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\chapter{Basic Concepts}

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\section{Introduction}

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This book is a tutorial on how to use the theorem prover Isabelle/HOL as a

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specification and verification system. Isabelle is a generic system for

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implementing logical formalisms, and Isabelle/HOL is the specialization

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of Isabelle for HOL, which abbreviates HigherOrder Logic. We introduce

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HOL step by step following the equation

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

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We do not assume that you are familiar with mathematical logic.

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However, we do assume that

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you are used to logical and set theoretic notation, as covered

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in a good discrete mathematics course~\cite{RosenDMA}, and

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that you are familiar with the basic concepts of functional

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programming~\cite{BirdHaskell,HudakHaskell,paulsonml2,ThompsonHaskell}.

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Although this tutorial initially concentrates on functional programming, do

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not be 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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Isabelle~\cite{paulsonisabook} is implemented in ML~\cite{SML}. This has

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influenced some of Isabelle/HOL's concrete syntax but is otherwise irrelevant

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for us: this tutorial is based on

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Isabelle/Isar~\cite{isabelleisarref}, an extension of Isabelle which hides

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the implementation language almost completely. Thus the full name of the

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system should be Isabelle/Isar/HOL, but that is a bit of a mouthful.

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There are other implementations of HOL, in particular the one by Mike Gordon

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\index{Gordon, Mike}%

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\emph{et al.}, which is usually referred to as ``the HOL system''

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\cite{mgordonhol}. For us, HOL refers to the logical system, and sometimes

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its incarnation Isabelle/HOL\@.

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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, in particular the ability to write readable and structured proofs,

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you need to consult the Isabelle/Isar Reference

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Manual~\cite{isabelleisarref} and Wenzel's PhD thesis~\cite{WenzelPhD}

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which discusses many proof patterns. 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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\index{theories(}%

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Working with Isabelle means creating theories. Roughly speaking, a

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\textbf{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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{\rmfamily\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 \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 theories} of~\texttt{T}\@.

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Everything defined in the parent theories (and their parents, recursively) is

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automatically visible. To avoid name clashes, identifiers can be

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\textbf{qualified}\indexbold{identifiers!qualified}

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by theory names as in \texttt{T.f} and~\texttt{B.f}.

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Each theory \texttt{T} must

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reside in a \textbf{theory file}\index{theory files} 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 the \textit{declarations, definitions, and

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proofs} above. A complete grammar of the basic

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constructs is found in the Isabelle/Isar Reference

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Manual~\cite{isabelleisarref}.

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HOL's theory collection is available online at

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\begin{center}\small

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

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\end{center}

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and is recommended browsing. Note that most of the theories

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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 \thydx{Main}, the union of all the basic

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predefined theories like arithmetic, lists, sets, etc.

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

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as a direct or indirect parent of all your theories.

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\end{warn}

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There is also a growing Library~\cite{isabellelibrary}\index{Library}

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of useful theories that are not part of \isa{Main} but can to be included

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among the parents of a theory and will then be included automatically.%

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\index{theories)}

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\section{Types, Terms and Formulae}

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\label{sec:TypesTermsForms}

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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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\index{types(}

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\begin{description}

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\item[base types,]

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in particular \tydx{bool}, the type of truth values,

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

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\item[type constructors,]\index{type constructors}

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in particular \tydx{list}, the type of

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lists, and \tydx{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,]\index{function types}

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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,]\index{type variables}\index{variables!type}

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denoted by \ttindexboldpos{'a}{$Isatype}, \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 that you set the flag\index{flags}

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\isa{show_types}\index{*show_types (flag)}.

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Isabelle will then 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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which we introduce as we go along, can be set and reset in the same manner.%

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\index{flags!setting and resetting}

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\end{warn}%

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\index{types)}

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\index{terms(}

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\textbf{Terms} 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, such as conditional expressions:

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\begin{description}

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

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Here $b$ 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 $u$}]\index{*let expressions}

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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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\index{*case expressions}

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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.\index{lambda@$\lambda$ expressions}

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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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\index{terms)}

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\index{formulae(}%

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\textbf{Formulae} are terms of type \tydx{bool}.

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

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

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

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\indexboldpos{\protect\isasymor}{$HOL0or}, and \indexboldpos{\protect\isasymimp}{$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\index{equality} is available in the form of the infix function

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\isa{=} 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. If $t@1$ and $t@2$ are of type

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\isa{bool} then \isa{=} acts as \rmindex{ifandonlyif}.

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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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Quantifiers\index{quantifiers} are written as

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\isa{\isasymforall{}x.~$P$} and \isa{\isasymexists{}x.~$P$}.

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There is even

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\isa{\isasymuniqex{}x.~$P$}, which

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

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Nested quantifications can be abbreviated:

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

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

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\index{formulae)}

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Despite type inference, it is sometimes necessary to attach explicit

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\bfindex{type constraints} 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{::}{$Isatype} binds weakly and should therefore be enclosed

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in parentheses. For instance,

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

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\isa{(x < y)::nat}. Type constraints may be needed to disambiguate

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expressions

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involving overloaded functions such as~\isa{+},

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\isa{*} and~\isa{<}. Section~\ref{sec:overloading}

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discusses overloading, while Table~\ref{tab:overloading} presents the most

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important overloaded function symbols.

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In general, HOL's concrete \rmindex{syntax} tries to follow the conventions of

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functional programming and mathematics. Here are the main rules that you

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should be familiar with to avoid certain syntactic traps:

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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{(\isasymlambda{}x.~x) = f}. This includes

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\isa{if},\index{*if expressions}

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\isa{let},\index{*let expressions}

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\isa{case},\index{*case expressions}

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\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 taken as a single qualified identifier. Write

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

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\item Identifiers\indexbold{identifiers} may contain the characters \isa{_}

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and~\isa{'}, except at the beginning.

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\end{itemize}

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For the sake of readability, we use the usual mathematical symbols throughout

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the tutorial. Their \textsc{ascii}equivalents are shown in table~\ref{tab:ascii} in

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the appendix.

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\begin{warn}

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A particular

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

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additional parentheses. In those cases where Isabelle echoes your

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input, you can see which parentheses are dropped  they 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 the flag\index{flags}

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\isa{show_brackets}\index{*show_brackets (flag)}:

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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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\section{Variables}

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\label{sec:variables}

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\index{variables(}

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Isabelle distinguishes free and bound variables, 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 \textbf{schematic

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variable}\index{variables!schematic} or \textbf{unknown}\index{unknowns},

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which must a~\isa{?} as its first character.

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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

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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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For historical reasons, Isabelle accepts \isa{?} as an ASCII representation

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of the \(\exists\) symbol. However, the \isa{?} character must then be followed

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by a space, as in \isa{?~x. f(x) = 0}. Otherwise, \isa{?x} is

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interpreted as a schematic variable. The preferred ASCII representation of

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the \(\exists\) symbol is \isa{EX}\@.

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\end{warn}%

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\index{variables)}

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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 interaction 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{proofgeneral,Aspinall:TACAS:2000}.

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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 \textsc{ascii}equivalents

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are shown in table~\ref{tab: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 \textsc{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 that you

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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 by typing \texttt{Isabelle}.
