lean/OAI/Probability/Ballisticity/Geometry/TubePrefix.lean

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