Update

thue-morse.v

@@ -978,10 +978,79 @@ Proof.
 Qed.
 
 
-Prouver l'énoncé du concours général
-(parmi les n premiers termes avec n pairs, il y en a autant de chaque)
+Lemma tm_step_cube2 : forall (n : nat) (a hd tl: list bool),
+  tm_step n = hd ++ a ++ a ++ a ++ tl
+  -> 0 < length a
+  -> even (length hd) = even (length tl).
+Proof.
+  intros n a hd tl. intros H J.
+  destruct n.
+  - assert (3 <= length (a ++ a ++ a)).
+    rewrite app_length. rewrite app_length.
+    rewrite <- Nat.le_succ_l in J.
+    assert (2 <= length a + length a).
+    generalize J. generalize J. apply Nat.add_le_mono.
+    generalize H0. generalize J. apply Nat.add_le_mono.
+    assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
+    rewrite H in H1 at 2.
+    assert (0 <= length hd). apply Nat.le_0_l.
+    assert (0 + 3 <= length hd + length (a ++ a ++ a)).
+    generalize H0. generalize H2. apply Nat.add_le_mono.
+    simpl in H3. rewrite <- app_length in H3.
+    assert (0 <= length tl). apply Nat.le_0_l.
+    assert (3 + 0 <= length (hd ++ a ++ a ++ a) + length (tl)).
+    generalize H4. generalize H3. apply Nat.add_le_mono.
+    rewrite <- app_length in H5. rewrite Nat.add_0_r in H5.
+    replace ((hd++a++a++a)++tl) with (hd++a++a++a++tl) in H5.
+    rewrite <- H1 in H5. simpl in H5. apply le_S_n in H5.
+    apply Nat.nle_succ_0 in H5. contradiction H5.
+    rewrite app_assoc_reverse. rewrite app_inv_head_iff.
+    rewrite app_assoc_reverse. rewrite app_inv_head_iff.
+    rewrite app_assoc_reverse. reflexivity.
+  - assert (Nat.Even (length (tm_step (S n)))).
+    rewrite tm_size_power2. rewrite Nat.pow_succ_r'.
+    apply Nat.Even_mul_l. apply Nat.EvenT_Even.
+    apply Nat.EvenT_2. apply Nat.EvenT_0.
+    rewrite H in H0. rewrite app_length in H0. rewrite app_length in H0.
+    rewrite Nat.add_shuffle3 in H0.
+    assert (I := H). apply tm_step_cube1 in I.
+    apply Nat.even_EvenT in I. apply Nat.EvenT_Even in I.
+    assert (Nat.Even (length hd + length (a ++ a ++ tl))).
+    generalize I. generalize H0. apply Nat.Even_add_Even_inv_r.
+    rewrite app_length in H1. rewrite Nat.add_shuffle3 in H1.
+    assert (Nat.Even (length hd + length (a ++ tl))).
+    generalize I. generalize H1. apply Nat.Even_add_Even_inv_r.
+    rewrite app_length in H2. rewrite Nat.add_shuffle3 in H2.
+    assert (Nat.Even (length hd + length tl)).
+    generalize I. generalize H2. apply Nat.Even_add_Even_inv_r.
+
+    apply Nat.Even_EvenT in H3. apply Nat.EvenT_even in H3.
+    rewrite Nat.even_add in H3.
+    destruct (even (length hd)); destruct (even (length tl)).
+    reflexivity. simpl in H3. inversion H3.
+    simpl in H3. inversion H3. reflexivity.
+Qed.
+
+
+
+
+
+
+
+
+Lemma tm_step_cube2 : forall (n : nat) (a hd tl: list bool),
+  tm_step n = hd ++ a ++ a ++ a ++ tl -> even (length hd) = false
+  -> 0 < length a
+  -> hd_error a = hd_error tl.
+Proof.
+
+
+
+   (hd + [a] + [a] + [a])
+     -> prouver que le premier caractère qui suit hd suit aussi le troisième a
+        quand hd est de taille impaire
+   [hd + false] + /a+false/
 
-Prouver que si un cube existe il est de taille paire
 
 Prouver que si un cube existe de taille k, il existe aussi un cube
 de même taille précédé par un préfixe de taille pair (par décalage éventuel de 1)