section ‹Quotient and remainder›
theory QuotRem
imports Util "HOL-Library.Realizers"
begin
text ‹Derivation of quotient and remainder using program extraction.›
theorem division: "∃r q. a = Suc b * q + r ∧ r ≤ b"
proof (induct a)
case 0
have "0 = Suc b * 0 + 0 ∧ 0 ≤ b" by simp
then show ?case by iprover
next
case (Suc a)
then obtain r q where I: "a = Suc b * q + r" and "r ≤ b" by iprover
from nat_eq_dec show ?case
proof
assume "r = b"
with I have "Suc a = Suc b * (Suc q) + 0 ∧ 0 ≤ b" by simp
then show ?case by iprover
next
assume "r ≠ b"
with ‹r ≤ b› have "r < b" by (simp add: order_less_le)
with I have "Suc a = Suc b * q + (Suc r) ∧ (Suc r) ≤ b" by simp
then show ?case by iprover
qed
qed
extract division
text ‹
The program extracted from the above proof looks as follows
@{thm [display] division_def [no_vars]}
The corresponding correctness theorem is
@{thm [display] division_correctness [no_vars]}
›
lemma "division 9 2 = (0, 3)" by eval
end