import Mathlib import OAI.Probability.Ballisticity.Geometry.TubePrefixMass 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 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 tubeEndpoint {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s : ℕ) (ω : Environment d) : Measure (Lattice d) := Measure.sum fun n => ((quenchedKernel (ω,x)).restrict (TubePrefixAt ℓ f x θ z s n)).map (fun X => X n) lemma tubeEndpoint_apply {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s : ℕ) (ω : Environment d) (U : Set (Lattice d)) : tubeEndpoint ℓ f x θ z s ω U = ∑' n, quenchedKernel (ω,x) ((fun X => X n) ⁻¹' U ∩ TubePrefixAt ℓ f x θ z s n) := by rw [tubeEndpoint,Measure.sum_apply _ U.to_countable.measurableSet] apply tsum_congr; intro n rw [Measure.map_apply (measurable_pi_apply n) U.to_countable.measurableSet, Measure.restrict_apply ((measurable_pi_apply n) U.to_countable.measurableSet)] lemma tubeEndpoint_total {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s : ℕ) (ω : Environment d) : tubeEndpoint ℓ f x θ z s ω Set.univ = quenchedKernel (ω,x) (TubePrefix ℓ f x θ z s) := by rw [tubeEndpoint_apply,TubePrefix,measure_iUnion (tubePrefixAt_disjoint ℓ f x θ z s) (measurableSet_tubePrefixAt ℓ f x θ z s)] simp only [Set.preimage_univ,Set.univ_inter] instance tubeEndpoint_finite {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s : ℕ) (ω : Environment d) : IsFiniteMeasure (tubeEndpoint ℓ f x θ z s ω) := ⟨by rw [tubeEndpoint_total]; exact measure_lt_top _ _⟩ lemma tubeEndpoint_rows {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) {s : ℕ} (hs : 0 < s) : @Measurable _ _ (rowSigma (Strip ℓ x s)) _ (tubeEndpoint ℓ f x θ z s) := by let : MeasurableSpace (Environment d) := rowSigma (Strip ℓ x s) apply Measure.measurable_of_measurable_coe intro U _ simp_rw [tubeEndpoint_apply] apply Measurable.tsum; intro n apply measurable_quenched_prefix_event_rows (Strip ℓ x s) x (show x ∈ Strip ℓ x s from ⟨le_rfl,lt_add_of_pos_right _ (by exact_mod_cast hs)⟩) n · exact ((measurable_pi_apply n) U.to_countable.measurableSet).inter (measurableSet_tubePrefixAt ℓ f x θ z s n) · intro X Y hXY exact and_congr (by rw [Set.mem_preimage,Set.mem_preimage,hXY n le_rfl]) (tubePrefixAt_prefix ℓ f x θ z s n X Y hXY) · exact fun _ hX => hX.2.1.2 noncomputable def tubeEndpointMixture {d : ℕ} (ℓ : Vector d) (f : Direction d) (θ z : ℝ) (s : ℕ) (π : Measure (Lattice d)) (ω : Environment d) : Measure (Lattice d) := Measure.sum fun x => π {x} • tubeEndpoint ℓ f x θ z s ω lemma tubeEndpointMixture_total {d : ℕ} (ℓ : Vector d) (f : Direction d) (θ z : ℝ) (s : ℕ) (π : Measure (Lattice d)) (ω : Environment d) : tubeEndpointMixture ℓ f θ z s π ω Set.univ = tubePrefixMass ℓ f θ z s π ω := by rw [tubeEndpointMixture,Measure.sum_apply _ MeasurableSet.univ,tubePrefixMass,lintegral_countable'] apply tsum_congr; intro x rw [Measure.smul_apply,smul_eq_mul,tubeEndpoint_total,mul_comm] lemma tubeEndpointMixture_total_le {d : ℕ} (ℓ : Vector d) (f : Direction d) (θ z : ℝ) (s : ℕ) (π : Measure (Lattice d)) (ω : Environment d) : tubeEndpointMixture ℓ f θ z s π ω Set.univ ≤ π Set.univ := by rw [tubeEndpointMixture_total,tubePrefixMass] calc _ ≤ ∫⁻ _, (1:ℝ≥0∞) ∂π := lintegral_mono (fun _ => prob_le_one) _ = _ := by simp instance tubeEndpointMixture_finite {d : ℕ} (ℓ : Vector d) (f : Direction