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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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | 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
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