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Thomas Baruchel 2023-01-27 10:05:25 +01:00
parent 7c7b67b8b6
commit 25a6956538
2 changed files with 71 additions and 1 deletions
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11
src/thue_morse.v
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@ -172,7 +172,6 @@ Proof.
rewrite negb_involutive. reflexivity.
Qed.
Lemma tm_build_negb : forall (l : list bool),
tm_morphism (map negb l) = map negb (tm_morphism l).
Proof.
@ -331,6 +330,16 @@ Proof.
rewrite app_inv_head_iff in L. assumption.
Qed.
Lemma tm_morphism_rev2 : forall (l : list bool),
map negb (rev (tm_morphism l)) = tm_morphism (rev l).
Proof.
intro l. induction l.
- reflexivity.
- simpl. rewrite tm_morphism_app. rewrite <- IHl.
rewrite map_app. rewrite map_app. rewrite <- app_assoc.
simpl. rewrite negb_involutive. reflexivity.
Qed.
(**
Lemmas and theorems below are related to the second function defined initially.

61
src/thue_morse3.v
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@ -1613,6 +1613,67 @@ Proof.
assert (K': even (length (map negb (rev a'))) = true).
rewrite map_length. rewrite rev_length. assumption.
assert (H0': even (length (hd' ++ a')) = true).
rewrite app_length. rewrite Nat.even_add.
rewrite I'. rewrite J'. reflexivity.
destruct n. inversion V. inversion H10.
rewrite app_assoc in H. rewrite <- tm_step_lemma in H.
assert (H1': hd' ++ a' = tm_morphism (firstn (Nat.div2 (length (hd' ++ a')))
(tm_step n))).
generalize H0'. generalize H. apply tm_morphism_app2.
rewrite <- app_assoc in H. symmetry in H1'.
assert (H2': even (length (hd' ++ a' ++ (map negb (rev a')))) = true).
rewrite app_length. rewrite Nat.even_add. rewrite I'.
rewrite app_length. rewrite Nat.even_add. rewrite J'. rewrite K'.
reflexivity.
assert (H3': tl' = tm_morphism (skipn
(Nat.div2 (length (hd' ++ a' ++ (map negb (rev a')))))
(tm_step n))).
replace (hd' ++ a' ++ (map negb (rev a')) ++ tl')
with ((hd' ++ a' ++ (map negb (rev a'))) ++ tl') in H.
generalize H2'. generalize H. apply tm_morphism_app3.
rewrite <- app_assoc. rewrite <- app_assoc. reflexivity.
assert (H4': hd' = tm_morphism (firstn (Nat.div2 (length hd')) (tm_step n))).
generalize I'. generalize H. apply tm_morphism_app2.
assert (H5': a' = tm_morphism (skipn (Nat.div2 (length hd'))
(firstn (Nat.div2 (length (hd' ++ a'))) (tm_step n)))).
generalize I'. generalize H1'. apply tm_morphism_app3.
rewrite skipn_firstn_comm in H5'. rewrite app_length in H5'.
replace (Nat.div2 (length hd' + length a'))
with ((length hd') / 2 + Nat.div2 (length a')) in H5'.
rewrite <- Nat.div2_div in H5'. rewrite Nat.add_sub_swap in H5'.
rewrite Nat.sub_diag in H5'. rewrite Nat.add_0_l in H5'.
rewrite H4' in H. rewrite H5' in H. rewrite H3' in H.
rewrite tm_morphism_rev2 in H.
rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_eq in H.
rewrite app_length in H. rewrite app_length in H.
rewrite map_length in H. rewrite rev_length in H.
replace (length a' + length a') with ((length a')*2) in H.
replace (Nat.div2 (length hd' + length a' * 2))
with ((length hd' + length a' * 2) / 2) in H.
rewrite Nat.div_add in H. rewrite <- Nat.div2_div in H.
Admitted.