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Mon, 13 Oct 2014 16:07:11 +0200
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theory Basics
imports Main
This chapter introduces HOL as a functional programming language and shows
how to prove properties of functional programs by induction.


\subsection{Types, Terms and Formulas}

HOL is a typed logic whose type system resembles that of functional
programming languages. Thus there are
\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., like in ML\@.
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>}.

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"}}.

HOL also supports some basic constructs from functional programming:
@{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)"}
The above three constructs must always be enclosed in parentheses
if they occur inside other constructs.
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{@{term[source] "m + (n::nat)"}}}. Type constraints may be
needed to
disambiguate terms involving overloaded functions such as @{text "+"}.

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.
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\footnote{To display implications in this style in
Isabelle/jedit you need to set Plugins $>$ Plugin Options $>$ Isabelle/General $>$ Print Mode to ``\texttt{brackets}'' and restart.}) 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}}}


Roughly speaking, a \concept{theory} is a named collection of types,
functions, and theorems, much like a module in a programming language.
All Isabelle text needs to go into a theory.
The general format of a theory @{text T} is
\indexed{\isacom{theory}}{theory} @{text T}\\
\indexed{\isacom{imports}}{imports} @{text "T\<^sub>1 \<dots> T\<^sub>n"}\\
\emph{definitions, theorems and proofs}\\
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"}.

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.

In addition to the theories that come with the Isabelle/HOL distribution
(see @{url ""})
there is also the \emph{Archive of Formal Proofs}
at @{url ""}, 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}.