d) (θ z : ℝ) (s : ℕ) (π : Measure (Lattice d)) [IsFiniteMeasure π] (ω : Environment d) : IsFiniteMeasure (tubeEndpointMixture ℓ f θ z s π ω) := ⟨(tubeEndpointMixture_total_le ℓ f θ z s π ω).trans_lt (measure_lt_top _ _)⟩ lemma tubeEndpointMixture_lower_rows {d : ℕ} (ℓ : Vector d) (f : Direction d) (a θ z : ℝ) {s : ℕ} (hs : 0 < s) (π : Measure (Lattice d)) (hπ : ∀ᵐ x ∂π, dot (realPosition x) ℓ = a) : @Measurable _ _ (rowSigma {y | dot (realPosition y) ℓ < a+s}) _ (tubeEndpointMixture ℓ f θ z s π) := by let : MeasurableSpace (Environment d) := rowSigma {y | dot (realPosition y) ℓ < a+s} apply Measure.measurable_of_measurable_coe intro U hU simp only [tubeEndpointMixture,Measure.sum_apply _ hU,Measure.smul_apply,smul_eq_mul] apply Measurable.tsum; intro x by_cases hx : dot (realPosition x) ℓ = a · have hm := (tubeEndpoint_rows ℓ f x θ z hs).mono (rowSigma_mono (show Strip ℓ x s ⊆ {y | dot (realPosition y) ℓ < a+s} from fun _ hy => by simpa only [hx,Set.mem_ofPred_eq] using hy.2)) le_rfl exact measurable_const.mul ((Measure.measurable_coe hU).comp hm) · have hh : π {x} = 0 := measure_mono_null (show {x} ⊆ {y | dot (realPosition y) ℓ ≠ a} from fun y hy => by simpa only [Set.mem_singleton_iff.mp hy,Set.mem_ofPred_eq] using hx) (ae_iff.mp hπ) simp only [hh,zero_mul] exact measurable_const end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical namespace DirectionalTransience lemma two_tube_threat_bound {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] (ℓ : Vector d) (f : Direction d) (a θ z δ : ℝ) (H : ℕ) (π : Measure (Lattice d)) (ρ : ℕ → Environment d → Measure (Lattice d)) (hρ : ∀ s : ℕ, @Measurable _ _ (rowSigma {y | dot (realPosition y) ℓ < a+s}) _ (ρ s)) (c : ℝ≥0∞) (hfirst : environmentLaw ν {ω | (π,ω) ∈ TubeThreat ℓ f a θ z δ H} ≤ c) (hsecond : ∀ s ≤ H, ∀ η ∈ FirstTubeThreat ℓ f a θ z δ s π, environmentLaw ν {ω | (ρ s η,ω) ∈ TubeThreat ℓ f (a+s) θ z δ H} ≤ c) : environmentLaw ν (⋃ s ∈ Finset.range (H+1), {ω | ω ∈ FirstTubeThreat ℓ f a θ z δ s π ∧ (ρ s ω,ω) ∈ TubeThreat ℓ f (a+s) θ z δ H}) ≤ c*c := by let A := fun s => FirstTubeThreat ℓ f a θ z δ s π have hA (s : ℕ) : MeasurableSet (A s) := rowSigma_le _ _ (firstTubeThreat_lower_rows ℓ f a θ z δ s π) have hdisj : (↑(Finset.range (H+1)) : Set ℕ).PairwiseDisjoint A := fun _ _ _ _ h => firstTubeThreat_disjoint ℓ f a θ z δ π h have he : (⋃ s ∈ Finset.range (H+1), A s) = {ω | (π,ω) ∈ TubeThreat ℓ f a θ z δ H} := by simpa only [Finset.mem_range,Nat.lt_succ_iff] using firstTubeThreat_iUnion ℓ f a θ z δ π H calc _ ≤ ∑ s ∈ Finset.range (H+1), environmentLaw ν {ω | ω ∈ A s ∧ (ρ s ω,ω) ∈ TubeThreat ℓ f (a+s) θ z δ H} := measure_biUnion_finset_le _ _ _ ≤ ∑ s ∈ Finset.range (H+1), c*environmentLaw ν (A s) := by apply Finset.sum_le_sum intro s hs apply fresh_random_tube_threat ν ℓ f (a+s) θ z δ H {y | dot (realPosition y) ℓ < a+s} (Set.disjoint_left.mpr (fun _ hy hz => (not_lt_of_ge (show a+(s:ℝ) ≤ _ from hz)) hy)) (ρ s) (hρ s) (A s) (firstTubeThreat_lower_rows ℓ f a θ z δ s π) c exact hsecond s (Nat.lt_succ_iff.mp (Finset.mem_range.mp hs)) _ = c*environmentLaw ν (⋃ s ∈ Finset.range (H+1), A s) := by rw [← Finset.mul_sum,measure_biUnion_finset hdisj (fun s _ => hA s)] _ ≤ c*c := by rw [he]; exact mul_le_mul_right hfirst c end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical end end end OAI