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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 | import OAI.Computability.DegreeRigidity.SetModels.InternalCheckedValue
import OAI.Computability.DegreeRigidity.CohenForcing.FullSetForcing
namespace OAI
namespace TuringRigidity.FullSetForcing
open RecursiveNames TransitiveNameModel BoundedSetTheory AtomicForcing CountableForcing
open ElementaryModel SentenceForm
universe u
variable {c : ZFSet.{u}} [Preorder (Conditions c)] [Top (Conditions c)]
noncomputable def checkedMembershipFormula : SentenceForm :=
.ex (.conj (checkedValueFormula.rename (fun i => match i with | 0 => 0 | 1 => 1 | _ => 6))
(code (.member 0 1) 2 3 4 (fun i => match i with | 0 => 0 | _ => 5)))
theorem checkedMembership_spec (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M)
{o : ZFSet.{u}} (ho : ∀ r s : Conditions c,
ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(t : Name (Conditions c)) (p : Conditions c) (x : ZFSet.{u})
(e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M)
(h0 : e 0 = x) (h1 : e 1 = c) (h2 : e 2 = o) (h3 : e 3 = label c p)
(h4 : e 4 = t.encode (label c)) (h5 : e 5 = label c ⊤) :
checkedMembershipFormula.Sat (M : Set ZFSet) e ↔ MemForces (Name.check x) t p := by
let v : ℕ → ℕ := fun i => match i with | 0 => 0 | _ => 5
let k : ℕ → ℕ := fun i => match i with | 0 => 0 | 1 => 1 | _ => 6
let names : ℕ → Name (Conditions c) := fun i => match i with | 0 => Name.check x | _ => t
have hc : c ∈ M := h1 ▸ he 1
have hx : x ∈ M := h0 ▸ he 0
have hcheck : (Name.check x : Name (Conditions c)).encode (label c) ∈ M :=
encoded_check_mem M c hM hT.pairing hT.union hT.powerSet
hT.separation.finitePrefix.bounded hT.replacement.finitePrefix hT.infinity hc hx
have henv (z : ZFSet.{u}) (hz : z ∈ M) : ∀ i, cons z e i ∈ M := by
intro i; cases i; exact hz; exact he _
have checked (z : ZFSet.{u}) (hz : z ∈ M) :
(checkedValueFormula.rename k).Sat (M : Set ZFSet) (cons z e) ↔
z = (Name.check x : Name (Conditions c)).encode (label c) := by
rw [sat_rename,checkedValue_spec M hM hT _ (fun i => henv z hz (k i))]
simp only [k,cons_zero,cons_succ,h0,h5,encode_check]
have forcing (z : ZFSet.{u}) (hz : z ∈ M)
(hzv : z = (Name.check x : Name (Conditions c)).encode (label c)) :
(code (.member 0 1) 2 3 4 v).Sat (M : Set ZFSet) (cons z e) ↔
MemForces (Name.check x) t p := by
exact realize_code M hM hT ho (.member 0 1) names p (cons z e) (henv z hz)
2 3 4 v h1 h2 h3 (by intro i; cases i; exact hzv; exact h4)
change (∃ z ∈ M, (checkedValueFormula.rename k).Sat _ (cons z e) ∧ _) ↔ _
constructor
· rintro ⟨z,hz,hzv,hf⟩
exact (forcing z hz ((checked z hz).mp hzv)).mp hf
· intro hf
exact ⟨_,hcheck,(checked _ hcheck).mpr rfl,(forcing _ hcheck rfl).mpr hf⟩
noncomputable def checkedValueSigma : SigmaFormula :=
.existsSet (.existsSet (.existsSet (.bounded (.conj (CheckFormula.graph 2 1 0 5)
(.conj (.member 4 2) (.conj (.member 3 1) (.pairMem 4 3 0)))))))
noncomputable def checkedMembershipSigma : SigmaFormula :=
.existsSet (.conj
(checkedValueSigma.rename (fun i => match i with | 0 => 0 | 1 => 1 | _ => 6))
(BoundedForcing.SigmaCode.forcing (.bounded (.member 0 1)) 2 3 4
(fun i => match i with | 0 => 0 | _ => 5)))
theorem checkedMembershipSigma_spec (M : ZFSet.{u}) (e : ℕ → ZFSet.{u}) :
checkedMembershipSigma.Realize M e ↔ checkedMembershipFormula.Sat (M : Set ZFSet) e := by
change (∃ z ∈ M, (SigmaFormula.conj _ _).Realize M (cons z e)) ↔
∃ z ∈ M, (SentenceForm.conj _ _).Sat (M : Set ZFSet) (cons z e)
apply exists_congr; intro z
apply and_congr_right; intro _
rw [SigmaFormula.realize_conj]
change (_ ∧ _) ↔ (_ ∧ _)
rw [SigmaFormula.realize_rename,sat_rename]
exact and_congr (sigma_sat checkedValueSigma M _).symm
(sigma_sat (BoundedForcing.SigmaCode.forcing (.bounded (.member 0 1)) 2 3 4
(fun i => match i with | 0 => 0 | _ => 5)) M _).symm
theorem internal_forced_members (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M)
(hc : c ∈ M) {o a : ZFSet.{u}}
(hoM : o ∈ M) (ha : a ∈ M)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(t : Name (Conditions c)) (ht : t.encode (label c) ∈ M) (p : Conditions c) :
∃ b ∈ M, ∀ x, x ∈ b ↔ x ∈ a ∧ MemForces (Name.check x) t p := by
let e := cons c (cons o (cons (label c p) (cons (t.encode (label c)) (fun _ => label c ⊤))))
have he : ∀ i, e i ∈ M := by
intro i; rcases i with _|_|_|_|i
exact hc
exact hoM
exact hM c hc _ (label_mem c p)
exact ht
exact hM c hc _ (label_mem c ⊤)
obtain ⟨b,hb,hdef⟩ := hT.separation.finitePrefix checkedMembershipSigma e he a ha
have spec (x : ZFSet.{u}) (hx : x ∈ M) := checkedMembership_spec M hM hT ho t p x
(cons x e) (by intro i; cases i; exact hx; exact he _) rfl rfl rfl rfl rfl rfl
refine ⟨b,hb,fun x => ?_⟩
constructor
· intro hx
have hxM := hM b hb x hx
obtain ⟨hxa,hf⟩ := (hdef x hxM).mp hx
exact ⟨hxa,(spec x hxM).mp ((checkedMembershipSigma_spec M _).mp hf)⟩
· rintro ⟨hxa,hf⟩
have hxM := hM a ha x hxa
exact (hdef x hxM).mpr ⟨hxa,(checkedMembershipSigma_spec M _).mpr ((spec x hxM).mpr hf)⟩
end TuringRigidity.FullSetForcing
end OAI
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