File size: 3,273 Bytes
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 | import OAI.Computability.DegreeRigidity.SetModels.ElementaryLanguage
namespace OAI
namespace TuringRigidity.RelativeConstructible
open ElementaryModel BoundedSetTheory TransitiveNameModel
universe u
noncomputable def tupleEnv {n : ℕ} (v : Fin n → ZFSet.{u}) (i : ℕ) : ZFSet.{u} :=
if h : i < n then v ⟨i,h⟩ else ∅
theorem tupleEnv_lt {n : ℕ} (v : Fin n → ZFSet.{u}) (i : Fin n) :
tupleEnv v i = v i := by simp [tupleEnv]
noncomputable def definedSubset (A : ZFSet.{u}) (p : SentenceForm)
{n : ℕ} (v : Fin n → ZFSet.{u}) : ZFSet.{u} :=
A.sep (fun x => p.Sat (A : Set ZFSet) (cons x (tupleEnv v)))
theorem mem_definedSubset (A x : ZFSet.{u}) (p : SentenceForm)
{n : ℕ} (v : Fin n → ZFSet.{u}) :
x ∈ definedSubset A p v ↔ x ∈ A ∧ p.Sat (A : Set ZFSet) (cons x (tupleEnv v)) :=
ZFSet.mem_sep
def DefinableOver (A S : ZFSet.{u}) : Prop :=
∃ (n : ℕ) (p : SentenceForm) (v : Fin n → ZFSet.{u}),
p.bound ≤ n + 1 ∧ (∀ i, v i ∈ A) ∧ S = definedSubset A p v
noncomputable def definablePower (A : ZFSet.{u}) : ZFSet.{u} :=
(ZFSet.powerset A).sep (DefinableOver A)
theorem definableOver_subset {A S : ZFSet.{u}} (h : DefinableOver A S) : S ⊆ A := by
obtain ⟨n,p,v,_,_,rfl⟩ := h
exact fun _ hx => (mem_definedSubset A _ p v).mp hx |>.1
theorem mem_definablePower (A S : ZFSet.{u}) :
S ∈ definablePower A ↔ DefinableOver A S := by
rw [definablePower,ZFSet.mem_sep]
exact ⟨And.right,fun h => ⟨ZFSet.mem_powerset.mpr (definableOver_subset h),h⟩⟩
theorem subset_of_mem_definablePower {A S : ZFSet.{u}}
(h : S ∈ definablePower A) : S ⊆ A :=
definableOver_subset ((mem_definablePower A S).mp h)
theorem separation_mem_definablePower (A : ZFSet.{u}) (p : SentenceForm)
(e : ℕ → ZFSet.{u}) (he : ∀ i, i < p.bound → e i ∈ A) :
A.sep (fun x => p.Sat (A : Set ZFSet) (cons x e)) ∈ definablePower A := by
apply (mem_definablePower A _).mpr
let v : Fin p.bound → ZFSet.{u} := fun i => e i
refine ⟨p.bound,p,v,by omega,fun i => he i i.isLt,?_⟩
apply ZFSet.ext; intro x
rw [ZFSet.mem_sep,mem_definedSubset]
apply and_congr_right; intro _
apply p.finite_support
intro i hi
cases i with
| zero => rfl
| succ i =>
change e i = tupleEnv v i
simp [tupleEnv,v,show i < p.bound by omega]
theorem self_mem_definablePower (A : ZFSet.{u}) : A ∈ definablePower A := by
apply (mem_definablePower A A).mpr
refine ⟨0,.equal 0 0,Fin.elim0,by decide,fun i => Fin.elim0 i,?_⟩
apply ZFSet.ext; intro x
simp [mem_definedSubset,SentenceForm.Sat]
theorem subset_definablePower {A : ZFSet.{u}} (hA : Transitive A) : A ⊆ definablePower A := by
intro S hS
apply (mem_definablePower A S).mpr
refine ⟨1,.member 0 1,fun _ => S,by decide,fun _ => hS,?_⟩
apply ZFSet.ext; intro x
simp only [mem_definedSubset,SentenceForm.Sat,cons_zero,cons_succ]
simp only [tupleEnv,show 0 < 1 by decide,dite_eq_left]
exact ⟨fun hx => ⟨hA S hS x hx,hx⟩,And.right⟩
theorem definablePower_transitive {A : ZFSet.{u}} (hA : Transitive A) :
Transitive (definablePower A) := by
intro S hS x hx
exact subset_definablePower hA (subset_of_mem_definablePower hS hx)
end TuringRigidity.RelativeConstructible
end OAI
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