import Mathlib
import OAI.Probability.LogConcave.Dynamics.TriangularEquiv
import OAI.Probability.LogConcave.Sampling.ProbabilityVelocity

namespace OAI

section
section
noncomputable section
open MeasureTheory Filter
open scoped ENNReal NNReal Topology

section UpperProof
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open Function MeasureTheory Set Filter
open scoped Topology NNReal

namespace LogConcaveSampling
open Set Function MeasureTheory Filter ProbabilityTheory
open scoped Topology

theorem probabilityFlow_integral {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
    (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0≤r)
    (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1)
    {φ : Point d → ℝ} (hφ : Differentiable ℝ φ) {L : ℝ≥0} (hφL : LipschitzWith L φ)
    {B : ℝ} (hφB : ∀ z,‖φ z‖≤B) :
    (∫ y,φ y ∂interpolationLaw F x r T) =
      ∫ z,φ (probabilityFlow hF x hr hl hT0.le hT1 z) ∂stdGaussian (Point d) := by
  let μ := gibbs (primitivePotential F x r)
  let ν := stdGaussian (Point d)
  let := probability_gibbs_of_partition
    (partition_pos_of_continuous (hF.continuous_potential x r)).ne'
    (partition_ne_top_of_integrable (hF.integrable_exp_neg_potential x hr (by linarith)))
  have hc := clampedProbabilityVelocity_continuous hF x hr hl hT0.le hT1
  have hL := fun t (_ : t∈Icc 0 T) => clampedProbabilityVelocity_lipschitz hF x hr hl hT0.le hT1 t
  obtain ⟨g,hg,hgb⟩ := interpolationPath_defect_bound hF x hr hl hT0.le hT1
  have hh := GlobalODE.transport_expectation_of_flux (μ:=μ.prod ν) hT0 hc hL
    (clampedProbabilityDerivative_continuous hF x hr hl hT0.le hT1)
    (clampedProbabilityVelocity_hasFDerivAt hF x hr hl hT0.le hT1)
    (X:=interpolationPath) (X':=interpolationPathDerivative)
    interpolationPath_measurable interpolationPath_continuous
    (fun t ht p => interpolationPath_hasDerivAt (by
      have hh := (probability_time ht.1.le (ht.2.trans hT1)).1; linarith) p)
    interpolationPathDerivative_measurable hg (fun t ht => hgb t ⟨ht.1.le,ht.2.le⟩)
    (fun t ht S hS ⟨C,hC⟩ => by
      simp_rw [clampedProbabilityVelocity_eq F x r T hT0.le ⟨ht.1.le,ht.2.le⟩]
      exact interpolationPath_flux hF x hr hl ht.1.le (ht.2.trans hT1) hS hC)
    hφ hφL hφB
  have hfc := probabilityFlow_continuous hF x hr hl hT0.le hT1
  have he := integral_map (μ:=μ.prod ν) measurable_snd.aemeasurable
    (hφ.continuous.comp hfc).aestronglyMeasurable
  rw [Measure.map_snd_prod,measure_univ,one_smul] at he
  change (∫ y,φ y ∂(μ.prod ν).map (interpolationPath T))=_
  rw [integral_map (interpolationPath_measurable T).aemeasurable hφ.continuous.aestronglyMeasurable]
  change (∫ p,φ (interpolationPath T p) ∂μ.prod ν)=_
  rw [hh]
  trans ∫ p,φ (probabilityFlow hF x hr hl hT0.le hT1 p.2) ∂μ.prod ν
  · apply integral_congr_ae
    filter_upwards [] with p
    rw [show interpolationPath 0 p=p.2 by simp [interpolationPath]]
    rfl
  · exact he.symm
end LogConcaveSampling

end UpperProof
end
end
end

end OAI