Remplacement des fichiers .v ancienne syntaxe de theories, contrib et states par les fichiers nouvelle syntaxe

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@5027 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 274 insertions, 275 deletions

diff --git a/theories/Bool/Bool.v b/theories/Bool/Bool.v
index 3d0a7a2f1..fa786550c 100755
--- a/theories/Bool/Bool.v
+++ b/theories/Bool/Bool.v
@@ -14,131 +14,130 @@
     [Inductive bool : Set := true : bool | false : bool] *)
 
 (** Interpretation of booleans as Proposition *)
-Definition Is_true := [b:bool](Cases b of
-                                 true  => True
-                               | false => False
-                               end).
-Hints Unfold Is_true : bool.
+Definition Is_true (b:bool) :=
+  match b with
+  | true => True
+  | false => False
+  end.
+Hint Unfold Is_true: bool.
 
-Lemma Is_true_eq_left : (x:bool)x=true -> (Is_true x).
+Lemma Is_true_eq_left : forall x:bool, x = true -> Is_true x.
 Proof.
-  Intros; Rewrite H; Auto with bool.
+  intros; rewrite H; auto with bool.
 Qed.
 
-Lemma Is_true_eq_right : (x:bool)true=x -> (Is_true x).
+Lemma Is_true_eq_right : forall x:bool, true = x -> Is_true x.
 Proof.
-  Intros; Rewrite <- H; Auto with bool.
+  intros; rewrite <- H; auto with bool.
 Qed.
 
-Hints Immediate Is_true_eq_right Is_true_eq_left : bool.
+Hint Immediate Is_true_eq_right Is_true_eq_left: bool.
 
 (*******************)
 (** Discrimination *)
 (*******************)
 
-Lemma diff_true_false : ~true=false.
+Lemma diff_true_false : true <> false.
 Proof.
-Unfold not; Intro contr; Change (Is_true false).
-Elim contr; Simpl; Trivial with bool.
+unfold not in |- *; intro contr; change (Is_true false) in |- *.
+elim contr; simpl in |- *; trivial with bool.
 Qed.
-Hints Resolve diff_true_false : bool v62.
+Hint Resolve diff_true_false: bool v62.
 
-Lemma diff_false_true : ~false=true.
+Lemma diff_false_true : false <> true.
 Proof.
-Red; Intros H; Apply diff_true_false.
-Symmetry.
-Assumption.
+red in |- *; intros H; apply diff_true_false.
+symmetry  in |- *.
+assumption.
 Qed.
-Hints Resolve diff_false_true : bool v62.
+Hint Resolve diff_false_true: bool v62.
 
-Lemma eq_true_false_abs : (b:bool)(b=true)->(b=false)->False.
-Intros b H; Rewrite H; Auto with bool.
+Lemma eq_true_false_abs : forall b:bool, b = true -> b = false -> False.
+intros b H; rewrite H; auto with bool.
 Qed.
-Hints Resolve eq_true_false_abs : bool.
+Hint Resolve eq_true_false_abs: bool.
 
-Lemma not_true_is_false : (b:bool)~b=true->b=false.
-NewDestruct b.
-Intros.
-Red in H; Elim H.
-Reflexivity.
-Intros abs.
-Reflexivity.
+Lemma not_true_is_false : forall b:bool, b <> true -> b = false.
+destruct b.
+intros.
+red in H; elim H.
+reflexivity.
+intros abs.
+reflexivity.
 Qed.
 
-Lemma not_false_is_true : (b:bool)~b=false->b=true.
-NewDestruct b.
-Intros.
-Reflexivity.
-Intro H; Red in H; Elim H.
-Reflexivity.
+Lemma not_false_is_true : forall b:bool, b <> false -> b = true.
+destruct b.
+intros.
+reflexivity.
+intro H; red in H; elim H.
+reflexivity.
 Qed.
 
 (**********************)
 (** Order on booleans *)
 (**********************)
 
-Definition leb := [b1,b2:bool]
-  Cases b1 of 
-  | true => b2=true
-  | false => True 
+Definition leb (b1 b2:bool) :=
+  match b1 with
+  | true => b2 = true
+  | false => True
   end.
-Hints Unfold leb : bool v62.
+Hint Unfold leb: bool v62.
 
