Plain-English statement
- Ambient maps agreeing locally relative to a feasible set have equal actual derivatives on the velocity of any differentiable curve locally in that set. The maps need only be genuinely differentiable at the actual curve point.
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 fderiv_eq_on_curve_velocity
{s : Set E} {f g : E → F} {c : ℝ → E} {t₀ : ℝ} {v : E}
(hf : DifferentiableAt ℝ f (c t₀))
(hg : DifferentiableAt ℝ g (c t₀))
(hc : HasDerivAt c v t₀)
(hcs : ∀ᶠ t in 𝓝 t₀, c t ∈ s)
(hfg : f =ᶠ[𝓝[s] (c t₀)] g) :
fderiv ℝ f (c t₀) v = fderiv ℝ g (c t₀) v := by
have hcts : Tendsto c (𝓝 t₀) (𝓝[s] (c t₀)) :=
tendsto_nhdsWithin_iff.mpr ⟨hc.continuousAt, hcs⟩
have hcomp : (f ∘ c) =ᶠ[𝓝 t₀] (g ∘ c) := hfg.comp_tendsto hcts
have hfc := hf.hasFDerivAt.comp_hasDerivAt t₀ hc
have hgc := hg.hasFDerivAt.comp_hasDerivAt t₀ hc
exact (hfc.congr_of_eventuallyEq hcomp.symm).unique hgc
end AutoSamplingTheory.TechnicalLemmas.Geometry.CurveExtensionOpen AutoSamplingTheory/TechnicalLemmas/Geometry/CurveExtension.lean:24published source at ed709a7f320d
Proof architecture
Shared derivative equality on actual nonlinear feasible velocities for embedded first-order geometry
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `have` creates a named intermediate mathematical fact.
- `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.
- Genuine differentiability is localized to the hypotheses shown in the Lean statement.