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, 114 insertions, 114 deletions

diff --git a/theories/ZArith/zarith_aux.v b/theories/ZArith/zarith_aux.v
index 0b7a92289..a49c60fb4 100644
--- a/theories/ZArith/zarith_aux.v
+++ b/theories/ZArith/zarith_aux.v
@@ -52,27 +52,27 @@ Definition Zabs [z:Z] : Z :=
 Lemma Zabs_eq : (x:Z) (Zle ZERO x) -> (Zabs x)=x.
 NewDestruct x; Auto with arith.
 Compute; Intros; Absurd SUPERIEUR=SUPERIEUR; Trivial with arith.
-Save.
+Qed.
 
 Lemma Zabs_non_eq : (x:Z) (Zle x ZERO) -> (Zabs x)=(Zopp x).
 Proof.
 NewDestruct x; Auto with arith.
 Compute; Intros; Absurd SUPERIEUR=SUPERIEUR; Trivial with arith.
-Save.
+Qed.
 
 Lemma Zabs_dec : (x:Z){x=(Zabs x)}+{x=(Zopp (Zabs x))}.
 Proof.
 NewDestruct x;Auto with arith.
-Save.
+Qed.
 
 Lemma Zabs_pos : (x:Z)(Zle ZERO (Zabs x)).
 NewDestruct x;Auto with arith; Compute; Intros H;Inversion H.
-Save.
+Qed.
 
 Lemma Zsgn_Zabs: (x:Z)(Zmult x (Zsgn x))=(Zabs x).
 Proof.
 Destruct x;Intros;Rewrite Zmult_sym;Auto with arith.
-Save.
+Qed.
 
 Lemma Zabs_Zsgn: (x:Z)(Zabs x)=(Zmult (Zsgn x) x).
 Proof.
@@ -95,7 +95,7 @@ Theorem Zgt_Sn_n : (n:Z)(Zgt (Zs n) n).
 
 Intros n; Unfold Zgt Zs; Pattern 2 n; Rewrite <- (Zero_right n); 
 Rewrite Zcompare_Zplus_compatible;Auto with arith.
-Save.
+Qed.
 
 (** Properties of the order *)
 Theorem Zle_gt_trans : (n,m,p:Z)(Zle m n)->(Zgt m p)->(Zgt n p).
@@ -106,7 +106,7 @@ Unfold Zle Zgt; Intros n m p H1 H2; (ElimCompare 'm 'n); [
     Elim (Zcompare_ANTISYM n m); Intros H4 H5;Apply H5; Assumption
   | Assumption ]
 | Intro H3; Absurd (Zcompare m n)=SUPERIEUR;Assumption ].
-Save.
+Qed.
 
 Theorem Zgt_le_trans : (n,m,p:Z)(Zgt n m)->(Zle p m)->(Zgt n p).
 
@@ -116,49 +116,49 @@ Unfold Zle Zgt ;Intros n m p H1 H2; (ElimCompare 'p 'm); [
     Assumption
   | Elim (Zcompare_ANTISYM m p); Auto with arith ]
 | Intro H3; Absurd (Zcompare p m)=SUPERIEUR;Assumption ].
-Save.
+Qed.
 
 Theorem Zle_S_gt : (n,m:Z) (Zle (Zs n) m) -> (Zgt m n).
 
 Intros n m H;Apply Zle_gt_trans with m:=(Zs n);[ Assumption | Apply Zgt_Sn_n ].
-Save.
+Qed.
 
 Theorem Zcompare_n_S : (n,m:Z)(Zcompare (Zs n) (Zs m)) = (Zcompare n m).
 Intros n m;Unfold Zs ;Do 2 Rewrite -> [t:Z](Zplus_sym t (POS xH));
 Rewrite -> Zcompare_Zplus_compatible;Auto with arith.
-Save.
+Qed.
  
 Theorem Zgt_n_S : (n,m:Z)(Zgt m n) -> (Zgt (Zs m) (Zs n)).
 
 Unfold Zgt; Intros n m H; Rewrite Zcompare_n_S; Auto with arith.
-Save.
+Qed.
 
 Lemma Zle_not_gt     : (n,m:Z)(Zle n m) -> ~(Zgt n m).
 
 Unfold Zle Zgt; Auto with arith.
-Save.
+Qed.
 
 Lemma Zgt_antirefl   : (n:Z)~(Zgt n n).
 
 Unfold Zgt ;Intros n; Elim (Zcompare_EGAL n n); Intros H1 H2;
 Rewrite H2; [ Discriminate | Trivial with arith ].
-Save.
+Qed.
 
 Lemma Zgt_not_sym    : (n,m:Z)(Zgt n m) -> ~(Zgt m n).
 
