Imported Upstream version 8.4dfsgupstream/8.4dfsg

64 files changed, 9663 insertions, 6504 deletions

diff --git a/theories/Reals/Alembert.v b/theories/Reals/Alembert.v
index 18612a68..13b33301 100644
--- a/theories/Reals/Alembert.v
+++ b/theories/Reals/Alembert.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -13,7 +13,7 @@ Require Import SeqProp.
 Require Import PartSum.
 Require Import Max.
 
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (***************************************************)
 (* Various versions of the criterion of D'Alembert *)
@@ -31,23 +31,23 @@ Proof.
       { l:R | Un_cv (fun N:nat => sum_f_R0 An N) l }).
   intro X; apply X.
   apply completeness.
-  unfold Un_cv in H0; unfold bound in |- *; cut (0 < / 2);
+  unfold Un_cv in H0; unfold bound; cut (0 < / 2);
     [ intro | apply Rinv_0_lt_compat; prove_sup0 ].
   elim (H0 (/ 2) H1); intros.
   exists (sum_f_R0 An x + 2 * An (S x)).
-  unfold is_upper_bound in |- *; intros; unfold EUn in H3; elim H3; intros.
+  unfold is_upper_bound; intros; unfold EUn in H3; elim H3; intros.
   rewrite H4; assert (H5 := lt_eq_lt_dec x1 x).
   elim H5; intros.
   elim a; intro.
   replace (sum_f_R0 An x) with
     (sum_f_R0 An x1 + sum_f_R0 (fun i:nat => An (S x1 + i)%nat) (x - S x1)).
-  pattern (sum_f_R0 An x1) at 1 in |- *; rewrite <- Rplus_0_r;
+  pattern (sum_f_R0 An x1) at 1; rewrite <- Rplus_0_r;
     rewrite Rplus_assoc; apply Rplus_le_compat_l.
   left; apply Rplus_lt_0_compat.
   apply tech1; intros; apply H.
   apply Rmult_lt_0_compat; [ prove_sup0 | apply H ].
-  symmetry  in |- *; apply tech2; assumption.
-  rewrite b; pattern (sum_f_R0 An x) at 1 in |- *; rewrite <- Rplus_0_r;
+  symmetry ; apply tech2; assumption.
+  rewrite b; pattern (sum_f_R0 An x) at 1; rewrite <- Rplus_0_r;
     apply Rplus_le_compat_l.
   left; apply Rmult_lt_0_compat; [ prove_sup0 | apply H ].
   replace (sum_f_R0 An x1) with
@@ -64,14 +64,14 @@ Proof.
   left; apply H.
   rewrite tech3.
   replace (1 - / 2) with (/ 2).
-  unfold Rdiv in |- *; rewrite Rinv_involutive.
-  pattern 2 at 3 in |- *; rewrite <- Rmult_1_r; rewrite <- (Rmult_comm 2);
+  unfold Rdiv; rewrite Rinv_involutive.
+  pattern 2 at 3; rewrite <- Rmult_1_r; rewrite <- (Rmult_comm 2);
     apply Rmult_le_compat_l.
   left; prove_sup0.
   left; apply Rplus_lt_reg_r with ((/ 2) ^ S (x1 - S x)).
   replace ((/ 2) ^ S (x1 - S x) + (1 - (/ 2) ^ S (x1 - S x))) with 1;
     [ idtac | ring ].
-  rewrite <- (Rplus_comm 1); pattern 1 at 1 in |- *; rewrite <- Rplus_0_r;
+  rewrite <- (Rplus_comm 1); pattern 1 at 1; rewrite <- Rplus_0_r;
     apply Rplus_lt_compat_l.
   apply pow_lt; apply Rinv_0_lt_compat; prove_sup0.
   discrR.
@@ -80,14 +80,14 @@ Proof.
   ring.
   discrR.
   discrR.
-  pattern 1 at 3 in |- *; replace 1 with (/ 1);
+  pattern 1 at 3; replace 1 with (/ 1);
     [ apply tech7; discrR | apply Rinv_1 ].
   replace (An (S x)) with (An (S x + 0)%nat).
   apply (tech6 (fun i:nat => An (S x + i)%nat) (/ 2)).
   left; apply Rinv_0_lt_compat; prove_sup0.
   intro; cut (forall n:nat, (n >= x)%nat -> An (S n) < / 2 * An n).
   intro; replace (S x + S i)%nat with (S (S x + i)).
-  apply H6; unfold ge in |- *; apply tech8.
+  apply H6; unfold ge; apply tech8.
   apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR; ring.
   intros; unfold R_dist in H2; apply Rmult_lt_reg_l with (/ An n).
   apply Rinv_0_lt_compat; apply H.
@@ -96,20 +96,20 @@ Proof.
   rewrite Rmult_1_r;
     replace (An (S n) * / An n) with (Rabs (Rabs (An (S n) / An n) - 0)).
   apply H2; assumption.
-  unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r;
+  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r;
     rewrite Rabs_Rabsolu; rewrite Rabs_right.
-  unfold Rdiv in |- *; reflexivity.
-  left; unfold Rdiv in |- *; change (0 < An (S n) * / An n) in |- *;
+  unfold Rdiv; reflexivity.
+  left; unfold Rdiv; change (0 < An (S n) * / An n);
     apply Rmult_lt_0_compat; [ apply H | apply Rinv_0_lt_compat; apply H ].
-  red in |- *; intro; assert (H8 := H n); rewrite H7 in H8;
+  red; intro; assert (H8 := H n); rewrite H7 in H8;
     elim (Rlt_irrefl _ H8).
   replace (S x + 0)%nat with (S x); [ reflexivity | ring ].
-  symmetry  in |- *; apply tech2; assumption.
-  exists (sum_f_R0 An 0); unfold EUn in |- *; exists 0%nat; reflexivity.
+  symmetry ; apply tech2; assumption.
+  exists (sum_f_R0 An 0); unfold EUn; exists 0%nat; reflexivity.
   intro X; elim X; intros.
   exists x; apply Un_cv_crit_lub;
-    [ unfold Un_growing in |- *; intro; rewrite tech5;
-      pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r;
+    [ unfold Un_growing; intro; rewrite tech5;
+      pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r;
 	apply Rplus_le_compat_l; left; apply H
       | apply p ].
 Defined.
@@ -131,14 +131,14 @@ Proof.
   assert (H6 := Alembert_C1 Wn H2 H4).
   elim H5; intros.
   elim H6; intros.
-  exists (x - x0); unfold Un_cv in |- *; unfold Un_cv in p;
+  exists (x - x0); unfold Un_cv; unfold Un_cv in p;
     unfold Un_cv in p0; intros; cut (0 < eps / 2).
   intro; elim (p (eps / 2) H8); clear p; intros.
   elim (p0 (eps / 2) H8); clear p0; intros.
   set (N := max x1 x2).
   exists N; intros;
     replace (sum_f_R0 An n) with (sum_f_R0 Vn n - sum_f_R0 Wn n).
-  unfold R_dist in |- *;
+  unfold R_dist;
     replace (sum_f_R0 Vn n - sum_f_R0 Wn n - (x - x0)) with
       (sum_f_R0 Vn n - x + - (sum_f_R0 Wn n - x0)); [ idtac | ring ];
       apply Rle_lt_trans with
@@ -146,29 +146,29 @@ Proof.
   apply Rabs_triang.
   rewrite Rabs_Ropp; apply Rlt_le_trans with (eps / 2 + eps / 2).
   apply Rplus_lt_compat.
-  unfold R_dist in H9; apply H9; unfold ge in |- *; apply le_trans with N;
-    [ unfold N in |- *; apply le_max_l | assumption ].
-  unfold R_dist in H10; apply H10; unfold ge in |- *; apply le_trans with N;
-    [ unfold N in |- *; apply le_max_r | assumption ].
-  right; symmetry  in |- *; apply double_var.
-  symmetry  in |- *; apply tech11; intro; unfold Vn, Wn in |- *;
-    unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ 2));
+  unfold R_dist in H9; apply H9; unfold ge; apply le_trans with N;
+    [ unfold N; apply le_max_l | assumption ].
+  unfold R_dist in H10; apply H10; unfold ge; apply le_trans with N;
+    [ unfold N; apply le_max_r | assumption ].
+  right; symmetry ; apply double_var.
+  symmetry ; apply tech11; intro; unfold Vn, Wn;
+    unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ 2));
       apply Rmult_eq_reg_l with 2.
   rewrite Rmult_minus_distr_l; repeat rewrite <- Rmult_assoc;
     rewrite <- Rinv_r_sym.
   ring.
   discrR.
   discrR.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   cut (forall n:nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)).
   intro; cut (forall n:nat, / Wn n <= 2 * / Rabs (An n)).
   intro; cut (forall n:nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)).
-  intro; unfold Un_cv in |- *; intros; unfold Un_cv in H0; cut (0 < eps / 3).
+  intro; unfold Un_cv; intros; unfold Un_cv in H0; cut (0 < eps / 3).
   intro; elim (H0 (eps / 3) H8); intros.
   exists x; intros.
   assert (H11 := H9 n H10).
-  unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
+  unfold R_dist; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold R_dist in H11;
       unfold Rminus in H11; rewrite Ropp_0 in H11; rewrite Rplus_0_r in H11;
 	rewrite Rabs_Rabsolu in H11; rewrite Rabs_right.
@@ -179,13 +179,13 @@ Proof.
   rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ];
     rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H11;
       exact H11.
-  left; change (0 < Wn (S n) / Wn n) in |- *; unfold Rdiv in |- *;
+  left; change (0 < Wn (S n) / Wn n); unfold Rdiv;
     apply Rmult_lt_0_compat.
   apply H2.
   apply Rinv_0_lt_compat; apply H2.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
-  intro; unfold Rdiv in |- *; rewrite Rabs_mult; rewrite <- Rmult_assoc;
+  intro; unfold Rdiv; rewrite Rabs_mult; rewrite <- Rmult_assoc;
     replace 3 with (2 * (3 * / 2));
       [ idtac | rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR ];
       apply Rle_trans with (Wn (S n) * 2 * / Rabs (An n)).
@@ -218,32 +218,32 @@ Proof.
   rewrite Rmult_1_l; elim (H4 n); intros; assumption.
   discrR.
   apply Rabs_no_R0; apply H.
-  red in |- *; intro; assert (H6 := H2 n); rewrite H5 in H6;
+  red; intro; assert (H6 := H2 n); rewrite H5 in H6;
     elim (Rlt_irrefl _ H6).
   intro; split.
-  unfold Wn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+  unfold Wn; unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
     apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; prove_sup0.
-  pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; rewrite double;
-    unfold Rminus in |- *; rewrite Rplus_assoc; apply Rplus_le_compat_l.
+  pattern (Rabs (An n)) at 1; rewrite <- Rplus_0_r; rewrite double;
+    unfold Rminus; rewrite Rplus_assoc; apply Rplus_le_compat_l.
   apply Rplus_le_reg_l with (An n).
   rewrite Rplus_0_r; rewrite (Rplus_comm (An n)); rewrite Rplus_assoc;
     rewrite Rplus_opp_l; rewrite Rplus_0_r; apply RRle_abs.
-  unfold Wn in |- *; unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2));
+  unfold Wn; unfold Rdiv; repeat rewrite <- (Rmult_comm (/ 2));
     repeat rewrite Rmult_assoc; apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; prove_sup0.
-  unfold Rminus in |- *; rewrite double;
+  unfold Rminus; rewrite double;
     replace (3 * Rabs (An n)) with (Rabs (An n) + Rabs (An n) + Rabs (An n));
       [ idtac | ring ]; repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l.
   rewrite <- Rabs_Ropp; apply RRle_abs.
   cut (forall n:nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)).
   intro; cut (forall n:nat, / Vn n <= 2 * / Rabs (An n)).
   intro; cut (forall n:nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)).
-  intro; unfold Un_cv in |- *; intros; unfold Un_cv in H1; cut (0 < eps / 3).
+  intro; unfold Un_cv; intros; unfold Un_cv in H1; cut (0 < eps / 3).
   intro; elim (H0 (eps / 3) H7); intros.
   exists x; intros.
   assert (H10 := H8 n H9).
-  unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
+  unfold R_dist; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold R_dist in H10;
       unfold Rminus in H10; rewrite Ropp_0 in H10; rewrite Rplus_0_r in H10;
 	rewrite Rabs_Rabsolu in H10; rewrite Rabs_right.
@@ -254,13 +254,13 @@ Proof.
   rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ];
     rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H10;
       exact H10.
-  left; change (0 < Vn (S n) / Vn n) in |- *; unfold Rdiv in |- *;
+  left; change (0 < Vn (S n) / Vn n); unfold Rdiv;
     apply Rmult_lt_0_compat.
   apply H1.
   apply Rinv_0_lt_compat; apply H1.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
-  intro; unfold Rdiv in |- *; rewrite Rabs_mult; rewrite <- Rmult_assoc;
+  intro; unfold Rdiv; rewrite Rabs_mult; rewrite <- Rmult_assoc;
     replace 3 with (2 * (3 * / 2));
       [ idtac | rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR ];
       apply Rle_trans with (Vn (S n) * 2 * / Rabs (An n)).
@@ -293,44 +293,44 @@ Proof.
   rewrite Rmult_1_l; elim (H3 n); intros; assumption.
   discrR.
   apply Rabs_no_R0; apply H.
-  red in |- *; intro; assert (H5 := H1 n); rewrite H4 in H5;
+  red; intro; assert (H5 := H1 n); rewrite H4 in H5;
     elim (Rlt_irrefl _ H5).
   intro; split.
-  unfold Vn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+  unfold Vn; unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
     apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; prove_sup0.
-  pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; rewrite double;
+  pattern (Rabs (An n)) at 1; rewrite <- Rplus_0_r; rewrite double;
     rewrite Rplus_assoc; apply Rplus_le_compat_l.
   apply Rplus_le_reg_l with (- An n); rewrite Rplus_0_r;
     rewrite <- (Rplus_comm (An n)); rewrite <- Rplus_assoc;
       rewrite Rplus_opp_l; rewrite Rplus_0_l; rewrite <- Rabs_Ropp;
 	apply RRle_abs.
-  unfold Vn in |- *; unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2));
+  unfold Vn; unfold Rdiv; repeat rewrite <- (Rmult_comm (/ 2));
     repeat rewrite Rmult_assoc; apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; prove_sup0.
-  unfold Rminus in |- *; rewrite double;
+  unfold Rminus; rewrite double;
     replace (3 * Rabs (An n)) with (Rabs (An n) + Rabs (An n) + Rabs (An n));
       [ idtac | ring ]; repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l;
 	apply RRle_abs.
-  intro; unfold Wn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_0_r (/ 2));
+  intro; unfold Wn; unfold Rdiv; rewrite <- (Rmult_0_r (/ 2));
     rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l.
   apply Rinv_0_lt_compat; prove_sup0.
-  apply Rplus_lt_reg_r with (An n); rewrite Rplus_0_r; unfold Rminus in |- *;
+  apply Rplus_lt_reg_r with (An n); rewrite Rplus_0_r; unfold Rminus;
     rewrite (Rplus_comm (An n)); rewrite Rplus_assoc;
       rewrite Rplus_opp_l; rewrite Rplus_0_r;
 	apply Rle_lt_trans with (Rabs (An n)).
   apply RRle_abs.
-  rewrite double; pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r;
+  rewrite double; pattern (Rabs (An n)) at 1; rewrite <- Rplus_0_r;
     apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H.
-  intro; unfold Vn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_0_r (/ 2));
+  intro; unfold Vn; unfold Rdiv; rewrite <- (Rmult_0_r (/ 2));
     rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l.
   apply Rinv_0_lt_compat; prove_sup0.
-  apply Rplus_lt_reg_r with (- An n); rewrite Rplus_0_r; unfold Rminus in |- *;
+  apply Rplus_lt_reg_r with (- An n); rewrite Rplus_0_r; unfold Rminus;
     rewrite (Rplus_comm (- An n)); rewrite Rplus_assoc;
       rewrite Rplus_opp_r; rewrite Rplus_0_r;
 	apply Rle_lt_trans with (Rabs (An n)).
   rewrite <- Rabs_Ropp; apply RRle_abs.
-  rewrite double; pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r;
+  rewrite double; pattern (Rabs (An n)) at 1; rewrite <- Rplus_0_r;
     apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H.
 Defined.
 
@@ -347,11 +347,11 @@ Proof.
   intro; assert (H4 := Alembert_C2 Bn H2 H3).
   elim H4; intros.
   exists x0; unfold Bn in p; apply tech12; assumption.
-  unfold Un_cv in |- *; intros; unfold Un_cv in H1; cut (0 < eps / Rabs x).
+  unfold Un_cv; intros; unfold Un_cv in H1; cut (0 < eps / Rabs x).
   intro; elim (H1 (eps / Rabs x) H4); intros.
-  exists x0; intros; unfold R_dist in |- *; unfold Rminus in |- *;
+  exists x0; intros; unfold R_dist; unfold Rminus;
     rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu;
-      unfold Bn in |- *;
+      unfold Bn;
 	replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x).
   rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs x).
   apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
@@ -360,22 +360,22 @@ Proof.
   rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H5;
     replace (Rabs (An (S n) / An n)) with (R_dist (Rabs (An (S n) * / An n)) 0).
   apply H5; assumption.
-  unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
-    rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold Rdiv in |- *;
+  unfold R_dist; unfold Rminus; rewrite Ropp_0;
+    rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold Rdiv;
       reflexivity.
   apply Rabs_no_R0; assumption.
   replace (S n) with (n + 1)%nat; [ idtac | ring ]; rewrite pow_add;
-    unfold Rdiv in |- *; rewrite Rinv_mult_distr.
+    unfold Rdiv; rewrite Rinv_mult_distr.
   replace (An (n + 1)%nat * (x ^ n * x ^ 1) * (/ An n * / x ^ n)) with
     (An (n + 1)%nat * x ^ 1 * / An n * (x ^ n * / x ^ n));
     [ idtac | ring ]; rewrite <- Rinv_r_sym.
-  simpl in |- *; ring.
+  simpl; ring.
   apply pow_nonzero; assumption.
   apply H0.
   apply pow_nonzero; assumption.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ].
-  intro; unfold Bn in |- *; apply prod_neq_R0;
+  intro; unfold Bn; apply prod_neq_R0;
     [ apply H0 | apply pow_nonzero; assumption ].
 Defined.
 
@@ -383,14 +383,14 @@ Lemma AlembertC3_step2 :
   forall (An:nat -> R) (x:R), x = 0 -> { l:R | Pser An x l }.
 Proof.
   intros; exists (An 0%nat).
-  unfold Pser in |- *; unfold infinite_sum in |- *; intros; exists 0%nat; intros;
+  unfold Pser; unfold infinite_sum; intros; exists 0%nat; intros;
     replace (sum_f_R0 (fun n0:nat => An n0 * x ^ n0) n) with (An 0%nat).
-  unfold R_dist in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
+  unfold R_dist; unfold Rminus; rewrite Rplus_opp_r;
     rewrite Rabs_R0; assumption.
   induction  n as [| n Hrecn].
-  simpl in |- *; ring.
+  simpl; ring.
   rewrite tech5; rewrite Hrecn;
-    [ rewrite H; simpl in |- *; ring | unfold ge in |- *; apply le_O_n ].
+    [ rewrite H; simpl; ring | unfold ge; apply le_O_n ].
 Qed.
 
 (** A useful criterion of convergence for power series *)
@@ -404,11 +404,11 @@ Proof.
   elim s; intro.
   cut (x <> 0).
   intro; apply AlembertC3_step1; assumption.
-  red in |- *; intro; rewrite H1 in a; elim (Rlt_irrefl _ a).
+  red; intro; rewrite H1 in a; elim (Rlt_irrefl _ a).
   apply AlembertC3_step2; assumption.
   cut (x <> 0).
   intro; apply AlembertC3_step1; assumption.
-  red in |- *; intro; rewrite H1 in r; elim (Rlt_irrefl _ r).
+  red; intro; rewrite H1 in r; elim (Rlt_irrefl _ r).
 Defined.
 
 Lemma Alembert_C4 :
@@ -428,8 +428,8 @@ Proof.
   elim H1; intros.
   elim H2; intros.
   elim H4; intros.
-  unfold bound in |- *; exists (sum_f_R0 An x0 + / (1 - x) * An (S x0)).
-  unfold is_upper_bound in |- *; intros; unfold EUn in H6.
+  unfold bound; exists (sum_f_R0 An x0 + / (1 - x) * An (S x0)).
+  unfold is_upper_bound; intros; unfold EUn in H6.
   elim H6; intros.
   rewrite H7.
   assert (H8 := lt_eq_lt_dec x2 x0).
@@ -437,7 +437,7 @@ Proof.
   elim a; intro.
   replace (sum_f_R0 An x0) with
     (sum_f_R0 An x2 + sum_f_R0 (fun i:nat => An (S x2 + i)%nat) (x0 - S x2)).
-  pattern (sum_f_R0 An x2) at 1 in |- *; rewrite <- Rplus_0_r.
+  pattern (sum_f_R0 An x2) at 1; rewrite <- Rplus_0_r.
   rewrite Rplus_assoc; apply Rplus_le_compat_l.
   left; apply Rplus_lt_0_compat.
   apply tech1.
@@ -446,8 +446,8 @@ Proof.
   apply Rinv_0_lt_compat; apply Rplus_lt_reg_r with x; rewrite Rplus_0_r;
     replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ].
   apply H.
-  symmetry  in |- *; apply tech2; assumption.
-  rewrite b; pattern (sum_f_R0 An x0) at 1 in |- *; rewrite <- Rplus_0_r;
+  symmetry ; apply tech2; assumption.
+  rewrite b; pattern (sum_f_R0 An x0) at 1; rewrite <- Rplus_0_r;
     apply Rplus_le_compat_l.
   left; apply Rmult_lt_0_compat.
   apply Rinv_0_lt_compat; apply Rplus_lt_reg_r with x; rewrite Rplus_0_r;
@@ -465,7 +465,7 @@ Proof.
   rewrite <- (Rmult_comm (An (S x0))); apply Rmult_le_compat_l.
   left; apply H.
   rewrite tech3.
-  unfold Rdiv in |- *; apply Rmult_le_reg_l with (1 - x).
+  unfold Rdiv; apply Rmult_le_reg_l with (1 - x).
   apply Rplus_lt_reg_r with x; rewrite Rplus_0_r.
   replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ].
   do 2 rewrite (Rmult_comm (1 - x)).
@@ -473,17 +473,17 @@ Proof.
   rewrite Rmult_1_r; apply Rplus_le_reg_l with (x ^ S (x2 - S x0)).
   replace (x ^ S (x2 - S x0) + (1 - x ^ S (x2 - S x0))) with 1;
     [ idtac | ring ].
-  rewrite <- (Rplus_comm 1); pattern 1 at 1 in |- *; rewrite <- Rplus_0_r;
+  rewrite <- (Rplus_comm 1); pattern 1 at 1; rewrite <- Rplus_0_r;
     apply Rplus_le_compat_l.
   left; apply pow_lt.
   apply Rle_lt_trans with k.
   elim Hyp; intros; assumption.
   elim H3; intros; assumption.
   apply Rminus_eq_contra.
-  red in |- *; intro.
+  red; intro.
   elim H3; intros.
   rewrite H10 in H12; elim (Rlt_irrefl _ H12).
-  red in |- *; intro.
+  red; intro.
   elim H3; intros.
   rewrite H10 in H12; elim (Rlt_irrefl _ H12).
   replace (An (S x0)) with (An (S x0 + 0)%nat).
@@ -496,7 +496,7 @@ Proof.
   intro.
   replace (S x0 + S i)%nat with (S (S x0 + i)).
   apply H9.
-  unfold ge in |- *.
+  unfold ge.
   apply tech8.
   apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR;
     ring.
@@ -510,21 +510,21 @@ Proof.
   replace (An (S n) * / An n) with (Rabs (An (S n) / An n)).
   apply H5; assumption.
   rewrite Rabs_right.
-  unfold Rdiv in |- *; reflexivity.
-  left; unfold Rdiv in |- *; change (0 < An (S n) * / An n) in |- *;
+  unfold Rdiv; reflexivity.
+  left; unfold Rdiv; change (0 < An (S n) * / An n);
     apply Rmult_lt_0_compat.
   apply H.
   apply Rinv_0_lt_compat; apply H.
-  red in |- *; intro.
+  red; intro.
   assert (H11 := H n).
   rewrite H10 in H11; elim (Rlt_irrefl _ H11).
   replace (S x0 + 0)%nat with (S x0); [ reflexivity | ring ].
-  symmetry  in |- *; apply tech2; assumption.
-  exists (sum_f_R0 An 0); unfold EUn in |- *; exists 0%nat; reflexivity.
+  symmetry ; apply tech2; assumption.
+  exists (sum_f_R0 An 0); unfold EUn; exists 0%nat; reflexivity.
   intro X; elim X; intros.
   exists x; apply Un_cv_crit_lub;
-    [ unfold Un_growing in |- *; intro; rewrite tech5;
-      pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r;
+    [ unfold Un_growing; intro; rewrite tech5;
+      pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r;
 	apply Rplus_le_compat_l; left; apply H
       | apply p ].
 Qed.
@@ -551,9 +551,9 @@ Proof.
   apply (Alembert_C4 (fun i:nat => Rabs (An i)) k).
   assumption.
   intro; apply Rabs_pos_lt; apply H0.
-  unfold Un_cv in |- *.
+  unfold Un_cv.
   unfold Un_cv in H1.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   intros.
   elim (H1 eps H2); intros.
   exists x; intros.
@@ -590,22 +590,22 @@ Lemma Alembert_C6 :
   elim s; intro.
   eapply Alembert_C5 with (k * Rabs x).
   split.
-  unfold Rdiv in |- *; apply Rmult_le_pos.
+  unfold Rdiv; apply Rmult_le_pos.
   left; assumption.
   left; apply Rabs_pos_lt.
-  red in |- *; intro; rewrite H3 in a; elim (Rlt_irrefl _ a).
+  red; intro; rewrite H3 in a; elim (Rlt_irrefl _ a).
   apply Rmult_lt_reg_l with (/ k).
   apply Rinv_0_lt_compat; assumption.
   rewrite <- Rmult_assoc.
   rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l.
   rewrite Rmult_1_r; assumption.
-  red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H3 in H; elim (Rlt_irrefl _ H).
   intro; apply prod_neq_R0.
   apply H0.
   apply pow_nonzero.
-  red in |- *; intro; rewrite H3 in a; elim (Rlt_irrefl _ a).
-  unfold Un_cv in |- *; unfold Un_cv in H1.
+  red; intro; rewrite H3 in a; elim (Rlt_irrefl _ a).
+  unfold Un_cv; unfold Un_cv in H1.
   intros.
   cut (0 < eps / Rabs x).
   intro.
@@ -613,7 +613,7 @@ Lemma Alembert_C6 :
   exists x0.
   intros.
   replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x).
-  unfold R_dist in |- *.
+  unfold R_dist.
   rewrite Rabs_mult.
   replace (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) with
     (Rabs x * (Rabs (An (S n) / An n) - k)); [ idtac | ring ].
@@ -621,18 +621,18 @@ Lemma Alembert_C6 :
   rewrite Rabs_Rabsolu.
   apply Rmult_lt_reg_l with (/ Rabs x).
   apply Rinv_0_lt_compat; apply Rabs_pos_lt.
-  red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a).
+  red; intro; rewrite H7 in a; elim (Rlt_irrefl _ a).
   rewrite <- Rmult_assoc.
   rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l.
   rewrite <- (Rmult_comm eps).
   unfold R_dist in H5.
-  unfold Rdiv in |- *; unfold Rdiv in H5; apply H5; assumption.
+  unfold Rdiv; unfold Rdiv in H5; apply H5; assumption.
   apply Rabs_no_R0.
-  red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a).
-  unfold Rdiv in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ].
+  red; intro; rewrite H7 in a; elim (Rlt_irrefl _ a).
+  unfold Rdiv; replace (S n) with (n + 1)%nat; [ idtac | ring ].
   rewrite pow_add.
-  simpl in |- *.
+  simpl.
   rewrite Rmult_1_r.
   rewrite Rinv_mult_distr.
   replace (An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)) with
@@ -641,46 +641,46 @@ Lemma Alembert_C6 :
   rewrite <- Rinv_r_sym.
   rewrite Rmult_1_r; reflexivity.
   apply pow_nonzero.
-  red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a).
+  red; intro; rewrite H7 in a; elim (Rlt_irrefl _ a).
   apply H0.
   apply pow_nonzero.
-  red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  red; intro; rewrite H7 in a; elim (Rlt_irrefl _ a).
+  unfold Rdiv; apply Rmult_lt_0_compat.
   assumption.
   apply Rinv_0_lt_compat; apply Rabs_pos_lt.
-  red in |- *; intro H7; rewrite H7 in a; elim (Rlt_irrefl _ a).
+  red; intro H7; rewrite H7 in a; elim (Rlt_irrefl _ a).
   exists (An 0%nat).
-  unfold Un_cv in |- *.
+  unfold Un_cv.
   intros.
   exists 0%nat.
   intros.
-  unfold R_dist in |- *.
+  unfold R_dist.
   replace (sum_f_R0 (fun i:nat => An i * x ^ i) n) with (An 0%nat).
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
   induction  n as [| n Hrecn].
-  simpl in |- *; ring.
+  simpl; ring.
   rewrite tech5.
   rewrite <- Hrecn.
-  rewrite b; simpl in |- *; ring.
-  unfold ge in |- *; apply le_O_n.
+  rewrite b; simpl; ring.
+  unfold ge; apply le_O_n.
   eapply Alembert_C5 with (k * Rabs x).
   split.
-  unfold Rdiv in |- *; apply Rmult_le_pos.
+  unfold Rdiv; apply Rmult_le_pos.
   left; assumption.
   left; apply Rabs_pos_lt.
-  red in |- *; intro; rewrite H3 in r; elim (Rlt_irrefl _ r).
+  red; intro; rewrite H3 in r; elim (Rlt_irrefl _ r).
   apply Rmult_lt_reg_l with (/ k).
   apply Rinv_0_lt_compat; assumption.
   rewrite <- Rmult_assoc.
   rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l.
   rewrite Rmult_1_r; assumption.
-  red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H3 in H; elim (Rlt_irrefl _ H).
   intro; apply prod_neq_R0.
   apply H0.
   apply pow_nonzero.
-  red in |- *; intro; rewrite H3 in r; elim (Rlt_irrefl _ r).
-  unfold Un_cv in |- *; unfold Un_cv in H1.
+  red; intro; rewrite H3 in r; elim (Rlt_irrefl _ r).
+  unfold Un_cv; unfold Un_cv in H1.
   intros.
   cut (0 < eps / Rabs x).
   intro.
@@ -688,7 +688,7 @@ Lemma Alembert_C6 :
   exists x0.
   intros.
   replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x).
-  unfold R_dist in |- *.
+  unfold R_dist.
   rewrite Rabs_mult.
   replace (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) with
     (Rabs x * (Rabs (An (S n) / An n) - k)); [ idtac | ring ].
@@ -696,18 +696,18 @@ Lemma Alembert_C6 :
   rewrite Rabs_Rabsolu.
   apply Rmult_lt_reg_l with (/ Rabs x).
   apply Rinv_0_lt_compat; apply Rabs_pos_lt.
-  red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r).
+  red; intro; rewrite H7 in r; elim (Rlt_irrefl _ r).
   rewrite <- Rmult_assoc.
   rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l.
   rewrite <- (Rmult_comm eps).
   unfold R_dist in H5.
-  unfold Rdiv in |- *; unfold Rdiv in H5; apply H5; assumption.
+  unfold Rdiv; unfold Rdiv in H5; apply H5; assumption.
   apply Rabs_no_R0.
-  red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r).
-  unfold Rdiv in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ].
+  red; intro; rewrite H7 in r; elim (Rlt_irrefl _ r).
+  unfold Rdiv; replace (S n) with (n + 1)%nat; [ idtac | ring ].
   rewrite pow_add.
-  simpl in |- *.
+  simpl.
   rewrite Rmult_1_r.
   rewrite Rinv_mult_distr.
   replace (An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)) with
@@ -716,12 +716,12 @@ Lemma Alembert_C6 :
   rewrite <- Rinv_r_sym.
   rewrite Rmult_1_r; reflexivity.
   apply pow_nonzero.
-  red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r).
+  red; intro; rewrite H7 in r; elim (Rlt_irrefl _ r).
   apply H0.
   apply pow_nonzero.
-  red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  red; intro; rewrite H7 in r; elim (Rlt_irrefl _ r).
+  unfold Rdiv; apply Rmult_lt_0_compat.
   assumption.
   apply Rinv_0_lt_compat; apply Rabs_pos_lt.
-  red in |- *; intro H7; rewrite H7 in r; elim (Rlt_irrefl _ r).
+  red; intro H7; rewrite H7 in r; elim (Rlt_irrefl _ r).
 Qed.
diff --git a/theories/Reals/AltSeries.v b/theories/Reals/AltSeries.v
index 07a26929..69f29781 100644
--- a/theories/Reals/AltSeries.v
+++ b/theories/Reals/AltSeries.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -12,7 +12,7 @@ Require Import Rseries.
 Require Import SeqProp.
 Require Import PartSum.
 Require Import Max.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (**********)
 (** * Formalization of alternated series *)
@@ -24,13 +24,13 @@ Lemma CV_ALT_step0 :
     Un_decreasing Un ->
     Un_growing (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))).
 Proof.
-  intros; unfold Un_growing in |- *; intro.
+  intros; unfold Un_growing; intro.
   cut ((2 * S n)%nat = S (S (2 * n))).
   intro; rewrite H0.
   do 4 rewrite tech5; repeat rewrite Rplus_assoc; apply Rplus_le_compat_l.
-  pattern (tg_alt Un (S (2 * n))) at 1 in |- *; rewrite <- Rplus_0_r.
+  pattern (tg_alt Un (S (2 * n))) at 1; rewrite <- Rplus_0_r.
   apply Rplus_le_compat_l.
-  unfold tg_alt in |- *; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even;
+  unfold tg_alt; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even;
     rewrite Rmult_1_l.
   apply Rplus_le_reg_l with (Un (S (2 * S n))).
   rewrite Rplus_0_r;
@@ -46,12 +46,12 @@ Lemma CV_ALT_step1 :
     Un_decreasing Un ->
     Un_decreasing (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N)).
 Proof.
-  intros; unfold Un_decreasing in |- *; intro.
+  intros; unfold Un_decreasing; intro.
   cut ((2 * S n)%nat = S (S (2 * n))).
   intro; rewrite H0; do 2 rewrite tech5; repeat rewrite Rplus_assoc.
-  pattern (sum_f_R0 (tg_alt Un) (2 * n)) at 2 in |- *; rewrite <- Rplus_0_r.
+  pattern (sum_f_R0 (tg_alt Un) (2 * n)) at 2; rewrite <- Rplus_0_r.
   apply Rplus_le_compat_l.
-  unfold tg_alt in |- *; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even;
+  unfold tg_alt; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even;
     rewrite Rmult_1_l.
   apply Rplus_le_reg_l with (Un (S (2 * n))).
   rewrite Rplus_0_r;
@@ -70,7 +70,7 @@ Lemma CV_ALT_step2 :
     sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N)) <= 0.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; unfold tg_alt in |- *; simpl in |- *; rewrite Rmult_1_r.
+  simpl; unfold tg_alt; simpl; rewrite Rmult_1_r.
   replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ].
   apply Rplus_le_reg_l with (Un 1%nat); rewrite Rplus_0_r.
   replace (Un 1%nat + (-1 * Un 1%nat + Un 2%nat)) with (Un 2%nat);
@@ -78,10 +78,10 @@ Proof.
   cut (S (2 * S N) = S (S (S (2 * N)))).
   intro; rewrite H1; do 2 rewrite tech5.
   apply Rle_trans with (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))).
-  pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))) at 2 in |- *;
+  pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))) at 2;
     rewrite <- Rplus_0_r.
   rewrite Rplus_assoc; apply Rplus_le_compat_l.
-  unfold tg_alt in |- *; rewrite <- H1.
+  unfold tg_alt; rewrite <- H1.
   rewrite pow_1_odd.
   cut (S (S (2 * S N)) = (2 * S (S N))%nat).
   intro; rewrite H2; rewrite pow_1_even; rewrite Rmult_1_l; rewrite <- H2.
@@ -102,7 +102,7 @@ Lemma CV_ALT_step3 :
     positivity_seq Un -> sum_f_R0 (fun i:nat => tg_alt Un (S i)) N <= 0.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; unfold tg_alt in |- *; simpl in |- *; rewrite Rmult_1_r.
+  simpl; unfold tg_alt; simpl; rewrite Rmult_1_r.
   apply Rplus_le_reg_l with (Un 1%nat).
   rewrite Rplus_0_r; replace (Un 1%nat + -1 * Un 1%nat) with 0;
     [ apply H0 | ring ].
@@ -112,10 +112,10 @@ Proof.
   rewrite H3; apply CV_ALT_step2; assumption.
   rewrite H3; rewrite tech5.
   apply Rle_trans with (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))).
-  pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))) at 2 in |- *;
+  pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))) at 2;
     rewrite <- Rplus_0_r.
   apply Rplus_le_compat_l.
-  unfold tg_alt in |- *; simpl in |- *.
+  unfold tg_alt; simpl.
   replace (x + (x + 0))%nat with (2 * x)%nat; [ idtac | ring ].
   rewrite pow_1_even.
   replace (-1 * (-1 * (-1 * 1)) * Un (S (S (S (2 * x))))) with
@@ -133,15 +133,15 @@ Lemma CV_ALT_step4 :
     positivity_seq Un ->
     has_ub (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))).
 Proof.
-  intros; unfold has_ub in |- *; unfold bound in |- *.
+  intros; unfold has_ub; unfold bound.
   exists (Un 0%nat).
-  unfold is_upper_bound in |- *; intros; elim H1; intros.
+  unfold is_upper_bound; intros; elim H1; intros.
   rewrite H2; rewrite decomp_sum.
   replace (tg_alt Un 0) with (Un 0%nat).
-  pattern (Un 0%nat) at 2 in |- *; rewrite <- Rplus_0_r.
+  pattern (Un 0%nat) at 2; rewrite <- Rplus_0_r.
   apply Rplus_le_compat_l.
   apply CV_ALT_step3; assumption.
-  unfold tg_alt in |- *; simpl in |- *; ring.
+  unfold tg_alt; simpl; ring.
   apply lt_O_Sn.
 Qed.
 
@@ -159,11 +159,11 @@ Proof.
   assert (X := growing_cv _ H2 H3).
   elim X; intros.
   exists x.
-  unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1;
+  unfold Un_cv; unfold R_dist; unfold Un_cv in H1;
     unfold R_dist in H1; unfold Un_cv in p; unfold R_dist in p.
   intros; cut (0 < eps / 2);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
 	[ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
   elim (H1 (eps / 2) H5); intros N2 H6.
   elim (p (eps / 2) H5); intros N1 H7.
@@ -180,32 +180,32 @@ Proof.
   apply Rabs_triang.
   rewrite (double_var eps); apply Rplus_lt_compat.
   rewrite H12; apply H7; assumption.
-  rewrite Rabs_Ropp; unfold tg_alt in |- *; rewrite Rabs_mult;
+  rewrite Rabs_Ropp; unfold tg_alt; rewrite Rabs_mult;
     rewrite pow_1_abs; rewrite Rmult_1_l; unfold Rminus in H6;
       rewrite Ropp_0 in H6; rewrite <- (Rplus_0_r (Un (S n)));
 	apply H6.
-  unfold ge in |- *; apply le_trans with n.
-  apply le_trans with N; [ unfold N in |- *; apply le_max_r | assumption ].
+  unfold ge; apply le_trans with n.
+  apply le_trans with N; [ unfold N; apply le_max_r | assumption ].
   apply le_n_Sn.
   rewrite tech5; ring.
   rewrite H12; apply Rlt_trans with (eps / 2).
   apply H7; assumption.
-  unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2.
+  unfold Rdiv; apply Rmult_lt_reg_l with 2.
   prove_sup0.
   rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
     [ rewrite Rmult_1_r | discrR ].
   rewrite double.
-  pattern eps at 1 in |- *; rewrite <- (Rplus_0_r eps); apply Rplus_lt_compat_l;
+  pattern eps at 1; rewrite <- (Rplus_0_r eps); apply Rplus_lt_compat_l;
     assumption.
   elim H10; intro; apply le_double.
   rewrite <- H11; apply le_trans with N.
-  unfold N in |- *; apply le_trans with (S (2 * N1));
+  unfold N; apply le_trans with (S (2 * N1));
     [ apply le_n_Sn | apply le_max_l ].
   assumption.
   apply lt_n_Sm_le.
   rewrite <- H11.
   apply lt_le_trans with N.
-  unfold N in |- *; apply lt_le_trans with (S (2 * N1)).
+  unfold N; apply lt_le_trans with (S (2 * N1)).
   apply lt_n_Sn.
   apply le_max_l.
   assumption.
@@ -222,7 +222,7 @@ Theorem alternated_series :
 Proof.
   intros; apply CV_ALT.
   assumption.
-  unfold positivity_seq in |- *; apply decreasing_ineq; assumption.
+  unfold positivity_seq; apply decreasing_ineq; assumption.
   assumption.
 Qed.
 
@@ -243,31 +243,31 @@ Proof.
   apply (decreasing_ineq (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N))).
   apply CV_ALT_step1; assumption.
   assumption.
-  unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1;
+  unfold Un_cv; unfold R_dist; unfold Un_cv in H1;
     unfold R_dist in H1; intros.
   elim (H1 eps H2); intros.
   exists x; intros.
   apply H3.
-  unfold ge in |- *; apply le_trans with (2 * n)%nat.
+  unfold ge; apply le_trans with (2 * n)%nat.
   apply le_trans with n.
   assumption.
   assert (H5 := mult_O_le n 2).
   elim H5; intro.
   cut (0%nat <> 2%nat);
-    [ intro; elim H7; symmetry  in |- *; assumption | discriminate ].
+    [ intro; elim H7; symmetry ; assumption | discriminate ].
   assumption.
   apply le_n_Sn.
-  unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1;
+  unfold Un_cv; unfold R_dist; unfold Un_cv in H1;
     unfold R_dist in H1; intros.
   elim (H1 eps H2); intros.
   exists x; intros.
   apply H3.
-  unfold ge in |- *; apply le_trans with n.
+  unfold ge; apply le_trans with n.
   assumption.
   assert (H5 := mult_O_le n 2).
   elim H5; intro.
   cut (0%nat <> 2%nat);
-    [ intro; elim H7; symmetry  in |- *; assumption | discriminate ].
+    [ intro; elim H7; symmetry ; assumption | discriminate ].
   assumption.
 Qed.
 
@@ -279,13 +279,13 @@ Definition PI_tg (n:nat) := / INR (2 * n + 1).
 
 Lemma PI_tg_pos : forall n:nat, 0 <= PI_tg n.
 Proof.
-  intro; unfold PI_tg in |- *; left; apply Rinv_0_lt_compat; apply lt_INR_0;
+  intro; unfold PI_tg; left; apply Rinv_0_lt_compat; apply lt_INR_0;
     replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ].
 Qed.
 
 Lemma PI_tg_decreasing : Un_decreasing PI_tg.
 Proof.
-  unfold PI_tg, Un_decreasing in |- *; intro.
+  unfold PI_tg, Un_decreasing; intro.
   apply Rmult_le_reg_l with (INR (2 * n + 1)).
   apply lt_INR_0.
   replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ].
@@ -306,7 +306,7 @@ Qed.
 
 Lemma PI_tg_cv : Un_cv PI_tg 0.
 Proof.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold Un_cv; unfold R_dist; intros.
   cut (0 < 2 * eps);
     [ intro | apply Rmult_lt_0_compat; [ prove_sup0 | assumption ] ].
   assert (H1 := archimed (/ (2 * eps))).
@@ -316,9 +316,9 @@ Proof.
   cut (0 < N)%nat.
   intro; exists N; intros.
   cut (0 < n)%nat.
-  intro; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r;
+  intro; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r;
     rewrite Rabs_right.
-  unfold PI_tg in |- *; apply Rlt_trans with (/ INR (2 * n)).
+  unfold PI_tg; apply Rlt_trans with (/ INR (2 * n)).
   apply Rmult_lt_reg_l with (INR (2 * n)).
   apply lt_INR_0.
   replace (2 * n)%nat with (n + n)%nat; [ idtac | ring ].
@@ -337,27 +337,27 @@ Proof.
     [ discriminate | ring ].
   replace n with (S (pred n)).
   apply not_O_INR; discriminate.
-  symmetry  in |- *; apply S_pred with 0%nat.
+  symmetry ; apply S_pred with 0%nat.
   assumption.
   apply Rle_lt_trans with (/ INR (2 * N)).
   apply Rmult_le_reg_l with (INR (2 * N)).
   rewrite mult_INR; apply Rmult_lt_0_compat;
-    [ simpl in |- *; prove_sup0 | apply lt_INR_0; assumption ].
+    [ simpl; prove_sup0 | apply lt_INR_0; assumption ].
   rewrite <- Rinv_r_sym.
   apply Rmult_le_reg_l with (INR (2 * n)).
   rewrite mult_INR; apply Rmult_lt_0_compat;
-    [ simpl in |- *; prove_sup0 | apply lt_INR_0; assumption ].
+    [ simpl; prove_sup0 | apply lt_INR_0; assumption ].
   rewrite (Rmult_comm (INR (2 * n))); rewrite Rmult_assoc;
     rewrite <- Rinv_l_sym.
   do 2 rewrite Rmult_1_r; apply le_INR.
   apply (fun m n p:nat => mult_le_compat_l p n m); assumption.
   replace n with (S (pred n)).
   apply not_O_INR; discriminate.
-  symmetry  in |- *; apply S_pred with 0%nat.
+  symmetry ; apply S_pred with 0%nat.
   assumption.
   replace N with (S (pred N)).
   apply not_O_INR; discriminate.
-  symmetry  in |- *; apply S_pred with 0%nat.
+  symmetry ; apply S_pred with 0%nat.
   assumption.
   rewrite mult_INR.
   rewrite Rinv_mult_distr.
@@ -374,17 +374,17 @@ Proof.
     replace (/ (2 * eps) * (INR N * (2 * eps))) with
       (INR N * (2 * eps * / (2 * eps))); [ idtac | ring ].
   rewrite <- Rinv_r_sym.
-  rewrite Rmult_1_r; replace (INR N) with (IZR (Z_of_nat N)).
+  rewrite Rmult_1_r; replace (INR N) with (IZR (Z.of_nat N)).
   rewrite <- H4.
   elim H1; intros; assumption.
-  symmetry  in |- *; apply INR_IZR_INZ.
+  symmetry ; apply INR_IZR_INZ.
   apply prod_neq_R0;
-    [ discrR | red in |- *; intro; rewrite H8 in H; elim (Rlt_irrefl _ H) ].
+    [ discrR | red; intro; rewrite H8 in H; elim (Rlt_irrefl _ H) ].
   apply not_O_INR.
-  red in |- *; intro; rewrite H8 in H5; elim (lt_irrefl _ H5).
+  red; intro; rewrite H8 in H5; elim (lt_irrefl _ H5).
   replace (INR 2) with 2; [ discrR | reflexivity ].
   apply not_O_INR.
-  red in |- *; intro; rewrite H8 in H5; elim (lt_irrefl _ H5).
+  red; intro; rewrite H8 in H5; elim (lt_irrefl _ H5).
   apply Rle_ge; apply PI_tg_pos.
   apply lt_le_trans with N; assumption.
   elim H1; intros H5 _.
@@ -399,7 +399,7 @@ Proof.
   elim (Rlt_irrefl _ (Rlt_trans _ _ _ H7 H5)).
   elim (lt_n_O _ b).
   apply le_IZR.
-  simpl in |- *.
+  simpl.
   left; apply Rlt_trans with (/ (2 * eps)).
   apply Rinv_0_lt_compat; assumption.
   elim H1; intros; assumption.
@@ -414,41 +414,41 @@ Proof.
 Qed.
 
 (** Now, PI is defined *)
-Definition PI : R := 4 * (let (a,_) := exist_PI in a).
+Definition Alt_PI : R := 4 * (let (a,_) := exist_PI in a).
 
 (** We can get an approximation of PI with the following inequality *)
-Lemma PI_ineq :
+Lemma Alt_PI_ineq :
   forall N:nat,
-    sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= PI / 4 <=
+    sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= Alt_PI / 4 <=
     sum_f_R0 (tg_alt PI_tg) (2 * N).
 Proof.
   intro; apply alternated_series_ineq.
   apply PI_tg_decreasing.
   apply PI_tg_cv.
-  unfold PI in |- *; case exist_PI; intro.
+  unfold Alt_PI; case exist_PI; intro.
   replace (4 * x / 4) with x.
   trivial.
-  unfold Rdiv in |- *; rewrite (Rmult_comm 4); rewrite Rmult_assoc;
+  unfold Rdiv; rewrite (Rmult_comm 4); rewrite Rmult_assoc;
     rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r; reflexivity | discrR ].
 Qed.
 
-Lemma PI_RGT_0 : 0 < PI.
+Lemma Alt_PI_RGT_0 : 0 < Alt_PI.
 Proof.
-  assert (H := PI_ineq 0).
+  assert (H := Alt_PI_ineq 0).
   apply Rmult_lt_reg_l with (/ 4).
   apply Rinv_0_lt_compat; prove_sup0.
   rewrite Rmult_0_r; rewrite Rmult_comm.
   elim H; clear H; intros H _.
   unfold Rdiv in H;
     apply Rlt_le_trans with (sum_f_R0 (tg_alt PI_tg) (S (2 * 0))).
-  simpl in |- *; unfold tg_alt in |- *; simpl in |- *; rewrite Rmult_1_l;
+  simpl; unfold tg_alt; simpl; rewrite Rmult_1_l;
     rewrite Rmult_1_r; apply Rplus_lt_reg_r with (PI_tg 1).
   rewrite Rplus_0_r;
     replace (PI_tg 1 + (PI_tg 0 + -1 * PI_tg 1)) with (PI_tg 0);
-      [ unfold PI_tg in |- * | ring ].
-  simpl in |- *; apply Rinv_lt_contravar.
+      [ unfold PI_tg | ring ].
+  simpl; apply Rinv_lt_contravar.
   rewrite Rmult_1_l; replace (2 + 1) with 3; [ prove_sup0 | ring ].
-  rewrite Rplus_comm; pattern 1 at 1 in |- *; rewrite <- Rplus_0_r;
+  rewrite Rplus_comm; pattern 1 at 1; rewrite <- Rplus_0_r;
     apply Rplus_lt_compat_l; prove_sup0.
   assumption.
 Qed.
diff --git a/theories/Reals/ArithProp.v b/theories/Reals/ArithProp.v
index 620561dc..c817bdfa 100644
--- a/theories/Reals/ArithProp.v
+++ b/theories/Reals/ArithProp.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -12,12 +12,12 @@ Require Import Even.
 Require Import Div2.
 Require Import ArithRing.
 
-Open Local Scope Z_scope.
-Open Local Scope R_scope.
+Local Open Scope Z_scope.
+Local Open Scope R_scope.
 
 Lemma minus_neq_O : forall n i:nat, (i < n)%nat -> (n - i)%nat <> 0%nat.
 Proof.
-  intros; red in |- *; intro.
+  intros; red; intro.
   cut (forall n m:nat, (m <= n)%nat -> (n - m)%nat = 0%nat -> n = m).
   intro; assert (H2 := H1 _ _ (lt_le_weak _ _ H) H0); rewrite H2 in H;
     elim (lt_irrefl _ H).
@@ -27,11 +27,11 @@ Proof.
       forall n0 m:nat, (m <= n0)%nat -> (n0 - m)%nat = 0%nat -> n0 = m).
   intro; apply H1.
   apply nat_double_ind.
-  unfold R in |- *; intros; inversion H2; reflexivity.
-  unfold R in |- *; intros; simpl in H3; assumption.
-  unfold R in |- *; intros; simpl in H4; assert (H5 := le_S_n _ _ H3);
+  unfold R; intros; inversion H2; reflexivity.
+  unfold R; intros; simpl in H3; assumption.
+  unfold R; intros; simpl in H4; assert (H5 := le_S_n _ _ H3);
     assert (H6 := H2 H5 H4); rewrite H6; reflexivity.
-  unfold R in |- *; intros; apply H1; assumption.
+  unfold R; intros; apply H1; assumption.
 Qed.
 
 Lemma le_minusni_n : forall n i:nat, (i <= n)%nat -> (n - i <= n)%nat.
@@ -41,20 +41,20 @@ Proof.
     ((forall m n:nat, R m n) -> forall n i:nat, (i <= n)%nat -> (n - i <= n)%nat).
   intro; apply H.
   apply nat_double_ind.
-  unfold R in |- *; intros; simpl in |- *; apply le_n.
-  unfold R in |- *; intros; simpl in |- *; apply le_n.
-  unfold R in |- *; intros; simpl in |- *; apply le_trans with n.
+  unfold R; intros; simpl; apply le_n.
+  unfold R; intros; simpl; apply le_n.
+  unfold R; intros; simpl; apply le_trans with n.
   apply H0; apply le_S_n; assumption.
   apply le_n_Sn.
-  unfold R in |- *; intros; apply H; assumption.
+  unfold R; intros; apply H; assumption.
 Qed.
 
 Lemma lt_minus_O_lt : forall m n:nat, (m < n)%nat -> (0 < n - m)%nat.
 Proof.
-  intros n m; pattern n, m in |- *; apply nat_double_ind;
+  intros n m; pattern n, m; apply nat_double_ind;
     [ intros; rewrite <- minus_n_O; assumption
       | intros; elim (lt_n_O _ H)
-      | intros; simpl in |- *; apply H; apply lt_S_n; assumption ].
+      | intros; simpl; apply H; apply lt_S_n; assumption ].
 Qed.
 
 Lemma even_odd_cor :
@@ -73,7 +73,7 @@ Proof.
   apply H3; assumption.
   right.
   apply H4; assumption.
-  unfold double in |- *;ring.
+  unfold double;ring.
 Qed.
 
   (* 2m <= 2n => m<=n *)
@@ -105,9 +105,9 @@ Proof.
   exists (x - IZR k0 * y).
   split.
   ring.
-  unfold k0 in |- *; case (Rcase_abs y); intro.
-  assert (H0 := archimed (x / - y)); rewrite <- Z_R_minus; simpl in |- *;
-    unfold Rminus in |- *.
+  unfold k0; case (Rcase_abs y); intro.
+  assert (H0 := archimed (x / - y)); rewrite <- Z_R_minus; simpl;
+    unfold Rminus.
   replace (- ((1 + - IZR (up (x / - y))) * y)) with
     ((IZR (up (x / - y)) - 1) * y); [ idtac | ring ].
   split.
@@ -118,7 +118,7 @@ Proof.
   rewrite Rmult_assoc; repeat rewrite Ropp_mult_distr_r_reverse;
     rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r | assumption ].
   apply Rplus_le_reg_l with (IZR (up (x / - y)) - x / - y).
-  rewrite Rplus_0_r; unfold Rdiv in |- *; pattern (/ - y) at 4 in |- *;
+  rewrite Rplus_0_r; unfold Rdiv; pattern (/ - y) at 4;
     rewrite <- Ropp_inv_permute; [ idtac | assumption ].
   replace
     (IZR (up (x * / - y)) - x * - / y +
@@ -138,11 +138,11 @@ Proof.
   replace (IZR (up (x / - y)) - 1 + (- (x * / y) + - (IZR (up (x / - y)) - 1)))
     with (- (x * / y)); [ idtac | ring ].
   rewrite <- Ropp_mult_distr_r_reverse; rewrite (Ropp_inv_permute _ H); elim H0;
-    unfold Rdiv in |- *; intros H1 _; exact H1.
+    unfold Rdiv; intros H1 _; exact H1.
   apply Ropp_neq_0_compat; assumption.
-  assert (H0 := archimed (x / y)); rewrite <- Z_R_minus; simpl in |- *;
+  assert (H0 := archimed (x / y)); rewrite <- Z_R_minus; simpl;
     cut (0 < y).
-  intro; unfold Rminus in |- *;
+  intro; unfold Rminus;
     replace (- ((IZR (up (x / y)) + -1) * y)) with ((1 - IZR (up (x / y))) * y);
       [ idtac | ring ].
   split.
@@ -152,7 +152,7 @@ Proof.
     rewrite Rmult_assoc; rewrite <- Rinv_r_sym;
       [ rewrite Rmult_1_r | assumption ];
       apply Rplus_le_reg_l with (IZR (up (x / y)) - x / y);
-	rewrite Rplus_0_r; unfold Rdiv in |- *;
+	rewrite Rplus_0_r; unfold Rdiv;
 	  replace
 	    (IZR (up (x * / y)) - x * / y + (x * / y + (1 - IZR (up (x * / y))))) with
 	    1; [ idtac | ring ]; elim H0; intros _ H2; unfold Rdiv in H2;
@@ -166,12 +166,12 @@ Proof.
 	replace (IZR (up (x / y)) - 1 + 1) with (IZR (up (x / y)));
 	  [ idtac | ring ];
 	  replace (IZR (up (x / y)) - 1 + (x * / y + (1 - IZR (up (x / y))))) with
-	    (x * / y); [ idtac | ring ]; elim H0; unfold Rdiv in |- *;
+	    (x * / y); [ idtac | ring ]; elim H0; unfold Rdiv;
 	      intros H2 _; exact H2.
   case (total_order_T 0 y); intro.
   elim s; intro.
   assumption.
-  elim H; symmetry  in |- *; exact b.
+  elim H; symmetry ; exact b.
   assert (H1 := Rge_le _ _ r); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 r0)).
 Qed.
 
diff --git a/theories/Reals/Binomial.v b/theories/Reals/Binomial.v
index 412f6442..ad076c48 100644
--- a/theories/Reals/Binomial.v
+++ b/theories/Reals/Binomial.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,14 +9,14 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import PartSum.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Definition C (n p:nat) : R :=
   INR (fact n) / (INR (fact p) * INR (fact (n - p))).
 
 Lemma pascal_step1 : forall n i:nat, (i <= n)%nat -> C n i = C n (n - i).
 Proof.
-  intros; unfold C in |- *; replace (n - (n - i))%nat with i.
+  intros; unfold C; replace (n - (n - i))%nat with i.
   rewrite Rmult_comm.
   reflexivity.
   apply plus_minus; rewrite plus_comm; apply le_plus_minus; assumption.
@@ -26,10 +26,10 @@ Lemma pascal_step2 :
   forall n i:nat,
     (i <= n)%nat -> C (S n) i = INR (S n) / INR (S n - i) * C n i.
 Proof.
-  intros; unfold C in |- *; replace (S n - i)%nat with (S (n - i)).
+  intros; unfold C; replace (S n - i)%nat with (S (n - i)).
   cut (forall n:nat, fact (S n) = (S n * fact n)%nat).
   intro; repeat rewrite H0.
-  unfold Rdiv in |- *; repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr.
+  unfold Rdiv; repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr.
   ring.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
@@ -46,13 +46,13 @@ Qed.
 Lemma pascal_step3 :
   forall n i:nat, (i < n)%nat -> C n (S i) = INR (n - i) / INR (S i) * C n i.
 Proof.
-  intros; unfold C in |- *.
+  intros; unfold C.
   cut (forall n:nat, fact (S n) = (S n * fact n)%nat).
   intro.
   cut ((n - i)%nat = S (n - S i)).
   intro.
-  pattern (n - i)%nat at 2 in |- *; rewrite H1.
-  repeat rewrite H0; unfold Rdiv in |- *; repeat rewrite mult_INR;
+  pattern (n - i)%nat at 2; rewrite H1.
+  repeat rewrite H0; unfold Rdiv; repeat rewrite mult_INR;
     repeat rewrite Rinv_mult_distr.
   rewrite <- H1; rewrite (Rmult_comm (/ INR (n - i)));
     repeat rewrite Rmult_assoc; rewrite (Rmult_comm (INR (n - i)));
@@ -68,7 +68,7 @@ Proof.
   apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].
   apply INR_fact_neq_0.
   rewrite minus_Sn_m.
-  simpl in |- *; reflexivity.
+  simpl; reflexivity.
   apply lt_le_S; assumption.
   intro; reflexivity.
 Qed.
@@ -95,13 +95,13 @@ Proof.
   rewrite <- minus_Sn_m.
   cut ((n - (n - i))%nat = i).
   intro; rewrite H0; reflexivity.
-  symmetry  in |- *; apply plus_minus.
+  symmetry ; apply plus_minus.
   rewrite plus_comm; rewrite le_plus_minus_r.
   reflexivity.
   apply lt_le_weak; assumption.
   apply le_minusni_n; apply lt_le_weak; assumption.
   apply lt_le_weak; assumption.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   repeat rewrite S_INR.
   rewrite minus_INR.
   cut (INR i + 1 <> 0).
@@ -125,18 +125,18 @@ Lemma binomial :
     (x + y) ^ n = sum_f_R0 (fun i:nat => C n i * x ^ i * y ^ (n - i)) n.
 Proof.
   intros; induction  n as [| n Hrecn].
-  unfold C in |- *; simpl in |- *; unfold Rdiv in |- *;
+  unfold C; simpl; unfold Rdiv;
     repeat rewrite Rmult_1_r; rewrite Rinv_1; ring.
-  pattern (S n) at 1 in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ].
+  pattern (S n) at 1; replace (S n) with (n + 1)%nat; [ idtac | ring ].
   rewrite pow_add; rewrite Hrecn.
-  replace ((x + y) ^ 1) with (x + y); [ idtac | simpl in |- *; ring ].
+  replace ((x + y) ^ 1) with (x + y); [ idtac | simpl; ring ].
   rewrite tech5.
   cut (forall p:nat, C p p = 1).
   cut (forall p:nat, C p 0 = 1).
   intros; rewrite H0; rewrite <- minus_n_n; rewrite Rmult_1_l.
-  replace (y ^ 0) with 1; [ rewrite Rmult_1_r | simpl in |- *; reflexivity ].
+  replace (y ^ 0) with 1; [ rewrite Rmult_1_r | simpl; reflexivity ].
   induction  n as [| n Hrecn0].
-  simpl in |- *; do 2 rewrite H; ring.
+  simpl; do 2 rewrite H; ring.
   (* N >= 1 *)
   set (N := S n).
   rewrite Rmult_plus_distr_l.
@@ -158,7 +158,7 @@ Proof.
   rewrite (Rplus_comm (sum_f_R0 An n)).
   repeat rewrite Rplus_assoc.
   rewrite <- tech5.
-  fold N in |- *.
+  fold N.
   set (Cn := fun i:nat => C N i * x ^ i * y ^ (S N - i)).
   cut (forall i:nat, (i < N)%nat -> Cn (S i) = Bn i).
   intro; replace (sum_f_R0 Bn n) with (sum_f_R0 (fun i:nat => Cn (S i)) n).
@@ -166,42 +166,42 @@ Proof.
   rewrite <- Rplus_assoc; rewrite (decomp_sum Cn N).
   replace (pred N) with n.
   ring.
-  unfold N in |- *; simpl in |- *; reflexivity.
-  unfold N in |- *; apply lt_O_Sn.
-  unfold Cn in |- *; rewrite H; simpl in |- *; ring.
+  unfold N; simpl; reflexivity.
+  unfold N; apply lt_O_Sn.
+  unfold Cn; rewrite H; simpl; ring.
   apply sum_eq.
   intros; apply H1.
-  unfold N in |- *; apply le_lt_trans with n; [ assumption | apply lt_n_Sn ].
-  intros; unfold Bn, Cn in |- *.
+  unfold N; apply le_lt_trans with n; [ assumption | apply lt_n_Sn ].
+  intros; unfold Bn, Cn.
   replace (S N - S i)%nat with (N - i)%nat; reflexivity.
-  unfold An in |- *; fold N in |- *; rewrite <- minus_n_n; rewrite H0;
-    simpl in |- *; ring.
+  unfold An; fold N; rewrite <- minus_n_n; rewrite H0;
+    simpl; ring.
   apply sum_eq.
-  intros; unfold An, Bn in |- *; replace (S N - S i)%nat with (N - i)%nat;
+  intros; unfold An, Bn; replace (S N - S i)%nat with (N - i)%nat;
     [ idtac | reflexivity ].
   rewrite <- pascal;
     [ ring
-      | apply le_lt_trans with n; [ assumption | unfold N in |- *; apply lt_n_Sn ] ].
-  unfold N in |- *; reflexivity.
-  unfold N in |- *; apply lt_O_Sn.
+      | apply le_lt_trans with n; [ assumption | unfold N; apply lt_n_Sn ] ].
+  unfold N; reflexivity.
+  unfold N; apply lt_O_Sn.
   rewrite <- (Rmult_comm y); rewrite scal_sum; apply sum_eq.
   intros; replace (S N - i)%nat with (S (N - i)).
   replace (S (N - i)) with (N - i + 1)%nat; [ idtac | ring ].
-  rewrite pow_add; replace (y ^ 1) with y; [ idtac | simpl in |- *; ring ];
+  rewrite pow_add; replace (y ^ 1) with y; [ idtac | simpl; ring ];
     ring.
   apply minus_Sn_m; assumption.
   rewrite <- (Rmult_comm x); rewrite scal_sum; apply sum_eq.
   intros; replace (S i) with (i + 1)%nat; [ idtac | ring ]; rewrite pow_add;
-    replace (x ^ 1) with x; [ idtac | simpl in |- *; ring ];
+    replace (x ^ 1) with x; [ idtac | simpl; ring ];
       ring.
-  intro; unfold C in |- *.
+  intro; unfold C.
   replace (INR (fact 0)) with 1; [ idtac | reflexivity ].
   replace (p - 0)%nat with p; [ idtac | apply minus_n_O ].
-  rewrite Rmult_1_l; unfold Rdiv in |- *; rewrite <- Rinv_r_sym;
+  rewrite Rmult_1_l; unfold Rdiv; rewrite <- Rinv_r_sym;
     [ reflexivity | apply INR_fact_neq_0 ].
-  intro; unfold C in |- *.
+  intro; unfold C.
   replace (p - p)%nat with 0%nat; [ idtac | apply minus_n_n ].
   replace (INR (fact 0)) with 1; [ idtac | reflexivity ].
-  rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym;
+  rewrite Rmult_1_r; unfold Rdiv; rewrite <- Rinv_r_sym;
     [ reflexivity | apply INR_fact_neq_0 ].
 Qed.
diff --git a/theories/Reals/Cauchy_prod.v b/theories/Reals/Cauchy_prod.v
index a9d5cde3..f6a48adc 100644
--- a/theories/Reals/Cauchy_prod.v
+++ b/theories/Reals/Cauchy_prod.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -10,7 +10,7 @@ Require Import Rbase.
 Require Import Rfunctions.
 Require Import Rseries.
 Require Import PartSum.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
   (**********)
 Lemma sum_N_predN :
@@ -21,7 +21,7 @@ Proof.
   replace N with (S (pred N)).
   rewrite tech5.
   reflexivity.
-  symmetry  in |- *; apply S_pred with 0%nat; assumption.
+  symmetry ; apply S_pred with 0%nat; assumption.
 Qed.
 
   (**********)
@@ -51,7 +51,7 @@ Proof.
   elim (lt_irrefl _ H).
   cut (N = 0%nat \/ (0 < N)%nat).
   intro; elim H0; intro.
-  rewrite H1; simpl in |- *; ring.
+  rewrite H1; simpl; ring.
   replace (pred (S N)) with (S (pred N)).
   do 5 rewrite tech5.
   rewrite Rmult_plus_distr_r; rewrite Rmult_plus_distr_l; rewrite (HrecN H1).
@@ -66,7 +66,7 @@ Proof.
       repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l.
   rewrite <- minus_n_n; cut (N = 1%nat \/ (2 <= N)%nat).
   intro; elim H2; intro.
-  rewrite H3; simpl in |- *; ring.
+  rewrite H3; simpl; ring.
   replace
     (sum_f_R0
       (fun k:nat =>
@@ -147,7 +147,7 @@ Proof.
       (pred (pred N))).
   repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l.
   replace (pred (N - pred N)) with 0%nat.
-  simpl in |- *; rewrite <- minus_n_O.
+  simpl; rewrite <- minus_n_O.
   replace (S (pred N)) with N.
   replace (sum_f_R0 (fun k:nat => An (S N) * Bn (S k)) (pred (pred N))) with
     (sum_f_R0 (fun k:nat => Bn (S k) * An (S N)) (pred (pred N))).
@@ -161,11 +161,11 @@ Proof.
   apply S_pred with 0%nat; assumption.
   replace (N - pred N)%nat with 1%nat.
   reflexivity.
-  pattern N at 1 in |- *; replace N with (S (pred N)).
+  pattern N at 1; replace N with (S (pred N)).
   rewrite <- minus_Sn_m.
   rewrite <- minus_n_n; reflexivity.
   apply le_n.
-  symmetry  in |- *; apply S_pred with 0%nat; assumption.
+  symmetry ; apply S_pred with 0%nat; assumption.
   apply sum_eq; intros;
     rewrite
       (sum_N_predN (fun l:nat => An (S (S (l + i))) * Bn (N - l)%nat)
@@ -259,7 +259,7 @@ Proof.
   apply le_n.
   apply (fun p n m:nat => plus_le_reg_l n m p) with 1%nat.
   rewrite le_plus_minus_r.
-  simpl in |- *; assumption.
+  simpl; assumption.
   apply le_trans with 2%nat; [ apply le_n_Sn | assumption ].
   apply le_trans with 2%nat; [ apply le_n_Sn | assumption ].
   simpl; ring.
@@ -274,7 +274,7 @@ Proof.
   apply le_trans with (pred (pred N)).
   assumption.
   apply le_pred_n.
-  symmetry  in |- *; apply S_pred with 0%nat; assumption.
+  symmetry ; apply S_pred with 0%nat; assumption.
   apply INR_eq; rewrite S_INR; rewrite plus_INR; reflexivity.
   apply le_trans with (pred (pred N)).
   assumption.
@@ -427,7 +427,7 @@ Proof.
   apply le_trans with (pred (pred N)).
   assumption.
   apply le_pred_n.
-  symmetry  in |- *; apply S_pred with 0%nat; assumption.
+  symmetry ; apply S_pred with 0%nat; assumption.
   apply INR_eq; rewrite S_INR; rewrite plus_INR; simpl; ring.
   apply le_trans with (pred (pred N)).
   assumption.
@@ -441,11 +441,11 @@ Proof.
   inversion H1.
   left; reflexivity.
   right; apply le_n_S; assumption.
-  simpl in |- *.
+  simpl.
   replace (S (pred N)) with N.
   reflexivity.
   apply S_pred with 0%nat; assumption.
-  simpl in |- *.
+  simpl.
   cut ((N - pred N)%nat = 1%nat).
   intro; rewrite H2; reflexivity.
   rewrite pred_of_minus.
@@ -453,7 +453,7 @@ Proof.
   simpl; ring.
   apply lt_le_S; assumption.
   rewrite <- pred_of_minus; apply le_pred_n.
-  simpl in |- *; symmetry  in |- *; apply S_pred with 0%nat; assumption.
+  simpl; symmetry ; apply S_pred with 0%nat; assumption.
   inversion H.
   left; reflexivity.
   right; apply lt_le_trans with 1%nat; [ apply lt_n_Sn | exact H1 ].
diff --git a/theories/Reals/Cos_plus.v b/theories/Reals/Cos_plus.v
index ec1eeddf..c296d427 100644
--- a/theories/Reals/Cos_plus.v
+++ b/theories/Reals/Cos_plus.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -12,8 +12,8 @@ Require Import SeqSeries.
 Require Import Rtrigo_def.
 Require Import Cos_rel.
 Require Import Max.
-Open Local Scope nat_scope.
-Open Local Scope R_scope.
+Local Open Scope nat_scope.
+Local Open Scope R_scope.
 
 Definition Majxy (x y:R) (n:nat) : R :=
   Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n) / INR (fact n).
@@ -29,23 +29,23 @@ Proof.
   intro.
   assert (H1 := cv_speed_pow_fact C0).
   unfold Un_cv in H1; unfold R_dist in H1.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold Un_cv; unfold R_dist; intros.
   cut (0 < eps / C0);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
 	[ assumption | apply Rinv_0_lt_compat; assumption ] ].
   elim (H1 (eps / C0) H3); intros N0 H4.
   exists N0; intros.
   replace (Majxy x y n) with (C0 ^ S n / INR (fact n)).
-  simpl in |- *.
+  simpl.
   apply Rmult_lt_reg_l with (Rabs (/ C0)).
   apply Rabs_pos_lt.
   apply Rinv_neq_0_compat.
-  red in |- *; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0).
+  red; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0).
   rewrite <- Rabs_mult.
-  unfold Rminus in |- *; rewrite Rmult_plus_distr_l.
+  unfold Rminus; rewrite Rmult_plus_distr_l.
   rewrite Ropp_0; rewrite Rmult_0_r.
-  unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
+  unfold Rdiv; repeat rewrite <- Rmult_assoc.
   rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l.
   rewrite (Rabs_right (/ C0)).
@@ -54,15 +54,15 @@ Proof.
     [ idtac | ring ].
   unfold Rdiv in H4; apply H4; assumption.
   apply Rle_ge; left; apply Rinv_0_lt_compat; assumption.
-  red in |- *; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0).
-  unfold Majxy in |- *.
-  unfold C0 in |- *.
+  red; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0).
+  unfold Majxy.
+  unfold C0.
   rewrite pow_mult.
-  unfold C in |- *; reflexivity.
-  unfold C0 in |- *; apply pow_lt; assumption.
+  unfold C; reflexivity.
+  unfold C0; apply pow_lt; assumption.
   apply Rlt_le_trans with 1.
   apply Rlt_0_1.
-  unfold C in |- *.
+  unfold C.
   apply RmaxLess1.
 Qed.
 
@@ -72,7 +72,7 @@ Lemma reste1_maj :
 Proof.
   intros.
   set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))).
-  unfold Reste1 in |- *.
+  unfold Reste1.
   apply Rle_trans with
     (sum_f_R0
       (fun k:nat =>
@@ -120,7 +120,7 @@ Proof.
           C ^ (2 * S (N + k))) (pred (N - k))) (pred N)).
   apply sum_Rle; intros.
   apply sum_Rle; intros.
-  unfold Rdiv in |- *; repeat rewrite Rabs_mult.
+  unfold Rdiv; repeat rewrite Rabs_mult.
   do 2 rewrite pow_1_abs.
   do 2 rewrite Rmult_1_l.
   rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n))))).
@@ -142,7 +142,7 @@ Proof.
   apply pow_incr.
   split.
   apply Rabs_pos.
-  unfold C in |- *.
+  unfold C.
   apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2.
   apply Rle_trans with (C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))).
   do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0)))).
@@ -150,11 +150,11 @@ Proof.
   apply pow_le.
   apply Rle_trans with 1.
   left; apply Rlt_0_1.
-  unfold C in |- *; apply RmaxLess1.
+  unfold C; apply RmaxLess1.
   apply pow_incr.
   split.
   apply Rabs_pos.
-  unfold C in |- *; apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
+  unfold C; apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
   apply RmaxLess1.
   apply RmaxLess2.
   right.
@@ -203,7 +203,7 @@ Proof.
   left; apply Rinv_0_lt_compat.
   rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0.
   apply Rle_pow.
-  unfold C in |- *; apply RmaxLess1.
+  unfold C; apply RmaxLess1.
   replace (4 * N)%nat with (2 * (2 * N))%nat; [ idtac | ring ].
   apply (fun m n p:nat => mult_le_compat_l p n m).
   replace (2 * N)%nat with (S (N + pred N)).
@@ -223,33 +223,33 @@ Proof.
   apply pow_le.
   left; apply Rlt_le_trans with 1.
   apply Rlt_0_1.
-  unfold C in |- *; apply RmaxLess1.
+  unfold C; apply RmaxLess1.
   replace (/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))) with
     (Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) / INR (fact (2 * S (N + n)))).
   apply Rle_trans with
     (Binomial.C (2 * S (N + n)) (S (N + n)) / INR (fact (2 * S (N + n)))).
-  unfold Rdiv in |- *;
+  unfold Rdiv;
     do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (N + n))))).
   apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
   apply C_maj.
   omega.
   right.
-  unfold Rdiv in |- *; rewrite Rmult_comm.
-  unfold Binomial.C in |- *.
-  unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
+  unfold Rdiv; rewrite Rmult_comm.
+  unfold Binomial.C.
+  unfold Rdiv; repeat rewrite <- Rmult_assoc.
   rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l.
   replace (2 * S (N + n) - S (N + n))%nat with (S (N + n)).
   rewrite Rinv_mult_distr.
-  unfold Rsqr in |- *; reflexivity.
+  unfold Rsqr; reflexivity.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
   omega.
   apply INR_fact_neq_0.
-  unfold Rdiv in |- *; rewrite Rmult_comm.
-  unfold Binomial.C in |- *.
-  unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
+  unfold Rdiv; rewrite Rmult_comm.
+  unfold Binomial.C.
+  unfold Rdiv; repeat rewrite <- Rmult_assoc.
   rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l.
   replace (2 * S (N + n) - 2 * S (n0 + n))%nat with (2 * (N - n0))%nat.
@@ -271,17 +271,17 @@ Proof.
   apply pow_le.
   left; apply Rlt_le_trans with 1.
   apply Rlt_0_1.
-  unfold C in |- *; apply RmaxLess1.
+  unfold C; apply RmaxLess1.
   apply Rle_trans with (Rsqr (/ INR (fact (S (N + n)))) * INR N).
   apply Rmult_le_compat_l.
   apply Rle_0_sqr.
   apply le_INR.
   omega.
-  rewrite Rmult_comm; unfold Rdiv in |- *; apply Rmult_le_compat_l.
+  rewrite Rmult_comm; unfold Rdiv; apply Rmult_le_compat_l.
   apply pos_INR.
   apply Rle_trans with (/ INR (fact (S (N + n)))).
-  pattern (/ INR (fact (S (N + n)))) at 2 in |- *; rewrite <- Rmult_1_r.
-  unfold Rsqr in |- *.
+  pattern (/ INR (fact (S (N + n)))) at 2; rewrite <- Rmult_1_r.
+  unfold Rsqr.
   apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
   apply Rmult_le_reg_l with (INR (fact (S (N + n)))).
@@ -313,14 +313,14 @@ Proof.
   rewrite sum_cte.
   apply Rle_trans with (C ^ (4 * N) / INR (fact (pred N))).
   rewrite <- (Rmult_comm (C ^ (4 * N))).
-  unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l.
+  unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
   apply pow_le.
   left; apply Rlt_le_trans with 1.
   apply Rlt_0_1.
-  unfold C in |- *; apply RmaxLess1.
+  unfold C; apply RmaxLess1.
   cut (S (pred N) = N).
   intro; rewrite H0.
-  pattern N at 2 in |- *; rewrite <- H0.
+  pattern N at 2; rewrite <- H0.
   do 2 rewrite fact_simpl.
   rewrite H0.
   repeat rewrite mult_INR.
@@ -329,7 +329,7 @@ Proof.
   repeat rewrite <- Rmult_assoc.
   rewrite <- Rinv_r_sym.
   rewrite Rmult_1_l.
-  pattern (/ INR (fact (pred N))) at 2 in |- *; rewrite <- Rmult_1_r.
+  pattern (/ INR (fact (pred N))) at 2; rewrite <- Rmult_1_r.
   rewrite Rmult_assoc.
   apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
@@ -340,19 +340,19 @@ Proof.
   apply le_INR; apply le_n_Sn.
   apply not_O_INR; discriminate.
   apply not_O_INR.
-  red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H).
+  red; intro; rewrite H1 in H; elim (lt_irrefl _ H).
   apply not_O_INR.
-  red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H).
+  red; intro; rewrite H1 in H; elim (lt_irrefl _ H).
   apply INR_fact_neq_0.
   apply not_O_INR; discriminate.
   apply prod_neq_R0.
   apply not_O_INR.
-  red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H).
+  red; intro; rewrite H1 in H; elim (lt_irrefl _ H).
   apply INR_fact_neq_0.
-  symmetry  in |- *; apply S_pred with 0%nat; assumption.
+  symmetry ; apply S_pred with 0%nat; assumption.
   right.
-  unfold Majxy in |- *.
-  unfold C in |- *.
+  unfold Majxy.
+  unfold C.
   replace (S (pred N)) with N.
   reflexivity.
   apply S_pred with 0%nat; assumption.
@@ -363,7 +363,7 @@ Lemma reste2_maj :
 Proof.
   intros.
   set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))).
-  unfold Reste2 in |- *.
+  unfold Reste2.
   apply Rle_trans with
     (sum_f_R0
       (fun k:nat =>
@@ -415,7 +415,7 @@ Proof.
 	    pred N)).
   apply sum_Rle; intros.
   apply sum_Rle; intros.
-  unfold Rdiv in |- *; repeat rewrite Rabs_mult.
+  unfold Rdiv; repeat rewrite Rabs_mult.
   do 2 rewrite pow_1_abs.
   do 2 rewrite Rmult_1_l.
   rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n) + 1)))).
@@ -437,7 +437,7 @@ Proof.
   apply pow_incr.
   split.
   apply Rabs_pos.
-  unfold C in |- *.
+  unfold C.
   apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2.
   apply Rle_trans with (C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)).
   do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0) + 1))).
@@ -445,11 +445,11 @@ Proof.
   apply pow_le.
   apply Rle_trans with 1.
   left; apply Rlt_0_1.
-  unfold C in |- *; apply RmaxLess1.
+  unfold C; apply RmaxLess1.
   apply pow_incr.
   split.
   apply Rabs_pos.
-  unfold C in |- *; apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
+  unfold C; apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
   apply RmaxLess1.
   apply RmaxLess2.
   right.
@@ -477,7 +477,7 @@ Proof.
   left; apply Rinv_0_lt_compat.
   rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0.
   apply Rle_pow.
-  unfold C in |- *; apply RmaxLess1.
+  unfold C; apply RmaxLess1.
   replace (4 * S N)%nat with (2 * (2 * S N))%nat; [ idtac | ring ].
   apply (fun m n p:nat => mult_le_compat_l p n m).
   replace (2 * S N)%nat with (S (S (N + N))).
@@ -500,14 +500,14 @@ Proof.
   apply pow_le.
   left; apply Rlt_le_trans with 1.
   apply Rlt_0_1.
-  unfold C in |- *; apply RmaxLess1.
+  unfold C; apply RmaxLess1.
   replace (/ INR (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))) with
     (Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
       INR (fact (2 * S (S (N + n))))).
   apply Rle_trans with
     (Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
       INR (fact (2 * S (S (N + n))))).
-  unfold Rdiv in |- *;
+  unfold Rdiv;
     do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (S (N + n)))))).
   apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
@@ -518,21 +518,21 @@ Proof.
   ring.
   omega.
   right.
-  unfold Rdiv in |- *; rewrite Rmult_comm.
-  unfold Binomial.C in |- *.
-  unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
+  unfold Rdiv; rewrite Rmult_comm.
+  unfold Binomial.C.
+  unfold Rdiv; repeat rewrite <- Rmult_assoc.
   rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l.
   replace (2 * S (S (N + n)) - S (S (N + n)))%nat with (S (S (N + n))).
   rewrite Rinv_mult_distr.
-  unfold Rsqr in |- *; reflexivity.
+  unfold Rsqr; reflexivity.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
   omega.
   apply INR_fact_neq_0.
-  unfold Rdiv in |- *; rewrite Rmult_comm.
-  unfold Binomial.C in |- *.
-  unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
+  unfold Rdiv; rewrite Rmult_comm.
+  unfold Binomial.C.
+  unfold Rdiv; repeat rewrite <- Rmult_assoc.
   rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l.
   replace (2 * S (S (N + n)) - (2 * S (n0 + n) + 1))%nat with
@@ -556,7 +556,7 @@ Proof.
   apply pow_le.
   left; apply Rlt_le_trans with 1.
   apply Rlt_0_1.
-  unfold C in |- *; apply RmaxLess1.
+  unfold C; apply RmaxLess1.
   apply Rle_trans with (Rsqr (/ INR (fact (S (S (N + n))))) * INR N).
   apply Rmult_le_compat_l.
   apply Rle_0_sqr.
@@ -564,11 +564,11 @@ Proof.
   apply le_INR.
   omega.
   omega.
-  rewrite Rmult_comm; unfold Rdiv in |- *; apply Rmult_le_compat_l.
+  rewrite Rmult_comm; unfold Rdiv; apply Rmult_le_compat_l.
   apply pos_INR.
   apply Rle_trans with (/ INR (fact (S (S (N + n))))).
-  pattern (/ INR (fact (S (S (N + n))))) at 2 in |- *; rewrite <- Rmult_1_r.
-  unfold Rsqr in |- *.
+  pattern (/ INR (fact (S (S (N + n))))) at 2; rewrite <- Rmult_1_r.
+  unfold Rsqr.
   apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
   apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))).
@@ -599,11 +599,11 @@ Proof.
   rewrite sum_cte.
   apply Rle_trans with (C ^ (4 * S N) / INR (fact N)).
   rewrite <- (Rmult_comm (C ^ (4 * S N))).
-  unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l.
+  unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
   apply pow_le.
   left; apply Rlt_le_trans with 1.
   apply Rlt_0_1.
-  unfold C in |- *; apply RmaxLess1.
+  unfold C; apply RmaxLess1.
   cut (S (pred N) = N).
   intro; rewrite H0.
   do 2 rewrite fact_simpl.
@@ -642,10 +642,10 @@ Proof.
   apply INR_fact_neq_0.
   apply not_O_INR; discriminate.
   apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].
-  symmetry  in |- *; apply S_pred with 0%nat; assumption.
+  symmetry ; apply S_pred with 0%nat; assumption.
   right.
-  unfold Majxy in |- *.
-  unfold C in |- *.
+  unfold Majxy.
+  unfold C.
   reflexivity.
 Qed.
 
@@ -654,10 +654,10 @@ Proof.
   intros.
   assert (H := Majxy_cv_R0 x y).
   unfold Un_cv in H; unfold R_dist in H.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold Un_cv; unfold R_dist; intros.
   elim (H eps H0); intros N0 H1.
   exists (S N0); intros.
-  unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r.
+  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r.
   apply Rle_lt_trans with (Rabs (Majxy x y (pred n))).
   rewrite (Rabs_right (Majxy x y (pred n))).
   apply reste1_maj.
@@ -665,8 +665,8 @@ Proof.
   apply lt_O_Sn.
   assumption.
   apply Rle_ge.
-  unfold Majxy in |- *.
-  unfold Rdiv in |- *; apply Rmult_le_pos.
+  unfold Majxy.
+  unfold Rdiv; apply Rmult_le_pos.
   apply pow_le.
   apply Rle_trans with 1.
   left; apply Rlt_0_1.
@@ -674,7 +674,7 @@ Proof.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
   replace (Majxy x y (pred n)) with (Majxy x y (pred n) - 0); [ idtac | ring ].
   apply H1.
-  unfold ge in |- *; apply le_S_n.
+  unfold ge; apply le_S_n.
   replace (S (pred n)) with n.
   assumption.
   apply S_pred with 0%nat.
@@ -686,10 +686,10 @@ Proof.
   intros.
   assert (H := Majxy_cv_R0 x y).
   unfold Un_cv in H; unfold R_dist in H.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold Un_cv; unfold R_dist; intros.
   elim (H eps H0); intros N0 H1.
   exists (S N0); intros.
-  unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r.
+  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r.
   apply Rle_lt_trans with (Rabs (Majxy x y n)).
   rewrite (Rabs_right (Majxy x y n)).
   apply reste2_maj.
@@ -697,8 +697,8 @@ Proof.
   apply lt_O_Sn.
   assumption.
   apply Rle_ge.
-  unfold Majxy in |- *.
-  unfold Rdiv in |- *; apply Rmult_le_pos.
+  unfold Majxy.
+  unfold Rdiv; apply Rmult_le_pos.
   apply pow_le.
   apply Rle_trans with 1.
   left; apply Rlt_0_1.
@@ -706,7 +706,7 @@ Proof.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
   replace (Majxy x y n) with (Majxy x y n - 0); [ idtac | ring ].
   apply H1.
-  unfold ge in |- *; apply le_trans with (S N0).
+  unfold ge; apply le_trans with (S N0).
   apply le_n_Sn.
   exact H2.
 Qed.
@@ -714,7 +714,7 @@ Qed.
 Lemma reste_cv_R0 : forall x y:R, Un_cv (Reste x y) 0.
 Proof.
   intros.
-  unfold Reste in |- *.
+  unfold Reste.
   set (An := fun n:nat => Reste2 x y n).
   set (Bn := fun n:nat => Reste1 x y (S n)).
   cut
@@ -723,21 +723,21 @@ Proof.
   intro.
   apply H.
   apply CV_minus.
-  unfold An in |- *.
+  unfold An.
   replace (fun n:nat => Reste2 x y n) with (Reste2 x y).
   apply reste2_cv_R0.
   reflexivity.
-  unfold Bn in |- *.
+  unfold Bn.
   assert (H0 := reste1_cv_R0 x y).
   unfold Un_cv in H0; unfold R_dist in H0.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold Un_cv; unfold R_dist; intros.
   elim (H0 eps H1); intros N0 H2.
   exists N0; intros.
   apply H2.
-  unfold ge in |- *; apply le_trans with (S N0).
+  unfold ge; apply le_trans with (S N0).
   apply le_n_Sn.
   apply le_n_S; assumption.
-  unfold An, Bn in |- *.
+  unfold An, Bn.
   intro.
   replace 0 with (0 - 0); [ idtac | ring ].
   exact H.
@@ -751,7 +751,7 @@ Proof.
   intros.
   apply UL_sequence with (C1 x y); assumption.
   apply C1_cvg.
-  unfold Un_cv in |- *; unfold R_dist in |- *.
+  unfold Un_cv; unfold R_dist.
   intros.
   assert (H0 := A1_cvg x).
   assert (H1 := A1_cvg y).
@@ -764,7 +764,7 @@ Proof.
   unfold R_dist in H4; unfold R_dist in H5; unfold R_dist in H6.
   cut (0 < eps / 3);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
 	[ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
   elim (H4 (eps / 3) H7); intros N1 H8.
   elim (H5 (eps / 3) H7); intros N2 H9.
@@ -788,8 +788,8 @@ Proof.
   replace eps with (eps / 3 + (eps / 3 + eps / 3)).
   apply Rplus_lt_compat.
   apply H8.
-  unfold ge in |- *; apply le_trans with N.
-  unfold N in |- *.
+  unfold ge; apply le_trans with N.
+  unfold N.
   apply le_trans with (max N1 N2).
   apply le_max_l.
   apply le_trans with (max (max N1 N2) N3).
@@ -804,12 +804,12 @@ Proof.
   rewrite <- Rabs_Ropp.
   rewrite Ropp_minus_distr.
   apply H9.
-  unfold ge in |- *; apply le_trans with (max N1 N2).
+  unfold ge; apply le_trans with (max N1 N2).
   apply le_max_r.
   apply le_S_n.
   rewrite <- H12.
   apply le_trans with N.
-  unfold N in |- *.
+  unfold N.
   apply le_n_S.
   apply le_trans with (max (max N1 N2) N3).
   apply le_max_l.
@@ -817,35 +817,35 @@ Proof.
   assumption.
   replace (Reste x y (pred n)) with (Reste x y (pred n) - 0).
   apply H10.
-  unfold ge in |- *.
+  unfold ge.
   apply le_S_n.
   rewrite <- H12.
   apply le_trans with N.
-  unfold N in |- *.
+  unfold N.
   apply le_n_S.
   apply le_trans with (max (max N1 N2) N3).
   apply le_max_r.
   apply le_n_Sn.
   assumption.
   ring.
-  pattern eps at 4 in |- *; replace eps with (3 * (eps / 3)).
+  pattern eps at 4; replace eps with (3 * (eps / 3)).
   ring.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   rewrite <- Rmult_assoc.
   apply Rinv_r_simpl_m.
   discrR.
   apply lt_le_trans with (pred N).
-  unfold N in |- *; simpl in |- *; apply lt_O_Sn.
+  unfold N; simpl; apply lt_O_Sn.
   apply le_S_n.
   rewrite <- H12.
   replace (S (pred N)) with N.
   assumption.
-  unfold N in |- *; simpl in |- *; reflexivity.
+  unfold N; simpl; reflexivity.
   cut (0 < N)%nat.
   intro.
   cut (0 < n)%nat.
   intro.
   apply S_pred with 0%nat; assumption.
   apply lt_le_trans with N; assumption.
-  unfold N in |- *; apply lt_O_Sn.
+  unfold N; apply lt_O_Sn.
 Qed.
diff --git a/theories/Reals/Cos_rel.v b/theories/Reals/Cos_rel.v
index 73f3c0c6..9c7472fe 100644
--- a/theories/Reals/Cos_rel.v
+++ b/theories/Reals/Cos_rel.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -10,7 +10,7 @@ Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
 Require Import Rtrigo_def.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Definition A1 (x:R) (N:nat) : R :=
   sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) N.
@@ -50,7 +50,7 @@ Theorem cos_plus_form :
    (0 < n)%nat ->
    A1 x (S n) * A1 y (S n) - B1 x n * B1 y n + Reste x y n = C1 x y (S n).
 intros.
-unfold A1, B1 in |- *.
+unfold A1, B1.
 rewrite
  (cauchy_finite (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k))
     (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * y ^ (2 * k)) (
@@ -60,7 +60,7 @@ rewrite
     (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1))
     (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * y ^ (2 * k + 1)) n H)
  .
-unfold Reste in |- *.
+unfold Reste.
 replace
  (sum_f_R0
     (fun k:nat =>
@@ -119,13 +119,13 @@ replace
             ((-1) ^ (k - p) / INR (fact (2 * (k - p) + 1)) *
              y ^ (2 * (k - p) + 1))) k) n) with (sum_f_R0 sin_nnn (S n)).
 rewrite <- sum_plus.
-unfold C1 in |- *.
+unfold C1.
 apply sum_eq; intros.
 induction  i as [| i Hreci].
-simpl in |- *.
-unfold C in |- *; simpl in |- *.
+simpl.
+unfold C; simpl.
 field; discrR.
-unfold sin_nnn in |- *.
+unfold sin_nnn.
 rewrite <- Rmult_plus_distr_l.
 apply Rmult_eq_compat_l.
 rewrite binomial.
@@ -141,13 +141,13 @@ replace
  (sum_f_R0 (fun l:nat => Wn (S (2 * l))) i).
 apply sum_decomposition.
 apply sum_eq; intros.
-unfold Wn in |- *.
+unfold Wn.
 apply Rmult_eq_compat_l.
 replace (2 * S i - S (2 * i0))%nat with (S (2 * (i - i0))).
 reflexivity.
 omega.
 apply sum_eq; intros.
-unfold Wn in |- *.
+unfold Wn.
 apply Rmult_eq_compat_l.
 replace (2 * S i - 2 * i0)%nat with (2 * (S i - i0))%nat.
 reflexivity.
@@ -177,11 +177,11 @@ change (pred (S n)) with n.
    (* replace (pred (S n)) with n; [ idtac | reflexivity ]. *)
 apply sum_eq; intros.
 rewrite Rmult_comm.
-unfold sin_nnn in |- *.
+unfold sin_nnn.
 rewrite scal_sum.
 rewrite scal_sum.
 apply sum_eq; intros.
-unfold Rdiv in |- *.
+unfold Rdiv.
 (*repeat rewrite Rmult_assoc.*)
 (* rewrite (Rmult_comm (/ INR (fact (2 * S i)))). *)
 repeat rewrite <- Rmult_assoc.
@@ -193,13 +193,13 @@ replace (S (2 * i0)) with (2 * i0 + 1)%nat; [ idtac | ring ].
 replace (S (2 * (i - i0))) with (2 * (i - i0) + 1)%nat; [ idtac | ring ].
 replace ((-1) ^ S i) with (-1 * (-1) ^ i0 * (-1) ^ (i - i0)).
 ring.
-simpl in |- *.
-pattern i at 2 in |- *; replace i with (i0 + (i - i0))%nat.
+simpl.
+pattern i at 2; replace i with (i0 + (i - i0))%nat.
 rewrite pow_add.
 ring.
-symmetry  in |- *; apply le_plus_minus; assumption.
-unfold C in |- *.
-unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
+symmetry ; apply le_plus_minus; assumption.
+unfold C.
+unfold Rdiv; repeat rewrite <- Rmult_assoc.
 rewrite <- Rinv_l_sym.
 rewrite Rmult_1_l.
 rewrite Rinv_mult_distr.
@@ -217,7 +217,7 @@ apply lt_O_Sn.
 apply sum_eq; intros.
 rewrite scal_sum.
 apply sum_eq; intros.
-unfold Rdiv in |- *.
+unfold Rdiv.
 repeat rewrite <- Rmult_assoc.
 rewrite <- (Rmult_comm (/ INR (fact (2 * i)))).
 repeat rewrite <- Rmult_assoc.
@@ -225,12 +225,12 @@ replace (/ INR (fact (2 * i)) * C (2 * i) (2 * i0)) with
  (/ INR (fact (2 * i0)) * / INR (fact (2 * (i - i0)))).
 replace ((-1) ^ i) with ((-1) ^ i0 * (-1) ^ (i - i0)).
 ring.
-pattern i at 2 in |- *; replace i with (i0 + (i - i0))%nat.
+pattern i at 2; replace i with (i0 + (i - i0))%nat.
 rewrite pow_add.
 ring.
-symmetry  in |- *; apply le_plus_minus; assumption.
-unfold C in |- *.
-unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc.
+symmetry ; apply le_plus_minus; assumption.
+unfold C.
+unfold Rdiv; repeat rewrite <- Rmult_assoc.
 rewrite <- Rinv_l_sym.
 rewrite Rmult_1_l.
 rewrite Rinv_mult_distr.
@@ -240,12 +240,12 @@ omega.
 apply INR_fact_neq_0.
 apply INR_fact_neq_0.
 apply INR_fact_neq_0.
-unfold Reste2 in |- *; apply sum_eq; intros.
+unfold Reste2; apply sum_eq; intros.
 apply sum_eq; intros.
-unfold Rdiv in |- *; ring.
-unfold Reste1 in |- *; apply sum_eq; intros.
+unfold Rdiv; ring.
+unfold Reste1; apply sum_eq; intros.
 apply sum_eq; intros.
-unfold Rdiv in |- *; ring.
+unfold Rdiv; ring.
 apply lt_O_Sn.
 Qed.
 
@@ -266,10 +266,10 @@ unfold R_dist in p.
 cut (cos x = x0).
 intro.
 rewrite H0.
-unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+unfold Un_cv; unfold R_dist; intros.
 elim (p eps H1); intros.
 exists x1; intros.
-unfold A1 in |- *.
+unfold A1.
 replace
  (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) n) with
  (sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i)) * (x * x) ^ i) n).
@@ -279,9 +279,9 @@ intros.
 replace ((x * x) ^ i) with (x ^ (2 * i)).
 reflexivity.
 apply pow_sqr.
-unfold cos in |- *.
+unfold cos.
 case (exist_cos (Rsqr x)).
-unfold Rsqr in |- *; intros.
+unfold Rsqr; intros.
 unfold cos_in in p_i.
 unfold cos_in in c.
 apply uniqueness_sum with (fun i:nat => cos_n i * (x * x) ^ i); assumption.
@@ -298,10 +298,10 @@ unfold R_dist in p.
 cut (cos (x + y) = x0).
 intro.
 rewrite H0.
-unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+unfold Un_cv; unfold R_dist; intros.
 elim (p eps H1); intros.
 exists x1; intros.
-unfold C1 in |- *.
+unfold C1.
 replace
  (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * (x + y) ^ (2 * k)) n)
  with
@@ -313,9 +313,9 @@ intros.
 replace (((x + y) * (x + y)) ^ i) with ((x + y) ^ (2 * i)).
 reflexivity.
 apply pow_sqr.
-unfold cos in |- *.
+unfold cos.
 case (exist_cos (Rsqr (x + y))).
-unfold Rsqr in |- *; intros.
+unfold Rsqr; intros.
 unfold cos_in in p_i.
 unfold cos_in in c.
 apply uniqueness_sum with (fun i:nat => cos_n i * ((x + y) * (x + y)) ^ i);
@@ -327,17 +327,17 @@ intro.
 case (Req_dec x 0); intro.
 rewrite H.
 rewrite sin_0.
-unfold B1 in |- *.
-unfold Un_cv in |- *; unfold R_dist in |- *; intros; exists 0%nat; intros.
+unfold B1.
+unfold Un_cv; unfold R_dist; intros; exists 0%nat; intros.
 replace
  (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * 0 ^ (2 * k + 1))
     n) with 0.
-unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
+unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
 induction  n as [| n Hrecn].
-simpl in |- *; ring.
+simpl; ring.
 rewrite tech5; rewrite <- Hrecn.
-simpl in |- *; ring.
-unfold ge in |- *; apply le_O_n.
+simpl; ring.
+unfold ge; apply le_O_n.
 assert (H0 := exist_sin (x * x)).
 elim H0; intros.
 assert (p_i := p).
@@ -347,14 +347,14 @@ unfold R_dist in p.
 cut (sin x = x * x0).
 intro.
 rewrite H1.
-unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+unfold Un_cv; unfold R_dist; intros.
 cut (0 < eps / Rabs x);
  [ intro
- | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+ | unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ] ].
 elim (p (eps / Rabs x) H3); intros.
 exists x1; intros.
-unfold B1 in |- *.
+unfold B1.
 replace
  (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1))
     n) with
@@ -380,11 +380,11 @@ apply sum_eq.
 intros.
 rewrite pow_add.
 rewrite pow_sqr.
-simpl in |- *.
+simpl.
 ring.
-unfold sin in |- *.
+unfold sin.
 case (exist_sin (Rsqr x)).
-unfold Rsqr in |- *; intros.
+unfold Rsqr; intros.
 unfold sin_in in p_i.
 unfold sin_in in s.
 assert
diff --git a/theories/Reals/DiscrR.v b/theories/Reals/DiscrR.v
index 144de09e..1ec399d1 100644
--- a/theories/Reals/DiscrR.v
+++ b/theories/Reals/DiscrR.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -8,7 +8,7 @@
 
 Require Import RIneq.
 Require Import Omega.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Lemma Rlt_R0_R2 : 0 < 2.
 change 2 with (INR 2); apply lt_INR_0; apply lt_O_Sn.
@@ -21,7 +21,7 @@ intros; rewrite H; reflexivity.
 Qed.
 
 Lemma IZR_neq : forall z1 z2:Z, z1 <> z2 -> IZR z1 <> IZR z2.
-intros; red in |- *; intro; elim H; apply eq_IZR; assumption.
+intros; red; intro; elim H; apply eq_IZR; assumption.
 Qed.
 
 Ltac discrR :=
@@ -45,7 +45,7 @@ Ltac prove_sup0 :=
       repeat
        (apply Rmult_lt_0_compat || apply Rplus_lt_pos;
          try apply Rlt_0_1 || apply Rlt_R0_R2)
-  |  |- (?X1 > 0) => change (0 < X1) in |- *; prove_sup0
+  |  |- (?X1 > 0) => change (0 < X1); prove_sup0
   end.
 
 Ltac omega_sup :=
@@ -59,7 +59,7 @@ Ltac omega_sup :=
 
 Ltac prove_sup :=
   match goal with
-  |  |- (?X1 > ?X2) => change (X2 < X1) in |- *; prove_sup
+  |  |- (?X1 > ?X2) => change (X2 < X1); prove_sup
   |  |- (0 < ?X1) => prove_sup0
   |  |- (- ?X1 < 0) => rewrite <- Ropp_0; prove_sup
   |  |- (- ?X1 < - ?X2) => apply Ropp_lt_gt_contravar; prove_sup
diff --git a/theories/Reals/Exp_prop.v b/theories/Reals/Exp_prop.v
index dd97b865..b65ab045 100644
--- a/theories/Reals/Exp_prop.v
+++ b/theories/Reals/Exp_prop.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,23 +9,23 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
-Require Import Rtrigo.
+Require Import Rtrigo1.
 Require Import Ranalysis1.
 Require Import PSeries_reg.
 Require Import Div2.
 Require Import Even.
 Require Import Max.
-Open Local Scope nat_scope.
-Open Local Scope R_scope.
+Local Open Scope nat_scope.
+Local Open Scope R_scope.
 
 Definition E1 (x:R) (N:nat) : R :=
   sum_f_R0 (fun k:nat => / INR (fact k) * x ^ k) N.
 
 Lemma E1_cvg : forall x:R, Un_cv (E1 x) (exp x).
 Proof.
-  intro; unfold exp in |- *; unfold projT1 in |- *.
+  intro; unfold exp; unfold projT1.
   case (exist_exp x); intro.
-  unfold exp_in, Un_cv in |- *; unfold infinite_sum, E1 in |- *; trivial.
+  unfold exp_in, Un_cv; unfold infinite_sum, E1; trivial.
 Qed.
 
 Definition Reste_E (x y:R) (N:nat) : R :=
@@ -41,14 +41,14 @@ Lemma exp_form :
   forall (x y:R) (n:nat),
     (0 < n)%nat -> E1 x n * E1 y n - Reste_E x y n = E1 (x + y) n.
 Proof.
-  intros; unfold E1 in |- *.
+  intros; unfold E1.
   rewrite cauchy_finite.
-  unfold Reste_E in |- *; unfold Rminus in |- *; rewrite Rplus_assoc;
+  unfold Reste_E; unfold Rminus; rewrite Rplus_assoc;
     rewrite Rplus_opp_r; rewrite Rplus_0_r; apply sum_eq;
       intros.
   rewrite binomial.
   rewrite scal_sum; apply sum_eq; intros.
-  unfold C in |- *; unfold Rdiv in |- *; repeat rewrite Rmult_assoc;
+  unfold C; unfold Rdiv; repeat rewrite Rmult_assoc;
     rewrite (Rmult_comm (INR (fact i))); repeat rewrite Rmult_assoc;
       rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r; rewrite Rinv_mult_distr.
@@ -64,27 +64,13 @@ Definition maj_Reste_E (x y:R) (N:nat) : R :=
   (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * N) /
     Rsqr (INR (fact (div2 (pred N))))).
 
-Lemma Rle_Rinv : forall x y:R, 0 < x -> 0 < y -> x <= y -> / y <= / x.
-Proof.
-  intros; apply Rmult_le_reg_l with x.
-  apply H.
-  rewrite <- Rinv_r_sym.
-  apply Rmult_le_reg_l with y.
-  apply H0.
-  rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc;
-    rewrite <- Rinv_l_sym.
-  rewrite Rmult_1_r; apply H1.
-  red in |- *; intro; rewrite H2 in H0; elim (Rlt_irrefl _ H0).
-  red in |- *; intro; rewrite H2 in H; elim (Rlt_irrefl _ H).
-Qed.
-
 (**********)
 Lemma div2_double : forall N:nat, div2 (2 * N) = N.
 Proof.
   intro; induction  N as [| N HrecN].
   reflexivity.
   replace (2 * S N)%nat with (S (S (2 * N))).
-  simpl in |- *; simpl in HrecN; rewrite HrecN; reflexivity.
+  simpl; simpl in HrecN; rewrite HrecN; reflexivity.
   ring.
 Qed.
 
@@ -93,7 +79,7 @@ Proof.
   intro; induction  N as [| N HrecN].
   reflexivity.
   replace (2 * S N)%nat with (S (S (2 * N))).
-  simpl in |- *; simpl in HrecN; rewrite HrecN; reflexivity.
+  simpl; simpl in HrecN; rewrite HrecN; reflexivity.
   ring.
 Qed.
 
@@ -107,7 +93,7 @@ Proof.
   elim H2; intro.
   rewrite <- (even_div2 _ a); apply HrecN; assumption.
   rewrite <- (odd_div2 _ b); apply lt_O_Sn.
-  rewrite H1; simpl in |- *; apply lt_O_Sn.
+  rewrite H1; simpl; apply lt_O_Sn.
   inversion H.
   right; reflexivity.
   left; apply lt_le_trans with 2%nat; [ apply lt_n_Sn | apply H1 ].
@@ -124,7 +110,7 @@ Proof.
       (fun k:nat =>
         sum_f_R0 (fun l:nat => / Rsqr (INR (fact (div2 (S N)))))
         (pred (N - k))) (pred N)).
-  unfold Reste_E in |- *.
+  unfold Reste_E.
   apply Rle_trans with
     (sum_f_R0
       (fun k:nat =>
@@ -203,25 +189,25 @@ Proof.
   apply Rabs_pos.
   apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
   apply RmaxLess1.
-  unfold M in |- *; apply RmaxLess2.
+  unfold M; apply RmaxLess2.
   apply Rle_trans with (M ^ S (n0 + n) * M ^ (N - n0)).
   apply Rmult_le_compat_l.
   apply pow_le; apply Rle_trans with 1.
   left; apply Rlt_0_1.
-  unfold M in |- *; apply RmaxLess1.
+  unfold M; apply RmaxLess1.
   apply pow_incr; split.
   apply Rabs_pos.
   apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
   apply RmaxLess2.
-  unfold M in |- *; apply RmaxLess2.
+  unfold M; apply RmaxLess2.
   rewrite <- pow_add; replace (S (n0 + n) + (N - n0))%nat with (N + S n)%nat.
   apply Rle_pow.
-  unfold M in |- *; apply RmaxLess1.
+  unfold M; apply RmaxLess1.
   replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ].
   apply plus_le_compat_l.
   replace N with (S (pred N)).
   apply le_n_S; apply H0.
-  symmetry  in |- *; apply S_pred with 0%nat; apply H.
+  symmetry ; apply S_pred with 0%nat; apply H.
   apply INR_eq; do 2 rewrite plus_INR; do 2 rewrite S_INR; rewrite plus_INR;
     rewrite minus_INR.
   ring.
@@ -260,7 +246,7 @@ Proof.
   apply pow_le.
   apply Rle_trans with 1.
   left; apply Rlt_0_1.
-  unfold M in |- *; apply RmaxLess1.
+  unfold M; apply RmaxLess1.
   assert (H2 := even_odd_cor N).
   elim H2; intros N0 H3.
   elim H3; intro.
@@ -276,9 +262,9 @@ Proof.
   apply le_n_Sn.
   replace (/ INR (fact n0) * / INR (fact (N - n0))) with
   (C N n0 / INR (fact N)).
-  pattern N at 1 in |- *; rewrite H4.
+  pattern N at 1; rewrite H4.
   apply Rle_trans with (C N N0 / INR (fact N)).
-  unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ INR (fact N))).
+  unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ INR (fact N))).
   apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
   rewrite H4.
@@ -308,7 +294,7 @@ Proof.
   apply le_pred_n.
   replace (C N N0 / INR (fact N)) with (/ Rsqr (INR (fact N0))).
   rewrite H4; rewrite div2_S_double; right; reflexivity.
-  unfold Rsqr, C, Rdiv in |- *.
+  unfold Rsqr, C, Rdiv.
   repeat rewrite Rinv_mult_distr.
   rewrite (Rmult_comm (INR (fact N))).
   repeat rewrite Rmult_assoc.
@@ -316,7 +302,7 @@ Proof.
   rewrite Rmult_1_r; replace (N - N0)%nat with N0.
   ring.
   replace N with (N0 + N0)%nat.
-  symmetry  in |- *; apply minus_plus.
+  symmetry ; apply minus_plus.
   rewrite H4.
   ring.
   apply INR_fact_neq_0.
@@ -324,7 +310,7 @@ Proof.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
-  unfold C, Rdiv in |- *.
+  unfold C, Rdiv.
   rewrite (Rmult_comm (INR (fact N))).
   repeat rewrite Rmult_assoc.
   rewrite <- Rinv_r_sym.
@@ -336,7 +322,7 @@ Proof.
   replace (/ INR (fact (S n0)) * / INR (fact (N - n0))) with
   (C (S N) (S n0) / INR (fact (S N))).
   apply Rle_trans with (C (S N) (S N0) / INR (fact (S N))).
-  unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ INR (fact (S N)))).
+  unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ INR (fact (S N)))).
   apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
   cut (S N = (2 * S N0)%nat).
@@ -371,7 +357,7 @@ Proof.
   replace (C (S N) (S N0) / INR (fact (S N))) with (/ Rsqr (INR (fact (S N0)))).
   rewrite H5; rewrite div2_double.
   right; reflexivity.
-  unfold Rsqr, C, Rdiv in |- *.
+  unfold Rsqr, C, Rdiv.
   repeat rewrite Rinv_mult_distr.
   replace (S N - S N0)%nat with (S N0).
   rewrite (Rmult_comm (INR (fact (S N)))).
@@ -380,14 +366,14 @@ Proof.
   rewrite Rmult_1_r; reflexivity.
   apply INR_fact_neq_0.
   replace (S N) with (S N0 + S N0)%nat.
-  symmetry  in |- *; apply minus_plus.
+  symmetry ; apply minus_plus.
   rewrite H5; ring.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
   rewrite H4; ring.
-  unfold C, Rdiv in |- *.
+  unfold C, Rdiv.
   rewrite (Rmult_comm (INR (fact (S N)))).
   rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
   rewrite Rmult_1_r; rewrite Rinv_mult_distr.
@@ -395,8 +381,8 @@ Proof.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
-  unfold maj_Reste_E in |- *.
-  unfold Rdiv in |- *; rewrite (Rmult_comm 4).
+  unfold maj_Reste_E.
+  unfold Rdiv; rewrite (Rmult_comm 4).
   rewrite Rmult_assoc.
   apply Rmult_le_compat_l.
   apply pow_le.
@@ -447,7 +433,7 @@ Proof.
   cut (INR N <= INR (2 * div2 (S N))).
   intro; apply Rmult_le_reg_l with (Rsqr (INR (div2 (S N)))).
   apply Rsqr_pos_lt.
-  apply not_O_INR; red in |- *; intro.
+  apply not_O_INR; red; intro.
   cut (1 < S N)%nat.
   intro; assert (H4 := div2_not_R0 _ H3).
   rewrite H2 in H4; elim (lt_n_O _ H4).
@@ -470,17 +456,17 @@ Proof.
   apply lt_INR_0; apply div2_not_R0.
   apply lt_n_S; apply H.
   cut (1 < S N)%nat.
-  intro; unfold Rsqr in |- *; apply prod_neq_R0; apply not_O_INR; intro;
+  intro; unfold Rsqr; apply prod_neq_R0; apply not_O_INR; intro;
     assert (H4 := div2_not_R0 _ H2); rewrite H3 in H4;
       elim (lt_n_O _ H4).
   apply lt_n_S; apply H.
   assert (H1 := even_odd_cor N).
   elim H1; intros N0 H2.
   elim H2; intro.
-  pattern N at 2 in |- *; rewrite H3.
+  pattern N at 2; rewrite H3.
   rewrite div2_S_double.
   right; rewrite H3; reflexivity.
-  pattern N at 2 in |- *; rewrite H3.
+  pattern N at 2; rewrite H3.
   replace (S (S (2 * N0))) with (2 * S N0)%nat.
   rewrite div2_double.
   rewrite H3.
@@ -489,12 +475,12 @@ Proof.
   rewrite Rmult_plus_distr_l.
   apply Rplus_le_compat_l.
   rewrite Rmult_1_r.
-  simpl in |- *.
-  pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
+  simpl.
+  pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
     apply Rlt_0_1.
   ring.
-  unfold Rsqr in |- *; apply prod_neq_R0; apply INR_fact_neq_0.
-  unfold Rsqr in |- *; apply prod_neq_R0; apply not_O_INR; discriminate.
+  unfold Rsqr; apply prod_neq_R0; apply INR_fact_neq_0.
+  unfold Rsqr; apply prod_neq_R0; apply not_O_INR; discriminate.
   assert (H0 := even_odd_cor N).
   elim H0; intros N0 H1.
   elim H1; intro.
@@ -520,15 +506,15 @@ Qed.
 Lemma maj_Reste_cv_R0 : forall x y:R, Un_cv (maj_Reste_E x y) 0.
 Proof.
   intros; assert (H := Majxy_cv_R0 x y).
-  unfold Un_cv in H; unfold Un_cv in |- *; intros.
+  unfold Un_cv in H; unfold Un_cv; intros.
   cut (0 < eps / 4);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
         [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
   elim (H _ H1); intros N0 H2.
   exists (max (2 * S N0) 2); intros.
-  unfold R_dist in H2; unfold R_dist in |- *; rewrite Rminus_0_r;
-    unfold Majxy in H2; unfold maj_Reste_E in |- *.
+  unfold R_dist in H2; unfold R_dist; rewrite Rminus_0_r;
+    unfold Majxy in H2; unfold maj_Reste_E.
   rewrite Rabs_right.
   apply Rle_lt_trans with
     (4 *
@@ -536,7 +522,7 @@ Proof.
         INR (fact (div2 (pred n))))).
   apply Rmult_le_compat_l.
   left; prove_sup0.
-  unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr.
+  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr.
   rewrite (Rmult_comm (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)));
     rewrite
       (Rmult_comm (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n)))))
@@ -544,7 +530,7 @@ Proof.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
   apply Rle_trans with (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)).
   rewrite Rmult_comm;
-    pattern (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)) at 2 in |- *;
+    pattern (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)) at 2;
       rewrite <- Rmult_1_r; apply Rmult_le_compat_l.
   apply pow_le; apply Rle_trans with 1.
   left; apply Rlt_0_1.
@@ -598,11 +584,11 @@ Proof.
   (Rabs
     (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) /
       INR (fact (div2 (pred n))) - 0)).
-  apply H2; unfold ge in |- *.
+  apply H2; unfold ge.
   cut (2 * S N0 <= n)%nat.
   intro; apply le_S_n.
   apply INR_le; apply Rmult_le_reg_l with (INR 2).
-  simpl in |- *; prove_sup0.
+  simpl; prove_sup0.
   do 2 rewrite <- mult_INR; apply le_INR.
   apply le_trans with n.
   apply H4.
@@ -620,12 +606,12 @@ Proof.
   apply S_pred with 0%nat; apply H8.
   replace (2 * N1)%nat with (S (S (2 * pred N1))).
   reflexivity.
-  pattern N1 at 2 in |- *; replace N1 with (S (pred N1)).
+  pattern N1 at 2; replace N1 with (S (pred N1)).
   ring.
-  symmetry  in |- *; apply S_pred with 0%nat; apply H8.
+  symmetry ; apply S_pred with 0%nat; apply H8.
   apply INR_lt.
   apply Rmult_lt_reg_l with (INR 2).
-  simpl in |- *; prove_sup0.
+  simpl; prove_sup0.
   rewrite Rmult_0_r; rewrite <- mult_INR.
   apply lt_INR_0.
   rewrite <- H7.
@@ -646,7 +632,7 @@ Proof.
   apply H3.
   rewrite Rminus_0_r; apply Rabs_right.
   apply Rle_ge.
-  unfold Rdiv in |- *; apply Rmult_le_pos. 
+  unfold Rdiv; apply Rmult_le_pos.
   apply pow_le.
   apply Rle_trans with 1.
   left; apply Rlt_0_1.
@@ -654,7 +640,7 @@ Proof.
   left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
   discrR.
   apply Rle_ge.
-  unfold Rdiv in |- *; apply Rmult_le_pos.
+  unfold Rdiv; apply Rmult_le_pos.
   left; prove_sup0.
   apply Rmult_le_pos.
   apply pow_le.
@@ -668,9 +654,9 @@ Qed.
 Lemma Reste_E_cv : forall x y:R, Un_cv (Reste_E x y) 0.
 Proof.
   intros; assert (H := maj_Reste_cv_R0 x y).
-  unfold Un_cv in H; unfold Un_cv in |- *; intros; elim (H _ H0); intros.
+  unfold Un_cv in H; unfold Un_cv; intros; elim (H _ H0); intros.
   exists (max x0 1); intros.
-  unfold R_dist in |- *; rewrite Rminus_0_r.
+  unfold R_dist; rewrite Rminus_0_r.
   apply Rle_lt_trans with (maj_Reste_E x y n).
   apply Reste_E_maj.
   apply lt_le_trans with 1%nat.
@@ -680,10 +666,10 @@ Proof.
   apply H2.
   replace (maj_Reste_E x y n) with (R_dist (maj_Reste_E x y n) 0).
   apply H1.
-  unfold ge in |- *; apply le_trans with (max x0 1).
+  unfold ge; apply le_trans with (max x0 1).
   apply le_max_l.
   apply H2.
-  unfold R_dist in |- *; rewrite Rminus_0_r; apply Rabs_right.
+  unfold R_dist; rewrite Rminus_0_r; apply Rabs_right.
   apply Rle_ge; apply Rle_trans with (Rabs (Reste_E x y n)).
   apply Rabs_pos.
   apply Reste_E_maj.
@@ -704,13 +690,13 @@ Proof.
   apply H1.
   assert (H2 := CV_mult _ _ _ _ H0 H).
   assert (H3 := CV_minus _ _ _ _ H2 (Reste_E_cv x y)).
-  unfold Un_cv in |- *; unfold Un_cv in H3; intros.
+  unfold Un_cv; unfold Un_cv in H3; intros.
   elim (H3 _ H4); intros.
   exists (S x0); intros.
   rewrite <- (exp_form x y n).
   rewrite Rminus_0_r in H5.
   apply H5.
-  unfold ge in |- *; apply le_trans with (S x0).
+  unfold ge; apply le_trans with (S x0).
   apply le_n_Sn.
   apply H6.
   apply lt_le_trans with (S x0).
@@ -724,15 +710,15 @@ Proof.
   intros; set (An := fun N:nat => / INR (fact N) * x ^ N).
   cut (Un_cv (fun n:nat => sum_f_R0 An n) (exp x)).
   intro; apply Rlt_le_trans with (sum_f_R0 An 0).
-  unfold An in |- *; simpl in |- *; rewrite Rinv_1; rewrite Rmult_1_r;
+  unfold An; simpl; rewrite Rinv_1; rewrite Rmult_1_r;
     apply Rlt_0_1.
   apply sum_incr.
   assumption.
-  intro; unfold An in |- *; left; apply Rmult_lt_0_compat.
+  intro; unfold An; left; apply Rmult_lt_0_compat.
   apply Rinv_0_lt_compat; apply INR_fact_lt_0.
   apply (pow_lt _ n H).
-  unfold exp in |- *; unfold projT1 in |- *; case (exist_exp x); intro.
-  unfold exp_in in |- *; unfold infinite_sum, Un_cv in |- *; trivial.
+  unfold exp; unfold projT1; case (exist_exp x); intro.
+  unfold exp_in; unfold infinite_sum, Un_cv; trivial.
 Qed.
 
 (**********)
@@ -743,12 +729,12 @@ Proof.
   apply (exp_pos_pos _ a).
   rewrite <- b; rewrite exp_0; apply Rlt_0_1.
   replace (exp x) with (1 / exp (- x)).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   apply Rlt_0_1.
   apply Rinv_0_lt_compat; apply exp_pos_pos.
   apply (Ropp_0_gt_lt_contravar _ r).
   cut (exp (- x) <> 0).
-  intro; unfold Rdiv in |- *; apply Rmult_eq_reg_l with (exp (- x)).
+  intro; unfold Rdiv; apply Rmult_eq_reg_l with (exp (- x)).
   rewrite Rmult_1_l; rewrite <- Rinv_r_sym.
   rewrite <- exp_plus.
   rewrite Rplus_opp_l; rewrite exp_0; reflexivity.
@@ -756,7 +742,7 @@ Proof.
   apply H.
   assert (H := exp_plus x (- x)).
   rewrite Rplus_opp_r in H; rewrite exp_0 in H.
-  red in |- *; intro; rewrite H0 in H.
+  red; intro; rewrite H0 in H.
   rewrite Rmult_0_r in H.
   elim R1_neq_R0; assumption.
 Qed.
@@ -764,7 +750,7 @@ Qed.
 (* ((exp h)-1)/h -> 0 quand h->0 *)
 Lemma derivable_pt_lim_exp_0 : derivable_pt_lim exp 0 1.
 Proof.
-  unfold derivable_pt_lim in |- *; intros.
+  unfold derivable_pt_lim; intros.
   set (fn := fun (N:nat) (x:R) => x ^ N / INR (fact (S N))).
   cut (CVN_R fn).
   intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }).
@@ -782,41 +768,41 @@ Proof.
   replace 1 with (SFL fn cv 0).
   apply H5.
   split.
-  unfold D_x, no_cond in |- *; split.
+  unfold D_x, no_cond; split.
   trivial.
-  apply (sym_not_eq H6).
+  apply (not_eq_sym H6).
   rewrite Rminus_0_r; apply H7.
-  unfold SFL in |- *.
+  unfold SFL.
   case (cv 0); intros.
   eapply UL_sequence.
   apply u.
-  unfold Un_cv, SP in |- *.
+  unfold Un_cv, SP.
   intros; exists 1%nat; intros.
-  unfold R_dist in |- *; rewrite decomp_sum.
+  unfold R_dist; rewrite decomp_sum.
   rewrite (Rplus_comm (fn 0%nat 0)).
   replace (fn 0%nat 0) with 1.
-  unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_r;
+  unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_r;
     rewrite Rplus_0_r.
   replace (sum_f_R0 (fun i:nat => fn (S i) 0) (pred n)) with 0.
   rewrite Rabs_R0; apply H8.
-  symmetry  in |- *; apply sum_eq_R0; intros.
-  unfold fn in |- *.
-  simpl in |- *.
-  unfold Rdiv in |- *; do 2 rewrite Rmult_0_l; reflexivity.
-  unfold fn in |- *; simpl in |- *.
-  unfold Rdiv in |- *; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity.
+  symmetry ; apply sum_eq_R0; intros.
+  unfold fn.
+  simpl.
+  unfold Rdiv; do 2 rewrite Rmult_0_l; reflexivity.
+  unfold fn; simpl.
+  unfold Rdiv; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity.
   apply lt_le_trans with 1%nat; [ apply lt_n_Sn | apply H9 ].
-  unfold SFL, exp in |- *.
+  unfold SFL, exp.
   case (cv h); case (exist_exp h); simpl; intros.
   eapply UL_sequence.
   apply u.
-  unfold Un_cv in |- *; intros.
+  unfold Un_cv; intros.
   unfold exp_in in e.
   unfold infinite_sum in e.
   cut (0 < eps0 * Rabs h).
   intro; elim (e _ H9); intros N0 H10.
   exists N0; intros.
-  unfold R_dist in |- *.
+  unfold R_dist.
   apply Rmult_lt_reg_l with (Rabs h).
   apply Rabs_pos_lt; assumption.
   rewrite <- Rabs_mult.
@@ -827,47 +813,47 @@ Proof.
   (sum_f_R0 (fun i:nat => / INR (fact i) * h ^ i) (S n) - x).
   rewrite (Rmult_comm (Rabs h)).
   apply H10.
-  unfold ge in |- *.
+  unfold ge.
   apply le_trans with (S N0).
   apply le_n_Sn.
   apply le_n_S; apply H11.
   rewrite decomp_sum.
   replace (/ INR (fact 0) * h ^ 0) with 1.
-  unfold Rminus in |- *.
+  unfold Rminus.
   rewrite Ropp_plus_distr.
   rewrite Ropp_involutive.
   rewrite <- (Rplus_comm (- x)).
   rewrite <- (Rplus_comm (- x + 1)).
   rewrite Rplus_assoc; repeat apply Rplus_eq_compat_l.
   replace (pred (S n)) with n; [ idtac | reflexivity ].
-  unfold SP in |- *.
+  unfold SP.
   rewrite scal_sum.
   apply sum_eq; intros.
-  unfold fn in |- *.
+  unfold fn.
   replace (h ^ S i) with (h * h ^ i).
-  unfold Rdiv in |- *; ring.
-  simpl in |- *; ring.
-  simpl in |- *; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity.
+  unfold Rdiv; ring.
+  simpl; ring.
+  simpl; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity.
   apply lt_O_Sn.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   rewrite <- Rmult_assoc.
-  symmetry  in |- *; apply Rinv_r_simpl_m.
+  symmetry ; apply Rinv_r_simpl_m.
   assumption.
   apply Rmult_lt_0_compat.
   apply H8.
   apply Rabs_pos_lt; assumption.
   apply SFL_continuity; assumption.
-  intro; unfold fn in |- *.
+  intro; unfold fn.
   replace (fun x:R => x ^ n / INR (fact (S n))) with
   (pow_fct n / fct_cte (INR (fact (S n))))%F; [ idtac | reflexivity ].
   apply continuity_div.
   apply derivable_continuous; apply (derivable_pow n).
   apply derivable_continuous; apply derivable_const.
-  intro; unfold fct_cte in |- *; apply INR_fact_neq_0.
+  intro; unfold fct_cte; apply INR_fact_neq_0.
   apply (CVN_R_CVS _ X).
   assert (H0 := Alembert_exp).
-  unfold CVN_R in |- *.
-  intro; unfold CVN_r in |- *.
+  unfold CVN_R.
+  intro; unfold CVN_r.
   exists (fun N:nat => r ^ N / INR (fact (S N))).
   cut
     { l:R |
@@ -879,10 +865,10 @@ Proof.
   exists x; intros.
   split.
   apply p.
-  unfold Boule in |- *; intros.
+  unfold Boule; intros.
   rewrite Rminus_0_r in H1.
-  unfold fn in |- *.
-  unfold Rdiv in |- *; rewrite Rabs_mult.
+  unfold fn.
+  unfold Rdiv; rewrite Rabs_mult.
   cut (0 < INR (fact (S n))).
   intro.
   rewrite (Rabs_right (/ INR (fact (S n)))).
@@ -897,14 +883,14 @@ Proof.
   cut ((r:R) <> 0).
   intro; apply Alembert_C2.
   intro; apply Rabs_no_R0.
-  unfold Rdiv in |- *; apply prod_neq_R0.
+  unfold Rdiv; apply prod_neq_R0.
   apply pow_nonzero; assumption.
   apply Rinv_neq_0_compat; apply INR_fact_neq_0.
   unfold Un_cv in H0.
-  unfold Un_cv in |- *; intros.
+  unfold Un_cv; intros.
   cut (0 < eps0 / r);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
         [ assumption | apply Rinv_0_lt_compat; apply (cond_pos r) ] ].
   elim (H0 _ H3); intros N0 H4.
   exists N0; intros.
@@ -913,7 +899,7 @@ Proof.
   assert (H6 := H4 _ hyp_sn).
   unfold R_dist in H6; rewrite Rminus_0_r in H6.
   rewrite Rabs_Rabsolu in H6.
-  unfold R_dist in |- *; rewrite Rminus_0_r.
+  unfold R_dist; rewrite Rminus_0_r.
   rewrite Rabs_Rabsolu.
   replace
   (Rabs (r ^ S n / INR (fact (S (S n)))) / Rabs (r ^ n / INR (fact (S n))))
@@ -927,7 +913,7 @@ Proof.
   apply H6.
   assumption.
   apply Rle_ge; left; apply (cond_pos r).
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   repeat rewrite Rabs_mult.
   repeat rewrite Rabs_Rinv.
   rewrite Rinv_mult_distr.
@@ -940,7 +926,7 @@ Proof.
   rewrite (Rmult_comm r).
   rewrite <- Rmult_assoc; rewrite <- (Rmult_comm (INR (fact (S n)))).
   apply Rmult_eq_compat_l.
-  simpl in |- *.
+  simpl.
   rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
   ring.
   apply pow_nonzero; assumption.
@@ -953,10 +939,10 @@ Proof.
   apply Rinv_neq_0_compat; apply Rabs_no_R0; apply INR_fact_neq_0.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
-  unfold ge in |- *; apply le_trans with n.
+  unfold ge; apply le_trans with n.
   apply H5.
   apply le_n_Sn.
-  assert (H1 := cond_pos r); red in |- *; intro; rewrite H2 in H1;
+  assert (H1 := cond_pos r); red; intro; rewrite H2 in H1;
     elim (Rlt_irrefl _ H1).
 Qed.
 
@@ -964,10 +950,10 @@ Qed.
 Lemma derivable_pt_lim_exp : forall x:R, derivable_pt_lim exp x (exp x).
 Proof.
   intro; assert (H0 := derivable_pt_lim_exp_0).
-  unfold derivable_pt_lim in H0; unfold derivable_pt_lim in |- *; intros.
+  unfold derivable_pt_lim in H0; unfold derivable_pt_lim; intros.
   cut (0 < eps / exp x);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
         [ apply H | apply Rinv_0_lt_compat; apply exp_pos ] ].
   elim (H0 _ H1); intros del H2.
   exists del; intros.
@@ -981,11 +967,11 @@ Proof.
   rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l; rewrite <- (Rmult_comm eps).
   apply H5.
-  assert (H6 := exp_pos x); red in |- *; intro; rewrite H7 in H6;
+  assert (H6 := exp_pos x); red; intro; rewrite H7 in H6;
     elim (Rlt_irrefl _ H6).
   apply Rle_ge; left; apply exp_pos.
   rewrite Rmult_minus_distr_l.
-  rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rmult_assoc;
+  rewrite Rmult_1_r; unfold Rdiv; rewrite <- Rmult_assoc;
     rewrite Rmult_minus_distr_l.
   rewrite Rmult_1_r; rewrite exp_plus; reflexivity.
 Qed.
diff --git a/theories/Reals/Integration.v b/theories/Reals/Integration.v
index da1742ca..d7b3ab04 100644
--- a/theories/Reals/Integration.v
+++ b/theories/Reals/Integration.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
diff --git a/theories/Reals/LegacyRfield.v b/theories/Reals/LegacyRfield.v
index 49a94021..c45d1c5f 100644
--- a/theories/Reals/LegacyRfield.v
+++ b/theories/Reals/LegacyRfield.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -17,9 +17,9 @@ Open Scope R_scope.
 Lemma RLegacyTheory : Ring_Theory Rplus Rmult 1 0 Ropp (fun x y:R => false).
   split.
   exact Rplus_comm.
-  symmetry  in |- *; apply Rplus_assoc.
+  symmetry ; apply Rplus_assoc.
   exact Rmult_comm.
-  symmetry  in |- *; apply Rmult_assoc.
+  symmetry ; apply Rmult_assoc.
   intro; apply Rplus_0_l.
   intro; apply Rmult_1_l.
   exact Rplus_opp_r.
diff --git a/theories/Reals/MVT.v b/theories/Reals/MVT.v
index 29ebd46d..2ee22b6d 100644
--- a/theories/Reals/MVT.v
+++ b/theories/Reals/MVT.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -10,7 +10,7 @@ Require Import Rbase.
 Require Import Rfunctions.
 Require Import Ranalysis1.
 Require Import Rtopology.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (* The Mean Value Theorem *)
 Theorem MVT :
@@ -55,13 +55,13 @@ Proof.
   split.
   apply Rmult_lt_reg_l with 2.
   prove_sup0.
-  unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
     rewrite <- Rinv_r_sym.
   rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H.
   discrR.
   apply Rmult_lt_reg_l with 2.
   prove_sup0.
-  unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
     rewrite <- Rinv_r_sym.
   rewrite Rmult_1_l; rewrite Rplus_comm; rewrite double;
     apply Rplus_lt_compat_l; apply H.
@@ -103,7 +103,7 @@ Proof.
   inversion H13.
   apply H14.
   rewrite H8 in H10; rewrite <- H14 in H10; elim H10; reflexivity.
-  intros; unfold h in |- *;
+  intros; unfold h;
     replace
     (derive_pt (fun y:R => (g b - g a) * f y - (f b - f a) * g y) c (X c P))
     with
@@ -115,11 +115,11 @@ Proof.
   rewrite derive_pt_minus; do 2 rewrite derive_pt_mult;
     do 2 rewrite derive_pt_const; do 2 rewrite Rmult_0_l;
       do 2 rewrite Rplus_0_l; reflexivity.
-  unfold h in |- *; ring.
-  intros; unfold h in |- *;
+  unfold h; ring.
+  intros; unfold h;
     change
       (continuity_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F)
-        c) in |- *.
+        c).
   apply continuity_pt_minus; apply continuity_pt_mult.
   apply derivable_continuous_pt; apply derivable_const.
   apply H0; apply H3.
@@ -128,7 +128,7 @@ Proof.
   intros;
     change
       (derivable_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F)
-        c) in |- *.
+        c).
   apply derivable_pt_minus; apply derivable_pt_mult.
   apply derivable_pt_const.
   apply (pr1 _ H3).
@@ -178,7 +178,7 @@ Proof.
   cut (derive_pt id x (X2 x x0) = 1).
   cut (derive_pt f x (X0 x x0) = f' x).
   intros; rewrite H4 in H3; rewrite H5 in H3; unfold id in H3;
-    rewrite Rmult_1_r in H3; rewrite Rmult_comm; symmetry  in |- *;
+    rewrite Rmult_1_r in H3; rewrite Rmult_comm; symmetry ;
       assumption.
   apply derive_pt_eq_0; apply H0; elim x0; intros; split; left; assumption.
   apply derive_pt_eq_0; apply derivable_pt_lim_id.
@@ -188,7 +188,7 @@ Proof.
   intros; apply derivable_pt_id.
   intros; apply derivable_continuous_pt; apply X; assumption.
   intros; elim H1; intros; apply X; split; left; assumption.
-  intros; unfold derivable_pt in |- *; exists (f' c); apply H0;
+  intros; unfold derivable_pt; exists (f' c); apply H0;
     apply H1.
 Qed.
 
@@ -221,7 +221,7 @@ Proof.
     unfold id in H6; unfold Rminus in H6; rewrite Rplus_opp_r in H6;
       rewrite Rmult_0_l in H6; apply Rmult_eq_reg_l with (b - a);
         [ rewrite Rmult_0_r; apply H6
-          | apply Rminus_eq_contra; red in |- *; intro; rewrite H7 in H0;
+          | apply Rminus_eq_contra; red; intro; rewrite H7 in H0;
             elim (Rlt_irrefl _ H0) ].
 Qed.
 
@@ -231,7 +231,7 @@ Lemma nonneg_derivative_1 :
     (forall x:R, 0 <= derive_pt f x (pr x)) -> increasing f.
 Proof.
   intros.
-  unfold increasing in |- *.
+  unfold increasing.
   intros.
   case (total_order_T x y); intro.
   elim s; intro.
@@ -268,12 +268,12 @@ Proof.
   intro; decompose [and] H8; intros; generalize (H7 (delta / 2) H9 H12);
     cut ((f (x + delta / 2) - f x) / (delta / 2) <= 0).
   intro; cut (0 < - ((f (x + delta / 2) - f x) / (delta / 2) - l)).
-  intro; unfold Rabs in |- *;
+  intro; unfold Rabs;
     case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)).
   intros;
     generalize
       (Rplus_lt_compat_r (- l) (- ((f (x + delta / 2) - f x) / (delta / 2) - l))
-        (l / 2) H14); unfold Rminus in |- *.
+        (l / 2) H14); unfold Rminus.
   replace (l / 2 + - l) with (- (l / 2)).
   replace (- ((f (x + delta / 2) + - f x) / (delta / 2) + - l) + - l) with
   (- ((f (x + delta / 2) + - f x) / (delta / 2))).
@@ -290,7 +290,7 @@ Proof.
     (Rlt_irrefl 0
       (Rlt_le_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) 0 H17 H10)).
   ring.
-  pattern l at 3 in |- *; rewrite double_var.
+  pattern l at 3; rewrite double_var.
   ring.
   intros.
   generalize
@@ -303,22 +303,22 @@ Proof.
         H15)).
   replace (- ((f (x + delta / 2) - f x) / (delta / 2) - l)) with
   ((f x - f (x + delta / 2)) / (delta / 2) + l).
-  unfold Rminus in |- *.
+  unfold Rminus.
   apply Rplus_le_lt_0_compat.
-  unfold Rdiv in |- *; apply Rmult_le_pos.
+  unfold Rdiv; apply Rmult_le_pos.
   cut (x <= x + delta * / 2).
   intro; generalize (H0 x (x + delta * / 2) H13); intro;
     generalize
       (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H14);
       rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
-  pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
+  pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
     left; assumption.
   left; apply Rinv_0_lt_compat; assumption.
   assumption.
   rewrite Ropp_minus_distr.
-  unfold Rminus in |- *.
+  unfold Rminus.
   rewrite (Rplus_comm l).
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   rewrite <- Ropp_mult_distr_l_reverse.
   rewrite Ropp_plus_distr.
   rewrite Ropp_involutive.
@@ -329,38 +329,38 @@ Proof.
   rewrite <- Ropp_0.
   apply Ropp_ge_le_contravar.
   apply Rle_ge.
-  unfold Rdiv in |- *; apply Rmult_le_pos.
+  unfold Rdiv; apply Rmult_le_pos.
   cut (x <= x + delta * / 2).
   intro; generalize (H0 x (x + delta * / 2) H10); intro.
   generalize
     (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H13);
     rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
-  pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
+  pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
     left; assumption.
   left; apply Rinv_0_lt_compat; assumption.
-  unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse.
+  unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse.
   rewrite Ropp_minus_distr.
   reflexivity.
   split.
-  unfold Rdiv in |- *; apply prod_neq_R0.
-  generalize (cond_pos delta); intro; red in |- *; intro H9; rewrite H9 in H8;
+  unfold Rdiv; apply prod_neq_R0.
+  generalize (cond_pos delta); intro; red; intro H9; rewrite H9 in H8;
     elim (Rlt_irrefl 0 H8).
   apply Rinv_neq_0_compat; discrR.
   split.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
   rewrite Rabs_right.
-  unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2.
+  unfold Rdiv; apply Rmult_lt_reg_l with 2.
   prove_sup0.
   rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
-  rewrite Rmult_1_l; rewrite double; pattern (pos delta) at 1 in |- *;
+  rewrite Rmult_1_l; rewrite double; pattern (pos delta) at 1;
     rewrite <- Rplus_0_r.
   apply Rplus_lt_compat_l; apply (cond_pos delta).
   discrR.
-  apply Rle_ge; unfold Rdiv in |- *; left; apply Rmult_lt_0_compat.
+  apply Rle_ge; unfold Rdiv; left; apply Rmult_lt_0_compat.
   apply (cond_pos delta).
   apply Rinv_0_lt_compat; prove_sup0.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply H4 | apply Rinv_0_lt_compat; prove_sup0 ].
 Qed.
 
@@ -368,7 +368,7 @@ Qed.
 Lemma increasing_decreasing_opp :
   forall f:R -> R, increasing f -> decreasing (- f)%F.
 Proof.
-  unfold increasing, decreasing, opp_fct in |- *; intros; generalize (H x y H0);
+  unfold increasing, decreasing, opp_fct; intros; generalize (H x y H0);
     intro; apply Ropp_ge_le_contravar; apply Rle_ge; assumption.
 Qed.
 
@@ -381,8 +381,8 @@ Proof.
   cut (forall h:R, - - f h = f h).
   intro.
   generalize (increasing_decreasing_opp (- f)%F).
-  unfold decreasing in |- *.
-  unfold opp_fct in |- *.
+  unfold decreasing.
+  unfold opp_fct.
   intros.
   rewrite <- (H0 x); rewrite <- (H0 y).
   apply H1.
@@ -410,7 +410,7 @@ Lemma positive_derivative :
     (forall x:R, 0 < derive_pt f x (pr x)) -> strict_increasing f.
 Proof.
   intros.
-  unfold strict_increasing in |- *.
+  unfold strict_increasing.
   intros.
   apply Rplus_lt_reg_r with (- f x).
   rewrite Rplus_opp_l; rewrite Rplus_comm.
@@ -429,7 +429,7 @@ Qed.
 Lemma strictincreasing_strictdecreasing_opp :
   forall f:R -> R, strict_increasing f -> strict_decreasing (- f)%F.
 Proof.
-  unfold strict_increasing, strict_decreasing, opp_fct in |- *; intros;
+  unfold strict_increasing, strict_decreasing, opp_fct; intros;
     generalize (H x y H0); intro; apply Ropp_lt_gt_contravar;
       assumption.
 Qed.
@@ -443,7 +443,7 @@ Proof.
   cut (forall h:R, - - f h = f h).
   intros.
   generalize (strictincreasing_strictdecreasing_opp (- f)%F).
-  unfold strict_decreasing, opp_fct in |- *.
+  unfold strict_decreasing, opp_fct.
   intros.
   rewrite <- (H0 x).
   rewrite <- (H0 y).
@@ -470,8 +470,8 @@ Proof.
   intros.
   unfold constant in H.
   apply derive_pt_eq_0.
-  intros; exists (mkposreal 1 Rlt_0_1); simpl in |- *; intros.
-  rewrite (H x (x + h)); unfold Rminus in |- *; unfold Rdiv in |- *;
+  intros; exists (mkposreal 1 Rlt_0_1); simpl; intros.
+  rewrite (H x (x + h)); unfold Rminus; unfold Rdiv;
     rewrite Rplus_opp_r; rewrite Rmult_0_l; rewrite Rplus_opp_r;
       rewrite Rabs_R0; assumption.
 Qed.
@@ -480,13 +480,13 @@ Qed.
 Lemma increasing_decreasing :
   forall f:R -> R, increasing f -> decreasing f -> constant f.
 Proof.
-  unfold increasing, decreasing, constant in |- *; intros;
+  unfold increasing, decreasing, constant; intros;
     case (Rtotal_order x y); intro.
   generalize (Rlt_le x y H1); intro;
     apply (Rle_antisym (f x) (f y) (H x y H2) (H0 x y H2)).
   elim H1; intro.
   rewrite H2; reflexivity.
-  generalize (Rlt_le y x H2); intro; symmetry  in |- *;
+  generalize (Rlt_le y x H2); intro; symmetry ;
     apply (Rle_antisym (f y) (f x) (H y x H3) (H0 y x H3)).
 Qed.
 
@@ -502,7 +502,7 @@ Proof.
   assert (H2 := nonneg_derivative_1 f pr H0).
   assert (H3 := nonpos_derivative_1 f pr H1).
   apply increasing_decreasing; assumption.
-  intro; right; symmetry  in |- *; apply (H x).
+  intro; right; symmetry ; apply (H x).
   intro; right; apply (H x).
 Qed.
 
@@ -601,7 +601,7 @@ Proof.
   elim H4; intros.
   split; left; assumption.
   rewrite b0.
-  unfold Rminus in |- *; do 2 rewrite Rplus_opp_r.
+  unfold Rminus; do 2 rewrite Rplus_opp_r.
   rewrite Rmult_0_r; right; reflexivity.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)).
 Qed.
@@ -648,7 +648,7 @@ Lemma null_derivative_loc :
     (forall (x:R) (P:a < x < b), derive_pt f x (pr x P) = 0) ->
     constant_D_eq f (fun x:R => a <= x <= b) (f a).
 Proof.
-  intros; unfold constant_D_eq in |- *; intros; case (total_order_T a b); intro.
+  intros; unfold constant_D_eq; intros; case (total_order_T a b); intro.
   elim s; intro.
   assert (H2 : forall y:R, a < y < x -> derivable_pt id y).
   intros; apply derivable_pt_id.
@@ -674,7 +674,7 @@ Proof.
   assert (H12 : derive_pt id x0 (H2 x0 x1) = 1).
   apply derive_pt_eq_0; apply derivable_pt_lim_id.
   rewrite H11 in H9; rewrite H12 in H9; rewrite Rmult_0_r in H9;
-    rewrite Rmult_1_r in H9; apply Rminus_diag_uniq; symmetry  in |- *;
+    rewrite Rmult_1_r in H9; apply Rminus_diag_uniq; symmetry ;
       assumption.
   rewrite H1; reflexivity.
   assert (H2 : x = a).
@@ -691,15 +691,15 @@ Lemma antiderivative_Ucte :
     antiderivative f g2 a b ->
     exists c : R, (forall x:R, a <= x <= b -> g1 x = g2 x + c).
 Proof.
-  unfold antiderivative in |- *; intros; elim H; clear H; intros; elim H0;
+  unfold antiderivative; intros; elim H; clear H; intros; elim H0;
     clear H0; intros H0 _; exists (g1 a - g2 a); intros;
       assert (H3 : forall x:R, a <= x <= b -> derivable_pt g1 x).
-  intros; unfold derivable_pt in |- *; exists (f x0); elim (H x0 H3);
-    intros; eapply derive_pt_eq_1; symmetry  in |- *;
+  intros; unfold derivable_pt; exists (f x0); elim (H x0 H3);
+    intros; eapply derive_pt_eq_1; symmetry ;
       apply H4.
   assert (H4 : forall x:R, a <= x <= b -> derivable_pt g2 x).
-  intros; unfold derivable_pt in |- *; exists (f x0);
-    elim (H0 x0 H4); intros; eapply derive_pt_eq_1; symmetry  in |- *;
+  intros; unfold derivable_pt; exists (f x0);
+    elim (H0 x0 H4); intros; eapply derive_pt_eq_1; symmetry ;
       apply H5.
   assert (H5 : forall x:R, a < x < b -> derivable_pt (g1 - g2) x).
   intros; elim H5; intros; apply derivable_pt_minus;
@@ -713,7 +713,7 @@ Proof.
   assert (H9 : a <= x0 <= b).
   split; left; assumption.
   apply derivable_pt_lim_minus; [ elim (H _ H9) | elim (H0 _ H9) ]; intros;
-    eapply derive_pt_eq_1; symmetry  in |- *; apply H10.
+    eapply derive_pt_eq_1; symmetry ; apply H10.
   assert (H8 := null_derivative_loc (g1 - g2)%F a b H5 H6 H7);
     unfold constant_D_eq in H8; assert (H9 := H8 _ H2);
       unfold minus_fct in H9; rewrite <- H9; ring.
diff --git a/theories/Reals/Machin.v b/theories/Reals/Machin.v
new file mode 100644
index 00000000..6b91719d
--- /dev/null
+++ b/theories/Reals/Machin.v
@@ -0,0 +1,168 @@
+Require Import Fourier.
+Require Import Rbase.
+Require Import Rtrigo1.
+Require Import Ranalysis_reg.
+Require Import Rfunctions.
+Require Import AltSeries.
+Require Import Rseries.
+Require Import SeqProp.
+Require Import PartSum.
+Require Import Ratan.
+
+Local Open Scope R_scope.
+
+(* Proving a few formulas in the style of John Machin to compute Pi *)
+
+Definition atan_sub u v := (u - v)/(1 + u * v).
+
+Lemma atan_sub_correct :
+ forall u v, 1 + u * v <> 0 -> -PI/2 < atan u - atan v < PI/2 ->
+   -PI/2 < atan (atan_sub u v) < PI/2 ->
+   atan u = atan v + atan (atan_sub u v).
+intros u v pn0 uvint aint.
+assert (cos (atan u) <> 0).
+ destruct (atan_bound u); apply Rgt_not_eq, cos_gt_0; auto.
+ rewrite <- Ropp_div; assumption.
+assert (cos (atan v) <> 0).
+ destruct (atan_bound v); apply Rgt_not_eq, cos_gt_0; auto.
+ rewrite <- Ropp_div; assumption.
+assert (t : forall a b c, a - b = c -> a = b + c) by (intros; subst; field).
+apply t, tan_is_inj; clear t; try assumption.
+rewrite tan_minus; auto.
+  rewrite !atan_right_inv; reflexivity.
+apply Rgt_not_eq, cos_gt_0; rewrite <- ?Ropp_div; tauto.
+rewrite !atan_right_inv; assumption.
+Qed.
+
+Lemma tech : forall x y , -1 <= x <= 1 -> -1 < y < 1 -> 
+  -PI/2 < atan x - atan y < PI/2.
+assert (ut := PI_RGT_0).
+intros x y [xm1 x1] [ym1 y1].
+assert (-(PI/4) <= atan x).
+ destruct xm1 as [xm1 | xm1].
+  rewrite <- atan_1, <- atan_opp; apply Rlt_le, atan_increasing.
+  assumption.
+ solve[rewrite <- xm1, atan_opp, atan_1; apply Rle_refl].
+assert (-(PI/4) < atan y).
+ rewrite <- atan_1, <- atan_opp; apply atan_increasing.
+ assumption.
+assert (atan x <= PI/4).
+ destruct x1 as [x1 | x1].
+  rewrite <- atan_1; apply Rlt_le, atan_increasing.
+  assumption.
+ solve[rewrite x1, atan_1; apply Rle_refl].
+assert (atan y < PI/4).
+  rewrite <- atan_1; apply atan_increasing.
+  assumption.
+rewrite Ropp_div; split; fourier.
+Qed.
+
+(* A simple formula, reasonably efficient. *)
+Lemma Machin_2_3 : PI/4 = atan(/2) + atan(/3).
+assert (utility : 0 < PI/2) by (apply PI2_RGT_0).
+rewrite <- atan_1.
+rewrite (atan_sub_correct 1 (/2)).
+   apply f_equal, f_equal; unfold atan_sub; field.
+  apply Rgt_not_eq; fourier.
+ apply tech; try split; try fourier.
+apply atan_bound.
+Qed.
+
+Lemma Machin_4_5_239 : PI/4 = 4 * atan (/5) - atan(/239).
+rewrite <- atan_1.
+rewrite (atan_sub_correct 1 (/5));
+ [ | apply Rgt_not_eq; fourier | apply tech; try split; fourier |
+     apply atan_bound ].
+replace (4 * atan (/5) - atan (/239)) with
+  (atan (/5) + (atan (/5) + (atan (/5) + (atan (/5) + -
+     atan (/239))))) by ring.
+apply f_equal.
+replace (atan_sub 1 (/5)) with (2/3) by
+  (unfold atan_sub; field).
+rewrite (atan_sub_correct (2/3) (/5));
+ [apply f_equal | apply Rgt_not_eq; fourier | apply tech; try split; fourier |
+     apply atan_bound ].
+replace (atan_sub (2/3) (/5)) with (7/17)  by
+  (unfold atan_sub; field).
+rewrite (atan_sub_correct (7/17) (/5));
+ [apply f_equal | apply Rgt_not_eq; fourier | apply tech; try split; fourier |
+     apply atan_bound ].
+replace (atan_sub (7/17) (/5)) with (9/46) by
+  (unfold atan_sub; field).
+rewrite (atan_sub_correct (9/46) (/5));
+ [apply f_equal | apply Rgt_not_eq; fourier | apply tech; try split; fourier |
+     apply atan_bound ].
+rewrite <- atan_opp; apply f_equal.
+unfold atan_sub; field.
+Qed.
+
+Lemma Machin_2_3_7 : PI/4 = 2 * atan(/3) + (atan (/7)).
+rewrite <- atan_1.
+rewrite (atan_sub_correct 1 (/3));
+ [ | apply Rgt_not_eq; fourier | apply tech; try split; fourier |
+     apply atan_bound ].
+replace (2 * atan (/3) + atan (/7)) with
+  (atan (/3) + (atan (/3) + atan (/7))) by ring.
+apply f_equal.
+replace (atan_sub 1 (/3)) with (/2) by
+  (unfold atan_sub; field).
+rewrite (atan_sub_correct (/2) (/3));
+ [apply f_equal | apply Rgt_not_eq; fourier | apply tech; try split; fourier |
+     apply atan_bound ].
+apply f_equal; unfold atan_sub; field.
+Qed.
+
+(* More efficient way to compute approximations of PI. *)
+
+Definition PI_2_3_7_tg n := 
+  2 * Ratan_seq (/3) n + Ratan_seq (/7) n.
+
+Lemma PI_2_3_7_ineq :
+  forall N : nat,
+    sum_f_R0 (tg_alt PI_2_3_7_tg) (S (2 * N)) <= PI / 4 <=
+    sum_f_R0 (tg_alt PI_2_3_7_tg) (2 * N).
+Proof.
+assert (dec3 : 0 <= /3 <= 1) by (split; fourier).
+assert (dec7 : 0 <= /7 <= 1) by (split; fourier).
+assert (decr : Un_decreasing PI_2_3_7_tg).
+ apply Ratan_seq_decreasing in dec3.
+ apply Ratan_seq_decreasing in dec7.
+ intros n; apply Rplus_le_compat.
+  apply Rmult_le_compat_l; [ fourier | exact (dec3 n)].
+ exact (dec7 n).
+assert (cv : Un_cv PI_2_3_7_tg 0).
+ apply Ratan_seq_converging in dec3.
+ apply Ratan_seq_converging in dec7.
+ intros eps ep.
+ assert (ep' : 0 < eps /3) by fourier.
+ destruct (dec3 _ ep') as [N1 Pn1]; destruct (dec7 _ ep') as [N2 Pn2].
+ exists (N1 + N2)%nat; intros n Nn.
+ unfold PI_2_3_7_tg.
+ rewrite <- (Rplus_0_l 0).
+ apply Rle_lt_trans with
+    (1 := R_dist_plus (2 * Ratan_seq (/3) n) 0 (Ratan_seq (/7) n) 0).
+ replace eps with (2 * eps/3 + eps/3) by field.
+ apply Rplus_lt_compat.
+  unfold R_dist, Rminus, Rdiv.
+  rewrite <- (Rmult_0_r 2), <- Ropp_mult_distr_r_reverse.
+  rewrite <- Rmult_plus_distr_l, Rabs_mult, (Rabs_pos_eq 2);[|fourier].
+  rewrite Rmult_assoc; apply Rmult_lt_compat_l;[fourier | ].
+   apply (Pn1 n); omega.
+  apply (Pn2 n); omega.
+rewrite Machin_2_3_7.
+rewrite !atan_eq_ps_atan; try (split; fourier).
+unfold ps_atan; destruct (in_int (/3)); destruct (in_int (/7));
+  try match goal with id : ~ _ |- _ => case id; split; fourier end.
+destruct (ps_atan_exists_1 (/3)) as [v3 Pv3].
+destruct (ps_atan_exists_1 (/7)) as [v7 Pv7].
+assert (main : Un_cv (sum_f_R0 (tg_alt PI_2_3_7_tg)) (2 * v3 + v7)).
+ assert (main :Un_cv (fun n => 2 * sum_f_R0 (tg_alt (Ratan_seq (/3))) n +
+                        sum_f_R0 (tg_alt (Ratan_seq (/7))) n) (2 * v3 + v7)).
+  apply CV_plus;[ | assumption].
+  apply CV_mult;[ | assumption].
+  exists 0%nat; intros; rewrite R_dist_eq; assumption.
+ apply Un_cv_ext with (2 := main).
+ intros n; rewrite scal_sum, <- plus_sum; apply sum_eq; intros.
+ rewrite Rmult_comm; unfold PI_2_3_7_tg, tg_alt; field.
+intros N; apply (alternated_series_ineq _ _ _ decr cv main).
+Qed.
diff --git a/theories/Reals/NewtonInt.v b/theories/Reals/NewtonInt.v
index a4233021..67e353ee 100644
--- a/theories/Reals/NewtonInt.v
+++ b/theories/Reals/NewtonInt.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,9 +9,9 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
-Require Import Rtrigo.
+Require Import Rtrigo1.
 Require Import Ranalysis.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (*******************************************)
 (*            Newton's Integral            *)
@@ -28,8 +28,8 @@ Lemma FTCN_step1 :
   forall (f:Differential) (a b:R),
     Newton_integrable (fun x:R => derive_pt f x (cond_diff f x)) a b.
 Proof.
-  intros f a b; unfold Newton_integrable in |- *; exists (d1 f);
-    unfold antiderivative in |- *; intros; case (Rle_dec a b);
+  intros f a b; unfold Newton_integrable; exists (d1 f);
+    unfold antiderivative; intros; case (Rle_dec a b);
       intro;
         [ left; split; [ intros; exists (cond_diff f x); reflexivity | assumption ]
           | right; split;
@@ -42,26 +42,26 @@ Lemma FTC_Newton :
     NewtonInt (fun x:R => derive_pt f x (cond_diff f x)) a b
     (FTCN_step1 f a b) = f b - f a.
 Proof.
-  intros; unfold NewtonInt in |- *; reflexivity.
+  intros; unfold NewtonInt; reflexivity.
 Qed.
 
 (* $\int_a^a f$ exists forall a:R and f:R->R *)
 Lemma NewtonInt_P1 : forall (f:R -> R) (a:R), Newton_integrable f a a.
 Proof.
-  intros f a; unfold Newton_integrable in |- *;
+  intros f a; unfold Newton_integrable;
     exists (fct_cte (f a) * id)%F; left;
-      unfold antiderivative in |- *; split.
+      unfold antiderivative; split.
   intros; assert (H1 : derivable_pt (fct_cte (f a) * id) x).
   apply derivable_pt_mult.
   apply derivable_pt_const.
   apply derivable_pt_id.
   exists H1; assert (H2 : x = a).
   elim H; intros; apply Rle_antisym; assumption.
-  symmetry  in |- *; apply derive_pt_eq_0;
+  symmetry ; apply derive_pt_eq_0;
     replace (f x) with (0 * id x + fct_cte (f a) x * 1);
     [ apply (derivable_pt_lim_mult (fct_cte (f a)) id x);
       [ apply derivable_pt_lim_const | apply derivable_pt_lim_id ]
-      | unfold id, fct_cte in |- *; rewrite H2; ring ].
+      | unfold id, fct_cte; rewrite H2; ring ].
   right; reflexivity.
 Defined.
 
@@ -69,8 +69,8 @@ Defined.
 Lemma NewtonInt_P2 :
   forall (f:R -> R) (a:R), NewtonInt f a a (NewtonInt_P1 f a) = 0.
 Proof.
-  intros; unfold NewtonInt in |- *; simpl in |- *;
-    unfold mult_fct, fct_cte, id in |- *; ring.
+  intros; unfold NewtonInt; simpl;
+    unfold mult_fct, fct_cte, id; ring.
 Qed.
 
 (* If $\int_a^b f$ exists, then $\int_b^a f$ exists too *)
@@ -78,7 +78,7 @@ Lemma NewtonInt_P3 :
   forall (f:R -> R) (a b:R) (X:Newton_integrable f a b),
     Newton_integrable f b a.
 Proof.
-  unfold Newton_integrable in |- *; intros; elim X; intros g H;
+  unfold Newton_integrable; intros; elim X; intros g H;
     exists g; tauto.
 Defined.
 
@@ -88,7 +88,7 @@ Lemma NewtonInt_P4 :
     NewtonInt f a b pr = - NewtonInt f b a (NewtonInt_P3 f a b pr).
 Proof.
   intros; unfold Newton_integrable in pr; elim pr; intros; elim p; intro.
-  unfold NewtonInt in |- *;
+  unfold NewtonInt;
     case
     (NewtonInt_P3 f a b
       (exist
@@ -106,7 +106,7 @@ Proof.
   assert (H4 : a <= b <= b).
   split; [ assumption | right; reflexivity ].
   assert (H5 := H2 _ H3); assert (H6 := H2 _ H4); rewrite H5; rewrite H6; ring.
-  unfold NewtonInt in |- *;
+  unfold NewtonInt;
     case
     (NewtonInt_P3 f a b
       (exist
@@ -132,37 +132,37 @@ Lemma NewtonInt_P5 :
     Newton_integrable g a b ->
     Newton_integrable (fun x:R => l * f x + g x) a b.
 Proof.
-  unfold Newton_integrable in |- *; intros f g l a b X X0;
+  unfold Newton_integrable; intros f g l a b X X0;
     elim X; intros; elim X0; intros;
       exists (fun y:R => l * x y + x0 y).
   elim p; intro.
   elim p0; intro.
-  left; unfold antiderivative in |- *; unfold antiderivative in H, H0; elim H;
+  left; unfold antiderivative; unfold antiderivative in H, H0; elim H;
     clear H; intros; elim H0; clear H0; intros H0 _.
   split.
   intros; elim (H _ H2); elim (H0 _ H2); intros.
   assert (H5 : derivable_pt (fun y:R => l * x y + x0 y) x1).
   reg.
-  exists H5; symmetry  in |- *; reg; rewrite <- H3; rewrite <- H4; reflexivity.
+  exists H5; symmetry ; reg; rewrite <- H3; rewrite <- H4; reflexivity.
   assumption.
   unfold antiderivative in H, H0; elim H; elim H0; intros; elim H4; intro.
   elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H5 H2)).
-  left; rewrite <- H5; unfold antiderivative in |- *; split.
+  left; rewrite <- H5; unfold antiderivative; split.
   intros; elim H6; intros; assert (H9 : x1 = a).
   apply Rle_antisym; assumption.
   assert (H10 : a <= x1 <= b).
-  split; right; [ symmetry  in |- *; assumption | rewrite <- H5; assumption ].
+  split; right; [ symmetry ; assumption | rewrite <- H5; assumption ].
   assert (H11 : b <= x1 <= a).
-  split; right; [ rewrite <- H5; symmetry  in |- *; assumption | assumption ].
+  split; right; [ rewrite <- H5; symmetry ; assumption | assumption ].
   assert (H12 : derivable_pt x x1).
-  unfold derivable_pt in |- *; exists (f x1); elim (H3 _ H10); intros;
-    eapply derive_pt_eq_1; symmetry  in |- *; apply H12.
+  unfold derivable_pt; exists (f x1); elim (H3 _ H10); intros;
+    eapply derive_pt_eq_1; symmetry ; apply H12.
   assert (H13 : derivable_pt x0 x1).
-  unfold derivable_pt in |- *; exists (g x1); elim (H1 _ H11); intros;
-    eapply derive_pt_eq_1; symmetry  in |- *; apply H13.
+  unfold derivable_pt; exists (g x1); elim (H1 _ H11); intros;
+    eapply derive_pt_eq_1; symmetry ; apply H13.
   assert (H14 : derivable_pt (fun y:R => l * x y + x0 y) x1).
   reg.
-  exists H14; symmetry  in |- *; reg.
+  exists H14; symmetry ; reg.
   assert (H15 : derive_pt x0 x1 H13 = g x1).
   elim (H1 _ H11); intros; rewrite H15; apply pr_nu.
   assert (H16 : derive_pt x x1 H12 = f x1).
@@ -172,34 +172,34 @@ Proof.
   elim p0; intro.
   unfold antiderivative in H, H0; elim H; elim H0; intros; elim H4; intro.
   elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H5 H2)).
-  left; rewrite H5; unfold antiderivative in |- *; split.
+  left; rewrite H5; unfold antiderivative; split.
   intros; elim H6; intros; assert (H9 : x1 = a).
   apply Rle_antisym; assumption.
   assert (H10 : a <= x1 <= b).
-  split; right; [ symmetry  in |- *; assumption | rewrite H5; assumption ].
+  split; right; [ symmetry ; assumption | rewrite H5; assumption ].
   assert (H11 : b <= x1 <= a).
-  split; right; [ rewrite H5; symmetry  in |- *; assumption | assumption ].
+  split; right; [ rewrite H5; symmetry ; assumption | assumption ].
   assert (H12 : derivable_pt x x1).
-  unfold derivable_pt in |- *; exists (f x1); elim (H3 _ H11); intros;
-    eapply derive_pt_eq_1; symmetry  in |- *; apply H12.
+  unfold derivable_pt; exists (f x1); elim (H3 _ H11); intros;
+    eapply derive_pt_eq_1; symmetry ; apply H12.
   assert (H13 : derivable_pt x0 x1).
-  unfold derivable_pt in |- *; exists (g x1); elim (H1 _ H10); intros;
-    eapply derive_pt_eq_1; symmetry  in |- *; apply H13.
+  unfold derivable_pt; exists (g x1); elim (H1 _ H10); intros;
+    eapply derive_pt_eq_1; symmetry ; apply H13.
   assert (H14 : derivable_pt (fun y:R => l * x y + x0 y) x1).
   reg.
-  exists H14; symmetry  in |- *; reg.
+  exists H14; symmetry ; reg.
   assert (H15 : derive_pt x0 x1 H13 = g x1).
   elim (H1 _ H10); intros; rewrite H15; apply pr_nu.
   assert (H16 : derive_pt x x1 H12 = f x1).
   elim (H3 _ H11); intros; rewrite H16; apply pr_nu.
   rewrite H15; rewrite H16; ring.
   right; reflexivity.
-  right; unfold antiderivative in |- *; unfold antiderivative in H, H0; elim H;
+  right; unfold antiderivative; unfold antiderivative in H, H0; elim H;
     clear H; intros; elim H0; clear H0; intros H0 _; split.
   intros; elim (H _ H2); elim (H0 _ H2); intros.
   assert (H5 : derivable_pt (fun y:R => l * x y + x0 y) x1).
   reg.
-  exists H5; symmetry  in |- *; reg; rewrite <- H3; rewrite <- H4; reflexivity.
+  exists H5; symmetry ; reg; rewrite <- H3; rewrite <- H4; reflexivity.
   assumption.
 Defined.
 
@@ -210,12 +210,12 @@ Lemma antiderivative_P1 :
     antiderivative g G a b ->
     antiderivative (fun x:R => l * f x + g x) (fun x:R => l * F x + G x) a b.
 Proof.
-  unfold antiderivative in |- *; intros; elim H; elim H0; clear H H0; intros;
+  unfold antiderivative; intros; elim H; elim H0; clear H H0; intros;
     split.
   intros; elim (H _ H3); elim (H1 _ H3); intros.
   assert (H6 : derivable_pt (fun x:R => l * F x + G x) x).
   reg.
-  exists H6; symmetry  in |- *; reg; rewrite <- H4; rewrite <- H5; ring.
+  exists H6; symmetry ; reg; rewrite <- H4; rewrite <- H5; ring.
   assumption.
 Qed.
 
@@ -226,7 +226,7 @@ Lemma NewtonInt_P6 :
     NewtonInt (fun x:R => l * f x + g x) a b (NewtonInt_P5 f g l a b pr1 pr2) =
     l * NewtonInt f a b pr1 + NewtonInt g a b pr2.
 Proof.
-  intros f g l a b pr1 pr2; unfold NewtonInt in |- *;
+  intros f g l a b pr1 pr2; unfold NewtonInt;
     case (NewtonInt_P5 f g l a b pr1 pr2); intros; case pr1;
       intros; case pr2; intros; case (total_order_T a b);
         intro.
@@ -277,7 +277,7 @@ Lemma antiderivative_P2 :
         | right _ => F1 x + (F0 b - F1 b)
       end) a c.
 Proof.
-  unfold antiderivative in |- *; intros; elim H; clear H; intros; elim H0;
+  unfold antiderivative; intros; elim H; clear H; intros; elim H0;
     clear H0; intros; split.
   2: apply Rle_trans with b; assumption.
   intros; elim H3; clear H3; intros; case (total_order_T x b); intro.
@@ -293,25 +293,25 @@ Proof.
             | left _ => F0 x
             | right _ => F1 x + (F0 b - F1 b)
           end) x (f x)).
-  unfold derivable_pt_lim in |- *; assert (H7 : derive_pt F0 x x0 = f x).
-  symmetry  in |- *; assumption.
+  unfold derivable_pt_lim; assert (H7 : derive_pt F0 x x0 = f x).
+  symmetry ; assumption.
   assert (H8 := derive_pt_eq_1 F0 x (f x) x0 H7); unfold derivable_pt_lim in H8;
     intros; elim (H8 _ H9); intros; set (D := Rmin x1 (b - x)).
   assert (H11 : 0 < D).
-  unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x1 (b - x)); intro.
+  unfold D; unfold Rmin; case (Rle_dec x1 (b - x)); intro.
   apply (cond_pos x1).
   apply Rlt_Rminus; assumption.
   exists (mkposreal _ H11); intros; case (Rle_dec x b); intro.
   case (Rle_dec (x + h) b); intro.
   apply H10.
   assumption.
-  apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_l ].
+  apply Rlt_le_trans with D; [ assumption | unfold D; apply Rmin_l ].
   elim n; left; apply Rlt_le_trans with (x + D).
   apply Rplus_lt_compat_l; apply Rle_lt_trans with (Rabs h).
   apply RRle_abs.
   apply H13.
   apply Rplus_le_reg_l with (- x); rewrite <- Rplus_assoc; rewrite Rplus_opp_l;
-    rewrite Rplus_0_l; rewrite Rplus_comm; unfold D in |- *;
+    rewrite Rplus_0_l; rewrite Rplus_comm; unfold D;
       apply Rmin_r.
   elim n; left; assumption.
   assert
@@ -322,16 +322,16 @@ Proof.
           | left _ => F0 x
           | right _ => F1 x + (F0 b - F1 b)
         end) x).
-  unfold derivable_pt in |- *; exists (f x); apply H7.
-  exists H8; symmetry  in |- *; apply derive_pt_eq_0; apply H7.
+  unfold derivable_pt; exists (f x); apply H7.
+  exists H8; symmetry ; apply derive_pt_eq_0; apply H7.
   assert (H5 : a <= x <= b).
   split; [ assumption | right; assumption ].
   assert (H6 : b <= x <= c).
-  split; [ right; symmetry  in |- *; assumption | assumption ].
+  split; [ right; symmetry ; assumption | assumption ].
   elim (H _ H5); elim (H0 _ H6); intros; assert (H9 : derive_pt F0 x x1 = f x).
-  symmetry  in |- *; assumption.
+  symmetry ; assumption.
   assert (H10 : derive_pt F1 x x0 = f x).
-  symmetry  in |- *; assumption.
+  symmetry ; assumption.
   assert (H11 := derive_pt_eq_1 F0 x (f x) x1 H9);
     assert (H12 := derive_pt_eq_1 F1 x (f x) x0 H10);
       assert
@@ -342,21 +342,21 @@ Proof.
               | left _ => F0 x
               | right _ => F1 x + (F0 b - F1 b)
             end) x (f x)).
-  unfold derivable_pt_lim in |- *; unfold derivable_pt_lim in H11, H12; intros;
+  unfold derivable_pt_lim; unfold derivable_pt_lim in H11, H12; intros;
     elim (H11 _ H13); elim (H12 _ H13); intros; set (D := Rmin x2 x3);
       assert (H16 : 0 < D).
-  unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x2 x3); intro.
+  unfold D; unfold Rmin; case (Rle_dec x2 x3); intro.
   apply (cond_pos x2).
   apply (cond_pos x3).
   exists (mkposreal _ H16); intros; case (Rle_dec x b); intro.
   case (Rle_dec (x + h) b); intro.
   apply H15.
   assumption.
-  apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_r ].
+  apply Rlt_le_trans with D; [ assumption | unfold D; apply Rmin_r ].
   replace (F1 (x + h) + (F0 b - F1 b) - F0 x) with (F1 (x + h) - F1 x).
   apply H14.
   assumption.
-  apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_l ].
+  apply Rlt_le_trans with D; [ assumption | unfold D; apply Rmin_l ].
   rewrite b0; ring.
   elim n; right; assumption.
   assert
@@ -367,8 +367,8 @@ Proof.
           | left _ => F0 x
           | right _ => F1 x + (F0 b - F1 b)
         end) x).
-  unfold derivable_pt in |- *; exists (f x); apply H13.
-  exists H14; symmetry  in |- *; apply derive_pt_eq_0; apply H13.
+  unfold derivable_pt; exists (f x); apply H13.
+  exists H14; symmetry ; apply derive_pt_eq_0; apply H13.
   assert (H5 : b <= x <= c).
   split; [ left; assumption | assumption ].
   assert (H6 := H0 _ H5); elim H6; clear H6; intros;
@@ -380,12 +380,12 @@ Proof.
             | left _ => F0 x
             | right _ => F1 x + (F0 b - F1 b)
           end) x (f x)).
-  unfold derivable_pt_lim in |- *; assert (H7 : derive_pt F1 x x0 = f x).
-  symmetry  in |- *; assumption.
+  unfold derivable_pt_lim; assert (H7 : derive_pt F1 x x0 = f x).
+  symmetry ; assumption.
   assert (H8 := derive_pt_eq_1 F1 x (f x) x0 H7); unfold derivable_pt_lim in H8;
     intros; elim (H8 _ H9); intros; set (D := Rmin x1 (x - b));
       assert (H11 : 0 < D).
-  unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x1 (x - b)); intro.
+  unfold D; unfold Rmin; case (Rle_dec x1 (x - b)); intro.
   apply (cond_pos x1).
   apply Rlt_Rminus; assumption.
   exists (mkposreal _ H11); intros; case (Rle_dec x b); intro.
@@ -399,13 +399,13 @@ Proof.
   rewrite <- Rabs_Ropp; apply RRle_abs.
   apply Rlt_le_trans with D.
   apply H13.
-  unfold D in |- *; apply Rmin_r.
+  unfold D; apply Rmin_r.
   replace (F1 (x + h) + (F0 b - F1 b) - (F1 x + (F0 b - F1 b))) with
   (F1 (x + h) - F1 x); [ idtac | ring ]; apply H10.
   assumption.
   apply Rlt_le_trans with D.
   assumption.
-  unfold D in |- *; apply Rmin_l.
+  unfold D; apply Rmin_l.
   assert
     (H8 :
       derivable_pt
@@ -414,8 +414,8 @@ Proof.
           | left _ => F0 x
           | right _ => F1 x + (F0 b - F1 b)
         end) x).
-  unfold derivable_pt in |- *; exists (f x); apply H7.
-  exists H8; symmetry  in |- *; apply derive_pt_eq_0; apply H7.
+  unfold derivable_pt; exists (f x); apply H7.
+  exists H8; symmetry ; apply derive_pt_eq_0; apply H7.
 Qed.
 
 Lemma antiderivative_P3 :
@@ -427,15 +427,15 @@ Proof.
   intros; unfold antiderivative in H, H0; elim H; clear H; elim H0; clear H0;
     intros; case (total_order_T a c); intro.
   elim s; intro.
-  right; unfold antiderivative in |- *; split.
+  right; unfold antiderivative; split.
   intros; apply H1; elim H3; intros; split;
     [ assumption | apply Rle_trans with c; assumption ].
   left; assumption.
-  right; unfold antiderivative in |- *; split.
+  right; unfold antiderivative; split.
   intros; apply H1; elim H3; intros; split;
     [ assumption | apply Rle_trans with c; assumption ].
   right; assumption.
-  left; unfold antiderivative in |- *; split.
+  left; unfold antiderivative; split.
   intros; apply H; elim H3; intros; split;
     [ assumption | apply Rle_trans with a; assumption ].
   left; assumption.
@@ -450,15 +450,15 @@ Proof.
   intros; unfold antiderivative in H, H0; elim H; clear H; elim H0; clear H0;
     intros; case (total_order_T c b); intro.
   elim s; intro.
-  right; unfold antiderivative in |- *; split.
+  right; unfold antiderivative; split.
   intros; apply H1; elim H3; intros; split;
     [ apply Rle_trans with c; assumption | assumption ].
   left; assumption.
-  right; unfold antiderivative in |- *; split.
+  right; unfold antiderivative; split.
   intros; apply H1; elim H3; intros; split;
     [ apply Rle_trans with c; assumption | assumption ].
   right; assumption.
-  left; unfold antiderivative in |- *; split.
+  left; unfold antiderivative; split.
   intros; apply H; elim H3; intros; split;
     [ apply Rle_trans with b; assumption | assumption ].
   left; assumption.
@@ -471,7 +471,7 @@ Lemma NewtonInt_P7 :
     Newton_integrable f a b ->
     Newton_integrable f b c -> Newton_integrable f a c.
 Proof.
-  unfold Newton_integrable in |- *; intros f a b c Hab Hbc X X0; elim X;
+  unfold Newton_integrable; intros f a b c Hab Hbc X X0; elim X;
     clear X; intros F0 H0; elim X0; clear X0; intros F1 H1;
       set
         (g :=
@@ -479,7 +479,7 @@ Proof.
             match Rle_dec x b with
               | left _ => F0 x
               | right _ => F1 x + (F0 b - F1 b)
-            end); exists g; left; unfold g in |- *;
+            end); exists g; left; unfold g;
         apply antiderivative_P2.
   elim H0; intro.
   assumption.
@@ -504,7 +504,7 @@ Proof.
   case (total_order_T b c); intro.
   elim s0; intro.
 (* a<b & b<c *)
-  unfold Newton_integrable in |- *;
+  unfold Newton_integrable;
     exists
       (fun x:R =>
         match Rle_dec x b with
@@ -523,7 +523,7 @@ Proof.
 (* a<b & b>c *)
   case (total_order_T a c); intro.
   elim s0; intro.
-  unfold Newton_integrable in |- *; exists F0.
+  unfold Newton_integrable; exists F0.
   left.
   elim H1; intro.
   unfold antiderivative in H; elim H; clear H; intros _ H.
@@ -537,7 +537,7 @@ Proof.
   unfold antiderivative in H2; elim H2; clear H2; intros _ H2.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)).
   rewrite b0; apply NewtonInt_P1.
-  unfold Newton_integrable in |- *; exists F1.
+  unfold Newton_integrable; exists F1.
   right.
   elim H1; intro.
   unfold antiderivative in H; elim H; clear H; intros _ H.
@@ -557,7 +557,7 @@ Proof.
 (* a>b & b<c *)
   case (total_order_T a c); intro.
   elim s0; intro.
-  unfold Newton_integrable in |- *; exists F1.
+  unfold Newton_integrable; exists F1.
   left.
   elim H1; intro.
 (*****************)
@@ -572,7 +572,7 @@ Proof.
   unfold antiderivative in H; elim H; clear H; intros _ H.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)).
   rewrite b0; apply NewtonInt_P1.
-  unfold Newton_integrable in |- *; exists F0.
+  unfold Newton_integrable; exists F0.
   right.
   elim H0; intro.
   unfold antiderivative in H; elim H; clear H; intros _ H.
@@ -601,7 +601,7 @@ Lemma NewtonInt_P9 :
     NewtonInt f a c (NewtonInt_P8 f a b c pr1 pr2) =
     NewtonInt f a b pr1 + NewtonInt f b c pr2.
 Proof.
-  intros; unfold NewtonInt in |- *.
+  intros; unfold NewtonInt.
   case (NewtonInt_P8 f a b c pr1 pr2); intros.
   case pr1; intros.
   case pr2; intros.
@@ -641,7 +641,7 @@ Proof.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)).
 (* a<b & b=c *)
   rewrite <- b0.
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rplus_0_r.
   rewrite <- b0 in o.
   elim o0; intro.
   elim o; intro.
@@ -759,7 +759,7 @@ Proof.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 a0)).
 (* a>b & b=c *)
   rewrite <- b0.
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rplus_0_r.
   rewrite <- b0 in o.
   elim o0; intro.
   unfold antiderivative in H; elim H; clear H; intros _ H.
diff --git a/theories/Reals/PSeries_reg.v b/theories/Reals/PSeries_reg.v
index aa588e38..d4d91137 100644
--- a/theories/Reals/PSeries_reg.v
+++ b/theories/Reals/PSeries_reg.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -12,7 +12,7 @@ Require Import SeqSeries.
 Require Import Ranalysis1.
 Require Import Max.
 Require Import Even.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Definition Boule (x:R) (r:posreal) (y:R) : Prop := Rabs (y - x) < r.
 
