coqbooks/thue-morse.v

(*
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.

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.

(* ad hoc more or less general lemmas *)
Lemma negb_map_explode : forall (l1 l2 : list bool),
  map negb (l1 ++ l2) = map negb l1 ++ map negb l2.
Proof.
  intros l1 l2.
  induction l1.
  - reflexivity.
  - simpl. rewrite IHl1. reflexivity.
Qed.

Lemma negb_double_map : forall (l : list bool),
  map negb (map negb l) = l.
Proof.
  intro l.
  induction l.
  - reflexivity.
  - simpl. rewrite IHl. replace (negb (negb a)) with (a).
    reflexivity. rewrite negb_involutive. reflexivity.
Qed.

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 negb_map_explode.
    rewrite IHl. rewrite tm_morphism_concat.
    rewrite <- app_assoc. simpl.
    rewrite negb_involutive. reflexivity.
Qed.

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

(* Thue-Morse related lemmas and theorems *)
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.
  replace (tm_step n) with (rev (rev (tm_step n))).
  rewrite rev_involutive. rewrite tm_morphism_rev'. reflexivity.
  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.
  - simpl. rewrite tm_step_lemma. rewrite IHn. rewrite tm_morphism_concat.
    simpl in IHn. rewrite IHn. rewrite tm_build_negb. rewrite IHn.
    rewrite negb_map_explode. rewrite negb_double_map.
    reflexivity.
Qed.

Theorem tm_size_power2 : forall n : nat, length (tm_step n) = 2^n.
Proof.
  intro n.
  induction n.
  - reflexivity.
  - rewrite tm_build. rewrite app_length. rewrite map_length.
    replace (2^S n) with (2^n + 2^n). rewrite IHn. reflexivity.
    simpl. rewrite <- plus_n_O. reflexivity.
Qed.

Lemma tm_step_head_2 : forall (n : nat),
    tm_step (S n) = false :: true :: tl (tl (tm_step (S n))).
Proof.
  intro n.
  induction n.
  - reflexivity.
  - simpl. replace (tm_morphism (tm_step n)) with (tm_step (S n)).
    rewrite IHn. reflexivity. reflexivity.
Qed.

Lemma tm_step_end_2 : forall (n : nat),
  rev (tm_step (S n)) = even n :: odd n :: tl (tl (rev (tm_step (S n)))).
Proof.
  intro n. induction n.
  - reflexivity.
  - simpl tm_step. rewrite tm_morphism_rev.
    replace (tm_morphism (tm_step n)) with (tm_step (S n)).
    rewrite IHn. simpl tm_morphism. simpl tl.
    rewrite Nat.even_succ.
    rewrite Nat.odd_succ.
    rewrite negb_involutive.
    reflexivity. reflexivity.
Qed.

Lemma tm_step_head_1 : forall (n : nat),
    tm_step n = false :: tl (tm_step n).
Proof.
  intro n. 
  destruct n.
  - reflexivity.
  - rewrite tm_step_head_2. reflexivity.
Qed.

Lemma tm_step_end_1 : forall (n : nat),
  rev (tm_step n) = odd n :: tl (rev (tm_step n)).
Proof.
  intro n.
  destruct n.
  - reflexivity.
  - rewrite tm_step_end_2. simpl.
    rewrite Nat.odd_succ.
    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.
    rewrite tm_morphism_double_index. reflexivity.
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. rewrite tm_size_power2. easy.
Qed.

Lemma tm_step_repunit_index : forall (n : nat),
  nth_error (tm_step n) (2^n - 1) = Some (odd n).
Proof.
  intro n.
  assert (H: 2 ^ n - 1 < length (tm_step n)). rewrite tm_size_power2.
  rewrite Nat.sub_1_r. apply Nat.lt_pred_l. apply Nat.pow_nonzero. easy.

  rewrite nth_error_nth' with (d := false).
  replace (tm_step n) with (rev (rev (tm_step n))).
  rewrite rev_nth. rewrite rev_length. rewrite tm_size_power2.
  rewrite Nat.sub_1_r. rewrite Nat.succ_pred_pos. rewrite Nat.sub_diag.
  rewrite tm_step_end_1. reflexivity.
  rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
  rewrite rev_length. assumption. apply rev_involutive. assumption.
Qed.

