doc-src/TutorialI/Recdef/examples.thy

+(*<*)
+theory examples = Main:;
+(*>*)
+
+text{*
+Here is a simple example, the Fibonacci function:
+*}
+
+consts fib :: "nat \\<Rightarrow> nat";
+recdef fib "measure(\\<lambda>n. n)"
+  "fib 0 = 0"
+  "fib 1 = 1"
+  "fib (Suc(Suc x)) = fib x + fib (Suc x)";
+
+text{*\noindent
+The definition of \isa{fib} is accompanied by a \bfindex{measure function}
+\isa{\isasymlambda{}n.$\;$n} that maps the argument of \isa{fib} to a
+natural number. The requirement is that in each equation the measure of the
+argument on the left-hand side is strictly greater than the measure of the
+argument of each recursive call. In the case of \isa{fib} this is
+obviously true because the measure function is the identity and
+\isa{Suc(Suc~x)} is strictly greater than both \isa{x} and
+\isa{Suc~x}.
+
+Slightly more interesting is the insertion of a fixed element
+between any two elements of a list:
+*}
+
+consts sep :: "'a * 'a list \\<Rightarrow> 'a list";
+recdef sep "measure (\\<lambda>(a,xs). length xs)"
+  "sep(a, [])     = []"
+  "sep(a, [x])    = [x]"
+  "sep(a, x#y#zs) = x # a # sep(a,y#zs)";
+
+text{*\noindent
+This time the measure is the length of the list, which decreases with the
+recursive call; the first component of the argument tuple is irrelevant.
+
+Pattern matching need not be exhaustive:
+*}
+
+consts last :: "'a list \\<Rightarrow> 'a";
+recdef last "measure (\\<lambda>xs. length xs)"
+  "last [x]      = x"
+  "last (x#y#zs) = last (y#zs)";
+
+text{*
+Overlapping patterns are disambiguated by taking the order of equations into
+account, just as in functional programming:
+*}
+
+consts sep1 :: "'a * 'a list \\<Rightarrow> 'a list";
+recdef sep1 "measure (\\<lambda>(a,xs). length xs)"
+  "sep1(a, x#y#zs) = x # a # sep1(a,y#zs)"
+  "sep1(a, xs)     = xs";
+
+text{*\noindent
+This defines exactly the same function as \isa{sep} above, i.e.\
+\isa{sep1 = sep}.
+
+\begin{warn}
+  \isacommand{recdef} only takes the first argument of a (curried)
+  recursive function into account. This means both the termination measure
+  and pattern matching can only use that first argument. In general, you will
+  therefore have to combine several arguments into a tuple. In case only one
+  argument is relevant for termination, you can also rearrange the order of
+  arguments as in the following definition:
+\end{warn}
+*}
+consts sep2 :: "'a list \\<Rightarrow> 'a \\<Rightarrow> 'a list";
+recdef sep2 "measure length"
+  "sep2 (x#y#zs) = (\\<lambda>a. x # a # sep2 zs a)"
+  "sep2 xs       = (\\<lambda>a. xs)";
+
+text{*
+Because of its pattern-matching syntax, \isacommand{recdef} is also useful
+for the definition of non-recursive functions:
+*}
+
+consts swap12 :: "'a list \\<Rightarrow> 'a list";
+recdef swap12 "{}"
+  "swap12 (x#y#zs) = y#x#zs"
+  "swap12 zs       = zs";
+
+text{*\noindent
+In the non-recursive case the termination measure degenerates to the empty
+set \isa{\{\}}.
+*}
+
+(*<*)
+end
+(*>*)