baruchel
/
coqbooks
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
coqbooks/src/thue_morse4.v

1773 lines
76 KiB
Coq
Raw Blame History

(** * The Thue-Morse sequence (part 4)
TODO
*)
Require Import thue_morse.
Require Import thue_morse2.
Require Import thue_morse3.
Require Import Coq.Lists.List.
Require Import PeanoNat.
Require Import Nat.
Require Import Bool.
Require Import Arith.
Require Import Lia.
Import ListNotations.
Theorem tm_step_palindrome_power2_inverse :
forall (m n k : nat) (hd tl : list bool),
tm_step n = hd ++ tl
-> length hd = S (Nat.double k) * 2^m
-> odd m = true
-> tl <> nil
-> skipn (length hd - 2^m) hd = rev (firstn (2^m) tl).
Proof.
intros m n k hd tl. intros H I J K.
assert (L: 2^m <= length hd). rewrite I. lia.
assert (M: m < n). assert (length (tm_step n) = length (hd ++ tl)).
rewrite H. reflexivity. rewrite tm_size_power2 in H0.
rewrite app_length in H0. (* rewrite I in H0. *)
assert (n <= m \/ m < n). apply Nat.le_gt_cases. destruct H1.
apply Nat.pow_le_mono_r with (a := 2) in H1.
assert (2^n <= length hd). generalize L. generalize H1. apply Nat.le_trans.
rewrite H0 in H2. rewrite <- Nat.add_0_r in H2.
rewrite <- Nat.add_le_mono_l in H2. destruct tl. contradiction K.
reflexivity. simpl in H2. apply Nat.nle_succ_0 in H2. contradiction.
easy. assumption.
assert (N: length (tm_step n) mod 2^m = 0). rewrite tm_size_power2.
apply Nat.div_exact. apply Nat.pow_nonzero. easy. rewrite <- Nat.pow_sub_r.
rewrite <- Nat.pow_add_r. rewrite Nat.add_comm. rewrite Nat.sub_add.
reflexivity. apply Nat.lt_le_incl. assumption. easy.
apply Nat.lt_le_incl. assumption.
assert (O: length hd mod 2^m = 0).
apply Nat.div_exact. apply Nat.pow_nonzero. easy. rewrite I.
rewrite Nat.div_mul. rewrite Nat.mul_comm. reflexivity.
apply Nat.pow_nonzero. easy.
assert (P: length tl mod 2 ^ m = 0).
rewrite H in N. rewrite app_length in N.
rewrite <- Nat.add_mod_idemp_l in N. rewrite O in N. assumption.
apply Nat.pow_nonzero. easy.
assert (Q: 2^m <= length tl). apply Nat.div_exact in P. rewrite P.
destruct (length tl / 2^m). rewrite Nat.mul_0_r in P.
apply length_zero_iff_nil in P. rewrite P in K. contradiction K.
reflexivity.
rewrite <- Nat.mul_1_r at 1. apply Nat.mul_le_mono_l.
apply Nat.le_succ_l. apply Nat.lt_0_succ.
apply Nat.pow_nonzero. easy.
replace hd
with (firstn (length hd - 2^m) hd ++ skipn (length hd - 2^m) hd) in H.
replace tl with (firstn (2^m) tl ++ skipn (2^m) tl) in H.
rewrite <- app_assoc in H.
replace (
skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl ++ skipn (2 ^ m) tl )
with (
(skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl) ++ skipn (2 ^ m) tl )
in H.
assert (length (skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl) = 2^(S m)).
rewrite app_length. rewrite skipn_length. rewrite firstn_length_le.
replace (length hd) with (length hd -2^m + 2^m) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag. simpl. rewrite Nat.add_0_r.
reflexivity. apply Nat.le_refl. apply Nat.sub_add. assumption.
assumption.
assert (length (firstn (length hd - 2^m) hd) mod 2^(S m) = 0).
rewrite firstn_length_le. replace (2^m) with (2^m * 1).
rewrite I. rewrite Nat.mul_comm. rewrite <- Nat.mul_sub_distr_l.
rewrite Nat.sub_succ. rewrite Nat.sub_0_r. rewrite Nat.double_twice.
rewrite Nat.mul_assoc. replace (2^m * 2) with (2^(S m)).
rewrite Nat.mul_comm. rewrite Nat.mod_mul. reflexivity.
apply Nat.pow_nonzero. easy.
rewrite Nat.mul_comm. rewrite Nat.pow_succ_r. reflexivity.
apply Nat.le_0_l. apply Nat.mul_1_r. apply Nat.le_sub_l.
assert (
skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl = tm_step (S m)
\/
skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl = map negb (tm_step (S m))).
generalize H1. generalize H0. generalize H. apply tm_step_repeating_patterns2.
apply tm_step_palindromic_full in J.
destruct H2; rewrite J in H2.
- assert (skipn (length hd - 2^m) hd = tm_step m).
apply app_eq_length_head in H2. assumption.
rewrite skipn_length.
replace (length hd) with (length hd -2^m + 2^m) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag. rewrite tm_size_power2.
reflexivity. reflexivity. apply Nat.sub_add. assumption. rewrite H3.
assert (firstn (2^m) tl = rev (tm_step m)).
rewrite H3 in H2. apply app_inv_head in H2. rewrite <- H2. reflexivity.
rewrite H4. rewrite rev_involutive. reflexivity.
- assert (skipn (length hd - 2^m) hd = map negb (tm_step m)).
rewrite map_app in H2. apply app_eq_length_head in H2. assumption.
rewrite skipn_length. replace (length hd) with (length hd -2^m + 2^m) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite map_length. rewrite tm_size_power2.
reflexivity. reflexivity. apply Nat.sub_add. assumption. rewrite H3.
assert (firstn (2^m) tl = map negb (rev (tm_step m))).
rewrite H3 in H2. rewrite map_app in H2. apply app_inv_head in H2.
rewrite <- H2. reflexivity.
rewrite H4. rewrite map_rev. rewrite rev_involutive. reflexivity.
- rewrite <- app_assoc. reflexivity.
- rewrite firstn_skipn. reflexivity.
- rewrite firstn_skipn. reflexivity.
Qed.
Theorem tm_step_palindrome_power2_inverse' :
forall (m n k : nat) (hd tl : list bool),
tm_step n = hd ++ tl
-> length hd = S (Nat.double k) * 2^m
-> odd m = true
-> 0 < k
-> skipn (length hd - 2^m) hd = rev (firstn (2^m) tl).
Proof.
intros m n k hd tl. intros H I J L.
assert (K: {tl=nil} + {~ tl=nil}). apply list_eq_dec. apply bool_dec.
destruct K as [K|K]. rewrite K in H. rewrite app_nil_r in H.
rewrite <- H in I. rewrite tm_size_power2 in I.
assert (n <= m \/ m < n). apply Nat.le_gt_cases. destruct H0.
apply Nat.pow_le_mono_r with (a := 2) in H0. rewrite I in H0.
rewrite <- Nat.mul_1_l in H0. rewrite <- Nat.mul_le_mono_pos_r in H0.
rewrite <- Nat.succ_le_mono in H0. apply Nat.le_0_r in H0.
rewrite Nat.double_twice in H0. apply Nat.mul_eq_0_r in H0.
rewrite H0 in L. inversion L. easy.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy. easy.
replace n with (n-m+m) in I. rewrite Nat.pow_add_r in I.
rewrite Nat.mul_cancel_r in I.
assert (even (2^(n-m)) = true). apply Nat.sub_gt in H0.
apply Nat.neq_0_lt_0 in H0. destruct (n-m). inversion H0.
rewrite Nat.pow_succ_r. rewrite Nat.even_mul. reflexivity.
apply Nat.le_0_l. rewrite I in H1. rewrite Nat.even_succ in H1.
rewrite Nat.double_twice in H1. rewrite Nat.odd_mul in H1.
inversion H1. apply Nat.pow_nonzero. easy. lia.
generalize K. generalize J. generalize I. generalize H.
apply tm_step_palindrome_power2_inverse.
Qed.
Lemma tm_step_square_even_rev :
forall (j n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 2^(Nat.double j) \/ length a = 3 * 2^(Nat.double j)
-> a = rev a.
Proof.
intro j. induction j; intros n hd a tl; intros H I.
- destruct I.
+ destruct a. inversion H0. destruct a. reflexivity. inversion H0.
+ destruct a. inversion H0. destruct a. inversion H0. destruct a.
inversion H0. destruct a.
assert ({b=b1} + {~ b=b1}). apply bool_dec. destruct H1.
rewrite e. reflexivity.
assert ({b0=b1} + {~ b0=b1}). apply bool_dec. destruct H1.
rewrite e in H.
replace (hd ++ [b; b1; b1] ++ [b; b1; b1] ++ tl)
with ((hd ++ [b]) ++ [b1;b1] ++ [b] ++ [b1;b1] ++ tl) in H.
apply tm_step_consecutive_identical' in H. inversion H.
rewrite <- app_assoc. reflexivity.
assert (b = b0). destruct b; destruct b0; destruct b1;
reflexivity || contradiction n0 || contradiction n1; reflexivity.
rewrite H1 in H.
replace (hd ++ [b0; b0; b1] ++ [b0; b0; b1] ++ tl)
with (hd ++ [b0;b0] ++ [b1] ++ [b0;b0] ++ ([b1] ++ tl)) in H.
apply tm_step_consecutive_identical' in H. inversion H. reflexivity.
inversion H0.
