Update
This commit is contained in:
Thomas Baruchel
parent
210b3b8b79
commit
4ba41ee7ef
151
src/thue_morse.v
151
src/thue_morse.v
|
|
@ -719,6 +719,157 @@ Proof.
|
|||
generalize H2. generalize J. apply Nat.le_trans.
|
||||
Qed.
|
||||
|
||||
Lemma tm_step_repeating_patterns2 :
|
||||
forall (n m : nat) (hd pat tl : list bool),
|
||||
tm_step n = hd ++ pat ++ tl
|
||||
-> length pat = 2^m
|
||||
-> length hd mod (2^m) = 0
|
||||
-> pat = tm_step m \/ pat = map negb (tm_step m).
|
||||
Proof.
|
||||
intro n.
|
||||
induction n; intros m hd pat tl; intros H I J.
|
||||
- left. assert (K: length (tm_step 0) = length (tm_step 0)). reflexivity.
|
||||
rewrite H in K at 2. rewrite app_length in K. rewrite app_length in K.
|
||||
destruct hd; destruct tl.
|
||||
+ simpl in H. rewrite app_nil_r in H. rewrite <- H in I.
|
||||
destruct m. rewrite <- H. reflexivity.
|
||||
replace (length [false]) with (2^0) in I. apply Nat.pow_inj_r in I.
|
||||
inversion I. apply Nat.lt_1_2. reflexivity.
|
||||
+ simpl in K. rewrite Nat.add_succ_r in K. inversion K.
|
||||
symmetry in H1. apply Nat.eq_add_0 in H1. destruct H1.
|
||||
rewrite H0 in I. symmetry in I. apply Nat.pow_nonzero in I.
|
||||
contradiction. easy.
|
||||
+ simpl in K. inversion K.
|
||||
symmetry in H1. apply Nat.eq_add_0 in H1. destruct H1.
|
||||
rewrite Nat.add_0_r in H1. rewrite H1 in I.
|
||||
symmetry in I. apply Nat.pow_nonzero in I.
|
||||
contradiction. easy.
|
||||
+ simpl in K. rewrite Nat.add_succ_r in K. inversion K.
|
||||
symmetry in H1. apply Nat.eq_add_0 in H1. destruct H1.
|
||||
inversion H1.
|
||||
- assert (H' := H). rewrite tm_build in H.
|
||||
assert (K: length (hd ++ pat) <= 2^n \/ 2^n < length (hd ++ pat)).
|
||||
apply Nat.le_gt_cases. destruct K as [K | K].
|
||||
+ assert (tm_step n = hd ++ pat ++ firstn (2^n - length (hd ++ pat)) tl).
|
||||
replace tl with (firstn (2^n - length (hd ++ pat)) tl
|
||||
++ skipn (2^n - length (hd ++ pat)) tl) in H.
|
||||
rewrite app_assoc in H. rewrite app_assoc in H.
|
||||
apply app_eq_length_head in H. rewrite app_assoc. assumption.
|
||||
rewrite app_length. rewrite firstn_length_le.
|
||||
rewrite Nat.add_sub_assoc. rewrite Nat.add_sub_swap.
|
||||
rewrite Nat.sub_diag. rewrite tm_size_power2. reflexivity.
|
||||
apply Nat.le_refl. assumption.
|
||||
rewrite Nat.add_le_mono_r with (p := length (hd ++ pat)).
|
||||
rewrite Nat.sub_add. rewrite Nat.add_comm. rewrite <- app_length.
|
||||
rewrite <- app_assoc. rewrite <- H'. rewrite tm_size_power2.
|
||||
apply Nat.pow_le_mono_r. easy. apply Nat.le_succ_diag_r. assumption.
|
||||
apply firstn_skipn.
|
||||
generalize J. generalize I. generalize H0. apply IHn.
|
||||
+ assert (L: forall l1 l2, map negb l1 = l2 -> l1 = map negb l2).
|
||||
intro l1. induction l1; intros. rewrite <- H0. reflexivity.
|
||||
simpl in H0. destruct l2; inversion H0. apply IHl1 in H3.
|
||||
rewrite H0. rewrite <- H2. simpl. rewrite negb_involutive.
|
||||
rewrite <- H3. reflexivity.
|
||||
|
||||
assert (L': forall l1 l2, map negb l1 = map negb l2 -> l1 = l2).
|
||||
intro l1. induction l1; intros. destruct l2. reflexivity.
|
||||
inversion H0. destruct l2. inversion H0. inversion H0.
|
||||
apply IHl1 in H3. rewrite H3.
|
||||
destruct a; destruct b; try reflexivity || inversion H2.
|
||||
|
||||
assert (L'': forall l, map negb (map negb l) = l).
|
||||
intro l. induction l. reflexivity. simpl. rewrite IHl.
|
||||
rewrite negb_involutive. reflexivity.
|
||||
|
||||
assert (U: n < m \/ m <= n). apply Nat.lt_ge_cases. destruct U as [U | U].
|
||||
assert (V: length hd < 2^n \/ 2^n <= length hd). apply Nat.lt_ge_cases.
|
||||
destruct V as [V | V]. rewrite <- Nat.div_exact in J.
|
||||
assert (length hd / 2^ m = 0). apply Nat.div_small.
|
||||
rewrite Nat.pow_lt_mono_r_iff with (a := 2) in U.
|
||||
generalize U. generalize V. apply Nat.lt_trans. apply Nat.lt_1_2.
|
||||
rewrite H0 in J. rewrite Nat.mul_0_r in J.
|
||||
rewrite length_zero_iff_nil in J. rewrite J in H'. rewrite J in K.
|
||||
assert (length (tm_step (S n)) = length (pat ++ tl)).
|
||||
rewrite H'. reflexivity. rewrite tm_size_power2 in H1.
|
||||
rewrite app_length in H1. rewrite I in H1.
|
||||
rewrite <- Nat.le_succ_l in U.
|
||||
rewrite Nat.pow_le_mono_r_iff with (a := 2) in U.
|
||||
rewrite Nat.lt_eq_cases in U. destruct U. rewrite H1 in H2.
|
||||
(* contradiction en H2 *)
|
||||
assert (2^m <= 2^m + length tl). apply Nat.le_add_r.
|
||||
apply Nat.le_ngt in H3. contradiction.
|
||||
rewrite <- tm_build in H. rewrite app_nil_l in H'.
|
||||
rewrite <- app_nil_r in H' at 1.
|
||||
apply app_eq_length_head in H'. apply Nat.pow_inj_r in H2.
|
||||
rewrite H2 in H'. left. rewrite H'. reflexivity.
|
||||
apply Nat.lt_1_2. rewrite I. rewrite tm_size_power2. rewrite H2.
|
||||
reflexivity. apply Nat.lt_1_2. apply Nat.pow_nonzero. easy.
|
||||
|
||||
rewrite Nat.add_le_mono_r with (p := length pat) in V.
|
||||
rewrite Nat.pow_lt_mono_r_iff with (a := 2) in U.
|
||||
rewrite Nat.add_lt_mono_l with (p := 2^n) in U.
|
||||
rewrite <- I in U.
|
||||
assert (length (tm_step (S n)) < length (hd ++ pat)).
|
||||
rewrite app_length. rewrite tm_size_power2. simpl. rewrite Nat.add_0_r.
|
||||
generalize V. generalize U. apply Nat.lt_le_trans.
|
||||
rewrite H' in H0. rewrite app_assoc in H0. rewrite app_length in H0.
|
||||
rewrite <- Nat.add_0_r in H0. rewrite <- Nat.add_lt_mono_l in H0.
|
||||
apply Nat.nlt_0_r in H0. contradiction. apply Nat.lt_1_2.
|
||||
|
||||
assert ((2^n) mod (2^m) = 0). rewrite <- Nat.div_exact.
|
||||
rewrite <- Nat.pow_sub_r. rewrite <- Nat.pow_add_r.
