firstOrder_geodesicConvexity

Plain-English statement

The endpoint chord inequality for a geodesically convex functional yields its first-order supporting inequality at the initial endpoint.

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.

Mathematical statement

F(path(1)) >= F(path(0)) + d/dt|0 F(path(t)) + alpha*d(path(0),path(1))^2/2.

Intuition

Divide the chord inequality by a small positive interpolation time. The resulting secant slope converges to the directional derivative, while the curvature correction converges to its endpoint value.

Conditions

the selected path is a geodesic covered by IsAlphaGeodesicallyConvex
F composed with the path has a real derivative at time zero

Why these conditions cannot be dropped

the chord inequality supplies the finite-time comparison
the derivative identifies the limiting secant slope
a concrete Wasserstein use must separately identify this derivative with the gradient pairing

Proof route

specialize alpha-geodesic convexity at positive t less than one
rearrange it as an upper bound on the secant slope
take the positive-time limit of both sides
use closedness of the real order and rearrange the result

Lean interface notes

HasDerivAt.tendsto_slope supplies the punctured slope limit
le_of_tendsto_of_tendsto preserves the eventual inequality
gradientPairing is semantic data, not a supplied copy of the conclusion

Lean learning studio · mathematics → formal proof

Read the mathematics first, then descend into Lean

You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.

BeginnerWhy this theorem exists → intuition → statement → one hand calculation. Hide proof-engineering detail.

RigorousExpose assumptions, hidden measure/limit/domain issues, proof route, and rigorous references.

Lean learnerOpen the exact declaration, proof tree/network, syntax glossary, and line-by-line explanation.

Proof architecture

Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.

Loading source-derived dependency evidence…

Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.

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

When Samplinglib shows a short quotation from Chewi, it is labeled as a source excerpt and linked to the canonical book page. Intuition, expanded proof steps, hidden regularity assumptions, and Lean explanations are ASTIS-authored commentary. A quotation never substitutes for a formal proof, and an ASTIS explanation is never attributed to the textbook author.

Lean statement

theorem firstOrder_geodesicConvexity
    {M : Type*} [MetricSpace M]
    {isGeodesic : (ℝ → M) → Prop} {F : M → ℝ} {alpha : ℝ}
    (hconvex : IsAlphaGeodesicallyConvex isGeodesic F alpha)
    {path : ℝ → M} (hpath : isGeodesic path)
    {gradientPairing : ℝ}
    (hderiv : HasDerivAt (fun t => F (path t)) gradientPairing 0) :
    F (path 1) ≥
      F (path 0) + gradientPairing +
        alpha / 2 * dist (path 0) (path 1) ^ 2 := by
  let phi : ℝ → ℝ := fun t => F (path t)
  let upper : ℝ → ℝ := fun t =>
    F (path 1) - F (path 0) -
      alpha * (1 - t) / 2 * dist (path 0) (path 1) ^ 2
  have hslope :
      Tendsto (slope phi 0) (𝓝[>] 0) (𝓝 gradientPairing) := by
    apply hderiv.tendsto_slope.mono_left
    apply nhdsWithin_mono
    intro t ht
    simpa only [mem_compl_iff, mem_singleton_iff] using ne_of_gt ht
  have hupper :
      Tendsto upper (𝓝[>] 0) (𝓝 (upper 0)) := by
    have hcontinuous : ContinuousAt upper 0 := by
      dsimp only [upper]
      fun_prop
    have hfilter : 𝓝[>] (0 : ℝ) ≤ 𝓝 0 := inf_le_left
    exact hcontinuous.tendsto.mono_left hfilter
  have hslope_le : slope phi 0 ≤ᶠ[𝓝[>] 0] upper := by
    filter_upwards [self_mem_nhdsWithin,
      (eventually_lt_nhds zero_lt_one).filter_mono inf_le_left]
      with t ht htle
    have htpos : 0 < t := ht
    have hchord := hconvex path hpath t ⟨htpos.le, htle.le⟩
    have hquotient :
        (F (path t) - F (path 0)) / t ≤ upper t := by
      apply (div_le_iff₀ htpos).2
      dsimp only [upper]
      nlinarith
    rw [div_eq_inv_mul] at hquotient
    simpa [phi, slope, htpos.ne'] using hquotient
  have hfirst := le_of_tendsto_of_tendsto hslope hupper hslope_le
  dsimp only [upper, sub_zero, one_mul] at hfirst
  linarith

end GeodesicConvexity
end Geometry
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/Geometry/GeodesicConvexity.lean:40published source at 7bcd37294df1