Prove2Me

Positive proportional hazards and cumulative survival: cxSurvival_start

Proved
ActuarialValuation.cxSurvival_start

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

actuarial-sciencecox-survival-estimationformal-verification

Survival begins at certainty when accumulated hazard is zero. This original derived actuarial relation uses the parameter restrictions, endpoints and units stated in the separate formal theorem or definition.

Mathematical relation:

λ(t∣z)=λ0(t)eβz,S(t∣z)=e−Λ0(t)eβz\lambda(t\mid z)=\lambda_0(t)e^{\beta z},\quad S(t\mid z)=e^{-\Lambda_0(t)e^{\beta z}}
Preamble
import Mathlib.Data.Real.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Definitions.Def_actuarial_cxSurvival
Formal statement
namespace ActuarialValuation

theorem cxSurvival_start (beta z : ℝ) : cxSurvival 0 beta z=1 := by sorry

end ActuarialValuation
Source
Original derived actuarial mathematics, published source page 190. D. R. Cox (1972), Regression Models and Life-Tables, Journal of the Royal Statistical Society B 34:187–202, https://doi.org/10.1111/j.2517-6161.1972.tb00899.x; S. David Promislow (2015), Fundamentals of Actuarial Mathematics, lifetime failure-time models. Parent topic: Multiplicative baseline hazard, finite at-risk sets, partial likelihood, censored selection, score derivatives and survival prediction. 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://doi.org/10.1111/j.2517-6161.1972.tb00899.x
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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