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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** 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. *)
Local Open 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_split : forall n m,
(even (n + m) -> even n /\ even m \/ odd n /\ odd m)
with odd_plus_split : forall n m,
odd (n + m) -> odd n /\ even m \/ even n /\ odd m.
Proof.
intros. clear even_plus_split. destruct n; simpl in *.
auto with arith.
inversion_clear H;
apply odd_plus_split in H0 as [(H0,?)|(H0,?)]; auto with arith.
intros. clear odd_plus_split. destruct n; simpl in *.
auto with arith.
inversion_clear H;
apply even_plus_split in H0 as [(H0,?)|(H0,?)]; auto with arith.
Qed.
Lemma even_even_plus : forall n m, even n -> even m -> even (n + m)
with odd_plus_l : forall n m, odd n -> even m -> odd (n + m).
Proof.
intros n m [|] ?. trivial. apply even_S, odd_plus_l; trivial.
intros n m [] ?. apply odd_S, even_even_plus; trivial.
Qed.
Lemma odd_plus_r : forall n m, even n -> odd m -> odd (n + m)
with odd_even_plus : forall n m, odd n -> odd m -> even (n + m).
Proof.
intros n m [|] ?. trivial. apply odd_S, odd_even_plus; trivial.
intros n m [] ?. apply even_S, odd_plus_r; trivial.
Qed.
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.
split; split; auto using odd_plus_split, even_plus_split.
intros [[]|[]]; auto using odd_plus_r, odd_plus_l.
intros [[]|[]]; auto using even_even_plus, odd_even_plus.
Qed.
Lemma even_plus_even_inv_r : forall n m, even (n + m) -> even n -> even m.
Proof.
intros n m H; destruct (even_plus_split n m) as [[]|[]]; auto.
intro; destruct (not_even_and_odd n); auto.
Qed.
Lemma even_plus_even_inv_l : forall n m, even (n + m) -> even m -> even n.
Proof.
intros n m H; destruct (even_plus_split n m) as [[]|[]]; auto.
intro; destruct (not_even_and_odd m); auto.
Qed.
Lemma even_plus_odd_inv_r : forall n m, even (n + m) -> odd n -> odd m.
Proof.
intros n m H; destruct (even_plus_split n m) as [[]|[]]; auto.
intro; destruct (not_even_and_odd n); auto.
Qed.
Lemma even_plus_odd_inv_l : forall n m, even (n + m) -> odd m -> odd n.
Proof.
intros n m H; destruct (even_plus_split n m) as [[]|[]]; auto.
intro; destruct (not_even_and_odd m); auto.
Qed.
Hint Resolve even_even_plus odd_even_plus: arith.
Lemma odd_plus_even_inv_l : forall n m, odd (n + m) -> odd m -> even n.
Proof.
intros n m H; destruct (odd_plus_split n m) as [[]|[]]; auto.
intro; destruct (not_even_and_odd m); auto.
Qed.
Lemma odd_plus_even_inv_r : forall n m, odd (n + m) -> odd n -> even m.
Proof.
intros n m H; destruct (odd_plus_split n m) as [[]|[]]; auto.
intro; destruct (not_even_and_odd n); auto.
Qed.
Lemma odd_plus_odd_inv_l : forall n m, odd (n + m) -> even m -> odd n.
Proof.
intros n m H; destruct (odd_plus_split n m) as [[]|[]]; auto.
intro; destruct (not_even_and_odd m); auto.
Qed.
Lemma odd_plus_odd_inv_r : forall n m, odd (n + m) -> even n -> odd m.
Proof.
intros n m H; destruct (odd_plus_split n m) as [[]|[]]; auto.
intro; destruct (not_even_and_odd n); 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; 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.
|
