Prove2Me

First and second finite difference smoothness: whSecondDifference_shift

Proved
ActuarialValuation.whSecondDifference_shift

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

actuarial-scienceformal-verificationmortality-graduation-smoothing

Curvature penalties are invariant to uniform changes of mortality level. The full Lean declaration specifies the finite boundaries, exact units and any positivity, independence or regularity assumptions. This is an original derived Actuarial mathematical statement, not a claim of a numbered previously published theorem.

Mathematical relation:

Δ2(g+c)=Δ2g\Delta^2(g+c)=\Delta^2g
Preamble
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Definitions.Def_actuarial_whSecondDifference
import Definitions.Def_actuarial_whShiftedCurve
Formal statement
namespace ActuarialValuation

theorem whSecondDifference_shift (g : ℕ → ℝ) (i : ℕ) (s : ℝ) : whSecondDifference (fun j => whShiftedCurve g s j) i = whSecondDifference g i := by sorry

end ActuarialValuation
Source
Original derived actuarial mathematics, published source page 156. Casualty Actuarial Society (1968), Proceedings of the Casualty Actuarial Society Volume LV, Graduation section, https://www.casact.org/sites/default/files/database/proceed_proceed68_1968.pdf; Nels M. Valerius, Notes on Whittaker-Henderson Formula A, Proceedings of the Casualty Actuarial Society LIV (1967), p. 218, https://www.casact.org/sites/default/files/database/proceed_proceed67_67218.pdf. Parent topic: Actuarial graduation of crude mortality estimates, exposure-weighted data fidelity, first and second order finite difference roughness penalties. 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/sites/default/files/database/proceed_proceed68_1968.pdf
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).

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