1 \chapter{Functional Programming in HOL}
3 This chapter describes how to write
4 functional programs in HOL and how to verify them. However,
5 most of the constructs and
6 proof procedures introduced are general and recur in any specification
7 or verification task. We really should speak of functional
8 \emph{modelling} rather than functional \emph{programming}:
10 to write programs but to design abstract models of systems. HOL is
11 a specification language that goes well beyond what can be expressed as a
12 program. However, for the time being we concentrate on the computable.
14 If you are a purist functional programmer, please note that all functions
16 they must terminate for all inputs. Lazy data structures are not
19 \section{An Introductory Theory}
20 \label{sec:intro-theory}
22 Functional programming needs datatypes and functions. Both of them can be
23 defined in a theory with a syntax reminiscent of languages like ML or
24 Haskell. As an example consider the theory in figure~\ref{fig:ToyList}.
25 We will now examine it line by line.
28 \begin{ttbox}\makeatother
29 \input{ToyList2/ToyList1}\end{ttbox}
30 \caption{A Theory of Lists}
34 \index{*ToyList example|(}
35 {\makeatother\medskip\input{ToyList/document/ToyList.tex}}
37 The complete proof script is shown in Fig.\ts\ref{fig:ToyList-proofs}. The
38 concatenation of Figs.\ts\ref{fig:ToyList} and~\ref{fig:ToyList-proofs}
39 constitutes the complete theory \texttt{ToyList} and should reside in file
41 % It is good practice to present all declarations and
42 %definitions at the beginning of a theory to facilitate browsing.%
43 \index{*ToyList example|)}
46 \begin{ttbox}\makeatother
47 \input{ToyList2/ToyList2}\end{ttbox}
48 \caption{Proofs about Lists}
49 \label{fig:ToyList-proofs}
52 \subsubsection*{Review}
54 This is the end of our toy proof. It should have familiarized you with
56 \item the standard theorem proving procedure:
57 state a goal (lemma or theorem); proceed with proof until a separate lemma is
58 required; prove that lemma; come back to the original goal.
59 \item a specific procedure that works well for functional programs:
60 induction followed by all-out simplification via \isa{auto}.
61 \item a basic repertoire of proof commands.
65 It is tempting to think that all lemmas should have the \isa{simp} attribute
66 just because this was the case in the example above. However, in that example
67 all lemmas were equations, and the right-hand side was simpler than the
68 left-hand side --- an ideal situation for simplification purposes. Unless
69 this is clearly the case, novices should refrain from awarding a lemma the
70 \isa{simp} attribute, which has a global effect. Instead, lemmas can be
71 applied locally where they are needed, which is discussed in the following
75 \section{Some Helpful Commands}
76 \label{sec:commands-and-hints}
78 This section discusses a few basic commands for manipulating the proof state
79 and can be skipped by casual readers.
81 There are two kinds of commands used during a proof: the actual proof
82 commands and auxiliary commands for examining the proof state and controlling
83 the display. Simple proof commands are of the form
84 \commdx{apply}(\textit{method}), where \textit{method} is typically
85 \isa{induct_tac} or \isa{auto}. All such theorem proving operations
86 are referred to as \bfindex{methods}, and further ones are
87 introduced throughout the tutorial. Unless stated otherwise, you may
88 assume that a method attacks merely the first subgoal. An exception is
89 \isa{auto}, which tries to solve all subgoals.
91 The most useful auxiliary commands are as follows:
93 \item[Modifying the order of subgoals:]
94 \commdx{defer} moves the first subgoal to the end and
95 \commdx{prefer}~$n$ moves subgoal $n$ to the front.
96 \item[Printing theorems:]
97 \commdx{thm}~\textit{name}$@1$~\dots~\textit{name}$@n$
98 prints the named theorems.
99 \item[Reading terms and types:] \commdx{term}
100 \textit{string} reads, type-checks and prints the given string as a term in
101 the current context; the inferred type is output as well.
