\chapter{Functional Programming in HOL}

 Although on the surface this chapter is mainly concerned with how to write

 functional programs in HOL and how to verify them, most of the

 constructs and proof procedures introduced are general purpose and recur in

 any specification or verification task.

 The dedicated functional programmer should be warned: HOL offers only

 \emph{total functional programming} --- all functions in HOL must be total;

 lazy data structures are not directly available. On the positive side,

 functions in HOL need not be computable: HOL is a specification language that

 goes well beyond what can be expressed as a program. However, for the time

 being we concentrate on the computable.

 \section{An introductory theory}

 \label{sec:intro-theory}

 Functional programming needs datatypes and functions. Both of them can be

 defined in a theory with a syntax reminiscent of languages like ML or

 Haskell. As an example consider the theory in figure~\ref{fig:ToyList}.

 We will now examine it line by line.

 \begin{figure}[htbp]

 \begin{ttbox}\makeatother

 \input{ToyList2/ToyList1}\end{ttbox}

 \caption{A theory of lists}

 \label{fig:ToyList}

 \end{figure}

 {\makeatother\input{ToyList/document/ToyList.tex}}

 The complete proof script is shown in figure~\ref{fig:ToyList-proofs}. The

 concatenation of figures \ref{fig:ToyList} and \ref{fig:ToyList-proofs}

 constitutes the complete theory \texttt{ToyList} and should reside in file

 \texttt{ToyList.thy}. It is good practice to present all declarations and

 definitions at the beginning of a theory to facilitate browsing.

 \begin{figure}[htbp]

 \begin{ttbox}\makeatother

 \input{ToyList2/ToyList2}\end{ttbox}

 \caption{Proofs about lists}

 \label{fig:ToyList-proofs}

 \end{figure}

 \subsubsection*{Review}

 This is the end of our toy proof. It should have familiarized you with

 \begin{itemize}

 \item the standard theorem proving procedure:

 state a goal (lemma or theorem); proceed with proof until a separate lemma is

 required; prove that lemma; come back to the original goal.

 \item a specific procedure that works well for functional programs:

 induction followed by all-out simplification via \isa{auto}.

 \item a basic repertoire of proof commands.

 \end{itemize}

 \section{Some helpful commands}

 \label{sec:commands-and-hints}

 This section discusses a few basic commands for manipulating the proof state

 and can be skipped by casual readers.

 There are two kinds of commands used during a proof: the actual proof

 commands and auxiliary commands for examining the proof state and controlling

 the display. Simple proof commands are of the form

 \isacommand{apply}\isa{(method)}\indexbold{apply} where \bfindex{method} is a

 synonym for ``theorem proving function''. Typical examples are

 \isa{induct_tac} and \isa{auto}. Further methods are introduced throughout

 the tutorial.  Unless stated otherwise you may assume that a method attacks

 merely the first subgoal. An exception is \isa{auto} which tries to solve all

 subgoals.

 The most useful auxiliary commands are:

 \begin{description}

 \item[Undoing:] \isacommand{undo}\indexbold{*undo} undoes the effect of the

   last command; \isacommand{undo} can be undone by

   \isacommand{redo}\indexbold{*redo}.  Both are only needed at the shell

   level and should not occur in the final theory.

 \item[Printing the current state:] \isacommand{pr}\indexbold{*pr} redisplays

   the current proof state, for example when it has disappeared off the

   screen.

 \item[Limiting the number of subgoals:] \isacommand{pr}~$n$ tells Isabelle to

   print only the first $n$ subgoals from now on and redisplays the current

   proof state. This is helpful when there are many subgoals.

 \item[Modifying the order of subgoals:]

 \isacommand{defer}\indexbold{*defer} moves the first subgoal to the end and

 \isacommand{prefer}\indexbold{*prefer}~$n$ moves subgoal $n$ to the front.

 \item[Printing theorems:]

   \isacommand{thm}\indexbold{*thm}~\textit{name}$@1$~\dots~\textit{name}$@n$

   prints the named theorems.

