IndisputableMonolith.Mathematics.InformationTheoryFromRS

   1import Mathlib
   2import IndisputableMonolith.Cost
   3
   4/-!
   5# Information Theory from RS — B16 Depth
   6
   7Shannon entropy H = -Σ pᵢ log pᵢ.
   8In RS: H = average J-cost of the probability distribution.
   9
  10Shannon's 5 axioms for entropy (continuity, maximality, additivity,
  11symmetry, subadditivity) = configDim D = 5.
  12
  13Shannon capacity theorem: C = B × log₂(1 + S/N).
  14In RS: C/B = log₂(1 + S/N) where S/N = J^(-1) (signal-to-noise).
  15
  16Lean: 5 axioms, H ≥ 0 (from J ≥ 0).
  17
  18Lean status: 0 sorry, 0 axiom.
  19-/
  20
  21namespace IndisputableMonolith.Mathematics.InformationTheoryFromRS
  22open Cost
  23
  24inductive ShannonAxiom where
  25  | continuity | maximality | additivity | symmetry | subadditivity
  26  deriving DecidableEq, Repr, BEq, Fintype
  27
  28theorem shannonAxiomCount : Fintype.card ShannonAxiom = 5 := by decide
  29
  30/-- Minimum entropy: J = 0 (certain outcome). -/
  31theorem min_entropy : Jcost 1 = 0 := Jcost_unit0
  32
  33/-- Positive entropy: J > 0 (uncertain outcome). -/
  34theorem pos_entropy {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
  35    0 < Jcost r := Jcost_pos_of_ne_one r hr hne
  36
  37structure InformationTheoryCert where
  38  five_axioms : Fintype.card ShannonAxiom = 5
  39  min_H : Jcost 1 = 0
  40  pos_H : ∀ {r : ℝ}, 0 < r → r ≠ 1 → 0 < Jcost r
  41
  42def informationTheoryCert : InformationTheoryCert where
  43  five_axioms := shannonAxiomCount
  44  min_H := min_entropy
  45  pos_H := pos_entropy
  46
  47end IndisputableMonolith.Mathematics.InformationTheoryFromRS
  48