| import OAI.Geometry.Kahler.BaseNegativeThreshold |
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| namespace OAI |
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| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology |
| open scoped ContDiff |
| open Set Filter Topology MeasureTheory |
| noncomputable section |
|
|
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
|
|
| lemma logarithmicTest_scaled (a ε r : ℝ) (P : Finset Sphere) (k : ℕ) (ξ : Sphere) : |
| logarithmicTest a ε P k ((r:ℂ) • (ξ:Base)) = |
| regularizedLog 1 ε (((a:ℂ)*(r:ℂ)^k)*peakPolynomial P k ξ) := by |
| rw [logarithmicTest,peakPolynomial_homogeneous,regularizedLog_scale] |
| congr 1 |
| ring |
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| lemma logarithmicTest_later_comparison {a : ℝ} (ha : 0 ≤ a) (pr : RadialProfiles a) |
| {Q M i N : ℕ} (hQ : 2 ≤ Q) (hM : M+2 < Q) (hi : 1 ≤ i) (hiN : i ≤ N) |
| (P : ℕ → Finset Sphere) {D : ℝ} (hD : 1 ≤ D) |
| (hP : ProjectivelySeparated (D/Real.sqrt (Q^i:ℕ)) (P (Q^i))) |
| {ε θ r : ℝ} (he : ε ≠ 0) (hθ : 0 ≤ θ) (hθ1 : θ < 1) (hr : 0 ≤ r) (hr1 : r ≤ 1) |
| (hb : ∀ q ∈ Icc 0 (a*gaussianConstant), q/logFactor q ε ≤ θ) |
| (hposi : ∀ ξ : Sphere, 0 ≤ density Q pr.R pr.f pr.b P i ξ) |
| (hposN : ∀ ξ : Sphere, 0 ≤ density Q pr.R pr.f pr.b P N ξ) |
| [IsProbabilityMeasure (densityMeasure (density Q pr.R pr.f pr.b P i))] |
| [IsProbabilityMeasure (densityMeasure (density Q pr.R pr.f pr.b P N))] : |
| |(∫ ξ : Sphere, logarithmicTest a ε (P (Q^i)) (Q^i) ((r:ℂ) • (ξ:Base)) ∂densityMeasure (density Q pr.R pr.f pr.b P N))- |
| (∫ ξ : Sphere, logarithmicTest a ε (P (Q^i)) (Q^i) ((r:ℂ) • (ξ:Base)) ∂densityMeasure (density Q pr.R pr.f pr.b P i))| ≤ |
| 4*θ^(M+1)/(1-θ) := by |
| simp_rw [logarithmicTest_scaled] |
| apply density_log_comparison pr hQ hM hiN P he hθ hθ1 _ _ hposi hposN |
| intro ξ |
| rw [logNormalized,norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (logFactor_pos _ _)] |
| apply hb _ |
| constructor |
| · exact norm_nonneg _ |
| · rw [norm_mul,norm_mul,norm_pow,Complex.norm_real,Complex.norm_real, |
| Real.norm_eq_abs,Real.norm_eq_abs,abs_of_nonneg ha,abs_of_nonneg hr] |
| have hk : 1 ≤ Q^i := (show 1 ≤ Q by omega).trans |
| (le_self_pow (by omega) (Nat.ne_of_gt hi)) |
| calc a*r^(Q^i)*‖peakPolynomial (P (Q^i)) (Q^i) ξ‖ ≤ a*1*gaussianConstant := |
| mul_le_mul (mul_le_mul_of_nonneg_left (pow_le_one₀ hr hr1) ha) |
| (peak_sphere_bound hD hk hP ξ) (norm_nonneg _) (by positivity) |
| _ = _ := by ring |
|
|
| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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|
