proposition: Lemma euclid_next i j : ~~ euclid_cont i && (i <= j) ==> ~~ euclid_cont j.
proof: Proof. apply/implyP => /andP [H1 H2]. apply: contra H1 => /forallP /= H1. apply/forallP => /= x. have @b : 'I_j.+1 by exists x; rewrite ltnS (leq_trans _ H2) // -ltnS. by rewrite (_ : r x = r b). Qed.
imports: mathcomp(all_ssreflect ssralg poly polydiv matrix)