logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_convexOn

Plain-English statement

Normalizing exp(-V) by a finite nonzero constant preserves log-concavity whenever V is convex.

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.

Mathematical statement

ConvexOn(s,V), Z != 0, and Z != infinity imply LogConcaveOn(s, x -> (Z^{-1} g_V(x)).toReal).

Intuition

The negative of a convex potential is concave, exponentiation makes it log-concave, and multiplication by a positive constant only shifts the logarithm.

Conditions

V is convex on the target set.
The ENNReal normalizer is nonzero and finite.

Why these conditions cannot be dropped

ENNReal.toReal sends zero and infinity to degenerate real values.
Degenerate normalization would violate the strict positivity built into LogConcaveOn.

Proof route

Convert finite nonzero Z to a positive real scalar.
Apply the constant-times-exp-negative-convex-potential closure lemma.
Normalize ENNReal.toReal and inverse expressions by simp.

Lean interface notes

ENNReal.toReal_pos requires both Z != 0 and Z != infinity.
simpa performs only representation rewrites after the mathematical closure lemma is supplied.

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 logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_convexOn
    {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {V : E → ℝ} {Z : ℝ≥0∞}
    (hV : ConvexOn ℝ s V) (hZ0 : Z ≠ 0) (hZtop : Z ≠ ∞) :
    Geometry.LogConcavity.LogConcaveOn s
      (fun x : E => (Z⁻¹ * Measure.Gibbs.gibbsDensityENNReal V x).toReal) := by
  have hZreal_pos : 0 < Z.toReal := ENNReal.toReal_pos hZ0 hZtop
  have hshape :
      Geometry.LogConcavity.LogConcaveOn s
        (fun x : E => Z.toReal⁻¹ * Real.exp (-V x)) :=
    Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_of_convexOn hV
      (inv_pos.mpr hZreal_pos)
  simpa [Measure.Gibbs.gibbsDensityENNReal, ENNReal.toReal_mul, ENNReal.toReal_inv,
    ENNReal.toReal_ofReal (Real.exp_nonneg _)] using hshape

/-- Strong-convexity wrapper for the real-valued normalized Gibbs-density shape
associated with an `ℝ≥0∞` density and a finite nonzero normalizer. -/

Open AutoSamplingTheory/TechnicalLemmas/Measure/GibbsLogConcavity.lean:29published source at 7bcd37294df1