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7c7bc16120
coqbooks/thue-morse.v
Thomas Baruchel 7c7bc16120 Update
2022-12-28 18:47:14 +01:00

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(*
https://oeis.org/A010060
https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence
*)
Require Import Coq.Lists.List.
Require Import PeanoNat.
Require Import Nat.
Require Import Bool.
Import ListNotations.
Fixpoint tm_morphism (l:list bool) : list bool :=
match l with
| nil => []
| h :: t => h :: (negb h) :: (tm_morphism t)
end.
Fixpoint tm_step (n: nat) : list bool :=
match n with
| 0 => false :: nil
| S n' => tm_morphism (tm_step n')
end.
(* ad hoc more or less general lemmas *)
Lemma negb_map_explode : forall (l1 l2 : list bool),
map negb (l1 ++ l2) = map negb l1 ++ map negb l2.
Proof.
intros l1 l2.
induction l1.
- reflexivity.
- simpl. rewrite IHl1. reflexivity.
Qed.
Lemma negb_double_map : forall (l : list bool),
map negb (map negb l) = l.
Proof.
intro l.
induction l.
- reflexivity.
- simpl. rewrite IHl. replace (negb (negb a)) with (a).
reflexivity. rewrite negb_involutive. reflexivity.
Qed.
Lemma lt_split_bits : forall n m k j,
0 < k -> j < 2^m -> k*2^m < 2^n -> k*2^m+j < 2^n.
Proof.
intros n m k j. intros H I J.
assert (K: 2^m <= k*2^m). rewrite <- Nat.mul_1_l at 1.
apply Nat.mul_le_mono_r. rewrite Nat.le_succ_l. assumption.
assert (L:2^m < 2^n). generalize J. generalize K. apply Nat.le_lt_trans.
assert (k < 2^(n-m)). rewrite Nat.mul_lt_mono_pos_r with (p := 2^m).
rewrite <- Nat.pow_add_r. rewrite Nat.sub_add. assumption.
apply Nat.pow_lt_mono_r_iff in L. apply Nat.lt_le_incl. assumption.
apply Nat.lt_1_2. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
replace (2^(n-m)) with (S (2^(n-m)-1)) in H0. rewrite Nat.lt_succ_r in H0.
apply Nat.mul_le_mono_r with (p := 2^m) in H0.
rewrite Nat.mul_sub_distr_r in H0. rewrite Nat.mul_1_l in H0.
rewrite <- Nat.pow_add_r in H0. rewrite Nat.sub_add in H0.
rewrite Nat.add_le_mono_r with (p := j) in H0.
assert (2^n - 2^m + j < 2^n).
rewrite Nat.add_lt_mono_l with (p := 2^n) in I.
rewrite Nat.add_lt_mono_r with (p := 2^m).
rewrite <- Nat.add_assoc. rewrite <- Nat.add_sub_swap.
rewrite Nat.add_assoc. rewrite Nat.add_sub. assumption.
apply Nat.lt_le_incl. assumption.
generalize H1. generalize H0. apply Nat.le_lt_trans.
apply Nat.lt_le_incl. rewrite <- Nat.pow_lt_mono_r_iff in L. assumption.
apply Nat.lt_1_2. rewrite <- Nat.add_1_r. apply Nat.sub_add.
rewrite Nat.le_succ_l. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
Qed.
Lemma tm_morphism_concat : forall (l1 l2 : list bool),
tm_morphism (l1 ++ l2) = tm_morphism l1 ++ tm_morphism l2.
Proof.
intros l1 l2.
induction l1.
- reflexivity.
- simpl. rewrite IHl1. reflexivity.
Qed.
Lemma tm_morphism_rev : forall (l : list bool),
rev (tm_morphism l) = tm_morphism (map negb (rev l)).
Proof.
intro l. induction l.
- reflexivity.
- simpl. rewrite negb_map_explode.
rewrite IHl. rewrite tm_morphism_concat.
rewrite <- app_assoc. simpl.
rewrite negb_involutive. reflexivity.
Qed.
Lemma tm_morphism_rev' : forall (l : list bool),
rev (tm_morphism l) = map negb (tm_morphism (rev l)).
Proof.
intro l. rewrite tm_morphism_rev. induction l.
- reflexivity.
- simpl. rewrite map_app. rewrite tm_morphism_concat. rewrite IHl.
rewrite tm_morphism_concat. rewrite map_app. reflexivity.
Qed.
Lemma tm_morphism_double_index : forall (l : list bool) (k : nat),
nth_error l k = nth_error (tm_morphism l) (2*k).
Proof.
intro l.
induction l.
- intro k.
simpl. replace (nth_error [] k) with (None : option bool).
rewrite Nat.add_0_r. replace (nth_error [] (k+k)) with (None : option bool).
reflexivity. symmetry. apply nth_error_None. simpl. apply Nat.le_0_l.
symmetry. apply nth_error_None. simpl. apply Nat.le_0_l.
- intro k. induction k.
+ rewrite Nat.mul_0_r. reflexivity.
+ simpl. rewrite Nat.add_0_r. rewrite Nat.add_succ_r. simpl.
replace (k+k) with (2*k). apply IHl. simpl. rewrite Nat.add_0_r.
reflexivity.
Qed.
Lemma tm_build_negb : forall (l : list bool),
tm_morphism (map negb l) = map negb (tm_morphism l).
Proof.
intro l.
induction l.
- reflexivity.
- simpl. rewrite IHl. reflexivity.
Qed.
(* Thue-Morse related lemmas and theorems *)
Lemma tm_step_lemma : forall n : nat,
tm_morphism (tm_step n) = tm_step (S n).
Proof.
intro n. reflexivity.
Qed.
Theorem tm_step_negb : forall (n : nat),
map negb (tm_step (S n)) = rev (tm_morphism (rev (tm_step n))).
Proof.
intro n.
rewrite <- tm_step_lemma. symmetry.
replace (tm_step n) with (rev (rev (tm_step n))).
rewrite rev_involutive. rewrite tm_morphism_rev'. reflexivity.
rewrite rev_involutive. reflexivity.
Qed.
Theorem tm_build : forall (n : nat),
tm_step (S n) = tm_step n ++ map negb (tm_step n).
Proof.
intro n.
induction n.
- reflexivity.
- simpl. rewrite tm_step_lemma. rewrite IHn. rewrite tm_morphism_concat.
simpl in IHn. rewrite IHn. rewrite tm_build_negb. rewrite IHn.
rewrite negb_map_explode. rewrite negb_double_map.
reflexivity.
Qed.
Theorem tm_size_power2 : forall n : nat, length (tm_step n) = 2^n.
Proof.
intro n.
induction n.
- reflexivity.
- rewrite tm_build. rewrite app_length. rewrite map_length.
replace (2^S n) with (2^n + 2^n). rewrite IHn. reflexivity.
simpl. rewrite <- plus_n_O. reflexivity.
Qed.
Lemma tm_step_head_2 : forall (n : nat),
tm_step (S n) = false :: true :: tl (tl (tm_step (S n))).
Proof.
intro n.
induction n.
- reflexivity.
- simpl. replace (tm_morphism (tm_step n)) with (tm_step (S n)).
rewrite IHn. reflexivity. reflexivity.
Qed.
