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diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/Functions.thy
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+++ b/doc-src/Functions/Functions.thy	Mon Aug 27 22:14:17 2012 +0200
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+(*  Title:      doc-src/IsarAdvanced/Functions/Thy/Fundefs.thy
+    Author:     Alexander Krauss, TU Muenchen
+
+Tutorial for function definitions with the new "function" package.
+*)
+
+theory Functions
+imports Main
+begin
+
+section {* Function Definitions for Dummies *}
+
+text {*
+  In most cases, defining a recursive function is just as simple as other definitions:
+*}
+
+fun fib :: "nat \<Rightarrow> nat"
+where
+  "fib 0 = 1"
+| "fib (Suc 0) = 1"
+| "fib (Suc (Suc n)) = fib n + fib (Suc n)"
+
+text {*
+  The syntax is rather self-explanatory: We introduce a function by
+  giving its name, its type, 
+  and a set of defining recursive equations.
+  If we leave out the type, the most general type will be
+  inferred, which can sometimes lead to surprises: Since both @{term
+  "1::nat"} and @{text "+"} are overloaded, we would end up
+  with @{text "fib :: nat \<Rightarrow> 'a::{one,plus}"}.
+*}
+
+text {*
+  The function always terminates, since its argument gets smaller in
+  every recursive call. 
+  Since HOL is a logic of total functions, termination is a
+  fundamental requirement to prevent inconsistencies\footnote{From the
+  \qt{definition} @{text "f(n) = f(n) + 1"} we could prove 
+  @{text "0 = 1"} by subtracting @{text "f(n)"} on both sides.}.
+  Isabelle tries to prove termination automatically when a definition
+  is made. In \S\ref{termination}, we will look at cases where this
+  fails and see what to do then.
+*}
+
+subsection {* Pattern matching *}
+
+text {* \label{patmatch}
+  Like in functional programming, we can use pattern matching to
+  define functions. At the moment we will only consider \emph{constructor
+  patterns}, which only consist of datatype constructors and
+  variables. Furthermore, patterns must be linear, i.e.\ all variables
+  on the left hand side of an equation must be distinct. In
+  \S\ref{genpats} we discuss more general pattern matching.
+
+  If patterns overlap, the order of the equations is taken into
+  account. The following function inserts a fixed element between any
+  two elements of a list:
+*}
+
+fun sep :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+  "sep a (x#y#xs) = x # a # sep a (y # xs)"
+| "sep a xs       = xs"
+
+text {* 
+  Overlapping patterns are interpreted as \qt{increments} to what is
+  already there: The second equation is only meant for the cases where
+  the first one does not match. Consequently, Isabelle replaces it
+  internally by the remaining cases, making the patterns disjoint:
+*}
+
+thm sep.simps
+
+text {* @{thm [display] sep.simps[no_vars]} *}
+
+text {* 
+  \noindent The equations from function definitions are automatically used in
+  simplification:
+*}
+
+lemma "sep 0 [1, 2, 3] = [1, 0, 2, 0, 3]"
+by simp
+
+subsection {* Induction *}
+
+text {*
+
+  Isabelle provides customized induction rules for recursive
+  functions. These rules follow the recursive structure of the
+  definition. Here is the rule @{text sep.induct} arising from the
+  above definition of @{const sep}:
+
+  @{thm [display] sep.induct}
+  
+  We have a step case for list with at least two elements, and two
+  base cases for the zero- and the one-element list. Here is a simple
+  proof about @{const sep} and @{const map}
+*}
+
+lemma "map f (sep x ys) = sep (f x) (map f ys)"
+apply (induct x ys rule: sep.induct)
+
+txt {*
+  We get three cases, like in the definition.
+
+  @{subgoals [display]}
+*}
+
+apply auto 
+done
+text {*
+
+  With the \cmd{fun} command, you can define about 80\% of the
+  functions that occur in practice. The rest of this tutorial explains
+  the remaining 20\%.
+*}
+
+
+section {* fun vs.\ function *}
+
+text {* 
+  The \cmd{fun} command provides a
+  convenient shorthand notation for simple function definitions. In
+  this mode, Isabelle tries to solve all the necessary proof obligations
+  automatically. If any proof fails, the definition is
+  rejected. This can either mean that the definition is indeed faulty,
+  or that the default proof procedures are just not smart enough (or
+  rather: not designed) to handle the definition.
+
+  By expanding the abbreviation to the more verbose \cmd{function} command, these proof obligations become visible and can be analyzed or
+  solved manually. The expansion from \cmd{fun} to \cmd{function} is as follows:
+
+\end{isamarkuptext}
+
+
+\[\left[\;\begin{minipage}{0.25\textwidth}\vspace{6pt}
+\cmd{fun} @{text "f :: \<tau>"}\\%
+\cmd{where}\\%
+\hspace*{2ex}{\it equations}\\%
+\hspace*{2ex}\vdots\vspace*{6pt}
+\end{minipage}\right]
+\quad\equiv\quad
+\left[\;\begin{minipage}{0.48\textwidth}\vspace{6pt}
+\cmd{function} @{text "("}\cmd{sequential}@{text ") f :: \<tau>"}\\%
+\cmd{where}\\%
+\hspace*{2ex}{\it equations}\\%
+\hspace*{2ex}\vdots\\%
+\cmd{by} @{text "pat_completeness auto"}\\%
+\cmd{termination by} @{text "lexicographic_order"}\vspace{6pt}
+\end{minipage}
+\right]\]
+
+\begin{isamarkuptext}
+  \vspace*{1em}
+  \noindent Some details have now become explicit:
+
+  \begin{enumerate}
+  \item The \cmd{sequential} option enables the preprocessing of
+  pattern overlaps which we already saw. Without this option, the equations
+  must already be disjoint and complete. The automatic completion only
+  works with constructor patterns.
+
+  \item A function definition produces a proof obligation which
+  expresses completeness and compatibility of patterns (we talk about
+  this later). The combination of the methods @{text "pat_completeness"} and
+  @{text "auto"} is used to solve this proof obligation.
+
+  \item A termination proof follows the definition, started by the
+  \cmd{termination} command. This will be explained in \S\ref{termination}.
+ \end{enumerate}
+  Whenever a \cmd{fun} command fails, it is usually a good idea to
+  expand the syntax to the more verbose \cmd{function} form, to see
+  what is actually going on.
+ *}
+
+
+section {* Termination *}
+
+text {*\label{termination}
+  The method @{text "lexicographic_order"} is the default method for
+  termination proofs. It can prove termination of a
+  certain class of functions by searching for a suitable lexicographic
+  combination of size measures. Of course, not all functions have such
+  a simple termination argument. For them, we can specify the termination
+  relation manually.
+*}
+
+subsection {* The {\tt relation} method *}
+text{*
+  Consider the following function, which sums up natural numbers up to
+  @{text "N"}, using a counter @{text "i"}:
+*}
+
+function sum :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+  "sum i N = (if i > N then 0 else i + sum (Suc i) N)"
+by pat_completeness auto
+
+text {*
+  \noindent The @{text "lexicographic_order"} method fails on this example, because none of the
+  arguments decreases in the recursive call, with respect to the standard size ordering.
+  To prove termination manually, we must provide a custom wellfounded relation.
+
+  The termination argument for @{text "sum"} is based on the fact that
+  the \emph{difference} between @{text "i"} and @{text "N"} gets
+  smaller in every step, and that the recursion stops when @{text "i"}
+  is greater than @{text "N"}. Phrased differently, the expression 
+  @{text "N + 1 - i"} always decreases.
+
+  We can use this expression as a measure function suitable to prove termination.
+*}
+
+termination sum
+apply (relation "measure (\<lambda>(i,N). N + 1 - i)")
+
+txt {*
+  The \cmd{termination} command sets up the termination goal for the
+  specified function @{text "sum"}. If the function name is omitted, it
+  implicitly refers to the last function definition.
+
+  The @{text relation} method takes a relation of
+  type @{typ "('a \<times> 'a) set"}, where @{typ "'a"} is the argument type of
+  the function. If the function has multiple curried arguments, then
+  these are packed together into a tuple, as it happened in the above
+  example.
+
+  The predefined function @{term[source] "measure :: ('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set"} constructs a
+  wellfounded relation from a mapping into the natural numbers (a
+  \emph{measure function}). 
+
+  After the invocation of @{text "relation"}, we must prove that (a)
+  the relation we supplied is wellfounded, and (b) that the arguments
+  of recursive calls indeed decrease with respect to the
+  relation:
+
+  @{subgoals[display,indent=0]}
+
+  These goals are all solved by @{text "auto"}:
+*}
+
+apply auto
+done
+
+text {*
+  Let us complicate the function a little, by adding some more
+  recursive calls: 
+*}
+
+function foo :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+  "foo i N = (if i > N 
+              then (if N = 0 then 0 else foo 0 (N - 1))
+              else i + foo (Suc i) N)"
+by pat_completeness auto
+
+text {*
+  When @{text "i"} has reached @{text "N"}, it starts at zero again
+  and @{text "N"} is decremented.
+  This corresponds to a nested
+  loop where one index counts up and the other down. Termination can
+  be proved using a lexicographic combination of two measures, namely
+  the value of @{text "N"} and the above difference. The @{const
+  "measures"} combinator generalizes @{text "measure"} by taking a
+  list of measure functions.  
+*}
+
+termination 
+by (relation "measures [\<lambda>(i, N). N, \<lambda>(i,N). N + 1 - i]") auto
+
+subsection {* How @{text "lexicographic_order"} works *}
+
+(*fun fails :: "nat \<Rightarrow> nat list \<Rightarrow> nat"
+where
+  "fails a [] = a"
+| "fails a (x#xs) = fails (x + a) (x # xs)"
+*)
+
+text {*
+  To see how the automatic termination proofs work, let's look at an
+  example where it fails\footnote{For a detailed discussion of the
+  termination prover, see \cite{bulwahnKN07}}:
+
+\end{isamarkuptext}  
+\cmd{fun} @{text "fails :: \"nat \<Rightarrow> nat list \<Rightarrow> nat\""}\\%
+\cmd{where}\\%
+\hspace*{2ex}@{text "\"fails a [] = a\""}\\%
+|\hspace*{1.5ex}@{text "\"fails a (x#xs) = fails (x + a) (x#xs)\""}\\
+\begin{isamarkuptext}
+
+\noindent Isabelle responds with the following error:
+
+\begin{isabelle}
+*** Unfinished subgoals:\newline
+*** (a, 1, <):\newline
+*** \ 1.~@{text "\<And>x. x = 0"}\newline
+*** (a, 1, <=):\newline
+*** \ 1.~False\newline
+*** (a, 2, <):\newline
+*** \ 1.~False\newline
+*** Calls:\newline
+*** a) @{text "(a, x # xs) -->> (x + a, x # xs)"}\newline
+*** Measures:\newline
+*** 1) @{text "\<lambda>x. size (fst x)"}\newline
+*** 2) @{text "\<lambda>x. size (snd x)"}\newline
+*** Result matrix:\newline
+*** \ \ \ \ 1\ \ 2  \newline
+*** a:  ?   <= \newline
+*** Could not find lexicographic termination order.\newline
+*** At command "fun".\newline
+\end{isabelle}
+*}
+text {*
+  The key to this error message is the matrix at the bottom. The rows
+  of that matrix correspond to the different recursive calls (In our
+  case, there is just one). The columns are the function's arguments 
+  (expressed through different measure functions, which map the
+  argument tuple to a natural number). 
+
+  The contents of the matrix summarize what is known about argument
+  descents: The second argument has a weak descent (@{text "<="}) at the
+  recursive call, and for the first argument nothing could be proved,
+  which is expressed by @{text "?"}. In general, there are the values
+  @{text "<"}, @{text "<="} and @{text "?"}.
+
+  For the failed proof attempts, the unfinished subgoals are also
+  printed. Looking at these will often point to a missing lemma.
+*}
+
+subsection {* The @{text size_change} method *}
+
+text {*
+  Some termination goals that are beyond the powers of
+  @{text lexicographic_order} can be solved automatically by the
+  more powerful @{text size_change} method, which uses a variant of
+  the size-change principle, together with some other
+  techniques. While the details are discussed
+  elsewhere\cite{krauss_phd},
+  here are a few typical situations where
+  @{text lexicographic_order} has difficulties and @{text size_change}
+  may be worth a try:
+  \begin{itemize}
+  \item Arguments are permuted in a recursive call.
+  \item Several mutually recursive functions with multiple arguments.
+  \item Unusual control flow (e.g., when some recursive calls cannot
+  occur in sequence).
+  \end{itemize}
+
+  Loading the theory @{text Multiset} makes the @{text size_change}
+  method a bit stronger: it can then use multiset orders internally.
+*}
+
+section {* Mutual Recursion *}
+
+text {*
+  If two or more functions call one another mutually, they have to be defined
+  in one step. Here are @{text "even"} and @{text "odd"}:
+*}
+
+function even :: "nat \<Rightarrow> bool"
+    and odd  :: "nat \<Rightarrow> bool"
+where
+  "even 0 = True"
+| "odd 0 = False"
+| "even (Suc n) = odd n"
+| "odd (Suc n) = even n"
+by pat_completeness auto
+
+text {*
+  To eliminate the mutual dependencies, Isabelle internally
+  creates a single function operating on the sum
+  type @{typ "nat + nat"}. Then, @{const even} and @{const odd} are
+  defined as projections. Consequently, termination has to be proved
+  simultaneously for both functions, by specifying a measure on the
+  sum type: 
+*}
+
+termination 
+by (relation "measure (\<lambda>x. case x of Inl n \<Rightarrow> n | Inr n \<Rightarrow> n)") auto
+
+text {* 
+  We could also have used @{text lexicographic_order}, which
+  supports mutual recursive termination proofs to a certain extent.
+*}
+
+subsection {* Induction for mutual recursion *}
+
+text {*
+
+  When functions are mutually recursive, proving properties about them
+  generally requires simultaneous induction. The induction rule @{text "even_odd.induct"}
+  generated from the above definition reflects this.
+
+  Let us prove something about @{const even} and @{const odd}:
+*}
+
+lemma even_odd_mod2:
+  "even n = (n mod 2 = 0)"
+  "odd n = (n mod 2 = 1)"
+
+txt {* 
+  We apply simultaneous induction, specifying the induction variable
+  for both goals, separated by \cmd{and}:  *}
+
+apply (induct n and n rule: even_odd.induct)
+
+txt {* 
+  We get four subgoals, which correspond to the clauses in the
+  definition of @{const even} and @{const odd}:
+  @{subgoals[display,indent=0]}
+  Simplification solves the first two goals, leaving us with two
+  statements about the @{text "mod"} operation to prove:
+*}
+
+apply simp_all
+
+txt {* 
+  @{subgoals[display,indent=0]} 
+
+  \noindent These can be handled by Isabelle's arithmetic decision procedures.
+  
+*}
+
+apply arith
+apply arith
+done
+
+text {*
+  In proofs like this, the simultaneous induction is really essential:
+  Even if we are just interested in one of the results, the other
+  one is necessary to strengthen the induction hypothesis. If we leave
+  out the statement about @{const odd} and just write @{term True} instead,
+  the same proof fails:
+*}
+
+lemma failed_attempt:
+  "even n = (n mod 2 = 0)"
+  "True"
+apply (induct n rule: even_odd.induct)
+
+txt {*
+  \noindent Now the third subgoal is a dead end, since we have no
+  useful induction hypothesis available:
+
+  @{subgoals[display,indent=0]} 
+*}
+
+oops
+
+section {* General pattern matching *}
+text{*\label{genpats} *}
+
+subsection {* Avoiding automatic pattern splitting *}
+
+text {*
+
+  Up to now, we used pattern matching only on datatypes, and the
+  patterns were always disjoint and complete, and if they weren't,
+  they were made disjoint automatically like in the definition of
+  @{const "sep"} in \S\ref{patmatch}.
+
+  This automatic splitting can significantly increase the number of
+  equations involved, and this is not always desirable. The following
+  example shows the problem:
+  
+  Suppose we are modeling incomplete knowledge about the world by a
+  three-valued datatype, which has values @{term "T"}, @{term "F"}
+  and @{term "X"} for true, false and uncertain propositions, respectively. 
+*}
+
+datatype P3 = T | F | X
+
+text {* \noindent Then the conjunction of such values can be defined as follows: *}
+
+fun And :: "P3 \<Rightarrow> P3 \<Rightarrow> P3"
+where
+  "And T p = p"
+| "And p T = p"
+| "And p F = F"
+| "And F p = F"
+| "And X X = X"
+
+
+text {* 
+  This definition is useful, because the equations can directly be used
+  as simplification rules. But the patterns overlap: For example,
+  the expression @{term "And T T"} is matched by both the first and
+  the second equation. By default, Isabelle makes the patterns disjoint by
+  splitting them up, producing instances:
+*}
+
+thm And.simps
+
+text {*
+  @{thm[indent=4] And.simps}
+  
+  \vspace*{1em}
+  \noindent There are several problems with this:
+
+  \begin{enumerate}
+  \item If the datatype has many constructors, there can be an
+  explosion of equations. For @{const "And"}, we get seven instead of
+  five equations, which can be tolerated, but this is just a small
+  example.
+
+  \item Since splitting makes the equations \qt{less general}, they
+  do not always match in rewriting. While the term @{term "And x F"}
+  can be simplified to @{term "F"} with the original equations, a
+  (manual) case split on @{term "x"} is now necessary.
+
+  \item The splitting also concerns the induction rule @{text
+  "And.induct"}. Instead of five premises it now has seven, which
+  means that our induction proofs will have more cases.
+
+  \item In general, it increases clarity if we get the same definition
+  back which we put in.
+  \end{enumerate}
+
+  If we do not want the automatic splitting, we can switch it off by
+  leaving out the \cmd{sequential} option. However, we will have to
+  prove that our pattern matching is consistent\footnote{This prevents
+  us from defining something like @{term "f x = True"} and @{term "f x
+  = False"} simultaneously.}:
+*}
+
+function And2 :: "P3 \<Rightarrow> P3 \<Rightarrow> P3"
+where
+  "And2 T p = p"
+| "And2 p T = p"
+| "And2 p F = F"
+| "And2 F p = F"
+| "And2 X X = X"
+
+txt {*
+  \noindent Now let's look at the proof obligations generated by a
+  function definition. In this case, they are:
+
+  @{subgoals[display,indent=0]}\vspace{-1.2em}\hspace{3cm}\vdots\vspace{1.2em}
+
+  The first subgoal expresses the completeness of the patterns. It has
+  the form of an elimination rule and states that every @{term x} of
+  the function's input type must match at least one of the patterns\footnote{Completeness could
+  be equivalently stated as a disjunction of existential statements: 
+@{term "(\<exists>p. x = (T, p)) \<or> (\<exists>p. x = (p, T)) \<or> (\<exists>p. x = (p, F)) \<or>
+  (\<exists>p. x = (F, p)) \<or> (x = (X, X))"}, and you can use the method @{text atomize_elim} to get that form instead.}. If the patterns just involve
+  datatypes, we can solve it with the @{text "pat_completeness"}
+  method:
+*}
+
+apply pat_completeness
+
+txt {*
+  The remaining subgoals express \emph{pattern compatibility}. We do
+  allow that an input value matches multiple patterns, but in this
+  case, the result (i.e.~the right hand sides of the equations) must
+  also be equal. For each pair of two patterns, there is one such
+  subgoal. Usually this needs injectivity of the constructors, which
+  is used automatically by @{text "auto"}.
+*}
+
+by auto
+termination by (relation "{}") simp
+
+
+subsection {* Non-constructor patterns *}
+
+text {*
+  Most of Isabelle's basic types take the form of inductive datatypes,
+  and usually pattern matching works on the constructors of such types. 
+  However, this need not be always the case, and the \cmd{function}
+  command handles other kind of patterns, too.
+
+  One well-known instance of non-constructor patterns are
+  so-called \emph{$n+k$-patterns}, which are a little controversial in
+  the functional programming world. Here is the initial fibonacci
+  example with $n+k$-patterns:
+*}
+
+function fib2 :: "nat \<Rightarrow> nat"
+where
+  "fib2 0 = 1"
+| "fib2 1 = 1"
+| "fib2 (n + 2) = fib2 n + fib2 (Suc n)"
+
+txt {*
+  This kind of matching is again justified by the proof of pattern
+  completeness and compatibility. 
+  The proof obligation for pattern completeness states that every natural number is
+  either @{term "0::nat"}, @{term "1::nat"} or @{term "n +
+  (2::nat)"}:
+
+  @{subgoals[display,indent=0,goals_limit=1]}
+
+  This is an arithmetic triviality, but unfortunately the
+  @{text arith} method cannot handle this specific form of an
+  elimination rule. However, we can use the method @{text
+  "atomize_elim"} to do an ad-hoc conversion to a disjunction of
+  existentials, which can then be solved by the arithmetic decision procedure.
+  Pattern compatibility and termination are automatic as usual.
+*}
+apply atomize_elim
+apply arith
+apply auto
+done
+termination by lexicographic_order
+text {*
+  We can stretch the notion of pattern matching even more. The
+  following function is not a sensible functional program, but a
+  perfectly valid mathematical definition:
+*}
+
+function ev :: "nat \<Rightarrow> bool"
+where
+  "ev (2 * n) = True"
+| "ev (2 * n + 1) = False"
+apply atomize_elim
+by arith+
+termination by (relation "{}") simp
+
+text {*
+  This general notion of pattern matching gives you a certain freedom
+  in writing down specifications. However, as always, such freedom should
+  be used with care:
+
+  If we leave the area of constructor
+  patterns, we have effectively departed from the world of functional
+  programming. This means that it is no longer possible to use the
+  code generator, and expect it to generate ML code for our
+  definitions. Also, such a specification might not work very well together with
+  simplification. Your mileage may vary.
+*}
+
+
+subsection {* Conditional equations *}
+
+text {* 
+  The function package also supports conditional equations, which are
+  similar to guards in a language like Haskell. Here is Euclid's
+  algorithm written with conditional patterns\footnote{Note that the
+  patterns are also overlapping in the base case}:
+*}
+
+function gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat"
+where
+  "gcd x 0 = x"
+| "gcd 0 y = y"
+| "x < y \<Longrightarrow> gcd (Suc x) (Suc y) = gcd (Suc x) (y - x)"
+| "\<not> x < y \<Longrightarrow> gcd (Suc x) (Suc y) = gcd (x - y) (Suc y)"
+by (atomize_elim, auto, arith)
+termination by lexicographic_order
+
+text {*
+  By now, you can probably guess what the proof obligations for the
+  pattern completeness and compatibility look like. 
+
+  Again, functions with conditional patterns are not supported by the
+  code generator.
+*}
+
+
+subsection {* Pattern matching on strings *}
+
+text {*
+  As strings (as lists of characters) are normal datatypes, pattern
+  matching on them is possible, but somewhat problematic. Consider the
+  following definition:
+
+\end{isamarkuptext}
+\noindent\cmd{fun} @{text "check :: \"string \<Rightarrow> bool\""}\\%
+\cmd{where}\\%
+\hspace*{2ex}@{text "\"check (''good'') = True\""}\\%
+@{text "| \"check s = False\""}
+\begin{isamarkuptext}
+
+  \noindent An invocation of the above \cmd{fun} command does not
+  terminate. What is the problem? Strings are lists of characters, and
+  characters are a datatype with a lot of constructors. Splitting the
+  catch-all pattern thus leads to an explosion of cases, which cannot
+  be handled by Isabelle.
+
+  There are two things we can do here. Either we write an explicit
+  @{text "if"} on the right hand side, or we can use conditional patterns:
+*}
+
+function check :: "string \<Rightarrow> bool"
+where
+  "check (''good'') = True"
+| "s \<noteq> ''good'' \<Longrightarrow> check s = False"
+by auto
+termination by (relation "{}") simp
+
+
+section {* Partiality *}
+
+text {* 
+  In HOL, all functions are total. A function @{term "f"} applied to
+  @{term "x"} always has the value @{term "f x"}, and there is no notion
+  of undefinedness. 
+  This is why we have to do termination
+  proofs when defining functions: The proof justifies that the
+  function can be defined by wellfounded recursion.
+
+  However, the \cmd{function} package does support partiality to a
+  certain extent. Let's look at the following function which looks
+  for a zero of a given function f. 
+*}
+
+function (*<*)(domintros)(*>*)findzero :: "(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat"
+where
+  "findzero f n = (if f n = 0 then n else findzero f (Suc n))"
+by pat_completeness auto
+
+text {*
+  \noindent Clearly, any attempt of a termination proof must fail. And without
+  that, we do not get the usual rules @{text "findzero.simps"} and 
+  @{text "findzero.induct"}. So what was the definition good for at all?
+*}
+
+subsection {* Domain predicates *}
+
+text {*
+  The trick is that Isabelle has not only defined the function @{const findzero}, but also
+  a predicate @{term "findzero_dom"} that characterizes the values where the function
+  terminates: the \emph{domain} of the function. If we treat a
+  partial function just as a total function with an additional domain
+  predicate, we can derive simplification and
+  induction rules as we do for total functions. They are guarded
+  by domain conditions and are called @{text psimps} and @{text
+  pinduct}: 
+*}
+
+text {*
+  \noindent\begin{minipage}{0.79\textwidth}@{thm[display,margin=85] findzero.psimps}\end{minipage}
+  \hfill(@{text "findzero.psimps"})
+  \vspace{1em}
+
+  \noindent\begin{minipage}{0.79\textwidth}@{thm[display,margin=85] findzero.pinduct}\end{minipage}
+  \hfill(@{text "findzero.pinduct"})
+*}
+
+text {*
+  Remember that all we
+  are doing here is use some tricks to make a total function appear
+  as if it was partial. We can still write the term @{term "findzero
+  (\<lambda>x. 1) 0"} and like any other term of type @{typ nat} it is equal
+  to some natural number, although we might not be able to find out
+  which one. The function is \emph{underdefined}.
+
+  But it is defined enough to prove something interesting about it. We
+  can prove that if @{term "findzero f n"}
+  terminates, it indeed returns a zero of @{term f}:
+*}
+
+lemma findzero_zero: "findzero_dom (f, n) \<Longrightarrow> f (findzero f n) = 0"
+
+txt {* \noindent We apply induction as usual, but using the partial induction
+  rule: *}
+
+apply (induct f n rule: findzero.pinduct)
+
+txt {* \noindent This gives the following subgoals:
+
+  @{subgoals[display,indent=0]}
+
+  \noindent The hypothesis in our lemma was used to satisfy the first premise in
+  the induction rule. However, we also get @{term
+  "findzero_dom (f, n)"} as a local assumption in the induction step. This
+  allows unfolding @{term "findzero f n"} using the @{text psimps}
+  rule, and the rest is trivial.
+ *}
+apply (simp add: findzero.psimps)
+done
+
+text {*
+  Proofs about partial functions are often not harder than for total
+  functions. Fig.~\ref{findzero_isar} shows a slightly more
+  complicated proof written in Isar. It is verbose enough to show how
+  partiality comes into play: From the partial induction, we get an
+  additional domain condition hypothesis. Observe how this condition
+  is applied when calls to @{term findzero} are unfolded.
+*}
+
+text_raw {*
+\begin{figure}
+\hrule\vspace{6pt}
+\begin{minipage}{0.8\textwidth}
+\isabellestyle{it}
+\isastyle\isamarkuptrue
+*}
+lemma "\<lbrakk>findzero_dom (f, n); x \<in> {n ..< findzero f n}\<rbrakk> \<Longrightarrow> f x \<noteq> 0"
+proof (induct rule: findzero.pinduct)
+  fix f n assume dom: "findzero_dom (f, n)"
+               and IH: "\<lbrakk>f n \<noteq> 0; x \<in> {Suc n ..< findzero f (Suc n)}\<rbrakk> \<Longrightarrow> f x \<noteq> 0"
+               and x_range: "x \<in> {n ..< findzero f n}"
+  have "f n \<noteq> 0"
+  proof 
+    assume "f n = 0"
+    with dom have "findzero f n = n" by (simp add: findzero.psimps)
+    with x_range show False by auto
+  qed
+  
+  from x_range have "x = n \<or> x \<in> {Suc n ..< findzero f n}" by auto
+  thus "f x \<noteq> 0"
+  proof
+    assume "x = n"
+    with `f n \<noteq> 0` show ?thesis by simp
+  next
+    assume "x \<in> {Suc n ..< findzero f n}"
+    with dom and `f n \<noteq> 0` have "x \<in> {Suc n ..< findzero f (Suc n)}" by (simp add: findzero.psimps)
+    with IH and `f n \<noteq> 0`
+    show ?thesis by simp
+  qed
+qed
+text_raw {*
+\isamarkupfalse\isabellestyle{tt}
+\end{minipage}\vspace{6pt}\hrule
+\caption{A proof about a partial function}\label{findzero_isar}
+\end{figure}
+*}
+
+subsection {* Partial termination proofs *}
+
+text {*
+  Now that we have proved some interesting properties about our
+  function, we should turn to the domain predicate and see if it is
+  actually true for some values. Otherwise we would have just proved
+  lemmas with @{term False} as a premise.
+
+  Essentially, we need some introduction rules for @{text
+  findzero_dom}. The function package can prove such domain
+  introduction rules automatically. But since they are not used very
+  often (they are almost never needed if the function is total), this
+  functionality is disabled by default for efficiency reasons. So we have to go
+  back and ask for them explicitly by passing the @{text
+  "(domintros)"} option to the function package:
+
+\vspace{1ex}
+\noindent\cmd{function} @{text "(domintros) findzero :: \"(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat\""}\\%
+\cmd{where}\isanewline%
+\ \ \ldots\\
+
+  \noindent Now the package has proved an introduction rule for @{text findzero_dom}:
+*}
+
+thm findzero.domintros
+
+text {*
+  @{thm[display] findzero.domintros}
+
+  Domain introduction rules allow to show that a given value lies in the
+  domain of a function, if the arguments of all recursive calls
+  are in the domain as well. They allow to do a \qt{single step} in a
+  termination proof. Usually, you want to combine them with a suitable
+  induction principle.
+
+  Since our function increases its argument at recursive calls, we
+  need an induction principle which works \qt{backwards}. We will use
+  @{text inc_induct}, which allows to do induction from a fixed number
+  \qt{downwards}:
+
+  \begin{center}@{thm inc_induct}\hfill(@{text "inc_induct"})\end{center}
+
+  Figure \ref{findzero_term} gives a detailed Isar proof of the fact
+  that @{text findzero} terminates if there is a zero which is greater
+  or equal to @{term n}. First we derive two useful rules which will
+  solve the base case and the step case of the induction. The
+  induction is then straightforward, except for the unusual induction
+  principle.
+
+*}
+
+text_raw {*
+\begin{figure}
+\hrule\vspace{6pt}
+\begin{minipage}{0.8\textwidth}
+\isabellestyle{it}
+\isastyle\isamarkuptrue
+*}
+lemma findzero_termination:
+  assumes "x \<ge> n" and "f x = 0"
+  shows "findzero_dom (f, n)"
+proof - 
+  have base: "findzero_dom (f, x)"
+    by (rule findzero.domintros) (simp add:`f x = 0`)
+
+  have step: "\<And>i. findzero_dom (f, Suc i) 
+    \<Longrightarrow> findzero_dom (f, i)"
+    by (rule findzero.domintros) simp
+
+  from `x \<ge> n` show ?thesis
+  proof (induct rule:inc_induct)
+    show "findzero_dom (f, x)" by (rule base)
+  next
+    fix i assume "findzero_dom (f, Suc i)"
+    thus "findzero_dom (f, i)" by (rule step)
+  qed
+qed      
+text_raw {*
+\isamarkupfalse\isabellestyle{tt}
+\end{minipage}\vspace{6pt}\hrule
+\caption{Termination proof for @{text findzero}}\label{findzero_term}
+\end{figure}
+*}
+      
+text {*
+  Again, the proof given in Fig.~\ref{findzero_term} has a lot of
+  detail in order to explain the principles. Using more automation, we
+  can also have a short proof:
+*}
+
+lemma findzero_termination_short:
+  assumes zero: "x >= n" 
+  assumes [simp]: "f x = 0"
+  shows "findzero_dom (f, n)"
+using zero
+by (induct rule:inc_induct) (auto intro: findzero.domintros)
+    
+text {*
+  \noindent It is simple to combine the partial correctness result with the
+  termination lemma:
+*}
+
+lemma findzero_total_correctness:
+  "f x = 0 \<Longrightarrow> f (findzero f 0) = 0"
+by (blast intro: findzero_zero findzero_termination)
+
+subsection {* Definition of the domain predicate *}
+
+text {*
+  Sometimes it is useful to know what the definition of the domain
+  predicate looks like. Actually, @{text findzero_dom} is just an
+  abbreviation:
+
+  @{abbrev[display] findzero_dom}
+
+  The domain predicate is the \emph{accessible part} of a relation @{const
+  findzero_rel}, which was also created internally by the function
+  package. @{const findzero_rel} is just a normal
+  inductive predicate, so we can inspect its definition by
+  looking at the introduction rules @{text findzero_rel.intros}.
+  In our case there is just a single rule:
+
+  @{thm[display] findzero_rel.intros}
+
+  The predicate @{const findzero_rel}
+  describes the \emph{recursion relation} of the function
+  definition. The recursion relation is a binary relation on
+  the arguments of the function that relates each argument to its
+  recursive calls. In general, there is one introduction rule for each
+  recursive call.
+
+  The predicate @{term "accp findzero_rel"} is the accessible part of
+  that relation. An argument belongs to the accessible part, if it can
+  be reached in a finite number of steps (cf.~its definition in @{text
+  "Wellfounded.thy"}).
+
+  Since the domain predicate is just an abbreviation, you can use
+  lemmas for @{const accp} and @{const findzero_rel} directly. Some
+  lemmas which are occasionally useful are @{text accpI}, @{text
+  accp_downward}, and of course the introduction and elimination rules
+  for the recursion relation @{text "findzero.intros"} and @{text "findzero.cases"}.
+*}
+
+section {* Nested recursion *}
+
+text {*
+  Recursive calls which are nested in one another frequently cause
+  complications, since their termination proof can depend on a partial
+  correctness property of the function itself. 
+
+  As a small example, we define the \qt{nested zero} function:
+*}
+
+function nz :: "nat \<Rightarrow> nat"
+where
+  "nz 0 = 0"
+| "nz (Suc n) = nz (nz n)"
+by pat_completeness auto
+
+text {*
+  If we attempt to prove termination using the identity measure on
+  naturals, this fails:
+*}
+
+termination
+  apply (relation "measure (\<lambda>n. n)")
+  apply auto
+
+txt {*
+  We get stuck with the subgoal
+
+  @{subgoals[display]}
+
+  Of course this statement is true, since we know that @{const nz} is
+  the zero function. And in fact we have no problem proving this
+  property by induction.
+*}
+(*<*)oops(*>*)
+lemma nz_is_zero: "nz_dom n \<Longrightarrow> nz n = 0"
+  by (induct rule:nz.pinduct) (auto simp: nz.psimps)
+
+text {*
+  We formulate this as a partial correctness lemma with the condition
+  @{term "nz_dom n"}. This allows us to prove it with the @{text
+  pinduct} rule before we have proved termination. With this lemma,
+  the termination proof works as expected:
+*}
+
+termination
+  by (relation "measure (\<lambda>n. n)") (auto simp: nz_is_zero)
+
+text {*
+  As a general strategy, one should prove the statements needed for
+  termination as a partial property first. Then they can be used to do
+  the termination proof. This also works for less trivial
+  examples. Figure \ref{f91} defines the 91-function, a well-known
+  challenge problem due to John McCarthy, and proves its termination.
+*}
+
+text_raw {*
+\begin{figure}
+\hrule\vspace{6pt}
+\begin{minipage}{0.8\textwidth}
+\isabellestyle{it}
+\isastyle\isamarkuptrue
+*}
+
+function f91 :: "nat \<Rightarrow> nat"
+where
+  "f91 n = (if 100 < n then n - 10 else f91 (f91 (n + 11)))"
+by pat_completeness auto
+
+lemma f91_estimate: 
+  assumes trm: "f91_dom n" 
+  shows "n < f91 n + 11"
+using trm by induct (auto simp: f91.psimps)
+
+termination
+proof
+  let ?R = "measure (\<lambda>x. 101 - x)"
+  show "wf ?R" ..
+
+  fix n :: nat assume "\<not> 100 < n" -- "Assumptions for both calls"
+
+  thus "(n + 11, n) \<in> ?R" by simp -- "Inner call"
+
+  assume inner_trm: "f91_dom (n + 11)" -- "Outer call"
+  with f91_estimate have "n + 11 < f91 (n + 11) + 11" .
+  with `\<not> 100 < n` show "(f91 (n + 11), n) \<in> ?R" by simp
+qed
+
+text_raw {*
+\isamarkupfalse\isabellestyle{tt}
+\end{minipage}
+\vspace{6pt}\hrule
+\caption{McCarthy's 91-function}\label{f91}
+\end{figure}
+*}
+
+
+section {* Higher-Order Recursion *}
+
+text {*
+  Higher-order recursion occurs when recursive calls
+  are passed as arguments to higher-order combinators such as @{const
+  map}, @{term filter} etc.
+  As an example, imagine a datatype of n-ary trees:
+*}
+
+datatype 'a tree = 
+  Leaf 'a 
+| Branch "'a tree list"
+
+
+text {* \noindent We can define a function which swaps the left and right subtrees recursively, using the 
+  list functions @{const rev} and @{const map}: *}
+
+fun mirror :: "'a tree \<Rightarrow> 'a tree"
+where
+  "mirror (Leaf n) = Leaf n"
+| "mirror (Branch l) = Branch (rev (map mirror l))"
+
+text {*
+  Although the definition is accepted without problems, let us look at the termination proof:
+*}
+
+termination proof
+  txt {*
+
+  As usual, we have to give a wellfounded relation, such that the
+  arguments of the recursive calls get smaller. But what exactly are
+  the arguments of the recursive calls when mirror is given as an
+  argument to @{const map}? Isabelle gives us the
+  subgoals
+
+  @{subgoals[display,indent=0]} 
+
+  So the system seems to know that @{const map} only
+  applies the recursive call @{term "mirror"} to elements
+  of @{term "l"}, which is essential for the termination proof.
+
+  This knowledge about @{const map} is encoded in so-called congruence rules,
+  which are special theorems known to the \cmd{function} command. The
+  rule for @{const map} is
+
+  @{thm[display] map_cong}
+
+  You can read this in the following way: Two applications of @{const
+  map} are equal, if the list arguments are equal and the functions
+  coincide on the elements of the list. This means that for the value 
+  @{term "map f l"} we only have to know how @{term f} behaves on
+  the elements of @{term l}.
+
+  Usually, one such congruence rule is
+  needed for each higher-order construct that is used when defining
+  new functions. In fact, even basic functions like @{const
+  If} and @{const Let} are handled by this mechanism. The congruence
+  rule for @{const If} states that the @{text then} branch is only
+  relevant if the condition is true, and the @{text else} branch only if it
+  is false:
+
+  @{thm[display] if_cong}
+  
+  Congruence rules can be added to the
+  function package by giving them the @{term fundef_cong} attribute.
+
+  The constructs that are predefined in Isabelle, usually
+  come with the respective congruence rules.
+  But if you define your own higher-order functions, you may have to
+  state and prove the required congruence rules yourself, if you want to use your
+  functions in recursive definitions. 
+*}
+(*<*)oops(*>*)
+
+subsection {* Congruence Rules and Evaluation Order *}
+
+text {* 
+  Higher order logic differs from functional programming languages in
+  that it has no built-in notion of evaluation order. A program is
+  just a set of equations, and it is not specified how they must be
+  evaluated. 
+
+  However for the purpose of function definition, we must talk about
+  evaluation order implicitly, when we reason about termination.
+  Congruence rules express that a certain evaluation order is
+  consistent with the logical definition. 
+
+  Consider the following function.
+*}
+
+function f :: "nat \<Rightarrow> bool"
+where
+  "f n = (n = 0 \<or> f (n - 1))"
+(*<*)by pat_completeness auto(*>*)
+
+text {*
+  For this definition, the termination proof fails. The default configuration
+  specifies no congruence rule for disjunction. We have to add a
+  congruence rule that specifies left-to-right evaluation order:
+
+  \vspace{1ex}
+  \noindent @{thm disj_cong}\hfill(@{text "disj_cong"})
+  \vspace{1ex}
+
+  Now the definition works without problems. Note how the termination
+  proof depends on the extra condition that we get from the congruence
+  rule.
+
+  However, as evaluation is not a hard-wired concept, we
+  could just turn everything around by declaring a different
+  congruence rule. Then we can make the reverse definition:
+*}
+
+lemma disj_cong2[fundef_cong]: 
+  "(\<not> Q' \<Longrightarrow> P = P') \<Longrightarrow> (Q = Q') \<Longrightarrow> (P \<or> Q) = (P' \<or> Q')"
+  by blast
+
+fun f' :: "nat \<Rightarrow> bool"
+where
+  "f' n = (f' (n - 1) \<or> n = 0)"
+
+text {*
+  \noindent These examples show that, in general, there is no \qt{best} set of
+  congruence rules.
+
+  However, such tweaking should rarely be necessary in
+  practice, as most of the time, the default set of congruence rules
+  works well.
+*}
+
+end
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/Makefile
--- a/doc-src/Functions/Makefile	Mon Aug 27 22:00:04 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,35 +0,0 @@
-
-## targets
-
-default: dvi
-
-
-## dependencies
-
-include ../Makefile.in
-
-NAME = functions
-
-FILES = $(NAME).tex Thy/document/Functions.tex intro.tex conclusion.tex \
-  style.sty ../iman.sty ../extra.sty ../isar.sty \
-  ../../lib/texinputs/isabelle.sty ../../lib/texinputs/isabellesym.sty ../pdfsetup.sty \
-  ../manual.bib ../proof.sty
-
-dvi: $(NAME).dvi
-
-$(NAME).dvi: $(FILES) isabelle_isar.eps
-	$(LATEX) $(NAME)
-	$(BIBTEX) $(NAME)
-	$(LATEX) $(NAME)
-	$(LATEX) $(NAME)
-
-pdf: $(NAME).pdf
-
-$(NAME).pdf: $(FILES) isabelle_isar.pdf
-	$(PDFLATEX) $(NAME)
-	$(BIBTEX) $(NAME)
-	$(PDFLATEX) $(NAME)
-	$(PDFLATEX) $(NAME)
-	$(FIXBOOKMARKS) $(NAME).out
-	$(PDFLATEX) $(NAME)
-	$(PDFLATEX) $(NAME)
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/Thy/Functions.thy
--- a/doc-src/Functions/Thy/Functions.thy	Mon Aug 27 22:00:04 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1190 +0,0 @@
-(*  Title:      doc-src/IsarAdvanced/Functions/Thy/Fundefs.thy
-    Author:     Alexander Krauss, TU Muenchen
-
-Tutorial for function definitions with the new "function" package.
-*)
-
-theory Functions
-imports Main
-begin
-
-section {* Function Definitions for Dummies *}
-
-text {*
-  In most cases, defining a recursive function is just as simple as other definitions:
-*}
-
-fun fib :: "nat \<Rightarrow> nat"
-where
-  "fib 0 = 1"
-| "fib (Suc 0) = 1"
-| "fib (Suc (Suc n)) = fib n + fib (Suc n)"
-
-text {*
-  The syntax is rather self-explanatory: We introduce a function by
-  giving its name, its type, 
-  and a set of defining recursive equations.
-  If we leave out the type, the most general type will be
-  inferred, which can sometimes lead to surprises: Since both @{term
-  "1::nat"} and @{text "+"} are overloaded, we would end up
-  with @{text "fib :: nat \<Rightarrow> 'a::{one,plus}"}.
-*}
-
-text {*
-  The function always terminates, since its argument gets smaller in
-  every recursive call. 
-  Since HOL is a logic of total functions, termination is a
-  fundamental requirement to prevent inconsistencies\footnote{From the
-  \qt{definition} @{text "f(n) = f(n) + 1"} we could prove 
-  @{text "0 = 1"} by subtracting @{text "f(n)"} on both sides.}.
-  Isabelle tries to prove termination automatically when a definition
-  is made. In \S\ref{termination}, we will look at cases where this
-  fails and see what to do then.
-*}
-
-subsection {* Pattern matching *}
-
-text {* \label{patmatch}
-  Like in functional programming, we can use pattern matching to
-  define functions. At the moment we will only consider \emph{constructor
-  patterns}, which only consist of datatype constructors and
-  variables. Furthermore, patterns must be linear, i.e.\ all variables
-  on the left hand side of an equation must be distinct. In
-  \S\ref{genpats} we discuss more general pattern matching.
