Plain-English statement
- Actual unnormalized finite-sector weighted substitution.
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Lean statement
theorem actual_weighted_sector_lintegral
{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_weighted_sector_statement hα hαβ hV hH hη hβη := by
classical
unfold actual_weighted_sector_statement
intro c Φ S rate Λ Rec Hist Δ Prefix zAt K qReal q liveTime d weight Path Ψ Γ ρ R U w ell m n T
obtain ⟨rho, hrho, h97⟩ :=
AutoSamplingTheory.ExampleCases.ProximalBPS.ActualEndpointTimeReversal.actual_endpoint_time_reversal_laws
hα hαβ hV hH hη hβη n T
obtain ⟨rho101, hrho101, h101⟩ :=
AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedWeightedDensity.actual_prescribed_weighted_density_balance
hα hαβ hV hH hη hβη n T
have halign : rho101 = rho := by
funext t
apply Subtype.ext
funext i
exact (hrho101 t i).trans (hrho t i).symm
subst rho101
refine ⟨rho, hrho, ?_⟩
intro y r A Theta W F hF
have hTheta : MeasurePreserving Theta (ell.prod (m n T)) (ell.prod (m n T)) :=
h97.2.2.2.2.2.2.2.2.2 y r
have hbalance (q : (E × E) × Prefix n T) : W q = W (Theta q) :=
(h101 y r q.1 q.2).2
obtain ⟨Z, hZ⟩ :=
AutoSamplingTheory.ExampleCases.ProximalBPS.ActualFiniteCountPath.actual_finite_count_path_law
hα hαβ hV hH hη hβη
have hρjoint : Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
ρ p.1.1.1 p.1.1.2 p.1.2 n T p.2) := (hZ.2.2.2.2 n T).2.2.1
have hemb : Measurable (fun q : (E × E) × Prefix n T => (((y, r), q.1), q.2)) :=
((measurable_const.prodMk measurable_fst).prodMk measurable_snd)
have hρ := hρjoint.comp hemb
have hU : Continuous (U y) := by
exact ((hV.continuous.comp continuous_fst).add
(((continuous_fst.sub continuous_const).norm.pow 2).div_const (2 * η))).add
((continuous_snd.norm.pow 2).div_const 2)
have hw : Continuous (w y) := Real.continuous_exp.comp hU.neg
have hW : Measurable W :=
(ENNReal.measurable_ofReal.comp (hw.measurable.comp measurable_fst)).mul hρ
calc
(∫⁻ q, W q * F (Theta q) ∂(ell.prod (m n T))) =
∫⁻ q, W (Theta q) * F (Theta q) ∂(ell.prod (m n T)) := by
apply lintegral_congr
intro q
rw [hbalance q]
_ = ∫⁻ q, W q * F q ∂(ell.prod (m n T)) := hTheta.lintegral_comp (hW.mul hF)
#print axioms actual_weighted_sector_lintegral
end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualWeightedSectorOpen AutoSamplingTheory/ExampleCases/ProximalBPS/ActualWeightedSector.lean:109published source at ed709a7f320d
Proof architecture
Actual finite PBPS weighted sector substitution; no SALD claim
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.
- `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.
- Density statements retain normalization and absolute-continuity prerequisites.