IndisputableMonolith.Foundation.UniversalForcing.Invariance.Universal

   1import IndisputableMonolith.Foundation.UniversalForcing.Invariance.TwoCases
   2import IndisputableMonolith.Foundation.UniversalForcing.ModularRealization
   3import IndisputableMonolith.Foundation.UniversalForcing.OrderRealization
   4import IndisputableMonolith.Foundation.UniversalForcing.CategoricalRealization
   5
   6/-!
   7  Universal.lean
   8
   9  General Universal Forcing theorem: every Law-of-Logic realization carries
  10  canonically equivalent forced arithmetic.
  11-/
  12
  13namespace IndisputableMonolith
  14namespace Foundation
  15namespace UniversalForcing
  16namespace Invariance
  17namespace Universal
  18
  19/-- Every realization's forced arithmetic is canonically equivalent to the
  20reference `LogicNat` Peano object. -/
  21noncomputable def arith_universal_initial' (R : LogicRealization) :
  22    (arithmeticOf R).peano.carrier ≃ ArithmeticFromLogic.LogicNat :=
  23  R.orbitEquivLogicNat
  24
  25/-- **Universal Forcing Meta-Theorem.**
  26
  27For any two Law-of-Logic realizations, the forced arithmetic objects are
  28canonically equivalent. -/
  29noncomputable def universal_forcing (R S : LogicRealization) :
  30    (arithmeticOf R).peano.carrier ≃ (arithmeticOf S).peano.carrier :=
  31  ArithmeticOf.equivOfInitial (arithmeticOf R) (arithmeticOf S)
  32
  33/-- Peano surface is available for every realization's forced arithmetic. -/
  34theorem universal_peano_surface (R : LogicRealization) :
  35    ArithmeticOf.PeanoSurface (arithmeticOf R) :=
  36  peano_surface R
  37
  38end Universal
  39end Invariance
  40end UniversalForcing
  41end Foundation
  42end IndisputableMonolith
  43