openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Localization /BarrierConditionalBarrier.lean
| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open MeasureTheory Set Filter | |
| open scoped ENNReal Topology | |
| namespace CoulombBarrier | |
| def reaction (k t : ℝ) : ℝ := 4*Real.pi*k*(max t 0)^(3/2:ℝ) | |
| lemma reaction_nonneg {k : ℝ} (hk : 0 ≤ k) (t : ℝ) : 0 ≤ reaction k t := by | |
| unfold reaction | |
| positivity | |
| lemma reaction_monotone {k : ℝ} (hk : 0 ≤ k) : Monotone (reaction k) := by | |
| intro s t hst | |
| exact mul_le_mul_of_nonneg_left | |
| (Real.rpow_le_rpow (le_max_right _ _) (max_le_max hst le_rfl) (by norm_num)) | |
| (by positivity) | |
| lemma reaction_continuous (k : ℝ) : Continuous (reaction k) := by | |
| apply continuous_const.mul | |
| exact (continuous_id.max continuous_const).rpow_const (fun _ => Or.inr (by norm_num)) | |
| lemma reaction_convex {k : ℝ} (hk : 0 ≤ k) : ConvexOn ℝ univ (reaction k) := by | |
| have hm : ConvexOn ℝ (univ : Set ℝ) (fun t : ℝ => max t 0) := | |
| (convexOn_id convex_univ).sup (convexOn_const 0 convex_univ) | |
| have him : (fun t : ℝ => max t 0) '' univ = Ici 0 := by | |
| ext t | |
| constructor | |
| · rintro ⟨s,_,rfl⟩ | |
| exact le_max_right (s : ℝ) 0 | |
| · intro ht | |
| exact ⟨t,mem_univ _,max_eq_left ht⟩ | |
| have hp : ConvexOn ℝ ((fun t : ℝ => max t 0) '' univ) | |
| (fun t : ℝ => t^(3/2:ℝ)) := by | |
| rw [him] | |
| exact convexOn_rpow (by norm_num) | |
| have hmono : MonotoneOn (fun t : ℝ => t^(3/2:ℝ)) | |
| ((fun t : ℝ => max t 0) '' univ) := by | |
| rw [him] | |
| intro s hs t _ hst | |
| exact Real.rpow_le_rpow hs hst (by norm_num) | |
| exact (hp.comp hm hmono).smul (by positivity : 0 ≤ 4*Real.pi*k) | |
| variable {Ω : Type*} {m mΩ : MeasurableSpace Ω} {P : Measure Ω} | |
| variable [IsProbabilityMeasure P] (hm : m ≤ mΩ) | |
| include hm | |
| theorem reaction_condExp {k : ℝ} (hk : 0 ≤ k) {u : Ω → ℝ} | |
| (hu : Integrable u P) (hfu : Integrable (reaction k ∘ u) P) : | |
| ∀ᵐ sample ∂P, reaction k (P[u | m] sample) ≤ P[reaction k ∘ u | m] sample := by | |
| exact (reaction_convex hk).map_condExp_le_univ hm (reaction_continuous k).lowerSemicontinuous hu hfu | |
| theorem offset_condExp {u : Ω → ℝ} (hu : Integrable u P) (V : ℝ) : | |
| (fun sample => V + P[fun sample => u sample - V | m] sample) =ᵐ[P] P[u | m] := by | |
| have he := condExp_sub hu (integrable_const V) m | |
| rw [condExp_const hm] at he | |
| filter_upwards [he] with sample hω | |
| change (P[fun sample => u sample - V | m]) sample = (P[u | m]) sample - V at hω | |
| rw [hω] | |
| ring | |
| theorem reaction_offset_condExp {k : ℝ} (hk : 0 ≤ k) {u : Ω → ℝ} | |
| (hu : Integrable u P) (hfu : Integrable (reaction k ∘ u) P) (V : ℝ) : | |
| ∀ᵐ sample ∂P, reaction k (V + P[fun sample => u sample-V | m] sample) ≤ | |
| P[reaction k ∘ u | m] sample := by | |
| filter_upwards [offset_condExp hm hu V,reaction_condExp hm hk hu hfu] with sample he hj | |
| rwa [he] | |
| theorem offset_condExp_bounds {u : Ω → ℝ} (hu : Integrable u P) | |
| {a b : ℝ} (ha : ∀ᵐ sample ∂P, a ≤ u sample) (hb : ∀ᵐ sample ∂P, u sample ≤ b) (V : ℝ) : | |
| ∀ᵐ sample ∂P, a ≤ V + P[fun sample => u sample-V | m] sample ∧ | |
| V + P[fun sample => u sample-V | m] sample ≤ b := by | |
| have h1 := condExp_mono (integrable_const a) hu ha (m := m) | |
| have h2 := condExp_mono hu (integrable_const b) hb (m := m) | |
| rw [condExp_const hm] at h1 h2 | |
| filter_upwards [offset_condExp hm hu V,h1,h2] with sample he h1 h2 | |
| rw [he] | |
| exact ⟨h1,h2⟩ | |
| end CoulombBarrier | |
| end | |
| end OAI | |