--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/IsaMakefile Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,36 @@
+## targets
+
+default: HOL-ProgProve
+images:
+test: HOL-ProgProve
+
+all: images test
+
+
+## global settings
+
+SRC = $(ISABELLE_HOME)/src
+OUT = $(ISABELLE_OUTPUT)
+LOG = $(OUT)/log
+
+USEDIR = $(ISABELLE_TOOL) usedir -v true -i false -d false -C false -D document
+
+
+## sessions
+
+HOL-ProgProve: $(LOG)/HOL-ProgProve.gz
+
+$(LOG)/HOL-ProgProve.gz: Thys/*.thy Thys/ROOT.ML
+ @$(USEDIR) -s ProgProve HOL Thys
+ @rm -f Thys/document/MyList.tex
+ @rm -f Thys/document/isabelle.sty
+ @rm -f Thys/document/isabellesym.sty
+ @rm -f Thys/document/pdfsetup.sty
+ @rm -f Thys/document/railsetup.sty
+ @rm -f Thys/document/session.tex
+
+
+## clean
+
+clean:
+ @rm -f $(LOG)/HOL-ProgProve.gz
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Makefile Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,32 @@
+## targets
+
+default: dvi
+
+
+## dependencies
+
+include ../Makefile.in
+
+NAME = prog-prove
+
+FILES = prog-prove.tex prog-prove.bib Thys/document/*.tex prelude.tex \
+ svmono.cls mathpartir.sty isabelle.sty isabellesym.sty
+
+dvi: $(NAME).dvi
+
+$(NAME).dvi: $(FILES) isabelle_hol.eps bang.eps
+ $(LATEX) $(NAME)
+ $(BIBTEX) $(NAME)
+ $(LATEX) $(NAME)
+ $(LATEX) $(NAME)
+ $(LATEX) $(NAME)
+
+pdf: $(NAME).pdf
+
+$(NAME).pdf: $(FILES) isabelle_hol.pdf bang.pdf
+ $(PDFLATEX) $(NAME)
+ $(BIBTEX) $(NAME)
+ $(PDFLATEX) $(NAME)
+ $(PDFLATEX) $(NAME)
+ $(FIXBOOKMARKS) $(NAME).out
+ $(PDFLATEX) $(NAME)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/Basics.thy Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,146 @@
+(*<*)
+theory Basics
+imports Main
+begin
+(*>*)
+text{*
+This chapter introduces HOL as a functional programming language and shows
+how to prove properties of functional programs by induction.
+
+\section{Basics}
+
+\subsection{Types, Terms and Formulae}
+\label{sec:TypesTermsForms}
+
+HOL is a typed logic whose type system resembles that of functional
+programming languages. Thus there are
+\begin{description}
+\item[base types,]
+in particular @{typ bool}, the type of truth values,
+@{typ nat}, the type of natural numbers ($\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}.
+\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}
+*}
+(*<*)
+end
+(*>*)
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/Bool_nat_list.thy Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,404 @@
+(*<*)
+theory Bool_nat_list
+imports Main
+begin
+(*>*)
+
+text{*
+\vspace{-4ex}
+\section{\texorpdfstring{Types @{typ bool}, @{typ nat} and @{text list}}{Types bool, nat and list}}
+
+These are the most important predefined types. We go through them one by one.
+Based on examples we learn how to define (possibly recursive) functions and
+prove theorems about them by induction and simplification.
+
+\subsection{Type @{typ bool}}
+
+The type of boolean values is a predefined datatype
+@{datatype[display] bool}
+with the two values @{const True} and @{const False} and
+with many predefined functions: @{text "\<not>"}, @{text "\<and>"}, @{text "\<or>"}, @{text
+"\<longrightarrow>"} etc. Here is how conjunction could be defined by pattern matching:
+*}
+
+fun conj :: "bool \<Rightarrow> bool \<Rightarrow> bool" where
+"conj True True = True" |
+"conj _ _ = False"
+
+text{* Both the datatype and function definitions roughly follow the syntax
+of functional programming languages.
+
+\subsection{Type @{typ nat}}
+
+Natural numbers are another predefined datatype:
+@{datatype[display] nat}
+All values of type @{typ nat} are generated by the constructors
+@{text 0} and @{const Suc}. Thus the values of type @{typ nat} are
+@{text 0}, @{term"Suc 0"}, @{term"Suc(Suc 0)"} etc.
+There are many predefined functions: @{text "+"}, @{text "*"}, @{text
+"\<le>"}, etc. Here is how you could define your own addition:
+*}
+
+fun add :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
+"add 0 n = n" |
+"add (Suc m) n = Suc(add m n)"
+
+text{* And here is a proof of the fact that @{prop"add m 0 = m"}: *}
+
+lemma add_02: "add m 0 = m"
+apply(induction m)
+apply(auto)
+done
+(*<*)
+lemma "add m 0 = m"
+apply(induction m)
+(*>*)
+txt{* The \isacom{lemma} command starts the proof and gives the lemma
+a name, @{text add_02}. Properties of recursively defined functions
+need to be established by induction in most cases.
+Command \isacom{apply}@{text"(induction m)"} instructs Isabelle to
+start a proof by induction on @{text m}. In response, it will show the
+following proof state:
+@{subgoals[display,indent=0]}
+The numbered lines are known as \emph{subgoals}.
+The first subgoal is the base case, the second one the induction step.
+The prefix @{text"\<And>m."} is Isabelle's way of saying ``for an arbitrary but fixed @{text m}''. The @{text"\<Longrightarrow>"} separates assumptions from the conclusion.
+The command \isacom{apply}@{text"(auto)"} instructs Isabelle to try
+and prove all subgoals automatically, essentially by simplifying them.
+Because both subgoals are easy, Isabelle can do it.
+The base case @{prop"add 0 0 = 0"} holds by definition of @{const add},
+and the induction step is almost as simple:
+@{text"add\<^raw:~>(Suc m) 0 = Suc(add m 0) = Suc m"}
+using first the definition of @{const add} and then the induction hypothesis.
+In summary, both subproofs rely on simplification with function definitions and
+the induction hypothesis.
+As a result of that final \isacom{done}, Isabelle associates the lemma
+just proved with its name. You can now inspect the lemma with the command
+*}
+
+thm add_02
+
+txt{* which displays @{thm[show_question_marks,display] add_02} The free
+variable @{text m} has been replaced by the \concept{unknown}
+@{text"?m"}. There is no logical difference between the two but an
+operational one: unknowns can be instantiated, which is what you want after
+some lemma has been proved.
+
+Note that there is also a proof method @{text induct}, which behaves almost
+like @{text induction}; the difference is explained in \autoref{ch:Isar}.
+
+\begin{warn}
+Terminology: We use \concept{lemma}, \concept{theorem} and \concept{rule}
+interchangeably for propositions that have been proved.
+\end{warn}
+\begin{warn}
+ Numerals (@{text 0}, @{text 1}, @{text 2}, \dots) and most of the standard
+ arithmetic operations (@{text "+"}, @{text "-"}, @{text "*"}, @{text"\<le>"},
+ @{text"<"} etc) are overloaded: they are available
+ not just for natural numbers but for other types as well.
+ For example, given the goal @{text"x + 0 = x"}, there is nothing to indicate
+ that you are talking about natural numbers. Hence Isabelle can only infer
+ that @{term x} is of some arbitrary type where @{text 0} and @{text"+"}
+ exist. As a consequence, you will be unable to prove the
+ goal. To alert you to such pitfalls, Isabelle flags numerals without a
+ fixed type in its output: @{prop"x+0 = x"}. In this particular example,
+ you need to include
+ an explicit type constraint, for example @{text"x+0 = (x::nat)"}. If there
+ is enough contextual information this may not be necessary: @{prop"Suc x =
+ x"} automatically implies @{text"x::nat"} because @{term Suc} is not
+ overloaded.
+\end{warn}
+
+\subsubsection{An informal proof}
+
+Above we gave some terse informal explanation of the proof of
+@{prop"add m 0 = m"}. A more detailed informal exposition of the lemma
+might look like this:
+\bigskip
+
+\noindent
+\textbf{Lemma} @{prop"add m 0 = m"}
+
+\noindent
+\textbf{Proof} by induction on @{text m}.
+\begin{itemize}
+\item Case @{text 0} (the base case): @{prop"add 0 0 = 0"}
+ holds by definition of @{const add}.
+\item Case @{term"Suc m"} (the induction step):
+ We assume @{prop"add m 0 = m"}, the induction hypothesis (IH),
+ and we need to show @{text"add (Suc m) 0 = Suc m"}.
+ The proof is as follows:\smallskip
+
+ \begin{tabular}{@ {}rcl@ {\quad}l@ {}}
+ @{term "add (Suc m) 0"} &@{text"="}& @{term"Suc(add m 0)"}
+ & by definition of @{text add}\\
+ &@{text"="}& @{term "Suc m"} & by IH
+ \end{tabular}
+\end{itemize}
+Throughout this book, \concept{IH} will stand for ``induction hypothesis''.
+
+We have now seen three proofs of @{prop"add m 0 = 0"}: the Isabelle one, the
+terse 4 lines explaining the base case and the induction step, and just now a
+model of a traditional inductive proof. The three proofs differ in the level
+of detail given and the intended reader: the Isabelle proof is for the
+machine, the informal proofs are for humans. Although this book concentrates
+of Isabelle proofs, it is important to be able to rephrase those proofs
+as informal text comprehensible to a reader familiar with traditional
+mathematical proofs. Later on we will introduce an Isabelle proof language
+that is closer to traditional informal mathematical language and is often
+directly readable.
+
+\subsection{Type @{text list}}
+
+Although lists are already predefined, we define our own copy just for
+demonstration purposes:
+*}
+(*<*)
+apply(auto)
+done
+declare [[names_short]]
+(*>*)
+datatype 'a list = Nil | Cons "'a" "('a list)"
+
+text{* This means that all lists are built up from @{const Nil}, the empty
+list, and @{const Cons}, the operation of putting an element in front of a
+list. Hence all lists are of the form @{const Nil}, or @{term"Cons x Nil"},
+or @{term"Cons x (Cons y Nil)"} etc.
+
+We also define two standard functions, append and reverse: *}
+
+fun app :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+"app Nil ys = ys" |
+"app (Cons x xs) ys = Cons x (app xs ys)"
+
+fun rev :: "'a list \<Rightarrow> 'a list" where
+"rev Nil = Nil" |
+"rev (Cons x xs) = app (rev xs) (Cons x Nil)"
+
+text{* By default, variables @{text xs}, @{text ys} and @{text zs} are of
+@{text list} type.
+
+Command \isacom{value} evaluates a term. For example, *}
+
+value "rev(Cons True (Cons False Nil))"
+
+text{* yields the result @{value "rev(Cons True (Cons False Nil))"}. This works symbolically, too: *}
+
+value "rev(Cons a (Cons b Nil))"
+
+text{* yields @{value "rev(Cons a (Cons b Nil))"}.
+\medskip
+
+Figure~\ref{fig:MyList} shows the theory created so far. Notice where the
+quotations marks are needed that we mostly sweep under the carpet. In
+particular, notice that \isacom{datatype} requires no quotation marks on the
+left-hand side, but that on the right-hand side each of the argument
+types of a constructor needs to be enclosed in quotation marks.
+
+\begin{figure}[htbp]
+\begin{alltt}
+\input{Thys/MyList.thy}\end{alltt}
+\caption{A Theory of Lists}
+\label{fig:MyList}
+\end{figure}
+
+\subsubsection{Structural Induction for Lists}
+
+Just as for natural numbers, there is a proof principle of induction for
+lists. Induction over a list is essentially induction over the length of
+the list, although the length remains implicit. To prove that some property
+@{text P} holds for all lists @{text xs}, i.e.\ \mbox{@{prop"P(xs)"}},
+you need to prove
+\begin{enumerate}
+\item the base case @{prop"P(Nil)"} and
+\item the inductive case @{prop"P(Cons x xs)"} under the assumption @{prop"P(xs)"}, for some arbitrary but fixed @{text xs}.
+\end{enumerate}
+This is often called \concept{structural induction}.
+
+\subsection{The Proof Process}
+
+We will now demonstrate the typical proof process, which involves
+the formulation and proof of auxiliary lemmas.
+Our goal is to show that reversing a list twice produces the original
+list. *}
+
+theorem rev_rev [simp]: "rev(rev xs) = xs"
+
+txt{* Commands \isacom{theorem} and \isacom{lemma} are
+interchangeable and merely indicate the importance we attach to a
+proposition. Via the bracketed attribute @{text simp} we also tell Isabelle
+to make the eventual theorem a \concept{simplification rule}: future proofs
+involving simplification will replace occurrences of @{term"rev(rev xs)"} by
+@{term"xs"}. The proof is by induction: *}
+
+apply(induction xs)
+
+txt{*
+As explained above, we obtain two subgoals, namely the base case (@{const Nil}) and the induction step (@{const Cons}):
+@{subgoals[display,indent=0,margin=65]}
+Let us try to solve both goals automatically:
+*}
+
+apply(auto)
+
+txt{*Subgoal~1 is proved, and disappears; the simplified version
+of subgoal~2 becomes the new subgoal~1:
+@{subgoals[display,indent=0,margin=70]}
+In order to simplify this subgoal further, a lemma suggests itself.
+
+\subsubsection{A First Lemma}
+
+We insert the following lemma in front of the main theorem:
+*}
+(*<*)
+oops
+(*>*)
+lemma rev_app [simp]: "rev(app xs ys) = app (rev ys) (rev xs)"
+
+txt{* There are two variables that we could induct on: @{text xs} and
+@{text ys}. Because @{const app} is defined by recursion on
+the first argument, @{text xs} is the correct one:
+*}
+
+apply(induction xs)
+
+txt{* This time not even the base case is solved automatically: *}
+apply(auto)
+txt{*
+\vspace{-5ex}
+@{subgoals[display,goals_limit=1]}
+Again, we need to abandon this proof attempt and prove another simple lemma
+first.
+
+\subsubsection{A Second Lemma}
+
+We again try the canonical proof procedure:
+*}
+(*<*)
+oops
+(*>*)
+lemma app_Nil2 [simp]: "app xs Nil = xs"
+apply(induction xs)
+apply(auto)
+done
+
+text{*
+Thankfully, this worked.
+Now we can continue with our stuck proof attempt of the first lemma:
+*}
+
+lemma rev_app [simp]: "rev(app xs ys) = app (rev ys) (rev xs)"
+apply(induction xs)
+apply(auto)
+
+txt{*
+We find that this time @{text"auto"} solves the base case, but the
+induction step merely simplifies to
+@{subgoals[display,indent=0,goals_limit=1]}
+The the missing lemma is associativity of @{const app},
+which we insert in front of the failed lemma @{text rev_app}.
+
+\subsubsection{Associativity of @{const app}}
+
+The canonical proof procedure succeeds without further ado:
+*}
+(*<*)oops(*>*)
+lemma app_assoc [simp]: "app (app xs ys) zs = app xs (app ys zs)"
+apply(induction xs)
+apply(auto)
+done
+(*<*)
+lemma rev_app [simp]: "rev(app xs ys) = app (rev ys)(rev xs)"
+apply(induction xs)
+apply(auto)
+done
+
+theorem rev_rev [simp]: "rev(rev xs) = xs"
+apply(induction xs)
+apply(auto)
+done
+(*>*)
+text{*
+Finally the proofs of @{thm[source] rev_app} and @{thm[source] rev_rev}
+succeed, too.
+
+\subsubsection{Another informal proof}
+
+Here is the informal proof of associativity of @{const app}
+corresponding to the Isabelle proof above.
+\bigskip
+
+\noindent
+\textbf{Lemma} @{prop"app (app xs ys) zs = app xs (app ys zs)"}
+
+\noindent
+\textbf{Proof} by induction on @{text xs}.
+\begin{itemize}
+\item Case @{text Nil}: \ @{prop"app (app Nil ys) zs = app ys zs"} @{text"="}
+ \mbox{@{term"app Nil (app ys zs)"}} \ holds by definition of @{text app}.
+\item Case @{text"Cons x xs"}: We assume
+ \begin{center} \hfill @{term"app (app xs ys) zs"} @{text"="}
+ @{term"app xs (app ys zs)"} \hfill (IH) \end{center}
+ and we need to show
+ \begin{center} @{prop"app (app (Cons x xs) ys) zs = app (Cons x xs) (app ys zs)"}.\end{center}
+ The proof is as follows:\smallskip
+
+ \begin{tabular}{@ {}l@ {\quad}l@ {}}
+ @{term"app (app (Cons x xs) ys) zs"}\\
+ @{text"= app (Cons x (app xs ys)) zs"} & by definition of @{text app}\\
+ @{text"= Cons x (app (app xs ys) zs)"} & by definition of @{text app}\\
+ @{text"= Cons x (app xs (app ys zs))"} & by IH\\
+ @{text"= app (Cons x xs) (app ys zs)"} & by definition of @{text app}
+ \end{tabular}
+\end{itemize}
+\medskip
+
+\noindent Didn't we say earlier that all proofs are by simplification? But
+in both cases, going from left to right, the last equality step is not a
+simplification at all! In the base case it is @{prop"app ys zs = app Nil (app
+ys zs)"}. It appears almost mysterious because we suddenly complicate the
+term by appending @{text Nil} on the left. What is really going on is this:
+when proving some equality \mbox{@{prop"s = t"}}, both @{text s} and @{text t} are
+simplified to some common term @{text u}. This heuristic for equality proofs
+works well for a functional programming context like ours. In the base case
+@{text s} is @{term"app (app Nil ys) zs"}, @{text t} is @{term"app Nil (app
+ys zs)"}, and @{text u} is @{term"app ys zs"}.
+
+\subsection{Predefined lists}
+\label{sec:predeflists}
+
+Isabelle's predefined lists are the same as the ones above, but with
+more syntactic sugar:
+\begin{itemize}
+\item @{text "[]"} is @{const Nil},
+\item @{term"x # xs"} is @{term"Cons x xs"},
+\item @{text"[x\<^isub>1, \<dots>, x\<^isub>n]"} is @{text"x\<^isub>1 # \<dots> # x\<^isub>n # []"}, and
+\item @{term "xs @ ys"} is @{term"app xs ys"}.
+\end{itemize}
+There is also a large library of predefined functions.
+The most important ones are the length function
+@{text"length :: 'a list \<Rightarrow> nat"} (with the obvious definition),
+and the map function that applies a function to all elements of a list:
+\begin{isabelle}
+\isacom{fun} @{text "map :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list"}\\
+@{thm map.simps(1)} @{text"|"}\\
+@{thm map.simps(2)}
+\end{isabelle}
+Also useful are the \concept{head} of a list, its first element,
+and the \concept{tail}, the rest of the list:
+\begin{isabelle}
+\isacom{fun} @{text"hd :: 'a list \<Rightarrow> 'a"}\\
+@{prop"hd(x#xs) = x"}
+\end{isabelle}
+\begin{isabelle}
+\isacom{fun} @{text"tl :: 'a list \<Rightarrow> 'a list"}\\
+@{prop"tl [] = []"} @{text"|"}\\
+@{prop"tl(x#xs) = xs"}
+\end{isabelle}
+Note that since HOL is a logic of total functions, @{term"hd []"} is defined,
+but we do now know what the result is. That is, @{term"hd []"} is not undefined
+but underdefined.
+*}
+(*<*)
+end
+(*>*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/Isar.thy Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,995 @@
+(*<*)
+theory Isar
+imports LaTeXsugar
+begin
+ML{* quick_and_dirty := true *}
+(*>*)
+text{*
+Apply-scripts are unreadable and hard to maintain. The language of choice
+for larger proofs is \concept{Isar}. The two key features of Isar are:
+\begin{itemize}
+\item It is structured, not linear.
+\item It is readable without running it because
+you need to state what you are proving at any given point.
+\end{itemize}
+Whereas apply-scripts are like assembly language programs, Isar proofs
+are like structured programs with comments. A typical Isar proof looks like this:
+*}text{*
+\begin{tabular}{@ {}l}
+\isacom{proof}\\
+\quad\isacom{assume} @{text"\""}$\mathit{formula}_0$@{text"\""}\\
+\quad\isacom{have} @{text"\""}$\mathit{formula}_1$@{text"\""} \quad\isacom{by} @{text simp}\\
+\quad\vdots\\
+\quad\isacom{have} @{text"\""}$\mathit{formula}_n$@{text"\""} \quad\isacom{by} @{text blast}\\
+\quad\isacom{show} @{text"\""}$\mathit{formula}_{n+1}$@{text"\""} \quad\isacom{by} @{text \<dots>}\\
+\isacom{qed}
+\end{tabular}
+*}text{*
+It proves $\mathit{formula}_0 \Longrightarrow \mathit{formula}_{n+1}$
+(provided provided each proof step succeeds).
+The intermediate \isacom{have} statements are merely stepping stones
+on the way towards the \isacom{show} statement that proves the actual
+goal. In more detail, this is the Isar core syntax:
+\medskip
+
+\begin{tabular}{@ {}lcl@ {}}
+\textit{proof} &=& \isacom{by} \textit{method}\\
+ &$\mid$& \isacom{proof} [\textit{method}] \ \textit{step}$^*$ \ \isacom{qed}
+\end{tabular}
+\medskip
+
+\begin{tabular}{@ {}lcl@ {}}
+\textit{step} &=& \isacom{fix} \textit{variables} \\
+ &$\mid$& \isacom{assume} \textit{proposition} \\
+ &$\mid$& [\isacom{from} \textit{fact}$^+$] (\isacom{have} $\mid$ \isacom{show}) \ \textit{proposition} \ \textit{proof}
+\end{tabular}
+\medskip
+
+\begin{tabular}{@ {}lcl@ {}}
+\textit{proposition} &=& [\textit{name}:] @{text"\""}\textit{formula}@{text"\""}
+\end{tabular}
+\medskip
+
+\begin{tabular}{@ {}lcl@ {}}
+\textit{fact} &=& \textit{name} \ $\mid$ \ \dots
+\end{tabular}
+\medskip
+
+\noindent A proof can either be an atomic \isacom{by} with a single proof
+method which must finish off the statement being proved, for example @{text
+auto}. Or it can be a \isacom{proof}--\isacom{qed} block of multiple
+steps. Such a block can optionally begin with a proof method that indicates
+how to start off the proof, e.g.\ \mbox{@{text"(induction xs)"}}.
+
+A step either assumes a proposition or states a proposition
+together with its proof. The optional \isacom{from} clause
+indicates which facts are to be used in the proof.
+Intermediate propositions are stated with \isacom{have}, the overall goal
+with \isacom{show}. A step can also introduce new local variables with
+\isacom{fix}. Logically, \isacom{fix} introduces @{text"\<And>"}-quantified
+variables, \isacom{assume} introduces the assumption of an implication
+(@{text"\<Longrightarrow>"}) and \isacom{have}/\isacom{show} the conclusion.
+
+Propositions are optionally named formulas. These names can be referred to in
+later \isacom{from} clauses. In the simplest case, a fact is such a name.
+But facts can also be composed with @{text OF} and @{text of} as shown in
+\S\ref{sec:forward-proof}---hence the \dots\ in the above grammar. Note
+that assumptions, intermediate \isacom{have} statements and global lemmas all
+have the same status and are thus collectively referred to as
+\concept{facts}.
+
+Fact names can stand for whole lists of facts. For example, if @{text f} is
+defined by command \isacom{fun}, @{text"f.simps"} refers to the whole list of
+recursion equations defining @{text f}. Individual facts can be selected by
+writing @{text"f.simps(2)"}, whole sublists by @{text"f.simps(2-4)"}.
+
+
+\section{Isar by example}
+
+We show a number of proofs of Cantors theorem that a function from a set to
+its powerset cannot be surjective, illustrating various features of Isar. The
+constant @{const surj} is predefined.
+*}
+
+lemma "\<not> surj(f :: 'a \<Rightarrow> 'a set)"
+proof
+ assume 0: "surj f"
+ from 0 have 1: "\<forall>A. \<exists>a. A = f a" by(simp add: surj_def)
+ from 1 have 2: "\<exists>a. {x. x \<notin> f x} = f a" by blast
+ from 2 show "False" by blast
+qed
+
+text{*
+The \isacom{proof} command lacks an explicit method how to perform
+the proof. In such cases Isabelle tries to use some standard introduction
+rule, in the above case for @{text"\<not>"}:
+\[
+\inferrule{
+\mbox{@{thm (prem 1) notI}}}
+{\mbox{@{thm (concl) notI}}}
+\]
+In order to prove @{prop"~ P"}, assume @{text P} and show @{text False}.
+Thus we may assume @{prop"surj f"}. The proof shows that names of propositions
+may be (single!) digits---meaningful names are hard to invent and are often
+not necessary. Both \isacom{have} steps are obvious. The second one introduces
+the diagonal set @{term"{x. x \<notin> f x}"}, the key idea in the proof.
+If you wonder why @{text 2} directly implies @{text False}: from @{text 2}
+it follows that @{prop"a \<notin> f a \<longleftrightarrow> a \<in> f a"}.
+
+\subsection{@{text this}, @{text then}, @{text hence} and @{text thus}}
+
+Labels should be avoided. They interrupt the flow of the reader who has to
+scan the context for the point where the label was introduced. Ideally, the
+proof is a linear flow, where the output of one step becomes the input of the
+next step, piping the previously proved fact into the next proof, just like
+in a UNIX pipe. In such cases the predefined name @{text this} can be used
+to refer to the proposition proved in the previous step. This allows us to
+eliminate all labels from our proof (we suppress the \isacom{lemma} statement):
+*}
+(*<*)
+lemma "\<not> surj(f :: 'a \<Rightarrow> 'a set)"
+(*>*)
+proof
+ assume "surj f"
+ from this have "\<exists>a. {x. x \<notin> f x} = f a" by(auto simp: surj_def)
+ from this show "False" by blast
+qed
+
+text{* We have also taken the opportunity to compress the two \isacom{have}
+steps into one.
+
+To compact the text further, Isar has a few convenient abbreviations:
+\medskip
+
+\begin{tabular}{rcl}
+\isacom{then} &=& \isacom{from} @{text this}\\
+\isacom{thus} &=& \isacom{then} \isacom{show}\\
+\isacom{hence} &=& \isacom{then} \isacom{have}
+\end{tabular}
+\medskip
+
+\noindent
+With the help of these abbreviations the proof becomes
+*}
+(*<*)
+lemma "\<not> surj(f :: 'a \<Rightarrow> 'a set)"
+(*>*)
+proof
+ assume "surj f"
+ hence "\<exists>a. {x. x \<notin> f x} = f a" by(auto simp: surj_def)
+ thus "False" by blast
+qed
+text{*
+
+There are two further linguistic variations:
+\medskip
+
+\begin{tabular}{rcl}
+(\isacom{have}$\mid$\isacom{show}) \ \textit{prop} \ \isacom{using} \ \textit{facts}
+&=&
+\isacom{from} \ \textit{facts} \ (\isacom{have}$\mid$\isacom{show}) \ \textit{prop}\\
+\isacom{with} \ \textit{facts} &=& \isacom{from} \ \textit{facts} \isa{this}
+\end{tabular}
+\medskip
+
+\noindent The \isacom{using} idiom de-emphasises the used facts by moving them
+behind the proposition.
+
+\subsection{Structured lemma statements: \isacom{fixes}, \isacom{assumes}, \isacom{shows}}
+
+Lemmas can also be stated in a more structured fashion. To demonstrate this
+feature with Cantor's theorem, we rephrase \noquotes{@{prop[source]"\<not> surj f"}}
+a little:
+*}
+
+lemma
+ fixes f :: "'a \<Rightarrow> 'a set"
+ assumes s: "surj f"
+ shows "False"
+
+txt{* The optional \isacom{fixes} part allows you to state the types of
+variables up front rather than by decorating one of their occurrences in the
+formula with a type constraint. The key advantage of the structured format is
+the \isacom{assumes} part that allows you to name each assumption. The
+\isacom{shows} part gives the goal. The actual theorem that will come out of
+the proof is @{prop"surj f \<Longrightarrow> False"}, but during the proof the assumption
+@{prop"surj f"} is available under the name @{text s} like any other fact.
+*}
+
+proof -
+ have "\<exists> a. {x. x \<notin> f x} = f a" using s
+ by(auto simp: surj_def)
+ thus "False" by blast
+qed
+
+text{* In the \isacom{have} step the assumption @{prop"surj f"} is now
+referenced by its name @{text s}. The duplication of @{prop"surj f"} in the
+above proofs (once in the statement of the lemma, once in its proof) has been
+eliminated.
+
+\begin{warn}
+Note the dash after the \isacom{proof}
+command. It is the null method that does nothing to the goal. Leaving it out
+would ask Isabelle to try some suitable introduction rule on the goal @{const
+False}---but there is no suitable introduction rule and \isacom{proof}
+would fail.
+\end{warn}
+
+Stating a lemmas with \isacom{assumes}-\isacom{shows} implicitly introduces the
+name @{text assms} that stands for the list of all assumptions. You can refer
+to individual assumptions by @{text"assms(1)"}, @{text"assms(2)"} etc,
+thus obviating the need to name them individually.
+
+\section{Proof patterns}
+
+We show a number of important basic proof patterns. Many of them arise from
+the rules of natural deduction that are applied by \isacom{proof} by
+default. The patterns are phrased in terms of \isacom{show} but work for
+\isacom{have} and \isacom{lemma}, too.
+
+We start with two forms of \concept{case distinction}:
+starting from a formula @{text P} we have the two cases @{text P} and
+@{prop"~P"}, and starting from a fact @{prop"P \<or> Q"}
+we have the two cases @{text P} and @{text Q}:
+*}text_raw{*
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "R" proof-(*>*)
+show "R"
+proof cases
+ assume "P"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "R" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+next
+ assume "\<not> P"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "R" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+qed(*<*)oops(*>*)
+text_raw {* }
+\end{minipage}
+&
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "R" proof-(*>*)
+have "P \<or> Q" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+then show "R"
+proof
+ assume "P"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "R" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+next
+ assume "Q"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "R" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+qed(*<*)oops(*>*)
+
+text_raw {* }
+\end{minipage}
+\end{tabular}
+\medskip
+\begin{isamarkuptext}%
+How to prove a logical equivalence:
+\end{isamarkuptext}%
+\isa{%
+*}
+(*<*)lemma "P\<longleftrightarrow>Q" proof-(*>*)
+show "P \<longleftrightarrow> Q"
+proof
+ assume "P"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "Q" (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
+next
+ assume "Q"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "P" (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
+qed(*<*)qed(*>*)
+text_raw {* }
+\medskip
+\begin{isamarkuptext}%
+Proofs by contradiction:
+\end{isamarkuptext}%
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "\<not> P" proof-(*>*)
+show "\<not> P"
+proof
+ assume "P"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "False" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+qed(*<*)oops(*>*)
+
+text_raw {* }
+\end{minipage}
+&
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "P" proof-(*>*)
+show "P"
+proof (rule ccontr)
+ assume "\<not>P"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "False" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+qed(*<*)oops(*>*)
+
+text_raw {* }
+\end{minipage}
+\end{tabular}
+\medskip
+\begin{isamarkuptext}%
+The name @{thm[source] ccontr} stands for ``classical contradiction''.
+
+How to prove quantified formulas:
+\end{isamarkuptext}%
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "ALL x. P x" proof-(*>*)
+show "\<forall>x. P(x)"
+proof
+ fix x
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "P(x)" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+qed(*<*)oops(*>*)
+
+text_raw {* }
+\end{minipage}
+&
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "EX x. P(x)" proof-(*>*)
+show "\<exists>x. P(x)"
+proof
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "P(witness)" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+qed
+(*<*)oops(*>*)
+
+text_raw {* }
+\end{minipage}
+\end{tabular}
+\medskip
+\begin{isamarkuptext}%
+In the proof of \noquotes{@{prop[source]"\<forall>x. P(x)"}},
+the step \isacom{fix}~@{text x} introduces a locale fixed variable @{text x}
+into the subproof, the proverbial ``arbitrary but fixed value''.
+Instead of @{text x} we could have chosen any name in the subproof.
+In the proof of \noquotes{@{prop[source]"\<exists>x. P(x)"}},
+@{text witness} is some arbitrary
+term for which we can prove that it satisfies @{text P}.
+
+How to reason forward from \noquotes{@{prop[source] "\<exists>x. P(x)"}}:
+\end{isamarkuptext}%
+*}
+(*<*)lemma True proof- assume 1: "EX x. P x"(*>*)
+have "\<exists>x. P(x)" (*<*)by(rule 1)(*>*)txt_raw{*\ $\dots$\\*}
+then obtain x where p: "P(x)" by blast
+(*<*)oops(*>*)
+text{*
+After the \isacom{obtain} step, @{text x} (we could have chosen any name)
+is a fixed local
+variable, and @{text p} is the name of the fact
+\noquotes{@{prop[source] "P(x)"}}.
+This pattern works for one or more @{text x}.
+As an example of the \isacom{obtain} command, here is the proof of
+Cantor's theorem in more detail:
+*}
+
+lemma "\<not> surj(f :: 'a \<Rightarrow> 'a set)"
+proof
+ assume "surj f"
+ hence "\<exists>a. {x. x \<notin> f x} = f a" by(auto simp: surj_def)
+ then obtain a where "{x. x \<notin> f x} = f a" by blast
+ hence "a \<notin> f a \<longleftrightarrow> a \<in> f a" by blast
+ thus "False" by blast
+qed
+
+text_raw{*
+\begin{isamarkuptext}%
+How to prove set equality and subset relationship:
+\end{isamarkuptext}%
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "A = (B::'a set)" proof-(*>*)
+show "A = B"
+proof
+ show "A \<subseteq> B" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+next
+ show "B \<subseteq> A" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+qed(*<*)qed(*>*)
+
+text_raw {* }
+\end{minipage}
+&
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "A <= (B::'a set)" proof-(*>*)
+show "A \<subseteq> B"
+proof
+ fix x
+ assume "x \<in> A"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "x \<in> B" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+qed(*<*)qed(*>*)
+
+text_raw {* }
+\end{minipage}
+\end{tabular}
+\begin{isamarkuptext}%
+\section{Streamlining proofs}
+
+\subsection{Pattern matching and quotations}
+
+In the proof patterns shown above, formulas are often duplicated.
+This can make the text harder to read, write and maintain. Pattern matching
+is an abbreviation mechanism to avoid such duplication. Writing
+\begin{quote}
+\isacom{show} \ \textit{formula} @{text"("}\isacom{is} \textit{pattern}@{text")"}
+\end{quote}
+matches the pattern against the formula, thus instantiating the unknowns in
+the pattern for later use. As an example, consider the proof pattern for
+@{text"\<longleftrightarrow>"}:
+\end{isamarkuptext}%
+*}
+(*<*)lemma "formula\<^isub>1 \<longleftrightarrow> formula\<^isub>2" proof-(*>*)
+show "formula\<^isub>1 \<longleftrightarrow> formula\<^isub>2" (is "?L \<longleftrightarrow> ?R")
+proof
+ assume "?L"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "?R" (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
+next
+ assume "?R"
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show "?L" (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
+qed(*<*)qed(*>*)
+
+text{* Instead of duplicating @{text"formula\<^isub>i"} in the text, we introduce
+the two abbreviations @{text"?L"} and @{text"?R"} by pattern matching.
+Pattern matching works wherever a formula is stated, in particular
+with \isacom{have} and \isacom{lemma}.
+
+The unknown @{text"?thesis"} is implicitly matched against any goal stated by
+\isacom{lemma} or \isacom{show}. Here is a typical example: *}
+
+lemma "formula"
+proof -
+ txt_raw{*\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}*}
+ show ?thesis (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
+qed
+
+text{*
+Unknowns can also be instantiated with \isacom{let} commands
+\begin{quote}
+\isacom{let} @{text"?t"} = @{text"\""}\textit{some-big-term}@{text"\""}
+\end{quote}
+Later proof steps can refer to @{text"?t"}:
+\begin{quote}
+\isacom{have} @{text"\""}\dots @{text"?t"} \dots@{text"\""}
+\end{quote}
+\begin{warn}
+Names of facts are introduced with @{text"name:"} and refer to proved
+theorems. Unknowns @{text"?X"} refer to terms or formulas.
+\end{warn}
+
+Although abbreviations shorten the text, the reader needs to remember what
+they stand for. Similarly for names of facts. Names like @{text 1}, @{text 2}
+and @{text 3} are not helpful and should only be used in short proofs. For
+longer proof, descriptive names are better. But look at this example:
+\begin{quote}
+\isacom{have} \ @{text"x_gr_0: \"x > 0\""}\\
+$\vdots$\\
+\isacom{from} @{text "x_gr_0"} \dots
+\end{quote}
+The name is longer than the fact it stands for! Short facts do not need names,
+one can refer to them easily by quoting them:
+\begin{quote}
+\isacom{have} \ @{text"\"x > 0\""}\\
+$\vdots$\\
+\isacom{from} @{text "`x>0`"} \dots
+\end{quote}
+Note that the quotes around @{text"x>0"} are \concept{back quotes}.
+They refer to the fact not by name but by value.
+
+\subsection{\isacom{moreover}}
+
+Sometimes one needs a number of facts to enable some deduction. Of course
+one can name these facts individually, as shown on the right,
+but one can also combine them with \isacom{moreover}, as shown on the left:
+*}text_raw{*
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "P" proof-(*>*)
+have "P\<^isub>1" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+moreover have "P\<^isub>2" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+moreover
+txt_raw{*\\$\vdots$\\\hspace{-1.4ex}*}(*<*)have "True" ..(*>*)
+moreover have "P\<^isub>n" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+ultimately have "P" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+(*<*)oops(*>*)
+
+text_raw {* }
+\end{minipage}
+&
+\qquad
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "P" proof-(*>*)
+have lab\<^isub>1: "P\<^isub>1" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+have lab\<^isub>2: "P\<^isub>2" (*<*)sorry(*>*)txt_raw{*\ $\dots$*}
+txt_raw{*\\$\vdots$\\\hspace{-1.4ex}*}
+have lab\<^isub>n: "P\<^isub>n" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+from lab\<^isub>1 lab\<^isub>2 txt_raw{*\ $\dots$\\*}
+have "P" (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+(*<*)oops(*>*)
+
+text_raw {* }
+\end{minipage}
+\end{tabular}
+\begin{isamarkuptext}%
+The \isacom{moreover} version is no shorter but expresses the structure more
+clearly and avoids new names.
+
+\subsection{Raw proof blocks}
+
+Sometimes one would like to prove some lemma locally with in a proof.
+A lemma that shares the current context of assumptions but that
+has its own assumptions and is generalised over its locally fixed
+variables at the end. This is what a \concept{raw proof block} does:
+\begin{quote}
+@{text"{"} \isacom{fix} @{text"x\<^isub>1 \<dots> x\<^isub>n"}\\
+\mbox{}\ \ \ \isacom{assume} @{text"A\<^isub>1 \<dots> A\<^isub>m"}\\
+\mbox{}\ \ \ $\vdots$\\
+\mbox{}\ \ \ \isacom{have} @{text"B"}\\
+@{text"}"}
+\end{quote}
+proves @{text"\<lbrakk> A\<^isub>1; \<dots> ; A\<^isub>m \<rbrakk> \<Longrightarrow> B"}
+where all @{text"x\<^isub>i"} have been replaced by unknowns @{text"?x\<^isub>i"}.
+\begin{warn}
+The conclusion of a raw proof block is \emph{not} indicated by \isacom{show}
+but is simply the final \isacom{have}.
+\end{warn}
+
+As an example we prove a simple fact about divisibility on integers.
+The definition of @{text "dvd"} is @{thm dvd_def}.
+\end{isamarkuptext}%
+*}
+
+lemma fixes a b :: int assumes "b dvd (a+b)" shows "b dvd a"
+proof -
+ { fix k assume k: "a+b = b*k"
+ have "\<exists>k'. a = b*k'"
+ proof
+ show "a = b*(k - 1)" using k by(simp add: algebra_simps)
+ qed }
+ then show ?thesis using assms by(auto simp add: dvd_def)
+qed
+
+text{* Note that the result of a raw proof block has no name. In this example
+it was directly piped (via \isacom{then}) into the final proof, but it can
+also be named for later reference: you simply follow the block directly by a
+\isacom{note} command:
+\begin{quote}
+\isacom{note} \ @{text"name = this"}
+\end{quote}
+This introduces a new name @{text name} that refers to @{text this},
+the fact just proved, in this case the preceding block. In general,
+\isacom{note} introduces a new name for one or more facts.
+
+\section{Case distinction and induction}
+
+\subsection{Datatype case distinction}
+
+We have seen case distinction on formulas. Now we want to distinguish
+which form some term takes: is it @{text 0} or of the form @{term"Suc n"},
+is it @{term"[]"} or of the form @{term"x#xs"}, etc. Here is a typical example
+proof by case distinction on the form of @{text xs}:
+*}
+
+lemma "length(tl xs) = length xs - 1"
+proof (cases xs)
+ assume "xs = []"
+ thus ?thesis by simp
+next
+ fix y ys assume "xs = y#ys"
+ thus ?thesis by simp
+qed
+
+text{* Function @{text tl} (''tail'') is defined by @{thm tl.simps(1)} and
+@{thm tl.simps(2)}. Note that the result type of @{const length} is @{typ nat}
+and @{prop"0 - 1 = (0::nat)"}.
+
+This proof pattern works for any term @{text t} whose type is a datatype.
+The goal has to be proved for each constructor @{text C}:
+\begin{quote}
+\isacom{fix} \ @{text"x\<^isub>1 \<dots> x\<^isub>n"} \isacom{assume} @{text"\"t = C x\<^isub>1 \<dots> x\<^isub>n\""}
+\end{quote}
+Each case can be written in a more compact form by means of the \isacom{case}
+command:
+\begin{quote}
+\isacom{case} @{text "(C x\<^isub>1 \<dots> x\<^isub>n)"}
+\end{quote}
+This is equivalent to the explicit \isacom{fix}-\isacom{assumen} line
+but also gives the assumption @{text"\"t = C x\<^isub>1 \<dots> x\<^isub>n\""} a name: @{text C},
+like the constructor.
+Here is the \isacom{case} version of the proof above:
+*}
+(*<*)lemma "length(tl xs) = length xs - 1"(*>*)
+proof (cases xs)
+ case Nil
+ thus ?thesis by simp
+next
+ case (Cons y ys)
+ thus ?thesis by simp
+qed
+
+text{* Remember that @{text Nil} and @{text Cons} are the alphanumeric names
+for @{text"[]"} and @{text"#"}. The names of the assumptions
+are not used because they are directly piped (via \isacom{thus})
+into the proof of the claim.
+
+\subsection{Structural induction}
+
+We illustrate structural induction with an example based on natural numbers:
+the sum (@{text"\<Sum>"}) of the first @{text n} natural numbers
+(@{text"{0..n::nat}"}) is equal to \mbox{@{term"n*(n+1) div 2::nat"}}.
+Never mind the details, just focus on the pattern:
+*}
+
+lemma "\<Sum>{0..n::nat} = n*(n+1) div 2" (is "?P n")
+proof (induction n)
+ show "\<Sum>{0..0::nat} = 0*(0+1) div 2" by simp
+next
+ fix n assume "\<Sum>{0..n::nat} = n*(n+1) div 2"
+ thus "\<Sum>{0..Suc n::nat} = Suc n*(Suc n+1) div 2" by simp
+qed
+
+text{* Except for the rewrite steps, everything is explicitly given. This
+makes the proof easily readable, but the duplication means it is tedious to
+write and maintain. Here is how pattern
+matching can completely avoid any duplication: *}
+
+lemma "\<Sum>{0..n::nat} = n*(n+1) div 2" (is "?P n")
+proof (induction n)
+ show "?P 0" by simp
+next
+ fix n assume "?P n"
+ thus "?P(Suc n)" by simp
+qed
+
+text{* The first line introduces an abbreviation @{text"?P n"} for the goal.
+Pattern matching @{text"?P n"} with the goal instantiates @{text"?P"} to the
+function @{term"\<lambda>n. \<Sum>{0..n::nat} = n*(n+1) div 2"}. Now the proposition to
+be proved in the base case can be written as @{text"?P 0"}, the induction
+hypothesis as @{text"?P n"}, and the conclusion of the induction step as
+@{text"?P(Suc n)"}.
