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Thomas Baruchel
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@ -301,15 +301,6 @@ Proof.
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Qed.
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(**
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In this notebook I call "proper palindrome" in the Thue-Morse sequence,
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any palindromic subsequence such that the middle (of the palindromic
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subsequence) is not also the middle of a wider palindromic subsequence.
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In ther words, a proper palindrome can not be enlarged by adding more
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termes on both sides.
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*)
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(* palidrome 2*4 : soit centré en 4n soit pas plus de 2*6 *)
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(* modifier l'énoncé : ajouter le modulo = 2 ET la différence sur le 7ème
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ET existence d'un palindrome 2 * - *)
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@ -2707,123 +2698,3 @@ Proof.
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apply tm_step_palindromic_power2_odd with (n := n) (tl := tl).
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assumption. assumption. rewrite <- E'. assumption.
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Qed.
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Theorem tm_step_palindrome_power2' :
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forall (m n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ (rev a) ++ tl
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-> length a = 2^m
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-> 2 < m
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-> length (hd ++ a) mod 2^ (Nat.double (Nat.div2 (S m))) = 2^ (pred (Nat.double (Nat.div2 (S m)))).
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Proof.
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intros m n hd a tl. intros H I J.
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assert (K: length (hd ++ a) mod 2^ (pred (Nat.double (Nat.div2 (S m)))) = 0).
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generalize J. generalize I. generalize H. apply tm_step_palindrome_power2.
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rewrite <- Nat.div_exact in K.
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assert (L: Nat.Even
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(length (hd ++ a) / 2 ^ pred (Nat.double (Nat.div2 (S m))))
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\/ Nat.Odd
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(length (hd ++ a) / 2 ^ pred (Nat.double (Nat.div2 (S m))))).
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apply Nat.Even_or_Odd.
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destruct L.
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- assert (length (hd ++ a) mod 2 ^ Nat.double (Nat.div2 (S m)) = 0).
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rewrite K. apply Nat.Even_double in H0. symmetry in H0.
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rewrite Nat.double_twice in H0. rewrite <- H0. rewrite Nat.mul_assoc.
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rewrite Nat.mul_shuffle0. rewrite Nat.mul_comm. rewrite Nat.mul_assoc.
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rewrite <- Nat.pow_succ_r. rewrite Nat.succ_pred_pos. rewrite Nat.mul_comm.
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apply Nat.mod_mul. apply Nat.pow_nonzero. easy.
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assert (Nat.Even (S m) \/ Nat.Odd (S m)). apply Nat.Even_or_Odd.
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destruct H1.
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apply Nat.Even_double in H1. rewrite <- H1. apply Nat.lt_0_succ.
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apply Nat.Odd_double in H1. apply Nat.succ_inj in H1.
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rewrite <- H1. apply Nat.lt_succ_l in J. apply Nat.lt_succ_l in J.
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assumption. apply Nat.le_0_l.
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(* on cherche une contradiction à partir de H1 *)
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assert (Nat.Even (S m) \/ Nat.Odd (S m)). apply Nat.Even_or_Odd.
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destruct H2.
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+ assert (E := H2). apply Nat.Even_double in H2. rewrite <- H2 in H1.
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(*
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assert (length (hd ++ a ++ rev a) mod 2^(S m) = 2^m).
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rewrite app_assoc. rewrite app_length.
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rewrite <- Nat.add_mod_idemp_l. rewrite H1. rewrite rev_length.
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rewrite I. rewrite Nat.mod_small_iff. simpl.
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apply Nat.lt_add_pos_r. rewrite Nat.add_0_r.
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rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
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apply Nat.pow_nonzero. easy. apply Nat.pow_nonzero. easy.
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*)
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(*
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apply Nat.Even_double in H0.
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*)
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(* montrer que sur les repeating patterns de taille 2^(S (S m))
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les deux valeurs centrales sont distinctes *)
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rewrite <- Nat.div_exact in H1.
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(* on prouve que (length (hd ++ a) / 2^(S m)) est nn nul *)
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destruct (length (hd ++ a) / 2^(S m)).
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rewrite Nat.mul_0_r in H1. rewrite app_length in H1.
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apply Nat.eq_add_0 in H1. destruct H1. rewrite H3 in I.
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symmetry in I. apply Nat.pow_nonzero in I. contradiction. easy.
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Abort.
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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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Lemma tm_step_repeating_patterns2 :
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forall (n m : nat) (hd pat tl : list bool),
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tm_step n = hd ++ pat ++ tl
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-> length pat = 2^m
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-> length hd mod (2^m) = 0
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-> pat = tm_step m \/ pat = map negb (tm_step m).
