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7d9f027 | 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 65 66 67 68 69 70 71 72 | 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
end
end OAI
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