Plain-English statement
Compiled generic library interface. Exact hypotheses, actual integrals and remaining global-law boundaries are exposed in the authored lesson. Current public context-bound independent semantic reviews are accepted; serialized stabilization remains pending; no stochastic algorithm or model theorem is claimed.
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
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.
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 quadratic_feature_mean_calculus
{W H : Type*} [MeasurableSpace W]
[NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H]
(μ ν : Measure W) (feature : W → H) (label : H)
(hμ : Integrable feature μ) (hν : Integrable feature ν) :
let m : H := ∫ w, feature w ∂μ
let n : H := ∫ w, feature w ∂ν
let potential : W → ℝ := fun w => 2 * ⟪m - label, feature w⟫_ℝ
Integrable potential μ ∧ Integrable potential ν ∧
((∫ w, potential w ∂ν) - (∫ w, potential w ∂μ) =
2 * ⟪m - label, n - m⟫_ℝ) ∧
(‖n - label‖ ^ 2 - ‖m - label‖ ^ 2 -
((∫ w, potential w ∂ν) - (∫ w, potential w ∂μ)) = ‖n - m‖ ^ 2) ∧
(‖m - label‖ ^ 2 + ((∫ w, potential w ∂ν) - (∫ w, potential w ∂μ)) ≤
‖n - label‖ ^ 2) ∧
HasDerivAt (fun t : ℝ => ‖m + t • (n - m) - label‖ ^ 2)
((∫ w, potential w ∂ν) - (∫ w, potential w ∂μ)) 0 := by
dsimp only
let m : H := ∫ w, feature w ∂μ
let n : H := ∫ w, feature w ∂ν
let potential : W → ℝ := fun w => 2 * ⟪m - label, feature w⟫_ℝ
change Integrable potential μ ∧ Integrable potential ν ∧ _
have hpotμ : Integrable potential μ :=
(Integrable.const_inner (m - label) hμ).const_mul 2
have hpotν : Integrable potential ν :=
(Integrable.const_inner (m - label) hν).const_mul 2
have hpμ : (∫ w, potential w ∂μ) = 2 * ⟪m - label, m⟫_ℝ := by
dsimp only [potential, m]
rw [integral_const_mul, integral_inner hμ]
have hpν : (∫ w, potential w ∂ν) = 2 * ⟪m - label, n⟫_ℝ := by
dsimp only [potential, n]
rw [integral_const_mul, integral_inner hν]
have hpair : (∫ w, potential w ∂ν) - (∫ w, potential w ∂μ) =
2 * ⟪m - label, n - m⟫_ℝ := by
rw [hpν, hpμ, inner_sub_right]
ring
have hbre : ‖n - label‖ ^ 2 - ‖m - label‖ ^ 2 -
((∫ w, potential w ∂ν) - (∫ w, potential w ∂μ)) = ‖n - m‖ ^ 2 := by
rw [hpair]
have hsplit : n - label = (m - label) + (n - m) := by abel
rw [hsplit, norm_add_sq_real]
ring
refine ⟨hpotμ, hpotν, hpair, hbre, ?_, ?_⟩
· linarith [sq_nonneg ‖n - m‖]
· rw [hpair]
have hcurve : HasDerivAt (fun t : ℝ => m + t • (n - m) - label) (n - m) 0 := by
have hsmul : HasDerivAt (fun t : ℝ => t • (n - m)) (n - m) 0 := by
simpa only [one_smul, id_eq] using (hasDerivAt_id (0 : ℝ)).smul_const (n - m)
have hadd : HasDerivAt (fun t : ℝ => m + t • (n - m)) (n - m) 0 :=
hsmul.const_add m
exact hadd.sub_const label
simpa only [zero_smul, add_zero] using hcurve.norm_sqOpen AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticFeatureRisk.lean:171published source at ed709a7f320d
Proof architecture
Reusable squared-loss, genuine integral and indexed replacement calculations
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `have` creates a named intermediate mathematical fact.
- `rw` rewrites by an established identity.
- `refine` instantiates a reusable theorem while leaving explicit subgoals.
- `exact` closes the current goal with an already typed term.
- `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
- Measurability is represented explicitly or must be supplied by a dependency.
- Integrability is an input or proved output; a displayed integral alone does not supply it.