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Thomas Baruchel
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54cdff7f66
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4fcaba8c73
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@ -1650,3 +1650,54 @@ Proof.
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generalize H1. generalize H0. apply IHn.
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- assumption.
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Qed.
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(*
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Theorem tm_step_repeating_patterns_no_square :
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forall (n m : nat) (hd a b tl : list bool),
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tm_step n = hd ++ a ++ b ++ tl -> length a = 2^m
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-> (forall k, length hd = k * 2 ^ S m -> a <> b).
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Proof.
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intros n m hd a b tl. intros H I.
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induction n.
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- assert (m = 0).
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destruct m. reflexivity.
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assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
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rewrite H in H0 at 2. rewrite app_length in H0. rewrite Nat.add_comm in H0.
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rewrite app_length in H0. rewrite <- Nat.add_assoc in H0.
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rewrite I in H0. rewrite Nat.pow_succ_r in H0. symmetry in H0.
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apply Nat.eq_add_1 in H0. destruct H0; destruct H0.
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apply Nat.mul_eq_1 in H0. destruct H0. inversion H0.
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apply Nat.mul_eq_0 in H0. destruct H0. inversion H0.
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apply Nat.pow_nonzero in H0. contradiction H0. easy. apply le_0_n.
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intro k. intro J. rewrite H0 in J. rewrite Nat.pow_1_r in J.
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assert (k = 0).
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assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
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rewrite H in H1 at 2. rewrite app_length in H1. rewrite J in H1.
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symmetry in H1. apply Nat.eq_add_1 in H1. destruct H1; destruct H1.
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apply Nat.mul_eq_1 in H1. destruct H1. inversion H3.
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apply Nat.mul_eq_0 in H1. destruct H1. assumption. inversion H1.
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rewrite H1 in J. rewrite Nat.mul_0_l in J. rewrite length_zero_iff_nil in J.
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assert ({a=b} + {~ a=b}). apply list_eq_dec. apply bool_dec.
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destruct H2. rewrite <- e in H.
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assert (0 < length a). rewrite I. rewrite H0. apply Nat.lt_0_1.
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assert (hd <> nil). generalize H2. generalize H.
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apply tm_step_square_prefix_not_nil. rewrite J in H3.
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contradiction H3. reflexivity. assumption.
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-
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Reformulation:
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forall (n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ a ++ tl
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-> 0 < length a
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-> (length hd) mod (length (a ++ a)) <> 0.
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PB : manque l'idée d'une puissance de 2 !
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Theorem tm_step_square_prefix_not_nil :
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forall (n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ a ++ tl -> 0 < length a -> hd <> nil.
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*)
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@ -1819,6 +1819,80 @@ Proof.
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Qed.
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Lemma tm_step_palindromic_power2_even :
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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)
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-> length (hd ++ a) mod (2 ^ (pred (Nat.double m))) = 0
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<-> (length (hd ++ a) / 4) mod (2^(pred (Nat.double (pred m)))) = 0.
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Proof.
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intros m n k hd a tl. intros H I J.
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destruct m. rewrite J in I. inversion I. inversion H1.
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destruct m. rewrite J in I. inversion I. inversion H1.
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inversion H3. inversion H5. inversion H7.
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assert (W: (length a) mod 4 = 0).
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rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
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rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
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rewrite Nat.mul_assoc in J.
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replace (2*2) with 4 in J. rewrite J.
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rewrite <- Nat.mul_mod_idemp_l. reflexivity.
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easy. reflexivity. apply le_0_n. apply le_0_n.
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assert (length (hd ++ a) mod 4 = 0).
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assert (W' := W).
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rewrite tm_step_palindromic_length_12_prefix
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with (hd := hd) (n := n) (tl := tl) in W'.
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rewrite app_length. rewrite Nat.add_mod.
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rewrite W. rewrite W'. reflexivity. easy. assumption. assumption.
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split.
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- intro P.
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rewrite <- Nat.mul_cancel_l with (p := 4).
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rewrite <- Nat.mul_mod_distr_l.
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replace 4 with (2*2) at 3. rewrite <- Nat.mul_assoc.
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rewrite <- Nat.pow_succ_r. rewrite <- Nat.pow_succ_r.
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replace (S (S (pred (Nat.double (pred (S (S m)))))))
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with (pred (Nat.double (S (S m)))).
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replace (4 * (length (hd ++ a) / 4))
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with (4 * (length (hd ++ a) / 4) + length (hd ++ a) mod 4).
