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Thomas Baruchel
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thue-morse.v
297
thue-morse.v
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@ -556,8 +556,7 @@ Proof.
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apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. apply Nat.lt_succ_diag_r.
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assumption.
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assert (G: k = 0 \/ 0 < k). apply Nat.eq_0_gt_0_cases.
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destruct G as [G1|G2].
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assert (G: k = 0 \/ 0 < k). apply Nat.eq_0_gt_0_cases. destruct G as [G1|G2].
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(* k = 0 *)
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rewrite G1. rewrite Nat.mul_0_l. rewrite Nat.add_0_l.
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@ -631,6 +630,293 @@ Qed.
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Lemma tm_step_pred : forall (n k m : nat),
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S (2*k) * 2^m < 2^n -> (
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nth_error (tm_step n) (S (2*k) * 2^m)
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= nth_error (tm_step n) (S (2*k) * 2^m - 1)
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<-> odd m = true).
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Proof.
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intros n k m.
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generalize n. induction m. rewrite Nat.pow_0_r. rewrite Nat.mul_1_r.
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destruct n0. rewrite Nat.pow_0_r. rewrite Nat.lt_1_r.
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intro H. apply Nat.neq_succ_0 in H. contradiction H.
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rewrite Nat.odd_0. rewrite Nat.sub_succ. rewrite Nat.sub_0_r.
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rewrite <- tm_step_double_index. intro H. split. intro I.
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symmetry in I. apply tm_step_succ_double_index in I. contradiction I.
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apply Nat.lt_succ_l in H. rewrite Nat.pow_succ_r' in H.
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rewrite <- Nat.mul_lt_mono_pos_l in H. assumption. apply Nat.lt_0_2.
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intro I. apply diff_false_true in I. contradiction I.
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intro n0. intro I.
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destruct n0. rewrite Nat.pow_0_r in I. rewrite Nat.lt_1_r in I.
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rewrite Nat.mul_eq_0 in I. destruct I.
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apply Nat.neq_succ_0 in H. contradiction H.
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apply Nat.pow_nonzero in H. contradiction H. easy.
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rewrite Nat.pow_succ_r'. rewrite Nat.mul_assoc.
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replace (S (2*k) * 2) with (2* (S (2*k))).
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rewrite <- Nat.mul_assoc. rewrite <- tm_step_double_index.
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(* réécriture de I *)
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rewrite Nat.pow_succ_r' in I.
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rewrite Nat.pow_succ_r' in I.
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rewrite Nat.mul_assoc in I.
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replace (S (2*k) * 2) with (2* (S (2*k))) in I.
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rewrite <- Nat.mul_assoc in I.
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rewrite <- Nat.mul_lt_mono_pos_l in I.
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assert (J := I). apply IHm in J.
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assert (M: 0 < 2^m). destruct m. simpl. apply Nat.lt_0_1.
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replace (0) with (0 ^ (S m)) at 1.
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apply Nat.pow_lt_mono_l. apply Nat.neq_succ_0. apply Nat.lt_0_2.
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apply Nat.pow_0_l. apply Nat.neq_succ_0.
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assert (L: S (2 * k) * 2 ^ m - 1 < S (2 * k) * 2 ^ m).
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replace (S (2 * k) * 2 ^ m) with (S (S (2 * k) * 2 ^ m - 1)) at 2.
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apply Nat.lt_succ_diag_r.
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rewrite <- Nat.sub_succ_l. rewrite Nat.sub_1_r. rewrite pred_Sn.
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reflexivity. rewrite Nat.le_succ_l. apply Nat.mul_pos_pos.
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apply Nat.lt_0_succ. assumption.
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assert (N: 2 * (S (2 * k) * 2 ^ m) - 1 < length (tm_step (S n0))).
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rewrite tm_size_power2.
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rewrite <- Nat.le_succ_l. rewrite <- Nat.add_1_r.
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rewrite Nat.sub_add. rewrite Nat.pow_succ_r'.
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rewrite <- Nat.mul_le_mono_pos_l.
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apply Nat.lt_le_incl. assumption. apply Nat.lt_0_2.
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apply Nat.le_succ_l. apply Nat.mul_pos_pos. apply Nat.lt_0_2.
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apply Nat.mul_pos_pos. apply Nat.lt_0_succ.
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assumption.
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assert (nth_error (tm_step n0) ((S (2 * k) * 2 ^ m) - 1)
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<> nth_error (tm_step (S n0)) (S (2 * (S (2 * k) * 2 ^ m - 1)))).
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apply tm_step_succ_double_index.
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generalize I. generalize L. apply Nat.lt_trans.
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replace (S (2 * (S (2 * k) * 2 ^ m - 1))) with (2 * (S (2*k) * 2^m) - 1) in H.
