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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 | import OAI.Computability.DegreeRigidity.SetModels.ModelDegreeUniverse
import OAI.Computability.DegreeRigidity.SetModels.InternalPresentationEnumeration
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel EncodedForcing
universe u
namespace Formula
def selectedEnumerationBit (o Q I E F v : ℕ) : Formula :=
.existsMem o (.existsMem (o+1) (.conj (naturalPair (o+2) (Q+2) 1 0 (v+2))
(.existsMem (I+2) (.conj (.pairMem 2 0 (E+3))
(.existsMem 0 (.conj (.pairMem 1 0 (F+4)) (.member 2 0)))))))
theorem selectedEnumerationBit_spec (o Q I E F v : ℕ) (e : ℕ → ZFSet.{u})
(ho : e o = ZFSet.omega) (hQ : ∀ f : ℕ → ℕ, ∀ k, finiteNaturalGraph f k ∈ e Q)
(N K : ℕ) (hv : e v = natSet (Nat.pair N K)) :
(selectedEnumerationBit o Q I E F v).Eval e ↔
∃ D ∈ e I, ZFSet.pair (natSet N) D ∈ e E ∧
∃ A ∈ D, ZFSet.pair D A ∈ e F ∧ natSet K ∈ A := by
simp only [selectedEnumerationBit,Formula.Eval,eval_pairMem,cons_zero,cons_succ]
rw [ho]
constructor
· rintro ⟨n,hn,k,hk,hp,hrest⟩
obtain ⟨n,rfl⟩ := (mem_omega n).mp hn
obtain ⟨k,rfl⟩ := (mem_omega k).mp hk
have hp' := (naturalPair_spec (o+2) (Q+2) 1 0 (v+2)
(cons (natSet k) (cons (natSet n) e)) ho hQ n k rfl rfl).mp hp
have hnk : Nat.pair N K = Nat.pair n k := natSet_injective (hv.symm.trans hp')
have he := congrArg Nat.unpair hnk
simp only [Nat.unpair_pair,Prod.mk.injEq] at he
obtain ⟨rfl,rfl⟩ := he
exact hrest
· intro h
refine ⟨natSet N,(mem_omega _).mpr ⟨N,rfl⟩,natSet K,(mem_omega _).mpr ⟨K,rfl⟩,?_,h⟩
exact (naturalPair_spec (o+2) (Q+2) 1 0 (v+2)
(cons (natSet K) (cons (natSet N) e)) ho hQ N K rfl rfl).mpr hv
end Formula
theorem internal_selected_enumeration_real (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M)
{I E F : ZFSet.{u}} (hIM : I ∈ M) (hEM : E ∈ M) (hFM : F ∈ M)
(hE : TransitiveNameModel.FunctionGraph ZFSet.omega I E) (hF : ChoiceGraph I F) :
∃ H ∈ modelReals M, ∀ n, ∀ D ∈ I, ZFSet.pair (natSet n) D ∈ E →
∀ A : Oracle, realCode A ∈ D → ZFSet.pair D (realCode A) ∈ F → columns H n = A := by
have hS := hT.separation.finitePrefix.bounded
have hω := sourceT_omega_mem M hM hT
obtain ⟨Q,hQM,_,hQ⟩ := internal_finite_natural_graph_bound M hM hT.pairing hT.union hT.powerSet hS hω
let e := cons ZFSet.omega (cons Q (cons I (cons E (fun _ => F))))
have he : ∀ i, e i ∈ M := by
intro i
rcases i with _|i; exact hω
rcases i with _|i; exact hQM
rcases i with _|i; exact hIM
rcases i with _|i; exact hEM
exact hFM
let B := ZFSet.sep (fun v => (Formula.selectedEnumerationBit 1 2 3 4 5 0).Eval (cons v e)) ZFSet.omega
