(** * The Thue-Morse sequence

   Some proofs related to the Thue-Morse sequence.
   See:
   - https://oeis.org/A010060
   - https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence
*)  

Require Import Coq.Lists.List.
Require Import PeanoNat.
Require Import Nat.
Require Import Bool.

Import ListNotations.


(**
   This whole notebook contains proofs related to the following two functions.
   Nothing else is defined afterwards in the current notebook.
*)

Fixpoint tm_morphism (l:list bool) : list bool :=
  match l with
  | nil => []
  | h :: t => h :: (negb h) :: (tm_morphism t)
  end.

Fixpoint tm_step (n: nat) : list bool :=
  match n with
  | 0 => false :: nil
  | S n' => tm_morphism (tm_step n')
  end.


(**
   More or less "general" lemmas, which are not directly related to
   the previous functions are proved below.
*)

Lemma lt_split_bits : forall n m k j,
  0 < k -> j < 2^m -> k*2^m < 2^n -> k*2^m+j < 2^n.
Proof.
  intros n m k j. intros H I J.

  assert (K: 2^m <= k*2^m). rewrite <- Nat.mul_1_l at 1.
  apply Nat.mul_le_mono_r. rewrite Nat.le_succ_l. assumption.

  assert (L:2^m < 2^n). generalize J. generalize K. apply Nat.le_lt_trans.

  assert (k < 2^(n-m)). rewrite Nat.mul_lt_mono_pos_r with (p := 2^m).
  rewrite <- Nat.pow_add_r. rewrite Nat.sub_add. assumption.
  apply Nat.pow_lt_mono_r_iff in L. apply Nat.lt_le_incl. assumption.
  apply Nat.lt_1_2. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.

  replace (2^(n-m)) with (S (2^(n-m)-1)) in H0. rewrite Nat.lt_succ_r in H0.
  apply Nat.mul_le_mono_r with (p := 2^m) in H0.
  rewrite Nat.mul_sub_distr_r in H0. rewrite Nat.mul_1_l in H0.
  rewrite <- Nat.pow_add_r in H0. rewrite Nat.sub_add in H0.

  rewrite Nat.add_le_mono_r with (p := j) in H0.
  assert (2^n - 2^m + j < 2^n).
  rewrite Nat.add_lt_mono_l with (p := 2^n) in I.
  rewrite Nat.add_lt_mono_r with (p := 2^m).
  rewrite <- Nat.add_assoc. rewrite <- Nat.add_sub_swap.
  rewrite Nat.add_assoc. rewrite Nat.add_sub. assumption.

  apply Nat.lt_le_incl. assumption.
  generalize H1. generalize H0. apply Nat.le_lt_trans.
  apply Nat.lt_le_incl. rewrite <- Nat.pow_lt_mono_r_iff in L. assumption.
  apply Nat.lt_1_2. rewrite <- Nat.add_1_r. apply Nat.sub_add.
  rewrite Nat.le_succ_l. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
Qed.


Lemma count_occ_bool_list : forall (a: list bool),
  length a = (count_occ bool_dec a true) + (count_occ bool_dec a false).
Proof.
  intro a.
  induction a.
  - reflexivity.
  - destruct a.
    + rewrite count_occ_cons_eq. rewrite count_occ_cons_neq.
      simpl. rewrite IHa. reflexivity. easy. reflexivity.
    + rewrite count_occ_cons_neq. rewrite count_occ_cons_eq.
      simpl. rewrite Nat.add_succ_r. rewrite IHa. reflexivity. reflexivity. easy.
Qed.


Lemma count_occ_bool_list2 : forall (a: list bool),
  count_occ bool_dec a false = count_occ bool_dec a true -> even (length a) = true.
Proof.
  intro a. intro H.
  rewrite count_occ_bool_list. rewrite H.
  replace (count_occ bool_dec a true) with (1 * (count_occ bool_dec a true)).
  rewrite <- Nat.mul_add_distr_r. rewrite Nat.add_1_r.
  replace (2 * count_occ bool_dec a true) with (0 + 2 * count_occ bool_dec a true).
  apply Nat.even_add_mul_2. apply Nat.add_0_l. apply Nat.mul_1_l.
Qed.

Lemma app_eq_length_head : forall x0 x1 y0 y1 : list bool,
    x0 ++ x1 = y0 ++ y1 -> length x0 = length y0 -> x0 = y0.
Proof.
  intro x0. induction x0. destruct y0. intros. reflexivity.
  intros. inversion H0. destruct y0. intros. inversion H0.
  intros. simpl in H0. apply Nat.succ_inj in H0.
  rewrite <- app_comm_cons in H.
  rewrite <- app_comm_cons in H.

  destruct a; destruct b. inversion H.
  apply IHx0 with (x1 := x1) (y1 := y1) in H2. rewrite H2. reflexivity.
  assumption. inversion H. inversion H. inversion H.
  apply IHx0 with (x1 := x1) (y1 := y1) in H2. rewrite H2. reflexivity.
  assumption.
Qed.

Lemma even_mod4 : forall n : nat,
  even n = true -> n mod 4 = 0 \/ n mod 4 = 2.
Proof.
  intro n. intro H.
  assert (K: 0 = n mod 2).
  apply Nat.mod_unique with (q := Nat.div2 n). apply Nat.lt_0_2.
  replace (0) with (Nat.b2n (Nat.odd n)) at 2.
  apply Nat.div2_odd. rewrite <- Nat.negb_even. rewrite H. reflexivity.
  assert (L: n mod (2*2) = n mod 2 + 2 * ((n / 2) mod 2)).
  apply Nat.mod_mul_r; easy. rewrite <- K in L.
  replace (2*2) with (4) in L. rewrite <- Nat.bit0_mod in L.
  rewrite L.
  destruct (Nat.testbit (n / 2) 0) ; [right | left] ; reflexivity.
  reflexivity.
Qed.

Lemma odd_mod4 : forall n : nat,
  odd n = true -> n mod 4 = 1 \/ n mod 4 = 3.
Proof.
  intro n. intro H.
  assert (K: 1 = n mod 2).
  apply Nat.mod_unique with (q := Nat.div2 n). apply Nat.lt_1_2.
  replace (1) with (Nat.b2n (Nat.odd n)) at 2.
  apply Nat.div2_odd. rewrite H. reflexivity.
  assert (L: n mod (2*2) = n mod 2 + 2 * ((n / 2) mod 2)).
  apply Nat.mod_mul_r; easy. rewrite <- K in L.
  replace (2*2) with (4) in L. rewrite <- Nat.bit0_mod in L.
  rewrite L.
  destruct (Nat.testbit (n / 2) 0) ; [right | left] ; reflexivity.
  reflexivity.
Qed.


(**
   Some lemmas related to the first function (tm_morphism) are proved here.
   They have a wider scope than the following ones since they don't focus on
   the Thue-Morse sequence by itself.
*)

Lemma tm_morphism_concat : forall (l1 l2 : list bool),
  tm_morphism (l1 ++ l2) = tm_morphism l1 ++ tm_morphism l2.
Proof.
  intros l1 l2.
  induction l1.
  - reflexivity.
  - simpl. rewrite IHl1. reflexivity.
Qed.

Lemma tm_morphism_rev : forall (l : list bool),
  rev (tm_morphism l) = tm_morphism (map negb (rev l)).
Proof.
  intro l. induction l.
  - reflexivity.
  - simpl. rewrite map_app.
    rewrite IHl. rewrite tm_morphism_concat.
    rewrite <- app_assoc. simpl.
    rewrite negb_involutive. reflexivity.
Qed.

Lemma tm_build_negb : forall (l : list bool),
  tm_morphism (map negb l) = map negb (tm_morphism l).
Proof.
  intro l.
  induction l.
  - reflexivity.
  - simpl. rewrite IHl. reflexivity.
Qed.


Lemma tm_morphism_double_index : forall (l : list bool) (k : nat),
  nth_error l k = nth_error (tm_morphism l) (2*k).
Proof.
  intro l.
  induction l.
  - intro k.
    simpl. replace (nth_error [] k) with (None : option bool).
    rewrite Nat.add_0_r. replace (nth_error [] (k+k)) with (None : option bool).
    reflexivity. symmetry. apply nth_error_None. simpl. apply Nat.le_0_l.
    symmetry. apply nth_error_None. simpl. apply Nat.le_0_l.
  - intro k. induction k.
    + rewrite Nat.mul_0_r. reflexivity.
    + simpl. rewrite Nat.add_0_r. rewrite Nat.add_succ_r. simpl.
      replace (k+k) with (2*k). apply IHl. simpl. rewrite Nat.add_0_r.
      reflexivity.
Qed.


