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Commit cbdd03bc authored by Mouhcine's avatar Mouhcine
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Add LagP1_is_affine and LagQ1_is_LagP1 lemmas.

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......@@ -105,7 +105,6 @@ contradict H.
admit.
Admitted.
(* TODO: Faut des hypothèses de plus sur x_i > 0 *)
Lemma LagP1_is_non_neg : forall (i : 'I_d.+1) (x : 'R^d),
convex_envelop d.+1 vtxP1 x -> 0 <= LagP1 i x.
Proof.
......@@ -119,6 +118,36 @@ intros.
unfold comb_lin.
Admitted.
Lemma LagP1_is_affine: forall i,
is_affine (LagP1 i).
Proof.
intros i.
pose (L := fun x => minus (LagP1 i x) (LagP1 i zero)).
apply is_affine_ext with
(fun x => plus (L x) (LagP1 i zero)).
intros x.
unfold L.
admit. (* easy*)
apply (Is_affine L).
unfold L, LagP1.
case (eq_nat_dec i 0); intros Hi.
rewrite bigop_op_idx; try easy.
apply is_linear_mapping_ext with
(fun x : 'R^d => \big[+%R/0]_(i < d) x i).
intros x.
admit. (* easy *)
apply is_linear_mapping_ext with
(fun x : 'R^d => comb_lin (fun => 1) x).
admit. (* easy *)
apply component_sum_is_linear_mapping.
(* *)
apply is_linear_mapping_ext with
(fun x : 'R^d =>
(x (Ordinal (n:=d) (m:=i - 1) (lt_minus_1 i Hi)))).
admit. (* easy *)
apply component_is_linear_mapping.
Admitted.
End Poly_Lagrange_Simplex.
......@@ -174,3 +203,16 @@ unfold comb_lin.
Admitted.
End Poly_Lagrange_Quad.
Section LagPQ.
(* TODO Faut le phi *)
Lemma LagQ1_is_LagP1 : forall (i : 'I_2) x,
LagQ1 1 i x = LagP1 1 i x.
Proof.
intros i x.
unfold LagQ1.
Admitted.
End LagPQ.
\ No newline at end of file
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