(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(*i $Id: Ranalysis2.v 5920 2004-07-16 20:01:26Z herbelin $ i*)

Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis1. Open Local Scope R_scope.

(**********)
Lemma formule :
 forall (x h l1 l2:R) (f1 f2:R -> R),
   h <> 0 ->
   f2 x <> 0 ->
   f2 (x + h) <> 0 ->
   (f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h -
   (l1 * f2 x - l2 * f1 x) / Rsqr (f2 x) =
   / f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) +
   l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) -
   f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2) +
   l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x).
intros; unfold Rdiv, Rminus, Rsqr in |- *.
repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l;
 repeat rewrite Rinv_mult_distr; try assumption.
replace (l1 * f2 x * (/ f2 x * / f2 x)) with (l1 * / f2 x * (f2 x * / f2 x));
 [ idtac | ring ].
replace (l1 * (/ f2 x * / f2 (x + h)) * f2 x) with
 (l1 * / f2 (x + h) * (f2 x * / f2 x)); [ idtac | ring ].
replace (l1 * (/ f2 x * / f2 (x + h)) * - f2 (x + h)) with
 (- (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))); [ idtac | ring ].
replace (f1 x * (/ f2 x * / f2 (x + h)) * (f2 (x + h) * / h)) with
 (f1 x * / f2 x * / h * (f2 (x + h) * / f2 (x + h))); 
 [ idtac | ring ].
replace (f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h)) with
 (- (f1 x * / f2 (x + h) * / h * (f2 x * / f2 x))); 
 [ idtac | ring ].
replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * f2 (x + h)) with
 (l2 * f1 x * / f2 x * / f2 x * (f2 (x + h) * / f2 (x + h)));
 [ idtac | ring ].
replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x) with
 (- (l2 * f1 x * / f2 x * / f2 (x + h) * (f2 x * / f2 x))); 
 [ idtac | ring ].
repeat rewrite <- Rinv_r_sym; try assumption || ring.
apply prod_neq_R0; assumption.
Qed.

Lemma Rmin_pos : forall x y:R, 0 < x -> 0 < y -> 0 < Rmin x y.
intros; unfold Rmin in |- *.
case (Rle_dec x y); intro; assumption.
Qed.

Lemma maj_term1 :
 forall (x h eps l1 alp_f2:R) (eps_f2 alp_f1d:posreal) 
   (f1 f2:R -> R),
   0 < eps ->
   f2 x <> 0 ->
   f2 (x + h) <> 0 ->
   (forall h:R,
      h <> 0 ->
      Rabs h < alp_f1d ->
      Rabs ((f1 (x + h) - f1 x) / h - l1) < Rabs (eps * f2 x / 8)) ->
   (forall a:R,
      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
   h <> 0 ->
   Rabs h < alp_f1d ->
   Rabs h < Rmin eps_f2 alp_f2 ->
   Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) < eps / 4.
intros.
assert (H7 := H3 h H6).
assert (H8 := H2 h H4 H5).
apply Rle_lt_trans with
 (2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1)).
rewrite Rabs_mult.
apply Rmult_le_compat_r.
apply Rabs_pos.
rewrite Rabs_Rinv; [ left; exact H7 | assumption ].
apply Rlt_le_trans with (2 / Rabs (f2 x) * Rabs (eps * f2 x / 8)).
apply Rmult_lt_compat_l.
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
 [ prove_sup0 | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ].
exact H8.
right; unfold Rdiv in |- *.
repeat rewrite Rabs_mult.
rewrite Rabs_Rinv; discrR.
replace (Rabs 8) with 8.
replace 8 with 8; [ idtac | ring ].
rewrite Rinv_mult_distr; [ idtac | discrR | discrR ].
replace (2 * / Rabs (f2 x) * (Rabs eps * Rabs (f2 x) * (/ 2 * / 4))) with
 (Rabs eps * / 4 * (2 * / 2) * (Rabs (f2 x) * / Rabs (f2 x)));
 [ idtac | ring ].
replace (Rabs eps) with eps.
repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
ring.
symmetry  in |- *; apply Rabs_right; left; assumption.
symmetry  in |- *; apply Rabs_right; left; prove_sup.
Qed.

