
55350 | 2697 |
|
2698 |
text {* \blankline *} |
|
2699 |
||
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2700 |
lift_definition map_fn :: "('a \<Rightarrow> 'b) \<Rightarrow> ('d, 'a) fn \<Rightarrow> ('d, 'b) fn" is "op \<circ>" . |
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2701 |
lift_definition set_fn :: "('d, 'a) fn \<Rightarrow> 'a set" is range . |
55350 | 2702 |
|
2703 |
text {* \blankline *} |
|
2704 |
||
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2705 |
lift_definition |
55350 | 2706 |
rel_fn :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('d, 'a) fn \<Rightarrow> ('d, 'b) fn \<Rightarrow> bool" |
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2707 |
is |
55945 | 2708 |
"rel_fun (op =)" . |
55350 | 2709 |
|
2710 |
text {* \blankline *} |
|
2711 |
||
2712 |
bnf "('d, 'a) fn" |
|
2713 |
map: map_fn |
|
2714 |
sets: set_fn |
|
2715 |
bd: "natLeq +c |UNIV :: 'd set|" |
|
2716 |
rel: rel_fn |
|
2717 |
proof - |
|
2718 |
show "map_fn id = id" |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2719 |
by transfer auto |
55350 | 2720 |
next |
60136 | 2721 |
fix f :: "'a \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'c" |
2722 |
show "map_fn (g \<circ> f) = map_fn g \<circ> map_fn f" |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2723 |
by transfer (auto simp add: comp_def) |
55350 | 2724 |
next |
60136 | 2725 |
fix F :: "('d, 'a) fn" and f g :: "'a \<Rightarrow> 'b" |
55350 | 2726 |
assume "\<And>x. x \<in> set_fn F \<Longrightarrow> f x = g x" |
2727 |
thus "map_fn f F = map_fn g F" |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2728 |
by transfer auto |
55350 | 2729 |
next |
60136 | 2730 |
fix f :: "'a \<Rightarrow> 'b" |
2731 |
show "set_fn \<circ> map_fn f = op ` f \<circ> set_fn" |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2732 |
by transfer (auto simp add: comp_def) |
55350 | 2733 |
next |
2734 |
show "card_order (natLeq +c |UNIV \<Colon> 'd set| )" |
|
2735 |
apply (rule card_order_csum) |
|
2736 |
apply (rule natLeq_card_order) |
|
2737 |
by (rule card_of_card_order_on) |
|
2738 |
next |
|
2739 |
show "cinfinite (natLeq +c |UNIV \<Colon> 'd set| )" |
|
2740 |
apply (rule cinfinite_csum) |
|
2741 |
apply (rule disjI1) |
|
2742 |
by (rule natLeq_cinfinite) |
|
2743 |
next |
|
2744 |
fix F :: "('d, 'a) fn" |
|
2745 |
have "|set_fn F| \<le>o |UNIV \<Colon> 'd set|" (is "_ \<le>o ?U") |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2746 |
by transfer (rule card_of_image) |
55350 | 2747 |
also have "?U \<le>o natLeq +c ?U" |
2748 |
by (rule ordLeq_csum2) (rule card_of_Card_order) |
|
2749 |
finally show "|set_fn F| \<le>o natLeq +c |UNIV \<Colon> 'd set|" . |
|
2750 |
next |
|
60136 | 2751 |
fix R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and S :: "'b \<Rightarrow> 'c \<Rightarrow> bool" |
55350 | 2752 |
show "rel_fn R OO rel_fn S \<le> rel_fn (R OO S)" |
55945 | 2753 |
by (rule, transfer) (auto simp add: rel_fun_def) |
55350 | 2754 |
next |
60136 | 2755 |
fix R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" |
55350 | 2756 |
show "rel_fn R = |
57398 | 2757 |
(BNF_Def.Grp {x. set_fn x \<subseteq> Collect (split R)} (map_fn fst))\<inverse>\<inverse> OO |
2758 |
BNF_Def.Grp {x. set_fn x \<subseteq> Collect (split R)} (map_fn snd)" |
|
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2759 |
unfolding Grp_def fun_eq_iff relcompp.simps conversep.simps |
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2760 |
apply transfer |
55945 | 2761 |
unfolding rel_fun_def subset_iff image_iff |
55459
1cd927ca8296
cleaner, complete proof in documentation, contributed by Dmitriy T.
