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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | import OAI.Computability.DegreeRigidity.SetModels.ElementaryDefinableSets
import OAI.Computability.DegreeRigidity.Representation.AmbientDegreeSyntax
import OAI.Computability.DegreeRigidity.SetModels.InternalCollapseConditions
namespace OAI
namespace TuringRigidity.ElementaryModel
open BoundedSetTheory TransitiveNameModel RelationCollapse SentenceForm
open FullSetForcing InternalCollapse
universe u
theorem definedSet_univ (s : ℕ) (p : Formula) (e : ℕ → ZFSet.{u}) :
(definedSet s p).Sat Set.univ e ↔ ∀ x, x ∈ e s ↔ p.Eval (cons x e) := by
simp only [definedSet,sat_all,sat_iff,Sat,bounded_univ,cons_zero,cons_succ,
Set.mem_univ,forall_const]
theorem collapse_bounded_eval (a : ℕ → ZFSet.{u}) (p : Formula)
(e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ hullSet a) :
p.Eval (fun i => structureMap (hullSet a) (e i)) ↔ p.Eval e := by
have h := collapsed_hull_elementary a (fromBounded p) e he
rw [bounded_sat,Formula.absolute _ _ (collapsed_transitive _) _
(fun i => (mem_collapsed _ _).mpr ⟨e i,he i,rfl⟩),bounded_univ] at h
exact h
noncomputable def ambientConditions (A : ZFSet.{u}) : ZFSet.{u} :=
ZFSet.sep (Prefix A) (ZFSet.powerset (ZFSet.prod ZFSet.omega A))
theorem mem_ambientConditions (A p : ZFSet.{u}) : p ∈ ambientConditions A ↔ Prefix A p := by
rw [ambientConditions,ZFSet.mem_sep]
exact ⟨And.right,fun h => ⟨ZFSet.mem_powerset.mpr (prefix_subset h),h⟩⟩
theorem ambientConditions_def (A c : ZFSet.{u}) :
(definedSet 0 (prefixFormula 3 2 0)).Sat Set.univ
(cons c (cons A (fun _ => ZFSet.omega))) ↔ c = ambientConditions A := by
rw [definedSet_univ]
have hs (p : ZFSet.{u}) := eval_prefixFormula 3 2 0
(cons p (cons c (cons A (fun _ => ZFSet.omega)))) rfl
simp only [hs,cons_zero,cons_succ]
exact ⟨fun h => ZFSet.ext (fun p => (h p).trans (mem_ambientConditions A p).symm),
fun h p => h ▸ mem_ambientConditions A p⟩
theorem ambientConditions_mem_hull (a : ℕ → ZFSet.{u})
(hω : ZFSet.omega ∈ hullSet a) {A : ZFSet.{u}} (hA : A ∈ hullSet a) :
ambientConditions A ∈ hullSet a := by
exact unique_definable_mem a (definedSet 0 (prefixFormula 3 2 0))
(cons A (fun _ => ZFSet.omega)) (by intro i; cases i <;> assumption)
_ ((ambientConditions_def A _).mpr rfl) (fun c hc => (ambientConditions_def A c).mp hc)
theorem collapse_ambientConditions (a : ℕ → ZFSet.{u})
(hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a)
{A : ZFSet.{u}} (hA : A ∈ hullSet a) :
∀ p, p ∈ structureMap (hullSet a) (ambientConditions A) ↔
p ∈ collapsed (hullSet a) ∧ Prefix (structureMap (hullSet a) A) p := by
let e := cons (ambientConditions A) (cons A (fun _ => ZFSet.omega))
have hc := ambientConditions_mem_hull a hω hA
have he : ∀ i, e i ∈ hullSet a := by
intro i; rcases i with _|_|i
exact hc
exact hA
exact hω
have ht := (collapsed_hull_elementary a (definedSet 0 (prefixFormula 3 2 0)) e he).mpr
((ambientConditions_def A _).mpr rfl)
have hm := collapsed_transitive (hullSet a)
