Uniformisation (Qed/Save et Implicits Arguments)

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2650 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 212 insertions, 212 deletions

diff --git a/theories/Reals/Rbase.v b/theories/Reals/Rbase.v
index f20254f62..fd243969b 100644
--- a/theories/Reals/Rbase.v
+++ b/theories/Reals/Rbase.v
@@ -48,22 +48,22 @@ Add Field R Rplus Rmult R1 R0 Ropp [x,y:R]false Rinv RTheory Rinv_l
 (**********)
 Lemma Rlt_antirefl:(r:R)~``r<r``.
   Generalize Rlt_antisym. Intuition EAuto.
-Save.
+Qed.
 Hints Resolve Rlt_antirefl : real.
 
 Lemma Rlt_not_eq:(r1,r2:R)``r1<r2``->``r1<>r2``.
   Red; Intros r1 r2 H H0; Apply (Rlt_antirefl r1).
   Pattern 2 r1; Rewrite H0; Trivial.
-Save.
+Qed.
 
 Lemma Rgt_not_eq:(r1,r2:R)``r1>r2``->``r1<>r2``.
 Intros; Apply sym_not_eqT; Apply Rlt_not_eq; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma imp_not_Req:(r1,r2:R)(``r1<r2``\/ ``r1>r2``) -> ``r1<>r2``.
 Generalize Rlt_not_eq Rgt_not_eq. Intuition EAuto.
-Save.
+Qed.
 Hints Resolve imp_not_Req : real.
 
 (** Reasoning by case on equalities and order *)
@@ -71,18 +71,18 @@ Hints Resolve imp_not_Req : real.
 (**********)
 Lemma Req_EM:(r1,r2:R)(r1==r2)\/``r1<>r2``.
 Intros ; Generalize (total_order_T r1 r2) imp_not_Req ; Intuition EAuto 3. 
-Save.
+Qed.
 Hints Resolve Req_EM : real.
 
 (**********)
 Lemma total_order:(r1,r2:R)``r1<r2``\/(r1==r2)\/``r1>r2``.
 Intros;Generalize (total_order_T r1 r2);Tauto.
-Save.
+Qed.
 
 (**********)
 Lemma not_Req:(r1,r2:R)``r1<>r2``->(``r1<r2``\/``r1>r2``).
 Intros; Generalize (total_order_T r1 r2) ; Tauto.
-Save.
+Qed.
 
 
 (*********************************************************************************)
@@ -92,23 +92,23 @@ Save.
 (**********)
 Lemma Rlt_le:(r1,r2:R)``r1<r2``-> ``r1<=r2``.
 Intros ; Red ; Tauto.
-Save.
+Qed.
 Hints Resolve Rlt_le : real.
 
 (**********)
 Lemma Rle_ge : (r1,r2:R)``r1<=r2`` -> ``r2>=r1``.
 NewDestruct 1; Red; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rge_le : (r1,r2:R)``r1>=r2`` -> ``r2<=r1``.
 NewDestruct 1; Red; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma not_Rle:(r1,r2:R)~(``r1<=r2``)->``r1>r2``.
 Intros r1 r2 ; Generalize (total_order r1 r2) ; Unfold Rle; Tauto.
-Save.
+Qed.
 
 Hints Immediate Rle_ge Rge_le not_Rle : real.
 
@@ -116,7 +116,7 @@ Hints Immediate Rle_ge Rge_le not_Rle : real.
 Lemma Rlt_le_not:(r1,r2:R)``r2<r1``->~(``r1<=r2``).
 Generalize Rlt_antisym imp_not_Req ; Unfold Rle.
 Intuition EAuto 3.
-Save.
+Qed.
 
 Lemma Rle_not:(r1,r2:R)``r1>r2`` -> ~(``r1<=r2``).
 Proof Rlt_le_not.
@@ -126,109 +126,109 @@ Hints Immediate Rlt_le_not : real.
 Lemma Rle_not_lt: (r1, r2:R) ``r2 <= r1`` ->~ (``r1<r2``).
 Intros r1 r2. Generalize (Rlt_antisym r1 r2) (imp_not_Req r1 r2).
 Unfold Rle; Intuition.
-Save.
+Qed.
 
 (**********)
 Lemma Rlt_ge_not:(r1,r2:R)``r1<r2``->~(``r1>=r2``).
 Generalize Rlt_le_not. Unfold Rle Rge. Intuition EAuto 3.
-Save.
+Qed.
 
 Hints Immediate Rlt_ge_not : real.
 
 (**********)
 Lemma eq_Rle:(r1,r2:R)r1==r2->``r1<=r2``.
 Unfold Rle; Tauto.
-Save.
+Qed.
 Hints Immediate eq_Rle : real.
 
 Lemma eq_Rge:(r1,r2:R)r1==r2->``r1>=r2``.
 Unfold Rge; Tauto.
-Save.
+Qed.
 Hints Immediate eq_Rge : real.
 
 Lemma eq_Rle_sym:(r1,r2:R)r2==r1->``r1<=r2``.
 Unfold Rle; Auto.
-Save.
+Qed.
 Hints Immediate eq_Rle_sym : real.
 
 Lemma eq_Rge_sym:(r1,r2:R)r2==r1->``r1>=r2``.
 Unfold Rge; Auto.
-Save.
+Qed.
 Hints Immediate eq_Rge_sym : real.
 
 Lemma Rle_antisym : (r1,r2:R)``r1<=r2`` -> ``r2<=r1``-> r1==r2.
 Intros r1 r2; Generalize (Rlt_antisym r1 r2) ; Unfold Rle ; Intuition.
-Save.
+Qed.
 Hints Resolve Rle_antisym : real.
 
 (**********)
 Lemma Rle_le_eq:(r1,r2:R)(``r1<=r2``/\``r2<=r1``)<->(r1==r2).
 Intuition.
-Save.
+Qed.
 
 Lemma Rlt_rew : (x,x',y,y':R)``x==x'``->``x'<y'`` -> `` y' == y`` -> ``x < y``.
 Intros; Replace x with x'; Replace y with y'; Assumption.
-Save.
+Qed.
 
 (**********)
 Lemma Rle_trans:(r1,r2,r3:R) ``r1<=r2``->``r2<=r3``->``r1<=r3``.
 Generalize trans_eqT Rlt_trans Rlt_rew.
 Unfold Rle.
 Intuition EAuto 2.
-Save.
+Qed.
 
 (**********)
 Lemma Rle_lt_trans:(r1,r2,r3:R)``r1<=r2``->``r2<r3``->``r1<r3``.
 Generalize Rlt_trans Rlt_rew. 
 Unfold Rle.
 Intuition EAuto 2.
-Save.
+Qed.
 
 (**********)
 Lemma Rlt_le_trans:(r1,r2,r3:R)``r1<r2``->``r2<=r3``->``r1<r3``.
 Generalize Rlt_trans Rlt_rew; Unfold Rle; Intuition EAuto 2.
-Save.
+Qed.
 
 
 (** Decidability of the order *)
 Lemma total_order_Rlt:(r1,r2:R)(sumboolT ``r1<r2`` ~(``r1<r2``)).
 Intros;Generalize (total_order_T r1 r2) (imp_not_Req r1 r2) ; Intuition.
-Save.
+Qed.
 
 (**********)
 Lemma total_order_Rle:(r1,r2:R)(sumboolT ``r1<=r2`` ~(``r1<=r2``)).
 Intros r1 r2.
 Generalize (total_order_T r1 r2) (imp_not_Req r1 r2).
 Intuition EAuto 4 with real.
-Save.
+Qed.
 
 (**********)
 Lemma total_order_Rgt:(r1,r2:R)(sumboolT ``r1>r2`` ~(``r1>r2``)).
 Intros;Unfold Rgt;Intros;Apply total_order_Rlt.
-Save.
+Qed.
 
 (**********)
 Lemma total_order_Rge:(r1,r2:R)(sumboolT (``r1>=r2``) ~(``r1>=r2``)).
 Intros;Generalize (total_order_Rle r2 r1);Intuition.
-Save.
+Qed.
 
 Lemma total_order_Rlt_Rle:(r1,r2:R)(sumboolT ``r1<r2`` ``r2<=r1``).
 Intros;Generalize (total_order_T r1 r2); Intuition.
-Save.
+Qed.
 
