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theory Greatest_Common_Divisor(* Title: HOL/Extraction/Greatest_Common_Divisor.thy ID: $Id: Greatest_Common_Divisor.thy,v 1.1 2007/11/13 09:58:46 berghofe Exp $ Author: Stefan Berghofer, TU Muenchen Helmut Schwichtenberg, LMU Muenchen *) header {* Greatest common divisor *} theory Greatest_Common_Divisor imports QuotRem begin theorem greatest_common_divisor: "!!n::nat. Suc m < n ==> ∃k n1 m1. k * n1 = n ∧ k * m1 = Suc m ∧ (∀l l1 l2. l * l1 = n --> l * l2 = Suc m --> l ≤ k)" proof (induct m rule: nat_wf_ind) case (1 m n) from division obtain r q where h1: "n = Suc m * q + r" and h2: "r ≤ m" by iprover show ?case proof (cases r) case 0 with h1 have "Suc m * q = n" by simp moreover have "Suc m * 1 = Suc m" by simp moreover { fix l2 have "!!l l1. l * l1 = n ==> l * l2 = Suc m ==> l ≤ Suc m" by (cases l2) simp_all } ultimately show ?thesis by iprover next case (Suc nat) with h2 have h: "nat < m" by simp moreover from h have "Suc nat < Suc m" by simp ultimately have "∃k m1 r1. k * m1 = Suc m ∧ k * r1 = Suc nat ∧ (∀l l1 l2. l * l1 = Suc m --> l * l2 = Suc nat --> l ≤ k)" by (rule 1) then obtain k m1 r1 where h1': "k * m1 = Suc m" and h2': "k * r1 = Suc nat" and h3': "!!l l1 l2. l * l1 = Suc m ==> l * l2 = Suc nat ==> l ≤ k" by iprover have mn: "Suc m < n" by (rule 1) from h1 h1' h2' Suc have "k * (m1 * q + r1) = n" by (simp add: add_mult_distrib2 nat_mult_assoc [symmetric]) moreover have "!!l l1 l2. l * l1 = n ==> l * l2 = Suc m ==> l ≤ k" proof - fix l l1 l2 assume ll1n: "l * l1 = n" assume ll2m: "l * l2 = Suc m" moreover have "l * (l1 - l2 * q) = Suc nat" by (simp add: diff_mult_distrib2 h1 Suc [symmetric] mn ll1n ll2m [symmetric]) ultimately show "l ≤ k" by (rule h3') qed ultimately show ?thesis using h1' by iprover qed qed extract greatest_common_divisor text {* The extracted program for computing the greatest common divisor is @{thm [display] greatest_common_divisor_def} *} consts_code arbitrary ("(error \"arbitrary\")") code_module GCD contains test = "greatest_common_divisor 7 12" ML GCD.test end
theorem greatest_common_divisor:
Suc m < n
==> ∃k n1 m1.
k * n1 = n ∧
k * m1 = Suc m ∧ (∀l l1 l2. l * l1 = n --> l * l2 = Suc m --> l ≤ k)