| import Mathlib |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.Conditional2 |
| import OAI.Analysis.CoulombIonization.RadialBounds.BarrierForgottenInvariantBarrier |
| import OAI.Analysis.CoulombIonization.RadialBounds.BarrierInitialInvariantBarrier |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| open Set Filter MeasureTheory Metric ProbabilityTheory |
| open scoped Topology |
| namespace CoulombBarrier |
| open CoulombAtom CoulombAnalysis |
| section Conditional |
| variable {Ω D : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] |
| variable [MeasurableSpace D] (P : Measure Ω) [IsProbabilityMeasure P] |
| variable (X : Ω → D) (hX : Measurable X) |
|
|
| lemma conditionalField_plus_const {u : Ω → TFSpace → ℝ} |
| (hm : Measurable (Function.uncurry u)) (hb : DeterministicLocalBound u) |
| (c : ℝ) (d : D) (x : TFSpace) : |
| c+conditionalField P X u d x = ∫ sample, c+u sample x ∂condDistrib id X P d := by |
| rw [integral_add (integrable_const c) (deterministicBound_integrable_section hm hb x)] |
| simp [conditionalField] |
|
|
| lemma conditionalField_le {u : Ω → TFSpace → ℝ} |
| (hm : Measurable (Function.uncurry u)) (hb : DeterministicLocalBound u) |
| (c b : ℝ) (d : D) (x : TFSpace) (h : ∀ sample, c+u sample x ≤ b) : |
| c+conditionalField P X u d x ≤ b := by |
| rw [conditionalField_plus_const P X hm hb c d x] |
| calc |
| _ ≤ ∫ _ : Ω, b ∂condDistrib id X P d := integral_mono |
| ((integrable_const c).add (deterministicBound_integrable_section hm hb x)) (integrable_const b) h |
| _ = b := by simp |
|
|
| lemma le_conditionalField {u : Ω → TFSpace → ℝ} |
| (hm : Measurable (Function.uncurry u)) (hb : DeterministicLocalBound u) |
| (c b : ℝ) (d : D) (x : TFSpace) (h : ∀ sample, b ≤ c+u sample x) : |
| b ≤ c+conditionalField P X u d x := by |
| rw [conditionalField_plus_const P X hm hb c d x] |
| calc |
| b = ∫ _ : Ω, b ∂condDistrib id X P d := by simp |
| _ ≤ _ := integral_mono (integrable_const b) |
| ((integrable_const c).add (deterministicBound_integrable_section hm hb x)) h |
|
|
| include hX |
|
|
| theorem IsNuclearBarrier.forget {μ a p : Ω → TFSpace → ℝ} {Z κ B C R T η : ℝ} |
| (old : IsNuclearBarrier P μ Z κ B C R T η a p) (hR : 0 < R) (hκ : 0 ≤ κ) |
| (hμm : Measurable (Function.uncurry μ)) (hμb : DeterministicLocalBound μ) : |
| IsNuclearBarrier P (fun sample => conditionalField P X μ (X sample)) Z κ B C R T η |
| (fun sample => conditionalField P X a (X sample)) (fun sample => conditionalField P X p (X sample)) := by |
| obtain ⟨L,hpL⟩ := old.bounded_error |
| have hpb := nonneg_bounded_deterministicBound old.nonneg_error hpL |
| have hpm := old.measurable_error |
| have ham := old.measurable_offset |
| have hab := old.bound_offset |
| refine ⟨(conditionalField_measurable P X ham).comp ((hX.comp measurable_fst).prodMk measurable_snd), |
| (conditionalField_deterministicLip P X ham hab old.uniform_lipschitz_offset).comp X, |
| (conditionalField_bound P X hab).comp X, |
| (conditionalField_measurable P X hpm).comp ((hX.comp measurable_fst).prodMk measurable_snd), |
| (fun sample x => conditionalField_nonneg P X old.nonneg_error (X sample) x),⟨L,?_⟩, |
| (fun sample x hx => conditionalField_support P X old.support_error (X sample) x hx),?_,?_,?_,?_,?_⟩ |
| · intro sample x |
| simpa only [zero_add] using conditionalField_le P X hpm hpb 0 L (X sample) x (fun t => by simpa using hpL t x) |
| · have hw := weak_nuclear_conditional P X hX Z (g := fun sample => innerSource R (μ sample) (p sample)) ham hab |
| (ae_of_all _ (fun sample => (old.locallyLipschitz_offset sample).continuous)) |
| (innerSource_measurable hμm hpm R) (innerSource_bound hμb hpb R) |
| (outerCoefficient_measurable R) (outerCoefficient_bound R) (outerCoefficient_nonneg R) |
| hR hκ (outerCoefficient_zero R) old.weak |
| filter_upwards [ae_of_ae_map hX.aemeasurable hw] with sample hω |
| apply hω.congr_source |
| intro x _ |
| rw [conditionalField_innerSource P X hμm hμb hpm hpb R] |
| · intro sample x hx |
| exact le_conditionalField P X ham hab _ _ (X sample) x (fun t => old.lower t x hx) |
| · intro sample x hx |
| exact conditionalField_le P X ham hab _ _ (X sample) x (fun t => old.upper t x hx) |
| · intro sample x hx |
| apply le_antisymm |
| · exact conditionalField_le P X ham hab _ _ (X sample) x (fun t => (old.tail t x hx).le) |
| · exact le_conditionalField P X ham hab _ _ (X sample) x (fun t => (old.tail t x hx).ge) |
| · rw [conditionalField_totalMass P X hX hpm hpb old.support_error] |
| exact old.mass |
|
|
| end Conditional |
|
|
| lemma IsNuclearBarrier.congr_density {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
| {μ ν a p : Ω → TFSpace → ℝ} {Z κ B C R T η : ℝ} |
| (old : IsNuclearBarrier P μ Z κ B C R T η a p) (he : ∀ᵐ sample ∂P, ∀ x, μ sample x = ν sample x) : |
| IsNuclearBarrier P ν Z κ B C R T η a p := by |
| refine ⟨old.measurable_offset,old.uniform_lipschitz_offset,old.bound_offset, |
| old.measurable_error,old.nonneg_error,old.bounded_error,old.support_error,?_, |
| old.lower,old.upper,old.tail,old.mass⟩ |
| filter_upwards [old.weak,he] with sample hw he |
| apply hw.congr_source |
| intro x _ |
| simp only [innerSource,he x] |
|
|
| end CoulombBarrier |
|
|
| end |
|
|
| end OAI |
|
|
