Plain-English statement
- Product rule for algebra-valued line derivatives. This is a direct wrapper around Mathlib's `HasDerivAt.mul` applied to the one-dimensional curve `t ↦ x + t • v`. In the Langevin tree, the real-valued specialization supplies only the product-rule part of a coordinate calculation such as `∂ᵢ (rho * g) = rho * ∂ᵢ g + (∂ᵢ rho) * g`. It does not identify `g` with a coordinate derivative of a test function, prove the derivative of `g`, define a divergence operator, or prove an integration-by-parts identity.
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Proof architecture
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How to read the exact Lean declaration
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- NameWhat reusable mathematical fact is being created?
- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
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- 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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Lean statement
theorem hasLineDerivAt_mul
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [AddCommGroup E] [Module 𝕜 E]
{𝔸 : Type*} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸]
{f g : E → 𝔸} {x v : E} {f' g' : 𝔸}
(hf : HasLineDerivAt 𝕜 f f' x v)
(hg : HasLineDerivAt 𝕜 g g' x v) :
HasLineDerivAt 𝕜 (fun y : E => f y * g y)
(f' * g x + f x * g') x v := by
unfold HasLineDerivAt at hf hg ⊢
simpa only [Pi.mul_apply, zero_smul, add_zero] using! hf.mul hg
/-- Real-valued `rho * g` specialization of `hasLineDerivAt_mul`, in the
summand order used by finite-coordinate weighted-divergence algebra. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:37published source at 7bcd37294df1
Proof architecture
Chewi SDE/ANALYSIS root: expose the generic algebra-valued line-derivative product rule before finite Euclidean weighted-divergence algebra
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `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
- A Gibbs expression is not automatically a probability law or an invariant law.