-
-  If patterns overlap, the order of the equations is taken into
-  account. The following function inserts a fixed element between any
-  two elements of a list:
-*}
-
-fun sep :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where
-  "sep a (x#y#xs) = x # a # sep a (y # xs)"
-| "sep a xs       = xs"
-
-text {* 
-  Overlapping patterns are interpreted as \qt{increments} to what is
-  already there: The second equation is only meant for the cases where
-  the first one does not match. Consequently, Isabelle replaces it
-  internally by the remaining cases, making the patterns disjoint:
-*}
-
-thm sep.simps
-
-text {* @{thm [display] sep.simps[no_vars]} *}
-
-text {* 
-  \noindent The equations from function definitions are automatically used in
-  simplification:
-*}
-
-lemma "sep 0 [1, 2, 3] = [1, 0, 2, 0, 3]"
-by simp
-
-subsection {* Induction *}
-
-text {*
-
-  Isabelle provides customized induction rules for recursive
-  functions. These rules follow the recursive structure of the
-  definition. Here is the rule @{text sep.induct} arising from the
-  above definition of @{const sep}:
-
-  @{thm [display] sep.induct}
-  
-  We have a step case for list with at least two elements, and two
-  base cases for the zero- and the one-element list. Here is a simple
-  proof about @{const sep} and @{const map}
-*}
-
-lemma "map f (sep x ys) = sep (f x) (map f ys)"
-apply (induct x ys rule: sep.induct)
-
-txt {*
-  We get three cases, like in the definition.
-
-  @{subgoals [display]}
-*}
-
-apply auto 
-done
-text {*
-
-  With the \cmd{fun} command, you can define about 80\% of the
-  functions that occur in practice. The rest of this tutorial explains
-  the remaining 20\%.
-*}
-
-
-section {* fun vs.\ function *}
-
-text {* 
-  The \cmd{fun} command provides a
-  convenient shorthand notation for simple function definitions. In
-  this mode, Isabelle tries to solve all the necessary proof obligations
-  automatically. If any proof fails, the definition is
-  rejected. This can either mean that the definition is indeed faulty,
-  or that the default proof procedures are just not smart enough (or
-  rather: not designed) to handle the definition.
-
-  By expanding the abbreviation to the more verbose \cmd{function} command, these proof obligations become visible and can be analyzed or
-  solved manually. The expansion from \cmd{fun} to \cmd{function} is as follows:
-
-\end{isamarkuptext}
-
-
-\[\left[\;\begin{minipage}{0.25\textwidth}\vspace{6pt}
-\cmd{fun} @{text "f :: \<tau>"}\\%
-\cmd{where}\\%
-\hspace*{2ex}{\it equations}\\%
-\hspace*{2ex}\vdots\vspace*{6pt}
-\end{minipage}\right]
-\quad\equiv\quad
-\left[\;\begin{minipage}{0.48\textwidth}\vspace{6pt}
-\cmd{function} @{text "("}\cmd{sequential}@{text ") f :: \<tau>"}\\%
-\cmd{where}\\%
-\hspace*{2ex}{\it equations}\\%
-\hspace*{2ex}\vdots\\%
-\cmd{by} @{text "pat_completeness auto"}\\%
-\cmd{termination by} @{text "lexicographic_order"}\vspace{6pt}
-\end{minipage}
-\right]\]
-
-\begin{isamarkuptext}
-  \vspace*{1em}
-  \noindent Some details have now become explicit:
-
-  \begin{enumerate}
-  \item The \cmd{sequential} option enables the preprocessing of
-  pattern overlaps which we already saw. Without this option, the equations
-  must already be disjoint and complete. The automatic completion only
-  works with constructor patterns.
-
-  \item A function definition produces a proof obligation which
-  expresses completeness and compatibility of patterns (we talk about
-  this later). The combination of the methods @{text "pat_completeness"} and
-  @{text "auto"} is used to solve this proof obligation.
-
-  \item A termination proof follows the definition, started by the
-  \cmd{termination} command. This will be explained in \S\ref{termination}.
- \end{enumerate}
-  Whenever a \cmd{fun} command fails, it is usually a good idea to
-  expand the syntax to the more verbose \cmd{function} form, to see
-  what is actually going on.
- *}
-
-
-section {* Termination *}
-
-text {*\label{termination}
-  The method @{text "lexicographic_order"} is the default method for
-  termination proofs. It can prove termination of a
-  certain class of functions by searching for a suitable lexicographic
-  combination of size measures. Of course, not all functions have such
-  a simple termination argument. For them, we can specify the termination
-  relation manually.
-*}
-
-subsection {* The {\tt relation} method *}
-text{*
-  Consider the following function, which sums up natural numbers up to
-  @{text "N"}, using a counter @{text "i"}:
-*}
-
-function sum :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-  "sum i N = (if i > N then 0 else i + sum (Suc i) N)"
-by pat_completeness auto
-
-text {*
-  \noindent The @{text "lexicographic_order"} method fails on this example, because none of the
-  arguments decreases in the recursive call, with respect to the standard size ordering.
-  To prove termination manually, we must provide a custom wellfounded relation.
-
-  The termination argument for @{text "sum"} is based on the fact that
-  the \emph{difference} between @{text "i"} and @{text "N"} gets
-  smaller in every step, and that the recursion stops when @{text "i"}
-  is greater than @{text "N"}. Phrased differently, the expression 
-  @{text "N + 1 - i"} always decreases.
-
-  We can use this expression as a measure function suitable to prove termination.
-*}
-
-termination sum
-apply (relation "measure (\<lambda>(i,N). N + 1 - i)")
-
-txt {*
-  The \cmd{termination} command sets up the termination goal for the
-  specified function @{text "sum"}. If the function name is omitted, it
-  implicitly refers to the last function definition.
-
-  The @{text relation} method takes a relation of
-  type @{typ "('a \<times> 'a) set"}, where @{typ "'a"} is the argument type of
-  the function. If the function has multiple curried arguments, then
-  these are packed together into a tuple, as it happened in the above
-  example.
-
-  The predefined function @{term[source] "measure :: ('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set"} constructs a
-  wellfounded relation from a mapping into the natural numbers (a
-  \emph{measure function}). 
-
-  After the invocation of @{text "relation"}, we must prove that (a)
-  the relation we supplied is wellfounded, and (b) that the arguments
-  of recursive calls indeed decrease with respect to the
-  relation:
-
-  @{subgoals[display,indent=0]}
-
-  These goals are all solved by @{text "auto"}:
-*}
-
-apply auto
-done
-
-text {*
-  Let us complicate the function a little, by adding some more
-  recursive calls: 
-*}
-
-function foo :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-  "foo i N = (if i > N 
-              then (if N = 0 then 0 else foo 0 (N - 1))
-              else i + foo (Suc i) N)"
-by pat_completeness auto
-
-text {*
-  When @{text "i"} has reached @{text "N"}, it starts at zero again
-  and @{text "N"} is decremented.
-  This corresponds to a nested
-  loop where one index counts up and the other down. Termination can
-  be proved using a lexicographic combination of two measures, namely
-  the value of @{text "N"} and the above difference. The @{const
-  "measures"} combinator generalizes @{text "measure"} by taking a
-  list of measure functions.  
-*}
-
-termination 
-by (relation "measures [\<lambda>(i, N). N, \<lambda>(i,N). N + 1 - i]") auto
-
-subsection {* How @{text "lexicographic_order"} works *}
-
-(*fun fails :: "nat \<Rightarrow> nat list \<Rightarrow> nat"
-where
-  "fails a [] = a"
-| "fails a (x#xs) = fails (x + a) (x # xs)"
-*)
-
-text {*
-  To see how the automatic termination proofs work, let's look at an
-  example where it fails\footnote{For a detailed discussion of the
-  termination prover, see \cite{bulwahnKN07}}:
-
-\end{isamarkuptext}  
-\cmd{fun} @{text "fails :: \"nat \<Rightarrow> nat list \<Rightarrow> nat\""}\\%
-\cmd{where}\\%
-\hspace*{2ex}@{text "\"fails a [] = a\""}\\%
-|\hspace*{1.5ex}@{text "\"fails a (x#xs) = fails (x + a) (x#xs)\""}\\
-\begin{isamarkuptext}
-
-\noindent Isabelle responds with the following error:
-
-\begin{isabelle}
-*** Unfinished subgoals:\newline
-*** (a, 1, <):\newline
-*** \ 1.~@{text "\<And>x. x = 0"}\newline
-*** (a, 1, <=):\newline
-*** \ 1.~False\newline
-*** (a, 2, <):\newline
-*** \ 1.~False\newline
-*** Calls:\newline
-*** a) @{text "(a, x # xs) -->> (x + a, x # xs)"}\newline
-*** Measures:\newline
-*** 1) @{text "\<lambda>x. size (fst x)"}\newline
-*** 2) @{text "\<lambda>x. size (snd x)"}\newline
-*** Result matrix:\newline
-*** \ \ \ \ 1\ \ 2  \newline
-*** a:  ?   <= \newline
-*** Could not find lexicographic termination order.\newline
-*** At command "fun".\newline
-\end{isabelle}
-*}
-text {*
-  The key to this error message is the matrix at the bottom. The rows
-  of that matrix correspond to the different recursive calls (In our
-  case, there is just one). The columns are the function's arguments 
-  (expressed through different measure functions, which map the
-  argument tuple to a natural number). 
-
-  The contents of the matrix summarize what is known about argument
-  descents: The second argument has a weak descent (@{text "<="}) at the
-  recursive call, and for the first argument nothing could be proved,
-  which is expressed by @{text "?"}. In general, there are the values
-  @{text "<"}, @{text "<="} and @{text "?"}.
-
-  For the failed proof attempts, the unfinished subgoals are also
-  printed. Looking at these will often point to a missing lemma.
-*}
-
-subsection {* The @{text size_change} method *}
-
-text {*
-  Some termination goals that are beyond the powers of
-  @{text lexicographic_order} can be solved automatically by the
-  more powerful @{text size_change} method, which uses a variant of
-  the size-change principle, together with some other
-  techniques. While the details are discussed
-  elsewhere\cite{krauss_phd},
-  here are a few typical situations where
-  @{text lexicographic_order} has difficulties and @{text size_change}
-  may be worth a try:
-  \begin{itemize}
-  \item Arguments are permuted in a recursive call.
-  \item Several mutually recursive functions with multiple arguments.
-  \item Unusual control flow (e.g., when some recursive calls cannot
-  occur in sequence).
-  \end{itemize}
-
-  Loading the theory @{text Multiset} makes the @{text size_change}
-  method a bit stronger: it can then use multiset orders internally.
-*}
-
-section {* Mutual Recursion *}
-
-text {*
-  If two or more functions call one another mutually, they have to be defined
-  in one step. Here are @{text "even"} and @{text "odd"}:
-*}
-
-function even :: "nat \<Rightarrow> bool"
-    and odd  :: "nat \<Rightarrow> bool"
-where
-  "even 0 = True"
-| "odd 0 = False"
-| "even (Suc n) = odd n"
-| "odd (Suc n) = even n"
-by pat_completeness auto
-
-text {*
-  To eliminate the mutual dependencies, Isabelle internally
-  creates a single function operating on the sum
-  type @{typ "nat + nat"}. Then, @{const even} and @{const odd} are
-  defined as projections. Consequently, termination has to be proved
-  simultaneously for both functions, by specifying a measure on the
-  sum type: 
-*}
-
-termination 
-by (relation "measure (\<lambda>x. case x of Inl n \<Rightarrow> n | Inr n \<Rightarrow> n)") auto
-
-text {* 
-  We could also have used @{text lexicographic_order}, which
-  supports mutual recursive termination proofs to a certain extent.
-*}
-
-subsection {* Induction for mutual recursion *}
-
-text {*
-
-  When functions are mutually recursive, proving properties about them
-  generally requires simultaneous induction. The induction rule @{text "even_odd.induct"}
-  generated from the above definition reflects this.
-
-  Let us prove something about @{const even} and @{const odd}:
-*}
-
-lemma even_odd_mod2:
-  "even n = (n mod 2 = 0)"
-  "odd n = (n mod 2 = 1)"
-
-txt {* 
-  We apply simultaneous induction, specifying the induction variable
-  for both goals, separated by \cmd{and}:  *}
-
-apply (induct n and n rule: even_odd.induct)
-
-txt {* 
-  We get four subgoals, which correspond to the clauses in the
-  definition of @{const even} and @{const odd}:
-  @{subgoals[display,indent=0]}
-  Simplification solves the first two goals, leaving us with two
-  statements about the @{text "mod"} operation to prove:
-*}
-
-apply simp_all
-
-txt {* 
-  @{subgoals[display,indent=0]} 
-
-  \noindent These can be handled by Isabelle's arithmetic decision procedures.
-  
-*}
-
-apply arith
-apply arith
-done
-
-text {*
-  In proofs like this, the simultaneous induction is really essential:
-  Even if we are just interested in one of the results, the other
-  one is necessary to strengthen the induction hypothesis. If we leave
-  out the statement about @{const odd} and just write @{term True} instead,
-  the same proof fails:
-*}
-
-lemma failed_attempt:
-  "even n = (n mod 2 = 0)"
-  "True"
-apply (induct n rule: even_odd.induct)
-
-txt {*
-  \noindent Now the third subgoal is a dead end, since we have no
-  useful induction hypothesis available:
-
-  @{subgoals[display,indent=0]} 
-*}
-
-oops
-
-section {* General pattern matching *}
-text{*\label{genpats} *}
-
-subsection {* Avoiding automatic pattern splitting *}
-
-text {*
-
-  Up to now, we used pattern matching only on datatypes, and the
-  patterns were always disjoint and complete, and if they weren't,
-  they were made disjoint automatically like in the definition of
-  @{const "sep"} in \S\ref{patmatch}.
-
-  This automatic splitting can significantly increase the number of
-  equations involved, and this is not always desirable. The following
-  example shows the problem:
-  
-  Suppose we are modeling incomplete knowledge about the world by a
-  three-valued datatype, which has values @{term "T"}, @{term "F"}
-  and @{term "X"} for true, false and uncertain propositions, respectively. 
-*}
-
-datatype P3 = T | F | X
-
-text {* \noindent Then the conjunction of such values can be defined as follows: *}
-
-fun And :: "P3 \<Rightarrow> P3 \<Rightarrow> P3"
-where
-  "And T p = p"
-| "And p T = p"
-| "And p F = F"
-| "And F p = F"
-| "And X X = X"
-
-
-text {* 
-  This definition is useful, because the equations can directly be used
-  as simplification rules. But the patterns overlap: For example,
-  the expression @{term "And T T"} is matched by both the first and
-  the second equation. By default, Isabelle makes the patterns disjoint by
-  splitting them up, producing instances:
-*}
-
-thm And.simps
-
-text {*
-  @{thm[indent=4] And.simps}
-  
-  \vspace*{1em}
-  \noindent There are several problems with this:
-
-  \begin{enumerate}
-  \item If the datatype has many constructors, there can be an
-  explosion of equations. For @{const "And"}, we get seven instead of
-  five equations, which can be tolerated, but this is just a small
-  example.
-
-  \item Since splitting makes the equations \qt{less general}, they
-  do not always match in rewriting. While the term @{term "And x F"}
-  can be simplified to @{term "F"} with the original equations, a
-  (manual) case split on @{term "x"} is now necessary.
-
-  \item The splitting also concerns the induction rule @{text
-  "And.induct"}. Instead of five premises it now has seven, which
-  means that our induction proofs will have more cases.
-
-  \item In general, it increases clarity if we get the same definition
-  back which we put in.
-  \end{enumerate}
-
-  If we do not want the automatic splitting, we can switch it off by
-  leaving out the \cmd{sequential} option. However, we will have to
-  prove that our pattern matching is consistent\footnote{This prevents
-  us from defining something like @{term "f x = True"} and @{term "f x
-  = False"} simultaneously.}:
-*}
-
-function And2 :: "P3 \<Rightarrow> P3 \<Rightarrow> P3"
-where
-  "And2 T p = p"
-| "And2 p T = p"
-| "And2 p F = F"
-| "And2 F p = F"
-| "And2 X X = X"
-
-txt {*
-  \noindent Now let's look at the proof obligations generated by a
-  function definition. In this case, they are:
-
-  @{subgoals[display,indent=0]}\vspace{-1.2em}\hspace{3cm}\vdots\vspace{1.2em}
-
-  The first subgoal expresses the completeness of the patterns. It has
-  the form of an elimination rule and states that every @{term x} of
-  the function's input type must match at least one of the patterns\footnote{Completeness could
-  be equivalently stated as a disjunction of existential statements: 
-@{term "(\<exists>p. x = (T, p)) \<or> (\<exists>p. x = (p, T)) \<or> (\<exists>p. x = (p, F)) \<or>
-  (\<exists>p. x = (F, p)) \<or> (x = (X, X))"}, and you can use the method @{text atomize_elim} to get that form instead.}. If the patterns just involve
-  datatypes, we can solve it with the @{text "pat_completeness"}
-  method:
-*}
-
-apply pat_completeness
-
-txt {*
-  The remaining subgoals express \emph{pattern compatibility}. We do
-  allow that an input value matches multiple patterns, but in this
-  case, the result (i.e.~the right hand sides of the equations) must
-  also be equal. For each pair of two patterns, there is one such
-  subgoal. Usually this needs injectivity of the constructors, which
-  is used automatically by @{text "auto"}.
-*}
-
-by auto
-termination by (relation "{}") simp
-
-
-subsection {* Non-constructor patterns *}
-
-text {*
-  Most of Isabelle's basic types take the form of inductive datatypes,
-  and usually pattern matching works on the constructors of such types. 
-  However, this need not be always the case, and the \cmd{function}
-  command handles other kind of patterns, too.
-
-  One well-known instance of non-constructor patterns are
-  so-called \emph{$n+k$-patterns}, which are a little controversial in
-  the functional programming world. Here is the initial fibonacci
-  example with $n+k$-patterns:
-*}
-
-function fib2 :: "nat \<Rightarrow> nat"
-where
-  "fib2 0 = 1"
-| "fib2 1 = 1"
-| "fib2 (n + 2) = fib2 n + fib2 (Suc n)"
-
-txt {*
-  This kind of matching is again justified by the proof of pattern
-  completeness and compatibility. 
-  The proof obligation for pattern completeness states that every natural number is
-  either @{term "0::nat"}, @{term "1::nat"} or @{term "n +
-  (2::nat)"}:
-
-  @{subgoals[display,indent=0,goals_limit=1]}
-
-  This is an arithmetic triviality, but unfortunately the
-  @{text arith} method cannot handle this specific form of an
-  elimination rule. However, we can use the method @{text
-  "atomize_elim"} to do an ad-hoc conversion to a disjunction of
-  existentials, which can then be solved by the arithmetic decision procedure.
-  Pattern compatibility and termination are automatic as usual.
-*}
-apply atomize_elim
-apply arith
-apply auto
-done
-termination by lexicographic_order
-text {*
-  We can stretch the notion of pattern matching even more. The
-  following function is not a sensible functional program, but a
-  perfectly valid mathematical definition:
-*}
-
-function ev :: "nat \<Rightarrow> bool"
-where
-  "ev (2 * n) = True"
-| "ev (2 * n + 1) = False"
-apply atomize_elim
-by arith+
-termination by (relation "{}") simp
-
-text {*
-  This general notion of pattern matching gives you a certain freedom
-  in writing down specifications. However, as always, such freedom should
-  be used with care:
-
-  If we leave the area of constructor
-  patterns, we have effectively departed from the world of functional
-  programming. This means that it is no longer possible to use the
-  code generator, and expect it to generate ML code for our
-  definitions. Also, such a specification might not work very well together with
-  simplification. Your mileage may vary.
-*}
-
-
-subsection {* Conditional equations *}
-
-text {* 
-  The function package also supports conditional equations, which are
-  similar to guards in a language like Haskell. Here is Euclid's
-  algorithm written with conditional patterns\footnote{Note that the
-  patterns are also overlapping in the base case}:
-*}
-
-function gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat"
-where
-  "gcd x 0 = x"
-| "gcd 0 y = y"
-| "x < y \<Longrightarrow> gcd (Suc x) (Suc y) = gcd (Suc x) (y - x)"
-| "\<not> x < y \<Longrightarrow> gcd (Suc x) (Suc y) = gcd (x - y) (Suc y)"
-by (atomize_elim, auto, arith)
-termination by lexicographic_order
-
-text {*
-  By now, you can probably guess what the proof obligations for the
-  pattern completeness and compatibility look like. 
-
-  Again, functions with conditional patterns are not supported by the
-  code generator.
-*}
-
-
-subsection {* Pattern matching on strings *}
-
-text {*
-  As strings (as lists of characters) are normal datatypes, pattern
-  matching on them is possible, but somewhat problematic. Consider the
-  following definition:
-
-\end{isamarkuptext}
-\noindent\cmd{fun} @{text "check :: \"string \<Rightarrow> bool\""}\\%
-\cmd{where}\\%
-\hspace*{2ex}@{text "\"check (''good'') = True\""}\\%
-@{text "| \"check s = False\""}
-\begin{isamarkuptext}
-
-  \noindent An invocation of the above \cmd{fun} command does not
-  terminate. What is the problem? Strings are lists of characters, and
-  characters are a datatype with a lot of constructors. Splitting the
-  catch-all pattern thus leads to an explosion of cases, which cannot
-  be handled by Isabelle.
-
-  There are two things we can do here. Either we write an explicit
-  @{text "if"} on the right hand side, or we can use conditional patterns:
-*}
-
-function check :: "string \<Rightarrow> bool"
-where
-  "check (''good'') = True"
-| "s \<noteq> ''good'' \<Longrightarrow> check s = False"
-by auto
-termination by (relation "{}") simp
-
-
-section {* Partiality *}
-
-text {* 
-  In HOL, all functions are total. A function @{term "f"} applied to
-  @{term "x"} always has the value @{term "f x"}, and there is no notion
-  of undefinedness. 
-  This is why we have to do termination
-  proofs when defining functions: The proof justifies that the
-  function can be defined by wellfounded recursion.
-
-  However, the \cmd{function} package does support partiality to a
-  certain extent. Let's look at the following function which looks
-  for a zero of a given function f. 
-*}
-
-function (*<*)(domintros)(*>*)findzero :: "(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat"
-where
-  "findzero f n = (if f n = 0 then n else findzero f (Suc n))"
-by pat_completeness auto
-
-text {*
-  \noindent Clearly, any attempt of a termination proof must fail. And without
-  that, we do not get the usual rules @{text "findzero.simps"} and 
-  @{text "findzero.induct"}. So what was the definition good for at all?
-*}
-
-subsection {* Domain predicates *}
-
-text {*
-  The trick is that Isabelle has not only defined the function @{const findzero}, but also
-  a predicate @{term "findzero_dom"} that characterizes the values where the function
-  terminates: the \emph{domain} of the function. If we treat a
-  partial function just as a total function with an additional domain
-  predicate, we can derive simplification and
-  induction rules as we do for total functions. They are guarded
-  by domain conditions and are called @{text psimps} and @{text
-  pinduct}: 
-*}
-
-text {*
-  \noindent\begin{minipage}{0.79\textwidth}@{thm[display,margin=85] findzero.psimps}\end{minipage}
-  \hfill(@{text "findzero.psimps"})
-  \vspace{1em}
-
-  \noindent\begin{minipage}{0.79\textwidth}@{thm[display,margin=85] findzero.pinduct}\end{minipage}
-  \hfill(@{text "findzero.pinduct"})
-*}
-
-text {*
-  Remember that all we
-  are doing here is use some tricks to make a total function appear
-  as if it was partial. We can still write the term @{term "findzero
-  (\<lambda>x. 1) 0"} and like any other term of type @{typ nat} it is equal
-  to some natural number, although we might not be able to find out
-  which one. The function is \emph{underdefined}.
-
-  But it is defined enough to prove something interesting about it. We
-  can prove that if @{term "findzero f n"}
-  terminates, it indeed returns a zero of @{term f}:
-*}
-
-lemma findzero_zero: "findzero_dom (f, n) \<Longrightarrow> f (findzero f n) = 0"
-
-txt {* \noindent We apply induction as usual, but using the partial induction
-  rule: *}
-
-apply (induct f n rule: findzero.pinduct)
-
-txt {* \noindent This gives the following subgoals:
-
-  @{subgoals[display,indent=0]}
-
-  \noindent The hypothesis in our lemma was used to satisfy the first premise in
-  the induction rule. However, we also get @{term
-  "findzero_dom (f, n)"} as a local assumption in the induction step. This
-  allows unfolding @{term "findzero f n"} using the @{text psimps}
-  rule, and the rest is trivial.
- *}
-apply (simp add: findzero.psimps)
-done
-
-text {*
-  Proofs about partial functions are often not harder than for total
-  functions. Fig.~\ref{findzero_isar} shows a slightly more
-  complicated proof written in Isar. It is verbose enough to show how
-  partiality comes into play: From the partial induction, we get an
-  additional domain condition hypothesis. Observe how this condition
-  is applied when calls to @{term findzero} are unfolded.
-*}
-
-text_raw {*
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-*}
-lemma "\<lbrakk>findzero_dom (f, n); x \<in> {n ..< findzero f n}\<rbrakk> \<Longrightarrow> f x \<noteq> 0"
-proof (induct rule: findzero.pinduct)
-  fix f n assume dom: "findzero_dom (f, n)"
-               and IH: "\<lbrakk>f n \<noteq> 0; x \<in> {Suc n ..< findzero f (Suc n)}\<rbrakk> \<Longrightarrow> f x \<noteq> 0"
-               and x_range: "x \<in> {n ..< findzero f n}"
-  have "f n \<noteq> 0"
-  proof 
-    assume "f n = 0"
-    with dom have "findzero f n = n" by (simp add: findzero.psimps)
-    with x_range show False by auto
-  qed
-  
-  from x_range have "x = n \<or> x \<in> {Suc n ..< findzero f n}" by auto
-  thus "f x \<noteq> 0"
-  proof
-    assume "x = n"
-    with `f n \<noteq> 0` show ?thesis by simp
-  next
-    assume "x \<in> {Suc n ..< findzero f n}"
-    with dom and `f n \<noteq> 0` have "x \<in> {Suc n ..< findzero f (Suc n)}" by (simp add: findzero.psimps)
-    with IH and `f n \<noteq> 0`
-    show ?thesis by simp
-  qed
-qed
-text_raw {*
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}\vspace{6pt}\hrule
-\caption{A proof about a partial function}\label{findzero_isar}
-\end{figure}
-*}
-
-subsection {* Partial termination proofs *}
-
-text {*
-  Now that we have proved some interesting properties about our
-  function, we should turn to the domain predicate and see if it is
-  actually true for some values. Otherwise we would have just proved
-  lemmas with @{term False} as a premise.
-
-  Essentially, we need some introduction rules for @{text
-  findzero_dom}. The function package can prove such domain
-  introduction rules automatically. But since they are not used very
-  often (they are almost never needed if the function is total), this
-  functionality is disabled by default for efficiency reasons. So we have to go
-  back and ask for them explicitly by passing the @{text
-  "(domintros)"} option to the function package:
-
-\vspace{1ex}
-\noindent\cmd{function} @{text "(domintros) findzero :: \"(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat\""}\\%
-\cmd{where}\isanewline%
-\ \ \ldots\\
-
-  \noindent Now the package has proved an introduction rule for @{text findzero_dom}:
-*}
-
-thm findzero.domintros
-
-text {*
-  @{thm[display] findzero.domintros}
-
-  Domain introduction rules allow to show that a given value lies in the
-  domain of a function, if the arguments of all recursive calls
-  are in the domain as well. They allow to do a \qt{single step} in a
-  termination proof. Usually, you want to combine them with a suitable
-  induction principle.
-
-  Since our function increases its argument at recursive calls, we
-  need an induction principle which works \qt{backwards}. We will use
-  @{text inc_induct}, which allows to do induction from a fixed number
-  \qt{downwards}:
-
-  \begin{center}@{thm inc_induct}\hfill(@{text "inc_induct"})\end{center}
-
-  Figure \ref{findzero_term} gives a detailed Isar proof of the fact
-  that @{text findzero} terminates if there is a zero which is greater
-  or equal to @{term n}. First we derive two useful rules which will
-  solve the base case and the step case of the induction. The
-  induction is then straightforward, except for the unusual induction
-  principle.
-
-*}
-
-text_raw {*
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-*}
-lemma findzero_termination:
-  assumes "x \<ge> n" and "f x = 0"
-  shows "findzero_dom (f, n)"
-proof - 
-  have base: "findzero_dom (f, x)"
-    by (rule findzero.domintros) (simp add:`f x = 0`)
-
-  have step: "\<And>i. findzero_dom (f, Suc i) 
-    \<Longrightarrow> findzero_dom (f, i)"
-    by (rule findzero.domintros) simp
-
-  from `x \<ge> n` show ?thesis
-  proof (induct rule:inc_induct)
-    show "findzero_dom (f, x)" by (rule base)
-  next
-    fix i assume "findzero_dom (f, Suc i)"
-    thus "findzero_dom (f, i)" by (rule step)
-  qed
-qed      
-text_raw {*
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}\vspace{6pt}\hrule
-\caption{Termination proof for @{text findzero}}\label{findzero_term}
-\end{figure}
-*}
-      
-text {*
-  Again, the proof given in Fig.~\ref{findzero_term} has a lot of
-  detail in order to explain the principles. Using more automation, we
-  can also have a short proof:
-*}
-
-lemma findzero_termination_short:
-  assumes zero: "x >= n" 
-  assumes [simp]: "f x = 0"
-  shows "findzero_dom (f, n)"
-using zero
-by (induct rule:inc_induct) (auto intro: findzero.domintros)
-    
-text {*
-  \noindent It is simple to combine the partial correctness result with the
-  termination lemma:
-*}
-
-lemma findzero_total_correctness:
-  "f x = 0 \<Longrightarrow> f (findzero f 0) = 0"
-by (blast intro: findzero_zero findzero_termination)
-
-subsection {* Definition of the domain predicate *}
-
-text {*
-  Sometimes it is useful to know what the definition of the domain
-  predicate looks like. Actually, @{text findzero_dom} is just an
-  abbreviation:
-
-  @{abbrev[display] findzero_dom}
-
-  The domain predicate is the \emph{accessible part} of a relation @{const
-  findzero_rel}, which was also created internally by the function
-  package. @{const findzero_rel} is just a normal
-  inductive predicate, so we can inspect its definition by
-  looking at the introduction rules @{text findzero_rel.intros}.
-  In our case there is just a single rule:
-
-  @{thm[display] findzero_rel.intros}
-
-  The predicate @{const findzero_rel}
-  describes the \emph{recursion relation} of the function
-  definition. The recursion relation is a binary relation on
-  the arguments of the function that relates each argument to its
-  recursive calls. In general, there is one introduction rule for each
-  recursive call.
-
-  The predicate @{term "accp findzero_rel"} is the accessible part of
-  that relation. An argument belongs to the accessible part, if it can
-  be reached in a finite number of steps (cf.~its definition in @{text
-  "Wellfounded.thy"}).
-
-  Since the domain predicate is just an abbreviation, you can use
-  lemmas for @{const accp} and @{const findzero_rel} directly. Some
-  lemmas which are occasionally useful are @{text accpI}, @{text
-  accp_downward}, and of course the introduction and elimination rules
-  for the recursion relation @{text "findzero.intros"} and @{text "findzero.cases"}.
-*}
-
-section {* Nested recursion *}
-
-text {*
-  Recursive calls which are nested in one another frequently cause
-  complications, since their termination proof can depend on a partial
-  correctness property of the function itself. 
-
-  As a small example, we define the \qt{nested zero} function:
-*}
-
-function nz :: "nat \<Rightarrow> nat"
-where
-  "nz 0 = 0"
-| "nz (Suc n) = nz (nz n)"
-by pat_completeness auto
-
-text {*
-  If we attempt to prove termination using the identity measure on
-  naturals, this fails:
-*}
-
-termination
-  apply (relation "measure (\<lambda>n. n)")
-  apply auto
-
-txt {*
-  We get stuck with the subgoal
-
-  @{subgoals[display]}
-
-  Of course this statement is true, since we know that @{const nz} is
-  the zero function. And in fact we have no problem proving this
-  property by induction.
-*}
-(*<*)oops(*>*)
-lemma nz_is_zero: "nz_dom n \<Longrightarrow> nz n = 0"
-  by (induct rule:nz.pinduct) (auto simp: nz.psimps)
-
-text {*
-  We formulate this as a partial correctness lemma with the condition
-  @{term "nz_dom n"}. This allows us to prove it with the @{text
-  pinduct} rule before we have proved termination. With this lemma,
-  the termination proof works as expected:
-*}
-
-termination
-  by (relation "measure (\<lambda>n. n)") (auto simp: nz_is_zero)
-
-text {*
-  As a general strategy, one should prove the statements needed for
-  termination as a partial property first. Then they can be used to do
-  the termination proof. This also works for less trivial
-  examples. Figure \ref{f91} defines the 91-function, a well-known
-  challenge problem due to John McCarthy, and proves its termination.
-*}
-
-text_raw {*
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-*}
-
-function f91 :: "nat \<Rightarrow> nat"
-where
-  "f91 n = (if 100 < n then n - 10 else f91 (f91 (n + 11)))"
-by pat_completeness auto
-
-lemma f91_estimate: 
-  assumes trm: "f91_dom n" 
-  shows "n < f91 n + 11"
-using trm by induct (auto simp: f91.psimps)
-
-termination
-proof
-  let ?R = "measure (\<lambda>x. 101 - x)"
-  show "wf ?R" ..
-
-  fix n :: nat assume "\<not> 100 < n" -- "Assumptions for both calls"
-
-  thus "(n + 11, n) \<in> ?R" by simp -- "Inner call"
-
-  assume inner_trm: "f91_dom (n + 11)" -- "Outer call"
-  with f91_estimate have "n + 11 < f91 (n + 11) + 11" .
-  with `\<not> 100 < n` show "(f91 (n + 11), n) \<in> ?R" by simp
-qed
-
-text_raw {*
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}
-\vspace{6pt}\hrule
-\caption{McCarthy's 91-function}\label{f91}
-\end{figure}
-*}
-
-
-section {* Higher-Order Recursion *}
-
-text {*
-  Higher-order recursion occurs when recursive calls
-  are passed as arguments to higher-order combinators such as @{const
-  map}, @{term filter} etc.
-  As an example, imagine a datatype of n-ary trees:
-*}
-
-datatype 'a tree = 
-  Leaf 'a 
-| Branch "'a tree list"
-
-
-text {* \noindent We can define a function which swaps the left and right subtrees recursively, using the 
-  list functions @{const rev} and @{const map}: *}
-
-fun mirror :: "'a tree \<Rightarrow> 'a tree"
-where
-  "mirror (Leaf n) = Leaf n"
-| "mirror (Branch l) = Branch (rev (map mirror l))"
-
-text {*
-  Although the definition is accepted without problems, let us look at the termination proof:
-*}
-
-termination proof
-  txt {*
-
-  As usual, we have to give a wellfounded relation, such that the
-  arguments of the recursive calls get smaller. But what exactly are
-  the arguments of the recursive calls when mirror is given as an
-  argument to @{const map}? Isabelle gives us the
-  subgoals
-
-  @{subgoals[display,indent=0]} 
-
-  So the system seems to know that @{const map} only
-  applies the recursive call @{term "mirror"} to elements
-  of @{term "l"}, which is essential for the termination proof.
-
-  This knowledge about @{const map} is encoded in so-called congruence rules,
-  which are special theorems known to the \cmd{function} command. The
-  rule for @{const map} is
-
-  @{thm[display] map_cong}
-
-  You can read this in the following way: Two applications of @{const
-  map} are equal, if the list arguments are equal and the functions
-  coincide on the elements of the list. This means that for the value 
-  @{term "map f l"} we only have to know how @{term f} behaves on
-  the elements of @{term l}.
-
-  Usually, one such congruence rule is
-  needed for each higher-order construct that is used when defining
-  new functions. In fact, even basic functions like @{const
-  If} and @{const Let} are handled by this mechanism. The congruence
-  rule for @{const If} states that the @{text then} branch is only
-  relevant if the condition is true, and the @{text else} branch only if it
-  is false:
-
-  @{thm[display] if_cong}
-  
-  Congruence rules can be added to the
-  function package by giving them the @{term fundef_cong} attribute.
-
-  The constructs that are predefined in Isabelle, usually
-  come with the respective congruence rules.
-  But if you define your own higher-order functions, you may have to
-  state and prove the required congruence rules yourself, if you want to use your
-  functions in recursive definitions. 
-*}
-(*<*)oops(*>*)
-
-subsection {* Congruence Rules and Evaluation Order *}
-
-text {* 
-  Higher order logic differs from functional programming languages in
-  that it has no built-in notion of evaluation order. A program is
-  just a set of equations, and it is not specified how they must be
-  evaluated. 
-
-  However for the purpose of function definition, we must talk about
-  evaluation order implicitly, when we reason about termination.
-  Congruence rules express that a certain evaluation order is
-  consistent with the logical definition. 
-
-  Consider the following function.
-*}
-
-function f :: "nat \<Rightarrow> bool"
-where
-  "f n = (n = 0 \<or> f (n - 1))"
-(*<*)by pat_completeness auto(*>*)
-
-text {*
-  For this definition, the termination proof fails. The default configuration
-  specifies no congruence rule for disjunction. We have to add a
-  congruence rule that specifies left-to-right evaluation order:
-
-  \vspace{1ex}
-  \noindent @{thm disj_cong}\hfill(@{text "disj_cong"})
-  \vspace{1ex}
-
-  Now the definition works without problems. Note how the termination
-  proof depends on the extra condition that we get from the congruence
-  rule.
-
-  However, as evaluation is not a hard-wired concept, we
-  could just turn everything around by declaring a different
-  congruence rule. Then we can make the reverse definition:
-*}
-
-lemma disj_cong2[fundef_cong]: 
-  "(\<not> Q' \<Longrightarrow> P = P') \<Longrightarrow> (Q = Q') \<Longrightarrow> (P \<or> Q) = (P' \<or> Q')"
-  by blast
-
-fun f' :: "nat \<Rightarrow> bool"
-where
-  "f' n = (f' (n - 1) \<or> n = 0)"
-
-text {*
-  \noindent These examples show that, in general, there is no \qt{best} set of
-  congruence rules.
-
-  However, such tweaking should rarely be necessary in
-  practice, as most of the time, the default set of congruence rules
-  works well.
-*}
-
-end
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/Thy/document/Functions.tex
--- a/doc-src/Functions/Thy/document/Functions.tex	Mon Aug 27 22:00:04 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1920 +0,0 @@
-%
-\begin{isabellebody}%
-\def\isabellecontext{Functions}%
-%
-\isadelimtheory
-\isanewline
-\isanewline
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{theory}\isamarkupfalse%
-\ Functions\isanewline
-\isakeyword{imports}\ Main\isanewline
-\isakeyword{begin}%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isamarkupsection{Function Definitions for Dummies%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In most cases, defining a recursive function is just as simple as other definitions:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ fib\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}fib\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}fib\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}fib\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ fib\ n\ {\isaliteral{2B}{\isacharplus}}\ fib\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-The syntax is rather self-explanatory: We introduce a function by
-  giving its name, its type, 
-  and a set of defining recursive equations.
-  If we leave out the type, the most general type will be
-  inferred, which can sometimes lead to surprises: Since both \isa{{\isadigit{1}}} and \isa{{\isaliteral{2B}{\isacharplus}}} are overloaded, we would end up
-  with \isa{fib\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{7B}{\isacharbraceleft}}one{\isaliteral{2C}{\isacharcomma}}plus{\isaliteral{7D}{\isacharbraceright}}}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The function always terminates, since its argument gets smaller in
-  every recursive call. 
-  Since HOL is a logic of total functions, termination is a
-  fundamental requirement to prevent inconsistencies\footnote{From the
-  \qt{definition} \isa{f{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}} we could prove 
-  \isa{{\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}} by subtracting \isa{f{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{29}{\isacharparenright}}} on both sides.}.
-  Isabelle tries to prove termination automatically when a definition
-  is made. In \S\ref{termination}, we will look at cases where this
-  fails and see what to do then.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Pattern matching%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{patmatch}
-  Like in functional programming, we can use pattern matching to
-  define functions. At the moment we will only consider \emph{constructor
-  patterns}, which only consist of datatype constructors and
-  variables. Furthermore, patterns must be linear, i.e.\ all variables
-  on the left hand side of an equation must be distinct. In
-  \S\ref{genpats} we discuss more general pattern matching.
-
-  If patterns overlap, the order of the equations is taken into
-  account. The following function inserts a fixed element between any
-  two elements of a list:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ sep\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ list{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}sep\ a\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{23}{\isacharhash}}y{\isaliteral{23}{\isacharhash}}xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{23}{\isacharhash}}\ a\ {\isaliteral{23}{\isacharhash}}\ sep\ a\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}sep\ a\ xs\ \ \ \ \ \ \ {\isaliteral{3D}{\isacharequal}}\ xs{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-Overlapping patterns are interpreted as \qt{increments} to what is
-  already there: The second equation is only meant for the cases where
-  the first one does not match. Consequently, Isabelle replaces it
-  internally by the remaining cases, making the patterns disjoint:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ sep{\isaliteral{2E}{\isachardot}}simps%
-\begin{isamarkuptext}%
-\begin{isabelle}%
-sep\ a\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ y\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ x\ {\isaliteral{23}{\isacharhash}}\ a\ {\isaliteral{23}{\isacharhash}}\ sep\ a\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\isasep\isanewline%
-sep\ a\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\isasep\isanewline%
-sep\ a\ {\isaliteral{5B}{\isacharbrackleft}}v{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}v{\isaliteral{5D}{\isacharbrackright}}%
-\end{isabelle}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent The equations from function definitions are automatically used in
-  simplification:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}sep\ {\isadigit{0}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{2}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{3}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{2}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{3}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Induction%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Isabelle provides customized induction rules for recursive
-  functions. These rules follow the recursive structure of the
-  definition. Here is the rule \isa{sep{\isaliteral{2E}{\isachardot}}induct} arising from the
-  above definition of \isa{sep}:
-
-  \begin{isabelle}%
-{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{5C3C416E643E}{\isasymAnd}}a\ x\ y\ xs{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}P\ a\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}P\ a\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ y\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}P\ a\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ v{\isaliteral{2E}{\isachardot}}\ {\isaliteral{3F}{\isacharquery}}P\ a\ {\isaliteral{5B}{\isacharbrackleft}}v{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
-{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{3F}{\isacharquery}}a{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isadigit{0}}\ {\isaliteral{3F}{\isacharquery}}a{\isadigit{1}}{\isaliteral{2E}{\isachardot}}{\isadigit{0}}%
-\end{isabelle}
-  
-  We have a step case for list with at least two elements, and two
-  base cases for the zero- and the one-element list. Here is a simple
-  proof about \isa{sep} and \isa{map}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}map\ f\ {\isaliteral{28}{\isacharparenleft}}sep\ x\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ sep\ {\isaliteral{28}{\isacharparenleft}}f\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}map\ f\ ys{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}induct\ x\ ys\ rule{\isaliteral{3A}{\isacharcolon}}\ sep{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-We get three cases, like in the definition.
-
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ x\ y\ xs{\isaliteral{2E}{\isachardot}}\isanewline
-\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }map\ f\ {\isaliteral{28}{\isacharparenleft}}sep\ a\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ sep\ {\isaliteral{28}{\isacharparenleft}}f\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}map\ f\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
-\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ \ \ \ }map\ f\ {\isaliteral{28}{\isacharparenleft}}sep\ a\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ y\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ sep\ {\isaliteral{28}{\isacharparenleft}}f\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}map\ f\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{23}{\isacharhash}}\ y\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a{\isaliteral{2E}{\isachardot}}\ map\ f\ {\isaliteral{28}{\isacharparenleft}}sep\ a\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ sep\ {\isaliteral{28}{\isacharparenleft}}f\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}map\ f\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{3}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}a\ v{\isaliteral{2E}{\isachardot}}\ map\ f\ {\isaliteral{28}{\isacharparenleft}}sep\ a\ {\isaliteral{5B}{\isacharbrackleft}}v{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ sep\ {\isaliteral{28}{\isacharparenleft}}f\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}map\ f\ {\isaliteral{5B}{\isacharbrackleft}}v{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ auto\ \isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-With the \cmd{fun} command, you can define about 80\% of the
-  functions that occur in practice. The rest of this tutorial explains
-  the remaining 20\%.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{fun vs.\ function%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The \cmd{fun} command provides a
-  convenient shorthand notation for simple function definitions. In
-  this mode, Isabelle tries to solve all the necessary proof obligations
-  automatically. If any proof fails, the definition is
-  rejected. This can either mean that the definition is indeed faulty,
-  or that the default proof procedures are just not smart enough (or
-  rather: not designed) to handle the definition.