Lemma tm_step_end_2 : forall (n : nat),
rev (tm_step (S n)) = even n :: odd n :: tl (tl (rev (tm_step (S n)))).
Proof.
intro n. induction n.
- reflexivity.
- simpl tm_step. rewrite tm_morphism_rev.
replace (tm_morphism (tm_step n)) with (tm_step (S n)).
rewrite IHn. simpl tm_morphism. simpl tl.
rewrite Nat.even_succ.
rewrite Nat.odd_succ.
rewrite negb_involutive.
reflexivity. reflexivity.
Qed.
Lemma tm_step_head_1 : forall (n : nat),
tm_step n = false :: tl (tm_step n).
Proof.
intro n.
destruct n.
- reflexivity.
- rewrite tm_step_head_2. reflexivity.
Qed.
Lemma tm_step_end_1 : forall (n : nat),
rev (tm_step n) = odd n :: tl (rev (tm_step n)).
Proof.
intro n.
destruct n.
- reflexivity.
- rewrite tm_step_end_2. simpl.
rewrite Nat.odd_succ.
reflexivity.
Qed.
Theorem tm_step_double_index : forall (n k : nat),
nth_error (tm_step n) k = nth_error (tm_step (S n)) (2*k).
Proof.
intros n k.
destruct k.
- rewrite Nat.mul_0_r. rewrite tm_step_head_1. simpl.
rewrite tm_step_head_1. reflexivity.
- rewrite <- tm_step_lemma.
rewrite tm_morphism_double_index. reflexivity.
Qed.
Lemma tm_step_single_bit_index : forall (n : nat),
nth_error (tm_step (S n)) (2^n) = Some true.
Proof.
intro n.
rewrite tm_build.
rewrite nth_error_app2. rewrite tm_size_power2. rewrite Nat.sub_diag.
replace (true) with (negb false). apply map_nth_error.
rewrite tm_step_head_1. reflexivity.
reflexivity. rewrite tm_size_power2. easy.
Qed.
Lemma tm_step_repunit_index : forall (n : nat),
nth_error (tm_step n) (2^n - 1) = Some (odd n).
Proof.
intro n.
assert (H: 2 ^ n - 1 < length (tm_step n)). rewrite tm_size_power2.
rewrite Nat.sub_1_r. apply Nat.lt_pred_l. apply Nat.pow_nonzero. easy.
rewrite nth_error_nth' with (d := false).
replace (tm_step n) with (rev (rev (tm_step n))).
rewrite rev_nth. rewrite rev_length. rewrite tm_size_power2.
rewrite Nat.sub_1_r. rewrite Nat.succ_pred_pos. rewrite Nat.sub_diag.
rewrite tm_step_end_1. reflexivity.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
rewrite rev_length. assumption. apply rev_involutive. assumption.
Qed.
Lemma list_app_length_lt : forall (l l1 l2 : list bool) (b : bool),
l = l1 ++ b :: l2 -> length l1 < length l.
Proof.
intros l l1 l2 b. intro H. rewrite H.
rewrite app_length. simpl. apply Nat.lt_add_pos_r.
apply Nat.lt_0_succ.
Qed.
Lemma tm_step_next_range :
forall (n k : nat) (b : bool),
nth_error (tm_step n) k = Some b -> nth_error (tm_step (S n)) k = Some b.
Proof.
intros n k b. intro H. assert (I := H).
apply nth_error_split in H. destruct H. destruct H. inversion H.
rewrite tm_build. rewrite nth_error_app1. apply I.
apply list_app_length_lt in H0. rewrite H1 in H0. apply H0.
Qed.
Lemma tm_step_add_range : forall (n m k : nat),
k < 2^n -> nth_error (tm_step n) k = nth_error (tm_step (n+m)) k.
Proof.
intros n m k. intro H.
induction m.
- rewrite Nat.add_comm. reflexivity.
- rewrite Nat.add_succ_r. rewrite <- tm_size_power2 in H.
assert (nth_error (tm_step n) k = Some (nth k (tm_step n) false)).
generalize H. apply nth_error_nth'.
rewrite H0 in IHm. symmetry in IHm.
rewrite H0. symmetry. generalize IHm.
apply tm_step_next_range.
Qed.
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.
Proof.
intros n m k. intros.
assert (I: n < m /\ max n m = m \/ m <= n /\ max n m = n).
apply Nat.max_spec. destruct I.
- destruct H1. replace (m) with (n + (m-n)). apply tm_step_add_range. apply H.
apply Nat.lt_le_incl in H1. replace (m) with (n + m - n) at 2. generalize H1.
apply Nat.add_sub_assoc. rewrite Nat.add_comm.
assert (n <= n). apply le_n. symmetry.
replace (m) with (m + (n-n)) at 1. generalize H3.
apply Nat.add_sub_assoc. rewrite Nat.sub_diag. rewrite Nat.add_comm.
reflexivity.
- destruct H1. symmetry. replace (n) with (m + (n - m)). apply tm_step_add_range.
apply H0. replace (n) with (m + n - m) at 2. generalize H1.
apply Nat.add_sub_assoc. rewrite Nat.add_comm.
assert (m <= m). apply le_n. symmetry.
replace (n) with (n + (m-m)) at 1. generalize H3.
apply Nat.add_sub_assoc. rewrite Nat.sub_diag. rewrite Nat.add_comm.
reflexivity.
Qed.
Lemma tm_step_next_range2 :
forall (n k : nat),
k < 2^n -> nth_error (tm_step n) k <> nth_error (tm_step (S n)) (k + 2^n).
Proof.
intros n k. intro H.
rewrite tm_build.
rewrite nth_error_app2. rewrite tm_size_power2. rewrite Nat.add_sub.
assert (nth_error (tm_step n) k = Some (nth k (tm_step n) false)).
generalize H. rewrite <- tm_size_power2. apply nth_error_nth'.
rewrite H0.
assert (nth_error (map negb (tm_step n)) k
= Some (negb (nth k (tm_step n) false))).
generalize H0. apply map_nth_error. rewrite H1.
destruct (nth k (tm_step n) false). easy. easy.
rewrite tm_size_power2. apply Nat.le_add_l.
Qed.
Lemma tm_step_next_range2_flip_two : forall (n k m : nat),
k < 2^n -> m < 2^n ->
nth_error (tm_step n) k = nth_error (tm_step n) m
<->
nth_error (tm_step (S n)) (k + 2^n)
= nth_error (tm_step (S n)) (m + 2^n).
Proof.
intros n k m. intro H. intro I.
(* Part 1: preamble *)
assert (nth_error (tm_step n) k <> nth_error (tm_step (S n)) (k + 2^n)).
generalize H. apply tm_step_next_range2.
assert (nth_error (tm_step n) m <> nth_error (tm_step (S n)) (m + 2^n)).
generalize I. apply tm_step_next_range2.
assert (K: k + 2^n < 2^(S n)). simpl. rewrite Nat.add_0_r.
generalize H. apply Nat.add_lt_mono_r.
assert (J: m + 2^n < 2^(S n)). simpl. rewrite Nat.add_0_r.
generalize I. apply Nat.add_lt_mono_r.