- assert (even (length a) = true).
destruct I; rewrite H0; rewrite Nat.double_S;
rewrite Nat.pow_succ_r.
rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l.
rewrite Nat.even_mul. rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l.
assert (0 < length a). destruct I; rewrite H1.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
apply Nat.mul_pos_pos. lia.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
assert (even (length (hd ++ a)) = true). generalize H1. generalize H.
apply tm_step_square_pos.
assert (even (length hd) = true). rewrite app_length in H2.
rewrite Nat.even_add in H2. rewrite H0 in H2.
destruct (even (length hd)). reflexivity. inversion H2.
assert (2 <= n).
assert (length (tm_step n) = length (tm_step n)). reflexivity.
rewrite H in H4 at 2. rewrite app_length in H4. rewrite Nat.add_comm in H4.
rewrite app_length in H4. destruct I; rewrite H5 in H4;
rewrite tm_size_power2 in H4; rewrite Nat.double_S in H4;
symmetry in H4; apply Nat.eq_le_incl in H4.
assert (2^(S (S (Nat.double j))) <= 2^n).
assert (2^(S (S (Nat.double j)))
<= 2 ^ (S (S (Nat.double j))) + length (a ++ tl) + length hd).
rewrite <- Nat.add_assoc. apply Nat.le_add_r. generalize H4.
generalize H6. apply Nat.le_trans. rewrite <- Nat.pow_le_mono_r_iff in H6.
assert (2 <= S (S (Nat.double j))). lia. generalize H6. generalize H7.
apply Nat.le_trans. apply Nat.lt_1_2.
assert (3 * 2^(S (S (Nat.double j))) <= 2^n).
assert (3 * 2^(S (S (Nat.double j)))
<= 3 * 2 ^ (S (S (Nat.double j))) + length (a ++ tl) + length hd).
rewrite <- Nat.add_assoc. apply Nat.le_add_r. generalize H4.
generalize H6. apply Nat.le_trans.
assert (2^(S (S (Nat.double j))) <= 3 * 2^(S (S (Nat.double j)))).
rewrite <- Nat.mul_1_l at 1. apply Nat.mul_le_mono_pos_r.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy. lia.
assert (2^(S (S (Nat.double j))) <= 2^n).
generalize H6. generalize H7. apply Nat.le_trans.
rewrite <- Nat.pow_le_mono_r_iff in H8.
assert (2 <= S (S (Nat.double j))). lia. generalize H8. generalize H9.
apply Nat.le_trans. apply Nat.lt_1_2.
destruct n. inversion H4. destruct n. inversion H4. inversion H6.
assert(
tm_step (S (S n)) = tm_morphism
(firstn (Nat.div2 (length hd)) (tm_step (S n)) ++
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length hd)) (tm_step (S n))) ++
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length (hd ++ a))) (tm_step (S n))) ++
skipn (Nat.div2 (length (hd ++ a ++ a))) (tm_step (S n)))).
generalize H0. generalize H0. generalize H3. generalize H.
apply tm_step_morphism4.
pose (hd' := firstn (Nat.div2 (length hd)) (tm_step (S n))).
pose (a' := firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length hd)) (tm_step (S n)))).
pose (tl' := skipn (Nat.div2 (length (hd ++ a ++ a))) (tm_step (S n))).
fold hd' in H5. fold a' in H5. fold tl' in H5.
assert (length hd' = Nat.div2 (length hd)). unfold hd'.
rewrite firstn_length_le. reflexivity.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite <- Nat.pow_succ_r.
rewrite <- tm_size_power2. rewrite <- Nat.Even_double.
rewrite H. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption. lia. lia.
assert (length hd = length (tm_morphism hd')).
rewrite tm_morphism_length. rewrite H6. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
assert (length a' = Nat.div2 (length a)). unfold a'.
rewrite firstn_length_le. reflexivity. rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (length a = length (tm_morphism a')).
rewrite tm_morphism_length. rewrite H8. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite H in H5. rewrite tm_morphism_app in H5.
assert (hd = tm_morphism hd'). generalize H7. generalize H5.
apply app_eq_length_head. rewrite <- H10 in H5.
apply app_inv_head in H5. rewrite tm_morphism_app in H5.
assert (a = tm_morphism a'). generalize H9. generalize H5.
apply app_eq_length_head. rewrite <- H11 in H5.
apply app_inv_head in H5. rewrite tm_morphism_app in H5.
assert (length a = length (tm_morphism (
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length (hd ++ a))) (tm_step (S n)))))).
rewrite tm_morphism_length. rewrite firstn_length_le.
rewrite <- Nat.double_twice. rewrite <- Nat.Even_double.
reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H. rewrite app_assoc.
rewrite app_length. rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite app_length. lia. apply Nat.le_refl.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (a = tm_morphism (
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length (hd ++ a))) (tm_step (S n))))).
generalize H12. generalize H5. apply app_eq_length_head.
rewrite <- H13 in H5. apply app_inv_head in H5.
assert (H' := H).
rewrite H10 in H'. rewrite H11 in H'. rewrite H5 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_morphism_eq in H'.
assert (even (length a') = true). unfold a'.
rewrite firstn_length_le.
destruct I; rewrite H14; rewrite Nat.double_S.
rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.div2_double.
rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l. apply Nat.le_0_l.
rewrite Nat.mul_comm. rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite <- Nat.mul_assoc.
rewrite Nat.div2_double.
rewrite Nat.even_mul. rewrite Nat.even_mul.
reflexivity. apply Nat.le_0_l. apply Nat.le_0_l.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (0 < length a'). unfold a'. rewrite firstn_length_le.
destruct (length a). destruct I. inversion H1. inversion H1.
destruct n0. inversion H0. replace (S (S n0)) with (n0 + 1*2).
rewrite Nat.div2_div. rewrite Nat.div_add. rewrite Nat.add_1_r. lia.
easy. lia.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (even (length (hd' ++ a')) = true). generalize H15.
generalize H'. apply tm_step_square_pos.
assert (even (length hd') = true). rewrite app_length in H16.
rewrite Nat.even_add in H16. rewrite H14 in H16.
destruct (even (length hd')). reflexivity. inversion H16.
assert (
tm_step (S n) = tm_morphism
(firstn (Nat.div2 (length hd')) (tm_step n) ++
firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length hd')) (tm_step n)) ++
firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length (hd' ++ a'))) (tm_step n)) ++
skipn (Nat.div2 (length (hd' ++ a' ++ a'))) (tm_step n))).
generalize H14. generalize H14. generalize H17. generalize H'.
apply tm_step_morphism4.
pose (hd'' := firstn (Nat.div2 (length hd')) (tm_step n)).
pose (a'' := firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length hd')) (tm_step n))).
pose (tl'' := skipn (Nat.div2 (length (hd' ++ a' ++ a'))) (tm_step n)).
fold hd'' in H18. fold a'' in H18. fold tl'' in H18.
assert (length hd'' = Nat.div2 (length hd')). unfold hd''.
rewrite firstn_length_le. reflexivity.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite <- Nat.pow_succ_r.
rewrite <- tm_size_power2. rewrite <- Nat.Even_double.
rewrite H'. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption. lia. lia.
assert (length hd' = length (tm_morphism hd'')).
rewrite tm_morphism_length. rewrite H19. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
assert (length a'' = Nat.div2 (length a')). unfold a''.
rewrite firstn_length_le. reflexivity. rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H'. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (length a' = length (tm_morphism a'')).
rewrite tm_morphism_length. rewrite H21. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite H' in H18. rewrite tm_morphism_app in H18.
assert (hd' = tm_morphism hd''). generalize H20. generalize H18.
apply app_eq_length_head. rewrite <- H23 in H18.
apply app_inv_head in H18. rewrite tm_morphism_app in H18.
assert (a' = tm_morphism a''). generalize H22. generalize H18.
apply app_eq_length_head. rewrite <- H24 in H18.
apply app_inv_head in H18. rewrite tm_morphism_app in H18.
assert (length a' = length (tm_morphism (
firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length (hd' ++ a'))) (tm_step n))))).
rewrite tm_morphism_length. rewrite firstn_length_le.
rewrite <- Nat.double_twice. rewrite <- Nat.Even_double.
reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H'. rewrite app_assoc.
rewrite app_length. rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite app_length. lia. apply Nat.le_refl.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (a' = tm_morphism (
firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length (hd' ++ a'))) (tm_step n)))).
generalize H25. generalize H18. apply app_eq_length_head.
rewrite <- H26 in H18. apply app_inv_head in H18.
assert (H'' := H').
rewrite H23 in H''. rewrite H24 in H''. rewrite H18 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_morphism_eq in H''.
assert (0 < length a''). unfold a''. rewrite firstn_length_le.
destruct (length a'). inversion H15. destruct n0. inversion H14.
replace (S (S n0)) with (n0 + 1*2).
rewrite Nat.div2_div. rewrite Nat.div_add. rewrite Nat.add_1_r. lia.
easy. lia.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H'. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (length a = 4 * length a''). rewrite H9. rewrite H24.
rewrite tm_morphism_length. rewrite tm_morphism_length. lia.
assert (
length a'' = 2 ^ Nat.double j \/ length a'' = 3 * 2 ^ Nat.double j ).
destruct I. left.
rewrite <- Nat.mul_cancel_l with (p := 4). rewrite <- H28.
replace 4 with (2*2). rewrite <- Nat.mul_assoc.
rewrite <- Nat.pow_succ_r. rewrite <- Nat.pow_succ_r.
rewrite <- Nat.double_S. assumption. lia. lia. lia. lia.
right.
rewrite <- Nat.mul_cancel_l with (p := 4). rewrite <- H28.
rewrite Nat.mul_assoc. replace (4*3) with (3*2*2).
rewrite <- Nat.mul_assoc. rewrite <- Nat.pow_succ_r.
rewrite <- Nat.mul_assoc. rewrite <- Nat.pow_succ_r.
rewrite <- Nat.double_S. assumption. lia. lia. lia. lia.
assert (a'' = rev a''). generalize H29. generalize H''. apply IHj.
assert (Z := H11).
rewrite H24 in H11. rewrite H30 in H11.
rewrite <- tm_morphism_twice_rev in H11.
rewrite <- H24 in H11. rewrite <- Z in H11. assumption.
Qed.
Lemma tm_step_square_odd_rev :
forall (j n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 2^(S (Nat.double j)) \/ length a = 3 * 2^(S (Nat.double j))
-> a = map negb (rev a).
Proof.
intros j n hd a tl. intros H I.
assert (even (length a) = true).
destruct I; rewrite H0; rewrite Nat.pow_succ_r.
rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l.
rewrite Nat.even_mul. rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l.
assert (0 < length a). destruct I; rewrite H1.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
apply Nat.mul_pos_pos. lia.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
assert (even (length (hd ++ a)) = true). generalize H1. generalize H.
apply tm_step_square_pos.
assert (even (length hd) = true). rewrite app_length in H2.
rewrite Nat.even_add in H2. rewrite H0 in H2.
destruct (even (length hd)). reflexivity. inversion H2.
assert (0 < n).
assert (length (tm_step n) = length (tm_step n)). reflexivity.
rewrite H in H4 at 2. rewrite app_length in H4.
rewrite Nat.add_comm in H4.
destruct a. inversion H1. simpl in H4. rewrite app_length in H4.
simpl in H4. rewrite Nat.add_succ_r in H4.
destruct n. inversion H4. lia.
destruct n. inversion H4.
assert(
tm_step (S n) = tm_morphism
(firstn (Nat.div2 (length hd)) (tm_step n) ++
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length hd)) (tm_step n)) ++
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length (hd ++ a))) (tm_step n)) ++
skipn (Nat.div2 (length (hd ++ a ++ a))) (tm_step n))).
generalize H0. generalize H0. generalize H3. generalize H.
apply tm_step_morphism4.
pose (hd' := firstn (Nat.div2 (length hd)) (tm_step n)).
pose (a' := firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length hd)) (tm_step n))).
pose (tl' := skipn (Nat.div2 (length (hd ++ a ++ a))) (tm_step n)).
fold hd' in H5. fold a' in H5. fold tl' in H5.
assert (length hd' = Nat.div2 (length hd)). unfold hd'.
rewrite firstn_length_le. reflexivity.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite <- Nat.pow_succ_r.
rewrite <- tm_size_power2. rewrite <- Nat.Even_double.
rewrite H. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption. lia. lia.
assert (length hd = length (tm_morphism hd')).
rewrite tm_morphism_length. rewrite H6. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
assert (length a' = Nat.div2 (length a)). unfold a'.
rewrite firstn_length_le. reflexivity. rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (length a = length (tm_morphism a')).
rewrite tm_morphism_length. rewrite H8. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite H in H5. rewrite tm_morphism_app in H5.
assert (hd = tm_morphism hd'). generalize H7. generalize H5.
apply app_eq_length_head. rewrite <- H10 in H5.
apply app_inv_head in H5. rewrite tm_morphism_app in H5.
assert (a = tm_morphism a'). generalize H9. generalize H5.
apply app_eq_length_head. rewrite <- H11 in H5.
apply app_inv_head in H5. rewrite tm_morphism_app in H5.
assert (length a = length (tm_morphism (
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length (hd ++ a))) (tm_step n))))).
rewrite tm_morphism_length. rewrite firstn_length_le.
rewrite <- Nat.double_twice. rewrite <- Nat.Even_double.
reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H. rewrite app_assoc.
rewrite app_length. rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite app_length. lia. apply Nat.le_refl.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (a = tm_morphism (
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
generalize H12. generalize H5. apply app_eq_length_head.
rewrite <- H13 in H5. apply app_inv_head in H5.
assert (H' := H).
rewrite H10 in H'. rewrite H11 in H'. rewrite H5 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_morphism_eq in H'.
assert (length a = 2 * length a'). rewrite tm_morphism_length in H9.
assumption.
assert (length a' = 2^(Nat.double j) \/ length a' = 3 * 2^(Nat.double j)).
destruct I; [left | right]; rewrite <- Nat.mul_cancel_l with (p := 2).
rewrite <- tm_morphism_length. rewrite <- H11.
rewrite <- Nat.pow_succ_r. assumption. lia. lia.
rewrite <- tm_morphism_length. rewrite <- H11.
rewrite Nat.mul_assoc. replace (2*3) with (3*2).
rewrite <- Nat.mul_assoc.
rewrite <- Nat.pow_succ_r. assumption. lia. lia. easy.
assert (Z := H11).
assert (a' = rev a'). generalize H15. generalize H'.
apply tm_step_square_even_rev. rewrite H16 in H11.
rewrite <- tm_morphism_rev2 in H11. rewrite <- Z in H11.
assumption.
Qed.
Theorem tm_step_square_rev :
forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> 0 < length a
-> ( (a = rev a /\ exists j,
length a = 2^(Nat.double j) \/ length a = 3 * 2^(Nat.double j))
\/ (a = map negb (rev a) /\ exists j,
length a = 2^(S (Nat.double j)) \/ length a = 3 * 2^(S (Nat.double j)))).
Proof.
intros n hd a tl. intros H I.
assert (exists k j, length a = S (Nat.double k) * 2^j).
apply trailing_zeros; assumption. destruct H0. destruct H0.
assert (0 < n).
assert (length (tm_step n) = length (tm_step n)). reflexivity.
rewrite H in H1 at 2. rewrite app_length in H1.
rewrite Nat.add_comm in H1.
destruct a. inversion I. simpl in H1. rewrite app_length in H1.
simpl in H1. rewrite Nat.add_succ_r in H1.
destruct n. inversion H1. lia.
destruct n. inversion H1.
assert (x = 0 \/ x = 1). generalize H0. generalize H.
apply tm_step_square_size.
assert (Nat.Even x0 \/ Nat.Odd x0). apply Nat.Even_or_Odd.
destruct H3. apply Nat.Even_double in H3. rewrite H3 in H0.
assert (length a = 2^(Nat.double (Nat.div2 x0))
\/ length a = 3 * 2^(Nat.double (Nat.div2 x0))).
destruct H2; [left|right]; rewrite H2 in H0; rewrite H0. simpl. lia.
reflexivity.
left. split.
generalize H4. generalize H. apply tm_step_square_even_rev.
exists (Nat.div2 x0). assumption.
apply Nat.Odd_double in H3. rewrite H3 in H0.
assert (length a = 2^(S (Nat.double (Nat.div2 x0)))
\/ length a = 3 * 2^(S (Nat.double (Nat.div2 x0)))).
destruct H2; [left|right]; rewrite H2 in H0; rewrite H0. simpl. lia.
reflexivity.
right. split.
generalize H4. generalize H. apply tm_step_square_odd_rev.
exists (Nat.div2 x0). assumption.
Qed.
Lemma tm_step_square_rev_even :
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 2^m
-> a = rev a
-> even m = true.
Proof.
intros m n hd a tl. intros H I J.
assert (0 < length a). rewrite I.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
assert ( (a = rev a /\ exists j,
length a = 2^(Nat.double j) \/ length a = 3 * 2^(Nat.double j))
\/ (a = map negb (rev a) /\ exists j,
length a = 2^(S (Nat.double j)) \/ length a = 3 * 2^(S (Nat.double j)))).
generalize H0. generalize H. apply tm_step_square_rev.
destruct H1. destruct H1. destruct H2. destruct H2.
rewrite I in H2. apply Nat.pow_inj_r in H2. rewrite H2.
rewrite Nat.double_twice. rewrite Nat.even_mul. reflexivity. lia.
rewrite I in H2.
assert (Nat.log2 (2^m) = Nat.log2 (2^m)). reflexivity. rewrite H2 in H3 at 2.
rewrite Nat.log2_pow2 in H3.
rewrite Nat.log2_mul_pow2 in H3. replace (Nat.log2 3) with 1 in H3.
rewrite H3 in H2. rewrite Nat.add_succ_r in H2.
rewrite Nat.add_0_r in H2. rewrite Nat.pow_succ_r in H2.
rewrite Nat.mul_cancel_r in H2. inversion H2.
apply Nat.pow_nonzero. easy. lia. reflexivity. lia. lia. lia.
destruct H1. rewrite J in H1 at 1.
destruct a. inversion H0. simpl in H1.
rewrite map_app in H1. apply app_inj_tail in H1.
destruct H1. destruct b. inversion H3. inversion H3.
Qed.
Lemma tm_step_square_rev_odd :
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 2^m
-> a = map negb (rev a)
-> odd m = true.
