Require Import Nat.
Require Import PeanoNat.
Require Import List.
Import ListNotations.


Fixpoint Subsequence {X: Type} (l s : list X) : Prop :=
  match s with
  | nil    => True
  | hd::tl => exists l1 l2, l = l1 ++ (hd::l2) /\ Subsequence l2 tl
  end.



Theorem Subsequence_nil_r {X: Type} : forall (l : list X), Subsequence l nil.
Proof.
  intro l. reflexivity.
Qed.


Theorem Subsequence_nil_cons_r {X: Type}: forall (l: list X) (a:X),
  ~ Subsequence nil (a::l).
Proof.
  intros l a. intro H. destruct H. destruct H. destruct H.
  destruct x; apply nil_cons in H; contradiction H.
Qed.


Theorem Subsequence_cons_l {X: Type} : forall (l s: list X) (a: X),
  Subsequence l s -> Subsequence (a::l) s.
Proof.
  intros l s a. intro H. destruct s. apply Subsequence_nil_r.
  destruct H. destruct H. destruct H. exists (a::x0). exists x1. rewrite H.
  split; easy.
Qed.


Theorem Subsequence_cons_r {X: Type} : forall (l s: list X) (a: X),
  Subsequence l (a::s) -> Subsequence l s.
Proof.
  intros l s a. intro H. simpl in H. destruct H. destruct H.
  destruct s. apply Subsequence_nil_r. destruct H. simpl in H0.
  destruct H0. destruct H0. destruct H0.
  exists (x ++ a::x2). exists x3.
  rewrite H. rewrite H0. split. rewrite <- app_assoc.
  rewrite app_comm_cons. reflexivity. assumption.
Qed.


Theorem Subsequence_cons_eq {X: Type} : forall (l1 l2: list X) (a: X),
  Subsequence (a::l1) (a::l2) <-> Subsequence l1 l2.
Proof.
  intros l s a. split; intro H.
  destruct H. destruct H. destruct H.
  destruct x. inversion H. assumption.
  destruct s. apply Subsequence_nil_r.
  destruct H0. destruct H0. destruct H0.
  exists (x1 ++ (a::x3)). exists x4.
  inversion H. rewrite H0. rewrite <- app_assoc.
  rewrite app_comm_cons. split; easy.
  exists nil. exists l. split; easy.
Qed.


Lemma Subsequence_alt_defs {X: Type} :
  forall l s : list X,
  (exists (l1: list X) (l2 : list (list X)),
    length s = length l2
    /\ l = l1 ++ flat_map (fun e => (fst e) :: (snd e)) (combine s l2))
    ->
  (exists (t: list bool),
    length t = length l /\ s = map snd (filter fst (combine t l))).
Proof.
  intros l s. intro H. destruct H. destruct H. destruct H.

  exists (
    (repeat false (length x)) ++
    (flat_map (fun e => true :: (repeat false (length e))) x0)).
  split.
  rewrite H0. rewrite app_length. rewrite app_length. rewrite repeat_length.
  rewrite Nat.add_cancel_l. rewrite flat_map_length. rewrite flat_map_length.

  assert (forall v (u: list X),
    length u = length v
    -> map (fun e => length (fst e :: snd e)) (combine u v)
       = map (fun e => length (true :: repeat false (length e))) v).
  intro v. induction v; intro u; intro I.
  apply length_zero_iff_nil in I. rewrite I. reflexivity.
  destruct u. apply O_S in I. contradiction I.
  simpl. rewrite IHv. rewrite repeat_length. reflexivity.
  inversion I. reflexivity.

  rewrite H1; inversion H; reflexivity. rewrite H0.
  assert (forall (u: list X) v w,
    filter fst (combine (repeat false (length u) ++ v) (u ++ w))
      = filter fst (combine v w)).
  intro u. induction u; intros v w. reflexivity. simpl. apply IHu.

  assert (forall (v: list (list X)) (u : list X),
    length u = length v
    -> u = map snd (filter fst (combine
         (flat_map (fun e => true:: repeat false (length e)) v)
         (flat_map (fun e => fst e :: snd e) (combine u v))))).
  intro v. induction v; intro u; intro I.
  apply length_zero_iff_nil in I. rewrite I. reflexivity.
  destruct u. apply O_S in I. contradiction I.
  simpl. rewrite H1. rewrite <- IHv; inversion I; reflexivity.
  rewrite H1. rewrite <- H2; inversion H; reflexivity.
Qed.