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Proof.
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Nat.mul_mod_distr_l:
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forall a b c : nat, b <> 0 -> c <> 0 -> (c * a) mod (c * b) = c * (a mod b)
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Nat.mul_mod_distr_r:
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forall a b c : nat, b <> 0 -> c <> 0 -> (a * c) mod (b * c) = a mod b * c
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Nat.mod_mul_r:
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forall a b c : nat,
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b <> 0 -> c <> 0 -> a mod (b * c) = a mod b + b * ((a / b) mod c)
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(*
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Theorem tm_step_palindrome_power2_reciprocal :
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forall (m n k : nat) (hd tl : list bool),
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tm_step n = hd ++ tl
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-> length hd = S (Nat.double k) * 2^m
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-> odd m = true
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-> skipn ((length hd) - 2^m) hd = rev (firstn (2^m) tl).
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Proof.
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intros m n.
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assert (A: odd m = true
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-> (forall k hd tl, tm_step n = hd ++ tl
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->length hd = S (Nat.double k) * 2^m
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-> skipn ((length hd) - 2^m) hd = rev (firstn (2^m) tl))
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-> (forall k hd tl, tm_step (S n) = hd ++ tl
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-> length hd = S (Nat.double k) * 2^m
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-> skipn ((length hd) - 2^m) hd = rev (firstn (2^m) tl))).
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intros H I.
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intros k hd tl. intros J K.
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rewrite tm_build in J.
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*)
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(*
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Lemma tm_step_proper_palindrome_center :
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forall (m n k : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ (rev a) ++ tl
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-> 6 < length a
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-> length a = 2^(Nat.double m) (* palindrome non propre *)
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-> skipn (length hd - length a) hd = rev (firstn (length a) tl).
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*)
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@ -783,5 +783,47 @@ Proof.
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assert (even m = true). generalize J. generalize I. generalize H.
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apply tm_step_square_rev_even.
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(*
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TODO: voir s'il faut remplacer la dernière implication
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par une équivalence
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*)
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Abort.
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Theorem tm_step_palindrome_power2' :
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forall (m n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ (rev a) ++ tl
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-> length a = 2^m
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-> 2 < m
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-> length (hd ++ a) mod 2^ (Nat.double (Nat.div2 (S m))) = 2^ (pred (Nat.double (Nat.div2 (S m)))).
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Proof.
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intros m n hd a tl. intros H I J.
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assert (K: length (hd ++ a) mod 2^ (pred (Nat.double (Nat.div2 (S m)))) = 0).
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generalize J. generalize I. generalize H. apply tm_step_palindrome_power2.
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rewrite <- Nat.div_exact in K.
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assert (L: Nat.Even
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(length (hd ++ a) / 2 ^ pred (Nat.double (Nat.div2 (S m))))
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\/ Nat.Odd
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(length (hd ++ a) / 2 ^ pred (Nat.double (Nat.div2 (S m))))).
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apply Nat.Even_or_Odd.
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destruct L.
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- assert (length (hd ++ a) mod 2 ^ Nat.double (Nat.div2 (S m)) = 0).
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rewrite K. apply Nat.Even_double in H0. symmetry in H0.
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rewrite Nat.double_twice in H0. rewrite <- H0. rewrite Nat.mul_assoc.
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rewrite Nat.mul_shuffle0. rewrite Nat.mul_comm. rewrite Nat.mul_assoc.
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rewrite <- Nat.pow_succ_r. rewrite Nat.succ_pred_pos. rewrite Nat.mul_comm.
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apply Nat.mod_mul. apply Nat.pow_nonzero. easy.
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assert (Nat.Even (S m) \/ Nat.Odd (S m)). apply Nat.Even_or_Odd.
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destruct H1.
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apply Nat.Even_double in H1. rewrite <- H1. apply Nat.lt_0_succ.
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apply Nat.Odd_double in H1. apply Nat.succ_inj in H1.
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rewrite <- H1. apply Nat.lt_succ_l in J. apply Nat.lt_succ_l in J.
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assumption. apply Nat.le_0_l.
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(* on cherche une contradiction à partir de H1 *)
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assert (Nat.Even (S m) \/ Nat.Odd (S m)). apply Nat.Even_or_Odd.
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destruct H2.
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+ assert (E := H2). apply Nat.Even_double in H2. rewrite <- H2 in H1.
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Abort.
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