Proof.
intros m n hd a tl. intros H I J.
assert (0 < length a). rewrite I.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
assert ( (a = rev a /\ exists j,
length a = 2^(Nat.double j) \/ length a = 3 * 2^(Nat.double j))
\/ (a = map negb (rev a) /\ exists j,
length a = 2^(S (Nat.double j)) \/ length a = 3 * 2^(S (Nat.double j)))).
generalize H0. generalize H. apply tm_step_square_rev.
destruct H1. destruct H1. rewrite J in H1 at 1.
destruct a. inversion H0. simpl in H1.
rewrite map_app in H1. apply app_inj_tail in H1.
destruct H1. destruct b. inversion H3. inversion H3.
destruct H1. destruct H2. destruct H2.
rewrite I in H2. apply Nat.pow_inj_r in H2. rewrite H2.
rewrite Nat.odd_succ. rewrite Nat.double_twice.
rewrite Nat.even_mul. reflexivity. lia.
rewrite I in H2.
assert (Nat.log2 (2^m) = Nat.log2 (2^m)). reflexivity. rewrite H2 in H3 at 2.
rewrite Nat.log2_pow2 in H3.
rewrite Nat.log2_mul_pow2 in H3. replace (Nat.log2 3) with 1 in H3.
rewrite H3 in H2. rewrite Nat.add_succ_r in H2.
rewrite Nat.add_0_r in H2. rewrite Nat.pow_succ_r in H2.
rewrite Nat.mul_cancel_r in H2. inversion H2.
apply Nat.pow_nonzero. easy. lia. reflexivity. lia. lia. lia.
Qed.
Lemma tm_step_square_even_rev'_alpha :
forall (j n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 2^(Nat.double j) \/ length a = 3 * 2^(Nat.double j)
-> skipn (length (hd ++ a) - 2^(S (Nat.log2 (length a)))) (hd ++ a)
= rev (firstn (2^S (Nat.log2 (length a))) (a ++ tl))
/\ 2^(S (Nat.log2 (length a))) <= length (hd ++ a)
/\ 2^(S (Nat.log2 (length a))) <= length (a ++ tl).
Proof.
intro j. induction j; intros n hd a tl; intros H I.
- destruct n. assert (length a = 0 \/ 0 < length a).
apply Nat.eq_0_gt_0_cases. destruct H0. rewrite H0 in I.
destruct I. symmetry in H1. apply Nat.pow_nonzero in H1. contradiction.
easy. symmetry in H1. apply Nat.eq_mul_0 in H1. destruct H1.
inversion H1. apply Nat.pow_nonzero in H1. contradiction. easy.
assert (length (tm_step 0) = length (hd ++ a ++ a ++ tl)).
rewrite H. reflexivity. rewrite app_length in H1.
rewrite Nat.add_comm in H1.
rewrite app_length in H1. rewrite app_length in H1.
rewrite Nat.add_assoc in H1. symmetry in H1.
apply Nat.eq_le_incl in H1.
assert (2 <= length a + length a + length tl + length hd).
assert (2 <= length a + length a). replace 2 with (1+1).
rewrite <- Nat.le_succ_l in H0.
apply Nat.add_le_mono; assumption. reflexivity.
assert (length a + length a
<= (length a + length a) + (length tl + length hd)).
apply Nat.le_add_r. generalize H3. generalize H2.
rewrite Nat.add_assoc. apply Nat.le_trans.
assert (2 <= length (tm_step 0)).
generalize H1. generalize H2. apply Nat.le_trans.
inversion H3. apply Nat.nle_succ_0 in H5. contradiction.
destruct I.
+ destruct a. inversion H0. destruct a.
destruct hd.
assert (nth_error (tm_step (S n)) 0
<> nth_error (tm_step (S (S n))) (S (2*0))).
apply tm_step_succ_double_index.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
rewrite tm_step_stable with (n := S (S n)) (m := S n) in H1.
rewrite H in H1. simpl in H1. contradiction H1. reflexivity.
replace (S (2*0)) with (2^0).
rewrite <- Nat.pow_lt_mono_r_iff. lia. lia.
replace (S (2*0)) with (2^0).
rewrite <- Nat.pow_lt_mono_r_iff. lia. lia.
rewrite app_removelast_last with (l := b0::hd) (d := false) at 2.
destruct tl.
assert (tm_step (S n) = rev (tm_step (S n))
\/ tm_step (S n) = map negb (rev (tm_step (S n)))).
apply tm_step_rev. rewrite H in H1 at 2. rewrite H in H1 at 3.
rewrite rev_app_distr in H1. rewrite rev_app_distr in H1.
rewrite <- app_nil_end in H1.
assert (nth_error (tm_step (S n)) 0
<> nth_error (tm_step (S (S n))) (S (2*0))).
apply tm_step_succ_double_index.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
rewrite tm_step_stable with (n := S (S n)) (m := S n) in H2.
destruct H1.
rewrite H1 in H2. simpl in H2. contradiction H2. reflexivity.
rewrite H1 in H2. simpl in H2. contradiction H2. reflexivity.
replace (S (2*0)) with (2^0).
rewrite <- Nat.pow_lt_mono_r_iff. lia. lia.
replace (S (2*0)) with (2^0).
rewrite <- Nat.pow_lt_mono_r_iff. lia. lia.
assert ({last (b0::hd) false=b} + {~ last (b0::hd) false=b}).
apply bool_dec. destruct H1.
rewrite app_removelast_last with (l := b0::hd) (d := false) in H.
rewrite e in H. rewrite <- app_assoc in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_1. easy.
assert ({b=b1} + {~ b=b1}). apply bool_dec. destruct H1.
rewrite <- e in H. replace (b::tl) with ([b] ++ tl) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_1. easy.
assert (last (b0::hd) false = b1).
destruct (last (b0::hd) false); destruct b; destruct b1;
reflexivity || contradiction n0 || contradiction n1; reflexivity.
rewrite H1.
rewrite app_length.
replace (2 ^ S (Nat.log2 (length [b]))) with (1+1).
rewrite Nat.sub_add_distr. rewrite <- Nat.add_sub_assoc.
rewrite Nat.sub_diag. rewrite Nat.add_0_r.
rewrite <- app_assoc. rewrite skipn_app.
replace (length (b0::hd) - 1) with (length (removelast (b0::hd))).
rewrite skipn_all. rewrite Nat.sub_diag.
split. reflexivity. rewrite <- Nat.add_le_mono_r.
split. simpl. rewrite <- Nat.succ_le_mono. lia.
simpl. rewrite <- Nat.succ_le_mono. lia.
rewrite app_removelast_last with (l := b0::hd) (d := false) at 2.
rewrite app_length. rewrite Nat.add_sub. reflexivity.
easy. apply Nat.le_refl. reflexivity. easy.
inversion H0.
+ destruct a. inversion H0. destruct a. inversion H0. destruct a.
inversion H0. destruct a.
assert ({b=b0} + {~ b=b0}). apply bool_dec. destruct H1.
rewrite e in H.
replace (hd ++ [b0; b0; b1] ++ [b0; b0; b1] ++ tl)
with (hd ++ [b0;b0] ++ [b1] ++ [b0;b0] ++ b1::tl) in H.
apply tm_step_consecutive_identical' in H. inversion H. reflexivity.
assert ({b0=b1} + {~ b0=b1}). apply bool_dec. destruct H1.
rewrite e in H.
replace (hd ++ [b; b1; b1] ++ [b; b1; b1] ++ tl)
with ((hd ++ [b]) ++ [b1;b1] ++ [b] ++ [b1;b1] ++ tl) in H.
apply tm_step_consecutive_identical' in H. inversion H.
rewrite <- app_assoc. reflexivity.
assert (b = b1). destruct b; destruct b0; destruct b1;
reflexivity || contradiction n0 || contradiction n1; reflexivity.
rewrite H1 in H. rewrite H1.
destruct n.
assert (length (tm_step 1) = length (tm_step 1)). reflexivity.
rewrite H in H2 at 2.
rewrite app_length in H2. rewrite Nat.add_comm in H2.
rewrite app_length in H2. inversion H2.
destruct hd.
assert (nth_error (tm_step (S n)) 1
= nth_error (tm_step (S (S n))) (2 * 1)).
apply tm_step_double_index.
rewrite tm_step_stable with (n := S n) (m := S (S n)) in H2.
rewrite H in H2. simpl in H2. inversion H2. rewrite H4 in H.
replace ([] ++ [b1; b1; b1] ++ [b1; b1; b1] ++ tl)
with ([b1;b1;b1] ++ [b1] ++ [b1] ++ [b1] ++ tl) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_1. reflexivity. replace 1 with (2^0) at 1.
apply Nat.pow_lt_mono_r. lia. lia. reflexivity.
replace 1 with (2^0) at 1.
apply Nat.pow_lt_mono_r. lia. lia. reflexivity.
rewrite app_removelast_last with (l := b2::hd) (d := false) at 2.
destruct tl.
assert (tm_step (S (S n)) = rev (tm_step (S (S n)))
\/ tm_step (S (S n)) = map negb (rev (tm_step (S (S n))))).
apply tm_step_rev. rewrite H in H2 at 2. rewrite H in H2 at 3.
destruct H2. rewrite rev_app_distr in H2.
assert (nth_error (tm_step (S n)) 1
= nth_error (tm_step (S (S n))) (2 * 1)).
apply tm_step_double_index.
rewrite tm_step_stable with (n := S n) (m := S (S n)) in H3.
rewrite H2 in H3. simpl in H3. inversion H3. rewrite H5 in H.
replace ((b2 :: hd) ++ [b1; b1; b1] ++ [b1; b1; b1] ++ [])
with ((b2::hd) ++ [b1] ++ [b1] ++ [b1] ++ [b1;b1;b1]) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_1. reflexivity.
replace 1 with (2^0) at 1.
apply Nat.pow_lt_mono_r. lia. lia. reflexivity.
replace 1 with (2^0) at 1.
apply Nat.pow_lt_mono_r. lia. lia. reflexivity.
rewrite rev_app_distr in H2.
assert (nth_error (tm_step (S n)) 1
= nth_error (tm_step (S (S n))) (2 * 1)).
