actual_fixed_reference_finite_jump_recursion

Plain-English statement

Independent exactSCI76 e1f1d85d34426954829a97a46b563ea8e1dab8f1. Original six analytic callers, eleven literal definitions and ten conclusion groups: actual finite stopped postjump recursion(A.2) using actual73 flow, actual74 bounce/rate and actual75 first clock; joint Borel update/records/event times; monotone and absorbing stopping; original initial energy on records and pre-jump arcs; original-energy uniform waiting increment including stopped records; zero-cap/zero-threshold and strictly increasing finite-successor guards. No phase at infinity or global physical-time path for arbitrary zero thresholds. iid Exp1 realization, a.s. positivity/divergence and nonaccumulation remain next independent boundaries; full PDMP/Markov/invariance/kernel/hypocoercivity/main/errors/expected-cost/composition, Exposition/PURIFIED/main/live/Goal remain open.

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.

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Lean statement

theorem actual_fixed_reference_finite_jump_recursion
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : ℝ≥0} {η : ℝ}
    (hα : 0 < (α : ℝ)) (hαβ : α ≤ β) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x v : E,
      (α : ℝ) * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v ∧
      (fderiv ℝ (fderiv ℝ V) x v) v ≤ (β : ℝ) * ‖v‖ ^ 2)
    (hη : 0 < η) (hβη : (β : ℝ) * η ≤ 1) :
    actual_fixed_reference_finite_jump_recursion_statement hα hαβ hV hH hη hβη := by
  let c : E → E → E := fun y xRef => y - η • gradient V xRef
  let Φ : E → E → ℝ → (E × E) → (E × E) := fun y xRef t z =>
    (c y xRef + Real.cos t • (z.1 - c y xRef) +
        (Real.sqrt η * Real.sin t) • z.2,
     (-Real.sin t / Real.sqrt η) • (z.1 - c y xRef) + Real.cos t • z.2)
  let S : E → (E × E) → (E × E) := fun xRef z =>
    (z.1, z.2 - (2 * inner ℝ z.2 (gradient V z.1 - gradient V xRef) /
      ‖gradient V z.1 - gradient V xRef‖ ^ 2) • (gradient V z.1 - gradient V xRef))
  let rate : E → (E × E) → ℝ := fun xRef z =>
    Real.sqrt η * max 0 (inner ℝ z.2 (gradient V z.1 - gradient V xRef))
  let H : E → E → (E × E) → ℝ := fun y xRef z =>
    (η⁻¹ * ‖z.1 - c y xRef‖ ^ 2 + ‖z.2‖ ^ 2) / 2
  let C : E → E → (E × E) → ℝ := fun y xRef z =>
    Real.sqrt η * (β : ℝ) * Real.sqrt (2 * H y xRef z) *
      (Real.sqrt (2 * η * H y xRef z) + ‖c y xRef - xRef‖)
  let Λ : E → E → (E × E) → ℝ≥0 → ℝ := fun y xRef z t =>
    ∫ s in (0 : ℝ)..(t : ℝ), rate xRef (Φ y xRef s z)
  let τ : E → E → (E × E) → ℝ≥0 → WithTop ℝ≥0 := fun y xRef z e =>
    MeasureTheory.hittingAfter
      (fun t : ℝ≥0 => fun a : (E × E) × (E × E) × ℝ≥0 =>
        Λ a.1.1 a.1.2 a.2.1 t - (a.2.2 : ℝ))
      (Set.Ici (0 : ℝ)) 0 ((y, xRef), z, e)
  let next : E → E → ℝ≥0 → ((ℝ≥0 × (E × E)) ⊕ Unit) →