 Lemma Rle_or_lt: (n, m:R)(Rle n m) \/ (Rlt m n).
 Intros n m; Elim (total_order_Rlt_Rle m n);Auto with real.
-Save.
+Qed.
 
 Lemma total_order_Rle_Rlt_eq :(r1,r2:R)``r1<=r2``-> 
                               (sumboolT ``r1<r2`` ``r1==r2``).
 Intros r1 r2 H;Generalize (total_order_T r1 r2); Intuition.
-Save.
+Qed.
 
 (**********)
 Lemma inser_trans_R:(n,m,p,q:R)``n<=m<p``-> (sumboolT ``n<=m<q`` ``q<=m<p``).
 Intros; Generalize (total_order_Rlt_Rle m q); Intuition.
-Save.
+Qed.
 
 (****************************************************************)
 (**        Field Lemmas                                         *)
@@ -240,18 +240,18 @@ Save.
 
 Lemma Rplus_ne:(r:R)``r+0==r``/\``0+r==r``.
 Intro;Split;Ring.
-Save.
+Qed.
 Hints Resolve Rplus_ne : real v62.
 
 Lemma Rplus_Or:(r:R)``r+0==r``.
 Intro; Ring.
-Save.
+Qed.
 Hints Resolve Rplus_Or : real.
 
 (**********)
 Lemma Rplus_Ropp_l:(r:R)``(-r)+r==0``.
   Intro; Ring.
-Save.
+Qed.
 Hints Resolve Rplus_Ropp_l : real.
 
 
@@ -260,14 +260,14 @@ Lemma Rplus_Ropp:(x,y:R)``x+y==0``->``y== -x``.
   Intros; Replace y with ``(-x+x)+y``;
   [ Rewrite -> Rplus_assoc; Rewrite -> H; Ring
   | Ring ].
-Save.
+Qed.
 
 (*i New i*)
 Hint eqT_R_congr : real := Resolve (congr_eqT R).
 
 Lemma Rplus_plus_r:(r,r1,r2:R)(r1==r2)->``r+r1==r+r2``.
   Auto with real.
-Save.
+Qed.
 
 (*i Old i*)Hints Resolve Rplus_plus_r : v62.
 
@@ -278,14 +278,14 @@ Lemma r_Rplus_plus:(r,r1,r2:R)``r+r1==r+r2``->r1==r2.
   Transitivity ``(-r+r)+r2``.
   Repeat Rewrite -> Rplus_assoc; Rewrite <- H; Reflexivity.
   Ring.
-Save.
+Qed.
 Hints Resolve r_Rplus_plus : real.
 
 (**********)
 Lemma Rplus_ne_i:(r,b:R)``r+b==r`` -> ``b==0``.
   Intros r b; Pattern 2 r; Replace r with ``r+0``;
   EAuto with real.
-Save.
+Qed.
 
 (***********************************************************)       
 (**       Multiplication                                   *)
@@ -294,47 +294,47 @@ Save.
 (**********)
 Lemma Rinv_r:(r:R)``r<>0``->``r* (/r)==1``.
   Intros; Rewrite -> Rmult_sym; Auto with real.
-Save.
+Qed.
 Hints Resolve Rinv_r : real.
 
 Lemma Rinv_l_sym:(r:R)``r<>0``->``1==(/r) * r``.
   Symmetry; Auto with real.
-Save.
+Qed.
 
 Lemma Rinv_r_sym:(r:R)``r<>0``->``1==r* (/r)``.
   Symmetry; Auto with real.
-Save.
+Qed.
 Hints Resolve Rinv_l_sym Rinv_r_sym : real.
 
 
 (**********)
 Lemma Rmult_Or :(r:R) ``r*0==0``.
 Intro; Ring.
-Save.
+Qed.
 Hints Resolve Rmult_Or : real v62.
 
 (**********)
 Lemma Rmult_Ol:(r:R)(``0*r==0``).
 Intro; Ring.
-Save.
+Qed.
 Hints Resolve Rmult_Ol : real v62.
 
 (**********)
 Lemma Rmult_ne:(r:R)``r*1==r``/\``1*r==r``.
 Intro;Split;Ring.
-Save.
+Qed.
 Hints Resolve Rmult_ne : real v62.
 
 (**********)
 Lemma Rmult_1r:(r:R)(``r*1==r``).
 Intro; Ring.
-Save.
+Qed.
 Hints Resolve Rmult_1r : real.
 
 (**********)
 Lemma Rmult_mult_r:(r,r1,r2:R)r1==r2->``r*r1==r*r2``.
   Auto with real.
-Save.
+Qed.
 
 (*i OLD i*)Hints Resolve Rmult_mult_r : v62.
 
@@ -345,7 +345,7 @@ Lemma r_Rmult_mult:(r,r1,r2:R)(``r*r1==r*r2``)->``r<>0``->(r1==r2).
   Transitivity ``(/r * r)*r2``.
   Repeat Rewrite Rmult_assoc; Rewrite H; Trivial.
   Rewrite Rinv_l; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma without_div_Od:(r1,r2:R)``r1*r2==0`` -> ``r1==0`` \/ ``r2==0``.
@@ -353,43 +353,43 @@ Lemma without_div_Od:(r1,r2:R)``r1*r2==0`` -> ``r1==0`` \/ ``r2==0``.
   Auto.
   Right; Apply r_Rmult_mult with r1; Trivial.
   Rewrite H; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma without_div_Oi:(r1,r2:R) ``r1==0``\/``r2==0`` -> ``r1*r2==0``.
   Intros r1 r2 [H | H]; Rewrite H; Auto with real.
-Save.
+Qed.
 
 Hints Resolve without_div_Oi : real.
 
 (**********)
 Lemma without_div_Oi1:(r1,r2:R) ``r1==0`` -> ``r1*r2==0``.
   Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma without_div_Oi2:(r1,r2:R) ``r2==0`` -> ``r1*r2==0``.
   Auto with real.
-Save.
+Qed.
 
 
 (**********) 
 Lemma without_div_O_contr:(r1,r2:R)``r1*r2<>0`` -> ``r1<>0`` /\ ``r2<>0``.
 Intros r1 r2 H; Split; Red; Intro; Apply H; Auto with real.
-Save.
+Qed.
 
 (**********) 
 Lemma mult_non_zero :(r1,r2:R)``r1<>0`` /\ ``r2<>0`` -> ``r1*r2<>0``.
 Red; Intros r1 r2 (H1,H2) H.
 Case (without_div_Od r1 r2); Auto with real.
-Save.
+Qed.
 Hints Resolve mult_non_zero : real.
 
 (**********) 
 Lemma Rmult_Rplus_distrl:
    (r1,r2,r3:R) ``(r1+r2)*r3 == (r1*r3)+(r2*r3)``.
 Intros; Ring.
-Save.
+Qed.
 
 (** Square function *)
 
@@ -399,12 +399,12 @@ Definition Rsqr:R->R:=[r:R]``r*r``.
 (***********)
 Lemma Rsqr_O:(Rsqr ``0``)==``0``.
   Unfold Rsqr; Auto with real.
-Save.
+Qed.
 
 (***********)
 Lemma Rsqr_r_R0:(r:R)(Rsqr r)==``0``->``r==0``.
 Unfold Rsqr;Intros;Elim (without_div_Od r r H);Trivial.
-Save.
+Qed.
 
 (*********************************************************)
 (**      Opposite                                        *)
@@ -413,25 +413,25 @@ Save.
 (**********)
 Lemma eq_Ropp:(r1,r2:R)(r1==r2)->``-r1 == -r2``.
   Auto with real.
-Save.
+Qed.
 Hints Resolve eq_Ropp : real.
 
 (**********)
 Lemma Ropp_O:``-0==0``.
   Ring.
-Save.
+Qed.
 Hints Resolve Ropp_O : real v62.
 
 (**********)
 Lemma eq_RoppO:(r:R)``r==0``-> ``-r==0``.
   Intros; Rewrite -> H; Auto with real.
-Save.
+Qed.
 Hints Resolve eq_RoppO : real.
 
