| import Mathlib |
| import OAI.Probability.Ballisticity.Coupling.FreshBudgetEvent |
|
|
| namespace OAI |
|
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| section |
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|
| 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 |
|
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| 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) |
|
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| 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 |
|
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| end DirectionalTransience |
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| end |
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| end OAI |
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