theory ToyList = PreList:

 text{*\noindent

 HOL already has a predefined theory of lists called @{text"List"} ---

 @{text"ToyList"} is merely a small fragment of it chosen as an example. In

 contrast to what is recommended in \S\ref{sec:Basic:Theories},

 @{text"ToyList"} is not based on @{text"Main"} but on @{text"PreList"}, a

 theory that contains pretty much everything but lists, thus avoiding

 ambiguities caused by defining lists twice.

 *}

 datatype 'a list = Nil                          ("[]")

                  | Cons 'a "'a list"            (infixr "#" 65);

 text{*\noindent

 The datatype\index{*datatype} \isaindexbold{list} introduces two

 constructors \isaindexbold{Nil} and \isaindexbold{Cons}, the

 empty~list and the operator that adds an element to the front of a list. For

 example, the term \isa{Cons True (Cons False Nil)} is a value of

 type @{typ"bool list"}, namely the list with the elements @{term"True"} and

 @{term"False"}. Because this notation becomes unwieldy very quickly, the

 datatype declaration is annotated with an alternative syntax: instead of

 @{term[source]Nil} and \isa{Cons x xs} we can write

 @{term"[]"}\index{$HOL2list@\texttt{[]}|bold} and

 @{term"x # xs"}\index{$HOL2list@\texttt{\#}|bold}. In fact, this

 alternative syntax is the standard syntax. Thus the list \isa{Cons True

 (Cons False Nil)} becomes @{term"True # False # []"}. The annotation

 \isacommand{infixr}\indexbold{*infixr} means that @{text"#"} associates to

 the right, i.e.\ the term @{term"x # y # z"} is read as @{text"x # (y # z)"}

 and not as @{text"(x # y) # z"}.

 \begin{warn}

   Syntax annotations are a powerful but completely optional feature. You

   could drop them from theory @{text"ToyList"} and go back to the identifiers

   @{term[source]Nil} and @{term[source]Cons}. However, lists are such a

   central datatype

   that their syntax is highly customized. We recommend that novices should

   not use syntax annotations in their own theories.

 \end{warn}

 Next, two functions @{text"app"} and \isaindexbold{rev} are declared:

 *}

 consts app :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"   (infixr "@" 65)

        rev :: "'a list \<Rightarrow> 'a list";

 text{*

 \noindent

 In contrast to ML, Isabelle insists on explicit declarations of all functions

 (keyword \isacommand{consts}).  (Apart from the declaration-before-use

 restriction, the order of items in a theory file is unconstrained.) Function

 @{term"app"} is annotated with concrete syntax too. Instead of the prefix

 syntax \isa{app xs ys} the infix

 @{term"xs @ ys"}\index{$HOL2list@\texttt{\at}|bold} becomes the preferred

 form. Both functions are defined recursively:

 *}

 primrec

 "[] @ ys       = ys"

 "(x # xs) @ ys = x # (xs @ ys)";

 primrec

 "rev []        = []"

 "rev (x # xs)  = (rev xs) @ (x # [])";

 text{*

 \noindent

 The equations for @{term"app"} and @{term"rev"} hardly need comments:

 @{term"app"} appends two lists and @{term"rev"} reverses a list.  The keyword

 \isacommand{primrec}\index{*primrec} indicates that the recursion is of a

 particularly primitive kind where each recursive call peels off a datatype

 constructor from one of the arguments.  Thus the

 recursion always terminates, i.e.\ the function is \bfindex{total}.

 The termination requirement is absolutely essential in HOL, a logic of total

 functions. If we were to drop it, inconsistencies would quickly arise: the

 ``definition'' $f(n) = f(n)+1$ immediately leads to $0 = 1$ by subtracting

 $f(n)$ on both sides.

 % However, this is a subtle issue that we cannot discuss here further.

 \begin{warn}

   As we have indicated, the desire for total functions is not a gratuitously

   imposed restriction but an essential characteristic of HOL. It is only

   because of totality that reasoning in HOL is comparatively easy.  More

   generally, the philosophy in HOL is not to allow arbitrary axioms (such as

   function definitions whose totality has not been proved) because they

   quickly lead to inconsistencies. Instead, fixed constructs for introducing

   types and functions are offered (such as \isacommand{datatype} and

   \isacommand{primrec}) which are guaranteed to preserve consistency.

 \end{warn}

 A remark about syntax.  The textual definition of a theory follows a fixed

 syntax with keywords like \isacommand{datatype} and \isacommand{end} (see

 Fig.~\ref{fig:keywords} in Appendix~\ref{sec:Appendix} for a full list).

 Embedded in this syntax are the types and formulae of HOL, whose syntax is

 extensible, e.g.\ by new user-defined infix operators

 (see~\ref{sec:infix-syntax}). To distinguish the two levels, everything

 HOL-specific (terms and types) should be enclosed in

 \texttt{"}\dots\texttt{"}. 

