(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(** Properties of the greatest common divisor *)
Module Type NGcdProp
(Import A : NAxiomsSig')
(Import B : NSubProp A).
Module Import Private_NZGcdProp := NZGcdProp A A B.
(** Properties of divide *)
#[global] Instance divide_wd : Proper (eq==>eq==>iff) divide := divide_wd.
Definition divide_1_r n := divide_1_r_nonneg n (le_0_l n).
Definition divide_1_l := divide_1_l.
Definition divide_0_r := divide_0_r.
Definition divide_0_l := divide_0_l.
Definition divide_refl := divide_refl.
Definition divide_trans := divide_trans.
#[global] Instance divide_reflexive : Reflexive divide | 5 := divide_refl.
#[global] Instance divide_transitive : Transitive divide | 5 := divide_trans.
Definition divide_antisym n m := divide_antisym_nonneg n m (le_0_l n) (le_0_l m).
Definition mul_divide_mono_l := mul_divide_mono_l.
Definition mul_divide_mono_r := mul_divide_mono_r.
Definition mul_divide_cancel_l := mul_divide_cancel_l.
Definition mul_divide_cancel_r := mul_divide_cancel_r.
Definition divide_add_r := divide_add_r.
Definition divide_mul_l := divide_mul_l.
Definition divide_mul_r := divide_mul_r.
Definition divide_factor_l := divide_factor_l.
Definition divide_factor_r := divide_factor_r.
Definition divide_pos_le := divide_pos_le.
(** Properties of gcd *)
Definition gcd_0_l n : gcd 0 n == n := gcd_0_l_nonneg n (le_0_l n).
Definition gcd_0_r n : gcd n 0 == n := gcd_0_r_nonneg n (le_0_l n).
Definition gcd_diag n : gcd n n == n := gcd_diag_nonneg n (le_0_l n).
Definition gcd_unique n m p := gcd_unique n m p (le_0_l p).
Definition gcd_unique_alt n m p := gcd_unique_alt n m p (le_0_l p).
Definition divide_gcd_iff n m := divide_gcd_iff n m (le_0_l n).
#[global] Instance gcd_wd : Proper (eq==>eq==>eq) gcd := gcd_wd.
Definition gcd_comm := gcd_comm.
Definition gcd_assoc := gcd_assoc.
Definition gcd_eq_0_l := gcd_eq_0_l.
Definition gcd_eq_0_r := gcd_eq_0_r.
Definition gcd_eq_0 := gcd_eq_0.
Definition gcd_mul_diag_l n m := gcd_mul_diag_l n m (le_0_l n).
#[deprecated(since="8.17",note="Use divide_antisym instead.")]
Notation divide_antisym_nonneg := divide_antisym_nonneg (only parsing).
#[deprecated(since="8.17",note="Use gcd_unique instead.")]
Notation gcd_unique' n m p := gcd_unique (only parsing).
#[deprecated(since="8.17",note="Use gcd_unique_alt instead.")]
Notation gcd_unique_alt' := gcd_unique_alt.
#[deprecated(since="8.17",note="Use divide_gcd_iff instead.")]
Notation divide_gcd_iff' := divide_gcd_iff.

Lemma gcd_sub_diag_r : forall n m, n<=m -> gcd n (m-n) == gcd n m.
Definition Bezout n m p := exists a b, a*n == p + b*m.