-
-  By expanding the abbreviation to the more verbose \cmd{function} command, these proof obligations become visible and can be analyzed or
-  solved manually. The expansion from \cmd{fun} to \cmd{function} is as follows:
-
-\end{isamarkuptext}
-
-
-\[\left[\;\begin{minipage}{0.25\textwidth}\vspace{6pt}
-\cmd{fun} \isa{f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}}\\%
-\cmd{where}\\%
-\hspace*{2ex}{\it equations}\\%
-\hspace*{2ex}\vdots\vspace*{6pt}
-\end{minipage}\right]
-\quad\equiv\quad
-\left[\;\begin{minipage}{0.48\textwidth}\vspace{6pt}
-\cmd{function} \isa{{\isaliteral{28}{\isacharparenleft}}}\cmd{sequential}\isa{{\isaliteral{29}{\isacharparenright}}\ f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}}\\%
-\cmd{where}\\%
-\hspace*{2ex}{\it equations}\\%
-\hspace*{2ex}\vdots\\%
-\cmd{by} \isa{pat{\isaliteral{5F}{\isacharunderscore}}completeness\ auto}\\%
-\cmd{termination by} \isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order}\vspace{6pt}
-\end{minipage}
-\right]\]
-
-\begin{isamarkuptext}
-  \vspace*{1em}
-  \noindent Some details have now become explicit:
-
-  \begin{enumerate}
-  \item The \cmd{sequential} option enables the preprocessing of
-  pattern overlaps which we already saw. Without this option, the equations
-  must already be disjoint and complete. The automatic completion only
-  works with constructor patterns.
-
-  \item A function definition produces a proof obligation which
-  expresses completeness and compatibility of patterns (we talk about
-  this later). The combination of the methods \isa{pat{\isaliteral{5F}{\isacharunderscore}}completeness} and
-  \isa{auto} is used to solve this proof obligation.
-
-  \item A termination proof follows the definition, started by the
-  \cmd{termination} command. This will be explained in \S\ref{termination}.
- \end{enumerate}
-  Whenever a \cmd{fun} command fails, it is usually a good idea to
-  expand the syntax to the more verbose \cmd{function} form, to see
-  what is actually going on.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Termination%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{termination}
-  The method \isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order} is the default method for
-  termination proofs. It can prove termination of a
-  certain class of functions by searching for a suitable lexicographic
-  combination of size measures. Of course, not all functions have such
-  a simple termination argument. For them, we can specify the termination
-  relation manually.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{The {\tt relation} method%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Consider the following function, which sums up natural numbers up to
-  \isa{N}, using a counter \isa{i}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ sum\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}sum\ i\ N\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ i\ {\isaliteral{3E}{\isachargreater}}\ N\ then\ {\isadigit{0}}\ else\ i\ {\isaliteral{2B}{\isacharplus}}\ sum\ {\isaliteral{28}{\isacharparenleft}}Suc\ i{\isaliteral{29}{\isacharparenright}}\ N{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isaliteral{5F}{\isacharunderscore}}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent The \isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order} method fails on this example, because none of the
-  arguments decreases in the recursive call, with respect to the standard size ordering.
-  To prove termination manually, we must provide a custom wellfounded relation.
-
-  The termination argument for \isa{sum} is based on the fact that
-  the \emph{difference} between \isa{i} and \isa{N} gets
-  smaller in every step, and that the recursion stops when \isa{i}
-  is greater than \isa{N}. Phrased differently, the expression 
-  \isa{N\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}\ {\isaliteral{2D}{\isacharminus}}\ i} always decreases.
-
-  We can use this expression as a measure function suitable to prove termination.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ sum\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}relation\ {\isaliteral{22}{\isachardoublequoteopen}}measure\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}i{\isaliteral{2C}{\isacharcomma}}N{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ N\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}\ {\isaliteral{2D}{\isacharminus}}\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-The \cmd{termination} command sets up the termination goal for the
-  specified function \isa{sum}. If the function name is omitted, it
-  implicitly refers to the last function definition.
-
-  The \isa{relation} method takes a relation of
-  type \isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ set}, where \isa{{\isaliteral{27}{\isacharprime}}a} is the argument type of
-  the function. If the function has multiple curried arguments, then
-  these are packed together into a tuple, as it happened in the above
-  example.
-
-  The predefined function \isa{{\isaliteral{22}{\isachardoublequote}}measure\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ set{\isaliteral{22}{\isachardoublequote}}} constructs a
-  wellfounded relation from a mapping into the natural numbers (a
-  \emph{measure function}). 
-
-  After the invocation of \isa{relation}, we must prove that (a)
-  the relation we supplied is wellfounded, and (b) that the arguments
-  of recursive calls indeed decrease with respect to the
-  relation:
-
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ wf\ {\isaliteral{28}{\isacharparenleft}}measure\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}i{\isaliteral{2C}{\isacharcomma}}\ N{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ N\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}\ {\isaliteral{2D}{\isacharminus}}\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}i\ N{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ N\ {\isaliteral{3C}{\isacharless}}\ i\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Suc\ i{\isaliteral{2C}{\isacharcomma}}\ N{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ i{\isaliteral{2C}{\isacharcomma}}\ N{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ measure\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}i{\isaliteral{2C}{\isacharcomma}}\ N{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ N\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}\ {\isaliteral{2D}{\isacharminus}}\ i{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-
-  These goals are all solved by \isa{auto}:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Let us complicate the function a little, by adding some more
-  recursive calls:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ foo\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}foo\ i\ N\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ i\ {\isaliteral{3E}{\isachargreater}}\ N\ \isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ then\ {\isaliteral{28}{\isacharparenleft}}if\ N\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ then\ {\isadigit{0}}\ else\ foo\ {\isadigit{0}}\ {\isaliteral{28}{\isacharparenleft}}N\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ else\ i\ {\isaliteral{2B}{\isacharplus}}\ foo\ {\isaliteral{28}{\isacharparenleft}}Suc\ i{\isaliteral{29}{\isacharparenright}}\ N{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isaliteral{5F}{\isacharunderscore}}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-When \isa{i} has reached \isa{N}, it starts at zero again
-  and \isa{N} is decremented.
-  This corresponds to a nested
-  loop where one index counts up and the other down. Termination can
-  be proved using a lexicographic combination of two measures, namely
-  the value of \isa{N} and the above difference. The \isa{measures} combinator generalizes \isa{measure} by taking a
-  list of measure functions.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ \isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}relation\ {\isaliteral{22}{\isachardoublequoteopen}}measures\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}i{\isaliteral{2C}{\isacharcomma}}\ N{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ N{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{28}{\isacharparenleft}}i{\isaliteral{2C}{\isacharcomma}}N{\isaliteral{29}{\isacharparenright}}{\isaliteral{2E}{\isachardot}}\ N\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}\ {\isaliteral{2D}{\isacharminus}}\ i{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{How \isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order} works%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-To see how the automatic termination proofs work, let's look at an
-  example where it fails\footnote{For a detailed discussion of the
-  termination prover, see \cite{bulwahnKN07}}:
-
-\end{isamarkuptext}  
-\cmd{fun} \isa{fails\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequote}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequote}}}\\%
-\cmd{where}\\%
-\hspace*{2ex}\isa{{\isaliteral{22}{\isachardoublequote}}fails\ a\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{22}{\isachardoublequote}}}\\%
-|\hspace*{1.5ex}\isa{{\isaliteral{22}{\isachardoublequote}}fails\ a\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{23}{\isacharhash}}xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ fails\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{2B}{\isacharplus}}\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{23}{\isacharhash}}xs{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}\\
-\begin{isamarkuptext}
-
-\noindent Isabelle responds with the following error:
-
-\begin{isabelle}
-*** Unfinished subgoals:\newline
-*** (a, 1, <):\newline
-*** \ 1.~\isa{{\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}}\newline
-*** (a, 1, <=):\newline
-*** \ 1.~False\newline
-*** (a, 2, <):\newline
-*** \ 1.~False\newline
-*** Calls:\newline
-*** a) \isa{{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ x\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2D}{\isacharminus}}{\isaliteral{2D}{\isacharminus}}{\isaliteral{3E}{\isachargreater}}{\isaliteral{3E}{\isachargreater}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{2B}{\isacharplus}}\ a{\isaliteral{2C}{\isacharcomma}}\ x\ {\isaliteral{23}{\isacharhash}}\ xs{\isaliteral{29}{\isacharparenright}}}\newline
-*** Measures:\newline
-*** 1) \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ size\ {\isaliteral{28}{\isacharparenleft}}fst\ x{\isaliteral{29}{\isacharparenright}}}\newline
-*** 2) \isa{{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ size\ {\isaliteral{28}{\isacharparenleft}}snd\ x{\isaliteral{29}{\isacharparenright}}}\newline
-*** Result matrix:\newline
-*** \ \ \ \ 1\ \ 2  \newline
-*** a:  ?   <= \newline
-*** Could not find lexicographic termination order.\newline
-*** At command "fun".\newline
-\end{isabelle}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The key to this error message is the matrix at the bottom. The rows
-  of that matrix correspond to the different recursive calls (In our
-  case, there is just one). The columns are the function's arguments 
-  (expressed through different measure functions, which map the
-  argument tuple to a natural number). 
-
-  The contents of the matrix summarize what is known about argument
-  descents: The second argument has a weak descent (\isa{{\isaliteral{3C}{\isacharless}}{\isaliteral{3D}{\isacharequal}}}) at the
-  recursive call, and for the first argument nothing could be proved,
-  which is expressed by \isa{{\isaliteral{3F}{\isacharquery}}}. In general, there are the values
-  \isa{{\isaliteral{3C}{\isacharless}}}, \isa{{\isaliteral{3C}{\isacharless}}{\isaliteral{3D}{\isacharequal}}} and \isa{{\isaliteral{3F}{\isacharquery}}}.
-
-  For the failed proof attempts, the unfinished subgoals are also
-  printed. Looking at these will often point to a missing lemma.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{The \isa{size{\isaliteral{5F}{\isacharunderscore}}change} method%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Some termination goals that are beyond the powers of
-  \isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order} can be solved automatically by the
-  more powerful \isa{size{\isaliteral{5F}{\isacharunderscore}}change} method, which uses a variant of
-  the size-change principle, together with some other
-  techniques. While the details are discussed
-  elsewhere\cite{krauss_phd},
-  here are a few typical situations where
-  \isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order} has difficulties and \isa{size{\isaliteral{5F}{\isacharunderscore}}change}
-  may be worth a try:
-  \begin{itemize}
-  \item Arguments are permuted in a recursive call.
-  \item Several mutually recursive functions with multiple arguments.
-  \item Unusual control flow (e.g., when some recursive calls cannot
-  occur in sequence).
-  \end{itemize}
-
-  Loading the theory \isa{Multiset} makes the \isa{size{\isaliteral{5F}{\isacharunderscore}}change}
-  method a bit stronger: it can then use multiset orders internally.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Mutual Recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-If two or more functions call one another mutually, they have to be defined
-  in one step. Here are \isa{even} and \isa{odd}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ even\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isakeyword{and}\ odd\ \ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}even\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ True{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}odd\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ False{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}even\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ odd\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}odd\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ even\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isaliteral{5F}{\isacharunderscore}}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-To eliminate the mutual dependencies, Isabelle internally
-  creates a single function operating on the sum
-  type \isa{nat\ {\isaliteral{2B}{\isacharplus}}\ nat}. Then, \isa{even} and \isa{odd} are
-  defined as projections. Consequently, termination has to be proved
-  simultaneously for both functions, by specifying a measure on the
-  sum type:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\ \isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}relation\ {\isaliteral{22}{\isachardoublequoteopen}}measure\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ case\ x\ of\ Inl\ n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ n\ {\isaliteral{7C}{\isacharbar}}\ Inr\ n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We could also have used \isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order}, which
-  supports mutual recursive termination proofs to a certain extent.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Induction for mutual recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-When functions are mutually recursive, proving properties about them
-  generally requires simultaneous induction. The induction rule \isa{even{\isaliteral{5F}{\isacharunderscore}}odd{\isaliteral{2E}{\isachardot}}induct}
-  generated from the above definition reflects this.
-
-  Let us prove something about \isa{even} and \isa{odd}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ even{\isaliteral{5F}{\isacharunderscore}}odd{\isaliteral{5F}{\isacharunderscore}}mod{\isadigit{2}}{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}even\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}odd\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-We apply simultaneous induction, specifying the induction variable
-  for both goals, separated by \cmd{and}:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}induct\ n\ \isakeyword{and}\ n\ rule{\isaliteral{3A}{\isacharcolon}}\ even{\isaliteral{5F}{\isacharunderscore}}odd{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-We get four subgoals, which correspond to the clauses in the
-  definition of \isa{even} and \isa{odd}:
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ even\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ odd\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{3}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ odd\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ even\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{4}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ even\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ odd\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-  Simplification solves the first two goals, leaving us with two
-  statements about the \isa{mod} operation to prove:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ simp{\isaliteral{5F}{\isacharunderscore}}all%
-\begin{isamarkuptxt}%
-\begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ odd\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ even\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle} 
-
-  \noindent These can be handled by Isabelle's arithmetic decision procedures.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ arith\isanewline
-\isacommand{apply}\isamarkupfalse%
-\ arith\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-In proofs like this, the simultaneous induction is really essential:
-  Even if we are just interested in one of the results, the other
-  one is necessary to strengthen the induction hypothesis. If we leave
-  out the statement about \isa{odd} and just write \isa{True} instead,
-  the same proof fails:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ failed{\isaliteral{5F}{\isacharunderscore}}attempt{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}even\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}True{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}induct\ n\ rule{\isaliteral{3A}{\isacharcolon}}\ even{\isaliteral{5F}{\isacharunderscore}}odd{\isaliteral{2E}{\isachardot}}induct{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-\noindent Now the third subgoal is a dead end, since we have no
-  useful induction hypothesis available:
-
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ even\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ True\isanewline
-\ {\isadigit{3}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ True\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ even\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ {\isadigit{4}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ even\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ mod\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ True%
-\end{isabelle}%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{oops}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsection{General pattern matching%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\label{genpats}%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Avoiding automatic pattern splitting%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Up to now, we used pattern matching only on datatypes, and the
-  patterns were always disjoint and complete, and if they weren't,
-  they were made disjoint automatically like in the definition of
-  \isa{sep} in \S\ref{patmatch}.
-
-  This automatic splitting can significantly increase the number of
-  equations involved, and this is not always desirable. The following
-  example shows the problem:
-  
-  Suppose we are modeling incomplete knowledge about the world by a
-  three-valued datatype, which has values \isa{T}, \isa{F}
-  and \isa{X} for true, false and uncertain propositions, respectively.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ P{\isadigit{3}}\ {\isaliteral{3D}{\isacharequal}}\ T\ {\isaliteral{7C}{\isacharbar}}\ F\ {\isaliteral{7C}{\isacharbar}}\ X%
-\begin{isamarkuptext}%
-\noindent Then the conjunction of such values can be defined as follows:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{fun}\isamarkupfalse%
-\ And\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isadigit{3}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ P{\isadigit{3}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ P{\isadigit{3}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}And\ T\ p\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}And\ p\ T\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}And\ p\ F\ {\isaliteral{3D}{\isacharequal}}\ F{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}And\ F\ p\ {\isaliteral{3D}{\isacharequal}}\ F{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}And\ X\ X\ {\isaliteral{3D}{\isacharequal}}\ X{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-This definition is useful, because the equations can directly be used
-  as simplification rules. But the patterns overlap: For example,
-  the expression \isa{And\ T\ T} is matched by both the first and
-  the second equation. By default, Isabelle makes the patterns disjoint by
-  splitting them up, producing instances:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ And{\isaliteral{2E}{\isachardot}}simps%
-\begin{isamarkuptext}%
-\isa{And\ T\ {\isaliteral{3F}{\isacharquery}}p\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}p\isasep\isanewline%
-And\ F\ T\ {\isaliteral{3D}{\isacharequal}}\ F\isasep\isanewline%
-And\ X\ T\ {\isaliteral{3D}{\isacharequal}}\ X\isasep\isanewline%
-And\ F\ F\ {\isaliteral{3D}{\isacharequal}}\ F\isasep\isanewline%
-And\ X\ F\ {\isaliteral{3D}{\isacharequal}}\ F\isasep\isanewline%
-And\ F\ X\ {\isaliteral{3D}{\isacharequal}}\ F\isasep\isanewline%
-And\ X\ X\ {\isaliteral{3D}{\isacharequal}}\ X}
-  
-  \vspace*{1em}
-  \noindent There are several problems with this:
-
-  \begin{enumerate}
-  \item If the datatype has many constructors, there can be an
-  explosion of equations. For \isa{And}, we get seven instead of
-  five equations, which can be tolerated, but this is just a small
-  example.
-
-  \item Since splitting makes the equations \qt{less general}, they
-  do not always match in rewriting. While the term \isa{And\ x\ F}
-  can be simplified to \isa{F} with the original equations, a
-  (manual) case split on \isa{x} is now necessary.
-
-  \item The splitting also concerns the induction rule \isa{And{\isaliteral{2E}{\isachardot}}induct}. Instead of five premises it now has seven, which
-  means that our induction proofs will have more cases.
-
-  \item In general, it increases clarity if we get the same definition
-  back which we put in.
-  \end{enumerate}
-
-  If we do not want the automatic splitting, we can switch it off by
-  leaving out the \cmd{sequential} option. However, we will have to
-  prove that our pattern matching is consistent\footnote{This prevents
-  us from defining something like \isa{f\ x\ {\isaliteral{3D}{\isacharequal}}\ True} and \isa{f\ x\ {\isaliteral{3D}{\isacharequal}}\ False} simultaneously.}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ And{\isadigit{2}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}P{\isadigit{3}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ P{\isadigit{3}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ P{\isadigit{3}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}And{\isadigit{2}}\ T\ p\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}And{\isadigit{2}}\ p\ T\ {\isaliteral{3D}{\isacharequal}}\ p{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}And{\isadigit{2}}\ p\ F\ {\isaliteral{3D}{\isacharequal}}\ F{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}And{\isadigit{2}}\ F\ p\ {\isaliteral{3D}{\isacharequal}}\ F{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}And{\isadigit{2}}\ X\ X\ {\isaliteral{3D}{\isacharequal}}\ X{\isaliteral{22}{\isachardoublequoteclose}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-\noindent Now let's look at the proof obligations generated by a
-  function definition. In this case, they are:
-
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}P\ x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{5C3C416E643E}{\isasymAnd}}p{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}T{\isaliteral{2C}{\isacharcomma}}\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}p{\isaliteral{2C}{\isacharcomma}}\ T{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}p{\isaliteral{2C}{\isacharcomma}}\ F{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{3B}{\isacharsemicolon}}\isanewline
-\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}P\ x{\isaliteral{2E}{\isachardot}}\ \ }{\isaliteral{5C3C416E643E}{\isasymAnd}}p{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}F{\isaliteral{2C}{\isacharcomma}}\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{3B}{\isacharsemicolon}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}X{\isaliteral{2C}{\isacharcomma}}\ X{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
-\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}P\ x{\isaliteral{2E}{\isachardot}}\ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p\ pa{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}T{\isaliteral{2C}{\isacharcomma}}\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}T{\isaliteral{2C}{\isacharcomma}}\ pa{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ p\ {\isaliteral{3D}{\isacharequal}}\ pa\isanewline
-\ {\isadigit{3}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p\ pa{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}T{\isaliteral{2C}{\isacharcomma}}\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}pa{\isaliteral{2C}{\isacharcomma}}\ T{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ p\ {\isaliteral{3D}{\isacharequal}}\ pa\isanewline
-\ {\isadigit{4}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p\ pa{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}T{\isaliteral{2C}{\isacharcomma}}\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}pa{\isaliteral{2C}{\isacharcomma}}\ F{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ p\ {\isaliteral{3D}{\isacharequal}}\ F\isanewline
-\ {\isadigit{5}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p\ pa{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}T{\isaliteral{2C}{\isacharcomma}}\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}F{\isaliteral{2C}{\isacharcomma}}\ pa{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ p\ {\isaliteral{3D}{\isacharequal}}\ F\isanewline
-\ {\isadigit{6}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}T{\isaliteral{2C}{\isacharcomma}}\ p{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}X{\isaliteral{2C}{\isacharcomma}}\ X{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ p\ {\isaliteral{3D}{\isacharequal}}\ X\isanewline
-\ {\isadigit{7}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p\ pa{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}p{\isaliteral{2C}{\isacharcomma}}\ T{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}pa{\isaliteral{2C}{\isacharcomma}}\ T{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ p\ {\isaliteral{3D}{\isacharequal}}\ pa\isanewline
-\ {\isadigit{8}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p\ pa{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}p{\isaliteral{2C}{\isacharcomma}}\ T{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}pa{\isaliteral{2C}{\isacharcomma}}\ F{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ p\ {\isaliteral{3D}{\isacharequal}}\ F\isanewline
-\ {\isadigit{9}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p\ pa{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}p{\isaliteral{2C}{\isacharcomma}}\ T{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}F{\isaliteral{2C}{\isacharcomma}}\ pa{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ p\ {\isaliteral{3D}{\isacharequal}}\ F\isanewline
-\ {\isadigit{1}}{\isadigit{0}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}p{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}p{\isaliteral{2C}{\isacharcomma}}\ T{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}X{\isaliteral{2C}{\isacharcomma}}\ X{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ p\ {\isaliteral{3D}{\isacharequal}}\ X%
-\end{isabelle}\vspace{-1.2em}\hspace{3cm}\vdots\vspace{1.2em}
-
-  The first subgoal expresses the completeness of the patterns. It has
-  the form of an elimination rule and states that every \isa{x} of
-  the function's input type must match at least one of the patterns\footnote{Completeness could
-  be equivalently stated as a disjunction of existential statements: 
-\isa{{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}p{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}T{\isaliteral{2C}{\isacharcomma}}\ p{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}p{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}p{\isaliteral{2C}{\isacharcomma}}\ T{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}p{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}p{\isaliteral{2C}{\isacharcomma}}\ F{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}p{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}F{\isaliteral{2C}{\isacharcomma}}\ p{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}X{\isaliteral{2C}{\isacharcomma}}\ X{\isaliteral{29}{\isacharparenright}}}, and you can use the method \isa{atomize{\isaliteral{5F}{\isacharunderscore}}elim} to get that form instead.}. If the patterns just involve
-  datatypes, we can solve it with the \isa{pat{\isaliteral{5F}{\isacharunderscore}}completeness}
-  method:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ pat{\isaliteral{5F}{\isacharunderscore}}completeness%
-\begin{isamarkuptxt}%
-The remaining subgoals express \emph{pattern compatibility}. We do
-  allow that an input value matches multiple patterns, but in this
-  case, the result (i.e.~the right hand sides of the equations) must
-  also be equal. For each pair of two patterns, there is one such
-  subgoal. Usually this needs injectivity of the constructors, which
-  is used automatically by \isa{auto}.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}relation\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Non-constructor patterns%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Most of Isabelle's basic types take the form of inductive datatypes,
-  and usually pattern matching works on the constructors of such types. 
-  However, this need not be always the case, and the \cmd{function}
-  command handles other kind of patterns, too.
-
-  One well-known instance of non-constructor patterns are
-  so-called \emph{$n+k$-patterns}, which are a little controversial in
-  the functional programming world. Here is the initial fibonacci
-  example with $n+k$-patterns:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ fib{\isadigit{2}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}fib{\isadigit{2}}\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}fib{\isadigit{2}}\ {\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}fib{\isadigit{2}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ fib{\isadigit{2}}\ n\ {\isaliteral{2B}{\isacharplus}}\ fib{\isadigit{2}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-This kind of matching is again justified by the proof of pattern
-  completeness and compatibility. 
-  The proof obligation for pattern completeness states that every natural number is
-  either \isa{{\isadigit{0}}}, \isa{{\isadigit{1}}} or \isa{n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}}:
-
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}P\ x{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}x\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{3B}{\isacharsemicolon}}\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P%
-\end{isabelle}
-
-  This is an arithmetic triviality, but unfortunately the
-  \isa{arith} method cannot handle this specific form of an
-  elimination rule. However, we can use the method \isa{atomize{\isaliteral{5F}{\isacharunderscore}}elim} to do an ad-hoc conversion to a disjunction of
-  existentials, which can then be solved by the arithmetic decision procedure.
-  Pattern compatibility and termination are automatic as usual.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ atomize{\isaliteral{5F}{\isacharunderscore}}elim\isanewline
-\isacommand{apply}\isamarkupfalse%
-\ arith\isanewline
-\isacommand{apply}\isamarkupfalse%
-\ auto\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ lexicographic{\isaliteral{5F}{\isacharunderscore}}order%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We can stretch the notion of pattern matching even more. The
-  following function is not a sensible functional program, but a
-  perfectly valid mathematical definition:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ ev\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}ev\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ True{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}ev\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ False{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ atomize{\isaliteral{5F}{\isacharunderscore}}elim\isanewline
-\isacommand{by}\isamarkupfalse%
-\ arith{\isaliteral{2B}{\isacharplus}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}relation\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-This general notion of pattern matching gives you a certain freedom
-  in writing down specifications. However, as always, such freedom should
-  be used with care:
-
-  If we leave the area of constructor
-  patterns, we have effectively departed from the world of functional
-  programming. This means that it is no longer possible to use the
-  code generator, and expect it to generate ML code for our
-  definitions. Also, such a specification might not work very well together with
-  simplification. Your mileage may vary.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Conditional equations%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The function package also supports conditional equations, which are
-  similar to guards in a language like Haskell. Here is Euclid's
-  algorithm written with conditional patterns\footnote{Note that the
-  patterns are also overlapping in the base case}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ gcd\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}gcd\ x\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}gcd\ {\isadigit{0}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3C}{\isacharless}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ gcd\ {\isaliteral{28}{\isacharparenleft}}Suc\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ gcd\ {\isaliteral{28}{\isacharparenleft}}Suc\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{2D}{\isacharminus}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ x\ {\isaliteral{3C}{\isacharless}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ gcd\ {\isaliteral{28}{\isacharparenleft}}Suc\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ gcd\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{2D}{\isacharminus}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}Suc\ y{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}atomize{\isaliteral{5F}{\isacharunderscore}}elim{\isaliteral{2C}{\isacharcomma}}\ auto{\isaliteral{2C}{\isacharcomma}}\ arith{\isaliteral{29}{\isacharparenright}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ lexicographic{\isaliteral{5F}{\isacharunderscore}}order%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-By now, you can probably guess what the proof obligations for the
-  pattern completeness and compatibility look like. 
-
-  Again, functions with conditional patterns are not supported by the
-  code generator.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Pattern matching on strings%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-As strings (as lists of characters) are normal datatypes, pattern
-  matching on them is possible, but somewhat problematic. Consider the
-  following definition:
-
-\end{isamarkuptext}
-\noindent\cmd{fun} \isa{check\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequote}}string\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequote}}}\\%
-\cmd{where}\\%
-\hspace*{2ex}\isa{{\isaliteral{22}{\isachardoublequote}}check\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}{\isaliteral{27}{\isacharprime}}good{\isaliteral{27}{\isacharprime}}{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ True{\isaliteral{22}{\isachardoublequote}}}\\%
-\isa{{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequote}}check\ s\ {\isaliteral{3D}{\isacharequal}}\ False{\isaliteral{22}{\isachardoublequote}}}
-\begin{isamarkuptext}
-
-  \noindent An invocation of the above \cmd{fun} command does not
-  terminate. What is the problem? Strings are lists of characters, and
-  characters are a datatype with a lot of constructors. Splitting the
-  catch-all pattern thus leads to an explosion of cases, which cannot
-  be handled by Isabelle.
-
-  There are two things we can do here. Either we write an explicit
-  \isa{if} on the right hand side, or we can use conditional patterns:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ check\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}string\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}check\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}{\isaliteral{27}{\isacharprime}}good{\isaliteral{27}{\isacharprime}}{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ True{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isaliteral{27}{\isacharprime}}{\isaliteral{27}{\isacharprime}}good{\isaliteral{27}{\isacharprime}}{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ check\ s\ {\isaliteral{3D}{\isacharequal}}\ False{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}relation\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ simp%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsection{Partiality%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-In HOL, all functions are total. A function \isa{f} applied to
-  \isa{x} always has the value \isa{f\ x}, and there is no notion
-  of undefinedness. 
-  This is why we have to do termination
-  proofs when defining functions: The proof justifies that the
-  function can be defined by wellfounded recursion.
-
-  However, the \cmd{function} package does support partiality to a
-  certain extent. Let's look at the following function which looks
-  for a zero of a given function f.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ findzero\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}findzero\ f\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ f\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ then\ n\ else\ findzero\ f\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isaliteral{5F}{\isacharunderscore}}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent Clearly, any attempt of a termination proof must fail. And without
-  that, we do not get the usual rules \isa{findzero{\isaliteral{2E}{\isachardot}}simps} and 
-  \isa{findzero{\isaliteral{2E}{\isachardot}}induct}. So what was the definition good for at all?%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsubsection{Domain predicates%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-The trick is that Isabelle has not only defined the function \isa{findzero}, but also
-  a predicate \isa{findzero{\isaliteral{5F}{\isacharunderscore}}dom} that characterizes the values where the function
-  terminates: the \emph{domain} of the function. If we treat a
-  partial function just as a total function with an additional domain
-  predicate, we can derive simplification and
-  induction rules as we do for total functions. They are guarded
-  by domain conditions and are called \isa{psimps} and \isa{pinduct}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-\noindent\begin{minipage}{0.79\textwidth}\begin{isabelle}%
-findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}f{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{3F}{\isacharquery}}n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\isanewline
-findzero\ {\isaliteral{3F}{\isacharquery}}f\ {\isaliteral{3F}{\isacharquery}}n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{3F}{\isacharquery}}f\ {\isaliteral{3F}{\isacharquery}}n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ then\ {\isaliteral{3F}{\isacharquery}}n\ else\ findzero\ {\isaliteral{3F}{\isacharquery}}f\ {\isaliteral{28}{\isacharparenleft}}Suc\ {\isaliteral{3F}{\isacharquery}}n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}\end{minipage}
-  \hfill(\isa{findzero{\isaliteral{2E}{\isachardot}}psimps})
-  \vspace{1em}
-
-  \noindent\begin{minipage}{0.79\textwidth}\begin{isabelle}%
-{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}a{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{3F}{\isacharquery}}a{\isadigit{1}}{\isaliteral{2E}{\isachardot}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\isanewline
-\isaindent{\ }{\isaliteral{5C3C416E643E}{\isasymAnd}}f\ n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ f\ n\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}P\ f\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}P\ f\ n{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
-{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{3F}{\isacharquery}}a{\isadigit{0}}{\isaliteral{2E}{\isachardot}}{\isadigit{0}}\ {\isaliteral{3F}{\isacharquery}}a{\isadigit{1}}{\isaliteral{2E}{\isachardot}}{\isadigit{0}}%
-\end{isabelle}\end{minipage}
-  \hfill(\isa{findzero{\isaliteral{2E}{\isachardot}}pinduct})%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Remember that all we
-  are doing here is use some tricks to make a total function appear
-  as if it was partial. We can still write the term \isa{findzero\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isadigit{0}}} and like any other term of type \isa{nat} it is equal
-  to some natural number, although we might not be able to find out
-  which one. The function is \emph{underdefined}.
-
-  But it is defined enough to prove something interesting about it. We
-  can prove that if \isa{findzero\ f\ n}
-  terminates, it indeed returns a zero of \isa{f}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ f\ {\isaliteral{28}{\isacharparenleft}}findzero\ f\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\begin{isamarkuptxt}%
-\noindent We apply induction as usual, but using the partial induction
-  rule:%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}induct\ f\ n\ rule{\isaliteral{3A}{\isacharcolon}}\ findzero{\isaliteral{2E}{\isachardot}}pinduct{\isaliteral{29}{\isacharparenright}}%
-\begin{isamarkuptxt}%
-\noindent This gives the following subgoals:
-
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}f\ n{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ f\ n\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ f\ {\isaliteral{28}{\isacharparenleft}}findzero\ f\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
-\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}f\ n{\isaliteral{2E}{\isachardot}}\ }{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ f\ {\isaliteral{28}{\isacharparenleft}}findzero\ f\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}%
-\end{isabelle}
-
-  \noindent The hypothesis in our lemma was used to satisfy the first premise in
-  the induction rule. However, we also get \isa{findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}} as a local assumption in the induction step. This
-  allows unfolding \isa{findzero\ f\ n} using the \isa{psimps}
-  rule, and the rest is trivial.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-\isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ findzero{\isaliteral{2E}{\isachardot}}psimps{\isaliteral{29}{\isacharparenright}}\isanewline
-\isacommand{done}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-Proofs about partial functions are often not harder than for total
-  functions. Fig.~\ref{findzero_isar} shows a slightly more
-  complicated proof written in Isar. It is verbose enough to show how
-  partiality comes into play: From the partial induction, we get an
-  additional domain condition hypothesis. Observe how this condition
-  is applied when calls to \isa{findzero} are unfolded.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-\isacommand{lemma}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{7B}{\isacharbraceleft}}n\ {\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{3C}{\isacharless}}\ findzero\ f\ n{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ f\ x\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}induct\ rule{\isaliteral{3A}{\isacharcolon}}\ findzero{\isaliteral{2E}{\isachardot}}pinduct{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ f\ n\ \isacommand{assume}\isamarkupfalse%
-\ dom{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isakeyword{and}\ IH{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}f\ n\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}{\isaliteral{3B}{\isacharsemicolon}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{7B}{\isacharbraceleft}}Suc\ n\ {\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{3C}{\isacharless}}\ findzero\ f\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ f\ x\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \isakeyword{and}\ x{\isaliteral{5F}{\isacharunderscore}}range{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{7B}{\isacharbraceleft}}n\ {\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{3C}{\isacharless}}\ findzero\ f\ n{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}f\ n\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ \isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}f\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ dom\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}findzero\ f\ n\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ findzero{\isaliteral{2E}{\isachardot}}psimps{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ x{\isaliteral{5F}{\isacharunderscore}}range\ \isacommand{show}\isamarkupfalse%
-\ False\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\ \ \isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ x{\isaliteral{5F}{\isacharunderscore}}range\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{5C3C6F723E}{\isasymor}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{7B}{\isacharbraceleft}}Suc\ n\ {\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{3C}{\isacharless}}\ findzero\ f\ n{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ auto\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}f\ x\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ {\isaliteral{60}{\isacharbackquoteopen}}f\ n\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{7B}{\isacharbraceleft}}Suc\ n\ {\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{3C}{\isacharless}}\ findzero\ f\ n{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ dom\ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}f\ n\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{7B}{\isacharbraceleft}}Suc\ n\ {\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{3C}{\isacharless}}\ findzero\ f\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ findzero{\isaliteral{2E}{\isachardot}}psimps{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{with}\isamarkupfalse%
-\ IH\ \isakeyword{and}\ {\isaliteral{60}{\isacharbackquoteopen}}f\ n\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}{\isaliteral{60}{\isacharbackquoteclose}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}\vspace{6pt}\hrule
-\caption{A proof about a partial function}\label{findzero_isar}
-\end{figure}
-%
-\isamarkupsubsection{Partial termination proofs%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Now that we have proved some interesting properties about our
-  function, we should turn to the domain predicate and see if it is
-  actually true for some values. Otherwise we would have just proved
-  lemmas with \isa{False} as a premise.
-
-  Essentially, we need some introduction rules for \isa{findzero{\isaliteral{5F}{\isacharunderscore}}dom}. The function package can prove such domain
-  introduction rules automatically. But since they are not used very
-  often (they are almost never needed if the function is total), this
-  functionality is disabled by default for efficiency reasons. So we have to go
-  back and ask for them explicitly by passing the \isa{{\isaliteral{28}{\isacharparenleft}}domintros{\isaliteral{29}{\isacharparenright}}} option to the function package:
-
-\vspace{1ex}
-\noindent\cmd{function} \isa{{\isaliteral{28}{\isacharparenleft}}domintros{\isaliteral{29}{\isacharparenright}}\ findzero\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequote}}}\\%
-\cmd{where}\isanewline%
-\ \ \ldots\\
-
-  \noindent Now the package has proved an introduction rule for \isa{findzero{\isaliteral{5F}{\isacharunderscore}}dom}:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{thm}\isamarkupfalse%
-\ findzero{\isaliteral{2E}{\isachardot}}domintros%
-\begin{isamarkuptext}%
-\begin{isabelle}%
-{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ {\isaliteral{3F}{\isacharquery}}f\ {\isaliteral{3F}{\isacharquery}}n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}f{\isaliteral{2C}{\isacharcomma}}\ Suc\ {\isaliteral{3F}{\isacharquery}}n{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}f{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{3F}{\isacharquery}}n{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-
-  Domain introduction rules allow to show that a given value lies in the
-  domain of a function, if the arguments of all recursive calls
-  are in the domain as well. They allow to do a \qt{single step} in a
-  termination proof. Usually, you want to combine them with a suitable
-  induction principle.
-
-  Since our function increases its argument at recursive calls, we
-  need an induction principle which works \qt{backwards}. We will use
-  \isa{inc{\isaliteral{5F}{\isacharunderscore}}induct}, which allows to do induction from a fixed number
-  \qt{downwards}:
-
-  \begin{center}\isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{3F}{\isacharquery}}i\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{3F}{\isacharquery}}j{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{3F}{\isacharquery}}j{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}i{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}i\ {\isaliteral{3C}{\isacharless}}\ {\isaliteral{3F}{\isacharquery}}j{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{28}{\isacharparenleft}}Suc\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}P\ i{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{3F}{\isacharquery}}i}\hfill(\isa{inc{\isaliteral{5F}{\isacharunderscore}}induct})\end{center}
-
-  Figure \ref{findzero_term} gives a detailed Isar proof of the fact
-  that \isa{findzero} terminates if there is a zero which is greater
-  or equal to \isa{n}. First we derive two useful rules which will
-  solve the base case and the step case of the induction. The
-  induction is then straightforward, except for the unusual induction
-  principle.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isaliteral{5F}{\isacharunderscore}}termination{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \isakeyword{assumes}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{5C3C67653E}{\isasymge}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\ {\isaliteral{22}{\isachardoublequoteopen}}f\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{2D}{\isacharminus}}\ \isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ base{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ findzero{\isaliteral{2E}{\isachardot}}domintros{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}{\isaliteral{60}{\isacharbackquoteopen}}f\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{60}{\isacharbackquoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\isanewline
-\ \ \isacommand{have}\isamarkupfalse%
-\ step{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C416E643E}{\isasymAnd}}i{\isaliteral{2E}{\isachardot}}\ findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ Suc\ i{\isaliteral{29}{\isacharparenright}}\ \isanewline
-\ \ \ \ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ findzero{\isaliteral{2E}{\isachardot}}domintros{\isaliteral{29}{\isacharparenright}}\ simp\isanewline
-\isanewline
-\ \ \isacommand{from}\isamarkupfalse%
-\ {\isaliteral{60}{\isacharbackquoteopen}}x\ {\isaliteral{5C3C67653E}{\isasymge}}\ n{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}thesis\isanewline
-\ \ \isacommand{proof}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}induct\ rule{\isaliteral{3A}{\isacharcolon}}inc{\isaliteral{5F}{\isacharunderscore}}induct{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ base{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \isacommand{next}\isamarkupfalse%
-\isanewline
-\ \ \ \ \isacommand{fix}\isamarkupfalse%
-\ i\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ Suc\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \ \ \isacommand{thus}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}rule\ step{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \isacommand{qed}\isamarkupfalse%
-\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}\vspace{6pt}\hrule
-\caption{Termination proof for \isa{findzero}}\label{findzero_term}
-\end{figure}
-%
-\begin{isamarkuptext}%
-Again, the proof given in Fig.~\ref{findzero_term} has a lot of
-  detail in order to explain the principles. Using more automation, we
-  can also have a short proof:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isaliteral{5F}{\isacharunderscore}}termination{\isaliteral{5F}{\isacharunderscore}}short{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ \isakeyword{assumes}\ zero{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3E}{\isachargreater}}{\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\ \isanewline
-\ \ \isakeyword{assumes}\ {\isaliteral{5B}{\isacharbrackleft}}simp{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}f\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ zero\isanewline
-\isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}induct\ rule{\isaliteral{3A}{\isacharcolon}}inc{\isaliteral{5F}{\isacharunderscore}}induct{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}auto\ intro{\isaliteral{3A}{\isacharcolon}}\ findzero{\isaliteral{2E}{\isachardot}}domintros{\isaliteral{29}{\isacharparenright}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-\noindent It is simple to combine the partial correctness result with the
-  termination lemma:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ findzero{\isaliteral{5F}{\isacharunderscore}}total{\isaliteral{5F}{\isacharunderscore}}correctness{\isaliteral{3A}{\isacharcolon}}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}f\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ f\ {\isaliteral{28}{\isacharparenleft}}findzero\ f\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}blast\ intro{\isaliteral{3A}{\isacharcolon}}\ findzero{\isaliteral{5F}{\isacharunderscore}}zero\ findzero{\isaliteral{5F}{\isacharunderscore}}termination{\isaliteral{29}{\isacharparenright}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Definition of the domain predicate%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Sometimes it is useful to know what the definition of the domain
-  predicate looks like. Actually, \isa{findzero{\isaliteral{5F}{\isacharunderscore}}dom} is just an
-  abbreviation:
-
-  \begin{isabelle}%
-findzero{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ accp\ findzero{\isaliteral{5F}{\isacharunderscore}}rel%
-\end{isabelle}
-
-  The domain predicate is the \emph{accessible part} of a relation \isa{findzero{\isaliteral{5F}{\isacharunderscore}}rel}, which was also created internally by the function
-  package. \isa{findzero{\isaliteral{5F}{\isacharunderscore}}rel} is just a normal
-  inductive predicate, so we can inspect its definition by
-  looking at the introduction rules \isa{findzero{\isaliteral{5F}{\isacharunderscore}}rel{\isaliteral{2E}{\isachardot}}intros}.
-  In our case there is just a single rule:
-
-  \begin{isabelle}%
-{\isaliteral{3F}{\isacharquery}}f\ {\isaliteral{3F}{\isacharquery}}n\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ findzero{\isaliteral{5F}{\isacharunderscore}}rel\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}f{\isaliteral{2C}{\isacharcomma}}\ Suc\ {\isaliteral{3F}{\isacharquery}}n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}f{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{3F}{\isacharquery}}n{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-
-  The predicate \isa{findzero{\isaliteral{5F}{\isacharunderscore}}rel}
-  describes the \emph{recursion relation} of the function
-  definition. The recursion relation is a binary relation on
-  the arguments of the function that relates each argument to its
-  recursive calls. In general, there is one introduction rule for each
-  recursive call.
-
-  The predicate \isa{findzero{\isaliteral{5F}{\isacharunderscore}}dom} is the accessible part of
-  that relation. An argument belongs to the accessible part, if it can
-  be reached in a finite number of steps (cf.~its definition in \isa{Wellfounded{\isaliteral{2E}{\isachardot}}thy}).
-
-  Since the domain predicate is just an abbreviation, you can use
-  lemmas for \isa{accp} and \isa{findzero{\isaliteral{5F}{\isacharunderscore}}rel} directly. Some
-  lemmas which are occasionally useful are \isa{accpI}, \isa{accp{\isaliteral{5F}{\isacharunderscore}}downward}, and of course the introduction and elimination rules
-  for the recursion relation \isa{findzero{\isaliteral{2E}{\isachardot}}intros} and \isa{findzero{\isaliteral{2E}{\isachardot}}cases}.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isamarkupsection{Nested recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Recursive calls which are nested in one another frequently cause
-  complications, since their termination proof can depend on a partial
-  correctness property of the function itself. 