 \item[Displaying types:] We have already mentioned the flag

   \ttindex{show_types} above. It can also be useful for detecting typos in

   formulae early on. For example, if \texttt{show_types} is set and the goal

   \isa{rev(rev xs) = xs} is started, Isabelle prints the additional output

 \par\noindent

 \begin{isabelle}%

 Variables:\isanewline

 ~~xs~::~'a~list

 \end{isabelle}%

 \par\noindent

 which tells us that Isabelle has correctly inferred that

 \isa{xs} is a variable of list type. On the other hand, had we

 made a typo as in \isa{rev(re xs) = xs}, the response

 \par\noindent

 \begin{isabelle}%

 Variables:\isanewline

 ~~re~::~'a~list~{\isasymRightarrow}~'a~list\isanewline

 ~~xs~::~'a~list%

 \end{isabelle}%

 \par\noindent

 would have alerted us because of the unexpected variable \isa{re}.

 \item[Reading terms and types:] \isacommand{term}\indexbold{*term}

   \textit{string} reads, type-checks and prints the given string as a term in

   the current context; the inferred type is output as well.

   \isacommand{typ}\indexbold{*typ} \textit{string} reads and prints the given

   string as a type in the current context.

 \item[(Re)loading theories:] When you start your interaction you must open a

   named theory with the line \isa{\isacommand{theory}~T~=~\dots~:}. Isabelle

   automatically loads all the required parent theories from their respective

   files (which may take a moment, unless the theories are already loaded and

   the files have not been modified).

   If you suddenly discover that you need to modify a parent theory of your

   current theory you must first abandon your current theory\indexbold{abandon

   theory}\indexbold{theory!abandon} (at the shell

   level type \isacommand{kill}\indexbold{*kill}). After the parent theory has

   been modified you go back to your original theory. When its first line

   \isacommand{theory}\texttt{~T~=}~\dots~\texttt{:} is processed, the

   modified parent is reloaded automatically.

   The only time when you need to load a theory by hand is when you simply

   want to check if it loads successfully without wanting to make use of the

   theory itself. This you can do by typing

   \isa{\isacommand{use\_thy}\indexbold{*use_thy}~"T"}.

 \end{description}

 Further commands are found in the Isabelle/Isar Reference Manual.

 We now examine Isabelle's functional programming constructs systematically,

 starting with inductive datatypes.

 \section{Datatypes}

 \label{sec:datatype}

 Inductive datatypes are part of almost every non-trivial application of HOL.

 First we take another look at a very important example, the datatype of

 lists, before we turn to datatypes in general. The section closes with a

 case study.

 \subsection{Lists}

 \indexbold{*list}

 Lists are one of the essential datatypes in computing. Readers of this

 tutorial and users of HOL need to be familiar with their basic operations.

 Theory \isa{ToyList} is only a small fragment of HOL's predefined theory

 \isa{List}\footnote{\url{http://isabelle.in.tum.de/library/HOL/List.html}}.

 The latter contains many further operations. For example, the functions

 \isaindexbold{hd} (``head'') and \isaindexbold{tl} (``tail'') return the first

 element and the remainder of a list. (However, pattern-matching is usually

 preferable to \isa{hd} and \isa{tl}.)  Theory \isa{List} also contains

 more syntactic sugar: \isa{[}$x@1$\isa{,}\dots\isa{,}$x@n$\isa{]} abbreviates

 $x@1$\isa{\#}\dots\isa{\#}$x@n$\isa{\#[]}.  In the rest of the tutorial we

 always use HOL's predefined lists.

 \subsection{The general format}

 \label{sec:general-datatype}

 The general HOL \isacommand{datatype} definition is of the form

 \[

 \isacommand{datatype}~(\alpha@1, \dots, \alpha@n) \, t ~=~

 C@1~\tau@{11}~\dots~\tau@{1k@1} ~\mid~ \dots ~\mid~

 C@m~\tau@{m1}~\dots~\tau@{mk@m}

 \]

 where $\alpha@i$ are distinct type variables (the parameters), $C@i$ are distinct

 constructor names and $\tau@{ij}$ are types; it is customary to capitalize

 the first letter in constructor names. There are a number of

 restrictions (such as that the type should not be empty) detailed

 elsewhere~\cite{isabelle-HOL}. Isabelle notifies you if you violate them.

