Plain-English statement
- A strongly convex function lies above its first-order model by the quadratic term `m / 2 * ‖y - x‖²`. This is the ASTIS-owned shared port of Optlib's pinned `Strong_Convex_second_lower`. The statement is deliberately domain-local and uses a supplied Mathlib gradient. Chewi's whole-space `C¹` formulation is a downstream specialization, not a silent relabeling of this more general shared interface. The proof factors through `Geometry.GeodesicConvexity.firstOrder_geodesicConvexity`: strong convexity supplies the chord inequality on the affine segment and the supplied gradient supplies the derivative of that segment at its initial point.
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 firstOrder_lower_bound_of_strongConvexOn
{s : Set E} {f : E → ℝ} {m : ℝ} {grad : E → E}
(hsc : StrongConvexOn s m f)
(hgrad : ∀ z ∈ s, HasGradientAt f (grad z) z)
{x y : E} (hx : x ∈ s) (hy : y ∈ s) :
f y ≥ f x + inner ℝ (grad x) (y - x) + m / 2 * ‖y - x‖ ^ 2 := by
let path : ℝ → E := fun t => x + t • (y - x)
let isSelected : (ℝ → E) → Prop := fun q => q = path
have hpath : isSelected path := rfl
have hconvex :
Geometry.GeodesicConvexity.IsAlphaGeodesicallyConvex isSelected f m := by
intro q hq t ht
subst q
have hnonneg_left : 0 ≤ 1 - t := sub_nonneg.mpr ht.2
have hnonneg_right : 0 ≤ t := ht.1
have hstrong := hsc.2 hx hy hnonneg_left hnonneg_right (by ring)
have hpoint : path t = (1 - t) • x + t • y := by
dsimp only [path]
rw [smul_sub, sub_smul, one_smul]
abel
calc
f (path t) = f ((1 - t) • x + t • y) := congrArg f hpoint
_ ≤ (1 - t) * f x + t * f y -
(1 - t) * t * (m / 2 * ‖x - y‖ ^ 2) := by
simpa [smul_eq_mul] using hstrong
_ = (1 - t) * f (path 0) + t * f (path 1) -
(m * t * (1 - t) / 2) * dist (path 0) (path 1) ^ 2 := by
simp [path, dist_eq_norm]
ring
have hline :
HasLineDerivAt ℝ f (inner ℝ (grad x) (y - x)) x (y - x) := by
have h := (hgrad x hx).hasFDerivAt.hasLineDerivAt (y - x)
simpa [InnerProductSpace.toDual_apply_apply] using h
have hderiv :
HasDerivAt (fun t : ℝ => f (path t)) (inner ℝ (grad x) (y - x)) 0 := by
simpa [HasLineDerivAt, path] using hline
have hfirst :=
Geometry.GeodesicConvexity.firstOrder_geodesicConvexity
hconvex hpath hderiv
simpa [path, dist_eq_norm, norm_sub_rev] using hfirst
/-- The gradient of a strongly convex function has the corresponding inner-product
lower bound on its domain.
Mathematical provenance: Optlib `Strong_Convex_lower`, commit
`5da27c5f95aa6a8a45b8c14b968ade4c13ff18c3`,
`Optlib/Convex/StronglyConvex.lean` (Apache-2.0; Chenyi Li and Ziyu Wang).
This proof reuses the local first-order bound in both directions, as in Chewi,
arXiv:2605.07006v1, Proposition 1.6, (1.4) implies (1.5).
The domain-local supplied-gradient interface retains arbitrary real `m`;
Chewi's whole-space `C¹`, nonnegative-modulus statement is a specialization.
-/Open AutoSamplingTheory/TechnicalLemmas/Analysis/StrongConvexFirstOrder.lean:45published source at 0e31a3cda412
Proof architecture
Parent of gradient_inner_lower_bound_of_strongConvexOn; intended Optimization Proposition 1.6 (1.4) and sampling convex-potential adapters
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.
- `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
- Totalized `fderiv` values must not be read as a differentiability theorem.