Lemma tm_morphism_count_occ : forall (l : list bool),
  count_occ Bool.bool_dec (tm_morphism l) true
    = count_occ Bool.bool_dec (tm_morphism l) false.
Proof.
  intro l. induction l.
  - reflexivity.
  - destruct a.
    + simpl. apply eq_S. assumption.
    + simpl. apply eq_S. assumption.
Qed.

Lemma tm_morphism_eq : forall (l1 l2 : list bool),
  l1 = l2 <-> tm_morphism l1 = tm_morphism l2.
Proof.
  intros l1 l2. split.
  - intro H. rewrite H. reflexivity.
  - generalize l2. induction l1.
    + intro l. intro H. simpl in H.
      induction l. reflexivity. simpl in H. inversion H.
    + simpl. intro l0. intro H.
      induction l0. simpl in H. inversion H.
      simpl in H. inversion H. apply IHl1 in H3. rewrite H3.
      reflexivity.
Qed.

Lemma tm_morphism_length : forall (l : list bool),
  length (tm_morphism l) = 2 * (length l).
Proof.
  induction l.
  - reflexivity.
  - simpl. rewrite Nat.add_0_r. rewrite IHl.
    rewrite Nat.add_succ_r.
    simpl. rewrite Nat.add_0_r. reflexivity.
Qed.

Lemma tm_morphism_length_half : forall (u : list bool),
  Nat.div2 (length (tm_morphism u)) = length u.
Proof.
  intro u.
  induction u. reflexivity.
  simpl. rewrite IHu. reflexivity.
Qed.

Lemma tm_morphism_app : forall (l1 l2 : list bool),
  tm_morphism (l1 ++ l2) = tm_morphism l1 ++ tm_morphism l2.
Proof.
  intros l1 l2.
  induction l1.
  - reflexivity.
  - simpl. rewrite IHl1. reflexivity.
Qed.

Lemma tm_morphism_app2 : forall (l hd tl : list bool),
  tm_morphism l = hd ++ tl
  -> even (length hd) = true
  -> hd = tm_morphism (firstn (Nat.div2 (length hd)) l).
Proof.
  intros l hd tl. intros H I.
  assert (J : l = (firstn (Nat.div2 (length hd)) l) ++ (skipn (Nat.div2 (length hd)) l)).
  symmetry. apply firstn_skipn.
  rewrite tm_morphism_eq in J.
  rewrite tm_morphism_app in J. rewrite H in J.
  apply app_eq_app in J.

  assert (H2: length (tm_morphism l) = length hd + length tl).
  rewrite H. apply app_length.

  assert (H3: length hd <= length (tm_morphism l)).
  rewrite H2. apply Nat.le_add_r.
  rewrite tm_morphism_length in H3.

  assert (H4: Nat.div2 (length hd) <= length l).
  rewrite Nat.mul_le_mono_pos_l with (p := 2).
  rewrite <- Nat.add_0_r at 1.
  replace (0) with (Nat.b2n (Nat.odd (length hd))) at 2.
  rewrite <- Nat.div2_odd. assumption.
  rewrite <- Nat.negb_even. rewrite I. reflexivity. apply Nat.lt_0_2.

  destruct J. destruct H0; destruct H0.

  assert (L: length hd = length hd). reflexivity.
  rewrite H0 in L at 2. rewrite app_length in L.
  rewrite tm_morphism_length in L.

  apply firstn_length_le in H4. rewrite H4 in L.
  replace (2 * Nat.div2 (length hd))
    with (2 * Nat.div2 (length hd) + Nat.b2n (Nat.odd (length hd))) in L.
  rewrite <- Nat.div2_odd in L.
  rewrite <- Nat.add_0_r in L at 1.
  apply Nat.add_cancel_l in L. symmetry in L.
  apply length_zero_iff_nil in L.
  rewrite L in H0. rewrite <- app_nil_end in H0. assumption.
  rewrite <- Nat.negb_even. rewrite I. simpl.
  rewrite Nat.add_0_r. reflexivity.

  assert (L: length (tm_morphism (firstn (Nat.div2 (length hd)) l))
      = length (tm_morphism (firstn (Nat.div2 (length hd)) l))).
  reflexivity. rewrite H0 in L at 2. rewrite app_length in L.
  rewrite tm_morphism_length in L.

  apply firstn_length_le in H4. rewrite H4 in L.
  replace (2 * Nat.div2 (length hd))
    with (2 * Nat.div2 (length hd) + Nat.b2n (Nat.odd (length hd))) in L.
  rewrite <- Nat.div2_odd in L.
  rewrite <- Nat.add_0_r in L at 1.
  apply Nat.add_cancel_l in L. symmetry in L.
  apply length_zero_iff_nil in L.
  rewrite L in H0. rewrite <- app_nil_end in H0. symmetry. assumption.
  rewrite <- Nat.negb_even. rewrite I. simpl.
  rewrite Nat.add_0_r. reflexivity.
Qed.


Lemma tm_morphism_app3 : forall (l hd tl : list bool),
  tm_morphism l = hd ++ tl
  -> even (length hd) = true
  -> tl = tm_morphism (skipn (Nat.div2 (length hd)) l).
Proof.
  intros l hd tl. intros H I.
  assert (J: hd = tm_morphism (firstn (Nat.div2 (length hd)) l)).
  generalize I. generalize H. apply tm_morphism_app2.
  assert (K: (firstn (Nat.div2 (length hd)) l) ++ (skipn (Nat.div2 (length hd)) l) = l).
  apply firstn_skipn.
  assert (L: tm_morphism l = (tm_morphism (firstn (Nat.div2 (length hd)) l))
                          ++ (tm_morphism (skipn (Nat.div2 (length hd)) l))).
  rewrite <- K at 1. apply tm_morphism_app.
  rewrite H in L. rewrite <- J in L.
  rewrite app_inv_head_iff in L. assumption.
Qed.

Lemma tm_morphism_rev2 : forall (l : list bool),
  map negb (rev (tm_morphism l)) = tm_morphism (rev l).
Proof.
  intro l. induction l.
  - reflexivity.
  - simpl. rewrite tm_morphism_app. rewrite <- IHl.
    rewrite map_app. rewrite map_app. rewrite <- app_assoc.
    simpl. rewrite negb_involutive. reflexivity.
Qed.

Lemma tm_morphism_twice_rev : forall (l : list bool),
  rev (tm_morphism (tm_morphism l)) = tm_morphism (tm_morphism (rev l)).
Proof.
  intro l. induction l.
  - reflexivity.
  - simpl. rewrite tm_morphism_app. rewrite tm_morphism_app.
    rewrite <- IHl. rewrite <- app_assoc. rewrite <- app_assoc.
    rewrite <- app_assoc. simpl. rewrite negb_involutive. reflexivity.
Qed.


(**
   Lemmas and theorems below are related to the second function defined initially.
   They focus on the Thue-Morse sequence.
 *)

Lemma tm_step_lemma : forall n : nat,
  tm_morphism (tm_step n) = tm_step (S n).
Proof.
  intro n. reflexivity.
Qed.

Theorem tm_step_negb : forall (n : nat),
  map negb (tm_step (S n)) = rev (tm_morphism (rev (tm_step n))).
Proof.
  intro n.
  rewrite <- tm_step_lemma. symmetry.
  rewrite tm_morphism_rev. rewrite tm_build_negb. rewrite rev_involutive.
  reflexivity.
Qed.

Theorem tm_build : forall (n : nat),
  tm_step (S n) = tm_step n ++ map negb (tm_step n).
Proof.
  intro n.
  induction n.
  - reflexivity.
  - rewrite <- tm_step_lemma. rewrite IHn. rewrite tm_morphism_concat.
    rewrite tm_step_lemma. rewrite IHn. rewrite tm_build_negb.
    rewrite <- IHn. rewrite tm_step_lemma. reflexivity.
Qed.

Theorem tm_size_power2 : forall n : nat, length (tm_step n) = 2^n.
Proof.
  intro n.
  induction n.
  - reflexivity.
  - rewrite <- tm_step_lemma. rewrite tm_morphism_length.
    rewrite Nat.pow_succ_r. rewrite Nat.mul_cancel_l.
    assumption. easy. apply Nat.le_0_l.
Qed.


Lemma tm_step_head_1 : forall (n : nat),
    tm_step n = false :: tl (tm_step n).
Proof.
  intro n. 
  induction n.
  - reflexivity.
  - rewrite <- tm_step_lemma. rewrite IHn. reflexivity.
Qed.


Lemma tm_step_end_1 : forall (n : nat),
  rev (tm_step n) = odd n :: tl (rev (tm_step n)).
Proof.
  intro n.
  induction n.
  - reflexivity.
  - rewrite <- tm_step_lemma. rewrite tm_morphism_rev.
    rewrite IHn. simpl. rewrite <- Nat.negb_even at 1. rewrite Nat.odd_succ.
    rewrite negb_involutive. reflexivity.
Qed.

Lemma tm_step_odd_step : forall (n : nat),
  rev (tm_step (S (Nat.double n))) = map negb (tm_step (S (Nat.double n))).
Proof.
  intro n. induction n.
  - reflexivity.
  - rewrite <- tm_step_lemma. rewrite tm_morphism_rev.
    rewrite Nat.double_S. rewrite tm_build. rewrite rev_app_distr.
    rewrite tm_build_negb. rewrite IHn. rewrite <- IHn at 1.
    rewrite rev_involutive. reflexivity.
Qed.

Theorem tm_step_double_index : forall (n k : nat),
  nth_error (tm_step n) k = nth_error (tm_step (S n)) (2*k).
Proof.
  intros n k.
  destruct k.
  - rewrite Nat.mul_0_r. rewrite tm_step_head_1. simpl.
    rewrite tm_step_head_1. reflexivity.
  - rewrite <- tm_step_lemma. apply tm_morphism_double_index.
Qed.

Lemma tm_step_single_bit_index : forall (n : nat),
  nth_error (tm_step (S n)) (2^n) = Some true.
Proof.
  intro n.
  rewrite tm_build. rewrite nth_error_app2; rewrite tm_size_power2.
  - rewrite Nat.sub_diag.
    replace (true) with (negb false). apply map_nth_error.
    rewrite tm_step_head_1. reflexivity. reflexivity.
  - apply Nat.le_refl.
Qed.

Lemma tm_step_repunit_index : forall (n : nat),
  nth_error (tm_step n) (pred (2^n)) = Some (odd n).
Proof.
  intro n.
  rewrite <- Nat.sub_1_r. rewrite <- tm_size_power2.
  rewrite nth_error_nth' with (d := false).
  - rewrite <- rev_nth. rewrite tm_step_end_1. simpl. reflexivity.
    rewrite tm_size_power2.
    rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
  - rewrite Nat.sub_1_r. apply Nat.lt_pred_l.
    rewrite tm_size_power2. apply Nat.pow_nonzero. easy.
Qed.

Theorem tm_step_count_occ : forall (hd tl : list bool) (n : nat),
  tm_step n = hd ++ tl -> even (length hd) = true
  -> count_occ Bool.bool_dec hd true = count_occ Bool.bool_dec hd false.
Proof.
  intros hd tl n. intros H I.
  destruct n.
  - assert (J: hd = nil). destruct hd. reflexivity.
    simpl in H. inversion H.
    symmetry in H2. apply app_eq_nil in H2.
    destruct H2. rewrite H0 in I. simpl in I. inversion I.
    rewrite J. reflexivity.
  - rewrite <- tm_step_lemma in H.
    assert (J: hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
    generalize I. generalize H. apply tm_morphism_app2.
    rewrite J. apply tm_morphism_count_occ.
Qed.

Lemma tm_step_consecutive_identical :
  forall (n : nat) (hd tl : list bool) (b : bool),
  tm_step n = hd ++ (b::b::nil) ++ tl
  -> odd (length hd) = true.
Proof.
  intros n hd tl b. intro H.
  assert (J: {even (length hd) = false} + { ~ (even (length hd)) = false}).
  apply bool_dec. destruct J.
  - rewrite <- Nat.negb_even. rewrite e. reflexivity.
  - apply not_false_is_true in n0.
    assert (K: count_occ Bool.bool_dec hd true
             = count_occ Bool.bool_dec hd false).
    generalize n0. generalize H. apply tm_step_count_occ.
    assert (L: count_occ Bool.bool_dec (hd ++ [b;b]) true
             = count_occ Bool.bool_dec (hd ++ [b;b]) false).
    assert (even (length (hd ++ [b;b])) = true).
    rewrite app_length. rewrite Nat.even_add_even. assumption.
    simpl. apply Nat.EvenT_Even. apply Nat.even_EvenT. reflexivity.
    generalize H0. rewrite app_assoc in H. generalize H.
    apply tm_step_count_occ.
    rewrite count_occ_app in L. rewrite count_occ_app in L.
    rewrite K in L. rewrite Nat.add_cancel_l in L.
    destruct b; inversion L.
Qed.

Lemma tm_step_consecutive_identical' :
  forall (n : nat) (hd a tl : list bool) (b1 b2 : bool),
  tm_step n = hd ++ (b1::b1::nil) ++ a ++ (b2::b2::nil) ++ tl
  -> even (length a) = true.
Proof.
  intros n hd a tl b1 b2. intros H.
  assert (Nat.odd (length hd) = true).
  generalize H. apply tm_step_consecutive_identical.
  rewrite app_assoc in H. rewrite app_assoc in H.
  assert (Nat.odd (length ((hd ++ [b1;b1])++a)) = true).
  generalize H. apply tm_step_consecutive_identical.
  rewrite app_length in H1. rewrite Nat.odd_add in H1.
  rewrite app_length in H1. rewrite Nat.odd_add in H1. rewrite H0 in H1.
  replace (Nat.odd (length a)) with (negb (Nat.even (length a))) in H1.
  destruct (even (length a)). reflexivity. inversion H1.
  reflexivity.
Qed.

Theorem tm_step_length_even : forall (n : nat),
  0 < n -> even (length (tm_step n)) = true.
Proof.
  intro n. intro H.
  rewrite tm_size_power2. destruct n. inversion H.
  rewrite Nat.pow_succ_r'. rewrite Nat.even_mul. reflexivity.
Qed.

Theorem tm_step_not_nil_factor_even : forall (n : nat) (hd a tl : list bool),
  tm_step n = hd ++ a ++ tl
  -> 0 < length a
  -> even (length a) = true
  -> 0 < n.
Proof.
  intros n hd a tl. intros H I J.
  assert (M: 1 < length (tm_step n)). rewrite H.
  rewrite app_length. rewrite app_length.
  assert (0 <= length hd). apply le_0_n.
  assert (2 <= length a + length tl). rewrite <- Nat.add_0_r at 1.
  apply Nat.add_le_mono. destruct a. inversion I.
  destruct a. inversion J. simpl. apply le_n_S. apply le_n_S. apply le_0_n.
  apply le_0_n. rewrite <- Nat.add_0_l at 1. rewrite <- Nat.le_succ_l.
  rewrite plus_n_Sm. apply Nat.add_le_mono; assumption.
  rewrite tm_size_power2 in M.
  rewrite Nat.pow_lt_mono_r_iff with (a := 2). simpl. assumption.
  apply Nat.lt_1_2.
Qed.

Theorem tm_step_tail_not_nil_prefix_odd : forall (n : nat) (hd a tl : list bool),
  tm_step n = hd ++ a ++ tl
  -> 0 < length a
  -> odd (length hd) = true
  -> even (length a) = true
  -> nil <> tl.
Proof.
  intros n hd a tl. intros H I J K.
  assert (L: 0 < n). generalize K. generalize I. generalize H.
  apply tm_step_not_nil_factor_even.
  assert (M: even (length (tm_step n)) = true). generalize L.
  apply tm_step_length_even.
  destruct tl. rewrite app_nil_r in H.
  rewrite H in M. rewrite app_length in M.
  rewrite Nat.even_add in M. rewrite K in M. rewrite <- Nat.negb_odd in M.
  rewrite J in M. inversion M. apply nil_cons.
Qed.


(**
   The following lemmas and theorems focus on the stability of the sequence:
   while lists in the sequence get longer and longer, the beginning of a
   longer list contain the same elements as the previous shorter lists.
*)
Lemma tm_step_next_range :
  forall (n k : nat) (b : bool),
  nth_error (tm_step n) k = Some b -> nth_error (tm_step (S n)) k = Some b.
Proof.
  intros n k b. intro H. assert (I := H).
  apply nth_error_split in H. destruct H. destruct H. inversion H.
  rewrite tm_build. rewrite nth_error_app1. apply I.
  rewrite H0. rewrite app_length. rewrite <- H1. simpl.
  apply Nat.lt_add_pos_r. apply Nat.lt_0_succ.
Qed.

Lemma tm_step_add_range : forall (n m k : nat),
  k < 2^n -> nth_error (tm_step n) k = nth_error (tm_step (n+m)) k.
Proof.
  intros n m k. intro H.
  induction m.
  - rewrite Nat.add_0_r. reflexivity.
  - rewrite Nat.add_succ_r. rewrite <- tm_size_power2 in H.
    assert (nth_error (tm_step n) k = Some (nth k (tm_step n) false)).
    generalize H. apply nth_error_nth'.
    rewrite H0 in IHm. symmetry in IHm.
    rewrite H0. symmetry. generalize IHm.
    apply tm_step_next_range.
Qed.

