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Thomas Baruchel 2023-01-31 22:03:20 +01:00
parent 4fcaba8c73
commit 36b7d26757
2 changed files with 283 additions and 82 deletions
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83
src/thue_morse2.v
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@ -10,9 +10,11 @@
But some powerful theorems are provided here:
- tm_step_cubefree (there is no cube in the sequence)
- tm_step_square_size (about length of squared factrs)
- tm_step_square_size (about length of squared factors)
- tm_step_square_pos (length of the prefix has the same
parity than the length of the factor being squared)
- tm_step_square_prefix_not_nil (no square at the beginning
of the sequence)
*)
Require Import thue_morse.
@ -1622,82 +1624,3 @@ Proof.
simpl. rewrite Nat.add_0_r. reflexivity.
left. assumption.
Qed.
Theorem tm_step_square_prefix_not_nil :
forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl -> 0 < length a -> hd <> nil.
Proof.
intros n hd a tl. intros H I.
assert (W: {hd=nil} + {~ hd=nil}). apply list_eq_dec. apply bool_dec.
destruct W. rewrite e in H. rewrite app_nil_l in H.
generalize dependent a.
generalize dependent tl.
induction n.
- intros tl a. intros H I.
destruct a. inversion I. inversion H.
assert (length (nil : list bool) = length (nil : list bool)).
reflexivity. rewrite H2 in H0 at 2. rewrite app_length in H0.
rewrite Nat.add_comm in H0. inversion H0.
- intros tl a. intros H I.
assert (hd <> nil
\/ exists a' tl', tm_step (pred (S n)) = a' ++ a' ++ tl' /\ 0 < length a').
generalize I. generalize H. replace (a++a++tl) with (hd ++ a ++ a ++ tl).
apply tm_step_square_prefix_not_nil0. rewrite e. reflexivity.
destruct H0. assumption. destruct H0. destruct H0.
rewrite <- pred_Sn in H0.
destruct H0.
generalize H1. generalize H0. apply IHn.
- assumption.
Qed.
(*
Theorem tm_step_repeating_patterns_no_square :
forall (n m : nat) (hd a b tl : list bool),
tm_step n = hd ++ a ++ b ++ tl -> length a = 2^m
-> (forall k, length hd = k * 2 ^ S m -> a <> b).
Proof.
intros n m hd a b tl. intros H I.
induction n.
- assert (m = 0).
destruct m. reflexivity.
assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
rewrite H in H0 at 2. rewrite app_length in H0. rewrite Nat.add_comm in H0.
rewrite app_length in H0. rewrite <- Nat.add_assoc in H0.
rewrite I in H0. rewrite Nat.pow_succ_r in H0. symmetry in H0.
apply Nat.eq_add_1 in H0. destruct H0; destruct H0.
apply Nat.mul_eq_1 in H0. destruct H0. inversion H0.
apply Nat.mul_eq_0 in H0. destruct H0. inversion H0.
apply Nat.pow_nonzero in H0. contradiction H0. easy. apply le_0_n.
intro k. intro J. rewrite H0 in J. rewrite Nat.pow_1_r in J.
assert (k = 0).
assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
rewrite H in H1 at 2. rewrite app_length in H1. rewrite J in H1.
symmetry in H1. apply Nat.eq_add_1 in H1. destruct H1; destruct H1.
apply Nat.mul_eq_1 in H1. destruct H1. inversion H3.
apply Nat.mul_eq_0 in H1. destruct H1. assumption. inversion H1.
rewrite H1 in J. rewrite Nat.mul_0_l in J. rewrite length_zero_iff_nil in J.
assert ({a=b} + {~ a=b}). apply list_eq_dec. apply bool_dec.
destruct H2. rewrite <- e in H.
assert (0 < length a). rewrite I. rewrite H0. apply Nat.lt_0_1.
assert (hd <> nil). generalize H2. generalize H.
apply tm_step_square_prefix_not_nil. rewrite J in H3.
contradiction H3. reflexivity. assumption.
-
Reformulation:
forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> 0 < length a
-> (length hd) mod (length (a ++ a)) <> 0.
PB : manque l'idée d'une puissance de 2 !
Theorem tm_step_square_prefix_not_nil :
forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl -> 0 < length a -> hd <> nil.
*)

282
src/thue_morse3.v
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@ -1820,14 +1820,14 @@ Qed.
