# Library mathcomp.ssreflect.div

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.
Distributed under the terms of CeCILL-B. *)
From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq.
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].
Set Implicit Arguments.
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).
Lemma edivnP m d : edivn_spec m d (edivn m d).
Lemma edivn_eq d q r : r < d → edivn (q × d + r) 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.
Lemma modn_def m d : m %% d = (edivn m d).2.
Lemma edivn_def m d : edivn m d = (m %/ d, m %% d).
Lemma divn_eq m d : m = m %/ d × d + m %% d.
Lemma div0n d : 0 %/ d = 0.
Lemma divn0 m : m %/ 0 = 0.
Lemma mod0n d : 0 %% d = 0.
Lemma modn0 m : m %% 0 = m.
Lemma divn_small m d : m < d → m %/ d = 0.
Lemma divnMDl q m d : 0 < d → (q × d + m) %/ d = q + m %/ d.
Lemma mulnK m d : 0 < d → m × d %/ d = m.
Lemma mulKn m d : 0 < d → d × m %/ d = m.
Lemma expnB p m n : p > 0 → m ≥ n → p ^ (m - n) = p ^ m %/ p ^ n.
Lemma modn1 m : m %% 1 = 0.
Lemma divn1 m : m %/ 1 = m.
Lemma divnn d : d %/ d = (0 < d).
Lemma divnMl p m d : p > 0 → p × m %/ (p × d) = m %/ d.
Arguments divnMl [p m d].
Lemma divnMr p m d : p > 0 → m × p %/ (d × p) = m %/ d.
Arguments divnMr [p m d].
Lemma ltn_mod m d : (m %% d < d) = (0 < d).
Lemma ltn_pmod m d : 0 < d → m %% d < d.
Lemma leq_trunc_div m d : m %/ d × d ≤ m.
Lemma leq_mod m d : m %% d ≤ m.
Lemma leq_div m d : m %/ d ≤ m.
Lemma ltn_ceil m d : 0 < d → m < (m %/ d).+1 × d.
Lemma ltn_divLR m n d : d > 0 → (m %/ d < n) = (m < n × d).
Lemma leq_divRL m n d : d > 0 → (m ≤ n %/ d) = (m × d ≤ n).
Lemma ltn_Pdiv m d : 1 < d → 0 < m → m %/ d < m.
Lemma divn_gt0 d m : 0 < d → (0 < m %/ d) = (d ≤ m).
Lemma leq_div2r d m n : m ≤ n → m %/ d ≤ n %/ d.
Lemma leq_div2l m d e : 0 < d → d ≤ e → m %/ e ≤ m %/ d.
Lemma edivnD m n d (offset := m %% d + n %% d ≥ d) : 0 < d →
edivn (m + n) d = (m %/ d + n %/ d + offset, m %% d + n %% d - offset × d).
Lemma divnD m n d : 0 < d →
(m + n) %/ d = (m %/ d) + (n %/ d) + (m %% d + n %% d ≥ d).
Lemma modnD m n d : 0 < d →
(m + n) %% d = m %% d + n %% d - (m %% d + n %% d ≥ d) × d.
Lemma leqDmod m n d : 0 < d →
(d ≤ m %% d + n %% d) = ((m + n) %% d < n %% d).
Lemma divnB n m d : 0 < d →
(m - n) %/ d = (m %/ d) - (n %/ d) - (m %% d < n %% d).
Lemma modnB m n d : 0 < d → n ≤ m →
(m - n) %% d = (m %% d < n %% d) × d + m %% d - n %% d.
Lemma edivnB m n d (offset := m %% d < n %% d) : 0 < d → n ≤ m →
edivn (m - n) d = (m %/ d - n %/ d - offset, offset × d + m %% d - n %% d).
Lemma leq_divDl p m n : (m + n) %/ p ≤ m %/ p + n %/ p + 1.
Lemma geq_divBl k m p : k %/ p - m %/ p ≤ (k - m) %/ p + 1.
Lemma divnMA m n p : m %/ (n × p) = m %/ n %/ p.
Lemma divnAC m n p : m %/ n %/ p = m %/ p %/ n.
Lemma modn_small m d : m < d → m %% d = m.