Theorem tm_step_stable : forall (n m k : nat),
  k < 2^n -> k < 2^m -> nth_error (tm_step n) k = nth_error (tm_step m) k.
Proof.
  intros n m k. intros.
  assert (I: n < m /\ max n m = m \/ m <= n /\ max n m = n).
  apply Nat.max_spec. destruct I.
  - destruct H1. replace (m) with (n + (m-n)). apply tm_step_add_range. apply H.
    apply Nat.lt_le_incl in H1. replace (m) with (n + m - n) at 2. generalize H1.
    apply Nat.add_sub_assoc. rewrite Nat.add_comm.
    assert (n <= n). apply le_n. symmetry.
    replace (m) with (m + (n-n)) at 1. generalize H3.
    apply Nat.add_sub_assoc. rewrite Nat.sub_diag. apply Nat.add_0_r.
  - destruct H1. symmetry. replace (n) with (m + (n - m)). apply tm_step_add_range.
    apply H0. replace (n) with (m + n - m) at 2. generalize H1.
    apply Nat.add_sub_assoc. rewrite Nat.add_comm.
    assert (m <= m). apply le_n. symmetry.
    replace (n) with (n + (m-m)) at 1. generalize H3.
    apply Nat.add_sub_assoc. rewrite Nat.sub_diag. apply Nat.add_0_r.
Qed.

(**
   The following lemma states that a block of terms in the Thue-Morse
   sequence having a size being a power of 2 is repeated, either
   as an identical version or with all values together flipped.
*)

Lemma tm_step_repeating_patterns :
  forall (n m i j : nat),
  i < 2^m -> j < 2^m
  -> forall k, k < 2^n -> nth_error (tm_step m) i
                        = nth_error (tm_step m) j
                      <-> nth_error (tm_step (m+n)) (k * 2^m + i)
                        = nth_error (tm_step (m+n)) (k * 2^m + j).
Proof.
  intros n m i j. intros H I.
  induction n.
  - rewrite Nat.add_0_r. intro k. simpl. rewrite Nat.lt_1_r. intro.
    rewrite H0. simpl. easy.
  - rewrite Nat.add_succ_r. intro k. intro.
    rewrite tm_build.
    assert (S: k < 2^n \/ 2^n <= k). apply Nat.lt_ge_cases.
    destruct S.

    assert (k*2^m < 2^(m+n)).
    destruct k.
    + simpl. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
    + rewrite Nat.mul_lt_mono_pos_r with (p := 2^m) in H1.
      rewrite <- Nat.pow_add_r in H1. rewrite Nat.add_comm.
      assumption. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
    + assert (L: forall l, l < 2^m -> k*2^m + l < length (tm_step (m+n))).
      intro l. intro M. rewrite tm_size_power2.
      destruct k. simpl. assert (2^m <= 2^(m+n)). 
      apply Nat.pow_le_mono_r. easy. apply Nat.le_add_r.
      generalize H3. generalize M. apply Nat.lt_le_trans.
      apply lt_split_bits. apply Nat.lt_0_succ. assumption.
      assumption.

      rewrite nth_error_app1. rewrite nth_error_app1.
      generalize H1. apply IHn.
      apply L. assumption. apply L. assumption.

    + assert (J: 2 ^ (m + n) <= k * 2 ^ m).  
      rewrite Nat.pow_add_r. rewrite Nat.mul_comm.
      apply Nat.mul_le_mono_r. assumption.

      rewrite nth_error_app2. rewrite nth_error_app2. rewrite tm_size_power2.

      assert (forall a b, option_map negb a = option_map negb b <-> a = b).
      intros a b. destruct a. destruct b. destruct b0; destruct b; simpl.
      split; intro; reflexivity. split; intro; inversion H2.
      split; intro; inversion H2. split; intro; reflexivity.
      split; intro; inversion H2.
      destruct b. split; intro; inversion H2. split; intro; reflexivity.

      replace (k * 2 ^ m + i - 2^(m + n)) with ((k-2^n)*2^m + i).
      replace (k * 2 ^ m + j - 2^(m + n)) with ((k-2^n)*2^m + j).
      rewrite nth_error_map. rewrite nth_error_map.

      rewrite H2. apply IHn.
      rewrite Nat.add_lt_mono_r with (p := 2^n). rewrite Nat.sub_add.
      rewrite Nat.pow_succ_r in H0. rewrite <- Nat.add_1_r in H0 at 1.
      rewrite Nat.mul_add_distr_r in H0. simpl in H0.
      rewrite Nat.add_0_r in H0. assumption.
      apply Nat.le_0_l. assumption.

      rewrite Nat.mul_sub_distr_r. rewrite <- Nat.pow_add_r.
      rewrite Nat.add_sub_swap. replace (n+m) with (m+n). reflexivity.
      apply Nat.add_comm. assumption.

      rewrite Nat.mul_sub_distr_r. rewrite <- Nat.pow_add_r.
      rewrite Nat.add_sub_swap. replace (n+m) with (m+n). reflexivity.
      apply Nat.add_comm. assumption.

      rewrite tm_size_power2.
      assert (k*2^m <= k*2^m + j). apply Nat.le_add_r.
      generalize H2. generalize J. apply Nat.le_trans.

      rewrite tm_size_power2.
      assert (k*2^m <= k*2^m + i). apply Nat.le_add_r.
      generalize H2. generalize J. apply Nat.le_trans.
Qed.

(**
   The following lemmas and theorems are related to flipping the
   most significant bit in the numerical value of indices (of terms
   in the lists).
*)

Lemma tm_step_next_range2 :
  forall (n k : nat),
  k < 2^n -> nth_error (tm_step n) k <> nth_error (tm_step (S n)) (k + 2^n).
Proof.
  intros n k. intro H.
  rewrite tm_build.
  rewrite nth_error_app2. rewrite tm_size_power2. rewrite Nat.add_sub.
  assert (nth_error (tm_step n) k = Some (nth k (tm_step n) false)).
  generalize H. rewrite <- tm_size_power2. apply nth_error_nth'.
  rewrite H0.
  assert (nth_error (map negb (tm_step n)) k
    = Some (negb (nth k (tm_step n) false))).
  generalize H0. apply map_nth_error. rewrite H1.
  destruct (nth k (tm_step n) false). easy. easy.
  rewrite tm_size_power2. apply Nat.le_add_l.
Qed.

Lemma tm_step_next_range2_flip_two : forall (n k m : nat),
  k < 2^n -> m < 2^n ->
    nth_error (tm_step n) k = nth_error (tm_step n) m
    <->
    nth_error (tm_step (S n)) (k + 2^n)
      = nth_error (tm_step (S n)) (m + 2^n).
Proof.
  intros n k m. intro H. intro I.
  (* Part 1: preamble *)

  assert (nth_error (tm_step n) k <> nth_error (tm_step (S n)) (k + 2^n)).
  generalize H. apply tm_step_next_range2.
  assert (nth_error (tm_step n) m <> nth_error (tm_step (S n)) (m + 2^n)).
  generalize I. apply tm_step_next_range2.

  assert (K: k + 2^n < 2^(S n)). simpl. rewrite Nat.add_0_r.
  generalize H. apply Nat.add_lt_mono_r.
  assert (J: m + 2^n < 2^(S n)). simpl. rewrite Nat.add_0_r.
  generalize I. apply Nat.add_lt_mono_r.

  (* Part 2: main proof *)
  assert (nth_error (tm_step n) k = Some (nth k (tm_step n) false)).
  generalize H. rewrite <- tm_size_power2. apply nth_error_nth'.
  assert (nth_error (tm_step n) m = Some (nth m (tm_step n) false)).
  generalize I. rewrite <- tm_size_power2. apply nth_error_nth'.
  assert (nth_error (tm_step (S n)) (k + 2 ^ n) =
    Some (nth (k + 2^n) (tm_step (S n)) false)).
  generalize K. rewrite <- tm_size_power2. rewrite <- tm_size_power2.
  apply nth_error_nth'.
  assert (nth_error (tm_step (S n)) (m + 2 ^ n) =
    Some (nth (m + 2^n) (tm_step (S n)) false)).
  generalize J. rewrite <- tm_size_power2. rewrite <- tm_size_power2.
  apply nth_error_nth'.
  rewrite H2. rewrite H3. rewrite H4. rewrite H5.

