IndisputableMonolith.Foundation.UniversalForcing.Strict.Invariance

   1import IndisputableMonolith.Foundation.UniversalForcing.Strict.Categorical
   2
   3/-!
   4  Strict/Invariance.lean
   5
   6  Strict Universal Forcing theorem: native Law-of-Logic data determines a
   7  derived free orbit, and all derived free orbits are canonically equivalent.
   8-/
   9
  10namespace IndisputableMonolith
  11namespace Foundation
  12namespace UniversalForcing
  13namespace Strict
  14namespace Invariance
  15
  16/-- Every strict realization's derived forced arithmetic is canonically
  17equivalent to `LogicNat`. -/
  18def strict_arith_universal_initial (R : StrictLogicRealization) :
  19    (StrictLogicRealization.arith R).peano.carrier ≃ ArithmeticFromLogic.LogicNat :=
  20  (StrictLogicRealization.toLightweight R).orbitEquivLogicNat
  21
  22/-- **Strict Universal Forcing.**
  23
  24For any two strict Law-of-Logic realizations, the arithmetic derived from
  25their native law data is canonically equivalent. -/
  26noncomputable def strict_universal_forcing (R S : StrictLogicRealization) :
  27    (StrictLogicRealization.arith R).peano.carrier
  28      ≃ (StrictLogicRealization.arith S).peano.carrier :=
  29  ArithmeticOf.equivOfInitial (StrictLogicRealization.arith R)
  30    (StrictLogicRealization.arith S)
  31
  32/-- The Peano surface for every strict realization. -/
  33theorem strict_peano_surface (R : StrictLogicRealization) :
  34    ArithmeticOf.PeanoSurface (StrictLogicRealization.arith R) :=
  35  StrictLogicRealization.peano_surface R
  36
  37end Invariance
  38end Strict
  39end UniversalForcing
  40end Foundation
  41end IndisputableMonolith
  42