| import Mathlib |
| import OAI.Probability.Ballisticity.Geometry.TubePrefixMass |
|
|
| namespace OAI |
|
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| 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 |
| |
