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'").


Lemma forall_and_dist T (P Q: T → Prop):
  (∀ x, P x ∧ Q x) ↔ (∀ x, P x) ∧ (∀ x, Q x).
Proof.
  repeat split; ins; try specialize (H x); desf.
Qed.

Lemma list_destruct_end: ∀ A (l: list A), l = nil ∨ ∃ l´ a, l = l´ ++ a :: nil.
  induction l; desf; eauto.
  by right; eexists nil, _; rewrite app_nil_l.
  eby right; eexists,; rewrite app_comm_cons.
Qed.

Ltac des_list_tail l l´ a :=
  let F := fresh in destruct (list_destruct_end l) as [ | [l´ F]]; [ | destruct F as [a]]; subst l.

Ltac find_in_list := (by repeat progress (rewrite ?In_app; ins); eauto 8).

Lemma map_is_nil: ∀ T1 T2 (f: T1 → T2) l, map f l = nil → l = nil.
Proof.
  by destruct l.
Qed.

Lemma not_or_and_iff : ∀ P Q, ¬ (P ∨ Q) ↔ ¬ P ∧ ¬ Q.
Proof.
  split; ins.
    by apply not_or_and.
  intro; desf.
Qed.

Lemma true_imp_b: ∀ (b : bool), (true → b) ↔ b = true.
Proof.
   by split; ins; rewrite H.
Qed.

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_two A: ∀ (x y: A), NoDup (x :: y :: nil) ↔ x ≠ y.
Proof.
  ins; rewrite nodup_cons; firstorder.
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 In_implies_perm A (x : A) l (IN: In x l) :
  ∃ l´, Permutation l (x :: l´).
Proof.
  induction l; ins; desf; eauto.
  destruct IHl; eauto using Permutation.
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 Permutation_filter_split T:
  ∀ (l: list T) (f: T → bool), Permutation l ((filter f l) ++ (filter (fun x ⇒ negb (f x)) l)).
Proof.
  ins; induction l; ins; desf; try done; ins.
  by apply perm_skip.
  by apply Permutation_cons_app.
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 NoDup_neq: ∀ {A} a b (l: list A), NoDup (a :: b :: l) → a ≠ b.
Proof.
  ins; intro; subst.
  rewrite <- app_nil_l in H; apply NoDup_remove_2 in H.
  by destruct H; rewrite app_nil_l; left.
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.

Lemma list_not_nil: ∀ {A} (l: list A), l ≠ nil ↔ ∃ a, In a l.
Proof.
  split.
  by induction l as [ | head l´]; ins; eexists; eauto.
  by ins; intro; desf.
Qed.

Lemma list_zero_one: ∀ {A} a (l: list A) (ND: NoDup l),
                     (∀ x (IN: In x l), x = a) ↔ l = nil ∨ l = a :: nil.
Proof.
  split; ins.
  × destruct l as [ | a´ l´]; [by left | right].
    f_equal.
    by apply H.
    destruct l´ as [ | b l´´]; [done | exfalso].
    exploit (H a´); exploit (H b); ins; try tauto.
    apply NoDup_neq in ND; congruence.
  × by desf; red in IN; desf.
Qed.

Lemma neq_contra: ∀ (x: nat), ¬ (x != x).
Proof.
  ins; intro.
  rewrite eqnn in H.
  unfold is_true in H.
  rewrite Bool.negb_true_iff in H.
  congruence.
Qed.

Lemma subset_perm: ∀ {A} (X Y: list A) (NDX: NoDup X) (NDY: NoDup Y)
                          (SUBSET: ∀ a, In a Y → In a X),
                   ∃ Y´, Permutation X (Y ++ Y´).
Proof.
  induction X as [ | x X´].
  × ins; ∃ nil.
    assert (YN: Y = nil).
    {
      apply NNPP; intro CONTRA.
      apply list_not_nil in CONTRA; desf.
      eby eapply SUBSET.
    }
    by subst; simpl; apply perm_nil.
  × ins.
    destruct (classic (In x Y)) as [IN | NotIN].
    + apply In_implies_perm in IN as [Y´ PERM]; desf.
      assert (SUBSET´: ∀ a, In a Y´ → In a X´).
      {
        ins.
        destruct (SUBSET a); try done.
        + symmetry in PERM; eapply Permutation_in; eauto.
          by right.
        + subst; exfalso.
          apply (Permutation_NoDup PERM ) in NDY.
          by rewrite nodup_cons in NDY; desf.
      }
      rewrite nodup_cons in NDX; desf.
      apply (Permutation_NoDup PERM ) in NDY.
      rewrite nodup_cons in NDY; desf.
      specialize (IHX´ _ NDX0 NDY0 SUBSET´); desf.
      ∃ Y´0.
      apply Permutation_cons with (a := x) in IHX´.
      assert (P := Permutation_app PERM (Permutation_refl Y´0)); simpls.
      by symmetry in P; eapply Permutation_trans; eauto.
    + assert (SUBSET´: ∀ a : A, In a Y → In a X´) by (ins; specialize (SUBSET a H); desf).
      rewrite nodup_cons in NDX; desf.
      specialize (IHX´ _ NDX0 NDY SUBSET´); desf.
      apply Permutation_cons_app with (a := x) in IHX´.
      by eexists; eauto.
Qed.

Require Import Wf Wf_nat.

Lemma first_exists:
  ∀ A (l: list A) phi (EX: ∃ a, In a l ∧ phi a),
  ∃ a´ l1 l2, l = l1 ++ a´ :: l2 ∧ phi a´ ∧ ∀ b (IN: In b l1), ¬ phi b.
Proof.
  intro.
  induction l using (well_founded_induction (well_founded_ltof _ (@length A))).
  ins; desf.
  apply In_split in EX; desf.
  destruct (classic (∃ a1, In a1 l1 ∧ phi a1)).
  × desf.
    exploit (H l1); eauto.
      by red; rewrite length_app; ins; omega.
    ins; desf.
    ∃ a´, l0, (l3 ++ a :: l2); split; eauto.
    by rewrite <- app_assoc, app_comm_cons.
  × eexists,,,; split; eauto.
Qed.

Lemma last_exists:
  ∀ A (l: list A) phi (EX: ∃ a, In a l ∧ phi a),
  ∃ a´ l1 l2, l = l1 ++ a´ :: l2 ∧ phi a´ ∧ ∀ b (IN: In b l2), ¬ phi b.
Proof.
  intro.
  induction l using (well_founded_induction (well_founded_ltof _ (@length A))).
  ins; desf.
  apply In_split in EX; desf.
  destruct (classic (∃ a2, In a2 l2 ∧ phi a2)).
  × desf.
    exploit (H l2); eauto.
      by red; rewrite length_app; ins; omega.
    ins; desf.
    ∃ a´, (l1 ++ a :: l0), l3; split; eauto.
    by rewrite <- app_assoc, app_comm_cons.
  × eexists,,,; split; eauto.
Qed.

Lemma map_eq A B:
  ∀ (f g: A → B) (l: list A) (fEQ: ∀ x (IN: In x l), f x = g x), map f l = map g l.
Proof.
  induction l; ins.
  f_equal.
    by apply fEQ; eauto.
  apply IHl; ins.
  by apply fEQ; eauto.
Qed.