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     1 \chapter{Functional Programming in HOL}
     3 Although on the surface this chapter is mainly concerned with how to write
     4 functional programs in HOL and how to verify them, most of the
     5 constructs and proof procedures introduced are general purpose and recur in
     6 any specification or verification task.
     8 The dedicated functional programmer should be warned: HOL offers only
     9 \emph{total functional programming} --- all functions in HOL must be total;
    10 lazy data structures are not directly available. On the positive side,
    11 functions in HOL need not be computable: HOL is a specification language that
    12 goes well beyond what can be expressed as a program. However, for the time
    13 being we concentrate on the computable.
    15 \section{An introductory theory}
    16 \label{sec:intro-theory}
    18 Functional programming needs datatypes and functions. Both of them can be
    19 defined in a theory with a syntax reminiscent of languages like ML or
    20 Haskell. As an example consider the theory in figure~\ref{fig:ToyList}.
    21 We will now examine it line by line.
    23 \begin{figure}[htbp]
    24 \begin{ttbox}\makeatother
    25 \input{ToyList2/ToyList1}\end{ttbox}
    26 \caption{A theory of lists}
    27 \label{fig:ToyList}
    28 \end{figure}
    30 {\makeatother\input{ToyList/document/ToyList.tex}}
    32 The complete proof script is shown in figure~\ref{fig:ToyList-proofs}. The
    33 concatenation of figures \ref{fig:ToyList} and \ref{fig:ToyList-proofs}
    34 constitutes the complete theory \texttt{ToyList} and should reside in file
    35 \texttt{ToyList.thy}. It is good practice to present all declarations and
    36 definitions at the beginning of a theory to facilitate browsing.
    38 \begin{figure}[htbp]
    39 \begin{ttbox}\makeatother
    40 \input{ToyList2/ToyList2}\end{ttbox}
    41 \caption{Proofs about lists}
    42 \label{fig:ToyList-proofs}
    43 \end{figure}
    45 \subsubsection*{Review}
    47 This is the end of our toy proof. It should have familiarized you with
    48 \begin{itemize}
    49 \item the standard theorem proving procedure:
    50 state a goal (lemma or theorem); proceed with proof until a separate lemma is
    51 required; prove that lemma; come back to the original goal.
    52 \item a specific procedure that works well for functional programs:
    53 induction followed by all-out simplification via \isa{auto}.
    54 \item a basic repertoire of proof commands.
    55 \end{itemize}
    58 \section{Some helpful commands}
    59 \label{sec:commands-and-hints}
    61 This section discusses a few basic commands for manipulating the proof state
    62 and can be skipped by casual readers.
    64 There are two kinds of commands used during a proof: the actual proof
    65 commands and auxiliary commands for examining the proof state and controlling
    66 the display. Simple proof commands are of the form
    67 \isacommand{apply}\isa{(method)}\indexbold{apply} where \bfindex{method} is a
    68 synonym for ``theorem proving function''. Typical examples are
    69 \isa{induct_tac} and \isa{auto}. Further methods are introduced throughout
    70 the tutorial.  Unless stated otherwise you may assume that a method attacks
    71 merely the first subgoal. An exception is \isa{auto} which tries to solve all
    72 subgoals.
    74 The most useful auxiliary commands are:
    75 \begin{description}
    76 \item[Undoing:] \isacommand{undo}\indexbold{*undo} undoes the effect of the
    77   last command; \isacommand{undo} can be undone by
    78   \isacommand{redo}\indexbold{*redo}.  Both are only needed at the shell
    79   level and should not occur in the final theory.
    80 \item[Printing the current state:] \isacommand{pr}\indexbold{*pr} redisplays
    81   the current proof state, for example when it has disappeared off the
    82   screen.
    83 \item[Limiting the number of subgoals:] \isacommand{pr}~$n$ tells Isabelle to
    84   print only the first $n$ subgoals from now on and redisplays the current
    85   proof state. This is helpful when there are many subgoals.
    86 \item[Modifying the order of subgoals:]
    87 \isacommand{defer}\indexbold{*defer} moves the first subgoal to the end and
    88 \isacommand{prefer}\indexbold{*prefer}~$n$ moves subgoal $n$ to the front.
    89 \item[Printing theorems:]
    90   \isacommand{thm}\indexbold{*thm}~\textit{name}$@1$~\dots~\textit{name}$@n$
    91   prints the named theorems.
