openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Variational /ContinuousUniversalEvent.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.OriginalFieldOscillationBarrier | |
| import OAI.Analysis.CoulombIonization.Localization.InverseNetEventsBarrier | |
| namespace OAI | |
| noncomputable section | |
| namespace CoulombAtom | |
| open Set MeasureTheory | |
| theorem measurableSet_forall_closed_parameter | |
| {Ω X : Type*} [MeasurableSpace Ω] [TopologicalSpace X] [TopologicalSpace.SeparableSpace X] | |
| {P : Ω → X → Prop} (hm : ∀ x, MeasurableSet {ω | P ω x}) | |
| (hc : ∀ ω, IsClosed {x | P ω x}) : MeasurableSet {ω | ∀ x, P ω x} := by | |
| obtain ⟨D,hD,hd⟩ := TopologicalSpace.exists_countable_dense X | |
| have he : {ω | ∀ x, P ω x} = ⋂ x ∈ D, {ω | P ω x} := by | |
| ext ω | |
| simp only [mem_ofPred_eq,mem_iInter] | |
| constructor | |
| · intro h x _ | |
| exact h x | |
| · intro h x | |
| exact closure_minimal h (hc ω) (hd x) | |
| rw [he] | |
| exact MeasurableSet.biInter hD (fun x _ => hm x) | |
| open Set MeasureTheory | |
| open CoulombAnalysis CoulombObservation CoulombBarrier | |
| lemma inverseComparisonAt_closed {X : Type*} [TopologicalSpace X] | |
| {d H μ : X → ℝ} (hd : Continuous d) (hH : Continuous H) (hμ : Continuous μ) | |
| (κ lo hi xi C : ℝ) : IsClosed {x | InverseComparisonAt (d x) (H x) (μ x) κ lo hi xi C} := by | |
| have hf := (hd.pow 4).mul hH | |
| have hr := (hd.pow 6).mul hμ | |
| have h₀ := isClosed_le hf continuous_const (g := fun _ => C) | |
| have h₁ := isClosed_iInter (fun h : Icc lo hi => | |
| (isClosed_le hr continuous_const (g := fun _ => κ*(h:ℝ)^(3/2:ℝ))).union | |
| (isClosed_le continuous_const hf (f := fun _ => (h:ℝ)-xi))) | |
| have h₂ := isClosed_iInter (fun h : Icc lo hi => | |
| (isClosed_le continuous_const hr (f := fun _ => κ*(h:ℝ)^(3/2:ℝ))).union | |
| (isClosed_le hf continuous_const (g := fun _ => (h:ℝ)+xi))) | |
| convert h₀.inter (h₁.inter h₂) using 1 | |
| ext x | |
| simp only [mem_inter_iff,mem_iInter,mem_union,mem_ofPred_eq,InverseComparisonAt, | |
| imp_iff_not_or,not_lt,Subtype.forall,Pi.mul_apply,Pi.pow_apply] | |
| lemma inverseComparisonAt_measurable {Ω : Type*} [MeasurableSpace Ω] | |
| {H μ : Ω → ℝ} (hH : Measurable H) (hμ : Measurable μ) | |
| (d κ lo hi xi C : ℝ) : MeasurableSet {q | InverseComparisonAt d (H q) (μ q) κ lo hi xi C} := by | |
| have hf := hH.const_mul (d^4) | |
| have hr := hμ.const_mul (d^6) | |
| have h₀ := measurableSet_le hf measurable_const (g := fun _ => C) | |
| have ht : Continuous (fun h : Icc lo hi => κ*(h:ℝ)^(3/2:ℝ)) := | |
| ((Real.continuous_rpow_const (by norm_num : (0:ℝ) ≤ 3/2)).comp continuous_subtype_val).const_mul κ | |
| have h₁ : MeasurableSet {q | ∀ h : Icc lo hi, | |
| d^6*μ q ≤ κ*(h:ℝ)^(3/2:ℝ) ∨ (h:ℝ)-xi ≤ d^4*H q} := by | |
| apply measurableSet_forall_closed_parameter | |
| · intro h | |
| exact (measurableSet_le hr measurable_const).union (measurableSet_le measurable_const hf) | |
