doc-src/TutorialI/basics.tex
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 nipkow@8743  1 \chapter{Basic Concepts}  nipkow@8743  2 nipkow@8743  3 \section{Introduction}  nipkow@8743  4 nipkow@10971  5 This is a tutorial on how to use the theorem prover Isabelle/HOL as a specification and  nipkow@8743  6 verification system. Isabelle is a generic system for implementing logical  nipkow@8743  7 formalisms, and Isabelle/HOL is the specialization of Isabelle for  nipkow@8743  8 HOL, which abbreviates Higher-Order Logic. We introduce HOL step by step  nipkow@8743  9 following the equation  nipkow@8743  10 $\mbox{HOL} = \mbox{Functional Programming} + \mbox{Logic}.$  nipkow@8743  11 We assume that the reader is familiar with the basic concepts of both fields.  nipkow@8743  12 For excellent introductions to functional programming consult the textbooks  nipkow@8743  13 by Bird and Wadler~\cite{Bird-Wadler} or Paulson~\cite{paulson-ml2}. Although  nipkow@8743  14 this tutorial initially concentrates on functional programming, do not be  nipkow@8743  15 misled: HOL can express most mathematical concepts, and functional  nipkow@8743  16 programming is just one particularly simple and ubiquitous instance.  nipkow@8743  17 nipkow@10971  18 Isabelle~\cite{paulson-isa-book} is implemented in ML~\cite{SML}.  nipkow@10971  19 This has influenced some of HOL's concrete syntax but is otherwise  nipkow@10971  20 irrelevant for us because this tutorial is based on  nipkow@10971  21 Isabelle/Isar~\cite{isabelle-isar-ref}, an extension of Isabelle  nipkow@10971  22 with structured proofs.\footnote{Thus the full name of the system should be  nipkow@10971  23  Isabelle/Isar/HOL, but that is a bit of a mouthful.}  nipkow@8743  24 nipkow@8743  25 A tutorial is by definition incomplete. Currently the tutorial only  nipkow@8743  26 introduces the rudiments of Isar's proof language. To fully exploit the power  nipkow@8743  27 of Isar you need to consult the Isabelle/Isar Reference  nipkow@8743  28 Manual~\cite{isabelle-isar-ref}. If you want to use Isabelle's ML level  nipkow@8743  29 directly (for example for writing your own proof procedures) see the Isabelle  nipkow@8743  30 Reference Manual~\cite{isabelle-ref}; for details relating to HOL see the  nipkow@8743  31 Isabelle/HOL manual~\cite{isabelle-HOL}. All manuals have a comprehensive  nipkow@8743  32 index.  nipkow@8743  33 nipkow@8743  34 \section{Theories}  nipkow@8743  35 \label{sec:Basic:Theories}  nipkow@8743  36 nipkow@8743  37 Working with Isabelle means creating theories. Roughly speaking, a  nipkow@8743  38 \bfindex{theory} is a named collection of types, functions, and theorems,  nipkow@8743  39 much like a module in a programming language or a specification in a  nipkow@8743  40 specification language. In fact, theories in HOL can be either. The general  nipkow@8743  41 format of a theory \texttt{T} is  nipkow@8743  42 \begin{ttbox}  nipkow@8743  43 theory T = B$$@1$$ + $$\cdots$$ + B$$@n$$:  nipkow@8743  44 $$\textit{declarations, definitions, and proofs}$$  nipkow@8743  45 end  nipkow@8743  46 \end{ttbox}  nipkow@8743  47 where \texttt{B}$@1$, \dots, \texttt{B}$@n$ are the names of existing  nipkow@8743  48 theories that \texttt{T} is based on and \texttt{\textit{declarations,  nipkow@8743  49  definitions, and proofs}} represents the newly introduced concepts  nipkow@8771  50 (types, functions etc.) and proofs about them. The \texttt{B}$@i$ are the  nipkow@8743  51 direct \textbf{parent theories}\indexbold{parent theory} of \texttt{T}.  