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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import BinInt.
Open Scope Z_scope.
(*******************************************************************)
(** About parity: even and odd predicates on Z, division by 2 on Z *)
(***************************************************)
(** * [Zeven], [Zodd] and their related properties *)
Definition Zeven (z:Z) :=
match z with
| Z0 => True
| Zpos (xO _) => True
| Zneg (xO _) => True
| _ => False
end.
Definition Zodd (z:Z) :=
match z with
| Zpos xH => True
| Zneg xH => True
| Zpos (xI _) => True
| Zneg (xI _) => True
| _ => False
end.
Definition Zeven_bool (z:Z) :=
match z with
| Z0 => true
| Zpos (xO _) => true
| Zneg (xO _) => true
| _ => false
end.
Definition Zodd_bool (z:Z) :=
match z with
| Z0 => false
| Zpos (xO _) => false
| Zneg (xO _) => false
| _ => true
end.
Lemma Zeven_bool_iff : forall n, Zeven_bool n = true <-> Zeven n.
Proof.
destruct n as [|p|p]; try destruct p; simpl in *; split; easy.
Qed.
Lemma Zodd_bool_iff : forall n, Zodd_bool n = true <-> Zodd n.
Proof.
destruct n as [|p|p]; try destruct p; simpl in *; split; easy.
Qed.
Lemma Zodd_even_bool : forall n, Zodd_bool n = negb (Zeven_bool n).
Proof.
destruct n as [|p|p]; trivial; now destruct p.
Qed.
Lemma Zeven_odd_bool : forall n, Zeven_bool n = negb (Zodd_bool n).
Proof.
destruct n as [|p|p]; trivial; now destruct p.
Qed.
Definition Zeven_odd_dec : forall z:Z, {Zeven z} + {Zodd z}.
Proof.
intro z. case z;
[ left; compute; trivial
| intro p; case p; intros;
(right; compute; exact I) || (left; compute; exact I)
| intro p; case p; intros;
(right; compute; exact I) || (left; compute; exact I) ].
Defined.
Definition Zeven_dec : forall z:Z, {Zeven z} + {~ Zeven z}.
Proof.
intro z. case z;
[ left; compute; trivial
| intro p; case p; intros;
(left; compute; exact I) || (right; compute; trivial)
| intro p; case p; intros;
(left; compute; exact I) || (right; compute; trivial) ].
Defined.
Definition Zodd_dec : forall z:Z, {Zodd z} + {~ Zodd z}.
Proof.
intro z. case z;
[ right; compute; trivial
| intro p; case p; intros;
(left; compute; exact I) || (right; compute; trivial)
| intro p; case p; intros;
(left; compute; exact I) || (right; compute; trivial) ].
Defined.
Lemma Zeven_not_Zodd : forall n:Z, Zeven n -> ~ Zodd n.
Proof.
intro z; destruct z; [ idtac | destruct p | destruct p ]; compute;
trivial.
Qed.
Lemma Zodd_not_Zeven : forall n:Z, Zodd n -> ~ Zeven n.
Proof.
intro z; destruct z; [ idtac | destruct p | destruct p ]; compute;
trivial.
Qed.
Lemma Zeven_Sn : forall n:Z, Zodd n -> Zeven (Zsucc n).
Proof.
intro z; destruct z; unfold Zsucc;
[ idtac | destruct p | destruct p ]; simpl;
trivial.
unfold Pdouble_minus_one; case p; simpl; auto.
Qed.
Lemma Zodd_Sn : forall n:Z, Zeven n -> Zodd (Zsucc n).
Proof.
intro z; destruct z; unfold Zsucc;
[ idtac | destruct p | destruct p ]; simpl;
trivial.
unfold Pdouble_minus_one; case p; simpl; auto.
Qed.
Lemma Zeven_pred : forall n:Z, Zodd n -> Zeven (Zpred n).
Proof.
intro z; destruct z; unfold Zpred;
[ idtac | destruct p | destruct p ]; simpl;
trivial.
unfold Pdouble_minus_one; case p; simpl; auto.
Qed.
Lemma Zodd_pred : forall n:Z, Zeven n -> Zodd (Zpred n).
Proof.
intro z; destruct z; unfold Zpred;
[ idtac | destruct p | destruct p ]; simpl;
trivial.
unfold Pdouble_minus_one; case p; simpl; auto.
Qed.
Hint Unfold Zeven Zodd: zarith.
Lemma Zeven_bool_succ : forall n, Zeven_bool (Zsucc n) = Zodd_bool n.
Proof.
destruct n as [ |p|p]; trivial; destruct p as [p|p| ]; trivial.
now destruct p.
Qed.
Lemma Zeven_bool_pred : forall n, Zeven_bool (Zpred n) = Zodd_bool n.
Proof.
destruct n as [ |p|p]; trivial; destruct p as [p|p| ]; trivial.
now destruct p.
Qed.
