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Thomas Baruchel 2023-02-11 16:02:22 +01:00
parent 6a9c177d2f
commit 8d067d4ae7
3 changed files with 1122 additions and 31 deletions
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1
src/thue_morse.v
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@ -1287,6 +1287,7 @@ Lemma tm_step_factor4_1mod4 : forall (n' : nat) (w z y : list bool),
tm_step n' = w ++ z ++ y
-> length z = 4 -> length w mod 4 = 1
-> exists (x : bool), firstn 2 z = [x;x].
Proof.
intros n' w z y. intros G0 G1 G2.
assert (U: 4 * (length w / 4) <= length w).
apply Nat.mul_div_le. easy.

144
src/thue_morse3.v
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@ -308,8 +308,14 @@ Lemma tm_step_palindromic_length_8 :
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
\/ nth_error hd (length hd - 3) <> nth_error tl 2.
-> (
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.
@ -450,30 +456,44 @@ Proof.
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. assumption. (* ici on postule le modulo = 2 *)
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 H1.
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 H2.
assert (length [b1;b;b1;b1] = 4). reflexivity. generalize H3.
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 :: b2 :: b2 :: b1 :: b1 :: b :: b4 :: tl )
[b1] ++ b :: b1 :: b1 :: b :: b :: b1 :: b1 :: b :: b4 :: tl )
with (
removelast (b3 :: hd) ++ [b1;b;b1;b1]
++ (b2 :: b2 :: b1 :: b1 :: b :: b4 :: tl )) in H.
++ (b :: b :: b1 :: b1 :: b :: b4 :: tl )) in H.
generalize H. apply tm_step_factor4_1mod4. reflexivity.
destruct H3. inversion H3. rewrite <- H6 in H5. rewrite H5 in n1.
destruct H4. inversion H4. rewrite <- H7 in H6. rewrite H6 in n1.
contradiction n1. reflexivity.
rewrite app_length in H1. rewrite app_length in H1.
rewrite <- Nat.add_assoc in H1. rewrite <- Nat.add_mod_idemp_l in H1.
rewrite H2 in H1. inversion H1. easy. easy. easy.
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
@ -483,23 +503,35 @@ Proof.
(* un sous-cas :
(T F) | F T T F || F T T F | (T F) impossible à droite
*)
destruct M. left. assumption. (* ici on postule le modulo = 2 *)
rewrite e in H1.
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 H1.
rewrite app_length. rewrite <- Nat.add_mod_idemp_l. rewrite H2.
reflexivity. easy.
replace(
removelast (b3 :: hd) ++ [last (b3 :: hd) false] ++
b :: b1 :: b1 :: b2 :: b2 :: b1 :: b1 :: b :: b1 :: tl )
b :: b1 :: b1 :: b :: b :: b1 :: b1 :: b :: b1 :: tl )
with (
(((b3 :: hd) ++ [b; b1; b1; b2]) ++ [b2]) ++ [b1; b1;b;b1] ++ tl) in H.
(((b3 :: hd) ++ [b; b1; b1; b]) ++ [b]) ++ [b1; b1;b;b1] ++ tl) in H.
assert (exists x, skipn 2 [b1;b1;b;b1] = [x;x]). generalize H2.
assert (length [b1;b1;b;b1] = 4). reflexivity. generalize H3.
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 H3. inversion H3. rewrite <- H6 in H5. rewrite H5 in n1.
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.
@ -601,8 +633,28 @@ reflexivity. contradiction n0. reflexivity. reflexivity. rewrite H1 in H.
apply bool_dec. destruct H1. rewrite e0 in H.
(* problème à gauche *)
destruct M. left. assumption. (* ici on postule le modulo = 2 *)
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.
@ -635,8 +687,27 @@ reflexivity. contradiction n0. reflexivity. reflexivity. rewrite H1 in H.
(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. left. assumption. (* ici on postule le modulo = 2 *)
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.
@ -851,10 +922,6 @@ Le lemme repeating_patterns se base sur les huit premiers termes de TM :
rewrite <- Nat.add_0_l at 1. apply Nat.add_le_mono.
apply Nat.le_0_l. apply Nat.le_refl.
(* inutile
assert (V: nth_error (b4 :: b6 :: b9 :: tl) 2 = Some b9). reflexivity.
*)
(* analyse finale *)
assert ({b8=b9} + {~ b8=b9}). apply bool_dec. destruct H7. rewrite e0 in H.
@ -938,7 +1005,7 @@ Le lemme repeating_patterns se base sur les huit premiers termes de TM :
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.
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.
@ -1193,7 +1260,15 @@ Le lemme repeating_patterns se base sur les huit premiers termes de TM :
rewrite <- length_zero_iff_nil. rewrite Y''. easy.
(* fin de la preuve, on a b8 <> b9 *)
right. rewrite U. simpl. injection. inversion H7. rewrite H9 in n6.
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.
@ -1262,11 +1337,17 @@ Proof.
rewrite <- app_assoc in H.
pose (tl' := rev (firstn (length a - 4) a) ++ tl). fold tl' in H.
assert (K: length (hd' ++ a') mod 4 = 0
\/ nth_error hd' (length hd' - 3) <> nth_error tl' 2).
assert (K: (
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
)).
fold a' in H1. generalize H1. generalize H.
apply tm_step_palindromic_length_8.
destruct K.
destruct K. destruct H2.
unfold hd' in H2. unfold a' in H2. rewrite <- app_assoc in H2.
rewrite firstn_skipn in H2. assumption.
@ -1280,6 +1361,7 @@ Proof.
apply Nat.succ_lt_mono in I. apply Nat.succ_lt_mono in I.
inversion I.
destruct H2 as [H2' H2'']. destruct H2'' as [H2'' H2].
unfold hd' in H2. unfold tl' in H2.
rewrite nth_error_app2 in H2. rewrite nth_error_app1 in H2.
rewrite app_length in H2. rewrite Nat.add_comm in H2.

1008
src/thue_morse4.v
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