  (* Part 3: handling 16 different cases *)
  destruct (nth k (tm_step n) false).
  destruct (nth m (tm_step n) false).
  destruct (nth (k + 2 ^ n) (tm_step (S n)) false).
  destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
  easy.
  rewrite H2 in H0. rewrite H4 in H0. contradiction H0. reflexivity.
  destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
  rewrite H3 in H1. rewrite H5 in H1. contradiction H1. reflexivity.
  easy.
  destruct (nth (k + 2 ^ n) (tm_step (S n)) false).
  destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
  rewrite H2 in H0. rewrite H4 in H0. contradiction H0. reflexivity.
  easy.
  destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
  easy.
  rewrite H3 in H1. rewrite H5 in H1. contradiction H1. reflexivity.
  destruct (nth m (tm_step n) false).
  destruct (nth (k + 2 ^ n) (tm_step (S n)) false).
  destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
  rewrite H3 in H1. rewrite H5 in H1. contradiction H1. reflexivity.
  easy.
  destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
  easy.
  rewrite H2 in H0. rewrite H4 in H0. contradiction H0. reflexivity.
  destruct (nth (k + 2 ^ n) (tm_step (S n)) false).
  destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
  easy.
  rewrite H3 in H1. rewrite H5 in H1. contradiction H1. reflexivity.
  destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
  rewrite H2 in H0. rewrite H4 in H0. contradiction H0. reflexivity.
  easy.
Qed.

Lemma tm_step_add_range2_flip_two : forall (n m j k : nat),
  k < 2^n -> m < 2^n ->
    nth_error (tm_step n) k = nth_error (tm_step n) m
    <->
    nth_error (tm_step (S n+j)) (k + 2^(n+j))
      = nth_error (tm_step (S n+j)) (m + 2^(n+j)).
Proof.
  intros n m j k. intros H I.
  assert (K: k + 2^n < 2^(S n)). simpl. rewrite Nat.add_0_r.
  generalize H. apply Nat.add_lt_mono_r.
  assert (J: m + 2^n < 2^(S n)). simpl. rewrite Nat.add_0_r.
  generalize I. apply Nat.add_lt_mono_r.

  split.
  - induction j.
    + intros L. rewrite Nat.add_0_r. rewrite Nat.add_0_r.
    rewrite <- tm_step_next_range2_flip_two. apply L. apply H. apply I.
    + intros L.

      rewrite Nat.add_succ_r. rewrite Nat.add_succ_r.

      assert (2^n < 2^(S n + j)).
      assert (n < S n + j). assert (n < S n). apply Nat.lt_succ_diag_r.
      generalize H0. apply Nat.lt_lt_add_r.
      generalize H0. apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
      assert (k < 2^(S n + j)).
      generalize H0. generalize H. apply Nat.lt_trans.
      assert (m < 2^(S n + j)).
      generalize H0. generalize I. apply Nat.lt_trans.

      rewrite <- tm_step_next_range2_flip_two.

      assert (nth_error (tm_step n) k = nth_error (tm_step (S n + j)) k).
      generalize H1. generalize H. apply tm_step_stable.
      assert (nth_error (tm_step n) m = nth_error (tm_step (S n + j)) m).
      generalize H2. generalize I. apply tm_step_stable.
      rewrite <- H3. rewrite <- H4. apply L.

      apply H1. apply H2.
  - induction j.
    + intro L. rewrite Nat.add_0_r in L. rewrite Nat.add_0_r in L.
      apply tm_step_next_range2_flip_two. apply H. apply I. apply L.
    + intro L.
      rewrite Nat.add_succ_r in L. rewrite Nat.add_succ_r in L.

      assert (2^n < 2^(S n + j)).
      assert (n < S n + j). assert (n < S n). apply Nat.lt_succ_diag_r.
      generalize H0. apply Nat.lt_lt_add_r.
      generalize H0. apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
      assert (k < 2^(S n + j)).
      generalize H0. generalize H. apply Nat.lt_trans.
      assert (m < 2^(S n + j)).
      generalize H0. generalize I. apply Nat.lt_trans.

      rewrite <- tm_step_next_range2_flip_two in L.

      assert (nth_error (tm_step n) k = nth_error (tm_step (S n + j)) k).
      generalize H1. generalize H. apply tm_step_stable.
      assert (nth_error (tm_step n) m = nth_error (tm_step (S n + j)) m).
      generalize H2. generalize I. apply tm_step_stable.
      rewrite <- H3 in L. rewrite <- H4 in L. apply L.

      apply H1. apply H2.
Qed.

(**
   The following lemmas and theorems are related to flipping the
   least significant bit in the numerical value of indices (of terms
   in the lists).
*)


(* Note: a first version included the  0 < k  hypothesis but in the very
   unorthodox case where k=0 it happens to be true anyway, and the hypothesis
   was removed in order to make the theorem more general *)
Theorem tm_step_flip_low_bit :
  forall (n m k j : nat),
  j < m -> k * 2^m < 2^n
  -> nth_error (tm_step n) (k * 2^m) <> nth_error (tm_step n) (k * 2^m + 2^j).
Proof.
  intros n m k j. intros H I.

  assert (L: nth_error (tm_step m) 0 = Some false).
  rewrite tm_step_head_1. reflexivity.

  assert (M: 2^j < 2^m).
  apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. assumption.

  assert (N: nth_error (tm_step m) (2^j) = Some true).
  replace (nth_error (tm_step m) (2^j)) with (nth_error (tm_step (S j)) (2^j)).
  apply tm_step_single_bit_index.
  apply tm_step_stable.
  apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. apply Nat.lt_succ_diag_r.
  assumption.

  assert (G: k = 0 \/ 0 < k). apply Nat.eq_0_gt_0_cases. destruct G as [G1|G2].

  (* k = 0 *)
  rewrite G1. rewrite Nat.mul_0_l. rewrite Nat.add_0_l.
  rewrite tm_step_head_1 at 1.
  assert (S: n < S j \/ S j <= n). apply Nat.lt_ge_cases.
  destruct S as [S1|S2]. rewrite Nat.lt_succ_r in S1.
  apply Nat.pow_le_mono_r with (a := 2) in S1.
  rewrite <- tm_size_power2 in S1. rewrite <- nth_error_None in S1.
  rewrite S1. easy. easy.
  rewrite Nat.le_succ_l in S2. apply Nat.pow_lt_mono_r with (a := 2) in S2.
  rewrite tm_step_stable with (m := m). rewrite N. easy. assumption. assumption.
  apply Nat.lt_1_2.

  (* k < 0 *)
  assert (O: k*2^m + 2^j < 2^n). apply lt_split_bits; assumption.
  assert (P: 2^n <= 2^(m+n)). apply Nat.pow_le_mono_r. easy. apply Nat.le_add_l.

  assert ( nth_error (tm_step m) 0
         = nth_error (tm_step m) (2^j)
       <-> nth_error (tm_step (m+n)) (k * 2^m + 0)
         = nth_error (tm_step (m+n)) (k * 2^m + (2^j)) ).
  apply tm_step_repeating_patterns.
  rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
  apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. assumption.

  assert (1 * k <= (2^m) * k). apply Nat.mul_le_mono_nonneg.
  apply Nat.le_0_1. rewrite Nat.le_succ_l.
  rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
  apply Nat.le_0_l. apply Nat.le_refl. rewrite Nat.mul_1_l in H0.
  rewrite Nat.mul_comm in H0.
  generalize I. generalize H0. apply Nat.le_lt_trans.
  rewrite Nat.add_0_r in H0.

  replace (nth_error (tm_step (m + n)) (k * 2 ^ m))
    with (nth_error (tm_step n) (k * 2 ^ m)) in H0.
  replace (nth_error (tm_step (m + n)) (k * 2 ^ m + 2^j))
    with (nth_error (tm_step n) (k * 2 ^ m + 2^j)) in H0.
  rewrite L in H0. rewrite N in H0.

  destruct (nth_error (tm_step n) (k * 2 ^ m)).
  destruct (nth_error (tm_step n) (k * 2 ^ m + 2^j)).
  destruct b; destruct b0; destruct H0.
  assert (Some true = Some true). reflexivity.
  apply H1 in H2. rewrite <- H2 at 1. easy. easy. easy.
  assert (Some false = Some false). reflexivity.
  apply H1 in H2. rewrite H2 at 1. easy. easy.
  rewrite nth_error_nth' with (d := false). easy.
  rewrite tm_size_power2. assumption.

  apply tm_step_stable. assumption.
  generalize P. generalize O. apply Nat.lt_le_trans.

  apply tm_step_stable. assumption.
  generalize P. generalize I. apply Nat.lt_le_trans.
Qed.