 To lessen this burden, quotation marks around a single identifier can be

 dropped, unless the identifier happens to be a keyword, as in

 *}

 consts "end" :: "'a list \<Rightarrow> 'a"

 text{*\noindent

 When Isabelle prints a syntax error message, it refers to the HOL syntax as

 the \bfindex{inner syntax} and the enclosing theory language as the \bfindex{outer syntax}.

 \section{An introductory proof}

 \label{sec:intro-proof}

 Assuming you have input the declarations and definitions of \texttt{ToyList}

 presented so far, we are ready to prove a few simple theorems. This will

 illustrate not just the basic proof commands but also the typical proof

 process.

 \subsubsection*{Main goal: @{text"rev(rev xs) = xs"}}

 Our goal is to show that reversing a list twice produces the original

 list. The input line

 *}

 theorem rev_rev [simp]: "rev(rev xs) = xs";

 txt{*\index{*theorem|bold}\index{*simp (attribute)|bold}

 \begin{itemize}

 \item

 establishes a new theorem to be proved, namely @{prop"rev(rev xs) = xs"},

 \item

 gives that theorem the name @{text"rev_rev"} by which it can be

 referred to,

 \item

 and tells Isabelle (via @{text"[simp]"}) to use the theorem (once it has been

 proved) as a simplification rule, i.e.\ all future proofs involving

 simplification will replace occurrences of @{term"rev(rev xs)"} by

 @{term"xs"}.

 The name and the simplification attribute are optional.

 \end{itemize}

 Isabelle's response is to print

 \begin{isabelle}

 proof(prove):~step~0\isanewline

 \isanewline

 goal~(theorem~rev\_rev):\isanewline

 rev~(rev~xs)~=~xs\isanewline

 ~1.~rev~(rev~xs)~=~xs

 \end{isabelle}

 The first three lines tell us that we are 0 steps into the proof of

 theorem @{text"rev_rev"}; for compactness reasons we rarely show these

 initial lines in this tutorial. The remaining lines display the current

 proof state.

 Until we have finished a proof, the proof state always looks like this:

 \begin{isabelle}

 $G$\isanewline

 ~1.~$G\sb{1}$\isanewline

 ~~\vdots~~\isanewline

 ~$n$.~$G\sb{n}$

 \end{isabelle}

 where $G$

 is the overall goal that we are trying to prove, and the numbered lines

 contain the subgoals $G\sb{1}$, \dots, $G\sb{n}$ that we need to prove to

 establish $G$. At @{text"step 0"} there is only one subgoal, which is

 identical with the overall goal.  Normally $G$ is constant and only serves as

 a reminder. Hence we rarely show it in this tutorial.

 Let us now get back to @{prop"rev(rev xs) = xs"}. Properties of recursively

 defined functions are best established by induction. In this case there is

 not much choice except to induct on @{term"xs"}:

 *}

 apply(induct_tac xs);

 txt{*\noindent\index{*induct_tac}%

 This tells Isabelle to perform induction on variable @{term"xs"}. The suffix

 @{term"tac"} stands for ``tactic'', a synonym for ``theorem proving function''.

 By default, induction acts on the first subgoal. The new proof state contains

 two subgoals, namely the base case (@{term[source]Nil}) and the induction step

 (@{term[source]Cons}):

 \begin{isabelle}

 ~1.~rev~(rev~[])~=~[]\isanewline

 ~2.~{\isasymAnd}a~list.~rev(rev~list)~=~list~{\isasymLongrightarrow}~rev(rev(a~\#~list))~=~a~\#~list

 \end{isabelle}

 The induction step is an example of the general format of a subgoal:

 \begin{isabelle}

 ~$i$.~{\isasymAnd}$x\sb{1}$~\dots~$x\sb{n}$.~{\it assumptions}~{\isasymLongrightarrow}~{\it conclusion}

 \end{isabelle}\index{$IsaAnd@\isasymAnd|bold}

 The prefix of bound variables \isasymAnd$x\sb{1}$~\dots~$x\sb{n}$ can be

 ignored most of the time, or simply treated as a list of variables local to

 this subgoal. Their deeper significance is explained in Chapter~\ref{chap:rules}.

 The {\it assumptions} are the local assumptions for this subgoal and {\it

   conclusion} is the actual proposition to be proved. Typical proof steps

 that add new assumptions are induction or case distinction. In our example

 the only assumption is the induction hypothesis @{term"rev (rev list) =

   list"}, where @{term"list"} is a variable name chosen by Isabelle. If there

 are multiple assumptions, they are enclosed in the bracket pair

 \indexboldpos{\isasymlbrakk}{$Isabrl} and

 \indexboldpos{\isasymrbrakk}{$Isabrr} and separated by semicolons.

 Let us try to solve both goals automatically:

 *}

 apply(auto);

 txt{*\noindent

 This command tells Isabelle to apply a proof strategy called

 @{text"auto"} to all subgoals. Essentially, @{text"auto"} tries to

 ``simplify'' the subgoals.  In our case, subgoal~1 is solved completely (thanks

 to the equation @{prop"rev [] = []"}) and disappears; the simplified version

 of subgoal~2 becomes the new subgoal~1:

 \begin{isabelle}

 ~1.~\dots~rev(rev~list)~=~list~{\isasymLongrightarrow}~rev(rev~list~@~a~\#~[])~=~a~\#~list

 \end{isabelle}

 In order to simplify this subgoal further, a lemma suggests itself.