-
-  As a small example, we define the \qt{nested zero} function:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ nz\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}nz\ {\isadigit{0}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}nz\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ nz\ {\isaliteral{28}{\isacharparenleft}}nz\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isaliteral{5F}{\isacharunderscore}}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-If we attempt to prove termination using the identity measure on
-  naturals, this fails:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{apply}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}relation\ {\isaliteral{22}{\isachardoublequoteopen}}measure\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}n{\isaliteral{2E}{\isachardot}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
-\ \ \isacommand{apply}\isamarkupfalse%
-\ auto%
-\begin{isamarkuptxt}%
-We get stuck with the subgoal
-
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}n{\isaliteral{2E}{\isachardot}}\ nz{\isaliteral{5F}{\isacharunderscore}}dom\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ nz\ n\ {\isaliteral{3C}{\isacharless}}\ Suc\ n%
-\end{isabelle}
-
-  Of course this statement is true, since we know that \isa{nz} is
-  the zero function. And in fact we have no problem proving this
-  property by induction.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-\isacommand{lemma}\isamarkupfalse%
-\ nz{\isaliteral{5F}{\isacharunderscore}}is{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nz{\isaliteral{5F}{\isacharunderscore}}dom\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ nz\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}induct\ rule{\isaliteral{3A}{\isacharcolon}}nz{\isaliteral{2E}{\isachardot}}pinduct{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ nz{\isaliteral{2E}{\isachardot}}psimps{\isaliteral{29}{\isacharparenright}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-We formulate this as a partial correctness lemma with the condition
-  \isa{nz{\isaliteral{5F}{\isacharunderscore}}dom\ n}. This allows us to prove it with the \isa{pinduct} rule before we have proved termination. With this lemma,
-  the termination proof works as expected:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ {\isaliteral{28}{\isacharparenleft}}relation\ {\isaliteral{22}{\isachardoublequoteopen}}measure\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}n{\isaliteral{2E}{\isachardot}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ nz{\isaliteral{5F}{\isacharunderscore}}is{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{29}{\isacharparenright}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-As a general strategy, one should prove the statements needed for
-  termination as a partial property first. Then they can be used to do
-  the termination proof. This also works for less trivial
-  examples. Figure \ref{f91} defines the 91-function, a well-known
-  challenge problem due to John McCarthy, and proves its termination.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\begin{figure}
-\hrule\vspace{6pt}
-\begin{minipage}{0.8\textwidth}
-\isabellestyle{it}
-\isastyle\isamarkuptrue
-\isacommand{function}\isamarkupfalse%
-\ f{\isadigit{9}}{\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}f{\isadigit{9}}{\isadigit{1}}\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ n\ then\ n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isadigit{0}}\ else\ f{\isadigit{9}}{\isadigit{1}}\ {\isaliteral{28}{\isacharparenleft}}f{\isadigit{9}}{\isadigit{1}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ pat{\isaliteral{5F}{\isacharunderscore}}completeness\ auto%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{lemma}\isamarkupfalse%
-\ f{\isadigit{9}}{\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}estimate{\isaliteral{3A}{\isacharcolon}}\ \isanewline
-\ \ \isakeyword{assumes}\ trm{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}f{\isadigit{9}}{\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}dom\ n{\isaliteral{22}{\isachardoublequoteclose}}\ \isanewline
-\ \ \isakeyword{shows}\ {\isaliteral{22}{\isachardoublequoteopen}}n\ {\isaliteral{3C}{\isacharless}}\ f{\isadigit{9}}{\isadigit{1}}\ n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{using}\isamarkupfalse%
-\ trm\ \isacommand{by}\isamarkupfalse%
-\ induct\ {\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ f{\isadigit{9}}{\isadigit{1}}{\isaliteral{2E}{\isachardot}}psimps{\isaliteral{29}{\isacharparenright}}%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{termination}\isamarkupfalse%
-\isanewline
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{let}\isamarkupfalse%
-\ {\isaliteral{3F}{\isacharquery}}R\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}measure\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{1}}\ {\isaliteral{2D}{\isacharminus}}\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\ \ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}wf\ {\isaliteral{3F}{\isacharquery}}R{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\isanewline
-\ \ \isacommand{fix}\isamarkupfalse%
-\ n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ nat\ \isacommand{assume}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\ %
-\isamarkupcmt{Assumptions for both calls%
-}
-\isanewline
-\isanewline
-\ \ \isacommand{thus}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{3F}{\isacharquery}}R{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ simp\ %
-\isamarkupcmt{Inner call%
-}
-\isanewline
-\isanewline
-\ \ \isacommand{assume}\isamarkupfalse%
-\ inner{\isaliteral{5F}{\isacharunderscore}}trm{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}f{\isadigit{9}}{\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}dom\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ %
-\isamarkupcmt{Outer call%
-}
-\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ f{\isadigit{9}}{\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}estimate\ \isacommand{have}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isadigit{1}}\ {\isaliteral{3C}{\isacharless}}\ f{\isadigit{9}}{\isadigit{1}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{{\isaliteral{2E}{\isachardot}}}\isamarkupfalse%
-\isanewline
-\ \ \isacommand{with}\isamarkupfalse%
-\ {\isaliteral{60}{\isacharbackquoteopen}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isadigit{1}}{\isadigit{0}}{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ n{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{show}\isamarkupfalse%
-\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}f{\isadigit{9}}{\isadigit{1}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{3F}{\isacharquery}}R{\isaliteral{22}{\isachardoublequoteclose}}\ \isacommand{by}\isamarkupfalse%
-\ simp\isanewline
-\isacommand{qed}\isamarkupfalse%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupfalse\isabellestyle{tt}
-\end{minipage}
-\vspace{6pt}\hrule
-\caption{McCarthy's 91-function}\label{f91}
-\end{figure}
-%
-\isamarkupsection{Higher-Order Recursion%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Higher-order recursion occurs when recursive calls
-  are passed as arguments to higher-order combinators such as \isa{map}, \isa{filter} etc.
-  As an example, imagine a datatype of n-ary trees:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{datatype}\isamarkupfalse%
-\ {\isaliteral{27}{\isacharprime}}a\ tree\ {\isaliteral{3D}{\isacharequal}}\ \isanewline
-\ \ Leaf\ {\isaliteral{27}{\isacharprime}}a\ \isanewline
-{\isaliteral{7C}{\isacharbar}}\ Branch\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ tree\ list{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\noindent We can define a function which swaps the left and right subtrees recursively, using the 
-  list functions \isa{rev} and \isa{map}:%
-\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}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}mirror\ {\isaliteral{28}{\isacharparenleft}}Leaf\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Leaf\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-{\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}mirror\ {\isaliteral{28}{\isacharparenleft}}Branch\ l{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Branch\ {\isaliteral{28}{\isacharparenleft}}rev\ {\isaliteral{28}{\isacharparenleft}}map\ mirror\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-Although the definition is accepted without problems, let us look at the termination proof:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{termination}\isamarkupfalse%
-%
-\isadelimproof
-\ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{proof}\isamarkupfalse%
-%
-\begin{isamarkuptxt}%
-As usual, we have to give a wellfounded relation, such that the
-  arguments of the recursive calls get smaller. But what exactly are
-  the arguments of the recursive calls when mirror is given as an
-  argument to \isa{map}? Isabelle gives us the
-  subgoals
-
-  \begin{isabelle}%
-\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ wf\ {\isaliteral{3F}{\isacharquery}}R\isanewline
-\ {\isadigit{2}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}l\ x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ l\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ Branch\ l{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ {\isaliteral{3F}{\isacharquery}}R%
-\end{isabelle} 
-
-  So the system seems to know that \isa{map} only
-  applies the recursive call \isa{mirror} to elements
-  of \isa{l}, which is essential for the termination proof.
-
-  This knowledge about \isa{map} is encoded in so-called congruence rules,
-  which are special theorems known to the \cmd{function} command. The
-  rule for \isa{map} is
-
-  \begin{isabelle}%
-{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{3F}{\isacharquery}}xs\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}ys{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ set\ {\isaliteral{3F}{\isacharquery}}ys\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}f\ x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}g\ x{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ map\ {\isaliteral{3F}{\isacharquery}}f\ {\isaliteral{3F}{\isacharquery}}xs\ {\isaliteral{3D}{\isacharequal}}\ map\ {\isaliteral{3F}{\isacharquery}}g\ {\isaliteral{3F}{\isacharquery}}ys%
-\end{isabelle}
-
-  You can read this in the following way: Two applications of \isa{map} are equal, if the list arguments are equal and the functions
-  coincide on the elements of the list. This means that for the value 
-  \isa{map\ f\ l} we only have to know how \isa{f} behaves on
-  the elements of \isa{l}.
-
-  Usually, one such congruence rule is
-  needed for each higher-order construct that is used when defining
-  new functions. In fact, even basic functions like \isa{If} and \isa{Let} are handled by this mechanism. The congruence
-  rule for \isa{If} states that the \isa{then} branch is only
-  relevant if the condition is true, and the \isa{else} branch only if it
-  is false:
-
-  \begin{isabelle}%
-{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{3F}{\isacharquery}}b\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}c{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{3F}{\isacharquery}}c\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}x\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}u{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{3F}{\isacharquery}}c\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}y\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}v{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\isanewline
-{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{3F}{\isacharquery}}b\ then\ {\isaliteral{3F}{\isacharquery}}x\ else\ {\isaliteral{3F}{\isacharquery}}y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ {\isaliteral{3F}{\isacharquery}}c\ then\ {\isaliteral{3F}{\isacharquery}}u\ else\ {\isaliteral{3F}{\isacharquery}}v{\isaliteral{29}{\isacharparenright}}%
-\end{isabelle}
-  
-  Congruence rules can be added to the
-  function package by giving them the \isa{fundef{\isaliteral{5F}{\isacharunderscore}}cong} attribute.
-
-  The constructs that are predefined in Isabelle, usually
-  come with the respective congruence rules.
-  But if you define your own higher-order functions, you may have to
-  state and prove the required congruence rules yourself, if you want to use your
-  functions in recursive definitions.%
-\end{isamarkuptxt}%
-\isamarkuptrue%
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isamarkupsubsection{Congruence Rules and Evaluation Order%
-}
-\isamarkuptrue%
-%
-\begin{isamarkuptext}%
-Higher order logic differs from functional programming languages in
-  that it has no built-in notion of evaluation order. A program is
-  just a set of equations, and it is not specified how they must be
-  evaluated. 
-
-  However for the purpose of function definition, we must talk about
-  evaluation order implicitly, when we reason about termination.
-  Congruence rules express that a certain evaluation order is
-  consistent with the logical definition. 
-
-  Consider the following function.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{function}\isamarkupfalse%
-\ f\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}f\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ f\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\isadelimproof
-%
-\endisadelimproof
-%
-\isatagproof
-%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-%
-\endisadelimproof
-%
-\begin{isamarkuptext}%
-For this definition, the termination proof fails. The default configuration
-  specifies no congruence rule for disjunction. We have to add a
-  congruence rule that specifies left-to-right evaluation order:
-
-  \vspace{1ex}
-  \noindent \isa{{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}{\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}P{\isaliteral{27}{\isacharprime}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{3F}{\isacharquery}}P{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{3F}{\isacharquery}}Q\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{3F}{\isacharquery}}Q{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{3F}{\isacharquery}}Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{3F}{\isacharquery}}P{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{3F}{\isacharquery}}Q{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}}\hfill(\isa{disj{\isaliteral{5F}{\isacharunderscore}}cong})
-  \vspace{1ex}
-
-  Now the definition works without problems. Note how the termination
-  proof depends on the extra condition that we get from the congruence
-  rule.
-
-  However, as evaluation is not a hard-wired concept, we
-  could just turn everything around by declaring a different
-  congruence rule. Then we can make the reverse definition:%
-\end{isamarkuptext}%
-\isamarkuptrue%
-\isacommand{lemma}\isamarkupfalse%
-\ disj{\isaliteral{5F}{\isacharunderscore}}cong{\isadigit{2}}{\isaliteral{5B}{\isacharbrackleft}}fundef{\isaliteral{5F}{\isacharunderscore}}cong{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ \isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ Q{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{3D}{\isacharequal}}\ P{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}Q\ {\isaliteral{3D}{\isacharequal}}\ Q{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}P{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-%
-\isadelimproof
-\ \ %
-\endisadelimproof
-%
-\isatagproof
-\isacommand{by}\isamarkupfalse%
-\ blast%
-\endisatagproof
-{\isafoldproof}%
-%
-\isadelimproof
-\isanewline
-%
-\endisadelimproof
-\isanewline
-\isacommand{fun}\isamarkupfalse%
-\ f{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
-\isakeyword{where}\isanewline
-\ \ {\isaliteral{22}{\isachardoublequoteopen}}f{\isaliteral{27}{\isacharprime}}\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}f{\isaliteral{27}{\isacharprime}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
-\begin{isamarkuptext}%
-\noindent These examples show that, in general, there is no \qt{best} set of
-  congruence rules.
-
-  However, such tweaking should rarely be necessary in
-  practice, as most of the time, the default set of congruence rules
-  works well.%
-\end{isamarkuptext}%
-\isamarkuptrue%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-%
-\isatagtheory
-\isacommand{end}\isamarkupfalse%
-%
-\endisatagtheory
-{\isafoldtheory}%
-%
-\isadelimtheory
-%
-\endisadelimtheory
-\isanewline
-\end{isabellebody}%
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/conclusion.tex
--- a/doc-src/Functions/conclusion.tex	Mon Aug 27 22:00:04 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,7 +0,0 @@
-\section{Conclusion}
-
-\fixme{}
-
-
-
-
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/document/build
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/document/build	Mon Aug 27 22:14:17 2012 +0200
@@ -0,0 +1,19 @@
+#!/bin/bash
+
+set -e
+
+FORMAT="$1"
+VARIANT="$2"
+
+cp "$ISABELLE_HOME/doc-src/iman.sty" .
+cp "$ISABELLE_HOME/doc-src/extra.sty" .
+cp "$ISABELLE_HOME/doc-src/isar.sty" .
+cp "$ISABELLE_HOME/doc-src/manual.bib" .
+
+"$ISABELLE_TOOL" latex -o sty
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+"$ISABELLE_TOOL" latex -o bbl
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+[ -f root.out ] && "$ISABELLE_HOME/doc-src/fixbookmarks" root.out
+"$ISABELLE_TOOL" latex -o "$FORMAT"
+"$ISABELLE_TOOL" latex -o "$FORMAT"
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/document/conclusion.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/document/conclusion.tex	Mon Aug 27 22:14:17 2012 +0200
@@ -0,0 +1,7 @@
+\section{Conclusion}
+
+\fixme{}
+
+
+
+
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/document/intro.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/document/intro.tex	Mon Aug 27 22:14:17 2012 +0200
@@ -0,0 +1,55 @@
+\section{Introduction}
+
+Starting from Isabelle 2007, new facilities for recursive
+function definitions~\cite{krauss2006} are available. They provide
+better support for general recursive definitions than previous
+packages.  But despite all tool support, function definitions can
+sometimes be a difficult thing. 
+
+This tutorial is an example-guided introduction to the practical use
+of the package and related tools. It should help you get started with
+defining functions quickly. For the more difficult definitions we will
+discuss what problems can arise, and how they can be solved.
+
+We assume that you have mastered the fundamentals of Isabelle/HOL
+and are able to write basic specifications and proofs. To start out
+with Isabelle in general, consult the Isabelle/HOL tutorial
+\cite{isa-tutorial}.
+
+
+
+\paragraph{Structure of this tutorial.}
+Section 2 introduces the syntax and basic operation of the \cmd{fun}
+command, which provides full automation with reasonable default
+behavior.  The impatient reader can stop after that
+section, and consult the remaining sections only when needed.
+Section 3 introduces the more verbose \cmd{function} command which
+gives fine-grained control. This form should be used
+whenever the short form fails.
+After that we discuss more specialized issues:
+termination, mutual, nested and higher-order recursion, partiality, pattern matching
+and others.
+
+
+\paragraph{Some background.}
+Following the LCF tradition, the package is realized as a definitional
+extension: Recursive definitions are internally transformed into a
+non-recursive form, such that the function can be defined using
+standard definition facilities. Then the recursive specification is
+derived from the primitive definition.  This is a complex task, but it
+is fully automated and mostly transparent to the user. Definitional
+extensions are valuable because they are conservative by construction:
+The \qt{new} concept of general wellfounded recursion is completely reduced
+to existing principles.
+
+
+
+
+The new \cmd{function} command, and its short form \cmd{fun} have mostly
+replaced the traditional \cmd{recdef} command \cite{slind-tfl}. They solve
+a few of technical issues around \cmd{recdef}, and allow definitions
+which were not previously possible.
+
+
+
+
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/document/mathpartir.sty
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/document/mathpartir.sty	Mon Aug 27 22:14:17 2012 +0200
@@ -0,0 +1,421 @@
+%  Mathpartir --- Math Paragraph for Typesetting Inference Rules
+%
+%  Copyright (C) 2001, 2002, 2003, 2004, 2005 Didier Rémy
+%
+%  Author         : Didier Remy 
+%  Version        : 1.2.0
+%  Bug Reports    : to author
+%  Web Site       : http://pauillac.inria.fr/~remy/latex/
+% 
+%  Mathpartir 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}
+    [2005/12/20 version 1.2.0 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
+}
+
+\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}
+
+%% Part III -- Type inference rules
+
+\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@item #1{$\displaystyle #1$}
+\def \mpr@sep{2em}
+\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 T 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{\mpr@item {##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
+  }{}
+
+%% The default
+
+\def \mpr@@fraction #1#2{\hbox {\advance \hsize by -0.5em
+    $\displaystyle {#1\mpr@over #2}$}}
+\let \mpr@fraction \mpr@@fraction
+
+%% A generic solution to arrow
+
+\def \mpr@make@fraction #1#2#3#4#5{\hbox {%
+     \def \mpr@tail{#1}%
+     \def \mpr@body{#2}%
+     \def \mpr@head{#3}%
+     \setbox1=\hbox{$#4$}\setbox2=\hbox{$#5$}%
+     \setbox3=\hbox{$\mkern -3mu\mpr@body\mkern -3mu$}%
+     \setbox3=\hbox{$\mkern -3mu \mpr@body\mkern -3mu$}%
+     \dimen0=\dp1\advance\dimen0 by \ht3\relax\dp1\dimen0\relax
+     \dimen0=\ht2\advance\dimen0 by \dp3\relax\ht2\dimen0\relax
+     \setbox0=\hbox {$\box1 \@@atop \box2$}%
+     \dimen0=\wd0\box0
+     \box0 \hskip -\dimen0\relax
+     \hbox to \dimen0 {$%
+       \mathrel{\mpr@tail}\joinrel
+       \xleaders\hbox{\copy3}\hfil\joinrel\mathrel{\mpr@head}%
+     $}}}
+
+%% Old stuff should be removed in next version
+\def \mpr@@reduce #1#2{\hbox
+    {$\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$}}
+\def \mpr@infercenter #1{\vcenter {\mpr@hovbox{T}{#1}}}
+
+\def \mpr@empty {}
+\def \mpr@inferrule
+  {\bgroup
+     \ifnum \linewidth<\hsize \hsize \linewidth\fi
+     \mpr@rulelineskip
+     \let \and \qquad
+     \let \hva \mpr@hva
+     \let \@rulename \mpr@empty
+     \let \@rule@options \mpr@empty
+     \let \mpr@over \@@over
+     \mpr@inferrule@}
+\newcommand {\mpr@inferrule@}[3][]
+  {\everymath={\displaystyle}%       
+   \def \@test {#2}\ifx \empty \@test
+      \setbox0 \hbox {$\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. 
+%  
+
+%% Keys that make sence in all kinds of rules
+\def \mprset #1{\setkeys{mprset}{#1}}
+\define@key {mprset}{flushleft}[]{\mpr@centerfalse}
+\define@key {mprset}{center}[]{\mpr@centertrue}
+\define@key {mprset}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
+\define@key {mprset}{myfraction}[]{\let \mpr@fraction #1}
+\define@key {mprset}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
+
+\newbox \mpr@right
+\define@key {mpr}{flushleft}[]{\mpr@centerfalse}
+\define@key {mpr}{center}[]{\mpr@centertrue}
+\define@key {mpr}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
+\define@key {mpr}{myfraction}[]{\let \mpr@fraction #1}
+\define@key {mpr}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
+\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 \TirNameStyle #1{\small \textsc{#1}}
+\def \tir@name #1{\hbox {\small \TirNameStyle{#1}}}
+\let \TirName \tir@name
+\let \DefTirName \TirName
+\let \RefTirName \TirName
+
+%%% Other Exports
+
+% \let \listcons \mpr@cons
+% \let \listsnoc \mpr@snoc
+% \let \listhead \mpr@head
+% \let \listmake \mpr@makelist
+
+
+
+
+\endinput
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/document/root.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/document/root.tex	Mon Aug 27 22:14:17 2012 +0200
@@ -0,0 +1,90 @@
+
+\documentclass[a4paper,fleqn]{article}
+
+\usepackage{latexsym,graphicx}
+\usepackage[refpage]{nomencl}
+\usepackage{iman,extra,isar}
+\usepackage{isabelle,isabellesym}
+\usepackage{style}
+\usepackage{mathpartir}
+\usepackage{amsthm}
+\usepackage{pdfsetup}
+
+\newcommand{\cmd}[1]{\isacommand{#1}}
+
+\newcommand{\isasymINFIX}{\cmd{infix}}
+\newcommand{\isasymLOCALE}{\cmd{locale}}
+\newcommand{\isasymINCLUDES}{\cmd{includes}}
+\newcommand{\isasymDATATYPE}{\cmd{datatype}}
+\newcommand{\isasymDEFINES}{\cmd{defines}}
+\newcommand{\isasymNOTES}{\cmd{notes}}
+\newcommand{\isasymCLASS}{\cmd{class}}
+\newcommand{\isasymINSTANCE}{\cmd{instance}}
+\newcommand{\isasymLEMMA}{\cmd{lemma}}
+\newcommand{\isasymPROOF}{\cmd{proof}}
+\newcommand{\isasymQED}{\cmd{qed}}
+\newcommand{\isasymFIX}{\cmd{fix}}
+\newcommand{\isasymASSUME}{\cmd{assume}}
+\newcommand{\isasymSHOW}{\cmd{show}}
+\newcommand{\isasymNOTE}{\cmd{note}}
+\newcommand{\isasymCODEGEN}{\cmd{code\_gen}}
+\newcommand{\isasymPRINTCODETHMS}{\cmd{print\_codethms}}
+\newcommand{\isasymFUN}{\cmd{fun}}
+\newcommand{\isasymFUNCTION}{\cmd{function}}
+\newcommand{\isasymPRIMREC}{\cmd{primrec}}
+\newcommand{\isasymRECDEF}{\cmd{recdef}}
+
+\newcommand{\qt}[1]{``#1''}
+\newcommand{\qtt}[1]{"{}{#1}"{}}
+\newcommand{\qn}[1]{\emph{#1}}
+\newcommand{\strong}[1]{{\bfseries #1}}
+\newcommand{\fixme}[1][!]{\strong{FIXME: #1}}
+
+\newtheorem{exercise}{Exercise}{\bf}{\itshape}
+%\newtheorem*{thmstar}{Theorem}{\bf}{\itshape}
+
+\hyphenation{Isabelle}
+\hyphenation{Isar}
+
+\isadroptag{theory}
+\title{Defining Recursive Functions in Isabelle/HOL}
+\author{Alexander Krauss}
+
+\isabellestyle{tt}
+\renewcommand{\isastyletxt}{\isastyletext}% use same formatting for txt and text
+
+\begin{document}
+
+\date{\ \\}
+\maketitle
+
+\begin{abstract}
+  This tutorial describes the use of the new \emph{function} package,
+	which provides general recursive function definitions for Isabelle/HOL.
+	We start with very simple examples and then gradually move on to more
+	advanced topics such as manual termination proofs, nested recursion,
+	partiality, tail recursion and congruence rules.
+\end{abstract}
+
+%\thispagestyle{empty}\clearpage
+
+%\pagenumbering{roman}
+%\clearfirst
+
+\input{intro.tex}
+\input{Functions.tex}
+%\input{conclusion.tex}
+
+\begingroup
+%\tocentry{\bibname}
+\bibliographystyle{plain} \small\raggedright\frenchspacing
+\bibliography{manual}
+\endgroup
+
+\end{document}
+
+
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: t
+%%% End: 
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/document/style.sty
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Functions/document/style.sty	Mon Aug 27 22:14:17 2012 +0200
@@ -0,0 +1,46 @@
+%% toc
+\newcommand{\tocentry}[1]{\cleardoublepage\phantomsection\addcontentsline{toc}{chapter}{#1}
+\@mkboth{\MakeUppercase{#1}}{\MakeUppercase{#1}}}
+
+%% references
+\newcommand{\secref}[1]{\S\ref{#1}}
+\newcommand{\chref}[1]{chapter~\ref{#1}}
+\newcommand{\figref}[1]{figure~\ref{#1}}
+
+%% math
+\newcommand{\text}[1]{\mbox{#1}}
+\newcommand{\isasymvartheta}{\isamath{\theta}}
+\newcommand{\isactrlvec}[1]{\emph{$\overline{#1}$}}
+
+\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
+
+\pagestyle{headings}
+\sloppy
+\binperiod
+\underscoreon
+
+\renewcommand{\isadigit}[1]{\isamath{#1}}
+
+\newenvironment{mldecls}{\par\noindent\begingroup\footnotesize\def\isanewline{\\}\begin{tabular}{l}}{\end{tabular}\smallskip\endgroup}
+
+\isafoldtag{FIXME}
+\isakeeptag{mlref}
+\renewcommand{\isatagmlref}{\subsection*{\makebox[0pt][r]{\fbox{\ML}~~}Reference}\begingroup\def\isastyletext{\rm}\small}
+\renewcommand{\endisatagmlref}{\endgroup}
+
+\newcommand{\isasymGUESS}{\isakeyword{guess}}
+\newcommand{\isasymOBTAIN}{\isakeyword{obtain}}
+\newcommand{\isasymTHEORY}{\isakeyword{theory}}
+\newcommand{\isasymUSES}{\isakeyword{uses}}
+\newcommand{\isasymEND}{\isakeyword{end}}
+\newcommand{\isasymCONSTS}{\isakeyword{consts}}
+\newcommand{\isasymDEFS}{\isakeyword{defs}}
+\newcommand{\isasymTHEOREM}{\isakeyword{theorem}}
+\newcommand{\isasymDEFINITION}{\isakeyword{definition}}
+
+\isabellestyle{it}
+
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: "implementation"
+%%% End: 
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/functions.tex
--- a/doc-src/Functions/functions.tex	Mon Aug 27 22:00:04 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,90 +0,0 @@
-
-\documentclass[a4paper,fleqn]{article}
-
-\usepackage{latexsym,graphicx}
-\usepackage[refpage]{nomencl}
-\usepackage{../iman,../extra,../isar,../proof}
-\usepackage{../../lib/texinputs/isabelle,../../lib/texinputs/isabellesym}
-\usepackage{style}
-\usepackage{mathpartir}
-\usepackage{amsthm}
-\usepackage{../pdfsetup}
-
-\newcommand{\cmd}[1]{\isacommand{#1}}
-
-\newcommand{\isasymINFIX}{\cmd{infix}}
-\newcommand{\isasymLOCALE}{\cmd{locale}}
-\newcommand{\isasymINCLUDES}{\cmd{includes}}
-\newcommand{\isasymDATATYPE}{\cmd{datatype}}
-\newcommand{\isasymDEFINES}{\cmd{defines}}
-\newcommand{\isasymNOTES}{\cmd{notes}}
-\newcommand{\isasymCLASS}{\cmd{class}}
-\newcommand{\isasymINSTANCE}{\cmd{instance}}
-\newcommand{\isasymLEMMA}{\cmd{lemma}}
-\newcommand{\isasymPROOF}{\cmd{proof}}
-\newcommand{\isasymQED}{\cmd{qed}}
-\newcommand{\isasymFIX}{\cmd{fix}}
-\newcommand{\isasymASSUME}{\cmd{assume}}
-\newcommand{\isasymSHOW}{\cmd{show}}
-\newcommand{\isasymNOTE}{\cmd{note}}
-\newcommand{\isasymCODEGEN}{\cmd{code\_gen}}
-\newcommand{\isasymPRINTCODETHMS}{\cmd{print\_codethms}}
-\newcommand{\isasymFUN}{\cmd{fun}}
-\newcommand{\isasymFUNCTION}{\cmd{function}}
-\newcommand{\isasymPRIMREC}{\cmd{primrec}}
-\newcommand{\isasymRECDEF}{\cmd{recdef}}
-
-\newcommand{\qt}[1]{``#1''}
-\newcommand{\qtt}[1]{"{}{#1}"{}}
-\newcommand{\qn}[1]{\emph{#1}}
-\newcommand{\strong}[1]{{\bfseries #1}}
-\newcommand{\fixme}[1][!]{\strong{FIXME: #1}}
-
-\newtheorem{exercise}{Exercise}{\bf}{\itshape}
-%\newtheorem*{thmstar}{Theorem}{\bf}{\itshape}
-
-\hyphenation{Isabelle}
-\hyphenation{Isar}
-
-\isadroptag{theory}
-\title{Defining Recursive Functions in Isabelle/HOL}
-\author{Alexander Krauss}
-
-\isabellestyle{tt}
-\renewcommand{\isastyletxt}{\isastyletext}% use same formatting for txt and text
-
-\begin{document}
-
-\date{\ \\}
-\maketitle
-
-\begin{abstract}
-  This tutorial describes the use of the new \emph{function} package,
-	which provides general recursive function definitions for Isabelle/HOL.
-	We start with very simple examples and then gradually move on to more
-	advanced topics such as manual termination proofs, nested recursion,
-	partiality, tail recursion and congruence rules.
-\end{abstract}
-
-%\thispagestyle{empty}\clearpage
-
-%\pagenumbering{roman}
-%\clearfirst
-
-\input{intro.tex}
-\input{Thy/document/Functions.tex}
-%\input{conclusion.tex}
-
-\begingroup
-%\tocentry{\bibname}
-\bibliographystyle{plain} \small\raggedright\frenchspacing
-\bibliography{../manual}
-\endgroup
-
-\end{document}
-
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: t
-%%% End: 
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/intro.tex
--- a/doc-src/Functions/intro.tex	Mon Aug 27 22:00:04 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,55 +0,0 @@
-\section{Introduction}
-
-Starting from Isabelle 2007, new facilities for recursive
-function definitions~\cite{krauss2006} are available. They provide
-better support for general recursive definitions than previous
-packages.  But despite all tool support, function definitions can
-sometimes be a difficult thing. 
-
-This tutorial is an example-guided introduction to the practical use
-of the package and related tools. It should help you get started with
-defining functions quickly. For the more difficult definitions we will
-discuss what problems can arise, and how they can be solved.
-
-We assume that you have mastered the fundamentals of Isabelle/HOL
-and are able to write basic specifications and proofs. To start out
-with Isabelle in general, consult the Isabelle/HOL tutorial
-\cite{isa-tutorial}.
-
-
-
-\paragraph{Structure of this tutorial.}
-Section 2 introduces the syntax and basic operation of the \cmd{fun}
-command, which provides full automation with reasonable default
-behavior.  The impatient reader can stop after that
-section, and consult the remaining sections only when needed.
-Section 3 introduces the more verbose \cmd{function} command which
-gives fine-grained control. This form should be used
-whenever the short form fails.
-After that we discuss more specialized issues:
-termination, mutual, nested and higher-order recursion, partiality, pattern matching
-and others.
-
-
-\paragraph{Some background.}
-Following the LCF tradition, the package is realized as a definitional
-extension: Recursive definitions are internally transformed into a
-non-recursive form, such that the function can be defined using
-standard definition facilities. Then the recursive specification is
-derived from the primitive definition.  This is a complex task, but it
-is fully automated and mostly transparent to the user. Definitional
-extensions are valuable because they are conservative by construction:
-The \qt{new} concept of general wellfounded recursion is completely reduced
-to existing principles.
-
-
-
-
-The new \cmd{function} command, and its short form \cmd{fun} have mostly
-replaced the traditional \cmd{recdef} command \cite{slind-tfl}. They solve
-a few of technical issues around \cmd{recdef}, and allow definitions
-which were not previously possible.
-
-
-
-
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/isabelle_isar.eps
--- a/doc-src/Functions/isabelle_isar.eps	Mon Aug 27 22:00:04 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,6488 +0,0 @@
-%!PS-Adobe-2.0 EPSF-1.2
-%%Title: isabelle_any
-%%Creator: FreeHand 5.5
-%%CreationDate: 24.09.1998 21:04 Uhr
-%%BoundingBox: 0 0 202 178
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-%ALDOriginalFile:MacSystem:Home:Markus:TUM:Isabelle Logo:export:isabelle_any
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-%%FHPageNum:1
-%%DocumentSuppliedResources: procset Altsys_header 4 0
-%%ColorUsage: Color
-%%DocumentProcessColors: Cyan Magenta Yellow Black
-%%DocumentNeededResources: font Symbol
-%%+ font ZapfHumanist601BT-Bold
-%%DocumentFonts: Symbol
-%%+ ZapfHumanist601BT-Bold
-%%DocumentNeededFonts: Symbol
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-%%EndComments
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-%%CreationDate: Mon Jun 22 16:09:28 1992
-%%VMusage: 35200 38400   
-% Bitstream Type 1 Font Program
-% Copyright 1990-1992 as an unpublished work by Bitstream Inc., Cambridge, MA.
-% All rights reserved.
-% Confidential and proprietary to Bitstream Inc.
-% U.S. GOVERNMENT RESTRICTED RIGHTS
-% This software typeface product is provided with RESTRICTED RIGHTS. Use,
-% duplication or disclosure by the Government is subject to restrictions
-% as set forth in the license agreement and in FAR 52.227-19 (c) (2) (May, 1987),
-% when applicable, or the applicable provisions of the DOD FAR supplement
-% 252.227-7013 subdivision (a) (15) (April, 1988) or subdivision (a) (17)
-% (April, 1988).  Contractor/manufacturer is Bitstream Inc.,
-% 215 First Street, Cambridge, MA 02142.