      ((ℝ≥0 × (E × E)) ⊕ Unit) := fun y xRef e r =>
    match r with
    | Sum.inr _ => Sum.inr ()
    | Sum.inl a =>
      if τ y xRef a.2 e = ⊤ then Sum.inr () else
        Sum.inl (a.1 + (τ y xRef a.2 e).untopD 0,
          S xRef (Φ y xRef (((τ y xRef a.2 e).untopD 0 : ℝ≥0) : ℝ) a.2))
  let record : E → E → (E × E) → (ℕ → ℝ≥0) → ℕ →
      ((ℝ≥0 × (E × E)) ⊕ Unit) := fun y xRef z₀ e n =>
    Nat.rec (Sum.inl (0, z₀)) (fun k r => next y xRef (e k) r) n
  let eventTime : E → E → (E × E) → (ℕ → ℝ≥0) → ℕ → WithTop ℝ≥0 :=
    fun y xRef z₀ e n => (record y xRef z₀ e n).elim
      (fun a => (a.1 : WithTop ℝ≥0)) (fun _ => ⊤)
  change
  (∀ y xRef : E, ∀ z₀ : E × E, ∀ e : ℕ → ℝ≥0,
    record y xRef z₀ e 0 = Sum.inl (0, z₀)) ∧
  (∀ y xRef : E, ∀ z₀ : E × E, ∀ e : ℕ → ℝ≥0, ∀ n : ℕ,
    record y xRef z₀ e (n + 1) = next y xRef (e n) (record y xRef z₀ e n) ∧
    (∀ a : ℝ≥0 × (E × E), record y xRef z₀ e n = Sum.inl a →
      (record y xRef z₀ e (n + 1) = Sum.inr () ↔ τ y xRef a.2 (e n) = ⊤) ∧
      (τ y xRef a.2 (e n) ≠ ⊤ → record y xRef z₀ e (n + 1) =
        Sum.inl (a.1 + (τ y xRef a.2 (e n)).untopD 0,
          S xRef (Φ y xRef (((τ y xRef a.2 (e n)).untopD 0 : ℝ≥0) : ℝ) a.2))))) ∧
  Measurable (fun a : (E × E) × ℝ≥0 × ((ℝ≥0 × (E × E)) ⊕ Unit) =>
    next a.1.1 a.1.2 a.2.1 a.2.2) ∧
  (∀ n : ℕ, Measurable (fun a : (E × E) × (E × E) × (ℕ → ℝ≥0) =>
    record a.1.1 a.1.2 a.2.1 a.2.2 n)) ∧
  (∀ n : ℕ, Measurable (fun a : (E × E) × (E × E) × (ℕ → ℝ≥0) =>
    eventTime a.1.1 a.1.2 a.2.1 a.2.2 n)) ∧
  (∀ y xRef : E, ∀ z₀ : E × E, ∀ e : ℕ → ℝ≥0,
    eventTime y xRef z₀ e 0 = 0 ∧ Monotone (eventTime y xRef z₀ e)) ∧
  (∀ y xRef : E, ∀ z₀ : E × E, ∀ e : ℕ → ℝ≥0, ∀ n k : ℕ,
    record y xRef z₀ e n = Sum.inr () →
      record y xRef z₀ e (n + k) = Sum.inr ()) ∧
  (∀ y xRef : E, ∀ z₀ : E × E, ∀ e : ℕ → ℝ≥0, ∀ n : ℕ,
    ∀ a : ℝ≥0 × (E × E), record y xRef z₀ e n = Sum.inl a →
      H y xRef a.2 = H y xRef z₀ ∧
      ∀ t : ℝ≥0, (t : WithTop ℝ≥0) < τ y xRef a.2 (e n) →
        H y xRef (Φ y xRef (t : ℝ) a.2) = H y xRef z₀ ∧
        0 ≤ rate xRef (Φ y xRef (t : ℝ) a.2) ∧
        rate xRef (Φ y xRef (t : ℝ) a.2) ≤ C y xRef z₀) ∧
  (∀ y xRef : E, ∀ z₀ : E × E, ∀ e : ℕ → ℝ≥0,
    0 ≤ C y xRef z₀ ∧
    (0 < C y xRef z₀ → ∀ n : ℕ,
      eventTime y xRef z₀ e n +
        (Real.toNNReal ((e n : ℝ) / C y xRef z₀) : WithTop ℝ≥0) ≤
      eventTime y xRef z₀ e (n + 1)) ∧
    (C y xRef z₀ = 0 → ∀ n : ℕ, ∀ a : ℝ≥0 × (E × E),
      record y xRef z₀ e n = Sum.inl a → 0 < e n →
        record y xRef z₀ e (n + 1) = Sum.inr ())) ∧
  (∀ y xRef : E, ∀ z₀ : E × E, ∀ e : ℕ → ℝ≥0, ∀ n : ℕ,
    ∀ a : ℝ≥0 × (E × E), record y xRef z₀ e n = Sum.inl a →
      (e n = 0 → record y xRef z₀ e (n + 1) = Sum.inl (a.1, S xRef a.2)) ∧
      (∀ b : ℝ≥0 × (E × E), 0 < e n →
        record y xRef z₀ e (n + 1) = Sum.inl b → a.1 < b.1))