Lemma maj_term2 :
 forall (x h eps l1 alp_f2 alp_f2t2:R) (eps_f2:posreal) 
   (f2:R -> R),
   0 < eps ->
   f2 x <> 0 ->
   f2 (x + h) <> 0 ->
   (forall a:R,
      Rabs a < alp_f2t2 ->
      Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))) ->
   (forall a:R,
      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
   h <> 0 ->
   Rabs h < alp_f2t2 ->
   Rabs h < Rmin eps_f2 alp_f2 ->
   l1 <> 0 -> Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) < eps / 4.
intros.
assert (H8 := H3 h H6).
assert (H9 := H2 h H5).
apply Rle_lt_trans with
 (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).
rewrite Rabs_mult; apply Rmult_le_compat_l.
apply Rabs_pos.
rewrite <- (Rabs_Ropp (f2 x - f2 (x + h))); rewrite Ropp_minus_distr.
left; apply H9.
apply Rlt_le_trans with
 (Rabs (2 * (l1 / (f2 x * f2 x))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))).
apply Rmult_lt_compat_r.
apply Rabs_pos_lt.
unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
 try assumption || discrR.
red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
apply Rinv_neq_0_compat; apply prod_neq_R0; try assumption || discrR.
unfold Rdiv in |- *.
repeat rewrite Rinv_mult_distr; try assumption.
repeat rewrite Rabs_mult.
replace (Rabs 2) with 2.
rewrite (Rmult_comm 2).
replace (Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with
 (Rabs l1 * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2))); 
 [ idtac | ring ].
repeat apply Rmult_lt_compat_l.
apply Rabs_pos_lt; assumption.
apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.
repeat rewrite Rabs_Rinv; try assumption.
rewrite <- (Rmult_comm 2).
unfold Rdiv in H8; exact H8.
symmetry  in |- *; apply Rabs_right; left; prove_sup0.
right.
unfold Rsqr, Rdiv in |- *.
do 1 rewrite Rinv_mult_distr; try assumption || discrR.
do 1 rewrite Rinv_mult_distr; try assumption || discrR.
repeat rewrite Rabs_mult.
repeat rewrite Rabs_Rinv; try assumption || discrR.
replace (Rabs eps) with eps.
replace (Rabs 8) with 8.
replace (Rabs 2) with 2.
replace 8 with (4 * 2); [ idtac | ring ].
rewrite Rinv_mult_distr; discrR.
replace
 (2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) *
  (eps * (Rabs (f2 x) * Rabs (f2 x)) * (/ 4 * / 2 * / Rabs l1))) with
 (eps * / 4 * (Rabs l1 * / Rabs l1) * (Rabs (f2 x) * / Rabs (f2 x)) *
  (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ].
repeat rewrite <- Rinv_r_sym; try (apply Rabs_no_R0; assumption) || discrR.
ring.
symmetry  in |- *; apply Rabs_right; left; prove_sup0.
symmetry  in |- *; apply Rabs_right; left; prove_sup.
symmetry  in |- *; apply Rabs_right; left; assumption.
Qed.

Lemma maj_term3 :
 forall (x h eps l2 alp_f2:R) (eps_f2 alp_f2d:posreal) 
   (f1 f2:R -> R),
   0 < eps ->
   f2 x <> 0 ->
   f2 (x + h) <> 0 ->
   (forall h:R,
      h <> 0 ->
      Rabs h < alp_f2d ->
      Rabs ((f2 (x + h) - f2 x) / h - l2) <
      Rabs (Rsqr (f2 x) * eps / (8 * f1 x))) ->
   (forall a:R,
      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
   h <> 0 ->
   Rabs h < alp_f2d ->
   Rabs h < Rmin eps_f2 alp_f2 ->
   f1 x <> 0 ->
   Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) <
   eps / 4.
intros.
assert (H8 := H2 h H4 H5).
assert (H9 := H3 h H6).
apply Rle_lt_trans with
 (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))).
rewrite Rabs_mult.
apply Rmult_le_compat_l.
apply Rabs_pos.
left; apply H8.
apply Rlt_le_trans with
 (Rabs (2 * (f1 x / (f2 x * f2 x))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))).
apply Rmult_lt_compat_r.
apply Rabs_pos_lt.
unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
 try assumption.
red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption.
unfold Rdiv in |- *.
repeat rewrite Rinv_mult_distr; try assumption.
repeat rewrite Rabs_mult.
replace (Rabs 2) with 2.
rewrite (Rmult_comm 2).
replace (Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with
 (Rabs (f1 x) * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2))); 
 [ idtac | ring ].
repeat apply Rmult_lt_compat_l.
apply Rabs_pos_lt; assumption.
apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption.
repeat rewrite Rabs_Rinv; assumption || idtac.
rewrite <- (Rmult_comm 2).
unfold Rdiv in H9; exact H9.
symmetry  in |- *; apply Rabs_right; left; prove_sup0.
right.
unfold Rsqr, Rdiv in |- *.
rewrite Rinv_mult_distr; try assumption || discrR.
rewrite Rinv_mult_distr; try assumption || discrR.
repeat rewrite Rabs_mult.
repeat rewrite Rabs_Rinv; try assumption || discrR.
replace (Rabs eps) with eps.
replace (Rabs 8) with 8.
replace (Rabs 2) with 2.
replace 8 with (4 * 2); [ idtac | ring ].
rewrite Rinv_mult_distr; discrR.
replace
 (2 * (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) *
  (Rabs (f2 x) * Rabs (f2 x) * eps * (/ 4 * / 2 * / Rabs (f1 x)))) with
 (eps * / 4 * (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) *
  (Rabs (f1 x) * / Rabs (f1 x)) * (2 * / 2)); [ idtac | ring ].
repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
ring.
symmetry  in |- *; apply Rabs_right; left; prove_sup0.
symmetry  in |- *; apply Rabs_right; left; prove_sup.
symmetry  in |- *; apply Rabs_right; left; assumption.
Qed.