apply tm_step_double_index.
rewrite tm_step_stable with (n := S n) (m := S (S n)) in H3.
rewrite H2 in H3. simpl in H3. inversion H3.
assert (b0=b1). destruct b0; destruct b1; reflexivity || inversion H5.
rewrite H4 in H.
replace ((b2 :: hd) ++ [b1; b1; b1] ++ [b1; b1; b1] ++ [])
with ((b2::hd) ++ [b1] ++ [b1] ++ [b1] ++ [b1;b1;b1]) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_1. reflexivity.
replace 1 with (2^0) at 1.
apply Nat.pow_lt_mono_r. lia. lia. reflexivity.
replace 1 with (2^0) at 1.
apply Nat.pow_lt_mono_r. lia. lia. reflexivity.
assert ({last (b2::hd) false=b1} + {~ last (b2::hd) false=b1}).
apply bool_dec. destruct H2.
rewrite app_removelast_last with (l := b2::hd) (d := false) in H.
rewrite e in H. rewrite <- app_assoc in H.
replace (
removelast (b2 :: hd) ++ [b1]
++ [b1; b0; b1] ++ [b1; b0; b1] ++ b3 :: tl)
with (removelast (b2::hd) ++ [b1;b1] ++ [b0] ++ [b1;b1]
++ b0::b1::b3::tl) in H.
apply tm_step_consecutive_identical' in H. inversion H.
reflexivity. easy.
assert ({b3=b1} + {~ b3=b1}). apply bool_dec. destruct H2.
rewrite <- e in H.
replace ((b2 :: hd) ++ [b3; b0; b3] ++ [b3; b0; b3] ++ b3 :: tl)
with (((b2::hd) ++ [b3;b0]) ++ [b3;b3] ++ [b0] ++ [b3;b3] ++ tl) in H.
apply tm_step_consecutive_identical' in H. inversion H.
rewrite <- app_assoc. reflexivity.
assert (last (b2::hd) false = b3).
destruct (last (b2::hd) false); destruct b3; destruct b1;
reflexivity || contradiction n2 || contradiction n3; reflexivity.
rewrite H2.
rewrite app_length.
replace (2 ^ S (Nat.log2 (length [b1;b0;b1]))) with (3+1).
rewrite Nat.sub_add_distr. rewrite <- Nat.add_sub_assoc.
rewrite Nat.sub_diag. rewrite Nat.add_0_r.
rewrite <- app_assoc. rewrite skipn_app.
replace (length (b2::hd) - 1) with (length (removelast (b2::hd))).
rewrite skipn_all. rewrite Nat.sub_diag.
split. reflexivity. simpl. rewrite Nat.add_succ_r.
rewrite Nat.add_succ_r. rewrite Nat.add_succ_r.
split. lia. lia.
rewrite app_removelast_last with (l := b2::hd) (d := false) at 2.
rewrite app_length. rewrite Nat.add_sub. reflexivity.
easy. apply Nat.le_refl. reflexivity. easy.
inversion H0.
- assert (even (length a) = true).
destruct I; rewrite H0; rewrite Nat.double_S;
rewrite Nat.pow_succ_r.
rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l.
rewrite Nat.even_mul. rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l.
assert (0 < length a). destruct I; rewrite H1.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
apply Nat.mul_pos_pos. lia.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
assert (even (length (hd ++ a)) = true). generalize H1. generalize H.
apply tm_step_square_pos.
assert (even (length hd) = true). rewrite app_length in H2.
rewrite Nat.even_add in H2. rewrite H0 in H2.
destruct (even (length hd)). reflexivity. inversion H2.
assert (2 <= n).
assert (length (tm_step n) = length (tm_step n)). reflexivity.
rewrite H in H4 at 2. rewrite app_length in H4. rewrite Nat.add_comm in H4.
rewrite app_length in H4. destruct I; rewrite H5 in H4;
rewrite tm_size_power2 in H4; rewrite Nat.double_S in H4;
symmetry in H4; apply Nat.eq_le_incl in H4.
assert (2^(S (S (Nat.double j))) <= 2^n).
assert (2^(S (S (Nat.double j)))
<= 2 ^ (S (S (Nat.double j))) + length (a ++ tl) + length hd).
rewrite <- Nat.add_assoc. apply Nat.le_add_r. generalize H4.
generalize H6. apply Nat.le_trans. rewrite <- Nat.pow_le_mono_r_iff in H6.
assert (2 <= S (S (Nat.double j))). lia. generalize H6. generalize H7.
apply Nat.le_trans. apply Nat.lt_1_2.
assert (3 * 2^(S (S (Nat.double j))) <= 2^n).
assert (3 * 2^(S (S (Nat.double j)))
<= 3 * 2 ^ (S (S (Nat.double j))) + length (a ++ tl) + length hd).
rewrite <- Nat.add_assoc. apply Nat.le_add_r. generalize H4.
generalize H6. apply Nat.le_trans.
assert (2^(S (S (Nat.double j))) <= 3 * 2^(S (S (Nat.double j)))).
rewrite <- Nat.mul_1_l at 1. apply Nat.mul_le_mono_pos_r.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy. lia.
assert (2^(S (S (Nat.double j))) <= 2^n).
generalize H6. generalize H7. apply Nat.le_trans.
rewrite <- Nat.pow_le_mono_r_iff in H8.
assert (2 <= S (S (Nat.double j))). lia. generalize H8. generalize H9.
apply Nat.le_trans. apply Nat.lt_1_2.
destruct n. inversion H4. destruct n. inversion H4. inversion H6.
assert(
tm_step (S (S n)) = tm_morphism
(firstn (Nat.div2 (length hd)) (tm_step (S n)) ++
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length hd)) (tm_step (S n))) ++
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length (hd ++ a))) (tm_step (S n))) ++
skipn (Nat.div2 (length (hd ++ a ++ a))) (tm_step (S n)))).
generalize H0. generalize H0. generalize H3. generalize H.
apply tm_step_morphism4.
pose (hd' := firstn (Nat.div2 (length hd)) (tm_step (S n))).
pose (a' := firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length hd)) (tm_step (S n)))).
pose (tl' := skipn (Nat.div2 (length (hd ++ a ++ a))) (tm_step (S n))).
fold hd' in H5. fold a' in H5. fold tl' in H5.
assert (length hd' = Nat.div2 (length hd)). unfold hd'.
rewrite firstn_length_le. reflexivity.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite <- Nat.pow_succ_r.
rewrite <- tm_size_power2. rewrite <- Nat.Even_double.
rewrite H. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption. lia. lia.
assert (length hd = length (tm_morphism hd')).
rewrite tm_morphism_length. rewrite H6. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
assert (length a' = Nat.div2 (length a)). unfold a'.
rewrite firstn_length_le. reflexivity. rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (length a = length (tm_morphism a')).
rewrite tm_morphism_length. rewrite H8. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite H in H5. rewrite tm_morphism_app in H5.
assert (hd = tm_morphism hd'). generalize H7. generalize H5.
apply app_eq_length_head. rewrite <- H10 in H5.
apply app_inv_head in H5. rewrite tm_morphism_app in H5.
assert (a = tm_morphism a'). generalize H9. generalize H5.
apply app_eq_length_head. rewrite <- H11 in H5.
apply app_inv_head in H5. rewrite tm_morphism_app in H5.
assert (length a = length (tm_morphism (
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length (hd ++ a))) (tm_step (S n)))))).
rewrite tm_morphism_length. rewrite firstn_length_le.
rewrite <- Nat.double_twice. rewrite <- Nat.Even_double.
reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H. rewrite app_assoc.
rewrite app_length. rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite app_length. lia. apply Nat.le_refl.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (a = tm_morphism (
firstn (Nat.div2 (length a))
(skipn (Nat.div2 (length (hd ++ a))) (tm_step (S n))))).
generalize H12. generalize H5. apply app_eq_length_head.
rewrite <- H13 in H5. apply app_inv_head in H5.
assert (H' := H).
rewrite H10 in H'. rewrite H11 in H'. rewrite H5 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_morphism_eq in H'.
assert (even (length a') = true). unfold a'.
rewrite firstn_length_le.
destruct I; rewrite H14; rewrite Nat.double_S.
rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.div2_double.
rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l. apply Nat.le_0_l.
rewrite Nat.mul_comm. rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite <- Nat.mul_assoc.
rewrite Nat.div2_double.
rewrite Nat.even_mul. rewrite Nat.even_mul.
reflexivity. apply Nat.le_0_l. apply Nat.le_0_l.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (0 < length a'). unfold a'. rewrite firstn_length_le.
destruct (length a). destruct I. inversion H1. inversion H1.
destruct n0. inversion H0. replace (S (S n0)) with (n0 + 1*2).
rewrite Nat.div2_div. rewrite Nat.div_add. rewrite Nat.add_1_r. lia.
easy. lia.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (even (length (hd' ++ a')) = true). generalize H15.
generalize H'. apply tm_step_square_pos.
assert (even (length hd') = true). rewrite app_length in H16.
rewrite Nat.even_add in H16. rewrite H14 in H16.
destruct (even (length hd')). reflexivity. inversion H16.
assert (
tm_step (S n) = tm_morphism
(firstn (Nat.div2 (length hd')) (tm_step n) ++
firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length hd')) (tm_step n)) ++
firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length (hd' ++ a'))) (tm_step n)) ++
skipn (Nat.div2 (length (hd' ++ a' ++ a'))) (tm_step n))).
generalize H14. generalize H14. generalize H17. generalize H'.
apply tm_step_morphism4.
pose (hd'' := firstn (Nat.div2 (length hd')) (tm_step n)).
