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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 96 97 98 99 100 101 102 103 104 105 106 107 | 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
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