openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Separation /FiniteProjection.lean
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Separation.Fourier | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Separation.NoWrap | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Separation.SquareWeights | |
| import Mathlib.Algebra.Polynomial.Degree.Lemmas | |
| namespace OAI | |
| /-! | |
| # Exact finite Fourier projection and square filtering | |
| The `5 * M`-term Fourier average is applied to an arbitrary branch coefficient. | |
| The label bounds eliminate modular wrap. The weights on the three tensor legs | |
| then give a nonnegative square degree on every surviving branch. Taking the | |
| constant coefficient retains exactly the branches with `g = h` and `u = v`. | |
| The finite-family statements permit arbitrary coefficients and arbitrary | |
| repetitions of labels; they use only linearity of the Fourier projection and | |
| coefficient extraction, without assumptions on an ambient tensor. | |
| -/ | |
| open scoped BigOperators | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| /-- A normalized finite Fourier average of one branch coefficient. -/ | |
| noncomputable def finiteFourierCoefficient (ζ : ℂ) (L : ℕ) (e : ℤ) (c : ℂ) : ℂ := | |
| (L : ℂ)⁻¹ * ∑ r ∈ Finset.range L, ζ ^ ((r : ℤ) * e) * c | |
| /-- The Fourier average is a scalar support indicator, for every coefficient. -/ | |
| theorem finiteFourierCoefficient_eq_ite {ζ : ℂ} {L : ℕ} | |
| (hL : L ≠ 0) (hζ : IsPrimitiveRoot ζ L) (e : ℤ) (c : ℂ) : | |
| finiteFourierCoefficient ζ L e c = if (L : ℤ) ∣ e then c else 0 := by | |
| unfold finiteFourierCoefficient | |
| rw [← Finset.sum_mul, ← mul_assoc, normalized_sum_zpow_primitive_root hL hζ] | |
| split_ifs <;> simp | |
| /-- Averaging a finite family of branches agrees with averaging each branch. -/ | |
| theorem finiteFourierCoefficient_sum {ι : Type*} (s : Finset ι) | |
| (ζ : ℂ) (L : ℕ) (e : ι → ℤ) (c : ι → ℂ) : | |
| (L : ℂ)⁻¹ * (∑ r ∈ Finset.range L, ∑ i ∈ s, ζ ^ ((r : ℤ) * e i) * c i) = | |
| ∑ i ∈ s, finiteFourierCoefficient ζ L (e i) (c i) := by | |
| rw [Finset.sum_comm, Finset.mul_sum] | |
| rfl | |
| /-- Exactly `5 * M` phases select the integer support equation. -/ | |
| theorem fiveMFourierCoefficient_eq_ite {ζ : ℂ} {M : ℕ} {g h u v : ℤ} | |
| (hM : 0 < M) (hζ : IsPrimitiveRoot ζ (5 * M)) | |
| (hg : 1 ≤ g ∧ g ≤ M) (hh : 1 ≤ h ∧ h ≤ M) | |
| (hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M) (c : ℂ) : | |
| finiteFourierCoefficient ζ (5 * M) (phase g h u v) c = | |
| if phase g h u v = 0 then c else 0 := by | |
| rw [finiteFourierCoefficient_eq_ite (by omega) hζ] | |
| have hnoWrap := fourierPhase_dvd_iff_eq_zero (M := (M : ℤ)) | |
| (by omega) hh hg hu hv | |
| simp only [phase, Nat.cast_mul, Nat.cast_ofNat, hnoWrap] | |
| rfl | |
| /-- A projected branch, recorded as an ordinary polynomial after Fourier averaging. -/ | |
| noncomputable def projectedBranchPolynomial (ζ : ℂ) (M : ℕ) | |
| (g h u v : ℤ) (c : ℂ) : Polynomial ℂ := | |
| Polynomial.monomial (totalDegree g h u v) | |
| (finiteFourierCoefficient ζ (5 * M) (phase g h u v) c) | |
| /-- The surviving polynomial weight is the square `(g - h)^2`. -/ | |
