\section{The set of even numbers}

The set of even numbers can be inductively defined as the least set
containing 0 and closed under the operation ${\cdots}+2$.  Obviously,
\emph{even} can also be expressed using the divides relation (\isa{dvd}). 
We shall prove below that the two formulations coincide.  On the way we
shall examine the primary means of reasoning about inductively defined
sets: rule induction.

\subsection{Making an inductive definition}

Using \isacommand{consts}, we declare the constant \isa{even} to be a set
of natural numbers. The \isacommand{inductive} declaration gives it the
desired properties.
\begin{isabelle}
\isacommand{consts}\ even\ ::\ "nat\ set"\isanewline
\isacommand{inductive}\ even\isanewline
\isakeyword{intros}\isanewline
zero[intro!]:\ "0\ \isasymin \ even"\isanewline
step[intro!]:\ "n\ \isasymin \ even\ \isasymLongrightarrow \ (Suc\ (Suc\
n))\ \isasymin \ even"
\end{isabelle}

An inductive definition consists of introduction rules.  The first one
above states that 0 is even; the second states that if $n$ is even, then so
is
$n+2$.  Given this declaration, Isabelle generates a fixed point definition
for \isa{even} and proves theorems about it.  These theorems include the
introduction rules specified in the declaration, an elimination rule for case
analysis and an induction rule.  We can refer to these theorems by
automatically-generated names.  Here are two examples:
%
\begin{isabelle}
0\ \isasymin \ even
\rulename{even.zero}
\par\smallskip
n\ \isasymin \ even\ \isasymLongrightarrow \ Suc\ (Suc\ n)\ \isasymin \
even%
\rulename{even.step}
\end{isabelle}

The introduction rules can be given attributes.  Here both rules are
specified as \isa{intro!}, directing the classical reasoner to 
apply them aggressively. Obviously, regarding 0 as even is safe.  The
\isa{step} rule is also safe because $n+2$ is even if and only if $n$ is
even.  We prove this equivalence later.

\subsection{Using introduction rules}

Our first lemma states that numbers of the form $2\times k$ are even.
Introduction rules are used to show that specific values belong to the
inductive set.  Such proofs typically involve 
induction, perhaps over some other inductive set.
\begin{isabelle}
\isacommand{lemma}\ two_times_even[intro!]:\ "\#2*k\ \isasymin \ even"
\isanewline
\isacommand{apply}\ (induct\ "k")\isanewline
\ \isacommand{apply}\ auto\isanewline
\isacommand{done}
\end{isabelle}
%
The first step is induction on the natural number \isa{k}, which leaves
two subgoals:
\begin{isabelle}
\ 1.\ \#2\ *\ 0\ \isasymin \ even\isanewline
\ 2.\ \isasymAnd n.\ \#2\ *\ n\ \isasymin \ even\ \isasymLongrightarrow \ \#2\ *\ Suc\ n\ \isasymin \ even
\end{isabelle}
%
Here \isa{auto} simplifies both subgoals so that they match the introduction
rules, which are then applied automatically.

Our ultimate goal is to prove the equivalence between the traditional
definition of \isa{even} (using the divides relation) and our inductive
definition.  One direction of this equivalence is immediate by the lemma
just proved, whose \isa{intro!} attribute ensures it will be used.
\begin{isabelle}
\isacommand{lemma}\ dvd_imp_even:\ "\#2\ dvd\ n\ \isasymLongrightarrow \ n\ \isasymin \ even"\isanewline
\isacommand{apply}\ (auto\ simp\ add:\ dvd_def)\isanewline
\isacommand{done}
\end{isabelle}

\subsection{Rule induction}

From the definition of the set
\isa{even}, Isabelle has
generated an induction rule:
\begin{isabelle}
\isasymlbrakk xa\ \isasymin \ even;\isanewline
\ P\ 0;\isanewline
\ \isasymAnd n.\ \isasymlbrakk n\ \isasymin \ even;\ P\ n\isasymrbrakk \
\isasymLongrightarrow \ P\ (Suc\ (Suc\ n))\isasymrbrakk\isanewline
\ \isasymLongrightarrow \ P\ xa%
\rulename{even.induct}
\end{isabelle}
A property \isa{P} holds for every even number provided it
holds for~\isa{0} and is closed under the operation
\isa{Suc(Suc\(\cdots\))}.  Then \isa{P} is closed under the introduction
rules for \isa{even}, which is the least set closed under those rules. 
This type of inductive argument is called \textbf{rule induction}. 

Apart from the double application of \isa{Suc}, the induction rule above
resembles the familiar mathematical induction, which indeed is an instance
of rule induction; the natural numbers can be defined inductively to be
the least set containing \isa{0} and closed under~\isa{Suc}.