+
+Induction also provides the \isacom{case} idiom that abbreviates
+the \isacom{fix}-\isacom{assume} step. The above proof becomes
+*}
+(*<*)lemma "\<Sum>{0..n::nat} = n*(n+1) div 2"(*>*)
+proof (induction n)
+ case 0
+ show ?case by simp
+next
+ case (Suc n)
+ thus ?case by simp
+qed
+
+text{*
+The unknown @{text "?case"} is set in each case to the required
+claim, i.e.\ @{text"?P 0"} and \mbox{@{text"?P(Suc n)"}} in the above proof,
+without requiring the user to define a @{text "?P"}. The general
+pattern for induction over @{typ nat} is shown on the left-hand side:
+*}text_raw{*
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+*}
+(*<*)lemma "P(n::nat)" proof -(*>*)
+show "P(n)"
+proof (induction n)
+ case 0
+ txt_raw{*\\\mbox{}\ \ $\vdots$\\\mbox{}\hspace{-1ex}*}
+ show ?case (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
+next
+ case (Suc n)
+ txt_raw{*\\\mbox{}\ \ $\vdots$\\\mbox{}\hspace{-1ex}*}
+ show ?case (*<*)sorry(*>*) txt_raw{*\ $\dots$\\*}
+qed(*<*)qed(*>*)
+
+text_raw {* }
+\end{minipage}
+&
+\begin{minipage}[t]{.4\textwidth}
+~\\
+~\\
+\isacom{let} @{text"?case = \"P(0)\""}\\
+~\\
+~\\
+~\\[1ex]
+\isacom{fix} @{text n} \isacom{assume} @{text"Suc: \"P(n)\""}\\
+\isacom{let} @{text"?case = \"P(Suc n)\""}\\
+\end{minipage}
+\end{tabular}
+\medskip
+*}
+text{*
+On the right side you can see what the \isacom{case} command
+on the left stands for.
+
+In case the goal is an implication, induction does one more thing: the
+proposition to be proved in each case is not the whole implication but only
+its conclusion; the premises of the implication are immediately made
+assumptions of that case. That is, if in the above proof we replace
+\isacom{show}~@{text"P(n)"} by
+\mbox{\isacom{show}~@{text"A(n) \<Longrightarrow> P(n)"}}
+then \isacom{case}~@{text 0} stands for
+\begin{quote}
+\isacom{assume} \ @{text"0: \"A(0)\""}\\
+\isacom{let} @{text"?case = \"P(0)\""}
+\end{quote}
+and \isacom{case}~@{text"(Suc n)"} stands for
+\begin{quote}
+\isacom{fix} @{text n}\\
+\isacom{assume} @{text"Suc:"}
+ \begin{tabular}[t]{l}@{text"A(n) \<Longrightarrow> P(n)"}\\@{text"A(Suc n)"}\end{tabular}\\
+\isacom{let} @{text"?case = \"P(Suc n)\""}
+\end{quote}
+The list of assumptions @{text Suc} is actually subdivided
+into @{text"Suc.IH"}, the induction hypotheses (here @{text"A(n) \<Longrightarrow> P(n)"})
+and @{text"Suc.prems"}, the premises of the goal being proved
+(here @{text"A(Suc n)"}).
+
+Induction works for any datatype.
+Proving a goal @{text"\<lbrakk> A\<^isub>1(x); \<dots>; A\<^isub>k(x) \<rbrakk> \<Longrightarrow> P(x)"}
+by induction on @{text x} generates a proof obligation for each constructor
+@{text C} of the datatype. The command @{text"case (C x\<^isub>1 \<dots> x\<^isub>n)"}
+performs the following steps:
+\begin{enumerate}
+\item \isacom{fix} @{text"x\<^isub>1 \<dots> x\<^isub>n"}
+\item \isacom{assume} the induction hypotheses (calling them @{text C.IH})
+ and the premises \mbox{@{text"A\<^isub>i(C x\<^isub>1 \<dots> x\<^isub>n)"}} (calling them @{text"C.prems"})
+ and calling the whole list @{text C}
+\item \isacom{let} @{text"?case = \"P(C x\<^isub>1 \<dots> x\<^isub>n)\""}
+\end{enumerate}
+
+\subsection{Rule induction}
+
+Recall the inductive and recursive definitions of even numbers in
+\autoref{sec:inductive-defs}:
+*}
+
+inductive ev :: "nat \<Rightarrow> bool" where
+ev0: "ev 0" |
+evSS: "ev n \<Longrightarrow> ev(Suc(Suc n))"
+
+fun even :: "nat \<Rightarrow> bool" where
+"even 0 = True" |
+"even (Suc 0) = False" |
+"even (Suc(Suc n)) = even n"
+
+text{* We recast the proof of @{prop"ev n \<Longrightarrow> even n"} in Isar. The
+left column shows the actual proof text, the right column shows
+the implicit effect of the two \isacom{case} commands:*}text_raw{*
+\begin{tabular}{@ {}l@ {\qquad}l@ {}}
+\begin{minipage}[t]{.5\textwidth}
+\isa{%
+*}
+
+lemma "ev n \<Longrightarrow> even n"
+proof(induction rule: ev.induct)
+ case ev0
+ show ?case by simp
+next
+ case evSS
+
+
+
+ thus ?case by simp
+qed
+
+text_raw {* }
+\end{minipage}
+&
+\begin{minipage}[t]{.5\textwidth}
+~\\
+~\\
+\isacom{let} @{text"?case = \"even 0\""}\\
+~\\
+~\\
+\isacom{fix} @{text n}\\
+\isacom{assume} @{text"evSS:"}
+ \begin{tabular}[t]{l} @{text"ev n"}\\@{text"even n"}\end{tabular}\\
+\isacom{let} @{text"?case = even(Suc(Suc n))"}\\
+\end{minipage}
+\end{tabular}
+\medskip
+*}
+text{*
+The proof resembles structural induction, but the induction rule is given
+explicitly and the names of the cases are the names of the rules in the
+inductive definition.
+Let us examine the two assumptions named @{thm[source]evSS}:
+@{prop "ev n"} is the premise of rule @{thm[source]evSS}, which we may assume
+because we are in the case where that rule was used; @{prop"even n"}
+is the induction hypothesis.
+\begin{warn}
+Because each \isacom{case} command introduces a list of assumptions
+named like the case name, which is the name of a rule of the inductive
+definition, those rules now need to be accessed with a qualified name, here
+@{thm[source] ev.ev0} and @{thm[source] ev.evSS}
+\end{warn}
+
+In the case @{thm[source]evSS} of the proof above we have pretended that the
+system fixes a variable @{text n}. But unless the user provides the name
+@{text n}, the system will just invent its own name that cannot be referred
+to. In the above proof, we do not need to refer to it, hence we do not give
+it a specific name. In case one needs to refer to it one writes
+\begin{quote}
+\isacom{case} @{text"(evSS m)"}
+\end{quote}
+just like \isacom{case}~@{text"(Suc n)"} in earlier structural inductions.
+The name @{text m} is an arbitrary choice. As a result,
+case @{thm[source] evSS} is derived from a renamed version of
+rule @{thm[source] evSS}: @{text"ev m \<Longrightarrow> ev(Suc(Suc m))"}.
+Here is an example with a (contrived) intermediate step that refers to @{text m}:
+*}
+
+lemma "ev n \<Longrightarrow> even n"
+proof(induction rule: ev.induct)
+ case ev0 show ?case by simp
+next
+ case (evSS m)
+ have "even(Suc(Suc m)) = even m" by simp
+ thus ?case using `even m` by blast
+qed
+
+text{*
+\indent
+In general, let @{text I} be a (for simplicity unary) inductively defined
+predicate and let the rules in the definition of @{text I}
+be called @{text "rule\<^isub>1"}, \dots, @{text "rule\<^isub>n"}. A proof by rule
+induction follows this pattern:
+*}
+
+(*<*)
+inductive I where rule\<^isub>1: "I()" | rule\<^isub>2: "I()" | rule\<^isub>n: "I()"
+lemma "I x \<Longrightarrow> P x" proof-(*>*)
+show "I x \<Longrightarrow> P x"
+proof(induction rule: I.induct)
+ case rule\<^isub>1
+ txt_raw{*\\[-.4ex]\mbox{}\ \ $\vdots$\\[-.4ex]\mbox{}\hspace{-1ex}*}
+ show ?case (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+next
+ txt_raw{*\\[-.4ex]$\vdots$\\[-.4ex]\mbox{}\hspace{-1ex}*}
+(*<*)
+ case rule\<^isub>2
+ show ?case sorry
+(*>*)
+next
+ case rule\<^isub>n
+ txt_raw{*\\[-.4ex]\mbox{}\ \ $\vdots$\\[-.4ex]\mbox{}\hspace{-1ex}*}
+ show ?case (*<*)sorry(*>*)txt_raw{*\ $\dots$\\*}
+qed(*<*)qed(*>*)
+
+text{*
+One can provide explicit variable names by writing
+\isacom{case}~@{text"(rule\<^isub>i x\<^isub>1 \<dots> x\<^isub>k)"}, thus renaming the first @{text k}
+free variables in rule @{text i} to @{text"x\<^isub>1 \<dots> x\<^isub>k"},
+going through rule @{text i} from left to right.
+
+\subsection{Assumption naming}
+
+In any induction, \isacom{case}~@{text name} sets up a list of assumptions
+also called @{text name}, which is subdivided into three parts:
+\begin{description}
+\item[@{text name.IH}] contains the induction hypotheses.
+\item[@{text name.hyps}] contains all the other hypotheses of this case in the
+induction rule. For rule inductions these are the hypotheses of rule
+@{text name}, for structural inductions these are empty.
+\item[@{text name.prems}] contains the (suitably instantiated) premises
+of the statement being proved, i.e. the @{text A\<^isub>i} when
+proving @{text"\<lbrakk> A\<^isub>1; \<dots>; A\<^isub>n \<rbrakk> \<Longrightarrow> A"}.
+\end{description}
+\begin{warn}
+Proof method @{text induct} differs from @{text induction}
+only in this naming policy: @{text induct} does not distinguish
+@{text IH} from @{text hyps} but subsumes @{text IH} under @{text hyps}.
+\end{warn}
+
+More complicated inductive proofs than the ones we have seen so far
+often need to refer to specific assumptions---just @{text name} or even
+@{text name.prems} and @{text name.IH} can be too unspecific.
+This is where the indexing of fact lists comes in handy, e.g.\
+@{text"name.IH(2)"} or @{text"name.prems(1-2)"}.
+
+\subsection{Rule inversion}
+
+Rule inversion is case distinction on which rule could have been used to
+derive some fact. The name \concept{rule inversion} emphasizes that we are
+reasoning backwards: by which rules could some given fact have been proved?
+For the inductive definition of @{const ev}, rule inversion can be summarized
+like this:
+@{prop[display]"ev n \<Longrightarrow> n = 0 \<or> (EX k. n = Suc(Suc k) \<and> ev k)"}
+The realisation in Isabelle is a case distinction.
+A simple example is the proof that @{prop"ev n \<Longrightarrow> ev (n - 2)"}. We
+already went through the details informally in \autoref{sec:Logic:even}. This
+is the Isar proof:
+*}
+(*<*)
+notepad
+begin fix n
+(*>*)
+ assume "ev n"
+ from this have "ev(n - 2)"
+ proof cases
+ case ev0 thus "ev(n - 2)" by (simp add: ev.ev0)
+ next
+ case (evSS k) thus "ev(n - 2)" by (simp add: ev.evSS)
+ qed
+(*<*)
+end
+(*>*)
+
+text{* The key point here is that a case distinction over some inductively
+defined predicate is triggered by piping the given fact
+(here: \isacom{from}~@{text this}) into a proof by @{text cases}.
+Let us examine the assumptions available in each case. In case @{text ev0}
+we have @{text"n = 0"} and in case @{text evSS} we have @{prop"n = Suc(Suc k)"}
+and @{prop"ev k"}. In each case the assumptions are available under the name
+of the case; there is no fine grained naming schema like for induction.
+
+Sometimes some rules could not have beed used to derive the given fact
+because constructors clash. As an extreme example consider
+rule inversion applied to @{prop"ev(Suc 0)"}: neither rule @{text ev0} nor
+rule @{text evSS} can yield @{prop"ev(Suc 0)"} because @{text"Suc 0"} unifies
+neither with @{text 0} nor with @{term"Suc(Suc n)"}. Impossible cases do not
+have to be proved. Hence we can prove anything from @{prop"ev(Suc 0)"}:
+*}
+(*<*)
+notepad begin fix P
+(*>*)
+ assume "ev(Suc 0)" then have P by cases
+(*<*)
+end
+(*>*)
+
+text{* That is, @{prop"ev(Suc 0)"} is simply not provable: *}
+
+lemma "\<not> ev(Suc 0)"
+proof
+ assume "ev(Suc 0)" then show False by cases
+qed
+
+text{* Normally not all cases will be impossible. As a simple exercise,
+prove that \mbox{@{prop"\<not> ev(Suc(Suc(Suc 0)))"}.}
+*}
+
+(*
+lemma "\<not> ev(Suc(Suc(Suc 0)))"
+proof
+ assume "ev(Suc(Suc(Suc 0)))"
+ then show False
+ proof cases
+ case evSS
+ from `ev(Suc 0)` show False by cases
+ qed
+qed
+*)
+
+(*<*)
+end
+(*>*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/LaTeXsugar.thy Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,47 @@
+(* Title: HOL/Library/LaTeXsugar.thy
+ Author: Gerwin Klain, Tobias Nipkow, Norbert Schirmer
+ Copyright 2005 NICTA and TUM
+*)
+
+(*<*)
+theory LaTeXsugar
+imports Main
+begin
+
+(* DUMMY *)
+consts DUMMY :: 'a ("\<^raw:\_>")
+
+(* THEOREMS *)
+notation (Rule output)
+ "==>" ("\<^raw:\mbox{}\inferrule{\mbox{>_\<^raw:}}>\<^raw:{\mbox{>_\<^raw:}}>")
+
+syntax (Rule output)
+ "_bigimpl" :: "asms \<Rightarrow> prop \<Rightarrow> prop"
+ ("\<^raw:\mbox{}\inferrule{>_\<^raw:}>\<^raw:{\mbox{>_\<^raw:}}>")
+
+ "_asms" :: "prop \<Rightarrow> asms \<Rightarrow> asms"
+ ("\<^raw:\mbox{>_\<^raw:}\\>/ _")
+
+ "_asm" :: "prop \<Rightarrow> asms" ("\<^raw:\mbox{>_\<^raw:}>")
+
+notation (Axiom output)
+ "Trueprop" ("\<^raw:\mbox{}\inferrule{\mbox{}}{\mbox{>_\<^raw:}}>")
+
+notation (IfThen output)
+ "==>" ("\<^raw:{\normalsize{}>If\<^raw:\,}> _/ \<^raw:{\normalsize \,>then\<^raw:\,}>/ _.")
+syntax (IfThen output)
+ "_bigimpl" :: "asms \<Rightarrow> prop \<Rightarrow> prop"
+ ("\<^raw:{\normalsize{}>If\<^raw:\,}> _ /\<^raw:{\normalsize \,>then\<^raw:\,}>/ _.")
+ "_asms" :: "prop \<Rightarrow> asms \<Rightarrow> asms" ("\<^raw:\mbox{>_\<^raw:}> /\<^raw:{\normalsize \,>and\<^raw:\,}>/ _")
+ "_asm" :: "prop \<Rightarrow> asms" ("\<^raw:\mbox{>_\<^raw:}>")
+
+notation (IfThenNoBox output)
+ "==>" ("\<^raw:{\normalsize{}>If\<^raw:\,}> _/ \<^raw:{\normalsize \,>then\<^raw:\,}>/ _.")
+syntax (IfThenNoBox output)
+ "_bigimpl" :: "asms \<Rightarrow> prop \<Rightarrow> prop"
+ ("\<^raw:{\normalsize{}>If\<^raw:\,}> _ /\<^raw:{\normalsize \,>then\<^raw:\,}>/ _.")
+ "_asms" :: "prop \<Rightarrow> asms \<Rightarrow> asms" ("_ /\<^raw:{\normalsize \,>and\<^raw:\,}>/ _")
+ "_asm" :: "prop \<Rightarrow> asms" ("_")
+
+end
+(*>*)
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/Logic.thy Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,707 @@
+(*<*)
+theory Logic
+imports LaTeXsugar
+begin
+(*>*)
+text{*
+\vspace{-5ex}
+\section{Logic and Proof Beyond Equality}
+\label{sec:Logic}
+
+\subsection{Formulas}
+
+The basic syntax of formulas (\textit{form} below)
+provides the standard logical constructs, in decreasing precedence:
+\[
+\begin{array}{rcl}
+
+\mathit{form} & ::= &
+ @{text"(form)"} ~\mid~
+ @{const True} ~\mid~
+ @{const False} ~\mid~
+ @{prop "term = term"}\\
+ &\mid& @{prop"\<not> form"} ~\mid~
+ @{prop "form \<and> form"} ~\mid~
+ @{prop "form \<or> form"} ~\mid~
+ @{prop "form \<longrightarrow> form"}\\
+ &\mid& @{prop"\<forall>x. form"} ~\mid~ @{prop"\<exists>x. form"}
+\end{array}
+\]
+Terms are the ones we have seen all along, built from constant, variables,
+function application and @{text"\<lambda>"}-abstraction, including all the syntactic
+sugar like infix symbols, @{text "if"}, @{text "case"} etc.
+\begin{warn}
+Remember that formulas are simply terms of type @{text bool}. Hence
+@{text "="} also works for formulas. Beware that @{text"="} has a higher
+precedence than the other logical operators. Hence @{prop"s = t \<and> A"} means
+@{text"(s = t) \<and> A"}, and @{prop"A\<and>B = B\<and>A"} means @{text"A \<and> (B = A) \<and> B"}.
+Logical equivalence can also be written with
+@{text "\<longleftrightarrow>"} instead of @{text"="}, where @{text"\<longleftrightarrow>"} has the same low
+precedence as @{text"\<longrightarrow>"}. Hence @{text"A \<and> B \<longleftrightarrow> B \<and> A"} really means
+@{text"(A \<and> B) \<longleftrightarrow> (B \<and> A)"}.
+\end{warn}
+\begin{warn}
+Quantifiers need to be enclosed in parentheses if they are nested within
+other constructs (just like @{text "if"}, @{text case} and @{text let}).
+\end{warn}
+The most frequent logical symbols have the following ASCII representations:
+\begin{center}
+\begin{tabular}{l@ {\qquad}l@ {\qquad}l}
+@{text "\<forall>"} & \xsymbol{forall} & \texttt{ALL}\\
+@{text "\<exists>"} & \xsymbol{exists} & \texttt{EX}\\
+@{text "\<lambda>"} & \xsymbol{lambda} & \texttt{\%}\\
+@{text "\<longrightarrow>"} & \texttt{-{}->}\\
+@{text "\<longleftrightarrow>"} & \texttt{<-{}->}\\
+@{text "\<and>"} & \texttt{/\char`\\} & \texttt{\&}\\
+@{text "\<or>"} & \texttt{\char`\\/} & \texttt{|}\\
+@{text "\<not>"} & \xsymbol{not} & \texttt{\char`~}\\
+@{text "\<noteq>"} & \xsymbol{noteq} & \texttt{\char`~=}
+\end{tabular}
+\end{center}
+The first column shows the symbols, the second column ASCII representations
+that Isabelle interfaces convert into the corresponding symbol,
+and the third column shows ASCII representations that stay fixed.
+\begin{warn}
+The implication @{text"\<Longrightarrow>"} is part of the Isabelle framework. It structures
+theorems and proof states, separating assumptions from conclusion.
+The implication @{text"\<longrightarrow>"} is part of the logic HOL and can occur inside the
+formulas that make up the assumptions and conclusion.
+Theorems should be of the form @{text"\<lbrakk> A\<^isub>1; \<dots>; A\<^isub>n \<rbrakk> \<Longrightarrow> A"},
+not @{text"A\<^isub>1 \<and> \<dots> \<and> A\<^isub>n \<longrightarrow> A"}. Both are logically equivalent
+but the first one works better when using the theorem in further proofs.
+\end{warn}
+
+\subsection{Sets}
+
+Sets of elements of type @{typ 'a} have type @{typ"'a set"}.
+They can be finite or infinite. Sets come with the usual notations:
+\begin{itemize}
+\item @{term"{}"},\quad @{text"{e\<^isub>1,\<dots>,e\<^isub>n}"}
+\item @{prop"e \<in> A"},\quad @{prop"A \<subseteq> B"}
+\item @{term"A \<union> B"},\quad @{term"A \<inter> B"},\quad @{term"A - B"},\quad @{term"-A"}
+\end{itemize}
+and much more. @{const UNIV} is the set of all elements of some type.
+Set comprehension is written @{term"{x. P}"}
+rather than @{text"{x | P}"}, to emphasize the variable binding nature
+of the construct.
+\begin{warn}
+In @{term"{x. P}"} the @{text x} must be a variable. Set comprehension
+involving a proper term @{text t} must be written
+@{term[source]"{t |x y z. P}"},
+where @{text "x y z"} are the free variables in @{text t}.
+This is just a shorthand for @{term"{v. EX x y z. v = t \<and> P}"}, where
+@{text v} is a new variable.
+\end{warn}
+
+Here are the ASCII representations of the mathematical symbols:
+\begin{center}
+\begin{tabular}{l@ {\quad}l@ {\quad}l}
+@{text "\<in>"} & \texttt{\char`\\\char`\<in>} & \texttt{:}\\
+@{text "\<subseteq>"} & \texttt{\char`\\\char`\<subseteq>} & \texttt{<=}\\
+@{text "\<union>"} & \texttt{\char`\\\char`\<union>} & \texttt{Un}\\
+@{text "\<inter>"} & \texttt{\char`\\\char`\<inter>} & \texttt{Int}
+\end{tabular}
+\end{center}
+Sets also allow bounded quantifications @{prop"ALL x : A. P"} and
+@{prop"EX x : A. P"}.
+
+\subsection{Proof automation}
+
+So far we have only seen @{text simp} and @{text auto}: Both perform
+rewriting, both can also prove linear arithmetic facts (no multiplication),
+and @{text auto} is also able to prove simple logical or set-theoretic goals:
+*}
+
+lemma "\<forall>x. \<exists>y. x = y"
+by auto
+
+lemma "A \<subseteq> B \<inter> C \<Longrightarrow> A \<subseteq> B \<union> C"
+by auto
+
+text{* where
+\begin{quote}
+\isacom{by} \textit{proof-method}
+\end{quote}
+is short for
+\begin{quote}
+\isacom{apply} \textit{proof-method}\\
+\isacom{done}
+\end{quote}
+The key characteristics of both @{text simp} and @{text auto} are
+\begin{itemize}
+\item They show you were they got stuck, giving you an idea how to continue.
+\item They perform the obvious steps but are highly incomplete.
+\end{itemize}
+A proof method is \concept{complete} if it can prove all true formulas.
+There is no complete proof method for HOL, not even in theory.
+Hence all our proof methods only differ in how incomplete they are.
+
+A proof method that is still incomplete but tries harder than @{text auto} is
+@{text fastforce}. It either succeeds or fails, it acts on the first
+subgoal only, and it can be modified just like @{text auto}, e.g.\
+with @{text "simp add"}. Here is a typical example of what @{text fastforce}
+can do:
+*}
+
+lemma "\<lbrakk> \<forall>xs \<in> A. \<exists>ys. xs = ys @ ys; us \<in> A \<rbrakk>
+ \<Longrightarrow> \<exists>n. length us = n+n"
+by fastforce
+
+text{* This lemma is out of reach for @{text auto} because of the
+quantifiers. Even @{text fastforce} fails when the quantifier structure
+becomes more complicated. In a few cases, its slow version @{text force}
+succeeds where @{text fastforce} fails.
+
+The method of choice for complex logical goals is @{text blast}. In the
+following example, @{text T} and @{text A} are two binary predicates, and it
+is shown that @{text T} is total, @{text A} is antisymmetric and @{text T} is
+a subset of @{text A}, then @{text A} is a subset of @{text T}:
+*}
+
+lemma
+ "\<lbrakk> \<forall>x y. T x y \<or> T y x;
+ \<forall>x y. A x y \<and> A y x \<longrightarrow> x = y;
+ \<forall>x y. T x y \<longrightarrow> A x y \<rbrakk>
+ \<Longrightarrow> \<forall>x y. A x y \<longrightarrow> T x y"
+by blast
+
+text{*
+We leave it to the reader to figure out why this lemma is true.
+Method @{text blast}
+\begin{itemize}
+\item is (in principle) a complete proof procedure for first-order formulas,
+ a fragment of HOL. In practice there is a search bound.
+\item does no rewriting and knows very little about equality.
+\item covers logic, sets and relations.
+\item either succeeds or fails.
+\end{itemize}
+Because of its strength in logic and sets and its weakness in equality reasoning, it complements the earlier proof methods.
+
+
+\subsubsection{Sledgehammer}
+
+Command \isacom{sledgehammer} calls a number of external automatic
+theorem provers (ATPs) that run for up to 30 seconds searching for a
+proof. Some of these ATPs are part of the Isabelle installation, others are
+queried over the internet. If successful, a proof command is generated and can
+be inserted into your proof. The biggest win of \isacom{sledgehammer} is
+that it will take into account the whole lemma library and you do not need to
+feed in any lemma explicitly. For example,*}
+
+lemma "\<lbrakk> xs @ ys = ys @ xs; length xs = length ys \<rbrakk> \<Longrightarrow> xs = ys"
+
+txt{* cannot be solved by any of the standard proof methods, but
+\isacom{sledgehammer} finds the following proof: *}
+
+by (metis append_eq_conv_conj)
+
+text{* We do not explain how the proof was found but what this command
+means. For a start, Isabelle does not trust external tools (and in particular
+not the translations from Isabelle's logic to those tools!)
+and insists on a proof that it can check. This is what @{text metis} does.
+It is given a list of lemmas and tries to find a proof just using those lemmas
+(and pure logic). In contrast to @{text simp} and friends that know a lot of
+lemmas already, using @{text metis} manually is tedious because one has
+to find all the relevant lemmas first. But that is precisely what
+\isacom{sledgehammer} does for us.
+In this case lemma @{thm[source]append_eq_conv_conj} alone suffices:
+@{thm[display] append_eq_conv_conj}
+We leave it to the reader to figure out why this lemma suffices to prove
+the above lemma, even without any knowledge of what the functions @{const take}
+and @{const drop} do. Keep in mind that the variables in the two lemmas
+are independent of each other, despite the same names, and that you can
+substitute arbitrary values for the free variables in a lemma.
+
+Just as for the other proof methods we have seen, there is no guarantee that
+\isacom{sledgehammer} will find a proof if it exists. Nor is
+\isacom{sledgehammer} superior to the other proof methods. They are
+incomparable. Therefore it is recommended to apply @{text simp} or @{text
+auto} before invoking \isacom{sledgehammer} on what is left.
+
+\subsubsection{Arithmetic}
+
+By arithmetic formulas we mean formulas involving variables, numbers, @{text
+"+"}, @{text"-"}, @{text "="}, @{text "<"}, @{text "\<le>"} and the usual logical
+connectives @{text"\<not>"}, @{text"\<and>"}, @{text"\<or>"}, @{text"\<longrightarrow>"},
+@{text"\<longleftrightarrow>"}. Strictly speaking, this is known as \concept{linear arithmetic}
+because it does not involve multiplication, although multiplication with
+numbers, e.g.\ @{text"2*n"} is allowed. Such formulas can be proved by
+@{text arith}:
+*}
+
+lemma "\<lbrakk> (a::nat) \<le> x + b; 2*x < c \<rbrakk> \<Longrightarrow> 2*a + 1 \<le> 2*b + c"
+by arith
+
+text{* In fact, @{text auto} and @{text simp} can prove many linear
+arithmetic formulas already, like the one above, by calling a weak but fast
+version of @{text arith}. Hence it is usually not necessary to invoke
+@{text arith} explicitly.
+
+The above example involves natural numbers, but integers (type @{typ int})
+and real numbers (type @{text real}) are supported as well. As are a number
+of further operators like @{const min} and @{const max}. On @{typ nat} and
+@{typ int}, @{text arith} can even prove theorems with quantifiers in them,
+but we will not enlarge on that here.
+
+
+\subsection{Single step proofs}
+
+Although automation is nice, it often fails, at least initially, and you need
+to find out why. When @{text fastforce} or @{text blast} simply fail, you have
+no clue why. At this point, the stepwise
+application of proof rules may be necessary. For example, if @{text blast}
+fails on @{prop"A \<and> B"}, you want to attack the two
+conjuncts @{text A} and @{text B} separately. This can
+be achieved by applying \emph{conjunction introduction}
+\[ @{thm[mode=Rule,show_question_marks]conjI}\ @{text conjI}
+\]
+to the proof state. We will now examine the details of this process.
+
+\subsubsection{Instantiating unknowns}
+
+We had briefly mentioned earlier that after proving some theorem,
+Isabelle replaces all free variables @{text x} by so called \concept{unknowns}
+@{text "?x"}. We can see this clearly in rule @{thm[source] conjI}.
+These unknowns can later be instantiated explicitly or implicitly:
+\begin{itemize}
+\item By hand, using @{text of}.
+The expression @{text"conjI[of \"a=b\" \"False\"]"}
+instantiates the unknowns in @{thm[source] conjI} from left to right with the
+two formulas @{text"a=b"} and @{text False}, yielding the rule
+@{thm[display,mode=Rule]conjI[of "a=b" False]}
+
+In general, @{text"th[of string\<^isub>1 \<dots> string\<^isub>n]"} instantiates
+the unknowns in the theorem @{text th} from left to right with the terms
+@{text string\<^isub>1} to @{text string\<^isub>n}.
+
+\item By unification. \concept{Unification} is the process of making two
+terms syntactically equal by suitable instantiations of unknowns. For example,
+unifying @{text"?P \<and> ?Q"} with \mbox{@{prop"a=b \<and> False"}} instantiates
+@{text "?P"} with @{prop "a=b"} and @{text "?Q"} with @{prop False}.
+\end{itemize}
+We need not instantiate all unknowns. If we want to skip a particular one we
+can just write @{text"_"} instead, for example @{text "conjI[of _ \"False\"]"}.
+Unknowns can also be instantiated by name, for example
+@{text "conjI[where ?P = \"a=b\" and ?Q = \"False\"]"}.
+
+
+\subsubsection{Rule application}
+
+\concept{Rule application} means applying a rule backwards to a proof state.
+For example, applying rule @{thm[source]conjI} to a proof state
+\begin{quote}
+@{text"1. \<dots> \<Longrightarrow> A \<and> B"}
+\end{quote}
+results in two subgoals, one for each premise of @{thm[source]conjI}:
+\begin{quote}
+@{text"1. \<dots> \<Longrightarrow> A"}\\
+@{text"2. \<dots> \<Longrightarrow> B"}
+\end{quote}
+In general, the application of a rule @{text"\<lbrakk> A\<^isub>1; \<dots>; A\<^isub>n \<rbrakk> \<Longrightarrow> A"}
+to a subgoal \mbox{@{text"\<dots> \<Longrightarrow> C"}} proceeds in two steps:
+\begin{enumerate}
+\item
+Unify @{text A} and @{text C}, thus instantiating the unknowns in the rule.
+\item
+Replace the subgoal @{text C} with @{text n} new subgoals @{text"A\<^isub>1"} to @{text"A\<^isub>n"}.
+\end{enumerate}
+This is the command to apply rule @{text xyz}:
+\begin{quote}
+\isacom{apply}@{text"(rule xyz)"}
+\end{quote}
+This is also called \concept{backchaining} with rule @{text xyz}.
+
+\subsubsection{Introduction rules}
+
+Conjunction introduction (@{thm[source] conjI}) is one example of a whole
+class of rules known as \concept{introduction rules}. They explain under which
+premises some logical construct can be introduced. Here are some further
+useful introduction rules:
+\[
+\inferrule*[right=\mbox{@{text impI}}]{\mbox{@{text"?P \<Longrightarrow> ?Q"}}}{\mbox{@{text"?P \<longrightarrow> ?Q"}}}
+\qquad
+\inferrule*[right=\mbox{@{text allI}}]{\mbox{@{text"\<And>x. ?P x"}}}{\mbox{@{text"\<forall>x. ?P x"}}}
+\]
+\[
+\inferrule*[right=\mbox{@{text iffI}}]{\mbox{@{text"?P \<Longrightarrow> ?Q"}} \\ \mbox{@{text"?Q \<Longrightarrow> ?P"}}}
+ {\mbox{@{text"?P = ?Q"}}}
+\]
+These rules are part of the logical system of \concept{natural deduction}
+(e.g.\ \cite{HuthRyan}). Although we intentionally de-emphasize the basic rules
+of logic in favour of automatic proof methods that allow you to take bigger
+steps, these rules are helpful in locating where and why automation fails.
+When applied backwards, these rules decompose the goal:
+\begin{itemize}
+\item @{thm[source] conjI} and @{thm[source]iffI} split the goal into two subgoals,
+\item @{thm[source] impI} moves the left-hand side of a HOL implication into the list of assumptions,
+\item and @{thm[source] allI} removes a @{text "\<forall>"} by turning the quantified variable into a fixed local variable of the subgoal.
+\end{itemize}
+Isabelle knows about these and a number of other introduction rules.
+The command
+\begin{quote}
+\isacom{apply} @{text rule}
+\end{quote}
+automatically selects the appropriate rule for the current subgoal.
+
+You can also turn your own theorems into introduction rules by giving them
+them @{text"intro"} attribute, analogous to the @{text simp} attribute. In
+that case @{text blast}, @{text fastforce} and (to a limited extent) @{text
+auto} will automatically backchain with those theorems. The @{text intro}
+attribute should be used with care because it increases the search space and
+can lead to nontermination. Sometimes it is better to use it only in a
+particular calls of @{text blast} and friends. For example,
+@{thm[source] le_trans}, transitivity of @{text"\<le>"} on type @{typ nat},
+is not an introduction rule by default because of the disastrous effect
+on the search space, but can be useful in specific situations:
+*}
+
+lemma "\<lbrakk> (a::nat) \<le> b; b \<le> c; c \<le> d; d \<le> e \<rbrakk> \<Longrightarrow> a \<le> e"
+by(blast intro: le_trans)
+
+text{*
+Of course this is just an example and could be proved by @{text arith}, too.
+
+\subsubsection{Forward proof}
+\label{sec:forward-proof}
+
+Forward proof means deriving new theorems from old theorems. We have already
+seen a very simple form of forward proof: the @{text of} operator for
+instantiating unknowns in a theorem. The big brother of @{text of} is @{text
+OF} for applying one theorem to others. Given a theorem @{prop"A \<Longrightarrow> B"} called
+@{text r} and a theorem @{text A'} called @{text r'}, the theorem @{text
+"r[OF r']"} is the result of applying @{text r} to @{text r'}, where @{text
+r} should be viewed as a function taking a theorem @{text A} and returning
+@{text B}. More precisely, @{text A} and @{text A'} are unified, thus
+instantiating the unknowns in @{text B}, and the result is the instantiated
+@{text B}. Of course, unification may also fail.
+\begin{warn}
+Application of rules to other rules operates in the forward direction: from
+the premises to the conclusion of the rule; application of rules to proof
+states operates in the backward direction, from the conclusion to the
+premises.
+\end{warn}
+
+In general @{text r} can be of the form @{text"\<lbrakk> A\<^isub>1; \<dots>; A\<^isub>n \<rbrakk> \<Longrightarrow> A"}
+and there can be multiple argument theorems @{text r\<^isub>1} to @{text r\<^isub>m}
+(with @{text"m \<le> n"}), in which case @{text "r[OF r\<^isub>1 \<dots> r\<^isub>m]"} is obtained
+by unifying and thus proving @{text "A\<^isub>i"} with @{text "r\<^isub>i"}, @{text"i = 1\<dots>m"}.
+Here is an example, where @{thm[source]refl} is the theorem
+@{thm[show_question_marks] refl}:
+*}
+
+thm conjI[OF refl[of "a"] refl[of "b"]]
+
+text{* yields the theorem @{thm conjI[OF refl[of "a"] refl[of "b"]]}.
+The command \isacom{thm} merely displays the result.
+
+Forward reasoning also makes sense in connection with proof states.
+Therefore @{text blast}, @{text fastforce} and @{text auto} support a modifier
+@{text dest} which instructs the proof method to use certain rules in a
+forward fashion. If @{text r} is of the form \mbox{@{text "A \<Longrightarrow> B"}}, the modifier
+\mbox{@{text"dest: r"}}
+allows proof search to reason forward with @{text r}, i.e.\
+to replace an assumption @{text A'}, where @{text A'} unifies with @{text A},
+with the correspondingly instantiated @{text B}. For example, @{thm[source,show_question_marks] Suc_leD} is the theorem \mbox{@{thm Suc_leD}}, which works well for forward reasoning:
+*}
+
+lemma "Suc(Suc(Suc a)) \<le> b \<Longrightarrow> a \<le> b"
+by(blast dest: Suc_leD)
+
+text{* In this particular example we could have backchained with
+@{thm[source] Suc_leD}, too, but because the premise is more complicated than the conclusion this can easily lead to nontermination.
+
+\begin{warn}
+To ease readability we will drop the question marks
+in front of unknowns from now on.
+\end{warn}
+
+\section{Inductive definitions}
+\label{sec:inductive-defs}
+
+Inductive definitions are the third important definition facility, after
+datatypes and recursive function. In fact, they are the key construct in the
+definition of operational semantics in the second part of the book.
+
+\subsection{An example: even numbers}
+\label{sec:Logic:even}
+
+Here is a simple example of an inductively defined predicate:
+\begin{itemize}
+\item 0 is even
+\item If $n$ is even, so is $n+2$.
+\end{itemize}
+The operative word ``inductive'' means that these are the only even numbers.
+In Isabelle we give the two rules the names @{text ev0} and @{text evSS}
+and write
+*}
+
+inductive ev :: "nat \<Rightarrow> bool" where
+ev0: "ev 0" |
+evSS: (*<*)"ev n \<Longrightarrow> ev (Suc(Suc n))"(*>*)
+text_raw{* @{prop"ev n \<Longrightarrow> ev (n + 2)"} *}
+
+text{* To get used to inductive definitions, we will first prove a few
+properties of @{const ev} informally before we descend to the Isabelle level.
+
+How do we prove that some number is even, e.g.\ @{prop "ev 4"}? Simply by combining the defining rules for @{const ev}:
+\begin{quote}
+@{text "ev 0 \<Longrightarrow> ev (0 + 2) \<Longrightarrow> ev((0 + 2) + 2) = ev 4"}
+\end{quote}
+
+\subsubsection{Rule induction}
+
+Showing that all even numbers have some property is more complicated. For
+example, let us prove that the inductive definition of even numbers agrees
+with the following recursive one:*}
+
+fun even :: "nat \<Rightarrow> bool" where
+"even 0 = True" |
+"even (Suc 0) = False" |
+"even (Suc(Suc n)) = even n"
+
+text{* We prove @{prop"ev m \<Longrightarrow> even m"}. That is, we
+assume @{prop"ev m"} and by induction on the form of its derivation
+prove @{prop"even m"}. There are two cases corresponding to the two rules
+for @{const ev}:
+\begin{description}
+\item[Case @{thm[source]ev0}:]
+ @{prop"ev m"} was derived by rule @{prop "ev 0"}: \\
+ @{text"\<Longrightarrow>"} @{prop"m=(0::nat)"} @{text"\<Longrightarrow>"} @{text "even m = even 0 = True"}
+\item[Case @{thm[source]evSS}:]
+ @{prop"ev m"} was derived by rule @{prop "ev n \<Longrightarrow> ev(n+2)"}: \\
+@{text"\<Longrightarrow>"} @{prop"m=n+(2::nat)"} and by induction hypothesis @{prop"even n"}\\
+@{text"\<Longrightarrow>"} @{text"even m = even(n + 2) = even n = True"}
+\end{description}
+
+What we have just seen is a special case of \concept{rule induction}.
+Rule induction applies to propositions of this form
+\begin{quote}
+@{prop "ev n \<Longrightarrow> P n"}
+\end{quote}
+That is, we want to prove a property @{prop"P n"}
+for all even @{text n}. But if we assume @{prop"ev n"}, then there must be
+some derivation of this assumption using the two defining rules for
+@{const ev}. That is, we must prove
+\begin{description}
+\item[Case @{thm[source]ev0}:] @{prop"P(0::nat)"}
+\item[Case @{thm[source]evSS}:] @{prop"\<lbrakk> ev n; P n \<rbrakk> \<Longrightarrow> P(n + 2::nat)"}
+\end{description}
+The corresponding rule is called @{thm[source] ev.induct} and looks like this:
+\[
+\inferrule{
+\mbox{@{thm (prem 1) ev.induct[of "n"]}}\\
+\mbox{@{thm (prem 2) ev.induct}}\\
+\mbox{@{prop"!!n. \<lbrakk> ev n; P n \<rbrakk> \<Longrightarrow> P(n+2)"}}}
+{\mbox{@{thm (concl) ev.induct[of "n"]}}}
+\]
+The first premise @{prop"ev n"} enforces that this rule can only be applied
+in situations where we know that @{text n} is even.
+
+Note that in the induction step we may not just assume @{prop"P n"} but also
+\mbox{@{prop"ev n"}}, which is simply the premise of rule @{thm[source]
+evSS}. Here is an example where the local assumption @{prop"ev n"} comes in
+handy: we prove @{prop"ev m \<Longrightarrow> ev(m - 2)"} by induction on @{prop"ev m"}.
+Case @{thm[source]ev0} requires us to prove @{prop"ev(0 - 2)"}, which follows
+from @{prop"ev 0"} because @{prop"0 - 2 = (0::nat)"} on type @{typ nat}. In
+case @{thm[source]evSS} we have \mbox{@{prop"m = n+(2::nat)"}} and may assume
+@{prop"ev n"}, which implies @{prop"ev (m - 2)"} because @{text"m - 2 = (n +
+2) - 2 = n"}. We did not need the induction hypothesis at all for this proof,
+it is just a case distinction on which rule was used, but having @{prop"ev
+n"} at our disposal in case @{thm[source]evSS} was essential.
+This case distinction over rules is also called ``rule inversion''
+and is discussed in more detail in \autoref{ch:Isar}.
+
+\subsubsection{In Isabelle}
+
+Let us now recast the above informal proofs in Isabelle. For a start,
+we use @{const Suc} terms instead of numerals in rule @{thm[source]evSS}:
+@{thm[display] evSS}
+This avoids the difficulty of unifying @{text"n+2"} with some numeral,
+which is not automatic.
+
+The simplest way to prove @{prop"ev(Suc(Suc(Suc(Suc 0))))"} is in a forward
+direction: @{text "evSS[OF evSS[OF ev0]]"} yields the theorem @{thm evSS[OF
+evSS[OF ev0]]}. Alternatively, you can also prove it as a lemma in backwards
+fashion. Although this is more verbose, it allows us to demonstrate how each
+rule application changes the proof state: *}
+
+lemma "ev(Suc(Suc(Suc(Suc 0))))"
+txt{*
+@{subgoals[display,indent=0,goals_limit=1]}
+*}
+apply(rule evSS)
+txt{*
+@{subgoals[display,indent=0,goals_limit=1]}
+*}
+apply(rule evSS)
+txt{*
+@{subgoals[display,indent=0,goals_limit=1]}
+*}
+apply(rule ev0)
+done
+
+text{* \indent
+Rule induction is applied by giving the induction rule explicitly via the
+@{text"rule:"} modifier:*}
+
+lemma "ev m \<Longrightarrow> even m"
+apply(induction rule: ev.induct)
+by(simp_all)
+
+text{* Both cases are automatic. Note that if there are multiple assumptions
+of the form @{prop"ev t"}, method @{text induction} will induct on the leftmost
+one.
+
+As a bonus, we also prove the remaining direction of the equivalence of
+@{const ev} and @{const even}:
+*}
+
+lemma "even n \<Longrightarrow> ev n"
+apply(induction n rule: even.induct)
+
+txt{* This is a proof by computation induction on @{text n} (see
+\autoref{sec:recursive-funs}) that sets up three subgoals corresponding to
+the three equations for @{const even}:
+@{subgoals[display,indent=0]}
+The first and third subgoals follow with @{thm[source]ev0} and @{thm[source]evSS}, and the second subgoal is trivially true because @{prop"even(Suc 0)"} is @{const False}:
+*}
+
+by (simp_all add: ev0 evSS)
+
+text{* The rules for @{const ev} make perfect simplification and introduction
+rules because their premises are always smaller than the conclusion. It
+makes sense to turn them into simplification and introduction rules
+permanently, to enhance proof automation: *}
+
+declare ev.intros[simp,intro]
+
+text{* The rules of an inductive definition are not simplification rules by
+default because, in contrast to recursive functions, there is no termination
+requirement for inductive definitions.