Lemma tm_step_palindromic_power2_even :
forall (m n k : nat) (hd a tl : list bool),
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ (rev a) ++ tl
-> 6 < length a
-> length a = 2^(Nat.double m)
-> length (hd ++ a) mod (2 ^ (pred (Nat.double m))) = 0
<-> (length (hd ++ a) / 4) mod (2^(pred (Nat.double (pred m)))) = 0.
Proof.
intros m n k hd a tl. intros H I J.
intros m n hd a tl. intros H I J.
destruct m. rewrite J in I. inversion I. inversion H1.
destruct m. rewrite J in I. inversion I. inversion H1.
@ -1902,11 +1902,15 @@ Lemma xxx :
-> length (hd ++ a) mod (2 ^ (pred (Nat.double m))) = 0.
Proof.
intros m n k hd a tl. intros H I J.
assert (R := H).
destruct m. rewrite J in I. inversion I. inversion H1.
destruct m. rewrite J in I. inversion I. inversion H1.
inversion H3. inversion H5. inversion H7.
generalize dependent a. generalize dependent hd.
induction m. intros hd a H I J H'.
assert (W: (length a) mod 4 = 0).
rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
@ -1946,6 +1950,15 @@ Proof.
pose (tl' := (skipn (length hd / 4 + Nat.div2 (length a)) (tm_step n))).
fold hd' in H. fold a' in H. fold tl' in H.
(*
assert (U: even (length (hd ++ a)) = true).
apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
generalize I. generalize H'.
apply tm_step_palindromic_even_center.
*)
rewrite Nat.pred_succ in K. rewrite Nat.pred_succ in K.
rewrite Nat.pred_succ in L. rewrite Nat.pred_succ in L.
fold hd' in K. fold a' in L.
@ -1956,6 +1969,271 @@ Proof.
rewrite tm_morphism_length in O. rewrite tm_morphism_length in O.
rewrite Nat.mul_assoc in O. replace (2*2) with 4 in O.
replace (2^ pred (Nat.double 2)) with (4*2).
rewrite app_length. rewrite N. rewrite O.
rewrite <- Nat.mul_add_distr_l. rewrite Nat.mul_mod_distr_l.
rewrite Nat.mul_eq_0. right. rewrite <- app_length.
assert (U: even (length (hd' ++ a')) = true).
assert (0 < length a').
apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
rewrite N in I.
rewrite <- Nat.mul_1_r in I at 1.
rewrite <- Nat.mul_lt_mono_pos_l in I.
apply Nat.lt_succ_l in I. assumption. apply Nat.lt_0_succ.
generalize H0. generalize H. apply tm_step_palindromic_even_center.
replace (length (hd' ++ a'))
with (2 * Nat.div2 (length (hd' ++ a'))
+ Nat.b2n (Nat.odd (length (hd' ++ a')))).
rewrite <- Nat.negb_even. rewrite U. rewrite Nat.add_0_r.
rewrite Nat.mul_mod. reflexivity. easy. symmetry. apply Nat.div2_odd.
easy. easy. reflexivity. reflexivity. reflexivity.
intros hd a H I J H'. rewrite Nat.double_S.
replace (pred (S (S (Nat.double (S (S m))))))
with (S (S (pred (Nat.double (S (S m)))))).
rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.mul_assoc. replace (2*2) with 4.
assert (W: (length a) mod 4 = 0).
rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
rewrite Nat.mul_assoc in J.
replace (2*2) with 4 in J. rewrite J.
rewrite <- Nat.mul_mod_idemp_l. reflexivity.
easy. reflexivity. apply le_0_n. apply le_0_n.
assert (
hd = tm_morphism (tm_morphism (firstn (length hd / 4)
(tm_step (pred (pred n)))))
/\ a = tm_morphism (tm_morphism
(firstn (length a / 4) (skipn (length hd / 4)
(tm_step (pred (pred n))))))
/\ tl = tm_morphism (tm_morphism
(skipn (length hd / 4 + Nat.div2 (length a))
(tm_step (pred (pred n)))))).
generalize W. generalize I. generalize H.
apply tm_step_palindromic_even_morphism2.
destruct H0 as [K H0]. destruct H0 as [L M].
assert (V: 3 < n). generalize I. generalize H.
apply tm_step_palindromic_length_12_n.
destruct n. inversion V. destruct n. inversion V. inversion H1.
rewrite K in H. rewrite L in H. rewrite M in H.
rewrite tm_morphism_twice_rev in H.
rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
rewrite <- tm_step_lemma in H. rewrite <- tm_step_lemma in H.
rewrite <- tm_morphism_eq in H. rewrite <- tm_morphism_eq in H.
rewrite Nat.pred_succ in H. rewrite Nat.pred_succ in H.
