openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Separation /Fourier.lean
| import Mathlib.Analysis.Complex.Basic | |
| import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | |
| namespace OAI | |
| /-! | |
| # Finite root-of-unity filters | |
| Averaging the integer powers of a primitive root over one period detects | |
| exactly the exponents divisible by that period. This is the finite Fourier | |
| filter used for auxiliary separation. | |
| -/ | |
| open scoped BigOperators | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| /-- The sum over a complete period of a root of unity is zero unless the root is one. -/ | |
| lemma sum_pow_root_of_unity (z : ℂ) (L : ℕ) (hz : z ^ L = 1) : | |
| (∑ r ∈ Finset.range L, z ^ r) = if z = 1 then (L : ℂ) else 0 := by | |
| split_ifs with h | |
| · simp [h] | |
| · have hsum : (1 - z) * ∑ r ∈ Finset.range L, z ^ r = 0 := by | |
| rw [mul_neg_geom_sum, hz, sub_self] | |
| exact (mul_eq_zero.mp hsum).resolve_left (sub_ne_zero.mpr (Ne.symm h)) | |
| /-- An unnormalized finite Fourier filter, with an integer exponent. -/ | |
| theorem sum_zpow_primitive_root {ζ : ℂ} {L : ℕ} | |
| (hζ : IsPrimitiveRoot ζ L) (e : ℤ) : | |
| (∑ r ∈ Finset.range L, ζ ^ ((r : ℤ) * e)) = | |
| if (L : ℤ) ∣ e then (L : ℂ) else 0 := by | |
| have hpow : (ζ ^ e) ^ L = 1 := by | |
| rw [← zpow_natCast, ← zpow_mul, mul_comm, zpow_mul, hζ.zpow_eq_one, one_zpow] | |
| have hterm (r : ℕ) : ζ ^ ((r : ℤ) * e) = (ζ ^ e) ^ r := by | |
| rw [← zpow_natCast, ← zpow_mul, mul_comm] | |
| simp_rw [hterm] | |
| rw [sum_pow_root_of_unity _ _ hpow] | |
| simp only [hζ.zpow_eq_one_iff_dvd] | |
| /-- Averaging over a primitive root's period selects exactly its divisible exponents. -/ | |
| theorem normalized_sum_zpow_primitive_root {ζ : ℂ} {L : ℕ} | |
| (hL : L ≠ 0) (hζ : IsPrimitiveRoot ζ L) (e : ℤ) : | |
| (L : ℂ)⁻¹ * (∑ r ∈ Finset.range L, ζ ^ ((r : ℤ) * e)) = | |
| if (L : ℤ) ∣ e then 1 else 0 := by | |
| rw [sum_zpow_primitive_root hζ e] | |
| split_ifs <;> simp [hL] | |
| /-- Divisibility by a positive period is exact equality to zero inside one period. -/ | |
| lemma int_dvd_iff_eq_zero_of_abs_lt {L : ℕ} {e : ℤ} (he : |e| < (L : ℤ)) : | |
| (L : ℤ) ∣ e ↔ e = 0 := by | |
| constructor | |
| · intro hd | |
| apply abs_eq_zero.mp | |
| exact Int.eq_zero_of_dvd_of_nonneg_of_lt (abs_nonneg e) he | |
| ((dvd_abs (L : ℤ) e).mpr hd) | |
| · rintro rfl | |
| exact dvd_zero _ | |
| /-- Within one period, the finite Fourier filter selects exactly exponent zero. -/ | |
| theorem normalized_sum_zpow_primitive_root_of_abs_lt {ζ : ℂ} {L : ℕ} {e : ℤ} | |
| (hL : L ≠ 0) (hζ : IsPrimitiveRoot ζ L) (he : |e| < (L : ℤ)) : | |
| (L : ℂ)⁻¹ * (∑ r ∈ Finset.range L, ζ ^ ((r : ℤ) * e)) = | |
| if e = 0 then 1 else 0 := by | |
| rw [normalized_sum_zpow_primitive_root hL hζ e] | |
| simp only [int_dvd_iff_eq_zero_of_abs_lt he] | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |