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\chapter{Basic Concepts}

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\section{Introduction}

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This is a tutorial on how to use Isabelle/HOL as a specification and

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verification system. Isabelle is a generic system for implementing logical

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formalisms, and Isabelle/HOL is the specialization of Isabelle for

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HOL, which abbreviates HigherOrder Logic. We introduce HOL step by step

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following the equation

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\[ \mbox{HOL} = \mbox{Functional Programming} + \mbox{Logic}. \]

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We assume that the reader is familiar with the basic concepts of both fields.

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For excellent introductions to functional programming consult the textbooks

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by Bird and Wadler~\cite{BirdWadler} or Paulson~\cite{paulsonml2}. Although

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this tutorial initially concentrates on functional programming, do not be

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misled: HOL can express most mathematical concepts, and functional

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programming is just one particularly simple and ubiquitous instance.

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This tutorial introduces HOL via Isabelle/Isar~\cite{isabelleisarref},

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which is an extension of Isabelle~\cite{paulsonisabook} with structured

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proofs.\footnote{Thus the full name of the system should be

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Isabelle/Isar/HOL, but that is a bit of a mouthful.} The most noticeable

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difference to classical Isabelle (which is the basis of another version of

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this tutorial) is the replacement of the ML level by a dedicated language for

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definitions and proofs.

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A tutorial is by definition incomplete. Currently the tutorial only

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introduces the rudiments of Isar's proof language. To fully exploit the power

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of Isar you need to consult the Isabelle/Isar Reference

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Manual~\cite{isabelleisarref}. If you want to use Isabelle's ML level

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directly (for example for writing your own proof procedures) see the Isabelle

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Reference Manual~\cite{isabelleref}; for details relating to HOL see the

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Isabelle/HOL manual~\cite{isabelleHOL}. All manuals have a comprehensive

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index.

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\section{Theories}

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\label{sec:Basic:Theories}

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Working with Isabelle means creating theories. Roughly speaking, a

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\bfindex{theory} is a named collection of types, functions, and theorems,

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much like a module in a programming language or a specification in a

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specification language. In fact, theories in HOL can be either. The general

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format of a theory \texttt{T} is

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\begin{ttbox}

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theory T = B\(@1\) + \(\cdots\) + B\(@n\):

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\(\textit{declarations, definitions, and proofs}\)

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end

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\end{ttbox}

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where \texttt{B}$@1$, \dots, \texttt{B}$@n$ are the names of existing

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theories that \texttt{T} is based on and \texttt{\textit{declarations,

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definitions, and proofs}} represents the newly introduced concepts

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(types, functions etc.) and proofs about them. The \texttt{B}$@i$ are the

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direct \textbf{parent theories}\indexbold{parent theory} of \texttt{T}.

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Everything defined in the parent theories (and their parents \dots) is

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automatically visible. To avoid name clashes, identifiers can be

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\textbf{qualified} by theory names as in \texttt{T.f} and

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\texttt{B.f}.\indexbold{identifier!qualified} Each theory \texttt{T} must

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reside in a \bfindex{theory file} named \texttt{T.thy}.

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This tutorial is concerned with introducing you to the different linguistic

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constructs that can fill \textit{\texttt{declarations, definitions, and

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proofs}} in the above theory template. A complete grammar of the basic

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constructs is found in the Isabelle/Isar Reference Manual.

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HOL's theory library is available online at

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\begin{center}\small

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\url{http://isabelle.in.tum.de/library/}

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\end{center}

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and is recommended browsing. Note that most of the theories in the library

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are based on classical Isabelle without the Isar extension. This means that

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they look slightly different than the theories in this tutorial, and that all

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proofs are in separate ML files.

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\begin{warn}

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HOL contains a theory \isaindexbold{Main}, the union of all the basic

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predefined theories like arithmetic, lists, sets, etc.\ (see the online

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library). Unless you know what you are doing, always include \isa{Main}

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as a direct or indirect parent theory of all your theories.

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\end{warn}

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\section{Types, terms and formulae}

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\label{sec:TypesTermsForms}

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\indexbold{type}

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Embedded in a theory are the types, terms and formulae of HOL. HOL is a typed

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logic whose type system resembles that of functional programming languages

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like ML or Haskell. Thus there are

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\begin{description}

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\item[base types,] in particular \isaindex{bool}, the type of truth values,

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and \isaindex{nat}, the type of natural numbers.

