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Thomas Baruchel 2023-01-17 17:12:20 +01:00
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(** * The Thue-Morse sequence (part 3)
TODO
*)
Require Import thue_morse.
Require Import thue_morse2.
Require Import Coq.Lists.List.
Require Import PeanoNat.
Require Import Nat.
Require Import Bool.
Import ListNotations.
Lemma tm_step_palindromic_odd : forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ tl
-> a = rev a
-> odd (length a) = true
-> length a <> 5.
Proof.
intros n hd a tl. intros H I J.
destruct a. easy. destruct a. easy.
destruct a. easy. destruct a. easy.
destruct a. easy.
destruct a. (* case of length 5 *)
assert (exists (u v : list bool) (d: bool),
b::b0::b1::b2::b3::nil = u ++ [d;d] ++ v).
assert (4 < length (b::b0::b1::b2::b3::nil)). 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. generalize H0. generalize H.
apply tm_step_consecutive_identical_length.
destruct H0. destruct H0. destruct H0.
assert (K: {b=b0} + {~ b=b0}). apply bool_dec.
assert (L: {b0=b1} + {~ b0=b1}). apply bool_dec.
assert (M: {b1=b2} + {~ b1=b2}). apply bool_dec.
assert (N: {b2=b3} + {~ b2=b3}). apply bool_dec.
destruct K. rewrite e in H. rewrite e in I.
destruct N. rewrite e0 in H.
replace ([b0;b0;b1;b3;b3] ++ tl) with ([b0;b0] ++ [b1] ++ [b3;b3] ++ tl) in H.
apply tm_step_consecutive_identical' in H. inversion H. reflexivity.
inversion I. rewrite H3 in n0. contradiction n0.
destruct L. rewrite e in H. rewrite e in I.
destruct M. rewrite <- e0 in H.
replace (hd ++ [b; b1; b1; b1; b3] ++ tl)
with ((hd ++ [b]) ++ [b1] ++ [b1] ++ [b1] ++ ([b3] ++ tl)) in H.
apply tm_step_cubefree in H. contradiction H. reflexivity.
apply Nat.lt_0_succ. rewrite <- app_assoc. simpl. reflexivity.
inversion I. rewrite H3 in n1. contradiction n1.
destruct M. rewrite <- e in I. inversion I.
rewrite H3 in n1. contradiction n1.
destruct N. rewrite e in I. inversion I.
rewrite H2 in n0. rewrite H3 in n0. contradiction n0.
destruct x. inversion H0.
rewrite H2 in n0. rewrite H3 in n0. contradiction n0.
reflexivity.
destruct x. inversion H0.
rewrite H3 in n1. rewrite H4 in n1. contradiction n1.
reflexivity.
destruct x. inversion H0.
rewrite H4 in n2. rewrite H5 in n2. contradiction n2.
reflexivity.
destruct x. inversion H0.
rewrite H5 in n3. rewrite H6 in n3. contradiction n3.
reflexivity.
assert (length [b;b0;b1;b2;b3] = length [b;b0;b1;b2;b3]).
reflexivity. rewrite H0 in H1 at 2. simpl in H1.
apply Nat.succ_inj in H1. apply Nat.succ_inj in H1.
apply Nat.succ_inj in H1. apply Nat.succ_inj in H1.
rewrite app_length in H1. simpl in H1.
rewrite Nat.add_succ_r in H1. rewrite Nat.add_succ_r in H1.
apply Nat.succ_inj in H1. apply O_S in H1. contradiction H1.
simpl. apply not_eq_S. apply not_eq_S. apply not_eq_S.
apply not_eq_S. apply not_eq_S. apply Nat.neq_succ_0.
Qed.
Lemma tm_step_palindromic_odd' : forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ tl
-> a = rev a
-> odd (length a) = true
-> length a < 4.
Proof.
intros n hd a tl. intros H I J.
assert (length a <= 5 \/ 5 < length a). apply Nat.le_gt_cases.
destruct H0.
apply Nat.lt_eq_cases in H0. destruct H0.
rewrite Nat.lt_succ_r in H0.
apply Nat.lt_eq_cases in H0. destruct H0. assumption.
rewrite H0 in J. inversion J.
assert (length a <> 5). generalize J. generalize I. generalize H.
apply tm_step_palindromic_odd. rewrite H0 in H1. contradiction H1.
reflexivity.
(* main part of the proof:
each odd palindromic factor greater than 5
contains a palindromic subfactor of length 5 *)