Lemma list_app_length_lt : forall (l l1 l2 : list bool) (b : bool),
  l = l1 ++ b :: l2 -> length l1 < length l.
Proof.
  intros l l1 l2 b. intro H. rewrite H.
  rewrite app_length. simpl. apply Nat.lt_add_pos_r.
  apply Nat.lt_0_succ.
Qed.

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.
  apply list_app_length_lt in H0. rewrite H1 in H0. apply H0.
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_comm. 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. rewrite Nat.add_comm.
    reflexivity.
  - 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. rewrite Nat.add_comm.
    reflexivity.
Qed.

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.

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 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. replace (2) with (1+1) in H0.
      rewrite Nat.mul_add_distr_r in H0. simpl in H0.
      rewrite Nat.add_0_r in H0. assumption. reflexivity.
      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.
      rewrite Nat.add_comm. reflexivity. assumption.

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

(* 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)).
  rewrite tm_step_single_bit_index. reflexivity.
  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.

Lemma tm_step_succ_double_index : forall (n k : nat),
  2 * k < 2^n -> nth_error (tm_step n) k <> nth_error (tm_step n) (S (2*k)).
Proof.
  intros n k. intro H. rewrite tm_step_double_index.
  replace (nth_error (tm_step (S n)) (2*k)) with (nth_error (tm_step n) (2*k)).
  rewrite Nat.mul_comm. replace (2) with (2^1).
  replace (S (k*2^1)) with (k*2^1 + 2^0).
  apply tm_step_flip_low_bit. apply Nat.lt_0_1.
  simpl. rewrite Nat.mul_comm. assumption.
  rewrite Nat.add_1_r. reflexivity. easy.
  apply tm_step_stable. assumption.
  assert (2^n < 2^(S n)). apply Nat.pow_lt_mono_r.
  apply Nat.lt_1_2. apply Nat.lt_succ_diag_r.
  generalize H0. generalize H. apply Nat.lt_trans.
Qed.





  (* TODO: supprimer head_2 *)

(* vérifier si les deux sont nécessaires *)
Require Import BinPosDef.
Require Import BinPos.

Require Import BinNat.


(* Hamming weight of a positive; argument can not be 0! *)
Fixpoint hamming_weight_positive (x: positive) :=
  match x with
  | xH   => 1
  | xO p => hamming_weight_positive p
  | xI p => 1 + hamming_weight_positive p
  end.

Definition hamming_weight_n (x: N) :=
  match x with
  | N0 => 0
  | Npos x => hamming_weight_positive x
  end.

Lemma hamming_weight_remove_high : forall (x : N),
  (0 < x)%N -> hamming_weight_n x = S (hamming_weight_n (x - 2^(N.log2 x))).
Proof.
  intros x.






N.shiftl_spec_alt:
  forall a n m : N, N.testbit (N.shiftl a n) (m + n) = N.testbit a m



Lemma hamming_weight_positive_nonzero : forall (x: positive),
  hamming_weight_positive x > 0.
Proof.
  intros x.
  induction x.
  - simpl. apply Arith_prebase.gt_Sn_O_stt.
  - simpl. apply IHx.
  - simpl. apply Arith_prebase.gt_Sn_O_stt.
Qed.



Lemma size_double_bin : forall (x: positive),
  Pos.size_nat (Pos.succ (Pos.pred_double x)) = S (Pos.size_nat x).
Proof.
  intros x.
  rewrite Pos.succ_pred_double.
  reflexivity.
Qed.

Lemma succ_succ_pred_double_succ : forall (x: positive),
  Pos.pred_double (Pos.succ x) = Pos.succ (Pos.succ (Pos.pred_double x)).
Proof.
  intros x.
  rewrite <- Pos.add_1_l.
  rewrite Pos.add_xO_pred_double.
  rewrite <- Pos.add_1_l.
  rewrite <- Pos.add_1_l.
  rewrite Pos.add_assoc.
  reflexivity.
Qed.                                                                                  
Lemma size_double_bin2 : forall (x: positive),
  Pos.size_nat (Pos.pred_double (Pos.succ x)) = S (Pos.size_nat x).
Proof.
  intros x.
  rewrite succ_succ_pred_double_succ. rewrite Pos.succ_pred_double.
  reflexivity.
Qed.

