openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Localization /ConditionalSandwich.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.Localization.UnorderedObservations | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open MeasureTheory Filter | |
| namespace CoulombObservation | |
| lemma condExp_observable_sandwich {Ω : Type*} {m₀ : MeasurableSpace Ω} | |
| {m : MeasurableSpace Ω} (hm : m ≤ m₀) (μ : @Measure Ω m₀) [IsFiniteMeasure μ] | |
| {f h d : Ω → ℝ} (hf : Integrable f μ) (hh : Integrable h μ) (hd : Integrable d μ) | |
| (hhm : StronglyMeasurable[m] h) (hdm : StronglyMeasurable[m] d) | |
| (hb : ∀ᵐ z ∂μ, |f z-h z| ≤ d z) : | |
| ∀ᵐ z ∂μ, |μ[f | m] z-h z| ≤ d z := by | |
| have hlow : (fun z => h z-d z) ≤ᵐ[μ] f := hb.mono (fun z hz => by | |
| have := (abs_le.mp hz).1 | |
| linarith) | |
| have hupp : f ≤ᵐ[μ] (fun z => h z+d z) := hb.mono (fun z hz => by | |
| have := (abs_le.mp hz).2 | |
| linarith) | |
| have hlo := condExp_mono (m := m) (hh.sub hd) hf hlow | |
| have hup := condExp_mono (m := m) hf (hh.add hd) hupp | |
| rw [condExp_of_stronglyMeasurable (m₀ := m₀) (μ := μ) hm (hhm.sub hdm) (hh.sub hd)] at hlo | |
| rw [condExp_of_stronglyMeasurable (m₀ := m₀) (μ := μ) hm (hhm.add hdm) (hh.add hd)] at hup | |
| filter_upwards [hlo,hup] with z hl hu | |
| simp only [Pi.sub_apply,Pi.add_apply] at hl hu | |
| exact abs_le.mpr ⟨by linarith,by linarith⟩ | |
| lemma raw_condExp_observable_error {Ω : Type*} {m₀ : MeasurableSpace Ω} | |
| {m : MeasurableSpace Ω} (hm : m ≤ m₀) (μ : @Measure Ω m₀) [IsFiniteMeasure μ] | |
| {f h d : Ω → ℝ} (hf : Integrable f μ) (hh : Integrable h μ) (hd : Integrable d μ) | |
| (hhm : StronglyMeasurable[m] h) (hdm : StronglyMeasurable[m] d) | |
| (hb : ∀ᵐ z ∂μ, |f z-h z| ≤ d z) : | |
| ∀ᵐ z ∂μ, |f z-μ[f | m] z| ≤ 2*d z := by | |
| filter_upwards [hb,condExp_observable_sandwich hm μ hf hh hd hhm hdm hb] with z hz hc | |
| have he : f z-μ[f | m] z = (f z-h z)+(h z-μ[f | m] z) := by ring | |
| rw [he] | |
| calc | |
| _ ≤ |f z-h z|+|h z-μ[f | m] z| := abs_add_le _ _ | |
| _ ≤ d z+d z := add_le_add hz (by rwa [abs_sub_comm]) | |
| _ = _ := by ring | |
| end CoulombObservation | |
| open MeasureTheory Filter Set Metric | |
| open scoped BigOperators NNReal | |
| namespace CoulombAtom | |
| open CoulombObservation | |
| lemma integrable_configuration_test {Ω : Type*} [MeasurableSpace Ω] | |
| (μ : Measure Ω) [IsFiniteMeasure μ] {N : ℕ} {X : Ω → Configuration N} | |
| (hX : Measurable X) {g : Space → ℝ} (hg : Continuous g) (hc : HasCompactSupport g) : | |
| Integrable (fun z => ∑ i, g (X z i)) μ := by | |
| obtain ⟨v,hv⟩ := hg.norm.exists_forall_ge_of_hasCompactSupport hc.norm | |
| apply integrable_finsetSum | |
| intro i _ | |
| apply (integrable_const ‖g v‖).mono' | |
| ((hg.measurable.comp ((measurable_pi_apply i).comp hX)).aestronglyMeasurable) | |
| exact ae_of_all _ (fun z => hv (X z i)) | |
| lemma observedLocalCount_integrable {N K : ℕ} (μ : Measure (Configuration N)) | |
| [IsFiniteMeasure μ] (ell : Fin K → ℝ) (k : Fin K) (y : Space) (R : ℝ) : | |
| Integrable (observedLocalCount (N := N) ell k y R) (physicalObservationLaw μ K) := by | |
| have h := (observedLocalCount_information_measurable (N := N) ell k (Nat.zero_le _) y R).mono | |
| (observationInformation_le ell 0) (le_refl _) | |
| apply (integrable_const (N:ℝ)).mono' h.aestronglyMeasurable | |
| exact ae_of_all _ (fun z => by | |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (observedLocalCount_nonneg ell k y R z)] using | |
| observedLocalCount_le ell k y R z) | |
| def posteriorTestValue {N K : ℕ} (μ : Measure (Configuration N)) (ell : Fin K → ℝ) | |
| (j : ℕ) (g : Space → ℝ) : | |