Lemma modn_mod m d : m %% d = m %[mod d].
Lemma modnMDl p m d : p × d + m = m %[mod d].
Lemma muln_modr p m d : p × (m %% d) = (p × m) %% (p × d).
Lemma muln_modl p m d : (m %% d) × p = (m × p) %% (d × p).
Lemma modn_divl m n d : (m %/ d) %% n = m %% (n × d) %/ d.
Lemma modnDl m d : d + m = m %[mod d].
Lemma modnDr m d : m + d = m %[mod d].
Lemma modnn d : d %% d = 0.
Lemma modnMl p d : p × d %% d = 0.
Lemma modnMr p d : d × p %% d = 0.
Lemma modnDml m n d : m %% d + n = m + n %[mod d].
Lemma modnDmr m n d : m + n %% d = m + n %[mod d].
Lemma modnDm m n d : m %% d + n %% d = m + n %[mod d].
Lemma eqn_modDl p m n d : (p + m == p + n %[mod d]) = (m == n %[mod d]).
Lemma eqn_modDr p m n d : (m + p == n + p %[mod d]) = (m == n %[mod d]).
Lemma modnMml m n d : m %% d × n = m × n %[mod d].
Lemma modnMmr m n d : m × (n %% d) = m × n %[mod d].
Lemma modnMm m n d : m %% d × (n %% d) = m × n %[mod d].
Lemma modn2 m : m %% 2 = odd m.
Lemma divn2 m : m %/ 2 = m./2.
Lemma odd_mod m d : odd d = false → odd (m %% d) = odd m.
Lemma modnXm m n a : (a %% n) ^ m = a ^ m %[mod n].
Divisibility *
Definition dvdn d m := m %% d == 0.
Notation "m %| d" := (dvdn m d) : nat_scope.
Lemma dvdnP d m : reflect (∃ k, m = k × d) (d %| m).
Arguments dvdnP {d m}.
Lemma dvdn0 d : d %| 0.
Lemma dvd0n n : (0 %| n) = (n == 0).
Lemma dvdn1 d : (d %| 1) = (d == 1).
Lemma dvd1n m : 1 %| m.
Lemma dvdn_gt0 d m : m > 0 → d %| m → d > 0.
Lemma dvdnn m : m %| m.
Lemma dvdn_mull d m n : d %| n → d %| m × n.
Lemma dvdn_mulr d m n : d %| m → d %| m × n.
#[global] Hint Resolve dvdn0 dvd1n dvdnn dvdn_mull dvdn_mulr : core.
Lemma dvdn_mul d1 d2 m1 m2 : d1 %| m1 → d2 %| m2 → d1 × d2 %| m1 × m2.
Lemma dvdn_trans n d m : d %| n → n %| m → d %| m.
Lemma dvdn_eq d m : (d %| m) = (m %/ d × d == m).
Lemma dvdn2 n : (2 %| n) = ~~ odd n.
Lemma dvdn_odd m n : m %| n → odd n → odd m.
Lemma divnK d m : d %| m → m %/ d × d = m.
Lemma leq_divLR d m n : d %| m → (m %/ d ≤ n) = (m ≤ n × d).
Lemma ltn_divRL d m n : d %| m → (n < m %/ d) = (n × d < m).
Lemma eqn_div d m n : d > 0 → d %| m → (n == m %/ d) = (n × d == m).
Lemma eqn_mul d m n : d > 0 → d %| m → (m == n × d) = (m %/ d == n).
Lemma divn_mulAC d m n : d %| m → m %/ d × n = m × n %/ d.
Lemma muln_divA d m n : d %| n → m × (n %/ d) = m × n %/ d.
Lemma muln_divCA d m n : d %| m → d %| n → m × (n %/ d) = n × (m %/ d).
Lemma divnA m n p : p %| n → m %/ (n %/ p) = m × p %/ n.
Lemma modn_dvdm m n d : d %| m → n %% m = n %[mod d].
Lemma dvdn_leq d m : 0 < m → d %| m → d ≤ m.
Lemma gtnNdvd n d : 0 < n → n < d → (d %| n) = false.
Lemma eqn_dvd m n : (m == n) = (m %| n) && (n %| m).
Lemma dvdn_pmul2l p d m : 0 < p → (p × d %| p × m) = (d %| m).
Arguments dvdn_pmul2l [p d m].
Lemma dvdn_pmul2r p d m : 0 < p → (d × p %| m × p) = (d %| m).
Arguments dvdn_pmul2r [p d m].
Lemma dvdn_divLR p d m : 0 < p → p %| d → (d %/ p %| m) = (d %| m × p).
Lemma dvdn_divRL p d m : p %| m → (d %| m %/ p) = (d × p %| m).
Lemma dvdn_div d m : d %| m → m %/ d %| m.
Lemma dvdn_exp2l p m n : m ≤ n → p ^ m %| p ^ n.
Lemma dvdn_Pexp2l p m n : p > 1 → (p ^ m %| p ^ n) = (m ≤ n).
Lemma dvdn_exp2r m n k : m %| n → m ^ k %| n ^ k.
Lemma divn_modl m n d : d %| n → (m %% n) %/ d = (m %/ d) %% (n %/ d).
Lemma dvdn_addr m d n : d %| m → (d %| m + n) = (d %| n).
Lemma dvdn_addl n d m : d %| n → (d %| m + n) = (d %| m).
Lemma dvdn_add d m n : d %| m → d %| n → d %| m + n.
Lemma dvdn_add_eq d m n : d %| m + n → (d %| m) = (d %| n).
Lemma dvdn_subr d m n : n ≤ m → d %| m → (d %| m - n) = (d %| n).
Lemma dvdn_subl d m n : n ≤ m → d %| n → (d %| m - n) = (d %| m).
Lemma dvdn_sub d m n : d %| m → d %| n → d %| m - n.
Lemma dvdn_exp k d m : 0 < k → d %| m → d %| (m ^ k).
Lemma dvdn_fact m n : 0 < m ≤ n → m %| n`!.
#[global] Hint Resolve dvdn_add dvdn_sub dvdn_exp : core.
Lemma eqn_mod_dvd d m n : n ≤ m → (m == n %[mod d]) = (d %| m - n).
Lemma divnDMl q m d : 0 < d → (m + q × d) %/ d = (m %/ d) + q.
Lemma divnMBl q m d : 0 < d → (q × d - m) %/ d = q - (m %/ d) - (~~ (d %| m)).
Lemma divnBMl q m d : (m - q × d) %/ d = (m %/ d) - q.
Lemma divnDl m n d : d %| m → (m + n) %/ d = m %/ d + n %/ d.
Lemma divnDr m n d : d %| n → (m + n) %/ d = m %/ d + n %/ d.
Lemma divnBl m n d : d %| m → (m - n) %/ d = m %/ d - (n %/ d) - (~~ (d %| n)).
Lemma divnBr m n d : d %| n → (m - n) %/ d = m %/ d - n %/ d.
Lemma edivnS m d : 0 < d → edivn m.+1 d =
if d %| m.+1 then ((m %/ d).+1, 0) else (m %/ d, (m %% d).+1).
Lemma modnS m d : m.+1 %% d = if d %| m.+1 then 0 else (m %% d).+1.
Lemma divnS m d : 0 < d → m.+1 %/ d = (d %| m.+1) + m %/ d.
Lemma divn_pred m d : m.-1 %/ d = (m %/ d) - (d %| m).
Lemma modn_pred m d : d != 1 → 0 < m →
m.-1 %% d = if d %| m then d.-1 else (m %% d).-1.
Lemma edivn_pred m d : d != 1 → 0 < m →
edivn m.-1 d = if d %| m then ((m %/ d).-1, d.-1) else (m %/ d, (m %% d).-1).
A function that computes the gcd of 2 numbers
Fixpoint gcdn m n :=
let n' := n %% m in if n' is 0 then m else
if m - n'.-1 is m'.+1 then gcdn (m' %% n') n' else n'.
Arguments gcdn : simpl never.
Lemma gcdnE m n : gcdn m n = if m == 0 then n else gcdn (n %% m) m.
Lemma gcdnn : idempotent_op gcdn.
Lemma gcdnC : commutative gcdn.
Lemma gcd0n : left_id 0 gcdn.
Lemma gcdn0 : right_id 0 gcdn.
Lemma gcd1n : left_zero 1 gcdn.
Lemma gcdn1 : right_zero 1 gcdn.
Lemma dvdn_gcdr m n : gcdn m n %| n.
Lemma dvdn_gcdl m n : gcdn m n %| m.
Lemma gcdn_gt0 m n : (0 < gcdn m n) = (0 < m) || (0 < n).
Lemma gcdnMDl k m n : gcdn m (k × m + n) = gcdn m n.
Lemma gcdnDl m n : gcdn m (m + n) = gcdn m n.
Lemma gcdnDr m n : gcdn m (n + m) = gcdn m n.
Lemma gcdnMl n m : gcdn n (m × n) = n.
Lemma gcdnMr n m : gcdn n (n × m) = n.
Lemma gcdn_idPl {m n} : reflect (gcdn m n = m) (m %| n).
Lemma gcdn_idPr {m n} : reflect (gcdn m n = n) (n %| m).
Lemma expn_min e m n : e ^ minn m n = gcdn (e ^ m) (e ^ n).
Lemma gcdn_modr m n : gcdn m (n %% m) = gcdn m n.
Lemma gcdn_modl m n : gcdn (m %% n) n = gcdn m n.
Extended gcd, which computes Bezout coefficients.
Fixpoint Bezout_rec km kn qs :=
if qs is q :: qs' then Bezout_rec kn (NatTrec.add_mul q kn km) qs'
else (km, kn).
Fixpoint egcdn_rec m n s qs :=
if s is s'.+1 then
let: (q, r) := edivn m n in
if r > 0 then egcdn_rec n r s' (q :: qs) else
if odd (size qs) then qs else q.-1 :: qs
else [::0].
Definition egcdn m n := Bezout_rec 0 1 (egcdn_rec m n n [::]).
Variant egcdn_spec m n : nat × nat → Type :=
EgcdnSpec km kn of km × m = kn × n + gcdn m n & kn × gcdn m n < m :
egcdn_spec m n (km, kn).
Lemma egcd0n n : egcdn 0 n = (1, 0).
Lemma egcdnP m n : m > 0 → egcdn_spec m n (egcdn m n).
Lemma Bezoutl m n : m > 0 → {a | a < m & m %| gcdn m n + a × n}.
Lemma Bezoutr m n : n > 0 → {a | a < n & n %| gcdn m n + a × m}.
Back to the gcd.
Lemma dvdn_gcd p m n : p %| gcdn m n = (p %| m) && (p %| n).
Lemma gcdnAC : right_commutative gcdn.
Lemma gcdnA : associative gcdn.
Lemma gcdnCA : left_commutative gcdn.
Lemma gcdnACA : interchange gcdn gcdn.
Lemma muln_gcdr : right_distributive muln gcdn.
Lemma muln_gcdl : left_distributive muln gcdn.
Lemma gcdn_def d m n :
d %| m → d %| n → (∀ d', d' %| m → d' %| n → d' %| d) →
gcdn m n = d.
Lemma muln_divCA_gcd n m : n × (m %/ gcdn n m) = m × (n %/ gcdn n m).
We derive the lcm directly.
Definition lcmn m n := m × n %/ gcdn m n.
Lemma lcmnC : commutative lcmn.
Lemma lcm0n : left_zero 0 lcmn.
Lemma lcmn0 : right_zero 0 lcmn.
Lemma lcm1n : left_id 1 lcmn.
Lemma lcmn1 : right_id 1 lcmn.
Lemma muln_lcm_gcd m n : lcmn m n × gcdn m n = m × n.
Lemma lcmn_gt0 m n : (0 < lcmn m n) = (0 < m) && (0 < n).
Lemma muln_lcmr : right_distributive muln lcmn.
Lemma muln_lcml : left_distributive muln lcmn.
Lemma lcmnA : associative lcmn.
Lemma lcmnCA : left_commutative lcmn.
Lemma lcmnAC : right_commutative lcmn.
Lemma lcmnACA : interchange lcmn lcmn.
Lemma dvdn_lcml d1 d2 : d1 %| lcmn d1 d2.
Lemma dvdn_lcmr d1 d2 : d2 %| lcmn d1 d2.
Lemma dvdn_lcm d1 d2 m : lcmn d1 d2 %| m = (d1 %| m) && (d2 %| m).
Lemma lcmnMl m n : lcmn m (m × n) = m × n.
Lemma lcmnMr m n : lcmn n (m × n) = m × n.
Lemma lcmn_idPr {m n} : reflect (lcmn m n = n) (m %| n).
Lemma lcmn_idPl {m n} : reflect (lcmn m n = m) (n %| m).
Lemma expn_max e m n : e ^ maxn m n = lcmn (e ^ m) (e ^ n).
Coprime factors
Definition coprime m n := gcdn m n == 1.
Lemma coprime1n n : coprime 1 n.
Lemma coprimen1 n : coprime n 1.
Lemma coprime_sym m n : coprime m n = coprime n m.
Lemma coprime_modl m n : coprime (m %% n) n = coprime m n.
Lemma coprime_modr m n : coprime m (n %% m) = coprime m n.
Lemma coprime2n n : coprime 2 n = odd n.
Lemma coprimen2 n : coprime n 2 = odd n.
Lemma coprimeSn n : coprime n.+1 n.
Lemma coprimenS n : coprime n n.+1.
Lemma coprimePn n : n > 0 → coprime n.-1 n.
Lemma coprimenP n : n > 0 → coprime n n.-1.
Lemma coprimeP n m :
n > 0 → reflect (∃ u, u.1 × n - u.2 × m = 1) (coprime n m).
Lemma modn_coprime k n : 0 < k → (∃ u, (k × u) %% n = 1) → coprime k n.
Lemma Gauss_dvd m n p : coprime m n → (m × n %| p) = (m %| p) && (n %| p).
Lemma Gauss_dvdr m n p : coprime m n → (m %| n × p) = (m %| p).
Lemma Gauss_dvdl m n p : coprime m p → (m %| n × p) = (m %| n).
Lemma dvdn_double_leq m n : m %| n → odd m → ~~ odd n → 0 < n → m.*2 ≤ n.
Lemma dvdn_double_ltn m n : m %| n.-1 → odd m → odd n → 1 < n → m.*2 < n.
Lemma Gauss_gcdr p m n : coprime p m → gcdn p (m × n) = gcdn p n.
Lemma Gauss_gcdl p m n : coprime p n → gcdn p (m × n) = gcdn p m.
Lemma coprimeMr p m n : coprime p (m × n) = coprime p m && coprime p n.
Lemma coprimeMl p m n : coprime (m × n) p = coprime m p && coprime n p.
Lemma coprime_pexpl k m n : 0 < k → coprime (m ^ k) n = coprime m n.
Lemma coprime_pexpr k m n : 0 < k → coprime m (n ^ k) = coprime m n.
Lemma coprimeXl k m n : coprime m n → coprime (m ^ k) n.
Lemma coprimeXr k m n : coprime m n → coprime m (n ^ k).
Lemma coprime_dvdl m n p : m %| n → coprime n p → coprime m p.
Lemma coprime_dvdr m n p : m %| n → coprime p n → coprime p m.
Lemma coprime_egcdn n m : n > 0 → coprime (egcdn n m).1 (egcdn n m).2.
Lemma dvdn_pexp2r m n k : k > 0 → (m ^ k %| n ^ k) = (m %| n).
Section Chinese.
The chinese remainder theorem
Variables m1 m2 : nat.
Hypothesis co_m12 : coprime m1 m2.
Lemma chinese_remainder x y :
(x == y %[mod m1 × m2]) = (x == y %[mod m1]) && (x == y %[mod m2]).
A function that solves the chinese remainder problem
Definition chinese r1 r2 :=
r1 × m2 × (egcdn m2 m1).1 + r2 × m1 × (egcdn m1 m2).1.
Lemma chinese_modl r1 r2 : chinese r1 r2 = r1 %[mod m1].
Lemma chinese_modr r1 r2 : chinese r1 r2 = r2 %[mod m2].
Lemma chinese_mod x : x = chinese (x %% m1) (x %% m2) %[mod m1 × m2].
End Chinese.