Lemma matrix_mult_zero_vec_eq {m n : nat} {mat : Matrix R m n} : mat ** col_vec (const_vec 0%rig) = col_vec (const_vec 0%rig). Proof. rewrite matrix_mult_eq; unfold matrix_mult_unf. apply funextfun; intros i. unfold col_vec. apply funextfun; intros _. apply vecsum_eq_zero. intros k. apply rigmultx0. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.Combinatorics.StandardFiniteSets. Require Import UniMath.Combinatorics.Vectors. Require Import UniMath.Algebra.RigsAndRings. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Algebra.Matrix. Require Import UniMath.Algebra.Domains_and_Fields. Require Import UniMath.Algebra.GaussianElimination.Auxiliary. Require Import UniMath.Algebra.GaussianElimination.Vectors.