remplace Zarith par ZArith

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-(***********************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
-(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
-(*   \VV/  *************************************************************)
-(*    //   *      This file is distributed under the terms of the      *)
-(*         *       GNU Lesser General Public License Version 2.1       *)
-(***********************************************************************)
-
-(*i $Id$ i*)
-
-(********************************************************)
-(* Module Zmisc.v :           				*)
-(* Definitions et lemmes complementaires		*)
-(* Division euclidienne					*)
-(* Patrick Loiseleur, avril 1997			*)
-(********************************************************)
-
-Require fast_integer.
-Require zarith_aux.
-Require auxiliary.
-Require Zsyntax.
-Require Bool.
-
-(**********************************************************************
- Overview of the sections of this file :
-
- - logic : Logic complements.
- - numbers : a few very simple lemmas for manipulating the
-   constructors [POS], [NEG], [ZERO] and [xI], [xO], [xH]
- - registers : defining arrays of bits and their relation with integers.
- - iter : the n-th iterate of a function is defined for n:nat and n:positive.
-   The two notions are identified and an invariant conservation theorem
-   is proved.
- - recursors : Here a nat-like recursor is built.
- - arith : lemmas about [< <= ?= + *] ...
-
-************************************************************************)
-
-Section logic.
-
-Lemma rename : (A:Set)(P:A->Prop)(x:A) ((y:A)(x=y)->(P y)) -> (P x).
-Auto with arith. 
-Save.
-
-End logic.
-
-Section numbers.
-
-Definition entier_of_Z := [z:Z]Case z of Nul Pos Pos end.
-Definition Z_of_entier := [x:entier]Case x of ZERO POS end.
- 
-(*i Coercion Z_of_entier : entier >-> Z. i*)
-
-Lemma POS_xI : (p:positive) (POS (xI p))=`2*(POS p) + 1`.
-Intro; Apply refl_equal.
-Save.
-Lemma POS_xO : (p:positive) (POS (xO p))=`2*(POS p)`.
-Intro; Apply refl_equal.
-Save.
-Lemma NEG_xI : (p:positive) (NEG (xI p))=`2*(NEG p) - 1`.
-Intro; Apply refl_equal.
-Save.
-Lemma NEG_xO : (p:positive) (NEG (xO p))=`2*(NEG p)`.
-Intro; Apply refl_equal.
-Save.
-
-Lemma POS_add : (p,p':positive)`(POS (add p p'))=(POS p)+(POS p')`.
-Induction p; Induction p'; Simpl; Auto with arith.
-Save.
-
-Lemma NEG_add : (p,p':positive)`(NEG (add p p'))=(NEG p)+(NEG p')`.
-Induction p; Induction p'; Simpl; Auto with arith.
-Save.
-
-Definition Zle_bool := [x,y:Z]Case `x ?= y` of true true false end.
-Definition Zge_bool := [x,y:Z]Case `x ?= y` of true false true end.
-Definition Zlt_bool := [x,y:Z]Case `x ?= y` of false true false end.
-Definition Zgt_bool := [x,y:Z]Case ` x ?= y` of false false true end.
-Definition Zeq_bool := [x,y:Z]Cases `x ?= y` of EGAL => true | _ => false end.
-Definition Zneq_bool := [x,y:Z]Cases `x ?= y` of EGAL =>false | _ => true end.
-
-End numbers.
-
-Section iterate.
-
-(* l'itere n-ieme d'une fonction f*)
-Fixpoint iter_nat[n:nat]  : (A:Set)(f:A->A)A->A :=
-  [A:Set][f:A->A][x:A]
-    Cases n of
-      O => x
-    | (S n') => (f (iter_nat n' A f x))
-    end.
-
-Fixpoint iter_pos[n:positive] : (A:Set)(f:A->A)A->A :=
-  [A:Set][f:A->A][x:A]
-    Cases n of
-     	xH => (f x)
-      | (xO n') => (iter_pos n' A f (iter_pos n' A f x))
-      | (xI n') => (f (iter_pos n' A f (iter_pos n' A f x)))
-    end.
-
-Definition iter :=
-  [n:Z][A:Set][f:A->A][x:A]Cases n of
-    ZERO => x
-  | (POS p) => (iter_pos p A f x)
-  | (NEG p) => x
-  end.
-
-Theorem iter_nat_plus :
-  (n,m:nat)(A:Set)(f:A->A)(x:A)
-    (iter_nat (plus n m) A f x)=(iter_nat n A f (iter_nat m A f x)).
-    
-Induction n;
-[ Simpl; Auto with arith
-| Intros; Simpl; Apply f_equal with f:=f; Apply H
-].  
-Save.
-
-Theorem iter_convert : (n:positive)(A:Set)(f:A->A)(x:A)
-  (iter_pos n A f x) = (iter_nat (convert n) A f x).
-
-Induction n;
-[ Intros; Simpl; Rewrite -> (H A f x);
-  Rewrite -> (H A f (iter_nat (convert p) A f x));
-  Rewrite -> (ZL6 p); Symmetry; Apply f_equal with f:=f;
-  Apply iter_nat_plus
-| Intros; Unfold convert; Simpl; Rewrite -> (H A f x);
-  Rewrite -> (H A f (iter_nat (convert p) A f x));
-  Rewrite -> (ZL6 p); Symmetry;
-  Apply iter_nat_plus
-| Simpl; Auto with arith
-].
-Save.
-
-Theorem iter_pos_add :
-  (n,m:positive)(A:Set)(f:A->A)(x:A)
-    (iter_pos (add n m) A f x)=(iter_pos n A f (iter_pos m A f x)).
-
-Intros.
-Rewrite -> (iter_convert m A f x).
-Rewrite -> (iter_convert n A f (iter_nat (convert m) A f x)).
-Rewrite -> (iter_convert (add n m) A f x).
-Rewrite -> (convert_add n m).
-Apply iter_nat_plus.
-Save.
-
-(* Preservation of invariants : if f : A->A preserves the invariant Inv, 
-  then the iterates of f also preserve it. *)
-
-Theorem iter_nat_invariant :
-  (n:nat)(A:Set)(f:A->A)(Inv:A->Prop)
-  ((x:A)(Inv x)->(Inv (f x)))->(x:A)(Inv x)->(Inv (iter_nat n A f x)).
-Induction n; Intros;
-[ Trivial with arith
-| Simpl; Apply H0 with x:=(iter_nat n0 A f x); Apply H; Trivial with arith].
-Save.
-
-Theorem iter_pos_invariant :
-  (n:positive)(A:Set)(f:A->A)(Inv:A->Prop)
-  ((x:A)(Inv x)->(Inv (f x)))->(x:A)(Inv x)->(Inv (iter_pos n A f x)).
-Intros; Rewrite iter_convert; Apply iter_nat_invariant; Trivial with arith.
-Save.
-
-End iterate.
-
-
-Section arith.
-
-Lemma ZERO_le_POS : (p:positive) `0 <= (POS p)`.
-Intro; Unfold Zle; Unfold Zcompare; Discriminate.
-Save.
-
-Lemma POS_gt_ZERO : (p:positive) `(POS p) > 0`.
-Intro; Unfold Zgt; Simpl; Trivial with arith.
-Save.
-
-Lemma Zlt_ZERO_pred_le_ZERO : (x:Z) `0 < x` -> `0 <= (Zpred x)`.
-Intros.
-Rewrite (Zs_pred x) in H.
-Apply Zgt_S_le.
-Apply Zlt_gt.
-Assumption.
-Save.
-
-(* 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.
-
-Lemma 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) ].
-  (*i was 
-  Realizer Zeven_bool.
-  Repeat Program; Compute; Trivial.
-  i*)
-Save.
-
-Lemma 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) ].