+
+\subsubsection{Inductive versus recursive}
+
+We have seen two definitions of the notion of evenness, an inductive and a
+recursive one. Which one is better? Much of the time, the recursive one is
+more convenient: it allows us to do rewriting in the middle of terms, and it
+expresses both the positive information (which numbers are even) and the
+negative information (which numbers are not even) directly. An inductive
+definition only expresses the positive information directly. The negative
+information, for example, that @{text 1} is not even, has to be proved from
+it (by induction or rule inversion). On the other hand, rule induction is
+Taylor made for proving \mbox{@{prop"ev n \<Longrightarrow> P n"}} because it only asks you
+to prove the positive cases. In the proof of @{prop"even n \<Longrightarrow> P n"} by
+computation induction via @{thm[source]even.induct}, we are also presented
+with the trivial negative cases. If you want the convenience of both
+rewriting and rule induction, you can make two definitions and show their
+equivalence (as above) or make one definition and prove additional properties
+from it, for example rule induction from computation induction.
+
+But many concepts do not admit a recursive definition at all because there is
+no datatype for the recursion (for example, the transitive closure of a
+relation), or the recursion would not terminate (for example, the operational
+semantics in the second part of this book). Even if there is a recursive
+definition, if we are only interested in the positive information, the
+inductive definition may be much simpler.
+
+\subsection{The reflexive transitive closure}
+\label{sec:star}
+
+Evenness is really more conveniently expressed recursively than inductively.
+As a second and very typical example of an inductive definition we define the
+reflexive transitive closure. It will also be an important building block for
+some of the semantics considered in the second part of the book.
+
+The reflexive transitive closure, called @{text star} below, is a function
+that maps a binary predicate to another binary predicate: if @{text r} is of
+type @{text"\<tau> \<Rightarrow> \<tau> \<Rightarrow> bool"} then @{term "star r"} is again of type @{text"\<tau> \<Rightarrow>
+\<tau> \<Rightarrow> bool"}, and @{prop"star r x y"} means that @{text x} and @{text y} are in
+the relation @{term"star r"}. Think @{term"r^*"} when you see @{term"star
+r"}, because @{text"star r"} is meant to be the reflexive transitive closure.
+That is, @{prop"star r x y"} is meant to be true if from @{text x} we can
+reach @{text y} in finitely many @{text r} steps. This concept is naturally
+defined inductively: *}
+
+inductive star :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" for r where
+refl: "star r x x" |
+step: "r x y \<Longrightarrow> star r y z \<Longrightarrow> star r x z"
+
+text{* The base case @{thm[source] refl} is reflexivity: @{term "x=y"}. The
+step case @{thm[source]step} combines an @{text r} step (from @{text x} to
+@{text y}) and a @{const star} step (from @{text y} to @{text z}) into a
+@{const star} step (from @{text x} to @{text z}).
+The ``\isacom{for}~@{text r}'' in the header is merely a hint to Isabelle
+that @{text r} is a fixed parameter of @{const star}, in contrast to the
+further parameters of @{const star}, which change. As a result, Isabelle
+generates a simpler induction rule.
+
+By definition @{term"star r"} is reflexive. It is also transitive, but we
+need rule induction to prove that: *}
+
+lemma star_trans: "star r x y \<Longrightarrow> star r y z \<Longrightarrow> star r x z"
+apply(induction rule: star.induct)
+(*<*)
+defer
+apply(rename_tac u x y)
+defer
+(*>*)
+txt{* The induction is over @{prop"star r x y"} and we try to prove
+\mbox{@{prop"star r y z \<Longrightarrow> star r x z"}},
+which we abbreviate by @{prop"P x y"}. These are our two subgoals:
+@{subgoals[display,indent=0]}
+The first one is @{prop"P x x"}, the result of case @{thm[source]refl},
+and it is trivial.
+*}
+apply(assumption)
+txt{* Let us examine subgoal @{text 2}, case @{thm[source] step}.
+Assumptions @{prop"r u x"} and \mbox{@{prop"star r x y"}}
+are the premises of rule @{thm[source]step}.
+Assumption @{prop"star r y z \<Longrightarrow> star r x z"} is \mbox{@{prop"P x y"}},
+the IH coming from @{prop"star r x y"}. We have to prove @{prop"P u y"},
+which we do by assuming @{prop"star r y z"} and proving @{prop"star r u z"}.
+The proof itself is straightforward: from \mbox{@{prop"star r y z"}} the IH
+leads to @{prop"star r x z"} which, together with @{prop"r u x"},
+leads to \mbox{@{prop"star r u z"}} via rule @{thm[source]step}:
+*}
+apply(metis step)
+done
+
+text{*
+
+\subsection{The general case}
+
+Inductive definitions have approximately the following general form:
+\begin{quote}
+\isacom{inductive} @{text"I :: \"\<tau> \<Rightarrow> bool\""} \isacom{where}
+\end{quote}
+followed by a sequence of (possibly named) rules of the form
+\begin{quote}
+@{text "\<lbrakk> I a\<^isub>1; \<dots>; I a\<^isub>n \<rbrakk> \<Longrightarrow> I a"}
+\end{quote}
+separated by @{text"|"}. As usual, @{text n} can be 0.
+The corresponding rule induction principle
+@{text I.induct} applies to propositions of the form
+\begin{quote}
+@{prop "I x \<Longrightarrow> P x"}
+\end{quote}
+where @{text P} may itself be a chain of implications.
+\begin{warn}
+Rule induction is always on the leftmost premise of the goal.
+Hence @{text "I x"} must be the first premise.
+\end{warn}
+Proving @{prop "I x \<Longrightarrow> P x"} by rule induction means proving
+for every rule of @{text I} that @{text P} is invariant:
+\begin{quote}
+@{text "\<lbrakk> I a\<^isub>1; P a\<^isub>1; \<dots>; I a\<^isub>n; P a\<^isub>n \<rbrakk> \<Longrightarrow> P a"}
+\end{quote}
+
+The above format for inductive definitions is simplified in a number of
+respects. @{text I} can have any number of arguments and each rule can have
+additional premises not involving @{text I}, so-called \concept{side
+conditions}. In rule inductions, these side-conditions appear as additional
+assumptions. The \isacom{for} clause seen in the definition of the reflexive
+transitive closure merely simplifies the form of the induction rule.
+*}
+(*<*)
+end
+(*>*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/MyList.thy Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,17 @@
+theory MyList
+imports Main
+begin
+
+datatype 'a list = Nil | Cons "'a" "('a list)"
+
+fun app :: "'a list => 'a list => 'a list" where
+"app Nil ys = ys" |
+"app (Cons x xs) ys = Cons x (app xs ys)"
+
+fun rev :: "'a list => 'a list" where
+"rev Nil = Nil" |
+"rev (Cons x xs) = app (rev xs) (Cons x Nil)"
+
+value "rev(Cons True (Cons False Nil))"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/ROOT.ML Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,10 @@
+Printer.show_question_marks_default := false;
+
+use_thys [
+ "Basics",
+ "Bool_nat_list",
+ "MyList",
+ "Types_and_funs",
+ "Logic",
+ "Isar"
+];
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/Types_and_funs.thy Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,479 @@
+(*<*)
+theory Types_and_funs
+imports Main
+begin
+(*>*)
+text{*
+\vspace{-5ex}
+\section{Type and function definitions}
+
+Type synonyms are abbreviations for existing types, for example
+*}
+
+type_synonym string = "char list"
+
+text{*
+Type synonyms are expanded after parsing and are not present in internal representation and output. They are mere conveniences for the reader.
+
+\subsection{Datatypes}
+
+The general form of a datatype definition looks like this:
+\begin{quote}
+\begin{tabular}{@ {}rclcll}
+\isacom{datatype} @{text "('a\<^isub>1,\<dots>,'a\<^isub>n)t"}
+ & = & $C_1 \ @{text"\""}\tau_{1,1}@{text"\""} \dots @{text"\""}\tau_{1,n_1}@{text"\""}$ \\
+ & $|$ & \dots \\
+ & $|$ & $C_k \ @{text"\""}\tau_{k,1}@{text"\""} \dots @{text"\""}\tau_{k,n_k}@{text"\""}$
+\end{tabular}
+\end{quote}
+It introduces the constructors \
+$C_i :: \tau_{i,1}\Rightarrow \cdots \Rightarrow \tau_{i,n_i} \Rightarrow$~@{text "('a\<^isub>1,\<dots>,'a\<^isub>n)t"} \ and expresses that any value of this type is built from these constructors in a unique manner. Uniqueness is implied by the following
+properties of the constructors:
+\begin{itemize}
+\item \emph{Distinctness:} $C_i\ \ldots \neq C_j\ \dots$ \quad if $i \neq j$
+\item \emph{Injectivity:}
+\begin{tabular}[t]{l}
+ $(C_i \ x_1 \dots x_{n_i} = C_i \ y_1 \dots y_{n_i}) =$\\
+ $(x_1 = y_1 \land \dots \land x_{n_i} = y_{n_i})$
+\end{tabular}
+\end{itemize}
+The fact that any value of the datatype is built from the constructors implies
+the structural induction rule: to show
+$P~x$ for all $x$ of type @{text "('a\<^isub>1,\<dots>,'a\<^isub>n)t"},
+one needs to show $P(C_i\ x_1 \dots x_{n_i})$ (for each $i$) assuming
+$P(x_j)$ for all $j$ where $\tau_{i,j} =$~@{text "('a\<^isub>1,\<dots>,'a\<^isub>n)t"}.
+Distinctness and injectivity are applied automatically by @{text auto}
+and other proof methods. Induction must be applied explicitly.
+
+Datatype values can be taken apart with case-expressions, for example
+\begin{quote}
+\noquotes{@{term[source] "(case xs of [] \<Rightarrow> 0 | x # _ \<Rightarrow> Suc x)"}}
+\end{quote}
+just like in functional programming languages. Case expressions
+must be enclosed in parentheses.
+
+As an example, consider binary trees:
+*}
+
+datatype 'a tree = Tip | Node "('a tree)" 'a "('a tree)"
+
+text{* with a mirror function: *}
+
+fun mirror :: "'a tree \<Rightarrow> 'a tree" where
+"mirror Tip = Tip" |
+"mirror (Node l a r) = Node (mirror r) a (mirror l)"
+
+text{* The following lemma illustrates induction: *}
+
+lemma "mirror(mirror t) = t"
+apply(induction t)
+
+txt{* yields
+@{subgoals[display]}
+The induction step contains two induction hypotheses, one for each subtree.
+An application of @{text auto} finishes the proof.
+
+A very simple but also very useful datatype is the predefined
+@{datatype[display] option}
+Its sole purpose is to add a new element @{const None} to an existing
+type @{typ 'a}. To make sure that @{const None} is distinct from all the
+elements of @{typ 'a}, you wrap them up in @{const Some} and call
+the new type @{typ"'a option"}. A typical application is a lookup function
+on a list of key-value pairs, often called an association list:
+*}
+(*<*)
+apply auto
+done
+(*>*)
+fun lookup :: "('a * 'b) list \<Rightarrow> 'a \<Rightarrow> 'b option" where
+"lookup [] x' = None" |
+"lookup ((x,y) # ps) x' = (if x = x' then Some y else lookup ps x')"
+
+text{*
+Note that @{text"\<tau>\<^isub>1 * \<tau>\<^isub>2"} is the type of pairs, also written @{text"\<tau>\<^isub>1 \<times> \<tau>\<^isub>2"}.
+
+\subsection{Definitions}
+
+Non recursive functions can be defined as in the following example:
+*}
+
+definition sq :: "nat \<Rightarrow> nat" where
+"sq n = n * n"
+
+text{* Such definitions do not allow pattern matching but only
+@{text"f x\<^isub>1 \<dots> x\<^isub>n = t"}, where @{text f} does not occur in @{text t}.
+
+\subsection{Abbreviations}
+
+Abbreviations are similar to definitions:
+*}
+
+abbreviation sq' :: "nat \<Rightarrow> nat" where
+"sq' n == n * n"
+
+text{* The key difference is that @{const sq'} is only syntactic sugar:
+@{term"sq' t"} is replaced by \mbox{@{term"t*t"}} after parsing, and every
+occurrence of a term @{term"u*u"} is replaced by \mbox{@{term"sq' u"}} before
+printing. Internally, @{const sq'} does not exist. This is also the
+advantage of abbreviations over definitions: definitions need to be expanded
+explicitly (see \autoref{sec:rewr-defs}) whereas abbreviations are already
+expanded upon parsing. However, abbreviations should be introduced sparingly:
+if abused, they can lead to a confusing discrepancy between the internal and
+external view of a term.
+
+\subsection{Recursive functions}
+\label{sec:recursive-funs}
+
+Recursive functions are defined with \isacom{fun} by pattern matching
+over datatype constructors. The order of equations matters. Just as in
+functional programming languages. However, all HOL functions must be
+total. This simplifies the logic---terms are always defined---but means
+that recursive functions must terminate. Otherwise one could define a
+function @{prop"f n = f n + (1::nat)"} and conclude \mbox{@{prop"(0::nat) = 1"}} by
+subtracting @{term"f n"} on both sides.
+
+Isabelle automatic termination checker requires that the arguments of
+recursive calls on the right-hand side must be strictly smaller than the
+arguments on the left-hand side. In the simplest case, this means that one
+fixed argument position decreases in size with each recursive call. The size
+is measured as the number of constructors (excluding 0-ary ones, e.g.\ @{text
+Nil}). Lexicographic combinations are also recognised. In more complicated
+situations, the user may have to prove termination by hand. For details
+see~\cite{Krauss}.
+
+Functions defined with \isacom{fun} come with their own induction schema
+that mirrors the recursion schema and is derived from the termination
+order. For example,
+*}
+
+fun div2 :: "nat \<Rightarrow> nat" where
+"div2 0 = 0" |
+"div2 (Suc 0) = Suc 0" |
+"div2 (Suc(Suc n)) = Suc(div2 n)"
+
+text{* does not just define @{const div2} but also proves a
+customised induction rule:
+\[
+\inferrule{
+\mbox{@{thm (prem 1) div2.induct}}\\
+\mbox{@{thm (prem 2) div2.induct}}\\
+\mbox{@{thm (prem 3) div2.induct}}}
+{\mbox{@{thm (concl) div2.induct[of _ "m"]}}}
+\]
+This customised induction rule can simplify inductive proofs. For example,
+*}
+
+lemma "div2(n+n) = n"
+apply(induction n rule: div2.induct)
+
+txt{* yields the 3 subgoals
+@{subgoals[display,margin=65]}
+An application of @{text auto} finishes the proof.
+Had we used ordinary structural induction on @{text n},
+the proof would have needed an additional
+case distinction in the induction step.
+
+The general case is often called \concept{computation induction},
+because the induction follows the (terminating!) computation.
+For every defining equation
+\begin{quote}
+@{text "f(e) = \<dots> f(r\<^isub>1) \<dots> f(r\<^isub>k) \<dots>"}
+\end{quote}
+where @{text"f(r\<^isub>i)"}, @{text"i=1\<dots>k"}, are all the recursive calls,
+the induction rule @{text"f.induct"} contains one premise of the form
+\begin{quote}
+@{text"P(r\<^isub>1) \<Longrightarrow> \<dots> \<Longrightarrow> P(r\<^isub>k) \<Longrightarrow> P(e)"}
+\end{quote}
+If @{text "f :: \<tau>\<^isub>1 \<Rightarrow> \<dots> \<Rightarrow> \<tau>\<^isub>n \<Rightarrow> \<tau>"} then @{text"f.induct"} is applied like this:
+\begin{quote}
+\isacom{apply}@{text"(induction x\<^isub>1 \<dots> x\<^isub>n rule: f.induct)"}
+\end{quote}
+where typically there is a call @{text"f x\<^isub>1 \<dots> x\<^isub>n"} in the goal.
+But note that the induction rule does not mention @{text f} at all,
+except in its name, and is applicable independently of @{text f}.
+
+\section{Induction heuristics}
+
+We have already noted that theorems about recursive functions are proved by
+induction. In case the function has more than one argument, we have followed
+the following heuristic in the proofs about the append function:
+\begin{quote}
+\emph{Perform induction on argument number $i$\\
+ if the function is defined by recursion on argument number $i$.}
+\end{quote}
+The key heuristic, and the main point of this section, is to
+\emph{generalise the goal before induction}.
+The reason is simple: if the goal is
+too specific, the induction hypothesis is too weak to allow the induction
+step to go through. Let us illustrate the idea with an example.
+
+Function @{const rev} has quadratic worst-case running time
+because it calls append for each element of the list and
+append is linear in its first argument. A linear time version of
+@{const rev} requires an extra argument where the result is accumulated
+gradually, using only~@{text"#"}:
+*}
+(*<*)
+apply auto
+done
+(*>*)
+fun itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
+"itrev [] ys = ys" |
+"itrev (x#xs) ys = itrev xs (x#ys)"
+
+text{* The behaviour of @{const itrev} is simple: it reverses
+its first argument by stacking its elements onto the second argument,
+and returning that second argument when the first one becomes
+empty. Note that @{const itrev} is tail-recursive: it can be
+compiled into a loop, no stack is necessary for executing it.
+
+Naturally, we would like to show that @{const itrev} does indeed reverse
+its first argument provided the second one is empty:
+*}
+
+lemma "itrev xs [] = rev xs"
+
+txt{* There is no choice as to the induction variable:
+*}
+
+apply(induction xs)
+apply(auto)
+
+txt{*
+Unfortunately, this attempt does not prove
+the induction step:
+@{subgoals[display,margin=70]}
+The induction hypothesis is too weak. The fixed
+argument,~@{term"[]"}, prevents it from rewriting the conclusion.
+This example suggests a heuristic:
+\begin{quote}
+\emph{Generalise goals for induction by replacing constants by variables.}
+\end{quote}
+Of course one cannot do this na\"{\i}vely: @{prop"itrev xs ys = rev xs"} is
+just not true. The correct generalisation is
+*};
+(*<*)oops;(*>*)
+lemma "itrev xs ys = rev xs @ ys"
+(*<*)apply(induction xs, auto)(*>*)
+txt{*
+If @{text ys} is replaced by @{term"[]"}, the right-hand side simplifies to
+@{term"rev xs"}, as required.
+In this instance it was easy to guess the right generalisation.
+Other situations can require a good deal of creativity.
+
+Although we now have two variables, only @{text xs} is suitable for
+induction, and we repeat our proof attempt. Unfortunately, we are still
+not there:
+@{subgoals[display,margin=65]}
+The induction hypothesis is still too weak, but this time it takes no
+intuition to generalise: the problem is that the @{text ys}
+in the induction hypothesis is fixed,
+but the induction hypothesis needs to be applied with
+@{term"a # ys"} instead of @{text ys}. Hence we prove the theorem
+for all @{text ys} instead of a fixed one. We can instruct induction
+to perform this generalisation for us by adding @{text "arbitrary: ys"}.
+*}
+(*<*)oops
+lemma "itrev xs ys = rev xs @ ys"
+(*>*)
+apply(induction xs arbitrary: ys)
+
+txt{* The induction hypothesis in the induction step is now universally quantified over @{text ys}:
+@{subgoals[display,margin=65]}
+Thus the proof succeeds:
+*}
+
+apply auto
+done
+
+text{*
+This leads to another heuristic for generalisation:
+\begin{quote}
+\emph{Generalise induction by generalising all free
+variables\\ {\em(except the induction variable itself)}.}
+\end{quote}
+Generalisation is best performed with @{text"arbitrary: y\<^isub>1 \<dots> y\<^isub>k"}.
+This heuristic prevents trivial failures like the one above.
+However, it should not be applied blindly.
+It is not always required, and the additional quantifiers can complicate
+matters in some cases. The variables that need to be quantified are typically
+those that change in recursive calls.
+
+\section{Simplification}
+
+So far we have talked a lot about simplifying terms without explaining the concept. \concept{Simplification} means
+\begin{itemize}
+\item using equations $l = r$ from left to right (only),
+\item as long as possible.
+\end{itemize}
+To emphasise the directionality, equations that have been given the
+@{text"simp"} attribute are called \concept{simplification}
+rules. Logically, they are still symmetric, but proofs by
+simplification use them only in the left-to-right direction. The proof tool
+that performs simplifications is called the \concept{simplifier}. It is the
+basis of @{text auto} and other related proof methods.
+
+The idea of simplification is best explained by an example. Given the
+simplification rules
+\[
+\begin{array}{rcl@ {\quad}l}
+@{term"0 + n::nat"} &@{text"="}& @{text n} & (1) \\
+@{term"(Suc m) + n"} &@{text"="}& @{term"Suc (m + n)"} & (2) \\
+@{text"(Suc m \<le> Suc n)"} &@{text"="}& @{text"(m \<le> n)"} & (3)\\
+@{text"(0 \<le> m)"} &@{text"="}& @{const True} & (4)
+\end{array}
+\]
+the formula @{prop"0 + Suc 0 \<le> Suc 0 + x"} is simplified to @{const True}
+as follows:
+\[
+\begin{array}{r@ {}c@ {}l@ {\quad}l}
+@{text"(0 + Suc 0"} & \leq & @{text"Suc 0 + x)"} & \stackrel{(1)}{=} \\
+@{text"(Suc 0"} & \leq & @{text"Suc 0 + x)"} & \stackrel{(2)}{=} \\
+@{text"(Suc 0"} & \leq & @{text"Suc (0 + x)"} & \stackrel{(3)}{=} \\
+@{text"(0"} & \leq & @{text"0 + x)"} & \stackrel{(4)}{=} \\[1ex]
+ & @{const True}
+\end{array}
+\]
+Simplification is often also called \concept{rewriting}
+and simplification rules \concept{rewrite rules}.
+
+\subsection{Simplification rules}
+
+The attribute @{text"simp"} declares theorems to be simplification rules,
+which the simplifier will use automatically. In addition,
+\isacom{datatype} and \isacom{fun} commands implicitly declare some
+simplification rules: \isacom{datatype} the distinctness and injectivity
+rules, \isacom{fun} the defining equations. Definitions are not declared
+as simplification rules automatically! Nearly any theorem can become a
+simplification rule. The simplifier will try to transform it into an
+equation. For example, the theorem @{prop"\<not> P"} is turned into @{prop"P = False"}.
+
+Only equations that really simplify, like @{prop"rev (rev xs) = xs"} and
+@{prop"xs @ [] = xs"}, should be declared as simplification
+rules. Equations that may be counterproductive as simplification rules
+should only be used in specific proof steps (see \S\ref{sec:simp} below).
+Distributivity laws, for example, alter the structure of terms
+and can produce an exponential blow-up.
+
+\subsection{Conditional simplification rules}
+
+Simplification rules can be conditional. Before applying such a rule, the
+simplifier will first try to prove the preconditions, again by
+simplification. For example, given the simplification rules
+\begin{quote}
+@{prop"p(0::nat) = True"}\\
+@{prop"p(x) \<Longrightarrow> f(x) = g(x)"},
+\end{quote}
+the term @{term"f(0::nat)"} simplifies to @{term"g(0::nat)"}
+but @{term"f(1::nat)"} does not simplify because @{prop"p(1::nat)"}
+is not provable.
+
+\subsection{Termination}
+
+Simplification can run forever, for example if both @{prop"f x = g x"} and
+@{prop"g(x) = f(x)"} are simplification rules. It is the user's
+responsibility not to include simplification rules that can lead to
+nontermination, either on their own or in combination with other
+simplification rules. The right-hand side of a simplification rule should
+always be ``simpler'' than the left-hand side---in some sense. But since
+termination is undecidable, such a check cannot be automated completely
+and Isabelle makes little attempt to detect nontermination.
+
+When conditional simplification rules are applied, their preconditions are
+proved first. Hence all preconditions need to be
+simpler than the left-hand side of the conclusion. For example
+\begin{quote}
+@{prop "n < m \<Longrightarrow> (n < Suc m) = True"}
+\end{quote}
+is suitable as a simplification rule: both @{prop"n<m"} and @{const True}
+are simpler than \mbox{@{prop"n < Suc m"}}. But
+\begin{quote}
+@{prop "Suc n < m \<Longrightarrow> (n < m) = True"}
+\end{quote}
+leads to nontermination: when trying to rewrite @{prop"n<m"} to @{const True}
+one first has to prove \mbox{@{prop"Suc n < m"}}, which can be rewritten to @{const True} provided @{prop"Suc(Suc n) < m"}, \emph{ad infinitum}.
+
+\subsection{The @{text simp} proof method}
+\label{sec:simp}
+
+So far we have only used the proof method @{text auto}. Method @{text simp}
+is the key component of @{text auto}, but @{text auto} can do much more. In
+some cases, @{text auto} is overeager and modifies the proof state too much.
+In such cases the more predictable @{text simp} method should be used.
+Given a goal
+\begin{quote}
+@{text"1. \<lbrakk> P\<^isub>1; \<dots>; P\<^isub>m \<rbrakk> \<Longrightarrow> C"}
+\end{quote}
+the command
+\begin{quote}
+\isacom{apply}@{text"(simp add: th\<^isub>1 \<dots> th\<^isub>n)"}
+\end{quote}
+simplifies the assumptions @{text "P\<^isub>i"} and the conclusion @{text C} using
+\begin{itemize}
+\item all simplification rules, including the ones coming from \isacom{datatype} and \isacom{fun},
+\item the additional lemmas @{text"th\<^isub>1 \<dots> th\<^isub>n"}, and
+\item the assumptions.
+\end{itemize}
+In addition to or instead of @{text add} there is also @{text del} for removing
+simplification rules temporarily. Both are optional. Method @{text auto}
+can be modified similarly:
+\begin{quote}
+\isacom{apply}@{text"(auto simp add: \<dots> simp del: \<dots>)"}
+\end{quote}
+Here the modifiers are @{text"simp add"} and @{text"simp del"}
+instead of just @{text add} and @{text del} because @{text auto}
+does not just perform simplification.
+
+Note that @{text simp} acts only on subgoal 1, @{text auto} acts on all
+subgoals. There is also @{text simp_all}, which applies @{text simp} to
+all subgoals.
+
+\subsection{Rewriting with definitions}
+\label{sec:rewr-defs}
+
+Definitions introduced by the command \isacom{definition}
+can also be used as simplification rules,
+but by default they are not: the simplifier does not expand them
+automatically. Definitions are intended for introducing abstract concepts and
+not merely as abbreviations. Of course, we need to expand the definition
+initially, but once we have proved enough abstract properties of the new
+constant, we can forget its original definition. This style makes proofs more
+robust: if the definition has to be changed, only the proofs of the abstract
+properties will be affected.
+
+The definition of a function @{text f} is a theorem named @{text f_def}
+and can be added to a call of @{text simp} just like any other theorem:
+\begin{quote}
+\isacom{apply}@{text"(simp add: f_def)"}
+\end{quote}
+In particular, let-expressions can be unfolded by
+making @{thm[source] Let_def} a simplification rule.
+
+\subsection{Case splitting with @{text simp}}
+
+Goals containing if-expressions are automatically split into two cases by
+@{text simp} using the rule
+\begin{quote}
+@{prop"P(if A then s else t) = ((A \<longrightarrow> P(s)) \<and> (~A \<longrightarrow> P(t)))"}
+\end{quote}
+For example, @{text simp} can prove
+\begin{quote}
+@{prop"(A \<and> B) = (if A then B else False)"}
+\end{quote}
+because both @{prop"A \<longrightarrow> (A & B) = B"} and @{prop"~A \<longrightarrow> (A & B) = False"}
+simplify to @{const True}.
+
+We can split case-expressions similarly. For @{text nat} the rule looks like this:
+@{prop[display,margin=65,indent=4]"P(case e of 0 \<Rightarrow> a | Suc n \<Rightarrow> b n) = ((e = 0 \<longrightarrow> P a) & (ALL n. e = Suc n \<longrightarrow> P(b n)))"}
+Case expressions are not split automatically by @{text simp}, but @{text simp}
+can be instructed to do so:
+\begin{quote}
+\isacom{apply}@{text"(simp split: nat.split)"}
+\end{quote}
+splits all case-expressions over natural numbers. For an arbitrary
+datatype @{text t} it is @{text "t.split"} instead of @{thm[source] nat.split}.
+Method @{text auto} can be modified in exactly the same way.
+*}
+(*<*)
+end
+(*>*)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/document/Basics.tex Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,173 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Basics}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\begin{isamarkuptext}%
+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 \isa{bool}, the type of truth values,
+\isa{nat}, the type of natural numbers ($\mathbb{N}$), and \isa{int},
+the type of mathematical integers ($\mathbb{Z}$).
+\item[type constructors,]
+ in particular \isa{list}, the type of
+lists, and \isa{set}, the type of sets. Type constructors are written
+postfix, e.g.\ \isa{nat\ list} is the type of lists whose elements are
+natural numbers.
+\item[function types,]
+denoted by \isa{{\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}}.
+\item[type variables,]
+ denoted by \isa{{\isaliteral{27}{\isacharprime}}a}, \isa{{\isaliteral{27}{\isacharprime}}b} etc., just like in ML\@.
+\end{description}
+
+\concept{Terms} are formed as in functional programming by
+applying functions to arguments. If \isa{f} is a function of type
+\isa{{\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}} and \isa{t} is a term of type
+\isa{{\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}} then \isa{f\ t} is a term of type \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}. We write \isa{t\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}} to mean that term \isa{t} has type \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}}.
+
+\begin{warn}
+There are many predefined infix symbols like \isa{{\isaliteral{2B}{\isacharplus}}} and \isa{{\isaliteral{5C3C6C653E}{\isasymle}}}.
+The name of the corresponding binary function is \isa{op\ {\isaliteral{2B}{\isacharplus}}},
+not just \isa{{\isaliteral{2B}{\isacharplus}}}. That is, \isa{x\ {\isaliteral{2B}{\isacharplus}}\ y} is syntactic sugar for
+\noquotes{\isa{{\isaliteral{22}{\isachardoublequote}}op\ {\isaliteral{2B}{\isacharplus}}\ x\ y{\isaliteral{22}{\isachardoublequote}}}}.
+\end{warn}
+
+HOL also supports some basic constructs from functional programming:
+\begin{quote}
+\isa{{\isaliteral{28}{\isacharparenleft}}if\ b\ then\ t\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ else\ t\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}\\
+\isa{{\isaliteral{28}{\isacharparenleft}}let\ x\ {\isaliteral{3D}{\isacharequal}}\ t\ in\ u{\isaliteral{29}{\isacharparenright}}}\\
+\isa{{\isaliteral{28}{\isacharparenleft}}case\ t\ of\ pat\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ t\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{7C}{\isacharbar}}\ pat\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ t\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}}
+\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 \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}}-abstractions. For example,
+\isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ x} is the identity function.
+
+\concept{Formulae} are terms of type \isa{bool}.
+There are the basic constants \isa{True} and \isa{False} and
+the usual logical connectives (in decreasing order of precedence):
+\isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}}, \isa{{\isaliteral{5C3C616E643E}{\isasymand}}}, \isa{{\isaliteral{5C3C6F723E}{\isasymor}}}, \isa{{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}}.
+
+\concept{Equality} is available in the form of the infix function \isa{{\isaliteral{3D}{\isacharequal}}}
+of type \isa{{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool}. It also works for formulas, where
+it means ``if and only if''.
+
+\concept{Quantifiers} are written \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ P} and \isa{{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ 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 \isa{t\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}} as in
+\mbox{\noquotes{\isa{{\isaliteral{22}{\isachardoublequote}}m\ {\isaliteral{3C}{\isacharless}}\ {\isaliteral{28}{\isacharparenleft}}n{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}}. Type constraints may be
+needed to
+disambiguate terms involving overloaded functions such as \isa{{\isaliteral{2B}{\isacharplus}}}, \isa{{\isaliteral{2A}{\isacharasterisk}}} and \isa{{\isaliteral{5C3C6C653E}{\isasymle}}}.
+
+Finally there are the universal quantifier \isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}} and the implication
+\isa{{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}}. They are part of the Isabelle framework, not the logic
+HOL. Logically, they agree with their HOL counterparts \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}} and
+\isa{{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}}, 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
+\isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{3}}} means \isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{3}}{\isaliteral{29}{\isacharparenright}}}.
+The (Isabelle specific) notation \mbox{\isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{3B}{\isacharsemicolon}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A}}
+is short for the iterated implication \mbox{\isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A}}.
+Sometimes we also employ inference rule notation:
+\inferrule{\mbox{\isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}}\\ \mbox{\isa{{\isaliteral{5C3C646F74733E}{\isasymdots}}}}\\ \mbox{\isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub n}}}
+{\mbox{\isa{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 \isa{T} is
+\begin{quote}
+\isacom{theory} \isa{T}\\
+\isacom{imports} \isa{T\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ T\isaliteral{5C3C5E697375623E}{}\isactrlisub n}\\
+\isacom{begin}\\
+\emph{definitions, theorems and proofs}\\
+\isacom{end}
+\end{quote}
+where \isa{T\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ T\isaliteral{5C3C5E697375623E}{}\isactrlisub n} are the names of existing
+theories that \isa{T} is based on. The \isa{T\isaliteral{5C3C5E697375623E}{}\isactrlisub i} are the
+direct \concept{parent theories} of \isa{T}.
+Everything defined in the parent theories (and their parents, recursively) is
+automatically visible. Each theory \isa{T} must
+reside in a \concept{theory file} named \isa{T{\isaliteral{2E}{\isachardot}}thy}.
+
+\begin{warn}
+HOL contains a theory \isa{Main}, the union of all the basic
+predefined theories like arithmetic, lists, sets, etc.
+Unless you know what you are doing, always include \isa{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}.
+\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}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/document/Bool_nat_list.tex Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,591 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Bool{\isaliteral{5F}{\isacharunderscore}}nat{\isaliteral{5F}{\isacharunderscore}}list}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\begin{isamarkuptext}%
+\vspace{-4ex}
+\section{\texorpdfstring{Types \isa{bool}, \isa{nat} and \isa{list}}{Types bool, nat and list}}
+
+These are the most important predefined types. We go through them one by one.
+Based on examples we learn how to define (possibly recursive) functions and
+prove theorems about them by induction and simplification.
+
+\subsection{Type \isa{bool}}
+
+The type of boolean values is a predefined datatype
+\begin{isabelle}%
+\isacommand{datatype}\ bool\ {\isaliteral{3D}{\isacharequal}}\ True\ {\isaliteral{7C}{\isacharbar}}\ False%
+\end{isabelle}
+with the two values \isa{True} and \isa{False} and
+with many predefined functions: \isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}}, \isa{{\isaliteral{5C3C616E643E}{\isasymand}}}, \isa{{\isaliteral{5C3C6F723E}{\isasymor}}}, \isa{{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}} etc. Here is how conjunction could be defined by pattern matching:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ conj\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}bool\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}conj\ True\ True\ {\isaliteral{3D}{\isacharequal}}\ True{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}conj\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{3D}{\isacharequal}}\ False{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+Both the datatype and function definitions roughly follow the syntax
+of functional programming languages.
+
+\subsection{Type \isa{nat}}
+
+Natural numbers are another predefined datatype:
+\begin{isabelle}%
+\isacommand{datatype}\ nat\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ {\isaliteral{7C}{\isacharbar}}\ Suc\ nat%
+\end{isabelle}
+All values of type \isa{nat} are generated by the constructors
+\isa{{\isadigit{0}}} and \isa{Suc}. Thus the values of type \isa{nat} are
+\isa{{\isadigit{0}}}, \isa{Suc\ {\isadigit{0}}}, \isa{Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}} etc.
+There are many predefined functions: \isa{{\isaliteral{2B}{\isacharplus}}}, \isa{{\isaliteral{2A}{\isacharasterisk}}}, \isa{{\isaliteral{5C3C6C653E}{\isasymle}}}, etc. Here is how you could define your own addition:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ add\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}add\ {\isadigit{0}}\ n\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}add\ {\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}\ n\ {\isaliteral{3D}{\isacharequal}}\ Suc{\isaliteral{28}{\isacharparenleft}}add\ m\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+And here is a proof of the fact that \isa{add\ m\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ m}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ add{\isaliteral{5F}{\isacharunderscore}}{\isadigit{0}}{\isadigit{2}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}add\ m\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ m{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ m{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+The \isacom{lemma} command starts the proof and gives the lemma
+a name, \isa{add{\isaliteral{5F}{\isacharunderscore}}{\isadigit{0}}{\isadigit{2}}}. Properties of recursively defined functions
+need to be established by induction in most cases.
+Command \isacom{apply}\isa{{\isaliteral{28}{\isacharparenleft}}induction\ m{\isaliteral{29}{\isacharparenright}}} instructs Isabelle to
+start a proof by induction on \isa{m}. In response, it will show the
+following proof state:
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ add\ {\isadigit{0}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\isanewline
+\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}m{\isaliteral{2E}{\isachardot}}\ add\ m\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ m\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ add\ {\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ m%
+\end{isabelle}
+The numbered lines are known as \emph{subgoals}.
+The first subgoal is the base case, the second one the induction step.
+The prefix \isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}m{\isaliteral{2E}{\isachardot}}} is Isabelle's way of saying ``for an arbitrary but fixed \isa{m}''. The \isa{{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}} separates assumptions from the conclusion.
+The command \isacom{apply}\isa{{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}} instructs Isabelle to try
+and prove all subgoals automatically, essentially by simplifying them.
+Because both subgoals are easy, Isabelle can do it.
+The base case \isa{add\ {\isadigit{0}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}} holds by definition of \isa{add},
+and the induction step is almost as simple:
+\isa{add~{\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ Suc{\isaliteral{28}{\isacharparenleft}}add\ m\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ m}
+using first the definition of \isa{add} and then the induction hypothesis.
+In summary, both subproofs rely on simplification with function definitions and
+the induction hypothesis.
+As a result of that final \isacom{done}, Isabelle associates the lemma
+just proved with its name. You can now inspect the lemma with the command%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{thm}\isamarkupfalse%
+\ add{\isaliteral{5F}{\isacharunderscore}}{\isadigit{0}}{\isadigit{2}}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+which displays \begin{isabelle}%
+add\ {\isaliteral{3F}{\isacharquery}}m\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}m%
+\end{isabelle} The free
+variable \isa{m} has been replaced by the \concept{unknown}
+\isa{{\isaliteral{3F}{\isacharquery}}m}. There is no logical difference between the two but an
+operational one: unknowns can be instantiated, which is what you want after
+some lemma has been proved.
+
+Note that there is also a proof method \isa{induct}, which behaves almost
+like \isa{induction}; the difference is explained in \autoref{ch:Isar}.
+
+\begin{warn}
+Terminology: We use \concept{lemma}, \concept{theorem} and \concept{rule}
+interchangeably for propositions that have been proved.
+\end{warn}
+\begin{warn}
+ Numerals (\isa{{\isadigit{0}}}, \isa{{\isadigit{1}}}, \isa{{\isadigit{2}}}, \dots) and most of the standard
+ arithmetic operations (\isa{{\isaliteral{2B}{\isacharplus}}}, \isa{{\isaliteral{2D}{\isacharminus}}}, \isa{{\isaliteral{2A}{\isacharasterisk}}}, \isa{{\isaliteral{5C3C6C653E}{\isasymle}}},
+ \isa{{\isaliteral{3C}{\isacharless}}} etc) are overloaded: they are available
+ not just for natural numbers but for other types as well.
+ For example, given the goal \isa{x\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ x}, there is nothing to indicate
+ that you are talking about natural numbers. Hence Isabelle can only infer
+ that \isa{x} is of some arbitrary type where \isa{{\isadigit{0}}} and \isa{{\isaliteral{2B}{\isacharplus}}}
+ exist. As a consequence, you will be unable to prove the
+ goal. To alert you to such pitfalls, Isabelle flags numerals without a
+ fixed type in its output: \isa{x\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x}. In this particular example,
+ you need to include
+ an explicit type constraint, for example \isa{x{\isaliteral{2B}{\isacharplus}}{\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}}. If there
+ is enough contextual information this may not be necessary: \isa{Suc\ x\ {\isaliteral{3D}{\isacharequal}}\ x} automatically implies \isa{x{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat} because \isa{Suc} is not
+ overloaded.
+\end{warn}
+
+\subsubsection{An informal proof}
+
+Above we gave some terse informal explanation of the proof of
+\isa{add\ m\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ m}. A more detailed informal exposition of the lemma
+might look like this:
+\bigskip
+
+\noindent
+\textbf{Lemma} \isa{add\ m\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ m}
+
+\noindent
+\textbf{Proof} by induction on \isa{m}.
+\begin{itemize}
+\item Case \isa{{\isadigit{0}}} (the base case): \isa{add\ {\isadigit{0}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}}
+ holds by definition of \isa{add}.
+\item Case \isa{Suc\ m} (the induction step):
+ We assume \isa{add\ m\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ m}, the induction hypothesis (IH),
+ and we need to show \isa{add\ {\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ m}.
+ The proof is as follows:\smallskip
+
+ \begin{tabular}{@ {}rcl@ {\quad}l@ {}}
+ \isa{add\ {\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}\ {\isadigit{0}}} &\isa{{\isaliteral{3D}{\isacharequal}}}& \isa{Suc\ {\isaliteral{28}{\isacharparenleft}}add\ m\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}}
+ & by definition of \isa{add}\\
+ &\isa{{\isaliteral{3D}{\isacharequal}}}& \isa{Suc\ m} & by IH
+ \end{tabular}
+\end{itemize}
+Throughout this book, \concept{IH} will stand for ``induction hypothesis''.
+
+We have now seen three proofs of \isa{add\ m\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}}: the Isabelle one, the
+terse 4 lines explaining the base case and the induction step, and just now a
+model of a traditional inductive proof. The three proofs differ in the level
+of detail given and the intended reader: the Isabelle proof is for the
+machine, the informal proofs are for humans. Although this book concentrates
+of Isabelle proofs, it is important to be able to rephrase those proofs
+as informal text comprehensible to a reader familiar with traditional
+mathematical proofs. Later on we will introduce an Isabelle proof language
+that is closer to traditional informal mathematical language and is often
+directly readable.
+
+\subsection{Type \isa{list}}
+
+Although lists are already predefined, we define our own copy just for
+demonstration purposes:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{datatype}\isamarkupfalse%
+\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{3D}{\isacharequal}}\ Nil\ {\isaliteral{7C}{\isacharbar}}\ Cons\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ list{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+This means that all lists are built up from \isa{Nil}, the empty
+list, and \isa{Cons}, the operation of putting an element in front of a
+list. Hence all lists are of the form \isa{Nil}, or \isa{Cons\ x\ Nil},
+or \isa{Cons\ x\ {\isaliteral{28}{\isacharparenleft}}Cons\ y\ Nil{\isaliteral{29}{\isacharparenright}}} etc.
+
+We also define two standard functions, append and reverse:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\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}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}app\ Nil\ ys\ {\isaliteral{3D}{\isacharequal}}\ ys{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}app\ {\isaliteral{28}{\isacharparenleft}}Cons\ x\ xs{\isaliteral{29}{\isacharparenright}}\ ys\ {\isaliteral{3D}{\isacharequal}}\ Cons\ x\ {\isaliteral{28}{\isacharparenleft}}app\ xs\ ys{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isanewline
+\isacommand{fun}\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\ Nil\ {\isaliteral{3D}{\isacharequal}}\ Nil{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}rev\ {\isaliteral{28}{\isacharparenleft}}Cons\ x\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ app\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}Cons\ x\ Nil{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+By default, variables \isa{xs}, \isa{ys} and \isa{zs} are of
+\isa{list} type.
+
+Command \isacom{value} evaluates a term. For example,%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{value}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}rev{\isaliteral{28}{\isacharparenleft}}Cons\ True\ {\isaliteral{28}{\isacharparenleft}}Cons\ False\ Nil{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+yields the result \isa{Cons\ False\ {\isaliteral{28}{\isacharparenleft}}Cons\ True\ Nil{\isaliteral{29}{\isacharparenright}}}. This works symbolically, too:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{value}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}rev{\isaliteral{28}{\isacharparenleft}}Cons\ a\ {\isaliteral{28}{\isacharparenleft}}Cons\ b\ Nil{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+yields \isa{Cons\ b\ {\isaliteral{28}{\isacharparenleft}}Cons\ a\ Nil{\isaliteral{29}{\isacharparenright}}}.
+\medskip
+
+Figure~\ref{fig:MyList} shows the theory created so far. Notice where the
+quotations marks are needed that we mostly sweep under the carpet. In
+particular, notice that \isacom{datatype} requires no quotation marks on the
+left-hand side, but that on the right-hand side each of the argument
+types of a constructor needs to be enclosed in quotation marks.
+
+\begin{figure}[htbp]
+\begin{alltt}
+\input{Thys/MyList.thy}\end{alltt}
+\caption{A Theory of Lists}
+\label{fig:MyList}
+\end{figure}
+
+\subsubsection{Structural Induction for Lists}
+
+Just as for natural numbers, there is a proof principle of induction for
+lists. Induction over a list is essentially induction over the length of
+the list, although the length remains implicit. To prove that some property
+\isa{P} holds for all lists \isa{xs}, i.e.\ \mbox{\isa{P\ xs}},
+you need to prove
+\begin{enumerate}
+\item the base case \isa{P\ Nil} and
+\item the inductive case \isa{P\ {\isaliteral{28}{\isacharparenleft}}Cons\ x\ xs{\isaliteral{29}{\isacharparenright}}} under the assumption \isa{P\ xs}, for some arbitrary but fixed \isa{xs}.