Lemma size_succ_double_bin : forall (x: positive),
  Pos.size_nat (Pos.succ (Pos.succ (Pos.pred_double x))) = S (Pos.size_nat x).
Proof.
  intros x.
  rewrite <- succ_succ_pred_double_succ. rewrite size_double_bin2.
  reflexivity.
Qed.

Lemma nat_size_pos_size : forall (x: positive),
  N.size_nat (N.pos x) = Pos.size_nat x.
Proof.
  intros x. reflexivity.
Qed.

Lemma hamming_weight_increase : forall (x: positive),
  hamming_weight_positive x~1 = S (hamming_weight_positive x).
Proof.
  intros x. reflexivity.
Qed.


Fixpoint bin2power (x : positive) (k : N) :=
  match x with
  | xH   => (2^k)%N
  | xO y => bin2power y (N.succ k)
  | xI y => (2^k + bin2power y (N.succ k))%N
  end.

Lemma bin2power_same : forall  (x : positive),
  Npos x = bin2power x 0.
Proof.
  intros x. 
  induction x.
  - simpl.



(*
Fixpoint bin2power (x : positive) (k : nat) :=
  match x with
  | xH   => 2^k
  | xO y => bin2power y (S k)
  | xI y => 2^k + bin2power y (S k)
  end.
 *)

Lemma bin2power_double : forall (x : positive) (k : N),
  bin2power x~0 k = bin2power x (N.succ k).
Proof.
  intros x. simpl. reflexivity.
Qed.

Lemma bin2power_double2 : forall (x : positive),
  bin2power x~0 0%N = (2 * bin2power x 0)%N.
Proof.
  intros x.
  destruct x.
  - simpl.


N.testbit_even_succ:
  forall a n : N, (0 <= n)%N -> N.testbit (2 * a) (N.succ n) = N.testbit a n







  simpl. reflexivity.
Qed.

Lemma bin2power_succ : forall (x : positive) (k : N),
  bin2power (x~1) k = ((2^k)%N + (bin2power (x~0) k))%N.
Proof.
  intros x k.
  reflexivity.
Qed.

Lemma bin2power_double2 : forall (x : positive),
  bin2power x 1%N = (2 * bin2power x 0)%N.
Proof.
  intros x.
  induction x.
  - rewrite bin2power_succ. rewrite bin2power_succ. rewrite N.pow_0_r.
    rewrite N.mul_add_distr_l. rewrite N.pow_1_r. rewrite N.mul_1_r.
    rewrite N.add_cancel_l.
    replace (bin2power x~0 0) with (bin2power x 1%N).





    assert ((2 * bin2power (Pos.succ x~0) 0)%N = (2 + 2*binpower (x~0) 0)%N).

    replace (x~0) with ((N.pos (N.shiftl (x) 1) 1)%N).
  



  induction x.
  - simpl. rewrite Nat.add_0_r. apply eq_S. rewrite <- plus_n_Sm.
    apply eq_S.
    destruct x.
    + simpl.




    rewrite Pos.xI_succ_xO.
    rewrite <- bin2power_double.


  rewrite bin2power_double.



  - rewrite bin2power_double.
    assert ( I: bin2power x~1 0 = S (bin2power x~0 0) ).
    { simpl. reflexivity. } rewrite I.
    assert ( J: bin2power x~1 1 = S (S (S (bin2power x~0 1) ))).
    { rewrite <- I.






  simpl. rewrite Nat.add_0_r.






Qed.

Lemma bin2power_same : forall (x: positive),
  bin2power x 0 = Pos.to_nat x.
Proof.
  intros x.
  induction x.
  - simpl. rewrite <- Pos.pred_double_succ.
    rewrite succ_succ_pred_double_succ. rewrite Pos.succ_pred_double.

    rewrite <- Pos.xI_succ_xO.
    rewrite <- bin2power_double.

    assert (I: bin2power x 1 = 2 * (bin2power x 0)).
    { induction x. simpl. rewrite Nat.add_0_r. rewrite <- plus_n_Sm.
      simpl in IHx.


    assert (I: bin2power x 1 = 2 * (bin2power x 0)).
    { induction x. simpl. rewrite Nat.add_0_r. rewrite <- plus_n_Sm.
      simpl in IHx.


