diff --git a/src/thue_morse3.v b/src/thue_morse3.v
index 22433e3..8644b04 100644
--- a/src/thue_morse3.v
+++ b/src/thue_morse3.v
@@ -1141,12 +1141,37 @@ Le lemme repeating_patterns se base sur les huit premiers termes de TM :
   reflexivity. contradiction n8. reflexivity. reflexivity.
   rewrite H10 in H.
 
+  (* simplify notations *)
+  replace ( hd'' ++ [b0] ++ [b1] ++ [b0] ++ [b1] ++ b1
+    :: b0 :: b1 :: b0 :: b0 :: b1 :: b0 :: b1 :: b1 :: b0 :: b1 :: b0 :: tl)
+  with (hd'' ++ [b0;b1;b0;b1;b1;b0;b1;b0;b0;b1;b0;b1;b1;b0;b1;b0] ++ tl) in H.
+  pose (s := [b0;b1;b0;b1;b1;b0;b1;b0;b0;b1;b0;b1;b1;b0;b1;b0]). fold s in H.
+  assert (even (length hd'') = true). unfold hd''. 
+
+  rewrite removelast_firstn_len. rewrite Y''.
+  replace (pred (length (b10::hd))) with (length hd).
+  rewrite firstn_length_le. simpl in Q. rewrite Q. reflexivity.
+  rewrite Y''. apply Nat.le_succ_diag_r. reflexivity.
+
+
+
+
+
+Lemma tm_morphism_app : forall (l1 l2 : list bool),
+  tm_morphism (l1 ++ l2) = tm_morphism l1 ++ tm_morphism l2.
+Lemma tm_morphism_app2 : forall (l hd tl : list bool),
+  tm_morphism l = hd ++ tl
+  -> even (length hd) = true
+  -> hd = tm_morphism (firstn (Nat.div2 (length hd)) l).
+Lemma tm_morphism_app3 : forall (l hd tl : list bool),
+  tm_morphism l = hd ++ tl
+  -> even (length hd) = true
+  -> tl = tm_morphism (skipn (Nat.div2 (length hd)) l).
   
 
 
 
 
-False, True, True, False, True, False, False, True, True, False, False, True, False, True, True, False]'