ASTIS mathematical exposition

Positive rescaling of the shifted quadratic-norm Gibbs shape

AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_shifted_quadratic_norm · theorem · Teaching coverage

Statement

On a real normed vector space E, let a,b,c∈ℝ with a≥0 and c>0, and let m∈E be fixed. With V=a‖x-m‖²+b, the function c exp(−V) is positive log-concave on the whole domain. The constant c is not required or proved to normalize an integral.

\[V=a\|x-m\|^2+b,\qquad a\ge0,\ c>0\Longrightarrow\operatorname{LC}_{E}(ce^{-V}).\]

All objects and hypotheses

E : Type* with [NormedAddCommGroup E] and [NormedSpace ℝ E]. No finite-dimensionality, completeness or inner-product structure is required.
a,b : ℝ; ha : 0≤a; no sign condition on b.
m : E is an arbitrary fixed center.
c : ℝ; hc : 0<c.

{'term': 'Positive log-concavity', 'text': 'LC_s(f) means strict positivity of f at every point of s and concavity of log f on the convex domain s. No zero-valued points are included in this convention.', 'formula': '\\operatorname{LC}_s(f)\\iff(\\forall x\\in s,\\ f(x)>0)\\land\\operatorname{Concave}_s(\\log f).'}
{'term': 'Convex combinations and Jensen inequalities', 'text': 'Throughout, x,y lie in the stated domain; λ,θ are nonnegative real weights with λ+θ=1. The Lean code often names these weights a,b, independently of the a,b coefficients in specialized potentials.', 'formula': 'z=\\lambda x+\\theta y,\\quad \\lambda,\\theta\\ge0,\\quad\\lambda+\\theta=1;\\qquad V(z)\\le\\lambda V(x)+\\theta V(y)\\text{ for convex }V.'}
{'term': 'Geometry, not probability normalization', 'text': 'The module proves shapes and convexity properties of real-valued functions. It has no reference measure in its declarations. In particular, names containing normalized_density do not themselves prove normalization, and the quadratic prefactor is not certified as the integral of an arbitrary norm-based shape.', 'formula': '\\operatorname{LC}(Z^{-1}e^{-V})\\quad\\text{does not assert}\\quad Z=\\int e^{-V}\\,d\\mu\\quad\\text{or}\\quad \\int Z^{-1}e^{-V}\\,d\\mu=1.'}

Mathematical proof

1. Verify the potential's convexity

Reuse the corresponding convexity theorem for V; it requires a≥0 and imposes no bound on b.

\[\operatorname{Convex}_{E}(V).\]

Corresponding Lean step

convexOn_univ_const_mul_norm_sub_sq_add (E := E) (a := a) (b := b) m ha

The identifiers refer to the assumptions or previously proved facts described in this step; the full original proof below supplies their exact context.

2. Apply the positive-rescaled Gibbs-shape theorem

For any positive c, log(c exp(−V))=log c−V is concave and the function itself is positive. Apply the generic result to the verified V and the supplied c>0.

\[\log(ce^{-V})=\log c-V,\qquad ce^{-V}>0.\]

Corresponding Lean step

simpa using logConcaveOn_const_mul_exp_neg_of_convexOn
  (convexOn_univ_const_mul_norm_sub_sq_add (E := E) (a := a) (b := b) m ha) hc

The identifiers refer to the assumptions or previously proved facts described in this step; the full original proof below supplies their exact context.

Lean statement · logConcaveOn_const_mul_exp_neg_shifted_quadratic_norm

Braces name inputs Lean can infer, and bracketed classes state the ambient structures listed above. The assumptions before the final colon are inputs; the expression after it is the exact property this declaration establishes. There are two sign assumptions with different roles: ha lets the potential be convex, and hc keeps the rescaled function strictly positive. The theorem accepts any such c.

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

theorem logConcaveOn_const_mul_exp_neg_shifted_quadratic_norm
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    {a b c : ℝ} (m : E) (ha : 0 ≤ a) (hc : 0 < c) :
    LogConcaveOn (Set.univ : Set E)
      (fun x : E => c * Real.exp (-(a * ‖x - m‖ ^ 2 + b)))

Exact module and namespace context

Lean proof · logConcaveOn_const_mul_exp_neg_shifted_quadratic_norm

There are two sign assumptions with different roles: ha lets the potential be convex, and hc keeps the rescaled function strictly positive. The theorem accepts any such c.

theorem logConcaveOn_const_mul_exp_neg_shifted_quadratic_norm
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    {a b c : ℝ} (m : E) (ha : 0 ≤ a) (hc : 0 < c) :
    LogConcaveOn (Set.univ : Set E)
      (fun x : E => c * Real.exp (-(a * ‖x - m‖ ^ 2 + b))) := by
  simpa using logConcaveOn_const_mul_exp_neg_of_convexOn
    (convexOn_univ_const_mul_norm_sub_sq_add (E := E) (a := a) (b := b) m ha) hc

/-- The explicitly normalized shifted finite-dimensional quadratic Gibbs density is log-concave. -/

Exact module and namespace context

Scope and omitted-condition boundaries

Positive rescaling does not certify a partition function or total mass one.
This documentation adds no Lean theorem, compilation evidence, source-equivalence verdict, or new source-fidelity certification.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

No direct Mathlib call recorded; see the ASTIS parents.

Mathematical sources

Existing ASTIS declaration and exact proof — Directly read current local source; no Lean edit or fresh build.
Existing curated module card — Local API/source-boundary memory; not independent primary textbook verification.

ASTIS prose is not a quotation or a source-equivalence certificate. Definitions and aliases are explained as constructions, not counted as new mathematical proofs.