    92 \item[Displaying types:] We have already mentioned the flag
    93   \ttindex{show_types} above. It can also be useful for detecting typos in
    94   formulae early on. For example, if \texttt{show_types} is set and the goal
    95   \isa{rev(rev xs) = xs} is started, Isabelle prints the additional output
    96 \par\noindent
    97 \begin{isabelle}%
    98 Variables:\isanewline
    99 ~~xs~::~'a~list
   100 \end{isabelle}%
   101 \par\noindent
   102 which tells us that Isabelle has correctly inferred that
   103 \isa{xs} is a variable of list type. On the other hand, had we
   104 made a typo as in \isa{rev(re xs) = xs}, the response
   105 \par\noindent
   106 \begin{isabelle}%
   107 Variables:\isanewline
   108 ~~re~::~'a~list~{\isasymRightarrow}~'a~list\isanewline
   109 ~~xs~::~'a~list%
   110 \end{isabelle}%
   111 \par\noindent
   112 would have alerted us because of the unexpected variable \isa{re}.
   113 \item[Reading terms and types:] \isacommand{term}\indexbold{*term}
   114   \textit{string} reads, type-checks and prints the given string as a term in
   115   the current context; the inferred type is output as well.
   116   \isacommand{typ}\indexbold{*typ} \textit{string} reads and prints the given
   117   string as a type in the current context.
   118 \item[(Re)loading theories:] When you start your interaction you must open a
   119   named theory with the line \isa{\isacommand{theory}~T~=~\dots~:}. Isabelle
   120   automatically loads all the required parent theories from their respective
   121   files (which may take a moment, unless the theories are already loaded and
   122   the files have not been modified).
   124   If you suddenly discover that you need to modify a parent theory of your
   125   current theory you must first abandon your current theory\indexbold{abandon
   126   theory}\indexbold{theory!abandon} (at the shell
   127   level type \isacommand{kill}\indexbold{*kill}). After the parent theory has
   128   been modified you go back to your original theory. When its first line
   129   \isacommand{theory}\texttt{~T~=}~\dots~\texttt{:} is processed, the
   130   modified parent is reloaded automatically.
   132   The only time when you need to load a theory by hand is when you simply
   133   want to check if it loads successfully without wanting to make use of the
   134   theory itself. This you can do by typing
   135   \isa{\isacommand{use\_thy}\indexbold{*use_thy}~"T"}.
   136 \end{description}
   137 Further commands are found in the Isabelle/Isar Reference Manual.
   139 We now examine Isabelle's functional programming constructs systematically,
   140 starting with inductive datatypes.
   143 \section{Datatypes}
   144 \label{sec:datatype}
   146 Inductive datatypes are part of almost every non-trivial application of HOL.
   147 First we take another look at a very important example, the datatype of
   148 lists, before we turn to datatypes in general. The section closes with a
   149 case study.
   152 \subsection{Lists}
   153 \indexbold{*list}
   155 Lists are one of the essential datatypes in computing. Readers of this
   156 tutorial and users of HOL need to be familiar with their basic operations.
   157 Theory \isa{ToyList} is only a small fragment of HOL's predefined theory
   158 \isa{List}\footnote{\url{}}.
   159 The latter contains many further operations. For example, the functions
   160 \isaindexbold{hd} (``head'') and \isaindexbold{tl} (``tail'') return the first
   161 element and the remainder of a list. (However, pattern-matching is usually
   162 preferable to \isa{hd} and \isa{tl}.)  Theory \isa{List} also contains
   163 more syntactic sugar: \isa{[}$x@1$\isa{,}\dots\isa{,}$x@n$\isa{]} abbreviates
   164 $x@1$\isa{\#}\dots\isa{\#}$x@n$\isa{\#[]}.  In the rest of the tutorial we
   165 always use HOL's predefined lists.
   168 \subsection{The general format}
   169 \label{sec:general-datatype}
   171 The general HOL \isacommand{datatype} definition is of the form
   172 \[
   173 \isacommand{datatype}~(\alpha@1, \dots, \alpha@n) \, t ~=~
   174 C@1~\tau@{11}~\dots~\tau@{1k@1} ~\mid~ \dots ~\mid~
   175 C@m~\tau@{m1}~\dots~\tau@{mk@m}
   176 \]
   177 where $\alpha@i$ are distinct type variables (the parameters), $C@i$ are distinct
   178 constructor names and $\tau@{ij}$ are types; it is customary to capitalize
   179 the first letter in constructor names. There are a number of
   180 restrictions (such as that the type should not be empty) detailed
   181 elsewhere~\cite{isabelle-HOL}. Isabelle notifies you if you violate them.
