sum_smulRight_apply_pi_single_eq_apply

Plain-English statement

- The trace contribution of `χ'.smulRight G` over the standard finite Pi basis is exactly the scalar derivative `χ'` applied to `G`. This is pure finite-dimensional linear algebra. It identifies the cutoff cross term used by the divergence product rule but proves no measurability, integrability, convergence, or boundary result.

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.

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Lean statement

theorem sum_smulRight_apply_pi_single_eq_apply
    {ι : Type*} [Fintype ι] [DecidableEq ι]
    (χ' : (ι → ℝ) →L[ℝ] ℝ) (G : ι → ℝ) :
    ∑ i, ((χ'.smulRight G) (Pi.single i (1 : ℝ))) i = χ' G := by
  calc
    ∑ i, ((χ'.smulRight G) (Pi.single i (1 : ℝ))) i =
        ∑ i, χ' ((G i) • Pi.single (M := fun _ : ι => ℝ) i (1 : ℝ)) := by
      apply Finset.sum_congr rfl
      intro i _hi
      simp [ContinuousLinearMap.smulRight_apply, Pi.smul_apply, smul_eq_mul,
        mul_comm]
    _ = χ' (∑ i, (G i) • Pi.single (M := fun _ : ι => ℝ) i (1 : ℝ)) := by
      rw [map_sum]
    _ = χ' G := by rw [← pi_eq_sum_univ' G]

/-- 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. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:403published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 divergence product route: collapse the standard-basis trace of the cutoff cross derivative to the scalar derivative applied to the vector field

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.
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`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
`apply` reduces the goal to the hypotheses of a reusable theorem.

Why the statement has this shape

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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.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
Almost-everywhere hypotheses depend on the stated measure and representative.
A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.