doc-src/TutorialI/fp.tex
author paulson
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\chapter{Functional Programming in HOL}
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Although on the surface this chapter is mainly concerned with how to write
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functional programs in HOL and how to verify them, most of the
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constructs and proof procedures introduced are general purpose and recur in
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any specification or verification task.
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The dedicated functional programmer should be warned: HOL offers only
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\emph{total functional programming} --- all functions in HOL must be total;
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lazy data structures are not directly available. On the positive side,
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functions in HOL need not be computable: HOL is a specification language that
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goes well beyond what can be expressed as a program. However, for the time
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being we concentrate on the computable.
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\section{An introductory theory}
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\label{sec:intro-theory}
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Functional programming needs datatypes and functions. Both of them can be
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defined in a theory with a syntax reminiscent of languages like ML or
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Haskell. As an example consider the theory in figure~\ref{fig:ToyList}.
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We will now examine it line by line.
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\begin{figure}[htbp]
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\begin{ttbox}\makeatother
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\input{ToyList2/ToyList1}\end{ttbox}
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\caption{A theory of lists}
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\label{fig:ToyList}
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\end{figure}
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{\makeatother\input{ToyList/document/ToyList.tex}}
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The complete proof script is shown in figure~\ref{fig:ToyList-proofs}. The
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concatenation of figures \ref{fig:ToyList} and \ref{fig:ToyList-proofs}
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constitutes the complete theory \texttt{ToyList} and should reside in file
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\texttt{ToyList.thy}. It is good practice to present all declarations and
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definitions at the beginning of a theory to facilitate browsing.
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\begin{figure}[htbp]
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\begin{ttbox}\makeatother
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\input{ToyList2/ToyList2}\end{ttbox}
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\caption{Proofs about lists}
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\label{fig:ToyList-proofs}
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\end{figure}
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\subsubsection*{Review}
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This is the end of our toy proof. It should have familiarized you with
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\begin{itemize}
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\item the standard theorem proving procedure:
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state a goal (lemma or theorem); proceed with proof until a separate lemma is
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required; prove that lemma; come back to the original goal.
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\item a specific procedure that works well for functional programs:
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induction followed by all-out simplification via \isa{auto}.
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\item a basic repertoire of proof commands.
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\end{itemize}
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\section{Some helpful commands}
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\label{sec:commands-and-hints}
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This section discusses a few basic commands for manipulating the proof state
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and can be skipped by casual readers.
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There are two kinds of commands used during a proof: the actual proof
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commands and auxiliary commands for examining the proof state and controlling
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the display. Simple proof commands are of the form
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\isacommand{apply}\isa{(method)}\indexbold{apply} where \bfindex{method} is a
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synonym for ``theorem proving function''. Typical examples are
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\isa{induct_tac} and \isa{auto}. Further methods are introduced throughout
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the tutorial.  Unless stated otherwise you may assume that a method attacks
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merely the first subgoal. An exception is \isa{auto} which tries to solve all
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subgoals.
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The most useful auxiliary commands are:
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\begin{description}
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\item[Undoing:] \isacommand{undo}\indexbold{*undo} undoes the effect of the
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  last command; \isacommand{undo} can be undone by
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  \isacommand{redo}\indexbold{*redo}.  Both are only needed at the shell
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  level and should not occur in the final theory.
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\item[Printing the current state:] \isacommand{pr}\indexbold{*pr} redisplays
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  the current proof state, for example when it has disappeared off the
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  screen.
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\item[Limiting the number of subgoals:] \isacommand{pr}~$n$ tells Isabelle to
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  print only the first $n$ subgoals from now on and redisplays the current
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  proof state. This is helpful when there are many subgoals.
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\item[Modifying the order of subgoals:]
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\isacommand{defer}\indexbold{*defer} moves the first subgoal to the end and
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\isacommand{prefer}\indexbold{*prefer}~$n$ moves subgoal $n$ to the front.
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\item[Printing theorems:]
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  \isacommand{thm}\indexbold{*thm}~\textit{name}$@1$~\dots~\textit{name}$@n$
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  prints the named theorems.