  have hf := ActualHarmonicFlow.actual_harmonic_flow_laws hα hαβ hV hH hη hβη
  have hb := ActualBounceRate.actual_bounce_rate_energy_laws hα hαβ hV hH hη hβη
  have hk := ActualHazardClock.actual_integrated_hazard_clock_laws hα hαβ hV hH hη hβη
  have hΦmeas : Measurable (fun a : (E × E) × ℝ × (E × E) =>
      Φ a.1.1 a.1.2 a.2.1 a.2.2) := hf.2.1
  have hΦzero (y xRef : E) (z : E × E) : Φ y xRef 0 z = z := hf.2.2.1 y xRef z
  have hΦH (y xRef : E) (t : ℝ) (z : E × E) :
      H y xRef (Φ y xRef t z) = H y xRef z := hf.2.2.2.2.2.2.2.1 y xRef t z
  have hSmeas : Measurable (fun a : E × (E × E) => S a.1 a.2) := hb.1
  have hSH (y xRef : E) (z : E × E) : H y xRef (S xRef z) = H y xRef z :=
    hb.2.2.2.2.2.2.1 y xRef z
  have hrate0 (xRef : E) (z : E × E) : 0 ≤ rate xRef z :=
    (hb.2.2.2.2.2.2.2.1 xRef z).1
  have hratecap (y xRef : E) (z₀ z : E × E) (hz : H y xRef z = H y xRef z₀) :
      rate xRef z ≤ C y xRef z₀ :=
    ((hb.2.2.2.2.2.2.2.2.2 y xRef z₀).2 z hz).2.2
  have hτpositive (y xRef : E) (z : E × E) (e : ℝ≥0) :
      0 < e → (0 : WithTop ℝ≥0) < τ y xRef z e :=
    (hk.2.2.2.2.2.1 y xRef z e).2.2.1
  have hτzero (y xRef : E) (z : E × E) : τ y xRef z 0 = 0 :=
    (hk.2.2.2.2.2.1 y xRef z 0).2.2.2
  have hτmeas : Measurable (fun a : (E × E) × (E × E) × ℝ≥0 =>
      τ a.1.1 a.1.2 a.2.1 a.2.2) := hk.2.2.2.2.2.2.1
  have hC0 (y xRef : E) (z : E × E) : 0 ≤ C y xRef z :=
    (hk.2.2.2.2.2.2.2.2.1 y xRef z).1
  have hwait (y xRef : E) (z : E × E) (e : ℝ≥0) :
      (0 < C y xRef z →
        (Real.toNNReal ((e : ℝ) / C y xRef z) : WithTop ℝ≥0) ≤ τ y xRef z e) ∧
      (C y xRef z = 0 → 0 < e → τ y xRef z e = ⊤) :=
    hk.2.2.2.2.2.2.2.2.2 y xRef z e
  have hfiniteval (q : ℝ≥0) : (q : WithTop ℝ≥0).untopD 0 = q := rfl
  have hzeroval : (0 : WithTop ℝ≥0).untopD 0 = 0 := rfl
  have hrec0 (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0) :
      record y xRef z₀ e 0 = Sum.inl (0, z₀) := rfl
  have hrecstep (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0) (n : ℕ) :
      record y xRef z₀ e (n + 1) = next y xRef (e n) (record y xRef z₀ e n) := rfl
  have hbranches (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0) (n : ℕ)
      (a : ℝ≥0 × (E × E)) (ha : record y xRef z₀ e n = Sum.inl a) :
      (record y xRef z₀ e (n + 1) = Sum.inr () ↔ τ y xRef a.2 (e n) = ⊤) ∧
      (τ y xRef a.2 (e n) ≠ ⊤ → record y xRef z₀ e (n + 1) =
        Sum.inl (a.1 + (τ y xRef a.2 (e n)).untopD 0,
          S xRef (Φ y xRef (((τ y xRef a.2 (e n)).untopD 0 : ℝ≥0) : ℝ) a.2))) := by
    rw [hrecstep, ha]
    dsimp only [next]
    split_ifs with ht <;> simp [ht]
  have hnextmeas : Measurable
      (fun a : (E × E) × ℝ≥0 × ((ℝ≥0 × (E × E)) ⊕ Unit) =>
        next a.1.1 a.1.2 a.2.1 a.2.2) := by