blanchet
parents:
55410
diff
changeset
|
2762 |
by auto (force, metis pair_collapse) |
55350 | 2763 |
qed |
2764 |
||
2765 |
text {* \blankline *} |
|
2766 |
||
2767 |
print_theorems |
|
2768 |
print_bnfs |
|
2769 |
||
2770 |
text {* |
|
2771 |
\noindent |
|
58931 | 2772 |
Using \keyw{print_theorems} and @{command print_bnfs}, we can contemplate and |
55350 | 2773 |
show the world what we have achieved. |
2774 |
||
2775 |
This particular example does not need any nonemptiness witness, because the |
|
2776 |
one generated by default is good enough, but in general this would be |
|
2777 |
necessary. See \verb|~~/src/HOL/Basic_BNFs.thy|, |
|
2778 |
\verb|~~/src/HOL/Library/FSet.thy|, and \verb|~~/src/HOL/Library/Multiset.thy| |
|
2779 |
for further examples of BNF registration, some of which feature custom |
|
2780 |
witnesses. |
|
2781 |
||
57542 | 2782 |
The next example declares a BNF axiomatically. This can be convenient for |
2783 |
reasoning abstractly about an arbitrary BNF. The @{command bnf_axiomatization} |
|
2784 |
command below introduces a type @{text "('a, 'b, 'c) F"}, three set constants, |
|
2785 |
a map function, a relator, and a nonemptiness witness that depends only on |
|
59284 | 2786 |
@{typ 'a}. The type @{text "'a \<Rightarrow> ('a, 'b, 'c) F"} of the witness can be read |
2787 |
as an implication: Given a witness for @{typ 'a}, we can construct a witness for |
|
2788 |
@{text "('a, 'b, 'c) F"}. The BNF properties are postulated as axioms. |
|
55350 | 2789 |
*} |
2790 |
||
57542 | 2791 |
bnf_axiomatization (setA: 'a, setB: 'b, setC: 'c) F |
2792 |
[wits: "'a \<Rightarrow> ('a, 'b, 'c) F"] |
|
55350 | 2793 |
|
2794 |
text {* \blankline *} |
|
2795 |
||
2796 |
print_theorems |
|
2797 |
print_bnfs |
|
52794 | 2798 |
|
52824 | 2799 |
|
53617 | 2800 |
subsection {* Command Syntax |
2801 |
\label{ssec:bnf-command-syntax} *} |
|
2802 |
||
53621 | 2803 |
subsubsection {* \keyw{bnf} |
2804 |
\label{sssec:bnf} *} |
|
52794 | 2805 |
|
53028 | 2806 |
text {* |
53829 | 2807 |
\begin{matharray}{rcl} |
2808 |
@{command_def "bnf"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
|
2809 |
\end{matharray} |
|
2810 |
||
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2811 |
@{rail \<open> |
55474 | 2812 |
@@{command bnf} target? (name ':')? type \<newline> |
55029
61a6bf7d4b02
clarified @{rail} syntax: prefer explicit \<newline> symbol;
wenzelm
parents:
54958
diff
changeset
|
2813 |
'map:' term ('sets:' (term +))? 'bd:' term \<newline> |
58190 | 2814 |
('wits:' (term +))? ('rel:' term)? \<newline> |
2815 |
@{syntax plugins}? |
|
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2816 |
\<close>} |
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2817 |
|
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2818 |
\medskip |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2819 |
|
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2820 |
\noindent |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2821 |
The @{command bnf} command registers an existing type as a bounded natural |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2822 |
functor (BNF). The type must be equipped with an appropriate map function |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2823 |
(functorial action). In addition, custom set functions, relators, and |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2824 |
nonemptiness witnesses can be specified; otherwise, default versions are used. |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2825 |
|
55474 | 2826 |
The syntactic entity \synt{target} can be used to specify a local context, |
2827 |
\synt{type} denotes a HOL type, and \synt{term} denotes a HOL term |
