actual_same_root_inverse_commutation

Plain-English statement

Independently exact-science67 VERIFIED 3da29415011a971a65f749502a625e416213f487. Original C2/two Hessians/positive capped eta produce SAME actual Gamma/T/A/HP0/Inv. Internal I+T positivity yields root commutation; exact A0 restriction is selfadjoint, commutes with Gamma0 and Inv, and A0²+Gamma0²=I. The SAME inverse is selfadjoint. Genuine original-input Test gives K=A0Inv selfadjoint and I+K²=Inv². Rank0/alphaeta1 retained. Two literal private Props are representations, never providers/premises. Sharp B23/LemmaB3 energy/B21/H1/dynamics/main/errors/cost/composition remain open.

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Lean statement

theorem actual_same_root_inverse_commutation
    {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_same_root_inverse_commutation_statement (E := E) (V := V) (α := α) (β := β) (η := η) hα hαβ hV hH hη hβη := by
  classical
  unfold actual_same_root_inverse_commutation_statement
  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.AmbientAdjointCorrector.actual_ambient_adjoint_centered_decomposition
    (E:=E) (V:=V) (α:=α) (β:=β) (η:=η) hα hαβ hV hH hη hβη
  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,R,hR,hAmbient,hGlobal⟩
  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
  change Lp ℝ 2 J →L[ℝ] Hperp at R
  change (∀ g : Lp ℝ 2 J, (R g : Lp ℝ 2 J)=g-P g) at hR
  change (∀ g : Lp ℝ 2 J,
    B.adjoint g=HP0.subtypeL (B0.adjoint (R g))) at hAmbient
  change (∀ f : Lp ℝ 2 J, (∫ z, f z ∂J)=0 →
    ∃ fP : HP0,
      HP0.subtypeL fP=condExpL2 ℝ ℝ (μ:=J) measurable_snd.comap_le f ∧
      let fperp : Hperp := R f
      let fV : HP0 := V0.adjoint fperp
      B.adjoint (fperp : Lp ℝ 2 J)=HP0.subtypeL (ΓP0 fV) ∧
      HP0.subtypeL (ΓP0 fV)=ΓP (HP0.subtypeL fV) ∧
      ‖f‖^2=‖fP‖^2+‖fperp‖^2 ∧ ‖fV‖≤‖fperp‖) at hGlobal
  have hAeqLocal : A=e.conjStarAlgEquiv T := hAeq
  have hAsLocal : IsSelfAdjoint A := hAs
  have hLeftLocal : Inv*ΓP0=(1 : HP0 →L[ℝ] HP0) := hLeft
  have hPosLocal : ΓP0.IsPositive := hPos0
  have hDpos : ((1 : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν)+T).IsPositive := by
    refine ⟨((IsSelfAdjoint.one (Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν)).add hTs).isSymmetric,?_⟩
    intro u
    change 0 ≤ inner ℝ (u+T u) u
    rw [inner_add_left,real_inner_self_eq_norm_sq]
    have hn : ‖T u‖≤‖u‖ := (hAll u).1
    have hi := (abs_le.mp (abs_real_inner_le_norm (T u) u)).1
    nlinarith only [hn,hi,norm_nonneg u]
  have hCommSquare : Commute (Γ*Γ) ((1 : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν)+T) := by
    rw [hΓSq]
    show (1-T*T)*(1+T)=(1+T)*(1-T*T)
    noncomm_ring
  have hCommRootPlus := AutoSamplingTheory.TechnicalLemmas.Measure.L2RealSquareCommute.positive_square_commutation
    ν Γ (1+T) hΓ hDpos hCommSquare
  have hCommRoot : Commute Γ T := by
    have h := hCommRootPlus.eq
    simp only [mul_add,add_mul,mul_one,one_mul] at h
    exact add_left_cancel h
  have hCommP : Commute ΓP A := by
    change ΓP*A=A*ΓP
    change (e.conjStarAlgEquiv Γ)*A=A*(e.conjStarAlgEquiv Γ)
    rw [hAeqLocal,← map_mul,← map_mul,hCommRoot.eq]
  have hAq : A qP=qP := by
    rw [hAeqLocal]
    change e (T (e.symm (e q)))=e q
    rw [e.symm_apply_apply,hTq]
  have hAPreserves (u : HP0) : A (u : HP) ∈ HP0 := by
    change inner ℝ qP (A (u : HP))=0
    calc
      inner ℝ qP (A (u : HP))=inner ℝ (A qP) (u : HP) :=
        (hAsLocal.isSymmetric qP (u : HP)).symm
