# Theory Greatest_Common_Divisor

theory Greatest_Common_Divisor
imports QuotRem
```(*  Title:      HOL/Proofs/Extraction/Greatest_Common_Divisor.thy
Author:     Stefan Berghofer, TU Muenchen
Author:     Helmut Schwichtenberg, LMU Muenchen
*)

section ‹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 have "l * l1 = n ⟹ l * l2 = Suc m ⟹ l ≤ Suc m" for l l1 l2
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"
moreover have "l ≤ k" if ll1n: "l * l1 = n" and ll2m: "l * l2 = Suc m" for l l1 l2
proof -
have "l * (l1 - l2 * q) = Suc nat"
by (simp add: diff_mult_distrib2 h1 Suc [symmetric] mn ll1n ll2m [symmetric])
with ll2m 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}
›

instantiation nat :: default
begin

definition "default = (0::nat)"

instance ..

end

instantiation prod :: (default, default) default
begin

definition "default = (default, default)"

instance ..

end

instantiation "fun" :: (type, default) default
begin

definition "default = (λx. default)"

instance ..

end

lemma "greatest_common_divisor 7 12 = (4, 3, 2)" by eval

end
```