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thue-morse.v
144
thue-morse.v
@ -1117,12 +1117,147 @@ Proof.
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+ simpl in H. symmetry in H. apply Bool.diff_true_false in H. contradiction H.
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+ simpl in H. symmetry in H. apply Bool.diff_true_false in H. contradiction H.
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Qed.
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Qed.
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Lemma lt_split_bits : forall n m k j,
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0 < k -> j < m -> k * 2^m < 2^n -> k * 2^m +2^j < 2^n.
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Proof.
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intros n m k j. intros H I J.
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assert (N0: 2^m < 2^n). assert (2^m <= k * 2^m).
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replace (k * 2^m) with (1*2^m + (k-1)*2^m).
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simpl. rewrite Nat.add_0_r. apply Nat.le_add_r.
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rewrite <- Nat.mul_add_distr_r. rewrite Nat.add_1_l.
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rewrite <- Nat.sub_succ_l. rewrite <- Nat.add_1_r. rewrite Nat.add_sub.
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reflexivity. rewrite Nat.le_succ_l. assumption.
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generalize J. generalize H0. apply Nat.le_lt_trans.
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assert (N1: m < n). apply Nat.pow_lt_mono_r_iff in N0. assumption.
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apply Nat.lt_1_2.
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assert (N2: 0 < m). assert (0 <= j). apply Nat.le_0_l.
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generalize I. generalize H0. apply Nat.le_lt_trans.
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assert (N4: 0 < m-j). rewrite Nat.add_lt_mono_r with (p := j). simpl.
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replace (m-j+j) with (m). assumption. symmetry. apply Nat.sub_add.
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generalize I. apply Nat.lt_le_incl.
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assert (N5: j < n). generalize N1. generalize I. apply Nat.lt_trans.
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assert (N6: 0 < n-j). rewrite Nat.add_lt_mono_r with (p := j). simpl.
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replace (n-j+j) with (n). apply N5. symmetry. apply Nat.sub_add.
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generalize N5. apply Nat.lt_le_incl.
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replace (k*2^m + 2^j < 2^n) with ((k*2^(m-j)+1)*(2^j) < 2^(n-j)*2^j).
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apply Nat.mul_lt_mono_pos_r.
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rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
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rewrite Nat.add_1_r. apply lt_even_even.
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- destruct (m-j).
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+ apply Nat.lt_irrefl in N4. contradiction N4.
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+ rewrite Nat.pow_succ_r. rewrite Nat.even_mul. rewrite Nat.even_mul.
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rewrite Nat.even_2. rewrite orb_true_r. reflexivity. apply Nat.le_0_l.
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- destruct (n-j).
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+ apply Nat.lt_irrefl in N6. contradiction N6.
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+ rewrite Nat.pow_succ_r. rewrite Nat.even_mul.
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rewrite Nat.even_2. rewrite orb_true_l. reflexivity. apply Nat.le_0_l.
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- apply Nat.mul_lt_mono_pos_r with (p := 2^j).
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rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
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rewrite <- Nat.pow_add_r.
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rewrite <- Nat.mul_assoc. rewrite <- Nat.pow_add_r.
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rewrite Nat.sub_add. rewrite Nat.sub_add. assumption.
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apply Nat.lt_le_incl. assumption.
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apply Nat.lt_le_incl. assumption.
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- rewrite Nat.mul_add_distr_r.
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rewrite <- Nat.mul_assoc.
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rewrite <- Nat.pow_add_r. rewrite Nat.mul_1_l. rewrite Nat.sub_add.
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rewrite <- Nat.pow_add_r. rewrite Nat.sub_add. reflexivity.
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apply Nat.lt_le_incl. assumption.
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apply Nat.lt_le_incl. assumption.
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Qed.
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Lemma tm_step_repeating_patterns :
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forall (n m i j : nat),
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i < m -> j < m
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-> forall k, k < 2^n -> nth_error (tm_step m) (2^i)
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= nth_error (tm_step m) (2^j)
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<-> nth_error (tm_step (m+n)) (k * 2^m + (2^i))
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= nth_error (tm_step (m+n)) (k * 2^m + (2^j)).
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Proof.
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intros n m i j. intros H I.
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induction n.
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- rewrite Nat.add_0_r. intro k. simpl. rewrite Nat.lt_1_r. intro.
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rewrite H0. simpl. easy.
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- rewrite Nat.add_succ_r. intro k. intro.
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rewrite tm_build.
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assert (S: k < 2^n \/ 2^n <= k). apply Nat.lt_ge_cases.
