| import OAI.LinearAlgebra.Barker.Model |
| import Mathlib.Tactic.NormNum |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| namespace CirculantHadamard.Barker |
|
|
| open scoped BigOperators |
|
|
| def negativeBit {n : ℕ} (h : Fin n → ℤ) (j : Fin n) : ℤ := |
| if h j = -1 then 1 else 0 |
|
|
| theorem sign_eq_one_sub_two_bit {n : ℕ} (h : Fin n → ℤ) |
| (hs : ∀ j, IsSign (h j)) (j : Fin n) : |
| h j = 1 - 2 * negativeBit h j := by |
| rcases hs j with hj | hj <;> norm_num [negativeBit, hj] |
|
|
| def cyclicSubtractEquiv {n : ℕ} [NeZero n] (a : Fin n) : Fin n ≃ Fin n where |
| toFun j := j - a |
| invFun j := j + a |
| left_inv j := sub_add_cancel j a |
| right_inv j := add_sub_cancel_right j a |
|
|
| theorem sum_negativeBit_shift {n : ℕ} [NeZero n] (h : Fin n → ℤ) (a : Fin n) : |
| (∑ j, negativeBit h (j - a)) = ∑ j, negativeBit h j := by |
| exact (cyclicSubtractEquiv a).sum_comp (negativeBit h) |
|
|
| theorem periodic_bit_identity {n : ℕ} [NeZero n] (h : Fin n → ℤ) |
| (hs : ∀ j, IsSign (h j)) (a : Fin n) : |
| periodic h a = (n : ℤ) + 4 * |
| ((∑ j, negativeBit h j * negativeBit h (j - a)) - ∑ j, negativeBit h j) := by |
| have hterm (j : Fin n) : h j * h (j - a) = |
| 1 - 2 * negativeBit h j - 2 * negativeBit h (j - a) + |
| 4 * (negativeBit h j * negativeBit h (j - a)) := by |
| rw [sign_eq_one_sub_two_bit h hs j, sign_eq_one_sub_two_bit h hs (j - a)] |
| ring |
| calc |
| periodic h a = ∑ j : Fin n, |
| (1 - 2 * negativeBit h j - 2 * negativeBit h (j - a) + |
| 4 * (negativeBit h j * negativeBit h (j - a))) := by |
| unfold periodic |
| exact Finset.sum_congr rfl (fun j _ => hterm j) |
| _ = (n : ℤ) - 2 * (∑ j, negativeBit h j) - |
| 2 * (∑ j, negativeBit h (j - a)) + |
| 4 * (∑ j, negativeBit h j * negativeBit h (j - a)) := by |
| simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.mul_sum] |
| simp |
| _ = _ := by rw [sum_negativeBit_shift]; ring |
|
|
| theorem periodic_mod_four {n : ℕ} [NeZero n] (h : Fin n → ℤ) |
| (hs : ∀ j, IsSign (h j)) (a : Fin n) : |
| (4 : ℤ) ∣ periodic h a - (n : ℤ) := by |
| refine ⟨(∑ j, negativeBit h j * negativeBit h (j - a)) - |
| ∑ j, negativeBit h j, ?_⟩ |
| rw [periodic_bit_identity h hs a] |
| ring |
|
|
| end CirculantHadamard.Barker |
|
|
| end OAI |
|
|
