doc-src/TutorialI/Advanced/Partial.thy

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(*<*)theory Partial = While_Combinator:(*>*)

text{*\noindent
Throughout the tutorial we have emphasized the fact that all functions
in HOL are total. Hence we cannot hope to define truly partial
functions. The best we can do are functions that are
\emph{underdefined}\index{underdefined function}:
for certain arguments we only know that a result
exists, but we don't know what it is. When defining functions that are
normally considered partial, underdefinedness turns out to be a very
reasonable alternative.

We have already seen an instance of underdefinedness by means of
non-exhaustive pattern matching: the definition of @{term last} in
\S\ref{sec:recdef-examples}. The same is allowed for \isacommand{primrec}
*}

consts hd :: "'a list \<Rightarrow> 'a"
primrec "hd (x#xs) = x"

text{*\noindent
although it generates a warning.

Even ordinary definitions allow underdefinedness, this time by means of
preconditions:
*}

constdefs minus :: "nat \<Rightarrow> nat \<Rightarrow> nat"
"n \<le> m \<Longrightarrow> minus m n \<equiv> m - n"

text{*
The rest of this section is devoted to the question of how to define
partial recursive functions by other means that non-exhaustive pattern
matching.
*}

subsubsection{*Guarded Recursion*}

text{* Neither \isacommand{primrec} nor \isacommand{recdef} allow to
prefix an equation with a condition in the way ordinary definitions do
(see @{term minus} above). Instead we have to move the condition over
to the right-hand side of the equation. Given a partial function $f$
that should satisfy the recursion equation $f(x) = t$ over its domain
$dom(f)$, we turn this into the \isacommand{recdef}
@{prop[display]"f(x) = (if x \<in> dom(f) then t else arbitrary)"}
where @{term arbitrary} is a predeclared constant of type @{typ 'a}
which has no definition. Thus we know nothing about its value,
which is ideal for specifying underdefined functions on top of it.

As a simple example we define division on @{typ nat}:
*}

consts divi :: "nat \<times> nat \<Rightarrow> nat"
recdef divi "measure(\<lambda>(m,n). m)"
  "divi(m,n) = (if n = 0 then arbitrary else
                if m < n then 0 else divi(m-n,n)+1)"

text{*\noindent Of course we could also have defined
@{term"divi(m,0)"} to be some specific number, for example 0. The
latter option is chosen for the predefined @{text div} function, which
simplifies proofs at the expense of deviating from the
standard mathematical division function.

As a more substantial example we consider the problem of searching a graph.
For simplicity our graph is given by a function (@{term f}) of
type @{typ"'a \<Rightarrow> 'a"} which
maps each node to its successor, and the task is to find the end of a chain,
i.e.\ a node pointing to itself. Here is a first attempt:
@{prop[display]"find(f,x) = (if f x = x then x else find(f, f x))"}
This may be viewed as a fixed point finder or as one half of the well known
\emph{Union-Find} algorithm.
The snag is that it may not terminate if @{term f} has non-trivial cycles.
Phrased differently, the relation
*}

constdefs step1 :: "('a \<Rightarrow> 'a) \<Rightarrow> ('a \<times> 'a)set"
  "step1 f \<equiv> {(y,x). y = f x \<and> y \<noteq> x}"

text{*\noindent
must be well-founded. Thus we make the following definition:
*}

consts find :: "('a \<Rightarrow> 'a) \<times> 'a \<Rightarrow> 'a"
recdef find "same_fst (\<lambda>f. wf(step1 f)) step1"
  "find(f,x) = (if wf(step1 f)
                then if f x = x then x else find(f, f x)
                else arbitrary)"
(hints recdef_simp:same_fst_def step1_def)

text{*\noindent
The recursion equation itself should be clear enough: it is our aborted
first attempt augmented with a check that there are no non-trivial loops.

What complicates the termination proof is that the argument of
@{term find} is a pair. To express the required well-founded relation
we employ the predefined combinator @{term same_fst} of type
@{text[display]"('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> ('b\<times>'b)set) \<Rightarrow> (('a\<times>'b) \<times> ('a\<times>'b))set"}
defined as
@{thm[display]same_fst_def[no_vars]}
This combinator is designed for recursive functions on pairs where the first
component of the argument is passed unchanged to all recursive
calls. Given a constraint on the first component and a relation on the second
component, @{term same_fst} builds the required relation on pairs.
The theorem @{thm[display]wf_same_fst[no_vars]}
is known to the well-foundedness prover of \isacommand{recdef}.
Thus well-foundedness of the given relation is immediate.
Furthermore, each recursive call descends along the given relation:
the first argument stays unchanged and the second one descends along
@{term"step1 f"}. The proof merely requires unfolding of some definitions.

