openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Syntax /MembershipInterpretation.lean
| 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 | |