Plain-English statement
- A minimum gradient norm is attained on the supplied time interval, and is bounded by the square root of the initial objective gap divided by elapsed time.
Read the mathematics first, then descend into Lean
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Proof architecture
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Syntax used on this page
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Lean statement
theorem exists_min_norm_le {f : E → ℝ} {X : ℝ → E} {z : E} {T : ℝ}
(hT : 0 < T) (hf : ContDiff ℝ 1 f) (hz : IsMinOn f univ z)
(hX : ContinuousOn X (Icc 0 T))
(hflow : ∀ t ∈ Ico 0 T, HasDerivWithinAt X (-gradient f (X t)) (Ici t) t) :
∃ s ∈ Icc 0 T, IsMinOn (fun u => ‖gradient f (X u)‖) (Icc 0 T) s ∧
‖gradient f (X s)‖ ≤ Real.sqrt ((f (X 0) - f z) / T) := by
have hdf : Differentiable ℝ f := hf.differentiable (by norm_num)
have hcg : Continuous (gradient f) :=
(InnerProductSpace.toDual ℝ E).symm.continuous.comp (hf.continuous_fderiv (by norm_num))
obtain ⟨s, hs, hmin⟩ := isCompact_Icc.exists_isMinOn
(nonempty_Icc.mpr hT.le) (hcg.comp_continuousOn hX).norm
refine ⟨s, hs, hmin, ?_⟩
have hd (u : ℝ) (hu : u ∈ Ico 0 T) :
HasDerivWithinAt (fun v => f (X v)) (-‖gradient f (X u)‖ ^ 2) (Ici u) u := by
have hg : HasFDerivAt f (InnerProductSpace.toDual ℝ E (gradient f (X u))) (X u) :=
(hdf (X u)).hasGradientAt
simpa only [Function.comp_def, InnerProductSpace.toDual_apply_apply,
inner_neg_right, real_inner_self_eq_norm_sq] using
hg.comp_hasDerivWithinAt u (hflow u hu)
have hb := le_gronwallBound_of_liminf_deriv_right_le
(f' := fun u => -‖gradient f (X u)‖ ^ 2) (δ := f (X 0))
(K := 0) (ε := -‖gradient f (X s)‖ ^ 2)
(hf.continuous.comp_continuousOn hX)
(fun u hu r hr => by simpa [slope] using (hd u hu).liminf_right_slope_le hr)
le_rfl (fun u hu => by
have hm := hmin (show u ∈ Icc 0 T from ⟨hu.1, hu.2.le⟩)
have hsq := sq_le_sq₀ (norm_nonneg (gradient f (X s))) (norm_nonneg (gradient f (X u)))
simpa using (neg_le_neg (hsq.mpr hm))) T ⟨hT.le, le_rfl⟩
simp only [gronwallBound_K0, sub_zero] at hb
apply (Real.le_sqrt (norm_nonneg _) (div_nonneg
(sub_nonneg.mpr (hz (mem_univ _))) hT.le)).mpr
apply (le_div_iff₀ hT).mpr
have hzT : f z ≤ f (X T) := hz (mem_univ (X T))
change f (X T) ≤ _ at hb
nlinarith
end AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowStationarityOpen AutoSamplingTheory/TechnicalLemmas/Analysis/GradientFlowStationarity.lean:23published source at 0e31a3cda412
Proof architecture
Attained minimum of actual gradient norm over a positive time interval, bounded by the square root of initial objective gap divided by time.
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.
- `apply` reduces the goal to the hypotheses of a reusable theorem.
- `refine` instantiates a reusable theorem while leaving explicit subgoals.
- `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
- Totalized `fderiv` values must not be read as a differentiability theorem.
- Genuine differentiability is localized to the hypotheses shown in the Lean statement.