`BookProof.ChapterAbelianClassificationList.diffuse_finite_multiplication_model_inc` (hne : nu Set.univ ≠ 0) : ∃ U : Lp ℂ 2 (volume.restrict (Set.Icc (0 : ℝ) 1)) ≃ₗᵢ[ℂ] Lp ℂ 2...

Proved

BookProof.ChapterAbelianClassificationList.diffuse_finite_multiplication_model_inc

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterAbelianClassificationList.

BookProof.ChapterAbelianClassificationList.diffuse_finite_multiplication_model_inc (hne : nu Set.univ ≠ 0) : ∃ U : Lp ℂ 2 (volume.restrict (Set.Icc (0 : ℝ) 1)) ≃ₗᵢ[ℂ] Lp ℂ 2 nu, ∀ (g : ℝ → ℂ) (hg : MemLp g ⊤ (volume.restrict (Set.Icc (0 : ℝ) 1))) (u : Lp ℂ 2 (volume.restrict (Set.Icc (0 : ℝ) 1))), (U (multOp g hg u) : ℝ → ℂ) =ᵐ[nu] fun x => g (cdf (normalized nu) x) * (U u : ℝ → ℂ) x

Formalization note: Lean 4 identifier BookProof.ChapterAbelianClassificationList.diffuse_finite_multiplication_model_inc.

Replaces the unuploadable node BookProof_ChapterAbelianClassificationList_diffuse_finite_multiplication_model, whose statement name is already declared inside its published definition bundle; the _inc suffix publishes the same statement as its own node.

Preamble

-- Generated from ChapterAbelianClassificationList.lean — theorem BookProof.ChapterAbelianClassificationList.diffuse_finite_multiplication_model_inc
import Definitions.Def_ChapterMeasureAtomicDiffuse
import Definitions.Def_ChapterAtomicDiagonalModel
import Definitions.Def_ChapterDiffuseUnitaryModel
import Definitions.Def_ChapterLpRestrictSplit
import Mathlib
import Definitions.Def_ChapterAbelianClassificationList
import Definitions.Def_ChapterLinftyMultiplication
import Definitions.Def_ChapterLpScaleMeasure
open BookProof.ChapterLinftyMultiplication
open BookProof.ChapterLpScaleMeasure
open BookProof.ChapterAbelianClassificationList


noncomputable section

open MeasureTheory ProbabilityTheory


open BookProof.ChapterMeasureAtomicDiffuse BookProof.ChapterAtomicDiagonalModel
open BookProof.ChapterDiffuseUnitaryModel BookProof.ChapterLinftyMultiplication
open BookProof.ChapterLpRestrictSplit BookProof.ChapterLpScaleMeasure

variable {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α]
variable (nu : Measure ℝ) [IsFiniteMeasure nu] [NullSingletonClass nu]

Formal statement

theorem BookProof.ChapterAbelianClassificationList.diffuse_finite_multiplication_model_inc (hne : nu Set.univ ≠ 0) :
    ∃ U : Lp ℂ 2 (volume.restrict (Set.Icc (0 : ℝ) 1)) ≃ₗᵢ[ℂ] Lp ℂ 2 nu,
      ∀ (g : ℝ → ℂ) (hg : MemLp g ⊤ (volume.restrict (Set.Icc (0 : ℝ) 1)))
        (u : Lp ℂ 2 (volume.restrict (Set.Icc (0 : ℝ) 1))),
        (U (multOp g hg u) : ℝ → ℂ)
          =ᵐ[nu] fun x => g (cdf (normalized nu) x) * (U u : ℝ → ℂ) x := by sorry

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