| · intro q | |
| exact (isClosed_le continuous_const ht).union | |
| (isClosed_le (continuous_subtype_val.sub continuous_const) continuous_const) | |
| have h₂ : MeasurableSet {q | ∀ h : Icc lo hi, | |
| κ*(h:ℝ)^(3/2:ℝ) ≤ d^6*μ q ∨ d^4*H q ≤ (h:ℝ)+xi} := by | |
| apply measurableSet_forall_closed_parameter | |
| · intro h | |
| exact (measurableSet_le measurable_const hr).union (measurableSet_le hf measurable_const) | |
| · intro q | |
| exact (isClosed_le ht continuous_const).union | |
| (isClosed_le continuous_const (continuous_subtype_val.add_const xi)) | |
| convert h₀.inter (h₁.inter h₂) using 1 | |
| ext q | |
| simp only [mem_inter_iff,mem_ofPred_eq,InverseComparisonAt,imp_iff_not_or,not_lt,Subtype.forall] | |
| lemma original_inverse_fullAnnulus_measurable {N K : ℕ} (F : fermionGraph N) | |
| (Z lam r : ℝ) (j : ℕ) {c₁ r₀ s u B κ lo hi xi C : ℝ} | |
| (hc : 0 < c₁) (hr : 0 < r₀) (hs : 0 < s) (hrs : r₀ ≤ s) (hu : 0 < u) : | |
| MeasurableSet[observationInformation (fun k : Fin K => dyadicObservationWidth r k) j] | |
| {q | ∀ y : Space, u ≤ ‖y‖ → ‖y‖ ≤ B*u → | |
| InverseComparisonAt ‖y‖ | |
| (originalQueryField F Z lam r j c₁ r₀ s q y) | |
| (originalQueryDensity F r j c₁ r₀ s q y) κ lo hi xi C} := by | |
| let Y := {y : Space // u ≤ ‖y‖ ∧ ‖y‖ ≤ B*u} | |
| have he : MeasurableSet[observationInformation (fun k : Fin K => dyadicObservationWidth r k) j] | |
| {q | ∀ y : Y, InverseComparisonAt ‖(y:Space)‖ | |
| (originalQueryField F Z lam r j c₁ r₀ s q y) | |
| (originalQueryDensity F r j c₁ r₀ s q y) κ lo hi xi C} := by | |
| apply _ Y | |
| (observationInformation (fun k : Fin K => dyadicObservationWidth r k) j) _ _ | |
| · intro y | |
| exact _ | |
| (observationInformation (fun k : Fin K => dyadicObservationWidth r k) j) _ _ | |
| (originalQueryField_info_measurable F Z lam r j hc hr hs y) | |
| (originalQueryDensity_info_measurable F r j hc hr hs y) _ _ _ _ _ _ | |
| · intro q | |
| have hfield : Continuous (fun y : Y => originalQueryField F Z lam r j c₁ r₀ s q y) := by | |
| apply (originalQueryField_continuousOn_punctured F Z lam r j hc hr hs hrs q).comp_continuous | |
| continuous_subtype_val | |
| intro y | |
| change (y:Space) ≠ 0 | |
| exact norm_pos_iff.mp (hu.trans_le y.property.1) | |
| have hden : Continuous (fun y : Y => originalQueryDensity F r j c₁ r₀ s q y) := | |
| (jointMasterPosterior_continuous (graphRawLaw F) | |
| (fun k : Fin K => dyadicObservationWidth r k) j hc hr hs hrs | |
| canonicalRealPacket_smooth.continuous canonicalRealPacket_compact | |
| (originalDatum (fun k : Fin K => dyadicObservationWidth r k) j q)).comp continuous_subtype_val | |
| exact inverseComparisonAt_closed continuous_subtype_val.norm hfield hden _ _ _ _ _ | |
| convert he using 1 | |
| ext q | |
| simp only [mem_ofPred_eq,Subtype.forall,Y] | |
| simp only [and_imp] | |
| end CoulombAtom | |
| end | |
| end OAI | |