-  (*i was 
-  Realizer Zeven_bool.
-  Repeat Program; Compute; Trivial.
-  i*)
-Save.
-
-Lemma 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) ].
-  (*i was 
-  Realizer Zodd_bool.
-  Repeat Program; Compute; Trivial.
-  i*)
-Save.
-
-Lemma Zeven_not_Zodd : (z:Z)(Zeven z) -> ~(Zodd z).
-Proof.
-  Destruct z; [ Idtac | Destruct p | Destruct p  ]; Compute; Trivial.
-Save.
-
-Lemma Zodd_not_Zeven : (z:Z)(Zodd z) -> ~(Zeven z).
-Proof.
-  Destruct z; [ Idtac | Destruct p | Destruct p  ]; Compute; Trivial.
-Save.
-
-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_pos :=
-  [z:positive]Cases z of xH => xH
-                       | (xO p) => p
-		       | (xI p) => p
-		      end.
-
-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.
-Destruct x.
-Auto with arith.
-Destruct p; Auto with arith.
-Intros. Absurd (Zeven (POS (xI p0))); Red; Auto with arith.
-Intros. Absurd (Zeven `1`); Red; Auto with arith.
-Destruct p; Auto with arith.
-Intros. Absurd (Zeven (NEG (xI p0))); Red; Auto with arith.
-Intros. Absurd (Zeven `-1`); Red; Auto with arith.
-Save.
-
-Lemma Zodd_div2 : (x:Z) `x >= 0` -> (Zodd x) -> `x = 2*(Zdiv2 x)+1`.
-Proof.
-Destruct x.
-Intros. Absurd (Zodd `0`); Red; Auto with arith.
-Destruct p; Auto with arith.
-Intros. Absurd (Zodd (POS (xO p0))); Red; Auto with arith.
-Intros. Absurd `(NEG p) >= 0`; Red; Auto with arith.
-Save.
-
-Lemma Z_modulo_2 : (x:Z) `x >= 0` -> { y:Z | `x=2*y` }+{ y:Z | `x=2*y+1` }.
-Proof.
-Intros x Hx.
-Elim (Zeven_odd_dec x); Intro.
-Left. Split with (Zdiv2 x). Exact (Zeven_div2 x a).
-Right. Split with (Zdiv2 x). Exact (Zodd_div2 x Hx b).
-Save.
-
-(* Very simple *)
-Lemma Zminus_Zplus_compatible :
-  (x,y,n:Z) `(x+n) - (y+n) = x - y`.
-Intros.
-Unfold Zminus.
-Rewrite -> Zopp_Zplus.
-Rewrite -> (Zplus_sym (Zopp y) (Zopp n)).
-Rewrite -> Zplus_assoc.
-Rewrite <- (Zplus_assoc x n (Zopp n)).
-Rewrite -> (Zplus_inverse_r n).
-Rewrite <- Zplus_n_O.
-Reflexivity.
-Save.
-
-(* Decompose an egality between two ?= relations into 3 implications *)
-Theorem Zcompare_egal_dec :
-   (x1,y1,x2,y2:Z)
-    (`x1 < y1`->`x2 < y2`)
-     ->(`x1 ?= y1`=EGAL -> `x2 ?= y2`=EGAL)
-        ->(`x1 > y1`->`x2 > y2`)->`x1 ?= y1`=`x2 ?= y2`.
-Intros x1 y1 x2 y2.
-Unfold Zgt; Unfold Zlt;
-Case `x1 ?= y1`; Case `x2 ?= y2`; Auto with arith; Symmetry; Auto with arith.
-Save.
-
-Theorem Zcompare_elim :
-  (c1,c2,c3:Prop)(x,y:Z)
-    ((x=y) -> c1) ->(`x < y` -> c2) ->(`x > y`-> c3)
-       -> Case `x ?= y`of c1 c2 c3 end.
-
-Intros.
-Apply rename with x:=`x ?= y`; Intro r; Elim r;
-[ Intro; Apply H; Apply (let (h1, h2)=(Zcompare_EGAL x y) in h1); Assumption
-| Unfold Zlt in H0; Assumption
-| Unfold Zgt in H1; Assumption ].
-Save.
-
-Lemma Zcompare_x_x : (x:Z) `x ?= x` = EGAL.
-Intro; Apply Case (Zcompare_EGAL x x) of [h1,h2: ?]h2 end.
-Apply refl_equal.
-Save.
-
-Lemma Zlt_not_eq : (x,y:Z)`x < y` -> ~x=y.
-Proof.
-Intros.
-Unfold Zlt in H.
-Unfold not.
-Intro.
-Generalize (proj2 ? ? (Zcompare_EGAL x y) H0).
-Intro.
-Rewrite H1 in H.
-Discriminate H.
-Save.
-
-Lemma Zcompare_eq_case : 
-  (c1,c2,c3:Prop)(x,y:Z) c1 -> x=y -> (Case `x ?= y` of c1 c2 c3 end).
-Intros.
-Rewrite -> (Case (Zcompare_EGAL x y) of [h1,h2: ?]h2 end H0).
-Assumption.
-Save.
-
-(* Four very basic lemmas about Zle, Zlt, Zge, Zgt *)
-Lemma Zle_Zcompare :
-  (x,y:Z)`x <= y` -> Case `x ?= y` of True True False end.
-Intros x y; Unfold Zle; Elim `x ?=y`; Auto with arith.
-Save.
-
-Lemma Zlt_Zcompare :
-  (x,y:Z)`x < y`  -> Case `x ?= y` of False True False end.
-Intros x y; Unfold Zlt; Elim `x ?=y`; Intros; Discriminate Orelse Trivial with arith.
-Save.
-
-Lemma Zge_Zcompare :
-  (x,y:Z)` x >= y`-> Case `x ?= y` of True False True end.
-Intros x y; Unfold Zge; Elim `x ?=y`; Auto with arith. 
-Save.
-
-Lemma Zgt_Zcompare :
-  (x,y:Z)`x > y` -> Case `x ?= y` of False False True end.
-Intros x y; Unfold Zgt; Elim `x ?= y`; Intros; Discriminate Orelse Trivial with arith.
-Save.
-
-(* Lemmas about Zmin *)
-
-Lemma Zmin_plus : (x,y,n:Z) `(Zmin (x+n)(y+n))=(Zmin x y)+n`.
-Intros; Unfold Zmin.
-Rewrite (Zplus_sym x n);
-Rewrite (Zplus_sym y n);
-Rewrite (Zcompare_Zplus_compatible x y n).
-Case `x ?= y`; Apply Zplus_sym.
-Save.
-
-(* Lemmas about absolu *)
-
-Lemma absolu_lt : (x,y:Z) `0 <= x < y` -> (lt (absolu x) (absolu y)).
-Proof.
-Intros x y. Case x; Simpl. Case y; Simpl.
-
-Intro. Absurd `0 < 0`. Compute. Intro H0. Discriminate H0. Intuition.
-Intros. Elim (ZL4 p). Intros. Rewrite H0. Auto with arith.
-Intros. Elim (ZL4 p). Intros. Rewrite H0. Auto with arith.
-
-Case y; Simpl.
-Intros. Absurd `(POS p) < 0`. Compute. Intro H0. Discriminate H0. Intuition.
-Intros. Change (gt (convert p) (convert p0)).
-Apply compare_convert_SUPERIEUR.
-Elim H; Auto with arith. Intro. Exact (ZC2 p0 p).
-
-Intros. Absurd `(POS p0) < (NEG p)`.
-Compute. Intro H0. Discriminate H0. Intuition.
-
-Intros. Absurd `0 <= (NEG p)`. Compute. Auto with arith. Intuition.
-Save.
-
-
-End arith.
-