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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 99 100 101 102 103 104 105 | import OAI.Computability.DegreeRigidity.SetModels.ElementaryRealModel
import OAI.Computability.DegreeRigidity.Representation.NativeModelCompatibility
namespace OAI
namespace TuringRigidity.ElementaryModel
open BoundedSetTheory TransitiveNameModel RelationCollapse SentenceForm SetDegreeDecoding
universe u
theorem unique_definable_mem (a : ℕ → ZFSet.{u}) (p : SentenceForm) (e : ℕ → ZFSet.{u})
(he : ∀ i, e i ∈ hullSet a) (b : ZFSet.{u}) (hb : p.Sat Set.univ (cons b e))
(hu : ∀ x, p.Sat Set.univ (cons x e) → x = b) : b ∈ hullSet a := by
have hen : ∀ i, e i ∈ hull a := fun i => (mem_hullSet a _).mp (he i)
have hs := (hull_elementary a (.ex p) e hen).mpr ⟨b,Set.mem_univ _,hb⟩
obtain ⟨x,hx,hp⟩ := hs
have ht := (hull_elementary a p (cons x e)
(by intro i; cases i <;> simp [cons,hen,hx])).mp hp
exact hu x ht ▸ (mem_hullSet a x).mpr hx
theorem unorderedPair_mem_hull (a : ℕ → ZFSet.{u}) {x y : ZFSet.{u}}
(hx : x ∈ hullSet a) (hy : y ∈ hullSet a) : ({x,y} : ZFSet.{u}) ∈ hullSet a := by
apply unique_definable_mem a (fromBounded (.unorderedPair 0 1 2)) (cons x (fun _ => y))
(by intro i; cases i <;> assumption)
· exact (bounded_univ _ _).mpr ((Formula.eval_unorderedPair _ _ _ _).mpr rfl)
· intro z hz
exact (Formula.eval_unorderedPair _ _ _ _).mp ((bounded_univ _ _).mp hz)
theorem orderedPair_mem_hull (a : ℕ → ZFSet.{u}) {x y : ZFSet.{u}}
(hx : x ∈ hullSet a) (hy : y ∈ hullSet a) : ZFSet.pair x y ∈ hullSet a := by
have hxx := unorderedPair_mem_hull a hx hx
have hxy := unorderedPair_mem_hull a hx hy
simpa [ZFSet.pair] using unorderedPair_mem_hull a hxx hxy
theorem collapse_unorderedPair (a : ℕ → ZFSet.{u}) {x y : ZFSet.{u}}
(hx : x ∈ hullSet a) (hy : y ∈ hullSet a) :
structureMap (hullSet a) ({x,y} : ZFSet.{u}) =
({structureMap (hullSet a) x,structureMap (hullSet a) y} : ZFSet.{u}) := by
apply ZFSet.ext; intro z
rw [structureMap,mem_value,ZFSet.mem_pair]
constructor
· rintro ⟨w,_,hw,hzw⟩
have hwp := ((membershipRelation_pair _ _ _).mp hw).2.2
rcases ZFSet.mem_pair.mp hwp with rfl|rfl
· exact Or.inl hzw
· exact Or.inr hzw
· intro hz
have hp := unorderedPair_mem_hull a hx hy
rcases hz with hxz|hyz
· exact ⟨x,hx,(membershipRelation_pair _ _ _).mpr ⟨hx,hp,ZFSet.mem_pair.mpr (Or.inl rfl)⟩,hxz⟩
· exact ⟨y,hy,(membershipRelation_pair _ _ _).mpr ⟨hy,hp,ZFSet.mem_pair.mpr (Or.inr rfl)⟩,hyz⟩
theorem collapse_orderedPair (a : ℕ → ZFSet.{u}) {x y : ZFSet.{u}}
(hx : x ∈ hullSet a) (hy : y ∈ hullSet a) :
structureMap (hullSet a) (ZFSet.pair x y) =
ZFSet.pair (structureMap (hullSet a) x) (structureMap (hullSet a) y) := by
have hxx := unorderedPair_mem_hull a hx hx
have hxy := unorderedPair_mem_hull a hx hy
have hh := collapse_unorderedPair a hxx hxy
rw [collapse_unorderedPair a hx hx,collapse_unorderedPair a hx hy] at hh
simpa [ZFSet.pair] using hh
noncomputable def ambientRealSet : ZFSet.{u} := ZFSet.powerset ZFSet.omega
noncomputable def ambientGraphBound : ZFSet.{u} := ZFSet.powerset (ZFSet.prod ZFSet.omega ZFSet.omega)
theorem ambient_real_decode (x : ZFSet.{u}) (hx : x ∈ ambientRealSet) :
∃ B : Oracle, realCode B = x :=
⟨decodeReal x,realCode_decodeReal (ZFSet.mem_powerset.mp hx)⟩
theorem ambient_graph_bound (f : ℕ → ℕ) (k : ℕ) : finiteNaturalGraph.{u} f k ∈ ambientGraphBound := by
apply ZFSet.mem_powerset.mpr
intro x hx
obtain ⟨n,_,rfl⟩ := (mem_finiteNaturalGraph f k x).mp hx
exact ZFSet.mem_prod.mpr ⟨natSet n,(mem_omega _).mpr ⟨n,rfl⟩,
natSet (f n),(mem_omega _).mpr ⟨f n,rfl⟩,rfl⟩
theorem ambient_degreeCode (A : Oracle) : degreeCode ambientRealSet.{u} A = degreeSet (degree A) := by
apply ZFSet.ext; intro x
rw [degreeCode,ZFSet.mem_sep,mem_degreeSet]
exact ⟨And.right,fun ⟨B,hB,hBA⟩ =>
⟨hB.symm ▸ ZFSet.mem_powerset.mpr (realCode_subset B),B,hB,hBA⟩⟩
theorem degreeSet_mem_hull (a : ℕ → ZFSet.{u})
(hω : ZFSet.omega ∈ hullSet a) (h0 : natSet 0 ∈ hullSet a)
(hR : ambientRealSet ∈ hullSet a) (hQ : ambientGraphBound ∈ hullSet a)
{A : Oracle} (hA : realCode A ∈ hullSet a) : degreeSet (degree A) ∈ hullSet a := by
obtain ⟨φ,hφ⟩ := degree_equality_bounded.{u}
let e := cons ZFSet.omega (cons ambientGraphBound (cons (natSet 0) (cons ambientRealSet (fun _ => realCode A))))
have he : ∀ i, e i ∈ hullSet a := by
intro i
rcases i with _|i; exact hω
rcases i with _|i; exact hQ
rcases i with _|i; exact h0
rcases i with _|i; exact hR
exact hA
let p := fromBounded (Formula.degreeSet φ 1 2 3 4 5 0)
have hp (x : ZFSet.{u}) : p.Sat Set.univ (cons x e) ↔ x = degreeSet (degree A) := by
rw [bounded_univ,Formula.degreeSet_spec hφ 1 2 3 4 5 0 _ rfl ambient_graph_bound rfl ambient_real_decode A rfl]
simp only [cons_zero,cons_succ,e,ambient_degreeCode]
exact unique_definable_mem a p e he _ ((hp _).mpr rfl) (fun x hx => (hp x).mp hx)
end TuringRigidity.ElementaryModel
end OAI
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