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\item[type constructors,] in particular \isaindex{list}, the type of

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lists, and \isaindex{set}, the type of sets. Type constructors are written

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postfix, e.g.\ \isa{(nat)list} is the type of lists whose elements are

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natural numbers. Parentheses around single arguments can be dropped (as in

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\isa{nat list}), multiple arguments are separated by commas (as in

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\isa{(bool,nat)ty}).

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\item[function types,] denoted by \isasymFun\indexbold{$IsaFun@\isasymFun}.

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In HOL \isasymFun\ represents \emph{total} functions only. As is customary,

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\isa{$\tau@1$ \isasymFun~$\tau@2$ \isasymFun~$\tau@3$} means

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\isa{$\tau@1$ \isasymFun~($\tau@2$ \isasymFun~$\tau@3$)}. Isabelle also

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supports the notation \isa{[$\tau@1,\dots,\tau@n$] \isasymFun~$\tau$}

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which abbreviates \isa{$\tau@1$ \isasymFun~$\cdots$ \isasymFun~$\tau@n$

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\isasymFun~$\tau$}.

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\item[type variables,]\indexbold{type variable}\indexbold{variable!type}

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denoted by \isaindexbold{'a}, \isa{'b} etc., just like in ML. They give rise

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to polymorphic types like \isa{'a \isasymFun~'a}, the type of the identity

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function.

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\end{description}

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\begin{warn}

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Types are extremely important because they prevent us from writing

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nonsense. Isabelle insists that all terms and formulae must be welltyped

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and will print an error message if a type mismatch is encountered. To

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reduce the amount of explicit type information that needs to be provided by

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the user, Isabelle infers the type of all variables automatically (this is

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called \bfindex{type inference}) and keeps quiet about it. Occasionally

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this may lead to misunderstandings between you and the system. If anything

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strange happens, we recommend to set the \rmindex{flag}

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\isaindexbold{show_types} that tells Isabelle to display type information

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that is usually suppressed: simply type

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\begin{ttbox}

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ML "set show_types"

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\end{ttbox}

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\noindent

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This can be reversed by \texttt{ML "reset show_types"}. Various other flags

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can be set and reset in the same manner.\indexbold{flag!(re)setting}

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\end{warn}

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\textbf{Terms}\indexbold{term} are formed as in functional programming by

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applying functions to arguments. If \isa{f} is a function of type

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\isa{$\tau@1$ \isasymFun~$\tau@2$} and \isa{t} is a term of type

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$\tau@1$ then \isa{f~t} is a term of type $\tau@2$. HOL also supports

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infix functions like \isa{+} and some basic constructs from functional

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programming:

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\begin{description}

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\item[\isa{if $b$ then $t@1$ else $t@2$}]\indexbold{*if}

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means what you think it means and requires that

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$b$ is of type \isa{bool} and $t@1$ and $t@2$ are of the same type.

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\item[\isa{let $x$ = $t$ in $u$}]\indexbold{*let}

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is equivalent to $u$ where all occurrences of $x$ have been replaced by

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$t$. For example,

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\isa{let x = 0 in x+x} is equivalent to \isa{0+0}. Multiple bindings are separated

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by semicolons: \isa{let $x@1$ = $t@1$; \dots; $x@n$ = $t@n$ in $u$}.

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\item[\isa{case $e$ of $c@1$ \isasymFun~$e@1$ ~\dots~ $c@n$ \isasymFun~$e@n$}]

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\indexbold{*case}

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evaluates to $e@i$ if $e$ is of the form $c@i$.

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\end{description}

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Terms may also contain

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\isasymlambdaabstractions\indexbold{$Isalam@\isasymlambda}. For example,

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\isa{\isasymlambda{}x.~x+1} is the function that takes an argument \isa{x} and

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returns \isa{x+1}. Instead of

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\isa{\isasymlambda{}x.\isasymlambda{}y.\isasymlambda{}z.~$t$} we can write

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\isa{\isasymlambda{}x~y~z.~$t$}.

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\textbf{Formulae}\indexbold{formula} are terms of type \isaindexbold{bool}.