      _ = inner ℝ qP (u : HP) := by rw [hAq]
      _ = 0 := u.property
  let A0 : HP0 →L[ℝ] HP0 := (A.comp HP0.subtypeL).codRestrict HP0 hAPreserves
  have hA0 (u : HP0) : (A0 u : HP)=A (u : HP) := rfl
  have hA0self : IsSelfAdjoint A0 := by
    apply ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric.mpr
    intro u v
    exact hAsLocal.isSymmetric (u : HP) (v : HP)
  have hComm0 : Commute A0 ΓP0 := by
    apply ContinuousLinearMap.ext
    intro u
    apply Subtype.ext
    change (A0 (ΓP0 u) : HP)=(ΓP0 (A0 u) : HP)
    rw [hA0,hΓP0,hΓP0,hA0]
    exact congrArg (fun D : HP →L[ℝ] HP => D (u : HP)) hCommP.eq.symm
  have hSquare0 : A0*A0+ΓP0*ΓP0=(1 : HP0 →L[ℝ] HP0) := by
    apply ContinuousLinearMap.ext
    intro u
    apply Subtype.ext
    change (A0 (A0 u) : HP)+(ΓP0 (ΓP0 u) : HP)=(u : HP)
    rw [hA0,hA0,hΓP0,hΓP0]
    have h : ΓP (ΓP (u : HP))=(u : HP)-A (A (u : HP)) :=
      congrArg (fun D : HP →L[ℝ] HP => D (u : HP)) hΓPSq
    rw [h]
    abel
  have hStarInv : ΓP0*star Inv=(1 : HP0 →L[ℝ] HP0) := by
    have h := congrArg (fun D : HP0 →L[ℝ] HP0 => star D) hLeftLocal
    simpa only [star_mul,hPosLocal.isSelfAdjoint.star_eq,star_one] using h
  have hInvSelf : IsSelfAdjoint Inv := by
    change star Inv=Inv
    calc
      star Inv=(Inv*ΓP0)*star Inv := by rw [hLeft,one_mul]
      _ = Inv*(ΓP0*star Inv) := mul_assoc _ _ _
      _ = Inv := by rw [hStarInv,mul_one]
  have hCommInv : Commute A0 Inv := by
    change A0*Inv=Inv*A0
    calc
      A0*Inv=(Inv*ΓP0)*(A0*Inv) := by rw [hLeft,one_mul]
      _ = Inv*((ΓP0*A0)*Inv) := by noncomm_ring
      _ = Inv*((A0*ΓP0)*Inv) := by rw [hComm0.eq.symm]
      _ = Inv*A0 := by simp only [mul_assoc,hRight,mul_one]
  refine ⟨hμ, ?_⟩
  refine ⟨hJ, ?_⟩
  refine ⟨hν, ?_⟩
  refine ⟨hRange, ?_⟩
  refine ⟨hPf, ?_⟩
  refine ⟨S, ?_⟩
  refine ⟨hS, ?_⟩
  refine ⟨hSd, ?_⟩
  refine ⟨hSc, ?_⟩
  refine ⟨hSf, ?_⟩
  refine ⟨hSs, ?_⟩
  refine ⟨e, ?_⟩
  refine ⟨he, ?_⟩
  refine ⟨U, ?_⟩
  refine ⟨hU, ?_⟩
  refine ⟨hUi, ?_⟩
  refine ⟨hUs, ?_⟩
  refine ⟨T, ?_⟩
  refine ⟨hTs, ?_⟩
  refine ⟨hAll, ?_⟩
  refine ⟨hAeq, ?_⟩
  refine ⟨hAs, ?_⟩
  refine ⟨hAn, ?_⟩
  refine ⟨hPB, ?_⟩
  refine ⟨Γ, ?_⟩
  refine ⟨hΓ, ?_⟩
  refine ⟨hΓSq, ?_⟩
  refine ⟨hCommRoot, ?_⟩
  refine ⟨hΓP, ?_⟩
  refine ⟨hΓPSq, ?_⟩
  refine ⟨hCommP, ?_⟩
  refine ⟨hGram, ?_⟩
  refine ⟨q, ?_⟩
  refine ⟨hqa, ?_⟩
  refine ⟨hTq, ?_⟩
  refine ⟨hq, ?_⟩
  refine ⟨hCenter, ?_⟩
  refine ⟨hΓPq, ?_⟩
  refine ⟨hγ, ?_⟩
  refine ⟨ΓP0, ?_⟩
  refine ⟨hΓP0, ?_⟩
  refine ⟨hPos0, ?_⟩
  refine ⟨hOrder0, ?_⟩
  refine ⟨hUnit, ?_⟩
  refine ⟨Inv, ?_⟩
  refine ⟨hLeft, ?_⟩
  refine ⟨hRight, ?_⟩
  refine ⟨hNormInv, ?_⟩
  refine ⟨hInvSelf,A0,hA0,hA0self,hComm0,hSquare0,hCommInv,?_⟩
  refine ⟨B0, ?_⟩
  refine ⟨hB0, ?_⟩
  refine ⟨V0, ?_⟩
  refine ⟨hVdef, ?_⟩
  refine ⟨hFactor, ?_⟩
  refine ⟨hVadj, ?_⟩
  refine ⟨hVnorm, ?_⟩
  refine ⟨R, ?_⟩
  refine ⟨hR, ?_⟩
  refine ⟨hAmbient, ?_⟩
  exact hGlobal

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualRootCommutation

#print axioms AutoSamplingTheory.ExampleCases.ProximalBPS.ActualRootCommutation.actual_same_root_inverse_commutation

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualRootCommutation.lean:115published source at ed709a7f320d

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Hidden assumptions and non-claims

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Totalized `fderiv` values must not be read as a differentiability theorem.
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