+\end{enumerate}
+This is often called \concept{structural induction}.
+
+\subsection{The Proof Process}
+
+We will now demonstrate the typical proof process, which involves
+the formulation and proof of auxiliary lemmas.
+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}%
+Commands \isacom{theorem} and \isacom{lemma} are
+interchangeable and merely indicate the importance we attach to a
+proposition. Via the bracketed attribute \isa{simp} we also tell Isabelle
+to make the eventual theorem a \concept{simplification rule}: future proofs
+involving simplification will replace occurrences of \isa{rev\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}} by
+\isa{xs}. The proof is by induction:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ xs{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+As explained above, we obtain 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\ Nil{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Nil\isanewline
+\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ xs{\isaliteral{2E}{\isachardot}}\ rev\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ xs\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ rev\ {\isaliteral{28}{\isacharparenleft}}rev\ {\isaliteral{28}{\isacharparenleft}}Cons\ a\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Cons\ a\ xs%
+\end{isabelle}
+Let us try to solve both goals automatically:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+Subgoal~1 is proved, and disappears; the simplified version
+of subgoal~2 becomes the new subgoal~1:
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ xs{\isaliteral{2E}{\isachardot}}\isanewline
+\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }rev\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ xs\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
+\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }rev\ {\isaliteral{28}{\isacharparenleft}}app\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}Cons\ a\ Nil{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Cons\ a\ xs%
+\end{isabelle}
+In order to simplify this subgoal further, a lemma suggests itself.
+
+\subsubsection{A First Lemma}
+
+We insert the following lemma in front of the main theorem:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\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}}app\ xs\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ app\ {\isaliteral{28}{\isacharparenleft}}rev\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+There are two variables that we could induct on: \isa{xs} and
+\isa{ys}. Because \isa{app} is defined by recursion on
+the first argument, \isa{xs} is the correct one:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ xs{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+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}%
+\vspace{-5ex}
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ rev\ ys\ {\isaliteral{3D}{\isacharequal}}\ app\ {\isaliteral{28}{\isacharparenleft}}rev\ ys{\isaliteral{29}{\isacharparenright}}\ Nil%
+\end{isabelle}
+Again, we need to abandon this proof attempt and prove another simple lemma
+first.
+
+\subsubsection{A Second Lemma}
+
+We again try the canonical proof procedure:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{lemma}\isamarkupfalse%
+\ app{\isaliteral{5F}{\isacharunderscore}}Nil{\isadigit{2}}\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}app\ xs\ Nil\ {\isaliteral{3D}{\isacharequal}}\ xs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ 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}%
+Thankfully, this worked.
+Now we can continue with our stuck proof attempt 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}}app\ xs\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ app\ {\isaliteral{28}{\isacharparenleft}}rev\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ xs{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+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\ xs{\isaliteral{2E}{\isachardot}}\isanewline
+\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }rev\ {\isaliteral{28}{\isacharparenleft}}app\ xs\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ app\ {\isaliteral{28}{\isacharparenleft}}rev\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
+\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }app\ {\isaliteral{28}{\isacharparenleft}}app\ {\isaliteral{28}{\isacharparenleft}}rev\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}Cons\ a\ Nil{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline
+\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }app\ {\isaliteral{28}{\isacharparenleft}}rev\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}app\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}Cons\ a\ Nil{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}%
+\end{isabelle}
+The the missing lemma is associativity of \isa{app},
+which we insert in front of the failed lemma \isa{rev{\isaliteral{5F}{\isacharunderscore}}app}.
+
+\subsubsection{Associativity of \isa{app}}
+
+The canonical proof procedure succeeds without further ado:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{lemma}\isamarkupfalse%
+\ app{\isaliteral{5F}{\isacharunderscore}}assoc\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}app\ {\isaliteral{28}{\isacharparenleft}}app\ xs\ ys{\isaliteral{29}{\isacharparenright}}\ zs\ {\isaliteral{3D}{\isacharequal}}\ app\ xs\ {\isaliteral{28}{\isacharparenleft}}app\ ys\ zs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ xs{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Finally the proofs of \isa{rev{\isaliteral{5F}{\isacharunderscore}}app} and \isa{rev{\isaliteral{5F}{\isacharunderscore}}rev}
+succeed, too.
+
+\subsubsection{Another informal proof}
+
+Here is the informal proof of associativity of \isa{app}
+corresponding to the Isabelle proof above.
+\bigskip
+
+\noindent
+\textbf{Lemma} \isa{app\ {\isaliteral{28}{\isacharparenleft}}app\ xs\ ys{\isaliteral{29}{\isacharparenright}}\ zs\ {\isaliteral{3D}{\isacharequal}}\ app\ xs\ {\isaliteral{28}{\isacharparenleft}}app\ ys\ zs{\isaliteral{29}{\isacharparenright}}}
+
+\noindent
+\textbf{Proof} by induction on \isa{xs}.
+\begin{itemize}
+\item Case \isa{Nil}: \ \isa{app\ {\isaliteral{28}{\isacharparenleft}}app\ Nil\ ys{\isaliteral{29}{\isacharparenright}}\ zs\ {\isaliteral{3D}{\isacharequal}}\ app\ ys\ zs} \isa{{\isaliteral{3D}{\isacharequal}}}
+ \mbox{\isa{app\ Nil\ {\isaliteral{28}{\isacharparenleft}}app\ ys\ zs{\isaliteral{29}{\isacharparenright}}}} \ holds by definition of \isa{app}.
+\item Case \isa{Cons\ x\ xs}: We assume
+ \begin{center} \hfill \isa{app\ {\isaliteral{28}{\isacharparenleft}}app\ xs\ ys{\isaliteral{29}{\isacharparenright}}\ zs} \isa{{\isaliteral{3D}{\isacharequal}}}
+ \isa{app\ xs\ {\isaliteral{28}{\isacharparenleft}}app\ ys\ zs{\isaliteral{29}{\isacharparenright}}} \hfill (IH) \end{center}
+ and we need to show
+ \begin{center} \isa{app\ {\isaliteral{28}{\isacharparenleft}}app\ {\isaliteral{28}{\isacharparenleft}}Cons\ x\ xs{\isaliteral{29}{\isacharparenright}}\ ys{\isaliteral{29}{\isacharparenright}}\ zs\ {\isaliteral{3D}{\isacharequal}}\ app\ {\isaliteral{28}{\isacharparenleft}}Cons\ x\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}app\ ys\ zs{\isaliteral{29}{\isacharparenright}}}.\end{center}
+ The proof is as follows:\smallskip
+
+ \begin{tabular}{@ {}l@ {\quad}l@ {}}
+ \isa{app\ {\isaliteral{28}{\isacharparenleft}}app\ {\isaliteral{28}{\isacharparenleft}}Cons\ x\ xs{\isaliteral{29}{\isacharparenright}}\ ys{\isaliteral{29}{\isacharparenright}}\ zs}\\
+ \isa{{\isaliteral{3D}{\isacharequal}}\ app\ {\isaliteral{28}{\isacharparenleft}}Cons\ x\ {\isaliteral{28}{\isacharparenleft}}app\ xs\ ys{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ zs} & by definition of \isa{app}\\
+ \isa{{\isaliteral{3D}{\isacharequal}}\ Cons\ x\ {\isaliteral{28}{\isacharparenleft}}app\ {\isaliteral{28}{\isacharparenleft}}app\ xs\ ys{\isaliteral{29}{\isacharparenright}}\ zs{\isaliteral{29}{\isacharparenright}}} & by definition of \isa{app}\\
+ \isa{{\isaliteral{3D}{\isacharequal}}\ Cons\ x\ {\isaliteral{28}{\isacharparenleft}}app\ xs\ {\isaliteral{28}{\isacharparenleft}}app\ ys\ zs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}} & by IH\\
+ \isa{{\isaliteral{3D}{\isacharequal}}\ app\ {\isaliteral{28}{\isacharparenleft}}Cons\ x\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}app\ ys\ zs{\isaliteral{29}{\isacharparenright}}} & by definition of \isa{app}
+ \end{tabular}
+\end{itemize}
+\medskip
+
+\noindent Didn't we say earlier that all proofs are by simplification? But
+in both cases, going from left to right, the last equality step is not a
+simplification at all! In the base case it is \isa{app\ ys\ zs\ {\isaliteral{3D}{\isacharequal}}\ app\ Nil\ {\isaliteral{28}{\isacharparenleft}}app\ ys\ zs{\isaliteral{29}{\isacharparenright}}}. It appears almost mysterious because we suddenly complicate the
+term by appending \isa{Nil} on the left. What is really going on is this:
+when proving some equality \mbox{\isa{s\ {\isaliteral{3D}{\isacharequal}}\ t}}, both \isa{s} and \isa{t} are
+simplified to some common term \isa{u}. This heuristic for equality proofs
+works well for a functional programming context like ours. In the base case
+\isa{s} is \isa{app\ {\isaliteral{28}{\isacharparenleft}}app\ Nil\ ys{\isaliteral{29}{\isacharparenright}}\ zs}, \isa{t} is \isa{app\ Nil\ {\isaliteral{28}{\isacharparenleft}}app\ ys\ zs{\isaliteral{29}{\isacharparenright}}}, and \isa{u} is \isa{app\ ys\ zs}.
+
+\subsection{Predefined lists}
+\label{sec:predeflists}
+
+Isabelle's predefined lists are the same as the ones above, but with
+more syntactic sugar:
+\begin{itemize}
+\item \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} is \isa{Nil},
+\item \isa{x\ {\isaliteral{23}{\isacharhash}}\ xs} is \isa{Cons\ x\ xs},
+\item \isa{{\isaliteral{5B}{\isacharbrackleft}}x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{5D}{\isacharbrackright}}} is \isa{x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{23}{\isacharhash}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}}, and
+\item \isa{xs\ {\isaliteral{40}{\isacharat}}\ ys} is \isa{app\ xs\ ys}.
+\end{itemize}
+There is also a large library of predefined functions.
+The most important ones are the length function
+\isa{length\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat} (with the obvious definition),
+and the map function that applies a function to all elements of a list:
+\begin{isabelle}
+\isacom{fun} \isa{map\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ list}\\
+\isa{map\ f\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} \isa{{\isaliteral{7C}{\isacharbar}}}\\
+\isa{map\ f\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ x\ {\isaliteral{23}{\isacharhash}}\ map\ f\ xs}
+\end{isabelle}
+Also useful are the \concept{head} of a list, its first element,
+and the \concept{tail}, the rest of the list:
+\begin{isabelle}
+\isacom{fun} \isa{hd\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a}\\
+\isa{hd\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x}
+\end{isabelle}
+\begin{isabelle}
+\isacom{fun} \isa{tl\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list}\\
+\isa{tl\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} \isa{{\isaliteral{7C}{\isacharbar}}}\\
+\isa{tl\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ xs}
+\end{isabelle}
+Note that since HOL is a logic of total functions, \isa{hd\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} is defined,
+but we do now know what the result is. That is, \isa{hd\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} is not undefined
+but underdefined.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/document/Isar.tex Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,1680 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Isar}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isadelimML
+%
+\endisadelimML
+%
+\isatagML
+%
+\endisatagML
+{\isafoldML}%
+%
+\isadelimML
+%
+\endisadelimML
+%
+\begin{isamarkuptext}%
+Apply-scripts are unreadable and hard to maintain. The language of choice
+for larger proofs is \concept{Isar}. The two key features of Isar are:
+\begin{itemize}
+\item It is structured, not linear.
+\item It is readable without running it because
+you need to state what you are proving at any given point.
+\end{itemize}
+Whereas apply-scripts are like assembly language programs, Isar proofs
+are like structured programs with comments. A typical Isar proof looks like this:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{tabular}{@ {}l}
+\isacom{proof}\\
+\quad\isacom{assume} \isa{{\isaliteral{22}{\isachardoublequote}}}$\mathit{formula}_0$\isa{{\isaliteral{22}{\isachardoublequote}}}\\
+\quad\isacom{have} \isa{{\isaliteral{22}{\isachardoublequote}}}$\mathit{formula}_1$\isa{{\isaliteral{22}{\isachardoublequote}}} \quad\isacom{by} \isa{simp}\\
+\quad\vdots\\
+\quad\isacom{have} \isa{{\isaliteral{22}{\isachardoublequote}}}$\mathit{formula}_n$\isa{{\isaliteral{22}{\isachardoublequote}}} \quad\isacom{by} \isa{blast}\\
+\quad\isacom{show} \isa{{\isaliteral{22}{\isachardoublequote}}}$\mathit{formula}_{n+1}$\isa{{\isaliteral{22}{\isachardoublequote}}} \quad\isacom{by} \isa{{\isaliteral{5C3C646F74733E}{\isasymdots}}}\\
+\isacom{qed}
+\end{tabular}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+It proves $\mathit{formula}_0 \Longrightarrow \mathit{formula}_{n+1}$
+(provided provided each proof step succeeds).
+The intermediate \isacom{have} statements are merely stepping stones
+on the way towards the \isacom{show} statement that proves the actual
+goal. In more detail, this is the Isar core syntax:
+\medskip
+
+\begin{tabular}{@ {}lcl@ {}}
+\textit{proof} &=& \isacom{by} \textit{method}\\
+ &$\mid$& \isacom{proof} [\textit{method}] \ \textit{step}$^*$ \ \isacom{qed}
+\end{tabular}
+\medskip
+
+\begin{tabular}{@ {}lcl@ {}}
+\textit{step} &=& \isacom{fix} \textit{variables} \\
+ &$\mid$& \isacom{assume} \textit{proposition} \\
+ &$\mid$& [\isacom{from} \textit{fact}$^+$] (\isacom{have} $\mid$ \isacom{show}) \ \textit{proposition} \ \textit{proof}
+\end{tabular}
+\medskip
+
+\begin{tabular}{@ {}lcl@ {}}
+\textit{proposition} &=& [\textit{name}:] \isa{{\isaliteral{22}{\isachardoublequote}}}\textit{formula}\isa{{\isaliteral{22}{\isachardoublequote}}}
+\end{tabular}
+\medskip
+
+\begin{tabular}{@ {}lcl@ {}}
+\textit{fact} &=& \textit{name} \ $\mid$ \ \dots
+\end{tabular}
+\medskip
+
+\noindent A proof can either be an atomic \isacom{by} with a single proof
+method which must finish off the statement being proved, for example \isa{auto}. Or it can be a \isacom{proof}--\isacom{qed} block of multiple
+steps. Such a block can optionally begin with a proof method that indicates
+how to start off the proof, e.g.\ \mbox{\isa{{\isaliteral{28}{\isacharparenleft}}induction\ xs{\isaliteral{29}{\isacharparenright}}}}.
+
+A step either assumes a proposition or states a proposition
+together with its proof. The optional \isacom{from} clause
+indicates which facts are to be used in the proof.
+Intermediate propositions are stated with \isacom{have}, the overall goal
+with \isacom{show}. A step can also introduce new local variables with
+\isacom{fix}. Logically, \isacom{fix} introduces \isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}}-quantified
+variables, \isacom{assume} introduces the assumption of an implication
+(\isa{{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}}) and \isacom{have}/\isacom{show} the conclusion.
+
+Propositions are optionally named formulas. These names can be referred to in
+later \isacom{from} clauses. In the simplest case, a fact is such a name.
+But facts can also be composed with \isa{OF} and \isa{of} as shown in
+\S\ref{sec:forward-proof}---hence the \dots\ in the above grammar. Note
+that assumptions, intermediate \isacom{have} statements and global lemmas all
+have the same status and are thus collectively referred to as
+\concept{facts}.
+
+Fact names can stand for whole lists of facts. For example, if \isa{f} is
+defined by command \isacom{fun}, \isa{f{\isaliteral{2E}{\isachardot}}simps} refers to the whole list of
+recursion equations defining \isa{f}. Individual facts can be selected by
+writing \isa{f{\isaliteral{2E}{\isachardot}}simps{\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}, whole sublists by \isa{f{\isaliteral{2E}{\isachardot}}simps{\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}{\isaliteral{2D}{\isacharminus}}{\isadigit{4}}{\isaliteral{29}{\isacharparenright}}}.
+
+
+\section{Isar by example}
+
+We show a number of proofs of Cantors theorem that a function from a set to
+its powerset cannot be surjective, illustrating various features of Isar. The
+constant \isa{surj} is predefined.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ surj{\isaliteral{28}{\isacharparenleft}}f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ set{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isadigit{0}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}surj\ f{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{from}\isamarkupfalse%
+\ {\isadigit{0}}\ \isacommand{have}\isamarkupfalse%
+\ {\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}A{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}a{\isaliteral{2E}{\isachardot}}\ A\ {\isaliteral{3D}{\isacharequal}}\ f\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ surj{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{from}\isamarkupfalse%
+\ {\isadigit{1}}\ \isacommand{have}\isamarkupfalse%
+\ {\isadigit{2}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}a{\isaliteral{2E}{\isachardot}}\ {\isaliteral{7B}{\isacharbraceleft}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ f\ x{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ blast\isanewline
+\ \ \isacommand{from}\isamarkupfalse%
+\ {\isadigit{2}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}False{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ blast\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The \isacom{proof} command lacks an explicit method how to perform
+the proof. In such cases Isabelle tries to use some standard introduction
+rule, in the above case for \isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}}:
+\[
+\inferrule{
+\mbox{\isa{P\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ False}}}
+{\mbox{\isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P}}}
+\]
+In order to prove \isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P}, assume \isa{P} and show \isa{False}.
+Thus we may assume \isa{surj\ f}. The proof shows that names of propositions
+may be (single!) digits---meaningful names are hard to invent and are often
+not necessary. Both \isacom{have} steps are obvious. The second one introduces
+the diagonal set \isa{{\isaliteral{7B}{\isacharbraceleft}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ f\ x{\isaliteral{7D}{\isacharbraceright}}}, the key idea in the proof.
+If you wonder why \isa{{\isadigit{2}}} directly implies \isa{False}: from \isa{{\isadigit{2}}}
+it follows that \isa{{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ f\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{5C3C696E3E}{\isasymin}}\ f\ a{\isaliteral{29}{\isacharparenright}}}.
+
+\subsection{\isa{this}, \isa{then}, \isa{hence} and \isa{thus}}
+
+Labels should be avoided. They interrupt the flow of the reader who has to
+scan the context for the point where the label was introduced. Ideally, the
+proof is a linear flow, where the output of one step becomes the input of the
+next step, piping the previously proved fact into the next proof, just like
+in a UNIX pipe. In such cases the predefined name \isa{this} can be used
+to refer to the proposition proved in the previous step. This allows us to
+eliminate all labels from our proof (we suppress the \isacom{lemma} statement):%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}surj\ f{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{from}\isamarkupfalse%
+\ this\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}a{\isaliteral{2E}{\isachardot}}\ {\isaliteral{7B}{\isacharbraceleft}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ f\ x{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ surj{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{from}\isamarkupfalse%
+\ this\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}False{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ blast\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+We have also taken the opportunity to compress the two \isacom{have}
+steps into one.
+
+To compact the text further, Isar has a few convenient abbreviations:
+\medskip
+
+\begin{tabular}{rcl}
+\isacom{then} &=& \isacom{from} \isa{this}\\
+\isacom{thus} &=& \isacom{then} \isacom{show}\\
+\isacom{hence} &=& \isacom{then} \isacom{have}
+\end{tabular}
+\medskip
+
+\noindent
+With the help of these abbreviations the proof becomes%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}surj\ f{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{hence}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}a{\isaliteral{2E}{\isachardot}}\ {\isaliteral{7B}{\isacharbraceleft}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ f\ x{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ surj{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}False{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ blast\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+There are two further linguistic variations:
+\medskip
+
+\begin{tabular}{rcl}
+(\isacom{have}$\mid$\isacom{show}) \ \textit{prop} \ \isacom{using} \ \textit{facts}
+&=&
+\isacom{from} \ \textit{facts} \ (\isacom{have}$\mid$\isacom{show}) \ \textit{prop}\\
+\isacom{with} \ \textit{facts} &=& \isacom{from} \ \textit{facts} \isa{this}
+\end{tabular}
+\medskip
+
+\noindent The \isacom{using} idiom de-emphasises the used facts by moving them
+behind the proposition.
+
+\subsection{Structured lemma statements: \isacom{fixes}, \isacom{assumes}, \isacom{shows}}
+
+Lemmas can also be stated in a more structured fashion. To demonstrate this
+feature with Cantor's theorem, we rephrase \noquotes{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ surj\ f{\isaliteral{22}{\isachardoublequote}}}}
+a little:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\isanewline
+\ \ \isakeyword{fixes}\ f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isakeyword{assumes}\ s{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}surj\ f{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}False{\isaliteral{22}{\isachardoublequoteclose}}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+The optional \isacom{fixes} part allows you to state the types of
+variables up front rather than by decorating one of their occurrences in the
+formula with a type constraint. The key advantage of the structured format is
+the \isacom{assumes} part that allows you to name each assumption. The
+\isacom{shows} part gives the goal. The actual theorem that will come out of
+the proof is \isa{surj\ f\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ False}, but during the proof the assumption
+\isa{surj\ f} is available under the name \isa{s} like any other fact.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{2D}{\isacharminus}}\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}\ a{\isaliteral{2E}{\isachardot}}\ {\isaliteral{7B}{\isacharbraceleft}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ f\ x{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{using}\isamarkupfalse%
+\ s\isanewline
+\ \ \ \ \isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ surj{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}False{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ blast\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+In the \isacom{have} step the assumption \isa{surj\ f} is now
+referenced by its name \isa{s}. The duplication of \isa{surj\ f} in the
+above proofs (once in the statement of the lemma, once in its proof) has been
+eliminated.
+
+\begin{warn}
+Note the dash after the \isacom{proof}
+command. It is the null method that does nothing to the goal. Leaving it out
+would ask Isabelle to try some suitable introduction rule on the goal \isa{False}---but there is no suitable introduction rule and \isacom{proof}
+would fail.
+\end{warn}
+
+Stating a lemmas with \isacom{assumes}-\isacom{shows} implicitly introduces the
+name \isa{assms} that stands for the list of all assumptions. You can refer
+to individual assumptions by \isa{assms{\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}}, \isa{assms{\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}} etc,
+thus obviating the need to name them individually.
+
+\section{Proof patterns}
+
+We show a number of important basic proof patterns. Many of them arise from
+the rules of natural deduction that are applied by \isacom{proof} by
+default. The patterns are phrased in terms of \isacom{show} but work for
+\isacom{have} and \isacom{lemma}, too.
+
+We start with two forms of \concept{case distinction}:
+starting from a formula \isa{P} we have the two cases \isa{P} and
+\isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P}, and starting from a fact \isa{P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q}
+we have the two cases \isa{P} and \isa{Q}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}R{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\ cases\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}R{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}R{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+&
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}R{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}R{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Q{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}R{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+\end{tabular}
+\medskip
+\begin{isamarkuptext}%
+How to prove a logical equivalence:
+\end{isamarkuptext}%
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ Q{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Q{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Q{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\medskip
+\begin{isamarkuptext}%
+Proofs by contradiction:
+\end{isamarkuptext}%
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}False{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+&
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}rule\ ccontr{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}P{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}False{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+\end{tabular}
+\medskip
+\begin{isamarkuptext}%
+The name \isa{ccontr} stands for ``classical contradiction''.
+
+How to prove quantified formulas:
+\end{isamarkuptext}%
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ P{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ x%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+&
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ P{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{28}{\isacharparenleft}}witness{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+\end{tabular}
+\medskip
+\begin{isamarkuptext}%
+In the proof of \noquotes{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ P{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}},
+the step \isacom{fix}~\isa{x} introduces a locale fixed variable \isa{x}
+into the subproof, the proverbial ``arbitrary but fixed value''.
+Instead of \isa{x} we could have chosen any name in the subproof.
+In the proof of \noquotes{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ P{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}},
+\isa{witness} is some arbitrary
+term for which we can prove that it satisfies \isa{P}.
+
+How to reason forward from \noquotes{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ P{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}:
+\end{isamarkuptext}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ P{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{then}\isamarkupfalse%
+\ \isacommand{obtain}\isamarkupfalse%
+\ x\ \isakeyword{where}\ p{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ blast%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+After the \isacom{obtain} step, \isa{x} (we could have chosen any name)
+is a fixed local
+variable, and \isa{p} is the name of the fact
+\noquotes{\isa{{\isaliteral{22}{\isachardoublequote}}P{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}.
+This pattern works for one or more \isa{x}.
+As an example of the \isacom{obtain} command, here is the proof of
+Cantor's theorem in more detail:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ surj{\isaliteral{28}{\isacharparenleft}}f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ set{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}surj\ f{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{hence}\isamarkupfalse%
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}a{\isaliteral{2E}{\isachardot}}\ {\isaliteral{7B}{\isacharbraceleft}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ f\ x{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ surj{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{obtain}\isamarkupfalse%
+\ a\ \isakeyword{where}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{7B}{\isacharbraceleft}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ f\ x{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \ \isacommand{by}\isamarkupfalse%
+\ blast\isanewline
+\ \ \isacommand{hence}\isamarkupfalse%
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{5C3C6E6F74696E3E}{\isasymnotin}}\ f\ a\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ a\ {\isaliteral{5C3C696E3E}{\isasymin}}\ f\ a{\isaliteral{22}{\isachardoublequoteclose}}\ \ \isacommand{by}\isamarkupfalse%
+\ blast\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}False{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ blast\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+How to prove set equality and subset relationship:
+\end{isamarkuptext}%
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{3D}{\isacharequal}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}B\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ A{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+&
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ x\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ B{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+\end{tabular}
+\begin{isamarkuptext}%
+\section{Streamlining proofs}
+
+\subsection{Pattern matching and quotations}
+
+In the proof patterns shown above, formulas are often duplicated.
+This can make the text harder to read, write and maintain. Pattern matching
+is an abbreviation mechanism to avoid such duplication. Writing
+\begin{quote}
+\isacom{show} \ \textit{formula} \isa{{\isaliteral{28}{\isacharparenleft}}}\isacom{is} \textit{pattern}\isa{{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+matches the pattern against the formula, thus instantiating the unknowns in
+the pattern for later use. As an example, consider the proof pattern for
+\isa{{\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}}:
+\end{isamarkuptext}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}formula\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ formula\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{is}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}L\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ {\isaliteral{3F}{\isacharquery}}R{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}L{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}R{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}R{\isaliteral{22}{\isachardoublequoteclose}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}L{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Instead of duplicating \isa{formula\isaliteral{5C3C5E697375623E}{}\isactrlisub i} in the text, we introduce
+the two abbreviations \isa{{\isaliteral{3F}{\isacharquery}}L} and \isa{{\isaliteral{3F}{\isacharquery}}R} by pattern matching.
+Pattern matching works wherever a formula is stated, in particular
+with \isacom{have} and \isacom{lemma}.
+
+The unknown \isa{{\isaliteral{3F}{\isacharquery}}thesis} is implicitly matched against any goal stated by
+\isacom{lemma} or \isacom{show}. Here is a typical example:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}formula{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{2D}{\isacharminus}}%
+\\\mbox{}\quad$\vdots$\\\mbox{}\hspace{-1.4ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Unknowns can also be instantiated with \isacom{let} commands
+\begin{quote}
+\isacom{let} \isa{{\isaliteral{3F}{\isacharquery}}t} = \isa{{\isaliteral{22}{\isachardoublequote}}}\textit{some-big-term}\isa{{\isaliteral{22}{\isachardoublequote}}}
+\end{quote}
+Later proof steps can refer to \isa{{\isaliteral{3F}{\isacharquery}}t}:
+\begin{quote}
+\isacom{have} \isa{{\isaliteral{22}{\isachardoublequote}}}\dots \isa{{\isaliteral{3F}{\isacharquery}}t} \dots\isa{{\isaliteral{22}{\isachardoublequote}}}
+\end{quote}
+\begin{warn}
+Names of facts are introduced with \isa{name{\isaliteral{3A}{\isacharcolon}}} and refer to proved
+theorems. Unknowns \isa{{\isaliteral{3F}{\isacharquery}}X} refer to terms or formulas.
+\end{warn}
+
+Although abbreviations shorten the text, the reader needs to remember what
+they stand for. Similarly for names of facts. Names like \isa{{\isadigit{1}}}, \isa{{\isadigit{2}}}
+and \isa{{\isadigit{3}}} are not helpful and should only be used in short proofs. For
+longer proof, descriptive names are better. But look at this example:
+\begin{quote}
+\isacom{have} \ \isa{x{\isaliteral{5F}{\isacharunderscore}}gr{\isaliteral{5F}{\isacharunderscore}}{\isadigit{0}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3E}{\isachargreater}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequote}}}\\
+$\vdots$\\
+\isacom{from} \isa{x{\isaliteral{5F}{\isacharunderscore}}gr{\isaliteral{5F}{\isacharunderscore}}{\isadigit{0}}} \dots
+\end{quote}
+The name is longer than the fact it stands for! Short facts do not need names,
+one can refer to them easily by quoting them:
+\begin{quote}
+\isacom{have} \ \isa{{\isaliteral{22}{\isachardoublequote}}x\ {\isaliteral{3E}{\isachargreater}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequote}}}\\
+$\vdots$\\
+\isacom{from} \isa{{\isaliteral{60}{\isacharbackquote}}x{\isaliteral{3E}{\isachargreater}}{\isadigit{0}}{\isaliteral{60}{\isacharbackquote}}} \dots
+\end{quote}
+Note that the quotes around \isa{x{\isaliteral{3E}{\isachargreater}}{\isadigit{0}}} are \concept{back quotes}.
+They refer to the fact not by name but by value.
+
+\subsection{\isacom{moreover}}
+
+Sometimes one needs a number of facts to enable some deduction. Of course
+one can name these facts individually, as shown on the right,
+but one can also combine them with \isacom{moreover}, as shown on the left:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{moreover}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{moreover}\isamarkupfalse%
+%
+\\$\vdots$\\\hspace{-1.4ex}
+\isacommand{moreover}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{ultimately}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{22}{\isachardoublequoteclose}}\ \ %
+\ $\dots$\\
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+&
+\qquad
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{have}\isamarkupfalse%
+\ lab\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}P\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{have}\isamarkupfalse%
+\ lab\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}P\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$
+%
+\\$\vdots$\\\hspace{-1.4ex}
+\isacommand{have}\isamarkupfalse%
+\ lab\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}P\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequoteclose}}\ %
+\ $\dots$\\
+\isacommand{from}\isamarkupfalse%
+\ lab\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ lab\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}%
+\ $\dots$\\
+\isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{22}{\isachardoublequoteclose}}\ \ %
+\ $\dots$\\
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+\end{tabular}
+\begin{isamarkuptext}%
+The \isacom{moreover} version is no shorter but expresses the structure more
+clearly and avoids new names.
+
+\subsection{Raw proof blocks}
+
+Sometimes one would like to prove some lemma locally with in a proof.
+A lemma that shares the current context of assumptions but that
+has its own assumptions and is generalised over its locally fixed
+variables at the end. This is what a \concept{raw proof block} does:
+\begin{quote}
+\isa{{\isaliteral{7B}{\isacharbraceleft}}} \isacom{fix} \isa{x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n}\\
+\mbox{}\ \ \ \isacom{assume} \isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub m}\\
+\mbox{}\ \ \ $\vdots$\\
+\mbox{}\ \ \ \isacom{have} \isa{B}\\
+\isa{{\isaliteral{7D}{\isacharbraceright}}}
+\end{quote}
+proves \isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3B}{\isacharsemicolon}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub m\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B}
+where all \isa{x\isaliteral{5C3C5E697375623E}{}\isactrlisub i} have been replaced by unknowns \isa{{\isaliteral{3F}{\isacharquery}}x\isaliteral{5C3C5E697375623E}{}\isactrlisub i}.
+\begin{warn}
+The conclusion of a raw proof block is \emph{not} indicated by \isacom{show}
+but is simply the final \isacom{have}.
+\end{warn}
+
+As an example we prove a simple fact about divisibility on integers.
+The definition of \isa{dvd} is \isa{{\isaliteral{28}{\isacharparenleft}}b\ dvd\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}k{\isaliteral{2E}{\isachardot}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2A}{\isacharasterisk}}\ k{\isaliteral{29}{\isacharparenright}}}.
+\end{isamarkuptext}%
+\isacommand{lemma}\isamarkupfalse%
+\ \isakeyword{fixes}\ a\ b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int\ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}b\ dvd\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2B}{\isacharplus}}b{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}b\ dvd\ a{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{2D}{\isacharminus}}\isanewline
+\ \ \isacommand{{\isaliteral{7B}{\isacharbraceleft}}}\isamarkupfalse%
+\ \isacommand{fix}\isamarkupfalse%
+\ k\ \isacommand{assume}\isamarkupfalse%
+\ k{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}a{\isaliteral{2B}{\isacharplus}}b\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2A}{\isacharasterisk}}k{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \ \ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}k{\isaliteral{27}{\isacharprime}}{\isaliteral{2E}{\isachardot}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2A}{\isacharasterisk}}k{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \ \ \isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \ \ \ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{using}\isamarkupfalse%
+\ k\ \isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ algebra{\isaliteral{5F}{\isacharunderscore}}simps{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \ \ \isacommand{qed}\isamarkupfalse%
+\ \isacommand{{\isaliteral{7D}{\isacharbraceright}}}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{using}\isamarkupfalse%
+\ assms\ \isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}auto\ simp\ add{\isaliteral{3A}{\isacharcolon}}\ dvd{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Note that the result of a raw proof block has no name. In this example
+it was directly piped (via \isacom{then}) into the final proof, but it can
+also be named for later reference: you simply follow the block directly by a
+\isacom{note} command:
+\begin{quote}
+\isacom{note} \ \isa{name\ {\isaliteral{3D}{\isacharequal}}\ this}
+\end{quote}
+This introduces a new name \isa{name} that refers to \isa{this},
+the fact just proved, in this case the preceding block. In general,
+\isacom{note} introduces a new name for one or more facts.
+
+\section{Case distinction and induction}
+
+\subsection{Datatype case distinction}
+
+We have seen case distinction on formulas. Now we want to distinguish
+which form some term takes: is it \isa{{\isadigit{0}}} or of the form \isa{Suc\ n},
+is it \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} or of the form \isa{x\ {\isaliteral{23}{\isacharhash}}\ xs}, etc. Here is a typical example
+proof by case distinction on the form of \isa{xs}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}length{\isaliteral{28}{\isacharparenleft}}tl\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ length\ xs\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}cases\ xs{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}xs\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ y\ ys\ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}xs\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{23}{\isacharhash}}ys{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Function \isa{tl} (''tail'') is defined by \isa{tl\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} and
+\isa{tl\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ xs}. Note that the result type of \isa{length} is \isa{nat}
+and \isa{{\isadigit{0}}\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}}.
+
+This proof pattern works for any term \isa{t} whose type is a datatype.
+The goal has to be proved for each constructor \isa{C}:
+\begin{quote}
+\isacom{fix} \ \isa{x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n} \isacom{assume} \isa{{\isaliteral{22}{\isachardoublequote}}t\ {\isaliteral{3D}{\isacharequal}}\ C\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}}
+\end{quote}
+Each case can be written in a more compact form by means of the \isacom{case}
+command:
+\begin{quote}
+\isacom{case} \isa{{\isaliteral{28}{\isacharparenleft}}C\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+This is equivalent to the explicit \isacom{fix}-\isacom{assumen} line
+but also gives the assumption \isa{{\isaliteral{22}{\isachardoublequote}}t\ {\isaliteral{3D}{\isacharequal}}\ C\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}} a name: \isa{C},
+like the constructor.
+Here is the \isacom{case} version of the proof above:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}cases\ xs{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ Nil\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Cons\ y\ ys{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Remember that \isa{Nil} and \isa{Cons} are the alphanumeric names
+for \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}} and \isa{{\isaliteral{23}{\isacharhash}}}. The names of the assumptions
+are not used because they are directly piped (via \isacom{thus})
+into the proof of the claim.
+
+\subsection{Structural induction}
+
+We illustrate structural induction with an example based on natural numbers:
+the sum (\isa{{\isaliteral{5C3C53756D3E}{\isasymSum}}}) of the first \isa{n} natural numbers
+(\isa{{\isaliteral{7B}{\isacharbraceleft}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}n{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{7D}{\isacharbraceright}}}) is equal to \mbox{\isa{n\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}}}.
+Never mind the details, just focus on the pattern:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C53756D3E}{\isasymSum}}{\isaliteral{7B}{\isacharbraceleft}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}n{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{is}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}P\ n{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induction\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C53756D3E}{\isasymSum}}{\isaliteral{7B}{\isacharbraceleft}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isadigit{0}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ n\ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C53756D3E}{\isasymSum}}{\isaliteral{7B}{\isacharbraceleft}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}n{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C53756D3E}{\isasymSum}}{\isaliteral{7B}{\isacharbraceleft}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}Suc\ n{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ n{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Except for the rewrite steps, everything is explicitly given. This
+makes the proof easily readable, but the duplication means it is tedious to
+write and maintain. Here is how pattern
+matching can completely avoid any duplication:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C53756D3E}{\isasymSum}}{\isaliteral{7B}{\isacharbraceleft}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}n{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{28}{\isacharparenleft}}\isakeyword{is}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}P\ n{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induction\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}P\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{fix}\isamarkupfalse%
+\ n\ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}P\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{3F}{\isacharquery}}P{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The first line introduces an abbreviation \isa{{\isaliteral{3F}{\isacharquery}}P\ n} for the goal.
+Pattern matching \isa{{\isaliteral{3F}{\isacharquery}}P\ n} with the goal instantiates \isa{{\isaliteral{3F}{\isacharquery}}P} to the
+function \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C53756D3E}{\isasymSum}}{\isaliteral{7B}{\isacharbraceleft}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}n{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ div\ {\isadigit{2}}}. Now the proposition to
+be proved in the base case can be written as \isa{{\isaliteral{3F}{\isacharquery}}P\ {\isadigit{0}}}, the induction
+hypothesis as \isa{{\isaliteral{3F}{\isacharquery}}P\ n}, and the conclusion of the induction step as
+\isa{{\isaliteral{3F}{\isacharquery}}P{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}}.
+
+Induction also provides the \isacom{case} idiom that abbreviates
+the \isacom{fix}-\isacom{assume} step. The above proof becomes%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induction\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ {\isadigit{0}}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The unknown \isa{{\isaliteral{3F}{\isacharquery}}case} is set in each case to the required
+claim, i.e.\ \isa{{\isaliteral{3F}{\isacharquery}}P\ {\isadigit{0}}} and \mbox{\isa{{\isaliteral{3F}{\isacharquery}}P{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}}} in the above proof,
+without requiring the user to define a \isa{{\isaliteral{3F}{\isacharquery}}P}. The general
+pattern for induction over \isa{nat} is shown on the left-hand side:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{tabular}{@ {}ll@ {}}
+\begin{minipage}[t]{.4\textwidth}
+\isa{%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}induction\ n{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ {\isadigit{0}}%
+\\\mbox{}\ \ $\vdots$\\\mbox{}\hspace{-1ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ %
+\ $\dots$\\
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}%
+\\\mbox{}\ \ $\vdots$\\\mbox{}\hspace{-1ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+&
+\begin{minipage}[t]{.4\textwidth}
+~\\
+~\\
+\isacom{let} \isa{{\isaliteral{3F}{\isacharquery}}case\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequote}}P{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}\\
+~\\
+~\\
+~\\[1ex]
+\isacom{fix} \isa{n} \isacom{assume} \isa{Suc{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequote}}P{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}\\
+\isacom{let} \isa{{\isaliteral{3F}{\isacharquery}}case\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequote}}P{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}\\
+\end{minipage}
+\end{tabular}
+\medskip
+%
+\begin{isamarkuptext}%
+On the right side you can see what the \isacom{case} command
+on the left stands for.
+
+In case the goal is an implication, induction does one more thing: the
+proposition to be proved in each case is not the whole implication but only
+its conclusion; the premises of the implication are immediately made
+assumptions of that case. That is, if in the above proof we replace
+\isacom{show}~\isa{P{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}} by
+\mbox{\isacom{show}~\isa{A{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}}}
+then \isacom{case}~\isa{{\isadigit{0}}} stands for
+\begin{quote}
+\isacom{assume} \ \isa{{\isadigit{0}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequote}}A{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}\\
+\isacom{let} \isa{{\isaliteral{3F}{\isacharquery}}case\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequote}}P{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}
+\end{quote}
+and \isacom{case}~\isa{{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}} stands for
+\begin{quote}
+\isacom{fix} \isa{n}\\
+\isacom{assume} \isa{Suc{\isaliteral{3A}{\isacharcolon}}}
+ \begin{tabular}[t]{l}\isa{A{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}}\\\isa{A{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}}\end{tabular}\\
+\isacom{let} \isa{{\isaliteral{3F}{\isacharquery}}case\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequote}}P{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}
+\end{quote}
+The list of assumptions \isa{Suc} is actually subdivided
+into \isa{Suc{\isaliteral{2E}{\isachardot}}IH}, the induction hypotheses (here \isa{A{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}})
+and \isa{Suc{\isaliteral{2E}{\isachardot}}prems}, the premises of the goal being proved
+(here \isa{A{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}}).
+
+Induction works for any datatype.
+Proving a goal \isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{3B}{\isacharsemicolon}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub k{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{29}{\isacharparenright}}}
+by induction on \isa{x} generates a proof obligation for each constructor
+\isa{C} of the datatype. The command \isa{case\ {\isaliteral{28}{\isacharparenleft}}C\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}}
+performs the following steps:
+\begin{enumerate}
+\item \isacom{fix} \isa{x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n}
+\item \isacom{assume} the induction hypotheses (calling them \isa{C{\isaliteral{2E}{\isachardot}}IH})
+ and the premises \mbox{\isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub i{\isaliteral{28}{\isacharparenleft}}C\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}}} (calling them \isa{C{\isaliteral{2E}{\isachardot}}prems})
+ and calling the whole list \isa{C}
+\item \isacom{let} \isa{{\isaliteral{3F}{\isacharquery}}case\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequote}}P{\isaliteral{28}{\isacharparenleft}}C\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}
+\end{enumerate}
+
+\subsection{Rule induction}
+
+Recall the inductive and recursive definitions of even numbers in
+\autoref{sec:inductive-defs}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{inductive}\isamarkupfalse%
+\ ev\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+ev{\isadigit{0}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}ev\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+evSS{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev{\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isanewline
+\isacommand{fun}\isamarkupfalse%
+\ even\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}even\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ True{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}even\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ False{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}even\ {\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ even\ n{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+We recast the proof of \isa{ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ even\ n} in Isar. The
+left column shows the actual proof text, the right column shows
+the implicit effect of the two \isacom{case} commands:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\begin{tabular}{@ {}l@ {\qquad}l@ {}}
+\begin{minipage}[t]{.5\textwidth}
+\isa{%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ even\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ rule{\isaliteral{3A}{\isacharcolon}}\ ev{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ ev{\isadigit{0}}\isanewline
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ evSS\isanewline
+\isanewline
+\isanewline
+\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+}
+\end{minipage}
+&
+\begin{minipage}[t]{.5\textwidth}
+~\\
+~\\
+\isacom{let} \isa{{\isaliteral{3F}{\isacharquery}}case\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequote}}even\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequote}}}\\
+~\\
+~\\
+\isacom{fix} \isa{n}\\
+\isacom{assume} \isa{evSS{\isaliteral{3A}{\isacharcolon}}}
+ \begin{tabular}[t]{l} \isa{ev\ n}\\\isa{even\ n}\end{tabular}\\
+\isacom{let} \isa{{\isaliteral{3F}{\isacharquery}}case\ {\isaliteral{3D}{\isacharequal}}\ even{\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}}\\
+\end{minipage}
+\end{tabular}
+\medskip
+%
+\begin{isamarkuptext}%
+The proof resembles structural induction, but the induction rule is given
+explicitly and the names of the cases are the names of the rules in the
+inductive definition.
+Let us examine the two assumptions named \isa{evSS}:
+\isa{ev\ n} is the premise of rule \isa{evSS}, which we may assume
+because we are in the case where that rule was used; \isa{even\ n}
+is the induction hypothesis.
+\begin{warn}
+Because each \isacom{case} command introduces a list of assumptions
+named like the case name, which is the name of a rule of the inductive
+definition, those rules now need to be accessed with a qualified name, here
+\isa{ev{\isaliteral{2E}{\isachardot}}ev{\isadigit{0}}} and \isa{ev{\isaliteral{2E}{\isachardot}}evSS}
+\end{warn}
+
+In the case \isa{evSS} of the proof above we have pretended that the
+system fixes a variable \isa{n}. But unless the user provides the name
+\isa{n}, the system will just invent its own name that cannot be referred
+to. In the above proof, we do not need to refer to it, hence we do not give
+it a specific name. In case one needs to refer to it one writes
+\begin{quote}
+\isacom{case} \isa{{\isaliteral{28}{\isacharparenleft}}evSS\ m{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+just like \isacom{case}~\isa{{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}} in earlier structural inductions.
+The name \isa{m} is an arbitrary choice. As a result,
+case \isa{evSS} is derived from a renamed version of
+rule \isa{evSS}: \isa{ev\ m\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev{\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}}.
+Here is an example with a (contrived) intermediate step that refers to \isa{m}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ even\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ rule{\isaliteral{3A}{\isacharcolon}}\ ev{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ ev{\isadigit{0}}\ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}evSS\ m{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}even{\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ m{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ even\ m{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ simp\isanewline
+\ \ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ \isacommand{using}\isamarkupfalse%
+\ {\isaliteral{60}{\isacharbackquoteopen}}even\ m{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ blast\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\indent
+In general, let \isa{I} be a (for simplicity unary) inductively defined
+predicate and let the rules in the definition of \isa{I}
+be called \isa{rule\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}, \dots, \isa{rule\isaliteral{5C3C5E697375623E}{}\isactrlisub n}. A proof by rule
+induction follows this pattern:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{show}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}I\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\isacommand{proof}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ rule{\isaliteral{3A}{\isacharcolon}}\ I{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ rule\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}%
+\\[-.4ex]\mbox{}\ \ $\vdots$\\[-.4ex]\mbox{}\hspace{-1ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ %
+\ $\dots$\\
+\isacommand{next}\isamarkupfalse%
+%
+\\[-.4ex]$\vdots$\\[-.4ex]\mbox{}\hspace{-1ex}
+\isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{case}\isamarkupfalse%
+\ rule\isaliteral{5C3C5E697375623E}{}\isactrlisub n%
+\\[-.4ex]\mbox{}\ \ $\vdots$\\[-.4ex]\mbox{}\hspace{-1ex}
+\ \ \isacommand{show}\isamarkupfalse%
+\ {\isaliteral{3F}{\isacharquery}}case\ %
+\ $\dots$\\
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+One can provide explicit variable names by writing
+\isacom{case}~\isa{{\isaliteral{28}{\isacharparenleft}}rule\isaliteral{5C3C5E697375623E}{}\isactrlisub i\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub k{\isaliteral{29}{\isacharparenright}}}, thus renaming the first \isa{k}
+free variables in rule \isa{i} to \isa{x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub k},
+going through rule \isa{i} from left to right.
+
+\subsection{Assumption naming}
+
+In any induction, \isacom{case}~\isa{name} sets up a list of assumptions
+also called \isa{name}, which is subdivided into three parts:
+\begin{description}
+\item[\isa{name{\isaliteral{2E}{\isachardot}}IH}] contains the induction hypotheses.
+\item[\isa{name{\isaliteral{2E}{\isachardot}}hyps}] contains all the other hypotheses of this case in the
+induction rule. For rule inductions these are the hypotheses of rule
+\isa{name}, for structural inductions these are empty.
+\item[\isa{name{\isaliteral{2E}{\isachardot}}prems}] contains the (suitably instantiated) premises
+of the statement being proved, i.e. the \isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub i} when
+proving \isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{3B}{\isacharsemicolon}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A}.
+\end{description}
+\begin{warn}
+Proof method \isa{induct} differs from \isa{induction}
+only in this naming policy: \isa{induct} does not distinguish
+\isa{IH} from \isa{hyps} but subsumes \isa{IH} under \isa{hyps}.
+\end{warn}
+
+More complicated inductive proofs than the ones we have seen so far
+often need to refer to specific assumptions---just \isa{name} or even
+\isa{name{\isaliteral{2E}{\isachardot}}prems} and \isa{name{\isaliteral{2E}{\isachardot}}IH} can be too unspecific.
+This is where the indexing of fact lists comes in handy, e.g.\
+\isa{name{\isaliteral{2E}{\isachardot}}IH{\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}} or \isa{name{\isaliteral{2E}{\isachardot}}prems{\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isaliteral{2D}{\isacharminus}}{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}.
+
+\subsection{Rule inversion}
+
+Rule inversion is case distinction on which rule could have been used to
+derive some fact. The name \concept{rule inversion} emphasizes that we are
+reasoning backwards: by which rules could some given fact have been proved?
+For the inductive definition of \isa{ev}, rule inversion can be summarized
+like this:
+\begin{isabelle}%
+ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}k{\isaliteral{2E}{\isachardot}}\ n\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ k{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ ev\ k{\isaliteral{29}{\isacharparenright}}%
+\end{isabelle}
+The realisation in Isabelle is a case distinction.
+A simple example is the proof that \isa{ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}. We
+already went through the details informally in \autoref{sec:Logic:even}. This
+is the Isar proof:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}ev\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{from}\isamarkupfalse%
+\ this\ \isacommand{have}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}ev{\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+\ \ \isacommand{proof}\isamarkupfalse%
+\ cases\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ ev{\isadigit{0}}\ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}ev{\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ ev{\isaliteral{2E}{\isachardot}}ev{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{next}\isamarkupfalse%
+\isanewline
+\ \ \ \ \isacommand{case}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}evSS\ k{\isaliteral{29}{\isacharparenright}}\ \isacommand{thus}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}ev{\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ ev{\isaliteral{2E}{\isachardot}}evSS{\isaliteral{29}{\isacharparenright}}\isanewline
+\ \ \isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The key point here is that a case distinction over some inductively
+defined predicate is triggered by piping the given fact
+(here: \isacom{from}~\isa{this}) into a proof by \isa{cases}.
+Let us examine the assumptions available in each case. In case \isa{ev{\isadigit{0}}}
+we have \isa{n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}} and in case \isa{evSS} we have \isa{n\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ k{\isaliteral{29}{\isacharparenright}}}
+and \isa{ev\ k}. In each case the assumptions are available under the name
+of the case; there is no fine grained naming schema like for induction.
+
+Sometimes some rules could not have beed used to derive the given fact
+because constructors clash. As an extreme example consider
+rule inversion applied to \isa{ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}}: neither rule \isa{ev{\isadigit{0}}} nor
+rule \isa{evSS} can yield \isa{ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}} because \isa{Suc\ {\isadigit{0}}} unifies
+neither with \isa{{\isadigit{0}}} nor with \isa{Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}}. Impossible cases do not
+have to be proved. Hence we can prove anything from \isa{ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}ev{\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{then}\isamarkupfalse%
+\ \isacommand{have}\isamarkupfalse%
+\ P\ \isacommand{by}\isamarkupfalse%
+\ cases%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+That is, \isa{ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}} is simply not provable:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ ev{\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{proof}\isamarkupfalse%
+\isanewline
+\ \ \isacommand{assume}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}ev{\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{then}\isamarkupfalse%
+\ \isacommand{show}\isamarkupfalse%
+\ False\ \isacommand{by}\isamarkupfalse%
+\ cases\isanewline
+\isacommand{qed}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Normally not all cases will be impossible. As a simple exercise,
+prove that \mbox{\isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}}.}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/document/LaTeXsugar.tex Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,35 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{LaTeXsugar}%
+%
+\isadelimtheory
+\isanewline
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/document/Logic.tex Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,944 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Logic}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\begin{isamarkuptext}%
+\vspace{-5ex}
+\section{Logic and Proof Beyond Equality}
+\label{sec:Logic}
+
+\subsection{Formulas}
+
+The basic syntax of formulas (\textit{form} below)
+provides the standard logical constructs, in decreasing precedence:
+\[
+\begin{array}{rcl}
+
+\mathit{form} & ::= &
+ \isa{{\isaliteral{28}{\isacharparenleft}}form{\isaliteral{29}{\isacharparenright}}} ~\mid~
+ \isa{True} ~\mid~
+ \isa{False} ~\mid~
+ \isa{term\ {\isaliteral{3D}{\isacharequal}}\ term}\\
+ &\mid& \isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ form} ~\mid~
+ \isa{form\ {\isaliteral{5C3C616E643E}{\isasymand}}\ form} ~\mid~
+ \isa{form\ {\isaliteral{5C3C6F723E}{\isasymor}}\ form} ~\mid~
+ \isa{form\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ form}\\
+ &\mid& \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ form} ~\mid~ \isa{{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ form}
+\end{array}
+\]
+Terms are the ones we have seen all along, built from constant, variables,
+function application and \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}}-abstraction, including all the syntactic
+sugar like infix symbols, \isa{if}, \isa{case} etc.
+\begin{warn}
+Remember that formulas are simply terms of type \isa{bool}. Hence
+\isa{{\isaliteral{3D}{\isacharequal}}} also works for formulas. Beware that \isa{{\isaliteral{3D}{\isacharequal}}} has a higher
+precedence than the other logical operators. Hence \isa{s\ {\isaliteral{3D}{\isacharequal}}\ t\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A} means
+\isa{{\isaliteral{28}{\isacharparenleft}}s\ {\isaliteral{3D}{\isacharequal}}\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A}, and \isa{A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{3D}{\isacharequal}}\ B\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A} means \isa{A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}B\ {\isaliteral{3D}{\isacharequal}}\ A{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B}.
+Logical equivalence can also be written with
+\isa{{\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}} instead of \isa{{\isaliteral{3D}{\isacharequal}}}, where \isa{{\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}} has the same low
+precedence as \isa{{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}}. Hence \isa{A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ B\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A} really means
+\isa{{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}B\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A{\isaliteral{29}{\isacharparenright}}}.
+\end{warn}
+\begin{warn}
+Quantifiers need to be enclosed in parentheses if they are nested within
+other constructs (just like \isa{if}, \isa{case} and \isa{let}).
+\end{warn}
+The most frequent logical symbols have the following ASCII representations:
+\begin{center}
+\begin{tabular}{l@ {\qquad}l@ {\qquad}l}
+\isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}} & \xsymbol{forall} & \texttt{ALL}\\
+\isa{{\isaliteral{5C3C6578697374733E}{\isasymexists}}} & \xsymbol{exists} & \texttt{EX}\\
+\isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}} & \xsymbol{lambda} & \texttt{\%}\\
+\isa{{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}} & \texttt{-{}->}\\
+\isa{{\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}} & \texttt{<-{}->}\\
+\isa{{\isaliteral{5C3C616E643E}{\isasymand}}} & \texttt{/\char`\\} & \texttt{\&}\\
+\isa{{\isaliteral{5C3C6F723E}{\isasymor}}} & \texttt{\char`\\/} & \texttt{|}\\
+\isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}} & \xsymbol{not} & \texttt{\char`~}\\
+\isa{{\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}} & \xsymbol{noteq} & \texttt{\char`~=}
+\end{tabular}
+\end{center}
+The first column shows the symbols, the second column ASCII representations
+that Isabelle interfaces convert into the corresponding symbol,
+and the third column shows ASCII representations that stay fixed.
+\begin{warn}
+The implication \isa{{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}} is part of the Isabelle framework. It structures
+theorems and proof states, separating assumptions from conclusion.
+The implication \isa{{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}} is part of the logic HOL and can occur inside the
+formulas that make up the assumptions and conclusion.
+Theorems should be of the form \isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{3B}{\isacharsemicolon}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A},
+not \isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ A}. Both are logically equivalent
+but the first one works better when using the theorem in further proofs.
+\end{warn}
+
+\subsection{Sets}
+
+Sets of elements of type \isa{{\isaliteral{27}{\isacharprime}}a} have type \isa{{\isaliteral{27}{\isacharprime}}a\ set}.
+They can be finite or infinite. Sets come with the usual notations:
+\begin{itemize}
+\item \isa{{\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}},\quad \isa{{\isaliteral{7B}{\isacharbraceleft}}e\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}e\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{7D}{\isacharbraceright}}}
+\item \isa{e\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A},\quad \isa{A\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ B}
+\item \isa{A\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ B},\quad \isa{A\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ B},\quad \isa{A\ {\isaliteral{2D}{\isacharminus}}\ B},\quad \isa{{\isaliteral{2D}{\isacharminus}}\ A}
+\end{itemize}
+and much more. \isa{UNIV} is the set of all elements of some type.
+Set comprehension is written \isa{{\isaliteral{7B}{\isacharbraceleft}}x{\isaliteral{2E}{\isachardot}}\ P{\isaliteral{7D}{\isacharbraceright}}}
+rather than \isa{{\isaliteral{7B}{\isacharbraceleft}}x\ {\isaliteral{7C}{\isacharbar}}\ P{\isaliteral{7D}{\isacharbraceright}}}, to emphasize the variable binding nature
+of the construct.
+\begin{warn}
+In \isa{{\isaliteral{7B}{\isacharbraceleft}}x{\isaliteral{2E}{\isachardot}}\ P{\isaliteral{7D}{\isacharbraceright}}} the \isa{x} must be a variable. Set comprehension
+involving a proper term \isa{t} must be written
+\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{7B}{\isacharbraceleft}}t\ {\isaliteral{7C}{\isacharbar}}x\ y\ z{\isaliteral{2E}{\isachardot}}\ P{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequote}}},
+where \isa{x\ y\ z} are the free variables in \isa{t}.
+This is just a shorthand for \isa{{\isaliteral{7B}{\isacharbraceleft}}v{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}x\ y\ z{\isaliteral{2E}{\isachardot}}\ v\ {\isaliteral{3D}{\isacharequal}}\ t\ {\isaliteral{5C3C616E643E}{\isasymand}}\ P{\isaliteral{7D}{\isacharbraceright}}}, where
+\isa{v} is a new variable.
+\end{warn}
+
+Here are the ASCII representations of the mathematical symbols:
+\begin{center}
+\begin{tabular}{l@ {\quad}l@ {\quad}l}
+\isa{{\isaliteral{5C3C696E3E}{\isasymin}}} & \texttt{\char`\\\char`\<in>} & \texttt{:}\\
+\isa{{\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}} & \texttt{\char`\\\char`\<subseteq>} & \texttt{<=}\\
+\isa{{\isaliteral{5C3C756E696F6E3E}{\isasymunion}}} & \texttt{\char`\\\char`\<union>} & \texttt{Un}\\
+\isa{{\isaliteral{5C3C696E7465723E}{\isasyminter}}} & \texttt{\char`\\\char`\<inter>} & \texttt{Int}
+\end{tabular}
+\end{center}
+Sets also allow bounded quantifications \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{5C3C696E3E}{\isasymin}}A{\isaliteral{2E}{\isachardot}}\ P} and
+\isa{{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{5C3C696E3E}{\isasymin}}A{\isaliteral{2E}{\isachardot}}\ P}.
+
+\subsection{Proof automation}
+
+So far we have only seen \isa{simp} and \isa{auto}: Both perform
+rewriting, both can also prove linear arithmetic facts (no multiplication),
+and \isa{auto} is also able to prove simple logical or set-theoretic goals:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}y{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+\isanewline
+%
+\endisadelimproof
+\isanewline
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ B\ {\isaliteral{5C3C696E7465723E}{\isasyminter}}\ C\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C73756273657465713E}{\isasymsubseteq}}\ B\ {\isaliteral{5C3C756E696F6E3E}{\isasymunion}}\ C{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ auto%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+where
+\begin{quote}
+\isacom{by} \textit{proof-method}
+\end{quote}
+is short for
+\begin{quote}
+\isacom{apply} \textit{proof-method}\\
+\isacom{done}
+\end{quote}
+The key characteristics of both \isa{simp} and \isa{auto} are
+\begin{itemize}
+\item They show you were they got stuck, giving you an idea how to continue.
+\item They perform the obvious steps but are highly incomplete.
+\end{itemize}
+A proof method is \concept{complete} if it can prove all true formulas.
+There is no complete proof method for HOL, not even in theory.
+Hence all our proof methods only differ in how incomplete they are.
+
+A proof method that is still incomplete but tries harder than \isa{auto} is
+\isa{fastforce}. It either succeeds or fails, it acts on the first
+subgoal only, and it can be modified just like \isa{auto}, e.g.\
+with \isa{simp\ add}. Here is a typical example of what \isa{fastforce}
+can do:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}xs\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}ys{\isaliteral{2E}{\isachardot}}\ xs\ {\isaliteral{3D}{\isacharequal}}\ ys\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{3B}{\isacharsemicolon}}\ \ us\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
+\ \ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}n{\isaliteral{2E}{\isachardot}}\ length\ us\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{2B}{\isacharplus}}n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ fastforce%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+This lemma is out of reach for \isa{auto} because of the
+quantifiers. Even \isa{fastforce} fails when the quantifier structure
+becomes more complicated. In a few cases, its slow version \isa{force}
+succeeds where \isa{fastforce} fails.
+
+The method of choice for complex logical goals is \isa{blast}. In the
+following example, \isa{T} and \isa{A} are two binary predicates, and it
+is shown that \isa{T} is total, \isa{A} is antisymmetric and \isa{T} is
+a subset of \isa{A}, then \isa{A} is a subset of \isa{T}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\isanewline
+\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x\ y{\isaliteral{2E}{\isachardot}}\ T\ x\ y\ {\isaliteral{5C3C6F723E}{\isasymor}}\ T\ y\ x{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+\ \ \ \ \ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x\ y{\isaliteral{2E}{\isachardot}}\ A\ x\ y\ {\isaliteral{5C3C616E643E}{\isasymand}}\ A\ y\ x\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ x\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{3B}{\isacharsemicolon}}\isanewline
+\ \ \ \ \ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x\ y{\isaliteral{2E}{\isachardot}}\ T\ x\ y\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ A\ x\ y\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
+\ \ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x\ y{\isaliteral{2E}{\isachardot}}\ A\ x\ y\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ T\ x\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ blast%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+We leave it to the reader to figure out why this lemma is true.
+Method \isa{blast}
+\begin{itemize}
+\item is (in principle) a complete proof procedure for first-order formulas,
+ a fragment of HOL. In practice there is a search bound.
+\item does no rewriting and knows very little about equality.
+\item covers logic, sets and relations.
+\item either succeeds or fails.
+\end{itemize}
+Because of its strength in logic and sets and its weakness in equality reasoning, it complements the earlier proof methods.
+
+
+\subsubsection{Sledgehammer}
+
+Command \isacom{sledgehammer} calls a number of external automatic
+theorem provers (ATPs) that run for up to 30 seconds searching for a
+proof. Some of these ATPs are part of the Isabelle installation, others are
+queried over the internet. If successful, a proof command is generated and can
+be inserted into your proof. The biggest win of \isacom{sledgehammer} is
+that it will take into account the whole lemma library and you do not need to
+feed in any lemma explicitly. For example,%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ xs\ {\isaliteral{40}{\isacharat}}\ ys\ {\isaliteral{3D}{\isacharequal}}\ ys\ {\isaliteral{40}{\isacharat}}\ xs{\isaliteral{3B}{\isacharsemicolon}}\ \ length\ xs\ {\isaliteral{3D}{\isacharequal}}\ length\ ys\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ xs\ {\isaliteral{3D}{\isacharequal}}\ ys{\isaliteral{22}{\isachardoublequoteclose}}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+cannot be solved by any of the standard proof methods, but
+\isacom{sledgehammer} finds the following proof:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{by}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}metis\ append{\isaliteral{5F}{\isacharunderscore}}eq{\isaliteral{5F}{\isacharunderscore}}conv{\isaliteral{5F}{\isacharunderscore}}conj{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+We do not explain how the proof was found but what this command
+means. For a start, Isabelle does not trust external tools (and in particular
+not the translations from Isabelle's logic to those tools!)
+and insists on a proof that it can check. This is what \isa{metis} does.
+It is given a list of lemmas and tries to find a proof just using those lemmas
+(and pure logic). In contrast to \isa{simp} and friends that know a lot of
+lemmas already, using \isa{metis} manually is tedious because one has
+to find all the relevant lemmas first. But that is precisely what
+\isacom{sledgehammer} does for us.
+In this case lemma \isa{append{\isaliteral{5F}{\isacharunderscore}}eq{\isaliteral{5F}{\isacharunderscore}}conv{\isaliteral{5F}{\isacharunderscore}}conj} alone suffices:
+\begin{isabelle}%
+{\isaliteral{28}{\isacharparenleft}}xs\ {\isaliteral{40}{\isacharat}}\ ys\ {\isaliteral{3D}{\isacharequal}}\ zs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}xs\ {\isaliteral{3D}{\isacharequal}}\ take\ {\isaliteral{28}{\isacharparenleft}}length\ xs{\isaliteral{29}{\isacharparenright}}\ zs\ {\isaliteral{5C3C616E643E}{\isasymand}}\ ys\ {\isaliteral{3D}{\isacharequal}}\ drop\ {\isaliteral{28}{\isacharparenleft}}length\ xs{\isaliteral{29}{\isacharparenright}}\ zs{\isaliteral{29}{\isacharparenright}}%
+\end{isabelle}
+We leave it to the reader to figure out why this lemma suffices to prove
+the above lemma, even without any knowledge of what the functions \isa{take}
+and \isa{drop} do. Keep in mind that the variables in the two lemmas
+are independent of each other, despite the same names, and that you can
+substitute arbitrary values for the free variables in a lemma.
+
+Just as for the other proof methods we have seen, there is no guarantee that
+\isacom{sledgehammer} will find a proof if it exists. Nor is
+\isacom{sledgehammer} superior to the other proof methods. They are
+incomparable. Therefore it is recommended to apply \isa{simp} or \isa{auto} before invoking \isacom{sledgehammer} on what is left.
+
+\subsubsection{Arithmetic}
+
+By arithmetic formulas we mean formulas involving variables, numbers, \isa{{\isaliteral{2B}{\isacharplus}}}, \isa{{\isaliteral{2D}{\isacharminus}}}, \isa{{\isaliteral{3D}{\isacharequal}}}, \isa{{\isaliteral{3C}{\isacharless}}}, \isa{{\isaliteral{5C3C6C653E}{\isasymle}}} and the usual logical
+connectives \isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}}, \isa{{\isaliteral{5C3C616E643E}{\isasymand}}}, \isa{{\isaliteral{5C3C6F723E}{\isasymor}}}, \isa{{\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}},
+\isa{{\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}}. Strictly speaking, this is known as \concept{linear arithmetic}
+because it does not involve multiplication, although multiplication with
+numbers, e.g.\ \isa{{\isadigit{2}}{\isaliteral{2A}{\isacharasterisk}}n} is allowed. Such formulas can be proved by
+\isa{arith}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ x\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{3B}{\isacharsemicolon}}\ {\isadigit{2}}{\isaliteral{2A}{\isacharasterisk}}x\ {\isaliteral{3C}{\isacharless}}\ c\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isadigit{2}}{\isaliteral{2A}{\isacharasterisk}}a\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isadigit{2}}{\isaliteral{2A}{\isacharasterisk}}b\ {\isaliteral{2B}{\isacharplus}}\ c{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+\ arith%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+In fact, \isa{auto} and \isa{simp} can prove many linear
+arithmetic formulas already, like the one above, by calling a weak but fast
+version of \isa{arith}. Hence it is usually not necessary to invoke
+\isa{arith} explicitly.
+
+The above example involves natural numbers, but integers (type \isa{int})
+and real numbers (type \isa{real}) are supported as well. As are a number
+of further operators like \isa{min} and \isa{max}. On \isa{nat} and
+\isa{int}, \isa{arith} can even prove theorems with quantifiers in them,
+but we will not enlarge on that here.
+
+
+\subsection{Single step proofs}
+
+Although automation is nice, it often fails, at least initially, and you need
+to find out why. When \isa{fastforce} or \isa{blast} simply fail, you have
+no clue why. At this point, the stepwise
+application of proof rules may be necessary. For example, if \isa{blast}
+fails on \isa{A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B}, you want to attack the two
+conjuncts \isa{A} and \isa{B} separately. This can
+be achieved by applying \emph{conjunction introduction}
+\[ \isa{\mbox{}\inferrule{\mbox{{\isaliteral{3F}{\isacharquery}}P}\\\ \mbox{{\isaliteral{3F}{\isacharquery}}Q}}{\mbox{{\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{3F}{\isacharquery}}Q}}}\ \isa{conjI}
+\]
+to the proof state. We will now examine the details of this process.
+
+\subsubsection{Instantiating unknowns}
+
+We had briefly mentioned earlier that after proving some theorem,
+Isabelle replaces all free variables \isa{x} by so called \concept{unknowns}
+\isa{{\isaliteral{3F}{\isacharquery}}x}. We can see this clearly in rule \isa{conjI}.
+These unknowns can later be instantiated explicitly or implicitly:
+\begin{itemize}
+\item By hand, using \isa{of}.
+The expression \isa{conjI{\isaliteral{5B}{\isacharbrackleft}}of\ {\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3D}{\isacharequal}}b{\isaliteral{22}{\isachardoublequote}}\ {\isaliteral{22}{\isachardoublequote}}False{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5D}{\isacharbrackright}}}
+instantiates the unknowns in \isa{conjI} from left to right with the
+two formulas \isa{a{\isaliteral{3D}{\isacharequal}}b} and \isa{False}, yielding the rule
+\begin{isabelle}%
+\mbox{}\inferrule{\mbox{a\ {\isaliteral{3D}{\isacharequal}}\ b}\\\ \mbox{False}}{\mbox{a\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{5C3C616E643E}{\isasymand}}\ False}}%
+\end{isabelle}
+
+In general, \isa{th{\isaliteral{5B}{\isacharbrackleft}}of\ string\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ string\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{5D}{\isacharbrackright}}} instantiates
+the unknowns in the theorem \isa{th} from left to right with the terms
+\isa{string\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}} to \isa{string\isaliteral{5C3C5E697375623E}{}\isactrlisub n}.
+
+\item By unification. \concept{Unification} is the process of making two
+terms syntactically equal by suitable instantiations of unknowns. For example,
+unifying \isa{{\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{3F}{\isacharquery}}Q} with \mbox{\isa{a\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{5C3C616E643E}{\isasymand}}\ False}} instantiates
+\isa{{\isaliteral{3F}{\isacharquery}}P} with \isa{a\ {\isaliteral{3D}{\isacharequal}}\ b} and \isa{{\isaliteral{3F}{\isacharquery}}Q} with \isa{False}.
+\end{itemize}
+We need not instantiate all unknowns. If we want to skip a particular one we
+can just write \isa{{\isaliteral{5F}{\isacharunderscore}}} instead, for example \isa{conjI{\isaliteral{5B}{\isacharbrackleft}}of\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{22}{\isachardoublequote}}False{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5D}{\isacharbrackright}}}.
+Unknowns can also be instantiated by name, for example
+\isa{conjI{\isaliteral{5B}{\isacharbrackleft}}where\ {\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequote}}a{\isaliteral{3D}{\isacharequal}}b{\isaliteral{22}{\isachardoublequote}}\ and\ {\isaliteral{3F}{\isacharquery}}Q\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequote}}False{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5D}{\isacharbrackright}}}.
+
+
+\subsubsection{Rule application}
+
+\concept{Rule application} means applying a rule backwards to a proof state.
+For example, applying rule \isa{conjI} to a proof state
+\begin{quote}
+\isa{{\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B}
+\end{quote}
+results in two subgoals, one for each premise of \isa{conjI}:
+\begin{quote}
+\isa{{\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A}\\
+\isa{{\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ \ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B}
+\end{quote}
+In general, the application of a rule \isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{3B}{\isacharsemicolon}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A}
+to a subgoal \mbox{\isa{{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C}} proceeds in two steps:
+\begin{enumerate}
+\item
+Unify \isa{A} and \isa{C}, thus instantiating the unknowns in the rule.
+\item
+Replace the subgoal \isa{C} with \isa{n} new subgoals \isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}} to \isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub n}.
+\end{enumerate}
+This is the command to apply rule \isa{xyz}:
+\begin{quote}
+\isacom{apply}\isa{{\isaliteral{28}{\isacharparenleft}}rule\ xyz{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+This is also called \concept{backchaining} with rule \isa{xyz}.
+
+\subsubsection{Introduction rules}
+
+Conjunction introduction (\isa{conjI}) is one example of a whole
+class of rules known as \concept{introduction rules}. They explain under which
+premises some logical construct can be introduced. Here are some further
+useful introduction rules:
+\[
+\inferrule*[right=\mbox{\isa{impI}}]{\mbox{\isa{{\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}Q}}}{\mbox{\isa{{\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}Q}}}
+\qquad
+\inferrule*[right=\mbox{\isa{allI}}]{\mbox{\isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}P\ x}}}{\mbox{\isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}P\ x}}}
+\]
+\[
+\inferrule*[right=\mbox{\isa{iffI}}]{\mbox{\isa{{\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}Q}} \\ \mbox{\isa{{\isaliteral{3F}{\isacharquery}}Q\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}P}}}
+ {\mbox{\isa{{\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}Q}}}
+\]
+These rules are part of the logical system of \concept{natural deduction}
+(e.g.\ \cite{HuthRyan}). Although we intentionally de-emphasize the basic rules
+of logic in favour of automatic proof methods that allow you to take bigger
+steps, these rules are helpful in locating where and why automation fails.
+When applied backwards, these rules decompose the goal:
+\begin{itemize}
+\item \isa{conjI} and \isa{iffI} split the goal into two subgoals,
+\item \isa{impI} moves the left-hand side of a HOL implication into the list of assumptions,
+\item and \isa{allI} removes a \isa{{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}} by turning the quantified variable into a fixed local variable of the subgoal.
+\end{itemize}
+Isabelle knows about these and a number of other introduction rules.
+The command
+\begin{quote}
+\isacom{apply} \isa{rule}
+\end{quote}
+automatically selects the appropriate rule for the current subgoal.
+
+You can also turn your own theorems into introduction rules by giving them
+them \isa{intro} attribute, analogous to the \isa{simp} attribute. In
+that case \isa{blast}, \isa{fastforce} and (to a limited extent) \isa{auto} will automatically backchain with those theorems. The \isa{intro}
+attribute should be used with care because it increases the search space and
+can lead to nontermination. Sometimes it is better to use it only in a
+particular calls of \isa{blast} and friends. For example,
+\isa{le{\isaliteral{5F}{\isacharunderscore}}trans}, transitivity of \isa{{\isaliteral{5C3C6C653E}{\isasymle}}} on type \isa{nat},
+is not an introduction rule by default because of the disastrous effect
+on the search space, but can be useful in specific situations:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ b{\isaliteral{3B}{\isacharsemicolon}}\ b\ {\isaliteral{5C3C6C653E}{\isasymle}}\ c{\isaliteral{3B}{\isacharsemicolon}}\ c\ {\isaliteral{5C3C6C653E}{\isasymle}}\ d{\isaliteral{3B}{\isacharsemicolon}}\ d\ {\isaliteral{5C3C6C653E}{\isasymle}}\ e\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ {\isaliteral{5C3C6C653E}{\isasymle}}\ e{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}blast\ intro{\isaliteral{3A}{\isacharcolon}}\ le{\isaliteral{5F}{\isacharunderscore}}trans{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Of course this is just an example and could be proved by \isa{arith}, too.
+
+\subsubsection{Forward proof}
+\label{sec:forward-proof}
+
+Forward proof means deriving new theorems from old theorems. We have already
+seen a very simple form of forward proof: the \isa{of} operator for
+instantiating unknowns in a theorem. The big brother of \isa{of} is \isa{OF} for applying one theorem to others. Given a theorem \isa{A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B} called
+\isa{r} and a theorem \isa{A{\isaliteral{27}{\isacharprime}}} called \isa{r{\isaliteral{27}{\isacharprime}}}, the theorem \isa{r{\isaliteral{5B}{\isacharbrackleft}}OF\ r{\isaliteral{27}{\isacharprime}}{\isaliteral{5D}{\isacharbrackright}}} is the result of applying \isa{r} to \isa{r{\isaliteral{27}{\isacharprime}}}, where \isa{r} should be viewed as a function taking a theorem \isa{A} and returning
+\isa{B}. More precisely, \isa{A} and \isa{A{\isaliteral{27}{\isacharprime}}} are unified, thus
+instantiating the unknowns in \isa{B}, and the result is the instantiated
+\isa{B}. Of course, unification may also fail.
+\begin{warn}
+Application of rules to other rules operates in the forward direction: from
+the premises to the conclusion of the rule; application of rules to proof
+states operates in the backward direction, from the conclusion to the
+premises.
+\end{warn}
+
+In general \isa{r} can be of the form \isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{3B}{\isacharsemicolon}}\ A\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ A}
+and there can be multiple argument theorems \isa{r\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}} to \isa{r\isaliteral{5C3C5E697375623E}{}\isactrlisub m}
+(with \isa{m\ {\isaliteral{5C3C6C653E}{\isasymle}}\ n}), in which case \isa{r{\isaliteral{5B}{\isacharbrackleft}}OF\ r\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ r\isaliteral{5C3C5E697375623E}{}\isactrlisub m{\isaliteral{5D}{\isacharbrackright}}} is obtained
+by unifying and thus proving \isa{A\isaliteral{5C3C5E697375623E}{}\isactrlisub i} with \isa{r\isaliteral{5C3C5E697375623E}{}\isactrlisub i}, \isa{i\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{5C3C646F74733E}{\isasymdots}}m}.
+Here is an example, where \isa{refl} is the theorem
+\isa{{\isaliteral{3F}{\isacharquery}}t\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}t}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{thm}\isamarkupfalse%
+\ conjI{\isaliteral{5B}{\isacharbrackleft}}OF\ refl{\isaliteral{5B}{\isacharbrackleft}}of\ {\isaliteral{22}{\isachardoublequoteopen}}a{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{5D}{\isacharbrackright}}\ refl{\isaliteral{5B}{\isacharbrackleft}}of\ {\isaliteral{22}{\isachardoublequoteopen}}b{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{5D}{\isacharbrackright}}%
+\begin{isamarkuptext}%
+yields the theorem \isa{a\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{5C3C616E643E}{\isasymand}}\ b\ {\isaliteral{3D}{\isacharequal}}\ b}.
+The command \isacom{thm} merely displays the result.
+
+Forward reasoning also makes sense in connection with proof states.
+Therefore \isa{blast}, \isa{fastforce} and \isa{auto} support a modifier
+\isa{dest} which instructs the proof method to use certain rules in a
+forward fashion. If \isa{r} is of the form \mbox{\isa{A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B}}, the modifier
+\mbox{\isa{dest{\isaliteral{3A}{\isacharcolon}}\ r}}
+allows proof search to reason forward with \isa{r}, i.e.\
+to replace an assumption \isa{A{\isaliteral{27}{\isacharprime}}}, where \isa{A{\isaliteral{27}{\isacharprime}}} unifies with \isa{A},
+with the correspondingly instantiated \isa{B}. For example, \isa{Suc{\isaliteral{5F}{\isacharunderscore}}leD} is the theorem \mbox{\isa{Suc\ m\ {\isaliteral{5C3C6C653E}{\isasymle}}\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ m\ {\isaliteral{5C3C6C653E}{\isasymle}}\ n}}, which works well for forward reasoning:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}Suc{\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ a{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ b\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ {\isaliteral{5C3C6C653E}{\isasymle}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}blast\ dest{\isaliteral{3A}{\isacharcolon}}\ Suc{\isaliteral{5F}{\isacharunderscore}}leD{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+In this particular example we could have backchained with
+\isa{Suc{\isaliteral{5F}{\isacharunderscore}}leD}, too, but because the premise is more complicated than the conclusion this can easily lead to nontermination.
+
+\begin{warn}
+To ease readability we will drop the question marks
+in front of unknowns from now on.
+\end{warn}
+
+\section{Inductive definitions}
+\label{sec:inductive-defs}
+
+Inductive definitions are the third important definition facility, after
+datatypes and recursive function. In fact, they are the key construct in the
+definition of operational semantics in the second part of the book.
+
+\subsection{An example: even numbers}
+\label{sec:Logic:even}
+
+Here is a simple example of an inductively defined predicate:
+\begin{itemize}
+\item 0 is even
+\item If $n$ is even, so is $n+2$.
+\end{itemize}
+The operative word ``inductive'' means that these are the only even numbers.
+In Isabelle we give the two rules the names \isa{ev{\isadigit{0}}} and \isa{evSS}
+and write%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{inductive}\isamarkupfalse%
+\ ev\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+ev{\isadigit{0}}{\isaliteral{3A}{\isacharcolon}}\ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}ev\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+evSS{\isaliteral{3A}{\isacharcolon}}\ \ %
+\isa{ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}
+%
+\begin{isamarkuptext}%
+To get used to inductive definitions, we will first prove a few
+properties of \isa{ev} informally before we descend to the Isabelle level.
+
+How do we prove that some number is even, e.g.\ \isa{ev\ {\isadigit{4}}}? Simply by combining the defining rules for \isa{ev}:
+\begin{quote}
+\isa{ev\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ ev\ {\isadigit{4}}}
+\end{quote}
+
+\subsubsection{Rule induction}
+
+Showing that all even numbers have some property is more complicated. For
+example, let us prove that the inductive definition of even numbers agrees
+with the following recursive one:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ even\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}even\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ True{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}even\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ False{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}even\ {\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ even\ n{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+We prove \isa{ev\ m\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ even\ m}. That is, we
+assume \isa{ev\ m} and by induction on the form of its derivation
+prove \isa{even\ m}. There are two cases corresponding to the two rules
+for \isa{ev}:
+\begin{description}
+\item[Case \isa{ev{\isadigit{0}}}:]
+ \isa{ev\ m} was derived by rule \isa{ev\ {\isadigit{0}}}: \\
+ \isa{{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}} \isa{m\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}} \isa{{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}} \isa{even\ m\ {\isaliteral{3D}{\isacharequal}}\ even\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ True}
+\item[Case \isa{evSS}:]
+ \isa{ev\ m} was derived by rule \isa{ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}: \\
+\isa{{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}} \isa{m\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}} and by induction hypothesis \isa{even\ n}\\
+\isa{{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}} \isa{even\ m\ {\isaliteral{3D}{\isacharequal}}\ even{\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ even\ n\ {\isaliteral{3D}{\isacharequal}}\ True}
+\end{description}
+
+What we have just seen is a special case of \concept{rule induction}.
+Rule induction applies to propositions of this form
+\begin{quote}
+\isa{ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ n}
+\end{quote}
+That is, we want to prove a property \isa{P\ n}
+for all even \isa{n}. But if we assume \isa{ev\ n}, then there must be
+some derivation of this assumption using the two defining rules for
+\isa{ev}. That is, we must prove
+\begin{description}
+\item[Case \isa{ev{\isadigit{0}}}:] \isa{P\ {\isadigit{0}}}
+\item[Case \isa{evSS}:] \isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}ev\ n{\isaliteral{3B}{\isacharsemicolon}}\ P\ n{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}
+\end{description}
+The corresponding rule is called \isa{ev{\isaliteral{2E}{\isachardot}}induct} and looks like this:
+\[
+\inferrule{
+\mbox{\isa{ev\ n}}\\
+\mbox{\isa{P\ {\isadigit{0}}}}\\
+\mbox{\isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}ev\ n{\isaliteral{3B}{\isacharsemicolon}}\ P\ n{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}}}
+{\mbox{\isa{P\ n}}}
+\]
+The first premise \isa{ev\ n} enforces that this rule can only be applied
+in situations where we know that \isa{n} is even.
+
+Note that in the induction step we may not just assume \isa{P\ n} but also
+\mbox{\isa{ev\ n}}, which is simply the premise of rule \isa{evSS}. Here is an example where the local assumption \isa{ev\ n} comes in
+handy: we prove \isa{ev\ m\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}} by induction on \isa{ev\ m}.
+Case \isa{ev{\isadigit{0}}} requires us to prove \isa{ev\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}}, which follows
+from \isa{ev\ {\isadigit{0}}} because \isa{{\isadigit{0}}\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}} on type \isa{nat}. In
+case \isa{evSS} we have \mbox{\isa{m\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}}} and may assume
+\isa{ev\ n}, which implies \isa{ev\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}} because \isa{m\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ n}. We did not need the induction hypothesis at all for this proof,
+it is just a case distinction on which rule was used, but having \isa{ev\ n} at our disposal in case \isa{evSS} was essential.
+This case distinction over rules is also called ``rule inversion''
+and is discussed in more detail in \autoref{ch:Isar}.
+
+\subsubsection{In Isabelle}
+
+Let us now recast the above informal proofs in Isabelle. For a start,
+we use \isa{Suc} terms instead of numerals in rule \isa{evSS}:
+\begin{isabelle}%
+ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}%
+\end{isabelle}
+This avoids the difficulty of unifying \isa{n{\isaliteral{2B}{\isacharplus}}{\isadigit{2}}} with some numeral,
+which is not automatic.
+
+The simplest way to prove \isa{ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}} is in a forward
+direction: \isa{evSS{\isaliteral{5B}{\isacharbrackleft}}OF\ evSS{\isaliteral{5B}{\isacharbrackleft}}OF\ ev{\isadigit{0}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{5D}{\isacharbrackright}}} yields the theorem \isa{ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}}. Alternatively, you can also prove it as a lemma in backwards
+fashion. Although this is more verbose, it allows us to demonstrate how each
+rule application changes the proof state:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}ev{\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}%
+\end{isabelle}%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}rule\ evSS{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}%
+\end{isabelle}%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}rule\ evSS{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ ev\ {\isadigit{0}}%
+\end{isabelle}%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}rule\ ev{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\indent
+Rule induction is applied by giving the induction rule explicitly via the
+\isa{rule{\isaliteral{3A}{\isacharcolon}}} modifier:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}ev\ m\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ even\ m{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ rule{\isaliteral{3A}{\isacharcolon}}\ ev{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{by}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}simp{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+Both cases are automatic. Note that if there are multiple assumptions
+of the form \isa{ev\ t}, method \isa{induction} will induct on the leftmost
+one.
+
+As a bonus, we also prove the remaining direction of the equivalence of
+\isa{ev} and \isa{even}:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}even\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ n\ rule{\isaliteral{3A}{\isacharcolon}}\ even{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+This is a proof by computation induction on \isa{n} (see
+\autoref{sec:recursive-funs}) that sets up three subgoals corresponding to
+the three equations for \isa{even}:
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ even\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ {\isadigit{0}}\isanewline
+\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ even\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
+\ {\isadigit{3}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}even\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ n{\isaliteral{3B}{\isacharsemicolon}}\ even\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ ev\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}%
+\end{isabelle}
+The first and third subgoals follow with \isa{ev{\isadigit{0}}} and \isa{evSS}, and the second subgoal is trivially true because \isa{even\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}} is \isa{False}:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{by}\isamarkupfalse%
+\ {\isaliteral{28}{\isacharparenleft}}simp{\isaliteral{5F}{\isacharunderscore}}all\ add{\isaliteral{3A}{\isacharcolon}}\ ev{\isadigit{0}}\ evSS{\isaliteral{29}{\isacharparenright}}%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+The rules for \isa{ev} make perfect simplification and introduction
+rules because their premises are always smaller than the conclusion. It
+makes sense to turn them into simplification and introduction rules
+permanently, to enhance proof automation:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{declare}\isamarkupfalse%
+\ ev{\isaliteral{2E}{\isachardot}}intros{\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{2C}{\isacharcomma}}intro{\isaliteral{5D}{\isacharbrackright}}%
+\begin{isamarkuptext}%
+The rules of an inductive definition are not simplification rules by
+default because, in contrast to recursive functions, there is no termination
+requirement for inductive definitions.
+
+\subsubsection{Inductive versus recursive}
+
+We have seen two definitions of the notion of evenness, an inductive and a
+recursive one. Which one is better? Much of the time, the recursive one is
+more convenient: it allows us to do rewriting in the middle of terms, and it
+expresses both the positive information (which numbers are even) and the
+negative information (which numbers are not even) directly. An inductive
+definition only expresses the positive information directly. The negative
+information, for example, that \isa{{\isadigit{1}}} is not even, has to be proved from
+it (by induction or rule inversion). On the other hand, rule induction is
+Taylor made for proving \mbox{\isa{ev\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ n}} because it only asks you
+to prove the positive cases. In the proof of \isa{even\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ n} by
+computation induction via \isa{even{\isaliteral{2E}{\isachardot}}induct}, we are also presented
+with the trivial negative cases. If you want the convenience of both
+rewriting and rule induction, you can make two definitions and show their
+equivalence (as above) or make one definition and prove additional properties
+from it, for example rule induction from computation induction.
+
+But many concepts do not admit a recursive definition at all because there is
+no datatype for the recursion (for example, the transitive closure of a
+relation), or the recursion would not terminate (for example, the operational
+semantics in the second part of this book). Even if there is a recursive
+definition, if we are only interested in the positive information, the
+inductive definition may be much simpler.
+
+\subsection{The reflexive transitive closure}
+\label{sec:star}
+
+Evenness is really more conveniently expressed recursively than inductively.