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\item[Displaying types:] We have already mentioned the flag
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  \ttindex{show_types} above. It can also be useful for detecting typos in
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  formulae early on. For example, if \texttt{show_types} is set and the goal
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  \isa{rev(rev xs) = xs} is started, Isabelle prints the additional output
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\par\noindent
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\begin{isabelle}%
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Variables:\isanewline
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~~xs~::~'a~list
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\end{isabelle}%
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\par\noindent
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which tells us that Isabelle has correctly inferred that
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\isa{xs} is a variable of list type. On the other hand, had we
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made a typo as in \isa{rev(re xs) = xs}, the response
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\par\noindent
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\begin{isabelle}%
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Variables:\isanewline
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~~re~::~'a~list~{\isasymRightarrow}~'a~list\isanewline
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~~xs~::~'a~list%
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\end{isabelle}%
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\par\noindent
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would have alerted us because of the unexpected variable \isa{re}.
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\item[Reading terms and types:] \isacommand{term}\indexbold{*term}
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  \textit{string} reads, type-checks and prints the given string as a term in
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  the current context; the inferred type is output as well.
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  \isacommand{typ}\indexbold{*typ} \textit{string} reads and prints the given
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  string as a type in the current context.
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\item[(Re)loading theories:] When you start your interaction you must open a
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  named theory with the line \isa{\isacommand{theory}~T~=~\dots~:}. Isabelle
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  automatically loads all the required parent theories from their respective
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  files (which may take a moment, unless the theories are already loaded and
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  the files have not been modified).
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  If you suddenly discover that you need to modify a parent theory of your
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  current theory you must first abandon your current theory\indexbold{abandon
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  theory}\indexbold{theory!abandon} (at the shell
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  level type \isacommand{kill}\indexbold{*kill}). After the parent theory has
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  been modified you go back to your original theory. When its first line
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  \isacommand{theory}\texttt{~T~=}~\dots~\texttt{:} is processed, the
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  modified parent is reloaded automatically.
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  The only time when you need to load a theory by hand is when you simply
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  want to check if it loads successfully without wanting to make use of the
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  theory itself. This you can do by typing
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  \isa{\isacommand{use\_thy}\indexbold{*use_thy}~"T"}.
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\end{description}
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Further commands are found in the Isabelle/Isar Reference Manual.
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We now examine Isabelle's functional programming constructs systematically,
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starting with inductive datatypes.
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\section{Datatypes}
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\label{sec:datatype}
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Inductive datatypes are part of almost every non-trivial application of HOL.
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First we take another look at a very important example, the datatype of
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lists, before we turn to datatypes in general. The section closes with a
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case study.
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\subsection{Lists}
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\indexbold{*list}
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Lists are one of the essential datatypes in computing. Readers of this
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tutorial and users of HOL need to be familiar with their basic operations.
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Theory \isa{ToyList} is only a small fragment of HOL's predefined theory
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\isa{List}\footnote{\url{http://isabelle.in.tum.de/library/HOL/List.html}}.
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The latter contains many further operations. For example, the functions
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\isaindexbold{hd} (``head'') and \isaindexbold{tl} (``tail'') return the first
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element and the remainder of a list. (However, pattern-matching is usually
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preferable to \isa{hd} and \isa{tl}.)  Theory \isa{List} also contains
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more syntactic sugar: \isa{[}$x@1$\isa{,}\dots\isa{,}$x@n$\isa{]} abbreviates
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$x@1$\isa{\#}\dots\isa{\#}$x@n$\isa{\#[]}.  In the rest of the tutorial we
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always use HOL's predefined lists.
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\subsection{The general format}
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\label{sec:general-datatype}
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The general HOL \isacommand{datatype} definition is of the form
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\[
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\isacommand{datatype}~(\alpha@1, \dots, \alpha@n) \, t ~=~
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C@1~\tau@{11}~\dots~\tau@{1k@1} ~\mid~ \dots ~\mid~
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C@m~\tau@{m1}~\dots~\tau@{mk@m}
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\]
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where $\alpha@i$ are distinct type variables (the parameters), $C@i$ are distinct
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constructor names and $\tau@{ij}$ are types; it is customary to capitalize
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the first letter in constructor names. There are a number of
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restrictions (such as that the type should not be empty) detailed
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elsewhere~\cite{isabelle-HOL}. Isabelle notifies you if you violate them.