|
58620 | 2828 |
@{cite "isabelle-isar-ref"}. |
55474 | 2829 |
|
59280 | 2830 |
The @{syntax plugins} option indicates which plugins should be enabled |
2831 |
(@{text only}) or disabled (@{text del}). By default, all plugins are enabled. |
|
2832 |
||
55474 | 2833 |
%%% TODO: elaborate on proof obligations |
53028 | 2834 |
*} |
52805 | 2835 |
|
53617 | 2836 |
|
57542 | 2837 |
subsubsection {* \keyw{bnf_axiomatization} |
56948 | 2838 |
\label{sssec:bnf-axiomatization} *} |
54187 | 2839 |
|
2840 |
text {* |
|
2841 |
\begin{matharray}{rcl} |
|
56942 | 2842 |
@{command_def "bnf_axiomatization"} & : & @{text "local_theory \<rightarrow> local_theory"} |
54187 | 2843 |
\end{matharray} |
2844 |
||
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2845 |
@{rail \<open> |
59280 | 2846 |
@@{command bnf_axiomatization} target? ('(' @{syntax plugins} ')')? \<newline> |
58190 | 2847 |
@{syntax tyargs}? name @{syntax wit_types}? \<newline> |
2848 |
mixfix? @{syntax map_rel}? |
|
54602 | 2849 |
; |
55350 | 2850 |
@{syntax_def wit_types}: '[' 'wits' ':' types ']' |
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2851 |
\<close>} |
54602 | 2852 |
|
55351 | 2853 |
\medskip |
2854 |
||
55350 | 2855 |
\noindent |
56942 | 2856 |
The @{command bnf_axiomatization} command declares a new type and associated constants |
55350 | 2857 |
(map, set, relator, and cardinal bound) and asserts the BNF properties for |
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2858 |
these constants as axioms. |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2859 |
|
55474 | 2860 |
The syntactic entity \synt{target} can be used to specify a local context, |
2861 |
\synt{name} denotes an identifier, \synt{typefree} denotes fixed type variable |
|
2862 |
(@{typ 'a}, @{typ 'b}, \ldots), and \synt{mixfix} denotes the usual |
|
58620 | 2863 |
parenthesized mixfix notation @{cite "isabelle-isar-ref"}. |
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2864 |
|
59280 | 2865 |
The @{syntax plugins} option indicates which plugins should be enabled |
2866 |
(@{text only}) or disabled (@{text del}). By default, all plugins are enabled. |
|
2867 |
||
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2868 |
Type arguments are live by default; they can be marked as dead by entering |
58220 | 2869 |
@{text dead} in front of the type variable (e.g., @{text "(dead 'a)"}) |
57092 | 2870 |
instead of an identifier for the corresponding set function. Witnesses can be |
2871 |
specified by their types. Otherwise, the syntax of @{command bnf_axiomatization} |
|
58310 | 2872 |
is identical to the left-hand side of a @{command datatype} or |
57092 | 2873 |
@{command codatatype} definition. |
55350 | 2874 |
|
2875 |
The command is useful to reason abstractly about BNFs. The axioms are safe |
|
57575 | 2876 |
because there exist BNFs of arbitrary large arities. Applications must import |
56942 | 2877 |
the theory @{theory BNF_Axiomatization}, located in the directory |
57542 | 2878 |
\verb|~~/src/|\allowbreak\verb|HOL/Library|, to use this functionality. |
54187 | 2879 |
*} |
2880 |
||
2881 |
||
57542 | 2882 |
subsubsection {* \keyw{print_bnfs} |
53621 | 2883 |
\label{sssec:print-bnfs} *} |
53617 | 2884 |
|
2885 |
text {* |
|
53829 | 2886 |
\begin{matharray}{rcl} |
2887 |
@{command_def "print_bnfs"} & : & @{text "local_theory \<rightarrow>"} |
|
2888 |
\end{matharray} |
|
2889 |
||
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2890 |
@{rail \<open> |
53829 | 2891 |
@@{command print_bnfs} |
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2892 |
\<close>} |
53617 | 2893 |
*} |
2894 |
||
2895 |
||
2896 |
