Definition back_sub_step { n : nat } ( row : (⟦ n ⟧)%stn ) (mat : Matrix F n n) (x : Vector F n) (b : Vector F n) : Vector F n. Proof. intros i. destruct (nat_eq_or_neq row i). - exact (((b i) * fldmultinv' (mat i i)) - ((Σ (mat i *pw x) - (x i)* (mat i i)) * (fldmultinv' (mat i i))))%ring. - exact (x i). Defined.
Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.MoreFoundations.Nat. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.Matrix. Require Import UniMath.Algebra.Domains_and_Fields. Require Import UniMath.Algebra.Matrix. Require Import UniMath.Algebra.RigsAndRings. Require Import UniMath.Algebra.IteratedBinaryOperations. Require Import UniMath.Algebra.GaussianElimination.Auxiliary. Require Import UniMath.Algebra.GaussianElimination.Vectors. Require Import UniMath.Algebra.GaussianElimination.Matrices. Require Import UniMath.Algebra.GaussianElimination.Elimination.