@@ -44,7 +44,7 @@ Lemma CVN_CVU :
     (cv:forall x:R, {l:R | Un_cv (fun N:nat => SP fn N x) l })
     (r:posreal), CVN_r fn r -> CVU (fun n:nat => SP fn n) (SFL fn cv) 0 r.
 Proof.
-  intros; unfold CVU in |- *; intros.
+  intros; unfold CVU; intros.
   unfold CVN_r in X.
   elim X; intros An X0.
   elim X0; intros s H0.
@@ -58,7 +58,7 @@ Proof.
     rewrite Ropp_minus_distr';
       rewrite (Rabs_right (s - sum_f_R0 (fun k:nat => Rabs (An k)) n)).
   eapply sum_maj1.
-  unfold SFL in |- *; case (cv y); intro.
+  unfold SFL; case (cv y); intro.
   trivial.
   apply H1.
   intro; elim H0; intros.
@@ -69,7 +69,7 @@ Proof.
   apply H8; apply H6.
   apply Rle_ge;
     apply Rplus_le_reg_l with (sum_f_R0 (fun k:nat => Rabs (An k)) n).
-  rewrite Rplus_0_r; unfold Rminus in |- *; rewrite (Rplus_comm s);
+  rewrite Rplus_0_r; unfold Rminus; rewrite (Rplus_comm s);
     rewrite <- Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_l;
       apply sum_incr.
   apply H1.
@@ -77,10 +77,10 @@ Proof.
   unfold R_dist in H4; unfold Rminus in H4; rewrite Ropp_0 in H4.
   assert (H7 := H4 n H5).
   rewrite Rplus_0_r in H7; apply H7.
-  unfold Un_cv in H1; unfold Un_cv in |- *; intros.
+  unfold Un_cv in H1; unfold Un_cv; intros.
   elim (H1 _ H3); intros.
   exists x; intros.
-  unfold R_dist in |- *; unfold R_dist in H4.
+  unfold R_dist; unfold R_dist in H4.
   rewrite Rminus_0_r; apply H4; assumption.
 Qed.
 
@@ -91,13 +91,13 @@ Lemma CVU_continuity :
     (forall (n:nat) (y:R), Boule x r y -> continuity_pt (fn n) y) ->
     forall y:R, Boule x r y -> continuity_pt f y.
 Proof.
-  intros; unfold continuity_pt in |- *; unfold continue_in in |- *;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
-      simpl in |- *; unfold R_dist in |- *; intros.
+  intros; unfold continuity_pt; unfold continue_in;
+    unfold limit1_in; unfold limit_in;
+      simpl; unfold R_dist; intros.
   unfold CVU in H.
   cut (0 < eps / 3);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
         [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
   elim (H _ H3); intros N0 H4.
   assert (H5 := H0 N0 y H1).
@@ -110,7 +110,7 @@ Proof.
   set (del := Rmin del1 del2).
   exists del; intros.
   split.
-  unfold del in |- *; unfold Rmin in |- *; case (Rle_dec del1 del2); intro.
+  unfold del; unfold Rmin; case (Rle_dec del1 del2); intro.
   apply (cond_pos del1).
   elim H8; intros; assumption.
   intros;
@@ -130,27 +130,27 @@ Proof.
   elim H9; intros.
   apply Rlt_le_trans with del.
   assumption.
-  unfold del in |- *; apply Rmin_l.
+  unfold del; apply Rmin_l.
   elim H8; intros.
   apply H11.
   split.
   elim H9; intros; assumption.
   elim H9; intros; apply Rlt_le_trans with del.
   assumption.
-  unfold del in |- *; apply Rmin_r.
+  unfold del; apply Rmin_r.
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply H4.
   apply le_n.
   assumption.
   apply Rmult_eq_reg_l with 3.
-  do 2 rewrite Rmult_plus_distr_l; unfold Rdiv in |- *; rewrite <- Rmult_assoc;
+  do 2 rewrite Rmult_plus_distr_l; unfold Rdiv; rewrite <- Rmult_assoc;
     rewrite Rinv_r_simpl_m.
   ring.
   discrR.
   discrR.
   cut (0 < r - Rabs (x - y)).
   intro; exists (mkposreal _ H6).
-  simpl in |- *; intros.
-  unfold Boule in |- *; replace (y + h - x) with (h + (y - x));
+  simpl; intros.
+  unfold Boule; replace (y + h - x) with (h + (y - x));
     [ idtac | ring ]; apply Rle_lt_trans with (Rabs h + Rabs (y - x)).
   apply Rabs_triang.
   apply Rplus_lt_reg_r with (- Rabs (x - y)).
@@ -173,8 +173,8 @@ Lemma continuity_pt_finite_SF :
     continuity_pt (fun y:R => sum_f_R0 (fun k:nat => fn k y) N) x.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; apply (H 0%nat); apply le_n.
-  simpl in |- *;
+  simpl; apply (H 0%nat); apply le_n.
+  simpl;
     replace (fun y:R => sum_f_R0 (fun k:nat => fn k y) N + fn (S N) y) with
     ((fun y:R => sum_f_R0 (fun k:nat => fn k y) N) + (fun y:R => fn (S N) y))%F;
     [ idtac | reflexivity ].
@@ -197,7 +197,7 @@ Proof.
   intros; eapply CVU_continuity.
   apply CVN_CVU.
   apply X.
-  intros; unfold SP in |- *; apply continuity_pt_finite_SF.
+  intros; unfold SP; apply continuity_pt_finite_SF.
   intros; apply H.
   apply H1.
   apply H0.
@@ -208,7 +208,7 @@ Lemma SFL_continuity :
     (cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }),
     CVN_R fn -> (forall n:nat, continuity (fn n)) -> continuity (SFL fn cv).
 Proof.
-  intros; unfold continuity in |- *; intro.
+  intros; unfold continuity; intro.
   cut (0 < Rabs x + 1);
     [ intro | apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ] ].
   cut (Boule 0 (mkposreal _ H0) x).
@@ -216,8 +216,8 @@ Proof.
   apply X.
   intros; apply (H n y).
   apply H1.
-  unfold Boule in |- *; simpl in |- *; rewrite Rminus_0_r;
-    pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r;
+  unfold Boule; simpl; rewrite Rminus_0_r;
+    pattern (Rabs x) at 1; rewrite <- Rplus_0_r;
       apply Rplus_lt_compat_l; apply Rlt_0_1.
 Qed.
 
@@ -227,10 +227,10 @@ Lemma CVN_R_CVS :
     CVN_R fn -> forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }.
 Proof.
   intros; apply R_complete.
-  unfold SP in |- *; set (An := fun N:nat => fn N x).
-  change (Cauchy_crit_series An) in |- *.
+  unfold SP; set (An := fun N:nat => fn N x).
+  change (Cauchy_crit_series An).
   apply cauchy_abs.
-  unfold Cauchy_crit_series in |- *; apply CV_Cauchy.
+  unfold Cauchy_crit_series; apply CV_Cauchy.
   unfold CVN_R in X; cut (0 < Rabs x + 1).
   intro; assert (H0 := X (mkposreal _ H)).
   unfold CVN_r in H0; elim H0; intros Bn H1.
@@ -239,13 +239,13 @@ Proof.
   apply Rseries_CV_comp with Bn.
   intro; split.
   apply Rabs_pos.
-  unfold An in |- *; apply H4; unfold Boule in |- *; simpl in |- *;
+  unfold An; apply H4; unfold Boule; simpl;
     rewrite Rminus_0_r.
-  pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
+  pattern (Rabs x) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
     apply Rlt_0_1.
   exists l.
   cut (forall n:nat, 0 <= Bn n).
-  intro; unfold Un_cv in H3; unfold Un_cv in |- *; intros.
+  intro; unfold Un_cv in H3; unfold Un_cv; intros.
   elim (H3 _ H6); intros.
   exists x0; intros.
   replace (sum_f_R0 Bn n) with (sum_f_R0 (fun k:nat => Rabs (Bn k)) n).
@@ -253,8 +253,8 @@ Proof.
   apply sum_eq; intros; apply Rabs_right; apply Rle_ge; apply H5.
   intro; apply Rle_trans with (Rabs (An n)).
   apply Rabs_pos.
-  unfold An in |- *; apply H4; unfold Boule in |- *; simpl in |- *;
-    rewrite Rminus_0_r; pattern (Rabs x) at 1 in |- *;
+  unfold An; apply H4; unfold Boule; simpl;
+    rewrite Rminus_0_r; pattern (Rabs x) at 1;
       rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1.
   apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ].
 Qed.
diff --git a/theories/Reals/PartSum.v b/theories/Reals/PartSum.v
index 3f90f15a..d765cf78 100644
--- a/theories/Reals/PartSum.v
+++ b/theories/Reals/PartSum.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -11,15 +11,15 @@ Require Import Rfunctions.
 Require Import Rseries.
 Require Import Rcomplete.
 Require Import Max.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Lemma tech1 :
   forall (An:nat -> R) (N:nat),
     (forall n:nat, (n <= N)%nat -> 0 < An n) -> 0 < sum_f_R0 An N.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; apply H; apply le_n.
-  simpl in |- *; apply Rplus_lt_0_compat.
+  simpl; apply H; apply le_n.
+  simpl; apply Rplus_lt_0_compat.
   apply HrecN; intros; apply H; apply le_S; assumption.
   apply H; apply le_n.
 Qed.
@@ -52,7 +52,7 @@ Proof.
   repeat rewrite S_INR; ring.
   apply le_n_S; apply lt_le_weak; assumption.
   apply lt_le_S; assumption.
-  rewrite H1; rewrite <- minus_n_n; simpl in |- *.
+  rewrite H1; rewrite <- minus_n_n; simpl.
   replace (n + 0)%nat with n; [ reflexivity | ring ].
   inversion H.
   right; reflexivity.
@@ -66,7 +66,7 @@ Lemma tech3 :
 Proof.
   intros; cut (1 - k <> 0).
   intro; induction  N as [| N HrecN].
-  simpl in |- *; rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym.
+  simpl; rewrite Rmult_1_r; unfold Rdiv; rewrite <- Rinv_r_sym.
   reflexivity.
   apply H0.
   replace (sum_f_R0 (fun i:nat => k ^ i) (S N)) with
@@ -75,15 +75,15 @@ Proof.
     replace ((1 - k ^ S N) / (1 - k) + k ^ S N) with
     ((1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k)).
   apply Rmult_eq_reg_l with (1 - k).
-  unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ (1 - k)));
+  unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ (1 - k)));
     repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
-      [ do 2 rewrite Rmult_1_l; simpl in |- *; ring | apply H0 ].
+      [ do 2 rewrite Rmult_1_l; simpl; ring | apply H0 ].
   apply H0.
-  unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; rewrite (Rmult_comm (1 - k));
+  unfold Rdiv; rewrite Rmult_plus_distr_r; rewrite (Rmult_comm (1 - k));
     repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
   rewrite Rmult_1_r; reflexivity.
   apply H0.
-  apply Rminus_eq_contra; red in |- *; intro; elim H; symmetry  in |- *;
+  apply Rminus_eq_contra; red; intro; elim H; symmetry ;
     assumption.
 Qed.
 
@@ -92,11 +92,11 @@ Lemma tech4 :
     0 <= k -> (forall i:nat, An (S i) < k * An i) -> An N <= An 0%nat * k ^ N.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; right; ring.
+  simpl; right; ring.
   apply Rle_trans with (k * An N).
   left; apply (H0 N).
   replace (S N) with (N + 1)%nat; [ idtac | ring ].
-  rewrite pow_add; simpl in |- *; rewrite Rmult_1_r;
+  rewrite pow_add; simpl; rewrite Rmult_1_r;
     replace (An 0%nat * (k ^ N * k)) with (k * (An 0%nat * k ^ N));
     [ idtac | ring ]; apply Rmult_le_compat_l.
   assumption.
@@ -116,7 +116,7 @@ Lemma tech6 :
     sum_f_R0 An N <= An 0%nat * sum_f_R0 (fun i:nat => k ^ i) N.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; right; ring.
+  simpl; right; ring.
   apply Rle_trans with (An 0%nat * sum_f_R0 (fun i:nat => k ^ i) N + An (S N)).
   rewrite tech5; do 2 rewrite <- (Rplus_comm (An (S N)));
     apply Rplus_le_compat_l.
@@ -127,13 +127,13 @@ Qed.
 
 Lemma tech7 : forall r1 r2:R, r1 <> 0 -> r2 <> 0 -> r1 <> r2 -> / r1 <> / r2.
 Proof.
-  intros; red in |- *; intro.
+  intros; red; intro.
   assert (H3 := Rmult_eq_compat_l r1 _ _ H2).
   rewrite <- Rinv_r_sym in H3; [ idtac | assumption ].
   assert (H4 := Rmult_eq_compat_l r2 _ _ H3).
   rewrite Rmult_1_r in H4; rewrite <- Rmult_assoc in H4.
   rewrite Rinv_r_simpl_m in H4; [ idtac | assumption ].
-  elim H1; symmetry  in |- *; assumption.
+  elim H1; symmetry ; assumption.
 Qed.
 
 Lemma tech11 :
@@ -142,7 +142,7 @@ Lemma tech11 :
     sum_f_R0 An N = sum_f_R0 Bn N - sum_f_R0 Cn N.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; apply H.
+  simpl; apply H.
   do 3 rewrite tech5; rewrite HrecN; rewrite (H (S N)); ring.
 Qed.
 
@@ -151,7 +151,7 @@ Lemma tech12 :
     Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l ->
     Pser An x l.
 Proof.
-  intros; unfold Pser in |- *; unfold infinite_sum in |- *; unfold Un_cv in H;
+  intros; unfold Pser; unfold infinite_sum; unfold Un_cv in H;
     assumption.
 Qed.
 
@@ -160,7 +160,7 @@ Lemma scal_sum :
     x * sum_f_R0 An N = sum_f_R0 (fun i:nat => An i * x) N.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; ring.
+  simpl; ring.
   do 2 rewrite tech5.
   rewrite Rmult_plus_distr_l; rewrite <- HrecN; ring.
 Qed.
@@ -179,14 +179,14 @@ Proof.
   do 2 rewrite tech5.
   replace (S (S (pred N))) with (S N).
   rewrite (HrecN H1); ring.
-  rewrite H2; simpl in |- *; reflexivity.
+  rewrite H2; simpl; reflexivity.
   assert (H2 := O_or_S N).
   elim H2; intros.
   elim a; intros.
   rewrite <- p.
-  simpl in |- *; reflexivity.
+  simpl; reflexivity.
   rewrite <- b in H1; elim (lt_irrefl _ H1).
-  rewrite H1; simpl in |- *; reflexivity.
+  rewrite H1; simpl; reflexivity.
   inversion H.
   right; reflexivity.
   left; apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ].
@@ -197,7 +197,7 @@ Lemma plus_sum :
     sum_f_R0 (fun i:nat => An i + Bn i) N = sum_f_R0 An N + sum_f_R0 Bn N.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; ring.
+  simpl; ring.
   do 3 rewrite tech5; rewrite HrecN; ring.
 Qed.
 
@@ -207,7 +207,7 @@ Lemma sum_eq :
     sum_f_R0 An N = sum_f_R0 Bn N.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; apply H; apply le_n.
+  simpl; apply H; apply le_n.
   do 2 rewrite tech5; rewrite HrecN.
   rewrite (H (S N)); [ reflexivity | apply le_n ].
   intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ].
@@ -218,7 +218,7 @@ Lemma uniqueness_sum :
   forall (An:nat -> R) (l1 l2:R),
     infinite_sum An l1 -> infinite_sum An l2 -> l1 = l2.
 Proof.
-  unfold infinite_sum in |- *; intros.
+  unfold infinite_sum; intros.
   case (Req_dec l1 l2); intro.
   assumption.
   cut (0 < Rabs ((l1 - l2) / 2)); [ intro | apply Rabs_pos_lt ].
@@ -235,19 +235,19 @@ Proof.
   intro; rewrite H12 in H11; assert (H13 := double_var); unfold Rdiv in H13;
     rewrite <- H13 in H11.
   elim (Rlt_irrefl _ H11).
-  apply Rabs_right; left; change (0 < / 2) in |- *; apply Rinv_0_lt_compat;
+  apply Rabs_right; left; change (0 < / 2); apply Rinv_0_lt_compat;
     cut (0%nat <> 2%nat);
-      [ intro H20; generalize (lt_INR_0 2 (neq_O_lt 2 H20)); unfold INR in |- *;
+      [ intro H20; generalize (lt_INR_0 2 (neq_O_lt 2 H20)); unfold INR;
         intro; assumption
         | discriminate ].
-  unfold R_dist in |- *; rewrite <- (Rabs_Ropp (sum_f_R0 An N - l1));
+  unfold R_dist; rewrite <- (Rabs_Ropp (sum_f_R0 An N - l1));
     rewrite Ropp_minus_distr'.
   replace (l1 - l2) with (l1 - sum_f_R0 An N + (sum_f_R0 An N - l2));
   [ idtac | ring ].
   apply Rabs_triang.
-  unfold ge in |- *; unfold N in |- *; apply le_max_r.
-  unfold ge in |- *; unfold N in |- *; apply le_max_l.
-  unfold Rdiv in |- *; apply prod_neq_R0.
+  unfold ge; unfold N; apply le_max_r.
+  unfold ge; unfold N; apply le_max_l.
+  unfold Rdiv; apply prod_neq_R0.
   apply Rminus_eq_contra; assumption.
   apply Rinv_neq_0_compat; discrR.
 Qed.
@@ -257,7 +257,7 @@ Lemma minus_sum :
     sum_f_R0 (fun i:nat => An i - Bn i) N = sum_f_R0 An N - sum_f_R0 Bn N.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; ring.
+  simpl; ring.
   do 3 rewrite tech5; rewrite HrecN; ring.
 Qed.
 
@@ -268,7 +268,7 @@ Lemma sum_decomposition :
 Proof.
   intros.
   induction  N as [| N HrecN].
-  simpl in |- *; ring.
+  simpl; ring.
   rewrite tech5.
   rewrite (tech5 (fun l:nat => An (S (2 * l))) N).
   replace (2 * S (S N))%nat with (S (S (2 * S N))).
@@ -286,7 +286,7 @@ Lemma sum_Rle :
 Proof.
   intros.
   induction  N as [| N HrecN].
-  simpl in |- *; apply H.
+  simpl; apply H.
   apply le_n.
   do 2 rewrite tech5.
   apply Rle_trans with (sum_f_R0 An N + Bn (S N)).
@@ -306,7 +306,7 @@ Lemma Rsum_abs :
 Proof.
   intros.
   induction  N as [| N HrecN].
-  simpl in |- *.
+  simpl.
   right; reflexivity.
   do 2 rewrite tech5.
   apply Rle_trans with (Rabs (sum_f_R0 An N) + Rabs (An (S N))).
@@ -321,7 +321,7 @@ Lemma sum_cte :
 Proof.
   intros.
   induction  N as [| N HrecN].
-  simpl in |- *; ring.
+  simpl; ring.
   rewrite tech5.
   rewrite HrecN; repeat rewrite S_INR; ring.
 Qed.
@@ -333,7 +333,7 @@ Lemma sum_growing :
 Proof.
   intros.
   induction  N as [| N HrecN].
-  simpl in |- *; apply H.
+  simpl; apply H.
   do 2 rewrite tech5.
   apply Rle_trans with (sum_f_R0 An N + Bn (S N)).
   apply Rplus_le_compat_l; apply H.
@@ -348,7 +348,7 @@ Lemma Rabs_triang_gen :
 Proof.
   intros.
   induction  N as [| N HrecN].
-  simpl in |- *.
+  simpl.
   right; reflexivity.
   do 2 rewrite tech5.
   apply Rle_trans with (Rabs (sum_f_R0 An N) + Rabs (An (S N))).
@@ -364,7 +364,7 @@ Lemma cond_pos_sum :
 Proof.
   intros.
   induction  N as [| N HrecN].
-  simpl in |- *; apply H.
+  simpl; apply H.
   rewrite tech5.
   apply Rplus_le_le_0_compat.
   apply HrecN.
@@ -380,7 +380,7 @@ Lemma cauchy_abs :
   forall An:nat -> R,
     Cauchy_crit_series (fun i:nat => Rabs (An i)) -> Cauchy_crit_series An.
 Proof.
-  unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *.
+  unfold Cauchy_crit_series; unfold Cauchy_crit.
   intros.
   elim (H eps H0); intros.
   exists x.
@@ -400,8 +400,8 @@ Proof.
   elim a; intro.
   rewrite (tech2 An n m); [ idtac | assumption ].
   rewrite (tech2 (fun i:nat => Rabs (An i)) n m); [ idtac | assumption ].
-  unfold R_dist in |- *.
-  unfold Rminus in |- *.
+  unfold R_dist.
+  unfold Rminus.
   do 2 rewrite Ropp_plus_distr.
   do 2 rewrite <- Rplus_assoc.
   do 2 rewrite Rplus_opp_r.
@@ -414,18 +414,18 @@ Proof.
   replace (fun i:nat => Rabs (An (S n + i)%nat)) with
   (fun i:nat => Rabs (Bn i)).
   apply Rabs_triang_gen.
-  unfold Bn in |- *; reflexivity.
+  unfold Bn; reflexivity.
   apply Rle_ge.
   apply cond_pos_sum.
   intro; apply Rabs_pos.
   rewrite b.
-  unfold R_dist in |- *.
-  unfold Rminus in |- *; do 2 rewrite Rplus_opp_r.
+  unfold R_dist.
+  unfold Rminus; do 2 rewrite Rplus_opp_r.
   rewrite Rabs_R0; right; reflexivity.
   rewrite (tech2 An m n); [ idtac | assumption ].
   rewrite (tech2 (fun i:nat => Rabs (An i)) m n); [ idtac | assumption ].
-  unfold R_dist in |- *.
-  unfold Rminus in |- *.
+  unfold R_dist.
+  unfold Rminus.
   do 2 rewrite Rplus_assoc.
   rewrite (Rplus_comm (sum_f_R0 An m)).
   rewrite (Rplus_comm (sum_f_R0 (fun i:nat => Rabs (An i)) m)).
@@ -439,7 +439,7 @@ Proof.
   replace (fun i:nat => Rabs (An (S m + i)%nat)) with
   (fun i:nat => Rabs (Bn i)).
   apply Rabs_triang_gen.
-  unfold Bn in |- *; reflexivity.
+  unfold Bn; reflexivity.
   apply Rle_ge.
   apply cond_pos_sum.
   intro; apply Rabs_pos.
@@ -454,7 +454,7 @@ Proof.
   intros An X.
   elim X; intros.
   unfold Un_cv in p.
-  unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *.
+  unfold Cauchy_crit_series; unfold Cauchy_crit.
   intros.
   cut (0 < eps / 2).
   intro.
@@ -462,7 +462,7 @@ Proof.
   exists x0.
   intros.
   apply Rle_lt_trans with (R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x).
-  unfold R_dist in |- *.
+  unfold R_dist.
   replace (sum_f_R0 An n - sum_f_R0 An m) with
   (sum_f_R0 An n - x + - (sum_f_R0 An m - x)); [ idtac | ring ].
   rewrite <- (Rabs_Ropp (sum_f_R0 An m - x)).
@@ -471,8 +471,8 @@ Proof.
   apply Rplus_lt_compat.
   apply H1; assumption.
   apply H1; assumption.
-  right; symmetry  in |- *; apply double_var.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  right; symmetry ; apply double_var.
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
 Qed.
 
@@ -493,7 +493,7 @@ Lemma sum_eq_R0 :
     (forall n:nat, (n <= N)%nat -> An n = 0) -> sum_f_R0 An N = 0.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; apply H; apply le_n.
+  simpl; apply H; apply le_n.
   rewrite tech5; rewrite HrecN;
     [ rewrite Rplus_0_l; apply H; apply le_n
       | intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ] ].
@@ -530,15 +530,15 @@ Proof.
     [ idtac | ring ]; apply Rle_trans with l1.
   left; apply r.
   apply H6.
-  unfold l1 in |- *; apply Rge_le;
+  unfold l1; apply Rge_le;
     apply (growing_prop (fun k:nat => sum_f_R0 An k)).
   apply H1.
-  unfold ge, N0 in |- *; apply le_max_r.
-  unfold ge, N0 in |- *; apply le_max_l.
+  unfold ge, N0; apply le_max_r.
+  unfold ge, N0; apply le_max_l.
   apply Rplus_lt_reg_r with l; rewrite Rplus_0_r;
     replace (l + (l1 - l)) with l1; [ apply r | ring ].
-  unfold Un_growing in |- *; intro; simpl in |- *;
-    pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r;
+  unfold Un_growing; intro; simpl;
+    pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r;
       apply Rplus_le_compat_l; apply H0.
 Qed.
 
@@ -572,7 +572,7 @@ Proof.
   apply Rlt_trans with (Rabs l1).
   apply Rmult_lt_reg_l with 2.
   prove_sup0.
-  unfold Rdiv in |- *; rewrite (Rmult_comm 2); rewrite Rmult_assoc;
+  unfold Rdiv; rewrite (Rmult_comm 2); rewrite Rmult_assoc;
     rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r; rewrite double; apply Rplus_lt_compat_l; apply r.
   discrR.
@@ -581,18 +581,18 @@ Proof.
   apply Rplus_lt_reg_r with ((Rabs l1 - l2) / 2 - Rabs (SP fn N x)).
   replace ((Rabs l1 - l2) / 2 - Rabs (SP fn N x) + (Rabs l1 + l2) / 2) with
   (Rabs l1 - Rabs (SP fn N x)).
-  unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l;
+  unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l;
     rewrite Rplus_0_r; apply H7.
-  unfold Rdiv in |- *; rewrite Rmult_plus_distr_r;
+  unfold Rdiv; rewrite Rmult_plus_distr_r;
     rewrite <- (Rmult_comm (/ 2)); rewrite Rmult_minus_distr_l;
-      repeat rewrite (Rmult_comm (/ 2)); pattern (Rabs l1) at 1 in |- *;
-        rewrite double_var; unfold Rdiv in |- *; ring.
+      repeat rewrite (Rmult_comm (/ 2)); pattern (Rabs l1) at 1;
+        rewrite double_var; unfold Rdiv; ring.
   case (Rcase_abs (sum_f_R0 An N - l2)); intro.
   apply Rlt_trans with l2.
   apply (Rminus_lt _ _ r0).
   apply Rmult_lt_reg_l with 2.
   prove_sup0.
-  rewrite (double l2); unfold Rdiv in |- *; rewrite (Rmult_comm 2);
+  rewrite (double l2); unfold Rdiv; rewrite (Rmult_comm 2);
     rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r; rewrite (Rplus_comm (Rabs l1)); apply Rplus_lt_compat_l;
     apply r.
@@ -600,23 +600,23 @@ Proof.
   rewrite (Rabs_right _ r0) in H6; apply Rplus_lt_reg_r with (- l2).
   replace (- l2 + (Rabs l1 + l2) / 2) with ((Rabs l1 - l2) / 2).
   rewrite Rplus_comm; apply H6.
-  unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+  unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
     rewrite Rmult_minus_distr_l; rewrite Rmult_plus_distr_r;
-      pattern l2 at 2 in |- *; rewrite double_var;
+      pattern l2 at 2; rewrite double_var;
         repeat rewrite (Rmult_comm (/ 2)); rewrite Ropp_plus_distr;
-          unfold Rdiv in |- *; ring.
+          unfold Rdiv; ring.
   apply Rle_lt_trans with (Rabs (SP fn N x - l1)).
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply Rabs_triang_inv2.
-  apply H4; unfold ge, N in |- *; apply le_max_l.
-  apply H5; unfold ge, N in |- *; apply le_max_r.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  apply H4; unfold ge, N; apply le_max_l.
+  apply H5; unfold ge, N; apply le_max_r.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   apply Rplus_lt_reg_r with l2.
   rewrite Rplus_0_r; replace (l2 + (Rabs l1 - l2)) with (Rabs l1);
     [ apply r | ring ].
   apply Rinv_0_lt_compat; prove_sup0.
   intros; induction  n0 as [| n0 Hrecn0].
-  unfold SP in |- *; simpl in |- *; apply H1.
-  unfold SP in |- *; simpl in |- *.
+  unfold SP; simpl; apply H1.
+  unfold SP; simpl.
   apply Rle_trans with
     (Rabs (sum_f_R0 (fun k:nat => fn k x) n0) + Rabs (fn (S n0) x)).
   apply Rabs_triang.
diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v
index 70f4ff0d..5fc7d8fb 100644
--- a/theories/Reals/RIneq.v
+++ b/theories/Reals/RIneq.v
@@ -1,7 +1,7 @@
 (* -*- coding: utf-8 -*- *)
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -52,13 +52,13 @@ Proof. exact Rlt_irrefl. Qed.
 
 Lemma Rlt_not_eq : forall r1 r2, r1 < r2 -> r1 <> r2.
 Proof.
-  red in |- *; intros r1 r2 H H0; apply (Rlt_irrefl r1).
-  pattern r1 at 2 in |- *; rewrite H0; trivial.
+  red; intros r1 r2 H H0; apply (Rlt_irrefl r1).
+  pattern r1 at 2; rewrite H0; trivial.
 Qed.
 
 Lemma Rgt_not_eq : forall r1 r2, r1 > r2 -> r1 <> r2.
 Proof.
-  intros; apply sym_not_eq; apply Rlt_not_eq; auto with real.
+  intros; apply not_eq_sym; apply Rlt_not_eq; auto with real.
 Qed.
 
 (**********)
@@ -102,7 +102,7 @@ Qed.
 
 Lemma Rlt_le : forall r1 r2, r1 < r2 -> r1 <= r2.
 Proof.
-  intros; red in |- *; tauto.
+  intros; red; tauto.
 Qed.
 Hint Resolve Rlt_le: real.
 
@@ -114,14 +114,14 @@ Qed.
 (**********)
 Lemma Rle_ge : forall r1 r2, r1 <= r2 -> r2 >= r1.
 Proof.
-  destruct 1; red in |- *; auto with real.
+  destruct 1; red; auto with real.
 Qed.
 Hint Immediate Rle_ge: real.
 Hint Resolve Rle_ge: rorders.
 
 Lemma Rge_le : forall r1 r2, r1 >= r2 -> r2 <= r1.
 Proof.
-  destruct 1; red in |- *; auto with real.
+  destruct 1; red; auto with real.
 Qed.
 Hint Resolve Rge_le: real.
 Hint Immediate Rge_le: rorders.
@@ -143,7 +143,7 @@ Hint Immediate Rgt_lt: rorders.
 
 Lemma Rnot_le_lt : forall r1 r2, ~ r1 <= r2 -> r2 < r1.
 Proof.
-  intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rle in |- *; tauto.
+  intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rle; tauto.
 Qed.
 Hint Immediate Rnot_le_lt: real.
 
@@ -174,7 +174,7 @@ Proof. eauto using Rnot_gt_ge with rorders. Qed.
 (**********)
 Lemma Rlt_not_le : forall r1 r2, r2 < r1 -> ~ r1 <= r2.
 Proof.
-  generalize Rlt_asym Rlt_dichotomy_converse; unfold Rle in |- *.
+  generalize Rlt_asym Rlt_dichotomy_converse; unfold Rle.
   intuition eauto 3.
 Qed.
 Hint Immediate Rlt_not_le: real.
@@ -192,7 +192,7 @@ Proof. exact Rlt_not_ge. Qed.
 Lemma Rle_not_lt : forall r1 r2, r2 <= r1 -> ~ r1 < r2.
 Proof.
   intros r1 r2. generalize (Rlt_asym r1 r2) (Rlt_dichotomy_converse r1 r2).
-  unfold Rle in |- *; intuition.
+  unfold Rle; intuition.
 Qed.
 
 Lemma Rge_not_lt : forall r1 r2, r1 >= r2 -> ~ r1 < r2.
@@ -207,25 +207,25 @@ Proof. do 2 intro; apply Rge_not_lt. Qed.
 (**********)
 Lemma Req_le : forall r1 r2, r1 = r2 -> r1 <= r2.
 Proof.
-  unfold Rle in |- *; tauto.
+  unfold Rle; tauto.
 Qed.
 Hint Immediate Req_le: real.
 
 Lemma Req_ge : forall r1 r2, r1 = r2 -> r1 >= r2.
 Proof.
-  unfold Rge in |- *; tauto.
+  unfold Rge; tauto.
 Qed.
 Hint Immediate Req_ge: real.
 
 Lemma Req_le_sym : forall r1 r2, r2 = r1 -> r1 <= r2.
 Proof.
-  unfold Rle in |- *; auto.
+  unfold Rle; auto.
 Qed.
 Hint Immediate Req_le_sym: real.
 
 Lemma Req_ge_sym : forall r1 r2, r2 = r1 -> r1 >= r2.
 Proof.
-  unfold Rge in |- *; auto.
+  unfold Rge; auto.
 Qed.
 Hint Immediate Req_ge_sym: real.
 
@@ -240,7 +240,7 @@ Proof. do 2 intro; apply Rlt_asym. Qed.
 
 Lemma Rle_antisym : forall r1 r2, r1 <= r2 -> r2 <= r1 -> r1 = r2.
 Proof.
-  intros r1 r2; generalize (Rlt_asym r1 r2); unfold Rle in |- *; intuition.
+  intros r1 r2; generalize (Rlt_asym r1 r2); unfold Rle; intuition.
 Qed.
 Hint Resolve Rle_antisym: real.
 
@@ -276,8 +276,8 @@ Proof. intros; red; apply Rlt_eq_compat with (r2:=r4) (r4:=r2); auto. Qed.
 
 Lemma Rle_trans : forall r1 r2 r3, r1 <= r2 -> r2 <= r3 -> r1 <= r3.
 Proof.
-  generalize trans_eq Rlt_trans Rlt_eq_compat.
-  unfold Rle in |- *.
+  generalize eq_trans Rlt_trans Rlt_eq_compat.
+  unfold Rle.
   intuition eauto 2.
 Qed.
 
@@ -291,13 +291,13 @@ Proof. eauto using Rlt_trans with rorders. Qed.
 Lemma Rle_lt_trans : forall r1 r2 r3, r1 <= r2 -> r2 < r3 -> r1 < r3.
 Proof.
   generalize Rlt_trans Rlt_eq_compat.
-  unfold Rle in |- *.
+  unfold Rle.
   intuition eauto 2.
 Qed.
 
 Lemma Rlt_le_trans : forall r1 r2 r3, r1 < r2 -> r2 <= r3 -> r1 < r3.
 Proof.
-  generalize Rlt_trans Rlt_eq_compat; unfold Rle in |- *; intuition eauto 2.
+  generalize Rlt_trans Rlt_eq_compat; unfold Rle; intuition eauto 2.
 Qed.
 
 Lemma Rge_gt_trans : forall r1 r2 r3, r1 >= r2 -> r2 > r3 -> r1 > r3.
@@ -430,7 +430,7 @@ Hint Resolve Rplus_eq_reg_l: real.
 (**********)
 Lemma Rplus_0_r_uniq : forall r r1, r + r1 = r -> r1 = 0.
 Proof.
-  intros r b; pattern r at 2 in |- *; replace r with (r + 0); eauto with real.
+  intros r b; pattern r at 2; replace r with (r + 0); eauto with real.
 Qed.
 
 (***********)
@@ -441,7 +441,7 @@ Proof.
     absurd (0 < a + b).
       rewrite H1; auto with real.
       apply Rle_lt_trans with (a + 0).
-        rewrite Rplus_0_r in |- *; assumption.
+        rewrite Rplus_0_r; assumption.
         auto using Rplus_lt_compat_l with real.
     rewrite <- H0, Rplus_0_r in H1; assumption.
 Qed.
@@ -570,14 +570,14 @@ Qed.
 (**********)
 Lemma Rmult_neq_0_reg : forall r1 r2, r1 * r2 <> 0 -> r1 <> 0 /\ r2 <> 0.
 Proof.
-  intros r1 r2 H; split; red in |- *; intro; apply H; auto with real.
+  intros r1 r2 H; split; red; intro; apply H; auto with real.
 Qed.
 
 (**********)
 Lemma Rmult_integral_contrapositive :
   forall r1 r2, r1 <> 0 /\ r2 <> 0 -> r1 * r2 <> 0.
 Proof.
-  red in |- *; intros r1 r2 [H1 H2] H.
+  red; intros r1 r2 [H1 H2] H.
   case (Rmult_integral r1 r2); auto with real.
 Qed.
 Hint Resolve Rmult_integral_contrapositive: real.
@@ -604,12 +604,12 @@ Notation "r ²" := (Rsqr r) (at level 1, format "r ²") : R_scope.
 
 (***********)
 Lemma Rsqr_0 : Rsqr 0 = 0.
-  unfold Rsqr in |- *; auto with real.
+  unfold Rsqr; auto with real.
 Qed.
 
 (***********)
 Lemma Rsqr_0_uniq : forall r, Rsqr r = 0 -> r = 0.
-  unfold Rsqr in |- *; intros; elim (Rmult_integral r r H); trivial.
+  unfold Rsqr; intros; elim (Rmult_integral r r H); trivial.
 Qed.
 
 (*********************************************************)
@@ -647,7 +647,7 @@ Hint Resolve Ropp_involutive: real.
 (*********)
 Lemma Ropp_neq_0_compat : forall r, r <> 0 -> - r <> 0.
 Proof.
-  red in |- *; intros r H H0.
+  red; intros r H H0.
   apply H.
   transitivity (- - r); auto with real.
 Qed.
@@ -720,7 +720,7 @@ Hint Resolve Rminus_diag_eq: real.
 (**********)
 Lemma Rminus_diag_uniq : forall r1 r2, r1 - r2 = 0 -> r1 = r2.
 Proof.
-  intros r1 r2; unfold Rminus in |- *; rewrite Rplus_comm; intro.
+  intros r1 r2; unfold Rminus; rewrite Rplus_comm; intro.
   rewrite <- (Ropp_involutive r2); apply (Rplus_opp_r_uniq (- r2) r1 H).
 Qed.
 Hint Immediate Rminus_diag_uniq: real.
@@ -741,20 +741,20 @@ Hint Resolve Rplus_minus: real.
 (**********)
 Lemma Rminus_eq_contra : forall r1 r2, r1 <> r2 -> r1 - r2 <> 0.
 Proof.
-  red in |- *; intros r1 r2 H H0.
+  red; intros r1 r2 H H0.
   apply H; auto with real.
 Qed.
 Hint Resolve Rminus_eq_contra: real.
 
 Lemma Rminus_not_eq : forall r1 r2, r1 - r2 <> 0 -> r1 <> r2.
 Proof.
-  red in |- *; intros; elim H; apply Rminus_diag_eq; auto.
+  red; intros; elim H; apply Rminus_diag_eq; auto.
 Qed.
 Hint Resolve Rminus_not_eq: real.
 
 Lemma Rminus_not_eq_right : forall r1 r2, r2 - r1 <> 0 -> r1 <> r2.
 Proof.
-  red in |- *; intros; elim H; rewrite H0; ring.
+  red; intros; elim H; rewrite H0; ring.
 Qed.
 Hint Resolve Rminus_not_eq_right: real.
 
@@ -778,7 +778,7 @@ Hint Resolve Rinv_1: real.
 (*********)
 Lemma Rinv_neq_0_compat : forall r, r <> 0 -> / r <> 0.
 Proof.
-  red in |- *; intros; apply R1_neq_R0.
+  red; intros; apply R1_neq_R0.
   replace 1 with (/ r * r); auto with real.
 Qed.
 Hint Resolve Rinv_neq_0_compat: real.
@@ -858,7 +858,7 @@ Proof. do 3 intro; apply Rplus_lt_compat_r. Qed.
 (**********)
 Lemma Rplus_le_compat_l : forall r r1 r2, r1 <= r2 -> r + r1 <= r + r2.
 Proof.
-  unfold Rle in |- *; intros; elim H; intro.
+  unfold Rle; intros; elim H; intro.
   left; apply (Rplus_lt_compat_l r r1 r2 H0).
   right; rewrite <- H0; auto with zarith real.
 Qed.
@@ -870,7 +870,7 @@ Hint Resolve Rplus_ge_compat_l: real.
 (**********)
 Lemma Rplus_le_compat_r : forall r r1 r2, r1 <= r2 -> r1 + r <= r2 + r.
 Proof.
-  unfold Rle in |- *; intros; elim H; intro.
+  unfold Rle; intros; elim H; intro.
   left; apply (Rplus_lt_compat_r r r1 r2 H0).
   right; rewrite <- H0; auto with real.
 Qed.
@@ -931,7 +931,7 @@ Lemma Rplus_lt_0_compat : forall r1 r2, 0 < r1 -> 0 < r2 -> 0 < r1 + r2.
 Proof.
   intros x y; intros; apply Rlt_trans with x;
     [ assumption
-      | pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l;
+      | pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l;
         assumption ].
 Qed.
 
@@ -939,7 +939,7 @@ Lemma Rplus_le_lt_0_compat : forall r1 r2, 0 <= r1 -> 0 < r2 -> 0 < r1 + r2.
 Proof.
   intros x y; intros; apply Rle_lt_trans with x;
     [ assumption
-      | pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l;
+      | pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l;
         assumption ].
 Qed.
 
@@ -953,7 +953,7 @@ Lemma Rplus_le_le_0_compat : forall r1 r2, 0 <= r1 -> 0 <= r2 -> 0 <= r1 + r2.
 Proof.
   intros x y; intros; apply Rle_trans with x;
     [ assumption
-      | pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
+      | pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
         assumption ].
 Qed.
 
@@ -981,7 +981,7 @@ Qed.
 
 Lemma Rplus_le_reg_l : forall r r1 r2, r + r1 <= r + r2 -> r1 <= r2.
 Proof.
-  unfold Rle in |- *; intros; elim H; intro.
+  unfold Rle; intros; elim H; intro.
   left; apply (Rplus_lt_reg_r r r1 r2 H0).
   right; apply (Rplus_eq_reg_l r r1 r2 H0).
 Qed.
@@ -995,7 +995,7 @@ Qed.
 
 Lemma Rplus_gt_reg_l : forall r r1 r2, r + r1 > r + r2 -> r1 > r2.
 Proof.
-  unfold Rgt in |- *; intros; apply (Rplus_lt_reg_r r r2 r1 H).
+  unfold Rgt; intros; apply (Rplus_lt_reg_r r r2 r1 H).
 Qed.
 
 Lemma Rplus_ge_reg_l : forall r r1 r2, r + r1 >= r + r2 -> r1 >= r2.
@@ -1046,7 +1046,7 @@ Qed.
 
 Lemma Ropp_gt_lt_contravar : forall r1 r2, r1 > r2 -> - r1 < - r2.
 Proof.
-  unfold Rgt in |- *; intros.
+  unfold Rgt; intros.
   apply (Rplus_lt_reg_r (r2 + r1)).
   replace (r2 + r1 + - r1) with r2.
   replace (r2 + r1 + - r2) with r1.
@@ -1058,7 +1058,7 @@ Hint Resolve Ropp_gt_lt_contravar.
 
 Lemma Ropp_lt_gt_contravar : forall r1 r2, r1 < r2 -> - r1 > - r2.
 Proof.
-  unfold Rgt in |- *; auto with real.
+  unfold Rgt; auto with real.
 Qed.
 Hint Resolve Ropp_lt_gt_contravar: real.
 
@@ -1183,7 +1183,7 @@ Proof. eauto using Rmult_lt_compat_l with rorders. Qed.
 Lemma Rmult_le_compat_l :
   forall r r1 r2, 0 <= r -> r1 <= r2 -> r * r1 <= r * r2.
 Proof.
-  intros r r1 r2 H H0; destruct H; destruct H0; unfold Rle in |- *;
+  intros r r1 r2 H H0; destruct H; destruct H0; unfold Rle;
     auto with real.
   right; rewrite <- H; do 2 rewrite Rmult_0_l; reflexivity.
 Qed.
@@ -1342,7 +1342,7 @@ Qed.
 (**********)
 Lemma Rle_minus : forall r1 r2, r1 <= r2 -> r1 - r2 <= 0.
 Proof.
-  destruct 1; unfold Rle in |- *; auto with real.
+  destruct 1; unfold Rle; auto with real.
 Qed.
 
 Lemma Rge_minus : forall r1 r2, r1 >= r2 -> r1 - r2 >= 0.
@@ -1356,7 +1356,7 @@ Qed.
 Lemma Rminus_lt : forall r1 r2, r1 - r2 < 0 -> r1 < r2.
 Proof.
   intros; replace r1 with (r1 - r2 + r2).
-  pattern r2 at 3 in |- *; replace r2 with (0 + r2); auto with real.
+  pattern r2 at 3; replace r2 with (0 + r2); auto with real.
   ring.
 Qed.
 
@@ -1372,7 +1372,7 @@ Qed.
 Lemma Rminus_le : forall r1 r2, r1 - r2 <= 0 -> r1 <= r2.
 Proof.
   intros; replace r1 with (r1 - r2 + r2).
-  pattern r2 at 3 in |- *; replace r2 with (0 + r2); auto with real.
+  pattern r2 at 3; replace r2 with (0 + r2); auto with real.
   ring.
 Qed.
 
@@ -1387,7 +1387,7 @@ Qed.
 (**********)
 Lemma tech_Rplus : forall r (s:R), 0 <= r -> 0 < s -> r + s <> 0.
 Proof.
-  intros; apply sym_not_eq; apply Rlt_not_eq.
+  intros; apply not_eq_sym; apply Rlt_not_eq.
   rewrite Rplus_comm; replace 0 with (0 + 0); auto with real.
 Qed.
 Hint Immediate tech_Rplus: real.
@@ -1398,7 +1398,7 @@ Hint Immediate tech_Rplus: real.
 
 Lemma Rle_0_sqr : forall r, 0 <= Rsqr r.
 Proof.
-  intro; case (Rlt_le_dec r 0); unfold Rsqr in |- *; intro.
+  intro; case (Rlt_le_dec r 0); unfold Rsqr; intro.
   replace (r * r) with (- r * - r); auto with real.
   replace 0 with (- r * 0); auto with real.
   replace 0 with (0 * r); auto with real.
@@ -1407,7 +1407,7 @@ Qed.
 (***********)
 Lemma Rlt_0_sqr : forall r, r <> 0 -> 0 < Rsqr r.
 Proof.
-  intros; case (Rdichotomy r 0); trivial; unfold Rsqr in |- *; intro.
+  intros; case (Rdichotomy r 0); trivial; unfold Rsqr; intro.
   replace (r * r) with (- r * - r); auto with real.
   replace 0 with (- r * 0); auto with real.
   replace 0 with (0 * r); auto with real.
@@ -1437,7 +1437,7 @@ Qed.
 Lemma Rlt_0_1 : 0 < 1.
 Proof.
   replace 1 with (Rsqr 1); auto with real.
-  unfold Rsqr in |- *; auto with real.
+  unfold Rsqr; auto with real.
 Qed.
 Hint Resolve Rlt_0_1: real.
 
@@ -1453,7 +1453,7 @@ Qed.
 
 Lemma Rinv_0_lt_compat : forall r, 0 < r -> 0 < / r.
 Proof.
-  intros; apply Rnot_le_lt; red in |- *; intros.
+  intros; apply Rnot_le_lt; red; intros.
   absurd (1 <= 0); auto with real.
   replace 1 with (r * / r); auto with real.
   replace 0 with (r * 0); auto with real.
@@ -1463,7 +1463,7 @@ Hint Resolve Rinv_0_lt_compat: real.
 (*********)
 Lemma Rinv_lt_0_compat : forall r, r < 0 -> / r < 0.
 Proof.
-  intros; apply Rnot_le_lt; red in |- *; intros.
+  intros; apply Rnot_le_lt; red; intros.
   absurd (1 <= 0); auto with real.
   replace 1 with (r * / r); auto with real.
   replace 0 with (r * 0); auto with real.
@@ -1477,8 +1477,8 @@ Proof.
   case (Rmult_neq_0_reg r1 r2); intros; auto with real.
   replace (r1 * r2 * / r2) with r1.
   replace (r1 * r2 * / r1) with r2; trivial.
-  symmetry  in |- *; auto with real.
-  symmetry  in |- *; auto with real.
+  symmetry ; auto with real.
+  symmetry ; auto with real.
 Qed.
 
 Lemma Rinv_1_lt_contravar : forall r1 r2, 1 <= r1 -> r1 < r2 -> / r2 < / r1.
@@ -1495,7 +1495,7 @@ Proof.
   rewrite (Rmult_comm x); rewrite <- Rmult_assoc; rewrite (Rmult_comm y (/ y));
     rewrite Rinv_l; auto with real.
   apply Rlt_dichotomy_converse; right.
-  red in |- *; apply Rlt_trans with (r2 := x); auto with real.
+  red; apply Rlt_trans with (r2 := x); auto with real.
 Qed.
 Hint Resolve Rinv_1_lt_contravar: real.
 
@@ -1508,7 +1508,7 @@ Lemma Rle_lt_0_plus_1 : forall r, 0 <= r -> 0 < r + 1.
 Proof.
   intros.
   apply Rlt_le_trans with 1; auto with real.
-  pattern 1 at 1 in |- *; replace 1 with (0 + 1); auto with real.
+  pattern 1 at 1; replace 1 with (0 + 1); auto with real.
 Qed.
 Hint Resolve Rle_lt_0_plus_1: real.
 
@@ -1516,15 +1516,15 @@ Hint Resolve Rle_lt_0_plus_1: real.
 Lemma Rlt_plus_1 : forall r, r < r + 1.
 Proof.
   intros.
-  pattern r at 1 in |- *; replace r with (r + 0); auto with real.
+  pattern r at 1; replace r with (r + 0); auto with real.
 Qed.
 Hint Resolve Rlt_plus_1: real.
 
 (**********)
 Lemma tech_Rgt_minus : forall r1 r2, 0 < r2 -> r1 > r1 - r2.
 Proof.
-  red in |- *; unfold Rminus in |- *; intros.
-  pattern r1 at 2 in |- *; replace r1 with (r1 + 0); auto with real.
+  red; unfold Rminus; intros.
+  pattern r1 at 2; replace r1 with (r1 + 0); auto with real.
 Qed.
 
 (*********************************************************)
@@ -1540,14 +1540,14 @@ Qed.
 (**********)
 Lemma S_O_plus_INR : forall n:nat, INR (1 + n) = INR 1 + INR n.
 Proof.
-  intro; simpl in |- *; case n; intros; auto with real.
+  intro; simpl; case n; intros; auto with real.
 Qed.
 
 (**********)
 Lemma plus_INR : forall n m:nat, INR (n + m) = INR n + INR m.
 Proof.
   intros n m; induction  n as [| n Hrecn].
-  simpl in |- *; auto with real.
+  simpl; auto with real.
   replace (S n + m)%nat with (S (n + m)); auto with arith.
   repeat rewrite S_INR.
   rewrite Hrecn; ring.
@@ -1557,9 +1557,9 @@ Hint Resolve plus_INR: real.
 (**********)
 Lemma minus_INR : forall n m:nat, (m <= n)%nat -> INR (n - m) = INR n - INR m.
 Proof.
-  intros n m le; pattern m, n in |- *; apply le_elim_rel; auto with real.
+  intros n m le; pattern m, n; apply le_elim_rel; auto with real.
   intros; rewrite <- minus_n_O; auto with real.
-  intros; repeat rewrite S_INR; simpl in |- *.
+  intros; repeat rewrite S_INR; simpl.
   rewrite H0; ring.
 Qed.
 Hint Resolve minus_INR: real.
@@ -1568,8 +1568,8 @@ Hint Resolve minus_INR: real.
 Lemma mult_INR : forall n m:nat, INR (n * m) = INR n * INR m.
 Proof.
   intros n m; induction  n as [| n Hrecn].
-  simpl in |- *; auto with real.
-  intros; repeat rewrite S_INR; simpl in |- *.
+  simpl; auto with real.
+  intros; repeat rewrite S_INR; simpl.
   rewrite plus_INR; rewrite Hrecn; ring.
 Qed.
 Hint Resolve mult_INR: real.
@@ -1597,11 +1597,11 @@ Qed.
 Hint Resolve lt_1_INR: real.
 
 (**********)
-Lemma pos_INR_nat_of_P : forall p:positive, 0 < INR (nat_of_P p).
+Lemma pos_INR_nat_of_P : forall p:positive, 0 < INR (Pos.to_nat p).
 Proof.
   intro; apply lt_0_INR.
-  simpl in |- *; auto with real.
-  apply nat_of_P_pos.
+  simpl; auto with real.
+  apply Pos2Nat.is_pos.
 Qed.
 Hint Resolve pos_INR_nat_of_P: real.
 
@@ -1609,7 +1609,7 @@ Hint Resolve pos_INR_nat_of_P: real.
 Lemma pos_INR : forall n:nat, 0 <= INR n.
 Proof.
   intro n; case n.
-  simpl in |- *; auto with real.
+  simpl; auto with real.
   auto with arith real.
 Qed.
 Hint Resolve pos_INR: real.
@@ -1617,10 +1617,10 @@ Hint Resolve pos_INR: real.
 Lemma INR_lt : forall n m:nat, INR n < INR m -> (n < m)%nat.
 Proof.
   double induction n m; intros.
-  simpl in |- *; exfalso; apply (Rlt_irrefl 0); auto.
+  simpl; exfalso; apply (Rlt_irrefl 0); auto.
   auto with arith.
   generalize (pos_INR (S n0)); intro; cut (INR 0 = 0);
-    [ intro H2; rewrite H2 in H0; idtac | simpl in |- *; trivial ].
+    [ intro H2; rewrite H2 in H0; idtac | simpl; trivial ].
   generalize (Rle_lt_trans 0 (INR (S n0)) 0 H1 H0); intro; exfalso;
     apply (Rlt_irrefl 0); auto.
   do 2 rewrite S_INR in H1; cut (INR n1 < INR n0).
@@ -1642,7 +1642,7 @@ Hint Resolve le_INR: real.
 (**********)
 Lemma INR_not_0 : forall n:nat, INR n <> 0 -> n <> 0%nat.
 Proof.
-  red in |- *; intros n H H1.
+  red; intros n H H1.
   apply H.
   rewrite H1; trivial.
 Qed.
@@ -1654,7 +1654,7 @@ Proof.
   intro n; case n.
   intro; absurd (0%nat = 0%nat); trivial.
   intros; rewrite S_INR.
-  apply Rgt_not_eq; red in |- *; auto with real.
+  apply Rgt_not_eq; red; auto with real.
 Qed.
 Hint Resolve not_0_INR: real.
 
@@ -1664,7 +1664,7 @@ Proof.
   case (le_lt_or_eq _ _ H1); intros H2.
   apply Rlt_dichotomy_converse; auto with real.
   exfalso; auto.
-  apply sym_not_eq; apply Rlt_dichotomy_converse; auto with real.
+  apply not_eq_sym; apply Rlt_dichotomy_converse; auto with real.
 Qed.
 Hint Resolve not_INR: real.
 
@@ -1675,7 +1675,7 @@ Proof.
   cut (n <> m).
   intro H3; generalize (not_INR n m H3); intro H4; exfalso; auto.
   omega.
-  symmetry  in |- *; cut (m <> n).
+  symmetry ; cut (m <> n).
   intro H3; generalize (not_INR m n H3); intro H4; exfalso; auto.
   omega.
 Qed.
@@ -1701,16 +1701,16 @@ Hint Resolve not_1_INR: real.
 
 
 (**********)
-Lemma IZN : forall n:Z, (0 <= n)%Z ->  exists m : nat, n = Z_of_nat m.
+Lemma IZN : forall n:Z, (0 <= n)%Z ->  exists m : nat, n = Z.of_nat m.
 Proof.
   intros z; idtac; apply Z_of_nat_complete; assumption.
 Qed.
 
 (**********)
-Lemma INR_IZR_INZ : forall n:nat, INR n = IZR (Z_of_nat n).
+Lemma INR_IZR_INZ : forall n:nat, INR n = IZR (Z.of_nat n).
 Proof.
   simple induction n; auto with real.
-  intros; simpl in |- *; rewrite nat_of_P_of_succ_nat;
+  intros; simpl; rewrite SuccNat2Pos.id_succ;
     auto with real.
 Qed.
 
@@ -1718,13 +1718,13 @@ Lemma plus_IZR_NEG_POS :
   forall p q:positive, IZR (Zpos p + Zneg q) = IZR (Zpos p) + IZR (Zneg q).
 Proof.
   intros p q; simpl. rewrite Z.pos_sub_spec.
-  case Pcompare_spec; intros H; simpl.
+  case Pos.compare_spec; intros H; simpl.
   subst. ring.
-  rewrite Pminus_minus by trivial.
-  rewrite minus_INR by (now apply lt_le_weak, Plt_lt).
+  rewrite Pos2Nat.inj_sub by trivial.
+  rewrite minus_INR by (now apply lt_le_weak, Pos2Nat.inj_lt).
   ring.
-  rewrite Pminus_minus by trivial.
-  rewrite minus_INR by (now apply lt_le_weak, Plt_lt).
+  rewrite Pos2Nat.inj_sub by trivial.
+  rewrite minus_INR by (now apply lt_le_weak, Pos2Nat.inj_lt).
   ring.
 Qed.
 
@@ -1732,55 +1732,55 @@ Qed.
 Lemma plus_IZR : forall n m:Z, IZR (n + m) = IZR n + IZR m.
 Proof.
   intro z; destruct z; intro t; destruct t; intros; auto with real.
-  simpl; intros; rewrite Pplus_plus; auto with real.
+  simpl; intros; rewrite Pos2Nat.inj_add; auto with real.
   apply plus_IZR_NEG_POS.
-  rewrite Zplus_comm; rewrite Rplus_comm; apply plus_IZR_NEG_POS.
-  simpl; intros; rewrite Pplus_plus; rewrite plus_INR;
+  rewrite Z.add_comm; rewrite Rplus_comm; apply plus_IZR_NEG_POS.
+  simpl; intros; rewrite Pos2Nat.inj_add; rewrite plus_INR;
     auto with real.
 Qed.
 
 (**********)
 Lemma mult_IZR : forall n m:Z, IZR (n * m) = IZR n * IZR m.
 Proof.
-  intros z t; case z; case t; simpl in |- *; auto with real.
-  intros t1 z1; rewrite Pmult_mult; auto with real.
-  intros t1 z1; rewrite Pmult_mult; auto with real.
+  intros z t; case z; case t; simpl; auto with real.
+  intros t1 z1; rewrite Pos2Nat.inj_mul; auto with real.
+  intros t1 z1; rewrite Pos2Nat.inj_mul; auto with real.
   rewrite Rmult_comm.
   rewrite Ropp_mult_distr_l_reverse; auto with real.
   apply Ropp_eq_compat; rewrite mult_comm; auto with real.
-  intros t1 z1; rewrite Pmult_mult; auto with real.
+  intros t1 z1; rewrite Pos2Nat.inj_mul; auto with real.
   rewrite Ropp_mult_distr_l_reverse; auto with real.
-  intros t1 z1; rewrite Pmult_mult; auto with real.
+  intros t1 z1; rewrite Pos2Nat.inj_mul; auto with real.
   rewrite Rmult_opp_opp; auto with real.
 Qed.
 
-Lemma pow_IZR : forall z n, pow (IZR z) n = IZR (Zpower z (Z_of_nat n)).
+Lemma pow_IZR : forall z n, pow (IZR z) n = IZR (Z.pow z (Z.of_nat n)).
 Proof.
  intros z [|n];simpl;trivial.
  rewrite Zpower_pos_nat.
- rewrite nat_of_P_of_succ_nat. unfold Zpower_nat;simpl.
+ rewrite SuccNat2Pos.id_succ. unfold Zpower_nat;simpl.
  rewrite mult_IZR.
  induction n;simpl;trivial.
  rewrite mult_IZR;ring[IHn].
 Qed.
 
 (**********)
-Lemma succ_IZR : forall n:Z, IZR (Zsucc n) = IZR n + 1.
+Lemma succ_IZR : forall n:Z, IZR (Z.succ n) = IZR n + 1.
 Proof.
-  intro; change 1 with (IZR 1); unfold Zsucc; apply plus_IZR.
+  intro; change 1 with (IZR 1); unfold Z.succ; apply plus_IZR.
 Qed.
 
 (**********)
 Lemma opp_IZR : forall n:Z, IZR (- n) = - IZR n.
 Proof.
-  intro z; case z; simpl in |- *; auto with real.
+  intro z; case z; simpl; auto with real.
 Qed.
 
 Definition Ropp_Ropp_IZR := opp_IZR.
 
 Lemma minus_IZR : forall n m:Z, IZR (n - m) = IZR n - IZR m.
 Proof.
-  intros; unfold Zminus, Rminus.
+  intros; unfold Z.sub, Rminus.
   rewrite <- opp_IZR.
   apply plus_IZR.
 Qed.
@@ -1788,16 +1788,16 @@ Qed.
 (**********)
 Lemma Z_R_minus : forall n m:Z, IZR n - IZR m = IZR (n - m).
 Proof.
-  intros z1 z2; unfold Rminus in |- *; unfold Zminus in |- *.
-  rewrite <- (Ropp_Ropp_IZR z2); symmetry  in |- *; apply plus_IZR.
+  intros z1 z2; unfold Rminus; unfold Z.sub.
+  rewrite <- (Ropp_Ropp_IZR z2); symmetry ; apply plus_IZR.
 Qed.
 
 (**********)
 Lemma lt_0_IZR : forall n:Z, 0 < IZR n -> (0 < n)%Z.
 Proof.
-  intro z; case z; simpl in |- *; intros.
+  intro z; case z; simpl; intros.
   absurd (0 < 0); auto with real.
-  unfold Zlt in |- *; simpl in |- *; trivial.
+  unfold Z.lt; simpl; trivial.
   case Rlt_not_le with (1 := H).
   replace 0 with (-0); auto with real.
 Qed.
@@ -1805,7 +1805,7 @@ Qed.
 (**********)
 Lemma lt_IZR : forall n m:Z, IZR n < IZR m -> (n < m)%Z.
 Proof.
-  intros z1 z2 H; apply Zlt_0_minus_lt.
+  intros z1 z2 H; apply Z.lt_0_sub.
   apply lt_0_IZR.
   rewrite <- Z_R_minus.
   exact (Rgt_minus (IZR z2) (IZR z1) H).
@@ -1814,10 +1814,10 @@ Qed.
 (**********)
 Lemma eq_IZR_R0 : forall n:Z, IZR n = 0 -> n = 0%Z.
 Proof.
-  intro z; destruct z; simpl in |- *; intros; auto with zarith.
-  case (Rlt_not_eq 0 (INR (nat_of_P p))); auto with real.
-  case (Rlt_not_eq (- INR (nat_of_P p)) 0); auto with real.
-  apply Ropp_lt_gt_0_contravar. unfold Rgt in |- *; apply pos_INR_nat_of_P.
+  intro z; destruct z; simpl; intros; auto with zarith.
+  case (Rlt_not_eq 0 (INR (Pos.to_nat p))); auto with real.
+  case (Rlt_not_eq (- INR (Pos.to_nat p)) 0); auto with real.
+  apply Ropp_lt_gt_0_contravar. unfold Rgt; apply pos_INR_nat_of_P.
 Qed.
 
 (**********)
@@ -1831,23 +1831,23 @@ Qed.
 (**********)
 Lemma not_0_IZR : forall n:Z, n <> 0%Z -> IZR n <> 0.
 Proof.
-  intros z H; red in |- *; intros H0; case H.
+  intros z H; red; intros H0; case H.
   apply eq_IZR; auto.
 Qed.
 
 (*********)
 Lemma le_0_IZR : forall n:Z, 0 <= IZR n -> (0 <= n)%Z.
 Proof.
-  unfold Rle in |- *; intros z [H| H].
-  red in |- *; intro; apply (Zlt_le_weak 0 z (lt_0_IZR z H)); assumption.
+  unfold Rle; intros z [H| H].
+  red; intro; apply (Z.lt_le_incl 0 z (lt_0_IZR z H)); assumption.
   rewrite (eq_IZR_R0 z); auto with zarith real.
 Qed.
 
 (**********)
 Lemma le_IZR : forall n m:Z, IZR n <= IZR m -> (n <= m)%Z.
 Proof.
-  unfold Rle in |- *; intros z1 z2 [H| H].
-  apply (Zlt_le_weak z1 z2); auto with real.
+  unfold Rle; intros z1 z2 [H| H].
+  apply (Z.lt_le_incl z1 z2); auto with real.
   apply lt_IZR; trivial.
   rewrite (eq_IZR z1 z2); auto with zarith real.
 Qed.
@@ -1855,20 +1855,20 @@ Qed.
 (**********)
 Lemma le_IZR_R1 : forall n:Z, IZR n <= 1 -> (n <= 1)%Z.
 Proof.
-  pattern 1 at 1 in |- *; replace 1 with (IZR 1); intros; auto.
+  pattern 1 at 1; replace 1 with (IZR 1); intros; auto.
   apply le_IZR; trivial.
 Qed.
 
 (**********)
 Lemma IZR_ge : forall n m:Z, (n >= m)%Z -> IZR n >= IZR m.
 Proof.
-  intros m n H; apply Rnot_lt_ge; red in |- *; intro.
+  intros m n H; apply Rnot_lt_ge; red; intro.
   generalize (lt_IZR m n H0); intro; omega.
 Qed.
 
 Lemma IZR_le : forall n m:Z, (n <= m)%Z -> IZR n <= IZR m.
 Proof.
-  intros m n H; apply Rnot_gt_le; red in |- *; intro.
+  intros m n H; apply Rnot_gt_le; red; intro.
   unfold Rgt in H0; generalize (lt_IZR n m H0); intro; omega.
 Qed.
 
@@ -1883,10 +1883,10 @@ Qed.
 Lemma one_IZR_lt1 : forall n:Z, -1 < IZR n < 1 -> n = 0%Z.
 Proof.
   intros z [H1 H2].
-  apply Zle_antisym.
-  apply Zlt_succ_le; apply lt_IZR; trivial.
-  replace 0%Z with (Zsucc (-1)); trivial.
-  apply Zlt_le_succ; apply lt_IZR; trivial.
+  apply Z.le_antisymm.
+  apply Z.lt_succ_r; apply lt_IZR; trivial.
+  replace 0%Z with (Z.succ (-1)); trivial.
+  apply Z.le_succ_l; apply lt_IZR; trivial.
 Qed.
 
 Lemma one_IZR_r_R1 :
@@ -1897,10 +1897,10 @@ Proof.
   apply one_IZR_lt1.
   rewrite <- Z_R_minus; split.
   replace (-1) with (r - (r + 1)).
-  unfold Rminus in |- *; apply Rplus_lt_le_compat; auto with real.
+  unfold Rminus; apply Rplus_lt_le_compat; auto with real.
   ring.
   replace 1 with (r + 1 - r).
-  unfold Rminus in |- *; apply Rplus_le_lt_compat; auto with real.
+  unfold Rminus; apply Rplus_le_lt_compat; auto with real.
   ring.
 Qed.
 
@@ -1931,6 +1931,20 @@ Proof.
     apply (Rmult_le_compat_l x 0 y H H0).
 Qed.
 
+Lemma Rle_Rinv : forall x y:R, 0 < x -> 0 < y -> x <= y -> / y <= / x.
+Proof.
+  intros; apply Rmult_le_reg_l with x.
+  apply H.
+  rewrite <- Rinv_r_sym.
+  apply Rmult_le_reg_l with y.
+  apply H0.
+  rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc;
+    rewrite <- Rinv_l_sym.
+  rewrite Rmult_1_r; apply H1.
+  red; intro; rewrite H2 in H0; elim (Rlt_irrefl _ H0).
+  red; intro; rewrite H2 in H; elim (Rlt_irrefl _ H).
+Qed.
+
 Lemma double : forall r1, 2 * r1 = r1 + r1.
 Proof.
   intro; ring.
@@ -1938,10 +1952,10 @@ Qed.
 
 Lemma double_var : forall r1, r1 = r1 / 2 + r1 / 2.
 Proof.
-  intro; rewrite <- double; unfold Rdiv in |- *; rewrite <- Rmult_assoc;
-    symmetry  in |- *; apply Rinv_r_simpl_m.
+  intro; rewrite <- double; unfold Rdiv; rewrite <- Rmult_assoc;
+    symmetry ; apply Rinv_r_simpl_m.
   replace 2 with (INR 2);
-  [ apply not_0_INR; discriminate | unfold INR in |- *; ring ].
+  [ apply not_0_INR; discriminate | unfold INR; ring ].
 Qed.
 
 (*********************************************************)
@@ -1976,22 +1990,22 @@ Proof.
   rewrite (Rplus_comm y); intro H5; apply Rplus_le_reg_l with x; assumption.
   ring.
   replace 2 with (INR 2); [ apply not_0_INR; discriminate | reflexivity ].
-  pattern y at 2 in |- *; replace y with (y / 2 + y / 2).
-  unfold Rminus, Rdiv in |- *.
+  pattern y at 2; replace y with (y / 2 + y / 2).
+  unfold Rminus, Rdiv.
   repeat rewrite Rmult_plus_distr_r.
   ring.
   cut (forall z:R, 2 * z = z + z).
   intro.
   rewrite <- (H4 (y / 2)).
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   rewrite <- Rmult_assoc; apply Rinv_r_simpl_m.
   replace 2 with (INR 2).
   apply not_0_INR.
   discriminate.
-  unfold INR in |- *; reflexivity.
+  unfold INR; reflexivity.
   intro; ring.
   cut (0%nat <> 2%nat);
-    [ intro H0; generalize (lt_0_INR 2 (neq_O_lt 2 H0)); unfold INR in |- *;
+    [ intro H0; generalize (lt_0_INR 2 (neq_O_lt 2 H0)); unfold INR;
       intro; assumption
       | discriminate ].
 Qed.
diff --git a/theories/Reals/RList.v b/theories/Reals/RList.v
index dbd2e52f..6d42434a 100644
--- a/theories/Reals/RList.v
+++ b/theories/Reals/RList.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -8,7 +8,7 @@
 
 Require Import Rbase.
 Require Import Rfunctions.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Inductive Rlist : Type :=
 | nil : Rlist
@@ -52,19 +52,19 @@ Proof.
   simpl in H; elim H.
   induction  l as [| r0 l Hrecl0].
   simpl in H; elim H; intro.
-  simpl in |- *; right; assumption.
+  simpl; right; assumption.
   elim H0.
   replace (MaxRlist (cons r (cons r0 l))) with (Rmax r (MaxRlist (cons r0 l))).
   simpl in H; decompose [or] H.
   rewrite H0; apply RmaxLess1.
-  unfold Rmax in |- *; case (Rle_dec r (MaxRlist (cons r0 l))); intro.
-  apply Hrecl; simpl in |- *; tauto.
+  unfold Rmax; case (Rle_dec r (MaxRlist (cons r0 l))); intro.
+  apply Hrecl; simpl; tauto.
   apply Rle_trans with (MaxRlist (cons r0 l));
-    [ apply Hrecl; simpl in |- *; tauto | left; auto with real ].
-  unfold Rmax in |- *; case (Rle_dec r (MaxRlist (cons r0 l))); intro.
-  apply Hrecl; simpl in |- *; tauto.
+    [ apply Hrecl; simpl; tauto | left; auto with real ].
+  unfold Rmax; case (Rle_dec r (MaxRlist (cons r0 l))); intro.
+  apply Hrecl; simpl; tauto.
   apply Rle_trans with (MaxRlist (cons r0 l));
-    [ apply Hrecl; simpl in |- *; tauto | left; auto with real ].
+    [ apply Hrecl; simpl; tauto | left; auto with real ].
   reflexivity.
 Qed.
 
@@ -80,19 +80,19 @@ Proof.
   simpl in H; elim H.
   induction  l as [| r0 l Hrecl0].
   simpl in H; elim H; intro.
-  simpl in |- *; right; symmetry  in |- *; assumption.
+  simpl; right; symmetry ; assumption.
   elim H0.
   replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))).
   simpl in H; decompose [or] H.
   rewrite H0; apply Rmin_l.
-  unfold Rmin in |- *; case (Rle_dec r (MinRlist (cons r0 l))); intro.
+  unfold Rmin; case (Rle_dec r (MinRlist (cons r0 l))); intro.
   apply Rle_trans with (MinRlist (cons r0 l)).
   assumption.
-  apply Hrecl; simpl in |- *; tauto.
-  apply Hrecl; simpl in |- *; tauto.
+  apply Hrecl; simpl; tauto.
+  apply Hrecl; simpl; tauto.
   apply Rle_trans with (MinRlist (cons r0 l)).
   apply Rmin_r.
-  apply Hrecl; simpl in |- *; tauto.
+  apply Hrecl; simpl; tauto.
   reflexivity.
 Qed.
 
@@ -101,7 +101,7 @@ Lemma AbsList_P1 :
 Proof.
   intros; induction  l as [| r l Hrecl].
   elim H.
-  simpl in |- *; simpl in H; elim H; intro.
+  simpl; simpl in H; elim H; intro.
   left; rewrite H0; reflexivity.
   right; apply Hrecl; assumption.
 Qed.
@@ -112,11 +112,11 @@ Proof.
   intros; induction  l as [| r l Hrecl].
   apply Rlt_0_1.
   induction  l as [| r0 l Hrecl0].
-  simpl in |- *; apply H; simpl in |- *; tauto.
+  simpl; apply H; simpl; tauto.
   replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))).
-  unfold Rmin in |- *; case (Rle_dec r (MinRlist (cons r0 l))); intro.
-  apply H; simpl in |- *; tauto.
-  apply Hrecl; intros; apply H; simpl in |- *; simpl in H0; tauto.
+  unfold Rmin; case (Rle_dec r (MinRlist (cons r0 l))); intro.
+  apply H; simpl; tauto.
+  apply Hrecl; intros; apply H; simpl; simpl in H0; tauto.
   reflexivity.
 Qed.
 
@@ -128,10 +128,10 @@ Proof.
   elim H.
   elim H; intro.
   exists r; split.
-  simpl in |- *; tauto.
+  simpl; tauto.
   assumption.
   assert (H1 := Hrecl H0); elim H1; intros; elim H2; clear H2; intros;
-    exists x0; simpl in |- *; simpl in H2; tauto.
+    exists x0; simpl; simpl in H2; tauto.
 Qed.
 
 Lemma MaxRlist_P2 :
@@ -140,9 +140,9 @@ Proof.
   intros; induction  l as [| r l Hrecl].
   simpl in H; elim H; trivial.
   induction  l as [| r0 l Hrecl0].
-  simpl in |- *; left; reflexivity.
-  change (In (Rmax r (MaxRlist (cons r0 l))) (cons r (cons r0 l))) in |- *;
-    unfold Rmax in |- *; case (Rle_dec r (MaxRlist (cons r0 l)));
+  simpl; left; reflexivity.
+  change (In (Rmax r (MaxRlist (cons r0 l))) (cons r (cons r0 l)));
+    unfold Rmax; case (Rle_dec r (MaxRlist (cons r0 l)));
       intro.
   right; apply Hrecl; exists r0; left; reflexivity.
   left; reflexivity.
@@ -164,7 +164,7 @@ Lemma pos_Rl_P1 :
 Proof.
   intros; induction  l as [| r l Hrecl];
     [ elim (lt_n_O _ H)
-      | simpl in |- *; case (Rlength l); [ reflexivity | intro; reflexivity ] ].
+      | simpl; case (Rlength l); [ reflexivity | intro; reflexivity ] ].
 Qed.
 
 Lemma pos_Rl_P2 :
@@ -177,14 +177,14 @@ Proof.
   split; intro.
   elim H; intro.
   exists 0%nat; split;
-    [ simpl in |- *; apply lt_O_Sn | simpl in |- *; apply H0 ].
+    [ simpl; apply lt_O_Sn | simpl; apply H0 ].
   elim Hrecl; intros; assert (H3 := H1 H0); elim H3; intros; elim H4; intros;
     exists (S x0); split;
-      [ simpl in |- *; apply lt_n_S; assumption | simpl in |- *; assumption ].
+      [ simpl; apply lt_n_S; assumption | simpl; assumption ].
   elim H; intros; elim H0; intros; elim (zerop x0); intro.
   rewrite a in H2; simpl in H2; left; assumption.
   right; elim Hrecl; intros; apply H4; assert (H5 : S (pred x0) = x0).
-  symmetry  in |- *; apply S_pred with 0%nat; assumption.
+  symmetry ; apply S_pred with 0%nat; assumption.
   exists (pred x0); split;
     [ simpl in H1; apply lt_S_n; rewrite H5; assumption
       | rewrite <- H5 in H2; simpl in H2; assumption ].
@@ -201,18 +201,18 @@ Proof.
   exists nil; intros; split;
     [ reflexivity | intros; simpl in H0; elim (lt_n_O _ H0) ].
   assert (H0 : In r (cons r l)).
-  simpl in |- *; left; reflexivity.
+  simpl; left; reflexivity.
   assert (H1 := H _ H0);
     assert (H2 : forall x:R, In x l ->  exists y : R, P x y).
-  intros; apply H; simpl in |- *; right; assumption.
+  intros; apply H; simpl; right; assumption.
   assert (H3 := Hrecl H2); elim H1; intros; elim H3; intros; exists (cons x x0);
     intros; elim H5; clear H5; intros; split.
-  simpl in |- *; rewrite H5; reflexivity.
+  simpl; rewrite H5; reflexivity.
   intros; elim (zerop i); intro.
-  rewrite a; simpl in |- *; assumption.
+  rewrite a; simpl; assumption.
   assert (H8 : i = S (pred i)).
   apply S_pred with 0%nat; assumption.
-  rewrite H8; simpl in |- *; apply H6; simpl in H7; apply lt_S_n; rewrite <- H8;
+  rewrite H8; simpl; apply H6; simpl in H7; apply lt_S_n; rewrite <- H8;
     assumption.
 Qed.
 
@@ -271,7 +271,7 @@ Lemma RList_P0 :
 Proof.
   intros; induction  l as [| r l Hrecl];
     [ left; reflexivity
-      | simpl in |- *; case (Rle_dec r a); intro;
+      | simpl; case (Rle_dec r a); intro;
         [ right; reflexivity | left; reflexivity ] ].
 Qed.
 
@@ -279,41 +279,41 @@ Lemma RList_P1 :
   forall (l:Rlist) (a:R), ordered_Rlist l -> ordered_Rlist (insert l a).
 Proof.
   intros; induction  l as [| r l Hrecl].
-  simpl in |- *; unfold ordered_Rlist in |- *; intros; simpl in H0;
+  simpl; unfold ordered_Rlist; intros; simpl in H0;
     elim (lt_n_O _ H0).
-  simpl in |- *; case (Rle_dec r a); intro.
+  simpl; case (Rle_dec r a); intro.
   assert (H1 : ordered_Rlist l).
-  unfold ordered_Rlist in |- *; unfold ordered_Rlist in H; intros;
+  unfold ordered_Rlist; unfold ordered_Rlist in H; intros;
     assert (H1 : (S i < pred (Rlength (cons r l)))%nat);
-      [ simpl in |- *; replace (Rlength l) with (S (pred (Rlength l)));
+      [ simpl; replace (Rlength l) with (S (pred (Rlength l)));
         [ apply lt_n_S; assumption
-          | symmetry  in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *;
+          | symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
             intro; rewrite <- H1 in H0; simpl in H0; elim (lt_n_O _ H0) ]
         | apply (H _ H1) ].
-  assert (H2 := Hrecl H1); unfold ordered_Rlist in |- *; intros;
+  assert (H2 := Hrecl H1); unfold ordered_Rlist; intros;
     induction  i as [| i Hreci].
-  simpl in |- *; assert (H3 := RList_P0 l a); elim H3; intro.
+  simpl; assert (H3 := RList_P0 l a); elim H3; intro.
   rewrite H4; assumption.
   induction  l as [| r1 l Hrecl0];
-    [ simpl in |- *; assumption
-      | rewrite H4; apply (H 0%nat); simpl in |- *; apply lt_O_Sn ].
-  simpl in |- *; apply H2; simpl in H0; apply lt_S_n;
+    [ simpl; assumption
+      | rewrite H4; apply (H 0%nat); simpl; apply lt_O_Sn ].
+  simpl; apply H2; simpl in H0; apply lt_S_n;
     replace (S (pred (Rlength (insert l a)))) with (Rlength (insert l a));
     [ assumption
-      | apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+      | apply S_pred with 0%nat; apply neq_O_lt; red; intro;
         rewrite <- H3 in H0; elim (lt_n_O _ H0) ].
-  unfold ordered_Rlist in |- *; intros; induction  i as [| i Hreci];
-    [ simpl in |- *; auto with real
-      | change (pos_Rl (cons r l) i <= pos_Rl (cons r l) (S i)) in |- *; apply H;
-        simpl in H0; simpl in |- *; apply (lt_S_n _ _ H0) ].
+  unfold ordered_Rlist; intros; induction  i as [| i Hreci];
+    [ simpl; auto with real
+      | change (pos_Rl (cons r l) i <= pos_Rl (cons r l) (S i)); apply H;
+        simpl in H0; simpl; apply (lt_S_n _ _ H0) ].
 Qed.
 
 Lemma RList_P2 :
   forall l1 l2:Rlist, ordered_Rlist l2 -> ordered_Rlist (cons_ORlist l1 l2).
 Proof.
   simple induction l1;
-    [ intros; simpl in |- *; apply H
-      | intros; simpl in |- *; apply H; apply RList_P1; assumption ].
+    [ intros; simpl; apply H
+      | intros; simpl; apply H; apply RList_P1; assumption ].
 Qed.
 
 Lemma RList_P3 :
@@ -324,11 +324,11 @@ Proof.
     [ induction  l as [| r l Hrecl] | induction  l as [| r l Hrecl] ].
   elim H.
   elim H; intro;
-    [ exists 0%nat; split; [ apply H0 | simpl in |- *; apply lt_O_Sn ]
+    [ exists 0%nat; split; [ apply H0 | simpl; apply lt_O_Sn ]
       | elim (Hrecl H0); intros; elim H1; clear H1; intros; exists (S x0); split;
-        [ apply H1 | simpl in |- *; apply lt_n_S; assumption ] ].
+        [ apply H1 | simpl; apply lt_n_S; assumption ] ].
   elim H; intros; elim H0; intros; elim (lt_n_O _ H2).
-  simpl in |- *; elim H; intros; elim H0; clear H0; intros;
+  simpl; elim H; intros; elim H0; clear H0; intros;
     induction  x0 as [| x0 Hrecx0];
       [ left; apply H0
         | right; apply Hrecl; exists x0; split;
@@ -338,10 +338,10 @@ Qed.
 Lemma RList_P4 :
   forall (l1:Rlist) (a:R), ordered_Rlist (cons a l1) -> ordered_Rlist l1.
 Proof.
-  intros; unfold ordered_Rlist in |- *; intros; apply (H (S i)); simpl in |- *;
+  intros; unfold ordered_Rlist; intros; apply (H (S i)); simpl;
     replace (Rlength l1) with (S (pred (Rlength l1)));
     [ apply lt_n_S; assumption
-      | symmetry  in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *;
+      | symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
         intro; rewrite <- H1 in H0; elim (lt_n_O _ H0) ].
 Qed.
 
@@ -350,11 +350,11 @@ Lemma RList_P5 :
 Proof.
   intros; induction  l as [| r l Hrecl];
     [ elim H0
-      | simpl in |- *; elim H0; intro;
+      | simpl; elim H0; intro;
         [ rewrite H1; right; reflexivity
           | apply Rle_trans with (pos_Rl l 0);
-            [ apply (H 0%nat); simpl in |- *; induction  l as [| r0 l Hrecl0];
-              [ elim H1 | simpl in |- *; apply lt_O_Sn ]
+            [ apply (H 0%nat); simpl; induction  l as [| r0 l Hrecl0];
+              [ elim H1 | simpl; apply lt_O_Sn ]
               | apply Hrecl; [ eapply RList_P4; apply H | assumption ] ] ] ].
 Qed.
 
@@ -366,13 +366,13 @@ Lemma RList_P6 :
 Proof.
   simple induction l; split; intro.
   intros; right; reflexivity.
-  unfold ordered_Rlist in |- *; intros; simpl in H0; elim (lt_n_O _ H0).
+  unfold ordered_Rlist; intros; simpl in H0; elim (lt_n_O _ H0).
   intros; induction  i as [| i Hreci];
     [ induction  j as [| j Hrecj];
       [ right; reflexivity
-        | simpl in |- *; apply Rle_trans with (pos_Rl r0 0);
-          [ apply (H0 0%nat); simpl in |- *; simpl in H2; apply neq_O_lt;
-            red in |- *; intro; rewrite <- H3 in H2;
+        | simpl; apply Rle_trans with (pos_Rl r0 0);
+          [ apply (H0 0%nat); simpl; simpl in H2; apply neq_O_lt;
+            red; intro; rewrite <- H3 in H2;
               assert (H4 := lt_S_n _ _ H2); elim (lt_n_O _ H4)
             | elim H; intros; apply H3;
               [ apply RList_P4 with r; assumption
@@ -380,12 +380,12 @@ Proof.
                 | simpl in H2; apply lt_S_n; assumption ] ] ]
       | induction  j as [| j Hrecj];
         [ elim (le_Sn_O _ H1)
-          | simpl in |- *; elim H; intros; apply H3;
+          | simpl; elim H; intros; apply H3;
             [ apply RList_P4 with r; assumption
               | apply le_S_n; assumption
               | simpl in H2; apply lt_S_n; assumption ] ] ].
-  unfold ordered_Rlist in |- *; intros; apply H0;
-    [ apply le_n_Sn | simpl in |- *; simpl in H1; apply lt_n_S; assumption ].
+  unfold ordered_Rlist; intros; apply H0;
+    [ apply le_n_Sn | simpl; simpl in H1; apply lt_n_S; assumption ].
 Qed.
 
 Lemma RList_P7 :
@@ -397,11 +397,11 @@ Proof.
       clear H1; intros; assert (H4 := H1 H0); elim H4; clear H4;
         intros; elim H4; clear H4; intros; rewrite H4;
           assert (H6 : Rlength l = S (pred (Rlength l))).
-  apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
     rewrite <- H6 in H5; elim (lt_n_O _ H5).
   apply H3;
     [ rewrite H6 in H5; apply lt_n_Sm_le; assumption
-      | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H7 in H5;
+      | apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H7 in H5;
         elim (lt_n_O _ H5) ].
 Qed.
 
@@ -420,7 +420,7 @@ Proof.
           [ left; assumption
             | right; left; assumption
             | right; right; assumption ] ]
-      | simpl in |- *; case (Rle_dec r a); intro;
+      | simpl; case (Rle_dec r a); intro;
         [ simpl in H0; decompose [or] H0;
           [ right; elim (H a x); intros; apply H3; left
             | left
@@ -435,14 +435,14 @@ Proof.
   simple induction l1.
   intros; split; intro;
     [ simpl in H; right; assumption
-      | simpl in |- *; elim H; intro; [ elim H0 | assumption ] ].
+      | simpl; elim H; intro; [ elim H0 | assumption ] ].
   intros; split.
-  simpl in |- *; intros; elim (H (insert l2 r) x); intros; assert (H3 := H1 H0);
+  simpl; intros; elim (H (insert l2 r) x); intros; assert (H3 := H1 H0);
     elim H3; intro;
       [ left; right; assumption
         | elim (RList_P8 l2 r x); intros H5 _; assert (H6 := H5 H4); elim H6; intro;
           [ left; left; assumption | right; assumption ] ].
-  intro; simpl in |- *; elim (H (insert l2 r) x); intros _ H1; apply H1;
+  intro; simpl; elim (H (insert l2 r) x); intros _ H1; apply H1;
     elim H0; intro;
       [ elim H2; intro;
         [ right; elim (RList_P8 l2 r x); intros _ H4; apply H4; left; assumption
@@ -455,8 +455,8 @@ Lemma RList_P10 :
 Proof.
   intros; induction  l as [| r l Hrecl];
     [ reflexivity
-      | simpl in |- *; case (Rle_dec r a); intro;
-        [ simpl in |- *; rewrite Hrecl; reflexivity | reflexivity ] ].
+      | simpl; case (Rle_dec r a); intro;
+        [ simpl; rewrite Hrecl; reflexivity | reflexivity ] ].
 Qed.
 
 Lemma RList_P11 :
@@ -465,7 +465,7 @@ Lemma RList_P11 :
 Proof.
   simple induction l1;
     [ intro; reflexivity
-      | intros; simpl in |- *; rewrite (H (insert l2 r)); rewrite RList_P10;
+      | intros; simpl; rewrite (H (insert l2 r)); rewrite RList_P10;
         apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR;
           rewrite S_INR; ring ].
 Qed.
@@ -477,7 +477,7 @@ Proof.
   simple induction l;
     [ intros; elim (lt_n_O _ H)
       | intros; induction  i as [| i Hreci];
-        [ reflexivity | simpl in |- *; apply H; apply lt_S_n; apply H0 ] ].
+        [ reflexivity | simpl; apply H; apply lt_S_n; apply H0 ] ].
 Qed.
 
 Lemma RList_P13 :
@@ -494,13 +494,13 @@ Proof.
   change
     (pos_Rl (mid_Rlist (cons r1 r2) r) (S i) =
       (pos_Rl (cons r1 r2) i + pos_Rl (cons r1 r2) (S i)) / 2)
-    in |- *; apply H0; simpl in |- *; apply lt_S_n; assumption.
+   ; apply H0; simpl; apply lt_S_n; assumption.
 Qed.
 
 Lemma RList_P14 : forall (l:Rlist) (a:R), Rlength (mid_Rlist l a) = Rlength l.
 Proof.
   simple induction l; intros;
-    [ reflexivity | simpl in |- *; rewrite (H r); reflexivity ].
+    [ reflexivity | simpl; rewrite (H r); reflexivity ].
 Qed.
 
 Lemma RList_P15 :
@@ -511,7 +511,7 @@ Lemma RList_P15 :
 Proof.
   intros; apply Rle_antisym.
   induction  l1 as [| r l1 Hrecl1];
-    [ simpl in |- *; simpl in H1; right; symmetry  in |- *; assumption
+    [ simpl; simpl in H1; right; symmetry ; assumption
       | elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) 0)); intros;
         assert
           (H4 :
@@ -520,7 +520,7 @@ Proof.
             | assert (H5 := H3 H4); apply RList_P5;
               [ apply RList_P2; assumption | assumption ] ] ].
   induction  l1 as [| r l1 Hrecl1];
-    [ simpl in |- *; simpl in H1; right; assumption
+    [ simpl; simpl in H1; right; assumption
       | assert
         (H2 :
           In (pos_Rl (cons_ORlist (cons r l1) l2) 0) (cons_ORlist (cons r l1) l2));
@@ -528,7 +528,7 @@ Proof.
           (RList_P3 (cons_ORlist (cons r l1) l2)
             (pos_Rl (cons_ORlist (cons r l1) l2) 0));
           intros; apply H3; exists 0%nat; split;
-            [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_O_Sn ]
+            [ reflexivity | rewrite RList_P11; simpl; apply lt_O_Sn ]
           | elim (RList_P9 (cons r l1) l2 (pos_Rl (cons_ORlist (cons r l1) l2) 0));
             intros; assert (H5 := H3 H2); elim H5; intro;
               [ apply RList_P5; assumption
@@ -545,7 +545,7 @@ Lemma RList_P16 :
 Proof.
   intros; apply Rle_antisym.
   induction  l1 as [| r l1 Hrecl1].
-  simpl in |- *; simpl in H1; right; symmetry  in |- *; assumption.
+  simpl; simpl in H1; right; symmetry ; assumption.
   assert
     (H2 :
       In
@@ -557,7 +557,7 @@ Proof.
         (pos_Rl (cons_ORlist (cons r l1) l2)
           (pred (Rlength (cons_ORlist (cons r l1) l2)))));
       intros; apply H3; exists (pred (Rlength (cons_ORlist (cons r l1) l2)));
-        split; [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_n_Sn ]
+        split; [ reflexivity | rewrite RList_P11; simpl; apply lt_n_Sn ]
       | elim
         (RList_P9 (cons r l1) l2
           (pos_Rl (cons_ORlist (cons r l1) l2)
@@ -565,7 +565,7 @@ Proof.
         intros; assert (H5 := H3 H2); elim H5; intro;
           [ apply RList_P7; assumption | rewrite H1; apply RList_P7; assumption ] ].
   induction  l1 as [| r l1 Hrecl1].
-  simpl in |- *; simpl in H1; right; assumption.
+  simpl; simpl in H1; right; assumption.
   elim
     (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))));
     intros;
@@ -573,10 +573,10 @@ Proof.
         (H4 :
           In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) (cons r l1) \/
           In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) l2);
-        [ left; change (In (pos_Rl (cons r l1) (Rlength l1)) (cons r l1)) in |- *;
+        [ left; change (In (pos_Rl (cons r l1) (Rlength l1)) (cons r l1));
           elim (RList_P3 (cons r l1) (pos_Rl (cons r l1) (Rlength l1)));
             intros; apply H5; exists (Rlength l1); split;
-              [ reflexivity | simpl in |- *; apply lt_n_Sn ]
+              [ reflexivity | simpl; apply lt_n_Sn ]
           | assert (H5 := H3 H4); apply RList_P7;
             [ apply RList_P2; assumption
               | elim
@@ -587,7 +587,7 @@ Proof.
                     (RList_P3 (cons r l1)
                       (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))));
                     intros; apply H9; exists (pred (Rlength (cons r l1)));
-                      split; [ reflexivity | simpl in |- *; apply lt_n_Sn ] ] ].
+                      split; [ reflexivity | simpl; apply lt_n_Sn ] ] ].
 Qed.
 
 Lemma RList_P17 :
@@ -599,14 +599,14 @@ Proof.
   simple induction l1.
   intros; elim H0.
   intros; induction  i as [| i Hreci].
-  simpl in |- *; elim H1; intro;
+  simpl; elim H1; intro;
     [ simpl in H2; rewrite H4 in H2; elim (Rlt_irrefl _ H2)
       | apply RList_P5; [ apply RList_P4 with r; assumption | assumption ] ].
-  simpl in |- *; simpl in H2; elim H1; intro.
+  simpl; simpl in H2; elim H1; intro.
   rewrite H4 in H2; assert (H5 : r <= pos_Rl r0 i);
     [ apply Rle_trans with (pos_Rl r0 0);
-      [ apply (H0 0%nat); simpl in |- *; simpl in H3; apply neq_O_lt;
-        red in |- *; intro; rewrite <- H5 in H3; elim (lt_n_O _ H3)
+      [ apply (H0 0%nat); simpl; simpl in H3; apply neq_O_lt;
+        red; intro; rewrite <- H5 in H3; elim (lt_n_O _ H3)
         | elim (RList_P6 r0); intros; apply H5;
           [ apply RList_P4 with r; assumption
             | apply le_O_n
@@ -618,7 +618,7 @@ Proof.
       | simpl in H3; apply lt_S_n;
         replace (S (pred (Rlength r0))) with (Rlength r0);
         [ apply H3
-          | apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+          | apply S_pred with 0%nat; apply neq_O_lt; red; intro;
             rewrite <- H5 in H3; elim (lt_n_O _ H3) ] ].
 Qed.
 
@@ -626,7 +626,7 @@ Lemma RList_P18 :
   forall (l:Rlist) (f:R -> R), Rlength (app_Rlist l f) = Rlength l.
 Proof.
   simple induction l; intros;
-    [ reflexivity | simpl in |- *; rewrite H; reflexivity ].
+    [ reflexivity | simpl; rewrite H; reflexivity ].
 Qed.
 
 Lemma RList_P19 :
@@ -666,7 +666,7 @@ Lemma RList_P23 :
     Rlength (cons_Rlist l1 l2) = (Rlength l1 + Rlength l2)%nat.
 Proof.
   simple induction l1;
-    [ intro; reflexivity | intros; simpl in |- *; rewrite H; reflexivity ].
+    [ intro; reflexivity | intros; simpl; rewrite H; reflexivity ].
 Qed.
 
 Lemma RList_P24 :
@@ -685,9 +685,9 @@ Proof.
     [ replace (Rlength r0 + Rlength (cons r1 l2))%nat with
       (S (Rlength r0 + Rlength l2));
       [ reflexivity
-        | simpl in |- *; apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR;
+        | simpl; apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR;
           rewrite S_INR; ring ]
-      | simpl in |- *; apply INR_eq; do 3 rewrite S_INR; do 2 rewrite plus_INR;
+      | simpl; apply INR_eq; do 3 rewrite S_INR; do 2 rewrite plus_INR;
         rewrite S_INR; ring ].
 Qed.
 
@@ -699,27 +699,27 @@ Lemma RList_P25 :
     ordered_Rlist (cons_Rlist l1 l2).
 Proof.
   simple induction l1.
-  intros; simpl in |- *; assumption.
+  intros; simpl; assumption.
   simple induction r0.
-  intros; simpl in |- *; simpl in H2; unfold ordered_Rlist in |- *; intros;
+  intros; simpl; simpl in H2; unfold ordered_Rlist; intros;
     simpl in H3.
   induction  i as [| i Hreci].
-  simpl in |- *; assumption.
-  change (pos_Rl l2 i <= pos_Rl l2 (S i)) in |- *; apply (H1 i); apply lt_S_n;
+  simpl; assumption.
+  change (pos_Rl l2 i <= pos_Rl l2 (S i)); apply (H1 i); apply lt_S_n;
     replace (S (pred (Rlength l2))) with (Rlength l2);
     [ assumption
-      | apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+      | apply S_pred with 0%nat; apply neq_O_lt; red; intro;
         rewrite <- H4 in H3; elim (lt_n_O _ H3) ].
   intros; clear H; assert (H : ordered_Rlist (cons_Rlist (cons r1 r2) l2)).
   apply H0; try assumption.
   apply RList_P4 with r; assumption.
-  unfold ordered_Rlist in |- *; intros; simpl in H4;
+  unfold ordered_Rlist; intros; simpl in H4;
     induction  i as [| i Hreci].
-  simpl in |- *; apply (H1 0%nat); simpl in |- *; apply lt_O_Sn.
+  simpl; apply (H1 0%nat); simpl; apply lt_O_Sn.
   change
     (pos_Rl (cons_Rlist (cons r1 r2) l2) i <=
-      pos_Rl (cons_Rlist (cons r1 r2) l2) (S i)) in |- *;
-    apply (H i); simpl in |- *; apply lt_S_n; assumption.
+      pos_Rl (cons_Rlist (cons r1 r2) l2) (S i));
+    apply (H i); simpl; apply lt_S_n; assumption.
 Qed.
 
 Lemma RList_P26 :
@@ -738,13 +738,13 @@ Lemma RList_P27 :
     cons_Rlist l1 (cons_Rlist l2 l3) = cons_Rlist (cons_Rlist l1 l2) l3.
 Proof.
   simple induction l1; intros;
-    [ reflexivity | simpl in |- *; rewrite (H l2 l3); reflexivity ].
+    [ reflexivity | simpl; rewrite (H l2 l3); reflexivity ].
 Qed.
 
 Lemma RList_P28 : forall l:Rlist, cons_Rlist l nil = l.
 Proof.
   simple induction l;
-    [ reflexivity | intros; simpl in |- *; rewrite H; reflexivity ].
+    [ reflexivity | intros; simpl; rewrite H; reflexivity ].
 Qed.
 
 Lemma RList_P29 :
@@ -759,23 +759,23 @@ Proof.
     replace (cons_Rlist l1 (cons r r0)) with
     (cons_Rlist (cons_Rlist l1 (cons r nil)) r0).
   inversion H0.
-  rewrite <- minus_n_n; simpl in |- *; rewrite RList_P26.
+  rewrite <- minus_n_n; simpl; rewrite RList_P26.
   clear l2 r0 H i H0 H1 H2; induction  l1 as [| r0 l1 Hrecl1].
   reflexivity.
-  simpl in |- *; assumption.
-  rewrite RList_P23; rewrite plus_comm; simpl in |- *; apply lt_n_Sn.
+  simpl; assumption.
+  rewrite RList_P23; rewrite plus_comm; simpl; apply lt_n_Sn.
   replace (S m - Rlength l1)%nat with (S (S m - S (Rlength l1))).
-  rewrite H3; simpl in |- *;
+  rewrite H3; simpl;
     replace (S (Rlength l1)) with (Rlength (cons_Rlist l1 (cons r nil))).
   apply (H (cons_Rlist l1 (cons r nil)) i).
-  rewrite RList_P23; rewrite plus_comm; simpl in |- *; rewrite <- H3;
+  rewrite RList_P23; rewrite plus_comm; simpl; rewrite <- H3;
     apply le_n_S; assumption.
-  repeat rewrite RList_P23; simpl in |- *; rewrite RList_P23 in H1;
+  repeat rewrite RList_P23; simpl; rewrite RList_P23 in H1;
     rewrite plus_comm in H1; simpl in H1; rewrite (plus_comm (Rlength l1));
-      simpl in |- *; rewrite plus_comm; apply H1.
+      simpl; rewrite plus_comm; apply H1.
   rewrite RList_P23; rewrite plus_comm; reflexivity.
-  change (S (m - Rlength l1) = (S m - Rlength l1)%nat) in |- *;
+  change (S (m - Rlength l1) = (S m - Rlength l1)%nat);
     apply minus_Sn_m; assumption.
   replace (cons r r0) with (cons_Rlist (cons r nil) r0);
-  [ symmetry  in |- *; apply RList_P27 | reflexivity ].
+  [ symmetry ; apply RList_P27 | reflexivity ].
 Qed.
diff --git a/theories/Reals/ROrderedType.v b/theories/Reals/ROrderedType.v
index 0a8d89c7..726f1204 100644
--- a/theories/Reals/ROrderedType.v
+++ b/theories/Reals/ROrderedType.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
diff --git a/theories/Reals/R_Ifp.v b/theories/Reals/R_Ifp.v
index 9e04a7da..8364e986 100644
--- a/theories/Reals/R_Ifp.v
+++ b/theories/Reals/R_Ifp.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -13,7 +13,7 @@
 
 Require Import Rbase.
 Require Import Omega.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (*********************************************************)
 (** *    Fractional part                                 *)
@@ -45,7 +45,7 @@ Proof.
   intros; generalize (Rplus_le_compat_l 1 (IZR z) r H); intro; clear H;
     rewrite (Rplus_comm 1 (IZR z)) in H1; rewrite (Rplus_comm 1 r) in H1;
       cut (1 = IZR 1); auto with zarith real.
-  intro; generalize H1; pattern 1 at 1 in |- *; rewrite H; intro; clear H H1;
+  intro; generalize H1; pattern 1 at 1; rewrite H; intro; clear H H1;
     rewrite <- (plus_IZR z 1) in H2; apply (tech_up r (z + 1));
       auto with zarith real.
 Qed.
@@ -53,12 +53,12 @@ Qed.
 (**********)
 Lemma fp_R0 : frac_part 0 = 0.
 Proof.
-  unfold frac_part in |- *; unfold Int_part in |- *; elim (archimed 0); intros;
-    unfold Rminus in |- *; elim (Rplus_ne (- IZR (up 0 - 1)));
+  unfold frac_part; unfold Int_part; elim (archimed 0); intros;
+    unfold Rminus; elim (Rplus_ne (- IZR (up 0 - 1)));
       intros a b; rewrite b; clear a b; rewrite <- Z_R_minus;
         cut (up 0 = 1%Z).
   intro; rewrite H1;
-    rewrite (Rminus_diag_eq (IZR 1) (IZR 1) (refl_equal (IZR 1)));
+    rewrite (Rminus_diag_eq (IZR 1) (IZR 1) (eq_refl (IZR 1)));
       apply Ropp_0.
   elim (archimed 0); intros; clear H2; unfold Rgt in H1;
     rewrite (Rminus_0_r (IZR (up 0))) in H0; generalize (lt_O_IZR (up 0) H1);
@@ -81,21 +81,21 @@ Qed.
 (**********)
 Lemma base_fp : forall r:R, frac_part r >= 0 /\ frac_part r < 1.
 Proof.
-  intro; unfold frac_part in |- *; unfold Int_part in |- *; split.
+  intro; unfold frac_part; unfold Int_part; split.
      (*sup a O*)
   cut (r - IZR (up r) >= -1).
-  rewrite <- Z_R_minus; simpl in |- *; intro; unfold Rminus in |- *;
+  rewrite <- Z_R_minus; simpl; intro; unfold Rminus;
     rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
-      fold (r - IZR (up r)) in |- *; fold (r - IZR (up r) - -1) in |- *;
+      fold (r - IZR (up r)); fold (r - IZR (up r) - -1);
         apply Rge_minus; auto with zarith real.
   rewrite <- Ropp_minus_distr; apply Ropp_le_ge_contravar; elim (for_base_fp r);
     auto with zarith real.
     (*inf a 1*)
   cut (r - IZR (up r) < 0).
-  rewrite <- Z_R_minus; simpl in |- *; intro; unfold Rminus in |- *;
+  rewrite <- Z_R_minus; simpl; intro; unfold Rminus;
     rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
-      fold (r - IZR (up r)) in |- *; rewrite Ropp_involutive;
-        elim (Rplus_ne 1); intros a b; pattern 1 at 2 in |- *;
+      fold (r - IZR (up r)); rewrite Ropp_involutive;
+        elim (Rplus_ne 1); intros a b; pattern 1 at 2;
           rewrite <- a; clear a b; rewrite (Rplus_comm (r - IZR (up r)) 1);
             apply Rplus_lt_compat_l; auto with zarith real.
   elim (for_base_fp r); intros; rewrite <- Ropp_0; rewrite <- Ropp_minus_distr;
@@ -110,8 +110,8 @@ Qed.
 Lemma base_Int_part :
   forall r:R, IZR (Int_part r) <= r /\ IZR (Int_part r) - r > -1.
 Proof.
-  intro; unfold Int_part in |- *; elim (archimed r); intros.
-  split; rewrite <- (Z_R_minus (up r) 1); simpl in |- *.
+  intro; unfold Int_part; elim (archimed r); intros.
+  split; rewrite <- (Z_R_minus (up r) 1); simpl.
   generalize (Rle_minus (IZR (up r) - r) 1 H0); intro; unfold Rminus in H1;
     rewrite (Rplus_assoc (IZR (up r)) (- r) (-1)) in H1;
       rewrite (Rplus_comm (- r) (-1)) in H1;
@@ -130,31 +130,31 @@ Proof.
 Qed.
 
 (**********)
-Lemma Int_part_INR : forall n:nat, Int_part (INR n) = Z_of_nat n.
+Lemma Int_part_INR : forall n:nat, Int_part (INR n) = Z.of_nat n.
 Proof.
-  intros n; unfold Int_part in |- *.
-  cut (up (INR n) = (Z_of_nat n + Z_of_nat 1)%Z).
-  intros H'; rewrite H'; simpl in |- *; ring.
-  apply sym_equal; apply tech_up; auto.
-  replace (Z_of_nat n + Z_of_nat 1)%Z with (Z_of_nat (S n)).
+  intros n; unfold Int_part.
+  cut (up (INR n) = (Z.of_nat n + Z.of_nat 1)%Z).
+  intros H'; rewrite H'; simpl; ring.
+  symmetry; apply tech_up; auto.
+  replace (Z.of_nat n + Z.of_nat 1)%Z with (Z.of_nat (S n)).
   repeat rewrite <- INR_IZR_INZ.
   apply lt_INR; auto.
-  rewrite Zplus_comm; rewrite <- Znat.inj_plus; simpl in |- *; auto.
-  rewrite plus_IZR; simpl in |- *; auto with real.
+  rewrite Z.add_comm; rewrite <- Znat.Nat2Z.inj_add; simpl; auto.
+  rewrite plus_IZR; simpl; auto with real.
   repeat rewrite <- INR_IZR_INZ; auto with real.
 Qed.
 
 (**********)
 Lemma fp_nat : forall r:R, frac_part r = 0 ->  exists c : Z, r = IZR c.
 Proof.
-  unfold frac_part in |- *; intros; split with (Int_part r);
+  unfold frac_part; intros; split with (Int_part r);
     apply Rminus_diag_uniq; auto with zarith real.
 Qed.
 
 (**********)
 Lemma R0_fp_O : forall r:R, 0 <> frac_part r -> 0 <> r.
 Proof.
-  red in |- *; intros; rewrite <- H0 in H; generalize fp_R0; intro;
+  red; intros; rewrite <- H0 in H; generalize fp_R0; intro;
     auto with zarith real.
 Qed.
 
@@ -243,7 +243,7 @@ Proof.
   intro; rewrite H1 in H; clear H1;
     rewrite <- (plus_IZR (Int_part r1 - Int_part r2) 1) in H;
       generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2) H0 H);
-        intros; clear H H0; unfold Int_part at 1 in |- *;
+        intros; clear H H0; unfold Int_part at 1;
           omega.
 Qed.
 
@@ -336,7 +336,7 @@ Proof.
           generalize (Rlt_le (IZR (Int_part r1 - Int_part r2 - 1)) (r1 - r2) H);
             intro; clear H;
               generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2 - 1) H1 H0);
-                intros; clear H0 H1; unfold Int_part at 1 in |- *;
+                intros; clear H0 H1; unfold Int_part at 1;
                   omega.
 Qed.
 
@@ -346,9 +346,9 @@ Lemma Rminus_fp1 :
     frac_part r1 >= frac_part r2 ->
     frac_part (r1 - r2) = frac_part r1 - frac_part r2.
 Proof.
-  intros; unfold frac_part in |- *; generalize (Rminus_Int_part1 r1 r2 H);
+  intros; unfold frac_part; generalize (Rminus_Int_part1 r1 r2 H);
     intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));
-      unfold Rminus in |- *;
+      unfold Rminus;
         rewrite (Ropp_plus_distr (IZR (Int_part r1)) (- IZR (Int_part r2)));
           rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2)));
             rewrite (Ropp_involutive (IZR (Int_part r2)));
@@ -366,17 +366,17 @@ Lemma Rminus_fp2 :
     frac_part r1 < frac_part r2 ->
     frac_part (r1 - r2) = frac_part r1 - frac_part r2 + 1.
 Proof.
-  intros; unfold frac_part in |- *; generalize (Rminus_Int_part2 r1 r2 H);
+  intros; unfold frac_part; generalize (Rminus_Int_part2 r1 r2 H);
     intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1 - Int_part r2) 1);
       rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));
-        unfold Rminus in |- *;
+        unfold Rminus;
           rewrite
             (Ropp_plus_distr (IZR (Int_part r1) + - IZR (Int_part r2)) (- IZR 1))
             ; rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2)));
               rewrite (Ropp_involutive (IZR 1));
                 rewrite (Ropp_involutive (IZR (Int_part r2)));
                   rewrite (Ropp_plus_distr (IZR (Int_part r1)));
-                    rewrite (Ropp_involutive (IZR (Int_part r2))); simpl in |- *;
+                    rewrite (Ropp_involutive (IZR (Int_part r2))); simpl;
                       rewrite <-
                         (Rplus_assoc (r1 + - r2) (- IZR (Int_part r1) + IZR (Int_part r2)) 1)
                         ; rewrite (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)));
@@ -451,7 +451,7 @@ Proof.
           rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H0;
             rewrite <- (plus_IZR (Int_part r1 + Int_part r2 + 1) 1) in H0;
               generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2 + 1) H H0);
-                intro; clear H H0; unfold Int_part at 1 in |- *; omega.
+                intro; clear H H0; unfold Int_part at 1; omega.
 Qed.
 
 (**********)
@@ -514,7 +514,7 @@ Proof.
       rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H1;
         rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H1;
           generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2) H0 H1);
-            intro; clear H0 H1; unfold Int_part at 1 in |- *;
+            intro; clear H0 H1; unfold Int_part at 1;
               omega.
 Qed.
 
@@ -524,17 +524,17 @@ Lemma plus_frac_part1 :
     frac_part r1 + frac_part r2 >= 1 ->
     frac_part (r1 + r2) = frac_part r1 + frac_part r2 - 1.
 Proof.
-  intros; unfold frac_part in |- *; generalize (plus_Int_part1 r1 r2 H); intro;
+  intros; unfold frac_part; generalize (plus_Int_part1 r1 r2 H); intro;
     rewrite H0; rewrite (plus_IZR (Int_part r1 + Int_part r2) 1);
-      rewrite (plus_IZR (Int_part r1) (Int_part r2)); simpl in |- *;
-        unfold Rminus at 3 4 in |- *;
+      rewrite (plus_IZR (Int_part r1) (Int_part r2)); simpl;
+        unfold Rminus at 3 4;
           rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)));
             rewrite (Rplus_comm r2 (- IZR (Int_part r2)));
               rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2);
                 rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2);
                   rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)));
                     rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2)));
-                      unfold Rminus in |- *;
+                      unfold Rminus;
                         rewrite
                           (Rplus_assoc (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))) (-1))
                           ; rewrite <- (Ropp_plus_distr (IZR (Int_part r1) + IZR (Int_part r2)) 1);
@@ -547,14 +547,14 @@ Lemma plus_frac_part2 :
     frac_part r1 + frac_part r2 < 1 ->
     frac_part (r1 + r2) = frac_part r1 + frac_part r2.
 Proof.
-  intros; unfold frac_part in |- *; generalize (plus_Int_part2 r1 r2 H); intro;
+  intros; unfold frac_part; generalize (plus_Int_part2 r1 r2 H); intro;
     rewrite H0; rewrite (plus_IZR (Int_part r1) (Int_part r2));
-      unfold Rminus at 2 3 in |- *;
+      unfold Rminus at 2 3;
         rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)));
           rewrite (Rplus_comm r2 (- IZR (Int_part r2)));
             rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2);
               rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2);
                 rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)));
                   rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2)));
-                    unfold Rminus in |- *; trivial with zarith real.
+                    unfold Rminus; trivial with zarith real.
 Qed.
diff --git a/theories/Reals/R_sqr.v b/theories/Reals/R_sqr.v
index f23b9f17..d6e18d9d 100644
--- a/theories/Reals/R_sqr.v
+++ b/theories/Reals/R_sqr.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -8,13 +8,13 @@
 
 Require Import Rbase.
 Require Import Rbasic_fun.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (****************************************************)
 (** Rsqr : some results                             *)
 (****************************************************)
 
-Ltac ring_Rsqr := unfold Rsqr in |- *; ring.
+Ltac ring_Rsqr := unfold Rsqr; ring.
 
 Lemma Rsqr_neg : forall x:R, Rsqr x = Rsqr (- x).
 Proof.
@@ -48,25 +48,25 @@ Qed.
 
 Lemma Rsqr_gt_0_0 : forall x:R, 0 < Rsqr x -> x <> 0.
 Proof.
-  intros; red in |- *; intro; rewrite H0 in H; rewrite Rsqr_0 in H;
+  intros; red; intro; rewrite H0 in H; rewrite Rsqr_0 in H;
     elim (Rlt_irrefl 0 H).
 Qed.
 
 Lemma Rsqr_pos_lt : forall x:R, x <> 0 -> 0 < Rsqr x.
 Proof.
   intros; case (Rtotal_order 0 x); intro;
-    [ unfold Rsqr in |- *; apply Rmult_lt_0_compat; assumption
+    [ unfold Rsqr; apply Rmult_lt_0_compat; assumption
       | elim H0; intro;
-        [ elim H; symmetry  in |- *; exact H1
+        [ elim H; symmetry ; exact H1
           | rewrite Rsqr_neg; generalize (Ropp_lt_gt_contravar x 0 H1);
-            rewrite Ropp_0; intro; unfold Rsqr in |- *;
+            rewrite Ropp_0; intro; unfold Rsqr;
               apply Rmult_lt_0_compat; assumption ] ].
 Qed.
 
 Lemma Rsqr_div : forall x y:R, y <> 0 -> Rsqr (x / y) = Rsqr x / Rsqr y.
 Proof.
-  intros; unfold Rsqr in |- *.
-  unfold Rdiv in |- *.
+  intros; unfold Rsqr.
+  unfold Rdiv.
   rewrite Rinv_mult_distr.
   repeat rewrite Rmult_assoc.
   apply Rmult_eq_compat_l.
@@ -80,7 +80,7 @@ Qed.
 
 Lemma Rsqr_eq_0 : forall x:R, Rsqr x = 0 -> x = 0.
 Proof.
-  unfold Rsqr in |- *; intros; generalize (Rmult_integral x x H); intro;
+  unfold Rsqr; intros; generalize (Rmult_integral x x H); intro;
     elim H0; intro; assumption.
 Qed.
 
@@ -122,7 +122,7 @@ Qed.
 Lemma Rsqr_incr_1 :
   forall x y:R, x <= y -> 0 <= x -> 0 <= y -> Rsqr x <= Rsqr y.
 Proof.
-  intros; unfold Rsqr in |- *; apply Rmult_le_compat; assumption.
+  intros; unfold Rsqr; apply Rmult_le_compat; assumption.
 Qed.
 
 Lemma Rsqr_incrst_0 :
@@ -140,7 +140,7 @@ Qed.
 Lemma Rsqr_incrst_1 :
   forall x y:R, x < y -> 0 <= x -> 0 <= y -> Rsqr x < Rsqr y.
 Proof.
-  intros; unfold Rsqr in |- *; apply Rmult_le_0_lt_compat; assumption.
+  intros; unfold Rsqr; apply Rmult_le_0_lt_compat; assumption.
 Qed.
 
 Lemma Rsqr_neg_pos_le_0 :
@@ -183,7 +183,7 @@ Qed.
 
 Lemma Rsqr_abs : forall x:R, Rsqr x = Rsqr (Rabs x).
 Proof.
-  intro; unfold Rabs in |- *; case (Rcase_abs x); intro;
+  intro; unfold Rabs; case (Rcase_abs x); intro;
     [ apply Rsqr_neg | reflexivity ].
 Qed.
 
@@ -220,7 +220,7 @@ Qed.
 
 Lemma Rsqr_eq_abs_0 : forall x y:R, Rsqr x = Rsqr y -> Rabs x = Rabs y.
 Proof.
-  intros; unfold Rabs in |- *; case (Rcase_abs x); case (Rcase_abs y); intros.
+  intros; unfold Rabs; case (Rcase_abs x); case (Rcase_abs y); intros.
   rewrite (Rsqr_neg x) in H; rewrite (Rsqr_neg y) in H;
     generalize (Ropp_lt_gt_contravar y 0 r);
       generalize (Ropp_lt_gt_contravar x 0 r0); rewrite Ropp_0;
@@ -288,7 +288,7 @@ Qed.
 
 Lemma Rsqr_inv : forall x:R, x <> 0 -> Rsqr (/ x) = / Rsqr x.
 Proof.
-  intros; unfold Rsqr in |- *.
+  intros; unfold Rsqr.
   rewrite Rinv_mult_distr; try reflexivity || assumption.
 Qed.
 
@@ -302,7 +302,7 @@ Proof.
   repeat rewrite Rmult_plus_distr_l.
   repeat rewrite Rplus_assoc.
   apply Rplus_eq_compat_l.
-  unfold Rdiv, Rminus in |- *.
+  unfold Rdiv, Rminus.
   replace (2 * 1 + 2 * 1) with 4; [ idtac | ring ].
   rewrite (Rmult_plus_distr_r (4 * a * c) (- Rsqr b) (/ (4 * a))).
   rewrite Rsqr_mult.
@@ -332,7 +332,7 @@ Proof.
   rewrite (Rmult_comm x).
   apply Rplus_eq_compat_l.
   rewrite (Rmult_comm (/ a)).
-  unfold Rsqr in |- *; repeat rewrite Rmult_assoc.
+  unfold Rsqr; repeat rewrite Rmult_assoc.
   rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r.
   ring.
@@ -357,7 +357,7 @@ Proof.
       rewrite Rplus_opp_l; replace (- (y * y) + x * x) with ((x - y) * (x + y)).
   intro; generalize (Rmult_integral (x - y) (x + y) H0); intro; elim H1; intros.
   left; apply Rminus_diag_uniq; assumption.
-  right; apply Rminus_diag_uniq; unfold Rminus in |- *; rewrite Ropp_involutive;
+  right; apply Rminus_diag_uniq; unfold Rminus; rewrite Ropp_involutive;
     assumption.
   ring.
 Qed.
diff --git a/theories/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v
index 2c5ede23..2d9419bd 100644
--- a/theories/Reals/R_sqrt.v
+++ b/theories/Reals/R_sqrt.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,7 +9,7 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import Rsqrt_def.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (** * Continuous extension of Rsqrt on R *)
 Definition sqrt (x:R) : R :=
@@ -36,7 +36,7 @@ Qed.
 Lemma sqrt_sqrt : forall x:R, 0 <= x -> sqrt x * sqrt x = x.
 Proof.
   intros.
-  unfold sqrt in |- *.
+  unfold sqrt.
   case (Rcase_abs x); intro.
   elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ r H)).
   rewrite Rsqrt_Rsqrt; reflexivity.
@@ -44,7 +44,7 @@ Qed.
 
 Lemma sqrt_0 : sqrt 0 = 0.
 Proof.
-  apply Rsqr_eq_0; unfold Rsqr in |- *; apply sqrt_sqrt; right; reflexivity.
+  apply Rsqr_eq_0; unfold Rsqr; apply sqrt_sqrt; right; reflexivity.
 Qed.
 
 Lemma sqrt_1 : sqrt 1 = 1.
@@ -52,7 +52,7 @@ Proof.
   apply (Rsqr_inj (sqrt 1) 1);
     [ apply sqrt_positivity; left
       | left
-      | unfold Rsqr in |- *; rewrite sqrt_sqrt; [ ring | left ] ];
+      | unfold Rsqr; rewrite sqrt_sqrt; [ ring | left ] ];
     apply Rlt_0_1.
 Qed.
 
@@ -73,7 +73,7 @@ Proof.
   intros; apply Rsqr_inj;
     [ apply (sqrt_positivity x H)
       | assumption
-      | unfold Rsqr in |- *; rewrite H1; apply (sqrt_sqrt x H) ].
+      | unfold Rsqr; rewrite H1; apply (sqrt_sqrt x H) ].
 Qed.
 
 Lemma sqrt_def : forall x:R, 0 <= x -> sqrt x * sqrt x = x.
@@ -86,12 +86,12 @@ Proof.
   intros;
     apply
       (Rsqr_inj (sqrt (Rsqr x)) x (sqrt_positivity (Rsqr x) (Rle_0_sqr x)) H);
-      unfold Rsqr in |- *; apply (sqrt_sqrt (Rsqr x) (Rle_0_sqr x)).
+      unfold Rsqr; apply (sqrt_sqrt (Rsqr x) (Rle_0_sqr x)).
 Qed.
 
 Lemma sqrt_Rsqr : forall x:R, 0 <= x -> sqrt (Rsqr x) = x.
 Proof.
-  intros; unfold Rsqr in |- *; apply sqrt_square; assumption.
+  intros; unfold Rsqr; apply sqrt_square; assumption.
 Qed.
 
 Lemma sqrt_Rsqr_abs : forall x:R, sqrt (Rsqr x) = Rabs x.
@@ -101,7 +101,7 @@ Qed.
 
 Lemma Rsqr_sqrt : forall x:R, 0 <= x -> Rsqr (sqrt x) = x.
 Proof.
-  intros x H1; unfold Rsqr in |- *; apply (sqrt_sqrt x H1).
+  intros x H1; unfold Rsqr; apply (sqrt_sqrt x H1).
 Qed.
 
 Lemma sqrt_mult_alt :
@@ -300,7 +300,7 @@ Proof.
   intros x H1 H2;
     generalize (sqrt_lt_1 x 1 (Rlt_le 0 x H1) (Rlt_le 0 1 Rlt_0_1) H2);
       intro H3; rewrite sqrt_1 in H3; generalize (Rmult_ne (sqrt x));
-        intro H4; elim H4; intros H5 H6; rewrite <- H5; pattern x at 1 in |- *;
+        intro H4; elim H4; intros H5 H6; rewrite <- H5; pattern x at 1;
           rewrite <- (sqrt_def x (Rlt_le 0 x H1));
             apply (Rmult_lt_compat_l (sqrt x) (sqrt x) 1 (sqrt_lt_R0 x H1) H3).
 Qed.
@@ -310,7 +310,7 @@ Lemma sqrt_cauchy :
     a * c + b * d <= sqrt (Rsqr a + Rsqr b) * sqrt (Rsqr c + Rsqr d).
 Proof.
   intros a b c d; apply Rsqr_incr_0_var;
-    [ rewrite Rsqr_mult; repeat rewrite Rsqr_sqrt; unfold Rsqr in |- *;
+    [ rewrite Rsqr_mult; repeat rewrite Rsqr_sqrt; unfold Rsqr;
       [ replace ((a * c + b * d) * (a * c + b * d)) with
         (a * a * c * c + b * b * d * d + 2 * a * b * c * d);
         [ replace ((a * a + b * b) * (c * c + d * d)) with
@@ -319,11 +319,11 @@ Proof.
             replace (a * a * d * d + b * b * c * c) with
             (2 * a * b * c * d +
               (a * a * d * d + b * b * c * c - 2 * a * b * c * d));
-            [ pattern (2 * a * b * c * d) at 1 in |- *; rewrite <- Rplus_0_r;
+            [ pattern (2 * a * b * c * d) at 1; rewrite <- Rplus_0_r;
               apply Rplus_le_compat_l;
                 replace (a * a * d * d + b * b * c * c - 2 * a * b * c * d)
               with (Rsqr (a * d - b * c));
-                [ apply Rle_0_sqr | unfold Rsqr in |- *; ring ]
+                [ apply Rle_0_sqr | unfold Rsqr; ring ]
               | ring ]
             | ring ]
           | ring ]
@@ -355,16 +355,16 @@ Lemma Rsqr_sol_eq_0_1 :
     x = sol_x1 a b c \/ x = sol_x2 a b c -> a * Rsqr x + b * x + c = 0.
 Proof.
   intros; elim H0; intro.
-  unfold sol_x1 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv in |- *;
+  unfold sol_x1 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv;
     repeat rewrite Rsqr_mult; rewrite Rsqr_plus; rewrite <- Rsqr_neg;
       rewrite Rsqr_sqrt.
   rewrite Rsqr_inv.
-  unfold Rsqr in |- *; repeat rewrite Rinv_mult_distr.
+  unfold Rsqr; repeat rewrite Rinv_mult_distr.
   repeat rewrite Rmult_assoc; rewrite (Rmult_comm a).
   repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r; rewrite Rmult_plus_distr_r.
   repeat rewrite Rmult_assoc.
-  pattern 2 at 2 in |- *; rewrite (Rmult_comm 2).
+  pattern 2 at 2; rewrite (Rmult_comm 2).
   repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r.
   rewrite
@@ -376,7 +376,7 @@ Proof.
     (b * (- b * (/ 2 * / a)) +
       (b * (sqrt (b * b - 2 * (2 * (a * c))) * (/ 2 * / a)) + c))) with
   (b * (- b * (/ 2 * / a)) + c).
-  unfold Rminus in |- *; repeat rewrite <- Rplus_assoc.
+  unfold Rminus; repeat rewrite <- Rplus_assoc.
   replace (b * b + b * b) with (2 * (b * b)).
   rewrite Rmult_plus_distr_r; repeat rewrite Rmult_assoc.
   rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc.
@@ -407,17 +407,17 @@ Proof.
   apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ].
   apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ].
   assumption.
-  unfold sol_x2 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv in |- *;
+  unfold sol_x2 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv;
     repeat rewrite Rsqr_mult; rewrite Rsqr_minus; rewrite <- Rsqr_neg;
       rewrite Rsqr_sqrt.
   rewrite Rsqr_inv.
-  unfold Rsqr in |- *; repeat rewrite Rinv_mult_distr;
+  unfold Rsqr; repeat rewrite Rinv_mult_distr;
     repeat rewrite Rmult_assoc.
   rewrite (Rmult_comm a); repeat rewrite Rmult_assoc.
   rewrite <- Rinv_l_sym.
-  rewrite Rmult_1_r; unfold Rminus in |- *; rewrite Rmult_plus_distr_r.
+  rewrite Rmult_1_r; unfold Rminus; rewrite Rmult_plus_distr_r.
   rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc;
-    pattern 2 at 2 in |- *; rewrite (Rmult_comm 2).
+    pattern 2 at 2; rewrite (Rmult_comm 2).
   repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r;
     rewrite
@@ -480,23 +480,23 @@ Proof.
   intro;
     generalize (Rsqr_eq (x + b / (2 * a)) (sqrt (Delta a b c) / (2 * a)) H3);
       intro; elim H4; intro.
-  left; unfold sol_x1 in |- *;
+  left; unfold sol_x1;
     generalize
       (Rplus_eq_compat_l (- (b / (2 * a))) (x + b / (2 * a))
         (sqrt (Delta a b c) / (2 * a)) H5);
       replace (- (b / (2 * a)) + (x + b / (2 * a))) with x.
-  intro; rewrite H6; unfold Rdiv in |- *; ring.
+  intro; rewrite H6; unfold Rdiv; ring.
   ring.
-  right; unfold sol_x2 in |- *;
+  right; unfold sol_x2;
     generalize
       (Rplus_eq_compat_l (- (b / (2 * a))) (x + b / (2 * a))
         (- (sqrt (Delta a b c) / (2 * a))) H5);
       replace (- (b / (2 * a)) + (x + b / (2 * a))) with x.
-  intro; rewrite H6; unfold Rdiv in |- *; ring.
+  intro; rewrite H6; unfold Rdiv; ring.
   ring.
   rewrite Rsqr_div.
   rewrite Rsqr_sqrt.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   repeat rewrite Rmult_assoc.
   rewrite (Rmult_comm (/ a)).
   rewrite Rmult_assoc.
@@ -510,9 +510,9 @@ Proof.
   assumption.
   apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ].
   rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
-  symmetry  in |- *; apply Rmult_1_l.
+  symmetry ; apply Rmult_1_l.
   apply (cond_nonzero a).
-  unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse.
+  unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse.
   rewrite Ropp_minus_distr.
   reflexivity.
   reflexivity.
diff --git a/theories/Reals/Ranalysis.v b/theories/Reals/Ranalysis.v
index 01715cf3..ad86a197 100644
--- a/theories/Reals/Ranalysis.v
+++ b/theories/Reals/Ranalysis.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -26,775 +26,4 @@ Require Export RList.
 Require Export Sqrt_reg.
 Require Export Ranalysis4.
 Require Export Rpower.
-Open Local Scope R_scope.
-
-Axiom AppVar : R.
-
-(**********)
-Ltac intro_hyp_glob trm :=
-  match constr:trm with
-    | (?X1 + ?X2)%F =>
-      match goal with
-        |  |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
-        |  |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
-        | _ => idtac
-      end
-    | (?X1 - ?X2)%F =>
-      match goal with
-        |  |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
-        |  |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
-        | _ => idtac
-      end
-    | (?X1 * ?X2)%F =>
-      match goal with
-        |  |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
-        |  |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
-        | _ => idtac
-      end
-    | (?X1 / ?X2)%F =>
-      let aux := constr:X2 in
-        match goal with
-          | _:(forall x0:R, aux x0 <> 0) |- (derivable _) =>
-            intro_hyp_glob X1; intro_hyp_glob X2
-          | _:(forall x0:R, aux x0 <> 0) |- (continuity _) =>
-            intro_hyp_glob X1; intro_hyp_glob X2
-          |  |- (derivable _) =>
-            cut (forall x0:R, aux x0 <> 0);
-              [ intro; intro_hyp_glob X1; intro_hyp_glob X2 | try assumption ]
-          |  |- (continuity _) =>
-            cut (forall x0:R, aux x0 <> 0);
-              [ intro; intro_hyp_glob X1; intro_hyp_glob X2 | try assumption ]
-          | _ => idtac
-        end
-    | (comp ?X1 ?X2) =>
-      match goal with
-        |  |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
-        |  |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
-        | _ => idtac
-      end
-    | (- ?X1)%F =>
-      match goal with
-        |  |- (derivable _) => intro_hyp_glob X1
-        |  |- (continuity _) => intro_hyp_glob X1
-        | _ => idtac
-      end
-    | (/ ?X1)%F =>
-      let aux := constr:X1 in
-        match goal with
-          | _:(forall x0:R, aux x0 <> 0) |- (derivable _) =>
-            intro_hyp_glob X1
-          | _:(forall x0:R, aux x0 <> 0) |- (continuity _) =>
-            intro_hyp_glob X1
-          |  |- (derivable _) =>
-            cut (forall x0:R, aux x0 <> 0);
-              [ intro; intro_hyp_glob X1 | try assumption ]
-          |  |- (continuity _) =>
-            cut (forall x0:R, aux x0 <> 0);
-              [ intro; intro_hyp_glob X1 | try assumption ]
-          | _ => idtac
-        end
-    | cos => idtac
-    | sin => idtac
-    | cosh => idtac
-    | sinh => idtac
-    | exp => idtac
-    | Rsqr => idtac
-    | sqrt => idtac
-    | id => idtac
-    | (fct_cte _) => idtac
-    | (pow_fct _) => idtac
-    | Rabs => idtac
-    | ?X1 =>
-      let p := constr:X1 in
-        match goal with
-          | _:(derivable p) |- _ => idtac
-          |  |- (derivable p) => idtac
-          |  |- (derivable _) =>
-            cut (True -> derivable p);
-              [ intro HYPPD; cut (derivable p);
-                [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
-                | idtac ]
-          | _:(continuity p) |- _ => idtac
-          |  |- (continuity p) => idtac
-          |  |- (continuity _) =>
-            cut (True -> continuity p);
-              [ intro HYPPD; cut (continuity p);
-                [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
-                | idtac ]
-          | _ => idtac
-        end
-  end.
-
-(**********)
-Ltac intro_hyp_pt trm pt :=
-  match constr:trm with
-    | (?X1 + ?X2)%F =>
-      match goal with
-        |  |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-        |  |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-        |  |- (derive_pt _ _ _ = _) =>
-          intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-        | _ => idtac
-      end
-    | (?X1 - ?X2)%F =>
-      match goal with
-        |  |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-        |  |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-        |  |- (derive_pt _ _ _ = _) =>
-          intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-        | _ => idtac
-      end
-    | (?X1 * ?X2)%F =>
-      match goal with
-        |  |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-        |  |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-        |  |- (derive_pt _ _ _ = _) =>
-          intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-        | _ => idtac
-      end
-    | (?X1 / ?X2)%F =>
-      let aux := constr:X2 in
-        match goal with
-          | _:(aux pt <> 0) |- (derivable_pt _ _) =>
-            intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-          | _:(aux pt <> 0) |- (continuity_pt _ _) =>
-            intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-          | _:(aux pt <> 0) |- (derive_pt _ _ _ = _) =>
-            intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-          | id:(forall x0:R, aux x0 <> 0) |- (derivable_pt _ _) =>
-            generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-          | id:(forall x0:R, aux x0 <> 0) |- (continuity_pt _ _) =>
-            generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-          | id:(forall x0:R, aux x0 <> 0) |- (derive_pt _ _ _ = _) =>
-            generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
-          |  |- (derivable_pt _ _) =>
-            cut (aux pt <> 0);
-              [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ]
-          |  |- (continuity_pt _ _) =>
-            cut (aux pt <> 0);
-              [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ]
-          |  |- (derive_pt _ _ _ = _) =>
-            cut (aux pt <> 0);
-              [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ]
-          | _ => idtac
-        end
-    | (comp ?X1 ?X2) =>
-      match goal with
-        |  |- (derivable_pt _ _) =>
-          let pt_f1 := eval cbv beta in (X2 pt) in
-            (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt)
-        |  |- (continuity_pt _ _) =>
-          let pt_f1 := eval cbv beta in (X2 pt) in
-            (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt)
-        |  |- (derive_pt _ _ _ = _) =>
-          let pt_f1 := eval cbv beta in (X2 pt) in
-            (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt)
-        | _ => idtac
-      end
-    | (- ?X1)%F =>
-      match goal with
-        |  |- (derivable_pt _ _) => intro_hyp_pt X1 pt
-        |  |- (continuity_pt _ _) => intro_hyp_pt X1 pt
-        |  |- (derive_pt _ _ _ = _) => intro_hyp_pt X1 pt
-        | _ => idtac
-      end
-    | (/ ?X1)%F =>
-      let aux := constr:X1 in
-        match goal with
-          | _:(aux pt <> 0) |- (derivable_pt _ _) =>
-            intro_hyp_pt X1 pt
-          | _:(aux pt <> 0) |- (continuity_pt _ _) =>
-            intro_hyp_pt X1 pt
-          | _:(aux pt <> 0) |- (derive_pt _ _ _ = _) =>
-            intro_hyp_pt X1 pt
-          | id:(forall x0:R, aux x0 <> 0) |- (derivable_pt _ _) =>
-            generalize (id pt); intro; intro_hyp_pt X1 pt
-          | id:(forall x0:R, aux x0 <> 0) |- (continuity_pt _ _) =>
-            generalize (id pt); intro; intro_hyp_pt X1 pt
-          | id:(forall x0:R, aux x0 <> 0) |- (derive_pt _ _ _ = _) =>
-            generalize (id pt); intro; intro_hyp_pt X1 pt
-          |  |- (derivable_pt _ _) =>
-            cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ]
-          |  |- (continuity_pt _ _) =>
-            cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ]
-          |  |- (derive_pt _ _ _ = _) =>
-            cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ]
-          | _ => idtac
-        end
-    | cos => idtac
-    | sin => idtac
-    | cosh => idtac
-    | sinh => idtac
-    | exp => idtac
-    | Rsqr => idtac
-    | id => idtac
-    | (fct_cte _) => idtac
-    | (pow_fct _) => idtac
-    | sqrt =>
-      match goal with
-        |  |- (derivable_pt _ _) => cut (0 < pt); [ intro | try assumption ]
-        |  |- (continuity_pt _ _) =>
-          cut (0 <= pt); [ intro | try assumption ]
-        |  |- (derive_pt _ _ _ = _) =>
-          cut (0 < pt); [ intro | try assumption ]
-        | _ => idtac
-      end
-    | Rabs =>
-      match goal with
-        |  |- (derivable_pt _ _) =>
-          cut (pt <> 0); [ intro | try assumption ]
-        | _ => idtac
-      end
-    | ?X1 =>
-      let p := constr:X1 in
-        match goal with
-          | _:(derivable_pt p pt) |- _ => idtac
-          |  |- (derivable_pt p pt) => idtac
-          |  |- (derivable_pt _ _) =>
-            cut (True -> derivable_pt p pt);
-              [ intro HYPPD; cut (derivable_pt p pt);
-                [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
-                | idtac ]
-          | _:(continuity_pt p pt) |- _ => idtac
-          |  |- (continuity_pt p pt) => idtac
-          |  |- (continuity_pt _ _) =>
-            cut (True -> continuity_pt p pt);
-              [ intro HYPPD; cut (continuity_pt p pt);
-                [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
-                | idtac ]
-          |  |- (derive_pt _ _ _ = _) =>
-            cut (True -> derivable_pt p pt);
-              [ intro HYPPD; cut (derivable_pt p pt);
-                [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
-                | idtac ]
-          | _ => idtac
-        end
-  end.
-
-(**********)
-Ltac is_diff_pt :=
-  match goal with
-    |  |- (derivable_pt Rsqr _) =>
-
-  (* fonctions de base *)
-      apply derivable_pt_Rsqr
-    |  |- (derivable_pt id ?X1) => apply (derivable_pt_id X1)
-    |  |- (derivable_pt (fct_cte _) _) => apply derivable_pt_const
-    |  |- (derivable_pt sin _) => apply derivable_pt_sin
-    |  |- (derivable_pt cos _) => apply derivable_pt_cos
-    |  |- (derivable_pt sinh _) => apply derivable_pt_sinh
-    |  |- (derivable_pt cosh _) => apply derivable_pt_cosh
-    |  |- (derivable_pt exp _) => apply derivable_pt_exp
-    |  |- (derivable_pt (pow_fct _) _) =>
-      unfold pow_fct in |- *; apply derivable_pt_pow
-    |  |- (derivable_pt sqrt ?X1) =>
-      apply (derivable_pt_sqrt X1);
-        assumption ||
-          unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct,
-            comp, id, fct_cte, pow_fct in |- *
-    |  |- (derivable_pt Rabs ?X1) =>
-      apply (Rderivable_pt_abs X1);
-        assumption ||
-          unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct,
-            comp, id, fct_cte, pow_fct in |- *
-        (* regles de differentiabilite *)
-        (* PLUS *)
-    |  |- (derivable_pt (?X1 + ?X2) ?X3) =>
-      apply (derivable_pt_plus X1 X2 X3); is_diff_pt
-       (* MOINS *)
-    |  |- (derivable_pt (?X1 - ?X2) ?X3) =>
-      apply (derivable_pt_minus X1 X2 X3); is_diff_pt
-       (* OPPOSE *)
-    |  |- (derivable_pt (- ?X1) ?X2) =>
-      apply (derivable_pt_opp X1 X2);
-        is_diff_pt
-       (* MULTIPLICATION PAR UN SCALAIRE *)
-    |  |- (derivable_pt (mult_real_fct ?X1 ?X2) ?X3) =>
-      apply (derivable_pt_scal X2 X1 X3); is_diff_pt
-       (* MULTIPLICATION *)
-    |  |- (derivable_pt (?X1 * ?X2) ?X3) =>
-      apply (derivable_pt_mult X1 X2 X3); is_diff_pt
-       (* DIVISION *)
-    |  |- (derivable_pt (?X1 / ?X2) ?X3) =>
-      apply (derivable_pt_div X1 X2 X3);
-        [ is_diff_pt
-          | is_diff_pt
-          | try
-            assumption ||
-              unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
-                comp, pow_fct, id, fct_cte in |- * ]
-    |  |- (derivable_pt (/ ?X1) ?X2) =>
-
-       (* INVERSION *)
-      apply (derivable_pt_inv X1 X2);
-        [ assumption ||
-          unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
-            comp, pow_fct, id, fct_cte in |- *
-          | is_diff_pt ]
-    |  |- (derivable_pt (comp ?X1 ?X2) ?X3) =>
-
-       (* COMPOSITION *)
-      apply (derivable_pt_comp X2 X1 X3); is_diff_pt
-    | _:(derivable_pt ?X1 ?X2) |- (derivable_pt ?X1 ?X2) =>
-      assumption
-    | _:(derivable ?X1) |- (derivable_pt ?X1 ?X2) =>
-      cut (derivable X1); [ intro HypDDPT; apply HypDDPT | assumption ]
-    |  |- (True -> derivable_pt _ _) =>
-      intro HypTruE; clear HypTruE; is_diff_pt
-    | _ =>
-      try
-        unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
-          fct_cte, comp, pow_fct in |- *
-  end.
-
-(**********)
-Ltac is_diff_glob :=
-  match goal with
-    |  |- (derivable Rsqr) =>
-  (* fonctions de base *)
-      apply derivable_Rsqr
-    |  |- (derivable id) => apply derivable_id
-    |  |- (derivable (fct_cte _)) => apply derivable_const
-    |  |- (derivable sin) => apply derivable_sin
-    |  |- (derivable cos) => apply derivable_cos
-    |  |- (derivable cosh) => apply derivable_cosh
-    |  |- (derivable sinh) => apply derivable_sinh
-    |  |- (derivable exp) => apply derivable_exp
-    |  |- (derivable (pow_fct _)) =>
-      unfold pow_fct in |- *;
-        apply derivable_pow
-        (* regles de differentiabilite *)
-        (* PLUS *)
-    |  |- (derivable (?X1 + ?X2)) =>
-      apply (derivable_plus X1 X2); is_diff_glob
-       (* MOINS *)
-    |  |- (derivable (?X1 - ?X2)) =>
-      apply (derivable_minus X1 X2); is_diff_glob
-       (* OPPOSE *)
-    |  |- (derivable (- ?X1)) =>
-      apply (derivable_opp X1);
-        is_diff_glob
-       (* MULTIPLICATION PAR UN SCALAIRE *)
-    |  |- (derivable (mult_real_fct ?X1 ?X2)) =>
-      apply (derivable_scal X2 X1); is_diff_glob
-       (* MULTIPLICATION *)
-    |  |- (derivable (?X1 * ?X2)) =>
-      apply (derivable_mult X1 X2); is_diff_glob
-       (* DIVISION *)
-    |  |- (derivable (?X1 / ?X2)) =>
-      apply (derivable_div X1 X2);
-        [ is_diff_glob
-          | is_diff_glob
-          | try
-            assumption ||
-              unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
-                id, fct_cte, comp, pow_fct in |- * ]
-    |  |- (derivable (/ ?X1)) =>
-
-       (* INVERSION *)
-      apply (derivable_inv X1);
-        [ try
-          assumption ||
-            unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
-              id, fct_cte, comp, pow_fct in |- *
-          | is_diff_glob ]
-    |  |- (derivable (comp sqrt _)) =>
-
-       (* COMPOSITION *)
-      unfold derivable in |- *; intro; try is_diff_pt
-    |  |- (derivable (comp Rabs _)) =>
-      unfold derivable in |- *; intro; try is_diff_pt
-    |  |- (derivable (comp ?X1 ?X2)) =>
-      apply (derivable_comp X2 X1); is_diff_glob
-    | _:(derivable ?X1) |- (derivable ?X1) => assumption
-    |  |- (True -> derivable _) =>
-      intro HypTruE; clear HypTruE; is_diff_glob
-    | _ =>
-      try
-        unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
-          fct_cte, comp, pow_fct in |- *
-  end.
-
-(**********)
-Ltac is_cont_pt :=
-  match goal with
-    |  |- (continuity_pt Rsqr _) =>
-
-       (* fonctions de base *)
-      apply derivable_continuous_pt; apply derivable_pt_Rsqr
-    |  |- (continuity_pt id ?X1) =>
-      apply derivable_continuous_pt; apply (derivable_pt_id X1)
-    |  |- (continuity_pt (fct_cte _) _) =>
-      apply derivable_continuous_pt; apply derivable_pt_const
-    |  |- (continuity_pt sin _) =>
-      apply derivable_continuous_pt; apply derivable_pt_sin
-    |  |- (continuity_pt cos _) =>
-      apply derivable_continuous_pt; apply derivable_pt_cos
-    |  |- (continuity_pt sinh _) =>
-      apply derivable_continuous_pt; apply derivable_pt_sinh
-    |  |- (continuity_pt cosh _) =>
-      apply derivable_continuous_pt; apply derivable_pt_cosh
-    |  |- (continuity_pt exp _) =>
-      apply derivable_continuous_pt; apply derivable_pt_exp
-    |  |- (continuity_pt (pow_fct _) _) =>
-      unfold pow_fct in |- *; apply derivable_continuous_pt;
-        apply derivable_pt_pow
-    |  |- (continuity_pt sqrt ?X1) =>
-      apply continuity_pt_sqrt;
-        assumption ||
-          unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct,
-            comp, id, fct_cte, pow_fct in |- *
-    |  |- (continuity_pt Rabs ?X1) =>
-      apply (Rcontinuity_abs X1)
-       (* regles de differentiabilite *)
-       (* PLUS *)
-    |  |- (continuity_pt (?X1 + ?X2) ?X3) =>
-      apply (continuity_pt_plus X1 X2 X3); is_cont_pt
-       (* MOINS *)
-    |  |- (continuity_pt (?X1 - ?X2) ?X3) =>
-      apply (continuity_pt_minus X1 X2 X3); is_cont_pt
-       (* OPPOSE *)
-    |  |- (continuity_pt (- ?X1) ?X2) =>
-      apply (continuity_pt_opp X1 X2);
-        is_cont_pt
-       (* MULTIPLICATION PAR UN SCALAIRE *)
-    |  |- (continuity_pt (mult_real_fct ?X1 ?X2) ?X3) =>
-      apply (continuity_pt_scal X2 X1 X3); is_cont_pt
-       (* MULTIPLICATION *)
-    |  |- (continuity_pt (?X1 * ?X2) ?X3) =>
-      apply (continuity_pt_mult X1 X2 X3); is_cont_pt
-       (* DIVISION *)
-    |  |- (continuity_pt (?X1 / ?X2) ?X3) =>
-      apply (continuity_pt_div X1 X2 X3);
-        [ is_cont_pt
-          | is_cont_pt
-          | try
-            assumption ||
-              unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
-                comp, id, fct_cte, pow_fct in |- * ]
-    |  |- (continuity_pt (/ ?X1) ?X2) =>
-
-       (* INVERSION *)
-      apply (continuity_pt_inv X1 X2);
-        [ is_cont_pt
-          | assumption ||
-            unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
-              comp, id, fct_cte, pow_fct in |- * ]
-    |  |- (continuity_pt (comp ?X1 ?X2) ?X3) =>
-
-       (* COMPOSITION *)
-      apply (continuity_pt_comp X2 X1 X3); is_cont_pt
-    | _:(continuity_pt ?X1 ?X2) |- (continuity_pt ?X1 ?X2) =>
-      assumption
-    | _:(continuity ?X1) |- (continuity_pt ?X1 ?X2) =>
-      cut (continuity X1); [ intro HypDDPT; apply HypDDPT | assumption ]
-    | _:(derivable_pt ?X1 ?X2) |- (continuity_pt ?X1 ?X2) =>
-      apply derivable_continuous_pt; assumption
-    | _:(derivable ?X1) |- (continuity_pt ?X1 ?X2) =>
-      cut (continuity X1);
-        [ intro HypDDPT; apply HypDDPT
-          | apply derivable_continuous; assumption ]
-    |  |- (True -> continuity_pt _ _) =>
-      intro HypTruE; clear HypTruE; is_cont_pt
-    | _ =>
-      try
-        unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
-          fct_cte, comp, pow_fct in |- *
-  end.
-
-(**********)
-Ltac is_cont_glob :=
-  match goal with
-    |  |- (continuity Rsqr) =>
-
-       (* fonctions de base *)
-      apply derivable_continuous; apply derivable_Rsqr
-    |  |- (continuity id) => apply derivable_continuous; apply derivable_id
-    |  |- (continuity (fct_cte _)) =>
-      apply derivable_continuous; apply derivable_const
-    |  |- (continuity sin) => apply derivable_continuous; apply derivable_sin
-    |  |- (continuity cos) => apply derivable_continuous; apply derivable_cos
-    |  |- (continuity exp) => apply derivable_continuous; apply derivable_exp
-    |  |- (continuity (pow_fct _)) =>
-      unfold pow_fct in |- *; apply derivable_continuous; apply derivable_pow
-    |  |- (continuity sinh) =>
-      apply derivable_continuous; apply derivable_sinh
-    |  |- (continuity cosh) =>
-      apply derivable_continuous; apply derivable_cosh
-    |  |- (continuity Rabs) =>
-      apply Rcontinuity_abs
-       (* regles de continuite *)
-       (* PLUS *)
-    |  |- (continuity (?X1 + ?X2)) =>
-      apply (continuity_plus X1 X2);
-        try is_cont_glob || assumption
-            (* MOINS *)
-    |  |- (continuity (?X1 - ?X2)) =>
-      apply (continuity_minus X1 X2);
-        try is_cont_glob || assumption
-            (* OPPOSE *)
-    |  |- (continuity (- ?X1)) =>
-      apply (continuity_opp X1); try is_cont_glob || assumption
-                                      (* INVERSE *)
-    |  |- (continuity (/ ?X1)) =>
-      apply (continuity_inv X1);
-        try is_cont_glob || assumption
-            (* MULTIPLICATION PAR UN SCALAIRE *)
-    |  |- (continuity (mult_real_fct ?X1 ?X2)) =>
-      apply (continuity_scal X2 X1);
-        try is_cont_glob || assumption
-            (* MULTIPLICATION *)
-    |  |- (continuity (?X1 * ?X2)) =>
-      apply (continuity_mult X1 X2);
-        try is_cont_glob || assumption
-            (* DIVISION *)
-    |  |- (continuity (?X1 / ?X2)) =>
-      apply (continuity_div X1 X2);
-        [ try is_cont_glob || assumption
-          | try is_cont_glob || assumption
-          | try
-            assumption ||
-              unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
-                id, fct_cte, pow_fct in |- * ]
-    |  |- (continuity (comp sqrt _)) =>
-
-       (* COMPOSITION *)
-      unfold continuity_pt in |- *; intro; try is_cont_pt
-    |  |- (continuity (comp ?X1 ?X2)) =>
-      apply (continuity_comp X2 X1); try is_cont_glob || assumption
-    | _:(continuity ?X1) |- (continuity ?X1) => assumption
-    |  |- (True -> continuity _) =>
-      intro HypTruE; clear HypTruE; is_cont_glob
-    | _:(derivable ?X1) |- (continuity ?X1) =>
-      apply derivable_continuous; assumption
-    | _ =>
-      try
-        unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
-          fct_cte, comp, pow_fct in |- *
-  end.
-
-(**********)
-Ltac rew_term trm :=
-  match constr:trm with
-    | (?X1 + ?X2) =>
-      let p1 := rew_term X1 with p2 := rew_term X2 in
-        match constr:p1 with
-          | (fct_cte ?X3) =>
-            match constr:p2 with
-              | (fct_cte ?X4) => constr:(fct_cte (X3 + X4))
-              | _ => constr:(p1 + p2)%F
-            end
-          | _ => constr:(p1 + p2)%F
-        end
-    | (?X1 - ?X2) =>
-      let p1 := rew_term X1 with p2 := rew_term X2 in
-        match constr:p1 with
-          | (fct_cte ?X3) =>
-            match constr:p2 with
-              | (fct_cte ?X4) => constr:(fct_cte (X3 - X4))
-              | _ => constr:(p1 - p2)%F
-            end
-          | _ => constr:(p1 - p2)%F
-        end
-    | (?X1 / ?X2) =>
-      let p1 := rew_term X1 with p2 := rew_term X2 in
-        match constr:p1 with
-          | (fct_cte ?X3) =>
-            match constr:p2 with
-              | (fct_cte ?X4) => constr:(fct_cte (X3 / X4))
-              | _ => constr:(p1 / p2)%F
-            end
-          | _ =>
-            match constr:p2 with
-              | (fct_cte ?X4) => constr:(p1 * fct_cte (/ X4))%F
-              | _ => constr:(p1 / p2)%F
-            end
-        end
-    | (?X1 * / ?X2) =>
-      let p1 := rew_term X1 with p2 := rew_term X2 in
-        match constr:p1 with
-          | (fct_cte ?X3) =>
-            match constr:p2 with
-              | (fct_cte ?X4) => constr:(fct_cte (X3 / X4))
-              | _ => constr:(p1 / p2)%F
-            end
-          | _ =>
-            match constr:p2 with
-              | (fct_cte ?X4) => constr:(p1 * fct_cte (/ X4))%F
-              | _ => constr:(p1 / p2)%F
-            end
-        end
-    | (?X1 * ?X2) =>
-      let p1 := rew_term X1 with p2 := rew_term X2 in
-        match constr:p1 with
-          | (fct_cte ?X3) =>
-            match constr:p2 with
-              | (fct_cte ?X4) => constr:(fct_cte (X3 * X4))
-              | _ => constr:(p1 * p2)%F
-            end
-          | _ => constr:(p1 * p2)%F
-        end
-    | (- ?X1) =>
-      let p := rew_term X1 in
-        match constr:p with
-          | (fct_cte ?X2) => constr:(fct_cte (- X2))
-          | _ => constr:(- p)%F
-        end
-    | (/ ?X1) =>
-      let p := rew_term X1 in
-        match constr:p with
-          | (fct_cte ?X2) => constr:(fct_cte (/ X2))
-          | _ => constr:(/ p)%F
-        end
-    | (?X1 AppVar) => constr:X1
-    | (?X1 ?X2) =>
-      let p := rew_term X2 in
-        match constr:p with
-          | (fct_cte ?X3) => constr:(fct_cte (X1 X3))
-          | _ => constr:(comp X1 p)
-        end
-    | AppVar => constr:id
-    | (AppVar ^ ?X1) => constr:(pow_fct X1)
-    | (?X1 ^ ?X2) =>
-      let p := rew_term X1 in
-        match constr:p with
-          | (fct_cte ?X3) => constr:(fct_cte (pow_fct X2 X3))
-          | _ => constr:(comp (pow_fct X2) p)
-        end
-    | ?X1 => constr:(fct_cte X1)
-  end.
-
-(**********)
-Ltac deriv_proof trm pt :=
-  match constr:trm with
-    | (?X1 + ?X2)%F =>
-      let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
-        constr:(derivable_pt_plus X1 X2 pt p1 p2)
-    | (?X1 - ?X2)%F =>
-      let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
-        constr:(derivable_pt_minus X1 X2 pt p1 p2)
-    | (?X1 * ?X2)%F =>
-      let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
-        constr:(derivable_pt_mult X1 X2 pt p1 p2)
-    | (?X1 / ?X2)%F =>
-      match goal with
-        | id:(?X2 pt <> 0) |- _ =>
-          let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
-            constr:(derivable_pt_div X1 X2 pt p1 p2 id)
-        | _ => constr:False
-      end
-    | (/ ?X1)%F =>
-      match goal with
-        | id:(?X1 pt <> 0) |- _ =>
-          let p1 := deriv_proof X1 pt in
-            constr:(derivable_pt_inv X1 pt p1 id)
-        | _ => constr:False
-      end
-    | (comp ?X1 ?X2) =>
-      let pt_f1 := eval cbv beta in (X2 pt) in
-        let p1 := deriv_proof X1 pt_f1 with p2 := deriv_proof X2 pt in
-          constr:(derivable_pt_comp X2 X1 pt p2 p1)
-    | (- ?X1)%F =>
-      let p1 := deriv_proof X1 pt in
-        constr:(derivable_pt_opp X1 pt p1)
-    | sin => constr:(derivable_pt_sin pt)
-    | cos => constr:(derivable_pt_cos pt)
-    | sinh => constr:(derivable_pt_sinh pt)
-    | cosh => constr:(derivable_pt_cosh pt)
-    | exp => constr:(derivable_pt_exp pt)
-    | id => constr:(derivable_pt_id pt)
-    | Rsqr => constr:(derivable_pt_Rsqr pt)
-    | sqrt =>
-      match goal with
-        | id:(0 < pt) |- _ => constr:(derivable_pt_sqrt pt id)
-        | _ => constr:False
-      end
-    | (fct_cte ?X1) => constr:(derivable_pt_const X1 pt)
-    | ?X1 =>
-      let aux := constr:X1 in
-        match goal with
-          | id:(derivable_pt aux pt) |- _ => constr:id
-          | id:(derivable aux) |- _ => constr:(id pt)
-          | _ => constr:False
-        end
-  end.
-
-(**********)
-Ltac simplify_derive trm pt :=
-  match constr:trm with
-    | (?X1 + ?X2)%F =>
-      try rewrite derive_pt_plus; simplify_derive X1 pt;
-        simplify_derive X2 pt
-    | (?X1 - ?X2)%F =>
-      try rewrite derive_pt_minus; simplify_derive X1 pt;
-        simplify_derive X2 pt
-    | (?X1 * ?X2)%F =>
-      try rewrite derive_pt_mult; simplify_derive X1 pt;
-        simplify_derive X2 pt
-    | (?X1 / ?X2)%F =>
-      try rewrite derive_pt_div; simplify_derive X1 pt; simplify_derive X2 pt
-    | (comp ?X1 ?X2) =>
-      let pt_f1 := eval cbv beta in (X2 pt) in
-        (try rewrite derive_pt_comp; simplify_derive X1 pt_f1;
-          simplify_derive X2 pt)
-    | (- ?X1)%F => try rewrite derive_pt_opp; simplify_derive X1 pt
-    | (/ ?X1)%F =>
-      try rewrite derive_pt_inv; simplify_derive X1 pt
-    | (fct_cte ?X1) => try rewrite derive_pt_const
-    | id => try rewrite derive_pt_id
-    | sin => try rewrite derive_pt_sin
-    | cos => try rewrite derive_pt_cos
-    | sinh => try rewrite derive_pt_sinh
-    | cosh => try rewrite derive_pt_cosh
-    | exp => try rewrite derive_pt_exp
-    | Rsqr => try rewrite derive_pt_Rsqr
-    | sqrt => try rewrite derive_pt_sqrt
-    | ?X1 =>
-      let aux := constr:X1 in
-        match goal with
-          | id:(derive_pt aux pt ?X2 = _),H:(derivable aux) |- _ =>
-            try replace (derive_pt aux pt (H pt)) with (derive_pt aux pt X2);
-              [ rewrite id | apply pr_nu ]
-          | id:(derive_pt aux pt ?X2 = _),H:(derivable_pt aux pt) |- _ =>
-            try replace (derive_pt aux pt H) with (derive_pt aux pt X2);
-              [ rewrite id | apply pr_nu ]
-          | _ => idtac
-        end
-    | _ => idtac
-  end.
-
-(**********)
-Ltac reg :=
-  match goal with
-    |  |- (derivable_pt ?X1 ?X2) =>
-      let trm := eval cbv beta in (X1 AppVar) in
-        let aux := rew_term trm in
-          (intro_hyp_pt aux X2;
-            try (change (derivable_pt aux X2) in |- *; is_diff_pt) || is_diff_pt)
-    |  |- (derivable ?X1) =>
-      let trm := eval cbv beta in (X1 AppVar) in
-        let aux := rew_term trm in
-          (intro_hyp_glob aux;
-            try (change (derivable aux) in |- *; is_diff_glob) || is_diff_glob)
-    |  |- (continuity ?X1) =>
-      let trm := eval cbv beta in (X1 AppVar) in
-        let aux := rew_term trm in
-          (intro_hyp_glob aux;
-            try (change (continuity aux) in |- *; is_cont_glob) || is_cont_glob)
-    |  |- (continuity_pt ?X1 ?X2) =>
-      let trm := eval cbv beta in (X1 AppVar) in
-        let aux := rew_term trm in
-          (intro_hyp_pt aux X2;
-            try (change (continuity_pt aux X2) in |- *; is_cont_pt) || is_cont_pt)
-    |  |- (derive_pt ?X1 ?X2 ?X3 = ?X4) =>
-      let trm := eval cbv beta in (X1 AppVar) in
-      let aux := rew_term trm in
-      intro_hyp_pt aux X2;
-      (let aux2 := deriv_proof aux X2 in
-       try
-         (replace (derive_pt X1 X2 X3) with (derive_pt aux X2 aux2);
-           [ simplify_derive aux X2;
-             try unfold plus_fct, minus_fct, mult_fct, div_fct, id, fct_cte,
-                        inv_fct, opp_fct in |- *; ring || ring_simplify
-           | try apply pr_nu ]) || is_diff_pt)
-  end.
+Require Export Ranalysis_reg.
\ No newline at end of file
diff --git a/theories/Reals/Ranalysis1.v b/theories/Reals/Ranalysis1.v
index 3075bee8..2f54ee94 100644
--- a/theories/Reals/Ranalysis1.v
+++ b/theories/Reals/Ranalysis1.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -10,7 +10,7 @@ Require Import Rbase.
 Require Import Rfunctions.
 Require Export Rlimit.
 Require Export Rderiv.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 Implicit Type f : R -> R.
 
 (****************************************************)
@@ -43,7 +43,7 @@ Notation "- x" := (opp_fct x) : Rfun_scope.
 Infix "*" := mult_fct : Rfun_scope.
 Infix "-" := minus_fct : Rfun_scope.
 Infix "/" := div_fct : Rfun_scope.
-Notation Local "f1 'o' f2" := (comp f1 f2)
+Local Notation "f1 'o' f2" := (comp f1 f2)
   (at level 20, right associativity) : Rfun_scope.
 Notation "/ x" := (inv_fct x) : Rfun_scope.
 
@@ -82,14 +82,14 @@ Lemma continuity_pt_plus :
   forall f1 f2 (x0:R),
     continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 + f2) x0.
 Proof.
-  unfold continuity_pt, plus_fct in |- *; unfold continue_in in |- *; intros;
+  unfold continuity_pt, plus_fct; unfold continue_in; intros;
     apply limit_plus; assumption.
 Qed.
 
 Lemma continuity_pt_opp :
   forall f (x0:R), continuity_pt f x0 -> continuity_pt (- f) x0.
 Proof.
-  unfold continuity_pt, opp_fct in |- *; unfold continue_in in |- *; intros;
+  unfold continuity_pt, opp_fct; unfold continue_in; intros;
     apply limit_Ropp; assumption.
 Qed.
 
@@ -97,7 +97,7 @@ Lemma continuity_pt_minus :
   forall f1 f2 (x0:R),
     continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 - f2) x0.
 Proof.
-  unfold continuity_pt, minus_fct in |- *; unfold continue_in in |- *; intros;
+  unfold continuity_pt, minus_fct; unfold continue_in; intros;
     apply limit_minus; assumption.
 Qed.
 
@@ -105,17 +105,17 @@ Lemma continuity_pt_mult :
   forall f1 f2 (x0:R),
     continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 * f2) x0.
 Proof.
-  unfold continuity_pt, mult_fct in |- *; unfold continue_in in |- *; intros;
+  unfold continuity_pt, mult_fct; unfold continue_in; intros;
     apply limit_mul; assumption.
 Qed.
 
 Lemma continuity_pt_const : forall f (x0:R), constant f -> continuity_pt f x0.
 Proof.
-  unfold constant, continuity_pt in |- *; unfold continue_in in |- *;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
+  unfold constant, continuity_pt; unfold continue_in;
+    unfold limit1_in; unfold limit_in;
       intros; exists 1; split;
         [ apply Rlt_0_1
-          | intros; generalize (H x x0); intro; rewrite H2; simpl in |- *;
+          | intros; generalize (H x x0); intro; rewrite H2; simpl;
             rewrite R_dist_eq; assumption ].
 Qed.
 
@@ -123,9 +123,9 @@ Lemma continuity_pt_scal :
   forall f (a x0:R),
     continuity_pt f x0 -> continuity_pt (mult_real_fct a f) x0.
 Proof.
-  unfold continuity_pt, mult_real_fct in |- *; unfold continue_in in |- *;
+  unfold continuity_pt, mult_real_fct; unfold continue_in;
     intros; apply (limit_mul (fun x:R => a) f (D_x no_cond x0) a (f x0) x0).
-  unfold limit1_in in |- *; unfold limit_in in |- *; intros; exists 1; split.
+  unfold limit1_in; unfold limit_in; intros; exists 1; split.
   apply Rlt_0_1.
   intros; rewrite R_dist_eq; assumption.
   assumption.
@@ -136,9 +136,9 @@ Lemma continuity_pt_inv :
 Proof.
   intros.
   replace (/ f)%F with (fun x:R => / f x).
-  unfold continuity_pt in |- *; unfold continue_in in |- *; intros;
+  unfold continuity_pt; unfold continue_in; intros;
     apply limit_inv; assumption.
-  unfold inv_fct in |- *; reflexivity.
+  unfold inv_fct; reflexivity.
 Qed.
 
 Lemma div_eq_inv : forall f1 f2, (f1 / f2)%F = (f1 * / f2)%F.
@@ -159,8 +159,8 @@ Lemma continuity_pt_comp :
   forall f1 f2 (x:R),
     continuity_pt f1 x -> continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x.
 Proof.
-  unfold continuity_pt in |- *; unfold continue_in in |- *; intros;
-    unfold comp in |- *.
+  unfold continuity_pt; unfold continue_in; intros;
+    unfold comp.
   cut
     (limit1_in (fun x0:R => f2 (f1 x0))
       (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1) (
@@ -170,23 +170,23 @@ Proof.
   eapply limit_comp.
   apply H.
   apply H0.
-  unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *;
-    simpl in |- *; unfold R_dist in |- *; intros.
+  unfold limit1_in; unfold limit_in; unfold dist;
+    simpl; unfold R_dist; intros.
   assert (H3 := H1 eps H2).
   elim H3; intros.
   exists x0.
   split.
   elim H4; intros; assumption.
   intros; case (Req_dec (f1 x) (f1 x1)); intro.
-  rewrite H6; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  rewrite H6; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     assumption.
   elim H4; intros; apply H8.
   split.
-  unfold Dgf, D_x, no_cond in |- *.
+  unfold Dgf, D_x, no_cond.
   split.
   split.
   trivial.
-  elim H5; unfold D_x, no_cond in |- *; intros.
+  elim H5; unfold D_x, no_cond; intros.
   elim H9; intros; assumption.
   split.
   trivial.
@@ -198,44 +198,44 @@ Qed.
 Lemma continuity_plus :
   forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 + f2).
 Proof.
-  unfold continuity in |- *; intros;
+  unfold continuity; intros;
     apply (continuity_pt_plus f1 f2 x (H x) (H0 x)).
 Qed.
 
 Lemma continuity_opp : forall f, continuity f -> continuity (- f).
 Proof.
-  unfold continuity in |- *; intros; apply (continuity_pt_opp f x (H x)).
+  unfold continuity; intros; apply (continuity_pt_opp f x (H x)).
 Qed.
 
 Lemma continuity_minus :
   forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 - f2).
 Proof.
-  unfold continuity in |- *; intros;
+  unfold continuity; intros;
     apply (continuity_pt_minus f1 f2 x (H x) (H0 x)).
 Qed.
 
 Lemma continuity_mult :
   forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 * f2).
 Proof.
-  unfold continuity in |- *; intros;
+  unfold continuity; intros;
     apply (continuity_pt_mult f1 f2 x (H x) (H0 x)).
 Qed.
 
 Lemma continuity_const : forall f, constant f -> continuity f.
 Proof.
-  unfold continuity in |- *; intros; apply (continuity_pt_const f x H).
+  unfold continuity; intros; apply (continuity_pt_const f x H).
 Qed.
 
 Lemma continuity_scal :
   forall f (a:R), continuity f -> continuity (mult_real_fct a f).
 Proof.
-  unfold continuity in |- *; intros; apply (continuity_pt_scal f a x (H x)).
+  unfold continuity; intros; apply (continuity_pt_scal f a x (H x)).
 Qed.
 
 Lemma continuity_inv :
   forall f, continuity f -> (forall x:R, f x <> 0) -> continuity (/ f).
 Proof.
-  unfold continuity in |- *; intros; apply (continuity_pt_inv f x (H x) (H0 x)).
+  unfold continuity; intros; apply (continuity_pt_inv f x (H x) (H0 x)).
 Qed.
 
 Lemma continuity_div :
@@ -243,14 +243,14 @@ Lemma continuity_div :
     continuity f1 ->
     continuity f2 -> (forall x:R, f2 x <> 0) -> continuity (f1 / f2).
 Proof.
-  unfold continuity in |- *; intros;
+  unfold continuity; intros;
     apply (continuity_pt_div f1 f2 x (H x) (H0 x) (H1 x)).
 Qed.
 
 Lemma continuity_comp :
   forall f1 f2, continuity f1 -> continuity f2 -> continuity (f2 o f1).
 Proof.
-  unfold continuity in |- *; intros.
+  unfold continuity; intros.
   apply (continuity_pt_comp f1 f2 x (H x) (H0 (f1 x))).
 Qed.
 
@@ -307,23 +307,23 @@ Proof.
     apply
       (single_limit (fun h:R => (f (x + h) - f x) / h) (
         fun h:R => h <> 0) l1 l2 0); try assumption.
-  unfold adhDa in |- *; intros; exists (alp / 2).
+  unfold adhDa; intros; exists (alp / 2).
   split.
-  unfold Rdiv in |- *; apply prod_neq_R0.
-  red in |- *; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1).
+  unfold Rdiv; apply prod_neq_R0.
+  red; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1).
   apply Rinv_neq_0_compat; discrR.
-  unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
-    rewrite Rplus_0_r; unfold Rdiv in |- *; rewrite Rabs_mult.
+  unfold R_dist; unfold Rminus; rewrite Ropp_0;
+    rewrite Rplus_0_r; unfold Rdiv; rewrite Rabs_mult.
   replace (Rabs (/ 2)) with (/ 2).
   replace (Rabs alp) with alp.
   apply Rmult_lt_reg_l with 2.
   prove_sup0.
   rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
     [ idtac | discrR ]; rewrite Rmult_1_r; rewrite double;
-      pattern alp at 1 in |- *; replace alp with (alp + 0);
+      pattern alp at 1; replace alp with (alp + 0);
         [ idtac | ring ]; apply Rplus_lt_compat_l; assumption.
-  symmetry  in |- *; apply Rabs_right; left; assumption.
-  symmetry  in |- *; apply Rabs_right; left; change (0 < / 2) in |- *;
+  symmetry ; apply Rabs_right; left; assumption.
+  symmetry ; apply Rabs_right; left; change (0 < / 2);
     apply Rinv_0_lt_compat; prove_sup0.
 Qed.
 
@@ -332,14 +332,14 @@ Lemma uniqueness_step2 :
     derivable_pt_lim f x l ->
     limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0.
 Proof.
-  unfold derivable_pt_lim in |- *; intros; unfold limit1_in in |- *;
-    unfold limit_in in |- *; intros.
+  unfold derivable_pt_lim; intros; unfold limit1_in;
+    unfold limit_in; intros.
   assert (H1 := H eps H0).
   elim H1; intros.
   exists (pos x0).
   split.
   apply (cond_pos x0).
-  simpl in |- *; unfold R_dist in |- *; intros.
+  simpl; unfold R_dist; intros.
   elim H3; intros.
   apply H2;
     [ assumption
@@ -352,15 +352,15 @@ Lemma uniqueness_step3 :
     limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 ->
     derivable_pt_lim f x l.
 Proof.
-  unfold limit1_in, derivable_pt_lim in |- *; unfold limit_in in |- *;
-    unfold dist in |- *; simpl in |- *; intros.
+  unfold limit1_in, derivable_pt_lim; unfold limit_in;
+    unfold dist; simpl; intros.
   elim (H eps H0).
   intros; elim H1; intros.
   exists (mkposreal x0 H2).
-  simpl in |- *; intros; unfold R_dist in H3; apply (H3 h).
+  simpl; intros; unfold R_dist in H3; apply (H3 h).
   split;
     [ assumption
-      | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; assumption ].
+      | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; assumption ].
 Qed.
 
 Lemma uniqueness_limite :
@@ -383,8 +383,8 @@ Proof.
     assumption.
   intro; assert (H1 := proj2_sig pr); unfold derivable_pt_abs in H1.
   assert (H2 := uniqueness_limite _ _ _ _ H H1).
-  unfold derive_pt in |- *; unfold derivable_pt_abs in |- *.
-  symmetry  in |- *; assumption.
+  unfold derive_pt; unfold derivable_pt_abs.
+  symmetry ; assumption.
 Qed.
 
 (**********)
@@ -414,25 +414,25 @@ Lemma derive_pt_D_in :
     D_in f df no_cond x <-> derive_pt f x pr = df x.
 Proof.
   intros; split.
-  unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *;
-    simpl in |- *; unfold R_dist in |- *; intros.
+  unfold D_in; unfold limit1_in; unfold limit_in;
+    simpl; unfold R_dist; intros.
   apply derive_pt_eq_0.
-  unfold derivable_pt_lim in |- *.
+  unfold derivable_pt_lim.
   intros; elim (H eps H0); intros alpha H1; elim H1; intros;
     exists (mkposreal alpha H2); intros; generalize (H3 (x + h));
       intro; cut (x + h - x = h);
         [ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha);
           [ intro; generalize (H6 H8); rewrite H7; intro; assumption
             | split;
-              [ unfold D_x in |- *; split;
-                [ unfold no_cond in |- *; trivial
+              [ unfold D_x; split;
+                [ unfold no_cond; trivial
                   | apply Rminus_not_eq_right; rewrite H7; assumption ]
                 | rewrite H7; assumption ] ]
           | ring ].
   intro.
   assert (H0 := derive_pt_eq_1 f x (df x) pr H).
-  unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *;
-    unfold dist in |- *; simpl in |- *; unfold R_dist in |- *;
+  unfold D_in; unfold limit1_in; unfold limit_in;
+    unfold dist; simpl; unfold R_dist;
       intros.
   elim (H0 eps H1); intros alpha H2; exists (pos alpha); split.
   apply (cond_pos alpha).
@@ -448,24 +448,24 @@ Lemma derivable_pt_lim_D_in :
     D_in f df no_cond x <-> derivable_pt_lim f x (df x).
 Proof.
   intros; split.
-  unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *;
-    simpl in |- *; unfold R_dist in |- *; intros.
-  unfold derivable_pt_lim in |- *.
+  unfold D_in; unfold limit1_in; unfold limit_in;
+    simpl; unfold R_dist; intros.
+  unfold derivable_pt_lim.
   intros; elim (H eps H0); intros alpha H1; elim H1; intros;
     exists (mkposreal alpha H2); intros; generalize (H3 (x + h));
       intro; cut (x + h - x = h);
         [ intro; cut (D_x no_cond x (x + h) /\ Rabs (x + h - x) < alpha);
           [ intro; generalize (H6 H8); rewrite H7; intro; assumption
             | split;
-              [ unfold D_x in |- *; split;
-                [ unfold no_cond in |- *; trivial
+              [ unfold D_x; split;
+                [ unfold no_cond; trivial
                   | apply Rminus_not_eq_right; rewrite H7; assumption ]
                 | rewrite H7; assumption ] ]
           | ring ].
   intro.
   unfold derivable_pt_lim in H.
-  unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *;
-    unfold dist in |- *; simpl in |- *; unfold R_dist in |- *;
+  unfold D_in; unfold limit1_in; unfold limit_in;
+    unfold dist; simpl; unfold R_dist;
       intros.
   elim (H eps H0); intros alpha H2; exists (pos alpha); split.
   apply (cond_pos alpha).
@@ -486,7 +486,7 @@ Lemma derivable_derive :
   forall f (x:R) (pr:derivable_pt f x),  exists l : R, derive_pt f x pr = l.
 Proof.
   intros; exists (proj1_sig pr).
-  unfold derive_pt in |- *; reflexivity.
+  unfold derive_pt; reflexivity.
 Qed.
 
 Theorem derivable_continuous_pt :
@@ -501,14 +501,14 @@ Proof.
   generalize (derive_pt_D_in f (fct_cte l) x); intro.
   elim (H2 X); intros.
   generalize (H4 H1); intro.
-  unfold continuity_pt in |- *.
+  unfold continuity_pt.
   apply (cont_deriv f (fct_cte l) no_cond x H5).
-  unfold fct_cte in |- *; reflexivity.
+  unfold fct_cte; reflexivity.
 Qed.
 
 Theorem derivable_continuous : forall f, derivable f -> continuity f.
 Proof.
-  unfold derivable, continuity in |- *; intros f X x.
+  unfold derivable, continuity; intros f X x.
   apply (derivable_continuous_pt f x (X x)).
 Qed.
 
@@ -524,7 +524,7 @@ Lemma derivable_pt_lim_plus :
   apply uniqueness_step3.
   assert (H1 := uniqueness_step2 _ _ _ H).
   assert (H2 := uniqueness_step2 _ _ _ H0).
-  unfold plus_fct in |- *.
+  unfold plus_fct.
   cut
     (forall h:R,
       (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h =
@@ -533,15 +533,15 @@ Lemma derivable_pt_lim_plus :
   generalize
     (limit_plus (fun h':R => (f1 (x + h') - f1 x) / h')
       (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2).
-  unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *;
-    simpl in |- *; unfold R_dist in |- *; intros.
+  unfold limit1_in; unfold limit_in; unfold dist;
+    simpl; unfold R_dist; intros.
   elim (H4 eps H5); intros.
   exists x0.
   elim H6; intros.
   split.
   assumption.
   intros; rewrite H3; apply H8; assumption.
-  intro; unfold Rdiv in |- *; ring.
+  intro; unfold Rdiv; ring.
 Qed.
 
 Lemma derivable_pt_lim_opp :
@@ -550,20 +550,20 @@ Proof.
   intros.
   apply uniqueness_step3.
   assert (H1 := uniqueness_step2 _ _ _ H).
-  unfold opp_fct in |- *.
+  unfold opp_fct.
   cut (forall h:R, (- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)).
   intro.
   generalize
     (limit_Ropp (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 H1).
-  unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *;
-    simpl in |- *; unfold R_dist in |- *; intros.
+  unfold limit1_in; unfold limit_in; unfold dist;
+    simpl; unfold R_dist; intros.
   elim (H2 eps H3); intros.
   exists x0.
   elim H4; intros.
   split.
   assumption.
   intros; rewrite H0; apply H6; assumption.
-  intro; unfold Rdiv in |- *; ring.
+  intro; unfold Rdiv; ring.
 Qed.
 
 Lemma derivable_pt_lim_minus :
@@ -575,7 +575,7 @@ Proof.
   apply uniqueness_step3.
   assert (H1 := uniqueness_step2 _ _ _ H).
   assert (H2 := uniqueness_step2 _ _ _ H0).
-  unfold minus_fct in |- *.
+  unfold minus_fct.
   cut
     (forall h:R,
       (f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h =
@@ -584,15 +584,15 @@ Proof.
   generalize
     (limit_minus (fun h':R => (f1 (x + h') - f1 x) / h')
       (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2).
-  unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *;
-    simpl in |- *; unfold R_dist in |- *; intros.
+  unfold limit1_in; unfold limit_in; unfold dist;
+    simpl; unfold R_dist; intros.
   elim (H4 eps H5); intros.
   exists x0.
   elim H6; intros.
   split.
   assumption.
   intros; rewrite <- H3; apply H8; assumption.
-  intro; unfold Rdiv in |- *; ring.
+  intro; unfold Rdiv; ring.
 Qed.
 
 Lemma derivable_pt_lim_mult :
@@ -615,15 +615,15 @@ Proof.
   elim H1; intros.
   clear H1 H3.
   apply H2.
-  unfold mult_fct in |- *.
+  unfold mult_fct.
   apply (Dmult no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x); assumption.
 Qed.
 