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There are the basic constants \isaindexbold{True} and \isaindexbold{False} and

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the usual logical connectives (in decreasing order of priority):

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\indexboldpos{\isasymnot}{$HOL0not}, \indexboldpos{\isasymand}{$HOL0and},

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\indexboldpos{\isasymor}{$HOL0or}, and \indexboldpos{\isasymimp}{$HOL0imp},

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all of which (except the unary \isasymnot) associate to the right. In

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particular \isa{A \isasymimp~B \isasymimp~C} means \isa{A \isasymimp~(B

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\isasymimp~C)} and is thus logically equivalent to \isa{A \isasymand~B

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\isasymimp~C} (which is \isa{(A \isasymand~B) \isasymimp~C}).

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Equality is available in the form of the infix function

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\isa{=}\indexbold{$HOL0eq@\texttt{=}} of type \isa{'a \isasymFun~'a

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\isasymFun~bool}. Thus \isa{$t@1$ = $t@2$} is a formula provided $t@1$

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and $t@2$ are terms of the same type. In case $t@1$ and $t@2$ are of type

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\isa{bool}, \isa{=} acts as ifandonlyif. The formula

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\isa{$t@1$~\isasymnoteq~$t@2$} is merely an abbreviation for

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\isa{\isasymnot($t@1$ = $t@2$)}.

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The syntax for quantifiers is

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\isa{\isasymforall{}x.~$P$}\indexbold{$HOL0All@\isasymforall} and

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\isa{\isasymexists{}x.~$P$}\indexbold{$HOL0Ex@\isasymexists}. There is

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even \isa{\isasymuniqex{}x.~$P$}\index{$HOL0ExU@\isasymuniqexbold}, which

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means that there exists exactly one \isa{x} that satisfies \isa{$P$}. Nested

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quantifications can be abbreviated: \isa{\isasymforall{}x~y~z.~$P$} means

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\isa{\isasymforall{}x.\isasymforall{}y.\isasymforall{}z.~$P$}.

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Despite type inference, it is sometimes necessary to attach explicit

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\textbf{type constraints}\indexbold{type constraint} to a term. The syntax is

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\isa{$t$::$\tau$} as in \isa{x < (y::nat)}. Note that

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\ttindexboldpos{::}{$Isalamtc} binds weakly and should therefore be enclosed

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in parentheses: \isa{x < y::nat} is illtyped because it is interpreted as

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\isa{(x < y)::nat}. The main reason for type constraints are overloaded

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functions like \isa{+}, \isa{*} and \isa{<}. (See \S\ref{sec:TypeClasses} for

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a full discussion of overloading.)

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\begin{warn}

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In general, HOL's concrete syntax tries to follow the conventions of

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functional programming and mathematics. Below we list the main rules that you

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should be familiar with to avoid certain syntactic traps. A particular

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problem for novices can be the priority of operators. If you are unsure, use

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more rather than fewer parentheses. In those cases where Isabelle echoes your

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input, you can see which parentheses are droppedthey were superfluous. If

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you are unsure how to interpret Isabelle's output because you don't know

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where the (dropped) parentheses go, set (and possibly reset) the \rmindex{flag}

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\isaindexbold{show_brackets}:

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\begin{ttbox}

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ML "set show_brackets"; \(\dots\); ML "reset show_brackets";

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\end{ttbox}

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\end{warn}

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\begin{itemize}

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\item

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Remember that \isa{f t u} means \isa{(f t) u} and not \isa{f(t u)}!

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\item

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Isabelle allows infix functions like \isa{+}. The prefix form of function

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application binds more strongly than anything else and hence \isa{f~x + y}

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means \isa{(f~x)~+~y} and not \isa{f(x+y)}.

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\item Remember that in HOL ifandonlyif is expressed using equality. But

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equality has a high priority, as befitting a relation, while ifandonlyif

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typically has the lowest priority. Thus, \isa{\isasymnot~\isasymnot~P =

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P} means \isa{\isasymnot\isasymnot(P = P)} and not

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\isa{(\isasymnot\isasymnot P) = P}. When using \isa{=} to mean

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logical equivalence, enclose both operands in parentheses, as in \isa{(A

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\isasymand~B) = (B \isasymand~A)}.

