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-(*<*)
-theory Basics
-imports Main
-begin
-(*>*)
-text{*
-This chapter introduces HOL as a functional programming language and shows
-how to prove properties of functional programs by induction.
-
-\section{Basics}
-
-\subsection{Types, Terms and Formulas}
-\label{sec:TypesTermsForms}
-
-HOL is a typed logic whose type system resembles that of functional
-programming languages. Thus there are
-\begin{description}
-\item[base types,] 
-in particular @{typ bool}, the type of truth values,
-@{typ nat}, the type of natural numbers ($\mathbb{N}$), and \indexed{@{typ int}}{int},
-the type of mathematical integers ($\mathbb{Z}$).
-\item[type constructors,]
- in particular @{text list}, the type of
-lists, and @{text set}, the type of sets. Type constructors are written
-postfix, i.e., after their arguments. For example,
-@{typ "nat list"} is the type of lists whose elements are natural numbers.
-\item[function types,]
-denoted by @{text"\<Rightarrow>"}.
-\item[type variables,]
-  denoted by @{typ 'a}, @{typ 'b} etc., just like in ML\@.
-\end{description}
-Note that @{typ"'a \<Rightarrow> 'b list"} means @{typ[source]"'a \<Rightarrow> ('b list)"},
-not @{typ"('a \<Rightarrow> 'b) list"}: postfix type constructors have precedence
-over @{text"\<Rightarrow>"}.
-
-\conceptidx{Terms}{term} are formed as in functional programming by
-applying functions to arguments. If @{text f} is a function of type
-@{text"\<tau>\<^sub>1 \<Rightarrow> \<tau>\<^sub>2"} and @{text t} is a term of type
-@{text"\<tau>\<^sub>1"} then @{term"f t"} is a term of type @{text"\<tau>\<^sub>2"}. We write @{text "t :: \<tau>"} to mean that term @{text t} has type @{text \<tau>}.
-
-\begin{warn}
-There are many predefined infix symbols like @{text "+"} and @{text"\<le>"}.
-The name of the corresponding binary function is @{term"op +"},
-not just @{text"+"}. That is, @{term"x + y"} is nice surface syntax
-(``syntactic sugar'') for \noquotes{@{term[source]"op + x y"}}.
-\end{warn}
-
-HOL also supports some basic constructs from functional programming:
-\begin{quote}
-@{text "(if b then t\<^sub>1 else t\<^sub>2)"}\\
-@{text "(let x = t in u)"}\\
-@{text "(case t of pat\<^sub>1 \<Rightarrow> t\<^sub>1 | \<dots> | pat\<^sub>n \<Rightarrow> t\<^sub>n)"}
-\end{quote}
-\begin{warn}
-The above three constructs must always be enclosed in parentheses
-if they occur inside other constructs.
-\end{warn}
-Terms may also contain @{text "\<lambda>"}-abstractions. For example,
-@{term "\<lambda>x. x"} is the identity function.
-
-\conceptidx{Formulas}{formula} are terms of type @{text bool}.
-There are the basic constants @{term True} and @{term False} and
-the usual logical connectives (in decreasing order of precedence):
-@{text"\<not>"}, @{text"\<and>"}, @{text"\<or>"}, @{text"\<longrightarrow>"}.
-
-\conceptidx{Equality}{equality} is available in the form of the infix function @{text "="}
-of type @{typ "'a \<Rightarrow> 'a \<Rightarrow> bool"}. It also works for formulas, where
-it means ``if and only if''.
-
-\conceptidx{Quantifiers}{quantifier} are written @{prop"\<forall>x. P"} and @{prop"\<exists>x. P"}.
-
-Isabelle automatically computes the type of each variable in a term. This is
-called \concept{type inference}.  Despite type inference, it is sometimes
-necessary to attach an explicit \concept{type constraint} (or \concept{type
-annotation}) to a variable or term.  The syntax is @{text "t :: \<tau>"} as in
-\mbox{\noquotes{@{prop[source] "m < (n::nat)"}}}. Type constraints may be
-needed to
-disambiguate terms involving overloaded functions such as @{text "+"}, @{text
-"*"} and @{text"\<le>"}.