Lemma maj_term4 :
 forall (x h eps l2 alp_f2 alp_f2c:R) (eps_f2:posreal) 
   (f1 f2:R -> R),
   0 < eps ->
   f2 x <> 0 ->
   f2 (x + h) <> 0 ->
   (forall a:R,
      Rabs a < alp_f2c ->
      Rabs (f2 (x + a) - f2 x) <
      Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) ->
   (forall a:R,
      Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) ->
   h <> 0 ->
   Rabs h < alp_f2c ->
   Rabs h < Rmin eps_f2 alp_f2 ->
   f1 x <> 0 ->
   l2 <> 0 ->
   Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x)) <
   eps / 4.
intros.
assert (H9 := H2 h H5).
assert (H10 := H3 h H6).
apply Rle_lt_trans with
 (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) *
  Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
rewrite Rabs_mult.
apply Rmult_le_compat_l.
apply Rabs_pos.
left; apply H9.
apply Rlt_le_trans with
 (Rabs (2 * l2 * (f1 x / (Rsqr (f2 x) * f2 x))) *
  Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
apply Rmult_lt_compat_r.
apply Rabs_pos_lt.
unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0;
 assumption || idtac.
red in |- *; intro H11; rewrite H11 in H; elim (Rlt_irrefl _ H).
apply Rinv_neq_0_compat; apply prod_neq_R0.
apply prod_neq_R0.
discrR.
assumption.
assumption.
unfold Rdiv in |- *.
repeat rewrite Rinv_mult_distr;
 try assumption || (unfold Rsqr in |- *; apply prod_neq_R0; assumption).
repeat rewrite Rabs_mult.
replace (Rabs 2) with 2.
replace
 (2 * Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 x)))) with
 (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * (Rabs (/ f2 x) * 2))));
 [ idtac | ring ].
replace
 (Rabs l2 * Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))) with
 (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))));
 [ idtac | ring ].
repeat apply Rmult_lt_compat_l.
apply Rabs_pos_lt; assumption.
apply Rabs_pos_lt; assumption.
apply Rabs_pos_lt; apply Rinv_neq_0_compat; unfold Rsqr in |- *;
 apply prod_neq_R0; assumption.
repeat rewrite Rabs_Rinv; [ idtac | assumption | assumption ].
rewrite <- (Rmult_comm 2).
unfold Rdiv in H10; exact H10.
symmetry  in |- *; apply Rabs_right; left; prove_sup0.
right; unfold Rsqr, Rdiv in |- *.
rewrite Rinv_mult_distr; try assumption || discrR.
rewrite Rinv_mult_distr; try assumption || discrR.
rewrite Rinv_mult_distr; try assumption || discrR.
rewrite Rinv_mult_distr; try assumption || discrR.
repeat rewrite Rabs_mult.
repeat rewrite Rabs_Rinv; try assumption || discrR.
replace (Rabs eps) with eps.
replace (Rabs 8) with 8.
replace (Rabs 2) with 2.
replace 8 with (4 * 2); [ idtac | ring ].
rewrite Rinv_mult_distr; discrR.
replace
 (2 * Rabs l2 *
  (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) *
  (Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps *
   (/ 4 * / 2 * / Rabs (f1 x) * / Rabs l2))) with
 (eps * / 4 * (Rabs l2 * / Rabs l2) * (Rabs (f1 x) * / Rabs (f1 x)) *
  (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) *
  (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ].
repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption).
ring.
symmetry  in |- *; apply Rabs_right; left; prove_sup0.
symmetry  in |- *; apply Rabs_right; left; prove_sup.
symmetry  in |- *; apply Rabs_right; left; assumption.
apply prod_neq_R0; assumption || discrR.
apply prod_neq_R0; assumption.
Qed.

Lemma D_x_no_cond : forall x a:R, a <> 0 -> D_x no_cond x (x + a).
intros.
unfold D_x, no_cond in |- *.
split.
trivial.
apply Rminus_not_eq.
unfold Rminus in |- *.
rewrite Ropp_plus_distr.
rewrite <- Rplus_assoc.
rewrite Rplus_opp_r.
rewrite Rplus_0_l.
apply Ropp_neq_0_compat; assumption.
Qed.