Lemma tm_step_hamming : forall (x: N),
  nth_error (tm_step (N.size_nat x)) (N.to_nat x)
  = Some (odd (hamming_weight_n x)).
Proof.
  intros x.
  destruct x.
  - reflexivity.
  - induction p.
    + rewrite <- Pos.pred_double_succ.

      rewrite nat_size_pos_size.
      rewrite succ_succ_pred_double_succ at 1.
      rewrite size_succ_double_bin.

      unfold hamming_weight_n. rewrite Pos.pred_double_succ.
      rewrite hamming_weight_increase. rewrite Nat.odd_succ.
      rewrite <- Nat.negb_odd.

      rewrite tm_build. rewrite nth_error_app1.






      apply map_nth_error.




      rewrite succ_succ_pred_double_succ.







      simpl. rewrite Pos.succ_pred_double.
      rewrite size_double_bin.


      unfold N.to_nat at 2. unfold Pos.to_nat. simpl.
      
      rewrite Pos.xI_succ_xO.   transforme p~1 en Pos.succ p~0
      Lemma pred_double_succ p : pred_double (succ p) = p~1.
      Lemma pred_of_succ_nat (n:nat) : pred (of_succ_nat n) = of_nat n.
      Lemma succ_of_nat (n:nat) : n<>O -> succ (of_nat n) = of_succ_nat n.




Theorem tm_step_hamming_index : forall (n : N),
  nth_error (tm_step (N.to_nat n)) (N.to_nat n)
     = Some (odd (hamming_weight_n n)).
Proof.
  intros n.
  destruct n.
  - reflexivity.
  - induction p. simpl.




Theorem tm_step_hamming_index : forall (n m : nat) (k j: N),
  (N.to_nat k) < 2^n -> (N.to_nat j) < 2^m ->
    hamming_weight_n k = hamming_weight_n j ->
    nth_error (tm_step n) (N.to_nat k) = nth_error (tm_step m) (N.to_nat j).
Proof.
  intros n m k j. intros H I K.
  induction k.
  - simpl in K. assert (j = N0). induction j. reflexivity.
    rewrite <- N.succ_pos_pred. 
    unfold hamming_weight_n in K.
    assert (L: hamming_weight_positive p > 0).
    apply hamming_weight_positive_nonzero. rewrite <- K in L.
    apply Arith_prebase.gt_irrefl_stt in L. contradiction L.
    rewrite H0. rewrite H0 in I.
    generalize I. generalize H. apply tm_step_stable.
  - induction j. simpl in K. assert (L: hamming_weight_positive p > 0).
    apply hamming_weight_positive_nonzero. rewrite K in L.
    apply Arith_prebase.gt_irrefl_stt in L. contradiction L.
    (* coeur de l'induction *)


    induction p. simpl in K.
    symmetry in K. apply Nat.neq_succ_0 in K. contradiction K.



    Theorem N. succ_0_discr n : succ n <> 0.
  Nat.neq_succ_0: forall n : nat, S n <> 0






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






Lemma tm_step_index_split :
  forall (a b n m : nat),
  (* every natural number k can be written according to the following form *)
  a < 2^n -> (a + 2^n * b) < 2^m
    -> nth_error (tm_step m) (a + 2^n * b)
        = nth_error (tm_step (S m)) (a + 2^S n * b).
Proof.
  intros a b n m. intros H I.







Lemma tm_step_cancel_high_bits :
  forall (k b n m : nat),
  (* every natural number k can be written according to the following form *)
  S k < 2^m -> k = 2^n - 1 + 2^S n * b
    -> nth_error (tm_step m) k = nth_error (tm_step m) (S k)
      <-> odd n = true.
Proof.
  intros k b n m. intros H I.

  assert (L: 2^n - 1 < 2^n). apply Nat.sub_lt. replace (1) with (2^0) at 1.
  apply Nat.pow_le_mono_r. easy.
  apply le_0_n. simpl. reflexivity. apply Nat.lt_0_1.

  assert (M: S(2^n - 1) = 2^n). rewrite Nat.sub_1_r. apply Nat.succ_pred.
  apply Nat.pow_nonzero. apply Nat.neq_succ_0.

