Plain-English statement
- A source-backed Hessian representative supplies the operator-norm bound on `fderiv (fderiv f)`. This is the reusable version of a SALD Brownian/Ito bridge: once source correspondence gives a Hessian field and a uniform bound for it, downstream proofs should consume this lemma rather than re-assuming the final operator bound.
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
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
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Lean statement
theorem hessianOpNormOfSourceHessianField
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(f : E → ℝ)
(sourceHessian : E → E →L[ℝ] (E →L[ℝ] ℝ))
(C : ℝ)
(hSourceHasHessian :
∀ z : E, HasFDerivAt (fderiv ℝ f) (sourceHessian z) z)
(hSourceHessianBound :
∀ z : E, ‖sourceHessian z‖ ≤ C) :
∀ z : E, ‖fderiv ℝ (fderiv ℝ f) z‖ ≤ C := by
intro z
rw [(hSourceHasHessian z).fderiv]
exact hSourceHessianBound z
/-- Convert an operator-norm bound on `fderiv (fderiv f)` to the corresponding
Mathlib `iteratedFDeriv` bound of order two. -/Proof architecture
convert source-supplied selected-test Hessian representative into downstream Hessian operator-norm bound
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.
- `rw` rewrites by an established identity.
- `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.