Update

src/thue_morse2.v

@@ -1430,6 +1430,39 @@ Proof.
     apply IHn in H0. assumption.
 Qed.
 
+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.
+Proof.
+  intros n hd a tl. intros H I.
+  assert (J: Nat.even (length a) || Nat.odd (length a) = true).
+  apply Nat.orb_even_odd. apply orb_prop in J.
+  assert (K: Nat.even (length hd) || Nat.odd (length hd) = true).
+  apply Nat.orb_even_odd. apply orb_prop in K.
+  destruct J; destruct K.
+  - rewrite app_length. rewrite Nat.even_add. rewrite H0. rewrite H1.
+    reflexivity.
+  - assert (exists (hd' a' tl' : list bool),
+      tm_step 0 = hd' ++ a' ++ a' ++ tl' /\ 0 < length a'
+        /\ odd (length hd') = true /\ even (length a') = true).
+    assert (exists (hd' a' tl' : list bool),
+      tm_step n = hd' ++ a' ++ a' ++ tl' /\ 0 < length a'
+        /\ odd (length hd') = true /\ even (length a') = true).
+    exists hd. exists a. exists tl. split. assumption.
+    split. assumption. split. assumption. assumption.
+    generalize H2. apply tm_step_square_pos_even'''.
+    destruct H2. destruct H2. destruct H2.
+    destruct H2. destruct H3. destruct H4.
+
+    assert (0 < 0). generalize H5. generalize H3. generalize H2.
+    apply tm_step_not_nil_factor_even. inversion H6.
+  - assert (even (length hd) = false).
+    generalize H0. generalize H. apply tm_step_odd_prefix_square.
+    rewrite H1 in H2. inversion H2.
+  - rewrite app_length. rewrite Nat.even_add.
+    rewrite <- Nat.negb_odd. rewrite <- Nat.negb_odd.
+    rewrite H0. rewrite H1. reflexivity.
+Qed.
 
 
     (* TODO: la taille du facteur divise la taille du préfixe ? *)