(* Part 2: main proof *)
assert (nth_error (tm_step n) k = Some (nth k (tm_step n) false)).
generalize H. rewrite <- tm_size_power2. apply nth_error_nth'.
assert (nth_error (tm_step n) m = Some (nth m (tm_step n) false)).
generalize I. rewrite <- tm_size_power2. apply nth_error_nth'.
assert (nth_error (tm_step (S n)) (k + 2 ^ n) =
Some (nth (k + 2^n) (tm_step (S n)) false)).
generalize K. rewrite <- tm_size_power2. rewrite <- tm_size_power2.
apply nth_error_nth'.
assert (nth_error (tm_step (S n)) (m + 2 ^ n) =
Some (nth (m + 2^n) (tm_step (S n)) false)).
generalize J. rewrite <- tm_size_power2. rewrite <- tm_size_power2.
apply nth_error_nth'.
rewrite H2. rewrite H3. rewrite H4. rewrite H5.
(* Part 3: handling 16 different cases *)
destruct (nth k (tm_step n) false).
destruct (nth m (tm_step n) false).
destruct (nth (k + 2 ^ n) (tm_step (S n)) false).
destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
easy.
rewrite H2 in H0. rewrite H4 in H0. contradiction H0. reflexivity.
destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
rewrite H3 in H1. rewrite H5 in H1. contradiction H1. reflexivity.
easy.
destruct (nth (k + 2 ^ n) (tm_step (S n)) false).
destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
rewrite H2 in H0. rewrite H4 in H0. contradiction H0. reflexivity.
easy.
destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
easy.
rewrite H3 in H1. rewrite H5 in H1. contradiction H1. reflexivity.
destruct (nth m (tm_step n) false).
destruct (nth (k + 2 ^ n) (tm_step (S n)) false).
destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
rewrite H3 in H1. rewrite H5 in H1. contradiction H1. reflexivity.
easy.
destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
easy.
rewrite H2 in H0. rewrite H4 in H0. contradiction H0. reflexivity.
destruct (nth (k + 2 ^ n) (tm_step (S n)) false).
destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
easy.
rewrite H3 in H1. rewrite H5 in H1. contradiction H1. reflexivity.
destruct (nth (m + 2 ^ n) (tm_step (S n)) false).
rewrite H2 in H0. rewrite H4 in H0. contradiction H0. reflexivity.
easy.
Qed.
Lemma tm_step_add_range2_flip_two : forall (n m j k : nat),
k < 2^n -> m < 2^n ->
nth_error (tm_step n) k = nth_error (tm_step n) m
<->
nth_error (tm_step (S n+j)) (k + 2^(n+j))
= nth_error (tm_step (S n+j)) (m + 2^(n+j)).
Proof.
intros n m j k. intros H I.
assert (K: k + 2^n < 2^(S n)). simpl. rewrite Nat.add_0_r.
generalize H. apply Nat.add_lt_mono_r.
assert (J: m + 2^n < 2^(S n)). simpl. rewrite Nat.add_0_r.
generalize I. apply Nat.add_lt_mono_r.
split.
- induction j.
+ intros L. rewrite Nat.add_0_r. rewrite Nat.add_0_r.
rewrite <- tm_step_next_range2_flip_two. apply L. apply H. apply I.
+ intros L.
rewrite Nat.add_succ_r. rewrite Nat.add_succ_r.
assert (2^n < 2^(S n + j)).
assert (n < S n + j). 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.
assert (k < 2^(S n + j)).
generalize H0. generalize H. apply Nat.lt_trans.
assert (m < 2^(S n + j)).
generalize H0. generalize I. apply Nat.lt_trans.
rewrite <- tm_step_next_range2_flip_two.
assert (nth_error (tm_step n) k = nth_error (tm_step (S n + j)) k).
generalize H1. generalize H. apply tm_step_stable.
assert (nth_error (tm_step n) m = nth_error (tm_step (S n + j)) m).
generalize H2. generalize I. apply tm_step_stable.
rewrite <- H3. rewrite <- H4. apply L.
apply H1. apply H2.
- induction j.
+ intro L. rewrite Nat.add_0_r in L. rewrite Nat.add_0_r in L.
apply tm_step_next_range2_flip_two. apply H. apply I. apply L.
+ intro L.
rewrite Nat.add_succ_r in L. rewrite Nat.add_succ_r in L.
assert (2^n < 2^(S n + j)).
assert (n < S n + j). 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.
assert (k < 2^(S n + j)).
generalize H0. generalize H. apply Nat.lt_trans.
assert (m < 2^(S n + j)).
generalize H0. generalize I. apply Nat.lt_trans.
rewrite <- tm_step_next_range2_flip_two in L.
assert (nth_error (tm_step n) k = nth_error (tm_step (S n + j)) k).
generalize H1. generalize H. apply tm_step_stable.
assert (nth_error (tm_step n) m = nth_error (tm_step (S n + j)) m).
generalize H2. generalize I. apply tm_step_stable.
rewrite <- H3 in L. rewrite <- H4 in L. apply L.
apply H1. apply H2.
Qed.
Lemma tm_step_repeating_patterns :
forall (n m i j : nat),
i < 2^m -> j < 2^m
-> forall k, k < 2^n -> nth_error (tm_step m) i
= nth_error (tm_step m) j
<-> nth_error (tm_step (m+n)) (k * 2^m + i)
= nth_error (tm_step (m+n)) (k * 2^m + j).
Proof.
intros n m i j. intros H I.
induction n.
- rewrite Nat.add_0_r. intro k. simpl. rewrite Nat.lt_1_r. intro.
rewrite H0. simpl. easy.
- rewrite Nat.add_succ_r. intro k. intro.
rewrite tm_build.
assert (S: k < 2^n \/ 2^n <= k). apply Nat.lt_ge_cases.
destruct S.
assert (k*2^m < 2^(m+n)).
destruct k.
+ simpl. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
+ rewrite Nat.mul_lt_mono_pos_r with (p := 2^m) in H1.
rewrite <- Nat.pow_add_r in H1. rewrite Nat.add_comm.
assumption. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
+ assert (L: forall l, l < 2^m -> k*2^m + l < length (tm_step (m+n))).
intro l. intro M. rewrite tm_size_power2.
destruct k. simpl. assert (2^m <= 2^(m+n)).
apply Nat.pow_le_mono_r. easy. apply Nat.le_add_r.
generalize H3. generalize M. apply Nat.lt_le_trans.
apply lt_split_bits. apply Nat.lt_0_succ. assumption.
assumption.
rewrite nth_error_app1. rewrite nth_error_app1.
generalize H1. apply IHn.
apply L. assumption. apply L. assumption.