   183 Laws about datatypes, such as \isa{[] \isasymnoteq~x\#xs} and
   184 \isa{(x\#xs = y\#ys) = (x=y \isasymand~xs=ys)}, are used automatically
   185 during proofs by simplification.  The same is true for the equations in
   186 primitive recursive function definitions.
   188 Every datatype $t$ comes equipped with a \isa{size} function from $t$ into
   189 the natural numbers (see~{\S}\ref{sec:nat} below). For lists, \isa{size} is
   190 just the length, i.e.\ \isa{size [] = 0} and \isa{size(x \# xs) = size xs +
   191   1}.  In general, \isaindexbold{size} returns \isa{0} for all constructors
   192 that do not have an argument of type $t$, and for all other constructors
   193 \isa{1 +} the sum of the sizes of all arguments of type $t$. Note that because
   194 \isa{size} is defined on every datatype, it is overloaded; on lists
   195 \isa{size} is also called \isaindexbold{length}, which is not overloaded.
   196 Isbelle will always show \isa{size} on lists as \isa{length}.
   199 \subsection{Primitive recursion}
   201 Functions on datatypes are usually defined by recursion. In fact, most of the
   202 time they are defined by what is called \bfindex{primitive recursion}.
   203 The keyword \isacommand{primrec}\indexbold{*primrec} is followed by a list of
   204 equations
   205 \[ f \, x@1 \, \dots \, (C \, y@1 \, \dots \, y@k)\, \dots \, x@n = r \]
   206 such that $C$ is a constructor of the datatype $t$ and all recursive calls of
   207 $f$ in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$. Thus
   208 Isabelle immediately sees that $f$ terminates because one (fixed!) argument
   209 becomes smaller with every recursive call. There must be at most one equation
   210 for each constructor.  Their order is immaterial.
   211 A more general method for defining total recursive functions is introduced in
   212 {\S}\ref{sec:recdef}.
   214 \begin{exercise}\label{ex:Tree}
   215 \input{Misc/document/Tree.tex}%
   216 \end{exercise}
   218 \input{Misc/document/case_exprs.tex}
   220 \begin{warn}
   221   Induction is only allowed on free (or \isasymAnd-bound) variables that
   222   should not occur among the assumptions of the subgoal; see
   223   {\S}\ref{sec:ind-var-in-prems} for details. Case distinction
   224   (\isa{case_tac}) works for arbitrary terms, which need to be
   225   quoted if they are non-atomic.
   226 \end{warn}
   229 \input{Ifexpr/document/Ifexpr.tex}
   231 \section{Some basic types}
   234 \subsection{Natural numbers}
   235 \label{sec:nat}
   236 \index{arithmetic|(}
   238 \input{Misc/document/fakenat.tex}
   239 \input{Misc/document/natsum.tex}
   241 \index{arithmetic|)}
   244 \subsection{Pairs}
   245 \input{Misc/document/pairs.tex}
   247 \subsection{Datatype {\tt\slshape option}}
   248 \label{sec:option}
   249 \input{Misc/document/Option2.tex}
   251 \section{Definitions}
   252 \label{sec:Definitions}
   254 A definition is simply an abbreviation, i.e.\ a new name for an existing
   255 construction. In particular, definitions cannot be recursive. Isabelle offers
   256 definitions on the level of types and terms. Those on the type level are
   257 called type synonyms, those on the term level are called (constant)
   258 definitions.
   261 \subsection{Type synonyms}
   262 \indexbold{type synonym}
   264 Type synonyms are similar to those found in ML. Their syntax is fairly self
   265 explanatory:
   267 \input{Misc/document/types.tex}%
   269 Note that pattern-matching is not allowed, i.e.\ each definition must be of
   270 the form $f\,x@1\,\dots\,x@n~\isasymequiv~t$.
   271 Section~{\S}\ref{sec:Simplification} explains how definitions are used
   272 in proofs.
   274 \input{Misc/document/prime_def.tex}
   277 \chapter{More Functional Programming}
   279 The purpose of this chapter is to deepen the reader's understanding of the
   280 concepts encountered so far and to introduce advanced forms of datatypes and
   281 recursive functions. The first two sections give a structured presentation of
   282 theorem proving by simplification ({\S}\ref{sec:Simplification}) and discuss
   283 important heuristics for induction ({\S}\ref{sec:InductionHeuristics}). They can
   284 be skipped by readers less interested in proofs. They are followed by a case
   285 study, a compiler for expressions ({\S}\ref{sec:ExprCompiler}). Advanced
   286 datatypes, including those involving function spaces, are covered in
   287 {\S}\ref{sec:advanced-datatypes}, which closes with another case study, search
   288 trees (``tries'').  Finally we introduce \isacommand{recdef}, a very general
   289 form of recursive function definition which goes well beyond what
   290 \isacommand{primrec} allows ({\S}\ref{sec:recdef}).