Normally you will then derive the following conditional variant of and from
the recursion equation
*}

lemma [simp]:
  "wf(step1 f) \<Longrightarrow> find(f,x) = (if f x = x then x else find(f, f x))"
by simp

text{*\noindent and then disable the original recursion equation:*}

declare find.simps[simp del]

text{*
We can reason about such underdefined functions just like about any other
recursive function. Here is a simple example of recursion induction:
*}

lemma "wf(step1 f) \<longrightarrow> f(find(f,x)) = find(f,x)"
apply(induct_tac f x rule:find.induct);
apply simp
done

subsubsection{*The {\tt\slshape while} Combinator*}

text{*If the recursive function happens to be tail recursive, its
definition becomes a triviality if based on the predefined \isaindexbold{while}
combinator.  The latter lives in the Library theory
\isa{While_Combinator}, which is not part of @{text Main} but needs to
be included explicitly among the ancestor theories.

Constant @{term while} is of type @{text"('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a"}
and satisfies the recursion equation @{thm[display]while_unfold[no_vars]}
That is, @{term"while b c s"} is equivalent to the imperative program
\begin{verbatim}
     x := s; while b(x) do x := c(x); return x
\end{verbatim}
In general, @{term s} will be a tuple (better still: a record). As an example
consider the following definition of function @{term find} above:
*}

constdefs find2 :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
  "find2 f x \<equiv>
   fst(while (\<lambda>(x,x'). x' \<noteq> x) (\<lambda>(x,x'). (x',f x')) (x,f x))"

text{*\noindent
The loop operates on two ``local variables'' @{term x} and @{term x'}
containing the ``current'' and the ``next'' value of function @{term f}.
They are initalized with the global @{term x} and @{term"f x"}. At the
end @{term fst} selects the local @{term x}.

This looks like we can define at least tail recursive functions
without bothering about termination after all. But there is no free
lunch: when proving properties of functions defined by @{term while},
termination rears its ugly head again. Here is
@{thm[source]while_rule}, the well known proof rule for total
correctness of loops expressed with @{term while}:
@{thm[display,margin=50]while_rule[no_vars]} @{term P} needs to be
true of the initial state @{term s} and invariant under @{term c}
(premises 1 and~2). The post-condition @{term Q} must become true when
leaving the loop (premise~3). And each loop iteration must descend
along a well-founded relation @{term r} (premises 4 and~5).

Let us now prove that @{term find2} does indeed find a fixed point. Instead
of induction we apply the above while rule, suitably instantiated.
Only the final premise of @{thm[source]while_rule} is left unproved
by @{text auto} but falls to @{text simp}:
*}

lemma lem: "\<lbrakk> wf(step1 f); x' = f x \<rbrakk> \<Longrightarrow> 
   \<exists>y. while (\<lambda>(x,x'). x' \<noteq> x) (\<lambda>(x,x'). (x',f x')) (x,x') = (y,y) \<and>
       f y = y"
apply(rule_tac P = "\<lambda>(x,x'). x' = f x" and
               r = "inv_image (step1 f) fst" in while_rule);
apply auto
apply(simp add:inv_image_def step1_def)
done

text{*
The theorem itself is a simple consequence of this lemma:
*}

theorem "wf(step1 f) \<Longrightarrow> f(find2 f x) = find2 f x"
apply(drule_tac x = x in lem)
apply(auto simp add:find2_def)
done

text{* Let us conclude this section on partial functions by a
discussion of the merits of the @{term while} combinator. We have
already seen that the advantage (if it is one) of not having to
provide a termintion argument when defining a function via @{term
while} merely puts off the evil hour. On top of that, tail recursive
functions tend to be more complicated to reason about. So why use
@{term while} at all? The only reason is executability: the recursion
equation for @{term while} is a directly executable functional
program. This is in stark contrast to guarded recursion as introduced
above which requires an explicit test @{prop"x \<in> dom f"} in the
function body.  Unless @{term dom} is trivial, this leads to a
definition that is impossible to execute (or prohibitively slow).
Thus, if you are aiming for an efficiently executable definition
of a partial function, you are likely to need @{term while}.
*}

(*<*)end(*>*)