|
||||
rewrite Nat.add_sub_assoc. rewrite Nat.add_sub_swap.
|
||||
rewrite Nat.sub_diag. reflexivity. apply Nat.le_refl.
|
||||
assumption. easy. assumption. apply Nat.pow_nonzero. easy.
|
||||
|
||||
assert (K': 2^n <= length hd \/ length hd < 2^n).
|
||||
apply Nat.le_gt_cases. destruct K' as [K'|K'].
|
||||
rewrite <- firstn_skipn with (n := 2^n) (l := hd) in H.
|
||||
rewrite <- app_assoc in H.
|
||||
assert (tm_step n = firstn (2^n) hd).
|
||||
apply app_eq_length_head
|
||||
with (x1 := map negb (tm_step n))
|
||||
(y1 := skipn (2 ^ n) hd ++ pat ++ tl). assumption.
|
||||
rewrite firstn_length_le. apply tm_size_power2. assumption.
|
||||
rewrite H1 in H at 1. apply app_inv_head_iff in H.
|
||||
|
||||
apply L in H. rewrite map_app in H. rewrite map_app in H.
|
||||
assert (length (map negb pat) = 2^m). rewrite map_length. assumption.
|
||||
assert (length (map negb (skipn (2^n) hd)) mod (2^m) = 0).
|
||||
rewrite map_length. rewrite skipn_length.
|
||||
|
||||
replace (length hd - 2^n) with (length hd - 2^n + (2^n mod 2^m)).
|
||||
rewrite Nat.add_mod_idemp_r. rewrite Nat.sub_add. assumption.
|
||||
assumption. apply Nat.pow_nonzero. easy. rewrite H0.
|
||||
apply Nat.add_0_r.
|
||||
|
||||
assert (forall l1 l2,
|
||||
(map negb l1 = map negb l2 \/ map negb l1 = map negb (map negb l2))
|
||||
-> l1 = l2 \/ l1 = map negb l2).
|
||||
intros. destruct H4. left. apply L'. assumption. apply L' in H4.
|
||||
right. assumption. apply H4. rewrite L''. rewrite or_comm.
|
||||
generalize H3. generalize H2. generalize H. apply IHn.
|
||||
|
||||
(* cas impossible *)
|
||||
rewrite <- Nat.div_exact in J. rewrite J in K'.
|
||||
rewrite <- Nat.div_exact in H0. rewrite H0 in K'.
|
||||
rewrite <- Nat.mul_lt_mono_pos_l in K'.
|
||||
apply Nat.lt_le_pred in K'.
|
||||
rewrite Nat.mul_le_mono_pos_l with (p := 2^m) in K'.
|
||||
rewrite <- J in K'.
|
||||
rewrite <- Nat.sub_1_r in K'. rewrite Nat.mul_sub_distr_l in K'.
|
||||
rewrite Nat.mul_1_r in K'.
|
||||
rewrite Nat.add_le_mono_r with (p := length pat) in K'.
|
||||
rewrite <- app_length in K'. rewrite I in K'.
|
||||
rewrite Nat.sub_add in K'. rewrite <- H0 in K'.
|
||||
apply Nat.le_ngt in K'. apply K' in K. contradiction.
|
||||
rewrite <- H0. apply Nat.pow_le_mono_r. easy. assumption.
|
||||
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
|
||||
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
|
||||
apply Nat.pow_nonzero. easy.
|
||||
apply Nat.pow_nonzero. easy.
|
||||
Qed.
|
||||
|
||||
(**
|
||||
The following lemmas and theorems are related to flipping the
|
||||
most significant bit in the numerical value of indices (of terms
|
||||
|
|
|
|||
|
|
@ -13,223 +13,111 @@ Require Import PeanoNat.
|
|||
Require Import Nat.
|
||||
Require Import Bool.
|
||||
|
||||
Require Import Lia.
|
||||
Require Import Arith.
|
||||
Require Import Lia.
|
||||
|
||||
Import ListNotations.