 (**********)
 Lemma Ropp_Ropp:(r:R)``-(-r)==r``.
   Intro; Ring.
-Save.
+Qed.
 Hints Resolve Ropp_Ropp : real.
 
 (*********)
@@ -439,115 +439,115 @@ Lemma Ropp_neq:(r:R)``r<>0``->``-r<>0``.
 Red;Intros r H H0.
 Apply H.
 Transitivity ``-(-r)``; Auto with real.
-Save.
+Qed.
 Hints Resolve Ropp_neq : real.
 
 (**********)
 Lemma Ropp_distr1:(r1,r2:R)``-(r1+r2)==(-r1 + -r2)``.
   Intros; Ring.
-Save.
+Qed.
 Hints Resolve Ropp_distr1 : real.
 
 (** Opposite and multiplication *)
 
 Lemma Ropp_mul1:(r1,r2:R)``(-r1)*r2 == -(r1*r2)``.
   Intros; Ring.
-Save.
+Qed.
 Hints Resolve Ropp_mul1 : real.
 
 (**********)
 Lemma Ropp_mul2:(r1,r2:R)``(-r1)*(-r2)==r1*r2``.
   Intros; Ring.
-Save.
+Qed.
 Hints Resolve Ropp_mul2 : real.
 
 (** Substraction *)
 
 Lemma minus_R0:(r:R)``r-0==r``.
 Intro;Ring.
-Save.
+Qed.
 Hints Resolve minus_R0 : real.
 
 Lemma Rminus_Ropp:(r:R)``0-r==-r``.
 Intro;Ring.
-Save.
+Qed.
 Hints Resolve Rminus_Ropp : real.
 
 (**********)
 Lemma Ropp_distr2:(r1,r2:R)``-(r1-r2)==r2-r1``.
   Intros; Ring.
-Save.
+Qed.
 Hints Resolve Ropp_distr2 : real.
 
 Lemma Ropp_distr3:(r1,r2:R)``-(r2-r1)==r1-r2``.
 Intros; Ring.
-Save.
+Qed.
 Hints Resolve Ropp_distr3 : real. 
 
 (**********)
 Lemma eq_Rminus:(r1,r2:R)(r1==r2)->``r1-r2==0``.
   Intros; Rewrite H; Ring.
-Save.
+Qed.
 Hints Resolve eq_Rminus : real.
 
 (**********)
 Lemma Rminus_eq:(r1,r2:R)``r1-r2==0`` -> r1==r2.
   Intros r1 r2; Unfold Rminus; Rewrite -> Rplus_sym; Intro.
   Rewrite <- (Ropp_Ropp r2); Apply (Rplus_Ropp (Ropp r2) r1 H).
-Save.
+Qed.
 Hints Immediate Rminus_eq : real.
 
 Lemma Rminus_eq_right:(r1,r2:R)``r2-r1==0`` -> r1==r2.
 Intros;Generalize (Rminus_eq r2 r1 H);Clear H;Intro H;Rewrite H;Ring.
-Save.
+Qed.
 Hints Immediate Rminus_eq_right : real.
 
 Lemma Rplus_Rminus: (p,q:R)``p+(q-p)``==q.
 Intros; Ring.
-Save.
+Qed.
 Hints Resolve Rplus_Rminus:real.
 
 (**********)
 Lemma Rminus_eq_contra:(r1,r2:R)``r1<>r2``->``r1-r2<>0``.
 Red; Intros r1 r2 H H0.
 Apply H; Auto with real.
-Save.
+Qed.
 Hints Resolve Rminus_eq_contra : real.
 
 Lemma Rminus_not_eq:(r1,r2:R)``r1-r2<>0``->``r1<>r2``.
 Red; Intros; Elim H; Apply eq_Rminus; Auto.
-Save.
+Qed.
 Hints Resolve Rminus_not_eq : real.
 
 Lemma Rminus_not_eq_right:(r1,r2:R)``r2-r1<>0`` -> ``r1<>r2``.
 Red; Intros;Elim H;Rewrite H0; Ring.
-Save.
+Qed.
 Hints Resolve Rminus_not_eq_right : real. 
 
 (**********)
 Lemma Rminus_distr:  (x,y,z:R) ``x*(y-z)==(x*y) - (x*z)``.
 Intros; Ring.
-Save.
+Qed.
 
 (** Inverse *)
 Lemma Rinv_R1:``/1==1``.
 Apply (r_Rmult_mult ``1`` ``/1`` ``1``); Auto with real.
 Rewrite (Rinv_r R1 R1_neq_R0);Auto with real.
-Save.
+Qed.
 Hints Resolve Rinv_R1 : real.
 
 (*********)
 Lemma Rinv_neq_R0:(r:R)``r<>0``->``(/r)<>0``.
 Red; Intros; Apply R1_neq_R0.
 Replace ``1`` with ``(/r) * r``; Auto with real.
-Save.
+Qed.
 Hints Resolve Rinv_neq_R0 : real.
 
 (*********)
 Lemma Rinv_Rinv:(r:R)``r<>0``->``/(/r)==r``.
 Intros;Apply (r_Rmult_mult ``/r``); Auto with real.
 Transitivity ``1``; Auto with real.
-Save.
+Qed.
 Hints Resolve Rinv_Rinv : real.
 
 (*********)
@@ -558,30 +558,30 @@ Transitivity ``(r1*/r1)*(r2*(/r2))``; Auto with real.
 Rewrite Rinv_r; Trivial.
 Rewrite Rinv_r; Auto with real.
 Ring.
-Save.
+Qed.
 
 (*********)
 Lemma Ropp_Rinv:(r:R)``r<>0``->``-(/r)==/(-r)``.
 Intros; Apply (r_Rmult_mult ``-r``);Auto with real.
 Transitivity ``1``; Auto with real.
 Rewrite (Ropp_mul2 r ``/r``); Auto with real.
-Save.
+Qed.
 
 Lemma Rinv_r_simpl_r : (r1,r2:R)``r1<>0``->``r1*(/r1)*r2==r2``.
 Intros; Transitivity ``1*r2``; Auto with real.
 Rewrite Rinv_r; Auto with real.
-Save.
+Qed.
 
 Lemma Rinv_r_simpl_l : (r1,r2:R)``r1<>0``->``r2*r1*(/r1)==r2``.
 Intros; Transitivity ``r2*1``; Auto with real.
 Transitivity ``r2*(r1*/r1)``; Auto with real.
-Save.
+Qed.
 
 Lemma Rinv_r_simpl_m : (r1,r2:R)``r1<>0``->``r1*r2*(/r1)==r2``.
 Intros; Transitivity ``r2*1``; Auto with real.
 Transitivity ``r2*(r1*/r1)``; Auto with real.
 Ring.
-Save.
+Qed.
 Hints Resolve Rinv_r_simpl_l Rinv_r_simpl_r Rinv_r_simpl_m : real.
 
 (*********)
@@ -589,14 +589,14 @@ Lemma Rinv_Rmult_simpl:(a,b,c:R)``a<>0``->``(a*(/b))*(c*(/a))==c*(/b)``.
 Intros.
 Transitivity ``(a*/a)*(c*(/b))``; Auto with real.
 Ring.
-Save.
+Qed.
 
 (** Order and addition *)
 
 Lemma Rlt_compatibility_r:(r,r1,r2:R)``r1<r2``->``r1+r<r2+r``.
 Intros.
 Rewrite (Rplus_sym r1 r); Rewrite (Rplus_sym r2 r); Auto with real.
-Save.
+Qed.
 
 Hints Resolve Rlt_compatibility_r : real.
 
@@ -608,21 +608,21 @@ Elim (Rplus_ne r1); Elim (Rplus_ne r2); Intros; Rewrite <- H3;
  Rewrite <- H1; Auto with zarith real.
 Rewrite -> Rplus_assoc; Rewrite -> Rplus_assoc;
  Apply (Rlt_compatibility ``-r`` ``r+r1`` ``r+r2`` H).
-Save.
+Qed.
 
 (**********)
 Lemma Rle_compatibility:(r,r1,r2:R)``r1<=r2`` -> ``r+r1 <= r+r2 ``.
 Unfold Rle; Intros; Elim H; Intro.
 Left; Apply (Rlt_compatibility r r1 r2 H0).
 Right; Rewrite <- H0; Auto with zarith real.
-Save.
+Qed.
 
