gradient_sub_inner_eq_integral_fderiv2

Plain-English statement

- The genuine Hessian integrates to the gradient difference paired with the segment direction. C² supplies continuity and integrability.

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 gradient_sub_inner_eq_integral_fderiv2
    {f : E → ℝ} (hf : ContDiff ℝ 2 f) (x v : E) :
    inner ℝ (gradient f (x + v) - gradient f x) v =
      ∫ t : ℝ in 0..1, (fderiv ℝ (fderiv ℝ f) (x + t • v) v) v := by
  have hfd : ContDiff ℝ 1 (fderiv ℝ f) := hf.fderiv_right (by norm_num)
  have hd (t : ℝ) : HasDerivAt (fun s : ℝ => fderiv ℝ f (x + s • v) v)
      ((fderiv ℝ (fderiv ℝ f) (x + t • v) v) v) t := by
    convert ((hfd.differentiable_one _).hasFDerivAt.comp_hasDerivAt t
      (((hasDerivAt_id t).smul_const v).const_add x)).clm_apply
      (hasDerivAt_const t v) using 1 <;> first | rfl | simp
  have hc : Continuous (fun t : ℝ => (fderiv ℝ (fderiv ℝ f) (x + t • v) v) v) :=
    (((hfd.continuous_fderiv (by norm_num)).comp
      (continuous_const.add (continuous_id.smul continuous_const))).clm_apply
        continuous_const).clm_apply continuous_const
  simpa [inner_sub_left, gradient, Function.comp_def] using
    (intervalIntegral.integral_eq_sub_of_hasDerivAt
      (fun t _ => hd t) (hc.intervalIntegrable 0 1)).symm

/-- Global quantitative gradient monotonicity is equivalent to the genuine
Hessian diagonal lower bound, by a right derivative limit and the FTC. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexityC2.lean:26published source at 0e31a3cda412

Proof architecture

Actual parent of gradient-Hessian equivalence; reversed quadratic segment test

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.
`simp` normalizes through registered definitional and theorem rewrites.
`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

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.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.