 (*************)
 (** Equality *)
 (*************)
 
-Definition eqb : bool->bool->bool :=
- [b1,b2:bool]
-    Cases b1 b2 of
-      true  true  => true
-    | true  false => false
-    | false true  => false
-    | false false => true
-    end.
+Definition eqb (b1 b2:bool) : bool :=
+  match b1, b2 with
+  | true, true => true
+  | true, false => false
+  | false, true => false
+  | false, false => true
+  end.
 
-Lemma eqb_refl : (x:bool)(Is_true (eqb x x)).
-NewDestruct x; Simpl; Auto with bool.
+Lemma eqb_refl : forall x:bool, Is_true (eqb x x).
+destruct x; simpl in |- *; auto with bool.
 Qed.
 
-Lemma eqb_eq : (x,y:bool)(Is_true (eqb x y))->x=y.
-NewDestruct x; NewDestruct y; Simpl; Tauto.
+Lemma eqb_eq : forall x y:bool, Is_true (eqb x y) -> x = y.
+destruct x; destruct y; simpl in |- *; tauto.
 Qed.
 
-Lemma Is_true_eq_true : (x:bool) (Is_true x) -> x=true.
-NewDestruct x; Simpl; Tauto.
+Lemma Is_true_eq_true : forall x:bool, Is_true x -> x = true.
+destruct x; simpl in |- *; tauto.
 Qed.
  
-Lemma Is_true_eq_true2 : (x:bool)  x=true -> (Is_true x).
-NewDestruct x; Simpl; Auto with bool.
+Lemma Is_true_eq_true2 : forall x:bool, x = true -> Is_true x.
+destruct x; simpl in |- *; auto with bool.
 Qed.
 
-Lemma eqb_subst : 
-  (P:bool->Prop)(b1,b2:bool)(eqb b1 b2)=true->(P b1)->(P b2).
-Unfold eqb .
-Intros P b1.
-Intros b2.
-Case b1.
-Case b2.
-Trivial with bool.
-Intros H.
-Inversion_clear H.
-Case b2.
-Intros H.
-Inversion_clear H.
-Trivial with bool.
+Lemma eqb_subst :
+ forall (P:bool -> Prop) (b1 b2:bool), eqb b1 b2 = true -> P b1 -> P b2.
+unfold eqb in |- *.
+intros P b1.
+intros b2.
+case b1.
+case b2.
+trivial with bool.
+intros H.
+inversion_clear H.
+case b2.
+intros H.
+inversion_clear H.
+trivial with bool.
 Qed.
 
-Lemma eqb_reflx : (b:bool)(eqb b b)=true.
-Intro b.
-Case b.
-Trivial with bool.
-Trivial with bool.
+Lemma eqb_reflx : forall b:bool, eqb b b = true.
+intro b.
+case b.
+trivial with bool.
+trivial with bool.
 Qed.
 
-Lemma eqb_prop : (a,b:bool)(eqb a b)=true -> a=b.
-NewDestruct a; NewDestruct b; Simpl; Intro;
-  Discriminate H Orelse Reflexivity.
+Lemma eqb_prop : forall a b:bool, eqb a b = true -> a = b.
+destruct a; destruct b; simpl in |- *; intro; discriminate H || reflexivity.
 Qed.
 
 
@@ -146,36 +145,34 @@ Qed.
 (** Logical combinators *)
 (************************)
  
-Definition ifb : bool -> bool -> bool -> bool
-     := [b1,b2,b3:bool](Cases b1 of true => b2 | false => b3 end).
+Definition ifb (b1 b2 b3:bool) : bool :=
+  match b1 with
+  | true => b2
+  | false => b3
+  end.
 
-Definition andb : bool -> bool -> bool
-     := [b1,b2:bool](ifb b1 b2 false).
+Definition andb (b1 b2:bool) : bool := ifb b1 b2 false.
 
-Definition orb : bool -> bool -> bool
-     := [b1,b2:bool](ifb b1 true b2).
+Definition orb (b1 b2:bool) : bool := ifb b1 true b2.
 
-Definition implb : bool -> bool -> bool
-     := [b1,b2:bool](ifb b1 b2 true).
+Definition implb (b1 b2:bool) : bool := ifb b1 b2 true.
 
-Definition xorb :  bool -> bool -> bool
-     := [b1,b2:bool]
-    Cases b1 b2 of
-      true  true  => false
-    | true  false => true
-    | false true  => true
-    | false false => false
-    end.
+Definition xorb (b1 b2:bool) : bool :=
+  match b1, b2 with
+  | true, true => false
+  | true, false => true
+  | false, true => true
+  | false, false => false
+  end.
 
