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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 $Id$ i*)
(** Here we define the predicates [even] and [odd] by mutual induction
and we prove the decidability and the exclusion of those predicates.
The main results about parity are proved in the module Div2. *)
Open Local Scope nat_scope.
Implicit Types m n : nat.
(** * Definition of [even] and [odd], and basic facts *)
Inductive even : nat -> Prop :=
| even_O : even 0
| even_S : forall n, odd n -> even (S n)
with odd : nat -> Prop :=
odd_S : forall n, even n -> odd (S n).
Hint Constructors even: arith.
Hint Constructors odd: arith.
Lemma even_or_odd : forall n, even n \/ odd n.
Proof.
induction n.
auto with arith.
elim IHn; auto with arith.
Qed.
Lemma even_odd_dec : forall n, {even n} + {odd n}.
Proof.
induction n.
auto with arith.
elim IHn; auto with arith.
Defined.
Lemma not_even_and_odd : forall n, even n -> odd n -> False.
Proof.
induction n.
intros even_0 odd_0. inversion odd_0.
intros even_Sn odd_Sn. inversion even_Sn. inversion odd_Sn. auto with arith.
Qed.
(** * Facts about [even] & [odd] wrt. [plus] *)
Lemma even_plus_aux :
forall n m,
(odd (n + m) <-> odd n /\ even m \/ even n /\ odd m) /\
(even (n + m) <-> even n /\ even m \/ odd n /\ odd m).
Proof.
intros n; elim n; simpl in |- *; auto with arith.
intros m; split; auto.
split.
intros H; right; split; auto with arith.
intros H'; case H'; auto with arith.
intros H'0; elim H'0; intros H'1 H'2; inversion H'1.
intros H; elim H; auto.
split; auto with arith.
intros H'; elim H'; auto with arith.
intros H; elim H; auto.
intros H'0; elim H'0; intros H'1 H'2; inversion H'1.
intros n0 H' m; elim (H' m); intros H'1 H'2; elim H'1; intros E1 E2; elim H'2;
intros E3 E4; clear H'1 H'2.
split; split.
intros H'0; case E3.
inversion H'0; auto.
intros H; elim H; intros H0 H1; clear H; auto with arith.
intros H; elim H; intros H0 H1; clear H; auto with arith.
intros H'0; case H'0; intros C0; case C0; intros C1 C2.
apply odd_S.
apply E4; left; split; auto with arith.
inversion C1; auto.
apply odd_S.
apply E4; right; split; auto with arith.
inversion C1; auto.
intros H'0.
case E1.
inversion H'0; auto.
intros H; elim H; intros H0 H1; clear H; auto with arith.
intros H; elim H; intros H0 H1; clear H; auto with arith.
intros H'0; case H'0; intros C0; case C0; intros C1 C2.
apply even_S.
apply E2; left; split; auto with arith.
inversion C1; auto.
apply even_S.
apply E2; right; split; auto with arith.
inversion C1; auto.
Qed.
Lemma even_even_plus : forall n m, even n -> even m -> even (n + m).
Proof.
intros n m; case (even_plus_aux n m).
intros H H0; case H0; auto.
Qed.
Lemma odd_even_plus : forall n m, odd n -> odd m -> even (n + m).
Proof.
intros n m; case (even_plus_aux n m).
intros H H0; case H0; auto.
Qed.
Lemma even_plus_even_inv_r : forall n m, even (n + m) -> even n -> even m.
Proof.
intros n m H; case (even_plus_aux n m).
intros H' H'0; elim H'0.
intros H'1; case H'1; auto.
intros H0; elim H0; auto.
intros H0 H1 H2; case (not_even_and_odd n); auto.
case H0; auto.
Qed.
Lemma even_plus_even_inv_l : forall n m, even (n + m) -> even m -> even n.
Proof.
intros n m H; case (even_plus_aux n m).
intros H' H'0; elim H'0.
intros H'1; case H'1; auto.
intros H0; elim H0; auto.
intros H0 H1 H2; case (not_even_and_odd m); auto.
case H0; auto.
Qed.
Lemma even_plus_odd_inv_r : forall n m, even (n + m) -> odd n -> odd m.
Proof.
intros n m H; case (even_plus_aux n m).
intros H' H'0; elim H'0.
intros H'1; case H'1; auto.
intros H0 H1 H2; case (not_even_and_odd n); auto.
case H0; auto.
intros H0; case H0; auto.
Qed.
Lemma even_plus_odd_inv_l : forall n m, even (n + m) -> odd m -> odd n.
Proof.
intros n m H; case (even_plus_aux n m).
intros H' H'0; elim H'0.
intros H'1; case H'1; auto.
intros H0 H1 H2; case (not_even_and_odd m); auto.
case H0; auto.
intros H0; case H0; auto.
Qed.
Hint Resolve even_even_plus odd_even_plus: arith.
Lemma odd_plus_l : forall n m, odd n -> even m -> odd (n + m).
Proof.
intros n m; case (even_plus_aux n m).
intros H; case H; auto.
Qed.
Lemma odd_plus_r : forall n m, even n -> odd m -> odd (n + m).
Proof.
intros n m; case (even_plus_aux n m).
intros H; case H; auto.
Qed.
Lemma odd_plus_even_inv_l : forall n m, odd (n + m) -> odd m -> even n.