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destruct S.
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assert (k*2^m < 2^(m+n)).
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destruct k.
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+ simpl. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
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+ rewrite Nat.mul_lt_mono_pos_r with (p := 2^m) in H1.
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rewrite <- Nat.pow_add_r in H1. rewrite Nat.add_comm.
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assumption. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
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+ rewrite nth_error_app1. rewrite nth_error_app1.
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generalize H1. apply IHn.
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rewrite tm_size_power2.
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assert (k * 2^m + 2^j < 2^(m+n)).
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destruct k. simpl. apply Nat.log2_lt_cancel.
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rewrite Nat.log2_pow2. rewrite Nat.log2_pow2.
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apply Nat.lt_lt_add_r. assumption. apply Nat.le_0_l. apply Nat.le_0_l.
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generalize H2. apply lt_split_bits.
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apply Nat.lt_0_succ. assumption. assumption.
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rewrite tm_size_power2.
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destruct k. simpl. apply Nat.log2_lt_cancel.
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rewrite Nat.log2_pow2. rewrite Nat.log2_pow2.
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apply Nat.lt_lt_add_r. assumption. apply Nat.le_0_l. apply Nat.le_0_l.
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generalize H2. apply lt_split_bits.
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apply Nat.lt_0_succ. assumption.
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+ assert (J: 2 ^ (m + n) <= k * 2 ^ m).
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rewrite Nat.pow_add_r. rewrite Nat.mul_comm.
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apply Nat.mul_le_mono_r. assumption.
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rewrite nth_error_app2. rewrite nth_error_app2. rewrite tm_size_power2.
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assert (forall a b, option_map negb a = option_map negb b <-> a = b).
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intros a b. destruct a. destruct b. destruct b0. destruct b.
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simpl. split. intro. reflexivity. intro. reflexivity.
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simpl. split. intro. inversion H2. intro. inversion H2.
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destruct b. simpl. split. intro. inversion H2. intro. inversion H2.
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simpl. split. intro. reflexivity. intro. reflexivity.
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split. intro. inversion H2. intro. inversion H2.
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destruct b. split. intro. inversion H2. intro. inversion H2.
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split. intro. reflexivity. intro. reflexivity.
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replace (k * 2 ^ m + 2^i - 2^(m + n)) with ((k-2^n)*2^m + 2^i).
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replace (k * 2 ^ m + 2^j - 2^(m + n)) with ((k-2^n)*2^m + 2^j).
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rewrite nth_error_map. rewrite nth_error_map.
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rewrite H2. apply IHn.
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rewrite Nat.add_lt_mono_r with (p := 2^n). rewrite Nat.sub_add.
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rewrite Nat.pow_succ_r in H0. replace (2) with (1+1) in H0.
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rewrite Nat.mul_add_distr_r in H0. simpl in H0.
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rewrite Nat.add_0_r in H0. assumption. reflexivity.
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apply Nat.le_0_l. assumption.
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rewrite Nat.mul_sub_distr_r. rewrite <- Nat.pow_add_r.
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rewrite Nat.add_sub_swap. replace (n+m) with (m+n). reflexivity.
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rewrite Nat.add_comm. reflexivity. assumption.
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rewrite Nat.mul_sub_distr_r. rewrite <- Nat.pow_add_r.
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rewrite Nat.add_sub_swap. replace (n+m) with (m+n). reflexivity.
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rewrite Nat.add_comm. reflexivity. assumption.
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rewrite tm_size_power2.
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assert (k*2^m <= k*2^m + 2^j). apply Nat.le_add_r.
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generalize H2. generalize J. apply Nat.le_trans.
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rewrite tm_size_power2.
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assert (k*2^m <= k*2^m + 2^i). apply Nat.le_add_r.
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generalize H2. generalize J. apply Nat.le_trans.
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Qed.
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(* TODO: écrire deux lemmes, avec
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- une implication vers la première moitié de la construction suivante
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- une implication vers la seconde moitié de la construction suivante
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ENSUITE : voir si on peut itérer sur le facteur multiplicatif k *)
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Lemma tm_step_add_small_power :
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Lemma tm_step_add_small_power :
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forall (n m k j : nat),
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forall (n m k j : nat),
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@ -1203,6 +1338,7 @@ Proof.
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rewrite Nat.mul_1_l. reflexivity. rewrite Nat.mul_1_l. reflexivity.
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rewrite Nat.mul_1_l. reflexivity. rewrite Nat.mul_1_l. reflexivity.
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apply N0.
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apply N0.
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TODO
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