-
-Finally there are the universal quantifier @{text"\<And>"}\index{$4@\isasymAnd} and the implication
-@{text"\<Longrightarrow>"}\index{$3@\isasymLongrightarrow}. They are part of the Isabelle framework, not the logic
-HOL. Logically, they agree with their HOL counterparts @{text"\<forall>"} and
-@{text"\<longrightarrow>"}, but operationally they behave differently. This will become
-clearer as we go along.
-\begin{warn}
-Right-arrows of all kinds always associate to the right. In particular,
-the formula
-@{text"A\<^sub>1 \<Longrightarrow> A\<^sub>2 \<Longrightarrow> A\<^sub>3"} means @{text "A\<^sub>1 \<Longrightarrow> (A\<^sub>2 \<Longrightarrow> A\<^sub>3)"}.
-The (Isabelle specific) notation \mbox{@{text"\<lbrakk> A\<^sub>1; \<dots>; A\<^sub>n \<rbrakk> \<Longrightarrow> A"}}
-is short for the iterated implication \mbox{@{text"A\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^sub>n \<Longrightarrow> A"}}.
-Sometimes we also employ inference rule notation:
-\inferrule{\mbox{@{text "A\<^sub>1"}}\\ \mbox{@{text "\<dots>"}}\\ \mbox{@{text "A\<^sub>n"}}}
-{\mbox{@{text A}}}
-\end{warn}
-
-
-\subsection{Theories}
-\label{sec:Basic:Theories}
-
-Roughly speaking, a \concept{theory} is a named collection of types,
-functions, and theorems, much like a module in a programming language.
-All the Isabelle text that you ever type needs to go into a theory.
-The general format of a theory @{text T} is
-\begin{quote}
-\indexed{\isacom{theory}}{theory} @{text T}\\
-\indexed{\isacom{imports}}{imports} @{text "T\<^sub>1 \<dots> T\<^sub>n"}\\
-\isacom{begin}\\
-\emph{definitions, theorems and proofs}\\
-\isacom{end}
-\end{quote}
-where @{text "T\<^sub>1 \<dots> T\<^sub>n"} are the names of existing
-theories that @{text T} is based on. The @{text "T\<^sub>i"} are the
-direct \conceptidx{parent theories}{parent theory} of @{text T}.
-Everything defined in the parent theories (and their parents, recursively) is
-automatically visible. Each theory @{text T} must
-reside in a \concept{theory file} named @{text "T.thy"}.
-
-\begin{warn}
-HOL contains a theory @{theory Main}\index{Main@@{theory Main}}, the union of all the basic
-predefined theories like arithmetic, lists, sets, etc.
-Unless you know what you are doing, always include @{text Main}
-as a direct or indirect parent of all your theories.
-\end{warn}
-
-In addition to the theories that come with the Isabelle/HOL distribution
-(see @{url "http://isabelle.in.tum.de/library/HOL/"})
-there is also the \emph{Archive of Formal Proofs}
-at @{url "http://afp.sourceforge.net"}, a growing collection of Isabelle theories
-that everybody can contribute to.
-
-\subsection{Quotation Marks}
-
-The textual definition of a theory follows a fixed syntax with keywords like
-\isacommand{begin} and \isacommand{datatype}.  Embedded in this syntax are
-the types and formulas of HOL.  To distinguish the two levels, everything
-HOL-specific (terms and types) must be enclosed in quotation marks:
-\texttt{"}\dots\texttt{"}. To lessen this burden, quotation marks around a
-single identifier can be dropped.  When Isabelle prints a syntax error
-message, it refers to the HOL syntax as the \concept{inner syntax} and the
-enclosing theory language as the \concept{outer syntax}.
-*}
-(*<*)
-end
-(*>*)
\ No newline at end of file




