coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt

Plain-English statement

- Pointwise bridge from Mathlib's Pi-space derivative to ASTIS `EuclideanSpace` coordinate divergence for a wrapped vector field. This is the pointwise core needed to discharge the `hdiv_ae` assumption in the box face-term wrapper when differentiability is available almost everywhere. It does not prove that differentiability holds a.e. on a box, prove integrability, prove boundary-null facts, perform integration by parts, or prove invariant-law consequences.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Lean learning studio · mathematics → formal proof

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.

BeginnerWhy this theorem exists → intuition → statement → one hand calculation. Hide proof-engineering detail.

RigorousExpose assumptions, hidden measure/limit/domain issues, proof route, and rigorous references.

Lean learnerOpen the exact declaration, proof tree/network, syntax glossary, and line-by-line explanation.

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.

Loading source-derived dependency evidence…

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 coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt
    {ι : Type*} [Fintype ι] [DecidableEq ι]
    {F : (ι → ℝ) → ι → ℝ}
    {F' : (ι → ℝ) →L[ℝ] (ι → ℝ)}
    {x : ι → ℝ}
    (hF : HasFDerivAt F F' x) :
    coordinateDivergence
        (fun y : EuclideanSpace ℝ ι =>
          (WithLp.toLp 2 (F (WithLp.ofLp y)) : EuclideanSpace ℝ ι))
        (WithLp.toLp 2 x : EuclideanSpace ℝ ι) =
      ∑ i, F' (Pi.single i (1 : ℝ)) i := by
  let e : EuclideanSpace ℝ ι ≃L[ℝ] (ι → ℝ) :=
    PiLp.continuousLinearEquiv 2 ℝ (fun _ : ι => ℝ)
  have hofLp : HasFDerivAt (fun y : EuclideanSpace ℝ ι => WithLp.ofLp y)
      e.toContinuousLinearMap (WithLp.toLp 2 x : EuclideanSpace ℝ ι) := by
    simpa [e] using
      (PiLp.hasFDerivAt_ofLp (𝕜 := ℝ) (E := fun _ : ι => ℝ) 2
        (WithLp.toLp 2 x : EuclideanSpace ℝ ι))
  have hF_ofLp : HasFDerivAt (fun y : EuclideanSpace ℝ ι => F (WithLp.ofLp y))
      (F'.comp e.toContinuousLinearMap) (WithLp.toLp 2 x : EuclideanSpace ℝ ι) := by
    simpa [Function.comp_def] using hF.comp (WithLp.toLp 2 x : EuclideanSpace ℝ ι) hofLp
  have htoLp : HasFDerivAt (fun z : ι → ℝ =>
        (WithLp.toLp 2 z : EuclideanSpace ℝ ι))
      e.symm.toContinuousLinearMap (F x) := by
    simpa [e] using
      (PiLp.hasFDerivAt_toLp (𝕜 := ℝ) (E := fun _ : ι => ℝ) 2 (F x))
  have hwrapped : HasFDerivAt
      (fun y : EuclideanSpace ℝ ι =>
        (WithLp.toLp 2 (F (WithLp.ofLp y)) : EuclideanSpace ℝ ι))
      (e.symm.toContinuousLinearMap.comp (F'.comp e.toContinuousLinearMap))
      (WithLp.toLp 2 x : EuclideanSpace ℝ ι) := by
    simpa [Function.comp_def] using htoLp.comp (WithLp.toLp 2 x : EuclideanSpace ℝ ι)
      hF_ofLp
  have htrace := coordinateDivergence_eq_sum_fderiv_apply_of_hasFDerivAt hwrapped
  trans ∑ i, (e.symm.toContinuousLinearMap.comp (F'.comp e.toContinuousLinearMap))
      (EuclideanSpace.single i (1 : ℝ)) i
  · exact htrace
  · refine Finset.sum_congr rfl ?_
    intro i _hi
    have hsingle : e (EuclideanSpace.single i (1 : ℝ)) = Pi.single i (1 : ℝ) := by
      simpa [e] using continuousLinearEquiv_apply_euclideanSpace_single (ι := ι) i
    simp [ContinuousLinearMap.comp_apply, hsingle, e]

/-- If two functions on a finite-dimensional box agree on the open box away
from a countable exceptional set, then they agree a.e. on the closed box with
respect to restricted volume.

This is a reusable measure-theoretic transfer leaf.  It packages Mathlib's
fact that the open Pi-box is a.e. equal to the closed Pi-box, plus countable
sets have zero volume. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:426published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 calculus root: transport a Pi-space `HasFDerivAt F F' x` into the ASTIS wrapped `EuclideanSpace` coordinate divergence trace formula

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.
`simp` normalizes through registered definitional and theorem rewrites.
`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.
Totalized `fderiv` values must not be read as a differentiability theorem.
A formal generator display does not establish a closed operator domain.