+ assert (J: 2 ^ (m + n) <= k * 2 ^ m).
rewrite Nat.pow_add_r. rewrite Nat.mul_comm.
apply Nat.mul_le_mono_r. assumption.
rewrite nth_error_app2. rewrite nth_error_app2. rewrite tm_size_power2.
assert (forall a b, option_map negb a = option_map negb b <-> a = b).
intros a b. destruct a. destruct b. destruct b0; destruct b; simpl.
split; intro; reflexivity. split; intro; inversion H2.
split; intro; inversion H2. split; intro; reflexivity.
split; intro; inversion H2.
destruct b. split; intro; inversion H2. split; intro; reflexivity.
replace (k * 2 ^ m + i - 2^(m + n)) with ((k-2^n)*2^m + i).
replace (k * 2 ^ m + j - 2^(m + n)) with ((k-2^n)*2^m + j).
rewrite nth_error_map. rewrite nth_error_map.
rewrite H2. apply IHn.
rewrite Nat.add_lt_mono_r with (p := 2^n). rewrite Nat.sub_add.
rewrite Nat.pow_succ_r in H0. replace (2) with (1+1) in H0.
rewrite Nat.mul_add_distr_r in H0. simpl in H0.
rewrite Nat.add_0_r in H0. assumption. reflexivity.
apply Nat.le_0_l. assumption.
rewrite Nat.mul_sub_distr_r. rewrite <- Nat.pow_add_r.
rewrite Nat.add_sub_swap. replace (n+m) with (m+n). reflexivity.
rewrite Nat.add_comm. reflexivity. assumption.
rewrite Nat.mul_sub_distr_r. rewrite <- Nat.pow_add_r.
rewrite Nat.add_sub_swap. replace (n+m) with (m+n). reflexivity.
rewrite Nat.add_comm. reflexivity. assumption.
rewrite tm_size_power2.
assert (k*2^m <= k*2^m + j). apply Nat.le_add_r.
generalize H2. generalize J. apply Nat.le_trans.
rewrite tm_size_power2.
assert (k*2^m <= k*2^m + i). apply Nat.le_add_r.
generalize H2. generalize J. apply Nat.le_trans.
Qed.
(* Note: a first version included the 0 < k hypothesis but in the very
unorthodox case where k=0 it happens to be true anyway, and the hypothesis
was removed in order to make the theorem more general *)
Theorem tm_step_flip_low_bit :
forall (n m k j : nat),
j < m -> k * 2^m < 2^n
-> nth_error (tm_step n) (k * 2^m) <> nth_error (tm_step n) (k * 2^m + 2^j).
Proof.
intros n m k j. intros H I.
assert (L: nth_error (tm_step m) 0 = Some false).
rewrite tm_step_head_1. reflexivity.
assert (M: 2^j < 2^m).
apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. assumption.
assert (N: nth_error (tm_step m) (2^j) = Some true).
replace (nth_error (tm_step m) (2^j)) with (nth_error (tm_step (S j)) (2^j)).
rewrite tm_step_single_bit_index. reflexivity.
apply tm_step_stable.
apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. apply Nat.lt_succ_diag_r.
assumption.
assert (G: k = 0 \/ 0 < k). apply Nat.eq_0_gt_0_cases. destruct G as [G1|G2].
(* k = 0 *)
rewrite G1. rewrite Nat.mul_0_l. rewrite Nat.add_0_l.
rewrite tm_step_head_1 at 1.
assert (S: n < S j \/ S j <= n). apply Nat.lt_ge_cases.
destruct S as [S1|S2]. rewrite Nat.lt_succ_r in S1.
apply Nat.pow_le_mono_r with (a := 2) in S1.
rewrite <- tm_size_power2 in S1. rewrite <- nth_error_None in S1.
rewrite S1. easy. easy.
rewrite Nat.le_succ_l in S2. apply Nat.pow_lt_mono_r with (a := 2) in S2.
rewrite tm_step_stable with (m := m). rewrite N. easy. assumption. assumption.
apply Nat.lt_1_2.
(* k < 0 *)
assert (O: k*2^m + 2^j < 2^n). apply lt_split_bits; assumption.
assert (P: 2^n <= 2^(m+n)). apply Nat.pow_le_mono_r. easy. apply Nat.le_add_l.
assert ( nth_error (tm_step m) 0
= nth_error (tm_step m) (2^j)
<-> nth_error (tm_step (m+n)) (k * 2^m + 0)
= nth_error (tm_step (m+n)) (k * 2^m + (2^j)) ).
apply tm_step_repeating_patterns.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. assumption.
assert (1 * k <= (2^m) * k). apply Nat.mul_le_mono_nonneg.
apply Nat.le_0_1. rewrite Nat.le_succ_l.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
apply Nat.le_0_l. apply Nat.le_refl. rewrite Nat.mul_1_l in H0.
rewrite Nat.mul_comm in H0.
generalize I. generalize H0. apply Nat.le_lt_trans.
rewrite Nat.add_0_r in H0.
replace (nth_error (tm_step (m + n)) (k * 2 ^ m))
with (nth_error (tm_step n) (k * 2 ^ m)) in H0.
replace (nth_error (tm_step (m + n)) (k * 2 ^ m + 2^j))
with (nth_error (tm_step n) (k * 2 ^ m + 2^j)) in H0.
rewrite L in H0. rewrite N in H0.
destruct (nth_error (tm_step n) (k * 2 ^ m)).
destruct (nth_error (tm_step n) (k * 2 ^ m + 2^j)).
destruct b; destruct b0; destruct H0.
assert (Some true = Some true). reflexivity.
apply H1 in H2. rewrite <- H2 at 1. easy. easy. easy.
assert (Some false = Some false). reflexivity.
apply H1 in H2. rewrite H2 at 1. easy. easy.
rewrite nth_error_nth' with (d := false). easy.
rewrite tm_size_power2. assumption.
apply tm_step_stable. assumption.
generalize P. generalize O. apply Nat.lt_le_trans.
apply tm_step_stable. assumption.
generalize P. generalize I. apply Nat.lt_le_trans.
Qed.
Theorem tm_step_succ_double_index : forall (n k : nat),
k < 2^n -> nth_error (tm_step n) k <> nth_error (tm_step (S n)) (S (2*k)).
Proof.
intros n k. intro H. rewrite tm_step_double_index.
rewrite Nat.mul_comm. replace (2) with (2^1).
replace (S (k*2^1)) with (k*2^1 + 2^0).
apply tm_step_flip_low_bit. apply Nat.lt_0_1.
simpl. rewrite Nat.add_0_r. replace (k*2) with (k+k).
apply Nat.add_lt_mono; assumption.
rewrite Nat.mul_comm. simpl. rewrite Nat.add_0_r. reflexivity.
simpl. rewrite Nat.add_1_r. reflexivity. reflexivity.
Qed.
Lemma tm_step_pred : forall (n k m : nat),
S (2*k) * 2^m < 2^n -> (
nth_error (tm_step n) (S (2*k) * 2^m)
= nth_error (tm_step n) (S (2*k) * 2^m - 1)
<-> odd m = true).
Proof.
intros n k m.
generalize n. induction m. rewrite Nat.pow_0_r. rewrite Nat.mul_1_r.
destruct n0. rewrite Nat.pow_0_r. rewrite Nat.lt_1_r.
intro H. apply Nat.neq_succ_0 in H. contradiction H.
rewrite Nat.odd_0. rewrite Nat.sub_succ. rewrite Nat.sub_0_r.
rewrite <- tm_step_double_index. intro H. split. intro I.
symmetry in I. apply tm_step_succ_double_index in I. contradiction I.
apply Nat.lt_succ_l in H. rewrite Nat.pow_succ_r' in H.
rewrite <- Nat.mul_lt_mono_pos_l in H. assumption. apply Nat.lt_0_2.
intro I. apply diff_false_true in I. contradiction I.
intro n0. intro I.
destruct n0. rewrite Nat.pow_0_r in I. rewrite Nat.lt_1_r in I.
rewrite Nat.mul_eq_0 in I. destruct I.
apply Nat.neq_succ_0 in H. contradiction H.
apply Nat.pow_nonzero in H. contradiction H. easy.
rewrite Nat.pow_succ_r'. rewrite Nat.mul_assoc.
replace (S (2*k) * 2) with (2* (S (2*k))).
rewrite <- Nat.mul_assoc. rewrite <- tm_step_double_index.
