Update

src/thue_morse2.v

@@ -1650,3 +1650,54 @@ Proof.
     generalize H1. generalize H0. apply IHn.
   - assumption.
 Qed.
+
+(*
+Theorem tm_step_repeating_patterns_no_square :
+  forall (n m : nat) (hd a b tl : list bool),
+  tm_step n = hd ++ a ++ b ++ tl -> length a = 2^m
+  -> (forall k, length hd = k * 2 ^ S m -> a <> b).
+Proof.
+  intros n m hd a b tl. intros H I.
+  induction n.
+  - assert (m = 0).
+    destruct m. reflexivity.
+    assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
+    rewrite H in H0 at 2. rewrite app_length in H0. rewrite Nat.add_comm in H0.
+    rewrite app_length in H0. rewrite <- Nat.add_assoc in H0.
+    rewrite I in H0. rewrite Nat.pow_succ_r in H0. symmetry in H0.
+    apply Nat.eq_add_1 in H0. destruct H0; destruct H0.
+    apply Nat.mul_eq_1 in H0. destruct H0. inversion H0.
+    apply Nat.mul_eq_0 in H0. destruct H0. inversion H0.
+    apply Nat.pow_nonzero in H0. contradiction H0. easy. apply le_0_n.
+    intro k. intro J. rewrite H0 in J. rewrite Nat.pow_1_r in J.
+    assert (k = 0).
+    assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
+    rewrite H in H1 at 2. rewrite app_length in H1. rewrite J in H1.
+    symmetry in H1. apply Nat.eq_add_1 in H1. destruct H1; destruct H1.
+    apply Nat.mul_eq_1 in H1. destruct H1. inversion H3.
+    apply Nat.mul_eq_0 in H1. destruct H1. assumption. inversion H1.
+    rewrite H1 in J. rewrite Nat.mul_0_l in J. rewrite length_zero_iff_nil in J.
+    assert ({a=b} + {~ a=b}). apply list_eq_dec. apply bool_dec.
+    destruct H2. rewrite <- e in H.
+
+    assert (0 < length a). rewrite I. rewrite H0. apply Nat.lt_0_1.
+
+    assert (hd <> nil). generalize H2. generalize H.
+    apply tm_step_square_prefix_not_nil. rewrite J in H3.
+    contradiction H3. reflexivity. assumption.
+  -
+
+    Reformulation:
+forall (n : nat) (hd a tl : list bool),
+tm_step n = hd ++ a ++ a ++ tl
+  -> 0 < length a
+  -> (length hd) mod (length (a ++ a)) <> 0.
+    PB : manque l'idée d'une puissance de 2 !
+
+
+
+
+Theorem tm_step_square_prefix_not_nil :
+  forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ a ++ tl -> 0 < length a -> hd <> nil.
+ *)

src/thue_morse3.v

@@ -1819,6 +1819,80 @@ Proof.
 Qed.
 
 
+Lemma tm_step_palindromic_power2_even :
+  forall (m n k : nat) (hd a tl : list bool),
+    tm_step n = hd ++ a ++ (rev a) ++ tl
+    -> 6 < length a
+    -> length a = 2^(Nat.double m)
+    -> length (hd ++ a) mod (2 ^ (pred (Nat.double m))) = 0
+    <-> (length (hd ++ a) / 4) mod (2^(pred (Nat.double (pred m)))) = 0.
+Proof.
+  intros m n k hd a tl. intros H I J.
+
+  destruct m. rewrite J in I. inversion I. inversion H1.
+  destruct m. rewrite J in I. inversion I. inversion H1.
+  inversion H3. inversion H5. inversion H7.
+
+  assert (W: (length a) mod 4 = 0).
+  rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
+  rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
+  rewrite Nat.mul_assoc in J.
+  replace (2*2) with 4 in J. rewrite J.
+  rewrite <- Nat.mul_mod_idemp_l. reflexivity.
+  easy. reflexivity. apply le_0_n. apply le_0_n.
+
+  assert (length (hd ++ a) mod 4 = 0).
+  assert (W' := W).
+  rewrite tm_step_palindromic_length_12_prefix
+    with (hd := hd) (n := n) (tl := tl) in W'.
+  rewrite app_length. rewrite Nat.add_mod.
+  rewrite W. rewrite W'. reflexivity. easy. assumption. assumption.
+
+  split.
+  - intro P.
+    rewrite <- Nat.mul_cancel_l with (p := 4).
+    rewrite <- Nat.mul_mod_distr_l.
+    replace 4 with (2*2) at 3. rewrite <- Nat.mul_assoc.
+    rewrite <- Nat.pow_succ_r. rewrite <- Nat.pow_succ_r.
+
+    replace (S (S (pred (Nat.double (pred (S (S m)))))))
+      with (pred (Nat.double (S (S m)))).
+    replace (4 * (length (hd ++ a) / 4))
+      with (4 * (length (hd ++ a) / 4) + length (hd ++ a) mod 4).
+    rewrite <- Nat.div_mod_eq. assumption.
+    rewrite H0. rewrite Nat.add_0_r. reflexivity.
+
+    rewrite <- pred_Sn. rewrite Nat.double_S.
+    rewrite <- pred_Sn. rewrite Nat.succ_pred. reflexivity.
+    rewrite Nat.neq_0_lt_0. rewrite Nat.double_S.
+    apply Nat.lt_0_succ. apply le_0_n. apply le_0_n.
+    reflexivity.
+
+    apply Nat.pow_nonzero. easy. easy. easy.
+  - intro P.
+    rewrite <- Nat.mul_cancel_l with (p := 4) in P.
+    rewrite <- Nat.mul_mod_distr_l in P.
+    replace 4 with (2*2) in P at 3. rewrite <- Nat.mul_assoc in P.
+    rewrite <- Nat.pow_succ_r in P. rewrite <- Nat.pow_succ_r in P.
+
+    replace (S (S (pred (Nat.double (pred (S (S m)))))))
+      with (pred (Nat.double (S (S m)))) in P.
+    replace (4 * (length (hd ++ a) / 4))
+      with (4 * (length (hd ++ a) / 4) + length (hd ++ a) mod 4) in P.
+    rewrite <- Nat.div_mod_eq in P. assumption.
+    rewrite H0. rewrite Nat.add_0_r. reflexivity.
+
+    rewrite <- pred_Sn. rewrite Nat.double_S.
+    rewrite <- pred_Sn. rewrite Nat.succ_pred. reflexivity.
+    rewrite Nat.neq_0_lt_0. rewrite Nat.double_S.
+    apply Nat.lt_0_succ. apply le_0_n. apply le_0_n.
+    reflexivity.
+
+    apply Nat.pow_nonzero. easy. easy. easy.
+Qed.
+
+
+
 
