Plain-English statement
- Integrating a derivative bound with a backward exponential kernel gives the exact endpoint convolution factor. The quantities `source` and `energy` may have either sign: the theorem assumes the required derivative inequality directly. Algebraically, a nonzero rate is enough for the proof; strict positivity records the intended decay regime.
Read the mathematics first, then descend into Lean
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Proof architecture
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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
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Lean statement
theorem endpoint_le_of_deriv_le_exponential_kernel
{H H' : ℝ → ℝ} {rate source t energy : ℝ}
(hrate : 0 < rate) (ht : 0 ≤ t)
(hH : ∀ s ∈ Icc (0 : ℝ) t, HasDerivAt H (H' s) s)
(hzero : H 0 = 0)
(hderiv : ∀ s ∈ Ioo (0 : ℝ) t,
H' s ≤ source * Real.exp (-rate * (t - s)) * energy) :
H t ≤ (source / rate) * (1 - Real.exp (-rate * t)) * energy := by
let B : ℝ → ℝ := fun s =>
(source / rate) *
(Real.exp (-rate * (t - s)) - Real.exp (-rate * t)) * energy
have hB : ∀ s : ℝ,
HasDerivAt B
(source * Real.exp (-rate * (t - s)) * energy) s := by
intro s
have hinner : HasDerivAt (fun u : ℝ => -rate * (t - u)) rate s := by
have hcoeff : -(-rate) = rate := by ring
simpa only [Pi.sub_apply, id_eq, zero_sub, mul_neg, mul_one, hcoeff] using
(((hasDerivAt_const s t).sub (hasDerivAt_id s)).const_mul (-rate))
have hexp := (Real.hasDerivAt_exp (-rate * (t - s))).comp s hinner
have hsub := hexp.sub_const (Real.exp (-rate * t))
have hmul := (hsub.const_mul (source / rate)).mul_const energy
convert hmul using 1 <;> try rfl
field_simp [ne_of_gt hrate]
have hmono : MonotoneOn (fun s => B s - H s) (Icc (0 : ℝ) t) := by
apply monotoneOn_of_deriv_nonneg (convex_Icc (0 : ℝ) t)
· exact continuousOn_of_forall_continuousAt fun s hs =>
((hB s).sub (hH s hs)).continuousAt
· intro s hs
exact ((hB s).sub (hH s (interior_subset hs))).differentiableAt.differentiableWithinAt
· intro s hs
change 0 ≤ deriv (B - H) s
rw [((hB s).sub (hH s (interior_subset hs))).deriv]
exact sub_nonneg.mpr (hderiv s (by simpa only [interior_Icc] using hs))
have hend := hmono ⟨le_rfl, ht⟩ ⟨ht, le_rfl⟩ ht
dsimp [B] at hend
have hBzero :
(source / rate) *
(Real.exp (-rate * (t - 0)) - Real.exp (-rate * t)) * energy = 0 := by
ring_nf
rw [hBzero, hzero] at hend
have hBt :
(source / rate) *
(Real.exp (-rate * (t - t)) - Real.exp (-rate * t)) * energy =
(source / rate) * (1 - Real.exp (-rate * t)) * energy := by
rw [sub_self, mul_zero, Real.exp_zero]
rw [hBt] at hend
exact sub_nonneg.mp (by simpa only [sub_self] using hend)
end
end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalitiesOpen AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/ExponentialKernelInterpolation.lean:26published source at ed709a7f320d
Proof architecture
integrate a backward exponential derivative bound into its exact endpoint convolution factor
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.
- `apply` reduces the goal to the hypotheses of a reusable theorem.
- `exact` closes the current goal with an already typed term.
- `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
- Genuine differentiability is localized to the hypotheses shown in the Lean statement.