have hei : ∀ i, structureMap (hullSet a) (e i) ∈ collapsed (hullSet a) :=
fun i => (mem_collapsed _ _).mpr ⟨e i,he i,rfl⟩
have h := (definedSet_spec _ hm 0 _ _ hei).mp ht
have hω' := hull_omega_fixed a hn hω
have hs (p : ZFSet.{u}) (hp : p ∈ collapsed (hullSet a)) :
p ∈ structureMap (hullSet a) (ambientConditions A) ↔
Prefix (structureMap (hullSet a) A) p :=
(h p hp).trans (eval_prefixFormula 3 2 0
(cons p (fun i => structureMap (hullSet a) (e i))) hω')
intro p
exact ⟨fun hp => ⟨hm _ (hei 0) _ hp,(hs p (hm _ (hei 0) _ hp)).mp hp⟩,
fun hp => (hs p hp.1).mpr hp.2⟩
def orderMember (c z : ℕ) : Formula :=
.existsMem c (.existsMem (c+1) (.conj (.orderedPair (z+2) 1 0) (.subset 0 1)))
theorem eval_orderMember (c z : ℕ) (e : ℕ → ZFSet.{u}) :
(orderMember c z).Eval e ↔ e z ∈ orderSet (e c) := by
simp only [orderMember,Formula.Eval,Formula.eval_orderedPair,Formula.eval_subset,
cons_zero,cons_succ,orderSet,ZFSet.mem_sep]
exact ⟨fun ⟨p,hp,q,hq,hz,hpq⟩ =>
⟨hz ▸ ZFSet.mem_prod.mpr ⟨p,hp,q,hq,rfl⟩,p,hp,q,hq,hz,hpq⟩,And.right⟩
theorem orderSet_mem_hull (a : ℕ → ZFSet.{u}) {c : ZFSet.{u}} (hc : c ∈ hullSet a) :
orderSet c ∈ hullSet a := by
have hs (x : ZFSet.{u}) : (definedSet 0 (orderMember 2 0)).Sat Set.univ
(cons x (fun _ => c)) ↔ x = orderSet c := by
rw [definedSet_univ]
simp only [eval_orderMember,cons_zero,cons_succ]
exact ZFSet.ext_iff.symm
exact unique_definable_mem a (definedSet 0 (orderMember 2 0))
(fun _ => c) (fun _ => hc) _ ((hs _).mpr rfl) (fun x hx => (hs x).mp hx)
theorem collapse_orderSet (a : ℕ → ZFSet.{u}) {c : ZFSet.{u}} (hc : c ∈ hullSet a) :
structureMap (hullSet a) (orderSet c) = orderSet (structureMap (hullSet a) c) := by
let e := cons (orderSet c) (fun _ => c)
have he : ∀ i, e i ∈ hullSet a := by
intro i; cases i
exact orderSet_mem_hull a hc
exact hc
have hi : ∀ i, structureMap (hullSet a) (e i) ∈ collapsed (hullSet a) :=
fun i => (mem_collapsed _ _).mpr ⟨e i,he i,rfl⟩
have hs : (definedSet 0 (orderMember 2 0)).Sat Set.univ e := by
rw [definedSet_univ]
intro x
exact (eval_orderMember 2 0 (cons x e)).symm
have ht := (collapsed_hull_elementary a _ e he).mpr hs
exact (definedSet_unique _ (collapsed_transitive _) 0 _ _ hi
(orderSet (structureMap (hullSet a) c))
(orderSet_mem _ (collapsed_transitive _) (collapsed_hull_sourceT a) (hi 1))
(fun x _ => eval_orderMember 2 0 _)).mp ht
theorem collapse_selected_condition (a : ℕ → ZFSet.{u})
(hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a)
{A p : ZFSet.{u}} (hA : A ∈ hullSet a) (hp : p ∈ hullSet a)
(hpc : p ∈ ambientConditions A) :
structureMap (hullSet a) p ∈ structureMap (hullSet a) (ambientConditions A) ∧
Prefix (structureMap (hullSet a) A) (structureMap (hullSet a) p) := by
have h := (collapse_membership _ (hullSet_extensional a) _ hp _
(ambientConditions_mem_hull a hω hA)).mpr hpc
exact ⟨h,((collapse_ambientConditions a hn hω hA _).mp h).2⟩
end TuringRigidity.ElementaryModel
end OAI
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