Detection of dead code.

FEM/poly_LagPd1.v

@@ -60,11 +60,13 @@ Proof. easy. Qed.
 Lemma LagPd1_eqF : LagPd1 = compF_l LagPd1_ref (T_geom_inv Hvtx).
 Proof. easy. Qed.
 
+(* Unused!
 Lemma LagPd1_ref_eq : forall i, LagPd1_ref i = LagPd1 i \o (T_geom vtx).
-Proof. intro; rewrite LagPd1_eq -comp_assoc T_geom_inv_id_l; easy. Qed.
+Proof. intro; rewrite LagPd1_eq -comp_assoc T_geom_inv_id_l; easy. Qed.*)
 
+(* Unused!
 Lemma LagPd1_ref_eqF : LagPd1_ref = compF_l LagPd1 (T_geom vtx).
-Proof. extF; apply LagPd1_ref_eq. Qed.
+Proof. extF; apply LagPd1_ref_eq. Qed.*)
 
 Lemma LagPd1_0 :
   forall i, i = ord0 -> LagPd1 i = fun x => 1 - sum (T_geom_inv Hvtx x).
@@ -84,16 +86,19 @@ intro; extF; unfold liftF_S.
 rewrite (LagPd1_S (lift_S_not_first _)) lower_lift_S; easy.
 Qed.
 
+(* Unused!
 Lemma LagPd1_S_eq_f : (fun x => liftF_S (LagPd1^~ x)) = T_geom_inv Hvtx.
-Proof. fun_ext; apply LagPd1_S_eq. Qed.
+Proof. fun_ext; apply LagPd1_S_eq. Qed.*)
 
+(* Unused!
 Lemma LagPd1_0_eq :
   forall {i} x, i = ord0 -> LagPd1 i x = 1 - sum (liftF_S (LagPd1^~ x)).
-Proof. intros; subst; rewrite LagPd1_0// LagPd1_S_eq; easy. Qed.
+Proof. intros; subst; rewrite LagPd1_0// LagPd1_S_eq; easy. Qed.*)
 
+(* Unused!
 Lemma LagPd1_0_eq_f :
   forall {i}, i = ord0 -> LagPd1 i = 1 - fun x => sum (liftF_S (LagPd1^~ x)).
-Proof. intros; fun_ext; apply LagPd1_0_eq; easy. Qed.
+Proof. intros; fun_ext; apply LagPd1_0_eq; easy. Qed.*)
 
 (**
  #<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
@@ -107,18 +112,20 @@ Proof. intro; rewrite LagPd1_S_eq; apply T_geom_inv_can_l. Qed.
 Lemma LagPd1_S_can_r : forall x, T_geom vtx (liftF_S (LagPd1^~ x)) = x.
 Proof. intro; rewrite LagPd1_S_eq; apply T_geom_inv_can_r. Qed.
 
+(* Unused!
 (* Could be: injective (gather LagPd1). *)
 Lemma LagPd1_inj :
   forall (x y : 'R^d), (forall i, LagPd1 i x = LagPd1 i y) -> x = y.
-Proof. move=>> /LagPd1_ref_inj /T_geom_inv_inj; easy. Qed.
+Proof. move=>> /LagPd1_ref_inj /T_geom_inv_inj; easy. Qed.*)
 
+(* Unused!
 (* Could be: surjS fullset (fun L => sum L = 1) (fun x => LagPd1^~ x). *)
 Lemma LagPd1_surjS :
   forall (L : 'R^d.+1), sum L = 1 -> exists x, LagPd1^~ x = L.
 Proof.
 intros L HL; exists (T_geom vtx (liftF_S L)).
 unfold LagPd1; rewrite T_geom_inv_can_l LagPd1_ref_liftF_S; easy.
-Qed.
+Qed.*)
 
 (**
  #<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
@@ -126,22 +133,25 @@ Qed.
 Lemma LagPd1_sum : forall x, sum (LagPd1^~ x) = 1.
 Proof. intros; apply LagPd1_ref_sum. Qed.
 
+(* Unused!
 Lemma LagPd1_sum_f : sum LagPd1 = 1.
-Proof. fun_ext; rewrite fct_sum_eq LagPd1_sum; easy. Qed.
+Proof. fun_ext; rewrite fct_sum_eq LagPd1_sum; easy. Qed.*)
 
 (**
  #<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
  Lem 1554, Eq (9.45), p. 85. *)
-Lemma LagPd1_kron_vtx : forall i j, LagPd1 j (vtx i) = kronR i j.
+Lemma LagPd1_kron_vtx : forall i, LagPd1^~ (vtx i) = kronR i.
 Proof.
-intros; unfold LagPd1; rewrite T_geom_inv_vtx; apply LagPd1_ref_kron_vtx.
+intros; extF; unfold LagPd1; rewrite T_geom_inv_vtx; apply LagPd1_ref_kron_vtx.
 Qed.
 
+(* Unused!
 Lemma LagPd1_kron_vtx_r : forall i, LagPd1^~ (vtx i) = kronR i.
-Proof. intro; extF; apply LagPd1_kron_vtx. Qed.
+Proof. intro; extF; apply LagPd1_kron_vtx. Qed.*)
 
+(* Unused!
 Lemma LagPd1_kron_vtx_l : forall j, mapF (LagPd1 j) vtx = kronR^~ j.
-Proof. intro; extF; rewrite mapF_correct; apply LagPd1_kron_vtx. Qed.
+Proof. intro; extF; rewrite mapF_correct; apply LagPd1_kron_vtx. Qed.*)
 
 Lemma LagPd1_am : forall i, aff_map_ms (LagPd1 i).
 Proof.
@@ -162,10 +172,11 @@ intros; eapply am_ext.
 easy.
 Qed.
 
+(* Unused!
 Lemma LagPd1_am_lc : forall L i, sum L = 1 -> LagPd1 i (lin_comb L vtx) = L i.
 Proof.
 intros; unfold LagPd1; rewrite T_geom_inv_am_lc// LagPd1_ref_baryc_lc; easy.
-Qed.
+Qed.*)
 
 Let K_geom : 'R^d -> Prop := convex_envelop vtx.
 
@@ -184,26 +195,30 @@ Proof.
 intros x; rewrite -{2}(LagPd1_S_can_r x); unfold T_geom; rewrite LagPd1_S_eq//.
 Qed.
 
+(* Unused!
 Lemma LagPd1_baryc_rev :
   forall L, sum L = 1 -> LagPd1^~ (barycenter_ms L vtx) = L.
 Proof.
 intros; extF j; rewrite (am_normal_rev (LagPd1_am j))//.
 rewrite LagPd1_kron_vtx_l; apply baryc_r_kron_l; easy.
-Qed.
+Qed.*)
 
+(* Unused!
 Lemma LagPd1_baryc_alt : forall x L, L = LagPd1^~ x -> barycenter_ms L vtx = x.
-Proof. intros; subst; apply LagPd1_baryc. Qed.
+Proof. intros; subst; apply LagPd1_baryc. Qed.*)
 
+(* Unused!
 Lemma LagPd1_baryc_rev_alt :
   forall x L, sum L = 1 -> x = barycenter_ms L vtx -> LagPd1^~ x = L.
-Proof. intros; subst; apply LagPd1_baryc_rev; easy. Qed.
+Proof. intros; subst; apply LagPd1_baryc_rev; easy. Qed.*)
 
+(* Unused!
 Lemma LagPd1_baryc_equiv :
   forall x L, sum L = 1 -> x = barycenter_ms L vtx <-> LagPd1^~ x = L.
 Proof.
 intros; split; intros; subst;
     [apply LagPd1_baryc_rev_alt | apply eq_sym, LagPd1_baryc_alt]; easy.
-Qed.
+Qed.*)
 
 Lemma LagPd1_is_Pd1 : inclF LagPd1 (Pdk d 1).
 Proof.
@@ -214,7 +229,7 @@ Qed.
 Lemma LagPd1_lin_indep : lin_indep LagPd1.
 Proof.
 intros L HL; extF; rewrite -lc_r_kron_r.
-rewrite -LagPd1_kron_vtx_r -fct_lc_r_eq HL; easy.
+rewrite -LagPd1_kron_vtx -fct_lc_r_eq HL; easy.
 Qed.
 
 (**
@@ -233,7 +248,7 @@ Lemma LagPd1_decomp_vtx :
   forall {p}, Pdk d 1 p -> p = lin_comb (mapF p vtx) LagPd1.
 Proof.
 intros p Hp; destruct (basis_decomp_ex LagPd1_basis p Hp) as [L ->].
-apply lc_ext_l; intro; rewrite mapF_correct fct_lc_r_eq LagPd1_kron_vtx_r.
+apply lc_ext_l; intro; rewrite mapF_correct fct_lc_r_eq LagPd1_kron_vtx.
 apply eq_sym, lc_r_kron_r.
 Qed.
 
@@ -479,6 +494,7 @@ rewrite (LagPd1_eq_transpF _ i) transpF_correct_l1//
     -f_mult_compat_comp_l -Hq2; easy.
 Qed.
 
+(* Unused!
 (**
  #<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
  Lem 1623, reverse way, p. 105. *)
@@ -489,8 +505,9 @@ Lemma factor_zero_on_Hface_rev :
 Proof.
 move=>> Hp [q [Hq ->]] =>>; rewrite Hface_is_Ker => Hx.
 rewrite fct_mult_eq Hx mult_zero_l; easy.
-Qed.
+Qed.*)
 
+(* Unused!
 Lemma factor_zero_on_Hface_equiv :
   forall {p} i, Pdk d k.+1 p ->
     (forall x, Hface vtx i x -> p x = 0) <->
@@ -498,7 +515,7 @@ Lemma factor_zero_on_Hface_equiv :
 Proof.
 intros; split; [apply factor_zero_on_Hface |
     apply factor_zero_on_Hface_rev]; easy.
-Qed.
+Qed.*)
 
 End Pdk_Facts.