(* réécriture de I *)
rewrite Nat.pow_succ_r' in I.
rewrite Nat.pow_succ_r' in I.
rewrite Nat.mul_assoc in I.
replace (S (2*k) * 2) with (2* (S (2*k))) in I.
rewrite <- Nat.mul_assoc in I.
rewrite <- Nat.mul_lt_mono_pos_l in I.
assert (J := I). apply IHm in J.
assert (M: 0 < 2^m). destruct m. simpl. apply Nat.lt_0_1.
replace (0) with (0 ^ (S m)) at 1.
apply Nat.pow_lt_mono_l. apply Nat.neq_succ_0. apply Nat.lt_0_2.
apply Nat.pow_0_l. apply Nat.neq_succ_0.
assert (L: S (2 * k) * 2 ^ m - 1 < S (2 * k) * 2 ^ m).
replace (S (2 * k) * 2 ^ m) with (S (S (2 * k) * 2 ^ m - 1)) at 2.
apply Nat.lt_succ_diag_r.
rewrite <- Nat.sub_succ_l. rewrite Nat.sub_1_r. rewrite pred_Sn.
reflexivity. rewrite Nat.le_succ_l. apply Nat.mul_pos_pos.
apply Nat.lt_0_succ. assumption.
assert (N: 2 * (S (2 * k) * 2 ^ m) - 1 < length (tm_step (S n0))).
rewrite tm_size_power2.
rewrite <- Nat.le_succ_l. rewrite <- Nat.add_1_r.
rewrite Nat.sub_add. rewrite Nat.pow_succ_r'.
rewrite <- Nat.mul_le_mono_pos_l.
apply Nat.lt_le_incl. assumption. apply Nat.lt_0_2.
apply Nat.le_succ_l. apply Nat.mul_pos_pos. apply Nat.lt_0_2.
apply Nat.mul_pos_pos. apply Nat.lt_0_succ.
assumption.
assert (nth_error (tm_step n0) ((S (2 * k) * 2 ^ m) - 1)
<> nth_error (tm_step (S n0)) (S (2 * (S (2 * k) * 2 ^ m - 1)))).
apply tm_step_succ_double_index.
generalize I. generalize L. apply Nat.lt_trans.
replace (S (2 * (S (2 * k) * 2 ^ m - 1))) with (2 * (S (2*k) * 2^m) - 1) in H.
rewrite Nat.odd_succ. rewrite <- Nat.negb_odd. destruct (odd m).
split. intro K. rewrite <- K in H.
destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m));
destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m - 1)).
destruct J. replace (Some b) with (Some b0) in H. contradiction H.
reflexivity. symmetry. apply H1. reflexivity.
destruct J. replace (Some b) with (None : option bool) in H.
contradiction H. reflexivity. symmetry. apply H1. reflexivity.
destruct J. replace (None : option bool) with (Some b) in H.
contradiction H. reflexivity. symmetry. apply H1. reflexivity.
contradiction H. reflexivity.
destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m));
destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m - 1)).
intro K. simpl in K. apply diff_false_true in K. contradiction K.
intro K. simpl in K. apply diff_false_true in K. contradiction K.
intro K. simpl in K. apply diff_false_true in K. contradiction K.
intro K. simpl in K. apply diff_false_true in K. contradiction K.
rewrite nth_error_nth' with (d := false) in J.
rewrite nth_error_nth' with (d := false) in J.
rewrite nth_error_nth' with (d := false) in H.
rewrite nth_error_nth' with (d := false) in H.
rewrite nth_error_nth' with (d := false).
rewrite nth_error_nth' with (d := false).
destruct (nth (S (2 * k) * 2 ^ m) (tm_step n0) false);
destruct (nth (S (2 * k) * 2 ^ m - 1) (tm_step n0) false);
destruct (nth (2 * (S (2 * k) * 2 ^ m) - 1) (tm_step (S n0)) false).
simpl; split; reflexivity.
destruct J. rewrite H0 in H. contradiction H. reflexivity.
reflexivity.
simpl; split; reflexivity. contradiction H. reflexivity.
contradiction H. reflexivity.
simpl; split; reflexivity.
destruct J. rewrite H0 in H. contradiction H. reflexivity.
reflexivity.
simpl; split; reflexivity.
assumption.
rewrite tm_size_power2. assumption.
assumption.
rewrite tm_size_power2. generalize I. generalize L.
apply Nat.lt_trans.
rewrite tm_size_power2. generalize I. generalize L.
apply Nat.lt_trans.
rewrite tm_size_power2. assumption.
rewrite Nat.mul_sub_distr_l. rewrite Nat.mul_1_r.
rewrite <- Nat.sub_succ_l.
replace (S (2 * (S (2 * k) * 2 ^ m))) with (2 * (S (2 * k) * 2 ^ m) + 1).
rewrite <- Nat.add_cancel_r with (p := 1).
rewrite Nat.sub_add.
destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m));
destruct (nth_error (tm_step n0) (S (2 * k) * 2 ^ m - 1));
destruct (nth_error (tm_step (S n0)) (2 * (S (2 * k) * 2 ^ m) - 1)).
destruct (b); destruct (b0); destruct (b1).
split. reflexivity. reflexivity.
destruct J. rewrite H0 in H. contradiction H. reflexivity.
reflexivity. simpl. split; reflexivity.
contradiction H. reflexivity. contradiction H. reflexivity.
simpl. split; reflexivity.
destruct J. rewrite H0 in H. contradiction H. reflexivity.
reflexivity. contradiction H. reflexivity.
destruct (b); destruct (b0). destruct J. rewrite H0.
split. intro J. inversion J. destruct H0. reflexivity.
intro J. simpl in J. symmetry in J. apply diff_false_true in J.
contradiction J. reflexivity.
split. intro K. inversion K.
apply Nat.lt_0_2. apply Nat.mul_comm.
Lemma tm_step_repeating_patterns :
forall (n m i j : nat),
i < 2^m -> j < 2^m
-> forall k, k < 2^n -> nth_error (tm_step m) i
= nth_error (tm_step m) j
<-> nth_error (tm_step (m+n)) (k * 2^m + i)
= nth_error (tm_step (m+n)) (k * 2^m + j).
induction m. rewrite Nat.pow_0_r. rewrite Nat.mul_1_r.
rewrite Nat.odd_0. rewrite Nat.sub_succ. rewrite Nat.sub_0_r.
destruct n. rewrite Nat.pow_0_r. rewrite Nat.lt_1_r.
intro H. apply Nat.neq_succ_0 in H. contradiction H.
rewrite <- tm_step_double_index. intro H. split. intro I.
symmetry in I. apply tm_step_succ_double_index in I. contradiction I.
apply Nat.lt_succ_l in H. rewrite Nat.pow_succ_r' in H.
rewrite <- Nat.mul_lt_mono_pos_l in H. assumption. apply Nat.lt_0_2.
intro I. apply diff_false_true in I. contradiction I.
intro H.
