Update

src/thue_morse2.v

@@ -10,8 +10,7 @@
    But some powerful theorems are provided here:
 
    - tm_step_cubefree (there is no cube in the sequence)
-   - tm_step_square_size (all squared factors of odd
-     length have length 1 or 3).
+   - tm_step_square_size (about length of squared factrs)
    - tm_step_square_pos (length of the prefix has the same
      parity than the length of the factor being squared)
 *)  
@@ -742,7 +741,6 @@ Proof.
 Qed.
 
 
-
 Theorem tm_step_square_size : forall (n k j : nat) (hd a tl : list bool),
   tm_step (S n) = hd ++ a ++ a ++ tl
   -> length a = (S (Nat.double k)) * 2^j

src/thue_morse3.v

@@ -1058,8 +1058,48 @@ Le lemme repeating_patterns se base sur les huit premiers termes de TM :
   (* le premier nouveau sous cas est  lh mod 4 = 2 en H1
               FTFT TFTF FTFT TFTF (µ de FF TT FF TT) pourquoi impossible ?
        c'est un carré ???
+       c'est un palindrome fait avec un carré de palindrome ?
+       morphisme de FFTTFFTT   sans doute impossible (testé)
    *)
+  
+  unfold hd' in H. destruct hd. inversion P.
+  rewrite app_removelast_last
+    with (l := (removelast (removelast (removelast (b3::b5::b7::b10::hd)))))
+         (d := false) in H.
+  pose (b11 := last (removelast (removelast
+           (removelast (b3 :: b5 :: b7 :: b10 :: hd)))) false).
+  fold b11 in H. rewrite <- app_assoc in H.
+  pose (hd'' := removelast (removelast (removelast
+          (removelast (b3 :: b5 :: b7 :: b10 :: hd))))).
+  fold hd'' in H.
+    (* proof that b11 <> b1 *)
+  assert ({b11=b1} + {~ b11=b1}). apply bool_dec. destruct H9. rewrite e1 in H.
+  assert (even (length hd'') = false).
+  replace ( hd'' ++ [b1] ++ [b1] ++ [b0] ++ [b1] ++
+    b1 :: b0 :: b1 :: b0 :: b0 :: b1 :: b0 :: b1 :: b1 :: b0 :: b1 :: tl)
+    with (hd'' ++ [b1;b1;b0] ++ [b1;b1;b0]
+              ++ b1::b0::b0::b1::b0::b1::b1::b0::b1::tl) in H.
+  assert (odd (length [b1;b1;b0]) = true). reflexivity.
+  generalize H9. generalize H. apply tm_step_odd_prefix_square.
+  reflexivity. unfold hd'' in H9.
+  rewrite removelast_firstn_len in H9. rewrite Y'' in H9.
+  rewrite firstn_length_le in H9. simpl in H9.
+  replace (b3::b5::b7::b10::hd) with ([b3;b5;b7;b10] ++ hd) in Q.
+  rewrite app_length in Q. rewrite Nat.even_add in Q. rewrite H9 in Q.
+  inversion Q. reflexivity. rewrite Y''. apply Nat.le_pred_l.
 
+   
+
+  
+  
+
+
+Lemma tm_step_odd_prefix_square : forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ a ++ tl
+  -> odd (length a) = true -> even (length hd) = false.
+Theorem tm_step_square_pos : forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ a ++ tl
+  -> 0 < length a -> even (length (hd ++ a)) = true.