| import OAI.LinearAlgebra.Barker.Parity |
| import OAI.LinearAlgebra.Barker.Wraparound |
| import OAI.LinearAlgebra.Barker.PeriodicModFour |
| import OAI.LinearAlgebra.CirculantHadamard.Main |
|
|
| namespace OAI |
|
|
| namespace CirculantHadamard.Barker |
|
|
| theorem four_dvd_length {n : ℕ} [NeZero n] {h : Fin n → ℤ} |
| (hb : IsBarker h) (hn : Even n) (hnlarge : 2 < n) : (4 : ℤ) ∣ (n : ℤ) := by |
| let a : Fin n := ⟨2, hnlarge⟩ |
| have ha : a ≠ 0 := by |
| intro heq |
| have hv := congrArg Fin.val heq |
| change 2 = 0 at hv |
| omega |
| have ht : aperiodic h 2 = 0 := |
| aperiodic_even_shift_zero hb hn 2 (by decide) (by omega) hnlarge |
| have hnsub : Even (n - 2) := by |
| obtain ⟨u, hu⟩ := hn |
| exact ⟨u - 1, by omega⟩ |
| have ht' : aperiodic h (n - 2) = 0 := |
| aperiodic_even_shift_zero hb hn (n - 2) hnsub (by omega) (by omega) |
| have hpa : periodic h a = 0 := by |
| rw [periodic_eq_aperiodic_add h a ha] |
| change aperiodic h 2 + aperiodic h (n - 2) = 0 |
| rw [ht, ht', add_zero] |
| have hd := periodic_mod_four h hb.1 a |
| rw [hpa, zero_sub] at hd |
| exact dvd_neg.mp hd |
|
|
| theorem periodic_offpeak_zero {n : ℕ} [NeZero n] {h : Fin n → ℤ} |
| (hb : IsBarker h) (hn : Even n) (hnlarge : 2 < n) |
| (a : Fin n) (ha : a ≠ 0) : periodic h a = 0 := by |
| have ha0 : 0 < a.val := by |
| by_contra hnot |
| have hz : a.val = 0 := by omega |
| apply ha |
| apply Fin.ext |
| simpa using hz |
| have hab := aperiodic_bounds hb a.val ha0 a.isLt |
| have hac := aperiodic_bounds hb (n - a.val) (by omega) (by omega) |
| have hp := periodic_eq_aperiodic_add h a ha |
| obtain ⟨u, hu⟩ := periodic_mod_four h hb.1 a |
| obtain ⟨v, hv⟩ := four_dvd_length hb hn hnlarge |
| omega |
|
|
| theorem existsRealCirculantHadamard_of_even_barker {n : ℕ} {h : Fin n → ℤ} |
| (hb : IsBarker h) (hn : Even n) (hnlarge : 2 < n) : |
| ExistsRealCirculantHadamard n := by |
| let : NeZero n := ⟨by omega⟩ |
| apply existsReal_iff_integer_autocorrelation.mpr |
| refine ⟨h, hb.1, ?_⟩ |
| intro a |
| change periodic h a = if a = 0 then (n : ℤ) else 0 |
| by_cases ha : a = 0 |
| · subst a |
| simp [periodic_zero h hb.1] |
| · rw [ite_eq_right ha] |
| exact periodic_offpeak_zero hb hn hnlarge a ha |
|
|
| theorem even_length_eq_two_or_four {n : ℕ} (h : Fin n → ℤ) |
| (hnpos : 0 < n) (hn : Even n) (hb : IsBarker h) : n = 2 ∨ n = 4 := by |
| by_cases hsmall : n ≤ 2 |
| · left |
| obtain ⟨u, hu⟩ := hn |
| omega |
| · have hnlarge : 2 < n := by omega |
| have hH := existsRealCirculantHadamard_of_even_barker hb hn hnlarge |
| have hc := (exists_iff_order_one_or_four n hnpos).mp hH |
| right |
| omega |
|
|
| end CirculantHadamard.Barker |
|
|
| end OAI |
|
|
