| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialCurvature |
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| namespace OAI |
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| /-! Directional control by coordinate-plane derivatives. -/ |
| noncomputable section |
| open Matrix MvPolynomial |
| open scoped BigOperators |
| namespace InvariantIsing |
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|
| lemma plane_functional_abs_bound {N : ℕ} |
| (ell : Matrix (Fin N) (Fin N) ℝ →ₗ[ℝ] ℝ) |
| (A : Matrix (Fin N) (Fin N) ℝ) (hA : A.transpose = -A) |
| (δ : ℝ) (hδ : ∀ i j, |ell (planeGenerator i j)| ≤ δ) : |
| 2*|ell A| ≤ (∑ i, ∑ j, |A i j|)*δ := by |
| have hr := plane_functional_representation ell A hA |
| rw [matrixFrobeniusPair_eq_sum] at hr |
| change (∑ i, ∑ j, ell (planeGenerator i j)*A i j) = 2*ell A at hr |
| have hs : |∑ i, ∑ j, ell (planeGenerator i j)*A i j| ≤ |
| (∑ i, ∑ j, |A i j|)*δ := by |
| calc |
| |∑ i, ∑ j, ell (planeGenerator i j)*A i j| ≤ |
| ∑ i, |∑ j, ell (planeGenerator i j)*A i j| := Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ i, ∑ j, |ell (planeGenerator i j)*A i j| := by |
| apply Finset.sum_le_sum; intro i _ |
| exact Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ i, ∑ j, |A i j| * δ := by |
| apply Finset.sum_le_sum; intro i _ |
| apply Finset.sum_le_sum; intro j _ |
| rw [abs_mul,mul_comm] |
| exact mul_le_mul_of_nonneg_left (hδ i j) (abs_nonneg _) |
| _ = (∑ i, ∑ j, |A i j|)*δ := by simp only [Finset.sum_mul] |
| rw [hr,abs_mul,abs_of_nonneg (by norm_num : (0:ℝ) ≤ 2)] at hs |
| exact hs |
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|
| lemma matrixPolynomial_direction_bound {N : ℕ} |
| (p : MatrixPolynomial N) (M A : Matrix (Fin N) (Fin N) ℝ) |
| (hA : A.transpose = -A) (δ : ℝ) (hδ : 0 ≤ δ) |
| (hp : matrixPolynomialEval M (haarPolynomialGamma p p) ≤ δ^2) : |
| 2*|matrixPolynomialEval M (matrixPolynomialDerivation A p)| ≤ |
| (∑ i, ∑ j, |A i j|)*δ := by |
| apply plane_functional_abs_bound (matrixPolynomialDifferential p M) A hA δ |
| intro i j |
| change |matrixPolynomialEval M (matrixPolynomialDerivation (planeGenerator i j) p)| ≤ δ |
| have hpos (i j : Fin N) : |
| 0 ≤ (matrixPolynomialEval M (matrixPolynomialDerivation (planeGenerator i j) p))^2 := |
| sq_nonneg _ |
| have htotal : matrixPolynomialEval M (haarPolynomialGamma p p) = |
| ∑ i : Fin N, ∑ j : Fin N, |
| (matrixPolynomialEval M (matrixPolynomialDerivation (planeGenerator i j) p))^2 := by |
| simp only [haarPolynomialGamma,map_sum,map_mul,pow_two] |
| have hs : (matrixPolynomialEval M (matrixPolynomialDerivation (planeGenerator i j) p))^2 ≤ δ^2 := by |
| apply le_trans _ hp |
| rw [htotal] |
| exact (Finset.single_le_sum (fun k _ => hpos i k) (Finset.mem_univ j)).trans |
| (Finset.single_le_sum (fun k _ => Finset.sum_nonneg (fun l _ => hpos k l)) (Finset.mem_univ i)) |
| exact (sq_le_sq₀ (abs_nonneg _) hδ).mp (by simpa only [sq_abs] using hs) |
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| end InvariantIsing |
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| end |
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| end OAI |
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