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coqbooks/src/thue_morse4.v

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(** * 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 xxx :
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 (0 < length a). rewrite I.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
assert (even m = true). generalize J. generalize I. generalize H.
apply tm_step_square_rev_even.
(*
TODO: voir s'il faut remplacer la dernière implication
par une équivalence
*)
Abort.
Theorem tm_step_palindrome_power2' :
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ (rev a) ++ tl
-> length a = 2^m
-> 2 < m
-> length (hd ++ a) mod 2^ (Nat.double (Nat.div2 (S m))) = 2^ (pred (Nat.double (Nat.div2 (S m)))).
Proof.
intros m n hd a tl. intros H I J.
assert (K: length (hd ++ a) mod 2^ (pred (Nat.double (Nat.div2 (S m)))) = 0).
generalize J. generalize I. generalize H. apply tm_step_palindrome_power2.
rewrite <- Nat.div_exact in K.
assert (L: Nat.Even
(length (hd ++ a) / 2 ^ pred (Nat.double (Nat.div2 (S m))))
\/ Nat.Odd
(length (hd ++ a) / 2 ^ pred (Nat.double (Nat.div2 (S m))))).
apply Nat.Even_or_Odd.
destruct L.
- assert (length (hd ++ a) mod 2 ^ Nat.double (Nat.div2 (S m)) = 0).
rewrite K. apply Nat.Even_double in H0. symmetry in H0.
rewrite Nat.double_twice in H0. rewrite <- H0. rewrite Nat.mul_assoc.
rewrite Nat.mul_shuffle0. rewrite Nat.mul_comm. rewrite Nat.mul_assoc.
rewrite <- Nat.pow_succ_r. rewrite Nat.succ_pred_pos. rewrite Nat.mul_comm.
apply Nat.mod_mul. apply Nat.pow_nonzero. easy.
assert (Nat.Even (S m) \/ Nat.Odd (S m)). apply Nat.Even_or_Odd.
destruct H1.
apply Nat.Even_double in H1. rewrite <- H1. apply Nat.lt_0_succ.
apply Nat.Odd_double in H1. apply Nat.succ_inj in H1.
rewrite <- H1. apply Nat.lt_succ_l in J. apply Nat.lt_succ_l in J.
assumption. apply Nat.le_0_l.
(* on cherche une contradiction à partir de H1 *)
assert (Nat.Even (S m) \/ Nat.Odd (S m)). apply Nat.Even_or_Odd.
destruct H2.
+ assert (E := H2). apply Nat.Even_double in H2. rewrite <- H2 in H1.
Abort.
(* palidrome 2*4 : soit centré en 4n soit pas plus de 2*6 *)
(* modifier l'énoncé : ajouter le modulo = 2 ET la différence sur le 7ème
ET existence d'un palindrome 2 * - *)
Lemma tm_step_palindromic_length_8_bis :
forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ (rev a) ++ tl
-> length a = 4
-> (
length (hd ++ a) mod 4 = 0
/\ exists b, a = [b; negb b; negb b; b]
) \/ (
length (hd ++ a) mod 4 = 2
/\ (exists b, a = [b; negb b; b; negb b])
/\ nth_error hd (length hd - 3) <> nth_error tl 2
).
Proof.
intros n hd a tl. intros H I.
(* proof that length hd++a is even *)
assert (P: even (length (hd ++ a)) = true).
assert (0 < length a). rewrite I. apply Nat.lt_0_succ. generalize H0.
generalize H. apply tm_step_palindromic_even_center.
assert (M: length (hd ++ a) mod 4 = 0 \/ length (hd ++ a) mod 4 = 2).
generalize P. apply even_mod4.
(* proof that length hd is even *)
assert (Q: even (length hd) = true). rewrite app_length in P.
rewrite Nat.even_add_even in P. assumption. rewrite I.
apply Nat.EvenT_Even. apply Nat.even_EvenT. reflexivity.
(* construction de a *)
destruct a. inversion I. destruct a. inversion I.
destruct a. inversion I. destruct a. inversion I.
destruct a. simpl in H.
(* proof that b1 <> b2 *)
assert ({b1=b2} + {~ b1=b2}). apply bool_dec. destruct H0.
replace (hd ++ b :: b0 :: b1 :: b2 :: b2 :: b1 :: b0 :: b :: tl)
with ((hd ++ b :: b0 :: nil)
++ [b1] ++ [b2] ++ [b2] ++ b1 :: b0 :: b :: tl) in H.
rewrite e in H. apply tm_step_cubefree in H. contradiction H.
reflexivity. apply Nat.lt_0_1. rewrite <- app_assoc. reflexivity.
(* proof that n > 2 *)
assert (2 < n).
assert (J: 0 + length (b :: b0 :: b1 :: b2 :: b2 :: b1 :: b0 :: b :: tl)
<= length hd
+ length (b :: b0 :: b1 :: b2 :: b2 :: b1 :: b0 :: b :: tl)).
apply Nat.add_le_mono. apply Nat.le_0_l. apply Nat.le_refl.
rewrite Nat.add_0_l in J. rewrite <- app_length in J. rewrite <- H in J.
rewrite tm_size_power2 in J.
destruct n. inversion J. apply Nat.nle_succ_0 in H1. contradiction H1.
destruct n. inversion J. inversion H1.
apply Nat.nle_succ_0 in H3. contradiction H3.
destruct n. inversion J. inversion H1. inversion H3. inversion H5.
inversion H7. rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
apply Nat.lt_0_succ.
(* proof that hd <> nil *)
destruct hd.
assert (Z: 4 < 2^n). replace 4 with (2^2).
apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. assumption.
assert (Some b2 = nth_error (tm_step n) 3). rewrite H. reflexivity.
replace (nth_error (tm_step n) 3) with (nth_error (tm_step 3) 3) in H1.
simpl in H1.
assert (Some b2 = nth_error (tm_step n) 4). rewrite H. reflexivity.
replace (nth_error (tm_step n) 4) with (nth_error (tm_step 3) 4) in H2.
simpl in H2.
rewrite H1 in H2. inversion H2. apply tm_step_stable. simpl.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
apply Nat.lt_0_succ. assumption. apply tm_step_stable. simpl.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
assert (3 < 4).
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
generalize Z. generalize H2. apply Nat.lt_trans.
(* proof that tl <> nil *)
destruct tl.
assert (Z: 4 < 2^n). replace 4 with (2^2).
apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. assumption.
assert (Y: tm_step n = rev (tm_step n)
\/ tm_step n = map negb (rev (tm_step n))). apply tm_step_rev.
destruct Y; rewrite H in H1 at 2; rewrite rev_app_distr in H1.
assert (Some b2 = nth_error (tm_step n) 3). rewrite H1. reflexivity.
replace (nth_error (tm_step n) 3) with (nth_error (tm_step 3) 3) in H2.
simpl in H2.
assert (Some b2 = nth_error (tm_step n) 4). rewrite H1. reflexivity.
replace (nth_error (tm_step n) 4) with (nth_error (tm_step 3) 4) in H3.
simpl in H3.
rewrite H2 in H3. inversion H3. apply tm_step_stable. simpl.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
apply Nat.lt_0_succ. assumption. apply tm_step_stable. simpl.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
assert (3 < 4).
