prod · Samplinglib

Plain-English statement

- Product-domain tensorization: the product of log-concave factors on convex domains is log-concave on the Cartesian product.

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.

Lean learning studio · mathematics → formal proof

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BeginnerWhy this theorem exists → intuition → statement → one hand calculation. Hide proof-engineering detail.

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Proof architecture

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Syntax used on this page

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Lean statement

theorem LogConcaveOn.prod {E F : Type*}
    [AddCommMonoid E] [Module ℝ E] [AddCommMonoid F] [Module ℝ F]
    {s : Set E} {t : Set F} {f : E → ℝ} {g : F → ℝ}
    (hf : LogConcaveOn s f) (hg : LogConcaveOn t g) :
    LogConcaveOn (s ×ˢ t) (fun x : E × F => f x.1 * g x.2) := by
  refine ⟨fun x hx => mul_pos (hf.pos (x := x.1) hx.1) (hg.pos (x := x.2) hx.2), ?_⟩
  refine ⟨hf.convex_domain.prod hg.convex_domain, ?_⟩
  intro x hx y hy a b ha hb hab
  have hF := hf.concaveOn_log.2 hx.1 hy.1 ha hb hab
  have hG := hg.concaveOn_log.2 hx.2 hy.2 ha hb hab
  have hsum := add_le_add hF hG
  have hmid : a • x + b • y ∈ s ×ˢ t :=
    (hf.convex_domain.prod hg.convex_domain) hx hy ha hb hab
  calc
    a • Real.log (f x.1 * g x.2) + b • Real.log (f y.1 * g y.2)
        = (a • Real.log (f x.1) + b • Real.log (f y.1)) +
            (a • Real.log (g x.2) + b • Real.log (g y.2)) := by
          rw [Real.log_mul (hf.pos (x := x.1) hx.1).ne' (hg.pos (x := x.2) hx.2).ne',
            Real.log_mul (hf.pos (x := y.1) hy.1).ne' (hg.pos (x := y.2) hy.2).ne']
          simp [smul_eq_mul]
          ring
    _ ≤ Real.log (f (a • x.1 + b • y.1)) + Real.log (g (a • x.2 + b • y.2)) := hsum
    _ = Real.log (f (a • x.1 + b • y.1) * g (a • x.2 + b • y.2)) := by
          rw [Real.log_mul (hf.pos (x := a • x.1 + b • y.1) hmid.1).ne'
            (hg.pos (x := a • x.2 + b • y.2) hmid.2).ne']
    _ = Real.log (f (a • x + b • y).1 * g (a • x + b • y).2) := by simp

/-- Multiplication by a positive constant preserves log-concavity. -/

Open AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:182published source at 7bcd37294df1

Proof architecture

Chewi CONV/DENS root: tensorize log-concave density factors over Cartesian product domains

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`calc` records an equality or inequality chain matching a paper calculation.
`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
`refine` instantiates a reusable theorem while leaving explicit subgoals.

Why the statement has this shape

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Hidden assumptions and non-claims

Density statements retain normalization and absolute-continuity prerequisites.