|
||||
|
||||
|
||||
Lemma tm_step_repeating_patterns2 :
|
||||
forall (n m : nat) (hd pat tl : list bool),
|
||||
tm_step n = hd ++ pat ++ tl
|
||||
-> length pat = 2^m
|
||||
-> length hd mod (2^m) = 0
|
||||
-> pat = tm_step m \/ pat = map negb (tm_step m).
|
||||
Theorem tm_step_palindrome_power2_inverse :
|
||||
forall (m n k : nat) (hd tl : list bool),
|
||||
tm_step n = hd ++ tl
|
||||
-> length hd = S (Nat.double k) * 2^m
|
||||
-> odd m = true
|
||||
-> tl <> nil
|
||||
-> skipn (length hd - 2^m) hd = rev (firstn (2^m) tl).
|
||||
Proof.
|
||||
intro n.
|
||||
induction n; intros m hd pat tl; intros H I J.
|
||||
- left. assert (K: length (tm_step 0) = length (tm_step 0)). reflexivity.
|
||||
rewrite H in K at 2. rewrite app_length in K. rewrite app_length in K.
|
||||
destruct hd; destruct tl.
|
||||
+ simpl in H. rewrite app_nil_r in H. rewrite <- H in I.
|
||||
destruct m. rewrite <- H. reflexivity.
|
||||
replace (length [false]) with (2^0) in I. apply Nat.pow_inj_r in I.
|
||||
inversion I. apply Nat.lt_1_2. reflexivity.
|
||||
+ simpl in K. rewrite Nat.add_succ_r in K. inversion K.
|
||||
symmetry in H1. apply Nat.eq_add_0 in H1. destruct H1.
|
||||
rewrite H0 in I. symmetry in I. apply Nat.pow_nonzero in I.
|
||||
contradiction. easy.
|
||||
+ simpl in K. inversion K.
|
||||
symmetry in H1. apply Nat.eq_add_0 in H1. destruct H1.
|
||||
rewrite Nat.add_0_r in H1. rewrite H1 in I.
|
||||
symmetry in I. apply Nat.pow_nonzero in I.
|
||||
contradiction. easy.
|
||||
+ simpl in K. rewrite Nat.add_succ_r in K. inversion K.
|
||||
symmetry in H1. apply Nat.eq_add_0 in H1. destruct H1.
|
||||
inversion H1.
|
||||
- assert (H' := H). rewrite tm_build in H.
|
||||
assert (K: length (hd ++ pat) <= 2^n \/ 2^n < length (hd ++ pat)).
|
||||
apply Nat.le_gt_cases. destruct K as [K | K].
|
||||
+ assert (tm_step n = hd ++ pat ++ firstn (2^n - length (hd ++ pat)) tl).
|
||||
replace tl with (firstn (2^n - length (hd ++ pat)) tl
|
||||
++ skipn (2^n - length (hd ++ pat)) tl) in H.
|
||||
rewrite app_assoc in H. rewrite app_assoc in H.
|
||||
apply app_eq_length_head in H. rewrite app_assoc. assumption.
|
||||
rewrite app_length. rewrite firstn_length_le.
|
||||
rewrite Nat.add_sub_assoc. rewrite Nat.add_sub_swap.
|
||||
rewrite Nat.sub_diag. rewrite tm_size_power2. reflexivity.
|
||||
apply Nat.le_refl. assumption.
|
||||
rewrite Nat.add_le_mono_r with (p := length (hd ++ pat)).
|
||||
rewrite Nat.sub_add. rewrite Nat.add_comm. rewrite <- app_length.
|
||||
rewrite <- app_assoc. rewrite <- H'. rewrite tm_size_power2.
|
||||
apply Nat.pow_le_mono_r. easy. apply Nat.le_succ_diag_r. assumption.
|
||||
apply firstn_skipn.
|
||||
generalize J. generalize I. generalize H0. apply IHn.
|
||||
+ assert (L: forall l1 l2, map negb l1 = l2 -> l1 = map negb l2).
|
||||
intro l1. induction l1; intros. rewrite <- H0. reflexivity.
