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