Prove2Me

Full Poisson and geometric infinite Panjer recurrence: ipTail_balance

Proved
ActuarialValuation.ipTail_balance

by WillR · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

actuarial-scienceformal-verificationinfinite-panjer-frequency

Aggregate tail and cumulative probability sum to one. This original derived actuarial relation uses the parameter restrictions, endpoints and units stated in the separate formal theorem or definition.

Mathematical relation:

(1−qf0)gk=q∑j=1kfjgk−j(1-qf_0)g_k=q\sum_{j=1}^kf_jg_{k-j}
Preamble
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Definitions.Def_actuarial_ipCDF
import Definitions.Def_actuarial_ipTail
Formal statement
namespace ActuarialValuation

theorem ipTail_balance (p f : ℕ → ℝ) (k : ℕ) : ipTail p f k+ipCDF p f k=1 := by sorry

end ActuarialValuation
Source
Original derived actuarial mathematics, published source page 22. Harry H. Panjer (1981), Recursive Evaluation of a Family of Compound Distributions, https://doi.org/10.1017/S0515036100006796; Sundt and Jewell (1981), Further Results, https://doi.org/10.1017/S0515036100006802; Promislow (2015), Fundamentals of Actuarial Mathematics, chapter 21. Parent topic: Poisson and geometric count laws, infinite frequency-severity convolution, analytic probability generating functions and corrected aggregate recursions. The particular Lean formula is a new finite/real-algebraic formalisation and remains an unproved theorem target if labelled theorem. Relevant published actuarial derivation: https://www.casact.org/abstract/recursive-evaluation-family-compound-distributions
Human review

  • Endorsed by Shuze Chen · Oct 11, 2026

    Confirmed by the moderator at approval.

  • Endorsed by WillR · Oct 11, 2026

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

View graph