header {* Defining natural numbers in FOL \label{sec:ex-natclass} *};
theory NatClass = FOL:;
text {*
\medskip\noindent Axiomatic type classes abstract over exactly one
type argument. Thus, any \emph{axiomatic} theory extension where each
axiom refers to at most one type variable, may be trivially turned
into a \emph{definitional} one.
We illustrate this with the natural numbers in
Isabelle/FOL.\footnote{See also
\url{http://isabelle.in.tum.de/library/FOL/ex/NatClass.html}}
*};
consts
zero :: 'a ("0")
Suc :: "'a \\<Rightarrow> 'a"
rec :: "'a \\<Rightarrow> 'a \\<Rightarrow> ('a \\<Rightarrow> 'a \\<Rightarrow> 'a) \\<Rightarrow> 'a";
axclass
nat < "term"
induct: "P(0) \\<Longrightarrow> (\\<And>x. P(x) \\<Longrightarrow> P(Suc(x))) \\<Longrightarrow> P(n)"
Suc_inject: "Suc(m) = Suc(n) \\<Longrightarrow> m = n"
Suc_neq_0: "Suc(m) = 0 \\<Longrightarrow> R"
rec_0: "rec(0, a, f) = a"
rec_Suc: "rec(Suc(m), a, f) = f(m, rec(m, a, f))";
constdefs
add :: "'a::nat \\<Rightarrow> 'a \\<Rightarrow> 'a" (infixl "+" 60)
"m + n \\<equiv> rec(m, n, \\<lambda>x y. Suc(y))";
text {*
This is an abstract version of the plain $Nat$ theory in
FOL.\footnote{See
\url{http://isabelle.in.tum.de/library/FOL/ex/Nat.html}} Basically,
we have just replaced all occurrences of type $nat$ by $\alpha$ and
used the natural number axioms to define class $nat$. There is only
a minor snag, that the original recursion operator $rec$ had to be
made monomorphic.
Thus class $nat$ contains exactly those types $\tau$ that are
isomorphic to ``the'' natural numbers (with signature $0$, $Suc$,
$rec$).
\medskip What we have done here can be also viewed as \emph{type
specification}. Of course, it still remains open if there is some
type at all that meets the class axioms. Now a very nice property of
axiomatic type classes is that abstract reasoning is always possible
--- independent of satisfiability. The meta-logic won't break, even
if some classes (or general sorts) turns out to be empty later ---
``inconsistent'' class definitions may be useless, but do not cause
any harm.
Theorems of the abstract natural numbers may be derived in the same
way as for the concrete version. The original proof scripts may be
re-used with some trivial changes only (mostly adding some type
constraints).
*};
end;