Lemma Zodd_bool_succ : forall n, Zodd_bool (Zsucc n) = Zeven_bool n.
Proof.
destruct n as [ |p|p]; trivial; destruct p as [p|p| ]; trivial.
now destruct p.
Qed.
Lemma Zodd_bool_pred : forall n, Zodd_bool (Zpred n) = Zeven_bool n.
Proof.
destruct n as [ |p|p]; trivial; destruct p as [p|p| ]; trivial.
now destruct p.
Qed.
(******************************************************************)
(** * Definition of [Zdiv2] and properties wrt [Zeven] and [Zodd] *)
(** [Zdiv2] is defined on all [Z], but notice that for odd negative
integers we have [n = 2*(Zdiv2 n)-1], hence it does not
correspond to the usual Coq division [Zdiv], for which we would
have here [n = 2*(n/2)+1]. Since [Zdiv2] performs rounding
toward zero, it is rather a particular case of the alternative
division [Zquot].
*)
Definition Zdiv2 (z:Z) :=
match z with
| 0 => 0
| Zpos 1 => 0
| Zpos p => Zpos (Pdiv2 p)
| Zneg 1 => 0
| Zneg p => Zneg (Pdiv2 p)
end.
(** We also provide an alternative [Zdiv2'] performing round toward
bottom, and hence corresponding to [Zdiv]. *)
Definition Zdiv2' a :=
match a with
| 0 => 0
| Zpos 1 => 0
| Zpos p => Zpos (Pdiv2 p)
| Zneg p => Zneg (Pdiv2_up p)
end.
Lemma Zdiv2'_odd : forall a,
a = 2*(Zdiv2' a) + if Zodd_bool a then 1 else 0.
Proof.
intros [ |[p|p| ]|[p|p| ]]; simpl; trivial.
f_equal. now rewrite xO_succ_permute, <-Ppred_minus, Ppred_succ.
Qed.
Lemma Zeven_div2 : forall n:Z, Zeven n -> n = 2 * Zdiv2 n.
Proof.
intro x; destruct x.
auto with arith.
destruct p; auto with arith.
intros. absurd (Zeven (Zpos (xI p))); red; auto with arith.
intros. absurd (Zeven 1); red; auto with arith.
destruct p; auto with arith.
intros. absurd (Zeven (Zneg (xI p))); red; auto with arith.
intros. absurd (Zeven (-1)); red; auto with arith.
Qed.
Lemma Zodd_div2 : forall n:Z, n >= 0 -> Zodd n -> n = 2 * Zdiv2 n + 1.
Proof.
intro x; destruct x.
intros. absurd (Zodd 0); red; auto with arith.
destruct p; auto with arith.
intros. absurd (Zodd (Zpos (xO p))); red; auto with arith.
intros. absurd (Zneg p >= 0); red; auto with arith.
Qed.
Lemma Zodd_div2_neg :
forall n:Z, n <= 0 -> Zodd n -> n = 2 * Zdiv2 n - 1.
Proof.
intro x; destruct x.
intros. absurd (Zodd 0); red; auto with arith.
intros. absurd (Zneg p >= 0); red; auto with arith.
destruct p; auto with arith.
intros. absurd (Zodd (Zneg (xO p))); red; auto with arith.
Qed.
Lemma Z_modulo_2 :
forall n:Z, {y : Z | n = 2 * y} + {y : Z | n = 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 (Zpos p)). apply (Zodd_div2 (Zpos p)); trivial.
unfold Zge, Zcompare; simpl; discriminate.
intro p; split with (Zdiv2 (Zpred (Zneg p))).
pattern (Zneg p) at 1; rewrite (Zsucc_pred (Zneg p)).
pattern (Zpred (Zneg p)) at 1; rewrite (Zeven_div2 (Zpred (Zneg p))).
reflexivity.
apply Zeven_pred; assumption.
Qed.
Lemma Zsplit2 :
forall n:Z,
{p : Z * Z |
let (x1, x2) := p in n = x1 + x2 /\ (x1 = x2 \/ x2 = x1 + 1)}.
Proof.
intros x.
elim (Z_modulo_2 x); intros [y Hy]; rewrite Zmult_comm in Hy;
rewrite <- Zplus_diag_eq_mult_2 in Hy.
exists (y, y); split.
assumption.
left; reflexivity.
exists (y, (y + 1)%Z); split.
rewrite Zplus_assoc; assumption.
right; reflexivity.
Qed.
Theorem Zeven_ex: forall n, Zeven n -> exists m, n = 2 * m.
Proof.
intro n; exists (Zdiv2 n); apply Zeven_div2; auto.
Qed.
Theorem Zodd_ex: forall n, Zodd n -> exists m, n = 2 * m + 1.