Lemma Subsequence_alt_defs2 {X: Type} :
  forall l s : list X,
  (exists (t: list bool),
    length t = length l /\ s = map snd (filter fst (combine t l)))
  -> Subsequence l s.
Proof.
  intros l s. intro H. destruct H. destruct H.

  assert (I: forall u (v w: list X),
    length u = length w
    -> v = map snd (filter fst (combine u w))
    -> Subsequence w v).
  intro u. induction u; intros v w; intros I J.
  rewrite J. apply Subsequence_nil_r.
  destruct v. apply Subsequence_nil_r.
  destruct w. apply Nat.neq_succ_0 in I. contradiction I.
  destruct a. exists nil. exists w. inversion J.
  apply IHu in H3. split; inversion J; try rewrite <- H5; easy.
  inversion I. reflexivity.
  assert (Subsequence w (x0::v)). apply IHu; inversion I; easy.
  apply Subsequence_cons_l; assumption.
  apply I with (u := x); assumption.
Qed.


Theorem Subsequence_flat_map {X: Type} :
  forall l s : list X, Subsequence l s
  <-> exists (l1: list X) (l2 : list (list X)),
    length s = length l2
    /\ l = l1 ++ flat_map (fun e => (fst e) :: (snd e)) (combine s l2).
Proof.
  intros l s. split. generalize l. induction s; intro l0; intro H.
  exists l0. exists nil. split. reflexivity. rewrite app_nil_r.
  reflexivity. destruct H. destruct H. destruct H. apply IHs in H0.
  destruct H0. destruct H0. destruct H0. exists x. exists (x1::x2).
  split; simpl; [rewrite H0 | rewrite <- H1]; easy.
  intro. apply Subsequence_alt_defs in H. apply Subsequence_alt_defs2.
  assumption.
Qed.


Theorem Subsequence_bools {X: Type} :
  forall l s : list X, Subsequence l s
  <-> (exists (t: list bool),
    length t = length l /\ s = map snd (filter fst (combine t l))).
Proof.
  intros l s. split; intro. apply Subsequence_alt_defs.
  apply Subsequence_flat_map. assumption.
  apply Subsequence_alt_defs2. assumption.
Qed.


Theorem Subsequence_dec {X: Type}:
  (forall x y : X, {x = y} + {x <> y})
  -> forall (l s : list X), { Subsequence l s } + { ~ Subsequence l s }.
Proof.
  intro H. intro l. induction l; intro s. destruct s. left.
  apply Subsequence_nil_r. right. apply Subsequence_nil_cons_r.

  assert ({ Subsequence l s} + { ~ Subsequence l s }).
  apply IHl. destruct H0.

  rewrite <- Subsequence_cons_eq with (a := a) in s0.
  apply Subsequence_cons_r in s0. apply Subsequence_bools in s0.
  left. apply Subsequence_bools. assumption.

  destruct s. left. apply Subsequence_nil_r.
  assert ({x=a}+{x<>a}). apply H. destruct H0. rewrite e.
  destruct IHl with (s := s); [ left | right ];
  rewrite Subsequence_cons_eq. assumption. assumption.

  right. intro I. apply Subsequence_bools in I.
  destruct I. destruct H0. destruct x0.
  symmetry in H1. apply nil_cons in H1. contradiction H1.
  destruct b. inversion H1. rewrite H3 in n0. easy.
  apply n. apply Subsequence_bools. exists x0; split; inversion H0; easy.
Qed.


(** * Various general properties
*)

Theorem Subsequence_id {X: Type} :
  forall u : list X, Subsequence u u.
Proof.
  intro u. induction u. easy. exists nil. exists u. split; easy.
Qed.


