\chapter{Functional Programming in HOL}

This chapter describes how to write

functional programs in HOL and how to verify them.  However, 

most of the constructs and

proof procedures introduced are general and recur in any specification

or verification task.  We really should speak of functional

\emph{modelling} rather than functional \emph{programming}: 

our primary aim is not

to write programs but to design abstract models of systems.  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.

If you are a purist functional programmer, please note that all functions

in HOL must be total:

they must terminate for all inputs.  Lazy data structures are not

directly available.

\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}

\index{*ToyList example|(}

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

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

concatenation of Figs.\ts\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.%

\index{*ToyList example|)}

\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

\commdx{apply}(\textit{method}), where \textit{method} is typically 

\isa{induct_tac} or \isa{auto}.  All such theorem proving operations

are referred to as \bfindex{methods}, and further ones 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 as follows:

\begin{description}

\item[Undoing:] \commdx{undo} undoes the effect of

the

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

  \commdx{redo}.  Both are only needed at the shell

  level and should not occur in the final theory.

\item[Printing the current state:] \commdx{pr}

redisplays

  the current proof state, for example when it has scrolled past the top of

  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:]

\commdx{defer} moves the first subgoal to the end and

\commdx{prefer}~$n$ moves subgoal $n$ to the front.

\item[Printing theorems:]

  \commdx{thm}~\textit{name}$@1$~\dots~\textit{name}$@n$

  prints the named theorems.

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

  \texttt{show_types} above.\index{*show_types (flag)}

  It can also be useful for detecting misspellings in

  formulae. 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:] \commdx{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.

  \commdx{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 \commdx{theory}~\isa{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{abandoning a theory}\indexbold{theories!abandoning} 

  (at the shell

  level type \commdx{kill}). After the parent theory has

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

  \isa{\isacommand{theory}~T~=~\dots~:} 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{\commdx{use\_thy}~"T"}.

\end{description}

Further commands are found in the Isabelle/Isar Reference

Manual~\cite{isabelle-isar-ref}.

We now examine Isabelle's functional programming constructs systematically,

starting with inductive datatypes.

\section{Datatypes}

\label{sec:datatype}

\index{datatypes|(}%

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

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

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

case study.

\subsection{Lists}

\index{*list (type)}%

Lists are one of the essential datatypes in computing.  We expect that you

are already familiar with their basic operations.

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

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

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

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

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

preferable to \isa{hd} and \isa{tl}.)  

Also available are higher-order functions like \isa{map} and \isa{filter}.

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 by building on theory \isa{Main}.

\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\footnote{Except for advanced datatypes where the recursion involves

``\isasymRightarrow'' as in {\S}\ref{sec:nested-fun-datatype}.} 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,

\cdx{size} returns

\begin{itemize}

\item zero for all constructors

that do not have an argument of type $t$\\

\item one plus the sum of the sizes of all arguments of type~$t$,

for all other constructors

\end{itemize}

Note that because

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

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

Isabelle will always show \isa{size} on lists as \isa{length}.

\subsection{Primitive Recursion}

\index{recursion!primitive}%

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

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

The keyword \commdx{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 at most 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}

\input{Ifexpr/document/Ifexpr.tex}

\index{datatypes|)}

\section{Some Basic Types}

This section introduces the types of natural numbers and ordered pairs.  Also

described is type \isa{option}, which is useful for modelling exceptional

cases. 

\subsection{Natural Numbers}

\label{sec:nat}\index{natural numbers}%

\index{linear arithmetic|(}

\input{Misc/document/fakenat.tex}

\input{Misc/document/natsum.tex}

\index{linear arithmetic|)}

\subsection{Pairs}

\input{Misc/document/pairs.tex}

\subsection{Datatype {\tt\slshape option}}

\label{sec:option}

\input{Misc/document/Option2.tex}

\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 \textbf{type synonyms}; those on the term level are simply called 

definitions.

\subsection{Type Synonyms}

\index{type synonyms}%

Type synonyms are similar to those found in ML\@. They are created by a 

\commdx{types} command:

\input{Misc/document/types.tex}

\input{Misc/document/prime_def.tex}

\input{Misc/document/Translations.tex}

\section{The Definitional Approach}

\label{sec:definitional}

\index{Definitional Approach}%

As we pointed out at the beginning of the chapter, asserting arbitrary

axioms such as $f(n) = f(n) + 1$ can easily lead to contradictions. In order

to avoid this danger, we advocate the definitional rather than

the axiomatic approach: introduce new concepts by definitions. However,  Isabelle/HOL seems to

support many richer definitional constructs, such as

\isacommand{primrec}. The point is that Isabelle reduces such constructs to first principles. For example, each

\isacommand{primrec} function definition is turned into a proper

(nonrecursive!) definition from which the user-supplied recursion equations are

automatically proved.  This process is

hidden from the user, who does not have to understand the details.  Other commands described

later, like \isacommand{recdef} and \isacommand{inductive}, work similarly.  

This strict adherence to the definitional approach reduces the risk of 

soundness errors.

\chapter{More Functional Programming}

The purpose of this chapter is to deepen your 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}).  You can

skip them if you are not planning to perform proofs yourself.

We then present 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}; it closes with another case study, search

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

form of recursive function definition that goes well beyond 

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

\section{Simplification}

\label{sec:Simplification}

\index{simplification|(}

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

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

To go beyond toy examples, you

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

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

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

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

This section introduces the many features of the simplifier

and is required reading if you intend to perform proofs.  Later on,

{\S}\ref{sec:simplification-II} explains some more advanced features and a

little bit of how the simplifier works. The serious student should read that

section as well, in particular to understand why the simplifier did

something unexpected.

\subsection{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}\indexbold{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.

The simplifier proves arithmetic goals as described in

{\S}\ref{sec:nat} above.  Arithmetic expressions are simplified using built-in

procedures that go beyond mere rewrite rules.  New simplification procedures

can be coded and installed, but they are definitely not a matter for this

tutorial. 

\input{Misc/document/simp.tex}

\index{simplification|)}

\input{Misc/document/Itrev.tex}

\begin{exercise}

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

\end{exercise}

\input{CodeGen/document/CodeGen.tex}

\section{Advanced Datatypes}

\label{sec:advanced-datatypes}

\index{datatype@\isacommand {datatype} (command)|(}

\index{primrec@\isacommand {primrec} (command)|(}

%|)

This section presents advanced forms of datatypes: mutual and nested

recursion.  A series of examples will culminate in a treatment of the trie

data structure.

\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}

\label{sec:nested-fun-datatype}

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{isabelle}

\isacommand{datatype} t = C "t \isasymRightarrow\ bool"

\end{isabelle}

This declaration 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 

\index{infinitely branching trees}%

infinitely branching tree is accepted:

\smallskip

\input{Datatype/document/Fundata.tex}

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

alternatives to pure HOL\@.  In the Logic for Computable Functions 

(\rmindex{LCF}), types like

\begin{isabelle}

\isacommand{datatype} lam = C "lam \isasymrightarrow\ lam"

\end{isabelle}

do indeed make sense~\cite{paulson87}.  Note the different arrow,

\isa{\isasymrightarrow} instead of \isa{\isasymRightarrow},

expressing the type of \emph{continuous} functions. 

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

called \rmindex{HOLCF}~\cite{MuellerNvOS99}.

\index{datatype@\isacommand {datatype} (command)|)}

\index{primrec@\isacommand {primrec} (command)|)}

\subsection{Case Study: Tries}

\label{sec:Trie}

\index{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 modeled with the help

of the predefined datatype \isa{option} (see {\S}\ref{sec:option}).

\input{Trie/document/Trie.tex}

\index{tries|)}

\section{Total Recursive Functions}

\label{sec:recdef}

\index{recdef@\isacommand {recdef} (command)|(}\index{functions!total|(}

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. In this section we restrict ourselves to measure functions;

more advanced termination proofs are discussed in {\S}\ref{sec:beyond-measure}.

\subsection{Defining Recursive Functions}

\label{sec:recdef-examples}

\input{Recdef/document/examples.tex}

\subsection{Proving Termination}

\input{Recdef/document/termination.tex}

\subsection{Simplification and Recursive Functions}

\label{sec:recdef-simplification}

\input{Recdef/document/simplification.tex}

\subsection{Induction and Recursive Functions}

\index{induction!recursion|(}

\index{recursion induction|(}

\input{Recdef/document/Induction.tex}

\label{sec:recdef-induction}

\index{induction!recursion|)}

\index{recursion induction|)}

\index{recdef@\isacommand {recdef} (command)|)}\index{functions!total|)}