baruchel/coqbooks
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Update

Browse Source
This commit is contained in:
Thomas Baruchel 2023-01-15 16:24:02 +01:00
parent 141c4ba029
commit 3e96d39b15
1 changed files with 9 additions and 73 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

82
src/thue_morse2.v
View File

@ -1061,17 +1061,17 @@ Proof.
Qed.
Lemma tm_step_square_pos_even'' : forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
-> 0 < length a
-> odd (length hd) = true
-> even (length a) = true
-> exists (hd' a' tl' : list bool),
tm_step n = hd' ++ a' ++ a' ++ tl'
tm_step (Nat.pred n) = hd' ++ a' ++ a' ++ tl'
/\ odd (length hd') = true
/\ even (length a') = true
/\ length a' < length a
/\ 0 < length a'.
/\ length a' < length a.
Proof.
intros n hd a tl. intros H I J K.
@ -1166,11 +1166,8 @@ Proof.
exists (firstn (Nat.div2 (length a''))
(skipn (Nat.div2 (length hd'')) (tm_step n))).
exists (skipn (Nat.div2 (length (a'' ++ a'')))
(skipn (Nat.div2 (length hd'')) (tm_step n))
++ map negb (tm_step n)).
split. rewrite tm_build. rewrite M1 at 1.
rewrite <- app_assoc. rewrite <- app_assoc.
rewrite <- app_assoc. reflexivity.
(skipn (Nat.div2 (length hd'')) (tm_step n))).
split. rewrite <- pred_Sn. rewrite M1 at 1. reflexivity.
assert (Nat.div2 (length hd'') <= length (tm_step n)).
rewrite Nat.mul_le_mono_pos_l with (p := 2).
@ -1218,8 +1215,6 @@ Proof.
apply Nat.lt_0_2. apply Nat.eq_le_incl. reflexivity.
assumption.
split.
rewrite firstn_length.
assert (min (Nat.div2 (length a''))
(length (skipn (Nat.div2 (length hd'')) (tm_step n)))
@ -1229,41 +1224,13 @@ Proof.
apply Nat.lt_div2. assumption.
generalize H3. generalize H2. apply Nat.le_lt_trans.
rewrite firstn_length_le. rewrite M3.
rewrite Nat.mul_lt_mono_pos_l with (p := 2).
rewrite Nat.mul_0_r. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.lt_0_2.
rewrite skipn_length.
rewrite Nat.add_le_mono_l with (p := Nat.div2 (length hd'')).
rewrite Nat.add_sub_assoc. rewrite Nat.add_sub_swap.
rewrite Nat.sub_diag. rewrite Nat.add_0_l.
rewrite Nat.mul_le_mono_pos_l with (p := 2).
rewrite tm_size_power2. rewrite Nat.mul_add_distr_l.
rewrite <- Nat.double_twice. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. rewrite <- Nat.Even_double.
rewrite <- Nat.pow_succ_r'. rewrite <- tm_size_power2. rewrite M0.
rewrite app_assoc. rewrite app_length. rewrite <- Nat.add_0_r at 1.
apply Nat.add_le_mono. rewrite app_length. apply Nat.eq_le_incl.
reflexivity.
apply le_0_n.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.lt_0_2. apply Nat.eq_le_incl. reflexivity.
assumption.
+ (* case Nat.div2 (length hd') has an odd length *)
exists (firstn (Nat.div2 (length hd')) (tm_step n)).
exists (firstn (Nat.div2 (length a'))
(skipn (Nat.div2 (length hd')) (tm_step n))).
exists (skipn (Nat.div2 (length (a' ++ a')))
(skipn (Nat.div2 (length hd')) (tm_step n))
++ map negb (tm_step n)).
split. rewrite tm_build. rewrite L1 at 1.
rewrite <- app_assoc. rewrite <- app_assoc.
rewrite <- app_assoc. reflexivity.
(skipn (Nat.div2 (length hd')) (tm_step n))).
split. rewrite <- pred_Sn. rewrite L1 at 1. reflexivity.
assert (Nat.div2 (length hd') <= length (tm_step n)).
rewrite Nat.mul_le_mono_pos_l with (p := 2).
@ -1302,8 +1269,6 @@ Proof.
apply Nat.lt_0_2. apply Nat.eq_le_incl. reflexivity.
assumption.
split.
rewrite firstn_length.
assert (min (Nat.div2 (length a'))
(length (skipn (Nat.div2 (length hd')) (tm_step n)))
@ -1312,31 +1277,6 @@ Proof.
assert (Nat.div2 (length a') < length a). rewrite L3.
apply Nat.lt_div2. assumption.
generalize H3. generalize H2. apply Nat.le_lt_trans.
rewrite firstn_length_le. rewrite L3.
rewrite Nat.mul_lt_mono_pos_l with (p := 2).
rewrite Nat.mul_0_r. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.lt_0_2.
rewrite skipn_length.
rewrite Nat.add_le_mono_l with (p := Nat.div2 (length hd')).
rewrite Nat.add_sub_assoc. rewrite Nat.add_sub_swap.
rewrite Nat.sub_diag. rewrite Nat.add_0_l.
rewrite Nat.mul_le_mono_pos_l with (p := 2).
rewrite tm_size_power2. rewrite Nat.mul_add_distr_l.
rewrite <- Nat.double_twice. rewrite <- Nat.double_twice.
rewrite <- Nat.Even_double. rewrite <- Nat.Even_double.
rewrite <- Nat.pow_succ_r'. rewrite <- tm_size_power2. rewrite L0.
rewrite app_assoc. rewrite app_length. rewrite <- Nat.add_0_r at 1.
apply Nat.add_le_mono. rewrite app_length. apply Nat.eq_le_incl.
reflexivity.
apply le_0_n.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
apply Nat.lt_0_2. apply Nat.eq_le_incl. reflexivity.
assumption.
- (* second case, a'0 is odd and we are looking for an even prefix *)
assert (Nat.Even (Nat.div2 (length hd')) \/ Nat.Odd (Nat.div2 (length hd'))).
apply Nat.Even_or_Odd. destruct H0.
@ -1434,9 +1374,6 @@ Qed.
(*
Lemma tm_step_square_pos_even'' : forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl
@ -1444,11 +1381,10 @@ Lemma tm_step_square_pos_even'' : forall (n : nat) (hd a tl : list bool),
-> odd (length hd) = true
-> even (length a) = true
-> exists (hd' a' tl' : list bool),
tm_step n = hd' ++ a' ++ a' ++ tl'
tm_step (Nat.pred n) = hd' ++ a' ++ a' ++ tl'
/\ odd (length hd') = true
/\ even (length a') = true
/\ length a' < length a
/\ 0 < length a'.
/\ length a' < length a.
*)