Update

src/thue_morse3.v

@@ -2479,43 +2479,62 @@ Proof.
 Qed.
 
 
-
-
-
-Lemma tm_step_palindrome_8_mod8 :
-  forall (n : nat) (hd a tl : list bool),
-  tm_step n = hd ++ a ++ (rev a) ++ tl
-  -> length a = 8
-  -> length (hd ++ a) mod 8 = 0.
+Lemma tm_step_palindromic_power2_odd_alpha :
+  forall (m n : nat) (hd a tl : list bool),
+    tm_step n = hd ++ a ++ (rev a) ++ tl
+    -> 6 < length a
+    -> length a = 2^(S (Nat.double m))
+    -> length (hd ++ a) mod (2 ^ (S (Nat.double m))) = 0
+    <-> (length (hd ++ a) / 4) mod (2^(S (Nat.double (pred m)))) = 0.
 Proof.
-  intros n hd a tl. intros H I.
+  intros m n hd a tl. intros H I J.
 
-  assert (J: length a mod 4 = 0). rewrite I. reflexivity.
-  assert (K := J).
-  rewrite tm_step_palindromic_length_12_prefix
-    with (n := n) (hd := hd) (tl := tl) in K.
-  assert (L: even (length hd) = true).
-  rewrite <- Nat.div_exact in K. rewrite K.
-  rewrite Nat.even_mul. reflexivity. easy.
+  destruct m. rewrite J in I. inversion I. inversion H1. inversion H3.
 
-  assert (M: exists x, a = [x; negb x; negb x; x; negb x; x; x; negb x]).
-  generalize I. generalize H. apply tm_step_palindrome_8_destruct.
-  destruct M as [x M].
+  assert (length (hd ++ a) mod 4 = 0).
+  apply tm_step_palindromic_length_12 with (n := n) (tl := tl); assumption.
 
-  assert (N: length (hd ++ a) mod 4 = 0).
-  rewrite app_length. rewrite Nat.add_mod.
-  rewrite J. rewrite K. reflexivity. easy.
+  assert (length (hd ++ a) mod 8 = 0).
+  apply tm_step_palindrome_mod8 with (n := n) (tl := tl); assumption.
+
+  split.
+  - intro P.
+    rewrite <- Nat.mul_cancel_l with (p := 4).
+    rewrite <- Nat.mul_mod_distr_l.
+    replace 4 with (2*2) at 3. rewrite <- Nat.mul_assoc.
+    rewrite <- Nat.pow_succ_r. rewrite <- Nat.pow_succ_r.
+
+    replace (S (S (S (Nat.double (pred (S m))))))
+      with (S (Nat.double (S m))).
+    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 <- pred_Sn. rewrite Nat.double_S. reflexivity.
+    apply le_0_n. apply le_0_n. reflexivity.
+
+    apply Nat.pow_nonzero. easy. easy. easy.
+  - intro P.
+    rewrite <- Nat.mul_cancel_l with (p := 4) in P.
+    rewrite <- Nat.mul_mod_distr_l in P.
+    replace 4 with (2*2) in P at 3. rewrite <- Nat.mul_assoc in P.
+    rewrite <- Nat.pow_succ_r in P. rewrite <- Nat.pow_succ_r in P.
+
+    replace (S (S (S (Nat.double (pred (S m))))))
+      with (S (Nat.double (S m))) in P.
+    replace (4 * (length (hd ++ a) / 4))
+      with (4 * (length (hd ++ a) / 4) + length (hd ++ a) mod 4) in P.
+    rewrite <- Nat.div_mod_eq in P. assumption.
+    rewrite H0. rewrite Nat.add_0_r. reflexivity.
+
+    rewrite <- pred_Sn. rewrite Nat.double_S. reflexivity.
+    apply le_0_n. apply le_0_n. reflexivity.
+
+    apply Nat.pow_nonzero. easy. easy. easy.
+Qed.
 
-  assert (O: length (hd++a) mod 8 = 0 \/ length (hd++a) mod 8 = 4).
-  assert (length (hd ++ a) mod (4*2)
-  = (length (hd ++ a) mod 4 + 4 * (length (hd ++ a) / 4 mod 2))).
-  apply Nat.mod_mul_r. easy. easy.
-  rewrite N in H0. rewrite <- Nat.bit0_mod in H0.
-  replace (4*2) with 8 in H0. rewrite Nat.add_0_l in H0.
-  destruct (Nat.testbit (length (hd ++ a) / 4)).
-  right. apply H0. left. apply H0. reflexivity.
 
-  destruct O as [O | O]. assumption.