Library Coq.Numbers.Integer.Abstract.ZLcm

Require Import ZAxioms ZMulOrder ZSgnAbs ZGcd ZDivTrunc ZDivFloor.

Least Common Multiple

Unlike other functions around, we will define lcm below instead of axiomatizing it. Indeed, there is no "prior art" about lcm in the standard library to be compliant with, and the generic definition of lcm via gcd is quite reasonable.

By the way, we also state here some combined properties of div/mod and quot/rem and gcd.

Module Type ZLcmProp
(Import A : ZAxiomsSig')
(Import B : ZMulOrderProp A)
(Import C : ZSgnAbsProp A B)
(Import D : ZDivProp A B C)
(Import E : ZQuotProp A B C)
(Import F : ZGcdProp A B C).

The two notions of division are equal on non-negative numbers

Lemma quot_div_nonneg : forall a b, 0<=a -> 0<b -> a ÷b == a /b.
Proof.
intros. apply div_unique_pos with (a rem b).
now apply rem_bound_pos.
apply quot_rem. order.
Qed.

Lemma rem_mod_nonneg : forall a b, 0<=a -> 0<b -> a rem b == a mod b.
Proof.
intros. apply mod_unique_pos with (a÷b).
now apply rem_bound_pos.
apply quot_rem. order.
Qed.

We can use the sign rule to have an relation between divisions.

Lemma quot_div : forall a b, b ~=0 ->
a ÷b == (sgn a )*(sgn b )*(abs a / abs b ).
Proof.
assert (AUX : forall a b, 0<b -> a ÷b == (sgn a )*(sgn b )*(abs a / abs b )).
  intros a b Hb. rewrite (sgn_pos b), (abs_eq b), mul_1_r by order.
  destruct (lt_trichotomy 0 a) as [Ha|[Ha|Ha]].
  rewrite sgn_pos, abs_eq, mul_1_l, quot_div_nonneg; order.
  rewrite <- Ha, abs_0, sgn_0, quot_0_l, div_0_l, mul_0_l; order.
  rewrite sgn_neg, abs_neq, mul_opp_l, mul_1_l, eq_opp_r, <-quot_opp_l
    by order.
   apply quot_div_nonneg; trivial. apply opp_nonneg_nonpos; order.
intros a b Hb.
apply neg_pos_cases in Hb. destruct Hb as [Hb|Hb]; [|now apply AUX].
rewrite <- (opp_involutive b) at 1. rewrite quot_opp_r.
rewrite AUX, abs_opp, sgn_opp, mul_opp_r, mul_opp_l, opp_involutive.
reflexivity.
now apply opp_pos_neg.
rewrite eq_opp_l, opp_0; order.
Qed.

Lemma rem_mod : forall a b, b ~=0 ->
a rem b == (sgn a ) * ((abs a ) mod (abs b )).
Proof.
intros a b Hb.
rewrite <- rem_abs_r by trivial.
assert (Hb' := proj2 (abs_pos b) Hb).
destruct (lt_trichotomy 0 a) as [Ha|[Ha|Ha]].
rewrite (abs_eq a), sgn_pos, mul_1_l, rem_mod_nonneg; order.
rewrite <- Ha, abs_0, sgn_0, mod_0_l, rem_0_l, mul_0_l; order.
rewrite sgn_neg, (abs_neq a), mul_opp_l, mul_1_l, eq_opp_r, <-rem_opp_l
    by order.
  apply rem_mod_nonneg; trivial. apply opp_nonneg_nonpos; order.
Qed.

Modulo and remainder are null at the same place, and this correspond to the divisibility relation.

Lemma mod_divide : forall a b, b ~=0 -> (a mod b == 0 <-> (b |a )).
Proof.
intros a b Hb. split.
intros Hab. exists (a/b). rewrite mul_comm.
rewrite (div_mod a b Hb) at 1. rewrite Hab; now nzsimpl.
intros (c,Hc). rewrite Hc. now apply mod_mul.
Qed.

Lemma rem_divide : forall a b, b ~=0 -> (a rem b == 0 <-> (b |a )).
Proof.
intros a b Hb. split.
intros Hab. exists (a÷b). rewrite mul_comm.
rewrite (quot_rem a b Hb) at 1. rewrite Hab; now nzsimpl.
intros (c,Hc). rewrite Hc. now apply rem_mul.
Qed.

Lemma rem_mod_eq_0 : forall a b, b ~=0 -> (a rem b == 0 <-> a mod b == 0).
Proof.
intros a b Hb. now rewrite mod_divide, rem_divide.
Qed.

When division is exact, div and quot agree

Lemma quot_div_exact : forall a b, b ~=0 -> (b |a ) -> a ÷b == a /b.
Proof.
intros a b Hb H.
apply mul_cancel_l with b; trivial.
assert (H':=H).
apply rem_divide, quot_exact in H; trivial.
apply mod_divide, div_exact in H'; trivial.
now rewrite <-H,<-H'.
Qed.

Lemma divide_div_mul_exact : forall a b c, b ~=0 -> (b |a ) ->
(c *a )/b == c *(a /b ).
Proof.
intros a b c Hb H.
apply mul_cancel_l with b; trivial.
rewrite mul_assoc, mul_shuffle0.
assert (H':=H). apply mod_divide, div_exact in H'; trivial.
rewrite <- H', (mul_comm a c).
symmetry. apply div_exact; trivial.
apply mod_divide; trivial.
now apply divide_mul_r.
Qed.

Lemma divide_quot_mul_exact : forall a b c, b ~=0 -> (b |a ) ->
(c *a )÷b == c *(a ÷b ).
Proof.
intros a b c Hb H.
rewrite 2 quot_div_exact; trivial.
apply divide_div_mul_exact; trivial.
now apply divide_mul_r.
Qed.

Gcd of divided elements, for exact divisions

Lemma gcd_div_factor : forall a b c, 0<c -> (c |a ) -> (c |b ) ->
gcd (a /c) (b /c) == (gcd a b )/c.
Proof.
intros a b c Hc Ha Hb.
apply mul_cancel_l with c; try order.
assert (H:=gcd_greatest _ _ _ Ha Hb).
apply mod_divide, div_exact in H; try order.
rewrite <- H.
rewrite <- gcd_mul_mono_l_nonneg; try order.
f_equiv; symmetry; apply div_exact; try order;
apply mod_divide; trivial; try order.
Qed.

Lemma gcd_quot_factor : forall a b c, 0<c -> (c |a ) -> (c |b ) ->
gcd (a ÷c) (b ÷c) == (gcd a b )÷c.
Proof.
intros a b c Hc Ha Hb. rewrite !quot_div_exact; trivial; try order.
now apply gcd_div_factor. now apply gcd_greatest.
Qed.

Lemma gcd_div_gcd : forall a b g, g ~=0 -> g == gcd a b ->
gcd (a /g) (b /g) == 1.
Proof.
intros a b g NZ EQ. rewrite gcd_div_factor.
now rewrite <- EQ, div_same.
generalize (gcd_nonneg a b); order.
rewrite EQ; apply gcd_divide_l.
rewrite EQ; apply gcd_divide_r.
Qed.

Lemma gcd_quot_gcd : forall a b g, g ~=0 -> g == gcd a b ->
gcd (a ÷g) (b ÷g) == 1.
Proof.
intros a b g NZ EQ. rewrite !quot_div_exact; trivial.
now apply gcd_div_gcd.
rewrite EQ; apply gcd_divide_r.
rewrite EQ; apply gcd_divide_l.
Qed.

The following equality is crucial for Euclid algorithm

Lemma gcd_mod : forall a b, b ~=0 -> gcd (a mod b) b == gcd b a.
Proof.
intros a b Hb. rewrite mod_eq; trivial.
rewrite <- add_opp_r, mul_comm, <- mul_opp_l.
rewrite (gcd_comm _ b).
apply gcd_add_mult_diag_r.
Qed.

Lemma gcd_rem : forall a b, b ~=0 -> gcd (a rem b) b == gcd b a.
Proof.
intros a b Hb. rewrite rem_eq; trivial.
rewrite <- add_opp_r, mul_comm, <- mul_opp_l.
rewrite (gcd_comm _ b).
apply gcd_add_mult_diag_r.
Qed.

We now define lcm thanks to gcd:

lcm a b = a * (b / gcd a b) = (a / gcd a b) * b = (a*b) / gcd a b

We had an abs in order to have an always-nonnegative lcm, in the spirit of gcd. Nota: lcm 0 0 should be 0, which isn't garantee with the third equation above.

Definition lcm a b := abs (a *(b /gcd a b )).

Instance lcm_wd : Proper (eq ==>eq ==>eq) lcm.
Proof. unfold lcm. solve_proper. Qed.

Lemma lcm_equiv1 : forall a b, gcd a b ~= 0 ->
  a * (b / gcd a b ) == (a *b )/gcd a b.
Proof.
intros a b H. rewrite divide_div_mul_exact; try easy. apply gcd_divide_r.
Qed.

Lemma lcm_equiv2 : forall a b, gcd a b ~= 0 ->
  (a / gcd a b ) * b == (a *b )/gcd a b.
Proof.
intros a b H. rewrite 2 (mul_comm _ b).
rewrite divide_div_mul_exact; try easy. apply gcd_divide_l.
Qed.

Lemma gcd_div_swap : forall a b,
(a / gcd a b ) * b == a * (b / gcd a b ).
Proof.
intros a b. destruct (eq_decidable (gcd a b) 0) as [EQ|NEQ].
apply gcd_eq_0 in EQ. destruct EQ as (EQ,EQ'). rewrite EQ, EQ'. now nzsimpl.
now rewrite lcm_equiv1, <-lcm_equiv2.
Qed.

Lemma divide_lcm_l : forall a b, (a | lcm a b ).
Proof.
unfold lcm. intros a b. apply divide_abs_r, divide_factor_l.
Qed.

Lemma divide_lcm_r : forall a b, (b | lcm a b ).
Proof.
unfold lcm. intros a b. apply divide_abs_r. rewrite <- gcd_div_swap.
apply divide_factor_r.
Qed.

Lemma divide_div : forall a b c, a ~=0 -> (a |b ) -> (b |c ) -> (b /a |c /a ).
Proof.
intros a b c Ha Hb (c',Hc). exists c'.
now rewrite <- divide_div_mul_exact, <- Hc.
Qed.

Lemma lcm_least : forall a b c,
(a | c ) -> (b | c ) -> (lcm a b | c ).
Proof.
intros a b c Ha Hb. unfold lcm. apply divide_abs_l.
destruct (eq_decidable (gcd a b) 0) as [EQ|NEQ].
apply gcd_eq_0 in EQ. destruct EQ as (EQ,EQ'). rewrite EQ in *. now nzsimpl.
assert (Ga := gcd_divide_l a b).
assert (Gb := gcd_divide_r a b).
set (g:=gcd a b) in *.
assert (Ha' := divide_div g a c NEQ Ga Ha).
assert (Hb' := divide_div g b c NEQ Gb Hb).
destruct Ha' as (a',Ha'). rewrite Ha', mul_comm in Hb'.
apply gauss in Hb'; [|apply gcd_div_gcd; unfold g; trivial using gcd_comm].
destruct Hb' as (b',Hb').
exists b'.
rewrite mul_shuffle3, <- Hb'.
rewrite (proj2 (div_exact c g NEQ)).
rewrite Ha', mul_shuffle3, (mul_comm a a'). f_equiv.
symmetry. apply div_exact; trivial.
apply mod_divide; trivial.
apply mod_divide; trivial. transitivity a; trivial.
Qed.

Lemma lcm_nonneg : forall a b, 0 <= lcm a b.
Proof.
intros a b. unfold lcm. apply abs_nonneg.
Qed.

Lemma lcm_comm : forall a b, lcm a b == lcm b a.
Proof.
intros a b. unfold lcm. rewrite (gcd_comm b), (mul_comm b).
now rewrite <- gcd_div_swap.
Qed.

Lemma lcm_divide_iff : forall n m p,
  (lcm n m | p ) <-> (n | p ) /\ (m | p ).
Proof.
intros. split. split.
transitivity (lcm n m); trivial using divide_lcm_l.
transitivity (lcm n m); trivial using divide_lcm_r.
intros (H,H'). now apply lcm_least.
Qed.

Lemma lcm_unique : forall n m p,
0<=p -> (n |p ) -> (m |p ) ->
(forall q, (n |q ) -> (m |q ) -> (p |q )) ->
lcm n m == p.
Proof.
intros n m p Hp Hn Hm H.
apply divide_antisym_nonneg; trivial. apply lcm_nonneg.
now apply lcm_least.
apply H. apply divide_lcm_l. apply divide_lcm_r.
Qed.

Lemma lcm_unique_alt : forall n m p, 0<=p ->
(forall q, (p |q ) <-> (n |q ) /\ (m |q )) ->
lcm n m == p.
Proof.
intros n m p Hp H.
apply lcm_unique; trivial.
apply H, divide_refl.
apply H, divide_refl.
intros. apply H. now split.
Qed.

Lemma lcm_assoc : forall n m p, lcm n (lcm m p) == lcm (lcm n m) p.
Proof.
intros. apply lcm_unique_alt; try apply lcm_nonneg.
intros. now rewrite !lcm_divide_iff, and_assoc.
Qed.

Lemma lcm_0_l : forall n, lcm 0 n == 0.
Proof.
intros. apply lcm_unique; trivial. order.
apply divide_refl.
apply divide_0_r.
Qed.

Lemma lcm_0_r : forall n, lcm n 0 == 0.
Proof.
intros. now rewrite lcm_comm, lcm_0_l.
Qed.

Lemma lcm_1_l_nonneg : forall n, 0<=n -> lcm 1 n == n.
Proof.
intros. apply lcm_unique; trivial using divide_1_l, le_0_1, divide_refl.
Qed.

Lemma lcm_1_r_nonneg : forall n, 0<=n -> lcm n 1 == n.
Proof.
intros. now rewrite lcm_comm, lcm_1_l_nonneg.
Qed.

Lemma lcm_diag_nonneg : forall n, 0<=n -> lcm n n == n.
Proof.
intros. apply lcm_unique; trivial using divide_refl.
Qed.

Lemma lcm_eq_0 : forall n m, lcm n m == 0 <-> n == 0 \/ m == 0.
Proof.
intros. split.
intros EQ.
apply eq_mul_0.
apply divide_0_l. rewrite <- EQ. apply lcm_least.
  apply divide_factor_l. apply divide_factor_r.
destruct 1 as [EQ|EQ]; rewrite EQ. apply lcm_0_l. apply lcm_0_r.
Qed.

Lemma divide_lcm_eq_r : forall n m, 0<=m -> (n |m ) -> lcm n m == m.
Proof.
intros n m Hm H. apply lcm_unique_alt; trivial.
intros q. split. split; trivial. now transitivity m.
now destruct 1.
Qed.

Lemma divide_lcm_iff : forall n m, 0<=m -> ((n |m ) <-> lcm n m == m ).
Proof.
intros n m Hn. split. now apply divide_lcm_eq_r.
intros EQ. rewrite <- EQ. apply divide_lcm_l.
Qed.

Lemma lcm_opp_l : forall n m, lcm (-n) m == lcm n m.
Proof.
intros. apply lcm_unique_alt; try apply lcm_nonneg.
intros. rewrite divide_opp_l. apply lcm_divide_iff.
Qed.

Lemma lcm_opp_r : forall n m, lcm n (-m) == lcm n m.
Proof.
intros. now rewrite lcm_comm, lcm_opp_l, lcm_comm.
Qed.

Lemma lcm_abs_l : forall n m, lcm (abs n) m == lcm n m.
Proof.
intros. destruct (abs_eq_or_opp n) as [H|H]; rewrite H.
easy. apply lcm_opp_l.
Qed.

Lemma lcm_abs_r : forall n m, lcm n (abs m) == lcm n m.
Proof.
intros. now rewrite lcm_comm, lcm_abs_l, lcm_comm.
Qed.

Lemma lcm_1_l : forall n, lcm 1 n == abs n.
Proof.
intros. rewrite <- lcm_abs_r. apply lcm_1_l_nonneg, abs_nonneg.
Qed.

Lemma lcm_1_r : forall n, lcm n 1 == abs n.
Proof.
intros. now rewrite lcm_comm, lcm_1_l.
Qed.

Lemma lcm_diag : forall n, lcm n n == abs n.
Proof.
intros. rewrite <- lcm_abs_l, <- lcm_abs_r.
apply lcm_diag_nonneg, abs_nonneg.
Qed.

Lemma lcm_mul_mono_l :
  forall n m p, lcm (p * n) (p * m) == abs p * lcm n m.
Proof.
intros n m p.
destruct (eq_decidable p 0) as [Hp|Hp].
  rewrite Hp. nzsimpl. rewrite lcm_0_l, abs_0. now nzsimpl.
destruct (eq_decidable (gcd n m) 0) as [Hg|Hg].
  apply gcd_eq_0 in Hg. destruct Hg as (Hn,Hm); rewrite Hn, Hm.
  nzsimpl. rewrite lcm_0_l. now nzsimpl.
unfold lcm.
rewrite gcd_mul_mono_l.
rewrite !abs_mul, mul_assoc. f_equiv.
rewrite <- (abs_sgn p) at 1. rewrite <- mul_assoc.
rewrite div_mul_cancel_l; trivial.
rewrite divide_div_mul_exact; trivial. rewrite abs_mul.
rewrite <- (sgn_abs (sgn p)), sgn_sgn.
destruct (sgn_spec p) as [(_,EQ)|[(EQ,_)|(_,EQ)]].
  rewrite EQ. now nzsimpl. order.
  rewrite EQ. rewrite mul_opp_l, mul_opp_r, opp_involutive. now nzsimpl.
apply gcd_divide_r.
contradict Hp. now apply abs_0_iff.
Qed.

Lemma lcm_mul_mono_l_nonneg :
forall n m p, 0<=p -> lcm (p *n) (p *m) == p * lcm n m.
Proof.
intros. rewrite <- (abs_eq p) at 3; trivial. apply lcm_mul_mono_l.
Qed.

Lemma lcm_mul_mono_r :
forall n m p, lcm (n * p) (m * p) == lcm n m * abs p.
Proof.
intros n m p. now rewrite !(mul_comm _ p), lcm_mul_mono_l, mul_comm.
Qed.

Lemma lcm_mul_mono_r_nonneg :
forall n m p, 0<=p -> lcm (n *p) (m *p) == lcm n m * p.
Proof.
intros. rewrite <- (abs_eq p) at 3; trivial. apply lcm_mul_mono_r.
Qed.

Lemma gcd_1_lcm_mul : forall n m, n ~=0 -> m ~=0 ->
(gcd n m == 1 <-> lcm n m == abs (n *m)).
Proof.
intros n m Hn Hm. split; intros H.
unfold lcm. rewrite H. now rewrite div_1_r.
unfold lcm in *.
rewrite !abs_mul in H. apply mul_cancel_l in H; [|now rewrite abs_0_iff].
assert (H' := gcd_divide_r n m).
assert (Hg : gcd n m ~= 0) by (red; rewrite gcd_eq_0; destruct 1; order).
apply mod_divide in H'; trivial. apply div_exact in H'; trivial.
assert (m / gcd n m ~=0) by (contradict Hm; rewrite H', Hm; now nzsimpl).
rewrite <- (mul_1_l (abs (_/_))) in H.
rewrite H' in H at 3. rewrite abs_mul in H.
apply mul_cancel_r in H; [|now rewrite abs_0_iff].
rewrite abs_eq in H. order. apply gcd_nonneg.
Qed.

End ZLcmProp.