diff --git a/src/thue_morse2.v b/src/thue_morse2.v
index 440300f..5a9baa3 100644
--- a/src/thue_morse2.v
+++ b/src/thue_morse2.v
@@ -10,9 +10,11 @@
    But some powerful theorems are provided here:
 
    - tm_step_cubefree (there is no cube in the sequence)
-   - tm_step_square_size (about length of squared factrs)
+   - tm_step_square_size (about length of squared factors)
    - tm_step_square_pos (length of the prefix has the same
      parity than the length of the factor being squared)
+   - tm_step_square_prefix_not_nil (no square at the beginning
+     of the sequence)
 *)  
 
 Require Import thue_morse.
@@ -1622,82 +1624,3 @@ Proof.
   simpl. rewrite Nat.add_0_r. reflexivity.
   left. assumption.
 Qed.
-
-Theorem tm_step_square_prefix_not_nil :
-  forall (n : nat) (hd a tl : list bool),
-  tm_step n = hd ++ a ++ a ++ tl -> 0 < length a -> hd <> nil.
-Proof.
-  intros n hd a tl. intros H I.
-
-  assert (W: {hd=nil} + {~ hd=nil}). apply list_eq_dec. apply bool_dec.
-  destruct W. rewrite e in H. rewrite app_nil_l in H.
-  generalize dependent a.
-  generalize dependent tl.
-  induction n.
-  - intros tl a. intros H I.
-    destruct a. inversion I. inversion H.
-    assert (length (nil : list bool) = length (nil : list bool)).
-    reflexivity. rewrite H2 in H0 at 2. rewrite app_length in H0.
-    rewrite Nat.add_comm in H0. inversion H0.
-  - intros tl a. intros H I.
-    assert (hd <> nil
-      \/ exists a' tl', tm_step (pred (S n)) = a' ++ a' ++ tl' /\ 0 < length a').
-    generalize I. generalize H. replace (a++a++tl) with (hd ++ a ++ a ++ tl).
-    apply tm_step_square_prefix_not_nil0. rewrite e. reflexivity.
-    destruct H0. assumption. destruct H0. destruct H0.
-    rewrite <- pred_Sn in H0.
-    destruct H0.
-    generalize H1. generalize H0. apply IHn.
-  - assumption.
-Qed.
-
-(*
-Theorem tm_step_repeating_patterns_no_square :
-  forall (n m : nat) (hd a b tl : list bool),
-  tm_step n = hd ++ a ++ b ++ tl -> length a = 2^m
-  -> (forall k, length hd = k * 2 ^ S m -> a <> b).
-Proof.
-  intros n m hd a b tl. intros H I.
-  induction n.
-  - assert (m = 0).
-    destruct m. reflexivity.
-    assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
-    rewrite H in H0 at 2. rewrite app_length in H0. rewrite Nat.add_comm in H0.
-    rewrite app_length in H0. rewrite <- Nat.add_assoc in H0.
-    rewrite I in H0. rewrite Nat.pow_succ_r in H0. symmetry in H0.
-    apply Nat.eq_add_1 in H0. destruct H0; destruct H0.
-    apply Nat.mul_eq_1 in H0. destruct H0. inversion H0.
-    apply Nat.mul_eq_0 in H0. destruct H0. inversion H0.
-    apply Nat.pow_nonzero in H0. contradiction H0. easy. apply le_0_n.
-    intro k. intro J. rewrite H0 in J. rewrite Nat.pow_1_r in J.
-    assert (k = 0).
-    assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
-    rewrite H in H1 at 2. rewrite app_length in H1. rewrite J in H1.
-    symmetry in H1. apply Nat.eq_add_1 in H1. destruct H1; destruct H1.
-    apply Nat.mul_eq_1 in H1. destruct H1. inversion H3.
-    apply Nat.mul_eq_0 in H1. destruct H1. assumption. inversion H1.
-    rewrite H1 in J. rewrite Nat.mul_0_l in J. rewrite length_zero_iff_nil in J.
-    assert ({a=b} + {~ a=b}). apply list_eq_dec. apply bool_dec.
-    destruct H2. rewrite <- e in H.
-
-    assert (0 < length a). rewrite I. rewrite H0. apply Nat.lt_0_1.
-
-    assert (hd <> nil). generalize H2. generalize H.
-    apply tm_step_square_prefix_not_nil. rewrite J in H3.
-    contradiction H3. reflexivity. assumption.
-  -
-
-    Reformulation:
-forall (n : nat) (hd a tl : list bool),
-tm_step n = hd ++ a ++ a ++ tl
-  -> 0 < length a
-  -> (length hd) mod (length (a ++ a)) <> 0.
-    PB : manque l'idée d'une puissance de 2 !
-
-
-
-
-Theorem tm_step_square_prefix_not_nil :
-  forall (n : nat) (hd a tl : list bool),
-  tm_step n = hd ++ a ++ a ++ tl -> 0 < length a -> hd <> nil.
- *)
diff --git a/src/thue_morse3.v b/src/thue_morse3.v
index 35aceb7..6d4af5d 100644
--- a/src/thue_morse3.v
+++ b/src/thue_morse3.v
@@ -1820,14 +1820,14 @@ Qed.
 