Theorem Subsequence_app {X: Type} :
  forall l1 s1 l2 s2 : list X,
  Subsequence l1 s1 -> Subsequence l2 s2 -> Subsequence (l1++l2) (s1++s2).
Proof.
  intros l1 s1 l2 s2. intros H I. apply Subsequence_bools.
  apply Subsequence_bools in H. apply Subsequence_bools in I.
  destruct H. destruct I. exists (x++x0). destruct H. destruct H0.
  split. rewrite app_length. rewrite app_length.
  rewrite H. rewrite H0. reflexivity.
  rewrite H1. rewrite H2.

  assert (J: forall (t : list bool) (u : list X) (v: list bool) (w : list X),
    length t = length u
    -> combine (t++v) (u++w) = (combine t u) ++ (combine v w)).
  intros t u v w. generalize u. induction t; intro u0; intro K.
  replace u0 with (nil : list X). reflexivity.
  destruct u0. reflexivity. apply O_S in K. contradiction K.
  destruct u0. apply PeanoNat.Nat.neq_succ_0 in K. contradiction K.
  simpl. rewrite IHt. reflexivity. inversion K. reflexivity.

  rewrite J. rewrite filter_app. rewrite map_app. reflexivity.
  assumption.
Qed.


Theorem Subsequence_trans {X: Type} :
  forall (l1 l2 l3: list X),
  Subsequence l1 l2 -> Subsequence l2 l3 -> Subsequence l1 l3.
Proof.
  intros l1 l2 l3. intros H I. apply Subsequence_bools.
  apply Subsequence_flat_map in H. apply Subsequence_bools in I.
  destruct H. destruct H. destruct H. destruct I. destruct H1.
  exists (
    (repeat false (length x)) ++
    (flat_map (fun e => (fst e) :: (repeat false (length (snd e))))
              (combine x1 x0))).
  split.

  rewrite app_length. rewrite repeat_length.
  rewrite H0. rewrite app_length. rewrite Nat.add_cancel_l.
  rewrite flat_map_length. rewrite flat_map_length.
  assert (forall (u: list X) (v: list (list X)),
    length u = length v
    -> list_sum (map (fun z => length (fst z :: snd z))
                     (combine u v))
       = list_sum (map (fun z => S (length z)) v)).
  intros u v. generalize u. induction v; intro u0; intro I.
  apply length_zero_iff_nil in I. rewrite I. reflexivity.
  destruct u0. apply O_S in I. contradiction I.
  simpl. rewrite IHv. reflexivity. inversion I. reflexivity.
  rewrite H3.
  assert (forall (u: list bool) (v: list (list X)),
    length u = length v
    -> list_sum (map (fun z => length (fst z :: repeat false (length (snd z))))
                     (combine u v))
       = list_sum (map (fun z => S (length z)) v)).
  intros u v. generalize u.
  induction v; intro u0; intro I; destruct u0. reflexivity.
  apply PeanoNat.Nat.neq_succ_0 in I. contradiction I.
  apply O_S in I. contradiction I.
  simpl. rewrite IHv. rewrite repeat_length. reflexivity.
  inversion I. reflexivity. rewrite H4. reflexivity.
  rewrite H1. rewrite H. reflexivity. assumption.

  assert (K: forall (w v: list X) u,
    filter fst (combine ((repeat false (length w)) ++ u)
                        (w ++ v)) = filter fst (combine u v)).
  intro w0. induction w0. reflexivity. apply IHw0.
  rewrite H2. rewrite H0.
  assert (forall (u v: list X) (w: list bool),
    map snd (filter fst (combine ((repeat false (length u)) ++ w) (u ++ v)))
      = map snd (filter fst (combine w v))).
  intro u. induction u; intros v w. reflexivity.
  apply IHu. rewrite H3.
  assert (forall (u: list bool) (v: list X) (w: list (list X)),
    length u = length w
    -> length u = length v
    -> filter fst (combine
                      (flat_map
                         (fun e => fst e:: (repeat false (length (snd e))))
                         (combine u w))
                      (flat_map (fun e => fst e :: snd e) (combine v w)))
         = filter fst (combine u v)).
  intros u v w. generalize u. generalize v.
  induction w; intros v0 u0; intros I J.
  apply length_zero_iff_nil in I. rewrite I. reflexivity.
  destruct u0. reflexivity. destruct v0.
  apply PeanoNat.Nat.neq_succ_0 in J; contradiction J.
  destruct b; simpl; rewrite K; rewrite IHw;
  inversion I; inversion J; reflexivity.
  rewrite H4; try rewrite H1; try rewrite H; reflexivity.
Qed.