pose (a'' := firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length hd')) (tm_step n))).
pose (tl'' := skipn (Nat.div2 (length (hd' ++ a' ++ a'))) (tm_step n)).
fold hd'' in H18. fold a'' in H18. fold tl'' in H18.
assert (length hd'' = Nat.div2 (length hd')). unfold hd''.
rewrite firstn_length_le. reflexivity.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite <- Nat.pow_succ_r.
rewrite <- tm_size_power2. rewrite <- Nat.Even_double.
rewrite H'. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption. lia. lia.
assert (length hd' = length (tm_morphism hd'')).
rewrite tm_morphism_length. rewrite H19. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
assert (length a'' = Nat.div2 (length a')). unfold a''.
rewrite firstn_length_le. reflexivity. rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H'. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (length a' = length (tm_morphism a'')).
rewrite tm_morphism_length. rewrite H21. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite H' in H18. rewrite tm_morphism_app in H18.
assert (hd' = tm_morphism hd''). generalize H20. generalize H18.
apply app_eq_length_head. rewrite <- H23 in H18.
apply app_inv_head in H18. rewrite tm_morphism_app in H18.
assert (a' = tm_morphism a''). generalize H22. generalize H18.
apply app_eq_length_head. rewrite <- H24 in H18.
apply app_inv_head in H18. rewrite tm_morphism_app in H18.
assert (length a' = length (tm_morphism (
firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length (hd' ++ a'))) (tm_step n))))).
rewrite tm_morphism_length. rewrite firstn_length_le.
rewrite <- Nat.double_twice. rewrite <- Nat.Even_double.
reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H'. rewrite app_assoc.
rewrite app_length. rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite app_length. lia. apply Nat.le_refl.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (a' = tm_morphism (
firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length (hd' ++ a'))) (tm_step n)))).
generalize H25. generalize H18. apply app_eq_length_head.
rewrite <- H26 in H18. apply app_inv_head in H18.
assert (H'' := H').
rewrite H23 in H''. rewrite H24 in H''. rewrite H18 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_morphism_eq in H''.
assert (0 < length a''). unfold a''. rewrite firstn_length_le.
destruct (length a'). inversion H15. destruct n0. inversion H14.
replace (S (S n0)) with (n0 + 1*2).
rewrite Nat.div2_div. rewrite Nat.div_add. rewrite Nat.add_1_r. lia.
easy. lia.
rewrite skipn_length.
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double.
rewrite H'. rewrite app_length. rewrite app_length. lia.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
lia. lia.
assert (length a = 4 * length a''). rewrite H9. rewrite H24.
rewrite tm_morphism_length. rewrite tm_morphism_length. lia.
assert (G: length hd = 4 * length hd''). rewrite H7. rewrite H23.
rewrite tm_morphism_length. rewrite tm_morphism_length. lia.
assert (G': length tl = 4 * length tl''). rewrite H5. rewrite H18.
rewrite tm_morphism_length. rewrite tm_morphism_length. lia.
assert (
length a'' = 2 ^ Nat.double j \/ length a'' = 3 * 2 ^ Nat.double j ).
destruct I. left.
rewrite <- Nat.mul_cancel_l with (p := 4). rewrite <- H28.
replace 4 with (2*2). rewrite <- Nat.mul_assoc.
rewrite <- Nat.pow_succ_r. rewrite <- Nat.pow_succ_r.
rewrite <- Nat.double_S. assumption. lia. lia. lia. lia.
right.
rewrite <- Nat.mul_cancel_l with (p := 4). rewrite <- H28.
rewrite Nat.mul_assoc. replace (4*3) with (3*2*2).
rewrite <- Nat.mul_assoc. rewrite <- Nat.pow_succ_r.
rewrite <- Nat.mul_assoc. rewrite <- Nat.pow_succ_r.
rewrite <- Nat.double_S. assumption. lia. lia. lia. lia.
assert (
skipn (length (hd'' ++ a'')
- 2 ^ S (Nat.log2 (length a''))) (hd'' ++ a'')
= rev (firstn (2 ^ S (Nat.log2 (length a''))) (a'' ++ tl''))
/\ 2 ^ S (Nat.log2 (length a'')) <= length (hd'' ++ a'')
/\ 2 ^ S (Nat.log2 (length a'')) <= length (a'' ++ tl'')).
generalize H29. generalize H''. apply IHj.
destruct H30 as [H30 H30']. destruct H30' as [H30' H30''].
rewrite app_assoc in H''.
rewrite <- firstn_skipn with (l := hd'' ++ a'')
(n := length (hd'' ++ a'') - 2^(S (Nat.log2 (length a'')))) in H''.
rewrite <- firstn_skipn with (l := a'' ++ tl'')
(n := 2^(S (Nat.log2 (length a'')))) in H''.
assert (rev ( skipn (length (hd'' ++ a'') - 2 ^ S (Nat.log2 (length a'')))
(hd'' ++ a'')) = firstn (2 ^ S (Nat.log2 (length a''))) (a'' ++ tl'')).
rewrite H30. rewrite rev_involutive. reflexivity.
rewrite <- H31 in H''.
assert (tm_step (S n) = tm_step (S n)). reflexivity.
rewrite <- tm_step_lemma in H32 at 2. rewrite H'' in H32.
rewrite tm_morphism_app in H32.
rewrite tm_morphism_app in H32.
rewrite tm_morphism_app in H32.
assert (tm_step (S (S n)) = tm_step (S (S n))). reflexivity.
rewrite <- tm_step_lemma in H33 at 2. rewrite H32 in H33.
rewrite tm_morphism_app in H33.
rewrite tm_morphism_app in H33.
rewrite tm_morphism_app in H33.
rewrite <- tm_morphism_twice_rev in H33. rewrite <- app_assoc in H33.
rewrite app_assoc in H.
rewrite <- firstn_skipn with (l := hd ++ a)
(n := length (hd++a) - 2^(S (Nat.log2 (length a)))) in H.
rewrite <- firstn_skipn
with (l := a ++ tl) (n := 2^(S (Nat.log2 (length a)))) in H.
rewrite <- app_assoc in H. rewrite H in H33.
assert (length (
firstn (length (hd ++ a) - 2 ^ S (Nat.log2 (length a))) (hd ++ a))
= length (tm_morphism
(tm_morphism
(firstn (length (hd'' ++ a'') - 2 ^ S (Nat.log2 (length a'')))
(hd'' ++ a''))))).
rewrite tm_morphism_length. rewrite tm_morphism_length.
rewrite firstn_length_le. rewrite firstn_length_le.
rewrite app_length. rewrite H28. rewrite G.
rewrite <- Nat.mul_add_distr_l. rewrite <- app_length.
replace (4 * length a'') with (length a'' * 2^2).
rewrite Nat.log2_mul_pow2.
rewrite Nat.add_succ_l. rewrite Nat.add_succ_l.
rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.mul_assoc. rewrite Nat.add_0_l.
replace (2*2) with 4.
rewrite <- Nat.mul_sub_distr_l.
rewrite Nat.mul_assoc. reflexivity.
reflexivity. lia. lia.
destruct a''. rewrite H28 in H1. inversion H1. simpl.
apply Nat.lt_0_succ. lia. rewrite Nat.mul_comm. reflexivity.
apply Nat.le_sub_l. apply Nat.le_sub_l.
assert (length (
skipn (length (hd ++ a) - 2 ^ S (Nat.log2 (length a))) (hd ++ a))
= length (
tm_morphism
(tm_morphism
(skipn (length (hd'' ++ a'') - 2 ^ S (Nat.log2 (length a'')))
(hd'' ++ a''))))).
rewrite tm_morphism_length. rewrite tm_morphism_length.
rewrite skipn_length. rewrite skipn_length.
replace (length (hd++a))
with ((length (hd++a) - 2^S (Nat.log2 (length a)))
+ 2^S (Nat.log2 (length a))) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
replace (length (hd''++a''))
with ((length (hd''++a'') - 2^S (Nat.log2 (length a'')))
+ 2^S (Nat.log2 (length a''))) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite H28. replace 4 with (2^2). rewrite Nat.mul_comm.
rewrite Nat.log2_mul_pow2.
rewrite Nat.add_0_l. rewrite Nat.add_0_l.
rewrite Nat.add_succ_l. rewrite Nat.add_succ_l.
rewrite Nat.add_0_l. reflexivity.
destruct a''. rewrite H28 in H1. inversion H1. simpl.
apply Nat.lt_0_succ. lia. reflexivity.
apply Nat.le_refl. apply Nat.sub_add.
assumption. apply Nat.le_refl. apply Nat.sub_add.
rewrite app_length. rewrite H28. rewrite G.
rewrite <- Nat.mul_add_distr_l. rewrite <- app_length.
rewrite Nat.mul_comm. replace 4 with (2^2) at 1.
rewrite Nat.log2_mul_pow2.
rewrite Nat.add_succ_l. rewrite Nat.add_succ_l.
rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.mul_assoc. rewrite Nat.add_0_l.
replace (2*2) with 4. apply Nat.mul_le_mono_l. assumption.
reflexivity. lia. lia.
destruct a''. rewrite H28 in H1. inversion H1. simpl.
apply Nat.lt_0_succ. lia. reflexivity.
assert (
firstn (length (hd ++ a) - 2 ^ S (Nat.log2 (length a))) (hd ++ a)
= tm_morphism (tm_morphism
(firstn (length (hd'' ++ a'') - 2 ^ S (Nat.log2 (length a'')))
(hd'' ++ a'')))).
