Library prosa.util.lcmseq

From mathcomp Require Export ssreflect seq div ssrbool ssrnat eqtype ssrfun.
Require Export prosa.util.tactics.

A function to calculate the least common multiple of all integers in a sequence xs, denoted by lcml xs.

Definition lcml (xs : seq nat) : nat := foldr lcmn 1 xs.

First we show that x divides lcml (x :: xs) for any x and xs.

Lemma int_divides_lcm_in_seq:
  ∀ (x : nat) (xs : seq nat), x %| lcml (x :: xs).
Proof.
  move⇒ x xs; induction xs.
  - by apply dvdn_lcml.
  - rewrite /lcml -cat1s foldr_cat /foldr.
    by apply dvdn_lcml.
Qed.

Similarly, lcml xs divides lcml (x :: xs) for any x and xs.

Lemma lcm_seq_divides_lcm_super:
  ∀ (x : nat) (xs : seq nat),
  lcml xs %| lcml (x :: xs).
Proof.
  move⇒ x xs; induction xs; first by auto.
  rewrite /lcml -cat1s foldr_cat /foldr.
  by apply dvdn_lcmr.
Qed.

Given a sequence xs, any integer x \in xs divides lcml xs.

Lemma lcm_seq_is_mult_of_all_ints:
  ∀ (x : nat) (xs: seq nat), x \in xs → x %| lcml xs.
Proof.
  intros x xs IN; apply/dvdnP.
  induction xs as [ | z sq IH_DIV]; first by done.
  rewrite in_cons in IN.
  move : IN ⇒ /orP [/eqP EQ | IN].
  - apply /dvdnP.
    rewrite EQ /lcml.
    by apply int_divides_lcm_in_seq.
  - move : (IH_DIV IN) ⇒ [k EQ].
    ∃ ((foldr lcmn 1 (z :: sq)) %/ (foldr lcmn 1 sq) × k).
    rewrite -mulnA -EQ divnK /lcml //.
    by apply lcm_seq_divides_lcm_super.
Qed.

The LCM of all elements in a sequence with only positive elements is positive.

Lemma all_pos_implies_lcml_pos:
  ∀ (xs : seq nat),
    (∀ x, x \in xs → x > 0) →
    lcml xs > 0.
Proof.
  elim⇒ [//|x xs IHxs] POS.
  rewrite /lcml -cat1s //= lcmn_gt0.
  apply/andP; split ⇒ //.
  - by apply POS; rewrite in_cons eq_refl.
  - apply: IHxs; intros b B_IN.
    by apply POS; rewrite in_cons; apply /orP; right ⇒ //.
Qed.