Tests.ProximalBPSCenteredRootOrderInverse

import AutoSamplingTheory.ExampleCases.ProximalBPS.CenteredRootOrderInverse

open MeasureTheory ProbabilityTheory InnerProductSpace
open scoped RealInnerProductSpace ContDiff NNReal Topology ENNReal
namespace Tests.ProximalBPSCenteredRootOrderInverse
noncomputable section
set_option autoImplicit false
set_option maxHeartbeats 2000000

-- Original-input consumer: normalized actual leakage preserves the norm and lies in ker P.
-- This tests the same bounded inverse without assuming a gap, root or polar isometry.
theorem genuine_actual_centered_inverse_consumer
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : ℝ≥0} {η : ℝ}
    (hα : 0 < (α : ℝ)) (hαβ : α ≤ β) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x v : E,
      (α : ℝ)*‖v‖^2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v ∧
      (fderiv ℝ (fderiv ℝ V) x v) v ≤ (β : ℝ)*‖v‖^2)
    (hη : 0 < η) (hβη : (β : ℝ)*η ≤ 1) :
    let μ := (volume : Measure E).tilted (fun x => -V x)
    let J := (μ.prod (stdGaussian E)).map
      (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
    let ν := J.snd
    let Λ := J.map (fun p : E × E => (p.2,(2 : ℝ) • p.1-p.2))
    let F := fun p : E × E => (p.1,(2 : ℝ) • p.1-p.2)
    let mY := MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)
    letI : MeasurableSpace (E × E) := Prod.instMeasurableSpace
    letI : Fact (mY ≤ (inferInstance : MeasurableSpace (E × E))) :=
      ⟨measurable_snd.comap_le⟩
    let HP := lpMeas ℝ ℝ mY 2 J
    let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
      HP.subtypeL ∘L condExpL2 ℝ ℝ (μ := J) measurable_snd.comap_le
    let hp : MeasurePreserving (Prod.snd : E × E → E) J ν := ⟨measurable_snd,rfl⟩
    let M : Lp ℝ 2 ν →ₗᵢ[ℝ] Lp ℝ 2 J := Lp.compMeasurePreservingₗᵢ ℝ Prod.snd hp
    IsProbabilityMeasure μ ∧ IsProbabilityMeasure J ∧ IsProbabilityMeasure ν ∧
    HP = P.toLinearMap.range ∧
    (∀ f : HP, P (f : Lp ℝ 2 J) = f) ∧
    ∃ S : Kernel E E, IsMarkovKernel S ∧
      (∀ y, S y = (volume : Measure E).tilted
        (fun x => -V ((1/2 : ℝ) • (y+x)) - ‖y-x‖^2/(8*η))) ∧
      Λ.IsCondKernel S ∧ Λ.fst = ν ∧ Λ.snd = ν ∧
      ∃ e : Lp ℝ 2 ν ≃ₗᵢ[ℝ] HP,
        (∀ u : Lp ℝ 2 ν, (e u : Lp ℝ 2 J) = M u) ∧
        ∃ U : Lp ℝ 2 J →ₗᵢ[ℝ] Lp ℝ 2 J,
          (∀ g : Lp ℝ 2 J, (U g : E × E → ℝ) =ᵐ[J] g ∘ F) ∧
          Function.Involutive U ∧ IsSelfAdjoint U.toContinuousLinearMap ∧
          ∃ T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν,
            IsSelfAdjoint T ∧
            (∀ u : Lp ℝ 2 ν,
              ‖T u‖ ≤ ‖u‖ ∧
              (T u : E → ℝ) =ᵐ[ν] (fun y => ∫ x, u x ∂S y) ∧
              (∫ y, T u y ∂ν) = ∫ y, u y ∂ν) ∧
            let A : HP →L[ℝ] HP :=
              HP.orthogonalProjectionOnto ∘L U.toContinuousLinearMap ∘L HP.subtypeL
            let B : HP →L[ℝ] Lp ℝ 2 J :=
              (((1 : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J)-P)*U.toContinuousLinearMap*P) ∘L HP.subtypeL
            A = e.conjStarAlgEquiv T ∧ IsSelfAdjoint A ∧
            (∀ f : HP, ‖A f‖ ≤ ‖f‖) ∧
            (∀ f : HP, P (B f) = 0) ∧
            ∃ Γ : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν,
              Γ.IsPositive ∧ Γ*Γ=(1 : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν)-T*T ∧
              let ΓP : HP →L[ℝ] HP := e.conjStarAlgEquiv Γ
              ΓP.IsPositive ∧ ΓP*ΓP=(1 : HP →L[ℝ] HP)-A*A ∧
              B.adjoint ∘L B = ΓP*ΓP ∧
              ∃ q : Lp ℝ 2 ν,
                (q : E → ℝ) =ᵐ[ν] (fun _ => (1 : ℝ)) ∧ T q = q ∧
                (∀ u : Lp ℝ 2 ν, inner ℝ q u = ∫ y, u y ∂ν) ∧
                let qP : HP := e q
                let HP0 := (innerSL ℝ qP).ker
                letI : NormedAddCommGroup HP0 := HP0.normedAddCommGroup