(**
   The following lemmas and theorems are of general interest; they
   come here because they use more specific previously defined
   lemmas and theorems.
*)

Theorem tm_step_succ_double_index : forall (n k : nat),
  k < 2^n -> nth_error (tm_step n) k <> nth_error (tm_step (S n)) (S (2*k)).
Proof.
  intros n k. intro H. rewrite tm_step_double_index.
  rewrite Nat.mul_comm. rewrite <- Nat.pow_1_r with (a := 2).
  replace (S (k*2^1)) with (k*2^1 + 2^0).
  apply tm_step_flip_low_bit. apply Nat.lt_0_1.
  rewrite Nat.mul_comm. simpl. rewrite Nat.add_0_r. rewrite Nat.add_0_r.
  apply Nat.add_lt_mono; assumption.
  simpl. apply Nat.add_1_r.
Qed.


Lemma tm_step_pred : forall (n k m : nat),
    S (2*k) * 2^m < 2^n ->
    nth_error (tm_step n) (S (2*k) * 2^m)
     = nth_error (tm_step n) (pred (S (2*k) * 2^m))
     <-> odd m = true.
Proof.
  intros n k m.

  generalize n. induction m. rewrite Nat.pow_0_r. rewrite Nat.mul_1_r.
  destruct n0. rewrite Nat.pow_0_r. rewrite Nat.lt_1_r.
  intro H. apply Nat.neq_succ_0 in H. contradiction H.
  rewrite Nat.odd_0. rewrite Nat.pred_succ.
  rewrite <- tm_step_double_index. intro H. split; intro I.
  symmetry in I. apply tm_step_succ_double_index in I. contradiction I.
  apply Nat.lt_succ_l in H. rewrite Nat.pow_succ_r' in H.
  rewrite <- Nat.mul_lt_mono_pos_l in H. assumption. apply Nat.lt_0_2.
  inversion I.

  intro n0. intro I.
  destruct n0. rewrite Nat.pow_0_r in I. rewrite Nat.lt_1_r in I.
  rewrite Nat.mul_eq_0 in I. destruct I.
  apply Nat.neq_succ_0 in H. contradiction H.
  apply Nat.pow_nonzero in H. contradiction H. easy.
  rewrite Nat.pow_succ_r'. rewrite Nat.mul_assoc.
  replace (S (2*k) * 2) with (2* (S (2*k))).
  rewrite <- Nat.mul_assoc. rewrite <- tm_step_double_index.

  rewrite Nat.pow_succ_r' in I.
  rewrite Nat.pow_succ_r' in I.
  rewrite Nat.mul_assoc in I.
  replace (S (2*k) * 2) with (2* (S (2*k))) in I.
  rewrite <- Nat.mul_assoc in I.
  rewrite <- Nat.mul_lt_mono_pos_l in I.
  assert (J := I). apply IHm in J.

  assert (M: 0 < 2^m). rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.

  assert (L: pred( S (2 * k) * 2 ^ m) < S (2 * k) * 2 ^ m).
  apply Nat.lt_pred_l. apply Nat.neq_mul_0.
  split. apply Nat.neq_succ_0. rewrite Nat.neq_0_lt_0. assumption.

  assert (N: pred (2 * (S (2 * k) * 2 ^ m)) < length (tm_step (S n0))).
  rewrite tm_size_power2.
  rewrite <- Nat.le_succ_l. rewrite Nat.succ_pred_pos. rewrite Nat.pow_succ_r.
  apply Nat.mul_le_mono_l. generalize I. apply Nat.lt_le_incl.
  apply Nat.le_0_l. apply Nat.mul_pos_pos. apply Nat.lt_0_2.
  apply Nat.mul_pos_pos. apply Nat.lt_0_succ. assumption.

  assert(T: pred (S (2 * k) * 2 ^ m) < 2 ^ n0).
  generalize I. generalize L. apply Nat.lt_trans.

  assert (nth_error (tm_step n0) (pred (S (2 * k) * 2 ^ m))
    <> nth_error (tm_step (S n0)) (S (2 * (pred(S (2 * k) * 2 ^ m))))).
  apply tm_step_succ_double_index. assumption.

  replace (S (2 * (pred(S (2 * k) * 2 ^ m)))) with (pred(2 * (S (2*k) * 2^m))) in H.
  rewrite Nat.odd_succ. rewrite <- Nat.negb_odd. destruct (odd m).

  split. intro K. rewrite <- K in H.
  destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m));
  destruct (nth_error (tm_step n0) (pred(S (2 * k) * 2 ^ m))).
  destruct J. replace (Some b) with (Some b0) in H. contradiction H.
  reflexivity. symmetry. apply H1. reflexivity.
  destruct J. replace (Some b) with (None : option bool) in H.
  contradiction H. reflexivity. symmetry. apply H1. reflexivity.
  destruct J. replace (None : option bool) with (Some b) in H.
  contradiction H. reflexivity. symmetry. apply H1. reflexivity.
  contradiction H. reflexivity.

  destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m));
  destruct (nth_error (tm_step n0) (pred(S (2 * k) * 2 ^ m)));
  intro K; simpl in K; inversion K.

  rewrite nth_error_nth' with (d := false) in J.
  rewrite nth_error_nth' with (d := false) in J.
  rewrite nth_error_nth' with (d := false) in H.
  rewrite nth_error_nth' with (d := false) in H.
  rewrite nth_error_nth' with (d := false).
  rewrite nth_error_nth' with (d := false).
  destruct (nth (S (2 * k) * 2 ^ m) (tm_step n0) false);
  destruct (nth (pred (S (2 * k) * 2 ^ m)) (tm_step n0) false);
  destruct (nth (pred (2 * (S (2 * k) * 2 ^ m))) (tm_step (S n0)) false).
  simpl; split; reflexivity.
  destruct J. rewrite H0 in H. contradiction H. reflexivity.
  reflexivity.
  simpl; split; reflexivity. contradiction H. reflexivity.
  contradiction H. reflexivity.
  simpl; split; reflexivity.
  destruct J. rewrite H0 in H. contradiction H. reflexivity.
  reflexivity.
  simpl; split; reflexivity.

  assumption. rewrite tm_size_power2. assumption. assumption.
  rewrite tm_size_power2. assumption.
  rewrite tm_size_power2. assumption.
  rewrite tm_size_power2. assumption.

  rewrite <- Nat.add_1_r at 4.
  rewrite Nat.mul_add_distr_r. rewrite Nat.mul_1_l.
  rewrite <- Nat.add_1_r at 1.
  rewrite Nat.mul_add_distr_r. rewrite Nat.mul_1_l.
  rewrite <- Nat.add_succ_l. rewrite Nat.succ_pred_pos. rewrite <- Nat.add_pred_r.
  reflexivity. apply Nat.neq_mul_0. split. apply Nat.neq_succ_0. rewrite Nat.neq_0_lt_0.
  assumption. apply Nat.mul_pos_pos. apply Nat.lt_0_succ. assumption.
  apply Nat.lt_0_2.

  apply Nat.mul_comm. apply Nat.mul_comm.
Qed.