+As a second and very typical example of an inductive definition we define the
+reflexive transitive closure. It will also be an important building block for
+some of the semantics considered in the second part of the book.
+
+The reflexive transitive closure, called \isa{star} below, is a function
+that maps a binary predicate to another binary predicate: if \isa{r} is of
+type \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool} then \isa{star\ r} is again of type \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool}, and \isa{star\ r\ x\ y} means that \isa{x} and \isa{y} are in
+the relation \isa{star\ r}. Think \isa{r\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}} when you see \isa{star\ r}, because \isa{star\ r} is meant to be the reflexive transitive closure.
+That is, \isa{star\ r\ x\ y} is meant to be true if from \isa{x} we can
+reach \isa{y} in finitely many \isa{r} steps. This concept is naturally
+defined inductively:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{inductive}\isamarkupfalse%
+\ star\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \ \isakeyword{for}\ r\ \isakeyword{where}\isanewline
+refl{\isaliteral{3A}{\isacharcolon}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}star\ r\ x\ x{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+step{\isaliteral{3A}{\isacharcolon}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}r\ x\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ star\ r\ y\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ star\ r\ x\ z{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The base case \isa{refl} is reflexivity: \isa{x\ {\isaliteral{3D}{\isacharequal}}\ y}. The
+step case \isa{step} combines an \isa{r} step (from \isa{x} to
+\isa{y}) and a \isa{star} step (from \isa{y} to \isa{z}) into a
+\isa{star} step (from \isa{x} to \isa{z}).
+The ``\isacom{for}~\isa{r}'' in the header is merely a hint to Isabelle
+that \isa{r} is a fixed parameter of \isa{star}, in contrast to the
+further parameters of \isa{star}, which change. As a result, Isabelle
+generates a simpler induction rule.
+
+By definition \isa{star\ r} is reflexive. It is also transitive, but we
+need rule induction to prove that:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ star{\isaliteral{5F}{\isacharunderscore}}trans{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}star\ r\ x\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ star\ r\ y\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ star\ r\ x\ z{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ rule{\isaliteral{3A}{\isacharcolon}}\ star{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+The induction is over \isa{star\ r\ x\ y} and we try to prove
+\mbox{\isa{star\ r\ y\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ star\ r\ x\ z}},
+which we abbreviate by \isa{P\ x\ y}. These are our two subgoals:
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ star\ r\ x\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ star\ r\ x\ z\isanewline
+\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}u\ x\ y{\isaliteral{2E}{\isachardot}}\isanewline
+\isaindent{\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}r\ u\ x{\isaliteral{3B}{\isacharsemicolon}}\ star\ r\ x\ y{\isaliteral{3B}{\isacharsemicolon}}\ star\ r\ y\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ star\ r\ x\ z{\isaliteral{3B}{\isacharsemicolon}}\ star\ r\ y\ z{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
+\isaindent{\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ star\ r\ u\ z%
+\end{isabelle}
+The first one is \isa{P\ x\ x}, the result of case \isa{refl},
+and it is trivial.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}assumption{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+Let us examine subgoal \isa{{\isadigit{2}}}, case \isa{step}.
+Assumptions \isa{r\ u\ x} and \mbox{\isa{star\ r\ x\ y}}
+are the premises of rule \isa{step}.
+Assumption \isa{star\ r\ y\ z\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ star\ r\ x\ z} is \mbox{\isa{P\ x\ y}},
+the IH coming from \isa{star\ r\ x\ y}. We have to prove \isa{P\ u\ y},
+which we do by assuming \isa{star\ r\ y\ z} and proving \isa{star\ r\ u\ z}.
+The proof itself is straightforward: from \mbox{\isa{star\ r\ y\ z}} the IH
+leads to \isa{star\ r\ x\ z} which, together with \isa{r\ u\ x},
+leads to \mbox{\isa{star\ r\ u\ z}} via rule \isa{step}:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}metis\ step{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+\subsection{The general case}
+
+Inductive definitions have approximately the following general form:
+\begin{quote}
+\isacom{inductive} \isa{I\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7461753E}{\isasymtau}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequote}}} \isacom{where}
+\end{quote}
+followed by a sequence of (possibly named) rules of the form
+\begin{quote}
+\isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ I\ a\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{3B}{\isacharsemicolon}}\ I\ a\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ I\ a}
+\end{quote}
+separated by \isa{{\isaliteral{7C}{\isacharbar}}}. As usual, \isa{n} can be 0.
+The corresponding rule induction principle
+\isa{I{\isaliteral{2E}{\isachardot}}induct} applies to propositions of the form
+\begin{quote}
+\isa{I\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ x}
+\end{quote}
+where \isa{P} may itself be a chain of implications.
+\begin{warn}
+Rule induction is always on the leftmost premise of the goal.
+Hence \isa{I\ x} must be the first premise.
+\end{warn}
+Proving \isa{I\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ x} by rule induction means proving
+for every rule of \isa{I} that \isa{P} is invariant:
+\begin{quote}
+\isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ I\ a\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ P\ a\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{3B}{\isacharsemicolon}}\ I\ a\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{3B}{\isacharsemicolon}}\ P\ a\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ a}
+\end{quote}
+
+The above format for inductive definitions is simplified in a number of
+respects. \isa{I} can have any number of arguments and each rule can have
+additional premises not involving \isa{I}, so-called \concept{side
+conditions}. In rule inductions, these side-conditions appear as additional
+assumptions. The \isacom{for} clause seen in the definition of the reflexive
+transitive closure merely simplifies the form of the induction rule.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/Thys/document/Types_and_funs.tex Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,605 @@
+%
+\begin{isabellebody}%
+\def\isabellecontext{Types{\isaliteral{5F}{\isacharunderscore}}and{\isaliteral{5F}{\isacharunderscore}}funs}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\begin{isamarkuptext}%
+\vspace{-5ex}
+\section{Type and function definitions}
+
+Type synonyms are abbreviations for existing types, for example%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}\isamarkupfalse%
+\ string\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}char\ list{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+Type synonyms are expanded after parsing and are not present in internal representation and output. They are mere conveniences for the reader.
+
+\subsection{Datatypes}
+
+The general form of a datatype definition looks like this:
+\begin{quote}
+\begin{tabular}{@ {}rclcll}
+\isacom{datatype} \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}a\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}t}
+ & = & $C_1 \ \isa{{\isaliteral{22}{\isachardoublequote}}}\tau_{1,1}\isa{{\isaliteral{22}{\isachardoublequote}}} \dots \isa{{\isaliteral{22}{\isachardoublequote}}}\tau_{1,n_1}\isa{{\isaliteral{22}{\isachardoublequote}}}$ \\
+ & $|$ & \dots \\
+ & $|$ & $C_k \ \isa{{\isaliteral{22}{\isachardoublequote}}}\tau_{k,1}\isa{{\isaliteral{22}{\isachardoublequote}}} \dots \isa{{\isaliteral{22}{\isachardoublequote}}}\tau_{k,n_k}\isa{{\isaliteral{22}{\isachardoublequote}}}$
+\end{tabular}
+\end{quote}
+It introduces the constructors \
+$C_i :: \tau_{i,1}\Rightarrow \cdots \Rightarrow \tau_{i,n_i} \Rightarrow$~\isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}a\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}t} \ and expresses that any value of this type is built from these constructors in a unique manner. Uniqueness is implied by the following
+properties of the constructors:
+\begin{itemize}
+\item \emph{Distinctness:} $C_i\ \ldots \neq C_j\ \dots$ \quad if $i \neq j$
+\item \emph{Injectivity:}
+\begin{tabular}[t]{l}
+ $(C_i \ x_1 \dots x_{n_i} = C_i \ y_1 \dots y_{n_i}) =$\\
+ $(x_1 = y_1 \land \dots \land x_{n_i} = y_{n_i})$
+\end{tabular}
+\end{itemize}
+The fact that any value of the datatype is built from the constructors implies
+the structural induction rule: to show
+$P~x$ for all $x$ of type \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}a\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}t},
+one needs to show $P(C_i\ x_1 \dots x_{n_i})$ (for each $i$) assuming
+$P(x_j)$ for all $j$ where $\tau_{i,j} =$~\isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{27}{\isacharprime}}a\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}t}.
+Distinctness and injectivity are applied automatically by \isa{auto}
+and other proof methods. Induction must be applied explicitly.
+
+Datatype values can be taken apart with case-expressions, for example
+\begin{quote}
+\noquotes{\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}case\ xs\ of\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isadigit{0}}\ {\isaliteral{7C}{\isacharbar}}\ x\ {\isaliteral{23}{\isacharhash}}\ {\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ Suc\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}}
+\end{quote}
+just like in functional programming languages. Case expressions
+must be enclosed in parentheses.
+
+As an example, consider binary trees:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{datatype}\isamarkupfalse%
+\ {\isaliteral{27}{\isacharprime}}a\ tree\ {\isaliteral{3D}{\isacharequal}}\ Tip\ {\isaliteral{7C}{\isacharbar}}\ Node\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ tree{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ tree{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+with a mirror function:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ mirror\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ tree\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ tree{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}mirror\ Tip\ {\isaliteral{3D}{\isacharequal}}\ Tip{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}mirror\ {\isaliteral{28}{\isacharparenleft}}Node\ l\ a\ r{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Node\ {\isaliteral{28}{\isacharparenleft}}mirror\ r{\isaliteral{29}{\isacharparenright}}\ a\ {\isaliteral{28}{\isacharparenleft}}mirror\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The following lemma illustrates induction:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}mirror{\isaliteral{28}{\isacharparenleft}}mirror\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ t{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ t{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+yields
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ mirror\ {\isaliteral{28}{\isacharparenleft}}mirror\ Tip{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Tip\isanewline
+\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}t{\isadigit{1}}\ a\ t{\isadigit{2}}{\isaliteral{2E}{\isachardot}}\isanewline
+\isaindent{\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}mirror\ {\isaliteral{28}{\isacharparenleft}}mirror\ t{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ t{\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ mirror\ {\isaliteral{28}{\isacharparenleft}}mirror\ t{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ t{\isadigit{2}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
+\isaindent{\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ mirror\ {\isaliteral{28}{\isacharparenleft}}mirror\ {\isaliteral{28}{\isacharparenleft}}Node\ t{\isadigit{1}}\ a\ t{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Node\ t{\isadigit{1}}\ a\ t{\isadigit{2}}%
+\end{isabelle}
+The induction step contains two induction hypotheses, one for each subtree.
+An application of \isa{auto} finishes the proof.
+
+A very simple but also very useful datatype is the predefined
+\begin{isabelle}%
+\isacommand{datatype}\ {\isaliteral{27}{\isacharprime}}a\ option\ {\isaliteral{3D}{\isacharequal}}\ None\ {\isaliteral{7C}{\isacharbar}}\ Some\ {\isaliteral{27}{\isacharprime}}a%
+\end{isabelle}
+Its sole purpose is to add a new element \isa{None} to an existing
+type \isa{{\isaliteral{27}{\isacharprime}}a}. To make sure that \isa{None} is distinct from all the
+elements of \isa{{\isaliteral{27}{\isacharprime}}a}, you wrap them up in \isa{Some} and call
+the new type \isa{{\isaliteral{27}{\isacharprime}}a\ option}. A typical application is a lookup function
+on a list of key-value pairs, often called an association list:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{fun}\isamarkupfalse%
+\ lookup\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ option{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}lookup\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ x{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ None{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}lookup\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{23}{\isacharhash}}\ ps{\isaliteral{29}{\isacharparenright}}\ x{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{27}{\isacharprime}}\ then\ Some\ y\ else\ lookup\ ps\ x{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+Note that \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}} is the type of pairs, also written \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{2}}}.
+
+\subsection{Definitions}
+
+Non recursive functions can be defined as in the following example:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{definition}\isamarkupfalse%
+\ sq\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}sq\ n\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ n{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+Such definitions do not allow pattern matching but only
+\isa{f\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{3D}{\isacharequal}}\ t}, where \isa{f} does not occur in \isa{t}.
+
+\subsection{Abbreviations}
+
+Abbreviations are similar to definitions:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{abbreviation}\isamarkupfalse%
+\ sq{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}sq{\isaliteral{27}{\isacharprime}}\ n\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ n{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The key difference is that \isa{sq{\isaliteral{27}{\isacharprime}}} is only syntactic sugar:
+\isa{sq{\isaliteral{27}{\isacharprime}}\ t} is replaced by \mbox{\isa{t\ {\isaliteral{2A}{\isacharasterisk}}\ t}} after parsing, and every
+occurrence of a term \isa{u\ {\isaliteral{2A}{\isacharasterisk}}\ u} is replaced by \mbox{\isa{sq{\isaliteral{27}{\isacharprime}}\ u}} before
+printing. Internally, \isa{sq{\isaliteral{27}{\isacharprime}}} does not exist. This is also the
+advantage of abbreviations over definitions: definitions need to be expanded
+explicitly (see \autoref{sec:rewr-defs}) whereas abbreviations are already
+expanded upon parsing. However, abbreviations should be introduced sparingly:
+if abused, they can lead to a confusing discrepancy between the internal and
+external view of a term.
+
+\subsection{Recursive functions}
+\label{sec:recursive-funs}
+
+Recursive functions are defined with \isacom{fun} by pattern matching
+over datatype constructors. The order of equations matters. Just as in
+functional programming languages. However, all HOL functions must be
+total. This simplifies the logic---terms are always defined---but means
+that recursive functions must terminate. Otherwise one could define a
+function \isa{f\ n\ {\isaliteral{3D}{\isacharequal}}\ f\ n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}} and conclude \mbox{\isa{{\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}}} by
+subtracting \isa{f\ n} on both sides.
+
+Isabelle automatic termination checker requires that the arguments of
+recursive calls on the right-hand side must be strictly smaller than the
+arguments on the left-hand side. In the simplest case, this means that one
+fixed argument position decreases in size with each recursive call. The size
+is measured as the number of constructors (excluding 0-ary ones, e.g.\ \isa{Nil}). Lexicographic combinations are also recognised. In more complicated
+situations, the user may have to prove termination by hand. For details
+see~\cite{Krauss}.
+
+Functions defined with \isacom{fun} come with their own induction schema
+that mirrors the recursion schema and is derived from the termination
+order. For example,%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{fun}\isamarkupfalse%
+\ div{\isadigit{2}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}div{\isadigit{2}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}div{\isadigit{2}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}div{\isadigit{2}}\ {\isaliteral{28}{\isacharparenleft}}Suc{\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Suc{\isaliteral{28}{\isacharparenleft}}div{\isadigit{2}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+does not just define \isa{div{\isadigit{2}}} but also proves a
+customised induction rule:
+\[
+\inferrule{
+\mbox{\isa{P\ {\isadigit{0}}}}\\
+\mbox{\isa{P\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}}}\\
+\mbox{\isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ P\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}}}}
+{\mbox{\isa{P\ m}}}
+\]
+This customised induction rule can simplify inductive proofs. For example,%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}div{\isadigit{2}}{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{2B}{\isacharplus}}n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ n\ rule{\isaliteral{3A}{\isacharcolon}}\ div{\isadigit{2}}{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+yields the 3 subgoals
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ div{\isadigit{2}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\isanewline
+\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ div{\isadigit{2}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isadigit{0}}\isanewline
+\ {\isadigit{3}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ div{\isadigit{2}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
+\isaindent{\ {\isadigit{3}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ }div{\isadigit{2}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}%
+\end{isabelle}
+An application of \isa{auto} finishes the proof.
+Had we used ordinary structural induction on \isa{n},
+the proof would have needed an additional
+case distinction in the induction step.
+
+The general case is often called \concept{computation induction},
+because the induction follows the (terminating!) computation.
+For every defining equation
+\begin{quote}
+\isa{f{\isaliteral{28}{\isacharparenleft}}e{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ f{\isaliteral{28}{\isacharparenleft}}r\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ f{\isaliteral{28}{\isacharparenleft}}r\isaliteral{5C3C5E697375623E}{}\isactrlisub k{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}}
+\end{quote}
+where \isa{f{\isaliteral{28}{\isacharparenleft}}r\isaliteral{5C3C5E697375623E}{}\isactrlisub i{\isaliteral{29}{\isacharparenright}}}, \isa{i{\isaliteral{3D}{\isacharequal}}{\isadigit{1}}{\isaliteral{5C3C646F74733E}{\isasymdots}}k}, are all the recursive calls,
+the induction rule \isa{f{\isaliteral{2E}{\isachardot}}induct} contains one premise of the form
+\begin{quote}
+\isa{P{\isaliteral{28}{\isacharparenleft}}r\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{28}{\isacharparenleft}}r\isaliteral{5C3C5E697375623E}{}\isactrlisub k{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{28}{\isacharparenleft}}e{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+If \isa{f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}} then \isa{f{\isaliteral{2E}{\isachardot}}induct} is applied like this:
+\begin{quote}
+\isacom{apply}\isa{{\isaliteral{28}{\isacharparenleft}}induction\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ rule{\isaliteral{3A}{\isacharcolon}}\ f{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+where typically there is a call \isa{f\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E697375623E}{}\isactrlisub n} in the goal.
+But note that the induction rule does not mention \isa{f} at all,
+except in its name, and is applicable independently of \isa{f}.
+
+\section{Induction heuristics}
+
+We have already noted that theorems about recursive functions are proved by
+induction. In case the function has more than one argument, we have followed
+the following heuristic in the proofs about the append function:
+\begin{quote}
+\emph{Perform induction on argument number $i$\\
+ if the function is defined by recursion on argument number $i$.}
+\end{quote}
+The key heuristic, and the main point of this section, is to
+\emph{generalise the goal before induction}.
+The reason is simple: if the goal is
+too specific, the induction hypothesis is too weak to allow the induction
+step to go through. Let us illustrate the idea with an example.
+
+Function \isa{rev} has quadratic worst-case running time
+because it calls append for each element of the list and
+append is linear in its first argument. A linear time version of
+\isa{rev} requires an extra argument where the result is accumulated
+gradually, using only~\isa{{\isaliteral{23}{\isacharhash}}}:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{fun}\isamarkupfalse%
+\ itrev\ {\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}}\ \isakeyword{where}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}itrev\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ \ \ \ \ ys\ {\isaliteral{3D}{\isacharequal}}\ ys{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\isanewline
+{\isaliteral{22}{\isachardoublequoteopen}}itrev\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{23}{\isacharhash}}xs{\isaliteral{29}{\isacharparenright}}\ ys\ {\isaliteral{3D}{\isacharequal}}\ itrev\ xs\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{23}{\isacharhash}}ys{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
+\begin{isamarkuptext}%
+The behaviour of \isa{itrev} is simple: it reverses
+its first argument by stacking its elements onto the second argument,
+and returning that second argument when the first one becomes
+empty. Note that \isa{itrev} is tail-recursive: it can be
+compiled into a loop, no stack is necessary for executing it.
+
+Naturally, we would like to show that \isa{itrev} does indeed reverse
+its first argument provided the second one is empty:%
+\end{isamarkuptext}%
+\isamarkuptrue%
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}itrev\ xs\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs{\isaliteral{22}{\isachardoublequoteclose}}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+There is no choice as to the induction variable:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ xs{\isaliteral{29}{\isacharparenright}}\isanewline
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}auto{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+Unfortunately, this attempt does not prove
+the induction step:
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ xs{\isaliteral{2E}{\isachardot}}\ itrev\ xs\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ itrev\ xs\ {\isaliteral{5B}{\isacharbrackleft}}a{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}a{\isaliteral{5D}{\isacharbrackright}}%
+\end{isabelle}
+The induction hypothesis is too weak. The fixed
+argument,~\isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}}, prevents it from rewriting the conclusion.
+This example suggests a heuristic:
+\begin{quote}
+\emph{Generalise goals for induction by replacing constants by variables.}
+\end{quote}
+Of course one cannot do this na\"{\i}vely: \isa{itrev\ xs\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs} is
+just not true. The correct generalisation is%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+\isacommand{lemma}\isamarkupfalse%
+\ {\isaliteral{22}{\isachardoublequoteopen}}itrev\ xs\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{22}{\isachardoublequoteclose}}%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+%
+\begin{isamarkuptxt}%
+If \isa{ys} is replaced by \isa{{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}}, the right-hand side simplifies to
+\isa{rev\ xs}, as required.
+In this instance it was easy to guess the right generalisation.
+Other situations can require a good deal of creativity.
+
+Although we now have two variables, only \isa{xs} is suitable for
+induction, and we repeat our proof attempt. Unfortunately, we are still
+not there:
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ xs{\isaliteral{2E}{\isachardot}}\isanewline
+\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }itrev\ xs\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs\ {\isaliteral{40}{\isacharat}}\ ys\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
+\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }itrev\ xs\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{23}{\isacharhash}}\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs\ {\isaliteral{40}{\isacharat}}\ a\ {\isaliteral{23}{\isacharhash}}\ ys%
+\end{isabelle}
+The induction hypothesis is still too weak, but this time it takes no
+intuition to generalise: the problem is that the \isa{ys}
+in the induction hypothesis is fixed,
+but the induction hypothesis needs to be applied with
+\isa{a\ {\isaliteral{23}{\isacharhash}}\ ys} instead of \isa{ys}. Hence we prove the theorem
+for all \isa{ys} instead of a fixed one. We can instruct induction
+to perform this generalisation for us by adding \isa{arbitrary{\isaliteral{3A}{\isacharcolon}}\ ys}.%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\isatagproof
+\isacommand{apply}\isamarkupfalse%
+{\isaliteral{28}{\isacharparenleft}}induction\ xs\ arbitrary{\isaliteral{3A}{\isacharcolon}}\ ys{\isaliteral{29}{\isacharparenright}}%
+\begin{isamarkuptxt}%
+The induction hypothesis in the induction step is now universally quantified over \isa{ys}:
+\begin{isabelle}%
+\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}ys{\isaliteral{2E}{\isachardot}}\ itrev\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{40}{\isacharat}}\ ys\isanewline
+\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ xs\ ys{\isaliteral{2E}{\isachardot}}\isanewline
+\isaindent{\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}ys{\isaliteral{2E}{\isachardot}}\ itrev\ xs\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ xs\ {\isaliteral{40}{\isacharat}}\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
+\isaindent{\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }itrev\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ ys\ {\isaliteral{3D}{\isacharequal}}\ rev\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{40}{\isacharat}}\ ys%
+\end{isabelle}
+Thus the proof succeeds:%
+\end{isamarkuptxt}%
+\isamarkuptrue%
+\isacommand{apply}\isamarkupfalse%
+\ auto\isanewline
+\isacommand{done}\isamarkupfalse%
+%
+\endisatagproof
+{\isafoldproof}%
+%
+\isadelimproof
+%
+\endisadelimproof
+%
+\begin{isamarkuptext}%
+This leads to another heuristic for generalisation:
+\begin{quote}
+\emph{Generalise induction by generalising all free
+variables\\ {\em(except the induction variable itself)}.}
+\end{quote}
+Generalisation is best performed with \isa{arbitrary{\isaliteral{3A}{\isacharcolon}}\ y\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ y\isaliteral{5C3C5E697375623E}{}\isactrlisub k}.
+This heuristic prevents trivial failures like the one above.
+However, it should not be applied blindly.
+It is not always required, and the additional quantifiers can complicate
+matters in some cases. The variables that need to be quantified are typically
+those that change in recursive calls.
+
+\section{Simplification}
+
+So far we have talked a lot about simplifying terms without explaining the concept. \concept{Simplification} means
+\begin{itemize}
+\item using equations $l = r$ from left to right (only),
+\item as long as possible.
+\end{itemize}
+To emphasise the directionality, equations that have been given the
+\isa{simp} attribute are called \concept{simplification}
+rules. Logically, they are still symmetric, but proofs by
+simplification use them only in the left-to-right direction. The proof tool
+that performs simplifications is called the \concept{simplifier}. It is the
+basis of \isa{auto} and other related proof methods.
+
+The idea of simplification is best explained by an example. Given the
+simplification rules
+\[
+\begin{array}{rcl@ {\quad}l}
+\isa{{\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ n} &\isa{{\isaliteral{3D}{\isacharequal}}}& \isa{n} & (1) \\
+\isa{Suc\ m\ {\isaliteral{2B}{\isacharplus}}\ n} &\isa{{\isaliteral{3D}{\isacharequal}}}& \isa{Suc\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2B}{\isacharplus}}\ n{\isaliteral{29}{\isacharparenright}}} & (2) \\
+\isa{{\isaliteral{28}{\isacharparenleft}}Suc\ m\ {\isaliteral{5C3C6C653E}{\isasymle}}\ Suc\ n{\isaliteral{29}{\isacharparenright}}} &\isa{{\isaliteral{3D}{\isacharequal}}}& \isa{{\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{5C3C6C653E}{\isasymle}}\ n{\isaliteral{29}{\isacharparenright}}} & (3)\\
+\isa{{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ m{\isaliteral{29}{\isacharparenright}}} &\isa{{\isaliteral{3D}{\isacharequal}}}& \isa{True} & (4)
+\end{array}
+\]
+the formula \isa{{\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ Suc\ {\isadigit{0}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ Suc\ {\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ x} is simplified to \isa{True}
+as follows:
+\[
+\begin{array}{r@ {}c@ {}l@ {\quad}l}
+\isa{{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ Suc\ {\isadigit{0}}} & \leq & \isa{Suc\ {\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ x{\isaliteral{29}{\isacharparenright}}} & \stackrel{(1)}{=} \\
+\isa{{\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}} & \leq & \isa{Suc\ {\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ x{\isaliteral{29}{\isacharparenright}}} & \stackrel{(2)}{=} \\
+\isa{{\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}} & \leq & \isa{Suc\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ x{\isaliteral{29}{\isacharparenright}}} & \stackrel{(3)}{=} \\
+\isa{{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}} & \leq & \isa{{\isadigit{0}}\ {\isaliteral{2B}{\isacharplus}}\ x{\isaliteral{29}{\isacharparenright}}} & \stackrel{(4)}{=} \\[1ex]
+ & \isa{True}
+\end{array}
+\]
+Simplification is often also called \concept{rewriting}
+and simplification rules \concept{rewrite rules}.
+
+\subsection{Simplification rules}
+
+The attribute \isa{simp} declares theorems to be simplification rules,
+which the simplifier will use automatically. In addition,
+\isacom{datatype} and \isacom{fun} commands implicitly declare some
+simplification rules: \isacom{datatype} the distinctness and injectivity
+rules, \isacom{fun} the defining equations. Definitions are not declared
+as simplification rules automatically! Nearly any theorem can become a
+simplification rule. The simplifier will try to transform it into an
+equation. For example, the theorem \isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P} is turned into \isa{P\ {\isaliteral{3D}{\isacharequal}}\ False}.
+
+Only equations that really simplify, like \isa{rev\ {\isaliteral{28}{\isacharparenleft}}rev\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ xs} and
+\isa{xs\ {\isaliteral{40}{\isacharat}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ xs}, should be declared as simplification
+rules. Equations that may be counterproductive as simplification rules
+should only be used in specific proof steps (see \S\ref{sec:simp} below).
+Distributivity laws, for example, alter the structure of terms
+and can produce an exponential blow-up.
+
+\subsection{Conditional simplification rules}
+
+Simplification rules can be conditional. Before applying such a rule, the
+simplifier will first try to prove the preconditions, again by
+simplification. For example, given the simplification rules
+\begin{quote}
+\isa{p\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ True}\\
+\isa{p\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ f\ x\ {\isaliteral{3D}{\isacharequal}}\ g\ x},
+\end{quote}
+the term \isa{f\ {\isadigit{0}}} simplifies to \isa{g\ {\isadigit{0}}}
+but \isa{f\ {\isadigit{1}}} does not simplify because \isa{p\ {\isadigit{1}}}
+is not provable.
+
+\subsection{Termination}
+
+Simplification can run forever, for example if both \isa{f\ x\ {\isaliteral{3D}{\isacharequal}}\ g\ x} and
+\isa{g\ x\ {\isaliteral{3D}{\isacharequal}}\ f\ x} are simplification rules. It is the user's
+responsibility not to include simplification rules that can lead to
+nontermination, either on their own or in combination with other
+simplification rules. The right-hand side of a simplification rule should
+always be ``simpler'' than the left-hand side---in some sense. But since
+termination is undecidable, such a check cannot be automated completely
+and Isabelle makes little attempt to detect nontermination.
+
+When conditional simplification rules are applied, their preconditions are
+proved first. Hence all preconditions need to be
+simpler than the left-hand side of the conclusion. For example
+\begin{quote}
+\isa{n\ {\isaliteral{3C}{\isacharless}}\ m\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{3C}{\isacharless}}\ Suc\ m{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ True}
+\end{quote}
+is suitable as a simplification rule: both \isa{n\ {\isaliteral{3C}{\isacharless}}\ m} and \isa{True}
+are simpler than \mbox{\isa{n\ {\isaliteral{3C}{\isacharless}}\ Suc\ m}}. But
+\begin{quote}
+\isa{Suc\ n\ {\isaliteral{3C}{\isacharless}}\ m\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{3C}{\isacharless}}\ m{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ True}
+\end{quote}
+leads to nontermination: when trying to rewrite \isa{n\ {\isaliteral{3C}{\isacharless}}\ m} to \isa{True}
+one first has to prove \mbox{\isa{Suc\ n\ {\isaliteral{3C}{\isacharless}}\ m}}, which can be rewritten to \isa{True} provided \isa{Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3C}{\isacharless}}\ m}, \emph{ad infinitum}.
+
+\subsection{The \isa{simp} proof method}
+\label{sec:simp}
+
+So far we have only used the proof method \isa{auto}. Method \isa{simp}
+is the key component of \isa{auto}, but \isa{auto} can do much more. In
+some cases, \isa{auto} is overeager and modifies the proof state too much.
+In such cases the more predictable \isa{simp} method should be used.
+Given a goal
+\begin{quote}
+\isa{{\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}\ P\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{3B}{\isacharsemicolon}}\ P\isaliteral{5C3C5E697375623E}{}\isactrlisub m\ {\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ C}
+\end{quote}
+the command
+\begin{quote}
+\isacom{apply}\isa{{\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ th\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ th\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+simplifies the assumptions \isa{P\isaliteral{5C3C5E697375623E}{}\isactrlisub i} and the conclusion \isa{C} using
+\begin{itemize}
+\item all simplification rules, including the ones coming from \isacom{datatype} and \isacom{fun},
+\item the additional lemmas \isa{th\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ th\isaliteral{5C3C5E697375623E}{}\isactrlisub n}, and
+\item the assumptions.
+\end{itemize}
+In addition to or instead of \isa{add} there is also \isa{del} for removing
+simplification rules temporarily. Both are optional. Method \isa{auto}
+can be modified similarly:
+\begin{quote}
+\isacom{apply}\isa{{\isaliteral{28}{\isacharparenleft}}auto\ simp\ add{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ simp\ del{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+Here the modifiers are \isa{simp\ add} and \isa{simp\ del}
+instead of just \isa{add} and \isa{del} because \isa{auto}
+does not just perform simplification.
+
+Note that \isa{simp} acts only on subgoal 1, \isa{auto} acts on all
+subgoals. There is also \isa{simp{\isaliteral{5F}{\isacharunderscore}}all}, which applies \isa{simp} to
+all subgoals.
+
+\subsection{Rewriting with definitions}
+\label{sec:rewr-defs}
+
+Definitions introduced by the command \isacom{definition}
+can also be used as simplification rules,
+but by default they are not: the simplifier does not expand them
+automatically. Definitions are intended for introducing abstract concepts and
+not merely as abbreviations. Of course, we need to expand the definition
+initially, but once we have proved enough abstract properties of the new
+constant, we can forget its original definition. This style makes proofs more
+robust: if the definition has to be changed, only the proofs of the abstract
+properties will be affected.
+
+The definition of a function \isa{f} is a theorem named \isa{f{\isaliteral{5F}{\isacharunderscore}}def}
+and can be added to a call of \isa{simp} just like any other theorem:
+\begin{quote}
+\isacom{apply}\isa{{\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ f{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+In particular, let-expressions can be unfolded by
+making \isa{Let{\isaliteral{5F}{\isacharunderscore}}def} a simplification rule.
+
+\subsection{Case splitting with \isa{simp}}
+
+Goals containing if-expressions are automatically split into two cases by
+\isa{simp} using the rule
+\begin{quote}
+\isa{P\ {\isaliteral{28}{\isacharparenleft}}if\ A\ then\ s\ else\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ s{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ t{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+For example, \isa{simp} can prove
+\begin{quote}
+\isa{{\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ A\ then\ B\ else\ False{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+because both \isa{A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ B} and \isa{{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}A\ {\isaliteral{5C3C616E643E}{\isasymand}}\ B{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ False}
+simplify to \isa{True}.
+
+We can split case-expressions similarly. For \isa{nat} the rule looks like this:
+\begin{isabelle}%
+\ \ \ \ P\ {\isaliteral{28}{\isacharparenleft}}case\ e\ of\ {\isadigit{0}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ a\ {\isaliteral{7C}{\isacharbar}}\ Suc\ n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ b\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline
+\isaindent{\ \ \ \ }{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}e\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}n{\isaliteral{2E}{\isachardot}}\ e\ {\isaliteral{3D}{\isacharequal}}\ Suc\ n\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}b\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}%
+\end{isabelle}
+Case expressions are not split automatically by \isa{simp}, but \isa{simp}
+can be instructed to do so:
+\begin{quote}
+\isacom{apply}\isa{{\isaliteral{28}{\isacharparenleft}}simp\ split{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{2E}{\isachardot}}split{\isaliteral{29}{\isacharparenright}}}
+\end{quote}
+splits all case-expressions over natural numbers. For an arbitrary
+datatype \isa{t} it is \isa{t{\isaliteral{2E}{\isachardot}}split} instead of \isa{nat{\isaliteral{2E}{\isachardot}}split}.
+Method \isa{auto} can be modified in exactly the same way.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
+%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+\end{isabellebody}%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "root"
+%%% End:
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/intro-isabelle.tex Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,38 @@
+Isabelle is a generic system for
+implementing logical formalisms, and Isabelle/HOL is the specialization
+of Isabelle for HOL, which abbreviates Higher-Order Logic. We introduce
+HOL step by step following the equation
+\[ \mbox{HOL} = \mbox{Functional Programming} + \mbox{Logic}. \]
+We assume that the reader is familiar with the basic concepts of functional
+programming and is used to logical and set theoretic notation.
+
+\autoref{sec:FP} introduces HOL as a functional programming language and
+explains how to write simple inductive proofs of mostly equational properties
+of recursive functions.
+\sem
+\autoref{sec:CaseStudyExp} contains a
+little case study: arithmetic and boolean expressions, their evaluation,
+optimization and compilation.
+\endsem
+\autoref{ch:Logic} introduces the rest of HOL: the
+language of formulas beyond equality, automatic proof tools, single
+step proofs, and inductive definitions, an essential specification construct.
+\autoref{ch:Isar} introduces Isar, Isabelle's language for writing structured
+proofs.
+
+%Further material (slides, demos etc) can be found online at
+%\url{http://www.in.tum.de/~nipkow}.
+
+% Relics:
+% We aim to minimise the amount of background knowledge of logic we expect
+% from the reader
+% We have focussed on the core material
+% in the intersection of computation and logic.
+
+This introduction to the core of Isabelle is intentionally concrete and
+example-based: we concentrate on examples that illustrate the typical cases;
+we do not explain the general case if it can be inferred from the examples.
+For a comprehensive treatment of all things Isabelle we recommend the
+\emph{Isabelle/Isar Reference Manual}~\cite{IsarRef}, which comes with the
+Isabelle distribution.
+The tutorial by Nipkow, Paulson and Wenzel~\cite{LNCS2283} (in its updated version that comes with the Isabelle distribution) is still recommended for the wealth of examples and material, but its proof style is outdated. In particular it fails to cover the structured proof language Isar.
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/isabelle.sty Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,1 @@
+../../lib/texinputs/isabelle.sty
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/isabellesym.sty Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,1 @@
+../../lib/texinputs/isabellesym.sty
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/mathpartir.sty Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,388 @@
+% Mathpartir --- Math Paragraph for Typesetting Inference Rules
+%
+% Copyright (C) 2001, 2002, 2003 Didier Rémy
+%
+% Author : Didier Remy
+% Version : 1.1.1
+% Bug Reports : to author
+% Web Site : http://pauillac.inria.fr/~remy/latex/
+%
+% WhizzyTeX is free software; you can redistribute it and/or modify
+% it under the terms of the GNU General Public License as published by
+% the Free Software Foundation; either version 2, or (at your option)
+% any later version.
+%
+% Mathpartir is distributed in the hope that it will be useful,
+% but WITHOUT ANY WARRANTY; without even the implied warranty of
+% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+% GNU General Public License for more details
+% (http://pauillac.inria.fr/~remy/license/GPL).
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% File mathpartir.sty (LaTeX macros)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{mathpartir}
+ [2003/07/10 version 1.1.1 Math Paragraph for Typesetting Inference Rules]
+
+%%
+
+%% Identification
+%% Preliminary declarations
+
+\RequirePackage {keyval}
+
+%% Options
+%% More declarations
+
+%% PART I: Typesetting maths in paragraphe mode
+
+\newdimen \mpr@tmpdim
+
+% To ensure hevea \hva compatibility, \hva should expands to nothing
+% in mathpar or in inferrule
+\let \mpr@hva \empty
+
+%% normal paragraph parametters, should rather be taken dynamically
+\def \mpr@savepar {%
+ \edef \MathparNormalpar
+ {\noexpand \lineskiplimit \the\lineskiplimit
+ \noexpand \lineskip \the\lineskip}%
+ }
+
+\def \mpr@rulelineskip {\lineskiplimit=0.3em\lineskip=0.2em plus 0.1em}
+\def \mpr@lesslineskip {\lineskiplimit=0.6em\lineskip=0.5em plus 0.2em}
+\def \mpr@lineskip {\lineskiplimit=1.2em\lineskip=1.2em plus 0.2em}
+\let \MathparLineskip \mpr@lineskip
+\def \mpr@paroptions {\MathparLineskip}
+\let \mpr@prebindings \relax
+
+\newskip \mpr@andskip \mpr@andskip 2em plus 0.5fil minus 0.5em
+
+\def \mpr@goodbreakand
+ {\hskip -\mpr@andskip \penalty -1000\hskip \mpr@andskip}
+\def \mpr@and {\hskip \mpr@andskip}
+\def \mpr@andcr {\penalty 50\mpr@and}
+\def \mpr@cr {\penalty -10000\mpr@and}
+\def \mpr@eqno #1{\mpr@andcr #1\hskip 0em plus -1fil \penalty 10}
+
+\def \mpr@bindings {%
+ \let \and \mpr@andcr
+ \let \par \mpr@andcr
+ \let \\\mpr@cr
+ \let \eqno \mpr@eqno
+ \let \hva \mpr@hva
+ }
+\let \MathparBindings \mpr@bindings
+
+% \@ifundefined {ignorespacesafterend}
+% {\def \ignorespacesafterend {\aftergroup \ignorespaces}
+
+\newenvironment{mathpar}[1][]
+ {$$\mpr@savepar \parskip 0em \hsize \linewidth \centering
+ \vbox \bgroup \mpr@prebindings \mpr@paroptions #1\ifmmode $\else
+ \noindent $\displaystyle\fi
+ \MathparBindings}
+ {\unskip \ifmmode $\fi\egroup $$\ignorespacesafterend}
+
+% \def \math@mathpar #1{\setbox0 \hbox {$\displaystyle #1$}\ifnum
+% \wd0 < \hsize $$\box0$$\else \bmathpar #1\emathpar \fi}
+
+%%% HOV BOXES
+
+\def \mathvbox@ #1{\hbox \bgroup \mpr@normallineskip
+ \vbox \bgroup \tabskip 0em \let \\ \cr
+ \halign \bgroup \hfil $##$\hfil\cr #1\crcr \egroup \egroup
+ \egroup}
+
+\def \mathhvbox@ #1{\setbox0 \hbox {\let \\\qquad $#1$}\ifnum \wd0 < \hsize
+ \box0\else \mathvbox {#1}\fi}
+
+
+%% Part II -- operations on lists
+
+\newtoks \mpr@lista
+\newtoks \mpr@listb
+
+\long \def\mpr@cons #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
+{#2}\edef #2{\the \mpr@lista \the \mpr@listb}}
+
+\long \def\mpr@snoc #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
+{#2}\edef #2{\the \mpr@listb\the\mpr@lista}}
+
+\long \def \mpr@concat#1=#2\mpr@to#3{\mpr@lista \expandafter {#2}\mpr@listb
+\expandafter {#3}\edef #1{\the \mpr@listb\the\mpr@lista}}
+
+\def \mpr@head #1\mpr@to #2{\expandafter \mpr@head@ #1\mpr@head@ #1#2}
+\long \def \mpr@head@ #1#2\mpr@head@ #3#4{\def #4{#1}\def#3{#2}}
+
+\def \mpr@flatten #1\mpr@to #2{\expandafter \mpr@flatten@ #1\mpr@flatten@ #1#2}
+\long \def \mpr@flatten@ \\#1\\#2\mpr@flatten@ #3#4{\def #4{#1}\def #3{\\#2}}
+
+\def \mpr@makelist #1\mpr@to #2{\def \mpr@all {#1}%
+ \mpr@lista {\\}\mpr@listb \expandafter {\mpr@all}\edef \mpr@all {\the
+ \mpr@lista \the \mpr@listb \the \mpr@lista}\let #2\empty
+ \def \mpr@stripof ##1##2\mpr@stripend{\def \mpr@stripped{##2}}\loop
+ \mpr@flatten \mpr@all \mpr@to \mpr@one
+ \expandafter \mpr@snoc \mpr@one \mpr@to #2\expandafter \mpr@stripof
+ \mpr@all \mpr@stripend
+ \ifx \mpr@stripped \empty \let \mpr@isempty 0\else \let \mpr@isempty 1\fi
+ \ifx 1\mpr@isempty
+ \repeat
+}
+
+%% Part III -- Type inference rules
+
+\def \mpr@rev #1\mpr@to #2{\let \mpr@tmp \empty
+ \def \\##1{\mpr@cons ##1\mpr@to \mpr@tmp}#1\let #2\mpr@tmp}
+
+\newif \if@premisse
+\newbox \mpr@hlist
+\newbox \mpr@vlist
+\newif \ifmpr@center \mpr@centertrue
+\def \mpr@htovlist {%
+ \setbox \mpr@hlist
+ \hbox {\strut
+ \ifmpr@center \hskip -0.5\wd\mpr@hlist\fi
+ \unhbox \mpr@hlist}%
+ \setbox \mpr@vlist
+ \vbox {\if@premisse \box \mpr@hlist \unvbox \mpr@vlist
+ \else \unvbox \mpr@vlist \box \mpr@hlist
+ \fi}%
+}
+% OLD version
+% \def \mpr@htovlist {%
+% \setbox \mpr@hlist
+% \hbox {\strut \hskip -0.5\wd\mpr@hlist \unhbox \mpr@hlist}%
+% \setbox \mpr@vlist
+% \vbox {\if@premisse \box \mpr@hlist \unvbox \mpr@vlist
+% \else \unvbox \mpr@vlist \box \mpr@hlist
+% \fi}%
+% }
+
+\def \mpr@sep{1.5em}
+\def \mpr@blank { }
+\def \mpr@hovbox #1#2{\hbox
+ \bgroup
+ \ifx #1T\@premissetrue
+ \else \ifx #1B\@premissefalse
+ \else
+ \PackageError{mathpartir}
+ {Premisse orientation should either be P or B}
+ {Fatal error in Package}%
+ \fi \fi
+ \def \@test {#2}\ifx \@test \mpr@blank\else
+ \setbox \mpr@hlist \hbox {}%
+ \setbox \mpr@vlist \vbox {}%
+ \if@premisse \let \snoc \mpr@cons \else \let \snoc \mpr@snoc \fi
+ \let \@hvlist \empty \let \@rev \empty
+ \mpr@tmpdim 0em
+ \expandafter \mpr@makelist #2\mpr@to \mpr@flat
+ \if@premisse \mpr@rev \mpr@flat \mpr@to \@rev \else \let \@rev \mpr@flat \fi
+ \def \\##1{%
+ \def \@test {##1}\ifx \@test \empty
+ \mpr@htovlist
+ \mpr@tmpdim 0em %%% last bug fix not extensively checked
+ \else
+ \setbox0 \hbox{$\displaystyle {##1}$}\relax
+ \advance \mpr@tmpdim by \wd0
+ %\mpr@tmpdim 1.02\mpr@tmpdim
+ \ifnum \mpr@tmpdim < \hsize
+ \ifnum \wd\mpr@hlist > 0
+ \if@premisse
+ \setbox \mpr@hlist
+ \hbox {\unhbox0 \hskip \mpr@sep \unhbox \mpr@hlist}%
+ \else
+ \setbox \mpr@hlist
+ \hbox {\unhbox \mpr@hlist \hskip \mpr@sep \unhbox0}%
+ \fi
+ \else
+ \setbox \mpr@hlist \hbox {\unhbox0}%
+ \fi
+ \else
+ \ifnum \wd \mpr@hlist > 0
+ \mpr@htovlist
+ \mpr@tmpdim \wd0
+ \fi
+ \setbox \mpr@hlist \hbox {\unhbox0}%
+ \fi
+ \advance \mpr@tmpdim by \mpr@sep
+ \fi
+ }%
+ \@rev
+ \mpr@htovlist
+ \ifmpr@center \hskip \wd\mpr@vlist\fi \box \mpr@vlist
+ \fi
+ \egroup
+}
+
+%%% INFERENCE RULES
+
+\@ifundefined{@@over}{%
+ \let\@@over\over % fallback if amsmath is not loaded
+ \let\@@overwithdelims\overwithdelims
+ \let\@@atop\atop \let\@@atopwithdelims\atopwithdelims
+ \let\@@above\above \let\@@abovewithdelims\abovewithdelims
+ }{}
+
+
+\def \mpr@@fraction #1#2{\hbox {\advance \hsize by -0.5em
+ $\displaystyle {#1\@@over #2}$}}
+\let \mpr@fraction \mpr@@fraction
+\def \mpr@@reduce #1#2{\hbox
+ {$\lower 0.01pt \mpr@@fraction {#1}{#2}\mkern -15mu\rightarrow$}}
+\def \mpr@@rewrite #1#2#3{\hbox
+ {$\lower 0.01pt \mpr@@fraction {#2}{#3}\mkern -8mu#1$}}
+\def \mpr@infercenter #1{\vcenter {\mpr@hovbox{T}{#1}}}
+
+\def \mpr@empty {}
+\def \mpr@inferrule
+ {\bgroup
+ \ifnum \linewidth<\hsize \hsize \linewidth\fi
+ \mpr@rulelineskip
+ \let \and \qquad
+ \let \hva \mpr@hva
+ \let \@rulename \mpr@empty
+ \let \@rule@options \mpr@empty
+ \mpr@inferrule@}
+\newcommand {\mpr@inferrule@}[3][]
+ {\everymath={\displaystyle}%
+ \def \@test {#2}\ifx \empty \@test
+ \setbox0 \hbox {$\vcenter {\mpr@hovbox{B}{#3}}$}%
+ \else
+ \def \@test {#3}\ifx \empty \@test
+ \setbox0 \hbox {$\vcenter {\mpr@hovbox{T}{#2}}$}%
+ \else
+ \setbox0 \mpr@fraction {\mpr@hovbox{T}{#2}}{\mpr@hovbox{B}{#3}}%
+ \fi \fi
+ \def \@test {#1}\ifx \@test\empty \box0
+ \else \vbox
+%%% Suggestion de Francois pour les etiquettes longues
+%%% {\hbox to \wd0 {\RefTirName {#1}\hfil}\box0}\fi
+ {\hbox {\RefTirName {#1}}\box0}\fi
+ \egroup}
+
+\def \mpr@vdotfil #1{\vbox to #1{\leaders \hbox{$\cdot$} \vfil}}
+
+% They are two forms
+% \inferrule [label]{[premisses}{conclusions}
+% or
+% \inferrule* [options]{[premisses}{conclusions}
+%
+% Premisses and conclusions are lists of elements separated by \\
+% Each \\ produces a break, attempting horizontal breaks if possible,
+% and vertical breaks if needed.
