# Theory Abstract_NAT

theory Abstract_NAT
imports Main
```(*  Title:      HOL/ex/Abstract_NAT.thy
Author:     Makarius
*)

section ‹Abstract Natural Numbers primitive recursion›

theory Abstract_NAT
imports Main
begin

text ‹Axiomatic Natural Numbers (Peano) -- a monomorphic theory.›

locale NAT =
fixes zero :: 'n
and succ :: "'n ⇒ 'n"
assumes succ_inject [simp]: "succ m = succ n ⟷ m = n"
and succ_neq_zero [simp]: "succ m ≠ zero"
and induct [case_names zero succ, induct type: 'n]:
"P zero ⟹ (⋀n. P n ⟹ P (succ n)) ⟹ P n"
begin

lemma zero_neq_succ [simp]: "zero ≠ succ m"
by (rule succ_neq_zero [symmetric])

text ‹┉ Primitive recursion as a (functional) relation -- polymorphic!›

inductive Rec :: "'a ⇒ ('n ⇒ 'a ⇒ 'a) ⇒ 'n ⇒ 'a ⇒ bool"
for e :: 'a and r :: "'n ⇒ 'a ⇒ 'a"
where
Rec_zero: "Rec e r zero e"
| Rec_succ: "Rec e r m n ⟹ Rec e r (succ m) (r m n)"

lemma Rec_functional:
fixes x :: 'n
shows "∃!y::'a. Rec e r x y"
proof -
let ?R = "Rec e r"
show ?thesis
proof (induct x)
case zero
show "∃!y. ?R zero y"
proof
show "?R zero e" ..
show "y = e" if "?R zero y" for y
using that by cases simp_all
qed
next
case (succ m)
from ‹∃!y. ?R m y›
obtain y where y: "?R m y" and yy': "⋀y'. ?R m y' ⟹ y = y'"
by blast
show "∃!z. ?R (succ m) z"
proof
from y show "?R (succ m) (r m y)" ..
next
fix z
assume "?R (succ m) z"
then obtain u where "z = r m u" and "?R m u"
by cases simp_all
with yy' show "z = r m y"
by (simp only:)
qed
qed
qed

text ‹┉ The recursion operator -- polymorphic!›

definition rec :: "'a ⇒ ('n ⇒ 'a ⇒ 'a) ⇒ 'n ⇒ 'a"
where "rec e r x = (THE y. Rec e r x y)"

lemma rec_eval:
assumes Rec: "Rec e r x y"
shows "rec e r x = y"
unfolding rec_def
using Rec_functional and Rec by (rule the1_equality)

lemma rec_zero [simp]: "rec e r zero = e"
proof (rule rec_eval)
show "Rec e r zero e" ..
qed

lemma rec_succ [simp]: "rec e r (succ m) = r m (rec e r m)"
proof (rule rec_eval)
let ?R = "Rec e r"
have "?R m (rec e r m)"
unfolding rec_def using Rec_functional by (rule theI')
then show "?R (succ m) (r m (rec e r m))" ..
qed

definition add :: "'n ⇒ 'n ⇒ 'n"
where "add m n = rec n (λ_ k. succ k) m"

by (induct k) simp_all

by (induct m) simp_all

by (induct m) simp_all

lemma "add (succ (succ (succ zero))) (succ (succ zero)) =
succ (succ (succ (succ (succ zero))))"
by simp

text ‹┉ Example: replication (polymorphic)›

definition repl :: "'n ⇒ 'a ⇒ 'a list"
where "repl n x = rec [] (λ_ xs. x # xs) n"

lemma repl_zero [simp]: "repl zero x = []"
and repl_succ [simp]: "repl (succ n) x = x # repl n x"
unfolding repl_def by simp_all

lemma "repl (succ (succ (succ zero))) True = [True, True, True]"
by simp

end

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 ≠ 0" by simp
show "P n"
if zero: "P 0"
and succ: "⋀n. P n ⟹ P (Suc n)"
for P
proof (induct n)
case 0
show ?case by (rule zero)
next
case Suc
then show ?case by (rule succ)
qed
qed

end
```