{R : Ring} {m n p} (M : Matrix R m n) (N : Matrix R n p) : matrix_transpose (matrix_mult M N) = matrix_mult (R:=rng_op R) (matrix_transpose N) (matrix_transpose M). Proof. apply path_matrix. intros i j Hi Hj. rewrite 3 entry_Build_Matrix. apply path_ab_sum. intros k Hk. rewrite 2 entry_Build_Matrix. reflexivity. Defined.
Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.