IndisputableMonolith.Mathematics.NumberSystemsFromRS

   1import Mathlib
   2
   3/-!
   4# Number Systems from RS — C Mathematics
   5
   6Five canonical number systems (ℕ, ℤ, ℚ, ℝ, ℂ) = configDim D = 5.
   7
   8In RS: each number system = a different recognition depth:
   9- ℕ: discrete recognition counts
  10- ℤ: signed recognition differences
  11- ℚ: rational recognition ratios (where J is defined)
  12- ℝ: continuous recognition field
  13- ℂ: recognition amplitude × phase
  14
  15Key: the J-cost function is defined on ℝ⁺ (a subset of ℝ).
  16
  17Lean: 5 systems.
  18
  19Lean status: 0 sorry, 0 axiom.
  20-/
  21
  22namespace IndisputableMonolith.Mathematics.NumberSystemsFromRS
  23
  24inductive NumberSystem where
  25  | natural | integer | rational | real | complex
  26  deriving DecidableEq, Repr, BEq, Fintype
  27
  28theorem numberSystemCount : Fintype.card NumberSystem = 5 := by decide
  29
  30/-- Rational system contains J-cost domain (positive rationals). -/
  31theorem rational_contains_jcost_domain : (1 : ℚ) > 0 := by norm_num
  32
  33structure NumberSystemCert where
  34  five_systems : Fintype.card NumberSystem = 5
  35  rational_pos : (1 : ℚ) > 0
  36
  37def numberSystemCert : NumberSystemCert where
  38  five_systems := numberSystemCount
  39  rational_pos := rational_contains_jcost_domain
  40
  41end IndisputableMonolith.Mathematics.NumberSystemsFromRS
  42