File size: 3,489 Bytes
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 | 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
|