Alternate def for dual_basis.

Rename bidual_nat_isom* -> bidual_isom*.

FEM/Algebra/Finite_dim_R.v

@@ -1494,6 +1494,15 @@ Definition dual_basis : '(PE_ms -> R)^n :=
   fun i x =>
     let Hx := eq_ind_r (@^~ (val x)) (in_sub x) (eq_sym (proj1 HB)) in
     proj1_sig (span_EX _ _ Hx) i.
+(*
+Proof.
+intros i [x Hx].
+rewrite (proj1 HB) in Hx.
+apply span_EX in Hx.
+destruct Hx as [L HL].
+apply (L i).
+Defined.
+*)
 
 End Dual_basis_Def.
 
@@ -1727,22 +1736,22 @@ Lemma bidual_pt_eval : forall (x : PE_ms), bidual (fun f => val f x).
 Proof. intros x; apply dual_lin_map_equiv; easy. Qed.
 
 (* This is the natural isomorphism between a subspace and its bidual. *)
-Definition bidual_nat_isom (x : PE_ms) : PE''_ms :=
+Definition bidual_isom (x : PE_ms) : PE''_ms :=
   mk_sub_ms (bidual_pt_eval x).
 
-Lemma bidual_nat_isom_correct :
-  forall (x : PE_ms) (f : PE'_ms), val (bidual_nat_isom x) f = val f x.
+Lemma bidual_isom_correct :
+  forall (x : PE_ms) (f : PE'_ms), val (bidual_isom x) f = val f x.
 Proof. easy. Qed.
 
-Lemma bidual_nat_isom_lin_map : is_linear_mapping bidual_nat_isom.
+Lemma bidual_isom_lin_map : is_linear_mapping bidual_isom.
 Proof.
 Admitted.
 
-Lemma bidual_nat_isom_inj : injective bidual_nat_isom.
+Lemma bidual_isom_inj : injective bidual_isom.
 Proof.
 Admitted.
 
-Lemma bidual_nat_isom_bij : bijective bidual_nat_isom.
+Lemma bidual_isom_bij : bijective bidual_isom.
 Proof.
 Admitted.
 
@@ -1774,7 +1783,7 @@ Let B'' := dual_basis HPE' HB'.
 Let HB'' := dual_basis_is_basis HPE' HB'.
 
 Definition predual_basis (i : 'I_n) : E :=
-  val (f_inv (bidual_nat_isom_bij HPE)
+  val (f_inv (bidual_isom_bij HPE)
     (mk_sub_ms (is_basis_inclF HB'' i))).
 
 (*