section {* Deriving Destructors and Theorems for Free Constructors |
|
2897 |
\label{sec:deriving-destructors-and-theorems-for-free-constructors} *} |
|
52794 | 2898 |
|
52805 | 2899 |
text {* |
53623 | 2900 |
The derivation of convenience theorems for types equipped with free constructors, |
58310 | 2901 |
as performed internally by @{command datatype} and @{command codatatype}, |
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
2902 |
is available as a stand-alone command called @{command free_constructors}. |
52794 | 2903 |
|
53617 | 2904 |
% * need for this is rare but may arise if you want e.g. to add destructors to |
2905 |
% a type not introduced by ... |
|
2906 |
% |
|
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
2907 |
% * @{command free_constructors} |
58192 | 2908 |
% * @{text plugins}, @{text discs_sels} |
53617 | 2909 |
% * hack to have both co and nonco view via locale (cf. ext nats) |
52805 | 2910 |
*} |
52792 | 2911 |
|
52824 | 2912 |
|
53619 | 2913 |
(* |
53617 | 2914 |
subsection {* Introductory Example |
2915 |
\label{ssec:ctors-introductory-example} *} |
|
53619 | 2916 |
*) |
52794 | 2917 |
|
52824 | 2918 |
|
53617 | 2919 |
subsection {* Command Syntax |
2920 |
\label{ssec:ctors-command-syntax} *} |
|
2921 |
||
57542 | 2922 |
subsubsection {* \keyw{free_constructors} |
55472
990651bfc65b
updated docs to reflect the new 'free_constructors' syntax
blanchet
parents:
55468
diff
changeset
|
2923 |
\label{sssec:free-constructors} *} |
52828 | 2924 |
|
53018 | 2925 |
text {* |
53829 | 2926 |
\begin{matharray}{rcl} |
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
2927 |
@{command_def "free_constructors"} & : & @{text "local_theory \<rightarrow> proof(prove)"} |
53829 | 2928 |
\end{matharray} |
53018 | 2929 |
|
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2930 |
@{rail \<open> |
55468
98b25c51e9e5
renamed 'wrap_free_constructors' to 'free_constructors' (cf. 'functor', 'bnf', etc.)
blanchet
parents:
55460
diff
changeset
|
2931 |
@@{command free_constructors} target? @{syntax dt_options} \<newline> |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
2932 |
name 'for' (@{syntax fc_ctor} + '|') \<newline> |
57206
d9be905d6283
changed syntax of map: and rel: arguments to BNF-based datatypes
blanchet
parents:
57200
diff
changeset
|
2933 |
(@'where' (prop + '|'))? |
53018 | 2934 |
; |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
2935 |
@{syntax_def fc_ctor}: (name ':')? term (name * ) |
55112
b1a5d603fd12
prefer rail cartouche -- avoid back-slashed quotes;
wenzelm
parents:
55029
diff
changeset
|
2936 |
\<close>} |
53018 | 2937 |
|
55351 | 2938 |
\medskip |
2939 |
||
55460
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2940 |
\noindent |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2941 |
The @{command free_constructors} command generates destructor constants for |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2942 |
freely constructed types as well as properties about constructors and |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2943 |
destructors. It also registers the constants and theorems in a data structure |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2944 |
that is queried by various tools (e.g., \keyw{function}). |
3f4efd7d950d
added a bit of documentation for the different commands
blanchet
parents:
55459
diff
changeset
|
2945 |
|
55474 | 2946 |
The syntactic entity \synt{target} can be used to specify a local context, |
57200
aab87ffa60cc
use 'where' clause for selector default value syntax
blanchet
parents:
57153
diff
changeset
|
2947 |
\synt{name} denotes an identifier, \synt{prop} denotes a HOL proposition, and |
58620 | 2948 |