generalize H34. generalize H33. apply app_eq_length_head.
rewrite H36 in H33. rewrite app_inv_head_iff in H33.
assert (
skipn (length (hd ++ a) - 2 ^ S (Nat.log2 (length a))) (hd ++ a)
= (tm_morphism (tm_morphism
(skipn (length (hd'' ++ a'') - 2 ^ S (Nat.log2 (length a'')))
(hd'' ++ a''))))).
generalize H35. generalize H33. apply app_eq_length_head.
rewrite H37 in H33. rewrite app_inv_head_iff in H33. rewrite H37.
assert (length (firstn (2 ^ S (Nat.log2 (length a))) (a ++ tl))
= length (rev (tm_morphism (tm_morphism
(skipn (length (hd'' ++ a'') - 2 ^ S (Nat.log2 (length a'')))
(hd'' ++ a'')))))).
rewrite rev_length. rewrite tm_morphism_length. rewrite tm_morphism_length.
rewrite firstn_length_le. rewrite skipn_length.
replace (length (hd''++a''))
with ((length (hd''++a'') - 2^S (Nat.log2 (length a'')))
+ 2^S (Nat.log2 (length a''))) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite H28. rewrite Nat.mul_comm. replace 4 with (2^2).
rewrite Nat.log2_mul_pow2. rewrite Nat.add_succ_l. rewrite Nat.add_succ_l.
rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.add_0_l. rewrite Nat.add_0_l. reflexivity.
lia. lia. destruct a''. rewrite H28 in H1. inversion H1. simpl.
apply Nat.lt_0_succ. lia. reflexivity. apply Nat.le_refl.
apply Nat.sub_add. assumption.
rewrite H28. rewrite Nat.mul_comm. replace 4 with (2^2).
rewrite Nat.log2_mul_pow2. rewrite Nat.add_succ_l. rewrite Nat.add_succ_l.
rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.add_0_l. rewrite app_length. rewrite H28. rewrite G'.
rewrite <- Nat.mul_add_distr_l. rewrite <- app_length.
rewrite Nat.mul_assoc. apply Nat.mul_le_mono_l. assumption.
lia. lia. destruct a''. rewrite H28 in H1. inversion H1. simpl.
apply Nat.lt_0_succ. lia. reflexivity.
assert (firstn (2 ^ S (Nat.log2 (length a))) (a ++ tl)
= rev (tm_morphism (tm_morphism
(skipn (length (hd'' ++ a'') - 2 ^ S (Nat.log2 (length a'')))
(hd'' ++ a''))))).
generalize H38. generalize H33. apply app_eq_length_head.
rewrite H39. rewrite rev_involutive.
split. reflexivity.
split; rewrite app_length; rewrite H28; [rewrite G | rewrite G'];
rewrite <- Nat.mul_add_distr_l; rewrite <- app_length;
replace 4 with (2^2) at 1; rewrite Nat.mul_comm || reflexivity;
rewrite Nat.log2_mul_pow2; lia || rewrite Nat.add_succ_l;
rewrite Nat.add_succ_l; rewrite Nat.add_0_l;
rewrite Nat.pow_succ_r; lia || rewrite Nat.pow_succ_r; lia.
Qed.
Lemma tm_step_square2_rev_even :
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 2^m
-> a = rev a
-> length (hd ++ a) mod 2^(S m) = 0.
Proof.
intros m n hd a tl. intros H I J.
assert (K: even m = true). generalize J. generalize I. generalize H.
apply tm_step_square_rev_even.
assert (length a = 2^m \/ length a = 3*2^m). left. assumption.
assert (K' := K). apply Nat.even_EvenT in K. apply Nat.EvenT_Even in K.
apply Nat.Even_double in K. rewrite K in H0.
assert (
skipn (length (hd ++ a) - 2^(S (Nat.log2 (length a)))) (hd ++ a)
= rev (firstn (2^S (Nat.log2 (length a))) (a ++ tl))
/\ 2^(S (Nat.log2 (length a))) <= length (hd ++ a)
/\ 2^(S (Nat.log2 (length a))) <= length (a ++ tl)).
generalize H0. generalize H. apply tm_step_square_even_rev'_alpha.
destruct H1. rewrite I in H1. rewrite Nat.log2_pow2 in H1.
rewrite I in H2. rewrite Nat.log2_pow2 in H2.
(* cas : length a = 1 *)
destruct m.
assert (0 < length a). rewrite I. apply Nat.lt_0_1.
assert (even (length (hd ++ a)) = true).
generalize H3. generalize H. apply tm_step_square_pos.
apply Nat.even_EvenT in H4. apply Nat.EvenT_Even in H4.
apply Nat.Even_double in H4. rewrite Nat.double_twice in H4.
rewrite H4. rewrite Nat.mul_comm. rewrite Nat.mod_mul.
reflexivity. easy.
rewrite app_assoc in H.
rewrite <- firstn_skipn with (l := hd ++ a)
(n := length (hd++a) - 2^(S (S m))) in H.
rewrite <- firstn_skipn with (l := a ++ tl) (n := 2^(S (S m))) in H.
rewrite <- app_assoc in H.
assert (firstn (2^(S (S m))) (a ++ tl) =
rev (skipn (length (hd ++ a) - 2 ^ (S (S m))) (hd ++ a))).
rewrite H1. rewrite rev_involutive. reflexivity. rewrite H3 in H.
destruct m. inversion K'.
assert (6 < length (skipn (length (hd ++ a) - 2 ^ S (S (S m))) (hd ++ a))).
rewrite skipn_length.
replace (length (hd++a)) with ((length (hd++a) - 2^(S (S (S m))))
+ 2^(S (S (S m)))) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite Nat.add_0_l. rewrite Nat.pow_succ_r.
rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.mul_assoc. rewrite Nat.mul_assoc.
assert (8*1 <= 8*2^m). apply Nat.mul_le_mono_l.
apply Nat.le_succ_l.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
assert (6 < 8). lia. generalize H4. generalize H5.
apply Nat.lt_le_trans. lia. lia. lia. lia. lia.
assert (length (skipn (length (hd ++ a) - 2 ^ S (S (S m))) (hd ++ a))
= 2^(S (Nat.double (Nat.div2 (S (S m)))))).
rewrite skipn_length.
replace (length (hd++a))
with ((length (hd++a) - 2^(S (S (S m)))) + 2^(S (S (S m)))) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite K at 1. reflexivity. lia.
apply Nat.sub_add. destruct H2. assumption.
assert (length (
firstn (length (hd ++ a) - 2 ^ S (S (S m))) (hd ++ a) ++
skipn (length (hd ++ a) - 2 ^ S (S (S m))) (hd ++ a))
mod (2^ (S (Nat.double (Nat.div2 (S (S m)))))) = 0).
generalize H5. generalize H4. generalize H.
apply tm_step_palindromic_power2_odd.
rewrite firstn_skipn in H6. rewrite <- Nat.Even_double in H6.
assumption. apply Nat.EvenT_Even. apply Nat.even_EvenT.
assumption. lia. lia.
Qed.
Lemma tm_step_square2_rev_even_swap :
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 2^m
-> (even m = true <-> a = rev a).
Proof.
intros m n hd a tl. intros H I.
split. intro J.
assert (( (a = rev a /\ exists j,
length a = 2^(Nat.double j) \/ length a = 3 * 2^(Nat.double j))
\/ (a = map negb (rev a) /\ exists j,
length a = 2^(S (Nat.double j))
\/ length a = 3 * 2^(S (Nat.double j))))).
assert (0 < length a). destruct a.
symmetry in I. apply Nat.pow_nonzero in I. contradiction. easy.
simpl. lia. generalize H0. generalize H. apply tm_step_square_rev.
destruct H0. destruct H0. assumption. destruct H0. destruct H1.
destruct H1. rewrite I in H1. apply Nat.pow_inj_r in H1.
rewrite H1 in J. rewrite Nat.even_succ in J. rewrite Nat.double_twice in J.
rewrite Nat.odd_mul in J. inversion J. lia.
rewrite I in H1.
assert (Nat.log2 (2^m) = Nat.log2 (2^m)). reflexivity. rewrite H1 in H2 at 2.
rewrite Nat.log2_pow2 in H2.
rewrite Nat.log2_mul_pow2 in H2. replace (Nat.log2 3) with 1 in H2.
rewrite H2 in H1. rewrite Nat.add_succ_r in H1.
rewrite Nat.add_0_r in H1. rewrite Nat.pow_succ_r in H1.
rewrite Nat.mul_cancel_r in H1. inversion H1.
apply Nat.pow_nonzero. easy. lia. reflexivity. lia. lia. lia.
intro J.
generalize J. generalize I. generalize H.
apply tm_step_square_rev_even.
Qed.
Lemma tm_step_square2_rev_even' :
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 2^m
-> even m = true
-> length (hd ++ a) mod 2^(S m) = 0.
Proof.
intros m n hd a tl. intros H I J.
assert (a = rev a).
rewrite tm_step_square2_rev_even_swap
with (n := n) (hd := hd) (a := a) (tl := tl) (m := m) in J; assumption.
generalize H0. generalize I. generalize H.
apply tm_step_square2_rev_even.
Qed.
Lemma tm_step_square3_rev_even_swap :
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 3*2^m
-> (even m = true <-> a = rev a).
Proof.
intros m n hd a tl. intros H I.
split. intro J.
assert (( (a = rev a /\ exists j,
length a = 2^(Nat.double j) \/ length a = 3 * 2^(Nat.double j))
\/ (a = map negb (rev a) /\ exists j,
length a = 2^(S (Nat.double j))
\/ length a = 3 * 2^(S (Nat.double j))))).
assert (0 < length a). destruct a.
symmetry in I. rewrite Nat.eq_mul_0 in I. destruct I. inversion H0.
apply Nat.pow_nonzero in H0. contradiction. easy.
simpl. lia. generalize H0. generalize H. apply tm_step_square_rev.
destruct H0. destruct H0. assumption. destruct H0. destruct H1.
destruct H1.
assert (Nat.log2 (3*2^m) = Nat.log2 (3*2^m)). reflexivity.
rewrite <- I in H2 at 1. rewrite H1 in H2.
rewrite Nat.log2_pow2 in H2.
rewrite Nat.log2_mul_pow2 in H2. replace (Nat.log2 3) with 1 in H2.
rewrite H2 in H1. rewrite Nat.add_succ_r in H1.
rewrite Nat.add_0_r in H1. rewrite Nat.pow_succ_r in H1.
rewrite I in H1. apply Nat.mul_cancel_r in H1. inversion H1.
apply Nat.pow_nonzero. easy. lia. reflexivity. lia. lia. lia.
rewrite I in H1. apply Nat.mul_cancel_l in H1.
apply Nat.pow_inj_r in H1.
rewrite H1 in J. rewrite Nat.even_succ in J. rewrite Nat.double_twice in J.
rewrite Nat.odd_mul in J. inversion J. lia. easy.
intro J.
assert (( (a = rev a /\ exists j,
length a = 2^(Nat.double j) \/ length a = 3 * 2^(Nat.double j))
\/ (a = map negb (rev a) /\ exists j,
length a = 2^(S (Nat.double j))
\/ length a = 3 * 2^(S (Nat.double j))))).
assert (0 < length a). destruct a.
symmetry in I. rewrite Nat.eq_mul_0 in I. destruct I. inversion H0.
apply Nat.pow_nonzero in H0. contradiction. easy.
simpl. lia. generalize H0. generalize H. apply tm_step_square_rev.
destruct H0. destruct H0. destruct H1. destruct H1.
assert (Nat.log2 (3*2^m) = Nat.log2 (3*2^m)). reflexivity.
rewrite <- I in H2 at 1. rewrite H1 in H2.
rewrite Nat.log2_pow2 in H2.
rewrite Nat.log2_mul_pow2 in H2. replace (Nat.log2 3) with 1 in H2.
rewrite H2 in H1. rewrite Nat.add_succ_r in H1.
rewrite Nat.add_0_r in H1. rewrite Nat.pow_succ_r in H1.
rewrite I in H1. apply Nat.mul_cancel_r in H1. inversion H1.
apply Nat.pow_nonzero. easy. lia. reflexivity. lia. lia. lia.
rewrite I in H1. apply Nat.mul_cancel_l in H1.
apply Nat.pow_inj_r in H1. rewrite H1. rewrite Nat.double_twice.
rewrite Nat.even_mul. reflexivity. lia. easy.
destruct H0. rewrite <- J in H0.
destruct a. symmetry in I. rewrite Nat.eq_mul_0 in I.
destruct I. inversion H2.
apply Nat.pow_nonzero in H2. contradiction.
easy. simpl in H0. inversion H0.
destruct b; inversion H3.
Qed.
Lemma tm_step_square3_rev_even :
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 3 * 2^m
-> a = rev a
-> length (hd ++ a) mod 2^(S m) = 0.
Proof.
intros m n hd a tl. intros H I J.
assert (even m = true).
rewrite <- tm_step_square3_rev_even_swap
with (n := n) (hd := hd) (a := a) (tl := tl) (m := m) in J; assumption.
destruct m.
assert (even (length (hd ++ a)) = true).
assert (0 < length a). destruct a. inversion I.
simpl. lia. generalize H1. generalize H. apply tm_step_square_pos.
apply Nat.even_EvenT in H1. apply Nat.EvenT_Even in H1.
apply Nat.Even_double in H1. rewrite H1. rewrite Nat.double_twice.
rewrite Nat.mul_comm. apply Nat.mod_mul. easy.
destruct m. inversion H0.
rewrite app_assoc in H.
rewrite <- firstn_skipn with (l := hd ++ a)
(n := length (hd++a) - 2^(S (S (S m)))) in H.
rewrite <- firstn_skipn with (l := a ++ tl) (n := 2^(S (S (S m)))) in H.
rewrite <- firstn_skipn with (l := a)
(n := length (a) - 2^(S (S (S m)))) in J at 1.
rewrite <- firstn_skipn with (l := a)
(n := 2^(S (S (S m)))) in J at 5.
rewrite skipn_app in H. rewrite skipn_all2 in H.
replace (firstn (2^(S (S (S m)))) (a++tl))
with (firstn (2^(S (S (S m)))) a
++ firstn (2^(S (S (S m))) - length a) tl) in H.
rewrite rev_app_distr in J. assert (J' := J).
apply app_eq_length_head in J.
rewrite J in J'. apply app_inv_head in J'.
rewrite app_nil_l in H.
replace (length (hd ++ a) - 2 ^ (S (S (S m))) - length hd)
with (length a - 2^(S (S (S m)))) in H.
assert (firstn (2^(S (S (S m)))) a
= rev (skipn (length a - 2^(S (S (S m)))) a)).
rewrite J'. rewrite rev_involutive. reflexivity. rewrite H1 in H.
assert (length (skipn (length a - 2^(S (S (S m)))) a)
= 2^(S (Nat.double (Nat.div2 (S (S m)))))).
rewrite skipn_length.
replace (length a) with ((length a - 2^(S (S (S m)))) + 2^(S (S (S m)))) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite <- Nat.Even_double. reflexivity.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.le_refl. apply Nat.sub_add. rewrite I.
rewrite Nat.pow_succ_r. apply Nat.mul_le_mono_r. lia. lia.
assert (6 < length (skipn (length a - 2^(S (S (S m)))) a)).
rewrite skipn_length.
replace (length a) with ((length a - 2^(S (S (S m)))) + 2^(S (S (S m)))) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite Nat.add_0_l. rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.pow_succ_r. rewrite Nat.mul_assoc. rewrite Nat.mul_assoc.
replace (2*2*2) with 8.
assert (6 < 8). lia. assert (8 <= 8*2^m).
rewrite <- Nat.mul_1_r at 1. apply Nat.mul_le_mono_pos_l. lia.
apply Nat.le_succ_l.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
generalize H4. generalize H3. apply Nat.lt_le_trans. reflexivity.
lia. lia. lia. lia. apply Nat.sub_add. rewrite I.
rewrite Nat.pow_succ_r. apply Nat.mul_le_mono_r. lia. lia.
assert (length
(firstn (length (hd ++ a) - 2 ^ S (S (S m))) (hd ++ a) ++
skipn (length a - 2 ^ S (S (S m))) a) mod
(2 ^ S (Nat.double (Nat.div2 (S (S m))))) = 0).
generalize H2. generalize H3.
rewrite <- app_assoc in H. rewrite <- app_assoc in H. generalize H.
apply tm_step_palindromic_power2_odd.
rewrite <- Nat.Even_double in H4.
rewrite app_length in H4. rewrite <- Nat.add_mod_idemp_r in H4.
replace (
length (skipn (length a - 2 ^ S (S (S m))) a) mod 2 ^ S (S (S m))
) with (2^(S (S (S m))) mod (2^(S (S (S m))))) in H4.
rewrite Nat.add_mod_idemp_r in H4. rewrite firstn_length_le in H4.
rewrite Nat.sub_add in H4. assumption.
rewrite app_length. rewrite I. replace 3 with (1+2).
rewrite Nat.mul_add_distr_r. rewrite Nat.add_assoc.
rewrite <- Nat.pow_succ_r. apply Nat.le_add_l. lia. reflexivity.
lia. apply Nat.pow_nonzero. easy. rewrite Nat.mod_same.
rewrite skipn_length.
replace (length a) with ((length a - 2^(S (S (S m)))) + 2^(S (S (S m)))) at 1.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
rewrite Nat.mod_same. reflexivity.
apply Nat.pow_nonzero. easy. apply Nat.le_refl. rewrite Nat.sub_add.
reflexivity. rewrite I. replace 3 with (1+2).
rewrite Nat.mul_add_distr_r.
rewrite <- Nat.pow_succ_r. apply Nat.le_add_l. lia. reflexivity.
apply Nat.pow_nonzero. easy.
apply Nat.pow_nonzero. easy.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
rewrite <- Nat.sub_add_distr. rewrite Nat.add_comm.
rewrite Nat.sub_add_distr.
rewrite app_length. rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
reflexivity. apply Nat.le_refl.
rewrite rev_length. rewrite firstn_length_le.
rewrite skipn_length. reflexivity. apply Nat.le_sub_l.
rewrite <- firstn_app. reflexivity.
rewrite app_length. rewrite Nat.pow_succ_r. rewrite I.
rewrite <- Nat.add_sub_assoc. rewrite <- Nat.mul_sub_distr_r.
apply Nat.le_add_r. apply Nat.mul_le_mono_r. lia. lia.
Qed.
Lemma tm_step_square3_rev_even' :
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 3 * 2^m
-> even m = true
-> length (hd ++ a) mod 2^(S m) = 0.
Proof.
intros m n hd a tl. intros H I J.
assert (a = rev a).
rewrite tm_step_square3_rev_even_swap
with (n := n) (hd := hd) (a := a) (tl := tl) (m := m) in J; assumption.
generalize H0. generalize I. generalize H. apply tm_step_square3_rev_even.
Qed.
Reference in New Issue View Git Blame Copy Permalink