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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.InternalBoundedTruth
import OAI.Computability.DegreeRigidity.SetModels.NameSupport
namespace OAI
namespace TuringRigidity.BoundedForcing
open Set RecursiveNames TransitiveNameModel BoundedSetTheory AtomicForcing CountableForcing
universe u
variable {c : ZFSet.{u}} [Preorder (Conditions c)]
theorem separation_name (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) (hS : Separation M)
(hR : SigmaReplacement M) (hI : Infinity M) (hc : c ∈ M)
{o : ZFSet.{u}} (hoM : o ∈ M)
(ho : ∀ r s : Conditions c, ZFSet.pair (label c r) (label c s) ∈ o ↔ r ≤ s)
(G : GenericFilter (Conditions c)) (hG : GroundGeneric M G)
(φ : BoundedSetTheory.Formula) (a : Name (Conditions c)) (ha : a.encode (label c) ∈ M)
(e : ℕ → Name (Conditions c)) (he : ∀ i, (e i).encode (label c) ∈ M) :
∃ B ∈ M, B ⊆ a.support (label c) ∧ ∀ x,
x ∈ (a.restrict (label c) B).val G.carrier ↔ x ∈ a.val G.carrier ∧
φ.Eval (cons x (fun i => (e i).val G.carrier)) := by
obtain ⟨d,hd,hdT,had,hnd⟩ := finite_container M hM hP hU hS hR hI
(fun i => (push a e i).encode (label c))
(by intro i; cases i <;> simp only [push_zero,push_succ]; exact ha; exact he _) (bound φ+2)
let e' := truncate (bound φ) e
have he' (i) : (e' i).encode (label c) ∈ d := by
dsimp only [e',truncate]
split
· exact hnd (i+1) (by omega)
· exact hnd 1 (by omega)
have hai (i) : (a.child i).encode (label c) ∈ d :=
names_childClosed d c hdT a had _ (child_relation a i)
have hpush (i) (n) : (push (a.child i) e' n).encode (label c) ∈ d := by
cases n with
| zero => exact hai i
| succ n => exact he' n
obtain ⟨f,hf,hfG⟩ := internal_atomic_graph M hM hP hU hPow hS hd hc hoM
let k := ZFSet.prod d d
have hk : k ∈ M := product_mem M hM hP hU hPow hS hd hd
let env := cons d (cons c (cons k (cons o (cons f (fun i => (e' i).encode (label c))))))
have henv : ∀ i, env i ∈ M := by
intro i
rcases i with _|i; exact hd
rcases i with _|i; exact hc
rcases i with _|i; exact hk
rcases i with _|i; exact hoM
rcases i with _|i; exact hf
exact hM d hd _ (he' i)
let ψ := Formula.existsMem 1 (.existsMem 3 (.conj (Formula.orderedPair 2 1 0)
(code φ 3 4 5 6 7 (push 1 (fun i => i+8)) 0)))
let B := ZFSet.sep (fun z => ψ.Eval (cons z env)) (a.support (label c))
have hBM : B ∈ M := sep_mem M hM hS ψ env henv
(support_mem M hM hP hU hPow hS hc hoM ho a ha)
have hBsub : B ⊆ a.support (label c) := fun z hz => (ZFSet.mem_sep.mp hz).1
have query (i : a.arity) (q : Conditions c) (z : ZFSet.{u}) :=
eval_code φ hdT ho hfG (push (a.child i) e') (hpush i) q
(cons (label c q) (cons ((a.child i).encode (label c)) (cons z env)))
3 4 5 6 7 (push 1 (fun n => n+8)) 0 ⟨rfl,rfl,rfl,rfl,rfl,rfl⟩
(by intro n; cases n <;> rfl)
have hB (i : a.arity) (q : Conditions c) (hq : q ≤ a.tag i) :
ZFSet.pair ((a.child i).encode (label c)) (label c q) ∈ B ↔ Forces (push (a.child i) e') φ q := by
have hs : ZFSet.pair ((a.child i).encode (label c)) (label c q) ∈ a.support (label c) :=
ZFSet.mem_range.mpr ⟨⟨i,⟨q,hq⟩⟩,rfl⟩
change _ ∈ ZFSet.sep _ _ ↔ _
rw [ZFSet.mem_sep,and_iff_right hs]
change (Formula.existsMem 1 _).Eval _ ↔ _
simp only [Formula.Eval,Formula.eval_orderedPair,cons_zero,cons_succ,env]
constructor
· rintro ⟨x,hx,r,hr,hxr,hφ⟩
obtain ⟨rfl,rfl⟩ := ZFSet.pair_inj.mp hxr
exact (query i q _).mp hφ
· intro hφ
exact ⟨_,hai i,_,label_mem c q,rfl,(query i q _).mpr hφ⟩
have ht (i : a.arity) :
φ.Eval (cons ((a.child i).val G.carrier) (fun n => (e n).val G.carrier)) ↔
∃ p ∈ G.carrier, Forces (push (a.child i) e') φ p := by
have hfinite : φ.Eval (cons ((a.child i).val G.carrier) (fun n => (e n).val G.carrier)) ↔
φ.Eval (fun n => (push (a.child i) e' n).val G.carrier) := by
apply eval_congr
intro n hn
cases n with
| zero => rfl
| succ n =>
simp only [cons_succ,push_succ]
dsimp only [e',truncate]
rw [ite_eq_left (by omega)]
exact hfinite.trans (internal_bounded_truth M hM hP hU hPow hS hR hI hc hoM ho G hG φ _
(fun n => hM d hd _ (hpush i n)))
refine ⟨B,hBM,hBsub,fun x => ?_⟩
constructor
· intro hx
have hxa := a.val_restrict_subset (label c) B G hx
obtain ⟨i,q,hqi,hqb,hqG,hix⟩ := (a.mem_val_restrict (label c) B G.carrier x).mp hx
have hφ := (ht i).mpr ⟨q,hqG,(hB i q hqi).mp hqb⟩
exact ⟨hxa,hix ▸ hφ⟩
· rintro ⟨hxa,hφ⟩
obtain ⟨i,hi,hix⟩ := (a.mem_val_children G.carrier x).mp hxa
obtain ⟨p,hp,hf⟩ := (ht i).mp (hix ▸ hφ)
obtain ⟨q,hq,hqp,hqi⟩ := G.directed hp hi
exact (a.mem_val_restrict (label c) B G.carrier x).mpr
⟨i,q,hqi,(hB i q hqi).mpr (forces_mono φ _ hqp hf),hq,hix⟩
end TuringRigidity.BoundedForcing
end OAI
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