| import OAI.LinearAlgebra.Hadamard.Equivalence | |
| namespace OAI | |
| /-! Fourier kernel identities and symmetric polynomials of row ratios. -/ | |
| open scoped BigOperators Matrix | |
| namespace HadamardSix | |
| lemma character_sub (x : Index → ℂ) (hx : ∀ i, ‖x i‖ = 1) (a b : Charge) : | |
| character x (a - b) = character x a * star (character x b) := by | |
| rw [sub_eq_add_neg, character_add x (fun k => unit_ne_zero (hx k)), character_neg x hx] | |
| lemma column_kernel_identity {H : Matrix6} (hH : IsComplexHadamard H) | |
| (A B : Index → ℂ) (i : Index) : | |
| (∑ j, (∑ k, A k * H i k * star (H j k)) * | |
| (∑ l, B l * star (H i l) * H j l)) = 6 * ∑ k, A k * B k := by | |
| have hexpand : | |
| (∑ j, (∑ k, A k * H i k * star (H j k)) * | |
| (∑ l, B l * star (H i l) * H j l)) = | |
| ∑ k, ∑ l, A k * B l * H i k * star (H i l) * | |
| ∑ j, star (H j k) * H j l := by | |
| simp only [Finset.sum_mul, Finset.mul_sum] | |
| conv_rhs => rw [Finset.sum_comm] | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro l _ | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| ring | |
| rw [hexpand] | |
| simp_rw [hH.column_orthogonality] | |
| simp only [mul_ite, mul_zero, Finset.sum_ite_eq, Finset.mem_univ, ite_true] | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| calc | |
| A k * B k * H i k * star (H i k) * 6 = 6 * (A k * B k) * | |
| (H i k * star (H i k)) := by ring | |
| _ = 6 * (A k * B k) := by rw [unit_mul_star (hH.1 i k), mul_one] | |
| lemma same_matrix_identity {H : Matrix6} (hH : IsComplexHadamard H) | |
| (a b : Charge) (i : Index) : | |
| ∑ j, g H (a + differenceCharge 1 i j) * g H (b - differenceCharge 1 i j) = | |
| g H (a + b) := by | |
| have hplus (j : Index) : | |
| g H (a + differenceCharge 1 i j) = (6 : ℂ)⁻¹ * | |
| ∑ k, character (fun t => H t k) a * H i k * star (H j k) := by | |
| unfold g | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| rw [character_add _ (fun t => hH.1.entry_ne_zero t k), | |
| character_difference _ (fun t => hH.1 t k)] | |
| simp [mul_assoc] | |
| have hminus (j : Index) : | |
| g H (b - differenceCharge 1 i j) = (6 : ℂ)⁻¹ * | |
| ∑ k, character (fun t => H t k) b * star (H i k) * H j k := by | |
| unfold g | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| rw [character_sub _ (fun t => hH.1 t k), | |
| character_difference _ (fun t => hH.1 t k)] | |
| simp only [zpow_one, star_mul, star_star] | |
| ring | |
| simp_rw [hplus, hminus] | |
| have hscale (x y : ℂ) : (6 : ℂ)⁻¹ * x * ((6 : ℂ)⁻¹ * y) = | |
| (6 : ℂ)⁻¹ ^ 2 * (x * y) := by ring | |
| simp_rw [hscale] | |
| rw [← Finset.mul_sum, column_kernel_identity hH] | |
| unfold g | |
| simp_rw [character_add _ (fun t => hH.1.entry_ne_zero t _)] | |
| ring | |
| noncomputable def elementary (v : Index → ℂ) (k : ℕ) : ℂ := | |
| (Finset.univ.val.map v).esymm k | |
| lemma elementary_eq_sum (v : Index → ℂ) (k : ℕ) : | |
| elementary v k = ∑ s ∈ Finset.univ.powersetCard k, ∏ i ∈ s, v i := | |
| Finset.esymm_map_val v Finset.univ k | |
| lemma elementary_zero (v : Index → ℂ) : elementary v 0 = 1 := by | |