\synt{term} denotes a HOL term @{cite "isabelle-isar-ref"}. |
55474 | 2949 |
|
58310 | 2950 |
The syntax resembles that of @{command datatype} and @{command codatatype} |
55474 | 2951 |
definitions (Sections |
2952 |
\ref{ssec:datatype-command-syntax}~and~\ref{ssec:codatatype-command-syntax}). |
|
2953 |
A constructor is specified by an optional name for the discriminator, the |
|
2954 |
constructor itself (as a term), and a list of optional names for the selectors. |
|
53028 | 2955 |
|
53542 | 2956 |
Section~\ref{ssec:datatype-generated-theorems} lists the generated theorems. |
57542 | 2957 |
For bootstrapping reasons, the generally useful @{text "[fundef_cong]"} |
2958 |
attribute is not set on the generated @{text case_cong} theorem. It can be |
|
2959 |
added manually using \keyw{declare}. |
|
53018 | 2960 |
*} |
52828 | 2961 |
|
52794 | 2962 |
|
53617 | 2963 |
(* |
58245 | 2964 |
section {* Using the Standard ML Interface |
2965 |
\label{sec:using-the-standard-ml-interface} *} |
|
52792 | 2966 |
|
52805 | 2967 |
text {* |
53623 | 2968 |
The package's programmatic interface. |
52805 | 2969 |
*} |
53617 | 2970 |
*) |
52794 | 2971 |
|
2972 |
||
59282 | 2973 |
section {* Selecting Plugins |
2974 |
\label{sec:selecting-plugins} *} |
|
58190 | 2975 |
|
2976 |
text {* |
|
2977 |
Plugins extend the (co)datatype package to interoperate with other Isabelle |
|
58192 | 2978 |
packages and tools, such as the code generator, Transfer, Lifting, and |
2979 |
Quickcheck. They can be enabled or disabled individually using the |
|
58310 | 2980 |
@{syntax plugins} option to the commands @{command datatype}, |
59300 | 2981 |
@{command primrec}, @{command codatatype}, @{command primcorec}, |
2982 |
@{command primcorecursive}, @{command bnf}, @{command bnf_axiomatization}, and |
|
2983 |
@{command free_constructors}. For example: |
|
58190 | 2984 |
*} |
2985 |
||
58659
6c9821c32dd5
Local_Interpretation is superseded by Plugin with formal Plugin_Name management, avoiding undeclared strings;
wenzelm
parents:
58620
diff
changeset
|
2986 |
datatype (plugins del: code "quickcheck") color = Red | Black |
58192 | 2987 |
|
2988 |
||
2989 |
subsection {* Code Generator |
|
2990 |
\label{ssec:code-generator} *} |
|
2991 |
||
2992 |
text {* |
|
59299 | 2993 |
The \hthm{code} plugin registers freely generated types, including |
2994 |
(co)datatypes, and (co)recursive functions for code generation. No distinction |
|
2995 |
is made between datatypes and codatatypes. This means that for target languages |
|
2996 |
with a strict evaluation strategy (e.g., Standard ML), programs that attempt to |
|
2997 |
produce infinite codatatype values will not terminate. |
|
2998 |
||
2999 |
For types, the plugin derives the following properties: |
|
58244 | 3000 |
|
3001 |
\begin{indentblock} |
|
3002 |
\begin{description} |
|
3003 |
||
3004 |
\item[@{text "t."}\hthm{eq.refl} @{text "[code nbe]"}\rm:] ~ \\ |
|
3005 |
@{thm list.eq.refl[no_vars]} |
|
3006 |
||
3007 |
\item[@{text "t."}\hthm{eq.simps} @{text "[code]"}\rm:] ~ \\ |
|
3008 |
@{thm list.eq.simps(1)[no_vars]} \\ |
|
3009 |
@{thm list.eq.simps(2)[no_vars]} \\ |
|
3010 |
@{thm list.eq.simps(3)[no_vars]} \\ |
|
3011 |
@{thm list.eq.simps(4)[no_vars]} \\ |
|
3012 |
@{thm list.eq.simps(5)[no_vars]} \\ |
|
3013 |
@{thm list.eq.simps(6)[no_vars]} |
|
3014 |
||
3015 |
\end{description} |
|
3016 |
\end{indentblock} |
|
58509 | 3017 |
|
3018 |
In addition, the plugin sets the @{text "[code]"} attribute on a number of |
|
59299 | 3019 |
properties of freely generated types and of (co)recursive functions, as |
3020 |
documented in Sections \ref{ssec:datatype-generated-theorems}, |
|
3021 |
\ref{ssec:primrec-generated-theorems}, \ref{ssec:codatatype-generated-theorems}, |
|
3022 |
and~\ref{ssec:primcorec-generated-theorems}. |
|
58192 | 3023 |
*} |
58190 | 3024 |
|
3025 |
||
3026 |
subsection {* Size |
|
3027 |
\label{ssec:size} *} |
|
52794 | 3028 |
|
52805 | 3029 |
text {* |
58737 | 3030 |
For each datatype, the \hthm{size} plugin generates a generic size |
58190 | 3031 |
function @{text "t.size_t"} as well as a specific instance |
60134 | 3032 |
@{text "size \<Colon> t \<Rightarrow> nat"} belonging to the @{text size} type |
58190 | 3033 |
class. The \keyw{fun} command relies on @{const size} to prove termination of |
3034 |
recursive functions on datatypes. |
|
3035 |
||
3036 |
The plugin derives the following properties: |
|
3037 |
||
3038 |
\begin{indentblock} |
|
3039 |
\begin{description} |
|
3040 |
||
3041 |
\item[@{text "t."}\hthm{size} @{text "[simp, code]"}\rm:] ~ \\ |
|
3042 |
@{thm list.size(1)[no_vars]} \\ |
|
3043 |
@{thm list.size(2)[no_vars]} \\ |
|
3044 |
@{thm list.size(3)[no_vars]} \\ |
|
3045 |
@{thm list.size(4)[no_vars]} |
|
3046 |
||
58737 | 3047 |
\item[@{text "t."}\hthm{size_gen}\rm:] ~ \\ |
3048 |
@{thm list.size_gen(1)[no_vars]} \\ |
|
3049 |
@{thm list.size_gen(2)[no_vars]} |
|
3050 |
||
58739 | 3051 |
\item[@{text "t."}\hthm{size_gen_o_map}\rm:] ~ \\ |
3052 |
@{thm list.size_gen_o_map[no_vars]} |
|
58190 | 3053 |
|
58914 | 3054 |
\item[@{text "t."}\hthm{size_neq}\rm:] ~ \\ |
3055 |
This property is missing for @{typ "'a list"}. If the @{term size} function |
|
60134 | 3056 |
always evaluates to a non-zero value, this theorem has the form |
58914 | 3057 |
@{prop "\<not> size x = 0"}. |
3058 |
||
58190 | 3059 |
\end{description} |
3060 |
\end{indentblock} |
|
52805 | 3061 |
*} |
52794 | 3062 |
|
58190 | 3063 |
subsection {* Transfer |
3064 |
\label{ssec:transfer} *} |
|
3065 |
||
3066 |
text {* |
|
58192 | 3067 |
For each (co)datatype with live type arguments and each manually registered BNF, |
58245 | 3068 |
the \hthm{transfer} plugin generates a predicator @{text "t.pred_t"} and |
58244 | 3069 |
properties that guide the Transfer tool. |
3070 |
||
59824 | 3071 |
For types with no dead type arguments (and at least one live type argument), the |
3072 |
plugin derives the following properties: |
|
58244 | 3073 |
|
3074 |
\begin{indentblock} |
|
3075 |
\begin{description} |
|
3076 |
||
3077 |
\item[@{text "t."}\hthm{Domainp_rel} @{text "[relator_domain]"}\rm:] ~ \\ |
|
3078 |
@{thm list.Domainp_rel[no_vars]} |
|
3079 |
||
3080 |
\item[@{text "t."}\hthm{pred_inject} @{text "[simp]"}\rm:] ~ \\ |
|
3081 |
@{thm list.pred_inject(1)[no_vars]} \\ |
|
59579 | 3082 |
@{thm list.pred_inject(2)[no_vars]} \\ |
3083 |
This property is generated only for (co)datatypes. |
|
58244 | 3084 |
|
3085 |
\item[@{text "t."}\hthm{rel_eq_onp}\rm:] ~ \\ |
|
3086 |
@{thm list.rel_eq_onp[no_vars]} |
|
3087 |
||
58374
1b4d31b7bd10
fixed attribute name in docs (thanks to Andreas Lochbihler)
blanchet
parents:
58357
diff
changeset
|
3088 |
\item[@{text "t."}\hthm{left_total_rel} @{text "[transfer_rule]"}\rm:] ~ \\ |
58244 | 3089 |
@{thm list.left_total_rel[no_vars]} |
3090 |
||
58374
1b4d31b7bd10
fixed attribute name in docs (thanks to Andreas Lochbihler)
blanchet
parents:
58357
diff
changeset
|
3091 |
\item[@{text "t."}\hthm{left_unique_rel} @{text "[transfer_rule]"}\rm:] ~ \\ |
58244 | 3092 |
@{thm list.left_unique_rel[no_vars]} |
3093 |
||
58374
1b4d31b7bd10
fixed attribute name in docs (thanks to Andreas Lochbihler)
blanchet
parents:
58357
diff
changeset
|
3094 |
\item[@{text "t."}\hthm{right_total_rel} @{text "[transfer_rule]"}\rm:] ~ \\ |
58244 | 3095 |
@{thm list.right_total_rel[no_vars]} |
3096 |
||
58374
1b4d31b7bd10
fixed attribute name in docs (thanks to Andreas Lochbihler)
blanchet
parents:
58357
diff
changeset
|
3097 |
\item[@{text "t."}\hthm{right_unique_rel} @{text "[transfer_rule]"}\rm:] ~ \\ |
58244 | 3098 |
@{thm list.right_unique_rel[no_vars]} |
3099 |
||
58374
1b4d31b7bd10
fixed attribute name in docs (thanks to Andreas Lochbihler)
blanchet
parents:
58357
diff
changeset
|
3100 |
\item[@{text "t."}\hthm{bi_total_rel} @{text "[transfer_rule]"}\rm:] ~ \\ |
58244 | 3101 |
@{thm list.bi_total_rel[no_vars]} |
3102 |
||
58374
1b4d31b7bd10
fixed attribute name in docs (thanks to Andreas Lochbihler)
blanchet
parents:
58357
diff
changeset
|
3103 |
\item[@{text "t."}\hthm{bi_unique_rel} @{text "[transfer_rule]"}\rm:] ~ \\ |
58244 | 3104 |
@{thm list.bi_unique_rel[no_vars]} |
3105 |
||
3106 |
\end{description} |
|
3107 |
\end{indentblock} |
|
59282 | 3108 |
|
59579 | 3109 |
\begin{sloppy} |
3110 |
In addition, the plugin sets the @{text "[transfer_rule]"} attribute on the |
|
3111 |
following (co)datatypes properties: |
|
3112 |
@{text "t.case_transfer"}, @{text "t.sel_transfer"}, @{text "t.ctr_transfer"}, |
|
3113 |
@{text "t.disc_transfer"}, @{text "t.set_transfer"}, @{text "t.map_transfer"}, |
|
3114 |
@{text "t.rel_transfer"}, @{text "t.rec_transfer"}, and @{text "t.corec_transfer"}. |
|
3115 |
\end{sloppy} |
|
3116 |
||
59282 | 3117 |
For @{command primrec}, @{command primcorec}, and @{command primcorecursive}, |
59579 | 3118 |
the plugin implements the generation of the @{text "f.transfer"} property, |
3119 |
conditioned by the @{text transfer} option, and sets the |
|
3120 |
@{text "[transfer_rule]"} attribute on these. |
|
58190 | 3121 |
*} |
3122 |
||
3123 |
||
3124 |
subsection {* Lifting |
|
3125 |
\label{ssec:lifting} *} |
|
3126 |
||
3127 |
text {* |
|
59721 | 3128 |
For each (co)datatype and each manually registered BNF with at least one live |
3129 |
type argument and no dead type arguments, the \hthm{lifting} plugin generates |
|
3130 |
properties and attributes that guide the Lifting tool. |
|
58244 | 3131 |
|
3132 |
The plugin derives the following property: |
|
3133 |
||
3134 |
\begin{indentblock} |
|
3135 |
\begin{description} |
|
3136 |
||
3137 |
\item[@{text "t."}\hthm{Quotient} @{text "[quot_map]"}\rm:] ~ \\ |
|
3138 |
@{thm list.Quotient[no_vars]} |
|
3139 |
||
3140 |
\end{description} |
|
3141 |
\end{indentblock} |
|
3142 |
||
58245 | 3143 |
In addition, the plugin sets the @{text "[relator_eq_onp]"} attribute on a |
58508 | 3144 |
variant of the @{text t.rel_eq_onp} property generated by the @{text lifting} |
58244 | 3145 |
plugin, the @{text "[relator_mono]"} attribute on @{text t.rel_mono}, and the |
3146 |
@{text "[relator_distr]"} attribute on @{text t.rel_compp}. |
|
58190 | 3147 |
*} |
52794 | 3148 |
|
52824 | 3149 |
|
52828 | 3150 |
subsection {* Quickcheck |
3151 |
\label{ssec:quickcheck} *} |
|
52794 | 3152 |
|
58190 | 3153 |
text {* |
59280 | 3154 |
The integration of datatypes with Quickcheck is accomplished by the |
59282 | 3155 |
\hthm{quick\-check} plugin. It combines a number of subplugins that instantiate |
59280 | 3156 |
specific type classes. The subplugins can be enabled or disabled individually. |
3157 |
They are listed below: |
|
58245 | 3158 |
|
3159 |
\begin{indentblock} |
|
3160 |
\hthm{quickcheck_random} \\ |
|
3161 |
\hthm{quickcheck_exhaustive} \\ |
|
3162 |
\hthm{quickcheck_bounded_forall} \\ |
|
3163 |
\hthm{quickcheck_full_exhaustive} \\ |
|
3164 |
\hthm{quickcheck_narrowing} |
|
3165 |
\end{indentblock} |
|
58190 | 3166 |
*} |
52794 | 3167 |
|
52805 | 3168 |
|
58278 | 3169 |
subsection {* Program Extraction |
3170 |
\label{ssec:program-extraction} *} |
