import Mathlib
import OAI.Analysis.HardSphere.PathBounds.ShiftInteracting
import OAI.Analysis.HardSphere.PathBounds.PhysicalVisitTails

namespace OAI

section

section
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal Topology BigOperators
namespace HardSphere.ShiftVisits
open Wiener PathCutoff BathSlice HardSphere.Lattice CellOrbital HardSphere.FreeVisits

theorem ground_excessVisitors_tail :
    ∃ αstar : ℝ, 0<αstar ∧ ∀ αmax : ℝ, 0<αmax →
    ∃ H : ℝ, 0<H ∧ ∀ (n K : ℕ) [NeZero K] (L : Length), L.val=(K:ℝ) →
    ∀ r D : ℝ, D=(n:ℝ)/L.val^3 → 1≤D → αstar≤D*r → D*r≤αmax → 0<r →
      (n:ℝ)*(2*Real.pi*r)^3≤L.val^3/2 →
    ∀ (g : Jet n L), IsGroundVector n L r g → ∀ (x₀ : Torus L) (A : Finset (DiscreteTorus K)) {t : ℝ},
      0<t → t≤2 →
      groundPathLaw n L r t g {ω | ∀v∈A,H*D≤(visitorCount L x₀ Set.univ v ω:ℝ)}≤
        ENNReal.ofReal (Real.exp (-(D*(A.card:ℝ)))) := by
  obtain ⟨Cd,Cf,Cs,α,hCd,hCf,hCs,hα,hm⟩ := interacting_visit_mgf (η₀:=1) (by norm_num)
  refine ⟨α,hα,?_⟩
  intro αmax hαmax
  let B := Real.exp (2*Cd*αmax)*Cf
  have hB : 0<B := by dsimp [B]; positivity
  let η := min 1 B⁻¹
  have hη : 0<η := lt_min (by norm_num) (inv_pos.mpr hB)
  have hη1 : η≤1 := min_le_left _ _
  have hb : B*η≤1 := by
    calc
      B*η≤B*B⁻¹ := mul_le_mul_of_nonneg_left (min_le_right _ _) hB.le
      _ = 1 := mul_inv_cancel₀ hB.ne'
  let H := (Cs+1)/η
  have hH : 0<H := div_pos (by linarith) hη
  refine ⟨H,hH,?_⟩
  intro n K inst L hL r D hDeq hD hαr hαup hr hs g hg x₀ A t ht ht2
  have hD0 : 0≤D := by linarith
  have hxp : t*Cd*D*r≤2*Cd*αmax := by
    calc
      t*Cd*D*r = (t*Cd)*(D*r) := by ring
      _ ≤ (t*Cd)*αmax := mul_le_mul_of_nonneg_left hαup (by positivity)
      _ ≤ (2*Cd)*αmax := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right ht2 hCd.le) hαmax.le
  have hz : Real.exp (t*Cd*D*r)*(Cf*η*(1:ℝ)^(1/3:ℝ))≤1 := by
    simp only [Real.one_rpow,mul_one]
    calc
      _ ≤ Real.exp (2*Cd*αmax)*(Cf*η) := mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr hxp) (by positivity)
      _ = B*η := by dsimp [B]; ring
      _ ≤ 1 := hb
  have hp (y : Torus L) : coordinateWiener 1 {w | freeTrajectory 1 L t w (oneConfiguration y)∈(Set.univ : Set (ConfigurationPath 1 L))}≤ENNReal.ofReal 1 := by simp
  have hh := hm n K L hL r D hDeq hD hαr hr hs g hg x₀ A ht ht2 hη.le hη1
    (by norm_num : (0:ℝ)≤1) Set.univ MeasurableSet.univ (fun _ ↦ Iff.rfl) hp hz
  have hj := joint_visitors_markov L x₀ A MeasurableSet.univ (groundPathLaw n L r t g) hη.le hh (H:=H*D)
  apply hj.trans
  apply ENNReal.ofReal_le_ofReal
  apply Real.exp_le_exp.mpr
  rw [←neg_mul]
  apply mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg A.card)
  have hhH : η*H=Cs+1 := mul_div_cancel₀ _ hη.ne'
  have he : Cs*(Real.exp (t*Cd*D*r)*(Cf*η*(1:ℝ)^(1/3:ℝ)))*D≤Cs*D := by
    exact (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hz hCs.le) hD0).trans_eq (by ring)
  change _≤ -D
  nlinarith