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
generalize Z. generalize H3. apply Nat.lt_trans.
assert (Some (negb b2) = nth_error (tm_step n) 3). rewrite H1.
rewrite nth_error_map. reflexivity.
replace (nth_error (tm_step n) 3) with (nth_error (tm_step 3) 3) in H2.
simpl in H2.
assert (Some (negb b2) = nth_error (tm_step n) 4). rewrite H1.
rewrite nth_error_map. reflexivity.
replace (nth_error (tm_step n) 4) with (nth_error (tm_step 3) 4) in H3.
simpl in H3.
rewrite H2 in H3. inversion H3. apply tm_step_stable. simpl.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
apply Nat.lt_0_succ. assumption. apply tm_step_stable. simpl.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
assert (3 < 4).
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
generalize Z. generalize H3. apply Nat.lt_trans.
(* FIRST PART OF THE PROOF: case b0 = b1 *)
(* première hypothèse b0 = b1 mais alors on construit vers la
gauche jusqu'à (lest hd) et l'on a dans l'ordre jusqu'au centre :
b1 | (negb b1) b1 | b1 (negb b1 ||
et les quatre premiers termes vont impliquer que le centre soit en 4n+2 *)
assert ({b0=b1} + {~ b0=b1}). apply bool_dec. destruct H1.
rewrite e in H.
(* on a alors b = negb b1 (car dans le même bloc pair on a 01 ou 10 *)
assert ({b=b1} + {~ b=b1}). apply bool_dec. destruct H1.
rewrite e0 in H.
replace
((b3 :: hd) ++ b1 :: b1 :: b1 :: b2 :: b2 :: b1 :: b1 :: b1 :: b4 :: tl)
with
((b3 :: hd) ++ [b1] ++ [b1] ++ [b1]
++ b2 :: b2 :: b1 :: b1 :: b1 :: b4 :: tl) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_1. reflexivity.
(* à ce stade on a F T | T F || F T T F
trois cas :
(??? T) | F T T F || F T T F | ??? impossible à gauche
(T F) | F T T F || F T T F | (F T) cube
(T F) | F T T F || F T T F | (T F) OK, diff + impossible à droite
*)
rewrite app_removelast_last with (l := b3::hd) (d := false) in H.
rewrite <- app_assoc in H.
assert ({last (b3::hd) false=b1} + {~ last (b3::hd) false=b1}).
apply bool_dec. destruct H1. rewrite e0 in H.
assert (b2 = b). destruct b2; destruct b1; destruct b.
reflexivity. contradiction n0. reflexivity. reflexivity.
contradiction n1. reflexivity. contradiction n1. reflexivity.
reflexivity. contradiction n0. reflexivity. reflexivity. rewrite H1 in H.
assert (R: b1 = negb b). destruct b; destruct b1;
reflexivity || ( rewrite H1 in n0; contradiction n0; reflexivity).
(* un cas à étudier
(??? T) | F T T F || F T T F | ??? impossible à gauche
*)
destruct M. left.
split. assumption.
rewrite e. rewrite R. rewrite H1. exists b. reflexivity.
(* suite *)
rewrite app_removelast_last with (l := b3::hd) (d := false) in Q.
rewrite last_length in Q. rewrite Nat.even_succ in Q. apply odd_mod4 in Q.
rewrite app_removelast_last with (l := b3::hd) (d := false) in H2.
destruct Q.
assert (exists x, firstn 2 [b1;b;b1;b1] = [x;x]). generalize H3.
assert (length [b1;b;b1;b1] = 4). reflexivity. generalize H4.
replace (
removelast (b3 :: hd) ++
[b1] ++ b :: b1 :: b1 :: b :: b :: b1 :: b1 :: b :: b4 :: tl )
with (
removelast (b3 :: hd) ++ [b1;b;b1;b1]
++ (b :: b :: b1 :: b1 :: b :: b4 :: tl )) in H.
generalize H. apply tm_step_factor4_1mod4. reflexivity.
destruct H4. inversion H4. rewrite <- H7 in H6. rewrite H6 in n1.
contradiction n1. reflexivity.
rewrite app_length in H2. rewrite app_length in H2.
rewrite <- Nat.add_assoc in H2. rewrite <- Nat.add_mod_idemp_l in H2.
rewrite H3 in H2. inversion H2.
easy. easy. easy.
(* Deux cas
(T F) | F T T F || F T T F | (F T) cube
(T F) | F T T F || F T T F | (T F) impossible à droite
*)
assert ({b4=b1} + {~ b4=b1}). apply bool_dec. destruct H1. rewrite e0 in H.
(* un sous-cas :
(T F) | F T T F || F T T F | (T F) impossible à droite
*)
assert (b2 = b). destruct b2; destruct b1; destruct b.
reflexivity. contradiction n0. reflexivity. reflexivity.
contradiction n1. reflexivity. contradiction n1. reflexivity.
reflexivity. contradiction n0. reflexivity. reflexivity. rewrite H1 in H.
assert (R: b1 = negb b). destruct b; destruct b1;
reflexivity || ( rewrite H1 in n0; contradiction n0; reflexivity).
destruct M. left.
split. assumption. (* ici on postule le modulo = 2 *)
rewrite e. rewrite R. rewrite H1. exists b. reflexivity.
rewrite e in H2.
assert (length (((b3 :: hd) ++ [b; b1; b1; b2]) ++ [b2]) mod 4 = 3).
rewrite app_length. rewrite <- Nat.add_mod_idemp_l. rewrite H2.
reflexivity. easy.
replace(
removelast (b3 :: hd) ++ [last (b3 :: hd) false] ++
b :: b1 :: b1 :: b :: b :: b1 :: b1 :: b :: b1 :: tl )
with (
(((b3 :: hd) ++ [b; b1; b1; b]) ++ [b]) ++ [b1; b1;b;b1] ++ tl) in H.
rewrite H1 in H3.
assert (exists x, skipn 2 [b1;b1;b;b1] = [x;x]). generalize H3.
assert (length [b1;b1;b;b1] = 4). reflexivity. generalize H4.
generalize H. apply tm_step_factor4_3mod4.
destruct H4. inversion H4. rewrite <- H7 in H6. rewrite H6 in n1.
contradiction n1. reflexivity.
symmetry. rewrite app_assoc. rewrite <- app_removelast_last.
rewrite <- app_assoc. rewrite <- app_assoc. reflexivity. easy.
(* un sous-cas
(T F) | F T T F || F T T F | (F T) cube
*)
assert (b4 = b). destruct b4; destruct b1; destruct b;
reflexivity || contradiction n3 || contradiction n1; reflexivity.
rewrite H1 in H.
assert (b2 = b). destruct b2; destruct b1; destruct b.
reflexivity. contradiction n0. reflexivity. reflexivity.
contradiction n1. reflexivity. contradiction n1. reflexivity.
reflexivity. contradiction n0. reflexivity. reflexivity. rewrite H2 in H.
assert (last (b3::hd) false = b).
destruct (last (b3::hd) false); destruct b1; destruct b;
reflexivity || contradiction n2 || contradiction n1; reflexivity.
(* élargir hd et tl à l'aide des booléens b5 (gauche) et b6 (droite) *)
destruct hd. inversion Q. destruct tl.
assert (0 < n). apply Nat.lt_succ_l. apply Nat.lt_succ_l. assumption.
apply tm_step_length_even in H4. rewrite H in H4.
rewrite app_assoc in H4. rewrite <- app_removelast_last in H4.
rewrite app_length in H4. rewrite Nat.even_add in H4.
rewrite Q in H4. inversion H4. easy. rewrite H3 in H.
rewrite app_removelast_last
with (l := removelast (b3::b5::hd)) (d := false) in H.
rewrite <- app_assoc in H.