Induction is the usual way of proving a property of the elements of an
inductively defined set.  Let us prove that all members of the set
\isa{even} are multiples of two.  
\begin{isabelle}
\isacommand{lemma}\ even_imp_dvd:\ "n\ \isasymin \ even\ \isasymLongrightarrow \ \#2\ dvd\ n"\isanewline
\isacommand{apply}\ (erule\ even.induct)\isanewline
\ \isacommand{apply}\ simp\isanewline
\isacommand{apply}\ (simp\ add:\ dvd_def)\isanewline
\isacommand{apply}\ clarify\isanewline
\isacommand{apply}\ (rule_tac\ x\ =\ "Suc\ k"\ \isakeyword{in}\ exI)\isanewline
\isacommand{apply}\ simp\isanewline
\isacommand{done}
\end{isabelle}
%
We begin by applying induction.  Note that \isa{even.induct} has the form
of an elimination rule, so we use the method \isa{erule}.  We get two
subgoals:
\begin{isabelle}
\ 1.\ \#2\ dvd\ 0\isanewline
\ 2.\ \isasymAnd n.\ \isasymlbrakk n\ \isasymin \ even;\ \#2\ dvd\ n\isasymrbrakk \ \isasymLongrightarrow \ \#2\ dvd\ Suc\ (Suc\ n)
\end{isabelle}
%
The first subgoal is trivial (by \isa{simp}).  For the second
subgoal, we unfold the definition of \isa{dvd}:
\begin{isabelle}
\ 1.\ \isasymAnd n.\ \isasymlbrakk n\ \isasymin \ even;\ \isasymexists k.\
n\ =\ \#2\ *\ k\isasymrbrakk \ \isasymLongrightarrow \ \isasymexists k.\
Suc\ (Suc\ n)\ =\ \#2\ *\ k
\end{isabelle}
%
Then we use
\isa{clarify} to eliminate the existential quantifier from the assumption
and replace \isa{n} by \isa{\#2\ *\ k}.
\begin{isabelle}
\ 1.\ \isasymAnd n\ k.\ \#2\ *\ k\ \isasymin \ even\ \isasymLongrightarrow \ \isasymexists ka.\ Suc\ (Suc\ (\#2\ *\ k))\ =\ \#2\ *\ ka%
\end{isabelle}
%
The \isa{rule_tac\ldots exI} tells Isabelle that the desired
\isa{ka} is
\isa{Suc\ k}.  With this hint, the subgoal becomes trivial, and \isa{simp}
concludes the proof.

\medskip
Combining the previous two results yields our objective, the
equivalence relating \isa{even} and \isa{dvd}. 
%
%we don't want [iff]: discuss?
\begin{isabelle}
\isacommand{theorem}\ even_iff_dvd:\ "(n\ \isasymin \ even)\ =\ (\#2\ dvd\ n)"\isanewline
\isacommand{apply}\ (blast\ intro:\ dvd_imp_even\ even_imp_dvd)\isanewline
\isacommand{done}
\end{isabelle}

\subsection{Generalization and rule induction}
\label{sec:gen-rule-induction}

Everybody knows that when before applying induction we often must generalize
the induction formula.  This step is just as important with rule induction,
and the required generalizations can be complicated.  In this 
example, the obvious statement of the result is not inductive:
%
\begin{isabelle}
\isacommand{lemma}\ "Suc\ (Suc\ n)\ \isasymin \ even\
\isasymLongrightarrow \ n\ \isasymin \ even"\isanewline
\isacommand{apply}\ (erule\ even.induct)\isanewline
\isacommand{oops}
\end{isabelle}
%
Rule induction finds no occurrences of \isa{Suc\ (Suc\ n)} in the
conclusion, which it therefore leaves unchanged.  (Look at
\isa{even.induct} to see why this happens.)  We get these subgoals:
\begin{isabelle}
\ 1.\ n\ \isasymin \ even\isanewline
\ 2.\ \isasymAnd na.\ \isasymlbrakk na\ \isasymin \ even;\ n\ \isasymin \ even\isasymrbrakk \ \isasymLongrightarrow \ n\ \isasymin \ even%
\end{isabelle}
The first one is hopeless.  In general, rule inductions involving
non-trivial terms will probably go wrong. How to deal with such situations
in general is described in {\S}\ref{sec:ind-var-in-prems} below.
In the current case the solution is easy because
we have the necessary inverse, subtraction:
\begin{isabelle}
\isacommand{lemma}\ even_imp_even_minus_2:\ "n\ \isasymin \ even\ \isasymLongrightarrow \ n-\#2\ \isasymin \ even"\isanewline
\isacommand{apply}\ (erule\ even.induct)\isanewline
\ \isacommand{apply}\ auto\isanewline
\isacommand{done}
\end{isabelle}
%
This lemma is trivially inductive.  Here are the subgoals:
\begin{isabelle}
\ 1.\ 0\ -\ \#2\ \isasymin \ even\isanewline
\ 2.\ \isasymAnd n.\ \isasymlbrakk n\ \isasymin \ even;\ n\ -\ \#2\ \isasymin \ even\isasymrbrakk \ \isasymLongrightarrow \ Suc\ (Suc\ n)\ -\ \#2\ \isasymin \ even%
\end{isabelle}
The first is trivial because \isa{0\ -\ \#2} simplifies to \isa{0}, which is
even.  The second is trivial too: \isa{Suc\ (Suc\ n)\ -\ \#2} simplifies to
\isa{n}, matching the assumption.

\medskip
Using our lemma, we can easily prove the result we originally wanted:
\begin{isabelle}
\isacommand{lemma}\ Suc_Suc_even_imp_even:\ "Suc\ (Suc\ n)\ \isasymin \ even\ \isasymLongrightarrow \ n\ \isasymin \ even"\isanewline
\isacommand{apply}\ (drule\ even_imp_even_minus_2)\isanewline
\isacommand{apply}\ simp\isanewline
\isacommand{done}
\end{isabelle}

We have just proved the converse of the introduction rule \isa{even.step}. 
This suggests proving the following equivalence.  We give it the \isa{iff}
attribute because of its obvious value for simplification.
\begin{isabelle}
\isacommand{lemma}\ [iff]:\ "((Suc\ (Suc\ n))\ \isasymin \ even)\ =\ (n\
\isasymin \ even)"\isanewline
\isacommand{apply}\ (blast\ dest:\ Suc_Suc_even_imp_even)\isanewline
\isacommand{done}
\end{isabelle}

The even numbers example has shown how inductive definitions can be used. 
Later examples will show that they are actually worth using.
