| import OAI.Probability.InvariantIsing.Haar.HaarCurveTangent |
| import OAI.Probability.InvariantIsing.Haar.HaarPolynomialValue |
| import Mathlib.Analysis.Calculus.MeanValue |
|
|
| namespace OAI |
|
|
| /-! Integrating the coordinate-plane gradient bound along an orthogonal curve. -/ |
| noncomputable section |
| open Matrix MvPolynomial Set |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| lemma haarPolynomial_curve_bound {N : ℕ} (p : MatrixPolynomial N) |
| (U : ℝ → SpecialOrthogonal N) (B : ℝ → Matrix (Fin N) (Fin N) ℝ) |
| (K δ : ℝ) (hδ : 0 ≤ δ) |
| (hB : ∀ t ∈ Icc (0:ℝ) 1, ∀ i j, |
| HasDerivAt (fun s => (U s : Matrix (Fin N) (Fin N) ℝ) i j) (B t i j) t) |
| (hK : ∀ t ∈ Icc (0:ℝ) 1, |
| (∑ i, ∑ j, |(B t*(U t : Matrix (Fin N) (Fin N) ℝ).transpose) i j|) ≤ K) |
| (hp : ∀ V : SpecialOrthogonal N, haarPolynomialValue (haarPolynomialGamma p p) V ≤ δ^2) : |
| |haarPolynomialValue p (U 1)-haarPolynomialValue p (U 0)| ≤ K*δ/2 := by |
| let f' := fun t => matrixPolynomialEval (U t : Matrix (Fin N) (Fin N) ℝ) |
| (matrixPolynomialDerivation (B t*(U t : Matrix (Fin N) (Fin N) ℝ).transpose) p) |
| have hd (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : |
| HasDerivAt (fun s => haarPolynomialValue p (U s)) (f' t) t := |
| orthogonalCurve_polynomial_derivative p U (B t) t (hB t ht) |
| have hb (t : ℝ) (ht : t ∈ Ico (0:ℝ) 1) : ‖f' t‖ ≤ K*δ/2 := by |
| have ht' : t ∈ Icc (0:ℝ) 1 := ⟨ht.1,ht.2.le⟩ |
| have hx := matrixPolynomial_direction_bound p |
| (U t : Matrix (Fin N) (Fin N) ℝ) |
| (B t*(U t : Matrix (Fin N) (Fin N) ℝ).transpose) |
| (orthogonalCurve_tangent_skew U (B t) t (hB t ht')) δ hδ (hp (U t)) |
| have hy := mul_le_mul_of_nonneg_right (hK t ht') hδ |
| change |f' t| ≤ K*δ/2 |
| dsimp [f'] at * |
| linarith |
| simpa only [Real.norm_eq_abs] using |
| norm_image_sub_le_of_norm_deriv_le_segment_01' |
| (fun t ht => (hd t ht).hasDerivWithinAt) hb |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
