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Commit d5cf327e authored by François Clément's avatar François Clément
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Detection of dead code.

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......@@ -87,11 +87,12 @@ Qed.
Lemma T_geom_vtxF : mapF T_geom vtx_ref = vtx.
Proof. extF; apply T_geom_vtx. Qed.
(* Unused!
Lemma T_geom_am_lc :
forall L, sum L = 1 -> T_geom (lin_comb L vtx_ref) = lin_comb L vtx.
Proof.
intros; rewrite -!baryc_ms_eq// (am_normal_rev T_geom_am)// T_geom_vtxF; easy.
Qed.
Qed.*)
Lemma T_geom_baryc :
forall L, sum L = 1 ->
......@@ -161,11 +162,12 @@ Proof. apply f_inv_can_l. Qed.
Lemma T_geom_inv_can_r : cancel T_geom_inv (T_geom vtx).
Proof. apply f_inv_can_r. Qed.
(* Unused!
Lemma T_geom_inv_bij : bijective T_geom_inv.
Proof. apply f_inv_bij. Qed.
Lemma T_geom_inv_inj : injective T_geom_inv.
Proof. apply bij_inj, T_geom_inv_bij. Qed.
Proof. apply bij_inj, T_geom_inv_bij. Qed.*)
Lemma T_geom_inv_am : aff_map_ms T_geom_inv.
Proof. apply: am_bij_compat; apply T_geom_am. Qed.
......@@ -179,12 +181,13 @@ Proof. rewrite -(T_geom_vtxF vtx) -mapF_comp T_geom_inv_id_l; easy. Qed.
Lemma T_geom_inv_vtx : forall i, T_geom_inv (vtx i) = vtx_ref i.
Proof. intro; rewrite -T_geom_inv_vtxF; easy. Qed.
(* Unused!
Lemma T_geom_inv_am_lc :
forall L, sum L = 1 -> T_geom_inv (lin_comb L vtx) = lin_comb L vtx_ref.
Proof.
intros; rewrite -!baryc_ms_eq//
(am_normal_rev T_geom_inv_am)// T_geom_inv_vtxF; easy.
Qed.
Qed.*)
Lemma T_geom_inv_baryc :
forall L, sum L = 1 ->
......@@ -217,6 +220,7 @@ Proof.
move=>> [L HL1 HL2]; unfold K_geom, preimage; rewrite T_geom_inv_baryc; easy.
Qed.
(* Unused!
Lemma T_geom_bijS_K_geom_spec :
bijS_spec K_geom_ref K_geom (T_geom vtx) (T_geom_inv Hvtx).
Proof.
......@@ -241,7 +245,7 @@ intros x_ref _; apply T_geom_inv_can_l.
Qed.
Lemma T_geom_inv_bijS_K_geom : bijS K_geom K_geom_ref (T_geom_inv Hvtx).
Proof. apply (BijS _ T_geom_inv_bijS_K_geom_spec). Qed.
Proof. apply (BijS _ T_geom_inv_bijS_K_geom_spec). Qed.*)
(**
#<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
......@@ -339,17 +343,19 @@ Lemma T_geom_permutF_inv_can_r :
cancel T_geom_permutF_inv (T_geom_permutF vtx pi_d).
Proof. apply f_inv_can_r. Qed.
(* Unused!
Lemma T_geom_permutF_inv_id_l :
T_geom_permutF_inv \o T_geom_permutF vtx pi_d = id.
Proof. apply f_inv_id_l. Qed.
Lemma T_geom_permutF_inv_id_r :
T_geom_permutF vtx pi_d \o T_geom_permutF_inv = id.
Proof. apply f_inv_id_r. Qed.
Proof. apply f_inv_id_r. Qed.*)
(* Unused!
Lemma T_geom_permutF_inv_correct :
T_geom_permutF_inv = T_geom_inv (aff_indep_permutF pi_d_bij Hvtx).
Proof. apply f_inv_uniq_l, f_inv_can_l. Qed.
Proof. apply f_inv_uniq_l, f_inv_can_l. Qed.*)
(**
#<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
......@@ -381,10 +387,11 @@ intro; unfold T_geom_permutF, T_geom;
rewrite LagPd1_ref_sum; apply invertible_one.
Qed.
(* Unused!
Lemma T_geom_permutF_vtx :
forall i,
T_geom_permutF vtx pi_d (vtx_ref i) = T_geom vtx (vtx_ref (pi_d i)).
Proof. intro; unfold T_geom_permutF; rewrite !T_geom_vtx; easy. Qed.
Proof. intro; unfold T_geom_permutF; rewrite !T_geom_vtx; easy. Qed.*)
Lemma T_geom_permutF_skipF :
forall i j,
......@@ -405,9 +412,10 @@ Variable vtx : 'R^{d.+1,d}.
Definition T_geom_transpF (i0 : 'I_d.+1) : 'R^d -> 'R^d :=
T_geom_permutF vtx (transp_ord ord_max i0).
(* Unused!
Lemma T_geom_transpF_correct :
forall i0, T_geom_transpF i0 = T_geom (transpF ord_max i0 vtx).
Proof. easy. Qed.
Proof. easy. Qed.*)
End T_geom_transpF_Def.
......@@ -440,15 +448,18 @@ Variable vtx : 'R^{d.+1,d}.
Definition T_geom_Hface (i : 'I_d.+1) : 'R^d1 -> 'R^d :=
T_geom vtx \o inj_Hface_ref i.
(* Unused!
