(* Title: FOL/ex/Natural_Numbers.thy
Author: Markus Wenzel, TU Munich
*)
section \<open>Natural numbers\<close>
theory Natural_Numbers
imports FOL
begin
text \<open>
Theory of the natural numbers: Peano's axioms, primitive recursion.
(Modernized version of Larry Paulson's theory "Nat".) \medskip
\<close>
typedecl nat
instance nat :: \<open>term\<close> ..
axiomatization
Zero :: \<open>nat\<close> (\<open>0\<close>) and
Suc :: \<open>nat => nat\<close> and
rec :: \<open>[nat, 'a, [nat, 'a] => 'a] => 'a\<close>
where
induct [case_names 0 Suc, induct type: nat]:
\<open>P(0) ==> (!!x. P(x) ==> P(Suc(x))) ==> P(n)\<close> and
Suc_inject: \<open>Suc(m) = Suc(n) ==> m = n\<close> and
Suc_neq_0: \<open>Suc(m) = 0 ==> R\<close> and
rec_0: \<open>rec(0, a, f) = a\<close> and
rec_Suc: \<open>rec(Suc(m), a, f) = f(m, rec(m, a, f))\<close>
lemma Suc_n_not_n: \<open>Suc(k) \<noteq> k\<close>
proof (induct \<open>k\<close>)
show \<open>Suc(0) \<noteq> 0\<close>
proof
assume \<open>Suc(0) = 0\<close>
then show \<open>False\<close> by (rule Suc_neq_0)
qed
next
fix n assume hyp: \<open>Suc(n) \<noteq> n\<close>
show \<open>Suc(Suc(n)) \<noteq> Suc(n)\<close>
proof
assume \<open>Suc(Suc(n)) = Suc(n)\<close>
then have \<open>Suc(n) = n\<close> by (rule Suc_inject)
with hyp show \<open>False\<close> by contradiction
qed
qed
definition add :: \<open>nat => nat => nat\<close> (infixl \<open>+\<close> 60)
where \<open>m + n = rec(m, n, \<lambda>x y. Suc(y))\<close>
lemma add_0 [simp]: \<open>0 + n = n\<close>
unfolding add_def by (rule rec_0)
lemma add_Suc [simp]: \<open>Suc(m) + n = Suc(m + n)\<close>
unfolding add_def by (rule rec_Suc)
lemma add_assoc: \<open>(k + m) + n = k + (m + n)\<close>
by (induct \<open>k\<close>) simp_all
lemma add_0_right: \<open>m + 0 = m\<close>
by (induct \<open>m\<close>) simp_all
lemma add_Suc_right: \<open>m + Suc(n) = Suc(m + n)\<close>
by (induct \<open>m\<close>) simp_all
lemma
assumes \<open>!!n. f(Suc(n)) = Suc(f(n))\<close>
shows \<open>f(i + j) = i + f(j)\<close>
using assms by (induct \<open>i\<close>) simp_all
end