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Laws about datatypes, such as \isa{[] \isasymnoteq~x\#xs} and
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\isa{(x\#xs = y\#ys) = (x=y \isasymand~xs=ys)}, are used automatically
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during proofs by simplification.  The same is true for the equations in
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primitive recursive function definitions.
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Every datatype $t$ comes equipped with a \isa{size} function from $t$ into
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the natural numbers (see~\S\ref{sec:nat} below). For lists, \isa{size} is
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just the length, i.e.\ \isa{size [] = 0} and \isa{size(x \# xs) = size xs +
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  1}.  In general, \isa{size} returns \isa{0} for all constructors that do
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not have an argument of type $t$, and for all other constructors \isa{1 +}
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the sum of the sizes of all arguments of type $t$. Note that because
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\isa{size} is defined on every datatype, it is overloaded; on lists
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\isa{size} is also called \isa{length}, which is not overloaded.
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\subsection{Primitive recursion}
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Functions on datatypes are usually defined by recursion. In fact, most of the
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time they are defined by what is called \bfindex{primitive recursion}.
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The keyword \isacommand{primrec}\indexbold{*primrec} is followed by a list of
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equations
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\[ f \, x@1 \, \dots \, (C \, y@1 \, \dots \, y@k)\, \dots \, x@n = r \]
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such that $C$ is a constructor of the datatype $t$ and all recursive calls of
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$f$ in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$. Thus
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Isabelle immediately sees that $f$ terminates because one (fixed!) argument
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becomes smaller with every recursive call. There must be exactly one equation
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for each constructor.  Their order is immaterial.
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A more general method for defining total recursive functions is introduced in
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\S\ref{sec:recdef}.
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\begin{exercise}\label{ex:Tree}
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\input{Misc/document/Tree.tex}%
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\end{exercise}
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\input{Misc/document/case_exprs.tex}
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\begin{warn}
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  Induction is only allowed on free (or \isasymAnd-bound) variables that
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  should not occur among the assumptions of the subgoal; see
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  \S\ref{sec:ind-var-in-prems} for details. Case distinction
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  (\isa{case_tac}) works for arbitrary terms, which need to be
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  quoted if they are non-atomic.
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\end{warn}
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\input{Ifexpr/document/Ifexpr.tex}
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\section{Some basic types}
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\subsection{Natural numbers}
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\label{sec:nat}
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\index{arithmetic|(}
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\input{Misc/document/fakenat.tex}
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\input{Misc/document/natsum.tex}
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The usual arithmetic operations \ttindexboldpos{+}{$HOL2arithfun},
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\ttindexboldpos{-}{$HOL2arithfun}, \ttindexboldpos{*}{$HOL2arithfun},
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\isaindexbold{div}, \isaindexbold{mod}, \isaindexbold{min} and
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\isaindexbold{max} are predefined, as are the relations
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\indexboldpos{\isasymle}{$HOL2arithrel} and
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\ttindexboldpos{<}{$HOL2arithrel}. There is even a least number operation
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\isaindexbold{LEAST}. For example, \isa{(LEAST n.$\,$1 < n) = 2}, although
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Isabelle does not prove this completely automatically. Note that \isa{1} and
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\isa{2} are available as abbreviations for the corresponding
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\isa{Suc}-expressions. If you need the full set of numerals,
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see~\S\ref{nat-numerals}.
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\begin{warn}
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  The constant \ttindexbold{0} and the operations
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  \ttindexboldpos{+}{$HOL2arithfun}, \ttindexboldpos{-}{$HOL2arithfun},
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  \ttindexboldpos{*}{$HOL2arithfun}, \isaindexbold{min}, \isaindexbold{max},
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  \indexboldpos{\isasymle}{$HOL2arithrel} and
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  \ttindexboldpos{<}{$HOL2arithrel} are overloaded, i.e.\ they are available
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  not just for natural numbers but at other types as well (see
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  \S\ref{sec:TypeClasses}). For example, given the goal \isa{x+0 = x}, there
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  is nothing to indicate that you are talking about natural numbers.  Hence
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  Isabelle can only infer that \isa{x} is of some arbitrary type where
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  \isa{0} and \isa{+} are declared. As a consequence, you will be unable to
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  prove the goal (although it may take you some time to realize what has
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  happened if \texttt{show_types} is not set).  In this particular example,
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  you need to include an explicit type constraint, for example \isa{x+0 =
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    (x::nat)}.  If there is enough contextual information this may not be
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  necessary: \isa{Suc x = x} automatically implies \isa{x::nat} because
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  \isa{Suc} is not overloaded.
