import Mathlib import OAI.Probability.Ballisticity.Estimates.AlignedRecordPrefix 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 namespace DirectionalTransience lemma quenched_prefix_event_congr {d : ℕ} (S : Set (Lattice d)) (ω η : Environment d) (hωη : Set.EqOn ω η S) (x : Lattice d) (n : ℕ) (A : Set (Path d)) (hA : PrefixDetermined n A) (hAS : ∀ X ∈ A, ∀ j < n, X j ∈ S) : quenchedKernel (ω,x) A = quenchedKernel (η,x) A := by let C : Set ((i : Finset.Iic n) → Lattice d) := {f | extendPrefix n f ∈ A} have he : (fun X : Path d => Preorder.frestrictLe n X) ⁻¹' C = A := by ext X exact hA (extendPrefix n (Preorder.frestrictLe n X)) X (fun j hj => by rw [extendPrefix_apply _ _ _ hj]; rfl) have hm (f : (i : Finset.Iic n) → Lattice d) (_ : f ∈ C) : MeasurableSet ((fun X : Path d => Preorder.frestrictLe n X) ⁻¹' {f}) := (Preorder.measurable_frestrictLe n) (measurableSet_singleton f) rw [← he,← tsum_measure_preimage_singleton C.to_countable hm, ← tsum_measure_preimage_singleton C.to_countable hm] apply tsum_congr intro f rw [restrict_prefix_fiber] exact pathCylinder_weight_locality S ω η hωη x (extendPrefix n f.val) n (hAS _ f.property) lemma measurable_quenched_prefix_event_rows {d : ℕ} (S : Set (Lattice d)) (x : Lattice d) (hx : x ∈ S) (n : ℕ) (A : Set (Path d)) (hA : MeasurableSet A) (hprefix : PrefixDetermined n A) (hAS : ∀ X ∈ A, ∀ j < n, X j ∈ S) : @Measurable _ _ (rowSigma S) _ (fun ω : Environment d => quenchedKernel (ω,x) A) := by apply measurable_of_row_locality _ ((Kernel.measurable_coe _ hA).comp (measurable_id.prodMk measurable_const)) S x hx intro ω η h exact quenched_prefix_event_congr S ω η h x n A hprefix hAS def TubePrefixAt {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s n : ℕ) : Set (Path d) := HitAt (Strip ℓ x s) (Upper ℓ x s) n ∩ {X | ∀ j ≤ s, ∃ a ≤ n, (fun k => X k-x) ∈ RecordOrZeroPrefix ℓ j a ∧ |signedCoordinate f (X a-x)-(j:ℝ)*θ| ≤ z} lemma measurableSet_tubePrefixAt {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s n : ℕ) : MeasurableSet (TubePrefixAt ℓ f x θ z s n) := by apply (measurableSet_hitAt _ _ n).inter simp only [Set.ofPred_forall,Set.ofPred_exists,Set.ofPred_and] apply MeasurableSet.iInter; intro j apply MeasurableSet.iInter; intro _ apply MeasurableSet.iUnion; intro a refine (MeasurableSet.const (a ≤ n)).inter ?_ have hm : Measurable (fun X : Path d => fun k => X k-x) := by fun_prop have hc : Measurable (fun X : Path d => signedCoordinate f (X a-x)) := (measurable_of_countable (signedCoordinate f)).comp ((measurable_pi_apply a : Measurable (fun X : Path d => X a)).sub_const x) exact ((measurableSet_recordOrZeroPrefix ℓ j a).preimage hm).inter (measurableSet_le ((hc.sub_const ((j:ℝ)*θ)).abs) measurable_const) lemma tubePrefixAt_prefix {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s n : ℕ) : PrefixDetermined n (TubePrefixAt ℓ f x θ z s n) := by have hdir (X Y : Path d) (hXY : ∀ j ≤ n, X j=Y j) (hX : X ∈ TubePrefixAt ℓ f x θ z s n) : Y ∈ TubePrefixAt ℓ f x θ z s n := by refine ⟨⟨by simpa only [← hXY n le_rfl] using hX.1.1, fun j hj => by simpa only [← hXY j hj.le] using hX.1.2 j hj⟩,?_⟩ intro j hj obtain ⟨a,ha,hrec,hgood⟩ := hX.2 j hj refine ⟨a,ha,(recordOrZeroPrefix_prefix ℓ j a _ _ (fun k hk => by rw [hXY k (hk.trans ha)])).mp hrec,?_⟩ simpa only [← hXY a ha] using hgood intro X Y h exact ⟨hdir X Y h,hdir Y X (fun j hj => (h j hj).symm)⟩ def TubePrefix {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s : ℕ) : Set (Path d) := ⋃ n, TubePrefixAt ℓ f x θ z s n lemma measurableSet_tubePrefix {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s : ℕ) : MeasurableSet (TubePrefix ℓ f x θ z s) := MeasurableSet.iUnion (measurableSet_tubePrefixAt ℓ f x θ z s) lemma tubePrefixAt_disjoint {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s : ℕ) : Pairwise (fun n m => Disjoint (TubePrefixAt ℓ f x θ z s n) (TubePrefixAt ℓ f x θ z s m)) := by intro n m hnm exact (hitAt_pairwise_disjoint (disjoint_strip_upper ℓ x s) hnm).mono Set.inter_subset_left Set.inter_subset_left lemma quenched_tubePrefix_congr {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z : ℝ) (s : ℕ) (ω η : Environment d) (hωη : Set.EqOn ω η (Strip ℓ x s)) : quenchedKernel (ω,x) (TubePrefix ℓ f x θ z s) = quenchedKernel (η,x) (TubePrefix ℓ f x θ z s) := by rw [TubePrefix,measure_iUnion (tubePrefixAt_disjoint ℓ f x θ z s) (measurableSet_tubePrefixAt ℓ f x θ z s),measure_iUnion (tubePrefixAt_disjoint ℓ f x θ z s) (measurableSet_tubePrefixAt ℓ f x θ z s)] exact tsum_congr fun n => quenched_prefix_event_congr (Strip ℓ x s) ω η hωη x n _ (tubePrefixAt_prefix ℓ f x θ z s n) (fun _ hX => hX.1.2) lemma measurable_quenched_tubePrefix_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)) _ (fun ω : Environment d => quenchedKernel (ω,x) (TubePrefix ℓ f x θ z s)) := by apply measurable_of_row_locality _ ((Kernel.measurable_coe _ (measurableSet_tubePrefix ℓ f x θ z s)).comp (measurable_id.prodMk measurable_const)) (Strip ℓ x s) x (show x ∈ Strip ℓ x s from ⟨le_rfl,by exact lt_add_of_pos_right _ (by exact_mod_cast hs)⟩) intro ω η h exact quenched_tubePrefix_congr ℓ f x θ z s ω η h end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical end end end OAI