    let g : ((E × E) × ℝ≥0) × (ℝ≥0 × (E × E)) →
        ((ℝ≥0 × (E × E)) ⊕ Unit) := fun a =>
      if τ a.1.1.1 a.1.1.2 a.2.2 a.1.2 = ⊤ then Sum.inr () else
        Sum.inl (a.2.1 + (τ a.1.1.1 a.1.1.2 a.2.2 a.1.2).untopD 0,
          S a.1.1.2 (Φ a.1.1.1 a.1.1.2
            (((τ a.1.1.1 a.1.1.2 a.2.2 a.1.2).untopD 0 : ℝ≥0) : ℝ) a.2.2))
    have hparams : Measurable
        (fun a : ((E × E) × ℝ≥0) × (ℝ≥0 × (E × E)) =>
          (a.1.1, a.2.2, a.1.2)) :=
      measurable_fst.fst.prodMk (measurable_snd.snd.prodMk measurable_fst.snd)
    have ht0 := hτmeas.comp hparams
    have ht : Measurable (fun a : ((E × E) × ℝ≥0) × (ℝ≥0 × (E × E)) =>
        τ a.1.1.1 a.1.1.2 a.2.2 a.1.2) := ht0
    have hq := ht.untopD (0 : ℝ≥0)
    have hp := hΦmeas.comp (measurable_fst.fst.prodMk
      ((measurable_coe_nnreal_real.comp hq).prodMk measurable_snd.snd))
    have hs := hSmeas.comp (measurable_fst.fst.snd.prodMk hp)
    have hg : Measurable g := measurable_const.ite
      (measurableSet_eq_fun ht measurable_const)
      (measurable_inl.comp ((measurable_snd.fst.add hq).prodMk hs))
    have hsum := hg.sumElim
      (show Measurable (fun _ : ((E × E) × ℝ≥0) × Unit =>
        (Sum.inr () : (ℝ≥0 × (E × E)) ⊕ Unit)) from measurable_const)
    have hm := hsum.comp ((MeasurableEquiv.prodSumDistrib
      ((E × E) × ℝ≥0) (ℝ≥0 × (E × E)) Unit).measurable.comp
      ((measurable_fst.prodMk measurable_snd.fst).prodMk measurable_snd.snd))
    convert hm using 1
    funext a
    rcases a with ⟨⟨y, xRef⟩, e, r⟩
    cases r <;> rfl
  have hrecordmeas (n : ℕ) : Measurable
      (fun a : (E × E) × (E × E) × (ℕ → ℝ≥0) =>
        record a.1.1 a.1.2 a.2.1 a.2.2 n) := by
    induction n with
    | zero => exact measurable_inl.comp (measurable_const.prodMk measurable_snd.fst)
    | succ n ih =>
      exact hnextmeas.comp (measurable_fst.prodMk
        (((measurable_pi_apply n).comp measurable_snd.snd).prodMk ih))
  have htimemeas (n : ℕ) : Measurable
      (fun a : (E × E) × (E × E) × (ℕ → ℝ≥0) =>
        eventTime a.1.1 a.1.2 a.2.1 a.2.2 n) :=
    ((WithTop.measurable_coe.comp measurable_fst).sumElim measurable_const).comp
      (hrecordmeas n)
  have henergy (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0) (n : ℕ)
      (a : ℝ≥0 × (E × E)) (ha : record y xRef z₀ e n = Sum.inl a) :
      H y xRef a.2 = H y xRef z₀ := by
    induction n generalizing a with
    | zero =>
      have heq : (0, z₀) = a := Sum.inl.inj ha
      subst a
      rfl
    | succ n ih =>
      rw [hrecstep] at ha
      cases hr : record y xRef z₀ e n with
      | inr u => simp [hr, next] at ha
      | inl b =>
        rw [hr] at ha
        dsimp only [next] at ha
        by_cases ht : τ y xRef b.2 (e n) = ⊤