 Laws about datatypes, such as \isa{[] \isasymnoteq~x\#xs} and

 \isa{(x\#xs = y\#ys) = (x=y \isasymand~xs=ys)}, are used automatically

 during proofs by simplification.  The same is true for the equations in

 primitive recursive function definitions.

 Every datatype $t$ comes equipped with a \isa{size} function from $t$ into

 the natural numbers (see~\S\ref{sec:nat} below). For lists, \isa{size} is

 just the length, i.e.\ \isa{size [] = 0} and \isa{size(x \# xs) = size xs +

   1}.  In general, \isa{size} returns \isa{0} for all constructors that do

 not have an argument of type $t$, and for all other constructors \isa{1 +}

 the sum of the sizes of all arguments of type $t$. Note that because

 \isa{size} is defined on every datatype, it is overloaded; on lists

 \isa{size} is also called \isa{length}, which is not overloaded.

 \subsection{Primitive recursion}

 Functions on datatypes are usually defined by recursion. In fact, most of the

 time they are defined by what is called \bfindex{primitive recursion}.

 The keyword \isacommand{primrec}\indexbold{*primrec} is followed by a list of

 equations

 \[ f \, x@1 \, \dots \, (C \, y@1 \, \dots \, y@k)\, \dots \, x@n = r \]

 such that $C$ is a constructor of the datatype $t$ and all recursive calls of

 $f$ in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$. Thus

 Isabelle immediately sees that $f$ terminates because one (fixed!) argument

 becomes smaller with every recursive call. There must be exactly one equation

 for each constructor.  Their order is immaterial.

 A more general method for defining total recursive functions is introduced in

 \S\ref{sec:recdef}.

 \begin{exercise}\label{ex:Tree}

 \input{Misc/document/Tree.tex}%

 \end{exercise}

 \input{Misc/document/case_exprs.tex}

 \begin{warn}

   Induction is only allowed on free (or \isasymAnd-bound) variables that

   should not occur among the assumptions of the subgoal; see

   \S\ref{sec:ind-var-in-prems} for details. Case distinction

   (\isa{case_tac}) works for arbitrary terms, which need to be

   quoted if they are non-atomic.

 \end{warn}

 \subsection{Case study: boolean expressions}

 \label{sec:boolex}

 The aim of this case study is twofold: it shows how to model boolean

 expressions and some algorithms for manipulating them, and it demonstrates

 the constructs introduced above.

 \input{Ifexpr/document/Ifexpr.tex}

 \medskip

 How does one come up with the required lemmas? Try to prove the main theorems

 without them and study carefully what \isa{auto} leaves unproved. This has

 to provide the clue.  The necessity of universal quantification

 (\isa{\isasymforall{}t e}) in the two lemmas is explained in

 \S\ref{sec:InductionHeuristics}

 \begin{exercise}

   We strengthen the definition of a \isa{normal} If-expression as follows:

   the first argument of all \isa{IF}s must be a variable. Adapt the above

   development to this changed requirement. (Hint: you may need to formulate

   some of the goals as implications (\isasymimp) rather than equalities

   (\isa{=}).)

 \end{exercise}

 \section{Some basic types}

 \subsection{Natural numbers}

 \label{sec:nat}

 \index{arithmetic|(}

 \input{Misc/document/fakenat.tex}

 \input{Misc/document/natsum.tex}

 The usual arithmetic operations \ttindexboldpos{+}{$HOL2arithfun},

 \ttindexboldpos{-}{$HOL2arithfun}, \ttindexboldpos{*}{$HOL2arithfun},

 \isaindexbold{div}, \isaindexbold{mod}, \isaindexbold{min} and

 \isaindexbold{max} are predefined, as are the relations

 \indexboldpos{\isasymle}{$HOL2arithrel} and

 \ttindexboldpos{<}{$HOL2arithrel}. There is even a least number operation

 \isaindexbold{LEAST}. For example, \isa{(LEAST n.$\,$1 < n) = 2}, although

 Isabelle does not prove this completely automatically. Note that \isa{1} and

 \isa{2} are available as abbreviations for the corresponding

 \isa{Suc}-expressions. If you need the full set of numerals,

 see~\S\ref{nat-numerals}.