 
 Lemma tm_step_palindromic_power2_even :
-  forall (m n k : nat) (hd a tl : list bool),
+  forall (m n : nat) (hd a tl : list bool),
     tm_step n = hd ++ a ++ (rev a) ++ tl
     -> 6 < length a
     -> length a = 2^(Nat.double m)
     -> length (hd ++ a) mod (2 ^ (pred (Nat.double m))) = 0
     <-> (length (hd ++ a) / 4) mod (2^(pred (Nat.double (pred m)))) = 0.
 Proof.
-  intros m n k hd a tl. intros H I J.
+  intros m n hd a tl. intros H I J.
 
   destruct m. rewrite J in I. inversion I. inversion H1.
   destruct m. rewrite J in I. inversion I. inversion H1.
@@ -1902,11 +1902,15 @@ Lemma xxx :
     -> length (hd ++ a) mod (2 ^ (pred (Nat.double m))) = 0.
 Proof.
   intros m n k hd a tl. intros H I J.
+  assert (R := H).
 
   destruct m. rewrite J in I. inversion I. inversion H1.
   destruct m. rewrite J in I. inversion I. inversion H1.
   inversion H3. inversion H5. inversion H7.
 
+  generalize dependent a. generalize dependent hd.
+  induction m. intros hd a H I J H'.
+
   assert (W: (length a) mod 4 = 0).
   rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
   rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
@@ -1946,6 +1950,15 @@ Proof.
   pose (tl' := (skipn (length hd / 4 + Nat.div2 (length a)) (tm_step n))).
   fold hd' in H. fold a' in H. fold tl' in H.
 
+  (*
+  assert (U: even (length (hd ++ a)) = true).
+  apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
+  apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
+  apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
+  generalize I. generalize H'.
+  apply tm_step_palindromic_even_center.
+  *)
+
   rewrite Nat.pred_succ in K. rewrite Nat.pred_succ in K.
   rewrite Nat.pred_succ in L. rewrite Nat.pred_succ in L.
   fold hd' in K. fold a' in L.
@@ -1956,6 +1969,271 @@ Proof.
   rewrite tm_morphism_length in O. rewrite tm_morphism_length in O.
   rewrite Nat.mul_assoc in O. replace (2*2) with 4 in O.
 