(* assigner
last (removelast (b3 :: b5 :: hd)) false = b1
b6 = b1
*)
assert ({last (removelast (b3 :: b5 :: hd)) false=b}
+ {~ last (removelast (b3 :: b5 :: hd)) false=b}). apply bool_dec.
destruct H4. rewrite e0 in H.
replace (
removelast (removelast (b3 :: b5 :: hd)) ++ [b] ++ [b]
++ b :: b1 :: b1 :: b :: b :: b1 :: b1 :: b :: b :: b6 :: tl)
with (
removelast (removelast (b3 :: b5 :: hd)) ++ [b] ++ [b] ++ [b]
++ b1 :: b1 :: b :: b :: b1 :: b1 :: b :: b :: b6 :: tl) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_1. reflexivity.
assert (last (removelast (b3 :: b5 :: hd)) false = b1).
destruct (last (removelast (b3 :: b5 :: hd)) false); destruct b1; destruct b.
reflexivity. reflexivity. contradiction n4. reflexivity.
contradiction n1. reflexivity. contradiction n1. reflexivity.
contradiction n4. reflexivity. reflexivity. reflexivity. rewrite H4 in H.
assert ({b6=b} + {~ b6=b}). apply bool_dec. destruct H5. rewrite e0 in H.
replace (
removelast (removelast (b3 :: b5 :: hd)) ++ [b1] ++ [b]
++ b :: b1 :: b1 :: b :: b :: b1 :: b1 :: b :: b :: b :: tl)
with (
(removelast (removelast (b3 :: b5 :: hd)) ++ [b1] ++ [b]
++ b :: b1 :: b1 :: b :: b :: b1 :: b1 :: nil)
++ [b] ++ [b] ++ [b] ++ tl) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_1. rewrite <- app_assoc. reflexivity.
assert (b6 = b1). destruct (b6); destruct b1; destruct b;
reflexivity || contradiction n5 || contradiction n1; reflexivity.
rewrite H5 in H.
(* contradiction *)
replace (
removelast (removelast (b3 :: b5 :: hd)) ++
[b1] ++ [b] ++ b :: b1 :: b1 :: b :: b :: b1 :: b1 :: b :: b :: b1 :: tl)
with (removelast (removelast (b3 :: b5 :: hd)) ++
[b1;b;b;b1] ++ [b1;b;b;b1] ++ [b1;b;b;b1] ++ tl) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_succ. reflexivity. easy. easy.
assert (H' := H).
(* SECOND PART PF THE PROOF: case b0 <> b1 *)
(* sinon, sur la base de T F T F || F T F T
quatre cas :
(F T) | T F T F || F T F T | (F T) diff + impossible à droite
(F T) | T F T F || F T F T | (T F) 4n+2 a/rev a/a possible ?
empiriquement : n'apparaît jamais
remonter encore d'un cran et prouver la différence à ce stade,
TFFT TFTF FTFT TFFT (µ de TF TT FF TF) pourquoi impossible ?
FTFT TFTF FTFT TFTF (µ de FF TT FF TT) pourquoi impossible ?
revoir l'énoncé en fonction
(T F) | T F T F || F T F T | (F T) cube
(T F) | T F T F || F T F T | (T F) diff + impossible à gauche
*)
assert ({b=b1} + {~ b=b1}). apply bool_dec. destruct H1. rewrite e in H.
(*
assert (b2 = b0). destruct (b2); destruct b1; destruct b0.
reflexivity. contradiction n0. reflexivity. reflexivity.
contradiction n1. reflexivity. contradiction n1. reflexivity.
reflexivity. contradiction n0. reflexivity. reflexivity. rewrite H1 in H.
*)
rewrite app_removelast_last with (l := b3::hd) (d := false) in H.
rewrite <- app_assoc in H.
assert ({last (b3::hd) false=b0} + {~ last (b3::hd) false=b0}).
apply bool_dec. destruct H1. rewrite e0 in H.
(* problème à gauche *)
destruct M.
(* problème avec n1 !!! *)
assert (T1: b0 = b1).
replace (removelast (b3 :: hd) ++
[b0] ++ b1 :: b0 :: b1 :: b2 :: b2 :: b1 :: b0 :: b1 :: b4 :: tl)
with (((removelast (b3 :: hd) ++ [b0]) ++ [b1; b0; b1; b2; b2])
++ [b1; b0; b1; b4] ++ tl) in H.
assert (T2: exists (x : bool), firstn 2 [b1;b0;b1;b4] = [x;x]).
assert (length ((b3 :: hd) ++ [b; b0; b1; b2; b2]) mod 4 = 1).
replace ((b3 :: hd) ++ [b; b0; b1; b2; b2])
with (((b3 :: hd) ++ [b; b0; b1; b2]) ++ [b2]).
rewrite app_length. rewrite <- Nat.add_mod_idemp_l.
rewrite H1. reflexivity. easy. rewrite <- app_assoc. reflexivity.
generalize H2. assert (length ([b1;b0;b1;b4]) = 4). reflexivity.
generalize H3. generalize H. rewrite e. rewrite <- e0 at 1.
rewrite <- app_removelast_last. apply tm_step_factor4_1mod4.
easy. inversion T2. inversion H2. reflexivity.
rewrite <- app_assoc. rewrite <- app_assoc. reflexivity.
rewrite T1 in n1. contradiction n1. reflexivity.
(* ici on postule le modulo = 2 *)
rewrite app_removelast_last with (l := b3::hd) (d := false) in Q.
rewrite last_length in Q. rewrite Nat.even_succ in Q. apply odd_mod4 in Q.
destruct Q.
assert (exists x, firstn 2 [b0;b1;b0;b1] = [x;x]). generalize H2.
assert (length [b0;b1;b0;b1] = 4). reflexivity. generalize H3.
replace (
removelast (b3 :: hd) ++
[b0] ++ b1 :: b0 :: b1 :: b2 :: b2 :: b1 :: b0 :: b1 :: b4 :: tl )
with (
removelast (b3 :: hd) ++ [b0;b1;b0;b1]
++ (b2 :: b2 :: b1 :: b0 :: b1 :: b4 :: tl )) in H.
generalize H. apply tm_step_factor4_1mod4. reflexivity.
destruct H3. inversion H3. rewrite <- H6 in H5. rewrite H5 in n1.
contradiction n1. reflexivity.
assert (exists x, skipn 2 [b0;b1;b0;b1] = [x;x]). generalize H2.
assert (length [b0;b1;b0;b1] = 4). reflexivity. generalize H3.
replace (
removelast (b3 :: hd) ++
[b0] ++ b1 :: b0 :: b1 :: b2 :: b2 :: b1 :: b0 :: b1 :: b4 :: tl )
with (
removelast (b3 :: hd) ++ [b0;b1;b0;b1]
++ (b2 :: b2 :: b1 :: b0 :: b1 :: b4 :: tl )) in H.
generalize H. apply tm_step_factor4_3mod4. reflexivity.
destruct H3. inversion H3. rewrite <- H6 in H5. rewrite H5 in n1.
contradiction n1. reflexivity. easy.
(* nouveau cas : last (b3 :: hd) false <> b0
(F T) | T F T F || F T F T | (F T) diff + impossible à droite
(F T) | T F T F || F T F T | (T F) 4n+2 a/rev a/a possible ?
