--- a/doc-src/ProgProve/Thys/Basics.thy Mon Aug 27 21:46:00 2012 +0200
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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 Formulae}
-\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 @{typ 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, e.g.\ @{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}
-
-\concept{Terms} are formed as in functional programming by
-applying functions to arguments. If @{text f} is a function of type
-@{text"\<tau>\<^isub>1 \<Rightarrow> \<tau>\<^isub>2"} and @{text t} is a term of type
-@{text"\<tau>\<^isub>1"} then @{term"f t"} is a term of type @{text"\<tau>\<^isub>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 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\<^isub>1 else t\<^isub>2)"}\\
-@{text "(let x = t in u)"}\\
-@{text "(case t of pat\<^isub>1 \<Rightarrow> t\<^isub>1 | \<dots> | pat\<^isub>n \<Rightarrow> t\<^isub>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.
-
-\concept{Formulae} 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>"}.
-
-\concept{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''.
-
-\concept{Quantifiers} 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 explicit \concept{type constraints} (or \concept{type
-annotations}) 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>"} and the implication
-@{text"\<Longrightarrow>"}. 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\<^isub>1 \<Longrightarrow> A\<^isub>2 \<Longrightarrow> A\<^isub>3"} means @{text "A\<^isub>1 \<Longrightarrow> (A\<^isub>2 \<Longrightarrow> A\<^isub>3)"}.
-The (Isabelle specific) notation \mbox{@{text"\<lbrakk> A\<^isub>1; \<dots>; A\<^isub>n \<rbrakk> \<Longrightarrow> A"}}
-is short for the iterated implication \mbox{@{text"A\<^isub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^isub>n \<Longrightarrow> A"}}.
-Sometimes we also employ inference rule notation:
-\inferrule{\mbox{@{text "A\<^isub>1"}}\\ \mbox{@{text "\<dots>"}}\\ \mbox{@{text "A\<^isub>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}
-\isacom{theory} @{text T}\\
-\isacom{imports} @{text "T\<^isub>1 \<dots> T\<^isub>n"}\\
-\isacom{begin}\\
-\emph{definitions, theorems and proofs}\\
-\isacom{end}
-\end{quote}
-where @{text "T\<^isub>1 \<dots> T\<^isub>n"} are the names of existing
-theories that @{text T} is based on. The @{text "T\<^isub>i"} are the
-direct \concept{parent theories} 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 @{text 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 formulae 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}.
-\sem
-\begin{warn}
-For reasons of readability, we almost never show the quotation marks in this
-book. Consult the accompanying theory files to see where they need to go.
-\end{warn}
-\endsem
-%
-*}
-(*<*)
-end
-(*>*)
\ No newline at end of file