literal_source_mean_in_closed_gradient

Plain-English statement

Actual literal source S on SAME mu/J/nu produces compact-observer whole mean Tf and gradientTf L2 witnesses; true whole C1 and the uniform genuine compact-gradient core yield canonical closed-gradient graph membership via the existing C1 cutoff/mollifier adapter. Source C2/two-sided Hessian/capped eta retained; finite Hilbert/rank0 and explicit core closure interpretation disclosed. PUP source mean and noncentered rank0 canonical1/0 tested; no Tf compactness or caller law/analytic certificate. Full rough B13, separately defined weak-H1 equivalence, Gamma/dynamics/main/errors/cost/composition remain separate. Four real existing parents reused; no new generic-background copy. Independently verified 16797326f3e06a853e2d047f67fde924ac8c1647.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Lean learning studio · mathematics → formal proof

Read the mathematics first, then descend into Lean

You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.

BeginnerWhy this theorem exists → intuition → statement → one hand calculation. Hide proof-engineering detail.

RigorousExpose assumptions, hidden measure/limit/domain issues, proof route, and rigorous references.

Lean learnerOpen the exact declaration, proof tree/network, syntax glossary, and line-by-line explanation.

Proof architecture

Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.

Loading source-derived dependency evidence…

Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.

How to read the exact Lean declaration

Read a Lean theorem left-to-right exactly as you would unpack a mathematical sentence: name → ambient types → automatically inferred structures → explicit hypotheses → conclusion → proof. Then read the proof top-to-bottom as transformations of the current goal.

NameWhat reusable mathematical fact is being created?
ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
PropositionAfter the colon, translate the Lean expression back into a paper statement.
Proof actionsAfter by, ask what each tactic does to the mathematical goal—not only what syntax it uses.

Syntax used on this page

This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.

Source voice and ASTIS voice stay separate

When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.

Lean statement

theorem literal_source_mean_in_closed_gradient
    {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 S := fun y : E => (volume : Measure E).tilted
      (fun u => -V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))
    IsProbabilityMeasure ν ∧
      ∃ G : Lp ℝ 2 ν →ₗ.[ℝ] Lp E 2 ν,
        Dense (G.domain : Set (Lp ℝ 2 ν)) ∧ G.IsClosable ∧ G.closure.IsClosed ∧
        (∀ (u : Lp ℝ 2 ν) (v : Lp E 2 ν), (u,v) ∈ G.graph ↔
          ∃ φ : E → ℝ, ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧
            u =ᵐ[ν] φ ∧ v =ᵐ[ν] gradient φ) ∧
        ∀ f : E → ℝ, ContDiff ℝ ∞ f → HasCompactSupport f →
          let Tf := fun y : E => ∫ u, f u ∂S y
          ∃ hTf : MemLp Tf 2 ν, ∃ hGrad : MemLp (gradient Tf) 2 ν,
            (hTf.toLp Tf,hGrad.toLp (gradient Tf)) ∈ G.closure.graph := 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 S := fun y : E => (volume : Measure E).tilted
    (fun u => -V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))
  obtain ⟨hμ,hJ,hν,R,S₀,hR,hS₀,hcond,hSR,hSd,hΛcond,hfst,
    U,hU,hUi,hUs,hBB,hBD,hblock,henergy⟩ :=
    MacroscopicEnergy.actual_macroscopic_gradient_energy_blocks hα hαβ hV hH hη hβη
  letI : IsProbabilityMeasure μ := hμ
  letI : IsProbabilityMeasure ν := hν
  obtain ⟨G,hDense,hClosable,hClosed,hGraph⟩ :=
    AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianMarginalGradient.gaussian_marginal_gradient_closable μ hη
  refine ⟨hν,G,hDense,hClosable,hClosed,hGraph,?_⟩
  intro f hf hfc
  let Tf := fun y : E => ∫ u, f u ∂S y
  have hsame : (fun y : E => ∫ u, f u ∂S₀ y) = Tf := by
    funext y
    exact congrArg (fun m : Measure E => ∫ u, f u ∂m) (hSd y)
  obtain ⟨_,_,hTf₀,hGrad₀,_,_⟩ := henergy f hf hfc
  have hTf : MemLp Tf 2 ν := by simpa only [hsame] using hTf₀
  have hGrad : MemLp (gradient Tf) 2 ν := by simpa only [hsame] using hGrad₀
  have hf1 : ContDiff ℝ 1 f := hf.of_le (by simp)
  obtain ⟨_,hC1⟩ := LiteralReflectedMean.reflected_gibbs_mean_c1
    hα hV (fun x v => (hH x v).1) hη hf1 hfc
  change ContDiff ℝ 1 Tf at hC1
  refine ⟨hTf,hGrad,?_⟩
  exact AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedC1GradientDomain.c1_in_closed_gradient ν G hClosable hGraph hC1 hTf hGrad

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.SourceMeanGradientDomain

Open AutoSamplingTheory/ExampleCases/ProximalBPS/SourceMeanGradientDomain.lean:31published source at ed709a7f320d

Proof architecture

Source-specific PBPS integration; no SALD admission

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`simp` normalizes through registered definitional and theorem rewrites.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`exact` closes the current goal with an already typed term.
`simpa` closes the goal after a controlled simplification of a typed result.

Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Totalized `fderiv` values must not be read as a differentiability theorem.
Pointwise support, topological support, and compact support retain distinct meanings.
A Gibbs expression is not automatically a probability law or an invariant law.
A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.