openAI-math-dataset / lean /OAI /Probability /Ballisticity /Coupling /QuenchedConditionalProbability.lean
| 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 | |