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This commit is contained in:
Thomas Baruchel 2023-01-14 20:04:32 +01:00
parent 047ed4c527
commit e1238e5198
1 changed files with 5 additions and 37 deletions
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42
src/thue_morse2.v
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@ -329,37 +329,6 @@ Qed.
prefix being odd or even in front of a squared pattern.
*)
Lemma tm_step_odd_prefix_square_1 : forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 1 -> even (length hd) = false.
Proof.
intros n hd a tl. intros H I.
assert (J: Nat.even (length hd) || Nat.odd (length hd) = true).
apply Nat.orb_even_odd. apply orb_prop in J. destruct J.
- assert (K: count_occ Bool.bool_dec hd true
= count_occ Bool.bool_dec hd false).
generalize H0. generalize H. apply tm_step_count_occ.
assert (L: even (length (hd ++ a ++ a)) = true).
rewrite app_length. rewrite app_length. rewrite I.
rewrite Nat.even_add_even. assumption. rewrite Nat.add_1_r.
apply Nat.EvenT_Even. apply Nat.even_EvenT. easy.
assert (M: count_occ Bool.bool_dec (hd ++ a ++ a) true
= count_occ Bool.bool_dec (hd ++ a ++ a) false).
generalize L. generalize H.
replace (hd ++ a ++ a ++ tl) with ((hd ++ a ++ a) ++ tl).
apply tm_step_count_occ.
rewrite <- app_assoc. apply app_inv_head_iff.
rewrite <- app_assoc. reflexivity.
rewrite count_occ_app in M. symmetry in M.
rewrite count_occ_app in M. rewrite K in M.
rewrite Nat.add_cancel_l in M. destruct a.
+ inversion I.
+ destruct b; simpl in I; apply Nat.succ_inj in I;
rewrite length_zero_iff_nil in I; rewrite I in M;
simpl in M; inversion M.
- rewrite <- Nat.negb_odd. rewrite H0. reflexivity.
Qed.
Lemma tm_step_odd_prefix_square_3 : forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> length a = 3 -> even (length hd) = false.
@ -430,12 +399,10 @@ Lemma tm_step_odd_prefix_square : forall (n : nat) (hd a tl : list bool),
-> odd (length a) = true -> even (length hd) = false.
Proof.
intros n hd a tl. intros H I.
assert (length a < 4).
generalize I. generalize H. apply tm_step_odd_square.
destruct a. inversion I.
destruct a.
apply tm_step_odd_prefix_square_1 with (n := n) (a := [b]) (tl := tl).
assumption. reflexivity.
assert (length a < 4). generalize I. generalize H. apply tm_step_odd_square.
destruct a. inversion I. destruct a.
rewrite <- negb_true_iff. rewrite Nat.negb_even.
generalize H. apply tm_step_consecutive_identical. apply hd.
destruct a. inversion I.
destruct a.
apply tm_step_odd_prefix_square_3 with (n := n) (a := [b;b0;b1]) (tl := tl).
@ -549,3 +516,4 @@ Proof.
(* TODO: utiliser tm_step_odd_prefix_square pour le cas impair *)
(* TODO: la taille du facteur divise la taille du préfixe ? *)