 (**********)
 Lemma Rle_compatibility_r:(r,r1,r2:R)``r1<=r2`` -> ``r1+r<=r2+r``.
 Unfold Rle; Intros; Elim H; Intro.
 Left; Apply (Rlt_compatibility_r r r1 r2 H0).
 Right; Rewrite <- H0; Auto with real.
-Save.
+Qed.
 
 Hints Resolve Rle_compatibility Rle_compatibility_r : real.
 
@@ -631,7 +631,7 @@ Lemma Rle_anti_compatibility: (r,r1,r2:R)``r+r1<=r+r2`` -> ``r1<=r2``.
 Unfold Rle; Intros; Elim H; Intro.
 Left; Apply (Rlt_anti_compatibility r r1 r2 H0).
 Right; Apply (r_Rplus_plus r r1 r2 H0).
-Save.
+Qed.
 
 (**********)
 Lemma sum_inequa_Rle_lt:(a,x,b,c,y,d:R)``a<=x`` -> ``x<b`` ->
@@ -639,28 +639,28 @@ Lemma sum_inequa_Rle_lt:(a,x,b,c,y,d:R)``a<=x`` -> ``x<b`` ->
 Intros;Split.
 Apply Rlt_le_trans with ``a+y``; Auto with real.
 Apply Rlt_le_trans with ``b+y``; Auto with real.
-Save.
+Qed.
 
 (*********)
 Lemma Rplus_lt:(r1,r2,r3,r4:R)``r1<r2`` -> ``r3<r4`` -> ``r1+r3 < r2+r4``.
 Intros; Apply Rlt_trans with ``r2+r3``; Auto with real.
-Save.
+Qed.
 
 Lemma Rplus_le:(r1,r2,r3,r4:R)``r1<=r2`` -> ``r3<=r4`` -> ``r1+r3 <= r2+r4``.
 Intros; Apply Rle_trans with ``r2+r3``; Auto with real.
-Save.
+Qed.
 
 (*********)
 Lemma Rplus_lt_le_lt:(r1,r2,r3,r4:R)``r1<r2`` -> ``r3<=r4`` -> 
                      ``r1+r3 < r2+r4``.
 Intros; Apply Rlt_le_trans with ``r2+r3``; Auto with real.
-Save.
+Qed.
 
 (*********)
 Lemma Rplus_le_lt_lt:(r1,r2,r3,r4:R)``r1<=r2`` -> ``r3<r4`` -> 
                                     ``r1+r3 < r2+r4``.
 Intros; Apply Rle_lt_trans with ``r2+r3``; Auto with real.
-Save.
+Qed.
 
 Hints Immediate Rplus_lt Rplus_le Rplus_lt_le_lt Rplus_le_lt_lt : real.
 
@@ -675,30 +675,30 @@ Replace ``r2+r1+(-r2)`` with r1.
 Trivial.
 Ring.
 Ring.
-Save.
+Qed.
 Hints Resolve Rgt_Ropp.
 
 (**********)
 Lemma Rlt_Ropp:(r1,r2:R) ``r1 < r2`` -> ``-r1 > -r2``.
 Unfold Rgt; Auto with real.
-Save.
+Qed.
 Hints Resolve Rlt_Ropp : real.
 
 Lemma Ropp_Rlt: (x,y:R) ``-y < -x`` ->``x<y``.
 Intros x y H'.
 Rewrite <- (Ropp_Ropp x); Rewrite <- (Ropp_Ropp y); Auto with real.
-Save.
+Qed.
 Hints Resolve Ropp_Rlt : real.
 
 Lemma Rlt_Ropp1:(r1,r2:R) ``r2 < r1`` -> ``-r1 < -r2``.
 Auto with real.
-Save.
+Qed.
 Hints Resolve Rlt_Ropp1 : real.
 
 (**********)
 Lemma Rle_Ropp:(r1,r2:R) ``r1 <= r2`` -> ``-r1 >= -r2``.
 Unfold Rge; Intros r1 r2 [H|H]; Auto with real.
-Save.
+Qed.
 Hints Resolve Rle_Ropp : real.
 
 Lemma Ropp_Rle: (x,y:R) ``-y <= -x`` ->``x <= y``.
@@ -706,50 +706,50 @@ Intros x y H.
 Elim H;Auto with real.
 Intro H1;Rewrite <-(Ropp_Ropp x);Rewrite <-(Ropp_Ropp y);Rewrite H1;
   Auto with real.
-Save.
+Qed.
 Hints Resolve Ropp_Rle : real.
 
 Lemma Rle_Ropp1:(r1,r2:R) ``r2 <= r1`` -> ``-r1 <= -r2``.
 Intros r1 r2 H;Elim H;Auto with real.
 Intro H1;Rewrite H1;Auto with real.
-Save.
+Qed.
 Hints Resolve Rle_Ropp1 : real.
 
 (**********)
 Lemma Rge_Ropp:(r1,r2:R) ``r1 >= r2`` -> ``-r1 <= -r2``.
 Unfold Rge; Intros r1 r2 [H|H]; Auto with real.
-Save.
+Qed.
 Hints Resolve Rge_Ropp : real.
 
 (**********)
 Lemma Rlt_RO_Ropp:(r:R) ``0 < r`` -> ``0 > -r``.
 Intros; Replace ``0`` with ``-0``; Auto with real.
-Save.
+Qed.
 Hints Resolve Rlt_RO_Ropp : real.
 
 (**********)
 Lemma Rgt_RO_Ropp:(r:R) ``0 > r`` -> ``0 < -r``.
 Intros; Replace ``0`` with ``-0``; Auto with real.
-Save.
+Qed.
 Hints Resolve Rgt_RO_Ropp : real.
 
 (**********)
 Lemma Rle_RO_Ropp:(r:R) ``0 <= r`` -> ``0 >= -r``.
 Intros; Replace ``0`` with ``-0``; Auto with real.
-Save.
+Qed.
 Hints Resolve Rle_RO_Ropp : real.
 
 (**********)
 Lemma Rge_RO_Ropp:(r:R) ``0 >= r`` -> ``0 <= -r``.
 Intros; Replace ``0`` with ``-0``; Auto with real.
-Save.
+Qed.
 Hints Resolve Rge_RO_Ropp : real.
 
 (** Order and multiplication *)
 
 Lemma  Rlt_monotony_r:(r,r1,r2:R)``0<r`` -> ``r1 < r2`` -> ``r1*r < r2*r``.
 Intros; Rewrite (Rmult_sym r1 r); Rewrite (Rmult_sym r2 r); Auto with real.
-Save.
+Qed.
 Hints Resolve Rlt_monotony_r.
 
 Lemma Rlt_monotony_contra:
@@ -760,27 +760,27 @@ Case (total_order x y); Intros Eq0; Auto; Elim Eq0; Clear Eq0; Intros Eq0.
 Generalize (Rlt_monotony z y x H Eq0);Intro;ElimType False;
  Generalize (Rlt_trans ``z*x`` ``z*y`` ``z*x`` H0 H1);Intro;
  Apply (Rlt_antirefl ``z*x``);Auto.
-Save.
+Qed.
 
 Lemma Rlt_anti_monotony:(r,r1,r2:R)``r < 0`` -> ``r1 < r2`` -> ``r*r1 > r*r2``.
 Intros; Replace r with ``-(-r)``; Auto with real.
 Rewrite (Ropp_mul1 ``-r``); Rewrite (Ropp_mul1 ``-r``).
 Apply Rlt_Ropp; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rle_monotony: 
   (r,r1,r2:R)``0 <= r`` -> ``r1 <= r2`` -> ``r*r1 <= r*r2``.
 Intros r r1 r2;Unfold Rle;Intros H H0;Elim H;Elim H0;Intros; Auto with real.
 Right;Rewrite <- H2;Ring.
-Save.
+Qed.
 Hints Resolve Rle_monotony : real.
 
 Lemma Rle_monotony_r: 
   (r,r1,r2:R)``0 <= r`` -> ``r1 <= r2`` -> ``r1*r <= r2*r``.
 Intros r r1 r2 H;
 Rewrite (Rmult_sym r1 r); Rewrite (Rmult_sym r2 r); Auto with real.
-Save.
+Qed.
 Hints Resolve Rle_monotony_r : real.
 
 Lemma Rle_monotony_contra:
@@ -793,14 +793,14 @@ Replace y with (Rmult (Rinv z) (Rmult z y)).
  Rewrite H1;Auto with real.
 Rewrite <- Rmult_assoc; Rewrite Rinv_l; Auto with real.
 Rewrite <- Rmult_assoc; Rewrite Rinv_l; Auto with real.
-Save.
+Qed.
 
 Lemma Rle_anti_monotony
 	:(r,r1,r2:R)``r <= 0`` -> ``r1 <= r2`` -> ``r*r1 >= r*r2``.
 Intros; Replace r with ``-(-r)``; Auto with real.
 Rewrite (Ropp_mul1 ``-r``); Rewrite (Ropp_mul1 ``-r``).
 Apply Rle_Ropp; Auto with real.
-Save.
+Qed.
 Hints Resolve Rle_anti_monotony : real.
 
 Lemma Rle_Rmult_comp:
@@ -811,13 +811,13 @@ Apply Rle_trans with r2 := ``x*t``; Auto with real.
 Repeat Rewrite [x:?](Rmult_sym x t).
 Apply Rle_monotony; Auto.
 Apply Rle_trans with z; Auto.
-Save.
+Qed.
 Hints Resolve Rle_Rmult_comp :real.
 
 Lemma Rmult_lt:(r1,r2,r3,r4:R)``r3>0`` -> ``r2>0`` ->
   `` r1 < r2`` -> ``r3 < r4`` -> ``r1*r3 < r2*r4``.
 Intros; Apply Rlt_trans with ``r2*r3``; Auto with real.
-Save.
+Qed.
 
 (** Order and Substractions *)
 Lemma Rlt_minus:(r1,r2:R)``r1 < r2`` -> ``r1-r2 < 0``.
@@ -825,33 +825,33 @@ Intros; Apply (Rlt_anti_compatibility ``r2``).
 Replace ``r2+(r1-r2)`` with r1.
 Replace  ``r2+0`` with r2; Auto with real.
 Ring.
-Save.
+Qed.
 Hints Resolve Rlt_minus : real.
 
 (**********)
 Lemma Rle_minus:(r1,r2:R)``r1 <= r2`` -> ``r1-r2 <= 0``.
 Unfold Rle; Intros; Elim H; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rminus_lt:(r1,r2:R)``r1-r2 < 0`` -> ``r1 < r2``.
 Intros; Replace r1 with ``r1-r2+r2``.
 Pattern 3 r2; Replace r2 with ``0+r2``; Auto with real.
 Ring.
-Save.
+Qed.
 
 (**********)
 Lemma Rminus_le:(r1,r2:R)``r1-r2 <= 0`` -> ``r1 <= r2``.
 Intros; Replace r1 with ``r1-r2+r2``.
 Pattern 3 r2; Replace r2 with ``0+r2``; Auto with real.
 Ring.
-Save.
+Qed.
 
 (**********)
 Lemma tech_Rplus:(r,s:R)``0<=r`` -> ``0<s`` -> ``r+s<>0``.
 Intros; Apply sym_not_eqT; Apply Rlt_not_eq.
 Rewrite Rplus_sym; Replace ``0`` with ``0+0``; Auto with real.
-Save.
+Qed.
 Hints Immediate tech_Rplus : real.
 
 (** Order and the square function *)
@@ -860,7 +860,7 @@ Intro; Case (total_order_Rlt_Rle 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.
-Save.
+Qed.
 
 (***********)
 Lemma pos_Rsqr1:(r:R)``r<>0``->``0<(Rsqr r)``.
@@ -868,14 +868,14 @@ Intros; Case (not_Req 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.
-Save.
+Qed.
 Hints Resolve pos_Rsqr pos_Rsqr1 : real.
 
 (** Zero is less than one *)
 Lemma Rlt_R0_R1:``0<1``.
 Replace ``1`` with ``(Rsqr 1)``; Auto with real.
 Unfold Rsqr; Auto with real.
-Save.
+Qed.
 Hints Resolve Rlt_R0_R1 : real.
 
 (** Order and inverse *)
@@ -884,7 +884,7 @@ Intros; Change ``/r>0``; Apply not_Rle; Red; Intros.
 Absurd ``1<=0``; Auto with real.
 Replace ``1`` with ``r*(/r)``; Auto with real.
 Replace ``0`` with ``r*0``; Auto with real.
-Save.
+Qed.
 Hints Resolve Rlt_Rinv : real.
 
 (*********)
@@ -893,7 +893,7 @@ Intros; Change ``0>/r``; Apply not_Rle; Red; Intros.
 Absurd ``1<=0``; Auto with real.
 Replace ``1`` with ``r*(/r)``; Auto with real.
 Replace ``0`` with ``r*0``; Auto with real.
-Save.
+Qed.
 Hints Resolve Rlt_Rinv2 : real.
 
 (**********)
@@ -905,7 +905,7 @@ Transitivity ``r*/r*r2``; Auto with real.
 Ring.
 Transitivity ``r*/r*r1``; Auto with real.
 Ring.
-Save.
+Qed.
 
 (*********)
 Lemma Rinv_lt:(r1,r2:R)``0 < r1*r2`` -> ``r1 < r2`` -> ``/r2 < /r1``.
@@ -915,7 +915,7 @@ Replace ``r1*r2*/r2`` with r1.
 Replace ``r1*r2*/r1`` with r2; Trivial.
 Symmetry; Auto with real.
 Symmetry; Auto with real.
-Save.
+Qed.
 
 Lemma Rlt_Rinv_R1: (x, y:R) ``1 <= x`` -> ``x<y`` ->``/y< /x``.
 Intros x y H' H'0.
@@ -931,7 +931,7 @@ Rewrite (Rmult_sym x); Rewrite <- Rmult_assoc; Rewrite (Rmult_sym y (Rinv y));
  Rewrite Rinv_l; Auto with real.
 Apply imp_not_Req; Right.
 Red; Apply Rlt_trans with r2 := x; Auto with real.
-Save.
+Qed.
 Hints Resolve Rlt_Rinv_R1 :real.
 
 (*********************************************************)        
@@ -941,30 +941,30 @@ Hints Resolve Rlt_Rinv_R1 :real.
 (**********)
 Lemma Rge_ge_eq:(r1,r2:R)``r1 >= r2`` -> ``r2 >= r1`` -> r1==r2.
 Intros; Apply Rle_antisym; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rlt_not_ge:(r1,r2:R)~(``r1<r2``)->``r1>=r2``.
 Intros; Unfold Rge; Elim (total_order r1 r2); Intro.
 Absurd ``r1<r2``; Trivial.
 Case H0; Auto.
-Save.
+Qed.
 
 (**********)
 Lemma Rgt_not_le:(r1,r2:R)~(``r1>r2``)->``r1<=r2``.
 Intros r1 r2 H; Apply Rge_le.
 Exact (Rlt_not_ge r2 r1 H).
-Save.
+Qed.
 
 (**********)
 Lemma Rgt_ge:(r1,r2:R)``r1>r2`` -> ``r1 >= r2``.
 Red; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rlt_sym:(r1,r2:R)``r1<r2`` <-> ``r2>r1``.
 Split; Unfold Rgt; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rle_sym1:(r1,r2:R)``r1<=r2``->``r2>=r1``.
@@ -977,38 +977,38 @@ Proof [r1,r2](Rge_le r2 r1).
 (**********)
 Lemma Rle_sym:(r1,r2:R)``r1<=r2``<->``r2>=r1``.
 Split; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rge_gt_trans:(r1,r2,r3:R)``r1>=r2``->``r2>r3``->``r1>r3``.
 Unfold Rgt; Intros; Apply Rlt_le_trans with r2; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rgt_ge_trans:(r1,r2,r3:R)``r1>r2`` -> ``r2>=r3`` -> ``r1>r3``.
 Unfold Rgt; Intros; Apply Rle_lt_trans with r2; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rgt_trans:(r1,r2,r3:R)``r1>r2`` -> ``r2>r3`` -> ``r1>r3``.
 Unfold Rgt; Intros; Apply Rlt_trans with r2; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rge_trans:(r1,r2,r3:R)``r1>=r2`` -> ``r2>=r3`` -> ``r1>=r3``.
 Intros; Apply Rle_ge.
 Apply Rle_trans with r2; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rgt_RoppO:(r:R)``r>0``->``(-r)<0``.
 Intros; Rewrite <- Ropp_O; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rlt_RoppO:(r:R)``r<0``->``-r>0``.
 Intros; Rewrite <- Ropp_O; Auto with real.
-Save.
+Qed.
 Hints Resolve Rgt_RoppO Rlt_RoppO: real.
 
 (**********)
@@ -1016,87 +1016,87 @@ Lemma Rlt_r_plus_R1:(r:R)``0<=r`` -> ``0<r+1``.
 Intros.
 Apply Rlt_le_trans with ``1``; Auto with real.
 Pattern 1 ``1``; Replace ``1`` with ``0+1``; Auto with real.
-Save.
+Qed.
 Hints Resolve Rlt_r_plus_R1: real.
 
 (**********)
 Lemma Rlt_r_r_plus_R1:(r:R)``r<r+1``.
 Intros.
 Pattern 1 r; Replace r with ``r+0``; Auto with real.
-Save.
+Qed.
 Hints Resolve Rlt_r_r_plus_R1: real.
 
 (**********)
 Lemma tech_Rgt_minus:(r1,r2:R)``0<r2``->``r1>r1-r2``.
 Red; Unfold Rminus; Intros.
 Pattern 2 r1; Replace r1 with ``r1+0``; Auto with real.
-Save.
+Qed.
 
 (***********)
 Lemma Rgt_plus_plus_r:(r,r1,r2:R)``r1>r2``->``r+r1 > r+r2``.
 Unfold Rgt; Auto with real.
-Save.
+Qed.
 Hints Resolve Rgt_plus_plus_r : real.
 
 (***********)
 Lemma Rgt_r_plus_plus:(r,r1,r2:R)``r+r1 > r+r2`` -> ``r1 > r2``.
 Unfold Rgt; Intros; Apply (Rlt_anti_compatibility r r2 r1 H).
-Save.
+Qed.
 
 (***********)
 Lemma Rge_plus_plus_r:(r,r1,r2:R)``r1>=r2`` -> ``r+r1 >= r+r2``.
 Intros; Apply Rle_ge; Auto with real.
-Save.
+Qed.
 Hints Resolve Rge_plus_plus_r : real.
 
 (***********)
 Lemma Rge_r_plus_plus:(r,r1,r2:R)``r+r1 >= r+r2`` -> ``r1>=r2``.
 Intros; Apply Rle_ge; Apply Rle_anti_compatibility with r; Auto with real.
-Save.
+Qed.
 
 (***********)
 Lemma Rge_monotony:
  (x,y,z:R) ``z>=0`` -> ``x>=y`` -> ``x*z >= y*z``.
 Intros; Apply Rle_ge; Auto with real.
-Save.
+Qed.
 
 (***********)
 Lemma Rgt_minus:(r1,r2:R)``r1>r2`` -> ``r1-r2 > 0``.
 Intros; Replace ``0`` with ``r2-r2``; Auto with real.
 Unfold Rgt Rminus; Auto with real.
-Save.
+Qed.
 
 (*********)
 Lemma minus_Rgt:(r1,r2:R)``r1-r2 > 0`` -> ``r1>r2``.
 Intros; Replace r2 with ``r2+0``; Auto with real.
 Intros; Replace r1 with ``r2+(r1-r2)``; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Rge_minus:(r1,r2:R)``r1>=r2`` -> ``r1-r2 >= 0``.
 Unfold Rge; Intros; Elim H; Intro.
 Left; Apply (Rgt_minus r1 r2 H0).
 Right; Apply (eq_Rminus r1 r2 H0).
-Save.
+Qed.
 
 (*********)
 Lemma minus_Rge:(r1,r2:R)``r1-r2 >= 0`` -> ``r1>=r2``.
 Intros; Replace r2 with ``r2+0``; Auto with real.
 Intros; Replace r1 with ``r2+(r1-r2)``; Auto with real.
-Save.
+Qed.
 
 
 (*********)
 Lemma Rmult_gt:(r1,r2:R)``r1>0`` -> ``r2>0`` -> ``r1*r2>0``.
 Unfold Rgt;Intros.
 Replace ``0`` with ``0*r2``; Auto with real.
-Save.
+Qed.
 
 (*********)
 Lemma Rmult_lt_0
   :(r1,r2,r3,r4:R)``r3>=0``->``r2>0``->``r1<r2``->``r3<r4``->``r1*r3<r2*r4``.
 Intros; Apply Rle_lt_trans with ``r2*r3``; Auto with real.
-Save.
+Qed.
 
 (*********)
 Lemma Rmult_lt_pos:(x,y:R)``0<x`` -> ``0<y`` -> ``0<x*y``.
@@ -1108,7 +1108,7 @@ Intros a b [H|H] H0 H1; Auto with real.
 Absurd ``0<a+b``.
 Rewrite H1; Auto with real.
 Replace ``0`` with ``0+0``; Auto with real.
-Save.
+Qed.
 
 
 Lemma Rplus_eq_R0
@@ -1117,20 +1117,20 @@ Split.
 Apply Rplus_eq_R0_l with b; Auto with real.
 Apply Rplus_eq_R0_l with a; Auto with real.
 Rewrite Rplus_sym; Auto with real.
-Save.
+Qed.
 
 
 (***********)
 Lemma Rplus_Rsr_eq_R0_l:(a,b:R)``(Rsqr a)+(Rsqr b)==0``->``a==0``.
 Intros; Apply Rsqr_r_R0; Apply Rplus_eq_R0_l with (Rsqr b); Auto with real.
-Save.
+Qed.
 
 Lemma Rplus_Rsr_eq_R0:(a,b:R)``(Rsqr a)+(Rsqr b)==0``->``a==0``/\``b==0``.
 Split.
 Apply Rplus_Rsr_eq_R0_l with b; Auto with real.
 Apply Rplus_Rsr_eq_R0_l with a; Auto with real.
 Rewrite Rplus_sym; Auto with real.
-Save.
+Qed.
 
 
 (**********************************************************) 
@@ -1140,13 +1140,13 @@ Save.
 (**********)
 Lemma S_INR:(n:nat)(INR (S n))==``(INR n)+1``.
 Intro; Case n; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma S_O_plus_INR:(n:nat)
     (INR (plus (S O) n))==``(INR (S O))+(INR n)``.
 Intro; Simpl; Case n; Intros; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma plus_INR:(n,m:nat)(INR (plus n m))==``(INR n)+(INR m)``. 
@@ -1155,7 +1155,7 @@ Simpl; Auto with real.
 Replace (plus (S n) m) with (S (plus n m)); Auto with arith.
 Repeat Rewrite S_INR.
 Rewrite Hrecn; Ring.
-Save.
+Qed.
 
 (**********)
 Lemma minus_INR:(n,m:nat)(le m n)->(INR (minus n m))==``(INR n)-(INR m)``.
@@ -1163,7 +1163,7 @@ 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.
 Rewrite H0; Ring.
-Save.
+Qed.
 
 (*********)
 Lemma mult_INR:(n,m:nat)(INR (mult n m))==(Rmult (INR n) (INR m)).
@@ -1171,7 +1171,7 @@ Intros n m; Induction n.
 Simpl; Auto with real.
 Intros; Repeat Rewrite S_INR; Simpl.
 Rewrite plus_INR; Rewrite Hrecn; Ring.
-Save.
+Qed.
 
 Hints Resolve plus_INR minus_INR mult_INR : real.
 
@@ -1179,19 +1179,19 @@ Hints Resolve plus_INR minus_INR mult_INR : real.
 Lemma lt_INR_0:(n:nat)(lt O n)->``0 < (INR n)``.
 Induction 1; Intros; Auto with real.
 Rewrite S_INR; Auto with real.
-Save.
+Qed.
 Hints Resolve lt_INR_0: real.
 