Proof.
intros n m H; case (even_plus_aux n m).
intros H' H'0; elim H'.
intros H'1; case H'1; auto.
intros H0 H1 H2; case (not_even_and_odd m); auto.
case H0; auto.
intros H0; case H0; auto.
Qed.
Lemma odd_plus_even_inv_r : forall n m, odd (n + m) -> odd n -> even m.
Proof.
intros n m H; case (even_plus_aux n m).
intros H' H'0; elim H'.
intros H'1; case H'1; auto.
intros H0; case H0; auto.
intros H0 H1 H2; case (not_even_and_odd n); auto.
case H0; auto.
Qed.
Lemma odd_plus_odd_inv_l : forall n m, odd (n + m) -> even m -> odd n.
Proof.
intros n m H; case (even_plus_aux n m).
intros H' H'0; elim H'.
intros H'1; case H'1; auto.
intros H0; case H0; auto.
intros H0 H1 H2; case (not_even_and_odd m); auto.
case H0; auto.
Qed.
Lemma odd_plus_odd_inv_r : forall n m, odd (n + m) -> even n -> odd m.
Proof.
intros n m H; case (even_plus_aux n m).
intros H' H'0; elim H'.
intros H'1; case H'1; auto.
intros H0 H1 H2; case (not_even_and_odd n); auto.
case H0; auto.
intros H0; case H0; auto.
Qed.
Hint Resolve odd_plus_l odd_plus_r: arith.
(** * Facts about [even] and [odd] wrt. [mult] *)
Lemma even_mult_aux :
forall n m,
(odd (n * m) <-> odd n /\ odd m) /\ (even (n * m) <-> even n \/ even m).
Proof.
intros n; elim n; simpl in |- *; auto with arith.
intros m; split; split; auto with arith.
intros H'; inversion H'.
intros H'; elim H'; auto.
intros n0 H' m; split; split; auto with arith.
intros H'0.
elim (even_plus_aux m (n0 * m)); intros H'3 H'4; case H'3; intros H'1 H'2;
case H'1; auto.
intros H'5; elim H'5; intros H'6 H'7; auto with arith.
split; auto with arith.
case (H' m).
intros H'8 H'9; case H'9.
intros H'10; case H'10; auto with arith.
intros H'11 H'12; case (not_even_and_odd m); auto with arith.
intros H'5; elim H'5; intros H'6 H'7; case (not_even_and_odd (n0 * m)); auto.
case (H' m).
intros H'8 H'9; case H'9; auto.
intros H'0; elim H'0; intros H'1 H'2; clear H'0.
elim (even_plus_aux m (n0 * m)); auto.
intros H'0 H'3.
elim H'0.
intros H'4 H'5; apply H'5; auto.
left; split; auto with arith.
case (H' m).
intros H'6 H'7; elim H'7.
intros H'8 H'9; apply H'9.
left.
inversion H'1; auto.
intros H'0.
elim (even_plus_aux m (n0 * m)); intros H'3 H'4; case H'4.
intros H'1 H'2.
elim H'1; auto.
intros H; case H; auto.
intros H'5; elim H'5; intros H'6 H'7; auto with arith.
left.
case (H' m).
intros H'8; elim H'8.
intros H'9; elim H'9; auto with arith.
intros H'0; elim H'0; intros H'1.
case (even_or_odd m); intros H'2.
apply even_even_plus; auto.
case (H' m).
intros H H0; case H0; auto.
apply odd_even_plus; auto.
inversion H'1; case (H' m); auto.
intros H1; case H1; auto.
apply even_even_plus; auto.
case (H' m).
intros H H0; case H0; auto.
Qed.
Lemma even_mult_l : forall n m, even n -> even (n * m).
Proof.
intros n m; case (even_mult_aux n m); auto.
intros H H0; case H0; auto.
Qed.
Lemma even_mult_r : forall n m, even m -> even (n * m).
Proof.
intros n m; case (even_mult_aux n m); auto.
intros H H0; case H0; auto.
Qed.
Hint Resolve even_mult_l even_mult_r: arith.
Lemma even_mult_inv_r : forall n m, even (n * m) -> odd n -> even m.
Proof.
intros n m H' H'0.
case (even_mult_aux n m).
intros H'1 H'2; elim H'2.
intros H'3; elim H'3; auto.
intros H; case (not_even_and_odd n); auto.
Qed.
Lemma even_mult_inv_l : forall n m, even (n * m) -> odd m -> even n.
Proof.
intros n m H' H'0.
case (even_mult_aux n m).
intros H'1 H'2; elim H'2.
intros H'3; elim H'3; auto.
intros H; case (not_even_and_odd m); auto.
Qed.
Lemma odd_mult : forall n m, odd n -> odd m -> odd (n * m).
Proof.
intros n m; case (even_mult_aux n m); intros H; case H; auto.
Qed.
Hint Resolve even_mult_l even_mult_r odd_mult: arith.
Lemma odd_mult_inv_l : forall n m, odd (n * m) -> odd n.
Proof.
intros n m H'.
case (even_mult_aux n m).
intros H'1 H'2; elim H'1.
intros H'3; elim H'3; auto.
Qed.
Lemma odd_mult_inv_r : forall n m, odd (n * m) -> odd m.
Proof.
intros n m H'.
case (even_mult_aux n m).
intros H'1 H'2; elim H'1.
intros H'3; elim H'3; auto.
Qed.
|
