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

(*
   Definition of a permutation;
   We call a list a "permutation" iff it is a permutation
   of integers 0, 1, 2, 3, ...
   Eg. [1; 3; 2; 0]
*)
Definition permutation (l: list nat) :=
  forall k:nat, k < List.length l -> In k l.


Theorem perm_seq_1 : forall (l: list nat),
  permutation l -> incl (seq 0 (length l)) l.
Proof.
  intro l. intro H. unfold permutation in H.
  induction (length l). apply incl_nil_l. rewrite seq_S. simpl.
  apply incl_app. apply IHn. intro k. intro I.
  apply Nat.lt_lt_succ_r in I. apply H in I. assumption.
  apply incl_cons. apply H. apply Nat.lt_succ_diag_r. easy.
Qed.


Theorem perm_nodup : forall (l: list nat), permutation l -> NoDup l.
Proof.
  intro l. intro H. apply perm_seq_1 in H.
  apply NoDup_incl_NoDup with (l := seq 0 (length l)).
  apply seq_NoDup. rewrite seq_length. easy. assumption.
Qed.


Theorem perm_seq_2 : forall (l: list nat),
  permutation l -> incl l (seq 0 (length l)).
Proof.
  intro l. intro H. apply NoDup_length_incl.
  apply seq_NoDup. rewrite seq_length. easy.
  apply perm_seq_1. assumption.
Qed.


Lemma perm_incl_incl : forall (l: list nat) (n: nat),
  incl l (seq 0 n) -> incl (seq 0 n) l -> NoDup l -> permutation l.
Proof.
  intros l n. intros H I J.
  assert (length l <= n). replace n with (length (seq 0 n)).
  apply NoDup_incl_length; assumption. apply seq_length.
  assert (n <= length l). replace n with (length (seq 0 n)).
  apply NoDup_incl_length. apply seq_NoDup. assumption.
  apply seq_length. assert (length l = n).
  apply Nat.le_antisymm; assumption.
  apply incl_Forall_in_iff in I. rewrite Forall_forall in I.
  unfold permutation. intro k. intro K. apply I. rewrite H2 in K.
  assert (0 <= k < 0 + n). split. apply Nat.le_0_l.
  assumption. apply in_seq in H3. assumption.
Qed.


Lemma perm_eq : forall (l1 l2: list nat),
  permutation l1 -> permutation l2 -> length l1 = length l2
  -> incl l1 l2 /\ incl l2 l1.
Proof.
  intros l1 l2. intros H I J.
  assert (K1 := H). assert (K2 := H). assert (K3 := I). assert (K4 := I).
  apply perm_seq_1 in K1. apply perm_seq_2 in K2.
  apply perm_seq_1 in K3. apply perm_seq_2 in K4.
  rewrite J in K1. rewrite J in K2.
  split. apply incl_tran with (m := seq 0 (length l2)); assumption.
  apply incl_tran with (m := seq 0 (length l2)); assumption.
Qed.


Lemma perm_incl : forall (l1 l2: list nat),
  permutation l1 -> permutation l2 -> (incl l1 l2 \/ incl l2 l1).
Proof.
  intros l1 l2. intros H I.
  assert (length l1 < length l2 \/ length l2 <= length l1).
  apply Nat.lt_ge_cases. destruct H0. left.
  apply perm_seq_2 in H. apply perm_seq_1 in I.
  apply incl_appl with (m := seq (length l1) (length l2 - length l1)) in H.
  rewrite <- seq_app in H. rewrite Nat.add_sub_assoc in H.
  rewrite Nat.add_sub_swap in H. rewrite Nat.sub_diag in H.
  apply incl_tran with (m := seq 0 (length l2)); assumption.
  apply Nat.le_refl. apply Nat.lt_le_incl. assumption.
  apply Nat.le_lteq in H0. destruct H0. right.
  apply perm_seq_2 in I. apply perm_seq_1 in H.
  apply incl_appl with (m := seq (length l2) (length l1 - length l2)) in I.
  rewrite <- seq_app in I. rewrite Nat.add_sub_assoc in I.
  rewrite Nat.add_sub_swap in I. rewrite Nat.sub_diag in I.
  apply incl_tran with (m := seq 0 (length l1)); assumption.
  apply Nat.le_refl. apply Nat.lt_le_incl. assumption.
  left. assert (incl l1 l2 /\ incl l2 l1).
  symmetry in H0. apply perm_eq; assumption. destruct H1. assumption.
Qed.



Lemma perm_max : forall (l: list nat),
  permutation l -> l <> nil -> In (length l - 1) l.
Proof.
  intro l. intros H I. unfold permutation in H. apply H.
  rewrite Nat.sub_1_r. apply Nat.lt_pred_l.
  destruct l. contradiction I. reflexivity. easy.
Qed.


Lemma perm_max_alt : forall (l: list nat),
  permutation l -> (forall k, In k l -> k < length l).
Proof.
  intro l. intro H. intro k. intro I.
  apply perm_seq_2 in H. apply incl_Forall_in_iff in H.
  rewrite Forall_forall in H. apply H in I. apply in_seq in I.
  destruct I. assumption.
Qed.


