ASTIS mathematical exposition

Restrict to a convex smaller domain

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

Statement

Let E be a real module, f:E→ℝ positive log-concave on s⊆E, and t⊆s a convex set. Then the same function f is positive log-concave on t.

\[\operatorname{LC}_s(f),\quad t\subseteq s,\quad t\text{ convex}\Longrightarrow\operatorname{LC}_t(f).\]

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,t : Set E; f : E → ℝ; hf : LogConcaveOn s f.
hts : t⊆s and ht : Convex ℝ t are both explicit assumptions.

{'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. Restrict positivity

Every x∈t lies in s, so the supplied positivity on s gives f(x)>0.

\[x\in t\subseteq s\Longrightarrow f(x)>0.\]

Corresponding Lean step

fun x hx => hf.pos (x := x) (hts hx)

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

2. Restrict the concavity inequality on a convex domain

For x,y∈t, the original concavity inequality applies because both lie in s. Convexity of t supplies the required domain condition for the restricted predicate. Pair this concavity statement with positivity.

\[a\log f(x)+b\log f(y)\le\log f(ax+by),\qquad ax+by\in t.\]

Corresponding Lean step

⟨fun x hx => hf.pos (x := x) (hts hx), hf.concaveOn_log.subset hts ht⟩

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 · subset

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. Restricting a function's domain does not alter its values. The separate ht is necessary because LogConcaveOn includes convexity of the domain.

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.subset {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s t : Set E} {f : E → ℝ}
    (hf : LogConcaveOn s f) (hts : t ⊆ s) (ht : Convex ℝ t) :
    LogConcaveOn t f

Exact module and namespace context

Lean proof · subset

Restricting a function's domain does not alter its values. The separate ht is necessary because LogConcaveOn includes convexity of the domain.

theorem LogConcaveOn.subset {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s t : Set E} {f : E → ℝ}
    (hf : LogConcaveOn s f) (hts : t ⊆ s) (ht : Convex ℝ t) :
    LogConcaveOn t f :=
  ⟨fun x hx => hf.pos (x := x) (hts hx), hf.concaveOn_log.subset hts ht⟩

/-- Restricting a positive log-concave function to one of its superlevel sets
preserves log-concavity. -/

Exact module and namespace context

Scope and omitted-condition boundaries

Arbitrary nonconvex restrictions are not allowed by this predicate.
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)

ConcaveOn.subset

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.
ConcaveOn.subset — Exact inspected Mathlib definition or theorem used by this exposition.

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