124 lines
5.0 KiB
Coq
124 lines
5.0 KiB
Coq
(** * The Thue-Morse sequence (part 4)
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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 thue_morse3.
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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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Require Import Arith.
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Require Import Lia.
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Import ListNotations.
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Theorem tm_step_palindrome_power2_inverse :
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forall (m n k : nat) (hd tl : list bool),
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tm_step n = hd ++ tl
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-> length hd = S (Nat.double k) * 2^m
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-> odd m = true
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-> tl <> nil
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-> skipn (length hd - 2^m) hd = rev (firstn (2^m) tl).
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Proof.
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intros m n k hd tl. intros H I J K.
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assert (L: 2^m <= length hd). rewrite I. lia.
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assert (M: m < n). assert (length (tm_step n) = length (hd ++ tl)).
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rewrite H. reflexivity. rewrite tm_size_power2 in H0.
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rewrite app_length in H0. (* rewrite I in H0. *)
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assert (n <= m \/ m < n). apply Nat.le_gt_cases. destruct H1.
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apply Nat.pow_le_mono_r with (a := 2) in H1.
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assert (2^n <= length hd). generalize L. generalize H1. apply Nat.le_trans.
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rewrite H0 in H2. rewrite <- Nat.add_0_r in H2.
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rewrite <- Nat.add_le_mono_l in H2. destruct tl. contradiction K.
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reflexivity. simpl in H2. apply Nat.nle_succ_0 in H2. contradiction.
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easy. assumption.
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assert (N: length (tm_step n) mod 2^m = 0). rewrite tm_size_power2.
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apply Nat.div_exact. apply Nat.pow_nonzero. easy. rewrite <- Nat.pow_sub_r.
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rewrite <- Nat.pow_add_r. rewrite Nat.add_comm. rewrite Nat.sub_add.
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reflexivity. apply Nat.lt_le_incl. assumption. easy.
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apply Nat.lt_le_incl. assumption.
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assert (O: length hd mod 2^m = 0).
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apply Nat.div_exact. apply Nat.pow_nonzero. easy. rewrite I.
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rewrite Nat.div_mul. rewrite Nat.mul_comm. reflexivity.
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apply Nat.pow_nonzero. easy.
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assert (P: length tl mod 2 ^ m = 0).
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rewrite H in N. rewrite app_length in N.
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rewrite <- Nat.add_mod_idemp_l in N. rewrite O in N. assumption.
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apply Nat.pow_nonzero. easy.
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assert (Q: 2^m <= length tl). apply Nat.div_exact in P. rewrite P.
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destruct (length tl / 2^m). rewrite Nat.mul_0_r in P.
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apply length_zero_iff_nil in P. rewrite P in K. contradiction K.
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reflexivity.
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rewrite <- Nat.mul_1_r at 1. apply Nat.mul_le_mono_l.
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apply Nat.le_succ_l. apply Nat.lt_0_succ.
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apply Nat.pow_nonzero. easy.
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replace hd
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with (firstn (length hd - 2^m) hd ++ skipn (length hd - 2^m) hd) in H.
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replace tl with (firstn (2^m) tl ++ skipn (2^m) tl) in H.
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rewrite <- app_assoc in H.
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replace (
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skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl ++ skipn (2 ^ m) tl )
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with (
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(skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl) ++ skipn (2 ^ m) tl )
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in H.
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assert (length (skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl) = 2^(S m)).
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rewrite app_length. rewrite skipn_length. rewrite firstn_length_le.
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replace (length hd) with (length hd -2^m + 2^m) at 1.
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rewrite Nat.add_sub_swap. rewrite Nat.sub_diag. simpl. rewrite Nat.add_0_r.
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reflexivity. apply Nat.le_refl. apply Nat.sub_add. assumption.
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assumption.
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assert (length (firstn (length hd - 2^m) hd) mod 2^(S m) = 0).
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rewrite firstn_length_le. replace (2^m) with (2^m * 1).
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rewrite I. rewrite Nat.mul_comm. rewrite <- Nat.mul_sub_distr_l.
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rewrite Nat.sub_succ. rewrite Nat.sub_0_r. rewrite Nat.double_twice.
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rewrite Nat.mul_assoc. replace (2^m * 2) with (2^(S m)).
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rewrite Nat.mul_comm. rewrite Nat.mod_mul. reflexivity.
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apply Nat.pow_nonzero. easy.
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rewrite Nat.mul_comm. rewrite Nat.pow_succ_r. reflexivity.
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apply le_0_n. apply Nat.mul_1_r. apply Nat.le_sub_l.
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assert (
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skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl = tm_step (S m)
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\/
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skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl = map negb (tm_step (S m))).
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generalize H1. generalize H0. generalize H. apply tm_step_repeating_patterns2.
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apply tm_step_palindromic_full in J.
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destruct H2; rewrite J in H2.
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- assert (skipn (length hd - 2^m) hd = tm_step m).
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apply app_eq_length_head in H2. assumption.
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rewrite skipn_length.
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replace (length hd) with (length hd -2^m + 2^m) at 1.
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rewrite Nat.add_sub_swap. rewrite Nat.sub_diag. rewrite tm_size_power2.
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reflexivity. reflexivity. apply Nat.sub_add. assumption. rewrite H3.
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assert (firstn (2^m) tl = rev (tm_step m)).
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rewrite H3 in H2. apply app_inv_head in H2. rewrite <- H2. reflexivity.
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rewrite H4. rewrite rev_involutive. reflexivity.
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- assert (skipn (length hd - 2^m) hd = map negb (tm_step m)).
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rewrite map_app in H2. apply app_eq_length_head in H2. assumption.
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rewrite skipn_length. replace (length hd) with (length hd -2^m + 2^m) at 1.
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rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
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rewrite map_length. rewrite tm_size_power2.
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reflexivity. reflexivity. apply Nat.sub_add. assumption. rewrite H3.
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assert (firstn (2^m) tl = map negb (rev (tm_step m))).
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rewrite H3 in H2. rewrite map_app in H2. apply app_inv_head in H2.
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rewrite <- H2. reflexivity.
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rewrite H4. rewrite map_rev. rewrite rev_involutive. reflexivity.
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- rewrite <- app_assoc. reflexivity.
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- rewrite firstn_skipn. reflexivity.
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- rewrite firstn_skipn. reflexivity.
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Qed.
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