*)
(* régler d'abord la question du modulo au centre *)
destruct M.
assert (T1: b0 = b1).
replace (removelast (b3 :: hd) ++ [last (b3::hd) false]
++ b1 :: b0 :: b1 :: b2 :: b2 :: b1 :: b0 :: b1 :: b4 :: tl)
with (((removelast (b3 :: hd) ++ [last (b3::hd) false])
++ [b1; b0; b1; b2; b2])
++ [b1; b0; b1; b4] ++ tl) in H.
assert (T2: exists (x : bool), firstn 2 [b1;b0;b1;b4] = [x;x]).
assert (length ((b3 :: hd) ++ [b; b0; b1; b2; b2]) mod 4 = 1).
replace ((b3 :: hd) ++ [b; b0; b1; b2; b2])
with (((b3 :: hd) ++ [b; b0; b1; b2]) ++ [b2]).
rewrite app_length. rewrite <- Nat.add_mod_idemp_l.
rewrite H1. reflexivity. easy. rewrite <- app_assoc. reflexivity.
generalize H2. assert (length ([b1;b0;b1;b4]) = 4). reflexivity.
generalize H3. generalize H. rewrite e.
rewrite <- app_removelast_last. apply tm_step_factor4_1mod4.
easy. inversion T2. inversion H2. reflexivity.
rewrite <- app_assoc. rewrite <- app_assoc. reflexivity.
rewrite T1 in n1. contradiction n1. reflexivity.
(* on suppose maintenant le modulo = 2 *)
assert (last (b3::hd) false = b1).
destruct (last (b3::hd) false); destruct b1; destruct b0;
reflexivity || contradiction n2 || contradiction n1; reflexivity.
assert ({b4=b0} + {~ b4=b0}). apply bool_dec. destruct H3. rewrite e0 in H.
assert (exists x, skipn 2 [b1;b0;b1;b0] = [x;x]).
replace (removelast (b3 :: hd) ++ [last (b3 :: hd) false] ++
b1 :: b0 :: b1 :: b2 :: b2 :: b1 :: b0 :: b1 :: b0 :: tl)
with (((b3 :: hd) ++ [b1;b0;b1;b2;b2])
++ [b1;b0;b1;b0] ++ tl) in H.
assert (length ((b3::hd) ++ [b1; b0; b1; b2; b2]) mod 4 = 3). rewrite e in H1.
replace ([b1;b0;b1;b2;b2]) with ([b1;b0;b1;b2] ++ [b2]).
rewrite app_assoc. rewrite app_length. rewrite <- Nat.add_mod_idemp_l.
rewrite H1. reflexivity. easy. reflexivity.
assert (length [b1;b0;b1;b0] = 4). reflexivity.
generalize H3. generalize H4. generalize H. apply tm_step_factor4_3mod4.
symmetry. rewrite app_assoc. rewrite <- app_removelast_last.
rewrite <- app_assoc. reflexivity. easy.
destruct H3. inversion H3. rewrite <- H6 in H5. rewrite H5 in n1.
contradiction n1. reflexivity.
assert (b4 = b1). destruct b4; destruct b1; destruct b0;
reflexivity || contradiction n3 || contradiction n1; reflexivity.
rewrite H3 in H. rewrite H2 in H.
assert (b2 = b0). destruct b2; destruct b1; destruct b0.
reflexivity. contradiction n0. reflexivity. reflexivity.
contradiction n1. reflexivity. contradiction n1. reflexivity.
reflexivity. contradiction n0. reflexivity. reflexivity.
rewrite H4 in H.
(* dernier cas (difficile
(F T) | T F T F || F T F T | (T F) 4n+2 a/rev a/a possible ?
remarquer le schéma a ++ (rev a) ++ a
apparaît jusqu'à six termes à gauche et à droite
empiriquement : n'apparaît jamais avec un 7ème palindromique ajouté
remonter encore d'un cran et prouver la différence à ce stade,
TFFT TFTF FTFT TFFT (µ de TF TT FF TF) pourquoi impossible ?
FTFT TFTF FTFT TFTF (µ de FF TT FF TT) pourquoi impossible ?
FTTFTFFT FTTFTF (pb avec le repeating_pattern de 8 ???)
TFT TFT FF TFT TFT (si on part sur 2 mod 8)
(idem en partant de la droite pour 6 mod 8)
Le lemme repeating_patterns se base sur les huit premiers termes de TM :
[False, True, True, False, True, False, False, True]
--> il y a une contradiction à chaque fois
*)
(* on prouve 3 < n *)
assert (R: 2^3 < length (tm_step n)). rewrite H.
rewrite app_length. rewrite <- Nat.add_0_l at 1.
apply Nat.add_le_lt_mono. apply Nat.le_0_l. simpl.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
apply Nat.lt_0_succ. rewrite tm_size_power2 in R.
rewrite <- Nat.pow_lt_mono_r_iff in R.
(* on étend hd *)
destruct hd. inversion Q.
rewrite app_removelast_last
with (l := removelast (b3::b5::hd)) (d := false) in H.
assert ({last (removelast (b3 :: b5 :: hd)) false = b1}
+ {~ last (removelast (b3 :: b5 :: hd)) false = b1}).
apply bool_dec. destruct H5. rewrite e0 in H.
rewrite <- app_assoc in H.
replace (b1 :: b0 :: b1 :: b0 :: b0 :: b1 :: b0 :: b1 :: b1 :: tl)
with ([b1] ++ b0 :: b1 :: b0 :: b0 :: b1 :: b0 :: b1 :: b1 :: tl) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_1. reflexivity.
(* on étend tl*)
destruct tl.
assert (tm_step n = rev (tm_step n)
\/ tm_step n = map negb (rev (tm_step n))).
apply tm_step_rev. destruct H5; rewrite H in H5 at 2;
rewrite rev_app_distr in H5; simpl in H5;
assert (odd 0 = true).
assert (nth_error (tm_step n) (S (2*0) * 2^0) =
nth_error (tm_step n) (pred (S (2*0) * 2^0))).
rewrite H5. reflexivity. generalize H6.
apply tm_step_pred. simpl. rewrite <- Nat.pow_0_r with (a := 2) at 1.
apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
apply Nat.lt_succ_l. apply Nat.lt_succ_l. assumption. inversion H6.
assert (nth_error (tm_step n) (S (2*0) * 2^0) =
nth_error (tm_step n) (pred (S (2*0) * 2^0))).
rewrite H5. reflexivity. generalize H6.
apply tm_step_pred. simpl. rewrite <- Nat.pow_0_r with (a := 2) at 1.
apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
apply Nat.lt_succ_l. apply Nat.lt_succ_l. assumption. inversion H6.
assert ({b6=b1} + {~ b6=b1}). apply bool_dec. destruct H5. rewrite e0 in H.
replace (
(removelast (removelast (b3 :: b5 :: hd)) ++
[last (removelast (b3 :: b5 :: hd)) false]) ++
[b1] ++ b1 :: b0 :: b1 :: b0 :: b0 :: b1 :: b0 :: b1 :: b1 :: b1 :: tl)
with (
(removelast (removelast (b3 :: b5 :: hd)) ++
[last (removelast (b3 :: b5 :: hd)) false] ++
[b1] ++ b1 :: b0 :: b1 :: b0 :: b0 :: b1 :: b0 :: nil)
++ [b1] ++ [b1] ++ [b1] ++ tl) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_1.
rewrite <- app_assoc. rewrite <- app_assoc. rewrite <- app_assoc.
rewrite <- app_assoc. reflexivity.
(* on assigne les valeurs correctes aux deux extrémités *)
assert (b6 = b0). destruct b6; destruct b1; destruct b0;
reflexivity || contradiction n5 || contradiction n1; reflexivity.
rewrite H5 in H.
assert (last (removelast (b3 :: b5 :: hd)) false = b0).
destruct (last (removelast (b3 :: b5 :: hd)) false);
destruct b1; destruct b0;
reflexivity || contradiction n4 || contradiction n1; reflexivity.
rewrite H6 in H.
(* on étend hd *)
destruct hd. simpl in H. assert (odd 3 = true). reflexivity.
rewrite <- tm_step_pred with (n := n) (k := 0) in H7.
rewrite H in H7. simpl in H7. inversion H7. rewrite H9 in n1.
contradiction n1. reflexivity. simpl.
replace 8 with (2^3). rewrite <- Nat.pow_lt_mono_r_iff.
assumption. apply Nat.lt_1_2. reflexivity.
rewrite app_removelast_last
with (l := removelast (removelast (b3::b5::b7::hd))) (d := false) in H.
pose (b8 := last (removelast (removelast (b3 :: b5 :: b7 :: hd))) false).
fold b8 in H. rewrite <- app_assoc in H. rewrite <- app_assoc in H.
pose (hd' := removelast (removelast (removelast (b3 :: b5 :: b7 :: hd)))).
fold hd' in H.
(* on étend tl *)
destruct tl.
assert (tm_step n = rev (tm_step n)
\/ tm_step n = map negb (rev (tm_step n))).
apply tm_step_rev. destruct H7; rewrite H in H7 at 2;
rewrite rev_app_distr in H7; simpl in H5.
assert (odd 3 = true). reflexivity.
rewrite <- tm_step_pred with (n := n) (k := 0) in H8.
rewrite H7 in H8. simpl in H8. inversion H8. rewrite H10 in n1.
contradiction n1. reflexivity. simpl.
replace 8 with (2^3). rewrite <- Nat.pow_lt_mono_r_iff.
assumption. apply Nat.lt_1_2. reflexivity.
assert (odd 3 = true). reflexivity.
rewrite <- tm_step_pred with (n := n) (k := 0) in H8.
rewrite H7 in H8. simpl in H8. inversion H8.
destruct b0; destruct b1. contradiction n1. reflexivity.
inversion H10. inversion H10. contradiction n1. reflexivity. simpl.
replace 8 with (2^3). rewrite <- Nat.pow_lt_mono_r_iff.
assumption. apply Nat.lt_1_2. reflexivity.
(* termes à prouver *)
(* lemmes initiaux *)
assert (Y: forall (k : bool) (x : list bool),
length (removelast (k::x)) = length x).
intros k x. rewrite removelast_firstn_len.
replace (length (k::x)) with (S (length x)). rewrite Nat.pred_succ.
rewrite firstn_length. simpl. apply Nat.min_l. apply Nat.le_succ_diag_r.
reflexivity.
assert (Y': forall (k1 k2 : bool) (x : list bool),
length (removelast (removelast (k1::k2::x))) = length x).
intros k1 k2 x.
rewrite removelast_firstn_len. rewrite Y.
replace (length (k2::x)) with (S (length x)).
rewrite Nat.pred_succ. rewrite firstn_length. rewrite Y.
apply Nat.min_l. apply Nat.le_succ_diag_r. reflexivity.
assert (Y'': forall (k1 k2 k3 : bool) (x : list bool),
length (removelast (removelast (removelast (k1::k2::k3::x)))) = length x).
intros k1 k2 k3 x.
rewrite removelast_firstn_len. rewrite removelast_firstn_len.
rewrite Y. replace (length (k2::k3::x)) with (S (length (k3::x))).
rewrite Nat.pred_succ. rewrite firstn_length.
rewrite firstn_length. rewrite Y. rewrite Nat.min_l. rewrite Nat.min_l.
reflexivity. apply Nat.le_succ_diag_r. rewrite Nat.min_l.
replace (length (k3 :: x)) with (S (length x)). rewrite Nat.pred_succ.
apply Nat.le_succ_diag_r. reflexivity.
replace (length (k2 :: k3::x)) with (S (length (k3::x))).
apply Nat.le_succ_diag_r. reflexivity. reflexivity.
(* preuves *)
assert (U:
nth_error (b3 :: b5 :: b7 :: hd) (length (b3 :: b5 :: b7 :: hd) - 3)
= Some b8). unfold b8.
rewrite app_removelast_last with (l := b3::b5::b7::hd) (d := false) at 1.
rewrite app_removelast_last
with (l := (removelast (b3::b5::b7::hd))) (d := false) at 1.
rewrite app_removelast_last
with (l := (removelast (removelast (b3::b5::b7::hd)))) (d := false) at 1.
rewrite <- app_assoc. rewrite <- app_assoc. rewrite nth_error_app2.
rewrite Y''. rewrite <- Nat.sub_add_distr.
replace (length (b3::b5::b7::hd)) with (3 + length hd).
rewrite Nat.sub_diag. reflexivity. reflexivity. rewrite Y''.
replace (length (b3::b5::b7::hd)) with (length hd + 3).
rewrite Nat.add_sub. apply Nat.le_refl. rewrite Nat.add_comm.
reflexivity.
assert (0 < length (removelast (removelast (b3 :: b5 :: b7 :: hd)))).
rewrite Y'. simpl. apply Nat.lt_0_succ.
assert ({removelast (removelast (b3 :: b5 :: b7 :: hd))=nil}
+ {~ removelast (removelast (b3 :: b5 :: b7 :: hd))=nil}).
apply list_eq_dec. apply bool_dec. destruct H8.
rewrite e0 in H7. inversion H7. assumption.
assert (0 < length (removelast (b3 :: b5 :: b7 :: hd))).
rewrite Y. simpl. apply Nat.lt_0_succ.
assert ({removelast (b3 :: b5 :: b7 :: hd)=nil}
+ {~ removelast (b3 :: b5 :: b7 :: hd)=nil}).
apply list_eq_dec. apply bool_dec. destruct H8.
rewrite e0 in H7. inversion H7. assumption.
easy.
assert (T: 8 <= length ((b3 :: b5 :: b7 :: hd) ++ [b; b0; b1; b2])).
destruct hd. inversion Q. rewrite app_length. simpl.
rewrite <- Nat.add_succ_r. rewrite <- Nat.add_succ_r.
rewrite <- Nat.add_succ_r. rewrite <- Nat.add_succ_r.
rewrite <- Nat.add_0_l at 1. apply Nat.add_le_mono.
apply Nat.le_0_l. apply Nat.le_refl.
(* analyse finale *)
assert ({b8=b9} + {~ b8=b9}). apply bool_dec. destruct H7. rewrite e0 in H.
(* premier sous-cas : b8 = b9, contradiction lié à repeating_patterns *)
(* lemme initial *)
assert (forall n : nat, n mod 4 = 2 -> n mod 8 = 2 \/ n mod 8 = 6).
intro m. intro A.
assert (m mod (4*2) = m mod 4 + 4 * ((m / 4) mod 2)).
apply Nat.mod_mul_r; easy. rewrite A in H7.
rewrite <- Nat.bit0_mod in H7. replace (4*2) with 8 in H7.
rewrite H7.
destruct (Nat.testbit (m / 4) 0) ; [right | left] ; reflexivity.
reflexivity.
pose (lh := length ((b3 :: b5 :: b7 :: hd) ++ [b; b0; b1; b2])).
fold lh in H1. fold lh in T.
assert ({b9=b0} + {~ b9=b0}). apply bool_dec. destruct H8. rewrite e1 in H.
