proposition: Lemma zero_row_to_non_right_invertibility { m n : nat } (A : Matrix R m n) (i : ⟦ m ⟧%stn) (zero_row : A i = (const_vec 0%ring)) : (@matrix_right_inverse R m n A) -> empty.
proof: Proof. intros [inv isrightinv]. contradiction (@nonzeroax R). etrans. { apply pathsinv0, (@id_mat_ii R). } do 2 (apply toforallpaths in isrightinv; specialize (isrightinv i)). etrans. { apply pathsinv0, isrightinv. } refine (toforallpaths _ _ _ _ i). apply (@zero_row_product R). apply zero_row. Defined.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.Combinatorics.StandardFiniteSets UniMath.Combinatorics.Vectors UniMath.Algebra.RigsAndRings UniMath.Algebra.IteratedBinaryOperations UniMath.Algebra.Matrix UniMath.Algebra.Domains_and_Fields UniMath.Algebra.GaussianElimination.Auxiliary UniMath.Algebra.GaussianElimination.Vectors