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(* 02110-1301 USA                                                     *)
(** %\subsection*{ examples :  Matrix\_multiplication.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export vecspace_Mmn.
Require Export pointwise.
Require Export sums.
(** - Remember that (Fn F n) is the vectorspace $F^n$, based on the setoid (seq n F), and (Mmn F m n) is the vectorspace $M_{mn}$ of $m\times n$-matrices over $F$; based on the setoid (matrix F m n) *)
Section matrix_x_vector.
Definition mat_vec_mult_fun :
forall (F : field) (m n : Nat), Mmn F m n -> Fn F n -> Fn F m.
intros.
simpl in X, X0.
simpl in |- *.
apply
(Build_Map
(Ap:=fun i : fin m =>
sum (pointwise (uncurry (RING_comp (R:=F))) (Ap2_Map X i) X0))).
red in |- *; simpl in |- *.
intros.
apply sum_comp; auto with algebra;
(apply toMap; apply pointwise_comp; auto with algebra).
simpl in |- *.
red in |- *; simpl in |- *.
intro; destruct X; simpl in |- *; auto with algebra.
Defined.
Definition matXvec :
forall (F : field) (m n : Nat), Map2 (Mmn F m n) (Fn F n) (Fn F m).
intros; apply (Build_Map2 (Ap2:=mat_vec_mult_fun (F:=F) (m:=m) (n:=n))).
red in |- *; simpl in |- *; red in |- *; simpl in |- *.
intros.
apply sum_comp; auto with algebra;
(apply toMap; apply pointwise_comp; auto with algebra).
simpl in |- *.
red in |- *; simpl in |- *.
intro; (apply H; auto with algebra).
Defined.
End matrix_x_vector.
Section matrix_x_matrix.
Definition mat_mat_mult_fun :
forall (F : field) (m n p : Nat), Mmn F m n -> Mmn F n p -> Mmn F m p.
intros F m n p M N.
apply
(Build_Map2
(Ap2:=fun i j =>
sum (pointwise (uncurry (RING_comp (R:=F))) (row M i) (col N j)))).
red in |- *; simpl in |- *.
intros.
apply sum_comp; auto with algebra.
apply toMap; apply pointwise_comp; auto with algebra.
change (row M x =' row M x' in _) in |- *.
apply row_comp; auto with algebra.
change (col N y =' col N y' in _) in |- *.
apply col_comp; auto with algebra.
Defined.
Definition matXmat :
forall (F : field) (m n p : Nat), Map2 (Mmn F m n) (Mmn F n p) (Mmn F m p).
intros;
apply (Build_Map2 (Ap2:=mat_mat_mult_fun (F:=F) (m:=m) (n:=n) (p:=p))).
red in |- *; simpl in |- *.
intros.
apply sum_comp; auto with algebra;
(apply toMap; apply pointwise_comp; auto with algebra).
change (row x i =' row x' i' in _) in |- *.
apply row_comp; auto with algebra.
change (col y j =' col y' j' in _) in |- *.
apply col_comp; auto with algebra.
Defined.
End matrix_x_matrix.
Section facts.
Variable F : field.
Variable n m p : Nat.
Variable M : Mmn F m n.
Variable N : Mmn F n p.

Lemma matXmat_col :
forall i : fin p,
col (matXmat _ _ _ _ M N) i =' matXvec _ _ _ M (col N i) in _.