| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.AlignedRecordPrefix |
|
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| 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 |
| 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 |
|
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
|
|
| end |
| end |
|
|
| end OAI |
|
|
