File size: 4,030 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 86 | import OAI.Computability.DegreeRigidity.SetModels.CollapseWithoutChoice
import OAI.Computability.DegreeRigidity.SetModels.InternalCollapseDense
namespace OAI
namespace TuringRigidity.BoundedSetTheory.InternalCollapse
open TransitiveNameModel CountableForcing
universe u
attribute [local instance] order collapsePreorder
theorem extend_length_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M)
(hω : ZFSet.omega.{u} ∈ M)
{A p : ZFSet.{u}} (hA : A ∈ M) (hpM : p ∈ M) (hp : Prefix A p)
(hne : ∃ x, x ∈ A) (m : ℕ) :
∃ q ∈ M, ∃ k : ℕ, m ≤ k ∧ TransitiveNameModel.FunctionGraph (natSet k) A q ∧ p ⊆ q := by
obtain ⟨x,hx⟩ := hne
induction m with
| zero =>
obtain ⟨k,hk⟩ := hp
exact ⟨p,hpM,k,Nat.zero_le _,hk,fun _ h => h⟩
| succ m ih =>
obtain ⟨q,hqM,k,hk,hq,hpq⟩ := ih
by_cases hmk : m+1 ≤ k
· exact ⟨q,hqM,k,hmk,hq,hpq⟩
· refine ⟨append q k x,append_mem_without_choice M hM hP hU hω hqM (hM _ hA _ hx) k,k+1,by omega,
append_function hq hx,fun z hz => ?_⟩
exact (mem_append _ _ _ _).mpr (Or.inl (hpq hz))
theorem covers_mem_without_choice (M : ZFSet.{u}) (hM : Transitive M)
(hS : Separation M) (hω : ZFSet.omega.{u} ∈ M)
{c x : ZFSet.{u}} (hc : c ∈ M) (hx : x ∈ M) : covers c x ∈ M := by
let e := cons ZFSet.omega (fun _ => x)
have he : ∀ i, e i ∈ M := by intro i; cases i <;> first | exact hω | exact hx
simpa only [covers,Formula.Eval,Formula.eval_pairMem,cons_zero,cons_succ,e] using
sep_mem M hM hS (.existsMem 1 (.pairMem 0 3 1)) e he hc
theorem definedAt_mem_without_choice (M : ZFSet.{u}) (hM : Transitive M)
(hS : Separation M) (hω : ZFSet.omega.{u} ∈ M)
{c A : ZFSet.{u}} (hc : c ∈ M) (hA : A ∈ M) (n : ℕ) : definedAt c A n ∈ M := by
let e := cons A (fun _ => natSet.{u} n)
have he : ∀ i, e i ∈ M := by
intro i; cases i
· exact hA
· exact hM _ (hω) _ ((mem_omega _).mpr ⟨n,rfl⟩)
simpa only [definedAt,Formula.Eval,Formula.eval_pairMem,cons_zero,cons_succ,e] using
sep_mem M hM hS (.existsMem 1 (.pairMem 3 0 1)) e he hc
theorem covers_dense_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M)
(hω : ZFSet.omega.{u} ∈ M)
{A c : ZFSet.{u}} (hA : A ∈ M) (hc : ∀ p, p ∈ c ↔ p ∈ M ∧ Prefix A p)
{x : ZFSet.{u}} (hx : x ∈ A) : Dense {p : Conditions c | label c p ∈ covers c x} := by
intro p
obtain ⟨hpM,n,hp⟩ := (hc _).mp (label_mem c p)
let q := append (label c p) n x
have hq : q ∈ c := (hc q).mpr ⟨append_mem_without_choice M hM hP hU hω hpM (hM _ hA _ hx) n,n+1,append_function hp hx⟩
obtain ⟨r,hr⟩ := label_surjective c hq
refine ⟨r,?_,?_⟩
· change label c p ⊆ label c r
rw [hr]
exact fun z hz => (mem_append _ _ _ _).mpr (Or.inl hz)
· change label c r ∈ covers c x
rw [hr,covers,ZFSet.mem_sep]
exact ⟨hq,natSet n,(mem_omega _).mpr ⟨n,rfl⟩,(mem_append _ _ _ _).mpr (Or.inr rfl)⟩
theorem definedAt_dense_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M)
(hω : ZFSet.omega.{u} ∈ M)
{A c : ZFSet.{u}} (hA : A ∈ M) (hc : ∀ p, p ∈ c ↔ p ∈ M ∧ Prefix A p)
(hne : ∃ x, x ∈ A) (n : ℕ) : Dense {p : Conditions c | label c p ∈ definedAt c A n} := by
intro p
obtain ⟨hpM,hp⟩ := (hc _).mp (label_mem c p)
obtain ⟨q,hqM,k,hnk,hq,hpq⟩ := extend_length_without_choice M hM hP hU hω hA hpM hp hne (n+1)
have hqc : q ∈ c := (hc q).mpr ⟨hqM,k,hq⟩
obtain ⟨r,hr⟩ := label_surjective c hqc
obtain ⟨x,hx,hnx,_⟩ := hq.2 (natSet n) ((natSet_mem_natSet n k).mpr (by omega))
refine ⟨r,?_,?_⟩
· change label c p ⊆ label c r
rw [hr]; exact hpq
· change label c r ∈ definedAt c A n
rw [hr,definedAt,ZFSet.mem_sep]
exact ⟨hqc,x,hx,hnx⟩
end TuringRigidity.BoundedSetTheory.InternalCollapse
end OAI
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