actual_ambient_adjoint_centered_decomposition

Plain-English statement

Independently exact-science66 VERIFIED a115115d42b3fa2b67885d87fe4d5300af36fcd1. SAME original C2/two Hessians/positive capped eta inputs produce canonical R:L2(J)->kerP with inclusion Rg=g-Pg and all-vector Bambient*=i0 B0*R. Every actual globally centered input produces fP in HP0,fperp=Rf,fV=V0*fperp with B* fperp=i0 Gamma0 fV=GammaP i0 fV, exact Pythagoras and contraction. Genuine Test gives original-input reconstruction,B*f=GammaP i0 fV and squared norm budget. Rank0/alphaeta1 retained. Two literal private Prop expansions are representations only; no extra premise/provider. Full B20/B21,halfturn/H1/dynamics/main/errors/expected costs/composition remain open.

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Lean statement

theorem actual_ambient_adjoint_centered_decomposition
    {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) : actual_ambient_adjoint_centered_decomposition_statement (E := E) (V := V) (α := α) (β := β) (η := η) hα hαβ hV hH hη hβη := by
  unfold actual_ambient_adjoint_centered_decomposition_statement
  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.PolarIsometry.actual_centered_polar_isometry
    (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,B0,hB0,V0,hVdef,hFactor,hVadj,hVnorm⟩
  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
  letI : CompleteSpace HP0 := (innerSL ℝ qP).isClosed_ker.completeSpace_coe
  let Hperp := P.ker
  letI : NormedAddCommGroup Hperp := Hperp.normedAddCommGroup
  letI : InnerProductSpace ℝ Hperp := Hperp.innerProductSpace
  letI : CompleteSpace Hperp := P.isClosed_ker.completeSpace_coe
  change HP0 →L[ℝ] Hperp at B0 V0
  change HP0 →L[ℝ] HP0 at ΓP0 Inv
  letI : IsProbabilityMeasure J := hJ
  letI : IsProbabilityMeasure ν := hν
  let C : Lp ℝ 2 J →L[ℝ] HP := condExpL2 ℝ ℝ (μ:=J) measurable_snd.comap_le
  have hPself (g k : Lp ℝ 2 J) : inner ℝ (P g) k=inner ℝ g (P k) :=
    inner_condExpL2_left_eq_right measurable_snd.comap_le
  have hPid (g : Lp ℝ 2 J) : P (P g)=P g := hPf (C g)
  let R : Lp ℝ 2 J →L[ℝ] Hperp :=
    ((1 : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J)-P).codRestrict Hperp (by
      intro g
      change P (g-P g)=0
      rw [map_sub,hPid,sub_self])
  have hR (g : Lp ℝ 2 J) : (R g : Lp ℝ 2 J)=g-P g := rfl
  have hRself (g : Hperp) : R (g : Lp ℝ 2 J)=g := by
    apply Subtype.ext
    change (g : Lp ℝ 2 J)-P g=g
    rw [show P (g : Lp ℝ 2 J)=0 from g.property,sub_zero]
  have hGramLocal : B.adjoint ∘L B=ΓP*ΓP := hGram
  have hRootq : ΓP qP=0 := hΓPq
  have hBq : B qP=0 := by
    have hn := B.apply_norm_sq_eq_inner_adjoint_right qP
    rw [hGramLocal] at hn
    have hz : (ΓP*ΓP) qP=0 := by
      change ΓP (ΓP qP)=0
      rw [hRootq,map_zero]
    rw [hz,inner_zero_right] at hn
    exact norm_eq_zero.mp (sq_eq_zero_iff.mp hn)
  have hBCenter (g : Lp ℝ 2 J) : B.adjoint g ∈ HP0 := by
    change inner ℝ qP (B.adjoint g)=0
    rw [B.adjoint_inner_right,hBq,inner_zero_left]
  have hAmbient (g : Lp ℝ 2 J) :
      B.adjoint g=HP0.subtypeL (B0.adjoint (R g)) := by
    let b : HP0 := ⟨B.adjoint g,hBCenter g⟩
    have heq : b=B0.adjoint (R g) := by
      apply ext_inner_left ℝ
      intro u
      change inner ℝ (u : HP) (B.adjoint g)=inner ℝ u (B0.adjoint (R g))
      rw [B.adjoint_inner_right,B0.adjoint_inner_right]