nipkow@8743  52 Everything defined in the parent theories (and their parents \dots) is  nipkow@8743  53 automatically visible. To avoid name clashes, identifiers can be  nipkow@8743  54 \textbf{qualified} by theory names as in \texttt{T.f} and  nipkow@8743  55 \texttt{B.f}.\indexbold{identifier!qualified} Each theory \texttt{T} must  nipkow@8771  56 reside in a \bfindex{theory file} named \texttt{T.thy}.  nipkow@8743  57 nipkow@8743  58 This tutorial is concerned with introducing you to the different linguistic  nipkow@8743  59 constructs that can fill \textit{\texttt{declarations, definitions, and  nipkow@8743  60  proofs}} in the above theory template. A complete grammar of the basic  nipkow@8743  61 constructs is found in the Isabelle/Isar Reference Manual.  nipkow@8743  62 paulson@10885  63 HOL's theory collection is available online at  nipkow@8743  64 \begin{center}\small  nipkow@8743  65  \url{http://isabelle.in.tum.de/library/}  nipkow@8743  66 \end{center}  paulson@10885  67 and is recommended browsing. Note that most of the theories  nipkow@9541  68 are based on classical Isabelle without the Isar extension. This means that  nipkow@9541  69 they look slightly different than the theories in this tutorial, and that all  nipkow@9541  70 proofs are in separate ML files.  nipkow@9541  71 nipkow@8743  72 \begin{warn}  nipkow@9792  73  HOL contains a theory \isaindexbold{Main}, the union of all the basic  paulson@10885  74  predefined theories like arithmetic, lists, sets, etc.  paulson@10885  75  Unless you know what you are doing, always include \isa{Main}  nipkow@10971  76  as a direct or indirect parent of all your theories.  nipkow@8743  77 \end{warn}  nipkow@8743  78 nipkow@8743  79 paulson@10885  80 \section{Types, Terms and Formulae}  nipkow@8743  81 \label{sec:TypesTermsForms}  nipkow@8743  82 \indexbold{type}  nipkow@8743  83 paulson@10795  84 Embedded in a theory are the types, terms and formulae of HOL\@. HOL is a typed  nipkow@8771  85 logic whose type system resembles that of functional programming languages  nipkow@8771  86 like ML or Haskell. Thus there are  nipkow@8743  87 \begin{description}  nipkow@8771  88 \item[base types,] in particular \isaindex{bool}, the type of truth values,  nipkow@8771  89 and \isaindex{nat}, the type of natural numbers.  nipkow@8771  90 \item[type constructors,] in particular \isaindex{list}, the type of  nipkow@8771  91 lists, and \isaindex{set}, the type of sets. Type constructors are written  nipkow@8771  92 postfix, e.g.\ \isa{(nat)list} is the type of lists whose elements are  nipkow@8743  93 natural numbers. Parentheses around single arguments can be dropped (as in  nipkow@8771  94 \isa{nat list}), multiple arguments are separated by commas (as in  nipkow@8771  95 \isa{(bool,nat)ty}).  nipkow@8743  96 \item[function types,] denoted by \isasymFun\indexbold{$IsaFun@\isasymFun}.  nipkow@8771  97  In HOL \isasymFun\ represents \emph{total} functions only. As is customary,  nipkow@8771  98  \isa{$\tau@1$\isasymFun~$\tau@2$\isasymFun~$\tau@3$} means  nipkow@8771  99  \isa{$\tau@1$\isasymFun~($\tau@2$\isasymFun~$\tau@3$)}. Isabelle also  nipkow@8771  100  supports the notation \isa{[$\tau@1,\dots,\tau@n$] \isasymFun~$\tau$}  nipkow@8771  101  which abbreviates \isa{$\tau@1$\isasymFun~$\cdots$\isasymFun~$\tau@n$ nipkow@8743  102  \isasymFun~$\tau$}.  nipkow@8771  103 \item[type variables,]\indexbold{type variable}\indexbold{variable!type}  paulson@10795  104  denoted by \ttindexboldpos{'a}{$Isatype}, \isa{'b} etc., just like in ML\@. They give rise  nipkow@8771  105  to polymorphic types like \isa{'a \isasymFun~'a}, the type of the identity  nipkow@8771  106  function.  nipkow@8743  107 \end{description}  nipkow@8743  108 \begin{warn}  nipkow@8743  109  Types are extremely important because they prevent us from writing  nipkow@8743  110  nonsense. Isabelle insists that all terms and formulae must be well-typed  nipkow@8743  111  and will print an error message if a type mismatch is encountered. To  nipkow@8743  112  reduce the amount of explicit type information that needs to be provided by  nipkow@8743  113  the user, Isabelle infers the type of all variables automatically (this is  nipkow@8743  114  called \bfindex{type inference}) and keeps quiet about it. Occasionally  nipkow@8743  115  this may lead to misunderstandings between you and the system. If anything  nipkow@8743  116  strange happens, we recommend to set the \rmindex{flag}  nipkow@9792  117  \isaindexbold{show_types} that tells Isabelle to display type information  nipkow@8743  118  that is usually suppressed: simply type  nipkow@8743  119 \begin{ttbox}  nipkow@8743  120 ML "set show_types"  nipkow@8743  121 \end{ttbox}  nipkow@8743  122 nipkow@8743  123 \noindent  nipkow@10971  124 This can be reversed by \texttt{ML "reset show_types"}. Various other flags,  nipkow@10971  125 which we introduce as we go along,  nipkow@8771  126 can be set and reset in the same manner.\indexbold{flag!(re)setting}  nipkow@8743  127 \end{warn}  nipkow@8743  128 nipkow@8743  129 nipkow@8743  130 \textbf{Terms}\indexbold{term} are formed as in functional programming by  nipkow@8771  131 applying functions to arguments. If \isa{f} is a function of type  nipkow@8771  132 \isa{$\tau@1$ \isasymFun~$\tau@2$} and \isa{t} is a term of type  nipkow@8771  133 $\tau@1$ then \isa{f~t} is a term of type $\tau@2$. HOL also supports  nipkow@8771  134 infix functions like \isa{+} and some basic constructs from functional  nipkow@8743  135 programming:  nipkow@8743  136 \begin{description}  nipkow@8771  137 \item[\isa{if $b$ then $t@1$ else $t@2$}]\indexbold{*if}  nipkow@8743  138 means what you think it means and requires that  nipkow@8771  139 $b$ is of type \isa{bool} and $t@1$ and $t@2$ are of the same type.  nipkow@8771  140 \item[\isa{let $x$ = $t$ in $u$}]\indexbold{*let}  nipkow@8743  141 is equivalent to $u$ where all occurrences of $x$ have been replaced by  nipkow@8743  142 $t$. For example,  nipkow@8771  143 \isa{let x = 0 in x+x} is equivalent to \isa{0+0}. Multiple bindings are separated  nipkow@8771  144 by semicolons: \isa{let $x@1$ = $t@1$; \dots; $x@n$ = $t@n$ in $u$}.  nipkow@8771  145 \item[\isa{case $e$ of $c@1$ \isasymFun~$e@1$ |~\dots~| $c@n$ \isasymFun~$e@n$}]  nipkow@8743  146 \indexbold{*case}  nipkow@8771  147 evaluates to $e@i$ if $e$ is of the form $c@i$.  nipkow@8743  148 \end{description}  nipkow@8743  149 nipkow@8743  150 Terms may also contain  nipkow@8743  151 \isasymlambda-abstractions\indexbold{$Isalam@\isasymlambda}. For example,  nipkow@8771  152 \isa{\isasymlambda{}x.~x+1} is the function that takes an argument \isa{x} and  nipkow@8771  153 returns \isa{x+1}. Instead of  nipkow@8771  154 \isa{\isasymlambda{}x.\isasymlambda{}y.\isasymlambda{}z.~$t$} we can write  nipkow@8771  155 \isa{\isasymlambda{}x~y~z.~$t$}.  nipkow@8743  156 nipkow@8771  157 \textbf{Formulae}\indexbold{formula} are terms of type \isaindexbold{bool}.  