-Definition negb := [b:bool]Cases b of
-                             true  => false
-                           | false => true
-                           end.
+Definition negb (b:bool) := match b with
+                            | true => false
+                            | false => true
+                            end.
 
-Infix "||" orb  (at level 4, left associativity) : bool_scope.
-Infix "&&" andb (at level 3, no associativity) : bool_scope
-  V8only (at level 40, left associativity).
-V8Notation "- b" := (negb b) : bool_scope.
+Infix "||" := orb (at level 50, left associativity) : bool_scope.
+Infix "&&" := andb (at level 40, left associativity) : bool_scope.
+Notation "- b" := (negb b) : bool_scope.
 
 Open Local Scope bool_scope.
 
@@ -183,54 +180,55 @@ Open Local Scope bool_scope.
 (** Lemmas about [negb]   *)
 (**************************)
 
-Lemma negb_intro : (b:bool)b=(negb (negb b)).
+Lemma negb_intro : forall b:bool, b = - - b.
 Proof.
-NewDestruct b; Reflexivity.
+destruct b; reflexivity.
 Qed.
 
-Lemma negb_elim : (b:bool)(negb (negb b))=b.
+Lemma negb_elim : forall b:bool, - - b = b.
 Proof.
-NewDestruct b; Reflexivity.
+destruct b; reflexivity.
 Qed.
        
-Lemma negb_orb : (b1,b2:bool)
-  (negb (orb b1 b2)) = (andb (negb b1) (negb b2)).
+Lemma negb_orb : forall b1 b2:bool, - (b1 || b2) = - b1 && - b2.
 Proof.
-  NewDestruct b1; NewDestruct b2; Simpl; Reflexivity.
+  destruct b1; destruct b2; simpl in |- *; reflexivity.
 Qed.
 
-Lemma negb_andb : (b1,b2:bool)
-  (negb (andb b1 b2)) = (orb (negb b1) (negb b2)).
+Lemma negb_andb : forall b1 b2:bool, - (b1 && b2) = - b1 || - b2.
 Proof.
-  NewDestruct b1; NewDestruct b2; Simpl; Reflexivity.
+  destruct b1; destruct b2; simpl in |- *; reflexivity.
 Qed.
 
-Lemma negb_sym : (b,b':bool)(b'=(negb b))->(b=(negb b')).
+Lemma negb_sym : forall b b':bool, b' = - b -> b = - b'.
 Proof.
-NewDestruct b; NewDestruct b'; Intros; Simpl; Trivial with bool.
+destruct b; destruct b'; intros; simpl in |- *; trivial with bool.
 Qed.
 
-Lemma no_fixpoint_negb : (b:bool)~(negb b)=b.
+Lemma no_fixpoint_negb : forall b:bool, - b <> b.
 Proof.
-NewDestruct b; Simpl; Unfold not; Intro; Apply diff_true_false; Auto with bool.
+destruct b; simpl in |- *; unfold not in |- *; intro; apply diff_true_false;
+ auto with bool.
 Qed.
 
-Lemma eqb_negb1 : (b:bool)(eqb (negb b) b)=false.
-NewDestruct b.
-Trivial with bool.
-Trivial with bool.
+Lemma eqb_negb1 : forall b:bool, eqb (- b) b = false.
+destruct b.
+trivial with bool.
+trivial with bool.
 Qed.
  
-Lemma eqb_negb2 : (b:bool)(eqb b (negb b))=false.
-NewDestruct b.
-Trivial with bool.
-Trivial with bool.
+Lemma eqb_negb2 : forall b:bool, eqb b (- b) = false.
+destruct b.
+trivial with bool.
+trivial with bool.
 Qed.
 
 
-Lemma if_negb : (A:Set) (b:bool) (x,y:A) (if (negb b) then x else y)=(if b then y else x).
+Lemma if_negb :
+ forall (A:Set) (b:bool) (x y:A),
+   (if - b then x else y) = (if b then y else x).
 Proof.
-  NewDestruct b;Trivial.
+  destruct b; trivial.
 Qed.
 
 
@@ -238,304 +236,305 @@ Qed.
 (** A few lemmas about [or] *)
 (****************************)
 
-Lemma orb_prop : 
-  (a,b:bool)(orb a b)=true -> (a = true)\/(b = true).
-NewDestruct a; NewDestruct b; Simpl; Try (Intro H;Discriminate H); Auto with bool.
+Lemma orb_prop : forall a b:bool, a || b = true -> a = true \/ b = true.
+destruct a; destruct b; simpl in |- *; try (intro H; discriminate H);
+ auto with bool.
 Qed.
 
-Lemma orb_prop2 : 
-  (a,b:bool)(Is_true (orb a b)) -> (Is_true a)\/(Is_true b).
-NewDestruct a; NewDestruct b; Simpl; Try (Intro H;Discriminate H); Auto with bool.
+Lemma orb_prop2 : forall a b:bool, Is_true (a || b) -> Is_true a \/ Is_true b.
+destruct a; destruct b; simpl in |- *; try (intro H; discriminate H);
+ auto with bool.
 Qed.
 
-Lemma orb_true_intro 
-    : (b1,b2:bool)(b1=true)\/(b2=true)->(orb b1 b2)=true.
-NewDestruct b1; Auto with bool.
-NewDestruct 1; Intros.
-Elim diff_true_false; Auto with bool.
-Rewrite H; Trivial with bool.
+Lemma orb_true_intro :
+ forall b1 b2:bool, b1 = true \/ b2 = true -> b1 || b2 = true.
+destruct b1; auto with bool.
+destruct 1; intros.
+elim diff_true_false; auto with bool.
+rewrite H; trivial with bool.
 Qed.
-Hints Resolve orb_true_intro : bool v62.
+Hint Resolve orb_true_intro: bool v62.
 
-Lemma orb_b_true : (b:bool)(orb b true)=true.
-Auto with bool.
+Lemma orb_b_true : forall b:bool, b || true = true.
+auto with bool.
 Qed.
-Hints Resolve orb_b_true : bool v62.
+Hint Resolve orb_b_true: bool v62.
 
-Lemma orb_true_b : (b:bool)(orb true b)=true.
-Trivial with bool.
+Lemma orb_true_b : forall b:bool, true || b = true.
+trivial with bool.
 Qed.
 
-Definition orb_true_elim : (b1,b2:bool)(orb b1 b2)=true -> {b1=true}+{b2=true}.
-NewDestruct b1; Simpl; Auto with bool.
+Definition orb_true_elim :
+  forall b1 b2:bool, b1 || b2 = true -> {b1 = true} + {b2 = true}.
+destruct b1; simpl in |- *; auto with bool.
 Defined.
 
-Lemma orb_false_intro 
-    : (b1,b2:bool)(b1=false)->(b2=false)->(orb b1 b2)=false.
-Intros b1 b2 H1 H2; Rewrite H1; Rewrite H2; Trivial with bool.
+Lemma orb_false_intro :
+ forall b1 b2:bool, b1 = false -> b2 = false -> b1 || b2 = false.
+intros b1 b2 H1 H2; rewrite H1; rewrite H2; trivial with bool.
 Qed.
-Hints Resolve orb_false_intro : bool v62.
+Hint Resolve orb_false_intro: bool v62.
 
-Lemma orb_b_false : (b:bool)(orb b false)=b.
+Lemma orb_b_false : forall b:bool, b || false = b.
 Proof.
-  NewDestruct b; Trivial with bool.
+  destruct b; trivial with bool.
 Qed.
-Hints Resolve orb_b_false : bool v62.
+Hint Resolve orb_b_false: bool v62.
 
-Lemma orb_false_b : (b:bool)(orb false b)=b.
+Lemma orb_false_b : forall b:bool, false || b = b.
 Proof.
-  NewDestruct b; Trivial with bool.
+  destruct b; trivial with bool.
 Qed.
-Hints Resolve orb_false_b : bool v62.
+Hint Resolve orb_false_b: bool v62.
 
-Lemma orb_false_elim : 
-    (b1,b2:bool)(orb b1 b2)=false -> (b1=false)/\(b2=false).
+Lemma orb_false_elim :
+ forall b1 b2:bool, b1 || b2 = false -> b1 = false /\ b2 = false.
 Proof.
-  NewDestruct b1.
-  Intros; Elim diff_true_false; Auto with bool.
-  NewDestruct b2.
-  Intros; Elim diff_true_false; Auto with bool.
-  Auto with bool.
+  destruct b1.
+  intros; elim diff_true_false; auto with bool.
+  destruct b2.
+  intros; elim diff_true_false; auto with bool.
+  auto with bool.
 Qed.
 
-Lemma orb_neg_b :
-  (b:bool)(orb b (negb b))=true.
+Lemma orb_neg_b : forall b:bool, b || - b = true.
 Proof.
-  NewDestruct b; Reflexivity.
+  destruct b; reflexivity.
 Qed.
-Hints Resolve orb_neg_b : bool v62.
+Hint Resolve orb_neg_b: bool v62.
 
-Lemma orb_sym : (b1,b2:bool)(orb b1 b2)=(orb b2 b1).
-NewDestruct b1; NewDestruct b2; Reflexivity.
+Lemma orb_comm : forall b1 b2:bool, b1 || b2 = b2 || b1.
+destruct b1; destruct b2; reflexivity.
 Qed.
 
-Lemma orb_assoc : (b1,b2,b3:bool)(orb b1 (orb b2 b3))=(orb (orb b1 b2) b3).
+Lemma orb_assoc : forall b1 b2 b3:bool, b1 || (b2 || b3) = b1 || b2 || b3.
 Proof.
-  NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
+  destruct b1; destruct b2; destruct b3; reflexivity.
 Qed.
 
-Hints Resolve orb_sym orb_assoc orb_b_false orb_false_b : bool v62.
+Hint Resolve orb_comm orb_assoc orb_b_false orb_false_b: bool v62.
 
 (*****************************)
 (** A few lemmas about [and] *)
 (*****************************)
 
-Lemma andb_prop : 
-  (a,b:bool)(andb a b) = true -> (a = true)/\(b = true).
+Lemma andb_prop : forall a b:bool, a && b = true -> a = true /\ b = true.
 
 Proof.
-  NewDestruct a; NewDestruct b; Simpl; Try (Intro H;Discriminate H);
-  Auto with bool.
+  destruct a; destruct b; simpl in |- *; try (intro H; discriminate H);
+   auto with bool.
 Qed.
-Hints Resolve andb_prop : bool v62.
+Hint Resolve andb_prop: bool v62.
 
-Definition andb_true_eq : (a,b:bool) true = (andb a b) -> true = a /\ true = b.
+Definition andb_true_eq :
+  forall a b:bool, true = a && b -> true = a /\ true = b.
 Proof.
-  NewDestruct a; NewDestruct b; Auto.
+  destruct a; destruct b; auto.
 Defined.
 
-Lemma andb_prop2 : 
-  (a,b:bool)(Is_true (andb a b)) -> (Is_true a)/\(Is_true b).
+Lemma andb_prop2 :
+ forall a b:bool, Is_true (a && b) -> Is_true a /\ Is_true b.
 Proof.
-  NewDestruct a; NewDestruct b; Simpl; Try (Intro H;Discriminate H);
-  Auto with bool.
+  destruct a; destruct b; simpl in |- *; try (intro H; discriminate H);
+   auto with bool.
 Qed.
-Hints Resolve andb_prop2 : bool v62.
+Hint Resolve andb_prop2: bool v62.
 
-Lemma andb_true_intro : (b1,b2:bool)(b1=true)/\(b2=true)->(andb b1 b2)=true.
+Lemma andb_true_intro :
+ forall b1 b2:bool, b1 = true /\ b2 = true -> b1 && b2 = true.
 Proof.
-  NewDestruct b1; NewDestruct b2; Simpl; Tauto Orelse Auto with bool.
+  destruct b1; destruct b2; simpl in |- *; tauto || auto with bool.
 Qed.
-Hints Resolve andb_true_intro : bool v62.
+Hint Resolve andb_true_intro: bool v62.
 
-Lemma andb_true_intro2 : 
-  (b1,b2:bool)(Is_true b1)->(Is_true b2)->(Is_true (andb b1 b2)).
+Lemma andb_true_intro2 :
+ forall b1 b2:bool, Is_true b1 -> Is_true b2 -> Is_true (b1 && b2).
 Proof.
-  NewDestruct b1; NewDestruct b2; Simpl; Tauto.
+  destruct b1; destruct b2; simpl in |- *; tauto.
 Qed.
-Hints Resolve andb_true_intro2 : bool v62.
+Hint Resolve andb_true_intro2: bool v62.
 