Lemma perm_max_split : forall (l: list nat),
  permutation l -> l <> nil
  -> exists l1 l2, l = l1 ++ [length(l) - 1] ++ l2.
Proof.
  intro l. intros H I.
  assert (J: In (length l - 1) l). apply perm_max; assumption.
  apply In_nth with (d := 0) in J. destruct J. destruct H0.
  apply nth_split with (d := 0) in H0. destruct H0. destruct H0.
  destruct H0. exists x0. exists x1. rewrite H1 in H0. assumption.
Qed.


Lemma perm_max_remove : forall (l: list nat),
  permutation l -> l <> nil
  -> exists l1 l2, ((l = l1 ++ [length(l) - 1] ++ l2)
                   /\ permutation (l1++l2)).
Proof.
  intro l. intros H I.
  assert (J: exists l1 l2, l = l1 ++ [length l - 1] ++ l2).
  apply perm_max_split; assumption.
  destruct J. destruct H0. exists x. exists x0. split. assumption.
  assert (J: forall k, k < length l - 1 -> In k l -> In k (x ++ x0)).
  intro k. intros. rewrite H0 in H2. apply in_elt_inv in H2.
  destruct H2. rewrite H2 in H1. apply Nat.lt_irrefl in H1. contradiction H1.
  assumption. unfold permutation. intros.
  assert (k < length l - 1). rewrite H0.
  rewrite app_length. simpl. rewrite <- Nat.add_sub_assoc.
  rewrite Nat.sub_1_r. rewrite Nat.pred_succ. rewrite <- app_length.
  assumption. rewrite Nat.le_succ_l. apply Nat.lt_0_succ.
  apply J. assumption. apply H. apply Nat.lt_lt_succ_r in H2.
  rewrite Nat.sub_1_r in H2. rewrite Nat.succ_pred in H2.
  assumption. destruct l. contradiction I. reflexivity. easy.
Qed.

Lemma perm_max_insert : forall (l1 l2: list nat),
  permutation (l1 ++ l2) -> permutation (l1 ++ [ length (l1++l2) ] ++ l2).
Proof.
  intros l1 l2. intro H.
  (* first part of the proof *)
  unfold permutation. intro k. intro I.
  rewrite app_length in I. simpl in I. rewrite Nat.add_succ_r in I.
  rewrite Nat.lt_succ_r in I. rewrite <- app_length in I.
  apply Nat.le_lteq in I. destruct I. apply H in H0.
  apply in_app_iff in H0. destruct H0. rewrite in_app_iff. left.
  assumption. rewrite in_app_iff. right. apply in_cons. assumption.
  rewrite <- H0. rewrite in_app_iff. right. apply in_eq.
Qed.

Lemma perm_max_insert_alt : forall (l1 l2: list nat),
  permutation (l1 ++ [ length (l1++l2) ] ++ l2) -> permutation (l1 ++ l2).
Proof.
  intros l1 l2. intro H.

  assert (I: incl (seq 0 (length (l1++l2))) (l1++l2)).
  assert (incl ([length (l1++l2)]++l2++l1) (l1++[length (l1++l2)]++l2)).
  unfold incl. intro a. intro J. rewrite app_assoc in J.
  rewrite in_app_iff. rewrite in_app_iff in J. destruct J.
  right. assumption. left. assumption.
  apply incl_app_inv in H0. destruct H0. apply incl_app_inv in H1.
  destruct H1.
  assert (incl (l1++l2) (l1 ++ [length (l1 ++ l2)] ++ l2)).
  apply incl_app; assumption.
  assert (incl ([length (l1++l2)]++l1++l2) (l1 ++ [length (l1 ++ l2)] ++ l2)).
  apply incl_app; assumption.
  apply incl_Add_inv
      with (a := length (l1++l2)) (v := l1++[length (l1++l2)]++l2).
  rewrite in_seq. lia. (* TODO *)
  replace (length (l1 ++ l2) :: seq 0 (length (l1 ++ l2)))
    with ([length (l1 ++ l2)] ++ seq 0 (length (l1 ++ l2))).
  apply incl_app. easy.
  apply perm_seq_1 in H. rewrite app_length in H. simpl in H.
  rewrite Nat.add_succ_r in H. rewrite seq_S in H. apply incl_app_inv in H.
  destruct H. rewrite app_length at 1. assumption.
  easy. apply Add_app.

  unfold incl in I. unfold permutation.
  intro k. intro. apply I. apply in_seq. lia. (* TODO *)
Qed.






Example test1: permutation [1;2;3;0].
Proof.
  unfold permutation.
  simpl. intro k. intro H.

  rewrite PeanoNat.Nat.lt_succ_r in H.
  rewrite PeanoNat.Nat.le_lteq in H.
  rewrite PeanoNat.Nat.lt_succ_r in H.
  rewrite PeanoNat.Nat.le_lteq in H.
  rewrite PeanoNat.Nat.lt_succ_r in H.
  rewrite PeanoNat.Nat.le_lteq in H.
  rewrite PeanoNat.Nat.lt_succ_r in H.
  rewrite PeanoNat.Nat.le_lteq in H.
  destruct H. destruct H. destruct H. destruct H.
  apply PeanoNat.Nat.nlt_0_r in H. contradiction.
  right. right. right. left. easy.
  left. easy. right. left. easy. right. right. left. easy.
Qed.


Example test2: permutation [].
Proof.
  unfold permutation. intro k. simpl. easy.
Qed.