--- a/doc-src/TutorialI/ToyList/document/ToyList.tex Thu Jul 26 16:54:44 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,530 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{ToyList}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ ToyList\isanewline
-\isakeyword{imports}\ Datatype\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\begin{isamarkuptext}%
-\noindent
-HOL already has a predefined theory of lists called \isa{List} ---
-\isa{ToyList} is merely a small fragment of it chosen as an example. In
-contrast to what is recommended in \S\ref{sec:Basic:Theories},
-\isa{ToyList} is not based on \isa{Main} but on \isa{Datatype}, a
-theory that contains pretty much everything but lists, thus avoiding
-ambiguities caused by defining lists twice.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{3D}{\isacharequal}}\ Nil\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{7C}{\isacharbar}}\ Cons\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ list{\isaliteral{22}{\isachardoublequoteclose}}\ \ \ \ \ \ \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixr}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{23}{\isacharhash}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{6}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptext}%
-\noindent
-The datatype\index{datatype@\isacommand {datatype} (command)}
-\tydx{list} introduces two
-constructors \cdx{Nil} and \cdx{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 \isa{bool\ list}, namely the list with the elements \isa{True} and
-\isa{False}. Because this notation quickly becomes unwieldy, the
-datatype declaration is annotated with an alternative syntax: instead of
-\isa{Nil} and \isa{Cons x xs} we can write
-\isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}}\index{$HOL2list@\isa{[]}|bold} and
-\isa{x\ {\isaliteral{23}{\isacharhash}}\ xs}\index{$HOL2list@\isa{\#}|bold}. In fact, this
-alternative syntax is the familiar one. Thus the list \isa{Cons True
-(Cons False Nil)} becomes \isa{True\ {\isaliteral{23}{\isacharhash}}\ False\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}}. The annotation
-\isacommand{infixr}\index{infixr@\isacommand{infixr} (annotation)}
-means that \isa{{\isaliteral{23}{\isacharhash}}} associates to
-the right: the term \isa{x\ {\isaliteral{23}{\isacharhash}}\ y\ {\isaliteral{23}{\isacharhash}}\ z} is read as \isa{x\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{23}{\isacharhash}}\ z{\isaliteral{29}{\isacharparenright}}}
-and not as \isa{{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{23}{\isacharhash}}\ z}.
-The \isa{{\isadigit{6}}{\isadigit{5}}} is the priority of the infix \isa{{\isaliteral{23}{\isacharhash}}}.
-
-\begin{warn}
- Syntax annotations can be powerful, but they are difficult to master and
- are never necessary. You
- could drop them from theory \isa{ToyList} and go back to the identifiers
- \isa{Nil} and \isa{Cons}. Novices should avoid using
- syntax annotations in their own theories.
-\end{warn}
-Next, two functions \isa{app} and \cdx{rev} are defined recursively,
-in this order, because Isabelle insists on definition before use:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{primrec}\isamarkupfalse%
-\ app\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixr}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{40}{\isacharat}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{6}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}\ \isakeyword{where}\isanewline
-{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{40}{\isacharat}}\ ys\ \ \ \ \ \ \ {\isaliteral{3D}{\isacharequal}}\ ys{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ ys\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{28}{\isacharparenleft}}xs\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isanewline
-\isacommand{primrec}\isamarkupfalse%
-\ rev\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
-{\isaliteral{22}{\isachardoublequoteopen}}rev\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ \ \ \ \ \ \ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
-{\isaliteral{22}{\isachardoublequoteopen}}rev\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ \ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\noindent
-Each function definition is of the form
-\begin{center}
-\isacommand{primrec} \textit{name} \isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}} \textit{type} \textit{(optional syntax)} \isakeyword{where} \textit{equations}
-\end{center}
-The equations must be separated by \isa{{\isaliteral{7C}{\isacharbar}}}.
-%
-Function \isa{app} is annotated with concrete syntax. Instead of the
-prefix syntax \isa{app\ xs\ ys} the infix
-\isa{xs\ {\isaliteral{40}{\isacharat}}\ ys}\index{$HOL2list@\isa{\at}|bold} becomes the preferred
-form.
-
-\index{*rev (constant)|(}\index{append function|(}
-The equations for \isa{app} and \isa{rev} hardly need comments:
-\isa{app} appends two lists and \isa{rev} reverses a list. The
-keyword \commdx{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 \textbf{total}.
-\index{functions!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 requirement for total functions is 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 to refrain from asserting 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}
-
-\index{syntax}%
-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 (see \S\ref{sec:concrete-syntax}), e.g.\ by new user-defined infix operators.
-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, for example
-\isa{"end"}.
-When Isabelle prints a syntax error message, it refers to the HOL syntax as
-the \textbf{inner syntax} and the enclosing theory language as the \textbf{outer syntax}.