102 \commdx{typ} \textit{string} reads and prints the given
103 string as a type in the current context.
105 Further commands are found in the Isabelle/Isar Reference
106 Manual~\cite{isabelle-isar-ref}.
109 Clicking on the \pgmenu{State} button redisplays the current proof state.
110 This is helpful in case commands like \isacommand{thm} have overwritten it.
113 We now examine Isabelle's functional programming constructs systematically,
114 starting with inductive datatypes.
121 Inductive datatypes are part of almost every non-trivial application of HOL.
122 First we take another look at an important example, the datatype of
123 lists, before we turn to datatypes in general. The section closes with a
129 \index{*list (type)}%
130 Lists are one of the essential datatypes in computing. We expect that you
131 are already familiar with their basic operations.
132 Theory \isa{ToyList} is only a small fragment of HOL's predefined theory
133 \thydx{List}\footnote{\url{http://isabelle.in.tum.de/library/HOL/List.html}}.
134 The latter contains many further operations. For example, the functions
135 \cdx{hd} (``head'') and \cdx{tl} (``tail'') return the first
136 element and the remainder of a list. (However, pattern matching is usually
137 preferable to \isa{hd} and \isa{tl}.)
138 Also available are higher-order functions like \isa{map} and \isa{filter}.
139 Theory \isa{List} also contains
140 more syntactic sugar: \isa{[}$x@1$\isa{,}\dots\isa{,}$x@n$\isa{]} abbreviates
141 $x@1$\isa{\#}\dots\isa{\#}$x@n$\isa{\#[]}. In the rest of the tutorial we
142 always use HOL's predefined lists by building on theory \isa{Main}.
144 Looking ahead to sets and quanifiers in Part II:
145 The best way to express that some element \isa{x} is in a list \isa{xs} is
146 \isa{x $\in$ set xs}, where \isa{set} is a function that turns a list into the
148 By the same device you can also write bounded quantifiers like
149 \isa{$\forall$x $\in$ set xs} or embed lists in other set expressions.
153 \subsection{The General Format}
154 \label{sec:general-datatype}
156 The general HOL \isacommand{datatype} definition is of the form
158 \isacommand{datatype}~(\alpha@1, \dots, \alpha@n) \, t ~=~
159 C@1~\tau@{11}~\dots~\tau@{1k@1} ~\mid~ \dots ~\mid~
160 C@m~\tau@{m1}~\dots~\tau@{mk@m}
162 where $\alpha@i$ are distinct type variables (the parameters), $C@i$ are distinct
163 constructor names and $\tau@{ij}$ are types; it is customary to capitalize
164 the first letter in constructor names. There are a number of
165 restrictions (such as that the type should not be empty) detailed
166 elsewhere~\cite{isabelle-HOL}. Isabelle notifies you if you violate them.
168 Laws about datatypes, such as \isa{[] \isasymnoteq~x\#xs} and
169 \isa{(x\#xs = y\#ys) = (x=y \isasymand~xs=ys)}, are used automatically
170 during proofs by simplification. The same is true for the equations in
171 primitive recursive function definitions.
173 Every\footnote{Except for advanced datatypes where the recursion involves
174 ``\isasymRightarrow'' as in {\S}\ref{sec:nested-fun-datatype}.} datatype $t$
175 comes equipped with a \isa{size} function from $t$ into the natural numbers
176 (see~{\S}\ref{sec:nat} below). For lists, \isa{size} is just the length, i.e.\
177 \isa{size [] = 0} and \isa{size(x \# xs) = size xs + 1}. In general,
180 \item zero for all constructors that do not have an argument of type $t$,
181 \item one plus the sum of the sizes of all arguments of type~$t$,
182 for all other constructors.
185 \isa{size} is defined on every datatype, it is overloaded; on lists
186 \isa{size} is also called \sdx{length}, which is not overloaded.