-Lemma andb_false_intro1 
-    : (b1,b2:bool)(b1=false)->(andb b1 b2)=false.
-NewDestruct b1; NewDestruct b2; Simpl; Tauto Orelse Auto with bool.
+Lemma andb_false_intro1 : forall b1 b2:bool, b1 = false -> b1 && b2 = false.
+destruct b1; destruct b2; simpl in |- *; tauto || auto with bool.
 Qed.
 
-Lemma andb_false_intro2 
-    : (b1,b2:bool)(b2=false)->(andb b1 b2)=false.
-NewDestruct b1; NewDestruct b2; Simpl; Tauto Orelse Auto with bool.
+Lemma andb_false_intro2 : forall b1 b2:bool, b2 = false -> b1 && b2 = false.
+destruct b1; destruct b2; simpl in |- *; tauto || auto with bool.
 Qed.
 
-Lemma andb_b_false : (b:bool)(andb b false)=false.
-NewDestruct b; Auto with bool.
+Lemma andb_b_false : forall b:bool, b && false = false.
+destruct b; auto with bool.
 Qed.
 
-Lemma andb_false_b : (b:bool)(andb false b)=false.
-Trivial with bool.
+Lemma andb_false_b : forall b:bool, false && b = false.
+trivial with bool.
 Qed.
 
-Lemma andb_b_true : (b:bool)(andb b true)=b.
-NewDestruct b; Auto with bool.
+Lemma andb_b_true : forall b:bool, b && true = b.
+destruct b; auto with bool.
 Qed.
 
-Lemma andb_true_b : (b:bool)(andb true b)=b.
-Trivial with bool.
+Lemma andb_true_b : forall b:bool, true && b = b.
+trivial with bool.
 Qed.
 
-Definition andb_false_elim : 
-    (b1,b2:bool)(andb b1 b2)=false -> {b1=false}+{b2=false}.
-NewDestruct b1; Simpl; Auto with bool.
+Definition andb_false_elim :
+  forall b1 b2:bool, b1 && b2 = false -> {b1 = false} + {b2 = false}.
+destruct b1; simpl in |- *; auto with bool.
 Defined.
-Hints Resolve andb_false_elim : bool v62.
+Hint Resolve andb_false_elim: bool v62.
 
-Lemma andb_neg_b :
-   (b:bool)(andb b (negb b))=false.
-NewDestruct b; Reflexivity.
+Lemma andb_neg_b : forall b:bool, b && - b = false.
+destruct b; reflexivity.
 Qed.   
-Hints Resolve andb_neg_b : bool v62.
+Hint Resolve andb_neg_b: bool v62.
 
-Lemma andb_sym : (b1,b2:bool)(andb b1 b2)=(andb b2 b1).
-NewDestruct b1; NewDestruct b2; Reflexivity.
+Lemma andb_comm : forall b1 b2:bool, b1 && b2 = b2 && b1.
+destruct b1; destruct b2; reflexivity.
 Qed.
 
-Lemma andb_assoc : (b1,b2,b3:bool)(andb b1 (andb b2 b3))=(andb (andb b1 b2) b3).
-NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
+Lemma andb_assoc : forall b1 b2 b3:bool, b1 && (b2 && b3) = b1 && b2 && b3.
+destruct b1; destruct b2; destruct b3; reflexivity.
 Qed.
 
-Hints Resolve andb_sym andb_assoc : bool v62.
+Hint Resolve andb_comm andb_assoc: bool v62.
 
 (*******************************)
 (** Properties of [xorb]       *)
 (*******************************)
 
-Lemma xorb_false : (b:bool) (xorb b false)=b.
+Lemma xorb_false : forall b:bool, xorb b false = b.
 Proof.
-  NewDestruct b; Trivial.
+  destruct b; trivial.
 Qed.
 
-Lemma false_xorb : (b:bool) (xorb false b)=b.
+Lemma false_xorb : forall b:bool, xorb false b = b.
 Proof.
-  NewDestruct b; Trivial.
+  destruct b; trivial.
 Qed.
 
-Lemma xorb_true : (b:bool) (xorb b true)=(negb b).
+Lemma xorb_true : forall b:bool, xorb b true = - b.
 Proof.
-  Trivial.
+  trivial.
 Qed.
 
-Lemma true_xorb : (b:bool) (xorb true b)=(negb b).
+Lemma true_xorb : forall b:bool, xorb true b = - b.
 Proof.
-  NewDestruct b; Trivial.
+  destruct b; trivial.
 Qed.
 
-Lemma xorb_nilpotent : (b:bool) (xorb b b)=false.
+Lemma xorb_nilpotent : forall b:bool, xorb b b = false.
 Proof.
-  NewDestruct b; Trivial.
+  destruct b; trivial.
 Qed.
 
-Lemma xorb_comm : (b,b':bool) (xorb b b')=(xorb b' b).
+Lemma xorb_comm : forall b b':bool, xorb b b' = xorb b' b.
 Proof.
-  NewDestruct b; NewDestruct b'; Trivial.
+  destruct b; destruct b'; trivial.
 Qed.
 
-Lemma xorb_assoc : (b,b',b'':bool) (xorb (xorb b b') b'')=(xorb b (xorb b' b'')).
+Lemma xorb_assoc :
+ forall b b' b'':bool, xorb (xorb b b') b'' = xorb b (xorb b' b'').
 Proof.
-  NewDestruct b; NewDestruct b'; NewDestruct b''; Trivial.
+  destruct b; destruct b'; destruct b''; trivial.
 Qed.
 
-Lemma xorb_eq : (b,b':bool) (xorb b b')=false -> b=b'.
+Lemma xorb_eq : forall b b':bool, xorb b b' = false -> b = b'.
 Proof.
-  NewDestruct b; NewDestruct b'; Trivial.
-  Unfold xorb. Intros. Rewrite H. Reflexivity.
+  destruct b; destruct b'; trivial.
+  unfold xorb in |- *. intros. rewrite H. reflexivity.
 Qed.
 
-Lemma xorb_move_l_r_1 : (b,b',b'':bool) (xorb b b')=b'' -> b'=(xorb b b'').
+Lemma xorb_move_l_r_1 :
+ forall b b' b'':bool, xorb b b' = b'' -> b' = xorb b b''.
 Proof.
-  Intros. Rewrite <- (false_xorb b'). Rewrite <- (xorb_nilpotent b). Rewrite xorb_assoc.
-  Rewrite H. Reflexivity.
+  intros. rewrite <- (false_xorb b'). rewrite <- (xorb_nilpotent b). rewrite xorb_assoc.
+  rewrite H. reflexivity.
 Qed.
 
-Lemma xorb_move_l_r_2 : (b,b',b'':bool) (xorb b b')=b'' -> b=(xorb b'' b').
+Lemma xorb_move_l_r_2 :
+ forall b b' b'':bool, xorb b b' = b'' -> b = xorb b'' b'.
 Proof.
-  Intros. Rewrite xorb_comm in H. Rewrite (xorb_move_l_r_1 b' b b'' H). Apply xorb_comm.
+  intros. rewrite xorb_comm in H. rewrite (xorb_move_l_r_1 b' b b'' H). apply xorb_comm.
 Qed.
 
-Lemma xorb_move_r_l_1 : (b,b',b'':bool) b=(xorb b' b'') -> (xorb b' b)=b''.
+Lemma xorb_move_r_l_1 :
+ forall b b' b'':bool, b = xorb b' b'' -> xorb b' b = b''.
 Proof.
-  Intros. Rewrite H. Rewrite <- xorb_assoc. Rewrite xorb_nilpotent. Apply false_xorb.
+  intros. rewrite H. rewrite <- xorb_assoc. rewrite xorb_nilpotent. apply false_xorb.
 Qed.
 
-Lemma xorb_move_r_l_2 : (b,b',b'':bool) b=(xorb b' b'') -> (xorb b b'')=b'.
+Lemma xorb_move_r_l_2 :
+ forall b b' b'':bool, b = xorb b' b'' -> xorb b b'' = b'.
 Proof.
-  Intros. Rewrite H. Rewrite xorb_assoc. Rewrite xorb_nilpotent. Apply xorb_false.
+  intros. rewrite H. rewrite xorb_assoc. rewrite xorb_nilpotent. apply xorb_false.
 Qed.
 
 (*******************************)
 (** De Morgan's law            *)
 (*******************************)
 
-Lemma demorgan1 : (b1,b2,b3:bool)
-  (andb b1 (orb b2 b3)) = (orb (andb b1 b2) (andb b1 b3)).
-NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
+Lemma demorgan1 :
+ forall b1 b2 b3:bool, b1 && (b2 || b3) = b1 && b2 || b1 && b3.
+destruct b1; destruct b2; destruct b3; reflexivity.
 Qed.
 
-Lemma demorgan2 : (b1,b2,b3:bool)
-  (andb (orb b1 b2) b3) = (orb (andb b1 b3) (andb b2 b3)).
-NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
+Lemma demorgan2 :
+ forall b1 b2 b3:bool, (b1 || b2) && b3 = b1 && b3 || b2 && b3.
+destruct b1; destruct b2; destruct b3; reflexivity.
 Qed.
 
-Lemma demorgan3 : (b1,b2,b3:bool)
-  (orb b1 (andb b2 b3)) = (andb (orb b1 b2) (orb b1 b3)).
-NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
+Lemma demorgan3 :
+ forall b1 b2 b3:bool, b1 || b2 && b3 = (b1 || b2) && (b1 || b3).
+destruct b1; destruct b2; destruct b3; reflexivity.
 Qed.
 
-Lemma demorgan4 : (b1,b2,b3:bool)
-  (orb (andb b1 b2) b3) = (andb (orb b1 b3) (orb b2 b3)).
-NewDestruct b1; NewDestruct b2; NewDestruct b3; Reflexivity.
+Lemma demorgan4 :
+ forall b1 b2 b3:bool, b1 && b2 || b3 = (b1 || b3) && (b2 || b3).
+destruct b1; destruct b2; destruct b3; reflexivity.
 Qed.
 
-Lemma absoption_andb : (b1,b2:bool)
-  (andb b1 (orb b1 b2)) = b1.
+Lemma absoption_andb : forall b1 b2:bool, b1 && (b1 || b2) = b1.
 Proof.
-  NewDestruct b1; NewDestruct b2; Simpl; Reflexivity.
+  destruct b1; destruct b2; simpl in |- *; reflexivity.
 Qed.
 
-Lemma absoption_orb : (b1,b2:bool)
-  (orb b1 (andb b1 b2)) = b1.
+Lemma absoption_orb : forall b1 b2:bool, b1 || b1 && b2 = b1.
 Proof.
-  NewDestruct b1; NewDestruct b2; Simpl; Reflexivity.
+  destruct b1; destruct b2; simpl in |- *; reflexivity.
 Qed.
 
 
 (** Misc. equalities between booleans (to be used by Auto) *)
 
-Lemma bool_1 : (b1,b2:bool)(b1=true <-> b2=true) -> b1=b2. 
+Lemma bool_1 : forall b1 b2:bool, (b1 = true <-> b2 = true) -> b1 = b2. 
 Proof. 
- Intros b1 b2; Case b1; Case b2; Intuition.
+ intros b1 b2; case b1; case b2; intuition.
 Qed.
 
-Lemma bool_2 : (b1,b2:bool)b1=b2 -> b1=true -> b2=true.
+Lemma bool_2 : forall b1 b2:bool, b1 = b2 -> b1 = true -> b2 = true.
 Proof. 
- Intros b1 b2; Case b1; Case b2; Intuition.
+ intros b1 b2; case b1; case b2; intuition.
 Qed. 
 
-Lemma bool_3 : (b:bool) ~(negb b)=true -> b=true.
+Lemma bool_3 : forall b:bool, - b <> true -> b = true.
 Proof.
-  NewDestruct b; Intuition.
+  destruct b; intuition.
 Qed.
 
-Lemma bool_4 : (b:bool) b=true -> ~(negb b)=true.  
+Lemma bool_4 : forall b:bool, b = true -> - b <> true.  
 Proof.
-  NewDestruct b; Intuition.
+  destruct b; intuition.
 Qed.
 
-Lemma bool_5 : (b:bool) (negb b)=true -> ~b=true.
+Lemma bool_5 : forall b:bool, - b = true -> b <> true.
 Proof.
-  NewDestruct b; Intuition.
+  destruct b; intuition.
 Qed.
 
-Lemma bool_6 : (b:bool) ~b=true -> (negb b)=true.  
+Lemma bool_6 : forall b:bool, b <> true -> - b = true.  
 Proof.
-  NewDestruct b; Intuition.
+  destruct b; intuition.
 Qed.
 
-Hints Resolve bool_1 bool_2 bool_3 bool_4 bool_5 bool_6.
+Hint Resolve bool_1 bool_2 bool_3 bool_4 bool_5 bool_6.
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