lean/OAI/Probability/Ballisticity/Geometry/FiniteConditionalDensityTubeTransfer.lean

import Mathlib
import OAI.Probability.Ballisticity.Geometry.WeightedConditionedTubeTransfer
import OAI.Probability.Ballisticity.Estimates.RawConditionedWeight

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
namespace DirectionalTransience

lemma finiteConditionalDensity_tube_transfer {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν]
    (e f : Direction d) (htrans : DirectionallyTransient ν (realPosition (step e)))
    (b : ℕ → ℝ) (H : ℕ → ℕ) (z : ℕ → ℝ) (hz : Tendsto z atTop atTop)
    (hsmall : ∀ᶠ i in atTop, (conditionedLaw ν (realPosition (step e))).real
      (MedianTubeFailure (realPosition (step e)) f b (H i) (z i/10)) ≤ 3/10)
    (N : ℕ) {ε : ℝ} (hε : 0 < ε) :
    ∀ᶠ i in atTop, (rawConditionedWeight ν (realPosition (step e))
      (finiteConditionalDensity (realPosition (step e)) N)).real
        (MedianTubeFailure (realPosition (step e)) f b (H i) (z i)) ≤
      (rawConditionedWeight ν (realPosition (step e)) (finiteConditionalDensity (realPosition (step e)) N)).real
        Set.univ * (conditionedLaw ν (realPosition (step e))).real
          (MedianTubeFailure (realPosition (step e)) f b (H i) (z i/10))+ε := by
  let ℓ := realPosition (step e)
  let p := annealedLaw ν (NoDrop ℓ 0)
  have hp : p ≠ 0 := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans)
  have hpfin : p ≠ ∞ := measure_ne_top _ _
  let C : ℝ≥0∞ := (N+1:ℕ)
  have hC : C ≠ 0 := by simp [C]
  have hCfin : C ≠ ∞ := by simp [C]
  let g := fun ω => C⁻¹ * finiteConditionalDensity ℓ N ω
  have hg : @Measurable _ _ (rowSigma {x | dot (realPosition x) ℓ ≤ (N:ℝ)}) _ g :=
    measurable_const.mul (finiteConditionalDensity_lower_rows e N)
  have hgle : ∀ ω, g ω ≤ 1 := by
    intro ω
    exact (mul_le_mul' (le_refl C⁻¹) (finiteConditionalDensity_le ℓ N ω)).trans (ENNReal.inv_mul_le_one C)
  let η := weightedConditioned ν ℓ g
  let ρ := rawConditionedWeight ν ℓ (finiteConditionalDensity ℓ N)
  have heq : ρ = (C*p) • η := by
    have he : finiteConditionalDensity ℓ N = fun ω => C*g ω := by
      funext ω
      dsimp only [g]
      rw [← mul_assoc,ENNReal.mul_inv_cancel hC hCfin,one_mul]
    dsimp only [ρ]
    rw [he,rawConditionedWeight_smul ν ℓ g (hg.mono (rowSigma_le _) le_rfl) C,
      rawConditionedWeight_eq_smul ν ℓ hp g,smul_smul]
  have hM : 0 < (C*p).toReal := ENNReal.toReal_pos (mul_ne_zero hC hp) (ENNReal.mul_ne_top hCfin hpfin)
  have hh := weightedConditioned_tube_transfer ν ℓ f htrans b H z hz hsmall
    (N:ℝ) (Nat.cast_nonneg N) g hg hgle (div_pos hε hM)
  filter_upwards [hh] with i hi
  have hm (A : Set (Path d)) : ρ.real A = (C*p).toReal*η.real A := by
    rw [heq,Measure.real,Measure.smul_apply,smul_eq_mul,ENNReal.toReal_mul]
    rfl
  change ρ.real (MedianTubeFailure ℓ f b (H i) (z i)) ≤
    ρ.real Set.univ*(conditionedLaw ν ℓ).real (MedianTubeFailure ℓ f b (H i) (z i/10))+ε
  rw [hm,hm]
  have hi' := mul_le_mul_of_nonneg_left hi hM.le
  have hc : (C*p).toReal*(ε/(C*p).toReal) = ε := by field_simp
  rw [mul_add,hc] at hi'
  simpa only [mul_assoc,η] using hi'