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-11 dict begin
-/FontInfo 9 dict dup begin
-  /version (003.001) readonly def
-  /Notice (Copyright 1990-1992 as an unpublished work by Bitstream Inc.  All rights reserved.  Confidential.) readonly def
-  /FullName (Zapf Humanist 601 Bold) readonly def
-  /FamilyName (Zapf Humanist 601) readonly def
-  /Weight (Bold) readonly def
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-  /UnderlineThickness 85 def
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-/FontName /ZapfHumanist601BT-Bold def
-/PaintType 0 def
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-/FontMatrix [0.001 0 0 0.001 0 0] readonly def
-/Encoding StandardEncoding def
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-1979.6 2081.1 1981.6 2086.8 1985.3 2084 C
-1993.4 2079.3 2001.8 2075.8 2010 2071.7 C
-2010.5 2071.5 2010.5 2071.1 2010.8 2070.8 C
-2011.2 2064.3 2010.9 2057.5 2011 2050.8 C
-2015.8 2046.9 2022.2 2046.2 2026.6 2041.7 C
-2026.5 2032.5 2026.8 2022.9 2026.4 2014.1 C
-2020.4 2008.3 2015 2002.4 2008.8 1997.1 C
-2003.8 1996.8 2000.7 2001.2 1996.1 2002.1 C
-1995.2 1996.4 1996.9 1990.5 1995.6 1984.8 C
-1989.9 1979 1984.5 1973.9 1978.8 1967.8 C
-1977.7 1968.6 1976 1967.6 1974.5 1968.3 C
-1967.4 1972.5 1960.1 1976.1 1952.7 1979.3 C
-1946.8 1976.3 1943.4 1970.7 1938.5 1966.1 C
-1933.9 1966.5 1929.4 1968.8 1925.1 1970.7 C
-1917.2 1978.2 1906 1977.9 1897.2 1983.4 C
-1893.2 1985.6 1889.4 1988.6 1885 1990.1 C
-1884.6 1990.6 1883.9 1991 1883.8 1991.6 C
-1883.7 2000.4 1884 2009.9 1883.6 2018.9 C
-1887.7 2024 1893.2 2028.8 1898 2033.8 C
-1899.1 2035.5 1900.9 2036.8 1902.5 2037.9 C
-1903.9 2037.3 1905.2 2036.6 1906.4 2035.5 C
-1906.3 2039.7 1906.5 2044.6 1906.1 2048.9 C
-1906.3 2049.6 1906.7 2050.2 1907.1 2050.8 C
-1913.4 2056 1918.5 2062.7 1924.8 2068.1 C
-1926.6 2067.9 1928 2066.9 1929.4 2066 C
-1930.2 2071 1927.7 2077.1 1930.6 2081.6 C
-1936.6 2086.9 1941.5 2092.9 1947.9 2097.9 C
-1949 2098.1 1949.9 2097.5 1950.8 2097.2 C
-[0 0 0 0.18]  vc
-f 
-0.4 w
-S 
-n
-1975.2 2084.7 m
-1976.6 2083.4 1975.7 2081.1 1976 2079.4 C
-1979.3 2079.5 1980.9 2086.2 1984.8 2084 C
-1992.9 2078.9 2001.7 2075.6 2010 2071.2 C
-2011 2064.6 2010.2 2057.3 2010.8 2050.6 C
-2015.4 2046.9 2021.1 2045.9 2025.9 2042.4 C
-2026.5 2033.2 2026.8 2022.9 2025.6 2013.9 C
-2020.5 2008.1 2014.5 2003.1 2009.3 1997.6 C
-2004.1 1996.7 2000.7 2001.6 1995.9 2002.6 C
-1995.2 1996.7 1996.3 1990.2 1994.9 1984.6 C
-1989.8 1978.7 1983.6 1973.7 1978.4 1968 C
-1977.3 1969.3 1976 1967.6 1974.8 1968.5 C
-1967.7 1972.7 1960.4 1976.3 1952.9 1979.6 C
-1946.5 1976.9 1943.1 1970.5 1937.8 1966.1 C
-1928.3 1968.2 1920.6 1974.8 1911.6 1978.4 C
-1901.9 1979.7 1893.9 1986.6 1885 1990.6 C
-1884.3 1991 1884.3 1991.7 1884 1992.3 C
-1884.5 2001 1884.2 2011 1884.3 2019.9 C
-1890.9 2025.3 1895.9 2031.9 1902.3 2037.4 C
-1904.2 2037.9 1905.6 2034.2 1906.8 2035.7 C
-1907.4 2040.9 1905.7 2046.1 1907.3 2050.8 C
-1913.6 2056.2 1919.2 2062.6 1925.1 2067.9 C
-1926.9 2067.8 1928 2066.3 1929.6 2065.7 C
-1929.9 2070.5 1929.2 2076 1930.1 2080.8 C
-1936.5 2086.1 1941.6 2092.8 1948.4 2097.6 C
-1957.3 2093.3 1966.2 2088.8 1975.2 2084.7 C
-[0 0 0 0]  vc
-f 
-S 
-n
-1954.8 2093.8 m
-1961.6 2090.5 1968.2 2087 1975 2084 C
-1975 2082.8 1975.6 2080.9 1974.8 2080.6 C
-1974.3 2075.2 1974.6 2069.6 1974.5 2064 C
-1977.5 2059.7 1984.5 2060 1988.9 2056.4 C
-1989.5 2055.5 1990.5 2055.3 1990.8 2054.4 C
-1991.1 2045.7 1991.4 2036.1 1990.6 2027.8 C
-1990.7 2026.6 1992 2027.3 1992.8 2027.1 C
-1997 2032.4 2002.6 2037.8 2007.6 2042.2 C
-2008.7 2042.3 2007.8 2040.6 2007.4 2040 C
-2002.3 2035.6 1997.5 2030 1992.8 2025.2 C
-1991.6 2024.7 1990.8 2024.9 1990.1 2025.4 C
-1989.4 2024.9 1988.1 2025.2 1987.2 2024.4 C
-1987.1 2025.8 1988.3 2026.5 1989.4 2026.8 C
-1989.4 2026.6 1989.3 2026.2 1989.6 2026.1 C
-1989.9 2026.2 1989.9 2026.6 1989.9 2026.8 C
-1989.8 2026.6 1990 2026.5 1990.1 2026.4 C
-1990.2 2027 1991.1 2028.3 1990.1 2028 C
-1989.9 2037.9 1990.5 2044.1 1989.6 2054.2 C
-1985.9 2058 1979.7 2057.4 1976 2061.2 C
-1974.5 2061.6 1975.2 2059.9 1974.5 2059.5 C
-1973.9 2058 1975.6 2057.8 1975 2056.6 C
-1974.5 2057.1 1974.6 2055.3 1973.6 2055.9 C
-1971.9 2059.3 1974.7 2062.1 1973.1 2065.5 C
-1973.1 2071.2 1972.9 2077 1973.3 2082.5 C
-1967.7 2085.6 1962 2088 1956.3 2090.7 C
-1953.9 2092.4 1951 2093 1948.6 2094.8 C
-1943.7 2089.9 1937.9 2084.3 1933 2079.6 C
-1931.3 2076.1 1933.2 2071.3 1932.3 2067.2 C
-1931.3 2062.9 1933.3 2060.6 1932 2057.6 C
-1932.7 2056.5 1930.9 2053.3 1933.2 2051.8 C
-1936.8 2050.1 1940.1 2046.9 1944 2046.8 C
-1946.3 2049.7 1949.3 2051.9 1952 2054.4 C
-1954.5 2054.2 1956.4 2052.3 1958.7 2051.3 C
-1960.8 2050 1963.2 2049 1965.6 2048.4 C
-1968.3 2050.8 1970.7 2054.3 1973.6 2055.4 C
-1973 2052.2 1969.7 2050.4 1967.6 2048.2 C
-1967.1 2046.7 1968.8 2046.6 1969.5 2045.8 C
-1972.8 2043.3 1980.6 2043.4 1979.3 2038.4 C
-1979.4 2038.6 1979.2 2038.7 1979.1 2038.8 C
-1978.7 2038.6 1978.9 2038.1 1978.8 2037.6 C
-1978.9 2037.9 1978.7 2038 1978.6 2038.1 C
-1978.2 2032.7 1978.4 2027.1 1978.4 2021.6 C
-1979.3 2021.1 1980 2020.2 1981.5 2020.1 C
-1983.5 2020.5 1984 2021.8 1985.1 2023.5 C
-1985.7 2024 1987.4 2023.7 1986 2022.8 C
-1984.7 2021.7 1983.3 2020.8 1983.9 2018.7 C
-1987.2 2015.9 1993 2015.4 1994.9 2011.5 C
-1992.2 2004.9 1999.3 2005.2 2002.1 2002.4 C
-2005.9 2002.7 2004.8 1997.4 2009.1 1999 C
-2011 1999.3 2010 2002.9 2012.7 2002.4 C
-2010.2 2000.7 2009.4 1996.1 2005.5 1998.5 C
-2002.1 2000.3 1999 2002.5 1995.4 2003.8 C
-1995.2 2003.6 1994.9 2003.3 1994.7 2003.1 C
-1994.3 1997 1995.6 1991.1 1994.4 1985.3 C
-1994.3 1986 1993.8 1985 1994 1985.6 C
-1993.8 1995.4 1994.4 2001.6 1993.5 2011.7 C
-1989.7 2015.5 1983.6 2014.9 1979.8 2018.7 C
-1978.3 2019.1 1979.1 2017.4 1978.4 2017 C
-1977.8 2015.5 1979.4 2015.3 1978.8 2014.1 C
-1978.4 2014.6 1978.5 2012.8 1977.4 2013.4 C
-1975.8 2016.8 1978.5 2019.6 1976.9 2023 C
-1977 2028.7 1976.7 2034.5 1977.2 2040 C
-1971.6 2043.1 1965.8 2045.6 1960.1 2048.2 C
-1957.7 2049.9 1954.8 2050.5 1952.4 2052.3 C
-1947.6 2047.4 1941.8 2041.8 1936.8 2037.2 C
-1935.2 2033.6 1937.1 2028.8 1936.1 2024.7 C
-1935.1 2020.4 1937.1 2018.1 1935.9 2015.1 C
-1936.5 2014.1 1934.7 2010.8 1937.1 2009.3 C
-1944.4 2004.8 1952 2000.9 1959.9 1997.8 C
-1963.9 1997 1963.9 2001.9 1966.8 2003.3 C
-1970.3 2006.9 1973.7 2009.9 1976.9 2012.9 C
-1977.9 2013 1977.1 2011.4 1976.7 2010.8 C
-1971.6 2006.3 1966.8 2000.7 1962 1995.9 C
-1960 1995.2 1960.1 1996.6 1958.2 1995.6 C
-1957 1997 1955.1 1998.8 1953.2 1998 C
-1951.7 1994.5 1954.1 1993.4 1952.9 1991.1 C
-1952.1 1990.5 1953.3 1990.2 1953.2 1989.6 C
-1954.2 1986.8 1950.9 1981.4 1954.4 1981.2 C
-1954.7 1981.6 1954.7 1981.7 1955.1 1982 C
-1961.9 1979.1 1967.6 1975 1974.3 1971.6 C
-1974.7 1969.8 1976.7 1969.5 1978.4 1969.7 C
-1980.3 1970 1979.3 1973.6 1982 1973.1 C
-1975.8 1962.2 1968 1975.8 1960.8 1976.7 C
-1956.9 1977.4 1953.3 1982.4 1949.1 1978.8 C
-1946 1975.8 1941.2 1971 1939.5 1969.2 C
-1938.5 1968.6 1938.9 1967.4 1937.8 1966.8 C
-1928.7 1969.4 1920.6 1974.5 1912.4 1979.1 C
-1904 1980 1896.6 1985 1889.3 1989.4 C
-1887.9 1990.4 1885.1 1990.3 1885 1992.5 C
-1885.4 2000.6 1885.2 2012.9 1885.2 2019.9 C
-1886.1 2022 1889.7 2019.5 1888.4 2022.8 C
-1889 2023.3 1889.8 2024.4 1890.3 2024 C
-1891.2 2023.5 1891.8 2028.2 1893.4 2026.6 C
-1894.2 2026.3 1893.9 2027.3 1894.4 2027.6 C
-1893.4 2027.6 1894.7 2028.3 1894.1 2028.5 C
-1894.4 2029.6 1896 2030 1896 2029.2 C
-1896.2 2029 1896.3 2029 1896.5 2029.2 C
-1896.8 2029.8 1897.3 2030 1897 2030.7 C
-1896.5 2030.7 1896.9 2031.5 1897.2 2031.6 C
-1898.3 2034 1899.5 2030.6 1899.6 2033.3 C
-1898.5 2033 1899.6 2034.4 1900.1 2034.8 C
-1901.3 2035.8 1903.2 2034.6 1902.5 2036.7 C
-1904.4 2036.9 1906.1 2032.2 1907.6 2035.5 C
-1907.5 2040.1 1907.7 2044.9 1907.3 2049.4 C
-1908 2050.2 1908.3 2051.4 1909.5 2051.6 C
-1910.1 2051.1 1911.6 2051.1 1911.4 2052.3 C
-1909.7 2052.8 1912.4 2054 1912.6 2054.7 C
-1913.4 2055.2 1913 2053.7 1913.6 2054.4 C
-1913.6 2054.5 1913.6 2055.3 1913.6 2054.7 C
-1913.7 2054.4 1913.9 2054.4 1914 2054.7 C
-1914 2054.9 1914.1 2055.3 1913.8 2055.4 C
-1913.7 2056 1915.2 2057.6 1916 2057.6 C
-1915.9 2057.3 1916.1 2057.2 1916.2 2057.1 C
-1917 2056.8 1916.7 2057.7 1917.2 2058 C
-1917 2058.3 1916.7 2058.3 1916.4 2058.3 C
-1917.1 2059 1917.3 2060.1 1918.4 2060.4 C
-1918.1 2059.2 1919.1 2060.6 1919.1 2059.5 C
-1919 2060.6 1920.6 2060.1 1919.8 2061.2 C
-1919.6 2061.2 1919.3 2061.2 1919.1 2061.2 C
-1919.6 2061.9 1921.4 2064.2 1921.5 2062.6 C
-1922.4 2062.1 1921.6 2063.9 1922.2 2064.3 C
-1922.9 2067.3 1926.1 2064.3 1925.6 2067.2 C
-1927.2 2066.8 1928.4 2064.6 1930.1 2065.2 C
-1931.8 2067.8 1931 2071.8 1930.8 2074.8 C
-1930.6 2076.4 1930.1 2078.6 1930.6 2080.4 C
-1936.6 2085.4 1941.8 2091.6 1948.1 2096.9 C
-1950.7 2096.7 1952.6 2094.8 1954.8 2093.8 C
-[0 0.33 0.33 0.99]  vc
-f 
-S 
-n
-1989.4 2080.6 m
-1996.1 2077.3 2002.7 2073.8 2009.6 2070.8 C
-2009.6 2069.6 2010.2 2067.7 2009.3 2067.4 C
-2008.9 2062 2009.1 2056.4 2009.1 2050.8 C
-2012.3 2046.6 2019 2046.6 2023.5 2043.2 C
-2024 2042.3 2025.1 2042.1 2025.4 2041.2 C
-2025.3 2032.7 2025.6 2023.1 2025.2 2014.6 C
-2025 2015.3 2024.6 2014.2 2024.7 2014.8 C
-2024.5 2024.7 2025.1 2030.9 2024.2 2041 C
-2020.4 2044.8 2014.3 2044.2 2010.5 2048 C
-2009 2048.4 2009.8 2046.7 2009.1 2046.3 C
-2008.5 2044.8 2010.2 2044.6 2009.6 2043.4 C
-2009.1 2043.9 2009.2 2042.1 2008.1 2042.7 C
-2006.5 2046.1 2009.3 2048.9 2007.6 2052.3 C
-2007.7 2058 2007.5 2063.8 2007.9 2069.3 C
-2002.3 2072.4 1996.5 2074.8 1990.8 2077.5 C
-1988.4 2079.2 1985.6 2079.8 1983.2 2081.6 C
-1980.5 2079 1977.9 2076.5 1975.5 2074.1 C
-1975.5 2075.1 1975.5 2076.2 1975.5 2077.2 C
-1977.8 2079.3 1980.3 2081.6 1982.7 2083.7 C
-1985.3 2083.5 1987.1 2081.6 1989.4 2080.6 C
-f 
-S 
-n
-1930.1 2079.9 m
-1931.1 2075.6 1929.2 2071.1 1930.8 2067.2 C
-1930.3 2066.3 1930.1 2064.6 1928.7 2065.5 C
-1927.7 2066.4 1926.5 2067 1925.3 2067.4 C
-1924.5 2066.9 1925.6 2065.7 1924.4 2066 C
-1924.2 2067.2 1923.6 2065.5 1923.2 2065.7 C
-1922.3 2063.6 1917.8 2062.1 1919.6 2060.4 C
-1919.3 2060.5 1919.2 2060.3 1919.1 2060.2 C
-1919.7 2060.9 1918.2 2061 1917.6 2060.2 C
-1917 2059.6 1916.1 2058.8 1916.4 2058 C
-1915.5 2058 1917.4 2057.1 1915.7 2057.8 C
-1914.8 2057.1 1913.4 2056.2 1913.3 2054.9 C
-1913.1 2055.4 1911.3 2054.3 1910.9 2053.2 C
-1910.7 2052.9 1910.2 2052.5 1910.7 2052.3 C
-1911.1 2052.5 1910.9 2052 1910.9 2051.8 C
-1910.5 2051.2 1909.9 2052.6 1909.2 2051.8 C
-1908.2 2051.4 1907.8 2050.2 1907.1 2049.4 C
-1907.5 2044.8 1907.3 2040 1907.3 2035.2 C
-1905.3 2033 1902.8 2039.3 1902.3 2035.7 C
-1899.6 2036 1898.4 2032.5 1896.3 2030.7 C
-1895.7 2030.1 1897.5 2030 1896.3 2029.7 C
-1896.3 2030.6 1895 2029.7 1894.4 2029.2 C
-1892.9 2028.1 1894.2 2027.4 1893.6 2027.1 C
-1892.1 2027.9 1891.7 2025.6 1890.8 2024.9 C
-1891.1 2024.6 1889.1 2024.3 1888.4 2023 C
-1887.5 2022.6 1888.2 2021.9 1888.1 2021.3 C
-1886.7 2022 1885.2 2020.4 1884.8 2019.2 C
-1884.8 2010 1884.6 2000.2 1885 1991.8 C
-1886.9 1989.6 1889.9 1989.3 1892.2 1987.5 C
-1898.3 1982.7 1905.6 1980.1 1912.8 1978.6 C
-1921 1974.2 1928.8 1968.9 1937.8 1966.6 C
-1939.8 1968.3 1938.8 1968.3 1940.4 1970 C
-1945.4 1972.5 1947.6 1981.5 1954.6 1979.3 C
-1952.3 1981 1950.4 1978.4 1948.6 1977.9 C
-1945.1 1973.9 1941.1 1970.6 1938 1966.6 C
-1928.4 1968.5 1920.6 1974.8 1911.9 1978.8 C
-1907.1 1979.2 1902.6 1981.7 1898.2 1983.6 C
-1893.9 1986 1889.9 1989 1885.5 1990.8 C
-1884.9 1991.2 1884.8 1991.8 1884.5 1992.3 C
-1884.9 2001.3 1884.7 2011.1 1884.8 2019.6 C
-1890.6 2025 1896.5 2031.2 1902.3 2036.9 C
-1904.6 2037.6 1905 2033 1907.3 2035.5 C
-1907.2 2040.2 1907 2044.8 1907.1 2049.6 C
-1913.6 2055.3 1918.4 2061.5 1925.1 2067.4 C
-1927.3 2068.2 1929.6 2062.5 1930.6 2066.9 C
-1929.7 2070.7 1930.3 2076 1930.1 2080.1 C
-1935.6 2085.7 1941.9 2090.7 1947.2 2096.7 C
-1942.2 2091.1 1935.5 2085.2 1930.1 2079.9 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1930.8 2061.9 m
-1930.3 2057.8 1931.8 2053.4 1931.1 2050.4 C
-1931.3 2050.3 1931.7 2050.5 1931.6 2050.1 C
-1933 2051.1 1934.4 2049.5 1935.9 2048.7 C
-1937 2046.5 1939.5 2047.1 1941.2 2045.1 C
-1939.7 2042.6 1937.3 2041.2 1935.4 2039.3 C
-1934 2039.7 1934.5 2038.1 1933.7 2037.6 C
-1934 2033.3 1933.1 2027.9 1934.4 2024.4 C
-1934.3 2023.8 1933.9 2022.8 1933 2022.8 C
-1931.6 2023.1 1930.5 2024.4 1929.2 2024.9 C
-1928.4 2024.5 1929.8 2023.5 1928.7 2023.5 C
-1927.7 2024.1 1926.2 2022.6 1925.6 2021.6 C
-1926.9 2021.6 1924.8 2020.6 1925.6 2020.4 C
-1924.7 2021.7 1923.9 2019.6 1923.2 2019.2 C
-1923.3 2018.3 1923.8 2018.1 1923.2 2018 C
-1922.9 2017.8 1922.9 2017.5 1922.9 2017.2 C
-1922.8 2018.3 1921.3 2017.3 1920.3 2018 C
-1916.6 2019.7 1913 2022.1 1910 2024.7 C
-1910 2032.9 1910 2041.2 1910 2049.4 C
-1915.4 2055.2 1920 2058.7 1925.3 2064.8 C
-1927.2 2064 1929 2061.4 1930.8 2061.9 C
-[0 0 0 0]  vc
-f 
-S 
-n
-1907.6 2030.4 m
-1907.5 2027.1 1906.4 2021.7 1908.5 2019.9 C
-1908.8 2020.1 1908.9 2019 1909.2 2019.6 C
-1910 2019.6 1912 2019.2 1913.1 2018.2 C
-1913.7 2016.5 1920.2 2015.7 1917.4 2012.7 C
-1918.2 2011.2 1917 2013.8 1917.2 2012 C
-1916.9 2012.3 1916 2012.4 1915.2 2012 C
-1912.5 2010.5 1916.6 2008.8 1913.6 2009.6 C
-1912.6 2009.2 1911.1 2009 1910.9 2007.6 C
-1911 1999.2 1911.8 1989.8 1911.2 1982.2 C
-1910.1 1981.1 1908.8 1982.2 1907.6 1982.2 C
-1900.8 1986.5 1893.2 1988.8 1887.2 1994.2 C
-1887.2 2002.4 1887.2 2010.7 1887.2 2018.9 C
-1892.6 2024.7 1897.2 2028.2 1902.5 2034.3 C
-1904.3 2033.3 1906.2 2032.1 1907.6 2030.4 C
-f 
-S 
-n
-1910.7 2025.4 m
-1912.7 2022.4 1916.7 2020.8 1919.8 2018.9 C
-1920.2 2018.7 1920.6 2018.6 1921 2018.4 C
-1925 2020 1927.4 2028.5 1932 2024.2 C
-1932.3 2025 1932.5 2023.7 1932.8 2024.4 C
-1932.8 2028 1932.8 2031.5 1932.8 2035 C
-1931.9 2033.9 1932.5 2036.3 1932.3 2036.9 C
-1933.2 2036.4 1932.5 2038.5 1933 2038.4 C
-1933.1 2040.5 1935.6 2042.2 1936.6 2043.2 C
-1936.2 2042.4 1935.1 2040.8 1933.7 2040.3 C
-1932.2 2034.4 1933.8 2029.8 1933 2023.2 C
-1931.1 2024.9 1928.4 2026.4 1926.5 2023.5 C
-1925.1 2021.6 1923 2019.8 1921.5 2018.2 C
-1917.8 2018.9 1915.2 2022.5 1911.6 2023.5 C
-1910.8 2023.8 1911.2 2024.7 1910.4 2025.2 C
-1910.9 2031.8 1910.6 2039.1 1910.7 2045.6 C
-1910.1 2048 1910.7 2045.9 1911.2 2044.8 C
-1910.6 2038.5 1911.2 2031.8 1910.7 2025.4 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1910.7 2048.9 m
-1910.3 2047.4 1911.3 2046.5 1911.6 2045.3 C
-1912.9 2045.3 1913.9 2047.1 1915.2 2045.8 C
-1915.2 2044.9 1916.6 2043.3 1917.2 2042.9 C
-1918.7 2042.9 1919.4 2044.4 1920.5 2043.2 C
-1921.2 2042.2 1921.4 2040.9 1922.4 2040.3 C
-1924.5 2040.3 1925.7 2040.9 1926.8 2039.6 C
-1927.1 2037.9 1926.8 2038.1 1927.7 2037.6 C
-1929 2037.5 1930.4 2037 1931.6 2037.2 C
-1932.3 2038.2 1933.1 2038.7 1932.8 2040.3 C
-1935 2041.8 1935.9 2043.8 1938.5 2044.8 C
-1938.6 2045 1938.3 2045.5 1938.8 2045.3 C
-1939.1 2042.9 1935.4 2044.2 1935.4 2042.2 C
-1932.1 2040.8 1932.8 2037.2 1932 2034.8 C
-1932.3 2034 1932.7 2035.4 1932.5 2034.8 C
-1931.3 2031.8 1935.5 2020.1 1928.9 2025.9 C
-1924.6 2024.7 1922.6 2014.5 1917.4 2020.4 C
-1915.5 2022.8 1912 2022.6 1910.9 2025.4 C
-1911.5 2031.9 1910.9 2038.8 1911.4 2045.3 C
-1911.1 2046.5 1910 2047.4 1910.4 2048.9 C
-1915.1 2054.4 1920.4 2058.3 1925.1 2063.8 C
-1920.8 2058.6 1914.9 2054.3 1910.7 2048.9 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1934.7 2031.9 m
-1934.6 2030.7 1934.9 2029.5 1934.4 2028.5 C
-1934 2029.5 1934.3 2031.2 1934.2 2032.6 C
-1933.8 2031.7 1934.9 2031.6 1934.7 2031.9 C
-[0.92 0.92 0 0.67]  vc
-f 
-S 
-n
-vmrs
-1934.7 2019.4 m
-1934.1 2015.3 1935.6 2010.9 1934.9 2007.9 C
-1935.1 2007.8 1935.6 2008.1 1935.4 2007.6 C
-1936.8 2008.6 1938.2 2007 1939.7 2006.2 C
-1940.1 2004.3 1942.7 2005 1943.6 2003.8 C
-1945.1 2000.3 1954 2000.8 1950 1996.6 C
-1952.1 1993.3 1948.2 1989.2 1951.2 1985.6 C
-1953 1981.4 1948.4 1982.3 1947.9 1979.8 C
-1945.4 1979.6 1945.1 1975.5 1942.4 1975 C
-1942.4 1972.3 1938 1973.6 1938.5 1970.4 C
-1937.4 1969 1935.6 1970.1 1934.2 1970.2 C
-1927.5 1974.5 1919.8 1976.8 1913.8 1982.2 C
-1913.8 1990.4 1913.8 1998.7 1913.8 2006.9 C
-1919.3 2012.7 1923.8 2016.2 1929.2 2022.3 C
-1931.1 2021.6 1932.8 2018.9 1934.7 2019.4 C
-[0 0 0 0]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-2024.2 2038.1 m
-2024.1 2029.3 2024.4 2021.7 2024.7 2014.4 C
-2024.4 2013.6 2020.6 2013.4 2021.3 2011.2 C
-2020.5 2010.3 2018.4 2010.6 2018.9 2008.6 C
-2019 2008.8 2018.8 2009 2018.7 2009.1 C
-2018.2 2006.7 2015.2 2007.9 2015.3 2005.5 C
-2014.7 2004.8 2012.4 2005.1 2013.2 2003.6 C
-2012.3 2004.2 2012.8 2002.4 2012.7 2002.6 C
-2009.4 2003.3 2011.2 1998.6 2008.4 1999.2 C
-2007 1999.1 2006.1 1999.4 2005.7 2000.4 C
-2006.9 1998.5 2007.7 2000.5 2009.3 2000.2 C
-2009.2 2003.7 2012.4 2002.1 2012.9 2005.2 C
-2015.9 2005.6 2015.2 2008.6 2017.7 2008.8 C
-2018.4 2009.6 2018.3 2011.4 2019.6 2011 C
-2021.1 2011.7 2021.4 2014.8 2023.7 2015.1 C
-2023.7 2023.5 2023.9 2031.6 2023.5 2040.5 C
-2021.8 2041.7 2020.7 2043.6 2018.4 2043.9 C
-2020.8 2042.7 2025.5 2041.8 2024.2 2038.1 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-2023.5 2040 m
-2023.5 2031.1 2023.5 2023.4 2023.5 2015.1 C
-2020.2 2015 2021.8 2010.3 2018.4 2011 C
-2018.6 2007.5 2014.7 2009.3 2014.8 2006.4 C
-2011.8 2006.3 2012.2 2002.3 2009.8 2002.4 C
-2009.7 2001.5 2009.2 2000.1 2008.4 2000.2 C
-2008.7 2000.9 2009.7 2001.2 2009.3 2002.4 C
-2008.4 2004.2 2007.5 2003.1 2007.9 2005.5 C
-2007.9 2010.8 2007.7 2018.7 2008.1 2023.2 C
-2009 2024.3 2007.3 2023.4 2007.9 2024 C
-2007.7 2024.6 2007.3 2026.3 2008.6 2027.1 C
-2009.7 2026.8 2010 2027.6 2010.5 2028 C
-2010.5 2028.2 2010.5 2029.1 2010.5 2028.5 C
-2011.5 2028 2010.5 2030 2011.5 2030 C
-2014.2 2029.7 2012.9 2032.2 2014.8 2032.6 C
-2015.1 2033.6 2015.3 2033 2016 2033.3 C
-2017 2033.9 2016.6 2035.4 2017.2 2036.2 C
-2018.7 2036.4 2019.2 2039 2021.3 2038.4 C
-2021.6 2035.4 2019.7 2029.5 2021.1 2027.3 C
-2020.9 2023.5 2021.5 2018.5 2020.6 2016 C
-2020.9 2013.9 2021.5 2015.4 2022.3 2014.4 C
-2022.2 2015.1 2023.3 2014.8 2023.2 2015.6 C
-2022.7 2019.8 2023.3 2024.3 2022.8 2028.5 C
-2022.3 2028.2 2022.6 2027.6 2022.5 2027.1 C
-2022.5 2027.8 2022.5 2029.2 2022.5 2029.2 C
-2022.6 2029.2 2022.7 2029.1 2022.8 2029 C
-2023.9 2032.8 2022.6 2037 2023 2040.8 C
-2022.3 2041.2 2021.6 2041.5 2021.1 2042.2 C
-2022 2041.2 2022.9 2041.4 2023.5 2040 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-2009.1 1997.8 m
-2003.8 1997.7 2000.1 2002.4 1995.4 2003.1 C
-1995 1999.5 1995.2 1995 1995.2 1992 C
-1995.2 1995.8 1995 1999.7 1995.4 2003.3 C
-2000.3 2002.2 2003.8 1997.9 2009.1 1997.8 C
-2012.3 2001.2 2015.6 2004.8 2018.7 2008.1 C
-2021.6 2011.2 2027.5 2013.9 2025.9 2019.9 C
-2026.1 2017.9 2025.6 2016.2 2025.4 2014.4 C
-2020.2 2008.4 2014 2003.6 2009.1 1997.8 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-2009.3 1997.8 m
-2008.7 1997.4 2007.9 1997.6 2007.2 1997.6 C
-2007.9 1997.6 2008.9 1997.4 2009.6 1997.8 C
-2014.7 2003.6 2020.8 2008.8 2025.9 2014.8 C
-2025.8 2017.7 2026.1 2014.8 2025.6 2014.1 C
-2020.4 2008.8 2014.8 2003.3 2009.3 1997.8 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-2009.6 1997.6 m
-2009 1997.1 2008.1 1997.4 2007.4 1997.3 C
-2008.1 1997.4 2009 1997.1 2009.6 1997.6 C
-2014.8 2003.7 2021.1 2008.3 2025.9 2014.4 C
-2021.1 2008.3 2014.7 2003.5 2009.6 1997.6 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-2021.8 2011.5 m
-2021.9 2012.2 2022.3 2013.5 2023.7 2013.6 C
-2023.4 2012.7 2022.8 2011.8 2021.8 2011.5 C
-[0 0.33 0.33 0.99]  vc
-f 
-S 
-n
-2021.1 2042 m
-2022.1 2041.1 2020.9 2040.2 2020.6 2039.6 C
-2018.4 2039.5 2018.1 2036.9 2016.3 2036.4 C
-2015.8 2035.5 2015.3 2033.8 2014.8 2033.6 C
-2012.4 2033.8 2013 2030.4 2010.5 2030.2 C
-2009.6 2028.9 2009.6 2028.3 2008.4 2028 C
-2006.9 2026.7 2007.5 2024.3 2006 2023.2 C
-2006.6 2023.2 2005.7 2023.3 2005.7 2023 C
-2006.4 2022.5 2006.3 2021.1 2006.7 2020.6 C
-2006.6 2015 2006.9 2009 2006.4 2003.8 C
-2006.9 2002.5 2007.6 2001.1 2006.9 2000.7 C
-2004.6 2003.6 2003 2002.9 2000.2 2004.3 C
-1999.3 2005.8 1997.9 2006.3 1996.1 2006.7 C
-1995.7 2008.9 1996 2011.1 1995.9 2012.9 C
-1993.4 2015.1 1990.5 2016.2 1987.7 2017.7 C
-1987.1 2019.3 1991.1 2019.4 1990.4 2021.3 C
-1990.5 2021.5 1991.9 2022.3 1992 2023 C
-1994.8 2024.4 1996.2 2027.5 1998.5 2030 C
-2002.4 2033 2005.2 2037.2 2008.8 2041 C
-2010.2 2041.3 2011.6 2042 2011 2043.9 C
-2011.2 2044.8 2010.1 2045.3 2010.5 2046.3 C
-2013.8 2044.8 2017.5 2043.4 2021.1 2042 C
-[0 0.5 0.5 0.2]  vc
-f 
-S 
-n
-2019.4 2008.8 m
-2018.9 2009.2 2019.3 2009.9 2019.6 2010.3 C
-2022.2 2011.5 2020.3 2009.1 2019.4 2008.8 C
-[0 0.33 0.33 0.99]  vc
-f 
-S 
-n
-2018 2007.4 m
-2015.7 2006.7 2015.3 2003.6 2012.9 2002.8 C
-2013.5 2003.7 2013.5 2005.1 2015.6 2005.2 C
-2016.4 2006.1 2015.7 2007.7 2018 2007.4 C
-f 
-S 
-n
-vmrs
-1993.5 2008.8 m
-1993.4 2000 1993.7 1992.5 1994 1985.1 C
-1993.7 1984.3 1989.9 1984.1 1990.6 1982 C
-1989.8 1981.1 1987.7 1981.4 1988.2 1979.3 C
-1988.3 1979.6 1988.1 1979.7 1988 1979.8 C
-1987.5 1977.5 1984.5 1978.6 1984.6 1976.2 C
-1983.9 1975.5 1981.7 1975.8 1982.4 1974.3 C
-1981.6 1974.9 1982.1 1973.1 1982 1973.3 C
-1979 1973.7 1980 1968.8 1976.9 1969.7 C
-1975.9 1969.8 1975.3 1970.3 1975 1971.2 C
-1976.2 1969.2 1977 1971.2 1978.6 1970.9 C
-1978.5 1974.4 1981.7 1972.8 1982.2 1976 C
-1985.2 1976.3 1984.5 1979.3 1987 1979.6 C
-1987.7 1980.3 1987.5 1982.1 1988.9 1981.7 C
-1990.4 1982.4 1990.7 1985.5 1993 1985.8 C
-1992.9 1994.3 1993.2 2002.3 1992.8 2011.2 C
-1991.1 2012.4 1990 2014.4 1987.7 2014.6 C
-1990.1 2013.4 1994.7 2012.6 1993.5 2008.8 C
-[0 0.87 0.91 0.83]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1992.8 2010.8 m
-1992.8 2001.8 1992.8 1994.1 1992.8 1985.8 C
-1989.5 1985.7 1991.1 1981.1 1987.7 1981.7 C
-1987.9 1978.2 1983.9 1980 1984.1 1977.2 C
-1981.1 1977 1981.5 1973 1979.1 1973.1 C
-1979 1972.2 1978.5 1970.9 1977.6 1970.9 C
-1977.9 1971.6 1979 1971.9 1978.6 1973.1 C
-1977.6 1974.9 1976.8 1973.9 1977.2 1976.2 C
-1977.2 1981.5 1977 1989.4 1977.4 1994 C
-1978.3 1995 1976.6 1994.1 1977.2 1994.7 C
-1977 1995.3 1976.6 1997 1977.9 1997.8 C
-1979 1997.5 1979.3 1998.3 1979.8 1998.8 C
-1979.8 1998.9 1979.8 1999.8 1979.8 1999.2 C
-1980.8 1998.7 1979.7 2000.7 1980.8 2000.7 C
-1983.5 2000.4 1982.1 2003 1984.1 2003.3 C
-1984.4 2004.3 1984.5 2003.7 1985.3 2004 C
-1986.3 2004.6 1985.9 2006.1 1986.5 2006.9 C
-1988 2007.1 1988.4 2009.7 1990.6 2009.1 C
-1990.9 2006.1 1989 2000.2 1990.4 1998 C
-1990.2 1994.3 1990.8 1989.2 1989.9 1986.8 C
-1990.2 1984.7 1990.8 1986.2 1991.6 1985.1 C
-1991.5 1985.9 1992.6 1985.5 1992.5 1986.3 C
-1992 1990.5 1992.6 1995 1992 1999.2 C
-1991.6 1998.9 1991.9 1998.3 1991.8 1997.8 C
-1991.8 1998.5 1991.8 2000 1991.8 2000 C
-1991.9 1999.9 1992 1999.8 1992 1999.7 C
-1993.2 2003.5 1991.9 2007.7 1992.3 2011.5 C
-1991.6 2012 1990.9 2012.2 1990.4 2012.9 C
-1991.3 2011.9 1992.2 2012.1 1992.8 2010.8 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1978.4 1968.5 m
-1977 1969.2 1975.8 1968.2 1974.5 1969 C
-1968.3 1973 1961.6 1976 1955.1 1979.1 C
-1962 1975.9 1968.8 1972.5 1975.5 1968.8 C
-1976.5 1968.8 1977.6 1968.8 1978.6 1968.8 C
-1981.7 1972.1 1984.8 1975.7 1988 1978.8 C
-1990.9 1981.9 1996.8 1984.6 1995.2 1990.6 C
-1995.3 1988.6 1994.9 1986.9 1994.7 1985.1 C
-1989.5 1979.1 1983.3 1974.3 1978.4 1968.5 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1978.4 1968.3 m
-1977.9 1968.7 1977.1 1968.5 1976.4 1968.5 C
-1977.3 1968.8 1978.1 1967.9 1978.8 1968.5 C
-1984 1974.3 1990.1 1979.5 1995.2 1985.6 C
-1995.1 1988.4 1995.3 1985.6 1994.9 1984.8 C
-1989.5 1979.4 1983.9 1973.8 1978.4 1968.3 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1978.6 1968 m
-1977.9 1968 1977.4 1968.6 1978.4 1968 C
-1983.9 1973.9 1990.1 1979.1 1995.2 1985.1 C
-1990.2 1979 1983.8 1974.1 1978.6 1968 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1991.1 1982.2 m
-1991.2 1982.9 1991.6 1984.2 1993 1984.4 C
-1992.6 1983.5 1992.1 1982.5 1991.1 1982.2 C
-[0 0.33 0.33 0.99]  vc
-f 
-S 
-n
-1990.4 2012.7 m
-1991.4 2011.8 1990.2 2010.9 1989.9 2010.3 C
-1987.7 2010.2 1987.4 2007.6 1985.6 2007.2 C
-1985.1 2006.2 1984.6 2004.5 1984.1 2004.3 C
-1981.7 2004.5 1982.3 2001.2 1979.8 2000.9 C
-1978.8 1999.6 1978.8 1999.1 1977.6 1998.8 C
-1976.1 1997.4 1976.7 1995 1975.2 1994 C
-1975.8 1994 1975 1994 1975 1993.7 C
-1975.7 1993.2 1975.6 1991.8 1976 1991.3 C
-1975.9 1985.7 1976.1 1979.7 1975.7 1974.5 C
-1976.2 1973.3 1976.9 1971.8 1976.2 1971.4 C
-1973.9 1974.3 1972.2 1973.6 1969.5 1975 C
-1967.9 1977.5 1963.8 1977.1 1961.8 1980 C
-1959 1980 1957.6 1983 1954.8 1982.9 C
-1953.8 1984.2 1954.8 1985.7 1955.1 1987.2 C
-1956.2 1989.5 1959.7 1990.1 1959.9 1991.8 C
-1965.9 1998 1971.8 2005.2 1978.1 2011.7 C
-1979.5 2012 1980.9 2012.7 1980.3 2014.6 C
-1980.5 2015.6 1979.4 2016 1979.8 2017 C
-1983 2015.6 1986.8 2014.1 1990.4 2012.7 C
-[0 0.5 0.5 0.2]  vc
-f 
-S 
-n
-1988.7 1979.6 m
-1988.2 1979.9 1988.6 1980.6 1988.9 1981 C
-1991.4 1982.2 1989.6 1979.9 1988.7 1979.6 C
-[0 0.33 0.33 0.99]  vc
-f 
-S 
-n
-1987.2 1978.1 m
-1985 1977.5 1984.6 1974.3 1982.2 1973.6 C
-1982.7 1974.5 1982.8 1975.8 1984.8 1976 C
-1985.7 1976.9 1985 1978.4 1987.2 1978.1 C
-f 
-S 
-n
-1975.5 2084 m
-1975.5 2082 1975.3 2080 1975.7 2078.2 C
-1978.8 2079 1980.9 2085.5 1984.8 2083.5 C
-1993 2078.7 2001.6 2075 2010 2070.8 C
-2010.1 2064 2009.9 2057.2 2010.3 2050.6 C
-2014.8 2046.2 2020.9 2045.7 2025.6 2042 C
-2026.1 2035.1 2025.8 2028 2025.9 2021.1 C
-2025.8 2027.8 2026.1 2034.6 2025.6 2041.2 C
-2022.2 2044.9 2017.6 2046.8 2012.9 2048 C
-2012.5 2049.5 2010.4 2049.4 2009.8 2051.1 C
-2009.9 2057.6 2009.6 2064.2 2010 2070.5 C
-2001.2 2075.4 1992 2079.1 1983.2 2084 C
-1980.3 2082.3 1977.8 2079.2 1975.2 2077.5 C
-1974.9 2079.9 1977.2 2084.6 1973.3 2085.2 C
-1964.7 2088.6 1956.8 2093.7 1948.1 2097.2 C
-1949 2097.3 1949.6 2096.9 1950.3 2096.7 C
-1958.4 2091.9 1967.1 2088.2 1975.5 2084 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-vmrs
-1948.6 2094.5 m
-1950.2 2093.7 1951.8 2092.9 1953.4 2092.1 C
-1951.8 2092.9 1950.2 2093.7 1948.6 2094.5 C
-[0 0.87 0.91 0.83]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1971.6 2082.3 m
-1971.6 2081.9 1970.7 2081.1 1970.9 2081.3 C
-1970.7 2081.6 1970.6 2081.6 1970.4 2081.3 C
-1970.8 2080.1 1968.7 2081.7 1968.3 2080.8 C
-1966.6 2080.9 1966.7 2078 1964.2 2078.2 C
-1964.8 2075 1960.1 2075.8 1960.1 2072.9 C
-1958 2072.3 1957.5 2069.3 1955.3 2069.3 C
-1953.9 2070.9 1948.8 2067.8 1950 2072 C
-1949 2074 1943.2 2070.6 1944 2074.8 C
-1942.2 2076.6 1937.6 2073.9 1938 2078.2 C
-1936.7 2078.6 1935 2078.6 1933.7 2078.2 C
-1933.5 2080 1936.8 2080.7 1937.3 2082.8 C
-1939.9 2083.5 1940.6 2086.4 1942.6 2088 C
-1945.2 2089.2 1946 2091.3 1948.4 2093.6 C
-1956 2089.5 1963.9 2086.1 1971.6 2082.3 C
-[0 0.01 1 0]  vc
-f 
-S 
-n
-1958.2 2089.7 m
-1956.4 2090 1955.6 2091.3 1953.9 2091.9 C
-1955.6 2091.9 1956.5 2089.7 1958.2 2089.7 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-1929.9 2080.4 m
-1929.5 2077.3 1929.7 2073.9 1929.6 2070.8 C
-1929.8 2074.1 1929.2 2077.8 1930.1 2080.8 C
-1935.8 2085.9 1941.4 2091.3 1946.9 2096.9 C
-1941.2 2091 1935.7 2086 1929.9 2080.4 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1930.1 2080.4 m
-1935.8 2086 1941.5 2090.7 1946.9 2096.7 C
-1941.5 2090.9 1935.7 2085.8 1930.1 2080.4 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1940.9 2087.1 m
-1941.7 2088 1944.8 2090.6 1943.6 2089.2 C
-1942.5 2089 1941.6 2087.7 1940.9 2087.1 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-1972.8 2082.8 m
-1973 2075.3 1972.4 2066.9 1973.3 2059.5 C
-1972.5 2058.9 1972.8 2057.3 1973.1 2056.4 C
-1974.8 2055.2 1973.4 2055.5 1972.4 2055.4 C
-1970.1 2053.2 1967.9 2050.9 1965.6 2048.7 C
-1960.9 2049.9 1956.9 2052.7 1952.4 2054.7 C
-1949.3 2052.5 1946.3 2049.5 1943.6 2046.8 C
-1939.9 2047.7 1936.8 2050.1 1933.5 2051.8 C
-1930.9 2054.9 1933.5 2056.2 1932.3 2059.7 C
-1933.2 2059.7 1932.2 2060.5 1932.5 2060.2 C
-1933.2 2062.5 1931.6 2064.6 1932.5 2067.4 C
-1932.9 2069.7 1932.7 2072.2 1932.8 2074.6 C
-1933.6 2070.6 1932.2 2066.3 1933 2062.6 C
-1934.4 2058.2 1929.8 2053.5 1935.2 2051.1 C
-1937.7 2049.7 1940.2 2048 1942.8 2046.8 C
-1945.9 2049.2 1948.8 2052 1951.7 2054.7 C
-1952.7 2054.7 1953.6 2054.6 1954.4 2054.2 C
-1958.1 2052.5 1961.7 2049.3 1965.9 2049.2 C
-1968.2 2052.8 1975.2 2055 1972.6 2060.9 C
-1973.3 2062.4 1972.2 2065.2 1972.6 2067.6 C
-1972.7 2072.6 1972.4 2077.7 1972.8 2082.5 C
-1968.1 2084.9 1963.5 2087.5 1958.7 2089.5 C
-1963.5 2087.4 1968.2 2085 1972.8 2082.8 C
-f 
-S 
-n
-1935.2 2081.1 m
-1936.8 2083.4 1938.6 2084.6 1940.4 2086.6 C
-1938.8 2084.4 1936.7 2083.4 1935.2 2081.1 C
-f 
-S 
-n
-1983.2 2081.3 m
-1984.8 2080.5 1986.3 2079.7 1988 2078.9 C
-1986.3 2079.7 1984.8 2080.5 1983.2 2081.3 C
-f 
-S 
-n
-2006.2 2069.1 m
-2006.2 2068.7 2005.2 2067.9 2005.5 2068.1 C
-2005.3 2068.4 2005.2 2068.4 2005 2068.1 C
-2005.4 2066.9 2003.3 2068.5 2002.8 2067.6 C
-2001.2 2067.7 2001.2 2064.8 1998.8 2065 C
-1999.4 2061.8 1994.7 2062.6 1994.7 2059.7 C
-1992.4 2059.5 1992.4 2055.8 1990.1 2056.8 C
-1985.9 2059.5 1981.1 2061 1976.9 2063.8 C
-1977.2 2067.6 1974.9 2074.2 1978.8 2075.8 C
-1979.6 2077.8 1981.7 2078.4 1982.9 2080.4 C
-1990.6 2076.3 1998.5 2072.9 2006.2 2069.1 C
-[0 0.01 1 0]  vc
-f 
-S 
-n
-vmrs
-1992.8 2076.5 m
-1991 2076.8 1990.2 2078.1 1988.4 2078.7 C
-1990.2 2078.7 1991 2076.5 1992.8 2076.5 C
-[0 0.87 0.91 0.83]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1975.5 2073.4 m
-1976.1 2069.7 1973.9 2064.6 1977.4 2062.4 C