                letI : InnerProductSpace ℝ HP0 := HP0.innerProductSpace
                let γ : ℝ := 2*Real.sqrt ((α : ℝ)*η)/(1+(α : ℝ)*η)
                (∀ f : HP, f ∈ HP0 ↔ (∫ y, (e.symm f) y ∂ν) = 0) ∧
                ΓP qP = 0 ∧ 0 < γ ∧
                ∃ ΓP0 : HP0 →L[ℝ] HP0,
                  (∀ f : HP0, (ΓP0 f : HP) = ΓP (f : HP)) ∧
                  ΓP0.IsPositive ∧ (ΓP0-γ • (1 : HP0 →L[ℝ] HP0)).IsPositive ∧
                  IsUnit ΓP0 ∧
                  ∃ Inv : HP0 →L[ℝ] HP0,
                    Inv*ΓP0=(1 : HP0 →L[ℝ] HP0) ∧
                    ΓP0*Inv=(1 : HP0 →L[ℝ] HP0) ∧ ‖Inv‖ ≤ 1/γ ∧
                    (∀ f : HP0,
                      ‖B (HP0.subtypeL (Inv f))‖ = ‖f‖ ∧
                      P (B (HP0.subtypeL (Inv f))) = 0) := by
  classical
  dsimp only
  let μ := (volume : Measure E).tilted (fun x => -V x)
  let J := (μ.prod (stdGaussian E)).map
    (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
  let ν := J.snd
  let mY : MeasurableSpace (E × E) := MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)
  letI : MeasurableSpace (E × E) := Prod.instMeasurableSpace
  letI : Fact (mY ≤ (inferInstance : MeasurableSpace (E × E))) := ⟨measurable_snd.comap_le⟩
  let HP := lpMeas ℝ ℝ mY 2 J
  letI : NormedAddCommGroup HP := HP.normedAddCommGroup
  letI : InnerProductSpace ℝ HP := HP.innerProductSpace
  let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
    HP.subtypeL ∘L condExpL2 ℝ ℝ (μ := J) measurable_snd.comap_le
  have hBase := AutoSamplingTheory.ExampleCases.ProximalBPS.CenteredRootOrderInverse.actual_centered_root_order_inverse
    (E:=E) (V:=V) (α:=α) (β:=β) (η:=η) hα hαβ hV hH hη hβη
  dsimp only at hBase
  rcases hBase with
    ⟨hμ,hJ,hν,hRange,hPf,S,hS,hSd,hSc,hSf,hSs,e,he,U,hU,hUi,hUs,
      T,hTs,hAll,hAeq,hAs,hAn,hPB,Γ,hΓ,hΓSq,hΓP,hΓPSq,hGram,
      q,hqa,hTq,hq,hCenter,hΓPq,hγ,ΓP0,hΓP0,hPos0,hOrder0,hUnit,
      Inv,hLeft,hRight,hNormInv⟩
  let A : HP →L[ℝ] HP := HP.orthogonalProjectionOnto ∘L U.toContinuousLinearMap ∘L HP.subtypeL
  let B : HP →L[ℝ] Lp ℝ 2 J :=
    (((1 : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J)-P)*U.toContinuousLinearMap*P) ∘L HP.subtypeL
  let ΓP : HP →L[ℝ] HP := e.conjStarAlgEquiv Γ
  let qP : HP := e q
  let HP0 := (innerSL ℝ qP).ker
  letI : NormedAddCommGroup HP0 := HP0.normedAddCommGroup
  letI : InnerProductSpace ℝ HP0 := HP0.innerProductSpace
  have hGramLocal : B.adjoint ∘L B=ΓP*ΓP := hGram
  have hPosLocal : ΓP.IsPositive := hΓP
  refine ⟨hμ,hJ,hν,hRange,hPf,S,hS,hSd,hSc,hSf,hSs,e,he,U,hU,hUi,hUs,T,hTs,hAll,
    hAeq,hAs,hAn,hPB,Γ,hΓ,hΓSq,hΓP,hΓPSq,hGram,q,hqa,hTq,hq,hCenter,hΓPq,hγ,
    ΓP0,hΓP0,hPos0,hOrder0,hUnit,Inv,hLeft,hRight,hNormInv,?_⟩
  intro f
  let g : HP0 := Inv f
  have hg : ΓP0 g=f := congrArg (fun C : HP0 →L[ℝ] HP0 => C f) hRight
  have hb := B.apply_norm_sq_eq_inner_adjoint_right (g : HP)
  have hr := ΓP.apply_norm_sq_eq_inner_adjoint_right (g : HP)
  rw [hGramLocal] at hb
  rw [hPosLocal.isSelfAdjoint.adjoint_eq] at hr
  have hNorm : ‖B (g : HP)‖=‖ΓP (g : HP)‖ :=
    (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp (hb.trans hr.symm)
  refine ⟨?_,hPB (g : HP)⟩
  calc
    ‖B (HP0.subtypeL (Inv f))‖=‖ΓP (g : HP)‖ := hNorm
    _ = ‖ΓP0 g‖ := congrArg (fun z : HP => ‖z‖) (hΓP0 g).symm
    _ = ‖f‖ := congrArg (fun z : HP0 => ‖z‖) hg

end
end Tests.ProximalBPSCenteredRootOrderInverse

#print axioms AutoSamplingTheory.TechnicalLemmas.Measure.L2RealSquareOrder.positive_square_order
#print axioms AutoSamplingTheory.ExampleCases.ProximalBPS.CenteredRootOrderInverse.actual_centered_root_order_inverse
#print axioms Tests.ProximalBPSCenteredRootOrderInverse.genuine_actual_centered_inverse_consumer