 Unfold Zgt ;Intros n m H; Elim (Zcompare_ANTISYM n m); Intros H1 H2;
 Rewrite -> H1; [ Discriminate | Assumption ].
-Save.
+Qed.
 
 Lemma Zgt_not_le     : (n,m:Z)(Zgt n m) -> ~(Zle n m).
 
 Unfold Zgt Zle not; Auto with arith.
-Save.
+Qed.
 
 Lemma Zgt_trans      : (n,m,p:Z)(Zgt n m)->(Zgt m p)->(Zgt n p).
 
 Unfold Zgt; Exact Zcompare_trans_SUPERIEUR.
-Save.
+Qed.
 
 Lemma Zle_gt_S       : (n,p:Z)(Zle n p)->(Zgt (Zs p) n).
 
@@ -171,7 +171,7 @@ Unfold Zle Zgt ;Intros n p H; (ElimCompare 'n 'p); [
     Exact (Zgt_Sn_n p)
   | Elim (Zcompare_ANTISYM p n); Auto with arith ]
 | Intros H1;Absurd (Zcompare n p)=SUPERIEUR;Assumption ].
-Save.
+Qed.
 
 Lemma Zgt_pred       
 	: (n,p:Z)(Zgt p (Zs n))->(Zgt (Zpred p) n).
@@ -180,14 +180,14 @@ Unfold Zgt Zs Zpred ;Intros n p H;
 Rewrite <- [x,y:Z](Zcompare_Zplus_compatible x y (POS xH));
 Rewrite (Zplus_sym p); Rewrite Zplus_assoc; Rewrite [x:Z](Zplus_sym x n);
 Simpl; Assumption.
-Save.
+Qed.
 
 Lemma Zsimpl_gt_plus_l 
 	: (n,m,p:Z)(Zgt (Zplus p n) (Zplus p m))->(Zgt n m).
 
 Unfold Zgt; Intros n m p H; 
 	Rewrite <- (Zcompare_Zplus_compatible n m p); Assumption.
-Save.
+Qed.
 
 Lemma Zsimpl_gt_plus_r
 	: (n,m,p:Z)(Zgt (Zplus n p) (Zplus m p))->(Zgt n m).
@@ -195,18 +195,18 @@ Lemma Zsimpl_gt_plus_r
 Intros n m p H; Apply Zsimpl_gt_plus_l with p.
 Rewrite (Zplus_sym p n); Rewrite (Zplus_sym p m); Trivial.
 
-Save.
+Qed.
 
 Lemma Zgt_reg_l      
 	: (n,m,p:Z)(Zgt n m)->(Zgt (Zplus p n) (Zplus p m)).
 
 Unfold Zgt; Intros n m p H; Rewrite (Zcompare_Zplus_compatible n m p); 
 Assumption.
-Save.
+Qed.
 
 Lemma Zgt_reg_r : (n,m,p:Z)(Zgt n m)->(Zgt (Zplus n p) (Zplus m p)).
 Intros n m p H; Rewrite (Zplus_sym n p); Rewrite (Zplus_sym m p); Apply Zgt_reg_l; Trivial.
-Save.
+Qed.
 
 Theorem Zcompare_et_un: 
   (x,y:Z) (Zcompare x y)=SUPERIEUR <-> 
@@ -235,20 +235,20 @@ Intros x y; Split; [
   | Intros H1;Absurd (Zcompare x (Zplus y (POS xH)))=INFERIEUR;Assumption
   | Intros H1; Apply Zcompare_trans_SUPERIEUR with y:=(Zs y); 
       [ Exact H1 | Exact (Zgt_Sn_n y) ]]].
-Save.
+Qed.
 
 Lemma Zgt_S_n        : (n,p:Z)(Zgt (Zs p) (Zs n))->(Zgt p n).
 
 Unfold Zs Zgt;Intros n p;Do 2 Rewrite -> [m:Z](Zplus_sym m (POS xH));
 Rewrite -> (Zcompare_Zplus_compatible p n (POS xH));Trivial with arith.
-Save.
+Qed.
 
 Lemma Zle_S_n     : (n,m:Z) (Zle (Zs m) (Zs n)) -> (Zle m n).
 
 Unfold Zle not ;Intros m n H1 H2;Apply H1;
 Unfold Zs ;Do 2 Rewrite <- (Zplus_sym (POS xH));
 Rewrite -> (Zcompare_Zplus_compatible n m (POS xH));Assumption.
-Save.
+Qed.
 
 Lemma Zgt_le_S       : (n,p:Z)(Zgt p n)->(Zle (Zs n) p).
 
