| import OAI.LinearAlgebra.Barker.Model |
| import Mathlib.Algebra.BigOperators.Fin |
|
|
| namespace OAI |
|
|
| namespace CirculantHadamard.Barker |
| open scoped BigOperators |
|
|
| theorem periodic_eq_aperiodic_add {n : ℕ} [NeZero n] |
| (h : Fin n → ℤ) (a : Fin n) (_ : a ≠ 0) : |
| periodic h a = aperiodic h a.val + aperiodic h (n - a.val) := by |
| have ha_le : a.val ≤ n := Nat.le_of_lt a.isLt |
| have hn : a.val + (n - a.val) = n := Nat.add_sub_of_le ha_le |
| let f : Fin n → ℤ := fun j => h j * h (j - a) |
| have split := Fin.sum_univ_add |
| (fun j : Fin (a.val + (n - a.val)) => f (j.cast hn)) |
| rw [Fin.sum_congr' f hn] at split |
| have low : |
| (∑ j : Fin a.val, f ((Fin.castAdd (n - a.val) j).cast hn)) = |
| aperiodic h (n - a.val) := by |
| have hc : n - (n - a.val) = a.val := by omega |
| have reindex : aperiodic h (n - a.val) = |
| ∑ j : Fin a.val, |
| h ⟨j.val, by omega⟩ * h ⟨j.val + (n - a.val), by omega⟩ := by |
| unfold aperiodic |
| apply Fintype.sum_equiv (finCongr hc) |
| intro j |
| rfl |
| rw [reindex] |
| apply Finset.sum_congr rfl |
| intro j _ |
| have hsub : ((Fin.castAdd (n - a.val) j).cast hn - a) = |
| (⟨j.val + (n - a.val), by omega⟩ : Fin n) := by |
| apply Fin.ext |
| have hlt : (Fin.castAdd (n - a.val) j).cast hn < a := by |
| change j.val < a.val |
| exact j.isLt |
| rw [Fin.coe_sub_iff_lt.mpr hlt] |
| dsimp only [Fin.val_cast, Fin.val_castAdd] |
| omega |
| change h ((Fin.castAdd (n - a.val) j).cast hn) * |
| h ((Fin.castAdd (n - a.val) j).cast hn - a) = _ |
| rw [hsub] |
| rfl |
| have high : |
| (∑ j : Fin (n - a.val), f ((Fin.natAdd a.val j).cast hn)) = |
| aperiodic h a.val := by |
| unfold aperiodic |
| apply Finset.sum_congr rfl |
| intro j _ |
| have hsub : ((Fin.natAdd a.val j).cast hn - a) = |
| (⟨j.val, by omega⟩ : Fin n) := by |
| apply Fin.ext |
| have hle : a ≤ (Fin.natAdd a.val j).cast hn := by |
| change a.val ≤ a.val + j.val |
| omega |
| rw [Fin.sub_val_of_le hle] |
| dsimp only [Fin.val_cast, Fin.val_natAdd] |
| omega |
| change h ((Fin.natAdd a.val j).cast hn) * |
| h ((Fin.natAdd a.val j).cast hn - a) = _ |
| rw [hsub, mul_comm] |
| congr 1 |
| apply congrArg h |
| apply Fin.ext |
| change a.val + j.val = j.val + a.val |
| omega |
| change (∑ j : Fin n, f j) = _ |
| rw [split, low, high, add_comm] |
|
|
| end CirculantHadamard.Barker |
|
|
| end OAI |
|
|