(* si centre = 8n + 2, alors les cinq premiers sont absurdes *)
(* si centre = 8n + 6, alors les cinq derniers sont absurdes *)
apply H7 in H1. destruct H1.
(* on commence par supposer le centre en 8n+2 : hypothèse H1 *)
assert (lh = 8 * (lh / 8) + lh mod 8). apply Nat.div_mod. easy.
rewrite H1 in H8. rewrite <- Nat.succ_pred_pos with (n := lh/8) in H8.
assert (lh - 8 = 8 * Nat.pred (lh / 8) + 2).
rewrite <- Nat.add_cancel_r with (p := 8). rewrite <- Nat.add_sub_swap.
rewrite Nat.add_sub. rewrite Nat.add_shuffle0. rewrite <- Nat.mul_succ_r.
assumption. assumption.
assert (nth_error (tm_step (3 + (n-3))) (Nat.pred (lh/8) * 8 + 3)
= nth_error (tm_step (3 + (n-3))) (Nat.pred (lh/8) * 8 + 4)).
rewrite Nat.add_comm. rewrite <- Nat.add_sub_swap. rewrite Nat.add_sub.
rewrite H. rewrite Nat.mul_comm.
apply eq_S in H9. symmetry in H9. rewrite <- Nat.add_1_r in H9.
rewrite <- Nat.add_assoc in H9. rewrite Nat.add_1_r in H9.
rewrite H9.
apply eq_S in H9. rewrite <- Nat.add_1_r in H9.
rewrite <- Nat.add_assoc in H9. rewrite Nat.add_1_r in H9.
rewrite H9. unfold lh. unfold hd'.
rewrite nth_error_app2. symmetry. rewrite nth_error_app2. rewrite Y''.
rewrite app_length.
replace (length (b3::b5::b7::hd)) with (S (S (S (length hd)))).
rewrite Nat.add_succ_comm. rewrite Nat.add_succ_comm.
rewrite Nat.add_succ_comm. rewrite <- Nat.sub_succ_l.
rewrite <- Nat.add_succ_r. rewrite Nat.add_sub.
rewrite Nat.sub_succ_l. rewrite Nat.sub_diag. reflexivity.
apply Nat.le_refl. unfold lh in T.
rewrite <- Nat.add_succ_comm. rewrite <- Nat.add_succ_comm.
rewrite <- Nat.add_succ_comm. rewrite app_length in T.
assumption. reflexivity. rewrite Y''.
rewrite app_length.
replace (length (b3::b5::b7::hd)) with (S (S (S (length hd)))).
rewrite Nat.add_succ_comm. rewrite Nat.add_succ_comm.
rewrite Nat.add_succ_comm. rewrite <- Nat.sub_succ_l.
rewrite <- Nat.add_succ_r. rewrite Nat.add_sub.
apply Nat.le_succ_diag_r. rewrite Nat.add_succ_r.
rewrite Nat.add_succ_r. rewrite Nat.add_succ_r.
rewrite <- Nat.succ_le_mono. rewrite <- Nat.succ_le_mono.
rewrite <- Nat.succ_le_mono. replace 5 with (1 + 4).
rewrite <- Nat.add_le_mono_r. destruct hd. inversion Q.
simpl. apply le_n_S. apply Nat.le_0_l. reflexivity. reflexivity.
rewrite Y''. rewrite app_length.
replace (length (b3::b5::b7::hd)) with (S (S (S (length hd)))).
rewrite Nat.add_succ_comm. rewrite Nat.add_succ_comm.
rewrite Nat.add_succ_comm. rewrite <- Nat.sub_succ_l.
rewrite <- Nat.add_succ_r. rewrite Nat.add_sub. apply Nat.le_refl.
rewrite Nat.add_succ_r. rewrite Nat.add_succ_r. rewrite Nat.add_succ_r.
rewrite <- Nat.succ_le_mono. rewrite <- Nat.succ_le_mono.
rewrite <- Nat.succ_le_mono. replace 5 with (1 + 4).
rewrite <- Nat.add_le_mono_r. destruct hd. inversion Q.
simpl. apply le_n_S. apply Nat.le_0_l. reflexivity. reflexivity.
apply Nat.lt_le_incl. assumption.
rewrite <- tm_step_repeating_patterns in H10. inversion H10.
simpl. rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
simpl. rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
rewrite Nat.mul_lt_mono_pos_l with (p := 8).
rewrite Nat.add_lt_mono_r with (p := 2). rewrite <- H9.
rewrite Nat.add_lt_mono_r with (p := 8).
rewrite <- Nat.add_sub_swap. rewrite Nat.add_sub.
rewrite <- Nat.add_assoc. replace (2+8) with 10.
replace 8 with (2^3) at 1. rewrite <- Nat.pow_add_r.
rewrite Nat.add_sub_assoc. rewrite Nat.add_sub_swap. simpl.
rewrite <- tm_size_power2. rewrite H'. unfold lh.
rewrite app_length. rewrite app_length. rewrite <- Nat.add_assoc.
rewrite <- Nat.add_lt_mono_l. simpl.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
apply Nat.lt_0_succ. apply Nat.le_refl.
apply Nat.lt_le_incl. assumption. reflexivity. reflexivity.
assumption. apply Nat.lt_0_succ.
apply Nat.div_le_mono with (c := 8) in T. rewrite Nat.div_same in T.
rewrite Nat.le_succ_l in T. assumption. easy. easy.
(* on change de modulo ; on travaille sur 8k+6 maintenant *)
(* si centre = 8n + 6, alors les cinq derniers sont absurdes *)
assert (lh = 8 * (lh / 8) + lh mod 8). apply Nat.div_mod. easy.
rewrite H1 in H8. rewrite <- Nat.pred_succ with (n := lh/8) in H8.
rewrite Nat.mul_pred_r in H8.
assert (lh + 8 = 8 * S (lh / 8) + 6).
rewrite <- Nat.add_cancel_r with (p := 8) in H8.
rewrite <- Nat.add_sub_swap in H8. rewrite Nat.sub_add in H8. assumption.
rewrite <- Nat.add_0_r at 1. apply Nat.add_le_mono.
assert (1 <= S (lh/8)). rewrite Nat.le_succ_l. apply Nat.lt_0_succ.
apply Nat.mul_le_mono_l with (p := 8) in H9.
rewrite Nat.mul_1_r in H9. assumption. apply Nat.le_0_l.
assert (1 <= S (lh/8)). rewrite Nat.le_succ_l. apply Nat.lt_0_succ.
apply Nat.mul_le_mono_l with (p := 8) in H9.
rewrite Nat.mul_1_r in H9. assumption.
assert (nth_error (tm_step (3 + (n-3))) (S (lh/8) * 8 + 3)
= nth_error (tm_step (3 + (n-3))) (S (lh/8) * 8 + 4)).
rewrite Nat.add_comm. rewrite <- Nat.add_sub_swap. rewrite Nat.add_sub.
rewrite Nat.add_succ_r in H9. rewrite Nat.add_succ_r in H9.
symmetry in H9. rewrite Nat.add_succ_r in H9. rewrite Nat.add_succ_r in H9.
apply Nat.succ_inj in H9. apply Nat.succ_inj in H9.
rewrite Nat.mul_comm in H9. rewrite H9.
rewrite Nat.add_succ_r in H9. symmetry in H9. rewrite Nat.add_succ_r in H9.
apply Nat.succ_inj in H9. rewrite <- H9. rewrite H. unfold lh. unfold hd'.
rewrite nth_error_app2. symmetry. rewrite nth_error_app2. rewrite Y''.
rewrite app_length.
replace (length (b3::b5::b7::hd)) with (S (S (S (length hd)))).
rewrite Nat.add_succ_comm. rewrite Nat.add_succ_comm.
rewrite Nat.add_succ_comm.
rewrite Nat.add_comm. rewrite <- Nat.add_sub_assoc.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag. symmetry.
rewrite Nat.add_comm. rewrite <- Nat.add_sub_assoc.
