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This commit is contained in:
Thomas Baruchel 2023-01-27 09:38:54 +01:00
parent a1186e0654
commit 7c7b67b8b6
2 changed files with 44 additions and 28 deletions
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8
src/thue_morse.v
View File

@ -236,6 +236,14 @@ Proof.
simpl. rewrite Nat.add_0_r. reflexivity.
Qed.
Lemma tm_morphism_length_half : forall (u : list bool),
Nat.div2 (length (tm_morphism u)) = length u.
Proof.
intro u.
induction u. reflexivity.
simpl. rewrite IHu. reflexivity.
Qed.
Lemma tm_morphism_app : forall (l1 l2 : list bool),
tm_morphism (l1 ++ l2) = tm_morphism l1 ++ tm_morphism l2.
Proof.

64
src/thue_morse3.v
View File

@ -1575,38 +1575,46 @@ Proof.
pose (tl' := skipn (Nat.div2 (length hd) + length a) (tm_step n)).
fold hd' in H. fold a' in H. fold tl' in H.
assert (EVEN2: forall m, m mod 4 = 0 -> even (Nat.div2 m) = true).
intro m. intro U. rewrite <- Nat.div_exact in U.
rewrite U. replace 4 with (2*2). rewrite <- Nat.mul_assoc.
rewrite Nat.div2_double. rewrite Nat.even_mul. reflexivity. easy. easy.
assert (I': even (length hd') = true). unfold hd'. rewrite firstn_length.
assert (C: Nat.div2 (length hd) <= length (tm_step n)
\/ length (tm_step n) < Nat.div2 (length hd)).
apply Nat.le_gt_cases. destruct C.
apply Nat.min_r in H6. rewrite Nat.min_comm. rewrite H6.
apply EVEN2. assumption. apply Nat.lt_le_incl in H6.
apply Nat.min_l in H6. rewrite Nat.min_comm. rewrite H6.
rewrite tm_size_power2. destruct n. inversion V. inversion H8.
rewrite Nat.pow_succ_r. rewrite Nat.even_mul. reflexivity.
apply Nat.le_0_l.
assert (J': even (length a') = true). unfold a'. rewrite firstn_length.
rewrite skipn_length.
assert (C: Nat.div2 (length a) <= length (tm_step n) - Nat.div2 (length hd)
\/ length (tm_step n) - Nat.div2 (length hd) < Nat.div2 (length a)).
apply Nat.le_gt_cases. destruct C.
apply Nat.min_r in H6. rewrite Nat.min_comm. rewrite H6. apply EVEN2.
assumption. apply Nat.lt_le_incl in H6.
apply Nat.min_l in H6. rewrite Nat.min_comm. rewrite H6.
rewrite tm_size_power2. destruct n. inversion V. inversion H8.
rewrite Nat.pow_succ_r. rewrite Nat.even_sub. rewrite Nat.even_mul.
assert (Nat.even (Nat.div2 (length hd)) = true). apply EVEN2. assumption.
rewrite H7. reflexivity.
assert (length hd' = Nat.div2 (length hd)). unfold hd'. rewrite H4.
rewrite tm_morphism_length_half. rewrite firstn_length. rewrite firstn_length.
replace (min (Nat.div2 (length hd)) (length (tm_step n)))
with (min (length (tm_step n)) (Nat.div2 (length hd))).
rewrite Nat.min_comm. rewrite Nat.min_assoc. rewrite Nat.min_id.
reflexivity. rewrite Nat.min_comm. reflexivity.
assert (length a' = Nat.div2 (length a)). unfold a'. rewrite H5.
rewrite tm_morphism_length_half. rewrite firstn_length. rewrite firstn_length.
rewrite Nat.min_comm. rewrite <- H6.
replace
(min (Nat.div2 (length a)) (length (skipn (length hd') (tm_step n))))
with
(min (length (skipn (length hd') (tm_step n))) (Nat.div2 (length a))).
rewrite Nat.min_assoc. rewrite Nat.min_id. reflexivity.
rewrite Nat.min_comm. reflexivity.
assert (length tl' = Nat.div2 (length tl)). unfold tl'. rewrite H3.
rewrite tm_morphism_length_half. rewrite app_length.
symmetry. rewrite Nat.div2_div.
rewrite app_length. rewrite rev_length.
replace (length a + length a) with (length a * 2).
rewrite Nat.div_add. rewrite <- Nat.div2_div. reflexivity. easy.
rewrite Nat.mul_comm. simpl. rewrite Nat.add_0_r. reflexivity.
assert (I': even (length hd') = true). rewrite H6.
rewrite <- Nat.div_exact in J1. rewrite J1.
replace 4 with (2*2). rewrite <- Nat.mul_assoc. rewrite Nat.div2_double.
rewrite Nat.even_mul. reflexivity. reflexivity. easy.
assert (J': even (length a') = true). rewrite H7.
rewrite <- Nat.div_exact in J0. rewrite J0.
replace 4 with (2*2). rewrite <- Nat.mul_assoc. rewrite Nat.div2_double.
rewrite Nat.even_mul. reflexivity. reflexivity. easy.
assert (K': even (length (map negb (rev a'))) = true).
rewrite map_length. rewrite rev_length. assumption.
Admitted.