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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
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(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
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(** Decidability results about lists *)

Require Import List Decidable.
Set Implicit Arguments.

Definition decidable_eq A := forall x y:A, decidable (x=y).

Section Dec_in_Prop.
Variables (A:Type)(dec:decidable_eq A).

Lemma In_decidable x (l:list A) : decidable (In x l).
Proof using A dec.
 induction l as [|a l IH].
 - now right.
 - destruct (dec a x).
   + left. now left.
   + destruct IH; simpl; [left|right]; tauto.
Qed.

Lemma incl_decidable (l l':list A) : decidable (incl l l').
Proof using A dec.
 induction l as [|a l IH].
 - left. inversion 1.
 - destruct (In_decidable a l') as [IN|IN].
   + destruct IH as [IC|IC].
     * left. destruct 1; subst; auto.
     * right. contradict IC. intros x H. apply IC; now right.
   + right. contradict IN. apply IN; now left.
Qed.

Lemma NoDup_decidable (l:list A) : decidable (NoDup l).
Proof using A dec.
 induction l as [|a l IH].
 - left; now constructor.
 - destruct (In_decidable a l).
   + right. inversion_clear 1. tauto.
   + destruct IH.
     * left. now constructor.
     * right. inversion_clear 1. tauto.
Qed.

End Dec_in_Prop.

Section Dec_in_Type.
Variables (A:Type)(dec : forall x y:A, {x=y}+{x<>y}).

Definition In_dec := List.In_dec dec. (* Already in List.v *)

Lemma incl_dec (l l':list A) : {incl l l'}+{~incl l l'}.
Proof using A dec.
 induction l as [|a l IH].
 - left. inversion 1.
 - destruct (In_dec a l') as [IN|IN].
   + destruct IH as [IC|IC].
     * left. destruct 1; subst; auto.
     * right. contradict IC. intros x H. apply IC; now right.
   + right. contradict IN. apply IN; now left.
Qed.

Lemma NoDup_dec (l:list A) : {NoDup l}+{~NoDup l}.
Proof using A dec.
 induction l as [|a l IH].
 - left; now constructor.
 - destruct (In_dec a l).
   + right. inversion_clear 1. tauto.
   + destruct IH.
     * left. now constructor.
     * right. inversion_clear 1. tauto.
Qed.

End Dec_in_Type.

(** An extra result: thanks to decidability, a list can be purged
    from redundancies. *)

Lemma uniquify_map A B (d:decidable_eq B)(f:A->B)(l:list A) :
 exists l', NoDup (map f l') /\ incl (map f l) (map f l').
Proof.
 induction l.
 - exists nil. simpl. split; [now constructor | red; trivial].
 - destruct IHl as (l' & N & I).
   destruct (In_decidable d (f a) (map f l')).
   + exists l'; simpl; split; trivial.
     intros x [Hx|Hx]. now subst. now apply I.
   + exists (a::l'); simpl; split.
     * now constructor.
     * intros x [Hx|Hx]. subst; now left. right; now apply I.
Qed.

Lemma uniquify A (d:decidable_eq A)(l:list A) :
 exists l', NoDup l' /\ incl l l'.
Proof.
 destruct (uniquify_map d id l) as (l',H).
 exists l'. now rewrite !map_id in H.
Qed.
