Update

src/thue_morse2.v

@@ -559,6 +559,50 @@ Qed.
 
 
 
+Lemma xxx : 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).
+Proof.
+  intros n k j hd a tl.
+  generalize hd. generalize a. generalize tl. induction j.
+  - intros tl0 a0 hd0. intros H I. simpl in I. rewrite Nat.mul_1_r in I.
+    assert (odd (length a0) = true). rewrite I.
+    rewrite Nat.odd_succ. rewrite Nat.double_twice.
+    apply Nat.even_mul.
+    assert (J: length a0 < 4). generalize H0. generalize H.
+    apply tm_step_odd_square.
+    rewrite I in J.
+    destruct k. left. reflexivity.
+    destruct k. right. reflexivity.
+    apply Nat.succ_lt_mono in J.
+    apply Nat.lt_lt_succ_r in J.
+    rewrite Nat.double_twice in J. replace (4) with (2*2) in J.
+    rewrite <- Nat.mul_lt_mono_pos_l in J.
+    apply Nat.succ_lt_mono in J.
+    apply Nat.succ_lt_mono in J.
+    apply Nat.nlt_0_r in J. contradiction J.
+    apply Nat.lt_0_2. reflexivity.
+  - intros tl0 a0 hd0. intros H I.
+    assert (exists (hd' a' tl' : list bool), tm_step n = hd' ++ a' ++ a' ++ tl'
+          /\ length a0 = Nat.double (length a')).
+    assert (even (length a0) = true).
+    rewrite I. rewrite Nat.pow_succ_r'. rewrite Nat.mul_comm.
+    rewrite <- Nat.mul_assoc. apply Nat.even_mul.
+    generalize H0. generalize H. apply tm_step_square_half.
+    destruct H0. destruct H0. destruct H0. destruct H0.
+    rewrite H1 in I. rewrite Nat.pow_succ_r' in I. rewrite Nat.mul_comm in I.
+    rewrite <- Nat.mul_assoc in I. rewrite Nat.double_twice in I.
+    rewrite Nat.mul_cancel_l in I. rewrite Nat.mul_comm in I.
+    generalize I.
+    assert (tm_step (S n) = x ++ x0 ++ x0 ++ x1 ++ (map negb (tm_step n))).
+    rewrite tm_build. rewrite H0.
+    rewrite <- app_assoc. apply app_inv_head_iff.
+    rewrite <- app_assoc. apply app_inv_head_iff.
+    rewrite <- app_assoc. reflexivity.
+    generalize H2. apply IHj. easy.
+Qed.
+
 
 
 (*