import Mathlib
import OAI.Probability.LogConcave.Dynamics.ProbabilityFlowIntegral
import OAI.Probability.LogConcave.Sampling.MeasureSmoothLipschitz

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 MeasureTheory ProbabilityTheory

theorem probabilityFlow_pushforward {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) :
    (stdGaussian (Point d)).map (probabilityFlow hF x hr hl hT0.le hT1)=
      interpolationLaw F x r T := by
  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 := probabilityFlow_continuous hF x hr hl hT0.le hT1
  let hprob : IsProbabilityMeasure (interpolationLaw F x r T) := by
    unfold interpolationLaw
    infer_instance
  apply measure_eq_of_smooth_lipschitz
  intro φ hφ ⟨L,hL⟩ ⟨B,hB⟩
  rw [integral_map hc.measurable.aemeasurable hφ.continuous.aestronglyMeasurable]
  exact (probabilityFlow_integral hF x hr hl hT0 hT1 (hφ.differentiable (by norm_num)) hL hB).symm
end LogConcaveSampling

end UpperProof
end
end
end

end OAI