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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Nota : this file is OBSOLETE, and left only for compatibility.
Please consider instead predicates [Nat.Even] and [Nat.Odd]
and Boolean functions [Nat.even] and [Nat.odd]. *)
(** 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. *)
Require Import PeanoNat.
Local Open Scope nat_scope.
Implicit Types m n : nat.
(** * Inductive definition of [even] and [odd] *)
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.
(** * Equivalence with predicates [Nat.Even] and [Nat.odd] *)
Lemma even_equiv : forall n, even n <-> Nat.Even n.
Proof.
fix even_equiv 1.
destruct n as [|[|n]]; simpl.
- split; [now exists 0 | constructor].
- split.
+ inversion_clear 1. inversion_clear H0.
+ now rewrite <- Nat.even_spec.
- rewrite Nat.Even_succ_succ, <- even_equiv.
split.
+ inversion_clear 1. now inversion_clear H0.
+ now do 2 constructor.
Qed.
Lemma odd_equiv : forall n, odd n <-> Nat.Odd n.
Proof.
fix odd_equiv 1.
destruct n as [|[|n]]; simpl.
- split.
+ inversion_clear 1.
+ now rewrite <- Nat.odd_spec.
- split; [ now exists 0 | do 2 constructor ].
- rewrite Nat.Odd_succ_succ, <- odd_equiv.
split.
+ inversion_clear 1. now inversion_clear H0.
+ now do 2 constructor.
Qed.
(** Basic facts *)
Lemma even_or_odd n : even n \/ odd n.
Proof.
induction n.
- auto with arith.
- elim IHn; auto with arith.
Qed.
Lemma even_odd_dec n : {even n} + {odd n}.
Proof.
induction n.
- auto with arith.
- elim IHn; auto with arith.
Defined.
Lemma not_even_and_odd n : even n -> odd n -> False.
Proof.
induction n.
- intros Ev Od. inversion Od.
- intros Ev Od. inversion Ev. inversion Od. auto with arith.
Qed.
(** * Facts about [even] & [odd] wrt. [plus] *)
Ltac parity2bool :=
rewrite ?even_equiv, ?odd_equiv, <- ?Nat.even_spec, <- ?Nat.odd_spec.
Ltac parity_binop_spec :=
rewrite ?Nat.even_add, ?Nat.odd_add, ?Nat.even_mul, ?Nat.odd_mul.
Ltac parity_binop :=
parity2bool; parity_binop_spec; unfold Nat.odd;
do 2 destruct Nat.even; simpl; tauto.
Lemma even_plus_split n m :
even (n + m) -> even n /\ even m \/ odd n /\ odd m.
Proof. parity_binop. Qed.
Lemma odd_plus_split n m :
odd (n + m) -> odd n /\ even m \/ even n /\ odd m.
Proof. parity_binop. Qed.
Lemma even_even_plus n m : even n -> even m -> even (n + m).
Proof. parity_binop. Qed.
Lemma odd_plus_l n m : odd n -> even m -> odd (n + m).
Proof. parity_binop. Qed.
Lemma odd_plus_r n m : even n -> odd m -> odd (n + m).
Proof. parity_binop. Qed.
Lemma odd_even_plus n m : odd n -> odd m -> even (n + m).
Proof. parity_binop. Qed.
Lemma even_plus_aux 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. parity_binop. Qed.
Lemma even_plus_even_inv_r n m : even (n + m) -> even n -> even m.
Proof. parity_binop. Qed.
Lemma even_plus_even_inv_l n m : even (n + m) -> even m -> even n.
Proof. parity_binop. Qed.
Lemma even_plus_odd_inv_r n m : even (n + m) -> odd n -> odd m.
Proof. parity_binop. Qed.
Lemma even_plus_odd_inv_l n m : even (n + m) -> odd m -> odd n.
Proof. parity_binop. Qed.
Lemma odd_plus_even_inv_l n m : odd (n + m) -> odd m -> even n.
Proof. parity_binop. Qed.
Lemma odd_plus_even_inv_r n m : odd (n + m) -> odd n -> even m.
Proof. parity_binop. Qed.
Lemma odd_plus_odd_inv_l n m : odd (n + m) -> even m -> odd n.
Proof. parity_binop. Qed.
Lemma odd_plus_odd_inv_r n m : odd (n + m) -> even n -> odd m.
Proof. parity_binop. Qed.
(** * Facts about [even] and [odd] wrt. [mult] *)
Lemma even_mult_aux n m :
(odd (n * m) <-> odd n /\ odd m) /\ (even (n * m) <-> even n \/ even m).
Proof. parity_binop. Qed.
Lemma even_mult_l n m : even n -> even (n * m).
Proof. parity_binop. Qed.
Lemma even_mult_r n m : even m -> even (n * m).
Proof. parity_binop. Qed.
Lemma even_mult_inv_r n m : even (n * m) -> odd n -> even m.
Proof. parity_binop. Qed.
Lemma even_mult_inv_l n m : even (n * m) -> odd m -> even n.
Proof. parity_binop. Qed.
Lemma odd_mult n m : odd n -> odd m -> odd (n * m).
Proof. parity_binop. Qed.
Lemma odd_mult_inv_l n m : odd (n * m) -> odd n.
Proof. parity_binop. Qed.
Lemma odd_mult_inv_r n m : odd (n * m) -> odd m.
Proof. parity_binop. Qed.
Hint Resolve
even_even_plus odd_even_plus odd_plus_l odd_plus_r
even_mult_l even_mult_r even_mult_l even_mult_r odd_mult : arith.
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