import Mathlib import OAI.Probability.Ballisticity.Crossings.FirstLayerHitRecordIndexTime import OAI.Probability.Ballisticity.Estimates.GaussianAbsTail 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 namespace DirectionalTransience def RegularPath {d : ℕ} (ℓ : Vector d) (x : Lattice d) : Set (Path d) := {X | X 0 = x ∧ (∀ n, ∃ f, X (n+1) = X n+step f) ∧ X ∈ TransientPaths ℓ} lemma measurableSet_regularPath {d : ℕ} (ℓ : Vector d) (x : Lattice d) : MeasurableSet (RegularPath ℓ x) := by apply (measurableSet_eq_fun (measurable_pi_apply 0) measurable_const).inter apply MeasurableSet.inter _ (measurableSet_transientPaths ℓ) change MeasurableSet {X : Path d | ∀ n, ∃ f, X (n+1) = X n+step f} simp only [Set.ofPred_forall,Set.ofPred_exists] exact MeasurableSet.iInter fun n => MeasurableSet.iUnion fun f => measurableSet_eq_fun (by fun_prop : Measurable (fun X : Path d => X (n+1))) (by fun_prop : Measurable (fun X : Path d => X n+step f)) lemma annealedFrom_regularPath {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] (ℓ : Vector d) (htrans : DirectionallyTransient ν ℓ) (x : Lattice d) : ∀ᵐ X ∂annealedFrom ν x, X ∈ RegularPath ℓ x := by apply annealedFrom_ae_of_quenched ν x _ (measurableSet_regularPath ℓ x) filter_upwards [quenched_directionallyTransient_ae ν ℓ htrans] with ω hω filter_upwards [quenched_initial_ae (ω,x),quenched_nearest_neighbor (ω,x),hω x] with X h0 hnn ht exact ⟨h0,hnn,ht⟩ lemma sharedConditioned_regularPath {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] (ℓ : Vector d) (htrans : DirectionallyTransient ν ℓ) (x y : Lattice d) : ∀ᵐ Z ∂sharedConditionedPairLaw ν ℓ x y, (Z.1 ∈ RegularPath ℓ x ∧ Z.1 ∈ NoDrop ℓ x) ∧ (Z.2 ∈ RegularPath ℓ y ∧ Z.2 ∈ NoDrop ℓ y) := by have hh := shared_pair_ae ν x y _ _ (measurableSet_regularPath ℓ x) (measurableSet_regularPath ℓ y) (annealedFrom_regularPath ν ℓ htrans x) (annealedFrom_regularPath ν ℓ htrans y) filter_upwards [(sharedConditionedPairLaw_absolutelyContinuous ν ℓ x y).ae_le hh, sharedConditionedPairLaw_noDrop ν ℓ x y] with Z hZ hD exact ⟨⟨hZ.1,hD.1⟩,hZ.2,hD.2⟩ lemma conditionedFrom_regularPath {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] (ℓ : Vector d) (htrans : DirectionallyTransient ν ℓ) (x : Lattice d) : ∀ᵐ X ∂conditionedFrom ν ℓ x, X ∈ RegularPath ℓ x ∧ X ∈ NoDrop ℓ x := by have hac : conditionedFrom ν ℓ x ≪ annealedFrom ν x := Measure.smul_absolutelyContinuous.trans Measure.absolutelyContinuous_restrict have hD : ∀ᵐ X ∂conditionedFrom ν ℓ x, X ∈ NoDrop ℓ x := by exact Measure.smul_absolutelyContinuous.ae_le (ae_restrict_mem (measurableSet_noDrop ℓ x)) have hR : ∀ᵐ X ∂conditionedFrom ν ℓ x, X ∈ RegularPath ℓ x := hac.ae_le (annealedFrom_regularPath ν ℓ htrans x) exact hR.and hD lemma open_lane_record_iff {d : ℕ} (e f : Direction d) (x : Lattice d) (X : Path d) (hX : X ∈ RegularPath (realPosition (step e)) x) (hD : X ∈ NoDrop (realPosition (step e)) x) {k : ℕ} (hk : 0 < k) (θ r a : ℝ) (hr : 0 < r) : X ∈ SurvivingLane {z | dot (realPosition x) (realPosition (step e)) ≤ dot (realPosition z) (realPosition (step e))} (Upper (realPosition (step e)) x k) {z | |signedCoordinate f z-(signedCoordinate f x+(k:ℝ)*θ)| < a*r} ↔ |scaledRecordEndpoint (realPosition (step e)) f θ r k x X| < a := by rw [survivingLane_record_iff e x X hX.1 hD hX.2.1 hX.2.2 hk] change |signedCoordinate f (_+x)-(signedCoordinate f x+(k:ℝ)*θ)| < a*r ↔ _ rw [signedCoordinate_add] have he (z : Lattice d) : signedCoordinate f z+signedCoordinate f x-(signedCoordinate f x+(k:ℝ)*θ) = signedCoordinate f z-(k:ℝ)*θ := by ring rw [he] simp only [scaledRecordEndpoint,abs_div,abs_of_pos hr] exact (div_lt_iff₀ hr).symm end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical end end end OAI