  assert (N: 2^n < 2^(S n)). apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
  apply Nat.lt_succ_diag_r.

  assert (J: nth_error (tm_step (S n)) (2^n-1) = nth_error (tm_step (S n)) (2^n)
              <-> odd n = true).
  rewrite tm_step_single_bit_index.
  assert (nth_error (tm_step n) (2^n - 1) = nth_error (tm_step (S n)) (2^n-1)).
  apply tm_step_stable. apply L.
  assert (2^n < 2^(S n)). apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
  apply Nat.lt_succ_diag_r.
  generalize H0. generalize L. apply Nat.lt_trans. rewrite <- H0.
  rewrite tm_step_repunit_index. destruct (odd n). easy. easy.
  rewrite <- J. rewrite I.


  destruct b.

  (* case b = 0 *)
  rewrite Nat.mul_0_r. rewrite Nat.add_0_r.
  rewrite J. rewrite Nat.mul_0_r in I. rewrite Nat.add_0_r in I.
  rewrite I in H. rewrite M.

  assert (nth_error (tm_step m) (2^n-1) = nth_error (tm_step n) (2^n-1)).
  generalize L. apply tm_step_stable. apply Nat.lt_succ_l. apply H.
  rewrite H0. rewrite tm_step_repunit_index.

  assert (nth_error (tm_step m) (2^n) = nth_error (tm_step (S n)) (2 ^ n)).
  generalize N. rewrite M in H. generalize H. apply tm_step_stable.
  rewrite H1. rewrite tm_step_single_bit_index.

  split. intros. inversion H2. reflexivity. intros. inversion H2.
  reflexivity.

  (* case b > 0 *)
  assert (S b = Pos.to_nat (Pos.of_nat (S b))).




  destruct n.
  - rewrite Nat.pow_0_r. rewrite Nat.sub_diag. rewrite plus_O_n.
    rewrite Nat.pow_1_r.
    rewrite Nat.pow_0_r in I. simpl in I. rewrite Nat.add_0_r in I.
    induction m.
    + rewrite Nat.pow_0_r in H.
      assert (K := H). rewrite Nat.lt_1_r in K.
      apply Nat.neq_succ_0 in K. contradiction K.
    +





  induction k.
  - rewrite <- I.
    assert (b=0).
    { symmetry in I. rewrite Nat.eq_add_0 in I. destruct I.
      apply Nat.mul_eq_0_r in H1. apply H1. apply Nat.pow_nonzero.
      easy. }
    rewrite H0 in I. rewrite Nat.mul_0_r in I.
    rewrite Nat.add_0_r in I. rewrite <- I.

    replace (2^n) with (S (2^n - 1)). rewrite <- I.

    split.

    intros. rewrite tm_step_head_2 at 2. rewrite tm_step_head_1. simpl.
    replace (m) with (S (m-1)) in H1 at 2.
    rewrite tm_step_head_2 in H1. rewrite tm_step_head_1 in H1. simpl in H1.
    apply H1.
    destruct m. simpl in H. apply Nat.lt_irrefl in H. contradiction H.
    rewrite Nat.sub_1_r. simpl. reflexivity.

    intros. rewrite tm_step_head_2 in H1 at 2.
    rewrite tm_step_head_1 in H1. simpl in H1. inversion H1.
    rewrite <- I.
    apply eq_S in I. rewrite I at 1.
    apply Nat.pred_inj. apply Nat.neq_succ_0. apply Nat.pow_nonzero.
    easy. simpl. rewrite Nat.sub_1_r. reflexivity.

  - apply IHk. apply Nat.lt_succ_l in H. apply H.
    rewrite <- I.




    split. intros. apply H1.








    assert (K: S (2 ^ n - 1 + 2 ^ S n * b) = (2 ^ n + 2 ^ S n * b)).
    + rewrite Nat.sub_1_r. rewrite <- Nat.add_succ_l.
      rewrite Nat.succ_pred_pos. reflexivity.
      rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
    + rewrite K. symmetry.
      split. intros. symmetry.
      apply tm_step_add_range2_flip_two.


