import Mathlib import OAI.Probability.Ballisticity.Estimates.RecordPrefixControl import OAI.Probability.Ballisticity.Estimates.RawConditionedWeight namespace OAI section section open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Pointwise Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Pointwise Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical namespace DirectionalTransience noncomputable def quenchedConditionalProbability {d : ℕ} (ℓ : Vector d) (x : Lattice d) (A : Set (Path d)) (ω : Environment d) : ℝ≥0∞ := (noDropQuenched ℓ x ω)⁻¹*quenchedKernel (ω,x) (A ∩ NoDrop ℓ x) lemma measurable_quenchedConditionalProbability {d : ℕ} (ℓ : Vector d) (x : Lattice d) (A : Set (Path d)) (hA : MeasurableSet A) : Measurable (quenchedConditionalProbability ℓ x A) := (measurable_noDropQuenched ℓ x).inv.mul ((Kernel.measurable_coe quenchedKernel (hA.inter (measurableSet_noDrop ℓ x))).comp (measurable_id.prodMk measurable_const)) lemma quenchedConditionalProbability_le_one {d : ℕ} (ℓ : Vector d) (x : Lattice d) (A : Set (Path d)) (ω : Environment d) : quenchedConditionalProbability ℓ x A ω ≤ 1 := (mul_le_mul' le_rfl (measure_mono Set.inter_subset_right)).trans (ENNReal.inv_mul_le_one _) lemma quenchedConditionalProbability_real_integrable {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] (ℓ : Vector d) (x : Lattice d) (A : Set (Path d)) (hA : MeasurableSet A) : Integrable (fun ω => (quenchedConditionalProbability ℓ x A ω).toReal) (environmentLaw ν) := by apply (integrable_const (1:ℝ)).mono' (measurable_quenchedConditionalProbability ℓ x A hA).ennreal_toReal.aestronglyMeasurable exact ae_of_all _ fun ω => by rw [Real.norm_eq_abs,abs_of_nonneg ENNReal.toReal_nonneg] simpa only [ENNReal.toReal_one] using ENNReal.toReal_mono ENNReal.one_ne_top (quenchedConditionalProbability_le_one ℓ x A ω) lemma quenchedConditionalProbability_mean {d : ℕ} (ν : Measure (Row d)) (ℓ : Vector d) (A : Set (Path d)) (hA : MeasurableSet A) : (∫ ω, (quenchedConditionalProbability ℓ 0 A ω).toReal ∂environmentLaw ν) = (quenchedConditionedAverage ν ℓ).real A := by rw [integral_toReal (measurable_quenchedConditionalProbability ℓ 0 A hA).aemeasurable (ae_of_all _ fun ω => lt_of_le_of_lt (quenchedConditionalProbability_le_one ℓ 0 A ω) ENNReal.one_lt_top)] rw [Measure.real,quenchedConditionedAverage_apply ν ℓ A hA] rfl lemma quenched_good_mass_identity {d : ℕ} (ℓ : Vector d) (x : Lattice d) (A : Set (Path d)) (hA : MeasurableSet A) (ω : Environment d) (hq : 0 < noDropQuenched ℓ x ω) : (quenchedKernel (ω,x)).real (Aᶜ ∩ NoDrop ℓ x) = (noDropQuenched ℓ x ω).toReal*(1-(quenchedConditionalProbability ℓ x A ω).toReal) := by have hp : 0 < (noDropQuenched ℓ x ω).toReal := ENNReal.toReal_pos (ne_of_gt hq) (measure_ne_top _ _) have hsplit := measureReal_inter_add_sdiff (μ := quenchedKernel (ω,x)) (s := NoDrop ℓ x) hA have he : NoDrop ℓ x \ A = Aᶜ ∩ NoDrop ℓ x := by ext X; simp only [Set.mem_sdiff,Set.mem_inter_iff,Set.mem_compl_iff]; tauto rw [he,Set.inter_comm (NoDrop ℓ x) A] at hsplit rw [quenchedConditionalProbability,ENNReal.toReal_mul,ENNReal.toReal_inv] change (quenchedKernel (ω,x)).real (Aᶜ ∩ NoDrop ℓ x) = (noDropQuenched ℓ x ω).toReal*(1-(noDropQuenched ℓ x ω).toReal⁻¹*(quenchedKernel (ω,x)).real (A ∩ NoDrop ℓ x)) rw [mul_sub,mul_one,← mul_assoc,mul_inv_cancel₀ (ne_of_gt hp),one_mul] change (quenchedKernel (ω,x)).real (A ∩ NoDrop ℓ x)+ (quenchedKernel (ω,x)).real (Aᶜ ∩ NoDrop ℓ x) = (noDropQuenched ℓ x ω).toReal at hsplit linarith theorem quenched_good_mass_bad_probability {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] (hue : UniformElliptic ν) (ℓ : Vector d) (hℓ : dot ℓ ℓ = 1) (htrans : DirectionallyTransient ν ℓ) (A : Set (Path d)) (hA : MeasurableSet A) {δ : ℝ} (hδ : 0 < δ) : (environmentLaw ν).real {ω | (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) < 2*δ} ≤ (environmentLaw ν).real {ω | (noDropQuenched ℓ 0 ω).toReal < 4*δ}+ 2*(quenchedConditionedAverage ν ℓ).real A := by let Q := fun ω => (quenchedConditionalProbability ℓ 0 A ω).toReal let B := {ω | (noDropQuenched ℓ 0 ω).toReal < 4*δ} let C := {ω | (1:ℝ)/2 ≤ Q ω} have hsub : ∀ᵐ ω ∂environmentLaw ν, ω ∈ {ω | (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) < 2*δ} → ω ∈ B ∪ C := by filter_upwards [quenched_noDrop_positive_of_directionallyTransient ν hue ℓ hℓ htrans] with ω hp intro hbad change (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) < 2*δ at hbad by_contra hn have hq : 4*δ ≤ (noDropQuenched ℓ 0 ω).toReal := by by_contra! h; exact hn (Or.inl h) have hQ : Q ω < 1/2 := by by_contra! h; exact hn (Or.inr h) have hg := quenched_good_mass_identity ℓ 0 A hA ω (hp 0) change (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) = (noDropQuenched ℓ 0 ω).toReal*(1-Q ω) at hg have hpos : 0 < (noDropQuenched ℓ 0 ω).toReal := by linarith nlinarith have hm := ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hsub) have hu := measureReal_union_le (μ := environmentLaw ν) B C have hmark := mul_meas_ge_le_integral_of_nonneg (ae_of_all (environmentLaw ν) fun ω => (show 0 ≤ Q ω from ENNReal.toReal_nonneg)) (quenchedConditionalProbability_real_integrable ν ℓ 0 A hA) (1/2) rw [quenchedConditionalProbability_mean ν ℓ A hA] at hmark change (1/2)*(environmentLaw ν).real C ≤ (quenchedConditionedAverage ν ℓ).real A at hmark change (environmentLaw ν).real {ω | (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) < 2*δ} ≤ (environmentLaw ν).real (B ∪ C) at hm change (environmentLaw ν).real {ω | (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) < 2*δ} ≤ (environmentLaw ν).real B+2*(quenchedConditionedAverage ν ℓ).real A linarith lemma positive_lower_quantile {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] (F : Ω → ℝ) (hF : Measurable F) (hpos : ∀ᵐ x ∂μ, 0 < F x) {ε : ℝ} (hε : 0 < ε) : ∃ δ : ℝ, 0 < δ ∧ δ < 1/4 ∧ μ.real {x | F x < 4*δ} ≤ ε := by have ht := fixed_real_tail_small μ (fun x => (F x)⁻¹) hF.inv (fun n : ℕ => (n:ℝ)+1) (tendsto_atTop_add_const_right _ _ tendsto_natCast_atTop_atTop) obtain ⟨N,hN⟩ := (ht.eventually_lt_const hε).exists let R : ℝ := (N:ℝ)+1 have hR : 0 < R := by dsimp [R]; positivity let δ : ℝ := 1/(8*R) have hδ : 0 < δ := by dsimp [δ]; positivity have hδsmall : δ < 1/4 := by dsimp [δ,R] apply (div_lt_iff₀ (by positivity)).mpr have hn : 0 ≤ (N:ℝ) := Nat.cast_nonneg N linarith refine ⟨δ,hδ,hδsmall,?_⟩ suffices hs : ∀ᵐ x ∂μ, x ∈ {x | F x < 4*δ} → x ∈ {x | R ≤ |(F x)⁻¹|} from (ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hs)).trans hN.le filter_upwards [hpos] with x hx hbad change F x < 4*δ at hbad rw [abs_of_pos (inv_pos.mpr hx)] rw [← one_div] apply (le_div_iff₀ hx).mpr have heq : 4*δ = 1/(2*R) := by dsimp [δ]; field_simp; ring rw [heq] at hbad have hh := (lt_div_iff₀ (by positivity : 0 < 2*R)).mp hbad nlinarith end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical end end end OAI