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Thomas Baruchel
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Require Import Nat.
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Require Import PeanoNat.
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Require Import List.
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Require Import Lia.
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Import ListNotations.
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(*
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@ -41,73 +42,6 @@ Proof.
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Qed.
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Lemma perm_max_alt : forall (l: list nat),
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permutation l -> (forall k, In k l -> k < length l).
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Proof.
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intro l. intro H. intro k. intro I.
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apply perm_seq_2 in H. apply incl_Forall_in_iff in H.
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rewrite Forall_forall in H. apply H in I. apply in_seq in I.
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destruct I. assumption.
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Qed.
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Lemma perm_max : forall (l: list nat),
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permutation l -> l <> nil -> In (length l - 1) l.
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Proof.
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intro l. intros H I. unfold permutation in H. apply H.
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rewrite Nat.sub_1_r. apply Nat.lt_pred_l.
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destruct l. contradiction I. reflexivity. easy.
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Qed.
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Lemma perm_max_split : forall (l: list nat),
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permutation l -> l <> nil
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-> exists l1 l2, l = l1 ++ [length(l) - 1] ++ l2.
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Proof.
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intro l. intros H I.
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assert (J: In (length l - 1) l). apply perm_max; assumption.
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apply In_nth with (d := 0) in J. destruct J. destruct H0.
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apply nth_split with (d := 0) in H0. destruct H0. destruct H0.
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destruct H0. exists x0. exists x1. rewrite H1 in H0. assumption.
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Qed.
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Lemma perm_max_remove : forall (l: list nat),
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permutation l -> l <> nil
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-> exists l1 l2, ((l = l1 ++ [length(l) - 1] ++ l2)
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/\ permutation (l1++l2)).
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Proof.
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intro l. intros H I.
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assert (J: exists l1 l2, l = l1 ++ [length l - 1] ++ l2).
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apply perm_max_split; assumption.
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destruct J. destruct H0. exists x. exists x0. split. assumption.
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assert (J: forall k, k < length l - 1 -> In k l -> In k (x ++ x0)).
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intro k. intros. rewrite H0 in H2. apply in_elt_inv in H2.
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destruct H2. rewrite H2 in H1. apply Nat.lt_irrefl in H1. contradiction H1.
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assumption. unfold permutation. intros.
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assert (k < length l - 1). rewrite H0.
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rewrite app_length. simpl. rewrite <- Nat.add_sub_assoc.
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rewrite Nat.sub_1_r. rewrite Nat.pred_succ. rewrite <- app_length.
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assumption. rewrite Nat.le_succ_l. apply Nat.lt_0_succ.
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apply J. assumption. apply H. apply Nat.lt_lt_succ_r in H2.
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rewrite Nat.sub_1_r in H2. rewrite Nat.succ_pred in H2.
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assumption. destruct l. contradiction I. reflexivity. easy.
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Qed.
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Lemma perm_max_insert : forall (l1 l2: list nat),
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permutation (l1 ++ l2) -> permutation (l1 ++ [ length (l1++l2) ] ++ l2).
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Proof.
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intros l1 l2. intro H. unfold permutation. intro k. intro I.
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rewrite app_length in I. simpl in I. rewrite Nat.add_succ_r in I.
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rewrite Nat.lt_succ_r in I. rewrite <- app_length in I.
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apply Nat.le_lteq in I. destruct I. apply H in H0.
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apply in_app_iff in H0. destruct H0. rewrite in_app_iff. left.
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assumption. rewrite in_app_iff. right. apply in_cons. assumption.
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rewrite <- H0. rewrite in_app_iff. right. apply in_eq.
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Qed.
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Lemma perm_incl_incl : forall (l: list nat) (n: nat),
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incl l (seq 0 n) -> incl (seq 0 n) l -> NoDup l -> permutation l.
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Proof.
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@ -164,12 +98,103 @@ Qed.
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Lemma perm_max : forall (l: list nat),
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permutation l -> l <> nil -> In (length l - 1) l.
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Proof.
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intro l. intros H I. unfold permutation in H. apply H.
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rewrite Nat.sub_1_r. apply Nat.lt_pred_l.
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destruct l. contradiction I. reflexivity. easy.
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Qed.
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Lemma perm_max_alt : forall (l: list nat),
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permutation l -> (forall k, In k l -> k < length l).
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Proof.
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intro l. intro H. intro k. intro I.
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apply perm_seq_2 in H. apply incl_Forall_in_iff in H.
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rewrite Forall_forall in H. apply H in I. apply in_seq in I.
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destruct I. assumption.