Proof.
destruct n; intros.
inversion H.
exists (Zdiv2 (Zpos p)).
apply Zodd_div2; simpl; auto; compute; inversion 1.
exists (Zdiv2 (Zneg p) - 1).
unfold Zminus.
rewrite Zmult_plus_distr_r.
rewrite <- Zplus_assoc.
assert (Zneg p <= 0) by (compute; inversion 1).
exact (Zodd_div2_neg _ H0 H).
Qed.
Theorem Zeven_2p: forall p, Zeven (2 * p).
Proof.
destruct p; simpl; auto.
Qed.
Theorem Zodd_2p_plus_1: forall p, Zodd (2 * p + 1).
Proof.
destruct p; simpl; auto.
destruct p; simpl; auto.
Qed.
Theorem Zeven_ex_iff : forall n, Zeven n <-> exists m, n = 2*m.
Proof.
split. apply Zeven_ex. intros (m,->). apply Zeven_2p.
Qed.
Theorem Zodd_ex_iff : forall n, Zodd n <-> exists m, n = 2*m + 1.
Proof.
split. apply Zodd_ex. intros (m,->). apply Zodd_2p_plus_1.
Qed.
Theorem Zeven_plus_Zodd: forall a b,
Zeven a -> Zodd b -> Zodd (a + b).
Proof.
intros a b H1 H2; case Zeven_ex with (1 := H1); intros x H3; try rewrite H3; auto.
case Zodd_ex with (1 := H2); intros y H4; try rewrite H4; auto.
replace (2 * x + (2 * y + 1)) with (2 * (x + y) + 1); try apply Zodd_2p_plus_1; auto with zarith.
rewrite Zmult_plus_distr_r, Zplus_assoc; auto.
Qed.
Theorem Zeven_plus_Zeven: forall a b,
Zeven a -> Zeven b -> Zeven (a + b).
Proof.
intros a b H1 H2; case Zeven_ex with (1 := H1); intros x H3; try rewrite H3; auto.
case Zeven_ex with (1 := H2); intros y H4; try rewrite H4; auto.
replace (2 * x + 2 * y) with (2 * (x + y)); try apply Zeven_2p; auto with zarith.
apply Zmult_plus_distr_r; auto.
Qed.
Theorem Zodd_plus_Zeven: forall a b,
Zodd a -> Zeven b -> Zodd (a + b).
Proof.
intros a b H1 H2; rewrite Zplus_comm; apply Zeven_plus_Zodd; auto.
Qed.
Theorem Zodd_plus_Zodd: forall a b,
Zodd a -> Zodd b -> Zeven (a + b).
Proof.
intros a b H1 H2; case Zodd_ex with (1 := H1); intros x H3; try rewrite H3; auto.
case Zodd_ex with (1 := H2); intros y H4; try rewrite H4; auto.
replace ((2 * x + 1) + (2 * y + 1)) with (2 * (x + y + 1)); try apply Zeven_2p; auto.
(* ring part *)
do 2 rewrite Zmult_plus_distr_r; auto.
repeat rewrite <- Zplus_assoc; f_equal.
rewrite (Zplus_comm 1).
repeat rewrite <- Zplus_assoc; auto.
Qed.
Theorem Zeven_mult_Zeven_l: forall a b,
Zeven a -> Zeven (a * b).
Proof.
intros a b H1; case Zeven_ex with (1 := H1); intros x H3; try rewrite H3; auto.
replace (2 * x * b) with (2 * (x * b)); try apply Zeven_2p; auto with zarith.
(* ring part *)
apply Zmult_assoc.
Qed.
Theorem Zeven_mult_Zeven_r: forall a b,
Zeven b -> Zeven (a * b).
Proof.
intros a b H1; case Zeven_ex with (1 := H1); intros x H3; try rewrite H3; auto.
replace (a * (2 * x)) with (2 * (x * a)); try apply Zeven_2p; auto.
(* ring part *)
rewrite (Zmult_comm x a).
do 2 rewrite Zmult_assoc.
rewrite (Zmult_comm 2 a); auto.
Qed.
Hint Rewrite Zmult_plus_distr_r Zmult_plus_distr_l
Zplus_assoc Zmult_1_r Zmult_1_l : Zexpand.
Theorem Zodd_mult_Zodd: forall a b,
Zodd a -> Zodd b -> Zodd (a * b).
Proof.
intros a b H1 H2; case Zodd_ex with (1 := H1); intros x H3; try rewrite H3; auto.
case Zodd_ex with (1 := H2); intros y H4; try rewrite H4; auto.
replace ((2 * x + 1) * (2 * y + 1)) with (2 * (2 * x * y + x + y) + 1); try apply Zodd_2p_plus_1; auto.
(* ring part *)
autorewrite with Zexpand; f_equal.
repeat rewrite <- Zplus_assoc; f_equal.
repeat rewrite <- Zmult_assoc; f_equal.
repeat rewrite Zmult_assoc; f_equal; apply Zmult_comm.
Qed.
(* for compatibility *)
Close Scope Z_scope.
|