have hBM : B ∈ M := sep_mem M hM hS _ e he hω
have hsub : B ⊆ ZFSet.omega := fun _ h => (ZFSet.mem_sep.mp h).1
have hB : realCode (decodeReal B) = B := realCode_decodeReal hsub
refine ⟨decodeReal B,?_,?_⟩
· change realCode (decodeReal B) ∈ M
rw [hB]; exact hBM
· intro n D hD hn A hA hDA
funext k
have hbit : columns (decodeReal B) n k = true ↔ A k = true := by
change decodeReal B (Nat.pair n k) = true ↔ A k = true
rw [decodeReal_true]
change natSet (Nat.pair n k) ∈ ZFSet.sep _ ZFSet.omega ↔ _
rw [ZFSet.mem_sep,Formula.selectedEnumerationBit_spec 1 2 3 4 5 0
(cons (natSet (Nat.pair n k)) e) rfl hQ n k rfl]
constructor
· rintro ⟨_,D',_,hn',A',_,hDA',hk⟩
have hDD := hE.functional ((mem_omega _).mpr ⟨n,rfl⟩) hn hn'
subst D'
have hAA := hF.functional hDA hDA'
rw [←hAA,natSet_mem_realCode] at hk
exact hk
· intro hk
exact ⟨(mem_omega _).mpr ⟨Nat.pair n k,rfl⟩,D,hD,hn,realCode A,hA,hDA,
(natSet_mem_realCode A k).mpr hk⟩
cases hx : columns (decodeReal B) n k <;> cases hy : A k <;> simp_all
def InternallyCountable (M I : ZFSet.{u}) : Prop :=
∃ E ∈ M, TransitiveNameModel.FunctionGraph ZFSet.omega I E ∧
∀ D ∈ I, ∃ n ∈ ZFSet.omega, ZFSet.pair n D ∈ E
theorem internal_countable_degree_presentation (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M)
(R : ZFSet.{u}) (hR : ∀ w ∈ R, ∃ A : Oracle, realCode A = w)
{I : ZFSet.{u}} (hIM : I ∈ M) (hI : I ⊆ degreeUniverse R)
(hcount : InternallyCountable M I) :
∃ H ∈ modelReals M, presentationSet R H = I := by
obtain ⟨E,hEM,hE,honto⟩ := hcount
obtain ⟨F,hFM,hF⟩ := hT.choice I hIM (fun D hD => by
obtain ⟨A,hA,_⟩ := degreeUniverse_nonempty_member R D (hI hD)
exact ⟨realCode A,hA⟩)
obtain ⟨H,hHM,hH⟩ := internal_selected_enumeration_real M hM hT hIM hEM hFM hE hF
have hentry (n : ℕ) (D : ZFSet.{u}) (hD : D ∈ I) (hn : ZFSet.pair (natSet n) D ∈ E) :
degreeCode R (columns H n) = D := by
obtain ⟨a,ha,hDa,_⟩ := hF.2 D hD
obtain ⟨B,hB,hDB⟩ := (mem_degreeUniverse R D).mp (hI hD)
have har : a ∈ R := degreeCode_subset R B (hDB ▸ ha)
obtain ⟨A,rfl⟩ := hR a har
rw [hH n D hD hn A ha hDa]
have heq := ((realCode_mem_degreeCode R B A).mp (hDB ▸ ha)).2
exact ((degreeCode_equal R har).mpr heq).trans hDB.symm
refine ⟨H,hHM,?_⟩
apply ZFSet.ext; intro D
constructor
· intro hD
obtain ⟨n,rfl⟩ := (mem_presentationSet R H D).mp hD
obtain ⟨D,hD,hn,_⟩ := hE.2 (natSet n) ((mem_omega _).mpr ⟨n,rfl⟩)
rw [hentry n D hD hn]
exact hD
· intro hD
obtain ⟨n,hn,hDn⟩ := honto D hD
obtain ⟨n,rfl⟩ := (mem_omega n).mp hn
exact (mem_presentationSet R H D).mpr ⟨n,(hentry n D hD hDn).symm⟩
end TuringRigidity.BoundedSetTheory
end OAI
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