Update

src/thue_morse3.v

@@ -228,8 +228,58 @@ Proof.
   reflexivity.
 Qed.
 
+Lemma tm_step_palindromic_even_center' :
+  forall (n k m: nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ (rev a) ++ tl
+  -> 0 < length a
+  -> length (hd ++ a) = S (2 * k) * 2^m
+  -> odd m = true.
+Proof.
+  intros n k m hd a tl. intros H I J.
+  assert (Z := H).
+  assert (K: {a=nil} + {~ a=nil}). apply list_eq_dec. apply bool_dec.
+  destruct K. rewrite e in I. inversion I.
+  assert (L: a = removelast a ++ [ last a false ]).
+  apply app_removelast_last. assumption.
+  rewrite L in H. rewrite rev_app_distr in H.
+  assert (nth_error (tm_step n) (length (hd++a))
+    = nth_error (tm_step n) (pred (length (hd++a)))).
+  rewrite H.
+  rewrite app_assoc. rewrite nth_error_app2.
+  symmetry. rewrite <- app_assoc. rewrite <- app_assoc.
+  rewrite app_assoc. rewrite nth_error_app2.
+  rewrite <- app_removelast_last. rewrite Nat.sub_diag.
+  replace (hd++a) with ((hd ++ removelast a) ++ [last a false]).
+  rewrite app_length. rewrite Nat.add_1_r. rewrite <- pred_Sn.
+  rewrite Nat.sub_diag. reflexivity. rewrite L at 3.
+  rewrite <- app_assoc. reflexivity. assumption.
+
+  rewrite L at 2. rewrite app_assoc.
+  replace (length ((hd ++ removelast a) ++ [last a false]))
+    with (length (hd ++ removelast a) + length [last a false]).
+  rewrite Nat.add_1_r. rewrite <- pred_Sn. apply Nat.le_refl.
+
+  symmetry. rewrite app_length. reflexivity.
+  rewrite <- app_removelast_last. apply Nat.le_refl.
+  assumption.
+
+  generalize H0. rewrite J. apply tm_step_pred.
+
+  rewrite <- J. rewrite <- tm_size_power2. rewrite Z.
+  rewrite app_length. rewrite app_length.
+  apply Nat.add_lt_mono_l. rewrite app_length.
+  rewrite <- Nat.add_0_r at 1.
+  apply Nat.add_lt_mono_l. rewrite app_length. rewrite rev_length.
+  assert (0 <= length tl). apply le_0_n. rewrite <- Nat.add_0_r at 1.
+  apply Nat.add_lt_le_mono; assumption.
+Qed.
 
 
+  
+
+
+
+  (* TODO: voir si une preuve du lemme précédent est plus rapide comme ceci *)
 
 
 (*