Plain-English statement
- The source Lyapunov decreases along an actual convex gradient trajectory, yielding last-time gradient and improved objective upper bounds.
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.
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.
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 lyapunov_and_rates {f : E → ℝ} {X : ℝ → E} {z : E} {T : ℝ}
(hT : 0 ≤ T) (hf : ContDiff ℝ 2 f) (hc : ConvexOn ℝ univ 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) :
AntitoneOn (fun t => t ^ 2 * ‖gradient f (X t)‖ ^ 2 +
2 * t * (f (X t) - f z) + ‖X t - z‖ ^ 2) (Icc 0 T) ∧
∀ t ∈ Ioc 0 T, ‖gradient f (X t)‖ ^ 2 ≤ ‖X 0 - z‖ ^ 2 / t ^ 2 ∧
f (X t) - f z ≤ ‖X 0 - z‖ ^ 2 / (4 * t) := by
have hdf : Differentiable ℝ f := hf.differentiable (by norm_num)
have hsc : StrongConvexOn univ 0 f := strongConvexOn_zero.mpr hc
have hH := (ConvexityC2.gradient_mono_iff_fderiv2_lower hf).mp (fun x y =>
StrongConvexFirstOrder.gradient_inner_lower_bound_of_strongConvexOn hsc
(fun w _ => (hdf w).hasGradientAt) (mem_univ x) (mem_univ y))
let R : (E →L[ℝ] ℝ) →L[ℝ] E :=
{ toFun := (toDual ℝ E).symm
map_add' := (toDual ℝ E).symm.map_add
map_smul' := by intros; simp
cont := (toDual ℝ E).symm.continuous }
let H (x : E) : E →L[ℝ] E := R.comp (fderiv ℝ (fderiv ℝ f) x)
have hfd : ContDiff ℝ 1 (fderiv ℝ f) := hf.fderiv_right (by norm_num)
have hg (x : E) : HasFDerivAt (gradient f) (H x) x :=
R.hasFDerivAt.comp x (hfd.differentiable_one x).hasFDerivAt
have hpos (x v : E) : 0 ≤ inner ℝ (H x v) v := by
have hi : inner ℝ (H x v) v = (fderiv ℝ (fderiv ℝ f) x v) v := toDual_symm_apply
simpa only [hi, zero_mul] using hH x v
have hsupp (x : E) : f x - f z ≤ inner ℝ (gradient f x) (x - z) := by
have h := StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn hsc
(fun w _ => (hdf w).hasGradientAt) (mem_univ x) (mem_univ z)
rw [show z - x = -(x - z) by abel, inner_neg_right] at h
simp only [zero_div, zero_mul, add_zero] at h
linarith
let L : ℝ → ℝ := fun t => t ^ 2 * ‖gradient f (X t)‖ ^ 2 +
2 * t * (f (X t) - f z) + ‖X t - z‖ ^ 2
let D : ℝ → ℝ := fun t => -2 * t ^ 2 * inner ℝ (H (X t) (gradient f (X t)))
(gradient f (X t)) + 2 * (f (X t) - f z) - 2 * inner ℝ (gradient f (X t)) (X t - z)
have hcont : ContinuousOn L (Icc 0 T) :=
((continuousOn_id.pow 2).mul (((continuous_iff_continuousAt.mpr (fun x => (hg x).continuousAt)).comp_continuousOn hX).norm.pow 2)).add
(((continuousOn_const.mul continuousOn_id).mul ((hf.continuous.comp_continuousOn hX).sub continuousOn_const))) |>.add
((hX.sub continuousOn_const).norm.pow 2)
have hd (t : ℝ) (ht : t ∈ Ico 0 T) : HasDerivWithinAt L (D t) (Ici t) t := by
have hn := ((hg (X t)).comp_hasDerivWithinAt t (hflow t ht)).norm_sq
have he : HasDerivWithinAt (fun s => f (X s)) (-‖gradient f (X t)‖ ^ 2) (Ici t) t := by
have h : HasFDerivAt f (toDual ℝ E (gradient f (X t))) (X t) := (hdf (X t)).hasGradientAt
simpa only [Function.comp_def, toDual_apply_apply, inner_neg_right,
real_inner_self_eq_norm_sq] using h.comp_hasDerivWithinAt t (hflow t ht)
have hr := ((hflow t ht).sub_const z).norm_sq
have hid := (hasDerivAt_id t).hasDerivWithinAt (s := Ici t)
convert (((hid.pow 2).mul hn).add ((hid.const_mul 2).mul (he.sub_const (f z)))).add hr using 1 <;>
first | rfl | (simp only [D, Function.comp_def, id_eq, Pi.pow_apply, Nat.cast_ofNat,
show (2 : ℕ) - 1 = 1 by decide, pow_one, mul_one, ContinuousLinearMap.map_neg,
inner_neg_right, real_inner_comm]; ring)
have hneg (t : ℝ) : D t ≤ 0 := by
have hh := mul_nonneg (sq_nonneg t) (hpos (X t) (gradient f (X t)))
have hs := hsupp (X t)
dsimp [D]
nlinarith
have hm : AntitoneOn L (Icc 0 T) := by
intro a ha b hb hab
have h := le_gronwallBound_of_liminf_deriv_right_le (f' := D)
(δ := L a) (K := 0) (ε := 0)
(hcont.mono (show Icc a b ⊆ Icc 0 T from fun u hu => ⟨ha.1.trans hu.1, hu.2.trans hb.2⟩))
(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 simpa using hneg u) b ⟨hab, le_rfl⟩
simpa [gronwallBound_K0] using h
refine ⟨hm, ?_⟩
intro t ht
have hL := hm ⟨le_rfl, hT⟩ ⟨ht.1.le, ht.2⟩ ht.1.le
simp only [L, zero_pow (by decide : 2 ≠ 0), zero_mul, zero_add] at hL
have he : 0 ≤ f (X t) - f z := sub_nonneg.mpr (hz (mem_univ _))
have hte := mul_nonneg ht.1.le he
refine ⟨(le_div_iff₀ (sq_pos_of_pos ht.1)).mpr (by nlinarith [sq_nonneg ‖X t - z‖]), ?_⟩
have hs := hsupp (X t)
have hcs := real_inner_le_norm (gradient f (X t)) (X t - z)
have hts := mul_le_mul_of_nonneg_left (hs.trans hcs) ht.1.le
have hy := sq_nonneg (t * ‖gradient f (X t)‖ - ‖X t - z‖)
apply (le_div_iff₀ (mul_pos (by norm_num) ht.1)).mpr
nlinarith
end AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowLastIterateOpen AutoSamplingTheory/TechnicalLemmas/Analysis/GradientFlowLastIterate.lean:22published source at 0e31a3cda412
Proof architecture
Actual convex-flow Lyapunov antitonicity and last-time gradient-square/value upper bounds.
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.
- `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.
- `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.