diff --git a/src/thue_morse2.v b/src/thue_morse2.v
index e6ba48a..ff888f3 100644
--- a/src/thue_morse2.v
+++ b/src/thue_morse2.v
@@ -15,73 +15,47 @@ Import ListNotations.
 *)
 
 Lemma tm_step_consecutive_identical :
+  forall (n : nat) (hd a tl : list bool) (b : bool),
+  tm_step n = hd ++ (b::b::nil) ++ tl
+  -> odd (length hd) = true.
+Proof.
+  intros n hd a tl b. intro H.
+  assert (J: {even (length hd) = false} + { ~ (even (length hd)) = false}).
+  apply bool_dec. destruct J.
+  - rewrite <- Nat.negb_even. rewrite e. reflexivity.
+  - apply not_false_is_true in n0.
+    assert (K: count_occ Bool.bool_dec hd true
+             = count_occ Bool.bool_dec hd false).
+    generalize n0. generalize H. apply tm_step_count_occ.
+    assert (L: count_occ Bool.bool_dec (hd ++ [b;b]) true
+             = count_occ Bool.bool_dec (hd ++ [b;b]) false).
+    assert (even (length (hd ++ [b;b])) = true).
+    rewrite app_length. rewrite Nat.even_add_even. assumption.
+    simpl. apply Nat.EvenT_Even. apply Nat.even_EvenT. reflexivity.
+    generalize H0. rewrite app_assoc in H. generalize H.
+    apply tm_step_count_occ.
+    rewrite count_occ_app in L. rewrite count_occ_app in L.
+    rewrite K in L. rewrite Nat.add_cancel_l in L.
+    destruct b. simpl in L. inversion L. simpl in L. inversion L.
+Qed.
+
+
+Lemma tm_step_consecutive_identical' :
   forall (n : nat) (hd a tl : list bool) (b1 b2 : bool),
   tm_step n = hd ++ (b1::b1::nil) ++ a ++ (b2::b2::nil) ++ tl
   -> even (length a) = true.
 Proof.
   intros n hd a tl b1 b2. intros H.
-  assert (I: even (length hd) = false).
-  assert (J: {even (length hd) = false} + { ~ (even (length hd)) = false}).
-  apply bool_dec. destruct J. assumption. apply not_false_is_true in n0.
-  assert (K: count_occ Bool.bool_dec hd true
-           = count_occ Bool.bool_dec hd false).
-  generalize n0. generalize H. apply tm_step_count_occ.
-  rewrite app_assoc in H.
-  assert (L: count_occ Bool.bool_dec (hd ++ [b1;b1]) true
-           = count_occ Bool.bool_dec (hd ++ [b1;b1]) false).
-  assert (even (length (hd ++ [b1;b1])) = true).
-  rewrite app_length. rewrite Nat.even_add_even. assumption.
-  simpl. apply Nat.EvenT_Even. apply Nat.even_EvenT. reflexivity.
-  generalize H0. generalize H. apply tm_step_count_occ.
-  rewrite count_occ_app in L. rewrite count_occ_app in L.
-  rewrite K in L. rewrite Nat.add_cancel_l in L.
-  destruct b1. simpl in L. inversion L. simpl in L. inversion L.
-
-  assert (J: {even (length (hd ++ [b1;b1] ++ a))
-            = false} + { ~ (even (length (hd ++ [b1;b1] ++ a))) = false}).
-  apply bool_dec. destruct J.
-
-  rewrite app_length in e. rewrite app_length in e.
-  rewrite Nat.add_assoc in e. rewrite Nat.add_shuffle0 in e.
-  rewrite Nat.even_add_even in e. rewrite Nat.even_add in e.
-  rewrite I in e. destruct (Nat.even (length a)). reflexivity.
-  simpl in e. inversion e. simpl.
-  apply Nat.EvenT_Even. apply Nat.even_EvenT. reflexivity.
-
-  apply not_false_is_true in n0.
-  assert (K: even (length (hd ++ [b1;b1] ++ a ++ [b2;b2])) = true).
-  replace (hd ++ [b1;b1] ++ a ++ [b2;b2])
-    with ((hd ++ [b1;b1] ++ a) ++ [b2;b2]).
-  rewrite app_length. rewrite Nat.even_add.
-  rewrite n0. reflexivity.
-  rewrite <- app_assoc. apply app_inv_head_iff.
-  rewrite <- app_assoc. reflexivity.
-
-  assert (N: count_occ Bool.bool_dec (hd ++ [b1;b1] ++ a ++ [b2;b2]) true
-    = count_occ Bool.bool_dec (hd ++ [b1;b1] ++ a ++ [b2;b2]) false).
-  replace (hd ++ [b1;b1] ++ a ++ [b2;b2] ++ tl)
-    with ((hd ++ [b1;b1] ++ a ++ [b2;b2]) ++ tl) in H.
-  generalize K. generalize H. apply tm_step_count_occ.
-  rewrite <- app_assoc. apply app_inv_head_iff.
-  rewrite <- app_assoc. apply app_inv_head_iff.
-  rewrite <- app_assoc. reflexivity.
-
-  assert (M: count_occ Bool.bool_dec (hd ++ [b1;b1] ++ a) true
-    = count_occ Bool.bool_dec (hd ++ [b1;b1] ++ a) false).
-  replace (hd ++ [b1;b1] ++ a ++ [b2;b2] ++ tl)
-    with ((hd ++ [b1;b1] ++ a) ++ [b2;b2] ++ tl) in H.
-  generalize n0. generalize H. apply tm_step_count_occ.
-  rewrite <- app_assoc. apply app_inv_head_iff.
-  rewrite <- app_assoc. reflexivity.
-
-  replace (hd ++ [b1;b1] ++ a ++ [b2;b2])
-    with ((hd ++ [b1;b1] ++ a) ++ [b2;b2]) in N.
-  rewrite count_occ_app in N. symmetry in N.
-  rewrite count_occ_app in N. rewrite M in N.
-  rewrite Nat.add_cancel_l in N.
-  destruct b2. simpl in N. inversion N. simpl in N. inversion N.
-  rewrite <- app_assoc. apply app_inv_head_iff.
-  rewrite <- app_assoc. reflexivity.
+  assert (Nat.odd (length hd) = true).
+  generalize H. apply tm_step_consecutive_identical. apply hd.
+  rewrite app_assoc in H. rewrite app_assoc in H.
+  assert (Nat.odd (length ((hd ++ [b1;b1])++a)) = true).
+  generalize H. apply tm_step_consecutive_identical. apply hd.
+  rewrite app_length in H1. rewrite Nat.odd_add in H1.
+  rewrite app_length in H1. rewrite Nat.odd_add in H1. rewrite H0 in H1.
+  replace (Nat.odd (length a)) with (negb (Nat.even (length a))) in H1.
+  destruct (even (length a)). reflexivity. inversion H1.
+  reflexivity.
 Qed.
 
 
@@ -103,7 +77,7 @@ Proof.
 
   replace (hd ++ (x ++ [x1; x1] ++ x0) ++ (x ++ [x1; x1] ++ x0) ++ tl)
     with ((hd ++ x) ++ [x1;x1] ++ (x0++x) ++ [x1;x1] ++ (x0 ++ tl)) in H.
-  apply tm_step_consecutive_identical in H.
+  apply tm_step_consecutive_identical' in H.
 
   assert (J: even (length (x0 ++ x)) = false).
   rewrite H1 in I. rewrite app_length in I. rewrite app_length in I.
@@ -450,29 +424,29 @@ Proof.
   - destruct b; destruct b0; destruct b1.
     + replace ([true;true;true] ++ [true;true;true] ++ tl)
       with ([true;true] ++ [true] ++ [true;true] ++ [true] ++ tl) in H.
-      apply tm_step_consecutive_identical in H. inversion H.
+      apply tm_step_consecutive_identical' in H. inversion H.
       reflexivity.
     + replace ([true;true;false] ++ [true;true;false] ++ tl)
       with ([true;true] ++ [false] ++ [true;true] ++ [false] ++ tl) in H.
-      apply tm_step_consecutive_identical in H. inversion H.
+      apply tm_step_consecutive_identical' in H. inversion H.
       reflexivity.
     + left. reflexivity.
     + replace (hd ++ [true;false;false] ++ [true;false;false] ++ tl)
       with ((hd ++ [true]) ++ [false; false] ++ [true] ++ [false;false] ++ tl) in H.
-      apply tm_step_consecutive_identical in H. inversion H.
+      apply tm_step_consecutive_identical' in H. inversion H.
       rewrite <- app_assoc. reflexivity.
     + replace (hd ++ [false;true;true] ++ [false;true;true] ++ tl)
       with ((hd ++ [false]) ++ [true; true] ++ [false] ++ [true;true] ++ tl) in H.
-      apply tm_step_consecutive_identical in H. inversion H.
+      apply tm_step_consecutive_identical' in H. inversion H.
       rewrite <- app_assoc. reflexivity.
     + right. reflexivity.
     + replace ([false;false;true] ++ [false;false;true] ++ tl)
       with ([false;false] ++ [true] ++ [false;false] ++ [true] ++ tl) in H.
-      apply tm_step_consecutive_identical in H. inversion H.
+      apply tm_step_consecutive_identical' in H. inversion H.
       reflexivity.
     + replace ([false;false;false] ++ [false;false;false] ++ tl)
       with ([false;false] ++ [false] ++ [false;false] ++ [false] ++ tl) in H.
-      apply tm_step_consecutive_identical in H. inversion H.
+      apply tm_step_consecutive_identical' in H. inversion H.
       reflexivity.
   - inversion I.
   - assert (K: count_occ Bool.bool_dec hd true