diff --git a/src/permutations.v b/src/permutations.v
deleted file mode 100644
index a5e195b..0000000
--- a/src/permutations.v
+++ /dev/null
@@ -1,264 +0,0 @@
-Require Import Nat.
-Require Import PeanoNat.
-Require Import List.
-Require Import Lia.
-Require Import Arith.Factorial.
-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.
-
-
-Example example_0: permutation [].
-Proof.
-  unfold permutation. intro k. simpl. easy.
-Qed.
-
-
-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. split; 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.
-
-  (* second part of the proof *)
-  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 | 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.
-  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. split. apply Nat.le_0_l. assumption.
-Qed.
-
-
-Lemma perm_max_insert_alt : forall (l1 l2: list nat),
-  permutation l1 -> Add (length l1) l1 l2 -> permutation l2.
-Proof.
-  intros l1 l2. intros H I.
-  apply Add_split in I. destruct I. destruct H0. destruct H0.
-  rewrite H0 in H. rewrite H1. rewrite H0.
-  replace (length (x++x0) :: x0) with ([length (x++x0)] ++ x0).
-  rewrite <- perm_max_insert. assumption. easy.
-Qed.
-
-
-Lemma perm_max_insert_alt_2 : forall (l1 l2: list nat),
-  Add (length l1) l1 l2 -> permutation l2 -> permutation l1.
-Proof.
-  intros l1 l2. intros H I.
-  apply Add_split in H. destruct H. destruct H. destruct H.
-  rewrite H0 in I. rewrite H. apply perm_max_insert.
-  simpl. rewrite H in I. assumption.
-Qed.
-
-
-Lemma perm_count : forall (n: nat) (E: list (list nat)),
-  NoDup E -> (forall l, In l E <-> (length l = n /\ permutation l))
-  -> length E = fact n.
-Proof.
-  intro n. induction n; intro E; intros H1 H2. simpl.
-
-  assert (I: length (nil: list nat) = 0 /\ permutation nil).
-  easy. apply H2 in I.
-
-   apply In_nth with (d := nil) in I.
-  destruct I. destruct (length E). destruct H.
-  apply Nat.nlt_0_r in H. contradiction H.
-  destruct n. reflexivity. destruct H.
-
-  assert (J: 0 < length E). apply In_nth with (d := nil) in I.
-  destruct I. destruct (length E). destruct H.
-  apply Nat.nlt_0_r in H. contradiction H.
-  apply Arith_prebase.gt_Sn_O_stt.
-
-  assert (K: length E
-
-
-
-
-
-  (*
-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.
-   *)
diff --git a/src/subsequences.v b/src/subsequences.v
index 1e2aa95..cc01d8a 100644
--- a/src/subsequences.v
+++ b/src/subsequences.v
@@ -4,7 +4,7 @@ Require Import List.
 Import ListNotations.
 
 
-(** * Various definitions of a subsequences
+(** * Various definitions
 
    Different definitions of a subsequence are given; they are proved
    below to be equivalent, allowing to choose the most convenient at
@@ -27,8 +27,7 @@ Fixpoint subsequence3 {X: Type} (l s : list X) : Prop :=
   end.
 
 
-(**
-   Identical elementary properties of all definitions of a subsequence.
+(** * Elementary properties
 *)
 
 Theorem subsequence_nil_r {X: Type}: forall (l : list X), subsequence l nil.
@@ -187,8 +186,13 @@ Proof.
 Qed.
 
 
-(**
-   Equivalence of all definitions of a subsequence.
+(** * Equivalence of all definitions
+
+   The following three implications are proved:
+
+   - subsequence l s -> subsequence2 l s
+   - subsequence2 l s -> subsequence3 l s
+   - subsequence3 l s -> subsequence l s
 *)
 
 Theorem subsequence_eq_def_1 {X: Type} :
@@ -270,8 +274,7 @@ Proof.
 Qed.
 
 
-(**
-   Decidability of all definitions of a subsequence.
+(** * Decidability of all definitions
 *)
 
 Theorem subsequence2_dec {X: Type}:
@@ -333,8 +336,7 @@ Proof.
 Qed.
 
 
-(**
-   Various general properties of a subsequence.
+(** * Various general properties
 *)
 
 Theorem subsequence_app {X: Type} :
@@ -435,6 +437,8 @@ Proof.
 Qed.
  
 
+(** * Tests
+*)
 
 Example test1: subsequence [1;2;3;4;5] [1;3;5].
 Proof.