   293 \section{Simplification}
   294 \label{sec:Simplification}
   295 \index{simplification|(}
   297 So far we have proved our theorems by \isa{auto}, which ``simplifies''
   298 \emph{all} subgoals. In fact, \isa{auto} can do much more than that, except
   299 that it did not need to so far. However, when you go beyond toy examples, you
   300 need to understand the ingredients of \isa{auto}.  This section covers the
   301 method that \isa{auto} always applies first, namely simplification.
   303 Simplification is one of the central theorem proving tools in Isabelle and
   304 many other systems. The tool itself is called the \bfindex{simplifier}. The
   305 purpose of this section is introduce the many features of the simplifier.
   306 Anybody intending to use HOL should read this section. Later on
   307 ({\S}\ref{sec:simplification-II}) we explain some more advanced features and a
   308 little bit of how the simplifier works. The serious student should read that
   309 section as well, in particular in order to understand what happened if things
   310 do not simplify as expected.
   312 \subsubsection{What is simplification}
   314 In its most basic form, simplification means repeated application of
   315 equations from left to right. For example, taking the rules for \isa{\at}
   316 and applying them to the term \isa{[0,1] \at\ []} results in a sequence of
   317 simplification steps:
   318 \begin{ttbox}\makeatother
   319 (0#1#[]) @ []  \(\leadsto\)  0#((1#[]) @ [])  \(\leadsto\)  0#(1#([] @ []))  \(\leadsto\)  0#1#[]
   320 \end{ttbox}
   321 This is also known as \bfindex{term rewriting}\indexbold{rewriting} and the
   322 equations are referred to as \textbf{rewrite rules}\indexbold{rewrite rule}.
   323 ``Rewriting'' is more honest than ``simplification'' because the terms do not
   324 necessarily become simpler in the process.
   326 \input{Misc/document/simp.tex}
   328 \index{simplification|)}
   330 \input{Misc/document/Itrev.tex}
   332 \begin{exercise}
   333 \input{Misc/document/Tree2.tex}%
   334 \end{exercise}
   336 \input{CodeGen/document/CodeGen.tex}
   339 \section{Advanced datatypes}
   340 \label{sec:advanced-datatypes}
   341 \index{*datatype|(}
   342 \index{*primrec|(}
   343 %|)
   345 This section presents advanced forms of \isacommand{datatype}s.
   347 \subsection{Mutual recursion}
   348 \label{sec:datatype-mut-rec}
   350 \input{Datatype/document/ABexpr.tex}
   352 \subsection{Nested recursion}
   353 \label{sec:nested-datatype}
   355 {\makeatother\input{Datatype/document/Nested.tex}}
   358 \subsection{The limits of nested recursion}
   360 How far can we push nested recursion? By the unfolding argument above, we can
   361 reduce nested to mutual recursion provided the nested recursion only involves
   362 previously defined datatypes. This does not include functions:
   363 \begin{isabelle}
   364 \isacommand{datatype} t = C "t \isasymRightarrow\ bool"
   365 \end{isabelle}
   366 is a real can of worms: in HOL it must be ruled out because it requires a type
   367 \isa{t} such that \isa{t} and its power set \isa{t \isasymFun\ bool} have the
   368 same cardinality---an impossibility. For the same reason it is not possible
   369 to allow recursion involving the type \isa{set}, which is isomorphic to
   370 \isa{t \isasymFun\ bool}.
   372 Fortunately, a limited form of recursion
   373 involving function spaces is permitted: the recursive type may occur on the
   374 right of a function arrow, but never on the left. Hence the above can of worms
   375 is ruled out but the following example of a potentially infinitely branching tree is
   376 accepted:
   377 \smallskip
   379 \input{Datatype/document/Fundata.tex}
   380 \bigskip
   382 If you need nested recursion on the left of a function arrow, there are
   383 alternatives to pure HOL: LCF~\cite{paulson87} is a logic where types like
   384 \begin{isabelle}
   385 \isacommand{datatype} lam = C "lam \isasymrightarrow\ lam"
   386 \end{isabelle}
   387 do indeed make sense (but note the intentionally different arrow
   388 \isa{\isasymrightarrow}). There is even a version of LCF on top of HOL,
   389 called HOLCF~\cite{MuellerNvOS99}.
