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9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | 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
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