(*  Title:      HOL/Proofs/Extraction/QuotRem.thy
    Author:     Stefan Berghofer, TU Muenchen
*)

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