(*
ID: $Id$
Author: Makarius
*)
header {* Abstract Natural Numbers with polymorphic recursion *}
theory Abstract_NAT
imports Main
begin
text {* Axiomatic Natural Numbers (Peano) -- a monomorphic theory. *}
locale NAT =
fixes zero :: 'n
and succ :: "'n \<Rightarrow> 'n"
assumes succ_inject [simp]: "(succ m = succ n) = (m = n)"
and succ_neq_zero [simp]: "succ m \<noteq> zero"
and induct [case_names zero succ, induct type: 'n]:
"P zero \<Longrightarrow> (\<And>n. P n \<Longrightarrow> P (succ n)) \<Longrightarrow> P n"
lemma (in NAT) zero_neq_succ [simp]: "zero \<noteq> succ m"
by (rule succ_neq_zero [symmetric])
text {*
Primitive recursion as a (functional) relation -- polymorphic!
(We simulate a localized version of the inductive packages using
explicit premises + parameters, and an abbreviation.) *}
consts
REC :: "'n \<Rightarrow> ('n \<Rightarrow> 'n) \<Rightarrow> 'a \<Rightarrow> ('n \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('n * 'a) set"
inductive "REC zero succ e r"
intros
Rec_zero: "NAT zero succ \<Longrightarrow> (zero, e) \<in> REC zero succ e r"
Rec_succ: "NAT zero succ \<Longrightarrow> (m, n) \<in> REC zero succ e r \<Longrightarrow>
(succ m, r m n) \<in> REC zero succ e r"
abbreviation (in NAT)
"Rec == REC zero succ"
lemma (in NAT) Rec_functional:
fixes x :: 'n
shows "\<exists>!y::'a. (x, y) \<in> Rec e r" (is "\<exists>!y::'a. _ \<in> ?Rec")
proof (induct x)
case zero
show "\<exists>!y. (zero, y) \<in> ?Rec"
proof
show "(zero, e) \<in> ?Rec" by (rule Rec_zero)
fix y assume "(zero, y) \<in> ?Rec"
then show "y = e" by cases simp_all
qed
next
case (succ m)
from `\<exists>!y. (m, y) \<in> ?Rec`
obtain y where y: "(m, y) \<in> ?Rec"
and yy': "\<And>y'. (m, y') \<in> ?Rec \<Longrightarrow> y = y'" by blast
show "\<exists>!z. (succ m, z) \<in> ?Rec"
proof
from _ y show "(succ m, r m y) \<in> ?Rec" by (rule Rec_succ)
fix z assume "(succ m, z) \<in> ?Rec"
then obtain u where "z = r m u" and "(m, u) \<in> ?Rec" by cases simp_all
with yy' show "z = r m y" by (simp only:)
qed
qed
text {* The recursion operator -- polymorphic! *}
definition (in NAT)
"rec e r x = (THE y. (x, y) \<in> Rec e r)"
lemma (in NAT) rec_eval:
assumes Rec: "(x, y) \<in> Rec e r"
shows "rec e r x = y"
unfolding rec_def
using Rec_functional and Rec by (rule the1_equality)
lemma (in NAT) rec_zero: "rec e r zero = e"
proof (rule rec_eval)
show "(zero, e) \<in> Rec e r" by (rule Rec_zero)
qed
lemma (in NAT) rec_succ: "rec e r (succ m) = r m (rec e r m)"
proof (rule rec_eval)
let ?Rec = "Rec e r"
have "(m, rec e r m) \<in> ?Rec"
unfolding rec_def
using Rec_functional by (rule theI')
with _ show "(succ m, r m (rec e r m)) \<in> ?Rec" by (rule Rec_succ)
qed
text {* Just see that our abstract specification makes sense \dots *}
interpretation NAT [0 Suc]
proof (rule NAT.intro)
fix m n
show "(Suc m = Suc n) = (m = n)" by simp
show "Suc m \<noteq> 0" by simp
fix P
assume zero: "P 0"
and succ: "\<And>n. P n \<Longrightarrow> P (Suc n)"
show "P n"
proof (induct n)
case 0 show ?case by (rule zero)
next
case Suc then show ?case by (rule succ)
qed
qed
end