 Lemma lt_INR:(n,m:nat)(lt n m)->``(INR n) < (INR m)``.
 Induction 1; Intros; Auto with real.
 Rewrite S_INR; Auto with real.
 Rewrite S_INR; Apply Rlt_trans with (INR m0); Auto with real.
-Save.
+Qed.
 Hints Resolve lt_INR: real.
 
 Lemma INR_lt_1:(n:nat)(lt (S O) n)->``1 < (INR n)``.
 Intros;Replace ``1`` with (INR (S O));Auto with real.
-Save.
+Qed.
 Hints Resolve INR_lt_1: real.
 
 (**********)
@@ -1199,7 +1199,7 @@ Lemma INR_pos : (p:positive)``0<(INR (convert p))``.
 Intro; Apply lt_INR_0.
 Simpl; Auto with real.
 Apply compare_convert_O.
-Save.
+Qed.
 Hints Resolve INR_pos : real.
 
 (**********)
@@ -1207,7 +1207,7 @@ Lemma pos_INR:(n:nat)``0 <= (INR n)``.
 Intro n; Case n.
 Simpl; Auto with real.
 Auto with arith real.
-Save.
+Qed.
 Hints Resolve pos_INR: real.
 
 Lemma INR_lt:(n,m:nat)``(INR n) < (INR m)``->(lt n m).
@@ -1222,7 +1222,7 @@ Do 2 Rewrite S_INR in H1;Cut ``(INR n1) < (INR n0)``.
 Intro H2;Generalize (H0 n0 H2);Intro;Auto with arith.
 Apply (Rlt_anti_compatibility ``1`` (INR n1) (INR n0)).
 Rewrite Rplus_sym;Rewrite (Rplus_sym ``1`` (INR n0));Trivial.
-Save.
+Qed.
 Hints Resolve INR_lt: real.
 
 (*********)
@@ -1230,7 +1230,7 @@ Lemma le_INR:(n,m:nat)(le n m)->``(INR n)<=(INR m)``.
 Induction 1; Intros; Auto with real.
 Rewrite S_INR.
 Apply Rle_trans with (INR m0); Auto with real.
-Save.
+Qed.
 Hints Resolve le_INR: real.
 
 (**********)
@@ -1238,7 +1238,7 @@ Lemma not_INR_O:(n:nat)``(INR n)<>0``->~n=O.
 Red; Intros n H H1.
 Apply H.
 Rewrite H1; Trivial.
-Save.
+Qed.
 Hints Immediate not_INR_O : real.
 
 (**********)
@@ -1247,7 +1247,7 @@ Intro n; Case n.
 Intro; Absurd (0)=(0); Trivial.
 Intros; Rewrite S_INR.
 Apply Rgt_not_eq; Red; Auto with real.
-Save.
+Qed.
 Hints Resolve not_O_INR : real.
 
 Lemma not_nm_INR:(n,m:nat)~n=m->``(INR n)<>(INR m)``.
@@ -1256,7 +1256,7 @@ Case (le_lt_or_eq ? ? H1); Intros H2.
 Apply imp_not_Req; Auto with real.
 ElimType False;Auto.
 Apply sym_not_eqT; Apply imp_not_Req; Auto with real.
-Save.
+Qed.
 Hints Resolve not_nm_INR : real.
 
 Lemma INR_eq: (n,m:nat)(INR n)==(INR m)->n=m.
@@ -1270,19 +1270,19 @@ Symmetry;Cut ~m=n.
 Intro H3;Generalize (not_nm_INR m n H3);Intro H4;
   ElimType False;Auto.
 Omega.
-Save.
+Qed.
 Hints Resolve INR_eq : real.
 
 Lemma INR_le: (n, m : nat) (Rle (INR n) (INR m)) ->  (le n m).
 Intros;Elim H;Intro.
 Generalize (INR_lt n m H0);Intro;Auto with arith.
 Generalize (INR_eq n m H0);Intro;Rewrite H1;Auto.
-Save.
+Qed.
 Hints Resolve INR_le : real.
 
 Lemma not_1_INR:(n:nat)~n=(S O)->``(INR n)<>1``.
 Replace ``1``  with (INR (S O)); Auto with real.
-Save.
+Qed.
 Hints Resolve not_1_INR : real.
 
 (**********************************************************) 
@@ -1295,13 +1295,13 @@ Definition INZ:=inject_nat.
 (**********)
 Lemma IZN:(z:Z)(`0<=z`)->(Ex [n:nat] z=(INZ n)).
 Unfold INZ;Intros;Apply inject_nat_complete;Assumption.
-Save.
+Qed.
 
 (**********)
 Lemma INR_IZR_INZ:(n:nat)(INR n)==(IZR (INZ n)).
 Induction n; Auto with real.
 Intros; Simpl; Rewrite bij1; Auto with real.
-Save.
+Qed.
 
 Lemma plus_IZR_NEG_POS : 
   (p,q:positive)(IZR `(POS p)+(NEG q)`)==``(IZR (POS p))+(IZR (NEG q))``.
@@ -1319,7 +1319,7 @@ Rewrite convert_compare_SUPERIEUR; Simpl; Auto with arith.
 Rewrite (true_sub_convert p q).
 Rewrite minus_INR; Auto with arith; Ring.
 Apply ZC2; Apply convert_compare_INFERIEUR; Trivial.
-Save.
+Qed.
 
 (**********)
 Lemma plus_IZR:(z,t:Z)(IZR `z+t`)==``(IZR z)+(IZR t)``.
@@ -1328,7 +1328,7 @@ Simpl; Intros; Rewrite convert_add; Auto with real.
 Apply plus_IZR_NEG_POS.
 Rewrite Zplus_sym; Rewrite Rplus_sym; Apply plus_IZR_NEG_POS.
 Simpl; Intros; Rewrite convert_add; Rewrite plus_INR; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma mult_IZR:(z,t:Z)(IZR `z*t`)==``(IZR z)*(IZR t)``.
@@ -1342,18 +1342,18 @@ Intros t1 z1; Rewrite times_convert; Auto with real.
 Rewrite Ropp_mul1; Auto with real.
 Intros t1 z1; Rewrite times_convert; Auto with real.
 Rewrite Ropp_mul2; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Ropp_Ropp_IZR:(z:Z)(IZR (`-z`))==``-(IZR z)``.
 Intro z; Case z; Simpl; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma Z_R_minus:(z1,z2:Z)``(IZR z1)-(IZR z2)``==(IZR `z1-z2`).
 Intros; Unfold Rminus; Unfold Zminus.
 Rewrite <-(Ropp_Ropp_IZR z2); Symmetry; Apply plus_IZR.
-Save. 
+Qed. 
 
 (**********)
 Lemma lt_O_IZR:(z:Z)``0 < (IZR z)``->`0<z`.
@@ -1362,7 +1362,7 @@ Absurd ``0<0``; Auto with real.
 Unfold Zlt; Simpl; Trivial.
 Case Rlt_le_not with 1:=H.
 Replace ``0`` with ``-0``; Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma lt_IZR:(z1,z2:Z)``(IZR z1)<(IZR z2)``->`z1<z2`.
@@ -1370,34 +1370,34 @@ Intros; Apply Zlt_O_minus_lt.
 Apply lt_O_IZR.
 Rewrite <- Z_R_minus.
 Exact (Rgt_minus (IZR z2) (IZR z1) H).
-Save.
+Qed.
 
 (**********)
 Lemma eq_IZR_R0:(z:Z)``(IZR z)==0``->`z=0`.
 NewDestruct z; Simpl; Intros; Auto with zarith.
 Case (Rlt_not_eq ``0`` (INR (convert p))); Auto with real.
 Case (Rlt_not_eq ``-(INR (convert p))`` ``0`` ); Auto with real.
-Save.
+Qed.
 
 (**********)
 Lemma eq_IZR:(z1,z2:Z)(IZR z1)==(IZR z2)->z1=z2.
 Intros;Generalize (eq_Rminus (IZR z1) (IZR z2) H);
  Rewrite (Z_R_minus z1 z2);Intro;Generalize (eq_IZR_R0 `z1-z2` H0);
  Intro;Omega.
-Save.
+Qed.
 
 (**********)
 Lemma not_O_IZR:(z:Z)`z<>0`->``(IZR z)<>0``.
 Intros z H; Red; Intros H0; Case H.
 Apply eq_IZR; Auto.
-Save.
+Qed.
 
 (*********)
 Lemma le_O_IZR:(z:Z)``0<= (IZR z)``->`0<=z`.
 Unfold Rle; Intros z [H|H].
 Red;Intro;Apply (Zlt_le_weak `0` z (lt_O_IZR z H)); Assumption.
 Rewrite (eq_IZR_R0 z); Auto with zarith real.
-Save.
+Qed.
 
 (**********)
 Lemma le_IZR:(z1,z2:Z)``(IZR z1)<=(IZR z2)``->`z1<=z2`.
@@ -1405,31 +1405,31 @@ Unfold Rle; Intros z1 z2 [H|H].
 Apply (Zlt_le_weak z1 z2); Auto with real.
 Apply lt_IZR; Trivial.
 Rewrite (eq_IZR z1 z2); Auto with zarith real.
-Save.
+Qed.
 
 (**********)
 Lemma le_IZR_R1:(z:Z)``(IZR z)<=1``-> `z<=1`.
 Pattern 1 ``1``; Replace ``1`` with (IZR `1`); Intros; Auto.
 Apply le_IZR; Trivial.
-Save.
+Qed.
 
 (**********)
 Lemma IZR_ge: (m,n:Z) `m>= n` -> ``(IZR m)>=(IZR n)``.
 Intros;Apply Rlt_not_ge;Red;Intro.
 Generalize (lt_IZR m n H0); Intro; Omega.
-Save.
+Qed.
 
 Lemma IZR_le: (m,n:Z) `m<= n` -> ``(IZR m)<=(IZR n)``.
 Intros;Apply Rgt_not_le;Red;Intro.
 Unfold Rgt in H0;Generalize (lt_IZR n m H0); Intro; Omega.
-Save.
+Qed.
 
 Lemma IZR_lt: (m,n:Z) `m< n` -> ``(IZR m)<(IZR n)``.
 Intros;Cut `m<=n`.
 Intro H0;Elim (IZR_le m n H0);Intro;Auto.
 Generalize (eq_IZR m n H1);Intro;ElimType False;Omega.
 Omega.
-Save.
+Qed.
 
 Lemma one_IZR_lt1 : (z:Z)``-1<(IZR z)<1``->`z=0`.
 Intros z (H1,H2).
@@ -1437,7 +1437,7 @@ Apply Zle_antisym.
 Apply Zlt_n_Sm_le; Apply lt_IZR; Trivial.
 Replace `0` with (Zs `-1`); Trivial.
 Apply Zlt_le_S; Apply lt_IZR; Trivial.
-Save.
+Qed.
 
 Lemma one_IZR_r_R1
   : (r:R)(z,x:Z)``r<(IZR z)<=r+1``->``r<(IZR x)<=r+1``->z=x.
@@ -1451,7 +1451,7 @@ Ring.
 Replace ``1`` with ``(r+1)-r``.
 Unfold Rminus; Apply Rplus_le_lt_lt; Auto with real.
 Ring.
-Save.
+Qed.
 
 
 (**********)
@@ -1459,7 +1459,7 @@ Lemma single_z_r_R1:
    (r:R)(z,x:Z)``r<(IZR z)``->``(IZR z)<=r+1``->``r<(IZR x)``->
         ``(IZR x)<=r+1``->z=x.
 Intros; Apply one_IZR_r_R1 with r; Auto.
-Save.
+Qed.
 
 (**********)
 Lemma tech_single_z_r_R1
@@ -1467,7 +1467,7 @@ Lemma tech_single_z_r_R1
          -> (Ex [s:Z] (~s=z/\``r<(IZR s)``/\``(IZR s)<=r+1``))->False.
 Intros r z H1 H2 (s, (H3,(H4,H5))).
 Apply H3; Apply single_z_r_R1 with r; Trivial.
-Save.
+Qed.
 
 (*****************************************************************)
 (** Definitions of new types                                     *)
@@ -1496,12 +1496,12 @@ cond_nonzero : ~``nonzero==0`` }.
 (**********)
 Lemma prod_neq_R0 : (x,y:R) ~``x==0``->~``y==0``->~``x*y==0``.
 Intros; Red; Intro; Generalize (without_div_Od x y H1); Intro; Elim H2; Intro; [Rewrite H3 in H; Elim H | Rewrite H3 in H0; Elim H0]; Reflexivity.
-Save.
+Qed.
 
 (*********)
 Lemma Rmult_le_pos : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``0<=x*y``.
 Intros; Rewrite <- (Rmult_Ol x); Rewrite <- (Rmult_sym x); Apply (Rle_monotony x R0 y H H0).
-Save.
+Qed.
 
 (**********************************************************)
 (** Other rules about < and <=                            *)
@@ -1509,31 +1509,31 @@ Save.
 
 Lemma gt0_plus_gt0_is_gt0 : (x,y:R) ``0<x`` -> ``0<y`` -> ``0<x+y``.
 Intros; Apply Rlt_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rlt_compatibility; Assumption].
-Save.
+Qed.
 
 Lemma ge0_plus_gt0_is_gt0 : (x,y:R) ``0<=x`` -> ``0<y`` -> ``0<x+y``.
 Intros; Apply Rle_lt_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rlt_compatibility; Assumption].
-Save.
+Qed.
 
 Lemma  gt0_plus_ge0_is_gt0 : (x,y:R) ``0<x`` -> ``0<=y`` -> ``0<x+y``.
 Intros; Rewrite <- Rplus_sym; Apply ge0_plus_gt0_is_gt0; Assumption.
-Save.
+Qed.
 
 Lemma ge0_plus_ge0_is_ge0 : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``0<=x+y``.
 Intros; Apply Rle_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption].
-Save.
+Qed.
 
 Lemma plus_le_is_le : (x,y,z:R) ``0<=y`` -> ``x+y<=z`` -> ``x<=z``.
 Intros; Apply Rle_trans with ``x+y``; [Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption | Assumption].
-Save.
+Qed.
 
 Lemma plus_lt_is_lt : (x,y,z:R) ``0<=y`` -> ``x+y<z`` -> ``x<z``.
 Intros; Apply Rle_lt_trans with ``x+y``; [Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption | Assumption].
-Save.
+Qed.
 
 Lemma Rmult_lt2 : (r1,r2,r3,r4:R) ``0<=r1`` -> ``0<=r3`` -> ``r1<r2`` -> ``r3<r4`` -> ``r1*r3<r2*r4``.
 Intros; Apply Rle_lt_trans with ``r2*r3``; [Apply Rle_monotony_r; [Assumption | Left; Assumption] | Apply Rlt_monotony; [Apply Rle_lt_trans with r1; Assumption | Assumption]].
-Save.
+Qed.
 
 Lemma le_epsilon : (x,y:R) ((eps : R) ``0<eps``->``x<=y+eps``) -> ``x<=y``.
 Intros; Elim (total_order x y); Intro.
@@ -1564,7 +1564,7 @@ Discriminate.
 Unfold INR; Reflexivity.
 Intro; Ring.
 Cut ~(O=(2)); [Intro H0; Generalize (lt_INR_0 (2) (neq_O_lt (2) H0)); Unfold INR; Intro; Assumption | Discriminate].
-Save.
+Qed.
 
 (*****************************************************)
 (** Complementary results about [INR]                *)
@@ -1580,8 +1580,8 @@ end.
 
 Theorem INR_eq_INR2 : (n:nat) (INR n)==(INR2 n).
 Induction n; [Unfold INR INR2; Reflexivity | Intros; Unfold INR INR2; Fold INR INR2; Rewrite H; Case n0; [Reflexivity | Intros; Ring]].
-Save.
+Qed.
 
 Lemma add_auto : (p,q:nat) ``(INR2 (S p))+(INR2 q)==(INR2 p)+(INR2 (S q))``.
 Intros; Repeat Rewrite <- INR_eq_INR2; Repeat Rewrite S_INR; Ring.
-Save.
\ No newline at end of file
+Qed.
\ No newline at end of file