openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /ConstructibleSuccessorCode.lean
| import OAI.Computability.DegreeRigidity.Constructibility.ConstructibleSatisfaction | |
| import OAI.Computability.DegreeRigidity.Constructibility.ConstructibleStages | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel | |
| universe u | |
| theorem tupleEnv_cons {n : ℕ} (x : ZFSet.{u}) (v : Fin n → ZFSet.{u}) : | |
| tupleEnv (Fin.cases x v) = cons x (tupleEnv v) := by | |
| funext i | |
| cases i with | |
| | zero => simp [tupleEnv,cons] | |
| | succ i => | |
| by_cases hi : i < n | |
| · simp only [tupleEnv,cons_succ,Nat.succ_lt_succ_iff,hi,dite_eq_left] | |
| rfl | |
| · simp [tupleEnv,cons_succ,hi] | |
| theorem definedSubset_record (A x : ZFSet.{u}) (p : SentenceForm) | |
| {n : ℕ} (v : Fin n → ZFSet.{u}) (hv : ∀ i, v i ∈ A) (hb : p.bound ≤ n+1) : | |
| x ∈ definedSubset A p v ↔ x ∈ A ∧ | |
| ZFSet.pair (natSet (Encodable.encode p)) (tupleCode (Fin.cases x v)) ∈ satisfactionSet A := by | |
| rw [mem_definedSubset] | |
| apply and_congr_right; intro hx | |
| rw [satisfactionSet_spec A p _ (fun i => Fin.cases hx hv i) hb,tupleEnv_cons] | |
| theorem definablePower_records (A S : ZFSet.{u}) : | |
| S ∈ definablePower A ↔ | |
| ∃ (n : ℕ) (p : SentenceForm) (v : Fin n → ZFSet.{u}), | |
| p.bound ≤ n+1 ∧ (∀ i, v i ∈ A) ∧ ∀ x, | |
| (x ∈ S ↔ x ∈ A ∧ ZFSet.pair (natSet (Encodable.encode p)) | |
| (tupleCode (Fin.cases x v)) ∈ satisfactionSet A) := by | |
| rw [mem_definablePower] | |
| constructor | |
| · rintro ⟨n,p,v,hb,hv,rfl⟩ | |
| exact ⟨n,p,v,hb,hv,fun x => definedSubset_record A x p v hv hb⟩ | |
| · rintro ⟨n,p,v,hb,hv,hS⟩ | |
| refine ⟨n,p,v,hb,hv,?_⟩ | |
| apply ZFSet.ext; intro x | |
| exact (hS x).trans (definedSubset_record A x p v hv hb).symm | |
| theorem level_succ_records (R S : ZFSet.{u}) (o : Ordinal.{u}) : | |
| S ∈ level R (o+1) ↔ | |
| ∃ (n : ℕ) (p : SentenceForm) (v : Fin n → ZFSet.{u}), | |
| p.bound ≤ n+1 ∧ (∀ i, v i ∈ level R o) ∧ ∀ x, | |
| (x ∈ S ↔ x ∈ level R o ∧ ZFSet.pair (natSet (Encodable.encode p)) | |
| (tupleCode (Fin.cases x v)) ∈ satisfactionSet (level R o)) := by | |
| rw [level_succ,definablePower_records] | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |