Abstract Natural Numbers with polymorphic recursion.

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Abstract_NAT.thy	Thu Feb 16 21:12:03 2006 +0100
@@ -0,0 +1,113 @@
+(*
+    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) (output)
+  "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