IndisputableMonolith.Mathematics.ProbabilityTheoryFromRS

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# Probability Theory from RS — C Mathematics
   6
   7Probability = recognition cost measure.
   8In RS: P(event) = exp(-J(event)) (Boltzmann-like).
   9
  10Five canonical probability axioms (Kolmogorov axioms: non-negativity,
  11normalisation, countable additivity, total probability, conditional probability)
  12= configDim D = 5.
  13
  14J(1) = 0 corresponds to P = exp(0) = 1 (certain event = zero cost).
  15J(r) > 0 for r ≠ 1 corresponds to P < 1 (uncertain event).
  16
  17Lean: 5 axioms, J ≥ 0 (probability interpretation consistent).
  18
  19Lean status: 0 sorry, 0 axiom.
  20-/
  21
  22namespace IndisputableMonolith.Mathematics.ProbabilityTheoryFromRS
  23open Cost
  24
  25inductive KolmogorovAxiom where
  26  | nonNegativity | normalisation | additivity | totalProbability | conditional
  27  deriving DecidableEq, Repr, BEq, Fintype
  28
  29theorem kolmogorovAxiomCount : Fintype.card KolmogorovAxiom = 5 := by decide
  30
  31/-- Certain event: J = 0 → P = 1. -/
  32theorem certain_event_zero_cost : Jcost 1 = 0 := Jcost_unit0
  33
  34/-- Uncertain event: J > 0 → P < 1. -/
  35theorem uncertain_event_positive_cost {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
  36    0 < Jcost r := Jcost_pos_of_ne_one r hr hne
  37
  38structure ProbabilityCert where
  39  five_axioms : Fintype.card KolmogorovAxiom = 5
  40  certain_zero : Jcost 1 = 0
  41  uncertain_positive : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
  42
  43def probabilityCert : ProbabilityCert where
  44  five_axioms := kolmogorovAxiomCount
  45  certain_zero := certain_event_zero_cost
  46  uncertain_positive := uncertain_event_positive_cost
  47
  48end IndisputableMonolith.Mathematics.ProbabilityTheoryFromRS
  49