|
||||
simpl in H0. destruct l2; inversion H0. apply IHl1 in H3.
|
||||
rewrite H0. rewrite <- H2. simpl. rewrite negb_involutive.
|
||||
rewrite <- H3. reflexivity.
|
||||
intros m n k hd tl. intros H I J K.
|
||||
assert (L: 2^m <= length hd). rewrite I. lia.
|
||||
assert (M: m < n). assert (length (tm_step n) = length (hd ++ tl)).
|
||||
rewrite H. reflexivity. rewrite tm_size_power2 in H0.
|
||||
rewrite app_length in H0. (* rewrite I in H0. *)
|
||||
assert (n <= m \/ m < n). apply Nat.le_gt_cases. destruct H1.
|
||||
apply Nat.pow_le_mono_r with (a := 2) in H1.
|
||||
assert (2^n <= length hd). generalize L. generalize H1. apply Nat.le_trans.
|
||||
rewrite H0 in H2. rewrite <- Nat.add_0_r in H2.
|
||||
rewrite <- Nat.add_le_mono_l in H2. destruct tl. contradiction K.
|
||||
reflexivity. simpl in H2. apply Nat.nle_succ_0 in H2. contradiction.
|
||||
easy. assumption.
|
||||
|
||||
assert (L': forall l1 l2, map negb l1 = map negb l2 -> l1 = l2).
|
||||
intro l1. induction l1; intros. destruct l2. reflexivity.
|
||||
inversion H0. destruct l2. inversion H0. inversion H0.
|
||||
apply IHl1 in H3. rewrite H3.
|
||||
destruct a; destruct b; try reflexivity || inversion H2.
|
||||
assert (N: length (tm_step n) mod 2^m = 0). rewrite tm_size_power2.
|
||||
apply Nat.div_exact. apply Nat.pow_nonzero. easy. rewrite <- Nat.pow_sub_r.
|
||||
rewrite <- Nat.pow_add_r. rewrite Nat.add_comm. rewrite Nat.sub_add.
|
||||
reflexivity. apply Nat.lt_le_incl. assumption. easy.
|
||||
apply Nat.lt_le_incl. assumption.
|
||||
|
||||
assert (L'': forall l, map negb (map negb l) = l).
|
||||
intro l. induction l. reflexivity. simpl. rewrite IHl.
|
||||
rewrite negb_involutive. reflexivity.
|
||||
assert (O: length hd mod 2^m = 0).
|
||||
apply Nat.div_exact. apply Nat.pow_nonzero. easy. rewrite I.
|
||||
rewrite Nat.div_mul. rewrite Nat.mul_comm. reflexivity.
|
||||
apply Nat.pow_nonzero. easy.
|
||||
|
||||
assert (U: n < m \/ m <= n). apply Nat.lt_ge_cases. destruct U as [U | U].
|
||||
assert (V: length hd < 2^n \/ 2^n <= length hd). apply Nat.lt_ge_cases.
|
||||
destruct V as [V | V]. rewrite <- Nat.div_exact in J.
|
||||
assert (length hd / 2^ m = 0). apply Nat.div_small.
|
||||
rewrite Nat.pow_lt_mono_r_iff with (a := 2) in U.
|
||||
generalize U. generalize V. apply Nat.lt_trans. apply Nat.lt_1_2.
|
||||
rewrite H0 in J. rewrite Nat.mul_0_r in J.
|
||||
rewrite length_zero_iff_nil in J. rewrite J in H'. rewrite J in K.
|
||||
assert (length (tm_step (S n)) = length (pat ++ tl)).
|
||||
rewrite H'. reflexivity. rewrite tm_size_power2 in H1.
|
||||
rewrite app_length in H1. rewrite I in H1.
|
||||
rewrite <- Nat.le_succ_l in U.
|
||||
rewrite Nat.pow_le_mono_r_iff with (a := 2) in U.
|
||||
rewrite Nat.lt_eq_cases in U. destruct U. rewrite H1 in H2.
|
||||
(* contradiction en H2 *)
|
||||
assert (2^m <= 2^m + length tl). apply Nat.le_add_r.