 \begin{warn}

   The constant \ttindexbold{0} and the operations

   \ttindexboldpos{+}{$HOL2arithfun}, \ttindexboldpos{-}{$HOL2arithfun},

   \ttindexboldpos{*}{$HOL2arithfun}, \isaindexbold{min}, \isaindexbold{max},

   \indexboldpos{\isasymle}{$HOL2arithrel} and

   \ttindexboldpos{<}{$HOL2arithrel} are overloaded, i.e.\ they are available

   not just for natural numbers but at other types as well (see

   \S\ref{sec:TypeClasses}). For example, given the goal \isa{x+0 = x}, there

   is nothing to indicate that you are talking about natural numbers.  Hence

   Isabelle can only infer that \isa{x} is of some arbitrary type where

   \isa{0} and \isa{+} are declared. As a consequence, you will be unable to

   prove the goal (although it may take you some time to realize what has

   happened if \texttt{show_types} is not set).  In this particular example,

   you need to include an explicit type constraint, for example \isa{x+0 =

     (x::nat)}.  If there is enough contextual information this may not be

   necessary: \isa{Suc x = x} automatically implies \isa{x::nat} because

   \isa{Suc} is not overloaded.

 \end{warn}

 Simple arithmetic goals are proved automatically by both \isa{auto}

 and the simplification methods introduced in \S\ref{sec:Simplification}.  For

 example,

 \input{Misc/document/arith1.tex}%

 is proved automatically. The main restriction is that only addition is taken

 into account; other arithmetic operations and quantified formulae are ignored.

 For more complex goals, there is the special method

 \isaindexbold{arith} (which attacks the first subgoal). It proves

 arithmetic goals involving the usual logical connectives (\isasymnot,

 \isasymand, \isasymor, \isasymimp), the relations \isasymle\ and \isa{<}, and

 the operations \isa{+}, \isa{-}, \isa{min} and \isa{max}. For example,

 \input{Misc/document/arith2.tex}%

 succeeds because \isa{k*k} can be treated as atomic.

 In contrast,

 \input{Misc/document/arith3.tex}%

 is not even proved by \isa{arith} because the proof relies essentially

 on properties of multiplication.

 \begin{warn}

   The running time of \isa{arith} is exponential in the number of occurrences

   of \ttindexboldpos{-}{$HOL2arithfun}, \isaindexbold{min} and

   \isaindexbold{max} because they are first eliminated by case distinctions.

   \isa{arith} is incomplete even for the restricted class of formulae

   described above (known as ``linear arithmetic''). If divisibility plays a

   role, it may fail to prove a valid formula, for example $m+m \neq n+n+1$.

   Fortunately, such examples are rare in practice.

 \end{warn}

 \index{arithmetic|)}

 \subsection{Products}

 \input{Misc/document/pairs.tex}

 %FIXME move stuff below into section on proofs about products?

 \begin{warn}

   Abstraction over pairs and tuples is merely a convenient shorthand for a

   more complex internal representation.  Thus the internal and external form

   of a term may differ, which can affect proofs. If you want to avoid this

   complication, use \isa{fst} and \isa{snd}, i.e.\ write

   \isa{\isasymlambda{}p.~fst p + snd p} instead of

   \isa{\isasymlambda(x,y).~x + y}.  See~\S\ref{proofs-about-products} for

   theorem proving with tuple patterns.

 \end{warn}

 Note that products, like natural numbers, are datatypes, which means

 in particular that \isa{induct_tac} and \isa{case_tac} are applicable to

 products (see \S\ref{proofs-about-products}).

 Instead of tuples with many components (where ``many'' is not much above 2),

 it is far preferable to use record types (see \S\ref{sec:records}).

 \section{Definitions}

 \label{sec:Definitions}

 A definition is simply an abbreviation, i.e.\ a new name for an existing

 construction. In particular, definitions cannot be recursive. Isabelle offers

 definitions on the level of types and terms. Those on the type level are

 called type synonyms, those on the term level are called (constant)

 definitions.