 Lemma xxx :
   forall (m n k : nat) (hd a tl : list bool),
@@ -1912,33 +1986,64 @@ Proof.
       with (S (S (pred (Nat.double (S (S m)))))).
     rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
     rewrite Nat.mul_assoc. replace (2*2) with 4.
+
+    rewrite Nat.mod_mul_r.
+    assert (length (hd ++ a) mod 4 = 0).
+    assert (length hd mod 4 = 0). rewrite O. rewrite Nat.mul_comm.
+    rewrite Nat.mod_mul. reflexivity. easy.
+    rewrite app_length. rewrite Nat.add_mod.
+    rewrite H0. rewrite W. reflexivity. easy. rewrite H0.
+    rewrite Nat.add_0_l. rewrite Nat.mul_eq_0. right.
+
+
+Nat.mod_mul_r:
+  forall a b c : nat,
+  b <> 0 -> c <> 0 -> a mod (b * c) = a mod b + b * ((a / b) mod c)
+
+
+
+
+    rewrite Nat.mod_mul_r. 
+    assert (length (hd ++ a) mod 4 = 0).
+    assert (length hd mod 4 = 0). rewrite O. rewrite Nat.mul_comm.
+    rewrite Nat.mod_mul. reflexivity. easy.
+    rewrite app_length. rewrite Nat.add_mod.
+    rewrite H0. rewrite W. reflexivity. easy. rewrite H0.
+    rewrite Nat.add_0_l. rewrite Nat.mul_eq_0. right.
+
+
+Nat.mod_mul_r:
+  forall a b c : nat,
+  b <> 0 -> c <> 0 -> a mod (b * c) = a mod b + b * ((a / b) mod c)
+
+Theorem tm_step_palindromic_length_12_prefix :
+  forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ (rev a) ++ tl
+  -> 6 < length a
+  -> length a mod 4 = 0 <-> length hd mod 4 = 0.
+
+
+
     rewrite app_length. rewrite N. rewrite O.
     rewrite <- Nat.mul_add_distr_l. rewrite Nat.mul_mod_distr_l.
     apply Nat.mul_eq_0. right. rewrite <- app_length.
+    apply IHm with (tl := tl').
 
 
 
 
 
-m, n, k : nat
-hd, a, tl : list bool
-hd' := firstn (length hd / 4) (tm_step n) : list bool
-a' := firstn (length a / 4) (skipn (length hd / 4) (tm_step n)) : list bool
-tl' := skipn (length hd / 4 + Nat.div2 (length a)) (tm_step n) : list bool
-H : tm_step n = hd' ++ a' ++ rev a' ++ tl'
-I : 6 < length a
-J : length a = 2 ^ Nat.double (S (S m))
-W : length a mod 4 = 0
-K : hd = tm_morphism (tm_morphism hd')
-L : a = tm_morphism (tm_morphism a')
-M : tl =
-    tm_morphism
-      (tm_morphism
-         (skipn (length hd / 4 + Nat.div2 (length a))
-            (tm_step (pred (pred (S (S n)))))))
-V : 3 < S (S n)
-N : length a = 4 * length a'
-O : length hd = 4 * length hd'
+Theorem tm_step_palindromic_length_12_prefix :
+  forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ (rev a) ++ tl
+  -> 6 < length a
+  -> length a mod 4 = 0 <-> length hd mod 4 = 0.
+
+
+
+
+    unfold hd' at 2.
+