-1973.9 2064.5 1976.1 2069.9 1975.5 2073.6 C
-1976 2074.8 1979.3 2077.4 1978.1 2076 C
-1977 2075.7 1975.8 2074.5 1975.5 2073.4 C
-f 
-S 
-n
-2007.4 2069.6 m
-2007.6 2062.1 2007 2053.7 2007.9 2046.3 C
-2007.1 2045.7 2007.3 2044.1 2007.6 2043.2 C
-2009.4 2042 2007.9 2042.3 2006.9 2042.2 C
-2002.2 2037.4 1996.7 2032.4 1992.5 2027.3 C
-1992 2027.3 1991.6 2027.3 1991.1 2027.3 C
-1991.4 2035.6 1991.4 2045.6 1991.1 2054.4 C
-1990.5 2055.5 1988.4 2056.6 1990.6 2055.4 C
-1991.6 2055.4 1991.6 2054.1 1991.6 2053.2 C
-1990.8 2044.7 1991.9 2035.4 1991.6 2027.6 C
-1991.8 2027.6 1992 2027.6 1992.3 2027.6 C
-1997 2032.8 2002.5 2037.7 2007.2 2042.9 C
-2007.3 2044.8 2006.7 2047.4 2007.6 2048.4 C
-2006.9 2055.1 2007.1 2062.5 2007.4 2069.3 C
-2002.7 2071.7 1998.1 2074.3 1993.2 2076.3 C
-1998 2074.2 2002.7 2071.8 2007.4 2069.6 C
-f 
-S 
-n
-2006.7 2069.1 m
-2006.3 2068.6 2005.9 2067.7 2005.7 2066.9 C
-2005.7 2059.7 2005.9 2051.4 2005.5 2045.1 C
-2004.9 2045.3 2004.7 2044.5 2004.3 2045.3 C
-2005.1 2045.3 2004.2 2045.8 2004.8 2046 C
-2004.8 2052.2 2004.8 2059.2 2004.8 2064.5 C
-2005.7 2065.7 2005.1 2065.7 2005 2066.7 C
-2003.8 2067 2002.7 2067.2 2001.9 2066.4 C
-2001.3 2064.6 1998 2063.1 1998 2061.9 C
-1996.1 2062.3 1996.6 2058.3 1994.2 2058.8 C
-1992.6 2057.7 1992.7 2054.8 1989.9 2056.6 C
-1985.6 2059.3 1980.9 2060.8 1976.7 2063.6 C
-1976 2066.9 1976 2071.2 1976.7 2074.6 C
-1977.6 2070.8 1973.1 2062.1 1980.5 2061.2 C
-1984.3 2060.3 1987.5 2058.2 1990.8 2056.4 C
-1991.7 2056.8 1992.9 2057.2 1993.5 2059.2 C
-1994.3 2058.6 1994.4 2060.6 1994.7 2059.2 C
-1995.3 2062.7 1999.2 2061.4 1998.8 2064.8 C
-2001.8 2065.4 2002.5 2068.4 2005.2 2067.4 C
-2004.9 2067.9 2006 2068 2006.4 2069.1 C
-2001.8 2071.1 1997.4 2073.9 1992.8 2075.8 C
-1997.5 2073.8 2002 2071.2 2006.7 2069.1 C
-[0 0.2 1 0]  vc
-f 
-S 
-n
-1988.7 2056.6 m
-1985.1 2058.7 1981.1 2060.1 1977.6 2061.9 C
-1981.3 2060.5 1985.6 2058.1 1988.7 2056.6 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-1977.9 2059.5 m
-1975.7 2064.5 1973.7 2054.7 1975.2 2060.9 C
-1976 2060.6 1977.6 2059.7 1977.9 2059.5 C
-f 
-S 
-n
-1989.6 2051.3 m
-1990.1 2042.3 1989.8 2036.6 1989.9 2028 C
-1989.8 2027 1990.8 2028.3 1990.1 2027.3 C
-1988.9 2026.7 1986.7 2026.9 1986.8 2024.7 C
-1987.4 2023 1985.9 2024.6 1985.1 2023.7 C
-1984.1 2021.4 1982.5 2020.5 1980.3 2020.6 C
-1979.9 2020.8 1979.5 2021.1 1979.3 2021.6 C
-1979.7 2025.8 1978.4 2033 1979.6 2038.1 C
-1983.7 2042.9 1968.8 2044.6 1978.8 2042.7 C
-1979.3 2042.3 1979.6 2041.9 1980 2041.5 C
-1980 2034.8 1980 2027 1980 2021.6 C
-1981.3 2020.5 1981.7 2021.5 1982.9 2021.8 C
-1983.6 2024.7 1986.1 2023.8 1986.8 2026.4 C
-1987.1 2027.7 1988.6 2027.1 1989.2 2028.3 C
-1989.1 2036.7 1989.3 2044.8 1988.9 2053.7 C
-1987.2 2054.9 1986.2 2056.8 1983.9 2057.1 C
-1986.3 2055.9 1990.9 2055 1989.6 2051.3 C
-f 
-S 
-n
-1971.6 2078.9 m
-1971.4 2070.5 1972.1 2062.2 1971.6 2055.9 C
-1969.9 2053.7 1967.6 2051.7 1965.6 2049.6 C
-1961.4 2050.4 1957.6 2053.6 1953.4 2055.2 C
-1949.8 2055.6 1948.2 2051.2 1945.5 2049.6 C
-1945.1 2048.8 1944.5 2047.9 1943.6 2047.5 C
-1940.1 2047.8 1937.3 2051 1934 2052.3 C
-1933.7 2052.6 1933.7 2053 1933.2 2053.2 C
-1933.7 2060.8 1933.4 2067.2 1933.5 2074.6 C
-1933.8 2068.1 1934 2060.9 1933.2 2054 C
-1935.3 2050.9 1939.3 2049.6 1942.4 2047.5 C
-1942.8 2047.5 1943.4 2047.4 1943.8 2047.7 C
-1947.1 2050.2 1950.3 2057.9 1955.3 2054.4 C
-1955.4 2054.4 1955.5 2054.3 1955.6 2054.2 C
-1955.9 2057.6 1956.1 2061.8 1955.3 2064.8 C
-1955.4 2064.3 1955.1 2063.8 1955.6 2063.6 C
-1956 2066.6 1955.3 2068.7 1958.7 2069.8 C
-1959.2 2071.7 1961.4 2071.7 1962 2074.1 C
-1964.4 2074.2 1964 2077.7 1967.3 2078.4 C
-1967 2079.7 1968.1 2079.9 1969 2080.1 C
-1971.1 2079.9 1970 2079.2 1970.4 2078 C
-1969.5 2077.2 1970.3 2075.9 1969.7 2075.1 C
-1970.1 2069.8 1970.1 2063.6 1969.7 2058.8 C
-1969.2 2058.5 1970 2058.1 1970.2 2057.8 C
-1970.4 2058.3 1971.2 2057.7 1971.4 2058.3 C
-1971.5 2065.3 1971.2 2073.6 1971.6 2081.1 C
-1974.1 2081.4 1969.8 2084.3 1972.4 2082.5 C
-1971.9 2081.4 1971.6 2080.2 1971.6 2078.9 C
-[0 0.4 1 0]  vc
-f 
-S 
-n
-1952.4 2052 m
-1954.1 2051.3 1955.6 2050.4 1957.2 2049.6 C
-1955.6 2050.4 1954.1 2051.3 1952.4 2052 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-1975.5 2039.8 m
-1975.5 2039.4 1974.5 2038.7 1974.8 2038.8 C
-1974.6 2039.1 1974.5 2039.1 1974.3 2038.8 C
-1974.6 2037.6 1972.5 2039.3 1972.1 2038.4 C
-1970.4 2038.4 1970.5 2035.5 1968 2035.7 C
-1968.6 2032.5 1964 2033.3 1964 2030.4 C
-1961.9 2029.8 1961.4 2026.8 1959.2 2026.8 C
-1957.7 2028.5 1952.6 2025.3 1953.9 2029.5 C
-1952.9 2031.5 1947 2028.2 1947.9 2032.4 C
-1946 2034.2 1941.5 2031.5 1941.9 2035.7 C
-1940.6 2036.1 1938.9 2036.1 1937.6 2035.7 C
-1937.3 2037.5 1940.7 2038.2 1941.2 2040.3 C
-1943.7 2041.1 1944.4 2043.9 1946.4 2045.6 C
-1949.1 2046.7 1949.9 2048.8 1952.2 2051.1 C
-1959.9 2047.1 1967.7 2043.6 1975.5 2039.8 C
-[0 0.01 1 0]  vc
-f 
-S 
-n
-vmrs
-1962 2047.2 m
-1960.2 2047.5 1959.5 2048.9 1957.7 2049.4 C
-1959.5 2049.5 1960.3 2047.2 1962 2047.2 C
-[0 0.87 0.91 0.83]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-2012.4 2046.3 m
-2010.3 2051.3 2008.3 2041.5 2009.8 2047.7 C
-2010.5 2047.4 2012.2 2046.5 2012.4 2046.3 C
-f 
-S 
-n
-1944.8 2044.6 m
-1945.5 2045.6 1948.6 2048.1 1947.4 2046.8 C
-1946.3 2046.5 1945.5 2045.2 1944.8 2044.6 C
-f 
-S 
-n
-1987.2 2054.9 m
-1983.7 2057.3 1979.6 2058 1976 2060.2 C
-1974.7 2058.2 1977.2 2055.8 1974.3 2054.9 C
-1973.1 2052 1970.4 2050.2 1968 2048 C
-1968 2047.7 1968 2047.4 1968.3 2047.2 C
-1969.5 2046.1 1983 2040.8 1972.4 2044.8 C
-1971.2 2046.6 1967.9 2046 1968 2048.2 C
-1970.5 2050.7 1973.8 2052.6 1974.3 2055.6 C
-1975.1 2055 1975.7 2056.7 1975.7 2057.1 C
-1975.7 2058.2 1974.8 2059.3 1975.5 2060.4 C
-1979.3 2058.2 1983.9 2057.7 1987.2 2054.9 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1967.8 2047.5 m
-1968.5 2047 1969.1 2046.5 1969.7 2046 C
-1969.1 2046.5 1968.5 2047 1967.8 2047.5 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-1976.7 2040.3 m
-1976.9 2032.8 1976.3 2024.4 1977.2 2017 C
-1976.4 2016.5 1976.6 2014.8 1976.9 2013.9 C
-1978.7 2012.7 1977.2 2013 1976.2 2012.9 C
-1971.5 2008.1 1965.9 2003.1 1961.8 1998 C
-1960.9 1998 1960.1 1998 1959.2 1998 C
-1951.5 2001.1 1944.3 2005.5 1937.1 2009.6 C
-1935 2012.9 1937 2013.6 1936.1 2017.2 C
-1937.1 2017.2 1936 2018 1936.4 2017.7 C
-1937 2020.1 1935.5 2022.1 1936.4 2024.9 C
-1936.8 2027.2 1936.5 2029.7 1936.6 2032.1 C
-1937.4 2028.2 1936 2023.8 1936.8 2020.1 C
-1938.3 2015.7 1933.6 2011 1939 2008.6 C
-1945.9 2004.5 1953.1 2000.3 1960.6 1998.3 C
-1960.9 1998.3 1961.3 1998.3 1961.6 1998.3 C
-1966.2 2003.5 1971.8 2008.4 1976.4 2013.6 C
-1976.6 2015.5 1976 2018.1 1976.9 2019.2 C
-1976.1 2025.8 1976.4 2033.2 1976.7 2040 C
-1971.9 2042.4 1967.4 2045 1962.5 2047 C
-1967.3 2044.9 1972 2042.6 1976.7 2040.3 C
-f 
-S 
-n
-1939 2038.6 m
-1940.6 2040.9 1942.5 2042.1 1944.3 2044.1 C
-1942.7 2041.9 1940.6 2040.9 1939 2038.6 C
-f 
-S 
-n
-2006.2 2065.7 m
-2006 2057.3 2006.7 2049 2006.2 2042.7 C
-2002.1 2038.4 1997.7 2033.4 1993 2030 C
-1992.9 2029.3 1992.5 2028.6 1992 2028.3 C
-1992.1 2036.6 1991.9 2046.2 1992.3 2054.9 C
-1990.8 2056.2 1989 2056.7 1987.5 2058 C
-1988.7 2057.7 1990.7 2054.4 1993 2056.4 C
-1993.4 2058.8 1996 2058.2 1996.6 2060.9 C
-1999 2061 1998.5 2064.5 2001.9 2065.2 C
-2001.5 2066.5 2002.7 2066.7 2003.6 2066.9 C
-2005.7 2066.7 2004.6 2066 2005 2064.8 C
-2004 2064 2004.8 2062.7 2004.3 2061.9 C
-2004.6 2056.6 2004.6 2050.4 2004.3 2045.6 C
-2003.7 2045.3 2004.6 2044.9 2004.8 2044.6 C
-2005 2045.1 2005.7 2044.5 2006 2045.1 C
-2006 2052.1 2005.8 2060.4 2006.2 2067.9 C
-2008.7 2068.2 2004.4 2071.1 2006.9 2069.3 C
-2006.4 2068.2 2006.2 2067 2006.2 2065.7 C
-[0 0.4 1 0]  vc
-f 
-S 
-n
-2021.8 2041.7 m
-2018.3 2044.1 2014.1 2044.8 2010.5 2047 C
-2009.3 2045 2011.7 2042.6 2008.8 2041.7 C
-2004.3 2035.1 1997.6 2030.9 1993 2024.4 C
-1992.1 2024 1991.5 2024.3 1990.8 2024 C
-1993.2 2023.9 1995.3 2027.1 1996.8 2029 C
-2000.4 2032.6 2004.9 2036.9 2008.4 2040.8 C
-2008.2 2043.1 2011.4 2042.8 2009.8 2045.8 C
-2009.8 2046.3 2009.7 2046.9 2010 2047.2 C
-2013.8 2045 2018.5 2044.5 2021.8 2041.7 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-2001.6 2034 m
-2000.7 2033.1 1999.9 2032.3 1999 2031.4 C
-1999.9 2032.3 2000.7 2033.1 2001.6 2034 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-vmrs
-1989.4 2024.4 m
-1989.5 2025.4 1988.6 2024.3 1988.9 2024.7 C
-1990.5 2025.8 1990.7 2024.2 1992.8 2024.9 C
-1993.8 2025.9 1995 2027.1 1995.9 2028 C
-1994.3 2026 1991.9 2023.4 1989.4 2024.4 C
-[0 0.87 0.91 0.83]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1984.8 2019.9 m
-1984.6 2018.6 1986.3 2017.2 1987.7 2016.8 C
-1987.2 2017.5 1982.9 2017.9 1984.4 2020.6 C
-1984.1 2019.9 1984.9 2020 1984.8 2019.9 C
-f 
-S 
-n
-1981.7 2017 m
-1979.6 2022 1977.6 2012.3 1979.1 2018.4 C
-1979.8 2018.1 1981.5 2017.2 1981.7 2017 C
-f 
-S 
-n
-1884.3 2019.2 m
-1884.7 2010.5 1884.5 2000.6 1884.5 1991.8 C
-1886.6 1989.3 1889.9 1988.9 1892.4 1987 C
-1890.8 1988.7 1886 1989.1 1884.3 1992.3 C
-1884.7 2001 1884.5 2011.3 1884.5 2019.9 C
-1891 2025.1 1895.7 2031.5 1902 2036.9 C
-1896.1 2031 1890 2024.9 1884.3 2019.2 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1884 2019.4 m
-1884.5 2010.6 1884.2 2000.4 1884.3 1991.8 C
-1884.8 1990.4 1887.8 1989 1884.8 1990.8 C
-1884.3 1991.3 1884.3 1992 1884 1992.5 C
-1884.5 2001.2 1884.2 2011.1 1884.3 2019.9 C
-1887.9 2023.1 1891.1 2026.4 1894.4 2030 C
-1891.7 2026.1 1887.1 2022.9 1884 2019.4 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1885 2011.7 m
-1885 2006.9 1885 2001.9 1885 1997.1 C
-1885 2001.9 1885 2006.9 1885 2011.7 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-1975.5 2036.4 m
-1975.2 2028 1976 2019.7 1975.5 2013.4 C
-1971.1 2008.5 1965.6 2003.6 1961.6 1999 C
-1958.8 1998 1956 2000 1953.6 2001.2 C
-1948.2 2004.7 1941.9 2006.5 1937.1 2010.8 C
-1937.5 2018.3 1937.3 2024.7 1937.3 2032.1 C
-1937.6 2025.6 1937.9 2018.4 1937.1 2011.5 C
-1937.3 2011 1937.6 2010.5 1937.8 2010 C
-1944.6 2005.7 1951.9 2002.3 1959.2 1999 C
-1960.1 1998.5 1960.1 1999.8 1960.4 2000.4 C
-1959.7 2006.9 1959.7 2014.2 1959.4 2021.1 C
-1959 2021.1 1959.2 2021.9 1959.2 2022.3 C
-1959.2 2021.9 1959 2021.3 1959.4 2021.1 C
-1959.8 2024.1 1959.2 2026.2 1962.5 2027.3 C
-1963 2029.2 1965.3 2029.2 1965.9 2031.6 C
-1968.3 2031.8 1967.8 2035.2 1971.2 2036 C
-1970.8 2037.2 1971.9 2037.5 1972.8 2037.6 C
-1974.9 2037.4 1973.9 2036.7 1974.3 2035.5 C
-1973.3 2034.7 1974.1 2033.4 1973.6 2032.6 C
-1973.9 2027.3 1973.9 2021.1 1973.6 2016.3 C
-1973 2016 1973.9 2015.6 1974 2015.3 C
-1974.3 2015.9 1975 2015.3 1975.2 2015.8 C
-1975.3 2022.8 1975.1 2031.2 1975.5 2038.6 C
-1977.9 2039 1973.7 2041.8 1976.2 2040 C
-1975.7 2039 1975.5 2037.8 1975.5 2036.4 C
-[0 0.4 1 0]  vc
-f 
-S 
-n
-1991.1 2012.4 m
-1987.5 2014.8 1983.4 2015.6 1979.8 2017.7 C
-1978.5 2015.7 1981 2013.3 1978.1 2012.4 C
-1973.6 2005.8 1966.8 2001.6 1962.3 1995.2 C
-1961.4 1994.7 1960.8 1995 1960.1 1994.7 C
-1962.5 1994.6 1964.6 1997.8 1966.1 1999.7 C
-1969.7 2003.3 1974.2 2007.6 1977.6 2011.5 C
-1977.5 2013.8 1980.6 2013.5 1979.1 2016.5 C
-1979.1 2017 1979 2017.6 1979.3 2018 C
-1983.1 2015.7 1987.8 2015.2 1991.1 2012.4 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1970.9 2004.8 m
-1970 2003.9 1969.2 2003 1968.3 2002.1 C
-1969.2 2003 1970 2003.9 1970.9 2004.8 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-1887.9 1994.9 m
-1888.5 1992.3 1891.4 1992.2 1893.2 1990.8 C
-1898.4 1987.5 1904 1984.8 1909.5 1982.2 C
-1909.7 1982.7 1910.3 1982.1 1910.4 1982.7 C
-1909.5 1990.5 1910.1 1996.4 1910 2004.5 C
-1909.1 2003.4 1909.7 2005.8 1909.5 2006.4 C
-1910.4 2006 1909.7 2008 1910.2 2007.9 C
-1911.3 2010.6 1912.5 2012.6 1915.7 2013.4 C
-1915.8 2013.7 1915.5 2014.4 1916 2014.4 C
-1916.3 2015 1915.4 2016 1915.2 2016 C
-1916.1 2015.5 1916.5 2014.5 1916 2013.6 C
-1913.4 2013.3 1913.1 2010.5 1910.9 2009.8 C
-1910.7 2008.8 1910.4 2007.9 1910.2 2006.9 C
-1911.1 1998.8 1909.4 1990.7 1910.7 1982.4 C
-1910 1982.1 1908.9 1982.1 1908.3 1982.4 C
-1901.9 1986.1 1895 1988.7 1888.8 1993 C
-1888 1993.4 1888.4 1994.3 1887.6 1994.7 C
-1888.1 2001.3 1887.8 2008.6 1887.9 2015.1 C
-1887.3 2017.5 1887.9 2015.4 1888.4 2014.4 C
-1887.8 2008 1888.4 2001.3 1887.9 1994.9 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-vmrs
-1887.9 2018.4 m
-1887.5 2016.9 1888.5 2016 1888.8 2014.8 C
-1890.1 2014.8 1891.1 2016.6 1892.4 2015.3 C
-1892.4 2014.4 1893.8 2012.9 1894.4 2012.4 C
-1895.9 2012.4 1896.6 2013.9 1897.7 2012.7 C
-1898.4 2011.7 1898.6 2010.4 1899.6 2009.8 C
-1901.7 2009.9 1902.9 2010.4 1904 2009.1 C
-1904.3 2007.4 1904 2007.6 1904.9 2007.2 C
-1906.2 2007 1907.6 2006.5 1908.8 2006.7 C
-1910.6 2008.2 1909.8 2011.5 1912.6 2012 C
-1912.4 2013 1913.8 2012.7 1914 2013.2 C
-1911.5 2011.1 1909.1 2007.9 1909.2 2004.3 C
-1909.5 2003.5 1909.9 2004.9 1909.7 2004.3 C
-1909.9 1996.2 1909.3 1990.5 1910.2 1982.7 C
-1909.5 1982.6 1909.5 1982.6 1908.8 1982.7 C
-1903.1 1985.7 1897 1987.9 1891.7 1992 C
-1890.5 1993 1888.2 1992.9 1888.1 1994.9 C
-1888.7 2001.4 1888.1 2008.4 1888.6 2014.8 C
-1888.3 2016 1887.2 2016.9 1887.6 2018.4 C
-1892.3 2023.9 1897.6 2027.9 1902.3 2033.3 C
-1898 2028.2 1892.1 2023.8 1887.9 2018.4 C
-[0.4 0.4 0 0]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1910.9 1995.2 m
-1910.4 1999.8 1911 2003.3 1910.9 2008.1 C
-1910.9 2003.8 1910.9 1999.2 1910.9 1995.2 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1911.2 2004.3 m
-1911.2 2001.9 1911.2 1999.7 1911.2 1997.3 C
-1911.2 1999.7 1911.2 2001.9 1911.2 2004.3 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-1958.7 1995.2 m
-1959 1995.6 1956.2 1995 1956.5 1996.8 C
-1955.8 1997.6 1954.2 1998.5 1953.6 1997.3 C
-1953.6 1990.8 1954.9 1989.6 1953.4 1983.9 C
-1953.4 1983.3 1953.3 1982.1 1954.4 1982 C
-1955.5 1982.6 1956.5 1981.3 1957.5 1981 C
-1956.3 1981.8 1954.7 1982.6 1953.9 1981.5 C
-1951.4 1983 1954.7 1988.8 1952.9 1990.6 C
-1953.8 1990.6 1953.2 1992.7 1953.4 1993.7 C
-1953.8 1994.5 1952.3 1996.1 1953.2 1997.8 C
-1956.3 1999.4 1957.5 1994 1959.9 1995.6 C
-1962 1994.4 1963.7 1997.7 1965.2 1998.8 C
-1963.5 1996.7 1961.2 1994.1 1958.7 1995.2 C
-f 
-S 
-n
-1945 2000.7 m
-1945.4 1998.7 1945.4 1997.9 1945 1995.9 C
-1944.5 1995.3 1944.2 1992.6 1945.7 1993.2 C
-1946 1992.2 1948.7 1992.5 1948.4 1990.6 C
-1947.5 1990.3 1948.1 1988.7 1947.9 1988.2 C
-1948.9 1987.8 1950.5 1986.8 1950.5 1984.6 C
-1951.5 1980.9 1946.7 1983 1947.2 1979.8 C
-1944.5 1979.9 1945.2 1976.6 1943.1 1976.7 C
-1941.8 1975.7 1942.1 1972.7 1939.2 1973.8 C
-1938.2 1974.6 1939.3 1971.6 1938.3 1970.9 C
-1938.8 1969.2 1933.4 1970.3 1937.3 1970 C
-1939.4 1971.2 1937.2 1973 1937.6 1974.3 C
-1937.2 1976.3 1937.1 1981.2 1937.8 1984.1 C
-1938.8 1982.3 1937.9 1976.6 1938.5 1973.1 C
-1938.9 1975 1938.5 1976.4 1939.7 1977.2 C
-1939.5 1983.5 1938.9 1991.3 1940.2 1997.3 C
-1939.4 1999.1 1938.6 1997.1 1937.8 1997.1 C
-1937.4 1996.7 1937.6 1996.1 1937.6 1995.6 C
-1936.5 1998.5 1940.1 1998.4 1940.9 2000.7 C
-1942.1 2000.4 1943.2 2001.3 1943.1 2002.4 C
-1943.6 2003.1 1941.1 2004.6 1942.8 2003.8 C
-1943.9 2002.5 1942.6 2000.6 1945 2000.7 C
-[0.65 0.65 0 0.42]  vc
-f 
-S 
-n
-1914.5 2006.4 m
-1914.1 2004.9 1915.2 2004 1915.5 2002.8 C
-1916.7 2002.8 1917.8 2004.6 1919.1 2003.3 C
-1919 2002.4 1920.4 2000.9 1921 2000.4 C
-1922.5 2000.4 1923.2 2001.9 1924.4 2000.7 C
-1925 1999.7 1925.3 1998.4 1926.3 1997.8 C
-1928.4 1997.9 1929.5 1998.4 1930.6 1997.1 C
-1930.9 1995.4 1930.7 1995.6 1931.6 1995.2 C
-1932.8 1995 1934.3 1994.5 1935.4 1994.7 C
-1936.1 1995.8 1936.9 1996.2 1936.6 1997.8 C
-1938.9 1999.4 1939.7 2001.3 1942.4 2002.4 C
-1942.4 2002.5 1942.2 2003 1942.6 2002.8 C
-1942.9 2000.4 1939.2 2001.8 1939.2 1999.7 C
-1936.2 1998.6 1937 1995.3 1935.9 1993.5 C
-1937.1 1986.5 1935.2 1977.9 1937.6 1971.2 C
-1937.6 1970.3 1936.6 1971 1936.4 1970.4 C
-1930.2 1973.4 1924 1976 1918.4 1980 C
-1917.2 1981 1914.9 1980.9 1914.8 1982.9 C
-1915.3 1989.4 1914.7 1996.4 1915.2 2002.8 C
-1914.9 2004 1913.9 2004.9 1914.3 2006.4 C
-1919 2011.9 1924.2 2015.9 1928.9 2021.3 C
-1924.6 2016.2 1918.7 2011.8 1914.5 2006.4 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1914.5 1982.9 m
-1915.1 1980.3 1918 1980.2 1919.8 1978.8 C
-1925 1975.5 1930.6 1972.8 1936.1 1970.2 C
-1939.4 1970.6 1936.1 1974.2 1936.6 1976.4 C
-1936.5 1981.9 1936.8 1987.5 1936.4 1992.8 C
-1935.9 1992.8 1936.2 1993.5 1936.1 1994 C
-1937.1 1993.6 1936.2 1995.9 1936.8 1995.9 C
-1937 1998 1939.5 1999.7 1940.4 2000.7 C
-1940.1 1998.6 1935 1997.2 1937.6 1993.7 C
-1938.3 1985.7 1935.9 1976.8 1937.8 1970.7 C
-1936.9 1969.8 1935.4 1970.3 1934.4 1970.7 C
-1928.3 1974.4 1921.4 1976.7 1915.5 1981 C
-1914.6 1981.4 1915.1 1982.3 1914.3 1982.7 C
-1914.7 1989.3 1914.5 1996.6 1914.5 2003.1 C
-1913.9 2005.5 1914.5 2003.4 1915 2002.4 C
-1914.5 1996 1915.1 1989.3 1914.5 1982.9 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1939.2 1994.9 m
-1939.3 1995 1939.4 1995.1 1939.5 1995.2 C
-1939.1 1989 1939.3 1981.6 1939 1976.7 C
-1938.6 1976.3 1938.6 1974.6 1938.5 1973.3 C
-1938.7 1976.1 1938.1 1979.4 1939 1981.7 C
-1937.3 1986 1937.7 1991.6 1938 1996.4 C
-1937.3 1994.3 1939.6 1996.2 1939.2 1994.9 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1938.3 1988.4 m
-1938.5 1990.5 1937.9 1994.1 1938.8 1994.7 C
-1937.9 1992.6 1939 1990.6 1938.3 1988.4 C
-[0 0.87 0.91 0.83]  vc
-f 
-S 
-n
-1938.8 1985.8 m
-1938.5 1985.9 1938.4 1985.7 1938.3 1985.6 C
-1938.4 1986.2 1938 1989.5 1938.8 1987.2 C
-1938.8 1986.8 1938.8 1986.3 1938.8 1985.8 C
-f 
-S 
-n
-vmrs
-1972.8 2062.1 m
-1971.9 2061 1972.5 2059.4 1972.4 2058 C
-1972.2 2063.8 1971.9 2073.7 1972.4 2081.3 C
-1972.5 2074.9 1971.9 2067.9 1972.8 2062.1 C
-[0 1 1 0.36]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1940.2 2071.7 m
-1941.3 2072 1943.1 2072.3 1944 2071.5 C
-1943.6 2069.9 1945.2 2069.1 1946 2068.8 C
-1950 2071.1 1948.7 2065.9 1951.7 2066.2 C
-1953.5 2063.9 1956.9 2069.4 1955.6 2063.8 C
-1955.5 2064.2 1955.7 2064.8 1955.3 2065 C
-1954.3 2063.7 1956.2 2063.6 1955.6 2062.1 C
-1954.5 2060 1958.3 2050.3 1952.2 2055.6 C
-1949.1 2053.8 1946 2051 1943.8 2048 C
-1940.3 2048 1937.5 2051.3 1934.2 2052.5 C
-1933.1 2054.6 1934.4 2057.3 1934 2060 C
-1934 2065.1 1934 2069.7 1934 2074.6 C
-1934.4 2069 1934.1 2061.5 1934.2 2054.9 C
-1934.6 2054.5 1935.3 2054.7 1935.9 2054.7 C
-1937 2055.3 1935.9 2056.1 1935.9 2056.8 C
-1936.5 2063 1935.6 2070.5 1935.9 2074.6 C
-1936.7 2074.4 1937.3 2075.2 1938 2074.6 C
-1937.9 2073.6 1939.1 2072.1 1940.2 2071.7 C
-[0 0.2 1 0]  vc
-f 
-S 
-n
-1933.2 2074.1 m
-1933.2 2071.5 1933.2 2069 1933.2 2066.4 C
-1933.2 2069 1933.2 2071.5 1933.2 2074.1 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-2007.4 2048.9 m
-2006.5 2047.8 2007.1 2046.2 2006.9 2044.8 C
-2006.7 2050.6 2006.5 2060.5 2006.9 2068.1 C
-2007.1 2061.7 2006.5 2054.7 2007.4 2048.9 C
-f 
-S 
-n
-1927.2 2062.4 m
-1925.8 2060.1 1928.1 2058.2 1927 2056.4 C
-1927.3 2055.5 1926.5 2053.5 1926.8 2051.8 C
-1926.8 2052.8 1926 2052.5 1925.3 2052.5 C
-1924.1 2052.8 1925 2050.5 1924.4 2050.1 C
-1925.3 2050.2 1925.4 2048.8 1926.3 2049.4 C
-1926.5 2052.3 1928.4 2047.2 1928.4 2051.1 C
-1928.9 2050.5 1929 2051.4 1928.9 2051.8 C
-1928.9 2052 1928.9 2052.3 1928.9 2052.5 C
-1929.4 2051.4 1928.9 2049 1930.1 2048.2 C
-1928.9 2047.1 1930.5 2047.1 1930.4 2046.5 C
-1931.9 2046.2 1933.1 2046.1 1934.7 2046.5 C
-1934.6 2046.9 1935.2 2047.9 1934.4 2048.4 C
-1936.9 2048.1 1933.6 2043.8 1935.9 2043.9 C
-1935.7 2043.9 1934.8 2041.3 1933.2 2041.7 C
-1932.5 2041.6 1932.4 2039.6 1932.3 2041 C
-1930.8 2042.6 1929 2040.6 1927.7 2042 C
-1927.5 2041.4 1927.1 2040.9 1927.2 2040.3 C
-1927.8 2040.6 1927.4 2039.1 1928.2 2038.6 C
-1929.4 2038 1930.5 2038.8 1931.3 2037.9 C
-1931.7 2039 1932.5 2038.6 1931.8 2037.6 C
-1930.9 2037 1928.7 2037.8 1928.2 2037.9 C
-1926.7 2037.8 1928 2039 1927 2038.8 C
-1927.4 2040.4 1925.6 2040.8 1925.1 2041 C
-1924.3 2040.4 1923.2 2040.5 1922.2 2040.5 C
-1921.4 2041.7 1921 2043.9 1919.3 2043.9 C
-1918.8 2043.4 1917.2 2043.3 1916.4 2043.4 C
-1915.9 2044.4 1915.7 2046 1914.3 2046.5 C
-1913.1 2046.6 1912 2044.5 1911.4 2046.3 C
-1912.8 2046.5 1913.8 2047.4 1915.7 2047 C
-1916.9 2047.7 1915.6 2048.8 1916 2049.4 C
-1915.4 2049.3 1913.9 2050.3 1913.3 2051.1 C
-1913.9 2054.1 1916 2050.2 1916.7 2053 C
-1916.9 2053.8 1915.5 2054.1 1916.7 2054.4 C
-1917 2054.7 1920.2 2054.3 1919.3 2056.6 C
-1918.8 2056.1 1920.2 2058.6 1920.3 2057.6 C
-1921.2 2057.9 1922.1 2057.5 1922.4 2059 C
-1922.3 2059.1 1922.2 2059.3 1922 2059.2 C
-1922.1 2059.7 1922.4 2060.3 1922.9 2060.7 C
-1923.2 2060.1 1923.8 2060.4 1924.6 2060.7 C
-1925.9 2062.6 1923.2 2062 1925.6 2063.6 C
-1926.1 2063.1 1927.3 2062.5 1927.2 2062.4 C
-[0.21 0.21 0 0]  vc
-f 
-S 
-n
-1933.2 2063.3 m
-1933.2 2060.7 1933.2 2058.2 1933.2 2055.6 C
-1933.2 2058.2 1933.2 2060.7 1933.2 2063.3 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-1965.2 2049.2 m
-1967.1 2050.1 1969.9 2053.7 1972.1 2056.4 C
-1970.5 2054 1967.6 2051.3 1965.2 2049.2 C
-f 
-S 
-n
-1991.8 2034.8 m
-1991.7 2041.5 1992 2048.5 1991.6 2055.2 C
-1990.5 2056.4 1991.9 2054.9 1991.8 2054.4 C
-1991.8 2047.9 1991.8 2041.3 1991.8 2034.8 C
-f 
-S 
-n
-1988.9 2053.2 m
-1988.9 2044.3 1988.9 2036.6 1988.9 2028.3 C
-1985.7 2028.2 1987.2 2023.5 1983.9 2024.2 C
-1983.9 2022.4 1982 2021.6 1981 2021.3 C
-1980.6 2021.1 1980.6 2021.7 1980.3 2021.6 C
-1980.3 2027 1980.3 2034.8 1980.3 2041.5 C
-1979.3 2043.2 1977.6 2043 1976.2 2043.6 C
-1977.1 2043.8 1978.5 2043.2 1978.8 2044.1 C
-1978.5 2045.3 1979.9 2045.3 1980.3 2045.8 C
-1980.5 2046.8 1980.7 2046.2 1981.5 2046.5 C
-1982.4 2047.1 1982 2048.6 1982.7 2049.4 C
-1984.2 2049.6 1984.6 2052.2 1986.8 2051.6 C
-1987.1 2048.6 1985.1 2042.7 1986.5 2040.5 C
-1986.3 2036.7 1986.9 2031.7 1986 2029.2 C
-1986.3 2027.1 1986.9 2028.6 1987.7 2027.6 C
-1987.7 2028.3 1988.7 2028 1988.7 2028.8 C
-1988.1 2033 1988.7 2037.5 1988.2 2041.7 C
-1987.8 2041.4 1988 2040.8 1988 2040.3 C
-1988 2041 1988 2042.4 1988 2042.4 C
-1988 2042.4 1988.1 2042.3 1988.2 2042.2 C
-1989.3 2046 1988 2050.2 1988.4 2054 C
-1987.8 2054.4 1987.1 2054.7 1986.5 2055.4 C
-1987.4 2054.4 1988.4 2054.6 1988.9 2053.2 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1950.8 2054.4 m
-1949.7 2053.4 1948.7 2052.3 1947.6 2051.3 C
-1948.7 2052.3 1949.7 2053.4 1950.8 2054.4 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-vmrs
-2006.7 2043.2 m
-2004.5 2040.8 2002.4 2038.4 2000.2 2036 C
-2002.4 2038.4 2004.5 2040.8 2006.7 2043.2 C
-[0 1 1 0.36]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1976.7 2019.6 m
-1975.8 2018.6 1976.4 2016.9 1976.2 2015.6 C
-1976 2021.3 1975.8 2031.2 1976.2 2038.8 C
-1976.4 2032.4 1975.8 2025.5 1976.7 2019.6 C
-f 
-S 
-n
-1988.4 2053.5 m
-1988.6 2049.2 1988.1 2042.8 1988 2040 C
-1988.4 2040.4 1988.1 2041 1988.2 2041.5 C
-1988.3 2037.2 1988 2032.7 1988.4 2028.5 C
-1987.6 2027.1 1987.2 2028.6 1986.8 2028 C
-1985.9 2028.5 1986.5 2029.7 1986.3 2030.4 C
-1986.9 2029.8 1986.6 2031 1987 2031.2 C
-1987.4 2039.6 1985 2043 1987.2 2050.4 C
-1987.2 2051.6 1985.9 2052.3 1984.6 2051.3 C
-1981.9 2049.7 1982.9 2047 1980.3 2046.5 C
-1980.3 2045.2 1978.1 2046.2 1978.6 2043.9 C
-1975.6 2043.3 1979.3 2045.6 1979.6 2046.5 C
-1980.8 2046.6 1981.5 2048.5 1982.2 2049.9 C
-1983.7 2050.8 1984.8 2052.8 1986.5 2053 C
-1986.7 2053.5 1987.5 2054.1 1987 2054.7 C
-1987.4 2053.9 1988.3 2054.3 1988.4 2053.5 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1988 2038.1 m
-1988 2036.7 1988 2035.4 1988 2034 C
-1988 2035.4 1988 2036.7 1988 2038.1 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-1999.7 2035.7 m
-1997.6 2033.5 1995.4 2031.2 1993.2 2029 C
-1995.4 2031.2 1997.6 2033.5 1999.7 2035.7 C
-f 
-S 
-n
-1944 2029.2 m
-1945.2 2029.5 1946.9 2029.8 1947.9 2029 C
-1947.4 2027.4 1949 2026.7 1949.8 2026.4 C
-1953.9 2028.6 1952.6 2023.4 1955.6 2023.7 C
-1957.4 2021.4 1960.7 2027 1959.4 2021.3 C
-1959.3 2021.7 1959.6 2022.3 1959.2 2022.5 C
-1958.1 2021.2 1960.1 2021.1 1959.4 2019.6 C
-1959.1 2012.7 1959.9 2005.1 1959.6 1999.2 C
-1955.3 2000.1 1951.3 2003.1 1947.2 2005 C
-1943.9 2006 1941.2 2008.7 1938 2010 C
-1936.9 2012.1 1938.2 2014.8 1937.8 2017.5 C
-1937.8 2022.6 1937.8 2027.3 1937.8 2032.1 C
-1938.2 2026.5 1938 2019 1938 2012.4 C
-1938.5 2012 1939.2 2012.3 1939.7 2012.2 C
-1940.8 2012.8 1939.7 2013.6 1939.7 2014.4 C
-1940.4 2020.5 1939.4 2028 1939.7 2032.1 C
-1940.6 2031.9 1941.2 2032.7 1941.9 2032.1 C
-1941.7 2031.2 1943 2029.7 1944 2029.2 C
-[0 0.2 1 0]  vc
-f 
-S 
-n
-1937.1 2031.6 m
-1937.1 2029.1 1937.1 2026.5 1937.1 2024 C
-1937.1 2026.5 1937.1 2029.1 1937.1 2031.6 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-1991.8 2028 m
-1992.5 2027.8 1993.2 2029.9 1994 2030.2 C
-1992.9 2029.6 1993.1 2028.1 1991.8 2028 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1991.8 2027.8 m
-1992.4 2027.6 1992.6 2028.3 1993 2028.5 C
-1992.6 2028.2 1992.2 2027.6 1991.6 2027.8 C
-1991.6 2028.5 1991.6 2029.1 1991.6 2029.7 C
-1991.6 2029.1 1991.4 2028.3 1991.8 2027.8 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-1985.8 2025.4 m
-1985.3 2025.2 1984.8 2024.7 1984.1 2024.9 C
-1983.3 2025.3 1983.6 2027.3 1983.9 2027.6 C
-1985 2028 1986.9 2026.9 1985.8 2025.4 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-vmrs
-1993.5 2024.4 m
-1992.4 2023.7 1991.3 2022.9 1990.1 2023.2 C
-1990.7 2023.7 1989.8 2023.8 1989.4 2023.7 C
-1989.1 2023.7 1988.6 2023.9 1988.4 2023.5 C
-1988.5 2023.2 1988.3 2022.7 1988.7 2022.5 C
-1989 2022.6 1988.9 2023 1988.9 2023.2 C
-1989.1 2022.8 1990.4 2022.3 1990.6 2021.3 C
-1990.4 2021.8 1990 2021.3 1990.1 2021.1 C
-1990.1 2020.9 1990.1 2020.1 1990.1 2020.6 C
-1989.9 2021.1 1989.5 2020.6 1989.6 2020.4 C
-1989.6 2019.8 1988.7 2019.6 1988.2 2019.2 C
-1987.5 2018.7 1987.7 2020.2 1987 2019.4 C
-1987.5 2020.4 1986 2021.1 1987.5 2021.8 C
-1986.8 2023.1 1986.6 2021.1 1986 2021.1 C
-1986.1 2020.1 1985.9 2019 1986.3 2018.2 C
-1986.7 2018.4 1986.5 2019 1986.5 2019.4 C
-1986.5 2018.7 1986.4 2017.8 1987.2 2017.7 C
-1986.5 2017.2 1985.5 2019.3 1985.3 2020.4 C
-1986.2 2022 1987.3 2023.5 1989.2 2024.2 C
-1990.8 2024.3 1991.6 2022.9 1993.2 2024.4 C
-1993.8 2025.4 1995 2026.6 1995.9 2027.1 C
-1995 2026.5 1994.1 2025.5 1993.5 2024.4 C
-[0 1 1 0.36]  vc
-f 
-0.4 w
-2 J
-2 M
-[0 0.5 0.5 0.2]  vc
-S 
-n
-2023 2040.3 m
-2023.2 2036 2022.7 2029.6 2022.5 2026.8 C
-2022.9 2027.2 2022.7 2027.8 2022.8 2028.3 C
-2022.8 2024 2022.6 2019.5 2023 2015.3 C
-2022.2 2013.9 2021.7 2015.4 2021.3 2014.8 C
-2020.4 2015.3 2021 2016.5 2020.8 2017.2 C
-2021.4 2016.6 2021.1 2017.8 2021.6 2018 C
-2022 2026.4 2019.6 2029.8 2021.8 2037.2 C
-2021.7 2038.4 2020.5 2039.1 2019.2 2038.1 C
-2016.5 2036.5 2017.5 2033.8 2014.8 2033.3 C
-2014.9 2032 2012.6 2033 2013.2 2030.7 C
-2011.9 2030.8 2011.2 2030.1 2010.8 2029.2 C
-2010.8 2029.1 2010.8 2028.2 2010.8 2028.8 C
-2010 2028.8 2010.4 2026.5 2008.6 2027.3 C
-2007.9 2026.6 2007.3 2025.9 2007.9 2027.1 C
-2009.7 2028 2010 2030.1 2012.2 2030.9 C
-2012.9 2032.1 2013.7 2033.6 2015.1 2033.6 C
-2015.7 2035.1 2016.9 2036.7 2018.4 2038.4 C
-2019.8 2039.3 2022 2039.4 2021.6 2041.5 C
-2021.9 2040.7 2022.9 2041.1 2023 2040.3 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-2022.5 2024.9 m
-2022.5 2023.5 2022.5 2022.2 2022.5 2020.8 C
-2022.5 2022.2 2022.5 2023.5 2022.5 2024.9 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-1983.2 2022.8 m
-1982.4 2022.5 1982.1 2021.6 1981.2 2022.3 C
-1981.1 2022.9 1980.5 2024 1981 2024.2 C
-1981.8 2024.6 1982.9 2024.4 1983.2 2022.8 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1931.1 2019.9 m
-1929.6 2017.7 1932 2015.7 1930.8 2013.9 C
-1931.1 2013 1930.3 2011 1930.6 2009.3 C
-1930.6 2010.3 1929.8 2010 1929.2 2010 C
-1928 2010.3 1928.8 2008.1 1928.2 2007.6 C
-1929.1 2007.8 1929.3 2006.3 1930.1 2006.9 C
-1930.3 2009.8 1932.2 2004.8 1932.3 2008.6 C
-1932.7 2008 1932.8 2009 1932.8 2009.3 C
-1932.8 2009.6 1932.8 2009.8 1932.8 2010 C
-1933.2 2009 1932.7 2006.6 1934 2005.7 C
-1932.7 2004.6 1934.3 2004.6 1934.2 2004 C
-1935.8 2003.7 1937 2003.6 1938.5 2004 C
-1938.5 2004.5 1939.1 2005.4 1938.3 2006 C
-1940.7 2005.7 1937.4 2001.3 1939.7 2001.4 C
-1939.5 2001.4 1938.6 1998.8 1937.1 1999.2 C
-1936.3 1999.1 1936.2 1997.1 1936.1 1998.5 C
-1934.7 2000.1 1932.9 1998.2 1931.6 1999.5 C
-1931.3 1998.9 1930.9 1998.5 1931.1 1997.8 C
-1931.6 1998.2 1931.3 1996.6 1932 1996.1 C
-1933.2 1995.5 1934.3 1996.4 1935.2 1995.4 C
-1935.5 1996.5 1936.3 1996.1 1935.6 1995.2 C
-1934.7 1994.5 1932.5 1995.3 1932 1995.4 C
-1930.5 1995.3 1931.9 1996.5 1930.8 1996.4 C
-1931.2 1997.9 1929.5 1998.3 1928.9 1998.5 C
-1928.1 1997.9 1927.1 1998 1926 1998 C
-1925.3 1999.2 1924.8 2001.4 1923.2 2001.4 C
-1922.6 2000.9 1921 2000.9 1920.3 2000.9 C
-1919.7 2001.9 1919.6 2003.5 1918.1 2004 C
-1916.9 2004.1 1915.8 2002 1915.2 2003.8 C
-1916.7 2004 1917.6 2004.9 1919.6 2004.5 C
-1920.7 2005.2 1919.4 2006.3 1919.8 2006.9 C
-1919.2 2006.9 1917.7 2007.8 1917.2 2008.6 C
-1917.8 2011.6 1919.8 2007.8 1920.5 2010.5 C
-1920.8 2011.3 1919.3 2011.6 1920.5 2012 C
-1920.8 2012.3 1924 2011.8 1923.2 2014.1 C
-1922.6 2013.6 1924.1 2016.1 1924.1 2015.1 C
-1925.1 2015.4 1925.9 2015 1926.3 2016.5 C
-1926.2 2016.6 1926 2016.8 1925.8 2016.8 C
-1925.9 2017.2 1926.2 2017.8 1926.8 2018.2 C
-1927.1 2017.6 1927.7 2018 1928.4 2018.2 C
-1929.7 2020.1 1927.1 2019.5 1929.4 2021.1 C
-1929.9 2020.7 1931.1 2020 1931.1 2019.9 C
-[0.21 0.21 0 0]  vc
-f 
-S 
-n
-1937.1 2020.8 m
-1937.1 2018.3 1937.1 2015.7 1937.1 2013.2 C