 *}

 (*<*)

 oops

 (*>*)

 subsubsection{*First lemma: @{text"rev(xs @ ys) = (rev ys) @ (rev xs)"}*}

 text{*

 After abandoning the above proof attempt\indexbold{abandon

 proof}\indexbold{proof!abandon} (at the shell level type

 \isacommand{oops}\indexbold{*oops}) we start a new proof:

 *}

 lemma rev_app [simp]: "rev(xs @ ys) = (rev ys) @ (rev xs)";

 txt{*\noindent The keywords \isacommand{theorem}\index{*theorem} and

 \isacommand{lemma}\indexbold{*lemma} are interchangable and merely indicate

 the importance we attach to a proposition. In general, we use the words

 \emph{theorem}\index{theorem} and \emph{lemma}\index{lemma} pretty much

 interchangeably.

 There are two variables that we could induct on: @{term"xs"} and

 @{term"ys"}. Because @{text"@"} is defined by recursion on

 the first argument, @{term"xs"} is the correct one:

 *}

 apply(induct_tac xs);

 txt{*\noindent

 This time not even the base case is solved automatically:

 *}

 apply(auto);

 txt{*

 \begin{isabelle}

 ~1.~rev~ys~=~rev~ys~@~[]\isanewline

 ~2. \dots

 \end{isabelle}

 Again, we need to abandon this proof attempt and prove another simple lemma first.

 In the future the step of abandoning an incomplete proof before embarking on

 the proof of a lemma usually remains implicit.

 *}

 (*<*)

 oops

 (*>*)

 subsubsection{*Second lemma: @{text"xs @ [] = xs"}*}

 text{*

 This time the canonical proof procedure

 *}

 lemma app_Nil2 [simp]: "xs @ [] = xs";

 apply(induct_tac xs);

 apply(auto);

 txt{*

 \noindent

 leads to the desired message @{text"No subgoals!"}:

 \begin{isabelle}

 xs~@~[]~=~xs\isanewline

 No~subgoals!

 \end{isabelle}

 We still need to confirm that the proof is now finished:

 *}

 done

 text{*\noindent\indexbold{done}%

 As a result of that final \isacommand{done}, Isabelle associates the lemma just proved

 with its name. In this tutorial, we sometimes omit to show that final \isacommand{done}

 if it is obvious from the context that the proof is finished.

 % Instead of \isacommand{apply} followed by a dot, you can simply write

 % \isacommand{by}\indexbold{by}, which we do most of the time.

 Notice that in lemma @{thm[source]app_Nil2}

 (as printed out after the final \isacommand{done}) the free variable @{term"xs"} has been

 replaced by the unknown @{text"?xs"}, just as explained in

 \S\ref{sec:variables}.

 Going back to the proof of the first lemma

 *}

 lemma rev_app [simp]: "rev(xs @ ys) = (rev ys) @ (rev xs)";

 apply(induct_tac xs);

 apply(auto);

 txt{*

 \noindent

 we find that this time @{text"auto"} solves the base case, but the

 induction step merely simplifies to

 \begin{isabelle}

 ~1.~{\isasymAnd}a~list.\isanewline

 ~~~~~~~rev~(list~@~ys)~=~rev~ys~@~rev~list~{\isasymLongrightarrow}\isanewline

 ~~~~~~~(rev~ys~@~rev~list)~@~a~\#~[]~=~rev~ys~@~rev~list~@~a~\#~[]

 \end{isabelle}

 Now we need to remember that @{text"@"} associates to the right, and that

 @{text"#"} and @{text"@"} have the same priority (namely the @{text"65"}

 in their \isacommand{infixr} annotation). Thus the conclusion really is

 \begin{isabelle}

 ~~~~~(rev~ys~@~rev~list)~@~(a~\#~[])~=~rev~ys~@~(rev~list~@~(a~\#~[]))

 \end{isabelle}

 and the missing lemma is associativity of @{text"@"}.

 *}

 (*<*)oops(*>*)

 subsubsection{*Third lemma: @{text"(xs @ ys) @ zs = xs @ (ys @ zs)"}*}

 text{*

 Abandoning the previous proof, the canonical proof procedure

 *}

 lemma app_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)";

 apply(induct_tac xs);

 apply(auto);

 done

 text{*

 \noindent

 succeeds without further ado.

 Now we can go back and prove the first lemma

 *}

 lemma rev_app [simp]: "rev(xs @ ys) = (rev ys) @ (rev xs)";

 apply(induct_tac xs);

 apply(auto);

 done

 text{*\noindent

 and then solve our main theorem:

 *}

 theorem rev_rev [simp]: "rev(rev xs) = xs";

 apply(induct_tac xs);

 apply(auto);

 done

 text{*\noindent

 The final \isacommand{end} tells Isabelle to close the current theory because

 we are finished with its development:

 *}

 end