nipkow@8771  158 There are the basic constants \isaindexbold{True} and \isaindexbold{False} and  nipkow@8771  159 the usual logical connectives (in decreasing order of priority):  nipkow@8771  160 \indexboldpos{\isasymnot}{$HOL0not}, \indexboldpos{\isasymand}{$HOL0and},  nipkow@8771  161 \indexboldpos{\isasymor}{$HOL0or}, and \indexboldpos{\isasymimp}{$HOL0imp},  nipkow@8743  162 all of which (except the unary \isasymnot) associate to the right. In  nipkow@8771  163 particular \isa{A \isasymimp~B \isasymimp~C} means \isa{A \isasymimp~(B  nipkow@8771  164  \isasymimp~C)} and is thus logically equivalent to \isa{A \isasymand~B  nipkow@8771  165  \isasymimp~C} (which is \isa{(A \isasymand~B) \isasymimp~C}).  nipkow@8743  166 nipkow@8743  167 Equality is available in the form of the infix function  nipkow@8771  168 \isa{=}\indexbold{$HOL0eq@\texttt{=}} of type \isa{'a \isasymFun~'a  nipkow@8771  169  \isasymFun~bool}. Thus \isa{$t@1$ = $t@2$} is a formula provided $t@1$  nipkow@8743  170 and $t@2$ are terms of the same type. In case $t@1$ and $t@2$ are of type  nipkow@8771  171 \isa{bool}, \isa{=} acts as if-and-only-if. The formula  nipkow@8771  172 \isa{$t@1$~\isasymnoteq~$t@2$} is merely an abbreviation for  nipkow@8771  173 \isa{\isasymnot($t@1$ = $t@2$)}.  nipkow@8743  174 paulson@10795  175 Quantifiers are written as  nipkow@8771  176 \isa{\isasymforall{}x.~$P$}\indexbold{$HOL0All@\isasymforall} and  nipkow@8771  177 \isa{\isasymexists{}x.~$P$}\indexbold{$HOL0Ex@\isasymexists}. There is  nipkow@8771  178 even \isa{\isasymuniqex{}x.~$P$}\index{$HOL0ExU@\isasymuniqex|bold}, which  nipkow@8771  179 means that there exists exactly one \isa{x} that satisfies \isa{$P$}. Nested  nipkow@8771  180 quantifications can be abbreviated: \isa{\isasymforall{}x~y~z.~$P$} means  nipkow@8771  181 \isa{\isasymforall{}x.\isasymforall{}y.\isasymforall{}z.~$P$}.  nipkow@8743  182 nipkow@8743  183 Despite type inference, it is sometimes necessary to attach explicit  nipkow@8771  184 \textbf{type constraints}\indexbold{type constraint} to a term. The syntax is  nipkow@8771  185 \isa{$t$::$\tau$} as in \isa{x < (y::nat)}. Note that  nipkow@10538  186 \ttindexboldpos{::}{$Isatype} binds weakly and should therefore be enclosed  nipkow@8771  187 in parentheses: \isa{x < y::nat} is ill-typed because it is interpreted as  paulson@10795  188 \isa{(x < y)::nat}. The main reason for type constraints is overloading of  nipkow@10538  189 functions like \isa{+}, \isa{*} and \isa{<}. See {\S}\ref{sec:overloading} for  nipkow@10695  190 a full discussion of overloading and Table~\ref{tab:overloading} for the most  nipkow@10695  191 important overloaded function symbols.  nipkow@8743  192 nipkow@8743  193 \begin{warn}  nipkow@8743  194 In general, HOL's concrete syntax tries to follow the conventions of  nipkow@8743  195 functional programming and mathematics. Below we list the main rules that you  nipkow@8743  196 should be familiar with to avoid certain syntactic traps. A particular  nipkow@8743  197 problem for novices can be the priority of operators. If you are unsure, use  paulson@10795  198 additional parentheses. In those cases where Isabelle echoes your  nipkow@10971  199 input, you can see which parentheses are dropped --- they were superfluous. If  nipkow@8743  200 you are unsure how to interpret Isabelle's output because you don't know  paulson@10795  201 where the (dropped) parentheses go, set the \rmindex{flag}  nipkow@9792  202 \isaindexbold{show_brackets}:  nipkow@8743  203 \begin{ttbox}  nipkow@8743  204 ML "set show_brackets"; $$\dots$$; ML "reset show_brackets";  nipkow@8743  205 \end{ttbox}  nipkow@8743  206 \end{warn}  nipkow@8743  207 nipkow@8743  208 \begin{itemize}  nipkow@8743  209 \item  nipkow@8771  210 Remember that \isa{f t u} means \isa{(f t) u} and not \isa{f(t u)}!  