Restructuration ZArith et déport de la partie sur 'positive' dans NArith, de la partie Omega dans contrib/omega

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@4801 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 0 insertions, 168 deletions

diff --git a/theories/ZArith/Zmisc.v b/theories/ZArith/Zmisc.v
index 7e153c8dd..d6d5cd3d3 100644
--- a/theories/ZArith/Zmisc.v
+++ b/theories/ZArith/Zmisc.v
@@ -224,172 +224,4 @@ Intros; Rewrite iter_convert; Apply iter_nat_invariant; Trivial with arith.
 Qed.
 
 
-(**********************************************************************)
-(** [Zeven], [Zodd], [Zdiv2] and their related properties *)
-
-Definition Zeven := 
-  [z:Z]Cases z of ZERO => True
-                | (POS (xO _)) => True
-		| (NEG (xO _)) => True
-		| _ => False
-               end.
-
-Definition Zodd := 
-  [z:Z]Cases z of (POS xH) => True
-                | (NEG xH) => True
-                | (POS (xI _)) => True
-		| (NEG (xI _)) => True
-		| _ => False
-               end.
-
-Definition Zeven_bool :=
-  [z:Z]Cases z of ZERO => true
-                | (POS (xO _)) => true
-		| (NEG (xO _)) => true
-		| _ => false
-               end.
-
-Definition Zodd_bool := 
-  [z:Z]Cases z of ZERO => false
-                | (POS (xO _)) => false
-		| (NEG (xO _)) => false
-		| _ => true
-               end.
-
-Definition Zeven_odd_dec : (z:Z) { (Zeven z) }+{ (Zodd z) }.
-Proof.
-  Intro z. Case z;
-  [ Left; Compute; Trivial
-  | Intro p; Case p; Intros; 
-    (Right; Compute; Exact I) Orelse (Left; Compute; Exact I)
-  | Intro p; Case p; Intros; 
-    (Right; Compute; Exact I) Orelse (Left; Compute; Exact I) ].
-Defined.
-
-Definition Zeven_dec : (z:Z) { (Zeven z) }+{ ~(Zeven z) }.
-Proof.
-  Intro z. Case z;
-  [ Left; Compute; Trivial
-  | Intro p; Case p; Intros; 
-    (Left; Compute; Exact I) Orelse (Right; Compute; Trivial) 
-  | Intro p; Case p; Intros; 
-    (Left; Compute; Exact I) Orelse (Right; Compute; Trivial) ].
-Defined.
-
-Definition Zodd_dec : (z:Z) { (Zodd z) }+{ ~(Zodd z) }.
-Proof.
-  Intro z. Case z;
-  [ Right; Compute; Trivial
-  | Intro p; Case p; Intros; 
-    (Left; Compute; Exact I) Orelse (Right; Compute; Trivial) 
-  | Intro p; Case p; Intros; 
-    (Left; Compute; Exact I) Orelse (Right; Compute; Trivial) ].
-Defined.
-
-Lemma Zeven_not_Zodd : (z:Z)(Zeven z) -> ~(Zodd z).
-Proof.
-  NewDestruct z; [ Idtac | NewDestruct p | NewDestruct p  ]; Compute; Trivial.
-Qed.
-
-Lemma Zodd_not_Zeven : (z:Z)(Zodd z) -> ~(Zeven z).
-Proof.
-  NewDestruct z; [ Idtac | NewDestruct p | NewDestruct p  ]; Compute; Trivial.
-Qed.
-
-Lemma Zeven_Sn : (z:Z)(Zodd z) -> (Zeven (Zs z)).
-Proof.
- NewDestruct z; Unfold Zs; [ Idtac | NewDestruct p | NewDestruct p  ]; Simpl; Trivial. 
- Unfold double_moins_un; Case p; Simpl; Auto.
-Qed.
-
-Lemma Zodd_Sn : (z:Z)(Zeven z) -> (Zodd (Zs z)).
-Proof.
- NewDestruct z; Unfold Zs; [ Idtac | NewDestruct p | NewDestruct p  ]; Simpl; Trivial. 
- Unfold double_moins_un; Case p; Simpl; Auto.
-Qed.
-
-Lemma Zeven_pred : (z:Z)(Zodd z) -> (Zeven (Zpred z)).
-Proof.
- NewDestruct z; Unfold Zpred; [ Idtac | NewDestruct p | NewDestruct p  ]; Simpl; Trivial. 
- Unfold double_moins_un; Case p; Simpl; Auto.
-Qed.
-
-Lemma Zodd_pred : (z:Z)(Zeven z) -> (Zodd (Zpred z)).
-Proof.
- NewDestruct z; Unfold Zpred; [ Idtac | NewDestruct p | NewDestruct p  ]; Simpl; Trivial. 
- Unfold double_moins_un; Case p; Simpl; Auto.
-Qed.
-
-Hints Unfold Zeven Zodd : zarith.
-
-(**********************************************************************)
-(** [Zdiv2] is defined on all [Z], but notice that for odd negative
-    integers it is not the euclidean quotient: in that case we have [n =
-    2*(n/2)-1] *)
-
-Definition Zdiv2 :=
-  [z:Z]Cases z of ZERO => ZERO
-                | (POS xH) => ZERO
-                | (POS p) => (POS (Zdiv2_pos p))
-		| (NEG xH) => ZERO
-		| (NEG p) => (NEG (Zdiv2_pos p))
-	       end.
-
-Lemma Zeven_div2 : (x:Z) (Zeven x) -> `x = 2*(Zdiv2 x)`.
-Proof.
-NewDestruct x.
-Auto with arith.
-NewDestruct p; Auto with arith.
-Intros. Absurd (Zeven (POS (xI p))); Red; Auto with arith.
-Intros. Absurd (Zeven `1`); Red; Auto with arith.
-NewDestruct p; Auto with arith.
-Intros. Absurd (Zeven (NEG (xI p))); Red; Auto with arith.
-Intros. Absurd (Zeven `-1`); Red; Auto with arith.
-Qed.
-
-Lemma Zodd_div2 : (x:Z) `x >= 0` -> (Zodd x) -> `x = 2*(Zdiv2 x)+1`.
-Proof.
-NewDestruct x.
-Intros. Absurd (Zodd `0`); Red; Auto with arith.
-NewDestruct p; Auto with arith.
-Intros. Absurd (Zodd (POS (xO p))); Red; Auto with arith.
-Intros. Absurd `(NEG p) >= 0`; Red; Auto with arith.
-Qed.
-
-Lemma Zodd_div2_neg : (x:Z) `x <= 0` -> (Zodd x) -> `x = 2*(Zdiv2 x)-1`.
-Proof.
-NewDestruct x.
-Intros. Absurd (Zodd `0`); Red; Auto with arith.
-Intros. Absurd `(NEG p) >= 0`; Red; Auto with arith.
-NewDestruct p; Auto with arith.
-Intros. Absurd (Zodd (NEG (xO p))); Red; Auto with arith.
-Qed.
-
-Lemma Z_modulo_2 : (x:Z) { y:Z | `x=2*y` }+{ y:Z | `x=2*y+1` }.
-Proof.
-Intros x.
-Elim (Zeven_odd_dec x); Intro.
-Left. Split with (Zdiv2 x). Exact (Zeven_div2 x a).
-Right. Generalize b; Clear b; Case x.
-Intro b; Inversion b.
-Intro p; Split with (Zdiv2 (POS p)). Apply (Zodd_div2 (POS p)); Trivial.
-Unfold Zge Zcompare; Simpl; Discriminate.
-Intro p; Split with (Zdiv2 (Zpred (NEG p))).
-Pattern 1 (NEG p); Rewrite (Zs_pred (NEG p)).
-Pattern 1 (Zpred (NEG p)); Rewrite (Zeven_div2 (Zpred (NEG p))).
-Reflexivity.
-Apply Zeven_pred; Assumption.
-Qed.
-
-Lemma Zsplit2 :  (x:Z) { p : Z*Z | let (x1,x2)=p in (`x=x1+x2` /\ (x1=x2 \/ `x2=x1+1`)) }.
-Proof.
-Intros x.
-Elim (Z_modulo_2 x); Intros (y,Hy); Rewrite Zmult_sym in Hy; Rewrite <- Zred_factor1 in Hy.
-Exists (y,y); Split. 
-Assumption.
-Left; Reflexivity.
-Exists (y,y+`1`); Split.
-Rewrite Zplus_assoc; Assumption.
-Right; Reflexivity.
-Qed.