|
||||
apply Nat.le_ngt in H3. contradiction.
|
||||
rewrite <- tm_build in H. rewrite app_nil_l in H'.
|
||||
rewrite <- app_nil_r in H' at 1.
|
||||
apply app_eq_length_head in H'. apply Nat.pow_inj_r in H2.
|
||||
rewrite H2 in H'. left. rewrite H'. reflexivity.
|
||||
apply Nat.lt_1_2. rewrite I. rewrite tm_size_power2. rewrite H2.
|
||||
reflexivity. apply Nat.lt_1_2. apply Nat.pow_nonzero. easy.
|
||||
|
||||
rewrite Nat.add_le_mono_r with (p := length pat) in V.
|
||||
rewrite Nat.pow_lt_mono_r_iff with (a := 2) in U.
|
||||
rewrite Nat.add_lt_mono_l with (p := 2^n) in U.
|
||||
rewrite <- I in U.
|
||||
assert (length (tm_step (S n)) < length (hd ++ pat)).
|
||||
rewrite app_length. rewrite tm_size_power2. simpl. rewrite Nat.add_0_r.
|
||||
generalize V. generalize U. apply Nat.lt_le_trans.
|
||||
rewrite H' in H0. rewrite app_assoc in H0. rewrite app_length in H0.
|
||||
rewrite <- Nat.add_0_r in H0. rewrite <- Nat.add_lt_mono_l in H0.
|
||||
apply Nat.nlt_0_r in H0. contradiction. apply Nat.lt_1_2.
|
||||
assert (P: length tl mod 2 ^ m = 0).
|
||||
rewrite H in N. rewrite app_length in N.
|
||||
rewrite <- Nat.add_mod_idemp_l in N. rewrite O in N. assumption.
|
||||
apply Nat.pow_nonzero. easy.
|
||||
|
||||
assert ((2^n) mod (2^m) = 0). rewrite <- Nat.div_exact.
|
||||
rewrite <- Nat.pow_sub_r. rewrite <- Nat.pow_add_r.
|
||||
rewrite Nat.add_sub_assoc. rewrite Nat.add_sub_swap.
|
||||
rewrite Nat.sub_diag. reflexivity. apply Nat.le_refl.
|
||||
assumption. easy. assumption. apply Nat.pow_nonzero. easy.
|
||||
assert (Q: 2^m <= length tl). apply Nat.div_exact in P. rewrite P.
|
||||
destruct (length tl / 2^m). rewrite Nat.mul_0_r in P.
|
||||
apply length_zero_iff_nil in P. rewrite P in K. contradiction K.
|
||||
reflexivity.
|
||||
rewrite <- Nat.mul_1_r at 1. apply Nat.mul_le_mono_l.
|
||||
apply Nat.le_succ_l. apply Nat.lt_0_succ.
|
||||
apply Nat.pow_nonzero. easy.
|
||||
|
||||
assert (K': 2^n <= length hd \/ length hd < 2^n).
|
||||
apply Nat.le_gt_cases. destruct K' as [K'|K'].
|
||||
rewrite <- firstn_skipn with (n := 2^n) (l := hd) in H.
|
||||
rewrite <- app_assoc in H.
|
||||
assert (tm_step n = firstn (2^n) hd).
|
||||
apply app_eq_length_head
|
||||
with (x1 := map negb (tm_step n))
|
||||
(y1 := skipn (2 ^ n) hd ++ pat ++ tl). assumption.
|
||||
rewrite firstn_length_le. apply tm_size_power2. assumption.
|
||||
rewrite H1 in H at 1. apply app_inv_head_iff in H.
|
||||
replace hd
|
||||
with (firstn (length hd - 2^m) hd ++ skipn (length hd - 2^m) hd) in H.
|
||||
replace tl with (firstn (2^m) tl ++ skipn (2^m) tl) in H.
|
||||
rewrite <- app_assoc in H.
|
||||
replace (
|
||||
skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl ++ skipn (2 ^ m) tl )
|
||||
with (
|
||||
(skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl) ++ skipn (2 ^ m) tl )
|
||||
in H.
|
||||
assert (length (skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl) = 2^(S m)).
|
||||
rewrite app_length. rewrite skipn_length. rewrite firstn_length_le.
|
||||
replace (length hd) with (length hd -2^m + 2^m) at 1.
|
||||
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag. simpl. rewrite Nat.add_0_r.
|
||||
reflexivity. apply Nat.le_refl. apply Nat.sub_add. assumption.
|
||||
assumption.
|
||||
|
||||
apply L in H. rewrite map_app in H. rewrite map_app in H.
|
||||
assert (length (map negb pat) = 2^m). rewrite map_length. assumption.
|
||||
assert (length (map negb (skipn (2^n) hd)) mod (2^m) = 0).
|
||||
rewrite map_length. rewrite skipn_length.
|
||||
assert (length (firstn (length hd - 2^m) hd) mod 2^(S m) = 0).
|
||||
rewrite firstn_length_le. replace (2^m) with (2^m * 1).
|
||||
rewrite I. rewrite Nat.mul_comm. rewrite <- Nat.mul_sub_distr_l.
|
||||
rewrite Nat.sub_succ. rewrite Nat.sub_0_r. rewrite Nat.double_twice.
|
||||
rewrite Nat.mul_assoc. replace (2^m * 2) with (2^(S m)).
|
||||
rewrite Nat.mul_comm. rewrite Nat.mod_mul. reflexivity.
|
||||
apply Nat.pow_nonzero. easy.
|
||||
rewrite Nat.mul_comm. rewrite Nat.pow_succ_r. reflexivity.
|
||||
apply le_0_n. apply Nat.mul_1_r. apply Nat.le_sub_l.
|
||||
|
||||
replace (length hd - 2^n) with (length hd - 2^n + (2^n mod 2^m)).
|
||||
rewrite Nat.add_mod_idemp_r. rewrite Nat.sub_add. assumption.
|
||||
assumption. apply Nat.pow_nonzero. easy. rewrite H0.
|
||||
apply Nat.add_0_r.
|
||||
assert (
|
||||
skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl = tm_step (S m)
|
||||
\/
|
||||
skipn (length hd - 2 ^ m) hd ++ firstn (2 ^ m) tl = map negb (tm_step (S m))).
|
||||
generalize H1. generalize H0. generalize H. apply tm_step_repeating_patterns2.
|
||||
|
||||
assert (forall l1 l2,
|
||||
(map negb l1 = map negb l2 \/ map negb l1 = map negb (map negb l2))
|
||||
-> l1 = l2 \/ l1 = map negb l2).
|
||||
intros. destruct H4. left. apply L'. assumption. apply L' in H4.
|
||||
right. assumption. apply H4. rewrite L''. rewrite or_comm.
|
||||
generalize H3. generalize H2. generalize H. apply IHn.
|
||||
|
||||
(* cas impossible *)
|
||||
rewrite <- Nat.div_exact in J. rewrite J in K'.
|
||||
rewrite <- Nat.div_exact in H0. rewrite H0 in K'.
|
||||
rewrite <- Nat.mul_lt_mono_pos_l in K'.
|
||||
apply Nat.lt_le_pred in K'.
|
||||
rewrite Nat.mul_le_mono_pos_l with (p := 2^m) in K'.
|
||||
rewrite <- J in K'.
|
||||
rewrite <- Nat.sub_1_r in K'. rewrite Nat.mul_sub_distr_l in K'.
|
||||
rewrite Nat.mul_1_r in K'.
|
||||
rewrite Nat.add_le_mono_r with (p := length pat) in K'.
|
||||
rewrite <- app_length in K'. rewrite I in K'.
|
||||
rewrite Nat.sub_add in K'. rewrite <- H0 in K'.
|
||||
apply Nat.le_ngt in K'. apply K' in K. contradiction.
|
||||
rewrite <- H0. apply Nat.pow_le_mono_r. easy. assumption.