-1937.1 2015.7 1937.1 2018.3 1937.1 2020.8 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-2020.4 2012.2 m
-2019.8 2012 2019.3 2011.5 2018.7 2011.7 C
-2017.9 2012.1 2018.1 2014.1 2018.4 2014.4 C
-2019.6 2014.8 2021.4 2013.7 2020.4 2012.2 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1976 2013.9 m
-1973.8 2011.5 1971.6 2009.1 1969.5 2006.7 C
-1971.6 2009.1 1973.8 2011.5 1976 2013.9 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-1995.4 2012.7 m
-1996.1 2010.3 1993.8 2006.2 1997.3 2005.7 C
-1998.9 2005.4 2000 2003.7 2001.4 2003.1 C
-2003.9 2003.1 2005.3 2001.3 2006.9 1999.7 C
-2004.5 2003.5 2000 2002.2 1997.6 2005.7 C
-1996.5 2005.9 1994.8 2006.1 1995.2 2007.6 C
-1995.7 2009.4 1995.2 2011.6 1994.7 2012.9 C
-1992 2015.8 1987.8 2015.7 1985.3 2018.7 C
-1988.3 2016.3 1992.3 2015.3 1995.4 2012.7 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1995.6 2012.4 m
-1995.6 2011.2 1995.6 2010 1995.6 2008.8 C
-1995.6 2010 1995.6 2011.2 1995.6 2012.4 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-vmrs
-2017.7 2009.6 m
-2016.9 2009.3 2016.7 2008.4 2015.8 2009.1 C
-2014.2 2010.6 2016 2010.6 2016.5 2011.5 C
-2017.2 2010.9 2018.1 2010.8 2017.7 2009.6 C
-[0 1 1 0.23]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-2014.4 2006.4 m
-2013.5 2006.8 2012.1 2005.6 2012 2006.7 C
-2013 2007.3 2011.9 2009.2 2012.9 2008.4 C
-2014.2 2008.3 2014.6 2007.8 2014.4 2006.4 C
-f 
-S 
-n
-1969 2006.4 m
-1966.5 2003.8 1964 2001.2 1961.6 1998.5 C
-1964 2001.2 1966.5 2003.8 1969 2006.4 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-2012 2005.2 m
-2012.2 2004.2 2011.4 2003.3 2010.3 2003.3 C
-2009 2003.6 2010 2004.7 2009.6 2004.8 C
-2009.3 2005.7 2011.4 2006.7 2012 2005.2 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1962.8 1995.2 m
-1961.7 1994.4 1960.6 1993.7 1959.4 1994 C
-1959.5 1994.9 1957.5 1994.1 1956.8 1994.7 C
-1955.9 1995.5 1956.7 1997 1955.1 1997.3 C
-1956.9 1996.7 1956.8 1994 1959.2 1994.7 C
-1961.1 1991 1968.9 2003.2 1962.8 1995.2 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-1954.6 1995.6 m
-1955.9 1994.7 1955.1 1989.8 1955.3 1988 C
-1954.5 1988.3 1954.9 1986.6 1954.4 1986 C
-1955.7 1989.2 1953.9 1991.1 1954.8 1994.2 C
-1954.5 1995.9 1953.5 1995.3 1953.9 1997.3 C
-1955.3 1998.3 1953.2 1995.5 1954.6 1995.6 C
-f 
-S 
-n
-1992.3 2011 m
-1992.5 2006.7 1992 2000.3 1991.8 1997.6 C
-1992.2 1997.9 1992 1998.5 1992 1999 C
-1992.1 1994.7 1991.9 1990.2 1992.3 1986 C
-1991.4 1984.6 1991 1986.1 1990.6 1985.6 C
-1989.7 1986 1990.3 1987.2 1990.1 1988 C
-1990.7 1987.4 1990.4 1988.5 1990.8 1988.7 C
-1991.3 1997.1 1988.9 2000.6 1991.1 2007.9 C
-1991 2009.1 1989.8 2009.9 1988.4 2008.8 C
-1985.7 2007.2 1986.8 2004.5 1984.1 2004 C
-1984.2 2002.7 1981.9 2003.7 1982.4 2001.4 C
-1981.2 2001.5 1980.5 2000.8 1980 2000 C
-1980 1999.8 1980 1998.9 1980 1999.5 C
-1979.3 1999.5 1979.7 1997.2 1977.9 1998 C
-1977.2 1997.3 1976.6 1996.7 1977.2 1997.8 C
-1979 1998.7 1979.3 2000.8 1981.5 2001.6 C
-1982.2 2002.8 1983 2004.3 1984.4 2004.3 C
-1985 2005.8 1986.2 2007.5 1987.7 2009.1 C
-1989 2010 1991.3 2010.2 1990.8 2012.2 C
-1991.2 2011.4 1992.2 2011.8 1992.3 2011 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1991.8 1995.6 m
-1991.8 1994.3 1991.8 1992.9 1991.8 1991.6 C
-1991.8 1992.9 1991.8 1994.3 1991.8 1995.6 C
-[0 1 1 0.36]  vc
-f 
-S 
-n
-1959.2 1994.2 m
-1958.8 1993.3 1960.7 1993.9 1961.1 1993.7 C
-1961.5 1993.9 1961.2 1994.4 1961.8 1994.2 C
-1960.9 1994 1960.8 1992.9 1959.9 1992.5 C
-1959.6 1993.5 1958.3 1993.5 1958.2 1994.2 C
-1958.1 1994.1 1958 1994 1958 1994 C
-1957.2 1994.9 1958 1993.4 1956.8 1993 C
-1955.6 1992.5 1956 1991 1956.3 1989.9 C
-1956.5 1989.8 1956.6 1990 1956.8 1990.1 C
-1957.1 1989 1956 1989.1 1955.8 1988.2 C
-1955.1 1990.4 1956.2 1995 1954.8 1995.9 C
-1954.1 1995.5 1954.5 1996.5 1954.4 1997.1 C
-1955 1996.8 1954.8 1997.4 1955.6 1996.8 C
-1956 1996 1956.3 1993.2 1958.7 1994.2 C
-1958.9 1994.2 1959.7 1994.2 1959.2 1994.2 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1958.2 1994 m
-1958.4 1993.5 1959.7 1993.1 1959.9 1992 C
-1959.7 1992.5 1959.3 1992 1959.4 1991.8 C
-1959.4 1991.6 1959.4 1990.8 1959.4 1991.3 C
-1959.2 1991.8 1958.8 1991.3 1958.9 1991.1 C
-1958.9 1990.5 1958 1990.3 1957.5 1989.9 C
-1956.8 1989.5 1956.9 1991 1956.3 1990.1 C
-1956.7 1991 1955.4 1992.1 1956.5 1992.3 C
-1956.8 1993.5 1958.3 1992.9 1957.2 1994 C
-1957.8 1994.3 1958.1 1992.4 1958.2 1994 C
-[0 0.5 0.5 0.2]  vc
-f 
-S 
-n
-vmrs
-1954.4 1982.7 m
-1956.1 1982.7 1954.1 1982.5 1953.9 1982.9 C
-1953.9 1983.7 1953.7 1984.7 1954.1 1985.3 C
-1954.4 1984.2 1953.6 1983.6 1954.4 1982.7 C
-[0 1 1 0.36]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1989.6 1982.9 m
-1989.1 1982.7 1988.6 1982.3 1988 1982.4 C
-1987.2 1982.8 1987.4 1984.8 1987.7 1985.1 C
-1988.9 1985.6 1990.7 1984.4 1989.6 1982.9 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1987 1980.3 m
-1986.2 1980 1986 1979.1 1985.1 1979.8 C
-1983.5 1981.4 1985.3 1981.4 1985.8 1982.2 C
-1986.5 1981.7 1987.4 1981.5 1987 1980.3 C
-f 
-S 
-n
-1983.6 1977.2 m
-1982.7 1977.5 1981.4 1976.3 1981.2 1977.4 C
-1982.3 1978 1981.2 1979.9 1982.2 1979.1 C
-1983.5 1979 1983.9 1978.5 1983.6 1977.2 C
-f 
-S 
-n
-1981.2 1976 m
-1981.5 1974.9 1980.6 1974 1979.6 1974 C
-1978.3 1974.3 1979.3 1975.4 1978.8 1975.5 C
-1978.6 1976.4 1980.7 1977.4 1981.2 1976 C
-f 
-S 
-n
-1972.1 2082.3 m
-1971.8 2081.8 1971.3 2080.9 1971.2 2080.1 C
-1971.1 2072.9 1971.3 2064.6 1970.9 2058.3 C
-1970.3 2058.5 1970.1 2057.7 1969.7 2058.5 C
-1970.6 2058.5 1969.7 2059 1970.2 2059.2 C
-1970.2 2065.4 1970.2 2072.4 1970.2 2077.7 C
-1971.1 2078.9 1970.6 2078.9 1970.4 2079.9 C
-1969.2 2080.2 1968.2 2080.4 1967.3 2079.6 C
-1966.8 2077.8 1963.4 2076.3 1963.5 2075.1 C
-1961.5 2075.5 1962 2071.5 1959.6 2072 C
-1959.2 2070 1956.5 2069.3 1955.8 2067.6 C
-1956 2068.4 1955.3 2069.7 1956.5 2069.8 C
-1958.6 2068.9 1958.1 2073.5 1960.1 2072.4 C
-1960.7 2075.9 1964.7 2074.6 1964.2 2078 C
-1967.2 2078.6 1967.9 2081.6 1970.7 2080.6 C
-1970.3 2081.1 1971.5 2081.2 1971.9 2082.3 C
-1967.2 2084.3 1962.9 2087.1 1958.2 2089 C
-1962.9 2087 1967.4 2084.4 1972.1 2082.3 C
-[0 0.2 1 0]  vc
-f 
-S 
-n
-1971.9 2080.1 m
-1971.9 2075.1 1971.9 2070 1971.9 2065 C
-1971.9 2070 1971.9 2075.1 1971.9 2080.1 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-2010.8 2050.6 m
-2013.2 2049 2010.5 2050.1 2010.5 2051.3 C
-2010.5 2057.7 2010.5 2064.1 2010.5 2070.5 C
-2008.7 2072.4 2006 2073.3 2003.6 2074.4 C
-2016.4 2073.7 2008 2058.4 2010.8 2050.6 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-2006.4 2066.9 m
-2006.4 2061.9 2006.4 2056.8 2006.4 2051.8 C
-2006.4 2056.8 2006.4 2061.9 2006.4 2066.9 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1971.9 2060.7 m
-1972.2 2060.3 1971.4 2068.2 1972.4 2061.9 C
-1971.8 2061.6 1972.4 2060.9 1971.9 2060.7 C
-f 
-S 
-n
-vmrs
-1986.5 2055.2 m
-1987.5 2054.3 1986.3 2053.4 1986 2052.8 C
-1983.8 2052.7 1983.6 2050.1 1981.7 2049.6 C
-1981.2 2048.7 1980.8 2047 1980.3 2046.8 C
-1978.5 2047 1978 2044.6 1976.7 2043.9 C
-1974 2044.4 1972 2046.6 1969.2 2047 C
-1969 2047.2 1968.8 2047.5 1968.5 2047.7 C
-1970.6 2049.6 1973.1 2051.3 1974.3 2054.2 C
-1975.7 2054.5 1977 2055.2 1976.4 2057.1 C
-1976.7 2058 1975.5 2058.5 1976 2059.5 C
-1979.2 2058 1983 2056.6 1986.5 2055.2 C
-[0 0.5 0.5 0.2]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1970.2 2054.2 m
-1971.5 2055.3 1972.5 2056.8 1972.1 2058.3 C
-1972.8 2056.5 1971.6 2055.6 1970.2 2054.2 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1992 2052.5 m
-1992 2053.4 1992.2 2054.4 1991.8 2055.2 C
-1992.2 2054.4 1992 2053.4 1992 2052.5 C
-f 
-S 
-n
-1957.2 2053 m
-1958.1 2052.6 1959 2052.2 1959.9 2051.8 C
-1959 2052.2 1958.1 2052.6 1957.2 2053 C
-f 
-S 
-n
-2006.4 2047.5 m
-2006.8 2047.1 2006 2055 2006.9 2048.7 C
-2006.4 2048.4 2007 2047.7 2006.4 2047.5 C
-f 
-S 
-n
-2004.8 2041 m
-2006.1 2042.1 2007.1 2043.6 2006.7 2045.1 C
-2007.3 2043.3 2006.2 2042.4 2004.8 2041 C
-f 
-S 
-n
-1976 2039.8 m
-1975.6 2039.3 1975.2 2038.4 1975 2037.6 C
-1974.9 2030.4 1975.2 2022.1 1974.8 2015.8 C
-1974.2 2016 1974 2015.3 1973.6 2016 C
-1974.4 2016 1973.5 2016.5 1974 2016.8 C
-1974 2022.9 1974 2030 1974 2035.2 C
-1974.9 2036.4 1974.4 2036.4 1974.3 2037.4 C
-1973.1 2037.7 1972 2037.9 1971.2 2037.2 C
-1970.6 2035.3 1967.3 2033.9 1967.3 2032.6 C
-1965.3 2033 1965.9 2029.1 1963.5 2029.5 C
-1963 2027.6 1960.4 2026.8 1959.6 2025.2 C
-1959.8 2025.9 1959.2 2027.2 1960.4 2027.3 C
-1962.5 2026.4 1961.9 2031 1964 2030 C
-1964.6 2033.4 1968.5 2032.1 1968 2035.5 C
-1971 2036.1 1971.8 2039.1 1974.5 2038.1 C
-1974.2 2038.7 1975.3 2038.7 1975.7 2039.8 C
-1971 2041.8 1966.7 2044.6 1962 2046.5 C
-1966.8 2044.5 1971.3 2041.9 1976 2039.8 C
-[0 0.2 1 0]  vc
-f 
-S 
-n
-1975.7 2037.6 m
-1975.7 2032.6 1975.7 2027.6 1975.7 2022.5 C
-1975.7 2027.6 1975.7 2032.6 1975.7 2037.6 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1992 2035.5 m
-1992 2034.2 1992 2032.9 1992 2031.6 C
-1992 2032.9 1992 2034.2 1992 2035.5 C
-f 
-S 
-n
-2015.3 2036 m
-2015.4 2034.1 2013.3 2034 2012.9 2033.3 C
-2011.5 2031 2009.3 2029.4 2007.4 2028 C
-2006.9 2027.1 2006.6 2023.8 2005 2024.9 C
-2004 2024.9 2002.9 2024.9 2001.9 2024.9 C
-2001.4 2026.5 2001 2028.4 2003.8 2028.3 C
-2006.6 2030.4 2008.9 2033.7 2011.2 2036.2 C
-2011.8 2036.4 2012.9 2035.8 2012.9 2036.7 C
-2013 2035.5 2015.3 2037.4 2015.3 2036 C
-[0 0 0 0]  vc
-f 
-S 
-n
-vmrs
-2009.1 2030.4 m
-2009.1 2029 2007.5 2029.4 2006.9 2028.3 C
-2007.2 2027.1 2006.5 2025.5 2005.7 2024.7 C
-2004.6 2025.1 2003.1 2024.9 2001.9 2024.9 C
-2001.8 2026.2 2000.9 2027 2002.4 2028 C
-2004.5 2027.3 2004.9 2029.4 2006.9 2029 C
-2007 2030.2 2007.6 2030.7 2008.4 2031.4 C
-2008.8 2031.5 2009.1 2031.1 2009.1 2030.4 C
-[0 0 0 0.18]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-2003.8 2029.5 m
-2003 2029.4 2001.9 2029.1 2002.4 2030.4 C
-2003.1 2031.3 2005.2 2030.3 2003.8 2029.5 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1999.2 2025.2 m
-1999.1 2025.6 1998 2025.7 1998.8 2026.6 C
-2000.9 2028.5 1999.5 2023.4 1999.2 2025.2 C
-f 
-S 
-n
-2007.6 2024.2 m
-2007.6 2022.9 2008.4 2024.2 2007.6 2022.8 C
-2007.6 2017.5 2007.8 2009.1 2007.4 2003.8 C
-2007.9 2003.7 2008.7 2002.8 2009.1 2002.1 C
-2009.6 2000.8 2008.3 2000.8 2007.9 2000.2 C
-2004.9 2000 2008.9 2001.3 2007.2 2002.1 C
-2006.7 2007.7 2007 2015.1 2006.9 2021.1 C
-2006.7 2022.1 2005.4 2022.8 2006.2 2023.5 C
-2006.6 2023.1 2008 2025.9 2007.6 2024.2 C
-f 
-S 
-n
-1989.9 2023.5 m
-1989.5 2022.6 1991.4 2023.2 1991.8 2023 C
-1992.2 2023.2 1991.9 2023.7 1992.5 2023.5 C
-1991.6 2023.2 1991.6 2022.2 1990.6 2021.8 C
-1990.4 2022.8 1989 2022.8 1988.9 2023.5 C
-1988.5 2023 1988.7 2022.6 1988.7 2023.5 C
-1989.1 2023.5 1990.2 2023.5 1989.9 2023.5 C
-f 
-[0 0.5 0.5 0.2]  vc
-S 
-n
-2003.3 2023.5 m
-2003.1 2023.3 2003.1 2023.2 2003.3 2023 C
-2003.7 2023.1 2003.9 2022.9 2003.8 2022.5 C
-2003.4 2022.2 2001.2 2022.3 2002.4 2023 C
-2002.6 2022.9 2002.7 2023.1 2002.8 2023.2 C
-2000.7 2023.7 2003.9 2023.4 2003.3 2023.5 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1986.8 2019.4 m
-1987.8 2019.8 1987.5 2018.6 1987.2 2018 C
-1986.2 2017.8 1987.3 2020.5 1986.3 2019.2 C
-1986.3 2017.7 1986.3 2020.6 1986.3 2021.3 C
-1988.5 2023.1 1985.6 2020.3 1986.8 2019.4 C
-f 
-S 
-n
-1975.7 2018.2 m
-1976.1 2017.8 1975.2 2025.7 1976.2 2019.4 C
-1975.7 2019.2 1976.3 2018.4 1975.7 2018.2 C
-f 
-S 
-n
-1974 2011.7 m
-1975.4 2012.8 1976.4 2014.3 1976 2015.8 C
-1976.6 2014 1975.5 2013.1 1974 2011.7 C
-f 
-S 
-n
-1984.6 2006.7 m
-1984.7 2004.8 1982.6 2004.8 1982.2 2004 C
-1980.8 2001.7 1978.6 2000.1 1976.7 1998.8 C
-1976.1 1997.8 1975.8 1994.5 1974.3 1995.6 C
-1973.3 1995.6 1972.2 1995.6 1971.2 1995.6 C
-1970.7 1997.2 1970.3 1999.1 1973.1 1999 C
-1975.8 2001.2 1978.2 2004.4 1980.5 2006.9 C
-1981.1 2007.1 1982.1 2006.5 1982.2 2007.4 C
-1982.3 2006.2 1984.5 2008.1 1984.6 2006.7 C
-[0 0 0 0]  vc
-f 
-S 
-n
-vmrs
-1978.4 2001.2 m
-1978.4 1999.7 1976.8 2000.1 1976.2 1999 C
-1976.5 1997.8 1975.8 1996.2 1975 1995.4 C
-1973.9 1995.8 1972.4 1995.6 1971.2 1995.6 C
-1971 1997 1970.2 1997.7 1971.6 1998.8 C
-1973.8 1998 1974.2 2000.1 1976.2 1999.7 C
-1976.3 2000.9 1976.9 2001.4 1977.6 2002.1 C
-1978.1 2002.2 1978.4 2001.8 1978.4 2001.2 C
-[0 0 0 0.18]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1973.1 2000.2 m
-1972.3 2000.1 1971.2 1999.8 1971.6 2001.2 C
-1972.4 2002 1974.5 2001 1973.1 2000.2 C
-[0 1 1 0.23]  vc
-f 
-S 
-n
-1960.8 1998.5 m
-1961.6 1998.2 1962.6 2000.3 1963.2 2000.9 C
-1962.3 2000.1 1962.2 1998.7 1960.8 1998.5 C
-f 
-S 
-n
-1968.5 1995.9 m
-1968.4 1996.4 1967.3 1996.4 1968 1997.3 C
-1970.1 1999.2 1968.8 1994.1 1968.5 1995.9 C
-f 
-S 
-n
-1976.9 1994.9 m
-1976.9 1993.7 1977.6 1994.9 1976.9 1993.5 C
-1976.9 1988.2 1977.1 1979.8 1976.7 1974.5 C
-1977.2 1974.5 1978 1973.5 1978.4 1972.8 C
-1978.8 1971.5 1977.6 1971.5 1977.2 1970.9 C
-1974.2 1970.7 1978.2 1972 1976.4 1972.8 C
-1976 1978.4 1976.3 1985.8 1976.2 1991.8 C
-1976 1992.8 1974.6 1993.5 1975.5 1994.2 C
-1975.9 1993.8 1977.3 1996.6 1976.9 1994.9 C
-f 
-S 
-n
-1972.6 1994.2 m
-1972.4 1994 1972.4 1993.9 1972.6 1993.7 C
-1973 1993.8 1973.1 1993.7 1973.1 1993.2 C
-1972.7 1992.9 1970.5 1993.1 1971.6 1993.7 C
-1971.9 1993.7 1972 1993.8 1972.1 1994 C
-1970 1994.4 1973.1 1994.1 1972.6 1994.2 C
-f 
-S 
-n
-1948.1 2093.8 m
-1947 2092.7 1945.9 2091.6 1944.8 2090.4 C
-1945.9 2091.6 1947 2092.7 1948.1 2093.8 C
-[0 0.4 1 0]  vc
-f 
-S 
-n
-1953.4 2091.4 m
-1954.8 2090.7 1956.3 2090 1957.7 2089.2 C
-1956.3 2090 1954.8 2090.7 1953.4 2091.4 C
-[0 0.2 1 0]  vc
-f 
-S 
-n
-1954.1 2091.4 m
-1956.6 2089.6 1957.2 2089.6 1954.1 2091.4 C
-[0 0.4 1 0]  vc
-f 
-S 
-n
-1962.3 2087.3 m
-1963.7 2086.6 1965.2 2085.9 1966.6 2085.2 C
-1965.2 2085.9 1963.7 2086.6 1962.3 2087.3 C
-f 
-S 
-n
-vmrs
-1967.1 2084.9 m
-1968.3 2084.4 1969.7 2083.8 1970.9 2083.2 C
-1969.7 2083.8 1968.3 2084.4 1967.1 2084.9 C
-[0 0.4 1 0]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1982.7 2080.6 m
-1981.5 2079.5 1980.5 2078.4 1979.3 2077.2 C
-1980.5 2078.4 1981.5 2079.5 1982.7 2080.6 C
-f 
-S 
-n
-1988 2078.2 m
-1989.4 2077.5 1990.8 2076.8 1992.3 2076 C
-1990.8 2076.8 1989.4 2077.5 1988 2078.2 C
-[0 0.2 1 0]  vc
-f 
-S 
-n
-1988.7 2078.2 m
-1991.1 2076.4 1991.8 2076.4 1988.7 2078.2 C
-[0 0.4 1 0]  vc
-f 
-S 
-n
-1976.2 2063.8 m
-1978.6 2062.2 1976 2063.3 1976 2064.5 C
-1976.1 2067.8 1975.5 2071.4 1976.4 2074.4 C
-1975.7 2071.1 1975.9 2067.2 1976.2 2063.8 C
-f 
-S 
-n
-1996.8 2074.1 m
-1998.3 2073.4 1999.7 2072.7 2001.2 2072 C
-1999.7 2072.7 1998.3 2073.4 1996.8 2074.1 C
-f 
-S 
-n
-2001.6 2071.7 m
-2002.9 2071.2 2004.2 2070.6 2005.5 2070 C
-2004.2 2070.6 2002.9 2071.2 2001.6 2071.7 C
-f 
-S 
-n
-1981.5 2060.7 m
-1980.2 2061.2 1978.9 2061.5 1977.9 2062.6 C
-1978.9 2061.5 1980.2 2061.2 1981.5 2060.7 C
-f 
-S 
-n
-1982 2060.4 m
-1982.7 2060.1 1983.6 2059.8 1984.4 2059.5 C
-1983.6 2059.8 1982.7 2060.1 1982 2060.4 C
-f 
-S 
-n
-1952 2051.3 m
-1950.8 2050.2 1949.7 2049.1 1948.6 2048 C
-1949.7 2049.1 1950.8 2050.2 1952 2051.3 C
-f 
-S 
-n
-vmrs
-1977.4 2047.7 m
-1975.8 2047.8 1974.8 2046.1 1974.5 2045.3 C
-1974.9 2044.4 1976 2044.5 1976.7 2044.8 C
-1977.9 2045 1977 2048.4 1979.3 2047.5 C
-1979.9 2047.5 1980.8 2048.6 1979.8 2049.2 C
-1978.2 2050.4 1980.8 2049.5 1980.3 2049.4 C
-1981.4 2049.8 1980.3 2048.4 1980.3 2048 C
-1979.8 2047.5 1979 2046.6 1978.4 2046.5 C
-1977.3 2045.9 1977.2 2043.3 1975.2 2044.6 C
-1974.7 2045.3 1973.6 2045 1973.3 2045.8 C
-1975 2046.3 1975.8 2049.8 1978.1 2049.4 C
-1978.4 2050.9 1978.7 2048.5 1977.9 2049.2 C
-1977.7 2048.7 1977.2 2047.8 1977.4 2047.7 C
-[0 0.5 0.5 0.2]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1957.2 2048.9 m
-1958.7 2048.2 1960.1 2047.5 1961.6 2046.8 C
-1960.1 2047.5 1958.7 2048.2 1957.2 2048.9 C
-[0 0.2 1 0]  vc
-f 
-S 
-n
-1958 2048.9 m
-1960.4 2047.1 1961.1 2047.1 1958 2048.9 C
-[0 0.4 1 0]  vc
-f 
-S 
-n
-1966.1 2044.8 m
-1967.6 2044.1 1969 2043.4 1970.4 2042.7 C
-1969 2043.4 1967.6 2044.1 1966.1 2044.8 C
-f 
-S 
-n
-1970.9 2042.4 m
-1972.2 2041.9 1973.5 2041.3 1974.8 2040.8 C
-1973.5 2041.3 1972.2 2041.9 1970.9 2042.4 C
-f 
-S 
-n
-2012 2034.5 m
-2010.4 2034.6 2009.3 2032.9 2009.1 2032.1 C
-2009.4 2031 2010.3 2031.3 2011.2 2031.6 C
-2012.5 2031.8 2011.6 2035.2 2013.9 2034.3 C
-2014.4 2034.3 2015.4 2035.4 2014.4 2036 C
-2012.7 2037.2 2015.3 2036.3 2014.8 2036.2 C
-2015.9 2036.6 2014.8 2035.2 2014.8 2034.8 C
-2014.4 2034.3 2013.6 2033.4 2012.9 2033.3 C
-2011.5 2031 2009.3 2029.4 2007.4 2028 C
-2007.5 2026.5 2007.3 2027.9 2007.2 2028.3 C
-2007.9 2028.8 2008.7 2029.1 2009.3 2030 C
-2009.6 2030.7 2009 2031.9 2008.4 2031.6 C
-2006.7 2031 2007.7 2028 2005 2028.8 C
-2004.8 2028.6 2004.3 2028.2 2003.8 2028.3 C
-2006.6 2030.4 2008.9 2033.7 2011.2 2036.2 C
-2011.8 2036.4 2012.9 2035.8 2012.9 2036.7 C
-2012.7 2036.1 2011.8 2035 2012 2034.5 C
-[0 0.5 0.5 0.2]  vc
-f 
-S 
-n
-1981.2 2005.2 m
-1979.7 2005.3 1978.6 2003.6 1978.4 2002.8 C
-1978.7 2001.8 1979.6 2002.1 1980.5 2002.4 C
-1981.8 2002.5 1980.9 2005.9 1983.2 2005 C
-1983.7 2005.1 1984.7 2006.1 1983.6 2006.7 C
-1982 2007.9 1984.6 2007 1984.1 2006.9 C
-1985.2 2007.3 1984.1 2006 1984.1 2005.5 C
-1983.6 2005 1982.9 2004.1 1982.2 2004 C
-1980.8 2001.7 1978.6 2000.1 1976.7 1998.8 C
-1976.7 1997.2 1976.6 1998.6 1976.4 1999 C
-1977.2 1999.5 1978 1999.8 1978.6 2000.7 C
-1978.8 2001.5 1978.3 2002.7 1977.6 2002.4 C
-1976 2001.8 1977 1998.7 1974.3 1999.5 C
-1974.1 1999.3 1973.6 1998.9 1973.1 1999 C
-1975.8 2001.2 1978.2 2004.4 1980.5 2006.9 C
-1981.1 2007.1 1982.1 2006.5 1982.2 2007.4 C
-1982 2006.8 1981.1 2005.7 1981.2 2005.2 C
-f 
-S 
-n
-1966.8 1976.4 m
-1969.4 1973 1974.4 1974.6 1976.2 1970.4 C
-1972.7 1974 1968 1975.1 1964 1977.4 C
-1960.9 1979.9 1957.1 1981.8 1953.9 1982.7 C
-1958.4 1981.1 1962.6 1978.8 1966.8 1976.4 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1948.4 2093.8 m
-1949.8 2093.1 1951.2 2092.5 1952.7 2091.9 C
-1951.2 2092.5 1949.8 2093.1 1948.4 2093.8 C
-[0 0.2 1 0]  vc
-f 
-S 
-n
-1948.1 2093.6 m
-1947.3 2092.8 1946.5 2091.9 1945.7 2091.2 C
-1946.5 2091.9 1947.3 2092.8 1948.1 2093.6 C
-f 
-S 
-n
-vmrs
-1942.1 2087.8 m
-1943.5 2088.4 1944.3 2089.5 1945.2 2090.7 C
-1944.8 2089.3 1943.3 2088.3 1942.1 2087.8 C
-[0 0.2 1 0]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1933.5 2078.4 m
-1933.5 2078 1933.2 2079 1933.7 2079.4 C
-1935 2080.4 1936.2 2081.3 1937.1 2082.8 C
-1936.7 2080.7 1933.7 2080.7 1933.5 2078.4 C
-f 
-S 
-n
-1982.9 2080.6 m
-1984.4 2079.9 1985.8 2079.3 1987.2 2078.7 C
-1985.8 2079.3 1984.4 2079.9 1982.9 2080.6 C
-f 
-S 
-n
-1982.7 2080.4 m
-1981.9 2079.6 1981.1 2078.7 1980.3 2078 C
-1981.1 2078.7 1981.9 2079.6 1982.7 2080.4 C
-f 
-S 
-n
-1977.4 2075.1 m
-1977.9 2075.3 1979.1 2076.4 1979.8 2077.5 C
-1979 2076.8 1978.7 2075.1 1977.4 2075.1 C
-f 
-S 
-n
-1952.2 2051.3 m
-1953.6 2050.7 1955.1 2050.1 1956.5 2049.4 C
-1955.1 2050.1 1953.6 2050.7 1952.2 2051.3 C
-f 
-S 
-n
-1952 2051.1 m
-1951.2 2050.3 1950.3 2049.5 1949.6 2048.7 C
-1950.3 2049.5 1951.2 2050.3 1952 2051.1 C
-f 
-S 
-n
-1946 2045.3 m
-1947.3 2045.9 1948.1 2047 1949.1 2048.2 C
-1948.6 2046.8 1947.1 2045.8 1946 2045.3 C
-f 
-S 
-n
-1937.3 2036 m
-1937.4 2035.5 1937 2036.5 1937.6 2036.9 C
-1938.8 2037.9 1940.1 2038.8 1940.9 2040.3 C
-1940.6 2038.2 1937.6 2038.2 1937.3 2036 C
-f 
-S 
-n
-1935.2 2073.2 m
-1936.4 2069.9 1935.8 2061.8 1935.6 2056.4 C
-1935.8 2055.9 1936.3 2055.7 1936.1 2055.2 C
-1935.7 2054.7 1935 2055 1934.4 2054.9 C
-1934.4 2061.5 1934.4 2068.7 1934.4 2074.6 C
-1935.7 2075.1 1936 2073.7 1935.2 2073.2 C
-[0 0.01 1 0]  vc
-f 
-S 
-n
-vmrs
-1939 2030.7 m
-1940.3 2027.4 1939.7 2019.3 1939.5 2013.9 C
-1939.7 2013.5 1940.1 2013.2 1940 2012.7 C
-1939.5 2012.3 1938.8 2012.5 1938.3 2012.4 C
-1938.3 2019 1938.3 2026.2 1938.3 2032.1 C
-1939.5 2032.7 1939.8 2031.2 1939 2030.7 C
-[0 0.01 1 0]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1975.2 2077.2 m
-1975.3 2077.3 1975.4 2077.4 1975.5 2077.5 C
-1974.7 2073.2 1974.9 2067.5 1975.2 2063.6 C
-1975.4 2064 1974.6 2063.9 1974.8 2064.3 C
-1974.9 2069.9 1974.3 2076.5 1975.2 2081.1 C
-1974.9 2079.9 1974.9 2078.4 1975.2 2077.2 C
-[0.92 0.92 0 0.67]  vc
-f 
-S 
-n
-1930.8 2067.4 m
-1931.5 2070.1 1929.6 2072.1 1930.6 2074.6 C
-1931 2072.6 1930.8 2069.8 1930.8 2067.4 C
-f 
-S 
-n
-2010 2050.1 m
-2009.8 2050.5 2009.5 2050.9 2009.3 2051.1 C
-2009.5 2056.7 2008.9 2063.3 2009.8 2067.9 C
-2009.5 2062.1 2009.3 2054.7 2010 2050.1 C
-f 
-S 
-n
-1930.1 2060.9 m
-1929.3 2057.1 1930.7 2054.8 1929.9 2051.3 C
-1930.2 2050.2 1931.1 2049.6 1931.8 2049.2 C
-1931.4 2049.6 1930.4 2049.5 1930.1 2050.1 C
-1928.4 2054.8 1933.4 2063.5 1925.3 2064.3 C
-1927.2 2063.9 1928.5 2062.1 1930.1 2060.9 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1929.6 2061.2 m
-1929.6 2057.6 1929.6 2054.1 1929.6 2050.6 C
-1930 2049.9 1930.5 2049.4 1931.1 2049.2 C
-1930 2048.6 1930.5 2050.2 1929.4 2049.6 C
-1928 2054.4 1932.8 2063 1925.3 2064 C
-1926.9 2063.3 1928.3 2062.4 1929.6 2061.2 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1930.8 2061.6 m
-1930.5 2058 1931.6 2054 1930.8 2051.3 C
-1930.3 2054.5 1930.9 2058.5 1930.4 2061.9 C
-1930.5 2061.2 1931 2062.2 1930.8 2061.6 C
-[0.92 0.92 0 0.67]  vc
-f 
-S 
-n
-1941.2 2045.1 m
-1939.7 2042.6 1937.3 2041.2 1935.4 2039.3 C
-1934.2 2040 1933.7 2036.4 1934 2039.3 C
-1934.9 2040.1 1936.1 2039.9 1936.8 2040.8 C
-1935.3 2044.2 1942.3 2041.7 1939.5 2046 C
-1937.1 2048.5 1940.5 2045.6 1941.2 2045.1 C
-f 
-S 
-n
-1910 2045.8 m
-1910 2039.4 1910 2033 1910 2026.6 C
-1910 2033 1910 2039.4 1910 2045.8 C
-f 
-S 
-n
-1978.8 2022.3 m
-1979.1 2021.7 1979.4 2020.4 1978.6 2021.6 C
-1978.6 2026.9 1978.6 2033 1978.6 2037.6 C
-1979.2 2037 1979.1 2038.2 1979.1 2038.6 C
-1978.7 2033.6 1978.9 2026.8 1978.8 2022.3 C
-f 
-S 
-n
-vmrs
-2026.1 2041.2 m
-2026.1 2034.8 2026.1 2028.3 2026.1 2021.8 C
-2026.1 2028.5 2026.3 2035.4 2025.9 2042 C
-2024.4 2042.9 2022.9 2044.1 2021.3 2044.8 C
-2023.1 2044 2025.1 2042.8 2026.1 2041.2 C
-[0.07 0.06 0 0.58]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-2026.4 2021.8 m
-2026.3 2028.5 2026.5 2035.4 2026.1 2042 C
-2025.6 2042.8 2024.7 2042.7 2024.2 2043.4 C
-2024.7 2042.7 2025.5 2042.7 2026.1 2042.2 C
-2026.5 2035.5 2026.3 2027.9 2026.4 2021.8 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-2025.6 2038.4 m
-2025.6 2033 2025.6 2027.6 2025.6 2022.3 C
-2025.6 2027.6 2025.6 2033 2025.6 2038.4 C
-[0.92 0.92 0 0.67]  vc
-f 
-S 
-n
-1934 2023.5 m
-1934 2024.7 1933.8 2026 1934.2 2027.1 C
-1934 2025.5 1934.7 2024.6 1934 2023.5 C
-f 
-S 
-n
-1928.2 2023.5 m
-1928 2024.6 1927.4 2023.1 1926.8 2023.2 C
-1926.2 2021 1921.4 2019.3 1923.2 2018 C
-1922.7 2016.5 1923.2 2019.3 1922.2 2018.2 C
-1924.4 2020.4 1926.2 2023.3 1928.9 2024.9 C
-1927.9 2024.2 1929.8 2023.5 1928.2 2023.5 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1934 2019.2 m
-1932 2019.6 1930.8 2022.6 1928.7 2021.8 C
-1924.5 2016.5 1918.2 2011.8 1914 2006.7 C
-1914 2005.7 1914 2004.6 1914 2003.6 C
-1913.6 2004.3 1913.9 2005.8 1913.8 2006.9 C
-1919 2012.4 1924.1 2016.5 1929.2 2022.3 C
-1931 2021.7 1932.2 2019.8 1934 2019.2 C
-f 
-S 
-n
-1928.7 2024.9 m
-1926.3 2022.7 1924.1 2020.4 1921.7 2018.2 C
-1924.1 2020.4 1926.3 2022.7 1928.7 2024.9 C
-[0.65 0.65 0 0.42]  vc
-f 
-S 
-n
-1914.3 2006.7 m
-1918.7 2011.8 1924.5 2016.4 1928.9 2021.6 C
-1924.2 2016.1 1919 2012.1 1914.3 2006.7 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1924.8 2020.8 m
-1921.2 2016.9 1925.6 2022.5 1926 2021.1 C
-1924.2 2021 1926.7 2019.6 1924.8 2020.8 C
-[0.92 0.92 0 0.67]  vc
-f 
-S 
-n
-1934 2018.4 m
-1933.2 2014.7 1934.5 2012.3 1933.7 2008.8 C
-1934 2007.8 1935 2007.2 1935.6 2006.7 C
-1935.3 2007.1 1934.3 2007 1934 2007.6 C
-1932.2 2012.3 1937.2 2021 1929.2 2021.8 C
-1931.1 2021.4 1932.3 2019.6 1934 2018.4 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-vmrs
-1933.5 2018.7 m
-1933.5 2015.1 1933.5 2011.7 1933.5 2008.1 C
-1933.8 2007.4 1934.3 2006.9 1934.9 2006.7 C
-1933.8 2006.1 1934.3 2007.7 1933.2 2007.2 C
-1931.9 2012 1936.7 2020.5 1929.2 2021.6 C
-1930.7 2020.8 1932.2 2019.9 1933.5 2018.7 C
-[0.4 0.4 0 0]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1934.7 2019.2 m
-1934.3 2015.6 1935.4 2011.5 1934.7 2008.8 C
-1934.1 2012 1934.7 2016 1934.2 2019.4 C
-1934.4 2018.7 1934.8 2019.8 1934.7 2019.2 C
-[0.92 0.92 0 0.67]  vc
-f 
-S 
-n
-1917.6 2013.6 m
-1917.8 2011.1 1916.8 2014.2 1917.2 2012.2 C
-1916.3 2012.9 1914.8 2011.8 1914.3 2010.8 C
-1914.2 2010.5 1914.4 2010.4 1914.5 2010.3 C
-1913.9 2008.8 1913.9 2011.9 1914.3 2012 C
-1916.3 2012 1917.6 2013.6 1916.7 2015.6 C
-1913.7 2017.4 1919.6 2014.8 1917.6 2013.6 C
-f 
-S 
-n
-1887.2 2015.3 m
-1887.2 2008.9 1887.2 2002.5 1887.2 1996.1 C
-1887.2 2002.5 1887.2 2008.9 1887.2 2015.3 C
-f 
-S 
-n
-1916.7 2014.4 m
-1917 2012.1 1913 2013 1913.8 2010.8 C
-1912.1 2009.8 1910.9 2009.4 1910.7 2007.9 C
-1910.4 2010.6 1913.4 2010.4 1914 2012.4 C
-1914.9 2012.8 1916.6 2012.9 1916.4 2014.4 C
-1916.9 2015.1 1914.5 2016.6 1916.2 2015.8 C
-1916.4 2015.3 1916.7 2015 1916.7 2014.4 C
-[0.65 0.65 0 0.42]  vc
-f 
-S 
-n
-1914 2009.3 m
-1912.8 2010.9 1909.6 2005.3 1911.9 2009.8 C
-1912.3 2009.6 1913.6 2010.2 1914 2009.3 C
-[0.92 0.92 0 0.67]  vc
-f 
-S 
-n
-1951.2 1998.8 m
-1949 1996.4 1951.5 1994 1950.3 1991.8 C
-1949.1 1989.1 1954 1982.7 1948.8 1981.2 C
-1949.2 1981.5 1951 1982.4 1950.8 1983.6 C
-1951.9 1988.6 1947.1 1986.5 1948.1 1990.4 C
-1948.5 1990.3 1948.7 1990.7 1948.6 1991.1 C
-1949 1992.5 1947.3 1991.9 1948.1 1992.5 C
-1947.1 1992.7 1945.7 1993.5 1945.2 1994.7 C
-1944.5 1996.8 1947.7 2000.5 1943.8 2001.4 C
-1943.4 2002 1943.7 2004 1942.4 2004.5 C
-1945.2 2002.2 1948.9 2000.9 1951.2 1998.8 C
-f 
-S 
-n
-1994.9 1993 m
-1995.1 1996.5 1994.5 2000.3 1995.4 2003.6 C
-1994.5 2000.3 1995.1 1996.5 1994.9 1993 C
-f 
-S 
-n
-1913.8 2003.3 m
-1913.8 1996.9 1913.8 1990.5 1913.8 1984.1 C
-1913.8 1990.5 1913.8 1996.9 1913.8 2003.3 C
-f 
-S 
-n
-1941.9 1998 m
-1940.5 1997.3 1940.7 1999.4 1940.7 2000 C
-1942.8 2001.3 1942.6 1998.8 1941.9 1998 C
-[0 0 0 0]  vc
-f 
-S 
-n
-vmrs
-1942.1 1999.2 m
-1942.2 1998.9 1941.8 1998.8 1941.6 1998.5 C
-1940.4 1998 1940.7 1999.7 1940.7 2000 C
-1941.6 2000.3 1942.6 2000.4 1942.1 1999.2 C
-[0.92 0.92 0 0.67]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1940 1997.1 m
-1939.8 1996 1939.7 1995.9 1939.2 1995.2 C
-1939.1 1995.3 1938.5 1997.9 1937.8 1996.4 C
-1938 1997.3 1939.4 1998.6 1940 1997.1 C
-f 
-S 
-n
-1911.2 1995.9 m
-1911.2 1991.6 1911.3 1987.2 1911.4 1982.9 C
-1911.3 1987.2 1911.2 1991.6 1911.2 1995.9 C
-f 
-S 
-n
-1947.2 1979.1 m
-1945.1 1978.8 1944.6 1975.7 1942.4 1975 C
-1940.5 1972.6 1942.2 1973.7 1942.4 1975.7 C
-1945.8 1975.5 1944.2 1979.8 1947.6 1979.6 C
-1948.3 1982.3 1948.5 1980 1947.2 1979.1 C
-f 
-S 
-n
-1939.5 1973.3 m
-1940.1 1972.6 1939.8 1974.2 1940.2 1973.1 C
-1939.1 1972.8 1938.8 1968.5 1935.9 1969.7 C
-1937.4 1969.2 1938.5 1970.6 1939 1971.4 C
-1939.2 1972.7 1938.6 1973.9 1939.5 1973.3 C
-f 
-S 
-n
-1975.2 2073.2 m
-1975.2 2070.2 1975.2 2067.2 1975.2 2064.3 C
-1975.2 2067.2 1975.2 2070.2 1975.2 2073.2 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1929.9 2065.7 m
-1928.1 2065.6 1926 2068.8 1924.1 2066.9 C
-1918.1 2060.9 1912.9 2055.7 1907.1 2049.9 C
-1906.7 2047.1 1906.9 2043.9 1906.8 2041 C
-1906.8 2043.9 1906.8 2046.8 1906.8 2049.6 C
-1913.2 2055.5 1918.7 2061.9 1925.1 2067.6 C
-1927.1 2067.9 1928.6 2064.4 1930.1 2066.2 C
-1929.7 2070.3 1929.9 2074.7 1929.9 2078.9 C
-1929.6 2074.4 1930.5 2070.1 1929.9 2065.7 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1930.1 2061.6 m
-1928.1 2062.1 1927 2065.1 1924.8 2064.3 C
-1920.7 2058.9 1914.4 2054.3 1910.2 2049.2 C
-1910.2 2048.1 1910.2 2047.1 1910.2 2046 C
-1909.8 2046.8 1910 2048.3 1910 2049.4 C
-1915.1 2054.9 1920.3 2059 1925.3 2064.8 C
-1927.1 2064.2 1928.4 2062.3 1930.1 2061.6 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1932 2049.9 m
-1932.3 2050.3 1932 2050.4 1932.8 2050.4 C
-1932 2050.4 1932.2 2049.2 1931.3 2049.6 C
-1931.4 2050.5 1930.3 2050.4 1930.4 2051.3 C
-1931.1 2051.1 1930.7 2049.4 1932 2049.9 C
-f 
-S 
-n
-1938.3 2046 m
-1936.3 2046.8 1935.2 2047.2 1934.2 2048.9 C
-1935.3 2047.7 1936.8 2046.2 1938.3 2046 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-vmrs
-1938.3 2047 m
-1937.9 2046.9 1936.6 2047.1 1936.1 2048 C
-1936.5 2047.5 1937.3 2046.7 1938.3 2047 C
-[0.18 0.18 0 0.78]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1910.2 2043.2 m
-1910.1 2037.5 1910 2031.8 1910 2026.1 C
-1910 2031.8 1910.1 2037.5 1910.2 2043.2 C
-f 
-S 
-n
-1933.5 2032.1 m
-1933.7 2035.2 1932.8 2035.8 1933.7 2038.6 C
-1933.3 2036.6 1934.6 2018 1933.5 2032.1 C
-f 
-S 
-n
-1907.3 2021.8 m
-1906.6 2025.9 1909.4 2032.6 1903.2 2034 C
-1902.8 2034.1 1902.4 2033.9 1902 2033.8 C
-1897.9 2028.5 1891.6 2023.8 1887.4 2018.7 C
-1887.4 2017.7 1887.4 2016.6 1887.4 2015.6 C
-1887 2016.3 1887.2 2017.8 1887.2 2018.9 C
-1892.3 2024.4 1897.5 2028.5 1902.5 2034.3 C
-1904.3 2033.6 1905.7 2032 1907.3 2030.9 C
-1907.3 2027.9 1907.3 2024.9 1907.3 2021.8 C
-f 
-S 
-n
-1933.7 2023.2 m
-1932 2021.7 1931.1 2024.9 1929.4 2024.9 C
-1931.2 2024.7 1932.4 2021.5 1933.7 2023.2 C
-f 
-S 
-n
-1989.2 2024.4 m
-1987.4 2023.7 1985.8 2022.2 1985.1 2020.4 C
-1984.6 2020.1 1986 2018.9 1985.1 2019.2 C
-1985.6 2020.8 1984.1 2019.4 1984.6 2021.1 C
-1986.3 2022.3 1988.1 2025.3 1989.2 2024.4 C
-f 
-S 
-n
-1904.4 2031.9 m
-1903 2029.7 1905.3 2027.7 1904.2 2025.9 C
-1904.5 2025 1903.7 2023 1904 2021.3 C