187 Isabelle will always show \isa{size} on lists as \isa{length}.
190 \subsection{Primitive Recursion}
192 \index{recursion!primitive}%
193 Functions on datatypes are usually defined by recursion. In fact, most of the
194 time they are defined by what is called \textbf{primitive recursion} over some
195 datatype $t$. This means that the recursion equations must be of the form
196 \[ f \, x@1 \, \dots \, (C \, y@1 \, \dots \, y@k)\, \dots \, x@n = r \]
197 such that $C$ is a constructor of $t$ and all recursive calls of
198 $f$ in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$. Thus
199 Isabelle immediately sees that $f$ terminates because one (fixed!) argument
200 becomes smaller with every recursive call. There must be at most one equation
201 for each constructor. Their order is immaterial.
202 A more general method for defining total recursive functions is introduced in
205 \begin{exercise}\label{ex:Tree}
206 \input{Misc/document/Tree.tex}%
209 \input{Misc/document/case_exprs.tex}
211 \input{Ifexpr/document/Ifexpr.tex}
215 \section{Some Basic Types}
217 This section introduces the types of natural numbers and ordered pairs. Also
218 described is type \isa{option}, which is useful for modelling exceptional
221 \subsection{Natural Numbers}
222 \label{sec:nat}\index{natural numbers}%
223 \index{linear arithmetic|(}
225 \input{Misc/document/fakenat.tex}\medskip
226 \input{Misc/document/natsum.tex}
228 \index{linear arithmetic|)}
232 \input{Misc/document/pairs.tex}
234 \subsection{Datatype {\tt\slshape option}}
236 \input{Misc/document/Option2.tex}
238 \section{Definitions}
239 \label{sec:Definitions}
241 A definition is simply an abbreviation, i.e.\ a new name for an existing
242 construction. In particular, definitions cannot be recursive. Isabelle offers
243 definitions on the level of types and terms. Those on the type level are
244 called \textbf{type synonyms}; those on the term level are simply called
248 \subsection{Type Synonyms}
250 \index{type synonyms}%
251 Type synonyms are similar to those found in ML\@. They are created by a
252 \commdx{types} command:
255 \input{Misc/document/types.tex}
257 \input{Misc/document/prime_def.tex}
260 \section{The Definitional Approach}
261 \label{sec:definitional}
263 \index{Definitional Approach}%
264 As we pointed out at the beginning of the chapter, asserting arbitrary
265 axioms such as $f(n) = f(n) + 1$ can easily lead to contradictions. In order
266 to avoid this danger, we advocate the definitional rather than
267 the axiomatic approach: introduce new concepts by definitions. However, Isabelle/HOL seems to
268 support many richer definitional constructs, such as
269 \isacommand{primrec}. The point is that Isabelle reduces such constructs to first principles. For example, each
270 \isacommand{primrec} function definition is turned into a proper
271 (nonrecursive!) definition from which the user-supplied recursion equations are
272 automatically proved. This process is
273 hidden from the user, who does not have to understand the details. Other commands described
274 later, like \isacommand{fun} and \isacommand{inductive}, work similarly.
275 This strict adherence to the definitional approach reduces the risk of
278 \chapter{More Functional Programming}
280 The purpose of this chapter is to deepen your understanding of the
281 concepts encountered so far and to introduce advanced forms of datatypes and
282 recursive functions. The first two sections give a structured presentation of
283 theorem proving by simplification ({\S}\ref{sec:Simplification}) and discuss
284 important heuristics for induction ({\S}\ref{sec:InductionHeuristics}). You can
285 skip them if you are not planning to perform proofs yourself.
286 We then present a case
287 study: a compiler for expressions ({\S}\ref{sec:ExprCompiler}). Advanced
288 datatypes, including those involving function spaces, are covered in
289 {\S}\ref{sec:advanced-datatypes}; it closes with another case study, search
290 trees (``tries''). Finally we introduce \isacommand{fun}, a general
291 form of recursive function definition that goes well beyond
292 \isacommand{primrec} ({\S}\ref{sec:fun}).
