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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 | import OAI.Computability.DegreeRigidity.SetModels.ExtensionBasicAxioms
import OAI.Computability.DegreeRigidity.SetModels.RecursiveNameSymmetry
namespace OAI
namespace TuringRigidity.BoundedSetTheory
universe u
def Formula.toMembership : Formula → RecursiveNames.Formula
| .equal i j => .equal i j
| .member i j => .member i j
| .conj φ ψ => .conj φ.toMembership ψ.toMembership
| .neg φ => .neg φ.toMembership
| .existsMem i φ => .existsSet (.conj (.member 0 (i+1)) φ.toMembership)
theorem Formula.realize_toMembership (φ : Formula) (M : ZFSet.{u})
(env : ℕ → ZFSet.{u}) :
φ.toMembership.Realize (M : Set ZFSet.{u}) env ↔ φ.Realize M env := by
induction φ generalizing env with
| equal => rfl
| member => rfl
| conj φ ψ ihφ ihψ => exact and_congr (ihφ env) (ihψ env)
| neg φ ih => exact not_congr (ih env)
| existsMem i φ ih =>
have hcons (x : ZFSet.{u}) : (fun n => if n = 0 then x else env (n-1)) = cons x env := by
funext n
cases n <;> simp [cons]
simp [toMembership,RecursiveNames.Formula.Realize,Formula.Realize,hcons,ih]
end TuringRigidity.BoundedSetTheory
end OAI
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