lemma measureReal_le_add_mass_defect {Ω : Type*} [MeasurableSpace Ω]
    (μ η : Measure Ω) [IsProbabilityMeasure μ] [IsFiniteMeasure η] (hη : η ≤ μ)
    (A : Set Ω) (hA : MeasurableSet A) : μ.real A ≤ η.real A+1-η.real Set.univ := by
  have hc : η.real Aᶜ ≤ μ.real Aᶜ := ENNReal.toReal_mono (measure_ne_top _ _) (hη Aᶜ)
  rw [measureReal_compl hA,measureReal_compl hA,show μ.real Set.univ = 1 from probReal_univ] at hc
  linarith

theorem quenchedConditioned_tube_transfer_small {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν]
    (hue : UniformElliptic ν) (e f : Direction d)
    (htrans : DirectionallyTransient ν (realPosition (step e)))
    (b : ℕ → ℝ) (H : ℕ → ℕ) (z : ℕ → ℝ) (hz : Tendsto z atTop atTop)
    (hsmall : ∀ᶠ i in atTop, (conditionedLaw ν (realPosition (step e))).real
      (MedianTubeFailure (realPosition (step e)) f b (H i) (z i/10)) ≤ 3/10)
    {ε : ℝ} (hε : 0 < ε) :
    ∀ᶠ i in atTop, (quenchedConditionedAverage ν (realPosition (step e))).real
      (MedianTubeFailure (realPosition (step e)) f b (H i) (z i)) ≤
        (conditionedLaw ν (realPosition (step e))).real
          (MedianTubeFailure (realPosition (step e)) f b (H i) (z i/10))+ε := by
  let ℓ := realPosition (step e)
  let μ := quenchedConditionedAverage ν ℓ
  let : IsProbabilityMeasure μ := quenchedConditionedAverage_probability ν hue ℓ (signed_direction_unit e) htrans
  let ρ := fun N => rawConditionedWeight ν ℓ (finiteConditionalDensity ℓ N)
  have hdom (N : ℕ) : ρ N ≤ μ := rawConditionedWeight_mono ν ℓ _ _
    (finiteConditionalDensity_measurable ℓ N) (measurable_noDropQuenched ℓ 0).inv
      (finiteConditionalDensity_le_inverse ℓ N)
  let : ∀ N, IsFiniteMeasure (ρ N) := fun N =>
    ⟨lt_of_le_of_lt (hdom N Set.univ) (measure_lt_top _ _)⟩
  have ht : Tendsto (fun N => (ρ N).real Set.univ) atTop (𝓝 1) := by
    have hh := (ENNReal.continuousAt_toReal (by simp : (1:ℝ≥0∞) ≠ ∞)).tendsto.comp
      (finiteConditionalDensity_mass_tendsto ν hue ℓ (signed_direction_unit e) htrans)
    simpa only [Function.comp_def,ENNReal.toReal_one,Measure.real,ρ] using hh
  obtain ⟨N,hN⟩ := (ht.eventually (lt_mem_nhds (show 1-ε/2 < (1:ℝ) by linarith))).exists
  have hh := finiteConditionalDensity_tube_transfer ν e f htrans b H z hz hsmall N
    (show 0 < ε/2 by positivity)
  filter_upwards [hh] with i hi
  let A := MedianTubeFailure ℓ f b (H i) (z i)
  let B := MedianTubeFailure ℓ f b (H i) (z i/10)
  have ha := measureReal_le_add_mass_defect μ (ρ N) (hdom N) A
    (measurableSet_medianTubeFailure ℓ f b (H i) (z i))
  have hmass : (ρ N).real Set.univ ≤ 1 := by
    have h := ENNReal.toReal_mono (measure_ne_top _ _) (hdom N Set.univ)
    simpa only [Measure.real,measure_univ,ENNReal.toReal_one] using h
  have hp := mul_le_mul_of_nonneg_right hmass (show 0 ≤ (conditionedLaw ν ℓ).real B from measureReal_nonneg)
  change (ρ N).real A ≤ (ρ N).real Set.univ*(conditionedLaw ν ℓ).real B+ε/2 at hi
  change μ.real A ≤ (conditionedLaw ν ℓ).real B+ε
  linarith

end DirectionalTransience

open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical

end
end

end OAI