(*  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

 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)"

 (*<*)ML_val "goals_limit := 1"(*>*)

 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]}

   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.

 *}

 (*<*)ML_val "goals_limit := 10"(*>*)

 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

 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, tailrec)(*>*)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

 (*<*)declare findzero.simps[simp del](*>*)

 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 to unfold @{term "findzero f n"} using the @{text psimps}

   rule, and the rest is trivial. Since the @{text psimps} rules carry the

   @{text "[simp]"} attribute by default, we just need a single step:

  *}

 apply simp

 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

     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

     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"}.

 *}

 (*lemma findzero_nicer_domintros:

   "f x = 0 \<Longrightarrow> findzero_dom (f, x)"

   "findzero_dom (f, Suc x) \<Longrightarrow> findzero_dom (f, x)"

 by (rule accpI, erule findzero_rel.cases, auto)+

 *)

 subsection {* A Useful Special Case: Tail recursion *}

 text {*

   The domain predicate is our trick that allows us to model partiality

   in a world of total functions. The downside of this is that we have

   to carry it around all the time. The termination proof above allowed

   us to replace the abstract @{term "findzero_dom (f, n)"} by the more

   concrete @{term "(x \<ge> n \<and> f x = (0::nat))"}, but the condition is still

   there and can only be discharged for special cases.

   In particular, the domain predicate guards the unfolding of our

   function, since it is there as a condition in the @{text psimp}

   rules. 

   Now there is an important special case: We can actually get rid

   of the condition in the simplification rules, \emph{if the function

   is tail-recursive}. The reason is that for all tail-recursive

   equations there is a total function satisfying them, even if they

   are non-terminating. 

 %  A function is tail recursive, if each call to the function is either

 %  equal

 %

 %  So the outer form of the 

 %

 %if it can be written in the following

 %  form:

 %  {term[display] "f x = (if COND x then BASE x else f (LOOP x))"}

   The function package internally does the right construction and can

   derive the unconditional simp rules, if we ask it to do so. Luckily,

   our @{const "findzero"} function is tail-recursive, so we can just go

   back and add another option to the \cmd{function} command:

 \vspace{1ex}

 \noindent\cmd{function} @{text "(domintros, tailrec) findzero :: \"(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat\""}\\%

 \cmd{where}\isanewline%

 \ \ \ldots\\%

   \noindent Now, we actually get unconditional simplification rules, even

   though the function is partial:

 *}

 thm findzero.simps

 text {*

   @{thm[display] findzero.simps}

   \noindent Of course these would make the simplifier loop, so we better remove

   them from the simpset:

 *}

 declare findzero.simps[simp del]

 text {* 

   Getting rid of the domain conditions in the simplification rules is

   not only useful because it simplifies proofs. It is also required in

   order to use Isabelle's code generator to generate ML code

   from a function definition.

   Since the code generator only works with equations, it cannot be

   used with @{text "psimp"} rules. Thus, in order to generate code for

   partial functions, they must be defined as a tail recursion.

   Luckily, many functions have a relatively natural tail recursive

   definition.

 *}

 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

 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

 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