Pointwise trace transport on an almost-everywhere differentiability set
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_wrapped_toPi_trace_ae_of_ae_hasFDerivAt · theorem · Teaching coverage
Statement
For raw P, endpoints a,b, F:P→P and derivative field A, if F has derivative A(x) for almost every x under volume restricted to K, then δF=τ_A almost everywhere on that same restricted measure.
All objects and hypotheses
- n∈ℕ, d=n+1≥1, P=(Fin d→ℝ) with its usual supremum norm, V=EuclideanSpace ℝ (Fin d) with its ℓ² norm. T:P→V is WithLp.toLp 2, a continuous linear equivalence, and e=T⁻¹=WithLp.ofLp.
- a,b∈P; K=[a,b]={x:∀i,a_i≤x_i≤b_i} is the closed box and O=∏_i(a_i,b_i) is the open box. All unspecified integrals and a.e. assertions use Lebesgue volume on P, restricted to K when indicated.
- F:P→P and A:P→(P→L[ℝ]P) is a supplied linear-map field. Write δF(x)=Div(T∘F∘T⁻¹)(Tx) and τ_A(x)=Σ_i(A(x)e_i)_i.
- HasFDerivAt F (A x) x for volume|K-almost every x; no a≤b or integrability assumption.
Notation and interpretation
- Notation used below
For a raw field F:P→P, W_F=T∘F∘T⁻¹ and δF(x) is coordinateDivergence W_F at Tx. For a supplied linear-map field A, τ_A is its coordinate trace.
\[W_F:=T\circ F\circ T^{-1},\qquad\delta F(x):=\operatorname{Div}W_F(Tx),\qquad\tau_A(x):=\sum_i(A(x)e_i)_i.\]
Mathematical proof
1. Apply the pointwise identity where its hypothesis holds
On the full-measure set of derivative points, the already-proved transport theorem identifies the two real values. Consequently the functions agree a.e. on K.
Corresponding Lean step
filter_upwards [hF_ae]; coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt hx.
Lean statement · coordinateDivergence_wrapped_toPi_trace_ae_of_ae_hasFDerivAt
This wrapper consumes an a.e. derivative hypothesis directly; it does not establish one from boundary or countability arguments.
Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.
theorem coordinateDivergence_wrapped_toPi_trace_ae_of_ae_hasFDerivAt
{n : ℕ}
(a b : Fin (n + 1) → ℝ)
(F : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
(F' : (Fin (n + 1) → ℝ) →
(Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → ℝ))
(hF_ae : ∀ᵐ x ∂volume.restrict (Set.Icc a b), HasFDerivAt F (F' x) x) :
(fun x => coordinateDivergence
(fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
(WithLp.toLp 2 (F (WithLp.ofLp y)) :
EuclideanSpace ℝ (Fin (n + 1))))
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
=ᵐ[volume.restrict (Set.Icc a b)]
fun x => ∑ i, F' x (Pi.single i (1 : ℝ)) iLean proof · coordinateDivergence_wrapped_toPi_trace_ae_of_ae_hasFDerivAt
On the full-measure set of derivative points, the already-proved transport theorem identifies the two real values. Consequently the functions agree a.e. on K. The Lean correspondence in that step identifies the exact existing rule or definitional reduction used.
Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.
theorem coordinateDivergence_wrapped_toPi_trace_ae_of_ae_hasFDerivAt
{n : ℕ}
(a b : Fin (n + 1) → ℝ)
(F : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
(F' : (Fin (n + 1) → ℝ) →
(Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → ℝ))
(hF_ae : ∀ᵐ x ∂volume.restrict (Set.Icc a b), HasFDerivAt F (F' x) x) :
(fun x => coordinateDivergence
(fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
(WithLp.toLp 2 (F (WithLp.ofLp y)) :
EuclideanSpace ℝ (Fin (n + 1))))
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
=ᵐ[volume.restrict (Set.Icc a b)]
fun x => ∑ i, F' x (Pi.single i (1 : ℝ)) i := by
filter_upwards [hF_ae] with x hx
exact coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt (ι := Fin (n + 1)) hx
/-- A.e. bridge from an open-box/off-countable `HasFDerivAt` hypothesis to the
`hdiv_ae` shape required by the finite-box face-term wrapper.
This discharges only the a.e. equality assumption. It does not prove
integrability of the divergence integrand, weighted IBP, no-boundary limits, or
invariant-law consequences. -/Scope and omitted-condition boundaries
- A.e. equality alone is not integrability.
Source and reuse
ASTIS parents called
Mathlib API called (external library)
No direct Mathlib call recorded; see the ASTIS parents.
Mathematical sources
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ASTIS prose is not a quotation or a source-equivalence certificate. Definitions and aliases are explained as constructions, not counted as new mathematical proofs.