- induction m.
+ rewrite Nat.pow_0_r. rewrite Nat.mul_1_r.
rewrite Nat.odd_0. rewrite Nat.sub_succ. rewrite Nat.sub_0_r.
destruct n. rewrite Nat.pow_0_r in H. rewrite Nat.mul_1_r in H.
rewrite Nat.lt_1_r in H. apply Nat.neq_succ_0 in H. contradiction H.
rewrite <- tm_step_double_index.
split.
intro I. symmetry in I.
apply tm_step_succ_double_index in I. contradiction I.
rewrite Nat.pow_0_r in H. rewrite Nat.mul_1_r in H.
apply Nat.lt_succ_l in H. rewrite Nat.pow_succ_r' in H.
rewrite <- Nat.mul_lt_mono_pos_l in H. assumption. apply Nat.lt_0_2.
intro I. apply diff_false_true in I. contradiction I.
+
split.
- induction m.
+ rewrite Nat.pow_0_r. rewrite Nat.mul_1_r.
rewrite Nat.odd_0. rewrite Nat.sub_succ. rewrite Nat.sub_0_r.
destruct n. rewrite Nat.pow_0_r in H. rewrite Nat.mul_1_r in H.
rewrite Nat.lt_1_r in H. apply Nat.neq_succ_0 in H. contradiction H.
rewrite <- tm_step_double_index. intro I. symmetry in I.
apply tm_step_succ_double_index in I. contradiction I.
rewrite Nat.pow_0_r in H. rewrite Nat.mul_1_r in H.
apply Nat.lt_succ_l in H. rewrite Nat.pow_succ_r' in H.
rewrite <- Nat.mul_lt_mono_pos_l in H. assumption. apply Nat.lt_0_2.
+
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.
Theorem tm_step_double_index : forall (n k : nat),
nth_error (tm_step n) k = nth_error (tm_step (S n)) (2*k).
Theorem tm_step_succ_double_index : forall (n k : nat),
k < 2^n -> nth_error (tm_step n) k <> nth_error (tm_step (S n)) (S (2*k)).
Require Import BinPosDef.
Require Import BinNatDef.
Lemma tm_step_double_index_pos : forall (p : positive),
nth_error (tm_step (Pos.size_nat p)) (Pos.to_nat p) =
nth_error (tm_step (S (Pos.size_nat p))) (Pos.to_nat (p~0)).
Proof.
intro p.
simpl.
Theorem tm_step_double_index : forall (n k : nat),
nth_error (tm_step n) k = nth_error (tm_step (S n)) (2*k).
(*
From a(0) to a(2n+1), there are n+1 terms equal to 0 and n+1 terms equal to 1 (see Hassan Tarfaoui link, Concours Général 1990). - Bernard Schott, Jan 21 2022
TODO Search "count_occ"
*)
Theorem tm_step_count_occ : forall (hd tl : list bool) (k : nat),
tm_step k = hd ++ tl -> even (length hd) = true
-> count_occ Bool.bool_dec hd true = count_occ Bool.bool_dec hd false.
Proof.
intros hd tl k. intros H I.
destruct k.
- assert (J: hd = nil). destruct hd. reflexivity.
simpl in H. inversion H.
symmetry in H2. apply app_eq_nil in H2.
destruct H2. rewrite H0 in I. simpl in I. inversion I.
rewrite J. reflexivity.
- rewrite <- tm_step_lemma in H.
generalize I. generalize H. induction hd.
reflexivity.
rewrite <- length_zero_iff_nil. simpl.
PeanoNat.Nat.neq_succ_0: forall n : nat, S n <> 0
(* TODO: supprimer head_2 *)
(* vérifier si les deux sont nécessaires *)
Require Import BinPosDef.
Require Import BinPos.
Require Import BinNat.
(* Hamming weight of a positive; argument can not be 0! *)
Fixpoint hamming_weight_positive (x: positive) :=
match x with
| xH => 1
| xO p => hamming_weight_positive p
| xI p => 1 + hamming_weight_positive p
end.
Definition hamming_weight_n (x: N) :=
match x with
| N0 => 0
| Npos x => hamming_weight_positive x
end.
Lemma hamming_weight_positive_nonzero : forall (x: positive),
hamming_weight_positive x > 0.
Proof.
intros x.
induction x.
- simpl. apply Arith_prebase.gt_Sn_O_stt.
- simpl. apply IHx.
- simpl. apply Arith_prebase.gt_Sn_O_stt.
Qed.
Lemma size_double_bin : forall (x: positive),
Pos.size_nat (Pos.succ (Pos.pred_double x)) = S (Pos.size_nat x).
Proof.
intros x.
rewrite Pos.succ_pred_double.
reflexivity.
Qed.
Lemma succ_succ_pred_double_succ : forall (x: positive),
Pos.pred_double (Pos.succ x) = Pos.succ (Pos.succ (Pos.pred_double x)).
Proof.
intros x.
rewrite <- Pos.add_1_l.
rewrite Pos.add_xO_pred_double.
rewrite <- Pos.add_1_l.
rewrite <- Pos.add_1_l.
rewrite Pos.add_assoc.
reflexivity.
Qed.
Lemma size_double_bin2 : forall (x: positive),
Pos.size_nat (Pos.pred_double (Pos.succ x)) = S (Pos.size_nat x).
Proof.
intros x.
rewrite succ_succ_pred_double_succ. rewrite Pos.succ_pred_double.
reflexivity.
Qed.
Lemma size_succ_double_bin : forall (x: positive),
Pos.size_nat (Pos.succ (Pos.succ (Pos.pred_double x))) = S (Pos.size_nat x).
Proof.
intros x.
rewrite <- succ_succ_pred_double_succ. rewrite size_double_bin2.
reflexivity.
Qed.
Lemma nat_size_pos_size : forall (x: positive),
N.size_nat (N.pos x) = Pos.size_nat x.
Proof.
intros x. reflexivity.
Qed.
Lemma hamming_weight_increase : forall (x: positive),
hamming_weight_positive x~1 = S (hamming_weight_positive x).
Proof.
intros x. reflexivity.
Qed.
Fixpoint bin2power (x : positive) (k : N) :=
match x with
| xH => (2^k)%N
| xO y => bin2power y (N.succ k)
| xI y => (2^k + bin2power y (N.succ k))%N
end.
Lemma bin2power_same : forall (x : positive),
Npos x = bin2power x 0.
Proof.
intros x.
induction x.
- simpl.
(*
Fixpoint bin2power (x : positive) (k : nat) :=
match x with
| xH => 2^k
| xO y => bin2power y (S k)
| xI y => 2^k + bin2power y (S k)
end.
*)
Lemma bin2power_double : forall (x : positive) (k : N),
bin2power x~0 k = bin2power x (N.succ k).
Proof.
intros x. simpl. reflexivity.
Qed.
Lemma bin2power_double2 : forall (x : positive),
bin2power x~0 0%N = (2 * bin2power x 0)%N.
Proof.
intros x.
destruct x.
- simpl.
N.testbit_even_succ:
forall a n : N, (0 <= n)%N -> N.testbit (2 * a) (N.succ n) = N.testbit a n
simpl. reflexivity.
Qed.
