header {* Defining natural numbers in FOL \label{sec:ex-natclass} *}
theory NatClass imports FOL begin
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 ("\<zero>")
Suc :: "'a \<Rightarrow> 'a"
rec :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a"
axclass nat \<subseteq> "term"
induct: "P(\<zero>) \<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) = \<zero> \<Longrightarrow> R"
rec_0: "rec(\<zero>, 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 @{text 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 @{text nat} by @{typ
'a} and used the natural number axioms to define class @{text nat}.
There is only a minor snag, that the original recursion operator
@{term rec} had to be made monomorphic.
Thus class @{text nat} contains exactly those types @{text \<tau>} that
are isomorphic to ``the'' natural numbers (with signature @{term
\<zero>}, @{term Suc}, @{term 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).
*}
(*<*)
lemma Suc_n_not_n: "Suc(k) ~= (k::'a::nat)"
apply (rule_tac n = k in induct)
apply (rule notI)
apply (erule Suc_neq_0)
apply (rule notI)
apply (erule notE)
apply (erule Suc_inject)
done
lemma "(k+m)+n = k+(m+n)"
apply (rule induct)
back
back
back
back
back
back
oops
lemma add_0 [simp]: "\<zero>+n = n"
apply (unfold add_def)
apply (rule rec_0)
done
lemma add_Suc [simp]: "Suc(m)+n = Suc(m+n)"
apply (unfold add_def)
apply (rule rec_Suc)
done
lemma add_assoc: "(k+m)+n = k+(m+n)"
apply (rule_tac n = k in induct)
apply simp
apply simp
done
lemma add_0_right: "m+\<zero> = m"
apply (rule_tac n = m in induct)
apply simp
apply simp
done
lemma add_Suc_right: "m+Suc(n) = Suc(m+n)"
apply (rule_tac n = m in induct)
apply simp_all
done
lemma
assumes prem: "!!n. f(Suc(n)) = Suc(f(n))"
shows "f(i+j) = i+f(j)"
apply (rule_tac n = i in induct)
apply simp
apply (simp add: prem)
done
(*>*)
end