      change inner ℝ (B (u : HP)) g=inner ℝ (B0 u : Lp ℝ 2 J) (R g : Lp ℝ 2 J)
      rw [hB0,hR,inner_sub_right]
      have ho : inner ℝ (B (u : HP)) (P g)=0 := by
        rw [← hPself,hPB,inner_zero_left]
      rw [ho,sub_zero]
    exact congrArg (fun z : HP0 => HP0.subtypeL z) heq
  have hPosLocal : ΓP0.IsPositive := hPos0
  have hBAdj : B0.adjoint=ΓP0 ∘L V0.adjoint := by
    calc
      B0.adjoint = (V0 ∘L ΓP0).adjoint :=
        congrArg (fun D : HP0 →L[ℝ] Hperp => D.adjoint) hFactor
      _ = ΓP0.adjoint ∘L V0.adjoint := ContinuousLinearMap.adjoint_comp V0 ΓP0
      _ = ΓP0 ∘L V0.adjoint :=
        congrArg (fun D : HP0 →L[ℝ] HP0 => D ∘L V0.adjoint)
          hPosLocal.isSelfAdjoint.adjoint_eq
  have hNormV : ‖V0‖ ≤ (1 : ℝ) :=
    ContinuousLinearMap.opNorm_le_bound _ zero_le_one (by intro u; rw [hVnorm u,one_mul])
  have hNormAdj : ‖V0.adjoint‖ ≤ (1 : ℝ) :=
    (ContinuousLinearMap.adjoint.norm_map V0).trans_le hNormV
  have hVcontract (g : Hperp) : ‖V0.adjoint g‖≤‖g‖ := by
    calc
      ‖V0.adjoint g‖ ≤ ‖V0.adjoint‖*‖g‖ := V0.adjoint.le_opNorm g
      _ ≤ 1*‖g‖ := mul_le_mul_of_nonneg_right hNormAdj (norm_nonneg _)
      _ = ‖g‖ := one_mul _
  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
  have hqOne : (qP : Lp ℝ 2 J)=
      AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation.one J := by
    rw [show (qP : Lp ℝ 2 J)=M q from he q]
    apply Lp.ext
    have hqaPull : (fun z : E × E => q z.2)=ᵐ[J] (fun _ => (1 : ℝ)) :=
      hp.quasiMeasurePreserving.ae hqa
    exact (Lp.coeFn_compMeasurePreserving q hp).trans
      (hqaPull.trans (Lp.coeFn_const (p:=(2 : ℝ≥0∞)) (μ:=J) (c:=(1 : ℝ))).symm)
  have hqIntegral (g : Lp ℝ 2 J) : inner ℝ (qP : Lp ℝ 2 J) g=∫ z, g z ∂J := by
    rw [hqOne]
    exact AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation.inner_one_eq_integral J g
  have hPq : P (qP : Lp ℝ 2 J)=qP := hPf qP
  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,B0,hB0,V0,hVdef,
    hFactor,hVadj,hVnorm,R,hR,hAmbient,?_⟩
  intro f hf
  have hCf : C f ∈ HP0 := by
    change inner ℝ qP (C f)=0
    change inner ℝ (qP : Lp ℝ 2 J) (P f)=0
    rw [← hPself,hPq,hqIntegral,hf]
  let fP : HP0 := ⟨C f,hCf⟩
  refine ⟨fP,rfl,?_,?_,?_,hVcontract (R f)⟩
  · calc
      B.adjoint (R f : Lp ℝ 2 J)=HP0.subtypeL (B0.adjoint (R (R f))) := hAmbient _
      _ = HP0.subtypeL (B0.adjoint (R f)) := congrArg
        (fun z : Hperp => HP0.subtypeL (B0.adjoint z)) (hRself (R f))
      _ = HP0.subtypeL (ΓP0 (V0.adjoint (R f))) :=
        congrArg (fun z : HP0 => HP0.subtypeL z)
          (congrArg (fun D : Hperp →L[ℝ] HP0 => D (R f)) hBAdj)
  · exact hΓP0 _
  · have hn := HP.norm_sq_eq_add_norm_sq_starProjection f
    have hProj : HP.starProjection=P := rfl
    rw [hProj,HP.starProjection_orthogonal_val] at hn
    change ‖f‖^2=‖(C f : Lp ℝ 2 J)‖^2+‖f-P f‖^2 at hn
    exact hn

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.AmbientAdjointCorrector

Open AutoSamplingTheory/ExampleCases/ProximalBPS/AmbientAdjointCorrector.lean:107published source at ed709a7f320d

Proof architecture

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Measurability is represented explicitly or must be supplied by a dependency.
Totalized `fderiv` values must not be read as a differentiability theorem.
Almost-everywhere hypotheses depend on the stated measure and representative.