295 \section{Simplification}
296 \label{sec:Simplification}
297 \index{simplification|(}
299 So far we have proved our theorems by \isa{auto}, which simplifies
300 all subgoals. In fact, \isa{auto} can do much more than that.
301 To go beyond toy examples, you
302 need to understand the ingredients of \isa{auto}. This section covers the
303 method that \isa{auto} always applies first, simplification.
305 Simplification is one of the central theorem proving tools in Isabelle and
306 many other systems. The tool itself is called the \textbf{simplifier}.
307 This section introduces the many features of the simplifier
308 and is required reading if you intend to perform proofs. Later on,
309 {\S}\ref{sec:simplification-II} explains some more advanced features and a
310 little bit of how the simplifier works. The serious student should read that
311 section as well, in particular to understand why the simplifier did
312 something unexpected.
314 \subsection{What is Simplification?}
316 In its most basic form, simplification means repeated application of
317 equations from left to right. For example, taking the rules for \isa{\at}
318 and applying them to the term \isa{[0,1] \at\ []} results in a sequence of
319 simplification steps:
320 \begin{ttbox}\makeatother
321 (0#1#[]) @ [] \(\leadsto\) 0#((1#[]) @ []) \(\leadsto\) 0#(1#([] @ [])) \(\leadsto\) 0#1#[]
323 This is also known as \bfindex{term rewriting}\indexbold{rewriting} and the
324 equations are referred to as \bfindex{rewrite rules}.
325 ``Rewriting'' is more honest than ``simplification'' because the terms do not
326 necessarily become simpler in the process.
328 The simplifier proves arithmetic goals as described in
329 {\S}\ref{sec:nat} above. Arithmetic expressions are simplified using built-in
330 procedures that go beyond mere rewrite rules. New simplification procedures
331 can be coded and installed, but they are definitely not a matter for this
334 \input{Misc/document/simp.tex}
336 \index{simplification|)}
338 \input{Misc/document/Itrev.tex}
340 \input{Misc/document/Plus.tex}%
343 \input{Misc/document/Tree2.tex}%
346 \input{CodeGen/document/CodeGen.tex}
349 \section{Advanced Datatypes}
350 \label{sec:advanced-datatypes}
351 \index{datatype@\isacommand {datatype} (command)|(}
352 \index{primrec@\isacommand {primrec} (command)|(}
355 This section presents advanced forms of datatypes: mutual and nested
356 recursion. A series of examples will culminate in a treatment of the trie
360 \subsection{Mutual Recursion}
361 \label{sec:datatype-mut-rec}
363 \input{Datatype/document/ABexpr.tex}
365 \subsection{Nested Recursion}
366 \label{sec:nested-datatype}
368 {\makeatother\input{Datatype/document/Nested.tex}}
371 \subsection{The Limits of Nested Recursion}
372 \label{sec:nested-fun-datatype}
374 How far can we push nested recursion? By the unfolding argument above, we can
375 reduce nested to mutual recursion provided the nested recursion only involves
376 previously defined datatypes. This does not include functions:
378 \isacommand{datatype} t = C "t \isasymRightarrow\ bool"
380 This declaration is a real can of worms.
381 In HOL it must be ruled out because it requires a type
382 \isa{t} such that \isa{t} and its power set \isa{t \isasymFun\ bool} have the
383 same cardinality --- an impossibility. For the same reason it is not possible
384 to allow recursion involving the type \isa{t set}, which is isomorphic to
385 \isa{t \isasymFun\ bool}.