   391 \index{*primrec|)}
   392 \index{*datatype|)}
   394 \subsection{Case study: Tries}
   395 \label{sec:Trie}
   397 Tries are a classic search tree data structure~\cite{Knuth3-75} for fast
   398 indexing with strings. Figure~\ref{fig:trie} gives a graphical example of a
   399 trie containing the words ``all'', ``an'', ``ape'', ``can'', ``car'' and
   400 ``cat''.  When searching a string in a trie, the letters of the string are
   401 examined sequentially. Each letter determines which subtrie to search next.
   402 In this case study we model tries as a datatype, define a lookup and an
   403 update function, and prove that they behave as expected.
   405 \begin{figure}[htbp]
   406 \begin{center}
   407 \unitlength1mm
   408 \begin{picture}(60,30)
   409 \put( 5, 0){\makebox(0,0)[b]{l}}
   410 \put(25, 0){\makebox(0,0)[b]{e}}
   411 \put(35, 0){\makebox(0,0)[b]{n}}
   412 \put(45, 0){\makebox(0,0)[b]{r}}
   413 \put(55, 0){\makebox(0,0)[b]{t}}
   414 %
   415 \put( 5, 9){\line(0,-1){5}}
   416 \put(25, 9){\line(0,-1){5}}
   417 \put(44, 9){\line(-3,-2){9}}
   418 \put(45, 9){\line(0,-1){5}}
   419 \put(46, 9){\line(3,-2){9}}
   420 %
   421 \put( 5,10){\makebox(0,0)[b]{l}}
   422 \put(15,10){\makebox(0,0)[b]{n}}
   423 \put(25,10){\makebox(0,0)[b]{p}}
   424 \put(45,10){\makebox(0,0)[b]{a}}
   425 %
   426 \put(14,19){\line(-3,-2){9}}
   427 \put(15,19){\line(0,-1){5}}
   428 \put(16,19){\line(3,-2){9}}
   429 \put(45,19){\line(0,-1){5}}
   430 %
   431 \put(15,20){\makebox(0,0)[b]{a}}
   432 \put(45,20){\makebox(0,0)[b]{c}}
   433 %
   434 \put(30,30){\line(-3,-2){13}}
   435 \put(30,30){\line(3,-2){13}}
   436 \end{picture}
   437 \end{center}
   438 \caption{A sample trie}
   439 \label{fig:trie}
   440 \end{figure}
   442 Proper tries associate some value with each string. Since the
   443 information is stored only in the final node associated with the string, many
   444 nodes do not carry any value. This distinction is modeled with the help
   445 of the predefined datatype \isa{option} (see {\S}\ref{sec:option}).
   446 \input{Trie/document/Trie.tex}
   448 \begin{exercise}
   449   Write an improved version of \isa{update} that does not suffer from the
   450   space leak in the version above. Prove the main theorem for your improved
   451   \isa{update}.
   452 \end{exercise}
   454 \begin{exercise}
   455   Write a function to \emph{delete} entries from a trie. An easy solution is
   456   a clever modification of \isa{update} which allows both insertion and
   457   deletion with a single function.  Prove (a modified version of) the main
   458   theorem above. Optimize you function such that it shrinks tries after
   459   deletion, if possible.
   460 \end{exercise}
   462 \section{Total recursive functions}
   463 \label{sec:recdef}
   464 \index{*recdef|(}
   466 Although many total functions have a natural primitive recursive definition,
   467 this is not always the case. Arbitrary total recursive functions can be
   468 defined by means of \isacommand{recdef}: you can use full pattern-matching,
   469 recursion need not involve datatypes, and termination is proved by showing
   470 that the arguments of all recursive calls are smaller in a suitable (user
   471 supplied) sense. In this section we ristrict ourselves to measure functions;
   472 more advanced termination proofs are discussed in {\S}\ref{sec:beyond-measure}.
   474 \subsection{Defining recursive functions}
   475 \label{sec:recdef-examples}
   476 \input{Recdef/document/examples.tex}
   478 \subsection{Proving termination}
   480 \input{Recdef/document/termination.tex}
   482 \subsection{Simplification with recdef}
   483 \label{sec:recdef-simplification}
   485 \input{Recdef/document/simplification.tex}
   487 \subsection{Induction}
   488 \index{induction!recursion|(}
   489 \index{recursion induction|(}
   491 \input{Recdef/document/Induction.tex}
   492 \label{sec:recdef-induction}
   494 \index{induction!recursion|)}
   495 \index{recursion induction|)}
   496 \index{*recdef|)}