nipkow@8743  211 \item  nipkow@8771  212 Isabelle allows infix functions like \isa{+}. The prefix form of function  nipkow@8771  213 application binds more strongly than anything else and hence \isa{f~x + y}  nipkow@8771  214 means \isa{(f~x)~+~y} and not \isa{f(x+y)}.  nipkow@8743  215 \item Remember that in HOL if-and-only-if is expressed using equality. But  nipkow@8743  216  equality has a high priority, as befitting a relation, while if-and-only-if  nipkow@8771  217  typically has the lowest priority. Thus, \isa{\isasymnot~\isasymnot~P =  nipkow@8771  218  P} means \isa{\isasymnot\isasymnot(P = P)} and not  nipkow@8771  219  \isa{(\isasymnot\isasymnot P) = P}. When using \isa{=} to mean  nipkow@8771  220  logical equivalence, enclose both operands in parentheses, as in \isa{(A  nipkow@8743  221  \isasymand~B) = (B \isasymand~A)}.  nipkow@8743  222 \item  nipkow@8743  223 Constructs with an opening but without a closing delimiter bind very weakly  nipkow@8743  224 and should therefore be enclosed in parentheses if they appear in subterms, as  nipkow@10971  225 in \isa{(\isasymlambda{}x.~x) = f}. This includes \isaindex{if},  nipkow@8771  226 \isaindex{let}, \isaindex{case}, \isa{\isasymlambda}, and quantifiers.  nipkow@8743  227 \item  nipkow@8771  228 Never write \isa{\isasymlambda{}x.x} or \isa{\isasymforall{}x.x=x}  nipkow@8771  229 because \isa{x.x} is always read as a single qualified identifier that  nipkow@8771  230 refers to an item \isa{x} in theory \isa{x}. Write  nipkow@8771  231 \isa{\isasymlambda{}x.~x} and \isa{\isasymforall{}x.~x=x} instead.  nipkow@8771  232 \item Identifiers\indexbold{identifier} may contain \isa{_} and \isa{'}.  nipkow@8743  233 \end{itemize}  nipkow@8743  234 nipkow@8771  235 For the sake of readability the usual mathematical symbols are used throughout  nipkow@8771  236 the tutorial. Their ASCII-equivalents are shown in figure~\ref{fig:ascii} in  nipkow@8771  237 the appendix.  nipkow@8771  238 nipkow@8743  239 nipkow@8743  240 \section{Variables}  nipkow@8743  241 \label{sec:variables}  nipkow@8743  242 \indexbold{variable}  nipkow@8743  243 nipkow@8743  244 Isabelle distinguishes free and bound variables just as is customary. Bound  nipkow@8743  245 variables are automatically renamed to avoid clashes with free variables. In  nipkow@8743  246 addition, Isabelle has a third kind of variable, called a \bfindex{schematic  nipkow@8743  247  variable}\indexbold{variable!schematic} or \bfindex{unknown}, which starts  nipkow@8771  248 with a \isa{?}. Logically, an unknown is a free variable. But it may be  nipkow@8743  249 instantiated by another term during the proof process. For example, the  nipkow@8771  250 mathematical theorem $x = x$ is represented in Isabelle as \isa{?x = ?x},  nipkow@8743  251 which means that Isabelle can instantiate it arbitrarily. This is in contrast  nipkow@8743  252 to ordinary variables, which remain fixed. The programming language Prolog  nipkow@8743  253 calls unknowns {\em logical\/} variables.  