Lemma Rabs_4 :
 forall a b c d:R, Rabs (a + b + c + d) <= Rabs a + Rabs b + Rabs c + Rabs d.
intros.
apply Rle_trans with (Rabs (a + b) + Rabs (c + d)).
replace (a + b + c + d) with (a + b + (c + d)); [ apply Rabs_triang | ring ].
apply Rle_trans with (Rabs a + Rabs b + Rabs (c + d)).
apply Rplus_le_compat_r.
apply Rabs_triang.
repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l.
apply Rabs_triang.
Qed.

Lemma Rlt_4 :
 forall a b c d e f g h:R,
   a < b -> c < d -> e < f -> g < h -> a + c + e + g < b + d + f + h.
intros; apply Rlt_trans with (b + c + e + g).
repeat apply Rplus_lt_compat_r; assumption.
repeat rewrite Rplus_assoc; apply Rplus_lt_compat_l.
apply Rlt_trans with (d + e + g).
rewrite Rplus_assoc; apply Rplus_lt_compat_r; assumption.
rewrite Rplus_assoc; apply Rplus_lt_compat_l; apply Rlt_trans with (f + g).
apply Rplus_lt_compat_r; assumption.
apply Rplus_lt_compat_l; assumption.
Qed.

Lemma Rmin_2 : forall a b c:R, a < b -> a < c -> a < Rmin b c.
intros; unfold Rmin in |- *; case (Rle_dec b c); intro; assumption.
Qed.

Lemma quadruple : forall x:R, 4 * x = x + x + x + x.
intro; ring.
Qed.

Lemma quadruple_var : forall x:R, x = x / 4 + x / 4 + x / 4 + x / 4.
intro; rewrite <- quadruple.
unfold Rdiv in |- *; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; discrR.
reflexivity.
Qed.

(**********)
Lemma continuous_neq_0 :
 forall (f:R -> R) (x0:R),
   continuity_pt f x0 ->
   f x0 <> 0 ->
    exists eps : posreal, (forall h:R, Rabs h < eps -> f (x0 + h) <> 0).
intros; unfold continuity_pt in H; unfold continue_in in H;
 unfold limit1_in in H; unfold limit_in in H; elim (H (Rabs (f x0 / 2))).
intros; elim H1; intros.
exists (mkposreal x H2).
intros; assert (H5 := H3 (x0 + h)).
cut
 (dist R_met (x0 + h) x0 < x ->
  dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)).
unfold dist in |- *; simpl in |- *; unfold R_dist in |- *;
 replace (x0 + h - x0) with h.
intros; assert (H7 := H6 H4).
red in |- *; intro.
rewrite H8 in H7; unfold Rminus in H7; rewrite Rplus_0_l in H7;
 rewrite Rabs_Ropp in H7; unfold Rdiv in H7; rewrite Rabs_mult in H7;
 pattern (Rabs (f x0)) at 1 in H7; rewrite <- Rmult_1_r in H7.
cut (0 < Rabs (f x0)).
intro; assert (H10 := Rmult_lt_reg_l _ _ _ H9 H7).
cut (Rabs (/ 2) = / 2).
assert (Hyp : 0 < 2).
prove_sup0.
intro; rewrite H11 in H10; assert (H12 := Rmult_lt_compat_l 2 _ _ Hyp H10);
 rewrite Rmult_1_r in H12; rewrite <- Rinv_r_sym in H12; 
 [ idtac | discrR ].
cut (IZR 1 < IZR 2).
unfold IZR in |- *; unfold INR, nat_of_P in |- *; simpl in |- *; intro;
 elim (Rlt_irrefl 1 (Rlt_trans _ _ _ H13 H12)).
apply IZR_lt; omega.
unfold Rabs in |- *; case (Rcase_abs (/ 2)); intro.
assert (Hyp : 0 < 2).
prove_sup0.
assert (H11 := Rmult_lt_compat_l 2 _ _ Hyp r); rewrite Rmult_0_r in H11;
 rewrite <- Rinv_r_sym in H11; [ idtac | discrR ].
elim (Rlt_irrefl 0 (Rlt_trans _ _ _ Rlt_0_1 H11)).
reflexivity.
apply (Rabs_pos_lt _ H0).
ring.
assert (H6 := Req_dec x0 (x0 + h)); elim H6; intro.
intro; rewrite <- H7; unfold dist, R_met in |- *; unfold R_dist in |- *;
 unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
 apply Rabs_pos_lt.
unfold Rdiv in |- *; apply prod_neq_R0;
 [ assumption | apply Rinv_neq_0_compat; discrR ].
intro; apply H5.
split.
unfold D_x, no_cond in |- *.
split; trivial || assumption.
assumption.
change (0 < Rabs (f x0 / 2)) in |- *.
apply Rabs_pos_lt; unfold Rdiv in |- *; apply prod_neq_R0.
assumption.
apply Rinv_neq_0_compat; discrR.
Qed.