diff --git a/src/thue_morse4.v b/src/thue_morse4.v
index f9f2552..5d2cd8a 100644
--- a/src/thue_morse4.v
+++ b/src/thue_morse4.v
@@ -124,8 +124,6 @@ Proof.
 Qed.
 
 
-
-
 Theorem tm_step_palindrome_power2_inverse' :
   forall (m n k : nat) (hd tl : list bool),
     tm_step n = hd ++ tl
@@ -161,3 +159,63 @@ Proof.
   apply tm_step_palindrome_power2_inverse.
 Qed.
 
+
+Lemma trailing_zeros :
+  forall n, 0 < n -> exists k j, n = S (Nat.double k) * 2^j.
+Proof.
+  intro n. intro H.
+  assert (exists m, 2^m <= n < 2^(S m)).
+  exists (Nat.log2 n). apply Nat.log2_spec; assumption.
+  destruct H0. generalize dependent n. induction x; intros n H I.
+  - exists 0. exists 0. simpl.
+    destruct n. inversion H. destruct I. destruct n. reflexivity.
+    simpl in H1. rewrite <- Nat.succ_lt_mono in H1.
+    rewrite <- Nat.succ_lt_mono in H1. apply Nat.nlt_0_r in H1.
+    contradiction.
+  - destruct I. apply Nat.div_le_mono with (c := 2) in H0.
+    rewrite Nat.pow_succ_r in H0. rewrite Nat.mul_comm in H0.
+    rewrite Nat.div_mul in H0.
+
+    assert (Nat.Even n \/ Nat.Odd n). apply Nat.Even_or_Odd.
+    destruct H2. assert (U := H2).
+    + apply Nat.Even_double in H2. rewrite Nat.double_twice in H2.
+      rewrite H2 in H1. rewrite Nat.pow_succ_r in H1.
+      rewrite <- Nat.mul_lt_mono_pos_l in H1. rewrite Nat.div2_div in H1.
+      assert (exists k j, n/2 = S (Nat.double k) * 2^j).
+      apply IHx. assert (0 < 2^x). rewrite <- Nat.neq_0_lt_0.
+      apply Nat.pow_nonzero. easy. generalize H0. generalize H3.
+      apply Nat.lt_le_trans. split; assumption.
+      destruct H3. destruct H3. exists x0. exists (S x1).
+      rewrite H2. rewrite Nat.pow_succ_r. symmetry. rewrite Nat.mul_comm.
+      rewrite <- Nat.mul_assoc. rewrite Nat.mul_cancel_l.
+      rewrite Nat.mul_comm. rewrite <- H3. rewrite Nat.div2_div.
+      reflexivity. easy. apply Nat.le_0_l. apply Nat.lt_0_2.
+      apply Nat.le_0_l.
+    + apply Nat.Odd_double in H2.
+      exists (Nat.div2 n). exists 0. rewrite H2 at 1.
+      rewrite Nat.mul_1_r. reflexivity.
+    + easy.
+    + apply Nat.le_0_l.
+    + easy.
+Qed.
+
+
+
+
+Lemma xxx :
+  forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ a ++ tl
+  -> a = rev a
+  -> 11 = 42.
+Proof.
+  intros n hd a tl. intros H I.
+
+
+
+
+
+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 -> k = 0 \/ k = 1
+