lean/OAI/Probability/Ballisticity/Coupling/QuenchedConditionalProbability.lean

import Mathlib
import OAI.Probability.Ballisticity.Estimates.RecordPrefixControl
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
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators 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 quenchedConditionalProbability {d : ℕ} (ℓ : Vector d) (x : Lattice d)
    (A : Set (Path d)) (ω : Environment d) : ℝ≥0∞ :=
  (noDropQuenched ℓ x ω)⁻¹*quenchedKernel (ω,x) (A ∩ NoDrop ℓ x)

lemma measurable_quenchedConditionalProbability {d : ℕ} (ℓ : Vector d) (x : Lattice d)
    (A : Set (Path d)) (hA : MeasurableSet A) : Measurable (quenchedConditionalProbability ℓ x A) :=
  (measurable_noDropQuenched ℓ x).inv.mul ((Kernel.measurable_coe quenchedKernel
    (hA.inter (measurableSet_noDrop ℓ x))).comp (measurable_id.prodMk measurable_const))

lemma quenchedConditionalProbability_le_one {d : ℕ} (ℓ : Vector d) (x : Lattice d)
    (A : Set (Path d)) (ω : Environment d) : quenchedConditionalProbability ℓ x A ω ≤ 1 :=
  (mul_le_mul' le_rfl (measure_mono Set.inter_subset_right)).trans (ENNReal.inv_mul_le_one _)

lemma quenchedConditionalProbability_real_integrable {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν]
    (ℓ : Vector d) (x : Lattice d) (A : Set (Path d)) (hA : MeasurableSet A) :
    Integrable (fun ω => (quenchedConditionalProbability ℓ x A ω).toReal) (environmentLaw ν) := by
  apply (integrable_const (1:ℝ)).mono'
    (measurable_quenchedConditionalProbability ℓ x A hA).ennreal_toReal.aestronglyMeasurable
  exact ae_of_all _ fun ω => by
    rw [Real.norm_eq_abs,abs_of_nonneg ENNReal.toReal_nonneg]
    simpa only [ENNReal.toReal_one] using ENNReal.toReal_mono ENNReal.one_ne_top (quenchedConditionalProbability_le_one ℓ x A ω)

lemma quenchedConditionalProbability_mean {d : ℕ} (ν : Measure (Row d))
    (ℓ : Vector d) (A : Set (Path d)) (hA : MeasurableSet A) :
    (∫ ω, (quenchedConditionalProbability ℓ 0 A ω).toReal ∂environmentLaw ν) =
      (quenchedConditionedAverage ν ℓ).real A := by
  rw [integral_toReal (measurable_quenchedConditionalProbability ℓ 0 A hA).aemeasurable
    (ae_of_all _ fun ω => lt_of_le_of_lt (quenchedConditionalProbability_le_one ℓ 0 A ω) ENNReal.one_lt_top)]
  rw [Measure.real,quenchedConditionedAverage_apply ν ℓ A hA]
  rfl

lemma quenched_good_mass_identity {d : ℕ} (ℓ : Vector d) (x : Lattice d)
    (A : Set (Path d)) (hA : MeasurableSet A) (ω : Environment d)
    (hq : 0 < noDropQuenched ℓ x ω) :
    (quenchedKernel (ω,x)).real (Aᶜ ∩ NoDrop ℓ x) =
      (noDropQuenched ℓ x ω).toReal*(1-(quenchedConditionalProbability ℓ x A ω).toReal) := by
  have hp : 0 < (noDropQuenched ℓ x ω).toReal := ENNReal.toReal_pos (ne_of_gt hq) (measure_ne_top _ _)
  have hsplit := measureReal_inter_add_sdiff (μ := quenchedKernel (ω,x)) (s := NoDrop ℓ x) hA
  have he : NoDrop ℓ x \ A = Aᶜ ∩ NoDrop ℓ x := by ext X; simp only [Set.mem_sdiff,Set.mem_inter_iff,Set.mem_compl_iff]; tauto
  rw [he,Set.inter_comm (NoDrop ℓ x) A] at hsplit
  rw [quenchedConditionalProbability,ENNReal.toReal_mul,ENNReal.toReal_inv]
  change (quenchedKernel (ω,x)).real (Aᶜ ∩ NoDrop ℓ x) =
    (noDropQuenched ℓ x ω).toReal*(1-(noDropQuenched ℓ x ω).toReal⁻¹*(quenchedKernel (ω,x)).real (A ∩ NoDrop ℓ x))
  rw [mul_sub,mul_one,← mul_assoc,mul_inv_cancel₀ (ne_of_gt hp),one_mul]
  change (quenchedKernel (ω,x)).real (A ∩ NoDrop ℓ x)+
    (quenchedKernel (ω,x)).real (Aᶜ ∩ NoDrop ℓ x) = (noDropQuenched ℓ x ω).toReal at hsplit
  linarith