 Lemma derivable_pt_lim_const : forall a x:R, derivable_pt_lim (fct_cte a) x 0.
 Proof.
-  intros; unfold fct_cte, derivable_pt_lim in |- *.
-  intros; exists (mkposreal 1 Rlt_0_1); intros; unfold Rminus in |- *;
-    rewrite Rplus_opp_r; unfold Rdiv in |- *; rewrite Rmult_0_l;
+  intros; unfold fct_cte, derivable_pt_lim.
+  intros; exists (mkposreal 1 Rlt_0_1); intros; unfold Rminus;
+    rewrite Rplus_opp_r; unfold Rdiv; rewrite Rmult_0_l;
       rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
 Qed.
 
@@ -636,34 +636,34 @@ Proof.
   replace (mult_real_fct a f) with (fct_cte a * f)%F.
   replace (a * l) with (0 * f x + a * l); [ idtac | ring ].
   apply (derivable_pt_lim_mult (fct_cte a) f x 0 l); assumption.
-  unfold mult_real_fct, mult_fct, fct_cte in |- *; reflexivity.
+  unfold mult_real_fct, mult_fct, fct_cte; reflexivity.
 Qed.
 
 Lemma derivable_pt_lim_id : forall x:R, derivable_pt_lim id x 1.
 Proof.
-  intro; unfold derivable_pt_lim in |- *.
+  intro; unfold derivable_pt_lim.
   intros eps Heps; exists (mkposreal eps Heps); intros h H1 H2;
-    unfold id in |- *; replace ((x + h - x) / h - 1) with 0.
+    unfold id; replace ((x + h - x) / h - 1) with 0.
   rewrite Rabs_R0; apply Rle_lt_trans with (Rabs h).
   apply Rabs_pos.
   assumption.
-  unfold Rminus in |- *; rewrite Rplus_assoc; rewrite (Rplus_comm x);
+  unfold Rminus; rewrite Rplus_assoc; rewrite (Rplus_comm x);
     rewrite Rplus_assoc.
-  rewrite Rplus_opp_l; rewrite Rplus_0_r; unfold Rdiv in |- *;
+  rewrite Rplus_opp_l; rewrite Rplus_0_r; unfold Rdiv;
     rewrite <- Rinv_r_sym.
-  symmetry  in |- *; apply Rplus_opp_r.
+  symmetry ; apply Rplus_opp_r.
   assumption.
 Qed.
 
 Lemma derivable_pt_lim_Rsqr : forall x:R, derivable_pt_lim Rsqr x (2 * x).
 Proof.
-  intro; unfold derivable_pt_lim in |- *.
-  unfold Rsqr in |- *; intros eps Heps; exists (mkposreal eps Heps);
+  intro; unfold derivable_pt_lim.
+  unfold Rsqr; intros eps Heps; exists (mkposreal eps Heps);
     intros h H1 H2; replace (((x + h) * (x + h) - x * x) / h - 2 * x) with h.
   assumption.
   replace ((x + h) * (x + h) - x * x) with (2 * x * h + h * h);
   [ idtac | ring ].
-  unfold Rdiv in |- *; rewrite Rmult_plus_distr_r.
+  unfold Rdiv; rewrite Rmult_plus_distr_r.
   repeat rewrite Rmult_assoc.
   repeat rewrite <- Rinv_r_sym; [ idtac | assumption ].
   ring.
@@ -684,7 +684,7 @@ Proof.
   assert (H1 := derivable_pt_lim_D_in (f2 o f1)%F (fun y:R => l2 * l1) x).
   elim H1; intros.
   clear H1 H3; apply H2.
-  unfold comp in |- *;
+  unfold comp;
     cut
       (D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1)
         (Dgf no_cond no_cond f1) x ->
@@ -693,14 +693,14 @@ Proof.
   rewrite Rmult_comm;
     apply (Dcomp no_cond no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x);
       assumption.
-  unfold Dgf, D_in, no_cond in |- *; unfold limit1_in in |- *;
-    unfold limit_in in |- *; unfold dist in |- *; simpl in |- *;
-      unfold R_dist in |- *; intros.
+  unfold Dgf, D_in, no_cond; unfold limit1_in;
+    unfold limit_in; unfold dist; simpl;
+      unfold R_dist; intros.
   elim (H1 eps H3); intros.
   exists x0; intros; split.
   elim H5; intros; assumption.
   intros; elim H5; intros; apply H9; split.
-  unfold D_x in |- *; split.
+  unfold D_x; split.
   split; trivial.
   elim H6; intros; unfold D_x in H10; elim H10; intros; assumption.
   elim H6; intros; assumption.
@@ -710,7 +710,7 @@ Lemma derivable_pt_plus :
   forall f1 f2 (x:R),
     derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 + f2) x.
 Proof.
-  unfold derivable_pt in |- *; intros f1 f2 x X X0.
+  unfold derivable_pt; intros f1 f2 x X X0.
   elim X; intros.
   elim X0; intros.
   exists (x0 + x1).
@@ -720,7 +720,7 @@ Qed.
 Lemma derivable_pt_opp :
   forall f (x:R), derivable_pt f x -> derivable_pt (- f) x.
 Proof.
-  unfold derivable_pt in |- *; intros f x X.
+  unfold derivable_pt; intros f x X.
   elim X; intros.
   exists (- x0).
   apply derivable_pt_lim_opp; assumption.
@@ -730,7 +730,7 @@ Lemma derivable_pt_minus :
   forall f1 f2 (x:R),
     derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 - f2) x.
 Proof.
-  unfold derivable_pt in |- *; intros f1 f2 x X X0.
+  unfold derivable_pt; intros f1 f2 x X X0.
   elim X; intros.
   elim X0; intros.
   exists (x0 - x1).
@@ -741,7 +741,7 @@ Lemma derivable_pt_mult :
   forall f1 f2 (x:R),
     derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 * f2) x.
 Proof.
-  unfold derivable_pt in |- *; intros f1 f2 x X X0.
+  unfold derivable_pt; intros f1 f2 x X X0.
   elim X; intros.
   elim X0; intros.
   exists (x0 * f2 x + f1 x * x1).
@@ -750,7 +750,7 @@ Qed.
 
 Lemma derivable_pt_const : forall a x:R, derivable_pt (fct_cte a) x.
 Proof.
-  intros; unfold derivable_pt in |- *.
+  intros; unfold derivable_pt.
   exists 0.
   apply derivable_pt_lim_const.
 Qed.
@@ -758,7 +758,7 @@ Qed.
 Lemma derivable_pt_scal :
   forall f (a x:R), derivable_pt f x -> derivable_pt (mult_real_fct a f) x.
 Proof.
-  unfold derivable_pt in |- *; intros f1 a x X.
+  unfold derivable_pt; intros f1 a x X.
   elim X; intros.
   exists (a * x0).
   apply derivable_pt_lim_scal; assumption.
@@ -766,14 +766,14 @@ Qed.
 
 Lemma derivable_pt_id : forall x:R, derivable_pt id x.
 Proof.
-  unfold derivable_pt in |- *; intro.
+  unfold derivable_pt; intro.
   exists 1.
   apply derivable_pt_lim_id.
 Qed.
 
 Lemma derivable_pt_Rsqr : forall x:R, derivable_pt Rsqr x.
 Proof.
-  unfold derivable_pt in |- *; intro; exists (2 * x).
+  unfold derivable_pt; intro; exists (2 * x).
   apply derivable_pt_lim_Rsqr.
 Qed.
 
@@ -781,7 +781,7 @@ Lemma derivable_pt_comp :
   forall f1 f2 (x:R),
     derivable_pt f1 x -> derivable_pt f2 (f1 x) -> derivable_pt (f2 o f1) x.
 Proof.
-  unfold derivable_pt in |- *; intros f1 f2 x X X0.
+  unfold derivable_pt; intros f1 f2 x X X0.
   elim X; intros.
   elim X0; intros.
   exists (x1 * x0).
@@ -791,57 +791,57 @@ Qed.
 Lemma derivable_plus :
   forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 + f2).
 Proof.
-  unfold derivable in |- *; intros f1 f2 X X0 x.
+  unfold derivable; intros f1 f2 X X0 x.
   apply (derivable_pt_plus _ _ x (X _) (X0 _)).
 Qed.
 
 Lemma derivable_opp : forall f, derivable f -> derivable (- f).
 Proof.
-  unfold derivable in |- *; intros f X x.
+  unfold derivable; intros f X x.
   apply (derivable_pt_opp _ x (X _)).
 Qed.
 
 Lemma derivable_minus :
   forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 - f2).
 Proof.
-  unfold derivable in |- *; intros f1 f2 X X0 x.
+  unfold derivable; intros f1 f2 X X0 x.
   apply (derivable_pt_minus _ _ x (X _) (X0 _)).
 Qed.
 
 Lemma derivable_mult :
   forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 * f2).
 Proof.
-  unfold derivable in |- *; intros f1 f2 X X0 x.
+  unfold derivable; intros f1 f2 X X0 x.
   apply (derivable_pt_mult _ _ x (X _) (X0 _)).
 Qed.
 
 Lemma derivable_const : forall a:R, derivable (fct_cte a).
 Proof.
-  unfold derivable in |- *; intros.
+  unfold derivable; intros.
   apply derivable_pt_const.
 Qed.
 
 Lemma derivable_scal :
   forall f (a:R), derivable f -> derivable (mult_real_fct a f).
 Proof.
-  unfold derivable in |- *; intros f a X x.
+  unfold derivable; intros f a X x.
   apply (derivable_pt_scal _ a x (X _)).
 Qed.
 
 Lemma derivable_id : derivable id.
 Proof.
-  unfold derivable in |- *; intro; apply derivable_pt_id.
+  unfold derivable; intro; apply derivable_pt_id.
 Qed.
 
 Lemma derivable_Rsqr : derivable Rsqr.
 Proof.
-  unfold derivable in |- *; intro; apply derivable_pt_Rsqr.
+  unfold derivable; intro; apply derivable_pt_Rsqr.
 Qed.
 
 Lemma derivable_comp :
   forall f1 f2, derivable f1 -> derivable f2 -> derivable (f2 o f1).
 Proof.
-  unfold derivable in |- *; intros f1 f2 X X0 x.
+  unfold derivable; intros f1 f2 X X0 x.
   apply (derivable_pt_comp _ _ x (X _) (X0 _)).
 Qed.
 
@@ -996,13 +996,13 @@ Proof.
   elim (lt_irrefl _ H).
   cut (n = 0%nat \/ (0 < n)%nat).
   intro; elim H0; intro.
-  rewrite H1; simpl in |- *.
+  rewrite H1; simpl.
   replace (fun y:R => y * 1) with (id * fct_cte 1)%F.
   replace (1 * 1) with (1 * fct_cte 1 x + id x * 0).
   apply derivable_pt_lim_mult.
   apply derivable_pt_lim_id.
   apply derivable_pt_lim_const.
-  unfold fct_cte, id in |- *; ring.
+  unfold fct_cte, id; ring.
   reflexivity.
   replace (fun y:R => y ^ S n) with (fun y:R => y * y ^ n).
   replace (pred (S n)) with n; [ idtac | reflexivity ].
@@ -1011,13 +1011,13 @@ Proof.
   replace (INR (S n) * x ^ n) with (1 * f x + id x * (INR n * x ^ pred n)).
   apply derivable_pt_lim_mult.
   apply derivable_pt_lim_id.
-  unfold f in |- *; apply Hrecn; assumption.
-  unfold f in |- *.
-  pattern n at 1 5 in |- *; replace n with (S (pred n)).
-  unfold id in |- *; rewrite S_INR; simpl in |- *.
+  unfold f; apply Hrecn; assumption.
+  unfold f.
+  pattern n at 1 5; replace n with (S (pred n)).
+  unfold id; rewrite S_INR; simpl.
   ring.
-  symmetry  in |- *; apply S_pred with 0%nat; assumption.
-  unfold mult_fct, id in |- *; reflexivity.
+  symmetry ; apply S_pred with 0%nat; assumption.
+  unfold mult_fct, id; reflexivity.
   reflexivity.
   inversion H.
   left; reflexivity.
@@ -1033,7 +1033,7 @@ Lemma derivable_pt_lim_pow :
 Proof.
   intros.
   induction  n as [| n Hrecn].
-  simpl in |- *.
+  simpl.
   rewrite Rmult_0_l.
   replace (fun _:R => 1) with (fct_cte 1);
   [ apply derivable_pt_lim_const | reflexivity ].
@@ -1044,14 +1044,14 @@ Qed.
 Lemma derivable_pt_pow :
   forall (n:nat) (x:R), derivable_pt (fun y:R => y ^ n) x.
 Proof.
-  intros; unfold derivable_pt in |- *.
+  intros; unfold derivable_pt.
   exists (INR n * x ^ pred n).
   apply derivable_pt_lim_pow.
 Qed.
 
 Lemma derivable_pow : forall n:nat, derivable (fun y:R => y ^ n).
 Proof.
-  intro; unfold derivable in |- *; intro; apply derivable_pt_pow.
+  intro; unfold derivable; intro; apply derivable_pt_pow.
 Qed.
 
 Lemma derive_pt_pow :
@@ -1073,7 +1073,7 @@ Proof.
   elim pr2; intros.
   unfold derivable_pt_abs in p.
   unfold derivable_pt_abs in p0.
-  simpl in |- *.
+  simpl.
   apply (uniqueness_limite f x x0 x1 p p0).
 Qed.
 
@@ -1094,7 +1094,7 @@ Proof.
   assert (H5 := derive_pt_eq_1 f c l pr H4).
   cut (0 < l / 2);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
         [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
   elim (H5 (l / 2) H6); intros delta H7.
   cut (0 < (b - c) / 2).
@@ -1119,7 +1119,7 @@ Proof.
       (Rabs
         ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
           Rmin (delta / 2) ((b + - c) / 2) + - l) < l / 2).
-  unfold Rabs in |- *;
+  unfold Rabs;
     case
     (Rcase_abs
       ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
@@ -1157,7 +1157,7 @@ Proof.
               (Rlt_le_trans 0
                 ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) /
                   Rmin (delta / 2) ((b + - c) / 2)) 0 H22 H16)).
-  pattern l at 2 in |- *; rewrite double_var.
+  pattern l at 2; rewrite double_var.
   ring.
   ring.
   intro.
@@ -1183,7 +1183,7 @@ Proof.
         l +
         -
         ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) /
-          Rmin (delta / 2) ((b + - c) / 2))) in |- *; apply Rplus_lt_le_0_compat;
+          Rmin (delta / 2) ((b + - c) / 2))); apply Rplus_lt_le_0_compat;
       [ assumption
         | rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; assumption ].
   unfold Rminus; ring.
@@ -1195,13 +1195,13 @@ Proof.
     ((f c - f (c + Rmin (delta / 2) ((b - c) / 2))) /
       Rmin (delta / 2) ((b - c) / 2))).
   rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge;
-    unfold Rdiv in |- *; apply Rmult_le_pos;
+    unfold Rdiv; apply Rmult_le_pos;
       [ generalize
         (Rplus_le_compat_r (- f (c + Rmin (delta * / 2) ((b - c) * / 2)))
           (f (c + Rmin (delta * / 2) ((b - c) * / 2))) (
             f c) H15); rewrite Rplus_opp_r; intro; assumption
         | left; apply Rinv_0_lt_compat; assumption ].
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   rewrite <- Ropp_mult_distr_l_reverse.
   repeat rewrite <- (Rmult_comm (/ Rmin (delta * / 2) ((b - c) * / 2))).
   apply Rmult_eq_reg_l with (Rmin (delta * / 2) ((b - c) * / 2)).
@@ -1209,9 +1209,9 @@ Proof.
   rewrite <- Rinv_r_sym.
   repeat rewrite Rmult_1_l.
   ring.
-  red in |- *; intro.
+  red; intro.
   unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12).
-  red in |- *; intro.
+  red; intro.
   unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12).
   assert (H14 := Rmin_r (delta / 2) ((b - c) / 2)).
   assert
@@ -1225,7 +1225,7 @@ Proof.
   replace (2 * b) with (b + b).
   apply Rplus_lt_compat_r; assumption.
   ring.
-  unfold Rdiv in |- *; rewrite Rmult_plus_distr_l.
+  unfold Rdiv; rewrite Rmult_plus_distr_l.
   repeat rewrite (Rmult_comm 2).
   rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r.
@@ -1233,51 +1233,51 @@ Proof.
   discrR.
   apply Rlt_trans with c.
   assumption.
-  pattern c at 1 in |- *; rewrite <- (Rplus_0_r c); apply Rplus_lt_compat_l;
+  pattern c at 1; rewrite <- (Rplus_0_r c); apply Rplus_lt_compat_l;
     assumption.
   cut (0 < delta / 2).
   intro;
     apply
       (Rmin_stable_in_posreal (mkposreal (delta / 2) H12)
         (mkposreal ((b - c) / 2) H8)).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
-  unfold Rabs in |- *; case (Rcase_abs (Rmin (delta / 2) ((b - c) / 2))).
+  unfold Rabs; case (Rcase_abs (Rmin (delta / 2) ((b - c) / 2))).
   intro.
   cut (0 < delta / 2).
   intro.
   generalize
     (Rmin_stable_in_posreal (mkposreal (delta / 2) H10)
-      (mkposreal ((b - c) / 2) H8)); simpl in |- *; intro;
+      (mkposreal ((b - c) / 2) H8)); simpl; intro;
     elim (Rlt_irrefl 0 (Rlt_trans 0 (Rmin (delta / 2) ((b - c) / 2)) 0 H11 r)).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
   intro; apply Rle_lt_trans with (delta / 2).
   apply Rmin_l.
-  unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2.
+  unfold Rdiv; apply Rmult_lt_reg_l with 2.
   prove_sup0.
   rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
   rewrite Rmult_1_l.
   replace (2 * delta) with (delta + delta).
-  pattern delta at 2 in |- *; rewrite <- (Rplus_0_r delta);
+  pattern delta at 2; rewrite <- (Rplus_0_r delta);
     apply Rplus_lt_compat_l.
   rewrite Rplus_0_r; apply (cond_pos delta).
-  symmetry  in |- *; apply double.
+  symmetry ; apply double.
   discrR.
   cut (0 < delta / 2).
   intro;
     generalize
       (Rmin_stable_in_posreal (mkposreal (delta / 2) H9)
-        (mkposreal ((b - c) / 2) H8)); simpl in |- *;
-      intro; red in |- *; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+        (mkposreal ((b - c) / 2) H8)); simpl;
+      intro; red; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10).
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   generalize (Rplus_lt_compat_r (- c) c b H0); rewrite Rplus_opp_r; intro;
     assumption.
   apply Rinv_0_lt_compat; prove_sup0.
   elim H2; intro.
-  symmetry  in |- *; assumption.
+  symmetry ; assumption.
   generalize (derivable_derive f c pr); intro; elim H4; intros l H5.
   rewrite H5 in H3; generalize (derive_pt_eq_1 f c l pr H5); intro;
     cut (0 < - (l / 2)).
@@ -1307,7 +1307,7 @@ Proof.
         ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
           Rmax (- (delta / 2)) ((a + - c) / 2) + - l) <
         - (l / 2)).
-  unfold Rabs in |- *;
+  unfold Rabs;
     case
     (Rcase_abs
       ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) /
@@ -1339,12 +1339,12 @@ Proof.
                 Rmax (- (delta / 2)) ((a - c) / 2)) 0 H17 H23)).
   rewrite <- (Ropp_involutive (l / 2)); rewrite <- Ropp_0;
     apply Ropp_lt_gt_contravar; assumption.
-  pattern l at 3 in |- *; rewrite double_var.
+  pattern l at 3; rewrite double_var.
   ring.
   assumption.
   apply Rplus_le_lt_0_compat; assumption.
   rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
-  unfold Rdiv in |- *;
+  unfold Rdiv;
     replace
     ((f (c + Rmax (- (delta * / 2)) ((a - c) * / 2)) - f c) *
       / Rmax (- (delta * / 2)) ((a - c) * / 2)) with
@@ -1361,7 +1361,7 @@ Proof.
   ring.
   left; apply Rinv_0_lt_compat; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar;
     assumption.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   rewrite <- Ropp_inv_permute.
   rewrite Rmult_opp_opp.
   reflexivity.
@@ -1380,7 +1380,7 @@ Proof.
   apply Rplus_lt_compat_l; assumption.
   field; discrR.
   assumption.
-  unfold Rabs in |- *; case (Rcase_abs (Rmax (- (delta / 2)) ((a - c) / 2))).
+  unfold Rabs; case (Rcase_abs (Rmax (- (delta / 2)) ((a - c) / 2))).
   intro; generalize (RmaxLess1 (- (delta / 2)) ((a - c) / 2)); intro;
     generalize
       (Ropp_le_ge_contravar (- (delta / 2)) (Rmax (- (delta / 2)) ((a - c) / 2))
@@ -1390,10 +1390,10 @@ Proof.
   assumption.
   apply Rmult_lt_reg_l with 2.
   prove_sup0.
-  unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
     rewrite <- Rinv_r_sym.
   rewrite Rmult_1_l; rewrite double.
-  pattern delta at 2 in |- *; rewrite <- (Rplus_0_r delta);
+  pattern delta at 2; rewrite <- (Rplus_0_r delta);
     apply Rplus_lt_compat_l; rewrite Rplus_0_r; apply (cond_pos delta).
   discrR.
   cut (- (delta / 2) < 0).
@@ -1401,7 +1401,7 @@ Proof.
   intros;
     generalize
       (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H13)
-        (mknegreal ((a - c) / 2) H12)); simpl in |- *;
+        (mknegreal ((a - c) / 2) H12)); simpl;
       intro; generalize (Rge_le (Rmax (- (delta / 2)) ((a - c) / 2)) 0 r);
         intro;
           elim
@@ -1410,41 +1410,41 @@ Proof.
   rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2));
     apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2).
   assumption.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   rewrite <- Ropp_mult_distr_l_reverse.
   rewrite (Ropp_minus_distr a c).
   reflexivity.
-  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv in |- *;
+  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv;
     apply Rmult_lt_0_compat;
       [ apply (cond_pos delta)
         | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].
-  red in |- *; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10).
+  red; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10).
   cut ((a - c) / 2 < 0).
   intro; cut (- (delta / 2) < 0).
   intro;
     apply
       (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H11)
         (mknegreal ((a - c) / 2) H10)).
-  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv in |- *;
+  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv;
     apply Rmult_lt_0_compat;
       [ apply (cond_pos delta)
         | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].
   rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2));
     apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2).
   assumption.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   rewrite <- Ropp_mult_distr_l_reverse.
   rewrite (Ropp_minus_distr a c).
   reflexivity.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ generalize (Rplus_lt_compat_r (- a) a c H); rewrite Rplus_opp_r; intro;
       assumption
       | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ].
   replace (- (l / 2)) with (- l / 2).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
   assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ].
-  unfold Rdiv in |- *; apply Ropp_mult_distr_l_reverse.
+  unfold Rdiv; apply Ropp_mult_distr_l_reverse.
 Qed.
 
 Theorem deriv_minimum :
@@ -1460,7 +1460,7 @@ Proof.
   cut (forall x:R, a < x -> x < b -> (- f)%F x <= (- f)%F c).
   intro.
   apply (deriv_maximum (- f)%F a b c (derivable_pt_opp _ _ pr) H H0 H2).
-  intros; unfold opp_fct in |- *; apply Ropp_ge_le_contravar; apply Rle_ge.
+  intros; unfold opp_fct; apply Ropp_ge_le_contravar; apply Rle_ge.
   apply (H1 x H2 H3).
 Qed.
 
@@ -1493,7 +1493,7 @@ Proof.
   intro; decompose [and] H7; intros; generalize (H6 (delta / 2) H8 H11);
     cut (0 <= (f (x + delta / 2) - f x) / (delta / 2)).
   intro; cut (0 <= (f (x + delta / 2) - f x) / (delta / 2) - l).
-  intro; unfold Rabs in |- *;
+  intro; unfold Rabs;
     case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)).
   intro;
     elim
@@ -1502,7 +1502,7 @@ Proof.
   intros;
     generalize
       (Rplus_lt_compat_r l ((f (x + delta / 2) - f x) / (delta / 2) - l)
-        (- (l / 2)) H13); unfold Rminus in |- *;
+        (- (l / 2)) H13); unfold Rminus;
       replace (- (l / 2) + l) with (l / 2).
   rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; intro;
     generalize
@@ -1512,50 +1512,50 @@ Proof.
   rewrite <- Ropp_0 in H5;
     generalize (Ropp_lt_gt_contravar (-0) (- (l / 2)) H5);
       repeat rewrite Ropp_involutive; intro; assumption.
-  pattern l at 3 in |- *; rewrite double_var.
+  pattern l at 3; rewrite double_var.
   ring.
-  unfold Rminus in |- *; apply Rplus_le_le_0_compat.
-  unfold Rdiv in |- *; apply Rmult_le_pos.
+  unfold Rminus; apply Rplus_le_le_0_compat.
+  unfold Rdiv; apply Rmult_le_pos.
   cut (x <= x + delta * / 2).
   intro; generalize (H x (x + delta * / 2) H12); intro;
     generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H13);
       rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
-  pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
+  pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
     left; assumption.
   left; apply Rinv_0_lt_compat; assumption.
   left; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption.
-  unfold Rdiv in |- *; apply Rmult_le_pos.
+  unfold Rdiv; apply Rmult_le_pos.
   cut (x <= x + delta * / 2).
   intro; generalize (H x (x + delta * / 2) H9); intro;
     generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H12);
       rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption.
-  pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
+  pattern x at 1; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l;
     left; assumption.
   left; apply Rinv_0_lt_compat; assumption.
   split.
-  unfold Rdiv in |- *; apply prod_neq_R0.
-  generalize (cond_pos delta); intro; red in |- *; intro H9; rewrite H9 in H7;
+  unfold Rdiv; apply prod_neq_R0.
+  generalize (cond_pos delta); intro; red; intro H9; rewrite H9 in H7;
     elim (Rlt_irrefl 0 H7).
   apply Rinv_neq_0_compat; discrR.
   split.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
   replace (Rabs (delta / 2)) with (delta / 2).
-  unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2.
+  unfold Rdiv; apply Rmult_lt_reg_l with 2.
   prove_sup0.
   rewrite (Rmult_comm 2).
   rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].
   rewrite Rmult_1_r.
   rewrite double.
-  pattern (pos delta) at 1 in |- *; rewrite <- Rplus_0_r.
+  pattern (pos delta) at 1; rewrite <- Rplus_0_r.
   apply Rplus_lt_compat_l; apply (cond_pos delta).
-  symmetry  in |- *; apply Rabs_right.
-  left; change (0 < delta / 2) in |- *; unfold Rdiv in |- *;
+  symmetry ; apply Rabs_right.
+  left; change (0 < delta / 2); unfold Rdiv;
     apply Rmult_lt_0_compat;
       [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ].
-  unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse;
+  unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse;
     apply Rmult_lt_0_compat.
   apply Rplus_lt_reg_r with l.
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r; assumption.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rplus_0_r; assumption.
   apply Rinv_0_lt_compat; prove_sup0.
 Qed.
diff --git a/theories/Reals/Ranalysis2.v b/theories/Reals/Ranalysis2.v
index ed80ac43..3c15a305 100644
--- a/theories/Reals/Ranalysis2.v
+++ b/theories/Reals/Ranalysis2.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,7 +9,7 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import Ranalysis1.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (**********)
 Lemma formule :
@@ -24,7 +24,7 @@ Lemma formule :
     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).
 Proof.
-  intros; unfold Rdiv, Rminus, Rsqr in |- *.
+  intros; unfold Rdiv, Rminus, Rsqr.
   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));
@@ -81,10 +81,10 @@ Proof.
   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;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ prove_sup0 | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ].
   exact H8.
-  right; unfold Rdiv in |- *.
+  right; unfold Rdiv.
   repeat rewrite Rabs_mult.
   rewrite Rabs_Rinv; discrR.
   replace (Rabs 8) with 8.
@@ -96,8 +96,8 @@ Proof.
   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.
+  symmetry ; apply Rabs_right; left; assumption.
+  symmetry ; apply Rabs_right; left; prove_sup.
 Qed.
 
 Lemma maj_term2 :
@@ -129,11 +129,11 @@ Proof.
     (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;
+  unfold Rdiv; unfold Rsqr; repeat apply prod_neq_R0;
     try assumption || discrR.
-  red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
+  red; 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 |- *.
+  unfold Rdiv.
   repeat rewrite Rinv_mult_distr; try assumption.
   repeat rewrite Rabs_mult.
   replace (Rabs 2) with 2.
@@ -147,9 +147,9 @@ Proof.
   repeat rewrite Rabs_Rinv; try assumption.
   rewrite <- (Rmult_comm 2).
   unfold Rdiv in H8; exact H8.
-  symmetry  in |- *; apply Rabs_right; left; prove_sup0.
+  symmetry ; apply Rabs_right; left; prove_sup0.
   right.
-  unfold Rsqr, Rdiv in |- *.
+  unfold Rsqr, Rdiv.
   do 1 rewrite Rinv_mult_distr; try assumption || discrR.
   do 1 rewrite Rinv_mult_distr; try assumption || discrR.
   repeat rewrite Rabs_mult.
@@ -166,9 +166,9 @@ Proof.
     (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.
+  symmetry ; apply Rabs_right; left; prove_sup0.
+  symmetry ; apply Rabs_right; left; prove_sup.
+  symmetry ; apply Rabs_right; left; assumption.
 Qed.
 
 Lemma maj_term3 :
@@ -204,11 +204,11 @@ Proof.
     (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;
+  unfold Rdiv; unfold Rsqr; repeat apply prod_neq_R0;
     try assumption.
-  red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
+  red; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H).
   apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   repeat rewrite Rinv_mult_distr; try assumption.
   repeat rewrite Rabs_mult.
   replace (Rabs 2) with 2.
@@ -222,9 +222,9 @@ Proof.
   repeat rewrite Rabs_Rinv; assumption || idtac.
   rewrite <- (Rmult_comm 2).
   unfold Rdiv in H9; exact H9.
-  symmetry  in |- *; apply Rabs_right; left; prove_sup0.
+  symmetry ; apply Rabs_right; left; prove_sup0.
   right.
-  unfold Rsqr, Rdiv in |- *.
+  unfold Rsqr, Rdiv.
   rewrite Rinv_mult_distr; try assumption || discrR.
   rewrite Rinv_mult_distr; try assumption || discrR.
   repeat rewrite Rabs_mult.
@@ -241,9 +241,9 @@ Proof.
     (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.
+  symmetry ; apply Rabs_right; left; prove_sup0.
+  symmetry ; apply Rabs_right; left; prove_sup.
+  symmetry ; apply Rabs_right; left; assumption.
 Qed.
 
 Lemma maj_term4 :
@@ -281,17 +281,17 @@ Proof.
       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;
+  unfold Rdiv; unfold Rsqr; repeat apply prod_neq_R0;
     assumption || idtac.
-  red in |- *; intro H11; rewrite H11 in H; elim (Rlt_irrefl _ H).
+  red; 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 |- *.
+  unfold Rdiv.
   repeat rewrite Rinv_mult_distr;
-    try assumption || (unfold Rsqr in |- *; apply prod_neq_R0; assumption).
+    try assumption || (unfold Rsqr; apply prod_neq_R0; assumption).
   repeat rewrite Rabs_mult.
   replace (Rabs 2) with 2.
   replace
@@ -305,13 +305,13 @@ Proof.
   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 Rabs_pos_lt; apply Rinv_neq_0_compat; unfold Rsqr;
     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 |- *.
+  symmetry ; apply Rabs_right; left; prove_sup0.
+  right; unfold Rsqr, Rdiv.
   rewrite Rinv_mult_distr; try assumption || discrR.
   rewrite Rinv_mult_distr; try assumption || discrR.
   rewrite Rinv_mult_distr; try assumption || discrR.
@@ -333,9 +333,9 @@ Proof.
     (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.
+  symmetry ; apply Rabs_right; left; prove_sup0.
+  symmetry ; apply Rabs_right; left; prove_sup.
+  symmetry ; apply Rabs_right; left; assumption.
   apply prod_neq_R0; assumption || discrR.
   apply prod_neq_R0; assumption.
 Qed.
@@ -343,11 +343,11 @@ Qed.
 Lemma D_x_no_cond : forall x a:R, a <> 0 -> D_x no_cond x (x + a).
 Proof.
   intros.
-  unfold D_x, no_cond in |- *.
+  unfold D_x, no_cond.
   split.
   trivial.
   apply Rminus_not_eq.
-  unfold Rminus in |- *.
+  unfold Rminus.
   rewrite Ropp_plus_distr.
   rewrite <- Rplus_assoc.
   rewrite Rplus_opp_r.
@@ -394,7 +394,7 @@ Qed.
 Lemma quadruple_var : forall x:R, x = x / 4 + x / 4 + x / 4 + x / 4.
 Proof.
   intro; rewrite <- quadruple.
-  unfold Rdiv in |- *; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; discrR.
+  unfold Rdiv; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; discrR.
   reflexivity.
 Qed.
 
@@ -413,10 +413,10 @@ Proof.
   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 |- *;
+  unfold dist; simpl; unfold R_dist;
     replace (x0 + h - x0) with h.
   intros; assert (H7 := H6 H4).
-  red in |- *; intro.
+  red; 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.
@@ -429,10 +429,10 @@ Proof.
     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;
+  unfold IZR; unfold INR, Pos.to_nat; simpl; intro;
     elim (Rlt_irrefl 1 (Rlt_trans _ _ _ H13 H12)).
   apply IZR_lt; omega.
-  unfold Rabs in |- *; case (Rcase_abs (/ 2)); intro.
+  unfold Rabs; 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;
@@ -442,18 +442,18 @@ Proof.
   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;
+  intro; rewrite <- H7; unfold dist, R_met; unfold R_dist;
+    unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
       apply Rabs_pos_lt.
-  unfold Rdiv in |- *; apply prod_neq_R0;
+  unfold Rdiv; apply prod_neq_R0;
     [ assumption | apply Rinv_neq_0_compat; discrR ].
   intro; apply H5.
   split.
-  unfold D_x, no_cond in |- *.
+  unfold D_x, no_cond.
   split; trivial || assumption.
   assumption.
-  change (0 < Rabs (f x0 / 2)) in |- *.
-  apply Rabs_pos_lt; unfold Rdiv in |- *; apply prod_neq_R0.
+  change (0 < Rabs (f x0 / 2)).
+  apply Rabs_pos_lt; unfold Rdiv; apply prod_neq_R0.
   assumption.
   apply Rinv_neq_0_compat; discrR.
 Qed.
diff --git a/theories/Reals/Ranalysis3.v b/theories/Reals/Ranalysis3.v
index afd4a4ee..5eaf5a57 100644
--- a/theories/Reals/Ranalysis3.v
+++ b/theories/Reals/Ranalysis3.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -10,7 +10,7 @@ Require Import Rbase.
 Require Import Rfunctions.
 Require Import Ranalysis1.
 Require Import Ranalysis2.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (** Division *)
 Theorem derivable_pt_lim_div :
@@ -22,17 +22,17 @@ Theorem derivable_pt_lim_div :
 Proof.
   intros f1 f2 x l1 l2 H H0 H1.
   cut (derivable_pt f2 x);
-    [ intro X | unfold derivable_pt in |- *; exists l2; exact H0 ].
+    [ intro X | unfold derivable_pt; exists l2; exact H0 ].
   assert (H2 := continuous_neq_0 _ _ (derivable_continuous_pt _ _ X) H1).
   elim H2; clear H2; intros eps_f2 H2.
-  unfold div_fct in |- *.
+  unfold div_fct.
   assert (H3 := derivable_continuous_pt _ _ X).
   unfold continuity_pt in H3; unfold continue_in in H3; unfold limit1_in in H3;
     unfold limit_in in H3; unfold dist in H3.
   simpl in H3; unfold R_dist in H3.
   elim (H3 (Rabs (f2 x) / 2));
     [ idtac
-      | unfold Rdiv in |- *; change (0 < Rabs (f2 x) * / 2) in |- *;
+      | unfold Rdiv; change (0 < Rabs (f2 x) * / 2);
         apply Rmult_lt_0_compat;
           [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
   clear H3; intros alp_f2 H3.
@@ -46,12 +46,12 @@ Proof.
     (forall a:R,
       Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)).
   intro Maj.
-  unfold derivable_pt_lim in |- *; intros.
+  unfold derivable_pt_lim; intros.
   elim (H (Rabs (eps * f2 x / 8)));
     [ idtac
-      | unfold Rdiv in |- *; change (0 < Rabs (eps * f2 x * / 8)) in |- *;
+      | unfold Rdiv; change (0 < Rabs (eps * f2 x * / 8));
         apply Rabs_pos_lt; repeat apply prod_neq_R0;
-          [ red in |- *; intro H7; rewrite H7 in H6; elim (Rlt_irrefl _ H6)
+          [ red; intro H7; rewrite H7 in H6; elim (Rlt_irrefl _ H6)
             | assumption
             | apply Rinv_neq_0_compat; discrR ] ].
   intros alp_f1d H7.
@@ -68,7 +68,7 @@ Proof.
           | elim H3; intros; assumption
           | apply (cond_pos alp_f1d) ] ].
   exists (mkposreal (Rmin eps_f2 (Rmin alp_f2 alp_f1d)) H10).
-  simpl in |- *; intros.
+  simpl; intros.
   assert (H13 := Rlt_le_trans _ _ _ H12 (Rmin_r _ _)).
   assert (H14 := Rlt_le_trans _ _ _ H12 (Rmin_l _ _)).
   assert (H15 := Rlt_le_trans _ _ _ H13 (Rmin_r _ _)).
@@ -80,7 +80,7 @@ Proof.
       Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
       Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
       Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
-  unfold Rminus in |- *.
+  unfold Rminus.
   rewrite <-
     (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
     .
@@ -98,15 +98,15 @@ Proof.
   intros.
   apply Rlt_4; assumption.
   rewrite H8.
-  unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
+  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
   rewrite Rabs_R0; rewrite Rmult_0_l.
   apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
   rewrite H8.
-  unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
+  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
   rewrite Rabs_R0; rewrite Rmult_0_l.
   apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
   rewrite H9.
-  unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
+  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
   rewrite Rabs_R0; rewrite Rmult_0_l.
   apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
   rewrite <- Rabs_mult.
@@ -114,7 +114,7 @@ Proof.
     try assumption || apply H2.
   apply H14.
   apply Rmin_2; assumption.
-  right; symmetry  in |- *; apply quadruple_var.
+  right; symmetry ; apply quadruple_var.
 (***********************************)
 (*           Second case           *)
 (*           (f1 x)=0  l1<>0       *)
@@ -137,7 +137,7 @@ Proof.
   cut (0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)).
   intro.
   exists (mkposreal (Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)) H12).
-  simpl in |- *.
+  simpl.
   intros.
   assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)).
   assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)).
@@ -152,7 +152,7 @@ Proof.
       Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
       Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
       Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
-  unfold Rminus in |- *.
+  unfold Rminus.
   rewrite <-
     (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
     .
@@ -170,11 +170,11 @@ Proof.
   intros.
   apply Rlt_4; assumption.
   rewrite H8.
-  unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
+  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
   rewrite Rabs_R0; rewrite Rmult_0_l.
   apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
   rewrite H8.
-  unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
+  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
   rewrite Rabs_R0; rewrite Rmult_0_l.
   apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
   rewrite <- Rabs_mult.
@@ -185,7 +185,7 @@ Proof.
   apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.
   apply H2; assumption.
   apply Rmin_2; assumption.
-  right; symmetry  in |- *; apply quadruple_var.
+  right; symmetry ; apply quadruple_var.
   apply H2; assumption.
   repeat apply Rmin_pos.
   apply (cond_pos eps_f2).
@@ -196,21 +196,21 @@ Proof.
   elim H10; intros.
   case (Req_dec a 0); intro.
   rewrite H14; rewrite Rplus_0_r.
-  unfold Rminus in |- *; rewrite Rplus_opp_r.
+  unfold Rminus; rewrite Rplus_opp_r.
   rewrite Rabs_R0.
   apply Rabs_pos_lt.
-  unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc.
+  unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc.
   repeat apply prod_neq_R0; try assumption.
-  red in |- *; intro; rewrite H15 in H6; elim (Rlt_irrefl _ H6).
+  red; intro; rewrite H15 in H6; elim (Rlt_irrefl _ H6).
   apply Rinv_neq_0_compat; repeat apply prod_neq_R0; discrR || assumption.
   apply H13.
   split.
   apply D_x_no_cond; assumption.
   replace (x + a - x) with a; [ assumption | ring ].
-  change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *.
-  apply Rabs_pos_lt; unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc;
+  change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))).
+  apply Rabs_pos_lt; unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc;
     repeat apply prod_neq_R0.
-  red in |- *; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6).
+  red; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6).
   assumption.
   assumption.
   apply Rinv_neq_0_compat; repeat apply prod_neq_R0;
@@ -223,17 +223,17 @@ Proof.
   case (Req_dec l2 0); intro.
   elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x))));
     [ idtac
-      | apply Rabs_pos_lt; unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc;
+      | apply Rabs_pos_lt; unfold Rdiv, Rsqr; repeat rewrite Rmult_assoc;
         repeat apply prod_neq_R0;
           [ assumption
             | assumption
-            | red in |- *; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6)
+            | red; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6)
             | apply Rinv_neq_0_compat; repeat apply prod_neq_R0; discrR || assumption ] ].
   intros alp_f2d H12.
   cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)).
   intro.
   exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) H11).
-  simpl in |- *.
+  simpl.
   intros.
   assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)).
   assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)).
@@ -248,7 +248,7 @@ Proof.
       Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
       Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
       Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
-  unfold Rminus in |- *.
+  unfold Rminus.
   rewrite <-
     (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
     .
@@ -266,7 +266,7 @@ Proof.
   intros.
   apply Rlt_4; assumption.
   rewrite H10.
-  unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
+  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
   rewrite Rabs_R0; rewrite Rmult_0_l.
   apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
   rewrite <- Rabs_mult.
@@ -274,14 +274,14 @@ Proof.
   apply H2; assumption.
   apply Rmin_2; assumption.
   rewrite H9.
-  unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
+  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
   rewrite Rabs_R0; rewrite Rmult_0_l.
   apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
   rewrite <- Rabs_mult.
   apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); assumption || idtac.
   apply H2; assumption.
   apply Rmin_2; assumption.
-  right; symmetry  in |- *; apply quadruple_var.
+  right; symmetry ; apply quadruple_var.
   apply H2; assumption.
   repeat apply Rmin_pos.
   apply (cond_pos eps_f2).
@@ -294,7 +294,7 @@ Proof.
 (***********************************)
   elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x))));
     [ idtac
-      | apply Rabs_pos_lt; unfold Rsqr, Rdiv in |- *;
+      | apply Rabs_pos_lt; unfold Rsqr, Rdiv;
         repeat rewrite Rinv_mult_distr; repeat apply prod_neq_R0;
           try assumption || discrR ].
   intros alp_f2d H11.
@@ -313,7 +313,7 @@ Proof.
   exists
     (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c)))
       H14).
-  simpl in |- *; intros.
+  simpl; intros.
   assert (H17 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)).
   assert (H18 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)).
   assert (H19 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)).
@@ -335,7 +335,7 @@ Proof.
       Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
       Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
       Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
-  unfold Rminus in |- *.
+  unfold Rminus.
   rewrite <-
     (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
     .
@@ -361,24 +361,24 @@ Proof.
   apply H2; assumption.
   apply Rmin_2; assumption.
   rewrite H9.
-  unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
+  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
   rewrite Rabs_R0; rewrite Rmult_0_l.
   apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
   rewrite <- Rabs_mult.
   apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.
   apply H2; assumption.
   apply Rmin_2; assumption.
-  right; symmetry  in |- *; apply quadruple_var.
+  right; symmetry ; apply quadruple_var.
   apply H2; assumption.
   intros.
   case (Req_dec a 0); intro.
   rewrite H17; rewrite Rplus_0_r.
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0.
   apply Rabs_pos_lt.
-  unfold Rdiv, Rsqr in |- *.
+  unfold Rdiv, Rsqr.
   repeat rewrite Rinv_mult_distr; try assumption.
   repeat apply prod_neq_R0; try assumption.
-  red in |- *; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6).
+  red; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6).
   apply Rinv_neq_0_compat; discrR.
   apply Rinv_neq_0_compat; discrR.
   apply Rinv_neq_0_compat; discrR.
@@ -401,19 +401,19 @@ Proof.
   apply (cond_pos alp_f1d).
   apply (cond_pos alp_f2d).
   elim H13; intros; assumption.
-  change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) in |- *.
+  change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))).
   apply Rabs_pos_lt.
-  unfold Rsqr, Rdiv in |- *.
+  unfold Rsqr, Rdiv.
   repeat rewrite Rinv_mult_distr; try assumption || discrR.
   repeat apply prod_neq_R0; try assumption.
-  red in |- *; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6).
+  red; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6).
   apply Rinv_neq_0_compat; discrR.
   apply Rinv_neq_0_compat; discrR.
   apply Rinv_neq_0_compat; discrR.
   apply Rinv_neq_0_compat; assumption.
   apply Rinv_neq_0_compat; assumption.
   apply prod_neq_R0; [ discrR | assumption ].
-  red in |- *; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6).
+  red; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6).
   apply Rinv_neq_0_compat; discrR.
   apply Rinv_neq_0_compat; discrR.
   apply Rinv_neq_0_compat; discrR.
@@ -440,7 +440,7 @@ Proof.
   exists
     (mkposreal
       (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))) H13).
-  simpl in |- *.
+  simpl.
   intros.
   cut
     (forall a:R,
@@ -462,7 +462,7 @@ Proof.
       Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
       Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
       Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
-  unfold Rminus in |- *.
+  unfold Rminus.
   rewrite <-
     (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
     .
@@ -480,7 +480,7 @@ Proof.
   intros.
   apply Rlt_4; assumption.
   rewrite H10.
-  unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
+  unfold Rdiv; repeat rewrite Rmult_0_r || rewrite Rmult_0_l.
   rewrite Rabs_R0; rewrite Rmult_0_l.
   apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ].
   rewrite <- Rabs_mult.
@@ -495,20 +495,20 @@ Proof.
   apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.
   apply H2; assumption.
   apply Rmin_2; assumption.
-  right; symmetry  in |- *; apply quadruple_var.
+  right; symmetry ; apply quadruple_var.
   apply H2; assumption.
   intros.
   case (Req_dec a 0); intro.
-  rewrite H17; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r;
+  rewrite H17; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r;
     rewrite Rabs_R0.
   apply Rabs_pos_lt.
-  unfold Rdiv in |- *; rewrite Rinv_mult_distr; try discrR || assumption.
-  unfold Rsqr in |- *.
+  unfold Rdiv; rewrite Rinv_mult_distr; try discrR || assumption.
+  unfold Rsqr.
   repeat apply prod_neq_R0;
     assumption ||
       (apply Rinv_neq_0_compat; assumption) ||
         (apply Rinv_neq_0_compat; discrR) ||
-          (red in |- *; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6)).
+          (red; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6)).
   elim H11; intros.
   apply H19.
   split.
@@ -521,20 +521,20 @@ Proof.
   apply (cond_pos alp_f2d).
   elim H11; intros; assumption.
   apply Rabs_pos_lt.
-  unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption.
+  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr; try discrR || assumption.
   repeat apply prod_neq_R0;
     assumption ||
       (apply Rinv_neq_0_compat; assumption) ||
         (apply Rinv_neq_0_compat; discrR) ||
-          (red in |- *; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)).
-  change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *.
+          (red; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)).
+  change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))).
   apply Rabs_pos_lt.
-  unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption.
+  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr; try discrR || assumption.
   repeat apply prod_neq_R0;
     assumption ||
       (apply Rinv_neq_0_compat; assumption) ||
         (apply Rinv_neq_0_compat; discrR) ||
-          (red in |- *; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)).
+          (red; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)).
 (***********************************)
 (*       Sixth case                *)
 (*    (f1 x)<>0  l1<>0  l2<>0      *)
@@ -562,7 +562,7 @@ Proof.
     (mkposreal
       (Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))
         (Rmin alp_f2c alp_f2t2)) H15).
-  simpl in |- *.
+  simpl.
   intros.
   assert (H18 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)).
   assert (H19 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)).
@@ -591,7 +591,7 @@ Proof.
       Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) +
       Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) +
       Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))).
-  unfold Rminus in |- *.
+  unfold Rminus.
   rewrite <-
     (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2)))
     .
@@ -624,18 +624,18 @@ Proof.
   apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption.
   apply H2; assumption.
   apply Rmin_2; assumption.
-  right; symmetry  in |- *; apply quadruple_var.
+  right; symmetry ; apply quadruple_var.
   apply H2; assumption.
   intros.
   case (Req_dec a 0); intro.
-  rewrite H18; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r;
+  rewrite H18; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r;
     rewrite Rabs_R0; apply Rabs_pos_lt.
-  unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr.
+  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr.
   repeat apply prod_neq_R0;
     assumption ||
       (apply Rinv_neq_0_compat; assumption) ||
         (apply Rinv_neq_0_compat; discrR) ||
-          (red in |- *; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)).
+          (red; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)).
   apply prod_neq_R0; [ discrR | assumption ].
   apply prod_neq_R0; [ discrR | assumption ].
   assumption.
@@ -646,20 +646,20 @@ Proof.
   replace (x + a - x) with a; [ assumption | ring ].
   intros.
   case (Req_dec a 0); intro.
-  rewrite H18; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r;
+  rewrite H18; rewrite Rplus_0_r; unfold Rminus; rewrite Rplus_opp_r;
     rewrite Rabs_R0; apply Rabs_pos_lt.
-  unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr.
+  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr.
   repeat apply prod_neq_R0;
     assumption ||
       (apply Rinv_neq_0_compat; assumption) ||
         (apply Rinv_neq_0_compat; discrR) ||
-          (red in |- *; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)).
+          (red; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)).
   discrR.
   assumption.
   elim H14; intros.
   apply H20.
   split.
-  unfold D_x, no_cond in |- *; split.
+  unfold D_x, no_cond; split.
   trivial.
   apply Rminus_not_eq_right.
   replace (x + a - x) with a; [ assumption | ring ].
@@ -671,34 +671,34 @@ Proof.
   apply (cond_pos alp_f2d).
   elim H13; intros; assumption.
   elim H14; intros; assumption.
-  change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *; apply Rabs_pos_lt.
-  unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption.
+  change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))); apply Rabs_pos_lt.
+  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr; try discrR || assumption.
   repeat apply prod_neq_R0;
     assumption ||
       (apply Rinv_neq_0_compat; assumption) ||
         (apply Rinv_neq_0_compat; discrR) ||
-          (red in |- *; intro H14; rewrite H14 in H6; elim (Rlt_irrefl _ H6)).
-  change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) in |- *;
+          (red; intro H14; rewrite H14 in H6; elim (Rlt_irrefl _ H6)).
+  change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)));
     apply Rabs_pos_lt.
-  unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr.
+  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr.
   repeat apply prod_neq_R0;
     assumption ||
       (apply Rinv_neq_0_compat; assumption) ||
         (apply Rinv_neq_0_compat; discrR) ||
-          (red in |- *; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6)).
+          (red; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6)).
   apply prod_neq_R0; [ discrR | assumption ].
   apply prod_neq_R0; [ discrR | assumption ].
   assumption.
   apply Rabs_pos_lt.
-  unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr;
+  unfold Rdiv, Rsqr; rewrite Rinv_mult_distr;
     [ idtac | discrR | assumption ].
   repeat apply prod_neq_R0;
     assumption ||
       (apply Rinv_neq_0_compat; assumption) ||
         (apply Rinv_neq_0_compat; discrR) ||
-          (red in |- *; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6)).
+          (red; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6)).
   intros.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   apply Rmult_lt_reg_l with (Rabs (f2 (x + a))).
   apply Rabs_pos_lt; apply H2.
   apply Rlt_le_trans with (Rmin eps_f2 alp_f2).
@@ -739,13 +739,13 @@ Proof.
   unfold Rminus in H7; assumption.
   intros.
   case (Req_dec x x0); intro.
-  rewrite <- H5; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
-    unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  rewrite <- H5; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
+    unfold Rdiv; apply Rmult_lt_0_compat;
       [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   elim H3; intros.
   apply H7.
   split.
-  unfold D_x, no_cond in |- *; split.
+  unfold D_x, no_cond; split.
   trivial.
   assumption.
   assumption.
@@ -756,7 +756,7 @@ Lemma derivable_pt_div :
     derivable_pt f1 x ->
     derivable_pt f2 x -> f2 x <> 0 -> derivable_pt (f1 / f2) x.
 Proof.
-  unfold derivable_pt in |- *.
+  unfold derivable_pt.
   intros f1 f2 x X X0 H.
   elim X; intros.
   elim X0; intros.
@@ -769,7 +769,7 @@ Lemma derivable_div :
     derivable f1 ->
     derivable f2 -> (forall x:R, f2 x <> 0) -> derivable (f1 / f2).
 Proof.
-  unfold derivable in |- *; intros f1 f2 X X0 H x.
+  unfold derivable; intros f1 f2 X X0 H x.
   apply (derivable_pt_div _ _ _ (X x) (X0 x) (H x)).
 Qed.
 
diff --git a/theories/Reals/Ranalysis4.v b/theories/Reals/Ranalysis4.v
index cc658fee..00c07592 100644
--- a/theories/Reals/Ranalysis4.v
+++ b/theories/Reals/Ranalysis4.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,11 +9,11 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
-Require Import Rtrigo.
+Require Import Rtrigo1.
 Require Import Ranalysis1.
 Require Import Ranalysis3.
 Require Import Exp_prop.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (**********)
 Lemma derivable_pt_inv :
@@ -26,12 +26,12 @@ Proof.
   apply derivable_pt_const.
   assumption.
   assumption.
-  unfold div_fct, inv_fct, fct_cte in |- *; intro X0; elim X0; intros;
-    unfold derivable_pt in |- *; exists x0;
-      unfold derivable_pt_abs in |- *; unfold derivable_pt_lim in |- *;
+  unfold div_fct, inv_fct, fct_cte; intro X0; elim X0; intros;
+    unfold derivable_pt; exists x0;
+      unfold derivable_pt_abs; unfold derivable_pt_lim;
         unfold derivable_pt_abs in p; unfold derivable_pt_lim in p;
           intros; elim (p eps H0); intros; exists x1; intros;
-            unfold Rdiv in H1; unfold Rdiv in |- *; rewrite <- (Rmult_1_l (/ f x));
+            unfold Rdiv in H1; unfold Rdiv; rewrite <- (Rmult_1_l (/ f x));
               rewrite <- (Rmult_1_l (/ f (x + h))).
   apply H1; assumption.
 Qed.
@@ -41,10 +41,10 @@ Lemma pr_nu_var :
   forall (f g:R -> R) (x:R) (pr1:derivable_pt f x) (pr2:derivable_pt g x),
     f = g -> derive_pt f x pr1 = derive_pt g x pr2.
 Proof.
-  unfold derivable_pt, derive_pt in |- *; intros.
+  unfold derivable_pt, derive_pt; intros.
   elim pr1; intros.
   elim pr2; intros.
-  simpl in |- *.
+  simpl.
   rewrite H in p.
   apply uniqueness_limite with g x; assumption.
 Qed.
@@ -54,17 +54,17 @@ Lemma pr_nu_var2 :
   forall (f g:R -> R) (x:R) (pr1:derivable_pt f x) (pr2:derivable_pt g x),
     (forall h:R, f h = g h) -> derive_pt f x pr1 = derive_pt g x pr2.
 Proof.
-  unfold derivable_pt, derive_pt in |- *; intros.
+  unfold derivable_pt, derive_pt; intros.
   elim pr1; intros.
   elim pr2; intros.
-  simpl in |- *.
+  simpl.
   assert (H0 := uniqueness_step2 _ _ _ p).
   assert (H1 := uniqueness_step2 _ _ _ p0).
   cut (limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) x1 0).
   intro; assert (H3 := uniqueness_step1 _ _ _ _ H0 H2).
   assumption.
-  unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *;
-    simpl in |- *; unfold R_dist in |- *; unfold limit1_in in H1;
+  unfold limit1_in; unfold limit_in; unfold dist;
+    simpl; unfold R_dist; unfold limit1_in in H1;
       unfold limit_in in H1; unfold dist in H1; simpl in H1;
         unfold R_dist in H1.
   intros; elim (H1 eps H2); intros.
@@ -80,7 +80,7 @@ Lemma derivable_inv :
   forall f:R -> R, (forall x:R, f x <> 0) -> derivable f -> derivable (/ f).
 Proof.
   intros f H X.
-  unfold derivable in |- *; intro x.
+  unfold derivable; intro x.
   apply derivable_pt_inv.
   apply (H x).
   apply (X x).
@@ -95,25 +95,25 @@ Proof.
     replace (derive_pt (/ f) x (derivable_pt_inv f x na pr)) with
     (derive_pt (fct_cte 1 / f) x
       (derivable_pt_div (fct_cte 1) f x (derivable_pt_const 1 x) pr na)).
-  rewrite derive_pt_div; rewrite derive_pt_const; unfold fct_cte in |- *;
-    rewrite Rmult_0_l; rewrite Rmult_1_r; unfold Rminus in |- *;
+  rewrite derive_pt_div; rewrite derive_pt_const; unfold fct_cte;
+    rewrite Rmult_0_l; rewrite Rmult_1_r; unfold Rminus;
       rewrite Rplus_0_l; reflexivity.
   apply pr_nu_var2.
-  intro; unfold div_fct, fct_cte, inv_fct in |- *.
-  unfold Rdiv in |- *; ring.
+  intro; unfold div_fct, fct_cte, inv_fct.
+  unfold Rdiv; ring.
 Qed.
 
 (** Rabsolu *)
 Lemma Rabs_derive_1 : forall x:R, 0 < x -> derivable_pt_lim Rabs x 1.
 Proof.
   intros.
-  unfold derivable_pt_lim in |- *; intros.
+  unfold derivable_pt_lim; intros.
   exists (mkposreal x H); intros.
   rewrite (Rabs_right x).
   rewrite (Rabs_right (x + h)).
   rewrite Rplus_comm.
-  unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_r.
-  rewrite Rplus_0_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym.
+  unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_r.
+  rewrite Rplus_0_r; unfold Rdiv; rewrite <- Rinv_r_sym.
   rewrite Rplus_opp_r; rewrite Rabs_R0; apply H0.
   apply H1.
   apply Rle_ge.
@@ -131,16 +131,16 @@ Qed.
 Lemma Rabs_derive_2 : forall x:R, x < 0 -> derivable_pt_lim Rabs x (-1).
 Proof.
   intros.
-  unfold derivable_pt_lim in |- *; intros.
+  unfold derivable_pt_lim; intros.
   cut (0 < - x).
   intro; exists (mkposreal (- x) H1); intros.
   rewrite (Rabs_left x).
   rewrite (Rabs_left (x + h)).
   rewrite Rplus_comm.
   rewrite Ropp_plus_distr.
-  unfold Rminus in |- *; rewrite Ropp_involutive; rewrite Rplus_assoc;
+  unfold Rminus; rewrite Ropp_involutive; rewrite Rplus_assoc;
     rewrite Rplus_opp_l.
-  rewrite Rplus_0_r; unfold Rdiv in |- *.
+  rewrite Rplus_0_r; unfold Rdiv.
   rewrite Ropp_mult_distr_l_reverse.
   rewrite <- Rinv_r_sym.
   rewrite Ropp_involutive; rewrite Rplus_opp_l; rewrite Rabs_R0; apply H0.
@@ -163,24 +163,24 @@ Proof.
   intros.
   case (total_order_T x 0); intro.
   elim s; intro.
-  unfold derivable_pt in |- *; exists (-1).
+  unfold derivable_pt; exists (-1).
   apply (Rabs_derive_2 x a).
   elim H; exact b.
-  unfold derivable_pt in |- *; exists 1.
+  unfold derivable_pt; exists 1.
   apply (Rabs_derive_1 x r).
 Qed.
 
 (** Rabsolu is continuous for all x *)
 Lemma Rcontinuity_abs : continuity Rabs.
 Proof.
-  unfold continuity in |- *; intro.
+  unfold continuity; intro.
   case (Req_dec x 0); intro.
-  unfold continuity_pt in |- *; unfold continue_in in |- *;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
-      simpl in |- *; unfold R_dist in |- *; intros; exists eps;
+  unfold continuity_pt; unfold continue_in;
+    unfold limit1_in; unfold limit_in;
+      simpl; unfold R_dist; intros; exists eps;
         split.
   apply H0.
-  intros; rewrite H; rewrite Rabs_R0; unfold Rminus in |- *; rewrite Ropp_0;
+  intros; rewrite H; rewrite Rabs_R0; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; rewrite Rabs_Rabsolu; elim H1;
       intros; rewrite H in H3; unfold Rminus in H3; rewrite Ropp_0 in H3;
         rewrite Rplus_0_r in H3; apply H3.
@@ -192,11 +192,11 @@ Lemma continuity_finite_sum :
   forall (An:nat -> R) (N:nat),
     continuity (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N).
 Proof.
-  intros; unfold continuity in |- *; intro.
+  intros; unfold continuity; intro.
   induction  N as [| N HrecN].
-  simpl in |- *.
+  simpl.
   apply continuity_pt_const.
-  unfold constant in |- *; intros; reflexivity.
+  unfold constant; intros; reflexivity.
   replace (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) (S N)) with
   ((fun y:R => sum_f_R0 (fun k:nat => (An k * y ^ k)%R) N) +
     (fun y:R => (An (S N) * y ^ S N)%R))%F.
@@ -222,7 +222,7 @@ Proof.
   cut (N = 0%nat \/ (0 < N)%nat).
   intro; elim H0; intro.
   rewrite H1.
-  simpl in |- *.
+  simpl.
   replace (fun y:R => An 0%nat * 1 + An 1%nat * (y * 1)) with
   (fct_cte (An 0%nat * 1) + mult_real_fct (An 1%nat) (id * fct_cte 1))%F.
   replace (1 * An 1%nat * 1) with (0 + An 1%nat * (1 * fct_cte 1 x + id x * 0)).
@@ -232,7 +232,7 @@ Proof.
   apply derivable_pt_lim_mult.
   apply derivable_pt_lim_id.
   apply derivable_pt_lim_const.
-  unfold fct_cte, id in |- *; ring.
+  unfold fct_cte, id; ring.
   reflexivity.
   replace (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) (S N)) with
   ((fun y:R => sum_f_R0 (fun k:nat => (An k * y ^ k)%R) N) +
@@ -248,7 +248,7 @@ Proof.
   (mult_real_fct (An (S N)) (fun y:R => y ^ S N)).
   apply derivable_pt_lim_scal.
   replace (pred (S N)) with N; [ idtac | reflexivity ].
-  pattern N at 3 in |- *; replace N with (pred (S N)).
+  pattern N at 3; replace N with (pred (S N)).
   apply derivable_pt_lim_pow.
   reflexivity.
   reflexivity.
@@ -259,10 +259,10 @@ Proof.
   rewrite <- H2.
   replace (pred (S N)) with N; [ idtac | reflexivity ].
   ring.
-  simpl in |- *.
+  simpl.
   apply S_pred with 0%nat; assumption.
-  unfold plus_fct in |- *.
-  simpl in |- *; reflexivity.
+  unfold plus_fct.
+  simpl; reflexivity.
   inversion H.
   left; reflexivity.
   right; apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ].
@@ -278,7 +278,7 @@ Lemma derivable_pt_lim_finite_sum :
 Proof.
   intros.
   induction  N as [| N HrecN].
-  simpl in |- *.
+  simpl.
   rewrite Rmult_1_r.
   replace (fun _:R => An 0%nat) with (fct_cte (An 0%nat));
   [ apply derivable_pt_lim_const | reflexivity ].
@@ -290,7 +290,7 @@ Lemma derivable_pt_finite_sum :
     derivable_pt (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N) x.
 Proof.
   intros.
-  unfold derivable_pt in |- *.
+  unfold derivable_pt.
   assert (H := derivable_pt_lim_finite_sum An x N).
   induction  N as [| N HrecN].
   exists 0; apply H.
@@ -303,14 +303,14 @@ Lemma derivable_finite_sum :
   forall (An:nat -> R) (N:nat),
     derivable (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N).
 Proof.
-  intros; unfold derivable in |- *; intro; apply derivable_pt_finite_sum.
+  intros; unfold derivable; intro; apply derivable_pt_finite_sum.
 Qed.
 
 (** Regularity of hyperbolic functions *)
 Lemma derivable_pt_lim_cosh : forall x:R, derivable_pt_lim cosh x (sinh x).
 Proof.
   intro.
-  unfold cosh, sinh in |- *; unfold Rdiv in |- *.
+  unfold cosh, sinh; unfold Rdiv.
   replace (fun x0:R => (exp x0 + exp (- x0)) * / 2) with
   ((exp + comp exp (- id)) * fct_cte (/ 2))%F; [ idtac | reflexivity ].
   replace ((exp x - exp (- x)) * / 2) with
@@ -324,13 +324,13 @@ Proof.
   apply derivable_pt_lim_id.
   apply derivable_pt_lim_exp.
   apply derivable_pt_lim_const.
-  unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte in |- *; ring.
+  unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte; ring.
 Qed.
 
 Lemma derivable_pt_lim_sinh : forall x:R, derivable_pt_lim sinh x (cosh x).
 Proof.
   intro.
-  unfold cosh, sinh in |- *; unfold Rdiv in |- *.
+  unfold cosh, sinh; unfold Rdiv.
   replace (fun x0:R => (exp x0 - exp (- x0)) * / 2) with
   ((exp - comp exp (- id)) * fct_cte (/ 2))%F; [ idtac | reflexivity ].
   replace ((exp x + exp (- x)) * / 2) with
@@ -344,13 +344,13 @@ Proof.
   apply derivable_pt_lim_id.
   apply derivable_pt_lim_exp.
   apply derivable_pt_lim_const.
-  unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte in |- *; ring.
+  unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte; ring.
 Qed.
 
 Lemma derivable_pt_exp : forall x:R, derivable_pt exp x.
 Proof.
   intro.
-  unfold derivable_pt in |- *.
+  unfold derivable_pt.
   exists (exp x).
   apply derivable_pt_lim_exp.
 Qed.
@@ -358,7 +358,7 @@ Qed.
 Lemma derivable_pt_cosh : forall x:R, derivable_pt cosh x.
 Proof.
   intro.
-  unfold derivable_pt in |- *.
+  unfold derivable_pt.
   exists (sinh x).
   apply derivable_pt_lim_cosh.
 Qed.
@@ -366,24 +366,24 @@ Qed.
 Lemma derivable_pt_sinh : forall x:R, derivable_pt sinh x.
 Proof.
   intro.
-  unfold derivable_pt in |- *.
+  unfold derivable_pt.
   exists (cosh x).
   apply derivable_pt_lim_sinh.
 Qed.
 
 Lemma derivable_exp : derivable exp.
 Proof.
-  unfold derivable in |- *; apply derivable_pt_exp.
+  unfold derivable; apply derivable_pt_exp.
 Qed.
 
 Lemma derivable_cosh : derivable cosh.
 Proof.
-  unfold derivable in |- *; apply derivable_pt_cosh.
+  unfold derivable; apply derivable_pt_cosh.
 Qed.
 
 Lemma derivable_sinh : derivable sinh.
 Proof.
-  unfold derivable in |- *; apply derivable_pt_sinh.
+  unfold derivable; apply derivable_pt_sinh.
 Qed.
 
 Lemma derive_pt_exp :
diff --git a/theories/Reals/Ranalysis5.v b/theories/Reals/Ranalysis5.v
new file mode 100644
index 00000000..c8a2e1a8
--- /dev/null
+++ b/theories/Reals/Ranalysis5.v
@@ -0,0 +1,1348 @@
+Require Import Rbase.
+Require Import Ranalysis_reg.
+Require Import Rfunctions.
+Require Import Rseries.
+Require Import Fourier.
+Require Import RiemannInt.
+Require Import SeqProp.
+Require Import Max.
+Local Open Scope R_scope.
+
+(** * Preliminaries lemmas *)
+
+Lemma f_incr_implies_g_incr_interv : forall f g:R->R, forall lb ub,
+       lb < ub ->
+       (forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) ->
+       (forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x) ->
+       (forall x , f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
+       (forall x y, f lb <= x -> x < y -> y <= f ub -> g x < g y).
+Proof.
+intros f g lb ub lb_lt_ub f_incr f_eq_g g_ok x y lb_le_x x_lt_y y_le_ub.
+ assert (x_encad : f lb <= x <= f ub).
+  split ; [assumption | apply Rle_trans with (r2:=y) ; [apply Rlt_le|] ; assumption].
+ assert (y_encad : f lb <= y <= f ub).
+  split ; [apply Rle_trans with (r2:=x) ; [|apply Rlt_le] ; assumption | assumption].
+  assert (Temp1 : lb <= lb) by intuition ; assert (Temp2 : ub <= ub) by intuition.
+ assert (gx_encad := g_ok _ (proj1 x_encad) (proj2 x_encad)).
+ assert (gy_encad := g_ok _ (proj1 y_encad) (proj2 y_encad)).
+ clear Temp1 Temp2.
+ case (Rlt_dec (g x) (g y)).
+  intuition.
+  intros Hfalse.
+   assert (Temp := Rnot_lt_le _ _ Hfalse).
+   assert (Hcontradiction : y <= x).
+   replace y with (id y) by intuition ; replace x with (id x) by intuition ;
+   rewrite <- f_eq_g. rewrite <- f_eq_g.
+   assert (f_incr2 : forall x y, lb <= x -> x <= y -> y < ub -> f x <= f y).
+    intros m n lb_le_m m_le_n n_lt_ub.
+    case (m_le_n).
+     intros ; apply Rlt_le ; apply f_incr ; [| | apply Rlt_le] ; assumption.
+     intros Hyp ; rewrite Hyp ; apply Req_le ; reflexivity.
+   apply f_incr2.
+   intuition. intuition.
+   Focus 3. intuition.
+   Focus 2. intuition.
+   Focus 2. intuition. Focus 2. intuition.
+   assert (Temp2 : g x <> ub).
+   intro Hf.
+    assert (Htemp : (comp f g) x = f ub).
+     unfold comp ; rewrite Hf ; reflexivity.
+     rewrite f_eq_g in Htemp ; unfold id in Htemp.
+    assert (Htemp2 : x < f ub).
+     apply Rlt_le_trans with (r2:=y) ; intuition.
+    clear -Htemp Htemp2. fourier.
+    intuition. intuition.
+   clear -Temp2 gx_encad.
+   case (proj2 gx_encad).
+    intuition.
+    intro Hfalse ; apply False_ind ; apply Temp2 ; assumption.
+   apply False_ind. clear - Hcontradiction x_lt_y. fourier.
+Qed.
+
+Lemma derivable_pt_id_interv : forall (lb ub x:R),
+       lb <= x <= ub ->
+       derivable_pt id x.
+Proof.
+intros.
+ reg.
+Qed.
+
+Lemma pr_nu_var2_interv : forall (f g : R -> R) (lb ub x : R) (pr1 : derivable_pt f x)
+       (pr2 : derivable_pt g x),
+       lb < ub ->
+       lb < x < ub ->
+       (forall h : R, lb < h < ub -> f h = g h) -> derive_pt f x pr1 = derive_pt g x pr2.
+Proof.
+intros f g lb ub x Prf Prg lb_lt_ub x_encad local_eq.
+assert (forall x l, lb < x < ub -> (derivable_pt_abs f x l <-> derivable_pt_abs g x l)).
+ intros a l a_encad.
+ unfold derivable_pt_abs, derivable_pt_lim.
+ split.
+ intros Hyp eps eps_pos.
+ elim (Hyp eps eps_pos) ; intros delta Hyp2.
+ assert (Pos_cond : Rmin delta (Rmin (ub - a) (a - lb)) > 0).
+  clear-a lb ub a_encad delta.
+  apply Rmin_pos ; [exact (delta.(cond_pos)) | apply Rmin_pos ] ; apply Rlt_Rminus ; intuition.
+ exists (mkposreal (Rmin delta (Rmin (ub - a) (a - lb))) Pos_cond).
+ intros h h_neq h_encad.
+ replace (g (a + h) - g a) with (f (a + h) - f a).
+ apply Hyp2 ; intuition.
+ apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))).
+ assumption. apply Rmin_l.
+ assert (local_eq2 : forall h : R, lb < h < ub -> - f h = - g h).
+  intros ; apply Ropp_eq_compat ; intuition.
+ rewrite local_eq ; unfold Rminus. rewrite local_eq2. reflexivity.
+ assumption.
+ assert (Sublemma2 : forall x y, Rabs x < Rabs y -> y > 0 -> x < y).
+  intros m n Hyp_abs y_pos. apply Rlt_le_trans with (r2:=Rabs n).
+   apply Rle_lt_trans with (r2:=Rabs m) ; [ | assumption] ; apply RRle_abs.
+   apply Req_le ; apply Rabs_right ; apply Rgt_ge ; assumption.
+ split.
+ assert (Sublemma : forall x y z, -z < y - x -> x < y + z).
+  intros ; fourier.
+ apply Sublemma.
+ apply Sublemma2. rewrite Rabs_Ropp.
+ apply Rlt_le_trans with (r2:=a-lb) ; [| apply RRle_abs] ;
+ apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_r] ;
+ apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
+ apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_r] ;
+ apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
+ assert (Sublemma : forall x y z, y < z - x -> x + y < z).
+  intros ; fourier.
+ apply Sublemma.
+ apply Sublemma2.
+ apply Rlt_le_trans with (r2:=ub-a) ; [| apply RRle_abs] ;
+ apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_l] ;
+ apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
+ apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_l] ;
+ apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
+ intros Hyp eps eps_pos.
+ elim (Hyp eps eps_pos) ; intros delta Hyp2.
+ assert (Pos_cond : Rmin delta (Rmin (ub - a) (a - lb)) > 0).
+  clear-a lb ub a_encad delta.
+  apply Rmin_pos ; [exact (delta.(cond_pos)) | apply Rmin_pos ] ; apply Rlt_Rminus ; intuition.
+ exists (mkposreal (Rmin delta (Rmin (ub - a) (a - lb))) Pos_cond).
+ intros h h_neq h_encad.
+ replace (f (a + h) - f a) with (g (a + h) - g a).
+ apply Hyp2 ; intuition.
+ apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))).
+ assumption. apply Rmin_l.
+ assert (local_eq2 : forall h : R, lb < h < ub -> - f h = - g h).
+  intros ; apply Ropp_eq_compat ; intuition.
+ rewrite local_eq ; unfold Rminus. rewrite local_eq2. reflexivity.
+ assumption.
+ assert (Sublemma2 : forall x y, Rabs x < Rabs y -> y > 0 -> x < y).
+  intros m n Hyp_abs y_pos. apply Rlt_le_trans with (r2:=Rabs n).
+   apply Rle_lt_trans with (r2:=Rabs m) ; [ | assumption] ; apply RRle_abs.
+   apply Req_le ; apply Rabs_right ; apply Rgt_ge ; assumption.
+ split.
+ assert (Sublemma : forall x y z, -z < y - x -> x < y + z).
+  intros ; fourier.
+ apply Sublemma.
+ apply Sublemma2. rewrite Rabs_Ropp.
+ apply Rlt_le_trans with (r2:=a-lb) ; [| apply RRle_abs] ;
+ apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_r] ;
+ apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
+ apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_r] ;
+ apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
+ assert (Sublemma : forall x y z, y < z - x -> x + y < z).
+  intros ; fourier.
+ apply Sublemma.
+ apply Sublemma2.
+ apply Rlt_le_trans with (r2:=ub-a) ; [| apply RRle_abs] ;
+ apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_l] ;
+ apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
+ apply Rlt_le_trans with (r2:=Rmin (ub - a) (a - lb)) ; [| apply Rmin_l] ;
+ apply Rlt_le_trans with (r2:=Rmin delta (Rmin (ub - a) (a - lb))) ; [| apply Rmin_r] ; assumption.
+ unfold derivable_pt in Prf.
+  unfold derivable_pt in Prg.
+  elim Prf; intros.
+  elim Prg; intros.
+  assert (Temp := p); rewrite H in Temp.
+  unfold derivable_pt_abs in p.
+  unfold derivable_pt_abs in p0.
+  simpl in |- *.
+  apply (uniqueness_limite g x x0 x1 Temp p0).
+  assumption.
+Qed.
+
+
+(* begin hide *)
+Lemma leftinv_is_rightinv : forall (f g:R->R),
+       (forall x y, x < y -> f x < f y) ->
+       (forall x, (comp f g) x = id x) ->
+       (forall x, (comp g f) x = id x).
+Proof.
+intros f g f_incr Hyp x.
+ assert (forall x, f (g (f x)) = f x).
+  intros ; apply Hyp.
+ assert(f_inj : forall x y, f x = f y -> x = y).
+  intros a b fa_eq_fb.
+  case(total_order_T a b).
+  intro s ; case s ; clear s.
+  intro Hf.
+  assert (Hfalse := f_incr a b Hf).
+  apply False_ind. apply (Rlt_not_eq (f a) (f b)) ; assumption.
+  intuition.
+  intro Hf. assert (Hfalse := f_incr b a Hf).
+  apply False_ind. apply (Rlt_not_eq (f b) (f a)) ; [|symmetry] ; assumption.
+ apply f_inj. unfold comp.
+ unfold comp in Hyp.
+ rewrite Hyp.
+ unfold id.
+ reflexivity.
+Qed.
+(* end hide *)
+
+Lemma leftinv_is_rightinv_interv : forall (f g:R->R) (lb ub:R),
+       (forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) ->
+       (forall y, f lb <= y -> y <= f ub -> (comp f g) y = id y) ->
+       (forall x, f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
+       forall x,
+       lb <= x <= ub ->
+       (comp g f) x = id x.
+Proof.
+intros f g lb ub f_incr_interv Hyp g_wf x x_encad.
+ assert(f_inj : forall x y, lb <= x <= ub -> lb <= y <= ub -> f x = f y -> x = y).
+  intros a b a_encad b_encad fa_eq_fb.
+  case(total_order_T a b).
+  intro s ; case s ; clear s.
+  intro Hf.
+  assert (Hfalse := f_incr_interv a b (proj1 a_encad) Hf (proj2 b_encad)).
+  apply False_ind. apply (Rlt_not_eq (f a) (f b)) ; assumption.
+  intuition.
+  intro Hf. assert (Hfalse := f_incr_interv b a (proj1 b_encad) Hf (proj2 a_encad)).
+  apply False_ind. apply (Rlt_not_eq (f b) (f a)) ; [|symmetry] ; assumption.
+ assert (f_incr_interv2 : forall x y, lb <= x -> x <= y -> y <= ub -> f x <= f y).
+  intros m n cond1 cond2 cond3.
+   case cond2.
+   intro cond. apply Rlt_le ; apply f_incr_interv ; assumption.
+   intro cond ; right ; rewrite cond ; reflexivity.
+ assert (Hyp2:forall x, lb <= x <= ub -> f (g (f x)) = f x).
+  intros ; apply Hyp.  apply f_incr_interv2 ; intuition. 
+ apply f_incr_interv2 ; intuition.
+ unfold comp ; unfold comp in Hyp.
+ apply f_inj.
+ apply g_wf ; apply f_incr_interv2 ; intuition.
+ unfold id ; assumption.
+ apply Hyp2 ; unfold id ; assumption.
+Qed.
+
+
+(** Intermediate Value Theorem on an Interval (Proof mainly taken from Reals.Rsqrt_def) and its corollary *)
+
+Lemma IVT_interv_prelim0 : forall (x y:R) (P:R->bool) (N:nat),
+       x < y ->
+       x <= Dichotomy_ub x y P N <= y /\ x <= Dichotomy_lb x y P N <= y.
+Proof.
+assert (Sublemma : forall x y lb ub, lb <= x <= ub /\ lb <= y <= ub -> lb <= (x+y) / 2 <= ub).
+ intros x y lb ub Hyp.
+  split.
+  replace lb with ((lb + lb) * /2) by field.
+  unfold Rdiv ; apply Rmult_le_compat_r ; intuition.
+  replace ub with ((ub + ub) * /2) by field.
+  unfold Rdiv ; apply Rmult_le_compat_r ; intuition.
+intros x y P N x_lt_y.
+induction N.
+ simpl ; intuition.
+ simpl.
+ case (P ((Dichotomy_lb x y P N + Dichotomy_ub x y P N) / 2)).
+ split. apply Sublemma ; intuition.
+ intuition.
+ split. intuition.
+ apply Sublemma ; intuition.
+Qed.
+
+Lemma IVT_interv_prelim1 : forall (x y x0:R) (D : R -> bool),
+       x < y ->
+       Un_cv (dicho_up x y D) x0 ->
+       x <= x0 <= y.
+Proof.
+intros x y x0 D x_lt_y bnd.
+ assert (Main : forall n, x <= dicho_up x y D n <= y).
+  intro n. unfold dicho_up.
+  apply (proj1 (IVT_interv_prelim0 x y D n x_lt_y)).
+ split.
+  apply Rle_cv_lim with (Vn:=dicho_up x y D) (Un:=fun n => x).
+  intro n ; exact (proj1 (Main n)).
+  unfold Un_cv ; intros ; exists 0%nat ; intros ; unfold R_dist ; replace (x -x) with 0 by field ; rewrite Rabs_R0 ; assumption.
+  assumption.
+  apply Rle_cv_lim with (Un:=dicho_up x y D) (Vn:=fun n => y).
+  intro n ; exact (proj2 (Main n)).
+  assumption.
+  unfold Un_cv ; intros ; exists 0%nat ; intros ; unfold R_dist ; replace (y -y) with 0 by field ; rewrite Rabs_R0 ; assumption.
+Qed.
+
+Lemma IVT_interv : forall (f : R -> R) (x y : R),
+       (forall a, x <= a <= y -> continuity_pt f a) ->
+       x < y ->
+       f x < 0 ->
+       0 < f y ->
+       {z : R | x <= z <= y /\ f z = 0}.
+Proof.
+intros. (* f x y f_cont_interv x_lt_y fx_neg fy_pos.*)
+  cut (x <= y).
+  intro.
+  generalize (dicho_lb_cv x y (fun z:R => cond_positivity (f z)) H3). 
+  generalize (dicho_up_cv x y (fun z:R => cond_positivity (f z)) H3). 
+  intros X X0.
+  elim X; intros.
+  elim X0; intros.
+  assert (H4 := cv_dicho _ _ _ _ _ H3 p0 p).
+  rewrite H4 in p0.
+  exists x0.
+  split.
+  split.
+  apply Rle_trans with (dicho_lb x y (fun z:R => cond_positivity (f z)) 0).
+  simpl in |- *.
+  right; reflexivity.
+  apply growing_ineq.
+  apply dicho_lb_growing; assumption.
+  assumption.
+  apply Rle_trans with (dicho_up x y (fun z:R => cond_positivity (f z)) 0).
+  apply decreasing_ineq.
+  apply dicho_up_decreasing; assumption.
+  assumption.
+  right; reflexivity.
+  2: left; assumption.
+  set (Vn := fun n:nat => dicho_lb x y (fun z:R => cond_positivity (f z)) n).
+  set (Wn := fun n:nat => dicho_up x y (fun z:R => cond_positivity (f z)) n).
+  cut ((forall n:nat, f (Vn n) <= 0) -> f x0 <= 0).
+  cut ((forall n:nat, 0 <= f (Wn n)) -> 0 <= f x0).
+  intros.
+  cut (forall n:nat, f (Vn n) <= 0).
+  cut (forall n:nat, 0 <= f (Wn n)).
+  intros.
+  assert (H9 := H6 H8).
+  assert (H10 := H5 H7).
+  apply Rle_antisym; assumption.
+  intro.
+  unfold Wn in |- *.
+  cut (forall z:R, cond_positivity z = true <-> 0 <= z).
+  intro.
+  assert (H8 := dicho_up_car x y (fun z:R => cond_positivity (f z)) n).
+  elim (H7 (f (dicho_up x y (fun z:R => cond_positivity (f z)) n))); intros.
+  apply H9.
+  apply H8.
+  elim (H7 (f y)); intros.
+  apply H12.
+  left; assumption.
+  intro.
+  unfold cond_positivity in |- *.
+  case (Rle_dec 0 z); intro.
+  split.
+  intro; assumption.
+  intro; reflexivity.
+  split.
+  intro feqt;discriminate feqt.
+  intro.
+  elim n0; assumption.
+  unfold Vn in |- *.
+  cut (forall z:R, cond_positivity z = false <-> z < 0).
+  intros.
+  assert (H8 := dicho_lb_car x y (fun z:R => cond_positivity (f z)) n).
+  left.
+  elim (H7 (f (dicho_lb x y (fun z:R => cond_positivity (f z)) n))); intros.
+  apply H9.
+  apply H8.
+  elim (H7 (f x)); intros.
+  apply H12.
+  assumption.
+  intro.
+  unfold cond_positivity in |- *.
+  case (Rle_dec 0 z); intro.
+  split.
+  intro feqt; discriminate feqt.
+  intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H7)).
+  split.
+  intro; auto with real.
+  intro; reflexivity.
+  cut (Un_cv Wn x0).
+  intros.
+  assert (Temp : x <= x0 <= y).
+   apply IVT_interv_prelim1 with (D:=(fun z : R => cond_positivity (f z))) ; assumption.
+  assert (H7 := continuity_seq f Wn x0 (H x0 Temp) H5).
+  case (total_order_T 0 (f x0)); intro.
+  elim s; intro.
+  left; assumption.
+  rewrite <- b; right; reflexivity.
+  unfold Un_cv in H7; unfold R_dist in H7.
+  cut (0 < - f x0).
+  intro.
+  elim (H7 (- f x0) H8); intros.
+  cut (x2 >= x2)%nat; [ intro | unfold ge in |- *; apply le_n ].
+  assert (H11 := H9 x2 H10).
+  rewrite Rabs_right in H11.
+  pattern (- f x0) at 1 in H11; rewrite <- Rplus_0_r in H11.
+  unfold Rminus in H11; rewrite (Rplus_comm (f (Wn x2))) in H11.
+  assert (H12 := Rplus_lt_reg_r _ _ _ H11).
+  assert (H13 := H6 x2).
+  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H13 H12)).
+  apply Rle_ge; left; unfold Rminus in |- *; apply Rplus_le_lt_0_compat.
+  apply H6.
+  exact H8.
+  apply Ropp_0_gt_lt_contravar; assumption.
+  unfold Wn in |- *; assumption.
+  cut (Un_cv Vn x0).
+  intros.
+  assert (Temp : x <= x0 <= y).
+   apply IVT_interv_prelim1 with (D:=(fun z : R => cond_positivity (f z))) ; assumption.
+  assert (H7 := continuity_seq f Vn x0 (H x0 Temp) H5).
+  case (total_order_T 0 (f x0)); intro.
+  elim s; intro.
+  unfold Un_cv in H7; unfold R_dist in H7.
+  elim (H7 (f x0) a); intros.
+  cut (x2 >= x2)%nat; [ intro | unfold ge in |- *; apply le_n ].
+  assert (H10 := H8 x2 H9).
+  rewrite Rabs_left in H10.
+  pattern (f x0) at 2 in H10; rewrite <- Rplus_0_r in H10.
+  rewrite Ropp_minus_distr' in H10.
+  unfold Rminus in H10.
+  assert (H11 := Rplus_lt_reg_r _ _ _ H10).
+  assert (H12 := H6 x2).
+  cut (0 < f (Vn x2)).
+  intro.
+  elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H13 H12)).
+  rewrite <- (Ropp_involutive (f (Vn x2))).
+  apply Ropp_0_gt_lt_contravar; assumption.
+  apply Rplus_lt_reg_r with (f x0 - f (Vn x2)).
+  rewrite Rplus_0_r; replace (f x0 - f (Vn x2) + (f (Vn x2) - f x0)) with 0;
+    [ unfold Rminus in |- *; apply Rplus_lt_le_0_compat | ring ].
+  assumption.
+  apply Ropp_0_ge_le_contravar; apply Rle_ge; apply H6.
+  right; rewrite <- b; reflexivity.
+  left; assumption.
+  unfold Vn in |- *; assumption.
+Qed.
+
+(* begin hide *)
+Ltac case_le H :=
+   let t := type of H in 
+   let h' := fresh in 
+      match t with ?x <= ?y => case (total_order_T x y);
+         [intros h'; case h'; clear h' | 
+          intros h'; clear -H h'; elimtype False; fourier ] end.
+(* end hide *)
+
+
+Lemma f_interv_is_interv : forall (f:R->R) (lb ub y:R),
+       lb < ub ->
+       f lb <= y <= f ub ->
+       (forall x, lb <=  x <= ub -> continuity_pt f x) ->
+       {x | lb <= x <= ub /\ f x = y}.
+Proof.
+intros f lb ub y lb_lt_ub y_encad f_cont_interv.
+ case y_encad ; intro y_encad1.
+   case_le y_encad1 ; intros y_encad2 y_encad3 ; case_le y_encad3.
+    intro y_encad4.
+     clear y_encad y_encad1 y_encad3.
+     assert (Cont : forall a : R, lb <= a <= ub -> continuity_pt (fun x => f x - y) a).
+      intros a a_encad. unfold continuity_pt, continue_in, limit1_in, limit_in ; simpl ; unfold R_dist.
+      intros eps eps_pos. elim (f_cont_interv a a_encad eps eps_pos).
+      intros alpha alpha_pos. destruct alpha_pos as (alpha_pos,Temp).
+      exists alpha. split.
+      assumption. intros x  x_cond.
+      replace (f x - y - (f a - y)) with (f x - f a) by field.
+      exact (Temp x x_cond).
+     assert (H1 : (fun x : R => f x - y) lb < 0).
+       apply Rlt_minus. assumption.
+      assert (H2 : 0 < (fun x : R => f x - y) ub).
+       apply Rgt_minus ; assumption.
+     destruct (IVT_interv (fun x => f x - y) lb ub Cont lb_lt_ub H1 H2) as (x,Hx).
+     exists x.
+     destruct Hx as (Hyp,Result).
+     intuition.
+     intro H ; exists ub ; intuition.
+     intro H ; exists lb ; intuition.
+     intro H ; exists ub ; intuition.
+Qed.
+
+(** ** The derivative of a reciprocal function       *)
+
+
+(** * Continuity of the reciprocal function *)
+
+Lemma continuity_pt_recip_prelim : forall (f g:R->R) (lb ub : R) (Pr1:lb < ub),
+       (forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) ->
+       (forall x, lb <= x <= ub -> (comp g f) x = id x) ->
+       (forall a, lb <= a <= ub -> continuity_pt f a) ->
+       forall b,
+       f lb < b < f ub ->
+       continuity_pt g b.
+Proof.
+assert (Sublemma : forall x y z, Rmax x y < z <-> x < z /\ y < z).
+ intros x y z. split.
+  unfold Rmax. case (Rle_dec x y) ; intros Hyp Hyp2.
+  split. apply Rle_lt_trans with (r2:=y) ; assumption. assumption.
+  split. assumption. apply Rlt_trans with (r2:=x).
+  assert (Temp : forall x y, ~ x <= y -> x > y).
+   intros m n Hypmn. intuition.
+  apply Temp ; clear Temp ; assumption.
+  assumption.
+  intros Hyp.
+  unfold Rmax. case (Rle_dec x y).
+  intro ; exact (proj2 Hyp).
+  intro ; exact (proj1 Hyp).
+assert (Sublemma2 : forall x y z, Rmin x y > z <-> x > z /\ y > z).
+ intros x y z. split.
+  unfold Rmin. case (Rle_dec x y) ; intros Hyp Hyp2.
+  split. assumption.
+  apply Rlt_le_trans with (r2:=x) ; intuition.
+  split.
+  apply Rlt_trans with (r2:=y). intuition.
+  assert (Temp : forall x y, ~ x <= y -> x > y).
+   intros m n Hypmn. intuition.
+  apply Temp ; clear Temp ; assumption.
+  assumption.
+  intros Hyp.
+  unfold Rmin. case (Rle_dec x y).
+  intro ; exact (proj1 Hyp).
+  intro ; exact (proj2 Hyp).
+assert (Sublemma3 : forall x y, x <= y /\ x <> y -> x < y).
+ intros m n Hyp. unfold Rle in Hyp.
+  destruct Hyp as (Hyp1,Hyp2).
+  case Hyp1.
+  intuition.
+  intro Hfalse ; apply False_ind ; apply Hyp2 ; exact Hfalse.
+intros f g lb ub lb_lt_ub f_incr_interv f_eq_g f_cont_interv b b_encad.
+ assert (f_incr_interv2 : forall x y, lb <= x -> x <= y -> y <= ub -> f x <= f y).
+  intros m n cond1 cond2 cond3.
+   case cond2.
+   intro cond. apply Rlt_le ; apply f_incr_interv ; assumption.
+   intro cond ; right ; rewrite cond ; reflexivity.
+ unfold continuity_pt, continue_in, limit1_in, limit_in ; intros eps eps_pos.
+ unfold dist ; simpl ; unfold R_dist.
+ assert (b_encad_e : f lb <= b <= f ub) by intuition.
+ elim (f_interv_is_interv f lb ub b lb_lt_ub b_encad_e f_cont_interv) ; intros x Temp.
+ destruct Temp as (x_encad,f_x_b).
+ assert (lb_lt_x : lb < x).
+ assert (Temp : x <> lb).
+  intro Hfalse.
+  assert (Temp' : b = f lb).
+   rewrite <- f_x_b ; rewrite Hfalse ; reflexivity.
+  assert (Temp'' : b <> f lb).
+   apply Rgt_not_eq ; exact (proj1 b_encad).
+  apply Temp'' ; exact Temp'.
+ apply Sublemma3.
+ split. exact (proj1 x_encad).
+ assert (Temp2 : forall x y:R, x <> y <-> y <> x).
+  intros m n. split ; intuition.
+ rewrite Temp2 ; assumption.
+ assert (x_lt_ub : x < ub).
+ assert (Temp : x <> ub).
+  intro Hfalse.
+  assert (Temp' : b = f ub).
+   rewrite <- f_x_b ; rewrite Hfalse ; reflexivity.
+  assert (Temp'' : b <> f ub).
+   apply Rlt_not_eq ; exact (proj2 b_encad).
+  apply Temp'' ; exact Temp'.
+ apply Sublemma3.
+ split ; [exact (proj2 x_encad) | assumption].
+ pose (x1 := Rmax (x - eps) lb).
+ pose (x2 := Rmin (x + eps) ub).
+ assert (Hx1 : x1 = Rmax (x - eps) lb) by intuition.
+ assert (Hx2 : x2 = Rmin (x + eps) ub) by intuition.
+ assert (x1_encad : lb <= x1 <= ub).
+  split. apply RmaxLess2.
+  apply Rlt_le. rewrite Hx1. rewrite Sublemma.
+  split. apply Rlt_trans with (r2:=x) ; fourier.
+  assumption.
+ assert (x2_encad : lb <= x2 <= ub).
+  split. apply Rlt_le ; rewrite Hx2 ; apply Rgt_lt ; rewrite Sublemma2.
+  split. apply Rgt_trans with (r2:=x) ; fourier.
+  assumption.
+  apply Rmin_r.
+ assert (x_lt_x2 : x < x2).
+  rewrite Hx2.
+  apply Rgt_lt. rewrite Sublemma2.
+  split ; fourier.
+ assert (x1_lt_x : x1 < x).
+  rewrite Hx1.
+  rewrite Sublemma.
+  split ; fourier.
+ exists (Rmin (f x - f x1) (f x2 - f x)).
+ split. apply Rmin_pos ; apply Rgt_minus. apply f_incr_interv ; [apply RmaxLess2 | | ] ; fourier.
+ apply f_incr_interv ; intuition.
+ intros y Temp.
+ destruct Temp as (_,y_cond).
+  rewrite <- f_x_b in y_cond.
+  assert (Temp : forall x y d1 d2, d1 > 0 -> d2 > 0 -> Rabs (y - x) < Rmin d1 d2 -> x - d1 <= y <= x + d2).
+   intros.
+    split. assert (H10 : forall x y z, x - y <= z -> x - z <= y). intuition. fourier.
+    apply H10. apply Rle_trans with (r2:=Rabs (y0 - x0)).
+    replace (Rabs (y0 - x0)) with (Rabs (x0 - y0)). apply RRle_abs.
+    rewrite <- Rabs_Ropp. unfold Rminus ; rewrite Ropp_plus_distr. rewrite Ropp_involutive.
+    intuition.
+    apply Rle_trans with (r2:= Rmin d1 d2). apply Rlt_le ; assumption.
+    apply Rmin_l.
+    assert (H10 : forall x y z, x - y <= z -> x <= y + z). intuition. fourier.
+    apply H10. apply Rle_trans with (r2:=Rabs (y0 - x0)). apply RRle_abs.
+    apply Rle_trans with (r2:= Rmin d1 d2). apply Rlt_le ; assumption.
+    apply Rmin_r.
+  assert (Temp' := Temp (f x) y (f x - f x1) (f x2 - f x)).
+  replace (f x - (f x - f x1)) with (f x1) in Temp' by field.
+  replace (f x + (f x2 - f x)) with (f x2) in Temp' by field.
+  assert (T : f x - f x1 > 0).
+   apply Rgt_minus. apply f_incr_interv ; intuition.
+  assert (T' : f x2 - f x > 0).
+   apply Rgt_minus. apply f_incr_interv ; intuition.
+  assert (Main := Temp' T T' y_cond).
+  clear Temp Temp' T T'.
+  assert (x1_lt_x2 : x1 < x2).
+   apply Rlt_trans with (r2:=x) ; assumption.
+   assert (f_cont_myinterv : forall a : R, x1 <= a <= x2 -> continuity_pt f a).
+    intros ; apply f_cont_interv ; split. 
+    apply Rle_trans with (r2 := x1) ; intuition.
+    apply Rle_trans with (r2 := x2) ; intuition.
+   elim (f_interv_is_interv f x1 x2 y x1_lt_x2 Main f_cont_myinterv) ; intros x' Temp.
+   destruct Temp as (x'_encad,f_x'_y).
+   rewrite <- f_x_b ; rewrite <- f_x'_y.
+   unfold comp in f_eq_g. rewrite f_eq_g. rewrite f_eq_g.
+   unfold id.
+   assert (x'_encad2 : x - eps <= x' <= x + eps).
+    split.
+    apply Rle_trans with (r2:=x1) ; [ apply RmaxLess1|] ; intuition.
+    apply Rle_trans with (r2:=x2) ; [ | apply Rmin_l] ; intuition.
+    assert (x1_lt_x' : x1 < x').
+     apply Sublemma3.
+     assert (x1_neq_x' : x1 <> x').
+      intro Hfalse. rewrite Hfalse, f_x'_y in y_cond.
+      assert (Hf : Rabs (y - f x) < f x - y).
+       apply Rlt_le_trans with (r2:=Rmin (f x - y) (f x2 - f x)). fourier.
+       apply Rmin_l.
+      assert(Hfin : f x - y < f x - y).
+       apply Rle_lt_trans with (r2:=Rabs (y - f x)).
+       replace (Rabs (y - f x)) with (Rabs (f x - y)). apply RRle_abs.
+       rewrite <- Rabs_Ropp. replace (- (f x - y)) with (y - f x) by field ; reflexivity. fourier.
+      apply (Rlt_irrefl (f x - y)) ; assumption.
+      split ; intuition.
+     assert (x'_lb : x - eps < x').
+      apply Sublemma3.
+      split. intuition. apply Rlt_not_eq.
+      apply Rle_lt_trans with (r2:=x1) ; [ apply RmaxLess1|] ; intuition.
+     assert (x'_lt_x2 : x' < x2).
+     apply Sublemma3.
+     assert (x1_neq_x' : x' <> x2).
+      intro Hfalse. rewrite <- Hfalse, f_x'_y in y_cond.
+      assert (Hf : Rabs (y - f x) < y - f x).
+       apply Rlt_le_trans with (r2:=Rmin (f x - f x1) (y - f x)). fourier.
+       apply Rmin_r.
+      assert(Hfin : y - f x < y - f x).
+       apply Rle_lt_trans with (r2:=Rabs (y - f x)). apply RRle_abs. fourier.
+      apply (Rlt_irrefl (y - f x)) ; assumption.
+      split ; intuition.
+     assert (x'_ub : x' < x + eps).
+      apply Sublemma3.
+      split. intuition. apply Rlt_not_eq.
+      apply Rlt_le_trans with (r2:=x2) ; [ |rewrite Hx2 ; apply Rmin_l] ; intuition.
+    apply Rabs_def1 ; fourier.
+    assumption.
+    split. apply Rle_trans with (r2:=x1) ; intuition.
+    apply Rle_trans with (r2:=x2) ; intuition.
+Qed.
+
+Lemma continuity_pt_recip_interv : forall (f g:R->R) (lb ub : R) (Pr1:lb < ub),
+       (forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) ->
+       (forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x) ->
+       (forall x, f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
+       (forall a, lb <= a <= ub -> continuity_pt f a) ->
+       forall b,
+       f lb < b < f ub ->
+       continuity_pt g b.
+Proof.
+intros f g lb ub lb_lt_ub f_incr_interv f_eq_g g_wf.
+assert (g_eq_f_prelim := leftinv_is_rightinv_interv f g lb ub f_incr_interv f_eq_g).
+assert (g_eq_f : forall x, lb <= x <= ub -> (comp g f) x = id x).
+intro x ; apply g_eq_f_prelim ; assumption.
+apply (continuity_pt_recip_prelim f g lb ub lb_lt_ub f_incr_interv g_eq_f).
+Qed.
+
+(** *   Derivability of the reciprocal function        *)
+
+Lemma derivable_pt_lim_recip_interv : forall (f g:R->R) (lb ub x:R)
+       (Prf:forall a : R, g lb <= a <= g ub -> derivable_pt f a) (Prg : continuity_pt g x),
+       lb < ub ->
+       lb < x < ub ->
+       forall (Prg_incr:g lb <= g x <= g ub),
+       (forall x, lb <= x <= ub -> (comp f g) x = id x) ->
+       derive_pt f (g x) (Prf (g x) Prg_incr) <> 0 ->
+       derivable_pt_lim g x (1 / derive_pt f (g x) (Prf (g x) Prg_incr)).
+Proof.
+intros f g lb ub x Prf g_cont_pur lb_lt_ub x_encad Prg_incr f_eq_g df_neq.
+ assert (x_encad2 : lb <= x <= ub).
+  split ; apply Rlt_le ; intuition.
+ elim (Prf (g x)); simpl; intros l Hl.
+ unfold derivable_pt_lim.
+ intros eps eps_pos.
+  pose (y := g x).
+   assert (Hlinv := limit_inv).
+    assert (Hf_deriv : forall eps:R,
+    0 < eps ->
+    exists delta : posreal,
+    (forall h:R,
+    h <> 0 -> Rabs h < delta -> Rabs ((f (g x + h) - f (g x)) / h - l) < eps)).
+     intros eps0 eps0_pos.
+      red in Hl ; red in Hl. elim (Hl eps0 eps0_pos).
+      intros deltatemp Htemp.
+       exists deltatemp ; exact Htemp.
+    elim (Hf_deriv eps eps_pos).
+    intros deltatemp Htemp.
+     red in Hlinv ; red in Hlinv ; simpl dist in Hlinv ; unfold R_dist in Hlinv.
+     assert (Hlinv' := Hlinv (fun h => (f (y+h) - f y)/h) (fun h => h <>0) l 0).
+     unfold limit1_in, limit_in, dist in Hlinv' ; simpl in Hlinv'. unfold R_dist in Hlinv'.
+     assert (Premisse : (forall eps : R,
+     eps > 0 ->
+     exists alp : R,
+     alp > 0 /\
+     (forall x : R,
+     (fun h => h <>0) x /\ Rabs (x - 0) < alp ->
+     Rabs ((f (y + x) - f y) / x - l) < eps))).
+      intros eps0 eps0_pos.
+       elim (Hf_deriv eps0 eps0_pos).
+       intros deltatemp' Htemp'.
+        exists deltatemp'.
+        split.
+         exact deltatemp'.(cond_pos).
+         intros htemp cond.
+          apply (Htemp' htemp).
+          exact (proj1 cond).
+         replace (htemp) with (htemp - 0).
+         exact (proj2 cond).
+         intuition.
+     assert (Premisse2 : l <> 0).
+      intro l_null.
+       rewrite l_null in Hl.
+       apply df_neq.
+       rewrite derive_pt_eq.
+       exact Hl.                
+     elim (Hlinv' Premisse Premisse2 eps eps_pos).
+     intros alpha cond.
+      assert (alpha_pos := proj1 cond) ; assert (inv_cont := proj2 cond) ; clear cond.
+      unfold derivable, derivable_pt, derivable_pt_abs, derivable_pt_lim in Prf.
+      elim (Hl eps eps_pos).
+      intros delta f_deriv.
+       assert (g_cont := g_cont_pur).
+       unfold continuity_pt, continue_in, limit1_in, limit_in in g_cont.
+       pose (mydelta := Rmin delta alpha).
+       assert (mydelta_pos : mydelta > 0).
+        unfold mydelta, Rmin.
+        case (Rle_dec delta alpha).
+        intro ; exact (delta.(cond_pos)).
+        intro ; exact alpha_pos.
+       elim (g_cont mydelta mydelta_pos).
+       intros delta' new_g_cont.
+        assert(delta'_pos := proj1 (new_g_cont)).
+        clear g_cont ; assert (g_cont := proj2 (new_g_cont)) ; clear new_g_cont.
+        pose (mydelta'' := Rmin delta' (Rmin (x - lb) (ub - x))).
+        assert(mydelta''_pos : mydelta'' > 0).
+         unfold mydelta''.
+         apply Rmin_pos ; [intuition | apply Rmin_pos] ; apply Rgt_minus ; intuition.
+        pose (delta'' := mkposreal mydelta'' mydelta''_pos: posreal).
+        exists delta''.
+        intros h h_neq h_le_delta'.
+  assert (lb <= x +h <= ub).
+ assert (Sublemma2 : forall x y, Rabs x < Rabs y -> y > 0 -> x < y).
+  intros m n Hyp_abs y_pos. apply Rlt_le_trans with (r2:=Rabs n).
+   apply Rle_lt_trans with (r2:=Rabs m) ; [ | assumption] ; apply RRle_abs.
+   apply Req_le ; apply Rabs_right ; apply Rgt_ge ; assumption.
+ assert (lb <= x + h <= ub).
+ split.
+ assert (Sublemma : forall x y z, -z <= y - x -> x <= y + z).
+  intros ; fourier.
+ apply Sublemma.
+ apply Rlt_le ; apply Sublemma2. rewrite Rabs_Ropp.
+ apply Rlt_le_trans with (r2:=x-lb) ; [| apply RRle_abs] ;
+ apply Rlt_le_trans with (r2:=Rmin (x - lb) (ub - x)) ; [| apply Rmin_l] ;
+ apply Rlt_le_trans with (r2:=Rmin delta' (Rmin (x - lb) (ub - x))).
+ apply Rlt_le_trans with (r2:=delta''). assumption. intuition. apply Rmin_r.
+ apply Rgt_minus. intuition.
+ assert (Sublemma : forall x y z, y <= z - x -> x + y <= z).
+  intros ; fourier.
+ apply Sublemma.
+ apply Rlt_le ; apply Sublemma2.
+ apply Rlt_le_trans with (r2:=ub-x) ; [| apply RRle_abs] ;
+ apply Rlt_le_trans with (r2:=Rmin (x - lb) (ub - x)) ; [| apply Rmin_r] ;
+ apply Rlt_le_trans with (r2:=Rmin delta' (Rmin (x - lb) (ub - x))) ; [| apply Rmin_r] ; assumption.
+ apply Rlt_le_trans with (r2:=delta''). assumption.
+  apply Rle_trans with (r2:=Rmin delta' (Rmin (x - lb) (ub - x))). intuition.
+  apply Rle_trans with (r2:=Rmin (x - lb) (ub - x)). apply Rmin_r. apply Rmin_r.
+         replace ((g (x + h) - g x) / h) with (1/ (h / (g (x + h) - g x))).
+         assert (Hrewr : h = (comp f g ) (x+h) - (comp f g) x).
+          rewrite f_eq_g. rewrite f_eq_g ; unfold id. rewrite Rplus_comm ;
+          unfold Rminus ; rewrite Rplus_assoc ; rewrite Rplus_opp_r. intuition. intuition.    
+ assumption.
+         split ; [|intuition].
+         assert (Sublemma : forall x y z, - z <= y - x -> x <= y + z).
+          intros ; fourier.
+         apply Sublemma ; apply Rlt_le ; apply Sublemma2. rewrite Rabs_Ropp.
+ apply Rlt_le_trans with (r2:=x-lb) ; [| apply RRle_abs] ;
+ apply Rlt_le_trans with (r2:=Rmin (x - lb) (ub - x)) ; [| apply Rmin_l] ;
+ apply Rlt_le_trans with (r2:=Rmin delta' (Rmin (x - lb) (ub - x))) ; [| apply Rmin_r] ; assumption.
+ apply Rgt_minus. intuition.
+ field.
+ split. assumption.
+ intro Hfalse. assert (Hf : g (x+h) = g x) by intuition.
+ assert ((comp f g) (x+h) = (comp f g) x).
+  unfold comp ; rewrite Hf ; intuition.
+ assert (Main : x+h = x).
+ replace (x +h) with (id (x+h)) by intuition.
+ assert (Temp : x = id x) by intuition ; rewrite Temp at 2 ; clear Temp.
+ rewrite <- f_eq_g. rewrite <- f_eq_g. assumption.
+ intuition. assumption.
+ assert (h = 0).
+ apply Rplus_0_r_uniq with (r:=x) ; assumption.
+ apply h_neq ; assumption.
+         replace ((g (x + h) - g x) / h) with (1/ (h / (g (x + h) - g x))).
+         assert (Hrewr : h = (comp f g ) (x+h) - (comp f g) x).
+          rewrite f_eq_g. rewrite f_eq_g. unfold id ; rewrite Rplus_comm ;
+          unfold Rminus ; rewrite Rplus_assoc ; rewrite Rplus_opp_r ; intuition.
+          assumption. assumption.
+         rewrite Hrewr at 1. 
+         unfold comp.
+         replace (g(x+h)) with (g x + (g (x+h) - g(x))) by field.
+         pose (h':=g (x+h) - g x).
+         replace (g (x+h) - g x) with h' by intuition.
+         replace (g x + h' - g x) with h' by field.
+         assert (h'_neq : h' <> 0).
+          unfold h'.
+          intro Hfalse.
+          unfold Rminus in Hfalse ; apply Rminus_diag_uniq in Hfalse.
+          assert (Hfalse' : (comp f g) (x+h) = (comp f g) x).
+           intros ; unfold comp ; rewrite Hfalse ; trivial.
+          rewrite f_eq_g in Hfalse' ; rewrite f_eq_g in Hfalse'.
+          unfold id in Hfalse'.
+          apply Rplus_0_r_uniq in Hfalse'.
+          apply h_neq ; exact Hfalse'. assumption. assumption. assumption.
+         unfold Rdiv at 1 3; rewrite Rmult_1_l ; rewrite Rmult_1_l.
+         apply inv_cont.
+         split.
+          exact h'_neq.
+          rewrite Rminus_0_r. 
+          unfold continuity_pt, continue_in, limit1_in, limit_in in g_cont_pur.
+          elim (g_cont_pur mydelta mydelta_pos).
+          intros delta3 cond3.
+          unfold dist in cond3 ; simpl in cond3 ; unfold R_dist in cond3.
+          unfold h'.
+          assert (mydelta_le_alpha : mydelta <= alpha).
+           unfold mydelta, Rmin ; case (Rle_dec delta alpha).
+           trivial.
+           intro ; intuition.
+          apply Rlt_le_trans with (r2:=mydelta).
+          unfold dist in g_cont ; simpl in g_cont ; unfold R_dist in g_cont ; apply g_cont.
+          split.
+           unfold D_x ; simpl.
+           split.
+            unfold no_cond ; trivial.
+            intro Hfalse ; apply h_neq.
+             apply (Rplus_0_r_uniq x).
+             symmetry ; assumption.
+            replace (x + h - x) with h by field. 
+            apply Rlt_le_trans with (r2:=delta'').
+            assumption ; unfold delta''. intuition.
+            apply Rle_trans with (r2:=mydelta''). apply Req_le. unfold delta''. intuition.
+            apply Rmin_l. assumption.
+            field ; split.
+             assumption.
+             intro Hfalse ; apply h_neq.
+             apply (Rplus_0_r_uniq x).
+             assert (Hfin : (comp f g) (x+h) = (comp f g) x).
+              apply Rminus_diag_uniq in Hfalse.
+              unfold comp.
+              rewrite Hfalse ; reflexivity.
+              rewrite f_eq_g in Hfin. rewrite f_eq_g in Hfin. unfold id in Hfin. exact Hfin.
+              assumption. assumption.
+Qed.
+
+Lemma derivable_pt_recip_interv_prelim0 : forall (f g : R -> R) (lb ub x : R)
+       (Prf : forall a : R, g lb <= a <= g ub -> derivable_pt f a),
+       continuity_pt g x ->
+       lb < ub ->
+       lb < x < ub ->
+       forall Prg_incr : g lb <= g x <= g ub,
+       (forall x0 : R, lb <= x0 <= ub -> comp f g x0 = id x0) ->
+       derive_pt f (g x) (Prf (g x) Prg_incr) <> 0 ->
+       derivable_pt g x.
+Proof.
+intros f g lb ub x Prf g_cont_pt lb_lt_ub x_encad Prg_incr f_eq_g Df_neq.
+unfold derivable_pt, derivable_pt_abs.
+exists (1 / derive_pt f (g x) (Prf (g x) Prg_incr)).
+apply derivable_pt_lim_recip_interv ; assumption.
+Qed.
+
+Lemma derivable_pt_recip_interv_prelim1 :forall (f g:R->R) (lb ub x : R),
+         lb < ub ->
+         f lb < x < f ub ->
+         (forall x : R, f lb <= x -> x <= f ub -> comp f g x = id x) ->
+         (forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
+         (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) ->
+         (forall a : R, lb <= a <= ub -> derivable_pt f a) ->
+         derivable_pt f (g x).
+Proof.
+intros f g lb ub x lb_lt_ub x_encad f_eq_g g_ok f_incr f_derivable.
+ apply f_derivable.
+ assert (Left_inv := leftinv_is_rightinv_interv f g lb ub f_incr f_eq_g g_ok).
+ replace lb with ((comp g f) lb).
+ replace ub with ((comp g f) ub).
+ unfold comp.
+ assert (Temp:= f_incr_implies_g_incr_interv f g lb ub lb_lt_ub f_incr f_eq_g g_ok).
+ split ; apply Rlt_le ; apply Temp ; intuition.
+ apply Left_inv ; intuition.
+ apply Left_inv ; intuition.
+Qed.
+
+Lemma derivable_pt_recip_interv : forall (f g:R->R) (lb ub x : R)
+         (lb_lt_ub:lb < ub) (x_encad:f lb < x < f ub)
+         (f_eq_g:forall x : R, f lb <= x -> x <= f ub -> comp f g x = id x)
+         (g_wf:forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub)
+         (f_incr:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y)
+         (f_derivable:forall a : R, lb <= a <= ub -> derivable_pt f a),
+         derive_pt f (g x)
+              (derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub
+              x_encad f_eq_g g_wf f_incr f_derivable)
+         <> 0 ->
+         derivable_pt g x.
+Proof.
+intros f g lb ub x lb_lt_ub x_encad f_eq_g g_wf f_incr f_derivable Df_neq.
+ assert(g_incr : g (f lb) < g x < g (f ub)).
+  assert (Temp:= f_incr_implies_g_incr_interv f g lb ub lb_lt_ub f_incr f_eq_g g_wf).
+  split ; apply Temp ; intuition.
+  exact (proj1 x_encad). apply Rlt_le ; exact (proj2 x_encad).
+  apply Rlt_le ; exact (proj1 x_encad). exact (proj2 x_encad).
+ assert(g_incr2 :  g (f lb) <= g x <= g (f ub)).
+  split ; apply Rlt_le ; intuition.
+ assert (g_eq_f :=  leftinv_is_rightinv_interv f g lb ub f_incr f_eq_g g_wf).
+ unfold comp, id in g_eq_f.
+ assert (f_derivable2 : forall a : R, g (f lb) <= a <= g (f ub) -> derivable_pt f a).
+  intros a a_encad ; apply f_derivable.
+   rewrite g_eq_f in a_encad ; rewrite g_eq_f in a_encad ; intuition.
+ apply derivable_pt_recip_interv_prelim0 with (f:=f) (lb:=f lb) (ub:=f ub)
+     (Prf:=f_derivable2) (Prg_incr:=g_incr2).
+ apply continuity_pt_recip_interv with (f:=f) (lb:=lb) (ub:=ub) ; intuition.
+ apply derivable_continuous_pt ; apply f_derivable ; intuition.
+ exact (proj1 x_encad). exact (proj2 x_encad).  apply f_incr ; intuition.
+ assumption.
+ intros x0 x0_encad ; apply f_eq_g ; intuition.
+ rewrite pr_nu_var2_interv with (g:=f) (lb:=lb) (ub:=ub) (pr2:=derivable_pt_recip_interv_prelim1 f g lb ub x lb_lt_ub x_encad
+      f_eq_g g_wf f_incr f_derivable) ; [| |rewrite g_eq_f in g_incr ; rewrite g_eq_f in g_incr| ] ; intuition.
+Qed.
+
+(****************************************************)
+(** *   Value of the derivative of the reciprocal function        *)
+(****************************************************)
+
+Lemma derive_pt_recip_interv_prelim0 : forall (f g:R->R) (lb ub x:R)
+       (Prf:derivable_pt f (g x)) (Prg:derivable_pt g x),
+       lb < ub ->
+       lb < x < ub ->
+       (forall x, lb < x < ub -> (comp f g) x = id x) ->
+       derive_pt f (g x) Prf <> 0 ->
+       derive_pt g x Prg = 1 / (derive_pt f (g x) Prf).
+Proof.
+intros f g lb ub x Prf Prg lb_lt_ub x_encad local_recip Df_neq.
+ replace (derive_pt g x Prg) with
+  ((derive_pt g x Prg) * (derive_pt f (g x) Prf) * / (derive_pt f (g x) Prf)).
+ unfold Rdiv.
+ rewrite (Rmult_comm _ (/ derive_pt f (g x) Prf)).
+ rewrite (Rmult_comm _ (/ derive_pt f (g x) Prf)). 
+ apply Rmult_eq_compat_l. 
+ rewrite Rmult_comm.
+ rewrite <- derive_pt_comp.
+ assert (x_encad2 : lb <= x <= ub) by intuition.
+ rewrite pr_nu_var2_interv with (g:=id) (pr2:= derivable_pt_id_interv lb ub x x_encad2) (lb:=lb) (ub:=ub) ; [reg| | |] ; assumption.
+ rewrite Rmult_assoc, Rinv_r.
+ intuition.
+ assumption.
+Qed.
+
+Lemma derive_pt_recip_interv_prelim1_0 : forall (f g:R->R) (lb ub x:R), 
+       lb < ub ->
+       f lb < x < f ub ->
+       (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) ->
+       (forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
+       (forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x) ->
+       lb < g x < ub.
+Proof.
+intros f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g.
+ assert (Temp:= f_incr_implies_g_incr_interv f g lb ub lb_lt_ub f_incr f_eq_g g_wf).
+ assert (Left_inv := leftinv_is_rightinv_interv f g lb ub f_incr f_eq_g g_wf).
+ unfold comp, id in Left_inv.
+ split ; [rewrite <- Left_inv with (x:=lb) | rewrite <- Left_inv ].
+ apply Temp ; intuition.
+ intuition.
+ apply Temp ; intuition.
+ intuition.
+Qed.
+
+Lemma derive_pt_recip_interv_prelim1_1 : forall (f g:R->R) (lb ub x:R), 
+       lb < ub ->
+       f lb < x < f ub ->
+       (forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y) ->
+       (forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
+       (forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x) ->
+       lb <= g x <= ub.
+Proof.
+intros f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g.
+ assert (Temp := derive_pt_recip_interv_prelim1_0 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g).
+ split ; apply Rlt_le ; intuition.
+Qed.
+
+Lemma derive_pt_recip_interv : forall (f g:R->R) (lb ub x:R)
+       (lb_lt_ub:lb < ub) (x_encad:f lb < x < f ub)
+       (f_incr:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y)
+       (g_wf:forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub)
+       (Prf:forall a : R, lb <= a <= ub -> derivable_pt f a)
+       (f_eq_g:forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x)
+       (Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x
+                 lb_lt_ub x_encad f_eq_g g_wf f_incr Prf) <> 0),
+       derive_pt g x (derivable_pt_recip_interv f g lb ub x lb_lt_ub x_encad f_eq_g
+                 g_wf f_incr Prf Df_neq)
+       =
+       1 / (derive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x
+       lb_lt_ub x_encad f_incr g_wf f_eq_g))).
+Proof.
+intros.
+ assert(g_incr := (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub
+        x_encad f_incr g_wf f_eq_g)).
+ apply derive_pt_recip_interv_prelim0 with (lb:=f lb) (ub:=f ub) ;
+ [intuition |assumption | intuition |].
+ intro Hfalse ; apply Df_neq.  rewrite pr_nu_var2_interv with (g:=f) (lb:=lb) (ub:=ub)
+        (pr2:= (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x lb_lt_ub x_encad
+                 f_incr g_wf f_eq_g))) ;
+ [intuition | intuition | | intuition].
+ exact (derive_pt_recip_interv_prelim1_0 f g lb ub x lb_lt_ub x_encad f_incr g_wf f_eq_g).
+Qed.
+  
+(****************************************************)
+(** * Existence of the derivative of a function which is the limit of a sequence of functions *)
+(****************************************************)
+
+(* begin hide *)
+Lemma ub_lt_2_pos : forall x ub lb, lb < x -> x < ub -> 0 < (ub-lb)/2.
+Proof.
+intros x ub lb lb_lt_x x_lt_ub.
+ assert (T : 0 < ub - lb).
+  fourier.
+ unfold Rdiv ; apply Rlt_mult_inv_pos ; intuition.
+Qed.
+
+Definition mkposreal_lb_ub (x lb ub:R) (lb_lt_x:lb<x) (x_lt_ub:x<ub) : posreal.
+ apply (mkposreal ((ub-lb)/2) (ub_lt_2_pos x ub lb lb_lt_x x_lt_ub)).
+Defined.
+(* end hide *)
+
+Definition boule_of_interval x y (h : x < y) :
+  {c :R & {r : posreal | c - r = x /\ c + r = y}}.
+exists ((x + y)/2).
+assert (radius : 0 < (y - x)/2).
+ unfold Rdiv; apply Rmult_lt_0_compat; fourier.
+ exists (mkposreal _ radius).
+ simpl; split; unfold Rdiv; field.
+Qed.
+
+Definition boule_in_interval x y z (h : x < z < y) :
+  {c : R & {r | Boule c r z /\  x < c - r /\ c + r < y}}.
+Proof.
+assert (cmp : x * /2 + z * /2 < z * /2 + y * /2).
+destruct h as [h1 h2]; fourier.
+destruct (boule_of_interval _ _ cmp) as [c [r [P1 P2]]].
+exists c, r; split.
+ destruct h; unfold Boule; simpl; apply Rabs_def1; fourier.
+destruct h; split; fourier.
+Qed.
+
+Lemma Ball_in_inter : forall c1 c2 r1 r2 x,
+  Boule c1 r1 x -> Boule c2 r2 x ->
+  {r3 : posreal | forall y, Boule x r3 y -> Boule c1 r1 y /\ Boule c2 r2 y}.
+intros c1 c2 [r1 r1p] [r2 r2p] x; unfold Boule; simpl; intros in1 in2.
+assert (Rmax (c1 - r1)(c2 - r2) < x).
+ apply Rmax_lub_lt;[revert in1 | revert in2]; intros h;
+  apply Rabs_def2 in h; destruct h; fourier.
+assert (x < Rmin (c1 + r1) (c2 + r2)).
+ apply Rmin_glb_lt;[revert in1 | revert in2]; intros h;
+  apply Rabs_def2 in h; destruct h; fourier.
+assert (t: 0 < Rmin (x - Rmax (c1 - r1) (c2 - r2)) 
+              (Rmin (c1 + r1) (c2 + r2) - x)).
+ apply Rmin_glb_lt; fourier.
+exists (mkposreal _ t).
+apply Rabs_def2 in in1; destruct in1.
+apply Rabs_def2 in in2; destruct in2.
+assert (c1 - r1 <= Rmax (c1 - r1) (c2 - r2)) by apply Rmax_l.
+assert (c2 - r2 <= Rmax (c1 - r1) (c2 - r2)) by apply Rmax_r.
+assert (Rmin (c1 + r1) (c2 + r2) <= c1 + r1) by apply Rmin_l.
+assert (Rmin (c1 + r1) (c2 + r2) <= c2 + r2) by apply Rmin_r.
+assert (Rmin (x - Rmax (c1 - r1) (c2 - r2)) 
+         (Rmin (c1 + r1) (c2 + r2) - x) <= x - Rmax (c1 - r1) (c2 - r2))
+ by apply Rmin_l.
+assert (Rmin (x - Rmax (c1 - r1) (c2 - r2)) 
+         (Rmin (c1 + r1) (c2 + r2) - x) <= Rmin (c1 + r1) (c2 + r2) - x)
+ by apply Rmin_r.
+simpl.
+intros y h; apply Rabs_def2 in h; destruct h;split; apply Rabs_def1; fourier.
+Qed.
+
+Lemma Boule_center : forall x r, Boule x r x.
+Proof.
+intros x [r rpos]; unfold Boule, Rminus; simpl; rewrite Rplus_opp_r.
+rewrite Rabs_pos_eq;[assumption | apply Rle_refl].
+Qed.
+
+Lemma derivable_pt_lim_CVU : forall (fn fn':nat -> R -> R) (f g:R->R)
+      (x:R) c r, Boule c r x ->
+      (forall y n, Boule c r y -> derivable_pt_lim (fn n) y (fn' n y)) ->
+      (forall y, Boule c r y -> Un_cv (fun n => fn n y) (f y)) ->
+      (CVU fn' g c r) ->
+      (forall y, Boule c r y -> continuity_pt g y) ->
+      derivable_pt_lim f x (g x).
+Proof.
+intros fn fn' f g x c' r xinb Dfn_eq_fn' fn_CV_f fn'_CVU_g g_cont eps eps_pos.
+assert (eps_8_pos : 0 < eps / 8) by fourier.
+elim (g_cont x xinb _ eps_8_pos) ; clear g_cont ;
+intros delta1 (delta1_pos, g_cont).
+destruct (Ball_in_inter _ _ _ _ _ xinb
+   (Boule_center x (mkposreal _ delta1_pos)))
+   as [delta Pdelta].
+exists delta; intros h hpos hinbdelta.
+assert (eps'_pos : 0 < (Rabs h) * eps / 4).
+ unfold Rdiv ; rewrite Rmult_assoc ; apply Rmult_lt_0_compat.
+ apply Rabs_pos_lt ; assumption.
+fourier.
+destruct (fn_CV_f x xinb ((Rabs h) * eps / 4) eps'_pos) as [N2 fnx_CV_fx].
+assert (xhinbxdelta : Boule x delta (x + h)).
+ clear -hinbdelta; apply Rabs_def2 in hinbdelta; unfold Boule; simpl.
+ destruct hinbdelta; apply Rabs_def1; fourier.
+assert (t : Boule c' r (x + h)).
+ apply Pdelta in xhinbxdelta; tauto.
+destruct (fn_CV_f (x+h) t ((Rabs h) * eps / 4) eps'_pos) as [N1 fnxh_CV_fxh].
+clear fn_CV_f t.
+destruct (fn'_CVU_g (eps/8) eps_8_pos) as [N3 fn'c_CVU_gc].
+pose (N := ((N1 + N2) + N3)%nat).
+assert (Main : Rabs ((f (x+h) - fn N (x+h)) - (f x - fn N x) + (fn N (x+h) - fn N x - h * (g x))) < (Rabs h)*eps).
+ apply Rle_lt_trans with (Rabs (f (x + h) - fn N (x + h) - (f x - fn N x)) +  Rabs ((fn N (x + h) - fn N x - h * g x))).
+  solve[apply Rabs_triang].
+ apply Rle_lt_trans with (Rabs (f (x + h) - fn N (x + h)) + Rabs (- (f x - fn N x)) + Rabs (fn N (x + h) - fn N x - h * g x)).
+  solve[apply Rplus_le_compat_r ; apply Rabs_triang].
+ rewrite Rabs_Ropp.
+ case (Rlt_le_dec h 0) ; intro sgn_h.
+  assert (pr1 : forall c : R, x + h < c < x -> derivable_pt (fn N) c).
+   intros c c_encad ; unfold derivable_pt.
+   exists (fn' N c) ; apply Dfn_eq_fn'.
+   assert (t : Boule x delta c).
+    apply Rabs_def2 in xhinbxdelta; destruct xhinbxdelta; destruct c_encad.
+    apply Rabs_def2 in xinb; apply Rabs_def1; fourier.
+   apply Pdelta in t; tauto.
+  assert (pr2 : forall c : R, x + h < c < x -> derivable_pt id c).
+   solve[intros; apply derivable_id].
+  assert (xh_x : x+h < x) by fourier.
+  assert (pr3 : forall c : R, x + h <= c <= x -> continuity_pt (fn N) c).
+   intros c c_encad ; apply derivable_continuous_pt.
+   exists (fn' N c) ; apply Dfn_eq_fn' ; intuition.
+   assert (t : Boule x delta c).
+    apply Rabs_def2 in xhinbxdelta; destruct xhinbxdelta.
+    apply Rabs_def2 in xinb; apply Rabs_def1; fourier.
+   apply Pdelta in t; tauto.
+  assert (pr4 : forall c : R, x + h <= c <= x -> continuity_pt id c).
+   solve[intros; apply derivable_continuous ; apply derivable_id].
+  destruct (MVT (fn N) id (x+h) x pr1 pr2 xh_x pr3 pr4) as [c [P Hc]].
+  assert (Hc' : h * derive_pt (fn N) c (pr1 c P) = (fn N (x+h) - fn N x)).
+   apply Rmult_eq_reg_l with (-1).
+    replace (-1 * (h * derive_pt (fn N) c (pr1 c P))) with (-h * derive_pt (fn N) c (pr1 c P)) by field.
+    replace (-1 * (fn N (x + h) - fn N x)) with (- (fn N (x + h) - fn N x)) by field.
+    replace (-h) with (id x - id (x + h)) by (unfold id; field).
+    rewrite <- Rmult_1_r ; replace 1 with (derive_pt id c (pr2 c P)) by reg.
+    replace (- (fn N (x + h) - fn N x)) with (fn N x - fn N (x + h)) by field.
+    assumption.
+   solve[apply Rlt_not_eq ; intuition].
+  rewrite <- Hc'; clear Hc Hc'.
+  replace (derive_pt (fn N) c (pr1 c P)) with (fn' N c).
+  replace (h * fn' N c - h * g x) with (h * (fn' N c - g x)) by field.
+   rewrite Rabs_mult.
+   apply Rlt_trans with (Rabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x)).
+    apply Rplus_lt_compat_r ; apply Rplus_lt_compat_r ; unfold R_dist in fnxh_CV_fxh ;
+ rewrite Rabs_minus_sym ; apply fnxh_CV_fxh.
+   unfold N; omega.
+   apply Rlt_trans with (Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x)).
+    apply Rplus_lt_compat_r ; apply Rplus_lt_compat_l.
+    unfold R_dist in fnx_CV_fx ; rewrite Rabs_minus_sym ; apply fnx_CV_fx.
+    unfold N ; omega.
+   replace (fn' N c - g x)  with ((fn' N c - g c) +  (g c - g x)) by field.
+   apply Rle_lt_trans with (Rabs h * eps / 4 + Rabs h * eps / 4 +
+          Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x)).
+    rewrite Rplus_assoc ; rewrite Rplus_assoc ; rewrite Rplus_assoc ;
+    apply Rplus_le_compat_l ; apply Rplus_le_compat_l ;
+    rewrite <- Rmult_plus_distr_l ; apply Rmult_le_compat_l. 
+     solve[apply Rabs_pos].
+    solve[apply Rabs_triang].
+   apply Rlt_trans with (Rabs h * eps / 4 + Rabs h * eps / 4 +
+          Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x)).
+    apply Rplus_lt_compat_r; apply Rplus_lt_compat_l; apply Rmult_lt_compat_l.
+     apply Rabs_pos_lt ; assumption.
+    rewrite Rabs_minus_sym ; apply fn'c_CVU_gc.
+     unfold N ; omega.
+    assert (t : Boule x delta c).
+     destruct P.
+     apply Rabs_def2 in xhinbxdelta; destruct xhinbxdelta.
+     apply Rabs_def2 in xinb; apply Rabs_def1; fourier.
+    apply Pdelta in t; tauto.
+   apply Rlt_trans with (Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) +
+         Rabs h * (eps / 8)).
+    rewrite Rplus_assoc ; rewrite Rplus_assoc ; rewrite Rplus_assoc ; rewrite Rplus_assoc ;
+    apply Rplus_lt_compat_l ; apply Rplus_lt_compat_l ; rewrite <- Rmult_plus_distr_l ;
+    rewrite <- Rmult_plus_distr_l ; apply Rmult_lt_compat_l.
+     apply Rabs_pos_lt ; assumption.
+     apply Rplus_lt_compat_l ; simpl in g_cont ; apply g_cont ; split ; [unfold D_x ; split |].
+      solve[unfold no_cond ; intuition].
+     apply Rgt_not_eq ; exact (proj2 P).
+    apply Rlt_trans with (Rabs h).
+     apply Rabs_def1. 
+      apply Rlt_trans with 0.
+       destruct P; fourier.
+      apply Rabs_pos_lt ; assumption.
+     rewrite <- Rabs_Ropp, Rabs_pos_eq, Ropp_involutive;[ | fourier]. 
+     destruct P; fourier.
+    clear -Pdelta xhinbxdelta.
+    apply Pdelta in xhinbxdelta; destruct xhinbxdelta as [_ P'].
+    apply Rabs_def2 in P'; simpl in P'; destruct P';
+    apply Rabs_def1; fourier.
+   rewrite Rplus_assoc ; rewrite Rplus_assoc ; rewrite <- Rmult_plus_distr_l.
+   replace (Rabs h * eps / 4 + (Rabs h * eps / 4 + Rabs h * (eps / 8 + eps / 8))) with
+      (Rabs h * (eps / 4 + eps / 4 + eps / 8 + eps / 8)) by field.
+   apply Rmult_lt_compat_l. 
+    apply Rabs_pos_lt ; assumption.
+   fourier.
+  assert (H := pr1 c P) ; elim H ; clear H ; intros l Hl.
+  assert (Temp : l = fn' N c).
+   assert (bc'rc : Boule c' r c).
+    assert (t : Boule x delta c).
+     clear - xhinbxdelta P.
+     destruct P; apply Rabs_def2 in xhinbxdelta; destruct xhinbxdelta.
+     apply Rabs_def1; fourier.
+    apply Pdelta in t; tauto.
+   assert (Hl' := Dfn_eq_fn' c N bc'rc).
+   unfold derivable_pt_abs in Hl; clear -Hl Hl'.
+   apply uniqueness_limite with (f:= fn N) (x:=c) ; assumption.
+  rewrite <- Temp.
+  assert (Hl' : derivable_pt (fn N) c).
+  exists l ; apply Hl.
+  rewrite pr_nu_var with (g:= fn N) (pr2:=Hl').
+   elim Hl' ; clear Hl' ; intros l' Hl'.
+   assert (Main : l = l').
+    apply uniqueness_limite with (f:= fn N) (x:=c) ; assumption.
+   rewrite Main ; reflexivity.
+  reflexivity.
+ assert (h_pos : h > 0).
+  case sgn_h ; intro Hyp.
+  assumption.
+  apply False_ind ; apply hpos ; symmetry ; assumption.
+ clear sgn_h.
+ assert (pr1 : forall c : R, x < c < x + h -> derivable_pt (fn N) c).
+  intros c c_encad ; unfold derivable_pt.
+  exists (fn' N c) ; apply Dfn_eq_fn'.
+  assert (t : Boule x delta c).
+   apply Rabs_def2 in xhinbxdelta; destruct xhinbxdelta; destruct c_encad.
+   apply Rabs_def2 in xinb; apply Rabs_def1; fourier.
+  apply Pdelta in t; tauto.
+  assert (pr2 : forall c : R, x < c < x + h -> derivable_pt id c).
+   solve[intros; apply derivable_id].
+  assert (xh_x : x < x + h) by fourier.
+  assert (pr3 : forall c : R, x <= c <= x + h -> continuity_pt (fn N) c).
+   intros c c_encad ; apply derivable_continuous_pt.
+   exists (fn' N c) ; apply Dfn_eq_fn' ; intuition.
+   assert (t : Boule x delta c).
+    apply Rabs_def2 in xhinbxdelta; destruct xhinbxdelta.
+    apply Rabs_def2 in xinb; apply Rabs_def1; fourier.
+   apply Pdelta in t; tauto.
+  assert (pr4 : forall c : R, x <= c <= x + h -> continuity_pt id c).
+   solve[intros; apply derivable_continuous ; apply derivable_id].
+  destruct (MVT (fn N) id x (x+h) pr1 pr2 xh_x pr3 pr4) as [c [P Hc]].
+  assert (Hc' : h * derive_pt (fn N) c (pr1 c P) = fn N (x+h) - fn N x).
+    pattern h at 1; replace h with (id (x + h) - id x) by (unfold id; field).
+    rewrite <- Rmult_1_r ; replace 1 with (derive_pt id c (pr2 c P)) by reg.
+    assumption.
+  rewrite <- Hc'; clear Hc Hc'.
+  replace (derive_pt (fn N) c (pr1 c P)) with (fn' N c).
+  replace (h * fn' N c - h * g x) with (h * (fn' N c - g x)) by field.
+   rewrite Rabs_mult.
+   apply Rlt_trans with (Rabs h * eps / 4 + Rabs (f x - fn N x) + Rabs h * Rabs (fn' N c - g x)).
+    apply Rplus_lt_compat_r ; apply Rplus_lt_compat_r ; unfold R_dist in fnxh_CV_fxh ;
+ rewrite Rabs_minus_sym ; apply fnxh_CV_fxh.
+   unfold N; omega.
+   apply Rlt_trans with (Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * Rabs (fn' N c - g x)).
+    apply Rplus_lt_compat_r ; apply Rplus_lt_compat_l.
+    unfold R_dist in fnx_CV_fx ; rewrite Rabs_minus_sym ; apply fnx_CV_fx.
+    unfold N ; omega.
+   replace (fn' N c - g x)  with ((fn' N c - g c) +  (g c - g x)) by field.
+   apply Rle_lt_trans with (Rabs h * eps / 4 + Rabs h * eps / 4 +
+          Rabs h * Rabs (fn' N c - g c) + Rabs h * Rabs (g c - g x)).
+    rewrite Rplus_assoc ; rewrite Rplus_assoc ; rewrite Rplus_assoc ;
+    apply Rplus_le_compat_l ; apply Rplus_le_compat_l ;
+    rewrite <- Rmult_plus_distr_l ; apply Rmult_le_compat_l. 
+     solve[apply Rabs_pos].
+    solve[apply Rabs_triang].
+   apply Rlt_trans with (Rabs h * eps / 4 + Rabs h * eps / 4 +
+          Rabs h * (eps / 8) + Rabs h * Rabs (g c - g x)).
+    apply Rplus_lt_compat_r; apply Rplus_lt_compat_l; apply Rmult_lt_compat_l.
+     apply Rabs_pos_lt ; assumption.
+    rewrite Rabs_minus_sym ; apply fn'c_CVU_gc.
+     unfold N ; omega.
+    assert (t : Boule x delta c).
+     destruct P.
+    apply Rabs_def2 in xhinbxdelta; destruct xhinbxdelta.
+    apply Rabs_def2 in xinb; apply Rabs_def1; fourier.
+   apply Pdelta in t; tauto.
+  apply Rlt_trans with (Rabs h * eps / 4 + Rabs h * eps / 4 + Rabs h * (eps / 8) +
+         Rabs h * (eps / 8)).
+   rewrite Rplus_assoc ; rewrite Rplus_assoc ; rewrite Rplus_assoc ; rewrite Rplus_assoc ;
+    apply Rplus_lt_compat_l ; apply Rplus_lt_compat_l ; rewrite <- Rmult_plus_distr_l ;
+   rewrite <- Rmult_plus_distr_l ; apply Rmult_lt_compat_l.
+    apply Rabs_pos_lt ; assumption.
+   apply Rplus_lt_compat_l ; simpl in g_cont ; apply g_cont ; split ; [unfold D_x ; split |].
+     solve[unfold no_cond ; intuition].
+    apply Rlt_not_eq ; exact (proj1 P).
+   apply Rlt_trans with (Rabs h).
+    apply Rabs_def1.
+     destruct P; rewrite Rabs_pos_eq;fourier.
+    apply Rle_lt_trans with 0.
+     assert (t := Rabs_pos h); clear -t; fourier.
+    clear -P; destruct P; fourier.
+   clear -Pdelta xhinbxdelta.
+   apply Pdelta in xhinbxdelta; destruct xhinbxdelta as [_ P'].
+   apply Rabs_def2 in P'; simpl in P'; destruct P';
+    apply Rabs_def1; fourier.
+  rewrite Rplus_assoc ; rewrite Rplus_assoc ; rewrite <- Rmult_plus_distr_l.
+  replace (Rabs h * eps / 4 + (Rabs h * eps / 4 + Rabs h * (eps / 8 + eps / 8))) with
+      (Rabs h * (eps / 4 + eps / 4 + eps / 8 + eps / 8)) by field.
+  apply Rmult_lt_compat_l. 
+   apply Rabs_pos_lt ; assumption.
+  fourier.
+ assert (H := pr1 c P) ; elim H ; clear H ; intros l Hl.
+ assert (Temp : l = fn' N c).
+  assert (bc'rc : Boule c' r c).
+   assert (t : Boule x delta c).
+    clear - xhinbxdelta P.
+    destruct P; apply Rabs_def2 in xhinbxdelta; destruct xhinbxdelta.
+    apply Rabs_def1; fourier.
+   apply Pdelta in t; tauto.
+  assert (Hl' := Dfn_eq_fn' c N bc'rc).
+  unfold derivable_pt_abs in Hl; clear -Hl Hl'.
+  apply uniqueness_limite with (f:= fn N) (x:=c) ; assumption.
+ rewrite <- Temp.
+ assert (Hl' : derivable_pt (fn N) c).
+ exists l ; apply Hl.
+ rewrite pr_nu_var with (g:= fn N) (pr2:=Hl').
+  elim Hl' ; clear Hl' ; intros l' Hl'.
+  assert (Main : l = l').
+   apply uniqueness_limite with (f:= fn N) (x:=c) ; assumption.
+  rewrite Main ; reflexivity.
+ reflexivity.
+ replace ((f (x + h) - f x) / h - g x) with ((/h) * ((f (x + h) - f x) - h * g x)). 
+ rewrite Rabs_mult ; rewrite Rabs_Rinv.
+ replace eps with (/ Rabs h * (Rabs h * eps)).
+ apply Rmult_lt_compat_l.
+ apply Rinv_0_lt_compat ; apply Rabs_pos_lt ; assumption.
+ replace (f (x + h) - f x - h * g x) with (f (x + h) - fn N (x + h) - (f x - fn N x) +
+          (fn N (x + h) - fn N x - h * g x)) by field.
+ assumption.
+ field ; apply Rgt_not_eq ; apply Rabs_pos_lt ; assumption.
+ assumption.
+ field. assumption.
+Qed.
\ No newline at end of file
diff --git a/theories/Reals/Ranalysis_reg.v b/theories/Reals/Ranalysis_reg.v
new file mode 100644
index 00000000..a4b18288
--- /dev/null
+++ b/theories/Reals/Ranalysis_reg.v
@@ -0,0 +1,800 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+Require Import Rbase.
+Require Import Rfunctions.
+Require Import Rtrigo1.
+Require Import SeqSeries.
+Require Export Ranalysis1.
+Require Export Ranalysis2.
+Require Export Ranalysis3.
+Require Export Rtopology.
+Require Export MVT.
+Require Export PSeries_reg.
+Require Export Exp_prop.
+Require Export Rtrigo_reg.
+Require Export Rsqrt_def.
+Require Export R_sqrt.
+Require Export Rtrigo_calc.
+Require Export Rgeom.
+Require Export RList.
+Require Export Sqrt_reg.
+Require Export Ranalysis4.
+Require Export Rpower.
+Local Open Scope R_scope.
+
+Axiom AppVar : R.
+
+(**********)
+Ltac intro_hyp_glob trm :=
+  match constr:trm with
+    | (?X1 + ?X2)%F =>
+      match goal with
+        |  |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
+        |  |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
+        | _ => idtac
+      end
+    | (?X1 - ?X2)%F =>
+      match goal with
+        |  |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
+        |  |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
+        | _ => idtac
+      end
+    | (?X1 * ?X2)%F =>
+      match goal with
+        |  |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
+        |  |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
+        | _ => idtac
+      end
+    | (?X1 / ?X2)%F =>
+      let aux := constr:X2 in
+        match goal with
+          | _:(forall x0:R, aux x0 <> 0) |- (derivable _) =>
+            intro_hyp_glob X1; intro_hyp_glob X2
+          | _:(forall x0:R, aux x0 <> 0) |- (continuity _) =>
+            intro_hyp_glob X1; intro_hyp_glob X2
+          |  |- (derivable _) =>
+            cut (forall x0:R, aux x0 <> 0);
+              [ intro; intro_hyp_glob X1; intro_hyp_glob X2 | try assumption ]
+          |  |- (continuity _) =>
+            cut (forall x0:R, aux x0 <> 0);
+              [ intro; intro_hyp_glob X1; intro_hyp_glob X2 | try assumption ]
+          | _ => idtac
+        end
+    | (comp ?X1 ?X2) =>
+      match goal with
+        |  |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2
+        |  |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2
+        | _ => idtac
+      end
+    | (- ?X1)%F =>
+      match goal with
+        |  |- (derivable _) => intro_hyp_glob X1
+        |  |- (continuity _) => intro_hyp_glob X1
+        | _ => idtac
+      end
+    | (/ ?X1)%F =>
+      let aux := constr:X1 in
+        match goal with
+          | _:(forall x0:R, aux x0 <> 0) |- (derivable _) =>
+            intro_hyp_glob X1
+          | _:(forall x0:R, aux x0 <> 0) |- (continuity _) =>
+            intro_hyp_glob X1
+          |  |- (derivable _) =>
+            cut (forall x0:R, aux x0 <> 0);
+              [ intro; intro_hyp_glob X1 | try assumption ]
+          |  |- (continuity _) =>
+            cut (forall x0:R, aux x0 <> 0);
+              [ intro; intro_hyp_glob X1 | try assumption ]
+          | _ => idtac
+        end
+    | cos => idtac
+    | sin => idtac
+    | cosh => idtac
+    | sinh => idtac
+    | exp => idtac
+    | Rsqr => idtac
+    | sqrt => idtac
+    | id => idtac
+    | (fct_cte _) => idtac
+    | (pow_fct _) => idtac
+    | Rabs => idtac
+    | ?X1 =>
+      let p := constr:X1 in
+        match goal with
+          | _:(derivable p) |- _ => idtac
+          |  |- (derivable p) => idtac
+          |  |- (derivable _) =>
+            cut (True -> derivable p);
+              [ intro HYPPD; cut (derivable p);
+                [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
+                | idtac ]
+          | _:(continuity p) |- _ => idtac
+          |  |- (continuity p) => idtac
+          |  |- (continuity _) =>
+            cut (True -> continuity p);
+              [ intro HYPPD; cut (continuity p);
+                [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
+                | idtac ]
+          | _ => idtac
+        end
+  end.
+
+(**********)
+Ltac intro_hyp_pt trm pt :=
+  match constr:trm with
+    | (?X1 + ?X2)%F =>
+      match goal with
+        |  |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+        |  |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+        |  |- (derive_pt _ _ _ = _) =>
+          intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+        | _ => idtac
+      end
+    | (?X1 - ?X2)%F =>
+      match goal with
+        |  |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+        |  |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+        |  |- (derive_pt _ _ _ = _) =>
+          intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+        | _ => idtac
+      end
+    | (?X1 * ?X2)%F =>
+      match goal with
+        |  |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+        |  |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+        |  |- (derive_pt _ _ _ = _) =>
+          intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+        | _ => idtac
+      end
+    | (?X1 / ?X2)%F =>
+      let aux := constr:X2 in
+        match goal with
+          | _:(aux pt <> 0) |- (derivable_pt _ _) =>
+            intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+          | _:(aux pt <> 0) |- (continuity_pt _ _) =>
+            intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+          | _:(aux pt <> 0) |- (derive_pt _ _ _ = _) =>
+            intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+          | id:(forall x0:R, aux x0 <> 0) |- (derivable_pt _ _) =>
+            generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+          | id:(forall x0:R, aux x0 <> 0) |- (continuity_pt _ _) =>
+            generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+          | id:(forall x0:R, aux x0 <> 0) |- (derive_pt _ _ _ = _) =>
+            generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt
+          |  |- (derivable_pt _ _) =>
+            cut (aux pt <> 0);
+              [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ]
+          |  |- (continuity_pt _ _) =>
+            cut (aux pt <> 0);
+              [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ]
+          |  |- (derive_pt _ _ _ = _) =>
+            cut (aux pt <> 0);
+              [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ]
+          | _ => idtac
+        end
+    | (comp ?X1 ?X2) =>
+      match goal with
+        |  |- (derivable_pt _ _) =>
+          let pt_f1 := eval cbv beta in (X2 pt) in
+            (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt)
+        |  |- (continuity_pt _ _) =>
+          let pt_f1 := eval cbv beta in (X2 pt) in
+            (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt)
+        |  |- (derive_pt _ _ _ = _) =>
+          let pt_f1 := eval cbv beta in (X2 pt) in
+            (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt)
+        | _ => idtac
+      end
+    | (- ?X1)%F =>
+      match goal with
+        |  |- (derivable_pt _ _) => intro_hyp_pt X1 pt
+        |  |- (continuity_pt _ _) => intro_hyp_pt X1 pt
+        |  |- (derive_pt _ _ _ = _) => intro_hyp_pt X1 pt
+        | _ => idtac
+      end
+    | (/ ?X1)%F =>
+      let aux := constr:X1 in
+        match goal with
+          | _:(aux pt <> 0) |- (derivable_pt _ _) =>
+            intro_hyp_pt X1 pt
+          | _:(aux pt <> 0) |- (continuity_pt _ _) =>
+            intro_hyp_pt X1 pt
+          | _:(aux pt <> 0) |- (derive_pt _ _ _ = _) =>
+            intro_hyp_pt X1 pt
+          | id:(forall x0:R, aux x0 <> 0) |- (derivable_pt _ _) =>
+            generalize (id pt); intro; intro_hyp_pt X1 pt
+          | id:(forall x0:R, aux x0 <> 0) |- (continuity_pt _ _) =>
+            generalize (id pt); intro; intro_hyp_pt X1 pt
+          | id:(forall x0:R, aux x0 <> 0) |- (derive_pt _ _ _ = _) =>
+            generalize (id pt); intro; intro_hyp_pt X1 pt
+          |  |- (derivable_pt _ _) =>
+            cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ]
+          |  |- (continuity_pt _ _) =>
+            cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ]
+          |  |- (derive_pt _ _ _ = _) =>
+            cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ]
+          | _ => idtac
+        end
+    | cos => idtac
+    | sin => idtac
+    | cosh => idtac
+    | sinh => idtac
+    | exp => idtac
+    | Rsqr => idtac
+    | id => idtac
+    | (fct_cte _) => idtac
+    | (pow_fct _) => idtac
+    | sqrt =>
+      match goal with
+        |  |- (derivable_pt _ _) => cut (0 < pt); [ intro | try assumption ]
+        |  |- (continuity_pt _ _) =>
+          cut (0 <= pt); [ intro | try assumption ]
+        |  |- (derive_pt _ _ _ = _) =>
+          cut (0 < pt); [ intro | try assumption ]
+        | _ => idtac
+      end
+    | Rabs =>
+      match goal with
+        |  |- (derivable_pt _ _) =>
+          cut (pt <> 0); [ intro | try assumption ]
+        | _ => idtac
+      end
+    | ?X1 =>
+      let p := constr:X1 in
+        match goal with
+          | _:(derivable_pt p pt) |- _ => idtac
+          |  |- (derivable_pt p pt) => idtac
+          |  |- (derivable_pt _ _) =>
+            cut (True -> derivable_pt p pt);
+              [ intro HYPPD; cut (derivable_pt p pt);
+                [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
+                | idtac ]
+          | _:(continuity_pt p pt) |- _ => idtac
+          |  |- (continuity_pt p pt) => idtac
+          |  |- (continuity_pt _ _) =>
+            cut (True -> continuity_pt p pt);
+              [ intro HYPPD; cut (continuity_pt p pt);
+                [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
+                | idtac ]
+          |  |- (derive_pt _ _ _ = _) =>
+            cut (True -> derivable_pt p pt);
+              [ intro HYPPD; cut (derivable_pt p pt);
+                [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ]
+                | idtac ]
+          | _ => idtac
+        end
+  end.
+
+(**********)
+Ltac is_diff_pt :=
+  match goal with
+    |  |- (derivable_pt Rsqr _) =>
+
+  (* fonctions de base *)
+      apply derivable_pt_Rsqr
+    |  |- (derivable_pt id ?X1) => apply (derivable_pt_id X1)
+    |  |- (derivable_pt (fct_cte _) _) => apply derivable_pt_const
+    |  |- (derivable_pt sin _) => apply derivable_pt_sin
+    |  |- (derivable_pt cos _) => apply derivable_pt_cos
+    |  |- (derivable_pt sinh _) => apply derivable_pt_sinh
+    |  |- (derivable_pt cosh _) => apply derivable_pt_cosh
+    |  |- (derivable_pt exp _) => apply derivable_pt_exp
+    |  |- (derivable_pt (pow_fct _) _) =>
+      unfold pow_fct in |- *; apply derivable_pt_pow
+    |  |- (derivable_pt sqrt ?X1) =>
+      apply (derivable_pt_sqrt X1);
+        assumption ||
+          unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct,
+            comp, id, fct_cte, pow_fct in |- *
+    |  |- (derivable_pt Rabs ?X1) =>
+      apply (Rderivable_pt_abs X1);
+        assumption ||
+          unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct,
+            comp, id, fct_cte, pow_fct in |- *
+        (* regles de differentiabilite *)
+        (* PLUS *)
+    |  |- (derivable_pt (?X1 + ?X2) ?X3) =>
+      apply (derivable_pt_plus X1 X2 X3); is_diff_pt
+       (* MOINS *)
+    |  |- (derivable_pt (?X1 - ?X2) ?X3) =>
+      apply (derivable_pt_minus X1 X2 X3); is_diff_pt
+       (* OPPOSE *)
+    |  |- (derivable_pt (- ?X1) ?X2) =>
+      apply (derivable_pt_opp X1 X2);
+        is_diff_pt
+       (* MULTIPLICATION PAR UN SCALAIRE *)
+    |  |- (derivable_pt (mult_real_fct ?X1 ?X2) ?X3) =>
+      apply (derivable_pt_scal X2 X1 X3); is_diff_pt
+       (* MULTIPLICATION *)
+    |  |- (derivable_pt (?X1 * ?X2) ?X3) =>
+      apply (derivable_pt_mult X1 X2 X3); is_diff_pt
+       (* DIVISION *)
+    |  |- (derivable_pt (?X1 / ?X2) ?X3) =>
+      apply (derivable_pt_div X1 X2 X3);
+        [ is_diff_pt
+          | is_diff_pt
+          | try
+            assumption ||
+              unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+                comp, pow_fct, id, fct_cte in |- * ]
+    |  |- (derivable_pt (/ ?X1) ?X2) =>
+
+       (* INVERSION *)
+      apply (derivable_pt_inv X1 X2);
+        [ assumption ||
+          unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+            comp, pow_fct, id, fct_cte in |- *
+          | is_diff_pt ]
+    |  |- (derivable_pt (comp ?X1 ?X2) ?X3) =>
+
+       (* COMPOSITION *)
+      apply (derivable_pt_comp X2 X1 X3); is_diff_pt
+    | _:(derivable_pt ?X1 ?X2) |- (derivable_pt ?X1 ?X2) =>
+      assumption
+    | _:(derivable ?X1) |- (derivable_pt ?X1 ?X2) =>
+      cut (derivable X1); [ intro HypDDPT; apply HypDDPT | assumption ]
+    |  |- (True -> derivable_pt _ _) =>
+      intro HypTruE; clear HypTruE; is_diff_pt
+    | _ =>
+      try
+        unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
+          fct_cte, comp, pow_fct in |- *
+  end.
+
+(**********)
+Ltac is_diff_glob :=
+  match goal with
+    |  |- (derivable Rsqr) =>
+  (* fonctions de base *)
+      apply derivable_Rsqr
+    |  |- (derivable id) => apply derivable_id
+    |  |- (derivable (fct_cte _)) => apply derivable_const
+    |  |- (derivable sin) => apply derivable_sin
+    |  |- (derivable cos) => apply derivable_cos
+    |  |- (derivable cosh) => apply derivable_cosh
+    |  |- (derivable sinh) => apply derivable_sinh
+    |  |- (derivable exp) => apply derivable_exp
+    |  |- (derivable (pow_fct _)) =>
+      unfold pow_fct in |- *;
+        apply derivable_pow
+        (* regles de differentiabilite *)
+        (* PLUS *)
+    |  |- (derivable (?X1 + ?X2)) =>
+      apply (derivable_plus X1 X2); is_diff_glob
+       (* MOINS *)
+    |  |- (derivable (?X1 - ?X2)) =>
+      apply (derivable_minus X1 X2); is_diff_glob
+       (* OPPOSE *)
+    |  |- (derivable (- ?X1)) =>
+      apply (derivable_opp X1);
+        is_diff_glob
+       (* MULTIPLICATION PAR UN SCALAIRE *)
+    |  |- (derivable (mult_real_fct ?X1 ?X2)) =>
+      apply (derivable_scal X2 X1); is_diff_glob
+       (* MULTIPLICATION *)
+    |  |- (derivable (?X1 * ?X2)) =>
+      apply (derivable_mult X1 X2); is_diff_glob
+       (* DIVISION *)
+    |  |- (derivable (?X1 / ?X2)) =>
+      apply (derivable_div X1 X2);
+        [ is_diff_glob
+          | is_diff_glob
+          | try
+            assumption ||
+              unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+                id, fct_cte, comp, pow_fct in |- * ]
+    |  |- (derivable (/ ?X1)) =>
+
+       (* INVERSION *)
+      apply (derivable_inv X1);
+        [ try
+          assumption ||
+            unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+              id, fct_cte, comp, pow_fct in |- *
+          | is_diff_glob ]
+    |  |- (derivable (comp sqrt _)) =>
+
+       (* COMPOSITION *)
+      unfold derivable in |- *; intro; try is_diff_pt
+    |  |- (derivable (comp Rabs _)) =>
+      unfold derivable in |- *; intro; try is_diff_pt
+    |  |- (derivable (comp ?X1 ?X2)) =>
+      apply (derivable_comp X2 X1); is_diff_glob
+    | _:(derivable ?X1) |- (derivable ?X1) => assumption
+    |  |- (True -> derivable _) =>
+      intro HypTruE; clear HypTruE; is_diff_glob
+    | _ =>
+      try
+        unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
+          fct_cte, comp, pow_fct in |- *
+  end.
+
+(**********)
+Ltac is_cont_pt :=
+  match goal with
+    |  |- (continuity_pt Rsqr _) =>
+
+       (* fonctions de base *)
+      apply derivable_continuous_pt; apply derivable_pt_Rsqr
+    |  |- (continuity_pt id ?X1) =>
+      apply derivable_continuous_pt; apply (derivable_pt_id X1)
+    |  |- (continuity_pt (fct_cte _) _) =>
+      apply derivable_continuous_pt; apply derivable_pt_const
+    |  |- (continuity_pt sin _) =>
+      apply derivable_continuous_pt; apply derivable_pt_sin
+    |  |- (continuity_pt cos _) =>
+      apply derivable_continuous_pt; apply derivable_pt_cos
+    |  |- (continuity_pt sinh _) =>
+      apply derivable_continuous_pt; apply derivable_pt_sinh
+    |  |- (continuity_pt cosh _) =>
+      apply derivable_continuous_pt; apply derivable_pt_cosh
+    |  |- (continuity_pt exp _) =>
+      apply derivable_continuous_pt; apply derivable_pt_exp
+    |  |- (continuity_pt (pow_fct _) _) =>
+      unfold pow_fct in |- *; apply derivable_continuous_pt;
+        apply derivable_pt_pow
+    |  |- (continuity_pt sqrt ?X1) =>
+      apply continuity_pt_sqrt;
+        assumption ||
+          unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct,
+            comp, id, fct_cte, pow_fct in |- *
+    |  |- (continuity_pt Rabs ?X1) =>
+      apply (Rcontinuity_abs X1)
+       (* regles de differentiabilite *)
+       (* PLUS *)
+    |  |- (continuity_pt (?X1 + ?X2) ?X3) =>
+      apply (continuity_pt_plus X1 X2 X3); is_cont_pt
+       (* MOINS *)
+    |  |- (continuity_pt (?X1 - ?X2) ?X3) =>
+      apply (continuity_pt_minus X1 X2 X3); is_cont_pt
+       (* OPPOSE *)
+    |  |- (continuity_pt (- ?X1) ?X2) =>
+      apply (continuity_pt_opp X1 X2);
+        is_cont_pt
+       (* MULTIPLICATION PAR UN SCALAIRE *)
+    |  |- (continuity_pt (mult_real_fct ?X1 ?X2) ?X3) =>
+      apply (continuity_pt_scal X2 X1 X3); is_cont_pt
+       (* MULTIPLICATION *)
+    |  |- (continuity_pt (?X1 * ?X2) ?X3) =>
+      apply (continuity_pt_mult X1 X2 X3); is_cont_pt
+       (* DIVISION *)
+    |  |- (continuity_pt (?X1 / ?X2) ?X3) =>
+      apply (continuity_pt_div X1 X2 X3);
+        [ is_cont_pt
+          | is_cont_pt
+          | try
+            assumption ||
+              unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+                comp, id, fct_cte, pow_fct in |- * ]
+    |  |- (continuity_pt (/ ?X1) ?X2) =>
+
+       (* INVERSION *)
+      apply (continuity_pt_inv X1 X2);
+        [ is_cont_pt
+          | assumption ||
+            unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+              comp, id, fct_cte, pow_fct in |- * ]
+    |  |- (continuity_pt (comp ?X1 ?X2) ?X3) =>
+
+       (* COMPOSITION *)
+      apply (continuity_pt_comp X2 X1 X3); is_cont_pt
+    | _:(continuity_pt ?X1 ?X2) |- (continuity_pt ?X1 ?X2) =>
+      assumption
+    | _:(continuity ?X1) |- (continuity_pt ?X1 ?X2) =>
+      cut (continuity X1); [ intro HypDDPT; apply HypDDPT | assumption ]
+    | _:(derivable_pt ?X1 ?X2) |- (continuity_pt ?X1 ?X2) =>
+      apply derivable_continuous_pt; assumption
+    | _:(derivable ?X1) |- (continuity_pt ?X1 ?X2) =>
+      cut (continuity X1);
+        [ intro HypDDPT; apply HypDDPT
+          | apply derivable_continuous; assumption ]
+    |  |- (True -> continuity_pt _ _) =>
+      intro HypTruE; clear HypTruE; is_cont_pt
+    | _ =>
+      try
+        unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
+          fct_cte, comp, pow_fct in |- *
+  end.
+
+(**********)
+Ltac is_cont_glob :=
+  match goal with
+    |  |- (continuity Rsqr) =>
+
+       (* fonctions de base *)
+      apply derivable_continuous; apply derivable_Rsqr
+    |  |- (continuity id) => apply derivable_continuous; apply derivable_id
+    |  |- (continuity (fct_cte _)) =>
+      apply derivable_continuous; apply derivable_const
+    |  |- (continuity sin) => apply derivable_continuous; apply derivable_sin
+    |  |- (continuity cos) => apply derivable_continuous; apply derivable_cos
+    |  |- (continuity exp) => apply derivable_continuous; apply derivable_exp
+    |  |- (continuity (pow_fct _)) =>
+      unfold pow_fct in |- *; apply derivable_continuous; apply derivable_pow
+    |  |- (continuity sinh) =>
+      apply derivable_continuous; apply derivable_sinh
+    |  |- (continuity cosh) =>
+      apply derivable_continuous; apply derivable_cosh
+    |  |- (continuity Rabs) =>
+      apply Rcontinuity_abs
+       (* regles de continuite *)
+       (* PLUS *)
+    |  |- (continuity (?X1 + ?X2)) =>
+      apply (continuity_plus X1 X2);
+        try is_cont_glob || assumption
+            (* MOINS *)
+    |  |- (continuity (?X1 - ?X2)) =>
+      apply (continuity_minus X1 X2);
+        try is_cont_glob || assumption
+            (* OPPOSE *)
+    |  |- (continuity (- ?X1)) =>
+      apply (continuity_opp X1); try is_cont_glob || assumption
+                                      (* INVERSE *)
+    |  |- (continuity (/ ?X1)) =>
+      apply (continuity_inv X1);
+        try is_cont_glob || assumption
+            (* MULTIPLICATION PAR UN SCALAIRE *)
+    |  |- (continuity (mult_real_fct ?X1 ?X2)) =>
+      apply (continuity_scal X2 X1);
+        try is_cont_glob || assumption
+            (* MULTIPLICATION *)
+    |  |- (continuity (?X1 * ?X2)) =>
+      apply (continuity_mult X1 X2);
+        try is_cont_glob || assumption
+            (* DIVISION *)
+    |  |- (continuity (?X1 / ?X2)) =>
+      apply (continuity_div X1 X2);
+        [ try is_cont_glob || assumption
+          | try is_cont_glob || assumption
+          | try
+            assumption ||
+              unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct,
+                id, fct_cte, pow_fct in |- * ]
+    |  |- (continuity (comp sqrt _)) =>
+
+       (* COMPOSITION *)
+      unfold continuity_pt in |- *; intro; try is_cont_pt
+    |  |- (continuity (comp ?X1 ?X2)) =>
+      apply (continuity_comp X2 X1); try is_cont_glob || assumption
+    | _:(continuity ?X1) |- (continuity ?X1) => assumption
+    |  |- (True -> continuity _) =>
+      intro HypTruE; clear HypTruE; is_cont_glob
+    | _:(derivable ?X1) |- (continuity ?X1) =>
+      apply derivable_continuous; assumption
+    | _ =>
+      try
+        unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id,
+          fct_cte, comp, pow_fct in |- *
+  end.
+
+(**********)
+Ltac rew_term trm :=
+  match constr:trm with
+    | (?X1 + ?X2) =>
+      let p1 := rew_term X1 with p2 := rew_term X2 in
+        match constr:p1 with
+          | (fct_cte ?X3) =>
+            match constr:p2 with
+              | (fct_cte ?X4) => constr:(fct_cte (X3 + X4))
+              | _ => constr:(p1 + p2)%F
+            end
+          | _ => constr:(p1 + p2)%F
+        end
+    | (?X1 - ?X2) =>
+      let p1 := rew_term X1 with p2 := rew_term X2 in
+        match constr:p1 with
+          | (fct_cte ?X3) =>
+            match constr:p2 with
+              | (fct_cte ?X4) => constr:(fct_cte (X3 - X4))
+              | _ => constr:(p1 - p2)%F
+            end
+          | _ => constr:(p1 - p2)%F
+        end
+    | (?X1 / ?X2) =>
+      let p1 := rew_term X1 with p2 := rew_term X2 in
+        match constr:p1 with
+          | (fct_cte ?X3) =>
+            match constr:p2 with
+              | (fct_cte ?X4) => constr:(fct_cte (X3 / X4))
+              | _ => constr:(p1 / p2)%F
+            end
+          | _ =>
+            match constr:p2 with
+              | (fct_cte ?X4) => constr:(p1 * fct_cte (/ X4))%F
+              | _ => constr:(p1 / p2)%F
+            end
+        end
+    | (?X1 * / ?X2) =>
+      let p1 := rew_term X1 with p2 := rew_term X2 in
+        match constr:p1 with
+          | (fct_cte ?X3) =>
+            match constr:p2 with
+              | (fct_cte ?X4) => constr:(fct_cte (X3 / X4))
+              | _ => constr:(p1 / p2)%F
+            end
+          | _ =>
+            match constr:p2 with
+              | (fct_cte ?X4) => constr:(p1 * fct_cte (/ X4))%F
+              | _ => constr:(p1 / p2)%F
+            end
+        end
+    | (?X1 * ?X2) =>
+      let p1 := rew_term X1 with p2 := rew_term X2 in
+        match constr:p1 with
+          | (fct_cte ?X3) =>
+            match constr:p2 with
+              | (fct_cte ?X4) => constr:(fct_cte (X3 * X4))
+              | _ => constr:(p1 * p2)%F
+            end
+          | _ => constr:(p1 * p2)%F
+        end
+    | (- ?X1) =>
+      let p := rew_term X1 in
+        match constr:p with
+          | (fct_cte ?X2) => constr:(fct_cte (- X2))
+          | _ => constr:(- p)%F
+        end
+    | (/ ?X1) =>
+      let p := rew_term X1 in
+        match constr:p with
+          | (fct_cte ?X2) => constr:(fct_cte (/ X2))
+          | _ => constr:(/ p)%F
+        end
+    | (?X1 AppVar) => constr:X1
+    | (?X1 ?X2) =>
+      let p := rew_term X2 in
+        match constr:p with
+          | (fct_cte ?X3) => constr:(fct_cte (X1 X3))
+          | _ => constr:(comp X1 p)
+        end
+    | AppVar => constr:id
+    | (AppVar ^ ?X1) => constr:(pow_fct X1)
+    | (?X1 ^ ?X2) =>
+      let p := rew_term X1 in
+        match constr:p with
+          | (fct_cte ?X3) => constr:(fct_cte (pow_fct X2 X3))
+          | _ => constr:(comp (pow_fct X2) p)
+        end
+    | ?X1 => constr:(fct_cte X1)
+  end.
+
+(**********)
+Ltac deriv_proof trm pt :=
+  match constr:trm with
+    | (?X1 + ?X2)%F =>
+      let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
+        constr:(derivable_pt_plus X1 X2 pt p1 p2)
+    | (?X1 - ?X2)%F =>
+      let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
+        constr:(derivable_pt_minus X1 X2 pt p1 p2)
+    | (?X1 * ?X2)%F =>
+      let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
+        constr:(derivable_pt_mult X1 X2 pt p1 p2)
+    | (?X1 / ?X2)%F =>
+      match goal with
+        | id:(?X2 pt <> 0) |- _ =>
+          let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in
+            constr:(derivable_pt_div X1 X2 pt p1 p2 id)
+        | _ => constr:False
+      end
+    | (/ ?X1)%F =>
+      match goal with
+        | id:(?X1 pt <> 0) |- _ =>
+          let p1 := deriv_proof X1 pt in
+            constr:(derivable_pt_inv X1 pt p1 id)
+        | _ => constr:False
+      end
+    | (comp ?X1 ?X2) =>
+      let pt_f1 := eval cbv beta in (X2 pt) in
+        let p1 := deriv_proof X1 pt_f1 with p2 := deriv_proof X2 pt in
+          constr:(derivable_pt_comp X2 X1 pt p2 p1)
+    | (- ?X1)%F =>
+      let p1 := deriv_proof X1 pt in
+        constr:(derivable_pt_opp X1 pt p1)
+    | sin => constr:(derivable_pt_sin pt)
+    | cos => constr:(derivable_pt_cos pt)
+    | sinh => constr:(derivable_pt_sinh pt)
+    | cosh => constr:(derivable_pt_cosh pt)
+    | exp => constr:(derivable_pt_exp pt)
+    | id => constr:(derivable_pt_id pt)
+    | Rsqr => constr:(derivable_pt_Rsqr pt)
+    | sqrt =>
+      match goal with
+        | id:(0 < pt) |- _ => constr:(derivable_pt_sqrt pt id)
+        | _ => constr:False
+      end
+    | (fct_cte ?X1) => constr:(derivable_pt_const X1 pt)
+    | ?X1 =>
+      let aux := constr:X1 in
+        match goal with
+          | id:(derivable_pt aux pt) |- _ => constr:id
+          | id:(derivable aux) |- _ => constr:(id pt)
+          | _ => constr:False
+        end
+  end.
+
+(**********)
+Ltac simplify_derive trm pt :=
+  match constr:trm with
+    | (?X1 + ?X2)%F =>
+      try rewrite derive_pt_plus; simplify_derive X1 pt;
+        simplify_derive X2 pt
+    | (?X1 - ?X2)%F =>
+      try rewrite derive_pt_minus; simplify_derive X1 pt;
+        simplify_derive X2 pt
+    | (?X1 * ?X2)%F =>
+      try rewrite derive_pt_mult; simplify_derive X1 pt;
+        simplify_derive X2 pt
+    | (?X1 / ?X2)%F =>
+      try rewrite derive_pt_div; simplify_derive X1 pt; simplify_derive X2 pt
+    | (comp ?X1 ?X2) =>
+      let pt_f1 := eval cbv beta in (X2 pt) in
+        (try rewrite derive_pt_comp; simplify_derive X1 pt_f1;
+          simplify_derive X2 pt)
+    | (- ?X1)%F => try rewrite derive_pt_opp; simplify_derive X1 pt
+    | (/ ?X1)%F =>
+      try rewrite derive_pt_inv; simplify_derive X1 pt
+    | (fct_cte ?X1) => try rewrite derive_pt_const
+    | id => try rewrite derive_pt_id
+    | sin => try rewrite derive_pt_sin
+    | cos => try rewrite derive_pt_cos
+    | sinh => try rewrite derive_pt_sinh
+    | cosh => try rewrite derive_pt_cosh
+    | exp => try rewrite derive_pt_exp
+    | Rsqr => try rewrite derive_pt_Rsqr
+    | sqrt => try rewrite derive_pt_sqrt
+    | ?X1 =>
+      let aux := constr:X1 in
+        match goal with
+          | id:(derive_pt aux pt ?X2 = _),H:(derivable aux) |- _ =>
+            try replace (derive_pt aux pt (H pt)) with (derive_pt aux pt X2);
+              [ rewrite id | apply pr_nu ]
+          | id:(derive_pt aux pt ?X2 = _),H:(derivable_pt aux pt) |- _ =>
+            try replace (derive_pt aux pt H) with (derive_pt aux pt X2);
+              [ rewrite id | apply pr_nu ]
+          | _ => idtac
+        end
+    | _ => idtac
+  end.
+
+(**********)
+Ltac reg :=
+  match goal with
+    |  |- (derivable_pt ?X1 ?X2) =>
+      let trm := eval cbv beta in (X1 AppVar) in
+        let aux := rew_term trm in
+          (intro_hyp_pt aux X2;
+            try (change (derivable_pt aux X2) in |- *; is_diff_pt) || is_diff_pt)
+    |  |- (derivable ?X1) =>
+      let trm := eval cbv beta in (X1 AppVar) in
+        let aux := rew_term trm in
+          (intro_hyp_glob aux;
+            try (change (derivable aux) in |- *; is_diff_glob) || is_diff_glob)
+    |  |- (continuity ?X1) =>
+      let trm := eval cbv beta in (X1 AppVar) in
+        let aux := rew_term trm in
+          (intro_hyp_glob aux;
+            try (change (continuity aux) in |- *; is_cont_glob) || is_cont_glob)
+    |  |- (continuity_pt ?X1 ?X2) =>
+      let trm := eval cbv beta in (X1 AppVar) in
+        let aux := rew_term trm in
+          (intro_hyp_pt aux X2;
+            try (change (continuity_pt aux X2) in |- *; is_cont_pt) || is_cont_pt)
+    |  |- (derive_pt ?X1 ?X2 ?X3 = ?X4) =>
+      let trm := eval cbv beta in (X1 AppVar) in
+      let aux := rew_term trm in
+      intro_hyp_pt aux X2;
+      (let aux2 := deriv_proof aux X2 in
+       try
+         (replace (derive_pt X1 X2 X3) with (derive_pt aux X2 aux2);
+           [ simplify_derive aux X2;
+             try unfold plus_fct, minus_fct, mult_fct, div_fct, id, fct_cte,
+                        inv_fct, opp_fct in |- *; ring || ring_simplify
+           | try apply pr_nu ]) || is_diff_pt)
+  end.
diff --git a/theories/Reals/Ratan.v b/theories/Reals/Ratan.v
new file mode 100644
index 00000000..1a0ea969
--- /dev/null
+++ b/theories/Reals/Ratan.v
@@ -0,0 +1,1602 @@
+Require Import Fourier.
+Require Import Rbase.
+Require Import PSeries_reg.
+Require Import Rtrigo1.
+Require Import Ranalysis_reg.
+Require Import Rfunctions.
+Require Import AltSeries.
+Require Import Rseries.
+Require Import SeqProp.
+Require Import Ranalysis5.
+Require Import SeqSeries.
+Require Import PartSum.
+
+Local Open Scope R_scope.
+
+(** Tools *)
+
+Lemma Ropp_div : forall x y, -x/y = -(x/y).
+Proof.
+intros x y; unfold Rdiv; rewrite <-Ropp_mult_distr_l_reverse; reflexivity.
+Qed.
+
+Definition pos_half_prf : 0 < /2.
+Proof. fourier. Qed.
+
+Definition pos_half := mkposreal (/2) pos_half_prf.
+
+Lemma Boule_half_to_interval :
+  forall x , Boule (/2) pos_half x -> 0 <= x <= 1.
+Proof.
+unfold Boule, pos_half; simpl.
+intros x b; apply Rabs_def2 in b; destruct b; split; fourier.
+Qed.
+
+Lemma Boule_lt : forall c r x, Boule c r x -> Rabs x < Rabs c + r.
+Proof.
+unfold Boule; intros c r x h.
+apply Rabs_def2 in h; destruct h; apply Rabs_def1;
+ (destruct (Rle_lt_dec 0 c);[rewrite Rabs_pos_eq; fourier | 
+    rewrite <- Rabs_Ropp, Rabs_pos_eq; fourier]).
+Qed.
+
+(* The following lemma does not belong here. *)
+Lemma Un_cv_ext :
+  forall un vn, (forall n, un n = vn n) ->
+  forall l, Un_cv un l -> Un_cv vn l.
+Proof.
+intros un vn quv l P eps ep; destruct (P eps ep) as [N Pn]; exists N.
+intro n; rewrite <- quv; apply Pn.
+Qed.
+
+(* The following two lemmas are general purposes about alternated series.
+  They do not belong here. *)
+Lemma Alt_first_term_bound :forall f l N n,
+   Un_decreasing f -> Un_cv f 0 ->
+   Un_cv (sum_f_R0 (tg_alt f)) l ->
+   (N <= n)%nat ->
+   R_dist (sum_f_R0 (tg_alt f) n) l <= f N.
+Proof.
+intros f l.
+assert (WLOG :
+   forall n P, (forall k, (0 < k)%nat -> P k) ->
+         ((forall k, (0 < k)%nat -> P k) -> P 0%nat) -> P n).
+clear.
+intros [ | n] P Hs Ho;[solve[apply Ho, Hs] | apply Hs; auto with arith].
+intros N; pattern N; apply WLOG; clear N.
+intros [ | N] Npos n decr to0 cv nN.
+  clear -Npos; elimtype False; omega.
+ assert (decr' : Un_decreasing (fun i => f (S N + i)%nat)).
+  intros k; replace (S N+S k)%nat with (S (S N+k)) by ring.
+  apply (decr (S N + k)%nat).
+ assert (to' : Un_cv (fun i => f (S N + i)%nat) 0).
+  intros eps ep; destruct (to0 eps ep) as [M PM].
+  exists M; intros k kM; apply PM; omega.
+ assert (cv' : Un_cv
+         (sum_f_R0 (tg_alt (fun i => ((-1) ^ S N * f(S N + i)%nat))))
+             (l - sum_f_R0 (tg_alt f) N)).
+  intros eps ep; destruct (cv eps ep) as [M PM]; exists M.
+  intros n' nM. 
+  match goal with |- ?C => set (U := C) end.
+  assert (nM' : (n' + S N >= M)%nat) by omega.
+   generalize (PM _ nM'); unfold R_dist.
+  rewrite (tech2 (tg_alt f) N (n' + S N)).
+  assert (t : forall a b c, (a + b) - c = b - (c - a)) by (intros; ring).
+  rewrite t; clear t; unfold U, R_dist; clear U.
+  replace (n' + S N - S N)%nat with n' by omega.
+  rewrite <- (sum_eq (tg_alt (fun i => (-1) ^ S N * f(S N + i)%nat))).
+   tauto.
+   intros i _; unfold tg_alt.
+   rewrite <- Rmult_assoc, <- pow_add, !(plus_comm i); reflexivity.
+  omega.
+  assert (cv'' : Un_cv (sum_f_R0 (tg_alt (fun i => f (S N + i)%nat)))
+                   ((-1) ^ S N * (l - sum_f_R0 (tg_alt f) N))).
+  apply (Un_cv_ext (fun n => (-1) ^ S N * 
+            sum_f_R0 (tg_alt (fun i : nat => (-1) ^ S N * f (S N + i)%nat)) n)).
+   intros n0; rewrite scal_sum; apply sum_eq; intros i _.
+   unfold tg_alt; ring_simplify; replace (((-1) ^ S N) ^ 2) with 1.
+    ring.
+   rewrite <- pow_mult, mult_comm, pow_mult; replace ((-1) ^2) with 1 by ring.
+   rewrite pow1; reflexivity.
+  apply CV_mult.
+   solve[intros eps ep; exists 0%nat; intros; rewrite R_dist_eq; auto].
+  assumption.
+ destruct (even_odd_cor N) as [p [Neven | Nodd]].
+  rewrite Neven; destruct (alternated_series_ineq _ _ p decr to0 cv) as [B C].
+  case (even_odd_cor n) as [p' [neven | nodd]].
+   rewrite neven.
+   destruct (alternated_series_ineq _ _ p' decr to0 cv) as [D E].
+   unfold R_dist; rewrite Rabs_pos_eq;[ | fourier].
+   assert (dist : (p <= p')%nat) by omega.
+   assert (t := decreasing_prop _ _ _ (CV_ALT_step1 f decr) dist).
+   apply Rle_trans with (sum_f_R0 (tg_alt f) (2 * p) - l).
+    unfold Rminus; apply Rplus_le_compat_r; exact t.
+   match goal with _ : ?a <= l, _ : l <= ?b |- _ => 
+    replace (f (S (2 * p))) with (b - a) by
+     (rewrite tech5; unfold tg_alt; rewrite pow_1_odd; ring); fourier
+   end.
+  rewrite nodd; destruct (alternated_series_ineq _ _ p' decr to0 cv) as [D E].
+   unfold R_dist; rewrite <- Rabs_Ropp, Rabs_pos_eq, Ropp_minus_distr;
+     [ | fourier].
+   assert (dist : (p <= p')%nat) by omega.
+   apply Rle_trans with (l - sum_f_R0 (tg_alt f) (S (2 * p))).
+    unfold Rminus; apply Rplus_le_compat_l, Ropp_le_contravar.
+    solve[apply Rge_le, (growing_prop _ _ _ (CV_ALT_step0 f decr) dist)].
+   unfold Rminus; rewrite tech5, Ropp_plus_distr, <- Rplus_assoc.
+   unfold tg_alt at 2; rewrite pow_1_odd, Ropp_mult_distr_l_reverse; fourier.
+ rewrite Nodd; destruct (alternated_series_ineq _ _ p decr to0 cv) as [B _].
+ destruct (alternated_series_ineq _ _ (S p) decr to0 cv) as [_ C].
+ assert (keep : (2 * S p = S (S ( 2 * p)))%nat) by ring.
+ case (even_odd_cor n) as [p' [neven | nodd]].
+   rewrite neven;
+   destruct (alternated_series_ineq _ _ p' decr to0 cv) as [D E].
+   unfold R_dist; rewrite Rabs_pos_eq;[ | fourier].
+   assert (dist : (S p < S p')%nat) by omega.
+   apply Rle_trans with (sum_f_R0 (tg_alt f) (2 * S p) - l).
+   unfold Rminus; apply Rplus_le_compat_r,
+     (decreasing_prop _ _ _ (CV_ALT_step1 f decr)).
+   omega.
+  rewrite keep, tech5; unfold tg_alt at 2; rewrite <- keep, pow_1_even.
+  fourier.
+ rewrite nodd; destruct (alternated_series_ineq _ _ p' decr to0 cv) as [D E].
+ unfold R_dist; rewrite <- Rabs_Ropp, Rabs_pos_eq;[ | fourier].
+ rewrite Ropp_minus_distr.
+ apply Rle_trans with (l - sum_f_R0 (tg_alt f) (S (2 * p))).
+  unfold Rminus; apply Rplus_le_compat_l, Ropp_le_contravar, Rge_le,
+   (growing_prop _ _ _ (CV_ALT_step0 f decr)); omega.
+ generalize C; rewrite keep, tech5; unfold tg_alt.
+ rewrite <- keep, pow_1_even.
+ assert (t : forall a b c, a <= b + 1 * c -> a - b <= c) by (intros; fourier).
+ solve[apply t].
+clear WLOG; intros Hyp [ | n] decr to0 cv _.
+ generalize (alternated_series_ineq f l 0 decr to0 cv).
+ unfold R_dist, tg_alt; simpl; rewrite !Rmult_1_l, !Rmult_1_r.
+ assert (f 1%nat <= f 0%nat) by apply decr.
+ rewrite Ropp_mult_distr_l_reverse.
+ intros [A B]; rewrite Rabs_pos_eq; fourier.
+apply Rle_trans with (f 1%nat).
+ apply (Hyp 1%nat (le_n 1) (S n) decr to0 cv).
+ omega.
+solve[apply decr].
+Qed.
+
+Lemma Alt_CVU : forall (f : nat -> R -> R) g h c r,
+   (forall x, Boule c r x ->Un_decreasing (fun n => f n x)) -> 
+   (forall x, Boule c r x -> Un_cv (fun n => f n x) 0) ->
+   (forall x, Boule c r x -> 
+       Un_cv (sum_f_R0 (tg_alt  (fun i => f i x))) (g x)) ->
+   (forall x n, Boule c r x -> f n x <= h n) ->
+   (Un_cv h 0) ->
+   CVU (fun N x => sum_f_R0 (tg_alt (fun i => f i x)) N) g c r.
+Proof.
+intros f g h c r decr to0 to_g bound bound0 eps ep.  
+assert (ep' : 0 <eps/2) by fourier.
+destruct (bound0 _ ep) as [N Pn]; exists N.
+intros n y nN dy.
+rewrite <- Rabs_Ropp, Ropp_minus_distr; apply Rle_lt_trans with (f n y).
+ solve[apply (Alt_first_term_bound (fun i => f i y) (g y) n n); auto].
+apply Rle_lt_trans with (h n).
+ apply bound; assumption.
+clear - nN Pn.
+generalize (Pn _ nN); unfold R_dist; rewrite Rminus_0_r; intros t.
+apply Rabs_def2 in t; tauto.
+Qed.
+
+(* The following lemmas are general purpose lemmas about squares. 
+  They do not belong here *)
+
+Lemma pow2_ge_0 : forall x, 0 <= x ^ 2.
+Proof.
+intros x; destruct (Rle_lt_dec 0 x).
+ replace (x ^ 2) with (x * x) by field.
+ apply Rmult_le_pos; assumption.
+ replace (x ^ 2) with ((-x) * (-x)) by field.
+apply Rmult_le_pos; fourier.
+Qed.
+
+Lemma pow2_abs : forall x,  Rabs x ^ 2 = x ^ 2.
+Proof.
+intros x; destruct (Rle_lt_dec 0 x).
+ rewrite Rabs_pos_eq;[field | assumption].
+rewrite <- Rabs_Ropp, Rabs_pos_eq;[field | fourier].
+Qed.
+
+(** * Properties of tangent *)
+
+Lemma derivable_pt_tan : forall x, -PI/2 < x < PI/2 -> derivable_pt tan x.
+Proof.
+intros x xint.
+ unfold derivable_pt, tan. 
+ apply derivable_pt_div ; [reg | reg | ].
+ apply Rgt_not_eq.
+ unfold Rgt ; apply cos_gt_0;
+  [unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse; fold (-PI/2) |];tauto.
+Qed.
+
+Lemma derive_pt_tan : forall (x:R),
+ forall (Pr1: -PI/2 < x < PI/2),
+ derive_pt tan x (derivable_pt_tan x Pr1) = 1 + (tan x)^2.
+Proof.
+intros x pr.
+assert (cos x <> 0).
+ apply Rgt_not_eq, cos_gt_0; rewrite <- ?Ropp_div; tauto.
+unfold tan; reg; unfold pow, Rsqr; field; assumption.
+Qed.
+
+(** Proof that tangent is a bijection *)
+(* to be removed? *)
+
+Lemma derive_increasing_interv :
+  forall (a b:R) (f:R -> R),
+    a < b ->
+    forall (pr:forall x, a < x < b -> derivable_pt f x),
+    (forall t:R, forall (t_encad : a < t < b), 0 < derive_pt f t (pr t t_encad)) ->
+    forall x y:R, a < x < b -> a < y < b -> x < y -> f x < f y.
+Proof.
+intros a b f a_lt_b pr Df_gt_0 x y x_encad y_encad x_lt_y.
+ assert (derivable_id_interv : forall c : R, x < c < y -> derivable_pt id c).
+  intros ; apply derivable_pt_id.
+ assert (derivable_f_interv :  forall c : R, x < c < y -> derivable_pt f c).
+  intros c c_encad. apply pr. split.
+  apply Rlt_trans with (r2:=x) ; [exact (proj1 x_encad) | exact (proj1 c_encad)].
+  apply Rlt_trans with (r2:=y) ; [exact (proj2 c_encad) | exact (proj2 y_encad)].
+ assert (f_cont_interv : forall c : R, x <= c <= y -> continuity_pt f c).
+  intros c c_encad; apply derivable_continuous_pt ; apply pr. split.
+  apply Rlt_le_trans with (r2:=x) ; [exact (proj1 x_encad) | exact (proj1 c_encad)].
+  apply Rle_lt_trans with (r2:=y) ; [ exact (proj2 c_encad) | exact (proj2 y_encad)].
+ assert (id_cont_interv : forall c : R, x <= c <= y -> continuity_pt id c).
+  intros ; apply derivable_continuous_pt ; apply derivable_pt_id. 
+ elim (MVT f id x y derivable_f_interv derivable_id_interv x_lt_y f_cont_interv id_cont_interv).
+  intros c Temp ; elim Temp ; clear Temp ; intros Pr eq.
+   replace (id y - id x) with (y - x) in eq by intuition.
+   replace (derive_pt id c (derivable_id_interv c Pr)) with 1 in eq.
+   assert (Hyp : f y - f x > 0).
+    rewrite Rmult_1_r in eq. rewrite <- eq.
+    apply Rmult_gt_0_compat.
+    apply Rgt_minus ; assumption.
+    assert (c_encad2 : a <= c < b).
+     split.
+     apply Rlt_le ; apply Rlt_trans with (r2:=x) ; [exact (proj1 x_encad) | exact (proj1 Pr)].
+     apply Rle_lt_trans with (r2:=y) ; [apply Rlt_le ; exact (proj2 Pr) | exact (proj2 y_encad)].
+     assert (c_encad : a < c < b).
+      split.
+      apply Rlt_trans with (r2:=x) ; [exact (proj1 x_encad) | exact (proj1 Pr)].
+      apply Rle_lt_trans with (r2:=y) ; [apply Rlt_le ; exact (proj2 Pr) | exact (proj2 y_encad)].
+     assert (Temp := Df_gt_0 c c_encad).
+     assert (Temp2 := pr_nu f c (derivable_f_interv c Pr) (pr c c_encad)).
+     rewrite Temp2 ; apply Temp.
+   apply Rminus_gt ; exact Hyp.
+   symmetry ; rewrite derive_pt_eq ; apply derivable_pt_lim_id.
+Qed.
+
+(* begin hide *)
+Lemma plus_Rsqr_gt_0 : forall x, 1 + x ^ 2 > 0.
+Proof.
+intro m. replace 0 with (0+0) by intuition.
+ apply Rplus_gt_ge_compat. intuition.
+ elim (total_order_T m 0) ; intro s'. case s'.
+ intros m_cond. replace 0 with (0*0) by intuition.
+  replace (m ^ 2) with ((-m)^2).
+  apply Rle_ge ; apply Rmult_le_compat ; intuition ; apply Rlt_le ; rewrite Rmult_1_r ; intuition.
+  field.
+  intro H' ; rewrite H' ; right ; field.
+   left. intuition.
+Qed.
+(* end hide *)
+
+(* The following lemmas about PI should probably be in Rtrigo. *)
+
+Lemma PI2_lower_bound :
+  forall x, 0 < x < 2 -> 0 < cos x  -> x < PI/2.
+Proof.
+intros x [xp xlt2] cx.
+destruct (Rtotal_order x (PI/2)) as [xltpi2 | [xeqpi2 | xgtpi2]].
+  assumption.
+ now case (Rgt_not_eq _ _ cx); rewrite xeqpi2, cos_PI2.
+destruct (MVT_cor1 cos (PI/2) x derivable_cos xgtpi2) as
+   [c [Pc [cint1 cint2]]].
+revert Pc; rewrite cos_PI2, Rminus_0_r. 
+rewrite <- (pr_nu cos c (derivable_pt_cos c)), derive_pt_cos.
+assert (0 < c < 2) by (split; assert (t := PI2_RGT_0); fourier).
+assert (0 < sin c) by now apply sin_pos_tech.
+intros Pc.
+case (Rlt_not_le _ _ cx).
+rewrite <- (Rplus_0_l (cos x)), Pc, Ropp_mult_distr_l_reverse.
+apply Rle_minus, Rmult_le_pos;[apply Rlt_le; assumption | fourier ].
+Qed.
+
+Lemma PI2_3_2 : 3/2 < PI/2.
+Proof.
+apply PI2_lower_bound;[split; fourier | ].
+destruct (pre_cos_bound (3/2) 1) as [t _]; [fourier | fourier | ].
+apply Rlt_le_trans with (2 := t); clear t.
+unfold cos_approx; simpl; unfold cos_term.
+simpl mult; replace ((-1)^ 0) with 1 by ring; replace ((-1)^2) with 1 by ring;
+ replace ((-1)^4) with 1 by ring; replace ((-1)^1) with (-1) by ring;
+ replace ((-1)^3) with (-1) by ring; replace 3 with (IZR 3) by (simpl; ring);
+ replace 2 with (IZR 2) by (simpl; ring); simpl Z.of_nat;
+ rewrite !INR_IZR_INZ, Ropp_mult_distr_l_reverse, Rmult_1_l.
+match goal with |- _ < ?a =>
+replace a with ((- IZR 3 ^ 6 * IZR (Z.of_nat (fact 0)) * IZR (Z.of_nat (fact 2)) *
+    IZR (Z.of_nat (fact 4)) +
+    IZR 3 ^ 4 * IZR 2 ^ 2 * IZR (Z.of_nat (fact 0)) * IZR (Z.of_nat (fact 2)) *
+    IZR (Z.of_nat (fact 6)) -
+    IZR 3 ^ 2 * IZR 2 ^ 4 * IZR (Z.of_nat (fact 0)) * IZR (Z.of_nat (fact 4)) *
+    IZR (Z.of_nat (fact 6)) +
+    IZR 2 ^ 6 * IZR (Z.of_nat (fact 2)) * IZR (Z.of_nat (fact 4)) *
+    IZR (Z.of_nat (fact 6))) /
+   (IZR 2 ^ 6 * IZR (Z.of_nat (fact 0)) * IZR (Z.of_nat (fact 2)) *
+    IZR (Z.of_nat (fact 4)) * IZR (Z.of_nat (fact 6))));[ | field;
+    repeat apply conj;((rewrite <- INR_IZR_INZ; apply INR_fact_neq_0) ||
+         (apply Rgt_not_eq; apply (IZR_lt 0); reflexivity)) ]
+end.
+rewrite !fact_simpl, !Nat2Z.inj_mul; simpl Z.of_nat.
+unfold Rdiv; apply Rmult_lt_0_compat.
+unfold Rminus; rewrite !pow_IZR, <- !opp_IZR, <- !mult_IZR, <- !opp_IZR, 
+  <- !plus_IZR; apply (IZR_lt 0); reflexivity.
+apply Rinv_0_lt_compat; rewrite !pow_IZR, <- !mult_IZR; apply (IZR_lt 0).
+reflexivity.
+Qed.
+
+Lemma PI2_1 : 1 < PI/2.
+Proof. assert (t := PI2_3_2); fourier. Qed.
+
+Lemma tan_increasing :
+  forall x y:R,
+    -PI/2 < x ->
+    x < y -> 
+    y < PI/2 -> tan x < tan y.
+Proof.
+intros x y Z_le_x x_lt_y y_le_1.
+ assert (x_encad : -PI/2 < x < PI/2).
+  split ; [assumption | apply Rlt_trans with (r2:=y) ; assumption].
+ assert (y_encad : -PI/2 < y < PI/2).
+  split ; [apply Rlt_trans with (r2:=x) ; intuition | intuition ].
+ assert (local_derivable_pt_tan : forall x : R, -PI/2 < x < PI/2 ->
+          derivable_pt tan x).
+  intros ; apply derivable_pt_tan ; intuition.
+ apply derive_increasing_interv with (a:=-PI/2) (b:=PI/2) (pr:=local_derivable_pt_tan) ; intuition.
+ fourier.
+ assert (Temp := pr_nu tan t (derivable_pt_tan t t_encad) (local_derivable_pt_tan t t_encad)) ;
+ rewrite <- Temp ; clear Temp.
+ assert (Temp := derive_pt_tan t t_encad) ; rewrite Temp ; clear Temp.
+ apply plus_Rsqr_gt_0.
+Qed.
+
+Lemma tan_is_inj : forall x y, -PI/2 < x < PI/2 -> -PI/2 < y < PI/2 ->
+   tan x = tan y -> x = y.
+Proof.
+  intros a b a_encad b_encad fa_eq_fb.
+  case(total_order_T a b).
+  intro s ; case s ; clear s.
+  intro Hf.
+  assert (Hfalse := tan_increasing a b (proj1 a_encad) Hf (proj2 b_encad)).
+  case (Rlt_not_eq (tan a) (tan b)) ; assumption.
+  intuition.
+  intro Hf. assert (Hfalse := tan_increasing b a (proj1 b_encad) Hf (proj2 a_encad)).
+  case (Rlt_not_eq (tan b) (tan a)) ; [|symmetry] ; assumption.
+Qed.
+
+Lemma exists_atan_in_frame :
+ forall lb ub y, lb < ub -> -PI/2 < lb -> ub < PI/2 ->
+ tan lb < y < tan ub -> {x | lb < x < ub /\ tan x = y}.
+Proof.
+intros lb ub y lb_lt_ub lb_cond ub_cond y_encad.
+ case y_encad ; intros y_encad1 y_encad2.
+     assert (f_cont : forall a : R, lb <= a <= ub -> continuity_pt tan a).
+      intros a a_encad. apply derivable_continuous_pt ; apply derivable_pt_tan.
+      split. apply Rlt_le_trans with (r2:=lb) ; intuition.
+      apply Rle_lt_trans with (r2:=ub) ; intuition.
+     assert (Cont : forall a : R, lb <= a <= ub -> continuity_pt (fun x => tan x - y) a).
+      intros a a_encad. unfold continuity_pt, continue_in, limit1_in, limit_in ; simpl ; unfold R_dist.
+      intros eps eps_pos. elim (f_cont a a_encad eps eps_pos).
+      intros alpha alpha_pos. destruct alpha_pos as (alpha_pos,Temp).
+      exists alpha. split.
+      assumption. intros x  x_cond.
+      replace (tan x - y - (tan a - y)) with (tan x - tan a) by field.
+      exact (Temp x x_cond).
+     assert (H1 : (fun x : R => tan x - y) lb < 0).
+       apply Rlt_minus. assumption.
+      assert (H2 : 0 < (fun x : R => tan x - y) ub).
+       apply Rgt_minus. assumption.
+     destruct (IVT_interv (fun x => tan x - y) lb ub Cont lb_lt_ub H1 H2) as (x,Hx).
+     exists x.
+     destruct Hx as (Hyp,Result).
+     intuition.
+     assert (Temp2 : x <> lb).
+      intro Hfalse. rewrite Hfalse in Result.
+      assert (Temp2 : y <> tan lb).
+       apply Rgt_not_eq ; assumption.
+      clear - Temp2 Result. apply Temp2.
+      intuition.
+     clear -Temp2 H3.
+      case H3 ; intuition. apply False_ind ; apply Temp2 ; symmetry ; assumption.
+      assert (Temp : x <> ub).
+      intro Hfalse. rewrite Hfalse in Result.
+      assert (Temp2 : y <> tan ub).
+       apply Rlt_not_eq ; assumption.
+      clear - Temp2 Result. apply Temp2.
+      intuition.
+      case H4 ; intuition.
+Qed.
+
+(** * Definition of arctangent as the reciprocal function of tangent and proof of this status *)
+Lemma tan_1_gt_1 : tan 1 > 1.
+Proof.
+assert (0 < cos 1) by (apply cos_gt_0; assert (t:=PI2_1); fourier).
+assert (t1 : cos 1 <= 1 - 1/2 + 1/24).
+ destruct (pre_cos_bound 1 0) as [_ t]; try fourier; revert t.
+ unfold cos_approx, cos_term; simpl; intros t; apply Rle_trans with (1:=t).
+ clear t; apply Req_le; field.
+assert (t2 : 1 - 1/6 <= sin 1).
+ destruct (pre_sin_bound 1 0) as [t _]; try fourier; revert t.
+ unfold sin_approx, sin_term; simpl; intros t; apply Rle_trans with (2:=t).
+ clear t; apply Req_le; field.
+pattern 1 at 2; replace 1 with
+  (cos 1 / cos 1) by (field; apply Rgt_not_eq; fourier).
+apply Rlt_gt; apply (Rmult_lt_compat_r (/ cos 1) (cos 1) (sin 1)).
+ apply Rinv_0_lt_compat; assumption.
+apply Rle_lt_trans with (1 := t1); apply Rlt_le_trans with (2 := t2).
+fourier.
+Qed.
+
+Definition frame_tan y : {x | 0 < x < PI/2 /\ Rabs y < tan x}.
+destruct (total_order_T (Rabs y) 1).
+ assert (yle1 : Rabs y <= 1) by (destruct s; fourier).
+ clear s; exists 1; split;[split; [exact Rlt_0_1 | exact PI2_1] | ].
+ apply Rle_lt_trans with (1 := yle1); exact tan_1_gt_1.
+assert (0 < / (Rabs y + 1)).
+ apply Rinv_0_lt_compat; fourier.
+set (u := /2 * / (Rabs y + 1)).
+assert (0 < u).
+ apply Rmult_lt_0_compat; [fourier | assumption].
+assert (vlt1 : / (Rabs y + 1) < 1).
+ apply Rmult_lt_reg_r with (Rabs y + 1).
+  assert (t := Rabs_pos y); fourier.
+ rewrite Rinv_l; [rewrite Rmult_1_l | apply Rgt_not_eq]; fourier.
+assert (vlt2 : u < 1).
+ apply Rlt_trans with (/ (Rabs y + 1)).
+  rewrite double_var.
+  assert (t : forall x, 0 < x -> x < x + x) by (clear; intros; fourier).
+  unfold u; rewrite Rmult_comm; apply t.
+  unfold Rdiv; rewrite Rmult_comm; assumption.
+ assumption.
+assert(int : 0 < PI / 2 - u < PI / 2).
+ split.
+  assert (t := PI2_1); apply Rlt_Rminus, Rlt_trans with (2 := t); assumption.
+ assert (dumb : forall x y, 0 < y -> x - y < x) by (clear; intros; fourier).
+ apply dumb; clear dumb; assumption.
+exists (PI/2 - u).
+assert (tmp : forall x y, 0 < x -> y < 1 -> x * y < x).
+ clear; intros x y x0 y1; pattern x at 2; rewrite <- (Rmult_1_r x).
+ apply Rmult_lt_compat_l; assumption.
+assert (0 < sin u).
+ apply sin_gt_0;[ assumption | ].
+ assert (t := PI2_Rlt_PI); assert (t' := PI2_1).
+ apply Rlt_trans with (2 := Rlt_trans _ _ _ t' t); assumption.
+split.
+ assumption.
+ apply Rlt_trans with (/2 * / cos(PI / 2 - u)).
+ rewrite cos_shift.
+ assert (sin u < u).
+  assert (t1 : 0 <= u) by (apply Rlt_le; assumption).
+  assert (t2 : u <= 4) by
+   (apply Rle_trans with 1;[apply Rlt_le | fourier]; assumption).
+  destruct (pre_sin_bound u 0 t1 t2) as [_ t].
+  apply Rle_lt_trans with (1 := t); clear t1 t2 t.
+  unfold sin_approx; simpl; unfold sin_term; simpl ((-1) ^ 0);
+  replace ((-1) ^ 2) with 1 by ring; simpl ((-1) ^ 1);
+  rewrite !Rmult_1_r, !Rmult_1_l; simpl plus; simpl (INR (fact 1)).
+  rewrite <- (fun x => tech_pow_Rmult x 0), <- (fun x => tech_pow_Rmult x 2),
+    <- (fun x => tech_pow_Rmult x 4).
+  rewrite (Rmult_comm (-1)); simpl ((/(Rabs y + 1)) ^ 0).
+  unfold Rdiv; rewrite Rinv_1, !Rmult_assoc, <- !Rmult_plus_distr_l.
+  apply tmp;[assumption | ].
+  rewrite Rplus_assoc, Rmult_1_l; pattern 1 at 3; rewrite <- Rplus_0_r.
+  apply Rplus_lt_compat_l.
+  rewrite <- Rmult_assoc.
+  match goal with |- (?a * (-1)) + _ < 0 =>
+   rewrite <- (Rplus_opp_l a), Ropp_mult_distr_r_reverse, Rmult_1_r
+  end.
+  apply Rplus_lt_compat_l.
+  assert (0 < u ^ 2) by (apply pow_lt; assumption).
+  replace (u ^ 4) with (u ^ 2 * u ^ 2) by ring.
+ rewrite Rmult_assoc; apply Rmult_lt_compat_l; auto.
+  apply Rlt_trans with (u ^ 2 * /INR (fact 3)).
+   apply Rmult_lt_compat_l; auto.
+   apply Rinv_lt_contravar.
+    solve[apply Rmult_lt_0_compat; apply INR_fact_lt_0].
+   rewrite !INR_IZR_INZ; apply IZR_lt; reflexivity.
+  rewrite Rmult_comm; apply tmp.
+   solve[apply Rinv_0_lt_compat, INR_fact_lt_0].
+  apply Rlt_trans with (2 := vlt2).
+  simpl; unfold u; apply tmp; auto; rewrite Rmult_1_r; assumption.
+ apply Rlt_trans with (Rabs y + 1);[fourier | ]. 
+ pattern (Rabs y + 1) at 1; rewrite <- (Rinv_involutive (Rabs y + 1));
+  [ | apply Rgt_not_eq; fourier].
+ rewrite <- Rinv_mult_distr.
+   apply Rinv_lt_contravar.
+    apply Rmult_lt_0_compat.
+     apply Rmult_lt_0_compat;[fourier | assumption].
+    assumption.
+   replace (/(Rabs y + 1)) with (2 * u).
+    fourier.
+   unfold u; field; apply Rgt_not_eq; clear -r; fourier.
+  solve[discrR].
+ apply Rgt_not_eq; assumption.
+unfold tan.
+set (u' := PI / 2); unfold Rdiv; apply Rmult_lt_compat_r; unfold u'.
+ apply Rinv_0_lt_compat. 
+ rewrite cos_shift; assumption.
+assert (vlt3 : u < /4).
+ replace (/4) with (/2 * /2) by field.
+ unfold u; apply Rmult_lt_compat_l;[fourier | ].
+ apply Rinv_lt_contravar.
+  apply Rmult_lt_0_compat; fourier.
+ fourier.
+assert (1 < PI / 2 - u) by (assert (t := PI2_3_2); fourier).
+apply Rlt_trans with (sin 1).
+ assert (t' : 1 <= 4) by fourier.
+ destruct (pre_sin_bound 1 0 (Rlt_le _ _ Rlt_0_1) t') as [t _].
+ apply Rlt_le_trans with (2 := t); clear t.
+ simpl plus; replace (sin_approx 1 1) with (5/6);[fourier | ].
+ unfold sin_approx, sin_term; simpl; field.
+apply sin_increasing_1.
+    assert (t := PI2_1); fourier.
+   apply Rlt_le, PI2_1.
+  assert (t := PI2_1); fourier.
+ fourier.
+assumption.
+Qed.
+
+Lemma ub_opp : forall x, x < PI/2 -> -PI/2 < -x.
+Proof.
+intros x h; rewrite Ropp_div; apply Ropp_lt_contravar; assumption.
+Qed.
+
+Lemma pos_opp_lt : forall x, 0 < x -> -x < x.
+Proof. intros; fourier. Qed.
+
+Lemma tech_opp_tan : forall x y, -tan x < y -> tan (-x) < y.
+intros; rewrite tan_neg; assumption.
+Qed.
+
+Definition pre_atan (y : R) : {x : R | -PI/2 < x < PI/2 /\ tan x = y}.
+destruct (frame_tan y) as [ub [[ub0 ubpi2] Ptan_ub]].
+set (pr := (conj (tech_opp_tan _ _ (proj2 (Rabs_def2 _ _ Ptan_ub)))
+     (proj1 (Rabs_def2 _ _ Ptan_ub)))).
+destruct (exists_atan_in_frame (-ub) ub y (pos_opp_lt _ ub0) (ub_opp _ ubpi2)
+             ubpi2 pr) as [v [[vl vu] vq]].
+exists v; clear pr.
+split;[rewrite Ropp_div; split; fourier | assumption].
+Qed.
+
+Definition atan x := let (v, _) := pre_atan x in v.
+
+Lemma atan_bound : forall x, -PI/2 < atan x < PI/2.
+Proof.
+intros x; unfold atan; destruct (pre_atan x) as [v [int _]]; exact int.
+Qed.
+
+Lemma atan_right_inv : forall x, tan (atan x) = x.
+Proof.
+intros x; unfold atan; destruct (pre_atan x) as [v [_ q]]; exact q.
+Qed.
+
+Lemma atan_opp : forall x, atan (- x) = - atan x.
+Proof.
+intros x; generalize (atan_bound (-x)); rewrite Ropp_div;intros [a b].
+generalize (atan_bound x); rewrite Ropp_div; intros [c d].
+apply tan_is_inj; try rewrite Ropp_div; try split; try fourier.
+rewrite tan_neg, !atan_right_inv; reflexivity.
+Qed.
+
+Lemma derivable_pt_atan : forall x, derivable_pt atan x.
+Proof.
+intros x.
+destruct (frame_tan x) as [ub [[ub0 ubpi] P]].
+assert (lb_lt_ub : -ub < ub) by apply pos_opp_lt, ub0.
+assert (xint : tan(-ub) < x < tan ub).
+ assert (xint' : x < tan ub /\ -(tan ub) < x) by apply Rabs_def2, P.
+  rewrite tan_neg; tauto.
+assert (inv_p : forall x, tan(-ub) <= x -> x <= tan ub -> 
+                     comp tan atan x = id x).
+ intros; apply atan_right_inv.
+assert (int_tan : forall y, tan (- ub) <= y -> y <= tan ub ->
+                       -ub <= atan y <= ub).
+ clear -ub0 ubpi; intros y lo up; split.
+  destruct (Rle_lt_dec (-ub) (atan y)) as [h | abs]; auto.
+  assert (y < tan (-ub)).
+   rewrite <- (atan_right_inv y); apply tan_increasing.
+     destruct (atan_bound y); assumption.
+    assumption.
+   fourier.
+  fourier.
+ destruct (Rle_lt_dec (atan y) ub) as [h | abs]; auto.
+  assert (tan ub < y).
+   rewrite <- (atan_right_inv y); apply tan_increasing.
+    rewrite Ropp_div; fourier.
+   assumption.
+  destruct (atan_bound y); assumption.
+ fourier.
+assert (incr : forall x y, -ub <= x -> x < y -> y <= ub -> tan x < tan y).
+ intros y z l yz u; apply tan_increasing.
+   rewrite Ropp_div; fourier.
+  assumption.
+ fourier.
+assert (der : forall a, -ub <= a <= ub -> derivable_pt tan a).
+ intros a [la ua]; apply derivable_pt_tan.
+ rewrite Ropp_div; split; fourier.
+assert (df_neq : derive_pt tan (atan x)
+     (derivable_pt_recip_interv_prelim1 tan atan 
+        (- ub) ub x lb_lt_ub xint inv_p int_tan incr der) <> 0).
+ rewrite <- (pr_nu tan (atan x)
+              (derivable_pt_tan (atan x) (atan_bound x))).
+ rewrite derive_pt_tan.
+ solve[apply Rgt_not_eq, plus_Rsqr_gt_0].
+apply (derivable_pt_recip_interv tan atan (-ub) ub x
+      lb_lt_ub xint inv_p int_tan incr der).
+exact df_neq.
+Qed.
+
+Lemma atan_increasing : forall x y, x < y -> atan x < atan y.
+intros x y d.
+assert (t1 := atan_bound x).
+assert (t2 := atan_bound y).
+destruct (Rlt_le_dec (atan x) (atan y)) as [lt | bad].
+ assumption.
+apply Rlt_not_le in d.
+case d.
+rewrite <- (atan_right_inv y), <- (atan_right_inv x).
+destruct bad as [ylt | yx].
+ apply Rlt_le, tan_increasing; try tauto.
+solve[rewrite yx; apply Rle_refl].
+Qed.
+
+Lemma atan_0 : atan 0 = 0.
+apply tan_is_inj; try (apply atan_bound).
+ assert (t := PI_RGT_0); rewrite Ropp_div; split; fourier.
+rewrite atan_right_inv, tan_0.
+reflexivity.
+Qed.
+
+Lemma atan_1 : atan 1 = PI/4.
+assert (ut := PI_RGT_0).
+assert (-PI/2 < PI/4 < PI/2) by (rewrite Ropp_div; split; fourier).
+assert (t := atan_bound 1).
+apply tan_is_inj; auto.
+rewrite tan_PI4, atan_right_inv; reflexivity.
+Qed.
+
+(** atan's derivative value is the function 1 / (1+x²) *)
+
+Lemma derive_pt_atan : forall x,
+      derive_pt atan x (derivable_pt_atan x) =
+         1 / (1 + x²).
+Proof.
+intros x.
+destruct (frame_tan x) as [ub [[ub0 ubpi] Pub]].
+assert (lb_lt_ub : -ub < ub) by apply pos_opp_lt, ub0.
+assert (xint : tan(-ub) < x < tan ub).
+ assert (xint' : x < tan ub /\ -(tan ub) < x) by apply Rabs_def2, Pub.
+  rewrite tan_neg; tauto.
+assert (inv_p : forall x, tan(-ub) <= x -> x <= tan ub -> 
+                     comp tan atan x = id x).
+ intros; apply atan_right_inv.
+assert (int_tan : forall y, tan (- ub) <= y -> y <= tan ub ->
+                       -ub <= atan y <= ub).
+ clear -ub0 ubpi; intros y lo up; split.
+  destruct (Rle_lt_dec (-ub) (atan y)) as [h | abs]; auto.
+  assert (y < tan (-ub)).
+   rewrite <- (atan_right_inv y); apply tan_increasing.
+     destruct (atan_bound y); assumption.
+    assumption.
+   fourier.
+  fourier.
+ destruct (Rle_lt_dec (atan y) ub) as [h | abs]; auto.
+  assert (tan ub < y).
+   rewrite <- (atan_right_inv y); apply tan_increasing.
+    rewrite Ropp_div; fourier.
+   assumption.
+  destruct (atan_bound y); assumption.
+ fourier.
+assert (incr : forall x y, -ub <= x -> x < y -> y <= ub -> tan x < tan y).
+ intros y z l yz u; apply tan_increasing.
+   rewrite Ropp_div; fourier.
+  assumption.
+ fourier.
+assert (der : forall a, -ub <= a <= ub -> derivable_pt tan a).
+ intros a [la ua]; apply derivable_pt_tan.
+ rewrite Ropp_div; split; fourier.
+assert (df_neq : derive_pt tan (atan x)
+     (derivable_pt_recip_interv_prelim1 tan atan 
+        (- ub) ub x lb_lt_ub xint inv_p int_tan incr der) <> 0).
+ rewrite <- (pr_nu tan (atan x)
+              (derivable_pt_tan (atan x) (atan_bound x))).
+ rewrite derive_pt_tan.
+ solve[apply Rgt_not_eq, plus_Rsqr_gt_0].
+assert (t := derive_pt_recip_interv tan atan (-ub) ub x lb_lt_ub
+        xint incr int_tan der inv_p df_neq).
+rewrite <- (pr_nu atan x (derivable_pt_recip_interv tan atan (- ub) ub
+      x lb_lt_ub xint inv_p int_tan incr der df_neq)).
+rewrite t.
+assert (t' := atan_bound x).
+rewrite <- (pr_nu tan (atan x) (derivable_pt_tan _ t')). 
+rewrite derive_pt_tan, atan_right_inv.
+replace (Rsqr x) with (x ^ 2) by (unfold Rsqr; ring).
+reflexivity.
+Qed.
+
+(** * Definition of the arctangent function as the sum of the arctan power series *)
+(* Proof taken from Guillaume Melquiond's interval package for Coq *)
+
+Definition Ratan_seq x :=  fun n => (x ^ (2 * n + 1) / INR (2 * n + 1))%R.
+
+Lemma Ratan_seq_decreasing : forall x, (0 <= x <= 1)%R -> Un_decreasing (Ratan_seq x).
+Proof.
+intros x Hx n.
+ unfold Ratan_seq, Rdiv.
+ apply Rmult_le_compat. apply pow_le.
+ exact (proj1 Hx).
+ apply Rlt_le.
+ apply Rinv_0_lt_compat.
+ apply lt_INR_0.
+ omega.
+ destruct (proj1 Hx) as [Hx1|Hx1].
+ destruct (proj2 Hx) as [Hx2|Hx2].
+ (* . 0 < x < 1 *)
+ rewrite <- (Rinv_involutive x).
+ assert (/ x <> 0)%R by auto with real.
+ repeat rewrite <- Rinv_pow with (1 := H).
+ apply Rlt_le.
+ apply Rinv_lt_contravar.
+ apply Rmult_lt_0_compat ; apply pow_lt ; auto with real.
+ apply Rlt_pow.
+ rewrite <- Rinv_1.
+ apply Rinv_lt_contravar.
+ rewrite Rmult_1_r.
+ exact Hx1.
+ exact Hx2.
+ omega.
+ apply Rgt_not_eq.
+ exact Hx1.
+ (* . x = 1 *)
+ rewrite Hx2.
+ do 2 rewrite pow1.
+ apply Rle_refl.
+ (* . x = 0 *)
+ rewrite <- Hx1.
+ do 2 (rewrite pow_i ; [ idtac | omega ]).
+ apply Rle_refl.
+ apply Rlt_le.
+ apply Rinv_lt_contravar.
+ apply Rmult_lt_0_compat ; apply lt_INR_0 ; omega.
+ apply lt_INR.
+ omega.
+Qed.
+
+Lemma Ratan_seq_converging : forall x, (0 <= x <= 1)%R -> Un_cv (Ratan_seq x) 0.
+Proof.
+intros x Hx eps Heps.
+ destruct (archimed (/ eps)) as (HN,_).
+ assert (0 < up (/ eps))%Z.
+  apply lt_IZR.
+  apply Rlt_trans with (2 := HN).
+  apply Rinv_0_lt_compat.
+  exact Heps.
+ case_eq (up (/ eps)) ;
+  intros ; rewrite H0 in H ; try discriminate H.
+   rewrite H0 in HN.
+   simpl in HN.
+   pose (N := Pos.to_nat p).
+   fold N in HN.
+   clear H H0.
+   exists N.
+  intros n Hn.
+   unfold R_dist.
+   rewrite Rminus_0_r.
+   unfold Ratan_seq.
+   rewrite Rabs_right.
+   apply Rle_lt_trans with (1 ^ (2 * n + 1) / INR (2 * n + 1))%R.
+   unfold Rdiv.
+   apply Rmult_le_compat_r.
+   apply Rlt_le.
+   apply Rinv_0_lt_compat.
+   apply lt_INR_0.
+   omega.
+   apply pow_incr.
+   exact Hx.
+   rewrite pow1.
+   apply Rle_lt_trans with (/ INR (2 * N + 1))%R.
+   unfold Rdiv.
+   rewrite Rmult_1_l.
+   apply Rle_Rinv.
+   apply lt_INR_0.
+   omega.
+   replace 0 with (INR 0) by intuition.
+   apply lt_INR.
+   omega.
+   intuition.
+   rewrite <- (Rinv_involutive eps).
+   apply Rinv_lt_contravar.
+   apply Rmult_lt_0_compat.
+   auto with real.
+   apply lt_INR_0.
+   omega.
+   apply Rlt_trans with (INR N).
+   destruct (archimed (/ eps)) as (H,_).
+   assert (0 < up (/ eps))%Z.
+   apply lt_IZR.
+   apply Rlt_trans with (2 := H).
+   apply Rinv_0_lt_compat.
+   exact Heps.
+   exact HN.
+   apply lt_INR.
+   omega.
+   apply Rgt_not_eq.
+   exact Heps.
+   apply Rle_ge.
+   unfold Rdiv.
+   apply Rmult_le_pos.
+   apply pow_le.
+   exact (proj1 Hx).
+   apply Rlt_le.
+   apply Rinv_0_lt_compat.
+   apply lt_INR_0.
+   omega.
+Qed.
+
+Definition ps_atan_exists_01 (x : R) (Hx:0 <= x <= 1) :
+   {l : R | Un_cv (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}.
+exact (alternated_series (Ratan_seq x)
+  (Ratan_seq_decreasing _ Hx) (Ratan_seq_converging _ Hx)).
+Defined.
+
+Lemma Ratan_seq_opp : forall x n, Ratan_seq (-x) n = -Ratan_seq x n.
+Proof.
+intros x n; unfold Ratan_seq.
+rewrite !pow_add, !pow_mult, !pow_1.
+unfold Rdiv; replace ((-x) ^ 2) with (x ^ 2) by ring; ring.
+Qed.
+
+Lemma sum_Ratan_seq_opp : 
+  forall x n, sum_f_R0 (tg_alt (Ratan_seq (- x))) n =
+     - sum_f_R0 (tg_alt (Ratan_seq x)) n.
+Proof.
+intros x n; replace (-sum_f_R0 (tg_alt (Ratan_seq x)) n) with 
+  (-1 * sum_f_R0 (tg_alt (Ratan_seq x)) n) by ring.
+rewrite scal_sum; apply sum_eq; intros i _; unfold tg_alt.
+rewrite Ratan_seq_opp; ring.
+Qed.
+
+Definition ps_atan_exists_1 (x : R) (Hx : -1 <= x <= 1) :
+   {l : R | Un_cv (fun N : nat => sum_f_R0 (tg_alt (Ratan_seq x)) N) l}.
+destruct (Rle_lt_dec 0 x).
+ assert (pr : 0 <= x <= 1) by tauto.
+ exact (ps_atan_exists_01 x pr).
+assert (pr : 0 <= -x <= 1) by (destruct Hx; split; fourier).
+destruct (ps_atan_exists_01 _ pr) as [v Pv].
+exists (-v).
+ apply (Un_cv_ext (fun n => (- 1) * sum_f_R0 (tg_alt (Ratan_seq (- x))) n)).
+ intros n; rewrite sum_Ratan_seq_opp; ring.
+replace (-v) with (-1 * v) by ring.
+apply CV_mult;[ | assumption].
+solve[intros; exists 0%nat; intros; rewrite R_dist_eq; auto].
+Qed.
+
+Definition in_int (x : R) : {-1 <= x <= 1}+{~ -1 <= x <= 1}.
+destruct (Rle_lt_dec x 1).
+ destruct (Rle_lt_dec (-1) x).
+  left;split; auto.
+ right;intros [a1 a2]; fourier.
+right;intros [a1 a2]; fourier.
+Qed.
+
+Definition ps_atan (x : R) : R :=
+ match in_int x with
+   left h => let (v, _) := ps_atan_exists_1 x h in v
+ | right h => atan x
+ end.
+
+(** * Proof of the equivalence of the two definitions between -1 and 1 *)
+
+Lemma ps_atan0_0 : ps_atan 0 = 0.
+Proof.
+unfold ps_atan.
+ destruct (in_int 0) as [h1 | h2].
+ destruct (ps_atan_exists_1 0 h1) as [v P].
+ apply (UL_sequence _ _ _ P).
+  apply (Un_cv_ext (fun n => 0)).
+  symmetry;apply sum_eq_R0.
+  intros i _; unfold tg_alt, Ratan_seq; rewrite plus_comm; simpl.
+  unfold Rdiv; rewrite !Rmult_0_l, Rmult_0_r; reflexivity.
+ intros eps ep; exists 0%nat; intros n _; unfold R_dist.
+ rewrite Rminus_0_r, Rabs_pos_eq; auto with real.
+case h2; split; fourier.
+Qed.
+
+Lemma ps_atan_exists_1_opp :
+  forall x h h', proj1_sig (ps_atan_exists_1 (-x) h) = 
+     -(proj1_sig (ps_atan_exists_1 x h')).
+Proof.
+intros x h h'; destruct (ps_atan_exists_1 (-x) h) as [v Pv].
+destruct (ps_atan_exists_1 x h') as [u Pu]; simpl.
+assert (Pu' : Un_cv (fun N => (-1) * sum_f_R0 (tg_alt (Ratan_seq x)) N) (-1 * u)).
+ apply CV_mult;[ | assumption].
+ intros eps ep; exists 0%nat; intros; rewrite R_dist_eq; assumption.  
+assert (Pv' : Un_cv
+           (fun N : nat => -1 * sum_f_R0 (tg_alt (Ratan_seq x)) N) v).
+ apply Un_cv_ext with (2 := Pv); intros n; rewrite sum_Ratan_seq_opp; ring.
+replace (-u) with (-1 * u) by ring.
+apply UL_sequence with (1:=Pv') (2:= Pu').
+Qed.
+
+Lemma ps_atan_opp : forall x, ps_atan (-x) = -ps_atan x.
+Proof.
+intros x; unfold ps_atan.
+destruct (in_int (- x)) as [inside | outside].
+ destruct (in_int x) as [ins' | outs'].
+ generalize (ps_atan_exists_1_opp x inside ins').
+  intros h; exact h.
+ destruct inside; case outs'; split; fourier.   
+destruct (in_int x) as [ins' | outs'].
+ destruct outside; case ins'; split; fourier.   
+apply atan_opp.
+Qed.
+
+(** atan = ps_atan *)
+
+Lemma ps_atanSeq_continuity_pt_1 : forall (N:nat) (x:R),
+      0 <= x ->
+      x <= 1 ->
+      continuity_pt (fun x => sum_f_R0 (tg_alt (Ratan_seq x)) N) x.
+Proof.
+assert (Sublemma : forall (x:R) (N:nat), sum_f_R0 (tg_alt (Ratan_seq x)) N = x * (comp (fun x => sum_f_R0 (fun n => (fun i : nat => (-1) ^ i / INR (2 * i + 1)) n * x ^ n) N) (fun x => x ^ 2) x)).
+ intros x N.
+  induction N.
+   unfold tg_alt, Ratan_seq, comp ; simpl ; field.
+   simpl sum_f_R0 at 1.
+   rewrite IHN.
+   replace (comp (fun x => sum_f_R0 (fun n : nat => (-1) ^ n / INR (2 * n + 1) * x ^ n) (S N)) (fun x => x ^ 2))
+     with (comp (fun x => sum_f_R0 (fun n : nat => (-1) ^ n / INR (2 * n + 1) * x ^ n) N + (-1) ^ (S N) / INR (2 * (S N) + 1) * x ^ (S N)) (fun x => x ^ 2)).
+   unfold comp.
+   rewrite Rmult_plus_distr_l.
+   apply Rplus_eq_compat_l.
+   unfold tg_alt, Ratan_seq.
+   rewrite <- Rmult_assoc.
+   case (Req_dec x 0) ; intro Hyp.
+   rewrite Hyp ; rewrite pow_i. rewrite Rmult_0_l ; rewrite Rmult_0_l.
+   unfold Rdiv ; rewrite Rmult_0_l ; rewrite Rmult_0_r ; reflexivity.
+   intuition.
+   replace (x * ((-1) ^ S N / INR (2 * S N + 1)) * (x ^ 2) ^ S N) with (x ^ (2 * S N + 1) * ((-1) ^ S N / INR (2 * S N + 1))).
+   rewrite Rmult_comm ; unfold Rdiv at 1.
+   rewrite Rmult_assoc ; apply Rmult_eq_compat_l.
+   field. apply Rgt_not_eq ; intuition.
+   rewrite Rmult_assoc.
+   replace (x * ((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N)) with (((-1) ^ S N / INR (2 * S N + 1) * (x ^ 2) ^ S N) * x).
+   rewrite Rmult_assoc.
+   replace ((x ^ 2) ^ S N * x) with (x ^ (2 * S N + 1)).
+   rewrite Rmult_comm at 1 ; reflexivity.
+   rewrite <- pow_mult.
+   assert (Temp : forall x n, x ^ n * x = x ^ (n+1)).
+    intros a n ; induction n. rewrite pow_O. simpl ; intuition.
+    simpl ; rewrite Rmult_assoc ; rewrite IHn ; intuition.
+   rewrite Temp ; reflexivity.
+   rewrite Rmult_comm ; reflexivity.
+   intuition.
+intros N x x_lb x_ub.
+ intros eps eps_pos.
+ assert (continuity_id : continuity id).
+ apply derivable_continuous ; exact derivable_id.
+assert (Temp := continuity_mult id (comp
+        (fun x1 : R =>
+         sum_f_R0 (fun n : nat => (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
+        (fun x1 : R => x1 ^ 2))
+           continuity_id).
+assert (Temp2 : continuity
+    (comp
+       (fun x1 : R =>
+        sum_f_R0 (fun n : nat => (-1) ^ n / INR (2 * n + 1) * x1 ^ n) N)
+       (fun x1 : R => x1 ^ 2))).
+ apply continuity_comp.
+ reg.
+ apply continuity_finite_sum.
+ elim (Temp Temp2 x eps eps_pos) ; clear Temp Temp2 ; intros alpha T ; destruct T as (alpha_pos, T).
+ exists alpha ; split.
+ intuition.
+intros x0 x0_cond.
+ rewrite Sublemma ; rewrite Sublemma.
+apply T.
+intuition.
+Qed.
+
+(** Definition of ps_atan's derivative *)
+
+Definition Datan_seq := fun (x:R) (n:nat) => x ^ (2*n).
+
+Lemma pow_lt_1_compat : forall x n, 0 <= x < 1 -> (0 < n)%nat ->
+   0 <= x ^ n < 1.
+Proof.
+intros x n hx; induction 1; simpl.
+ rewrite Rmult_1_r; tauto.
+split.
+ apply Rmult_le_pos; tauto.
+rewrite <- (Rmult_1_r 1); apply Rmult_le_0_lt_compat; intuition.
+Qed.
+
+Lemma Datan_seq_Rabs : forall x n, Datan_seq (Rabs x) n = Datan_seq x n.
+Proof.
+intros x n; unfold Datan_seq; rewrite !pow_mult, pow2_abs; reflexivity.
+Qed.
+
+Lemma Datan_seq_pos : forall x n, 0 < x -> 0 < Datan_seq x n.
+Proof.
+intros x n x_lb ; unfold Datan_seq ; induction n.
+ simpl ; intuition.
+ replace (x ^ (2 * S n)) with ((x ^ 2) * (x ^ (2 * n))).
+ apply Rmult_gt_0_compat.
+ replace (x^2) with (x*x) by field ; apply Rmult_gt_0_compat ; assumption.
+ assumption.
+ replace (2 * S n)%nat with (S (S (2 * n))) by intuition.
+ simpl ; field.
+Qed.
+
+Lemma Datan_sum_eq :forall x n,
+  sum_f_R0 (tg_alt (Datan_seq x)) n = (1 - (- x ^ 2) ^ S n)/(1 + x ^ 2).
+Proof.
+intros x n.
+assert (dif : - x ^ 2 <> 1).
+apply Rlt_not_eq; apply Rle_lt_trans with 0;[ | apply Rlt_0_1].
+assert (t := pow2_ge_0 x); fourier.
+replace (1 + x ^ 2) with (1 - - (x ^ 2)) by ring; rewrite <- (tech3 _ n dif).
+apply sum_eq; unfold tg_alt, Datan_seq; intros i _.
+rewrite pow_mult, <- Rpow_mult_distr, Ropp_mult_distr_l_reverse, Rmult_1_l.
+reflexivity.
+Qed.
+
+Lemma Datan_seq_increasing : forall x y n, (n > 0)%nat -> 0 <= x < y -> Datan_seq x n < Datan_seq y n.
+Proof.
+intros x y n n_lb x_encad ; assert (x_pos : x >= 0) by intuition.
+ assert (y_pos : y > 0). apply Rle_lt_trans with (r2:=x) ; intuition.
+ induction n.
+ apply False_ind ; intuition.
+ clear -x_encad x_pos y_pos ; induction n ; unfold Datan_seq.
+ case x_pos ; clear x_pos ; intro x_pos.
+ simpl ; apply Rmult_gt_0_lt_compat ; intuition. fourier.
+ rewrite x_pos ; rewrite pow_i. replace (y ^ (2*1)) with (y*y).
+ apply Rmult_gt_0_compat ; assumption.
+ simpl ; field.
+ intuition.
+ assert (Hrew : forall a, a^(2 * S (S n)) = (a ^ 2) * (a ^ (2 * S n))).
+  clear ; intro a ; replace (2 * S (S n))%nat with (S (S (2 * S n)))%nat by intuition.
+  simpl ; field.
+ case x_pos ; clear x_pos ; intro x_pos.
+ rewrite Hrew ; rewrite Hrew.
+ apply Rmult_gt_0_lt_compat ; intuition.
+ apply Rmult_gt_0_lt_compat ; intuition ; fourier.
+ rewrite x_pos.
+ rewrite pow_i ; intuition.
+Qed.
+
+Lemma Datan_seq_decreasing : forall x,  -1 < x -> x < 1 -> Un_decreasing (Datan_seq x).
+Proof.
+intros x x_lb x_ub n.
+unfold Datan_seq.
+replace (2 * S n)%nat with (2 + 2 * n)%nat by ring.
+rewrite <- (Rmult_1_l (x ^ (2 * n))).
+rewrite pow_add.
+apply Rmult_le_compat_r.
+rewrite pow_mult; apply pow_le, pow2_ge_0.
+apply Rlt_le; rewrite <- pow2_abs.
+assert (intabs : 0 <= Rabs x < 1).
+ split;[apply Rabs_pos | apply Rabs_def1]; tauto.
+apply (pow_lt_1_compat (Rabs x) 2) in intabs.
+ tauto.
+omega.
+Qed.
+
+Lemma Datan_seq_CV_0 : forall x, -1 < x -> x < 1 -> Un_cv (Datan_seq x) 0.
+Proof.
+intros x x_lb x_ub eps eps_pos.
+assert (x_ub2 : Rabs (x^2) < 1).
+ rewrite Rabs_pos_eq;[ | apply pow2_ge_0].
+ rewrite <- pow2_abs.
+ assert (H: 0 <= Rabs x < 1)
+   by (split;[apply Rabs_pos | apply Rabs_def1; auto]).
+ apply (pow_lt_1_compat _ 2) in H;[tauto | omega].
+elim (pow_lt_1_zero (x^2) x_ub2 eps eps_pos) ; intros N HN ; exists N ; intros n Hn.
+unfold R_dist, Datan_seq.
+replace (x ^ (2 * n) - 0) with ((x ^ 2) ^ n). apply HN ; assumption.
+rewrite pow_mult ; field.
+Qed.
+
+Lemma Datan_lim : forall x, -1 < x -> x < 1 ->
+    Un_cv (fun N : nat => sum_f_R0 (tg_alt (Datan_seq x)) N) (/ (1 + x ^ 2)).
+Proof.
+intros x x_lb x_ub eps eps_pos.
+assert (Tool0 : 0 <= x ^ 2) by apply pow2_ge_0.
+assert (Tool1 : 0 < (1 + x ^ 2)).
+ solve[apply Rplus_lt_le_0_compat ; intuition].
+assert (Tool2 : / (1 + x ^ 2) > 0).
+ apply Rinv_0_lt_compat ; tauto.
+assert (x_ub2' : 0<= Rabs (x^2) < 1).
+ rewrite Rabs_pos_eq, <- pow2_abs;[ | apply pow2_ge_0].
+ apply pow_lt_1_compat;[split;[apply Rabs_pos | ] | omega].
+ apply Rabs_def1; assumption.
+assert (x_ub2 : Rabs (x^2) < 1) by tauto.
+assert (eps'_pos : ((1+x^2)*eps) > 0).
+  apply Rmult_gt_0_compat ; assumption.
+elim (pow_lt_1_zero _ x_ub2 _ eps'_pos) ; intros N HN ; exists N.
+intros n Hn.
+assert (H1 : - x^2 <> 1).
+ apply Rlt_not_eq; apply Rle_lt_trans with (2 := Rlt_0_1).
+assert (t := pow2_ge_0 x); fourier.
+rewrite Datan_sum_eq. 
+unfold R_dist.
+assert (tool : forall a b, a / b - /b = (-1 + a) /b).
+ intros a b; rewrite <- (Rmult_1_l (/b)); unfold Rdiv, Rminus.
+ rewrite <- Ropp_mult_distr_l_reverse, Rmult_plus_distr_r, Rplus_comm.
+ reflexivity.
+set (u := 1 + x ^ 2); rewrite tool; unfold Rminus; rewrite <- Rplus_assoc.
+unfold Rdiv, u.
+rewrite Rplus_opp_l, Rplus_0_l, Ropp_mult_distr_l_reverse, Rabs_Ropp.
+rewrite Rabs_mult; clear tool u.
+assert (tool : forall k, Rabs ((-x ^ 2) ^ k) = Rabs ((x ^ 2) ^ k)).
+ clear -Tool0; induction k;[simpl; rewrite Rabs_R1;tauto | ].
+ rewrite <- !(tech_pow_Rmult _ k), !Rabs_mult, Rabs_Ropp, IHk, Rabs_pos_eq.
+  reflexivity.
+ exact Tool0.
+rewrite tool, (Rabs_pos_eq (/ _)); clear tool;[ | apply Rlt_le; assumption].
+assert (tool : forall a b c, 0 < b -> a < b * c -> a * / b < c).
+ intros a b c bp h; replace c with (b * c * /b).
+  apply Rmult_lt_compat_r.  
+   apply Rinv_0_lt_compat; assumption.
+  assumption.
+ field; apply Rgt_not_eq; exact bp.
+apply tool;[exact Tool1 | ].
+apply HN; omega.
+Qed.
+
+Lemma Datan_CVU_prelim : forall c (r : posreal), Rabs c + r < 1 ->
+ CVU (fun N x => sum_f_R0 (tg_alt (Datan_seq x)) N)
+     (fun y : R => / (1 + y ^ 2)) c r.
+Proof.
+intros c r ub_ub eps eps_pos.
+apply (Alt_CVU (fun x n => Datan_seq n x) 
+         (fun x => /(1 + x ^ 2))
+         (Datan_seq (Rabs c + r)) c r).
+     intros x inb; apply Datan_seq_decreasing;
+      try (apply Boule_lt in inb; apply Rabs_def2 in inb;
+         destruct inb; fourier).
+    intros x inb; apply Datan_seq_CV_0;
+      try (apply Boule_lt in inb; apply Rabs_def2 in inb;
+         destruct inb; fourier).
+   intros x inb; apply (Datan_lim x);
+      try (apply Boule_lt in inb; apply Rabs_def2 in inb;
+         destruct inb; fourier).
+  intros x [ | n] inb.
+   solve[unfold Datan_seq; apply Rle_refl].
+  rewrite <- (Datan_seq_Rabs x); apply Rlt_le, Datan_seq_increasing.
+   omega.
+  apply Boule_lt in inb; intuition.
+  solve[apply Rabs_pos].
+ apply Datan_seq_CV_0.
+  apply Rlt_trans with 0;[fourier | ].
+  apply Rplus_le_lt_0_compat.
+   solve[apply Rabs_pos].
+  destruct r; assumption.
+ assumption.
+assumption.
+Qed.
+
+Lemma Datan_is_datan : forall (N:nat) (x:R),
+      -1 <= x ->
+      x < 1 ->
+derivable_pt_lim (fun x => sum_f_R0 (tg_alt (Ratan_seq x)) N) x (sum_f_R0 (tg_alt (Datan_seq x)) N).
+Proof.
+assert (Tool : forall N, (-1) ^ (S (2 * N))  = - 1).
+ intro n ; induction n.
+  simpl ; field.
+  replace ((-1) ^ S (2 * S n)) with ((-1) ^ 2 * (-1) ^ S (2*n)).
+  rewrite IHn ; field.
+  rewrite <- pow_add.
+  replace (2 + S (2 * n))%nat with (S (2 * S n))%nat.
+  reflexivity.
+  intuition.
+intros N x x_lb x_ub.
+ induction N.
+  unfold Datan_seq, Ratan_seq, tg_alt ; simpl.
+  intros eps eps_pos.
+   elim (derivable_pt_lim_id x eps eps_pos) ; intros delta Hdelta ; exists delta.
+   intros h hneq h_b.
+    replace (1 * ((x + h) * 1 / 1) - 1 * (x * 1 / 1)) with (id (x + h) - id x). 
+    rewrite Rmult_1_r.
+    apply Hdelta ; assumption.
+    unfold id ; field ; assumption.
+  intros eps eps_pos.
+  assert (eps_3_pos : (eps/3) > 0) by fourier.
+  elim (IHN (eps/3) eps_3_pos) ; intros delta1 Hdelta1.
+  assert (Main : derivable_pt_lim (fun x : R =>tg_alt (Ratan_seq x) (S N)) x ((tg_alt (Datan_seq x)) (S N))).
+   clear -Tool ; intros eps' eps'_pos.
+   elim (derivable_pt_lim_pow x (2 * (S N) + 1) eps' eps'_pos) ; intros delta Hdelta ; exists delta.
+   intros h h_neq h_b ; unfold tg_alt, Ratan_seq, Datan_seq.
+   replace (((-1) ^ S N * ((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
+    (-1) ^ S N * (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h -
+    (-1) ^ S N * x ^ (2 * S N)) 
+    with (((-1)^(S N)) * ((((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1)) -
+    (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h - x ^ (2 * S N))).
+   rewrite Rabs_mult ; rewrite pow_1_abs ; rewrite Rmult_1_l.
+   replace (((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) -
+    x ^ (2 * S N + 1) / INR (2 * S N + 1)) / h - x ^ (2 * S N))
+    with ((/INR (2* S N + 1)) * (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
+    INR (2 * S N + 1) * x ^ pred (2 * S N + 1))).
+   rewrite Rabs_mult.
+   case (Req_dec (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
+     INR (2 * S N + 1) * x ^ pred (2 * S N + 1)) 0) ; intro Heq.
+   rewrite Heq ; rewrite Rabs_R0 ; rewrite Rmult_0_r ; assumption.
+   apply Rlt_trans with (r2:=Rabs
+           (((x + h) ^ (2 * S N + 1) - x ^ (2 * S N + 1)) / h -
+            INR (2 * S N + 1) * x ^ pred (2 * S N + 1))).
+   rewrite <- Rmult_1_l ; apply Rmult_lt_compat_r.
+   apply Rabs_pos_lt ; assumption.
+   rewrite Rabs_right.
+   replace 1 with (/1) by field.
+   apply Rinv_1_lt_contravar ; intuition.
+   apply Rgt_ge ; replace (INR (2 * S N + 1)) with (INR (2*S N) + 1) ;
+   [apply RiemannInt.RinvN_pos | ].
+   replace (2 * S N + 1)%nat with (S (2 * S N))%nat by intuition ;
+   rewrite S_INR ; reflexivity.
+   apply Hdelta ; assumption.
+   rewrite Rmult_minus_distr_l.
+   replace (/ INR (2 * S N + 1) * (INR (2 * S N + 1) * x ^ pred (2 * S N + 1))) with (x ^ (2 * S N)).
+   unfold Rminus ; rewrite Rplus_comm.
+   replace (((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) +
+      - (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h + - x ^ (2 * S N))
+      with (- x ^ (2 * S N) + (((x + h) ^ (2 * S N + 1) / INR (2 * S N + 1) +
+      - (x ^ (2 * S N + 1) / INR (2 * S N + 1))) / h)) by intuition.
+   apply Rplus_eq_compat_l. field.
+   split ; [apply Rgt_not_eq|] ; intuition.
+   clear ; replace (pred (2 * S N + 1)) with (2 * S N)%nat by intuition.
+   field ; apply Rgt_not_eq ; intuition.
+   field ; split ; [apply Rgt_not_eq |] ; intuition.
+   elim (Main (eps/3) eps_3_pos) ; intros delta2 Hdelta2.
+   destruct delta1 as (delta1, delta1_pos) ; destruct delta2 as (delta2, delta2_pos).
+   pose (mydelta := Rmin delta1 delta2).
+   assert (mydelta_pos : mydelta > 0).
+    unfold mydelta ; rewrite Rmin_Rgt ; split ; assumption.
+   pose (delta := mkposreal mydelta mydelta_pos) ; exists delta ; intros h h_neq h_b.
+   clear Main IHN.
+   unfold Rminus at 1.
+   apply Rle_lt_trans with (r2:=eps/3 + eps / 3).
+   assert (Temp : (sum_f_R0 (tg_alt (Ratan_seq (x + h))) (S N) -
+    sum_f_R0 (tg_alt (Ratan_seq x)) (S N)) / h +
+    - sum_f_R0 (tg_alt (Datan_seq x)) (S N) = ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
+              sum_f_R0 (tg_alt (Ratan_seq x)) N) / h) + (-
+             sum_f_R0 (tg_alt (Datan_seq x)) N) + ((tg_alt (Ratan_seq (x + h)) (S N) - tg_alt (Ratan_seq x) (S N)) /
+             h - tg_alt (Datan_seq x) (S N))).
+    simpl ; field ; intuition.
+   apply Rle_trans with (r2:= Rabs ((sum_f_R0 (tg_alt (Ratan_seq (x + h))) N -
+        sum_f_R0 (tg_alt (Ratan_seq x)) N) / h +
+       - sum_f_R0 (tg_alt (Datan_seq x)) N) +
+       Rabs ((tg_alt (Ratan_seq (x + h)) (S N) - tg_alt (Ratan_seq x) (S N)) / h -
+        tg_alt (Datan_seq x) (S N))).
+   rewrite Temp ; clear Temp ; apply Rabs_triang.
+   apply Rplus_le_compat ; apply Rlt_le ; [apply Hdelta1 | apply Hdelta2] ;
+   intuition ; apply Rlt_le_trans with (r2:=delta) ; intuition unfold delta, mydelta.
+   apply Rmin_l.
+   apply Rmin_r.
+   fourier.
+Qed.
+
+Lemma Ratan_CVU' :
+  CVU (fun N x => sum_f_R0 (tg_alt (Ratan_seq x)) N)
+                     ps_atan (/2) (mkposreal (/2) pos_half_prf).
+Proof.
+apply (Alt_CVU (fun i r => Ratan_seq r i) ps_atan PI_tg (/2) pos_half);
+  lazy beta.
+    now intros; apply Ratan_seq_decreasing, Boule_half_to_interval.
+   now intros; apply Ratan_seq_converging, Boule_half_to_interval.
+  intros x b; apply Boule_half_to_interval in b.
+  unfold ps_atan; destruct (in_int x) as [inside | outside];
+   [ | destruct b; case outside; split; fourier].
+  destruct (ps_atan_exists_1 x inside) as [v Pv].
+  apply Un_cv_ext with (2 := Pv);[reflexivity].
+ intros x n b; apply Boule_half_to_interval in b.
+ rewrite <- (Rmult_1_l (PI_tg n)); unfold Ratan_seq, PI_tg. 
+ apply Rmult_le_compat_r.
+  apply Rlt_le, Rinv_0_lt_compat, (lt_INR 0); omega.
+ rewrite <- (pow1 (2 * n + 1)); apply pow_incr; assumption.
+exact PI_tg_cv.
+Qed.
+
+Lemma Ratan_CVU :
+  CVU (fun N x => sum_f_R0 (tg_alt (Ratan_seq x)) N)
+                     ps_atan 0  (mkposreal 1 Rlt_0_1).
+Proof.
+intros eps ep; destruct (Ratan_CVU' eps ep) as [N Pn].
+exists N; intros n x nN b_y.
+case (Rtotal_order 0 x) as [xgt0 | [x0 | x0]].
+  assert (Boule (/2) {| pos := / 2; cond_pos := pos_half_prf|} x).
+   revert b_y; unfold Boule; simpl; intros b_y; apply Rabs_def2 in b_y.
+   destruct b_y; unfold Boule; simpl; apply Rabs_def1; fourier.
+  apply Pn; assumption.
+ rewrite <- x0, ps_atan0_0.
+ rewrite <- (sum_eq (fun _ => 0)), sum_cte, Rmult_0_l, Rminus_0_r, Rabs_pos_eq.
+   assumption.
+  apply Rle_refl.
+ intros i _; unfold tg_alt, Ratan_seq, Rdiv; rewrite plus_comm; simpl.
+ solve[rewrite !Rmult_0_l, Rmult_0_r; auto].
+replace (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) n) with
+  (-(ps_atan (-x) - sum_f_R0 (tg_alt (Ratan_seq (-x))) n)).
+ rewrite Rabs_Ropp.
+ assert (Boule (/2) {| pos := / 2; cond_pos := pos_half_prf|} (-x)).
+  revert b_y; unfold Boule; simpl; intros b_y; apply Rabs_def2 in b_y.
+  destruct b_y; unfold Boule; simpl; apply Rabs_def1; fourier.
+ apply Pn; assumption.
+unfold Rminus; rewrite ps_atan_opp, Ropp_plus_distr, sum_Ratan_seq_opp.
+rewrite !Ropp_involutive; reflexivity.
+Qed.
+
+Lemma Alt_PI_tg : forall n, PI_tg n = Ratan_seq 1 n.
+Proof.
+intros n; unfold PI_tg, Ratan_seq, Rdiv; rewrite pow1, Rmult_1_l.
+reflexivity.
+Qed.
+
+Lemma Ratan_is_ps_atan : forall eps, eps > 0 ->
+       exists N, forall n, (n >= N)%nat -> forall x, -1 < x -> x < 1 ->
+       Rabs (sum_f_R0 (tg_alt (Ratan_seq x)) n - ps_atan x) < eps.
+Proof.
+intros eps ep.
+destruct (Ratan_CVU _ ep) as [N1 PN1].
+exists N1; intros n nN x xm1 x1; rewrite <- Rabs_Ropp, Ropp_minus_distr.
+apply PN1; [assumption | ].
+unfold Boule; simpl; rewrite Rminus_0_r; apply Rabs_def1; assumption.
+Qed.
+
+Lemma Datan_continuity : continuity (fun x => /(1+x ^ 2)).
+Proof.
+apply continuity_inv.
+apply continuity_plus.
+apply continuity_const ; unfold constant ; intuition.
+apply derivable_continuous ; apply derivable_pow.
+intro x ; apply Rgt_not_eq ; apply Rge_gt_trans with (1+0) ; [|fourier] ;
+ apply Rplus_ge_compat_l.
+ replace (x^2) with (x²).
+ apply Rle_ge ; apply Rle_0_sqr.
+ unfold Rsqr ; field.
+Qed.
+
+Lemma derivable_pt_lim_ps_atan : forall x, -1 < x < 1 ->
+  derivable_pt_lim ps_atan x ((fun y => /(1 + y ^ 2)) x).
+Proof.
+intros x x_encad.
+destruct (boule_in_interval (-1) 1 x x_encad) as [c [r [Pcr1 [P1 P2]]]].
+change (/ (1 + x ^ 2)) with ((fun u => /(1 + u ^ 2)) x).
+assert (t := derivable_pt_lim_CVU).
+apply derivable_pt_lim_CVU with 
+          (fn := (fun N x => sum_f_R0 (tg_alt (Ratan_seq x)) N))
+          (fn' := (fun N x => sum_f_R0 (tg_alt (Datan_seq x)) N))
+          (c := c) (r := r).
+    assumption.
+   intros y N inb; apply Rabs_def2 in inb; destruct inb.
+   apply Datan_is_datan.
+    fourier.
+   fourier.
+  intros y inb; apply Rabs_def2 in inb; destruct inb.
+  assert (y_gt_0 : -1 < y) by fourier.
+  assert (y_lt_1 : y < 1) by fourier.
+  intros eps eps_pos ; elim (Ratan_is_ps_atan eps eps_pos).
+  intros N HN ; exists N; intros n n_lb ; apply HN ; tauto.
+ apply Datan_CVU_prelim.
+ replace ((c - r + (c + r)) / 2) with c by field.
+ unfold mkposreal_lb_ub; simpl.
+ replace ((c + r - (c - r)) / 2) with (r :R) by field.
+ assert (Rabs c < 1 - r).
+  unfold Boule in Pcr1; destruct r; simpl in *; apply Rabs_def1;
+  apply Rabs_def2 in Pcr1; destruct Pcr1; fourier.
+ fourier.
+intros; apply Datan_continuity.
+Qed.
+
+Lemma derivable_pt_ps_atan :
+   forall x, -1 < x < 1 -> derivable_pt ps_atan x.
+Proof.
+intros x x_encad.
+exists (/(1+x^2)) ; apply derivable_pt_lim_ps_atan; assumption.
+Qed.
+
+Lemma ps_atan_continuity_pt_1 : forall eps : R,
+       eps > 0 ->
+       exists alp : R,
+       alp > 0 /\
+       (forall x, x < 1 -> 0 < x -> R_dist x 1 < alp ->
+       dist R_met (ps_atan x) (Alt_PI/4) < eps).
+Proof.
+intros eps eps_pos.
+assert (eps_3_pos : eps / 3 > 0) by fourier.
+elim (Ratan_is_ps_atan (eps / 3) eps_3_pos) ; intros N1 HN1.
+unfold Alt_PI.
+destruct exist_PI as [v Pv]; replace ((4 * v)/4) with v by field.
+assert (Pv' : Un_cv (sum_f_R0 (tg_alt (Ratan_seq 1))) v).
+ apply Un_cv_ext with (2:= Pv).
+ intros; apply sum_eq; intros; unfold tg_alt; rewrite Alt_PI_tg; tauto.
+destruct (Pv' (eps / 3) eps_3_pos) as [N2 HN2].
+set (N := (N1 + N2)%nat).
+assert (O_lb : 0 <= 1) by intuition ; assert (O_ub : 1 <= 1) by intuition ;
+ elim (ps_atanSeq_continuity_pt_1 N 1 O_lb O_ub (eps / 3) eps_3_pos) ; intros alpha Halpha ;
+ clear -HN1 HN2 Halpha eps_3_pos; destruct Halpha as (alpha_pos, Halpha).
+exists alpha ; split;[assumption | ].
+intros x x_ub x_lb x_bounds.
+simpl ; unfold R_dist.
+replace (ps_atan x - v) with ((ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N)
+     + (sum_f_R0 (tg_alt (Ratan_seq x)) N - sum_f_R0 (tg_alt (Ratan_seq 1)) N)
+     + (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)).
+apply Rle_lt_trans with (r2:=Rabs (ps_atan x - sum_f_R0 (tg_alt (Ratan_seq x)) N) +
+      Rabs ((sum_f_R0 (tg_alt (Ratan_seq x)) N - sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
+   (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v))).
+rewrite Rplus_assoc ; apply Rabs_triang.
+   replace eps with (2 / 3 * eps + eps / 3).
+   rewrite Rplus_comm.
+   apply Rplus_lt_compat.
+   apply Rle_lt_trans with (r2 := Rabs (sum_f_R0 (tg_alt (Ratan_seq x)) N - sum_f_R0 (tg_alt (Ratan_seq 1)) N) +
+      Rabs (sum_f_R0 (tg_alt (Ratan_seq 1)) N - v)).
+   apply Rabs_triang.
+   apply Rlt_le_trans with (r2:= eps / 3 + eps / 3).
+   apply Rplus_lt_compat.
+   simpl in Halpha ; unfold R_dist in Halpha.
+   apply Halpha ; split.
+   unfold D_x, no_cond ; split ; [ | apply Rgt_not_eq ] ; intuition.
+   intuition.
+   apply HN2; unfold N; omega.
+   fourier.
+  rewrite <- Rabs_Ropp, Ropp_minus_distr; apply HN1.
+     unfold N; omega.  
+    fourier.
+  assumption.
+ field.
+ring.
+Qed.
+
+Lemma Datan_eq_DatanSeq_interv : forall x, -1 < x < 1 ->
+ forall (Pratan:derivable_pt ps_atan x) (Prmymeta:derivable_pt atan x),
+      derive_pt ps_atan x Pratan = derive_pt atan x Prmymeta.
+Proof.
+assert (freq : 0 < tan 1) by apply (Rlt_trans _ _ _ Rlt_0_1 tan_1_gt_1).
+intros x x_encad Pratan Prmymeta.
+ rewrite pr_nu_var2_interv with (g:=ps_atan) (lb:=-1) (ub:=tan 1)
+   (pr2 := derivable_pt_ps_atan x x_encad).
+ rewrite pr_nu_var2_interv with (f:=atan) (g:=atan) (lb:=-1) (ub:= 1) (pr2:=derivable_pt_atan x).
+ assert (Temp := derivable_pt_lim_ps_atan x x_encad).
+ assert (Hrew1 : derive_pt ps_atan x (derivable_pt_ps_atan x x_encad) = (/(1+x^2))).
+  apply derive_pt_eq_0 ; assumption.
+  rewrite derive_pt_atan.
+  rewrite Hrew1.
+  replace (Rsqr x) with (x ^ 2) by (unfold Rsqr; ring).
+  unfold Rdiv; rewrite Rmult_1_l; reflexivity.
+     fourier.
+    assumption.
+   intros; reflexivity.
+  fourier.
+ assert (t := tan_1_gt_1); split;destruct x_encad; fourier.
+intros; reflexivity.
+Qed.
+
+Lemma atan_eq_ps_atan :
+ forall x, 0 < x < 1 -> atan x = ps_atan x.
+Proof.
+intros x x_encad.
+assert (pr1 : forall c : R, 0 < c < x -> derivable_pt (atan - ps_atan) c).
+ intros c c_encad.
+ apply derivable_pt_minus.
+  exact (derivable_pt_atan c).
+ apply derivable_pt_ps_atan.
+  destruct x_encad; destruct c_encad; split; fourier.
+assert (pr2 : forall c : R, 0 < c < x -> derivable_pt id c).
+ intros ; apply derivable_pt_id; fourier.
+assert (delta_cont : forall c : R, 0 <= c <= x -> continuity_pt (atan - ps_atan) c).
+ intros c [[c_encad1 | c_encad1 ] [c_encad2 | c_encad2]];
+        apply continuity_pt_minus.
+        apply derivable_continuous_pt ; apply derivable_pt_atan. 
+       apply derivable_continuous_pt ; apply derivable_pt_ps_atan.
+       split; destruct x_encad; fourier.
+      apply derivable_continuous_pt, derivable_pt_atan.
+     apply derivable_continuous_pt, derivable_pt_ps_atan.
+     subst c; destruct x_encad; split; fourier.
+    apply derivable_continuous_pt, derivable_pt_atan.
+   apply derivable_continuous_pt, derivable_pt_ps_atan.
+   subst c; split; fourier.
+  apply derivable_continuous_pt, derivable_pt_atan.
+ apply derivable_continuous_pt, derivable_pt_ps_atan.
+ subst c; destruct x_encad; split; fourier.
+assert (id_cont : forall c : R, 0 <= c <= x -> continuity_pt id c).
+    intros ; apply derivable_continuous ; apply derivable_id.
+assert (x_lb : 0 < x) by (destruct x_encad; fourier).
+elim (MVT (atan - ps_atan)%F id 0 x pr1 pr2 x_lb delta_cont id_cont) ; intros d Temp ; elim Temp ; intros d_encad Main.
+clear - Main x_encad.
+assert (Temp : forall (pr: derivable_pt (atan - ps_atan) d), derive_pt (atan - ps_atan) d pr = 0).
+ intro pr.
+ assert (d_encad3 : -1 < d < 1).
+  destruct d_encad; destruct x_encad; split; fourier.
+ pose (pr3 := derivable_pt_minus atan ps_atan d (derivable_pt_atan d) (derivable_pt_ps_atan d d_encad3)).
+ rewrite <- pr_nu_var2_interv with (f:=(atan - ps_atan)%F) (g:=(atan - ps_atan)%F) (lb:=0) (ub:=x) (pr1:=pr3) (pr2:=pr).
+    unfold pr3. rewrite derive_pt_minus.
+    rewrite Datan_eq_DatanSeq_interv with (Prmymeta := derivable_pt_atan d).
+     intuition.
+    assumption. 
+   destruct d_encad; fourier.
+  assumption.
+ reflexivity.
+assert (iatan0 : atan 0 = 0).
+ apply tan_is_inj.
+   apply atan_bound.
+  rewrite Ropp_div; assert (t := PI2_RGT_0); split; fourier.
+ rewrite tan_0, atan_right_inv; reflexivity.
+generalize Main; rewrite Temp, Rmult_0_r.
+replace ((atan - ps_atan)%F x) with (atan x - ps_atan x) by intuition.
+replace ((atan - ps_atan)%F 0) with (atan 0 - ps_atan 0) by intuition.
+rewrite iatan0, ps_atan0_0, !Rminus_0_r.
+replace (derive_pt id d (pr2 d d_encad)) with 1. 
+ rewrite Rmult_1_r.
+ solve[intros M; apply Rminus_diag_uniq; auto].
+rewrite pr_nu_var with (g:=id) (pr2:=derivable_pt_id d).
+ symmetry ; apply derive_pt_id.
+tauto.
+Qed.
+
+
+Theorem Alt_PI_eq : Alt_PI = PI.
+apply Rmult_eq_reg_r with (/4); fold (Alt_PI/4); fold (PI/4);
+  [ | apply Rgt_not_eq; fourier].
+assert (0 < PI/6) by (apply PI6_RGT_0).
+assert (t1:= PI2_1).
+assert (t2 := PI_4).
+assert (m := Alt_PI_RGT_0).
+assert (-PI/2 < 1 < PI/2) by (rewrite Ropp_div; split; fourier).
+apply cond_eq; intros eps ep.
+change (R_dist (Alt_PI/4) (PI/4) < eps).
+assert (ca : continuity_pt atan 1).
+  apply derivable_continuous_pt, derivable_pt_atan.
+assert (Xe : exists eps', exists eps'',
+  eps' + eps'' <= eps /\ 0 < eps' /\ 0 < eps'').
+ exists (eps/2); exists (eps/2); repeat apply conj; fourier.
+destruct Xe as [eps' [eps'' [eps_ineq [ep' ep'']]]].
+destruct (ps_atan_continuity_pt_1 _ ep') as [alpha [a0 Palpha]].
+destruct (ca _ ep'') as [beta [b0 Pbeta]].
+assert (Xa : exists a, 0 < a < 1 /\ R_dist a 1 < alpha /\
+                R_dist a 1 < beta).
+ exists (Rmax (/2) (Rmax (1 - alpha /2) (1 - beta /2))).
+ assert (/2 <= Rmax (/2) (Rmax (1 - alpha /2) (1 - beta /2))) by apply Rmax_l.
+ assert (Rmax (1 - alpha /2) (1 - beta /2) <=
+         Rmax (/2) (Rmax (1 - alpha /2) (1 - beta /2))) by apply Rmax_r.
+ assert ((1 - alpha /2) <= Rmax (1 - alpha /2) (1 - beta /2)) by apply Rmax_l.
+ assert ((1 - beta /2) <= Rmax (1 - alpha /2) (1 - beta /2)) by apply Rmax_r.
+ assert (Rmax (1 - alpha /2) (1 - beta /2) < 1)
+   by (apply Rmax_lub_lt; fourier).
+ split;[split;[ | apply Rmax_lub_lt]; fourier | ].
+ assert (0 <= 1 - Rmax (/ 2) (Rmax (1 - alpha / 2) (1 - beta / 2))).
+  assert (Rmax (/2) (Rmax (1 - alpha / 2) 
+            (1 - beta /2)) <= 1) by (apply Rmax_lub; fourier).
+  fourier.
+ split; unfold R_dist; rewrite <-Rabs_Ropp, Ropp_minus_distr,
+   Rabs_pos_eq;fourier.
+destruct Xa as [a [[Pa0 Pa1] [P1 P2]]].
+apply Rle_lt_trans with (1 := R_dist_tri _ _ (ps_atan a)).
+apply Rlt_le_trans with (2 := eps_ineq).
+apply Rplus_lt_compat.
+rewrite R_dist_sym; apply Palpha; assumption.
+rewrite <- atan_eq_ps_atan.
+ rewrite <- atan_1; apply (Pbeta a); auto.
+ split; [ | exact P2].
+split;[exact I | apply Rgt_not_eq; assumption].
+split; assumption.
+Qed.
+
+Lemma PI_ineq :
+  forall N : nat,
+    sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= PI / 4 <=
+    sum_f_R0 (tg_alt PI_tg) (2 * N).
+Proof.
+intros; rewrite <- Alt_PI_eq; apply Alt_PI_ineq.
+Qed.
+
diff --git a/theories/Reals/Raxioms.v b/theories/Reals/Raxioms.v
index 8f01d7d0..200019a8 100644
--- a/theories/Reals/Raxioms.v
+++ b/theories/Reals/Raxioms.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -12,7 +12,7 @@
 
 Require Export ZArith_base.
 Require Export Rdefinitions.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (*********************************************************)
 (** *            Field axioms                            *)
@@ -122,8 +122,8 @@ Arguments INR n%nat.
 Definition IZR (z:Z) : R :=
   match z with
   | Z0 => 0
-  | Zpos n => INR (nat_of_P n)
-  | Zneg n => - INR (nat_of_P n)
+  | Zpos n => INR (Pos.to_nat n)
+  | Zneg n => - INR (Pos.to_nat n)
   end.
 Arguments IZR z%Z.
 
diff --git a/theories/Reals/Rbase.v b/theories/Reals/Rbase.v
index dbf9ad71..29715ed9 100644
--- a/theories/Reals/Rbase.v
+++ b/theories/Reals/Rbase.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
diff --git a/theories/Reals/Rbasic_fun.v b/theories/Reals/Rbasic_fun.v
index 4bc7fd10..560f389b 100644
--- a/theories/Reals/Rbasic_fun.v
+++ b/theories/Reals/Rbasic_fun.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -45,10 +45,10 @@ Qed.
 (*********)
 Lemma Rmin_Rgt_l : forall r1 r2 r, Rmin r1 r2 > r -> r1 > r /\ r2 > r.
 Proof.
-  intros r1 r2 r; unfold Rmin in |- *; case (Rle_dec r1 r2); intros.
+  intros r1 r2 r; unfold Rmin; case (Rle_dec r1 r2); intros.
   split.
   assumption.
-  unfold Rgt in |- *; unfold Rgt in H; exact (Rlt_le_trans r r1 r2 H r0).
+  unfold Rgt; unfold Rgt in H; exact (Rlt_le_trans r r1 r2 H r0).
   split.
   generalize (Rnot_le_lt r1 r2 n); intro; exact (Rgt_trans r1 r2 r H0 H).
   assumption.
@@ -57,7 +57,7 @@ Qed.
 (*********)
 Lemma Rmin_Rgt_r : forall r1 r2 r, r1 > r /\ r2 > r -> Rmin r1 r2 > r.
 Proof.
-  intros; unfold Rmin in |- *; case (Rle_dec r1 r2); elim H; clear H; intros;
+  intros; unfold Rmin; case (Rle_dec r1 r2); elim H; clear H; intros;
     assumption.
 Qed.
 
@@ -72,14 +72,14 @@ Qed.
 (*********)
 Lemma Rmin_l : forall x y:R, Rmin x y <= x.
 Proof.
-  intros; unfold Rmin in |- *; case (Rle_dec x y); intro H1;
+  intros; unfold Rmin; case (Rle_dec x y); intro H1;
     [ right; reflexivity | auto with real ].
 Qed.
 
 (*********)
 Lemma Rmin_r : forall x y:R, Rmin x y <= y.
 Proof.
-  intros; unfold Rmin in |- *; case (Rle_dec x y); intro H1;
+  intros; unfold Rmin; case (Rle_dec x y); intro H1;
     [ assumption | auto with real ].
 Qed.
 
@@ -123,20 +123,20 @@ Qed.
 (*********)
 Lemma Rmin_pos : forall x y:R, 0 < x -> 0 < y -> 0 < Rmin x y.
 Proof.
-  intros; unfold Rmin in |- *.
+  intros; unfold Rmin.
   case (Rle_dec x y); intro; assumption.
 Qed.
 
 (*********)
 Lemma Rmin_glb : forall x y z:R, z <= x -> z <= y -> z <= Rmin x y.
 Proof.
-  intros; unfold Rmin in |- *; case (Rle_dec x y); intro; assumption.
+  intros; unfold Rmin; case (Rle_dec x y); intro; assumption.
 Qed.
 
 (*********)
 Lemma Rmin_glb_lt : forall x y z:R, z < x -> z < y -> z < Rmin x y.
 Proof.
-  intros; unfold Rmin in |- *; case (Rle_dec x y); intro; assumption.
+  intros; unfold Rmin; case (Rle_dec x y); intro; assumption.
 Qed.
 
 (*******************************)
@@ -167,8 +167,8 @@ Qed.
 Lemma Rmax_Rle : forall r1 r2 r, r <= Rmax r1 r2 <-> r <= r1 \/ r <= r2.
 Proof.
   intros; split.
-  unfold Rmax in |- *; case (Rle_dec r1 r2); intros; auto.
-  intro; unfold Rmax in |- *; case (Rle_dec r1 r2); elim H; clear H; intros;
+  unfold Rmax; case (Rle_dec r1 r2); intros; auto.
+  intro; unfold Rmax; case (Rle_dec r1 r2); elim H; clear H; intros;
     auto.
   apply (Rle_trans r r1 r2); auto.
   generalize (Rnot_le_lt r1 r2 n); clear n; intro; unfold Rgt in H0;
@@ -177,7 +177,7 @@ Qed.
 
 Lemma Rmax_comm : forall x y:R, Rmax x y = Rmax y x.
 Proof.
-  intros p q; unfold Rmax in |- *; case (Rle_dec p q); case (Rle_dec q p); auto;
+  intros p q; unfold Rmax; case (Rle_dec p q); case (Rle_dec q p); auto;
     intros H1 H2; apply Rle_antisym; auto with real.
 Qed.
 
@@ -188,14 +188,14 @@ Notation RmaxSym := Rmax_comm (only parsing).
 (*********)
 Lemma Rmax_l : forall x y:R, x <= Rmax x y.
 Proof.
-  intros; unfold Rmax in |- *; case (Rle_dec x y); intro H1;
+  intros; unfold Rmax; case (Rle_dec x y); intro H1;
     [ assumption | auto with real ].
 Qed.
 
 (*********)
 Lemma Rmax_r : forall x y:R, y <= Rmax x y.
 Proof.
-  intros; unfold Rmax in |- *; case (Rle_dec x y); intro H1;
+  intros; unfold Rmax; case (Rle_dec x y); intro H1;
     [ right; reflexivity | auto with real ].
 Qed.
 
@@ -232,7 +232,7 @@ Qed.
 Lemma RmaxRmult :
   forall (p q:R) r, 0 <= r -> Rmax (r * p) (r * q) = r * Rmax p q.
 Proof.
-  intros p q r H; unfold Rmax in |- *.
+  intros p q r H; unfold Rmax.
   case (Rle_dec p q); case (Rle_dec (r * p) (r * q)); auto; intros H1 H2; auto.
   case H; intros E1.
   case H1; auto with real.
@@ -246,7 +246,7 @@ Qed.
 (*********)
 Lemma Rmax_stable_in_negreal : forall x y:negreal, Rmax x y < 0.
 Proof.
-  intros; unfold Rmax in |- *; case (Rle_dec x y); intro;
+  intros; unfold Rmax; case (Rle_dec x y); intro;
     [ apply (cond_neg y) | apply (cond_neg x) ].
 Qed.
 
@@ -265,7 +265,7 @@ Qed.
 (*********)
 Lemma Rmax_neg : forall x y:R, x < 0 -> y < 0 -> Rmax x y < 0.
 Proof.
-  intros; unfold Rmax in |- *.
+  intros; unfold Rmax.
   case (Rle_dec x y); intro; assumption.
 Qed.
 
@@ -278,7 +278,7 @@ Lemma Rcase_abs : forall r, {r < 0} + {r >= 0}.
 Proof.
   intro; generalize (Rle_dec 0 r); intro X; elim X; intro; clear X.
   right; apply (Rle_ge 0 r a).
-  left; fold (0 > r) in |- *; apply (Rnot_le_lt 0 r b).
+  left; fold (0 > r); apply (Rnot_le_lt 0 r b).
 Qed.
 
 (*********)
@@ -291,27 +291,27 @@ Definition Rabs r : R :=
 (*********)
 Lemma Rabs_R0 : Rabs 0 = 0.
 Proof.
-  unfold Rabs in |- *; case (Rcase_abs 0); auto; intro.
+  unfold Rabs; case (Rcase_abs 0); auto; intro.
   generalize (Rlt_irrefl 0); intro; exfalso; auto.
 Qed.
 
 Lemma Rabs_R1 : Rabs 1 = 1.
 Proof.
-unfold Rabs in |- *; case (Rcase_abs 1); auto with real.
+unfold Rabs; case (Rcase_abs 1); auto with real.
 intros H; absurd (1 < 0); auto with real.
 Qed.
 
 (*********)
 Lemma Rabs_no_R0 : forall r, r <> 0 -> Rabs r <> 0.
 Proof.
-  intros; unfold Rabs in |- *; case (Rcase_abs r); intro; auto.
+  intros; unfold Rabs; case (Rcase_abs r); intro; auto.
   apply Ropp_neq_0_compat; auto.
 Qed.
 
 (*********)
 Lemma Rabs_left : forall r, r < 0 -> Rabs r = - r.
 Proof.
-  intros; unfold Rabs in |- *; case (Rcase_abs r); trivial; intro;
+  intros; unfold Rabs; case (Rcase_abs r); trivial; intro;
     absurd (r >= 0).
   exact (Rlt_not_ge r 0 H).
   assumption.
@@ -320,7 +320,7 @@ Qed.
 (*********)
 Lemma Rabs_right : forall r, r >= 0 -> Rabs r = r.
 Proof.
-  intros; unfold Rabs in |- *; case (Rcase_abs r); intro.
+  intros; unfold Rabs; case (Rcase_abs r); intro.
   absurd (r >= 0).
   exact (Rlt_not_ge r 0 r0).
   assumption.
@@ -331,21 +331,21 @@ Lemma Rabs_left1 : forall a:R, a <= 0 -> Rabs a = - a.
 Proof.
   intros a H; case H; intros H1.
   apply Rabs_left; auto.
-  rewrite H1; simpl in |- *; rewrite Rabs_right; auto with real.
+  rewrite H1; simpl; rewrite Rabs_right; auto with real.
 Qed.
 
 (*********)
 Lemma Rabs_pos : forall x:R, 0 <= Rabs x.
 Proof.
-  intros; unfold Rabs in |- *; case (Rcase_abs x); intro.
+  intros; unfold Rabs; case (Rcase_abs x); intro.
   generalize (Ropp_lt_gt_contravar x 0 r); intro; unfold Rgt in H;
-    rewrite Ropp_0 in H; unfold Rle in |- *; left; assumption.
+    rewrite Ropp_0 in H; unfold Rle; left; assumption.
   apply Rge_le; assumption.
 Qed.
 
 Lemma Rle_abs : forall x:R, x <= Rabs x.
 Proof.
-  intro; unfold Rabs in |- *; case (Rcase_abs x); intros; fourier.
+  intro; unfold Rabs; case (Rcase_abs x); intros; fourier.
 Qed.
 
 Definition RRle_abs := Rle_abs.
@@ -353,7 +353,7 @@ Definition RRle_abs := Rle_abs.
 (*********)
 Lemma Rabs_pos_eq : forall x:R, 0 <= x -> Rabs x = x.
 Proof.
-  intros; unfold Rabs in |- *; case (Rcase_abs x); intro;
+  intros; unfold Rabs; case (Rcase_abs x); intro;
     [ generalize (Rgt_not_le 0 x r); intro; exfalso; auto | trivial ].
 Qed.
 
@@ -368,7 +368,7 @@ Lemma Rabs_pos_lt : forall x:R, x <> 0 -> 0 < Rabs x.
 Proof.
   intros; generalize (Rabs_pos x); intro; unfold Rle in H0; elim H0; intro;
     auto.
-  exfalso; clear H0; elim H; clear H; generalize H1; unfold Rabs in |- *;
+  exfalso; clear H0; elim H; clear H; generalize H1; unfold Rabs;
     case (Rcase_abs x); intros; auto.
   clear r H1; generalize (Rplus_eq_compat_l x 0 (- x) H0);
     rewrite (let (H1, H2) := Rplus_ne x in H1); rewrite (Rplus_opp_r x);
@@ -378,7 +378,7 @@ Qed.
 (*********)
 Lemma Rabs_minus_sym : forall x y:R, Rabs (x - y) = Rabs (y - x).
 Proof.
-  intros; unfold Rabs in |- *; case (Rcase_abs (x - y));
+  intros; unfold Rabs; case (Rcase_abs (x - y));
     case (Rcase_abs (y - x)); intros.
   generalize (Rminus_lt y x r); generalize (Rminus_lt x y r0); intros;
     generalize (Rlt_asym x y H); intro; exfalso;
@@ -397,7 +397,7 @@ Qed.
 (*********)
 Lemma Rabs_mult : forall x y:R, Rabs (x * y) = Rabs x * Rabs y.
 Proof.
-  intros; unfold Rabs in |- *; case (Rcase_abs (x * y)); case (Rcase_abs x);
+  intros; unfold Rabs; case (Rcase_abs (x * y)); case (Rcase_abs x);
     case (Rcase_abs y); intros; auto.
   generalize (Rmult_lt_gt_compat_neg_l y x 0 r r0); intro;
     rewrite (Rmult_0_r y) in H; generalize (Rlt_asym (x * y) 0 r1);
@@ -448,7 +448,7 @@ Qed.
 (*********)
 Lemma Rabs_Rinv : forall r, r <> 0 -> Rabs (/ r) = / Rabs r.
 Proof.
-  intro; unfold Rabs in |- *; case (Rcase_abs r); case (Rcase_abs (/ r)); auto;
+  intro; unfold Rabs; case (Rcase_abs r); case (Rcase_abs (/ r)); auto;
     intros.
   apply Ropp_inv_permute; auto.
   generalize (Rinv_lt_0_compat r r1); intro; unfold Rge in r0; elim r0; intros.
@@ -470,7 +470,7 @@ Proof.
   cut (Rabs (-1) = 1).
   intros; rewrite H0.
   ring.
-  unfold Rabs in |- *; case (Rcase_abs (-1)).
+  unfold Rabs; case (Rcase_abs (-1)).
   intro; ring.
   intro H0; generalize (Rge_le (-1) 0 H0); intros.
   generalize (Ropp_le_ge_contravar 0 (-1) H1).
@@ -483,13 +483,13 @@ Qed.
 (*********)
 Lemma Rabs_triang : forall a b:R, Rabs (a + b) <= Rabs a + Rabs b.
 Proof.
-  intros a b; unfold Rabs in |- *; case (Rcase_abs (a + b)); case (Rcase_abs a);
+  intros a b; unfold Rabs; case (Rcase_abs (a + b)); case (Rcase_abs a);
     case (Rcase_abs b); intros.
   apply (Req_le (- (a + b)) (- a + - b)); rewrite (Ropp_plus_distr a b);
     reflexivity.
 (**)
   rewrite (Ropp_plus_distr a b); apply (Rplus_le_compat_l (- a) (- b) b);
-    unfold Rle in |- *; unfold Rge in r; elim r; intro.
+    unfold Rle; unfold Rge in r; elim r; intro.
   left; unfold Rgt in H; generalize (Rplus_lt_compat_l (- b) 0 b H); intro;
     elim (Rplus_ne (- b)); intros v w; rewrite v in H0;
       clear v w; rewrite (Rplus_opp_l b) in H0; apply (Rlt_trans (- b) 0 b H0 H).
@@ -497,7 +497,7 @@ Proof.
 (**)
   rewrite (Ropp_plus_distr a b); rewrite (Rplus_comm (- a) (- b));
     rewrite (Rplus_comm a (- b)); apply (Rplus_le_compat_l (- b) (- a) a);
-      unfold Rle in |- *; unfold Rge in r0; elim r0; intro.
+      unfold Rle; unfold Rge in r0; elim r0; intro.
   left; unfold Rgt in H; generalize (Rplus_lt_compat_l (- a) 0 a H); intro;
     elim (Rplus_ne (- a)); intros v w; rewrite v in H0;
       clear v w; rewrite (Rplus_opp_l a) in H0; apply (Rlt_trans (- a) 0 a H0 H).
@@ -521,27 +521,27 @@ Proof.
 (**)
   rewrite (Rplus_comm a b); rewrite (Rplus_comm (- a) b);
     apply (Rplus_le_compat_l b a (- a)); apply (Rminus_le a (- a));
-      unfold Rminus in |- *; rewrite (Ropp_involutive a);
+      unfold Rminus; rewrite (Ropp_involutive a);
         generalize (Rplus_lt_compat_l a a 0 r0); clear r r1;
           intro; elim (Rplus_ne a); intros v w; rewrite v in H;
             clear v w; generalize (Rlt_trans (a + a) a 0 H r0);
               intro; apply (Rlt_le (a + a) 0 H0).
 (**)
   apply (Rplus_le_compat_l a b (- b)); apply (Rminus_le b (- b));
-    unfold Rminus in |- *; rewrite (Ropp_involutive b);
+    unfold Rminus; rewrite (Ropp_involutive b);
       generalize (Rplus_lt_compat_l b b 0 r); clear r0 r1;
         intro; elim (Rplus_ne b); intros v w; rewrite v in H;
           clear v w; generalize (Rlt_trans (b + b) b 0 H r);
             intro; apply (Rlt_le (b + b) 0 H0).
 (**)
-  unfold Rle in |- *; right; reflexivity.
+  unfold Rle; right; reflexivity.
 Qed.
 
 (*********)
 Lemma Rabs_triang_inv : forall a b:R, Rabs a - Rabs b <= Rabs (a - b).
 Proof.
   intros; apply (Rplus_le_reg_l (Rabs b) (Rabs a - Rabs b) (Rabs (a - b)));
-    unfold Rminus in |- *; rewrite <- (Rplus_assoc (Rabs b) (Rabs a) (- Rabs b));
+    unfold Rminus; rewrite <- (Rplus_assoc (Rabs b) (Rabs a) (- Rabs b));
       rewrite (Rplus_comm (Rabs b) (Rabs a));
         rewrite (Rplus_assoc (Rabs a) (Rabs b) (- Rabs b));
           rewrite (Rplus_opp_r (Rabs b)); rewrite (proj1 (Rplus_ne (Rabs a)));
@@ -561,7 +561,7 @@ Proof.
   rewrite <- (Rabs_Ropp (Rabs a - Rabs b)); rewrite <- (Rabs_Ropp (a - b));
     do 2 rewrite Ropp_minus_distr.
   apply H; left; assumption.
-  rewrite Heq; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  rewrite Heq; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     apply Rabs_pos.
   apply H; left; assumption.
   intros; replace (Rabs (Rabs a - Rabs b)) with (Rabs a - Rabs b).
@@ -576,8 +576,8 @@ Qed.
 (*********)
 Lemma Rabs_def1 : forall x a:R, x < a -> - a < x -> Rabs x < a.
 Proof.
-  unfold Rabs in |- *; intros; case (Rcase_abs x); intro.
-  generalize (Ropp_lt_gt_contravar (- a) x H0); unfold Rgt in |- *;
+  unfold Rabs; intros; case (Rcase_abs x); intro.
+  generalize (Ropp_lt_gt_contravar (- a) x H0); unfold Rgt;
     rewrite Ropp_involutive; intro; assumption.
   assumption.
 Qed.
@@ -585,15 +585,15 @@ Qed.
 (*********)
 Lemma Rabs_def2 : forall x a:R, Rabs x < a -> x < a /\ - a < x.
 Proof.
-  unfold Rabs in |- *; intro x; case (Rcase_abs x); intros.
-  generalize (Ropp_gt_lt_0_contravar x r); unfold Rgt in |- *; intro;
+  unfold Rabs; intro x; case (Rcase_abs x); intros.
+  generalize (Ropp_gt_lt_0_contravar x r); unfold Rgt; intro;
     generalize (Rlt_trans 0 (- x) a H0 H); intro; split.
   apply (Rlt_trans x 0 a r H1).
   generalize (Ropp_lt_gt_contravar (- x) a H); rewrite (Ropp_involutive x);
-    unfold Rgt in |- *; trivial.
+    unfold Rgt; trivial.
   fold (a > x) in H; generalize (Rgt_ge_trans a x 0 H r); intro;
-    generalize (Ropp_lt_gt_0_contravar a H0); intro; fold (0 > - a) in |- *;
-      generalize (Rge_gt_trans x 0 (- a) r H1); unfold Rgt in |- *;
+    generalize (Ropp_lt_gt_0_contravar a H0); intro; fold (0 > - a);
+      generalize (Rge_gt_trans x 0 (- a) r H1); unfold Rgt;
         intro; split; assumption.
 Qed.
 
@@ -623,16 +623,16 @@ Proof.
   apply RmaxLess1; auto.
 Qed.
 
-Lemma Rabs_Zabs : forall z:Z, Rabs (IZR z) = IZR (Zabs z).
+Lemma Rabs_Zabs : forall z:Z, Rabs (IZR z) = IZR (Z.abs z).
 Proof.
-  intros z; case z; simpl in |- *; auto with real.
+  intros z; case z; simpl; auto with real.
   apply Rabs_right; auto with real.
   intros p0; apply Rabs_right; auto with real zarith.
   intros p0; rewrite Rabs_Ropp.
   apply Rabs_right; auto with real zarith.
 Qed.
 
-Lemma abs_IZR : forall z, IZR (Zabs z) = Rabs (IZR z).
+Lemma abs_IZR : forall z, IZR (Z.abs z) = Rabs (IZR z).
 Proof.
   intros.
   now rewrite Rabs_Zabs.
diff --git a/theories/Reals/Rcomplete.v b/theories/Reals/Rcomplete.v
index 77cb560c..8e0e0692 100644
--- a/theories/Reals/Rcomplete.v
+++ b/theories/Reals/Rcomplete.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -11,7 +11,7 @@ Require Import Rfunctions.
 Require Import Rseries.
 Require Import SeqProp.
 Require Import Max.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (****************************************************)
 (*              R is complete :                     *)
@@ -37,7 +37,7 @@ Proof.
   intros.
   exists x.
   rewrite <- H2 in p0.
-  unfold Un_cv in |- *.
+  unfold Un_cv.
   intros.
   unfold Un_cv in p; unfold Un_cv in p0.
   cut (0 < eps / 3).
@@ -46,7 +46,7 @@ Proof.
   elim (p0 (eps / 3) H4); intros.
   exists (max x1 x2).
   intros.
-  unfold R_dist in |- *.
+  unfold R_dist.
   apply Rle_lt_trans with (Rabs (Un n - Vn n) + Rabs (Vn n - x)).
   replace (Un n - x) with (Un n - Vn n + (Vn n - x));
   [ apply Rabs_triang | ring ].
@@ -54,14 +54,14 @@ Proof.
   do 2 rewrite <- (Rplus_comm (Rabs (Vn n - x))).
   apply Rplus_le_compat_l.
   repeat rewrite Rabs_right.
-  unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- Vn n));
+  unfold Rminus; do 2 rewrite <- (Rplus_comm (- Vn n));
     apply Rplus_le_compat_l.
   assert (H8 := Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)).
   fold Vn Wn in H8.
   elim (H8 n); intros.
   assumption.
   apply Rle_ge.
-  unfold Rminus in |- *; apply Rplus_le_reg_l with (Vn n).
+  unfold Rminus; apply Rplus_le_reg_l with (Vn n).
   rewrite Rplus_0_r.
   replace (Vn n + (Wn n + - Vn n)) with (Wn n); [ idtac | ring ].
   assert (H8 := Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)).
@@ -69,7 +69,7 @@ Proof.
   elim (H8 n); intros.
   apply Rle_trans with (Un n); assumption.
   apply Rle_ge.
-  unfold Rminus in |- *; apply Rplus_le_reg_l with (Vn n).
+  unfold Rminus; apply Rplus_le_reg_l with (Vn n).
   rewrite Rplus_0_r.
   replace (Vn n + (Un n + - Vn n)) with (Un n); [ idtac | ring ].
   assert (H8 := Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)).
@@ -85,26 +85,26 @@ Proof.
   repeat apply Rplus_lt_compat.
   unfold R_dist in H5.
   apply H5.
-  unfold ge in |- *; apply le_trans with (max x1 x2).
+  unfold ge; apply le_trans with (max x1 x2).
   apply le_max_l.
   assumption.
   rewrite <- Rabs_Ropp.
   replace (- (x - Vn n)) with (Vn n - x); [ idtac | ring ].
   unfold R_dist in H6.
   apply H6.
-  unfold ge in |- *; apply le_trans with (max x1 x2).
+  unfold ge; apply le_trans with (max x1 x2).
   apply le_max_r.
   assumption.
   unfold R_dist in H6.
   apply H6.
-  unfold ge in |- *; apply le_trans with (max x1 x2).
+  unfold ge; apply le_trans with (max x1 x2).
   apply le_max_r.
   assumption.
   right.
-  pattern eps at 4 in |- *; replace eps with (3 * (eps / 3)).
+  pattern eps at 4; replace eps with (3 * (eps / 3)).
   ring.
-  unfold Rdiv in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR.
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   apply cond_eq.
   intros.
@@ -130,10 +130,10 @@ Proof.
   repeat apply Rplus_lt_compat.
   rewrite <- Rabs_Ropp.
   replace (- (x - Wn N)) with (Wn N - x); [ apply H4 | ring ].
-  unfold ge, N in |- *.
+  unfold ge, N.
   apply le_trans with (max N1 N2); apply le_max_l.
-  unfold Wn, Vn in |- *.
-  unfold sequence_majorant, sequence_minorant in |- *.
+  unfold Wn, Vn.
+  unfold sequence_majorant, sequence_minorant.
   assert
     (H7 :=
       approx_maj (fun k:nat => Un (N + k)%nat) (maj_ss Un N (cauchy_maj Un H))).
@@ -169,13 +169,13 @@ Proof.
   [ repeat apply Rplus_lt_compat | ring ].
   assumption.
   apply H6.
-  unfold ge in |- *.
+  unfold ge.
   apply le_trans with N.
-  unfold N in |- *; apply le_max_r.
+  unfold N; apply le_max_r.
   apply le_plus_l.
-  unfold ge in |- *.
+  unfold ge.
   apply le_trans with N.
-  unfold N in |- *; apply le_max_r.
+  unfold N; apply le_max_r.
   apply le_plus_l.
   rewrite <- Rabs_Ropp.
   replace (- (Un (N + k1)%nat - Vn N)) with (Vn N - Un (N + k1)%nat);
@@ -183,14 +183,14 @@ Proof.
   reflexivity.
   reflexivity.
   apply H5.
-  unfold ge in |- *; apply le_trans with (max N1 N2).
+  unfold ge; apply le_trans with (max N1 N2).
   apply le_max_r.
-  unfold N in |- *; apply le_max_l.
-  pattern eps at 4 in |- *; replace eps with (5 * (eps / 5)).
+  unfold N; apply le_max_l.
+  pattern eps at 4; replace eps with (5 * (eps / 5)).
   ring.
-  unfold Rdiv in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m.
+  unfold Rdiv; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m.
   discrR.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   assumption.
   apply Rinv_0_lt_compat.
   prove_sup0; try apply lt_O_Sn.
diff --git a/theories/Reals/Rdefinitions.v b/theories/Reals/Rdefinitions.v
index 83c6b82d..f7d03ed8 100644
--- a/theories/Reals/Rdefinitions.v
+++ b/theories/Reals/Rdefinitions.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -21,7 +21,7 @@ Delimit Scope R_scope with R.
 (* Automatically open scope R_scope for arguments of type R *)
 Bind Scope R_scope with R.
 
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Parameter R0 : R.
 Parameter R1 : R.
diff --git a/theories/Reals/Rderiv.v b/theories/Reals/Rderiv.v
index 105d8347..e714f5f8 100644
--- a/theories/Reals/Rderiv.v
+++ b/theories/Reals/Rderiv.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -16,7 +16,7 @@ Require Import Rfunctions.
 Require Import Rlimit.
 Require Import Fourier.
 Require Import Omega.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (*********)
 Definition D_x (D:R -> Prop) (y x:R) : Prop := D x /\ y <> x.
@@ -34,18 +34,18 @@ Lemma cont_deriv :
   forall (f d:R -> R) (D:R -> Prop) (x0:R),
     D_in f d D x0 -> continue_in f D x0.
 Proof.
-  unfold continue_in in |- *; unfold D_in in |- *; unfold limit1_in in |- *;
-    unfold limit_in in |- *; unfold Rdiv in |- *; simpl in |- *;
+  unfold continue_in; unfold D_in; unfold limit1_in;
+    unfold limit_in; unfold Rdiv; simpl;
       intros; elim (H eps H0); clear H; intros; elim H;
         clear H; intros; elim (Req_dec (d x0) 0); intro.
   split with (Rmin 1 x); split.
   elim (Rmin_Rgt 1 x 0); intros a b; apply (b (conj Rlt_0_1 H)).
   intros; elim H3; clear H3; intros;
     generalize (let (H1, H2) := Rmin_Rgt 1 x (R_dist x1 x0) in H1);
-      unfold Rgt in |- *; intro; elim (H5 H4); clear H5;
+      unfold Rgt; intro; elim (H5 H4); clear H5;
         intros; generalize (H1 x1 (conj H3 H6)); clear H1;
           intro; unfold D_x in H3; elim H3; intros.
-  rewrite H2 in H1; unfold R_dist in |- *; unfold R_dist in H1;
+  rewrite H2 in H1; unfold R_dist; unfold R_dist in H1;
     cut (Rabs (f x1 - f x0) < eps * Rabs (x1 - x0)).
   intro; unfold R_dist in H5;
     generalize (Rmult_lt_compat_l eps (Rabs (x1 - x0)) 1 H0 H5);
@@ -68,7 +68,7 @@ Proof.
   intros; elim (Rmin_Rgt (Rmin (/ 2) x) (eps * / Rabs (2 * d x0)) 0);
     intros a b; apply (b (conj H4 H3)).
   apply Rmult_gt_0_compat; auto.
-  unfold Rgt in |- *; apply Rinv_0_lt_compat; apply Rabs_pos_lt;
+  unfold Rgt; apply Rinv_0_lt_compat; apply Rabs_pos_lt;
     apply Rmult_integral_contrapositive; split.
   discrR.
   assumption.
@@ -80,17 +80,17 @@ Proof.
     generalize
       (let (H1, H2) :=
         Rmin_Rgt (Rmin (/ 2) x) (eps * / Rabs (2 * d x0)) (R_dist x1 x0) in
-        H1); unfold Rgt in |- *; intro; elim (H5 H4); clear H5;
+        H1); unfold Rgt; intro; elim (H5 H4); clear H5;
       intros; generalize (let (H1, H2) := Rmin_Rgt (/ 2) x (R_dist x1 x0) in H1);
-        unfold Rgt in |- *; intro; elim (H7 H5); clear H7;
+        unfold Rgt; intro; elim (H7 H5); clear H7;
           intros; clear H4 H5; generalize (H1 x1 (conj H3 H8));
             clear H1; intro; unfold D_x in H3; elim H3; intros;
-              generalize (sym_not_eq H5); clear H5; intro H5;
+              generalize (not_eq_sym H5); clear H5; intro H5;
                 generalize (Rminus_eq_contra x1 x0 H5); intro; generalize H1;
-                  pattern (d x0) at 1 in |- *;
+                  pattern (d x0) at 1;
                     rewrite <- (let (H1, H2) := Rmult_ne (d x0) in H2);
-                      rewrite <- (Rinv_l (x1 - x0) H9); unfold R_dist in |- *;
-                        unfold Rminus at 1 in |- *; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0)));
+                      rewrite <- (Rinv_l (x1 - x0) H9); unfold R_dist;
+                        unfold Rminus at 1; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0)));
                           rewrite (Rmult_comm (/ (x1 - x0) * (x1 - x0)) (d x0));
                             rewrite <- (Ropp_mult_distr_l_reverse (d x0) (/ (x1 - x0) * (x1 - x0)));
                               rewrite (Rmult_comm (- d x0) (/ (x1 - x0) * (x1 - x0)));
@@ -113,7 +113,7 @@ Proof.
                                                   ; generalize (Rabs_triang_inv (f x1 - f x0) ((x1 - x0) * d x0));
                                                     intro; rewrite (Rmult_comm (x1 - x0) (- d x0));
                                                       rewrite (Ropp_mult_distr_l_reverse (d x0) (x1 - x0));
-                                                        fold (f x1 - f x0 - d x0 * (x1 - x0)) in |- *;
+                                                        fold (f x1 - f x0 - d x0 * (x1 - x0));
                                                           rewrite (Rmult_comm (x1 - x0) (d x0)) in H10; clear H1;
                                                             intro;
                                                               generalize
@@ -123,7 +123,7 @@ Proof.
                                                                   generalize
                                                                     (Rplus_lt_compat_l (Rabs (d x0 * (x1 - x0)))
                                                                       (Rabs (f x1 - f x0) - Rabs (d x0 * (x1 - x0))) (
-                                                                        Rabs (x1 - x0) * eps) H1); unfold Rminus at 2 in |- *;
+                                                                        Rabs (x1 - x0) * eps) H1); unfold Rminus at 2;
                                                                     rewrite (Rplus_comm (Rabs (f x1 - f x0)) (- Rabs (d x0 * (x1 - x0))));
                                                                       rewrite <-
                                                                         (Rplus_assoc (Rabs (d x0 * (x1 - x0))) (- Rabs (d x0 * (x1 - x0)))
@@ -162,7 +162,7 @@ Proof.
       (Rplus_lt_compat (Rabs (d x0 * (x1 - x0))) (eps * / 2)
         (Rabs (x1 - x0) * eps) (eps * / 2) H5 H3); intro;
       rewrite eps2 in H10; assumption.
-  unfold Rabs in |- *; case (Rcase_abs 2); auto.
+  unfold Rabs; case (Rcase_abs 2); auto.
   intro; cut (0 < 2).
   intro ; elim (Rlt_asym 0 2 H7 r).
   fourier.
@@ -174,14 +174,14 @@ Qed.
 Lemma Dconst :
   forall (D:R -> Prop) (y x0:R), D_in (fun x:R => y) (fun x:R => 0) D x0.
 Proof.
-  unfold D_in in |- *; intros; unfold limit1_in in |- *;
-    unfold limit_in in |- *; unfold Rdiv in |- *; intros;
-      simpl in |- *; split with eps; split; auto.
-  intros; rewrite (Rminus_diag_eq y y (refl_equal y)); rewrite Rmult_0_l;
-    unfold R_dist in |- *; rewrite (Rminus_diag_eq 0 0 (refl_equal 0));
-      unfold Rabs in |- *; case (Rcase_abs 0); intro.
+  unfold D_in; intros; unfold limit1_in;
+    unfold limit_in; unfold Rdiv; intros;
+      simpl; split with eps; split; auto.
+  intros; rewrite (Rminus_diag_eq y y (eq_refl y)); rewrite Rmult_0_l;
+    unfold R_dist; rewrite (Rminus_diag_eq 0 0 (eq_refl 0));
+      unfold Rabs; case (Rcase_abs 0); intro.
   absurd (0 < 0); auto.
-  red in |- *; intro; apply (Rlt_irrefl 0 H1).
+  red; intro; apply (Rlt_irrefl 0 H1).
   unfold Rgt in H0; assumption.
 Qed.
 
@@ -189,15 +189,15 @@ Qed.
 Lemma Dx :
   forall (D:R -> Prop) (x0:R), D_in (fun x:R => x) (fun x:R => 1) D x0.
 Proof.
-  unfold D_in in |- *; unfold Rdiv in |- *; intros; unfold limit1_in in |- *;
-    unfold limit_in in |- *; intros; simpl in |- *; split with eps;
+  unfold D_in; unfold Rdiv; intros; unfold limit1_in;
+    unfold limit_in; intros; simpl; split with eps;
       split; auto.
   intros; elim H0; clear H0; intros; unfold D_x in H0; elim H0; intros;
-    rewrite (Rinv_r (x - x0) (Rminus_eq_contra x x0 (sym_not_eq H3)));
-      unfold R_dist in |- *; rewrite (Rminus_diag_eq 1 1 (refl_equal 1));
-        unfold Rabs in |- *; case (Rcase_abs 0); intro.
+    rewrite (Rinv_r (x - x0) (Rminus_eq_contra x x0 (not_eq_sym H3)));
+      unfold R_dist; rewrite (Rminus_diag_eq 1 1 (eq_refl 1));
+        unfold Rabs; case (Rcase_abs 0); intro.
   absurd (0 < 0); auto.
-  red in |- *; intro; apply (Rlt_irrefl 0 r).
+  red; intro; apply (Rlt_irrefl 0 r).
   unfold Rgt in H; assumption.
 Qed.
 
@@ -208,12 +208,12 @@ Lemma Dadd :
     D_in g dg D x0 ->
     D_in (fun x:R => f x + g x) (fun x:R => df x + dg x) D x0.
 Proof.
-  unfold D_in in |- *; intros;
+  unfold D_in; intros;
     generalize
       (limit_plus (fun x:R => (f x - f x0) * / (x - x0))
         (fun x:R => (g x - g x0) * / (x - x0)) (D_x D x0) (
-          df x0) (dg x0) x0 H H0); clear H H0; unfold limit1_in in |- *;
-      unfold limit_in in |- *; simpl in |- *; intros; elim (H eps H0);
+          df x0) (dg x0) x0 H H0); clear H H0; unfold limit1_in;
+      unfold limit_in; simpl; intros; elim (H eps H0);
         clear H; intros; elim H; clear H; intros; split with x;
           split; auto; intros; generalize (H1 x1 H2); clear H1;
             intro; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))) in H1;
@@ -233,8 +233,8 @@ Lemma Dmult :
     D_in g dg D x0 ->
     D_in (fun x:R => f x * g x) (fun x:R => df x * g x + f x * dg x) D x0.
 Proof.
-  intros; unfold D_in in |- *; generalize H H0; intros; unfold D_in in H, H0;
-    generalize (cont_deriv f df D x0 H1); unfold continue_in in |- *;
+  intros; unfold D_in; generalize H H0; intros; unfold D_in in H, H0;
+    generalize (cont_deriv f df D x0 H1); unfold continue_in;
       intro;
         generalize
           (limit_mul (fun x:R => (g x - g x0) * / (x - x0)) (
@@ -250,8 +250,8 @@ Proof.
             (fun x:R => (g x - g x0) * / (x - x0) * f x) (
               D_x D x0) (df x0 * g x0) (dg x0 * f x0) x0 H H4);
           clear H4 H; intro; unfold limit1_in in H; unfold limit_in in H;
-            simpl in H; unfold limit1_in in |- *; unfold limit_in in |- *;
-              simpl in |- *; intros; elim (H eps H0); clear H; intros;
+            simpl in H; unfold limit1_in; unfold limit_in;
+              simpl; intros; elim (H eps H0); clear H; intros;
                 elim H; clear H; intros; split with x; split; auto;
                   intros; generalize (H1 x1 H2); clear H1; intro;
                     rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))) in H1;
@@ -268,9 +268,9 @@ Proof.
                                     ((f x1 - f x0) * g x0 + (g x1 - g x0) * f x1 = f x1 * g x1 - f x0 * g x0).
   intro; rewrite H3 in H1; assumption.
   ring.
-  unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros;
+  unfold limit1_in; unfold limit_in; simpl; intros;
     split with eps; split; auto; intros; elim (R_dist_refl (g x0) (g x0));
-      intros a b; rewrite (b (refl_equal (g x0))); unfold Rgt in H;
+      intros a b; rewrite (b (eq_refl (g x0))); unfold Rgt in H;
         assumption.
 Qed.
 
@@ -281,7 +281,7 @@ Lemma Dmult_const :
 Proof.
   intros;
     generalize (Dmult D (fun _:R => 0) df (fun _:R => a) f x0 (Dconst D a x0) H);
-      unfold D_in in |- *; intros; rewrite (Rmult_0_l (f x0)) in H0;
+      unfold D_in; intros; rewrite (Rmult_0_l (f x0)) in H0;
         rewrite (let (H1, H2) := Rplus_ne (a * df x0) in H2) in H0;
           assumption.
 Qed.
@@ -291,10 +291,10 @@ Lemma Dopp :
   forall (D:R -> Prop) (f df:R -> R) (x0:R),
     D_in f df D x0 -> D_in (fun x:R => - f x) (fun x:R => - df x) D x0.
 Proof.
-  intros; generalize (Dmult_const D f df x0 (-1) H); unfold D_in in |- *;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
+  intros; generalize (Dmult_const D f df x0 (-1) H); unfold D_in;
+    unfold limit1_in; unfold limit_in;
       intros; generalize (H0 eps H1); clear H0; intro; elim H0;
-        clear H0; intros; elim H0; clear H0; simpl in |- *;
+        clear H0; intros; elim H0; clear H0; simpl;
           intros; split with x; split; auto.
   intros; generalize (H2 x1 H3); clear H2; intro;
     rewrite Ropp_mult_distr_l_reverse in H2;
@@ -313,7 +313,7 @@ Lemma Dminus :
     D_in g dg D x0 ->
     D_in (fun x:R => f x - g x) (fun x:R => df x - dg x) D x0.
 Proof.
-  unfold Rminus in |- *; intros; generalize (Dopp D g dg x0 H0); intro;
+  unfold Rminus; intros; generalize (Dopp D g dg x0 H0); intro;
     apply (Dadd D df (fun x:R => - dg x) f (fun x:R => - g x) x0);
       assumption.
 Qed.
@@ -324,14 +324,14 @@ Lemma Dx_pow_n :
     D_in (fun x:R => x ^ n) (fun x:R => INR n * x ^ (n - 1)) D x0.
 Proof.
   simple induction n; intros.
-  simpl in |- *; rewrite Rmult_0_l; apply Dconst.
+  simpl; rewrite Rmult_0_l; apply Dconst.
   intros; cut (n0 = (S n0 - 1)%nat);
-    [ intro a; rewrite <- a; clear a | simpl in |- *; apply minus_n_O ].
+    [ intro a; rewrite <- a; clear a | simpl; apply minus_n_O ].
   generalize
     (Dmult D (fun _:R => 1) (fun x:R => INR n0 * x ^ (n0 - 1)) (
       fun x:R => x) (fun x:R => x ^ n0) x0 (Dx D x0) (
-        H D x0)); unfold D_in in |- *; unfold limit1_in in |- *;
-    unfold limit_in in |- *; simpl in |- *; intros; elim (H0 eps H1);
+        H D x0)); unfold D_in; unfold limit1_in;
+    unfold limit_in; simpl; intros; elim (H0 eps H1);
       clear H0; intros; elim H0; clear H0; intros; split with x;
         split; auto.
   intros; generalize (H2 x1 H3); clear H2 H3; intro;
@@ -340,7 +340,7 @@ Proof.
         rewrite (Rmult_comm (INR n0) (x0 ^ (n0 - 1))) in H2;
           rewrite <- (Rmult_assoc x0 (x0 ^ (n0 - 1)) (INR n0)) in H2;
             rewrite (tech_pow_Rmult x0 (n0 - 1)) in H2; elim (Peano_dec.eq_nat_dec n0 0) ; intros cond.
-  rewrite cond in H2; rewrite cond; simpl in H2; simpl in |- *;
+  rewrite cond in H2; rewrite cond; simpl in H2; simpl;
     cut (1 + x0 * 1 * 0 = 1 * 1);
       [ intro A; rewrite A in H2; assumption | ring ].
   cut (n0 <> 0%nat -> S (n0 - 1) = n0); [ intro | omega ];
@@ -355,8 +355,8 @@ Lemma Dcomp :
     D_in g dg Dg (f x0) ->
     D_in (fun x:R => g (f x)) (fun x:R => df x * dg (f x)) (Dgf Df Dg f) x0.
 Proof.
-  intros Df Dg df dg f g x0 H H0; generalize H H0; unfold D_in in |- *;
-    unfold Rdiv in |- *; intros;
+  intros Df Dg df dg f g x0 H H0; generalize H H0; unfold D_in;
+    unfold Rdiv; intros;
       generalize
         (limit_comp f (fun x:R => (g x - g (f x0)) * / (x - f x0)) (
           D_x Df x0) (D_x Dg (f x0)) (f x0) (dg (f x0)) x0);
@@ -376,8 +376,8 @@ Proof.
       (limit_mul (fun x:R => (f x - f x0) * / (x - x0)) (
         fun x:R => dg (f x0)) (D_x Df x0) (df x0) (dg (f x0)) x0 H1
       (limit_free (fun x:R => dg (f x0)) (D_x Df x0) x0 x0));
-      intro; unfold limit1_in in |- *; unfold limit_in in |- *;
-        simpl in |- *; unfold limit1_in in H5, H7; unfold limit_in in H5, H7;
+      intro; unfold limit1_in; unfold limit_in;
+        simpl; unfold limit1_in in H5, H7; unfold limit_in in H5, H7;
           simpl in H5, H7; intros; elim (H5 eps H8); elim (H7 eps H8);
             clear H5 H7; intros; elim H5; elim H7; clear H5 H7;
               intros; split with (Rmin x x1); split.
@@ -391,7 +391,7 @@ Proof.
       rewrite (Rminus_diag_eq (f x2) (f x0) H12) in H16;
         rewrite (Rmult_0_l (/ (x2 - x0))) in H16;
           rewrite (Rmult_0_l (dg (f x0))) in H16; rewrite H12;
-            rewrite (Rminus_diag_eq (g (f x0)) (g (f x0)) (refl_equal (g (f x0))));
+            rewrite (Rminus_diag_eq (g (f x0)) (g (f x0)) (eq_refl (g (f x0))));
               rewrite (Rmult_0_l (/ (x2 - x0))); assumption.
   clear H10 H5; elim H11; clear H11; intros; elim H5; clear H5; intros;
     cut
@@ -405,8 +405,8 @@ Proof.
               in H15; rewrite (Rinv_l (f x2 - f x0) H16) in H15;
                 rewrite (let (H1, H2) := Rmult_ne (/ (x2 - x0)) in H2) in H15;
                   rewrite (Rmult_comm (df x0) (dg (f x0))); assumption.
-  clear H5 H3 H4 H2; unfold limit1_in in |- *; unfold limit_in in |- *;
-    simpl in |- *; unfold limit1_in in H1; unfold limit_in in H1;
+  clear H5 H3 H4 H2; unfold limit1_in; unfold limit_in;
+    simpl; unfold limit1_in in H1; unfold limit_in in H1;
       simpl in H1; intros; elim (H1 eps H2); clear H1; intros;
         elim H1; clear H1; intros; split with x; split; auto;
           intros; unfold D_x, Dgf in H4, H3; elim H4; clear H4;
@@ -425,8 +425,8 @@ Proof.
     generalize
       (Dcomp D D dexpr (fun x:R => INR n * x ^ (n - 1)) expr (
         fun x:R => x ^ n) x0 H (Dx_pow_n n D (expr x0)));
-      intro; unfold D_in in |- *; unfold limit1_in in |- *;
-        unfold limit_in in |- *; simpl in |- *; intros; unfold D_in in H0;
+      intro; unfold D_in; unfold limit1_in;
+        unfold limit_in; simpl; intros; unfold D_in in H0;
           unfold limit1_in in H0; unfold limit_in in H0; simpl in H0;
             elim (H0 eps H1); clear H0; intros; elim H0; clear H0;
               intros; split with x; split; intros; auto.
diff --git a/theories/Reals/Reals.v b/theories/Reals/Reals.v
index a15e9949..03bf534d 100644
--- a/theories/Reals/Reals.v
+++ b/theories/Reals/Reals.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
diff --git a/theories/Reals/Rfunctions.v b/theories/Reals/Rfunctions.v
index c0cd7864..4724d0e5 100644
--- a/theories/Reals/Rfunctions.v
+++ b/theories/Reals/Rfunctions.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -25,8 +25,8 @@ Require Export SplitRmult.
 Require Export ArithProp.
 Require Import Omega.
 Require Import Zpower.
-Open Local Scope nat_scope.
-Open Local Scope R_scope.
+Local Open Scope nat_scope.
+Local Open Scope R_scope.
 
 (*******************************)
 (** * Lemmas about factorial   *)
@@ -82,6 +82,15 @@ Proof.
   intros n0 H' m; rewrite H'; auto with real.
 Qed.
 
+Lemma Rpow_mult_distr : forall (x y:R) (n:nat), (x * y) ^ n = x^n * y^n.
+Proof.
+intros x y n ; induction n.
+ field.
+ simpl.
+ repeat (rewrite Rmult_assoc) ; apply Rmult_eq_compat_l.
+ rewrite IHn ; field.
+Qed.
+
 Lemma pow_nonzero : forall (x:R) (n:nat), x <> 0 -> x ^ n <> 0.
 Proof.
   intro; simple induction n; simpl.
@@ -212,8 +221,8 @@ Qed.
 Lemma RPow_abs : forall (x:R) (n:nat), Rabs x ^ n = Rabs (x ^ n).
 Proof.
   intro; simple induction n; simpl.
-  apply sym_eq; apply Rabs_pos_eq; apply Rlt_le; apply Rlt_0_1.
-  intros; rewrite H; apply sym_eq; apply Rabs_mult.
+  symmetry; apply Rabs_pos_eq; apply Rlt_le; apply Rlt_0_1.
+  intros; rewrite H; symmetry; apply Rabs_mult.
 Qed.
 
 
@@ -517,16 +526,16 @@ Qed.
 (*i Due to L.Thery i*)
 
 Ltac case_eq name :=
-  generalize (refl_equal name); pattern name at -1; case name.
+  generalize (eq_refl name); pattern name at -1; case name.
 
 Definition powerRZ (x:R) (n:Z) :=
   match n with
     | Z0 => 1
-    | Zpos p => x ^ nat_of_P p
-    | Zneg p => / x ^ nat_of_P p
+    | Zpos p => x ^ Pos.to_nat p
+    | Zneg p => / x ^ Pos.to_nat p
   end.
 
-Infix Local "^Z" := powerRZ (at level 30, right associativity) : R_scope.
+Local Infix "^Z" := powerRZ (at level 30, right associativity) : R_scope.
 
 Lemma Zpower_NR0 :
   forall (x:Z) (n:nat), (0 <= x)%Z -> (0 <= Zpower_nat x n)%Z.
@@ -539,7 +548,7 @@ Proof.
   reflexivity.
 Qed.
 
-Lemma powerRZ_1 : forall x:R, x ^Z Zsucc 0 = x.
+Lemma powerRZ_1 : forall x:R, x ^Z Z.succ 0 = x.
 Proof.
   simpl; auto with real.
 Qed.
@@ -549,67 +558,63 @@ Proof.
   destruct z; simpl; auto with real.
 Qed.
 
+Lemma powerRZ_pos_sub (x:R) (n m:positive) : x <> 0 ->
+   x ^Z (Z.pos_sub n m) = x ^ Pos.to_nat n * / x ^ Pos.to_nat m.
+Proof.
+ intro Hx.
+ rewrite Z.pos_sub_spec.
+ case Pos.compare_spec; intro H; simpl.
+ - subst; auto with real.
+ - rewrite Pos2Nat.inj_sub by trivial.
+   rewrite Pos2Nat.inj_lt in H.
+   rewrite (pow_RN_plus x _ (Pos.to_nat n)) by auto with real.
+   rewrite plus_comm, le_plus_minus_r by auto with real.
+   rewrite Rinv_mult_distr, Rinv_involutive; auto with real.
+ - rewrite Pos2Nat.inj_sub by trivial.
+   rewrite Pos2Nat.inj_lt in H.
+   rewrite (pow_RN_plus x _ (Pos.to_nat m)) by auto with real.
+   rewrite plus_comm, le_plus_minus_r by auto with real.
+   reflexivity.
+Qed.
+
 Lemma powerRZ_add :
   forall (x:R) (n m:Z), x <> 0 -> x ^Z (n + m) = x ^Z n * x ^Z m.
 Proof.
-  intro x; destruct n as [| n1| n1]; destruct m as [| m1| m1]; simpl;
-    auto with real.
-(* POS/POS *)
-  rewrite Pplus_plus; auto with real.
-(* POS/NEG *)
-  rewrite Z.pos_sub_spec.
-  case Pcompare_spec; intros; simpl.
-  subst; auto with real.
-  rewrite Pminus_minus by trivial.
-  rewrite (pow_RN_plus x _ (nat_of_P n1)) by auto with real.
-  rewrite plus_comm, le_plus_minus_r by (now apply lt_le_weak, Plt_lt).
-  rewrite Rinv_mult_distr, Rinv_involutive; auto with real.
-  rewrite Pminus_minus by trivial.
-  rewrite (pow_RN_plus x _ (nat_of_P m1)) by auto with real.
-  rewrite plus_comm, le_plus_minus_r by (now apply lt_le_weak, Plt_lt).
-  reflexivity.
-(* NEG/POS *)
-  rewrite Z.pos_sub_spec.
-  case Pcompare_spec; intros; simpl.
-  subst; auto with real.
-  rewrite Pminus_minus by trivial.
-  rewrite (pow_RN_plus x _ (nat_of_P m1)) by auto with real.
-  rewrite plus_comm, le_plus_minus_r by (now apply lt_le_weak, Plt_lt).
-  rewrite Rinv_mult_distr, Rinv_involutive; auto with real.
-  rewrite Pminus_minus by trivial.
-  rewrite (pow_RN_plus x _ (nat_of_P n1)) by auto with real.
-  rewrite plus_comm, le_plus_minus_r by (now apply lt_le_weak, Plt_lt).
-  auto with real.
-(* NEG/NEG *)
-  rewrite Pplus_plus; auto with real.
-  intros H'; rewrite pow_add; auto with real.
-  apply Rinv_mult_distr; auto.
-  apply pow_nonzero; auto.
-  apply pow_nonzero; auto.
+  intros x [|n|n] [|m|m]; simpl; intros; auto with real.
+  - (* + + *)
+    rewrite Pos2Nat.inj_add; auto with real.
+  - (* + - *)
+    now apply powerRZ_pos_sub.
+  - (* - + *)
+    rewrite Rmult_comm. now apply powerRZ_pos_sub.
+  - (* - - *)
+    rewrite Pos2Nat.inj_add; auto with real.
+    rewrite pow_add; auto with real.
+    apply Rinv_mult_distr; apply pow_nonzero; auto.
 Qed.
 Hint Resolve powerRZ_O powerRZ_1 powerRZ_NOR powerRZ_add: real.
 
 Lemma Zpower_nat_powerRZ :
-  forall n m:nat, IZR (Zpower_nat (Z_of_nat n) m) = INR n ^Z Z_of_nat m.
+  forall n m:nat, IZR (Zpower_nat (Z.of_nat n) m) = INR n ^Z Z.of_nat m.
 Proof.
   intros n m; elim m; simpl; auto with real.
-  intros m1 H'; rewrite nat_of_P_of_succ_nat; simpl.
-  replace (Zpower_nat (Z_of_nat n) (S m1)) with
-  (Z_of_nat n * Zpower_nat (Z_of_nat n) m1)%Z.
+  intros m1 H'; rewrite SuccNat2Pos.id_succ; simpl.
+  replace (Zpower_nat (Z.of_nat n) (S m1)) with
+  (Z.of_nat n * Zpower_nat (Z.of_nat n) m1)%Z.
   rewrite mult_IZR; auto with real.
   repeat rewrite <- INR_IZR_INZ; simpl.
   rewrite H'; simpl.
   case m1; simpl; auto with real.
-  intros m2; rewrite nat_of_P_of_succ_nat; auto.
+  intros m2; rewrite SuccNat2Pos.id_succ; auto.
   unfold Zpower_nat; auto.
 Qed.
 
 Lemma Zpower_pos_powerRZ :
-  forall n m, IZR (Zpower_pos n m) = IZR n ^Z Zpos m.
+  forall n m, IZR (Z.pow_pos n m) = IZR n ^Z Zpos m.
 Proof.
   intros.
   rewrite Zpower_pos_nat; simpl.
-  induction (nat_of_P m).
+  induction (Pos.to_nat m).
   easy.
   unfold Zpower_nat; simpl.
   rewrite mult_IZR.
@@ -629,10 +634,10 @@ Qed.
 Hint Resolve powerRZ_le: real.
 
 Lemma Zpower_nat_powerRZ_absolu :
-  forall n m:Z, (0 <= m)%Z -> IZR (Zpower_nat n (Zabs_nat m)) = IZR n ^Z m.
+  forall n m:Z, (0 <= m)%Z -> IZR (Zpower_nat n (Z.abs_nat m)) = IZR n ^Z m.
 Proof.
   intros n m; case m; simpl; auto with zarith.
-  intros p H'; elim (nat_of_P p); simpl; auto with zarith.
+  intros p H'; elim (Pos.to_nat p); simpl; auto with zarith.
   intros n0 H'0; rewrite <- H'0; simpl; auto with zarith.
   rewrite <- mult_IZR; auto.
   intros p H'; absurd (0 <= Zneg p)%Z; auto with zarith.
@@ -641,9 +646,9 @@ Qed.
 Lemma powerRZ_R1 : forall n:Z, 1 ^Z n = 1.
 Proof.
   intros n; case n; simpl; auto.
-  intros p; elim (nat_of_P p); simpl; auto; intros n0 H'; rewrite H';
+  intros p; elim (Pos.to_nat p); simpl; auto; intros n0 H'; rewrite H';
     ring.
-  intros p; elim (nat_of_P p); simpl.
+  intros p; elim (Pos.to_nat p); simpl.
   exact Rinv_1.
   intros n1 H'; rewrite Rinv_mult_distr; try rewrite Rinv_1; try rewrite H';
     auto with real.
@@ -751,9 +756,9 @@ Qed.
 Lemma R_dist_refl : forall x y:R, R_dist x y = 0 <-> x = y.
 Proof.
   unfold R_dist; intros; split_Rabs; split; intros.
-  rewrite (Ropp_minus_distr x y) in H; apply sym_eq;
+  rewrite (Ropp_minus_distr x y) in H; symmetry;
     apply (Rminus_diag_uniq y x H).
-  rewrite (Ropp_minus_distr x y); generalize (sym_eq H); intro;
+  rewrite (Ropp_minus_distr x y); generalize (eq_sym H); intro;
     apply (Rminus_diag_eq y x H0).
   apply (Rminus_diag_uniq x y H).
   apply (Rminus_diag_eq x y H).
diff --git a/theories/Reals/Rgeom.v b/theories/Reals/Rgeom.v
index bda64e77..ffa11608 100644
--- a/theories/Reals/Rgeom.v
+++ b/theories/Reals/Rgeom.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,9 +9,9 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
-Require Import Rtrigo.
+Require Import Rtrigo1.
 Require Import R_sqrt.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (** * Distance *)
 
@@ -20,23 +20,23 @@ Definition dist_euc (x0 y0 x1 y1:R) : R :=
 
 Lemma distance_refl : forall x0 y0:R, dist_euc x0 y0 x0 y0 = 0.
 Proof.
-  intros x0 y0; unfold dist_euc in |- *; apply Rsqr_inj;
+  intros x0 y0; unfold dist_euc; apply Rsqr_inj;
     [ apply sqrt_positivity; apply Rplus_le_le_0_compat;
       [ apply Rle_0_sqr | apply Rle_0_sqr ]
       | right; reflexivity
       | rewrite Rsqr_0; rewrite Rsqr_sqrt;
-        [ unfold Rsqr in |- *; ring
+        [ unfold Rsqr; ring
           | apply Rplus_le_le_0_compat; [ apply Rle_0_sqr | apply Rle_0_sqr ] ] ].
 Qed.
 
 Lemma distance_symm :
   forall x0 y0 x1 y1:R, dist_euc x0 y0 x1 y1 = dist_euc x1 y1 x0 y0.
 Proof.
-  intros x0 y0 x1 y1; unfold dist_euc in |- *; apply Rsqr_inj;
+  intros x0 y0 x1 y1; unfold dist_euc; apply Rsqr_inj;
     [ apply sqrt_positivity; apply Rplus_le_le_0_compat
       | apply sqrt_positivity; apply Rplus_le_le_0_compat
       | repeat rewrite Rsqr_sqrt;
-        [ unfold Rsqr in |- *; ring
+        [ unfold Rsqr; ring
           | apply Rplus_le_le_0_compat
           | apply Rplus_le_le_0_compat ] ]; apply Rle_0_sqr.
 Qed.
@@ -49,8 +49,8 @@ Lemma law_cosines :
           a * c * cos ac = (x0 - x1) * (x2 - x1) + (y0 - y1) * (y2 - y1) ->
           Rsqr b = Rsqr c + Rsqr a - 2 * (a * c * cos ac).
 Proof.
-  unfold dist_euc in |- *; intros; repeat rewrite Rsqr_sqrt;
-    [ rewrite H; unfold Rsqr in |- *; ring
+  unfold dist_euc; intros; repeat rewrite Rsqr_sqrt;
+    [ rewrite H; unfold Rsqr; ring
       | apply Rplus_le_le_0_compat
       | apply Rplus_le_le_0_compat
       | apply Rplus_le_le_0_compat ]; apply Rle_0_sqr.
@@ -60,7 +60,7 @@ Lemma triangle :
   forall x0 y0 x1 y1 x2 y2:R,
     dist_euc x0 y0 x1 y1 <= dist_euc x0 y0 x2 y2 + dist_euc x2 y2 x1 y1.
 Proof.
-  intros; unfold dist_euc in |- *; apply Rsqr_incr_0;
+  intros; unfold dist_euc; apply Rsqr_incr_0;
     [ rewrite Rsqr_plus; repeat rewrite Rsqr_sqrt;
       [ replace (Rsqr (x0 - x1)) with
         (Rsqr (x0 - x2) + Rsqr (x2 - x1) + 2 * (x0 - x2) * (x2 - x1));
@@ -112,7 +112,7 @@ Definition yt (y ty:R) : R := y + ty.
 
 Lemma translation_0 : forall x y:R, xt x 0 = x /\ yt y 0 = y.
 Proof.
-  intros x y; split; [ unfold xt in |- * | unfold yt in |- * ]; ring.
+  intros x y; split; [ unfold xt | unfold yt ]; ring.
 Qed.
 
 Lemma isometric_translation :
@@ -120,7 +120,7 @@ Lemma isometric_translation :
     Rsqr (x1 - x2) + Rsqr (y1 - y2) =
     Rsqr (xt x1 tx - xt x2 tx) + Rsqr (yt y1 ty - yt y2 ty).
 Proof.
-  intros; unfold Rsqr, xt, yt in |- *; ring.
+  intros; unfold Rsqr, xt, yt; ring.
 Qed.
 
 (******************************************************************)
@@ -132,13 +132,13 @@ Definition yr (x y theta:R) : R := - x * sin theta + y * cos theta.
 
 Lemma rotation_0 : forall x y:R, xr x y 0 = x /\ yr x y 0 = y.
 Proof.
-  intros x y; unfold xr, yr in |- *; split; rewrite cos_0; rewrite sin_0; ring.
+  intros x y; unfold xr, yr; split; rewrite cos_0; rewrite sin_0; ring.
 Qed.
 
 Lemma rotation_PI2 :
   forall x y:R, xr x y (PI / 2) = y /\ yr x y (PI / 2) = - x.
 Proof.
-  intros x y; unfold xr, yr in |- *; split; rewrite cos_PI2; rewrite sin_PI2;
+  intros x y; unfold xr, yr; split; rewrite cos_PI2; rewrite sin_PI2;
     ring.
 Qed.
 
@@ -148,7 +148,7 @@ Lemma isometric_rotation_0 :
     Rsqr (xr x1 y1 theta - xr x2 y2 theta) +
     Rsqr (yr x1 y1 theta - yr x2 y2 theta).
 Proof.
-  intros; unfold xr, yr in |- *;
+  intros; unfold xr, yr;
     replace
     (x1 * cos theta + y1 * sin theta - (x2 * cos theta + y2 * sin theta)) with
     (cos theta * (x1 - x2) + sin theta * (y1 - y2));
@@ -168,7 +168,7 @@ Lemma isometric_rotation :
     dist_euc (xr x1 y1 theta) (yr x1 y1 theta) (xr x2 y2 theta)
     (yr x2 y2 theta).
 Proof.
-  unfold dist_euc in |- *; intros; apply Rsqr_inj;
+  unfold dist_euc; intros; apply Rsqr_inj;
     [ apply sqrt_positivity; apply Rplus_le_le_0_compat
       | apply sqrt_positivity; apply Rplus_le_le_0_compat
       | repeat rewrite Rsqr_sqrt;
diff --git a/theories/Reals/RiemannInt.v b/theories/Reals/RiemannInt.v
index 8acfd75b..0a00ca22 100644
--- a/theories/Reals/RiemannInt.v
+++ b/theories/Reals/RiemannInt.v
@@ -1,7 +1,7 @@
 (* -*- coding: utf-8 -*- *)
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,13 +9,13 @@
 
 Require Import Rfunctions.
 Require Import SeqSeries.
-Require Import Ranalysis.
+Require Import Ranalysis_reg.
 Require Import Rbase.
 Require Import RiemannInt_SF.
 Require Import Classical_Prop.
 Require Import Classical_Pred_Type.
 Require Import Max.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Set Implicit Arguments.
 
@@ -51,19 +51,19 @@ Lemma RiemannInt_P1 :
   forall (f:R -> R) (a b:R),
     Riemann_integrable f a b -> Riemann_integrable f b a.
 Proof.
-  unfold Riemann_integrable in |- *; intros; elim (X eps); clear X; intros;
+  unfold Riemann_integrable; intros; elim (X eps); clear X; intros;
     elim p; clear p; intros; exists (mkStepFun (StepFun_P6 (pre x)));
       exists (mkStepFun (StepFun_P6 (pre x0)));
         elim p; clear p; intros; split.
   intros; apply (H t); elim H1; clear H1; intros; split;
     [ apply Rle_trans with (Rmin b a); try assumption; right;
-      unfold Rmin in |- *
+      unfold Rmin
       | apply Rle_trans with (Rmax b a); try assumption; right;
-        unfold Rmax in |- * ];
+        unfold Rmax ];
     (case (Rle_dec a b); case (Rle_dec b a); intros;
       try reflexivity || apply Rle_antisym;
         [ assumption | assumption | auto with real | auto with real ]).
-  generalize H0; unfold RiemannInt_SF in |- *; case (Rle_dec a b);
+  generalize H0; unfold RiemannInt_SF; case (Rle_dec a b);
     case (Rle_dec b a); intros;
       (replace
         (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre x0))))
@@ -89,11 +89,11 @@ Lemma RiemannInt_P2 :
       Rabs (RiemannInt_SF (wn n)) < un n) ->
     { l:R | Un_cv (fun N:nat => RiemannInt_SF (vn N)) l }.
 Proof.
-  intros; apply R_complete; unfold Un_cv in H; unfold Cauchy_crit in |- *;
+  intros; apply R_complete; unfold Un_cv in H; unfold Cauchy_crit;
     intros; assert (H3 : 0 < eps / 2).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
-  elim (H _ H3); intros N0 H4; exists N0; intros; unfold R_dist in |- *;
+  elim (H _ H3); intros N0 H4; exists N0; intros; unfold R_dist;
     unfold R_dist in H4; elim (H1 n); elim (H1 m); intros;
       replace (RiemannInt_SF (vn n) - RiemannInt_SF (vn m)) with
       (RiemannInt_SF (vn n) + -1 * RiemannInt_SF (vn m));
@@ -105,15 +105,15 @@ Proof.
   apply Rle_lt_trans with
     (RiemannInt_SF (mkStepFun (StepFun_P28 1 (wn n) (wn m)))).
   apply StepFun_P37; try assumption.
-  intros; simpl in |- *;
+  intros; simpl;
     apply Rle_trans with (Rabs (vn n x - f x) + Rabs (f x - vn m x)).
   replace (vn n x + -1 * vn m x) with (vn n x - f x + (f x - vn m x));
   [ apply Rabs_triang | ring ].
   assert (H12 : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   assert (H13 : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   rewrite <- H12 in H11; pattern b at 2 in H11; rewrite <- H13 in H11;
     rewrite Rmult_1_l; apply Rplus_le_compat.
@@ -156,14 +156,14 @@ Proof.
   intro; elim (H0 n0); intros; split.
   intros; apply (H2 t); elim H4; clear H4; intros; split;
     [ apply Rle_trans with (Rmin b a); try assumption; right;
-      unfold Rmin in |- *
+      unfold Rmin
       | apply Rle_trans with (Rmax b a); try assumption; right;
-        unfold Rmax in |- * ];
+        unfold Rmax ];
     (case (Rle_dec a b); case (Rle_dec b a); intros;
       try reflexivity || apply Rle_antisym;
         [ assumption | assumption | auto with real | auto with real ]).
-  generalize H3; unfold RiemannInt_SF in |- *; case (Rle_dec a b);
-    case (Rle_dec b a); unfold wn' in |- *; intros;
+  generalize H3; unfold RiemannInt_SF; case (Rle_dec a b);
+    case (Rle_dec b a); unfold wn'; intros;
       (replace
         (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (wn n0)))))
           (subdivision (mkStepFun (StepFun_P6 (pre (wn n0)))))) with
@@ -178,19 +178,19 @@ Proof.
   rewrite Rabs_Ropp in H4; apply H4.
   apply H4.
   assert (H3 := RiemannInt_P2 _ _ _ _ H H1 H2); elim H3; intros;
-    exists (- x); unfold Un_cv in |- *; unfold Un_cv in p;
+    exists (- x); unfold Un_cv; unfold Un_cv in p;
       intros; elim (p _ H4); intros; exists x0; intros;
-        generalize (H5 _ H6); unfold R_dist, RiemannInt_SF in |- *;
+        generalize (H5 _ H6); unfold R_dist, RiemannInt_SF;
           case (Rle_dec b a); case (Rle_dec a b); intros.
   elim n; assumption.
   unfold vn' in H7;
     replace (Int_SF (subdivision_val (vn n0)) (subdivision (vn n0))) with
     (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (vn n0)))))
       (subdivision (mkStepFun (StepFun_P6 (pre (vn n0))))));
-    [ unfold Rminus in |- *; rewrite Ropp_involutive; rewrite <- Rabs_Ropp;
+    [ unfold Rminus; rewrite Ropp_involutive; rewrite <- Rabs_Ropp;
       rewrite Ropp_plus_distr; rewrite Ropp_involutive;
         apply H7
-      | symmetry  in |- *; apply StepFun_P17 with (fe (vn n0)) a b;
+      | symmetry ; apply StepFun_P17 with (fe (vn n0)) a b;
         [ apply StepFun_P1
           | apply StepFun_P2;
             apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (vn n0))))) ] ].
@@ -218,9 +218,9 @@ Lemma RiemannInt_P4 :
     Un_cv (fun N:nat => RiemannInt_SF (phi_sequence un pr1 N)) l ->
     Un_cv (fun N:nat => RiemannInt_SF (phi_sequence vn pr2 N)) l.
 Proof.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros f; intros;
+  unfold Un_cv; unfold R_dist; intros f; intros;
     assert (H3 : 0 < eps / 3).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   elim (H _ H3); clear H; intros N0 H; elim (H0 _ H3); clear H0; intros N1 H0;
     elim (H1 _ H3); clear H1; intros N2 H1; set (N := max (max N0 N1) N2);
@@ -255,7 +255,7 @@ Proof.
   apply StepFun_P34; assumption.
   apply Rle_lt_trans with
     (RiemannInt_SF (mkStepFun (StepFun_P28 1 psi_un psi_vn))).
-  apply StepFun_P37; try assumption; intros; simpl in |- *; rewrite Rmult_1_l;
+  apply StepFun_P37; try assumption; intros; simpl; rewrite Rmult_1_l;
     apply Rle_trans with
       (Rabs (phi_sequence vn pr2 n x - f x) +
         Rabs (f x - phi_sequence un pr1 n x)).
@@ -263,10 +263,10 @@ Proof.
   (phi_sequence vn pr2 n x - f x + (f x - phi_sequence un pr1 n x));
   [ apply Rabs_triang | ring ].
   assert (H10 : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   assert (H11 : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   rewrite (Rplus_comm (psi_un x)); apply Rplus_le_compat.
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim H5; intros; apply H8.
@@ -279,20 +279,20 @@ Proof.
   apply RRle_abs.
   assumption.
   replace (pos (un n)) with (Rabs (un n - 0));
-  [ apply H; unfold ge in |- *; apply le_trans with N; try assumption;
-    unfold N in |- *; apply le_trans with (max N0 N1);
+  [ apply H; unfold ge; apply le_trans with N; try assumption;
+    unfold N; apply le_trans with (max N0 N1);
       apply le_max_l
-    | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
+    | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
       apply Rle_ge; left; apply (cond_pos (un n)) ].
   apply Rlt_trans with (pos (vn n)).
   elim H5; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_vn)).
   apply RRle_abs; assumption.
   assumption.
   replace (pos (vn n)) with (Rabs (vn n - 0));
-  [ apply H0; unfold ge in |- *; apply le_trans with N; try assumption;
-    unfold N in |- *; apply le_trans with (max N0 N1);
+  [ apply H0; unfold ge; apply le_trans with N; try assumption;
+    unfold N; apply le_trans with (max N0 N1);
       [ apply le_max_r | apply le_max_l ]
-    | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
+    | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
       apply Rle_ge; left; apply (cond_pos (vn n)) ].
   rewrite StepFun_P39; rewrite Rabs_Ropp;
     apply Rle_lt_trans with
@@ -311,7 +311,7 @@ Proof.
       (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 psi_vn psi_un)))))).
   apply StepFun_P37.
   auto with real.
-  intros; simpl in |- *; rewrite Rmult_1_l;
+  intros; simpl; rewrite Rmult_1_l;
     apply Rle_trans with
       (Rabs (phi_sequence vn pr2 n x - f x) +
         Rabs (f x - phi_sequence un pr1 n x)).
@@ -319,10 +319,10 @@ Proof.
   (phi_sequence vn pr2 n x - f x + (f x - phi_sequence un pr1 n x));
   [ apply Rabs_triang | ring ].
   assert (H10 : Rmin a b = b).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ elim n0; assumption | reflexivity ].
   assert (H11 : Rmax a b = a).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ elim n0; assumption | reflexivity ].
   apply Rplus_le_compat.
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim H5; intros; apply H8.
@@ -341,10 +341,10 @@ Proof.
   rewrite <- Rabs_Ropp; apply RRle_abs.
   assumption.
   replace (pos (vn n)) with (Rabs (vn n - 0));
-  [ apply H0; unfold ge in |- *; apply le_trans with N; try assumption;
-    unfold N in |- *; apply le_trans with (max N0 N1);
+  [ apply H0; unfold ge; apply le_trans with N; try assumption;
+    unfold N; apply le_trans with (max N0 N1);
       [ apply le_max_r | apply le_max_l ]
-    | unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
+    | unfold R_dist; unfold Rminus; rewrite Ropp_0;
       rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge;
         left; apply (cond_pos (vn n)) ].
   apply Rlt_trans with (pos (un n)).
@@ -352,15 +352,15 @@ Proof.
   rewrite <- Rabs_Ropp; apply RRle_abs; assumption.
   assumption.
   replace (pos (un n)) with (Rabs (un n - 0));
-  [ apply H; unfold ge in |- *; apply le_trans with N; try assumption;
-    unfold N in |- *; apply le_trans with (max N0 N1);
+  [ apply H; unfold ge; apply le_trans with N; try assumption;
+    unfold N; apply le_trans with (max N0 N1);
       apply le_max_l
-    | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
+    | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
       apply Rle_ge; left; apply (cond_pos (un n)) ].
-  apply H1; unfold ge in |- *; apply le_trans with N; try assumption;
-    unfold N in |- *; apply le_max_r.
+  apply H1; unfold ge; apply le_trans with N; try assumption;
+    unfold N; apply le_max_r.
   apply Rmult_eq_reg_l with 3;
-    [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l;
+    [ unfold Rdiv; rewrite Rmult_plus_distr_l;
       do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
         rewrite <- Rinv_l_sym; [ ring | discrR ]
       | discrR ].
@@ -376,17 +376,17 @@ Definition RinvN (N:nat) : posreal := mkposreal _ (RinvN_pos N).
 
 Lemma RinvN_cv : Un_cv RinvN 0.
 Proof.
-  unfold Un_cv in |- *; intros; assert (H0 := archimed (/ eps)); elim H0;
+  unfold Un_cv; intros; assert (H0 := archimed (/ eps)); elim H0;
     clear H0; intros; assert (H2 : (0 <= up (/ eps))%Z).
   apply le_IZR; left; apply Rlt_trans with (/ eps);
     [ apply Rinv_0_lt_compat; assumption | assumption ].
-  elim (IZN _ H2); intros; exists x; intros; unfold R_dist in |- *;
-    simpl in |- *; unfold Rminus in |- *; rewrite Ropp_0;
+  elim (IZN _ H2); intros; exists x; intros; unfold R_dist;
+    simpl; unfold Rminus; rewrite Ropp_0;
       rewrite Rplus_0_r; assert (H5 : 0 < INR n + 1).
   apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ].
   rewrite Rabs_right;
     [ idtac
-      | left; change (0 < / (INR n + 1)) in |- *; apply Rinv_0_lt_compat;
+      | left; change (0 < / (INR n + 1)); apply Rinv_0_lt_compat;
         assumption ]; apply Rle_lt_trans with (/ (INR x + 1)).
   apply Rle_Rinv.
   apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ].
@@ -400,9 +400,9 @@ Proof.
   apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ].
   apply Rlt_trans with (INR x);
     [ rewrite INR_IZR_INZ; rewrite <- H3; apply H0
-      | pattern (INR x) at 1 in |- *; rewrite <- Rplus_0_r;
+      | pattern (INR x) at 1; rewrite <- Rplus_0_r;
         apply Rplus_lt_compat_l; apply Rlt_0_1 ].
-  red in |- *; intro; rewrite H6 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H6 in H; elim (Rlt_irrefl _ H).
 Qed.
 
 (**********)
@@ -413,7 +413,7 @@ Lemma RiemannInt_P5 :
   forall (f:R -> R) (a b:R) (pr1 pr2:Riemann_integrable f a b),
     RiemannInt pr1 = RiemannInt pr2.
 Proof.
-  intros; unfold RiemannInt in |- *;
+  intros; unfold RiemannInt;
     case (RiemannInt_exists pr1 RinvN RinvN_cv);
     case (RiemannInt_exists pr2 RinvN RinvN_cv); intros;
       eapply UL_sequence;
@@ -431,7 +431,7 @@ Lemma maxN :
 Proof.
   intros; set (I := fun n:nat => a + INR n * del < b);
     assert (H0 :  exists n : nat, I n).
-  exists 0%nat; unfold I in |- *; rewrite Rmult_0_l; rewrite Rplus_0_r;
+  exists 0%nat; unfold I; rewrite Rmult_0_l; rewrite Rplus_0_r;
     assumption.
   cut (Nbound I).
   intro; assert (H2 := Nzorn H0 H1); elim H2; intros; exists x; elim p; intros;
@@ -440,27 +440,27 @@ Proof.
   case (total_order_T (a + INR (S x) * del) b); intro.
   elim s; intro.
   assert (H5 := H4 (S x) a0); elim (le_Sn_n _ H5).
-  right; symmetry  in |- *; assumption.
+  right; symmetry ; assumption.
   left; apply r.
   assert (H1 : 0 <= (b - a) / del).
-  unfold Rdiv in |- *; apply Rmult_le_pos;
+  unfold Rdiv; apply Rmult_le_pos;
     [ apply Rge_le; apply Rge_minus; apply Rle_ge; left; apply H
       | left; apply Rinv_0_lt_compat; apply (cond_pos del) ].
   elim (archimed ((b - a) / del)); intros;
     assert (H4 : (0 <= up ((b - a) / del))%Z).
-  apply le_IZR; simpl in |- *; left; apply Rle_lt_trans with ((b - a) / del);
+  apply le_IZR; simpl; left; apply Rle_lt_trans with ((b - a) / del);
     assumption.
   assert (H5 := IZN _ H4); elim H5; clear H5; intros N H5;
-    unfold Nbound in |- *; exists N; intros; unfold I in H6;
+    unfold Nbound; exists N; intros; unfold I in H6;
       apply INR_le; rewrite H5 in H2; rewrite <- INR_IZR_INZ in H2;
         left; apply Rle_lt_trans with ((b - a) / del); try assumption;
           apply Rmult_le_reg_l with (pos del);
             [ apply (cond_pos del)
-              | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ del));
+              | unfold Rdiv; rewrite <- (Rmult_comm (/ del));
                 rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
                   [ rewrite Rmult_1_l; rewrite Rmult_comm; apply Rplus_le_reg_l with a;
                     replace (a + (b - a)) with b; [ left; assumption | ring ]
-                    | assert (H7 := cond_pos del); red in |- *; intro; rewrite H8 in H7;
+                    | assert (H7 := cond_pos del); red; intro; rewrite H8 in H7;
                       elim (Rlt_irrefl _ H7) ] ].
 Qed.
 
@@ -496,15 +496,15 @@ Proof.
             a <= x <= b ->
             a <= y <= b -> Rabs (x - y) < l -> Rabs (f x - f y) < eps));
       assert (H1 : bound E).
-  unfold bound in |- *; exists (b - a); unfold is_upper_bound in |- *; intros;
+  unfold bound; exists (b - a); unfold is_upper_bound; intros;
     unfold E in H1; elim H1; clear H1; intros H1 _; elim H1;
       intros; assumption.
   assert (H2 :  exists x : R, E x).
   assert (H2 := Heine f (fun x:R => a <= x <= b) (compact_P3 a b) H0 eps);
-    elim H2; intros; exists (Rmin x (b - a)); unfold E in |- *;
+    elim H2; intros; exists (Rmin x (b - a)); unfold E;
       split;
         [ split;
-          [ unfold Rmin in |- *; case (Rle_dec x (b - a)); intro;
+          [ unfold Rmin; case (Rle_dec x (b - a)); intro;
             [ apply (cond_pos x) | apply Rlt_Rminus; assumption ]
             | apply Rmin_r ]
           | intros; apply H3; try assumption; apply Rlt_le_trans with (Rmin x (b - a));
@@ -519,7 +519,7 @@ Proof.
     intros; apply H15; assumption.
   assert (H12 := not_ex_all_not _ (fun y:R => D < y /\ E y) H11);
     assert (H13 : is_upper_bound E D).
-  unfold is_upper_bound in |- *; intros; assert (H14 := H12 x1);
+  unfold is_upper_bound; intros; assert (H14 := H12 x1);
     elim (not_and_or (D < x1) (E x1) H14); intro.
   case (Rle_dec x1 D); intro.
   assumption.
@@ -551,7 +551,7 @@ Proof.
   exists (mkposreal _ Rlt_0_1); intros; assert (H3 : x = y);
     [ elim H0; elim H1; intros; rewrite b0 in H3; rewrite b0 in H5;
       apply Rle_antisym; apply Rle_trans with b; assumption
-      | rewrite H3; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+      | rewrite H3; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
         apply (cond_pos eps) ].
   exists (mkposreal _ Rlt_0_1); intros; elim H0; intros;
     elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H3 H4) r)).
@@ -560,14 +560,14 @@ Qed.
 Lemma SubEqui_P1 :
   forall (a b:R) (del:posreal) (h:a < b), pos_Rl (SubEqui del h) 0 = a.
 Proof.
-  intros; unfold SubEqui in |- *; case (maxN del h); intros; reflexivity.
+  intros; unfold SubEqui; case (maxN del h); intros; reflexivity.
 Qed.
 
 Lemma SubEqui_P2 :
   forall (a b:R) (del:posreal) (h:a < b),
     pos_Rl (SubEqui del h) (pred (Rlength (SubEqui del h))) = b.
 Proof.
-  intros; unfold SubEqui in |- *; case (maxN del h); intros; clear a0;
+  intros; unfold SubEqui; case (maxN del h); intros; clear a0;
     cut
       (forall (x:nat) (a:R) (del:posreal),
         pos_Rl (SubEquiN (S x) a b del)
@@ -579,14 +579,14 @@ Proof.
               change
                 (pos_Rl (SubEquiN (S n) (a0 + del0) b del0)
                   (pred (Rlength (SubEquiN (S n) (a0 + del0) b del0))) = b)
-                in |- *; apply H ] ].
+               ; apply H ] ].
 Qed.
 
 Lemma SubEqui_P3 :
   forall (N:nat) (a b:R) (del:posreal), Rlength (SubEquiN N a b del) = S N.
 Proof.
   simple induction N; intros;
-    [ reflexivity | simpl in |- *; rewrite H; reflexivity ].
+    [ reflexivity | simpl; rewrite H; reflexivity ].
 Qed.
 
 Lemma SubEqui_P4 :
@@ -594,36 +594,36 @@ Lemma SubEqui_P4 :
     (i < S N)%nat -> pos_Rl (SubEquiN (S N) a b del) i = a + INR i * del.
 Proof.
   simple induction N;
-    [ intros; inversion H; [ simpl in |- *; ring | elim (le_Sn_O _ H1) ]
+    [ intros; inversion H; [ simpl; ring | elim (le_Sn_O _ H1) ]
       | intros; induction  i as [| i Hreci];
-        [ simpl in |- *; ring
+        [ simpl; ring
           | change
             (pos_Rl (SubEquiN (S n) (a + del) b del) i = a + INR (S i) * del)
-            in |- *; rewrite H; [ rewrite S_INR; ring | apply lt_S_n; apply H0 ] ] ].
+           ; rewrite H; [ rewrite S_INR; ring | apply lt_S_n; apply H0 ] ] ].
 Qed.
 
 Lemma SubEqui_P5 :
   forall (a b:R) (del:posreal) (h:a < b),
     Rlength (SubEqui del h) = S (S (max_N del h)).
 Proof.
-  intros; unfold SubEqui in |- *; apply SubEqui_P3.
+  intros; unfold SubEqui; apply SubEqui_P3.
 Qed.
 
 Lemma SubEqui_P6 :
   forall (a b:R) (del:posreal) (h:a < b) (i:nat),
     (i < S (max_N del h))%nat -> pos_Rl (SubEqui del h) i = a + INR i * del.
 Proof.
-  intros; unfold SubEqui in |- *; apply SubEqui_P4; assumption.
+  intros; unfold SubEqui; apply SubEqui_P4; assumption.
 Qed.
 
 Lemma SubEqui_P7 :
   forall (a b:R) (del:posreal) (h:a < b), ordered_Rlist (SubEqui del h).
 Proof.
-  intros; unfold ordered_Rlist in |- *; intros; rewrite SubEqui_P5 in H;
+  intros; unfold ordered_Rlist; intros; rewrite SubEqui_P5 in H;
     simpl in H; inversion H.
   rewrite (SubEqui_P6 del h (i:=(max_N del h))).
   replace (S (max_N del h)) with (pred (Rlength (SubEqui del h))).
-  rewrite SubEqui_P2; unfold max_N in |- *; case (maxN del h); intros; left;
+  rewrite SubEqui_P2; unfold max_N; case (maxN del h); intros; left;
     elim a0; intros; assumption.
   rewrite SubEqui_P5; reflexivity.
   apply lt_n_Sn.
@@ -631,7 +631,7 @@ Proof.
   3: assumption.
   2: apply le_lt_n_Sm; assumption.
   apply Rplus_le_compat_l; rewrite S_INR; rewrite Rmult_plus_distr_r;
-    pattern (INR i * del) at 1 in |- *; rewrite <- Rplus_0_r;
+    pattern (INR i * del) at 1; rewrite <- Rplus_0_r;
       apply Rplus_le_compat_l; rewrite Rmult_1_l; left;
         apply (cond_pos del).
 Qed.
@@ -641,11 +641,11 @@ Lemma SubEqui_P8 :
     (i < Rlength (SubEqui del h))%nat -> a <= pos_Rl (SubEqui del h) i <= b.
 Proof.
   intros; split.
-  pattern a at 1 in |- *; rewrite <- (SubEqui_P1 del h); apply RList_P5.
+  pattern a at 1; rewrite <- (SubEqui_P1 del h); apply RList_P5.
   apply SubEqui_P7.
   elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); intros; apply H1;
     exists i; split; [ reflexivity | assumption ].
-  pattern b at 2 in |- *; rewrite <- (SubEqui_P2 del h); apply RList_P7;
+  pattern b at 2; rewrite <- (SubEqui_P2 del h); apply RList_P7;
     [ apply SubEqui_P7
       | elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); intros;
         apply H1; exists i; split; [ reflexivity | assumption ] ].
@@ -671,42 +671,42 @@ Lemma RiemannInt_P6 :
     a < b ->
     (forall x:R, a <= x <= b -> continuity_pt f x) -> Riemann_integrable f a b.
 Proof.
-  intros; unfold Riemann_integrable in |- *; intro;
+  intros; unfold Riemann_integrable; intro;
     assert (H1 : 0 < eps / (2 * (b - a))).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply (cond_pos eps)
       | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat;
         [ prove_sup0 | apply Rlt_Rminus; assumption ] ].
   assert (H2 : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; left; assumption ].
   assert (H3 : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; left; assumption ].
   elim (Heine_cor2 H0 (mkposreal _ H1)); intros del H4;
     elim (SubEqui_P9 del f H); intros phi [H5 H6]; split with phi;
       split with (mkStepFun (StepFun_P4 a b (eps / (2 * (b - a)))));
         split.
-  2: rewrite StepFun_P18; unfold Rdiv in |- *; rewrite Rinv_mult_distr.
+  2: rewrite StepFun_P18; unfold Rdiv; rewrite Rinv_mult_distr.
   2: do 2 rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   2: rewrite Rmult_1_r; rewrite Rabs_right.
   2: apply Rmult_lt_reg_l with 2.
   2: prove_sup0.
   2: rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
     rewrite <- Rinv_r_sym.
-  2: rewrite Rmult_1_l; pattern (pos eps) at 1 in |- *; rewrite <- Rplus_0_r;
+  2: rewrite Rmult_1_l; pattern (pos eps) at 1; rewrite <- Rplus_0_r;
     rewrite double; apply Rplus_lt_compat_l; apply (cond_pos eps).
   2: discrR.
   2: apply Rle_ge; left; apply Rmult_lt_0_compat.
   2: apply (cond_pos eps).
   2: apply Rinv_0_lt_compat; prove_sup0.
-  2: apply Rminus_eq_contra; red in |- *; intro; clear H6; rewrite H7 in H;
+  2: apply Rminus_eq_contra; red; intro; clear H6; rewrite H7 in H;
     elim (Rlt_irrefl _ H).
   2: discrR.
-  2: apply Rminus_eq_contra; red in |- *; intro; clear H6; rewrite H7 in H;
+  2: apply Rminus_eq_contra; red; intro; clear H6; rewrite H7 in H;
     elim (Rlt_irrefl _ H).
-  intros; rewrite H2 in H7; rewrite H3 in H7; simpl in |- *;
-    unfold fct_cte in |- *;
+  intros; rewrite H2 in H7; rewrite H3 in H7; simpl;
+    unfold fct_cte;
       cut
         (forall t:R,
           a <= t <= b ->
@@ -716,14 +716,14 @@ Proof.
             co_interval (pos_Rl (SubEqui del H) i) (pos_Rl (SubEqui del H) (S i))
             t)).
   intro; elim (H8 _ H7); intro.
-  rewrite H9; rewrite H5; unfold Rminus in |- *; rewrite Rplus_opp_r;
+  rewrite H9; rewrite H5; unfold Rminus; rewrite Rplus_opp_r;
     rewrite Rabs_R0; left; assumption.
   elim H9; clear H9; intros I [H9 H10]; assert (H11 := H6 I H9 t H10);
     rewrite H11; left; apply H4.
   assumption.
   apply SubEqui_P8; apply lt_trans with (pred (Rlength (SubEqui del H))).
   assumption.
-  apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H9;
+  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H12 in H9;
     elim (lt_n_O _ H9).
   unfold co_interval in H10; elim H10; clear H10; intros; rewrite Rabs_right.
   rewrite SubEqui_P5 in H9; simpl in H9; inversion H9.
@@ -738,7 +738,7 @@ Proof.
   rewrite H13 in H12; rewrite SubEqui_P2 in H12; apply H12.
   rewrite SubEqui_P6.
   2: apply lt_n_Sn.
-  unfold max_N in |- *; case (maxN del H); intros; elim a0; clear a0;
+  unfold max_N; case (maxN del H); intros; elim a0; clear a0;
     intros _ H13; replace (a + INR x * del + del) with (a + INR (S x) * del);
       [ assumption | rewrite S_INR; ring ].
   apply Rplus_lt_reg_r with (pos_Rl (SubEqui del H) I);
@@ -755,10 +755,10 @@ Proof.
   left; assumption.
   right; set (I := fun j:nat => a + INR j * del <= t0);
     assert (H1 :  exists n : nat, I n).
-  exists 0%nat; unfold I in |- *; rewrite Rmult_0_l; rewrite Rplus_0_r; elim H8;
+  exists 0%nat; unfold I; rewrite Rmult_0_l; rewrite Rplus_0_r; elim H8;
     intros; assumption.
   assert (H4 : Nbound I).
-  unfold Nbound in |- *; exists (S (max_N del H)); intros; unfold max_N in |- *;
+  unfold Nbound; exists (S (max_N del H)); intros; unfold max_N;
     case (maxN del H); intros; elim a0; clear a0; intros _ H5;
       apply INR_le; apply Rmult_le_reg_l with (pos del).
   apply (cond_pos del).
@@ -767,7 +767,7 @@ Proof.
       apply Rle_trans with b; try assumption; elim H8; intros;
         assumption.
   elim (Nzorn H1 H4); intros N [H5 H6]; assert (H7 : (N < S (max_N del H))%nat).
-  unfold max_N in |- *; case (maxN del H); intros; apply INR_lt;
+  unfold max_N; case (maxN del H); intros; apply INR_lt;
     apply Rmult_lt_reg_l with (pos del).
   apply (cond_pos del).
   apply Rplus_lt_reg_r with a; do 2 rewrite (Rmult_comm del);
@@ -778,8 +778,8 @@ Proof.
   assumption.
   elim H0; assumption.
   exists N; split.
-  rewrite SubEqui_P5; simpl in |- *; assumption.
-  unfold co_interval in |- *; split.
+  rewrite SubEqui_P5; simpl; assumption.
+  unfold co_interval; split.
   rewrite SubEqui_P6.
   apply H5.
   assumption.
@@ -799,13 +799,13 @@ Qed.
 