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rewrite Nat.odd_succ. rewrite <- Nat.negb_odd. destruct (odd m).
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split. intro K. rewrite <- K in H.
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destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m));
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destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m - 1)).
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destruct J. replace (Some b) with (Some b0) in H. contradiction H.
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reflexivity. symmetry. apply H1. reflexivity.
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destruct J. replace (Some b) with (None : option bool) in H.
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contradiction H. reflexivity. symmetry. apply H1. reflexivity.
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destruct J. replace (None : option bool) with (Some b) in H.
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contradiction H. reflexivity. symmetry. apply H1. reflexivity.
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contradiction H. reflexivity.
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destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m));
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destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m - 1)).
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intro K. simpl in K. apply diff_false_true in K. contradiction K.
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intro K. simpl in K. apply diff_false_true in K. contradiction K.
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intro K. simpl in K. apply diff_false_true in K. contradiction K.
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intro K. simpl in K. apply diff_false_true in K. contradiction K.
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rewrite nth_error_nth' with (d := false) in J.
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rewrite nth_error_nth' with (d := false) in J.
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rewrite nth_error_nth' with (d := false) in H.
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rewrite nth_error_nth' with (d := false) in H.
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rewrite nth_error_nth' with (d := false).
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rewrite nth_error_nth' with (d := false).
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destruct (nth (S (2 * k) * 2 ^ m) (tm_step n0) false);
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destruct (nth (S (2 * k) * 2 ^ m - 1) (tm_step n0) false);
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destruct (nth (2 * (S (2 * k) * 2 ^ m) - 1) (tm_step (S n0)) false).
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simpl; split; reflexivity.
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destruct J. rewrite H0 in H. contradiction H. reflexivity.
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reflexivity.
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simpl; split; reflexivity. contradiction H. reflexivity.
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contradiction H. reflexivity.
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simpl; split; reflexivity.
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destruct J. rewrite H0 in H. contradiction H. reflexivity.
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reflexivity.
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simpl; split; reflexivity.
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assumption.
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rewrite tm_size_power2. assumption.
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assumption.
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rewrite tm_size_power2. generalize I. generalize L.
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apply Nat.lt_trans.
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rewrite tm_size_power2. generalize I. generalize L.
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apply Nat.lt_trans.
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rewrite tm_size_power2. assumption.
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rewrite Nat.mul_sub_distr_l. rewrite Nat.mul_1_r.
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rewrite <- Nat.sub_succ_l.
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replace (S (2 * (S (2 * k) * 2 ^ m))) with (2 * (S (2 * k) * 2 ^ m) + 1).
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rewrite <- Nat.add_cancel_r with (p := 1).
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rewrite Nat.sub_add.
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destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m));
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destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m - 1));
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destruct (nth_error (tm_step (S n0)) (2 * (S (2 * k) * 2 ^ m) - 1)).
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destruct (b); destruct (b0); destruct (b1).
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split. reflexivity. reflexivity.
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destruct J. rewrite H0 in H. contradiction H. reflexivity.
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reflexivity. simpl. split; reflexivity.
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contradiction H. reflexivity. contradiction H. reflexivity.
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simpl. split; reflexivity.
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destruct J. rewrite H0 in H. contradiction H. reflexivity.
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reflexivity. contradiction H. reflexivity.
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destruct (b); destruct (b0). destruct J. rewrite H0.
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split. intro J. inversion J. destruct H0. reflexivity.
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intro J. simpl in J. symmetry in J. apply diff_false_true in J.
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contradiction J. reflexivity.
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split. intro K. inversion K.
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apply Nat.lt_0_2. apply Nat.mul_comm.
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Lemma tm_step_repeating_patterns :
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forall (n m i j : nat),
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i < 2^m -> j < 2^m
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-> forall k, k < 2^n -> nth_error (tm_step m) i
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= nth_error (tm_step m) j
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<-> nth_error (tm_step (m+n)) (k * 2^m + i)
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= nth_error (tm_step (m+n)) (k * 2^m + j).
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induction m. rewrite Nat.pow_0_r. rewrite Nat.mul_1_r.
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rewrite Nat.odd_0. rewrite Nat.sub_succ. rewrite Nat.sub_0_r.
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destruct n. rewrite Nat.pow_0_r. rewrite Nat.lt_1_r.
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intro H. apply Nat.neq_succ_0 in H. contradiction H.
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rewrite <- tm_step_double_index. intro H. split. intro I.
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symmetry in I. apply tm_step_succ_double_index in I. contradiction I.
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apply Nat.lt_succ_l in H. rewrite Nat.pow_succ_r' in H.