theorem ground_longVisitors_tail :
    ∃ αstar : ℝ, 0<αstar ∧ ∀ αmax θ : ℝ, 0<αmax → 0<θ →
    ∃ t₀ : ℝ, 0<t₀ ∧ ∀ (n K : ℕ) [NeZero K] (L : Length), L.val=(K:ℝ) →
    ∀ r D : ℝ, D=(n:ℝ)/L.val^3 → 1≤D → αstar≤D*r → D*r≤αmax → 0<r →
      (n:ℝ)*(2*Real.pi*r)^3≤L.val^3/2 →
    ∀ (g : Jet n L), IsGroundVector n L r g → ∀ (x₀ : Torus L) (A : Finset (DiscreteTorus K)) {t : ℝ},
      0<t → t ≤ min 2 t₀ →
      groundPathLaw n L r t g {ω | ∀v∈A,θ*D≤
        (visitorCount L x₀ {p | 1 < intervalDiameter 0 1 p} v ω:ℝ)}≤
        ENNReal.ofReal (Real.exp (-(θ*D/2)*(A.card:ℝ))) := by
  obtain ⟨Cd,Cf,Cs,α,hCd,hCf,hCs,hα,hm⟩ := interacting_visit_mgf (η₀:=1) (by norm_num)
  refine ⟨α,hα,?_⟩
  intro αmax θ hαmax hθ
  let B := Real.exp (2*Cd*αmax)*Cf
  have hB : 0<B := by dsimp [B]; positivity
  let δ := min 1 (θ/(2*Cs))
  have hδ : 0<δ := lt_min (by norm_num) (by positivity)
  let ε := (δ/B)^3
  have hε : 0<ε := by dsimp [ε]; positivity
  have hrp : ε^(1/3:ℝ)=δ/B := by
    simpa [ε] using Real.pow_rpow_inv_natCast (n:=3) (div_pos hδ hB).le (by norm_num : (3:ℕ)≠0)
  obtain ⟨t₀,ht₀,hsmall⟩ := free_long_probability_small hε
  refine ⟨t₀,ht₀,?_⟩
  intro n K inst L hL r D hDeq hD hαr hαup hr hs g hg x₀ A t ht htt
  have ht2 := (le_min_iff.mp htt).1
  have hD0 : 0≤D := by linarith
  have hxp : t*Cd*D*r≤2*Cd*αmax := by
    calc
      t*Cd*D*r = (t*Cd)*(D*r) := by ring
      _ ≤ (t*Cd)*αmax := mul_le_mul_of_nonneg_left hαup (by positivity)
      _ ≤ (2*Cd)*αmax := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right ht2 hCd.le) hαmax.le
  have hzδ : Real.exp (t*Cd*D*r)*(Cf*1*ε^(1/3:ℝ))≤δ := by
    rw [hrp,mul_one]
    calc
      _ ≤ Real.exp (2*Cd*αmax)*(Cf*(δ/B)) := mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr hxp) (by positivity)
      _ = B*(δ/B) := by dsimp [B]; ring
      _ = δ := mul_div_cancel₀ _ hB.ne'
  have hz1 := hzδ.trans (min_le_left 1 (θ/(2*Cs)))
  let E : Set (ConfigurationPath 1 L) := {p | 1 < intervalDiameter 0 1 p}
  have hE : MeasurableSet E := measurableSet_lt measurable_const (intervalDiameter_continuous L 0 1).measurable
  have hrev (p : ConfigurationPath 1 L) : reverseConfigurationPath 1 L p∈E↔p∈E := by
    simp only [E,Set.mem_ofPred_eq,intervalDiameter_reverse,reverseTime_zero,reverseTime_one]
  have hp (y : Torus L) : coordinateWiener 1 {w | freeTrajectory 1 L t w (oneConfiguration y)∈E}≤ENNReal.ofReal ε :=
    hsmall L ht (le_min_iff.mp htt).2 y
  have hh := hm n K L hL r D hDeq hD hαr hr hs g hg x₀ A ht ht2 (by norm_num : (0:ℝ)≤1)
    (le_refl 1) hε.le E hE hrev hp hz1
  have hj := joint_visitors_markov L x₀ A hE (groundPathLaw n L r t g) (by norm_num : (0:ℝ)≤1) hh (H:=θ*D)
  apply hj.trans
  apply ENNReal.ofReal_le_ofReal
  apply Real.exp_le_exp.mpr
  apply mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg A.card)
  have hδθ : Cs*δ≤θ/2 := by
    have hh := (le_div_iff₀ (by positivity : 0<(2:ℝ)*Cs)).mp (min_le_right 1 (θ/(2*Cs)))
    dsimp [δ]
    linarith
  have hc := mul_le_mul_of_nonneg_right ((mul_le_mul_of_nonneg_left hzδ hCs.le).trans hδθ) hD0
  nlinarith

end HardSphere.ShiftVisits

end
end

end

end OAI