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\end{warn}
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Simple arithmetic goals are proved automatically by both \isa{auto}
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and the simplification methods introduced in \S\ref{sec:Simplification}.  For
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example,
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\input{Misc/document/arith1.tex}%
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is proved automatically. The main restriction is that only addition is taken
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into account; other arithmetic operations and quantified formulae are ignored.
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For more complex goals, there is the special method
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\isaindexbold{arith} (which attacks the first subgoal). It proves
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arithmetic goals involving the usual logical connectives (\isasymnot,
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\isasymand, \isasymor, \isasymimp), the relations \isasymle\ and \isa{<}, and
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the operations \isa{+}, \isa{-}, \isa{min} and \isa{max}. For example,
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\input{Misc/document/arith2.tex}%
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succeeds because \isa{k*k} can be treated as atomic.
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In contrast,
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\input{Misc/document/arith3.tex}%
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is not even proved by \isa{arith} because the proof relies essentially
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on properties of multiplication.
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\begin{warn}
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  The running time of \isa{arith} is exponential in the number of occurrences
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  of \ttindexboldpos{-}{$HOL2arithfun}, \isaindexbold{min} and
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  \isaindexbold{max} because they are first eliminated by case distinctions.
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  \isa{arith} is incomplete even for the restricted class of formulae
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  described above (known as ``linear arithmetic''). If divisibility plays a
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  role, it may fail to prove a valid formula, for example $m+m \neq n+n+1$.
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  Fortunately, such examples are rare in practice.
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\end{warn}
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\index{arithmetic|)}
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\subsection{Products}
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\input{Misc/document/pairs.tex}
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%FIXME move stuff below into section on proofs about products?
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\begin{warn}
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  Abstraction over pairs and tuples is merely a convenient shorthand for a
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  more complex internal representation.  Thus the internal and external form
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  of a term may differ, which can affect proofs. If you want to avoid this
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  complication, use \isa{fst} and \isa{snd}, i.e.\ write
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  \isa{\isasymlambda{}p.~fst p + snd p} instead of
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  \isa{\isasymlambda(x,y).~x + y}.  See~\S\ref{proofs-about-products} for
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  theorem proving with tuple patterns.
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\end{warn}
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Note that products, like natural numbers, are datatypes, which means
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in particular that \isa{induct_tac} and \isa{case_tac} are applicable to
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products (see \S\ref{proofs-about-products}).
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Instead of tuples with many components (where ``many'' is not much above 2),
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it is far preferable to use record types (see \S\ref{sec:records}).
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\section{Definitions}
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\label{sec:Definitions}
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A definition is simply an abbreviation, i.e.\ a new name for an existing
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construction. In particular, definitions cannot be recursive. Isabelle offers
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definitions on the level of types and terms. Those on the type level are
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called type synonyms, those on the term level are called (constant)
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definitions.
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\subsection{Type synonyms}
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\indexbold{type synonym}
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Type synonyms are similar to those found in ML. Their syntax is fairly self
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explanatory:
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\input{Misc/document/types.tex}%
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Note that pattern-matching is not allowed, i.e.\ each definition must be of
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the form $f\,x@1\,\dots\,x@n~\isasymequiv~t$.
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Section~\S\ref{sec:Simplification} explains how definitions are used
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in proofs.
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\input{Misc/document/prime_def.tex}
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\chapter{More Functional Programming}
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The purpose of this chapter is to deepen the reader's understanding of the
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concepts encountered so far and to introduce advanced forms of datatypes and
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recursive functions. The first two sections give a structured presentation of
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theorem proving by simplification (\S\ref{sec:Simplification}) and discuss
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important heuristics for induction (\S\ref{sec:InductionHeuristics}). They can
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be skipped by readers less interested in proofs. They are followed by a case
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study, a compiler for expressions (\S\ref{sec:ExprCompiler}). Advanced
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datatypes, including those involving function spaces, are covered in
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\S\ref{sec:advanced-datatypes}, which closes with another case study, search
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trees (``tries'').  Finally we introduce \isacommand{recdef}, a very general
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form of recursive function definition which goes well beyond what
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\isacommand{primrec} allows (\S\ref{sec:recdef}).
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\section{Simplification}
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\label{sec:Simplification}
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\index{simplification|(}
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So far we have proved our theorems by \isa{auto}, which ``simplifies''
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\emph{all} subgoals. In fact, \isa{auto} can do much more than that, except
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that it did not need to so far. However, when you go beyond toy examples, you
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need to understand the ingredients of \isa{auto}.  This section covers the
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method that \isa{auto} always applies first, namely simplification.
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Simplification is one of the central theorem proving tools in Isabelle and
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many other systems. The tool itself is called the \bfindex{simplifier}. The
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purpose of this section is introduce the many features of the simplifier.
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Anybody intending to use HOL should read this section. Later on
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(\S\ref{sec:simplification-II}) we explain some more advanced features and a
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little bit of how the simplifier works. The serious student should read that
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section as well, in particular in order to understand what happened if things
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do not simplify as expected.
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\subsubsection{What is simplification}
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In its most basic form, simplification means repeated application of
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equations from left to right. For example, taking the rules for \isa{\at}
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and applying them to the term \isa{[0,1] \at\ []} results in a sequence of
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simplification steps:
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\begin{ttbox}\makeatother
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(0#1#[]) @ []  \(\leadsto\)  0#((1#[]) @ [])  \(\leadsto\)  0#(1#([] @ []))  \(\leadsto\)  0#1#[]
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\end{ttbox}
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This is also known as \bfindex{term rewriting}\indexbold{rewriting} and the
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equations are referred to as \textbf{rewrite rules}\indexbold{rewrite rule}.
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``Rewriting'' is more honest than ``simplification'' because the terms do not
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necessarily become simpler in the process.
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\input{Misc/document/simp.tex}
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\index{simplification|)}
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\input{Misc/document/Itrev.tex}
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\begin{exercise}
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\input{Misc/document/Tree2.tex}%
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\end{exercise}
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\input{CodeGen/document/CodeGen.tex}
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\section{Advanced datatypes}
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\label{sec:advanced-datatypes}
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\index{*datatype|(}
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\index{*primrec|(}
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%|)
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This section presents advanced forms of \isacommand{datatype}s.
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\subsection{Mutual recursion}
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\label{sec:datatype-mut-rec}
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\input{Datatype/document/ABexpr.tex}
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\subsection{Nested recursion}
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\label{sec:nested-datatype}
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{\makeatother\input{Datatype/document/Nested.tex}}
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\subsection{The limits of nested recursion}
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How far can we push nested recursion? By the unfolding argument above, we can
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reduce nested to mutual recursion provided the nested recursion only involves
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previously defined datatypes. This does not include functions:
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\begin{isabelle}
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\isacommand{datatype} t = C "t \isasymRightarrow\ bool"
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\end{isabelle}
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is a real can of worms: in HOL it must be ruled out because it requires a type
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\isa{t} such that \isa{t} and its power set \isa{t \isasymFun\ bool} have the
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same cardinality---an impossibility. For the same reason it is not possible
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to allow recursion involving the type \isa{set}, which is isomorphic to
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\isa{t \isasymFun\ bool}.
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Fortunately, a limited form of recursion
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involving function spaces is permitted: the recursive type may occur on the
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right of a function arrow, but never on the left. Hence the above can of worms
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is ruled out but the following example of a potentially infinitely branching tree is
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accepted:
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\smallskip
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\input{Datatype/document/Fundata.tex}
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\bigskip
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If you need nested recursion on the left of a function arrow, there are
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alternatives to pure HOL: LCF~\cite{paulson87} is a logic where types like
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\begin{isabelle}
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\isacommand{datatype} lam = C "lam \isasymrightarrow\ lam"
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\end{isabelle}
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do indeed make sense (but note the intentionally different arrow
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\isa{\isasymrightarrow}). There is even a version of LCF on top of HOL,
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called HOLCF~\cite{MuellerNvOS99}.
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\index{*primrec|)}
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\index{*datatype|)}
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\subsection{Case study: Tries}
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Tries are a classic search tree data structure~\cite{Knuth3-75} for fast
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indexing with strings. Figure~\ref{fig:trie} gives a graphical example of a
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trie containing the words ``all'', ``an'', ``ape'', ``can'', ``car'' and
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``cat''.  When searching a string in a trie, the letters of the string are
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examined sequentially. Each letter determines which subtrie to search next.
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In this case study we model tries as a datatype, define a lookup and an
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update function, and prove that they behave as expected.
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\begin{figure}[htbp]
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\begin{center}
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\unitlength1mm
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\begin{picture}(60,30)
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\put( 5, 0){\makebox(0,0)[b]{l}}
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\put(25, 0){\makebox(0,0)[b]{e}}
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\put(35, 0){\makebox(0,0)[b]{n}}
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\put(45, 0){\makebox(0,0)[b]{r}}
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\put(55, 0){\makebox(0,0)[b]{t}}
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%
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\put( 5, 9){\line(0,-1){5}}
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\put(25, 9){\line(0,-1){5}}
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\put(44, 9){\line(-3,-2){9}}
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\put(45, 9){\line(0,-1){5}}
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\put(46, 9){\line(3,-2){9}}
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%
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\put( 5,10){\makebox(0,0)[b]{l}}
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\put(15,10){\makebox(0,0)[b]{n}}
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\put(25,10){\makebox(0,0)[b]{p}}
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\put(45,10){\makebox(0,0)[b]{a}}
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%
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\put(14,19){\line(-3,-2){9}}
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\put(15,19){\line(0,-1){5}}
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\put(16,19){\line(3,-2){9}}
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\put(45,19){\line(0,-1){5}}
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%
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\put(15,20){\makebox(0,0)[b]{a}}
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\put(45,20){\makebox(0,0)[b]{c}}
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%
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\put(30,30){\line(-3,-2){13}}
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\put(30,30){\line(3,-2){13}}
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\end{picture}
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\end{center}
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\caption{A sample trie}
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\label{fig:trie}
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\end{figure}
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Proper tries associate some value with each string. Since the
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information is stored only in the final node associated with the string, many
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nodes do not carry any value. This distinction is captured by the
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following predefined datatype (from theory \isa{Option}, which is a parent
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of \isa{Main}):
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\smallskip
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\input{Trie/document/Option2.tex}
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\indexbold{*option}\indexbold{*None}\indexbold{*Some}%
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   526
\input{Trie/document/Trie.tex}
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   527
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   528
\begin{exercise}
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  Write an improved version of \isa{update} that does not suffer from the
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  space leak in the version above. Prove the main theorem for your improved
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   531
  \isa{update}.
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   532
\end{exercise}
nipkow@8743
   533
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   534
\begin{exercise}
nipkow@8743
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  Write a function to \emph{delete} entries from a trie. An easy solution is
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   536
  a clever modification of \isa{update} which allows both insertion and
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  deletion with a single function.  Prove (a modified version of) the main
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  theorem above. Optimize you function such that it shrinks tries after
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  deletion, if possible.
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   540
\end{exercise}
nipkow@8743
   541
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   542
\section{Total recursive functions}
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\label{sec:recdef}
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   544
\index{*recdef|(}
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   545
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   546
Although many total functions have a natural primitive recursive definition,
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this is not always the case. Arbitrary total recursive functions can be
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defined by means of \isacommand{recdef}: you can use full pattern-matching,
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recursion need not involve datatypes, and termination is proved by showing
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that the arguments of all recursive calls are smaller in a suitable (user
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supplied) sense.
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   552
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\subsection{Defining recursive functions}
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\input{Recdef/document/examples.tex}
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   556
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\subsection{Proving termination}
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\input{Recdef/document/termination.tex}
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\subsection{Simplification with recdef}
paulson@10181
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\label{sec:recdef-simplification}
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   563
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\input{Recdef/document/simplification.tex}
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   565
nipkow@8743
   566
\subsection{Induction}
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\index{induction!recursion|(}
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   568
\index{recursion induction|(}
nipkow@8743
   569
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   570
\input{Recdef/document/Induction.tex}
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\label{sec:recdef-induction}
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   572
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\index{induction!recursion|)}
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\index{recursion induction|)}
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\index{*recdef|)}