Lemma bin2power_succ : forall (x : positive) (k : N),
bin2power (x~1) k = ((2^k)%N + (bin2power (x~0) k))%N.
Proof.
intros x k.
reflexivity.
Qed.
Lemma bin2power_double2 : forall (x : positive),
bin2power x 1%N = (2 * bin2power x 0)%N.
Proof.
intros x.
induction x.
- rewrite bin2power_succ. rewrite bin2power_succ. rewrite N.pow_0_r.
rewrite N.mul_add_distr_l. rewrite N.pow_1_r. rewrite N.mul_1_r.
rewrite N.add_cancel_l.
replace (bin2power x~0 0) with (bin2power x 1%N).
assert ((2 * bin2power (Pos.succ x~0) 0)%N = (2 + 2*binpower (x~0) 0)%N).
replace (x~0) with ((N.pos (N.shiftl (x) 1) 1)%N).
induction x.
- simpl. rewrite Nat.add_0_r. apply eq_S. rewrite <- plus_n_Sm.
apply eq_S.
destruct x.
+ simpl.
rewrite Pos.xI_succ_xO.
rewrite <- bin2power_double.
rewrite bin2power_double.
- rewrite bin2power_double.
assert ( I: bin2power x~1 0 = S (bin2power x~0 0) ).
{ simpl. reflexivity. } rewrite I.
assert ( J: bin2power x~1 1 = S (S (S (bin2power x~0 1) ))).
{ rewrite <- I.
simpl. rewrite Nat.add_0_r.
Qed.
Lemma bin2power_same : forall (x: positive),
bin2power x 0 = Pos.to_nat x.
Proof.
intros x.
induction x.
- simpl. rewrite <- Pos.pred_double_succ.
rewrite succ_succ_pred_double_succ. rewrite Pos.succ_pred_double.
rewrite <- Pos.xI_succ_xO.
rewrite <- bin2power_double.
assert (I: bin2power x 1 = 2 * (bin2power x 0)).
{ induction x. simpl. rewrite Nat.add_0_r. rewrite <- plus_n_Sm.
simpl in IHx.
assert (I: bin2power x 1 = 2 * (bin2power x 0)).
{ induction x. simpl. rewrite Nat.add_0_r. rewrite <- plus_n_Sm.
simpl in IHx.
Lemma tm_step_hamming : forall (x: N),
nth_error (tm_step (N.size_nat x)) (N.to_nat x)
= Some (odd (hamming_weight_n x)).
Proof.
intros x.
destruct x.
- reflexivity.
- induction p.
+ rewrite <- Pos.pred_double_succ.
rewrite nat_size_pos_size.
rewrite succ_succ_pred_double_succ at 1.
rewrite size_succ_double_bin.
unfold hamming_weight_n. rewrite Pos.pred_double_succ.
rewrite hamming_weight_increase. rewrite Nat.odd_succ.
rewrite <- Nat.negb_odd.
rewrite tm_build. rewrite nth_error_app1.
apply map_nth_error.
rewrite succ_succ_pred_double_succ.
simpl. rewrite Pos.succ_pred_double.
rewrite size_double_bin.
unfold N.to_nat at 2. unfold Pos.to_nat. simpl.
rewrite Pos.xI_succ_xO. transforme p~1 en Pos.succ p~0
Lemma pred_double_succ p : pred_double (succ p) = p~1.
Lemma pred_of_succ_nat (n:nat) : pred (of_succ_nat n) = of_nat n.
Lemma succ_of_nat (n:nat) : n<>O -> succ (of_nat n) = of_succ_nat n.
Theorem tm_step_hamming_index : forall (n : N),
nth_error (tm_step (N.to_nat n)) (N.to_nat n)
= Some (odd (hamming_weight_n n)).
Proof.
intros n.
destruct n.
- reflexivity.
- induction p. simpl.
Theorem tm_step_hamming_index : forall (n m : nat) (k j: N),
(N.to_nat k) < 2^n -> (N.to_nat j) < 2^m ->
hamming_weight_n k = hamming_weight_n j ->
nth_error (tm_step n) (N.to_nat k) = nth_error (tm_step m) (N.to_nat j).
Proof.
intros n m k j. intros H I K.
induction k.
- simpl in K. assert (j = N0). induction j. reflexivity.
rewrite <- N.succ_pos_pred.
unfold hamming_weight_n in K.
assert (L: hamming_weight_positive p > 0).
apply hamming_weight_positive_nonzero. rewrite <- K in L.
apply Arith_prebase.gt_irrefl_stt in L. contradiction L.
rewrite H0. rewrite H0 in I.
generalize I. generalize H. apply tm_step_stable.
- induction j. simpl in K. assert (L: hamming_weight_positive p > 0).
apply hamming_weight_positive_nonzero. rewrite K in L.
apply Arith_prebase.gt_irrefl_stt in L. contradiction L.
(* coeur de l'induction *)
induction p. simpl in K.
symmetry in K. apply Nat.neq_succ_0 in K. contradiction K.
Theorem N. succ_0_discr n : succ n <> 0.
Nat.neq_succ_0: forall n : nat, S n <> 0
Fixpoint tm_step (n: nat) : list bool :=
match n with
| 0 => false :: nil
| S n' => tm_morphism (tm_step n')
Lemma tm_step_index_split :
forall (a b n m : nat),
(* every natural number k can be written according to the following form *)
a < 2^n -> (a + 2^n * b) < 2^m
-> nth_error (tm_step m) (a + 2^n * b)
= nth_error (tm_step (S m)) (a + 2^S n * b).
Proof.
intros a b n m. intros H I.
Lemma tm_step_cancel_high_bits :
forall (k b n m : nat),
(* every natural number k can be written according to the following form *)
S k < 2^m -> k = 2^n - 1 + 2^S n * b
-> nth_error (tm_step m) k = nth_error (tm_step m) (S k)
<-> odd n = true.
Proof.
intros k b n m. intros H I.
assert (L: 2^n - 1 < 2^n). apply Nat.sub_lt. replace (1) with (2^0) at 1.
apply Nat.pow_le_mono_r. easy.
apply le_0_n. simpl. reflexivity. apply Nat.lt_0_1.
assert (M: S(2^n - 1) = 2^n). rewrite Nat.sub_1_r. apply Nat.succ_pred.
apply Nat.pow_nonzero. apply Nat.neq_succ_0.
assert (N: 2^n < 2^(S n)). apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
apply Nat.lt_succ_diag_r.
assert (J: nth_error (tm_step (S n)) (2^n-1) = nth_error (tm_step (S n)) (2^n)
<-> odd n = true).
rewrite tm_step_single_bit_index.
assert (nth_error (tm_step n) (2^n - 1) = nth_error (tm_step (S n)) (2^n-1)).
apply tm_step_stable. apply L.
assert (2^n < 2^(S n)). apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
apply Nat.lt_succ_diag_r.
generalize H0. generalize L. apply Nat.lt_trans. rewrite <- H0.
rewrite tm_step_repunit_index. destruct (odd n). easy. easy.
rewrite <- J. rewrite I.
destruct b.
