openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /CanonicalOmegaSyntax.lean
| import OAI.Computability.DegreeRigidity.Syntax.SentenceInlining | |
| import OAI.Computability.DegreeRigidity.SetModels.SetModelAddition | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel | |
| universe u | |
| def inductiveSetFormula (w : ℕ) : Formula := | |
| .conj (.existsMem w (.empty 0)) | |
| (.allMem w (.existsMem (w+1) (.successor 0 1))) | |
| theorem inductiveSetFormula_spec (w : ℕ) (e : ℕ → ZFSet.{u}) : | |
| (inductiveSetFormula w).Eval e ↔ | |
| (∅ : ZFSet) ∈ e w ∧ ∀ x ∈ e w, insert x x ∈ e w := by | |
| simp only [inductiveSetFormula,Formula.Eval,Formula.eval_empty, | |
| Formula.eval_allMem,Formula.eval_successor,cons_zero,cons_succ] | |
| simp only [exists_eq_right] | |
| theorem omega_subset_inductive {W : ZFSet.{u}} | |
| (h : (∅ : ZFSet) ∈ W ∧ ∀ x ∈ W, insert x x ∈ W) : ZFSet.omega ⊆ W := by | |
| intro x hx | |
| obtain ⟨n,rfl⟩ := (mem_omega x).mp hx | |
| clear hx | |
| induction n with | |
| | zero => exact h.1 | |
| | succ n ih => exact h.2 _ ih | |
| noncomputable def canonicalOmega (w : ℕ) : SentenceForm := | |
| .conj (fromBounded (inductiveSetFormula w)) | |
| (SentenceForm.all (SentenceForm.imp (fromBounded (inductiveSetFormula 0)) | |
| (fromBounded (.subset (w+1) 0)))) | |
| theorem canonicalOmega_spec (M : ZFSet.{u}) (hM : Transitive M) | |
| (hω : ZFSet.omega.{u} ∈ M) (w : ℕ) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) : | |
| (canonicalOmega w).Sat (M : Set ZFSet) e ↔ e w = ZFSet.omega := by | |
| have habs (p : Formula) (x : ZFSet.{u}) (hx : x ∈ M) : | |
| (fromBounded p).Sat (M : Set ZFSet) (cons x e) ↔ p.Eval (cons x e) := by | |
| rw [bounded_sat,Formula.absolute p M hM _ | |
| (by intro i; cases i; exact hx; exact he _)] | |
| have hw : (∅ : ZFSet.{u}) ∈ ZFSet.omega ∧ | |
| ∀ x ∈ ZFSet.omega.{u}, insert x x ∈ ZFSet.omega := | |
| ⟨(mem_omega _).mpr ⟨0,rfl⟩,SetModelArithmetic.setSucc_omega⟩ | |
| change ((fromBounded (inductiveSetFormula w)).Sat (M : Set ZFSet) e ∧ | |
| (SentenceForm.all (SentenceForm.imp (fromBounded (inductiveSetFormula 0)) | |
| (fromBounded (.subset (w+1) 0)))).Sat (M : Set ZFSet) e) ↔ _ | |
| simp only [SentenceForm.sat_all,SentenceForm.sat_imp] | |
| rw [bounded_sat,Formula.absolute _ M hM e he,inductiveSetFormula_spec] | |
| have hmin : (∀ x ∈ (M : Set ZFSet), | |
| (fromBounded (inductiveSetFormula 0)).Sat (M : Set ZFSet) (cons x e) → | |
| (fromBounded (.subset (w+1) 0)).Sat (M : Set ZFSet) (cons x e)) ↔ | |
| ∀ x ∈ M, ((∅ : ZFSet) ∈ x ∧ ∀ y ∈ x, insert y y ∈ x) → e w ⊆ x := by | |
| apply forall_congr'; intro x | |
| apply forall_congr'; intro hx | |
| rw [habs _ x hx,habs _ x hx,inductiveSetFormula_spec,Formula.eval_subset] | |
| rfl | |
| rw [hmin] | |
| constructor | |
| · rintro ⟨hi,hminimal⟩ | |
| exact ZFSet.ext fun x => ⟨fun hx => hminimal _ hω hw hx,fun hx => omega_subset_inductive hi hx⟩ | |
| · intro h | |
| rw [h] | |
| exact ⟨hw,fun _ _ hi => omega_subset_inductive hi⟩ | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |