Library prosa.util.lcmseq

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

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.

Any integer a that is contained in the sequence xs divides lcml xs.

Lemma int_divides_lcm_in_seq :
  ∀ (a : nat) (xs : seq nat), a %| lcml (a :: xs).
Proof.
  intros.
  rewrite /lcml.
  induction xs.
  - rewrite /foldr.
    now apply dvdn_lcml.
  - rewrite -cat1s.
    rewrite foldr_cat /foldr.
    by apply dvdn_lcml.
Qed.

Also, lcml xs1 divides lcml xs2 if xs2 contains one extra element as compared to xs1.

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

All integers in a sequence xs divide lcml xs.

Lemma lcm_seq_is_mult_of_all_ints :
  ∀ (sq : seq nat) (a : nat), a \in sq → ∃ k, lcml sq = k × a.
Proof.
  intros xs x IN.
  induction xs as [ | z sq IH_DIVIDES]; first by easy.
  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_DIVIDES 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.