+%
+% An empty element obtained by \\\\ produces a vertical break in all cases.
+%
+% The former rule is aligned on the fraction bar.
+% The optional label appears on top of the rule
+% The second form to be used in a derivation tree is aligned on the last
+% line of its conclusion
+%
+% The second form can be parameterized, using the key=val interface. The
+% folloiwng keys are recognized:
+%
+% width set the width of the rule to val
+% narrower set the width of the rule to val\hsize
+% before execute val at the beginning/left
+% lab put a label [Val] on top of the rule
+% lskip add negative skip on the right
+% left put a left label [Val]
+% Left put a left label [Val], ignoring its width
+% right put a right label [Val]
+% Right put a right label [Val], ignoring its width
+% leftskip skip negative space on the left-hand side
+% rightskip skip negative space on the right-hand side
+% vdots lift the rule by val and fill vertical space with dots
+% after execute val at the end/right
+%
+% Note that most options must come in this order to avoid strange
+% typesetting (in particular leftskip must preceed left and Left and
+% rightskip must follow Right or right; vdots must come last
+% or be only followed by rightskip.
+%
+
+\define@key {mprset}{flushleft}[]{\mpr@centerfalse}
+\define@key {mprset}{center}[]{\mpr@centertrue}
+\def \mprset #1{\setkeys{mprset}{#1}}
+
+\newbox \mpr@right
+\define@key {mpr}{flushleft}[]{\mpr@centerfalse}
+\define@key {mpr}{center}[]{\mpr@centertrue}
+\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
+ \advance \hsize by -\wd0\box0}
+\define@key {mpr}{width}{\hsize #1}
+\define@key {mpr}{sep}{\def\mpr@sep{#1}}
+\define@key {mpr}{before}{#1}
+\define@key {mpr}{lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
+\define@key {mpr}{Lab}{\let \RefTirName \TirName \def \mpr@rulename {#1}}
+\define@key {mpr}{narrower}{\hsize #1\hsize}
+\define@key {mpr}{leftskip}{\hskip -#1}
+\define@key {mpr}{reduce}[]{\let \mpr@fraction \mpr@@reduce}
+\define@key {mpr}{rightskip}
+ {\setbox \mpr@right \hbox {\unhbox \mpr@right \hskip -#1}}
+\define@key {mpr}{LEFT}{\setbox0 \hbox {$#1$}\relax
+ \advance \hsize by -\wd0\box0}
+\define@key {mpr}{left}{\setbox0 \hbox {$\TirName {#1}\;$}\relax
+ \advance \hsize by -\wd0\box0}
+\define@key {mpr}{Left}{\llap{$\TirName {#1}\;$}}
+\define@key {mpr}{right}
+ {\setbox0 \hbox {$\;\TirName {#1}$}\relax \advance \hsize by -\wd0
+ \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
+\define@key {mpr}{RIGHT}
+ {\setbox0 \hbox {$#1$}\relax \advance \hsize by -\wd0
+ \setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
+\define@key {mpr}{Right}
+ {\setbox \mpr@right \hbox {\unhbox \mpr@right \rlap {$\;\TirName {#1}$}}}
+\define@key {mpr}{vdots}{\def \mpr@vdots {\@@atop \mpr@vdotfil{#1}}}
+\define@key {mpr}{after}{\edef \mpr@after {\mpr@after #1}}
+
+\newdimen \rule@dimen
+\newcommand \mpr@inferstar@ [3][]{\setbox0
+ \hbox {\let \mpr@rulename \mpr@empty \let \mpr@vdots \relax
+ \setbox \mpr@right \hbox{}%
+ $\setkeys{mpr}{#1}%
+ \ifx \mpr@rulename \mpr@empty \mpr@inferrule {#2}{#3}\else
+ \mpr@inferrule [{\mpr@rulename}]{#2}{#3}\fi
+ \box \mpr@right \mpr@vdots$}
+ \setbox1 \hbox {\strut}
+ \rule@dimen \dp0 \advance \rule@dimen by -\dp1
+ \raise \rule@dimen \box0}
+
+\def \mpr@infer {\@ifnextchar *{\mpr@inferstar}{\mpr@inferrule}}
+\newcommand \mpr@err@skipargs[3][]{}
+\def \mpr@inferstar*{\ifmmode
+ \let \@do \mpr@inferstar@
+ \else
+ \let \@do \mpr@err@skipargs
+ \PackageError {mathpartir}
+ {\string\inferrule* can only be used in math mode}{}%
+ \fi \@do}
+
+
+%%% Exports
+
+% Envirnonment mathpar
+
+\let \inferrule \mpr@infer
+
+% make a short name \infer is not already defined
+\@ifundefined {infer}{\let \infer \mpr@infer}{}
+
+\def \tir@name #1{\hbox {\small \sc #1}}
+\let \TirName \tir@name
+\let \RefTirName \tir@name
+
+%%% Other Exports
+
+% \let \listcons \mpr@cons
+% \let \listsnoc \mpr@snoc
+% \let \listhead \mpr@head
+% \let \listmake \mpr@makelist
+
+
+
+
+\endinput
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/prelude.tex Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,98 @@
+\usepackage{makeidx} % allows index generation
+\usepackage{graphicx} % standard LaTeX graphics tool
+ % when including figure files
+\usepackage{multicol} % used for the two-column index
+\usepackage[bottom]{footmisc}% places footnotes at page bottom
+\usepackage{alltt}
+
+\usepackage[T1]{fontenc}
+\usepackage{ccfonts}
+\usepackage[euler-digits]{eulervm}
+
+\let\bfdefaultold=\bfdefault
+\def\bfdefault{sbc} % make sans serif the default bold font
+
+\usepackage{isabelle,isabellesym}
+\usepackage{mathpartir}
+\usepackage{amssymb}
+
+% Enables fixmes
+\newif \ifDraft \Drafttrue
+
+\ifDraft
+ \newcommand{\FIXME}[1]{\textbf{\textsl{FIXME: #1}}}
+\else
+ \newcommand{\FIXME}[1]{\relax}
+\fi
+
+\renewcommand*\descriptionlabel[1]{\hspace\labelsep \textbf{#1}\hfil}
+
+% this should be the last package used
+\usepackage{color}
+\definecolor{linkcolor}{rgb}{0,0,0.4}
+\usepackage[colorlinks=true,linkcolor=linkcolor,citecolor=linkcolor,
+ filecolor=linkcolor,pagecolor=linkcolor,
+ urlcolor=linkcolor]{hyperref}
+
+% urls in roman style, theory text in math-similar italics
+\urlstyle{tt}
+\isabellestyle{it}
+
+\renewcommand{\isadigit}[1]{\ensuremath{#1}}
+
+% font size
+\renewcommand{\isastyle}{\isastyleminor}
+
+% indexing
+\usepackage{ifthen}
+\newcommand{\indexdef}[3]%
+{\ifthenelse{\equal{}{#1}}{\index{#3 (#2)|bold}}{\index{#3 (#1\ #2)|bold}}}
+
+% section commands
+\renewcommand{\chapterautorefname}{Chapter}
+\renewcommand{\sectionautorefname}{Section}
+
+\renewcommand{\isamarkupheader}[1]{{\rmfamily\subsection{#1}}}
+\renewcommand{\isamarkupsection}[1]{{\rmfamily\subsection{#1}}}
+\renewcommand{\isamarkupsubsection}[1]{{\rmfamily\subsubsection{#1}}}
+% isabelle in-text command font
+\newcommand{\isacom}[1]{\isa{\isacommand{#1}}}
+% isabelle keyword, adapted from isabelle.sty
+\renewcommand{\isakeyword}[1]
+{\emph{\def\isachardot{.}%
+\def\isacharbraceleft{\{}\def\isacharbraceright{\}}\textbf{#1}}}
+\renewcommand{\isacharunderscore}{\_}
+\renewcommand{\isacharunderscorekeyword}{\_}
+
+% add \noindent to text blocks automatically
+\renewenvironment{isamarkuptext}{\par\isastyletext\begin{isapar}\noindent}{\end{isapar}}
+\renewenvironment{isamarkuptxt}{\par\isastyletext\begin{isapar}\noindent}{\end{isapar}}
+
+\newcommand{\noquotes}[1]{{\renewcommand{\isachardoublequote}{}#1}}
+\newcommand{\concept}[1]{\textbf{#1}}
+\newcommand{\xsymbol}[1]{\texttt{\char`\\\char`<#1>}}
+
+\newcommand{\isabox}{\fbox}
+\newcommand{\bigisabox}[1]
+{\isabox{\renewcommand{\isanewline}{\\}%
+ \begin{tabular}{l}#1\end{tabular}}}
+
+%%%% ``WARNING'' environment: 2 ! characters separated by negative thin space
+%\def\warnbang{\vtop to 0pt{\vss\hbox{\let\bfdefault=\bfdefaultold\Huge\textbf{!$\!$!}}\vss}}
+
+\def\warnbang{\vtop to 0pt{\vss\hbox{\includegraphics[width=3ex, height=5.5ex]{bang}}\vss}}
+
+\newenvironment{warn}{\begin{trivlist}\small\item[]\noindent%
+ \begingroup\hangindent=\parindent\hangafter=-2%\clubpenalty=10000%
+ \def\par{\endgraf\endgroup}%
+ \hbox to0pt{\hskip-\hangindent\warnbang\hfill}\ignorespaces}
+ {\par\end{trivlist}}
+
+\renewcommand{\isachardoublequote}{}
+\renewcommand{\isachardoublequoteopen}{}
+\renewcommand{\isachardoublequoteclose}{}
+
+\renewcommand{\isacharbackquoteopen}{\isacharbackquote}
+\renewcommand{\isacharbackquoteclose}{\isacharbackquote}
+
+\hyphenation{Isa-belle}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/prog-prove.bib Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,19 @@
+@string{CUP="Cambridge University Press"}
+@string{LNCS="Lect.\ Notes in Comp.\ Sci."}
+@string{Springer="Springer-Verlag"}
+
+@book{HuthRyan,author="Michael Huth and Mark Ryan",
+title={Logic in Computer Science},publisher=CUP,year=2004}
+
+@manual{Krauss,author={Alexander Krauss},
+title={Defining Recursive Functions in Isabelle/HOL},
+note={\url{http://isabelle.in.tum.de/doc/functions.pdf}}}
+
+@book{LNCS2283,author={Tobias Nipkow and Lawrence Paulson and Markus Wenzel},
+title="Isabelle/HOL --- A Proof Assistant for Higher-Order Logic",
+publisher=Springer,series=LNCS,volume=2283,year=2002}
+
+@manual{IsarRef,author={Makarius Wenzel},
+title={The Isabelle/Isar Reference Manual},
+note={\url{http://isabelle.in.tum.de/doc/isar-ref.pdf}}}
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/prog-prove.tex Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,52 @@
+\documentclass[envcountsame,envcountchap]{svmono}
+
+\input{prelude}
+
+\excludecomment{sem}
+
+\begin{document}
+
+\title{Programming and Proving in Isabelle/HOL}
+\subtitle{\includegraphics[scale=.7]{isabelle_hol}}
+\author{Tobias Nipkow}
+\maketitle
+
+\frontmatter%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\setcounter{tocdepth}{1}
+\tableofcontents
+
+
+\mainmatter%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%\part{Isabelle}
+
+\chapter{Introduction}
+\input{intro-isabelle.tex}
+
+\chapter{Programming and Proving}
+\label{sec:FP}
+\input{Thys/document/Basics.tex}
+\input{Thys/document/Bool_nat_list}
+\input{Thys/document/Types_and_funs}
+
+%\chapter{Case Study: IMP Expressions}
+%\label{sec:CaseStudyExp}
+%\input{../generated/Expressions}
+
+\chapter{Logic}
+\label{ch:Logic}
+\input{Thys/document/Logic}
+
+\chapter{Isar: A Language for Structured Proofs}
+\label{ch:Isar}
+\input{Thys/document/Isar}
+
+\backmatter%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\bibliographystyle{plain}
+\bibliography{prog-prove}
+
+%\printindex
+
+\end{document}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/ProgProve/svmono.cls Mon Apr 02 10:49:03 2012 +0200
@@ -0,0 +1,1691 @@
+% SVMONO DOCUMENT CLASS -- version 4.17 (31-Oct-06)
+% Springer Verlag global LaTeX2e support for monographs
+%%
+%%
+%% \CharacterTable
+%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
+%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
+%% Digits \0\1\2\3\4\5\6\7\8\9
+%% Exclamation \! Double quote \" Hash (number) \#
+%% Dollar \$ Percent \% Ampersand \&
+%% Acute accent \' Left paren \( Right paren \)
+%% Asterisk \* Plus \+ Comma \,
+%% Minus \- Point \. Solidus \/
+%% Colon \: Semicolon \; Less than \<
+%% Equals \= Greater than \> Question mark \?
+%% Commercial at \@ Left bracket \[ Backslash \\
+%% Right bracket \] Circumflex \^ Underscore \_
+%% Grave accent \` Left brace \{ Vertical bar \|
+%% Right brace \} Tilde \~}
+%%
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+\ProvidesClass{svmono}[2006/10/31 v4.17
+^^JSpringer Verlag global LaTeX document class for monographs]
+
+% Options
+% citations
+\DeclareOption{natbib}{\ExecuteOptions{oribibl}%
+\AtEndOfClass{% Loading package 'NATBIB'
+\RequirePackage{natbib}
+% Changing some parameters of NATBIB
+\setlength{\bibhang}{\parindent}
+%\setlength{\bibsep}{0mm}
+\let\bibfont=\small
+\def\@biblabel#1{#1.}
+\newcommand{\etal}{\textit{et al}.}
+%\bibpunct[,]{(}{)}{;}{a}{}{,}}}
+}}
+% Springer environment
+\let\if@spthms\iftrue
+\DeclareOption{nospthms}{\let\if@spthms\iffalse}
+%
+\let\envankh\@empty % no anchor for "theorems"
+%
+\let\if@envcntreset\iffalse % environment counter is not reset
+\let\if@envcntresetsect=\iffalse % reset each section?
+\DeclareOption{envcountresetchap}{\let\if@envcntreset\iftrue}
+\DeclareOption{envcountresetsect}{\let\if@envcntreset\iftrue
+\let\if@envcntresetsect=\iftrue}
+%
+\let\if@envcntsame\iffalse % NOT all environments work like "Theorem",
+ % each using its own counter
+\DeclareOption{envcountsame}{\let\if@envcntsame\iftrue}
+%
+\let\if@envcntshowhiercnt=\iffalse % do not show hierarchy counter at all
+%
+% enhance theorem counter
+\DeclareOption{envcountchap}{\def\envankh{chapter}% show \thechapter along with theorem number
+\let\if@envcntshowhiercnt=\iftrue
+\ExecuteOptions{envcountreset}}
+%
+\DeclareOption{envcountsect}{\def\envankh{section}% show \thesection along with theorem number
+\let\if@envcntshowhiercnt=\iftrue
+\ExecuteOptions{envcountreset}}
+%
+% languages
+\let\switcht@@therlang\relax
+\let\svlanginfo\relax
+\def\ds@deutsch{\def\switcht@@therlang{\switcht@deutsch}%
+\gdef\svlanginfo{\typeout{Man spricht deutsch.}\global\let\svlanginfo\relax}}
+\def\ds@francais{\def\switcht@@therlang{\switcht@francais}%
+\gdef\svlanginfo{\typeout{On parle francais.}\global\let\svlanginfo\relax}}
+%
+\AtBeginDocument{\@ifpackageloaded{babel}{%
+\@ifundefined{extrasamerican}{}{\addto\extrasamerican{\switcht@albion}}%
+\@ifundefined{extrasaustralian}{}{\addto\extrasaustralian{\switcht@albion}}%
+\@ifundefined{extrasbritish}{}{\addto\extrasbritish{\switcht@albion}}%
+\@ifundefined{extrascanadian}{}{\addto\extrascanadian{\switcht@albion}}%
+\@ifundefined{extrasenglish}{}{\addto\extrasenglish{\switcht@albion}}%
+\@ifundefined{extrasnewzealand}{}{\addto\extrasnewzealand{\switcht@albion}}%
+\@ifundefined{extrasUKenglish}{}{\addto\extrasUKenglish{\switcht@albion}}%
+\@ifundefined{extrasUSenglish}{}{\addto\extrasUSenglish{\switcht@albion}}%
+\@ifundefined{captionsfrench}{}{\addto\captionsfrench{\switcht@francais}}%
+\@ifundefined{extrasgerman}{}{\addto\extrasgerman{\switcht@deutsch}}%
+\@ifundefined{extrasngerman}{}{\addto\extrasngerman{\switcht@deutsch}}%
+}{\switcht@@therlang}%
+}
+% numbering style of floats, equations
+\newif\if@numart \@numartfalse
+\DeclareOption{numart}{\@numarttrue}
+\def\set@numbering{\if@numart\else\num@book\fi}
+\AtEndOfClass{\set@numbering}
+% style for vectors
+\DeclareOption{vecphys}{\def\vec@style{phys}}
+\DeclareOption{vecarrow}{\def\vec@style{arrow}}
+% running heads
+\let\if@runhead\iftrue
+\DeclareOption{norunningheads}{\let\if@runhead\iffalse}
+% referee option
+\let\if@referee\iffalse
+\def\makereferee{\def\baselinestretch{2}\selectfont
+\newbox\refereebox
+\setbox\refereebox=\vbox to\z@{\vskip0.5cm%
+ \hbox to\textwidth{\normalsize\tt\hrulefill\lower0.5ex
+ \hbox{\kern5\p@ referee's copy\kern5\p@}\hrulefill}\vss}%
+\def\@oddfoot{\copy\refereebox}\let\@evenfoot=\@oddfoot}
+\DeclareOption{referee}{\let\if@referee\iftrue
+\AtBeginDocument{\makereferee\small\normalsize}}
+% modification of thebibliography
+\let\if@openbib\iffalse
+\DeclareOption{openbib}{\let\if@openbib\iftrue}
+% LaTeX standard, sectionwise references
+\DeclareOption{oribibl}{\let\oribibl=Y}
+\DeclareOption{sectrefs}{\let\secbibl=Y}
+%
+% footinfo option (provides an informatory line on every page)
+\def\SpringerMacroPackageNameA{svmono.cls}
+% \thetime, \thedate and \timstamp are macros to include
+% time, date (or both) of the TeX run in the document
+\def\maketimestamp{\count255=\time
+\divide\count255 by 60\relax
+\edef\thetime{\the\count255:}%
+\multiply\count255 by-60\relax
+\advance\count255 by\time
+\edef\thetime{\thetime\ifnum\count255<10 0\fi\the\count255}
+\edef\thedate{\number\day-\ifcase\month\or Jan\or Feb\or Mar\or
+ Apr\or May\or Jun\or Jul\or Aug\or Sep\or Oct\or
+ Nov\or Dec\fi-\number\year}
+\def\timstamp{\hbox to\hsize{\tt\hfil\thedate\hfil\thetime\hfil}}}
+\maketimestamp
+%
+% \footinfo generates a info footline on every page containing
+% pagenumber, jobname, macroname, and timestamp
+\DeclareOption{footinfo}{\AtBeginDocument{\maketimestamp
+ \def\ps@empty{\let\@mkboth\@gobbletwo
+ \let\@oddhead\@empty\let\@evenhead\@empty}%
+ \def\@oddfoot{\scriptsize\tt Page:\,\thepage\space\hfil
+ job:\,\jobname\space\hfil
+ macro:\,\SpringerMacroPackageNameA\space\hfil
+ date/time:\,\thedate/\thetime}%
+ \let\@evenfoot=\@oddfoot}}
+%
+% start new chapter on any page
+\newif\if@openright \@openrighttrue
+\DeclareOption{openany}{\@openrightfalse}
+%
+% no size changing allowed
+\DeclareOption{11pt}{\OptionNotUsed}
+\DeclareOption{12pt}{\OptionNotUsed}
+% options for the article class
+\def\@rticle@options{10pt,twoside}
+% fleqn
+\DeclareOption{fleqn}{\def\@rticle@options{10pt,twoside,fleqn}%
+\AtEndOfClass{\let\leftlegendglue\relax}%
+\AtBeginDocument{\mathindent\parindent}}
+% hanging sectioning titles
+\let\if@sechang\iffalse
+\DeclareOption{sechang}{\let\if@sechang\iftrue}
+\def\ClassInfoNoLine#1#2{%
+ \ClassInfo{#1}{#2\@gobble}%
+}
+\let\SVMonoOpt\@empty
+\DeclareOption*{\InputIfFileExists{sv\CurrentOption.clo}{%
+\global\let\SVMonoOpt\CurrentOption}{%
+\ClassWarning{Springer-SVMono}{Specified option or subpackage
+"\CurrentOption" \MessageBreak not found
+passing it to article class \MessageBreak
+-}\PassOptionsToClass{\CurrentOption}{article}%
+}}
+\ProcessOptions\relax
+\ifx\SVMonoOpt\@empty\relax
+\ClassInfoNoLine{Springer-SVMono}{extra/valid Springer sub-package
+\MessageBreak not found in option list - using "global" style}{}
+\fi
+\LoadClass[\@rticle@options]{article}
+\raggedbottom
+
+% various sizes and settings for monographs
+
+\setlength{\textwidth}{28pc} % 11.8cm
+%\setlength{\textheight}{12pt}\multiply\textheight by 45\relax
+\setlength{\textheight}{540\p@}
+\setlength{\topmargin}{0cm}
+\setlength\oddsidemargin {63\p@}
+\setlength\evensidemargin {63\p@}
+\setlength\marginparwidth{90\p@}
+\setlength\headsep {12\p@}
+
+\setlength{\parindent}{15\p@}
+\setlength{\parskip}{\z@ \@plus \p@}
+\setlength{\hfuzz}{2\p@}
+\setlength{\arraycolsep}{1.5\p@}
+
+\frenchspacing
+
+\tolerance=500
+
+\predisplaypenalty=0
+\clubpenalty=10000
+\widowpenalty=10000
+
+\setlength\footnotesep{7.7\p@}
+
+\newdimen\betweenumberspace % dimension for space between
+\betweenumberspace=5\p@ % number and text of titles
+\newdimen\headlineindent % dimension for space of
+\headlineindent=2.5cc % number and gap of running heads
+
+% fonts, sizes, and the like
+\renewcommand\small{%
+ \@setfontsize\small\@ixpt{11}%
+ \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@
+ \abovedisplayshortskip \z@ \@plus2\p@
+ \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@
+ \def\@listi{\leftmargin\leftmargini
+ \parsep \z@ \@plus\p@ \@minus\p@
+ \topsep 6\p@ \@plus2\p@ \@minus4\p@
+ \itemsep\z@}%
+ \belowdisplayskip \abovedisplayskip
+}
+%
+\let\footnotesize=\small
+%
+\newenvironment{petit}{\par\addvspace{6\p@}\small}{\par\addvspace{6\p@}}
+%
+
+% modification of automatic positioning of floating objects
+\setlength\@fptop{\z@ }
+\setlength\@fpsep{12\p@ }
+\setlength\@fpbot{\z@ \@plus 1fil }
+\def\textfraction{.01}
+\def\floatpagefraction{.8}
+\setlength{\intextsep}{20\p@ \@plus 2\p@ \@minus 2\p@}
+\setcounter{topnumber}{4}
+\def\topfraction{.9}
+\setcounter{bottomnumber}{2}
+\def\bottomfraction{.7}
+\setcounter{totalnumber}{6}
+%
+% size and style of headings
+\newcommand{\partsize}{\Large}
+\newcommand{\partstyle}{\bfseries\boldmath}
+\newcommand{\chapsize}{\Large}
+\newcommand{\chapstyle}{\bfseries\boldmath}
+\newcommand{\chapshooksize}{\small}
+\newcommand{\chapshookstyle}{\itshape\unboldmath}
+\newcommand{\secsize}{\large}
+\newcommand{\secstyle}{\bfseries\boldmath}
+\newcommand{\subsecsize}{\normalsize}
+\newcommand{\subsecstyle}{\bfseries\boldmath}
+%
+\def\cleardoublepage{\clearpage\if@twoside \ifodd\c@page\else
+ \hbox{}\newpage\if@twocolumn\hbox{}\newpage\fi\fi\fi}
+
+\newcommand{\clearemptydoublepage}{%
+ \clearpage{\pagestyle{empty}\cleardoublepage}}
+\newcommand{\startnewpage}{\if@openright\clearemptydoublepage\else\clearpage\fi}
+
+% redefinition of \part
+\renewcommand\part{\clearemptydoublepage
+ \thispagestyle{empty}
+ \if@twocolumn
+ \onecolumn
+ \@tempswatrue
+ \else
+ \@tempswafalse
+ \fi
+ \@ifundefined{thispagecropped}{}{\thispagecropped}
+ \secdef\@part\@spart}
+
+\def\@part[#1]#2{\ifnum \c@secnumdepth >-2\relax
+ \refstepcounter{part}
+ \addcontentsline{toc}{part}{\partname\
+ \thepart\thechapterend\hspace{\betweenumberspace}%
+ #1}\else
+ \addcontentsline{toc}{part}{#1}\fi
+ \markboth{}{}
+ {\raggedleft
+ \ifnum \c@secnumdepth >-2\relax
+ \normalfont\partstyle\partsize\vrule height 34pt width 0pt depth 0pt%
+ \partname\ \thepart\llap{\smash{\lower 5pt\hbox to\textwidth{\hrulefill}}}
+ \par
+ \vskip 128.3\p@ \fi
+ #2\par}\@endpart}
+%
+% \@endpart finishes the part page
+%
+\def\@endpart{\vfil\newpage
+ \if@twoside
+ \hbox{}
+ \thispagestyle{empty}
+ \newpage
+ \fi
+ \if@tempswa
+ \twocolumn
+ \fi}
+%
+\def\@spart#1{{\raggedleft
+ \normalfont\partsize\partstyle
+ #1\par}\@endpart}
+%
+\newenvironment{partbacktext}{\def\@endpart{\vfil\newpage}}
+{\thispagestyle{empty} \newpage \if@tempswa\twocolumn\fi}
+%
+% (re)define sectioning
+\setcounter{secnumdepth}{2}
+
+\def\seccounterend{}
+\def\seccountergap{\hskip\betweenumberspace}
+\def\@seccntformat#1{\csname the#1\endcsname\seccounterend\seccountergap\ignorespaces}
+%
+\let\firstmark=\botmark
+%
+\@ifundefined{thechapterend}{\def\thechapterend{}}{}
+%
+\if@sechang
+ \def\sec@hangfrom#1{\setbox\@tempboxa\hbox{#1}%
+ \hangindent\wd\@tempboxa\noindent\box\@tempboxa}
+\else
+ \def\sec@hangfrom#1{\setbox\@tempboxa\hbox{#1}%
+ \hangindent\z@\noindent\box\@tempboxa}
+\fi
+
+\def\chap@hangfrom#1{\noindent\vrule height 34pt width 0pt depth 0pt
+\rlap{\smash{\lower 5pt\hbox to\textwidth{\hrulefill}}}\hbox{#1}
+\vskip10pt}
+\def\schap@hangfrom{\chap@hangfrom{}}
+
+\newcounter{chapter}
+%
+\@addtoreset{section}{chapter}
+\@addtoreset{footnote}{chapter}
+
+\newif\if@mainmatter \@mainmattertrue
+\newcommand\frontmatter{\startnewpage
+ \@mainmatterfalse\pagenumbering{Roman}
+ \setcounter{page}{5}}
+%
+\newcommand\mainmatter{\clearemptydoublepage
+ \@mainmattertrue\pagenumbering{arabic}}
+%
+\newcommand\backmatter{\clearemptydoublepage\@mainmatterfalse}
+
+\def\@chapapp{\chaptername}
+
+\newdimen\chapstarthookwidth
+\newcommand\chapstarthook[2][0.66\textwidth]{%
+\setlength{\chapstarthookwidth}{#1}%
+\gdef\chapst@rthook{#2}}
+
+\newcommand{\processchapstarthook}{\@ifundefined{chapst@rthook}{}{%
+ \setbox0=\hbox{\vbox{\hyphenpenalty=50
+ \begin{flushright}
+ \begin{minipage}{\chapstarthookwidth}
+ \vrule\@width\z@\@height21\p@\@depth\z@
+ \normalfont\chapshooksize\chapshookstyle\chapst@rthook
+ \end{minipage}
+ \end{flushright}}}%
+ \@tempdima=\pagetotal
+ \advance\@tempdima by\ht0
+ \ifdim\@tempdima<106\p@
+ \multiply\@tempdima by-1
+ \advance\@tempdima by106\p@
+ \vskip\@tempdima
+ \fi
+ \box0\par
+ \global\let\chapst@rthook=\undefined}}
+
+\newcommand\chapter{\startnewpage
+ \@ifundefined{thispagecropped}{}{\thispagecropped}
+ \thispagestyle{empty}%
+ \global\@topnum\z@
+ \@afterindentfalse
+ \secdef\@chapter\@schapter}
+
+\def\@chapter[#1]#2{\ifnum \c@secnumdepth >\m@ne
+ \refstepcounter{chapter}%
+ \if@mainmatter
+ \typeout{\@chapapp\space\thechapter.}%
+ \addcontentsline{toc}{chapter}{\protect
+ \numberline{\thechapter\thechapterend}#1}%
+ \else
+ \addcontentsline{toc}{chapter}{#1}%
+ \fi
+ \else
+ \addcontentsline{toc}{chapter}{#1}%
+ \fi
+ \chaptermark{#1}%
+ \addtocontents{lof}{\protect\addvspace{10\p@}}%
+ \addtocontents{lot}{\protect\addvspace{10\p@}}%
+ \if@twocolumn
+ \@topnewpage[\@makechapterhead{#2}]%
+ \else
+ \@makechapterhead{#2}%
+ \@afterheading
+ \fi}
+
+\def\@schapter#1{\if@twocolumn
+ \@topnewpage[\@makeschapterhead{#1}]%
+ \else
+ \@makeschapterhead{#1}%
+ \@afterheading
+ \fi}
+
+%%changes position and layout of numbered chapter headings
+\def\@makechapterhead#1{{\parindent\z@\raggedright\normalfont
+ \hyphenpenalty \@M
+ \interlinepenalty\@M
+ \chapsize\chapstyle
+ \chap@hangfrom{\thechapter\thechapterend\hskip\betweenumberspace}%!!!
+ \ignorespaces#1\par\nobreak
+ \processchapstarthook
+ \ifdim\pagetotal>157\p@
+ \vskip 11\p@
+ \else
+ \@tempdima=168\p@\advance\@tempdima by-\pagetotal
+ \vskip\@tempdima
+ \fi}}
+
+%%changes position and layout of unnumbered chapter headings
+\def\@makeschapterhead#1{{\parindent \z@ \raggedright\normalfont
+ \hyphenpenalty \@M
+ \interlinepenalty\@M
+ \chapsize\chapstyle
+ \schap@hangfrom
+ \ignorespaces#1\par\nobreak
+ \processchapstarthook
+ \ifdim\pagetotal>157\p@
+ \vskip 11\p@
+ \else
+ \@tempdima=168\p@\advance\@tempdima by-\pagetotal
+ \vskip\@tempdima
+ \fi}}
+
+% predefined unnumbered headings
+\newcommand{\preface}[1][\prefacename]{\chapter*{#1}\markboth{#1}{#1}}
+% same with TOC entry
+\newcommand{\Preface}[1][\prefacename]{\chapter*{#1}\markboth{#1}{#1}%
+\addcontentsline{toc}{chapter}{#1}}
+
+% measures and setting of sections
+\renewcommand\section{\@startsection{section}{1}{\z@}%
+ {-24\p@ \@plus -4\p@ \@minus -4\p@}%
+ {12\p@ \@plus 4\p@ \@minus 4\p@}%
+ {\normalfont\secsize\secstyle
+ \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
+ {-17\p@ \@plus -4\p@ \@minus -4\p@}%
+ {10\p@ \@plus 4\p@ \@minus 4\p@}%
+ {\normalfont\subsecsize\subsecstyle
+ \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
+ {-17\p@ \@plus -4\p@ \@minus -4\p@}%
+ {10\p@ \@plus 4\p@ \@minus 4\p@}%
+ {\normalfont\normalsize\subsecstyle
+ \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
+ {-10\p@ \@plus -4\p@ \@minus -4\p@}%
+ {10\p@ \@plus 4\p@ \@minus 4\p@}%
+ {\normalfont\normalsize\itshape
+ \rightskip=\z@ \@plus 8em\pretolerance=10000 }}
+\def\subparagraph{\@startsection{subparagraph}{5}{\z@}%
+ {-5.388\p@ \@plus-4\p@ \@minus-4\p@}{-5\p@}{\normalfont\normalsize\itshape}}
+
+% Appendix
+\renewcommand\appendix{\par
+ \stepcounter{chapter}
+ \setcounter{chapter}{0}
+ \stepcounter{section}
+ \setcounter{section}{0}
+ \setcounter{equation}{0}
+ \setcounter{figure}{0}
+ \setcounter{table}{0}
+ \setcounter{footnote}{0}
+ \def\@chapapp{\appendixname}%
+ \renewcommand\thechapter{\@Alph\c@chapter}}
+
+% definition of sections
+% \hyphenpenalty and \raggedright added, so that there is no
+% hyphenation and the text is set ragged-right in sectioning
+
+\def\runinsep{}
+\def\aftertext{\unskip\runinsep}
+%
+\def\thesection{\thechapter.\arabic{section}}
+\def\thesubsection{\thesection.\arabic{subsection}}
+\def\thesubsubsection{\thesubsection.\arabic{subsubsection}}
+\def\theparagraph{\thesubsubsection.\arabic{paragraph}}
+\def\thesubparagraph{\theparagraph.\arabic{subparagraph}}
+\def\chaptermark#1{}
+%
+\def\@ssect#1#2#3#4#5{%
+ \@tempskipa #3\relax
+ \ifdim \@tempskipa>\z@
+ \begingroup
+ #4{%
+ \@hangfrom{\hskip #1}%
+ \raggedright
+ \hyphenpenalty \@M
+ \interlinepenalty \@M #5\@@par}%
+ \endgroup
+ \else
+ \def\@svsechd{#4{\hskip #1\relax #5}}%
+ \fi
+ \@xsect{#3}}
+%
+\def\@sect#1#2#3#4#5#6[#7]#8{%
+ \ifnum #2>\c@secnumdepth
+ \let\@svsec\@empty
+ \else
+ \refstepcounter{#1}%
+ \protected@edef\@svsec{\@seccntformat{#1}\relax}%
+ \fi
+ \@tempskipa #5\relax
+ \ifdim \@tempskipa>\z@
+ \begingroup #6\relax
+ \sec@hangfrom{\hskip #3\relax\@svsec}%
+ {\raggedright
+ \hyphenpenalty \@M
+ \interlinepenalty \@M #8\@@par}%
+ \endgroup
+ \csname #1mark\endcsname{#7\seccounterend}%
+ \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth
+ \else
+ \protect\numberline{\csname the#1\endcsname\seccounterend}%
+ \fi
+ #7}%
+ \else
+ \def\@svsechd{%
+ #6\hskip #3\relax
+ \@svsec #8\aftertext\ignorespaces
+ \csname #1mark\endcsname{#7}%
+ \addcontentsline{toc}{#1}{%
+ \ifnum #2>\c@secnumdepth \else
+ \protect\numberline{\csname the#1\endcsname\seccounterend}%
+ \fi
+ #7}}%
+ \fi
+ \@xsect{#5}}
+
+% figures and tables are processed in small print
+\def \@floatboxreset {%
+ \reset@font
+ \small
+ \@setnobreak
+ \@setminipage
+}
+\def\fps@figure{htbp}
+\def\fps@table{htbp}
+
+% Frame for paste-in figures or tables
+\def\mpicplace#1#2{% #1 =width #2 =height
+\vbox{\hbox to #1{\vrule\@width \fboxrule \@height #2\hfill}}}
+
+% labels of enumerate
+\renewcommand\labelenumii{\theenumii)}
+\renewcommand\theenumii{\@alph\c@enumii}
+
+% labels of itemize
+\renewcommand\labelitemi{\textbullet}
+\renewcommand\labelitemii{\textendash}
+\let\labelitemiii=\labelitemiv
+
+% labels of description
+\renewcommand*\descriptionlabel[1]{\hspace\labelsep #1\hfil}
+
+% fixed indentation for standard itemize-environment
+\newdimen\svitemindent \setlength{\svitemindent}{\parindent}
+
+
+% make indentations changeable
+
+\def\setitemindent#1{\settowidth{\labelwidth}{#1}%
+ \let\setit@m=Y%
+ \leftmargini\labelwidth
+ \advance\leftmargini\labelsep
+ \def\@listi{\leftmargin\leftmargini
+ \labelwidth\leftmargini\advance\labelwidth by -\labelsep
+ \parsep=\parskip
+ \topsep=\medskipamount
+ \itemsep=\parskip \advance\itemsep by -\parsep}}
+\def\setitemitemindent#1{\settowidth{\labelwidth}{#1}%
+ \let\setit@m=Y%
+ \leftmarginii\labelwidth
+ \advance\leftmarginii\labelsep
+\def\@listii{\leftmargin\leftmarginii
+ \labelwidth\leftmarginii\advance\labelwidth by -\labelsep
+ \parsep=\parskip
+ \topsep=\z@
+ \itemsep=\parskip \advance\itemsep by -\parsep}}
+%
+% adjusted environment "description"
+% if an optional parameter (at the first two levels of lists)
+% is present, its width is considered to be the widest mark
+% throughout the current list.