nipkow@8743  254 nipkow@8743  255 Most of the time you can and should ignore unknowns and work with ordinary  nipkow@8743  256 variables. Just don't be surprised that after you have finished the proof of  nipkow@8743  257 a theorem, Isabelle will turn your free variables into unknowns: it merely  nipkow@8743  258 indicates that Isabelle will automatically instantiate those unknowns  nipkow@8743  259 suitably when the theorem is used in some other proof.  nipkow@9689  260 Note that for readability we often drop the \isa{?}s when displaying a theorem.  nipkow@8743  261 \begin{warn}  nipkow@8771  262  If you use \isa{?}\index{$HOL0Ex@\texttt{?}} as an existential  nipkow@8771  263  quantifier, it needs to be followed by a space. Otherwise \isa{?x} is  nipkow@8743  264  interpreted as a schematic variable.  nipkow@8743  265 \end{warn}  nipkow@8743  266 paulson@10885  267 \section{Interaction and Interfaces}  nipkow@8771  268 nipkow@8771  269 Interaction with Isabelle can either occur at the shell level or through more  nipkow@8771  270 advanced interfaces. To keep the tutorial independent of the interface we  nipkow@8771  271 have phrased the description of the intraction in a neutral language. For  nipkow@8771  272 example, the phrase to abandon a proof'' means to type \isacommand{oops} at the  nipkow@8771  273 shell level, which is explained the first time the phrase is used. Other  nipkow@8771  274 interfaces perform the same act by cursor movements and/or mouse clicks.  nipkow@8771  275 Although shell-based interaction is quite feasible for the kind of proof  nipkow@8771  276 scripts currently presented in this tutorial, the recommended interface for  nipkow@8771  277 Isabelle/Isar is the Emacs-based \bfindex{Proof  nipkow@8771  278  General}~\cite{Aspinall:TACAS:2000,proofgeneral}.  nipkow@8771  279 nipkow@8771  280 Some interfaces (including the shell level) offer special fonts with  nipkow@8771  281 mathematical symbols. For those that do not, remember that ASCII-equivalents  nipkow@8771  282 are shown in figure~\ref{fig:ascii} in the appendix.  nipkow@8771  283 nipkow@9541  284 Finally, a word about semicolons.\indexbold{$Isar@\texttt{;}}  nipkow@9541  285 Commands may but need not be terminated by semicolons.  nipkow@9541  286 At the shell level it is advisable to use semicolons to enforce that a command  nipkow@8771  287 is executed immediately; otherwise Isabelle may wait for the next keyword  nipkow@9541  288 before it knows that the command is complete.  nipkow@8771  289 nipkow@8771  290 paulson@10885  291 \section{Getting Started}  nipkow@8743  292 nipkow@8743  293 Assuming you have installed Isabelle, you start it by typing \texttt{isabelle  nipkow@8743  294  -I HOL} in a shell window.\footnote{Simply executing \texttt{isabelle -I}  nipkow@8743  295  starts the default logic, which usually is already \texttt{HOL}. This is  nipkow@8743  296  controlled by the \texttt{ISABELLE_LOGIC} setting, see \emph{The Isabelle  nipkow@8743  297  System Manual} for more details.} This presents you with Isabelle's most  nipkow@8743  298 basic ASCII interface. In addition you need to open an editor window to  nipkow@8743  299 create theory files. While you are developing a theory, we recommend to  nipkow@8743  300 type each command into the file first and then enter it into Isabelle by  nipkow@8743  301 copy-and-paste, thus ensuring that you have a complete record of your theory.  nipkow@8771  302 As mentioned above, Proof General offers a much superior interface.  paulson@10795  303 If you have installed Proof General, you can start it by typing \texttt{Isabelle}.