Update

src/thue_morse3.v

@@ -1357,6 +1357,30 @@ Proof.
 Qed.
 
 
+Theorem tm_step_palindromic_length_12_n :
+  forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ (rev a) ++ tl
+  -> length a > 6
+  -> 1 < n.
+Proof.
+  intros n hd a tl. intros H W.
+
+  assert (length (tm_step n) <= length (tm_step n)).
+  apply Nat.le_refl. rewrite H in H0 at 1.
+  rewrite app_length in H0. rewrite app_length in H0.
+  rewrite Nat.add_comm in H0. rewrite <- Nat.add_assoc in H0.
+  rewrite <- Nat.add_0_r in H0. apply Nat.le_le_add_le in H0.
+  rewrite tm_size_power2 in H0. destruct n. destruct a.
+  inversion W. destruct a. inversion W. inversion H2.
+  destruct a. inversion W. inversion H2. inversion H4.
+  inversion H0. inversion H2. destruct a. inversion W.
+  destruct a. inversion W. inversion H2. destruct a. inversion W.
+  inversion H2. inversion H4. destruct n. inversion H0. inversion H2.
+  inversion H4. rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
+  apply le_0_n.
+Qed.
+
+
 Theorem tm_step_palindromic_length_12_prefix :
   forall (n : nat) (hd a tl : list bool),
   tm_step n = hd ++ a ++ (rev a) ++ tl
@@ -1373,9 +1397,20 @@ Proof.
   assumption. easy. assumption.
 Qed.
 
+Theorem tm_step_palindromic_length_12_prefix2 :
+  forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ (rev a) ++ tl
+  -> length a > 6
+  -> length a mod 4 = 0 -> length tl mod 4 = 0.
+Proof.
+  intros n hd a tl. intros H I J.
+  assert (K: length hd mod 4 = 0).
+  generalize J. generalize I. generalize H.
+  apply tm_step_palindromic_length_12_prefix.
+  assert (L: length (hd ++ a ++ rev a ++ tl) mod 4 = 0).
+  rewrite <- H. rewrite tm_size_power2.
 
-
-
+Admitted.
 
 Lemma tm_step_palindromic_even_morphism1 :
   forall (n : nat) (hd a tl : list bool),
@@ -1484,7 +1519,9 @@ Lemma tm_step_palindromic_even_morphism2 :
   tm_step n = hd ++ a ++ (rev a) ++ tl
   -> length a > 6
   -> (length a) mod 4 = 0
-  -> 11 = 42.
+  -> hd = tm_morphism (tm_morphism (firstn (length hd / 4)
+           (tm_step (pred (pred n)))))
+     /\ 11 = 42.
 Proof.
   intros n hd a tl. intros H W J0.
 
@@ -1493,19 +1530,9 @@ Proof.
   apply Nat.lt_succ_l in W. apply Nat.lt_succ_l in W. apply Nat.lt_succ_l in W.
   assumption.
 
-  assert (V: 1 < n). assert (length (tm_step n) <= length (tm_step n)).
-  apply Nat.le_refl. rewrite H in H0 at 1.
-  rewrite app_length in H0. rewrite app_length in H0.
-  rewrite Nat.add_comm in H0. rewrite <- Nat.add_assoc in H0.
-  rewrite <- Nat.add_0_r in H0. apply Nat.le_le_add_le in H0.
-  rewrite tm_size_power2 in H0. destruct n. destruct a.
-  inversion W. destruct a. inversion W. inversion H2.
-  destruct a. inversion W. inversion H2. inversion H4.
-  inversion H0. inversion H2. destruct a. inversion W.
-  destruct a. inversion W. inversion H2. destruct a. inversion W.
-  inversion H2. inversion H4. destruct n. inversion H0. inversion H2.
-  inversion H4. rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
-  apply le_0_n. destruct n. inversion V.
+  assert (V: 1 < n). generalize W. generalize H.
+  apply tm_step_palindromic_length_12_n.
+  destruct n. inversion V.
 
   assert (J1: length hd mod 4 = 0). generalize J0. generalize W.
   generalize H. apply tm_step_palindromic_length_12_prefix.
@@ -1704,6 +1731,30 @@ Proof.
 
 
 
+  split.
+
+  rewrite <- pred_Sn. rewrite <- pred_Sn.
+  rewrite H4 at 1. fold hd'. rewrite H4'. unfold hd'.
+  rewrite firstn_length_le. rewrite Nat.div2_div. rewrite Nat.div2_div.
+  rewrite Nat.div_div. reflexivity. easy. easy.
+
+
+
+
+
+
+
+
+
+
+  assert (hd = hd). reflexivity. rewrite H4 in H12 at 2. fold hd' in H12.
+  rewrite H4' in H12. unfold hd' in H12.
+  rewrite firstn_length_le in H12.
+  rewrite Nat.div2_div in H12.
+  rewrite Nat.div2_div in H12. rewrite Nat.div_div in H12.
+  replace (2*2) with 4 in H12.
+
+
 Admitted.