Plain-English statement
- Positive multiples of two-point quadratic Gibbs kernel shapes are log-concave.
Read the mathematics first, then descend into Lean
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Proof architecture
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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
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Source voice and ASTIS voice stay separate
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Lean statement
theorem logConcaveOn_const_mul_exp_neg_pair_sub_quadratic_norm
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{a b c : ℝ} (ha : 0 ≤ a) (hc : 0 < c) :
LogConcaveOn (Set.univ : Set (E × E))
(fun z : E × E => c * Real.exp (-(a * ‖z.1 - z.2‖ ^ 2 + b))) := by
simpa using logConcaveOn_const_mul_exp_neg_of_convexOn
(convexOn_univ_const_mul_norm_fst_sub_snd_sq_add (E := E) (a := a) (b := b) ha) hc
/-- The finite-dimensional Gaussian-kernel normalizing constant times
`exp (-(a‖x-y‖^2+b))` is log-concave as a function of `(x, y)`.
This is a geometry/kernel-shape leaf. It does not claim that the function is a
probability density on the full product space. -/Open AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:426published source at 7bcd37294df1
Proof architecture
Chewi DENS/CONV/GAUSS/DISC root: positive constants preserve two-point Gaussian-kernel log-concavity
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `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
- Density statements retain normalization and absolute-continuity prerequisites.
- A Gibbs expression is not automatically a probability law or an invariant law.