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# 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.