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rewrite <- Nat.div_mod_eq. assumption.
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rewrite H0. rewrite Nat.add_0_r. reflexivity.
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rewrite <- pred_Sn. rewrite Nat.double_S.
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rewrite <- pred_Sn. rewrite Nat.succ_pred. reflexivity.
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rewrite Nat.neq_0_lt_0. rewrite Nat.double_S.
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apply Nat.lt_0_succ. apply le_0_n. apply le_0_n.
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reflexivity.
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apply Nat.pow_nonzero. easy. easy. easy.
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- intro P.
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rewrite <- Nat.mul_cancel_l with (p := 4) in P.
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rewrite <- Nat.mul_mod_distr_l in P.
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replace 4 with (2*2) in P at 3. rewrite <- Nat.mul_assoc in P.
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rewrite <- Nat.pow_succ_r in P. rewrite <- Nat.pow_succ_r in P.
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replace (S (S (pred (Nat.double (pred (S (S m)))))))
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with (pred (Nat.double (S (S m)))) in P.
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replace (4 * (length (hd ++ a) / 4))
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with (4 * (length (hd ++ a) / 4) + length (hd ++ a) mod 4) in P.
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rewrite <- Nat.div_mod_eq in P. assumption.
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rewrite H0. rewrite Nat.add_0_r. reflexivity.
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rewrite <- pred_Sn. rewrite Nat.double_S.
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rewrite <- pred_Sn. rewrite Nat.succ_pred. reflexivity.
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rewrite Nat.neq_0_lt_0. rewrite Nat.double_S.
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apply Nat.lt_0_succ. apply le_0_n. apply le_0_n.
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reflexivity.
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apply Nat.pow_nonzero. easy. easy. easy.
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Qed.
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Lemma xxx :
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forall (m n k : nat) (hd a tl : list bool),
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@ -1912,33 +1986,64 @@ Proof.
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with (S (S (pred (Nat.double (S (S m)))))).
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rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
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rewrite Nat.mul_assoc. replace (2*2) with 4.
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rewrite Nat.mod_mul_r.
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assert (length (hd ++ a) mod 4 = 0).
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assert (length hd mod 4 = 0). rewrite O. rewrite Nat.mul_comm.
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rewrite Nat.mod_mul. reflexivity. easy.
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rewrite app_length. rewrite Nat.add_mod.
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rewrite H0. rewrite W. reflexivity. easy. rewrite H0.
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rewrite Nat.add_0_l. rewrite Nat.mul_eq_0. right.
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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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rewrite Nat.mod_mul_r.
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assert (length (hd ++ a) mod 4 = 0).
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assert (length hd mod 4 = 0). rewrite O. rewrite Nat.mul_comm.
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rewrite Nat.mod_mul. reflexivity. easy.
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rewrite app_length. rewrite Nat.add_mod.
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rewrite H0. rewrite W. reflexivity. easy. rewrite H0.
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rewrite Nat.add_0_l. rewrite Nat.mul_eq_0. right.
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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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Theorem tm_step_palindromic_length_12_prefix :
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forall (n : 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 mod 4 = 0 <-> length hd mod 4 = 0.
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rewrite app_length. rewrite N. rewrite O.
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rewrite <- Nat.mul_add_distr_l. rewrite Nat.mul_mod_distr_l.
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apply Nat.mul_eq_0. right. rewrite <- app_length.
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apply IHm with (tl := tl').
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m, n, k : nat
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hd, a, tl : list bool
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hd' := firstn (length hd / 4) (tm_step n) : list bool
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a' := firstn (length a / 4) (skipn (length hd / 4) (tm_step n)) : list bool
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tl' := skipn (length hd / 4 + Nat.div2 (length a)) (tm_step n) : list bool
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H : tm_step n = hd' ++ a' ++ rev a' ++ tl'
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I : 6 < length a
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J : length a = 2 ^ Nat.double (S (S m))
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W : length a mod 4 = 0
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K : hd = tm_morphism (tm_morphism hd')
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L : a = tm_morphism (tm_morphism a')
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M : tl =
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tm_morphism
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(tm_morphism
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(skipn (length hd / 4 + Nat.div2 (length a))
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(tm_step (pred (pred (S (S n)))))))
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V : 3 < S (S n)
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N : length a = 4 * length a'
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O : length hd = 4 * length hd'
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Theorem tm_step_palindromic_length_12_prefix :
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forall (n : 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 mod 4 = 0 <-> length hd mod 4 = 0.
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unfold hd' at 2.
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