| (Configuration N × (Fin K × (Fin N × Fin 3) → ℝ)) → ℝ := | |
| (physicalObservationLaw μ K)[(fun z => ∑ i, g (z.1 i)) | observationInformation ell j] | |
| lemma posteriorTestValue_tower {N K : ℕ} (μ : Measure (Configuration N)) | |
| [IsFiniteMeasure μ] (ell : Fin K → ℝ) {j k : ℕ} (hjk : j ≤ k) (g : Space → ℝ) : | |
| (physicalObservationLaw μ K)[posteriorTestValue μ ell j g | observationInformation ell k] =ᵐ[ | |
| physicalObservationLaw μ K] posteriorTestValue μ ell k g := | |
| condExp_condExp_of_le (observationInformation_antitone ell hjk) (observationInformation_le ell j) | |
| theorem posterior_observed_test_sandwich {N K : ℕ} (μ : Measure (Configuration N)) | |
| [IsFiniteMeasure μ] (ell : Fin K → ℝ) (hell : ∀ k, 0 ≤ ell k) | |
| {j : ℕ} (k : Fin K) (hk : j ≤ k.val) {g : Space → ℝ} {L : ℝ≥0} | |
| (hg : LipschitzWith L g) (y : Space) (R : ℝ) | |
| (hs : Function.support g ⊆ closedBall y R) : | |
| ∀ᵐ z ∂physicalObservationLaw μ K, | |
| |posteriorTestValue μ ell j g z-(∑ i, g (physicalObservedConfiguration ell k z i))| ≤ | |
| observedLocalCount ell k y (R+Real.sqrt 3*ell k) z*((L:ℝ)*(Real.sqrt 3*ell k)) := by | |
| have hc : HasCompactSupport g := HasCompactSupport.of_support_subset_isCompact (isCompact_closedBall y R) hs | |
| have hf := integrable_configuration_test (physicalObservationLaw μ K) | |
| measurable_fst hg.continuous hc | |
| have hh := integrable_configuration_test (physicalObservationLaw μ K) | |
| (physicalObservedConfiguration_measurable ell k) hg.continuous hc | |
| have hd := (observedLocalCount_integrable μ ell k y (R+Real.sqrt 3*ell k)).mul_const | |
| ((L:ℝ)*(Real.sqrt 3*ell k)) | |
| have hhm := (observedStatistic_information_measurable (N := N) ell k hk hg.continuous.measurable).stronglyMeasurable | |
| have hdm := ((observedLocalCount_information_measurable (N := N) ell k hk y (R+Real.sqrt 3*ell k)).mul_const | |
| ((L:ℝ)*(Real.sqrt 3*ell k))).stronglyMeasurable | |
| apply condExp_observable_sandwich (observationInformation_le ell j) | |
| (physicalObservationLaw μ K) hf hh hd hhm hdm | |
| filter_upwards [physicalObservationLaw_ae_displacement μ ell hell] with z hz | |
| exact matchedLocal_test_error z.1 (physicalObservedConfiguration ell k z) | |
| (mul_nonneg (Real.sqrt_nonneg _) (hell k)) (hz k) hg y hs | |
| theorem raw_posterior_local_test_error {N K : ℕ} (μ : Measure (Configuration N)) | |
| [IsFiniteMeasure μ] (ell : Fin K → ℝ) (hell : ∀ k, 0 ≤ ell k) | |
| {j : ℕ} (k : Fin K) (hk : j ≤ k.val) {g : Space → ℝ} {L : ℝ≥0} | |
| (hg : LipschitzWith L g) (y : Space) (R : ℝ) | |
| (hs : Function.support g ⊆ closedBall y R) : | |
| ∀ᵐ z ∂physicalObservationLaw μ K, | |
| |(∑ i, g (z.1 i))-posteriorTestValue μ ell j g z| ≤ | |
| 2*observedLocalCount ell k y (R+Real.sqrt 3*ell k) z*((L:ℝ)*(Real.sqrt 3*ell k)) := by | |
| filter_upwards [physicalObservationLaw_ae_displacement μ ell hell, | |
| posterior_observed_test_sandwich μ ell hell k hk hg y R hs] with z hz hp | |
| have hr := matchedLocal_test_error z.1 (physicalObservedConfiguration ell k z) | |
| (mul_nonneg (Real.sqrt_nonneg _) (hell k)) (hz k) hg y hs | |
| have he : (∑ i, g (z.1 i))-posteriorTestValue μ ell j g z = | |
| ((∑ i, g (z.1 i))-(∑ i, g (physicalObservedConfiguration ell k z i)))+ | |
| ((∑ i, g (physicalObservedConfiguration ell k z i))-posteriorTestValue μ ell j g z) := by ring | |
| rw [he] | |
| refine (abs_add_le _ _).trans ?_ | |
| rw [abs_sub_comm (∑ i, g (physicalObservedConfiguration ell k z i))] | |
| change _+_ ≤ 2*observedLocalCount ell k y (R+Real.sqrt 3*ell k) z*((L:ℝ)*(Real.sqrt 3*ell k)) | |
| dsimp only [observedLocalCount] at hp ⊢ | |
| linarith | |
| end CoulombAtom | |
| end | |
| end OAI | |