Require Import List Vbase Varith Vlistbase Vlist.
Require Import Permutation Classical.
Set Implicit Arguments.

Notation sval := (@proj1_sig _ _).
Notation "@ 'sval'" := (@proj1_sig) (at level 10, format "@ 'sval'").

Definition disjoint A (l1 l2 : list A) :=
  ∀ a (IN1: In a l1) (IN2: In a l2), False.

Lemma nodup_one A (x: A) : NoDup (x :: nil).
Proof. vauto. Qed.
Hint Resolve NoDup_nil nodup_one.

Lemma nodup_map:
  ∀ (A B: Type) (f: A → B) (l: list A),
  NoDup l →
  (∀ x y, In x l → In y l → x ≠ y → f x ≠ f y) →
  NoDup (map f l).
Proof.
  induction 1; ins; vauto.
  constructor; eauto.
  intro; rewrite In_map in *; desf.
  edestruct H1; try eapply H2; eauto.
  intro; desf.
Qed.

Lemma nodup_append_commut:
  ∀ (A: Type) (a b: list A),
  NoDup (a ++ b) → NoDup (b ++ a).
Proof.
  intro A.
  assert (∀ (x: A) (b: list A) (a: list A),
           NoDup (a ++ b) → ~(In x a) → ~(In x b) →
           NoDup (a ++ x :: b)).
    induction a; simpl; intros.
    constructor; auto.
    inversion H. constructor. red; intro.
    elim (in_app_or _ _ _ H6); intro.
    elim H4. apply in_or_app. tauto.
    elim H7; intro. subst a. elim H0. left. auto.
    elim H4. apply in_or_app. tauto.
    auto.
  induction a; simpl; intros.
  rewrite <- app_nil_end. auto.
  inversion H0. apply H. auto.
  red; intro; elim H3. apply in_or_app. tauto.
  red; intro; elim H3. apply in_or_app. tauto.
Qed.

Lemma nodup_cons A (x: A) l:
  NoDup (x :: l) ↔ ¬ In x l ∧ NoDup l.
Proof. split; inversion 1; vauto. Qed.

Lemma nodup_app:
  ∀ (A: Type) (l1 l2: list A),
  NoDup (l1 ++ l2) ↔
  NoDup l1 ∧ NoDup l2 ∧ disjoint l1 l2.
Proof.
  induction l1; ins.
    by split; ins; desf; vauto.
  rewrite !nodup_cons, IHl1, In_app; unfold disjoint.
  ins; intuition (subst; eauto).
Qed.

Lemma nodup_append:
  ∀ (A: Type) (l1 l2: list A),
  NoDup l1 → NoDup l2 → disjoint l1 l2 →
  NoDup (l1 ++ l2).
Proof.
  generalize nodup_app; firstorder.
Qed.

Lemma nodup_append_right:
  ∀ (A: Type) (l1 l2: list A),
  NoDup (l1 ++ l2) → NoDup l2.
Proof.
  generalize nodup_app; firstorder.
Qed.

Lemma nodup_append_left:
  ∀ (A: Type) (l1 l2: list A),
  NoDup (l1 ++ l2) → NoDup l1.
Proof.
  generalize nodup_app; firstorder.
Qed.



Definition mupd (A: eqType) B (h : A → B) y z :=
  fun x ⇒ if x == y then z else h x.
Arguments mupd [A B] h y z x.

Lemma mupds (A: eqType) B (f: A → B) x y : mupd f x y x = y.
Proof. by unfold mupd; desf; desf. Qed.

Lemma mupdo (A: eqType) B (f: A → B) x y z : x ≠ z → mupd f x y z = f z.
Proof. by unfold mupd; desf; desf. Qed.


Lemma In_perm A l l' (P: perm_eq (T:=A) l l') x : In x l ↔ In x l'.
Proof.
  by split; ins; apply/inP; instantiate;
     [rewrite <- (perm_eq_mem P)|rewrite (perm_eq_mem P)]; apply/inP.
Qed.

Lemma nodup_perm A l l' (P: perm_eq (T:=A) l l') : NoDup l ↔ NoDup l'.
Proof.
  by split; ins; apply/uniqP; instantiate;
     [rewrite <- (perm_eq_uniq P)|rewrite (perm_eq_uniq P)]; apply/uniqP.
Qed.

Lemma In_mem_eq (A: eqType) (l l': list A) (P: l =i l') x : In x l ↔ In x l'.
Proof.
  by split; ins; apply/inP; instantiate; [rewrite <- P | rewrite P]; apply/inP.
Qed.

Lemma NoDup_filter A (l: list A) (ND: NoDup l) f : NoDup (filter f l).
Proof.
  induction l; ins; inv ND; desf; eauto using NoDup.
  econstructor; eauto; rewrite In_filter; tauto.
Qed.

Hint Resolve NoDup_filter.

Lemma NoDup_eq_one A (x : A) l :
   NoDup l → In x l → (∀ y (IN: In y l), y = x) → l = x :: nil.
Proof.
  destruct l; ins; f_equal; eauto.
  inv H; desf; clear H H5; induction l; ins; desf; case H4; eauto using eq_sym.
  rewrite IHl in H0; ins; desf; eauto.
Qed.


Lemma map_perm :
  ∀ A B (f: A → B) l l', Permutation l l' → Permutation (map f l) (map f l').
Proof.
  induction 1; eauto using Permutation.
Qed.

Lemma perm_from_subset :
  ∀ A (l : list A) l',
    NoDup l' →
    (∀ x, In x l' → In x l) →
    ∃ l'', Permutation l (l' ++ l'').
Proof.
  induction l; ins; vauto.
    by destruct l'; ins; vauto; exfalso; eauto.
  destruct (classic (In a l')).

    eapply In_split in H1; desf; rewrite ?nodup_app, ?nodup_cons in *; desf.
    destruct (IHl (l1 ++ l2)); ins.
      by rewrite ?nodup_app, ?nodup_cons in *; desf; repeat split; ins; red; eauto using In_cons.
      by specialize (H0 x); rewrite In_app in *; ins; desf;
         destruct (classic (a = x)); subst; try tauto; exfalso; eauto using In_eq.
    eexists; rewrite appA in *; ins.
    by eapply Permutation_trans, Permutation_middle; eauto.

  destruct (IHl l'); eauto; ins.
    by destruct (H0 x); auto; ins; subst.
  by eexists (a :: _); eapply Permutation_trans, Permutation_middle; eauto.
Qed.

Lemma Permutation_NoDup A ( l l' : list A) :
  Permutation l l' → NoDup l → NoDup l'.
Proof.
  induction 1; eauto; rewrite !nodup_cons in *; ins; desf; intuition.
  eby symmetry in H; eapply H0; eapply Permutation_in.
Qed.

Lemma NoDup_mapD A B (f : A→ B) l :
  NoDup (map f l) → NoDup l.
Proof.
  induction l; ins; rewrite !nodup_cons, In_map in *; desf; eauto 8.
Qed.

Lemma olast_inv A l x :
  olast (T:=A) l = Some x → ∃ prefix, l = prefix ++ x :: nil.
Proof.
  destruct l; ins; desf; induction[a] l; ins; [by ∃ nil|].
  by specialize (IHl a); desf; ∃ (a0 :: prefix); ins; f_equal.
Qed.

Lemma In_flatten A (x:A) l :
  In x (flatten l) ↔ ∃ y, In x y ∧ In y l.
Proof.
  induction l; ins. by split; ins; desf.
  rewrite flatten_cons, In_app, IHl; clear; split; ins; desf; eauto.
Qed.