Plain-English statement
- The objective gap along an actual convex gradient trajectory, including the zero-curvature rate, at every positive time of a supplied forward interval.
Read the mathematics first, then descend into Lean
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Proof architecture
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Lean statement
theorem value_le {f : E → ℝ} {X : ℝ → E} {z : E} {α T : ℝ}
(hα : 0 ≤ α) (_hT : 0 ≤ T) (hf : Differentiable ℝ f)
(hsc : StrongConvexOn univ α 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) :
∀ t ∈ Ioc 0 T, 0 ≤ f (X t) - f z ∧
f (X t) - f z ≤ if α = 0 then ‖X 0 - z‖ ^ 2 / (2 * t)
else α / (2 * (Real.exp (α * t) - 1)) * ‖X 0 - z‖ ^ 2 := by
have hd (u : ℝ) (hu : u ∈ Ico 0 T) :
HasDerivWithinAt (fun s => f (X s)) (-‖gradient f (X u)‖ ^ 2) (Ici u) u := by
have hgrad : HasFDerivAt f (InnerProductSpace.toDual ℝ E (gradient f (X u))) (X u) :=
(hf (X u)).hasGradientAt
simpa only [Function.comp_def, InnerProductSpace.toDual_apply_apply,
inner_neg_right, real_inner_self_eq_norm_sq] using
hgrad.comp_hasDerivWithinAt u (hflow u hu)
have hmono : AntitoneOn (fun s => f (X s)) (Icc 0 T) := by
intro a ha b hb hab
have hc := (hf.continuous.comp_continuousOn hX).mono
(show Icc a b ⊆ Icc 0 T from fun u hu => ⟨ha.1.trans hu.1, hu.2.trans hb.2⟩)
have hg := le_gronwallBound_of_liminf_deriv_right_le
(f' := fun u => -‖gradient f (X u)‖ ^ 2) (δ := f (X a)) (K := 0) (ε := 0) hc
(fun u hu r hr => by
simpa [slope] using (hd u ⟨ha.1.trans hu.1, hu.2.trans_le hb.2⟩).liminf_right_slope_le hr)
le_rfl (fun u _ => by nlinarith [sq_nonneg ‖gradient f (X u)‖]) b ⟨hab, le_rfl⟩
simpa [gronwallBound_K0] using hg
have hdist (u : ℝ) (hu : u ∈ Ico 0 T) :
HasDerivWithinAt (fun s => ‖X s - z‖ ^ 2)
(-2 * inner ℝ (gradient f (X u)) (X u - z)) (Ici u) u := by
have h := ((hflow u hu).sub_const z).norm_sq
simp only [inner_neg_right, real_inner_comm] at h
convert h using 1; ring
intro t ht
refine ⟨sub_nonneg.mpr (hz (mem_univ _)), ?_⟩
have hbound := le_gronwallBound_of_liminf_deriv_right_le
(f := fun s => ‖X s - z‖ ^ 2)
(f' := fun u => -2 * inner ℝ (gradient f (X u)) (X u - z))
(δ := ‖X 0 - z‖ ^ 2) (K := -α) (ε := -2 * (f (X t) - f z))
(((hX.sub continuousOn_const).norm.pow 2).mono
(show Icc 0 t ⊆ Icc 0 T from fun u hu => ⟨hu.1, hu.2.trans ht.2⟩))
(fun u hu r hr => by
simpa [slope] using (hdist u ⟨hu.1, hu.2.trans_le ht.2⟩).liminf_right_slope_le hr)
le_rfl (fun u hu => by
have hs := StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn
hsc (fun x _ => (hf x).hasGradientAt) (mem_univ (X u)) (mem_univ z)
rw [show z - X u = -(X u - z) by abel, inner_neg_right, norm_neg] at hs
have hm := hmono ⟨hu.1, hu.2.le.trans ht.2⟩ ⟨ht.1.le, ht.2⟩ hu.2.le
nlinarith) t ⟨ht.1.le, le_rfl⟩
have hn := (sq_nonneg ‖X t - z‖).trans hbound
by_cases ha : α = 0
· simp only [ha, neg_zero, gronwallBound_K0, sub_zero] at hn ⊢
apply (le_div_iff₀ (mul_pos (by norm_num) ht.1)).mpr
nlinarith
· rw [if_neg ha]
have hap : 0 < α := lt_of_le_of_ne hα (Ne.symm ha)
rw [gronwallBound_of_K_ne_0 (neg_ne_zero.mpr ha), sub_zero] at hn
have hmul := mul_nonneg hn hap.le
have heq : (‖X 0 - z‖ ^ 2 * Real.exp (-α * t) +
(-2 * (f (X t) - f z)) / -α * (Real.exp (-α * t) - 1)) * α =
α * ‖X 0 - z‖ ^ 2 * Real.exp (-α * t) +
2 * (f (X t) - f z) * (Real.exp (-α * t) - 1) := by
field_simp
rw [heq] at hmul
have he : Real.exp (-α * t) * Real.exp (α * t) = 1 := by
rw [← Real.exp_add, show -α * t + α * t = 0 by ring, Real.exp_zero]
have hp := mul_nonneg hmul (Real.exp_pos (α * t)).le
have hcancel : (α * ‖X 0 - z‖ ^ 2 * Real.exp (-α * t) +
2 * (f (X t) - f z) * (Real.exp (-α * t) - 1)) * Real.exp (α * t) =
α * ‖X 0 - z‖ ^ 2 - 2 * (f (X t) - f z) * (Real.exp (α * t) - 1) := by
calc
_ = α * ‖X 0 - z‖ ^ 2 * (Real.exp (-α * t) * Real.exp (α * t)) +
2 * (f (X t) - f z) * (Real.exp (-α * t) * Real.exp (α * t)) -
2 * (f (X t) - f z) * Real.exp (α * t) := by ring
_ = _ := by rw [he]; ring
rw [hcancel] at hp
rw [div_mul_eq_mul_div]
apply (le_div_iff₀ (mul_pos (by norm_num) (sub_pos.mpr (Real.one_lt_exp_iff.mpr
(mul_pos hap ht.1))))).mpr
nlinarith
end AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowValueOpen AutoSamplingTheory/TechnicalLemmas/Analysis/GradientFlowValue.lean:22published source at 0e31a3cda412
Proof architecture
Actual convex gradient trajectory objective rate with direct zero-curvature branch on finite forward intervals.
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `intro` introduces quantified hypotheses into the local proof context.
- `have` creates a named intermediate mathematical fact.
- `calc` records an equality or inequality chain matching a paper calculation.
- `rw` rewrites by an established identity.
- `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.
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.