+  replace (2^ pred (Nat.double 2)) with (4*2).
+  rewrite app_length. rewrite N. rewrite O.
+  rewrite <- Nat.mul_add_distr_l. rewrite Nat.mul_mod_distr_l.
+  rewrite Nat.mul_eq_0. right. rewrite <- app_length.
+
+  assert (U: even (length (hd' ++ a')) = true).
+  assert (0 < length a').
+  apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
+  rewrite N in I.
+  rewrite <- Nat.mul_1_r in I at 1.
+  rewrite <- Nat.mul_lt_mono_pos_l in I.
+  apply Nat.lt_succ_l in I. assumption. apply Nat.lt_0_succ.
+  generalize H0. generalize H. apply tm_step_palindromic_even_center.
+
+  replace (length (hd' ++ a'))
+    with (2 * Nat.div2 (length (hd' ++ a'))
+         + Nat.b2n (Nat.odd (length (hd' ++ a')))).
+  rewrite <- Nat.negb_even. rewrite U. rewrite Nat.add_0_r.
+  rewrite Nat.mul_mod. reflexivity. easy. symmetry. apply Nat.div2_odd.
+  easy. easy. reflexivity. reflexivity. reflexivity.
+
+  intros hd a H I J H'. rewrite Nat.double_S.
+  replace (pred (S (S (Nat.double (S (S m))))))
+    with (S (S (pred (Nat.double (S (S m)))))).
+  rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
+  rewrite Nat.mul_assoc. replace (2*2) with 4.
+
+  assert (W: (length a) mod 4 = 0).
+  rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
+  rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
+  rewrite Nat.mul_assoc in J.
+  replace (2*2) with 4 in J. rewrite J.
+  rewrite <- Nat.mul_mod_idemp_l. reflexivity.
+  easy. reflexivity. apply le_0_n. apply le_0_n.
+
+  assert (
+     hd = tm_morphism (tm_morphism (firstn (length hd / 4)
+           (tm_step (pred (pred n)))))
+     /\ a = tm_morphism (tm_morphism
+           (firstn (length a / 4) (skipn (length hd / 4)
+           (tm_step (pred (pred n))))))
+     /\ tl = tm_morphism (tm_morphism
+           (skipn (length hd / 4 + Nat.div2 (length a))
+             (tm_step (pred (pred n)))))).
+  generalize W. generalize I. generalize H.
+  apply tm_step_palindromic_even_morphism2.
+  destruct H0 as [K H0]. destruct H0 as [L M].
+
+  assert (V: 3 < n). generalize I. generalize H.
+  apply tm_step_palindromic_length_12_n.
+  destruct n. inversion V. destruct n. inversion V. inversion H1.
+
+  rewrite K in H. rewrite L in H. rewrite M in H.
+  rewrite tm_morphism_twice_rev in H.
+  rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
+  rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
+  rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
+  rewrite <- tm_step_lemma in H. rewrite <- tm_step_lemma in H.
+  rewrite <- tm_morphism_eq in H. rewrite <- tm_morphism_eq in H.
+  rewrite Nat.pred_succ in H. rewrite Nat.pred_succ in H.
+  rewrite <- pred_Sn in K. rewrite <- pred_Sn in K.
+  rewrite <- pred_Sn in L. rewrite <- pred_Sn in L.
+
+  pose (hd' := (firstn (length hd / 4) (tm_step n))).
+  pose (a' := (firstn (length a / 4) (skipn (length hd / 4) (tm_step n)))).
+  pose (tl' := (skipn (length hd / 4 + Nat.div2 (length a)) (tm_step n))).
+  fold hd' in H. fold a' in H. fold tl' in H.
+  fold hd' in K. fold a' in L.
+
+  assert (N: length a = length a). reflexivity. rewrite L in N at 2.
+  rewrite tm_morphism_length in N. rewrite tm_morphism_length in N.
+  rewrite Nat.mul_assoc in N. replace (2*2) with 4 in N.
+  assert (O: length hd = length hd). reflexivity. rewrite K in O at 2.
+  rewrite tm_morphism_length in O. rewrite tm_morphism_length in O.
+  rewrite Nat.mul_assoc in O. replace (2*2) with 4 in O.
+
+  rewrite app_length. rewrite N. rewrite O.
