refactor(Analysis/Calculus/Gradient): ungate inner_gradient lemmas#39203
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refactor(Analysis/Calculus/Gradient): ungate inner_gradient lemmas#39203FordUniver wants to merge 1 commit into
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The four `inner_gradient[Within]_[left|right]` lemmas in `Mathlib/Analysis/Calculus/Gradient/Basic.lean` required differentiability hypotheses (`DifferentiableAt`, `DifferentiableWithinAt`, plus `UniqueDiffWithinAt` for the `Within` versions). These were never necessary: `gradient` is *defined* via the Riesz isomorphism from `fderiv`, so the identification `⟪∇ f x, y⟫ = fderiv 𝕜 f x y` holds unconditionally — both sides are zero when `f` is not differentiable. Removes the hypotheses and rewrites the proofs to go through `toDual_apply_apply` and `LinearIsometryEquiv.apply_symm_apply` directly. No call-site updates needed (no current mathlib callers pass the now-redundant arguments).
PR summary d59f7b3264Import changes for modified filesNo significant changes to the import graph Import changes for all files
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Removes superfluous differentiability hypotheses (plus
UniqueDiffWithinAtfor theWithinversions) from the fourinner_gradient[Within]_[left|right]lemmas:gradientis defined as(toDual 𝕜 F).symm (fderiv 𝕜 f x), so⟪∇ f x, y⟫ = fderiv 𝕜 f x yholds unconditionally since both sides are zero whenfis not differentiable atx.Genuinely unsure about this one because the only call I currently have in my code itself has the differentiability hypothesis. But including unused (or unnecessary) assumptions seems not intentional? My best guess is that the hypotheses were included by analogy with
HasGradientAt.fderiv_apply, which genuinely needs them. This PR drops the hypotheses and rewrites the proofs through the isomorphism directly; no existing mathlib callers pass the now-redundant arguments.