|
3171 |
||
3172 |
text {* |
|
3173 |
The \hthm{extraction} plugin provides realizers for induction and case analysis, |
|
3174 |
to enable program extraction from proofs involving datatypes. This functionality |
|
58395 | 3175 |
is only available with full proof objects, i.e., with the \emph{HOL-Proofs} |
58278 | 3176 |
session. |
3177 |
*} |
|
3178 |
||
3179 |
||
52827 | 3180 |
section {* Known Bugs and Limitations |
52805 | 3181 |
\label{sec:known-bugs-and-limitations} *} |
3182 |
||
3183 |
text {* |
|
58395 | 3184 |
This section lists the known bugs and limitations in the (co)datatype package at |
3185 |
the time of this writing. Many of them are expected to be addressed in future |
|
3186 |
releases. |
|
3187 |
||
3188 |
\begin{enumerate} |
|
3189 |
\setlength{\itemsep}{0pt} |
|
3190 |
||
3191 |
\item |
|
3192 |
\emph{Defining mutually (co)recursive (co)datatypes is slow.} Fortunately, |
|
3193 |
it is always possible to recast mutual specifications to nested ones, which are |
|
3194 |
processed more efficiently. |
|
3195 |
||
3196 |
\item |
|
3197 |
\emph{Locally fixed types cannot be used in (co)datatype specifications.} |
|
3198 |
This limitation can be circumvented by adding type arguments to the local |
|
3199 |
(co)datatypes to abstract over the locally fixed types. |
|
3200 |
||
3201 |
\item |
|
3202 |
\emph{The \emph{\keyw{primcorec}} command does not allow user-specified names and |
|
3203 |
attributes next to the entered formulas.} The less convenient syntax, using the |
|
3204 |
\keyw{lemmas} command, is available as an alternative. |
|
3205 |
||
3206 |
\item |
|
3207 |
\emph{There is no way to use an overloaded constant from a syntactic type |
|
3208 |
class, such as @{text 0}, as a constructor.} |
|
3209 |
||
3210 |
\item |
|
3211 |
\emph{There is no way to register the same type as both a datatype and a |
|
3212 |
codatatype.} This affects types such as the extended natural numbers, for which |
|
3213 |
both views would make sense (for a different set of constructors). |
|
3214 |
||
3215 |
\item |
|
3216 |
\emph{The names of variables are often suboptimal in the properties generated by |
|
3217 |
the package.} |
|
3218 |
||
3219 |
\end{enumerate} |
|
52805 | 3220 |
*} |
52794 | 3221 |
|
52822 | 3222 |
|
3223 |
text {* |
|
53863
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
3224 |
\section*{Acknowledgment} |
c7364dca96f2
textual improvements following Christian Sternagel's feedback
blanchet
parents:
53857
diff
changeset
|
3225 |
|
53749
b37db925b663
adapted primcorec documentation to reflect the three views
blanchet
parents:
53748
diff
changeset
|
3226 |
Tobias Nipkow and Makarius Wenzel encouraged us to implement the new |
53617 | 3227 |
(co)datatype package. Andreas Lochbihler provided lots of comments on earlier |
56655 | 3228 |
versions of the package, especially on the coinductive part. Brian Huffman |
58245 | 3229 |
suggested major simplifications to the internal constructions. Ond\v{r}ej |
3230 |
Kun\v{c}ar implemented the @{text transfer} and @{text lifting} plugins. |
|
60137 | 3231 |
Christian Sternagel and Ren\'e Thiemann ported the \keyw{derive} command |
3232 |
from the \emph{Archive of Formal Proofs} to the new datatypes. Florian Haftmann |
|
3233 |
and Christian Urban provided general advice on Isabelle and package writing. |
|
3234 |
Stefan Milius and Lutz Schr\"oder found an elegant proof that eliminated one of |
|
3235 |
the BNF proof obligations. Gerwin Klein, Andreas Lochbihler, Tobias Nipkow, and |
|
3236 |
Christian Sternagel suggested many textual improvements to this tutorial. |
|
52794 | 3237 |
*} |
53617 | 3238 |
|
52792 | 3239 |
end |