AbsoluteFloorWitness

formal source

  22open SpecifiabilityClosure
  23
  24/-- A fully named absolute-floor witness for a universe of discourse `K`. -/
  25structure AbsoluteFloorWitness (K : Type*) [Nonempty K] : Prop where
  26  meta_distinguishes : ∃ P Q : Prop, P ≠ Q
  27  nontrivial_specifiable : Nonempty (NontrivialSpecification K)
  28
  29/-- The absolute-floor witness forces bare distinguishability. -/
  30theorem bare_distinguishability_of_absolute_floor
  31    {K : Type*} [Nonempty K] (h : AbsoluteFloorWitness K) :
  32    ∃ x y : K, x ≠ y :=
  33  (distinguishability_iff_nontrivial_specifiability).mpr h.nontrivial_specifiable
  34
  35/-- Conversely, bare distinguishability supplies the non-trivial
  36specification part of the absolute-floor witness, while the meta-language
  37part is theorem-backed by Route A. -/
  38theorem absolute_floor_of_bare_distinguishability
  39    {K : Type*} [Nonempty K] (h : ∃ x y : K, x ≠ y) :
  40    AbsoluteFloorWitness K where
  41  meta_distinguishes := meta_language_distinguishes_props
  42  nontrivial_specifiable :=
  43    (distinguishability_iff_nontrivial_specifiability).mp h
  44
  45/-- Bare distinguishability and the absolute-floor witness are equivalent on
  46an inhabited carrier. -/
  47theorem absolute_floor_iff_bare_distinguishability
  48    {K : Type*} [Nonempty K] :
  49    AbsoluteFloorWitness K ↔ ∃ x y : K, x ≠ y :=
  50  ⟨bare_distinguishability_of_absolute_floor, absolute_floor_of_bare_distinguishability⟩
  51
  52/-- The minimal concrete carrier `Bool` realizes the absolute floor. -/
  53theorem bool_absolute_floor : AbsoluteFloorWitness Bool :=
  54  absolute_floor_of_bare_distinguishability ⟨false, true, bool_distinguishable⟩
  55