| simp [elementary_eq_sum] | |
| lemma elementary_one (v : Index → ℂ) : elementary v 1 = ∑ i, v i := by | |
| simp [elementary_eq_sum, Finset.powersetCard_one] | |
| lemma elementary_six (v : Index → ℂ) : elementary v 6 = ∏ i, v i := by | |
| rw [elementary_eq_sum] | |
| have hcard : (Finset.univ : Finset Index).card = 6 := by decide | |
| rw [show Finset.univ.powersetCard 6 = ({Finset.univ} : Finset (Finset Index)) by | |
| simpa only [hcard] using Finset.powersetCard_self (Finset.univ : Finset Index)] | |
| rw [Finset.sum_singleton] | |
| lemma elementary_two_of_power_sums (v : Index → ℂ) | |
| (h1 : ∑ i, v i = 0) (h2 : ∑ i, v i ^ 2 = 0) : elementary v 2 = 0 := by | |
| have ha : (Finset.HasAntidiagonal.antidiagonal 2 : Finset (ℕ × ℕ)) = | |
| {(0, 2), (1, 1), (2, 0)} := by decide | |
| have hnpoly := MvPolynomial.mul_esymm_eq_sum Index ℂ 2 | |
| rw [ha] at hnpoly | |
| norm_num [Finset.sum_filter] at hnpoly | |
| have hn := congrArg (MvPolynomial.aeval v : MvPolynomial Index ℂ →ₐ[ℂ] ℂ) hnpoly | |
| simp only [map_mul, map_ofNat, map_neg, map_add, map_sum, | |
| MvPolynomial.aeval_esymm_eq_multiset_esymm] at hn | |
| simp [MvPolynomial.psum, h1, h2] at hn | |
| exact hn | |
| lemma elementary_complement (v : Index → ℂ) (hv : ∀ i, ‖v i‖ = 1) | |
| (k : ℕ) (hk : k ≤ 6) : | |
| elementary v (6 - k) = elementary v 6 * star (elementary v k) := by | |
| rw [elementary_eq_sum, elementary_six, elementary_eq_sum, star_sum, Finset.mul_sum] | |
| refine Finset.sum_bij' (fun s _ => sᶜ) (fun s _ => sᶜ) ?_ ?_ ?_ ?_ ?_ | |
| · intro s hs | |
| rw [Finset.mem_powersetCard_univ] at hs ⊢ | |
| rw [Finset.card_compl, Fintype.card_fin, hs, Nat.sub_sub_self hk] | |
| · intro s hs | |
| rw [Finset.mem_powersetCard_univ] at hs ⊢ | |
| rw [Finset.card_compl, Fintype.card_fin, hs] | |
| · intro s hs | |
| exact compl_compl s | |
| · intro s hs | |
| exact compl_compl s | |
| · intro s hs | |
| have hunit : ‖∏ i ∈ sᶜ, v i‖ = 1 := by | |
| simp [norm_prod, hv] | |
| rw [← Finset.prod_mul_prod_compl s v, mul_assoc, unit_mul_star hunit, mul_one] | |
| noncomputable def ratioPolynomial (v : Index → ℂ) : Polynomial ℂ := | |
| ∏ i, (Polynomial.X - Polynomial.C (v i)) | |
| lemma ratioPolynomial_vieta (v : Index → ℂ) : | |
| ratioPolynomial v = ∑ k ∈ Finset.range 7, | |
| (-1) ^ k * (Polynomial.C (elementary v k) * Polynomial.X ^ (6 - k)) := by | |
| have h := Multiset.prod_X_sub_X_eq_sum_esymm (Finset.univ.val.map v) | |
| simp only [Multiset.card_map, ← Finset.card_def, | |
| Finset.card_univ, Fintype.card_fin] at h | |
| simpa only [ratioPolynomial, Finset.prod_eq_multiset_prod, Multiset.map_map, | |
| Function.comp_apply, elementary] using h | |
| lemma ratioPolynomial_of_power_sums (v : Index → ℂ) (hv : ∀ i, ‖v i‖ = 1) | |
| (h1 : ∑ i, v i = 0) (h2 : ∑ i, v i ^ 2 = 0) : | |
| ratioPolynomial v = Polynomial.X ^ 6 - | |
| Polynomial.C (elementary v 3) * Polynomial.X ^ 3 + | |
| Polynomial.C (elementary v 6) := by | |
| have e1 : elementary v 1 = 0 := by rw [elementary_one, h1] | |
| have e2 := elementary_two_of_power_sums v h1 h2 | |
| have e4 : elementary v 4 = 0 := by | |
| simpa only [e2, star_zero, mul_zero] using elementary_complement v hv 2 (by decide) | |
| have e5 : elementary v 5 = 0 := by | |
| simpa only [e1, star_zero, mul_zero] using elementary_complement v hv 1 (by decide) | |
| rw [ratioPolynomial_vieta] | |
| norm_num [Finset.sum_range_succ, elementary_zero, e1, e2, e4, e5, sub_eq_add_neg] | |
| lemma ratioPolynomial_eval (v : Index → ℂ) (i : Index) : | |
| (ratioPolynomial v).eval (v i) = 0 := by | |
| classical | |
| rw [ratioPolynomial, Polynomial.eval_prod] | |
| apply Finset.prod_eq_zero (Finset.mem_univ i) | |
| simp | |
| lemma ratio_power_six (v : Index → ℂ) (hv : ∀ i, ‖v i‖ = 1) | |
| (h1 : ∑ i, v i = 0) (h2 : ∑ i, v i ^ 2 = 0) (i : Index) : | |
| v i ^ 6 - elementary v 3 * v i ^ 3 + elementary v 6 = 0 := by | |
| have h := ratioPolynomial_eval v i | |
| rw [ratioPolynomial_of_power_sums v hv h1 h2] at h | |
| simpa using h | |
| lemma ratio_cubes_two_values (v : Index → ℂ) (hv : ∀ i, ‖v i‖ = 1) | |
| (h1 : ∑ i, v i = 0) (h2 : ∑ i, v i ^ 2 = 0) (i : Index) : | |
| v i ^ 3 = v 0 ^ 3 ∨ v i ^ 3 = elementary v 3 - v 0 ^ 3 := by | |
| have hz : (v i ^ 3 - v 0 ^ 3) * (v i ^ 3 - (elementary v 3 - v 0 ^ 3)) = 0 := by | |
| linear_combination ratio_power_six v hv h1 h2 i - ratio_power_six v hv h1 h2 0 | |
| simpa only [sub_eq_zero] using mul_eq_zero.mp hz | |
| lemma three_mul_elementary_three (v : Index → ℂ) | |
| (h1 : ∑ i, v i = 0) (h2 : ∑ i, v i ^ 2 = 0) : | |
| 3 * elementary v 3 = ∑ i, v i ^ 3 := by | |
| have ha : (Finset.HasAntidiagonal.antidiagonal 3 : Finset (ℕ × ℕ)) = | |
| {(0, 3), (1, 2), (2, 1), (3, 0)} := by decide | |
| have hnpoly := MvPolynomial.mul_esymm_eq_sum Index ℂ 3 | |
| rw [ha] at hnpoly | |
| norm_num [Finset.sum_filter] at hnpoly | |
| have hn := congrArg (MvPolynomial.aeval v : MvPolynomial Index ℂ →ₐ[ℂ] ℂ) hnpoly | |
| simp only [map_mul, map_ofNat, map_neg, map_add, map_sum, | |
| MvPolynomial.aeval_esymm_eq_multiset_esymm] at hn | |
| simpa [elementary, MvPolynomial.psum, h1, h2] using hn | |
| lemma fourth_power_sum_zero (v : Index → ℂ) (hv : ∀ i, ‖v i‖ = 1) | |
| (h1 : ∑ i, v i = 0) (h2 : ∑ i, v i ^ 2 = 0) : ∑ i, v i ^ 4 = 0 := by | |
| have ha : (Finset.HasAntidiagonal.antidiagonal 4 : Finset (ℕ × ℕ)) = | |
| {(0, 4), (1, 3), (2, 2), (3, 1), (4, 0)} := by decide | |
| have he4 : elementary v 4 = 0 := by | |
| simpa only [elementary_two_of_power_sums v h1 h2, star_zero, mul_zero] | |
| using elementary_complement v hv 2 (by decide) | |
| have hnpoly := MvPolynomial.mul_esymm_eq_sum Index ℂ 4 | |
| rw [ha] at hnpoly | |
| norm_num [Finset.sum_filter] at hnpoly | |
| have hn := congrArg (MvPolynomial.aeval v : MvPolynomial Index ℂ →ₐ[ℂ] ℂ) hnpoly | |
| simp only [map_mul, map_ofNat, map_neg, map_add, map_sum, | |
| MvPolynomial.aeval_esymm_eq_multiset_esymm] at hn | |
| have hn' : 4 * elementary v 4 = -(∑ i, v i ^ 4) := by | |
| simpa [elementary, MvPolynomial.psum, h1, h2] using hn | |
| rw [he4] at hn' | |
| simpa using hn'.symm | |
| end HadamardSix | |
| end OAI | |