Lemma tm_step_factor4_1mod4 : forall (n' : nat) (w z y : list bool),
    tm_step n' = w ++ z ++ y
    -> length z = 4 -> length w mod 4 = 1
    -> exists (x : bool), firstn 2 z = [x;x].
  intros n' w z y. intros G0 G1 G2.
  assert (U: 4 * (length w / 4) <= length w).
  apply Nat.mul_div_le. easy.
  assert (W: length w < length (tm_step n')). rewrite G0.
  rewrite app_length. rewrite <- Nat.add_0_r at 1. rewrite <- Nat.add_lt_mono_l.
  rewrite app_length. rewrite G1. apply Nat.lt_0_succ.
  assert (Q: length z <= length (tm_step n')). rewrite G0.
  rewrite app_length. rewrite app_length.
  rewrite Nat.add_assoc. rewrite Nat.add_shuffle0.
  apply Nat.le_add_l. rewrite tm_size_power2 in Q. rewrite G1 in Q.
  replace (4) with (2^2) in Q. rewrite <- Nat.pow_le_mono_r_iff in Q.
  assert (O: nth_error (tm_step 2) 1 = nth_error (tm_step 2) 2
    <-> nth_error (tm_step (2+(n'-2))) (length w / 4 * 2 ^ 2 + 1)
      = nth_error (tm_step (2+(n'-2))) (length w / 4 * 2 ^ 2 + 2)).
  apply tm_step_repeating_patterns.
  simpl. rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
  simpl. rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
  apply Nat.lt_0_succ.
  rewrite Nat.mul_lt_mono_pos_l with (p := 2^2). rewrite <- Nat.pow_add_r.
  rewrite Nat.add_comm. rewrite Nat.sub_add.
  replace (2^2) with 4.
  rewrite tm_size_power2 in W. generalize W. generalize U.
  apply Nat.le_lt_trans. reflexivity.
  assumption.
  rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
  rewrite Nat.mul_comm in O. replace (2^2) with 4 in O.
  rewrite <- G2 in O at 9. rewrite <- G2 in O at 14.
  replace (4 * (length w / 4) + S (length w mod 4))
       with (S (length w)) in O. rewrite <- Nat.div_mod_eq in O.
  replace (2+(n'-2)) with n' in O. rewrite G0 in O.
  rewrite nth_error_app2 in O. rewrite Nat.sub_diag in O.
  rewrite nth_error_app1 in O. rewrite nth_error_app2 in O.
  rewrite Nat.sub_succ_l in O. rewrite Nat.sub_diag in O.
  rewrite nth_error_app1 in O. destruct O.
  assert (nth_error z 0 = nth_error z 1). apply H. reflexivity.
  rewrite nth_error_nth' with (d := false) in H1.
  rewrite nth_error_nth' with (d := false) in H1.
  exists (nth 0 z false). inversion H1.
  rewrite H3 at 2.
  destruct z. inversion G1. destruct z. inversion G1. reflexivity.
  rewrite G1. rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
  rewrite G1. apply Nat.lt_0_succ.
  rewrite G1. rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
  apply Nat.le_refl. apply Nat.le_succ_diag_r.
  rewrite G1. apply Nat.lt_0_succ. apply Nat.le_refl.
  rewrite Nat.add_sub_assoc. rewrite Nat.add_sub_swap.
  reflexivity. easy. assumption.
  rewrite Nat.add_succ_r. rewrite <- Nat.div_mod_eq. reflexivity.
  reflexivity. apply Nat.lt_1_2. reflexivity.
Qed.

Lemma tm_step_factor4_3mod4 : forall (n' : nat) (w z y : list bool),
    tm_step n' = w ++ z ++ y
    -> length z = 4 -> length w mod 4 = 3
    -> exists (x : bool), skipn 2 z = [x;x].
Proof.
  intros n hd a tl. intros H I M.
  assert ((length (hd ++ (firstn 2 a))) mod 4 = 1).
  rewrite app_length. rewrite firstn_length. rewrite I.
  rewrite <- Nat.add_mod_idemp_l. rewrite M. reflexivity. easy.
  assert (tm_step (S n) = tm_step n ++ map negb (tm_step n)).
  apply tm_build. rewrite H in H1.
  assert (length ((skipn 2 a)
    ++ (firstn 2 (tl ++ (map negb (hd ++ a ++ tl))))) = 4).
  rewrite app_length. rewrite skipn_length. rewrite I.
  rewrite firstn_length. rewrite app_length. rewrite <- H.
  rewrite map_length.
  assert (4 <= length tl + length (tm_step n)).
  assert (Q: length a <= length (tm_step n)). rewrite H.
  rewrite app_length. rewrite app_length.
  rewrite Nat.add_assoc. rewrite Nat.add_shuffle0.
  apply Nat.le_add_l. rewrite I in Q.
  rewrite <- Nat.add_0_l at 1. apply Nat.add_le_mono.
  apply Nat.le_0_l. assumption. rewrite Nat.min_l. reflexivity.
  assert (2 <= 4). apply le_n_S. apply le_n_S. apply Nat.le_0_l.
  generalize H2. generalize H3. apply Nat.le_trans.
  replace ((hd ++ a ++ tl) ++ map negb (hd ++ a ++ tl))
    with ((hd ++ firstn 2 a)
      ++ (skipn 2 a ++ firstn 2 (tl ++ map negb (hd ++ a ++ tl)))
      ++ (skipn 2 (tl ++ map negb (hd ++ a ++ tl)))) in H1.
  assert (exists (x : bool), 
    firstn 2 (skipn 2 a ++ firstn 2 (tl ++ map negb (hd ++ a ++ tl)))
    = [x;x]).
  generalize H0. generalize H2. generalize H1. apply tm_step_factor4_1mod4.
  destruct H3. exists x.
  rewrite <- H3. rewrite firstn_app. rewrite skipn_length. rewrite I.
  replace (2 - (4 - 2)) with 0. rewrite firstn_O.
  rewrite app_nil_r.
  destruct a. inversion I. destruct a. inversion I.
  destruct a. inversion I. destruct a. inversion I.
  destruct a. reflexivity. inversion I.
  reflexivity.
  rewrite <- app_assoc. symmetry. rewrite <- app_assoc.
  apply app_inv_head_iff.
  rewrite <- app_assoc. symmetry. rewrite <- app_assoc.
  rewrite firstn_skipn.
  rewrite <- firstn_skipn with (n := 2) (l := a) at 4.
  rewrite <- app_assoc. reflexivity.
Admitted.


Lemma tm_step_factor4_odd_prefix : forall (n : nat) (hd a tl : list bool),
  tm_step n = hd ++ a ++ tl
  -> length a = 4 -> odd (length hd) = true
  -> exists (x : bool) (u v : list bool), a = u ++ [x;x] ++ v.
Proof.
  intros n hd a tl. intros H I M.
  apply odd_mod4 in M.
  destruct M as [M | M].
  - assert (exists (x : bool), firstn 2 a = [x;x]).
    generalize M. generalize I. generalize H. apply tm_step_factor4_1mod4.
    destruct H0. exists x. exists nil. exists (skipn 2 a).
    rewrite app_nil_l. rewrite <- H0. symmetry. apply firstn_skipn.
  - assert (exists (x : bool), skipn 2 a = [x;x]).
    generalize M. generalize I. generalize H. apply tm_step_factor4_3mod4.
    destruct H0. exists x. exists (firstn 2 a). exists nil.
    rewrite app_nil_r. rewrite <- H0. symmetry. apply firstn_skipn.
Qed.

Lemma tm_step_consecutive_identical_length :
  forall (n : nat) (hd a tl : list bool),
  tm_step n = hd ++ a ++ tl -> 4 < length a
  -> exists (b c : list bool) (d : bool), a = b ++ [d;d] ++ c.
Proof.
  intros n hd a tl. intros H I.
  assert (J: Nat.Even (length hd) \/ Nat.Odd (length hd)).
  apply Nat.Even_or_Odd. destruct J as [J | J].

  (* first case *)
  replace (hd++a++tl)
    with ((hd ++ (firstn 1 a)) ++ (skipn 1 (firstn 5 a))
                               ++ ((skipn 5 a) ++ tl)) in H.

  assert (length (skipn 1 (firstn 5 a)) = 4).
  rewrite skipn_length. rewrite firstn_length.
  rewrite <- Nat.le_succ_l in I. rewrite Nat.min_l.
  reflexivity. assumption.

  assert (odd (length (hd ++ firstn 1 a)) = true).
  rewrite app_length. rewrite Nat.odd_add. rewrite <- Nat.negb_even.
  apply Nat.Even_EvenT in J. apply Nat.EvenT_even in J. rewrite J.
  rewrite firstn_length. rewrite Nat.min_l. reflexivity.
  
  apply Nat.lt_le_incl.
  assert (1 < 4). rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
  generalize I. generalize H1. apply Nat.lt_trans.

  assert (exists (x : bool) (u v : list bool),
    skipn 1 (firstn 5 a) = u ++ [x;x] ++ v).
  generalize H1. generalize H0. generalize H. apply tm_step_factor4_odd_prefix.
  destruct H2. destruct H2. destruct H2.
  exists ((firstn 1 a) ++ x0). exists (x1 ++ (skipn 5 a)). exists x.
  replace ((firstn 1 a ++ x0) ++ [x; x] ++ x1 ++ skipn 5 a)
    with (firstn 1 a ++ (x0 ++ [x;x] ++ x1) ++ skipn 5 a). rewrite <- H2.
  rewrite app_assoc. replace (firstn 1 a) with (firstn 1 (firstn 5 a)).
  rewrite firstn_skipn. rewrite firstn_skipn. reflexivity.
  rewrite firstn_firstn. reflexivity.
  rewrite <- app_assoc. rewrite <- app_assoc. rewrite <- app_assoc.
  reflexivity.
  rewrite <- app_assoc. apply app_inv_head_iff.
  replace (firstn 1 a) with (firstn 1 (firstn 5 a)).
  rewrite app_assoc. rewrite firstn_skipn.
  rewrite app_assoc. rewrite firstn_skipn.
  reflexivity.
  rewrite firstn_firstn. reflexivity.

  (* second case *)
  replace (hd++a++tl) with (hd ++ (firstn 4 a) ++ ((skipn 4 a) ++ tl)) in H.

  assert (length (firstn 4 a) = 4). rewrite firstn_length.
  rewrite <- Nat.le_succ_l in I. rewrite Nat.min_l.
  reflexivity. assert (4 <= 5). apply Nat.le_succ_diag_r.
  generalize I. generalize H0. apply Nat.le_trans.

  apply Nat.Odd_OddT in J. apply Nat.OddT_odd in J.

  assert (exists (x : bool) (u v : list bool), firstn 4 a = u ++ [x;x] ++ v).
  generalize J. generalize H0. generalize H. apply tm_step_factor4_odd_prefix.
  destruct H1. destruct H1. destruct H1.
  exists x0. exists (x1++(skipn 4 a)). exists x.

  replace (x0 ++ [x;x] ++ x1 ++ (skipn 4 a))
  with ((x0++[x;x]++x1) ++ (skipn 4 a)).
  rewrite <- H1. rewrite firstn_skipn. reflexivity.
  rewrite <- app_assoc. rewrite <- app_assoc. reflexivity.
  apply app_inv_head_iff.
  rewrite app_assoc. rewrite firstn_skipn. reflexivity.
Qed.


(**
   Some more lemmas are added there as convenient tools for further
   investigations; they have been added quite late, after many
   following results were proved, but they are inserted here in
   case some proofs are re-written later.
*)

Lemma tm_step_morphism2 :
  forall (n : nat) (hd tl : list bool),
  tm_step (S n) = hd ++ tl
  -> even (length hd) = true
  -> tm_step (S n) = tm_morphism
      (firstn (Nat.div2 (length hd)) (tm_step n) ++
       skipn (Nat.div2 (length hd)) (tm_step n)).
Proof.
  intros n hd tl. intros H I.
  assert (hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
  generalize I. generalize H. apply tm_morphism_app2.
  assert (tl = tm_morphism (skipn (Nat.div2 (length hd)) (tm_step n))).
  generalize I. generalize H. apply tm_morphism_app3.
  rewrite H0 in H. rewrite H1 in H. rewrite <- tm_morphism_app in H.
  assumption.
Qed.


Lemma tm_step_morphism3 :
  forall (n : nat) (hd a tl : list bool),
  tm_step (S n) = hd ++ a ++ tl
  -> even (length hd) = true
  -> even (length a) = true
  -> tm_morphism (tm_step n) = tm_morphism
           (firstn (Nat.div2 (length hd)) (tm_step n) ++
            firstn (Nat.div2 (length a))
              (skipn (Nat.div2 (length hd)) (tm_step n)) ++
            skipn (Nat.div2 (length hd + length a)) (tm_step n)).
Proof.
  intros n hd a tl. intros H I J. rewrite <- tm_step_lemma in H.
  assert (even (length (hd ++ a)) = true). rewrite app_length.
  rewrite Nat.even_add. rewrite I. rewrite J. reflexivity.
  rewrite app_assoc in H.
  assert (hd ++ a = tm_morphism (firstn (Nat.div2 (length (hd ++ a)))
                         (tm_step n))).
  generalize H0. generalize H. apply tm_morphism_app2.
  assert (tl = tm_morphism (skipn (Nat.div2 (length (hd ++ a)))
                         (tm_step n))).
  generalize H0. generalize H. apply tm_morphism_app3.
  rewrite <- app_assoc in H.
  assert (hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
  generalize I. generalize H. apply tm_morphism_app2.
  assert (a = tm_morphism (skipn (Nat.div2 (length hd))
              (firstn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
  generalize I. symmetry in H1. generalize H1. apply tm_morphism_app3.
  rewrite skipn_firstn_comm in H4.
  rewrite H3 in H. rewrite H4 in H. rewrite H2 in H.
  rewrite app_length in H.

  rewrite <- tm_morphism_app in H.
  rewrite <- tm_morphism_app in H.

  replace (Nat.div2 (length hd + length a))
    with ((length hd) / 2 + Nat.div2 (length a)) in H.
  rewrite <- Nat.div2_div in H. rewrite Nat.add_sub_swap in H.
  rewrite Nat.sub_diag in H. rewrite Nat.add_0_l in H.

  rewrite Nat.div2_div in H at 3. rewrite <- Nat.div_add in H.
  rewrite Nat.mul_comm in H.

  replace (2 * Nat.div2 (length a))
    with (2 * Nat.div2 (length a) + Nat.b2n (Nat.odd (length a))) in H.
    rewrite <- Nat.div2_odd in H. rewrite <- Nat.div2_div in H.
  assumption.

  rewrite <- Nat.negb_even. rewrite J.
  rewrite <- Nat.add_0_r. reflexivity. easy.
  apply Nat.le_refl.

  replace (length a) with (Nat.div2 (length a) * 2) at 2.
  rewrite <- Nat.div_add. rewrite <- Nat.div2_div. reflexivity.
  easy. rewrite Nat.mul_comm.

  replace (2 * Nat.div2 (length a))
    with (2 * Nat.div2 (length a) + Nat.b2n (Nat.odd (length a))).
  rewrite Nat.div2_odd. reflexivity.
  rewrite <- Nat.negb_even. rewrite J.
  rewrite <- Nat.add_0_r. reflexivity.
Qed.


Lemma tm_step_morphism4 :
  forall (n : nat) (hd a b tl : list bool),
  tm_step (S n) = hd ++ a ++ b ++ tl
  -> even (length hd) = true
  -> even (length a) = true
  -> even (length b) = true
  -> tm_step (S n) = tm_morphism
      (firstn (Nat.div2 (length hd)) (tm_step n) ++
       firstn (Nat.div2 (length a))
         (skipn (Nat.div2 (length hd)) (tm_step n)) ++
       firstn (Nat.div2 (length b))
         (skipn (Nat.div2 (length (hd ++ a))) (tm_step n)) ++
       skipn (Nat.div2 (length (hd ++ a ++ b))) (tm_step n)).
Proof.
  intros n hd a b tl. intros H I J K.

  assert (even (length (hd ++ a)) = true).
  rewrite app_length. rewrite Nat.even_add.
  rewrite I. rewrite J. reflexivity.
  rewrite app_assoc in H. rewrite <- tm_step_lemma in H.

  assert (hd ++ a = tm_morphism (firstn (Nat.div2 (length (hd ++ a)))
                                  (tm_step n))).
  generalize H0. generalize H. apply tm_morphism_app2.

  assert (b ++ tl = tm_morphism (skipn (Nat.div2 (length (hd ++ a)))
                                  (tm_step n))).
  generalize H0. generalize H. apply tm_morphism_app3.

  rewrite <- app_assoc in H. symmetry in H1. symmetry in H2.
  assert (even (length (hd ++ a ++ b)) = true).
  rewrite app_length. rewrite Nat.even_add. rewrite I.
  rewrite app_length. rewrite Nat.even_add. rewrite J. rewrite K.
  reflexivity.

  assert (tl = tm_morphism (skipn (Nat.div2 (length (hd ++ a ++ b)))
                         (tm_step n))).
  replace (hd ++ a ++ b ++ tl) with ((hd ++ a ++ b) ++ tl) in H.
  generalize H3. generalize H. apply tm_morphism_app3.
  rewrite <- app_assoc. rewrite <- app_assoc. reflexivity.

  assert (hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
  generalize I. generalize H. apply tm_morphism_app2.

  assert (a = tm_morphism (skipn (Nat.div2 (length hd))
           (firstn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
  generalize I. generalize H1. apply tm_morphism_app3.

  assert (b = tm_morphism (firstn (Nat.div2 (length b))
          (skipn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
  generalize K. generalize H2. apply tm_morphism_app2.

  rewrite skipn_firstn_comm in H6. rewrite app_length in H6.

  replace (Nat.div2 (length hd + length a))
    with ((length hd) / 2 + Nat.div2 (length a)) in H6.
  rewrite <- Nat.div2_div in H6. rewrite Nat.add_sub_swap in H6.
  rewrite Nat.sub_diag in H6. rewrite Nat.add_0_l in H6.

  rewrite H5 in H. rewrite H6 in H. rewrite H7 in H. rewrite H4 in H.
  rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
  rewrite <- tm_morphism_app in H. rewrite tm_step_lemma in H.
  assumption.

  apply Nat.le_refl.
  rewrite <- Nat.div_add. rewrite <- Nat.div2_div.
  rewrite Nat.mul_comm.

  replace (2 * Nat.div2 (length a))
    with (2 * Nat.div2 (length a) + Nat.b2n (Nat.odd (length a))).
  rewrite <- Nat.div2_odd. reflexivity.

  rewrite <- Nat.negb_even. rewrite J.
  rewrite <- Nat.add_0_r. reflexivity.
  easy.
Qed.