| 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 |
| |