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Qed.
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Lemma perm_max_split : forall (l: list nat),
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permutation l -> l <> nil
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-> exists l1 l2, l = l1 ++ [length(l) - 1] ++ l2.
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Proof.
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intro l. intros H I.
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assert (J: In (length l - 1) l). apply perm_max; assumption.
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apply In_nth with (d := 0) in J. destruct J. destruct H0.
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apply nth_split with (d := 0) in H0. destruct H0. destruct H0.
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destruct H0. exists x0. exists x1. rewrite H1 in H0. assumption.
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Qed.
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Lemma perm_max_remove : forall (l: list nat),
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permutation l -> l <> nil
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-> exists l1 l2, ((l = l1 ++ [length(l) - 1] ++ l2)
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/\ permutation (l1++l2)).
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Proof.
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intro l. intros H I.
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assert (J: exists l1 l2, l = l1 ++ [length l - 1] ++ l2).
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apply perm_max_split; assumption.
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destruct J. destruct H0. exists x. exists x0. split. assumption.
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assert (J: forall k, k < length l - 1 -> In k l -> In k (x ++ x0)).
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intro k. intros. rewrite H0 in H2. apply in_elt_inv in H2.
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destruct H2. rewrite H2 in H1. apply Nat.lt_irrefl in H1. contradiction H1.
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assumption. unfold permutation. intros.
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assert (k < length l - 1). rewrite H0.
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rewrite app_length. simpl. rewrite <- Nat.add_sub_assoc.
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rewrite Nat.sub_1_r. rewrite Nat.pred_succ. rewrite <- app_length.
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assumption. rewrite Nat.le_succ_l. apply Nat.lt_0_succ.
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apply J. assumption. apply H. apply Nat.lt_lt_succ_r in H2.
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rewrite Nat.sub_1_r in H2. rewrite Nat.succ_pred in H2.
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assumption. destruct l. contradiction I. reflexivity. easy.
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Qed.
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Lemma perm_max_insert : forall (l1 l2: list nat),
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permutation (l1 ++ l2) -> permutation (l1 ++ [ length (l1++l2) ] ++ l2).
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Proof.
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intros l1 l2. intro H.
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(* first part of the proof *)
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unfold permutation. intro k. intro I.
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rewrite app_length in I. simpl in I. rewrite Nat.add_succ_r in I.
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rewrite Nat.lt_succ_r in I. rewrite <- app_length in I.
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apply Nat.le_lteq in I. destruct I. apply H in H0.
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apply in_app_iff in H0. destruct H0. rewrite in_app_iff. left.
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assumption. rewrite in_app_iff. right. apply in_cons. assumption.
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rewrite <- H0. rewrite in_app_iff. right. apply in_eq.
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Qed.
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Lemma perm_max_insert_alt : forall (l1 l2: list nat),
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permutation (l1 ++ [ length (l1++l2) ] ++ l2) -> permutation (l1 ++ l2).
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Proof.
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intros l1 l2. intro H.
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assert (I: incl (seq 0 (length (l1++l2))) (l1++l2)).
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assert (incl ([length (l1++l2)]++l2++l1) (l1++[length (l1++l2)]++l2)).
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unfold incl. intro a. intro J. rewrite app_assoc in J.
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rewrite in_app_iff. rewrite in_app_iff in J. destruct J.
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right. assumption. left. assumption.
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apply incl_app_inv in H0. destruct H0. apply incl_app_inv in H1.
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destruct H1.
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assert (incl (l1++l2) (l1 ++ [length (l1 ++ l2)] ++ l2)).
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apply incl_app; assumption.
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assert (incl ([length (l1++l2)]++l1++l2) (l1 ++ [length (l1 ++ l2)] ++ l2)).
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apply incl_app; assumption.
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apply incl_Add_inv
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with (a := length (l1++l2)) (v := l1++[length (l1++l2)]++l2).
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rewrite in_seq. lia. (* TODO *)
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replace (length (l1 ++ l2) :: seq 0 (length (l1 ++ l2)))
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with ([length (l1 ++ l2)] ++ seq 0 (length (l1 ++ l2))).
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apply incl_app. easy.
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apply perm_seq_1 in H. rewrite app_length in H. simpl in H.
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rewrite Nat.add_succ_r in H. rewrite seq_S in H. apply incl_app_inv in H.
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destruct H. rewrite app_length at 1. assumption.
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easy. apply Add_app.
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unfold incl in I. unfold permutation.
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intro k. intro. apply I. apply in_seq. lia. (* TODO *)
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Qed.
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