|
||||
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
|
||||
rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
|
||||
apply Nat.pow_nonzero. easy.
|
||||
apply Nat.pow_nonzero. easy.
|
||||
apply tm_step_palindromic_full in J.
|
||||
destruct H2; rewrite J in H2.
|
||||
- assert (skipn (length hd - 2^m) hd = tm_step m).
|
||||
apply app_eq_length_head in H2. assumption.
|
||||
rewrite skipn_length.
|
||||
replace (length hd) with (length hd -2^m + 2^m) at 1.
|
||||
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag. rewrite tm_size_power2.
|
||||
reflexivity. reflexivity. apply Nat.sub_add. assumption. rewrite H3.
|
||||
assert (firstn (2^m) tl = rev (tm_step m)).
|
||||
rewrite H3 in H2. apply app_inv_head in H2. rewrite <- H2. reflexivity.
|
||||
rewrite H4. rewrite rev_involutive. reflexivity.
|
||||
- assert (skipn (length hd - 2^m) hd = map negb (tm_step m)).
|
||||
rewrite map_app in H2. apply app_eq_length_head in H2. assumption.
|
||||
rewrite skipn_length. replace (length hd) with (length hd -2^m + 2^m) at 1.
|
||||
rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
|
||||
rewrite map_length. rewrite tm_size_power2.
|
||||
reflexivity. reflexivity. apply Nat.sub_add. assumption. rewrite H3.
|
||||
assert (firstn (2^m) tl = map negb (rev (tm_step m))).
|
||||
rewrite H3 in H2. rewrite map_app in H2. apply app_inv_head in H2.
|
||||
rewrite <- H2. reflexivity.
|
||||
rewrite H4. rewrite map_rev. rewrite rev_involutive. reflexivity.
|
||||
- rewrite <- app_assoc. reflexivity.
|
||||
- rewrite firstn_skipn. reflexivity.
|
||||
- rewrite firstn_skipn. reflexivity.
|
||||
Qed.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
Lemma tm_step_repeating_patterns2 :
|
||||
forall (n m k : nat), k < 2^n
|
||||
-> tm_step m = firstn (2^m) (skipn (k * 2^m) (tm_step (m + n)))
|
||||
\/ tm_step m = map negb (firstn (2^m) (skipn (k * 2^m) (tm_step (m + n)))).
|
||||
Proof.
|
||||
intros n m k. intro H.
|
||||
generalize dependent
|
||||
|
||||
|
||||
|
||||
forall n m i j : nat,
|
||||
i < 2 ^ m ->
|
||||
j < 2 ^ m ->
|
||||
forall k : nat,
|
||||
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)
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
Lemma tm_step_mod_palindromic :
|
||||
forall (n m k : nat) (hd a b tl : list bool),
|
||||
tm_step n = hd ++ a ++ b ++ tl
|
||||
-> length (hd ++ a) = (S (Nat.double k)) * (2^(S (Nat.double m)))
|
||||
-> 0 < m
|
||||
-> length a = 2^(S (Nat.double m))
|
||||
-> length a = length b
|
||||
-> a = rev b.
|
||||
Proof.
|
||||
intros n m k hd a b tl. intros H I J K L.
|
||||
destruct m. inversion J.
|
||||
generalize dependent n. generalize dependent hd.
|
||||
generalize dependent a. generalize dependent b.
|
||||
generalize dependent tl.
|
||||
induction k.
|
||||
|
||||
|
||||
|
||||
0 < k -> S (Nat.double k) < n ->
|
||||
firstn (2^(S (Nat.double k))) (tm_step n)
|
||||
= firstn (2^(S (Nat.double k))) (skipn (2^(S (Nat.double k))) (tm_step n)).
|
||||
Proof.
|
||||
|
||||
|
||||
tm_step_palindromic_full_even:
|
||||
forall n : nat, even n = true -> tm_step n = rev (tm_step n)
|
||||
|
||||
|
||||
5.10.3
|
||||
|
|
|
|||
Loading…
Reference in New Issue