theorem quenched_good_mass_bad_probability {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν]
    (hue : UniformElliptic ν) (ℓ : Vector d) (hℓ : dot ℓ ℓ = 1)
    (htrans : DirectionallyTransient ν ℓ) (A : Set (Path d)) (hA : MeasurableSet A)
    {δ : ℝ} (hδ : 0 < δ) :
    (environmentLaw ν).real {ω | (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) < 2*δ} ≤
      (environmentLaw ν).real {ω | (noDropQuenched ℓ 0 ω).toReal < 4*δ}+
        2*(quenchedConditionedAverage ν ℓ).real A := by
  let Q := fun ω => (quenchedConditionalProbability ℓ 0 A ω).toReal
  let B := {ω | (noDropQuenched ℓ 0 ω).toReal < 4*δ}
  let C := {ω | (1:ℝ)/2 ≤ Q ω}
  have hsub : ∀ᵐ ω ∂environmentLaw ν, ω ∈ {ω | (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) < 2*δ} → ω ∈ B ∪ C := by
    filter_upwards [quenched_noDrop_positive_of_directionallyTransient ν hue ℓ hℓ htrans] with ω hp
    intro hbad
    change (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) < 2*δ at hbad
    by_contra hn
    have hq : 4*δ ≤ (noDropQuenched ℓ 0 ω).toReal := by
      by_contra! h; exact hn (Or.inl h)
    have hQ : Q ω < 1/2 := by by_contra! h; exact hn (Or.inr h)
    have hg := quenched_good_mass_identity ℓ 0 A hA ω (hp 0)
    change (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) =
      (noDropQuenched ℓ 0 ω).toReal*(1-Q ω) at hg
    have hpos : 0 < (noDropQuenched ℓ 0 ω).toReal := by linarith
    nlinarith
  have hm := ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hsub)
  have hu := measureReal_union_le (μ := environmentLaw ν) B C
  have hmark := mul_meas_ge_le_integral_of_nonneg
    (ae_of_all (environmentLaw ν) fun ω => (show 0 ≤ Q ω from ENNReal.toReal_nonneg))
    (quenchedConditionalProbability_real_integrable ν ℓ 0 A hA) (1/2)
  rw [quenchedConditionalProbability_mean ν ℓ A hA] at hmark
  change (1/2)*(environmentLaw ν).real C ≤ (quenchedConditionedAverage ν ℓ).real A at hmark
  change (environmentLaw ν).real {ω | (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) < 2*δ} ≤
    (environmentLaw ν).real (B ∪ C) at hm
  change (environmentLaw ν).real {ω | (quenchedKernel (ω,0)).real (Aᶜ ∩ NoDrop ℓ 0) < 2*δ} ≤
    (environmentLaw ν).real B+2*(quenchedConditionedAverage ν ℓ).real A
  linarith

lemma positive_lower_quantile {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
    (F : Ω → ℝ) (hF : Measurable F) (hpos : ∀ᵐ x ∂μ, 0 < F x)
    {ε : ℝ} (hε : 0 < ε) : ∃ δ : ℝ, 0 < δ ∧ δ < 1/4 ∧ μ.real {x | F x < 4*δ} ≤ ε := by
  have ht := fixed_real_tail_small μ (fun x => (F x)⁻¹) hF.inv
    (fun n : ℕ => (n:ℝ)+1) (tendsto_atTop_add_const_right _ _ tendsto_natCast_atTop_atTop)
  obtain ⟨N,hN⟩ := (ht.eventually_lt_const hε).exists
  let R : ℝ := (N:ℝ)+1
  have hR : 0 < R := by dsimp [R]; positivity
  let δ : ℝ := 1/(8*R)
  have hδ : 0 < δ := by dsimp [δ]; positivity
  have hδsmall : δ < 1/4 := by
    dsimp [δ,R]
    apply (div_lt_iff₀ (by positivity)).mpr
    have hn : 0 ≤ (N:ℝ) := Nat.cast_nonneg N
    linarith
  refine ⟨δ,hδ,hδsmall,?_⟩
  suffices hs : ∀ᵐ x ∂μ, x ∈ {x | F x < 4*δ} → x ∈ {x | R ≤ |(F x)⁻¹|} from
    (ENNReal.toReal_mono (measure_ne_top _ _) (measure_mono_ae hs)).trans hN.le
  filter_upwards [hpos] with x hx hbad
  change F x < 4*δ at hbad
  rw [abs_of_pos (inv_pos.mpr hx)]
  rw [← one_div]
  apply (le_div_iff₀ hx).mpr
  have heq : 4*δ = 1/(2*R) := by dsimp [δ]; field_simp; ring
  rw [heq] at hbad
  have hh := (lt_div_iff₀ (by positivity : 0 < 2*R)).mp hbad
  nlinarith

end DirectionalTransience

open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical

end
end

end OAI