openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /FieldAnalysis /ExpandedAnnulusSmallEventIntegral.lean
| 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 | |