Prove2Me

Derivative of an identically-zero family is zero

Proved
ExactVacuumAtlas.Variation.eq_zero_of_hasDerivAt_onShell

by He Wang · Oct 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

einstein-maxwellexact-spacetime-atlas-ivexact-vacuum-atlasgeneral-relativity

Derivative of an identically-zero family is zero

If φ(ε) = 0 whenever (ε, x) ∈ W (open, (0, x) ∈ W) and HasDerivAt φ L 0, then L = 0.

Scope. Exactly the hypotheses printed in the formal statement.

中文摘要 恒零族的导数为零。

Preamble
import Mathlib
import Definitions.Def_Atlas_Core_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Basic
import Definitions.Def_Atlas_EinsteinMaxwell_Variation_Defs
import Definitions.Def_Atlas_Transport_MatrixCalc
open Filter Topology
open scoped ContDiff
open ExactVacuumAtlas.EinsteinMaxwell
open ExactVacuumAtlas
open ExactVacuumAtlas.Variation
variable {n : ℕ}
Formal statement
theorem ExactVacuumAtlas.Variation.eq_zero_of_hasDerivAt_onShell {W : Set (ℝ × Pt n)} (hW : IsOpen W) {x : Pt n}
    (hx : ((0 : ℝ), x) ∈ W) {φ : ℝ → ℝ} {L : ℝ} (hφ : HasDerivAt φ L 0)
    (hz : ∀ ε, (ε, x) ∈ W → φ ε = 0) : L = 0 := by sorry
Source
He Wang, Exact Spacetime Atlas IV: Certified Einstein–Maxwell Backgrounds and First Variations. Local source, no public repository; commit 543904bfcfa3bb169f43fc4d9dbd569999b71886; Atlas/EinsteinMaxwell/Variation/Bridge.lean#L256-L264; SHA-256 c0401e780ed4e7fdf93b3fceb547d092276cb438ae1a341a5d661c712a8a09c4. Registered definitions and literature: mission/PHYSICS_BOUNDARY.md (§H–§J); Müller–Grave arXiv:0904.4184; Newman et al., J. Math. Phys. 6 (1965) 918.
Human review

  • Endorsed by Shuze Chen · Oct 21, 2026

    Confirmed by the moderator at approval.

  • Endorsed by He Wang · Oct 21, 2026

    Confirmed by the mission captain (proposal self-audit).

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