ASTIS mathematical exposition

Restrict log-concavity to a superlevel set

AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.restrict_superlevel · theorem · Teaching coverage

Statement

Let E be a real module, f:E→ℝ positive log-concave on s⊆E, and c∈ℝ arbitrary. The function f remains positive log-concave on the domain {x∈s:c≤f(x)}.

\[\operatorname{LC}_s(f)\Longrightarrow\operatorname{LC}_{\{x\in s:c\le f(x)\}}(f).\]

All objects and hypotheses

E : Type* with [AddCommMonoid E] and [Module ℝ E]. These are the exact algebraic ambient structures; no topology, norm, measure or finite-dimensionality is assumed.
s : Set E, f : E → ℝ, hf : LogConcaveOn s f.
c : ℝ, with no sign restriction.

{'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. Check the restricted domain fits the restriction theorem

A superlevel point is in s by definition, and the superlevel set is convex by convex_superlevel. These are exactly the two requirements of LogConcaveOn.subset.

\[S_c=\{x\in s:c\le f(x)\}\subseteq s,\qquad S_c\text{ convex}.\]

Corresponding Lean step

(fun _ hx => hx.1)
(hf.convex_superlevel c)

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

2. Reuse convex-domain restriction

Apply the restriction theorem to those facts; it preserves the supplied positivity and logarithmic concavity on the smaller domain.

\[\operatorname{LC}_s(f)\Longrightarrow\operatorname{LC}_{S_c}(f).\]

Corresponding Lean step

hf.subset (fun _ hx => hx.1) (hf.convex_superlevel c)

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 · restrict_superlevel

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. hx.1 forgets the superlevel inequality and keeps only membership in s. The rest is a direct use of two previous results.

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.restrict_superlevel {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ}
    (hf : LogConcaveOn s f) (c : ℝ) :
    LogConcaveOn {x ∈ s | c ≤ f x} f

Exact module and namespace context

Lean proof · restrict_superlevel

hx.1 forgets the superlevel inequality and keeps only membership in s. The rest is a direct use of two previous results.

theorem LogConcaveOn.restrict_superlevel {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ}
    (hf : LogConcaveOn s f) (c : ℝ) :
    LogConcaveOn {x ∈ s | c ≤ f x} f :=
  hf.subset (fun _ hx => hx.1) (hf.convex_superlevel c)

/-- Precomposition by a linear map preserves log-concavity on the preimage domain. -/

Exact module and namespace context

Scope and omitted-condition boundaries

No new function is defined by setting f to zero outside the superlevel set; only its asserted domain is restricted.
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.
Existing usage in Tests.Basic — Read-only source example; no test was run.

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