Lemma T_geom_Hface_0 :
forall {i}, i = ord0 -> T_geom_Hface i = T_geom vtx \o part1F0.
Proof. intros; unfold T_geom_Hface; rewrite inj_Hface_ref_0; easy. Qed.
Proof. intros; unfold T_geom_Hface; rewrite inj_Hface_ref_0; easy. Qed.*)
(* Unused!
Lemma T_geom_Hface_S :
forall {i} (Hi : i <> ord0),
T_geom_Hface i = T_geom vtx \o (insert0F^~ (lower_S Hi)).
Proof. move=>>; unfold T_geom_Hface; rewrite inj_Hface_ref_S; easy. Qed.
Proof. move=>>; unfold T_geom_Hface; rewrite inj_Hface_ref_S; easy. Qed.*)
(* Unused!
Lemma T_geom_Hface_0_correct :
forall {i} (Hi : i = ord0) x_ref,
T_geom_Hface i x_ref = barycenter_ms (LagPd1_ref^~ x_ref) (skipF vtx i).
......@@ -462,8 +473,9 @@ destruct (ord_eq_dec j ord0) as [-> | Hj].
rewrite LagPd1_ref_0// insertF_correct_l; easy.
move: (proj1 (ord_n0_gt_equiv d) Hj) => Hj'.
rewrite LagPd1_ref_S insertF_correct_rr; f_equal; apply lower_S_ext.
Qed.
Qed.*)
(* Unused!
Lemma T_geom_Hface_S_correct :
forall {i} (Hi : i <> ord0) x_ref,
T_geom_Hface i x_ref = barycenter_ms (LagPd1_ref^~ x_ref) (skipF vtx i).
......@@ -486,9 +498,9 @@ rewrite (skip_ord_correct_l _ _ H'); apply ord_inj; simpl. admit.
(* *)
assert (H' : ~ (lower_S Hj < lower_S Hi)%coq_nat) by admit.
rewrite (skip_ord_correct_r _ _ H'); apply ord_inj; simpl. admit.
Admitted.
Admitted.*)
(* Hopefully, this should be useless! *)
(* Hopefully, this should be useless!
Lemma T_geom_Hface_0_lc :
forall x_ref,
T_geom_Hface ord0 x_ref = scal (1 - sum x_ref) (vtx ord1) +
......@@ -496,9 +508,9 @@ Lemma T_geom_Hface_0_lc :
Proof.
intro; rewrite T_geom_Hface_0_correct// baryc_ms_eq; [| apply LagPd1_ref_sum].
rewrite lc_ind_l LagPd1_ref_0// skipF_first liftF_S_0 LagPd1_ref_SF; easy.
Qed.
Qed.*)
(* Hopefully, this should be useless! *)
(* Hopefully, this should be useless!
Lemma T_geom_Hface_S_lc :
forall i x_ref, i <> ord0 ->
T_geom_Hface i x_ref = scal (1 - sum x_ref) (vtx ord0) +
......@@ -508,9 +520,9 @@ intros; rewrite T_geom_Hface_S_correct// baryc_ms_eq; [| apply LagPd1_ref_sum].
assert (Hi' : (@ord0 d1 < i)%coq_nat) by now apply ord_n0_gt_equiv.
rewrite lc_ind_l LagPd1_ref_0// skipF_correct_l//
widenF_S_0 LagPd1_ref_SF; easy.
Qed.
Qed.*)
(* Hopefully, this should be useless! *)
(* Hopefully, this should be useless!
(**
#<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
Lem 1584, Eq (9.72), pp. 94-95.#<BR>#
......@@ -524,9 +536,9 @@ Proof.
intros; rewrite T_geom_Hface_0_lc.
rewrite scal_minus_l scal_one -plus_assoc -minus_sym.
rewrite lc_minus_r lc_constF_r; easy.
Qed.
Qed.*)
(* Hopefully, this should be useless! *)
(* Hopefully, this should be useless!
(**
#<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
Lem 1584, Eq (9.73), pp. 94-95.#<BR>#
......@@ -540,7 +552,7 @@ Proof.
intros; rewrite T_geom_Hface_S_lc//.
rewrite scal_minus_l scal_one -plus_assoc -minus_sym.
rewrite lc_minus_r lc_constF_r; easy.
Qed.
Qed.*)
(**
#<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
......@@ -550,7 +562,7 @@ Proof.
intro; apply am_comp_ms; [apply inj_Hface_ref_am | apply T_geom_am].
Qed.
(* Hopefully, this should be useless! *)
(* Hopefully, this should be useless!
(**
#<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
Lem 1584, Eq (9.73), pp. 94-95.#<BR>#
......@@ -564,9 +576,9 @@ Proof.
intros x_ref; unfold fct_lm_ms; rewrite fct_lm_ms_eq.
rewrite plus_zero_l !T_geom_Hface_0_lc_alt lc_zero_l plus_zero_r.
apply: minus_plus_l.
Qed.
Qed.*)
(* Hopefully, this should be useless! *)
(* Hopefully, this should be useless!
(**
#<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
Lem 1584, Eq (9.73), pp. 94-95.#<BR>#
......@@ -580,7 +592,7 @@ Proof.
intros i x_ref Hi; unfold fct_lm_ms; rewrite fct_lm_ms_eq.
rewrite plus_zero_l !T_geom_Hface_S_lc_alt// lc_zero_l plus_zero_r.
apply: minus_plus_l.
Qed.
Qed.*)
(**
#<A HREF="##RR9557v1">#[[RR9557v1]]#</A>#
......
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