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Commit 8303aca0 authored by Mouhcine's avatar Mouhcine
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work on LagPQ_is_non_neg lemma

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......@@ -462,23 +462,30 @@ apply LagQ1_kron.
Qed.
(* TODO: Not really! we should add some hypothesis on x_i *)
(* ajouter une hyp qu'on est dans l'element de ref *)
Lemma LagPQ_is_non_neg : forall (i : 'I_nnvtx) (x : 'R^dd),
0 <= LagPQ i x.
K_geom_ref x -> 0 <= LagPQ i x.
Proof.
intros i x; unfold LagPQ.
intros i x Hx; unfold LagPQ.
case (nvtx_dec FE_ref); intros H.
apply LagP1_is_non_neg.
apply LagQ1_is_non_neg; easy.
Qed.
unfold K_geom_ref, K_geom in Hx.
unfold dd.
admit.
admit.
(*apply LagQ1_is_non_neg; easy.
Qed.*)
Admitted.
(* TODO verify if this lemma is true, we apply it to T_geom_is_bij_id *)
Lemma LagPQ_sum_1 : forall x,
comb_lin (fun i : 'I_nnvtx => LagPQ i x)(fun _ => 1) = 1.
Proof.
(*
intros x.
unfold LagPQ.
case (nvtx_dec FE_ref); intros H.
admit.
admit.*)
(*apply LagP1_sum_1.
apply LagQ1_sum_1.*)
......@@ -767,6 +774,7 @@ unfold range; intros x_cur H.
apply K_geom_ref_correct; easy.
(* *)
intros i; apply LagPQ_is_non_neg.
(*
apply LagPQ_sum_1.
(* *)
unfold identity; intros x_cur H; symmetry.
......@@ -774,7 +782,8 @@ apply T_geom_comp; easy.
(* *)
unfold identity; intros x_ref H; symmetry.
apply T_geom_inv_comp; easy.
Qed.
Qed.*)
Admitted.
Lemma theta_cur_correct : forall i : 'I_(ndof FE_cur),
theta_ref i = cur_to_ref (theta_cur i).
......
......@@ -107,9 +107,9 @@ Admitted.
(* TODO: Faut des hypothèses de plus sur x_i > 0 *)
Lemma LagP1_is_non_neg : forall (i : 'I_d.+1) (x : 'R^d),
0 <= LagP1 i x.
convex_envelop d.+1 vtxP1 x -> 0 <= LagP1 i x.
Proof.
intros i x; unfold LagP1.
intros i x Hx; unfold LagP1.
Admitted.
Lemma LagP1_sum_1 : forall x,
......@@ -160,9 +160,9 @@ Proof.
Admitted.
Lemma LagQ1_is_non_neg : forall (i : 'I_(2 ^ d)) (x : 'R^d),
0 <= LagQ1 i x.
convex_envelop (2^d) vtxQ1 x -> 0 <= LagQ1 i x.
Proof.
intros i x; unfold LagQ1.
intros i x Hx; unfold LagQ1.
Admitted.
......
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