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rewrite <- Nat.mul_lt_mono_pos_l in H. assumption. apply Nat.lt_0_2.
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intro I. apply diff_false_true in I. contradiction I.
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intro H.
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- induction m.
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+ rewrite Nat.pow_0_r. rewrite Nat.mul_1_r.
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rewrite Nat.odd_0. rewrite Nat.sub_succ. rewrite Nat.sub_0_r.
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destruct n. rewrite Nat.pow_0_r in H. rewrite Nat.mul_1_r in H.
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rewrite Nat.lt_1_r in H. apply Nat.neq_succ_0 in H. contradiction H.
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rewrite <- tm_step_double_index.
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split.
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intro I. symmetry in I.
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apply tm_step_succ_double_index in I. contradiction I.
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rewrite Nat.pow_0_r in H. rewrite Nat.mul_1_r in H.
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apply Nat.lt_succ_l in H. rewrite Nat.pow_succ_r' in H.
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rewrite <- Nat.mul_lt_mono_pos_l in H. assumption. apply Nat.lt_0_2.
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intro I. apply diff_false_true in I. contradiction I.
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+
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split.
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- induction m.
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+ rewrite Nat.pow_0_r. rewrite Nat.mul_1_r.
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rewrite Nat.odd_0. rewrite Nat.sub_succ. rewrite Nat.sub_0_r.
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destruct n. rewrite Nat.pow_0_r in H. rewrite Nat.mul_1_r in H.
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rewrite Nat.lt_1_r in H. apply Nat.neq_succ_0 in H. contradiction H.
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rewrite <- tm_step_double_index. intro I. symmetry in I.
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apply tm_step_succ_double_index in I. contradiction I.
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rewrite Nat.pow_0_r in H. rewrite Nat.mul_1_r in H.
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apply Nat.lt_succ_l in H. rewrite Nat.pow_succ_r' in H.
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rewrite <- Nat.mul_lt_mono_pos_l in H. assumption. apply Nat.lt_0_2.
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+
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Theorem tm_step_stable : forall (n m k : nat),
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k < 2^n -> k < 2^m -> nth_error (tm_step n) k = nth_error (tm_step m) k.
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Theorem tm_step_double_index : forall (n k : nat),
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nth_error (tm_step n) k = nth_error (tm_step (S n)) (2*k).
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Theorem tm_step_succ_double_index : forall (n k : nat),
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k < 2^n -> nth_error (tm_step n) k <> nth_error (tm_step (S n)) (S (2*k)).
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Require Import BinPosDef.
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Require Import BinNatDef.
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Lemma tm_step_double_index_pos : forall (p : positive),
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nth_error (tm_step (Pos.size_nat p)) (Pos.to_nat p) =
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nth_error (tm_step (S (Pos.size_nat p))) (Pos.to_nat (p~0)).
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Proof.
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intro p.
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simpl.
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Theorem tm_step_double_index : forall (n k : nat),
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nth_error (tm_step n) k = nth_error (tm_step (S n)) (2*k).
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(*
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From a(0) to a(2n+1), there are n+1 terms equal to 0 and n+1 terms equal to 1 (see Hassan Tarfaoui link, Concours Général 1990). - Bernard Schott, Jan 21 2022
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TODO Search "count_occ"
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*)
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Theorem tm_step_count_occ : forall (hd tl : list bool) (k : nat),
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tm_step k = hd ++ tl -> even (length hd) = true
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-> count_occ Bool.bool_dec hd true = count_occ Bool.bool_dec hd false.
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Proof.
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intros hd tl k. intros H I.
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destruct k.
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- assert (J: hd = nil). destruct hd. reflexivity.
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simpl in H. inversion H.
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symmetry in H2. apply app_eq_nil in H2.
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destruct H2. rewrite H0 in I. simpl in I. inversion I.
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rewrite J. reflexivity.
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- rewrite <- tm_step_lemma in H.
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generalize I. generalize H. induction hd.
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reflexivity.
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rewrite <- length_zero_iff_nil. simpl.
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PeanoNat.Nat.neq_succ_0: forall n : nat, S n <> 0
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(* TODO: supprimer head_2 *)
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(* vérifier si les deux sont nécessaires *)
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@ -654,19 +940,12 @@ Definition hamming_weight_n (x: N) :=
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| Npos x => hamming_weight_positive x
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end.
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Lemma hamming_weight_remove_high : forall (x : N),
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(0 < x)%N -> hamming_weight_n x = S (hamming_weight_n (x - 2^(N.log2 x))).
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Proof.
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intros x.
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N.shiftl_spec_alt:
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forall a n m : N, N.testbit (N.shiftl a n) (m + n) = N.testbit a m
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Lemma hamming_weight_positive_nonzero : forall (x: positive),
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