-1904 2022.3 1903.2 2022 1902.5 2022 C
-1901.3 2022.3 1902.2 2020.1 1901.6 2019.6 C
-1902.5 2019.8 1902.6 2018.3 1903.5 2018.9 C
-1903.7 2021.8 1905.6 2016.8 1905.6 2020.6 C
-1905.9 2020 1906.3 2020.8 1906.1 2021.1 C
-1905.8 2022.7 1906.7 2020.4 1906.4 2019.9 C
-1906.4 2018.5 1908.2 2017.8 1906.8 2016.5 C
-1906.9 2015.7 1907.7 2017.1 1907.1 2016.3 C
-1908.5 2015.8 1910.3 2015.1 1911.6 2016 C
-1912.2 2016.2 1911.9 2018 1911.6 2018 C
-1914.5 2017.1 1910.4 2013.6 1913.3 2013.4 C
-1912.4 2011.3 1910.5 2011.8 1909.5 2010 C
-1910 2010.5 1909 2010.8 1908.8 2011.2 C
-1907.5 2009.9 1906.1 2011.7 1904.9 2011.5 C
-1904.7 2010.9 1904.3 2010.5 1904.4 2009.8 C
-1905 2010.2 1904.6 2008.6 1905.4 2008.1 C
-1906.6 2007.5 1907.7 2008.4 1908.5 2007.4 C
-1908.9 2008.5 1909.7 2008.1 1909 2007.2 C
-1908.1 2006.5 1905.9 2007.3 1905.4 2007.4 C
-1903.9 2007.3 1905.2 2008.5 1904.2 2008.4 C
-1904.6 2009.9 1902.8 2010.3 1902.3 2010.5 C
-1901.5 2009.9 1900.4 2010 1899.4 2010 C
-1898.6 2011.2 1898.2 2013.4 1896.5 2013.4 C
-1896 2012.9 1894.4 2012.9 1893.6 2012.9 C
-1893.1 2013.9 1892.9 2015.5 1891.5 2016 C
-1890.3 2016.1 1889.2 2014 1888.6 2015.8 C
-1890 2016 1891 2016.9 1892.9 2016.5 C
-1894.1 2017.2 1892.8 2018.3 1893.2 2018.9 C
-1892.6 2018.9 1891.1 2019.8 1890.5 2020.6 C
-1891.1 2023.6 1893.2 2019.8 1893.9 2022.5 C
-1894.1 2023.3 1892.7 2023.6 1893.9 2024 C
-1894.2 2024.3 1897.4 2023.8 1896.5 2026.1 C
-1896 2025.6 1897.4 2028.1 1897.5 2027.1 C
-1898.4 2027.4 1899.3 2027 1899.6 2028.5 C
-1899.5 2028.6 1899.4 2028.8 1899.2 2028.8 C
-1899.3 2029.2 1899.6 2029.8 1900.1 2030.2 C
-1900.4 2029.6 1901 2030 1901.8 2030.2 C
-1903.1 2032.1 1900.4 2031.5 1902.8 2033.1 C
-1903.3 2032.7 1904.5 2032 1904.4 2031.9 C
-[0.21 0.21 0 0]  vc
-f 
-S 
-n
-1909.2 2019.4 m
-1908.8 2020.3 1910.2 2019.8 1909.2 2019.2 C
-1908.3 2019.3 1907.6 2020.2 1907.6 2021.3 C
-1908.5 2021 1907.6 2019 1909.2 2019.4 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1915.5 2015.6 m
-1913.5 2016.3 1912.4 2016.8 1911.4 2018.4 C
-1912.5 2017.2 1914 2015.7 1915.5 2015.6 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1915.5 2016.5 m
-1915.1 2016.4 1913.8 2016.6 1913.3 2017.5 C
-1913.7 2017 1914.5 2016.2 1915.5 2016.5 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-vmrs
-1887.4 2012.7 m
-1887.3 2007 1887.2 2001.3 1887.2 1995.6 C
-1887.2 2001.3 1887.3 2007 1887.4 2012.7 C
-[0.18 0.18 0 0.78]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1935.9 2007.4 m
-1936.2 2007.8 1935.8 2007.9 1936.6 2007.9 C
-1935.9 2007.9 1936.1 2006.7 1935.2 2007.2 C
-1935.2 2008.1 1934.1 2007.9 1934.2 2008.8 C
-1935 2008.7 1934.6 2006.9 1935.9 2007.4 C
-f 
-S 
-n
-1942.1 2003.6 m
-1940.1 2004.3 1939.1 2004.8 1938 2006.4 C
-1939.1 2005.2 1940.6 2003.7 1942.1 2003.6 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1942.1 2004.5 m
-1941.8 2004.4 1940.4 2004.6 1940 2005.5 C
-1940.4 2005 1941.2 2004.2 1942.1 2004.5 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-1914 2000.7 m
-1914 1995 1913.9 1989.3 1913.8 1983.6 C
-1913.9 1989.3 1914 1995 1914 2000.7 C
-f 
-S 
-n
-1941.6 1998.3 m
-1943.4 2001.9 1942.4 1996 1940.9 1998.3 C
-1941.2 1998.3 1941.4 1998.3 1941.6 1998.3 C
-f 
-S 
-n
-1954.8 1989.9 m
-1953.9 1989.6 1954.7 1991.6 1953.9 1991.1 C
-1954.5 1993.1 1953.6 1998 1954.6 1993.2 C
-1954 1992.2 1954.7 1990.7 1954.8 1989.9 C
-f 
-S 
-n
-1947.6 1992.5 m
-1946.2 1993.5 1944.9 1993 1944.8 1994.7 C
-1945.5 1994 1947 1992.2 1947.6 1992.5 C
-f 
-S 
-n
-1910.7 1982.2 m
-1910.3 1981.8 1909.7 1982 1909.2 1982 C
-1909.7 1982 1910.3 1981.9 1910.7 1982.2 C
-1911 1987.1 1910 1992.6 1910.7 1997.3 C
-1910.7 1992.3 1910.7 1987.2 1910.7 1982.2 C
-[0.65 0.65 0 0.42]  vc
-f 
-S 
-n
-1910.9 1992.8 m
-1910.9 1991.3 1910.9 1989.7 1910.9 1988.2 C
-1910.9 1989.7 1910.9 1991.3 1910.9 1992.8 C
-[0.18 0.18 0 0.78]  vc
-f 
-S 
-n
-vmrs
-1953.6 1983.6 m
-1954.1 1985.3 1953.2 1988.6 1954.8 1989.4 C
-1954.1 1987.9 1954.4 1985.4 1953.6 1983.6 C
-[0.18 0.18 0 0.78]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1910.7 1982 m
-1911.6 1982.9 1911 1984.4 1911.2 1985.6 C
-1911 1984.4 1911.6 1982.9 1910.7 1982 C
-f 
-S 
-n
-1947.2 1979.6 m
-1947.5 1980.6 1948.3 1980.6 1947.4 1979.6 C
-1946.2 1979.4 1945.7 1978.8 1947.2 1979.6 C
-f 
-S 
-n
-1930.4 2061.4 m
-1930.4 2058 1930.4 2053.5 1930.4 2051.1 C
-1930.7 2054.6 1929.8 2057.4 1930.1 2061.2 C
-1929.5 2061.9 1929.7 2061.2 1930.4 2061.4 C
-[0.65 0.65 0 0.42]  vc
-f 
-S 
-n
-1939.5 2044.8 m
-1940 2041.5 1935.2 2044.3 1936.4 2040.8 C
-1934.9 2040.9 1934.1 2039.7 1933.5 2038.6 C
-1933.3 2035.4 1933.2 2040 1934 2040.3 C
-1936.2 2040.6 1936.3 2043.6 1938.5 2043.4 C
-1939.7 2044.2 1939.4 2045.6 1938.3 2046.5 C
-1939.1 2046.6 1939.6 2045.6 1939.5 2044.8 C
-f 
-S 
-n
-1910.4 2045.3 m
-1910.4 2039.5 1910.4 2033.6 1910.4 2027.8 C
-1910.4 2033.6 1910.4 2039.5 1910.4 2045.3 C
-f 
-S 
-n
-1906.8 2030.9 m
-1907.6 2026.8 1905 2020.8 1909 2018.7 C
-1906.5 2018.9 1906.8 2022.4 1906.8 2024.7 C
-1906.4 2028.2 1907.9 2032 1903 2033.8 C
-1902.2 2034 1903.8 2033.4 1904.2 2033.1 C
-1905.1 2032.4 1905.9 2031.5 1906.8 2030.9 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1907.1 2030.7 m
-1907.1 2028.8 1907.1 2027 1907.1 2025.2 C
-1907.1 2027 1907.1 2028.8 1907.1 2030.7 C
-[0.65 0.65 0 0.42]  vc
-f 
-S 
-n
-1932 2023.2 m
-1932.2 2023.6 1931.7 2023.7 1931.6 2024 C
-1932 2023.7 1932.3 2022.8 1933 2023 C
-1933.9 2024.3 1933.3 2026.2 1933.5 2027.8 C
-1933.5 2026.4 1934.9 2022.2 1932 2023.2 C
-f 
-S 
-n
-2026.1 2021.6 m
-2026.1 2020.8 2026.1 2019.9 2026.1 2019.2 C
-2026.1 2019.9 2026.1 2020.8 2026.1 2021.6 C
-f 
-S 
-n
-vmrs
-1934.2 2018.9 m
-1934.2 2015.5 1934.2 2011 1934.2 2008.6 C
-1934.5 2012.1 1933.7 2014.9 1934 2018.7 C
-1933.4 2019.5 1933.5 2018.7 1934.2 2018.9 C
-[0.65 0.65 0 0.42]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1887.6 2014.8 m
-1887.6 2009 1887.6 2003.1 1887.6 1997.3 C
-1887.6 2003.1 1887.6 2009 1887.6 2014.8 C
-f 
-S 
-n
-1914.3 2002.8 m
-1914.3 1997 1914.3 1991.1 1914.3 1985.3 C
-1914.3 1991.1 1914.3 1997 1914.3 2002.8 C
-f 
-S 
-n
-1995.4 1992.3 m
-1995.4 1991.5 1995.4 1990.7 1995.4 1989.9 C
-1995.4 1990.7 1995.4 1991.5 1995.4 1992.3 C
-f 
-S 
-n
-1896 1988.4 m
-1896.9 1988 1897.8 1987.7 1898.7 1987.2 C
-1897.8 1987.7 1896.9 1988 1896 1988.4 C
-f 
-S 
-n
-1899.4 1986.8 m
-1900.4 1986.3 1901.3 1985.8 1902.3 1985.3 C
-1901.3 1985.8 1900.4 1986.3 1899.4 1986.8 C
-f 
-S 
-n
-1902.8 1985.1 m
-1905.2 1984 1905.2 1984 1902.8 1985.1 C
-f 
-S 
-n
-1949.1 1983.4 m
-1950.2 1984.4 1947.8 1984.6 1949.3 1985.1 C
-1949.5 1984.4 1949.6 1984.1 1949.1 1983.4 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1906.1 1983.4 m
-1908.6 1982 1908.6 1982 1906.1 1983.4 C
-[0.65 0.65 0 0.42]  vc
-f 
-S 
-n
-1922.7 1976.4 m
-1923.6 1976 1924.4 1975.7 1925.3 1975.2 C
-1924.4 1975.7 1923.6 1976 1922.7 1976.4 C
-f 
-S 
-n
-vmrs
-1926 1974.8 m
-1927 1974.3 1928 1973.8 1928.9 1973.3 C
-1928 1973.8 1927 1974.3 1926 1974.8 C
-[0.65 0.65 0 0.42]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1929.4 1973.1 m
-1931.9 1972 1931.9 1972 1929.4 1973.1 C
-f 
-S 
-n
-1932.8 1971.4 m
-1935.3 1970 1935.3 1970 1932.8 1971.4 C
-f 
-S 
-n
-1949.6 2097.2 m
-1951.1 2096.4 1952.6 2095.5 1954.1 2094.8 C
-1952.6 2095.5 1951.1 2096.4 1949.6 2097.2 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1955.1 2094.3 m
-1956.7 2093.5 1958.3 2092.7 1959.9 2091.9 C
-1958.3 2092.7 1956.7 2093.5 1955.1 2094.3 C
-f 
-S 
-n
-1960.4 2091.6 m
-1961.3 2091.2 1962.1 2090.9 1963 2090.4 C
-1962.1 2090.9 1961.3 2091.2 1960.4 2091.6 C
-f 
-S 
-n
-1963.5 2090.2 m
-1964.4 2089.7 1965.2 2089.2 1966.1 2088.8 C
-1965.2 2089.2 1964.4 2089.7 1963.5 2090.2 C
-f 
-S 
-n
-1966.6 2088.5 m
-1969.5 2087.1 1972.4 2085.8 1975.2 2084.4 C
-1972.4 2085.8 1969.5 2087.1 1966.6 2088.5 C
-f 
-S 
-n
-1965.2 2086.1 m
-1965.9 2085.7 1966.8 2085.3 1967.6 2084.9 C
-1966.8 2085.3 1965.9 2085.7 1965.2 2086.1 C
-f 
-S 
-n
-1968.3 2084.7 m
-1969.2 2084.3 1970 2083.9 1970.9 2083.5 C
-1970 2083.9 1969.2 2084.3 1968.3 2084.7 C
-f 
-S 
-n
-vmrs
-1984.1 2084 m
-1985.6 2083.2 1987.2 2082.3 1988.7 2081.6 C
-1987.2 2082.3 1985.6 2083.2 1984.1 2084 C
-[0.07 0.06 0 0.58]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1976 2078.7 m
-1978.1 2080.1 1980 2082 1982 2083.7 C
-1980 2081.9 1977.9 2080.3 1976 2078.2 C
-1975.5 2079.9 1975.8 2081.9 1975.7 2083.7 C
-1975.8 2082 1975.5 2080.2 1976 2078.7 C
-f 
-S 
-n
-1989.6 2081.1 m
-1991.3 2080.3 1992.8 2079.5 1994.4 2078.7 C
-1992.8 2079.5 1991.3 2080.3 1989.6 2081.1 C
-f 
-S 
-n
-1933.2 2074.6 m
-1932.4 2076.2 1932.8 2077.5 1933 2078.7 C
-1933 2077.6 1932.9 2074.8 1933.2 2074.6 C
-f 
-S 
-n
-1994.9 2078.4 m
-1995.8 2078 1996.7 2077.7 1997.6 2077.2 C
-1996.7 2077.7 1995.8 2078 1994.9 2078.4 C
-f 
-S 
-n
-1998 2077 m
-1998.9 2076.5 1999.8 2076 2000.7 2075.6 C
-1999.8 2076 1998.9 2076.5 1998 2077 C
-f 
-S 
-n
-2001.2 2075.3 m
-2004 2073.9 2006.9 2072.6 2009.8 2071.2 C
-2006.9 2072.6 2004 2073.9 2001.2 2075.3 C
-f 
-S 
-n
-1980.5 2060.7 m
-1979.9 2060.7 1976.7 2062.8 1975.7 2064.5 C
-1975.7 2067.5 1975.7 2070.5 1975.7 2073.4 C
-1976.3 2068.7 1973.9 2061.6 1980.5 2060.7 C
-f 
-S 
-n
-1999.7 2072.9 m
-2000.5 2072.5 2001.3 2072.1 2002.1 2071.7 C
-2001.3 2072.1 2000.5 2072.5 1999.7 2072.9 C
-f 
-S 
-n
-2002.8 2071.5 m
-2003.7 2071.1 2004.6 2070.7 2005.5 2070.3 C
-2004.6 2070.7 2003.7 2071.1 2002.8 2071.5 C
-f 
-S 
-n
-vmrs
-2015.1 2047.5 m
-2014.4 2047.5 2011.2 2049.6 2010.3 2051.3 C
-2010.3 2057.7 2010.3 2064.1 2010.3 2070.5 C
-2010.3 2063.9 2010.1 2057.1 2010.5 2050.6 C
-2012 2049.3 2013.5 2048.3 2015.1 2047.5 C
-[0.07 0.06 0 0.58]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1910.4 2049.2 m
-1914.8 2054.3 1920.7 2058.9 1925.1 2064 C
-1920.4 2058.6 1915.1 2054.6 1910.4 2049.2 C
-f 
-S 
-n
-1988.2 2057.3 m
-1989.1 2056.8 1989.9 2056.2 1990.8 2055.6 C
-1989.9 2056.2 1989.1 2056.8 1988.2 2057.3 C
-f 
-S 
-n
-1991.6 2051.3 m
-1991.6 2046.3 1991.6 2041.2 1991.6 2036.2 C
-1991.6 2041.2 1991.6 2046.3 1991.6 2051.3 C
-f 
-S 
-n
-1935.6 2047.5 m
-1932.9 2051.7 1939.7 2043.8 1935.6 2047.5 C
-f 
-S 
-n
-1938.8 2043.9 m
-1938.1 2043.3 1938.2 2043.7 1937.3 2043.4 C
-1938.7 2043 1938.2 2044.9 1939 2045.3 C
-1938.2 2045.3 1938.7 2046.6 1937.8 2046.5 C
-1939.1 2046.2 1939.1 2044.5 1938.8 2043.9 C
-f 
-S 
-n
-1972.4 2045.6 m
-1973.4 2045 1974.5 2044.4 1975.5 2043.9 C
-1974.5 2044.4 1973.4 2045 1972.4 2045.6 C
-f 
-S 
-n
-1969 2043.6 m
-1969.8 2043.2 1970.6 2042.9 1971.4 2042.4 C
-1970.6 2042.9 1969.8 2043.2 1969 2043.6 C
-f 
-S 
-n
-1972.1 2042.2 m
-1973 2041.8 1973.9 2041.4 1974.8 2041 C
-1973.9 2041.4 1973 2041.8 1972.1 2042.2 C
-f 
-S 
-n
-1906.6 2035 m
-1905 2034.7 1904.8 2036.6 1903.5 2036.9 C
-1904.9 2037 1905.8 2033.4 1907.1 2035.7 C
-1907.1 2037.2 1907.1 2038.6 1907.1 2040 C
-1906.9 2038.4 1907.5 2036.4 1906.6 2035 C
-f 
-S 
-n
-vmrs
-1937.1 2032.1 m
-1936.2 2033.7 1936.6 2035 1936.8 2036.2 C
-1936.8 2035.1 1936.8 2032.4 1937.1 2032.1 C
-[0.07 0.06 0 0.58]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1887.6 2018.7 m
-1892 2023.8 1897.9 2028.4 1902.3 2033.6 C
-1897.6 2028.1 1892.3 2024.1 1887.6 2018.7 C
-f 
-S 
-n
-1999.7 2031.4 m
-1998.7 2030.3 1997.6 2029.2 1996.6 2028 C
-1997.6 2029.2 1998.7 2030.3 1999.7 2031.4 C
-f 
-S 
-n
-1912.8 2017 m
-1910.6 2021.1 1913.6 2015.3 1914.5 2016 C
-1914 2016.3 1913.4 2016.7 1912.8 2017 C
-f 
-S 
-n
-1939.5 2005 m
-1936.7 2009.2 1943.6 2001.3 1939.5 2005 C
-f 
-S 
-n
-1942.6 2001.4 m
-1941.9 2000.8 1942 2001.2 1941.2 2000.9 C
-1942.5 2000.6 1942.1 2002.4 1942.8 2002.8 C
-1942 2002.8 1942.5 2004.1 1941.6 2004 C
-1943 2003.7 1942.9 2002.1 1942.6 2001.4 C
-f 
-S 
-n
-2006.2 2000.7 m
-2005.4 2001.5 2004 2002.8 2004 2002.8 C
-2004.5 2002.4 2005.5 2001.4 2006.2 2000.7 C
-f 
-S 
-n
-1998.5 2001.6 m
-1997.7 2002 1996.8 2002.4 1995.9 2002.6 C
-1995.5 1999.3 1995.7 1995.7 1995.6 1992.3 C
-1995.6 1995.7 1995.6 1999.2 1995.6 2002.6 C
-1996.6 2002.4 1997.7 2002.2 1998.5 2001.6 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1996.1 2002.8 m
-1995.9 2002.8 1995.8 2002.8 1995.6 2002.8 C
-1995.2 1999.5 1995.5 1995.9 1995.4 1992.5 C
-1995.4 1995.9 1995.4 1999.4 1995.4 2002.8 C
-1996.4 2003.1 1998.2 2001.6 1996.1 2002.8 C
-[0.07 0.06 0 0.58]  vc
-f 
-S 
-n
-1969 2002.1 m
-1968 2001 1966.9 1999.9 1965.9 1998.8 C
-1966.9 1999.9 1968 2001 1969 2002.1 C
-f 
-S 
-n
-vmrs
-2000 2001.2 m
-2002.1 2000 2004.1 1998.9 2006.2 1997.8 C
-2004.1 1998.9 2002.1 2000 2000 2001.2 C
-[0.07 0.06 0 0.58]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1895.8 1984.8 m
-1898.3 1983.6 1900.8 1982.3 1903.2 1981 C
-1900.8 1982.3 1898.3 1983.6 1895.8 1984.8 C
-f 
-S 
-n
-1905.2 1980.3 m
-1906.4 1979.9 1907.6 1979.5 1908.8 1979.1 C
-1907.6 1979.5 1906.4 1979.9 1905.2 1980.3 C
-f 
-S 
-n
-1964.7 1977.4 m
-1963.8 1977.5 1962.5 1980.2 1960.8 1980 C
-1962.5 1980.2 1963.3 1978 1964.7 1977.4 C
-f 
-S 
-n
-1952 1979.6 m
-1955.2 1979.2 1955.2 1979.2 1952 1979.6 C
-f 
-S 
-n
-1937.8 1966.4 m
-1941.2 1969.5 1946.1 1976.4 1951.5 1979.3 C
-1946.1 1976.7 1942.8 1970.4 1937.8 1966.4 C
-f 
-S 
-n
-1911.9 1978.6 m
-1914.3 1977.4 1916.7 1976.2 1919.1 1975 C
-1916.7 1976.2 1914.3 1977.4 1911.9 1978.6 C
-f 
-S 
-n
-1975.5 1971.4 m
-1974.6 1972.2 1973.3 1973.6 1973.3 1973.6 C
-1973.7 1973.1 1974.8 1972.1 1975.5 1971.4 C
-f 
-S 
-n
-1922.4 1972.8 m
-1924.9 1971.6 1927.4 1970.3 1929.9 1969 C
-1927.4 1970.3 1924.9 1971.6 1922.4 1972.8 C
-f 
-S 
-n
-1969.2 1971.9 m
-1971.1 1970.9 1972.9 1969.8 1974.8 1968.8 C
-1972.9 1969.8 1971.1 1970.9 1969.2 1971.9 C
-f 
-S 
-n
-vmrs
-1931.8 1968.3 m
-1933 1967.9 1934.2 1967.5 1935.4 1967.1 C
-1934.2 1967.5 1933 1967.9 1931.8 1968.3 C
-[0.07 0.06 0 0.58]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1940.7 2072.4 m
-1941.5 2072.4 1942.3 2072.3 1943.1 2072.2 C
-1942.3 2072.3 1941.5 2072.4 1940.7 2072.4 C
-[0 0 0 0.18]  vc
-f 
-S 
-n
-1948.6 2069.3 m
-1947 2069.5 1945.7 2068.9 1944.8 2069.8 C
-1945.9 2068.5 1948.4 2070.2 1948.6 2069.3 C
-f 
-S 
-n
-1954.6 2066.4 m
-1954.7 2067.9 1955.6 2067.3 1955.6 2068.8 C
-1955.4 2067.8 1956 2066.6 1954.6 2066.4 C
-f 
-S 
-n
-1929.2 2061.2 m
-1927.8 2062.1 1926.3 2064.1 1924.8 2063.3 C
-1926.3 2064.6 1928 2062 1929.2 2061.2 C
-f 
-S 
-n
-1924.4 2067.4 m
-1918.5 2061.6 1912.7 2055.9 1906.8 2050.1 C
-1912.7 2055.9 1918.5 2061.6 1924.4 2067.4 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1924.6 2062.8 m
-1923.9 2062.1 1923.2 2061.2 1922.4 2060.4 C
-1923.2 2061.2 1923.9 2062.1 1924.6 2062.8 C
-[0 0 0 0.18]  vc
-f 
-S 
-n
-1919.3 2057.3 m
-1917.5 2055.6 1915.7 2053.8 1913.8 2052 C
-1915.7 2053.8 1917.5 2055.6 1919.3 2057.3 C
-f 
-S 
-n
-1929.2 2055.2 m
-1929.2 2054.2 1929.2 2053.2 1929.2 2052.3 C
-1929.2 2053.2 1929.2 2054.2 1929.2 2055.2 C
-f 
-S 
-n
-1926.3 2049.6 m
-1925.4 2049 1925.4 2050.5 1924.4 2050.4 C
-1925.3 2051.3 1924.5 2051.9 1925.6 2052.5 C
-1926.9 2052.6 1926 2050.6 1926.3 2049.6 C
-f 
-S 
-n
-vmrs
-1911.2 2046.8 m
-1910.1 2048.9 1911.9 2050.1 1913.1 2051.3 C
-1912.1 2049.9 1910.6 2048.8 1911.2 2046.8 C
-[0 0 0 0.18]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1934 2048.7 m
-1932.6 2048.7 1930.1 2047.7 1929.6 2049.4 C
-1930.9 2048.6 1933.3 2049 1934 2048.7 C
-f 
-S 
-n
-1980 2048.4 m
-1979.5 2046.8 1976.3 2047.9 1977.2 2045.6 C
-1976.8 2045.1 1976.1 2044.7 1975.2 2044.8 C
-1973.7 2046 1976.3 2046.4 1976.7 2047.5 C
-1977.8 2047.2 1978.2 2050 1979.6 2049.2 C
-1980 2049 1979.6 2048.6 1980 2048.4 C
-f 
-S 
-n
-1938.3 2045.6 m
-1938.2 2044.4 1936.8 2043.8 1935.9 2043.4 C
-1936.4 2044.4 1939.1 2044.3 1937.6 2045.8 C
-1937 2046.1 1935.9 2046.1 1935.9 2046.8 C
-1936.7 2046.3 1937.8 2046.2 1938.3 2045.6 C
-f 
-S 
-n
-1932.5 2040 m
-1932.8 2038.1 1932 2038.9 1932.3 2040.3 C
-1933.1 2040.3 1932.7 2041.7 1933.7 2041.5 C
-1933.1 2041 1932.9 2040.5 1932.5 2040 C
-f 
-S 
-n
-2014.6 2035.2 m
-2014.1 2033.6 2010.9 2034.7 2011.7 2032.4 C
-2011.3 2031.9 2009.4 2030.7 2009.3 2032.1 C
-2009.5 2033.7 2012.9 2033.8 2012.4 2035.7 C
-2013 2036.4 2014.2 2036.5 2014.6 2035.2 C
-f 
-S 
-n
-1906.4 2030.7 m
-1905 2031.6 1903.5 2033.6 1902 2032.8 C
-1903.4 2034 1905.6 2031.4 1906.4 2030.7 C
-f 
-S 
-n
-1901.8 2037.2 m
-1899.5 2034.8 1897.2 2032.5 1894.8 2030.2 C
-1897.2 2032.5 1899.5 2034.8 1901.8 2037.2 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1901.8 2032.4 m
-1901.1 2031.6 1900.4 2030.7 1899.6 2030 C
-1900.4 2030.7 1901.1 2031.6 1901.8 2032.4 C
-[0 0 0 0.18]  vc
-f 
-S 
-n
-1944.5 2030 m
-1945.3 2029.9 1946.1 2029.8 1946.9 2029.7 C
-1946.1 2029.8 1945.3 2029.9 1944.5 2030 C
-f 
-S 
-n
-vmrs
-1997.8 2027.8 m
-1997.7 2027.9 1997.6 2028.1 1997.3 2028 C
-1997.4 2029.1 1998.5 2029.5 1999.2 2030 C
-2000.1 2029.5 1998.9 2028 1997.8 2027.8 C
-[0 0 0 0.18]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1906.4 2029.2 m
-1906.4 2026.6 1906.4 2024 1906.4 2021.3 C
-1906.4 2024 1906.4 2026.6 1906.4 2029.2 C
-f 
-S 
-n
-2006.2 2025.9 m
-2006 2025.9 2005.8 2025.8 2005.7 2025.6 C
-2005.7 2025.5 2005.7 2025.3 2005.7 2025.2 C
-2004.6 2025.8 2002.7 2024.7 2001.9 2026.1 C
-2001.9 2027.9 2007.8 2029.2 2006.2 2025.9 C
-[0 0 0 0]  vc
-f 
-S 
-n
-1952.4 2026.8 m
-1950.9 2027 1949.6 2026.4 1948.6 2027.3 C
-1949.7 2026.1 1952.2 2027.7 1952.4 2026.8 C
-[0 0 0 0.18]  vc
-f 
-S 
-n
-1896.5 2026.8 m
-1894.7 2025.1 1892.9 2023.3 1891 2021.6 C
-1892.9 2023.3 1894.7 2025.1 1896.5 2026.8 C
-f 
-S 
-n
-1958.4 2024 m
-1958.5 2025.5 1959.4 2024.8 1959.4 2026.4 C
-1959.3 2025.3 1959.8 2024.1 1958.4 2024 C
-f 
-S 
-n
-1903.5 2019.2 m
-1902.6 2018.6 1902.6 2020 1901.6 2019.9 C
-1902.5 2020.8 1901.7 2021.4 1902.8 2022 C
-1904.1 2022.2 1903.2 2020.1 1903.5 2019.2 C
-f 
-S 
-n
-1933 2018.7 m
-1931.7 2019.6 1930.1 2021.6 1928.7 2020.8 C
-1930.1 2022.1 1931.8 2019.5 1933 2018.7 C
-f 
-S 
-n
-1888.4 2016.3 m
-1887.3 2018.4 1889.1 2019.6 1890.3 2020.8 C
-1889.3 2019.5 1887.8 2018.3 1888.4 2016.3 C
-f 
-S 
-n
-1928.4 2020.4 m
-1927.7 2019.6 1927 2018.7 1926.3 2018 C
-1927 2018.7 1927.7 2019.6 1928.4 2020.4 C
-f 
-S 
-n
-vmrs
-1911.2 2018.2 m
-1909.8 2018.3 1907.3 2017.2 1906.8 2018.9 C
-1908.1 2018.1 1910.5 2018.6 1911.2 2018.2 C
-[0 0 0 0.18]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1915.5 2015.1 m
-1915.4 2013.9 1914 2013.3 1913.1 2012.9 C
-1913.6 2013.9 1916.3 2013.8 1914.8 2015.3 C
-1914.2 2015.6 1913.1 2015.6 1913.1 2016.3 C
-1913.9 2015.9 1915 2015.7 1915.5 2015.1 C
-f 
-S 
-n
-1923.2 2014.8 m
-1921.3 2013.1 1919.5 2011.3 1917.6 2009.6 C
-1919.5 2011.3 1921.3 2013.1 1923.2 2014.8 C
-f 
-S 
-n
-1933 2012.7 m
-1933 2011.7 1933 2010.8 1933 2009.8 C
-1933 2010.8 1933 2011.7 1933 2012.7 C
-f 
-S 
-n
-1909.7 2008.1 m
-1908.9 2009.2 1910.1 2009.9 1910.4 2011 C
-1911.1 2010.7 1908.9 2009.7 1909.7 2008.1 C
-f 
-S 
-n
-1930.1 2007.2 m
-1929.2 2006.6 1929.2 2008 1928.2 2007.9 C
-1929.1 2008.8 1928.4 2009.4 1929.4 2010 C
-1930.7 2010.2 1929.9 2008.1 1930.1 2007.2 C
-f 
-S 
-n
-1915 2004.3 m
-1914 2006.4 1915.7 2007.6 1916.9 2008.8 C
-1915.9 2007.5 1914.4 2006.3 1915 2004.3 C
-f 
-S 
-n
-1937.8 2006.2 m
-1936.4 2006.3 1934 2005.2 1933.5 2006.9 C
-1934.7 2006.1 1937.1 2006.6 1937.8 2006.2 C
-f 
-S 
-n
-1983.9 2006 m
-1983.3 2004.3 1980.2 2005.4 1981 2003.1 C
-1980.6 2002.7 1978.7 2001.5 1978.6 2002.8 C
-1978.8 2004.4 1982.1 2004.5 1981.7 2006.4 C
-1982.3 2007.2 1983.5 2007.2 1983.9 2006 C
-f 
-S 
-n
-1942.1 2003.1 m
-1942 2001.9 1940.6 2001.3 1939.7 2000.9 C
-1940.2 2001.9 1943 2001.8 1941.4 2003.3 C
-1940.9 2003.6 1939.7 2003.6 1939.7 2004.3 C
-1940.5 2003.9 1941.6 2003.7 1942.1 2003.1 C
-f 
-S 
-n
-vmrs
-1967.1 1998.5 m
-1967 1998.6 1966.8 1998.8 1966.6 1998.8 C
-1966.7 1999.8 1967.8 2000.2 1968.5 2000.7 C
-1969.4 2000.2 1968.2 1998.8 1967.1 1998.5 C
-[0 0 0 0.18]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1936.4 1997.6 m
-1936.7 1995.6 1935.8 1996.4 1936.1 1997.8 C
-1936.9 1997.9 1936.5 1999.2 1937.6 1999 C
-1937 1998.5 1936.8 1998 1936.4 1997.6 C
-f 
-S 
-n
-1975.5 1996.6 m
-1975.2 1996.7 1975.1 1996.5 1975 1996.4 C
-1975 1996.2 1975 1996.1 1975 1995.9 C
-1973.9 1996.5 1972 1995.5 1971.2 1996.8 C
-1971.2 1998.6 1977 1999.9 1975.5 1996.6 C
-[0 0 0 0]  vc
-f 
-S 
-n
-1949.3 2097.4 m
-1950.3 2096.9 1951.2 2096.4 1952.2 2096 C
-1951.2 2096.4 1950.3 2096.9 1949.3 2097.4 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1960.8 2091.6 m
-1961.7 2091.2 1962.6 2090.9 1963.5 2090.4 C
-1962.6 2090.9 1961.7 2091.2 1960.8 2091.6 C
-f 
-S 
-n
-1964.4 2090 m
-1965.7 2089.2 1967 2088.5 1968.3 2087.8 C
-1967 2088.5 1965.7 2089.2 1964.4 2090 C
-f 
-S 
-n
-1976 2083.7 m
-1976.3 2082.3 1975.2 2079.1 1976.9 2079.4 C
-1978.8 2080.7 1980.3 2082.9 1982.2 2084.2 C
-1980.6 2083.1 1978.2 2080.2 1976 2078.9 C
-1975.6 2081.2 1977 2084.9 1973.8 2085.4 C
-1972.2 2086.1 1970.7 2087 1969 2087.6 C
-1971.4 2086.5 1974.1 2085.6 1976 2083.7 C
-f 
-S 
-n
-1983.9 2084.2 m
-1984.8 2083.7 1985.8 2083.2 1986.8 2082.8 C
-1985.8 2083.2 1984.8 2083.7 1983.9 2084.2 C
-f 
-S 
-n
-1995.4 2078.4 m
-1996.3 2078 1997.1 2077.7 1998 2077.2 C
-1997.1 2077.7 1996.3 2078 1995.4 2078.4 C
-f 
-S 
-n
-1999 2076.8 m
-2000.3 2076 2001.6 2075.3 2002.8 2074.6 C
-2001.6 2075.3 2000.3 2076 1999 2076.8 C
-f 
-S 
-n
-vmrs
-1929.6 2065.7 m
-1930.1 2065.6 1929.8 2068.6 1929.9 2070 C
-1929.8 2068.6 1930.1 2067 1929.6 2065.7 C
-[0.4 0.4 0 0]  vc
-f 
-0.4 w
-2 J
-2 M
-S 
-n
-1906.6 2049.4 m
-1906.6 2046.7 1906.6 2043.9 1906.6 2041.2 C
-1906.6 2043.9 1906.6 2046.7 1906.6 2049.4 C
-f 
-S 
-n
-2016 2047.5 m
-2014.8 2048 2013.5 2048.3 2012.4 2049.4 C
-2013.5 2048.3 2014.8 2048 2016 2047.5 C
-f 
-S 
-n
-2016.5 2047.2 m
-2017.3 2046.9 2018.1 2046.6 2018.9 2046.3 C
-2018.1 2046.6 2017.3 2046.9 2016.5 2047.2 C
-f 
-S 
-n
-1912.4 2028.5 m
-1911.8 2032.4 1912.4 2037.2 1911.9 2041.2 C
-1911.5 2037.2 1911.7 2032.9 1911.6 2028.8 C
-1911.6 2033.5 1911.6 2038.9 1911.6 2042.9 C
-1912.5 2042.2 1911.6 2043.9 1912.6 2043.6 C
-1912.9 2039.3 1913.1 2033.3 1912.4 2028.5 C
-[0.21 0.21 0 0]  vc
-f 
-S 
-n
-1906.8 2040.8 m
-1906.8 2039 1906.8 2037.2 1906.8 2035.5 C
-1906.8 2037.2 1906.8 2039 1906.8 2040.8 C
-[0.4 0.4 0 0]  vc
-f 
-S 
-n
-1905.9 2035.2 m
-1904.9 2036.4 1903.7 2037.2 1902.3 2037.4 C
-1903.7 2037.2 1904.9 2036.4 1905.9 2035.2 C
-f 
-S 
-n
-1906.1 2031.2 m
-1907 2031.1 1906.4 2028 1906.6 2030.7 C
-1905.5 2032.1 1904 2032.8 1902.5 2033.6 C
-1903.9 2033.2 1905 2032.1 1906.1 2031.2 C
-f 
-S 
-n
-1908.3 2018.7 m
-1905.2 2018.6 1907.1 2023.2 1906.6 2025.4 C
-1906.8 2023 1905.9 2019.5 1908.3 2018.7 C
-f 
-S 
-n
-1889.6 1998 m
-1889 2001.9 1889.6 2006.7 1889.1 2010.8 C
-1888.7 2006.7 1888.9 2002.4 1888.8 1998.3 C
-1888.8 2003 1888.8 2008.4 1888.8 2012.4 C
-1889.7 2011.7 1888.8 2013.4 1889.8 2013.2 C
-1890.1 2008.8 1890.3 2002.8 1889.6 1998 C
-[0.21 0.21 0 0]  vc
-f 
-S 
-n
-vmrs
-1999 2001.4 m
-2001 2000.3 2003 1999.2 2005 1998 C
-2003 1999.2 2001 2000.3 1999 2001.4 C
-[0.4 0.4 0 0]  vc
-f 
-0.4 w
-2 J
-2 M
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diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/isabelle_isar.pdf
Binary file doc-src/Functions/isabelle_isar.pdf has changed
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/mathpartir.sty
--- a/doc-src/Functions/mathpartir.sty	Mon Aug 27 22:00:04 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,421 +0,0 @@
-%  Mathpartir --- Math Paragraph for Typesetting Inference Rules
-%
-%  Copyright (C) 2001, 2002, 2003, 2004, 2005 Didier Rémy
-%
-%  Author         : Didier Remy 
-%  Version        : 1.2.0
-%  Bug Reports    : to author
-%  Web Site       : http://pauillac.inria.fr/~remy/latex/
-% 
-%  Mathpartir 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}
-    [2005/12/20 version 1.2.0 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
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-
-\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
-}
-
-\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}
-
-%% Part III -- Type inference rules
-
-\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@item #1{$\displaystyle #1$}
-\def \mpr@sep{2em}
-\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 T 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{\mpr@item {##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
-  }{}
-
-%% The default
-
-\def \mpr@@fraction #1#2{\hbox {\advance \hsize by -0.5em
-    $\displaystyle {#1\mpr@over #2}$}}
-\let \mpr@fraction \mpr@@fraction
-
-%% A generic solution to arrow
-
-\def \mpr@make@fraction #1#2#3#4#5{\hbox {%
-     \def \mpr@tail{#1}%
-     \def \mpr@body{#2}%
-     \def \mpr@head{#3}%
-     \setbox1=\hbox{$#4$}\setbox2=\hbox{$#5$}%
-     \setbox3=\hbox{$\mkern -3mu\mpr@body\mkern -3mu$}%
-     \setbox3=\hbox{$\mkern -3mu \mpr@body\mkern -3mu$}%
-     \dimen0=\dp1\advance\dimen0 by \ht3\relax\dp1\dimen0\relax
-     \dimen0=\ht2\advance\dimen0 by \dp3\relax\ht2\dimen0\relax
-     \setbox0=\hbox {$\box1 \@@atop \box2$}%
-     \dimen0=\wd0\box0
-     \box0 \hskip -\dimen0\relax
-     \hbox to \dimen0 {$%
-       \mathrel{\mpr@tail}\joinrel
-       \xleaders\hbox{\copy3}\hfil\joinrel\mathrel{\mpr@head}%
-     $}}}
-
-%% Old stuff should be removed in next version
-\def \mpr@@reduce #1#2{\hbox
-    {$\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$}}
-\def \mpr@infercenter #1{\vcenter {\mpr@hovbox{T}{#1}}}
-
-\def \mpr@empty {}
-\def \mpr@inferrule
-  {\bgroup
-     \ifnum \linewidth<\hsize \hsize \linewidth\fi
-     \mpr@rulelineskip
-     \let \and \qquad
-     \let \hva \mpr@hva
-     \let \@rulename \mpr@empty
-     \let \@rule@options \mpr@empty
-     \let \mpr@over \@@over
-     \mpr@inferrule@}
-\newcommand {\mpr@inferrule@}[3][]
-  {\everymath={\displaystyle}%       
-   \def \@test {#2}\ifx \empty \@test
-      \setbox0 \hbox {$\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. 
-%  
-
-%% Keys that make sence in all kinds of rules
-\def \mprset #1{\setkeys{mprset}{#1}}
-\define@key {mprset}{flushleft}[]{\mpr@centerfalse}
-\define@key {mprset}{center}[]{\mpr@centertrue}
-\define@key {mprset}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
-\define@key {mprset}{myfraction}[]{\let \mpr@fraction #1}
-\define@key {mprset}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
-
-\newbox \mpr@right
-\define@key {mpr}{flushleft}[]{\mpr@centerfalse}
-\define@key {mpr}{center}[]{\mpr@centertrue}
-\define@key {mpr}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
-\define@key {mpr}{myfraction}[]{\let \mpr@fraction #1}
-\define@key {mpr}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
-\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 \TirNameStyle #1{\small \textsc{#1}}
-\def \tir@name #1{\hbox {\small \TirNameStyle{#1}}}
-\let \TirName \tir@name
-\let \DefTirName \TirName
-\let \RefTirName \TirName
-
-%%% Other Exports
-
-% \let \listcons \mpr@cons
-% \let \listsnoc \mpr@snoc
-% \let \listhead \mpr@head
-% \let \listmake \mpr@makelist
-
-
-
-
-\endinput
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/Functions/style.sty
--- a/doc-src/Functions/style.sty	Mon Aug 27 22:00:04 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,46 +0,0 @@
-%% toc
-\newcommand{\tocentry}[1]{\cleardoublepage\phantomsection\addcontentsline{toc}{chapter}{#1}
-\@mkboth{\MakeUppercase{#1}}{\MakeUppercase{#1}}}
-
-%% references
-\newcommand{\secref}[1]{\S\ref{#1}}
-\newcommand{\chref}[1]{chapter~\ref{#1}}
-\newcommand{\figref}[1]{figure~\ref{#1}}
-
-%% math
-\newcommand{\text}[1]{\mbox{#1}}
-\newcommand{\isasymvartheta}{\isamath{\theta}}
-\newcommand{\isactrlvec}[1]{\emph{$\overline{#1}$}}
-
-\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
-
-\pagestyle{headings}
-\sloppy
-\binperiod
-\underscoreon
-
-\renewcommand{\isadigit}[1]{\isamath{#1}}
-
-\newenvironment{mldecls}{\par\noindent\begingroup\footnotesize\def\isanewline{\\}\begin{tabular}{l}}{\end{tabular}\smallskip\endgroup}
-
-\isafoldtag{FIXME}
-\isakeeptag{mlref}
-\renewcommand{\isatagmlref}{\subsection*{\makebox[0pt][r]{\fbox{\ML}~~}Reference}\begingroup\def\isastyletext{\rm}\small}
-\renewcommand{\endisatagmlref}{\endgroup}
-
-\newcommand{\isasymGUESS}{\isakeyword{guess}}
-\newcommand{\isasymOBTAIN}{\isakeyword{obtain}}
-\newcommand{\isasymTHEORY}{\isakeyword{theory}}
-\newcommand{\isasymUSES}{\isakeyword{uses}}
-\newcommand{\isasymEND}{\isakeyword{end}}
-\newcommand{\isasymCONSTS}{\isakeyword{consts}}
-\newcommand{\isasymDEFS}{\isakeyword{defs}}
-\newcommand{\isasymTHEOREM}{\isakeyword{theorem}}
-\newcommand{\isasymDEFINITION}{\isakeyword{definition}}
-
-\isabellestyle{it}
-
-%%% Local Variables: 
-%%% mode: latex
-%%% TeX-master: "implementation"
-%%% End: 
diff -r 7eee8b2d2099 -r fa49f8890ef3 doc-src/ROOT
--- a/doc-src/ROOT	Mon Aug 27 22:00:04 2012 +0200
+++ b/doc-src/ROOT	Mon Aug 27 22:14:17 2012 +0200
@@ -18,10 +18,20 @@
     Adaptation
     Further
 
-session Functions (doc) in "Functions/Thy" = HOL +
-  options [browser_info = false, document = false,
-    document_dump = document, document_dump_mode = "tex"]
+session Functions (doc) in "Functions" = HOL +
+  options [document_variants = "functions"]
   theories Functions
+  files
+    "../iman.sty"
+    "../extra.sty"
+    "../isar.sty"
+    "../manual.bib"
+    "document/build"
+    "document/conclusion.tex"
+    "document/intro.tex"
+    "document/mathpartir.sty"
+    "document/root.tex"
+    "document/style.sty"
 
 session IsarImplementation (doc) in "IsarImplementation" = HOL +
   options [document_variants = "implementation"]