tendsto_integral_klFun_one_add_mul_div_sq

Plain-English statement

- Actual second-order compensated-entropy limit for a uniformly a.e.-bounded real measurable perturbation of a finite measure. The integral uses Mathlib's `klFun`, not an assumed KL asymptotic. The finite uniform bound yields eventual positivity and an integrable `f²` dominator; the scalar coefficient is obtained by genuine log differentiation and L'Hôpital's rule. No centering/probability normalization is needed for this compensated statement. Those are separate inputs to a probability-KL consumer.

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 tendsto_integral_klFun_one_add_mul_div_sq
    {X : Type*} [MeasurableSpace X] (mu : Measure X) [IsFiniteMeasure mu]
    (f : X → ℝ) (hf : AEStronglyMeasurable f mu) {M : ℝ}
    (hbound : ∀ᵐ x ∂mu, |f x| ≤ M) :
    Tendsto (fun e : ℝ => (∫ x, _root_.InformationTheory.klFun (1 + e * f x) ∂mu) / e ^ 2)
      (𝓝[≠] (0 : ℝ)) (𝓝 ((∫ x, f x ^ 2 ∂mu) / 2)) := by
  have hsq : Integrable (fun x => f x ^ 2) mu := by
    refine (integrable_const (|M| ^ 2)).mono' (hf.pow 2) ?_
    filter_upwards [hbound] with x hx
    simpa only [Real.norm_eq_abs, abs_pow] using
      pow_le_pow_left₀ (abs_nonneg (f x)) (hx.trans (le_abs_self M)) 2
  have hsmall : ∀ᶠ e : ℝ in 𝓝 (0 : ℝ), |e| * |M| < 1 :=
    (isOpen_lt (continuous_abs.mul continuous_const) continuous_const).mem_nhds (by simp)
  have hlimit := tendsto_integral_filter_of_dominated_convergence (μ := mu)
    (F := fun e : ℝ => fun x => _root_.InformationTheory.klFun (1 + e * f x) / e ^ 2)
    (f := fun x => f x ^ 2 / 2) (l := 𝓝[≠] (0 : ℝ)) (fun x => f x ^ 2)
    (Filter.Eventually.of_forall fun e =>
      (by simpa only [div_eq_mul_inv, Pi.add_apply] using
        (_root_.InformationTheory.continuous_klFun.comp_aestronglyMeasurable
          (aestronglyMeasurable_const.add (hf.const_mul e))).mul_const ((e ^ 2)⁻¹)))
    ?_ hsq (ae_of_all mu fun x => scalar_limit (f x))
  · simpa only [integral_div] using hlimit
  · filter_upwards [hsmall.filter_mono nhdsWithin_le_nhds, self_mem_nhdsWithin] with e he he0
    have hne : e ≠ 0 := by simpa using he0
    filter_upwards [hbound] with x hx
    have habs : |e * f x| < 1 := by
      rw [abs_mul]
      exact (mul_le_mul_of_nonneg_left (hx.trans (le_abs_self M)) (abs_nonneg e)).trans_lt he
    have hpos : 0 < 1 + e * f x := by linarith [neg_abs_le (e * f x)]
    have hn := _root_.InformationTheory.klFun_nonneg (le_of_lt hpos)
    have hu : _root_.InformationTheory.klFun (1 + e * f x) ≤ (e * f x) ^ 2 := by
      simpa using klFun_upper (le_of_lt hpos)
    rw [Real.norm_eq_abs, abs_of_nonneg (div_nonneg hn (sq_nonneg e))]
    calc
      _ ≤ (e * f x) ^ 2 / e ^ 2 := div_le_div_of_nonneg_right hu (sq_nonneg e)
      _ = f x ^ 2 := by rw [mul_pow, mul_div_cancel_left₀ _ (pow_ne_zero 2 hne)]

end AutoSamplingTheory.TechnicalLemmas.InformationTheory.BoundedDensityPerturbation

Open AutoSamplingTheory/TechnicalLemmas/InformationTheory/BoundedDensityPerturbation.lean:87published source at 9bb7a7394ab0

Proof architecture

Shared entropy expansion for actual numbered Chewi Exercise 1.7 and Corollary 2.1.2; not a SALD closure

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`have` creates a named intermediate mathematical fact.
`calc` records an equality or inequality chain matching a paper calculation.
`rw` rewrites by an established identity.
`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.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Almost-everywhere hypotheses depend on the stated measure and representative.
Density statements retain normalization and absolute-continuity prerequisites.