Update

src/thue_morse4.v

@@ -161,6 +161,51 @@ Qed.
 
 
 
+Lemma xxx :
+  forall (j n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ a ++ tl
+  -> length a = 2^(Nat.double j) \/ length a = 3 * 2^(Nat.double j)
+  -> a = rev a.
+Proof.
+  intro j. induction j; intros n hd a tl; intros H I.
+  - destruct I.
+    + destruct a. inversion H0. destruct a. reflexivity. inversion H0.
+    + destruct a. inversion H0. destruct a. inversion H0. destruct a.
+      inversion H0. destruct a.
+      assert ({b=b1} + {~ b=b1}). apply bool_dec. destruct H1.
+      rewrite e. reflexivity.
+      assert ({b0=b1} + {~ b0=b1}). apply bool_dec. destruct H1.
+      rewrite e in H.
+      replace (hd ++ [b; b1; b1] ++ [b; b1; b1] ++ tl)
+        with ((hd ++ [b]) ++ [b1;b1] ++ [b] ++ [b1;b1] ++ tl) in H.
+      apply tm_step_consecutive_identical' in H. inversion H.
+      rewrite <- app_assoc. reflexivity.
+      assert (b = b0). destruct b; destruct b0; destruct b1;
+      reflexivity || contradiction n0 || contradiction n1; reflexivity.
+      rewrite H1 in H.
+      replace (hd ++ [b0; b0; b1] ++ [b0; b0; b1] ++ tl)
+        with (hd ++ [b0;b0] ++ [b1] ++ [b0;b0] ++ ([b1] ++ tl)) in H.
+      apply tm_step_consecutive_identical' in H. inversion H. reflexivity.
+      inversion H0.
+  - assert (even (length a) = true).
+    destruct I; rewrite H0; rewrite Nat.double_S;
+    rewrite Nat.pow_succ_r.
+    rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l.
+    rewrite Nat.even_mul. rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l.
+    assert (0 < length a). destruct I; rewrite H1.
+    rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
+    apply Nat.mul_pos_pos. lia.
+    rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
+    assert (even (length (hd ++ a)) = true). generalize H1. generalize H.
+    apply tm_step_square_pos. 
+
+
+
+
+
+
+
+
 
 
 Lemma xxx :