| import OAI.Computability.DegreeRigidity.Representation.ParameterCoveringModels |
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| namespace OAI |
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| namespace TuringRigidity.ElementaryModel |
| open BoundedSetTheory TransitiveNameModel RelationCollapse SentenceForm |
| universe u |
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| theorem hull_real_fixed (a : ℕ → ZFSet.{u}) |
| (hn : ∀ n, natSet.{u} n ∈ hullSet a) {A : Oracle} (hA : realCode A ∈ hullSet a) : |
| structureMap (hullSet a) (realCode A) = realCode A := by |
| apply collapse_fixed _ _ hA |
| intro x hx |
| obtain ⟨n,rfl⟩ := (mem_omega x).mp (realCode_subset A hx) |
| refine ⟨hn n,structureMap_fixed omega_transitive ?_ _ ((mem_omega _).mpr ⟨n,rfl⟩)⟩ |
| intro y hy |
| obtain ⟨m,rfl⟩ := (mem_omega y).mp hy |
| exact hn m |
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| theorem hull_omega_fixed (a : ℕ → ZFSet.{u}) |
| (hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a) : |
| structureMap (hullSet a) ZFSet.omega = ZFSet.omega := by |
| apply collapse_fixed _ _ hω |
| intro x hx |
| obtain ⟨n,rfl⟩ := (mem_omega x).mp hx |
| refine ⟨hn n,structureMap_fixed omega_transitive ?_ _ ((mem_omega _).mpr ⟨n,rfl⟩)⟩ |
| intro y hy |
| obtain ⟨m,rfl⟩ := (mem_omega y).mp hy |
| exact hn m |
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| theorem collapsed_hull_reals (a : ℕ → ZFSet.{u}) |
| (hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a) (A : Oracle) : |
| A ∈ modelReals (collapsed (hullSet a)) ↔ realCode A ∈ hullSet a := by |
| constructor |
| · intro hA |
| obtain ⟨x,hx,heq⟩ := (mem_collapsed (hullSet a) (realCode A)).mp hA |
| let e := cons x (fun _ => ZFSet.omega) |
| have hen : ∀ i, e i ∈ hullSet a := by intro i; cases i <;> assumption |
| let p := all (imp (.member 0 1) (.member 0 2)) |
| have hs : p.Sat (collapsed (hullSet a) : Set ZFSet) (fun i => structureMap (hullSet a) (e i)) := by |
| simp only [p,sat_all,sat_imp,Sat,cons_zero,cons_succ,e] |
| intro y _ hy |
| rw [←heq] at hy |
| rw [hull_omega_fixed a hn hω] |
| exact realCode_subset A hy |
| have ht := (collapsed_hull_elementary a p e hen).mp hs |
| simp only [p,sat_all,sat_imp,Sat,cons_zero,cons_succ,e,Set.mem_univ,forall_const] at ht |
| have hxreal : realCode (decodeReal x) = x := realCode_decodeReal ht |
| have hfix : structureMap (hullSet a) x = x := by |
| rw [←hxreal] |
| exact hull_real_fixed a hn (hxreal.symm ▸ hx) |
| exact heq.symm ▸ hfix.symm ▸ hx |
| · intro hA |
| exact (mem_collapsed _ _).mpr ⟨realCode A,hA,(hull_real_fixed a hn hA).symm⟩ |
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| noncomputable def mixedSeed (a : ℕ → ZFSet.{u}) (A : ℕ → Oracle) (n : ℕ) : ZFSet.{u} := |
| match n.unpair.1 with |
| | 0 => natSet n.unpair.2 |
| | 1 => realCode (A n.unpair.2) |
| | 2 => a n.unpair.2 |
| | _ => ZFSet.omega |
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| theorem mixed_naturals (a : ℕ → ZFSet.{u}) (A : ℕ → Oracle) (n : ℕ) : |
| natSet.{u} n ∈ hullSet (mixedSeed a A) := by |
| apply (mem_hullSet _ _).mpr |
| simpa [mixedSeed] using parameter_mem (mixedSeed a A) (Nat.pair 0 n) |
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| theorem mixed_omega (a : ℕ → ZFSet.{u}) (A : ℕ → Oracle) : |
| ZFSet.omega.{u} ∈ hullSet (mixedSeed a A) := by |
| apply (mem_hullSet _ _).mpr |
| simpa [mixedSeed] using parameter_mem (mixedSeed a A) (Nat.pair 3 0) |
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| theorem parameter_models_with_elementarity (a : ℕ → ZFSet.{u}) (A : ℕ → Oracle) : |
| ∃ d M : ZFSet.{u}, Transitive M ∧ SourceT M ∧ Countable (Conditions M) ∧ |
| StructureExtensional d ∧ (∀ n, a n ∈ d) ∧ |
| (∀ n, realCode (A n) ∈ d ∧ structureMap d (realCode (A n)) = realCode (A n)) ∧ |
| (∀ n, structureMap d (a n) ∈ M) ∧ |
| (∀ B : Oracle, B ∈ modelReals M ↔ realCode B ∈ d) ∧ |
| (∀ p : SentenceForm, ∀ e : ℕ → ZFSet.{u}, (∀ i, e i ∈ d) → |
| (p.Sat (M : Set ZFSet) (fun i => structureMap d (e i)) ↔ p.Sat Set.univ e)) := by |
| let b := mixedSeed a A |
| have hn := mixed_naturals a A |
| have hω := mixed_omega a A |
| have ha (n : ℕ) : a n ∈ hullSet b := by |
| apply (mem_hullSet _ _).mpr |
| simpa [b,mixedSeed] using parameter_mem b (Nat.pair 2 n) |
| have hA (n : ℕ) : realCode (A n) ∈ hullSet b := by |
| apply (mem_hullSet _ _).mpr |
| simpa [b,mixedSeed] using parameter_mem b (Nat.pair 1 n) |
| refine ⟨hullSet b,collapsed (hullSet b),collapsed_transitive _,collapsed_hull_sourceT _, |
| inferInstance,hullSet_extensional _,ha,(fun n => ⟨hA n,hull_real_fixed b hn (hA n)⟩), |
| (fun n => (mem_collapsed _ _).mpr ⟨a n,ha n,rfl⟩),collapsed_hull_reals b hn hω,?_⟩ |
| exact collapsed_hull_elementary b |
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| end TuringRigidity.ElementaryModel |
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| end OAI |
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