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(* 02110-1301 USA                                                     *)
(** %\subsection*{ examples :  antisymmetric\_matrices.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export vecspace_Mmn.
Require Export subspaces.
Definition is_antisymmetric (F : field) (n : Nat) (M : Mmn F n n) :=
forall i j : fin n, M i j =' (min M j i) in _.
Definition is_antisymmetric_pred :
forall (F : field) (n : Nat), Predicate (Mmn F n n).
intros; apply (Build_Predicate (Pred_fun:=is_antisymmetric (F:=F) (n:=n))).
red in |- *; simpl in |- *.
intros.
red in |- *; red in H.
intros.
apply Trans with (x i j); auto with algebra.
apply Trans with (min x j i); auto with algebra.
apply GROUP_comp; auto with algebra.
apply Sym; auto with algebra.
Defined.

(* Specifications: *)
Lemma antisymm_matr_subspace :
forall (n : Nat) (F : field), is_subspace (is_antisymmetric_pred F n).