integral_exp_neg_add_mul_abs_eq

Plain-English statement

- Exact normalizer for one-dimensional absolute-linear Laplace tails.

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.

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 integral_exp_neg_add_mul_abs_eq {a b : ℝ} (ha : 0 < a) :
    ∫ x : ℝ, Real.exp (-(a * |x| + b)) ∂volume =
      2 * Real.exp (-b) / a := by
  let f : ℝ → ℝ := fun x => Real.exp (-(a * |x| + b))
  have hf : Integrable f volume :=
    integrable_exp_neg_add_mul_abs (a := a) (b := b) ha
  have hsplit :
      (∫ x, f x ∂volume) =
        ∫ x in Iic (0 : ℝ), f x ∂volume +
          ∫ x in Ioi (0 : ℝ), f x ∂volume := by
    rw [← setIntegral_univ, ← Iic_union_Ioi (a := (0 : ℝ))]
    exact setIntegral_union (Iic_disjoint_Ioi le_rfl) measurableSet_Ioi
      hf.integrableOn hf.integrableOn
  have hleft :
      ∫ x in Iic (0 : ℝ), f x ∂volume = Real.exp (-b) / a := by
    have hbase :
        ∫ x in Iic (0 : ℝ), Real.exp (-b) * Real.exp (a * x) ∂volume =
          Real.exp (-b) / a := by
      rw [integral_const_mul]
      rw [integral_exp_mul_Iic (a := a) ha (c := 0)]
      simp
      ring
    rw [← hbase]
    apply setIntegral_congr_fun measurableSet_Iic
    intro x hx
    have hxnonpos : x ≤ 0 := hx
    calc
      f x = Real.exp (-b + a * x) := by
        simp [f, abs_of_nonpos hxnonpos]
      _ = Real.exp (-b) * Real.exp (a * x) := by
        rw [Real.exp_add]
  have hright :
      ∫ x in Ioi (0 : ℝ), f x ∂volume = Real.exp (-b) / a := by
    have hbase :
        ∫ x in Ioi (0 : ℝ), Real.exp (-b) * Real.exp ((-a) * x) ∂volume =
          Real.exp (-b) / a := by
      rw [integral_const_mul]
      rw [integral_exp_mul_Ioi (a := -a) (by linarith) (c := 0)]
      simp
      ring
    rw [← hbase]
    apply setIntegral_congr_fun measurableSet_Ioi
    intro x hx
    have hxnonneg : 0 ≤ x := le_of_lt hx
    calc
      f x = Real.exp (-b + (-a) * x) := by
        simp [f, abs_of_nonneg hxnonneg]
      _ = Real.exp (-b) * Real.exp ((-a) * x) := by
        rw [Real.exp_add]
  rw [show (∫ x : ℝ, Real.exp (-(a * |x| + b)) ∂volume) =
      ∫ x : ℝ, f x ∂volume by rfl]
  rw [hsplit, hleft, hright]
  field_simp [ha.ne']
  ring

/-- Exact `ℝ≥0∞` normalizer for one-dimensional absolute-linear Laplace
tails. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:137published source at 7bcd37294df1

Proof architecture

Chewi DENS/CONV root: exact real integral `∫ exp (-(a|x|+b)) = 2 exp(-b)/a` for Laplace examples

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.
`exact` closes the current goal with an already typed term.

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

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.