Prove2Me

Last-survivor inclusion–exclusion

Proved
actuarial_d_last_inclusion

by WillR · Oct 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

actuarial-sciencejoint-lifelife-contingencymultiple-lives

Last-survivor inclusion–exclusion. Single-life survivor indicators count both-alive states twice. The settlement date, survivor entitlement, financial discount and relevant joint-law assumptions are those declared in the formal model.

Mathematical relation:

Ilast=Ix+Iy−IbothI_{\text{last}}=I_x+I_y-I_{\text{both}}
Preamble
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Definitions.Def_actuarial_d_alive
import Definitions.Def_actuarial_d_dead
import Definitions.Def_actuarial_d_both
import Definitions.Def_actuarial_d_onlyX
import Definitions.Def_actuarial_d_onlyY
import Definitions.Def_actuarial_d_neither
import Definitions.Def_actuarial_d_last

open ActuarialJointD
Formal statement
theorem actuarial_d_last_inclusion (i j t : ℕ) : last i j t = alive i t + alive j t - both i j t := by sorry
Source
Originally derived two-life finite-grid definition or theorem, motivated by S. David Promislow (2015), Fundamentals of Actuarial Mathematics, third edition, Chapter 10, Section 10.2, p. 144. The published reference supplies actuarial context, while the formal statement and joint-law conventions are new. https://www.oreilly.com/library/view/fundamentals-of-actuarial/9781118782521/
Human review

  • Endorsed by Shuze Chen · Oct 16, 2026

    Confirmed by the moderator at approval.

  • Endorsed by WillR · Oct 16, 2026

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

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