strongConvexOn_of_gradient_inner_lower_bound

Plain-English statement

- Quantitative monotonicity of genuine ambient gradients on a convex domain implies strong convexity, with the same (possibly signed) modulus.

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

Read the mathematics first, then descend into Lean

You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.

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

RigorousExpose assumptions, hidden measure/limit/domain issues, proof route, and rigorous references.

Lean learnerOpen the exact declaration, proof tree/network, syntax glossary, and line-by-line explanation.

Proof architecture

Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.

Loading source-derived dependency evidence…

Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.

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

This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.

Source voice and ASTIS voice stay separate

When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.

Lean statement

theorem strongConvexOn_of_gradient_inner_lower_bound
    {s : Set E} {f : E → ℝ} {m : ℝ} {grad : E → E}
    (hs : Convex ℝ s)
    (hgrad : ∀ z ∈ s, HasGradientAt f (grad z) z)
    (hmono : ∀ x ∈ s, ∀ y ∈ s,
      m * ‖y - x‖ ^ 2 ≤ inner ℝ (grad y - grad x) (y - x)) :
    StrongConvexOn s m f := by
  refine ⟨hs, ?_⟩
  intro x hx y hy a b ha hb hab
  let p : ℝ → E := fun t => x + t • (y - x)
  let q : ℝ → ℝ := fun t => f (p t) - m / 2 * t ^ 2 * ‖y - x‖ ^ 2
  let q' : ℝ → ℝ := fun t => inner ℝ (grad (p t)) (y - x) - m * t * ‖y - x‖ ^ 2
  have hp : ∀ t ∈ Icc (0 : ℝ) 1, p t ∈ s := by
    intro t ht
    exact hs.add_smul_sub_mem hx hy ht
  have hq : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivAt q (q' t) t := by
    intro t ht
    have hline : HasDerivAt (fun u => f (p u))
        (inner ℝ (grad (p t)) (y - x)) t := by
      have hpderiv : HasDerivAt p (y - x) t := by
        simpa [p] using ((hasDerivAt_id t).smul_const (y - x)).const_add x
      convert! (hgrad (p t) (hp t ht)).hasFDerivAt.comp_hasDerivAt t hpderiv using 1
    have hquad := (((hasDerivAt_id t).pow 2).const_mul (m / 2)).mul_const (‖y - x‖ ^ 2)
    convert! hline.sub hquad using 1
    simp only [q', id_eq]
    ring
  have hmon : MonotoneOn q' (Icc (0 : ℝ) 1) := by
    intro u hu v hv huv
    rcases eq_or_lt_of_le huv with rfl | huv
    · exact le_rfl
    have h := hmono (p u) (hp u hu) (p v) (hp v hv)
    have hdis : p v - p u = (v - u) • (y - x) := by
      simp only [p, add_sub_add_left_eq_sub, sub_smul]
    rw [hdis, inner_smul_right, norm_smul, Real.norm_eq_abs, mul_pow, sq_abs] at h
    have hscaled : (v - u) * (m * (v - u) * ‖y - x‖ ^ 2) ≤
        (v - u) * inner ℝ (grad (p v) - grad (p u)) (y - x) := by
      nlinarith [h]
    have hbound := le_of_mul_le_mul_left hscaled (sub_pos.mpr huv)
    rw [inner_sub_left] at hbound
    dsimp [q']
    linarith
  have hconv : ConvexOn ℝ (Icc (0 : ℝ) 1) q := by
    apply MonotoneOn.convexOn_of_deriv (convex_Icc (0 : ℝ) 1)
      (fun t ht => (hq t ht).continuousAt.continuousWithinAt)
      (fun t ht => (hq t (interior_subset ht)).differentiableAt.differentiableWithinAt)
    intro u hu v hv huv
    rw [(hq u (interior_subset hu)).deriv, (hq v (interior_subset hv)).deriv]
    exact hmon (interior_subset hu) (interior_subset hv) huv
  have hchord := hconv.2 (show (0 : ℝ) ∈ Icc 0 1 by norm_num)
    (show (1 : ℝ) ∈ Icc 0 1 by norm_num) ha hb hab
  have hpoint : a • x + b • y = p b := by
    have ha' : a = 1 - b := by linarith
    simp [p, ha', sub_smul, smul_sub]
    abel
  rw [hpoint]
  simp only [smul_eq_mul, mul_zero, mul_one, zero_add] at hchord ⊢
  norm_num [q, p] at hchord
  rw [norm_sub_rev] at hchord
  have ha' : a = 1 - b := by linarith
  rw [ha'] at hchord ⊢
  dsimp [p]
  nlinarith [hchord]

end AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexGradientConverse

Open AutoSamplingTheory/TechnicalLemmas/Analysis/StrongConvexGradientConverse.lean:31published source at 0e31a3cda412

Proof architecture

Shared converse for optimization gradient criteria; signed quadratic and closed-domain midpoint consumers in Tests.Shared.StrongConvexGradientConverse

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.
`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`exact` closes the current goal with an already typed term.

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

Totalized `fderiv` values must not be read as a differentiability theorem.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.