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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Zeven.v,v 1.1.2.1 2004/07/16 19:31:43 herbelin Exp $ i*)
Require BinInt.
Require Zsyntax.
(**********************************************************************)
(** About parity: even and odd predicates on Z, division by 2 on Z *)
(**********************************************************************)
(** [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.
Intro z; NewDestruct z; [ Idtac | NewDestruct p | NewDestruct p ]; Compute; Trivial.
Qed.
Lemma Zodd_not_Zeven : (z:Z)(Zodd z) -> ~(Zeven z).
Proof.
Intro z; NewDestruct z; [ Idtac | NewDestruct p | NewDestruct p ]; Compute; Trivial.
Qed.
Lemma Zeven_Sn : (z:Z)(Zodd z) -> (Zeven (Zs z)).
Proof.
Intro z; 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.
Intro z; 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.
Intro z; 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.
Intro z; 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.
Intro x; 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.
Intro x; 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.
Intro x; 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 <- Zplus_Zmult_2 in Hy.
Exists (y,y); Split.
Assumption.
Left; Reflexivity.
Exists (y,`y+1`); Split.
Rewrite Zplus_assoc; Assumption.
Right; Reflexivity.
Qed.
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