 Lemma RiemannInt_P7 : forall (f:R -> R) (a:R), Riemann_integrable f a a.
 Proof.
-  unfold Riemann_integrable in |- *; intro f; intros;
+  unfold Riemann_integrable; intro f; intros;
     split with (mkStepFun (StepFun_P4 a a (f a)));
       split with (mkStepFun (StepFun_P4 a a 0)); split.
-  intros; simpl in |- *; unfold fct_cte in |- *; replace t with a.
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; right;
+  intros; simpl; unfold fct_cte; replace t with a.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; right;
     reflexivity.
-  generalize H; unfold Rmin, Rmax in |- *; case (Rle_dec a a); intros; elim H0;
+  generalize H; unfold Rmin, Rmax; case (Rle_dec a a); intros; elim H0;
     intros; apply Rle_antisym; assumption.
   rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0; apply (cond_pos eps).
 Qed.
@@ -826,9 +826,9 @@ Lemma RiemannInt_P8 :
     (pr2:Riemann_integrable f b a), RiemannInt pr1 = - RiemannInt pr2.
 Proof.
   intro f; intros; eapply UL_sequence.
-  unfold RiemannInt in |- *; case (RiemannInt_exists pr1 RinvN RinvN_cv);
+  unfold RiemannInt; case (RiemannInt_exists pr1 RinvN RinvN_cv);
     intros; apply u.
-  unfold RiemannInt in |- *; case (RiemannInt_exists pr2 RinvN RinvN_cv);
+  unfold RiemannInt; case (RiemannInt_exists pr2 RinvN RinvN_cv);
     intros;
       cut
         (exists psi1 : nat -> StepFun a b,
@@ -845,9 +845,9 @@ Proof.
           Rabs (f t - phi_sequence RinvN pr2 n t) <= psi2 n t) /\
         Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
   intros; elim H; clear H; intros psi2 H; elim H0; clear H0; intros psi1 H0;
-    assert (H1 := RinvN_cv); unfold Un_cv in |- *; intros;
+    assert (H1 := RinvN_cv); unfold Un_cv; intros;
       assert (H3 : 0 < eps / 3).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   unfold Un_cv in H1; elim (H1 _ H3); clear H1; intros N0 H1;
     unfold R_dist in H1; simpl in H1;
@@ -855,10 +855,10 @@ Proof.
   intros; assert (H5 := H1 _ H4);
     replace (pos (RinvN n)) with (Rabs (/ (INR n + 1) - 0));
     [ assumption
-      | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
+      | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
         left; apply (cond_pos (RinvN n)) ].
   clear H1; unfold Un_cv in u; elim (u _ H3); clear u; intros N1 H1;
-    exists (max N0 N1); intros; unfold R_dist in |- *;
+    exists (max N0 N1); intros; unfold R_dist;
       apply Rle_lt_trans with
         (Rabs
           (RiemannInt_SF (phi_sequence RinvN pr1 n) +
@@ -895,7 +895,7 @@ Proof.
       (mkStepFun
         (StepFun_P28 1 (psi1 n) (mkStepFun (StepFun_P6 (pre (psi2 n))))))).
   apply StepFun_P37; try assumption.
-  intros; simpl in |- *; rewrite Rmult_1_l;
+  intros; simpl; rewrite Rmult_1_l;
     apply Rle_trans with
       (Rabs (phi_sequence RinvN pr1 n x0 - f x0) +
         Rabs (f x0 - phi_sequence RinvN pr2 n x0)).
@@ -903,10 +903,10 @@ Proof.
   (phi_sequence RinvN pr1 n x0 - f x0 + (f x0 - phi_sequence RinvN pr2 n x0));
   [ apply Rabs_triang | ring ].
   assert (H7 : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   assert (H8 : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   apply Rplus_le_compat.
   elim (H0 n); intros; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9;
@@ -919,7 +919,7 @@ Proof.
     [ apply RRle_abs
       | apply Rlt_trans with (pos (RinvN n));
         [ assumption
-          | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1);
+          | apply H4; unfold ge; apply le_trans with (max N0 N1);
             [ apply le_max_l | assumption ] ] ].
   elim (H n); intros;
     rewrite <-
@@ -929,7 +929,7 @@ Proof.
           [ rewrite <- Rabs_Ropp; apply RRle_abs
             | apply Rlt_trans with (pos (RinvN n));
               [ assumption
-                | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1);
+                | apply H4; unfold ge; apply le_trans with (max N0 N1);
                   [ apply le_max_l | assumption ] ] ].
   assert (Hyp : b <= a).
   auto with real.
@@ -948,7 +948,7 @@ Proof.
       (mkStepFun
         (StepFun_P28 1 (mkStepFun (StepFun_P6 (pre (psi1 n)))) (psi2 n)))).
   apply StepFun_P37; try assumption.
-  intros; simpl in |- *; rewrite Rmult_1_l;
+  intros; simpl; rewrite Rmult_1_l;
     apply Rle_trans with
       (Rabs (phi_sequence RinvN pr1 n x0 - f x0) +
         Rabs (f x0 - phi_sequence RinvN pr2 n x0)).
@@ -956,10 +956,10 @@ Proof.
   (phi_sequence RinvN pr1 n x0 - f x0 + (f x0 - phi_sequence RinvN pr2 n x0));
   [ apply Rabs_triang | ring ].
   assert (H7 : Rmin a b = b).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ elim n0; assumption | reflexivity ].
   assert (H8 : Rmax a b = a).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ elim n0; assumption | reflexivity ].
   apply Rplus_le_compat.
   elim (H0 n); intros; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9;
@@ -976,18 +976,18 @@ Proof.
           [ rewrite <- Rabs_Ropp; apply RRle_abs
             | apply Rlt_trans with (pos (RinvN n));
               [ assumption
-                | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1);
+                | apply H4; unfold ge; apply le_trans with (max N0 N1);
                   [ apply le_max_l | assumption ] ] ].
   elim (H n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n)));
     [ apply RRle_abs
       | apply Rlt_trans with (pos (RinvN n));
         [ assumption
-          | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1);
+          | apply H4; unfold ge; apply le_trans with (max N0 N1);
             [ apply le_max_l | assumption ] ] ].
-  unfold R_dist in H1; apply H1; unfold ge in |- *;
+  unfold R_dist in H1; apply H1; unfold ge;
     apply le_trans with (max N0 N1); [ apply le_max_r | assumption ].
   apply Rmult_eq_reg_l with 3;
-    [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l;
+    [ unfold Rdiv; rewrite Rmult_plus_distr_l;
       do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
         rewrite <- Rinv_l_sym; [ ring | discrR ]
       | discrR ].
@@ -1002,7 +1002,7 @@ Lemma RiemannInt_P9 :
   forall (f:R -> R) (a:R) (pr:Riemann_integrable f a a), RiemannInt pr = 0.
 Proof.
   intros; assert (H := RiemannInt_P8 pr pr); apply Rmult_eq_reg_l with 2;
-    [ rewrite Rmult_0_r; rewrite double; pattern (RiemannInt pr) at 2 in |- *;
+    [ rewrite Rmult_0_r; rewrite double; pattern (RiemannInt pr) at 2;
       rewrite H; apply Rplus_opp_r
       | discrR ].
 Qed.
@@ -1011,9 +1011,9 @@ Lemma Req_EM_T : forall r1 r2:R, {r1 = r2} + {r1 <> r2}.
 Proof.
   intros; elim (total_order_T r1 r2); intros;
     [ elim a; intro;
-      [ right; red in |- *; intro; rewrite H in a0; elim (Rlt_irrefl r2 a0)
+      [ right; red; intro; rewrite H in a0; elim (Rlt_irrefl r2 a0)
         | left; assumption ]
-      | right; red in |- *; intro; rewrite H in b; elim (Rlt_irrefl r2 b) ].
+      | right; red; intro; rewrite H in b; elim (Rlt_irrefl r2 b) ].
 Qed.
 
 (* L1([a,b]) is a vectorial space *)
@@ -1023,16 +1023,16 @@ Lemma RiemannInt_P10 :
     Riemann_integrable g a b ->
     Riemann_integrable (fun x:R => f x + l * g x) a b.
 Proof.
-  unfold Riemann_integrable in |- *; intros f g; intros; case (Req_EM_T l 0);
+  unfold Riemann_integrable; intros f g; intros; case (Req_EM_T l 0);
     intro.
   elim (X eps); intros; split with x; elim p; intros; split with x0; elim p0;
     intros; split; try assumption; rewrite e; intros;
       rewrite Rmult_0_l; rewrite Rplus_0_r; apply H; assumption.
   assert (H : 0 < eps / 2).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ].
   assert (H0 : 0 < eps / (2 * Rabs l)).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply (cond_pos eps)
       | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat;
         [ prove_sup0 | apply Rabs_pos_lt; assumption ] ].
@@ -1040,7 +1040,7 @@ Proof.
     split with (mkStepFun (StepFun_P28 l x x0)); elim p0;
       elim p; intros; split with (mkStepFun (StepFun_P28 (Rabs l) x1 x2));
         elim p1; elim p2; clear p1 p2 p0 p X X0; intros; split.
-  intros; simpl in |- *;
+  intros; simpl;
     apply Rle_trans with (Rabs (f t - x t) + Rabs (l * (g t - x0 t))).
   replace (f t + l * g t - (x t + l * x0 t)) with
   (f t - x t + l * (g t - x0 t)); [ apply Rabs_triang | ring ].
@@ -1060,7 +1060,7 @@ Proof.
     [ rewrite Rmult_1_l;
       replace (/ Rabs l * (eps / 2)) with (eps / (2 * Rabs l));
       [ apply H2
-        | unfold Rdiv in |- *; rewrite Rinv_mult_distr;
+        | unfold Rdiv; rewrite Rinv_mult_distr;
           [ ring | discrR | apply Rabs_no_R0; assumption ] ]
       | apply Rabs_no_R0; assumption ].
 Qed.
@@ -1080,14 +1080,14 @@ Lemma RiemannInt_P11 :
     Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) l ->
     Un_cv (fun N:nat => RiemannInt_SF (phi2 N)) l.
 Proof.
-  unfold Un_cv in |- *; intro f; intros; intros.
+  unfold Un_cv; intro f; intros; intros.
   case (Rle_dec a b); intro Hyp.
   assert (H4 : 0 < eps / 3).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   elim (H _ H4); clear H; intros N0 H.
   elim (H2 _ H4); clear H2; intros N1 H2.
-  set (N := max N0 N1); exists N; intros; unfold R_dist in |- *.
+  set (N := max N0 N1); exists N; intros; unfold R_dist.
   apply Rle_lt_trans with
     (Rabs (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) +
       Rabs (RiemannInt_SF (phi1 n) - l)).
@@ -1106,24 +1106,24 @@ Proof.
   apply StepFun_P34; assumption.
   apply Rle_lt_trans with
     (RiemannInt_SF (mkStepFun (StepFun_P28 1 (psi1 n) (psi2 n)))).
-  apply StepFun_P37; try assumption; intros; simpl in |- *; rewrite Rmult_1_l.
+  apply StepFun_P37; try assumption; intros; simpl; rewrite Rmult_1_l.
   apply Rle_trans with (Rabs (phi2 n x - f x) + Rabs (f x - phi1 n x)).
   replace (phi2 n x + -1 * phi1 n x) with (phi2 n x - f x + (f x - phi1 n x));
   [ apply Rabs_triang | ring ].
   rewrite (Rplus_comm (psi1 n x)); apply Rplus_le_compat.
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim (H1 n); intros; apply H7.
   assert (H10 : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   assert (H11 : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   rewrite H10; rewrite H11; elim H6; intros; split; left; assumption.
   elim (H0 n); intros; apply H7; assert (H10 : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   assert (H11 : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   rewrite H10; rewrite H11; elim H6; intros; split; left; assumption.
   rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat.
@@ -1132,9 +1132,9 @@ Proof.
   apply RRle_abs.
   assumption.
   replace (pos (un n)) with (R_dist (un n) 0).
-  apply H; unfold ge in |- *; apply le_trans with N; try assumption.
-  unfold N in |- *; apply le_max_l.
-  unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
+  apply H; unfold ge; apply le_trans with N; try assumption.
+  unfold N; apply le_max_l.
+  unfold R_dist; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; apply Rabs_right.
   apply Rle_ge; left; apply (cond_pos (un n)).
   apply Rlt_trans with (pos (un n)).
@@ -1142,24 +1142,24 @@ Proof.
   apply RRle_abs; assumption.
   assumption.
   replace (pos (un n)) with (R_dist (un n) 0).
-  apply H; unfold ge in |- *; apply le_trans with N; try assumption;
-    unfold N in |- *; apply le_max_l.
-  unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
+  apply H; unfold ge; apply le_trans with N; try assumption;
+    unfold N; apply le_max_l.
+  unfold R_dist; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge;
       left; apply (cond_pos (un n)).
-  unfold R_dist in H2; apply H2; unfold ge in |- *; apply le_trans with N;
-    try assumption; unfold N in |- *; apply le_max_r.
+  unfold R_dist in H2; apply H2; unfold ge; apply le_trans with N;
+    try assumption; unfold N; apply le_max_r.
   apply Rmult_eq_reg_l with 3;
-    [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l;
+    [ unfold Rdiv; rewrite Rmult_plus_distr_l;
       do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
         rewrite <- Rinv_l_sym; [ ring | discrR ]
       | discrR ].
   assert (H4 : 0 < eps / 3).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   elim (H _ H4); clear H; intros N0 H.
   elim (H2 _ H4); clear H2; intros N1 H2.
-  set (N := max N0 N1); exists N; intros; unfold R_dist in |- *.
+  set (N := max N0 N1); exists N; intros; unfold R_dist.
   apply Rle_lt_trans with
     (Rabs (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) +
       Rabs (RiemannInt_SF (phi1 n) - l)).
@@ -1189,24 +1189,24 @@ Proof.
       (mkStepFun
         (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 (psi1 n) (psi2 n))))))).
   apply StepFun_P37; try assumption.
-  intros; simpl in |- *; rewrite Rmult_1_l.
+  intros; simpl; rewrite Rmult_1_l.
   apply Rle_trans with (Rabs (phi2 n x - f x) + Rabs (f x - phi1 n x)).
   replace (phi2 n x + -1 * phi1 n x) with (phi2 n x - f x + (f x - phi1 n x));
   [ apply Rabs_triang | ring ].
   rewrite (Rplus_comm (psi1 n x)); apply Rplus_le_compat.
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim (H1 n); intros; apply H7.
   assert (H10 : Rmin a b = b).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ elim Hyp; assumption | reflexivity ].
   assert (H11 : Rmax a b = a).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ elim Hyp; assumption | reflexivity ].
   rewrite H10; rewrite H11; elim H6; intros; split; left; assumption.
   elim (H0 n); intros; apply H7; assert (H10 : Rmin a b = b).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ elim Hyp; assumption | reflexivity ].
   assert (H11 : Rmax a b = a).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ elim Hyp; assumption | reflexivity ].
   rewrite H10; rewrite H11; elim H6; intros; split; left; assumption.
   rewrite <-
@@ -1224,9 +1224,9 @@ Proof.
   rewrite <- Rabs_Ropp; apply RRle_abs.
   assumption.
   replace (pos (un n)) with (R_dist (un n) 0).
-  apply H; unfold ge in |- *; apply le_trans with N; try assumption.
-  unfold N in |- *; apply le_max_l.
-  unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
+  apply H; unfold ge; apply le_trans with N; try assumption.
+  unfold N; apply le_max_l.
+  unfold R_dist; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; apply Rabs_right.
   apply Rle_ge; left; apply (cond_pos (un n)).
   apply Rlt_trans with (pos (un n)).
@@ -1234,15 +1234,15 @@ Proof.
   rewrite <- Rabs_Ropp; apply RRle_abs; assumption.
   assumption.
   replace (pos (un n)) with (R_dist (un n) 0).
-  apply H; unfold ge in |- *; apply le_trans with N; try assumption;
-    unfold N in |- *; apply le_max_l.
-  unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
+  apply H; unfold ge; apply le_trans with N; try assumption;
+    unfold N; apply le_max_l.
+  unfold R_dist; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge;
       left; apply (cond_pos (un n)).
-  unfold R_dist in H2; apply H2; unfold ge in |- *; apply le_trans with N;
-    try assumption; unfold N in |- *; apply le_max_r.
+  unfold R_dist in H2; apply H2; unfold ge; apply le_trans with N;
+    try assumption; unfold N; apply le_max_r.
   apply Rmult_eq_reg_l with 3;
-    [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l;
+    [ unfold Rdiv; rewrite Rmult_plus_distr_l;
       do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
         rewrite <- Rinv_l_sym; [ ring | discrR ]
       | discrR ].
@@ -1255,8 +1255,8 @@ Lemma RiemannInt_P12 :
     a <= b -> RiemannInt pr3 = RiemannInt pr1 + l * RiemannInt pr2.
 Proof.
   intro f; intros; case (Req_dec l 0); intro.
-  pattern l at 2 in |- *; rewrite H0; rewrite Rmult_0_l; rewrite Rplus_0_r;
-    unfold RiemannInt in |- *; case (RiemannInt_exists pr3 RinvN RinvN_cv);
+  pattern l at 2; rewrite H0; rewrite Rmult_0_l; rewrite Rplus_0_r;
+    unfold RiemannInt; case (RiemannInt_exists pr3 RinvN RinvN_cv);
       case (RiemannInt_exists pr1 RinvN RinvN_cv); intros;
         eapply UL_sequence;
           [ apply u0
@@ -1278,18 +1278,18 @@ Proof.
                             [ apply H2; assumption | rewrite H0; ring ] ]
                     | assumption ] ].
   eapply UL_sequence.
-  unfold RiemannInt in |- *; case (RiemannInt_exists pr3 RinvN RinvN_cv);
+  unfold RiemannInt; case (RiemannInt_exists pr3 RinvN RinvN_cv);
     intros; apply u.
-  unfold Un_cv in |- *; intros; unfold RiemannInt in |- *;
+  unfold Un_cv; intros; unfold RiemannInt;
     case (RiemannInt_exists pr1 RinvN RinvN_cv);
-    case (RiemannInt_exists pr2 RinvN RinvN_cv); unfold Un_cv in |- *;
+    case (RiemannInt_exists pr2 RinvN RinvN_cv); unfold Un_cv;
       intros; assert (H2 : 0 < eps / 5).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   elim (u0 _ H2); clear u0; intros N0 H3; assert (H4 := RinvN_cv);
     unfold Un_cv in H4; elim (H4 _ H2); clear H4 H2; intros N1 H4;
       assert (H5 : 0 < eps / (5 * Rabs l)).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption
       | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat;
         [ prove_sup0 | apply Rabs_pos_lt; assumption ] ].
@@ -1298,17 +1298,17 @@ Proof.
       unfold R_dist in H3, H4, H5, H6; set (N := max (max N0 N1) (max N2 N3)).
   assert (H7 : forall n:nat, (n >= N1)%nat -> RinvN n < eps / 5).
   intros; replace (pos (RinvN n)) with (Rabs (RinvN n - 0));
-    [ unfold RinvN in |- *; apply H4; assumption
-      | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
+    [ unfold RinvN; apply H4; assumption
+      | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
         left; apply (cond_pos (RinvN n)) ].
   clear H4; assert (H4 := H7); clear H7;
     assert (H7 : forall n:nat, (n >= N3)%nat -> RinvN n < eps / (5 * Rabs l)).
   intros; replace (pos (RinvN n)) with (Rabs (RinvN n - 0));
-    [ unfold RinvN in |- *; apply H5; assumption
-      | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
+    [ unfold RinvN; apply H5; assumption
+      | unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
         left; apply (cond_pos (RinvN n)) ].
   clear H5; assert (H5 := H7); clear H7; exists N; intros;
-    unfold R_dist in |- *.
+    unfold R_dist.
   apply Rle_lt_trans with
     (Rabs
       (RiemannInt_SF (phi_sequence RinvN pr3 n) -
@@ -1381,10 +1381,10 @@ Proof.
         (RiemannInt_SF (phi_sequence RinvN pr1 n) +
           l * RiemannInt_SF (phi_sequence RinvN pr2 n)));
       [ idtac | ring ]; do 2 rewrite <- StepFun_P30; assert (H10 : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   assert (H11 : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   rewrite H10 in H7; rewrite H10 in H8; rewrite H10 in H9; rewrite H11 in H7;
     rewrite H11 in H8; rewrite H11 in H9;
@@ -1404,7 +1404,7 @@ Proof.
         (StepFun_P28 1 (psi3 n)
           (mkStepFun (StepFun_P28 (Rabs l) (psi1 n) (psi2 n)))))).
   apply StepFun_P37; try assumption.
-  intros; simpl in |- *; rewrite Rmult_1_l.
+  intros; simpl; rewrite Rmult_1_l.
   apply Rle_trans with
     (Rabs (phi_sequence RinvN pr3 n x1 - (f x1 + l * g x1)) +
       Rabs
@@ -1444,16 +1444,16 @@ Proof.
   apply Rlt_trans with (pos (RinvN n));
     [ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi3 n)));
       [ apply RRle_abs | elim (H9 n); intros; assumption ]
-      | apply H4; unfold ge in |- *; apply le_trans with N;
+      | apply H4; unfold ge; apply le_trans with N;
         [ apply le_trans with (max N0 N1);
-          [ apply le_max_r | unfold N in |- *; apply le_max_l ]
+          [ apply le_max_r | unfold N; apply le_max_l ]
           | assumption ] ].
   apply Rlt_trans with (pos (RinvN n));
     [ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n)));
       [ apply RRle_abs | elim (H7 n); intros; assumption ]
-      | apply H4; unfold ge in |- *; apply le_trans with N;
+      | apply H4; unfold ge; apply le_trans with N;
         [ apply le_trans with (max N0 N1);
-          [ apply le_max_r | unfold N in |- *; apply le_max_l ]
+          [ apply le_max_r | unfold N; apply le_max_l ]
           | assumption ] ].
   apply Rmult_lt_reg_l with (/ Rabs l).
   apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
@@ -1462,28 +1462,28 @@ Proof.
   apply Rlt_trans with (pos (RinvN n));
     [ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n)));
       [ apply RRle_abs | elim (H8 n); intros; assumption ]
-      | apply H5; unfold ge in |- *; apply le_trans with N;
+      | apply H5; unfold ge; apply le_trans with N;
         [ apply le_trans with (max N2 N3);
-          [ apply le_max_r | unfold N in |- *; apply le_max_r ]
+          [ apply le_max_r | unfold N; apply le_max_r ]
           | assumption ] ].
-  unfold Rdiv in |- *; rewrite Rinv_mult_distr;
+  unfold Rdiv; rewrite Rinv_mult_distr;
     [ ring | discrR | apply Rabs_no_R0; assumption ].
   apply Rabs_no_R0; assumption.
-  apply H3; unfold ge in |- *; apply le_trans with (max N0 N1);
+  apply H3; unfold ge; apply le_trans with (max N0 N1);
     [ apply le_max_l
-      | apply le_trans with N; [ unfold N in |- *; apply le_max_l | assumption ] ].
+      | apply le_trans with N; [ unfold N; apply le_max_l | assumption ] ].
   apply Rmult_lt_reg_l with (/ Rabs l).
   apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
   rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l; replace (/ Rabs l * (eps / 5)) with (eps / (5 * Rabs l)).
-  apply H6; unfold ge in |- *; apply le_trans with (max N2 N3);
+  apply H6; unfold ge; apply le_trans with (max N2 N3);
     [ apply le_max_l
-      | apply le_trans with N; [ unfold N in |- *; apply le_max_r | assumption ] ].
-  unfold Rdiv in |- *; rewrite Rinv_mult_distr;
+      | apply le_trans with N; [ unfold N; apply le_max_r | assumption ] ].
+  unfold Rdiv; rewrite Rinv_mult_distr;
     [ ring | discrR | apply Rabs_no_R0; assumption ].
   apply Rabs_no_R0; assumption.
   apply Rmult_eq_reg_l with 5;
-    [ unfold Rdiv in |- *; do 2 rewrite Rmult_plus_distr_l;
+    [ unfold Rdiv; do 2 rewrite Rmult_plus_distr_l;
       do 3 rewrite (Rmult_comm 5); repeat rewrite Rmult_assoc;
         rewrite <- Rinv_l_sym; [ ring | discrR ]
       | discrR ].
@@ -1500,11 +1500,11 @@ Proof.
       | assert (H : b <= a);
         [ auto with real
           | replace (RiemannInt pr3) with (- RiemannInt (RiemannInt_P1 pr3));
-            [ idtac | symmetry  in |- *; apply RiemannInt_P8 ];
+            [ idtac | symmetry ; apply RiemannInt_P8 ];
             replace (RiemannInt pr2) with (- RiemannInt (RiemannInt_P1 pr2));
-            [ idtac | symmetry  in |- *; apply RiemannInt_P8 ];
+            [ idtac | symmetry ; apply RiemannInt_P8 ];
             replace (RiemannInt pr1) with (- RiemannInt (RiemannInt_P1 pr1));
-            [ idtac | symmetry  in |- *; apply RiemannInt_P8 ];
+            [ idtac | symmetry ; apply RiemannInt_P8 ];
             rewrite
               (RiemannInt_P12 (RiemannInt_P1 pr1) (RiemannInt_P1 pr2)
                 (RiemannInt_P1 pr3) H); ring ] ].
@@ -1512,11 +1512,11 @@ Qed.
 
 Lemma RiemannInt_P14 : forall a b c:R, Riemann_integrable (fct_cte c) a b.
 Proof.
-  unfold Riemann_integrable in |- *; intros;
+  unfold Riemann_integrable; intros;
     split with (mkStepFun (StepFun_P4 a b c));
       split with (mkStepFun (StepFun_P4 a b 0)); split;
-        [ intros; simpl in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
-          rewrite Rabs_R0; unfold fct_cte in |- *; right;
+        [ intros; simpl; unfold Rminus; rewrite Rplus_opp_r;
+          rewrite Rabs_R0; unfold fct_cte; right;
             reflexivity
           | rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0;
             apply (cond_pos eps) ].
@@ -1526,11 +1526,11 @@ Lemma RiemannInt_P15 :
   forall (a b c:R) (pr:Riemann_integrable (fct_cte c) a b),
     RiemannInt pr = c * (b - a).
 Proof.
-  intros; unfold RiemannInt in |- *; case (RiemannInt_exists pr RinvN RinvN_cv);
+  intros; unfold RiemannInt; case (RiemannInt_exists pr RinvN RinvN_cv);
     intros; eapply UL_sequence.
   apply u.
   set (phi1 := fun N:nat => phi_sequence RinvN pr N);
-    change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) (c * (b - a))) in |- *;
+    change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) (c * (b - a)));
       set (f := fct_cte c);
         assert
           (H1 :
@@ -1549,13 +1549,13 @@ Proof.
           try assumption.
   apply RinvN_cv.
   intro; split.
-  intros; unfold f in |- *; simpl in |- *; unfold Rminus in |- *;
-    rewrite Rplus_opp_r; rewrite Rabs_R0; unfold fct_cte in |- *;
+  intros; unfold f; simpl; unfold Rminus;
+    rewrite Rplus_opp_r; rewrite Rabs_R0; unfold fct_cte;
       right; reflexivity.
-  unfold psi2 in |- *; rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0;
+  unfold psi2; rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0;
     apply (cond_pos (RinvN n)).
-  unfold Un_cv in |- *; intros; split with 0%nat; intros; unfold R_dist in |- *;
-    unfold phi2 in |- *; rewrite StepFun_P18; unfold Rminus in |- *;
+  unfold Un_cv; intros; split with 0%nat; intros; unfold R_dist;
+    unfold phi2; rewrite StepFun_P18; unfold Rminus;
       rewrite Rplus_opp_r; rewrite Rabs_R0; apply H.
 Qed.
 
@@ -1563,9 +1563,9 @@ Lemma RiemannInt_P16 :
   forall (f:R -> R) (a b:R),
     Riemann_integrable f a b -> Riemann_integrable (fun x:R => Rabs (f x)) a b.
 Proof.
-  unfold Riemann_integrable in |- *; intro f; intros; elim (X eps); clear X;
+  unfold Riemann_integrable; intro f; intros; elim (X eps); clear X;
     intros phi [psi [H H0]]; split with (mkStepFun (StepFun_P32 phi));
-      split with psi; split; try assumption; intros; simpl in |- *;
+      split with psi; split; try assumption; intros; simpl;
         apply Rle_trans with (Rabs (f t - phi t));
           [ apply Rabs_triang_inv2 | apply H; assumption ].
 Qed.
@@ -1579,9 +1579,9 @@ Proof.
   assert (H2 : l2 < l1).
   auto with real.
   clear n; assert (H3 : 0 < (l1 - l2) / 2).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply Rlt_Rminus; assumption | apply Rinv_0_lt_compat; prove_sup0 ].
-  elim (H1 _ H3); elim (H0 _ H3); clear H0 H1; unfold R_dist in |- *; intros;
+  elim (H1 _ H3); elim (H0 _ H3); clear H0 H1; unfold R_dist; intros;
     set (N := max x x0); cut (Vn N < Un N).
   intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (H N) H4)).
   apply Rlt_trans with ((l1 + l2) / 2).
@@ -1589,9 +1589,9 @@ Proof.
     replace (- l2 + (l1 + l2) / 2) with ((l1 - l2) / 2).
   rewrite Rplus_comm; apply Rle_lt_trans with (Rabs (Vn N - l2)).
   apply RRle_abs.
-  apply H1; unfold ge in |- *; unfold N in |- *; apply le_max_r.
+  apply H1; unfold ge; unfold N; apply le_max_r.
   apply Rmult_eq_reg_l with 2;
-    [ unfold Rdiv in |- *; do 2 rewrite (Rmult_comm 2);
+    [ unfold Rdiv; do 2 rewrite (Rmult_comm 2);
       rewrite (Rmult_plus_distr_r (- l2) ((l1 + l2) * / 2) 2);
         repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
           [ ring | discrR ]
@@ -1600,9 +1600,9 @@ Proof.
     replace (l1 + - ((l1 + l2) / 2)) with ((l1 - l2) / 2).
   apply Rle_lt_trans with (Rabs (Un N - l1)).
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply RRle_abs.
-  apply H0; unfold ge in |- *; unfold N in |- *; apply le_max_l.
+  apply H0; unfold ge; unfold N; apply le_max_l.
   apply Rmult_eq_reg_l with 2;
-    [ unfold Rdiv in |- *; do 2 rewrite (Rmult_comm 2);
+    [ unfold Rdiv; do 2 rewrite (Rmult_comm 2);
       rewrite (Rmult_plus_distr_r l1 (- ((l1 + l2) * / 2)) 2);
         rewrite <- Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc;
           rewrite <- Rinv_l_sym; [ ring | discrR ]
@@ -1614,7 +1614,7 @@ Lemma RiemannInt_P17 :
     (pr2:Riemann_integrable (fun x:R => Rabs (f x)) a b),
     a <= b -> Rabs (RiemannInt pr1) <= RiemannInt pr2.
 Proof.
-  intro f; intros; unfold RiemannInt in |- *;
+  intro f; intros; unfold RiemannInt;
     case (RiemannInt_exists pr1 RinvN RinvN_cv);
     case (RiemannInt_exists pr2 RinvN RinvN_cv); intros;
       set (phi1 := phi_sequence RinvN pr1) in u0;
@@ -1622,7 +1622,7 @@ Proof.
           apply Rle_cv_lim with
             (fun N:nat => Rabs (RiemannInt_SF (phi1 N)))
             (fun N:nat => RiemannInt_SF (phi2 N)).
-  intro; unfold phi2 in |- *; apply StepFun_P34; assumption.
+  intro; unfold phi2; apply StepFun_P34; assumption.
   apply (continuity_seq Rabs (fun N:nat => RiemannInt_SF (phi1 N)) x0);
     try assumption.
   apply Rcontinuity_abs.
@@ -1656,7 +1656,7 @@ Proof.
     apply (proj2_sig (phi_sequence_prop RinvN pr1 n)).
   elim H1; clear H1; intros psi2 H1; split with psi2; intros; elim (H1 n);
     clear H1; intros; split; try assumption.
-  intros; unfold phi2 in |- *; simpl in |- *;
+  intros; unfold phi2; simpl;
     apply Rle_trans with (Rabs (f t - phi1 n t)).
   apply Rabs_triang_inv2.
   apply H1; assumption.
@@ -1671,13 +1671,13 @@ Lemma RiemannInt_P18 :
     a <= b ->
     (forall x:R, a < x < b -> f x = g x) -> RiemannInt pr1 = RiemannInt pr2.
 Proof.
-  intro f; intros; unfold RiemannInt in |- *;
+  intro f; intros; unfold RiemannInt;
     case (RiemannInt_exists pr1 RinvN RinvN_cv);
     case (RiemannInt_exists pr2 RinvN RinvN_cv); intros;
       eapply UL_sequence.
   apply u0.
   set (phi1 := fun N:nat => phi_sequence RinvN pr1 N);
-    change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) x) in |- *;
+    change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) x);
       assert
         (H1 :
           exists psi1 : nat -> StepFun a b,
@@ -1717,45 +1717,45 @@ Proof.
       try assumption.
   apply RinvN_cv.
   intro; elim (H2 n); intros; split; try assumption.
-  intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *;
+  intros; unfold phi2_m; simpl; unfold phi2_aux;
     case (Req_EM_T t a); case (Req_EM_T t b); intros.
-  rewrite e0; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  rewrite e0; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     apply Rle_trans with (Rabs (g t - phi2 n t)).
   apply Rabs_pos.
-  pattern a at 3 in |- *; rewrite <- e0; apply H3; assumption.
-  rewrite e; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  pattern a at 3; rewrite <- e0; apply H3; assumption.
+  rewrite e; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     apply Rle_trans with (Rabs (g t - phi2 n t)).
   apply Rabs_pos.
-  pattern a at 3 in |- *; rewrite <- e; apply H3; assumption.
-  rewrite e; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  pattern a at 3; rewrite <- e; apply H3; assumption.
+  rewrite e; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     apply Rle_trans with (Rabs (g t - phi2 n t)).
   apply Rabs_pos.
-  pattern b at 3 in |- *; rewrite <- e; apply H3; assumption.
+  pattern b at 3; rewrite <- e; apply H3; assumption.
   replace (f t) with (g t).
   apply H3; assumption.
-  symmetry  in |- *; apply H0; elim H5; clear H5; intros.
+  symmetry ; apply H0; elim H5; clear H5; intros.
   assert (H7 : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n2; assumption ].
   assert (H8 : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n2; assumption ].
   rewrite H7 in H5; rewrite H8 in H6; split.
-  elim H5; intro; [ assumption | elim n1; symmetry  in |- *; assumption ].
+  elim H5; intro; [ assumption | elim n1; symmetry ; assumption ].
   elim H6; intro; [ assumption | elim n0; assumption ].
   cut (forall N:nat, RiemannInt_SF (phi2_m N) = RiemannInt_SF (phi2 N)).
-  intro; unfold Un_cv in |- *; intros; elim (u _ H4); intros; exists x1; intros;
+  intro; unfold Un_cv; intros; elim (u _ H4); intros; exists x1; intros;
     rewrite (H3 n); apply H5; assumption.
   intro; apply Rle_antisym.
   apply StepFun_P37; try assumption.
-  intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *;
+  intros; unfold phi2_m; simpl; unfold phi2_aux;
     case (Req_EM_T x1 a); case (Req_EM_T x1 b); intros.
   elim H3; intros; rewrite e0 in H4; elim (Rlt_irrefl _ H4).
   elim H3; intros; rewrite e in H4; elim (Rlt_irrefl _ H4).
   elim H3; intros; rewrite e in H5; elim (Rlt_irrefl _ H5).
   right; reflexivity.
   apply StepFun_P37; try assumption.
-  intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *;
+  intros; unfold phi2_m; simpl; unfold phi2_aux;
     case (Req_EM_T x1 a); case (Req_EM_T x1 b); intros.
   elim H3; intros; rewrite e0 in H4; elim (Rlt_irrefl _ H4).
   elim H3; intros; rewrite e in H4; elim (Rlt_irrefl _ H4).
@@ -1764,10 +1764,10 @@ Proof.
   intro; assert (H2 := pre (phi2 N)); unfold IsStepFun in H2;
     unfold is_subdivision in H2; elim H2; clear H2; intros l [lf H2];
       split with l; split with lf; unfold adapted_couple in H2;
-        decompose [and] H2; clear H2; unfold adapted_couple in |- *;
+        decompose [and] H2; clear H2; unfold adapted_couple;
           repeat split; try assumption.
   intros; assert (H9 := H8 i H2); unfold constant_D_eq, open_interval in H9;
-    unfold constant_D_eq, open_interval in |- *; intros;
+    unfold constant_D_eq, open_interval; intros;
       rewrite <- (H9 x1 H7); assert (H10 : a <= pos_Rl l i).
   replace a with (Rmin a b).
   rewrite <- H5; elim (RList_P6 l); intros; apply H10.
@@ -1775,7 +1775,7 @@ Proof.
   apply le_O_n.
   apply lt_trans with (pred (Rlength l)); [ assumption | apply lt_pred_n_n ].
   apply neq_O_lt; intro; rewrite <- H12 in H6; discriminate.
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   assert (H11 : pos_Rl l (S i) <= b).
   replace b with (Rmax a b).
@@ -1783,9 +1783,9 @@ Proof.
   assumption.
   apply lt_le_S; assumption.
   apply lt_pred_n_n; apply neq_O_lt; intro; rewrite <- H13 in H6; discriminate.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
-  elim H7; clear H7; intros; unfold phi2_aux in |- *; case (Req_EM_T x1 a);
+  elim H7; clear H7; intros; unfold phi2_aux; case (Req_EM_T x1 a);
     case (Req_EM_T x1 b); intros.
   rewrite e in H12; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H11 H12)).
   rewrite e in H7; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H10 H7)).
@@ -1852,12 +1852,12 @@ Proof.
   intros; replace (primitive h pr a) with 0.
   replace (RiemannInt pr0) with (primitive h pr b).
   ring.
-  unfold primitive in |- *; case (Rle_dec a b); case (Rle_dec b b); intros;
+  unfold primitive; case (Rle_dec a b); case (Rle_dec b b); intros;
     [ apply RiemannInt_P5
       | elim n; right; reflexivity
       | elim n; assumption
       | elim n0; assumption ].
-  symmetry  in |- *; unfold primitive in |- *; case (Rle_dec a a);
+  symmetry ; unfold primitive; case (Rle_dec a a);
     case (Rle_dec a b); intros;
       [ apply RiemannInt_P9
         | elim n; assumption
@@ -1872,9 +1872,9 @@ Lemma RiemannInt_P21 :
     Riemann_integrable f a b ->
     Riemann_integrable f b c -> Riemann_integrable f a c.
 Proof.
-  unfold Riemann_integrable in |- *; intros f a b c Hyp1 Hyp2 X X0 eps.
+  unfold Riemann_integrable; intros f a b c Hyp1 Hyp2 X X0 eps.
   assert (H : 0 < eps / 2).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ].
   elim (X (mkposreal _ H)); clear X; intros phi1 [psi1 H1];
     elim (X0 (mkposreal _ H)); clear X0; intros phi2 [psi2 H2].
@@ -1904,35 +1904,35 @@ Proof.
   intro; cut (IsStepFun psi3 a b).
   intro; cut (IsStepFun psi3 b c).
   intro; cut (IsStepFun psi3 a c).
-  intro; split with (mkStepFun X); split with (mkStepFun X2); simpl in |- *;
+  intro; split with (mkStepFun X); split with (mkStepFun X2); simpl;
     split.
-  intros; unfold phi3, psi3 in |- *; case (Rle_dec t b); case (Rle_dec a t);
+  intros; unfold phi3, psi3; case (Rle_dec t b); case (Rle_dec a t);
     intros.
   elim H1; intros; apply H3.
   replace (Rmin a b) with a.
   replace (Rmax a b) with b.
   split; assumption.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   elim n; replace a with (Rmin a c).
   elim H0; intros; assumption.
-  unfold Rmin in |- *; case (Rle_dec a c); intro;
+  unfold Rmin; case (Rle_dec a c); intro;
     [ reflexivity | elim n0; apply Rle_trans with b; assumption ].
   elim H2; intros; apply H3.
   replace (Rmax b c) with (Rmax a c).
   elim H0; intros; split; try assumption.
   replace (Rmin b c) with b.
   auto with real.
-  unfold Rmin in |- *; case (Rle_dec b c); intro;
+  unfold Rmin; case (Rle_dec b c); intro;
     [ reflexivity | elim n0; assumption ].
-  unfold Rmax in |- *; case (Rle_dec a c); case (Rle_dec b c); intros;
+  unfold Rmax; case (Rle_dec a c); case (Rle_dec b c); intros;
     try (elim n0; assumption || elim n0; apply Rle_trans with b; assumption).
   reflexivity.
   elim n; replace a with (Rmin a c).
   elim H0; intros; assumption.
-  unfold Rmin in |- *; case (Rle_dec a c); intro;
+  unfold Rmin; case (Rle_dec a c); intro;
     [ reflexivity | elim n1; apply Rle_trans with b; assumption ].
   rewrite <- (StepFun_P43 X0 X1 X2).
   apply Rle_lt_trans with
@@ -1946,14 +1946,14 @@ Proof.
   elim H2; intros; assumption.
   apply Rle_antisym.
   apply StepFun_P37; try assumption.
-  simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros;
+  simpl; intros; unfold psi3; elim H0; clear H0; intros;
     case (Rle_dec a x); case (Rle_dec x b); intros;
       [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H0))
         | right; reflexivity
         | elim n; apply Rle_trans with b; [ assumption | left; assumption ]
         | elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ].
   apply StepFun_P37; try assumption.
-  simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros;
+  simpl; intros; unfold psi3; elim H0; clear H0; intros;
     case (Rle_dec a x); case (Rle_dec x b); intros;
       [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H0))
         | right; reflexivity
@@ -1961,14 +1961,14 @@ Proof.
         | elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ].
   apply Rle_antisym.
   apply StepFun_P37; try assumption.
-  simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros;
+  simpl; intros; unfold psi3; elim H0; clear H0; intros;
     case (Rle_dec a x); case (Rle_dec x b); intros;
       [ right; reflexivity
         | elim n; left; assumption
         | elim n; left; assumption
         | elim n0; left; assumption ].
   apply StepFun_P37; try assumption.
-  simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros;
+  simpl; intros; unfold psi3; elim H0; clear H0; intros;
     case (Rle_dec a x); case (Rle_dec x b); intros;
       [ right; reflexivity
         | elim n; left; assumption
@@ -1978,19 +1978,19 @@ Proof.
   assert (H3 := pre psi2); unfold IsStepFun in H3; unfold is_subdivision in H3;
     elim H3; clear H3; intros l1 [lf1 H3]; split with l1;
       split with lf1; unfold adapted_couple in H3; decompose [and] H3;
-        clear H3; unfold adapted_couple in |- *; repeat split;
+        clear H3; unfold adapted_couple; repeat split;
           try assumption.
-  intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *;
+  intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval;
     unfold constant_D_eq, open_interval in H9; intros;
-      rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : b < x).
+      rewrite <- (H9 x H7); unfold psi3; assert (H10 : b < x).
   apply Rle_lt_trans with (pos_Rl l1 i).
   replace b with (Rmin b c).
   rewrite <- H5; elim (RList_P6 l1); intros; apply H10; try assumption.
   apply le_O_n.
   apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n;
-    apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6;
+    apply neq_O_lt; red; intro; rewrite <- H12 in H6;
       discriminate.
-  unfold Rmin in |- *; case (Rle_dec b c); intro;
+  unfold Rmin; case (Rle_dec b c); intro;
     [ reflexivity | elim n; assumption ].
   elim H7; intros; assumption.
   case (Rle_dec a x); case (Rle_dec x b); intros;
@@ -2001,18 +2001,18 @@ Proof.
   assert (H3 := pre psi1); unfold IsStepFun in H3; unfold is_subdivision in H3;
     elim H3; clear H3; intros l1 [lf1 H3]; split with l1;
       split with lf1; unfold adapted_couple in H3; decompose [and] H3;
-        clear H3; unfold adapted_couple in |- *; repeat split;
+        clear H3; unfold adapted_couple; repeat split;
           try assumption.
-  intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *;
+  intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval;
     unfold constant_D_eq, open_interval in H9; intros;
-      rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : x <= b).
+      rewrite <- (H9 x H7); unfold psi3; assert (H10 : x <= b).
   apply Rle_trans with (pos_Rl l1 (S i)).
   elim H7; intros; left; assumption.
   replace b with (Rmax a b).
   rewrite <- H4; elim (RList_P6 l1); intros; apply H10; try assumption.
-  apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6;
+  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H12 in H6;
     discriminate.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   assert (H11 : a <= x).
   apply Rle_trans with (pos_Rl l1 i).
@@ -2020,9 +2020,9 @@ Proof.
   rewrite <- H5; elim (RList_P6 l1); intros; apply H11; try assumption.
   apply le_O_n.
   apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n;
-    apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H6;
+    apply neq_O_lt; red; intro; rewrite <- H13 in H6;
       discriminate.
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   left; elim H7; intros; assumption.
   case (Rle_dec a x); case (Rle_dec x b); intros; reflexivity || elim n;
@@ -2031,18 +2031,18 @@ Proof.
   assert (H3 := pre phi1); unfold IsStepFun in H3; unfold is_subdivision in H3;
     elim H3; clear H3; intros l1 [lf1 H3]; split with l1;
       split with lf1; unfold adapted_couple in H3; decompose [and] H3;
-        clear H3; unfold adapted_couple in |- *; repeat split;
+        clear H3; unfold adapted_couple; repeat split;
           try assumption.
-  intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *;
+  intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval;
     unfold constant_D_eq, open_interval in H9; intros;
-      rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : x <= b).
+      rewrite <- (H9 x H7); unfold psi3; assert (H10 : x <= b).
   apply Rle_trans with (pos_Rl l1 (S i)).
   elim H7; intros; left; assumption.
   replace b with (Rmax a b).
   rewrite <- H4; elim (RList_P6 l1); intros; apply H10; try assumption.
-  apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6;
+  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H12 in H6;
     discriminate.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   assert (H11 : a <= x).
   apply Rle_trans with (pos_Rl l1 i).
@@ -2050,32 +2050,32 @@ Proof.
   rewrite <- H5; elim (RList_P6 l1); intros; apply H11; try assumption.
   apply le_O_n.
   apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n;
-    apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H6;
+    apply neq_O_lt; red; intro; rewrite <- H13 in H6;
       discriminate.
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   left; elim H7; intros; assumption.
-  unfold phi3 in |- *; case (Rle_dec a x); case (Rle_dec x b); intros;
+  unfold phi3; case (Rle_dec a x); case (Rle_dec x b); intros;
     reflexivity || elim n; assumption.
   assert (H3 := pre phi2); unfold IsStepFun in H3; unfold is_subdivision in H3;
     elim H3; clear H3; intros l1 [lf1 H3]; split with l1;
       split with lf1; unfold adapted_couple in H3; decompose [and] H3;
-        clear H3; unfold adapted_couple in |- *; repeat split;
+        clear H3; unfold adapted_couple; repeat split;
           try assumption.
-  intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *;
+  intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval;
     unfold constant_D_eq, open_interval in H9; intros;
-      rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : b < x).
+      rewrite <- (H9 x H7); unfold psi3; assert (H10 : b < x).
   apply Rle_lt_trans with (pos_Rl l1 i).
   replace b with (Rmin b c).
   rewrite <- H5; elim (RList_P6 l1); intros; apply H10; try assumption.
   apply le_O_n.
   apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n;
-    apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6;
+    apply neq_O_lt; red; intro; rewrite <- H12 in H6;
       discriminate.
-  unfold Rmin in |- *; case (Rle_dec b c); intro;
+  unfold Rmin; case (Rle_dec b c); intro;
     [ reflexivity | elim n; assumption ].
   elim H7; intros; assumption.
-  unfold phi3 in |- *; case (Rle_dec a x); case (Rle_dec x b); intros;
+  unfold phi3; case (Rle_dec a x); case (Rle_dec x b); intros;
     [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H10))
       | reflexivity
       | elim n; apply Rle_trans with b; [ assumption | left; assumption ]
@@ -2086,7 +2086,7 @@ Lemma RiemannInt_P22 :
   forall (f:R -> R) (a b c:R),
     Riemann_integrable f a b -> a <= c <= b -> Riemann_integrable f a c.
 Proof.
-  unfold Riemann_integrable in |- *; intros; elim (X eps); clear X;
+  unfold Riemann_integrable; intros; elim (X eps); clear X;
     intros phi [psi H0]; elim H; elim H0; clear H H0;
       intros; assert (H3 : IsStepFun phi a c).
   apply StepFun_P44 with b.
@@ -2097,18 +2097,18 @@ Proof.
   apply (pre psi).
   split; assumption.
   split with (mkStepFun H3); split with (mkStepFun H4); split.
-  simpl in |- *; intros; apply H.
+  simpl; intros; apply H.
   replace (Rmin a b) with (Rmin a c).
   elim H5; intros; split; try assumption.
   apply Rle_trans with (Rmax a c); try assumption.
   replace (Rmax a b) with b.
   replace (Rmax a c) with c.
   assumption.
-  unfold Rmax in |- *; case (Rle_dec a c); intro;
+  unfold Rmax; case (Rle_dec a c); intro;
     [ reflexivity | elim n; assumption ].
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; apply Rle_trans with c; assumption ].
-  unfold Rmin in |- *; case (Rle_dec a c); case (Rle_dec a b); intros;
+  unfold Rmin; case (Rle_dec a c); case (Rle_dec a b); intros;
     [ reflexivity
       | elim n; apply Rle_trans with c; assumption
       | elim n; assumption
@@ -2121,12 +2121,12 @@ Proof.
   replace (RiemannInt_SF (mkStepFun H4)) with
   (RiemannInt_SF psi - RiemannInt_SF (mkStepFun H5)).
   apply Rle_lt_trans with (RiemannInt_SF psi).
-  unfold Rminus in |- *; pattern (RiemannInt_SF psi) at 2 in |- *;
+  unfold Rminus; pattern (RiemannInt_SF psi) at 2;
     rewrite <- Rplus_0_r; apply Rplus_le_compat_l; rewrite <- Ropp_0;
       apply Ropp_ge_le_contravar; apply Rle_ge;
         replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b 0))).
   apply StepFun_P37; try assumption.
-  intros; simpl in |- *; unfold fct_cte in |- *;
+  intros; simpl; unfold fct_cte;
     apply Rle_trans with (Rabs (f x - phi x)).
   apply Rabs_pos.
   apply H.
@@ -2135,9 +2135,9 @@ Proof.
   elim H6; intros; split; left.
   apply Rle_lt_trans with c; assumption.
   assumption.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; apply Rle_trans with c; assumption ].
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; apply Rle_trans with c; assumption ].
   rewrite StepFun_P18; ring.
   apply Rle_lt_trans with (Rabs (RiemannInt_SF psi)).
@@ -2147,16 +2147,16 @@ Proof.
   apply (pre psi).
   replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)).
   rewrite <- (StepFun_P43 H4 H5 H6); ring.
-  unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
+  unfold RiemannInt_SF; case (Rle_dec a b); intro.
   eapply StepFun_P17.
   apply StepFun_P1.
-  simpl in |- *; apply StepFun_P1.
+  simpl; apply StepFun_P1.
   apply Ropp_eq_compat; eapply StepFun_P17.
   apply StepFun_P1.
-  simpl in |- *; apply StepFun_P1.
+  simpl; apply StepFun_P1.
   apply Rle_ge; replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c 0))).
   apply StepFun_P37; try assumption.
-  intros; simpl in |- *; unfold fct_cte in |- *;
+  intros; simpl; unfold fct_cte;
     apply Rle_trans with (Rabs (f x - phi x)).
   apply Rabs_pos.
   apply H.
@@ -2165,9 +2165,9 @@ Proof.
   elim H5; intros; split; left.
   assumption.
   apply Rlt_le_trans with c; assumption.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; apply Rle_trans with c; assumption ].
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; apply Rle_trans with c; assumption ].
   rewrite StepFun_P18; ring.
 Qed.
@@ -2176,7 +2176,7 @@ Lemma RiemannInt_P23 :
   forall (f:R -> R) (a b c:R),
     Riemann_integrable f a b -> a <= c <= b -> Riemann_integrable f c b.
 Proof.
-  unfold Riemann_integrable in |- *; intros; elim (X eps); clear X;
+  unfold Riemann_integrable; intros; elim (X eps); clear X;
     intros phi [psi H0]; elim H; elim H0; clear H H0;
       intros; assert (H3 : IsStepFun phi c b).
   apply StepFun_P45 with a.
@@ -2187,18 +2187,18 @@ Proof.
   apply (pre psi).
   split; assumption.
   split with (mkStepFun H3); split with (mkStepFun H4); split.
-  simpl in |- *; intros; apply H.
+  simpl; intros; apply H.
   replace (Rmax a b) with (Rmax c b).
   elim H5; intros; split; try assumption.
   apply Rle_trans with (Rmin c b); try assumption.
   replace (Rmin a b) with a.
   replace (Rmin c b) with c.
   assumption.
-  unfold Rmin in |- *; case (Rle_dec c b); intro;
+  unfold Rmin; case (Rle_dec c b); intro;
     [ reflexivity | elim n; assumption ].
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; apply Rle_trans with c; assumption ].
-  unfold Rmax in |- *; case (Rle_dec c b); case (Rle_dec a b); intros;
+  unfold Rmax; case (Rle_dec c b); case (Rle_dec a b); intros;
     [ reflexivity
       | elim n; apply Rle_trans with c; assumption
       | elim n; assumption
@@ -2211,12 +2211,12 @@ Proof.
   replace (RiemannInt_SF (mkStepFun H4)) with
   (RiemannInt_SF psi - RiemannInt_SF (mkStepFun H5)).
   apply Rle_lt_trans with (RiemannInt_SF psi).
-  unfold Rminus in |- *; pattern (RiemannInt_SF psi) at 2 in |- *;
+  unfold Rminus; pattern (RiemannInt_SF psi) at 2;
     rewrite <- Rplus_0_r; apply Rplus_le_compat_l; rewrite <- Ropp_0;
       apply Ropp_ge_le_contravar; apply Rle_ge;
         replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c 0))).
   apply StepFun_P37; try assumption.
-  intros; simpl in |- *; unfold fct_cte in |- *;
+  intros; simpl; unfold fct_cte;
     apply Rle_trans with (Rabs (f x - phi x)).
   apply Rabs_pos.
   apply H.
@@ -2225,9 +2225,9 @@ Proof.
   elim H6; intros; split; left.
   assumption.
   apply Rlt_le_trans with c; assumption.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; apply Rle_trans with c; assumption ].
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; apply Rle_trans with c; assumption ].
   rewrite StepFun_P18; ring.
   apply Rle_lt_trans with (Rabs (RiemannInt_SF psi)).
@@ -2237,16 +2237,16 @@ Proof.
   apply (pre psi).
   replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)).
   rewrite <- (StepFun_P43 H5 H4 H6); ring.
-  unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
+  unfold RiemannInt_SF; case (Rle_dec a b); intro.
   eapply StepFun_P17.
   apply StepFun_P1.
-  simpl in |- *; apply StepFun_P1. 
+  simpl; apply StepFun_P1.
   apply Ropp_eq_compat; eapply StepFun_P17.
   apply StepFun_P1.
-  simpl in |- *; apply StepFun_P1.
+  simpl; apply StepFun_P1.
   apply Rle_ge; replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b 0))).
   apply StepFun_P37; try assumption.
-  intros; simpl in |- *; unfold fct_cte in |- *;
+  intros; simpl; unfold fct_cte;
     apply Rle_trans with (Rabs (f x - phi x)).
   apply Rabs_pos.
   apply H.
@@ -2255,9 +2255,9 @@ Proof.
   elim H5; intros; split; left.
   apply Rle_lt_trans with c; assumption.
   assumption.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; apply Rle_trans with c; assumption ].
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; apply Rle_trans with c; assumption ].
   rewrite StepFun_P18; ring.
 Qed.
@@ -2290,14 +2290,14 @@ Lemma RiemannInt_P25 :
     (pr2:Riemann_integrable f b c) (pr3:Riemann_integrable f a c),
     a <= b -> b <= c -> RiemannInt pr1 + RiemannInt pr2 = RiemannInt pr3.
 Proof.
-  intros f a b c pr1 pr2 pr3 Hyp1 Hyp2; unfold RiemannInt in |- *;
+  intros f a b c pr1 pr2 pr3 Hyp1 Hyp2; unfold RiemannInt;
     case (RiemannInt_exists pr1 RinvN RinvN_cv);
     case (RiemannInt_exists pr2 RinvN RinvN_cv);
     case (RiemannInt_exists pr3 RinvN RinvN_cv); intros;
-      symmetry  in |- *; eapply UL_sequence.
+      symmetry ; eapply UL_sequence.
   apply u.
-  unfold Un_cv in |- *; intros; assert (H0 : 0 < eps / 3).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Un_cv; intros; assert (H0 : 0 < eps / 3).
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   elim (u1 _ H0); clear u1; intros N1 H1; elim (u0 _ H0); clear u0;
     intros N2 H2;
@@ -2309,7 +2309,7 @@ Proof.
               RiemannInt_SF (phi_sequence RinvN pr2 n))) 0).
   intro; elim (H3 _ H0); clear H3; intros N3 H3;
     set (N0 := max (max N1 N2) N3); exists N0; intros;
-      unfold R_dist in |- *;
+      unfold R_dist;
         apply Rle_lt_trans with
           (Rabs
             (RiemannInt_SF (phi_sequence RinvN pr3 n) -
@@ -2330,8 +2330,8 @@ Proof.
   unfold R_dist in H3; cut (n >= N3)%nat.
   intro; assert (H6 := H3 _ H5); unfold Rminus in H6; rewrite Ropp_0 in H6;
     rewrite Rplus_0_r in H6; apply H6.
-  unfold ge in |- *; apply le_trans with N0;
-    [ unfold N0 in |- *; apply le_max_r | assumption ].
+  unfold ge; apply le_trans with N0;
+    [ unfold N0; apply le_max_r | assumption ].
   apply Rle_lt_trans with
     (Rabs (RiemannInt_SF (phi_sequence RinvN pr1 n) - x1) +
       Rabs (RiemannInt_SF (phi_sequence RinvN pr2 n) - x0)).
@@ -2343,17 +2343,17 @@ Proof.
   [ apply Rabs_triang | ring ].
   apply Rplus_lt_compat.
   unfold R_dist in H1; apply H1.
-  unfold ge in |- *; apply le_trans with N0;
+  unfold ge; apply le_trans with N0;
     [ apply le_trans with (max N1 N2);
-      [ apply le_max_l | unfold N0 in |- *; apply le_max_l ]
+      [ apply le_max_l | unfold N0; apply le_max_l ]
       | assumption ].
   unfold R_dist in H2; apply H2.
-  unfold ge in |- *; apply le_trans with N0;
+  unfold ge; apply le_trans with N0;
     [ apply le_trans with (max N1 N2);
-      [ apply le_max_r | unfold N0 in |- *; apply le_max_l ]
+      [ apply le_max_r | unfold N0; apply le_max_l ]
       | assumption ].
   apply Rmult_eq_reg_l with 3;
-    [ unfold Rdiv in |- *; repeat rewrite Rmult_plus_distr_l;
+    [ unfold Rdiv; repeat rewrite Rmult_plus_distr_l;
       do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
         rewrite <- Rinv_l_sym; [ ring | discrR ]
       | discrR ].
@@ -2390,8 +2390,8 @@ Proof.
     apply (proj2_sig (phi_sequence_prop RinvN pr3 n)).
   elim H1; clear H1; intros psi1 H1; elim H2; clear H2; intros psi2 H2; elim H3;
     clear H3; intros psi3 H3; assert (H := RinvN_cv);
-      unfold Un_cv in |- *; intros; assert (H4 : 0 < eps / 3).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      unfold Un_cv; intros; assert (H4 : 0 < eps / 3).
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   elim (H _ H4); clear H; intros N0 H;
     assert (H5 : forall n:nat, (n >= N0)%nat -> RinvN n < eps / 3).
@@ -2399,11 +2399,11 @@ Proof.
     replace (pos (RinvN n)) with
     (R_dist (mkposreal (/ (INR n + 1)) (RinvN_pos n)) 0).
   apply H; assumption.
-  unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
+  unfold R_dist; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge;
       left; apply (cond_pos (RinvN n)).
   exists N0; intros; elim (H1 n); elim (H2 n); elim (H3 n); clear H1 H2 H3;
-    intros; unfold R_dist in |- *; unfold Rminus in |- *;
+    intros; unfold R_dist; unfold Rminus;
       rewrite Ropp_0; rewrite Rplus_0_r;
         set (phi1 := phi_sequence RinvN pr1 n) in H8 |- *;
           set (phi2 := phi_sequence RinvN pr2 n) in H3 |- *;
@@ -2469,7 +2469,7 @@ Proof.
         (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1)))) +
       RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H13) (psi2 n)))).
   apply Rplus_le_compat_l; apply StepFun_P37; try assumption.
-  intros; simpl in |- *; rewrite Rmult_1_l;
+  intros; simpl; rewrite Rmult_1_l;
     apply Rle_trans with (Rabs (f x - phi3 x) + Rabs (f x - phi2 x)).
   rewrite <- (Rabs_Ropp (f x - phi3 x)); rewrite Ropp_minus_distr;
     replace (phi3 x + -1 * phi2 x) with (phi3 x - f x + (f x - phi2 x));
@@ -2480,28 +2480,28 @@ Proof.
   replace (Rmin a c) with a.
   apply Rle_trans with b; try assumption.
   left; assumption.
-  unfold Rmin in |- *; case (Rle_dec a c); intro;
+  unfold Rmin; case (Rle_dec a c); intro;
     [ reflexivity | elim n0; apply Rle_trans with b; assumption ].
   replace (Rmax a c) with c.
   left; assumption.
-  unfold Rmax in |- *; case (Rle_dec a c); intro;
+  unfold Rmax; case (Rle_dec a c); intro;
     [ reflexivity | elim n0; apply Rle_trans with b; assumption ].
   apply H3.
   elim H14; intros; split.
   replace (Rmin b c) with b.
   left; assumption.
-  unfold Rmin in |- *; case (Rle_dec b c); intro;
+  unfold Rmin; case (Rle_dec b c); intro;
     [ reflexivity | elim n0; assumption ].
   replace (Rmax b c) with c.
   left; assumption.
-  unfold Rmax in |- *; case (Rle_dec b c); intro;
+  unfold Rmax; case (Rle_dec b c); intro;
     [ reflexivity | elim n0; assumption ].
   do 2
     rewrite <-
       (Rplus_comm
         (RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H13) (psi2 n)))))
       ; apply Rplus_le_compat_l; apply StepFun_P37; try assumption.
-  intros; simpl in |- *; rewrite Rmult_1_l;
+  intros; simpl; rewrite Rmult_1_l;
     apply Rle_trans with (Rabs (f x - phi3 x) + Rabs (f x - phi1 x)).
   rewrite <- (Rabs_Ropp (f x - phi3 x)); rewrite Ropp_minus_distr;
     replace (phi3 x + -1 * phi1 x) with (phi3 x - f x + (f x - phi1 x));
@@ -2511,23 +2511,23 @@ Proof.
   elim H14; intros; split.
   replace (Rmin a c) with a.
   left; assumption.
-  unfold Rmin in |- *; case (Rle_dec a c); intro;
+  unfold Rmin; case (Rle_dec a c); intro;
     [ reflexivity | elim n0; apply Rle_trans with b; assumption ].
   replace (Rmax a c) with c.
   apply Rle_trans with b.
   left; assumption.
   assumption.
-  unfold Rmax in |- *; case (Rle_dec a c); intro;
+  unfold Rmax; case (Rle_dec a c); intro;
     [ reflexivity | elim n0; apply Rle_trans with b; assumption ].
   apply H8.
   elim H14; intros; split.
   replace (Rmin a b) with a.
   left; assumption.
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   replace (Rmax a b) with b.
   left; assumption.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n0; assumption ].
   do 2 rewrite StepFun_P30.
   do 2 rewrite Rmult_1_l;
@@ -2553,7 +2553,7 @@ Proof.
   assumption.
   apply H5; assumption.
   apply Rmult_eq_reg_l with 3;
-    [ unfold Rdiv in |- *; repeat rewrite Rmult_plus_distr_l;
+    [ unfold Rdiv; repeat rewrite Rmult_plus_distr_l;
       do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
         rewrite <- Rinv_l_sym; [ ring | discrR ]
       | discrR ].
@@ -2608,13 +2608,13 @@ Lemma RiemannInt_P27 :
 Proof.
   intro f; intros; elim H; clear H; intros; assert (H1 : continuity_pt f x).
   apply C0; split; left; assumption.
-  unfold derivable_pt_lim in |- *; intros; assert (Hyp : 0 < eps / 2).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold derivable_pt_lim; intros; assert (Hyp : 0 < eps / 2).
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
-  elim (H1 _ Hyp); unfold dist, D_x, no_cond in |- *; simpl in |- *;
-    unfold R_dist in |- *; intros; set (del := Rmin x0 (Rmin (b - x) (x - a)));
+  elim (H1 _ Hyp); unfold dist, D_x, no_cond; simpl;
+    unfold R_dist; intros; set (del := Rmin x0 (Rmin (b - x) (x - a)));
       assert (H4 : 0 < del).
-  unfold del in |- *; unfold Rmin in |- *; case (Rle_dec (b - x) (x - a));
+  unfold del; unfold Rmin; case (Rle_dec (b - x) (x - a));
     intro.
   case (Rle_dec x0 (b - x)); intro;
     [ elim H3; intros; assumption | apply Rlt_Rminus; assumption ].
@@ -2631,22 +2631,22 @@ Proof.
   left; apply Rlt_le_trans with (x + del).
   apply Rplus_lt_compat_l; apply Rle_lt_trans with (Rabs h0);
     [ apply RRle_abs | apply H6 ].
-  unfold del in |- *; apply Rle_trans with (x + Rmin (b - x) (x - a)).
+  unfold del; apply Rle_trans with (x + Rmin (b - x) (x - a)).
   apply Rplus_le_compat_l; apply Rmin_r.
-  pattern b at 2 in |- *; replace b with (x + (b - x));
+  pattern b at 2; replace b with (x + (b - x));
     [ apply Rplus_le_compat_l; apply Rmin_l | ring ].
   apply RiemannInt_P1; apply continuity_implies_RiemannInt; auto with real.
   intros; apply C0; elim H7; intros; split.
   apply Rle_trans with (x + h0).
   left; apply Rle_lt_trans with (x - del).
-  unfold del in |- *; apply Rle_trans with (x - Rmin (b - x) (x - a)).
-  pattern a at 1 in |- *; replace a with (x + (a - x)); [ idtac | ring ].
-  unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel.
+  unfold del; apply Rle_trans with (x - Rmin (b - x) (x - a)).
+  pattern a at 1; replace a with (x + (a - x)); [ idtac | ring ].
+  unfold Rminus; apply Rplus_le_compat_l; apply Ropp_le_cancel.
   rewrite Ropp_involutive; rewrite Ropp_plus_distr; rewrite Ropp_involutive;
     rewrite (Rplus_comm x); apply Rmin_r.
-  unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel.
+  unfold Rminus; apply Rplus_le_compat_l; apply Ropp_le_cancel.
   do 2 rewrite Ropp_involutive; apply Rmin_r.
-  unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_cancel.
+  unfold Rminus; apply Rplus_lt_compat_l; apply Ropp_lt_cancel.
   rewrite Ropp_involutive; apply Rle_lt_trans with (Rabs h0);
     [ rewrite <- Rabs_Ropp; apply RRle_abs | apply H6 ].
   assumption.
@@ -2659,7 +2659,7 @@ Proof.
     with ((RiemannInt H7 - RiemannInt (RiemannInt_P14 x (x + h0) (f x))) / h0).
   replace (RiemannInt H7 - RiemannInt (RiemannInt_P14 x (x + h0) (f x))) with
   (RiemannInt (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))).
-  unfold Rdiv in |- *; rewrite Rabs_mult; case (Rle_dec x (x + h0)); intro.
+  unfold Rdiv; rewrite Rabs_mult; case (Rle_dec x (x + h0)); intro.
   apply Rle_lt_trans with
     (RiemannInt
       (RiemannInt_P16
@@ -2678,8 +2678,8 @@ Proof.
   apply Rabs_pos.
   apply RiemannInt_P19; try assumption.
   intros; replace (f x1 + -1 * fct_cte (f x) x1) with (f x1 - f x).
-  unfold fct_cte in |- *; case (Req_dec x x1); intro.
-  rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; left;
+  unfold fct_cte; case (Req_dec x x1); intro.
+  rewrite H9; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; left;
     assumption.
   elim H3; intros; left; apply H11.
   repeat split.
@@ -2690,16 +2690,16 @@ Proof.
   elim H8; intros; assumption.
   apply Rplus_le_compat_l; apply Rle_trans with del.
   left; apply Rle_lt_trans with (Rabs h0); [ apply RRle_abs | assumption ].
-  unfold del in |- *; apply Rmin_l.
+  unfold del; apply Rmin_l.
   apply Rge_minus; apply Rle_ge; left; elim H8; intros; assumption.
-  unfold fct_cte in |- *; ring.
+  unfold fct_cte; ring.
   rewrite RiemannInt_P15.
   rewrite Rmult_assoc; replace ((x + h0 - x) * Rabs (/ h0)) with 1.
-  rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2;
+  rewrite Rmult_1_r; unfold Rdiv; apply Rmult_lt_reg_l with 2;
     [ prove_sup0
       | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
-          [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r;
+          [ rewrite Rmult_1_l; pattern eps at 1; rewrite <- Rplus_0_r;
             rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
   rewrite Rabs_right.
@@ -2709,7 +2709,7 @@ Proof.
   apply Rle_ge; left; apply Rinv_0_lt_compat.
   elim r; intro.
   apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; assumption.
-  elim H5; symmetry  in |- *; apply Rplus_eq_reg_l with x; rewrite Rplus_0_r;
+  elim H5; symmetry ; apply Rplus_eq_reg_l with x; rewrite Rplus_0_r;
     assumption.
   apply Rle_lt_trans with
     (RiemannInt
@@ -2733,7 +2733,7 @@ Proof.
           (RiemannInt_P1
             (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))))));
       auto with real.
-  symmetry  in |- *; apply RiemannInt_P8.
+  symmetry ; apply RiemannInt_P8.
   apply Rle_lt_trans with
     (RiemannInt (RiemannInt_P14 (x + h0) x (eps / 2)) * Rabs (/ h0)).
   do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l.
@@ -2741,8 +2741,8 @@ Proof.
   apply RiemannInt_P19.
   auto with real.
   intros; replace (f x1 + -1 * fct_cte (f x) x1) with (f x1 - f x).
-  unfold fct_cte in |- *; case (Req_dec x x1); intro.
-  rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; left;
+  unfold fct_cte; case (Req_dec x x1); intro.
+  rewrite H9; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; left;
     assumption.
   elim H3; intros; left; apply H11.
   repeat split.
@@ -2752,22 +2752,22 @@ Proof.
     [ idtac | ring ].
   replace (x1 - x0 + - (x1 - x)) with (x - x0); [ idtac | ring ].
   apply Rle_lt_trans with (x + h0).
-  unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel.
+  unfold Rminus; apply Rplus_le_compat_l; apply Ropp_le_cancel.
   rewrite Ropp_involutive; apply Rle_trans with (Rabs h0).
   rewrite <- Rabs_Ropp; apply RRle_abs.
   apply Rle_trans with del;
-    [ left; assumption | unfold del in |- *; apply Rmin_l ].
+    [ left; assumption | unfold del; apply Rmin_l ].
   elim H8; intros; assumption.
   apply Rplus_lt_reg_r with x; rewrite Rplus_0_r;
     replace (x + (x1 - x)) with x1; [ elim H8; intros; assumption | ring ].
-  unfold fct_cte in |- *; ring.
+  unfold fct_cte; ring.
   rewrite RiemannInt_P15.
   rewrite Rmult_assoc; replace ((x - (x + h0)) * Rabs (/ h0)) with 1.
-  rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2;
+  rewrite Rmult_1_r; unfold Rdiv; apply Rmult_lt_reg_l with 2;
     [ prove_sup0
       | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
-          [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r;
+          [ rewrite Rmult_1_l; pattern eps at 1; rewrite <- Rplus_0_r;
             rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
   rewrite Rabs_left.
@@ -2784,14 +2784,14 @@ Proof.
       (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))))
     .
   ring.
-  unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring.
+  unfold Rdiv, Rminus; rewrite Rmult_plus_distr_r; ring.
   rewrite RiemannInt_P15; apply Rmult_eq_reg_l with h0;
-    [ unfold Rdiv in |- *; rewrite (Rmult_comm h0); repeat rewrite Rmult_assoc;
+    [ unfold Rdiv; rewrite (Rmult_comm h0); repeat rewrite Rmult_assoc;
       rewrite <- Rinv_l_sym; [ ring | assumption ]
       | assumption ].
   cut (a <= x + h0).
   cut (x + h0 <= b).
-  intros; unfold primitive in |- *.
+  intros; unfold primitive.
   case (Rle_dec a (x + h0)); case (Rle_dec (x + h0) b); case (Rle_dec a x);
   case (Rle_dec x b); intros; try (elim n; assumption || left; assumption).
   rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r0 r) H7 (FTC_P1 h C0 r2 r1)); ring.
@@ -2801,7 +2801,7 @@ Proof.
   apply RRle_abs.
   apply Rle_trans with del;
     [ left; assumption
-      | unfold del in |- *; apply Rle_trans with (Rmin (b - x) (x - a));
+      | unfold del; apply Rle_trans with (Rmin (b - x) (x - a));
         [ apply Rmin_r | apply Rmin_l ] ].
   apply Ropp_le_cancel; apply Rplus_le_reg_l with x;
     replace (x + - (x + h0)) with (- h0); [ idtac | ring ].
@@ -2809,7 +2809,7 @@ Proof.
     [ rewrite <- Rabs_Ropp; apply RRle_abs
       | apply Rle_trans with del;
         [ left; assumption
-          | unfold del in |- *; apply Rle_trans with (Rmin (b - x) (x - a));
+          | unfold del; apply Rle_trans with (Rmin (b - x) (x - a));
             apply Rmin_r ] ].
 Qed.
 
@@ -2826,14 +2826,14 @@ Proof.
     (f_b := fun x:R => f b * (x - b) + RiemannInt (FTC_P1 h C0 h (Rle_refl b)));
     rewrite H3.
   assert (H4 : derivable_pt_lim f_b b (f b)).
-  unfold f_b in |- *; pattern (f b) at 2 in |- *; replace (f b) with (f b + 0).
+  unfold f_b; pattern (f b) at 2; replace (f b) with (f b + 0).
   change
     (derivable_pt_lim
       ((fct_cte (f b) * (id - fct_cte b))%F +
         fct_cte (RiemannInt (FTC_P1 h C0 h (Rle_refl b)))) b (
-          f b + 0)) in |- *.
+          f b + 0)).
   apply derivable_pt_lim_plus.
-  pattern (f b) at 2 in |- *;
+  pattern (f b) at 2;
     replace (f b) with (0 * (id - fct_cte b)%F b + fct_cte (f b) b * 1).
   apply derivable_pt_lim_mult.
   apply derivable_pt_lim_const.
@@ -2841,26 +2841,26 @@ Proof.
   apply derivable_pt_lim_minus.
   apply derivable_pt_lim_id.
   apply derivable_pt_lim_const.
-  unfold fct_cte in |- *; ring.
+  unfold fct_cte; ring.
   apply derivable_pt_lim_const.
   ring.
-  unfold derivable_pt_lim in |- *; intros; elim (H4 _ H5); intros;
+  unfold derivable_pt_lim; intros; elim (H4 _ H5); intros;
     assert (H7 : continuity_pt f b).
   apply C0; split; [ left; assumption | right; reflexivity ].
   assert (H8 : 0 < eps / 2).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
-  elim (H7 _ H8); unfold D_x, no_cond, dist in |- *; simpl in |- *;
-    unfold R_dist in |- *; intros; set (del := Rmin x0 (Rmin x1 (b - a)));
+  elim (H7 _ H8); unfold D_x, no_cond, dist; simpl;
+    unfold R_dist; intros; set (del := Rmin x0 (Rmin x1 (b - a)));
       assert (H10 : 0 < del).
-  unfold del in |- *; unfold Rmin in |- *; case (Rle_dec x1 (b - a)); intros.
+  unfold del; unfold Rmin; case (Rle_dec x1 (b - a)); intros.
   case (Rle_dec x0 x1); intro;
     [ apply (cond_pos x0) | elim H9; intros; assumption ].
   case (Rle_dec x0 (b - a)); intro;
     [ apply (cond_pos x0) | apply Rlt_Rminus; assumption ].
   split with (mkposreal _ H10); intros; case (Rcase_abs h0); intro.
   assert (H14 : b + h0 < b).
-  pattern b at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
+  pattern b at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
     assumption.
   assert (H13 : Riemann_integrable f (b + h0) b).
   apply continuity_implies_RiemannInt.
@@ -2874,11 +2874,11 @@ Proof.
   apply Rle_trans with (Rabs h0).
   rewrite <- Rabs_Ropp; apply RRle_abs.
   left; assumption.
-  unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r.
+  unfold del; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r.
   replace (primitive h (FTC_P1 h C0) (b + h0) - primitive h (FTC_P1 h C0) b)
     with (- RiemannInt H13).
   replace (f b) with (- RiemannInt (RiemannInt_P14 (b + h0) b (f b)) / h0).
-  rewrite <- Rabs_Ropp; unfold Rminus in |- *; unfold Rdiv in |- *;
+  rewrite <- Rabs_Ropp; unfold Rminus; unfold Rdiv;
     rewrite Ropp_mult_distr_l_reverse; rewrite Ropp_plus_distr;
       repeat rewrite Ropp_involutive;
         replace
@@ -2887,7 +2887,7 @@ Proof.
         ((RiemannInt H13 - RiemannInt (RiemannInt_P14 (b + h0) b (f b))) / h0).
   replace (RiemannInt H13 - RiemannInt (RiemannInt_P14 (b + h0) b (f b))) with
   (RiemannInt (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b)))).
-  unfold Rdiv in |- *; rewrite Rabs_mult;
+  unfold Rdiv; rewrite Rabs_mult;
     apply Rle_lt_trans with
       (RiemannInt
         (RiemannInt_P16
@@ -2907,8 +2907,8 @@ Proof.
   apply RiemannInt_P19.
   left; assumption.
   intros; replace (f x2 + -1 * fct_cte (f b) x2) with (f x2 - f b).
-  unfold fct_cte in |- *; case (Req_dec b x2); intro.
-  rewrite H16; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  unfold fct_cte; case (Req_dec b x2); intro.
+  rewrite H16; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     left; assumption.
   elim H9; intros; left; apply H18.
   repeat split.
@@ -2919,22 +2919,22 @@ Proof.
   replace (x2 - x1 + x1) with x2; [ idtac | ring ].
   apply Rlt_le_trans with (b + h0).
   2: elim H15; intros; left; assumption.
-  unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_cancel;
+  unfold Rminus; apply Rplus_lt_compat_l; apply Ropp_lt_cancel;
     rewrite Ropp_involutive; apply Rle_lt_trans with (Rabs h0).
   rewrite <- Rabs_Ropp; apply RRle_abs.
   apply Rlt_le_trans with del;
     [ assumption
-      | unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a));
+      | unfold del; apply Rle_trans with (Rmin x1 (b - a));
         [ apply Rmin_r | apply Rmin_l ] ].
   apply Rle_ge; left; apply Rlt_Rminus; elim H15; intros; assumption.
-  unfold fct_cte in |- *; ring.
+  unfold fct_cte; ring.
   rewrite RiemannInt_P15.
   rewrite Rmult_assoc; replace ((b - (b + h0)) * Rabs (/ h0)) with 1.
-  rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2;
+  rewrite Rmult_1_r; unfold Rdiv; apply Rmult_lt_reg_l with 2;
     [ prove_sup0
       | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
-          [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r;
+          [ rewrite Rmult_1_l; pattern eps at 1; rewrite <- Rplus_0_r;
             rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
   rewrite Rabs_left.
@@ -2948,16 +2948,16 @@ Proof.
     (RiemannInt_P13 H13 (RiemannInt_P14 (b + h0) b (f b))
       (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b))))
     ; ring.
-  unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring.
+  unfold Rdiv, Rminus; rewrite Rmult_plus_distr_r; ring.
   rewrite RiemannInt_P15.
   rewrite <- Ropp_mult_distr_l_reverse; apply Rmult_eq_reg_l with h0;
-    [ repeat rewrite (Rmult_comm h0); unfold Rdiv in |- *;
+    [ repeat rewrite (Rmult_comm h0); unfold Rdiv;
       repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
         [ ring | assumption ]
       | assumption ].
   cut (a <= b + h0).
   cut (b + h0 <= b).
-  intros; unfold primitive in |- *; case (Rle_dec a (b + h0));
+  intros; unfold primitive; case (Rle_dec a (b + h0));
     case (Rle_dec (b + h0) b); case (Rle_dec a b); case (Rle_dec b b);
     intros; try (elim n; right; reflexivity) || (elim n; left; assumption).
   rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r3 r2) H13 (FTC_P1 h C0 r1 r0)); ring.
@@ -2970,26 +2970,26 @@ Proof.
   apply Rle_trans with (Rabs h0).
   rewrite <- Rabs_Ropp; apply RRle_abs.
   left; assumption.
-  unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r.
+  unfold del; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r.
   cut (primitive h (FTC_P1 h C0) b = f_b b).
   intro; cut (primitive h (FTC_P1 h C0) (b + h0) = f_b (b + h0)).
   intro; rewrite H13; rewrite H14; apply H6.
   assumption.
   apply Rlt_le_trans with del;
-    [ assumption | unfold del in |- *; apply Rmin_l ].
+    [ assumption | unfold del; apply Rmin_l ].
   assert (H14 : b < b + h0).
-  pattern b at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
+  pattern b at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
   assert (H14 := Rge_le _ _ r); elim H14; intro.
   assumption.
-  elim H11; symmetry  in |- *; assumption.
-  unfold primitive in |- *; case (Rle_dec a (b + h0));
+  elim H11; symmetry ; assumption.
+  unfold primitive; case (Rle_dec a (b + h0));
     case (Rle_dec (b + h0) b); intros;
       [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 H14))
-        | unfold f_b in |- *; reflexivity
+        | unfold f_b; reflexivity
         | elim n; left; apply Rlt_trans with b; assumption
         | elim n0; left; apply Rlt_trans with b; assumption ].
-  unfold f_b in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
-    rewrite Rmult_0_r; rewrite Rplus_0_l; unfold primitive in |- *;
+  unfold f_b; unfold Rminus; rewrite Rplus_opp_r;
+    rewrite Rmult_0_r; rewrite Rplus_0_l; unfold primitive;
       case (Rle_dec a b); case (Rle_dec b b); intros;
         [ apply RiemannInt_P5
           | elim n; right; reflexivity
@@ -2998,9 +2998,9 @@ Proof.
 (*****)
   set (f_a := fun x:R => f a * (x - a)); rewrite <- H2;
     assert (H3 : derivable_pt_lim f_a a (f a)).
-  unfold f_a in |- *;
+  unfold f_a;
     change (derivable_pt_lim (fct_cte (f a) * (id - fct_cte a)%F) a (f a))
-      in |- *; pattern (f a) at 2 in |- *;
+     ; pattern (f a) at 2;
         replace (f a) with (0 * (id - fct_cte a)%F a + fct_cte (f a) a * 1).
   apply derivable_pt_lim_mult.
   apply derivable_pt_lim_const.
@@ -3008,18 +3008,18 @@ Proof.
   apply derivable_pt_lim_minus.
   apply derivable_pt_lim_id.
   apply derivable_pt_lim_const.
-  unfold fct_cte in |- *; ring.
-  unfold derivable_pt_lim in |- *; intros; elim (H3 _ H4); intros.
+  unfold fct_cte; ring.
+  unfold derivable_pt_lim; intros; elim (H3 _ H4); intros.
   assert (H6 : continuity_pt f a).
   apply C0; split; [ right; reflexivity | left; assumption ].
   assert (H7 : 0 < eps / 2).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
-  elim (H6 _ H7); unfold D_x, no_cond, dist in |- *; simpl in |- *;
-    unfold R_dist in |- *; intros.
+  elim (H6 _ H7); unfold D_x, no_cond, dist; simpl;
+    unfold R_dist; intros.
   set (del := Rmin x0 (Rmin x1 (b - a))).
   assert (H9 : 0 < del).
-  unfold del in |- *; unfold Rmin in |- *.
+  unfold del; unfold Rmin.
   case (Rle_dec x1 (b - a)); intros.
   case (Rle_dec x0 x1); intro.
   apply (cond_pos x0).
@@ -3030,9 +3030,9 @@ Proof.
   split with (mkposreal _ H9).
   intros; case (Rcase_abs h0); intro.
   assert (H12 : a + h0 < a).
-  pattern a at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
+  pattern a at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
     assumption.
-  unfold primitive in |- *.
+  unfold primitive.
   case (Rle_dec a (a + h0)); case (Rle_dec (a + h0) b); case (Rle_dec a a);
   case (Rle_dec a b); intros;
     try (elim n; left; assumption) || (elim n; right; reflexivity).
@@ -3042,15 +3042,15 @@ Proof.
   replace (f a * (a + h0 - a)) with (f_a (a + h0)).
   apply H5; try assumption.
   apply Rlt_le_trans with del;
-    [ assumption | unfold del in |- *; apply Rmin_l ].
-  unfold f_a in |- *; ring.
-  unfold f_a in |- *; ring.
+    [ assumption | unfold del; apply Rmin_l ].
+  unfold f_a; ring.
+  unfold f_a; ring.
   elim n; left; apply Rlt_trans with a; assumption.
   assert (H12 : a < a + h0).
-  pattern a at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
+  pattern a at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
   assert (H12 := Rge_le _ _ r); elim H12; intro.
   assumption.
-  elim H10; symmetry  in |- *; assumption.
+  elim H10; symmetry ; assumption.
   assert (H13 : Riemann_integrable f a (a + h0)).
   apply continuity_implies_RiemannInt.
   left; assumption.
@@ -3062,7 +3062,7 @@ Proof.
   apply Ropp_le_cancel; rewrite Ropp_involutive; rewrite Ropp_minus_distr;
     apply Rle_trans with del.
   apply Rle_trans with (Rabs h0); [ apply RRle_abs | left; assumption ].
-  unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r.
+  unfold del; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r.
   replace (primitive h (FTC_P1 h C0) (a + h0) - primitive h (FTC_P1 h C0) a)
     with (RiemannInt H13).
   replace (f a) with (RiemannInt (RiemannInt_P14 a (a + h0) (f a)) / h0).
@@ -3071,7 +3071,7 @@ Proof.
     with ((RiemannInt H13 - RiemannInt (RiemannInt_P14 a (a + h0) (f a))) / h0).
   replace (RiemannInt H13 - RiemannInt (RiemannInt_P14 a (a + h0) (f a))) with
   (RiemannInt (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a)))).
-  unfold Rdiv in |- *; rewrite Rabs_mult;
+  unfold Rdiv; rewrite Rabs_mult;
     apply Rle_lt_trans with
       (RiemannInt
         (RiemannInt_P16
@@ -3091,8 +3091,8 @@ Proof.
   apply RiemannInt_P19.
   left; assumption.
   intros; replace (f x2 + -1 * fct_cte (f a) x2) with (f x2 - f a).
-  unfold fct_cte in |- *; case (Req_dec a x2); intro.
-  rewrite H15; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  unfold fct_cte; case (Req_dec a x2); intro.
+  rewrite H15; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     left; assumption.
   elim H8; intros; left; apply H17; repeat split.
   assumption.
@@ -3104,42 +3104,42 @@ Proof.
   apply RRle_abs.
   apply Rlt_le_trans with del;
     [ assumption
-      | unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a));
+      | unfold del; apply Rle_trans with (Rmin x1 (b - a));
         [ apply Rmin_r | apply Rmin_l ] ].
   apply Rle_ge; left; apply Rlt_Rminus; elim H14; intros; assumption.
-  unfold fct_cte in |- *; ring.
+  unfold fct_cte; ring.
   rewrite RiemannInt_P15.
   rewrite Rmult_assoc; replace ((a + h0 - a) * Rabs (/ h0)) with 1.
-  rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2;
+  rewrite Rmult_1_r; unfold Rdiv; apply Rmult_lt_reg_l with 2;
     [ prove_sup0
       | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
-          [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r;
+          [ rewrite Rmult_1_l; pattern eps at 1; rewrite <- Rplus_0_r;
             rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
   rewrite Rabs_right.
-  rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc;
+  rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc;
     rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite <- Rinv_r_sym;
       [ reflexivity | assumption ].
   apply Rle_ge; left; apply Rinv_0_lt_compat; assert (H14 := Rge_le _ _ r);
     elim H14; intro.
   assumption.
-  elim H10; symmetry  in |- *; assumption.
+  elim H10; symmetry ; assumption.
   rewrite
     (RiemannInt_P13 H13 (RiemannInt_P14 a (a + h0) (f a))
       (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a))))
     ; ring.
-  unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring.
+  unfold Rdiv, Rminus; rewrite Rmult_plus_distr_r; ring.
   rewrite RiemannInt_P15.
-  rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc;
-    rewrite Rplus_opp_r; rewrite Rplus_0_r; unfold Rdiv in |- *;
+  rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc;
+    rewrite Rplus_opp_r; rewrite Rplus_0_r; unfold Rdiv;
       rewrite Rmult_assoc; rewrite <- Rinv_r_sym; [ ring | assumption ].
   cut (a <= a + h0).
   cut (a + h0 <= b).
-  intros; unfold primitive in |- *; case (Rle_dec a (a + h0));
+  intros; unfold primitive; case (Rle_dec a (a + h0));
     case (Rle_dec (a + h0) b); case (Rle_dec a a); case (Rle_dec a b);
     intros; try (elim n; right; reflexivity) || (elim n; left; assumption).
-  rewrite RiemannInt_P9; unfold Rminus in |- *; rewrite Ropp_0;
+  rewrite RiemannInt_P9; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; apply RiemannInt_P5.
   elim n; assumption.
   elim n; assumption.
@@ -3148,15 +3148,15 @@ Proof.
     [ idtac | ring ].
   rewrite Rplus_comm; apply Rle_trans with del;
     [ apply Rle_trans with (Rabs h0); [ apply RRle_abs | left; assumption ]
-      | unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r ].
+      | unfold del; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r ].
 (*****)
   assert (H1 : x = a).
   rewrite <- H0 in H; elim H; intros; apply Rle_antisym; assumption.
   set (f_a := fun x:R => f a * (x - a)).
   assert (H2 : derivable_pt_lim f_a a (f a)).
-  unfold f_a in |- *;
+  unfold f_a;
     change (derivable_pt_lim (fct_cte (f a) * (id - fct_cte a)%F) a (f a))
-      in |- *; pattern (f a) at 2 in |- *;
+     ; pattern (f a) at 2;
         replace (f a) with (0 * (id - fct_cte a)%F a + fct_cte (f a) a * 1).
   apply derivable_pt_lim_mult.
   apply derivable_pt_lim_const.
@@ -3164,18 +3164,18 @@ Proof.
   apply derivable_pt_lim_minus.
   apply derivable_pt_lim_id.
   apply derivable_pt_lim_const.
-  unfold fct_cte in |- *; ring.
+  unfold fct_cte; ring.
   set
     (f_b := fun x:R => f b * (x - b) + RiemannInt (FTC_P1 h C0 h (Rle_refl b))).
   assert (H3 : derivable_pt_lim f_b b (f b)).
-  unfold f_b in |- *; pattern (f b) at 2 in |- *; replace (f b) with (f b + 0).
+  unfold f_b; pattern (f b) at 2; replace (f b) with (f b + 0).
   change
     (derivable_pt_lim
       ((fct_cte (f b) * (id - fct_cte b))%F +
         fct_cte (RiemannInt (FTC_P1 h C0 h (Rle_refl b)))) b (
-          f b + 0)) in |- *.
+          f b + 0)).
   apply derivable_pt_lim_plus.
-  pattern (f b) at 2 in |- *;
+  pattern (f b) at 2;
     replace (f b) with (0 * (id - fct_cte b)%F b + fct_cte (f b) b * 1).
   apply derivable_pt_lim_mult.
   apply derivable_pt_lim_const.
@@ -3183,20 +3183,20 @@ Proof.
   apply derivable_pt_lim_minus.
   apply derivable_pt_lim_id.
   apply derivable_pt_lim_const.
-  unfold fct_cte in |- *; ring.
+  unfold fct_cte; ring.
   apply derivable_pt_lim_const.
   ring.
-  unfold derivable_pt_lim in |- *; intros; elim (H2 _ H4); intros;
+  unfold derivable_pt_lim; intros; elim (H2 _ H4); intros;
     elim (H3 _ H4); intros; set (del := Rmin x0 x1).
   assert (H7 : 0 < del).
-  unfold del in |- *; unfold Rmin in |- *; case (Rle_dec x0 x1); intro.
+  unfold del; unfold Rmin; case (Rle_dec x0 x1); intro.
   apply (cond_pos x0).
   apply (cond_pos x1).
   split with (mkposreal _ H7); intros; case (Rcase_abs h0); intro.
   assert (H10 : a + h0 < a).
-  pattern a at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
+  pattern a at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
     assumption.
-  rewrite H1; unfold primitive in |- *; case (Rle_dec a (a + h0));
+  rewrite H1; unfold primitive; case (Rle_dec a (a + h0));
     case (Rle_dec (a + h0) b); case (Rle_dec a a); case (Rle_dec a b);
     intros; try (elim n; right; assumption || reflexivity).
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H10)).
@@ -3205,27 +3205,27 @@ Proof.
   replace (f a * (a + h0 - a)) with (f_a (a + h0)).
   apply H5; try assumption.
   apply Rlt_le_trans with del; try assumption.
-  unfold del in |- *; apply Rmin_l.
-  unfold f_a in |- *; ring.
-  unfold f_a in |- *; ring.
+  unfold del; apply Rmin_l.
+  unfold f_a; ring.
+  unfold f_a; ring.
   elim n; rewrite <- H0; left; assumption.
   assert (H10 : a < a + h0).
-  pattern a at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
+  pattern a at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
   assert (H10 := Rge_le _ _ r); elim H10; intro.
   assumption.
-  elim H8; symmetry  in |- *; assumption.
-  rewrite H0 in H1; rewrite H1; unfold primitive in |- *;
+  elim H8; symmetry ; assumption.
+  rewrite H0 in H1; rewrite H1; unfold primitive;
     case (Rle_dec a (b + h0)); case (Rle_dec (b + h0) b);
     case (Rle_dec a b); case (Rle_dec b b); intros;
       try (elim n; right; assumption || reflexivity).
   rewrite H0 in H10; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r2 H10)).
   repeat rewrite RiemannInt_P9.
   replace (RiemannInt (FTC_P1 h C0 r1 r0)) with (f_b b).
-  fold (f_b (b + h0)) in |- *.
+  fold (f_b (b + h0)).
   apply H6; try assumption.
   apply Rlt_le_trans with del; try assumption.
-  unfold del in |- *; apply Rmin_r.
-  unfold f_b in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
+  unfold del; apply Rmin_r.
+  unfold f_b; unfold Rminus; rewrite Rplus_opp_r;
     rewrite Rmult_0_r; rewrite Rplus_0_l; apply RiemannInt_P5.
   elim n; rewrite <- H0; left; assumption.
   elim n0; rewrite <- H0; left; assumption.
@@ -3236,11 +3236,11 @@ Lemma RiemannInt_P29 :
     (C0:forall x:R, a <= x <= b -> continuity_pt f x),
     antiderivative f (primitive h (FTC_P1 h C0)) a b.
 Proof.
-  intro f; intros; unfold antiderivative in |- *; split; try assumption; intros;
+  intro f; intros; unfold antiderivative; split; try assumption; intros;
     assert (H0 := RiemannInt_P28 h C0 H);
       assert (H1 : derivable_pt (primitive h (FTC_P1 h C0)) x);
-        [ unfold derivable_pt in |- *; split with (f x); apply H0
-          | split with H1; symmetry  in |- *; apply derive_pt_eq_0; apply H0 ].
+        [ unfold derivable_pt; split with (f x); apply H0
+          | split with H1; symmetry ; apply derive_pt_eq_0; apply H0 ].
 Qed.
 
 Lemma RiemannInt_P30 :
@@ -3259,7 +3259,7 @@ Lemma RiemannInt_P31 :
   forall (f:C1_fun) (a b:R),
     a <= b -> antiderivative (derive f (diff0 f)) f a b.
 Proof.
-  intro f; intros; unfold antiderivative in |- *; split; try assumption; intros;
+  intro f; intros; unfold antiderivative; split; try assumption; intros;
     split with (diff0 f x); reflexivity.
 Qed.
 
diff --git a/theories/Reals/RiemannInt_SF.v b/theories/Reals/RiemannInt_SF.v
index d16e7f2c..d523a1f4 100644
--- a/theories/Reals/RiemannInt_SF.v
+++ b/theories/Reals/RiemannInt_SF.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -8,9 +8,9 @@
 
 Require Import Rbase.
 Require Import Rfunctions.
-Require Import Ranalysis.
+Require Import Ranalysis_reg.
 Require Import Classical_Prop.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Set Implicit Arguments.
 
@@ -21,7 +21,7 @@ Set Implicit Arguments.
 Definition Nbound (I:nat -> Prop) : Prop :=
   exists n : nat, (forall i:nat, I i -> (i <= n)%nat).
 
-Lemma IZN_var : forall z:Z, (0 <= z)%Z -> {n : nat | z = Z_of_nat n}.
+Lemma IZN_var : forall z:Z, (0 <= z)%Z -> {n : nat | z = Z.of_nat n}.
 Proof.
   intros; apply Z_of_nat_complete_inf; assumption.
 Qed.
@@ -33,19 +33,19 @@ Lemma Nzorn :
 Proof.
   intros I H H0; set (E := fun x:R =>  exists i : nat, I i /\ INR i = x);
     assert (H1 : bound E).
-  unfold Nbound in H0; elim H0; intros N H1; unfold bound in |- *;
-    exists (INR N); unfold is_upper_bound in |- *; intros;
+  unfold Nbound in H0; elim H0; intros N H1; unfold bound;
+    exists (INR N); unfold is_upper_bound; intros;
       unfold E in H2; elim H2; intros; elim H3; intros;
         rewrite <- H5; apply le_INR; apply H1; assumption.
   assert (H2 :  exists x : R, E x).
-  elim H; intros; exists (INR x); unfold E in |- *; exists x; split;
+  elim H; intros; exists (INR x); unfold E; exists x; split;
     [ assumption | reflexivity ].
   assert (H3 := completeness E H1 H2); elim H3; intros; unfold is_lub in p;
     elim p; clear p; intros; unfold is_upper_bound in H4, H5;
       assert (H6 : 0 <= x).
   elim H2; intros; unfold E in H6; elim H6; intros; elim H7; intros;
     apply Rle_trans with x0;
-      [ rewrite <- H9; change (INR 0 <= INR x1) in |- *; apply le_INR;
+      [ rewrite <- H9; change (INR 0 <= INR x1); apply le_INR;
         apply le_O_n
         | apply H4; assumption ].
   assert (H7 := archimed x); elim H7; clear H7; intros;
@@ -88,7 +88,7 @@ Proof.
     [ idtac | reflexivity ]; rewrite <- minus_INR.
   replace (x0 - 1)%nat with (pred x0);
   [ reflexivity
-    | case x0; [ reflexivity | intro; simpl in |- *; apply minus_n_O ] ].
+    | case x0; [ reflexivity | intro; simpl; apply minus_n_O ] ].
   induction  x0 as [| x0 Hrecx0];
     [ rewrite p in H7; rewrite <- INR_IZR_INZ in H7; simpl in H7;
       elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 H7))
@@ -99,10 +99,10 @@ Proof.
       assert (H16 : INR x0 = INR x1 + 1).
   rewrite H15; ring.
   rewrite <- S_INR in H16; assert (H17 := INR_eq _ _ H16); rewrite H17;
-    simpl in |- *; split.
+    simpl; split.
   assumption.
   intros; apply INR_le; rewrite H15; rewrite <- H15; elim H12; intros;
-    rewrite H20; apply H4; unfold E in |- *; exists i;
+    rewrite H20; apply H4; unfold E; exists i;
       split; [ assumption | reflexivity ].
 Qed.
 
@@ -173,7 +173,7 @@ Lemma StepFun_P1 :
   forall (a b:R) (f:StepFun a b),
     adapted_couple f a b (subdivision f) (subdivision_val f).
 Proof.
-  intros a b f; unfold subdivision_val in |- *; case (projT2 (pre f)); intros;
+  intros a b f; unfold subdivision_val; case (projT2 (pre f)); intros;
     apply a0.
 Qed.
 
@@ -181,13 +181,13 @@ Lemma StepFun_P2 :
   forall (a b:R) (f:R -> R) (l lf:Rlist),
     adapted_couple f a b l lf -> adapted_couple f b a l lf.
 Proof.
-  unfold adapted_couple in |- *; intros; decompose [and] H; clear H;
+  unfold adapted_couple; intros; decompose [and] H; clear H;
     repeat split; try assumption.
-  rewrite H2; unfold Rmin in |- *; case (Rle_dec a b); intro;
+  rewrite H2; unfold Rmin; case (Rle_dec a b); intro;
     case (Rle_dec b a); intro; try reflexivity.
   apply Rle_antisym; assumption.
   apply Rle_antisym; auto with real.
-  rewrite H1; unfold Rmax in |- *; case (Rle_dec a b); intro;
+  rewrite H1; unfold Rmax; case (Rle_dec a b); intro;
     case (Rle_dec b a); intro; try reflexivity.
   apply Rle_antisym; assumption.
   apply Rle_antisym; auto with real.
@@ -198,23 +198,23 @@ Lemma StepFun_P3 :
     a <= b ->
     adapted_couple (fct_cte c) a b (cons a (cons b nil)) (cons c nil).
 Proof.
-  intros; unfold adapted_couple in |- *; repeat split.
-  unfold ordered_Rlist in |- *; intros; simpl in H0; inversion H0;
-    [ simpl in |- *; assumption | elim (le_Sn_O _ H2) ].
-  simpl in |- *; unfold Rmin in |- *; case (Rle_dec a b); intro;
+  intros; unfold adapted_couple; repeat split.
+  unfold ordered_Rlist; intros; simpl in H0; inversion H0;
+    [ simpl; assumption | elim (le_Sn_O _ H2) ].
+  simpl; unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
-  simpl in |- *; unfold Rmax in |- *; case (Rle_dec a b); intro;
+  simpl; unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
-  unfold constant_D_eq, open_interval in |- *; intros; simpl in H0;
+  unfold constant_D_eq, open_interval; intros; simpl in H0;
     inversion H0; [ reflexivity | elim (le_Sn_O _ H3) ].
 Qed.
 
 Lemma StepFun_P4 : forall a b c:R, IsStepFun (fct_cte c) a b.
 Proof.
-  intros; unfold IsStepFun in |- *; case (Rle_dec a b); intro.
-  apply existT with (cons a (cons b nil)); unfold is_subdivision in |- *;
+  intros; unfold IsStepFun; case (Rle_dec a b); intro.
+  apply existT with (cons a (cons b nil)); unfold is_subdivision;
     apply existT with (cons c nil); apply (StepFun_P3 c r).
-  apply existT with (cons b (cons a nil)); unfold is_subdivision in |- *;
+  apply existT with (cons b (cons a nil)); unfold is_subdivision;
     apply existT with (cons c nil); apply StepFun_P2;
       apply StepFun_P3; auto with real.
 Qed.
@@ -232,7 +232,7 @@ Qed.
 Lemma StepFun_P6 :
   forall (f:R -> R) (a b:R), IsStepFun f a b -> IsStepFun f b a.
 Proof.
-  unfold IsStepFun in |- *; intros; elim X; intros; apply existT with x;
+  unfold IsStepFun; intros; elim X; intros; apply existT with x;
     apply StepFun_P5; assumption.
 Qed.
 
@@ -242,26 +242,26 @@ Lemma StepFun_P7 :
     adapted_couple f a b (cons r1 (cons r2 l)) (cons r3 lf) ->
     adapted_couple f r2 b (cons r2 l) lf.
 Proof.
-  unfold adapted_couple in |- *; intros; decompose [and] H0; clear H0;
+  unfold adapted_couple; intros; decompose [and] H0; clear H0;
     assert (H5 : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   assert (H7 : r2 <= b).
   rewrite H5 in H2; rewrite <- H2; apply RList_P7;
-    [ assumption | simpl in |- *; right; left; reflexivity ].
+    [ assumption | simpl; right; left; reflexivity ].
   repeat split.
   apply RList_P4 with r1; assumption.
-  rewrite H5 in H2; unfold Rmin in |- *; case (Rle_dec r2 b); intro;
+  rewrite H5 in H2; unfold Rmin; case (Rle_dec r2 b); intro;
     [ reflexivity | elim n; assumption ].
-  unfold Rmax in |- *; case (Rle_dec r2 b); intro;
+  unfold Rmax; case (Rle_dec r2 b); intro;
     [ rewrite H5 in H2; rewrite <- H2; reflexivity | elim n; assumption ].
-  simpl in H4; simpl in |- *; apply INR_eq; apply Rplus_eq_reg_l with 1;
+  simpl in H4; simpl; apply INR_eq; apply Rplus_eq_reg_l with 1;
     do 2 rewrite (Rplus_comm 1); do 2 rewrite <- S_INR;
       rewrite H4; reflexivity.
-  intros; unfold constant_D_eq, open_interval in |- *; intros;
+  intros; unfold constant_D_eq, open_interval; intros;
     unfold constant_D_eq, open_interval in H6;
       assert (H9 : (S i < pred (Rlength (cons r1 (cons r2 l))))%nat).
-  simpl in |- *; simpl in H0; apply lt_n_S; assumption.
+  simpl; simpl in H0; apply lt_n_S; assumption.
   assert (H10 := H6 _ H9); apply H10; assumption.
 Qed.
 
@@ -278,19 +278,19 @@ Proof.
     discriminate.
   intros; induction  lf1 as [| r3 lf1 Hreclf1].
   reflexivity.
-  simpl in |- *; cut (r = r1).
+  simpl; cut (r = r1).
   intro; rewrite H3; rewrite (H0 lf1 r b).
   ring.
   rewrite H3; apply StepFun_P7 with a r r3; [ right; assumption | assumption ].
   clear H H0 Hreclf1 r0; unfold adapted_couple in H1; decompose [and] H1;
-    intros; simpl in H4; rewrite H4; unfold Rmin in |- *;
+    intros; simpl in H4; rewrite H4; unfold Rmin;
       case (Rle_dec a b); intro; [ assumption | reflexivity ].
   unfold adapted_couple in H1; decompose [and] H1; intros; apply Rle_antisym.
-  apply (H3 0%nat); simpl in |- *; apply lt_O_Sn.
+  apply (H3 0%nat); simpl; apply lt_O_Sn.
   simpl in H5; rewrite H2 in H5; rewrite H5; replace (Rmin b b) with (Rmax a b);
     [ rewrite <- H4; apply RList_P7;
-      [ assumption | simpl in |- *; right; left; reflexivity ]
-      | unfold Rmin, Rmax in |- *; case (Rle_dec b b); case (Rle_dec a b); intros;
+      [ assumption | simpl; right; left; reflexivity ]
+      | unfold Rmin, Rmax; case (Rle_dec b b); case (Rle_dec a b); intros;
         try assumption || reflexivity ].
 Qed.
 
@@ -303,10 +303,10 @@ Proof.
       [ simpl in H4; discriminate
         | induction  l as [| r0 l Hrecl0];
           [ simpl in H3; simpl in H2; generalize H3; generalize H2;
-            unfold Rmin, Rmax in |- *; case (Rle_dec a b);
+            unfold Rmin, Rmax; case (Rle_dec a b);
               intros; elim H0; rewrite <- H5; rewrite <- H7;
                 reflexivity
-            | simpl in |- *; do 2 apply le_n_S; apply le_O_n ] ].
+            | simpl; do 2 apply le_n_S; apply le_O_n ] ].
 Qed.
 
 Lemma StepFun_P10 :
@@ -320,12 +320,12 @@ Proof.
   intros; unfold adapted_couple in H0; decompose [and] H0; simpl in H4;
     discriminate.
   intros; case (Req_dec a b); intro.
-  exists (cons a nil); exists nil; unfold adapted_couple_opt in |- *;
-    unfold adapted_couple in |- *; unfold ordered_Rlist in |- *;
+  exists (cons a nil); exists nil; unfold adapted_couple_opt;
+    unfold adapted_couple; unfold ordered_Rlist;
       repeat split; try (intros; simpl in H3; elim (lt_n_O _ H3)).
-  simpl in |- *; rewrite <- H2; unfold Rmin in |- *; case (Rle_dec a a); intro;
+  simpl; rewrite <- H2; unfold Rmin; case (Rle_dec a a); intro;
     reflexivity.
-  simpl in |- *; rewrite <- H2; unfold Rmax in |- *; case (Rle_dec a a); intro;
+  simpl; rewrite <- H2; unfold Rmax; case (Rle_dec a a); intro;
     reflexivity.
   elim (RList_P20 _ (StepFun_P9 H1 H2)); intros t1 [t2 [t3 H3]];
     induction  lf as [| r1 lf Hreclf].
@@ -340,32 +340,32 @@ Proof.
   apply H6.
   rewrite <- Hyp_eq; rewrite H3 in H1; unfold adapted_couple in H1;
     decompose [and] H1; clear H1; simpl in H9; rewrite H9;
-      unfold Rmin in |- *; case (Rle_dec a b); intro;
+      unfold Rmin; case (Rle_dec a b); intro;
         [ reflexivity | elim n; assumption ].
   elim H6; clear H6; intros l' [lf' H6]; case (Req_dec t2 b); intro.
   exists (cons a (cons b nil)); exists (cons r1 nil);
-    unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *;
+    unfold adapted_couple_opt; unfold adapted_couple;
       repeat split.
-  unfold ordered_Rlist in |- *; intros; simpl in H8; inversion H8;
-    [ simpl in |- *; assumption | elim (le_Sn_O _ H10) ].
-  simpl in |- *; unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold ordered_Rlist; intros; simpl in H8; inversion H8;
+    [ simpl; assumption | elim (le_Sn_O _ H10) ].
+  simpl; unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
-  simpl in |- *; unfold Rmax in |- *; case (Rle_dec a b); intro;
+  simpl; unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   intros; simpl in H8; inversion H8.
-  unfold constant_D_eq, open_interval in |- *; intros; simpl in |- *;
+  unfold constant_D_eq, open_interval; intros; simpl;
     simpl in H9; rewrite H3 in H1; unfold adapted_couple in H1;
       decompose [and] H1; apply (H16 0%nat).
-  simpl in |- *; apply lt_O_Sn.
-  unfold open_interval in |- *; simpl in |- *; rewrite H7; simpl in H13;
-    rewrite H13; unfold Rmin in |- *; case (Rle_dec a b);
+  simpl; apply lt_O_Sn.
+  unfold open_interval; simpl; rewrite H7; simpl in H13;
+    rewrite H13; unfold Rmin; case (Rle_dec a b);
       intro; [ assumption | elim n; assumption ].
   elim (le_Sn_O _ H10).
   intros; simpl in H8; elim (lt_n_O _ H8).
   intros; simpl in H8; inversion H8;
-    [ simpl in |- *; assumption | elim (le_Sn_O _ H10) ].
+    [ simpl; assumption | elim (le_Sn_O _ H10) ].
   assert (Hyp_min : Rmin t2 b = t2).
-  unfold Rmin in |- *; case (Rle_dec t2 b); intro;
+  unfold Rmin; case (Rle_dec t2 b); intro;
     [ reflexivity | elim n; assumption ].
   unfold adapted_couple in H6; elim H6; clear H6; intros;
     elim (RList_P20 _ (StepFun_P9 H6 H7)); intros s1 [s2 [s3 H9]];
@@ -377,141 +377,141 @@ Proof.
   exists (cons t1 (cons s2 s3)); exists (cons r1 lf'); rewrite H3 in H1;
     rewrite H9 in H6; unfold adapted_couple in H6, H1;
       decompose [and] H1; decompose [and] H6; clear H1 H6;
-        unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *;
+        unfold adapted_couple_opt; unfold adapted_couple;
           repeat split.
-  unfold ordered_Rlist in |- *; intros; simpl in H1;
+  unfold ordered_Rlist; intros; simpl in H1;
     induction  i as [| i Hreci].
-  simpl in |- *; apply Rle_trans with s1.
+  simpl; apply Rle_trans with s1.
   replace s1 with t2.
   apply (H12 0%nat).
-  simpl in |- *; apply lt_O_Sn.
-  simpl in H19; rewrite H19; symmetry  in |- *; apply Hyp_min.
-  apply (H16 0%nat); simpl in |- *; apply lt_O_Sn.
-  change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i)) in |- *;
-    apply (H16 (S i)); simpl in |- *; assumption.
-  simpl in |- *; simpl in H14; rewrite H14; reflexivity.
-  simpl in |- *; simpl in H18; rewrite H18; unfold Rmax in |- *;
+  simpl; apply lt_O_Sn.
+  simpl in H19; rewrite H19; symmetry ; apply Hyp_min.
+  apply (H16 0%nat); simpl; apply lt_O_Sn.
+  change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i));
+    apply (H16 (S i)); simpl; assumption.
+  simpl; simpl in H14; rewrite H14; reflexivity.
+  simpl; simpl in H18; rewrite H18; unfold Rmax;
     case (Rle_dec a b); case (Rle_dec t2 b); intros; reflexivity || elim n;
       assumption.
-  simpl in |- *; simpl in H20; apply H20.
-  intros; simpl in H1; unfold constant_D_eq, open_interval in |- *; intros;
+  simpl; simpl in H20; apply H20.
+  intros; simpl in H1; unfold constant_D_eq, open_interval; intros;
     induction  i as [| i Hreci].
-  simpl in |- *; simpl in H6; case (total_order_T x t2); intro.
+  simpl; simpl in H6; case (total_order_T x t2); intro.
   elim s; intro.
   apply (H17 0%nat);
-    [ simpl in |- *; apply lt_O_Sn
-      | unfold open_interval in |- *; simpl in |- *; elim H6; intros; split;
+    [ simpl; apply lt_O_Sn
+      | unfold open_interval; simpl; elim H6; intros; split;
         assumption ].
   rewrite b0; assumption.
   rewrite H10; apply (H22 0%nat);
-    [ simpl in |- *; apply lt_O_Sn
-      | unfold open_interval in |- *; simpl in |- *; replace s1 with t2;
+    [ simpl; apply lt_O_Sn
+      | unfold open_interval; simpl; replace s1 with t2;
         [ elim H6; intros; split; assumption
           | simpl in H19; rewrite H19; rewrite Hyp_min; reflexivity ] ].
-  simpl in |- *; simpl in H6; apply (H22 (S i));
-    [ simpl in |- *; assumption
-      | unfold open_interval in |- *; simpl in |- *; apply H6 ].
+  simpl; simpl in H6; apply (H22 (S i));
+    [ simpl; assumption
+      | unfold open_interval; simpl; apply H6 ].
   intros; simpl in H1; rewrite H10;
     change
       (pos_Rl (cons r2 lf') i <> pos_Rl (cons r2 lf') (S i) \/
         f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <> pos_Rl (cons r2 lf') i)
-      in |- *; rewrite <- H9; elim H8; intros; apply H6;
-        simpl in |- *; apply H1.
+     ; rewrite <- H9; elim H8; intros; apply H6;
+        simpl; apply H1.
   intros; induction  i as [| i Hreci].
-  simpl in |- *; red in |- *; intro; elim Hyp_eq; apply Rle_antisym.
-  apply (H12 0%nat); simpl in |- *; apply lt_O_Sn.
+  simpl; red; intro; elim Hyp_eq; apply Rle_antisym.
+  apply (H12 0%nat); simpl; apply lt_O_Sn.
   rewrite <- Hyp_min; rewrite H6; simpl in H19; rewrite <- H19;
-    apply (H16 0%nat); simpl in |- *; apply lt_O_Sn.
-  elim H8; intros; rewrite H9 in H21; apply (H21 (S i)); simpl in |- *;
+    apply (H16 0%nat); simpl; apply lt_O_Sn.
+  elim H8; intros; rewrite H9 in H21; apply (H21 (S i)); simpl;
     simpl in H1; apply H1.
   exists (cons t1 l'); exists (cons r1 (cons r2 lf')); rewrite H9 in H6;
     rewrite H3 in H1; unfold adapted_couple in H1, H6;
       decompose [and] H6; decompose [and] H1; clear H6 H1;
-        unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *;
+        unfold adapted_couple_opt; unfold adapted_couple;
           repeat split.
-  rewrite H9; unfold ordered_Rlist in |- *; intros; simpl in H1;
+  rewrite H9; unfold ordered_Rlist; intros; simpl in H1;
     induction  i as [| i Hreci].
-  simpl in |- *; replace s1 with t2.
-  apply (H16 0%nat); simpl in |- *; apply lt_O_Sn.
+  simpl; replace s1 with t2.
+  apply (H16 0%nat); simpl; apply lt_O_Sn.
   simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity.
   change
     (pos_Rl (cons s1 (cons s2 s3)) i <= pos_Rl (cons s1 (cons s2 s3)) (S i))
-    in |- *; apply (H12 i); simpl in |- *; apply lt_S_n;
+   ; apply (H12 i); simpl; apply lt_S_n;
       assumption.
-  simpl in |- *; simpl in H19; apply H19.
-  rewrite H9; simpl in |- *; simpl in H13; rewrite H13; unfold Rmax in |- *;
+  simpl; simpl in H19; apply H19.
+  rewrite H9; simpl; simpl in H13; rewrite H13; unfold Rmax;
     case (Rle_dec t2 b); case (Rle_dec a b); intros; reflexivity || elim n;
       assumption.
-  rewrite H9; simpl in |- *; simpl in H15; rewrite H15; reflexivity.
-  intros; simpl in H1; unfold constant_D_eq, open_interval in |- *; intros;
+  rewrite H9; simpl; simpl in H15; rewrite H15; reflexivity.
+  intros; simpl in H1; unfold constant_D_eq, open_interval; intros;
     induction  i as [| i Hreci].
-  simpl in |- *; rewrite H9 in H6; simpl in H6; apply (H22 0%nat).
-  simpl in |- *; apply lt_O_Sn.
-  unfold open_interval in |- *; simpl in |- *.
+  simpl; rewrite H9 in H6; simpl in H6; apply (H22 0%nat).
+  simpl; apply lt_O_Sn.
+  unfold open_interval; simpl.
   replace t2 with s1.
   assumption.
   simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity.
-  change (f x = pos_Rl (cons r2 lf') i) in |- *; clear Hreci; apply (H17 i).
-  simpl in |- *; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1.
-  rewrite H9 in H6; unfold open_interval in |- *; apply H6.
+  change (f x = pos_Rl (cons r2 lf') i); clear Hreci; apply (H17 i).
+  simpl; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1.
+  rewrite H9 in H6; unfold open_interval; apply H6.
   intros; simpl in H1; induction  i as [| i Hreci].
-  simpl in |- *; rewrite H9; right; simpl in |- *; replace s1 with t2.
+  simpl; rewrite H9; right; simpl; replace s1 with t2.
   assumption.
   simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity.
   elim H8; intros; apply (H6 i).
-  simpl in |- *; apply lt_S_n; apply H1.
+  simpl; apply lt_S_n; apply H1.
   intros; rewrite H9; induction  i as [| i Hreci].
-  simpl in |- *; red in |- *; intro; elim Hyp_eq; apply Rle_antisym.
-  apply (H16 0%nat); simpl in |- *; apply lt_O_Sn.
+  simpl; red; intro; elim Hyp_eq; apply Rle_antisym.
+  apply (H16 0%nat); simpl; apply lt_O_Sn.
   rewrite <- Hyp_min; rewrite H6; simpl in H14; rewrite <- H14; right;
     reflexivity.
   elim H8; intros; rewrite <- H9; apply (H21 i); rewrite H9; rewrite H9 in H1;
-    simpl in |- *; simpl in H1; apply lt_S_n; apply H1.
+    simpl; simpl in H1; apply lt_S_n; apply H1.
   exists (cons t1 l'); exists (cons r1 (cons r2 lf')); rewrite H9 in H6;
     rewrite H3 in H1; unfold adapted_couple in H1, H6;
       decompose [and] H6; decompose [and] H1; clear H6 H1;
-        unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *;
+        unfold adapted_couple_opt; unfold adapted_couple;
           repeat split.
-  rewrite H9; unfold ordered_Rlist in |- *; intros; simpl in H1;
+  rewrite H9; unfold ordered_Rlist; intros; simpl in H1;
     induction  i as [| i Hreci].
-  simpl in |- *; replace s1 with t2.
-  apply (H15 0%nat); simpl in |- *; apply lt_O_Sn.
+  simpl; replace s1 with t2.
+  apply (H15 0%nat); simpl; apply lt_O_Sn.
   simpl in H13; rewrite H13; rewrite Hyp_min; reflexivity.
   change
     (pos_Rl (cons s1 (cons s2 s3)) i <= pos_Rl (cons s1 (cons s2 s3)) (S i))
-    in |- *; apply (H11 i); simpl in |- *; apply lt_S_n;
+   ; apply (H11 i); simpl; apply lt_S_n;
       assumption.
-  simpl in |- *; simpl in H18; apply H18.
-  rewrite H9; simpl in |- *; simpl in H12; rewrite H12; unfold Rmax in |- *;
+  simpl; simpl in H18; apply H18.
+  rewrite H9; simpl; simpl in H12; rewrite H12; unfold Rmax;
     case (Rle_dec t2 b); case (Rle_dec a b); intros; reflexivity || elim n;
       assumption.
-  rewrite H9; simpl in |- *; simpl in H14; rewrite H14; reflexivity.
-  intros; simpl in H1; unfold constant_D_eq, open_interval in |- *; intros;
+  rewrite H9; simpl; simpl in H14; rewrite H14; reflexivity.
+  intros; simpl in H1; unfold constant_D_eq, open_interval; intros;
     induction  i as [| i Hreci].
-  simpl in |- *; rewrite H9 in H6; simpl in H6; apply (H21 0%nat).
-  simpl in |- *; apply lt_O_Sn.
-  unfold open_interval in |- *; simpl in |- *; replace t2 with s1.
+  simpl; rewrite H9 in H6; simpl in H6; apply (H21 0%nat).
+  simpl; apply lt_O_Sn.
+  unfold open_interval; simpl; replace t2 with s1.
   assumption.
   simpl in H13; rewrite H13; rewrite Hyp_min; reflexivity.
-  change (f x = pos_Rl (cons r2 lf') i) in |- *; clear Hreci; apply (H16 i).
-  simpl in |- *; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1.
-  rewrite H9 in H6; unfold open_interval in |- *; apply H6.
+  change (f x = pos_Rl (cons r2 lf') i); clear Hreci; apply (H16 i).
+  simpl; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1.
+  rewrite H9 in H6; unfold open_interval; apply H6.
   intros; simpl in H1; induction  i as [| i Hreci].
-  simpl in |- *; left; assumption.
+  simpl; left; assumption.
   elim H8; intros; apply (H6 i).
-  simpl in |- *; apply lt_S_n; apply H1.
+  simpl; apply lt_S_n; apply H1.
   intros; rewrite H9; induction  i as [| i Hreci].
-  simpl in |- *; red in |- *; intro; elim Hyp_eq; apply Rle_antisym.
-  apply (H15 0%nat); simpl in |- *; apply lt_O_Sn.
+  simpl; red; intro; elim Hyp_eq; apply Rle_antisym.
+  apply (H15 0%nat); simpl; apply lt_O_Sn.
   rewrite <- Hyp_min; rewrite H6; simpl in H13; rewrite <- H13; right;
     reflexivity.
   elim H8; intros; rewrite <- H9; apply (H20 i); rewrite H9; rewrite H9 in H1;
-    simpl in |- *; simpl in H1; apply lt_S_n; apply H1.
+    simpl; simpl in H1; apply lt_S_n; apply H1.
   rewrite H3 in H1; clear H4; unfold adapted_couple in H1; decompose [and] H1;
     clear H1; clear H H7 H9; cut (Rmax a b = b);
       [ intro; rewrite H in H5; rewrite <- H5; apply RList_P7;
-        [ assumption | simpl in |- *; right; left; reflexivity ]
-        | unfold Rmax in |- *; case (Rle_dec a b); intro;
+        [ assumption | simpl; right; left; reflexivity ]
+        | unfold Rmax; case (Rle_dec a b); intro;
           [ reflexivity | elim n; assumption ] ].
 Qed.
 
@@ -534,7 +534,7 @@ Proof.
   simpl in H9; rewrite H9 in H16; cut (r1 <= Rmax a b).
   intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H17 H16)).
   rewrite <- H4; apply RList_P7;
-    [ assumption | simpl in |- *; right; left; reflexivity ].
+    [ assumption | simpl; right; left; reflexivity ].
   clear Hrecs3; induction  lf2 as [| r5 lf2 Hreclf2].
   simpl in H11; discriminate.
   clear Hreclf2; assert (H17 : r3 = r4).
@@ -544,31 +544,31 @@ Proof.
         simpl in H18; rewrite <- (H17 x).
   rewrite <- (H18 x).
   reflexivity.
-  rewrite <- H12; unfold x in |- *; split.
+  rewrite <- H12; unfold x; split.
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite (Rplus_comm r); rewrite double;
             apply Rplus_lt_compat_l; assumption
             | discrR ] ].
-  unfold x in |- *; split.
+  unfold x; split.
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
   apply Rlt_trans with s2;
     [ apply Rmult_lt_reg_l with 2;
       [ prove_sup0
-        | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+        | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
           rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
             [ rewrite Rmult_1_l; rewrite (Rplus_comm r); rewrite double;
               apply Rplus_lt_compat_l; assumption
@@ -576,8 +576,8 @@ Proof.
       | assumption ].
   assert (H18 : f s2 = r3).
   apply (H8 0%nat);
-    [ simpl in |- *; apply lt_O_Sn
-      | unfold open_interval in |- *; simpl in |- *; split; assumption ].
+    [ simpl; apply lt_O_Sn
+      | unfold open_interval; simpl; split; assumption ].
   assert (H19 : r3 = r5).
   assert (H19 := H7 1%nat); simpl in H19;
     assert (H20 := H19 (lt_n_S _ _ (lt_O_Sn _))); elim H20;
@@ -587,18 +587,18 @@ Proof.
       rewrite <- (H22 (lt_O_Sn _) x).
   rewrite <- (H23 (lt_n_S _ _ (lt_O_Sn _)) x).
   reflexivity.
-  unfold open_interval in |- *; simpl in |- *; unfold x in |- *; split.
+  unfold open_interval; simpl; unfold x; split.
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l;
-            unfold Rmin in |- *; case (Rle_dec r1 r0); intro;
+            unfold Rmin; case (Rle_dec r1 r0); intro;
               assumption
             | discrR ] ].
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double;
             apply Rlt_le_trans with (r0 + Rmin r1 r0);
@@ -606,20 +606,20 @@ Proof.
                 assumption
                 | apply Rplus_le_compat_l; apply Rmin_r ]
             | discrR ] ].
-  unfold open_interval in |- *; simpl in |- *; unfold x in |- *; split.
+  unfold open_interval; simpl; unfold x; split.
   apply Rlt_trans with s2;
     [ assumption
       | apply Rmult_lt_reg_l with 2;
         [ prove_sup0
-          | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+          | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
             rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
               [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l;
-                unfold Rmin in |- *; case (Rle_dec r1 r0);
+                unfold Rmin; case (Rle_dec r1 r0);
                   intro; assumption
                 | discrR ] ] ].
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double;
             apply Rlt_le_trans with (r1 + Rmin r1 r0);
@@ -636,20 +636,20 @@ Proof.
         | elim H24; rewrite <- H17; assumption ].
   elim H2; clear H2; intros; assert (H17 := H16 0%nat); simpl in H17;
     elim (H17 (lt_O_Sn _)); assumption.
-  rewrite <- H0; rewrite H12; apply (H7 0%nat); simpl in |- *; apply lt_O_Sn.
+  rewrite <- H0; rewrite H12; apply (H7 0%nat); simpl; apply lt_O_Sn.
 Qed.
 
 Lemma StepFun_P12 :
   forall (a b:R) (f:R -> R) (l lf:Rlist),
     adapted_couple_opt f a b l lf -> adapted_couple_opt f b a l lf.
 Proof.
-  unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *; intros;
+  unfold adapted_couple_opt; unfold adapted_couple; intros;
     decompose [and] H; clear H; repeat split; try assumption.
-  rewrite H0; unfold Rmin in |- *; case (Rle_dec a b); intro;
+  rewrite H0; unfold Rmin; case (Rle_dec a b); intro;
     case (Rle_dec b a); intro; try reflexivity.
   apply Rle_antisym; assumption.
   apply Rle_antisym; auto with real.
-  rewrite H3; unfold Rmax in |- *; case (Rle_dec a b); intro;
+  rewrite H3; unfold Rmax; case (Rle_dec a b); intro;
     case (Rle_dec b a); intro; try reflexivity.
   apply Rle_antisym; assumption.
   apply Rle_antisym; auto with real.
@@ -689,10 +689,10 @@ Proof.
     case (Req_dec a b); intro.
   rewrite (StepFun_P8 H2 H4); rewrite (StepFun_P8 H H4); reflexivity.
   assert (Hyp_min : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   assert (Hyp_max : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   elim (RList_P20 _ (StepFun_P9 H H4)); intros s1 [s2 [s3 H5]]; rewrite H5 in H;
     rewrite H5; induction  lf1 as [| r3 lf1 Hreclf1].
@@ -716,34 +716,34 @@ Proof.
   rewrite <- (H20 (lt_O_Sn _) x).
   reflexivity.
   assert (H21 := H13 0%nat (lt_O_Sn _)); simpl in H21; elim H21; intro;
-    [ idtac | elim H7; assumption ]; unfold x in |- *;
+    [ idtac | elim H7; assumption ]; unfold x;
       split.
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H
             | discrR ] ].
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite <- (Rplus_comm r1); rewrite double;
             apply Rplus_lt_compat_l; apply H
             | discrR ] ].
   rewrite <- H6; assert (H21 := H13 0%nat (lt_O_Sn _)); simpl in H21; elim H21;
-    intro; [ idtac | elim H7; assumption ]; unfold x in |- *;
+    intro; [ idtac | elim H7; assumption ]; unfold x;
       split.
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H
             | discrR ] ].
   apply Rlt_le_trans with r1;
     [ apply Rmult_lt_reg_l with 2;
       [ prove_sup0
-        | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+        | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
           rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
             [ rewrite Rmult_1_l; rewrite <- (Rplus_comm r1); rewrite double;
               apply Rplus_lt_compat_l; apply H
@@ -752,64 +752,64 @@ Proof.
   eapply StepFun_P13.
   apply H4.
   apply H2.
-  unfold adapted_couple_opt in |- *; split.
+  unfold adapted_couple_opt; split.
   apply H.
   rewrite H5 in H3; apply H3.
   assert (H8 : r1 <= s2).
   eapply StepFun_P13.
   apply H4.
   apply H2.
-  unfold adapted_couple_opt in |- *; split.
+  unfold adapted_couple_opt; split.
   apply H.
   rewrite H5 in H3; apply H3.
   elim H7; intro.
-  simpl in |- *; elim H8; intro.
+  simpl; elim H8; intro.
   replace (r4 * (s2 - s1)) with (r3 * (r1 - r) + r3 * (s2 - r1));
   [ idtac | rewrite H9; rewrite H6; ring ].
   rewrite Rplus_assoc; apply Rplus_eq_compat_l;
     change
       (Int_SF lf1 (cons r1 r2) = Int_SF (cons r3 lf2) (cons r1 (cons s2 s3)))
-      in |- *; apply H0 with r1 b.
+     ; apply H0 with r1 b.
   unfold adapted_couple in H2; decompose [and] H2; clear H2;
     replace b with (Rmax a b).
   rewrite <- H12; apply RList_P7;
-    [ assumption | simpl in |- *; right; left; reflexivity ].
+    [ assumption | simpl; right; left; reflexivity ].
   eapply StepFun_P7.
   apply H1.
   apply H2.
-  unfold adapted_couple_opt in |- *; split.
+  unfold adapted_couple_opt; split.
   apply StepFun_P7 with a a r3.
   apply H1.
   unfold adapted_couple in H2, H; decompose [and] H2; decompose [and] H;
     clear H H2; assert (H20 : r = a).
   simpl in H13; rewrite H13; apply Hyp_min.
-  unfold adapted_couple in |- *; repeat split.
-  unfold ordered_Rlist in |- *; intros; simpl in H; induction  i as [| i Hreci].
-  simpl in |- *; rewrite <- H20; apply (H11 0%nat).
-  simpl in |- *; apply lt_O_Sn.
+  unfold adapted_couple; repeat split.
+  unfold ordered_Rlist; intros; simpl in H; induction  i as [| i Hreci].
+  simpl; rewrite <- H20; apply (H11 0%nat).
+  simpl; apply lt_O_Sn.
   induction  i as [| i Hreci0].
-  simpl in |- *; assumption.
-  change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i)) in |- *;
-    apply (H15 (S i)); simpl in |- *; apply lt_S_n; assumption.
-  simpl in |- *; symmetry  in |- *; apply Hyp_min.
+  simpl; assumption.
+  change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i));
+    apply (H15 (S i)); simpl; apply lt_S_n; assumption.
+  simpl; symmetry ; apply Hyp_min.
   rewrite <- H17; reflexivity.
-  simpl in H19; simpl in |- *; rewrite H19; reflexivity.
-  intros; simpl in H; unfold constant_D_eq, open_interval in |- *; intros;
+  simpl in H19; simpl; rewrite H19; reflexivity.
+  intros; simpl in H; unfold constant_D_eq, open_interval; intros;
     induction  i as [| i Hreci].
-  simpl in |- *; apply (H16 0%nat).
-  simpl in |- *; apply lt_O_Sn.
-  simpl in H2; rewrite <- H20 in H2; unfold open_interval in |- *;
-    simpl in |- *; apply H2.
+  simpl; apply (H16 0%nat).
+  simpl; apply lt_O_Sn.
+  simpl in H2; rewrite <- H20 in H2; unfold open_interval;
+    simpl; apply H2.
   clear Hreci; induction  i as [| i Hreci].
-  simpl in |- *; simpl in H2; rewrite H9; apply (H21 0%nat).
-  simpl in |- *; apply lt_O_Sn.
-  unfold open_interval in |- *; simpl in |- *; elim H2; intros; split.
+  simpl; simpl in H2; rewrite H9; apply (H21 0%nat).
+  simpl; apply lt_O_Sn.
+  unfold open_interval; simpl; elim H2; intros; split.
   apply Rle_lt_trans with r1; try assumption; rewrite <- H6; apply (H11 0%nat);
-    simpl in |- *; apply lt_O_Sn.
+    simpl; apply lt_O_Sn.
   assumption.
-  clear Hreci; simpl in |- *; apply (H21 (S i)).
-  simpl in |- *; apply lt_S_n; assumption.
-  unfold open_interval in |- *; apply H2.
+  clear Hreci; simpl; apply (H21 (S i)).
+  simpl; apply lt_S_n; assumption.
+  unfold open_interval; apply H2.
   elim H3; clear H3; intros; split.
   rewrite H9;
     change
@@ -817,64 +817,64 @@ Proof.
         (i < pred (Rlength (cons r4 lf2)))%nat ->
         pos_Rl (cons r4 lf2) i <> pos_Rl (cons r4 lf2) (S i) \/
         f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <> pos_Rl (cons r4 lf2) i)
-      in |- *; rewrite <- H5; apply H3.
+     ; rewrite <- H5; apply H3.
   rewrite H5 in H11; intros; simpl in H12; induction  i as [| i Hreci].
-  simpl in |- *; red in |- *; intro; rewrite H13 in H10;
+  simpl; red; intro; rewrite H13 in H10;
     elim (Rlt_irrefl _ H10).
-  clear Hreci; apply (H11 (S i)); simpl in |- *; apply H12.
+  clear Hreci; apply (H11 (S i)); simpl; apply H12.
   rewrite H9; rewrite H10; rewrite H6; apply Rplus_eq_compat_l; rewrite <- H10;
     apply H0 with r1 b.
   unfold adapted_couple in H2; decompose [and] H2; clear H2;
     replace b with (Rmax a b).
   rewrite <- H12; apply RList_P7;
-    [ assumption | simpl in |- *; right; left; reflexivity ].
+    [ assumption | simpl; right; left; reflexivity ].
   eapply StepFun_P7.
   apply H1.
   apply H2.
-  unfold adapted_couple_opt in |- *; split.
+  unfold adapted_couple_opt; split.
   apply StepFun_P7 with a a r3.
   apply H1.
   unfold adapted_couple in H2, H; decompose [and] H2; decompose [and] H;
     clear H H2; assert (H20 : r = a).
   simpl in H13; rewrite H13; apply Hyp_min.
-  unfold adapted_couple in |- *; repeat split.
-  unfold ordered_Rlist in |- *; intros; simpl in H; induction  i as [| i Hreci].
-  simpl in |- *; rewrite <- H20; apply (H11 0%nat); simpl in |- *;
+  unfold adapted_couple; repeat split.
+  unfold ordered_Rlist; intros; simpl in H; induction  i as [| i Hreci].
+  simpl; rewrite <- H20; apply (H11 0%nat); simpl;
     apply lt_O_Sn.
-  rewrite H10; apply (H15 (S i)); simpl in |- *; assumption.
-  simpl in |- *; symmetry  in |- *; apply Hyp_min.
+  rewrite H10; apply (H15 (S i)); simpl; assumption.
+  simpl; symmetry ; apply Hyp_min.
   rewrite <- H17; rewrite H10; reflexivity.
-  simpl in H19; simpl in |- *; apply H19.
-  intros; simpl in H; unfold constant_D_eq, open_interval in |- *; intros;
+  simpl in H19; simpl; apply H19.
+  intros; simpl in H; unfold constant_D_eq, open_interval; intros;
     induction  i as [| i Hreci].
-  simpl in |- *; apply (H16 0%nat).
-  simpl in |- *; apply lt_O_Sn.
-  simpl in H2; rewrite <- H20 in H2; unfold open_interval in |- *;
-    simpl in |- *; apply H2.
-  clear Hreci; simpl in |- *; apply (H21 (S i)).
-  simpl in |- *; assumption.
-  rewrite <- H10; unfold open_interval in |- *; apply H2.
+  simpl; apply (H16 0%nat).
+  simpl; apply lt_O_Sn.
+  simpl in H2; rewrite <- H20 in H2; unfold open_interval;
+    simpl; apply H2.
+  clear Hreci; simpl; apply (H21 (S i)).
+  simpl; assumption.
+  rewrite <- H10; unfold open_interval; apply H2.
   elim H3; clear H3; intros; split.
   rewrite H5 in H3; intros; apply (H3 (S i)).
-  simpl in |- *; replace (Rlength lf2) with (S (pred (Rlength lf2))).
+  simpl; replace (Rlength lf2) with (S (pred (Rlength lf2))).
   apply lt_n_S; apply H12.
-  symmetry  in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *;
+  symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
     intro; rewrite <- H13 in H12; elim (lt_n_O _ H12).
   intros; simpl in H12; rewrite H10; rewrite H5 in H11; apply (H11 (S i));
-    simpl in |- *; apply lt_n_S; apply H12.
-  simpl in |- *; rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r;
+    simpl; apply lt_n_S; apply H12.
+  simpl; rewrite H9; unfold Rminus; rewrite Rplus_opp_r;
     rewrite Rmult_0_r; rewrite Rplus_0_l;
       change
         (Int_SF lf1 (cons r1 r2) = Int_SF (cons r4 lf2) (cons s1 (cons s2 s3)))
-        in |- *; eapply H0.
+       ; eapply H0.
   apply H1.
-  2: rewrite H5 in H3; unfold adapted_couple_opt in |- *; split; assumption.
+  2: rewrite H5 in H3; unfold adapted_couple_opt; split; assumption.
   assert (H10 : r = a).
   unfold adapted_couple in H2; decompose [and] H2; clear H2; simpl in H12;
     rewrite H12; apply Hyp_min.
   rewrite <- H9; rewrite H10; apply StepFun_P7 with a r r3;
     [ apply H1
-      | pattern a at 2 in |- *; rewrite <- H10; pattern r at 2 in |- *; rewrite H9;
+      | pattern a at 2; rewrite <- H10; pattern r at 2; rewrite H9;
         apply H2 ].
 Qed.
 
@@ -918,12 +918,12 @@ Qed.
 Lemma StepFun_P18 :
   forall a b c:R, RiemannInt_SF (mkStepFun (StepFun_P4 a b c)) = c * (b - a).
 Proof.
-  intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
+  intros; unfold RiemannInt_SF; case (Rle_dec a b); intro.
   replace
   (Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c)))
     (subdivision (mkStepFun (StepFun_P4 a b c)))) with
   (Int_SF (cons c nil) (cons a (cons b nil)));
-  [ simpl in |- *; ring
+  [ simpl; ring
     | apply StepFun_P17 with (fct_cte c) a b;
       [ apply StepFun_P3; assumption
         | apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c))) ] ].
@@ -931,7 +931,7 @@ Proof.
   (Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c)))
     (subdivision (mkStepFun (StepFun_P4 a b c)))) with
   (Int_SF (cons c nil) (cons b (cons a nil)));
-  [ simpl in |- *; ring
+  [ simpl; ring
     | apply StepFun_P17 with (fct_cte c) a b;
       [ apply StepFun_P2; apply StepFun_P3; auto with real
         | apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c))) ] ].
@@ -943,8 +943,8 @@ Lemma StepFun_P19 :
     Int_SF (FF l1 f) l1 + l * Int_SF (FF l1 g) l1.
 Proof.
   intros; induction  l1 as [| r l1 Hrecl1];
-    [ simpl in |- *; ring
-      | induction  l1 as [| r0 l1 Hrecl0]; simpl in |- *;
+    [ simpl; ring
+      | induction  l1 as [| r0 l1 Hrecl0]; simpl;
         [ ring | simpl in Hrecl1; rewrite Hrecl1; ring ] ].
 Qed.
 
@@ -954,38 +954,38 @@ Lemma StepFun_P20 :
 Proof.
   intros l f H; induction l;
     [ elim (lt_irrefl _ H)
-      | simpl in |- *; rewrite RList_P18; rewrite RList_P14; reflexivity ].
+      | simpl; rewrite RList_P18; rewrite RList_P14; reflexivity ].
 Qed.
 
 Lemma StepFun_P21 :
   forall (a b:R) (f:R -> R) (l:Rlist),
     is_subdivision f a b l -> adapted_couple f a b l (FF l f).
 Proof.
-  intros; unfold adapted_couple in |- *; unfold is_subdivision in X;
+  intros; unfold adapted_couple; unfold is_subdivision in X;
     unfold adapted_couple in X; elim X; clear X; intros;
       decompose [and] p; clear p; repeat split; try assumption.
   apply StepFun_P20; rewrite H2; apply lt_O_Sn.
   intros; assert (H5 := H4 _ H3); unfold constant_D_eq, open_interval in H5;
-    unfold constant_D_eq, open_interval in |- *; intros;
+    unfold constant_D_eq, open_interval; intros;
       induction  l as [| r l Hrecl].
   discriminate.
-  unfold FF in |- *; rewrite RList_P12.
-  simpl in |- *;
-    change (f x0 = f (pos_Rl (mid_Rlist (cons r l) r) (S i))) in |- *;
+  unfold FF; rewrite RList_P12.
+  simpl;
+    change (f x0 = f (pos_Rl (mid_Rlist (cons r l) r) (S i)));
       rewrite RList_P13; try assumption; rewrite (H5 x0 H6);
         rewrite H5.
   reflexivity.
   split.
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; elim H6;
             intros; apply Rlt_trans with x0; assumption
             | discrR ] ].
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double;
             rewrite (Rplus_comm (pos_Rl (cons r l) i));
@@ -1001,22 +1001,22 @@ Lemma StepFun_P22 :
     is_subdivision f a b lf ->
     is_subdivision g a b lg -> is_subdivision f a b (cons_ORlist lf lg).
 Proof.
-  unfold is_subdivision in |- *; intros a b f g lf lg Hyp X X0; elim X; elim X0;
+  unfold is_subdivision; intros a b f g lf lg Hyp X X0; elim X; elim X0;
     clear X X0; intros lg0 p lf0 p0; assert (Hyp_min : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   assert (Hyp_max : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   apply existT with (FF (cons_ORlist lf lg) f); unfold adapted_couple in p, p0;
     decompose [and] p; decompose [and] p0; clear p p0;
       rewrite Hyp_min in H6; rewrite Hyp_min in H1; rewrite Hyp_max in H0;
-        rewrite Hyp_max in H5; unfold adapted_couple in |- *;
+        rewrite Hyp_max in H5; unfold adapted_couple;
           repeat split.
   apply RList_P2; assumption.
-  rewrite Hyp_min; symmetry  in |- *; apply Rle_antisym.
+  rewrite Hyp_min; symmetry ; apply Rle_antisym.
   induction  lf as [| r lf Hreclf].
-  simpl in |- *; right; symmetry  in |- *; assumption.
+  simpl; right; symmetry ; assumption.
   assert
     (H10 :
       In (pos_Rl (cons_ORlist (cons r lf) lg) 0) (cons_ORlist (cons r lf) lg)).
@@ -1024,7 +1024,7 @@ Proof.
     (RList_P3 (cons_ORlist (cons r lf) lg)
       (pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros _ H10;
     apply H10; exists 0%nat; split;
-      [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_O_Sn ].
+      [ reflexivity | rewrite RList_P11; simpl; apply lt_O_Sn ].
   elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) 0));
     intros H12 _; assert (H13 := H12 H10); elim H13; intro.
   elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) 0));
@@ -1037,16 +1037,16 @@ Proof.
       clear H15; intros; rewrite H15; rewrite <- H1; elim (RList_P6 lg);
         intros; apply H17; [ assumption | apply le_O_n | assumption ].
   induction  lf as [| r lf Hreclf].
-  simpl in |- *; right; assumption.
+  simpl; right; assumption.
   assert (H8 : In a (cons_ORlist (cons r lf) lg)).
   elim (RList_P9 (cons r lf) lg a); intros; apply H10; left;
     elim (RList_P3 (cons r lf) a); intros; apply H12;
       exists 0%nat; split;
-        [ symmetry  in |- *; assumption | simpl in |- *; apply lt_O_Sn ].
+        [ symmetry ; assumption | simpl; apply lt_O_Sn ].
   apply RList_P5; [ apply RList_P2; assumption | assumption ].
   rewrite Hyp_max; apply Rle_antisym.
   induction  lf as [| r lf Hreclf].
-  simpl in |- *; right; assumption.
+  simpl; right; assumption.
   assert
     (H8 :
       In
@@ -1059,7 +1059,7 @@ Proof.
         (pred (Rlength (cons_ORlist (cons r lf) lg)))));
     intros _ H10; apply H10;
       exists (pred (Rlength (cons_ORlist (cons r lf) lg)));
-        split; [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_n_Sn ].
+        split; [ reflexivity | rewrite RList_P11; simpl; apply lt_n_Sn ].
   elim
     (RList_P9 (cons r lf) lg
       (pos_Rl (cons_ORlist (cons r lf) lg)
@@ -1074,8 +1074,8 @@ Proof.
       elim H15; clear H15; intros; rewrite H15; rewrite <- H5;
         elim (RList_P6 (cons r lf)); intros; apply H17;
           [ assumption
-            | simpl in |- *; simpl in H14; apply lt_n_Sm_le; assumption
-            | simpl in |- *; apply lt_n_Sn ].
+            | simpl; simpl in H14; apply lt_n_Sm_le; assumption
+            | simpl; apply lt_n_Sn ].
   elim
     (RList_P3 lg
       (pos_Rl (cons_ORlist (cons r lf) lg)
@@ -1083,23 +1083,23 @@ Proof.
     intros H13 _; assert (H14 := H13 H12); elim H14; intros;
       elim H15; clear H15; intros.
   rewrite H15; assert (H17 : Rlength lg = S (pred (Rlength lg))).
-  apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
     rewrite <- H17 in H16; elim (lt_n_O _ H16).
   rewrite <- H0; elim (RList_P6 lg); intros; apply H18;
     [ assumption
       | rewrite H17 in H16; apply lt_n_Sm_le; assumption
       | apply lt_pred_n_n; rewrite H17; apply lt_O_Sn ].
   induction  lf as [| r lf Hreclf].
-  simpl in |- *; right; symmetry  in |- *; assumption.
+  simpl; right; symmetry ; assumption.
   assert (H8 : In b (cons_ORlist (cons r lf) lg)).
   elim (RList_P9 (cons r lf) lg b); intros; apply H10; left;
     elim (RList_P3 (cons r lf) b); intros; apply H12;
       exists (pred (Rlength (cons r lf))); split;
-        [ symmetry  in |- *; assumption | simpl in |- *; apply lt_n_Sn ].
+        [ symmetry ; assumption | simpl; apply lt_n_Sn ].
   apply RList_P7; [ apply RList_P2; assumption | assumption ].
-  apply StepFun_P20; rewrite RList_P11; rewrite H2; rewrite H7; simpl in |- *;
+  apply StepFun_P20; rewrite RList_P11; rewrite H2; rewrite H7; simpl;
     apply lt_O_Sn.
-  intros; unfold constant_D_eq, open_interval in |- *; intros;
+  intros; unfold constant_D_eq, open_interval; intros;
     cut
       (exists l : R,
         constant_D_eq f
@@ -1109,10 +1109,10 @@ Proof.
     assert
       (Hyp_cons :
         exists r : R, (exists r0 : Rlist, cons_ORlist lf lg = cons r r0)).
-  apply RList_P19; red in |- *; intro; rewrite H13 in H8; elim (lt_n_O _ H8).
+  apply RList_P19; red; intro; rewrite H13 in H8; elim (lt_n_O _ H8).
   elim Hyp_cons; clear Hyp_cons; intros r [r0 Hyp_cons]; rewrite Hyp_cons;
-    unfold FF in |- *; rewrite RList_P12.
-  change (f x = f (pos_Rl (mid_Rlist (cons r r0) r) (S i))) in |- *;
+    unfold FF; rewrite RList_P12.
+  change (f x = f (pos_Rl (mid_Rlist (cons r r0) r) (S i)));
     rewrite <- Hyp_cons; rewrite RList_P13.
   assert (H13 := RList_P2 _ _ H _ H8); elim H13; intro.
   unfold constant_D_eq, open_interval in H11, H12; rewrite (H11 x H10);
@@ -1124,13 +1124,13 @@ Proof.
   split.
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double;
             rewrite (Rplus_comm (pos_Rl (cons_ORlist lf lg) i));
@@ -1149,7 +1149,7 @@ Proof.
   apply le_O_n.
   apply lt_trans with (pred (Rlength (cons_ORlist lf lg)));
     [ assumption
-      | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro;
+      | apply lt_pred_n_n; apply neq_O_lt; red; intro;
         rewrite <- H13 in H8; elim (lt_n_O _ H8) ].
   assumption.
   assumption.
@@ -1160,7 +1160,7 @@ Proof.
   elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11.
   apply RList_P2; assumption.
   apply lt_n_Sm_le; apply lt_n_S; assumption.
-  apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H8;
+  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H13 in H8;
     elim (lt_n_O _ H8).
   rewrite H0; assumption.
   set
@@ -1168,24 +1168,24 @@ Proof.
       fun j:nat =>
         pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\ (j < Rlength lf)%nat);
     assert (H12 : Nbound I).
-  unfold Nbound in |- *; exists (Rlength lf); intros; unfold I in H12; elim H12;
+  unfold Nbound; exists (Rlength lf); intros; unfold I in H12; elim H12;
     intros; apply lt_le_weak; assumption.
   assert (H13 :  exists n : nat, I n).
-  exists 0%nat; unfold I in |- *; split.
+  exists 0%nat; unfold I; split.
   apply Rle_trans with (pos_Rl (cons_ORlist lf lg) 0).
-  right; symmetry  in |- *.
+  right; symmetry .
   apply RList_P15; try assumption; rewrite H1; assumption.
   elim (RList_P6 (cons_ORlist lf lg)); intros; apply H13.
   apply RList_P2; assumption.
   apply le_O_n.
   apply lt_trans with (pred (Rlength (cons_ORlist lf lg))).
   assumption.
-  apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H15 in H8;
+  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H15 in H8;
     elim (lt_n_O _ H8).
-  apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H5;
+  apply neq_O_lt; red; intro; rewrite <- H13 in H5;
     rewrite <- H6 in H11; rewrite <- H5 in H11; elim (Rlt_irrefl _ H11).
   assert (H14 := Nzorn H13 H12); elim H14; clear H14; intros x0 H14;
-    exists (pos_Rl lf0 x0); unfold constant_D_eq, open_interval in |- *;
+    exists (pos_Rl lf0 x0); unfold constant_D_eq, open_interval;
       intros; assert (H16 := H9 x0); assert (H17 : (x0 < pred (Rlength lf))%nat).
   elim H14; clear H14; intros; unfold I in H14; elim H14; clear H14; intros;
     apply lt_S_n; replace (S (pred (Rlength lf))) with (Rlength lf).
@@ -1203,11 +1203,11 @@ Proof.
   elim (RList_P6 (cons_ORlist lf lg)); intros; apply H21.
   apply RList_P2; assumption.
   apply lt_n_Sm_le; apply lt_n_S; assumption.
-  apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H23 in H8;
+  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H23 in H8;
     elim (lt_n_O _ H8).
   right; apply RList_P16; try assumption; rewrite H0; assumption.
   rewrite <- H20; reflexivity.
-  apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
     rewrite <- H19 in H18; elim (lt_n_O _ H18).
   assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18;
     rewrite (H18 x1).
@@ -1219,11 +1219,11 @@ Proof.
   assert (H22 : (S x0 < Rlength lf)%nat).
   replace (Rlength lf) with (S (pred (Rlength lf)));
   [ apply lt_n_S; assumption
-    | symmetry  in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *;
+    | symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
       intro; rewrite <- H22 in H21; elim (lt_n_O _ H21) ].
   elim (Rle_dec (pos_Rl lf (S x0)) (pos_Rl (cons_ORlist lf lg) i)); intro.
   assert (H23 : (S x0 <= x0)%nat).
-  apply H20; unfold I in |- *; split; assumption.
+  apply H20; unfold I; split; assumption.
   elim (le_Sn_n _ H23).
   assert (H23 : pos_Rl (cons_ORlist lf lg) i < pos_Rl lf (S x0)).
   auto with real.
@@ -1253,22 +1253,22 @@ Lemma StepFun_P24 :
     is_subdivision f a b lf ->
     is_subdivision g a b lg -> is_subdivision g a b (cons_ORlist lf lg).
 Proof.
-  unfold is_subdivision in |- *; intros a b f g lf lg Hyp X X0; elim X; elim X0;
+  unfold is_subdivision; intros a b f g lf lg Hyp X X0; elim X; elim X0;
     clear X X0; intros lg0 p lf0 p0; assert (Hyp_min : Rmin a b = a).
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   assert (Hyp_max : Rmax a b = b).
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   apply existT with (FF (cons_ORlist lf lg) g); unfold adapted_couple in p, p0;
     decompose [and] p; decompose [and] p0; clear p p0;
       rewrite Hyp_min in H1; rewrite Hyp_min in H6; rewrite Hyp_max in H0;
-        rewrite Hyp_max in H5; unfold adapted_couple in |- *;
+        rewrite Hyp_max in H5; unfold adapted_couple;
           repeat split.
   apply RList_P2; assumption.
-  rewrite Hyp_min; symmetry  in |- *; apply Rle_antisym.
+  rewrite Hyp_min; symmetry ; apply Rle_antisym.
   induction  lf as [| r lf Hreclf].
-  simpl in |- *; right; symmetry  in |- *; assumption.
+  simpl; right; symmetry ; assumption.
   assert
     (H10 :
       In (pos_Rl (cons_ORlist (cons r lf) lg) 0) (cons_ORlist (cons r lf) lg)).
@@ -1276,7 +1276,7 @@ Proof.
     (RList_P3 (cons_ORlist (cons r lf) lg)
       (pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros _ H10;
     apply H10; exists 0%nat; split;
-      [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_O_Sn ].
+      [ reflexivity | rewrite RList_P11; simpl; apply lt_O_Sn ].
   elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) 0));
     intros H12 _; assert (H13 := H12 H10); elim H13; intro.
   elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) 0));
@@ -1289,16 +1289,16 @@ Proof.
       clear H15; intros; rewrite H15; rewrite <- H1; elim (RList_P6 lg);
         intros; apply H17; [ assumption | apply le_O_n | assumption ].
   induction  lf as [| r lf Hreclf].
-  simpl in |- *; right; assumption.
+  simpl; right; assumption.
   assert (H8 : In a (cons_ORlist (cons r lf) lg)).
   elim (RList_P9 (cons r lf) lg a); intros; apply H10; left;
     elim (RList_P3 (cons r lf) a); intros; apply H12;
       exists 0%nat; split;
-        [ symmetry  in |- *; assumption | simpl in |- *; apply lt_O_Sn ].
+        [ symmetry ; assumption | simpl; apply lt_O_Sn ].
   apply RList_P5; [ apply RList_P2; assumption | assumption ].
   rewrite Hyp_max; apply Rle_antisym.
   induction  lf as [| r lf Hreclf].
-  simpl in |- *; right; assumption.
+  simpl; right; assumption.
   assert
     (H8 :
       In
@@ -1311,7 +1311,7 @@ Proof.
         (pred (Rlength (cons_ORlist (cons r lf) lg)))));
     intros _ H10; apply H10;
       exists (pred (Rlength (cons_ORlist (cons r lf) lg)));
-        split; [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_n_Sn ].
+        split; [ reflexivity | rewrite RList_P11; simpl; apply lt_n_Sn ].
   elim
     (RList_P9 (cons r lf) lg
       (pos_Rl (cons_ORlist (cons r lf) lg)
@@ -1325,8 +1325,8 @@ Proof.
       elim H15; clear H15; intros; rewrite H15; rewrite <- H5;
         elim (RList_P6 (cons r lf)); intros; apply H17;
           [ assumption
-            | simpl in |- *; simpl in H14; apply lt_n_Sm_le; assumption
-            | simpl in |- *; apply lt_n_Sn ].
+            | simpl; simpl in H14; apply lt_n_Sm_le; assumption
+            | simpl; apply lt_n_Sn ].
   elim
     (RList_P3 lg
       (pos_Rl (cons_ORlist (cons r lf) lg)
@@ -1334,23 +1334,23 @@ Proof.
     intros H13 _; assert (H14 := H13 H12); elim H14; intros;
       elim H15; clear H15; intros; rewrite H15;
         assert (H17 : Rlength lg = S (pred (Rlength lg))).
-  apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
     rewrite <- H17 in H16; elim (lt_n_O _ H16).
   rewrite <- H0; elim (RList_P6 lg); intros; apply H18;
     [ assumption
       | rewrite H17 in H16; apply lt_n_Sm_le; assumption
       | apply lt_pred_n_n; rewrite H17; apply lt_O_Sn ].
   induction  lf as [| r lf Hreclf].
-  simpl in |- *; right; symmetry  in |- *; assumption.
+  simpl; right; symmetry ; assumption.
   assert (H8 : In b (cons_ORlist (cons r lf) lg)).
   elim (RList_P9 (cons r lf) lg b); intros; apply H10; left;
     elim (RList_P3 (cons r lf) b); intros; apply H12;
       exists (pred (Rlength (cons r lf))); split;
-        [ symmetry  in |- *; assumption | simpl in |- *; apply lt_n_Sn ].
+        [ symmetry ; assumption | simpl; apply lt_n_Sn ].
   apply RList_P7; [ apply RList_P2; assumption | assumption ].
-  apply StepFun_P20; rewrite RList_P11; rewrite H7; rewrite H2; simpl in |- *;
+  apply StepFun_P20; rewrite RList_P11; rewrite H7; rewrite H2; simpl;
     apply lt_O_Sn.
-  unfold constant_D_eq, open_interval in |- *; intros;
+  unfold constant_D_eq, open_interval; intros;
     cut
       (exists l : R,
         constant_D_eq g
@@ -1360,10 +1360,10 @@ Proof.
     assert
       (Hyp_cons :
         exists r : R, (exists r0 : Rlist, cons_ORlist lf lg = cons r r0)).
-  apply RList_P19; red in |- *; intro; rewrite H13 in H8; elim (lt_n_O _ H8).
+  apply RList_P19; red; intro; rewrite H13 in H8; elim (lt_n_O _ H8).
   elim Hyp_cons; clear Hyp_cons; intros r [r0 Hyp_cons]; rewrite Hyp_cons;
-    unfold FF in |- *; rewrite RList_P12.
-  change (g x = g (pos_Rl (mid_Rlist (cons r r0) r) (S i))) in |- *;
+    unfold FF; rewrite RList_P12.
+  change (g x = g (pos_Rl (mid_Rlist (cons r r0) r) (S i)));
     rewrite <- Hyp_cons; rewrite RList_P13.
   assert (H13 := RList_P2 _ _ H _ H8); elim H13; intro.
   unfold constant_D_eq, open_interval in H11, H12; rewrite (H11 x H10);
@@ -1375,13 +1375,13 @@ Proof.
   split.
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double;
             rewrite (Rplus_comm (pos_Rl (cons_ORlist lf lg) i));
@@ -1400,7 +1400,7 @@ Proof.
   apply le_O_n.
   apply lt_trans with (pred (Rlength (cons_ORlist lf lg)));
     [ assumption
-      | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro;
+      | apply lt_pred_n_n; apply neq_O_lt; red; intro;
         rewrite <- H13 in H8; elim (lt_n_O _ H8) ].
   rewrite H1; assumption.
   apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)).
@@ -1409,7 +1409,7 @@ Proof.
   elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11.
   apply RList_P2; assumption.
   apply lt_n_Sm_le; apply lt_n_S; assumption.
-  apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H8;
+  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H13 in H8;
     elim (lt_n_O _ H8).
   rewrite H0; assumption.
   set
@@ -1417,24 +1417,24 @@ Proof.
       fun j:nat =>
         pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\ (j < Rlength lg)%nat);
     assert (H12 : Nbound I).
-  unfold Nbound in |- *; exists (Rlength lg); intros; unfold I in H12; elim H12;
+  unfold Nbound; exists (Rlength lg); intros; unfold I in H12; elim H12;
     intros; apply lt_le_weak; assumption.
   assert (H13 :  exists n : nat, I n).
-  exists 0%nat; unfold I in |- *; split.
+  exists 0%nat; unfold I; split.
   apply Rle_trans with (pos_Rl (cons_ORlist lf lg) 0).
-  right; symmetry  in |- *; rewrite H1; rewrite <- H6; apply RList_P15;
+  right; symmetry ; rewrite H1; rewrite <- H6; apply RList_P15;
     try assumption; rewrite H1; assumption.
   elim (RList_P6 (cons_ORlist lf lg)); intros; apply H13;
     [ apply RList_P2; assumption
       | apply le_O_n
       | apply lt_trans with (pred (Rlength (cons_ORlist lf lg)));
         [ assumption
-          | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro;
+          | apply lt_pred_n_n; apply neq_O_lt; red; intro;
             rewrite <- H15 in H8; elim (lt_n_O _ H8) ] ].
-  apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H0;
+  apply neq_O_lt; red; intro; rewrite <- H13 in H0;
     rewrite <- H1 in H11; rewrite <- H0 in H11; elim (Rlt_irrefl _ H11).
   assert (H14 := Nzorn H13 H12); elim H14; clear H14; intros x0 H14;
-    exists (pos_Rl lg0 x0); unfold constant_D_eq, open_interval in |- *;
+    exists (pos_Rl lg0 x0); unfold constant_D_eq, open_interval;
       intros; assert (H16 := H4 x0); assert (H17 : (x0 < pred (Rlength lg))%nat).
   elim H14; clear H14; intros; unfold I in H14; elim H14; clear H14; intros;
     apply lt_S_n; replace (S (pred (Rlength lg))) with (Rlength lg).
@@ -1452,12 +1452,12 @@ Proof.
   elim (RList_P6 (cons_ORlist lf lg)); intros; apply H21.
   apply RList_P2; assumption.
   apply lt_n_Sm_le; apply lt_n_S; assumption.
-  apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H23 in H8;
+  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H23 in H8;
     elim (lt_n_O _ H8).
   right; rewrite H0; rewrite <- H5; apply RList_P16; try assumption.
   rewrite H0; assumption.
   rewrite <- H20; reflexivity.
-  apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
     rewrite <- H19 in H18; elim (lt_n_O _ H18).
   assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18;
     rewrite (H18 x1).
@@ -1469,11 +1469,11 @@ Proof.
   assert (H22 : (S x0 < Rlength lg)%nat).
   replace (Rlength lg) with (S (pred (Rlength lg))).
   apply lt_n_S; assumption.
-  symmetry  in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *;
+  symmetry ; apply S_pred with 0%nat; apply neq_O_lt; red;
     intro; rewrite <- H22 in H21; elim (lt_n_O _ H21).
   elim (Rle_dec (pos_Rl lg (S x0)) (pos_Rl (cons_ORlist lf lg) i)); intro.
   assert (H23 : (S x0 <= x0)%nat);
-    [ apply H20; unfold I in |- *; split; assumption | elim (le_Sn_n _ H23) ].
+    [ apply H20; unfold I; split; assumption | elim (le_Sn_n _ H23) ].
   assert (H23 : pos_Rl (cons_ORlist lf lg) i < pos_Rl lg (S x0)).
   auto with real.
   clear b0; apply RList_P17; try assumption;
@@ -1509,35 +1509,35 @@ Proof.
   intros i H8 x1 H10; unfold open_interval in H10, H9, H4;
     rewrite (H9 _ H8 _ H10); rewrite (H4 _ H8 _ H10);
       assert (H11 : l1 <> nil).
-  red in |- *; intro H11; rewrite H11 in H8; elim (lt_n_O _ H8).
+  red; intro H11; rewrite H11 in H8; elim (lt_n_O _ H8).
   destruct (RList_P19 _ H11) as (r,(r0,H12));
-    rewrite H12; unfold FF in |- *;
+    rewrite H12; unfold FF;
       change
         (pos_Rl x0 i + l * pos_Rl x i =
           pos_Rl
           (app_Rlist (mid_Rlist (cons r r0) r) (fun x2:R => f x2 + l * g x2))
-          (S i)) in |- *; rewrite RList_P12.
+          (S i)); rewrite RList_P12.
   rewrite RList_P13.
   rewrite <- H12; rewrite (H9 _ H8); try rewrite (H4 _ H8);
     reflexivity ||
       (elim H10; clear H10; intros; split;
         [ apply Rmult_lt_reg_l with 2;
           [ prove_sup0
-            | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+            | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
               rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
                 [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l;
                   apply Rlt_trans with x1; assumption
                   | discrR ] ]
           | apply Rmult_lt_reg_l with 2;
             [ prove_sup0
-              | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2));
+              | unfold Rdiv; rewrite <- (Rmult_comm (/ 2));
                 rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
                   [ rewrite Rmult_1_l; rewrite double;
                     rewrite (Rplus_comm (pos_Rl l1 i)); apply Rplus_lt_compat_l;
                       apply Rlt_trans with x1; assumption
                     | discrR ] ] ]).
   rewrite <- H12; assumption.
-  rewrite RList_P14; simpl in |- *; rewrite H12 in H8; simpl in H8;
+  rewrite RList_P14; simpl; rewrite H12 in H8; simpl in H8;
     apply lt_n_S; apply H8.
 Qed.
 
@@ -1556,7 +1556,7 @@ Qed.
 Lemma StepFun_P28 :
   forall (a b l:R) (f g:StepFun a b), IsStepFun (fun x:R => f x + l * g x) a b.
 Proof.
-  intros a b l f g; unfold IsStepFun in |- *; assert (H := pre f);
+  intros a b l f g; unfold IsStepFun; assert (H := pre f);
     assert (H0 := pre g); unfold IsStepFun in H, H0; elim H;
       elim H0; intros; apply existT with (cons_ORlist x0 x);
         apply StepFun_P27; assumption.
@@ -1565,7 +1565,7 @@ Qed.
 Lemma StepFun_P29 :
   forall (a b:R) (f:StepFun a b), is_subdivision f a b (subdivision f).
 Proof.
-  intros a b f; unfold is_subdivision in |- *;
+  intros a b f; unfold is_subdivision;
     apply existT with (subdivision_val f); apply StepFun_P1.
 Qed.
 
@@ -1574,7 +1574,7 @@ Lemma StepFun_P30 :
     RiemannInt_SF (mkStepFun (StepFun_P28 l f g)) =
     RiemannInt_SF f + l * RiemannInt_SF g.
 Proof.
-  intros a b l f g; unfold RiemannInt_SF in |- *; case (Rle_dec a b);
+  intros a b l f g; unfold RiemannInt_SF; case (Rle_dec a b);
     (intro;
       replace
       (Int_SF (subdivision_val (mkStepFun (StepFun_P28 l f g)))
@@ -1611,10 +1611,10 @@ Lemma StepFun_P31 :
     adapted_couple f a b l lf ->
     adapted_couple (fun x:R => Rabs (f x)) a b l (app_Rlist lf Rabs).
 Proof.
-  unfold adapted_couple in |- *; intros; decompose [and] H; clear H;
+  unfold adapted_couple; intros; decompose [and] H; clear H;
     repeat split; try assumption.
-  symmetry  in |- *; rewrite H3; rewrite RList_P18; reflexivity.
-  intros; unfold constant_D_eq, open_interval in |- *;
+  symmetry ; rewrite H3; rewrite RList_P18; reflexivity.
+  intros; unfold constant_D_eq, open_interval;
     unfold constant_D_eq, open_interval in H5; intros;
       rewrite (H5 _ H _ H4); rewrite RList_P12;
         [ reflexivity | rewrite H3 in H; simpl in H; apply H ].
@@ -1623,8 +1623,8 @@ Qed.
 Lemma StepFun_P32 :
   forall (a b:R) (f:StepFun a b), IsStepFun (fun x:R => Rabs (f x)) a b.
 Proof.
-  intros a b f; unfold IsStepFun in |- *; apply existT with (subdivision f);
-    unfold is_subdivision in |- *;
+  intros a b f; unfold IsStepFun; apply existT with (subdivision f);
+    unfold is_subdivision;
       apply existT with (app_Rlist (subdivision_val f) Rabs);
         apply StepFun_P31; apply StepFun_P1.
 Qed.
@@ -1634,8 +1634,8 @@ Lemma StepFun_P33 :
     ordered_Rlist l1 -> Rabs (Int_SF l2 l1) <= Int_SF (app_Rlist l2 Rabs) l1.
 Proof.
   simple induction l2; intros.
-  simpl in |- *; rewrite Rabs_R0; right; reflexivity.
-  simpl in |- *; induction  l1 as [| r1 l1 Hrecl1].
+  simpl; rewrite Rabs_R0; right; reflexivity.
+  simpl; induction  l1 as [| r1 l1 Hrecl1].
   rewrite Rabs_R0; right; reflexivity.
   induction  l1 as [| r2 l1 Hrecl0].
   rewrite Rabs_R0; right; reflexivity.
@@ -1643,7 +1643,7 @@ Proof.
   apply Rabs_triang.
   rewrite Rabs_mult; rewrite (Rabs_right (r2 - r1));
     [ apply Rplus_le_compat_l; apply H; apply RList_P4 with r1; assumption
-      | apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl in |- *;
+      | apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl;
         apply lt_O_Sn ].
 Qed.
 
@@ -1652,7 +1652,7 @@ Lemma StepFun_P34 :
     a <= b ->
     Rabs (RiemannInt_SF f) <= RiemannInt_SF (mkStepFun (StepFun_P32 f)).
 Proof.
-  intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
+  intros; unfold RiemannInt_SF; case (Rle_dec a b); intro.
   replace
   (Int_SF (subdivision_val (mkStepFun (StepFun_P32 f)))
     (subdivision (mkStepFun (StepFun_P32 f)))) with
@@ -1676,18 +1676,18 @@ Lemma StepFun_P35 :
 Proof.
   simple induction l; intros.
   right; reflexivity.
-  simpl in |- *; induction  r0 as [| r0 r1 Hrecr0].
+  simpl; induction  r0 as [| r0 r1 Hrecr0].
   right; reflexivity.
-  simpl in |- *; apply Rplus_le_compat.
+  simpl; apply Rplus_le_compat.
   case (Req_dec r r0); intro.
   rewrite H4; right; ring.
   do 2 rewrite <- (Rmult_comm (r0 - r)); apply Rmult_le_compat_l.
-  apply Rge_le; apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl in |- *;
+  apply Rge_le; apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl;
     apply lt_O_Sn.
   apply H3; split.
   apply Rmult_lt_reg_l with 2.
   prove_sup0.
-  unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
     rewrite <- Rinv_r_sym.
   assert (H5 : r = a).
   apply H1.
@@ -1700,7 +1700,7 @@ Proof.
   discrR.
   apply Rmult_lt_reg_l with 2.
   prove_sup0.
-  unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
     rewrite <- Rinv_r_sym.
   rewrite Rmult_1_l; rewrite double; assert (H5 : r0 <= b).
   replace b with
@@ -1708,9 +1708,9 @@ Proof.
   replace r0 with (pos_Rl (cons r (cons r0 r1)) 1).
   elim (RList_P6 (cons r (cons r0 r1))); intros; apply H5.
   assumption.
-  simpl in |- *; apply le_n_S.
+  simpl; apply le_n_S.
   apply le_O_n.
-  simpl in |- *; apply lt_n_Sn.
+  simpl; apply lt_n_Sn.
   reflexivity.
   apply Rle_lt_trans with (r + b).
   apply Rplus_le_compat_l; assumption.
@@ -1730,7 +1730,7 @@ Proof.
   intros; apply H3; elim H4; intros; split; try assumption.
   apply Rle_lt_trans with r0; try assumption.
   rewrite <- H1.
-  simpl in |- *; apply (H0 0%nat); simpl in |- *; apply lt_O_Sn.
+  simpl; apply (H0 0%nat); simpl; apply lt_O_Sn.
 Qed.
 
 Lemma StepFun_P36 :
@@ -1741,16 +1741,16 @@ Lemma StepFun_P36 :
     (forall x:R, a < x < b -> f x <= g x) ->
     RiemannInt_SF f <= RiemannInt_SF g.
 Proof.
-  intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
+  intros; unfold RiemannInt_SF; case (Rle_dec a b); intro.
   replace (Int_SF (subdivision_val f) (subdivision f)) with (Int_SF (FF l f) l).
   replace (Int_SF (subdivision_val g) (subdivision g)) with (Int_SF (FF l g) l).
   unfold is_subdivision in X; elim X; clear X; intros;
     unfold adapted_couple in p; decompose [and] p; clear p;
       assert (H5 : Rmin a b = a);
-        [ unfold Rmin in |- *; case (Rle_dec a b); intro;
+        [ unfold Rmin; case (Rle_dec a b); intro;
           [ reflexivity | elim n; assumption ]
           | assert (H7 : Rmax a b = b);
-            [ unfold Rmax in |- *; case (Rle_dec a b); intro;
+            [ unfold Rmax; case (Rle_dec a b); intro;
               [ reflexivity | elim n; assumption ]
               | rewrite H5 in H3; rewrite H7 in H2; eapply StepFun_P35 with a b;
                 assumption ] ].
@@ -1809,27 +1809,27 @@ Proof.
   assert (H7 : r1 <= b).
   rewrite <- H4; apply RList_P7; [ assumption | left; reflexivity ].
   assert (H8 : IsStepFun g' a b).
-  unfold IsStepFun in |- *; assert (H8 := pre g); unfold IsStepFun in H8;
+  unfold IsStepFun; assert (H8 := pre g); unfold IsStepFun in H8;
     elim H8; intros lg H9; unfold is_subdivision in H9;
       elim H9; clear H9; intros lg2 H9; split with (cons a lg);
-        unfold is_subdivision in |- *; split with (cons (f a) lg2);
+        unfold is_subdivision; split with (cons (f a) lg2);
           unfold adapted_couple in H9; decompose [and] H9; clear H9;
-            unfold adapted_couple in |- *; repeat split.
-  unfold ordered_Rlist in |- *; intros; simpl in H9;
+            unfold adapted_couple; repeat split.
+  unfold ordered_Rlist; intros; simpl in H9;
     induction  i as [| i Hreci].
-  simpl in |- *; rewrite H12; replace (Rmin r1 b) with r1.
-  simpl in H0; rewrite <- H0; apply (H 0%nat); simpl in |- *; apply lt_O_Sn.
-  unfold Rmin in |- *; case (Rle_dec r1 b); intro;
+  simpl; rewrite H12; replace (Rmin r1 b) with r1.
+  simpl in H0; rewrite <- H0; apply (H 0%nat); simpl; apply lt_O_Sn.
+  unfold Rmin; case (Rle_dec r1 b); intro;
     [ reflexivity | elim n; assumption ].
   apply (H10 i); apply lt_S_n.
   replace (S (pred (Rlength lg))) with (Rlength lg).
   apply H9.
   apply S_pred with 0%nat; apply neq_O_lt; intro; rewrite <- H14 in H9;
     elim (lt_n_O _ H9).
-  simpl in |- *; assert (H14 : a <= b).
+  simpl; assert (H14 : a <= b).
   rewrite <- H1; simpl in H0; rewrite <- H0; apply RList_P7;
     [ assumption | left; reflexivity ].
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   assert (H14 : a <= b).
   rewrite <- H1; simpl in H0; rewrite <- H0; apply RList_P7;
@@ -1838,30 +1838,30 @@ Proof.
   rewrite <- H11; induction  lg as [| r0 lg Hreclg].
   simpl in H13; discriminate.
   reflexivity.
-  unfold Rmax in |- *; case (Rle_dec a b); case (Rle_dec r1 b); intros;
+  unfold Rmax; case (Rle_dec a b); case (Rle_dec r1 b); intros;
     reflexivity || elim n; assumption.
-  simpl in |- *; rewrite H13; reflexivity.
+  simpl; rewrite H13; reflexivity.
   intros; simpl in H9; induction  i as [| i Hreci].
-  unfold constant_D_eq, open_interval in |- *; simpl in |- *; intros;
+  unfold constant_D_eq, open_interval; simpl; intros;
     assert (H16 : Rmin r1 b = r1).
-  unfold Rmin in |- *; case (Rle_dec r1 b); intro;
+  unfold Rmin; case (Rle_dec r1 b); intro;
     [ reflexivity | elim n; assumption ].
   rewrite H16 in H12; rewrite H12 in H14; elim H14; clear H14; intros _ H14;
-    unfold g' in |- *; case (Rle_dec r1 x); intro r3.
+    unfold g'; case (Rle_dec r1 x); intro r3.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H14)).
   reflexivity.
   change
     (constant_D_eq g' (open_interval (pos_Rl lg i) (pos_Rl lg (S i)))
-      (pos_Rl lg2 i)) in |- *; clear Hreci; assert (H16 := H15 i);
+      (pos_Rl lg2 i)); clear Hreci; assert (H16 := H15 i);
     assert (H17 : (i < pred (Rlength lg))%nat).
   apply lt_S_n.
   replace (S (pred (Rlength lg))) with (Rlength lg).
   assumption.
-  apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro;
+  apply S_pred with 0%nat; apply neq_O_lt; red; intro;
     rewrite <- H14 in H9; elim (lt_n_O _ H9).
   assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18;
-    unfold constant_D_eq, open_interval in |- *; intros;
-      assert (H19 := H18 _ H14); rewrite <- H19; unfold g' in |- *;
+    unfold constant_D_eq, open_interval; intros;
+      assert (H19 := H18 _ H14); rewrite <- H19; unfold g';
         case (Rle_dec r1 x); intro.
   reflexivity.
   elim n; replace r1 with (Rmin r1 b).
@@ -1872,17 +1872,17 @@ Proof.
   elim (RList_P3 lg (pos_Rl lg i)); intros; apply H21; exists i; split.
   reflexivity.
   apply lt_trans with (pred (Rlength lg)); try assumption.
-  apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H22 in H17;
+  apply lt_pred_n_n; apply neq_O_lt; red; intro; rewrite <- H22 in H17;
     elim (lt_n_O _ H17).
-  unfold Rmin in |- *; case (Rle_dec r1 b); intro;
+  unfold Rmin; case (Rle_dec r1 b); intro;
     [ reflexivity | elim n0; assumption ].
   exists (mkStepFun H8); split.
-  simpl in |- *; unfold g' in |- *; case (Rle_dec r1 b); intro.
+  simpl; unfold g'; case (Rle_dec r1 b); intro.
   assumption.
   elim n; assumption.
   intros; simpl in H9; induction  i as [| i Hreci].
-  unfold constant_D_eq, co_interval in |- *; simpl in |- *; intros; simpl in H0;
-    rewrite H0; elim H10; clear H10; intros; unfold g' in |- *;
+  unfold constant_D_eq, co_interval; simpl; intros; simpl in H0;
+    rewrite H0; elim H10; clear H10; intros; unfold g';
       case (Rle_dec r1 x); intro r3.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H11)).
   reflexivity.
@@ -1890,21 +1890,21 @@ Proof.
     change
       (constant_D_eq (mkStepFun H8)
         (co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i)))
-        (f (pos_Rl (cons r1 l) i))) in |- *; assert (H10 := H6 i);
+        (f (pos_Rl (cons r1 l) i))); assert (H10 := H6 i);
       assert (H11 : (i < pred (Rlength (cons r1 l)))%nat).
-  simpl in |- *; apply lt_S_n; assumption.
+  simpl; apply lt_S_n; assumption.
   assert (H12 := H10 H11); unfold constant_D_eq, co_interval in H12;
-    unfold constant_D_eq, co_interval in |- *; intros;
-      rewrite <- (H12 _ H13); simpl in |- *; unfold g' in |- *;
+    unfold constant_D_eq, co_interval; intros;
+      rewrite <- (H12 _ H13); simpl; unfold g';
         case (Rle_dec r1 x); intro.
   reflexivity.
   elim n; elim H13; clear H13; intros;
     apply Rle_trans with (pos_Rl (cons r1 l) i); try assumption;
-      change (pos_Rl (cons r1 l) 0 <= pos_Rl (cons r1 l) i) in |- *;
+      change (pos_Rl (cons r1 l) 0 <= pos_Rl (cons r1 l) i);
         elim (RList_P6 (cons r1 l)); intros; apply H15;
           [ assumption
             | apply le_O_n
-            | simpl in |- *; apply lt_trans with (Rlength l);
+            | simpl; apply lt_trans with (Rlength l);
               [ apply lt_S_n; assumption | apply lt_n_Sn ] ].
 Qed.
 
@@ -1912,7 +1912,7 @@ Lemma StepFun_P39 :
   forall (a b:R) (f:StepFun a b),
     RiemannInt_SF f = - RiemannInt_SF (mkStepFun (StepFun_P6 (pre f))).
 Proof.
-  intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); case (Rle_dec b a);
+  intros; unfold RiemannInt_SF; case (Rle_dec a b); case (Rle_dec b a);
     intros.
   assert (H : adapted_couple f a b (subdivision f) (subdivision_val f));
     [ apply StepFun_P1
@@ -1925,16 +1925,16 @@ Proof.
           | assert (H1 : a = b);
             [ apply Rle_antisym; assumption
               | rewrite (StepFun_P8 H H1); assert (H2 : b = a);
-                [ symmetry  in |- *; apply H1 | rewrite (StepFun_P8 H0 H2); ring ] ] ] ].
+                [ symmetry ; apply H1 | rewrite (StepFun_P8 H0 H2); ring ] ] ] ].
   rewrite Ropp_involutive; eapply StepFun_P17;
     [ apply StepFun_P1
       | apply StepFun_P2; set (H := StepFun_P6 (pre f)); unfold IsStepFun in H;
-        elim H; intros; unfold is_subdivision in |- *;
+        elim H; intros; unfold is_subdivision;
           elim p; intros; apply p0 ].
   apply Ropp_eq_compat; eapply StepFun_P17;
     [ apply StepFun_P1
       | apply StepFun_P2; set (H := StepFun_P6 (pre f)); unfold IsStepFun in H;
-        elim H; intros; unfold is_subdivision in |- *;
+        elim H; intros; unfold is_subdivision;
           elim p; intros; apply p0 ].
   assert (H : a < b);
     [ auto with real
@@ -1951,34 +1951,34 @@ Lemma StepFun_P40 :
     adapted_couple f a c (cons_Rlist l1 l2) (FF (cons_Rlist l1 l2) f).
 Proof.
   intros f a b c l1 l2 lf1 lf2 H H0 H1 H2; unfold adapted_couple in H1, H2;
-    unfold adapted_couple in |- *; decompose [and] H1;
+    unfold adapted_couple; decompose [and] H1;
       decompose [and] H2; clear H1 H2; repeat split.
   apply RList_P25; try assumption.
-  rewrite H10; rewrite H4; unfold Rmin, Rmax in |- *; case (Rle_dec a b);
+  rewrite H10; rewrite H4; unfold Rmin, Rmax; case (Rle_dec a b);
     case (Rle_dec b c); intros;
       (right; reflexivity) || (elim n; left; assumption).
   rewrite RList_P22.
-  rewrite H5; unfold Rmin, Rmax in |- *; case (Rle_dec a b); case (Rle_dec a c);
+  rewrite H5; unfold Rmin, Rmax; case (Rle_dec a b); case (Rle_dec a c);
     intros;
       [ reflexivity
         | elim n; apply Rle_trans with b; left; assumption
         | elim n; left; assumption
         | elim n0; left; assumption ].
-  red in |- *; intro; rewrite H1 in H6; discriminate.
+  red; intro; rewrite H1 in H6; discriminate.
   rewrite RList_P24.
-  rewrite H9; unfold Rmin, Rmax in |- *; case (Rle_dec b c); case (Rle_dec a c);
+  rewrite H9; unfold Rmin, Rmax; case (Rle_dec b c); case (Rle_dec a c);
     intros;
       [ reflexivity
         | elim n; apply Rle_trans with b; left; assumption
         | elim n; left; assumption
         | elim n0; left; assumption ].
-  red in |- *; intro; rewrite H1 in H11; discriminate.
+  red; intro; rewrite H1 in H11; discriminate.
   apply StepFun_P20.
-  rewrite RList_P23; apply neq_O_lt; red in |- *; intro.
+  rewrite RList_P23; apply neq_O_lt; red; intro.
   assert (H2 : (Rlength l1 + Rlength l2)%nat = 0%nat).
-  symmetry  in |- *; apply H1.
+  symmetry ; apply H1.
   elim (plus_is_O _ _ H2); intros; rewrite H12 in H6; discriminate.
-  unfold constant_D_eq, open_interval in |- *; intros;
+  unfold constant_D_eq, open_interval; intros;
     elim (le_or_lt (S (S i)) (Rlength l1)); intro.
   assert (H14 : pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i).
   apply RList_P26; apply lt_S_n; apply le_lt_n_Sm; apply le_S_n;
@@ -1991,28 +1991,28 @@ Proof.
   elim (RList_P20 _ H16); intros r1 [r2 [r3 H17]]; rewrite H17;
     change
       (f x = pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i)
-      in |- *; rewrite RList_P12.
+     ; rewrite RList_P12.
   induction  i as [| i Hreci].
-  simpl in |- *; assert (H18 := H8 0%nat);
+  simpl; assert (H18 := H8 0%nat);
     unfold constant_D_eq, open_interval in H18;
       assert (H19 : (0 < pred (Rlength l1))%nat).
-  rewrite H17; simpl in |- *; apply lt_O_Sn.
+  rewrite H17; simpl; apply lt_O_Sn.
   assert (H20 := H18 H19); repeat rewrite H20.
   reflexivity.
   assert (H21 : r1 <= r2).
   rewrite H17 in H3; apply (H3 0%nat).
-  simpl in |- *; apply lt_O_Sn.
+  simpl; apply lt_O_Sn.
   elim H21; intro.
   split.
-  rewrite H17; simpl in |- *; apply Rmult_lt_reg_l with 2;
+  rewrite H17; simpl; apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
-  rewrite H17; simpl in |- *; apply Rmult_lt_reg_l with 2;
+  rewrite H17; simpl; apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite (Rplus_comm r1); rewrite double;
             apply Rplus_lt_compat_l; assumption
@@ -2041,13 +2041,13 @@ Proof.
   split.
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite (Rplus_comm (pos_Rl l1 (S i)));
             rewrite double; apply Rplus_lt_compat_l; assumption
@@ -2055,21 +2055,21 @@ Proof.
   elim H2; intros; rewrite H22 in H23;
     elim (Rlt_irrefl _ (Rlt_trans _ _ _ H23 H24)).
   assumption.
-  simpl in |- *; rewrite H17 in H1; simpl in H1; apply lt_S_n; assumption.
+  simpl; rewrite H17 in H1; simpl in H1; apply lt_S_n; assumption.
   rewrite RList_P14; rewrite H17 in H1; simpl in H1; apply H1.
   inversion H12.
   assert (H16 : pos_Rl (cons_Rlist l1 l2) (S i) = b).
   rewrite RList_P29.
-  rewrite H15; rewrite <- minus_n_n; rewrite H10; unfold Rmin in |- *;
+  rewrite H15; rewrite <- minus_n_n; rewrite H10; unfold Rmin;
     case (Rle_dec b c); intro; [ reflexivity | elim n; left; assumption ].
   rewrite H15; apply le_n.
   induction  l1 as [| r l1 Hrecl1].
   simpl in H15; discriminate.
-  clear Hrecl1; simpl in H1; simpl in |- *; apply lt_n_S; assumption.
+  clear Hrecl1; simpl in H1; simpl; apply lt_n_S; assumption.
   assert (H17 : pos_Rl (cons_Rlist l1 l2) i = b).
   rewrite RList_P26.
   replace i with (pred (Rlength l1));
-  [ rewrite H4; unfold Rmax in |- *; case (Rle_dec a b); intro;
+  [ rewrite H4; unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; left; assumption ]
     | rewrite H15; reflexivity ].
   rewrite H15; apply lt_n_Sn.
@@ -2087,22 +2087,22 @@ Proof.
   apply le_S_n; apply le_trans with (S i); [ assumption | apply le_n_Sn ].
   induction  l1 as [| r l1 Hrecl1].
   simpl in H6; discriminate.
-  clear Hrecl1; simpl in H1; simpl in |- *; apply lt_n_S; assumption.
-  symmetry  in |- *; apply minus_Sn_m; apply le_S_n; assumption.
+  clear Hrecl1; simpl in H1; simpl; apply lt_n_S; assumption.
+  symmetry ; apply minus_Sn_m; apply le_S_n; assumption.
   assert (H18 : (2 <= Rlength l1)%nat).
   clear f c l2 lf2 H0 H3 H8 H7 H10 H9 H11 H13 i H1 x H2 H12 m H14 H15 H16 H17;
     induction  l1 as [| r l1 Hrecl1].
   discriminate.
   clear Hrecl1; induction  l1 as [| r0 l1 Hrecl1].
   simpl in H5; simpl in H4; assert (H0 : Rmin a b < Rmax a b).
-  unfold Rmin, Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmin, Rmax; case (Rle_dec a b); intro;
     [ assumption | elim n; left; assumption ].
   rewrite <- H5 in H0; rewrite <- H4 in H0; elim (Rlt_irrefl _ H0).
-  clear Hrecl1; simpl in |- *; repeat apply le_n_S; apply le_O_n.
+  clear Hrecl1; simpl; repeat apply le_n_S; apply le_O_n.
   elim (RList_P20 _ H18); intros r1 [r2 [r3 H19]]; rewrite H19;
     change
       (f x = pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i)
-      in |- *; rewrite RList_P12.
+     ; rewrite RList_P12.
   induction  i as [| i Hreci].
   assert (H20 := le_S_n _ _ H15); assert (H21 := le_trans _ _ _ H18 H20);
     elim (le_Sn_O _ H21).
@@ -2120,7 +2120,7 @@ Proof.
               assert (H21 : (S i - Rlength l1 < pred (Rlength l2))%nat).
   apply lt_pred; rewrite minus_Sn_m.
   apply plus_lt_reg_l with (Rlength l1); rewrite <- le_plus_minus.
-  rewrite H19 in H1; simpl in H1; rewrite H19; simpl in |- *;
+  rewrite H19 in H1; simpl in H1; rewrite H19; simpl;
     rewrite RList_P23 in H1; apply lt_n_S; assumption.
   apply le_trans with (S i); [ apply le_S_n; assumption | apply le_n_Sn ].
   apply le_S_n; assumption.
@@ -2132,7 +2132,7 @@ Proof.
   apply H7; apply lt_pred.
   rewrite minus_Sn_m.
   apply plus_lt_reg_l with (Rlength l1); rewrite <- le_plus_minus.
-  rewrite H19 in H1; simpl in H1; rewrite H19; simpl in |- *;
+  rewrite H19 in H1; simpl in H1; rewrite H19; simpl;
     rewrite RList_P23 in H1; apply lt_n_S; assumption.
   apply le_trans with (S i); [ apply le_S_n; assumption | apply le_n_Sn ].
   apply le_S_n; assumption.
@@ -2140,13 +2140,13 @@ Proof.
   split.
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption
             | discrR ] ].
   apply Rmult_lt_reg_l with 2;
     [ prove_sup0
-      | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+      | unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
         rewrite <- Rinv_r_sym;
           [ rewrite Rmult_1_l; rewrite (Rplus_comm (pos_Rl l2 (S i - Rlength l1)));
             rewrite double; apply Rplus_lt_compat_l; assumption
@@ -2157,14 +2157,14 @@ Proof.
         rewrite H17 in H26; simpl in H24; rewrite H24 in H25;
           elim (Rlt_irrefl _ (Rlt_trans _ _ _ H25 H26)).
   assert (H23 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l2 (S i - Rlength l1)).
-  rewrite H19; simpl in |- *; simpl in H16; apply H16.
+  rewrite H19; simpl; simpl in H16; apply H16.
   assert
     (H24 :
       pos_Rl (cons_Rlist l1 l2) (S (S i)) = pos_Rl l2 (S (S i - Rlength l1))).
-  rewrite H19; simpl in |- *; simpl in H17; apply H17.
+  rewrite H19; simpl; simpl in H17; apply H17.
   rewrite <- H23; rewrite <- H24; assumption.
-  simpl in |- *; rewrite H19 in H1; simpl in H1; apply lt_S_n; assumption.
-  rewrite RList_P14; rewrite H19 in H1; simpl in H1; simpl in |- *; apply H1.
+  simpl; rewrite H19 in H1; simpl in H1; apply lt_S_n; assumption.
+  rewrite RList_P14; rewrite H19 in H1; simpl in H1; simpl; apply H1.
 Qed.
 
 Lemma StepFun_P41 :
@@ -2189,11 +2189,11 @@ Lemma StepFun_P42 :
     Int_SF (FF l1 f) l1 + Int_SF (FF l2 f) l2.
 Proof.
   intros l1 l2 f; induction l1 as [| r l1 IHl1]; intros H;
-    [ simpl in |- *; ring
+    [ simpl; ring
       | destruct l1 as [| r0 r1];
-        [ simpl in H; simpl in |- *; destruct l2 as [| r0 r1];
-          [ simpl in |- *; ring | simpl in |- *; simpl in H; rewrite H; ring ]
-          | simpl in |- *; rewrite Rplus_assoc; apply Rplus_eq_compat_l; apply IHl1;
+        [ simpl in H; simpl; destruct l2 as [| r0 r1];
+          [ simpl; ring | simpl; simpl in H; rewrite H; ring ]
+          | simpl; rewrite Rplus_assoc; apply Rplus_eq_compat_l; apply IHl1;
             rewrite <- H; reflexivity ] ].
 Qed.
 
@@ -2229,27 +2229,27 @@ Proof.
   (Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)).
   replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).
   replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).
-  symmetry  in |- *; apply StepFun_P42.
+  symmetry ; apply StepFun_P42.
   unfold adapted_couple in H1, H2; decompose [and] H1; decompose [and] H2;
-    clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *;
+    clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin;
       case (Rle_dec a b); case (Rle_dec b c); intros; reflexivity || elim n;
         assumption.
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2;
+    [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2;
       assumption
       | assumption ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1
+    [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
       | assumption ].
   eapply StepFun_P17; [ apply (StepFun_P40 H H0 H1 H2) | apply H3 ].
   replace (Int_SF lf2 l2) with 0.
   rewrite Rplus_0_r; eapply StepFun_P17;
     [ apply H1 | rewrite <- H0 in H3; apply H3 ].
-  symmetry  in |- *; eapply StepFun_P8; [ apply H2 | assumption ].
+  symmetry ; eapply StepFun_P8; [ apply H2 | assumption ].
   replace (Int_SF lf1 l1) with 0.
   rewrite Rplus_0_l; eapply StepFun_P17;
     [ apply H2 | rewrite H in H3; apply H3 ].
-  symmetry  in |- *; eapply StepFun_P8; [ apply H1 | assumption ].
+  symmetry ; eapply StepFun_P8; [ apply H1 | assumption ].
   elim n; apply Rle_trans with b; assumption.
   apply Rplus_eq_reg_l with (Int_SF lf2 l2);
     replace (Int_SF lf2 l2 + (Int_SF lf1 l1 + - Int_SF lf2 l2)) with
@@ -2264,24 +2264,24 @@ Proof.
   replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).
   apply StepFun_P42.
   unfold adapted_couple in H2, H3; decompose [and] H2; decompose [and] H3;
-    clear H3 H2; rewrite H10; rewrite H6; unfold Rmax, Rmin in |- *;
+    clear H3 H2; rewrite H10; rewrite H6; unfold Rmax, Rmin;
       case (Rle_dec a c); case (Rle_dec b c); intros;
         [ elim n; assumption
           | reflexivity
           | elim n0; assumption
           | elim n1; assumption ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2
+    [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2
       | assumption ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3
+    [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
       | assumption ].
   eapply StepFun_P17;
     [ apply (StepFun_P40 H0 H H3 (StepFun_P2 H2)) | apply H1 ].
   replace (Int_SF lf3 l3) with 0.
   rewrite Rplus_0_r; eapply StepFun_P17;
     [ apply H1 | apply StepFun_P2; rewrite <- H0 in H2; apply H2 ].
-  symmetry  in |- *; eapply StepFun_P8; [ apply H3 | assumption ].
+  symmetry ; eapply StepFun_P8; [ apply H3 | assumption ].
   replace (Int_SF lf2 l2) with (Int_SF lf3 l3 + Int_SF lf1 l1).
   ring.
   elim r; intro.
@@ -2289,19 +2289,19 @@ Proof.
   (Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1)).
   replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3).
   replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).
-  symmetry  in |- *; apply StepFun_P42.
+  symmetry ; apply StepFun_P42.
   unfold adapted_couple in H1, H3; decompose [and] H1; decompose [and] H3;
-    clear H3 H1; rewrite H9; rewrite H5; unfold Rmax, Rmin in |- *;
+    clear H3 H1; rewrite H9; rewrite H5; unfold Rmax, Rmin;
       case (Rle_dec a c); case (Rle_dec a b); intros;
         [ elim n; assumption
           | elim n1; assumption
           | reflexivity
           | elim n1; assumption ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1
+    [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
       | assumption ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3
+    [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
       | assumption ].
   eapply StepFun_P17.
   assert (H0 : c < a).
@@ -2311,7 +2311,7 @@ Proof.
   replace (Int_SF lf1 l1) with 0.
   rewrite Rplus_0_r; eapply StepFun_P17;
     [ apply H3 | rewrite <- H in H2; apply H2 ].
-  symmetry  in |- *; eapply StepFun_P8; [ apply H1 | assumption ].
+  symmetry ; eapply StepFun_P8; [ apply H1 | assumption ].
   assert (H : b < a).
   auto with real.
   replace (Int_SF lf2 l2) with (Int_SF lf3 l3 + Int_SF lf1 l1).
@@ -2321,19 +2321,19 @@ Proof.
   (Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3)).
   replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3).
   replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).
-  symmetry  in |- *; apply StepFun_P42.
+  symmetry ; apply StepFun_P42.
   unfold adapted_couple in H1, H3; decompose [and] H1; decompose [and] H3;
-    clear H3 H1; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *;
+    clear H3 H1; rewrite H11; rewrite H5; unfold Rmax, Rmin;
       case (Rle_dec a c); case (Rle_dec a b); intros;
         [ elim n; assumption
           | reflexivity
           | elim n0; assumption
           | elim n1; assumption ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1
+    [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
       | assumption ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3
+    [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
       | assumption ].
   eapply StepFun_P17.
   apply (StepFun_P40 H H0 (StepFun_P2 H1) H3).
@@ -2341,7 +2341,7 @@ Proof.
   replace (Int_SF lf3 l3) with 0.
   rewrite Rplus_0_r; eapply StepFun_P17;
     [ apply H1 | rewrite <- H0 in H2; apply StepFun_P2; apply H2 ].
-  symmetry  in |- *; eapply StepFun_P8; [ apply H3 | assumption ].
+  symmetry ; eapply StepFun_P8; [ apply H3 | assumption ].
   assert (H : c < a).
   auto with real.
   replace (Int_SF lf1 l1) with (Int_SF lf2 l2 + Int_SF lf3 l3).
@@ -2351,19 +2351,19 @@ Proof.
   (Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3)).
   replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3).
   replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).
-  symmetry  in |- *; apply StepFun_P42.
+  symmetry ; apply StepFun_P42.
   unfold adapted_couple in H2, H3; decompose [and] H2; decompose [and] H3;
-    clear H3 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *;
+    clear H3 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin;
       case (Rle_dec a c); case (Rle_dec b c); intros;
         [ elim n; assumption
           | elim n1; assumption
           | reflexivity
           | elim n1; assumption ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2
+    [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2
       | assumption ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3
+    [ apply StepFun_P21; unfold is_subdivision; split with lf3; apply H3
       | assumption ].
   eapply StepFun_P17.
   apply (StepFun_P40 H0 H H2 (StepFun_P2 H3)).
@@ -2371,7 +2371,7 @@ Proof.
   replace (Int_SF lf2 l2) with 0.
   rewrite Rplus_0_l; eapply StepFun_P17;
     [ apply H3 | rewrite H0 in H1; apply H1 ].
-  symmetry  in |- *; eapply StepFun_P8; [ apply H2 | assumption ].
+  symmetry ; eapply StepFun_P8; [ apply H2 | assumption ].
   elim n; apply Rle_trans with a; try assumption.
   auto with real.
   assert (H : c < b).
@@ -2384,56 +2384,56 @@ Proof.
   (Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1)).
   replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1).
   replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2).
-  symmetry  in |- *; apply StepFun_P42.
+  symmetry ; apply StepFun_P42.
   unfold adapted_couple in H2, H1; decompose [and] H2; decompose [and] H1;
-    clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *;
+    clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin;
       case (Rle_dec a b); case (Rle_dec b c); intros;
         [ elim n1; assumption
           | elim n1; assumption
           | elim n0; assumption
           | reflexivity ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2
+    [ apply StepFun_P21; unfold is_subdivision; split with lf2; apply H2
       | assumption ].
   eapply StepFun_P17;
-    [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1
+    [ apply StepFun_P21; unfold is_subdivision; split with lf1; apply H1
       | assumption ].
   eapply StepFun_P17.
   apply (StepFun_P40 H H0 (StepFun_P2 H2) (StepFun_P2 H1)).
   apply StepFun_P2; apply H3.
-  unfold RiemannInt_SF in |- *; case (Rle_dec a c); intro.
+  unfold RiemannInt_SF; case (Rle_dec a c); intro.
   eapply StepFun_P17.
   apply H3.
   change
     (adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun pr3))
-      (subdivision_val (mkStepFun pr3))) in |- *; apply StepFun_P1.
+      (subdivision_val (mkStepFun pr3))); apply StepFun_P1.
   apply Ropp_eq_compat; eapply StepFun_P17.
   apply H3.
   change
     (adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun pr3))
-      (subdivision_val (mkStepFun pr3))) in |- *; apply StepFun_P1.
-  unfold RiemannInt_SF in |- *; case (Rle_dec b c); intro.
+      (subdivision_val (mkStepFun pr3))); apply StepFun_P1.
+  unfold RiemannInt_SF; case (Rle_dec b c); intro.
   eapply StepFun_P17.
   apply H2.
   change
     (adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun pr2))
-      (subdivision_val (mkStepFun pr2))) in |- *; apply StepFun_P1.
+      (subdivision_val (mkStepFun pr2))); apply StepFun_P1.
   apply Ropp_eq_compat; eapply StepFun_P17.
   apply H2.
   change
     (adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun pr2))
-      (subdivision_val (mkStepFun pr2))) in |- *; apply StepFun_P1.
-  unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
+      (subdivision_val (mkStepFun pr2))); apply StepFun_P1.
+  unfold RiemannInt_SF; case (Rle_dec a b); intro.
   eapply StepFun_P17.
   apply H1.
   change
     (adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun pr1))
-      (subdivision_val (mkStepFun pr1))) in |- *; apply StepFun_P1.
+      (subdivision_val (mkStepFun pr1))); apply StepFun_P1.
   apply Ropp_eq_compat; eapply StepFun_P17.
   apply H1.
   change
     (adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun pr1))
-      (subdivision_val (mkStepFun pr1))) in |- *; apply StepFun_P1.
+      (subdivision_val (mkStepFun pr1))); apply StepFun_P1.
 Qed.
 
 Lemma StepFun_P44 :
@@ -2449,7 +2449,7 @@ Proof.
           adapted_couple f a b l1 lf1 ->
           a <= c <= b ->
           { l:Rlist & { l0:Rlist & adapted_couple f a c l l0 } }).
-  intro X; unfold IsStepFun in |- *; unfold is_subdivision in |- *; eapply X.
+  intro X; unfold IsStepFun; unfold is_subdivision; eapply X.
   apply H2.
   split; assumption.
   clear f a b c H0 H H1 H2 l1 lf1; simple induction l1.
@@ -2461,11 +2461,11 @@ Proof.
     simpl in H2; assert (H7 : a <= b).
   elim H0; intros; apply Rle_trans with c; assumption.
   replace a with (Rmin a b).
-  pattern b at 2 in |- *; replace b with (Rmax a b).
+  pattern b at 2; replace b with (Rmax a b).
   rewrite <- H2; rewrite H3; reflexivity.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   split with (cons r nil); split with lf1; assert (H2 : c = b).
   rewrite H1 in H0; elim H0; intros; apply Rle_antisym; assumption.
@@ -2479,22 +2479,22 @@ Proof.
   split with (cons r (cons c nil)); split with (cons r3 nil);
     unfold adapted_couple in H; decompose [and] H; clear H;
       assert (H6 : r = a).
-  simpl in H4; rewrite H4; unfold Rmin in |- *; case (Rle_dec a b); intro;
+  simpl in H4; rewrite H4; unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity
       | elim n; elim H0; intros; apply Rle_trans with c; assumption ].
-  elim H0; clear H0; intros; unfold adapted_couple in |- *; repeat split.
-  rewrite H6; unfold ordered_Rlist in |- *; intros; simpl in H8; inversion H8;
-    [ simpl in |- *; assumption | elim (le_Sn_O _ H10) ].
-  simpl in |- *; unfold Rmin in |- *; case (Rle_dec a c); intro;
+  elim H0; clear H0; intros; unfold adapted_couple; repeat split.
+  rewrite H6; unfold ordered_Rlist; intros; simpl in H8; inversion H8;
+    [ simpl; assumption | elim (le_Sn_O _ H10) ].
+  simpl; unfold Rmin; case (Rle_dec a c); intro;
     [ assumption | elim n; assumption ].
-  simpl in |- *; unfold Rmax in |- *; case (Rle_dec a c); intro;
+  simpl; unfold Rmax; case (Rle_dec a c); intro;
     [ reflexivity | elim n; assumption ].
-  unfold constant_D_eq, open_interval in |- *; intros; simpl in H8;
+  unfold constant_D_eq, open_interval; intros; simpl in H8;
     inversion H8.
-  simpl in |- *; assert (H10 := H7 0%nat);
+  simpl; assert (H10 := H7 0%nat);
     assert (H12 : (0 < pred (Rlength (cons r (cons r1 r2))))%nat).
-  simpl in |- *; apply lt_O_Sn.
-  apply (H10 H12); unfold open_interval in |- *; simpl in |- *;
+  simpl; apply lt_O_Sn.
+  apply (H10 H12); unfold open_interval; simpl;
     rewrite H11 in H9; simpl in H9; elim H9; clear H9;
       intros; split; try assumption.
   apply Rlt_le_trans with c; assumption.
@@ -2508,42 +2508,42 @@ Proof.
         assert (H14 : a <= b).
   elim H0; intros; apply Rle_trans with c; assumption.
   assert (H16 : r = a).
-  simpl in H7; rewrite H7; unfold Rmin in |- *; case (Rle_dec a b); intro;
+  simpl in H7; rewrite H7; unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   induction  l1' as [| r4 l1' Hrecl1'].
   simpl in H13; discriminate.
-  clear Hrecl1'; unfold adapted_couple in |- *; repeat split.
-  unfold ordered_Rlist in |- *; intros; simpl in H; induction  i as [| i Hreci].
-  simpl in |- *; replace r4 with r1.
+  clear Hrecl1'; unfold adapted_couple; repeat split.
+  unfold ordered_Rlist; intros; simpl in H; induction  i as [| i Hreci].
+  simpl; replace r4 with r1.
   apply (H5 0%nat).
-  simpl in |- *; apply lt_O_Sn.
-  simpl in H12; rewrite H12; unfold Rmin in |- *; case (Rle_dec r1 c); intro;
+  simpl; apply lt_O_Sn.
+  simpl in H12; rewrite H12; unfold Rmin; case (Rle_dec r1 c); intro;
     [ reflexivity | elim n; left; assumption ].
-  apply (H9 i); simpl in |- *; apply lt_S_n; assumption.
-  simpl in |- *; unfold Rmin in |- *; case (Rle_dec a c); intro;
+  apply (H9 i); simpl; apply lt_S_n; assumption.
+  simpl; unfold Rmin; case (Rle_dec a c); intro;
     [ assumption | elim n; elim H0; intros; assumption ].
   replace (Rmax a c) with (Rmax r1 c).
   rewrite <- H11; reflexivity.
-  unfold Rmax in |- *; case (Rle_dec r1 c); case (Rle_dec a c); intros;
+  unfold Rmax; case (Rle_dec r1 c); case (Rle_dec a c); intros;
     [ reflexivity
       | elim n; elim H0; intros; assumption
       | elim n; left; assumption
       | elim n0; left; assumption ].
-  simpl in |- *; simpl in H13; rewrite H13; reflexivity.
-  intros; simpl in H; unfold constant_D_eq, open_interval in |- *; intros;
+  simpl; simpl in H13; rewrite H13; reflexivity.
+  intros; simpl in H; unfold constant_D_eq, open_interval; intros;
     induction  i as [| i Hreci].
-  simpl in |- *; assert (H17 := H10 0%nat);
+  simpl; assert (H17 := H10 0%nat);
     assert (H18 : (0 < pred (Rlength (cons r (cons r1 r2))))%nat).
-  simpl in |- *; apply lt_O_Sn.
-  apply (H17 H18); unfold open_interval in |- *; simpl in |- *; simpl in H4;
+  simpl; apply lt_O_Sn.
+  apply (H17 H18); unfold open_interval; simpl; simpl in H4;
     elim H4; clear H4; intros; split; try assumption;
       replace r1 with r4.
   assumption.
-  simpl in H12; rewrite H12; unfold Rmin in |- *; case (Rle_dec r1 c); intro;
+  simpl in H12; rewrite H12; unfold Rmin; case (Rle_dec r1 c); intro;
     [ reflexivity | elim n; left; assumption ].
-  clear Hreci; simpl in |- *; apply H15.
-  simpl in |- *; apply lt_S_n; assumption.
-  unfold open_interval in |- *; apply H4.
+  clear Hreci; simpl; apply H15.
+  simpl; apply lt_S_n; assumption.
+  unfold open_interval; apply H4.
   split.
   left; assumption.
   elim H0; intros; assumption.
@@ -2565,7 +2565,7 @@ Proof.
           adapted_couple f a b l1 lf1 ->
           a <= c <= b ->
           { l:Rlist & { l0:Rlist & adapted_couple f c b l l0 } }).
-  intro X; unfold IsStepFun in |- *; unfold is_subdivision in |- *; eapply X;
+  intro X; unfold IsStepFun; unfold is_subdivision; eapply X;
     [ apply H2 | split; assumption ].
   clear f a b c H0 H H1 H2 l1 lf1; simple induction l1.
   intros; unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4;
@@ -2576,11 +2576,11 @@ Proof.
     simpl in H2; assert (H7 : a <= b).
   elim H0; intros; apply Rle_trans with c; assumption.
   replace a with (Rmin a b).
-  pattern b at 2 in |- *; replace b with (Rmax a b).
+  pattern b at 2; replace b with (Rmax a b).
   rewrite <- H2; rewrite H3; reflexivity.
-  unfold Rmax in |- *; case (Rle_dec a b); intro;
+  unfold Rmax; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
-  unfold Rmin in |- *; case (Rle_dec a b); intro;
+  unfold Rmin; case (Rle_dec a b); intro;
     [ reflexivity | elim n; assumption ].
   split with (cons r nil); split with lf1; assert (H2 : c = b).
   rewrite H1 in H0; elim H0; intros; apply Rle_antisym; assumption.
@@ -2593,32 +2593,32 @@ Proof.
   elim H1; intro.
   split with (cons c (cons r1 r2)); split with (cons r3 lf1);
     unfold adapted_couple in H; decompose [and] H; clear H;
-      unfold adapted_couple in |- *; repeat split.
-  unfold ordered_Rlist in |- *; intros; simpl in H; induction  i as [| i Hreci].
-  simpl in |- *; assumption.
-  clear Hreci; apply (H2 (S i)); simpl in |- *; assumption.
-  simpl in |- *; unfold Rmin in |- *; case (Rle_dec c b); intro;
+      unfold adapted_couple; repeat split.
+  unfold ordered_Rlist; intros; simpl in H; induction  i as [| i Hreci].
+  simpl; assumption.
+  clear Hreci; apply (H2 (S i)); simpl; assumption.
+  simpl; unfold Rmin; case (Rle_dec c b); intro;
     [ reflexivity | elim n; elim H0; intros; assumption ].
   replace (Rmax c b) with (Rmax a b).
   rewrite <- H3; reflexivity.
-  unfold Rmax in |- *; case (Rle_dec a b); case (Rle_dec c b); intros;
+  unfold Rmax; case (Rle_dec a b); case (Rle_dec c b); intros;
     [ reflexivity
       | elim n; elim H0; intros; assumption
       | elim n; elim H0; intros; apply Rle_trans with c; assumption
       | elim n0; elim H0; intros; apply Rle_trans with c; assumption ].
-  simpl in |- *; simpl in H5; apply H5.
+  simpl; simpl in H5; apply H5.
   intros; simpl in H; induction  i as [| i Hreci].
-  unfold constant_D_eq, open_interval in |- *; intros; simpl in |- *;
+  unfold constant_D_eq, open_interval; intros; simpl;
     apply (H7 0%nat).
-  simpl in |- *; apply lt_O_Sn.
-  unfold open_interval in |- *; simpl in |- *; simpl in H6; elim H6; clear H6;
+  simpl; apply lt_O_Sn.
+  unfold open_interval; simpl; simpl in H6; elim H6; clear H6;
     intros; split; try assumption; apply Rle_lt_trans with c;
       try assumption; replace r with a.
   elim H0; intros; assumption.
-  simpl in H4; rewrite H4; unfold Rmin in |- *; case (Rle_dec a b); intros;
+  simpl in H4; rewrite H4; unfold Rmin; case (Rle_dec a b); intros;
     [ reflexivity
       | elim n; elim H0; intros; apply Rle_trans with c; assumption ].
-  clear Hreci; apply (H7 (S i)); simpl in |- *; assumption.
+  clear Hreci; apply (H7 (S i)); simpl; assumption.
   cut (adapted_couple f r1 b (cons r1 r2) lf1).
   cut (r1 <= c <= b).
   intros; elim (X0 _ _ _ _ _ H3 H2); intros l1' [lf1' H4]; split with l1';
diff --git a/theories/Reals/Rlimit.v b/theories/Reals/Rlimit.v
index 5c864de3..c5ee828a 100644
--- a/theories/Reals/Rlimit.v
+++ b/theories/Reals/Rlimit.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -14,7 +14,7 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import Fourier.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (*******************************)
 (** *   Calculus               *)
@@ -31,7 +31,7 @@ Proof.
   intro esp.
   assert (H := double_var esp).
   unfold Rdiv in H.
-  symmetry  in |- *; exact H.
+  symmetry ; exact H.
 Qed.
 
 (*********)
@@ -39,9 +39,9 @@ Lemma eps4 : forall eps:R, eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2.
 Proof.
   intro eps.
   replace (2 + 2) with 4.
-  pattern eps at 3 in |- *; rewrite double_var.
+  pattern eps at 3; rewrite double_var.
   rewrite (Rmult_plus_distr_r (eps / 2) (eps / 2) (/ 2)).
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   repeat rewrite Rmult_assoc.
   rewrite <- Rinv_mult_distr.
   reflexivity.
@@ -54,7 +54,7 @@ Qed.
 Lemma Rlt_eps2_eps : forall eps:R, eps > 0 -> eps * / 2 < eps.
 Proof.
   intros.
-  pattern eps at 2 in |- *; rewrite <- Rmult_1_r.
+  pattern eps at 2; rewrite <- Rmult_1_r.
   repeat rewrite (Rmult_comm eps).
   apply Rmult_lt_compat_r.
   exact H.
@@ -70,7 +70,7 @@ Lemma Rlt_eps4_eps : forall eps:R, eps > 0 -> eps * / (2 + 2) < eps.
 Proof.
   intros.
   replace (2 + 2) with 4.
-  pattern eps at 2 in |- *; rewrite <- Rmult_1_r.
+  pattern eps at 2; rewrite <- Rmult_1_r.
   repeat rewrite (Rmult_comm eps).
   apply Rmult_lt_compat_r.
   exact H.
@@ -113,10 +113,10 @@ Qed.
 (*********)
 Lemma mul_factor_gt : forall eps l l':R, eps > 0 -> eps * mul_factor l l' > 0.
 Proof.
-  intros; unfold Rgt in |- *; rewrite <- (Rmult_0_r eps);
+  intros; unfold Rgt; rewrite <- (Rmult_0_r eps);
     apply Rmult_lt_compat_l.
   assumption.
-  unfold mul_factor in |- *; apply Rinv_0_lt_compat;
+  unfold mul_factor; apply Rinv_0_lt_compat;
     cut (1 <= 1 + (Rabs l + Rabs l')).
   cut (0 < 1).
   exact (Rlt_le_trans _ _ _).
@@ -196,7 +196,7 @@ Proof.
   case (H0 (dist R_met (f x0) l)); auto.
   intros alpha1 [H2 H3]; apply H3; auto; split; auto.
   case (dist_refl R_met x0 x0); intros Hr1 Hr2; rewrite Hr2; auto.
-  case (dist_refl R_met (f x0) l); intros Hr1 Hr2; apply sym_eq; auto.
+  case (dist_refl R_met (f x0) l); intros Hr1 Hr2; symmetry; auto.
 Qed.
 
 (*********)
@@ -210,7 +210,7 @@ Qed.
 (*********)
 Lemma lim_x : forall (D:R -> Prop) (x0:R), limit1_in (fun x:R => x) D x0 x0.
 Proof.
-  unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros;
+  unfold limit1_in; unfold limit_in; simpl; intros;
     split with eps; split; auto; intros; elim H0; intros;
       auto.
 Qed.
@@ -221,9 +221,9 @@ Lemma limit_plus :
     limit1_in f D l x0 ->
     limit1_in g D l' x0 -> limit1_in (fun x:R => f x + g x) D (l + l') x0.
 Proof.
-  intros; unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *;
+  intros; unfold limit1_in; unfold limit_in; simpl;
     intros; elim (H (eps * / 2) (eps2_Rgt_R0 eps H1));
-      elim (H0 (eps * / 2) (eps2_Rgt_R0 eps H1)); simpl in |- *;
+      elim (H0 (eps * / 2) (eps2_Rgt_R0 eps H1)); simpl;
         clear H H0; intros; elim H; elim H0; clear H H0; intros;
           split with (Rmin x1 x); split.
   exact (Rmin_Rgt_r x1 x 0 (conj H H2)).
@@ -244,12 +244,12 @@ Lemma limit_Ropp :
   forall (f:R -> R) (D:R -> Prop) (l x0:R),
     limit1_in f D l x0 -> limit1_in (fun x:R => - f x) D (- l) x0.
 Proof.
-  unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros;
+  unfold limit1_in; unfold limit_in; simpl; intros;
     elim (H eps H0); clear H; intros; elim H; clear H;
       intros; split with x; split; auto; intros; generalize (H1 x1 H2);
-        clear H1; intro; unfold R_dist in |- *; unfold Rminus in |- *;
+        clear H1; intro; unfold R_dist; unfold Rminus;
           rewrite (Ropp_involutive l); rewrite (Rplus_comm (- f x1) l);
-            fold (l - f x1) in |- *; fold (R_dist l (f x1)) in |- *;
+            fold (l - f x1); fold (R_dist l (f x1));
               rewrite R_dist_sym; assumption.
 Qed.
 
@@ -259,7 +259,7 @@ Lemma limit_minus :
     limit1_in f D l x0 ->
     limit1_in g D l' x0 -> limit1_in (fun x:R => f x - g x) D (l - l') x0.
 Proof.
-  intros; unfold Rminus in |- *; generalize (limit_Ropp g D l' x0 H0); intro;
+  intros; unfold Rminus; generalize (limit_Ropp g D l' x0 H0); intro;
     exact (limit_plus f (fun x:R => - g x) D l (- l') x0 H H1).
 Qed.
 
@@ -268,9 +268,9 @@ Lemma limit_free :
   forall (f:R -> R) (D:R -> Prop) (x x0:R),
     limit1_in (fun h:R => f x) D (f x) x0.
 Proof.
-  unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros;
+  unfold limit1_in; unfold limit_in; simpl; intros;
     split with eps; split; auto; intros; elim (R_dist_refl (f x) (f x));
-      intros a b; rewrite (b (refl_equal (f x))); unfold Rgt in H;
+      intros a b; rewrite (b (eq_refl (f x))); unfold Rgt in H;
         assumption.
 Qed.
 
@@ -280,14 +280,14 @@ Lemma limit_mul :
     limit1_in f D l x0 ->
     limit1_in g D l' x0 -> limit1_in (fun x:R => f x * g x) D (l * l') x0.
 Proof.
-  intros; unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *;
+  intros; unfold limit1_in; unfold limit_in; simpl;
     intros;
       elim (H (Rmin 1 (eps * mul_factor l l')) (mul_factor_gt_f eps l l' H1));
         elim (H0 (eps * mul_factor l l') (mul_factor_gt eps l l' H1));
-          clear H H0; simpl in |- *; intros; elim H; elim H0;
+          clear H H0; simpl; intros; elim H; elim H0;
             clear H H0; intros; split with (Rmin x1 x); split.
   exact (Rmin_Rgt_r x1 x 0 (conj H H2)).
-  intros; elim H4; clear H4; intros; unfold R_dist in |- *;
+  intros; elim H4; clear H4; intros; unfold R_dist;
     replace (f x2 * g x2 - l * l') with (f x2 * (g x2 - l') + l' * (f x2 - l)).
   cut (Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) < eps).
   cut
@@ -309,7 +309,7 @@ Proof.
   apply Rmult_ge_0_gt_0_lt_compat.
   apply Rle_ge.
   exact (Rabs_pos (g x2 - l')).
-  rewrite (Rplus_comm 1 (Rabs l)); unfold Rgt in |- *; apply Rle_lt_0_plus_1;
+  rewrite (Rplus_comm 1 (Rabs l)); unfold Rgt; apply Rle_lt_0_plus_1;
     exact (Rabs_pos l).
   unfold R_dist in H9;
     apply (Rplus_lt_reg_r (- Rabs l) (Rabs (f x2)) (1 + Rabs l)).
@@ -323,13 +323,13 @@ Proof.
   generalize (H3 x2 (conj H4 H6)); trivial.
   apply Rmult_le_compat_l.
   exact (Rabs_pos l').
-  unfold Rle in |- *; left; assumption.
+  unfold Rle; left; assumption.
   rewrite (Rmult_comm (1 + Rabs l) (eps * mul_factor l l'));
     rewrite (Rmult_comm (Rabs l') (eps * mul_factor l l'));
       rewrite <-
         (Rmult_plus_distr_l (eps * mul_factor l l') (1 + Rabs l) (Rabs l'))
         ; rewrite (Rmult_assoc eps (mul_factor l l') (1 + Rabs l + Rabs l'));
-          rewrite (Rplus_assoc 1 (Rabs l) (Rabs l')); unfold mul_factor in |- *;
+          rewrite (Rplus_assoc 1 (Rabs l) (Rabs l')); unfold mul_factor;
             rewrite (Rinv_l (1 + (Rabs l + Rabs l')) (mul_factor_wd l l'));
               rewrite (proj1 (Rmult_ne eps)); apply Req_le; trivial.
   ring.
@@ -344,10 +344,10 @@ Lemma single_limit :
   forall (f:R -> R) (D:R -> Prop) (l l' x0:R),
     adhDa D x0 -> limit1_in f D l x0 -> limit1_in f D l' x0 -> l = l'.
 Proof.
-  unfold limit1_in in |- *; unfold limit_in in |- *; intros.
+  unfold limit1_in; unfold limit_in; intros.
   cut (forall eps:R, eps > 0 -> dist R_met l l' < 2 * eps).
-  clear H0 H1; unfold dist in |- *; unfold R_met in |- *; unfold R_dist in |- *;
-    unfold Rabs in |- *; case (Rcase_abs (l - l')); intros.
+  clear H0 H1; unfold dist; unfold R_met; unfold R_dist;
+    unfold Rabs; case (Rcase_abs (l - l')); intros.
   cut (forall eps:R, eps > 0 -> - (l - l') < eps).
   intro; generalize (prop_eps (- (l - l')) H1); intro;
     generalize (Ropp_gt_lt_0_contravar (l - l') r); intro;
@@ -358,10 +358,10 @@ Proof.
     rewrite <- (Rmult_assoc 2 (/ 2) eps); rewrite (Rinv_r 2).
   elim (Rmult_ne eps); intros a b; rewrite b; clear a b; trivial.
   apply (Rlt_dichotomy_converse 2 0); right; generalize Rlt_0_1; intro;
-    unfold Rgt in |- *; generalize (Rplus_lt_compat_l 1 0 1 H3);
+    unfold Rgt; generalize (Rplus_lt_compat_l 1 0 1 H3);
       intro; elim (Rplus_ne 1); intros a b; rewrite a in H4;
         clear a b; apply (Rlt_trans 0 1 2 H3 H4).
-  unfold Rgt in |- *; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2));
+  unfold Rgt; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2));
     rewrite <- (Rmult_0_r (/ 2)); apply (Rmult_lt_compat_l (/ 2) 0 eps);
       auto.
   apply (Rinv_0_lt_compat 2); cut (1 < 2).
@@ -380,10 +380,10 @@ Proof.
     rewrite <- (Rmult_assoc 2 (/ 2) eps); rewrite (Rinv_r 2).
   elim (Rmult_ne eps); intros a b; rewrite b; clear a b; trivial.
   apply (Rlt_dichotomy_converse 2 0); right; generalize Rlt_0_1; intro;
-    unfold Rgt in |- *; generalize (Rplus_lt_compat_l 1 0 1 H3);
+    unfold Rgt; generalize (Rplus_lt_compat_l 1 0 1 H3);
       intro; elim (Rplus_ne 1); intros a b; rewrite a in H4;
         clear a b; apply (Rlt_trans 0 1 2 H3 H4).
-  unfold Rgt in |- *; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2));
+  unfold Rgt; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2));
     rewrite <- (Rmult_0_r (/ 2)); apply (Rmult_lt_compat_l (/ 2) 0 eps);
       auto.
   apply (Rinv_0_lt_compat 2); cut (1 < 2).
@@ -393,7 +393,7 @@ Proof.
 (**)
   intros; unfold adhDa in H; elim (H0 eps H2); intros; elim (H1 eps H2); intros;
     clear H0 H1; elim H3; elim H4; clear H3 H4; intros;
-      simpl in |- *; simpl in H1, H4; generalize (Rmin_Rgt x x1 0);
+      simpl; simpl in H1, H4; generalize (Rmin_Rgt x x1 0);
         intro; elim H5; intros; clear H5; elim (H (Rmin x x1) (H7 (conj H3 H0)));
           intros; elim H5; intros; clear H5 H H6 H7;
             generalize (Rmin_Rgt x x1 (R_dist x2 x0)); intro;
@@ -403,10 +403,10 @@ Proof.
                     intros;
                       generalize
                         (Rplus_lt_compat (R_dist (f x2) l) eps (R_dist (f x2) l') eps H H0);
-                        unfold R_dist in |- *; intros; rewrite (Rabs_minus_sym (f x2) l) in H1;
+                        unfold R_dist; intros; rewrite (Rabs_minus_sym (f x2) l) in H1;
                           rewrite (Rmult_comm 2 eps); rewrite (Rmult_plus_distr_l eps 1 1);
                             elim (Rmult_ne eps); intros a b; rewrite a; clear a b;
-                              generalize (R_dist_tri l l' (f x2)); unfold R_dist in |- *;
+                              generalize (R_dist_tri l l' (f x2)); unfold R_dist;
                                 intros;
                                   apply
                                     (Rle_lt_trans (Rabs (l - l')) (Rabs (l - f x2) + Rabs (f x2 - l'))
@@ -419,7 +419,7 @@ Lemma limit_comp :
     limit1_in f Df l x0 ->
     limit1_in g Dg l' l -> limit1_in (fun x:R => g (f x)) (Dgf Df Dg f) l' x0.
 Proof.
-  unfold limit1_in, limit_in, Dgf in |- *; simpl in |- *.
+  unfold limit1_in, limit_in, Dgf; simpl.
   intros f g Df Dg l l' x0 Hf Hg eps eps_pos.
   elim (Hg eps eps_pos).
   intros alpg lg.
@@ -436,12 +436,12 @@ Lemma limit_inv :
   forall (f:R -> R) (D:R -> Prop) (l x0:R),
     limit1_in f D l x0 -> l <> 0 -> limit1_in (fun x:R => / f x) D (/ l) x0.
 Proof.
-  unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *;
-    unfold R_dist in |- *; intros; elim (H (Rabs l / 2)).
+  unfold limit1_in; unfold limit_in; simpl;
+    unfold R_dist; intros; elim (H (Rabs l / 2)).
   intros delta1 H2; elim (H (eps * (Rsqr l / 2))).
   intros delta2 H3; elim H2; elim H3; intros; exists (Rmin delta1 delta2);
     split.
-  unfold Rmin in |- *; case (Rle_dec delta1 delta2); intro; assumption.
+  unfold Rmin; case (Rle_dec delta1 delta2); intro; assumption.
   intro; generalize (H5 x); clear H5; intro H5; generalize (H7 x); clear H7;
     intro H7; intro H10; elim H10; intros; cut (D x /\ Rabs (x - x0) < delta1).
   cut (D x /\ Rabs (x - x0) < delta2).
@@ -455,7 +455,7 @@ Proof.
               (Rplus_lt_compat_l (Rabs (f x) - Rabs l / 2) (Rabs l - Rabs (f x))
                 (Rabs l / 2) H14);
               replace (Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))) with (Rabs l / 2).
-  unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l;
+  unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l;
     rewrite Rplus_0_r; intro; cut (f x <> 0).
   intro; replace (/ f x + - / l) with ((l - f x) * / (l * f x)).
   rewrite Rabs_mult; rewrite Rabs_Rinv.
@@ -467,7 +467,7 @@ Proof.
         (/ Rabs (l * f x)) (2 / Rsqr l) (Rabs_pos (l - f x)) H18 H5 H17);
       replace (eps * (Rsqr l / 2) * (2 / Rsqr l)) with eps.
   intro; assumption.
-  unfold Rdiv in |- *; unfold Rsqr in |- *; rewrite Rinv_mult_distr.
+  unfold Rdiv; unfold Rsqr; rewrite Rinv_mult_distr.
   repeat rewrite Rmult_assoc.
   rewrite (Rmult_comm l).
   repeat rewrite Rmult_assoc.
@@ -487,7 +487,7 @@ Proof.
   left; apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply prod_neq_R0;
     assumption.
   rewrite Rmult_comm; rewrite Rabs_mult; rewrite Rinv_mult_distr.
-  rewrite (Rsqr_abs l); unfold Rsqr in |- *; unfold Rdiv in |- *;
+  rewrite (Rsqr_abs l); unfold Rsqr; unfold Rdiv;
     rewrite Rinv_mult_distr.
   repeat rewrite <- Rmult_assoc; apply Rmult_lt_compat_r.
   apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
@@ -496,7 +496,7 @@ Proof.
   apply Rabs_pos_lt; assumption.
   apply Rabs_pos_lt; assumption.
   apply Rinv_0_lt_compat; cut (0%nat <> 2%nat);
-    [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR in |- *;
+    [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR;
       intro H18; assumption
       | discriminate ].
   replace (Rabs (f x) * Rabs l * / 2 * / Rabs (f x)) with (Rabs l / 2).
@@ -512,7 +512,7 @@ Proof.
   discrR.
   apply Rabs_no_R0.
   assumption.
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   repeat rewrite Rmult_assoc.
   rewrite (Rmult_comm (Rabs (f x))).
   repeat rewrite Rmult_assoc.
@@ -526,7 +526,7 @@ Proof.
   apply Rabs_no_R0; assumption.
   apply prod_neq_R0; assumption.
   rewrite (Rinv_mult_distr _ _ H0 H16).
-  unfold Rminus in |- *; rewrite Rmult_plus_distr_r.
+  unfold Rminus; rewrite Rmult_plus_distr_r.
   rewrite <- Rmult_assoc.
   rewrite <- Rinv_r_sym.
   rewrite Rmult_1_l.
@@ -538,16 +538,16 @@ Proof.
   reflexivity.
   assumption.
   assumption.
-  red in |- *; intro; rewrite H16 in H15; rewrite Rabs_R0 in H15;
+  red; intro; rewrite H16 in H15; rewrite Rabs_R0 in H15;
     cut (0 < Rabs l / 2).
   intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (Rabs l / 2) 0 H17 H15)).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   apply Rabs_pos_lt; assumption.
   apply Rinv_0_lt_compat; cut (0%nat <> 2%nat);
-    [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR in |- *;
+    [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR;
       intro; assumption
       | discriminate ].
-  pattern (Rabs l) at 3 in |- *; rewrite double_var.
+  pattern (Rabs l) at 3; rewrite double_var.
   ring.
   split;
     [ assumption
@@ -557,18 +557,18 @@ Proof.
     [ assumption
       | apply Rlt_le_trans with (Rmin delta1 delta2);
         [ assumption | apply Rmin_l ] ].
-  change (0 < eps * (Rsqr l / 2)) in |- *; unfold Rdiv in |- *;
+  change (0 < eps * (Rsqr l / 2)); unfold Rdiv;
     repeat rewrite Rmult_assoc; apply Rmult_lt_0_compat.
   assumption.
   apply Rmult_lt_0_compat. apply Rsqr_pos_lt; assumption.
   apply Rinv_0_lt_compat; cut (0%nat <> 2%nat);
-    [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR in |- *;
+    [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR;
       intro; assumption
       | discriminate ].
-  change (0 < Rabs l / 2) in |- *; unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  change (0 < Rabs l / 2); unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply Rabs_pos_lt; assumption
       | apply Rinv_0_lt_compat; cut (0%nat <> 2%nat);
-        [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR in |- *;
+        [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR;
           intro; assumption
           | discriminate ] ].
 Qed.
diff --git a/theories/Reals/Rlogic.v b/theories/Reals/Rlogic.v
index 2237ea6e..0b892a76 100644
--- a/theories/Reals/Rlogic.v
+++ b/theories/Reals/Rlogic.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -271,10 +271,10 @@ assert (H2 : ~ is_upper_bound E M').
   intro H5.
   assert (M <= M')%R by (apply H4; exact H5).
   apply (Rlt_not_le M M').
-    unfold M' in |- *.
-    pattern M at 2 in |- *.
+    unfold M'.
+    pattern M at 2.
     rewrite <- Rplus_0_l.
-    pattern (0 + M)%R in |- *.
+    pattern (0 + M)%R.
     rewrite Rplus_comm.
     rewrite <- (Rplus_opp_r 1).
     apply Rplus_lt_compat_l.
@@ -284,7 +284,7 @@ assert (H2 : ~ is_upper_bound E M').
 apply H2.
 intros N (n,H7).
 rewrite H7.
-unfold M' in |- *.
+unfold M'.
 assert (H5 : (INR (S n) <= M)%R) by (apply H3; exists (S n); reflexivity).
 rewrite S_INR in H5.
 assert (H6 : (INR n + 1 + -1 <= M + -1)%R).
diff --git a/theories/Reals/Rminmax.v b/theories/Reals/Rminmax.v
index 8f8207d7..da3c6ddd 100644
--- a/theories/Reals/Rminmax.v
+++ b/theories/Reals/Rminmax.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
diff --git a/theories/Reals/Rpow_def.v b/theories/Reals/Rpow_def.v
index 026153b7..cd94169f 100644
--- a/theories/Reals/Rpow_def.v
+++ b/theories/Reals/Rpow_def.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v
index 593e54c6..43f326a0 100644
--- a/theories/Reals/Rpower.v
+++ b/theories/Reals/Rpower.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -15,25 +15,25 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
-Require Import Rtrigo.
+Require Import Rtrigo1.
 Require Import Ranalysis1.
 Require Import Exp_prop.
 Require Import Rsqrt_def.
 Require Import R_sqrt.
 Require Import MVT.
 Require Import Ranalysis4.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Lemma P_Rmin : forall (P:R -> Prop) (x y:R), P x -> P y -> P (Rmin x y).
 Proof.
-  intros P x y H1 H2; unfold Rmin in |- *; case (Rle_dec x y); intro;
+  intros P x y H1 H2; unfold Rmin; case (Rle_dec x y); intro;
     assumption.
 Qed.
 
 Lemma exp_le_3 : exp 1 <= 3.
 Proof.
   assert (exp_1 : exp 1 <> 0).
-  assert (H0 := exp_pos 1); red in |- *; intro; rewrite H in H0;
+  assert (H0 := exp_pos 1); red; intro; rewrite H in H0;
     elim (Rlt_irrefl _ H0).
   apply Rmult_le_reg_l with (/ exp 1).
   apply Rinv_0_lt_compat; apply exp_pos.
@@ -43,7 +43,7 @@ Proof.
   rewrite Rmult_1_r; rewrite <- (Rmult_comm 3); rewrite <- Rmult_assoc;
     rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l; replace (/ exp 1) with (exp (-1)).
-  unfold exp in |- *; case (exist_exp (-1)); intros; simpl in |- *;
+  unfold exp; case (exist_exp (-1)); intros; simpl;
     unfold exp_in in e;
       assert (H := alternated_series_ineq (fun i:nat => / INR (fact i)) x 1).
   cut
@@ -73,7 +73,7 @@ Proof.
   ring.
   discrR.
   apply H.
-  unfold Un_decreasing in |- *; intros;
+  unfold Un_decreasing; intros;
     apply Rmult_le_reg_l with (INR (fact n)).
   apply INR_fact_lt_0.
   apply Rmult_le_reg_l with (INR (fact (S n))).
@@ -84,13 +84,13 @@ Proof.
   rewrite Rmult_1_r; apply le_INR; apply fact_le; apply le_n_Sn.
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
-  assert (H0 := cv_speed_pow_fact 1); unfold Un_cv in |- *; unfold Un_cv in H0;
+  assert (H0 := cv_speed_pow_fact 1); unfold Un_cv; unfold Un_cv in H0;
     intros; elim (H0 _ H1); intros; exists x0; intros;
-      unfold R_dist in H2; unfold R_dist in |- *;
+      unfold R_dist in H2; unfold R_dist;
         replace (/ INR (fact n)) with (1 ^ n / INR (fact n)).
   apply (H2 _ H3).
-  unfold Rdiv in |- *; rewrite pow1; rewrite Rmult_1_l; reflexivity.
-  unfold infinite_sum in e; unfold Un_cv, tg_alt in |- *; intros; elim (e _ H0);
+  unfold Rdiv; rewrite pow1; rewrite Rmult_1_l; reflexivity.
+  unfold infinite_sum in e; unfold Un_cv, tg_alt; intros; elim (e _ H0);
     intros; exists x0; intros;
       replace (sum_f_R0 (fun i:nat => (-1) ^ i * / INR (fact i)) n) with
       (sum_f_R0 (fun i:nat => / INR (fact i) * (-1) ^ i) n).
@@ -121,7 +121,7 @@ Proof.
   intro.
   replace (derive_pt exp x0 (H0 x0)) with (exp x0).
   apply exp_pos.
-  symmetry  in |- *; apply derive_pt_eq_0.
+  symmetry ; apply derive_pt_eq_0.
   apply (derivable_pt_lim_exp x0).
   apply H.
 Qed.
@@ -143,11 +143,11 @@ Proof.
         rewrite Ropp_0; rewrite Rplus_0_r;
           replace (derive_pt exp x0 (derivable_exp x0)) with (exp x0).
   rewrite exp_0; rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
-    pattern x at 1 in |- *; rewrite <- Rmult_1_r; rewrite (Rmult_comm (exp x0));
+    pattern x at 1; rewrite <- Rmult_1_r; rewrite (Rmult_comm (exp x0));
       apply Rmult_lt_compat_l.
   apply H.
   rewrite <- exp_0; apply exp_increasing; elim H3; intros; assumption.
-  symmetry  in |- *; apply derive_pt_eq_0; apply derivable_pt_lim_exp.
+  symmetry ; apply derive_pt_eq_0; apply derivable_pt_lim_exp.
 Qed.
 
 Lemma ln_exists1 : forall y:R, 1 <= y -> { z:R | y = exp z }.
@@ -160,18 +160,18 @@ Proof.
   cut (f 0 * f y <= 0); [intro H4|].
   pose proof (IVT_cor f 0 y H2 (Rlt_le _ _ H0) H4) as (t,(_,H7));
     exists t; unfold f in H7; apply Rminus_diag_uniq_sym; exact H7.
-  pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f y));
+  pattern 0 at 2; rewrite <- (Rmult_0_r (f y));
     rewrite (Rmult_comm (f 0)); apply Rmult_le_compat_l;
       assumption.
-  unfold f in |- *; apply Rplus_le_reg_l with y; left;
+  unfold f; apply Rplus_le_reg_l with y; left;
     apply Rlt_trans with (1 + y).
   rewrite <- (Rplus_comm y); apply Rplus_lt_compat_l; apply Rlt_0_1.
   replace (y + (exp y - y)) with (exp y); [ apply (exp_ineq1 y H0) | ring ].
-  unfold f in |- *; change (continuity (exp - fct_cte y)) in |- *;
+  unfold f; change (continuity (exp - fct_cte y));
     apply continuity_minus;
       [ apply derivable_continuous; apply derivable_exp
         | apply derivable_continuous; apply derivable_const ].
-  unfold f in |- *; rewrite exp_0; apply Rplus_le_reg_l with y;
+  unfold f; rewrite exp_0; apply Rplus_le_reg_l with y;
     rewrite Rplus_0_r; replace (y + (1 - y)) with 1; [ apply H | ring ].
 Qed.
 
@@ -185,18 +185,18 @@ Proof.
   apply H.
   rewrite <- Rinv_r_sym.
   rewrite Rmult_1_r; left; apply (Rnot_le_lt _ _ n).
-  red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H).
   destruct (ln_exists1 _ H0) as (x,p); exists (- x);
     apply Rmult_eq_reg_l with (exp x / y).
-  unfold Rdiv in |- *; rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
+  unfold Rdiv; rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r; rewrite <- (Rmult_comm (/ y)); rewrite Rmult_assoc;
     rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0;
-      rewrite Rmult_1_r; symmetry  in |- *; apply p.
-  red in |- *; intro H3; rewrite H3 in H; elim (Rlt_irrefl _ H).
-  unfold Rdiv in |- *; apply prod_neq_R0.
-  assert (H3 := exp_pos x); red in |- *; intro H4; rewrite H4 in H3;
+      rewrite Rmult_1_r; symmetry ; apply p.
+  red; intro H3; rewrite H3 in H; elim (Rlt_irrefl _ H).
+  unfold Rdiv; apply prod_neq_R0.
+  assert (H3 := exp_pos x); red; intro H4; rewrite H4 in H3;
     elim (Rlt_irrefl _ H3).
-  apply Rinv_neq_0_compat; red in |- *; intro H3; rewrite H3 in H;
+  apply Rinv_neq_0_compat; red; intro H3; rewrite H3 in H;
     elim (Rlt_irrefl _ H).
 Qed.
 
@@ -213,11 +213,11 @@ Definition ln (x:R) : R :=
 
 Lemma exp_ln : forall x:R, 0 < x -> exp (ln x) = x.
 Proof.
-  intros; unfold ln in |- *; case (Rlt_dec 0 x); intro.
-  unfold Rln in |- *;
+  intros; unfold ln; case (Rlt_dec 0 x); intro.
+  unfold Rln;
     case (ln_exists (mkposreal x r) (cond_pos (mkposreal x r)));
     intros.
-  simpl in e; symmetry  in |- *; apply e.
+  simpl in e; symmetry ; apply e.
   elim n; apply H.
 Qed.
 
@@ -231,7 +231,7 @@ Qed.
 Theorem exp_Ropp : forall x:R, exp (- x) = / exp x.
 Proof.
   intros x; assert (H : exp x <> 0).
-  assert (H := exp_pos x); red in |- *; intro; rewrite H0 in H;
+  assert (H := exp_pos x); red; intro; rewrite H0 in H;
     elim (Rlt_irrefl _ H).
   apply Rmult_eq_reg_l with (r := exp x).
   rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0.
@@ -306,11 +306,11 @@ Theorem ln_continue :
   forall y:R, 0 < y -> continue_in ln (fun x:R => 0 < x) y.
 Proof.
   intros y H.
-  unfold continue_in, limit1_in, limit_in in |- *; intros eps Heps.
+  unfold continue_in, limit1_in, limit_in; intros eps Heps.
   cut (1 < exp eps); [ intros H1 | idtac ].
   cut (exp (- eps) < 1); [ intros H2 | idtac ].
   exists (Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))); split.
-  red in |- *; apply P_Rmin.
+  red; apply P_Rmin.
   apply Rmult_lt_0_compat.
   assumption.
   apply Rplus_lt_reg_r with 1.
@@ -321,7 +321,7 @@ Proof.
   apply Rplus_lt_reg_r with (exp (- eps)).
   rewrite Rplus_0_r; replace (exp (- eps) + (1 - exp (- eps))) with 1;
     [ apply H2 | ring ].
-  unfold dist, R_met, R_dist in |- *; simpl in |- *.
+  unfold dist, R_met, R_dist; simpl.
   intros x [[H3 H4] H5].
   cut (y * (x * / y) = x).
   intro Hxyy.
@@ -351,7 +351,7 @@ Proof.
   rewrite Hxyy; rewrite Rmult_1_r; apply Hxy.
   rewrite Hxy; rewrite Rinv_r.
   rewrite ln_1; rewrite Rabs_R0; apply Heps.
-  red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H).
   rewrite Rabs_right.
   apply exp_lt_inv.
   rewrite exp_ln.
@@ -366,7 +366,7 @@ Proof.
   left; apply (Rgt_minus _ _ Hxy).
   apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ].
   rewrite <- ln_1.
-  apply Rgt_ge; red in |- *; apply ln_increasing.
+  apply Rgt_ge; red; apply ln_increasing.
   apply Rlt_0_1.
   apply Rmult_lt_reg_l with (r := y).
   apply H.
@@ -379,7 +379,7 @@ Proof.
   apply Rinv_0_lt_compat; assumption.
   rewrite (Rmult_comm x); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
   ring.
-  red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H0 in H; elim (Rlt_irrefl _ H).
   apply Rmult_lt_reg_l with (exp eps).
   apply exp_pos.
   rewrite <- exp_plus; rewrite Rmult_1_r; rewrite Rplus_opp_r; rewrite exp_0;
@@ -394,7 +394,7 @@ Qed.
 
 Definition Rpower (x y:R) := exp (y * ln x).
 
-Infix Local "^R" := Rpower (at level 30, right associativity) : R_scope.
+Local Infix "^R" := Rpower (at level 30, right associativity) : R_scope.
 
 (******************************************************************)
 (** *                     Properties of  Rpower                   *)
@@ -412,13 +412,13 @@ Infix Local "^R" := Rpower (at level 30, right associativity) : R_scope.
 
 Theorem Rpower_plus : forall x y z:R, z ^R (x + y) = z ^R x * z ^R y.
 Proof.
-  intros x y z; unfold Rpower in |- *.
+  intros x y z; unfold Rpower.
   rewrite Rmult_plus_distr_r; rewrite exp_plus; auto.
 Qed.
 
 Theorem Rpower_mult : forall x y z:R, (x ^R y) ^R z = x ^R (y * z).
 Proof.
-  intros x y z; unfold Rpower in |- *.
+  intros x y z; unfold Rpower.
   rewrite ln_exp.
   replace (z * (y * ln x)) with (y * z * ln x).
   reflexivity.
@@ -427,22 +427,22 @@ Qed.
 
 Theorem Rpower_O : forall x:R, 0 < x -> x ^R 0 = 1.
 Proof.
-  intros x _; unfold Rpower in |- *.
+  intros x _; unfold Rpower.
   rewrite Rmult_0_l; apply exp_0.
 Qed.
 
 Theorem Rpower_1 : forall x:R, 0 < x -> x ^R 1 = x.
 Proof.
-  intros x H; unfold Rpower in |- *.
+  intros x H; unfold Rpower.
   rewrite Rmult_1_l; apply exp_ln; apply H.
 Qed.
 
 Theorem Rpower_pow : forall (n:nat) (x:R), 0 < x -> x ^R INR n = x ^ n.
 Proof.
-  intros n; elim n; simpl in |- *; auto; fold INR in |- *.
+  intros n; elim n; simpl; auto; fold INR.
   intros x H; apply Rpower_O; auto.
   intros n1; case n1.
-  intros H x H0; simpl in |- *; rewrite Rmult_1_r; apply Rpower_1; auto.
+  intros H x H0; simpl; rewrite Rmult_1_r; apply Rpower_1; auto.
   intros n0 H x H0; rewrite Rpower_plus; rewrite H; try rewrite Rpower_1;
     try apply Rmult_comm || assumption.
 Qed.
@@ -451,7 +451,7 @@ Theorem Rpower_lt :
   forall x y z:R, 1 < x -> 0 <= y -> y < z -> x ^R y < x ^R z.
 Proof.
   intros x y z H H0 H1.
-  unfold Rpower in |- *.
+  unfold Rpower.
   apply exp_increasing.
   apply Rmult_lt_compat_r.
   rewrite <- ln_1; apply ln_increasing.
@@ -464,18 +464,18 @@ Theorem Rpower_sqrt : forall x:R, 0 < x -> x ^R (/ 2) = sqrt x.
 Proof.
   intros x H.
   apply ln_inv.
-  unfold Rpower in |- *; apply exp_pos.
+  unfold Rpower; apply exp_pos.
   apply sqrt_lt_R0; apply H.
   apply Rmult_eq_reg_l with (INR 2).
   apply exp_inv.
-  fold Rpower in |- *.
+  fold Rpower.
   cut ((x ^R (/ INR 2)) ^R INR 2 = sqrt x ^R INR 2).
-  unfold Rpower in |- *; auto.
+  unfold Rpower; auto.
   rewrite Rpower_mult.
   rewrite Rinv_l.
   replace 1 with (INR 1); auto.
-  repeat rewrite Rpower_pow; simpl in |- *.
-  pattern x at 1 in |- *; rewrite <- (sqrt_sqrt x (Rlt_le _ _ H)).
+  repeat rewrite Rpower_pow; simpl.
+  pattern x at 1; rewrite <- (sqrt_sqrt x (Rlt_le _ _ H)).
   ring.
   apply sqrt_lt_R0; apply H.
   apply H.
@@ -485,7 +485,7 @@ Qed.
 
 Theorem Rpower_Ropp : forall x y:R, x ^R (- y) = / x ^R y.
 Proof.
-  unfold Rpower in |- *.
+  unfold Rpower.
   intros x y; rewrite Ropp_mult_distr_l_reverse.
   apply exp_Ropp.
 Qed.
@@ -505,11 +505,11 @@ Proof.
   rewrite Rinv_r.
   apply exp_lt_inv.
   apply Rle_lt_trans with (1 := exp_le_3).
-  change (3 < 2 ^R 2) in |- *.
+  change (3 < 2 ^R 2).
   repeat rewrite Rpower_plus; repeat rewrite Rpower_1.
   repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l;
     repeat rewrite Rmult_1_l.
-  pattern 3 at 1 in |- *; rewrite <- Rplus_0_r; replace (2 + 2) with (3 + 1);
+  pattern 3 at 1; rewrite <- Rplus_0_r; replace (2 + 2) with (3 + 1);
     [ apply Rplus_lt_compat_l; apply Rlt_0_1 | ring ].
   prove_sup0.
   discrR.
@@ -523,7 +523,7 @@ Theorem limit1_ext :
   forall (f g:R -> R) (D:R -> Prop) (l x:R),
     (forall x:R, D x -> f x = g x) -> limit1_in f D l x -> limit1_in g D l x.
 Proof.
-  intros f g D l x H; unfold limit1_in, limit_in in |- *.
+  intros f g D l x H; unfold limit1_in, limit_in.
   intros H0 eps H1; case (H0 eps); auto.
   intros x0 [H2 H3]; exists x0; split; auto.
   intros x1 [H4 H5]; rewrite <- H; auto.
@@ -533,7 +533,7 @@ Theorem limit1_imp :
   forall (f:R -> R) (D D1:R -> Prop) (l x:R),
     (forall x:R, D1 x -> D x) -> limit1_in f D l x -> limit1_in f D1 l x.
 Proof.
-  intros f D D1 l x H; unfold limit1_in, limit_in in |- *.
+  intros f D D1 l x H; unfold limit1_in, limit_in.
   intros H0 eps H1; case (H0 eps H1); auto.
   intros alpha [H2 H3]; exists alpha; split; auto.
   intros d [H4 H5]; apply H3; split; auto.
@@ -541,7 +541,7 @@ Qed.
 
 Theorem Rinv_Rdiv : forall x y:R, x <> 0 -> y <> 0 -> / (x / y) = y / x.
 Proof.
-  intros x y H1 H2; unfold Rdiv in |- *; rewrite Rinv_mult_distr.
+  intros x y H1 H2; unfold Rdiv; rewrite Rinv_mult_distr.
   rewrite Rinv_involutive.
   apply Rmult_comm.
   assumption.
@@ -551,18 +551,18 @@ Qed.
 
 Theorem Dln : forall y:R, 0 < y -> D_in ln Rinv (fun x:R => 0 < x) y.
 Proof.
-  intros y Hy; unfold D_in in |- *.
+  intros y Hy; unfold D_in.
   apply limit1_ext with
     (f := fun x:R => / ((exp (ln x) - exp (ln y)) / (ln x - ln y))).
   intros x [HD1 HD2]; repeat rewrite exp_ln.
-  unfold Rdiv in |- *; rewrite Rinv_mult_distr.
+  unfold Rdiv; rewrite Rinv_mult_distr.
   rewrite Rinv_involutive.
   apply Rmult_comm.
   apply Rminus_eq_contra.
-  red in |- *; intros H2; case HD2.
-  symmetry  in |- *; apply (ln_inv _ _ HD1 Hy H2).
-  apply Rminus_eq_contra; apply (sym_not_eq HD2).
-  apply Rinv_neq_0_compat; apply Rminus_eq_contra; red in |- *; intros H2;
+  red; intros H2; case HD2.
+  symmetry ; apply (ln_inv _ _ HD1 Hy H2).
+  apply Rminus_eq_contra; apply (not_eq_sym HD2).
+  apply Rinv_neq_0_compat; apply Rminus_eq_contra; red; intros H2;
     case HD2; apply ln_inv; auto.
   assumption.
   assumption.
@@ -574,62 +574,62 @@ Proof.
   intros x [H1 H2]; split.
   split; auto.
   split; auto.
-  red in |- *; intros H3; case H2; apply ln_inv; auto.
+  red; intros H3; case H2; apply ln_inv; auto.
   apply limit_comp with
     (l := ln y) (g := fun x:R => (exp x - exp (ln y)) / (x - ln y)) (f := ln).
   apply ln_continue; auto.
   assert (H0 := derivable_pt_lim_exp (ln y)); unfold derivable_pt_lim in H0;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
-      simpl in |- *; unfold R_dist in |- *; intros; elim (H0 _ H);
+    unfold limit1_in; unfold limit_in;
+      simpl; unfold R_dist; intros; elim (H0 _ H);
         intros; exists (pos x); split.
   apply (cond_pos x).
-  intros; pattern y at 3 in |- *; rewrite <- exp_ln.
-  pattern x0 at 1 in |- *; replace x0 with (ln y + (x0 - ln y));
+  intros; pattern y at 3; rewrite <- exp_ln.
+  pattern x0 at 1; replace x0 with (ln y + (x0 - ln y));
     [ idtac | ring ].
   apply H1.
   elim H2; intros H3 _; unfold D_x in H3; elim H3; clear H3; intros _ H3;
-    apply Rminus_eq_contra; apply (sym_not_eq (A:=R));
+    apply Rminus_eq_contra; apply (not_eq_sym (A:=R));
       apply H3.
   elim H2; clear H2; intros _ H2; apply H2.
   assumption.
-  red in |- *; intro; rewrite H in Hy; elim (Rlt_irrefl _ Hy).
+  red; intro; rewrite H in Hy; elim (Rlt_irrefl _ Hy).
 Qed.
 
 Lemma derivable_pt_lim_ln : forall x:R, 0 < x -> derivable_pt_lim ln x (/ x).
 Proof.
   intros; assert (H0 := Dln x H); unfold D_in in H0; unfold limit1_in in H0;
     unfold limit_in in H0; simpl in H0; unfold R_dist in H0;
-      unfold derivable_pt_lim in |- *; intros; elim (H0 _ H1);
+      unfold derivable_pt_lim; intros; elim (H0 _ H1);
         intros; elim H2; clear H2; intros; set (alp := Rmin x0 (x / 2));
           assert (H4 : 0 < alp).
-  unfold alp in |- *; unfold Rmin in |- *; case (Rle_dec x0 (x / 2)); intro.
+  unfold alp; unfold Rmin; case (Rle_dec x0 (x / 2)); intro.
   apply H2.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
-  exists (mkposreal _ H4); intros; pattern h at 2 in |- *;
+  exists (mkposreal _ H4); intros; pattern h at 2;
     replace h with (x + h - x); [ idtac | ring ].
   apply H3; split.
-  unfold D_x in |- *; split.
+  unfold D_x; split.
   case (Rcase_abs h); intro.
   assert (H7 : Rabs h < x / 2).
   apply Rlt_le_trans with alp.
   apply H6.
-  unfold alp in |- *; apply Rmin_r.
+  unfold alp; apply Rmin_r.
   apply Rlt_trans with (x / 2).
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
   rewrite Rabs_left in H7.
   apply Rplus_lt_reg_r with (- h - x / 2).
   replace (- h - x / 2 + x / 2) with (- h); [ idtac | ring ].
-  pattern x at 2 in |- *; rewrite double_var.
+  pattern x at 2; rewrite double_var.
   replace (- h - x / 2 + (x / 2 + x / 2 + h)) with (x / 2); [ apply H7 | ring ].
   apply r.
   apply Rplus_lt_le_0_compat; [ assumption | apply Rge_le; apply r ].
-  apply (sym_not_eq (A:=R)); apply Rminus_not_eq; replace (x + h - x) with h;
+  apply (not_eq_sym (A:=R)); apply Rminus_not_eq; replace (x + h - x) with h;
     [ apply H5 | ring ].
   replace (x + h - x) with h;
   [ apply Rlt_le_trans with alp;
-    [ apply H6 | unfold alp in |- *; apply Rmin_l ]
+    [ apply H6 | unfold alp; apply Rmin_l ]
     | ring ].
 Qed.
 
@@ -637,7 +637,7 @@ Theorem D_in_imp :
   forall (f g:R -> R) (D D1:R -> Prop) (x:R),
     (forall x:R, D1 x -> D x) -> D_in f g D x -> D_in f g D1 x.
 Proof.
-  intros f g D D1 x H; unfold D_in in |- *.
+  intros f g D D1 x H; unfold D_in.
   intros H0; apply limit1_imp with (D := D_x D x); auto.
   intros x1 [H1 H2]; split; auto.
 Qed.
@@ -646,7 +646,7 @@ Theorem D_in_ext :
   forall (f g h:R -> R) (D:R -> Prop) (x:R),
     f x = g x -> D_in h f D x -> D_in h g D x.
 Proof.
-  intros f g h D x H; unfold D_in in |- *.
+  intros f g h D x H; unfold D_in.
   rewrite H; auto.
 Qed.
 
@@ -661,7 +661,7 @@ Proof.
   intros x H0; repeat split.
   assumption.
   apply D_in_ext with (f := fun x:R => / x * (z * exp (z * ln x))).
-  unfold Rminus in |- *; rewrite Rpower_plus; rewrite Rpower_Ropp;
+  unfold Rminus; rewrite Rpower_plus; rewrite Rpower_Ropp;
     rewrite (Rpower_1 _ H); unfold Rpower; ring.
   apply Dcomp with
     (f := ln)
@@ -674,7 +674,7 @@ Proof.
   intros x H1; repeat split; auto.
   apply
     (Dcomp (fun _:R => True) (fun _:R => True) (fun x => z) exp
-      (fun x:R => z * x) exp); simpl in |- *.
+      (fun x:R => z * x) exp); simpl.
   apply D_in_ext with (f := fun x:R => z * 1).
   apply Rmult_1_r.
   apply (Dmult_const (fun x => True) (fun x => x) (fun x => 1)); apply Dx.
@@ -687,16 +687,16 @@ Theorem derivable_pt_lim_power :
     0 < x -> derivable_pt_lim (fun x => x ^R y) x (y * x ^R (y - 1)).
 Proof.
   intros x y H.
-  unfold Rminus in |- *; rewrite Rpower_plus.
+  unfold Rminus; rewrite Rpower_plus.
   rewrite Rpower_Ropp.
   rewrite Rpower_1; auto.
   rewrite <- Rmult_assoc.
-  unfold Rpower in |- *.
+  unfold Rpower.
   apply derivable_pt_lim_comp with (f1 := ln) (f2 := fun x => exp (y * x)).
   apply derivable_pt_lim_ln; assumption.
   rewrite (Rmult_comm y).
   apply derivable_pt_lim_comp with (f1 := fun x => y * x) (f2 := exp).
-  pattern y at 2 in |- *; replace y with (0 * ln x + y * 1).
+  pattern y at 2; replace y with (0 * ln x + y * 1).
   apply derivable_pt_lim_mult with (f1 := fun x:R => y) (f2 := fun x:R => x).
   apply derivable_pt_lim_const with (a := y).
   apply derivable_pt_lim_id.
diff --git a/theories/Reals/Rprod.v b/theories/Reals/Rprod.v
index 12258d6b..88c4de23 100644
--- a/theories/Reals/Rprod.v
+++ b/theories/Reals/Rprod.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -12,7 +12,7 @@ Require Import Rfunctions.
 Require Import Rseries.
 Require Import PartSum.
 Require Import Binomial.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (** TT Ak; 0<=k<=N *)
 Fixpoint prod_f_R0 (f:nat -> R) (N:nat) : R :=
@@ -36,7 +36,7 @@ Proof.
   replace (S n - k - 1)%nat with O; [rewrite H1; simpl|omega].
   replace (n+1+0)%nat with (S n); ring.
   replace (S n - k-1)%nat with (S (n - k-1));[idtac|omega].
-  simpl in |- *; replace (k + S (n - k))%nat with (S n).
+  simpl; replace (k + S (n - k))%nat with (S n).
   replace (k + 1 + S (n - k - 1))%nat with (S n).
   rewrite Hrecn; [ ring | assumption ].
   omega.
@@ -49,8 +49,8 @@ Lemma prod_SO_pos :
     (forall n:nat, (n <= N)%nat -> 0 <= An n) -> 0 <= prod_f_R0 An N.
 Proof.
   intros; induction  N as [| N HrecN].
-  simpl in |- *; apply H; trivial.
-  simpl in |- *; apply Rmult_le_pos.
+  simpl; apply H; trivial.
+  simpl; apply Rmult_le_pos.
   apply HrecN; intros; apply H; apply le_trans with N;
     [ assumption | apply le_n_Sn ].
   apply H; apply le_n.
@@ -64,7 +64,7 @@ Lemma prod_SO_Rle :
 Proof.
   intros; induction  N as [| N HrecN].
   elim  H with O; trivial.
-  simpl in |- *; apply Rle_trans with (prod_f_R0 An N * Bn (S N)).
+  simpl; apply Rle_trans with (prod_f_R0 An N * Bn (S N)).
   apply Rmult_le_compat_l.
   apply prod_SO_pos; intros; elim (H n (le_trans _ _ _ H0 (le_n_Sn N))); intros;
     assumption.
@@ -114,7 +114,7 @@ Proof.
      (if eq_nat_dec n 0 then 1 else INR n) = INR n).
   intros n; case (eq_nat_dec n 0); auto with real.
   intros; absurd (0 < n)%nat; omega.
-  intros; unfold Rsqr in |- *; repeat rewrite fact_prodSO.
+  intros; unfold Rsqr; repeat rewrite fact_prodSO.
   cut ((k=N)%nat \/ (k < N)%nat \/ (N < k)%nat).
   intro H2; elim H2; intro H3.
   rewrite H3; replace (2*N-N)%nat with N;[right; ring|omega].
@@ -164,14 +164,14 @@ Qed.
 (**********)
 Lemma INR_fact_lt_0 : forall n:nat, 0 < INR (fact n).
 Proof.
-  intro; apply lt_INR_0; apply neq_O_lt; red in |- *; intro;
-    elim (fact_neq_0 n); symmetry  in |- *; assumption.
+  intro; apply lt_INR_0; apply neq_O_lt; red; intro;
+    elim (fact_neq_0 n); symmetry ; assumption.
 Qed.
 
 (** We have the following inequality : (C 2N k) <= (C 2N N) forall k in [|O;2N|] *)
 Lemma C_maj : forall N k:nat, (k <= 2 * N)%nat -> C (2 * N) k <= C (2 * N) N.
 Proof.
-  intros; unfold C in |- *; unfold Rdiv in |- *; apply Rmult_le_compat_l.
+  intros; unfold C; unfold Rdiv; apply Rmult_le_compat_l.
   apply pos_INR.
   replace (2 * N - N)%nat with N.
   apply Rmult_le_reg_l with (INR (fact N) * INR (fact N)).
diff --git a/theories/Reals/Rseries.v b/theories/Reals/Rseries.v
index 479d381d..3c10725b 100644
--- a/theories/Reals/Rseries.v
+++ b/theories/Reals/Rseries.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,7 +9,7 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import Compare.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Implicit Type r : R.
 
@@ -54,20 +54,20 @@ Section sequence.
 (*********)
   Lemma EUn_noempty :  exists r : R, EUn r.
   Proof.
-    unfold EUn in |- *; split with (Un 0); split with 0%nat; trivial.
+    unfold EUn; split with (Un 0); split with 0%nat; trivial.
   Qed.
 
 (*********)
   Lemma Un_in_EUn : forall n:nat, EUn (Un n).
   Proof.
-    intro; unfold EUn in |- *; split with n; trivial.
+    intro; unfold EUn; split with n; trivial.
   Qed.
 
 (*********)
   Lemma Un_bound_imp :
     forall x:R, (forall n:nat, Un n <= x) -> is_upper_bound EUn x.
   Proof.
-    intros; unfold is_upper_bound in |- *; intros; unfold EUn in H0; elim H0;
+    intros; unfold is_upper_bound; intros; unfold EUn in H0; elim H0;
       clear H0; intros; generalize (H x1); intro; rewrite <- H0 in H1;
         trivial.
   Qed.
@@ -77,7 +77,7 @@ Section sequence.
     forall n m:nat, Un_growing -> (n >= m)%nat -> Un n >= Un m.
   Proof.
     double induction n m; intros.
-    unfold Rge in |- *; right; trivial.
+    unfold Rge; right; trivial.
     exfalso; unfold ge in H1; generalize (le_Sn_O n0); intro; auto.
     cut (n0 >= 0)%nat.
     generalize H0; intros; unfold Un_growing in H0;
@@ -89,7 +89,7 @@ Section sequence.
     elim y; clear y; intro y.
     unfold ge in H2; generalize (le_not_lt n0 n1 (le_S_n n0 n1 H2)); intro;
       exfalso; auto.
-    rewrite y; unfold Rge in |- *; right; trivial.
+    rewrite y; unfold Rge; right; trivial.
     unfold ge in H0; generalize (H0 (S n0) H1 (lt_le_S n0 n1 y)); intro;
       unfold Un_growing in H1;
         apply
@@ -182,13 +182,13 @@ Section sequence.
 
     assert (Hs0: forall n, sum n = 0).
     intros n.
-    specialize (Hm1 (sum n) (ex_intro _ _ (refl_equal _))).
+    specialize (Hm1 (sum n) (ex_intro _ _ (eq_refl _))).
     apply Rle_antisym with (2 := proj1 (Hsum n)).
     now rewrite <- Hm.
 
     assert (Hub: forall n, Un n <= l - eps).
     intros n.
-    generalize (refl_equal (sum (S n))).
+    generalize (eq_refl (sum (S n))).
     simpl sum at 1.
     rewrite 2!Hs0, Rplus_0_l.
     unfold test.
@@ -238,7 +238,7 @@ Section sequence.
     rewrite (IHN H6), Rplus_0_l.
     unfold test.
     destruct Rle_lt_dec.
-    apply refl_equal.
+    apply eq_refl.
     now elim Rlt_not_le with (1 := r).
 
     destruct (le_or_lt N n) as [Hn|Hn].
@@ -272,20 +272,20 @@ Section sequence.
   Proof.
     intro; induction  N as [| N HrecN].
     split with (Un 0); intros; rewrite (le_n_O_eq n H);
-      apply (Req_le (Un n) (Un n) (refl_equal (Un n))).
+      apply (Req_le (Un n) (Un n) (eq_refl (Un n))).
     elim HrecN; clear HrecN; intros; split with (Rmax (Un (S N)) x); intros;
       elim (Rmax_Rle (Un (S N)) x (Un n)); intros; clear H1;
         inversion H0.
     rewrite <- H1; rewrite <- H1 in H2;
       apply
-        (H2 (or_introl (Un n <= x) (Req_le (Un n) (Un n) (refl_equal (Un n))))).
+        (H2 (or_introl (Un n <= x) (Req_le (Un n) (Un n) (eq_refl (Un n))))).
     apply (H2 (or_intror (Un n <= Un (S N)) (H n H3))).
   Qed.
 
 (*********)
   Lemma cauchy_bound : Cauchy_crit -> bound EUn.
   Proof.
-    unfold Cauchy_crit, bound in |- *; intros; unfold is_upper_bound in |- *;
+    unfold Cauchy_crit, bound; intros; unfold is_upper_bound;
       unfold Rgt in H; elim (H 1 Rlt_0_1); clear H; intros;
         generalize (H x); intro; generalize (le_dec x); intro;
           elim (finite_greater x); intros; split with (Rmax x0 (Un x + 1));
@@ -324,12 +324,12 @@ End Isequence.
 Lemma GP_infinite :
   forall x:R, Rabs x < 1 -> Pser (fun n:nat => 1) x (/ (1 - x)).
 Proof.
-  intros; unfold Pser in |- *; unfold infinite_sum in |- *; intros;
+  intros; unfold Pser; unfold infinite_sum; intros;
     elim (Req_dec x 0).
   intros; exists 0%nat; intros; rewrite H1; rewrite Rminus_0_r; rewrite Rinv_1;
     cut (sum_f_R0 (fun n0:nat => 1 * 0 ^ n0) n = 1).
   intros; rewrite H3; rewrite R_dist_eq; auto.
-  elim n; simpl in |- *.
+  elim n; simpl.
   ring.
   intros; rewrite H3; ring.
   intro; cut (0 < eps * (Rabs (1 - x) * Rabs (/ x))).
@@ -344,11 +344,11 @@ Proof.
   apply Rabs_pos_lt.
   apply Rminus_eq_contra.
   apply Rlt_dichotomy_converse.
-  right; unfold Rgt in |- *.
+  right; unfold Rgt.
   apply (Rle_lt_trans x (Rabs x) 1).
   apply RRle_abs.
   assumption.
-  unfold R_dist in |- *; rewrite <- Rabs_mult.
+  unfold R_dist; rewrite <- Rabs_mult.
   rewrite Rmult_minus_distr_l.
   cut
     ((1 - x) * sum_f_R0 (fun n0:nat => x ^ n0) n =
@@ -359,7 +359,7 @@ Proof.
   cut (- (x ^ (n + 1) - 1) - 1 = - x ^ (n + 1)).
   intro; rewrite H7.
   rewrite Rabs_Ropp; cut ((n + 1)%nat = S n); auto.
-  intro H8; rewrite H8; simpl in |- *; rewrite Rabs_mult;
+  intro H8; rewrite H8; simpl; rewrite Rabs_mult;
     apply
       (Rlt_le_trans (Rabs x * Rabs (x ^ n))
         (Rabs x * (eps * (Rabs (1 - x) * Rabs (/ x)))) (
@@ -373,7 +373,7 @@ Proof.
       Rabs x * Rabs (/ x) * (eps * Rabs (1 - x))).
   clear H8; intros; rewrite H8; rewrite <- Rabs_mult; rewrite Rinv_r.
   rewrite Rabs_R1; cut (1 * (eps * Rabs (1 - x)) = Rabs (1 - x) * eps).
-  intros; rewrite H9; unfold Rle in |- *; right; reflexivity.
+  intros; rewrite H9; unfold Rle; right; reflexivity.
   ring.
   assumption.
   ring.
@@ -381,12 +381,12 @@ Proof.
   ring.
   apply Rminus_eq_contra.
   apply Rlt_dichotomy_converse.
-  right; unfold Rgt in |- *.
+  right; unfold Rgt.
   apply (Rle_lt_trans x (Rabs x) 1).
   apply RRle_abs.
   assumption.
   ring; ring.
-  elim n; simpl in |- *.
+  elim n; simpl.
   ring.
   intros; rewrite H5.
   ring.
@@ -396,7 +396,7 @@ Proof.
   apply Rabs_pos_lt.
   apply Rminus_eq_contra.
   apply Rlt_dichotomy_converse.
-  right; unfold Rgt in |- *.
+  right; unfold Rgt.
   apply (Rle_lt_trans x (Rabs x) 1).
   apply RRle_abs.
   assumption.
diff --git a/theories/Reals/Rsigma.v b/theories/Reals/Rsigma.v
index 0027c274..76b44d96 100644
--- a/theories/Reals/Rsigma.v
+++ b/theories/Reals/Rsigma.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -10,7 +10,7 @@ Require Import Rbase.
 Require Import Rfunctions.
 Require Import Rseries.
 Require Import PartSum.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Set Implicit Arguments.
 
@@ -28,8 +28,8 @@ Section Sigma.
   Proof.
     intros; induction  k as [| k Hreck].
     cut (low = 0%nat).
-    intro; rewrite H1; unfold sigma in |- *; rewrite <- minus_n_n;
-      rewrite <- minus_n_O; simpl in |- *; replace (high - 1)%nat with (pred high).
+    intro; rewrite H1; unfold sigma; rewrite <- minus_n_n;
+      rewrite <- minus_n_O; simpl; replace (high - 1)%nat with (pred high).
     apply (decomp_sum (fun k:nat => f k)).
     assumption.
     apply pred_of_minus.
@@ -42,8 +42,8 @@ Section Sigma.
     apply Hreck.
     assumption.
     apply lt_trans with (S k); [ apply lt_n_Sn | assumption ].
-    unfold sigma in |- *; replace (high - S (S k))%nat with (pred (high - S k)).
-    pattern (S k) at 3 in |- *; replace (S k) with (S k + 0)%nat;
+    unfold sigma; replace (high - S (S k))%nat with (pred (high - S k)).
+    pattern (S k) at 3; replace (S k) with (S k + 0)%nat;
       [ idtac | ring ].
     replace (sum_f_R0 (fun k0:nat => f (S (S k) + k0)) (pred (high - S k))) with
     (sum_f_R0 (fun k0:nat => f (S k + S k0)) (pred (high - S k))).
@@ -55,12 +55,12 @@ Section Sigma.
     replace (high - S (S k))%nat with (high - S k - 1)%nat.
     apply pred_of_minus.
     omega.
-    unfold sigma in |- *; replace (S k - low)%nat with (S (k - low)).
-    pattern (S k) at 1 in |- *; replace (S k) with (low + S (k - low))%nat.
-    symmetry  in |- *; apply (tech5 (fun i:nat => f (low + i))).
+    unfold sigma; replace (S k - low)%nat with (S (k - low)).
+    pattern (S k) at 1; replace (S k) with (low + S (k - low))%nat.
+    symmetry ; apply (tech5 (fun i:nat => f (low + i))).
     omega.
     omega.
-    rewrite <- H2; unfold sigma in |- *; rewrite <- minus_n_n; simpl in |- *;
+    rewrite <- H2; unfold sigma; rewrite <- minus_n_n; simpl;
       replace (high - S low)%nat with (pred (high - low)).
     replace (sum_f_R0 (fun k0:nat => f (S (low + k0))) (pred (high - low))) with
     (sum_f_R0 (fun k0:nat => f (low + S k0)) (pred (high - low))).
@@ -79,7 +79,7 @@ Section Sigma.
       (low <= k)%nat ->
       (k < high)%nat -> sigma low high - sigma low k = sigma (S k) high.
   Proof.
-    intros low high k H1 H2; symmetry  in |- *; rewrite (sigma_split H1 H2); ring.
+    intros low high k H1 H2; symmetry ; rewrite (sigma_split H1 H2); ring.
   Qed.
 
   Theorem sigma_diff_neg :
@@ -100,8 +100,8 @@ Section Sigma.
     apply sigma_split.
     apply le_n.
     assumption.
-    unfold sigma in |- *; rewrite <- minus_n_n.
-    simpl in |- *.
+    unfold sigma; rewrite <- minus_n_n.
+    simpl.
     replace (low + 0)%nat with low; [ reflexivity | ring ].
   Qed.
 
@@ -113,20 +113,20 @@ Section Sigma.
       generalize (lt_le_weak low high H1); intro H3;
         replace (f high) with (sigma high high).
     rewrite Rplus_comm; cut (high = S (pred high)).
-    intro; pattern high at 3 in |- *; rewrite H.
+    intro; pattern high at 3; rewrite H.
     apply sigma_split.
     apply le_S_n; rewrite <- H; apply lt_le_S; assumption.
     apply lt_pred_n_n; apply le_lt_trans with low; [ apply le_O_n | assumption ].
     apply S_pred with 0%nat; apply le_lt_trans with low;
       [ apply le_O_n | assumption ].
-    unfold sigma in |- *; rewrite <- minus_n_n; simpl in |- *;
+    unfold sigma; rewrite <- minus_n_n; simpl;
       replace (high + 0)%nat with high; [ reflexivity | ring ].
   Qed.
 
   Theorem sigma_eq_arg : forall low:nat, sigma low low = f low.
   Proof.
-    intro; unfold sigma in |- *; rewrite <- minus_n_n.
-    simpl in |- *; replace (low + 0)%nat with low; [ reflexivity | ring ].
+    intro; unfold sigma; rewrite <- minus_n_n.
+    simpl; replace (low + 0)%nat with low; [ reflexivity | ring ].
   Qed.
 
 End Sigma.
diff --git a/theories/Reals/Rsqrt_def.v b/theories/Reals/Rsqrt_def.v
index 7c3b4699..a6e48f83 100644
--- a/theories/Reals/Rsqrt_def.v
+++ b/theories/Reals/Rsqrt_def.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -11,7 +11,7 @@ Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
 Require Import Ranalysis1.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Fixpoint Dichotomy_lb (x y:R) (P:R -> bool) (N:nat) {struct N} : R :=
   match N with
@@ -41,18 +41,18 @@ Lemma dicho_comp :
 Proof.
   intros.
   induction  n as [| n Hrecn].
-  simpl in |- *; assumption.
-  simpl in |- *.
+  simpl; assumption.
+  simpl.
   case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).
-  unfold Rdiv in |- *; apply Rmult_le_reg_l with 2.
+  unfold Rdiv; apply Rmult_le_reg_l with 2.
   prove_sup0.
-  pattern 2 at 1 in |- *; rewrite Rmult_comm.
+  pattern 2 at 1; rewrite Rmult_comm.
   rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].
   rewrite Rmult_1_r.
   rewrite double.
   apply Rplus_le_compat_l.
   assumption.
-  unfold Rdiv in |- *; apply Rmult_le_reg_l with 2.
+  unfold Rdiv; apply Rmult_le_reg_l with 2.
   prove_sup0.
   rewrite Rmult_comm.
   rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].
@@ -67,14 +67,14 @@ Lemma dicho_lb_growing :
   forall (x y:R) (P:R -> bool), x <= y -> Un_growing (dicho_lb x y P).
 Proof.
   intros.
-  unfold Un_growing in |- *.
+  unfold Un_growing.
   intro.
-  simpl in |- *.
+  simpl.
   case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).
   right; reflexivity.
-  unfold Rdiv in |- *; apply Rmult_le_reg_l with 2.
+  unfold Rdiv; apply Rmult_le_reg_l with 2.
   prove_sup0.
-  pattern 2 at 1 in |- *; rewrite Rmult_comm.
+  pattern 2 at 1; rewrite Rmult_comm.
   rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].
   rewrite Rmult_1_r.
   rewrite double.
@@ -87,11 +87,11 @@ Lemma dicho_up_decreasing :
   forall (x y:R) (P:R -> bool), x <= y -> Un_decreasing (dicho_up x y P).
 Proof.
   intros.
-  unfold Un_decreasing in |- *.
+  unfold Un_decreasing.
   intro.
-  simpl in |- *.
+  simpl.
   case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).
-  unfold Rdiv in |- *; apply Rmult_le_reg_l with 2.
+  unfold Rdiv; apply Rmult_le_reg_l with 2.
   prove_sup0.
   rewrite Rmult_comm.
   rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ].
@@ -112,17 +112,17 @@ Lemma dicho_lb_maj_y :
 Proof.
   intros.
   induction  n as [| n Hrecn].
-  simpl in |- *; assumption.
-  simpl in |- *.
+  simpl; assumption.
+  simpl.
   case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).
   assumption.
-  unfold Rdiv in |- *; apply Rmult_le_reg_l with 2.
+  unfold Rdiv; apply Rmult_le_reg_l with 2.
   prove_sup0.
   rewrite Rmult_comm.
   rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ rewrite Rmult_1_r | discrR ].
   rewrite double; apply Rplus_le_compat.
   assumption.
-  pattern y at 2 in |- *; replace y with (Dichotomy_ub x y P 0);
+  pattern y at 2; replace y with (Dichotomy_ub x y P 0);
     [ idtac | reflexivity ].
   apply decreasing_prop.
   assert (H0 := dicho_up_decreasing x y P H).
@@ -136,10 +136,10 @@ Proof.
   intros.
   cut (forall n:nat, dicho_lb x y P n <= y).
   intro.
-  unfold has_ub in |- *.
-  unfold bound in |- *.
+  unfold has_ub.
+  unfold bound.
   exists y.
-  unfold is_upper_bound in |- *.
+  unfold is_upper_bound.
   intros.
   elim H1; intros.
   rewrite H2; apply H0.
@@ -151,15 +151,15 @@ Lemma dicho_up_min_x :
 Proof.
   intros.
   induction  n as [| n Hrecn].
-  simpl in |- *; assumption.
-  simpl in |- *.
+  simpl; assumption.
+  simpl.
   case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).
-  unfold Rdiv in |- *; apply Rmult_le_reg_l with 2.
+  unfold Rdiv; apply Rmult_le_reg_l with 2.
   prove_sup0.
-  pattern 2 at 1 in |- *; rewrite Rmult_comm.
+  pattern 2 at 1; rewrite Rmult_comm.
   rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ rewrite Rmult_1_r | discrR ].
   rewrite double; apply Rplus_le_compat.
-  pattern x at 1 in |- *; replace x with (Dichotomy_lb x y P 0);
+  pattern x at 1; replace x with (Dichotomy_lb x y P 0);
     [ idtac | reflexivity ].
   apply tech9.
   assert (H0 := dicho_lb_growing x y P H).
@@ -175,14 +175,14 @@ Proof.
   intros.
   cut (forall n:nat, x <= dicho_up x y P n).
   intro.
-  unfold has_lb in |- *.
-  unfold bound in |- *.
+  unfold has_lb.
+  unfold bound.
   exists (- x).
-  unfold is_upper_bound in |- *.
+  unfold is_upper_bound.
   intros.
   elim H1; intros.
   rewrite H2.
-  unfold opp_seq in |- *.
+  unfold opp_seq.
   apply Ropp_le_contravar.
   apply H0.
   apply dicho_up_min_x; assumption.
@@ -214,35 +214,35 @@ Lemma dicho_lb_dicho_up :
 Proof.
   intros.
   induction  n as [| n Hrecn].
-  simpl in |- *.
-  unfold Rdiv in |- *; rewrite Rinv_1; ring.
-  simpl in |- *.
+  simpl.
+  unfold Rdiv; rewrite Rinv_1; ring.
+  simpl.
   case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)).
-  unfold Rdiv in |- *.
+  unfold Rdiv.
   replace
   ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 - Dichotomy_lb x y P n)
     with ((dicho_up x y P n - dicho_lb x y P n) / 2).
-  unfold Rdiv in |- *; rewrite Hrecn.
-  unfold Rdiv in |- *.
+  unfold Rdiv; rewrite Hrecn.
+  unfold Rdiv.
   rewrite Rinv_mult_distr.
   ring.
   discrR.
   apply pow_nonzero; discrR.
-  pattern (Dichotomy_lb x y P n) at 2 in |- *;
+  pattern (Dichotomy_lb x y P n) at 2;
     rewrite (double_var (Dichotomy_lb x y P n));
-      unfold dicho_up, dicho_lb, Rminus, Rdiv in |- *; ring.
+      unfold dicho_up, dicho_lb, Rminus, Rdiv; ring.
   replace
   (Dichotomy_ub x y P n - (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)
     with ((dicho_up x y P n - dicho_lb x y P n) / 2).
-  unfold Rdiv in |- *; rewrite Hrecn.
-  unfold Rdiv in |- *.
+  unfold Rdiv; rewrite Hrecn.
+  unfold Rdiv.
   rewrite Rinv_mult_distr.
   ring.
   discrR.
   apply pow_nonzero; discrR.
-  pattern (Dichotomy_ub x y P n) at 1 in |- *;
+  pattern (Dichotomy_ub x y P n) at 1;
     rewrite (double_var (Dichotomy_ub x y P n));
-      unfold dicho_up, dicho_lb, Rminus, Rdiv in |- *; ring.
+      unfold dicho_up, dicho_lb, Rminus, Rdiv; ring.
 Qed.
 
 Definition pow_2_n (n:nat) := 2 ^ n.
@@ -250,23 +250,23 @@ Definition pow_2_n (n:nat) := 2 ^ n.
 Lemma pow_2_n_neq_R0 : forall n:nat, pow_2_n n <> 0.
 Proof.
   intro.
-  unfold pow_2_n in |- *.
+  unfold pow_2_n.
   apply pow_nonzero.
   discrR.
 Qed.
 
 Lemma pow_2_n_growing : Un_growing pow_2_n.
 Proof.
-  unfold Un_growing in |- *.
+  unfold Un_growing.
   intro.
   replace (S n) with (n + 1)%nat;
-  [ unfold pow_2_n in |- *; rewrite pow_add | ring ].
-  pattern (2 ^ n) at 1 in |- *; rewrite <- Rmult_1_r.
+  [ unfold pow_2_n; rewrite pow_add | ring ].
+  pattern (2 ^ n) at 1; rewrite <- Rmult_1_r.
   apply Rmult_le_compat_l.
   left; apply pow_lt; prove_sup0.
-  simpl in |- *.
+  simpl.
   rewrite Rmult_1_r.
-  pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
+  pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
     apply Rlt_0_1.
 Qed.
 
@@ -274,7 +274,7 @@ Lemma pow_2_n_infty : cv_infty pow_2_n.
 Proof.
   cut (forall N:nat, INR N <= 2 ^ N).
   intros.
-  unfold cv_infty in |- *.
+  unfold cv_infty.
   intro.
   case (total_order_T 0 M); intro.
   elim s; intro.
@@ -287,41 +287,41 @@ Proof.
   apply Rlt_le_trans with (INR N0).
   rewrite INR_IZR_INZ.
   rewrite <- H1.
-  unfold N in |- *.
+  unfold N.
   assert (H3 := archimed M).
   elim H3; intros; assumption.
   apply Rle_trans with (pow_2_n N0).
-  unfold pow_2_n in |- *; apply H.
+  unfold pow_2_n; apply H.
   apply Rge_le.
   apply growing_prop.
   apply pow_2_n_growing.
   assumption.
   apply le_IZR.
-  unfold N in |- *.
-  simpl in |- *.
+  unfold N.
+  simpl.
   assert (H0 := archimed M); elim H0; intros.
   left; apply Rlt_trans with M; assumption.
   exists 0%nat; intros.
   rewrite <- b.
-  unfold pow_2_n in |- *; apply pow_lt; prove_sup0.
+  unfold pow_2_n; apply pow_lt; prove_sup0.
   exists 0%nat; intros.
   apply Rlt_trans with 0.
   assumption.
-  unfold pow_2_n in |- *; apply pow_lt; prove_sup0.
+  unfold pow_2_n; apply pow_lt; prove_sup0.
   simple induction N.
-  simpl in |- *.
+  simpl.
   left; apply Rlt_0_1.
   intros.
-  pattern (S n) at 2 in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ].
+  pattern (S n) at 2; replace (S n) with (n + 1)%nat; [ idtac | ring ].
   rewrite S_INR; rewrite pow_add.
-  simpl in |- *.
+  simpl.
   rewrite Rmult_1_r.
   apply Rle_trans with (2 ^ n).
   rewrite <- (Rplus_comm 1).
   rewrite <- (Rmult_1_r (INR n)).
   apply (poly n 1).
   apply Rlt_0_1.
-  pattern (2 ^ n) at 1 in |- *; rewrite <- Rplus_0_r.
+  pattern (2 ^ n) at 1; rewrite <- Rplus_0_r.
   rewrite <- (Rmult_comm 2).
   rewrite double.
   apply Rplus_le_compat_l.
@@ -338,8 +338,8 @@ Proof.
   cut (Un_cv (fun i:nat => dicho_lb x y P i - dicho_up x y P i) 0).
   intro.
   assert (H4 := UL_sequence _ _ _ H2 H3).
-  symmetry  in |- *; apply Rminus_diag_uniq_sym; assumption.
-  unfold Un_cv in |- *; unfold R_dist in |- *.
+  symmetry ; apply Rminus_diag_uniq_sym; assumption.
+  unfold Un_cv; unfold R_dist.
   intros.
   assert (H4 := cv_infty_cv_R0 pow_2_n pow_2_n_neq_R0 pow_2_n_infty).
   case (total_order_T x y); intro.
@@ -356,7 +356,7 @@ Proof.
   rewrite <- Rabs_Ropp.
   rewrite Ropp_minus_distr'.
   rewrite dicho_lb_dicho_up.
-  unfold Rdiv in |- *; rewrite Rabs_mult.
+  unfold Rdiv; rewrite Rabs_mult.
   rewrite (Rabs_right (y - x)).
   apply Rmult_lt_reg_l with (/ (y - x)).
   apply Rinv_0_lt_compat; assumption.
@@ -366,12 +366,12 @@ Proof.
   [ unfold pow_2_n, Rdiv in H6; rewrite <- (Rmult_comm eps); apply H6;
     assumption
     | ring ].
-  red in |- *; intro; rewrite H8 in Hyp; elim (Rlt_irrefl _ Hyp).
+  red; intro; rewrite H8 in Hyp; elim (Rlt_irrefl _ Hyp).
   apply Rle_ge.
   apply Rplus_le_reg_l with x; rewrite Rplus_0_r.
   replace (x + (y - x)) with y; [ assumption | ring ].
   assumption.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; assumption ].
   apply Rplus_lt_reg_r with x; rewrite Rplus_0_r.
   replace (x + (y - x)) with y; [ assumption | ring ].
@@ -382,7 +382,7 @@ Proof.
   rewrite Ropp_minus_distr'.
   rewrite dicho_lb_dicho_up.
   rewrite b.
-  unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r; rewrite Rmult_0_l;
+  unfold Rminus, Rdiv; rewrite Rplus_opp_r; rewrite Rmult_0_l;
     rewrite Rabs_R0; assumption.
   assumption.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)).
@@ -399,26 +399,26 @@ Lemma continuity_seq :
   forall (f:R -> R) (Un:nat -> R) (l:R),
     continuity_pt f l -> Un_cv Un l -> Un_cv (fun i:nat => f (Un i)) (f l).
 Proof.
-  unfold continuity_pt, Un_cv in |- *; unfold continue_in in |- *.
-  unfold limit1_in in |- *.
-  unfold limit_in in |- *.
-  unfold dist in |- *.
-  simpl in |- *.
-  unfold R_dist in |- *.
+  unfold continuity_pt, Un_cv; unfold continue_in.
+  unfold limit1_in.
+  unfold limit_in.
+  unfold dist.
+  simpl.
+  unfold R_dist.
   intros.
   elim (H eps H1); intros alp H2.
   elim H2; intros.
   elim (H0 alp H3); intros N H5.
   exists N; intros.
   case (Req_dec (Un n) l); intro.
-  rewrite H7; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  rewrite H7; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     assumption.
   apply H4.
   split.
-  unfold D_x, no_cond in |- *.
+  unfold D_x, no_cond.
   split.
   trivial.
-  apply (sym_not_eq (A:=R)); assumption.
+  apply (not_eq_sym (A:=R)); assumption.
   apply H5; assumption.
 Qed.
 
@@ -428,9 +428,9 @@ Lemma dicho_lb_car :
 Proof.
   intros.
   induction  n as [| n Hrecn].
-  simpl in |- *.
+  simpl.
   assumption.
-  simpl in |- *.
+  simpl.
   assert
     (X :=
       sumbool_of_bool (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2))).
@@ -447,9 +447,9 @@ Lemma dicho_up_car :
 Proof.
   intros.
   induction  n as [| n Hrecn].
-  simpl in |- *.
+  simpl.
   assumption.
-  simpl in |- *.
+  simpl.
   assert
     (X :=
       sumbool_of_bool (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2))).
@@ -480,7 +480,7 @@ Proof.
   split.
   split.
   apply Rle_trans with (dicho_lb x y (fun z:R => cond_positivity (f z)) 0).
-  simpl in |- *.
+  simpl.
   right; reflexivity.
   apply growing_ineq.
   apply dicho_lb_growing; assumption.
@@ -503,7 +503,7 @@ Proof.
   assert (H10 := H5 H7).
   apply Rle_antisym; assumption.
   intro.
-  unfold Wn in |- *.
+  unfold Wn.
   cut (forall z:R, cond_positivity z = true <-> 0 <= z).
   intro.
   assert (H8 := dicho_up_car x y (fun z:R => cond_positivity (f z)) n).
@@ -514,7 +514,7 @@ Proof.
   apply H12.
   left; assumption.
   intro.
-  unfold cond_positivity in |- *.
+  unfold cond_positivity.
   case (Rle_dec 0 z); intro.
   split.
   intro; assumption.
@@ -523,7 +523,7 @@ Proof.
   intro feqt;discriminate feqt.
   intro.
   elim n0; assumption.
-  unfold Vn in |- *.
+  unfold Vn.
   cut (forall z:R, cond_positivity z = false <-> z < 0).
   intros.
   assert (H8 := dicho_lb_car x y (fun z:R => cond_positivity (f z)) n).
@@ -535,7 +535,7 @@ Proof.
   apply H12.
   assumption.
   intro.
-  unfold cond_positivity in |- *.
+  unfold cond_positivity.
   case (Rle_dec 0 z); intro.
   split.
   intro feqt; discriminate feqt.
@@ -554,7 +554,7 @@ Proof.
   cut (0 < - f x0).
   intro.
   elim (H7 (- f x0) H8); intros.
-  cut (x2 >= x2)%nat; [ intro | unfold ge in |- *; apply le_n ].
+  cut (x2 >= x2)%nat; [ intro | unfold ge; apply le_n ].
   assert (H11 := H9 x2 H10).
   rewrite Rabs_right in H11.
   pattern (- f x0) at 1 in H11; rewrite <- Rplus_0_r in H11.
@@ -562,11 +562,11 @@ Proof.
   assert (H12 := Rplus_lt_reg_r _ _ _ H11).
   assert (H13 := H6 x2).
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H13 H12)).
-  apply Rle_ge; left; unfold Rminus in |- *; apply Rplus_le_lt_0_compat.
+  apply Rle_ge; left; unfold Rminus; apply Rplus_le_lt_0_compat.
   apply H6.
   exact H8.
   apply Ropp_0_gt_lt_contravar; assumption.
-  unfold Wn in |- *; assumption.
+  unfold Wn; assumption.
   cut (Un_cv Vn x0).
   intros.
   assert (H7 := continuity_seq f Vn x0 (H x0) H5).
@@ -574,7 +574,7 @@ Proof.
   elim s; intro.
   unfold Un_cv in H7; unfold R_dist in H7.
   elim (H7 (f x0) a); intros.
-  cut (x2 >= x2)%nat; [ intro | unfold ge in |- *; apply le_n ].
+  cut (x2 >= x2)%nat; [ intro | unfold ge; apply le_n ].
   assert (H10 := H8 x2 H9).
   rewrite Rabs_left in H10.
   pattern (f x0) at 2 in H10; rewrite <- Rplus_0_r in H10.
@@ -589,12 +589,12 @@ Proof.
   apply Ropp_0_gt_lt_contravar; assumption.
   apply Rplus_lt_reg_r with (f x0 - f (Vn x2)).
   rewrite Rplus_0_r; replace (f x0 - f (Vn x2) + (f (Vn x2) - f x0)) with 0;
-    [ unfold Rminus in |- *; apply Rplus_lt_le_0_compat | ring ].
+    [ unfold Rminus; apply Rplus_lt_le_0_compat | ring ].
   assumption.
   apply Ropp_0_ge_le_contravar; apply Rle_ge; apply H6.
   right; rewrite <- b; reflexivity.
   left; assumption.
-  unfold Vn in |- *; assumption.
+  unfold Vn; assumption.
 Qed.
 
 Lemma IVT_cor :
@@ -613,11 +613,11 @@ Proof.
   exists y.
   split.
   split; [ assumption | right; reflexivity ].
-  symmetry  in |- *; exact b.
+  symmetry ; exact b.
   exists x.
   split.
   split; [ right; reflexivity | assumption ].
-  symmetry  in |- *; exact b.
+  symmetry ; exact b.
   elim s; intro.
   cut (x < y).
   intro.
@@ -633,8 +633,8 @@ Proof.
   unfold opp_fct in H7.
   rewrite <- (Ropp_involutive (f x0)).
   apply Ropp_eq_0_compat; assumption.
-  unfold opp_fct in |- *; apply Ropp_0_gt_lt_contravar; assumption.
-  unfold opp_fct in |- *.
+  unfold opp_fct; apply Ropp_0_gt_lt_contravar; assumption.
+  unfold opp_fct.
   apply Rplus_lt_reg_r with (f x); rewrite Rplus_opp_r; rewrite Rplus_0_r;
     assumption.
   inversion H0.
@@ -644,7 +644,7 @@ Proof.
   exists x.
   split.
   split; [ right; reflexivity | assumption ].
-  symmetry  in |- *; assumption.
+  symmetry ; assumption.
   case (total_order_T 0 (f y)); intro.
   elim s; intro.
   cut (x < y).
@@ -657,7 +657,7 @@ Proof.
   exists y.
   split.
   split; [ assumption | right; reflexivity ].
-  symmetry  in |- *; assumption.
+  symmetry ; assumption.
   cut (0 < f x * f y).
   intro.
   elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H2 H1)).
@@ -690,18 +690,18 @@ Proof.
   elim H5; intros; assumption.
   unfold f in H6.
   apply Rminus_diag_uniq_sym; exact H6.
-  rewrite Rmult_comm; pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f 1)).
+  rewrite Rmult_comm; pattern 0 at 2; rewrite <- (Rmult_0_r (f 1)).
   apply Rmult_le_compat_l; assumption.
-  unfold f in |- *.
+  unfold f.
   rewrite Rsqr_1.
   apply Rplus_le_reg_l with y.
-  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *;
+  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus;
     rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
       left; assumption.
   exists 1.
   split.
   left; apply Rlt_0_1.
-  rewrite b; symmetry  in |- *; apply Rsqr_1.
+  rewrite b; symmetry ; apply Rsqr_1.
   cut (0 <= f y).
   intro.
   cut (f 0 * f y <= 0).
@@ -714,14 +714,14 @@ Proof.
   elim H5; intros; assumption.
   unfold f in H6.
   apply Rminus_diag_uniq_sym; exact H6.
-  rewrite Rmult_comm; pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f y)).
+  rewrite Rmult_comm; pattern 0 at 2; rewrite <- (Rmult_0_r (f y)).
   apply Rmult_le_compat_l; assumption.
-  unfold f in |- *.
+  unfold f.
   apply Rplus_le_reg_l with y.
-  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *;
+  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus;
     rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r.
-  pattern y at 1 in |- *; rewrite <- Rmult_1_r.
-  unfold Rsqr in |- *; apply Rmult_le_compat_l.
+  pattern y at 1; rewrite <- Rmult_1_r.
+  unfold Rsqr; apply Rmult_le_compat_l.
   assumption.
   left; exact r.
   replace f with (Rsqr - fct_cte y)%F.
@@ -729,8 +729,8 @@ Proof.
   apply derivable_continuous; apply derivable_Rsqr.
   apply derivable_continuous; apply derivable_const.
   reflexivity.
-  unfold f in |- *; rewrite Rsqr_0.
-  unfold Rminus in |- *; rewrite Rplus_0_l.
+  unfold f; rewrite Rsqr_0.
+  unfold Rminus; rewrite Rplus_0_l.
   apply Rge_le.
   apply Ropp_0_le_ge_contravar; assumption.
 Qed.
@@ -749,7 +749,7 @@ Proof.
   intros.
   elim p; intros.
   rewrite H in H0; assumption.
-  unfold Rsqrt in |- *.
+  unfold Rsqrt.
   case (Rsqrt_exists x (cond_nonneg x)).
   intros.
   elim p; elim a; intros.
@@ -770,7 +770,7 @@ Proof.
   rewrite <- H.
   elim p; intros.
   rewrite H1; reflexivity.
-  unfold Rsqrt in |- *.
+  unfold Rsqrt.
   case (Rsqrt_exists x (cond_nonneg x)).
   intros.
   elim p; elim a; intros.
diff --git a/theories/Reals/Rtopology.v b/theories/Reals/Rtopology.v
index f1142d24..51d0b99e 100644
--- a/theories/Reals/Rtopology.v
+++ b/theories/Reals/Rtopology.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -12,7 +12,7 @@ Require Import Ranalysis1.
 Require Import RList.
 Require Import Classical_Prop.
 Require Import Classical_Pred_Type.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (** * General definitions and propositions *)
 
@@ -30,16 +30,16 @@ Definition interior (D:R -> Prop) (x:R) : Prop := neighbourhood D x.
 
 Lemma interior_P1 : forall D:R -> Prop, included (interior D) D.
 Proof.
-  intros; unfold included in |- *; unfold interior in |- *; intros;
+  intros; unfold included; unfold interior; intros;
     unfold neighbourhood in H; elim H; intros; unfold included in H0;
-      apply H0; unfold disc in |- *; unfold Rminus in |- *;
+      apply H0; unfold disc; unfold Rminus;
         rewrite Rplus_opp_r; rewrite Rabs_R0; apply (cond_pos x0).
 Qed.
 
 Lemma interior_P2 : forall D:R -> Prop, open_set D -> included D (interior D).
 Proof.
-  intros; unfold open_set in H; unfold included in |- *; intros;
-    assert (H1 := H _ H0); unfold interior in |- *; apply H1.
+  intros; unfold open_set in H; unfold included; intros;
+    assert (H1 := H _ H0); unfold interior; apply H1.
 Qed.
 
 Definition point_adherent (D:R -> Prop) (x:R) : Prop :=
@@ -49,11 +49,11 @@ Definition adherence (D:R -> Prop) (x:R) : Prop := point_adherent D x.
 
 Lemma adherence_P1 : forall D:R -> Prop, included D (adherence D).
 Proof.
-  intro; unfold included in |- *; intros; unfold adherence in |- *;
-    unfold point_adherent in |- *; intros; exists x;
-      unfold intersection_domain in |- *; split.
+  intro; unfold included; intros; unfold adherence;
+    unfold point_adherent; intros; exists x;
+      unfold intersection_domain; split.
   unfold neighbourhood in H0; elim H0; intros; unfold included in H1; apply H1;
-    unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
+    unfold disc; unfold Rminus; rewrite Rplus_opp_r;
       rewrite Rabs_R0; apply (cond_pos x0).
   apply H.
 Qed.
@@ -62,29 +62,29 @@ Lemma included_trans :
   forall D1 D2 D3:R -> Prop,
     included D1 D2 -> included D2 D3 -> included D1 D3.
 Proof.
-  unfold included in |- *; intros; apply H0; apply H; apply H1.
+  unfold included; intros; apply H0; apply H; apply H1.
 Qed.
 
 Lemma interior_P3 : forall D:R -> Prop, open_set (interior D).
 Proof.
-  intro; unfold open_set, interior in |- *; unfold neighbourhood in |- *;
+  intro; unfold open_set, interior; unfold neighbourhood;
     intros; elim H; intros.
-  exists x0; unfold included in |- *; intros.
+  exists x0; unfold included; intros.
   set (del := x0 - Rabs (x - x1)).
   cut (0 < del).
   intro; exists (mkposreal del H2); intros.
   cut (included (disc x1 (mkposreal del H2)) (disc x x0)).
   intro; assert (H5 := included_trans _ _ _ H4 H0).
   apply H5; apply H3.
-  unfold included in |- *; unfold disc in |- *; intros.
+  unfold included; unfold disc; intros.
   apply Rle_lt_trans with (Rabs (x3 - x1) + Rabs (x1 - x)).
   replace (x3 - x) with (x3 - x1 + (x1 - x)); [ apply Rabs_triang | ring ].
   replace (pos x0) with (del + Rabs (x1 - x)).
   do 2 rewrite <- (Rplus_comm (Rabs (x1 - x))); apply Rplus_lt_compat_l;
     apply H4.
-  unfold del in |- *; rewrite <- (Rabs_Ropp (x - x1)); rewrite Ropp_minus_distr;
+  unfold del; rewrite <- (Rabs_Ropp (x - x1)); rewrite Ropp_minus_distr;
     ring.
-  unfold del in |- *; apply Rplus_lt_reg_r with (Rabs (x - x1));
+  unfold del; apply Rplus_lt_reg_r with (Rabs (x - x1));
     rewrite Rplus_0_r;
       replace (Rabs (x - x1) + (x0 - Rabs (x - x1))) with (pos x0);
       [ idtac | ring ].
@@ -95,7 +95,7 @@ Lemma complementary_P1 :
   forall D:R -> Prop,
     ~ (exists y : R, intersection_domain D (complementary D) y).
 Proof.
-  intro; red in |- *; intro; elim H; intros;
+  intro; red; intro; elim H; intros;
     unfold intersection_domain, complementary in H0; elim H0;
       intros; elim H2; assumption.
 Qed.
@@ -103,8 +103,8 @@ Qed.
 Lemma adherence_P2 :
   forall D:R -> Prop, closed_set D -> included (adherence D) D.
 Proof.
-  unfold closed_set in |- *; unfold open_set, complementary in |- *; intros;
-    unfold included, adherence in |- *; intros; assert (H1 := classic (D x));
+  unfold closed_set; unfold open_set, complementary; intros;
+    unfold included, adherence; intros; assert (H1 := classic (D x));
       elim H1; intro.
   assumption.
   assert (H3 := H _ H2); assert (H4 := H0 _ H3); elim H4; intros;
@@ -114,8 +114,8 @@ Qed.
 
 Lemma adherence_P3 : forall D:R -> Prop, closed_set (adherence D).
 Proof.
-  intro; unfold closed_set, adherence in |- *;
-    unfold open_set, complementary, point_adherent in |- *;
+  intro; unfold closed_set, adherence;
+    unfold open_set, complementary, point_adherent;
       intros;
         set
           (P :=
@@ -123,21 +123,21 @@ Proof.
               neighbourhood V x ->  exists y : R, intersection_domain V D y);
           assert (H0 := not_all_ex_not _ P H); elim H0; intros V0 H1;
             unfold P in H1; assert (H2 := imply_to_and _ _ H1);
-              unfold neighbourhood in |- *; elim H2; intros; unfold neighbourhood in H3;
-                elim H3; intros; exists x0; unfold included in |- *;
-                  intros; red in |- *; intro.
+              unfold neighbourhood; elim H2; intros; unfold neighbourhood in H3;
+                elim H3; intros; exists x0; unfold included;
+                  intros; red; intro.
   assert (H8 := H7 V0);
     cut (exists delta : posreal, (forall x:R, disc x1 delta x -> V0 x)).
   intro; assert (H10 := H8 H9); elim H4; assumption.
   cut (0 < x0 - Rabs (x - x1)).
   intro; set (del := mkposreal _ H9); exists del; intros;
-    unfold included in H5; apply H5; unfold disc in |- *;
+    unfold included in H5; apply H5; unfold disc;
       apply Rle_lt_trans with (Rabs (x2 - x1) + Rabs (x1 - x)).
   replace (x2 - x) with (x2 - x1 + (x1 - x)); [ apply Rabs_triang | ring ].
   replace (pos x0) with (del + Rabs (x1 - x)).
   do 2 rewrite <- (Rplus_comm (Rabs (x1 - x))); apply Rplus_lt_compat_l;
     apply H10.
-  unfold del in |- *; simpl in |- *; rewrite <- (Rabs_Ropp (x - x1));
+  unfold del; simpl; rewrite <- (Rabs_Ropp (x - x1));
     rewrite Ropp_minus_distr; ring.
   apply Rplus_lt_reg_r with (Rabs (x - x1)); rewrite Rplus_0_r;
     replace (Rabs (x - x1) + (x0 - Rabs (x - x1))) with (pos x0);
@@ -152,10 +152,10 @@ Infix "=_D" := eq_Dom (at level 70, no associativity).
 Lemma open_set_P1 : forall D:R -> Prop, open_set D <-> D =_D interior D.
 Proof.
   intro; split.
-  intro; unfold eq_Dom in |- *; split.
+  intro; unfold eq_Dom; split.
   apply interior_P2; assumption.
   apply interior_P1.
-  intro; unfold eq_Dom in H; elim H; clear H; intros; unfold open_set in |- *;
+  intro; unfold eq_Dom in H; elim H; clear H; intros; unfold open_set;
     intros; unfold included, interior in H; unfold included in H0;
       apply (H _ H1).
 Qed.
@@ -163,20 +163,20 @@ Qed.
 Lemma closed_set_P1 : forall D:R -> Prop, closed_set D <-> D =_D adherence D.
 Proof.
   intro; split.
-  intro; unfold eq_Dom in |- *; split.
+  intro; unfold eq_Dom; split.
   apply adherence_P1.
   apply adherence_P2; assumption.
-  unfold eq_Dom in |- *; unfold included in |- *; intros;
+  unfold eq_Dom; unfold included; intros;
     assert (H0 := adherence_P3 D); unfold closed_set in H0;
-      unfold closed_set in |- *; unfold open_set in |- *;
+      unfold closed_set; unfold open_set;
         unfold open_set in H0; intros; assert (H2 : complementary (adherence D) x).
-  unfold complementary in |- *; unfold complementary in H1; red in |- *; intro;
+  unfold complementary; unfold complementary in H1; red; intro;
     elim H; clear H; intros _ H; elim H1; apply (H _ H2).
-  assert (H3 := H0 _ H2); unfold neighbourhood in |- *;
+  assert (H3 := H0 _ H2); unfold neighbourhood;
     unfold neighbourhood in H3; elim H3; intros; exists x0;
-      unfold included in |- *; unfold included in H4; intros;
+      unfold included; unfold included in H4; intros;
         assert (H6 := H4 _ H5); unfold complementary in H6;
-          unfold complementary in |- *; red in |- *; intro;
+          unfold complementary; red; intro;
             elim H; clear H; intros H _; elim H6; apply (H _ H7).
 Qed.
 
@@ -184,8 +184,8 @@ Lemma neighbourhood_P1 :
   forall (D1 D2:R -> Prop) (x:R),
     included D1 D2 -> neighbourhood D1 x -> neighbourhood D2 x.
 Proof.
-  unfold included, neighbourhood in |- *; intros; elim H0; intros; exists x0;
-    intros; unfold included in |- *; unfold included in H1;
+  unfold included, neighbourhood; intros; elim H0; intros; exists x0;
+    intros; unfold included; unfold included in H1;
       intros; apply (H _ (H1 _ H2)).
 Qed.
 
@@ -193,12 +193,12 @@ Lemma open_set_P2 :
   forall D1 D2:R -> Prop,
     open_set D1 -> open_set D2 -> open_set (union_domain D1 D2).
 Proof.
-  unfold open_set in |- *; intros; unfold union_domain in H1; elim H1; intro.
+  unfold open_set; intros; unfold union_domain in H1; elim H1; intro.
   apply neighbourhood_P1 with D1.
-  unfold included, union_domain in |- *; tauto.
+  unfold included, union_domain; tauto.
   apply H; assumption.
   apply neighbourhood_P1 with D2.
-  unfold included, union_domain in |- *; tauto.
+  unfold included, union_domain; tauto.
   apply H0; assumption.
 Qed.
 
@@ -206,53 +206,53 @@ Lemma open_set_P3 :
   forall D1 D2:R -> Prop,
     open_set D1 -> open_set D2 -> open_set (intersection_domain D1 D2).
 Proof.
-  unfold open_set in |- *; intros; unfold intersection_domain in H1; elim H1;
+  unfold open_set; intros; unfold intersection_domain in H1; elim H1;
     intros.
   assert (H4 := H _ H2); assert (H5 := H0 _ H3);
-    unfold intersection_domain in |- *; unfold neighbourhood in H4, H5;
+    unfold intersection_domain; unfold neighbourhood in H4, H5;
       elim H4; clear H; intros del1 H; elim H5; clear H0;
         intros del2 H0; cut (0 < Rmin del1 del2).
   intro; set (del := mkposreal _ H6).
-  exists del; unfold included in |- *; intros; unfold included in H, H0;
+  exists del; unfold included; intros; unfold included in H, H0;
     unfold disc in H, H0, H7.
   split.
   apply H; apply Rlt_le_trans with (pos del).
   apply H7.
-  unfold del in |- *; simpl in |- *; apply Rmin_l.
+  unfold del; simpl; apply Rmin_l.
   apply H0; apply Rlt_le_trans with (pos del).
   apply H7.
-  unfold del in |- *; simpl in |- *; apply Rmin_r.
-  unfold Rmin in |- *; case (Rle_dec del1 del2); intro.
+  unfold del; simpl; apply Rmin_r.
+  unfold Rmin; case (Rle_dec del1 del2); intro.
   apply (cond_pos del1).
   apply (cond_pos del2).
 Qed.
 
 Lemma open_set_P4 : open_set (fun x:R => False).
 Proof.
-  unfold open_set in |- *; intros; elim H.
+  unfold open_set; intros; elim H.
 Qed.
 
 Lemma open_set_P5 : open_set (fun x:R => True).
 Proof.
-  unfold open_set in |- *; intros; unfold neighbourhood in |- *.
-  exists (mkposreal 1 Rlt_0_1); unfold included in |- *; intros; trivial.
+  unfold open_set; intros; unfold neighbourhood.
+  exists (mkposreal 1 Rlt_0_1); unfold included; intros; trivial.
 Qed.
 
 Lemma disc_P1 : forall (x:R) (del:posreal), open_set (disc x del).
 Proof.
   intros; assert (H := open_set_P1 (disc x del)).
   elim H; intros; apply H1.
-  unfold eq_Dom in |- *; split.
-  unfold included, interior, disc in |- *; intros;
+  unfold eq_Dom; split.
+  unfold included, interior, disc; intros;
     cut (0 < del - Rabs (x - x0)).
   intro; set (del2 := mkposreal _ H3).
-  exists del2; unfold included in |- *; intros.
+  exists del2; unfold included; intros.
   apply Rle_lt_trans with (Rabs (x1 - x0) + Rabs (x0 - x)).
   replace (x1 - x) with (x1 - x0 + (x0 - x)); [ apply Rabs_triang | ring ].
   replace (pos del) with (del2 + Rabs (x0 - x)).
   do 2 rewrite <- (Rplus_comm (Rabs (x0 - x))); apply Rplus_lt_compat_l.
   apply H4.
-  unfold del2 in |- *; simpl in |- *; rewrite <- (Rabs_Ropp (x - x0));
+  unfold del2; simpl; rewrite <- (Rabs_Ropp (x - x0));
     rewrite Ropp_minus_distr; ring.
   apply Rplus_lt_reg_r with (Rabs (x - x0)); rewrite Rplus_0_r;
     replace (Rabs (x - x0) + (del - Rabs (x - x0))) with (pos del);
@@ -278,19 +278,19 @@ Proof.
   elim H3; intros.
   exists (disc x (mkposreal del2 H4)).
   intros; unfold included in H1; split.
-  unfold neighbourhood, disc in |- *.
+  unfold neighbourhood, disc.
   exists (mkposreal del2 H4).
-  unfold included in |- *; intros; assumption.
-  intros; apply H1; unfold disc in |- *; case (Req_dec y x); intro.
-  rewrite H7; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  unfold included; intros; assumption.
+  intros; apply H1; unfold disc; case (Req_dec y x); intro.
+  rewrite H7; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     apply (cond_pos del1).
   apply H5; split.
-  unfold D_x, no_cond in |- *; split.
+  unfold D_x, no_cond; split.
   trivial.
-  apply (sym_not_eq (A:=R)); apply H7.
+  apply (not_eq_sym (A:=R)); apply H7.
   unfold disc in H6; apply H6.
-  intros; unfold continuity_pt in |- *; unfold continue_in in |- *;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
+  intros; unfold continuity_pt; unfold continue_in;
+    unfold limit1_in; unfold limit_in;
       intros.
   assert (H1 := H (disc (f x) (mkposreal eps H0))).
   cut (neighbourhood (disc (f x) (mkposreal eps H0)) (f x)).
@@ -299,10 +299,10 @@ Proof.
     intros del1 H7.
   exists (pos del1); split.
   apply (cond_pos del1).
-  intros; elim H8; intros; simpl in H10; unfold R_dist in H10; simpl in |- *;
-    unfold R_dist in |- *; apply (H6 _ (H7 _ H10)).
-  unfold neighbourhood, disc in |- *; exists (mkposreal eps H0);
-    unfold included in |- *; intros; assumption.
+  intros; elim H8; intros; simpl in H10; unfold R_dist in H10; simpl;
+    unfold R_dist; apply (H6 _ (H7 _ H10)).
+  unfold neighbourhood, disc; exists (mkposreal eps H0);
+    unfold included; intros; assumption.
 Qed.
 
 Definition image_rec (f:R -> R) (D:R -> Prop) (x:R) : Prop := D (f x).
@@ -312,13 +312,13 @@ Lemma continuity_P2 :
   forall (f:R -> R) (D:R -> Prop),
     continuity f -> open_set D -> open_set (image_rec f D).
 Proof.
-  intros; unfold open_set in H0; unfold open_set in |- *; intros;
+  intros; unfold open_set in H0; unfold open_set; intros;
     assert (H2 := continuity_P1 f x); elim H2; intros H3 _;
-      assert (H4 := H3 (H x)); unfold neighbourhood, image_rec in |- *;
+      assert (H4 := H3 (H x)); unfold neighbourhood, image_rec;
         unfold image_rec in H1; assert (H5 := H4 D (H0 (f x) H1));
           elim H5; intros V0 H6; elim H6; intros; unfold neighbourhood in H7;
             elim H7; intros del H9; exists del; unfold included in H9;
-              unfold included in |- *; intros; apply (H8 _ (H9 _ H10)).
+              unfold included; intros; apply (H8 _ (H9 _ H10)).
 Qed.
 
 (**********)
@@ -329,9 +329,9 @@ Lemma continuity_P3 :
 Proof.
   intros; split.
   intros; apply continuity_P2; assumption.
-  intros; unfold continuity in |- *; unfold continuity_pt in |- *;
-    unfold continue_in in |- *; unfold limit1_in in |- *;
-      unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *;
+  intros; unfold continuity; unfold continuity_pt;
+    unfold continue_in; unfold limit1_in;
+      unfold limit_in; simpl; unfold R_dist;
         intros; cut (open_set (disc (f x) (mkposreal _ H0))).
   intro; assert (H2 := H _ H1).
   unfold open_set, image_rec in H2; cut (disc (f x) (mkposreal _ H0) (f x)).
@@ -340,7 +340,7 @@ Proof.
   exists (pos del); split.
   apply (cond_pos del).
   intros; unfold included in H5; apply H5; elim H6; intros; apply H8.
-  unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
+  unfold disc; unfold Rminus; rewrite Rplus_opp_r;
     rewrite Rabs_R0; apply H0.
   apply disc_P1.
 Qed.
@@ -358,23 +358,23 @@ Proof.
   cut (0 < D / 2).
   intro; exists (disc x (mkposreal _ H)).
   exists (disc y (mkposreal _ H)); split.
-  unfold neighbourhood in |- *; exists (mkposreal _ H); unfold included in |- *;
+  unfold neighbourhood; exists (mkposreal _ H); unfold included;
     tauto.
   split.
-  unfold neighbourhood in |- *; exists (mkposreal _ H); unfold included in |- *;
+  unfold neighbourhood; exists (mkposreal _ H); unfold included;
     tauto.
-  red in |- *; intro; elim H0; intros; unfold intersection_domain in H1;
+  red; intro; elim H0; intros; unfold intersection_domain in H1;
     elim H1; intros.
   cut (D < D).
   intro; elim (Rlt_irrefl _ H4).
-  change (Rabs (x - y) < D) in |- *;
+  change (Rabs (x - y) < D);
     apply Rle_lt_trans with (Rabs (x - x0) + Rabs (x0 - y)).
   replace (x - y) with (x - x0 + (x0 - y)); [ apply Rabs_triang | ring ].
   rewrite (double_var D); apply Rplus_lt_compat.
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H2.
   apply H3.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
-  unfold D in |- *; apply Rabs_pos_lt; apply (Rminus_eq_contra _ _ Hsep).
+  unfold Rdiv; apply Rmult_lt_0_compat.
+  unfold D; apply Rabs_pos_lt; apply (Rminus_eq_contra _ _ Hsep).
   apply Rinv_0_lt_compat; prove_sup0.
 Qed.
 
@@ -404,7 +404,7 @@ Lemma restriction_family :
     (exists y : R, (fun z1 z2:R => f z1 z2 /\ D z1) x y) ->
     intersection_domain (ind f) D x.
 Proof.
-  intros; elim H; intros; unfold intersection_domain in |- *; elim H0; intros;
+  intros; elim H; intros; unfold intersection_domain; elim H0; intros;
     split.
   apply (cond_fam f0); exists x0; assumption.
   assumption.
@@ -424,19 +424,19 @@ Lemma family_P1 :
   forall (f:family) (D:R -> Prop),
     family_open_set f -> family_open_set (subfamily f D).
 Proof.
-  unfold family_open_set in |- *; intros; unfold subfamily in |- *;
-    simpl in |- *; assert (H0 := classic (D x)).
+  unfold family_open_set; intros; unfold subfamily;
+    simpl; assert (H0 := classic (D x)).
   elim H0; intro.
   cut (open_set (f0 x) -> open_set (fun y:R => f0 x y /\ D x)).
   intro; apply H2; apply H.
-  unfold open_set in |- *; unfold neighbourhood in |- *; intros; elim H3;
+  unfold open_set; unfold neighbourhood; intros; elim H3;
     intros; assert (H6 := H2 _ H4); elim H6; intros; exists x1;
-      unfold included in |- *; intros; split.
+      unfold included; intros; split.
   apply (H7 _ H8).
   assumption.
   cut (open_set (fun y:R => False) -> open_set (fun y:R => f0 x y /\ D x)).
   intro; apply H2; apply open_set_P4.
-  unfold open_set in |- *; unfold neighbourhood in |- *; intros; elim H3;
+  unfold open_set; unfold neighbourhood; intros; elim H3;
     intros; elim H1; assumption.
 Qed.
 
@@ -446,7 +446,7 @@ Definition bounded (D:R -> Prop) : Prop :=
 Lemma open_set_P6 :
   forall D1 D2:R -> Prop, open_set D1 -> D1 =_D D2 -> open_set D2.
 Proof.
-  unfold open_set in |- *; unfold neighbourhood in |- *; intros.
+  unfold open_set; unfold neighbourhood; intros.
   unfold eq_Dom in H0; elim H0; intros.
   assert (H4 := H _ (H3 _ H1)).
   elim H4; intros.
@@ -465,7 +465,7 @@ Proof.
   intro; assert (H3 := H1 H2); elim H3; intros D' H4;
     unfold covering_finite in H4; elim H4; intros; unfold family_finite in H6;
       unfold domain_finite in H6; elim H6; intros l H7;
-        unfold bounded in |- *; set (r := MaxRlist l).
+        unfold bounded; set (r := MaxRlist l).
   exists (- r); exists r; intros.
   unfold covering in H5; assert (H9 := H5 _ H8); elim H9; intros;
     unfold subfamily in H10; simpl in H10; elim H10; intros;
@@ -484,25 +484,25 @@ Proof.
   left; apply H11.
   assumption.
   apply (MaxRlist_P1 l x0 H16).
-  unfold intersection_domain, D in |- *; tauto.
-  unfold covering_open_set in |- *; split.
-  unfold covering in |- *; intros; simpl in |- *; exists (Rabs x + 1);
-    unfold g in |- *; pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r;
+  unfold intersection_domain, D; tauto.
+  unfold covering_open_set; split.
+  unfold covering; intros; simpl; exists (Rabs x + 1);
+    unfold g; pattern (Rabs x) at 1; rewrite <- Rplus_0_r;
       apply Rplus_lt_compat_l; apply Rlt_0_1.
-  unfold family_open_set in |- *; intro; case (Rtotal_order 0 x); intro.
+  unfold family_open_set; intro; case (Rtotal_order 0 x); intro.
   apply open_set_P6 with (disc 0 (mkposreal _ H2)).
   apply disc_P1.
-  unfold eq_Dom in |- *; unfold f0 in |- *; simpl in |- *;
-    unfold g, disc in |- *; split.
-  unfold included in |- *; intros; unfold Rminus in H3; rewrite Ropp_0 in H3;
+  unfold eq_Dom; unfold f0; simpl;
+    unfold g, disc; split.
+  unfold included; intros; unfold Rminus in H3; rewrite Ropp_0 in H3;
     rewrite Rplus_0_r in H3; apply H3.
-  unfold included in |- *; intros; unfold Rminus in |- *; rewrite Ropp_0;
+  unfold included; intros; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; apply H3.
   apply open_set_P6 with (fun x:R => False).
   apply open_set_P4.
-  unfold eq_Dom in |- *; split.
-  unfold included in |- *; intros; elim H3.
-  unfold included, f0 in |- *; simpl in |- *; unfold g in |- *; intros; elim H2;
+  unfold eq_Dom; split.
+  unfold included; intros; elim H3.
+  unfold included, f0; simpl; unfold g; intros; elim H2;
     intro;
       [ rewrite <- H4 in H3; assert (H5 := Rabs_pos x0);
         elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H5 H3))
@@ -515,10 +515,10 @@ Lemma compact_P2 : forall X:R -> Prop, compact X -> closed_set X.
 Proof.
   intros; assert (H0 := closed_set_P1 X); elim H0; clear H0; intros _ H0;
     apply H0; clear H0.
-  unfold eq_Dom in |- *; split.
+  unfold eq_Dom; split.
   apply adherence_P1.
-  unfold included in |- *; unfold adherence in |- *;
-    unfold point_adherent in |- *; intros; unfold compact in H;
+  unfold included; unfold adherence;
+    unfold point_adherent; intros; unfold compact in H;
       assert (H1 := classic (X x)); elim H1; clear H1; intro.
   assumption.
   cut (forall y:R, X y -> 0 < Rabs (y - x) / 2).
@@ -548,44 +548,44 @@ Proof.
   replace (y0 - x) with (y0 - y + (y - x)); [ apply Rabs_triang | ring ].
   rewrite (double_var (Rabs (y0 - x))); apply Rplus_lt_compat; assumption.
   apply (MinRlist_P1 (AbsList l x) (Rabs (y0 - x) / 2)); apply AbsList_P1;
-    elim (H8 y0); clear H8; intros; apply H8; unfold intersection_domain in |- *;
+    elim (H8 y0); clear H8; intros; apply H8; unfold intersection_domain;
       split; assumption.
   assert (H11 := disc_P1 x (mkposreal alp H9)); unfold open_set in H11;
     apply H11.
-  unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
+  unfold disc; unfold Rminus; rewrite Rplus_opp_r;
     rewrite Rabs_R0; apply H9.
-  unfold alp in |- *; apply MinRlist_P2; intros;
+  unfold alp; apply MinRlist_P2; intros;
     assert (H10 := AbsList_P2 _ _ _ H9); elim H10; clear H10;
       intros z H10; elim H10; clear H10; intros; rewrite H11;
         apply H2; elim (H8 z); clear H8; intros; assert (H13 := H12 H10);
           unfold intersection_domain, D in H13; elim H13; clear H13;
             intros; assumption.
-  unfold covering_open_set in |- *; split.
-  unfold covering in |- *; intros; exists x0; simpl in |- *; unfold g in |- *;
+  unfold covering_open_set; split.
+  unfold covering; intros; exists x0; simpl; unfold g;
     split.
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     unfold Rminus in H2; apply (H2 _ H5).
   apply H5.
-  unfold family_open_set in |- *; intro; simpl in |- *; unfold g in |- *;
+  unfold family_open_set; intro; simpl; unfold g;
     elim (classic (D x0)); intro.
   apply open_set_P6 with (disc x0 (mkposreal _ (H2 _ H5))).
   apply disc_P1.
-  unfold eq_Dom in |- *; split.
-  unfold included, disc in |- *; simpl in |- *; intros; split.
+  unfold eq_Dom; split.
+  unfold included, disc; simpl; intros; split.
   rewrite <- (Rabs_Ropp (x0 - x1)); rewrite Ropp_minus_distr; apply H6.
   apply H5.
-  unfold included, disc in |- *; simpl in |- *; intros; elim H6; intros;
+  unfold included, disc; simpl; intros; elim H6; intros;
     rewrite <- (Rabs_Ropp (x1 - x0)); rewrite Ropp_minus_distr;
       apply H7.
   apply open_set_P6 with (fun z:R => False).
   apply open_set_P4.
-  unfold eq_Dom in |- *; split.
-  unfold included in |- *; intros; elim H6.
-  unfold included in |- *; intros; elim H6; intros; elim H5; assumption.
+  unfold eq_Dom; split.
+  unfold included; intros; elim H6.
+  unfold included; intros; elim H6; intros; elim H5; assumption.
   intros; elim H3; intros; unfold g in H4; elim H4; clear H4; intros _ H4;
     apply H4.
-  intros; unfold Rdiv in |- *; apply Rmult_lt_0_compat.
-  apply Rabs_pos_lt; apply Rminus_eq_contra; red in |- *; intro;
+  intros; unfold Rdiv; apply Rmult_lt_0_compat.
+  apply Rabs_pos_lt; apply Rminus_eq_contra; red; intro;
     rewrite H3 in H2; elim H1; apply H2.
   apply Rinv_0_lt_compat; prove_sup0.
 Qed.
@@ -593,29 +593,29 @@ Qed.
 (**********)
 Lemma compact_EMP : compact (fun _:R => False).
 Proof.
-  unfold compact in |- *; intros; exists (fun x:R => False);
-    unfold covering_finite in |- *; split.
-  unfold covering in |- *; intros; elim H0.
-  unfold family_finite in |- *; unfold domain_finite in |- *; exists nil; intro.
+  unfold compact; intros; exists (fun x:R => False);
+    unfold covering_finite; split.
+  unfold covering; intros; elim H0.
+  unfold family_finite; unfold domain_finite; exists nil; intro.
   split.
-  simpl in |- *; unfold intersection_domain in |- *; intros; elim H0.
+  simpl; unfold intersection_domain; intros; elim H0.
   elim H0; clear H0; intros _ H0; elim H0.
-  simpl in |- *; intro; elim H0.
+  simpl; intro; elim H0.
 Qed.
 
 Lemma compact_eqDom :
   forall X1 X2:R -> Prop, compact X1 -> X1 =_D X2 -> compact X2.
 Proof.
-  unfold compact in |- *; intros; unfold eq_Dom in H0; elim H0; clear H0;
-    unfold included in |- *; intros; assert (H3 : covering_open_set X1 f0).
-  unfold covering_open_set in |- *; unfold covering_open_set in H1; elim H1;
+  unfold compact; intros; unfold eq_Dom in H0; elim H0; clear H0;
+    unfold included; intros; assert (H3 : covering_open_set X1 f0).
+  unfold covering_open_set; unfold covering_open_set in H1; elim H1;
     clear H1; intros; split.
-  unfold covering in H1; unfold covering in |- *; intros;
+  unfold covering in H1; unfold covering; intros;
     apply (H1 _ (H0 _ H4)).
   apply H3.
-  elim (H _ H3); intros D H4; exists D; unfold covering_finite in |- *;
+  elim (H _ H3); intros D H4; exists D; unfold covering_finite;
     unfold covering_finite in H4; elim H4; intros; split.
-  unfold covering in H5; unfold covering in |- *; intros;
+  unfold covering in H5; unfold covering; intros;
     apply (H5 _ (H2 _ H7)).
   apply H6.
 Qed.
@@ -624,7 +624,7 @@ Qed.
 Lemma compact_P3 : forall a b:R, compact (fun c:R => a <= c <= b).
 Proof.
   intros; case (Rle_dec a b); intro.
-  unfold compact in |- *; intros;
+  unfold compact; intros;
     set
       (A :=
         fun x:R =>
@@ -647,92 +647,92 @@ Proof.
   rewrite H11 in H10; rewrite H11 in H8; unfold A in H9; elim H9; clear H9;
     intros; elim H12; clear H12; intros Dx H12;
       set (Db := fun x:R => Dx x \/ x = y0); exists Db;
-        unfold covering_finite in |- *; split.
-  unfold covering in |- *; unfold covering_finite in H12; elim H12; clear H12;
+        unfold covering_finite; split.
+  unfold covering; unfold covering_finite in H12; elim H12; clear H12;
     intros; unfold covering in H12; case (Rle_dec x0 x);
       intro.
   cut (a <= x0 <= x).
   intro; assert (H16 := H12 x0 H15); elim H16; clear H16; intros; exists x1;
-    simpl in H16; simpl in |- *; unfold Db in |- *; elim H16;
+    simpl in H16; simpl; unfold Db; elim H16;
       clear H16; intros; split; [ apply H16 | left; apply H17 ].
   split.
   elim H14; intros; assumption.
   assumption.
-  exists y0; simpl in |- *; split.
-  apply H8; unfold disc in |- *; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr;
+  exists y0; simpl; split.
+  apply H8; unfold disc; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr;
     rewrite Rabs_right.
   apply Rlt_trans with (b - x).
-  unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_gt_contravar;
+  unfold Rminus; apply Rplus_lt_compat_l; apply Ropp_lt_gt_contravar;
     auto with real.
   elim H10; intros H15 _; apply Rplus_lt_reg_r with (x - eps);
     replace (x - eps + (b - x)) with (b - eps);
     [ replace (x - eps + eps) with x; [ apply H15 | ring ] | ring ].
   apply Rge_minus; apply Rle_ge; elim H14; intros _ H15; apply H15.
-  unfold Db in |- *; right; reflexivity.
-  unfold family_finite in |- *; unfold domain_finite in |- *;
+  unfold Db; right; reflexivity.
+  unfold family_finite; unfold domain_finite;
     unfold covering_finite in H12; elim H12; clear H12;
       intros; unfold family_finite in H13; unfold domain_finite in H13;
         elim H13; clear H13; intros l H13; exists (cons y0 l);
           intro; split.
   intro; simpl in H14; unfold intersection_domain in H14; elim (H13 x0);
     clear H13; intros; case (Req_dec x0 y0); intro.
-  simpl in |- *; left; apply H16.
-  simpl in |- *; right; apply H13.
-  simpl in |- *; unfold intersection_domain in |- *; unfold Db in H14;
+  simpl; left; apply H16.
+  simpl; right; apply H13.
+  simpl; unfold intersection_domain; unfold Db in H14;
     decompose [and or] H14.
   split; assumption.
   elim H16; assumption.
-  intro; simpl in H14; elim H14; intro; simpl in |- *;
-    unfold intersection_domain in |- *.
+  intro; simpl in H14; elim H14; intro; simpl;
+    unfold intersection_domain.
   split.
   apply (cond_fam f0); rewrite H15; exists m; apply H6.
-  unfold Db in |- *; right; assumption.
-  simpl in |- *; unfold intersection_domain in |- *; elim (H13 x0).
+  unfold Db; right; assumption.
+  simpl; unfold intersection_domain; elim (H13 x0).
   intros _ H16; assert (H17 := H16 H15); simpl in H17;
     unfold intersection_domain in H17; split.
   elim H17; intros; assumption.
-  unfold Db in |- *; left; elim H17; intros; assumption.
+  unfold Db; left; elim H17; intros; assumption.
   set (m' := Rmin (m + eps / 2) b); cut (A m').
   intro; elim H3; intros; unfold is_upper_bound in H13;
     assert (H15 := H13 m' H12); cut (m < m').
   intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H15 H16)).
-  unfold m' in |- *; unfold Rmin in |- *; case (Rle_dec (m + eps / 2) b); intro.
-  pattern m at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
-    unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold m'; unfold Rmin; case (Rle_dec (m + eps / 2) b); intro.
+  pattern m at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
+    unfold Rdiv; apply Rmult_lt_0_compat;
       [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ].
   elim H4; intros.
   elim H17; intro.
   assumption.
   elim H11; assumption.
-  unfold A in |- *; split.
+  unfold A; split.
   split.
   apply Rle_trans with m.
   elim H4; intros; assumption.
-  unfold m' in |- *; unfold Rmin in |- *; case (Rle_dec (m + eps / 2) b); intro.
-  pattern m at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
-    unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold m'; unfold Rmin; case (Rle_dec (m + eps / 2) b); intro.
+  pattern m at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
+    unfold Rdiv; apply Rmult_lt_0_compat;
       [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ].
   elim H4; intros.
   elim H13; intro.
   assumption.
   elim H11; assumption.
-  unfold m' in |- *; apply Rmin_r.
+  unfold m'; apply Rmin_r.
   unfold A in H9; elim H9; clear H9; intros; elim H12; clear H12; intros Dx H12;
     set (Db := fun x:R => Dx x \/ x = y0); exists Db;
-      unfold covering_finite in |- *; split.
-  unfold covering in |- *; unfold covering_finite in H12; elim H12; clear H12;
+      unfold covering_finite; split.
+  unfold covering; unfold covering_finite in H12; elim H12; clear H12;
     intros; unfold covering in H12; case (Rle_dec x0 x);
       intro.
   cut (a <= x0 <= x).
   intro; assert (H16 := H12 x0 H15); elim H16; clear H16; intros; exists x1;
-    simpl in H16; simpl in |- *; unfold Db in |- *.
+    simpl in H16; simpl; unfold Db.
   elim H16; clear H16; intros; split; [ apply H16 | left; apply H17 ].
   elim H14; intros; split; assumption.
-  exists y0; simpl in |- *; split.
-  apply H8; unfold disc in |- *; unfold Rabs in |- *; case (Rcase_abs (x0 - m));
+  exists y0; simpl; split.
+  apply H8; unfold disc; unfold Rabs; case (Rcase_abs (x0 - m));
     intro.
   rewrite Ropp_minus_distr; apply Rlt_trans with (m - x).
-  unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_gt_contravar;
+  unfold Rminus; apply Rplus_lt_compat_l; apply Ropp_lt_gt_contravar;
     auto with real.
   apply Rplus_lt_reg_r with (x - eps);
     replace (x - eps + (m - x)) with (m - eps).
@@ -741,56 +741,56 @@ Proof.
   ring.
   ring.
   apply Rle_lt_trans with (m' - m).
-  unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- m));
+  unfold Rminus; do 2 rewrite <- (Rplus_comm (- m));
     apply Rplus_le_compat_l; elim H14; intros; assumption.
   apply Rplus_lt_reg_r with m; replace (m + (m' - m)) with m'.
   apply Rle_lt_trans with (m + eps / 2).
-  unfold m' in |- *; apply Rmin_l.
+  unfold m'; apply Rmin_l.
   apply Rplus_lt_compat_l; apply Rmult_lt_reg_l with 2.
   prove_sup0.
-  unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
+  unfold Rdiv; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
     rewrite <- Rinv_r_sym.
-  rewrite Rmult_1_l; pattern (pos eps) at 1 in |- *; rewrite <- Rplus_0_r;
+  rewrite Rmult_1_l; pattern (pos eps) at 1; rewrite <- Rplus_0_r;
     rewrite double; apply Rplus_lt_compat_l; apply (cond_pos eps).
   discrR.
   ring.
-  unfold Db in |- *; right; reflexivity.
-  unfold family_finite in |- *; unfold domain_finite in |- *;
+  unfold Db; right; reflexivity.
+  unfold family_finite; unfold domain_finite;
     unfold covering_finite in H12; elim H12; clear H12;
       intros; unfold family_finite in H13; unfold domain_finite in H13;
         elim H13; clear H13; intros l H13; exists (cons y0 l);
           intro; split.
   intro; simpl in H14; unfold intersection_domain in H14; elim (H13 x0);
     clear H13; intros; case (Req_dec x0 y0); intro.
-  simpl in |- *; left; apply H16.
-  simpl in |- *; right; apply H13; simpl in |- *;
-    unfold intersection_domain in |- *; unfold Db in H14;
+  simpl; left; apply H16.
+  simpl; right; apply H13; simpl;
+    unfold intersection_domain; unfold Db in H14;
       decompose [and or] H14.
   split; assumption.
   elim H16; assumption.
-  intro; simpl in H14; elim H14; intro; simpl in |- *;
-    unfold intersection_domain in |- *.
+  intro; simpl in H14; elim H14; intro; simpl;
+    unfold intersection_domain.
   split.
   apply (cond_fam f0); rewrite H15; exists m; apply H6.
-  unfold Db in |- *; right; assumption.
+  unfold Db; right; assumption.
   elim (H13 x0); intros _ H16.
   assert (H17 := H16 H15).
   simpl in H17.
   unfold intersection_domain in H17.
   split.
   elim H17; intros; assumption.
-  unfold Db in |- *; left; elim H17; intros; assumption.
+  unfold Db; left; elim H17; intros; assumption.
   elim (classic (exists x : R, A x /\ m - eps < x <= m)); intro.
   assumption.
   elim H3; intros; cut (is_upper_bound A (m - eps)).
   intro; assert (H13 := H11 _ H12); cut (m - eps < m).
   intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H13 H14)).
-  pattern m at 2 in |- *; rewrite <- Rplus_0_r; unfold Rminus in |- *;
+  pattern m at 2; rewrite <- Rplus_0_r; unfold Rminus;
     apply Rplus_lt_compat_l; apply Ropp_lt_cancel; rewrite Ropp_involutive;
       rewrite Ropp_0; apply (cond_pos eps).
   set (P := fun n:R => A n /\ m - eps < n <= m);
     assert (H12 := not_ex_all_not _ P H9); unfold P in H12;
-      unfold is_upper_bound in |- *; intros;
+      unfold is_upper_bound; intros;
         assert (H14 := not_and_or _ _ (H12 x)); elim H14;
           intro.
   elim H15; apply H13.
@@ -803,44 +803,44 @@ Proof.
   unfold is_upper_bound in H3.
   split.
   apply (H3 _ H0).
-  apply (H4 b); unfold is_upper_bound in |- *; intros; unfold A in H5; elim H5;
+  apply (H4 b); unfold is_upper_bound; intros; unfold A in H5; elim H5;
     clear H5; intros H5 _; elim H5; clear H5; intros _ H5;
       apply H5.
   exists a; apply H0.
-  unfold bound in |- *; exists b; unfold is_upper_bound in |- *; intros;
+  unfold bound; exists b; unfold is_upper_bound; intros;
     unfold A in H1; elim H1; clear H1; intros H1 _; elim H1;
       clear H1; intros _ H1; apply H1.
-  unfold A in |- *; split.
+  unfold A; split.
   split; [ right; reflexivity | apply r ].
   unfold covering_open_set in H; elim H; clear H; intros; unfold covering in H;
     cut (a <= a <= b).
   intro; elim (H _ H1); intros y0 H2; set (D' := fun x:R => x = y0); exists D';
-    unfold covering_finite in |- *; split.
-  unfold covering in |- *; simpl in |- *; intros; cut (x = a).
+    unfold covering_finite; split.
+  unfold covering; simpl; intros; cut (x = a).
   intro; exists y0; split.
   rewrite H4; apply H2.
-  unfold D' in |- *; reflexivity.
+  unfold D'; reflexivity.
   elim H3; intros; apply Rle_antisym; assumption.
-  unfold family_finite in |- *; unfold domain_finite in |- *;
+  unfold family_finite; unfold domain_finite;
     exists (cons y0 nil); intro; split.
-  simpl in |- *; unfold intersection_domain in |- *; intro; elim H3; clear H3;
+  simpl; unfold intersection_domain; intro; elim H3; clear H3;
     intros; unfold D' in H4; left; apply H4.
-  simpl in |- *; unfold intersection_domain in |- *; intro; elim H3; intro.
+  simpl; unfold intersection_domain; intro; elim H3; intro.
   split; [ rewrite H4; apply (cond_fam f0); exists a; apply H2 | apply H4 ].
   elim H4.
   split; [ right; reflexivity | apply r ].
   apply compact_eqDom with (fun c:R => False).
   apply compact_EMP.
-  unfold eq_Dom in |- *; split.
-  unfold included in |- *; intros; elim H.
-  unfold included in |- *; intros; elim H; clear H; intros;
+  unfold eq_Dom; split.
+  unfold included; intros; elim H.
+  unfold included; intros; elim H; clear H; intros;
     assert (H1 := Rle_trans _ _ _ H H0); elim n; apply H1.
 Qed.
 
 Lemma compact_P4 :
   forall X F:R -> Prop, compact X -> closed_set F -> included F X -> compact F.
 Proof.
-  unfold compact in |- *; intros; elim (classic (exists z : R, F z));
+  unfold compact; intros; elim (classic (exists z : R, F z));
     intro Hyp_F_NE.
   set (D := ind f0); set (g := f f0); unfold closed_set in H0.
   set (g' := fun x y:R => f0 x y \/ complementary F y /\ D x).
@@ -848,61 +848,61 @@ Proof.
   cut (forall x:R, (exists y : R, g' x y) -> D' x).
   intro; set (f' := mkfamily D' g' H3); cut (covering_open_set X f').
   intro; elim (H _ H4); intros DX H5; exists DX.
-  unfold covering_finite in |- *; unfold covering_finite in H5; elim H5;
+  unfold covering_finite; unfold covering_finite in H5; elim H5;
     clear H5; intros.
   split.
-  unfold covering in |- *; unfold covering in H5; intros.
-  elim (H5 _ (H1 _ H7)); intros y0 H8; exists y0; simpl in H8; simpl in |- *;
+  unfold covering; unfold covering in H5; intros.
+  elim (H5 _ (H1 _ H7)); intros y0 H8; exists y0; simpl in H8; simpl;
     elim H8; clear H8; intros.
   split.
   unfold g' in H8; elim H8; intro.
   apply H10.
   elim H10; intros H11 _; unfold complementary in H11; elim H11; apply H7.
   apply H9.
-  unfold family_finite in |- *; unfold domain_finite in |- *;
+  unfold family_finite; unfold domain_finite;
     unfold family_finite in H6; unfold domain_finite in H6;
       elim H6; clear H6; intros l H6; exists l; intro; assert (H7 := H6 x);
         elim H7; clear H7; intros.
   split.
-  intro; apply H7; simpl in |- *; unfold intersection_domain in |- *;
-    simpl in H9; unfold intersection_domain in H9; unfold D' in |- *;
+  intro; apply H7; simpl; unfold intersection_domain;
+    simpl in H9; unfold intersection_domain in H9; unfold D';
       apply H9.
   intro; assert (H10 := H8 H9); simpl in H10; unfold intersection_domain in H10;
-    simpl in |- *; unfold intersection_domain in |- *;
+    simpl; unfold intersection_domain;
       unfold D' in H10; apply H10.
-  unfold covering_open_set in |- *; unfold covering_open_set in H2; elim H2;
+  unfold covering_open_set; unfold covering_open_set in H2; elim H2;
     clear H2; intros.
   split.
-  unfold covering in |- *; unfold covering in H2; intros.
+  unfold covering; unfold covering in H2; intros.
   elim (classic (F x)); intro.
-  elim (H2 _ H6); intros y0 H7; exists y0; simpl in |- *; unfold g' in |- *;
+  elim (H2 _ H6); intros y0 H7; exists y0; simpl; unfold g';
     left; assumption.
   cut (exists z : R, D z).
-  intro; elim H7; clear H7; intros x0 H7; exists x0; simpl in |- *;
-    unfold g' in |- *; right.
+  intro; elim H7; clear H7; intros x0 H7; exists x0; simpl;
+    unfold g'; right.
   split.
-  unfold complementary in |- *; apply H6.
+  unfold complementary; apply H6.
   apply H7.
   elim Hyp_F_NE; intros z0 H7.
   assert (H8 := H2 _ H7).
   elim H8; clear H8; intros t H8; exists t; apply (cond_fam f0); exists z0;
     apply H8.
-  unfold family_open_set in |- *; intro; simpl in |- *; unfold g' in |- *;
+  unfold family_open_set; intro; simpl; unfold g';
     elim (classic (D x)); intro.
   apply open_set_P6 with (union_domain (f0 x) (complementary F)).
   apply open_set_P2.
   unfold family_open_set in H4; apply H4.
   apply H0.
-  unfold eq_Dom in |- *; split.
-  unfold included, union_domain, complementary in |- *; intros.
+  unfold eq_Dom; split.
+  unfold included, union_domain, complementary; intros.
   elim H6; intro; [ left; apply H7 | right; split; assumption ].
-  unfold included, union_domain, complementary in |- *; intros.
+  unfold included, union_domain, complementary; intros.
   elim H6; intro; [ left; apply H7 | right; elim H7; intros; apply H8 ].
   apply open_set_P6 with (f0 x).
   unfold family_open_set in H4; apply H4.
-  unfold eq_Dom in |- *; split.
-  unfold included, complementary in |- *; intros; left; apply H6.
-  unfold included, complementary in |- *; intros.
+  unfold eq_Dom; split.
+  unfold included, complementary; intros; left; apply H6.
+  unfold included, complementary; intros.
   elim H6; intro.
   apply H7.
   elim H7; intros _ H8; elim H5; apply H8.
@@ -914,9 +914,9 @@ Proof.
   intro; apply (H3 f0 H2).
   apply compact_eqDom with (fun _:R => False).
   apply compact_EMP.
-  unfold eq_Dom in |- *; split.
-  unfold included in |- *; intros; elim H3.
-  assert (H3 := not_ex_all_not _ _ Hyp_F_NE); unfold included in |- *; intros;
+  unfold eq_Dom; split.
+  unfold included; intros; elim H3.
+  assert (H3 := not_ex_all_not _ _ Hyp_F_NE); unfold included; intros;
     elim (H3 x); apply H4.
 Qed.
 
@@ -947,7 +947,7 @@ Lemma continuity_compact :
   forall (f:R -> R) (X:R -> Prop),
     (forall x:R, continuity_pt f x) -> compact X -> compact (image_dir f X).
 Proof.
-  unfold compact in |- *; intros; unfold covering_open_set in H1.
+  unfold compact; intros; unfold covering_open_set in H1.
   elim H1; clear H1; intros.
   set (D := ind f1).
   set (g := fun x y:R => image_rec f0 (f1 x) y).
@@ -956,24 +956,24 @@ Proof.
   cut (covering_open_set X f').
   intro; elim (H0 f' H4); intros D' H5; exists D'.
   unfold covering_finite in H5; elim H5; clear H5; intros;
-    unfold covering_finite in |- *; split.
-  unfold covering, image_dir in |- *; simpl in |- *; unfold covering in H5;
+    unfold covering_finite; split.
+  unfold covering, image_dir; simpl; unfold covering in H5;
     intros; elim H7; intros y H8; elim H8; intros; assert (H11 := H5 _ H10);
       simpl in H11; elim H11; intros z H12; exists z; unfold g in H12;
         unfold image_rec in H12; rewrite H9; apply H12.
   unfold family_finite in H6; unfold domain_finite in H6;
-    unfold family_finite in |- *; unfold domain_finite in |- *;
+    unfold family_finite; unfold domain_finite;
       elim H6; intros l H7; exists l; intro; elim (H7 x);
         intros; split; intro.
-  apply H8; simpl in H10; simpl in |- *; apply H10.
+  apply H8; simpl in H10; simpl; apply H10.
   apply (H9 H10).
-  unfold covering_open_set in |- *; split.
-  unfold covering in |- *; intros; simpl in |- *; unfold covering in H1;
-    unfold image_dir in H1; unfold g in |- *; unfold image_rec in |- *;
+  unfold covering_open_set; split.
+  unfold covering; intros; simpl; unfold covering in H1;
+    unfold image_dir in H1; unfold g; unfold image_rec;
       apply H1.
   exists x; split; [ reflexivity | apply H4 ].
-  unfold family_open_set in |- *; unfold family_open_set in H2; intro;
-    simpl in |- *; unfold g in |- *;
+  unfold family_open_set; unfold family_open_set in H2; intro;
+    simpl; unfold g;
       cut ((fun y:R => image_rec f0 (f1 x) y) = image_rec f0 (f1 x)).
   intro; rewrite H4.
   apply (continuity_P2 f0 (f1 x) H (H2 x)).
@@ -1010,16 +1010,16 @@ Proof.
   assert (H2 : 0 < b - a).
   apply Rlt_Rminus; assumption.
   exists h; split.
-  unfold continuity in |- *; intro; case (Rtotal_order x a); intro.
-  unfold continuity_pt in |- *; unfold continue_in in |- *;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
-      simpl in |- *; unfold R_dist in |- *; intros; exists (a - x);
+  unfold continuity; intro; case (Rtotal_order x a); intro.
+  unfold continuity_pt; unfold continue_in;
+    unfold limit1_in; unfold limit_in;
+      simpl; unfold R_dist; intros; exists (a - x);
         split.
-  change (0 < a - x) in |- *; apply Rlt_Rminus; assumption.
-  intros; elim H5; clear H5; intros _ H5; unfold h in |- *.
+  change (0 < a - x); apply Rlt_Rminus; assumption.
+  intros; elim H5; clear H5; intros _ H5; unfold h.
   case (Rle_dec x a); intro.
   case (Rle_dec x0 a); intro.
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
   elim n; left; apply Rplus_lt_reg_r with (- x);
     do 2 rewrite (Rplus_comm (- x)); apply Rle_lt_trans with (Rabs (x0 - x)).
   apply RRle_abs.
@@ -1030,23 +1030,23 @@ Proof.
   split; [ right; reflexivity | left; assumption ].
   assert (H6 := H0 _ H5); unfold continuity_pt in H6; unfold continue_in in H6;
     unfold limit1_in in H6; unfold limit_in in H6; simpl in H6;
-      unfold R_dist in H6; unfold continuity_pt in |- *;
-        unfold continue_in in |- *; unfold limit1_in in |- *;
-          unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *;
+      unfold R_dist in H6; unfold continuity_pt;
+        unfold continue_in; unfold limit1_in;
+          unfold limit_in; simpl; unfold R_dist;
             intros; elim (H6 _ H7); intros; exists (Rmin x0 (b - a));
               split.
-  unfold Rmin in |- *; case (Rle_dec x0 (b - a)); intro.
+  unfold Rmin; case (Rle_dec x0 (b - a)); intro.
   elim H8; intros; assumption.
-  change (0 < b - a) in |- *; apply Rlt_Rminus; assumption.
+  change (0 < b - a); apply Rlt_Rminus; assumption.
   intros; elim H9; clear H9; intros _ H9; cut (x1 < b).
-  intro; unfold h in |- *; case (Rle_dec x a); intro.
+  intro; unfold h; case (Rle_dec x a); intro.
   case (Rle_dec x1 a); intro.
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
   case (Rle_dec x1 b); intro.
   elim H8; intros; apply H12; split.
-  unfold D_x, no_cond in |- *; split.
+  unfold D_x, no_cond; split.
   trivial.
-  red in |- *; intro; elim n; right; symmetry  in |- *; assumption.
+  red; intro; elim n; right; symmetry ; assumption.
   apply Rlt_le_trans with (Rmin x0 (b - a)).
   rewrite H4 in H9; apply H9.
   apply Rmin_l.
@@ -1063,9 +1063,9 @@ Proof.
   split; left; assumption.
   assert (H7 := H0 _ H6); unfold continuity_pt in H7; unfold continue_in in H7;
     unfold limit1_in in H7; unfold limit_in in H7; simpl in H7;
-      unfold R_dist in H7; unfold continuity_pt in |- *;
-        unfold continue_in in |- *; unfold limit1_in in |- *;
-          unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *;
+      unfold R_dist in H7; unfold continuity_pt;
+        unfold continue_in; unfold limit1_in;
+          unfold limit_in; simpl; unfold R_dist;
             intros; elim (H7 _ H8); intros; elim H9; clear H9;
               intros.
   assert (H11 : 0 < x - a).
@@ -1073,7 +1073,7 @@ Proof.
   assert (H12 : 0 < b - x).
   apply Rlt_Rminus; assumption.
   exists (Rmin x0 (Rmin (x - a) (b - x))); split.
-  unfold Rmin in |- *; case (Rle_dec (x - a) (b - x)); intro.
+  unfold Rmin; case (Rle_dec (x - a) (b - x)); intro.
   case (Rle_dec x0 (x - a)); intro.
   assumption.
   assumption.
@@ -1081,7 +1081,7 @@ Proof.
   assumption.
   assumption.
   intros; elim H13; clear H13; intros; cut (a < x1 < b).
-  intro; elim H15; clear H15; intros; unfold h in |- *; case (Rle_dec x a);
+  intro; elim H15; clear H15; intros; unfold h; case (Rle_dec x a);
     intro.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H4)).
   case (Rle_dec x b); intro.
@@ -1115,16 +1115,16 @@ Proof.
   split; [ left; assumption | right; reflexivity ].
   assert (H8 := H0 _ H7); unfold continuity_pt in H8; unfold continue_in in H8;
     unfold limit1_in in H8; unfold limit_in in H8; simpl in H8;
-      unfold R_dist in H8; unfold continuity_pt in |- *;
-        unfold continue_in in |- *; unfold limit1_in in |- *;
-          unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *;
+      unfold R_dist in H8; unfold continuity_pt;
+        unfold continue_in; unfold limit1_in;
+          unfold limit_in; simpl; unfold R_dist;
             intros; elim (H8 _ H9); intros; exists (Rmin x0 (b - a));
               split.
-  unfold Rmin in |- *; case (Rle_dec x0 (b - a)); intro.
+  unfold Rmin; case (Rle_dec x0 (b - a)); intro.
   elim H10; intros; assumption.
-  change (0 < b - a) in |- *; apply Rlt_Rminus; assumption.
+  change (0 < b - a); apply Rlt_Rminus; assumption.
   intros; elim H11; clear H11; intros _ H11; cut (a < x1).
-  intro; unfold h in |- *; case (Rle_dec x a); intro.
+  intro; unfold h; case (Rle_dec x a); intro.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H4)).
   case (Rle_dec x1 a); intro.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H12)).
@@ -1132,15 +1132,15 @@ Proof.
   case (Rle_dec x1 b); intro.
   rewrite H6; elim H10; intros; elim r0; intro.
   apply H14; split.
-  unfold D_x, no_cond in |- *; split.
+  unfold D_x, no_cond; split.
   trivial.
-  red in |- *; intro; rewrite <- H16 in H15; elim (Rlt_irrefl _ H15).
+  red; intro; rewrite <- H16 in H15; elim (Rlt_irrefl _ H15).
   rewrite H6 in H11; apply Rlt_le_trans with (Rmin x0 (b - a)).
   apply H11.
   apply Rmin_l.
-  rewrite H15; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  rewrite H15; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     assumption.
-  rewrite H6; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  rewrite H6; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     assumption.
   elim n1; right; assumption.
   rewrite H6 in H11; apply Ropp_lt_cancel; apply Rplus_lt_reg_r with b;
@@ -1149,18 +1149,18 @@ Proof.
   apply Rlt_le_trans with (Rmin x0 (b - a)).
   assumption.
   apply Rmin_r.
-  unfold continuity_pt in |- *; unfold continue_in in |- *;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
-      simpl in |- *; unfold R_dist in |- *; intros; exists (x - b);
+  unfold continuity_pt; unfold continue_in;
+    unfold limit1_in; unfold limit_in;
+      simpl; unfold R_dist; intros; exists (x - b);
         split.
-  change (0 < x - b) in |- *; apply Rlt_Rminus; assumption.
+  change (0 < x - b); apply Rlt_Rminus; assumption.
   intros; elim H8; clear H8; intros.
   assert (H10 : b < x0).
   apply Ropp_lt_cancel; apply Rplus_lt_reg_r with x;
     apply Rle_lt_trans with (Rabs (x0 - x)).
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply RRle_abs.
   assumption.
-  unfold h in |- *; case (Rle_dec x a); intro.
+  unfold h; case (Rle_dec x a); intro.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H4)).
   case (Rle_dec x b); intro.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H6)).
@@ -1168,8 +1168,8 @@ Proof.
   elim (Rlt_irrefl _ (Rlt_trans _ _ _ H1 (Rlt_le_trans _ _ _ H10 r))).
   case (Rle_dec x0 b); intro.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H10)).
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
-  intros; elim H3; intros; unfold h in |- *; case (Rle_dec c a); intro.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
+  intros; elim H3; intros; unfold h; case (Rle_dec c a); intro.
   elim r; intro.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 H6)).
   rewrite H6; reflexivity.
@@ -1210,7 +1210,7 @@ Proof.
   intros; rewrite <- (Heq c H10); rewrite <- (Heq Mxx H9); intros;
     rewrite <- H8; unfold is_lub in H7; elim H7; clear H7;
       intros H7 _; unfold is_upper_bound in H7; apply H7;
-        unfold image_dir in |- *; exists c; split; [ reflexivity | apply H10 ].
+        unfold image_dir; exists c; split; [ reflexivity | apply H10 ].
   apply H9.
   elim (classic (image_dir g (fun c:R => a <= c <= b) M)); intro.
   assumption.
@@ -1225,13 +1225,13 @@ Proof.
       cut (is_upper_bound (image_dir g (fun c:R => a <= c <= b)) (M - eps)).
   intro; assert (H12 := H10 _ H11); cut (M - eps < M).
   intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H12 H13)).
-  pattern M at 2 in |- *; rewrite <- Rplus_0_r; unfold Rminus in |- *;
+  pattern M at 2; rewrite <- Rplus_0_r; unfold Rminus;
     apply Rplus_lt_compat_l; apply Ropp_lt_cancel; rewrite Ropp_0;
       rewrite Ropp_involutive; apply (cond_pos eps).
-  unfold is_upper_bound, image_dir in |- *; intros; cut (x <= M).
+  unfold is_upper_bound, image_dir; intros; cut (x <= M).
   intro; case (Rle_dec x (M - eps)); intro.
   apply r.
-  elim (H9 x); unfold intersection_domain, disc, image_dir in |- *; split.
+  elim (H9 x); unfold intersection_domain, disc, image_dir; split.
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; rewrite Rabs_right.
   apply Rplus_lt_reg_r with (x - eps);
     replace (x - eps + (M - x)) with (M - eps).
@@ -1249,8 +1249,8 @@ Proof.
         ~ intersection_domain V (image_dir g (fun c:R => a <= c <= b)) y)).
   intro; elim H9; intros V H10; elim H10; clear H10; intros.
   unfold neighbourhood in H10; elim H10; intros del H12; exists del; intros;
-    red in |- *; intro; elim (H11 y).
-  unfold intersection_domain in |- *; unfold intersection_domain in H13;
+    red; intro; elim (H11 y).
+  unfold intersection_domain; unfold intersection_domain in H13;
     elim H13; clear H13; intros; split.
   apply (H12 _ H13).
   apply H14.
@@ -1268,18 +1268,18 @@ Proof.
   split.
   apply H12.
   apply (not_ex_all_not _ _ H13).
-  red in |- *; intro; cut (adherence (image_dir g (fun c:R => a <= c <= b)) M).
+  red; intro; cut (adherence (image_dir g (fun c:R => a <= c <= b)) M).
   intro; elim (closed_set_P1 (image_dir g (fun c:R => a <= c <= b)));
     intros H11 _; assert (H12 := H11 H3).
   elim H8.
   unfold eq_Dom in H12; elim H12; clear H12; intros.
   apply (H13 _ H10).
   apply H9.
-  exists (g a); unfold image_dir in |- *; exists a; split.
+  exists (g a); unfold image_dir; exists a; split.
   reflexivity.
   split; [ right; reflexivity | apply H ].
-  unfold bound in |- *; unfold bounded in H4; elim H4; clear H4; intros m H4;
-    elim H4; clear H4; intros M H4; exists M; unfold is_upper_bound in |- *;
+  unfold bound; unfold bounded in H4; elim H4; clear H4; intros m H4;
+    elim H4; clear H4; intros M H4; exists M; unfold is_upper_bound;
       intros; elim (H4 _ H5); intros _ H6; apply H6.
   apply prolongement_C0; assumption.
 Qed.
@@ -1327,8 +1327,8 @@ Proof.
   intros; elim H; intros; unfold f in H0; unfold adherence in H0;
     unfold point_adherent in H0;
       assert (H1 : neighbourhood (disc x0 (mkposreal _ Rlt_0_1)) x0).
-  unfold neighbourhood, disc in |- *; exists (mkposreal _ Rlt_0_1);
-    unfold included in |- *; trivial.
+  unfold neighbourhood, disc; exists (mkposreal _ Rlt_0_1);
+    unfold included; trivial.
   elim (H0 _ H1); intros; unfold intersection_domain in H2; elim H2; intros;
     elim H4; intros; apply H6.
 Qed.
@@ -1345,17 +1345,17 @@ Lemma ValAdh_un_prop :
   forall (un:nat -> R) (x:R), ValAdh un x <-> ValAdh_un un x.
 Proof.
   intros; split; intro.
-  unfold ValAdh in H; unfold ValAdh_un in |- *;
-    unfold intersection_family in |- *; simpl in |- *;
-      intros; elim H0; intros N H1; unfold adherence in |- *;
-        unfold point_adherent in |- *; intros; elim (H V N H2);
-          intros; exists (un x0); unfold intersection_domain in |- *;
+  unfold ValAdh in H; unfold ValAdh_un;
+    unfold intersection_family; simpl;
+      intros; elim H0; intros N H1; unfold adherence;
+        unfold point_adherent; intros; elim (H V N H2);
+          intros; exists (un x0); unfold intersection_domain;
             elim H3; clear H3; intros; split.
   assumption.
   split.
   exists x0; split; [ reflexivity | rewrite H1; apply (le_INR _ _ H3) ].
   exists N; assumption.
-  unfold ValAdh in |- *; intros; unfold ValAdh_un in H;
+  unfold ValAdh; intros; unfold ValAdh_un in H;
     unfold intersection_family in H; simpl in H;
       assert
         (H1 :
@@ -1376,8 +1376,8 @@ Qed.
 Lemma adherence_P4 :
   forall F G:R -> Prop, included F G -> included (adherence F) (adherence G).
 Proof.
-  unfold adherence, included in |- *; unfold point_adherent in |- *; intros;
-    elim (H0 _ H1); unfold intersection_domain in |- *;
+  unfold adherence, included; unfold point_adherent; intros;
+    elim (H0 _ H1); unfold intersection_domain;
       intros; elim H2; clear H2; intros; exists x0; split;
         [ assumption | apply (H _ H3) ].
 Qed.
@@ -1410,36 +1410,36 @@ Proof.
   intros; elim H2; intros; unfold f' in H3; elim H3; intros; assumption.
   set (f0 := mkfamily D' f' H2).
   unfold compact in H; assert (H3 : covering_open_set X f0).
-  unfold covering_open_set in |- *; split.
-  unfold covering in |- *; intros; unfold intersection_vide_in in H1;
+  unfold covering_open_set; split.
+  unfold covering; intros; unfold intersection_vide_in in H1;
     elim (H1 x); intros; unfold intersection_family in H5;
       assert
         (H6 := not_ex_all_not _ (fun y:R => forall y0:R, ind g y0 -> g y0 y) H5 x);
         assert (H7 := not_all_ex_not _ (fun y0:R => ind g y0 -> g y0 x) H6);
           elim H7; intros; exists x0; elim (imply_to_and _ _ H8);
-            intros; unfold f0 in |- *; simpl in |- *; unfold f' in |- *;
+            intros; unfold f0; simpl; unfold f';
               split; [ apply H10 | apply H9 ].
-  unfold family_open_set in |- *; intro; elim (classic (D' x)); intro.
+  unfold family_open_set; intro; elim (classic (D' x)); intro.
   apply open_set_P6 with (complementary (g x)).
   unfold family_closed_set in H0; unfold closed_set in H0; apply H0.
-  unfold f0 in |- *; simpl in |- *; unfold f' in |- *; unfold eq_Dom in |- *;
+  unfold f0; simpl; unfold f'; unfold eq_Dom;
     split.
-  unfold included in |- *; intros; split; [ apply H4 | apply H3 ].
-  unfold included in |- *; intros; elim H4; intros; assumption.
+  unfold included; intros; split; [ apply H4 | apply H3 ].
+  unfold included; intros; elim H4; intros; assumption.
   apply open_set_P6 with (fun _:R => False).
   apply open_set_P4.
-  unfold eq_Dom in |- *; unfold included in |- *; split; intros;
+  unfold eq_Dom; unfold included; split; intros;
     [ elim H4
       | simpl in H4; unfold f' in H4; elim H4; intros; elim H3; assumption ].
   elim (H _ H3); intros SF H4; exists SF;
-    unfold intersection_vide_finite_in in |- *; split.
-  unfold intersection_vide_in in |- *; simpl in |- *; intros; split.
-  intros; unfold included in |- *; intros; unfold intersection_vide_in in H1;
+    unfold intersection_vide_finite_in; split.
+  unfold intersection_vide_in; simpl; intros; split.
+  intros; unfold included; intros; unfold intersection_vide_in in H1;
     elim (H1 x); intros; elim H6; intros; apply H7.
   unfold intersection_domain in H5; elim H5; intros; assumption.
   assumption.
   elim (classic (exists y : R, intersection_domain (ind g) SF y)); intro Hyp'.
-  red in |- *; intro; elim H5; intros; unfold intersection_family in H6;
+  red; intro; elim H5; intros; unfold intersection_family in H6;
     simpl in H6.
   cut (X x0).
   intro; unfold covering_finite in H4; elim H4; clear H4; intros H4 _;
@@ -1462,16 +1462,16 @@ Proof.
     cut (exists z : R, X z).
   intro; elim H5; clear H5; intros; unfold covering in H4; elim (H4 x0 H5);
     intros; simpl in H6; elim Hyp'; exists x1; elim H6;
-      intros; unfold intersection_domain in |- *; split.
+      intros; unfold intersection_domain; split.
   apply (cond_fam f0); exists x0; apply H7.
   apply H8.
   apply Hyp.
   unfold covering_finite in H4; elim H4; clear H4; intros;
     unfold family_finite in H5; unfold domain_finite in H5;
-      unfold family_finite in |- *; unfold domain_finite in |- *;
+      unfold family_finite; unfold domain_finite;
         elim H5; clear H5; intros l H5; exists l; intro; elim (H5 x);
           intros; split; intro;
-            [ apply H6; simpl in |- *; simpl in H8; apply H8 | apply (H7 H8) ].
+            [ apply H6; simpl; simpl in H8; apply H8 | apply (H7 H8) ].
 Qed.
 
 Theorem Bolzano_Weierstrass :
@@ -1492,8 +1492,8 @@ Proof.
   intros; elim H2; intros; unfold g in H3; unfold adherence in H3;
     unfold point_adherent in H3.
   assert (H4 : neighbourhood (disc x0 (mkposreal _ Rlt_0_1)) x0).
-  unfold neighbourhood in |- *; exists (mkposreal _ Rlt_0_1);
-    unfold included in |- *; trivial.
+  unfold neighbourhood; exists (mkposreal _ Rlt_0_1);
+    unfold included; trivial.
   elim (H3 _ H4); intros; unfold intersection_domain in H5; decompose [and] H5;
     assumption.
   set (f0 := mkfamily D g H2).
@@ -1509,19 +1509,19 @@ Proof.
         unfold domain_finite in H9; elim H9; clear H9; intros l H9;
           set (r := MaxRlist l); cut (D r).
   intro; unfold D in H11; elim H11; intros; exists (un x);
-    unfold intersection_family in |- *; simpl in |- *;
-      unfold intersection_domain in |- *; intros; split.
-  unfold g in |- *; apply adherence_P1; split.
+    unfold intersection_family; simpl;
+      unfold intersection_domain; intros; split.
+  unfold g; apply adherence_P1; split.
   exists x; split;
     [ reflexivity
-      | rewrite <- H12; unfold r in |- *; apply MaxRlist_P1; elim (H9 y); intros;
-        apply H14; simpl in |- *; apply H13 ].
+      | rewrite <- H12; unfold r; apply MaxRlist_P1; elim (H9 y); intros;
+        apply H14; simpl; apply H13 ].
   elim H13; intros; assumption.
   elim H13; intros; assumption.
   elim (H9 r); intros.
   simpl in H12; unfold intersection_domain in H12; cut (In r l).
   intro; elim (H12 H13); intros; assumption.
-  unfold r in |- *; apply MaxRlist_P2;
+  unfold r; apply MaxRlist_P2;
     cut (exists z : R, intersection_domain (ind f0) SF z).
   intro; elim H13; intros; elim (H9 x); intros; simpl in H15;
     assert (H17 := H15 H14); exists x; apply H17.
@@ -1541,16 +1541,16 @@ Proof.
                 not_all_ex_not _ (fun y:R => intersection_domain D SF y -> g y x /\ SF y)
                 H18); elim H19; intros; assert (H21 := imply_to_and _ _ H20);
               elim (H17 x0); elim H21; intros; assumption.
-  unfold intersection_vide_in in |- *; intros; split.
-  intro; simpl in H6; unfold f0 in |- *; simpl in |- *; unfold g in |- *;
+  unfold intersection_vide_in; intros; split.
+  intro; simpl in H6; unfold f0; simpl; unfold g;
     apply included_trans with (adherence X).
   apply adherence_P4.
-  unfold included in |- *; intros; elim H7; intros; elim H8; intros; elim H10;
+  unfold included; intros; elim H7; intros; elim H8; intros; elim H10;
     intros; rewrite H11; apply H0.
   apply adherence_P2; apply compact_P2; assumption.
   apply H4.
-  unfold family_closed_set in |- *; unfold f0 in |- *; simpl in |- *;
-    unfold g in |- *; intro; apply adherence_P3.
+  unfold family_closed_set; unfold f0; simpl;
+    unfold g; intro; apply adherence_P3.
 Qed.
 
 (********************************************************)
@@ -1566,7 +1566,7 @@ Definition uniform_continuity (f:R -> R) (X:R -> Prop) : Prop :=
 Lemma is_lub_u :
   forall (E:R -> Prop) (x y:R), is_lub E x -> is_lub E y -> x = y.
 Proof.
-  unfold is_lub in |- *; intros; elim H; elim H0; intros; apply Rle_antisym;
+  unfold is_lub; intros; elim H; elim H0; intros; apply Rle_antisym;
     [ apply (H4 _ H1) | apply (H2 _ H3) ].
 Qed.
 
@@ -1581,7 +1581,7 @@ Proof.
   right; elim H1; intros; elim H2; intros; exists x; exists x0; intros.
   split;
     [ assumption
-      | split; [ assumption | apply (sym_not_eq (A:=R)); assumption ] ].
+      | split; [ assumption | apply (not_eq_sym (A:=R)); assumption ] ].
   left; exists x; split.
   assumption.
   intros; case (Req_dec x0 x); intro.
@@ -1597,14 +1597,14 @@ Theorem Heine :
 Proof.
   intros f0 X H0 H; elim (domain_P1 X); intro Hyp.
 (* X is empty *)
-  unfold uniform_continuity in |- *; intros; exists (mkposreal _ Rlt_0_1);
+  unfold uniform_continuity; intros; exists (mkposreal _ Rlt_0_1);
     intros; elim Hyp; exists x; assumption.
   elim Hyp; clear Hyp; intro Hyp.
 (* X has only one element *)
-  unfold uniform_continuity in |- *; intros; exists (mkposreal _ Rlt_0_1);
+  unfold uniform_continuity; intros; exists (mkposreal _ Rlt_0_1);
     intros; elim Hyp; clear Hyp; intros; elim H4; clear H4;
       intros; assert (H6 := H5 _ H1); assert (H7 := H5 _ H2);
-        rewrite H6; rewrite H7; unfold Rminus in |- *; rewrite Rplus_opp_r;
+        rewrite H6; rewrite H7; unfold Rminus; rewrite Rplus_opp_r;
           rewrite Rabs_R0; apply (cond_pos eps).
 (* X has at least two distinct elements *)
   assert
@@ -1624,9 +1624,9 @@ Proof.
   elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H13 H14) r)).
   elim X_enc; clear X_enc; intros m X_enc; elim X_enc; clear X_enc;
     intros M X_enc; elim X_enc; clear X_enc Hyp; intros X_enc Hyp;
-      unfold uniform_continuity in |- *; intro;
+      unfold uniform_continuity; intro;
         assert (H1 : forall t:posreal, 0 < t / 2).
-  intro; unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  intro; unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply (cond_pos t) | apply Rinv_0_lt_compat; prove_sup0 ].
   set
     (g :=
@@ -1644,8 +1644,8 @@ Proof.
     apply H3.
   set (f' := mkfamily X g H2); unfold compact in H0;
     assert (H3 : covering_open_set X f').
-  unfold covering_open_set in |- *; split.
-  unfold covering in |- *; intros; exists x; simpl in |- *; unfold g in |- *;
+  unfold covering_open_set; split.
+  unfold covering; intros; exists x; simpl; unfold g;
     split.
   assumption.
   assert (H4 := H _ H3); unfold continuity_pt in H4; unfold continue_in in H4;
@@ -1658,22 +1658,22 @@ Proof.
                 0 < zeta <= M - m /\
                 (forall z:R, Rabs (z - x) < zeta -> Rabs (f0 z - f0 x) < eps / 2));
             assert (H6 : bound E).
-  unfold bound in |- *; exists (M - m); unfold is_upper_bound in |- *;
-    unfold E in |- *; intros; elim H6; clear H6; intros H6 _;
+  unfold bound; exists (M - m); unfold is_upper_bound;
+    unfold E; intros; elim H6; clear H6; intros H6 _;
       elim H6; clear H6; intros _ H6; apply H6.
   assert (H7 :  exists x : R, E x).
-  elim H5; clear H5; intros; exists (Rmin x0 (M - m)); unfold E in |- *; intros;
+  elim H5; clear H5; intros; exists (Rmin x0 (M - m)); unfold E; intros;
     split.
   split.
-  unfold Rmin in |- *; case (Rle_dec x0 (M - m)); intro.
+  unfold Rmin; case (Rle_dec x0 (M - m)); intro.
   apply H5.
   apply Rlt_Rminus; apply Hyp.
   apply Rmin_r.
   intros; case (Req_dec x z); intro.
-  rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  rewrite H9; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     apply (H1 eps).
   apply H7; split.
-  unfold D_x, no_cond in |- *; split; [ trivial | assumption ].
+  unfold D_x, no_cond; split; [ trivial | assumption ].
   apply Rlt_le_trans with (Rmin x0 (M - m)); [ apply H8 | apply Rmin_l ].
   assert (H8 := completeness _ H6 H7); elim H8; clear H8; intros;
     cut (0 < x1 <= M - m).
@@ -1690,15 +1690,15 @@ Proof.
     unfold is_lub in p; elim p; intros; cut (is_upper_bound E (Rabs (z - x))).
   intro; assert (H16 := H14 _ H15);
     elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H10 H16)).
-  unfold is_upper_bound in |- *; intros; unfold is_upper_bound in H13;
+  unfold is_upper_bound; intros; unfold is_upper_bound in H13;
     assert (H16 := H13 _ H15); case (Rle_dec x2 (Rabs (z - x)));
       intro.
   assumption.
   elim (H12 x2); split; [ split; [ auto with real | assumption ] | assumption ].
   split.
   apply p.
-  unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
-    rewrite Rabs_R0; simpl in |- *; unfold Rdiv in |- *;
+  unfold disc; unfold Rminus; rewrite Rplus_opp_r;
+    rewrite Rabs_R0; simpl; unfold Rdiv;
       apply Rmult_lt_0_compat; [ apply H8 | apply Rinv_0_lt_compat; prove_sup0 ].
   elim H7; intros; unfold E in H8; elim H8; intros H9 _; elim H9; intros H10 _;
     unfold is_lub in p; elim p; intros; unfold is_upper_bound in H12;
@@ -1706,13 +1706,13 @@ Proof.
   apply Rlt_le_trans with x2; [ assumption | apply (H11 _ H8) ].
   apply H12; intros; unfold E in H13; elim H13; intros; elim H14; intros;
     assumption.
-  unfold family_open_set in |- *; intro; simpl in |- *; elim (classic (X x));
+  unfold family_open_set; intro; simpl; elim (classic (X x));
     intro.
-  unfold g in |- *; unfold open_set in |- *; intros; elim H4; clear H4;
+  unfold g; unfold open_set; intros; elim H4; clear H4;
     intros _ H4; elim H4; clear H4; intros; elim H4; clear H4;
-      intros; unfold neighbourhood in |- *; case (Req_dec x x0);
+      intros; unfold neighbourhood; case (Req_dec x x0);
         intro.
-  exists (mkposreal _ (H1 x1)); rewrite <- H6; unfold included in |- *; intros;
+  exists (mkposreal _ (H1 x1)); rewrite <- H6; unfold included; intros;
     split.
   assumption.
   exists x1; split.
@@ -1721,24 +1721,24 @@ Proof.
   elim H5; intros; apply H8.
   apply H7.
   set (d := x1 / 2 - Rabs (x0 - x)); assert (H7 : 0 < d).
-  unfold d in |- *; apply Rlt_Rminus; elim H5; clear H5; intros;
+  unfold d; apply Rlt_Rminus; elim H5; clear H5; intros;
     unfold disc in H7; apply H7.
-  exists (mkposreal _ H7); unfold included in |- *; intros; split.
+  exists (mkposreal _ H7); unfold included; intros; split.
   assumption.
   exists x1; split.
   apply H4.
   elim H5; intros; split.
   assumption.
-  unfold disc in H8; simpl in H8; unfold disc in |- *; simpl in |- *;
+  unfold disc in H8; simpl in H8; unfold disc; simpl;
     unfold disc in H10; simpl in H10;
       apply Rle_lt_trans with (Rabs (x2 - x0) + Rabs (x0 - x)).
   replace (x2 - x) with (x2 - x0 + (x0 - x)); [ apply Rabs_triang | ring ].
-  replace (x1 / 2) with (d + Rabs (x0 - x)); [ idtac | unfold d in |- *; ring ].
+  replace (x1 / 2) with (d + Rabs (x0 - x)); [ idtac | unfold d; ring ].
   do 2 rewrite <- (Rplus_comm (Rabs (x0 - x))); apply Rplus_lt_compat_l;
     apply H8.
   apply open_set_P6 with (fun _:R => False).
   apply open_set_P4.
-  unfold eq_Dom in |- *; unfold included in |- *; intros; split.
+  unfold eq_Dom; unfold included; intros; split.
   intros; elim H4.
   intros; unfold g in H4; elim H4; clear H4; intros H4 _; elim H3; apply H4.
   elim (H0 _ H3); intros DF H4; unfold covering_finite in H4; elim H4; clear H4;
@@ -1776,10 +1776,10 @@ Proof.
       apply Rlt_trans with (pos_Rl l' i / 2).
   apply H21.
   elim H13; clear H13; intros; assumption.
-  unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2.
+  unfold Rdiv; apply Rmult_lt_reg_l with 2.
   prove_sup0.
   rewrite Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
-  rewrite Rmult_1_r; pattern (pos_Rl l' i) at 1 in |- *; rewrite <- Rplus_0_r;
+  rewrite Rmult_1_r; pattern (pos_Rl l' i) at 1; rewrite <- Rplus_0_r;
     rewrite double; apply Rplus_lt_compat_l; apply H19.
   discrR.
   assert (H19 := H8 i H17); elim H19; clear H19; intros; rewrite <- H18 in H20;
@@ -1791,15 +1791,15 @@ Proof.
   rewrite (double_var (pos_Rl l' i)); apply Rplus_lt_compat.
   apply Rlt_le_trans with (D / 2).
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H12.
-  unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ 2));
+  unfold Rdiv; do 2 rewrite <- (Rmult_comm (/ 2));
     apply Rmult_le_compat_l.
   left; apply Rinv_0_lt_compat; prove_sup0.
-  unfold D in |- *; apply MinRlist_P1; elim (pos_Rl_P2 l' (pos_Rl l' i));
+  unfold D; apply MinRlist_P1; elim (pos_Rl_P2 l' (pos_Rl l' i));
     intros; apply H26; exists i; split;
       [ rewrite <- H7; assumption | reflexivity ].
   assumption.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
-    [ unfold D in |- *; apply MinRlist_P2; intros; elim (pos_Rl_P2 l' y); intros;
+  unfold Rdiv; apply Rmult_lt_0_compat;
+    [ unfold D; apply MinRlist_P2; intros; elim (pos_Rl_P2 l' y); intros;
       elim (H10 H9); intros; elim H12; intros; rewrite H14;
         rewrite <- H7 in H13; elim (H8 x H13); intros;
           apply H15
@@ -1811,25 +1811,25 @@ Proof.
           0 < zeta <= M - m /\
           (forall z:R, Rabs (z - x) < zeta -> Rabs (f0 z - f0 x) < eps / 2));
       assert (H11 : bound E).
-  unfold bound in |- *; exists (M - m); unfold is_upper_bound in |- *;
-    unfold E in |- *; intros; elim H11; clear H11; intros H11 _;
+  unfold bound; exists (M - m); unfold is_upper_bound;
+    unfold E; intros; elim H11; clear H11; intros H11 _;
       elim H11; clear H11; intros _ H11; apply H11.
   assert (H12 :  exists x : R, E x).
   assert (H13 := H _ H9); unfold continuity_pt in H13;
     unfold continue_in in H13; unfold limit1_in in H13;
       unfold limit_in in H13; simpl in H13; unfold R_dist in H13;
         elim (H13 _ (H1 eps)); intros; elim H12; clear H12;
-          intros; exists (Rmin x0 (M - m)); unfold E in |- *;
+          intros; exists (Rmin x0 (M - m)); unfold E;
             intros; split.
   split;
-    [ unfold Rmin in |- *; case (Rle_dec x0 (M - m)); intro;
+    [ unfold Rmin; case (Rle_dec x0 (M - m)); intro;
       [ apply H12 | apply Rlt_Rminus; apply Hyp ]
       | apply Rmin_r ].
   intros; case (Req_dec x z); intro.
-  rewrite H16; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
+  rewrite H16; unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0;
     apply (H1 eps).
   apply H14; split;
-    [ unfold D_x, no_cond in |- *; split; [ trivial | assumption ]
+    [ unfold D_x, no_cond; split; [ trivial | assumption ]
       | apply Rlt_le_trans with (Rmin x0 (M - m)); [ apply H15 | apply Rmin_l ] ].
   assert (H13 := completeness _ H11 H12); elim H13; clear H13; intros;
     cut (0 < x0 <= M - m).
@@ -1847,14 +1847,14 @@ Proof.
     unfold is_lub in p; elim p; intros; cut (is_upper_bound E (Rabs (z - x))).
   intro; assert (H21 := H19 _ H20);
     elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H15 H21)).
-  unfold is_upper_bound in |- *; intros; unfold is_upper_bound in H18;
+  unfold is_upper_bound; intros; unfold is_upper_bound in H18;
     assert (H21 := H18 _ H20); case (Rle_dec x1 (Rabs (z - x)));
       intro.
   assumption.
   elim (H17 x1); split.
   split; [ auto with real | assumption ].
   assumption.
-  unfold included, g in |- *; intros; elim H15; intros; elim H17; intros;
+  unfold included, g; intros; elim H15; intros; elim H17; intros;
     decompose [and] H18; cut (x0 = x2).
   intro; rewrite H20; apply H22.
   unfold E in p; eapply is_lub_u.
diff --git a/theories/Reals/Rtrigo.v b/theories/Reals/Rtrigo.v
index e45353b5..32c4d7d3 100644
--- a/theories/Reals/Rtrigo.v
+++ b/theories/Reals/Rtrigo.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -16,1789 +16,11 @@ Require Export Cos_rel.
 Require Export Cos_plus.
 Require Import ZArith_base.
 Require Import Zcomplements.
-Local Open Scope nat_scope.
-Local Open Scope R_scope.
-
-(** sin_PI2 is the only remaining axiom **)
-Axiom sin_PI2 : sin (PI / 2) = 1.
-
-(**********)
-Lemma PI_neq0 : PI <> 0.
-Proof.
-  red in |- *; intro; assert (H0 := PI_RGT_0); rewrite H in H0;
-    elim (Rlt_irrefl _ H0).
-Qed.
-
-(**********)
-Lemma cos_minus : forall x y:R, cos (x - y) = cos x * cos y + sin x * sin y.
-Proof.
-  intros; unfold Rminus in |- *; rewrite cos_plus.
-  rewrite <- cos_sym; rewrite sin_antisym; ring.
-Qed.
-
-(**********)
-Lemma sin2_cos2 : forall x:R, Rsqr (sin x) + Rsqr (cos x) = 1.
-Proof.
-  intro; unfold Rsqr in |- *; rewrite Rplus_comm; rewrite <- (cos_minus x x);
-    unfold Rminus in |- *; rewrite Rplus_opp_r; apply cos_0.
-Qed.
-
-Lemma cos2 : forall x:R, Rsqr (cos x) = 1 - Rsqr (sin x).
-Proof.
-  intro x; generalize (sin2_cos2 x); intro H1; rewrite <- H1;
-    unfold Rminus in |- *; rewrite <- (Rplus_comm (Rsqr (cos x)));
-      rewrite Rplus_assoc; rewrite Rplus_opp_r; symmetry  in |- *;
-        apply Rplus_0_r.
-Qed.
-
-(**********)
-Lemma cos_PI2 : cos (PI / 2) = 0.
-Proof.
-  apply Rsqr_eq_0; rewrite cos2; rewrite sin_PI2; rewrite Rsqr_1;
-    unfold Rminus in |- *; apply Rplus_opp_r.
-Qed.
-
-(**********)
-Lemma cos_PI : cos PI = -1.
-Proof.
-  replace PI with (PI / 2 + PI / 2).
-  rewrite cos_plus.
-  rewrite sin_PI2; rewrite cos_PI2.
-  ring.
-  symmetry  in |- *; apply double_var.
-Qed.
-
-Lemma sin_PI : sin PI = 0.
-Proof.
-  assert (H := sin2_cos2 PI).
-  rewrite cos_PI in H.
-  rewrite <- Rsqr_neg in H.
-  rewrite Rsqr_1 in H.
-  cut (Rsqr (sin PI) = 0).
-  intro; apply (Rsqr_eq_0 _ H0).
-  apply Rplus_eq_reg_l with 1.
-  rewrite Rplus_0_r; rewrite Rplus_comm; exact H.
-Qed.
-
-(**********)
-Lemma neg_cos : forall x:R, cos (x + PI) = - cos x.
-Proof.
-  intro x; rewrite cos_plus; rewrite sin_PI; rewrite cos_PI; ring.
-Qed.
-
-(**********)
-Lemma sin_cos : forall x:R, sin x = - cos (PI / 2 + x).
-Proof.
-  intro x; rewrite cos_plus; rewrite sin_PI2; rewrite cos_PI2; ring.
-Qed.
-
-(**********)
-Lemma sin_plus : forall x y:R, sin (x + y) = sin x * cos y + cos x * sin y.
-Proof.
-  intros.
-  rewrite (sin_cos (x + y)).
-  replace (PI / 2 + (x + y)) with (PI / 2 + x + y); [ rewrite cos_plus | ring ].
-  rewrite (sin_cos (PI / 2 + x)).
-  replace (PI / 2 + (PI / 2 + x)) with (x + PI).
-  rewrite neg_cos.
-  replace (cos (PI / 2 + x)) with (- sin x).
-  ring.
-  rewrite sin_cos; rewrite Ropp_involutive; reflexivity.
-  pattern PI at 1 in |- *; rewrite (double_var PI); ring.
-Qed.
-
-Lemma sin_minus : forall x y:R, sin (x - y) = sin x * cos y - cos x * sin y.
-Proof.
-  intros; unfold Rminus in |- *; rewrite sin_plus.
-  rewrite <- cos_sym; rewrite sin_antisym; ring.
-Qed.
-
-(**********)
-Definition tan (x:R) : R := sin x / cos x.
-
-Lemma tan_plus :
-  forall x y:R,
-    cos x <> 0 ->
-    cos y <> 0 ->
-    cos (x + y) <> 0 ->
-    1 - tan x * tan y <> 0 ->
-    tan (x + y) = (tan x + tan y) / (1 - tan x * tan y).
-Proof.
-  intros; unfold tan in |- *; rewrite sin_plus; rewrite cos_plus;
-    unfold Rdiv in |- *;
-      replace (cos x * cos y - sin x * sin y) with
-      (cos x * cos y * (1 - sin x * / cos x * (sin y * / cos y))).
-  rewrite Rinv_mult_distr.
-  repeat rewrite <- Rmult_assoc;
-    replace ((sin x * cos y + cos x * sin y) * / (cos x * cos y)) with
-    (sin x * / cos x + sin y * / cos y).
-  reflexivity.
-  rewrite Rmult_plus_distr_r; rewrite Rinv_mult_distr.
-  repeat rewrite Rmult_assoc; repeat rewrite (Rmult_comm (sin x));
-    repeat rewrite <- Rmult_assoc.
-  repeat rewrite Rinv_r_simpl_m; [ reflexivity | assumption | assumption ].
-  assumption.
-  assumption.
-  apply prod_neq_R0; assumption.
-  assumption.
-  unfold Rminus in |- *; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r;
-    apply Rplus_eq_compat_l; repeat rewrite Rmult_assoc;
-      rewrite (Rmult_comm (sin x)); rewrite (Rmult_comm (cos y));
-        rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite <- Rmult_assoc;
-          rewrite <- Rinv_r_sym.
-  rewrite Rmult_1_l; rewrite (Rmult_comm (sin x));
-    rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite Rmult_assoc;
-      apply Rmult_eq_compat_l; rewrite (Rmult_comm (/ cos y));
-        rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
-  apply Rmult_1_r.
-  assumption.
-  assumption.
-Qed.
-
-(*******************************************************)
-(** * Some properties of cos, sin and tan              *)
-(*******************************************************)
-
-Lemma sin2 : forall x:R, Rsqr (sin x) = 1 - Rsqr (cos x).
-Proof.
-  intro x; generalize (cos2 x); intro H1; rewrite H1.
-  unfold Rminus in |- *; rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
-    rewrite Rplus_opp_r; rewrite Rplus_0_l; symmetry  in |- *;
-      apply Ropp_involutive.
-Qed.
-
-Lemma sin_2a : forall x:R, sin (2 * x) = 2 * sin x * cos x.
-Proof.
-  intro x; rewrite double; rewrite sin_plus.
-  rewrite <- (Rmult_comm (sin x)); symmetry  in |- *; rewrite Rmult_assoc;
-    apply double.
-Qed.
-
-Lemma cos_2a : forall x:R, cos (2 * x) = cos x * cos x - sin x * sin x.
-Proof.
-  intro x; rewrite double; apply cos_plus.
-Qed.
-
-Lemma cos_2a_cos : forall x:R, cos (2 * x) = 2 * cos x * cos x - 1.
-Proof.
-  intro x; rewrite double; unfold Rminus in |- *; rewrite Rmult_assoc;
-    rewrite cos_plus; generalize (sin2_cos2 x); rewrite double;
-      intro H1; rewrite <- H1; ring_Rsqr.
-Qed.
-
-Lemma cos_2a_sin : forall x:R, cos (2 * x) = 1 - 2 * sin x * sin x.
-Proof.
-  intro x; rewrite Rmult_assoc; unfold Rminus in |- *; repeat rewrite double.
-  generalize (sin2_cos2 x); intro H1; rewrite <- H1; rewrite cos_plus;
-    ring_Rsqr.
-Qed.
-
-Lemma tan_2a :
-  forall x:R,
-    cos x <> 0 ->
-    cos (2 * x) <> 0 ->
-    1 - tan x * tan x <> 0 -> tan (2 * x) = 2 * tan x / (1 - tan x * tan x).
-Proof.
-  repeat rewrite double; intros; repeat rewrite double; rewrite double in H0;
-    apply tan_plus; assumption.
-Qed.
-
-Lemma sin_neg : forall x:R, sin (- x) = - sin x.
-Proof.
-  apply sin_antisym.
-Qed.
-
-Lemma cos_neg : forall x:R, cos (- x) = cos x.
-Proof.
-  intro; symmetry  in |- *; apply cos_sym.
-Qed.
-
-Lemma tan_0 : tan 0 = 0.
-Proof.
-  unfold tan in |- *; rewrite sin_0; rewrite cos_0.
-  unfold Rdiv in |- *; apply Rmult_0_l.
-Qed.
-
-Lemma tan_neg : forall x:R, tan (- x) = - tan x.
-Proof.
-  intros x; unfold tan in |- *; rewrite sin_neg; rewrite cos_neg;
-    unfold Rdiv in |- *.
-  apply Ropp_mult_distr_l_reverse.
-Qed.
-
-Lemma tan_minus :
-  forall x y:R,
-    cos x <> 0 ->
-    cos y <> 0 ->
-    cos (x - y) <> 0 ->
-    1 + tan x * tan y <> 0 ->
-    tan (x - y) = (tan x - tan y) / (1 + tan x * tan y).
-Proof.
-  intros; unfold Rminus in |- *; rewrite tan_plus.
-  rewrite tan_neg; unfold Rminus in |- *; rewrite <- Ropp_mult_distr_l_reverse;
-    rewrite Rmult_opp_opp; reflexivity.
-  assumption.
-  rewrite cos_neg; assumption.
-  assumption.
-  rewrite tan_neg; unfold Rminus in |- *; rewrite <- Ropp_mult_distr_l_reverse;
-    rewrite Rmult_opp_opp; assumption.
-Qed.
-
-Lemma cos_3PI2 : cos (3 * (PI / 2)) = 0.
-Proof.
-  replace (3 * (PI / 2)) with (PI + PI / 2).
-  rewrite cos_plus; rewrite sin_PI; rewrite cos_PI2; ring.
-  pattern PI at 1 in |- *; rewrite (double_var PI).
-  ring.
-Qed.
-
-Lemma sin_2PI : sin (2 * PI) = 0.
-Proof.
-  rewrite sin_2a; rewrite sin_PI; ring.
-Qed.
-
-Lemma cos_2PI : cos (2 * PI) = 1.
-Proof.
-  rewrite cos_2a; rewrite sin_PI; rewrite cos_PI; ring.
-Qed.
-
-Lemma neg_sin : forall x:R, sin (x + PI) = - sin x.
-Proof.
-  intro x; rewrite sin_plus; rewrite sin_PI; rewrite cos_PI; ring.
-Qed.
-
-Lemma sin_PI_x : forall x:R, sin (PI - x) = sin x.
-Proof.
-  intro x; rewrite sin_minus; rewrite sin_PI; rewrite cos_PI; rewrite Rmult_0_l;
-    unfold Rminus in |- *; rewrite Rplus_0_l; rewrite Ropp_mult_distr_l_reverse;
-      rewrite Ropp_involutive; apply Rmult_1_l.
-Qed.
-
-Lemma sin_period : forall (x:R) (k:nat), sin (x + 2 * INR k * PI) = sin x.
-Proof.
-  intros x k; induction  k as [| k Hreck].
-  simpl in |- *;  ring_simplify (x + 2 * 0 * PI).
-  trivial.
-
-  replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI).
-  rewrite sin_plus in |- *; rewrite sin_2PI in |- *; rewrite cos_2PI in |- *.
-  ring_simplify; trivial.
-  rewrite S_INR in |- *;  ring.
-Qed.
-
-Lemma cos_period : forall (x:R) (k:nat), cos (x + 2 * INR k * PI) = cos x.
-Proof.
-  intros x k; induction  k as [| k Hreck].
-  simpl in |- *;  ring_simplify (x + 2 * 0 * PI).
-  trivial.
-
-  replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI).
-  rewrite cos_plus in |- *; rewrite sin_2PI in |- *; rewrite cos_2PI in |- *.
-  ring_simplify; trivial.
-  rewrite S_INR in |- *;  ring.
-Qed.
-
-Lemma sin_shift : forall x:R, sin (PI / 2 - x) = cos x.
-Proof.
-  intro x; rewrite sin_minus; rewrite sin_PI2; rewrite cos_PI2; ring.
-Qed.
-
-Lemma cos_shift : forall x:R, cos (PI / 2 - x) = sin x.
-Proof.
-  intro x; rewrite cos_minus; rewrite sin_PI2; rewrite cos_PI2; ring.
-Qed.
-
-Lemma cos_sin : forall x:R, cos x = sin (PI / 2 + x).
-Proof.
-  intro x; rewrite sin_plus; rewrite sin_PI2; rewrite cos_PI2; ring.
-Qed.
-
-Lemma PI2_RGT_0 : 0 < PI / 2.
-Proof.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
-    [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup ].
-Qed.
-
-Lemma SIN_bound : forall x:R, -1 <= sin x <= 1.
-Proof.
-  intro; case (Rle_dec (-1) (sin x)); intro.
-  case (Rle_dec (sin x) 1); intro.
-  split; assumption.
-  cut (1 < sin x).
-  intro;
-    generalize
-      (Rsqr_incrst_1 1 (sin x) H (Rlt_le 0 1 Rlt_0_1)
-        (Rlt_le 0 (sin x) (Rlt_trans 0 1 (sin x) Rlt_0_1 H)));
-      rewrite Rsqr_1; intro; rewrite sin2 in H0; unfold Rminus in H0;
-        generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0);
-          repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l;
-            rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1;
-              generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1);
-                repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x));
-                  intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)).
-  auto with real.
-  cut (sin x < -1).
-  intro; generalize (Ropp_lt_gt_contravar (sin x) (-1) H);
-    rewrite Ropp_involutive; clear H; intro;
-      generalize
-        (Rsqr_incrst_1 1 (- sin x) H (Rlt_le 0 1 Rlt_0_1)
-          (Rlt_le 0 (- sin x) (Rlt_trans 0 1 (- sin x) Rlt_0_1 H)));
-        rewrite Rsqr_1; intro; rewrite <- Rsqr_neg in H0;
-          rewrite sin2 in H0; unfold Rminus in H0;
-            generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0);
-              repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l;
-                rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1;
-                  generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1);
-                    repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x));
-                      intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)).
-  auto with real.
-Qed.
-
-Lemma COS_bound : forall x:R, -1 <= cos x <= 1.
-Proof.
-  intro; rewrite <- sin_shift; apply SIN_bound.
-Qed.
-
-Lemma cos_sin_0 : forall x:R, ~ (cos x = 0 /\ sin x = 0).
-Proof.
-  intro; red in |- *; intro; elim H; intros; generalize (sin2_cos2 x); intro;
-    rewrite H0 in H2; rewrite H1 in H2; repeat rewrite Rsqr_0 in H2;
-      rewrite Rplus_0_r in H2; generalize Rlt_0_1; intro;
-        rewrite <- H2 in H3; elim (Rlt_irrefl 0 H3).
-Qed.
-
-Lemma cos_sin_0_var : forall x:R, cos x <> 0 \/ sin x <> 0.
-Proof.
-  intros x.
-  destruct (Req_dec (cos x) 0). 2: now left.
-  right. intros H'.
-  apply (cos_sin_0 x).
-  now split.
-Qed.
-
-(*****************************************************************)
-(** * Using series definitions of cos and sin                    *)
-(*****************************************************************)
-
-Definition sin_lb (a:R) : R := sin_approx a 3.
-Definition sin_ub (a:R) : R := sin_approx a 4.
-Definition cos_lb (a:R) : R := cos_approx a 3.
-Definition cos_ub (a:R) : R := cos_approx a 4.
-
-Lemma sin_lb_gt_0 : forall a:R, 0 < a -> a <= PI / 2 -> 0 < sin_lb a.
-Proof.
-  intros.
-  unfold sin_lb in |- *; unfold sin_approx in |- *; unfold sin_term in |- *.
-  set (Un := fun i:nat => a ^ (2 * i + 1) / INR (fact (2 * i + 1))).
-  replace
-  (sum_f_R0
-    (fun i:nat => (-1) ^ i * (a ^ (2 * i + 1) / INR (fact (2 * i + 1)))) 3)
-    with (sum_f_R0 (fun i:nat => (-1) ^ i * Un i) 3);
-      [ idtac | apply sum_eq; intros; unfold Un in |- *; reflexivity ].
-  cut (forall n:nat, Un (S n) < Un n).
-  intro; simpl in |- *.
-  repeat rewrite Rmult_1_l; repeat rewrite Rmult_1_r;
-    replace (-1 * Un 1%nat) with (- Un 1%nat); [ idtac | ring ];
-    replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ];
-    replace (-1 * (-1 * -1) * Un 3%nat) with (- Un 3%nat);
-    [ idtac | ring ];
-    replace (Un 0%nat + - Un 1%nat + Un 2%nat + - Un 3%nat) with
-    (Un 0%nat - Un 1%nat + (Un 2%nat - Un 3%nat)); [ idtac | ring ].
-  apply Rplus_lt_0_compat.
-  unfold Rminus in |- *; apply Rplus_lt_reg_r with (Un 1%nat);
-    rewrite Rplus_0_r; rewrite (Rplus_comm (Un 1%nat));
-      rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
-        apply H1.
-  unfold Rminus in |- *; apply Rplus_lt_reg_r with (Un 3%nat);
-    rewrite Rplus_0_r; rewrite (Rplus_comm (Un 3%nat));
-      rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
-        apply H1.
-  intro; unfold Un in |- *.
-  cut ((2 * S n + 1)%nat = (2 * n + 1 + 2)%nat).
-  intro; rewrite H1.
-  rewrite pow_add; unfold Rdiv in |- *; rewrite Rmult_assoc;
-    apply Rmult_lt_compat_l.
-  apply pow_lt; assumption.
-  rewrite <- H1; apply Rmult_lt_reg_l with (INR (fact (2 * n + 1))).
-  apply lt_INR_0; apply neq_O_lt.
-  assert (H2 := fact_neq_0 (2 * n + 1)).
-  red in |- *; intro; elim H2; symmetry  in |- *; assumption.
-  rewrite <- Rinv_r_sym.
-  apply Rmult_lt_reg_l with (INR (fact (2 * S n + 1))).
-  apply lt_INR_0; apply neq_O_lt.
-  assert (H2 := fact_neq_0 (2 * S n + 1)).
-  red in |- *; intro; elim H2; symmetry  in |- *; assumption.
-  rewrite (Rmult_comm (INR (fact (2 * S n + 1)))); repeat rewrite Rmult_assoc;
-    rewrite <- Rinv_l_sym.
-  do 2 rewrite Rmult_1_r; apply Rle_lt_trans with (INR (fact (2 * n + 1)) * 4).
-  apply Rmult_le_compat_l.
-  replace 0 with (INR 0); [ idtac | reflexivity ]; apply le_INR; apply le_O_n.
-  simpl in |- *; rewrite Rmult_1_r; replace 4 with (Rsqr 2);
-    [ idtac | ring_Rsqr ]; replace (a * a) with (Rsqr a);
-    [ idtac | reflexivity ]; apply Rsqr_incr_1.
-  apply Rle_trans with (PI / 2);
-    [ assumption
-      | unfold Rdiv in |- *; apply Rmult_le_reg_l with 2;
-        [ prove_sup0
-          | rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m;
-            [ replace 4 with 4; [ apply PI_4 | ring ] | discrR ] ] ].
-  left; assumption.
-  left; prove_sup0.
-  rewrite H1; replace (2 * n + 1 + 2)%nat with (S (S (2 * n + 1))).
-  do 2 rewrite fact_simpl; do 2 rewrite mult_INR.
-  repeat rewrite <- Rmult_assoc.
-  rewrite <- (Rmult_comm (INR (fact (2 * n + 1)))).
-  rewrite Rmult_assoc.
-  apply Rmult_lt_compat_l.
-  apply lt_INR_0; apply neq_O_lt.
-  assert (H2 := fact_neq_0 (2 * n + 1)).
-  red in |- *; intro; elim H2; symmetry  in |- *; assumption.
-  do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; set (x := INR n);
-    unfold INR in |- *.
-  replace ((2 * x + 1 + 1 + 1) * (2 * x + 1 + 1)) with (4 * x * x + 10 * x + 6);
-  [ idtac | ring ].
-  apply Rplus_lt_reg_r with (-4); rewrite Rplus_opp_l;
-    replace (-4 + (4 * x * x + 10 * x + 6)) with (4 * x * x + 10 * x + 2);
-    [ idtac | ring ].
-  apply Rplus_le_lt_0_compat.
-  cut (0 <= x).
-  intro; apply Rplus_le_le_0_compat; repeat apply Rmult_le_pos;
-    assumption || left; prove_sup.
-  unfold x in |- *; replace 0 with (INR 0);
-    [ apply le_INR; apply le_O_n | reflexivity ].
-  prove_sup0.
-  ring.
-  apply INR_fact_neq_0.
-  apply INR_fact_neq_0.
-  ring.
-Qed.
-
-Lemma SIN : forall a:R, 0 <= a -> a <= PI -> sin_lb a <= sin a <= sin_ub a.
-  intros; unfold sin_lb, sin_ub in |- *; apply (sin_bound a 1 H H0).
-Qed.
-
-Lemma COS :
-  forall a:R, - PI / 2 <= a -> a <= PI / 2 -> cos_lb a <= cos a <= cos_ub a.
-  intros; unfold cos_lb, cos_ub in |- *; apply (cos_bound a 1 H H0).
-Qed.
-
-(**********)
-Lemma _PI2_RLT_0 : - (PI / 2) < 0.
-Proof.
-  rewrite <- Ropp_0; apply Ropp_lt_contravar; apply PI2_RGT_0.
-Qed.
-
-Lemma PI4_RLT_PI2 : PI / 4 < PI / 2.
-Proof.
-  unfold Rdiv in |- *; apply Rmult_lt_compat_l.
-  apply PI_RGT_0.
-  apply Rinv_lt_contravar.
-  apply Rmult_lt_0_compat; prove_sup0.
-  pattern 2 at 1 in |- *; rewrite <- Rplus_0_r.
-  replace 4 with (2 + 2); [ apply Rplus_lt_compat_l; prove_sup0 | ring ].
-Qed.
-
-Lemma PI2_Rlt_PI : PI / 2 < PI.
-Proof.
-  unfold Rdiv in |- *; pattern PI at 2 in |- *; rewrite <- Rmult_1_r.
-  apply Rmult_lt_compat_l.
-  apply PI_RGT_0.
-  pattern 1 at 3 in |- *; rewrite <- Rinv_1; apply Rinv_lt_contravar.
-  rewrite Rmult_1_l; prove_sup0.
-  pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
-    apply Rlt_0_1.
-Qed.
-
-(***************************************************)
-(** * Increasing and decreasing of [cos] and [sin] *)
-(***************************************************)
-Theorem sin_gt_0 : forall x:R, 0 < x -> x < PI -> 0 < sin x.
-Proof.
-  intros; elim (SIN x (Rlt_le 0 x H) (Rlt_le x PI H0)); intros H1 _;
-    case (Rtotal_order x (PI / 2)); intro H2.
-  apply Rlt_le_trans with (sin_lb x).
-  apply sin_lb_gt_0; [ assumption | left; assumption ].
-  assumption.
-  elim H2; intro H3.
-  rewrite H3; rewrite sin_PI2; apply Rlt_0_1.
-  rewrite <- sin_PI_x; generalize (Ropp_gt_lt_contravar x (PI / 2) H3);
-    intro H4; generalize (Rplus_lt_compat_l PI (- x) (- (PI / 2)) H4).
-  replace (PI + - x) with (PI - x).
-  replace (PI + - (PI / 2)) with (PI / 2).
-  intro H5; generalize (Ropp_lt_gt_contravar x PI H0); intro H6;
-    change (- PI < - x) in H6; generalize (Rplus_lt_compat_l PI (- PI) (- x) H6).
-  rewrite Rplus_opp_r.
-  replace (PI + - x) with (PI - x).
-  intro H7;
-    elim
-      (SIN (PI - x) (Rlt_le 0 (PI - x) H7)
-        (Rlt_le (PI - x) PI (Rlt_trans (PI - x) (PI / 2) PI H5 PI2_Rlt_PI)));
-      intros H8 _;
-        generalize (sin_lb_gt_0 (PI - x) H7 (Rlt_le (PI - x) (PI / 2) H5));
-          intro H9; apply (Rlt_le_trans 0 (sin_lb (PI - x)) (sin (PI - x)) H9 H8).
-  reflexivity.
-  pattern PI at 2 in |- *; rewrite double_var; ring.
-  reflexivity.
-Qed.
-
-Theorem cos_gt_0 : forall x:R, - (PI / 2) < x -> x < PI / 2 -> 0 < cos x.
-Proof.
-  intros; rewrite cos_sin;
-    generalize (Rplus_lt_compat_l (PI / 2) (- (PI / 2)) x H).
-  rewrite Rplus_opp_r; intro H1;
-    generalize (Rplus_lt_compat_l (PI / 2) x (PI / 2) H0);
-      rewrite <- double_var; intro H2; apply (sin_gt_0 (PI / 2 + x) H1 H2).
-Qed.
-
-Lemma sin_ge_0 : forall x:R, 0 <= x -> x <= PI -> 0 <= sin x.
-Proof.
-  intros x H1 H2; elim H1; intro H3;
-    [ elim H2; intro H4;
-      [ left; apply (sin_gt_0 x H3 H4)
-        | rewrite H4; right; symmetry  in |- *; apply sin_PI ]
-      | rewrite <- H3; right; symmetry  in |- *; apply sin_0 ].
-Qed.
-
-Lemma cos_ge_0 : forall x:R, - (PI / 2) <= x -> x <= PI / 2 -> 0 <= cos x.
-Proof.
-  intros x H1 H2; elim H1; intro H3;
-    [ elim H2; intro H4;
-      [ left; apply (cos_gt_0 x H3 H4)
-        | rewrite H4; right; symmetry  in |- *; apply cos_PI2 ]
-      | rewrite <- H3; rewrite cos_neg; right; symmetry  in |- *; apply cos_PI2 ].
-Qed.
-
-Lemma sin_le_0 : forall x:R, PI <= x -> x <= 2 * PI -> sin x <= 0.
-Proof.
-  intros x H1 H2; apply Rge_le; rewrite <- Ropp_0;
-    rewrite <- (Ropp_involutive (sin x)); apply Ropp_le_ge_contravar;
-      rewrite <- neg_sin; replace (x + PI) with (x - PI + 2 * INR 1 * PI);
-        [ rewrite (sin_period (x - PI) 1); apply sin_ge_0;
-          [ replace (x - PI) with (x + - PI);
-            [ rewrite Rplus_comm; replace 0 with (- PI + PI);
-              [ apply Rplus_le_compat_l; assumption | ring ]
-              | ring ]
-            | replace (x - PI) with (x + - PI); rewrite Rplus_comm;
-              [ pattern PI at 2 in |- *; replace PI with (- PI + 2 * PI);
-                [ apply Rplus_le_compat_l; assumption | ring ]
-                | ring ] ]
-          | unfold INR in |- *; ring ].
-Qed.
-
-Lemma cos_le_0 : forall x:R, PI / 2 <= x -> x <= 3 * (PI / 2) -> cos x <= 0.
-Proof.
-  intros x H1 H2; apply Rge_le; rewrite <- Ropp_0;
-    rewrite <- (Ropp_involutive (cos x)); apply Ropp_le_ge_contravar;
-      rewrite <- neg_cos; replace (x + PI) with (x - PI + 2 * INR 1 * PI).
-  rewrite cos_period; apply cos_ge_0.
-  replace (- (PI / 2)) with (- PI + PI / 2).
-  unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_le_compat_l;
-    assumption.
-  pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
-    ring.
-  unfold Rminus in |- *; rewrite Rplus_comm;
-    replace (PI / 2) with (- PI + 3 * (PI / 2)).
-  apply Rplus_le_compat_l; assumption.
-  pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
-    ring.
-  unfold INR in |- *; ring.
-Qed.
-
-Lemma sin_lt_0 : forall x:R, PI < x -> x < 2 * PI -> sin x < 0.
-Proof.
-  intros x H1 H2; rewrite <- Ropp_0; rewrite <- (Ropp_involutive (sin x));
-    apply Ropp_lt_gt_contravar; rewrite <- neg_sin;
-      replace (x + PI) with (x - PI + 2 * INR 1 * PI);
-      [ rewrite (sin_period (x - PI) 1); apply sin_gt_0;
-        [ replace (x - PI) with (x + - PI);
-          [ rewrite Rplus_comm; replace 0 with (- PI + PI);
-            [ apply Rplus_lt_compat_l; assumption | ring ]
-            | ring ]
-          | replace (x - PI) with (x + - PI); rewrite Rplus_comm;
-            [ pattern PI at 2 in |- *; replace PI with (- PI + 2 * PI);
-              [ apply Rplus_lt_compat_l; assumption | ring ]
-              | ring ] ]
-        | unfold INR in |- *; ring ].
-Qed.
-
-Lemma sin_lt_0_var : forall x:R, - PI < x -> x < 0 -> sin x < 0.
-Proof.
-  intros; generalize (Rplus_lt_compat_l (2 * PI) (- PI) x H);
-    replace (2 * PI + - PI) with PI;
-    [ intro H1; rewrite Rplus_comm in H1;
-      generalize (Rplus_lt_compat_l (2 * PI) x 0 H0);
-        intro H2; rewrite (Rplus_comm (2 * PI)) in H2;
-          rewrite <- (Rplus_comm 0) in H2; rewrite Rplus_0_l in H2;
-            rewrite <- (sin_period x 1); unfold INR in |- *;
-              replace (2 * 1 * PI) with (2 * PI);
-              [ apply (sin_lt_0 (x + 2 * PI) H1 H2) | ring ]
-      | ring ].
-Qed.
-
-Lemma cos_lt_0 : forall x:R, PI / 2 < x -> x < 3 * (PI / 2) -> cos x < 0.
-Proof.
-  intros x H1 H2; rewrite <- Ropp_0; rewrite <- (Ropp_involutive (cos x));
-    apply Ropp_lt_gt_contravar; rewrite <- neg_cos;
-      replace (x + PI) with (x - PI + 2 * INR 1 * PI).
-  rewrite cos_period; apply cos_gt_0.
-  replace (- (PI / 2)) with (- PI + PI / 2).
-  unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_lt_compat_l;
-    assumption.
-  pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
-    ring.
-  unfold Rminus in |- *; rewrite Rplus_comm;
-    replace (PI / 2) with (- PI + 3 * (PI / 2)).
-  apply Rplus_lt_compat_l; assumption.
-  pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
-    ring.
-  unfold INR in |- *; ring.
-Qed.
-
-Lemma tan_gt_0 : forall x:R, 0 < x -> x < PI / 2 -> 0 < tan x.
-Proof.
-  intros x H1 H2; unfold tan in |- *; generalize _PI2_RLT_0;
-    generalize (Rlt_trans 0 x (PI / 2) H1 H2); intros;
-      generalize (Rlt_trans (- (PI / 2)) 0 x H0 H1); intro H5;
-        generalize (Rlt_trans x (PI / 2) PI H2 PI2_Rlt_PI);
-          intro H7; unfold Rdiv in |- *; apply Rmult_lt_0_compat.
-  apply sin_gt_0; assumption.
-  apply Rinv_0_lt_compat; apply cos_gt_0; assumption.
-Qed.
-
-Lemma tan_lt_0 : forall x:R, - (PI / 2) < x -> x < 0 -> tan x < 0.
-Proof.
-  intros x H1 H2; unfold tan in |- *;
-    generalize (cos_gt_0 x H1 (Rlt_trans x 0 (PI / 2) H2 PI2_RGT_0));
-      intro H3; rewrite <- Ropp_0;
-        replace (sin x / cos x) with (- (- sin x / cos x)).
-  rewrite <- sin_neg; apply Ropp_gt_lt_contravar;
-    change (0 < sin (- x) / cos x) in |- *; unfold Rdiv in |- *;
-      apply Rmult_lt_0_compat.
-  apply sin_gt_0.
-  rewrite <- Ropp_0; apply Ropp_gt_lt_contravar; assumption.
-  apply Rlt_trans with (PI / 2).
-  rewrite <- (Ropp_involutive (PI / 2)); apply Ropp_gt_lt_contravar; assumption.
-  apply PI2_Rlt_PI.
-  apply Rinv_0_lt_compat; assumption.
-  unfold Rdiv in |- *; ring.
-Qed.
-
-Lemma cos_ge_0_3PI2 :
-  forall x:R, 3 * (PI / 2) <= x -> x <= 2 * PI -> 0 <= cos x.
-Proof.
-  intros; rewrite <- cos_neg; rewrite <- (cos_period (- x) 1);
-    unfold INR in |- *; replace (- x + 2 * 1 * PI) with (2 * PI - x).
-  generalize (Ropp_le_ge_contravar x (2 * PI) H0); intro H1;
-    generalize (Rge_le (- x) (- (2 * PI)) H1); clear H1;
-      intro H1; generalize (Rplus_le_compat_l (2 * PI) (- (2 * PI)) (- x) H1).
-  rewrite Rplus_opp_r.
-  intro H2; generalize (Ropp_le_ge_contravar (3 * (PI / 2)) x H); intro H3;
-    generalize (Rge_le (- (3 * (PI / 2))) (- x) H3); clear H3;
-      intro H3;
-        generalize (Rplus_le_compat_l (2 * PI) (- x) (- (3 * (PI / 2))) H3).
-  replace (2 * PI + - (3 * (PI / 2))) with (PI / 2).
-  intro H4;
-    apply
-      (cos_ge_0 (2 * PI - x)
-        (Rlt_le (- (PI / 2)) (2 * PI - x)
-          (Rlt_le_trans (- (PI / 2)) 0 (2 * PI - x) _PI2_RLT_0 H2)) H4).
-  rewrite double; pattern PI at 2 3 in |- *; rewrite double_var; ring.
-  ring.
-Qed.
-
-Lemma form1 :
-  forall p q:R, cos p + cos q = 2 * cos ((p - q) / 2) * cos ((p + q) / 2).
-Proof.
-  intros p q; pattern p at 1 in |- *;
-    replace p with ((p - q) / 2 + (p + q) / 2).
-  rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2).
-  rewrite cos_plus; rewrite cos_minus; ring.
-  pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
-  pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
-Qed.
-
-Lemma form2 :
-  forall p q:R, cos p - cos q = -2 * sin ((p - q) / 2) * sin ((p + q) / 2).
-Proof.
-  intros p q; pattern p at 1 in |- *;
-    replace p with ((p - q) / 2 + (p + q) / 2).
-  rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2).
-  rewrite cos_plus; rewrite cos_minus; ring.
-  pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
-  pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
-Qed.
-
-Lemma form3 :
-  forall p q:R, sin p + sin q = 2 * cos ((p - q) / 2) * sin ((p + q) / 2).
-Proof.
-  intros p q; pattern p at 1 in |- *;
-    replace p with ((p - q) / 2 + (p + q) / 2).
-  pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2).
-  rewrite sin_plus; rewrite sin_minus; ring.
-  pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
-  pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
-Qed.
-
-Lemma form4 :
-  forall p q:R, sin p - sin q = 2 * cos ((p + q) / 2) * sin ((p - q) / 2).
-Proof.
-  intros p q; pattern p at 1 in |- *;
-    replace p with ((p - q) / 2 + (p + q) / 2).
-  pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2).
-  rewrite sin_plus; rewrite sin_minus; ring.
-  pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
-  pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
-
-Qed.
-
-Lemma sin_increasing_0 :
-  forall x y:R,
-    - (PI / 2) <= x ->
-    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x < sin y -> x < y.
-Proof.
-  intros; cut (sin ((x - y) / 2) < 0).
-  intro H4; case (Rtotal_order ((x - y) / 2) 0); intro H5.
-  assert (Hyp : 0 < 2).
-  prove_sup0.
-  generalize (Rmult_lt_compat_l 2 ((x - y) / 2) 0 Hyp H5).
-  unfold Rdiv in |- *.
-  rewrite <- Rmult_assoc.
-  rewrite Rinv_r_simpl_m.
-  rewrite Rmult_0_r.
-  clear H5; intro H5; apply Rminus_lt; assumption.
-  discrR.
-  elim H5; intro H6.
-  rewrite H6 in H4; rewrite sin_0 in H4; elim (Rlt_irrefl 0 H4).
-  change (0 < (x - y) / 2) in H6;
-    generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1).
-  rewrite Ropp_involutive.
-  intro H7; generalize (Rge_le (PI / 2) (- y) H7); clear H7; intro H7;
-    generalize (Rplus_le_compat x (PI / 2) (- y) (PI / 2) H0 H7).
-  rewrite <- double_var.
-  intro H8.
-  assert (Hyp : 0 < 2).
-  prove_sup0.
-  generalize
-    (Rmult_le_compat_l (/ 2) (x - y) PI
-      (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H8).
-  repeat rewrite (Rmult_comm (/ 2)).
-  intro H9;
-    generalize
-      (sin_gt_0 ((x - y) / 2) H6
-        (Rle_lt_trans ((x - y) / 2) (PI / 2) PI H9 PI2_Rlt_PI));
-      intro H10;
-        elim
-          (Rlt_irrefl (sin ((x - y) / 2))
-            (Rlt_trans (sin ((x - y) / 2)) 0 (sin ((x - y) / 2)) H4 H10)).
-  generalize (Rlt_minus (sin x) (sin y) H3); clear H3; intro H3;
-    rewrite form4 in H3;
-      generalize (Rplus_le_compat x (PI / 2) y (PI / 2) H0 H2).
-  rewrite <- double_var.
-  assert (Hyp : 0 < 2).
-  prove_sup0.
-  intro H4;
-    generalize
-      (Rmult_le_compat_l (/ 2) (x + y) PI
-        (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H4).
-  repeat rewrite (Rmult_comm (/ 2)).
-  clear H4; intro H4;
-    generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) y H H1);
-      replace (- (PI / 2) + - (PI / 2)) with (- PI).
-  intro H5;
-    generalize
-      (Rmult_le_compat_l (/ 2) (- PI) (x + y)
-        (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H5).
-  replace (/ 2 * (x + y)) with ((x + y) / 2).
-  replace (/ 2 * - PI) with (- (PI / 2)).
-  clear H5; intro H5; elim H4; intro H40.
-  elim H5; intro H50.
-  generalize (cos_gt_0 ((x + y) / 2) H50 H40); intro H6;
-    generalize (Rmult_lt_compat_l 2 0 (cos ((x + y) / 2)) Hyp H6).
-  rewrite Rmult_0_r.
-  clear H6; intro H6; case (Rcase_abs (sin ((x - y) / 2))); intro H7.
-  assumption.
-  generalize (Rge_le (sin ((x - y) / 2)) 0 H7); clear H7; intro H7;
-    generalize
-      (Rmult_le_pos (2 * cos ((x + y) / 2)) (sin ((x - y) / 2))
-        (Rlt_le 0 (2 * cos ((x + y) / 2)) H6) H7); intro H8;
-      generalize
-        (Rle_lt_trans 0 (2 * cos ((x + y) / 2) * sin ((x - y) / 2)) 0 H8 H3);
-        intro H9; elim (Rlt_irrefl 0 H9).
-  rewrite <- H50 in H3; rewrite cos_neg in H3; rewrite cos_PI2 in H3;
-    rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3;
-      elim (Rlt_irrefl 0 H3).
-  unfold Rdiv in H3.
-  rewrite H40 in H3; assert (H50 := cos_PI2); unfold Rdiv in H50;
-    rewrite H50 in H3; rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3;
-      elim (Rlt_irrefl 0 H3).
-  unfold Rdiv in |- *.
-  rewrite <- Ropp_mult_distr_l_reverse.
-  apply Rmult_comm.
-  unfold Rdiv in |- *; apply Rmult_comm.
-  pattern PI at 1 in |- *; rewrite double_var.
-  rewrite Ropp_plus_distr.
-  reflexivity.
-Qed.
-
-Lemma sin_increasing_1 :
-  forall x y:R,
-    - (PI / 2) <= x ->
-    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> x < y -> sin x < sin y.
-Proof.
-  intros; generalize (Rplus_lt_compat_l x x y H3); intro H4;
-    generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) x H H);
-      replace (- (PI / 2) + - (PI / 2)) with (- PI).
-  assert (Hyp : 0 < 2).
-  prove_sup0.
-  intro H5; generalize (Rle_lt_trans (- PI) (x + x) (x + y) H5 H4); intro H6;
-    generalize
-      (Rmult_lt_compat_l (/ 2) (- PI) (x + y) (Rinv_0_lt_compat 2 Hyp) H6);
-      replace (/ 2 * - PI) with (- (PI / 2)).
-  replace (/ 2 * (x + y)) with ((x + y) / 2).
-  clear H4 H5 H6; intro H4; generalize (Rplus_lt_compat_l y x y H3); intro H5;
-    rewrite Rplus_comm in H5;
-      generalize (Rplus_le_compat y (PI / 2) y (PI / 2) H2 H2).
-  rewrite <- double_var.
-  intro H6; generalize (Rlt_le_trans (x + y) (y + y) PI H5 H6); intro H7;
-    generalize (Rmult_lt_compat_l (/ 2) (x + y) PI (Rinv_0_lt_compat 2 Hyp) H7);
-      replace (/ 2 * PI) with (PI / 2).
-  replace (/ 2 * (x + y)) with ((x + y) / 2).
-  clear H5 H6 H7; intro H5; generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1);
-    rewrite Ropp_involutive; clear H1; intro H1;
-      generalize (Rge_le (PI / 2) (- y) H1); clear H1; intro H1;
-        generalize (Ropp_le_ge_contravar y (PI / 2) H2); clear H2;
-          intro H2; generalize (Rge_le (- y) (- (PI / 2)) H2);
-            clear H2; intro H2; generalize (Rplus_lt_compat_l (- y) x y H3);
-              replace (- y + x) with (x - y).
-  rewrite Rplus_opp_l.
-  intro H6;
-    generalize (Rmult_lt_compat_l (/ 2) (x - y) 0 (Rinv_0_lt_compat 2 Hyp) H6);
-      rewrite Rmult_0_r; replace (/ 2 * (x - y)) with ((x - y) / 2).
-  clear H6; intro H6;
-    generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) (- y) H H2);
-      replace (- (PI / 2) + - (PI / 2)) with (- PI).
-  replace (x + - y) with (x - y).
-  intro H7;
-    generalize
-      (Rmult_le_compat_l (/ 2) (- PI) (x - y)
-        (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H7);
-      replace (/ 2 * - PI) with (- (PI / 2)).
-  replace (/ 2 * (x - y)) with ((x - y) / 2).
-  clear H7; intro H7; clear H H0 H1 H2; apply Rminus_lt; rewrite form4;
-    generalize (cos_gt_0 ((x + y) / 2) H4 H5); intro H8;
-      generalize (Rmult_lt_0_compat 2 (cos ((x + y) / 2)) Hyp H8);
-        clear H8; intro H8; cut (- PI < - (PI / 2)).
-  intro H9;
-    generalize
-      (sin_lt_0_var ((x - y) / 2)
-        (Rlt_le_trans (- PI) (- (PI / 2)) ((x - y) / 2) H9 H7) H6);
-      intro H10;
-        generalize
-          (Rmult_lt_gt_compat_neg_l (sin ((x - y) / 2)) 0 (
-            2 * cos ((x + y) / 2)) H10 H8); intro H11; rewrite Rmult_0_r in H11;
-          rewrite Rmult_comm; assumption.
-  apply Ropp_lt_gt_contravar; apply PI2_Rlt_PI.
-  unfold Rdiv in |- *; apply Rmult_comm.
-  unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse; apply Rmult_comm.
-  reflexivity.
-  pattern PI at 1 in |- *; rewrite double_var.
-  rewrite Ropp_plus_distr.
-  reflexivity.
-  unfold Rdiv in |- *; apply Rmult_comm.
-  unfold Rminus in |- *; apply Rplus_comm.
-  unfold Rdiv in |- *; apply Rmult_comm.
-  unfold Rdiv in |- *; apply Rmult_comm.
-  unfold Rdiv in |- *; apply Rmult_comm.
-  unfold Rdiv in |- *.
-  rewrite <- Ropp_mult_distr_l_reverse.
-  apply Rmult_comm.
-  pattern PI at 1 in |- *; rewrite double_var.
-  rewrite Ropp_plus_distr.
-  reflexivity.
-Qed.
-
-Lemma sin_decreasing_0 :
-  forall x y:R,
-    x <= 3 * (PI / 2) ->
-    PI / 2 <= x -> y <= 3 * (PI / 2) -> PI / 2 <= y -> sin x < sin y -> y < x.
-Proof.
-  intros; rewrite <- (sin_PI_x x) in H3; rewrite <- (sin_PI_x y) in H3;
-    generalize (Ropp_lt_gt_contravar (sin (PI - x)) (sin (PI - y)) H3);
-      repeat rewrite <- sin_neg;
-        generalize (Rplus_le_compat_l (- PI) x (3 * (PI / 2)) H);
-          generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0);
-            generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1);
-              generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2);
-                replace (- PI + x) with (x - PI).
-  replace (- PI + PI / 2) with (- (PI / 2)).
-  replace (- PI + y) with (y - PI).
-  replace (- PI + 3 * (PI / 2)) with (PI / 2).
-  replace (- (PI - x)) with (x - PI).
-  replace (- (PI - y)) with (y - PI).
-  intros; change (sin (y - PI) < sin (x - PI)) in H8;
-    apply Rplus_lt_reg_r with (- PI); rewrite Rplus_comm;
-      replace (y + - PI) with (y - PI).
-  rewrite Rplus_comm; replace (x + - PI) with (x - PI).
-  apply (sin_increasing_0 (y - PI) (x - PI) H4 H5 H6 H7 H8).
-  reflexivity.
-  reflexivity.
-  unfold Rminus in |- *; rewrite Ropp_plus_distr.
-  rewrite Ropp_involutive.
-  apply Rplus_comm.
-  unfold Rminus in |- *; rewrite Ropp_plus_distr.
-  rewrite Ropp_involutive.
-  apply Rplus_comm.
-  pattern PI at 2 in |- *; rewrite double_var.
-  rewrite Ropp_plus_distr.
-  ring.
-  unfold Rminus in |- *; apply Rplus_comm.
-  pattern PI at 2 in |- *; rewrite double_var.
-  rewrite Ropp_plus_distr.
-  ring.
-  unfold Rminus in |- *; apply Rplus_comm.
-Qed.
-
-Lemma sin_decreasing_1 :
-  forall x y:R,
-    x <= 3 * (PI / 2) ->
-    PI / 2 <= x -> y <= 3 * (PI / 2) -> PI / 2 <= y -> x < y -> sin y < sin x.
-Proof.
-  intros; rewrite <- (sin_PI_x x); rewrite <- (sin_PI_x y);
-    generalize (Rplus_le_compat_l (- PI) x (3 * (PI / 2)) H);
-      generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0);
-        generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1);
-          generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2);
-            generalize (Rplus_lt_compat_l (- PI) x y H3);
-              replace (- PI + PI / 2) with (- (PI / 2)).
-  replace (- PI + y) with (y - PI).
-  replace (- PI + 3 * (PI / 2)) with (PI / 2).
-  replace (- PI + x) with (x - PI).
-  intros; apply Ropp_lt_cancel; repeat rewrite <- sin_neg;
-    replace (- (PI - x)) with (x - PI).
-  replace (- (PI - y)) with (y - PI).
-  apply (sin_increasing_1 (x - PI) (y - PI) H7 H8 H5 H6 H4).
-  unfold Rminus in |- *; rewrite Ropp_plus_distr.
-  rewrite Ropp_involutive.
-  apply Rplus_comm.
-  unfold Rminus in |- *; rewrite Ropp_plus_distr.
-  rewrite Ropp_involutive.
-  apply Rplus_comm.
-  unfold Rminus in |- *; apply Rplus_comm.
-  pattern PI at 2 in |- *; rewrite double_var; ring.
-  unfold Rminus in |- *; apply Rplus_comm.
-  pattern PI at 2 in |- *; rewrite double_var; ring.
-Qed.
-
-Lemma cos_increasing_0 :
-  forall x y:R,
-    PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x < cos y -> x < y.
-Proof.
-  intros x y H1 H2 H3 H4; rewrite <- (cos_neg x); rewrite <- (cos_neg y);
-    rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1);
-      unfold INR in |- *;
-        replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))).
-  replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))).
-  repeat rewrite cos_shift; intro H5;
-    generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1);
-      generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2);
-        generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3);
-          generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4).
-  replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)).
-  replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)).
-  replace (-3 * (PI / 2) + 2 * PI) with (PI / 2).
-  replace (-3 * (PI / 2) + PI) with (- (PI / 2)).
-  clear H1 H2 H3 H4; intros H1 H2 H3 H4;
-    apply Rplus_lt_reg_r with (-3 * (PI / 2));
-      replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)).
-  replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)).
-  apply (sin_increasing_0 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H4 H3 H2 H1 H5).
-  unfold Rminus in |- *.
-  rewrite Ropp_mult_distr_l_reverse.
-  apply Rplus_comm.
-  unfold Rminus in |- *.
-  rewrite Ropp_mult_distr_l_reverse.
-  apply Rplus_comm.
-  pattern PI at 3 in |- *; rewrite double_var.
-  ring.
-  rewrite double; pattern PI at 3 4 in |- *; rewrite double_var.
-  ring.
-  unfold Rminus in |- *.
-  rewrite Ropp_mult_distr_l_reverse.
-  apply Rplus_comm.
-  unfold Rminus in |- *.
-  rewrite Ropp_mult_distr_l_reverse.
-  apply Rplus_comm.
-  rewrite Rmult_1_r.
-  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
-  ring.
-  rewrite Rmult_1_r.
-  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
-  ring.
-Qed.
-
-Lemma cos_increasing_1 :
-  forall x y:R,
-    PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> x < y -> cos x < cos y.
-Proof.
-  intros x y H1 H2 H3 H4 H5;
-    generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1);
-      generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2);
-        generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3);
-          generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4);
-            generalize (Rplus_lt_compat_l (-3 * (PI / 2)) x y H5);
-              rewrite <- (cos_neg x); rewrite <- (cos_neg y);
-                rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1);
-                  unfold INR in |- *; replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)).
-  replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)).
-  replace (-3 * (PI / 2) + PI) with (- (PI / 2)).
-  replace (-3 * (PI / 2) + 2 * PI) with (PI / 2).
-  clear H1 H2 H3 H4 H5; intros H1 H2 H3 H4 H5;
-    replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))).
-  replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))).
-  repeat rewrite cos_shift;
-    apply
-      (sin_increasing_1 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H5 H4 H3 H2 H1).
-  rewrite Rmult_1_r.
-  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
-  ring.
-  rewrite Rmult_1_r.
-  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
-  ring.
-  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
-  ring.
-  pattern PI at 3 in |- *; rewrite double_var; ring.
-  unfold Rminus in |- *.
-  rewrite <- Ropp_mult_distr_l_reverse.
-  apply Rplus_comm.
-  unfold Rminus in |- *.
-  rewrite <- Ropp_mult_distr_l_reverse.
-  apply Rplus_comm.
-Qed.
-
-Lemma cos_decreasing_0 :
-  forall x y:R,
-    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> cos x < cos y -> y < x.
-Proof.
-  intros; generalize (Ropp_lt_gt_contravar (cos x) (cos y) H3);
-    repeat rewrite <- neg_cos; intro H4;
-      change (cos (y + PI) < cos (x + PI)) in H4; rewrite (Rplus_comm x) in H4;
-        rewrite (Rplus_comm y) in H4; generalize (Rplus_le_compat_l PI 0 x H);
-          generalize (Rplus_le_compat_l PI x PI H0);
-            generalize (Rplus_le_compat_l PI 0 y H1);
-              generalize (Rplus_le_compat_l PI y PI H2); rewrite Rplus_0_r.
-  rewrite <- double.
-  clear H H0 H1 H2 H3; intros; apply Rplus_lt_reg_r with PI;
-    apply (cos_increasing_0 (PI + y) (PI + x) H0 H H2 H1 H4).
-Qed.
-
-Lemma cos_decreasing_1 :
-  forall x y:R,
-    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> x < y -> cos y < cos x.
-Proof.
-  intros; apply Ropp_lt_cancel; repeat rewrite <- neg_cos;
-    rewrite (Rplus_comm x); rewrite (Rplus_comm y);
-      generalize (Rplus_le_compat_l PI 0 x H);
-        generalize (Rplus_le_compat_l PI x PI H0);
-          generalize (Rplus_le_compat_l PI 0 y H1);
-            generalize (Rplus_le_compat_l PI y PI H2); rewrite Rplus_0_r.
-  rewrite <- double.
-  generalize (Rplus_lt_compat_l PI x y H3); clear H H0 H1 H2 H3; intros;
-    apply (cos_increasing_1 (PI + x) (PI + y) H3 H2 H1 H0 H).
-Qed.
-
-Lemma tan_diff :
-  forall x y:R,
-    cos x <> 0 -> cos y <> 0 -> tan x - tan y = sin (x - y) / (cos x * cos y).
-Proof.
-  intros; unfold tan in |- *; rewrite sin_minus.
-  unfold Rdiv in |- *.
-  unfold Rminus in |- *.
-  rewrite Rmult_plus_distr_r.
-  rewrite Rinv_mult_distr.
-  repeat rewrite (Rmult_comm (sin x)).
-  repeat rewrite Rmult_assoc.
-  rewrite (Rmult_comm (cos y)).
-  repeat rewrite Rmult_assoc.
-  rewrite <- Rinv_l_sym.
-  rewrite Rmult_1_r.
-  rewrite (Rmult_comm (sin x)).
-  apply Rplus_eq_compat_l.
-  rewrite <- Ropp_mult_distr_l_reverse.
-  rewrite <- Ropp_mult_distr_r_reverse.
-  rewrite (Rmult_comm (/ cos x)).
-  repeat rewrite Rmult_assoc.
-  rewrite (Rmult_comm (cos x)).
-  repeat rewrite Rmult_assoc.
-  rewrite <- Rinv_l_sym.
-  rewrite Rmult_1_r.
-  reflexivity.
-  assumption.
-  assumption.
-  assumption.
-  assumption.
-Qed.
-
-Lemma tan_increasing_0 :
-  forall x y:R,
-    - (PI / 4) <= x ->
-    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x < tan y -> x < y.
-Proof.
-  intros; generalize PI4_RLT_PI2; intro H4;
-    generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4);
-      intro H5; change (- (PI / 2) < - (PI / 4)) in H5;
-        generalize
-          (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
-            (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)); intro HP1;
-          generalize
-            (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
-              (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)); intro HP2;
-            generalize
-              (sym_not_eq
-                (Rlt_not_eq 0 (cos x)
-                  (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
-                    (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4))));
-              intro H6;
-                generalize
-                  (sym_not_eq
-                    (Rlt_not_eq 0 (cos y)
-                      (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
-                        (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4))));
-                  intro H7; generalize (tan_diff x y H6 H7); intro H8;
-                    generalize (Rlt_minus (tan x) (tan y) H3); clear H3;
-                      intro H3; rewrite H8 in H3; cut (sin (x - y) < 0).
-  intro H9; generalize (Ropp_le_ge_contravar (- (PI / 4)) y H1);
-    rewrite Ropp_involutive; intro H10; generalize (Rge_le (PI / 4) (- y) H10);
-      clear H10; intro H10; generalize (Ropp_le_ge_contravar y (PI / 4) H2);
-        intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11);
-          clear H11; intro H11;
-            generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11);
-              generalize (Rplus_le_compat x (PI / 4) (- y) (PI / 4) H0 H10);
-                replace (x + - y) with (x - y).
-  replace (PI / 4 + PI / 4) with (PI / 2).
-  replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)).
-  intros; case (Rtotal_order 0 (x - y)); intro H14.
-  generalize
-    (sin_gt_0 (x - y) H14 (Rle_lt_trans (x - y) (PI / 2) PI H12 PI2_Rlt_PI));
-    intro H15; elim (Rlt_irrefl 0 (Rlt_trans 0 (sin (x - y)) 0 H15 H9)).
-  elim H14; intro H15.
-  rewrite <- H15 in H9; rewrite sin_0 in H9; elim (Rlt_irrefl 0 H9).
-  apply Rminus_lt; assumption.
-  pattern PI at 1 in |- *; rewrite double_var.
-  unfold Rdiv in |- *.
-  rewrite Rmult_plus_distr_r.
-  repeat rewrite Rmult_assoc.
-  rewrite <- Rinv_mult_distr.
-  rewrite Ropp_plus_distr.
-  replace 4 with 4.
-  reflexivity.
-  ring.
-  discrR.
-  discrR.
-  pattern PI at 1 in |- *; rewrite double_var.
-  unfold Rdiv in |- *.
-  rewrite Rmult_plus_distr_r.
-  repeat rewrite Rmult_assoc.
-  rewrite <- Rinv_mult_distr.
-  replace 4 with 4.
-  reflexivity.
-  ring.
-  discrR.
-  discrR.
-  reflexivity.
-  case (Rcase_abs (sin (x - y))); intro H9.
-  assumption.
-  generalize (Rge_le (sin (x - y)) 0 H9); clear H9; intro H9;
-    generalize (Rinv_0_lt_compat (cos x) HP1); intro H10;
-      generalize (Rinv_0_lt_compat (cos y) HP2); intro H11;
-        generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11);
-          replace (/ cos x * / cos y) with (/ (cos x * cos y)).
-  intro H12;
-    generalize
-      (Rmult_le_pos (sin (x - y)) (/ (cos x * cos y)) H9
-        (Rlt_le 0 (/ (cos x * cos y)) H12)); intro H13;
-      elim
-        (Rlt_irrefl 0 (Rle_lt_trans 0 (sin (x - y) * / (cos x * cos y)) 0 H13 H3)).
-  rewrite Rinv_mult_distr.
-  reflexivity.
-  assumption.
-  assumption.
-Qed.
-
-Lemma tan_increasing_1 :
-  forall x y:R,
-    - (PI / 4) <= x ->
-    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x < y -> tan x < tan y.
-Proof.
-  intros; apply Rminus_lt; generalize PI4_RLT_PI2; intro H4;
-    generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4);
-      intro H5; change (- (PI / 2) < - (PI / 4)) in H5;
-        generalize
-          (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
-            (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)); intro HP1;
-          generalize
-            (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
-              (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)); intro HP2;
-            generalize
-              (sym_not_eq
-                (Rlt_not_eq 0 (cos x)
-                  (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
-                    (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4))));
-              intro H6;
-                generalize
-                  (sym_not_eq
-                    (Rlt_not_eq 0 (cos y)
-                      (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
-                        (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4))));
-                  intro H7; rewrite (tan_diff x y H6 H7);
-                    generalize (Rinv_0_lt_compat (cos x) HP1); intro H10;
-                      generalize (Rinv_0_lt_compat (cos y) HP2); intro H11;
-                        generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11);
-                          replace (/ cos x * / cos y) with (/ (cos x * cos y)).
-  clear H10 H11; intro H8; generalize (Ropp_le_ge_contravar y (PI / 4) H2);
-    intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11);
-      clear H11; intro H11;
-        generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11);
-          replace (x + - y) with (x - y).
-  replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)).
-  clear H11; intro H9; generalize (Rlt_minus x y H3); clear H3; intro H3;
-    clear H H0 H1 H2 H4 H5 HP1 HP2; generalize PI2_Rlt_PI;
-      intro H1; generalize (Ropp_lt_gt_contravar (PI / 2) PI H1);
-        clear H1; intro H1;
-          generalize
-            (sin_lt_0_var (x - y) (Rlt_le_trans (- PI) (- (PI / 2)) (x - y) H1 H9) H3);
-            intro H2;
-              generalize
-                (Rmult_lt_gt_compat_neg_l (sin (x - y)) 0 (/ (cos x * cos y)) H2 H8);
-                rewrite Rmult_0_r; intro H4; assumption.
-  pattern PI at 1 in |- *; rewrite double_var.
-  unfold Rdiv in |- *.
-  rewrite Rmult_plus_distr_r.
-  repeat rewrite Rmult_assoc.
-  rewrite <- Rinv_mult_distr.
-  replace 4 with 4.
-  rewrite Ropp_plus_distr.
-  reflexivity.
-  ring.
-  discrR.
-  discrR.
-  reflexivity.
-  apply Rinv_mult_distr; assumption.
-Qed.
-
-Lemma sin_incr_0 :
-  forall x y:R,
-    - (PI / 2) <= x ->
-    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x <= sin y -> x <= y.
-Proof.
-  intros; case (Rtotal_order (sin x) (sin y)); intro H4;
-    [ left; apply (sin_increasing_0 x y H H0 H1 H2 H4)
-      | elim H4; intro H5;
-        [ case (Rtotal_order x y); intro H6;
-          [ left; assumption
-            | elim H6; intro H7;
-              [ right; assumption
-                | generalize (sin_increasing_1 y x H1 H2 H H0 H7); intro H8;
-                  rewrite H5 in H8; elim (Rlt_irrefl (sin y) H8) ] ]
-          | elim (Rlt_irrefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5)) ] ].
-Qed.
-
-Lemma sin_incr_1 :
-  forall x y:R,
-    - (PI / 2) <= x ->
-    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> x <= y -> sin x <= sin y.
-Proof.
-  intros; case (Rtotal_order x y); intro H4;
-    [ left; apply (sin_increasing_1 x y H H0 H1 H2 H4)
-      | elim H4; intro H5;
-        [ case (Rtotal_order (sin x) (sin y)); intro H6;
-          [ left; assumption
-            | elim H6; intro H7;
-              [ right; assumption
-                | generalize (sin_increasing_0 y x H1 H2 H H0 H7); intro H8;
-                  rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ]
-          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
-Qed.
-
-Lemma sin_decr_0 :
-  forall x y:R,
-    x <= 3 * (PI / 2) ->
-    PI / 2 <= x ->
-    y <= 3 * (PI / 2) -> PI / 2 <= y -> sin x <= sin y -> y <= x.
-Proof.
-  intros; case (Rtotal_order (sin x) (sin y)); intro H4;
-    [ left; apply (sin_decreasing_0 x y H H0 H1 H2 H4)
-      | elim H4; intro H5;
-        [ case (Rtotal_order x y); intro H6;
-          [ generalize (sin_decreasing_1 x y H H0 H1 H2 H6); intro H8;
-            rewrite H5 in H8; elim (Rlt_irrefl (sin y) H8)
-            | elim H6; intro H7;
-              [ right; symmetry  in |- *; assumption | left; assumption ] ]
-          | elim (Rlt_irrefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5)) ] ].
-Qed.
-
-Lemma sin_decr_1 :
-  forall x y:R,
-    x <= 3 * (PI / 2) ->
-    PI / 2 <= x ->
-    y <= 3 * (PI / 2) -> PI / 2 <= y -> x <= y -> sin y <= sin x.
-Proof.
-  intros; case (Rtotal_order x y); intro H4;
-    [ left; apply (sin_decreasing_1 x y H H0 H1 H2 H4)
-      | elim H4; intro H5;
-        [ case (Rtotal_order (sin x) (sin y)); intro H6;
-          [ generalize (sin_decreasing_0 x y H H0 H1 H2 H6); intro H8;
-            rewrite H5 in H8; elim (Rlt_irrefl y H8)
-            | elim H6; intro H7;
-              [ right; symmetry  in |- *; assumption | left; assumption ] ]
-          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
-Qed.
-
-Lemma cos_incr_0 :
-  forall x y:R,
-    PI <= x ->
-    x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x <= cos y -> x <= y.
-Proof.
-  intros; case (Rtotal_order (cos x) (cos y)); intro H4;
-    [ left; apply (cos_increasing_0 x y H H0 H1 H2 H4)
-      | elim H4; intro H5;
-        [ case (Rtotal_order x y); intro H6;
-          [ left; assumption
-            | elim H6; intro H7;
-              [ right; assumption
-                | generalize (cos_increasing_1 y x H1 H2 H H0 H7); intro H8;
-                  rewrite H5 in H8; elim (Rlt_irrefl (cos y) H8) ] ]
-          | elim (Rlt_irrefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5)) ] ].
-Qed.
-
-Lemma cos_incr_1 :
-  forall x y:R,
-    PI <= x ->
-    x <= 2 * PI -> PI <= y -> y <= 2 * PI -> x <= y -> cos x <= cos y.
-Proof.
-  intros; case (Rtotal_order x y); intro H4;
-    [ left; apply (cos_increasing_1 x y H H0 H1 H2 H4)
-      | elim H4; intro H5;
-        [ case (Rtotal_order (cos x) (cos y)); intro H6;
-          [ left; assumption
-            | elim H6; intro H7;
-              [ right; assumption
-                | generalize (cos_increasing_0 y x H1 H2 H H0 H7); intro H8;
-                  rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ]
-          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
-Qed.
-
-Lemma cos_decr_0 :
-  forall x y:R,
-    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> cos x <= cos y -> y <= x.
-Proof.
-  intros; case (Rtotal_order (cos x) (cos y)); intro H4;
-    [ left; apply (cos_decreasing_0 x y H H0 H1 H2 H4)
-      | elim H4; intro H5;
-        [ case (Rtotal_order x y); intro H6;
-          [ generalize (cos_decreasing_1 x y H H0 H1 H2 H6); intro H8;
-            rewrite H5 in H8; elim (Rlt_irrefl (cos y) H8)
-            | elim H6; intro H7;
-              [ right; symmetry  in |- *; assumption | left; assumption ] ]
-          | elim (Rlt_irrefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5)) ] ].
-Qed.
-
-Lemma cos_decr_1 :
-  forall x y:R,
-    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> x <= y -> cos y <= cos x.
-Proof.
-  intros; case (Rtotal_order x y); intro H4;
-    [ left; apply (cos_decreasing_1 x y H H0 H1 H2 H4)
-      | elim H4; intro H5;
-        [ case (Rtotal_order (cos x) (cos y)); intro H6;
-          [ generalize (cos_decreasing_0 x y H H0 H1 H2 H6); intro H8;
-            rewrite H5 in H8; elim (Rlt_irrefl y H8)
-            | elim H6; intro H7;
-              [ right; symmetry  in |- *; assumption | left; assumption ] ]
-          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
-Qed.
-
-Lemma tan_incr_0 :
-  forall x y:R,
-    - (PI / 4) <= x ->
-    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x <= tan y -> x <= y.
-Proof.
-  intros; case (Rtotal_order (tan x) (tan y)); intro H4;
-    [ left; apply (tan_increasing_0 x y H H0 H1 H2 H4)
-      | elim H4; intro H5;
-        [ case (Rtotal_order x y); intro H6;
-          [ left; assumption
-            | elim H6; intro H7;
-              [ right; assumption
-                | generalize (tan_increasing_1 y x H1 H2 H H0 H7); intro H8;
-                  rewrite H5 in H8; elim (Rlt_irrefl (tan y) H8) ] ]
-          | elim (Rlt_irrefl (tan x) (Rle_lt_trans (tan x) (tan y) (tan x) H3 H5)) ] ].
-Qed.
-
-Lemma tan_incr_1 :
-  forall x y:R,
-    - (PI / 4) <= x ->
-    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x <= y -> tan x <= tan y.
-Proof.
-  intros; case (Rtotal_order x y); intro H4;
-    [ left; apply (tan_increasing_1 x y H H0 H1 H2 H4)
-      | elim H4; intro H5;
-        [ case (Rtotal_order (tan x) (tan y)); intro H6;
-          [ left; assumption
-            | elim H6; intro H7;
-              [ right; assumption
-                | generalize (tan_increasing_0 y x H1 H2 H H0 H7); intro H8;
-                  rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ]
-          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
-Qed.
-
-(**********)
-Lemma sin_eq_0_1 : forall x:R, (exists k : Z, x = IZR k * PI) -> sin x = 0.
-Proof.
-  intros.
-  elim H; intros.
-  apply (Zcase_sign x0).
-  intro.
-  rewrite H1 in H0.
-  simpl in H0.
-  rewrite H0; rewrite Rmult_0_l; apply sin_0.
-  intro.
-  cut (0 <= x0)%Z.
-  intro.
-  elim (IZN x0 H2); intros.
-  rewrite H3 in H0.
-  rewrite <- INR_IZR_INZ in H0.
-  rewrite H0.
-  elim (even_odd_cor x1); intros.
-  elim H4; intro.
-  rewrite H5.
-  rewrite mult_INR.
-  simpl in |- *.
-  rewrite <- (Rplus_0_l (2 * INR x2 * PI)).
-  rewrite sin_period.
-  apply sin_0.
-  rewrite H5.
-  rewrite S_INR; rewrite mult_INR.
-  simpl in |- *.
-  rewrite Rmult_plus_distr_r.
-  rewrite Rmult_1_l; rewrite sin_plus.
-  rewrite sin_PI.
-  rewrite Rmult_0_r.
-  rewrite <- (Rplus_0_l (2 * INR x2 * PI)).
-  rewrite sin_period.
-  rewrite sin_0; ring.
-  apply le_IZR.
-  left; apply IZR_lt.
-  assert (H2 := Zorder.Zgt_iff_lt).
-  elim (H2 x0 0%Z); intros.
-  apply H3; assumption.
-  intro.
-  rewrite H0.
-  replace (sin (IZR x0 * PI)) with (- sin (- IZR x0 * PI)).
-  cut (0 <= - x0)%Z.
-  intro.
-  rewrite <- Ropp_Ropp_IZR.
-  elim (IZN (- x0) H2); intros.
-  rewrite H3.
-  rewrite <- INR_IZR_INZ.
-  elim (even_odd_cor x1); intros.
-  elim H4; intro.
-  rewrite H5.
-  rewrite mult_INR.
-  simpl in |- *.
-  rewrite <- (Rplus_0_l (2 * INR x2 * PI)).
-  rewrite sin_period.
-  rewrite sin_0; ring.
-  rewrite H5.
-  rewrite S_INR; rewrite mult_INR.
-  simpl in |- *.
-  rewrite Rmult_plus_distr_r.
-  rewrite Rmult_1_l; rewrite sin_plus.
-  rewrite sin_PI.
-  rewrite Rmult_0_r.
-  rewrite <- (Rplus_0_l (2 * INR x2 * PI)).
-  rewrite sin_period.
-  rewrite sin_0; ring.
-  apply le_IZR.
-  apply Rplus_le_reg_l with (IZR x0).
-  rewrite Rplus_0_r.
-  rewrite Ropp_Ropp_IZR.
-  rewrite Rplus_opp_r.
-  left; replace 0 with (IZR 0); [ apply IZR_lt | reflexivity ].
-  assumption.
-  rewrite <- sin_neg.
-  rewrite Ropp_mult_distr_l_reverse.
-  rewrite Ropp_involutive.
-  reflexivity.
-Qed.
-
-Lemma sin_eq_0_0 : forall x:R, sin x = 0 ->  exists k : Z, x = IZR k * PI.
-Proof.
-  intros.
-  assert (H0 := euclidian_division x PI PI_neq0).
-  elim H0; intros q H1.
-  elim H1; intros r H2.
-  exists q.
-  cut (r = 0).
-  intro.
-  elim H2; intros H4 _; rewrite H4; rewrite H3.
-  apply Rplus_0_r.
-  elim H2; intros.
-  rewrite H3 in H.
-  rewrite sin_plus in H.
-  cut (sin (IZR q * PI) = 0).
-  intro.
-  rewrite H5 in H.
-  rewrite Rmult_0_l in H.
-  rewrite Rplus_0_l in H.
-  assert (H6 := Rmult_integral _ _ H).
-  elim H6; intro.
-  assert (H8 := sin2_cos2 (IZR q * PI)).
-  rewrite H5 in H8; rewrite H7 in H8.
-  rewrite Rsqr_0 in H8.
-  rewrite Rplus_0_r in H8.
-  elim R1_neq_R0; symmetry  in |- *; assumption.
-  cut (r = 0 \/ 0 < r < PI).
-  intro; elim H8; intro.
-  assumption.
-  elim H9; intros.
-  assert (H12 := sin_gt_0 _ H10 H11).
-  rewrite H7 in H12; elim (Rlt_irrefl _ H12).
-  rewrite Rabs_right in H4.
-  elim H4; intros.
-  case (Rtotal_order 0 r); intro.
-  right; split; assumption.
-  elim H10; intro.
-  left; symmetry  in |- *; assumption.
-  elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H8 H11)).
-  apply Rle_ge.
-  left; apply PI_RGT_0.
-  apply sin_eq_0_1.
-  exists q; reflexivity.
-Qed.
-
-Lemma cos_eq_0_0 :
-  forall x:R, cos x = 0 ->  exists k : Z, x = IZR k * PI + PI / 2.
-Proof.
-  intros x H; rewrite cos_sin in H; generalize (sin_eq_0_0 (PI / INR 2 + x) H);
-    intro H2; elim H2; intros x0 H3; exists (x0 - Z_of_nat 1)%Z;
-      rewrite <- Z_R_minus; simpl.
-unfold INR in H3. field_simplify [(sym_eq H3)]. field.
-(**
-  ring_simplify.
- (* rewrite (Rmult_comm PI);*) (* old ring compat *)
-        rewrite <- H3; simpl;
-          field;repeat split; discrR.
-*)
-Qed.
-
-Lemma cos_eq_0_1 :
-  forall x:R, (exists k : Z, x = IZR k * PI + PI / 2) -> cos x = 0.
-Proof.
-  intros x H1; rewrite cos_sin; elim H1; intros x0 H2; rewrite H2;
-    replace (PI / 2 + (IZR x0 * PI + PI / 2)) with (IZR x0 * PI + PI).
-  rewrite neg_sin; rewrite <- Ropp_0.
-  apply Ropp_eq_compat; apply sin_eq_0_1; exists x0; reflexivity.
-  pattern PI at 2 in |- *; rewrite (double_var PI); ring.
-Qed.
-
-Lemma sin_eq_O_2PI_0 :
-  forall x:R,
-    0 <= x -> x <= 2 * PI -> sin x = 0 -> x = 0 \/ x = PI \/ x = 2 * PI.
-Proof.
-  intros; generalize (sin_eq_0_0 x H1); intro.
-  elim H2; intros k0 H3.
-  case (Rtotal_order PI x); intro.
-  rewrite H3 in H4; rewrite H3 in H0.
-  right; right.
-  generalize
-    (Rmult_lt_compat_r (/ PI) PI (IZR k0 * PI) (Rinv_0_lt_compat PI PI_RGT_0) H4);
-    rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym.
-  rewrite Rmult_1_r; intro;
-    generalize
-      (Rmult_le_compat_r (/ PI) (IZR k0 * PI) (2 * PI)
-        (Rlt_le 0 (/ PI) (Rinv_0_lt_compat PI PI_RGT_0)) H0);
-      repeat rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym.
-  repeat rewrite Rmult_1_r; intro;
-    generalize (Rplus_lt_compat_l (IZR (-2)) 1 (IZR k0) H5);
-      rewrite <- plus_IZR.
-  replace (IZR (-2) + 1) with (-1).
-  intro; generalize (Rplus_le_compat_l (IZR (-2)) (IZR k0) 2 H6);
-    rewrite <- plus_IZR.
-  replace (IZR (-2) + 2) with 0.
-  intro; cut (-1 < IZR (-2 + k0) < 1).
-  intro; generalize (one_IZR_lt1 (-2 + k0) H9); intro.
-  cut (k0 = 2%Z).
-  intro; rewrite H11 in H3; rewrite H3; simpl in |- *.
-  reflexivity.
-  rewrite <- (Zplus_opp_l 2) in H10; generalize (Zplus_reg_l (-2) k0 2 H10);
-    intro; assumption.
-  split.
-  assumption.
-  apply Rle_lt_trans with 0.
-  assumption.
-  apply Rlt_0_1.
-  simpl in |- *; ring.
-  simpl in |- *; ring.
-  apply PI_neq0.
-  apply PI_neq0.
-  elim H4; intro.
-  right; left.
-  symmetry  in |- *; assumption.
-  left.
-  rewrite H3 in H5; rewrite H3 in H;
-    generalize
-      (Rmult_lt_compat_r (/ PI) (IZR k0 * PI) PI (Rinv_0_lt_compat PI PI_RGT_0)
-        H5); rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym.
-  rewrite Rmult_1_r; intro;
-    generalize
-      (Rmult_le_compat_r (/ PI) 0 (IZR k0 * PI)
-        (Rlt_le 0 (/ PI) (Rinv_0_lt_compat PI PI_RGT_0)) H);
-      repeat rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym.
-  rewrite Rmult_1_r; rewrite Rmult_0_l; intro.
-  cut (-1 < IZR k0 < 1).
-  intro; generalize (one_IZR_lt1 k0 H8); intro; rewrite H9 in H3; rewrite H3;
-    simpl in |- *; apply Rmult_0_l.
-  split.
-  apply Rlt_le_trans with 0.
-  rewrite <- Ropp_0; apply Ropp_gt_lt_contravar; apply Rlt_0_1.
-  assumption.
-  assumption.
-  apply PI_neq0.
-  apply PI_neq0.
-Qed.
-
-Lemma sin_eq_O_2PI_1 :
-  forall x:R,
-    0 <= x -> x <= 2 * PI -> x = 0 \/ x = PI \/ x = 2 * PI -> sin x = 0.
-Proof.
-  intros x H1 H2 H3; elim H3; intro H4;
-    [ rewrite H4; rewrite sin_0; reflexivity
-      | elim H4; intro H5;
-        [ rewrite H5; rewrite sin_PI; reflexivity
-          | rewrite H5; rewrite sin_2PI; reflexivity ] ].
-Qed.
-
-Lemma cos_eq_0_2PI_0 :
-  forall x:R,
-    0 <= x -> x <= 2 * PI -> cos x = 0 -> x = PI / 2 \/ x = 3 * (PI / 2).
-Proof.
-  intros; case (Rtotal_order x (3 * (PI / 2))); intro.
-  rewrite cos_sin in H1.
-  cut (0 <= PI / 2 + x).
-  cut (PI / 2 + x <= 2 * PI).
-  intros; generalize (sin_eq_O_2PI_0 (PI / 2 + x) H4 H3 H1); intros.
-  decompose [or] H5.
-  generalize (Rplus_le_compat_l (PI / 2) 0 x H); rewrite Rplus_0_r; rewrite H6;
-    intro.
-  elim (Rlt_irrefl 0 (Rlt_le_trans 0 (PI / 2) 0 PI2_RGT_0 H7)).
-  left.
-  generalize (Rplus_eq_compat_l (- (PI / 2)) (PI / 2 + x) PI H7).
-  replace (- (PI / 2) + (PI / 2 + x)) with x.
-  replace (- (PI / 2) + PI) with (PI / 2).
-  intro; assumption.
-  pattern PI at 3 in |- *; rewrite (double_var PI); ring.
-  ring.
-  right.
-  generalize (Rplus_eq_compat_l (- (PI / 2)) (PI / 2 + x) (2 * PI) H7).
-  replace (- (PI / 2) + (PI / 2 + x)) with x.
-  replace (- (PI / 2) + 2 * PI) with (3 * (PI / 2)).
-  intro; assumption.
-  rewrite double; pattern PI at 3 4 in |- *; rewrite (double_var PI); ring.
-  ring.
-  left; replace (2 * PI) with (PI / 2 + 3 * (PI / 2)).
-  apply Rplus_lt_compat_l; assumption.
-  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite (double_var PI); ring.
-  apply Rplus_le_le_0_compat.
-  left; unfold Rdiv in |- *; apply Rmult_lt_0_compat.
-  apply PI_RGT_0.
-  apply Rinv_0_lt_compat; prove_sup0.
-  assumption.
-  elim H2; intro.
-  right; assumption.
-  generalize (cos_eq_0_0 x H1); intro; elim H4; intros k0 H5.
-  rewrite H5 in H3; rewrite H5 in H0;
-    generalize
-      (Rplus_lt_compat_l (- (PI / 2)) (3 * (PI / 2)) (IZR k0 * PI + PI / 2) H3);
-      generalize
-        (Rplus_le_compat_l (- (PI / 2)) (IZR k0 * PI + PI / 2) (2 * PI) H0).
-  replace (- (PI / 2) + 3 * (PI / 2)) with PI.
-  replace (- (PI / 2) + (IZR k0 * PI + PI / 2)) with (IZR k0 * PI).
-  replace (- (PI / 2) + 2 * PI) with (3 * (PI / 2)).
-  intros;
-    generalize
-      (Rmult_lt_compat_l (/ PI) PI (IZR k0 * PI) (Rinv_0_lt_compat PI PI_RGT_0)
-        H7);
-      generalize
-        (Rmult_le_compat_l (/ PI) (IZR k0 * PI) (3 * (PI / 2))
-          (Rlt_le 0 (/ PI) (Rinv_0_lt_compat PI PI_RGT_0)) H6).
-  replace (/ PI * (IZR k0 * PI)) with (IZR k0).
-  replace (/ PI * (3 * (PI / 2))) with (3 * / 2).
-  rewrite <- Rinv_l_sym.
-  intros; generalize (Rplus_lt_compat_l (IZR (-2)) 1 (IZR k0) H9);
-    rewrite <- plus_IZR.
-  replace (IZR (-2) + 1) with (-1).
-  intro; generalize (Rplus_le_compat_l (IZR (-2)) (IZR k0) (3 * / 2) H8);
-    rewrite <- plus_IZR.
-  replace (IZR (-2) + 2) with 0.
-  intro; cut (-1 < IZR (-2 + k0) < 1).
-  intro; generalize (one_IZR_lt1 (-2 + k0) H12); intro.
-  cut (k0 = 2%Z).
-  intro; rewrite H14 in H8.
-  assert (Hyp : 0 < 2).
-  prove_sup0.
-  generalize (Rmult_le_compat_l 2 (IZR 2) (3 * / 2) (Rlt_le 0 2 Hyp) H8);
-    simpl in |- *.
-  replace 4 with 4.
-  replace (2 * (3 * / 2)) with 3.
-  intro; cut (3 < 4).
-  intro; elim (Rlt_irrefl 3 (Rlt_le_trans 3 4 3 H16 H15)).
-  generalize (Rplus_lt_compat_l 3 0 1 Rlt_0_1); rewrite Rplus_0_r.
-  replace (3 + 1) with 4.
-  intro; assumption.
-  ring.
-  symmetry  in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m.
-  discrR.
-  ring.
-  rewrite <- (Zplus_opp_l 2) in H13; generalize (Zplus_reg_l (-2) k0 2 H13);
-    intro; assumption.
-  split.
-  assumption.
-  apply Rle_lt_trans with (IZR (-2) + 3 * / 2).
-  assumption.
-  simpl in |- *; replace (-2 + 3 * / 2) with (- (1 * / 2)).
-  apply Rlt_trans with 0.
-  rewrite <- Ropp_0; apply Ropp_lt_gt_contravar.
-  apply Rmult_lt_0_compat;
-    [ apply Rlt_0_1 | apply Rinv_0_lt_compat; prove_sup0 ].
-  apply Rlt_0_1.
-  rewrite Rmult_1_l; apply Rmult_eq_reg_l with 2.
-  rewrite Ropp_mult_distr_r_reverse; rewrite <- Rinv_r_sym.
-  rewrite Rmult_plus_distr_l; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m.
-  ring.
-  discrR.
-  discrR.
-  discrR.
-  simpl in |- *; ring.
-  simpl in |- *; ring.
-  apply PI_neq0.
-  unfold Rdiv in |- *; pattern 3 at 1 in |- *; rewrite (Rmult_comm 3);
-    repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
-  rewrite Rmult_1_l; apply Rmult_comm.
-  apply PI_neq0.
-  symmetry  in |- *; rewrite (Rmult_comm (/ PI)); rewrite Rmult_assoc;
-    rewrite <- Rinv_r_sym.
-  apply Rmult_1_r.
-  apply PI_neq0.
-  rewrite double; pattern PI at 3 4 in |- *; rewrite double_var; ring.
-  ring.
-  pattern PI at 1 in |- *; rewrite double_var; ring.
-Qed.
-
-Lemma cos_eq_0_2PI_1 :
-  forall x:R,
-    0 <= x -> x <= 2 * PI -> x = PI / 2 \/ x = 3 * (PI / 2) -> cos x = 0.
-Proof.
-  intros x H1 H2 H3; elim H3; intro H4;
-    [ rewrite H4; rewrite cos_PI2; reflexivity
-      | rewrite H4; rewrite cos_3PI2; reflexivity ].
-Qed.
+Require Import Classical_Prop.
+Require Import Fourier.
+Require Import Ranalysis1.
+Require Import Rsqrt_def. 
+Require Import PSeries_reg.
+Require Export Rtrigo1.
+Require Export Ratan.
+Require Export Machin.
\ No newline at end of file
diff --git a/theories/Reals/Rtrigo1.v b/theories/Reals/Rtrigo1.v
new file mode 100644
index 00000000..6174ef32
--- /dev/null
+++ b/theories/Reals/Rtrigo1.v
@@ -0,0 +1,1933 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+Require Import Rbase.
+Require Import Rfunctions.
+Require Import SeqSeries.
+Require Export Rtrigo_fun.
+Require Export Rtrigo_def.
+Require Export Rtrigo_alt.
+Require Export Cos_rel.
+Require Export Cos_plus.
+Require Import ZArith_base.
+Require Import Zcomplements.
+Require Import Classical_Prop.
+Require Import Fourier.
+Require Import Ranalysis1.
+Require Import Rsqrt_def. 
+Require Import PSeries_reg.
+
+Local Open Scope nat_scope.
+Local Open Scope R_scope.
+
+Lemma CVN_R_cos :
+  forall fn:nat -> R -> R,
+    fn = (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)) ->
+    CVN_R fn.
+Proof.
+  unfold CVN_R in |- *; intros.
+  cut ((r:R) <> 0).
+  intro hyp_r; unfold CVN_r in |- *.
+  exists (fun n:nat => / INR (fact (2 * n)) * r ^ (2 * n)).
+  cut
+    { l:R |
+        Un_cv
+        (fun n:nat =>
+          sum_f_R0 (fun k:nat => Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
+          n) l }.
+  intro X; elim X; intros.
+  exists x.
+  split.
+  apply p.
+  intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult.
+  rewrite pow_1_abs; rewrite Rmult_1_l.
+  cut (0 < / INR (fact (2 * n))).
+  intro; rewrite (Rabs_right _ (Rle_ge _ _ (Rlt_le _ _ H1))).
+  apply Rmult_le_compat_l.
+  left; apply H1.
+  rewrite <- RPow_abs; apply pow_maj_Rabs.
+  rewrite Rabs_Rabsolu.
+  unfold Boule in H0; rewrite Rminus_0_r in H0.
+  left; apply H0.
+  apply Rinv_0_lt_compat; apply INR_fact_lt_0.
+  apply Alembert_C2.
+  intro; apply Rabs_no_R0.
+  apply prod_neq_R0.
+  apply Rinv_neq_0_compat.
+  apply INR_fact_neq_0.
+  apply pow_nonzero; assumption.
+  assert (H0 := Alembert_cos).
+  unfold cos_n in H0; unfold Un_cv in H0; unfold Un_cv in |- *; intros.
+  cut (0 < eps / Rsqr r).
+  intro; elim (H0 _ H2); intros N0 H3.
+  exists N0; intros.
+  unfold R_dist in |- *; assert (H5 := H3 _ H4).
+  unfold R_dist in H5;
+    replace
+    (Rabs
+      (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
+        Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) with
+    (Rsqr r *
+      Rabs ((-1) ^ S n / INR (fact (2 * S n)) / ((-1) ^ n / INR (fact (2 * n))))).
+  apply Rmult_lt_reg_l with (/ Rsqr r).
+  apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption.
+  pattern (/ Rsqr r) at 1 in |- *; replace (/ Rsqr r) with (Rabs (/ Rsqr r)).
+  rewrite <- Rabs_mult; rewrite Rmult_minus_distr_l; rewrite Rmult_0_r;
+    rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
+  rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); apply H5.
+  unfold Rsqr in |- *; apply prod_neq_R0; assumption.
+  rewrite Rabs_Rinv.
+  rewrite Rabs_right.
+  reflexivity.
+  apply Rle_ge; apply Rle_0_sqr.
+  unfold Rsqr in |- *; apply prod_neq_R0; assumption.
+  rewrite (Rmult_comm (Rsqr r)); unfold Rdiv in |- *; repeat rewrite Rabs_mult;
+    rewrite Rabs_Rabsolu; rewrite pow_1_abs; rewrite Rmult_1_l;
+      repeat rewrite Rmult_assoc; apply Rmult_eq_compat_l.
+  rewrite Rabs_Rinv.
+  rewrite Rabs_mult; rewrite (pow_1_abs n); rewrite Rmult_1_l;
+    rewrite <- Rabs_Rinv.
+  rewrite Rinv_involutive.
+  rewrite Rinv_mult_distr.
+  rewrite Rabs_Rinv.
+  rewrite Rinv_involutive.
+  rewrite (Rmult_comm (Rabs (Rabs (r ^ (2 * S n))))); rewrite Rabs_mult;
+    rewrite Rabs_Rabsolu; rewrite Rmult_assoc; apply Rmult_eq_compat_l.
+  rewrite Rabs_Rinv.
+  do 2 rewrite Rabs_Rabsolu; repeat rewrite Rabs_right.
+  replace (r ^ (2 * S n)) with (r ^ (2 * n) * r * r).
+  repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
+  unfold Rsqr in |- *; ring.
+  apply pow_nonzero; assumption.
+  replace (2 * S n)%nat with (S (S (2 * n))).
+  simpl in |- *; ring.
+  ring.
+  apply Rle_ge; apply pow_le; left; apply (cond_pos r).
+  apply Rle_ge; apply pow_le; left; apply (cond_pos r).
+  apply Rabs_no_R0; apply pow_nonzero; assumption.
+  apply Rabs_no_R0; apply INR_fact_neq_0.
+  apply INR_fact_neq_0.
+  apply Rabs_no_R0; apply Rinv_neq_0_compat; apply INR_fact_neq_0.
+  apply Rabs_no_R0; apply pow_nonzero; assumption.
+  apply INR_fact_neq_0.
+  apply Rinv_neq_0_compat; apply INR_fact_neq_0.
+  apply prod_neq_R0.
+  apply pow_nonzero; discrR.
+  apply Rinv_neq_0_compat; apply INR_fact_neq_0.
+  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  apply H1.
+  apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption.
+  assert (H0 := cond_pos r); red in |- *; intro; rewrite H1 in H0;
+    elim (Rlt_irrefl _ H0).
+Qed.
+
+(**********)
+Lemma continuity_cos : continuity cos.
+Proof.
+  set (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)).
+  cut (CVN_R fn).
+  intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }).
+  intro cv; cut (forall n:nat, continuity (fn n)).
+  intro; cut (forall x:R, cos x = SFL fn cv x).
+  intro; cut (continuity (SFL fn cv) -> continuity cos).
+  intro; apply H1.
+  apply SFL_continuity; assumption.
+  unfold continuity in |- *; unfold continuity_pt in |- *;
+    unfold continue_in in |- *; unfold limit1_in in |- *;
+      unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *;
+        intros.
+  elim (H1 x _ H2); intros.
+  exists x0; intros.
+  elim H3; intros.
+  split.
+  apply H4.
+  intros; rewrite (H0 x); rewrite (H0 x1); apply H5; apply H6.
+  intro; unfold cos, SFL in |- *.
+  case (cv x); case (exist_cos (Rsqr x)); intros.
+  symmetry  in |- *; eapply UL_sequence.
+  apply u.
+  unfold cos_in in c; unfold infinite_sum in c; unfold Un_cv in |- *; intros.
+  elim (c _ H0); intros N0 H1.
+  exists N0; intros.
+  unfold R_dist in H1; unfold R_dist, SP in |- *.
+  replace (sum_f_R0 (fun k:nat => fn k x) n) with
+  (sum_f_R0 (fun i:nat => cos_n i * Rsqr x ^ i) n).
+  apply H1; assumption.
+  apply sum_eq; intros.
+  unfold cos_n, fn in |- *; apply Rmult_eq_compat_l.
+  unfold Rsqr in |- *; rewrite pow_sqr; reflexivity.
+  intro; unfold fn in |- *;
+    replace (fun x:R => (-1) ^ n / INR (fact (2 * n)) * x ^ (2 * n)) with
+    (fct_cte ((-1) ^ n / INR (fact (2 * n))) * pow_fct (2 * n))%F;
+    [ idtac | reflexivity ].
+  apply continuity_mult.
+  apply derivable_continuous; apply derivable_const.
+  apply derivable_continuous; apply (derivable_pow (2 * n)).
+  apply CVN_R_CVS; apply X.
+  apply CVN_R_cos; unfold fn in |- *; reflexivity.
+Qed.
+
+Lemma sin_gt_cos_7_8 : sin (7 / 8) > cos (7 / 8).
+Proof. 
+assert (lo1 : 0 <= 7/8) by fourier.
+assert (up1 : 7/8 <= 4) by fourier.
+assert (lo : -2 <= 7/8) by fourier.
+assert (up : 7/8 <= 2) by fourier.
+destruct (pre_sin_bound _ 0 lo1 up1) as [lower _ ].
+destruct (pre_cos_bound _ 0 lo up) as [_ upper].
+apply Rle_lt_trans with (1 := upper).
+apply Rlt_le_trans with (2 := lower).
+unfold cos_approx, sin_approx.
+simpl sum_f_R0; replace 7 with (IZR 7) by (simpl; field).
+replace 8 with (IZR 8) by (simpl; field).
+unfold cos_term, sin_term; simpl fact; rewrite !INR_IZR_INZ.
+simpl plus; simpl mult.
+field_simplify;
+  try (repeat apply conj; apply not_eq_sym, Rlt_not_eq, (IZR_lt 0); reflexivity).
+unfold Rminus; rewrite !pow_IZR, <- !mult_IZR, <- !opp_IZR, <- ?plus_IZR.
+match goal with 
+  |- IZR ?a / ?b < ?c / ?d =>
+  apply Rmult_lt_reg_r with d;[apply (IZR_lt 0); reflexivity |
+    unfold Rdiv at 2; rewrite Rmult_assoc, Rinv_l, Rmult_1_r, Rmult_comm;
+     [ |apply not_eq_sym, Rlt_not_eq, (IZR_lt 0); reflexivity ]];
+  apply Rmult_lt_reg_r with b;[apply (IZR_lt 0); reflexivity | ]
+end.
+unfold Rdiv; rewrite !Rmult_assoc, Rinv_l, Rmult_1_r;
+ [ | apply not_eq_sym, Rlt_not_eq, (IZR_lt 0); reflexivity].
+repeat (rewrite <- !plus_IZR || rewrite <- !mult_IZR).
+apply IZR_lt; reflexivity.
+Qed.
+
+Definition PI_2_aux : {z | 7/8 <= z <= 7/4 /\ -cos z = 0}.
+assert (cc : continuity (fun r =>- cos r)).
+ apply continuity_opp, continuity_cos.
+assert (cvp : 0 < cos (7/8)).
+ assert (int78 : -2 <= 7/8 <= 2) by (split; fourier).
+ destruct int78 as [lower upper].
+ case (pre_cos_bound _ 0 lower upper).
+ unfold cos_approx; simpl sum_f_R0; unfold cos_term.
+ intros cl _; apply Rlt_le_trans with (2 := cl); simpl.
+ fourier.
+assert (cun : cos (7/4) < 0).
+ replace (7/4) with (7/8 + 7/8) by field.
+ rewrite cos_plus.
+ apply Rlt_minus; apply Rsqr_incrst_1.
+   exact sin_gt_cos_7_8.
+  apply Rlt_le; assumption.
+ apply Rlt_le; apply Rlt_trans with (1 := cvp); exact sin_gt_cos_7_8.
+apply IVT; auto; fourier.
+Qed.
+
+Definition PI2 := proj1_sig PI_2_aux.
+
+Definition PI := 2 * PI2.
+
+Lemma cos_pi2 : cos PI2 = 0.
+unfold PI2; case PI_2_aux; simpl.
+intros x [_ q]; rewrite <- (Ropp_involutive (cos x)), q; apply Ropp_0.
+Qed.
+
+Lemma pi2_int : 7/8 <= PI2 <= 7/4.
+unfold PI2; case PI_2_aux; simpl; tauto.
+Qed.
+
+(**********)
+Lemma cos_minus : forall x y:R, cos (x - y) = cos x * cos y + sin x * sin y.
+Proof.
+  intros; unfold Rminus in |- *; rewrite cos_plus.
+  rewrite <- cos_sym; rewrite sin_antisym; ring.
+Qed.
+
+(**********)
+Lemma sin2_cos2 : forall x:R, Rsqr (sin x) + Rsqr (cos x) = 1.
+Proof.
+  intro; unfold Rsqr in |- *; rewrite Rplus_comm; rewrite <- (cos_minus x x);
+    unfold Rminus in |- *; rewrite Rplus_opp_r; apply cos_0.
+Qed.
+
+Lemma cos2 : forall x:R, Rsqr (cos x) = 1 - Rsqr (sin x).
+Proof.
+  intros x; rewrite <- (sin2_cos2 x); ring.
+Qed.
+
+Lemma sin2 : forall x:R, Rsqr (sin x) = 1 - Rsqr (cos x).
+Proof.
+  intro x; generalize (cos2 x); intro H1; rewrite H1.
+  unfold Rminus in |- *; rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
+    rewrite Rplus_opp_r; rewrite Rplus_0_l; symmetry  in |- *;
+      apply Ropp_involutive.
+Qed.
+
+(**********)
+Lemma cos_PI2 : cos (PI / 2) = 0.
+Proof.
+ unfold PI; generalize cos_pi2; replace ((2 * PI2)/2) with PI2 by field; tauto.
+Qed.
+
+Lemma sin_pos_tech : forall x, 0 < x < 2 -> 0 < sin x. 
+intros x [int1 int2].
+assert (lo : 0 <= x) by (apply Rlt_le; assumption).
+assert (up : x <= 4) by (apply Rlt_le, Rlt_trans with (1:=int2); fourier).
+destruct (pre_sin_bound _ 0 lo up) as [t _]; clear lo up.
+apply Rlt_le_trans with (2:= t); clear t.
+unfold sin_approx; simpl sum_f_R0; unfold sin_term; simpl.
+match goal with |- _ < ?a =>
+  replace a with (x * (1 - x^2/6)) by (simpl; field)
+end.
+assert (t' : x ^ 2 <= 4).
+ replace 4 with (2 ^ 2) by field.
+ apply (pow_incr x 2); split; apply Rlt_le; assumption.
+apply Rmult_lt_0_compat;[assumption | fourier ].
+Qed.
+
+Lemma sin_PI2 : sin (PI / 2) = 1.
+replace (PI / 2) with PI2 by (unfold PI; field).
+assert (int' : 0 < PI2 < 2).
+ destruct pi2_int; split; fourier.
+assert (lo2 := sin_pos_tech PI2 int').
+assert (t2 : Rabs (sin PI2) = 1).
+ rewrite <- Rabs_R1; apply Rsqr_eq_abs_0.
+ rewrite Rsqr_1, sin2, cos_pi2, Rsqr_0, Rminus_0_r; reflexivity.
+revert t2; rewrite Rabs_pos_eq;[| apply Rlt_le]; tauto.
+Qed.
+
+Lemma PI_RGT_0 : PI > 0.
+Proof. unfold PI; destruct pi2_int; fourier. Qed.
+
+Lemma PI_4 : PI <= 4.
+Proof. unfold PI; destruct pi2_int; fourier. Qed.
+
+(**********)
+Lemma PI_neq0 : PI <> 0.
+Proof.
+  red in |- *; intro; assert (H0 := PI_RGT_0); rewrite H in H0;
+    elim (Rlt_irrefl _ H0).
+Qed.
+
+
+(**********)
+Lemma cos_PI : cos PI = -1.
+Proof.
+  replace PI with (PI / 2 + PI / 2).
+  rewrite cos_plus.
+  rewrite sin_PI2; rewrite cos_PI2.
+  ring.
+  symmetry  in |- *; apply double_var.
+Qed.
+
+Lemma sin_PI : sin PI = 0.
+Proof.
+  assert (H := sin2_cos2 PI).
+  rewrite cos_PI in H.
+  rewrite <- Rsqr_neg in H.
+  rewrite Rsqr_1 in H.
+  cut (Rsqr (sin PI) = 0).
+  intro; apply (Rsqr_eq_0 _ H0).
+  apply Rplus_eq_reg_l with 1.
+  rewrite Rplus_0_r; rewrite Rplus_comm; exact H.
+Qed.
+
+Lemma sin_bound : forall (a : R) (n : nat), 0 <= a -> a <= PI ->
+       sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)).
+Proof.
+intros a n a0 api; apply pre_sin_bound.
+ assumption.
+apply Rle_trans with (1:= api) (2 := PI_4).
+Qed.
+
+Lemma cos_bound : forall (a : R) (n : nat), - PI / 2 <= a -> a <= PI / 2 ->
+       cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1)).
+Proof.
+intros a n lower upper; apply pre_cos_bound.
+ apply Rle_trans with (2 := lower).
+ apply Rmult_le_reg_r with 2; [fourier |].
+ replace ((-PI/2) * 2) with (-PI) by field.
+ assert (t := PI_4); fourier.
+apply Rle_trans with (1 := upper).
+apply Rmult_le_reg_r with 2; [fourier | ].
+replace ((PI/2) * 2) with PI by field.
+generalize PI_4; intros; fourier.
+Qed.
+(**********)
+Lemma neg_cos : forall x:R, cos (x + PI) = - cos x.
+Proof.
+  intro x; rewrite cos_plus; rewrite sin_PI; rewrite cos_PI; ring.
+Qed.
+
+(**********)
+Lemma sin_cos : forall x:R, sin x = - cos (PI / 2 + x).
+Proof.
+  intro x; rewrite cos_plus; rewrite sin_PI2; rewrite cos_PI2; ring.
+Qed.
+
+(**********)
+Lemma sin_plus : forall x y:R, sin (x + y) = sin x * cos y + cos x * sin y.
+Proof.
+  intros.
+  rewrite (sin_cos (x + y)).
+  replace (PI / 2 + (x + y)) with (PI / 2 + x + y); [ rewrite cos_plus | ring ].
+  rewrite (sin_cos (PI / 2 + x)).
+  replace (PI / 2 + (PI / 2 + x)) with (x + PI).
+  rewrite neg_cos.
+  replace (cos (PI / 2 + x)) with (- sin x).
+  ring.
+  rewrite sin_cos; rewrite Ropp_involutive; reflexivity.
+  pattern PI at 1 in |- *; rewrite (double_var PI); ring.
+Qed.
+
+Lemma sin_minus : forall x y:R, sin (x - y) = sin x * cos y - cos x * sin y.
+Proof.
+  intros; unfold Rminus in |- *; rewrite sin_plus.
+  rewrite <- cos_sym; rewrite sin_antisym; ring.
+Qed.
+
+(**********)
+Definition tan (x:R) : R := sin x / cos x.
+
+Lemma tan_plus :
+  forall x y:R,
+    cos x <> 0 ->
+    cos y <> 0 ->
+    cos (x + y) <> 0 ->
+    1 - tan x * tan y <> 0 ->
+    tan (x + y) = (tan x + tan y) / (1 - tan x * tan y).
+Proof.
+  intros; unfold tan in |- *; rewrite sin_plus; rewrite cos_plus;
+    unfold Rdiv in |- *;
+      replace (cos x * cos y - sin x * sin y) with
+      (cos x * cos y * (1 - sin x * / cos x * (sin y * / cos y))).
+  rewrite Rinv_mult_distr.
+  repeat rewrite <- Rmult_assoc;
+    replace ((sin x * cos y + cos x * sin y) * / (cos x * cos y)) with
+    (sin x * / cos x + sin y * / cos y).
+  reflexivity.
+  rewrite Rmult_plus_distr_r; rewrite Rinv_mult_distr.
+  repeat rewrite Rmult_assoc; repeat rewrite (Rmult_comm (sin x));
+    repeat rewrite <- Rmult_assoc.
+  repeat rewrite Rinv_r_simpl_m; [ reflexivity | assumption | assumption ].
+  assumption.
+  assumption.
+  apply prod_neq_R0; assumption.
+  assumption.
+  unfold Rminus in |- *; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r;
+    apply Rplus_eq_compat_l; repeat rewrite Rmult_assoc;
+      rewrite (Rmult_comm (sin x)); rewrite (Rmult_comm (cos y));
+        rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite <- Rmult_assoc;
+          rewrite <- Rinv_r_sym.
+  rewrite Rmult_1_l; rewrite (Rmult_comm (sin x));
+    rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite Rmult_assoc;
+      apply Rmult_eq_compat_l; rewrite (Rmult_comm (/ cos y));
+        rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
+  apply Rmult_1_r.
+  assumption.
+  assumption.
+Qed.
+
+(*******************************************************)
+(** * Some properties of cos, sin and tan              *)
+(*******************************************************)
+
+Lemma sin_2a : forall x:R, sin (2 * x) = 2 * sin x * cos x.
+Proof.
+  intro x; rewrite double; rewrite sin_plus.
+  rewrite <- (Rmult_comm (sin x)); symmetry  in |- *; rewrite Rmult_assoc;
+    apply double.
+Qed.
+
+Lemma cos_2a : forall x:R, cos (2 * x) = cos x * cos x - sin x * sin x.
+Proof.
+  intro x; rewrite double; apply cos_plus.
+Qed.
+
+Lemma cos_2a_cos : forall x:R, cos (2 * x) = 2 * cos x * cos x - 1.
+Proof.
+  intro x; rewrite double; unfold Rminus in |- *; rewrite Rmult_assoc;
+    rewrite cos_plus; generalize (sin2_cos2 x); rewrite double;
+      intro H1; rewrite <- H1; ring_Rsqr.
+Qed.
+
+Lemma cos_2a_sin : forall x:R, cos (2 * x) = 1 - 2 * sin x * sin x.
+Proof.
+  intro x; rewrite Rmult_assoc; unfold Rminus in |- *; repeat rewrite double.
+  generalize (sin2_cos2 x); intro H1; rewrite <- H1; rewrite cos_plus;
+    ring_Rsqr.
+Qed.
+
+Lemma tan_2a :
+  forall x:R,
+    cos x <> 0 ->
+    cos (2 * x) <> 0 ->
+    1 - tan x * tan x <> 0 -> tan (2 * x) = 2 * tan x / (1 - tan x * tan x).
+Proof.
+  repeat rewrite double; intros; repeat rewrite double; rewrite double in H0;
+    apply tan_plus; assumption.
+Qed.
+
+Lemma sin_neg : forall x:R, sin (- x) = - sin x.
+Proof.
+  apply sin_antisym.
+Qed.
+
+Lemma cos_neg : forall x:R, cos (- x) = cos x.
+Proof.
+  intro; symmetry  in |- *; apply cos_sym.
+Qed.
+
+Lemma tan_0 : tan 0 = 0.
+Proof.
+  unfold tan in |- *; rewrite sin_0; rewrite cos_0.
+  unfold Rdiv in |- *; apply Rmult_0_l.
+Qed.
+
+Lemma tan_neg : forall x:R, tan (- x) = - tan x.
+Proof.
+  intros x; unfold tan in |- *; rewrite sin_neg; rewrite cos_neg;
+    unfold Rdiv in |- *.
+  apply Ropp_mult_distr_l_reverse.
+Qed.
+
+Lemma tan_minus :
+  forall x y:R,
+    cos x <> 0 ->
+    cos y <> 0 ->
+    cos (x - y) <> 0 ->
+    1 + tan x * tan y <> 0 ->
+    tan (x - y) = (tan x - tan y) / (1 + tan x * tan y).
+Proof.
+  intros; unfold Rminus in |- *; rewrite tan_plus.
+  rewrite tan_neg; unfold Rminus in |- *; rewrite <- Ropp_mult_distr_l_reverse;
+    rewrite Rmult_opp_opp; reflexivity.
+  assumption.
+  rewrite cos_neg; assumption.
+  assumption.
+  rewrite tan_neg; unfold Rminus in |- *; rewrite <- Ropp_mult_distr_l_reverse;
+    rewrite Rmult_opp_opp; assumption.
+Qed.
+
+Lemma cos_3PI2 : cos (3 * (PI / 2)) = 0.
+Proof.
+  replace (3 * (PI / 2)) with (PI + PI / 2).
+  rewrite cos_plus; rewrite sin_PI; rewrite cos_PI2; ring.
+  pattern PI at 1 in |- *; rewrite (double_var PI).
+  ring.
+Qed.
+
+Lemma sin_2PI : sin (2 * PI) = 0.
+Proof.
+  rewrite sin_2a; rewrite sin_PI; ring.
+Qed.
+
+Lemma cos_2PI : cos (2 * PI) = 1.
+Proof.
+  rewrite cos_2a; rewrite sin_PI; rewrite cos_PI; ring.
+Qed.
+
+Lemma neg_sin : forall x:R, sin (x + PI) = - sin x.
+Proof.
+  intro x; rewrite sin_plus; rewrite sin_PI; rewrite cos_PI; ring.
+Qed.
+
+Lemma sin_PI_x : forall x:R, sin (PI - x) = sin x.
+Proof.
+  intro x; rewrite sin_minus; rewrite sin_PI; rewrite cos_PI; rewrite Rmult_0_l;
+    unfold Rminus in |- *; rewrite Rplus_0_l; rewrite Ropp_mult_distr_l_reverse;
+      rewrite Ropp_involutive; apply Rmult_1_l.
+Qed.
+
+Lemma sin_period : forall (x:R) (k:nat), sin (x + 2 * INR k * PI) = sin x.
+Proof.
+  intros x k; induction  k as [| k Hreck].
+  simpl in |- *;  ring_simplify (x + 2 * 0 * PI).
+  trivial.
+
+  replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI).
+  rewrite sin_plus in |- *; rewrite sin_2PI in |- *; rewrite cos_2PI in |- *.
+  ring_simplify; trivial.
+  rewrite S_INR in |- *;  ring.
+Qed.
+
+Lemma cos_period : forall (x:R) (k:nat), cos (x + 2 * INR k * PI) = cos x.
+Proof.
+  intros x k; induction  k as [| k Hreck].
+  simpl in |- *;  ring_simplify (x + 2 * 0 * PI).
+  trivial.
+
+  replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI).
+  rewrite cos_plus in |- *; rewrite sin_2PI in |- *; rewrite cos_2PI in |- *.
+  ring_simplify; trivial.
+  rewrite S_INR in |- *;  ring.
+Qed.
+
+Lemma sin_shift : forall x:R, sin (PI / 2 - x) = cos x.
+Proof.
+  intro x; rewrite sin_minus; rewrite sin_PI2; rewrite cos_PI2; ring.
+Qed.
+
+Lemma cos_shift : forall x:R, cos (PI / 2 - x) = sin x.
+Proof.
+  intro x; rewrite cos_minus; rewrite sin_PI2; rewrite cos_PI2; ring.
+Qed.
+
+Lemma cos_sin : forall x:R, cos x = sin (PI / 2 + x).
+Proof.
+  intro x; rewrite sin_plus; rewrite sin_PI2; rewrite cos_PI2; ring.
+Qed.
+
+Lemma PI2_RGT_0 : 0 < PI / 2.
+Proof.
+  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+    [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup ].
+Qed.
+
+Lemma SIN_bound : forall x:R, -1 <= sin x <= 1.
+Proof.
+  intro; case (Rle_dec (-1) (sin x)); intro.
+  case (Rle_dec (sin x) 1); intro.
+  split; assumption.
+  cut (1 < sin x).
+  intro;
+    generalize
+      (Rsqr_incrst_1 1 (sin x) H (Rlt_le 0 1 Rlt_0_1)
+        (Rlt_le 0 (sin x) (Rlt_trans 0 1 (sin x) Rlt_0_1 H)));
+      rewrite Rsqr_1; intro; rewrite sin2 in H0; unfold Rminus in H0;
+        generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0);
+          repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l;
+            rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1;
+              generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1);
+                repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x));
+                  intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)).
+  auto with real.
+  cut (sin x < -1).
+  intro; generalize (Ropp_lt_gt_contravar (sin x) (-1) H);
+    rewrite Ropp_involutive; clear H; intro;
+      generalize
+        (Rsqr_incrst_1 1 (- sin x) H (Rlt_le 0 1 Rlt_0_1)
+          (Rlt_le 0 (- sin x) (Rlt_trans 0 1 (- sin x) Rlt_0_1 H)));
+        rewrite Rsqr_1; intro; rewrite <- Rsqr_neg in H0;
+          rewrite sin2 in H0; unfold Rminus in H0;
+            generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0);
+              repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l;
+                rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1;
+                  generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1);
+                    repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x));
+                      intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)).
+  auto with real.
+Qed.
+
+Lemma COS_bound : forall x:R, -1 <= cos x <= 1.
+Proof.
+  intro; rewrite <- sin_shift; apply SIN_bound.
+Qed.
+
+Lemma cos_sin_0 : forall x:R, ~ (cos x = 0 /\ sin x = 0).
+Proof.
+  intro; red in |- *; intro; elim H; intros; generalize (sin2_cos2 x); intro;
+    rewrite H0 in H2; rewrite H1 in H2; repeat rewrite Rsqr_0 in H2;
+      rewrite Rplus_0_r in H2; generalize Rlt_0_1; intro;
+        rewrite <- H2 in H3; elim (Rlt_irrefl 0 H3).
+Qed.
+
+Lemma cos_sin_0_var : forall x:R, cos x <> 0 \/ sin x <> 0.
+Proof.
+  intros x.
+  destruct (Req_dec (cos x) 0). 2: now left.
+  right. intros H'.
+  apply (cos_sin_0 x).
+  now split.
+Qed.
+
+(*****************************************************************)
+(** * Using series definitions of cos and sin                    *)
+(*****************************************************************)
+
+Definition sin_lb (a:R) : R := sin_approx a 3.
+Definition sin_ub (a:R) : R := sin_approx a 4.
+Definition cos_lb (a:R) : R := cos_approx a 3.
+Definition cos_ub (a:R) : R := cos_approx a 4.
+
+Lemma sin_lb_gt_0 : forall a:R, 0 < a -> a <= PI / 2 -> 0 < sin_lb a.
+Proof.
+  intros.
+  unfold sin_lb in |- *; unfold sin_approx in |- *; unfold sin_term in |- *.
+  set (Un := fun i:nat => a ^ (2 * i + 1) / INR (fact (2 * i + 1))).
+  replace
+  (sum_f_R0
+    (fun i:nat => (-1) ^ i * (a ^ (2 * i + 1) / INR (fact (2 * i + 1)))) 3)
+    with (sum_f_R0 (fun i:nat => (-1) ^ i * Un i) 3);
+      [ idtac | apply sum_eq; intros; unfold Un in |- *; reflexivity ].
+  cut (forall n:nat, Un (S n) < Un n).
+  intro; simpl in |- *.
+  repeat rewrite Rmult_1_l; repeat rewrite Rmult_1_r;
+    replace (-1 * Un 1%nat) with (- Un 1%nat); [ idtac | ring ];
+    replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ];
+    replace (-1 * (-1 * -1) * Un 3%nat) with (- Un 3%nat);
+    [ idtac | ring ];
+    replace (Un 0%nat + - Un 1%nat + Un 2%nat + - Un 3%nat) with
+    (Un 0%nat - Un 1%nat + (Un 2%nat - Un 3%nat)); [ idtac | ring ].
+  apply Rplus_lt_0_compat.
+  unfold Rminus in |- *; apply Rplus_lt_reg_r with (Un 1%nat);
+    rewrite Rplus_0_r; rewrite (Rplus_comm (Un 1%nat));
+      rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
+        apply H1.
+  unfold Rminus in |- *; apply Rplus_lt_reg_r with (Un 3%nat);
+    rewrite Rplus_0_r; rewrite (Rplus_comm (Un 3%nat));
+      rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
+        apply H1.
+  intro; unfold Un in |- *.
+  cut ((2 * S n + 1)%nat = (2 * n + 1 + 2)%nat).
+  intro; rewrite H1.
+  rewrite pow_add; unfold Rdiv in |- *; rewrite Rmult_assoc;
+    apply Rmult_lt_compat_l.
+  apply pow_lt; assumption.
+  rewrite <- H1; apply Rmult_lt_reg_l with (INR (fact (2 * n + 1))).
+  apply lt_INR_0; apply neq_O_lt.
+  assert (H2 := fact_neq_0 (2 * n + 1)).
+  red in |- *; intro; elim H2; symmetry  in |- *; assumption.
+  rewrite <- Rinv_r_sym.
+  apply Rmult_lt_reg_l with (INR (fact (2 * S n + 1))).
+  apply lt_INR_0; apply neq_O_lt.
+  assert (H2 := fact_neq_0 (2 * S n + 1)).
+  red in |- *; intro; elim H2; symmetry  in |- *; assumption.
+  rewrite (Rmult_comm (INR (fact (2 * S n + 1)))); repeat rewrite Rmult_assoc;
+    rewrite <- Rinv_l_sym.
+  do 2 rewrite Rmult_1_r; apply Rle_lt_trans with (INR (fact (2 * n + 1)) * 4).
+  apply Rmult_le_compat_l.
+  replace 0 with (INR 0); [ idtac | reflexivity ]; apply le_INR; apply le_O_n.
+  simpl in |- *; rewrite Rmult_1_r; replace 4 with (Rsqr 2);
+    [ idtac | ring_Rsqr ]; replace (a * a) with (Rsqr a);
+    [ idtac | reflexivity ]; apply Rsqr_incr_1.
+  apply Rle_trans with (PI / 2);
+    [ assumption
+      | unfold Rdiv in |- *; apply Rmult_le_reg_l with 2;
+        [ prove_sup0
+          | rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m;
+            [ replace 4 with 4; [ apply PI_4 | ring ] | discrR ] ] ].
+  left; assumption.
+  left; prove_sup0.
+  rewrite H1; replace (2 * n + 1 + 2)%nat with (S (S (2 * n + 1))).
+  do 2 rewrite fact_simpl; do 2 rewrite mult_INR.
+  repeat rewrite <- Rmult_assoc.
+  rewrite <- (Rmult_comm (INR (fact (2 * n + 1)))).
+  rewrite Rmult_assoc.
+  apply Rmult_lt_compat_l.
+  apply lt_INR_0; apply neq_O_lt.
+  assert (H2 := fact_neq_0 (2 * n + 1)).
+  red in |- *; intro; elim H2; symmetry  in |- *; assumption.
+  do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; set (x := INR n);
+    unfold INR in |- *.
+  replace ((2 * x + 1 + 1 + 1) * (2 * x + 1 + 1)) with (4 * x * x + 10 * x + 6);
+  [ idtac | ring ].
+  apply Rplus_lt_reg_r with (-4); rewrite Rplus_opp_l;
+    replace (-4 + (4 * x * x + 10 * x + 6)) with (4 * x * x + 10 * x + 2);
+    [ idtac | ring ].
+  apply Rplus_le_lt_0_compat.
+  cut (0 <= x).
+  intro; apply Rplus_le_le_0_compat; repeat apply Rmult_le_pos;
+    assumption || left; prove_sup.
+  unfold x in |- *; replace 0 with (INR 0);
+    [ apply le_INR; apply le_O_n | reflexivity ].
+  prove_sup0.
+  ring.
+  apply INR_fact_neq_0.
+  apply INR_fact_neq_0.
+  ring.
+Qed.
+
+Lemma SIN : forall a:R, 0 <= a -> a <= PI -> sin_lb a <= sin a <= sin_ub a.
+  intros; unfold sin_lb, sin_ub in |- *; apply (sin_bound a 1 H H0).
+Qed.
+
+Lemma COS :
+  forall a:R, - PI / 2 <= a -> a <= PI / 2 -> cos_lb a <= cos a <= cos_ub a.
+  intros; unfold cos_lb, cos_ub in |- *; apply (cos_bound a 1 H H0).
+Qed.
+
+(**********)
+Lemma _PI2_RLT_0 : - (PI / 2) < 0.
+Proof.
+  rewrite <- Ropp_0; apply Ropp_lt_contravar; apply PI2_RGT_0.
+Qed.
+
+Lemma PI4_RLT_PI2 : PI / 4 < PI / 2.
+Proof.
+  unfold Rdiv in |- *; apply Rmult_lt_compat_l.
+  apply PI_RGT_0.
+  apply Rinv_lt_contravar.
+  apply Rmult_lt_0_compat; prove_sup0.
+  pattern 2 at 1 in |- *; rewrite <- Rplus_0_r.
+  replace 4 with (2 + 2); [ apply Rplus_lt_compat_l; prove_sup0 | ring ].
+Qed.
+
+Lemma PI2_Rlt_PI : PI / 2 < PI.
+Proof.
+  unfold Rdiv in |- *; pattern PI at 2 in |- *; rewrite <- Rmult_1_r.
+  apply Rmult_lt_compat_l.
+  apply PI_RGT_0.
+  pattern 1 at 3 in |- *; rewrite <- Rinv_1; apply Rinv_lt_contravar.
+  rewrite Rmult_1_l; prove_sup0.
+  pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
+    apply Rlt_0_1.
+Qed.
+
+(***************************************************)
+(** * Increasing and decreasing of [cos] and [sin] *)
+(***************************************************)
+Theorem sin_gt_0 : forall x:R, 0 < x -> x < PI -> 0 < sin x.
+Proof.
+  intros; elim (SIN x (Rlt_le 0 x H) (Rlt_le x PI H0)); intros H1 _;
+    case (Rtotal_order x (PI / 2)); intro H2.
+  apply Rlt_le_trans with (sin_lb x).
+  apply sin_lb_gt_0; [ assumption | left; assumption ].
+  assumption.
+  elim H2; intro H3.
+  rewrite H3; rewrite sin_PI2; apply Rlt_0_1.
+  rewrite <- sin_PI_x; generalize (Ropp_gt_lt_contravar x (PI / 2) H3);
+    intro H4; generalize (Rplus_lt_compat_l PI (- x) (- (PI / 2)) H4).
+  replace (PI + - x) with (PI - x).
+  replace (PI + - (PI / 2)) with (PI / 2).
+  intro H5; generalize (Ropp_lt_gt_contravar x PI H0); intro H6;
+    change (- PI < - x) in H6; generalize (Rplus_lt_compat_l PI (- PI) (- x) H6).
+  rewrite Rplus_opp_r.
+  replace (PI + - x) with (PI - x).
+  intro H7;
+    elim
+      (SIN (PI - x) (Rlt_le 0 (PI - x) H7)
+        (Rlt_le (PI - x) PI (Rlt_trans (PI - x) (PI / 2) PI H5 PI2_Rlt_PI)));
+      intros H8 _;
+        generalize (sin_lb_gt_0 (PI - x) H7 (Rlt_le (PI - x) (PI / 2) H5));
+          intro H9; apply (Rlt_le_trans 0 (sin_lb (PI - x)) (sin (PI - x)) H9 H8).
+  reflexivity.
+  pattern PI at 2 in |- *; rewrite double_var; ring.
+  reflexivity.
+Qed.
+
+Theorem cos_gt_0 : forall x:R, - (PI / 2) < x -> x < PI / 2 -> 0 < cos x.
+Proof.
+  intros; rewrite cos_sin;
+    generalize (Rplus_lt_compat_l (PI / 2) (- (PI / 2)) x H).
+  rewrite Rplus_opp_r; intro H1;
+    generalize (Rplus_lt_compat_l (PI / 2) x (PI / 2) H0);
+      rewrite <- double_var; intro H2; apply (sin_gt_0 (PI / 2 + x) H1 H2).
+Qed.
+
+Lemma sin_ge_0 : forall x:R, 0 <= x -> x <= PI -> 0 <= sin x.
+Proof.
+  intros x H1 H2; elim H1; intro H3;
+    [ elim H2; intro H4;
+      [ left; apply (sin_gt_0 x H3 H4)
+        | rewrite H4; right; symmetry  in |- *; apply sin_PI ]
+      | rewrite <- H3; right; symmetry  in |- *; apply sin_0 ].
+Qed.
+
+Lemma cos_ge_0 : forall x:R, - (PI / 2) <= x -> x <= PI / 2 -> 0 <= cos x.
+Proof.
+  intros x H1 H2; elim H1; intro H3;
+    [ elim H2; intro H4;
+      [ left; apply (cos_gt_0 x H3 H4)
+        | rewrite H4; right; symmetry  in |- *; apply cos_PI2 ]
+      | rewrite <- H3; rewrite cos_neg; right; symmetry  in |- *; apply cos_PI2 ].
+Qed.
+
+Lemma sin_le_0 : forall x:R, PI <= x -> x <= 2 * PI -> sin x <= 0.
+Proof.
+  intros x H1 H2; apply Rge_le; rewrite <- Ropp_0;
+    rewrite <- (Ropp_involutive (sin x)); apply Ropp_le_ge_contravar;
+      rewrite <- neg_sin; replace (x + PI) with (x - PI + 2 * INR 1 * PI);
+        [ rewrite (sin_period (x - PI) 1); apply sin_ge_0;
+          [ replace (x - PI) with (x + - PI);
+            [ rewrite Rplus_comm; replace 0 with (- PI + PI);
+              [ apply Rplus_le_compat_l; assumption | ring ]
+              | ring ]
+            | replace (x - PI) with (x + - PI); rewrite Rplus_comm;
+              [ pattern PI at 2 in |- *; replace PI with (- PI + 2 * PI);
+                [ apply Rplus_le_compat_l; assumption | ring ]
+                | ring ] ]
+          | unfold INR in |- *; ring ].
+Qed.
+
+Lemma cos_le_0 : forall x:R, PI / 2 <= x -> x <= 3 * (PI / 2) -> cos x <= 0.
+Proof.
+  intros x H1 H2; apply Rge_le; rewrite <- Ropp_0;
+    rewrite <- (Ropp_involutive (cos x)); apply Ropp_le_ge_contravar;
+      rewrite <- neg_cos; replace (x + PI) with (x - PI + 2 * INR 1 * PI).
+  rewrite cos_period; apply cos_ge_0.
+  replace (- (PI / 2)) with (- PI + PI / 2).
+  unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_le_compat_l;
+    assumption.
+  pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
+    ring.
+  unfold Rminus in |- *; rewrite Rplus_comm;
+    replace (PI / 2) with (- PI + 3 * (PI / 2)).
+  apply Rplus_le_compat_l; assumption.
+  pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
+    ring.
+  unfold INR in |- *; ring.
+Qed.
+
+Lemma sin_lt_0 : forall x:R, PI < x -> x < 2 * PI -> sin x < 0.
+Proof.
+  intros x H1 H2; rewrite <- Ropp_0; rewrite <- (Ropp_involutive (sin x));
+    apply Ropp_lt_gt_contravar; rewrite <- neg_sin;
+      replace (x + PI) with (x - PI + 2 * INR 1 * PI);
+      [ rewrite (sin_period (x - PI) 1); apply sin_gt_0;
+        [ replace (x - PI) with (x + - PI);
+          [ rewrite Rplus_comm; replace 0 with (- PI + PI);
+            [ apply Rplus_lt_compat_l; assumption | ring ]
+            | ring ]
+          | replace (x - PI) with (x + - PI); rewrite Rplus_comm;
+            [ pattern PI at 2 in |- *; replace PI with (- PI + 2 * PI);
+              [ apply Rplus_lt_compat_l; assumption | ring ]
+              | ring ] ]
+        | unfold INR in |- *; ring ].
+Qed.
+
+Lemma sin_lt_0_var : forall x:R, - PI < x -> x < 0 -> sin x < 0.
+Proof.
+  intros; generalize (Rplus_lt_compat_l (2 * PI) (- PI) x H);
+    replace (2 * PI + - PI) with PI;
+    [ intro H1; rewrite Rplus_comm in H1;
+      generalize (Rplus_lt_compat_l (2 * PI) x 0 H0);
+        intro H2; rewrite (Rplus_comm (2 * PI)) in H2;
+          rewrite <- (Rplus_comm 0) in H2; rewrite Rplus_0_l in H2;
+            rewrite <- (sin_period x 1); unfold INR in |- *;
+              replace (2 * 1 * PI) with (2 * PI);
+              [ apply (sin_lt_0 (x + 2 * PI) H1 H2) | ring ]
+      | ring ].
+Qed.
+
+Lemma cos_lt_0 : forall x:R, PI / 2 < x -> x < 3 * (PI / 2) -> cos x < 0.
+Proof.
+  intros x H1 H2; rewrite <- Ropp_0; rewrite <- (Ropp_involutive (cos x));
+    apply Ropp_lt_gt_contravar; rewrite <- neg_cos;
+      replace (x + PI) with (x - PI + 2 * INR 1 * PI).
+  rewrite cos_period; apply cos_gt_0.
+  replace (- (PI / 2)) with (- PI + PI / 2).
+  unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_lt_compat_l;
+    assumption.
+  pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
+    ring.
+  unfold Rminus in |- *; rewrite Rplus_comm;
+    replace (PI / 2) with (- PI + 3 * (PI / 2)).
+  apply Rplus_lt_compat_l; assumption.
+  pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr;
+    ring.
+  unfold INR in |- *; ring.
+Qed.
+
+Lemma tan_gt_0 : forall x:R, 0 < x -> x < PI / 2 -> 0 < tan x.
+Proof.
+  intros x H1 H2; unfold tan in |- *; generalize _PI2_RLT_0;
+    generalize (Rlt_trans 0 x (PI / 2) H1 H2); intros;
+      generalize (Rlt_trans (- (PI / 2)) 0 x H0 H1); intro H5;
+        generalize (Rlt_trans x (PI / 2) PI H2 PI2_Rlt_PI);
+          intro H7; unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  apply sin_gt_0; assumption.
+  apply Rinv_0_lt_compat; apply cos_gt_0; assumption.
+Qed.
+
+Lemma tan_lt_0 : forall x:R, - (PI / 2) < x -> x < 0 -> tan x < 0.
+Proof.
+  intros x H1 H2; unfold tan in |- *;
+    generalize (cos_gt_0 x H1 (Rlt_trans x 0 (PI / 2) H2 PI2_RGT_0));
+      intro H3; rewrite <- Ropp_0;
+        replace (sin x / cos x) with (- (- sin x / cos x)).
+  rewrite <- sin_neg; apply Ropp_gt_lt_contravar;
+    change (0 < sin (- x) / cos x) in |- *; unfold Rdiv in |- *;
+      apply Rmult_lt_0_compat.
+  apply sin_gt_0.
+  rewrite <- Ropp_0; apply Ropp_gt_lt_contravar; assumption.
+  apply Rlt_trans with (PI / 2).
+  rewrite <- (Ropp_involutive (PI / 2)); apply Ropp_gt_lt_contravar; assumption.
+  apply PI2_Rlt_PI.
+  apply Rinv_0_lt_compat; assumption.
+  unfold Rdiv in |- *; ring.
+Qed.
+
+Lemma cos_ge_0_3PI2 :
+  forall x:R, 3 * (PI / 2) <= x -> x <= 2 * PI -> 0 <= cos x.
+Proof.
+  intros; rewrite <- cos_neg; rewrite <- (cos_period (- x) 1);
+    unfold INR in |- *; replace (- x + 2 * 1 * PI) with (2 * PI - x).
+  generalize (Ropp_le_ge_contravar x (2 * PI) H0); intro H1;
+    generalize (Rge_le (- x) (- (2 * PI)) H1); clear H1;
+      intro H1; generalize (Rplus_le_compat_l (2 * PI) (- (2 * PI)) (- x) H1).
+  rewrite Rplus_opp_r.
+  intro H2; generalize (Ropp_le_ge_contravar (3 * (PI / 2)) x H); intro H3;
+    generalize (Rge_le (- (3 * (PI / 2))) (- x) H3); clear H3;
+      intro H3;
+        generalize (Rplus_le_compat_l (2 * PI) (- x) (- (3 * (PI / 2))) H3).
+  replace (2 * PI + - (3 * (PI / 2))) with (PI / 2).
+  intro H4;
+    apply
+      (cos_ge_0 (2 * PI - x)
+        (Rlt_le (- (PI / 2)) (2 * PI - x)
+          (Rlt_le_trans (- (PI / 2)) 0 (2 * PI - x) _PI2_RLT_0 H2)) H4).
+  rewrite double; pattern PI at 2 3 in |- *; rewrite double_var; ring.
+  ring.
+Qed.
+
+Lemma form1 :
+  forall p q:R, cos p + cos q = 2 * cos ((p - q) / 2) * cos ((p + q) / 2).
+Proof.
+  intros p q; pattern p at 1 in |- *;
+    replace p with ((p - q) / 2 + (p + q) / 2).
+  rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2).
+  rewrite cos_plus; rewrite cos_minus; ring.
+  pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
+  pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
+Qed.
+
+Lemma form2 :
+  forall p q:R, cos p - cos q = -2 * sin ((p - q) / 2) * sin ((p + q) / 2).
+Proof.
+  intros p q; pattern p at 1 in |- *;
+    replace p with ((p - q) / 2 + (p + q) / 2).
+  rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2).
+  rewrite cos_plus; rewrite cos_minus; ring.
+  pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
+  pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
+Qed.
+
+Lemma form3 :
+  forall p q:R, sin p + sin q = 2 * cos ((p - q) / 2) * sin ((p + q) / 2).
+Proof.
+  intros p q; pattern p at 1 in |- *;
+    replace p with ((p - q) / 2 + (p + q) / 2).
+  pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2).
+  rewrite sin_plus; rewrite sin_minus; ring.
+  pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
+  pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
+Qed.
+
+Lemma form4 :
+  forall p q:R, sin p - sin q = 2 * cos ((p + q) / 2) * sin ((p - q) / 2).
+Proof.
+  intros p q; pattern p at 1 in |- *;
+    replace p with ((p - q) / 2 + (p + q) / 2).
+  pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2).
+  rewrite sin_plus; rewrite sin_minus; ring.
+  pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
+  pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring.
+
+Qed.
+
+Lemma sin_increasing_0 :
+  forall x y:R,
+    - (PI / 2) <= x ->
+    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x < sin y -> x < y.
+Proof.
+  intros; cut (sin ((x - y) / 2) < 0).
+  intro H4; case (Rtotal_order ((x - y) / 2) 0); intro H5.
+  assert (Hyp : 0 < 2).
+  prove_sup0.
+  generalize (Rmult_lt_compat_l 2 ((x - y) / 2) 0 Hyp H5).
+  unfold Rdiv in |- *.
+  rewrite <- Rmult_assoc.
+  rewrite Rinv_r_simpl_m.
+  rewrite Rmult_0_r.
+  clear H5; intro H5; apply Rminus_lt; assumption.
+  discrR.
+  elim H5; intro H6.
+  rewrite H6 in H4; rewrite sin_0 in H4; elim (Rlt_irrefl 0 H4).
+  change (0 < (x - y) / 2) in H6;
+    generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1).
+  rewrite Ropp_involutive.
+  intro H7; generalize (Rge_le (PI / 2) (- y) H7); clear H7; intro H7;
+    generalize (Rplus_le_compat x (PI / 2) (- y) (PI / 2) H0 H7).
+  rewrite <- double_var.
+  intro H8.
+  assert (Hyp : 0 < 2).
+  prove_sup0.
+  generalize
+    (Rmult_le_compat_l (/ 2) (x - y) PI
+      (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H8).
+  repeat rewrite (Rmult_comm (/ 2)).
+  intro H9;
+    generalize
+      (sin_gt_0 ((x - y) / 2) H6
+        (Rle_lt_trans ((x - y) / 2) (PI / 2) PI H9 PI2_Rlt_PI));
+      intro H10;
+        elim
+          (Rlt_irrefl (sin ((x - y) / 2))
+            (Rlt_trans (sin ((x - y) / 2)) 0 (sin ((x - y) / 2)) H4 H10)).
+  generalize (Rlt_minus (sin x) (sin y) H3); clear H3; intro H3;
+    rewrite form4 in H3;
+      generalize (Rplus_le_compat x (PI / 2) y (PI / 2) H0 H2).
+  rewrite <- double_var.
+  assert (Hyp : 0 < 2).
+  prove_sup0.
+  intro H4;
+    generalize
+      (Rmult_le_compat_l (/ 2) (x + y) PI
+        (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H4).
+  repeat rewrite (Rmult_comm (/ 2)).
+  clear H4; intro H4;
+    generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) y H H1);
+      replace (- (PI / 2) + - (PI / 2)) with (- PI).
+  intro H5;
+    generalize
+      (Rmult_le_compat_l (/ 2) (- PI) (x + y)
+        (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H5).
+  replace (/ 2 * (x + y)) with ((x + y) / 2).
+  replace (/ 2 * - PI) with (- (PI / 2)).
+  clear H5; intro H5; elim H4; intro H40.
+  elim H5; intro H50.
+  generalize (cos_gt_0 ((x + y) / 2) H50 H40); intro H6;
+    generalize (Rmult_lt_compat_l 2 0 (cos ((x + y) / 2)) Hyp H6).
+  rewrite Rmult_0_r.
+  clear H6; intro H6; case (Rcase_abs (sin ((x - y) / 2))); intro H7.
+  assumption.
+  generalize (Rge_le (sin ((x - y) / 2)) 0 H7); clear H7; intro H7;
+    generalize
+      (Rmult_le_pos (2 * cos ((x + y) / 2)) (sin ((x - y) / 2))
+        (Rlt_le 0 (2 * cos ((x + y) / 2)) H6) H7); intro H8;
+      generalize
+        (Rle_lt_trans 0 (2 * cos ((x + y) / 2) * sin ((x - y) / 2)) 0 H8 H3);
+        intro H9; elim (Rlt_irrefl 0 H9).
+  rewrite <- H50 in H3; rewrite cos_neg in H3; rewrite cos_PI2 in H3;
+    rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3;
+      elim (Rlt_irrefl 0 H3).
+  unfold Rdiv in H3.
+  rewrite H40 in H3; assert (H50 := cos_PI2); unfold Rdiv in H50;
+    rewrite H50 in H3; rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3;
+      elim (Rlt_irrefl 0 H3).
+  unfold Rdiv in |- *.
+  rewrite <- Ropp_mult_distr_l_reverse.
+  apply Rmult_comm.
+  unfold Rdiv in |- *; apply Rmult_comm.
+  pattern PI at 1 in |- *; rewrite double_var.
+  rewrite Ropp_plus_distr.
+  reflexivity.
+Qed.
+
+Lemma sin_increasing_1 :
+  forall x y:R,
+    - (PI / 2) <= x ->
+    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> x < y -> sin x < sin y.
+Proof.
+  intros; generalize (Rplus_lt_compat_l x x y H3); intro H4;
+    generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) x H H);
+      replace (- (PI / 2) + - (PI / 2)) with (- PI).
+  assert (Hyp : 0 < 2).
+  prove_sup0.
+  intro H5; generalize (Rle_lt_trans (- PI) (x + x) (x + y) H5 H4); intro H6;
+    generalize
+      (Rmult_lt_compat_l (/ 2) (- PI) (x + y) (Rinv_0_lt_compat 2 Hyp) H6);
+      replace (/ 2 * - PI) with (- (PI / 2)).
+  replace (/ 2 * (x + y)) with ((x + y) / 2).
+  clear H4 H5 H6; intro H4; generalize (Rplus_lt_compat_l y x y H3); intro H5;
+    rewrite Rplus_comm in H5;
+      generalize (Rplus_le_compat y (PI / 2) y (PI / 2) H2 H2).
+  rewrite <- double_var.
+  intro H6; generalize (Rlt_le_trans (x + y) (y + y) PI H5 H6); intro H7;
+    generalize (Rmult_lt_compat_l (/ 2) (x + y) PI (Rinv_0_lt_compat 2 Hyp) H7);
+      replace (/ 2 * PI) with (PI / 2).
+  replace (/ 2 * (x + y)) with ((x + y) / 2).
+  clear H5 H6 H7; intro H5; generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1);
+    rewrite Ropp_involutive; clear H1; intro H1;
+      generalize (Rge_le (PI / 2) (- y) H1); clear H1; intro H1;
+        generalize (Ropp_le_ge_contravar y (PI / 2) H2); clear H2;
+          intro H2; generalize (Rge_le (- y) (- (PI / 2)) H2);
+            clear H2; intro H2; generalize (Rplus_lt_compat_l (- y) x y H3);
+              replace (- y + x) with (x - y).
+  rewrite Rplus_opp_l.
+  intro H6;
+    generalize (Rmult_lt_compat_l (/ 2) (x - y) 0 (Rinv_0_lt_compat 2 Hyp) H6);
+      rewrite Rmult_0_r; replace (/ 2 * (x - y)) with ((x - y) / 2).
+  clear H6; intro H6;
+    generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) (- y) H H2);
+      replace (- (PI / 2) + - (PI / 2)) with (- PI).
+  replace (x + - y) with (x - y).
+  intro H7;
+    generalize
+      (Rmult_le_compat_l (/ 2) (- PI) (x - y)
+        (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H7);
+      replace (/ 2 * - PI) with (- (PI / 2)).
+  replace (/ 2 * (x - y)) with ((x - y) / 2).
+  clear H7; intro H7; clear H H0 H1 H2; apply Rminus_lt; rewrite form4;
+    generalize (cos_gt_0 ((x + y) / 2) H4 H5); intro H8;
+      generalize (Rmult_lt_0_compat 2 (cos ((x + y) / 2)) Hyp H8);
+        clear H8; intro H8; cut (- PI < - (PI / 2)).
+  intro H9;
+    generalize
+      (sin_lt_0_var ((x - y) / 2)
+        (Rlt_le_trans (- PI) (- (PI / 2)) ((x - y) / 2) H9 H7) H6);
+      intro H10;
+        generalize
+          (Rmult_lt_gt_compat_neg_l (sin ((x - y) / 2)) 0 (
+            2 * cos ((x + y) / 2)) H10 H8); intro H11; rewrite Rmult_0_r in H11;
+          rewrite Rmult_comm; assumption.
+  apply Ropp_lt_gt_contravar; apply PI2_Rlt_PI.
+  unfold Rdiv in |- *; apply Rmult_comm.
+  unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse; apply Rmult_comm.
+  reflexivity.
+  pattern PI at 1 in |- *; rewrite double_var.
+  rewrite Ropp_plus_distr.
+  reflexivity.
+  unfold Rdiv in |- *; apply Rmult_comm.
+  unfold Rminus in |- *; apply Rplus_comm.
+  unfold Rdiv in |- *; apply Rmult_comm.
+  unfold Rdiv in |- *; apply Rmult_comm.
+  unfold Rdiv in |- *; apply Rmult_comm.
+  unfold Rdiv in |- *.
+  rewrite <- Ropp_mult_distr_l_reverse.
+  apply Rmult_comm.
+  pattern PI at 1 in |- *; rewrite double_var.
+  rewrite Ropp_plus_distr.
+  reflexivity.
+Qed.
+
+Lemma sin_decreasing_0 :
+  forall x y:R,
+    x <= 3 * (PI / 2) ->
+    PI / 2 <= x -> y <= 3 * (PI / 2) -> PI / 2 <= y -> sin x < sin y -> y < x.
+Proof.
+  intros; rewrite <- (sin_PI_x x) in H3; rewrite <- (sin_PI_x y) in H3;
+    generalize (Ropp_lt_gt_contravar (sin (PI - x)) (sin (PI - y)) H3);
+      repeat rewrite <- sin_neg;
+        generalize (Rplus_le_compat_l (- PI) x (3 * (PI / 2)) H);
+          generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0);
+            generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1);
+              generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2);
+                replace (- PI + x) with (x - PI).
+  replace (- PI + PI / 2) with (- (PI / 2)).
+  replace (- PI + y) with (y - PI).
+  replace (- PI + 3 * (PI / 2)) with (PI / 2).
+  replace (- (PI - x)) with (x - PI).
+  replace (- (PI - y)) with (y - PI).
+  intros; change (sin (y - PI) < sin (x - PI)) in H8;
+    apply Rplus_lt_reg_r with (- PI); rewrite Rplus_comm;
+      replace (y + - PI) with (y - PI).
+  rewrite Rplus_comm; replace (x + - PI) with (x - PI).
+  apply (sin_increasing_0 (y - PI) (x - PI) H4 H5 H6 H7 H8).
+  reflexivity.
+  reflexivity.
+  unfold Rminus in |- *; rewrite Ropp_plus_distr.
+  rewrite Ropp_involutive.
+  apply Rplus_comm.
+  unfold Rminus in |- *; rewrite Ropp_plus_distr.
+  rewrite Ropp_involutive.
+  apply Rplus_comm.
+  pattern PI at 2 in |- *; rewrite double_var.
+  rewrite Ropp_plus_distr.
+  ring.
+  unfold Rminus in |- *; apply Rplus_comm.
+  pattern PI at 2 in |- *; rewrite double_var.
+  rewrite Ropp_plus_distr.
+  ring.
+  unfold Rminus in |- *; apply Rplus_comm.
+Qed.
+
+Lemma sin_decreasing_1 :
+  forall x y:R,
+    x <= 3 * (PI / 2) ->
+    PI / 2 <= x -> y <= 3 * (PI / 2) -> PI / 2 <= y -> x < y -> sin y < sin x.
+Proof.
+  intros; rewrite <- (sin_PI_x x); rewrite <- (sin_PI_x y);
+    generalize (Rplus_le_compat_l (- PI) x (3 * (PI / 2)) H);
+      generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0);
+        generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1);
+          generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2);
+            generalize (Rplus_lt_compat_l (- PI) x y H3);
+              replace (- PI + PI / 2) with (- (PI / 2)).
+  replace (- PI + y) with (y - PI).
+  replace (- PI + 3 * (PI / 2)) with (PI / 2).
+  replace (- PI + x) with (x - PI).
+  intros; apply Ropp_lt_cancel; repeat rewrite <- sin_neg;
+    replace (- (PI - x)) with (x - PI).
+  replace (- (PI - y)) with (y - PI).
+  apply (sin_increasing_1 (x - PI) (y - PI) H7 H8 H5 H6 H4).
+  unfold Rminus in |- *; rewrite Ropp_plus_distr.
+  rewrite Ropp_involutive.
+  apply Rplus_comm.
+  unfold Rminus in |- *; rewrite Ropp_plus_distr.
+  rewrite Ropp_involutive.
+  apply Rplus_comm.
+  unfold Rminus in |- *; apply Rplus_comm.
+  pattern PI at 2 in |- *; rewrite double_var; ring.
+  unfold Rminus in |- *; apply Rplus_comm.
+  pattern PI at 2 in |- *; rewrite double_var; ring.
+Qed.
+
+Lemma cos_increasing_0 :
+  forall x y:R,
+    PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x < cos y -> x < y.
+Proof.
+  intros x y H1 H2 H3 H4; rewrite <- (cos_neg x); rewrite <- (cos_neg y);
+    rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1);
+      unfold INR in |- *;
+        replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))).
+  replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))).
+  repeat rewrite cos_shift; intro H5;
+    generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1);
+      generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2);
+        generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3);
+          generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4).
+  replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)).
+  replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)).
+  replace (-3 * (PI / 2) + 2 * PI) with (PI / 2).
+  replace (-3 * (PI / 2) + PI) with (- (PI / 2)).
+  clear H1 H2 H3 H4; intros H1 H2 H3 H4;
+    apply Rplus_lt_reg_r with (-3 * (PI / 2));
+      replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)).
+  replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)).
+  apply (sin_increasing_0 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H4 H3 H2 H1 H5).
+  unfold Rminus in |- *.
+  rewrite Ropp_mult_distr_l_reverse.
+  apply Rplus_comm.
+  unfold Rminus in |- *.
+  rewrite Ropp_mult_distr_l_reverse.
+  apply Rplus_comm.
+  pattern PI at 3 in |- *; rewrite double_var.
+  ring.
+  rewrite double; pattern PI at 3 4 in |- *; rewrite double_var.
+  ring.
+  unfold Rminus in |- *.
+  rewrite Ropp_mult_distr_l_reverse.
+  apply Rplus_comm.
+  unfold Rminus in |- *.
+  rewrite Ropp_mult_distr_l_reverse.
+  apply Rplus_comm.
+  rewrite Rmult_1_r.
+  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
+  ring.
+  rewrite Rmult_1_r.
+  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
+  ring.
+Qed.
+
+Lemma cos_increasing_1 :
+  forall x y:R,
+    PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> x < y -> cos x < cos y.
+Proof.
+  intros x y H1 H2 H3 H4 H5;
+    generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1);
+      generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2);
+        generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3);
+          generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4);
+            generalize (Rplus_lt_compat_l (-3 * (PI / 2)) x y H5);
+              rewrite <- (cos_neg x); rewrite <- (cos_neg y);
+                rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1);
+                  unfold INR in |- *; replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)).
+  replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)).
+  replace (-3 * (PI / 2) + PI) with (- (PI / 2)).
+  replace (-3 * (PI / 2) + 2 * PI) with (PI / 2).
+  clear H1 H2 H3 H4 H5; intros H1 H2 H3 H4 H5;
+    replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))).
+  replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))).
+  repeat rewrite cos_shift;
+    apply
+      (sin_increasing_1 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H5 H4 H3 H2 H1).
+  rewrite Rmult_1_r.
+  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
+  ring.
+  rewrite Rmult_1_r.
+  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
+  ring.
+  rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var.
+  ring.
+  pattern PI at 3 in |- *; rewrite double_var; ring.
+  unfold Rminus in |- *.
+  rewrite <- Ropp_mult_distr_l_reverse.
+  apply Rplus_comm.
+  unfold Rminus in |- *.
+  rewrite <- Ropp_mult_distr_l_reverse.
+  apply Rplus_comm.
+Qed.
+
+Lemma cos_decreasing_0 :
+  forall x y:R,
+    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> cos x < cos y -> y < x.
+Proof.
+  intros; generalize (Ropp_lt_gt_contravar (cos x) (cos y) H3);
+    repeat rewrite <- neg_cos; intro H4;
+      change (cos (y + PI) < cos (x + PI)) in H4; rewrite (Rplus_comm x) in H4;
+        rewrite (Rplus_comm y) in H4; generalize (Rplus_le_compat_l PI 0 x H);
+          generalize (Rplus_le_compat_l PI x PI H0);
+            generalize (Rplus_le_compat_l PI 0 y H1);
+              generalize (Rplus_le_compat_l PI y PI H2); rewrite Rplus_0_r.
+  rewrite <- double.
+  clear H H0 H1 H2 H3; intros; apply Rplus_lt_reg_r with PI;
+    apply (cos_increasing_0 (PI + y) (PI + x) H0 H H2 H1 H4).
+Qed.
+
+Lemma cos_decreasing_1 :
+  forall x y:R,
+    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> x < y -> cos y < cos x.
+Proof.
+  intros; apply Ropp_lt_cancel; repeat rewrite <- neg_cos;
+    rewrite (Rplus_comm x); rewrite (Rplus_comm y);
+      generalize (Rplus_le_compat_l PI 0 x H);
+        generalize (Rplus_le_compat_l PI x PI H0);
+          generalize (Rplus_le_compat_l PI 0 y H1);
+            generalize (Rplus_le_compat_l PI y PI H2); rewrite Rplus_0_r.
+  rewrite <- double.
+  generalize (Rplus_lt_compat_l PI x y H3); clear H H0 H1 H2 H3; intros;
+    apply (cos_increasing_1 (PI + x) (PI + y) H3 H2 H1 H0 H).
+Qed.
+
+Lemma tan_diff :
+  forall x y:R,
+    cos x <> 0 -> cos y <> 0 -> tan x - tan y = sin (x - y) / (cos x * cos y).
+Proof.
+  intros; unfold tan in |- *; rewrite sin_minus.
+  unfold Rdiv in |- *.
+  unfold Rminus in |- *.
+  rewrite Rmult_plus_distr_r.
+  rewrite Rinv_mult_distr.
+  repeat rewrite (Rmult_comm (sin x)).
+  repeat rewrite Rmult_assoc.
+  rewrite (Rmult_comm (cos y)).
+  repeat rewrite Rmult_assoc.
+  rewrite <- Rinv_l_sym.
+  rewrite Rmult_1_r.
+  rewrite (Rmult_comm (sin x)).
+  apply Rplus_eq_compat_l.
+  rewrite <- Ropp_mult_distr_l_reverse.
+  rewrite <- Ropp_mult_distr_r_reverse.
+  rewrite (Rmult_comm (/ cos x)).
+  repeat rewrite Rmult_assoc.
+  rewrite (Rmult_comm (cos x)).
+  repeat rewrite Rmult_assoc.
+  rewrite <- Rinv_l_sym.
+  rewrite Rmult_1_r.
+  reflexivity.
+  assumption.
+  assumption.
+  assumption.
+  assumption.
+Qed.
+
+Lemma tan_increasing_0 :
+  forall x y:R,
+    - (PI / 4) <= x ->
+    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x < tan y -> x < y.
+Proof.
+  intros; generalize PI4_RLT_PI2; intro H4;
+    generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4);
+      intro H5; change (- (PI / 2) < - (PI / 4)) in H5;
+        generalize
+          (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
+            (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)); intro HP1;
+          generalize
+            (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
+              (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)); intro HP2;
+            generalize
+              (not_eq_sym
+                (Rlt_not_eq 0 (cos x)
+                  (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
+                    (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4))));
+              intro H6;
+                generalize
+                  (not_eq_sym
+                    (Rlt_not_eq 0 (cos y)
+                      (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
+                        (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4))));
+                  intro H7; generalize (tan_diff x y H6 H7); intro H8;
+                    generalize (Rlt_minus (tan x) (tan y) H3); clear H3;
+                      intro H3; rewrite H8 in H3; cut (sin (x - y) < 0).
+  intro H9; generalize (Ropp_le_ge_contravar (- (PI / 4)) y H1);
+    rewrite Ropp_involutive; intro H10; generalize (Rge_le (PI / 4) (- y) H10);
+      clear H10; intro H10; generalize (Ropp_le_ge_contravar y (PI / 4) H2);
+        intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11);
+          clear H11; intro H11;
+            generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11);
+              generalize (Rplus_le_compat x (PI / 4) (- y) (PI / 4) H0 H10);
+                replace (x + - y) with (x - y).
+  replace (PI / 4 + PI / 4) with (PI / 2).
+  replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)).
+  intros; case (Rtotal_order 0 (x - y)); intro H14.
+  generalize
+    (sin_gt_0 (x - y) H14 (Rle_lt_trans (x - y) (PI / 2) PI H12 PI2_Rlt_PI));
+    intro H15; elim (Rlt_irrefl 0 (Rlt_trans 0 (sin (x - y)) 0 H15 H9)).
+  elim H14; intro H15.
+  rewrite <- H15 in H9; rewrite sin_0 in H9; elim (Rlt_irrefl 0 H9).
+  apply Rminus_lt; assumption.
+  pattern PI at 1 in |- *; rewrite double_var.
+  unfold Rdiv in |- *.
+  rewrite Rmult_plus_distr_r.
+  repeat rewrite Rmult_assoc.
+  rewrite <- Rinv_mult_distr.
+  rewrite Ropp_plus_distr.
+  replace 4 with 4.
+  reflexivity.
+  ring.
+  discrR.
+  discrR.
+  pattern PI at 1 in |- *; rewrite double_var.
+  unfold Rdiv in |- *.
+  rewrite Rmult_plus_distr_r.
+  repeat rewrite Rmult_assoc.
+  rewrite <- Rinv_mult_distr.
+  replace 4 with 4.
+  reflexivity.
+  ring.
+  discrR.
+  discrR.
+  reflexivity.
+  case (Rcase_abs (sin (x - y))); intro H9.
+  assumption.
+  generalize (Rge_le (sin (x - y)) 0 H9); clear H9; intro H9;
+    generalize (Rinv_0_lt_compat (cos x) HP1); intro H10;
+      generalize (Rinv_0_lt_compat (cos y) HP2); intro H11;
+        generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11);
+          replace (/ cos x * / cos y) with (/ (cos x * cos y)).
+  intro H12;
+    generalize
+      (Rmult_le_pos (sin (x - y)) (/ (cos x * cos y)) H9
+        (Rlt_le 0 (/ (cos x * cos y)) H12)); intro H13;
+      elim
+        (Rlt_irrefl 0 (Rle_lt_trans 0 (sin (x - y) * / (cos x * cos y)) 0 H13 H3)).
+  rewrite Rinv_mult_distr.
+  reflexivity.
+  assumption.
+  assumption.
+Qed.
+
+Lemma tan_increasing_1 :
+  forall x y:R,
+    - (PI / 4) <= x ->
+    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x < y -> tan x < tan y.
+Proof.
+  intros; apply Rminus_lt; generalize PI4_RLT_PI2; intro H4;
+    generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4);
+      intro H5; change (- (PI / 2) < - (PI / 4)) in H5;
+        generalize
+          (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
+            (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)); intro HP1;
+          generalize
+            (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
+              (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)); intro HP2;
+            generalize
+              (not_eq_sym
+                (Rlt_not_eq 0 (cos x)
+                  (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H)
+                    (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4))));
+              intro H6;
+                generalize
+                  (not_eq_sym
+                    (Rlt_not_eq 0 (cos y)
+                      (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1)
+                        (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4))));
+                  intro H7; rewrite (tan_diff x y H6 H7);
+                    generalize (Rinv_0_lt_compat (cos x) HP1); intro H10;
+                      generalize (Rinv_0_lt_compat (cos y) HP2); intro H11;
+                        generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11);
+                          replace (/ cos x * / cos y) with (/ (cos x * cos y)).
+  clear H10 H11; intro H8; generalize (Ropp_le_ge_contravar y (PI / 4) H2);
+    intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11);
+      clear H11; intro H11;
+        generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11);
+          replace (x + - y) with (x - y).
+  replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)).
+  clear H11; intro H9; generalize (Rlt_minus x y H3); clear H3; intro H3;
+    clear H H0 H1 H2 H4 H5 HP1 HP2; generalize PI2_Rlt_PI;
+      intro H1; generalize (Ropp_lt_gt_contravar (PI / 2) PI H1);
+        clear H1; intro H1;
+          generalize
+            (sin_lt_0_var (x - y) (Rlt_le_trans (- PI) (- (PI / 2)) (x - y) H1 H9) H3);
+            intro H2;
+              generalize
+                (Rmult_lt_gt_compat_neg_l (sin (x - y)) 0 (/ (cos x * cos y)) H2 H8);
+                rewrite Rmult_0_r; intro H4; assumption.
+  pattern PI at 1 in |- *; rewrite double_var.
+  unfold Rdiv in |- *.
+  rewrite Rmult_plus_distr_r.
+  repeat rewrite Rmult_assoc.
+  rewrite <- Rinv_mult_distr.
+  replace 4 with 4.
+  rewrite Ropp_plus_distr.
+  reflexivity.
+  ring.
+  discrR.
+  discrR.
+  reflexivity.
+  apply Rinv_mult_distr; assumption.
+Qed.
+
+Lemma sin_incr_0 :
+  forall x y:R,
+    - (PI / 2) <= x ->
+    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x <= sin y -> x <= y.
+Proof.
+  intros; case (Rtotal_order (sin x) (sin y)); intro H4;
+    [ left; apply (sin_increasing_0 x y H H0 H1 H2 H4)
+      | elim H4; intro H5;
+        [ case (Rtotal_order x y); intro H6;
+          [ left; assumption
+            | elim H6; intro H7;
+              [ right; assumption
+                | generalize (sin_increasing_1 y x H1 H2 H H0 H7); intro H8;
+                  rewrite H5 in H8; elim (Rlt_irrefl (sin y) H8) ] ]
+          | elim (Rlt_irrefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5)) ] ].
+Qed.
+
+Lemma sin_incr_1 :
+  forall x y:R,
+    - (PI / 2) <= x ->
+    x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> x <= y -> sin x <= sin y.
+Proof.
+  intros; case (Rtotal_order x y); intro H4;
+    [ left; apply (sin_increasing_1 x y H H0 H1 H2 H4)
+      | elim H4; intro H5;
+        [ case (Rtotal_order (sin x) (sin y)); intro H6;
+          [ left; assumption
+            | elim H6; intro H7;
+              [ right; assumption
+                | generalize (sin_increasing_0 y x H1 H2 H H0 H7); intro H8;
+                  rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ]
+          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
+Qed.
+
+Lemma sin_decr_0 :
+  forall x y:R,
+    x <= 3 * (PI / 2) ->
+    PI / 2 <= x ->
+    y <= 3 * (PI / 2) -> PI / 2 <= y -> sin x <= sin y -> y <= x.
+Proof.
+  intros; case (Rtotal_order (sin x) (sin y)); intro H4;
+    [ left; apply (sin_decreasing_0 x y H H0 H1 H2 H4)
+      | elim H4; intro H5;
+        [ case (Rtotal_order x y); intro H6;
+          [ generalize (sin_decreasing_1 x y H H0 H1 H2 H6); intro H8;
+            rewrite H5 in H8; elim (Rlt_irrefl (sin y) H8)
+            | elim H6; intro H7;
+              [ right; symmetry  in |- *; assumption | left; assumption ] ]
+          | elim (Rlt_irrefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5)) ] ].
+Qed.
+
+Lemma sin_decr_1 :
+  forall x y:R,
+    x <= 3 * (PI / 2) ->
+    PI / 2 <= x ->
+    y <= 3 * (PI / 2) -> PI / 2 <= y -> x <= y -> sin y <= sin x.
+Proof.
+  intros; case (Rtotal_order x y); intro H4;
+    [ left; apply (sin_decreasing_1 x y H H0 H1 H2 H4)
+      | elim H4; intro H5;
+        [ case (Rtotal_order (sin x) (sin y)); intro H6;
+          [ generalize (sin_decreasing_0 x y H H0 H1 H2 H6); intro H8;
+            rewrite H5 in H8; elim (Rlt_irrefl y H8)
+            | elim H6; intro H7;
+              [ right; symmetry  in |- *; assumption | left; assumption ] ]
+          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
+Qed.
+
+Lemma cos_incr_0 :
+  forall x y:R,
+    PI <= x ->
+    x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x <= cos y -> x <= y.
+Proof.
+  intros; case (Rtotal_order (cos x) (cos y)); intro H4;
+    [ left; apply (cos_increasing_0 x y H H0 H1 H2 H4)
+      | elim H4; intro H5;
+        [ case (Rtotal_order x y); intro H6;
+          [ left; assumption
+            | elim H6; intro H7;
+              [ right; assumption
+                | generalize (cos_increasing_1 y x H1 H2 H H0 H7); intro H8;
+                  rewrite H5 in H8; elim (Rlt_irrefl (cos y) H8) ] ]
+          | elim (Rlt_irrefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5)) ] ].
+Qed.
+
+Lemma cos_incr_1 :
+  forall x y:R,
+    PI <= x ->
+    x <= 2 * PI -> PI <= y -> y <= 2 * PI -> x <= y -> cos x <= cos y.
+Proof.
+  intros; case (Rtotal_order x y); intro H4;
+    [ left; apply (cos_increasing_1 x y H H0 H1 H2 H4)
+      | elim H4; intro H5;
+        [ case (Rtotal_order (cos x) (cos y)); intro H6;
+          [ left; assumption
+            | elim H6; intro H7;
+              [ right; assumption
+                | generalize (cos_increasing_0 y x H1 H2 H H0 H7); intro H8;
+                  rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ]
+          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
+Qed.
+
+Lemma cos_decr_0 :
+  forall x y:R,
+    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> cos x <= cos y -> y <= x.
+Proof.
+  intros; case (Rtotal_order (cos x) (cos y)); intro H4;
+    [ left; apply (cos_decreasing_0 x y H H0 H1 H2 H4)
+      | elim H4; intro H5;
+        [ case (Rtotal_order x y); intro H6;
+          [ generalize (cos_decreasing_1 x y H H0 H1 H2 H6); intro H8;
+            rewrite H5 in H8; elim (Rlt_irrefl (cos y) H8)
+            | elim H6; intro H7;
+              [ right; symmetry  in |- *; assumption | left; assumption ] ]
+          | elim (Rlt_irrefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5)) ] ].
+Qed.
+
+Lemma cos_decr_1 :
+  forall x y:R,
+    0 <= x -> x <= PI -> 0 <= y -> y <= PI -> x <= y -> cos y <= cos x.
+Proof.
+  intros; case (Rtotal_order x y); intro H4;
+    [ left; apply (cos_decreasing_1 x y H H0 H1 H2 H4)
+      | elim H4; intro H5;
+        [ case (Rtotal_order (cos x) (cos y)); intro H6;
+          [ generalize (cos_decreasing_0 x y H H0 H1 H2 H6); intro H8;
+            rewrite H5 in H8; elim (Rlt_irrefl y H8)
+            | elim H6; intro H7;
+              [ right; symmetry  in |- *; assumption | left; assumption ] ]
+          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
+Qed.
+
+Lemma tan_incr_0 :
+  forall x y:R,
+    - (PI / 4) <= x ->
+    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x <= tan y -> x <= y.
+Proof.
+  intros; case (Rtotal_order (tan x) (tan y)); intro H4;
+    [ left; apply (tan_increasing_0 x y H H0 H1 H2 H4)
+      | elim H4; intro H5;
+        [ case (Rtotal_order x y); intro H6;
+          [ left; assumption
+            | elim H6; intro H7;
+              [ right; assumption
+                | generalize (tan_increasing_1 y x H1 H2 H H0 H7); intro H8;
+                  rewrite H5 in H8; elim (Rlt_irrefl (tan y) H8) ] ]
+          | elim (Rlt_irrefl (tan x) (Rle_lt_trans (tan x) (tan y) (tan x) H3 H5)) ] ].
+Qed.
+
+Lemma tan_incr_1 :
+  forall x y:R,
+    - (PI / 4) <= x ->
+    x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x <= y -> tan x <= tan y.
+Proof.
+  intros; case (Rtotal_order x y); intro H4;
+    [ left; apply (tan_increasing_1 x y H H0 H1 H2 H4)
+      | elim H4; intro H5;
+        [ case (Rtotal_order (tan x) (tan y)); intro H6;
+          [ left; assumption
+            | elim H6; intro H7;
+              [ right; assumption
+                | generalize (tan_increasing_0 y x H1 H2 H H0 H7); intro H8;
+                  rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ]
+          | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ].
+Qed.
+
+(**********)
+Lemma sin_eq_0_1 : forall x:R, (exists k : Z, x = IZR k * PI) -> sin x = 0.
+Proof.
+  intros.
+  elim H; intros.
+  apply (Zcase_sign x0).
+  intro.
+  rewrite H1 in H0.
+  simpl in H0.
+  rewrite H0; rewrite Rmult_0_l; apply sin_0.
+  intro.
+  cut (0 <= x0)%Z.
+  intro.
+  elim (IZN x0 H2); intros.
+  rewrite H3 in H0.
+  rewrite <- INR_IZR_INZ in H0.
+  rewrite H0.
+  elim (even_odd_cor x1); intros.
+  elim H4; intro.
+  rewrite H5.
+  rewrite mult_INR.
+  simpl in |- *.
+  rewrite <- (Rplus_0_l (2 * INR x2 * PI)).
+  rewrite sin_period.
+  apply sin_0.
+  rewrite H5.
+  rewrite S_INR; rewrite mult_INR.
+  simpl in |- *.
+  rewrite Rmult_plus_distr_r.
+  rewrite Rmult_1_l; rewrite sin_plus.
+  rewrite sin_PI.
+  rewrite Rmult_0_r.
+  rewrite <- (Rplus_0_l (2 * INR x2 * PI)).
+  rewrite sin_period.
+  rewrite sin_0; ring.
+  apply le_IZR.
+  left; apply IZR_lt.
+  assert (H2 := Z.gt_lt_iff).
+  elim (H2 x0 0%Z); intros.
+  apply H3; assumption.
+  intro.
+  rewrite H0.
+  replace (sin (IZR x0 * PI)) with (- sin (- IZR x0 * PI)).
+  cut (0 <= - x0)%Z.
+  intro.
+  rewrite <- Ropp_Ropp_IZR.
+  elim (IZN (- x0) H2); intros.
+  rewrite H3.
+  rewrite <- INR_IZR_INZ.
+  elim (even_odd_cor x1); intros.
+  elim H4; intro.
+  rewrite H5.
+  rewrite mult_INR.
+  simpl in |- *.
+  rewrite <- (Rplus_0_l (2 * INR x2 * PI)).
+  rewrite sin_period.
+  rewrite sin_0; ring.
+  rewrite H5.
+  rewrite S_INR; rewrite mult_INR.
+  simpl in |- *.
+  rewrite Rmult_plus_distr_r.
+  rewrite Rmult_1_l; rewrite sin_plus.
+  rewrite sin_PI.
+  rewrite Rmult_0_r.
+  rewrite <- (Rplus_0_l (2 * INR x2 * PI)).
+  rewrite sin_period.
+  rewrite sin_0; ring.
+  apply le_IZR.
+  apply Rplus_le_reg_l with (IZR x0).
+  rewrite Rplus_0_r.
+  rewrite Ropp_Ropp_IZR.
+  rewrite Rplus_opp_r.
+  left; replace 0 with (IZR 0); [ apply IZR_lt | reflexivity ].
+  assumption.
+  rewrite <- sin_neg.
+  rewrite Ropp_mult_distr_l_reverse.
+  rewrite Ropp_involutive.
+  reflexivity.
+Qed.
+
+Lemma sin_eq_0_0 (x:R) : sin x = 0 ->  exists k : Z, x = IZR k * PI.
+Proof.
+  intros Hx.
+  destruct (euclidian_division x PI PI_neq0) as (q & r & EQ & Hr & Hr').
+  exists q.
+  rewrite <- (Rplus_0_r (_*_)). subst. apply Rplus_eq_compat_l.
+  rewrite sin_plus in Hx.
+  assert (H : sin (IZR q * PI) = 0) by (apply sin_eq_0_1; now exists q).
+  rewrite H, Rmult_0_l, Rplus_0_l in Hx.
+  destruct (Rmult_integral _ _ Hx) as [H'|H'].
+  - exfalso.
+    generalize (sin2_cos2 (IZR q * PI)).
+    rewrite H, H', Rsqr_0, Rplus_0_l.
+    intros; now apply R1_neq_R0.
+  - rewrite Rabs_right in Hr'; [|left; apply PI_RGT_0].
+    destruct Hr as [Hr | ->]; trivial.
+    exfalso.
+    generalize (sin_gt_0 r Hr Hr'). rewrite H'. apply Rlt_irrefl.
+Qed.
+
+Lemma cos_eq_0_0 (x:R) :
+  cos x = 0 ->  exists k : Z, x = IZR k * PI + PI / 2.
+Proof.
+  rewrite cos_sin. intros Hx.
+  destruct (sin_eq_0_0 (PI/2 + x) Hx) as (k,Hk). clear Hx.
+  exists (k-1)%Z. rewrite <- Z_R_minus; simpl.
+  symmetry in Hk. field_simplify [Hk]. field.
+Qed.
+
+Lemma cos_eq_0_1 (x:R) :
+  (exists k : Z, x = IZR k * PI + PI / 2) -> cos x = 0.
+Proof.
+  rewrite cos_sin. intros (k,->).
+  replace (_ + _) with (IZR k * PI + PI) by field.
+  rewrite neg_sin, <- Ropp_0. apply Ropp_eq_compat.
+  apply sin_eq_0_1. now exists k.
+Qed.
+
+Lemma sin_eq_O_2PI_0 (x:R) :
+  0 <= x -> x <= 2 * PI -> sin x = 0 ->
+  x = 0 \/ x = PI \/ x = 2 * PI.
+Proof.
+  intros Lo Hi Hx. destruct (sin_eq_0_0 x Hx) as (k,Hk). clear Hx.
+  destruct (Rtotal_order PI x) as [Hx|[Hx|Hx]].
+  - right; right.
+    clear Lo. subst.
+    f_equal. change 2 with (IZR (- (-2))). f_equal.
+    apply Z.add_move_0_l.
+    apply one_IZR_lt1.
+    rewrite plus_IZR; simpl.
+    split.
+    + replace (-1) with (-2 + 1) by ring.
+      apply Rplus_lt_compat_l.
+      apply Rmult_lt_reg_r with PI; [apply PI_RGT_0|].
+      now rewrite Rmult_1_l.
+    + apply Rle_lt_trans with 0; [|apply Rlt_0_1].
+      replace 0 with (-2 + 2) by ring.
+      apply Rplus_le_compat_l.
+      apply Rmult_le_reg_r with PI; [apply PI_RGT_0|].
+      trivial.
+  - right; left; auto.
+  - left.
+    clear Hi. subst.
+    replace 0 with (IZR 0 * PI) by (simpl; ring). f_equal. f_equal.
+    apply one_IZR_lt1.
+    split.
+    + apply Rlt_le_trans with 0;
+       [rewrite <- Ropp_0; apply Ropp_gt_lt_contravar, Rlt_0_1 | ].
+      apply Rmult_le_reg_r with PI; [apply PI_RGT_0|].
+      now rewrite Rmult_0_l.
+    + apply Rmult_lt_reg_r with PI; [apply PI_RGT_0|].
+      now rewrite Rmult_1_l.
+Qed.
+
+Lemma sin_eq_O_2PI_1 (x:R) :
+  0 <= x -> x <= 2 * PI ->
+  x = 0 \/ x = PI \/ x = 2 * PI -> sin x = 0.
+Proof.
+  intros _ _ [ -> |[ -> | -> ]].
+  - now rewrite sin_0.
+  - now rewrite sin_PI.
+  - now rewrite sin_2PI.
+Qed.
+
+Lemma cos_eq_0_2PI_0 (x:R) :
+  0 <= x -> x <= 2 * PI -> cos x = 0 ->
+  x = PI / 2 \/ x = 3 * (PI / 2).
+Proof.
+  intros Lo Hi Hx.
+  destruct (Rtotal_order x (3 * (PI / 2))) as [LT|[EQ|GT]].
+  - rewrite cos_sin in Hx.
+    assert (Lo' : 0 <= PI / 2 + x).
+    { apply Rplus_le_le_0_compat. apply Rlt_le, PI2_RGT_0. trivial. }
+    assert (Hi' : PI / 2 + x <= 2 * PI).
+    { apply Rlt_le.
+      replace (2 * PI) with (PI / 2 + 3 * (PI / 2)) by field.
+      now apply Rplus_lt_compat_l. }
+    destruct (sin_eq_O_2PI_0 (PI / 2 + x) Lo' Hi' Hx) as [H|[H|H]].
+    + exfalso.
+      apply (Rplus_le_compat_l (PI/2)) in Lo.
+      rewrite Rplus_0_r, H in Lo.
+      apply (Rlt_irrefl 0 (Rlt_le_trans 0 (PI / 2) 0 PI2_RGT_0 Lo)).
+    + left.
+      apply (Rplus_eq_compat_l (-(PI/2))) in H.
+      ring_simplify in H. rewrite H. field.
+    + right.
+      apply (Rplus_eq_compat_l (-(PI/2))) in H.
+      ring_simplify in H. rewrite H. field.
+  - now right.
+  - exfalso.
+    destruct (cos_eq_0_0 x Hx) as (k,Hk). clear Hx Lo.
+    subst.
+    assert (LT : (k < 2)%Z).
+    { apply lt_IZR. simpl.
+      apply (Rmult_lt_reg_r PI); [apply PI_RGT_0|].
+      apply Rlt_le_trans with (IZR k * PI + PI/2); trivial.
+      rewrite <- (Rplus_0_r (IZR k * PI)) at 1.
+      apply Rplus_lt_compat_l. apply PI2_RGT_0. }
+    assert (GT' : (1 < k)%Z).
+    { apply lt_IZR. simpl.
+      apply (Rmult_lt_reg_r PI); [apply PI_RGT_0|rewrite Rmult_1_l].
+      replace (3*(PI/2)) with (PI/2 + PI) in GT by field.
+      rewrite Rplus_comm in GT.
+      now apply Rplus_lt_reg_r in GT. }
+    omega.
+Qed.
+
+Lemma cos_eq_0_2PI_1 (x:R) :
+  0 <= x -> x <= 2 * PI ->
+  x = PI / 2 \/ x = 3 * (PI / 2) -> cos x = 0.
+Proof.
+ intros Lo Hi [ -> | -> ].
+ - now rewrite cos_PI2.
+ - now rewrite cos_3PI2.
+Qed.
diff --git a/theories/Reals/Rtrigo_alt.v b/theories/Reals/Rtrigo_alt.v
index 3ab7d598..23b8e847 100644
--- a/theories/Reals/Rtrigo_alt.v
+++ b/theories/Reals/Rtrigo_alt.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -10,7 +10,7 @@ Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
 Require Import Rtrigo_def.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (***************************************************************)
 (** Using series definitions of cos and sin                    *)
@@ -27,7 +27,8 @@ Definition sin_approx (a:R) (n:nat) : R := sum_f_R0 (sin_term a) n.
 Definition cos_approx (a:R) (n:nat) : R := sum_f_R0 (cos_term a) n.
 
