(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
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(* This file deals with divisibility for natural numbers.                     *)
(* It contains the definitions of:                                            *)
(*      edivn m d   == the pair composed of the quotient and remainder        *)
(*                     of the Euclidean division of m by d.                   *)
(*          m %/ d  == quotient of the Euclidean division of m by d.          *)
(*          m %% d  == remainder of the Euclidean division of m by d.         *)
(*  m = n %[mod d]  <-> m equals n modulo d.                                  *)
(*  m == n %[mod d] <=> m equals n modulo d (boolean version).                *)
(*  m <> n %[mod d] <-> m differs from n modulo d.                            *)
(*  m != n %[mod d] <=> m differs from n modulo d (boolean version).          *)
(*           d %| m <=> d divides m.                                          *)
(*         gcdn m n == the GCD of m and n.                                    *)
(*        egcdn m n == the extended GCD (Bezout coefficient pair) of m and n. *)
(*                     If egcdn m n = (u, v), then gcdn m n = m * u - n * v.  *)
(*         lcmn m n == the LCM of m and n.                                    *)
(*      coprime m n <=> m and n are coprime (:= gcdn m n == 1).               *)
(*  chinese m n r s == witness of the chinese remainder theorem.              *)
(* We adjoin an m to operator suffixes to indicate a nested %% (modn), as in  *)
(*   modnDml : m %% d + n = m + n %[mod d].                                   *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
(** Euclidean division *)
Definition edivn_rec d :=
fix loop m q := if m - d is m'.+1 then loop m' q.+1 else (q, m).
Definition edivn m d := if d > 0 then edivn_rec d.-1 m 0 else (0, m).
Variant edivn_spec m d : nat * nat -> Type :=
EdivnSpec q r of m = q * d + r & (d > 0) ==> (r < d) : edivn_spec m d (q, r).
Definition divn m d := (edivn m d).1.
Notation "m %/ d" := (divn m d) : nat_scope.
(* We redefine modn so that it is structurally decreasing. *)
Definition modn_rec d := fix loop m := if m - d is m'.+1 then loop m' else m.
Definition modn m d := if d > 0 then modn_rec d.-1 m else m.
Notation "m %% d" := (modn m d) : nat_scope.
Notation "m = n %[mod d ]" := (m %% d = n %% d) : nat_scope.
Notation "m == n %[mod d ]" := (m %% d == n %% d) : nat_scope.
Notation "m <> n %[mod d ]" := (m %% d <> n %% d) : nat_scope.
Notation "m != n %[mod d ]" := (m %% d != n %% d) : nat_scope.
Arguments divnMl [p m d].
Arguments dvdnP {d m}.
Hint Resolve dvdn_add dvdn_sub dvdn_exp : core.
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(*   A function that computes the gcd of 2 numbers                     *)
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Fixpoint gcdn_rec m n :=
let n' := n %% m in if n' is 0 then m else
if m - n'.-1 is m'.+1 then gcdn_rec (m' %% n') n' else n'.
Definition gcdn := nosimpl gcdn_rec.

Lemma modn_coprime k n : 0 < k -> (exists u, (k * u) %% n = 1) -> coprime k n.