+\def\description{\@ifnextchar[{\@describe}{\list{}{\labelwidth\z@
+ \itemindent-\leftmargin \let\makelabel\descriptionlabel}}}
+%
+\def\describelabel#1{#1\hfil}
+\def\@describe[#1]{\relax\ifnum\@listdepth=0
+\setitemindent{#1}\else\ifnum\@listdepth=1
+\setitemitemindent{#1}\fi\fi
+\list{--}{\let\makelabel\describelabel}}
+%
+\def\itemize{%
+ \ifnum \@itemdepth >\thr@@\@toodeep\else
+ \advance\@itemdepth\@ne
+ \ifx\setit@m\undefined
+ \ifnum \@itemdepth=1 \leftmargini=\svitemindent
+ \labelwidth\leftmargini\advance\labelwidth-\labelsep
+ \leftmarginii=\leftmargini \leftmarginiii=\leftmargini
+ \fi
+ \fi
+ \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
+ \expandafter\list
+ \csname\@itemitem\endcsname
+ {\def\makelabel##1{\rlap{##1}\hss}}%
+ \fi}
+%
+\newdimen\verbatimindent \verbatimindent\parindent
+\def\verbatim{\advance\@totalleftmargin by\verbatimindent
+\@verbatim \frenchspacing\@vobeyspaces \@xverbatim}
+
+%
+% special signs and characters
+\newcommand{\D}{\mathrm{d}}
+\newcommand{\E}{\mathrm{e}}
+\let\eul=\E
+\newcommand{\I}{{\rm i}}
+\let\imag=\I
+%
+% the definition of uppercase Greek characters
+% Springer likes them as italics to depict variables
+\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00}
+\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01}
+\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02}
+\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03}
+\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04}
+\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05}
+\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06}
+\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07}
+\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08}
+\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09}
+\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A}
+% the upright forms are defined here as \var<Character>
+\DeclareMathSymbol{\varGamma}{\mathalpha}{operators}{"00}
+\DeclareMathSymbol{\varDelta}{\mathalpha}{operators}{"01}
+\DeclareMathSymbol{\varTheta}{\mathalpha}{operators}{"02}
+\DeclareMathSymbol{\varLambda}{\mathalpha}{operators}{"03}
+\DeclareMathSymbol{\varXi}{\mathalpha}{operators}{"04}
+\DeclareMathSymbol{\varPi}{\mathalpha}{operators}{"05}
+\DeclareMathSymbol{\varSigma}{\mathalpha}{operators}{"06}
+\DeclareMathSymbol{\varUpsilon}{\mathalpha}{operators}{"07}
+\DeclareMathSymbol{\varPhi}{\mathalpha}{operators}{"08}
+\DeclareMathSymbol{\varPsi}{\mathalpha}{operators}{"09}
+\DeclareMathSymbol{\varOmega}{\mathalpha}{operators}{"0A}
+% Upright Lower Case Greek letters without using a new MathAlphabet
+\newcommand{\greeksym}[1]{\usefont{U}{psy}{m}{n}#1}
+\newcommand{\greeksymbold}[1]{{\usefont{U}{psy}{b}{n}#1}}
+\newcommand{\allmodesymb}[2]{\relax\ifmmode{\mathchoice
+{\mbox{\fontsize{\tf@size}{\tf@size}#1{#2}}}
+{\mbox{\fontsize{\tf@size}{\tf@size}#1{#2}}}
+{\mbox{\fontsize{\sf@size}{\sf@size}#1{#2}}}
+{\mbox{\fontsize{\ssf@size}{\ssf@size}#1{#2}}}}
+\else
+\mbox{#1{#2}}\fi}
+% Definition of lower case Greek letters
+\newcommand{\ualpha}{\allmodesymb{\greeksym}{a}}
+\newcommand{\ubeta}{\allmodesymb{\greeksym}{b}}
+\newcommand{\uchi}{\allmodesymb{\greeksym}{c}}
+\newcommand{\udelta}{\allmodesymb{\greeksym}{d}}
+\newcommand{\ugamma}{\allmodesymb{\greeksym}{g}}
+\newcommand{\umu}{\allmodesymb{\greeksym}{m}}
+\newcommand{\unu}{\allmodesymb{\greeksym}{n}}
+\newcommand{\upi}{\allmodesymb{\greeksym}{p}}
+\newcommand{\utau}{\allmodesymb{\greeksym}{t}}
+% redefines the \vec accent to a bold character - if desired
+\def\fig@type{arrow}% temporarily abused
+\ifx\vec@style\fig@type\else
+\@ifundefined{vec@style}{%
+ \def\vec#1{\ensuremath{\mathchoice
+ {\mbox{\boldmath$\displaystyle\mathbf{#1}$}}
+ {\mbox{\boldmath$\textstyle\mathbf{#1}$}}
+ {\mbox{\boldmath$\scriptstyle\mathbf{#1}$}}
+ {\mbox{\boldmath$\scriptscriptstyle\mathbf{#1}$}}}}%
+}
+{\def\vec#1{\ensuremath{\mathchoice
+ {\mbox{\boldmath$\displaystyle#1$}}
+ {\mbox{\boldmath$\textstyle#1$}}
+ {\mbox{\boldmath$\scriptstyle#1$}}
+ {\mbox{\boldmath$\scriptscriptstyle#1$}}}}%
+}
+\fi
+% tensor
+\def\tens#1{\relax\ifmmode\mathsf{#1}\else\textsf{#1}\fi}
+
+% end of proof symbol
+\newcommand\qedsymbol{\hbox{\rlap{$\sqcap$}$\sqcup$}}
+\newcommand\qed{\relax\ifmmode\else\unskip\quad\fi\qedsymbol}
+\newcommand\smartqed{\renewcommand\qed{\relax\ifmmode\qedsymbol\else
+ {\unskip\nobreak\hfil\penalty50\hskip1em\null\nobreak\hfil\qedsymbol
+ \parfillskip=\z@\finalhyphendemerits=0\endgraf}\fi}}
+%
+\def\num@book{%
+\renewcommand\thesection{\thechapter.\@arabic\c@section}%
+\renewcommand\thesubsection{\thesection.\@arabic\c@subsection}%
+\renewcommand\theequation{\thechapter.\@arabic\c@equation}%
+\renewcommand\thefigure{\thechapter.\@arabic\c@figure}%
+\renewcommand\thetable{\thechapter.\@arabic\c@table}%
+\@addtoreset{section}{chapter}%
+\@addtoreset{figure}{chapter}%
+\@addtoreset{table}{chapter}%
+\@addtoreset{equation}{chapter}}
+%
+% Ragged bottom for the actual page
+\def\thisbottomragged{\def\@textbottom{\vskip\z@ \@plus.0001fil
+\global\let\@textbottom\relax}}
+
+% This is texte.tex
+% it defines various texts and their translations
+% called up with documentstyle options
+\def\switcht@albion{%
+\def\abstractname{Summary.}%
+\def\ackname{Acknowledgement.}%
+\def\andname{and}%
+\def\bibname{References}%
+\def\lastandname{, and}%
+\def\appendixname{Appendix}%
+\def\chaptername{Chapter}%
+\def\claimname{Claim}%
+\def\conjecturename{Conjecture}%
+\def\contentsname{Contents}%
+\def\corollaryname{Corollary}%
+\def\definitionname{Definition}%
+\def\examplename{Example}%
+\def\exercisename{Exercise}%
+\def\figurename{Fig.}%
+\def\keywordname{{\bf Key words:}}%
+\def\indexname{Index}%
+\def\lemmaname{Lemma}%
+\def\contriblistname{List of Contributors}%
+\def\listfigurename{List of Figures}%
+\def\listtablename{List of Tables}%
+\def\mailname{{\it Correspondence to\/}:}%
+\def\noteaddname{Note added in proof}%
+\def\notename{Note}%
+\def\partname{Part}%
+\def\prefacename{Preface}%
+\def\problemname{Problem}%
+\def\proofname{Proof}%
+\def\propertyname{Property}%
+\def\propositionname{Proposition}%
+\def\questionname{Question}%
+\def\refname{References}%
+\def\remarkname{Remark}%
+\def\seename{see}%
+\def\solutionname{Solution}%
+\def\subclassname{{\it Subject Classifications\/}:}%
+\def\tablename{Table}%
+\def\theoremname{Theorem}}
+\switcht@albion
+% Names of theorem like environments are already defined
+% but must be translated if another language is chosen
+%
+% French section
+\def\switcht@francais{\svlanginfo
+ \def\abstractname{R\'esum\'e.}%
+ \def\ackname{Remerciements.}%
+ \def\andname{et}%
+ \def\lastandname{ et}%
+ \def\appendixname{Appendice}%
+ \def\bibname{Bibliographie}%
+ \def\chaptername{Chapitre}%
+ \def\claimname{Pr\'etention}%
+ \def\conjecturename{Hypoth\`ese}%
+ \def\contentsname{Table des mati\`eres}%
+ \def\corollaryname{Corollaire}%
+ \def\definitionname{D\'efinition}%
+ \def\examplename{Exemple}%
+ \def\exercisename{Exercice}%
+ \def\figurename{Fig.}%
+ \def\keywordname{{\bf Mots-cl\'e:}}%
+ \def\indexname{Index}%
+ \def\lemmaname{Lemme}%
+ \def\contriblistname{Liste des contributeurs}%
+ \def\listfigurename{Liste des figures}%
+ \def\listtablename{Liste des tables}%
+ \def\mailname{{\it Correspondence to\/}:}%
+ \def\noteaddname{Note ajout\'ee \`a l'\'epreuve}%
+ \def\notename{Remarque}%
+ \def\partname{Partie}%
+ \def\prefacename{Avant-propos}% ou Pr\'eface
+ \def\problemname{Probl\`eme}%
+ \def\proofname{Preuve}%
+ \def\propertyname{Caract\'eristique}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Question}%
+ \def\refname{Litt\'erature}%
+ \def\remarkname{Remarque}%
+ \def\seename{voir}%
+ \def\solutionname{Solution}%
+ \def\subclassname{{\it Subject Classifications\/}:}%
+ \def\tablename{Tableau}%
+ \def\theoremname{Th\'eor\`eme}%
+}
+%
+% German section
+\def\switcht@deutsch{\svlanginfo
+ \def\abstractname{Zusammenfassung.}%
+ \def\ackname{Danksagung.}%
+ \def\andname{und}%
+ \def\lastandname{ und}%
+ \def\appendixname{Anhang}%
+ \def\bibname{Literaturverzeichnis}%
+ \def\chaptername{Kapitel}%
+ \def\claimname{Behauptung}%
+ \def\conjecturename{Hypothese}%
+ \def\contentsname{Inhaltsverzeichnis}%
+ \def\corollaryname{Korollar}%
+%\def\definitionname{Definition}%
+ \def\examplename{Beispiel}%
+ \def\exercisename{\"Ubung}%
+ \def\figurename{Abb.}%
+ \def\keywordname{{\bf Schl\"usselw\"orter:}}%
+ \def\indexname{Sachverzeichnis}%
+%\def\lemmaname{Lemma}%
+ \def\contriblistname{Mitarbeiter}%
+ \def\listfigurename{Abbildungsverzeichnis}%
+ \def\listtablename{Tabellenverzeichnis}%
+ \def\mailname{{\it Correspondence to\/}:}%
+ \def\noteaddname{Nachtrag}%
+ \def\notename{Anmerkung}%
+ \def\partname{Teil}%
+ \def\prefacename{Vorwort}%
+%\def\problemname{Problem}%
+ \def\proofname{Beweis}%
+ \def\propertyname{Eigenschaft}%
+%\def\propositionname{Proposition}%
+ \def\questionname{Frage}%
+ \def\refname{Literaturverzeichnis}%
+ \def\remarkname{Anmerkung}%
+ \def\seename{siehe}%
+ \def\solutionname{L\"osung}%
+ \def\subclassname{{\it Subject Classifications\/}:}%
+ \def\tablename{Tabelle}%
+%\def\theoremname{Theorem}%
+}
+
+\def\getsto{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr\gets\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr\gets
+\cr\to\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+\gets\cr\to\cr}}}}}
+\def\lid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr<\cr\noalign{\vskip1.2\p@}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr<\cr
+\noalign{\vskip1.2\p@}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr<\cr
+\noalign{\vskip\p@}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+<\cr
+\noalign{\vskip0.9\p@}=\cr}}}}}
+\def\gid{\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip1.2\p@}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr>\cr
+\noalign{\vskip1.2\p@}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr>\cr
+\noalign{\vskip\p@}=\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr
+\noalign{\vskip0.9\p@}=\cr}}}}}
+\def\grole{\mathrel{\mathchoice {\vcenter{\offinterlineskip
+\halign{\hfil
+$\displaystyle##$\hfil\cr>\cr\noalign{\vskip-\p@}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\textstyle##$\hfil\cr
+>\cr\noalign{\vskip-\p@}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.8\p@}<\cr}}}
+{\vcenter{\offinterlineskip\halign{\hfil$\scriptscriptstyle##$\hfil\cr
+>\cr\noalign{\vskip-0.3\p@}<\cr}}}}}
+\def\bbbr{{\rm I\!R}} %reelle Zahlen
+\def\bbbm{{\rm I\!M}}
+\def\bbbn{{\rm I\!N}} %natuerliche Zahlen
+\def\bbbf{{\rm I\!F}}
+\def\bbbh{{\rm I\!H}}
+\def\bbbk{{\rm I\!K}}
+\def\bbbp{{\rm I\!P}}
+\def\bbbone{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
+{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
+\def\bbbc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
+to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox
+to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
+to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
+to\z@{\kern0.4\wd0\vrule\@height0.9\ht0\hss}\box0}}}}
+\def\bbbq{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+Q$}\hbox{\raise
+0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.8\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.7\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm Q$}\hbox{\raise
+0.15\ht0\hbox to\z@{\kern0.4\wd0\vrule\@height0.7\ht0\hss}\box0}}}}
+\def\bbbt{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm
+T$}\hbox{\hbox to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle\rm T$}\hbox{\hbox
+to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle\rm T$}\hbox{\hbox
+to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm T$}\hbox{\hbox
+to\z@{\kern0.3\wd0\vrule\@height0.9\ht0\hss}\box0}}}}
+\def\bbbs{{\mathchoice
+{\setbox0=\hbox{$\displaystyle \rm S$}\hbox{\raise0.5\ht0\hbox
+to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\hbox
+to\z@{\kern0.55\wd0\vrule\@height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\textstyle \rm S$}\hbox{\raise0.5\ht0\hbox
+to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\hbox
+to\z@{\kern0.55\wd0\vrule\@height0.5\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptstyle \rm S$}\hbox{\raise0.5\ht0\hbox
+to\z@{\kern0.35\wd0\vrule\@height0.45\ht0\hss}\raise0.05\ht0\hbox
+to\z@{\kern0.5\wd0\vrule\@height0.45\ht0\hss}\box0}}
+{\setbox0=\hbox{$\scriptscriptstyle\rm S$}\hbox{\raise0.5\ht0\hbox
+to\z@{\kern0.4\wd0\vrule\@height0.45\ht0\hss}\raise0.05\ht0\hbox
+to\z@{\kern0.55\wd0\vrule\@height0.45\ht0\hss}\box0}}}}
+\def\bbbz{{\mathchoice {\hbox{$\textstyle\sf Z\kern-0.4em Z$}}
+{\hbox{$\textstyle\sf Z\kern-0.4em Z$}}
+{\hbox{$\scriptstyle\sf Z\kern-0.3em Z$}}
+{\hbox{$\scriptscriptstyle\sf Z\kern-0.2em Z$}}}}
+
+\let\ts\,
+
+\setlength \labelsep {5\p@}
+\setlength\leftmargini {17\p@}
+\setlength\leftmargin {\leftmargini}
+\setlength\leftmarginii {\leftmargini}
+\setlength\leftmarginiii {\leftmargini}
+\setlength\leftmarginiv {\leftmargini}
+\setlength\labelwidth {\leftmargini}
+\addtolength\labelwidth{-\labelsep}
+
+\def\@listI{\leftmargin\leftmargini
+ \parsep=\parskip
+ \topsep=\medskipamount
+ \itemsep=\parskip \advance\itemsep by -\parsep}
+\let\@listi\@listI
+\@listi
+
+\def\@listii{\leftmargin\leftmarginii
+ \labelwidth\leftmarginii
+ \advance\labelwidth by -\labelsep
+ \parsep=\parskip
+ \topsep=\z@
+ \itemsep=\parskip
+ \advance\itemsep by -\parsep}
+
+\def\@listiii{\leftmargin\leftmarginiii
+ \labelwidth\leftmarginiii\advance\labelwidth by -\labelsep
+ \parsep=\parskip
+ \topsep=\z@
+ \itemsep=\parskip
+ \advance\itemsep by -\parsep
+ \partopsep=\topsep}
+
+\setlength\arraycolsep{1.5\p@}
+\setlength\tabcolsep{1.5\p@}
+
+\def\tableofcontents{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \fi\chapter*{\contentsname \@mkboth{{\contentsname}}{{\contentsname}}}
+ \@starttoc{toc}\if@restonecol\twocolumn\fi}
+
+\setcounter{tocdepth}{2}
+
+\def\l@part#1#2{\addpenalty{\@secpenalty}%
+ \addvspace{2em \@plus\p@}%
+ \begingroup
+ \parindent \z@
+ \rightskip \z@ \@plus 5em
+ \hrule\vskip5\p@
+ \bfseries\boldmath
+ \leavevmode
+ #1\par
+ \vskip5\p@
+ \hrule
+ \vskip\p@
+ \nobreak
+ \endgroup}
+
+\def\@dotsep{2}
+
+\def\addnumcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{\protect\numberline
+ {\thechapter}#3}{\thepage}}}
+\def\addcontentsmark#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage}}}
+\def\addcontentsmarkwop#1#2#3{%
+\addtocontents{#1}{\protect\contentsline{#2}{#3}{0}}}
+
+\def\@adcmk[#1]{\ifcase #1 \or
+\def\@gtempa{\addnumcontentsmark}%
+ \or \def\@gtempa{\addcontentsmark}%
+ \or \def\@gtempa{\addcontentsmarkwop}%
+ \fi\@gtempa{toc}{chapter}}
+\def\addtocmark{\@ifnextchar[{\@adcmk}{\@adcmk[3]}}
+
+\def\l@chapter#1#2{\par\addpenalty{-\@highpenalty}
+ \addvspace{1.0em \@plus \p@}
+ \@tempdima \tocchpnum \begingroup
+ \parindent \z@ \rightskip \@tocrmarg
+ \advance\rightskip by \z@ \@plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip
+ {\bfseries\boldmath#1}\ifx0#2\hfil\null
+ \else
+ \nobreak
+ \leaders\hbox{$\m@th \mkern \@dotsep mu\hbox{.}\mkern
+ \@dotsep mu$}\hfill
+ \nobreak\hbox to\@pnumwidth{\hfil #2}%
+ \fi\par
+ \penalty\@highpenalty \endgroup}
+
+\newdimen\tocchpnum
+\newdimen\tocsecnum
+\newdimen\tocsectotal
+\newdimen\tocsubsecnum
+\newdimen\tocsubsectotal
+\newdimen\tocsubsubsecnum
+\newdimen\tocsubsubsectotal
+\newdimen\tocparanum
+\newdimen\tocparatotal
+\newdimen\tocsubparanum
+\tocchpnum=20\p@ % chapter {\bf 88.} \@plus 5.3\p@
+\tocsecnum=22.5\p@ % section 88.8. plus 4.722\p@
+\tocsubsecnum=30.5\p@ % subsection 88.8.8 plus 4.944\p@
+\tocsubsubsecnum=38\p@ % subsubsection 88.8.8.8 plus 4.666\p@
+\tocparanum=45\p@ % paragraph 88.8.8.8.8 plus 3.888\p@
+\tocsubparanum=53\p@ % subparagraph 88.8.8.8.8.8 plus 4.11\p@
+\def\calctocindent{%
+\tocsectotal=\tocchpnum
+\advance\tocsectotal by\tocsecnum
+\tocsubsectotal=\tocsectotal
+\advance\tocsubsectotal by\tocsubsecnum
+\tocsubsubsectotal=\tocsubsectotal
+\advance\tocsubsubsectotal by\tocsubsubsecnum
+\tocparatotal=\tocsubsubsectotal
+\advance\tocparatotal by\tocparanum}
+\calctocindent
+
+\def\@dottedtocline#1#2#3#4#5{%
+ \ifnum #1>\c@tocdepth \else
+ \vskip \z@ \@plus.2\p@
+ {\leftskip #2\relax \rightskip \@tocrmarg \advance\rightskip by \z@ \@plus 2cm
+ \parfillskip -\rightskip \pretolerance=10000
+ \parindent #2\relax\@afterindenttrue
+ \interlinepenalty\@M
+ \leavevmode
+ \@tempdima #3\relax
+ \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip
+ {#4}\nobreak
+ \leaders\hbox{$\m@th
+ \mkern \@dotsep mu\hbox{.}\mkern \@dotsep
+ mu$}\hfill
+ \nobreak
+ \hb@xt@\@pnumwidth{\hfil\normalfont \normalcolor #5}%
+ \par}%
+ \fi}
+%
+\def\l@section{\@dottedtocline{1}{\tocchpnum}{\tocsecnum}}
+\def\l@subsection{\@dottedtocline{2}{\tocsectotal}{\tocsubsecnum}}
+\def\l@subsubsection{\@dottedtocline{3}{\tocsubsectotal}{\tocsubsubsecnum}}
+\def\l@paragraph{\@dottedtocline{4}{\tocsubsubsectotal}{\tocparanum}}
+\def\l@subparagraph{\@dottedtocline{5}{\tocparatotal}{\tocsubparanum}}
+
+\renewcommand\listoffigures{%
+ \chapter*{\listfigurename
+ \@mkboth{\listfigurename}{\listfigurename}}%
+ \@starttoc{lof}%
+ }
+
+\renewcommand\listoftables{%
+ \chapter*{\listtablename
+ \@mkboth{\listtablename}{\listtablename}}%
+ \@starttoc{lot}%
+ }
+
+\renewcommand\footnoterule{%
+ \kern-3\p@
+ \hrule\@width 50\p@
+ \kern2.6\p@}
+
+\newdimen\foot@parindent
+\foot@parindent 10.83\p@
+
+\AtBeginDocument{%
+\long\def\@makefntext#1{\@setpar{\@@par\@tempdima \hsize
+ \advance\@tempdima-\foot@parindent\parshape\@ne\foot@parindent
+ \@tempdima}\par
+ \parindent \foot@parindent\noindent \hbox to \z@{%
+ \hss\hss$^{\@thefnmark}$ }#1}}
+
+\if@spthms
+% Definition of the "\spnewtheorem" command.
+%
+% Usage:
+%
+% \spnewtheorem{env_nam}{caption}[within]{cap_font}{body_font}
+% or \spnewtheorem{env_nam}[numbered_like]{caption}{cap_font}{body_font}
+% or \spnewtheorem*{env_nam}{caption}{cap_font}{body_font}
+%
+% New is "cap_font" and "body_font". It stands for
+% fontdefinition of the caption and the text itself.
+%
+% "\spnewtheorem*" gives a theorem without number.
+%
+% A defined spnewthoerem environment is used as described
+% by Lamport.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\def\@thmcountersep{.}
+\def\@thmcounterend{.}
+\newcommand\nocaption{\noexpand\@gobble}
+\newdimen\spthmsep \spthmsep=3pt
+
+\def\spnewtheorem{\@ifstar{\@sthm}{\@Sthm}}
+
+% definition of \spnewtheorem with number
+
+\def\@spnthm#1#2{%
+ \@ifnextchar[{\@spxnthm{#1}{#2}}{\@spynthm{#1}{#2}}}
+\def\@Sthm#1{\@ifnextchar[{\@spothm{#1}}{\@spnthm{#1}}}
+
+\def\@spxnthm#1#2[#3]#4#5{\expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}\@addtoreset{#1}{#3}%
+ \expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
+ \csname the#3\endcsname \noexpand\@thmcountersep \@thmcounter{#1}}%
+ \expandafter\xdef\csname #1name\endcsname{#2}%
+ \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#4}{#5}}%
+ \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}%
+ \expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}%
+ \expandafter\xdef\csname #1name\endcsname{#2}%
+ \global\@namedef{#1}{\@spthm{#1}{\csname #1name\endcsname}{#3}{#4}}%
+ \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@spothm#1[#2]#3#4#5{%
+ \@ifundefined{c@#2}{\@latexerr{No theorem environment `#2' defined}\@eha}%
+ {\expandafter\@ifdefinable\csname #1\endcsname
+ {\global\@namedef{the#1}{\@nameuse{the#2}}%
+ \expandafter\xdef\csname #1name\endcsname{#3}%
+ \global\@namedef{#1}{\@spthm{#2}{\csname #1name\endcsname}{#4}{#5}}%
+ \global\@namedef{end#1}{\@endtheorem}}}}
+
+\def\@spthm#1#2#3#4{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\labelsep=\spthmsep\refstepcounter{#1}%
+\@ifnextchar[{\@spythm{#1}{#2}{#3}{#4}}{\@spxthm{#1}{#2}{#3}{#4}}}
+
+\def\@spxthm#1#2#3#4{\@spbegintheorem{#2}{\csname the#1\endcsname}{#3}{#4}%
+ \ignorespaces}
+
+\def\@spythm#1#2#3#4[#5]{\@spopargbegintheorem{#2}{\csname
+ the#1\endcsname}{#5}{#3}{#4}\ignorespaces}
+
+\def\normalthmheadings{\def\@spbegintheorem##1##2##3##4{\trivlist
+ \item[\hskip\labelsep{##3##1\ ##2\@thmcounterend}]##4}
+\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
+ \item[\hskip\labelsep{##4##1\ ##2}]{##4(##3)\@thmcounterend\ }##5}}
+\normalthmheadings
+
+\def\reversethmheadings{\def\@spbegintheorem##1##2##3##4{\trivlist
+ \item[\hskip\labelsep{##3##2\ ##1\@thmcounterend}]##4}
+\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
+ \item[\hskip\labelsep{##4##2\ ##1}]{##4(##3)\@thmcounterend\ }##5}}
+
+% definition of \spnewtheorem* without number
+
+\def\@sthm#1#2{\@Ynthm{#1}{#2}}
+
+\def\@Ynthm#1#2#3#4{\expandafter\@ifdefinable\csname #1\endcsname
+ {\global\@namedef{#1}{\@Thm{\csname #1name\endcsname}{#3}{#4}}%
+ \expandafter\xdef\csname #1name\endcsname{#2}%
+ \global\@namedef{end#1}{\@endtheorem}}}
+
+\def\@Thm#1#2#3{\topsep 7\p@ \@plus2\p@ \@minus4\p@
+\@ifnextchar[{\@Ythm{#1}{#2}{#3}}{\@Xthm{#1}{#2}{#3}}}
+
+\def\@Xthm#1#2#3{\@Begintheorem{#1}{#2}{#3}\ignorespaces}
+
+\def\@Ythm#1#2#3[#4]{\@Opargbegintheorem{#1}
+ {#4}{#2}{#3}\ignorespaces}
+
+\def\@Begintheorem#1#2#3{#3\trivlist
+ \item[\hskip\labelsep{#2#1\@thmcounterend}]}
+
+\def\@Opargbegintheorem#1#2#3#4{#4\trivlist
+ \item[\hskip\labelsep{#3#1}]{#3(#2)\@thmcounterend\ }}
+
+% initialize theorem environment
+
+\if@envcntshowhiercnt % show hierarchy counter
+ \def\@thmcountersep{.}
+ \spnewtheorem{theorem}{Theorem}[\envankh]{\bfseries}{\itshape}
+ \@addtoreset{theorem}{chapter}
+\else % theorem counter only
+ \spnewtheorem{theorem}{Theorem}{\bfseries}{\itshape}
+ \if@envcntreset
+ \@addtoreset{theorem}{chapter}
+ \if@envcntresetsect
+ \@addtoreset{theorem}{section}
+ \fi
+ \fi
+\fi
+
+%definition of divers theorem environments
+\spnewtheorem*{claim}{Claim}{\itshape}{\rmfamily}
+\spnewtheorem*{proof}{Proof}{\itshape}{\rmfamily}
+%
+\if@envcntsame % all environments like "Theorem" - using its counter
+ \def\spn@wtheorem#1#2#3#4{\@spothm{#1}[theorem]{#2}{#3}{#4}}
+\else % all environments with their own counter
+ \if@envcntshowhiercnt % show hierarchy counter
+ \def\spn@wtheorem#1#2#3#4{\@spxnthm{#1}{#2}[\envankh]{#3}{#4}}
+ \else % environment counter only
+ \if@envcntreset % environment counter is reset each section
+ \if@envcntresetsect
+ \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+ \@addtoreset{#1}{chapter}\@addtoreset{#1}{section}}
+ \else
+ \def\spn@wtheorem#1#2#3#4{\@spynthm{#1}{#2}{#3}{#4}
+ \@addtoreset{#1}{chapter}}
+ \fi
+ \else
+ \let\spn@wtheorem=\@spynthm
+ \fi
+ \fi
+\fi
+%
+\let\spdefaulttheorem=\spn@wtheorem
+%
+\spn@wtheorem{case}{Case}{\itshape}{\rmfamily}
+\spn@wtheorem{conjecture}{Conjecture}{\itshape}{\rmfamily}
+\spn@wtheorem{corollary}{Corollary}{\bfseries}{\itshape}
+\spn@wtheorem{definition}{Definition}{\bfseries}{\itshape}
+\spn@wtheorem{example}{Example}{\itshape}{\rmfamily}
+\spn@wtheorem{exercise}{Exercise}{\bfseries}{\rmfamily}
+\spn@wtheorem{lemma}{Lemma}{\bfseries}{\itshape}
+\spn@wtheorem{note}{Note}{\itshape}{\rmfamily}
+\spn@wtheorem{problem}{Problem}{\bfseries}{\rmfamily}
+\spn@wtheorem{property}{Property}{\itshape}{\rmfamily}
+\spn@wtheorem{proposition}{Proposition}{\bfseries}{\itshape}
+\spn@wtheorem{question}{Question}{\itshape}{\rmfamily}
+\spn@wtheorem{solution}{Solution}{\bfseries}{\rmfamily}
+\spn@wtheorem{remark}{Remark}{\itshape}{\rmfamily}
+%
+\newenvironment{theopargself}
+ {\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
+ \item[\hskip\labelsep{##4##1\ ##2}]{##4##3\@thmcounterend\ }##5}
+ \def\@Opargbegintheorem##1##2##3##4{##4\trivlist
+ \item[\hskip\labelsep{##3##1}]{##3##2\@thmcounterend\ }}}{}
+\newenvironment{theopargself*}
+ {\def\@spopargbegintheorem##1##2##3##4##5{\trivlist
+ \item[\hskip\labelsep{##4##1\ ##2}]{\hspace*{-\labelsep}##4##3\@thmcounterend}##5}
+ \def\@Opargbegintheorem##1##2##3##4{##4\trivlist
+ \item[\hskip\labelsep{##3##1}]{\hspace*{-\labelsep}##3##2\@thmcounterend}}}{}
+%
+\spnewtheorem{prob}{\nocaption}[chapter]{\bfseries}{\rmfamily}
+\newcommand{\probref}[1]{\textbf{\ref{#1}} }
+\newenvironment{sol}{\par\addvspace{6pt}\noindent\probref}{\par\addvspace{6pt}}
+%
+\fi
+
+\def\@takefromreset#1#2{%
+ \def\@tempa{#1}%
+ \let\@tempd\@elt
+ \def\@elt##1{%
+ \def\@tempb{##1}%
+ \ifx\@tempa\@tempb\else
+ \@addtoreset{##1}{#2}%
+ \fi}%
+ \expandafter\expandafter\let\expandafter\@tempc\csname cl@#2\endcsname
+ \expandafter\def\csname cl@#2\endcsname{}%
+ \@tempc
+ \let\@elt\@tempd}
+
+% redefininition of the captions for "figure" and "table" environments
+%
+\@ifundefined{floatlegendstyle}{\def\floatlegendstyle{\bfseries}}{}
+\def\floatcounterend{.\ }
+\def\capstrut{\vrule\@width\z@\@height\topskip}
+\@ifundefined{captionstyle}{\def\captionstyle{\normalfont\small}}{}
+\@ifundefined{instindent}{\newdimen\instindent}{}
+
+\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
+ ext@#1\endcsname}{#1}{\protect\numberline{\csname
+ the#1\endcsname}{\ignorespaces #2}}\begingroup
+ \@parboxrestore\if@minipage\@setminipage\fi
+ \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
+ \endgroup}
+
+\def\twocaptionwidth#1#2{\def\first@capwidth{#1}\def\second@capwidth{#2}}
+% Default: .46\textwidth
+\twocaptionwidth{.46\textwidth}{.46\textwidth}
+
+\def\leftcaption{\refstepcounter\@captype\@dblarg%
+ {\@leftcaption\@captype}}
+
+\def\rightcaption{\refstepcounter\@captype\@dblarg%
+ {\@rightcaption\@captype}}
+
+\long\def\@leftcaption#1[#2]#3{\addcontentsline{\csname
+ ext@#1\endcsname}{#1}{\protect\numberline{\csname
+ the#1\endcsname}{\ignorespaces #2}}\begingroup
+ \@parboxrestore
+ \vskip\figcapgap
+ \@maketwocaptions{\csname fnum@#1\endcsname}{\ignorespaces #3}%
+ {\first@capwidth}\ignorespaces\hspace{.073\textwidth}\hfill%
+ \endgroup}
+
+\long\def\@rightcaption#1[#2]#3{\addcontentsline{\csname
+ ext@#1\endcsname}{#1}{\protect\numberline{\csname
+ the#1\endcsname}{\ignorespaces #2}}\begingroup
+ \@parboxrestore
+ \@maketwocaptions{\csname fnum@#1\endcsname}{\ignorespaces #3}%
+ {\second@capwidth}\par
+ \endgroup}
+
+\long\def\@maketwocaptions#1#2#3{%
+ \parbox[t]{#3}{{\floatlegendstyle #1\floatcounterend}#2}}
+
+\def\fig@pos{l}
+\newcommand{\leftfigure}[2][\fig@pos]{\makebox[.4635\textwidth][#1]{#2}}
+\let\rightfigure\leftfigure
+
+\newdimen\figgap\figgap=0.5cm % hgap between figure and sidecaption
+%
+\long\def\@makesidecaption#1#2{%
+ \setbox0=\vbox{\hsize=\@tempdimb
+ \captionstyle{\floatlegendstyle
+ #1\floatcounterend}#2}%
+ \ifdim\instindent<\z@
+ \ifdim\ht0>-\instindent
+ \advance\instindent by\ht0
+ \typeout{^^JClass-Warning: Legend of \string\sidecaption\space for
+ \@captype\space\csname the\@captype\endcsname
+ ^^Jis \the\instindent\space taller than the corresponding float -
+ ^^Jyou'd better switch the environment. }%
+ \instindent\z@
+ \fi
+ \else
+ \ifdim\ht0<\instindent
+ \advance\instindent by-\ht0
+ \advance\instindent by-\dp0\relax
+ \advance\instindent by\topskip
+ \advance\instindent by-11\p@
+ \else
+ \advance\instindent by-\ht0
+ \instindent=-\instindent
+ \typeout{^^JClass-Warning: Legend of \string\sidecaption\space for
+ \@captype\space\csname the\@captype\endcsname
+ ^^Jis \the\instindent\space taller than the corresponding float -
+ ^^Jyou'd better switch the environment. }%
+ \instindent\z@
+ \fi
+ \fi
+ \parbox[b]{\@tempdimb}{\captionstyle{\floatlegendstyle
+ #1\floatcounterend}#2%
+ \ifdim\instindent>\z@ \\
+ \vrule\@width\z@\@height\instindent
+ \@depth\z@
+ \fi}}
+\def\sidecaption{\@ifnextchar[\sidec@ption{\sidec@ption[b]}}
+\def\sidec@ption[#1]#2\caption{%
+\setbox\@tempboxa=\hbox{\ignorespaces#2\unskip}%
+\if@twocolumn
+ \ifdim\hsize<\textwidth\else
+ \ifdim\wd\@tempboxa<\columnwidth
+ \typeout{Double column float fits into single column -
+ ^^Jyou'd better switch the environment. }%
+ \fi
+ \fi
+\fi
+ \instindent=\ht\@tempboxa
+ \advance\instindent by\dp\@tempboxa
+\if t#1
+\else
+ \instindent=-\instindent
+\fi
+\@tempdimb=\hsize
+\advance\@tempdimb by-\figgap
+\advance\@tempdimb by-\wd\@tempboxa
+\ifdim\@tempdimb<3cm
+ \ClassWarning{SVMono}{\string\sidecaption: No sufficient room for the legend;
+ ^^Jusing normal \string\caption}%
+ \unhbox\@tempboxa
+ \let\@capcommand=\@caption
+\else
+ \ifdim\@tempdimb<4.5cm
+ \ClassWarning{SVMono}{\string\sidecaption: Room for the legend very narrow;
+ ^^Jusing \string\raggedright}%
+ \toks@\expandafter{\captionstyle\sloppy
+ \rightskip=\z@\@plus6mm\relax}%
+ \def\captionstyle{\the\toks@}%
+ \fi
+ \let\@capcommand=\@sidecaption
+ \leavevmode
+ \unhbox\@tempboxa
+ \hfill
+\fi
+\refstepcounter\@captype
+\@dblarg{\@capcommand\@captype}}
+\long\def\@sidecaption#1[#2]#3{\addcontentsline{\csname
+ ext@#1\endcsname}{#1}{\protect\numberline{\csname
+ the#1\endcsname}{\ignorespaces #2}}\begingroup
+ \@parboxrestore
+ \@makesidecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
+ \endgroup}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\def\fig@type{figure}
+
+\def\leftlegendglue{\hfil}
+\newdimen\figcapgap\figcapgap=5\p@ % vgap between figure and caption
+\newdimen\tabcapgap\tabcapgap=5.5\p@ % vgap between caption and table
+
+\long\def\@makecaption#1#2{%
+ \captionstyle
+ \ifx\@captype\fig@type
+ \vskip\figcapgap
+ \fi
+ \setbox\@tempboxa\hbox{{\floatlegendstyle #1\floatcounterend}%
+ \capstrut #2}%
+ \ifdim \wd\@tempboxa >\hsize
+ {\floatlegendstyle #1\floatcounterend}\capstrut #2\par
+ \else
+ \hbox to\hsize{\leftlegendglue\unhbox\@tempboxa\hfil}%
+ \fi
+ \ifx\@captype\fig@type\else
+ \vskip\tabcapgap
+ \fi}
+
+\newcounter{merk}
+
+\def\endfigure{\resetsubfig\end@float}
+
+\@namedef{endfigure*}{\resetsubfig\end@dblfloat}
+
+\def\resetsubfig{\global\let\last@subfig=\undefined}
+
+\def\r@setsubfig{\xdef\last@subfig{\number\value{figure}}%
+\setcounter{figure}{\value{merk}}%
+\setcounter{merk}{0}}
+
+\def\subfigures{\refstepcounter{figure}%
+ \@tempcnta=\value{merk}%
+ \setcounter{merk}{\value{figure}}%
+ \setcounter{figure}{\the\@tempcnta}%
+ \def\thefigure{\if@numart\else\thechapter.\fi
+ \@arabic\c@merk\alph{figure}}%
+ \let\resetsubfig=\r@setsubfig}
+
+\def\samenumber{\addtocounter{\@captype}{-1}%
+\@ifundefined{last@subfig}{}{\setcounter{merk}{\last@subfig}}}
+
+% redefinition of the "bibliography" environment
+%
+\def\biblstarthook#1{\gdef\biblst@rthook{#1}}
+%
+\AtBeginDocument{%
+\ifx\secbibl\undefined
+ \def\bibsection{\chapter*{\refname}\markboth{\refname}{\refname}%
+ \addcontentsline{toc}{chapter}{\refname}%
+ \csname biblst@rthook\endcsname}
+\else
+ \def\bibsection{\section*{\refname}\markright{\refname}%
+ \addcontentsline{toc}{section}{\refname}%
+ \csname biblst@rthook\endcsname}
+\fi}
+\ifx\oribibl\undefined % Springer way of life
+ \renewenvironment{thebibliography}[1]{\bibsection
+ \global\let\biblst@rthook=\undefined
+ \def\@biblabel##1{##1.}
+ \small
+ \list{\@biblabel{\@arabic\c@enumiv}}%
+ {\settowidth\labelwidth{\@biblabel{#1}}%
+ \leftmargin\labelwidth
+ \advance\leftmargin\labelsep
+ \if@openbib
+ \advance\leftmargin\bibindent
+ \itemindent -\bibindent
+ \listparindent \itemindent
+ \parsep \z@
+ \fi
+ \usecounter{enumiv}%
+ \let\p@enumiv\@empty
+ \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+ \if@openbib
+ \renewcommand\newblock{\par}%
+ \else
+ \renewcommand\newblock{\hskip .11em \@plus.33em \@minus.07em}%
+ \fi
+ \sloppy\clubpenalty4000\widowpenalty4000%
+ \sfcode`\.=\@m}
+ {\def\@noitemerr
+ {\@latex@warning{Empty `thebibliography' environment}}%
+ \endlist}
+ \def\@lbibitem[#1]#2{\item[{[#1]}\hfill]\if@filesw
+ {\let\protect\noexpand\immediate
+ \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}
+\else % original bibliography is required
+ \let\bibname=\refname
+ \renewenvironment{thebibliography}[1]
+ {\chapter*{\bibname
+ \@mkboth{\bibname}{\bibname}}%
+ \list{\@biblabel{\@arabic\c@enumiv}}%
+ {\settowidth\labelwidth{\@biblabel{#1}}%
+ \leftmargin\labelwidth
+ \advance\leftmargin\labelsep
+ \@openbib@code
+ \usecounter{enumiv}%
+ \let\p@enumiv\@empty
+ \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+ \sloppy
+ \clubpenalty4000
+ \@clubpenalty \clubpenalty
+ \widowpenalty4000%
+ \sfcode`\.\@m}
+ {\def\@noitemerr
+ {\@latex@warning{Empty `thebibliography' environment}}%
+ \endlist}
+\fi
+
+\let\if@threecolind\iffalse
+\def\threecolindex{\let\if@threecolind\iftrue}
+\def\indexstarthook#1{\gdef\indexst@rthook{#1}}
+\renewenvironment{theindex}
+ {\if@twocolumn
+ \@restonecolfalse
+ \else
+ \@restonecoltrue
+ \fi
+ \columnseprule \z@
+ \columnsep 1cc
+ \@nobreaktrue
+ \if@threecolind
+ \begin{multicols}{3}[\chapter*{\indexname}%
+ \else
+ \begin{multicols}{2}[\chapter*{\indexname}%
+ \fi
+ {\csname indexst@rthook\endcsname}]%
+ \global\let\indexst@rthook=\undefined
+ \markboth{\indexname}{\indexname}%
+ \addcontentsline{toc}{chapter}{\indexname}%
+ \flushbottom
+ \parindent\z@
+ \rightskip\z@ \@plus 40\p@
+ \parskip\z@ \@plus .3\p@\relax
+ \flushbottom
+ \let\item\@idxitem
+ \def\,{\relax\ifmmode\mskip\thinmuskip
+ \else\hskip0.2em\ignorespaces\fi}%
+ \normalfont\small}
+ {\end{multicols}
+ \global\let\if@threecolind\iffalse
+ \if@restonecol\onecolumn\else\clearpage\fi}
+
+\def\idxquad{\hskip 10\p@}% space that divides entry from number
+
+\def\@idxitem{\par\setbox0=\hbox{--\,--\,--\enspace}%
+ \hangindent\wd0\relax}
+
+\def\subitem{\par\noindent\setbox0=\hbox{--\enspace}% second order
+ \kern\wd0\setbox0=\hbox{--\,--\,--\enspace}%
+ \hangindent\wd0\relax}% indexentry
+
+\def\subsubitem{\par\noindent\setbox0=\hbox{--\,--\enspace}% third order
+ \kern\wd0\setbox0=\hbox{--\,--\,--\enspace}%
+ \hangindent\wd0\relax}% indexentry
+
+\def\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax}
+
+\def\subtitle#1{\gdef\@subtitle{#1}}
+\def\@subtitle{}
+
+\def\maketitle{\par
+ \begingroup
+ \def\thefootnote{\fnsymbol{footnote}}%
+ \def\@makefnmark{\hbox
+ to\z@{$\m@th^{\@thefnmark}$\hss}}%
+ \if@twocolumn
+ \twocolumn[\@maketitle]%
+ \else \newpage
+ \global\@topnum\z@ % Prevents figures from going at top of page.
+ \@maketitle \fi\thispagestyle{empty}\@thanks
+ \par\penalty -\@M
+ \endgroup
+ \setcounter{footnote}{0}%
+ \let\maketitle\relax
+ \let\@maketitle\relax
+ \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax}
+
+\def\@maketitle{\newpage
+ \null
+ \vskip 2em % Vertical space above title.
+\begingroup
+ \def\and{\unskip, }
+ \parindent=\z@
+ \pretolerance=10000
+ \rightskip=\z@ \@plus 3cm
+ {\LARGE % each author set in \LARGE
+ \lineskip .5em
+ \@author
+ \par}%
+ \vskip 2cm % Vertical space after author.
+ {\Huge \@title \par}% % Title set in \Huge size.
+ \vskip 1cm % Vertical space after title.
+ \if!\@subtitle!\else
+ {\LARGE\ignorespaces\@subtitle \par}
+ \vskip 1cm % Vertical space after subtitle.
+ \fi
+ \if!\@date!\else
+ {\large \@date}% % Date set in \large size.
+ \par
+ \vskip 1.5em % Vertical space after date.
+ \fi
+ \vfill
+% {\Large Springer\par}
+%\vskip 5\p@
+%\large
+% Berlin\enspace Heidelberg\enspace New\kern0.1em York\\
+% Hong\thinspace Kong\enspace London\\
+% Milan\enspace Paris\enspace Tokyo\par
+\endgroup}
+
+% Useful environments
+\newenvironment{acknowledgement}{\par\addvspace{17\p@}\small\rm
+\trivlist\item[\hskip\labelsep{\it\ackname}]}
+{\endtrivlist\addvspace{6\p@}}
+%
+\newenvironment{noteadd}{\par\addvspace{17\p@}\small\rm
+\trivlist\item[\hskip\labelsep{\it\noteaddname}]}
+{\endtrivlist\addvspace{6\p@}}
+%
+\renewenvironment{abstract}{%
+ \advance\topsep by0.35cm\relax\small
+ \labelwidth=\z@
+ \listparindent=\z@
+ \itemindent\listparindent
+ \trivlist\item[\hskip\labelsep\bfseries\abstractname]%
+ \if!\abstractname!\hskip-\labelsep\fi
+ }
+ {\endtrivlist}
+
+% define the running headings of a twoside text
+\def\runheadsize{\small}
+\def\runheadstyle{\rmfamily\upshape}
+\def\customizhead{\hspace{\headlineindent}}
+
+\def\ps@headings{\let\@mkboth\markboth
+ \let\@oddfoot\@empty\let\@evenfoot\@empty
+ \def\@evenhead{\runheadsize\runheadstyle\rlap{\thepage}\customizhead
+ \leftmark\hfil}
+ \def\@oddhead{\runheadsize\runheadstyle\hfil\rightmark\customizhead
+ \llap{\thepage}}
+ \def\chaptermark##1{\markboth{{\ifnum\c@secnumdepth>\m@ne
+ \thechapter\thechapterend\hskip\betweenumberspace\fi ##1}}{{\ifnum %!!!
+ \c@secnumdepth>\m@ne\thechapter\thechapterend\hskip\betweenumberspace\fi ##1}}}%!!!
+ \def\sectionmark##1{\markright{{\ifnum\c@secnumdepth>\z@
+ \thesection\seccounterend\hskip\betweenumberspace\fi ##1}}}}
+
+\def\ps@myheadings{\let\@mkboth\@gobbletwo
+ \let\@oddfoot\@empty\let\@evenfoot\@empty
+ \def\@evenhead{\runheadsize\runheadstyle\rlap{\thepage}\customizhead
+ \leftmark\hfil}
+ \def\@oddhead{\runheadsize\runheadstyle\hfil\rightmark\customizhead
+ \llap{\thepage}}
+ \let\chaptermark\@gobble
+ \let\sectionmark\@gobble
+ \let\subsectionmark\@gobble}
+
+
+\ps@headings
+
+\endinput
+%end of file svmono.cls