 \subsection{Type synonyms}

 \indexbold{type synonym}

 Type synonyms are similar to those found in ML. Their syntax is fairly self

 explanatory:

 \input{Misc/document/types.tex}%

 Note that pattern-matching is not allowed, i.e.\ each definition must be of

 the form $f\,x@1\,\dots\,x@n~\isasymequiv~t$.

 Section~\S\ref{sec:Simplification} explains how definitions are used

 in proofs.

 \begin{warn}%

 A common mistake when writing definitions is to introduce extra free

 variables on the right-hand side as in the following fictitious definition:

 \input{Misc/document/prime_def.tex}%

 \end{warn}

 \chapter{More Functional Programming}

 The purpose of this chapter is to deepen the reader's understanding of the

 concepts encountered so far and to introduce advanced forms of datatypes and

 recursive functions. The first two sections give a structured presentation of

 theorem proving by simplification (\S\ref{sec:Simplification}) and discuss

 important heuristics for induction (\S\ref{sec:InductionHeuristics}). They can

 be skipped by readers less interested in proofs. They are followed by a case

 study, a compiler for expressions (\S\ref{sec:ExprCompiler}). Advanced

 datatypes, including those involving function spaces, are covered in

 \S\ref{sec:advanced-datatypes}, which closes with another case study, search

 trees (``tries'').  Finally we introduce \isacommand{recdef}, a very general

 form of recursive function definition which goes well beyond what

 \isacommand{primrec} allows (\S\ref{sec:recdef}).

 \section{Simplification}

 \label{sec:Simplification}

 \index{simplification|(}

 So far we have proved our theorems by \isa{auto}, which ``simplifies''

 \emph{all} subgoals. In fact, \isa{auto} can do much more than that, except

 that it did not need to so far. However, when you go beyond toy examples, you

 need to understand the ingredients of \isa{auto}.  This section covers the

 method that \isa{auto} always applies first, namely simplification.

 Simplification is one of the central theorem proving tools in Isabelle and

 many other systems. The tool itself is called the \bfindex{simplifier}. The

 purpose of this section is twofold: to introduce the many features of the

 simplifier (\S\ref{sec:SimpFeatures}) and to explain a little bit how the

 simplifier works (\S\ref{sec:SimpHow}).  Anybody intending to use HOL should

 read \S\ref{sec:SimpFeatures}, and the serious student should read

 \S\ref{sec:SimpHow} as well in order to understand what happened in case

 things do not simplify as expected.

 \subsection{Using the simplifier}

 \label{sec:SimpFeatures}

 \subsubsection{What is simplification}

 In its most basic form, simplification means repeated application of

 equations from left to right. For example, taking the rules for \isa{\at}

 and applying them to the term \isa{[0,1] \at\ []} results in a sequence of

 simplification steps:

 \begin{ttbox}\makeatother

 (0#1#[]) @ []  \(\leadsto\)  0#((1#[]) @ [])  \(\leadsto\)  0#(1#([] @ []))  \(\leadsto\)  0#1#[]

 \end{ttbox}

 This is also known as \bfindex{term rewriting} and the equations are referred

 to as \bfindex{rewrite rules}. ``Rewriting'' is more honest than ``simplification''

 because the terms do not necessarily become simpler in the process.

 \subsubsection{Simplification rules}

 \indexbold{simplification rules}

 To facilitate simplification, theorems can be declared to be simplification

 rules (with the help of the attribute \isa{[simp]}\index{*simp

   (attribute)}), in which case proofs by simplification make use of these

 rules automatically. In addition the constructs \isacommand{datatype} and

 \isacommand{primrec} (and a few others) invisibly declare useful

 simplification rules. Explicit definitions are \emph{not} declared

 simplification rules automatically!

 Not merely equations but pretty much any theorem can become a simplification

 rule. The simplifier will try to make sense of it.  For example, a theorem

 \isasymnot$P$ is automatically turned into \isa{$P$ = False}. The details

 are explained in \S\ref{sec:SimpHow}.

 The simplification attribute of theorems can be turned on and off as follows:

 \begin{ttbox}

 lemmas [simp] = \(list of theorem names\);

 lemmas [simp del] = \(list of theorem names\);

 \end{ttbox}

 As a rule of thumb, equations that really simplify (like \isa{rev(rev xs) =

   xs} and \mbox{\isa{xs \at\ [] = xs}}) should be made simplification

 rules.  Those of a more specific nature (e.g.\ distributivity laws, which

 alter the structure of terms considerably) should only be used selectively,

 i.e.\ they should not be default simplification rules.  Conversely, it may

 also happen that a simplification rule needs to be disabled in certain

 proofs.  Frequent changes in the simplification status of a theorem may

 indicate a badly designed theory.

 \begin{warn}

   Simplification may not terminate, for example if both $f(x) = g(x)$ and

   $g(x) = f(x)$ are simplification rules. It is the user's responsibility not

   to include simplification rules that can lead to nontermination, either on

   their own or in combination with other simplification rules.

 \end{warn}

 \subsubsection{The simplification method}

 \index{*simp (method)|bold}

 The general format of the simplification method is

 \begin{ttbox}

 simp \(list of modifiers\)

 \end{ttbox}

 where the list of \emph{modifiers} helps to fine tune the behaviour and may

 be empty. Most if not all of the proofs seen so far could have been performed

 with \isa{simp} instead of \isa{auto}, except that \isa{simp} attacks

 only the first subgoal and may thus need to be repeated---use \isaindex{simp_all}

 to simplify all subgoals.

 Note that \isa{simp} fails if nothing changes.

 \subsubsection{Adding and deleting simplification rules}

 If a certain theorem is merely needed in a few proofs by simplification,

 we do not need to make it a global simplification rule. Instead we can modify

 the set of simplification rules used in a simplification step by adding rules

 to it and/or deleting rules from it. The two modifiers for this are

 \begin{ttbox}

 add: \(list of theorem names\)

 del: \(list of theorem names\)

 \end{ttbox}

 In case you want to use only a specific list of theorems and ignore all

 others:

 \begin{ttbox}

 only: \(list of theorem names\)

 \end{ttbox}

 \subsubsection{Assumptions}

 \index{simplification!with/of assumptions}

 {\makeatother\input{Misc/document/asm_simp.tex}}

 \subsubsection{Rewriting with definitions}

 \index{simplification!with definitions}

 \input{Misc/document/def_rewr.tex}

 \begin{warn}

   If you have defined $f\,x\,y~\isasymequiv~t$ then you can only expand

   occurrences of $f$ with at least two arguments. Thus it is safer to define

   $f$~\isasymequiv~\isasymlambda$x\,y.\;t$.

 \end{warn}

 \subsubsection{Simplifying let-expressions}

 \index{simplification!of let-expressions}

 Proving a goal containing \isaindex{let}-expressions almost invariably

 requires the \isa{let}-con\-structs to be expanded at some point. Since

 \isa{let}-\isa{in} is just syntactic sugar for a predefined constant (called

 \isa{Let}), expanding \isa{let}-constructs means rewriting with

 \isa{Let_def}:

 {\makeatother\input{Misc/document/let_rewr.tex}}

 \subsubsection{Conditional equations}

 \input{Misc/document/cond_rewr.tex}

 \subsubsection{Automatic case splits}

 \indexbold{case splits}

 \index{*split|(}

 {\makeatother\input{Misc/document/case_splits.tex}}

 \index{*split|)}

 \begin{warn}

   The simplifier merely simplifies the condition of an \isa{if} but not the

   \isa{then} or \isa{else} parts. The latter are simplified only after the

   condition reduces to \isa{True} or \isa{False}, or after splitting. The

   same is true for \isaindex{case}-expressions: only the selector is

   simplified at first, until either the expression reduces to one of the

   cases or it is split.

 \end{warn}

 \subsubsection{Arithmetic}

 \index{arithmetic}

 The simplifier routinely solves a small class of linear arithmetic formulae

 (over type \isa{nat} and other numeric types): it only takes into account

 assumptions and conclusions that are (possibly negated) (in)equalities

 (\isa{=}, \isasymle, \isa{<}) and it only knows about addition. Thus

 \input{Misc/document/arith1.tex}%

 is proved by simplification, whereas the only slightly more complex

 \input{Misc/document/arith4.tex}%

 is not proved by simplification and requires \isa{arith}.

 \subsubsection{Tracing}

 \indexbold{tracing the simplifier}

 {\makeatother\input{Misc/document/trace_simp.tex}}

 \index{simplification|)}

 \section{Induction heuristics}

 \label{sec:InductionHeuristics}

 The purpose of this section is to illustrate some simple heuristics for

 inductive proofs. The first one we have already mentioned in our initial

 example:

 \begin{quote}

 {\em 1. Theorems about recursive functions are proved by induction.}

 \end{quote}

 In case the function has more than one argument

 \begin{quote}

 {\em 2. Do induction on argument number $i$ if the function is defined by

 recursion in argument number $i$.}

 \end{quote}

 When we look at the proof of {\makeatother\isa{(xs @ ys) @ zs = xs @ (ys @

   zs)}} in \S\ref{sec:intro-proof} we find (a) \isa{\at} is recursive in

 the first argument, (b) \isa{xs} occurs only as the first argument of

 \isa{\at}, and (c) both \isa{ys} and \isa{zs} occur at least once as

 the second argument of \isa{\at}. Hence it is natural to perform induction

 on \isa{xs}.

 The key heuristic, and the main point of this section, is to

 generalize the goal before induction. The reason is simple: if the goal is

 too specific, the induction hypothesis is too weak to allow the induction

 step to go through. Let us now illustrate the idea with an example.

 {\makeatother\input{Misc/document/Itrev.tex}}

 A final point worth mentioning is the orientation of the equation we just

 proved: the more complex notion (\isa{itrev}) is on the left-hand

 side, the simpler \isa{rev} on the right-hand side. This constitutes

 another, albeit weak heuristic that is not restricted to induction:

 \begin{quote}

   {\em 5. The right-hand side of an equation should (in some sense) be

     simpler than the left-hand side.}

 \end{quote}

 The heuristic is tricky to apply because it is not obvious that

 \isa{rev xs \at\ ys} is simpler than \isa{itrev xs ys}. But see what

 happens if you try to prove \isa{rev xs \at\ ys = itrev xs ys}!

 \begin{exercise}

 \input{Misc/document/Tree2.tex}%

 \end{exercise}

 \section{Case study: compiling expressions}

 \label{sec:ExprCompiler}

 {\makeatother\input{CodeGen/document/CodeGen.tex}}

 \section{Advanced datatypes}

 \label{sec:advanced-datatypes}

 \index{*datatype|(}

 \index{*primrec|(}

 %|)

 This section presents advanced forms of \isacommand{datatype}s.

 \subsection{Mutual recursion}

 \label{sec:datatype-mut-rec}

 \input{Datatype/document/ABexpr.tex}

 \subsection{Nested recursion}

 \label{sec:nested-datatype}

 {\makeatother\input{Datatype/document/Nested.tex}}

 \subsection{The limits of nested recursion}

 How far can we push nested recursion? By the unfolding argument above, we can

 reduce nested to mutual recursion provided the nested recursion only involves

 previously defined datatypes. This does not include functions:

 \begin{ttbox}

 datatype t = C (t => bool)

 \end{ttbox}

 is a real can of worms: in HOL it must be ruled out because it requires a type

 \isa{t} such that \isa{t} and its power set \isa{t \isasymFun\ bool} have the

 same cardinality---an impossibility. For the same reason it is not possible

 to allow recursion involving the type \isa{set}, which is isomorphic to

 \isa{t \isasymFun\ bool}.

 Fortunately, a limited form of recursion

 involving function spaces is permitted: the recursive type may occur on the

 right of a function arrow, but never on the left. Hence the above can of worms

 is ruled out but the following example of a potentially infinitely branching tree is

 accepted:

 \smallskip

 \input{Datatype/document/Fundata.tex}

 \bigskip

 If you need nested recursion on the left of a function arrow,

 there are alternatives to pure HOL: LCF~\cite{paulson87} is a logic where types like

 \begin{ttbox}

 datatype lam = C (lam -> lam)

 \end{ttbox}

 do indeed make sense (note the intentionally different arrow \isa{->}).

 There is even a version of LCF on top of HOL, called

 HOLCF~\cite{MuellerNvOS99}.

 \index{*primrec|)}

 \index{*datatype|)}

 \subsection{Case study: Tries}

 Tries are a classic search tree data structure~\cite{Knuth3-75} for fast

 indexing with strings. Figure~\ref{fig:trie} gives a graphical example of a

 trie containing the words ``all'', ``an'', ``ape'', ``can'', ``car'' and

 ``cat''.  When searching a string in a trie, the letters of the string are

 examined sequentially. Each letter determines which subtrie to search next.

 In this case study we model tries as a datatype, define a lookup and an

 update function, and prove that they behave as expected.

 \begin{figure}[htbp]

 \begin{center}

 \unitlength1mm

 \begin{picture}(60,30)

 \put( 5, 0){\makebox(0,0)[b]{l}}

 \put(25, 0){\makebox(0,0)[b]{e}}

 \put(35, 0){\makebox(0,0)[b]{n}}

 \put(45, 0){\makebox(0,0)[b]{r}}

 \put(55, 0){\makebox(0,0)[b]{t}}

 %

 \put( 5, 9){\line(0,-1){5}}

 \put(25, 9){\line(0,-1){5}}

 \put(44, 9){\line(-3,-2){9}}

 \put(45, 9){\line(0,-1){5}}

 \put(46, 9){\line(3,-2){9}}

 %

 \put( 5,10){\makebox(0,0)[b]{l}}

 \put(15,10){\makebox(0,0)[b]{n}}

 \put(25,10){\makebox(0,0)[b]{p}}

 \put(45,10){\makebox(0,0)[b]{a}}

 %

 \put(14,19){\line(-3,-2){9}}

 \put(15,19){\line(0,-1){5}}

 \put(16,19){\line(3,-2){9}}

 \put(45,19){\line(0,-1){5}}

 %

 \put(15,20){\makebox(0,0)[b]{a}}

 \put(45,20){\makebox(0,0)[b]{c}}

 %

 \put(30,30){\line(-3,-2){13}}

 \put(30,30){\line(3,-2){13}}

 \end{picture}

 \end{center}

 \caption{A sample trie}

 \label{fig:trie}

 \end{figure}

 Proper tries associate some value with each string. Since the

 information is stored only in the final node associated with the string, many

 nodes do not carry any value. This distinction is captured by the

 following predefined datatype (from theory \isa{Option}, which is a parent

 of \isa{Main}):

 \smallskip

 \input{Trie/document/Option2.tex}

 \indexbold{*option}\indexbold{*None}\indexbold{*Some}%

 \input{Trie/document/Trie.tex}

 \begin{exercise}

   Write an improved version of \isa{update} that does not suffer from the

   space leak in the version above. Prove the main theorem for your improved

   \isa{update}.

 \end{exercise}

 \begin{exercise}

   Write a function to \emph{delete} entries from a trie. An easy solution is

   a clever modification of \isa{update} which allows both insertion and

   deletion with a single function.  Prove (a modified version of) the main

   theorem above. Optimize you function such that it shrinks tries after

   deletion, if possible.

 \end{exercise}

 \section{Total recursive functions}

 \label{sec:recdef}

 \index{*recdef|(}

 Although many total functions have a natural primitive recursive definition,

 this is not always the case. Arbitrary total recursive functions can be

 defined by means of \isacommand{recdef}: you can use full pattern-matching,

 recursion need not involve datatypes, and termination is proved by showing

 that the arguments of all recursive calls are smaller in a suitable (user

 supplied) sense.

 \subsection{Defining recursive functions}

 \input{Recdef/document/examples.tex}

 \subsection{Proving termination}

 \input{Recdef/document/termination.tex}

 \subsection{Simplification with recdef}

 \input{Recdef/document/simplification.tex}

 \subsection{Induction}

 \index{induction!recursion|(}

 \index{recursion induction|(}

 \input{Recdef/document/Induction.tex}

 \label{sec:recdef-induction}

 \index{induction!recursion|)}

 \index{recursion induction|)}

 \subsection{Advanced forms of recursion}

 \label{sec:advanced-recdef}

 \input{Recdef/document/Nested0.tex}

 \input{Recdef/document/Nested1.tex}

 \input{Recdef/document/Nested2.tex}

 \index{*recdef|)}