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 2022-11-25 11:52:58 +01:00
parent a3f15b3984
commit ef846b4fdc
1 changed files with 15 additions and 157 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

172
thue-morse.v
View File

@ -1113,169 +1113,27 @@ Lemma tm_step_next_range2_neighbor' : forall (n k : nat),
- induction m. - induction m.
+ intros L. rewrite Nat.add_0_r in L. rewrite Nat.add_0_r in L. + intros L. rewrite Nat.add_0_r in L. rewrite Nat.add_0_r in L.
apply tm_step_next_range2_neighbor'. apply H. apply L. apply tm_step_next_range2_neighbor'. apply H. apply L.
+ + intros L.
rewrite Nat.add_succ_r in L. rewrite Nat.add_succ_r in L.
assert (S k < 2^(S n + m)).
assert (2^n < 2^(S n + m)).
assert (n < S n + m). assert (n < S n). apply Nat.lt_succ_diag_r.
generalize H0. apply Nat.lt_lt_add_r.
generalize H0. apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
generalize H0. generalize H. apply Nat.lt_trans.
assert (H1 := H0). apply Nat.lt_succ_l in H1.
rewrite <- tm_step_next_range2_neighbor' in L.
Theorem tm_step_stable : forall (n m k : nat),
k < 2^n -> k < 2^m -> nth_error(tm_step n) k = nth_error (tm_step m) k.
rewrite <- Nat.add_succ_l.
replace (2^(n + S m)) with (2^(n+m) + 2^(n+m)).
rewrite Nat.add_assoc.
rewrite Nat.add_assoc.
(* rewrite Nat.add_succ_r. *)
generalize L.
apply tm_step_next_range2_neighbor'. rewrite <- Nat.add_succ_l.
- assert (k < length (tm_step n)). rewrite tm_size_power2. apply I.
apply nth_error_nth' with (d :=false) in H0.
assert (S k < length (tm_step n)). rewrite tm_size_power2. apply H.
apply nth_error_nth' with (d :=false) in H1.
intros J. rewrite H0 in J. rewrite H1 in J.
apply tm_step_next_range2' in H0. apply tm_step_next_range2' in H1.
induction m.
+ rewrite Nat.add_0_r. rewrite Nat.add_0_r.
rewrite H0. rewrite H1. inversion J. rewrite H3. reflexivity.
+
assert (U: S k < 2^(S n+ m)).
assert (N: k < 2^n). apply I.
assert (L: 2^n < 2^(S n + m)).
assert (M: n < S n + m). rewrite Nat.add_succ_comm.
apply Nat.lt_add_pos_r. apply Nat.lt_0_succ.
apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. apply M.
generalize L. generalize H. apply Nat.lt_trans.
assert (K := U). apply Nat.lt_succ_l in K.
assert (nth_error (tm_step n) k = Some (nth k (tm_step n) false)).
apply nth_error_nth'. rewrite tm_size_power2. apply I.
assert (nth_error (tm_step n) (S k) = Some (nth (S k) (tm_step n) false)).
apply nth_error_nth'. rewrite tm_size_power2. apply H.
rewrite <- H2 in J. rewrite <- H3 in J.
assert (nth_error (tm_step n) k = nth_error (tm_step (S n + m)) k). assert (nth_error (tm_step n) k = nth_error (tm_step (S n + m)) k).
generalize K. generalize I. apply tm_step_stable. generalize H1. generalize I. apply tm_step_stable.
assert (nth_error (tm_step n) (S k) = nth_error (tm_step (S n + m)) (S k)). assert (nth_error (tm_step n) (S k) = nth_error (tm_step (S n + m)) (S k)).
generalize U. generalize H. apply tm_step_stable. generalize H0. generalize H. apply tm_step_stable.
rewrite <- H2 in L. rewrite <- H3 in L. apply L.
rewrite H4 in J. rewrite H5 in J. apply H0.