rewrite <- pred_Sn in K. rewrite <- pred_Sn in K.
rewrite <- pred_Sn in L. rewrite <- pred_Sn in L.
pose (hd' := (firstn (length hd / 4) (tm_step n))).
pose (a' := (firstn (length a / 4) (skipn (length hd / 4) (tm_step n)))).
pose (tl' := (skipn (length hd / 4 + Nat.div2 (length a)) (tm_step n))).
fold hd' in H. fold a' in H. fold tl' in H.
fold hd' in K. fold a' in L.
assert (N: length a = length a). reflexivity. rewrite L in N at 2.
rewrite tm_morphism_length in N. rewrite tm_morphism_length in N.
rewrite Nat.mul_assoc in N. replace (2*2) with 4 in N.
assert (O: length hd = length hd). reflexivity. rewrite K in O at 2.
rewrite tm_morphism_length in O. rewrite tm_morphism_length in O.
rewrite Nat.mul_assoc in O. replace (2*2) with 4 in O.
rewrite app_length. rewrite N. rewrite O.
rewrite <- Nat.mul_add_distr_l. rewrite Nat.mul_mod_distr_l.
rewrite Nat.mul_eq_0. right. rewrite <- app_length.
Lemma tm_step_palindromic_power2_even :
forall (m n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ (rev a) ++ tl
-> 6 < length a
-> length a = 2^(Nat.double m)
-> length (hd ++ a) mod (2 ^ (pred (Nat.double m))) = 0
<-> (length (hd ++ a) / 4) mod (2^(pred (Nat.double (pred m)))) = 0.
rewrite Nat.mul_comm. rewrite Nat.div_mul.
rewrite <- app_length. apply IHm with (n := S (S n)) (tl := tl').
(*
generalize dependent a. generalize dependent hd.
generalize dependent tl. generalize dependent n.
*)
induction m.
-
rewrite tm_step_palindromic_power2_even with (n := S (S n)) (tl := tl).
rewrite app_length. rewrite N.
rewrite Nat.mul_comm. rewrite Nat.div_add.
rewrite O. rewrite Nat.mul_comm. rewrite Nat.div_mul.
rewrite <- app_length.
assert (even (length (hd' ++ a')) = true).
assert (0 < length a').
apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
rewrite N in I.
rewrite <- Nat.mul_1_r in I at 1.
rewrite <- Nat.mul_lt_mono_pos_l in I.
apply Nat.lt_succ_l in I. assumption.
apply Nat.lt_0_succ. generalize H0. generalize H.
apply tm_step_palindromic_even_center.
replace (length (hd' ++ a'))
with (2 * Nat.div2 (length (hd' ++ a'))
+ Nat.b2n (Nat.odd (length (hd' ++ a')))).
rewrite <- Nat.negb_even. rewrite H0.
rewrite Nat.add_0_r.
rewrite Nat.mul_mod. reflexivity. simpl.
apply Nat.neq_succ_0. symmetry. apply Nat.div2_odd.
easy. easy. assumption. assumption. assumption.
(* intros n V tl hd hd' K O a a' tl' H I J R W L M N. *)
(*
replace (S (S (S m))) with (pred (S (S (S (S m))))).
rewrite Nat.double_S.
replace (S (S m)) with (pred (S (S (S m)))).
replace (pred (S (S (Nat.double (pred (S (S (S m))))))))
with (S (S (pred (Nat.double (pred (S (S (S m)))))))).
rewrite Nat.pow_succ_r.
rewrite Nat.pow_succ_r.
rewrite Nat.mul_assoc. replace (2*2) with 4.
replace (length (hd ++ a))
with ((length ((tm_morphism (tm_morphism hd))
++ (tm_morphism (tm_morphism a))))/4).
rewrite <- tm_step_palindromic_power2_even
with (n := S (S (S (S n)))) (tl := tm_morphism (tm_morphism tl)).
rewrite Nat.double_S. rewrite Nat.double_S.
*)
-
rewrite tm_step_palindromic_power2_even with (n := (S (S n))) (tl := tl).
rewrite <- pred_Sn.
rewrite app_length. rewrite N. rewrite O.
rewrite <- Nat.mul_add_distr_l.
rewrite Nat.mul_comm. rewrite Nat.div_mul.
rewrite <- app_length. apply IHm with (n := S (S n)) (tl := tl').
rewrite Nat.double_S.
replace (pred (S (S (Nat.double (S (S m))))))
with (S (S (pred (Nat.double (S (S m)))))).
rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.mul_assoc. replace (2*2) with 4.
rewrite app_length. rewrite O. rewrite N.
rewrite <- Nat.mul_add_distr_l. rewrite Nat.mul_mod_distr_l.
rewrite Nat.mul_eq_0. right. rewrite <- app_length.
apply IHm with (n := n) (tl := tl').
induction m.
