Plain-English statement
- Global quantitative gradient monotonicity is equivalent to the genuine Hessian diagonal lower bound, by a right derivative limit and the FTC.
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.
Proof architecture
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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 gradient_mono_iff_fderiv2_lower
{f : E → ℝ} {m : ℝ} (hf : ContDiff ℝ 2 f) :
(∀ x y : E, m * ‖y - x‖ ^ 2 ≤
inner ℝ (gradient f y - gradient f x) (y - x)) ↔
∀ x v : E, m * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ f) x v) v := by
have hfd : ContDiff ℝ 1 (fderiv ℝ f) := hf.fderiv_right (by norm_num)
constructor
· intro hm x v
let g : ℝ → ℝ := fun t => fderiv ℝ f (x + t • v) v
have hdAll (t : ℝ) : HasDerivAt g
((fderiv ℝ (fderiv ℝ f) (x + t • v) v) v) t := by
convert ((hfd.differentiable_one _).hasFDerivAt.comp_hasDerivAt t
(((hasDerivAt_id t).smul_const v).const_add x)).clm_apply
(hasDerivAt_const t v) using 1 <;> first | rfl | simp
have hd : HasDerivAt g ((fderiv ℝ (fderiv ℝ f) x v) v) 0 := by
simpa using hdAll 0
apply ge_of_tendsto hd.tendsto_slope_zero_right
filter_upwards [self_mem_nhdsWithin] with t ht
have ht : 0 < t := ht
have h := hm x (x + t • v)
rw [add_sub_cancel_left, norm_smul, Real.norm_eq_abs, mul_pow, sq_abs,
inner_smul_right, inner_sub_left] at h
have hscaled : t * (m * ‖v‖ ^ 2) ≤ g t - g 0 := by
apply le_of_mul_le_mul_left (a := t) _ ht
simpa [g, gradient, Function.comp_def, pow_two, mul_assoc, mul_left_comm, mul_comm] using h
simpa [smul_eq_mul, div_eq_inv_mul] using
(le_div_iff₀ ht).mpr (by simpa [mul_comm] using hscaled)
· intro hH x y
let v := y - x
have hc : Continuous (fun t : ℝ => (fderiv ℝ (fderiv ℝ f) (x + t • v) v) v) :=
(((hfd.continuous_fderiv (by norm_num)).comp
(continuous_const.add (continuous_id.smul continuous_const))).clm_apply
continuous_const).clm_apply continuous_const
have h := intervalIntegral.integral_mono_on (by norm_num : (0 : ℝ) ≤ 1)
(intervalIntegrable_const : IntervalIntegrable (fun _ : ℝ => m * ‖v‖ ^ 2) volume 0 1)
(hc.intervalIntegrable 0 1) (fun t _ => hH (x + t • v) v)
rw [← gradient_sub_inner_eq_integral_fderiv2 hf x v] at h
simpa [v] using h
/-- Proposition 1.6 part 2, with the source's whole Euclidean domain,
nonnegative modulus and C² regularity. Together with the C¹ theorem this
connects all four source conditions. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexityC2.lean:46published source at 0e31a3cda412
Proof architecture
Actual parent of source C2 specialization; signed quadratic curvature test
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.
- `simpa` closes the goal after a controlled simplification of a typed result.
- `filter_upwards` moves an almost-everywhere or eventual statement into a pointwise local context.
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
- Integrability is an input or proved output; a displayed integral alone does not supply it.
- Totalized `fderiv` values must not be read as a differentiability theorem.
- Genuine differentiability is localized to the hypotheses shown in the Lean statement.