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\item

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Constructs with an opening but without a closing delimiter bind very weakly

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and should therefore be enclosed in parentheses if they appear in subterms, as

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in \isa{f = (\isasymlambda{}x.~x)}. This includes \isaindex{if},

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\isaindex{let}, \isaindex{case}, \isa{\isasymlambda}, and quantifiers.

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\item

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Never write \isa{\isasymlambda{}x.x} or \isa{\isasymforall{}x.x=x}

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because \isa{x.x} is always read as a single qualified identifier that

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refers to an item \isa{x} in theory \isa{x}. Write

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\isa{\isasymlambda{}x.~x} and \isa{\isasymforall{}x.~x=x} instead.

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\item Identifiers\indexbold{identifier} may contain \isa{_} and \isa{'}.

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\end{itemize}

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For the sake of readability the usual mathematical symbols are used throughout

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the tutorial. Their ASCIIequivalents are shown in figure~\ref{fig:ascii} in

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the appendix.

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\section{Variables}

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\label{sec:variables}

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\indexbold{variable}

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Isabelle distinguishes free and bound variables just as is customary. Bound

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variables are automatically renamed to avoid clashes with free variables. In

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addition, Isabelle has a third kind of variable, called a \bfindex{schematic

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variable}\indexbold{variable!schematic} or \bfindex{unknown}, which starts

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with a \isa{?}. Logically, an unknown is a free variable. But it may be

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instantiated by another term during the proof process. For example, the

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mathematical theorem $x = x$ is represented in Isabelle as \isa{?x = ?x},

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which means that Isabelle can instantiate it arbitrarily. This is in contrast

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to ordinary variables, which remain fixed. The programming language Prolog

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calls unknowns {\em logical\/} variables.

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Most of the time you can and should ignore unknowns and work with ordinary

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variables. Just don't be surprised that after you have finished the proof of

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a theorem, Isabelle will turn your free variables into unknowns: it merely

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indicates that Isabelle will automatically instantiate those unknowns

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suitably when the theorem is used in some other proof.

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Note that for readability we often drop the \isa{?}s when displaying a theorem.

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\begin{warn}

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If you use \isa{?}\index{$HOL0Ex@\texttt{?}} as an existential

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quantifier, it needs to be followed by a space. Otherwise \isa{?x} is

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interpreted as a schematic variable.

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\end{warn}

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\section{Interaction and interfaces}

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Interaction with Isabelle can either occur at the shell level or through more

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advanced interfaces. To keep the tutorial independent of the interface we

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have phrased the description of the intraction in a neutral language. For

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example, the phrase ``to abandon a proof'' means to type \isacommand{oops} at the

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shell level, which is explained the first time the phrase is used. Other

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interfaces perform the same act by cursor movements and/or mouse clicks.

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Although shellbased interaction is quite feasible for the kind of proof

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scripts currently presented in this tutorial, the recommended interface for

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Isabelle/Isar is the Emacsbased \bfindex{Proof

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General}~\cite{Aspinall:TACAS:2000,proofgeneral}.

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Some interfaces (including the shell level) offer special fonts with

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mathematical symbols. For those that do not, remember that ASCIIequivalents

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are shown in figure~\ref{fig:ascii} in the appendix.

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Finally, a word about semicolons.\indexbold{$Isar@\texttt{;}}

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Commands may but need not be terminated by semicolons.

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At the shell level it is advisable to use semicolons to enforce that a command

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is executed immediately; otherwise Isabelle may wait for the next keyword

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before it knows that the command is complete.

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\section{Getting started}

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Assuming you have installed Isabelle, you start it by typing \texttt{isabelle

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I HOL} in a shell window.\footnote{Simply executing \texttt{isabelle I}

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starts the default logic, which usually is already \texttt{HOL}. This is

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controlled by the \texttt{ISABELLE_LOGIC} setting, see \emph{The Isabelle

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System Manual} for more details.} This presents you with Isabelle's most

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basic ASCII interface. In addition you need to open an editor window to

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create theory files. While you are developing a theory, we recommend to

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type each command into the file first and then enter it into Isabelle by

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copyandpaste, thus ensuring that you have a complete record of your theory.

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As mentioned above, Proof General offers a much superior interface.

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If you have installed Proof General, you can start it with \texttt{Isabelle}.