@@ -256,12 +256,12 @@ Unfold Zgt Zle; Intros n p H; Elim (Zcompare_et_un p n); Intros H1 H2;
 Unfold not ;Intros H3; Unfold not in H1; Apply H1; [
   Assumption
 | Elim (Zcompare_ANTISYM (Zplus n (POS xH)) p);Intros H4 H5;Apply H4;Exact H3].
-Save.
+Qed.
 
 Lemma Zgt_S_le       : (n,p:Z)(Zgt (Zs p) n)->(Zle n p).
 
 Intros n p H;Apply Zle_S_n; Apply Zgt_le_S; Assumption.
-Save.
+Qed.
 
 Theorem Zgt_S        : (n,m:Z)(Zgt (Zs n) m)->((Zgt n m)\/(<Z>m=n)).
 
@@ -270,22 +270,22 @@ Intros n m H; Unfold Zgt; (ElimCompare 'n 'm); [
 | Intros H1;Absurd (Zcompare m n)=SUPERIEUR; 
     [ Exact (Zgt_S_le m n H) | Elim (Zcompare_ANTISYM m n); Auto with arith ]
 | Auto with arith ].
-Save.
+Qed.
 
 Theorem Zgt_trans_S  : (n,m,p:Z)(Zgt (Zs n) m)->(Zgt m p)->(Zgt n p).
 
 Intros n m p H1 H2;Apply Zle_gt_trans with m:=m;
   [ Apply Zgt_S_le; Assumption | Assumption ].
-Save.
+Qed.
 
 Theorem Zeq_S : (n,m:Z) n=m -> (Zs n)=(Zs m).
 Intros n m H; Rewrite H; Auto with arith.
-Save.
+Qed.
 
 Theorem Zpred_Sn : (m:Z) m=(Zpred (Zs m)).
 Intros m; Unfold Zpred Zs; Rewrite <- Zplus_assoc; Simpl; 
 Rewrite Zplus_sym; Auto with arith.
-Save.
+Qed.
 
 Theorem Zeq_add_S : (n,m:Z) (Zs n)=(Zs m) -> n=m.
 Intros n m H.
@@ -293,81 +293,81 @@ Change (Zplus (Zplus (NEG xH) (POS xH)) n)=
        (Zplus (Zplus (NEG xH) (POS xH)) m);
 Do 2 Rewrite <- Zplus_assoc; Do 2 Rewrite (Zplus_sym (POS xH));
 Unfold Zs in H;Rewrite H; Trivial with arith.
-Save.
+Qed.
  
 Theorem Znot_eq_S : (n,m:Z) ~(n=m) -> ~((Zs n)=(Zs m)).
 
 Unfold not ;Intros n m H1 H2;Apply H1;Apply Zeq_add_S; Assumption.
-Save.
+Qed.
  
 Lemma Zsimpl_plus_l : (n,m,p:Z)(Zplus n m)=(Zplus n p)->m=p.
 Intros n m p H; Cut (Zplus (Zopp n) (Zplus n m))=(Zplus (Zopp n) (Zplus n p));[
   Do 2 Rewrite -> Zplus_assoc; Rewrite -> (Zplus_sym (Zopp n) n);
   Rewrite -> Zplus_inverse_r;Simpl; Trivial with arith
 | Rewrite -> H; Trivial with arith ].
-Save.
+Qed.
 
 Theorem Zn_Sn : (n:Z) ~(n=(Zs n)).
 Intros n;Cut ~ZERO=(POS xH);[
   Unfold not ;Intros H1 H2;Apply H1;Apply (Zsimpl_plus_l n);Rewrite Zero_right;
   Exact H2
 | Discriminate ].
-Save.
+Qed.
 
 Lemma Zplus_n_O : (n:Z) n=(Zplus n ZERO).
 Intro; Rewrite Zero_right; Trivial with arith.
-Save.
+Qed.
 
 Lemma Zplus_unit_left : (n,m:Z) (Zplus n ZERO)=m -> n=m.
 Intro; Rewrite Zero_right; Trivial with arith.
-Save.
+Qed.
 
 Lemma Zplus_unit_right : (n,m:Z) n=(Zplus m ZERO) -> n=m.
 Intros n m; Rewrite (Zero_right m); Trivial with arith.
-Save.
+Qed.
 
 Lemma Zplus_n_Sm : (n,m:Z) (Zs (Zplus n m))=(Zplus n (Zs m)).
 
 Intros n m; Unfold Zs; Rewrite Zplus_assoc; Trivial with arith.
-Save.
+Qed.
 
 Lemma Zmult_n_O : (n:Z) ZERO=(Zmult n ZERO).
 
 Intro;Rewrite Zmult_sym;Simpl; Trivial with arith.
-Save.
+Qed.
 
 Lemma Zmult_n_Sm : (n,m:Z) (Zplus (Zmult n m) n)=(Zmult n (Zs m)).
 
 Intros n m;Unfold Zs; Rewrite Zmult_plus_distr_r;
 Rewrite (Zmult_sym n (POS xH));Rewrite Zmult_one; Trivial with arith.
-Save.
+Qed.
 
 Theorem Zle_n : (n:Z) (Zle n n).
 Intros n;Elim (Zcompare_EGAL n n);Unfold Zle ;Intros H1 H2;Rewrite H2;
   [ Discriminate | Trivial with arith ].
-Save. 
+Qed. 
 
 Theorem Zle_refl : (n,m:Z) n=m -> (Zle n m).
 Intros; Rewrite H; Apply Zle_n.
-Save.
+Qed.
 
 Theorem Zle_trans : (n,m,p:Z)(Zle n m)->(Zle m p)->(Zle n p).
 
 Intros n m p;Unfold 1 3 Zle; Unfold not; Intros H1 H2 H3;Apply H1;
 Exact (Zgt_le_trans n p m H3 H2).
-Save.
+Qed.
 
 Theorem Zle_n_Sn : (n:Z)(Zle n (Zs n)).
 
 Intros n; Apply Zgt_S_le;Apply Zgt_trans with m:=(Zs n) ;Apply Zgt_Sn_n.
-Save.
+Qed.
 
 Lemma Zle_n_S : (n,m:Z) (Zle m n) -> (Zle (Zs m) (Zs n)).
 
 Unfold Zle not ;Intros m n H1 H2; Apply H1; 
 Rewrite <- (Zcompare_Zplus_compatible n m (POS xH));
 Do 2 Rewrite (Zplus_sym (POS xH)); Exact H2.
-Save.
+Qed.
 
 Hints Resolve Zle_n Zle_n_Sn Zle_trans Zle_n_S : zarith.
 Hints Immediate Zle_refl : zarith.
@@ -376,24 +376,24 @@ Lemma Zs_pred : (n:Z) n=(Zs (Zpred n)).
 
 Intros n; Unfold Zs Zpred ;Rewrite <- Zplus_assoc; Simpl; Rewrite Zero_right;
 Trivial with arith.
-Save. 
+Qed. 
 
 Hints Immediate Zs_pred : zarith.
  
 Theorem Zle_pred_n : (n:Z)(Zle (Zpred n) n).
 
 Intros n;Pattern 2 n ;Rewrite Zs_pred; Apply Zle_n_Sn.
-Save.
+Qed.
  
 Theorem Zle_trans_S : (n,m:Z)(Zle (Zs n) m)->(Zle n m).
 
 Intros n m H;Apply Zle_trans with m:=(Zs n); [ Apply Zle_n_Sn | Assumption ].
-Save.
+Qed.
 
 Theorem Zle_Sn_n : (n:Z)~(Zle (Zs n) n).
 
 Intros n; Apply Zgt_not_le; Apply Zgt_Sn_n.
-Save.
+Qed.
 
 Theorem Zle_antisym : (n,m:Z)(Zle n m)->(Zle m n)->(n=m).
 
@@ -403,96 +403,96 @@ Unfold Zle ;Intros n m H1 H2; (ElimCompare 'n 'm); [
     Assumption
   | Elim (Zcompare_ANTISYM m n);Auto with arith ]
 | Intros H3;Absurd (Zcompare n m)=SUPERIEUR;Assumption ].
-Save.
+Qed.
 
 Theorem Zgt_lt : (m,n:Z) (Zgt m n) -> (Zlt n m).
 Unfold Zgt Zlt ;Intros m n H; Elim (Zcompare_ANTISYM m n); Auto with arith.
-Save.
+Qed.
 
 Theorem Zlt_gt : (m,n:Z) (Zlt m n) -> (Zgt n m).
 Unfold Zgt Zlt ;Intros m n H; Elim (Zcompare_ANTISYM n m); Auto with arith.
-Save.
+Qed.
 
 Theorem Zge_le : (m,n:Z) (Zge m n) -> (Zle n m).
 Intros m n; Change ~(Zlt m n)-> ~(Zgt n m);
 Unfold not; Intros H1 H2; Apply H1; Apply Zgt_lt; Assumption.
-Save.
+Qed.
 
 Theorem Zle_ge : (m,n:Z) (Zle m n) -> (Zge n m).
 Intros m n; Change ~(Zgt m n)-> ~(Zlt n m);
 Unfold not; Intros H1 H2; Apply H1; Apply Zlt_gt; Assumption.
-Save.
+Qed.
 
 Theorem Zge_trans : (n, m, p : Z) (Zge n m) -> (Zge m p) -> (Zge n p).
 Intros n m p H1 H2.
 Apply Zle_ge.
 Apply Zle_trans with m; Apply Zge_le; Trivial.
-Save.
+Qed.
 
 Theorem Zlt_n_Sn : (n:Z)(Zlt n (Zs n)).
 Intro n; Apply Zgt_lt; Apply Zgt_Sn_n.
-Save.
+Qed.
 Theorem Zlt_S : (n,m:Z)(Zlt n m)->(Zlt n (Zs m)).
 Intros n m H;Apply Zgt_lt; Apply Zgt_trans with m:=m; [
   Apply Zgt_Sn_n
 | Apply Zlt_gt; Assumption ].
-Save.
+Qed.
 
 Theorem Zlt_n_S : (n,m:Z)(Zlt n m)->(Zlt (Zs n) (Zs m)).
 Intros n m H;Apply Zgt_lt;Apply Zgt_n_S;Apply Zlt_gt; Assumption.
-Save.
+Qed.
 
 Theorem Zlt_S_n : (n,m:Z)(Zlt (Zs n) (Zs m))->(Zlt n m).
 
 Intros n m H;Apply Zgt_lt;Apply Zgt_S_n;Apply Zlt_gt; Assumption.
-Save.
+Qed.
 
 Theorem Zlt_n_n : (n:Z)~(Zlt n n).
 
 Intros n;Elim (Zcompare_EGAL n n); Unfold Zlt ;Intros H1 H2;
 Rewrite H2; [ Discriminate | Trivial with arith ].
-Save.
+Qed.
 
 Lemma Zlt_pred : (n,p:Z)(Zlt (Zs n) p)->(Zlt n (Zpred p)).
 
 Intros n p H;Apply Zlt_S_n; Rewrite <- Zs_pred; Assumption.
-Save.
+Qed.
 
 Lemma Zlt_pred_n_n : (n:Z)(Zlt (Zpred n) n).
 
 Intros n; Apply Zlt_S_n; Rewrite <- Zs_pred; Apply Zlt_n_Sn.
-Save.
+Qed.
  
 Theorem Zlt_le_S : (n,p:Z)(Zlt n p)->(Zle (Zs n) p).
 Intros n p H; Apply Zgt_le_S; Apply Zlt_gt; Assumption.
-Save.
+Qed.
 
 Theorem Zlt_n_Sm_le : (n,m:Z)(Zlt n (Zs m))->(Zle n m).
 Intros n m H; Apply Zgt_S_le; Apply Zlt_gt; Assumption.
-Save.
+Qed.
 
 Theorem Zle_lt_n_Sm : (n,m:Z)(Zle n m)->(Zlt n (Zs m)).
 Intros n m H; Apply Zgt_lt; Apply Zle_gt_S; Assumption.
-Save.
+Qed.
 
 Theorem Zlt_le_weak : (n,m:Z)(Zlt n m)->(Zle n m).
 Unfold Zlt Zle ;Intros n m H;Rewrite H;Discriminate.
-Save.
+Qed.
 
 Theorem Zlt_trans : (n,m,p:Z)(Zlt n m)->(Zlt m p)->(Zlt n p).
 Intros n m p H1 H2; Apply Zgt_lt; Apply Zgt_trans with m:= m; 
 Apply Zlt_gt; Assumption.
-Save.
+Qed.
 Theorem Zlt_le_trans : (n,m,p:Z)(Zlt n m)->(Zle m p)->(Zlt n p).
 Intros n m p H1 H2;Apply Zgt_lt;Apply Zle_gt_trans with m:=m;
   [ Assumption | Apply Zlt_gt;Assumption ].
-Save.
+Qed.
 
 Theorem Zle_lt_trans : (n,m,p:Z)(Zle n m)->(Zlt m p)->(Zlt n p).
 
 Intros n m p H1 H2;Apply Zgt_lt;Apply Zgt_le_trans with m:=m; 
   [ Apply Zlt_gt;Assumption | Assumption ].
-Save.
+Qed.
  
 Theorem Zle_lt_or_eq : (n,m:Z)(Zle n m)->((Zlt n m) \/ n=m).
 
@@ -500,7 +500,7 @@ Unfold Zle Zlt ;Intros n m H; (ElimCompare 'n 'm); [
   Elim (Zcompare_EGAL n m);Auto with arith
 | Auto with arith
 | Intros H';Absurd (Zcompare n m)=SUPERIEUR;Assumption ].
-Save.
+Qed.
 
 Theorem Zle_or_lt : (n,m:Z)((Zle n m)\/(Zlt m n)).
 
@@ -508,27 +508,27 @@ Unfold Zle Zlt ;Intros n m; (ElimCompare 'n 'm); [
   Intros E;Rewrite -> E;Left;Discriminate
 | Intros E;Rewrite -> E;Left;Discriminate
 | Elim (Zcompare_ANTISYM n m); Auto with arith ].
-Save.
+Qed.
 
 Theorem Zle_not_lt : (n,m:Z)(Zle n m) -> ~(Zlt m n).
 
 Unfold Zle Zlt; Unfold not ;Intros n m H1 H2;Apply H1; 
 Elim (Zcompare_ANTISYM n m);Auto with arith.
-Save.
+Qed.
 
 Theorem Zlt_not_le : (n,m:Z)(Zlt n m) -> ~(Zle m n).
 Unfold Zlt Zle not ;Intros n m H1 H2; Apply H2; Elim (Zcompare_ANTISYM m n);
 Auto with arith.
-Save.
+Qed.
 
 Theorem Zlt_not_sym : (n,m:Z)(Zlt n m) -> ~(Zlt m n).
 Intros n m H;Apply Zle_not_lt; Apply Zlt_le_weak; Assumption.
-Save.
+Qed.
 
 Theorem Zle_le_S : (x,y:Z)(Zle x y)->(Zle x (Zs y)).
 Intros.
 Apply Zle_trans with y; Trivial with zarith.
-Save.
+Qed.
 
 Hints Resolve Zle_le_S : zarith.
 
@@ -543,13 +543,13 @@ Lemma Zmin_SS : (n,m:Z)((Zs (Zmin n m))=(Zmin (Zs n) (Zs m))).
 
 Intros n m;Unfold Zmin; Rewrite (Zcompare_n_S n m);
 (ElimCompare 'n 'm);Intros E;Rewrite E;Auto with arith.
-Save.
+Qed.
 
 Lemma Zle_min_l : (n,m:Z)(Zle (Zmin n m) n).
 
 Intros n m;Unfold Zmin ; (ElimCompare 'n 'm);Intros E;Rewrite -> E;
   [ Apply Zle_n | Apply Zle_n | Apply Zlt_le_weak; Apply Zgt_lt;Exact E ].
-Save.
+Qed.
 
 Lemma Zle_min_r : (n,m:Z)(Zle (Zmin n m) m).
 
@@ -557,93 +557,93 @@ Intros n m;Unfold Zmin ; (ElimCompare 'n 'm);Intros E;Rewrite -> E;[
   Unfold Zle ;Rewrite -> E;Discriminate
 | Unfold Zle ;Rewrite -> E;Discriminate
 | Apply Zle_n ].
-Save.
+Qed.
 
 Lemma Zmin_case : (n,m:Z)(P:Z->Set)(P n)->(P m)->(P (Zmin n m)).
 Intros n m P H1 H2; Unfold Zmin; Case (Zcompare n m);Auto with arith.
-Save.
+Qed.
 
 Lemma Zmin_or : (n,m:Z)(Zmin n m)=n \/ (Zmin n m)=m.
 Unfold Zmin; Intros; Elim (Zcompare n m); Auto.
-Save.
+Qed.
 
 Lemma Zmin_n_n : (n:Z) (Zmin n n)=n.
 Unfold Zmin; Intros; Elim (Zcompare n n); Auto.
-Save.
+Qed.
 
 Lemma Zplus_assoc_l : (n,m,p:Z)((Zplus n (Zplus m p))=(Zplus (Zplus n m) p)).
 
 Exact Zplus_assoc.
-Save.
+Qed.
 
 Lemma Zplus_assoc_r : (n,m,p:Z)(Zplus (Zplus n m) p) =(Zplus n (Zplus m p)).
 
 Intros; Symmetry; Apply Zplus_assoc.
-Save.
+Qed.
 
 Lemma Zplus_permute : (n,m,p:Z) (Zplus n (Zplus m p))=(Zplus m (Zplus n p)).
 
 Intros n m p;
 Rewrite Zplus_sym;Rewrite <- Zplus_assoc; Rewrite (Zplus_sym p n); Trivial with arith.
-Save.
+Qed.
 
 Lemma Zsimpl_le_plus_l : (p,n,m:Z)(Zle (Zplus p n) (Zplus p m))->(Zle n m).
 
 Intros p n m; Unfold Zle not ;Intros H1 H2;Apply H1; 
 Rewrite (Zcompare_Zplus_compatible n m p); Assumption.
-Save.
+Qed.
  
