Prove vtx_in_convex_envelop lemma.

move the comment to FE.v

FEM/geometry.v

@@ -52,72 +52,17 @@ intros i.
 replace (vtx i) with 
   (comb_lin (kronecker i) vtx).
 apply Cvx.
-
-
-(*; unfold elt_geom.
-exists (kronecker i).
-repeat split*)
-replace (vtx i) with (\big[plus/zero]_(j < nvtx) (scal (kronecker i j) (vtx j))).
-(*
-apply Cvx.
-intros j; apply kronecker_bound.*)
-
-(*
-apply kronecker_sum_r; easy.
-(* *)
-induction nvtx; try easy.
-rewrite sum_pn_Sn.
-case (lt_dec i n); intros Hn.
-(*case1*)
-rewrite <- IHn; try easy.
-replace (kronecker i n) with (@zero R_Ring).
-rewrite (scal_zero_l).
-rewrite plus_zero_r; easy.
-unfold kronecker.
-case (eq_nat_dec i n); try easy.
-intros; lia.
-(*case2*)
-unfold kronecker at 2.
-case (eq_nat_dec i n); try lia.
-intros H1.
-replace (sum_pn _ _) with (@zero E).
-replace 1 with (@one R_Ring) by easy.
-rewrite scal_one.
-rewrite H1.
-rewrite plus_zero_l; easy.
-rewrite sum_pn_zero; try easy.
-intros m Hm.
-unfold kronecker.
-case (eq_nat_dec i m); try lia.
-intros; replace 0 with (@zero R_Ring) by easy.
-apply (scal_zero_l (vtx m)).
+intros j; apply kronecker_bound.
+apply comb_lin_kronecker_r.
+apply comb_lin_kronecker_in_r.
 Qed.
-*)
-Admitted.
 
 (* TODO: hypothesis on vertices : geometrical form not degenerate *)
 
 (* TODO: define basic geometric shapes:
   (simplices) Seg Tria, Tetra,
-  (others) Quad, Hexa, Prism?...
-
-  Define the geometric transformation T_geom (from the reference mesh to the current mesh)
-  for all previous geometric shapes, and its Jacobian matrix and Jacobian determinant.
-
-  For the simplex in dim E = d, T_geom is affine : x^ -> x = a0 + J_geom x^
-  where J_geom = (a1-a0 a2-a0 ... ad-a0) is the Jacobian matrix of T_geom, made of the
-  column vectors ai-a0, where (a0,a1...ad) are the vertices of the current mesh.
-  The vertices of the reference mesh are a0^ = (0,0...0), a1^ = (1,0...0),
-  a2^ = (0,1,0...0),..., ad^ = (0,...0, 1).
-
-  In each case, prove that J_geom is indeed the Jacobian matrix of T_geom.
-
-  Prove that T_geom is invertible, ie its Jacobian determinant is nonzero.
-
-  Then, in the simple cases, T_geom{-1} : x -> x^ = J_geom{-1} (x-a0).
+  (others) Quad, Hexa, Prism?...*)
 
-  Maybe prove correction lemmas of the form:
-  forall x^, K_geom (T_geom x^) <-> K_geom^ x^, ie image T_geom K_geom^ = K_geom.
-  forall x, K_geom x <-> K_geom^ (T_geom{-1} x), ie image T_geom{-1} K_geom = K_geom^. *)
+ 
 
 End Geometry.