Prove2Me

Second Scale

Proved
ActuarialAnnuities.second_scale

by WillR · Oct 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

actuarial-scienceannuity-cashflowmomentssingle-life-annuities

Pathwise squares make the second moment quadratic in benefit scale.

Mathematical relation:

E[(cZ)2]=c2E[Z2]\mathbb E[(cZ)^2]=c^2\mathbb E[Z^2]

Timing and admissibility conditions are the exact conditions of the declaration. Cash amounts are per instalment and present values are at valuation time zero.

Preamble
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Definitions.Def_actuarial_bSecondMoment
Formal statement
namespace ActuarialAnnuities

theorem second_scale (h a n g delta : ℕ) (v c : ℝ) (b p : ℕ → ℝ) : bSecondMoment h a n g delta v (fun j => c*b j) p = c^2*bSecondMoment h a n g delta v b p := by sorry

end ActuarialAnnuities
Source
Original derived finite-cashflow result, contextual source page 143, Annual life annuities (Chapter 5, section 5.4). D. C. M. Dickson, M. R. Hardy and H. R. Waters (2020), Actuarial Mathematics for Life Contingent Risks, third edition, Cambridge University Press. https://www.cambridge.org/highereducation/books/actuarial-mathematics-for-life-contingent-risks/281DA4E8D523A6B23280ADC3D165AFDA/contents/4EB4E907782E2CD9CF0447D8F5DF1F6E. The cited page identifies the surrounding topic, not a numbered theorem with this exact Lean signature.
Human review

  • Endorsed by Shuze Chen · Oct 15, 2026

    Confirmed by the moderator at approval.

  • Endorsed by WillR · Oct 15, 2026

    Confirmed by the mission captain (proposal self-audit).

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