 Lemma Zsimpl_le_plus_r : (p,n,m:Z)(Zle (Zplus n p) (Zplus m p))->(Zle n m).
 
 Intros p n m H; Apply Zsimpl_le_plus_l with p.
 Rewrite (Zplus_sym p n); Rewrite (Zplus_sym p m); Trivial.
-Save.
+Qed.
 
 Lemma Zle_reg_l : (n,m,p:Z)(Zle n m)->(Zle (Zplus p n) (Zplus p m)).
 
 Intros n m p; Unfold Zle not ;Intros H1 H2;Apply H1; 
 Rewrite <- (Zcompare_Zplus_compatible n m p); Assumption.
-Save.
+Qed.
 
 Lemma Zle_reg_r : (a,b,c:Z) (Zle a b)->(Zle (Zplus a c) (Zplus b c)).
 
 Intros a b c;Do 2 Rewrite [n:Z](Zplus_sym n c); Exact (Zle_reg_l a b c).
-Save.
+Qed.
 
 Lemma Zle_plus_plus : 
  (n,m,p,q:Z) (Zle n m)->(Zle p q)->(Zle (Zplus n p) (Zplus m q)).
 
 Intros n m p q; Intros H1 H2;Apply Zle_trans with m:=(Zplus n q); [
   Apply Zle_reg_l;Assumption | Apply Zle_reg_r;Assumption ].
-Save.
+Qed.
 
 Lemma Zplus_Snm_nSm : (n,m:Z)(Zplus (Zs n) m)=(Zplus n (Zs m)).
 
 Unfold Zs ;Intros n m; Rewrite <- Zplus_assoc; Rewrite (Zplus_sym (POS xH));
 Trivial with arith.
-Save.
+Qed.
 
 Lemma Zsimpl_lt_plus_l 
 	: (n,m,p:Z)(Zlt (Zplus p n) (Zplus p m))->(Zlt n m).
 
 Unfold Zlt ;Intros n m p; 
 	Rewrite Zcompare_Zplus_compatible;Trivial with arith.
-Save.
+Qed.
  
 Lemma Zsimpl_lt_plus_r
 	: (n,m,p:Z)(Zlt (Zplus n p) (Zplus m p))->(Zlt n m).
 
 Intros n m p H; Apply Zsimpl_lt_plus_l with p.
 Rewrite (Zplus_sym p n); Rewrite (Zplus_sym p m); Trivial.
-Save.
+Qed.
  
 Lemma Zlt_reg_l : (n,m,p:Z)(Zlt n m)->(Zlt (Zplus p n) (Zplus p m)).
 Unfold Zlt ;Intros n m p; Rewrite Zcompare_Zplus_compatible;Trivial with arith.
-Save.
+Qed.
 
 Lemma Zlt_reg_r : (n,m,p:Z)(Zlt n m)->(Zlt (Zplus n p) (Zplus m p)).
 Intros n m p H; Rewrite (Zplus_sym n p); Rewrite (Zplus_sym m p); Apply Zlt_reg_l; Trivial.
-Save.
+Qed.
 
 Lemma Zlt_le_reg :
  (a,b,c,d:Z) (Zlt a b)->(Zle c d)->(Zlt (Zplus a c) (Zplus b d)).
@@ -651,7 +651,7 @@ Intros a b c d H0 H1.
 Apply Zlt_le_trans with (Zplus b c).
 Apply  Zlt_reg_r; Trivial.
 Apply  Zle_reg_l; Trivial.
-Save.
+Qed.
 
 
 Lemma Zle_lt_reg :
@@ -660,7 +660,7 @@ Intros a b c d H0 H1.
 Apply Zle_lt_trans with (Zplus b c).
 Apply  Zle_reg_r; Trivial.
 Apply  Zlt_reg_l; Trivial.
-Save.
+Qed.
 
 
 Definition Zminus := [m,n:Z](Zplus m (Zopp n)).
@@ -671,92 +671,92 @@ Lemma Zminus_plus_simpl :
 Intros n m p;Unfold Zminus; Rewrite Zopp_Zplus; Rewrite Zplus_assoc;
 Rewrite (Zplus_sym p); Rewrite <- (Zplus_assoc n p); Rewrite Zplus_inverse_r;
 Rewrite Zero_right; Trivial with arith.
-Save.
+Qed.
 
 Lemma Zminus_n_O : (n:Z)(n=(Zminus n ZERO)).
 
 Intro; Unfold Zminus; Simpl;Rewrite Zero_right; Trivial with arith.
-Save.
+Qed.
 
 Lemma Zminus_n_n : (n:Z)(ZERO=(Zminus n n)).
 Intro; Unfold Zminus; Rewrite Zplus_inverse_r; Trivial with arith.
-Save.
+Qed.
 
 Lemma Zplus_minus : (n,m,p:Z)(n=(Zplus m p))->(p=(Zminus n m)).
 
 Intros n m p H;Unfold Zminus;Apply (Zsimpl_plus_l m); 
 Rewrite (Zplus_sym m (Zplus n (Zopp m))); Rewrite <- Zplus_assoc;
 Rewrite Zplus_inverse_l; Rewrite Zero_right; Rewrite H; Trivial with arith.
-Save.
+Qed.
  
 Lemma Zminus_plus : (n,m:Z)(Zminus (Zplus n m) n)=m.
 Intros n m;Unfold Zminus ;Rewrite -> (Zplus_sym n m);Rewrite <- Zplus_assoc;
 Rewrite -> Zplus_inverse_r; Apply Zero_right.
-Save.
+Qed.
 
 Lemma Zle_plus_minus : (n,m:Z) (Zplus n (Zminus m n))=m.
 
 Unfold Zminus; Intros n m; Rewrite Zplus_permute; Rewrite Zplus_inverse_r;
 Apply Zero_right.
-Save.
+Qed.
 
 Lemma Zminus_Sn_m : (n,m:Z)((Zs (Zminus n m))=(Zminus (Zs n) m)).
 
 Intros n m;Unfold Zminus Zs; Rewrite (Zplus_sym n (Zopp m));
 Rewrite <- Zplus_assoc;Apply Zplus_sym.
-Save.
+Qed.
 
 Lemma Zlt_minus : (n,m:Z)(Zlt ZERO m)->(Zlt (Zminus n m) n).
 
 Intros n m H; Apply Zsimpl_lt_plus_l with p:=m; Rewrite Zle_plus_minus;
 Pattern 1 n ;Rewrite <- (Zero_right n); Rewrite (Zplus_sym m n);
 Apply Zlt_reg_l; Assumption.
-Save.
+Qed.
 
 Lemma Zlt_O_minus_lt : (n,m:Z)(Zlt ZERO (Zminus n m))->(Zlt m n).
 
 Intros n m H; Apply Zsimpl_lt_plus_l with p:=(Zopp m); Rewrite Zplus_inverse_l;
 Rewrite Zplus_sym;Exact H.
-Save.
+Qed.
 
 Lemma Zmult_plus_distr_l : 
   (n,m,p:Z)((Zmult (Zplus n m) p)=(Zplus (Zmult n p) (Zmult m p))).
 
 Intros n m p;Rewrite Zmult_sym;Rewrite Zmult_plus_distr_r; 
 Do 2 Rewrite -> (Zmult_sym p); Trivial with arith.
-Save.
+Qed.
 
 Lemma Zmult_minus_distr :
   (n,m,p:Z)((Zmult (Zminus n m) p)=(Zminus (Zmult n p) (Zmult m p))).
 Intros n m p;Unfold Zminus; Rewrite Zmult_plus_distr_l; Rewrite Zopp_Zmult;
 Trivial with arith.
-Save.
+Qed.
  
 Lemma Zmult_assoc_r : (n,m,p:Z)((Zmult (Zmult n m) p) = (Zmult n (Zmult m p))).
 Intros n m p; Rewrite Zmult_assoc; Trivial with arith.
-Save.
+Qed.
 
 Lemma Zmult_assoc_l : (n,m,p:Z)(Zmult n (Zmult m p)) = (Zmult (Zmult n m) p).
 Intros n m p; Rewrite Zmult_assoc; Trivial with arith.
-Save.
+Qed.
 
 Theorem Zmult_permute : (n,m,p:Z)(Zmult n (Zmult m p)) = (Zmult m (Zmult n p)).
 Intros; Rewrite -> (Zmult_assoc m n p); Rewrite -> (Zmult_sym m n).
 Apply Zmult_assoc.
-Save.
+Qed.
 
 Lemma Zmult_1_n : (n:Z)(Zmult (POS xH) n)=n.
 Exact Zmult_one.
-Save.
+Qed.
 
 Lemma Zmult_n_1 : (n:Z)(Zmult n (POS xH))=n.
 Intro; Rewrite Zmult_sym; Apply Zmult_one.
-Save.
+Qed.
 
 Lemma Zmult_Sm_n : (n,m:Z) (Zplus (Zmult n m) m)=(Zmult (Zs n) m).
 Intros n m; Unfold Zs; Rewrite Zmult_plus_distr_l; Rewrite Zmult_1_n;
 Trivial with arith.
-Save.
+Qed.
 
 
 (** Just for compatibility with previous versions.