- (* intros n V tl hd hd' K O a a' tl' H I J R W L M N. *)
rewrite tm_step_palindromic_power2_even with (n := S (S n)) (tl := tl).
rewrite app_length. rewrite N.
rewrite Nat.mul_comm. rewrite Nat.div_add.
rewrite O. rewrite Nat.mul_comm. rewrite Nat.div_mul.
rewrite <- app_length.
assert (even (length (hd' ++ a')) = true).
assert (0 < length a').
apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
rewrite N in I.
rewrite <- Nat.mul_1_r in I at 1.
rewrite <- Nat.mul_lt_mono_pos_l in I.
apply Nat.lt_succ_l in I. assumption.
apply Nat.lt_0_succ. generalize H0. generalize H.
apply tm_step_palindromic_even_center.
replace (length (hd' ++ a'))
with (2 * Nat.div2 (length (hd' ++ a'))
+ Nat.b2n (Nat.odd (length (hd' ++ a')))).
rewrite <- Nat.negb_even. rewrite H0.
rewrite Nat.add_0_r.
rewrite Nat.mul_mod. reflexivity. simpl.
apply Nat.neq_succ_0. symmetry. apply Nat.div2_odd.
easy. easy. assumption. assumption. assumption.
- (* intros n V tl hd hd' K O a a' tl' H I J R W L M N. *)
(*
replace (S (S (S m))) with (pred (S (S (S (S m))))).
rewrite Nat.double_S.
replace (S (S m)) with (pred (S (S (S m)))).
replace (pred (S (S (Nat.double (pred (S (S (S m))))))))
with (S (S (pred (Nat.double (pred (S (S (S m)))))))).
rewrite Nat.pow_succ_r.
rewrite Nat.pow_succ_r.
rewrite Nat.mul_assoc. replace (2*2) with 4.
replace (length (hd ++ a))
with ((length ((tm_morphism (tm_morphism hd))
++ (tm_morphism (tm_morphism a))))/4).
rewrite <- tm_step_palindromic_power2_even
with (n := S (S (S (S n)))) (tl := tm_morphism (tm_morphism tl)).
rewrite Nat.double_S. rewrite Nat.double_S.
*)
rewrite tm_step_palindromic_power2_even with (n := (S (S n))) (tl := tl).
rewrite <- pred_Sn.
rewrite app_length. rewrite N. rewrite O.
rewrite <- Nat.mul_add_distr_l.
rewrite Nat.mul_comm. rewrite Nat.div_mul.
rewrite <- app_length. apply IHm with (n := S (S n)) (tl := tl').
rewrite Nat.double_S.
replace (pred (S (S (Nat.double (S (S m))))))
with (S (S (pred (Nat.double (S (S m)))))).
rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
rewrite Nat.mul_assoc. replace (2*2) with 4.
rewrite app_length. rewrite O. rewrite N.
rewrite <- Nat.mul_add_distr_l. rewrite Nat.mul_mod_distr_l.
rewrite Nat.mul_eq_0. right. rewrite <- app_length.
apply IHm with (n := n) (tl := tl').
Nat.div2_odd: forall a : nat, a = 2 * Nat.div2 a + Nat.b2n (Nat.odd a)
Nat.div_str_pos: forall a b : nat, 0 < b <= a -> 0 < a / b
apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
apply Nat.lt_succ_l in I. apply Nat.lt_succ_l in I.
generalize I.
Theorem tm_step_palindromic_even_center :
forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ (rev a) ++ tl
-> 0 < length a
-> even (length (hd ++ a)) = true.
rewrite <- Nat.mul_cancel_l with (p := 4).
rewrite <- Nat.mul_mod_distr_l.
replace (4 * (length (hd ++ a) / 4))
with (4 * (length (hd ++ a) / 4) + length (hd ++ a) mod 4).
rewrite <- Nat.div_mod_eq. assumption.
rewrite H0. rewrite Nat.add_0_r. reflexivity.
rewrite app_length.
replace (4 * (length (hd ++ a) / 4))
with (4 * (length (hd ++ a) / 4) + length (hd ++ a) mod 4).
rewrite <- Nat.div_mod_eq. assumption.
rewrite H0. rewrite Nat.add_0_r. reflexivity.
generalize dependent a.
generalize dependent hd.