+  rewrite <- Nat.mul_add_distr_l. rewrite Nat.mul_mod_distr_l.
+  rewrite Nat.mul_eq_0. right. rewrite <- app_length.
+
+
+Lemma tm_step_palindromic_power2_even :
+  forall (m n : nat) (hd a tl : list bool),
+    tm_step n = hd ++ a ++ (rev a) ++ tl
+    -> 6 < length a
+    -> length a = 2^(Nat.double m)
+    -> length (hd ++ a) mod (2 ^ (pred (Nat.double m))) = 0
+    <-> (length (hd ++ a) / 4) mod (2^(pred (Nat.double (pred m)))) = 0.
+
+
+    rewrite Nat.mul_comm. rewrite Nat.div_mul.
+    rewrite <- app_length. apply IHm with (n := S (S n)) (tl := tl').
+
+
+  (*
+  generalize dependent a. generalize dependent hd.
+  generalize dependent tl. generalize dependent n.
+   *)
+  induction m.
+  - 
+    rewrite tm_step_palindromic_power2_even with (n := S (S n)) (tl := tl).
+    rewrite app_length. rewrite N.
+    rewrite Nat.mul_comm. rewrite Nat.div_add.
+    rewrite O. rewrite Nat.mul_comm. rewrite Nat.div_mul.
+    rewrite <- app_length.
+    assert (even (length (hd' ++ a')) = true).
+    assert (0 < length a').
+    apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
+    rewrite N in I.
+    rewrite <- Nat.mul_1_r in I at 1.
+    rewrite <- Nat.mul_lt_mono_pos_l in I.
+    apply Nat.lt_succ_l in I. assumption.
+    apply Nat.lt_0_succ. generalize H0. generalize H.
+    apply tm_step_palindromic_even_center.
+    replace (length (hd' ++ a'))
+      with (2 * Nat.div2 (length (hd' ++ a'))
+           + Nat.b2n (Nat.odd (length (hd' ++ a')))).
+    rewrite <- Nat.negb_even. rewrite H0.
+    rewrite Nat.add_0_r.
+    rewrite Nat.mul_mod. reflexivity. simpl.
+    apply Nat.neq_succ_0. symmetry. apply Nat.div2_odd.
+    easy. easy. assumption. assumption. assumption.
+    (* intros n V tl hd hd' K O a a' tl' H I J R W L M N. *)
+    (*
+    replace (S (S (S m))) with (pred (S (S (S (S m))))).
+    rewrite Nat.double_S.
+    replace (S (S m)) with (pred (S (S (S m)))).
+    replace (pred (S (S (Nat.double (pred (S (S (S m))))))))
+      with (S (S (pred (Nat.double (pred (S (S (S m)))))))).
+      rewrite Nat.pow_succ_r.
+      rewrite Nat.pow_succ_r.
+      rewrite Nat.mul_assoc. replace (2*2) with 4.
+    replace (length (hd ++ a))
+      with ((length ((tm_morphism (tm_morphism hd))
+            ++ (tm_morphism (tm_morphism a))))/4).
+    rewrite <- tm_step_palindromic_power2_even
+      with (n := S (S (S (S n)))) (tl := tm_morphism (tm_morphism tl)).
+    rewrite Nat.double_S. rewrite Nat.double_S.
+     *)
+  - 
+
+    rewrite tm_step_palindromic_power2_even with (n := (S (S n))) (tl := tl).
+    rewrite <- pred_Sn.
+    rewrite app_length. rewrite N. rewrite O.
+    rewrite <- Nat.mul_add_distr_l.
+    rewrite Nat.mul_comm. rewrite Nat.div_mul.
+    rewrite <- app_length. apply IHm with (n := S (S n)) (tl := tl').
+
+
+
+    rewrite Nat.double_S.
+    replace (pred (S (S (Nat.double (S (S m))))))
+      with (S (S (pred (Nat.double (S (S m)))))).
+    rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
+    rewrite Nat.mul_assoc. replace (2*2) with 4.
+    rewrite app_length. rewrite O. rewrite N.
+    rewrite <- Nat.mul_add_distr_l. rewrite Nat.mul_mod_distr_l.
+    rewrite Nat.mul_eq_0. right. rewrite <- app_length.
+    apply IHm with (n := n) (tl := tl').
+
+
+
+
+  induction m.
+  - (* intros n V tl hd hd' K O a a' tl' H I J R W L M N. *)
+    rewrite tm_step_palindromic_power2_even with (n := S (S n)) (tl := tl).
+    rewrite app_length. rewrite N.
+    rewrite Nat.mul_comm. rewrite Nat.div_add.
+    rewrite O. rewrite Nat.mul_comm. rewrite Nat.div_mul.
+    rewrite <- app_length.
+    assert (even (length (hd' ++ a')) = true).
+    assert (0 < length a').
+    apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
+    rewrite N in I.
+    rewrite <- Nat.mul_1_r in I at 1.
+    rewrite <- Nat.mul_lt_mono_pos_l in I.
+    apply Nat.lt_succ_l in I. assumption.
+    apply Nat.lt_0_succ. generalize H0. generalize H.
+    apply tm_step_palindromic_even_center.
+    replace (length (hd' ++ a'))
+      with (2 * Nat.div2 (length (hd' ++ a'))
+           + Nat.b2n (Nat.odd (length (hd' ++ a')))).
+    rewrite <- Nat.negb_even. rewrite H0.
+    rewrite Nat.add_0_r.
+    rewrite Nat.mul_mod. reflexivity. simpl.
+    apply Nat.neq_succ_0. symmetry. apply Nat.div2_odd.
+    easy. easy. assumption. assumption. assumption.
+  - (* intros n V tl hd hd' K O a a' tl' H I J R W L M N. *)
+    (*
+    replace (S (S (S m))) with (pred (S (S (S (S m))))).
+    rewrite Nat.double_S.
+    replace (S (S m)) with (pred (S (S (S m)))).
+    replace (pred (S (S (Nat.double (pred (S (S (S m))))))))
+      with (S (S (pred (Nat.double (pred (S (S (S m)))))))).
+      rewrite Nat.pow_succ_r.
+      rewrite Nat.pow_succ_r.
+      rewrite Nat.mul_assoc. replace (2*2) with 4.
+    replace (length (hd ++ a))
+      with ((length ((tm_morphism (tm_morphism hd))
+            ++ (tm_morphism (tm_morphism a))))/4).
+    rewrite <- tm_step_palindromic_power2_even
+      with (n := S (S (S (S n)))) (tl := tm_morphism (tm_morphism tl)).
+    rewrite Nat.double_S. rewrite Nat.double_S.
+     *)
+
+    rewrite tm_step_palindromic_power2_even with (n := (S (S n))) (tl := tl).
+    rewrite <- pred_Sn.
+    rewrite app_length. rewrite N. rewrite O.
+    rewrite <- Nat.mul_add_distr_l.
+    rewrite Nat.mul_comm. rewrite Nat.div_mul.
+    rewrite <- app_length. apply IHm with (n := S (S n)) (tl := tl').
+
+
+
+    rewrite Nat.double_S.
+    replace (pred (S (S (Nat.double (S (S m))))))
+      with (S (S (pred (Nat.double (S (S m)))))).
+    rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
+    rewrite Nat.mul_assoc. replace (2*2) with 4.
+    rewrite app_length. rewrite O. rewrite N.
+    rewrite <- Nat.mul_add_distr_l. rewrite Nat.mul_mod_distr_l.
+    rewrite Nat.mul_eq_0. right. rewrite <- app_length.
+    apply IHm with (n := n) (tl := tl').
+
+
+
+
+
+
+    
+Nat.div2_odd: forall a : nat, a = 2 * Nat.div2 a + Nat.b2n (Nat.odd a)
+
+
+
+Nat.div_str_pos: forall a b : nat, 0 < b <= a -> 0 < a / b
+      
+
+    apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
+    apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
+    apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
+    generalize I.
+Theorem tm_step_palindromic_even_center :
+  forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ (rev a) ++ tl
+  -> 0 < length a
+  -> even (length (hd ++ a)) = true.
+
+
+
+    rewrite <- Nat.mul_cancel_l with (p := 4).
+    rewrite <- Nat.mul_mod_distr_l.
+
+    replace (4 * (length (hd ++ a) / 4))
+      with (4 * (length (hd ++ a) / 4) + length (hd ++ a) mod 4).
+    rewrite <- Nat.div_mod_eq. assumption.
+    rewrite H0. rewrite Nat.add_0_r. reflexivity.
+    rewrite app_length.
+    replace (4 * (length (hd ++ a) / 4))
+      with (4 * (length (hd ++ a) / 4) + length (hd ++ a) mod 4).
+    rewrite <- Nat.div_mod_eq. assumption.
+    rewrite H0. rewrite Nat.add_0_r. reflexivity.
+
+
+
+
 
   generalize dependent a.
   generalize dependent hd.