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag. reflexivity.
apply Nat.le_refl. rewrite <- Nat.add_0_r at 1.
rewrite <- Nat.add_le_mono_l. apply Nat.le_0_l. apply Nat.le_refl.
rewrite <- Nat.add_0_r at 1.
rewrite <- Nat.add_le_mono_l. apply Nat.le_0_l. reflexivity.
rewrite Y''. rewrite app_length. simpl. rewrite <- Nat.add_succ_r.
rewrite <- Nat.add_succ_r. rewrite <- Nat.add_succ_r.
rewrite <- Nat.add_0_r at 1. rewrite <- Nat.add_assoc.
rewrite <- Nat.add_le_mono_l. apply Nat.le_0_l.
rewrite Y''. rewrite app_length. simpl. rewrite <- Nat.add_succ_r.
rewrite <- Nat.add_succ_r. rewrite <- Nat.add_succ_r.
rewrite <- Nat.add_0_r at 1. rewrite <- Nat.add_assoc.
rewrite <- Nat.add_le_mono_l. apply Nat.le_0_l.
apply Nat.lt_le_incl. assumption.
rewrite <- tm_step_repeating_patterns in H10. inversion H10.
simpl. rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
simpl. rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
rewrite Nat.mul_lt_mono_pos_l with (p := 8).
rewrite Nat.add_lt_mono_r with (p := 6). rewrite <- H9.
replace (8) with (2^3) at 2. rewrite <- Nat.pow_add_r.
rewrite Nat.add_sub_assoc. rewrite Nat.add_sub_swap.
rewrite Nat.sub_diag. rewrite Nat.add_0_l.
rewrite <- tm_size_power2. rewrite H'. unfold lh.
rewrite app_length. rewrite app_length. rewrite <- Nat.add_assoc.
rewrite <- Nat.add_assoc.
rewrite <- Nat.add_lt_mono_l. simpl.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
rewrite <- Nat.succ_lt_mono. rewrite Nat.add_succ_r.
rewrite <- Nat.succ_lt_mono. rewrite Nat.add_succ_r.
apply Nat.lt_0_succ. apply Nat.le_refl.
apply Nat.lt_le_incl. assumption. reflexivity. apply Nat.lt_0_succ.
(* on arrive à la suite du cas b8 = b9 avec cette fois b9 = b1
et non plus b9 = b0 *)
assert (b9 = b1). destruct b9; destruct b1; destruct b0.
reflexivity. reflexivity. contradiction n6. reflexivity.
contradiction n1. reflexivity. contradiction n1. reflexivity.
contradiction n6. reflexivity. reflexivity. reflexivity.
rewrite H8 in H.
(* le premier nouveau sous cas est lh mod 4 = 2 en H1
FTFT TFTF FTFT TFTF (µ de FF TT FF TT) pourquoi impossible ?
c'est un carré ???
c'est un palindrome fait avec un carré de palindrome ?
morphisme de FFTTFFTT sans doute impossible (testé)
*)
unfold hd' in H. destruct hd. inversion P.
rewrite app_removelast_last
with (l := (removelast (removelast (removelast (b3::b5::b7::b10::hd)))))
(d := false) in H.
pose (b11 := last (removelast (removelast
(removelast (b3 :: b5 :: b7 :: b10 :: hd)))) false).
fold b11 in H. rewrite <- app_assoc in H.
pose (hd'' := removelast (removelast (removelast
(removelast (b3 :: b5 :: b7 :: b10 :: hd))))).
fold hd'' in H.
(* proof that b11 <> b1 *)
assert ({b11=b1} + {~ b11=b1}). apply bool_dec. destruct H9. rewrite e1 in H.
assert (even (length hd'') = false).
replace ( hd'' ++ [b1] ++ [b1] ++ [b0] ++ [b1] ++
b1 :: b0 :: b1 :: b0 :: b0 :: b1 :: b0 :: b1 :: b1 :: b0 :: b1 :: tl)
with (hd'' ++ [b1;b1;b0] ++ [b1;b1;b0]
++ b1::b0::b0::b1::b0::b1::b1::b0::b1::tl) in H.
assert (odd (length [b1;b1;b0]) = true). reflexivity.
generalize H9. generalize H. apply tm_step_odd_prefix_square.
reflexivity. unfold hd'' in H9.
rewrite removelast_firstn_len in H9. rewrite Y'' in H9.
rewrite firstn_length_le in H9. simpl in H9.
replace (b3::b5::b7::b10::hd) with ([b3;b5;b7;b10] ++ hd) in Q.
rewrite app_length in Q. rewrite Nat.even_add in Q. rewrite H9 in Q.
inversion Q. reflexivity. rewrite Y''. apply Nat.le_pred_l.
(* proof ath b11 = b0 *)
assert (b11 = b0). destruct b11; destruct b1; destruct b0.
reflexivity. contradiction n7. reflexivity. reflexivity.
contradiction n1. reflexivity. contradiction n1. reflexivity.
reflexivity. contradiction n7. reflexivity. reflexivity.
rewrite H9 in H.
(* on élargit tl (adding b12 in front of tl) *)
destruct tl.
assert (tm_step n = rev (tm_step n)
\/ tm_step n = map negb (rev (tm_step n))).
apply tm_step_rev. destruct H10; rewrite H in H10 at 2;
rewrite rev_app_distr in H10;
assert (odd 1 = true). reflexivity.
rewrite <- tm_step_pred with (n := n) (k := 0) in H11.
rewrite H10 in H11. simpl in H11. inversion H11. rewrite H13 in n1.
contradiction n1. reflexivity. simpl.
replace 2 with (2^1). rewrite <- Nat.pow_lt_mono_r_iff.
apply Nat.lt_succ_l in R. apply Nat.lt_succ_l in R. assumption.
apply Nat.lt_1_2. reflexivity.
reflexivity.
rewrite <- tm_step_pred with (n := n) (k := 0) in H11.
rewrite H10 in H11. simpl in H11. inversion H11.
destruct b0; destruct b1. contradiction n1. reflexivity.
inversion H13. inversion H13. contradiction n1. reflexivity. simpl.
replace 2 with (2^1). rewrite <- Nat.pow_lt_mono_r_iff.
apply Nat.lt_succ_l in R. apply Nat.lt_succ_l in R. assumption.
apply Nat.lt_1_2. reflexivity.