 (**********)
-Lemma PI_4 : PI <= 4.
+(*
+Lemma Alt_PI_4 : Alt_PI <= 4.
 Proof.
   assert (H0 := PI_ineq 0).
   elim H0; clear H0; intros _ H0.
@@ -37,20 +38,20 @@ Proof.
   apply Rinv_0_lt_compat; prove_sup0.
   rewrite <- Rinv_l_sym; [ rewrite Rmult_comm; assumption | discrR ].
 Qed.
-
+*)
 (**********)
-Theorem sin_bound :
+Theorem pre_sin_bound :
   forall (a:R) (n:nat),
     0 <= a ->
-    a <= PI -> sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)).
+    a <= 4 -> sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)).
 Proof.
   intros; case (Req_dec a 0); intro Hyp_a.
-  rewrite Hyp_a; rewrite sin_0; split; right; unfold sin_approx in |- *;
-    apply sum_eq_R0 || (symmetry  in |- *; apply sum_eq_R0);
-      intros; unfold sin_term in |- *; rewrite pow_add;
-        simpl in |- *; unfold Rdiv in |- *; rewrite Rmult_0_l;
+  rewrite Hyp_a; rewrite sin_0; split; right; unfold sin_approx;
+    apply sum_eq_R0 || (symmetry ; apply sum_eq_R0);
+      intros; unfold sin_term; rewrite pow_add;
+        simpl; unfold Rdiv; rewrite Rmult_0_l;
           ring.
-  unfold sin_approx in |- *; cut (0 < a).
+  unfold sin_approx; cut (0 < a).
   intro Hyp_a_pos.
   rewrite (decomp_sum (sin_term a) (2 * n + 1)).
   rewrite (decomp_sum (sin_term a) (2 * (n + 1))).
@@ -75,22 +76,22 @@ Proof.
       - sum_f_R0 (tg_alt Un) (S (2 * n))).
   intro; apply H2.
   apply alternated_series_ineq.
-  unfold Un_decreasing, Un in |- *; intro;
+  unfold Un_decreasing, Un; intro;
     cut ((2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))).
   intro; rewrite H3.
   replace (a ^ S (S (2 * S n0 + 1))) with (a ^ (2 * S n0 + 1) * (a * a)).
-  unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l.
+  unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
   left; apply pow_lt; assumption.
   apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n0 + 1))))).
-  rewrite <- H3; apply lt_INR_0; apply neq_O_lt; red in |- *; intro;
-    assert (H5 := sym_eq H4); elim (fact_neq_0 _ H5).
+  rewrite <- H3; apply lt_INR_0; apply neq_O_lt; red; intro;
+    assert (H5 := eq_sym H4); elim (fact_neq_0 _ H5).
   rewrite <- H3; rewrite (Rmult_comm (INR (fact (2 * S (S n0) + 1))));
     rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r; rewrite H3; do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
     repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
   rewrite Rmult_1_r.
   do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR;
-    simpl in |- *;
+    simpl;
       replace
       (((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1 + 1) *
         ((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1)) with
@@ -100,12 +101,12 @@ Proof.
   replace 16 with (Rsqr 4); [ idtac | ring_Rsqr ].
   replace (a * a) with (Rsqr a); [ idtac | reflexivity ].
   apply Rsqr_incr_1.
-  apply Rle_trans with PI; [ assumption | apply PI_4 ].
+  assumption.
   assumption.
   left; prove_sup0.
   rewrite <- (Rplus_0_r 16); replace 20 with (16 + 4);
     [ apply Rplus_le_compat_l; left; prove_sup0 | ring ].
-  rewrite <- (Rplus_comm 20); pattern 20 at 1 in |- *; rewrite <- Rplus_0_r;
+  rewrite <- (Rplus_comm 20); pattern 20 at 1; rewrite <- Rplus_0_r;
     apply Rplus_le_compat_l.
   apply Rplus_le_le_0_compat.
   repeat apply Rmult_le_pos.
@@ -118,14 +119,14 @@ Proof.
   replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ].
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
-  simpl in |- *; ring.
+  simpl; ring.
   ring.
-  assert (H3 := cv_speed_pow_fact a); unfold Un in |- *; unfold Un_cv in H3;
-    unfold R_dist in H3; unfold Un_cv in |- *; unfold R_dist in |- *;
+  assert (H3 := cv_speed_pow_fact a); unfold Un; unfold Un_cv in H3;
+    unfold R_dist in H3; unfold Un_cv; unfold R_dist;
       intros; elim (H3 eps H4); intros N H5.
   exists N; intros; apply H5.
   replace (2 * S n0 + 1)%nat with (S (2 * S n0)).
-  unfold ge in |- *; apply le_trans with (2 * S n0)%nat.
+  unfold ge; apply le_trans with (2 * S n0)%nat.
   apply le_trans with (2 * S N)%nat.
   apply le_trans with (2 * N)%nat.
   apply le_n_2n.
@@ -136,49 +137,49 @@ Proof.
   assert (X := exist_sin (Rsqr a)); elim X; intros.
   cut (x = sin a / a).
   intro; rewrite H3 in p; unfold sin_in in p; unfold infinite_sum in p;
-    unfold R_dist in p; unfold Un_cv in |- *; unfold R_dist in |- *;
+    unfold R_dist in p; unfold Un_cv; unfold R_dist;
       intros.
   cut (0 < eps / Rabs a).
   intro; elim (p _ H5); intros N H6.
   exists N; intros.
   replace (sum_f_R0 (tg_alt Un) n0) with
   (a * (1 - sum_f_R0 (fun i:nat => sin_n i * Rsqr a ^ i) (S n0))).
-  unfold Rminus in |- *; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r;
+  unfold Rminus; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r;
     rewrite Ropp_plus_distr; rewrite Ropp_involutive;
       repeat rewrite Rplus_assoc; rewrite (Rplus_comm a);
         rewrite (Rplus_comm (- a)); repeat rewrite Rplus_assoc;
           rewrite Rplus_opp_l; rewrite Rplus_0_r; apply Rmult_lt_reg_l with (/ Rabs a).
   apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
-  pattern (/ Rabs a) at 1 in |- *; rewrite <- (Rabs_Rinv a Hyp_a).
+  pattern (/ Rabs a) at 1; rewrite <- (Rabs_Rinv a Hyp_a).
   rewrite <- Rabs_mult; rewrite Rmult_plus_distr_l; rewrite <- Rmult_assoc;
     rewrite <- Rinv_l_sym; [ rewrite Rmult_1_l | assumption ];
       rewrite (Rmult_comm (/ a)); rewrite (Rmult_comm (/ Rabs a));
         rewrite <- Rabs_Ropp; rewrite Ropp_plus_distr; rewrite Ropp_involutive;
-          unfold Rminus, Rdiv in H6; apply H6; unfold ge in |- *;
+          unfold Rminus, Rdiv in H6; apply H6; unfold ge;
             apply le_trans with n0; [ exact H7 | apply le_n_Sn ].
   rewrite (decomp_sum (fun i:nat => sin_n i * Rsqr a ^ i) (S n0)).
   replace (sin_n 0) with 1.
-  simpl in |- *; rewrite Rmult_1_r; unfold Rminus in |- *;
+  simpl; rewrite Rmult_1_r; unfold Rminus;
     rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r;
       rewrite Rplus_0_l; rewrite Ropp_mult_distr_r_reverse;
         rewrite <- Ropp_mult_distr_l_reverse; rewrite scal_sum;
           apply sum_eq.
-  intros; unfold sin_n, Un, tg_alt in |- *;
+  intros; unfold sin_n, Un, tg_alt;
     replace ((-1) ^ S i) with (- (-1) ^ i).
   replace (a ^ (2 * S i + 1)) with (Rsqr a * Rsqr a ^ i * a).
-  unfold Rdiv in |- *; ring.
-  rewrite pow_add; rewrite pow_Rsqr; simpl in |- *; ring.
-  simpl in |- *; ring.
-  unfold sin_n in |- *; unfold Rdiv in |- *; simpl in |- *; rewrite Rinv_1;
+  unfold Rdiv; ring.
+  rewrite pow_add; rewrite pow_Rsqr; simpl; ring.
+  simpl; ring.
+  unfold sin_n; unfold Rdiv; simpl; rewrite Rinv_1;
     rewrite Rmult_1_r; reflexivity.
   apply lt_O_Sn.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   assumption.
   apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
-  unfold sin in |- *; case (exist_sin (Rsqr a)).
+  unfold sin; case (exist_sin (Rsqr a)).
   intros; cut (x = x0).
-  intro; rewrite H3; unfold Rdiv in |- *.
-  symmetry  in |- *; apply Rinv_r_simpl_m; assumption.
+  intro; rewrite H3; unfold Rdiv.
+  symmetry ; apply Rinv_r_simpl_m; assumption.
   unfold sin_in in p; unfold sin_in in s; eapply uniqueness_sum.
   apply p.
   apply s.
@@ -187,16 +188,16 @@ Proof.
   split; apply Ropp_le_contravar; assumption.
   replace (- sum_f_R0 (tg_alt Un) (S (2 * n))) with
   (-1 * sum_f_R0 (tg_alt Un) (S (2 * n))); [ rewrite scal_sum | ring ].
-  apply sum_eq; intros; unfold sin_term, Un, tg_alt in |- *;
+  apply sum_eq; intros; unfold sin_term, Un, tg_alt;
     replace ((-1) ^ S i) with (-1 * (-1) ^ i).
-  unfold Rdiv in |- *; ring.
+  unfold Rdiv; ring.
   reflexivity.
   replace (- sum_f_R0 (tg_alt Un) (2 * n)) with
   (-1 * sum_f_R0 (tg_alt Un) (2 * n)); [ rewrite scal_sum | ring ].
   apply sum_eq; intros.
-  unfold sin_term, Un, tg_alt in |- *;
+  unfold sin_term, Un, tg_alt;
     replace ((-1) ^ S i) with (-1 * (-1) ^ i).
-  unfold Rdiv in |- *; ring.
+  unfold Rdiv; ring.
   reflexivity.
   replace (2 * (n + 1))%nat with (S (S (2 * n))).
   reflexivity.
@@ -212,7 +213,7 @@ Proof.
   apply Rplus_le_reg_l with (- a).
   rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
     rewrite (Rplus_comm (- a)); apply H3.
-  unfold sin_term in |- *; simpl in |- *; unfold Rdiv in |- *; rewrite Rinv_1;
+  unfold sin_term; simpl; unfold Rdiv; rewrite Rinv_1;
     ring.
   replace (2 * (n + 1))%nat with (S (S (2 * n))).
   apply lt_O_Sn.
@@ -220,27 +221,26 @@ Proof.
   replace (2 * n + 1)%nat with (S (2 * n)).
   apply lt_O_Sn.
   ring.
-  inversion H; [ assumption | elim Hyp_a; symmetry  in |- *; assumption ].
+  inversion H; [ assumption | elim Hyp_a; symmetry ; assumption ].
 Qed.
 
 (**********)
-Lemma cos_bound :
+Lemma pre_cos_bound :
   forall (a:R) (n:nat),
-    - PI / 2 <= a ->
-    a <= PI / 2 ->
+    - 2 <= a -> a <= 2 ->
     cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1)).
 Proof.
   cut
     ((forall (a:R) (n:nat),
       0 <= a ->
-      a <= PI / 2 ->
+      a <= 2 ->
       cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))) ->
     forall (a:R) (n:nat),
-      - PI / 2 <= a ->
-      a <= PI / 2 ->
+      - 2 <= a ->
+      a <= 2 ->
       cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))).
   intros H a n; apply H.
-  intros; unfold cos_approx in |- *.
+  intros; unfold cos_approx.
   rewrite (decomp_sum (cos_term a0) (2 * n0 + 1)).
   rewrite (decomp_sum (cos_term a0) (2 * (n0 + 1))).
   replace (cos_term a0 0) with 1.
@@ -266,21 +266,21 @@ Proof.
       - sum_f_R0 (tg_alt Un) (S (2 * n0))).
   intro; apply H3.
   apply alternated_series_ineq.
-  unfold Un_decreasing in |- *; intro; unfold Un in |- *.
+  unfold Un_decreasing; intro; unfold Un.
   cut ((2 * S (S n1))%nat = S (S (2 * S n1))).
   intro; rewrite H4;
     replace (a0 ^ S (S (2 * S n1))) with (a0 ^ (2 * S n1) * (a0 * a0)).
-  unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l.
+  unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
   apply pow_le; assumption.
   apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n1))))).
-  rewrite <- H4; apply lt_INR_0; apply neq_O_lt; red in |- *; intro;
-    assert (H6 := sym_eq H5); elim (fact_neq_0 _ H6).
+  rewrite <- H4; apply lt_INR_0; apply neq_O_lt; red; intro;
+    assert (H6 := eq_sym H5); elim (fact_neq_0 _ H6).
   rewrite <- H4; rewrite (Rmult_comm (INR (fact (2 * S (S n1)))));
     rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r; rewrite H4; do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
     repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
   rewrite Rmult_1_r; do 2 rewrite S_INR; rewrite mult_INR; repeat rewrite S_INR;
-    simpl in |- *;
+    simpl;
       replace
       (((0 + 1 + 1) * (INR n1 + 1) + 1 + 1) * ((0 + 1 + 1) * (INR n1 + 1) + 1))
     with (4 * INR n1 * INR n1 + 14 * INR n1 + 12); [ idtac | ring ].
@@ -289,18 +289,13 @@ Proof.
   replace 4 with (Rsqr 2); [ idtac | ring_Rsqr ].
   replace (a0 * a0) with (Rsqr a0); [ idtac | reflexivity ].
   apply Rsqr_incr_1.
-  apply Rle_trans with (PI / 2).
   assumption.
-  unfold Rdiv in |- *; apply Rmult_le_reg_l with 2.
-  prove_sup0.
-  rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m.
-  replace 4 with 4; [ apply PI_4 | ring ].
   discrR.
   assumption.
   left; prove_sup0.
-  pattern 4 at 1 in |- *; rewrite <- Rplus_0_r; replace 12 with (4 + 8);
+  pattern 4 at 1; rewrite <- Rplus_0_r; replace 12 with (4 + 8);
     [ apply Rplus_le_compat_l; left; prove_sup0 | ring ].
-  rewrite <- (Rplus_comm 12); pattern 12 at 1 in |- *; rewrite <- Rplus_0_r;
+  rewrite <- (Rplus_comm 12); pattern 12 at 1; rewrite <- Rplus_0_r;
     apply Rplus_le_compat_l.
   apply Rplus_le_le_0_compat.
   repeat apply Rmult_le_pos.
@@ -313,12 +308,12 @@ Proof.
   replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ].
   apply INR_fact_neq_0.
   apply INR_fact_neq_0.
-  simpl in |- *; ring.
+  simpl; ring.
   ring.
-  assert (H4 := cv_speed_pow_fact a0); unfold Un in |- *; unfold Un_cv in H4;
-    unfold R_dist in H4; unfold Un_cv in |- *; unfold R_dist in |- *;
+  assert (H4 := cv_speed_pow_fact a0); unfold Un; unfold Un_cv in H4;
+    unfold R_dist in H4; unfold Un_cv; unfold R_dist;
       intros; elim (H4 eps H5); intros N H6; exists N; intros.
-  apply H6; unfold ge in |- *; apply le_trans with (2 * S N)%nat.
+  apply H6; unfold ge; apply le_trans with (2 * S N)%nat.
   apply le_trans with (2 * N)%nat.
   apply le_n_2n.
   apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_Sn.
@@ -326,40 +321,40 @@ Proof.
   assert (X := exist_cos (Rsqr a0)); elim X; intros.
   cut (x = cos a0).
   intro; rewrite H4 in p; unfold cos_in in p; unfold infinite_sum in p;
-    unfold R_dist in p; unfold Un_cv in |- *; unfold R_dist in |- *;
+    unfold R_dist in p; unfold Un_cv; unfold R_dist;
       intros.
   elim (p _ H5); intros N H6.
   exists N; intros.
   replace (sum_f_R0 (tg_alt Un) n1) with
   (1 - sum_f_R0 (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)).
-  unfold Rminus in |- *; rewrite Ropp_plus_distr; rewrite Ropp_involutive;
+  unfold Rminus; rewrite Ropp_plus_distr; rewrite Ropp_involutive;
     repeat rewrite Rplus_assoc; rewrite (Rplus_comm 1);
       rewrite (Rplus_comm (-1)); repeat rewrite Rplus_assoc;
         rewrite Rplus_opp_l; rewrite Rplus_0_r; rewrite <- Rabs_Ropp;
           rewrite Ropp_plus_distr; rewrite Ropp_involutive;
             unfold Rminus in H6; apply H6.
-  unfold ge in |- *; apply le_trans with n1.
+  unfold ge; apply le_trans with n1.
   exact H7.
   apply le_n_Sn.
   rewrite (decomp_sum (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)).
   replace (cos_n 0) with 1.
-  simpl in |- *; rewrite Rmult_1_r; unfold Rminus in |- *;
+  simpl; rewrite Rmult_1_r; unfold Rminus;
     rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r;
       rewrite Rplus_0_l;
         replace (- sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1)
     with
       (-1 * sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1);
       [ idtac | ring ]; rewrite scal_sum; apply sum_eq;
-        intros; unfold cos_n, Un, tg_alt in |- *.
+        intros; unfold cos_n, Un, tg_alt.
   replace ((-1) ^ S i) with (- (-1) ^ i).
   replace (a0 ^ (2 * S i)) with (Rsqr a0 * Rsqr a0 ^ i).
-  unfold Rdiv in |- *; ring.
+  unfold Rdiv; ring.
   rewrite pow_Rsqr; reflexivity.
-  simpl in |- *; ring.
-  unfold cos_n in |- *; unfold Rdiv in |- *; simpl in |- *; rewrite Rinv_1;
+  simpl; ring.
+  unfold cos_n; unfold Rdiv; simpl; rewrite Rinv_1;
     rewrite Rmult_1_r; reflexivity.
   apply lt_O_Sn.
-  unfold cos in |- *; case (exist_cos (Rsqr a0)); intros; unfold cos_in in p;
+  unfold cos; case (exist_cos (Rsqr a0)); intros; unfold cos_in in p;
     unfold cos_in in c; eapply uniqueness_sum.
   apply p.
   apply c.
@@ -368,15 +363,15 @@ Proof.
   split; apply Ropp_le_contravar; assumption.
   replace (- sum_f_R0 (tg_alt Un) (S (2 * n0))) with
   (-1 * sum_f_R0 (tg_alt Un) (S (2 * n0))); [ rewrite scal_sum | ring ].
-  apply sum_eq; intros; unfold cos_term, Un, tg_alt in |- *;
+  apply sum_eq; intros; unfold cos_term, Un, tg_alt;
     replace ((-1) ^ S i) with (-1 * (-1) ^ i).
-  unfold Rdiv in |- *; ring.
+  unfold Rdiv; ring.
   reflexivity.
   replace (- sum_f_R0 (tg_alt Un) (2 * n0)) with
   (-1 * sum_f_R0 (tg_alt Un) (2 * n0)); [ rewrite scal_sum | ring ];
-  apply sum_eq; intros; unfold cos_term, Un, tg_alt in |- *;
+  apply sum_eq; intros; unfold cos_term, Un, tg_alt;
     replace ((-1) ^ S i) with (-1 * (-1) ^ i).
-  unfold Rdiv in |- *; ring.
+  unfold Rdiv; ring.
   reflexivity.
   replace (2 * (n0 + 1))%nat with (S (S (2 * n0))).
   reflexivity.
@@ -391,7 +386,7 @@ Proof.
   apply Rplus_le_reg_l with (-1).
   rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
     rewrite (Rplus_comm (-1)); apply H4.
-  unfold cos_term in |- *; simpl in |- *; unfold Rdiv in |- *; rewrite Rinv_1;
+  unfold cos_term; simpl; unfold Rdiv; rewrite Rinv_1;
     ring.
   replace (2 * (n0 + 1))%nat with (S (S (2 * n0))).
   apply lt_O_Sn.
@@ -407,11 +402,9 @@ Proof.
   intro; cut (forall (x:R) (n:nat), cos_approx x n = cos_approx (- x) n).
   intro; rewrite H3; rewrite (H3 a (2 * (n + 1))%nat); rewrite cos_sym; apply H.
   left; assumption.
-  rewrite <- (Ropp_involutive (PI / 2)); apply Ropp_le_contravar;
-    unfold Rdiv in |- *; unfold Rdiv in H0; rewrite <- Ropp_mult_distr_l_reverse;
-      exact H0.
-  intros; unfold cos_approx in |- *; apply sum_eq; intros;
-    unfold cos_term in |- *; do 2 rewrite pow_Rsqr; rewrite Rsqr_neg;
-      unfold Rdiv in |- *; reflexivity.
+  rewrite <- (Ropp_involutive 2); apply Ropp_le_contravar; exact H0.
+  intros; unfold cos_approx; apply sum_eq; intros;
+    unfold cos_term; do 2 rewrite pow_Rsqr; rewrite Rsqr_neg;
+      unfold Rdiv; reflexivity.
   apply Ropp_0_gt_lt_contravar; assumption.
 Qed.
diff --git a/theories/Reals/Rtrigo_calc.v b/theories/Reals/Rtrigo_calc.v
index 587c2424..a1a3b007 100644
--- a/theories/Reals/Rtrigo_calc.v
+++ b/theories/Reals/Rtrigo_calc.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,13 +9,13 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
-Require Import Rtrigo.
+Require Import Rtrigo1.
 Require Import R_sqrt.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 Lemma tan_PI : tan PI = 0.
 Proof.
-  unfold tan in |- *; rewrite sin_PI; rewrite cos_PI; unfold Rdiv in |- *;
+  unfold tan; rewrite sin_PI; rewrite cos_PI; unfold Rdiv;
     apply Rmult_0_l.
 Qed.
 
@@ -23,12 +23,12 @@ Lemma sin_3PI2 : sin (3 * (PI / 2)) = -1.
 Proof.
   replace (3 * (PI / 2)) with (PI + PI / 2).
   rewrite sin_plus; rewrite sin_PI; rewrite cos_PI; rewrite sin_PI2; ring.
-  pattern PI at 1 in |- *; rewrite (double_var PI); ring.
+  pattern PI at 1; rewrite (double_var PI); ring.
 Qed.
 
 Lemma tan_2PI : tan (2 * PI) = 0.
 Proof.
-  unfold tan in |- *; rewrite sin_2PI; unfold Rdiv in |- *; apply Rmult_0_l.
+  unfold tan; rewrite sin_2PI; unfold Rdiv; apply Rmult_0_l.
 Qed.
 
 Lemma sin_cos_PI4 : sin (PI / 4) = cos (PI / 4).
@@ -37,9 +37,9 @@ Proof with trivial.
   replace (PI / 2 + PI / 4) with (- (PI / 4) + PI)...
   rewrite neg_sin; rewrite sin_neg; ring...
   cut (PI = PI / 2 + PI / 2); [ intro | apply double_var ]...
-  pattern PI at 2 3 in |- *; rewrite H; pattern PI at 2 3 in |- *; rewrite H...
+  pattern PI at 2 3; rewrite H; pattern PI at 2 3; rewrite H...
   assert (H0 : 2 <> 0);
-    [ discrR | unfold Rdiv in |- *; rewrite Rinv_mult_distr; try ring ]...
+    [ discrR | unfold Rdiv; rewrite Rinv_mult_distr; try ring ]...
 Qed.
 
 Lemma sin_PI3_cos_PI6 : sin (PI / 3) = cos (PI / 6).
@@ -51,7 +51,7 @@ Proof with trivial.
   assert (H2 : 2 <> 0); [ discrR | idtac ]...
   apply Rmult_eq_reg_l with 6...
   rewrite Rmult_minus_distr_l; repeat rewrite (Rmult_comm 6)...
-  unfold Rdiv in |- *; repeat rewrite Rmult_assoc...
+  unfold Rdiv; repeat rewrite Rmult_assoc...
   rewrite <- Rinv_l_sym...
   rewrite (Rmult_comm (/ 3)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym...
   rewrite (Rmult_comm PI); repeat rewrite Rmult_1_r;
@@ -68,7 +68,7 @@ Proof with trivial.
   assert (H2 : 2 <> 0); [ discrR | idtac ]...
   apply Rmult_eq_reg_l with 6...
   rewrite Rmult_minus_distr_l; repeat rewrite (Rmult_comm 6)...
-  unfold Rdiv in |- *; repeat rewrite Rmult_assoc...
+  unfold Rdiv; repeat rewrite Rmult_assoc...
   rewrite <- Rinv_l_sym...
   rewrite (Rmult_comm (/ 3)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym...
   rewrite (Rmult_comm PI); repeat rewrite Rmult_1_r;
@@ -78,13 +78,13 @@ Qed.
 
 Lemma PI6_RGT_0 : 0 < PI / 6.
 Proof.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ].
 Qed.
 
 Lemma PI6_RLT_PI2 : PI / 6 < PI / 2.
 Proof.
-  unfold Rdiv in |- *; apply Rmult_lt_compat_l.
+  unfold Rdiv; apply Rmult_lt_compat_l.
   apply PI_RGT_0.
   apply Rinv_lt_contravar; prove_sup.
 Qed.
@@ -97,11 +97,11 @@ Proof with trivial.
   (2 * sin (PI / 6) * cos (PI / 6))...
   rewrite <- sin_2a; replace (2 * (PI / 6)) with (PI / 3)...
   rewrite sin_PI3_cos_PI6...
-  unfold Rdiv in |- *; rewrite Rmult_1_l; rewrite Rmult_assoc;
-    pattern 2 at 2 in |- *; rewrite (Rmult_comm 2); rewrite Rmult_assoc;
+  unfold Rdiv; rewrite Rmult_1_l; rewrite Rmult_assoc;
+    pattern 2 at 2; rewrite (Rmult_comm 2); rewrite Rmult_assoc;
       rewrite <- Rinv_l_sym...
   rewrite Rmult_1_r...
-  unfold Rdiv in |- *; rewrite Rinv_mult_distr...
+  unfold Rdiv; rewrite Rinv_mult_distr...
   rewrite (Rmult_comm (/ 2)); rewrite (Rmult_comm 2);
     repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym...
   rewrite Rmult_1_r...
@@ -119,7 +119,7 @@ Lemma sqrt2_neq_0 : sqrt 2 <> 0.
 Proof.
   assert (Hyp : 0 < 2);
     [ prove_sup0
-      | generalize (Rlt_le 0 2 Hyp); intro H1; red in |- *; intro H2;
+      | generalize (Rlt_le 0 2 Hyp); intro H1; red; intro H2;
         generalize (sqrt_eq_0 2 H1 H2); intro H; absurd (2 = 0);
           [ discrR | assumption ] ].
 Qed.
@@ -137,7 +137,7 @@ Proof.
     [ discrR
       | assert (Hyp : 0 < 3);
         [ prove_sup0
-          | generalize (Rlt_le 0 3 Hyp); intro H1; red in |- *; intro H2;
+          | generalize (Rlt_le 0 3 Hyp); intro H1; red; intro H2;
             generalize (sqrt_eq_0 3 H1 H2); intro H; absurd (3 = 0);
               [ discrR | assumption ] ] ].
 Qed.
@@ -150,7 +150,7 @@ Proof.
         intro H2;
           [ assumption
             | absurd (0 = sqrt 2);
-              [ apply (sym_not_eq (A:=R)); apply sqrt2_neq_0 | assumption ] ] ].
+              [ apply (not_eq_sym (A:=R)); apply sqrt2_neq_0 | assumption ] ] ].
 Qed.
 
 Lemma Rlt_sqrt3_0 : 0 < sqrt 3.
@@ -162,7 +162,7 @@ Proof.
           [ prove_sup0
             | generalize (Rlt_le 0 3 Hyp2); intro H2;
               generalize (lt_INR_0 1 (neq_O_lt 1 H0));
-                unfold INR in |- *; intro H3;
+                unfold INR; intro H3;
                   generalize (Rplus_lt_compat_l 2 0 1 H3);
                     rewrite Rplus_comm; rewrite Rplus_0_l; replace (2 + 1) with 3;
                       [ intro H4; generalize (sqrt_lt_1 2 3 H1 H2 H4); clear H3; intro H3;
@@ -173,7 +173,7 @@ Qed.
 
 Lemma PI4_RGT_0 : 0 < PI / 4.
 Proof.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ].
 Qed.
 
@@ -189,17 +189,17 @@ Proof with trivial.
   rewrite Rsqr_div...
   rewrite Rsqr_1; rewrite Rsqr_sqrt...
   assert (H : 2 <> 0); [ discrR | idtac ]...
-  unfold Rsqr in |- *; pattern (cos (PI / 4)) at 1 in |- *;
+  unfold Rsqr; pattern (cos (PI / 4)) at 1;
     rewrite <- sin_cos_PI4;
       replace (sin (PI / 4) * cos (PI / 4)) with
       (1 / 2 * (2 * sin (PI / 4) * cos (PI / 4)))...
   rewrite <- sin_2a; replace (2 * (PI / 4)) with (PI / 2)...
   rewrite sin_PI2...
   apply Rmult_1_r...
-  unfold Rdiv in |- *; rewrite (Rmult_comm 2); rewrite Rinv_mult_distr...
+  unfold Rdiv; rewrite (Rmult_comm 2); rewrite Rinv_mult_distr...
   repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym...
   rewrite Rmult_1_r...
-  unfold Rdiv in |- *; rewrite Rmult_1_l; repeat rewrite <- Rmult_assoc...
+  unfold Rdiv; rewrite Rmult_1_l; repeat rewrite <- Rmult_assoc...
   rewrite <- Rinv_l_sym...
   rewrite Rmult_1_l...
   left; prove_sup...
@@ -213,18 +213,18 @@ Qed.
 
 Lemma tan_PI4 : tan (PI / 4) = 1.
 Proof.
-  unfold tan in |- *; rewrite sin_cos_PI4.
-  unfold Rdiv in |- *; apply Rinv_r.
-  change (cos (PI / 4) <> 0) in |- *; rewrite cos_PI4; apply R1_sqrt2_neq_0.
+  unfold tan; rewrite sin_cos_PI4.
+  unfold Rdiv; apply Rinv_r.
+  change (cos (PI / 4) <> 0); rewrite cos_PI4; apply R1_sqrt2_neq_0.
 Qed.
 
 Lemma cos3PI4 : cos (3 * (PI / 4)) = -1 / sqrt 2.
 Proof with trivial.
   replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4))...
   rewrite cos_shift; rewrite sin_neg; rewrite sin_PI4...
-  unfold Rdiv in |- *; rewrite Ropp_mult_distr_l_reverse...
-  unfold Rminus in |- *; rewrite Ropp_involutive; pattern PI at 1 in |- *;
-    rewrite double_var; unfold Rdiv in |- *; rewrite Rmult_plus_distr_r;
+  unfold Rdiv; rewrite Ropp_mult_distr_l_reverse...
+  unfold Rminus; rewrite Ropp_involutive; pattern PI at 1;
+    rewrite double_var; unfold Rdiv; rewrite Rmult_plus_distr_r;
       repeat rewrite Rmult_assoc; rewrite <- Rinv_mult_distr;
         [ ring | discrR | discrR ]...
 Qed.
@@ -233,8 +233,8 @@ Lemma sin3PI4 : sin (3 * (PI / 4)) = 1 / sqrt 2.
 Proof with trivial.
   replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4))...
   rewrite sin_shift; rewrite cos_neg; rewrite cos_PI4...
-  unfold Rminus in |- *; rewrite Ropp_involutive; pattern PI at 1 in |- *;
-    rewrite double_var; unfold Rdiv in |- *; rewrite Rmult_plus_distr_r;
+  unfold Rminus; rewrite Ropp_involutive; pattern PI at 1;
+    rewrite double_var; unfold Rdiv; rewrite Rmult_plus_distr_r;
       repeat rewrite Rmult_assoc; rewrite <- Rinv_mult_distr;
         [ ring | discrR | discrR ]...
 Qed.
@@ -251,8 +251,8 @@ Proof with trivial.
   assert (H : 2 <> 0); [ discrR | idtac ]...
   assert (H1 : 4 <> 0); [ apply prod_neq_R0 | idtac ]...
   rewrite Rsqr_div...
-  rewrite cos2; unfold Rsqr in |- *; rewrite sin_PI6; rewrite sqrt_def...
-  unfold Rdiv in |- *; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4...
+  rewrite cos2; unfold Rsqr; rewrite sin_PI6; rewrite sqrt_def...
+  unfold Rdiv; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4...
   rewrite Rmult_minus_distr_l; rewrite (Rmult_comm 3);
     repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym...
   rewrite Rmult_1_l; rewrite Rmult_1_r...
@@ -265,14 +265,14 @@ Qed.
 
 Lemma tan_PI6 : tan (PI / 6) = 1 / sqrt 3.
 Proof.
-  unfold tan in |- *; rewrite sin_PI6; rewrite cos_PI6; unfold Rdiv in |- *;
+  unfold tan; rewrite sin_PI6; rewrite cos_PI6; unfold Rdiv;
     repeat rewrite Rmult_1_l; rewrite Rinv_mult_distr.
   rewrite Rinv_involutive.
   rewrite (Rmult_comm (/ 2)); rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
   apply Rmult_1_r.
   discrR.
   discrR.
-  red in |- *; intro; assert (H1 := Rlt_sqrt3_0); rewrite H in H1;
+  red; intro; assert (H1 := Rlt_sqrt3_0); rewrite H in H1;
     elim (Rlt_irrefl 0 H1).
   apply Rinv_neq_0_compat; discrR.
 Qed.
@@ -289,7 +289,7 @@ Qed.
 
 Lemma tan_PI3 : tan (PI / 3) = sqrt 3.
 Proof.
-  unfold tan in |- *; rewrite sin_PI3; rewrite cos_PI3; unfold Rdiv in |- *;
+  unfold tan; rewrite sin_PI3; rewrite cos_PI3; unfold Rdiv;
     rewrite Rmult_1_l; rewrite Rinv_involutive.
   rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   apply Rmult_1_r.
@@ -300,7 +300,7 @@ Qed.
 Lemma sin_2PI3 : sin (2 * (PI / 3)) = sqrt 3 / 2.
 Proof.
   rewrite double; rewrite sin_plus; rewrite sin_PI3; rewrite cos_PI3;
-    unfold Rdiv in |- *; repeat rewrite Rmult_1_l; rewrite (Rmult_comm (/ 2));
+    unfold Rdiv; repeat rewrite Rmult_1_l; rewrite (Rmult_comm (/ 2));
       repeat rewrite <- Rmult_assoc; rewrite double_var;
         reflexivity.
 Qed.
@@ -310,12 +310,12 @@ Proof with trivial.
   assert (H : 2 <> 0); [ discrR | idtac ]...
   assert (H0 : 4 <> 0); [ apply prod_neq_R0 | idtac ]...
   rewrite double; rewrite cos_plus; rewrite sin_PI3; rewrite cos_PI3;
-    unfold Rdiv in |- *; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4...
+    unfold Rdiv; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4...
   rewrite Rmult_minus_distr_l; repeat rewrite Rmult_assoc;
     rewrite (Rmult_comm 2)...
   repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym...
   rewrite Rmult_1_r; rewrite <- Rinv_r_sym...
-  pattern 2 at 4 in |- *; rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc;
+  pattern 2 at 4; rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc;
     rewrite <- Rinv_l_sym...
   rewrite Rmult_1_r; rewrite Ropp_mult_distr_r_reverse; rewrite Rmult_1_r...
   rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym...
@@ -329,7 +329,7 @@ Qed.
 Lemma tan_2PI3 : tan (2 * (PI / 3)) = - sqrt 3.
 Proof with trivial.
   assert (H : 2 <> 0); [ discrR | idtac ]...
-  unfold tan in |- *; rewrite sin_2PI3; rewrite cos_2PI3; unfold Rdiv in |- *;
+  unfold tan; rewrite sin_2PI3; rewrite cos_2PI3; unfold Rdiv;
     rewrite Ropp_mult_distr_l_reverse; rewrite Rmult_1_l;
       rewrite <- Ropp_inv_permute...
   rewrite Rinv_involutive...
@@ -341,21 +341,21 @@ Qed.
 Lemma cos_5PI4 : cos (5 * (PI / 4)) = -1 / sqrt 2.
 Proof with trivial.
   replace (5 * (PI / 4)) with (PI / 4 + PI)...
-  rewrite neg_cos; rewrite cos_PI4; unfold Rdiv in |- *;
+  rewrite neg_cos; rewrite cos_PI4; unfold Rdiv;
     rewrite Ropp_mult_distr_l_reverse...
-  pattern PI at 2 in |- *; rewrite double_var; pattern PI at 2 3 in |- *;
+  pattern PI at 2; rewrite double_var; pattern PI at 2 3;
     rewrite double_var; assert (H : 2 <> 0);
-      [ discrR | unfold Rdiv in |- *; repeat rewrite Rinv_mult_distr; try ring ]...
+      [ discrR | unfold Rdiv; repeat rewrite Rinv_mult_distr; try ring ]...
 Qed.
 
 Lemma sin_5PI4 : sin (5 * (PI / 4)) = -1 / sqrt 2.
 Proof with trivial.
   replace (5 * (PI / 4)) with (PI / 4 + PI)...
-  rewrite neg_sin; rewrite sin_PI4; unfold Rdiv in |- *;
+  rewrite neg_sin; rewrite sin_PI4; unfold Rdiv;
     rewrite Ropp_mult_distr_l_reverse...
-  pattern PI at 2 in |- *; rewrite double_var; pattern PI at 2 3 in |- *;
+  pattern PI at 2; rewrite double_var; pattern PI at 2 3;
     rewrite double_var; assert (H : 2 <> 0);
-      [ discrR | unfold Rdiv in |- *; repeat rewrite Rinv_mult_distr; try ring ]...
+      [ discrR | unfold Rdiv; repeat rewrite Rinv_mult_distr; try ring ]...
 Qed.
 
 Lemma sin_cos5PI4 : cos (5 * (PI / 4)) = sin (5 * (PI / 4)).
@@ -367,7 +367,7 @@ Lemma Rgt_3PI2_0 : 0 < 3 * (PI / 2).
 Proof.
   apply Rmult_lt_0_compat;
     [ prove_sup0
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
         [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ] ].
 Qed.
 
@@ -382,7 +382,7 @@ Proof.
     generalize (Rplus_lt_compat_l PI 0 (PI / 2) H1);
       replace (PI + PI / 2) with (3 * (PI / 2)).
   rewrite Rplus_0_r; intro H2; assumption.
-  pattern PI at 2 in |- *; rewrite double_var; ring.
+  pattern PI at 2; rewrite double_var; ring.
 Qed.
 
 Lemma Rlt_3PI2_2PI : 3 * (PI / 2) < 2 * PI.
@@ -391,7 +391,7 @@ Proof.
     generalize (Rplus_lt_compat_l (3 * (PI / 2)) 0 (PI / 2) H1);
       replace (3 * (PI / 2) + PI / 2) with (2 * PI).
   rewrite Rplus_0_r; intro H2; assumption.
-  rewrite double; pattern PI at 1 2 in |- *; rewrite double_var; ring.
+  rewrite double; pattern PI at 1 2; rewrite double_var; ring.
 Qed.
 
 (***************************************************************)
@@ -404,13 +404,13 @@ Definition toDeg (x:R) : R := x * plat * / PI.
 
 Lemma rad_deg : forall x:R, toRad (toDeg x) = x.
 Proof.
-  intro; unfold toRad, toDeg in |- *;
+  intro; unfold toRad, toDeg;
     replace (x * plat * / PI * PI * / plat) with
     (x * (plat * / plat) * (PI * / PI)); [ idtac | ring ].
   repeat rewrite <- Rinv_r_sym.
   ring.
   apply PI_neq0.
-  unfold plat in |- *; discrR.
+  unfold plat; discrR.
 Qed.
 
 Lemma toRad_inj : forall x y:R, toRad x = toRad y -> x = y.
@@ -420,7 +420,7 @@ Proof.
   apply Rmult_eq_reg_l with (/ plat).
   rewrite <- (Rmult_comm (x * PI)); rewrite <- (Rmult_comm (y * PI));
     assumption.
-  apply Rinv_neq_0_compat; unfold plat in |- *; discrR.
+  apply Rinv_neq_0_compat; unfold plat; discrR.
   apply PI_neq0.
 Qed.
 
@@ -435,7 +435,7 @@ Definition tand (x:R) : R := tan (toRad x).
 
 Lemma Rsqr_sin_cos_d_one : forall x:R, Rsqr (sind x) + Rsqr (cosd x) = 1.
 Proof.
-  intro x; unfold sind in |- *; unfold cosd in |- *; apply sin2_cos2.
+  intro x; unfold sind; unfold cosd; apply sin2_cos2.
 Qed.
 
 (***************************************************)
@@ -447,10 +447,10 @@ Proof.
   intros; case (Rtotal_order 0 a); intro.
   left; apply sin_lb_gt_0; assumption.
   elim H1; intro.
-  rewrite <- H2; unfold sin_lb in |- *; unfold sin_approx in |- *;
-    unfold sum_f_R0 in |- *; unfold sin_term in |- *;
+  rewrite <- H2; unfold sin_lb; unfold sin_approx;
+    unfold sum_f_R0; unfold sin_term;
       repeat rewrite pow_ne_zero.
-  unfold Rdiv in |- *; repeat rewrite Rmult_0_l; repeat rewrite Rmult_0_r;
+  unfold Rdiv; repeat rewrite Rmult_0_l; repeat rewrite Rmult_0_r;
     repeat rewrite Rplus_0_r; right; reflexivity.
   discriminate.
   discriminate.
diff --git a/theories/Reals/Rtrigo_def.v b/theories/Reals/Rtrigo_def.v
index c6493135..f3e69037 100644
--- a/theories/Reals/Rtrigo_def.v
+++ b/theories/Reals/Rtrigo_def.v
@@ -1,13 +1,13 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
 (************************************************************************)
 
 Require Import Rbase Rfunctions SeqSeries Rtrigo_fun Max.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (********************************)
 (** * Definition of exponential *)
@@ -27,7 +27,7 @@ Proof.
   intro;
     generalize
       (Alembert_C3 (fun n:nat => / INR (fact n)) x exp_cof_no_R0 Alembert_exp).
-  unfold Pser, exp_in in |- *.
+  unfold Pser, exp_in.
   trivial.
 Defined.
 
@@ -36,24 +36,24 @@ Definition exp (x:R) : R := proj1_sig (exist_exp x).
 Lemma pow_i : forall i:nat, (0 < i)%nat -> 0 ^ i = 0.
 Proof.
   intros; apply pow_ne_zero.
-  red in |- *; intro; rewrite H0 in H; elim (lt_irrefl _ H).
+  red; intro; rewrite H0 in H; elim (lt_irrefl _ H).
 Qed.
 
 Lemma exist_exp0 : { l:R | exp_in 0 l }.
 Proof.
   exists 1.
-  unfold exp_in in |- *; unfold infinite_sum in |- *; intros.
+  unfold exp_in; unfold infinite_sum; intros.
   exists 0%nat.
   intros; replace (sum_f_R0 (fun i:nat => / INR (fact i) * 0 ^ i) n) with 1.
-  unfold R_dist in |- *; replace (1 - 1) with 0;
+  unfold R_dist; replace (1 - 1) with 0;
     [ rewrite Rabs_R0; assumption | ring ].
   induction  n as [| n Hrecn].
-  simpl in |- *; rewrite Rinv_1; ring.
+  simpl; rewrite Rinv_1; ring.
   rewrite tech5.
   rewrite <- Hrecn.
-  simpl in |- *.
+  simpl.
   ring.
-  unfold ge in |- *; apply le_O_n.
+  unfold ge; apply le_O_n.
 Defined.
 
 (* Value of [exp 0] *)
@@ -61,7 +61,7 @@ Lemma exp_0 : exp 0 = 1.
 Proof.
   cut (exp_in 0 (exp 0)).
   cut (exp_in 0 1).
-  unfold exp_in in |- *; intros; eapply uniqueness_sum.
+  unfold exp_in; intros; eapply uniqueness_sum.
   apply H0.
   apply H.
   exact (proj2_sig exist_exp0).
@@ -77,14 +77,14 @@ Definition tanh (x:R) : R := sinh x / cosh x.
 
 Lemma cosh_0 : cosh 0 = 1.
 Proof.
-  unfold cosh in |- *; rewrite Ropp_0; rewrite exp_0.
-  unfold Rdiv in |- *; rewrite <- Rinv_r_sym; [ reflexivity | discrR ].
+  unfold cosh; rewrite Ropp_0; rewrite exp_0.
+  unfold Rdiv; rewrite <- Rinv_r_sym; [ reflexivity | discrR ].
 Qed.
 
 Lemma sinh_0 : sinh 0 = 0.
 Proof.
-  unfold sinh in |- *; rewrite Ropp_0; rewrite exp_0.
-  unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r; apply Rmult_0_l.
+  unfold sinh; rewrite Ropp_0; rewrite exp_0.
+  unfold Rminus, Rdiv; rewrite Rplus_opp_r; apply Rmult_0_l.
 Qed.
 
 Definition cos_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n)).
@@ -92,8 +92,8 @@ Definition cos_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n)).
 Lemma simpl_cos_n :
   forall n:nat, cos_n (S n) / cos_n n = - / INR (2 * S n * (2 * n + 1)).
 Proof.
-  intro; unfold cos_n in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ].
-  rewrite pow_add; unfold Rdiv in |- *; rewrite Rinv_mult_distr.
+  intro; unfold cos_n; replace (S n) with (n + 1)%nat; [ idtac | ring ].
+  rewrite pow_add; unfold Rdiv; rewrite Rinv_mult_distr.
   rewrite Rinv_involutive.
   replace
   ((-1) ^ n * (-1) ^ 1 * / INR (fact (2 * (n + 1))) *
@@ -101,7 +101,7 @@ Proof.
   ((-1) ^ n * / (-1) ^ n * / INR (fact (2 * (n + 1))) * INR (fact (2 * n)) *
     (-1) ^ 1); [ idtac | ring ].
   rewrite <- Rinv_r_sym.
-  rewrite Rmult_1_l; unfold pow in |- *; rewrite Rmult_1_r.
+  rewrite Rmult_1_l; unfold pow; rewrite Rmult_1_r.
   replace (2 * (n + 1))%nat with (S (S (2 * n))); [ idtac | ring ].
   do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
     repeat rewrite Rinv_mult_distr; try (apply not_O_INR; discriminate).
@@ -130,29 +130,29 @@ Proof.
   intro; cut (0 <= up (/ eps))%Z.
   intro; assert (H2 := IZN _ H1); elim H2; intros; exists (max x 1).
   split.
-  cut (0 < IZR (Z_of_nat x)).
-  intro; rewrite INR_IZR_INZ; apply Rle_lt_trans with (/ IZR (Z_of_nat x)).
-  apply Rmult_le_reg_l with (IZR (Z_of_nat x)).
+  cut (0 < IZR (Z.of_nat x)).
+  intro; rewrite INR_IZR_INZ; apply Rle_lt_trans with (/ IZR (Z.of_nat x)).
+  apply Rmult_le_reg_l with (IZR (Z.of_nat x)).
   assumption.
   rewrite <- Rinv_r_sym;
-    [ idtac | red in |- *; intro; rewrite H5 in H4; elim (Rlt_irrefl _ H4) ].
-  apply Rmult_le_reg_l with (IZR (Z_of_nat (max x 1))).
-  apply Rlt_le_trans with (IZR (Z_of_nat x)).
+    [ idtac | red; intro; rewrite H5 in H4; elim (Rlt_irrefl _ H4) ].
+  apply Rmult_le_reg_l with (IZR (Z.of_nat (max x 1))).
+  apply Rlt_le_trans with (IZR (Z.of_nat x)).
   assumption.
   repeat rewrite <- INR_IZR_INZ; apply le_INR; apply le_max_l.
-  rewrite Rmult_1_r; rewrite (Rmult_comm (IZR (Z_of_nat (max x 1))));
+  rewrite Rmult_1_r; rewrite (Rmult_comm (IZR (Z.of_nat (max x 1))));
     rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r; repeat rewrite <- INR_IZR_INZ; apply le_INR;
     apply le_max_l.
   rewrite <- INR_IZR_INZ; apply not_O_INR.
-  red in |- *; intro; assert (H6 := le_max_r x 1); cut (0 < 1)%nat;
+  red; intro; assert (H6 := le_max_r x 1); cut (0 < 1)%nat;
     [ intro | apply lt_O_Sn ]; assert (H8 := lt_le_trans _ _ _ H7 H6);
       rewrite H5 in H8; elim (lt_irrefl _ H8).
-  pattern eps at 1 in |- *; rewrite <- Rinv_involutive.
+  pattern eps at 1; rewrite <- Rinv_involutive.
   apply Rinv_lt_contravar.
   apply Rmult_lt_0_compat; [ apply Rinv_0_lt_compat; assumption | assumption ].
   rewrite H3 in H0; assumption.
-  red in |- *; intro; rewrite H5 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H5 in H; elim (Rlt_irrefl _ H).
   apply Rlt_trans with (/ eps).
   apply Rinv_0_lt_compat; assumption.
   rewrite H3 in H0; assumption.
@@ -166,10 +166,10 @@ Qed.
 
 Lemma Alembert_cos : Un_cv (fun n:nat => Rabs (cos_n (S n) / cos_n n)) 0.
 Proof.
-  unfold Un_cv in |- *; intros.
+  unfold Un_cv; intros.
   assert (H0 := archimed_cor1 eps H).
   elim H0; intros; exists x.
-  intros; rewrite simpl_cos_n; unfold R_dist in |- *; unfold Rminus in |- *;
+  intros; rewrite simpl_cos_n; unfold R_dist; unfold Rminus;
     rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu;
       rewrite Rabs_Ropp; rewrite Rabs_right.
   rewrite mult_INR; rewrite Rinv_mult_distr.
@@ -177,7 +177,7 @@ Proof.
   intro; cut (/ INR (2 * n + 1) < eps).
   intro; rewrite <- (Rmult_1_l eps).
   apply Rmult_gt_0_lt_compat; try assumption.
-  change (0 < / INR (2 * n + 1)) in |- *; apply Rinv_0_lt_compat;
+  change (0 < / INR (2 * n + 1)); apply Rinv_0_lt_compat;
     apply lt_INR_0.
   replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ].
   apply Rlt_0_1.
@@ -221,7 +221,7 @@ Proof.
 Qed.
 
 Lemma cosn_no_R0 : forall n:nat, cos_n n <> 0.
-  intro; unfold cos_n in |- *; unfold Rdiv in |- *; apply prod_neq_R0.
+  intro; unfold cos_n; unfold Rdiv; apply prod_neq_R0.
   apply pow_nonzero; discrR.
   apply Rinv_neq_0_compat.
   apply INR_fact_neq_0.
@@ -234,7 +234,7 @@ Definition cos_in (x l:R) : Prop :=
 (**********)
 Lemma exist_cos : forall x:R, { l:R | cos_in x l }.
   intro; generalize (Alembert_C3 cos_n x cosn_no_R0 Alembert_cos).
-  unfold Pser, cos_in in |- *; trivial.
+  unfold Pser, cos_in; trivial.
 Qed.
 
 
@@ -246,8 +246,8 @@ Definition sin_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n + 1)).
 Lemma simpl_sin_n :
   forall n:nat, sin_n (S n) / sin_n n = - / INR ((2 * S n + 1) * (2 * S n)).
 Proof.
-  intro; unfold sin_n in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ].
-  rewrite pow_add; unfold Rdiv in |- *; rewrite Rinv_mult_distr.
+  intro; unfold sin_n; replace (S n) with (n + 1)%nat; [ idtac | ring ].
+  rewrite pow_add; unfold Rdiv; rewrite Rinv_mult_distr.
   rewrite Rinv_involutive.
   replace
   ((-1) ^ n * (-1) ^ 1 * / INR (fact (2 * (n + 1) + 1)) *
@@ -255,7 +255,7 @@ Proof.
   ((-1) ^ n * / (-1) ^ n * / INR (fact (2 * (n + 1) + 1)) *
     INR (fact (2 * n + 1)) * (-1) ^ 1); [ idtac | ring ].
   rewrite <- Rinv_r_sym.
-  rewrite Rmult_1_l; unfold pow in |- *; rewrite Rmult_1_r;
+  rewrite Rmult_1_l; unfold pow; rewrite Rmult_1_r;
     replace (2 * (n + 1) + 1)%nat with (S (S (2 * n + 1))).
   do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
     repeat rewrite Rinv_mult_distr.
@@ -291,9 +291,9 @@ Qed.
 
 Lemma Alembert_sin : Un_cv (fun n:nat => Rabs (sin_n (S n) / sin_n n)) 0.
 Proof.
-  unfold Un_cv in |- *; intros; assert (H0 := archimed_cor1 eps H).
+  unfold Un_cv; intros; assert (H0 := archimed_cor1 eps H).
   elim H0; intros; exists x.
-  intros; rewrite simpl_sin_n; unfold R_dist in |- *; unfold Rminus in |- *;
+  intros; rewrite simpl_sin_n; unfold R_dist; unfold Rminus;
     rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu;
       rewrite Rabs_Ropp; rewrite Rabs_right.
   rewrite mult_INR; rewrite Rinv_mult_distr.
@@ -301,7 +301,7 @@ Proof.
   intro; cut (/ INR (2 * S n + 1) < eps).
   intro; rewrite <- (Rmult_1_l eps); rewrite (Rmult_comm (/ INR (2 * S n + 1)));
     apply Rmult_gt_0_lt_compat; try assumption.
-  change (0 < / INR (2 * S n + 1)) in |- *; apply Rinv_0_lt_compat;
+  change (0 < / INR (2 * S n + 1)); apply Rinv_0_lt_compat;
     apply lt_INR_0; replace (2 * S n + 1)%nat with (S (2 * S n));
       [ apply lt_O_Sn | ring ].
   apply Rlt_0_1.
@@ -329,7 +329,7 @@ Proof.
   apply not_O_INR; discriminate.
   apply not_O_INR; discriminate.
   apply not_O_INR; discriminate.
-  left; change (0 < / INR ((2 * S n + 1) * (2 * S n))) in |- *;
+  left; change (0 < / INR ((2 * S n + 1) * (2 * S n)));
     apply Rinv_0_lt_compat.
   apply lt_INR_0.
   replace ((2 * S n + 1) * (2 * S n))%nat with
@@ -342,7 +342,7 @@ Defined.
 
 Lemma sin_no_R0 : forall n:nat, sin_n n <> 0.
 Proof.
-  intro; unfold sin_n in |- *; unfold Rdiv in |- *; apply prod_neq_R0.
+  intro; unfold sin_n; unfold Rdiv; apply prod_neq_R0.
   apply pow_nonzero; discrR.
   apply Rinv_neq_0_compat; apply INR_fact_neq_0.
 Qed.
@@ -355,7 +355,7 @@ Definition sin_in (x l:R) : Prop :=
 Lemma exist_sin : forall x:R, { l:R | sin_in x l }.
 Proof.
   intro; generalize (Alembert_C3 sin_n x sin_no_R0 Alembert_sin).
-  unfold Pser, sin_n in |- *; trivial.
+  unfold Pser, sin_n; trivial.
 Defined.
 
 (***********************)
@@ -368,40 +368,40 @@ Definition sin (x:R) : R := let (a,_) := exist_sin (Rsqr x) in x * a.
 
 Lemma cos_sym : forall x:R, cos x = cos (- x).
 Proof.
-  intros; unfold cos in |- *; replace (Rsqr (- x)) with (Rsqr x).
+  intros; unfold cos; replace (Rsqr (- x)) with (Rsqr x).
   reflexivity.
   apply Rsqr_neg.
 Qed.
 
 Lemma sin_antisym : forall x:R, sin (- x) = - sin x.
 Proof.
-  intro; unfold sin in |- *; replace (Rsqr (- x)) with (Rsqr x);
+  intro; unfold sin; replace (Rsqr (- x)) with (Rsqr x);
     [ idtac | apply Rsqr_neg ].
   case (exist_sin (Rsqr x)); intros; ring.
 Qed.
 
 Lemma sin_0 : sin 0 = 0.
 Proof.
-  unfold sin in |- *; case (exist_sin (Rsqr 0)).
+  unfold sin; case (exist_sin (Rsqr 0)).
   intros; ring.
 Qed.
 
 Lemma exist_cos0 : { l:R | cos_in 0 l }.
 Proof.
   exists 1.
-  unfold cos_in in |- *; unfold infinite_sum in |- *; intros; exists 0%nat.
+  unfold cos_in; unfold infinite_sum; intros; exists 0%nat.
   intros.
-  unfold R_dist in |- *.
+  unfold R_dist.
   induction  n as [| n Hrecn].
-  unfold cos_n in |- *; simpl in |- *.
-  unfold Rdiv in |- *; rewrite Rinv_1.
+  unfold cos_n; simpl.
+  unfold Rdiv; rewrite Rinv_1.
   do 2 rewrite Rmult_1_r.
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
   rewrite tech5.
   replace (cos_n (S n) * 0 ^ S n) with 0.
   rewrite Rplus_0_r.
-  apply Hrecn; unfold ge in |- *; apply le_O_n.
-  simpl in |- *; ring.
+  apply Hrecn; unfold ge; apply le_O_n.
+  simpl; ring.
 Defined.
 
 (* Value of [cos 0] *)
@@ -409,10 +409,10 @@ Lemma cos_0 : cos 0 = 1.
 Proof.
   cut (cos_in 0 (cos 0)).
   cut (cos_in 0 1).
-  unfold cos_in in |- *; intros; eapply uniqueness_sum.
+  unfold cos_in; intros; eapply uniqueness_sum.
   apply H0.
   apply H.
   exact (proj2_sig exist_cos0).
-  assert (H := proj2_sig (exist_cos (Rsqr 0))); unfold cos in |- *;
-    pattern 0 at 1 in |- *; replace 0 with (Rsqr 0); [ exact H | apply Rsqr_0 ].
+  assert (H := proj2_sig (exist_cos (Rsqr 0))); unfold cos;
+    pattern 0 at 1; replace 0 with (Rsqr 0); [ exact H | apply Rsqr_0 ].
 Qed.
diff --git a/theories/Reals/Rtrigo_fun.v b/theories/Reals/Rtrigo_fun.v
index b7720141..b131b510 100644
--- a/theories/Reals/Rtrigo_fun.v
+++ b/theories/Reals/Rtrigo_fun.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,7 +9,7 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (*****************************************************************)
 (**           To define transcendental functions                 *)
@@ -20,8 +20,8 @@ Open Local Scope R_scope.
 Lemma Alembert_exp :
   Un_cv (fun n:nat => Rabs (/ INR (fact (S n)) * / / INR (fact n))) 0.
 Proof.
-  unfold Un_cv in |- *; intros; elim (Rgt_dec eps 1); intro.
-  split with 0%nat; intros; rewrite (simpl_fact n); unfold R_dist in |- *;
+  unfold Un_cv; intros; elim (Rgt_dec eps 1); intro.
+  split with 0%nat; intros; rewrite (simpl_fact n); unfold R_dist;
     rewrite (Rminus_0_r (Rabs (/ INR (S n))));
       rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0).
   intro; rewrite (Rabs_pos_eq (/ INR (S n))).
@@ -39,7 +39,7 @@ Proof.
               in H4; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H4;
                 rewrite (Rmult_comm (/ INR (S n))) in H4;
                   rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H4;
-                    rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H4;
+                    rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (not_eq_sym (O_S n)))) in H4;
                       rewrite (let (H1, H2) := Rmult_ne eps in H1) in H4;
                         assumption.
   apply Rlt_minus; unfold Rgt in a; rewrite <- Rinv_1;
@@ -47,11 +47,11 @@ Proof.
       rewrite (let (H1, H2) := Rmult_ne eps in H2); unfold Rgt in H;
         assumption.
   unfold Rgt in H1; apply Rlt_le; assumption.
-  unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
+  unfold Rgt; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
 (**)
   cut (0 <= up (/ eps - 1))%Z.
   intro; elim (IZN (up (/ eps - 1)) H0); intros; split with x; intros;
-    rewrite (simpl_fact n); unfold R_dist in |- *;
+    rewrite (simpl_fact n); unfold R_dist;
       rewrite (Rminus_0_r (Rabs (/ INR (S n))));
         rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0).
   intro; rewrite (Rabs_pos_eq (/ INR (S n))).
@@ -72,28 +72,28 @@ Proof.
               in H6; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H6;
                 rewrite (Rmult_comm (/ INR (S n))) in H6;
                   rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H6;
-                    rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H6;
+                    rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (not_eq_sym (O_S n)))) in H6;
                       rewrite (let (H1, H2) := Rmult_ne eps in H1) in H6;
                         assumption.
-  cut (IZR (up (/ eps - 1)) = IZR (Z_of_nat x));
+  cut (IZR (up (/ eps - 1)) = IZR (Z.of_nat x));
     [ intro | rewrite H1; trivial ].
   elim (archimed (/ eps - 1)); intros; clear H6; unfold Rgt in H5;
     rewrite H4 in H5; rewrite INR_IZR_INZ; assumption.
   unfold Rgt in H1; apply Rlt_le; assumption.
-  unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
+  unfold Rgt; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
   apply (le_O_IZR (up (/ eps - 1)));
     apply (Rle_trans 0 (/ eps - 1) (IZR (up (/ eps - 1)))).
-  generalize (Rnot_gt_le eps 1 b); clear b; unfold Rle in |- *; intro; elim H0;
+  generalize (Rnot_gt_le eps 1 b); clear b; unfold Rle; intro; elim H0;
     clear H0; intro.
   left; unfold Rgt in H;
     generalize (Rmult_lt_compat_l (/ eps) eps 1 (Rinv_0_lt_compat eps H) H0);
       rewrite
         (Rinv_l eps
-          (sym_not_eq (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H))))
+          (not_eq_sym (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H))))
         ; rewrite (let (H1, H2) := Rmult_ne (/ eps) in H1);
-          intro; fold (/ eps - 1 > 0) in |- *; apply Rgt_minus;
-            unfold Rgt in |- *; assumption.
-  right; rewrite H0; rewrite Rinv_1; apply sym_eq; apply Rminus_diag_eq; auto.
+          intro; fold (/ eps - 1 > 0); apply Rgt_minus;
+            unfold Rgt; assumption.
+  right; rewrite H0; rewrite Rinv_1; symmetry; apply Rminus_diag_eq; auto.
   elim (archimed (/ eps - 1)); intros; clear H1; unfold Rgt in H0; apply Rlt_le;
     assumption.
 Qed.
diff --git a/theories/Reals/Rtrigo_reg.v b/theories/Reals/Rtrigo_reg.v
index 100e0818..fff4fec9 100644
--- a/theories/Reals/Rtrigo_reg.v
+++ b/theories/Reals/Rtrigo_reg.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -9,169 +9,23 @@
 Require Import Rbase.
 Require Import Rfunctions.
 Require Import SeqSeries.
-Require Import Rtrigo.
+Require Import Rtrigo1.
 Require Import Ranalysis1.
 Require Import PSeries_reg.
-Open Local Scope nat_scope.
-Open Local Scope R_scope.
+Local Open Scope nat_scope.
+Local Open Scope R_scope.
 
-Lemma CVN_R_cos :
-  forall fn:nat -> R -> R,
-    fn = (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)) ->
-    CVN_R fn.
-Proof.
-  unfold CVN_R in |- *; intros.
-  cut ((r:R) <> 0).
-  intro hyp_r; unfold CVN_r in |- *.
-  exists (fun n:nat => / INR (fact (2 * n)) * r ^ (2 * n)).
-  cut
-    { l:R |
-        Un_cv
-        (fun n:nat =>
-          sum_f_R0 (fun k:nat => Rabs (/ INR (fact (2 * k)) * r ^ (2 * k)))
-          n) l }.
-  intro X; elim X; intros.
-  exists x.
-  split.
-  apply p.
-  intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult.
-  rewrite pow_1_abs; rewrite Rmult_1_l.
-  cut (0 < / INR (fact (2 * n))).
-  intro; rewrite (Rabs_right _ (Rle_ge _ _ (Rlt_le _ _ H1))).
-  apply Rmult_le_compat_l.
-  left; apply H1.
-  rewrite <- RPow_abs; apply pow_maj_Rabs.
-  rewrite Rabs_Rabsolu.
-  unfold Boule in H0; rewrite Rminus_0_r in H0.
-  left; apply H0.
-  apply Rinv_0_lt_compat; apply INR_fact_lt_0.
-  apply Alembert_C2.
-  intro; apply Rabs_no_R0.
-  apply prod_neq_R0.
-  apply Rinv_neq_0_compat.
-  apply INR_fact_neq_0.
-  apply pow_nonzero; assumption.
-  assert (H0 := Alembert_cos).
-  unfold cos_n in H0; unfold Un_cv in H0; unfold Un_cv in |- *; intros.
-  cut (0 < eps / Rsqr r).
-  intro; elim (H0 _ H2); intros N0 H3.
-  exists N0; intros.
-  unfold R_dist in |- *; assert (H5 := H3 _ H4).
-  unfold R_dist in H5;
-    replace
-    (Rabs
-      (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) /
-        Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) with
-    (Rsqr r *
-      Rabs ((-1) ^ S n / INR (fact (2 * S n)) / ((-1) ^ n / INR (fact (2 * n))))).
-  apply Rmult_lt_reg_l with (/ Rsqr r).
-  apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption.
-  pattern (/ Rsqr r) at 1 in |- *; replace (/ Rsqr r) with (Rabs (/ Rsqr r)).
-  rewrite <- Rabs_mult; rewrite Rmult_minus_distr_l; rewrite Rmult_0_r;
-    rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
-  rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); apply H5.
-  unfold Rsqr in |- *; apply prod_neq_R0; assumption.
-  rewrite Rabs_Rinv.
-  rewrite Rabs_right.
-  reflexivity.
-  apply Rle_ge; apply Rle_0_sqr.
-  unfold Rsqr in |- *; apply prod_neq_R0; assumption.
-  rewrite (Rmult_comm (Rsqr r)); unfold Rdiv in |- *; repeat rewrite Rabs_mult;
-    rewrite Rabs_Rabsolu; rewrite pow_1_abs; rewrite Rmult_1_l;
-      repeat rewrite Rmult_assoc; apply Rmult_eq_compat_l.
-  rewrite Rabs_Rinv.
-  rewrite Rabs_mult; rewrite (pow_1_abs n); rewrite Rmult_1_l;
-    rewrite <- Rabs_Rinv.
-  rewrite Rinv_involutive.
-  rewrite Rinv_mult_distr.
-  rewrite Rabs_Rinv.
-  rewrite Rinv_involutive.
-  rewrite (Rmult_comm (Rabs (Rabs (r ^ (2 * S n))))); rewrite Rabs_mult;
-    rewrite Rabs_Rabsolu; rewrite Rmult_assoc; apply Rmult_eq_compat_l.
-  rewrite Rabs_Rinv.
-  do 2 rewrite Rabs_Rabsolu; repeat rewrite Rabs_right.
-  replace (r ^ (2 * S n)) with (r ^ (2 * n) * r * r).
-  repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
-  unfold Rsqr in |- *; ring.
-  apply pow_nonzero; assumption.
-  replace (2 * S n)%nat with (S (S (2 * n))).
-  simpl in |- *; ring.
-  ring.
-  apply Rle_ge; apply pow_le; left; apply (cond_pos r).
-  apply Rle_ge; apply pow_le; left; apply (cond_pos r).
-  apply Rabs_no_R0; apply pow_nonzero; assumption.
-  apply Rabs_no_R0; apply INR_fact_neq_0.
-  apply INR_fact_neq_0.
-  apply Rabs_no_R0; apply Rinv_neq_0_compat; apply INR_fact_neq_0.
-  apply Rabs_no_R0; apply pow_nonzero; assumption.
-  apply INR_fact_neq_0.
-  apply Rinv_neq_0_compat; apply INR_fact_neq_0.
-  apply prod_neq_R0.
-  apply pow_nonzero; discrR.
-  apply Rinv_neq_0_compat; apply INR_fact_neq_0.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
-  apply H1.
-  apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption.
-  assert (H0 := cond_pos r); red in |- *; intro; rewrite H1 in H0;
-    elim (Rlt_irrefl _ H0).
-Qed.
-
-(**********)
-Lemma continuity_cos : continuity cos.
-Proof.
-  set (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)).
-  cut (CVN_R fn).
-  intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }).
-  intro cv; cut (forall n:nat, continuity (fn n)).
-  intro; cut (forall x:R, cos x = SFL fn cv x).
-  intro; cut (continuity (SFL fn cv) -> continuity cos).
-  intro; apply H1.
-  apply SFL_continuity; assumption.
-  unfold continuity in |- *; unfold continuity_pt in |- *;
-    unfold continue_in in |- *; unfold limit1_in in |- *;
-      unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *;
-        intros.
-  elim (H1 x _ H2); intros.
-  exists x0; intros.
-  elim H3; intros.
-  split.
-  apply H4.
-  intros; rewrite (H0 x); rewrite (H0 x1); apply H5; apply H6.
-  intro; unfold cos, SFL in |- *.
-  case (cv x); case (exist_cos (Rsqr x)); intros.
-  symmetry  in |- *; eapply UL_sequence.
-  apply u.
-  unfold cos_in in c; unfold infinite_sum in c; unfold Un_cv in |- *; intros.
-  elim (c _ H0); intros N0 H1.
-  exists N0; intros.
-  unfold R_dist in H1; unfold R_dist, SP in |- *.
-  replace (sum_f_R0 (fun k:nat => fn k x) n) with
-  (sum_f_R0 (fun i:nat => cos_n i * Rsqr x ^ i) n).
-  apply H1; assumption.
-  apply sum_eq; intros.
-  unfold cos_n, fn in |- *; apply Rmult_eq_compat_l.
-  unfold Rsqr in |- *; rewrite pow_sqr; reflexivity.
-  intro; unfold fn in |- *;
-    replace (fun x:R => (-1) ^ n / INR (fact (2 * n)) * x ^ (2 * n)) with
-    (fct_cte ((-1) ^ n / INR (fact (2 * n))) * pow_fct (2 * n))%F;
-    [ idtac | reflexivity ].
-  apply continuity_mult.
-  apply derivable_continuous; apply derivable_const.
-  apply derivable_continuous; apply (derivable_pow (2 * n)).
-  apply CVN_R_CVS; apply X.
-  apply CVN_R_cos; unfold fn in |- *; reflexivity.
-Qed.
 
 (**********)
 Lemma continuity_sin : continuity sin.
 Proof.
-  unfold continuity in |- *; intro.
+  unfold continuity; intro.
   assert (H0 := continuity_cos (PI / 2 - x)).
   unfold continuity_pt in H0; unfold continue_in in H0; unfold limit1_in in H0;
     unfold limit_in in H0; simpl in H0; unfold R_dist in H0;
-      unfold continuity_pt in |- *; unfold continue_in in |- *;
-        unfold limit1_in in |- *; unfold limit_in in |- *;
-          simpl in |- *; unfold R_dist in |- *; intros.
+      unfold continuity_pt; unfold continue_in;
+        unfold limit1_in; unfold limit_in;
+          simpl; unfold R_dist; intros.
   elim (H0 _ H); intros.
   exists x0; intros.
   elim H1; intros.
@@ -180,9 +34,9 @@ Proof.
   intros; rewrite <- (cos_shift x); rewrite <- (cos_shift x1); apply H3.
   elim H4; intros.
   split.
-  unfold D_x, no_cond in |- *; split.
+  unfold D_x, no_cond; split.
   trivial.
-  red in |- *; intro; unfold D_x, no_cond in H5; elim H5; intros _ H8; elim H8;
+  red; intro; unfold D_x, no_cond in H5; elim H5; intros _ H8; elim H8;
     rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive x1);
       apply Ropp_eq_compat; apply Rplus_eq_reg_l with (PI / 2);
         apply H7.
@@ -196,7 +50,7 @@ Lemma CVN_R_sin :
     (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)) ->
     CVN_R fn.
 Proof.
-  unfold CVN_R in |- *; unfold CVN_r in |- *; intros fn H r.
+  unfold CVN_R; unfold CVN_r; intros fn H r.
   exists (fun n:nat => / INR (fact (2 * n + 1)) * r ^ (2 * n)).
   cut
     { l:R |
@@ -209,7 +63,7 @@ Proof.
   exists x.
   split.
   apply p.
-  intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult;
+  intros; rewrite H; unfold Rdiv; do 2 rewrite Rabs_mult;
     rewrite pow_1_abs; rewrite Rmult_1_l.
   cut (0 < / INR (fact (2 * n + 1))).
   intro; rewrite (Rabs_right _ (Rle_ge _ _ (Rlt_le _ _ H1))).
@@ -226,11 +80,11 @@ Proof.
   apply Rinv_neq_0_compat; apply INR_fact_neq_0.
   apply pow_nonzero; assumption.
   assert (H1 := Alembert_sin).
-  unfold sin_n in H1; unfold Un_cv in H1; unfold Un_cv in |- *; intros.
+  unfold sin_n in H1; unfold Un_cv in H1; unfold Un_cv; intros.
   cut (0 < eps / Rsqr r).
   intro; elim (H1 _ H3); intros N0 H4.
   exists N0; intros.
-  unfold R_dist in |- *; assert (H6 := H4 _ H5).
+  unfold R_dist; assert (H6 := H4 _ H5).
   unfold R_dist in H5;
     replace
     (Rabs
@@ -242,15 +96,15 @@ Proof.
         ((-1) ^ n / INR (fact (2 * n + 1))))).
   apply Rmult_lt_reg_l with (/ Rsqr r).
   apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption.
-  pattern (/ Rsqr r) at 1 in |- *; rewrite <- (Rabs_right (/ Rsqr r)).
+  pattern (/ Rsqr r) at 1; rewrite <- (Rabs_right (/ Rsqr r)).
   rewrite <- Rabs_mult.
   rewrite Rmult_minus_distr_l.
   rewrite Rmult_0_r; rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l; rewrite <- (Rmult_comm eps).
   apply H6.
-  unfold Rsqr in |- *; apply prod_neq_R0; assumption.
+  unfold Rsqr; apply prod_neq_R0; assumption.
   apply Rle_ge; left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption.
-  unfold Rdiv in |- *; rewrite (Rmult_comm (Rsqr r)); repeat rewrite Rabs_mult;
+  unfold Rdiv; rewrite (Rmult_comm (Rsqr r)); repeat rewrite Rabs_mult;
     rewrite Rabs_Rabsolu; rewrite pow_1_abs.
   rewrite Rmult_1_l.
   repeat rewrite Rmult_assoc; apply Rmult_eq_compat_l.
@@ -272,10 +126,10 @@ Proof.
   replace (r ^ (2 * S n)) with (r ^ (2 * n) * r * r).
   do 2 rewrite <- Rmult_assoc.
   rewrite <- Rinv_l_sym.
-  unfold Rsqr in |- *; ring.
+  unfold Rsqr; ring.
   apply pow_nonzero; assumption.
   replace (2 * S n)%nat with (S (S (2 * n))).
-  simpl in |- *; ring.
+  simpl; ring.
   ring.
   apply Rle_ge; apply pow_le; left; apply (cond_pos r).
   apply Rle_ge; apply pow_le; left; apply (cond_pos r).
@@ -288,16 +142,16 @@ Proof.
   apply INR_fact_neq_0.
   apply pow_nonzero; discrR.
   apply Rinv_neq_0_compat; apply INR_fact_neq_0.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption ].
-  assert (H0 := cond_pos r); red in |- *; intro; rewrite H1 in H0;
+  assert (H0 := cond_pos r); red; intro; rewrite H1 in H0;
     elim (Rlt_irrefl _ H0).
 Qed.
 
 (** (sin h)/h -> 1 when h -> 0 *)
 Lemma derivable_pt_lim_sin_0 : derivable_pt_lim sin 0 1.
 Proof.
-  unfold derivable_pt_lim in |- *; intros.
+  unfold derivable_pt_lim; intros.
   set
     (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)).
   cut (CVN_R fn).
@@ -313,58 +167,58 @@ Proof.
   elim (H2 _ H); intros alp H3.
   elim H3; intros.
   exists (mkposreal _ H4).
-  simpl in |- *; intros.
-  rewrite sin_0; rewrite Rplus_0_l; unfold Rminus in |- *; rewrite Ropp_0;
+  simpl; intros.
+  rewrite sin_0; rewrite Rplus_0_l; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r.
   cut (Rabs (SFL fn cv h - SFL fn cv 0) < eps).
   intro; cut (SFL fn cv 0 = 1).
   intro; cut (SFL fn cv h = sin h / h).
   intro; rewrite H9 in H8; rewrite H10 in H8.
   apply H8.
-  unfold SFL, sin in |- *.
+  unfold SFL, sin.
   case (cv h); intros.
   case (exist_sin (Rsqr h)); intros.
-  unfold Rdiv in |- *; rewrite (Rinv_r_simpl_m h x0 H6).
+  unfold Rdiv; rewrite (Rinv_r_simpl_m h x0 H6).
   eapply UL_sequence.
   apply u.
   unfold sin_in in s; unfold sin_n, infinite_sum in s;
-    unfold SP, fn, Un_cv in |- *; intros.
+    unfold SP, fn, Un_cv; intros.
   elim (s _ H10); intros N0 H11.
   exists N0; intros.
-  unfold R_dist in |- *; unfold R_dist in H11.
+  unfold R_dist; unfold R_dist in H11.
   replace
   (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * h ^ (2 * k)) n)
     with
       (sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * Rsqr h ^ i) n).
   apply H11; assumption.
-  apply sum_eq; intros; apply Rmult_eq_compat_l; unfold Rsqr in |- *;
+  apply sum_eq; intros; apply Rmult_eq_compat_l; unfold Rsqr;
     rewrite pow_sqr; reflexivity.
-  unfold SFL, sin in |- *.
+  unfold SFL, sin.
   case (cv 0); intros.
   eapply UL_sequence.
   apply u.
-  unfold SP, fn in |- *; unfold Un_cv in |- *; intros; exists 1%nat; intros.
-  unfold R_dist in |- *;
+  unfold SP, fn; unfold Un_cv; intros; exists 1%nat; intros.
+  unfold R_dist;
     replace
     (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * 0 ^ (2 * k)) n)
     with 1.
-  unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
+  unfold Rminus; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption.
   rewrite decomp_sum.
-  simpl in |- *; rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite Rinv_1;
-    rewrite Rmult_1_r; pattern 1 at 1 in |- *; rewrite <- Rplus_0_r;
+  simpl; rewrite Rmult_1_r; unfold Rdiv; rewrite Rinv_1;
+    rewrite Rmult_1_r; pattern 1 at 1; rewrite <- Rplus_0_r;
       apply Rplus_eq_compat_l.
-  symmetry  in |- *; apply sum_eq_R0; intros.
+  symmetry ; apply sum_eq_R0; intros.
   rewrite Rmult_0_l; rewrite Rmult_0_r; reflexivity.
   unfold ge in H10; apply lt_le_trans with 1%nat; [ apply lt_n_Sn | apply H10 ].
   apply H5.
   split.
-  unfold D_x, no_cond in |- *; split.
+  unfold D_x, no_cond; split.
   trivial.
-  apply (sym_not_eq (A:=R)); apply H6.
-  unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply H7.
-  unfold Boule in |- *; unfold Rminus in |- *; rewrite Ropp_0;
+  apply (not_eq_sym (A:=R)); apply H6.
+  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; apply H7.
+  unfold Boule; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; rewrite Rabs_R0; apply (cond_pos r).
-  intros; unfold fn in |- *;
+  intros; unfold fn;
     replace (fun x:R => (-1) ^ n / INR (fact (2 * n + 1)) * x ^ (2 * n)) with
     (fct_cte ((-1) ^ n / INR (fact (2 * n + 1))) * pow_fct (2 * n))%F;
     [ idtac | reflexivity ].
@@ -375,13 +229,13 @@ Proof.
   apply (derivable_pt_pow (2 * n) y).
   apply (X r).
   apply (CVN_R_CVS _ X).
-  apply CVN_R_sin; unfold fn in |- *; reflexivity.
+  apply CVN_R_sin; unfold fn; reflexivity.
 Qed.
 
 (** ((cos h)-1)/h -> 0 when h -> 0 *)
 Lemma derivable_pt_lim_cos_0 : derivable_pt_lim cos 0 0.
 Proof.
-  unfold derivable_pt_lim in |- *; intros.
+  unfold derivable_pt_lim; intros.
   assert (H0 := derivable_pt_lim_sin_0).
   unfold derivable_pt_lim in H0.
   cut (0 < eps / 2).
@@ -396,8 +250,8 @@ Proof.
   intro; set (delta := mkposreal _ H6).
   exists delta; intros.
   rewrite Rplus_0_l; replace (cos h - cos 0) with (-2 * Rsqr (sin (h / 2))).
-  unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r.
-  unfold Rdiv in |- *; do 2 rewrite Ropp_mult_distr_l_reverse.
+  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r.
+  unfold Rdiv; do 2 rewrite Ropp_mult_distr_l_reverse.
   rewrite Rabs_Ropp.
   replace (2 * Rsqr (sin (h * / 2)) * / h) with
   (sin (h / 2) * (sin (h / 2) / (h / 2) - 1) + sin (h / 2)).
@@ -407,12 +261,12 @@ Proof.
   rewrite (double_var eps); apply Rplus_lt_compat.
   apply Rle_lt_trans with (Rabs (sin (h / 2) / (h / 2) - 1)).
   rewrite Rabs_mult; rewrite Rmult_comm;
-    pattern (Rabs (sin (h / 2) / (h / 2) - 1)) at 2 in |- *;
+    pattern (Rabs (sin (h / 2) / (h / 2) - 1)) at 2;
       rewrite <- Rmult_1_r; apply Rmult_le_compat_l.
   apply Rabs_pos.
   assert (H9 := SIN_bound (h / 2)).
-  unfold Rabs in |- *; case (Rcase_abs (sin (h / 2))); intro.
-  pattern 1 at 3 in |- *; rewrite <- (Ropp_involutive 1).
+  unfold Rabs; case (Rcase_abs (sin (h / 2))); intro.
+  pattern 1 at 3; rewrite <- (Ropp_involutive 1).
   apply Ropp_le_contravar.
   elim H9; intros; assumption.
   elim H9; intros; assumption.
@@ -421,50 +275,50 @@ Proof.
   intro; assert (H11 := H2 _ H10 H9).
   rewrite Rplus_0_l in H11; rewrite sin_0 in H11.
   rewrite Rminus_0_r in H11; apply H11.
-  unfold Rdiv in |- *; apply prod_neq_R0.
+  unfold Rdiv; apply prod_neq_R0.
   apply H7.
   apply Rinv_neq_0_compat; discrR.
   apply Rlt_trans with (del / 2).
-  unfold Rdiv in |- *; rewrite Rabs_mult.
+  unfold Rdiv; rewrite Rabs_mult.
   rewrite (Rabs_right (/ 2)).
   do 2 rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l.
   apply Rinv_0_lt_compat; prove_sup0.
   apply Rlt_le_trans with (pos delta).
   apply H8.
-  unfold delta in |- *; simpl in |- *; apply Rmin_l.
+  unfold delta; simpl; apply Rmin_l.
   apply Rle_ge; left; apply Rinv_0_lt_compat; prove_sup0.
-  rewrite <- (Rplus_0_r (del / 2)); pattern del at 1 in |- *;
+  rewrite <- (Rplus_0_r (del / 2)); pattern del at 1;
     rewrite (double_var del); apply Rplus_lt_compat_l;
-      unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+      unfold Rdiv; apply Rmult_lt_0_compat.
   apply (cond_pos del).
   apply Rinv_0_lt_compat; prove_sup0.
   elim H5; intros; assert (H11 := H10 (h / 2)).
   rewrite sin_0 in H11; do 2 rewrite Rminus_0_r in H11.
   apply H11.
   split.
-  unfold D_x, no_cond in |- *; split.
+  unfold D_x, no_cond; split.
   trivial.
-  apply (sym_not_eq (A:=R)); unfold Rdiv in |- *; apply prod_neq_R0.
+  apply (not_eq_sym (A:=R)); unfold Rdiv; apply prod_neq_R0.
   apply H7.
   apply Rinv_neq_0_compat; discrR.
   apply Rlt_trans with (del_c / 2).
-  unfold Rdiv in |- *; rewrite Rabs_mult.
+  unfold Rdiv; rewrite Rabs_mult.
   rewrite (Rabs_right (/ 2)).
   do 2 rewrite <- (Rmult_comm (/ 2)).
   apply Rmult_lt_compat_l.
   apply Rinv_0_lt_compat; prove_sup0.
   apply Rlt_le_trans with (pos delta).
   apply H8.
-  unfold delta in |- *; simpl in |- *; apply Rmin_r.
+  unfold delta; simpl; apply Rmin_r.
   apply Rle_ge; left; apply Rinv_0_lt_compat; prove_sup0.
-  rewrite <- (Rplus_0_r (del_c / 2)); pattern del_c at 2 in |- *;
+  rewrite <- (Rplus_0_r (del_c / 2)); pattern del_c at 2;
     rewrite (double_var del_c); apply Rplus_lt_compat_l.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   apply H9.
   apply Rinv_0_lt_compat; prove_sup0.
-  rewrite Rmult_minus_distr_l; rewrite Rmult_1_r; unfold Rminus in |- *;
+  rewrite Rmult_minus_distr_l; rewrite Rmult_1_r; unfold Rminus;
     rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
-      rewrite (Rmult_comm 2); unfold Rdiv, Rsqr in |- *.
+      rewrite (Rmult_comm 2); unfold Rdiv, Rsqr.
   repeat rewrite Rmult_assoc.
   repeat apply Rmult_eq_compat_l.
   rewrite Rinv_mult_distr.
@@ -473,16 +327,16 @@ Proof.
   discrR.
   apply H7.
   apply Rinv_neq_0_compat; discrR.
-  pattern h at 2 in |- *; replace h with (2 * (h / 2)).
+  pattern h at 2; replace h with (2 * (h / 2)).
   rewrite (cos_2a_sin (h / 2)).
-  rewrite cos_0; unfold Rsqr in |- *; ring.
-  unfold Rdiv in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m.
+  rewrite cos_0; unfold Rsqr; ring.
+  unfold Rdiv; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m.
   discrR.
-  unfold Rmin in |- *; case (Rle_dec del del_c); intro.
+  unfold Rmin; case (Rle_dec del del_c); intro.
   apply (cond_pos del).
   elim H5; intros; assumption.
   apply continuity_sin.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
 Qed.
 
@@ -492,10 +346,10 @@ Proof.
   intro; assert (H0 := derivable_pt_lim_sin_0).
   assert (H := derivable_pt_lim_cos_0).
   unfold derivable_pt_lim in H0, H.
-  unfold derivable_pt_lim in |- *; intros.
+  unfold derivable_pt_lim; intros.
   cut (0 < eps / 2);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
         [ apply H1 | apply Rinv_0_lt_compat; prove_sup0 ] ].
   elim (H0 _ H2); intros alp1 H3.
   elim (H _ H2); intros alp2 H4.
@@ -510,11 +364,11 @@ Proof.
   rewrite (double_var eps); apply Rplus_lt_compat.
   apply Rle_lt_trans with (Rabs ((cos h - 1) / h)).
   rewrite Rabs_mult; rewrite Rmult_comm;
-    pattern (Rabs ((cos h - 1) / h)) at 2 in |- *; rewrite <- Rmult_1_r;
+    pattern (Rabs ((cos h - 1) / h)) at 2; rewrite <- Rmult_1_r;
       apply Rmult_le_compat_l.
   apply Rabs_pos.
   assert (H8 := SIN_bound x); elim H8; intros.
-  unfold Rabs in |- *; case (Rcase_abs (sin x)); intro.
+  unfold Rabs; case (Rcase_abs (sin x)); intro.
   rewrite <- (Ropp_involutive 1).
   apply Ropp_le_contravar; assumption.
   assumption.
@@ -524,14 +378,14 @@ Proof.
     apply H9.
   apply Rlt_le_trans with alp.
   apply H7.
-  unfold alp in |- *; apply Rmin_r.
+  unfold alp; apply Rmin_r.
   apply Rle_lt_trans with (Rabs (sin h / h - 1)).
   rewrite Rabs_mult; rewrite Rmult_comm;
-    pattern (Rabs (sin h / h - 1)) at 2 in |- *; rewrite <- Rmult_1_r;
+    pattern (Rabs (sin h / h - 1)) at 2; rewrite <- Rmult_1_r;
       apply Rmult_le_compat_l.
   apply Rabs_pos.
   assert (H8 := COS_bound x); elim H8; intros.
-  unfold Rabs in |- *; case (Rcase_abs (cos x)); intro.
+  unfold Rabs; case (Rcase_abs (cos x)); intro.
   rewrite <- (Ropp_involutive 1); apply Ropp_le_contravar; assumption.
   assumption.
   cut (Rabs h < alp1).
@@ -540,8 +394,8 @@ Proof.
     apply H9.
   apply Rlt_le_trans with alp.
   apply H7.
-  unfold alp in |- *; apply Rmin_l.
-  rewrite sin_plus; unfold Rminus, Rdiv in |- *;
+  unfold alp; apply Rmin_l.
+  rewrite sin_plus; unfold Rminus, Rdiv;
     repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l;
       repeat rewrite Rmult_assoc; repeat rewrite Rplus_assoc;
         apply Rplus_eq_compat_l.
@@ -550,7 +404,7 @@ Proof.
   rewrite Ropp_mult_distr_r_reverse; rewrite Ropp_mult_distr_l_reverse;
     rewrite Rmult_1_r; rewrite Rmult_1_l; rewrite Ropp_mult_distr_r_reverse;
       rewrite <- Ropp_mult_distr_l_reverse; apply Rplus_comm.
-  unfold alp in |- *; unfold Rmin in |- *; case (Rle_dec alp1 alp2); intro.
+  unfold alp; unfold Rmin; case (Rle_dec alp1 alp2); intro.
   apply (cond_pos alp1).
   apply (cond_pos alp2).
 Qed.
@@ -565,7 +419,7 @@ Proof.
   intros; generalize (H0 _ _ _ H2 H1);
     replace (comp sin (id + fct_cte (PI / 2))%F) with
     (fun x:R => sin (x + PI / 2)); [ idtac | reflexivity ].
-  unfold derivable_pt_lim in |- *; intros.
+  unfold derivable_pt_lim; intros.
   elim (H3 eps H4); intros.
   exists x0.
   intros; rewrite <- (H (x + h)); rewrite <- (H x); apply H5; assumption.
@@ -579,26 +433,26 @@ Qed.
 
 Lemma derivable_pt_sin : forall x:R, derivable_pt sin x.
 Proof.
-  unfold derivable_pt in |- *; intro.
+  unfold derivable_pt; intro.
   exists (cos x).
   apply derivable_pt_lim_sin.
 Qed.
 
 Lemma derivable_pt_cos : forall x:R, derivable_pt cos x.
 Proof.
-  unfold derivable_pt in |- *; intro.
+  unfold derivable_pt; intro.
   exists (- sin x).
   apply derivable_pt_lim_cos.
 Qed.
 
 Lemma derivable_sin : derivable sin.
 Proof.
-  unfold derivable in |- *; intro; apply derivable_pt_sin.
+  unfold derivable; intro; apply derivable_pt_sin.
 Qed.
 
 Lemma derivable_cos : derivable cos.
 Proof.
-  unfold derivable in |- *; intro; apply derivable_pt_cos.
+  unfold derivable; intro; apply derivable_pt_cos.
 Qed.
 
 Lemma derive_pt_sin :
diff --git a/theories/Reals/SeqProp.v b/theories/Reals/SeqProp.v
index 75c57401..41e853cc 100644
--- a/theories/Reals/SeqProp.v
+++ b/theories/Reals/SeqProp.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -10,7 +10,7 @@ Require Import Rbase.
 Require Import Rfunctions.
 Require Import Rseries.
 Require Import Max.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (*****************************************************************)
 (**           Convergence properties of sequences                *)
@@ -36,7 +36,7 @@ Lemma decreasing_growing :
   forall Un:nat -> R, Un_decreasing Un -> Un_growing (opp_seq Un).
 Proof.
   intro.
-  unfold Un_growing, opp_seq, Un_decreasing in |- *.
+  unfold Un_growing, opp_seq, Un_decreasing.
   intros.
   apply Ropp_le_contravar.
   apply H.
@@ -58,8 +58,8 @@ Proof.
   unfold Un_cv in p.
   unfold R_dist in p.
   unfold opp_seq in p.
-  unfold Un_cv in |- *.
-  unfold R_dist in |- *.
+  unfold Un_cv.
+  unfold R_dist.
   intros.
   elim (p eps H1); intros.
   exists x0; intros.
@@ -77,7 +77,7 @@ Proof.
   apply completeness.
   assumption.
   exists (Un 0%nat).
-  unfold EUn in |- *.
+  unfold EUn.
   exists 0%nat; reflexivity.
 Qed.
 
@@ -114,9 +114,9 @@ Proof.
   unfold bound in H.
   elim H; intros.
   unfold is_upper_bound in H0.
-  unfold has_ub in |- *.
+  unfold has_ub.
   exists x.
-  unfold is_upper_bound in |- *.
+  unfold is_upper_bound.
   intros.
   apply H0.
   elim H1; intros.
@@ -132,9 +132,9 @@ Proof.
   unfold bound in H.
   elim H; intros.
   unfold is_upper_bound in H0.
-  unfold has_lb in |- *.
+  unfold has_lb.
   exists x.
-  unfold is_upper_bound in |- *.
+  unfold is_upper_bound.
   intros.
   apply H0.
   elim H1; intros.
@@ -155,9 +155,9 @@ Lemma Wn_decreasing :
   forall (Un:nat -> R) (pr:has_ub Un), Un_decreasing (sequence_ub Un pr).
 Proof.
   intros.
-  unfold Un_decreasing in |- *.
+  unfold Un_decreasing.
   intro.
-  unfold sequence_ub in |- *.
+  unfold sequence_ub.
   assert (H := ub_to_lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)).
   assert (H0 := ub_to_lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)).
   elim H; intros.
@@ -171,7 +171,7 @@ Proof.
   elim p; intros.
   apply H2.
   elim p0; intros.
-  unfold is_upper_bound in |- *.
+  unfold is_upper_bound.
   intros.
   unfold is_upper_bound in H3.
   apply H3.
@@ -190,7 +190,7 @@ Proof.
   assert
     (H7 := H3 (lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)) H4).
   apply Rle_antisym; assumption.
-  unfold lub in |- *.
+  unfold lub.
   case (ub_to_lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)).
   trivial.
   cut
@@ -204,7 +204,7 @@ Proof.
     (H7 :=
       H3 (lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)) H4).
   apply Rle_antisym; assumption.
-  unfold lub in |- *.
+  unfold lub.
   case (ub_to_lub (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)).
   trivial.
 Qed.
@@ -213,9 +213,9 @@ Lemma Vn_growing :
   forall (Un:nat -> R) (pr:has_lb Un), Un_growing (sequence_lb Un pr).
 Proof.
   intros.
-  unfold Un_growing in |- *.
+  unfold Un_growing.
   intro.
-  unfold sequence_lb in |- *.
+  unfold sequence_lb.
   assert (H := lb_to_glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)).
   assert (H0 := lb_to_glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)).
   elim H; intros.
@@ -230,14 +230,14 @@ Proof.
   apply Ropp_le_contravar.
   apply H2.
   elim p0; intros.
-  unfold is_upper_bound in |- *.
+  unfold is_upper_bound.
   intros.
   unfold is_upper_bound in H3.
   apply H3.
   elim H5; intros.
   exists (1 + x2)%nat.
   unfold opp_seq in H6.
-  unfold opp_seq in |- *.
+  unfold opp_seq.
   replace (n + (1 + x2))%nat with (S n + x2)%nat.
   assumption.
   replace (S n) with (1 + n)%nat; [ ring | ring ].
@@ -254,7 +254,7 @@ Proof.
     (Ropp_involutive (glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)))
     .
   apply Ropp_eq_compat; apply Rle_antisym; assumption.
-  unfold glb in |- *.
+  unfold glb.
   case (lb_to_glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)); simpl.
   intro; rewrite Ropp_involutive.
   trivial.
@@ -273,7 +273,7 @@ Proof.
       (glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)))
     .
   apply Ropp_eq_compat; apply Rle_antisym; assumption.
-  unfold glb in |- *.
+  unfold glb.
   case (lb_to_glb (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)); simpl.
   intro; rewrite Ropp_involutive.
   trivial.
@@ -286,7 +286,7 @@ Lemma Vn_Un_Wn_order :
 Proof.
   intros.
   split.
-  unfold sequence_lb in |- *.
+  unfold sequence_lb.
   cut { l:R | is_lub (EUn (opp_seq (fun i:nat => Un (n + i)%nat))) l }.
   intro X.
   elim X; intros.
@@ -298,7 +298,7 @@ Proof.
   apply Ropp_le_contravar.
   apply H.
   exists 0%nat.
-  unfold opp_seq in |- *.
+  unfold opp_seq.
   replace (n + 0)%nat with n; [ reflexivity | ring ].
   cut
     (is_lub (EUn (opp_seq (fun k:nat => Un (n + k)%nat)))
@@ -313,13 +313,13 @@ Proof.
     (Ropp_involutive (glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)))
     .
   apply Ropp_eq_compat; apply Rle_antisym; assumption.
-  unfold glb in |- *.
+  unfold glb.
   case (lb_to_glb (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)); simpl.
   intro; rewrite Ropp_involutive.
   trivial.
   apply lb_to_glb.
   apply min_ss; assumption.
-  unfold sequence_ub in |- *.
+  unfold sequence_ub.
   cut { l:R | is_lub (EUn (fun i:nat => Un (n + i)%nat)) l }.
   intro X.
   elim X; intros.
@@ -340,7 +340,7 @@ Proof.
   assert
     (H5 := H1 (lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)) H2).
   apply Rle_antisym; assumption.
-  unfold lub in |- *.
+  unfold lub.
   case (ub_to_lub (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)).
   intro; trivial.
   apply ub_to_lub.
@@ -353,13 +353,13 @@ Lemma min_maj :
 Proof.
   intros.
   assert (H := Vn_Un_Wn_order Un pr1 pr2).
-  unfold has_ub in |- *.
-  unfold bound in |- *.
+  unfold has_ub.
+  unfold bound.
   unfold has_ub in pr1.
   unfold bound in pr1.
   elim pr1; intros.
   exists x.
-  unfold is_upper_bound in |- *.
+  unfold is_upper_bound.
   intros.
   unfold is_upper_bound in H0.
   elim H1; intros.
@@ -376,20 +376,20 @@ Lemma maj_min :
 Proof.
   intros.
   assert (H := Vn_Un_Wn_order Un pr1 pr2).
-  unfold has_lb in |- *.
-  unfold bound in |- *.
+  unfold has_lb.
+  unfold bound.
   unfold has_lb in pr2.
   unfold bound in pr2.
   elim pr2; intros.
   exists x.
-  unfold is_upper_bound in |- *.
+  unfold is_upper_bound.
   intros.
   unfold is_upper_bound in H0.
   elim H1; intros.
   rewrite H2.
   apply Rle_trans with (opp_seq Un x1).
   assert (H3 := H x1); elim H3; intros.
-  unfold opp_seq in |- *; apply Ropp_le_contravar.
+  unfold opp_seq; apply Ropp_le_contravar.
   assumption.
   apply H0.
   exists x1; reflexivity.
@@ -399,7 +399,7 @@ Qed.
 Lemma cauchy_maj : forall Un:nat -> R, Cauchy_crit Un -> has_ub Un.
 Proof.
   intros.
-  unfold has_ub in |- *.
+  unfold has_ub.
   apply cauchy_bound.
   assumption.
 Qed.
@@ -409,12 +409,12 @@ Lemma cauchy_opp :
   forall Un:nat -> R, Cauchy_crit Un -> Cauchy_crit (opp_seq Un).
 Proof.
   intro.
-  unfold Cauchy_crit in |- *.
-  unfold R_dist in |- *.
+  unfold Cauchy_crit.
+  unfold R_dist.
   intros.
   elim (H eps H0); intros.
   exists x; intros.
-  unfold opp_seq in |- *.
+  unfold opp_seq.
   rewrite <- Rabs_Ropp.
   replace (- (- Un n - - Un m)) with (Un n - Un m);
   [ apply H1; assumption | ring ].
@@ -424,7 +424,7 @@ Qed.
 Lemma cauchy_min : forall Un:nat -> R, Cauchy_crit Un -> has_lb Un.
 Proof.
   intros.
-  unfold has_lb in |- *.
+  unfold has_lb.
   assert (H0 := cauchy_opp _ H).
   apply cauchy_bound.
   assumption.
@@ -485,7 +485,7 @@ Qed.
 
 Lemma not_Rlt : forall r1 r2:R, ~ r1 < r2 -> r1 >= r2.
 Proof.
-  intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rge in |- *.
+  intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rge.
   tauto.
 Qed.
 
@@ -595,11 +595,11 @@ Qed.
 Lemma UL_sequence :
   forall (Un:nat -> R) (l1 l2:R), Un_cv Un l1 -> Un_cv Un l2 -> l1 = l2.
 Proof.
-  intros Un l1 l2; unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  intros Un l1 l2; unfold Un_cv; unfold R_dist; intros.
   apply cond_eq.
   intros; cut (0 < eps / 2);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
         [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
   elim (H (eps / 2) H2); intros.
   elim (H0 (eps / 2) H2); intros.
@@ -609,8 +609,8 @@ Proof.
   [ apply Rabs_triang | ring ].
   rewrite (double_var eps); apply Rplus_lt_compat.
   rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H3;
-    unfold ge, N in |- *; apply le_max_l.
-  apply H4; unfold ge, N in |- *; apply le_max_r.
+    unfold ge, N; apply le_max_l.
+  apply H4; unfold ge, N; apply le_max_r.
 Qed.
 
 (**********)
@@ -618,10 +618,10 @@ Lemma CV_plus :
   forall (An Bn:nat -> R) (l1 l2:R),
     Un_cv An l1 -> Un_cv Bn l2 -> Un_cv (fun i:nat => An i + Bn i) (l1 + l2).
 Proof.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold Un_cv; unfold R_dist; intros.
   cut (0 < eps / 2);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
         [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
   elim (H (eps / 2) H2); intros.
   elim (H0 (eps / 2) H2); intros.
@@ -632,10 +632,10 @@ Proof.
   apply Rle_lt_trans with (Rabs (An n - l1) + Rabs (Bn n - l2)).
   apply Rabs_triang.
   rewrite (double_var eps); apply Rplus_lt_compat.
-  apply H3; unfold ge in |- *; apply le_trans with N;
-    [ unfold N in |- *; apply le_max_l | assumption ].
-  apply H4; unfold ge in |- *; apply le_trans with N;
-    [ unfold N in |- *; apply le_max_r | assumption ].
+  apply H3; unfold ge; apply le_trans with N;
+    [ unfold N; apply le_max_l | assumption ].
+  apply H4; unfold ge; apply le_trans with N;
+    [ unfold N; apply le_max_r | assumption ].
 Qed.
 
 (**********)
@@ -643,7 +643,7 @@ Lemma cv_cvabs :
   forall (Un:nat -> R) (l:R),
     Un_cv Un l -> Un_cv (fun i:nat => Rabs (Un i)) (Rabs l).
 Proof.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold Un_cv; unfold R_dist; intros.
   elim (H eps H0); intros.
   exists x; intros.
   apply Rle_lt_trans with (Rabs (Un n - l)).
@@ -656,15 +656,15 @@ Lemma CV_Cauchy :
   forall Un:nat -> R, { l:R | Un_cv Un l } -> Cauchy_crit Un.
 Proof.
   intros Un X; elim X; intros.
-  unfold Cauchy_crit in |- *; intros.
+  unfold Cauchy_crit; intros.
   unfold Un_cv in p; unfold R_dist in p.
   cut (0 < eps / 2);
     [ intro
-      | unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+      | unfold Rdiv; apply Rmult_lt_0_compat;
         [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
   elim (p (eps / 2) H0); intros.
   exists x0; intros.
-  unfold R_dist in |- *;
+  unfold R_dist;
     apply Rle_lt_trans with (Rabs (Un n - x) + Rabs (x - Un m)).
   replace (Un n - Un m) with (Un n - x + (x - Un m));
   [ apply Rabs_triang | ring ].
@@ -695,7 +695,7 @@ Proof.
   unfold is_upper_bound in H1.
   apply H1.
   exists n; reflexivity.
-  pattern x0 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
+  pattern x0 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
     apply Rlt_0_1.
   apply Rle_trans with (Rabs (Un 0%nat)).
   apply Rabs_pos.
@@ -717,7 +717,7 @@ Proof.
   assert (H1 := maj_by_pos An X).
   elim H1; intros M H2.
   elim H2; intros.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold Un_cv; unfold R_dist; intros.
   cut (0 < eps / (2 * M)).
   intro.
   case (Req_dec l2 0); intro.
@@ -744,24 +744,24 @@ Proof.
   rewrite Rmult_1_l; rewrite (Rmult_comm (/ M)).
   apply Rlt_trans with (eps / (2 * M)).
   apply H8; assumption.
-  unfold Rdiv in |- *; rewrite Rinv_mult_distr.
+  unfold Rdiv; rewrite Rinv_mult_distr.
   apply Rmult_lt_reg_l with 2.
   prove_sup0.
   replace (2 * (eps * (/ 2 * / M))) with (2 * / 2 * (eps * / M));
   [ idtac | ring ].
   rewrite <- Rinv_r_sym.
   rewrite Rmult_1_l; rewrite double.
-  pattern (eps * / M) at 1 in |- *; rewrite <- Rplus_0_r.
+  pattern (eps * / M) at 1; rewrite <- Rplus_0_r.
   apply Rplus_lt_compat_l; apply Rmult_lt_0_compat;
     [ assumption | apply Rinv_0_lt_compat; assumption ].
   discrR.
   discrR.
-  red in |- *; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3).
-  red in |- *; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3).
-  rewrite H7; do 2 rewrite Rmult_0_r; unfold Rminus in |- *;
+  red; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3).
+  red; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3).
+  rewrite H7; do 2 rewrite Rmult_0_r; unfold Rminus;
     rewrite Rplus_opp_r; rewrite Rabs_R0; reflexivity.
   replace (An n * Bn n - An n * l2) with (An n * (Bn n - l2)); [ idtac | ring ].
-  symmetry  in |- *; apply Rabs_mult.
+  symmetry ; apply Rabs_mult.
   cut (0 < eps / (2 * Rabs l2)).
   intro.
   unfold Un_cv in H; unfold R_dist in H; unfold Un_cv in H0;
@@ -790,36 +790,36 @@ Proof.
   rewrite Rmult_1_l; rewrite (Rmult_comm (/ M)).
   apply Rlt_le_trans with (eps / (2 * M)).
   apply H10.
-  unfold ge in |- *; apply le_trans with N.
-  unfold N in |- *; apply le_max_r.
+  unfold ge; apply le_trans with N.
+  unfold N; apply le_max_r.
   assumption.
-  unfold Rdiv in |- *; rewrite Rinv_mult_distr.
+  unfold Rdiv; rewrite Rinv_mult_distr.
   right; ring.
   discrR.
-  red in |- *; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3).
-  red in |- *; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3).
+  red; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3).
+  red; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3).
   apply Rmult_lt_reg_l with (/ Rabs l2).
   apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
   rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l; apply Rlt_le_trans with (eps / (2 * Rabs l2)).
   apply H9.
-  unfold ge in |- *; apply le_trans with N.
-  unfold N in |- *; apply le_max_l.
+  unfold ge; apply le_trans with N.
+  unfold N; apply le_max_l.
   assumption.
-  unfold Rdiv in |- *; right; rewrite Rinv_mult_distr.
+  unfold Rdiv; right; rewrite Rinv_mult_distr.
   ring.
   discrR.
   apply Rabs_no_R0; assumption.
   apply Rabs_no_R0; assumption.
   replace (An n * l2 - l1 * l2) with (l2 * (An n - l1));
-  [ symmetry  in |- *; apply Rabs_mult | ring ].
+  [ symmetry ; apply Rabs_mult | ring ].
   replace (An n * Bn n - An n * l2) with (An n * (Bn n - l2));
-  [ symmetry  in |- *; apply Rabs_mult | ring ].
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  [ symmetry ; apply Rabs_mult | ring ].
+  unfold Rdiv; apply Rmult_lt_0_compat.
   assumption.
   apply Rinv_0_lt_compat; apply Rmult_lt_0_compat;
     [ prove_sup0 | apply Rabs_pos_lt; assumption ].
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat;
+  unfold Rdiv; apply Rmult_lt_0_compat;
     [ assumption
       | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat;
         [ prove_sup0 | assumption ] ].
@@ -858,15 +858,15 @@ Proof.
   intros; exists (k + (1 - k) / 2).
   split.
   split.
-  pattern k at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  pattern k at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   apply Rplus_lt_reg_r with k; rewrite Rplus_0_r; replace (k + (1 - k)) with 1;
     [ elim H; intros; assumption | ring ].
   apply Rinv_0_lt_compat; prove_sup0.
   apply Rmult_lt_reg_l with 2.
   prove_sup0.
-  unfold Rdiv in |- *; rewrite Rmult_1_r; rewrite Rmult_plus_distr_l;
-    pattern 2 at 1 in |- *; rewrite Rmult_comm; rewrite Rmult_assoc;
+  unfold Rdiv; rewrite Rmult_1_r; rewrite Rmult_plus_distr_l;
+    pattern 2 at 1; rewrite Rmult_comm; rewrite Rmult_assoc;
       rewrite <- Rinv_l_sym; [ idtac | discrR ]; rewrite Rmult_1_r;
         replace (2 * k + (1 - k)) with (1 + k); [ idtac | ring ].
   elim H; intros.
@@ -885,7 +885,7 @@ Proof.
     repeat rewrite <- Rplus_assoc; rewrite Rplus_opp_l;
       repeat rewrite Rplus_0_l; apply H4.
   apply Rle_ge; elim H; intros; assumption.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   apply Rplus_lt_reg_r with k; rewrite Rplus_0_r; elim H; intros;
     replace (k + (1 - k)) with 1; [ assumption | ring ].
   apply Rinv_0_lt_compat; prove_sup0.
@@ -910,12 +910,12 @@ Proof.
   apply Rle_lt_trans with (Rabs (Un N - l)).
   apply RRle_abs.
   apply H2.
-  unfold ge, N in |- *; apply le_max_r.
-  unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- l));
+  unfold ge, N; apply le_max_r.
+  unfold Rminus; do 2 rewrite <- (Rplus_comm (- l));
     apply Rplus_le_compat_l.
   apply tech9.
   assumption.
-  unfold N in |- *; apply le_max_l.
+  unfold N; apply le_max_l.
   apply Rplus_lt_reg_r with l.
   rewrite Rplus_0_r.
   replace (l + (Un n - l)) with (Un n); [ assumption | ring ].
@@ -926,10 +926,10 @@ Lemma CV_opp :
   forall (An:nat -> R) (l:R), Un_cv An l -> Un_cv (opp_seq An) (- l).
 Proof.
   intros An l.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold Un_cv; unfold R_dist; intros.
   elim (H eps H0); intros.
   exists x; intros.
-  unfold opp_seq in |- *; replace (- An n - - l) with (- (An n - l));
+  unfold opp_seq; replace (- An n - - l) with (- (An n - l));
     [ rewrite Rabs_Ropp | ring ].
   apply H1; assumption.
 Qed.
@@ -954,10 +954,10 @@ Lemma CV_minus :
 Proof.
   intros.
   replace (fun i:nat => An i - Bn i) with (fun i:nat => An i + opp_seq Bn i).
-  unfold Rminus in |- *; apply CV_plus.
+  unfold Rminus; apply CV_plus.
   assumption.
   apply CV_opp; assumption.
-  unfold Rminus, opp_seq in |- *; reflexivity.
+  unfold Rminus, opp_seq; reflexivity.
 Qed.
 
 (** Un -> +oo *)
@@ -969,10 +969,10 @@ Lemma cv_infty_cv_R0 :
   forall Un:nat -> R,
     (forall n:nat, Un n <> 0) -> cv_infty Un -> Un_cv (fun n:nat => / Un n) 0.
 Proof.
-  unfold cv_infty, Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold cv_infty, Un_cv; unfold R_dist; intros.
   elim (H0 (/ eps)); intros N0 H2.
   exists N0; intros.
-  unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r;
+  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r;
     rewrite (Rabs_Rinv _ (H n)).
   apply Rmult_lt_reg_l with (Rabs (Un n)).
   apply Rabs_pos_lt; apply H.
@@ -984,7 +984,7 @@ Proof.
   rewrite Rmult_1_r; apply Rlt_le_trans with (Un n).
   apply H2; assumption.
   apply RRle_abs.
-  red in |- *; intro; rewrite H4 in H1; elim (Rlt_irrefl _ H1).
+  red; intro; rewrite H4 in H1; elim (Rlt_irrefl _ H1).
   apply Rabs_no_R0; apply H.
 Qed.
 
@@ -993,7 +993,7 @@ Lemma decreasing_prop :
   forall (Un:nat -> R) (m n:nat),
     Un_decreasing Un -> (m <= n)%nat -> Un n <= Un m.
 Proof.
-  unfold Un_decreasing in |- *; intros.
+  unfold Un_decreasing; intros.
   induction  n as [| n Hrecn].
   induction  m as [| m Hrecm].
   right; reflexivity.
@@ -1016,17 +1016,17 @@ Proof.
       (Un_cv (fun n:nat => Rabs x ^ n / INR (fact n)) 0 ->
         Un_cv (fun n:nat => x ^ n / INR (fact n)) 0).
   intro; apply H.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros; case (Req_dec x 0);
+  unfold Un_cv; unfold R_dist; intros; case (Req_dec x 0);
     intro.
   exists 1%nat; intros.
-  rewrite H1; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r;
+  rewrite H1; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r;
     rewrite Rabs_R0; rewrite pow_ne_zero;
-      [ unfold Rdiv in |- *; rewrite Rmult_0_l; rewrite Rabs_R0; assumption
-        | red in |- *; intro; rewrite H3 in H2; elim (le_Sn_n _ H2) ].
+      [ unfold Rdiv; rewrite Rmult_0_l; rewrite Rabs_R0; assumption
+        | red; intro; rewrite H3 in H2; elim (le_Sn_n _ H2) ].
   assert (H2 := Rabs_pos_lt x H1); set (M := up (Rabs x)); cut (0 <= M)%Z.
   intro; elim (IZN M H3); intros M_nat H4.
   set (Un := fun n:nat => Rabs x ^ (M_nat + n) / INR (fact (M_nat + n))).
-  cut (Un_cv Un 0); unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  cut (Un_cv Un 0); unfold Un_cv; unfold R_dist; intros.
   elim (H5 eps H0); intros N H6.
   exists (M_nat + N)%nat; intros;
     cut (exists p : nat, (p >= N)%nat /\ n = (M_nat + p)%nat).
@@ -1034,7 +1034,7 @@ Proof.
   elim H9; intros; rewrite H11; unfold Un in H6; apply H6; assumption.
   exists (n - M_nat)%nat.
   split.
-  unfold ge in |- *; apply (fun p n m:nat => plus_le_reg_l n m p) with M_nat;
+  unfold ge; apply (fun p n m:nat => plus_le_reg_l n m p) with M_nat;
     rewrite <- le_plus_minus.
   assumption.
   apply le_trans with (M_nat + N)%nat.
@@ -1048,43 +1048,43 @@ Proof.
   intro; cut (Un_decreasing Un).
   intro; cut (forall n:nat, Un (S n) <= Vn n).
   intro; cut (Un_cv Vn 0).
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros.
+  unfold Un_cv; unfold R_dist; intros.
   elim (H10 eps0 H5); intros N1 H11.
   exists (S N1); intros.
   cut (forall n:nat, 0 < Vn n).
   intro; apply Rle_lt_trans with (Rabs (Vn (pred n) - 0)).
   repeat rewrite Rabs_right.
-  unfold Rminus in |- *; rewrite Ropp_0; do 2 rewrite Rplus_0_r;
+  unfold Rminus; rewrite Ropp_0; do 2 rewrite Rplus_0_r;
     replace n with (S (pred n)).
   apply H9.
-  inversion H12; simpl in |- *; reflexivity.
-  apply Rle_ge; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; left;
+  inversion H12; simpl; reflexivity.
+  apply Rle_ge; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; left;
     apply H13.
-  apply Rle_ge; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; left;
+  apply Rle_ge; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; left;
     apply H7.
-  apply H11; unfold ge in |- *; apply le_S_n; replace (S (pred n)) with n;
-    [ unfold ge in H12; exact H12 | inversion H12; simpl in |- *; reflexivity ].
+  apply H11; unfold ge; apply le_S_n; replace (S (pred n)) with n;
+    [ unfold ge in H12; exact H12 | inversion H12; simpl; reflexivity ].
   intro; apply Rlt_le_trans with (Un (S n0)); [ apply H7 | apply H9 ].
   cut (cv_infty (fun n:nat => INR (S n))).
   intro; cut (Un_cv (fun n:nat => / INR (S n)) 0).
-  unfold Un_cv, R_dist in |- *; intros; unfold Vn in |- *.
+  unfold Un_cv, R_dist; intros; unfold Vn.
   cut (0 < eps1 / (Rabs x * Un 0%nat)).
   intro; elim (H11 _ H13); intros N H14.
   exists N; intros;
     replace (Rabs x * (Un 0%nat / INR (S n)) - 0) with
     (Rabs x * Un 0%nat * (/ INR (S n) - 0));
-    [ idtac | unfold Rdiv in |- *; ring ].
+    [ idtac | unfold Rdiv; ring ].
   rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs (Rabs x * Un 0%nat)).
   apply Rinv_0_lt_compat; apply Rabs_pos_lt.
   apply prod_neq_R0.
   apply Rabs_no_R0; assumption.
-  assert (H16 := H7 0%nat); red in |- *; intro; rewrite H17 in H16;
+  assert (H16 := H7 0%nat); red; intro; rewrite H17 in H16;
     elim (Rlt_irrefl _ H16).
   rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
   rewrite Rmult_1_l.
   replace (/ Rabs (Rabs x * Un 0%nat) * eps1) with (eps1 / (Rabs x * Un 0%nat)).
   apply H14; assumption.
-  unfold Rdiv in |- *; rewrite (Rabs_right (Rabs x * Un 0%nat)).
+  unfold Rdiv; rewrite (Rabs_right (Rabs x * Un 0%nat)).
   apply Rmult_comm.
   apply Rle_ge; apply Rmult_le_pos.
   apply Rabs_pos.
@@ -1092,9 +1092,9 @@ Proof.
   apply Rabs_no_R0.
   apply prod_neq_R0;
     [ apply Rabs_no_R0; assumption
-      | assert (H16 := H7 0%nat); red in |- *; intro; rewrite H17 in H16;
+      | assert (H16 := H7 0%nat); red; intro; rewrite H17 in H16;
         elim (Rlt_irrefl _ H16) ].
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   assumption.
   apply Rinv_0_lt_compat; apply Rmult_lt_0_compat.
   apply Rabs_pos_lt; assumption.
@@ -1102,7 +1102,7 @@ Proof.
   apply (cv_infty_cv_R0 (fun n:nat => INR (S n))).
   intro; apply not_O_INR; discriminate.
   assumption.
-  unfold cv_infty in |- *; intro; case (total_order_T M0 0); intro.
+  unfold cv_infty; intro; case (total_order_T M0 0); intro.
   elim s; intro.
   exists 0%nat; intros.
   apply Rlt_trans with 0; [ assumption | apply lt_INR_0; apply lt_O_Sn ].
@@ -1116,13 +1116,13 @@ Proof.
   elim H10; intros; assumption.
   rewrite H12; rewrite <- INR_IZR_INZ; apply le_INR.
   apply le_trans with n; [ assumption | apply le_n_Sn ].
-  apply le_IZR; left; simpl in |- *; unfold M0_z in |- *;
+  apply le_IZR; left; simpl; unfold M0_z;
     apply Rlt_trans with M0; [ assumption | elim H10; intros; assumption ].
   intro; apply Rle_trans with (Rabs x * Un n * / INR (S n)).
-  unfold Un in |- *; replace (M_nat + S n)%nat with (M_nat + n + 1)%nat.
+  unfold Un; replace (M_nat + S n)%nat with (M_nat + n + 1)%nat.
   rewrite pow_add; replace (Rabs x ^ 1) with (Rabs x);
-    [ idtac | simpl in |- *; ring ].
-  unfold Rdiv in |- *; rewrite <- (Rmult_comm (Rabs x));
+    [ idtac | simpl; ring ].
+  unfold Rdiv; rewrite <- (Rmult_comm (Rabs x));
     repeat rewrite Rmult_assoc; repeat apply Rmult_le_compat_l.
   apply Rabs_pos.
   left; apply pow_lt; assumption.
@@ -1130,33 +1130,33 @@ Proof.
   rewrite fact_simpl; rewrite mult_comm; rewrite mult_INR;
     rewrite Rinv_mult_distr.
   apply Rmult_le_compat_l.
-  left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *;
-    intro; assert (H10 := sym_eq H9); elim (fact_neq_0 _ H10).
+  left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red;
+    intro; assert (H10 := eq_sym H9); elim (fact_neq_0 _ H10).
   left; apply Rinv_lt_contravar.
   apply Rmult_lt_0_compat; apply lt_INR_0; apply lt_O_Sn.
   apply lt_INR; apply lt_n_S.
-  pattern n at 1 in |- *; replace n with (0 + n)%nat; [ idtac | reflexivity ].
+  pattern n at 1; replace n with (0 + n)%nat; [ idtac | reflexivity ].
   apply plus_lt_compat_r.
   apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ].
   apply INR_fact_neq_0.
   apply not_O_INR; discriminate.
   ring.
   ring.
-  unfold Vn in |- *; rewrite Rmult_assoc; unfold Rdiv in |- *;
+  unfold Vn; rewrite Rmult_assoc; unfold Rdiv;
     rewrite (Rmult_comm (Un 0%nat)); rewrite (Rmult_comm (Un n)).
   repeat apply Rmult_le_compat_l.
   apply Rabs_pos.
   left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
   apply decreasing_prop; [ assumption | apply le_O_n ].
-  unfold Un_decreasing in |- *; intro; unfold Un in |- *.
+  unfold Un_decreasing; intro; unfold Un.
   replace (M_nat + S n)%nat with (M_nat + n + 1)%nat.
-  rewrite pow_add; unfold Rdiv in |- *; rewrite Rmult_assoc;
+  rewrite pow_add; unfold Rdiv; rewrite Rmult_assoc;
     apply Rmult_le_compat_l.
   left; apply pow_lt; assumption.
-  replace (Rabs x ^ 1) with (Rabs x); [ idtac | simpl in |- *; ring ].
+  replace (Rabs x ^ 1) with (Rabs x); [ idtac | simpl; ring ].
   replace (M_nat + n + 1)%nat with (S (M_nat + n)).
   apply Rmult_le_reg_l with (INR (fact (S (M_nat + n)))).
-  apply lt_INR_0; apply neq_O_lt; red in |- *; intro; assert (H9 := sym_eq H8);
+  apply lt_INR_0; apply neq_O_lt; red; intro; assert (H9 := eq_sym H8);
     elim (fact_neq_0 _ H9).
   rewrite (Rmult_comm (Rabs x)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
   rewrite Rmult_1_l.
@@ -1170,37 +1170,37 @@ Proof.
   apply INR_fact_neq_0.
   ring.
   ring.
-  intro; unfold Un in |- *; unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  intro; unfold Un; unfold Rdiv; apply Rmult_lt_0_compat.
   apply pow_lt; assumption.
-  apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *; intro;
-    assert (H8 := sym_eq H7); elim (fact_neq_0 _ H8).
-  clear Un Vn; apply INR_le; simpl in |- *.
+  apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red; intro;
+    assert (H8 := eq_sym H7); elim (fact_neq_0 _ H8).
+  clear Un Vn; apply INR_le; simpl.
   induction  M_nat as [| M_nat HrecM_nat].
   assert (H6 := archimed (Rabs x)); fold M in H6; elim H6; intros.
   rewrite H4 in H7; rewrite <- INR_IZR_INZ in H7.
   simpl in H7; elim (Rlt_irrefl _ (Rlt_trans _ _ _ H2 H7)).
   replace 1 with (INR 1); [ apply le_INR | reflexivity ]; apply le_n_S;
     apply le_O_n.
-  apply le_IZR; simpl in |- *; left; apply Rlt_trans with (Rabs x).
+  apply le_IZR; simpl; left; apply Rlt_trans with (Rabs x).
   assumption.
   elim (archimed (Rabs x)); intros; assumption.
-  unfold Un_cv in |- *; unfold R_dist in |- *; intros; elim (H eps H0); intros.
+  unfold Un_cv; unfold R_dist; intros; elim (H eps H0); intros.
   exists x0; intros;
     apply Rle_lt_trans with (Rabs (Rabs x ^ n / INR (fact n) - 0)).
-  unfold Rminus in |- *; rewrite Ropp_0; do 2 rewrite Rplus_0_r;
+  unfold Rminus; rewrite Ropp_0; do 2 rewrite Rplus_0_r;
     rewrite (Rabs_right (Rabs x ^ n / INR (fact n))).
-  unfold Rdiv in |- *; rewrite Rabs_mult; rewrite (Rabs_right (/ INR (fact n))).
+  unfold Rdiv; rewrite Rabs_mult; rewrite (Rabs_right (/ INR (fact n))).
   rewrite RPow_abs; right; reflexivity.
   apply Rle_ge; left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt;
-    red in |- *; intro; assert (H4 := sym_eq H3); elim (fact_neq_0 _ H4).
-  apply Rle_ge; unfold Rdiv in |- *; apply Rmult_le_pos.
+    red; intro; assert (H4 := eq_sym H3); elim (fact_neq_0 _ H4).
+  apply Rle_ge; unfold Rdiv; apply Rmult_le_pos.
   case (Req_dec x 0); intro.
   rewrite H3; rewrite Rabs_R0.
   induction  n as [| n Hrecn];
-    [ simpl in |- *; left; apply Rlt_0_1
-      | simpl in |- *; rewrite Rmult_0_l; right; reflexivity ].
+    [ simpl; left; apply Rlt_0_1
+      | simpl; rewrite Rmult_0_l; right; reflexivity ].
   left; apply pow_lt; apply Rabs_pos_lt; assumption.
-  left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *;
-    intro; assert (H4 := sym_eq H3); elim (fact_neq_0 _ H4).
+  left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red;
+    intro; assert (H4 := eq_sym H3); elim (fact_neq_0 _ H4).
   apply H1; assumption.
 Qed.
diff --git a/theories/Reals/SeqSeries.v b/theories/Reals/SeqSeries.v
index 0d876be5..5140c29c 100644
--- a/theories/Reals/SeqSeries.v
+++ b/theories/Reals/SeqSeries.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -19,7 +19,7 @@ Require Export Rsigma.
 Require Export Rprod.
 Require Export Cauchy_prod.
 Require Export Alembert.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (**********)
 Lemma sum_maj1 :
@@ -41,21 +41,21 @@ Proof.
   intro; rewrite H4; rewrite H5.
   apply sum_cv_maj with
     (fun l:nat => An (S N + l)%nat) (fun (l:nat) (x:R) => fn (S N + l)%nat x) x.
-  unfold SP in |- *; apply H2.
+  unfold SP; apply H2.
   apply H3.
   intros; apply H1.
-  symmetry  in |- *; eapply UL_sequence.
+  symmetry ; eapply UL_sequence.
   apply H3.
-  unfold Un_cv in H0; unfold Un_cv in |- *; intros; elim (H0 eps H5);
+  unfold Un_cv in H0; unfold Un_cv; intros; elim (H0 eps H5);
     intros N0 H6.
   unfold R_dist in H6; exists N0; intros.
-  unfold R_dist in |- *;
+  unfold R_dist;
     replace (sum_f_R0 (fun l:nat => An (S N + l)%nat) n - (l2 - sum_f_R0 An N))
     with (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n - l2);
       [ idtac | ring ].
   replace (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n) with
   (sum_f_R0 An (S (N + n))).
-  apply H6; unfold ge in |- *; apply le_trans with n.
+  apply H6; unfold ge; apply le_trans with n.
   apply H7.
   apply le_trans with (N + n)%nat.
   apply le_plus_r.
@@ -80,12 +80,12 @@ Proof.
   reflexivity.
   apply le_lt_n_Sm; apply le_plus_l.
   apply le_O_n.
-  symmetry  in |- *; eapply UL_sequence.
+  symmetry ; eapply UL_sequence.
   apply H2.
-  unfold Un_cv in H; unfold Un_cv in |- *; intros.
+  unfold Un_cv in H; unfold Un_cv; intros.
   elim (H eps H4); intros N0 H5.
   unfold R_dist in H5; exists N0; intros.
-  unfold R_dist, SP in |- *;
+  unfold R_dist, SP;
     replace
     (sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n -
       (l1 - sum_f_R0 (fun k:nat => fn k x) N)) with
@@ -96,7 +96,7 @@ Proof.
   (sum_f_R0 (fun k:nat => fn k x) N +
     sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) with
   (sum_f_R0 (fun k:nat => fn k x) (S (N + n))).
-  unfold SP in H5; apply H5; unfold ge in |- *; apply le_trans with n.
+  unfold SP in H5; apply H5; unfold ge; apply le_trans with n.
   apply H6.
   apply le_trans with (N + n)%nat.
   apply le_plus_r.
@@ -124,16 +124,16 @@ Proof.
   apply le_plus_l.
   apply le_O_n.
   exists (l2 - sum_f_R0 An N).
-  unfold Un_cv in H0; unfold Un_cv in |- *; intros.
+  unfold Un_cv in H0; unfold Un_cv; intros.
   elim (H0 eps H2); intros N0 H3.
   unfold R_dist in H3; exists N0; intros.
-  unfold R_dist in |- *;
+  unfold R_dist;
     replace (sum_f_R0 (fun l:nat => An (S N + l)%nat) n - (l2 - sum_f_R0 An N))
     with (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n - l2);
       [ idtac | ring ].
   replace (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n) with
   (sum_f_R0 An (S (N + n))).
-  apply H3; unfold ge in |- *; apply le_trans with n.
+  apply H3; unfold ge; apply le_trans with n.
   apply H4.
   apply le_trans with (N + n)%nat.
   apply le_plus_r.
@@ -160,10 +160,10 @@ Proof.
   apply le_plus_l.
   apply le_O_n.
   exists (l1 - SP fn N x).
-  unfold Un_cv in H; unfold Un_cv in |- *; intros.
+  unfold Un_cv in H; unfold Un_cv; intros.
   elim (H eps H2); intros N0 H3.
   unfold R_dist in H3; exists N0; intros.
-  unfold R_dist, SP in |- *.
+  unfold R_dist, SP.
   replace
   (sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n -
     (l1 - sum_f_R0 (fun k:nat => fn k x) N)) with
@@ -175,7 +175,7 @@ Proof.
     sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) with
   (sum_f_R0 (fun k:nat => fn k x) (S (N + n))).
   unfold SP in H3; apply H3.
-  unfold ge in |- *; apply le_trans with n.
+  unfold ge; apply le_trans with n.
   apply H4.
   apply le_trans with (N + n)%nat.
   apply le_plus_r.
@@ -213,7 +213,7 @@ Lemma Rseries_CV_comp :
 Proof.
   intros An Bn H X; apply cv_cauchy_2.
   assert (H0 := cv_cauchy_1 _ X).
-  unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *.
+  unfold Cauchy_crit_series; unfold Cauchy_crit.
   intros; elim (H0 eps H1); intros.
   exists x; intros.
   cut
@@ -227,7 +227,7 @@ Proof.
   elim a; intro.
   rewrite (tech2 An n m); [ idtac | assumption ].
   rewrite (tech2 Bn n m); [ idtac | assumption ].
-  unfold R_dist in |- *; unfold Rminus in |- *; do 2 rewrite Ropp_plus_distr;
+  unfold R_dist; unfold Rminus; do 2 rewrite Ropp_plus_distr;
     do 2 rewrite <- Rplus_assoc; do 2 rewrite Rplus_opp_r;
       do 2 rewrite Rplus_0_l; do 2 rewrite Rabs_Ropp; repeat rewrite Rabs_right.
   apply sum_Rle; intros.
@@ -238,12 +238,12 @@ Proof.
   apply Rle_trans with (An (S n + n0)%nat); assumption.
   apply Rle_ge; apply cond_pos_sum; intro.
   elim (H (S n + n0)%nat); intros; assumption.
-  rewrite b; unfold R_dist in |- *; unfold Rminus in |- *;
+  rewrite b; unfold R_dist; unfold Rminus;
     do 2 rewrite Rplus_opp_r; rewrite Rabs_R0; right;
       reflexivity.
   rewrite (tech2 An m n); [ idtac | assumption ].
   rewrite (tech2 Bn m n); [ idtac | assumption ].
-  unfold R_dist in |- *; unfold Rminus in |- *; do 2 rewrite Rplus_assoc;
+  unfold R_dist; unfold Rminus; do 2 rewrite Rplus_assoc;
     rewrite (Rplus_comm (sum_f_R0 An m)); rewrite (Rplus_comm (sum_f_R0 Bn m));
       do 2 rewrite Rplus_assoc; do 2 rewrite Rplus_opp_l;
         do 2 rewrite Rplus_0_r; repeat rewrite Rabs_right.
@@ -266,13 +266,13 @@ Lemma Cesaro :
     Un_cv (fun n:nat => sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n)
     l.
 Proof with trivial.
-  unfold Un_cv in |- *; intros; assert (H3 : forall n:nat, 0 < sum_f_R0 An n)...
+  unfold Un_cv; intros; assert (H3 : forall n:nat, 0 < sum_f_R0 An n)...
   intro; apply tech1...
   assert (H4 : forall n:nat, sum_f_R0 An n <> 0)...
-  intro; red in |- *; intro; assert (H5 := H3 n); rewrite H4 in H5;
+  intro; red; intro; assert (H5 := H3 n); rewrite H4 in H5;
     elim (Rlt_irrefl _ H5)...
   assert (H5 := cv_infty_cv_R0 _ H4 H1); assert (H6 : 0 < eps / 2)...
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat...
+  unfold Rdiv; apply Rmult_lt_0_compat...
   apply Rinv_0_lt_compat; prove_sup...
   elim (H _ H6); clear H; intros N1 H;
     set (C := Rabs (sum_f_R0 (fun k:nat => An k * (Bn k - l)) N1));
@@ -282,10 +282,10 @@ Proof with trivial.
             (forall n:nat, (N <= n)%nat -> C / sum_f_R0 An n < eps / 2))...
   case (Req_dec C 0); intro...
   exists 0%nat; intros...
-  rewrite H7; unfold Rdiv in |- *; rewrite Rmult_0_l; apply Rmult_lt_0_compat...
+  rewrite H7; unfold Rdiv; rewrite Rmult_0_l; apply Rmult_lt_0_compat...
   apply Rinv_0_lt_compat; prove_sup...
   assert (H8 : 0 < eps / (2 * Rabs C))...
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat...
+  unfold Rdiv; apply Rmult_lt_0_compat...
   apply Rinv_0_lt_compat; apply Rmult_lt_0_compat...
   prove_sup...
   apply Rabs_pos_lt...
@@ -294,23 +294,23 @@ Proof with trivial.
       rewrite Rplus_0_r in H11...
   apply Rle_lt_trans with (Rabs (C / sum_f_R0 An n))...
   apply RRle_abs...
-  unfold Rdiv in |- *; rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs C)...
+  unfold Rdiv; rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs C)...
   apply Rinv_0_lt_compat; apply Rabs_pos_lt...
   rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym...
   rewrite Rmult_1_l; replace (/ Rabs C * (eps * / 2)) with (eps / (2 * Rabs C))...
-  unfold Rdiv in |- *; rewrite Rinv_mult_distr...
+  unfold Rdiv; rewrite Rinv_mult_distr...
   ring...
   discrR...
   apply Rabs_no_R0...
   apply Rabs_no_R0...
   elim H7; clear H7; intros N2 H7; set (N := max N1 N2); exists (S N); intros;
-    unfold R_dist in |- *;
+    unfold R_dist;
       replace (sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n - l) with
       (sum_f_R0 (fun k:nat => An k * (Bn k - l)) n / sum_f_R0 An n)...
   assert (H9 : (N1 < n)%nat)...
   apply lt_le_trans with (S N)...
-  apply le_lt_n_Sm; unfold N in |- *; apply le_max_l...
-  rewrite (tech2 (fun k:nat => An k * (Bn k - l)) _ _ H9); unfold Rdiv in |- *;
+  apply le_lt_n_Sm; unfold N; apply le_max_l...
+  rewrite (tech2 (fun k:nat => An k * (Bn k - l)) _ _ H9); unfold Rdiv;
     rewrite Rmult_plus_distr_r;
       apply Rle_lt_trans with
         (Rabs (sum_f_R0 (fun k:nat => An k * (Bn k - l)) N1 / sum_f_R0 An n) +
@@ -319,12 +319,12 @@ Proof with trivial.
             (n - S N1) / sum_f_R0 An n))...
   apply Rabs_triang...
   rewrite (double_var eps); apply Rplus_lt_compat...
-  unfold Rdiv in |- *; rewrite Rabs_mult; fold C in |- *; rewrite Rabs_right...
+  unfold Rdiv; rewrite Rabs_mult; fold C; rewrite Rabs_right...
   apply (H7 n); apply le_trans with (S N)...
-  apply le_trans with N; [ unfold N in |- *; apply le_max_r | apply le_n_Sn ]...
+  apply le_trans with N; [ unfold N; apply le_max_r | apply le_n_Sn ]...
   apply Rle_ge; left; apply Rinv_0_lt_compat...
 
-  unfold R_dist in H; unfold Rdiv in |- *; rewrite Rabs_mult;
+  unfold R_dist in H; unfold Rdiv; rewrite Rabs_mult;
     rewrite (Rabs_right (/ sum_f_R0 An n))...
   apply Rle_lt_trans with
     (sum_f_R0 (fun i:nat => Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)))
@@ -340,22 +340,22 @@ Proof with trivial.
   do 2 rewrite <- (Rmult_comm (/ sum_f_R0 An n)); apply Rmult_le_compat_l...
   left; apply Rinv_0_lt_compat...
   apply sum_Rle; intros; rewrite Rabs_mult;
-    pattern (An (S N1 + n0)%nat) at 2 in |- *;
+    pattern (An (S N1 + n0)%nat) at 2;
       rewrite <- (Rabs_right (An (S N1 + n0)%nat))...
   apply Rmult_le_compat_l...
   apply Rabs_pos...
-  left; apply H; unfold ge in |- *; apply le_trans with (S N1);
+  left; apply H; unfold ge; apply le_trans with (S N1);
     [ apply le_n_Sn | apply le_plus_l ]...
   apply Rle_ge; left...
   rewrite <- (scal_sum (fun i:nat => An (S N1 + i)%nat) (n - S N1) (eps / 2));
-    unfold Rdiv in |- *; repeat rewrite Rmult_assoc; apply Rmult_lt_compat_l...
-  pattern (/ 2) at 2 in |- *; rewrite <- Rmult_1_r; apply Rmult_lt_compat_l...
+    unfold Rdiv; repeat rewrite Rmult_assoc; apply Rmult_lt_compat_l...
+  pattern (/ 2) at 2; rewrite <- Rmult_1_r; apply Rmult_lt_compat_l...
   apply Rinv_0_lt_compat; prove_sup...
   rewrite Rmult_comm; apply Rmult_lt_reg_l with (sum_f_R0 An n)...
   rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym...
   rewrite Rmult_1_l; rewrite Rmult_1_r; rewrite (tech2 An N1 n)...
   rewrite Rplus_comm;
-    pattern (sum_f_R0 (fun i:nat => An (S N1 + i)%nat) (n - S N1)) at 1 in |- *;
+    pattern (sum_f_R0 (fun i:nat => An (S N1 + i)%nat) (n - S N1)) at 1;
       rewrite <- Rplus_0_r; apply Rplus_lt_compat_l...
   apply Rle_ge; left; apply Rinv_0_lt_compat...
   replace (sum_f_R0 (fun k:nat => An k * (Bn k - l)) n) with
@@ -371,41 +371,41 @@ Lemma Cesaro_1 :
 Proof with trivial.
   intros Bn l H; set (An := fun _:nat => 1)...
   assert (H0 : forall n:nat, 0 < An n)...
-  intro; unfold An in |- *; apply Rlt_0_1...
+  intro; unfold An; apply Rlt_0_1...
   assert (H1 : forall n:nat, 0 < sum_f_R0 An n)...
   intro; apply tech1...
   assert (H2 : cv_infty (fun n:nat => sum_f_R0 An n))...
-  unfold cv_infty in |- *; intro; case (Rle_dec M 0); intro...
+  unfold cv_infty; intro; case (Rle_dec M 0); intro...
   exists 0%nat; intros; apply Rle_lt_trans with 0...
   assert (H2 : 0 < M)...
   auto with real...
   clear n; set (m := up M); elim (archimed M); intros;
     assert (H5 : (0 <= m)%Z)...
-  apply le_IZR; unfold m in |- *; simpl in |- *; left; apply Rlt_trans with M...
-  elim (IZN _ H5); intros; exists x; intros; unfold An in |- *; rewrite sum_cte;
+  apply le_IZR; unfold m; simpl; left; apply Rlt_trans with M...
+  elim (IZN _ H5); intros; exists x; intros; unfold An; rewrite sum_cte;
     rewrite Rmult_1_l; apply Rlt_trans with (IZR (up M))...
   apply Rle_lt_trans with (INR x)...
-  rewrite INR_IZR_INZ; fold m in |- *; rewrite <- H6; right...
+  rewrite INR_IZR_INZ; fold m; rewrite <- H6; right...
   apply lt_INR; apply le_lt_n_Sm...
   assert (H3 := Cesaro _ _ _ H H0 H2)...
-  unfold Un_cv in |- *; unfold Un_cv in H3; intros; elim (H3 _ H4); intros;
-    exists (S x); intros; unfold R_dist in |- *; unfold R_dist in H5;
+  unfold Un_cv; unfold Un_cv in H3; intros; elim (H3 _ H4); intros;
+    exists (S x); intros; unfold R_dist; unfold R_dist in H5;
       apply Rle_lt_trans with
         (Rabs
           (sum_f_R0 (fun k:nat => An k * Bn k) (pred n) / sum_f_R0 An (pred n) - l))...
   right;
     replace (sum_f_R0 Bn (pred n) / INR n - l) with
     (sum_f_R0 (fun k:nat => An k * Bn k) (pred n) / sum_f_R0 An (pred n) - l)...
-  unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- l));
+  unfold Rminus; do 2 rewrite <- (Rplus_comm (- l));
     apply Rplus_eq_compat_l...
-  unfold An in |- *;
+  unfold An;
     replace (sum_f_R0 (fun k:nat => 1 * Bn k) (pred n)) with
     (sum_f_R0 Bn (pred n))...
   rewrite sum_cte; rewrite Rmult_1_l; replace (S (pred n)) with n...
   apply S_pred with 0%nat; apply lt_le_trans with (S x)...
   apply lt_O_Sn...
   apply sum_eq; intros; ring...
-  apply H5; unfold ge in |- *; apply le_S_n; replace (S (pred n)) with n...
+  apply H5; unfold ge; apply le_S_n; replace (S (pred n)) with n...
   apply S_pred with 0%nat; apply lt_le_trans with (S x)...
   apply lt_O_Sn...
 Qed.
diff --git a/theories/Reals/SplitAbsolu.v b/theories/Reals/SplitAbsolu.v
index 819606c4..d0de58b0 100644
--- a/theories/Reals/SplitAbsolu.v
+++ b/theories/Reals/SplitAbsolu.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -19,5 +19,5 @@ Ltac split_Rabs :=
   match goal with
     | id:context [(Rabs _)] |- _ => generalize id; clear id; try split_Rabs
     |  |- context [(Rabs ?X1)] =>
-      unfold Rabs in |- *; try split_case_Rabs; intros
+      unfold Rabs; try split_case_Rabs; intros
   end.
diff --git a/theories/Reals/SplitRmult.v b/theories/Reals/SplitRmult.v
index e554913c..09031fd6 100644
--- a/theories/Reals/SplitRmult.v
+++ b/theories/Reals/SplitRmult.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
diff --git a/theories/Reals/Sqrt_reg.v b/theories/Reals/Sqrt_reg.v
index d00ed178..89c17821 100644
--- a/theories/Reals/Sqrt_reg.v
+++ b/theories/Reals/Sqrt_reg.v
@@ -1,6 +1,6 @@
 (************************************************************************)
 (*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
 (*   \VV/  **************************************************************)
 (*    //   *      This file is distributed under the terms of the       *)
 (*         *       GNU Lesser General Public License Version 2.1        *)
@@ -10,7 +10,7 @@ Require Import Rbase.
 Require Import Rfunctions.
 Require Import Ranalysis1.
 Require Import R_sqrt.
-Open Local Scope R_scope.
+Local Open Scope R_scope.
 
 (**********)
 Lemma sqrt_var_maj :
@@ -21,67 +21,67 @@ Proof.
   case (total_order_T h 0); intro.
   elim s; intro.
   repeat rewrite Rabs_left.
-  unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (-1)).
+  unfold Rminus; do 2 rewrite <- (Rplus_comm (-1)).
   do 2 rewrite Ropp_plus_distr; rewrite Ropp_involutive;
     apply Rplus_le_compat_l.
   apply Ropp_le_contravar; apply sqrt_le_1.
   apply Rle_0_sqr.
   apply H0.
-  pattern (1 + h) at 2 in |- *; rewrite <- Rmult_1_r; unfold Rsqr in |- *;
+  pattern (1 + h) at 2; rewrite <- Rmult_1_r; unfold Rsqr;
     apply Rmult_le_compat_l.
   apply H0.
-  pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
+  pattern 1 at 2; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
     assumption.
   apply Rplus_lt_reg_r with 1; rewrite Rplus_0_r; rewrite Rplus_comm;
-    unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l;
+    unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l;
       rewrite Rplus_0_r.
-  pattern 1 at 2 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1.
+  pattern 1 at 2; rewrite <- sqrt_1; apply sqrt_lt_1.
   apply Rle_0_sqr.
   left; apply Rlt_0_1.
-  pattern 1 at 2 in |- *; rewrite <- Rsqr_1; apply Rsqr_incrst_1.
-  pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
+  pattern 1 at 2; rewrite <- Rsqr_1; apply Rsqr_incrst_1.
+  pattern 1 at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
     assumption.
   apply H0.
   left; apply Rlt_0_1.
   apply Rplus_lt_reg_r with 1; rewrite Rplus_0_r; rewrite Rplus_comm;
-    unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l;
+    unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l;
       rewrite Rplus_0_r.
-  pattern 1 at 2 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1.
+  pattern 1 at 2; rewrite <- sqrt_1; apply sqrt_lt_1.
   apply H0.
   left; apply Rlt_0_1.
-  pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
+  pattern 1 at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
     assumption.
   rewrite b; rewrite Rplus_0_r; rewrite Rsqr_1; rewrite sqrt_1; right;
     reflexivity.
   repeat rewrite Rabs_right.
-  unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (-1));
+  unfold Rminus; do 2 rewrite <- (Rplus_comm (-1));
     apply Rplus_le_compat_l.
   apply sqrt_le_1.
   apply H0.
   apply Rle_0_sqr.
-  pattern (1 + h) at 1 in |- *; rewrite <- Rmult_1_r; unfold Rsqr in |- *;
+  pattern (1 + h) at 1; rewrite <- Rmult_1_r; unfold Rsqr;
     apply Rmult_le_compat_l.
   apply H0.
-  pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
+  pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
     assumption.
   apply Rle_ge; apply Rplus_le_reg_l with 1.
-  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *;
+  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus;
     rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r.
-  pattern 1 at 1 in |- *; rewrite <- sqrt_1; apply sqrt_le_1.
+  pattern 1 at 1; rewrite <- sqrt_1; apply sqrt_le_1.
   left; apply Rlt_0_1.
   apply Rle_0_sqr.
-  pattern 1 at 1 in |- *; rewrite <- Rsqr_1; apply Rsqr_incr_1.
-  pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
+  pattern 1 at 1; rewrite <- Rsqr_1; apply Rsqr_incr_1.
+  pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
     assumption.
   left; apply Rlt_0_1.
   apply H0.
   apply Rle_ge; left; apply Rplus_lt_reg_r with 1.
-  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *;
+  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus;
     rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r.
-  pattern 1 at 1 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1.
+  pattern 1 at 1; rewrite <- sqrt_1; apply sqrt_lt_1.
   left; apply Rlt_0_1.
   apply H0.
-  pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
+  pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
     assumption.
   rewrite sqrt_Rsqr.
   replace (1 + h - 1) with h; [ right; reflexivity | ring ].
@@ -101,14 +101,14 @@ Qed.
 (** sqrt is continuous in 1 *)
 Lemma sqrt_continuity_pt_R1 : continuity_pt sqrt 1.
 Proof.
-  unfold continuity_pt in |- *; unfold continue_in in |- *;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
-      unfold dist in |- *; simpl in |- *; unfold R_dist in |- *;
+  unfold continuity_pt; unfold continue_in;
+    unfold limit1_in; unfold limit_in;
+      unfold dist; simpl; unfold R_dist;
         intros.
   set (alpha := Rmin eps 1).
   exists alpha; intros.
   split.
-  unfold alpha in |- *; unfold Rmin in |- *; case (Rle_dec eps 1); intro.
+  unfold alpha; unfold Rmin; case (Rle_dec eps 1); intro.
   assumption.
   apply Rlt_0_1.
   intros; elim H0; intros.
@@ -117,18 +117,18 @@ Proof.
   apply sqrt_var_maj.
   apply Rle_trans with alpha.
   left; apply H2.
-  unfold alpha in |- *; apply Rmin_r.
+  unfold alpha; apply Rmin_r.
   apply Rlt_le_trans with alpha;
-    [ apply H2 | unfold alpha in |- *; apply Rmin_l ].
+    [ apply H2 | unfold alpha; apply Rmin_l ].
 Qed.
 
 (** sqrt is continuous forall x>0 *)
 Lemma sqrt_continuity_pt : forall x:R, 0 < x -> continuity_pt sqrt x.
 Proof.
   intros; generalize sqrt_continuity_pt_R1.
-  unfold continuity_pt in |- *; unfold continue_in in |- *;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
-      unfold dist in |- *; simpl in |- *; unfold R_dist in |- *;
+  unfold continuity_pt; unfold continue_in;
+    unfold limit1_in; unfold limit_in;
+      unfold dist; simpl; unfold R_dist;
         intros.
   cut (0 < eps / sqrt x).
   intro; elim (H0 _ H2); intros alp_1 H3.
@@ -136,9 +136,9 @@ Proof.
   set (alpha := alp_1 * x).
   exists (Rmin alpha x); intros.
   split.
-  change (0 < Rmin alpha x) in |- *; unfold Rmin in |- *;
+  change (0 < Rmin alpha x); unfold Rmin;
     case (Rle_dec alpha x); intro.
-  unfold alpha in |- *; apply Rmult_lt_0_compat; assumption.
+  unfold alpha; apply Rmult_lt_0_compat; assumption.
   apply H.
   intros; replace x0 with (x + (x0 - x)); [ idtac | ring ];
     replace (sqrt (x + (x0 - x)) - sqrt x) with
@@ -150,7 +150,7 @@ Proof.
   rewrite Rmult_1_l; rewrite Rmult_comm.
   unfold Rdiv in H5.
   case (Req_dec x x0); intro.
-  rewrite H7; unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r;
+  rewrite H7; unfold Rminus, Rdiv; rewrite Rplus_opp_r;
     rewrite Rmult_0_l; rewrite Rplus_0_r; rewrite Rplus_opp_r;
       rewrite Rabs_R0.
   apply Rmult_lt_0_compat.
@@ -158,10 +158,10 @@ Proof.
   apply Rinv_0_lt_compat; rewrite <- H7; apply sqrt_lt_R0; assumption.
   apply H5.
   split.
-  unfold D_x, no_cond in |- *.
+  unfold D_x, no_cond.
   split.
   trivial.
-  red in |- *; intro.
+  red; intro.
   cut ((x0 - x) * / x = 0).
   intro.
   elim (Rmult_integral _ _ H9); intro.
@@ -170,35 +170,35 @@ Proof.
   assert (H11 := Rmult_eq_0_compat_r _ x H10).
   rewrite <- Rinv_l_sym in H11.
   elim R1_neq_R0; exact H11.
-  red in |- *; intro; rewrite H12 in H; elim (Rlt_irrefl _ H).
-  symmetry  in |- *; apply Rplus_eq_reg_l with 1; rewrite Rplus_0_r;
+  red; intro; rewrite H12 in H; elim (Rlt_irrefl _ H).
+  symmetry ; apply Rplus_eq_reg_l with 1; rewrite Rplus_0_r;
     unfold Rdiv in H8; exact H8.
-  unfold Rminus in |- *; rewrite Rplus_comm; rewrite <- Rplus_assoc;
+  unfold Rminus; rewrite Rplus_comm; rewrite <- Rplus_assoc;
     rewrite Rplus_opp_l; rewrite Rplus_0_l; elim H6; intros.
-  unfold Rdiv in |- *; rewrite Rabs_mult.
+  unfold Rdiv; rewrite Rabs_mult.
   rewrite Rabs_Rinv.
   rewrite (Rabs_right x).
   rewrite Rmult_comm; apply Rmult_lt_reg_l with x.
   apply H.
   rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
-  rewrite Rmult_1_l; rewrite Rmult_comm; fold alpha in |- *.
+  rewrite Rmult_1_l; rewrite Rmult_comm; fold alpha.
   apply Rlt_le_trans with (Rmin alpha x).
   apply H9.
   apply Rmin_l.
-  red in |- *; intro; rewrite H10 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H10 in H; elim (Rlt_irrefl _ H).
   apply Rle_ge; left; apply H.
-  red in |- *; intro; rewrite H10 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H10 in H; elim (Rlt_irrefl _ H).
   assert (H7 := sqrt_lt_R0 x H).
-  red in |- *; intro; rewrite H8 in H7; elim (Rlt_irrefl _ H7).
+  red; intro; rewrite H8 in H7; elim (Rlt_irrefl _ H7).
   apply Rle_ge; apply sqrt_positivity.
   left; apply H.
-  unfold Rminus in |- *; rewrite Rmult_plus_distr_l;
+  unfold Rminus; rewrite Rmult_plus_distr_l;
     rewrite Ropp_mult_distr_r_reverse; repeat rewrite <- sqrt_mult.
   rewrite Rmult_1_r; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r;
-    unfold Rdiv in |- *; rewrite Rmult_comm; rewrite Rmult_assoc;
+    unfold Rdiv; rewrite Rmult_comm; rewrite Rmult_assoc;
       rewrite <- Rinv_l_sym.
   rewrite Rmult_1_r; reflexivity.
-  red in |- *; intro; rewrite H7 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H7 in H; elim (Rlt_irrefl _ H).
   left; apply H.
   left; apply Rlt_0_1.
   left; apply H.
@@ -208,7 +208,7 @@ Proof.
   rewrite Rplus_comm.
   apply Rplus_le_reg_l with (- ((x0 - x) / x)).
   rewrite Rplus_0_r; rewrite <- Rplus_assoc; rewrite Rplus_opp_l;
-    rewrite Rplus_0_l; unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse.
+    rewrite Rplus_0_l; unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse.
   apply Rmult_le_reg_l with x.
   apply H.
   rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc;
@@ -216,13 +216,13 @@ Proof.
   rewrite Rmult_1_r; left; apply Rlt_le_trans with (Rmin alpha x).
   apply H8.
   apply Rmin_r.
-  red in |- *; intro; rewrite H9 in H; elim (Rlt_irrefl _ H).
+  red; intro; rewrite H9 in H; elim (Rlt_irrefl _ H).
   apply Rplus_le_le_0_compat.
   left; apply Rlt_0_1.
-  unfold Rdiv in |- *; apply Rmult_le_pos.
+  unfold Rdiv; apply Rmult_le_pos.
   apply Rge_le; exact r.
   left; apply Rinv_0_lt_compat; apply H.
-  unfold Rdiv in |- *; apply Rmult_lt_0_compat.
+  unfold Rdiv; apply Rmult_lt_0_compat.
   apply H1.
   apply Rinv_0_lt_compat; apply sqrt_lt_R0; apply H.
 Qed.
@@ -235,7 +235,7 @@ Proof.
   cut (continuity_pt g 0).
   intro; cut (g 0 <> 0).
   intro; assert (H2 := continuity_pt_inv g 0 H0 H1).
-  unfold derivable_pt_lim in |- *; intros; unfold continuity_pt in H2;
+  unfold derivable_pt_lim; intros; unfold continuity_pt in H2;
     unfold continue_in in H2; unfold limit1_in in H2;
       unfold limit_in in H2; simpl in H2; unfold R_dist in H2.
   elim (H2 eps H3); intros alpha H4.
@@ -247,29 +247,29 @@ Proof.
   unfold inv_fct, g in H6; replace (2 * sqrt x) with (sqrt x + sqrt (x + 0)).
   apply H6.
   split.
-  unfold D_x, no_cond in |- *.
+  unfold D_x, no_cond.
   split.
   trivial.
-  apply (sym_not_eq (A:=R)); exact H8.
-  unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r;
+  apply (not_eq_sym (A:=R)); exact H8.
+  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r;
     apply Rlt_le_trans with alpha1.
   exact H9.
-  unfold alpha1 in |- *; apply Rmin_l.
+  unfold alpha1; apply Rmin_l.
   rewrite Rplus_0_r; ring.
   cut (0 <= x + h).
   intro; cut (0 < sqrt x + sqrt (x + h)).
   intro; apply Rmult_eq_reg_l with (sqrt x + sqrt (x + h)).
   rewrite <- Rinv_r_sym.
-  rewrite Rplus_comm; unfold Rdiv in |- *; rewrite <- Rmult_assoc;
+  rewrite Rplus_comm; unfold Rdiv; rewrite <- Rmult_assoc;
     rewrite Rsqr_plus_minus; repeat rewrite Rsqr_sqrt.
-  rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc;
+  rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc;
     rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite <- Rinv_r_sym.
   reflexivity.
   apply H8.
   left; apply H.
   assumption.
-  red in |- *; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11).
-  red in |- *; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11).
+  red; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11).
+  red; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11).
   apply Rplus_lt_le_0_compat.
   apply sqrt_lt_R0; apply H.
   apply sqrt_positivity; apply H10.
@@ -279,35 +279,35 @@ Proof.
   rewrite Rplus_0_r; rewrite Rplus_comm; rewrite Rplus_assoc;
     rewrite Rplus_opp_r; rewrite Rplus_0_r; left; apply Rlt_le_trans with alpha1.
   apply H9.
-  unfold alpha1 in |- *; apply Rmin_r.
+  unfold alpha1; apply Rmin_r.
   apply Rplus_le_le_0_compat.
   left; assumption.
   apply Rge_le; apply r.
-  unfold alpha1 in |- *; unfold Rmin in |- *; case (Rle_dec alpha x); intro.
+  unfold alpha1; unfold Rmin; case (Rle_dec alpha x); intro.
   apply H5.
   apply H.
-  unfold g in |- *; rewrite Rplus_0_r.
+  unfold g; rewrite Rplus_0_r.
   cut (0 < sqrt x + sqrt x).
-  intro; red in |- *; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1).
+  intro; red; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1).
   apply Rplus_lt_0_compat; apply sqrt_lt_R0; apply H.
   replace g with (fct_cte (sqrt x) + comp sqrt (fct_cte x + id))%F;
   [ idtac | reflexivity ].
   apply continuity_pt_plus.
-  apply continuity_pt_const; unfold constant, fct_cte in |- *; intro;
+  apply continuity_pt_const; unfold constant, fct_cte; intro;
     reflexivity.
   apply continuity_pt_comp.
   apply continuity_pt_plus.
-  apply continuity_pt_const; unfold constant, fct_cte in |- *; intro;
+  apply continuity_pt_const; unfold constant, fct_cte; intro;
     reflexivity.
   apply derivable_continuous_pt; apply derivable_pt_id.
   apply sqrt_continuity_pt.
-  unfold plus_fct, fct_cte, id in |- *; rewrite Rplus_0_r; apply H.
+  unfold plus_fct, fct_cte, id; rewrite Rplus_0_r; apply H.
 Qed.
 
 (**********)
 Lemma derivable_pt_sqrt : forall x:R, 0 < x -> derivable_pt sqrt x.
 Proof.
-  unfold derivable_pt in |- *; intros.
+  unfold derivable_pt; intros.
   exists (/ (2 * sqrt x)).
   apply derivable_pt_lim_sqrt; assumption.
 Qed.
@@ -330,19 +330,19 @@ Proof.
   intros; case (Rtotal_order 0 x); intro.
   apply (sqrt_continuity_pt x H0).
   elim H0; intro.
-  unfold continuity_pt in |- *; unfold continue_in in |- *;
-    unfold limit1_in in |- *; unfold limit_in in |- *;
-      simpl in |- *; unfold R_dist in |- *; intros.
+  unfold continuity_pt; unfold continue_in;
+    unfold limit1_in; unfold limit_in;
+      simpl; unfold R_dist; intros.
   exists (Rsqr eps); intros.
   split.
-  change (0 < Rsqr eps) in |- *; apply Rsqr_pos_lt.
-  red in |- *; intro; rewrite H3 in H2; elim (Rlt_irrefl _ H2).
+  change (0 < Rsqr eps); apply Rsqr_pos_lt.
+  red; intro; rewrite H3 in H2; elim (Rlt_irrefl _ H2).
   intros; elim H3; intros.
-  rewrite <- H1; rewrite sqrt_0; unfold Rminus in |- *; rewrite Ropp_0;
+  rewrite <- H1; rewrite sqrt_0; unfold Rminus; rewrite Ropp_0;
     rewrite Rplus_0_r; rewrite <- H1 in H5; unfold Rminus in H5;
       rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5.
   case (Rcase_abs x0); intro.
-  unfold sqrt in |- *; case (Rcase_abs x0); intro.
+  unfold sqrt; case (Rcase_abs x0); intro.
   rewrite Rabs_R0; apply H2.
   assert (H6 := Rge_le _ _ r0); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 r)).
   rewrite Rabs_right.
diff --git a/theories/Reals/vo.itarget b/theories/Reals/vo.itarget
index bcd47a0b..36dd0f56 100644
--- a/theories/Reals/vo.itarget
+++ b/theories/Reals/vo.itarget
@@ -9,6 +9,7 @@ DiscrR.vo
 Exp_prop.vo
 Integration.vo
 LegacyRfield.vo
+Machin.vo
 MVT.vo
 NewtonInt.vo
 PartSum.vo
@@ -17,7 +18,10 @@ Ranalysis1.vo
 Ranalysis2.vo
 Ranalysis3.vo
 Ranalysis4.vo
+Ranalysis5.vo
 Ranalysis.vo
+Ranalysis_reg.vo
+Ratan.vo
 Raxioms.vo
 Rbase.vo
 Rbasic_fun.vo
@@ -48,6 +52,7 @@ Rtrigo_calc.vo
 Rtrigo_def.vo
 Rtrigo_fun.vo
 Rtrigo_reg.vo
+Rtrigo1.vo
 Rtrigo.vo
 SeqProp.vo
 SeqSeries.vo