feat(Analysis/Calculus/LipschitzSmooth): add descent lemma#28
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This was referenced May 18, 2026
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Adds the descent lemma in Fréchet / 1D `deriv` / Hilbert gradient form on top of the foundational `LipschitzSmoothWith` API: a differentiable function whose derivative (in any form) is `K`-Lipschitz is `K`-smooth. The proof routes through a segment-pointwise predicate `LipschitzSmoothOnSegmentWith` and the fundamental theorem of calculus along a line segment. Includes the Riesz isomorphism between Fréchet and gradient forms for Lipschitz constants and its 1D analogue.
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Establishes the descent lemma for
LipschitzSmoothWith: a differentiable functionf : E → Fbetween real normed spaces (Fcomplete) whose Fréchet derivative isK-Lipschitz isK-smooth. The 1Dderivand Hilbert-space gradient forms follow as corollaries. The proof routes through a segment-pointwise predicateLipschitzSmoothOnSegmentWithand the fundamental theorem of calculus along a line segment. Also adds the identification of Lipschitz constants between the Fréchet derivative and the gradient (via Riesz representation), and its 1Dderivanalogue.This is the substantive descent-lemma layer on top of the foundational
LipschitzSmoothWithpredicate API (leanprover-community#39574), whose codomain-generality it inherits. The Baillon–Haddad equivalence between cocoercivity and Lipschitz-smoothness under convexity is the final layer, deferred to a follow-up that additionally depends on the Convex first-order package (leanprover-community#41494).The Fréchet descent lemma proof uses segment-level curve integration (
curveIntegral_fderiv_segment) fromMathlib.MeasureTheory.Integral.CurveIntegral. The gradient form usestoDual_comp_gradientto identifyfderiv ℝ fand∇ fas related by aLinearIsometryEquiv; the 1D analogue uses the isometry(ContinuousLinearMap.toSpanSingletonLIE ℝ ℝ).symm(evaluation at1) in the same way.LipschitzSmoothWithandCocoerciveWithleanprover-community/mathlib4#39574AffineMap.lineMapleanprover-community/mathlib4#39206inner_gradientlemmas leanprover-community/mathlib4#39203toDual_gradientand companions leanprover-community/mathlib4#39202Diff: link