        · simp only [if_pos ht] at ha
          cases ha
        · rw [if_neg ht] at ha
          have heq := Sum.inl.inj ha
          rw [← heq]
          exact (hSH y xRef _).trans ((hΦH y xRef _ _).trans (ih b hr))
  have hmono (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0) :
      Monotone (eventTime y xRef z₀ e) := by
    apply monotone_nat_of_le_succ
    intro n
    dsimp only [eventTime]
    rw [hrecstep]
    cases hr : record y xRef z₀ e n with
    | inr u => simp [next]
    | inl a =>
      dsimp only [next]
      split_ifs with ht
      · exact le_top
      · exact WithTop.coe_le_coe.mpr (le_add_of_nonneg_right zero_le)
  have hstopped (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0) (n k : ℕ)
      (hn : record y xRef z₀ e n = Sum.inr ()) :
      record y xRef z₀ e (n + k) = Sum.inr () := by
    induction k with
    | zero => simpa using hn
    | succ k ih => simpa only [Nat.add_succ, hrecstep, ih, next]
  have harcs (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0) (n : ℕ)
      (a : ℝ≥0 × (E × E)) (ha : record y xRef z₀ e n = Sum.inl a) :
      H y xRef a.2 = H y xRef z₀ ∧
      ∀ t : ℝ≥0, (t : WithTop ℝ≥0) < τ y xRef a.2 (e n) →
        H y xRef (Φ y xRef (t : ℝ) a.2) = H y xRef z₀ ∧
        0 ≤ rate xRef (Φ y xRef (t : ℝ) a.2) ∧
        rate xRef (Φ y xRef (t : ℝ) a.2) ≤ C y xRef z₀ := by
    have hEa := henergy y xRef z₀ e n a ha
    refine ⟨hEa, fun t _ => ?_⟩
    have hEt := (hΦH y xRef t a.2).trans hEa
    exact ⟨hEt, hrate0 xRef _, hratecap y xRef z₀ _ hEt⟩
  have hcap_eq (y xRef : E) (z₀ z : E × E) (hz : H y xRef z = H y xRef z₀) :
      C y xRef z = C y xRef z₀ := by simp only [C, hz]
  have hincrement (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0)
      (hC : 0 < C y xRef z₀) (n : ℕ) :
      eventTime y xRef z₀ e n +
        (Real.toNNReal ((e n : ℝ) / C y xRef z₀) : WithTop ℝ≥0) ≤
      eventTime y xRef z₀ e (n + 1) := by
    cases hr : record y xRef z₀ e n with
    | inr u =>
      dsimp only [eventTime]
      rw [hrecstep, hr]
      change (⊤ : WithTop ℝ≥0) + _ ≤ ⊤
      exact le_top
    | inl a =>
      have hCa := hcap_eq y xRef z₀ a.2 (henergy y xRef z₀ e n a hr)
      have hw := (hwait y xRef a.2 (e n)).1 (by simpa only [hCa] using hC)
      rw [hCa] at hw
      dsimp only [eventTime]
      rw [hrecstep, hr]
      dsimp only [next]
      split_ifs with ht
      · exact le_top
      · have hq : (((τ y xRef a.2 (e n)).untopD 0 : ℝ≥0) : WithTop ℝ≥0) =
            τ y xRef a.2 (e n) := by
          induction hτ : τ y xRef a.2 (e n) using WithTop.recTopCoe with
          | top => exact False.elim (ht hτ)
          | coe q => exact congrArg (fun q : ℝ≥0 => (q : WithTop ℝ≥0)) (hfiniteval q)
        change (a.1 : WithTop ℝ≥0) + _ ≤
          ((a.1 + (τ y xRef a.2 (e n)).untopD 0 : ℝ≥0) : WithTop ℝ≥0)
        rw [WithTop.coe_add, hq]
        exact add_le_add le_rfl hw
  have hzerocap (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0)
      (hC : C y xRef z₀ = 0) (n : ℕ) (a : ℝ≥0 × (E × E))
      (ha : record y xRef z₀ e n = Sum.inl a) (he : 0 < e n) :
      record y xRef z₀ e (n + 1) = Sum.inr () := by
    apply (hbranches y xRef z₀ e n a ha).1.mpr
    apply (hwait y xRef a.2 (e n)).2 _ he
    exact (hcap_eq y xRef z₀ a.2 (henergy y xRef z₀ e n a ha)).trans hC
  have hzeroevent (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0) (n : ℕ)
      (a : ℝ≥0 × (E × E)) (ha : record y xRef z₀ e n = Sum.inl a)
      (he : e n = 0) :
      record y xRef z₀ e (n + 1) = Sum.inl (a.1, S xRef a.2) := by
    rw [hrecstep, ha]
    simp only [next, he, hτzero, WithTop.zero_ne_top, ↓reduceIte,
      hzeroval, add_zero, NNReal.coe_zero, hΦzero]
  have hstrictevent (y xRef : E) (z₀ : E × E) (e : ℕ → ℝ≥0) (n : ℕ)
      (a : ℝ≥0 × (E × E)) (ha : record y xRef z₀ e n = Sum.inl a)
      (b : ℝ≥0 × (E × E)) (he : 0 < e n)
      (hb : record y xRef z₀ e (n + 1) = Sum.inl b) : a.1 < b.1 := by
    have ht : τ y xRef a.2 (e n) ≠ ⊤ := by
      intro ht
      have hs := (hbranches y xRef z₀ e n a ha).1.mpr ht
      rw [hb] at hs
      cases hs
    have hq : 0 < (τ y xRef a.2 (e n)).untopD 0 := by
      have hp := hτpositive y xRef a.2 (e n) he
      induction hτ : τ y xRef a.2 (e n) using WithTop.recTopCoe with
      | top => exact False.elim (ht hτ)
      | coe q => simpa only [hτ, hfiniteval, ← WithTop.coe_zero,
          WithTop.coe_lt_coe] using hp
    have hupdate := (hbranches y xRef z₀ e n a ha).2 ht
    rw [hb] at hupdate
    have heq := Sum.inl.inj hupdate
    have hbtime := congrArg Prod.fst heq
    rw [hbtime]
    exact lt_add_of_pos_right _ hq
  exact ⟨hrec0, fun y xRef z₀ e n =>
    ⟨hrecstep y xRef z₀ e n, hbranches y xRef z₀ e n⟩,
    hnextmeas, hrecordmeas, htimemeas,
    fun y xRef z₀ e => ⟨rfl, hmono y xRef z₀ e⟩, hstopped, harcs,
    fun y xRef z₀ e => ⟨hC0 y xRef z₀, hincrement y xRef z₀ e,
      hzerocap y xRef z₀ e⟩,
    fun y xRef z₀ e n a ha => ⟨hzeroevent y xRef z₀ e n a ha,
      hstrictevent y xRef z₀ e n a ha⟩⟩

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteJumpRecursion

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualFiniteJumpRecursion.lean:104published source at ed709a7f320d

Proof architecture

PBPS consumer; no SALD admission

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.
`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.
Totalized `fderiv` values must not be read as a differentiability theorem.