387 Fortunately, a limited form of recursion
388 involving function spaces is permitted: the recursive type may occur on the
389 right of a function arrow, but never on the left. Hence the above can of worms
390 is ruled out but the following example of a potentially
391 \index{infinitely branching trees}%
392 infinitely branching tree is accepted:
395 \input{Datatype/document/Fundata.tex}
397 If you need nested recursion on the left of a function arrow, there are
398 alternatives to pure HOL\@. In the Logic for Computable Functions
399 (\rmindex{LCF}), types like
401 \isacommand{datatype} lam = C "lam \isasymrightarrow\ lam"
403 do indeed make sense~\cite{paulson87}. Note the different arrow,
404 \isa{\isasymrightarrow} instead of \isa{\isasymRightarrow},
405 expressing the type of \emph{continuous} functions.
406 There is even a version of LCF on top of HOL,
407 called \rmindex{HOLCF}~\cite{MuellerNvOS99}.
408 \index{datatype@\isacommand {datatype} (command)|)}
409 \index{primrec@\isacommand {primrec} (command)|)}
412 \subsection{Case Study: Tries}
416 Tries are a classic search tree data structure~\cite{Knuth3-75} for fast
417 indexing with strings. Figure~\ref{fig:trie} gives a graphical example of a
418 trie containing the words ``all'', ``an'', ``ape'', ``can'', ``car'' and
419 ``cat''. When searching a string in a trie, the letters of the string are
420 examined sequentially. Each letter determines which subtrie to search next.
421 In this case study we model tries as a datatype, define a lookup and an
422 update function, and prove that they behave as expected.
427 \begin{picture}(60,30)
428 \put( 5, 0){\makebox(0,0)[b]{l}}
429 \put(25, 0){\makebox(0,0)[b]{e}}
430 \put(35, 0){\makebox(0,0)[b]{n}}
431 \put(45, 0){\makebox(0,0)[b]{r}}
432 \put(55, 0){\makebox(0,0)[b]{t}}
434 \put( 5, 9){\line(0,-1){5}}
435 \put(25, 9){\line(0,-1){5}}
436 \put(44, 9){\line(-3,-2){9}}
437 \put(45, 9){\line(0,-1){5}}
438 \put(46, 9){\line(3,-2){9}}
440 \put( 5,10){\makebox(0,0)[b]{l}}
441 \put(15,10){\makebox(0,0)[b]{n}}
442 \put(25,10){\makebox(0,0)[b]{p}}
443 \put(45,10){\makebox(0,0)[b]{a}}
445 \put(14,19){\line(-3,-2){9}}
446 \put(15,19){\line(0,-1){5}}
447 \put(16,19){\line(3,-2){9}}
448 \put(45,19){\line(0,-1){5}}
450 \put(15,20){\makebox(0,0)[b]{a}}
451 \put(45,20){\makebox(0,0)[b]{c}}
453 \put(30,30){\line(-3,-2){13}}
454 \put(30,30){\line(3,-2){13}}
457 \caption{A Sample Trie}
461 Proper tries associate some value with each string. Since the
462 information is stored only in the final node associated with the string, many
463 nodes do not carry any value. This distinction is modeled with the help
464 of the predefined datatype \isa{option} (see {\S}\ref{sec:option}).
465 \input{Trie/document/Trie.tex}
468 \section{Total Recursive Functions: \isacommand{fun}}
470 \index{fun@\isacommand {fun} (command)|(}\index{functions!total|(}
472 Although many total functions have a natural primitive recursive definition,
473 this is not always the case. Arbitrary total recursive functions can be
474 defined by means of \isacommand{fun}: you can use full pattern matching,
475 recursion need not involve datatypes, and termination is proved by showing
476 that the arguments of all recursive calls are smaller in a suitable sense.
477 In this section we restrict ourselves to functions where Isabelle can prove
478 termination automatically. More advanced function definitions, including user
479 supplied termination proofs, nested recursion and partiality, are discussed
480 in a separate tutorial~\cite{isabelle-function}.
482 \input{Fun/document/fun0.tex}
484 \index{fun@\isacommand {fun} (command)|)}\index{functions!total|)}