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Thomas Baruchel 2023-01-02 14:37:17 +01:00
parent d7568ec506
commit 2099d9f563
1 changed files with 16 additions and 21 deletions
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thue-morse.v
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@ -799,19 +799,20 @@ Proof.
rewrite tm_morphism_eq in J. rewrite tm_morphism_eq in J.
rewrite tm_morphism_app in J. rewrite H in J. rewrite tm_morphism_app in J. rewrite H in J.
apply app_eq_app in J. apply app_eq_app in J.
assert (H2: length (tm_morphism l) = length hd + length tl).
rewrite H. apply app_length.
assert (H3: length hd <= length (tm_morphism l)).
rewrite H2. apply Nat.le_add_r.
rewrite tm_morphism_length in H3.
destruct J. destruct H0; destruct H0. destruct J. destruct H0; destruct H0.
assert (L: length hd = length hd). reflexivity. assert (L: length hd = length hd). reflexivity.
rewrite H0 in L at 2. rewrite app_length in L. rewrite H0 in L at 2. rewrite app_length in L.
rewrite tm_morphism_length in L. rewrite tm_morphism_length in L.
assert (length (tm_morphism l) = length hd + length tl).
rewrite H. apply app_length.
assert (length hd <= length (tm_morphism l)).
rewrite H2. apply Nat.le_add_r.
rewrite tm_morphism_length in H3.
assert (Nat.div2 (length hd) <= length l). assert (Nat.div2 (length hd) <= length l).
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite Nat.mul_le_mono_pos_l with (p := 2).
rewrite <- Nat.add_0_r at 1. rewrite <- Nat.add_0_r at 1.
@ -835,13 +836,6 @@ Proof.
reflexivity. rewrite H0 in L at 2. rewrite app_length in L. reflexivity. rewrite H0 in L at 2. rewrite app_length in L.
rewrite tm_morphism_length in L. rewrite tm_morphism_length in L.
assert (length (tm_morphism l) = length hd + length tl).
rewrite H. apply app_length.
assert (length hd <= length (tm_morphism l)).
rewrite H2. apply Nat.le_add_r.
rewrite tm_morphism_length in H3.
assert (Nat.div2 (length hd) <= length l). assert (Nat.div2 (length hd) <= length l).
rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite Nat.mul_le_mono_pos_l with (p := 2).
rewrite <- Nat.add_0_r at 1. rewrite <- Nat.add_0_r at 1.
@ -869,26 +863,27 @@ Qed.
From a(0) to a(2n+1), there are n+1 terms equal to 0 and n+1 terms equal to 1 (see Hassan Tarfaoui link, Concours Général 1990). - Bernard Schott, Jan 21 2022 From a(0) to a(2n+1), there are n+1 terms equal to 0 and n+1 terms equal to 1 (see Hassan Tarfaoui link, Concours Général 1990). - Bernard Schott, Jan 21 2022
TODO Search "count_occ" TODO Search "count_occ"
*) *)
Theorem tm_step_count_occ : forall (hd tl : list bool) (k : nat), Theorem tm_step_count_occ : forall (hd tl : list bool) (n : nat),
tm_step k = hd ++ tl -> even (length hd) = true tm_step n = hd ++ tl -> even (length hd) = true
-> count_occ Bool.bool_dec hd true = count_occ Bool.bool_dec hd false. -> count_occ Bool.bool_dec hd true = count_occ Bool.bool_dec hd false.
Proof. Proof.
intros hd tl k. intros H I. intros hd tl n. intros H I.
destruct k. destruct n.
- assert (J: hd = nil). destruct hd. reflexivity. - assert (J: hd = nil). destruct hd. reflexivity.
simpl in H. inversion H. simpl in H. inversion H.
symmetry in H2. apply app_eq_nil in H2. symmetry in H2. apply app_eq_nil in H2.
destruct H2. rewrite H0 in I. simpl in I. inversion I. destruct H2. rewrite H0 in I. simpl in I. inversion I.
rewrite J. reflexivity. rewrite J. reflexivity.
- rewrite <- tm_step_lemma in H. - rewrite <- tm_step_lemma in H.
assert (J: hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step k))). assert (J: hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
generalize I. generalize H. apply tm_morphism_app2. generalize I. generalize H. apply tm_morphism_app2.
rewrite J. apply tm_morphism_count_occ. rewrite J. apply tm_morphism_count_occ.
Qed. Qed.
Lemma tm_step_cube1 : forall (n : nat) (a hd tl: list bool),
tm_step n = hd ++ a ++ a ++ a ++ tl -> even (length a) = true.
Proof.