(* case b = 0 *)
rewrite Nat.mul_0_r. rewrite Nat.add_0_r.
rewrite J. rewrite Nat.mul_0_r in I. rewrite Nat.add_0_r in I.
rewrite I in H. rewrite M.
assert (nth_error (tm_step m) (2^n-1) = nth_error (tm_step n) (2^n-1)).
generalize L. apply tm_step_stable. apply Nat.lt_succ_l. apply H.
rewrite H0. rewrite tm_step_repunit_index.
assert (nth_error (tm_step m) (2^n) = nth_error (tm_step (S n)) (2 ^ n)).
generalize N. rewrite M in H. generalize H. apply tm_step_stable.
rewrite H1. rewrite tm_step_single_bit_index.
split. intros. inversion H2. reflexivity. intros. inversion H2.
reflexivity.
(* case b > 0 *)
assert (S b = Pos.to_nat (Pos.of_nat (S b))).
destruct n.
- rewrite Nat.pow_0_r. rewrite Nat.sub_diag. rewrite plus_O_n.
rewrite Nat.pow_1_r.
rewrite Nat.pow_0_r in I. simpl in I. rewrite Nat.add_0_r in I.
induction m.
+ rewrite Nat.pow_0_r in H.
assert (K := H). rewrite Nat.lt_1_r in K.
apply Nat.neq_succ_0 in K. contradiction K.
+
induction k.
- rewrite <- I.
assert (b=0).
{ symmetry in I. rewrite Nat.eq_add_0 in I. destruct I.
apply Nat.mul_eq_0_r in H1. apply H1. apply Nat.pow_nonzero.
easy. }
rewrite H0 in I. rewrite Nat.mul_0_r in I.
rewrite Nat.add_0_r in I. rewrite <- I.
replace (2^n) with (S (2^n - 1)). rewrite <- I.
split.
intros. rewrite tm_step_head_2 at 2. rewrite tm_step_head_1. simpl.
replace (m) with (S (m-1)) in H1 at 2.
rewrite tm_step_head_2 in H1. rewrite tm_step_head_1 in H1. simpl in H1.
apply H1.
destruct m. simpl in H. apply Nat.lt_irrefl in H. contradiction H.
rewrite Nat.sub_1_r. simpl. reflexivity.
intros. rewrite tm_step_head_2 in H1 at 2.
rewrite tm_step_head_1 in H1. simpl in H1. inversion H1.
rewrite <- I.
apply eq_S in I. rewrite I at 1.
apply Nat.pred_inj. apply Nat.neq_succ_0. apply Nat.pow_nonzero.
easy. simpl. rewrite Nat.sub_1_r. reflexivity.
- apply IHk. apply Nat.lt_succ_l in H. apply H.
rewrite <- I.
split. intros. apply H1.
assert (K: S (2 ^ n - 1 + 2 ^ S n * b) = (2 ^ n + 2 ^ S n * b)).
+ rewrite Nat.sub_1_r. rewrite <- Nat.add_succ_l.
rewrite Nat.succ_pred_pos. reflexivity.
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
+ rewrite K. symmetry.
split. intros. symmetry.
apply tm_step_add_range2_flip_two.
(*
assert (K: nth_error (tm_step n) a = Some (odd n)). rewrite I.
apply tm_step_repunit.
*)
Lemma tm_step_add_range2_flip_two : forall (n m j k : nat),
k < 2^n -> m < 2^n ->
nth_error (tm_step n) k = nth_error (tm_step n) m
<->
nth_error (tm_step (S n+j)) (k + 2^(n+j))
= nth_error (tm_step (S n+j)) (m + 2^(n+j)).
Déclagae vers la droite de k zéros pour un Bins :
Pos.iter xO xH 3. (ici k = 3)
--> on ajoute 3 zéros à gauche
Require Import BinPosDef.
(* Autre construction de la suite, ici n est le nombre de termes *)
(* la construction se fait à l'envers *)
(*
Fixpoint tm_bin_rev (n: nat) : list bool :=
match n with
| 0 => nil
| S n' => let t := tm_bin_rev n' in
let m := Pos.of_nat n' in
(xorb (hd true t) (odd (Pos.size_nat
match Pos.lxor m (Pos.pred m) with
| N0 => BinNums.xH
| Npos(p) => p
end))) :: t
end.
*)
Fixpoint tm_bin (n: nat) : list bool :=
match n with
| 0 => nil
| S n' => let t := tm_bin n' in
let m := Pos.of_nat n' in
t ++ [ xorb (last t true)
(odd (Pos.size_nat match Pos.lxor m (Pos.pred m) with
| N0 => BinNums.xH
| Npos(p) => p
end)) ]
end.
Theorem tm_morphism_eq_bin : forall (k : nat), tm_step k = tm_bin (2^k).
Proof.
Admitted.
Theorem tm_step_consecutive : forall (n : nat) (l1 l2 : list bool) (b1 b2 : bool),
tm_step n = l1 ++ b1 :: b2 :: l2 ->
(eqb b1 b2) =
let ind_b2 := Pos.of_nat (S (length l1)) in (* index of b2 *)
let ind_b1 := Pos.pred ind_b2 in (* index of b1 *)
even (Pos.size_nat
match Pos.lxor ind_b1 ind_b2 with
| N0 => BinNums.xH
| Npos(p) => p
end).
Proof.
intros n l1 l2 b1 b2.
destruct n.
- simpl. intros H. induction l1.
+ rewrite app_nil_l in H. discriminate.
+ destruct l1. rewrite app_nil_l in IHl1. discriminate. discriminate.
- rewrite tm_build.
Admitted.
(*
intros n l1 l2 b1 b2.
intros H.
induction l1.
- simpl. destruct n. discriminate.
rewrite app_nil_l in H. assert (I := H).
rewrite tm_step_head_2 in I. injection I.
assert (J: tl (tl (tm_morphism (tm_step n))) = l2).
{ replace (tm_morphism (tm_step n)) with (tm_step (S n)).
rewrite H. reflexivity. reflexivity. }
(* rewrite J *) intros. rewrite <- H1. rewrite <- H2. reflexivity.
- replace (S (length (a :: l1))) with (S (S (length l1))).
destruct b1 in H.
+ destruct b2 in H.
(* Case 1: b1 = true, b2 = true (false) *)
replace (l2) with (tl (tl (tm_step (S n)))) in H.
assert (J : tm_step (S n) = false :: true :: tl (tl (tm_step (S n)))).
apply tm_step_head_2. rewrite J in H. discriminate H. rewrite H. reflexivity.
(* Case 2: b1 = true, b2 = false (false) *)
replace (l2) with (tl (tl (tm_step (S n)))) in H.
assert (J : tm_step (S n) = false :: true :: tl (tl (tm_step (S n)))).
apply tm_step_head_2. rewrite J in H. discriminate H. rewrite H. reflexivity.
+ destruct b2.
(* Case 3: b1 = false, b2 = true (TRUE) *)
discriminate.
inversion H.
discriminate tm_step_head_2.
rewrite <- tm_step_head_2 in H.
discriminate H.
discriminate tm_step_head_2.
- simpl. simpl. simpl in H.
destruct n in H. discriminate H.
replace (l2) with (tl (tl (tm_step (S n)))) in H.
specialize (H tm_step_head_2).
rewrite <- tm_step_head_2 in H.
Lemma tm_step_head_2 : forall (n : nat),
tm_step (S n) = false :: true :: tl (tl (tm_step (S n))).
*)
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