| theorem projectedBranchPolynomial_eq_ite {ζ : ℂ} {M : ℕ} {g h u v : ℤ} | |
| (hM : 0 < M) (hζ : IsPrimitiveRoot ζ (5 * M)) | |
| (hg : 1 ≤ g ∧ g ≤ M) (hh : 1 ≤ h ∧ h ≤ M) | |
| (hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M) (c : ℂ) : | |
| projectedBranchPolynomial ζ M g h u v c = | |
| if phase g h u v = 0 then Polynomial.monomial ((g - h) ^ 2).toNat c else 0 := by | |
| unfold projectedBranchPolynomial | |
| rw [fiveMFourierCoefficient_eq_ite hM hζ hg hh hu hv] | |
| split_ifs with hp | |
| · rw [totalDegree, totalWeight_eq_sq_of_phase_eq_zero hp] | |
| · simp | |
| /-- A coefficient occurs exactly on the Fourier support at its integer total weight. -/ | |
| theorem projectedBranchPolynomial_coeff {ζ : ℂ} {M n : ℕ} {g h u v : ℤ} | |
| (hM : 0 < M) (hζ : IsPrimitiveRoot ζ (5 * M)) | |
| (hg : 1 ≤ g ∧ g ≤ M) (hh : 1 ≤ h ∧ h ≤ M) | |
| (hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M) (c : ℂ) : | |
| (projectedBranchPolynomial ζ M g h u v c).coeff n = | |
| if phase g h u v = 0 ∧ totalWeight g h u v = (n : ℤ) then c else 0 := by | |
| unfold projectedBranchPolynomial | |
| rw [fiveMFourierCoefficient_eq_ite hM hζ hg hh hu hv] | |
| by_cases hp : phase g h u v = 0 | |
| · have hd : totalDegree g h u v = n ↔ totalWeight g h u v = (n : ℤ) := by | |
| have hcast := totalDegree_cast_of_phase_eq_zero hp | |
| omega | |
| simp only [hp, ↓reduceIte, Polynomial.coeff_monomial, hd, true_and] | |
| · simp [hp] | |
| /-- Each projected branch has no polynomial coefficient above `(M - 1)^2`. -/ | |
| theorem projectedBranchPolynomial_coeff_eq_zero_of_lt | |
| {ζ : ℂ} {M n : ℕ} {g h u v : ℤ} | |
| (hM : 0 < M) (hζ : IsPrimitiveRoot ζ (5 * M)) | |
| (hg : 1 ≤ g ∧ g ≤ M) (hh : 1 ≤ h ∧ h ≤ M) | |
| (hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M) | |
| (hn : (M - 1) ^ 2 < n) (c : ℂ) : | |
| (projectedBranchPolynomial ζ M g h u v c).coeff n = 0 := by | |
| unfold projectedBranchPolynomial | |
| rw [fiveMFourierCoefficient_eq_ite hM hζ hg hh hu hv] | |
| split_ifs with hp | |
| · have hd := totalDegree_le_of_labels hg hh hp | |
| exact Polynomial.coeff_monomial_of_ne c (by omega) | |
| · simp | |
| /-- Constant coefficient extraction is the exact square filter, with no restriction | |
| on the original complex coefficient. -/ | |
| theorem projectedBranchPolynomial_coeff_zero {ζ : ℂ} {M : ℕ} {g h u v : ℤ} | |
| (hM : 0 < M) (hζ : IsPrimitiveRoot ζ (5 * M)) | |
| (hg : 1 ≤ g ∧ g ≤ M) (hh : 1 ≤ h ∧ h ≤ M) | |
| (hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M) (c : ℂ) : | |
| (projectedBranchPolynomial ζ M g h u v c).coeff 0 = | |
| if g = h ∧ u = v then c else 0 := by | |
| unfold projectedBranchPolynomial | |
| rw [fiveMFourierCoefficient_eq_ite hM hζ hg hh hu hv] | |
| by_cases hp : phase g h u v = 0 | |
| · simp only [hp, ↓reduceIte, Polynomial.coeff_monomial, | |
| totalDegree_eq_zero_iff_of_phase_eq_zero hp] | |
| · have hlabels : ¬ (g = h ∧ u = v) := by | |
| rintro ⟨rfl, rfl⟩ | |
| exact hp (by simp [phase]) | |
| simp [hp, hlabels] | |
| /-- Apply the finite Fourier projection and square weights branch by branch. -/ | |
| noncomputable def projectedPolynomial {ι : Type*} (s : Finset ι) | |
| (ζ : ℂ) (M : ℕ) (g h u v : ι → ℤ) (c : ι → ℂ) : Polynomial ℂ := | |
| ∑ i ∈ s, projectedBranchPolynomial ζ M (g i) (h i) (u i) (v i) (c i) | |
| /-- Every coefficient of the projected polynomial has the exact integer support. -/ | |
| theorem projectedPolynomial_coeff {ι : Type*} (s : Finset ι) | |
| {ζ : ℂ} {M n : ℕ} (g h u v : ι → ℤ) (c : ι → ℂ) | |
| (hM : 0 < M) (hζ : IsPrimitiveRoot ζ (5 * M)) | |
| (hg : ∀ i ∈ s, 1 ≤ g i ∧ g i ≤ M) (hh : ∀ i ∈ s, 1 ≤ h i ∧ h i ≤ M) | |
| (hu : ∀ i ∈ s, 1 ≤ u i ∧ u i ≤ M) (hv : ∀ i ∈ s, 1 ≤ v i ∧ v i ≤ M) : | |
| (projectedPolynomial s ζ M g h u v c).coeff n = | |
| ∑ i ∈ s, if phase (g i) (h i) (u i) (v i) = 0 ∧ | |
| totalWeight (g i) (h i) (u i) (v i) = (n : ℤ) then c i else 0 := by | |
| classical | |
| unfold projectedPolynomial | |
| simp only [Polynomial.finsetSum_coeff] | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| exact projectedBranchPolynomial_coeff hM hζ (hg i hi) (hh i hi) | |
| (hu i hi) (hv i hi) (c i) | |
| /-- Exact finite filtering for an arbitrary finite family of branch coefficients. -/ | |
| theorem projectedPolynomial_coeff_zero {ι : Type*} (s : Finset ι) | |
| {ζ : ℂ} {M : ℕ} (g h u v : ι → ℤ) (c : ι → ℂ) | |
| (hM : 0 < M) (hζ : IsPrimitiveRoot ζ (5 * M)) | |
| (hg : ∀ i ∈ s, 1 ≤ g i ∧ g i ≤ M) (hh : ∀ i ∈ s, 1 ≤ h i ∧ h i ≤ M) | |
| (hu : ∀ i ∈ s, 1 ≤ u i ∧ u i ≤ M) (hv : ∀ i ∈ s, 1 ≤ v i ∧ v i ≤ M) : | |
| (projectedPolynomial s ζ M g h u v c).coeff 0 = | |
| ∑ i ∈ s, if g i = h i ∧ u i = v i then c i else 0 := by | |
| classical | |
| unfold projectedPolynomial | |
| simp only [Polynomial.finsetSum_coeff] | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| exact projectedBranchPolynomial_coeff_zero hM hζ (hg i hi) (hh i hi) | |
| (hu i hi) (hv i hi) (c i) | |
| /-- The finite projected polynomial has support only in degrees at most `(M - 1)^2`. -/ | |
| theorem projectedPolynomial_coeff_eq_zero_of_lt {ι : Type*} (s : Finset ι) | |
| {ζ : ℂ} {M n : ℕ} (g h u v : ι → ℤ) (c : ι → ℂ) | |
| (hM : 0 < M) (hζ : IsPrimitiveRoot ζ (5 * M)) | |
| (hg : ∀ i ∈ s, 1 ≤ g i ∧ g i ≤ M) (hh : ∀ i ∈ s, 1 ≤ h i ∧ h i ≤ M) | |
| (hu : ∀ i ∈ s, 1 ≤ u i ∧ u i ≤ M) (hv : ∀ i ∈ s, 1 ≤ v i ∧ v i ≤ M) | |
| (hn : (M - 1) ^ 2 < n) : | |
| (projectedPolynomial s ζ M g h u v c).coeff n = 0 := by | |
| classical | |
| unfold projectedPolynomial | |
| simp only [Polynomial.finsetSum_coeff] | |
| apply Finset.sum_eq_zero | |
| intro i hi | |
| exact projectedBranchPolynomial_coeff_eq_zero_of_lt hM hζ (hg i hi) (hh i hi) | |
| (hu i hi) (hv i hi) hn (c i) | |
| /-- The degree bound is independent of the number or values of the coefficients. -/ | |
| theorem projectedPolynomial_natDegree_le {ι : Type*} (s : Finset ι) | |
| {ζ : ℂ} {M : ℕ} (g h u v : ι → ℤ) (c : ι → ℂ) | |
| (hM : 0 < M) (hζ : IsPrimitiveRoot ζ (5 * M)) | |
| (hg : ∀ i ∈ s, 1 ≤ g i ∧ g i ≤ M) (hh : ∀ i ∈ s, 1 ≤ h i ∧ h i ≤ M) | |
| (hu : ∀ i ∈ s, 1 ≤ u i ∧ u i ≤ M) (hv : ∀ i ∈ s, 1 ≤ v i ∧ v i ≤ M) : | |
| (projectedPolynomial s ζ M g h u v c).natDegree ≤ (M - 1) ^ 2 := by | |
| rw [Polynomial.natDegree_le_iff_coeff_eq_zero] | |
| intro n hn | |
| exact projectedPolynomial_coeff_eq_zero_of_lt s g h u v c hM hζ hg hh hu hv hn | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |