File size: 2,836 Bytes
a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | import OAI.Probability.InvariantIsing.Haar.HaarPolynomialCurvature
namespace OAI
/-! Directional control by coordinate-plane derivatives. -/
noncomputable section
open Matrix MvPolynomial
open scoped BigOperators
namespace InvariantIsing
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
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)
end InvariantIsing
end
end OAI
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