-
-Comments\index{comment} must be in enclosed in \texttt{(* }and\texttt{ *)}.
-
-\section{Evaluation}
-\index{evaluation}
-
-Assuming you have processed the declarations and definitions of
-\texttt{ToyList} presented so far, you may want to test your
-functions by running them. For example, what is the value of
-\isa{rev\ {\isaliteral{28}{\isacharparenleft}}True\ {\isaliteral{23}{\isacharhash}}\ False\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}}? Command%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{value}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}rev\ {\isaliteral{28}{\isacharparenleft}}True\ {\isaliteral{23}{\isacharhash}}\ False\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\noindent yields the correct result \isa{False\ {\isaliteral{23}{\isacharhash}}\ True\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}}.
-But we can go beyond mere functional programming and evaluate terms with
-variables in them, executing functions symbolically:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{value}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}rev\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{23}{\isacharhash}}\ b\ {\isaliteral{23}{\isacharhash}}\ c\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\noindent yields \isa{c\ {\isaliteral{23}{\isacharhash}}\ b\ {\isaliteral{23}{\isacharhash}}\ a\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}}.
-
-\section{An Introductory Proof}
-\label{sec:intro-proof}
-
-Having convinced ourselves (as well as one can by testing) that our
-definitions capture our intentions, 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.}
-
-Our goal is to show that reversing a list twice produces the original
-list.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isaliteral{5F}{\isacharunderscore}}rev\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}rev{\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ xs{\isaliteral{22}{\isachardoublequoteclose}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-\index{theorem@\isacommand {theorem} (command)|bold}%
-\noindent
-This \isacommand{theorem} command does several things:
-\begin{itemize}
-\item
-It establishes a new theorem to be proved, namely \isa{rev\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ xs}.
-\item
-It gives that theorem the name \isa{rev{\isaliteral{5F}{\isacharunderscore}}rev}, for later reference.
-\item
-It tells Isabelle (via the bracketed attribute \attrdx{simp}) to take the eventual theorem as a simplification rule: future proofs involving
-simplification will replace occurrences of \isa{rev\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}} by
-\isa{xs}.
-\end{itemize}
-The name and the simplification attribute are optional.
-Isabelle's response is to print the initial proof state consisting
-of some header information (like how many subgoals there are) followed by
-\begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ rev\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ xs%
-\end{isabelle}
-For compactness reasons we omit the header in this tutorial.
-Until we have finished a proof, the \rmindex{proof state} proper
-always looks like this:
-\begin{isabelle}
-~1.~$G\sb{1}$\isanewline
-~~\vdots~~\isanewline
-~$n$.~$G\sb{n}$
-\end{isabelle}
-The numbered lines contain the subgoals $G\sb{1}$, \dots, $G\sb{n}$
-that we need to prove to establish the main goal.\index{subgoals}
-Initially there is only one subgoal, which is identical with the
-main goal. (If you always want to see the main goal as well,
-set the flag \isa{Proof.show_main_goal}\index{*show_main_goal (flag)}
---- this flag used to be set by default.)
-
-Let us now get back to \isa{rev\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ xs}. Properties of recursively
-defined functions are best established by induction. In this case there is
-nothing obvious except induction on \isa{xs}:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ xs{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-\noindent\index{*induct_tac (method)}%
-This tells Isabelle to perform induction on variable \isa{xs}. The suffix
-\isa{tac} stands for \textbf{tactic},\index{tactics}
-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 (\isa{Nil}) and the induction step
-(\isa{Cons}):
-\begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ rev\ {\isaliteral{28}{\isacharparenleft}}rev\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ list{\isaliteral{2E}{\isachardot}}\isanewline
-\isaindent{\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }rev\ {\isaliteral{28}{\isacharparenleft}}rev\ list{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ list\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ rev\ {\isaliteral{28}{\isacharparenleft}}rev\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{23}{\isacharhash}}\ list{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{23}{\isacharhash}}\ list%
-\end{isabelle}
-
-The induction step is an example of the general format of a subgoal:\index{subgoals}
-\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}\index{assumptions!of subgoal}
-are the local assumptions for this subgoal and {\it
- conclusion}\index{conclusion!of subgoal} is the actual proposition to be proved.
-Typical proof steps
-that add new assumptions are induction and case distinction. In our example
-the only assumption is the induction hypothesis \isa{rev\ {\isaliteral{28}{\isacharparenleft}}rev\ list{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ list}, where \isa{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:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-\noindent
-This command tells Isabelle to apply a proof strategy called
-\isa{auto} to all subgoals. Essentially, \isa{auto} tries to
-simplify the subgoals. In our case, subgoal~1 is solved completely (thanks
-to the equation \isa{rev\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}}) and disappears; the simplified version
-of subgoal~2 becomes the new subgoal~1:
-\begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ list{\isaliteral{2E}{\isachardot}}\isanewline
-\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }rev\ {\isaliteral{28}{\isacharparenleft}}rev\ list{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ list\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ rev\ {\isaliteral{28}{\isacharparenleft}}rev\ list\ {\isaliteral{40}{\isacharat}}\ a\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{23}{\isacharhash}}\ list%
-\end{isabelle}
-In order to simplify this subgoal further, a lemma suggests itself.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsubsection{First Lemma%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\indexbold{abandoning a proof}\indexbold{proofs!abandoning}
-After abandoning the above proof attempt (at the shell level type
-\commdx{oops}) we start a new proof:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ rev{\isaliteral{5F}{\isacharunderscore}}app\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}rev{\isaliteral{28}{\isacharparenleft}}xs\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}rev\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-\noindent The keywords \commdx{theorem} and
-\commdx{lemma} are interchangeable and merely indicate
-the importance we attach to a proposition. Therefore we use the words
-\emph{theorem} and \emph{lemma} pretty much interchangeably, too.
-
-There are two variables that we could induct on: \isa{xs} and
-\isa{ys}. Because \isa{{\isaliteral{40}{\isacharat}}} is defined by recursion on
-the first argument, \isa{xs} is the correct one:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ xs{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-\noindent
-This time not even the base case is solved automatically:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-\begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ rev\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ ys\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}%
-\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.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsubsection{Second Lemma%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-We again try the canonical proof procedure:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ app{\isaliteral{5F}{\isacharunderscore}}Nil{\isadigit{2}}\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}xs\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ xs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ xs{\isaliteral{29}{\isacharparenright}}\isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-\noindent
-It works, yielding the desired message \isa{No\ subgoals{\isaliteral{21}{\isacharbang}}}:
-\begin{isabelle}%
-xs\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ xs\isanewline
-No\ subgoals{\isaliteral{21}{\isacharbang}}%
-\end{isabelle}
-We still need to confirm that the proof is now finished:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent
-As a result of that final \commdx{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 \isa{app{\isaliteral{5F}{\isacharunderscore}}Nil{\isadigit{2}}},
-as printed out after the final \isacommand{done}, the free variable \isa{xs} has been
-replaced by the unknown \isa{{\isaliteral{3F}{\isacharquery}}xs}, just as explained in
-\S\ref{sec:variables}.
-
-Going back to the proof of the first lemma%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ rev{\isaliteral{5F}{\isacharunderscore}}app\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}rev{\isaliteral{28}{\isacharparenleft}}xs\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}rev\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ xs{\isaliteral{29}{\isacharparenright}}\isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-\noindent
-we find that this time \isa{auto} solves the base case, but the
-induction step merely simplifies to
-\begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ list{\isaliteral{2E}{\isachardot}}\isanewline
-\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }rev\ {\isaliteral{28}{\isacharparenleft}}list\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ rev\ ys\ {\isaliteral{40}{\isacharat}}\ rev\ list\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
-\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{28}{\isacharparenleft}}rev\ ys\ {\isaliteral{40}{\isacharat}}\ rev\ list{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ a\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ rev\ ys\ {\isaliteral{40}{\isacharat}}\ rev\ list\ {\isaliteral{40}{\isacharat}}\ a\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}%
-\end{isabelle}
-Now we need to remember that \isa{{\isaliteral{40}{\isacharat}}} associates to the right, and that
-\isa{{\isaliteral{23}{\isacharhash}}} and \isa{{\isaliteral{40}{\isacharat}}} have the same priority (namely the \isa{{\isadigit{6}}{\isadigit{5}}}
-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 \isa{{\isaliteral{40}{\isacharat}}}.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsubsection{Third Lemma%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Abandoning the previous attempt, the canonical proof procedure
-succeeds without further ado.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ app{\isaliteral{5F}{\isacharunderscore}}assoc\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}xs\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ zs\ {\isaliteral{3D}{\isacharequal}}\ xs\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{28}{\isacharparenleft}}ys\ {\isaliteral{40}{\isacharat}}\ zs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ xs{\isaliteral{29}{\isacharparenright}}\isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent
-Now we can prove the first lemma:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ rev{\isaliteral{5F}{\isacharunderscore}}app\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}rev{\isaliteral{28}{\isacharparenleft}}xs\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}rev\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ xs{\isaliteral{29}{\isacharparenright}}\isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent
-Finally, we prove our main theorem:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{theorem}\isamarkupfalse%
-\ rev{\isaliteral{5F}{\isacharunderscore}}rev\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}rev{\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ xs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}induct{\isaliteral{5F}{\isacharunderscore}}tac\ xs{\isaliteral{29}{\isacharparenright}}\isanewline
-\isacommand{apply}\isamarkupfalse%
-{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent
-The final \commdx{end} tells Isabelle to close the current theory because
-we are finished with its development:%
-\index{*rev (constant)|)}\index{append function|)}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End: