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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 | import OAI.Computability.DegreeRigidity.SetModels.ElementaryCollapse
namespace OAI
namespace TuringRigidity.ElementaryModel
open BoundedSetTheory TransitiveNameModel RelationCollapse
universe u
theorem SentenceForm.collapse_sat (p : SentenceForm) (d : ZFSet.{u})
(hd : StructureExtensional d) (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ d) :
p.Sat (d : Set ZFSet) e ↔ p.Sat (collapsed d : Set ZFSet) (fun i => structureMap d (e i)) := by
induction p generalizing e with
| equal i j => exact ⟨congrArg (structureMap d),collapse_injective d hd _ (he i) _ (he j)⟩
| member i j => exact (collapse_membership d hd _ (he i) _ (he j)).symm
| conj p q ihp ihq => exact and_congr (ihp e he) (ihq e he)
| neg p ih => exact not_congr (ih e he)
| ex p ih =>
have hen (x : ZFSet.{u}) (hx : x ∈ d) : ∀ i, cons x e i ∈ d := by
intro i; cases i <;> simp [cons,he,hx]
constructor
· rintro ⟨x,hx,hp⟩
refine ⟨_,(mem_collapsed d _).mpr ⟨x,hx,rfl⟩,?_⟩
have h := (ih (cons x e) (hen x hx)).mp hp
rwa [map_cons] at h
· rintro ⟨y,hy,hp⟩
obtain ⟨x,hx,rfl⟩ := (mem_collapsed d y).mp hy
refine ⟨x,hx,(ih (cons x e) (hen x hx)).mpr ?_⟩
rwa [map_cons]
def Valid (M : ZFSet.{u}) : Prop :=
∀ p : SentenceForm, (∀ e : ℕ → ZFSet.{u}, p.Sat Set.univ e) →
∀ e : ℕ → ZFSet.{u}, (∀ i, e i ∈ M) → p.Sat (M : Set ZFSet) e
theorem collapsed_hull_valid (a : ℕ → ZFSet.{u}) : Valid (collapsed (hullSet a)) := by
intro p hp e he
choose f hf heq using fun i => (mem_collapsed (hullSet a) (e i)).mp (he i)
have hfe : (fun i => structureMap (hullSet a) (f i)) = e := by
funext i; exact (heq i).symm
have hh : (hullSet a : Set ZFSet) = hull a := by ext x; exact mem_hullSet a x
have hs : p.Sat (hullSet a : Set ZFSet) f := by
rw [hh]
exact (hull_elementary a p f (fun i => (mem_hullSet a _).mp (hf i))).mpr (hp f)
have ht := (p.collapse_sat (hullSet a) (hullSet_extensional a) f hf).mp hs
rwa [hfe] at ht
noncomputable def fromBounded : Formula → SentenceForm
| .equal i j => .equal i j
| .member i j => .member i j
| .conj p q => .conj (fromBounded p) (fromBounded q)
| .neg p => .neg (fromBounded p)
| .existsMem i p => .ex (.conj (.member 0 (i+1)) (fromBounded p))
theorem bounded_sat (p : Formula) (M : ZFSet.{u}) (e : ℕ → ZFSet.{u}) :
(fromBounded p).Sat (M : Set ZFSet) e ↔ p.Realize M e := by
induction p generalizing e with
| equal => rfl
| member => rfl
| conj p q ihp ihq => exact and_congr (ihp e) (ihq e)
| neg p ih => exact not_congr (ih e)
| existsMem i p ih =>
simp only [fromBounded,SentenceForm.Sat,Formula.Realize,cons_zero,cons_succ]
apply exists_congr; intro x
exact and_congr_right (fun _ => and_congr_right (fun _ => ih (cons x e)))
theorem bounded_univ (p : Formula) (e : ℕ → ZFSet.{u}) :
(fromBounded p).Sat Set.univ e ↔ p.Eval e := by
induction p generalizing e with
| equal => rfl
| member => rfl
| conj p q ihp ihq => exact and_congr (ihp e) (ihq e)
| neg p ih => exact not_congr (ih e)
| existsMem i p ih =>
simp only [fromBounded,SentenceForm.Sat,Formula.Eval,cons_zero,cons_succ,Set.mem_univ,true_and]
apply exists_congr; intro x
exact and_congr_right (fun _ => ih (cons x e))
noncomputable def fromSigma : SigmaFormula → SentenceForm
| .bounded p => fromBounded p
| .existsSet p => .ex (fromSigma p)
theorem sigma_sat (p : SigmaFormula) (M : ZFSet.{u}) (e : ℕ → ZFSet.{u}) :
(fromSigma p).Sat (M : Set ZFSet) e ↔ p.Realize M e := by
induction p generalizing e with
| bounded p => exact bounded_sat p M e
| existsSet p ih =>
apply exists_congr; intro x
exact and_congr_right (fun _ => ih (cons x e))
end TuringRigidity.ElementaryModel
end OAI
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