Theorem Subsequence_length {X: Type} :
  forall (u v: list X), Subsequence u v -> length v <= length u.
Proof.
  intros u v. generalize u. induction v; intro u0; intro H. apply Nat.le_0_l.
  destruct H. destruct H. destruct H.
  apply IHv in H0. simpl. apply Nat.le_succ_l. rewrite H.
  rewrite app_length. apply Nat.lt_lt_add_l.
  rewrite <- Nat.le_succ_l. simpl. rewrite <- Nat.succ_le_mono.
  assumption.
Qed.


Theorem Subsequence_eq {X: Type} :
  forall (u v: list X),
  Subsequence u v -> Subsequence v u -> u = v.
Proof.
  intro u. induction u; intro v; intros H I.

  destruct v. reflexivity.
  apply Subsequence_nil_cons_r in H. contradiction H.
  destruct v. apply Subsequence_nil_cons_r in I. contradiction I.

  apply Subsequence_bools in H. apply Subsequence_bools in I.

  destruct H. destruct H. destruct I. destruct H1.
  destruct x0. apply O_S in H. contradiction H.
  destruct x1. apply O_S in H1. contradiction H1.
  destruct b; destruct b0.

  assert (Subsequence u v). apply Subsequence_bools. exists x0.
  split; inversion H; inversion H0; reflexivity.
  assert (Subsequence v u). apply Subsequence_bools. exists x1.
  split; inversion H1; inversion H2; reflexivity.
  apply IHu in H4. rewrite H4. inversion H0. reflexivity. assumption.

  assert (Subsequence u v). apply Subsequence_bools. exists x0.
  split; inversion H; inversion H0; reflexivity.
  assert (Subsequence v (a::u)). apply Subsequence_bools. exists x1.
  split; inversion H1; inversion H2; reflexivity.
  assert (Subsequence u (a::u)).
  apply Subsequence_trans with (l2 := v); assumption.
  apply Subsequence_length in H5. apply Nat.nle_succ_diag_l in H5.
  contradiction.

  assert (Subsequence u (x::v)). apply Subsequence_bools. exists x0.
  split; inversion H; inversion H0; reflexivity.
  assert (Subsequence v u). apply Subsequence_bools. exists x1.
  split; inversion H1; inversion H2; reflexivity.
  assert (Subsequence v (x::v)). apply Subsequence_trans with (l2 := u);
  assumption.
  apply Subsequence_length in H5. apply Nat.nle_succ_diag_l in H5.
  contradiction H5.

  assert (Subsequence u (x::v)). apply Subsequence_bools. exists x0.
  split; inversion H; inversion H0; reflexivity.
  assert (Subsequence v (a::u)). apply Subsequence_bools. exists x1.
  split; inversion H1; inversion H2; reflexivity.
  apply Subsequence_cons_r in H3. apply Subsequence_cons_r in H4.
  apply IHu in H4. rewrite H4 in H0. rewrite H4 in H.
  simpl in H0.
  assert (Subsequence v (x::v)). apply Subsequence_bools. exists x0.
  split. inversion H. reflexivity. assumption.
  apply Subsequence_length in H5. apply Nat.nle_succ_diag_l in H5.
  contradiction H5. assumption.
Qed.


