openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /CanonicalRealsSyntax.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.CanonicalOmegaSyntax | |
| import OAI.Computability.DegreeRigidity.Representation.AmbientDegreeSyntax | |
| import OAI.Computability.DegreeRigidity.Constructibility.ConstructibleGroundReals | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel | |
| universe u | |
| noncomputable def canonicalReals (r : ℕ) : SentenceForm := | |
| .ex (.conj (canonicalOmega 0) (FullSetForcing.ownPower 0 (r+1))) | |
| theorem canonicalReals_spec (M : ZFSet.{u}) (hM : Transitive M) | |
| (hw : ZFSet.omega.{u} ∈ M) | |
| (r : ℕ) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) : | |
| (canonicalReals r).Sat (M : Set ZFSet) e ↔ e r = groundReals M := by | |
| have hc (w : ZFSet.{u}) (hwM : w ∈ M) : ∀ i, cons w e i ∈ M := by | |
| intro i; cases i; exact hwM; exact he _ | |
| have hpower : (FullSetForcing.ownPower 0 (r+1)).Sat (M : Set ZFSet) (cons ZFSet.omega e) ↔ | |
| e r = groundReals M := by | |
| rw [FullSetForcing.ownPower_spec M hM 0 (r+1) _ (hc _ hw)] | |
| simp only [cons_succ,cons_zero] | |
| constructor | |
| · intro h | |
| exact ZFSet.ext fun x => (h x).trans (mem_groundReals M x).symm | |
| · intro h x | |
| rw [h,mem_groundReals] | |
| change (∃ w ∈ M, (canonicalOmega 0).Sat (M : Set ZFSet) (cons w e) ∧ | |
| (FullSetForcing.ownPower 0 (r+1)).Sat (M : Set ZFSet) (cons w e)) ↔ _ | |
| constructor | |
| · rintro ⟨w,hwM,hwS,hp⟩ | |
| obtain rfl := (canonicalOmega_spec M hM hw 0 _ (hc w hwM)).mp hwS | |
| exact hpower.mp hp | |
| · intro h | |
| exact ⟨_,hw,(canonicalOmega_spec M hM hw 0 _ (hc _ hw)).mpr rfl,hpower.mpr h⟩ | |
| theorem groundReals_between (M N : ZFSet.{u}) (hN : Transitive N) | |
| (hNM : N ⊆ M) (hR : groundReals M ∈ N) : groundReals N = groundReals M := by | |
| apply ZFSet.ext; intro x | |
| rw [mem_groundReals,mem_groundReals] | |
| exact ⟨fun hx => ⟨hNM hx.1,hx.2⟩, | |
| fun hx => ⟨hN _ hR x ((mem_groundReals M x).mpr hx),hx.2⟩⟩ | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |