Update

2023-02-01 16:39:17 +01:00 · 2023-02-01 16:39:17 +01:00 · 837884e808
commit 837884e808
parent 36b7d26757
1 changed files with 287 additions and 0 deletions
--- a/src/thue_morse3.v
+++ b/src/thue_morse3.v
@ -1893,6 +1893,292 @@ Qed.



+Lemma xxx :
+  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
+       \/ 64 <= length a.
+Proof.
+  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.
+  inversion H3. inversion H5. inversion H7.
+
+  destruct m. left.
+
+  assert (W: length a mod 4 = 0). rewrite J. reflexivity.
+
+  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.
+
+  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.
+
+  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.
+  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.
+
+  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.
+
+  right. rewrite J. rewrite Nat.double_S. rewrite Nat.double_S.
+  rewrite Nat.double_S. rewrite Nat.pow_succ_r.
+  rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
+  rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
+  rewrite Nat.pow_succ_r. rewrite Nat.mul_assoc.
+  rewrite Nat.mul_assoc. rewrite Nat.mul_assoc.
+  rewrite Nat.mul_assoc. rewrite Nat.mul_assoc.
+
+  replace (2*2*2*2*2*2) with 64. rewrite <- Nat.mul_1_r at 1.
+  rewrite <- Nat.mul_le_mono_pos_l. rewrite Nat.le_succ_l.
+  rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
+  apply Nat.lt_0_succ. reflexivity.
+  apply le_0_n. apply le_0_n. apply le_0_n.
+  apply le_0_n. apply le_0_n. apply le_0_n.
+Qed.
+
+
+
+
+
+
+  destruct m. left.
+
+  assert (K: length a mod 4 = 0). rewrite J. reflexivity.
+  assert (L: length hd mod 4 = 0).
+  generalize K. generalize I. generalize H.
+  apply tm_step_palindromic_length_12_prefix.
+
+  assert (M: 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 K. generalize I. generalize H.
+  apply tm_step_palindromic_even_morphism2.
+
+  destruct M as [N O]. destruct O as [O P].
+  rewrite N in H. rewrite O in H. rewrite P 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.
+   *)
+
+  pose (hd' := (firstn (length hd / 4) (tm_step (pred (pred n))))).
+  pose (a' := (firstn (length a / 4) (skipn (length hd / 4) (tm_step (pred (pred n)))))).
+  pose (tl' := (skipn (length hd / 4 + Nat.div2 (length a)) (tm_step (pred (pred n))))).
+  fold hd' in H. fold a' in H. fold tl' in H.
+
+
+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.
+Proof.
+  intros m n hd a tl. intros H I J.
+
+
+
+
+
+Lemma xxx :
+  forall m : nat,
+    (forall (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)
+    -> (forall (n' : nat) (hd' a' tl' : list bool),
+        tm_step n' = hd' ++ a' ++ (rev a') ++ tl'
+        -> 6 < length a'
+         -> length a' = 2^(Nat.double (S m))
+         -> length (hd' ++ a') mod (2 ^ (pred (Nat.double (S m)))) = 0).
+Proof.
+  intro m. intro IH.
+  intros n hd a tl. intros H I J.
+  rewrite tm_step_palindromic_power2_even with (n := n) (tl := tl).
+  rewrite <- pred_Sn.
+
+  destruct m. rewrite J in I. inversion I. inversion H1.
+  inversion H3. inversion H5. inversion H7.
+
+  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.
+
+  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.
+
+  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.
+  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 <- app_length.
+  rewrite Nat.mul_comm. rewrite Nat.div_mul.
+  apply IH with (n := n) (tl := tl'). assumption.
+
+
+
+
+
+
+
+  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.
+
+
+
+
+Lemma tm_step_palindromic_even_morphism2 :
+  forall (n : nat) (hd a tl : list bool),
+  tm_step n = hd ++ a ++ (rev a) ++ tl
+  -> 6 < length a
+  -> (length a) mod 4 = 0
+  -> 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))))).
+
+
+  rewrite Nat.double_S.
+  replace (pred (S (S (Nat.double m)))) with (S (S (pred (Nat.double m)))).
+  rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r. rewrite Nat.mul_assoc.
+  replace (2*2) with 4.
+        
+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.
+
+  
+
+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
+
+
+
+

 Lemma xxx :
  forall (m n k : nat) (hd a tl : list bool),
@ -2048,6 +2334,7 @@ Proof.
  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.
+  apply IHm.


 Lemma tm_step_palindromic_power2_even :