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180bc03 | 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 | import Mathlib
import OAI.Probability.Ballisticity.Coupling.FreshBudgetEvent
namespace OAI
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
lemma fluctuationRadius_pos {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (S : Ω → ℝ) (s : ℝ) : 0<fluctuationRadius μ S s := by
unfold fluctuationRadius
split
· rename_i h
exact h.choose_spec.1
· norm_num
lemma seedSeparatedRaw_mass_antitone {d k : ℕ} (e f : Direction d) (H : ℕ)
{G F : ℝ} (hGF : G≤F) (π : Measure (Fin k → Lattice d)) (ω : Environment d) :
seedSeparatedRaw e f H F π ω Set.univ ≤ seedSeparatedRaw e f H G π ω Set.univ := by
rw [seedSeparatedRaw_univ,seedSeparatedRaw_univ]
exact measure_mono (fun x hx => tupleSeparated_mono f hGF x hx)
theorem adapted_separated_seeds_capped {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν]
(hue : UniformElliptic ν) (e f : Direction d) (hef : e.1≠f.1)
(htrans : DirectionallyTransient ν (realPosition (step e)))
(k : ℕ) (hk : 2≤k) (p : ℝ) (hp : 0<p) :
∃ C c₀ g₀ : ℝ, 0<C ∧ 0<c₀ ∧ c₀≤1 ∧ 0<g₀ ∧ ∃ H₀ : ℕ, ∀ H≥H₀,
∀ (a : ℝ) (π : Environment d → LayerTupleProfile (k:=k) e a),
@Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π →
∀ A : Set (Environment d), MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) a)] A →
environmentLaw ν (A∩{ω | seedSeparatedRaw e f H
(c₀*fluctuationRadius (independentConditionedPairLaw ν (realPosition (step e)))
(commonIncrementProcess (realPosition (step e)) f 0) ((H:ℝ)/C))
(π ω).val ω Set.univ<ENNReal.ofReal g₀})≤ENNReal.ofReal p*environmentLaw ν A := by
obtain ⟨C,c,g,hC,hc,hg,H₀,hh⟩ := adapted_separated_seeds ν hue e f hef htrans k hk p hp
refine ⟨C,min c 1,g,hC,lt_min hc (by norm_num),min_le_right _ _,hg,H₀,?_⟩
intro H hH a π hπ A hA
apply le_trans (measure_mono ?_) (hh H hH a π hπ A hA)
rintro ω ⟨hA,hfail⟩
refine ⟨hA,?_⟩
change seedSeparatedRaw e f H _ (π ω).val ω Set.univ < _
exact lt_of_le_of_lt (seedSeparatedRaw_mass_antitone e f H
(mul_le_mul_of_nonneg_right (min_le_left _ _) (fluctuationRadius_pos _ _ _).le) _ _) hfail
end DirectionalTransience
end
end OAI
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