| import OAI.Computability.DegreeRigidity.SetModels.CountableElementaryHull |
| import OAI.Computability.DegreeRigidity.SetModels.InternalStructureCollapse |
| import OAI.Computability.DegreeRigidity.SetModels.ModelReals |
|
|
| namespace OAI |
|
|
|
|
| namespace TuringRigidity.ElementaryModel |
| open BoundedSetTheory TransitiveNameModel RelationCollapse |
| universe u |
|
|
| noncomputable def hullSet (a : ℕ → ZFSet.{u}) : ZFSet.{u} := by |
| letI : Countable (hull a) := (hull_countable a).to_subtype |
| letI : Small.{u} (hull a) := Countable.toSmall _ |
| exact ZFSet.range (fun x : hull a => x.val) |
|
|
| theorem mem_hullSet (a : ℕ → ZFSet.{u}) (x : ZFSet.{u}) : x ∈ hullSet a ↔ x ∈ hull a := by |
| classical |
| simp only [hullSet,ZFSet.mem_range] |
| exact ⟨fun ⟨y,hy⟩ => hy ▸ y.property,fun hx => ⟨⟨x,hx⟩,rfl⟩⟩ |
|
|
| instance hullSet_countable (a : ℕ → ZFSet.{u}) : Countable (Conditions (hullSet a)) := by |
| let : Countable (hull a) := (hull_countable a).to_subtype |
| exact Function.Injective.countable (f := fun x : Conditions (hullSet a) => |
| (⟨label _ x,(mem_hullSet a _).mp (label_mem _ x)⟩ : hull a)) |
| (fun x y h => label_injective (hullSet a) (congrArg (fun z : hull a => z.val) h)) |
|
|
| theorem hullSet_extensional (a : ℕ → ZFSet.{u}) : StructureExtensional (hullSet a) := by |
| intro x hx y hy he |
| let e := cons x (fun _ => y) |
| have hen : ∀ i, e i ∈ hull a := by |
| intro i; cases i <;> simp only [e,cons_zero,cons_succ] |
| · exact (mem_hullSet a x).mp hx |
| · exact (mem_hullSet a y).mp hy |
| let p := SentenceForm.all (SentenceForm.iff (.member 0 1) (.member 0 2)) |
| have hs : p.Sat (hull a) e := by |
| simp only [p,SentenceForm.sat_all,SentenceForm.sat_iff,SentenceForm.Sat, |
| cons_zero,cons_succ,e] |
| exact fun z hz => he z ((mem_hullSet a z).mpr hz) |
| have ht := (hull_elementary a p e hen).mp hs |
| simp only [p,SentenceForm.sat_all,SentenceForm.sat_iff,SentenceForm.Sat, |
| cons_zero,cons_succ,e,Set.mem_univ,forall_const] at ht |
| exact ZFSet.ext ht |
|
|
| noncomputable def collapsed (d : ZFSet.{u}) : ZFSet.{u} := |
| ZFSet.range (fun x : Conditions d => structureMap d (label d x)) |
|
|
| theorem mem_collapsed (d z : ZFSet.{u}) : |
| z ∈ collapsed d ↔ ∃ x ∈ d, z = structureMap d x := by |
| rw [collapsed,ZFSet.mem_range] |
| constructor |
| · rintro ⟨x,rfl⟩; exact ⟨_,label_mem _ x,rfl⟩ |
| · rintro ⟨x,hx,rfl⟩ |
| obtain ⟨i,rfl⟩ := label_surjective d hx |
| exact ⟨i,rfl⟩ |
|
|
| theorem collapsed_transitive (d : ZFSet.{u}) : Transitive (collapsed d) := by |
| intro x hx y hy |
| obtain ⟨z,_,rfl⟩ := (mem_collapsed d x).mp hx |
| obtain ⟨w,hw,_,rfl⟩ := (mem_value d (membershipRelation d) _ z y).mp hy |
| exact (mem_collapsed d _).mpr ⟨w,hw,rfl⟩ |
|
|
| instance collapsed_countable (d : ZFSet.{u}) [Countable (Conditions d)] : |
| Countable (Conditions (collapsed d)) := by |
| let f : Conditions (collapsed d) → Conditions d := fun z => Classical.choose |
| (ZFSet.mem_range.mp (show label (collapsed d) z ∈ collapsed d from label_mem _ z)) |
| have hf (z : Conditions (collapsed d)) : structureMap d (label d (f z)) = label _ z := |
| Classical.choose_spec (ZFSet.mem_range.mp (label_mem _ z)) |
| exact Function.Injective.countable (f := f) (fun x y h => |
| label_injective _ ((hf x).symm.trans ((congrArg (fun z => structureMap d (label d z)) h).trans (hf y)))) |
|
|
| theorem collapse_injective (d : ZFSet.{u}) (he : StructureExtensional d) |
| (x : ZFSet.{u}) (hx : x ∈ d) (y : ZFSet.{u}) (hy : y ∈ d) |
| (h : structureMap d x = structureMap d y) : x = y := |
| value_injective (membershipRelation_wellFounded d) (membershipRelation_extensional he) x hx y hy h |
|
|
| theorem collapse_membership (d : ZFSet.{u}) (he : StructureExtensional d) |
| (x : ZFSet.{u}) (hx : x ∈ d) (y : ZFSet.{u}) (hy : y ∈ d) : |
| structureMap d x ∈ structureMap d y ↔ x ∈ y := by |
| rw [structureMap,value_mem_iff (membershipRelation_wellFounded d) |
| (membershipRelation_extensional he) hx,membershipRelation_pair] |
| simp only [hx,hy,true_and] |
|
|
| theorem collapse_fixed (d x : ZFSet.{u}) (hx : x ∈ d) |
| (h : ∀ y ∈ x, y ∈ d ∧ structureMap d y = y) : structureMap d x = x := by |
| apply ZFSet.ext; intro y |
| rw [structureMap,mem_value] |
| constructor |
| · rintro ⟨z,_,hz,hy⟩ |
| have hzx := ((membershipRelation_pair d z x).mp hz).2.2 |
| change y = structureMap d z at hy |
| rw [(h z hzx).2] at hy |
| exact hy ▸ hzx |
| · intro hy |
| refine ⟨y,(h y hy).1,(membershipRelation_pair d y x).mpr ⟨(h y hy).1,hx,hy⟩,?_⟩ |
| exact (h y hy).2.symm |
|
|
| end TuringRigidity.ElementaryModel |
|
|
| end OAI |
|
|
