sub_eq_integral_gradient

Plain-English statement

- The fundamental theorem of calculus along an affine segment. Continuity and interval integrability of the genuine gradient pairing follow from C¹.

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

/-- Quantitative gradient monotonicity implies the chord inequality by the
source's two affine-segment FTC identities and integration of their difference.
The modulus can be signed; no Hessian or extra integrability is assumed. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexityC1.lean:30published source at 0e31a3cda412

Proof architecture

Actual parent of the two-segment integral converse; reversed-segment quadratic 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

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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.