| import Mathlib |
| import OAI.Probability.SKRatio.Variational.CompactField |
| import OAI.Probability.SKRatio.Variational.ScalarCoherent |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped Topology |
| open MeasureTheory ProbabilityTheory Filter Real |
| namespace SKRatio.Bins |
| open Planted Scalar |
|
|
| lemma compact_coefficients_tolerance {a : ℝ} (ha : 0<a) : |
| ∃ δ : ℝ, 0<δ ∧ ∀ H : ℝ → ℝ, |
| (∀ x, dist (compactWeight x) (compactWeight (H x))<δ) → |
| (∀ x, |v x-v (H x)|≤a) ∧ |
| (∀ x, |m x*v x-m (H x)*v (H x)|≤a) ∧ |
| (∀ x y, |v y^2/(w x+w y)-v (H y)^2/(w (H x)+w (H y))|≤a) ∧ |
| (∀ x y, |kernel x y-kernel (H x) (H y)|≤a) := by |
| have cm : Continuous (fun u : FieldWeight => (1-(u:ℝ))*compactV u) := |
| (continuous_const.sub continuous_subtype_val).mul continuous_compactV |
| obtain ⟨δv,hδv,hv⟩ := Metric.uniformContinuous_iff.mp |
| (CompactSpace.uniformContinuous_of_continuous continuous_compactV) a ha |
| obtain ⟨δm,hδm,hm⟩ := Metric.uniformContinuous_iff.mp |
| (CompactSpace.uniformContinuous_of_continuous cm) a ha |
| obtain ⟨δf,hδf,hf⟩ := Metric.uniformContinuous_iff.mp |
| (CompactSpace.uniformContinuous_of_continuous continuous_compactF) a ha |
| obtain ⟨δk,hδk,hk⟩ := Metric.uniformContinuous_iff.mp |
| (CompactSpace.uniformContinuous_of_continuous continuous_compactK) a ha |
| let δ := min (min δv δm) (min δf δk) |
| have hδ : 0<δ := lt_min (lt_min hδv hδm) (lt_min hδf hδk) |
| have hδv' : δ≤δv := (min_le_left _ _).trans (min_le_left _ _) |
| have hδm' : δ≤δm := (min_le_left _ _).trans (min_le_right _ _) |
| have hδf' : δ≤δf := (min_le_right _ _).trans (min_le_left _ _) |
| have hδk' : δ≤δk := (min_le_right _ _).trans (min_le_right _ _) |
| refine ⟨δ,hδ,fun H hH => ⟨?_,?_,?_,?_⟩⟩ |
| · intro x |
| have he := hv ((hH x).trans_le hδv') |
| simpa only [Real.dist_eq,compactV_weight] using he.le |
| · intro x |
| have he := hm ((hH x).trans_le hδm') |
| have heq (x : ℝ) : 1-(compactWeight x:ℝ)=m x := by dsimp [compactWeight,w]; ring |
| simpa only [Real.dist_eq,compactV_weight,heq] using he.le |
| · intro x y |
| have hd : dist (compactWeight x,compactWeight y) (compactWeight (H x),compactWeight (H y))<δf := by |
| rw [Prod.dist_eq,max_lt_iff] |
| exact ⟨(hH x).trans_le hδf',(hH y).trans_le hδf'⟩ |
| have he := hf hd |
| simpa only [Real.dist_eq,compactF_weight] using he.le |
| · intro x y |
| have hd : dist (compactWeight x,compactWeight y) (compactWeight (H x),compactWeight (H y))<δk := by |
| rw [Prod.dist_eq,max_lt_iff] |
| exact ⟨(hH x).trans_le hδk',(hH y).trans_le hδk'⟩ |
| have he := hk hd |
| simpa only [Real.dist_eq,compactK_weight] using he.le |
|
|
| lemma field_error_tolerance (β T : ℝ) {e : ℝ} (he : 0<e) : |
| ∃ a : ℝ, 0<a ∧ a<1 ∧ 2*a+2*T*a+5*β^2*a+4*β*sqrt a<e := by |
| have hc : ContinuousAt (fun a : ℝ => 2*a+2*T*a+5*β^2*a+4*β*sqrt a) 0 := by fun_prop |
| simp only [ContinuousAt,mul_zero,add_zero,sqrt_zero] at hc |
| have hh := hc.eventually (gt_mem_nhds he) |
| have hu : ∀ᶠ a : ℝ in 𝓝 0, a<1 := gt_mem_nhds (by norm_num) |
| have hp : ∀ᶠ a : ℝ in 𝓝[>] 0, 0<a := self_mem_nhdsWithin |
| obtain ⟨a,ha,hua,hga⟩ := (hp.and ((hu.and hh).filter_mono nhdsWithin_le_nhds)).exists |
| exact ⟨a,ha,hua,hga⟩ |
|
|
| end SKRatio.Bins |
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| end |
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| end OAI |
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