(* now we have added b12 in front of tl *)
(* proof that b12 <> b1 *)
assert ({b12=b1} + {~ b12=b1}). apply bool_dec. destruct H10. rewrite e1 in H.
assert (even (length (hd'' ++ [b0;b1;b0;b1;b1; b0; b1; b0; b0; b1])) = false).
replace ( hd'' ++ [b0] ++ [b1] ++ [b0] ++ [b1] ++ b1
:: b0 :: b1 :: b0 :: b0 :: b1 :: b0 :: b1 :: b1 :: b0 :: b1 :: b1 :: tl)
with ((hd'' ++ [b0;b1;b0;b1;b1; b0; b1; b0; b0; b1])
++ [ b0;b1;b1] ++ [b0;b1;b1] ++ tl) in H.
assert (odd (length [b0;b1;b1]) = true). reflexivity.
generalize H10. generalize H. apply tm_step_odd_prefix_square.
rewrite <- app_assoc. reflexivity. unfold hd'' in H10.
rewrite removelast_firstn_len in H10. rewrite Y'' in H10.
rewrite app_length in H10. rewrite firstn_length_le in H10. simpl in H10.
rewrite Nat.even_add in H10.
simpl in Q. rewrite Q in H10. inversion H10.
rewrite Y''. apply Nat.le_pred_l.
(* now we know that b12 <> b1 *)
(* proof that b12 = b0 *)
assert (b12 = b0). destruct b12; destruct b1; destruct b0.
reflexivity. contradiction n8. reflexivity. reflexivity.
contradiction n1. reflexivity. contradiction n1. reflexivity.
reflexivity. contradiction n8. reflexivity. reflexivity.
rewrite H10 in H.
(* simplify notations *)
replace ( hd'' ++ [b0] ++ [b1] ++ [b0] ++ [b1] ++ b1
:: b0 :: b1 :: b0 :: b0 :: b1 :: b0 :: b1 :: b1 :: b0 :: b1 :: b0 :: tl)
with (hd'' ++ [b0;b1;b0;b1;b1;b0;b1;b0;b0;b1;b0;b1;b1;b0;b1;b0] ++ tl) in H.
pose (s := [b0;b1;b0;b1;b1;b0;b1;b0;b0;b1;b0;b1;b1;b0;b1;b0]). fold s in H.
assert (even (length hd'') = true). unfold hd''.
rewrite removelast_firstn_len. rewrite Y''.
replace (pred (length (b10::hd))) with (length hd).
rewrite firstn_length_le. simpl in Q. rewrite Q. reflexivity.
rewrite Y''. apply Nat.le_succ_diag_r. reflexivity.
(* destructuring n *)
destruct n. inversion H0. rewrite <- tm_step_lemma in H.
(* inverting tm_morphism in tm_step n *)
assert (hd'' = tm_morphism (firstn (Nat.div2 (length hd'')) (tm_step n))).
generalize H11. generalize H. apply tm_morphism_app2.
assert (s ++ tl = tm_morphism (skipn (Nat.div2 (length hd'')) (tm_step n))).
generalize H11. generalize H. apply tm_morphism_app3. symmetry in H13.
assert (even (length s) = true). unfold s. reflexivity.
assert (s = tm_morphism (firstn (Nat.div2 (length s))
(skipn (Nat.div2 (length hd'')) (tm_step n)))).
generalize H14. generalize H13. apply tm_morphism_app2.
assert (tl = tm_morphism (skipn (Nat.div2 (length s))
(skipn (Nat.div2 (length hd'')) (tm_step n)))).
generalize H14. generalize H13. apply tm_morphism_app3.
rewrite H12 in H. rewrite H15 in H. rewrite H16 in H.
rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
rewrite <- tm_morphism_eq in H.
pose (h0 := firstn (Nat.div2 (length hd'')) (tm_step n)).
pose (s0 := firstn (Nat.div2 (length s))
(skipn (Nat.div2 (length hd'')) (tm_step n))).
pose (t0 := skipn (Nat.div2 (length s))
(skipn (Nat.div2 (length hd'')) (tm_step n))).
fold h0 in H. fold s0 in H. fold t0 in H.
(* fin de la preuve pour cette partie : morphisme impossible *)
assert (s0 = [b0;b0;b1;b1;b0;b0;b1;b1]). fold s0 in H15.
unfold s in H15. destruct s0. inversion H15.
destruct s0. inversion H15. destruct s0. inversion H15.
destruct s0. inversion H15. destruct s0; inversion H15.
destruct s0. inversion H15. destruct s0. inversion H15.
destruct s0. inversion H15. destruct s0; inversion H15.
inversion H15. reflexivity.
rewrite H17 in H.
(* à ce stade H est contradictoire *)
assert (even (length h0) = false).
replace (h0 ++ [b0; b0; b1; b1; b0; b0; b1; b1] ++ t0)
with (h0 ++ [b0] ++ [b0] ++ ([b1; b1; b0; b0; b1; b1] ++ t0)) in H.
assert (odd (length [b0]) = true). reflexivity. generalize H18.
generalize H. apply tm_step_odd_prefix_square. reflexivity.
replace (h0 ++ [b0; b0; b1; b1; b0; b0; b1; b1] ++ t0)
with (h0 ++ [b0;b0;b1;b1] ++ [b0;b0;b1;b1] ++ t0) in H.
assert (even (length h0) = true).
assert (even (length (h0 ++ [b0;b0;b1;b1])) = true).
assert (0 < length [b0;b0;b1;b1]). simpl. apply Nat.lt_0_succ.
generalize H19. generalize H. apply tm_step_square_pos.
rewrite app_length in H19. rewrite Nat.even_add in H19.
rewrite H18 in H19. inversion H19. rewrite H18 in H19.
inversion H19. reflexivity. reflexivity.
rewrite <- length_zero_iff_nil. rewrite Y''. easy.
(* fin de la preuve, on a b8 <> b9 *)
right.
split. assumption.
split. rewrite H4. rewrite e.
assert (b0 = negb b1). destruct b0; destruct b1.
contradiction n1. reflexivity. reflexivity. reflexivity.
contradiction n1. reflexivity. rewrite H7. exists b1.
reflexivity.
rewrite U. simpl. injection. inversion H7. rewrite H9 in n6.
contradiction n6. reflexivity.
rewrite removelast_firstn_len. rewrite removelast_firstn_len. simpl.
rewrite <- length_zero_iff_nil. easy.
rewrite <- length_zero_iff_nil.
rewrite removelast_firstn_len. easy.
apply Nat.lt_1_2. easy.
(* désormais on a b <> b1 ; il suffit de montrer que b = b0 pour
arriver à un bloc de 2 contenant deux termes identiques *)
assert (b = b0). destruct b; destruct b1; destruct b0.
reflexivity. contradiction n2. reflexivity. reflexivity.
contradiction n1. reflexivity. contradiction n1. reflexivity.
reflexivity. contradiction n2. reflexivity. reflexivity.
rewrite H1 in H.
replace (
(b3 :: hd) ++ b0 :: b0 :: b1 :: b2 :: b2 :: b1 :: b0 :: b0 :: b4 :: tl)
with (
(b3 :: hd) ++ [b0] ++ [b0]
++ b1 :: b2 :: b2 :: b1 :: b0 :: b0 :: b4 :: tl) in H.
assert (even (length (b3 :: hd)) = false).
assert (odd (length [b0]) = true). reflexivity. generalize H2.
generalize H. apply tm_step_odd_prefix_square. rewrite H2 in Q.
inversion Q. reflexivity.
(* fin de la destructuration de a, désormais trop grand
cf. hypothèse I *)
simpl in I. apply eq_add_S in I. apply eq_add_S in I.
apply eq_add_S in I. apply eq_add_S in I.
symmetry in I. apply O_S in I. contradiction I.
Qed.
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