Theorem Subsequence_length_eq {X: Type} :
  forall (u v: list X),
  Subsequence u v -> length u = length v -> u = v.
Proof.
  intros u v. intros H I. apply Subsequence_bools in H.
  destruct H. destruct H. rewrite H0 in I.
  rewrite map_length in I.
  replace (length u) with (length (combine x u)) in I.
  symmetry in I. apply filter_length_forallb in I.
  apply forallb_filter_id in I. rewrite I in H0.

  assert (J: forall (a: list bool) (b: list X),
    length a = length b -> b = map snd (combine a b)).
  intro a. induction a; intro b; intro J.
  symmetry in J. apply length_zero_iff_nil in J. rewrite J. reflexivity.
  destruct b. inversion J. simpl. rewrite <- IHa. reflexivity.
  inversion J. reflexivity. rewrite H0. apply J. assumption.

  rewrite combine_length. rewrite H. apply PeanoNat.Nat.min_id.
Qed.


 Theorem Subsequence_in {X: Type} :
   forall (u: list X) a, In a u <-> Subsequence u [a].
 Proof.
   intros u a. split. induction u; intro H.
   apply in_nil in H. contradiction.
   destruct H. rewrite H. exists nil. exists u.
   split. reflexivity. apply Subsequence_nil_r.
   apply Subsequence_cons_l. apply IHu. assumption.
   intro H. destruct H. destruct H. destruct H. rewrite H.
   apply in_or_app. right. apply in_eq.
Qed.


Theorem Subsequence_rev {X: Type} :
  forall (u v: list X), Subsequence u v <-> Subsequence (rev u) (rev v).
Proof.
  assert (MAIN: forall (u v: list X),
      Subsequence u v -> Subsequence (rev u) (rev v)).
  intros u v. intro H.
  apply Subsequence_bools. apply Subsequence_bools in H.
  destruct H. destruct H.
  exists (rev x). split. rewrite rev_length. rewrite rev_length.
  assumption. rewrite H0.

  assert (LEMMA : forall (l1: list bool) (l2: list X),
    length l1 = length l2
    -> combine (rev l1) (rev l2) = rev (combine l1 l2)).
  intro l1. induction l1; intro l2; intro I. reflexivity.
  destruct l2. inversion I. inversion I. apply IHl1 in H2.
  simpl.
  assert (LEMMA2 : forall (u w : list bool) (v x : list X),
    length u = length v
    -> combine (u ++ w) (v ++ x) = (combine u v) ++ (combine w x)).
  intro u0. induction u0; intros w v0 x1; intro J.
  symmetry in J. apply length_zero_iff_nil in J. rewrite J.
  reflexivity. destruct v0. inversion J. inversion J.
  apply IHu0 with (w := w) (x := x1) in H3. simpl. rewrite H3.
  reflexivity. rewrite LEMMA2. rewrite IHl1. reflexivity.
  inversion I. reflexivity. rewrite rev_length. rewrite rev_length.
  inversion I. reflexivity. rewrite LEMMA. rewrite <- map_rev.
  assert (LEMMA3: forall v (w: list (bool * X)),
    rev (filter v w) = filter v (rev w)).
  intros v0 w. induction w. reflexivity. simpl. rewrite filter_app.
  simpl. destruct (v0 a). simpl. rewrite IHw. reflexivity.
  rewrite IHw. symmetry. apply app_nil_r. rewrite LEMMA3.
  reflexivity. assumption.
  split. apply MAIN. intro H. apply MAIN in H.
  rewrite rev_involutive in H. rewrite rev_involutive in H.
  assumption.
Qed.


Theorem Subsequence_map {X Y: Type} :
  forall (u v: list X) (f: X -> Y),
    Subsequence u v -> Subsequence (map f u) (map f v).
Proof.
  intros u v. generalize u. induction v; intros u0 f; intro H.
  apply Subsequence_nil_r. destruct H. destruct H. destruct H.
  exists (map f x). exists (map f x0). split.
  rewrite <- map_cons. rewrite <- map_app. rewrite H. reflexivity.
  apply IHv. assumption.
Qed.