Qed.
assert (nth_error (tm_step (S n + m)) k = Some (nth k (tm_step (S n + m)) false)).
generalize K. rewrite <- tm_size_power2. apply nth_error_nth'.
assert (nth_error (tm_step (S n + m)) (S k) = Some (nth (S k) (tm_step (S n + m)) false)).
generalize U. rewrite <- tm_size_power2. apply nth_error_nth'.
rewrite H6 in J. rewrite H7 in J.
rewrite Nat.add_succ_comm in J. inversion J.
assert ( S k + 2^(n + S m) < 2^(S n + S m)
S k < 2^n ->
eqb (nth k (tm_step n) false) (nth (S k) (tm_step n) false)
= eqb
partir de H9 et appliquer :
Lemma tm_step_next_range2_neighbor : forall (n k : nat),
S k < 2^n ->
eqb (nth k (tm_step n) false) (nth (S k) (tm_step n) false)
= eqb
(nth (k + 2^n) (tm_step (S n)) false) (nth (S k + 2^n) (tm_step (S n)) false).
assert (nth_error (tm_step (S n + m)) (k + 2 ^ (n + m))
= nth_error (tm_step (S n + m)) (k + 2 ^ (n + m))
Theorem tm_step_stable : forall (n m k : nat),
k < 2^n -> k < 2^m -> nth_error(tm_step n) k = nth_error (tm_step m) k.
assert (U: S k < 2^(S n+ m)).
assert (N: k < 2^n). apply I.
assert (L: 2^n < 2^(S n + m)).
assert (M: n < S n + m). rewrite Nat.add_succ_comm.
apply Nat.lt_add_pos_r. apply Nat.lt_0_succ.
apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. apply M.
generalize L. generalize H. apply Nat.lt_trans.
assert (K := U). apply Nat.lt_succ_l in K.
assert (k < length (tm_step (S n + m))).
rewrite tm_size_power2. apply K.
apply nth_error_nth' with (d :=false) in H2.
assert (S k < length (tm_step (S n + m))).
rewrite tm_size_power2. apply U.
apply nth_error_nth' with (d :=false) in H3.
apply tm_step_next_range2' in H2. apply tm_step_next_range2' in H3.
rewrite Nat.add_succ_comm in H2. rewrite <- Nat.add_succ_l in H2.
rewrite Nat.add_succ_comm in H3. rewrite <- Nat.add_succ_l in H3.
rewrite H2. rewrite H3.
assert (nth_error (tm_step (n + S m)) k
= Some (nth k (tm_step (n + S m)) false)).
rewrite <- Nat.add_succ_comm. generalize K. rewrite <- tm_size_power2.
apply nth_error_nth'.
assert (nth_error (tm_step (n + S m)) (S k)
= Some (nth (S k) (tm_step (n + S m)) false)).
rewrite <- Nat.add_succ_comm. generalize U. rewrite <- tm_size_power2.
apply nth_error_nth'.
Some (negb (nth (S k) (tm_step (n + S m)) false))
Theorem tm_step_stable : forall (n m k : nat),
k < 2^n -> k < 2^m -> nth_error(tm_step n) k = nth_error (tm_step m) k.
nth_error_nth':
forall [A : Type] (l : list A) [n : nat] (d : A),
n < length l -> nth_error l n = Some (nth n l d)
assert (V: S k < 2^(n+m)).
assert (N: k < 2^n). apply I.
assert (L: 2^n <= 2^(n + m)).
assert (M: n <= n + m). apply Nat.le_add_r.
apply Nat.pow_le_mono_r. easy. apply M.
generalize L. generalize H. apply Nat.lt_le_trans.
assert (L := V). apply Nat.lt_succ_l in L.
assert (k < length (tm_step (n + S m))).
rewrite tm_size_power2. apply L.
apply nth_error_nth' with (d :=false) in H2.
assert (S k < length (tm_step (n + S m))).
rewrite tm_size_power2. apply V.
apply nth_error_nth' with (d :=false) in H3.
rewrite <- Nat.add_succ_comm in V.
rewrite <- Nat.add_succ_comm in L.
assert (k < length (tm_step (S n + S m))).
rewrite tm_size_power2. apply K.
apply nth_error_nth' with (d :=false) in H2.
assert (S k < length (tm_step (S n + S m))).
rewrite tm_size_power2. apply U.
apply nth_error_nth' with (d :=false) in H3.
rewrite H2 in H3.
intros J. rewrite H0 in J. rewrite H1 in J. inversion J.
induction m. rewrite Nat.add_0_r. rewrite Nat.add_0_r.
rewrite Bool.eqb_true_iff
Lemma tm_step_next_range2_neighbor : forall (n k : nat), Lemma tm_step_next_range2_neighbor : forall (n k : nat),