  (*
  assert (K: nth_error (tm_step n) a = Some (odd n)). rewrite I.
  apply tm_step_repunit.
   *)
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)).
  






Déclagae vers la droite de k zéros pour un Bins :
Pos.iter xO xH 3.   (ici k = 3)
  --> on ajoute 3 zéros à gauche









Require Import BinPosDef.


(* Autre construction de la suite, ici n est le nombre de termes *)
(* la construction se fait à l'envers *)
(*
Fixpoint tm_bin_rev (n: nat) : list bool :=
  match n with
  | 0 => nil
  | S n' => let t := tm_bin_rev n' in
            let m := Pos.of_nat n' in
              (xorb (hd true t) (odd (Pos.size_nat
                match Pos.lxor m (Pos.pred m) with
                | N0 => BinNums.xH
                | Npos(p)  => p
                end))) :: t
  end.
*)

Fixpoint tm_bin (n: nat) : list bool :=
  match n with
  | 0 => nil
  | S n' => let t := tm_bin n' in
            let m := Pos.of_nat n' in
            t ++ [ xorb (last t true)
                     (odd (Pos.size_nat match Pos.lxor m (Pos.pred m) with
                                        | N0 => BinNums.xH
                                        | Npos(p)  => p
                                        end)) ]
  end.

Theorem tm_morphism_eq_bin : forall (k : nat), tm_step k = tm_bin (2^k).
Proof.
Admitted.





Theorem tm_step_consecutive : forall (n : nat) (l1 l2 : list bool) (b1 b2 : bool),
  tm_step n = l1 ++ b1 :: b2 :: l2 ->
    (eqb b1 b2) = 
      let ind_b2 := Pos.of_nat (S (length l1)) in (* index of b2 *)
      let ind_b1 := Pos.pred ind_b2 in            (* index of b1 *)
       even (Pos.size_nat
         match Pos.lxor ind_b1 ind_b2 with
         | N0 => BinNums.xH
         | Npos(p)  => p
         end).
Proof.
  intros n l1 l2 b1 b2.
  destruct n.
  - simpl. intros H. induction l1.
    + rewrite app_nil_l in H. discriminate.
    + destruct l1. rewrite app_nil_l in IHl1. discriminate. discriminate.
  - rewrite tm_build.
Admitted.




(*
  intros n l1 l2 b1 b2.
  intros H.
  induction l1.
  - simpl. destruct n. discriminate.
    rewrite app_nil_l in H. assert (I := H).
    rewrite tm_step_head_2 in I. injection I.
    assert (J: tl (tl (tm_morphism (tm_step n))) = l2).
    { replace (tm_morphism (tm_step n)) with (tm_step (S n)).
      rewrite H. reflexivity. reflexivity. }
    (* rewrite J *) intros. rewrite <- H1. rewrite <- H2. reflexivity.
  - replace (S (length (a :: l1))) with (S (S (length l1))).







    destruct b1 in H.
    + destruct b2 in H.
      (* Case 1: b1 = true, b2 = true  (false)   *)
      replace (l2) with (tl (tl (tm_step (S n)))) in H.
      assert (J : tm_step (S n) = false :: true :: tl (tl (tm_step (S n)))).
      apply tm_step_head_2. rewrite J in H. discriminate H. rewrite H. reflexivity.
      (* Case 2: b1 = true, b2 = false  (false)   *)
      replace (l2) with (tl (tl (tm_step (S n)))) in H.
      assert (J : tm_step (S n) = false :: true :: tl (tl (tm_step (S n)))).
      apply tm_step_head_2. rewrite J in H. discriminate H. rewrite H. reflexivity.
    + destruct b2.
      (* Case 3: b1 = false, b2 = true  (TRUE)   *)
      discriminate.






    inversion H.
    discriminate tm_step_head_2.
    rewrite <- tm_step_head_2 in H.
    discriminate H.
    discriminate tm_step_head_2.

  - simpl. simpl. simpl in H.
    destruct n in H. discriminate H.
    replace (l2) with (tl (tl (tm_step (S n)))) in H.
    specialize (H tm_step_head_2).
    rewrite <- tm_step_head_2 in H.



Lemma tm_step_head_2 : forall (n : nat),
    tm_step (S n) = false :: true :: tl (tl (tm_step (S n))).
 *)