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159357f | 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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import Mathlib
import OAI.Analysis.CoulombIonization.RadialBounds.PointInverseEventsCell
namespace OAI
open MeasureTheory Filter Set
open scoped Topology
noncomputable section
namespace CoulombAtom
open CoulombAnalysis CoulombObservation CoulombBarrier
attribute [local irreducible] graphComponent graphFormVector fermionGraph weakGraph fermionGraphValue
lemma tiny_floor_normalized_field_bound_half {Z u : ℝ} {y : Space}
(hZ : 0 ≤ Z) (hu : 0 < u) (hu1 : u ≤ 1) (huy : u/2 ≤ ‖y‖)
(hay : localCellRadius y ≤ 1) :
(localCellRadius y)^4*(tinyProbabilityFloor Z u*(Z/‖y‖)) ≤ 2*u^60 := by
have ha : 0 ≤ localCellRadius y := by unfold localCellRadius; positivity
have hn : 0 < ‖y‖ := by linarith
have htmp := tinyProbabilityFloor_nuclear_bound hZ hu (by linarith : u ≤ 2*‖y‖)
have hh : tinyProbabilityFloor Z u*(Z/‖y‖) ≤ 2*u^79 := by
have he : tinyProbabilityFloor Z u*(Z/‖y‖) =
2*(tinyProbabilityFloor Z u*(Z/(2*‖y‖))) := by ring
rw [he]
linarith
have h79 : u^79 ≤ u^60 := pow_le_pow_of_le_one hu.le hu1 (by norm_num : 60 ≤ 79)
calc
_ ≤ (localCellRadius y)^4*(2*u^79) := mul_le_mul_of_nonneg_left hh (pow_nonneg ha _)
_ ≤ 1^4*(2*u^79) := mul_le_mul_of_nonneg_right (pow_le_pow_left₀ ha hay 4) (by positivity)
_ ≤ _ := by norm_num; linarith
theorem OwnProbabilityTailTiltState.expandedAnnulus_small_event_integral {Z lam r : ℝ}
(hZ : 0 ≤ Z) (hlam : 0 < lam) (hr : 0 < r) {N K : ℕ} {F : fermionGraph N}
{δ c₁ r₀ s : ℝ}
(hF : OwnProbabilityTailTiltState Z lam r K
(fun j => tinyProbabilityFloor Z ((2:ℝ)^j.val*r)) δ F)
(hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹)
(hr₀ : 0 < r₀) (hs : 0 < s) (hs1 : s ≤ 1)
(j : Fin (K+1)) {y : Space}
(huy : ((2:ℝ)^j.val*r)/2 ≤ ‖y‖)
(hu1 : (2:ℝ)^j.val*r ≤ 1)
(hgeom : InverseFiniteGeometry c₁ r₀ s (((2:ℝ)^j.val*r)^(101/100:ℝ)) y)
(hcap : ∀ p : ℝ, 0 < p → p ≤ ((2:ℝ)^j.val*r)^80 →
(localCellRadius y)^4*(p*originalFieldCapBudget ((2:ℝ)^j.val*r) p δ c₁ r₀ s y
((localCellRadius y)^(6/5:ℝ))
(localCellRadius y*(localCellRadius y)^masterExponent)) ≤
p^(3/4:ℝ)*(tfPatchCapConstant+1))
(A : Set (Configuration N × (Fin K × (Fin N × Fin 3) → ℝ)))
(hA : MeasurableSet[observationInformation (fun k : Fin K => dyadicObservationWidth r k) j] A)
(hpA : (physicalObservationLaw (graphRawLaw F) K).real A ≤ ((2:ℝ)^j.val*r)^80) :
(localCellRadius y)^4*(∫ z in A, originalQueryField F Z lam r j c₁ r₀ s z y
∂physicalObservationLaw (graphRawLaw F) K) ≤
(tfPatchCapConstant+2)*((2:ℝ)^j.val*r)^60 := by
let u := (2:ℝ)^j.val*r
let p := (physicalObservationLaw (graphRawLaw F) K).real A
have hu : 0 < u := by dsimp [u]; positivity
have hy : y ≠ 0 := norm_pos_iff.mp (by linarith)
have ha := localCellRadius_pos hy
have hC := tfPatchCapConstant_pos
have hfloor : 0 < tinyProbabilityFloor Z u := tinyProbabilityFloor_pos hZ hu
by_cases hp : tinyProbabilityFloor Z u ≤ p
· have hpp : 0 < p := hfloor.trans_le hp
have hh := hF.event_integral_cap_of_cell hZ hlam
(fun _ => tinyProbabilityFloor_pos hZ (by positivity)) hc hcL hr₀ hs hs1 j A hA hp
hy hgeom.radius_le (by linarith [hgeom.master_fit,hgeom.cut_pos]) hgeom.cut_pos hgeom.cut_le hgeom.probe_pos
hgeom.probe_fit hgeom.probe_margin hgeom.collar
have hmul := mul_le_mul_of_nonneg_left hh (pow_nonneg ha.le 4)
have hweight := hcap p hpp hpA
have hpw := small_probability_three_quarters hpp.le hu hpA
calc _ ≤ _ := hmul
_ ≤ p^(3/4:ℝ)*(tfPatchCapConstant+1) := hweight
_ ≤ u^60*(tfPatchCapConstant+1) :=
mul_le_mul_of_nonneg_right hpw (by linarith)
_ ≤ _ := by dsimp only [u]; nlinarith [pow_nonneg hu.le 60]
· have hh : (∫ z in A, originalQueryField F Z lam r j c₁ r₀ s z y
∂physicalObservationLaw (graphRawLaw F) K) ≤ tinyProbabilityFloor Z u*(Z/‖y‖) := by
calc
_ ≤ ∫ _z in A, Z/‖y‖ ∂physicalObservationLaw (graphRawLaw F) K := by
apply integral_mono_ae
(originalQueryField_integrable F Z lam r j hc hr₀ hs y).integrableOn
(integrable_const _)
exact ae_of_all _ (fun z => originalQueryField_le_nuclear F Z r hlam.le j c₁ r₀ s z y)
_ = p*(Z/‖y‖) := by
simp only [integral_const,Measure.real,Measure.restrict_apply_univ,smul_eq_mul,p]
_ ≤ _ := mul_le_mul_of_nonneg_right (le_of_not_ge hp) (div_nonneg hZ (norm_nonneg y))
have hb := tiny_floor_normalized_field_bound_half hZ hu hu1 huy hgeom.radius_le
calc _ ≤ _ := mul_le_mul_of_nonneg_left hh (pow_nonneg ha.le 4)
_ ≤ 2*u^60 := hb
_ ≤ _ := by dsimp only [u]; nlinarith [pow_nonneg hu.le 60]
end CoulombAtom
end
end OAI
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