Theorem subsequence_double_cons {X: Type} :
  forall (u v: list X) a, subsequence (a::u) (a::v) <-> subsequence u v.
Proof.
  intros u v a. split; intro H. destruct H. destruct H. destruct H.
  destruct x; destruct x0. symmetry in H0. apply nil_cons in H0.
  contradiction. inversion H0.
  exists l. exists x0. split. inversion H. reflexivity. reflexivity.
  apply PeanoNat.Nat.neq_succ_0 in H. contradiction.
  inversion H0. exists (x1++x::l). exists x0. split. inversion H.
  reflexivity. rewrite <- app_assoc. reflexivity.
  apply subsequence_eq_def_3. exists nil. exists u.
  split. reflexivity. apply subsequence_eq_def_2. apply subsequence_eq_def_1.
  assumption.
Qed.


Theorem subsequence_filter {X: Type} :
  forall (u v: list X) f, subsequence (filter f u) v -> subsequence u v.
Proof.
  intro u. induction u; intros v f; intro H. assumption.
  simpl in H. destruct (f a). assert (K := H).
  apply subsequence_eq_def_1 in H. destruct H. destruct H.
  destruct x. apply O_S in H. contradiction. destruct b.
  destruct v. apply subsequence_nil_r. assert (x0 = a). inversion H0.
  reflexivity. rewrite H1.
  apply subsequence_eq_def_3. exists nil. exists u. split. reflexivity.
  apply subsequence_eq_def_2. apply subsequence_eq_def_1.
  apply IHu with (f := f). rewrite H1 in K.
  apply subsequence_double_cons in K. assumption.
  apply subsequence_cons_l. apply IHu with (f := f).
  apply subsequence_eq_def_3. apply subsequence_eq_def_2.
  exists x. split. inversion H. reflexivity. apply H0.
  apply subsequence_cons_l. apply IHu with (f := f). assumption.
Qed.


Theorem subsequence_incl {X: Type} :
  forall (u v: list X), subsequence u v -> incl v u.
Proof.
  intros u v. intro H. intro a. intro I.
  generalize I. generalize H. generalize u.
  induction v; intros u0; intros J K. inversion K.
  destruct K. rewrite H0 in J.
  apply subsequence_eq_def_1 in J.
  apply subsequence_eq_def_2 in J.
  destruct J. destruct H1. destruct H1. rewrite H1.

  assert (J: forall (u v : list X) w, In w (u++w::v)).
  intro  u1. induction u1; intros v0 w. left. reflexivity.
  right. apply IHu1. apply J. apply IHv.

  apply subsequence_cons_r in H. assumption. assumption.
  apply subsequence_cons_r in J. assumption. assumption.
Qed.


Theorem subsequence_split {X: Type} :
  forall (u v w: list X),
    subsequence (u++v) w 
    -> (exists a b, w = a++b /\ (subsequence u a) /\ subsequence v b).
Proof.
  intros u v w. intro H.
  apply subsequence_eq_def_1 in H. destruct H. destruct H.

  assert (forall (b: list bool) (u v: list X),
    length b = length (u++v)
    -> exists x y, b = x++y /\ length x = length u /\ length y = length v).
  intros b u0 v0. intro I. rewrite app_length in I.
  exists (firstn (length u0) b). exists (skipn (length u0) b).
  split. symmetry. apply firstn_skipn. split.
  rewrite firstn_length_le. reflexivity. rewrite I.
  apply PeanoNat.Nat.le_add_r. rewrite skipn_length.
  rewrite I. rewrite Nat.add_sub_swap. rewrite PeanoNat.Nat.sub_diag.
  reflexivity. apply le_n.
  apply H1 in H. destruct H. destruct H. destruct H. destruct H2.

  exists (map snd (filter fst (combine x0 u))).
  exists (map snd (filter fst (combine x1 v))).
  rewrite <- map_app. rewrite <- filter_app.

  assert (L: forall (g i: list bool) (h j: list X), length g = length h
    -> (combine g h) ++ (combine i j) = combine (g++i) (h++j)).
  intro g. induction g; intros i h j; intro I.
  symmetry in I. apply length_zero_iff_nil in I. rewrite I. reflexivity.
  destruct h. apply PeanoNat.Nat.neq_succ_0 in I. contradiction.
  simpl. rewrite IHg. reflexivity. inversion I. reflexivity.
  rewrite L. rewrite <- H. split. assumption.
  split; apply subsequence_eq_def_3; apply subsequence_eq_def_2;
  [ exists x0 | exists x1]; split; try assumption; reflexivity.

  assumption.
Qed.


Theorem subsequence_split_2 {X: Type} :
  forall (u v w: list X),
    subsequence u (v++w) 
    -> (exists a b, u = a++b /\ (subsequence a v) /\ subsequence b w).
Proof.
  intros u v w. intro H. destruct H. destruct H. destruct H.
  exists (x ++
     flat_map (fun e : X * list X => fst e :: snd e) (combine v x0)).
  exists (
     flat_map (fun e : X * list X => fst e :: snd e)
     (combine w (skipn (length v) x0))).
   split. rewrite H0. rewrite <- app_assoc. apply app_inv_head_iff.
   rewrite <- flat_map_app.

   assert (L: forall (w y: list X) (z : list (list X)),
     length z = length w + length y ->
     combine (w ++ y) z = combine w z ++ combine y (skipn (length w) z)).
   intros w0. induction w0; intros y z; intro I. reflexivity.
   destruct z. simpl. rewrite combine_nil. reflexivity.
   simpl. rewrite IHw0. reflexivity. inversion I. reflexivity.
   rewrite L. reflexivity. rewrite app_length in H. rewrite H.
   reflexivity. split.
   exists x. exists (firstn (length v) x0). split.
   rewrite firstn_length. rewrite <- H. rewrite app_length.
   rewrite min_l. reflexivity. apply PeanoNat.Nat.le_add_r.
   rewrite combine_firstn_l. reflexivity.

   exists nil. exists (skipn (length v) x0). split.
   rewrite app_length in H. rewrite skipn_length. rewrite <- H.
   rewrite PeanoNat.Nat.add_sub_swap. rewrite PeanoNat.Nat.sub_diag.
   reflexivity. reflexivity. reflexivity.
 Qed.


Theorem subsequence_filter_2 {X: Type} :
  forall (u v: list X) f,
    subsequence u v -> subsequence (filter f u) (filter f v).
Proof.
  intros u v. generalize u. induction v; intros u0 f; intro H.
  apply subsequence_nil_r.

  assert (f a = true \/ f a = false).
  destruct (f a); [ left | right ]; reflexivity. destruct H0.
  simpl. rewrite H0.
  replace (a::v) with ([a] ++ v) in H. apply subsequence_split_2 in H.
  destruct H. destruct H. destruct H. destruct H1.

  assert (subsequence ((filter f x) ++ (filter f x0))
                      ((filter f [a]) ++ (filter f v))).
  apply subsequence_app. simpl. rewrite H0.
  apply subsequence_incl in H1. apply incl_cons_inv in H1. destruct H1.
  assert (In a (filter f x)). apply filter_In. split; assumption.
  apply subsequence_in. assumption. apply IHv. assumption.
  rewrite <- filter_app in H3. rewrite <- H in H3. simpl in H3.
  rewrite H0 in H3. assumption. reflexivity.
  simpl. rewrite H0. apply IHv. apply subsequence_cons_r in H.
  assumption.
Qed.


Theorem subsequence_add {X: Type} :
  forall (u v w: list X) x, subsequence u v -> Add x u w -> subsequence w v.
Proof.
  intros u v w. generalize u v.
  induction w; intros u0 v0 x; intros H I; inversion I.
  rewrite <- H2. apply subsequence_cons_l. rewrite <- H3. assumption.

  assert (J := H). apply subsequence_eq_def_1 in H. rewrite <- H1 in H.
  rewrite H0 in H. destruct H. destruct H.
  destruct x1. apply O_S in H. contradiction. destruct b.
  rewrite H4. simpl. apply subsequence_double_cons.
  apply IHw with (u := l) (x := x).
  apply subsequence_eq_def_3. apply subsequence_eq_def_2. exists x1.
  inversion H. split; reflexivity. assumption.
  apply subsequence_cons_r with (a:=a).
  rewrite H4. simpl. apply subsequence_double_cons.
  apply IHw with (u := l) (x := x).
  apply subsequence_eq_def_3. apply subsequence_eq_def_2. exists x1.
  inversion H. split; reflexivity. assumption.
Qed.


Theorem subsequence_nodup {X: Type} :
  forall (u v: list X), subsequence u v -> NoDup u -> NoDup v.
Proof.
  intros u v. intros H I. apply subsequence_eq_def_1 in H.
  destruct H. destruct H. rewrite H0.

  assert (J: forall z (q: list X),
    NoDup q -> NoDup (map snd (filter fst (combine z q)))).
  intro z. induction z; intro q; intro J. apply NoDup_nil.
  destruct q. rewrite combine_nil. apply NoDup_nil.
  destruct a. simpl. rewrite NoDup_cons_iff in J. destruct J.
  apply IHz in H2. apply NoDup_cons.

  assert (K: forall (q: list X) g h,
    ~ In g q -> ~ In g (map snd (filter fst (combine h q)))).
  intro q0. induction q0; intros g h; intro J.
  rewrite combine_nil. easy. destruct h. easy. destruct b.
  replace (map snd (filter fst (combine (true :: h) (a :: q0))))
    with (a::(map snd (filter fst (combine h q0)))).
  rewrite not_in_cons. split;
  rewrite not_in_cons in J; destruct J. assumption. apply IHq0.
  assumption. reflexivity. apply IHq0.
  rewrite not_in_cons in J; destruct J. assumption. apply K. assumption.
  assumption. simpl. apply IHz.
  rewrite NoDup_cons_iff in J. destruct J. assumption. apply J.
  assumption.
Qed.


Theorem subsequence_in_2 {X: Type} :
  forall (u v: list X) a, subsequence u v -> In a v -> In a u.
Proof.
  intros u v. generalize u. induction v; intros u0 a0; intros H I.
  contradiction I. destruct I.
  apply subsequence_eq_def_1 in H. apply subsequence_eq_def_2 in H.
  destruct H. destruct H. destruct H. rewrite H. rewrite H0.
  apply in_elt. apply IHv. apply subsequence_cons_r in H.
  assumption. assumption.
Qed.


Theorem subsequence_not_in {X: Type} :
  forall (u v: list X) a, subsequence u v -> ~ In a u -> ~ In a v.
Proof.
  intros u v. generalize u. induction v; intros u0 a0; intros H I.
  easy. apply not_in_cons. split. apply subsequence_incl in H.
  apply incl_cons_inv in H. destruct H.

  assert (forall z (x y: X), In x z -> ~ In y z -> x <> y).
  intro z. induction z; intros x y; intros J K. inversion J.
  apply in_inv in J. destruct J. rewrite <- H1.
  apply not_in_cons in K. destruct K. apply not_eq_sym. assumption.
  apply IHz. assumption.
  apply not_in_cons in K. destruct K. assumption. apply not_eq_sym.
  generalize I. generalize H. apply H1. generalize I. apply IHv.
  apply subsequence_cons_r in H. assumption.
Qed.


Theorem subsequence_as_filter {X: Type} :
  forall (u: list X) f, subsequence u (filter f u).
Proof.
  intros u f. apply subsequence_eq_def_3. apply subsequence_eq_def_2.
  exists (map f u). split. apply map_length.
  induction u. reflexivity. simpl. destruct (f a).
  simpl. rewrite IHu. reflexivity. assumption.
Qed.


Theorem subsequence_as_partition {X: Type} :
  forall (u v w: list X) f,
    (v, w) = partition f u -> (subsequence u v) /\ (subsequence u w).
Proof.
  intros u v w f. intro H. rewrite partition_as_filter in H. split.
  replace v with (fst (v, w)). rewrite H. simpl.
  apply subsequence_as_filter. reflexivity.
  replace w with (snd (v, w)). rewrite H. simpl.
  apply subsequence_as_filter. reflexivity.
Qed.