openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /ElementaryCollapsePoset.lean
| import OAI.Computability.DegreeRigidity.SetModels.ElementaryDefinableSets | |
| import OAI.Computability.DegreeRigidity.Representation.AmbientDegreeSyntax | |
| import OAI.Computability.DegreeRigidity.SetModels.InternalCollapseConditions | |
| namespace OAI | |
| namespace TuringRigidity.ElementaryModel | |
| open BoundedSetTheory TransitiveNameModel RelationCollapse SentenceForm | |
| open FullSetForcing InternalCollapse | |
| universe u | |
| theorem definedSet_univ (s : ℕ) (p : Formula) (e : ℕ → ZFSet.{u}) : | |
| (definedSet s p).Sat Set.univ e ↔ ∀ x, x ∈ e s ↔ p.Eval (cons x e) := by | |
| simp only [definedSet,sat_all,sat_iff,Sat,bounded_univ,cons_zero,cons_succ, | |
| Set.mem_univ,forall_const] | |
| theorem collapse_bounded_eval (a : ℕ → ZFSet.{u}) (p : Formula) | |
| (e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ hullSet a) : | |
| p.Eval (fun i => structureMap (hullSet a) (e i)) ↔ p.Eval e := by | |
| have h := collapsed_hull_elementary a (fromBounded p) e he | |
| rw [bounded_sat,Formula.absolute _ _ (collapsed_transitive _) _ | |
| (fun i => (mem_collapsed _ _).mpr ⟨e i,he i,rfl⟩),bounded_univ] at h | |
| exact h | |
| noncomputable def ambientConditions (A : ZFSet.{u}) : ZFSet.{u} := | |
| ZFSet.sep (Prefix A) (ZFSet.powerset (ZFSet.prod ZFSet.omega A)) | |
| theorem mem_ambientConditions (A p : ZFSet.{u}) : p ∈ ambientConditions A ↔ Prefix A p := by | |
| rw [ambientConditions,ZFSet.mem_sep] | |
| exact ⟨And.right,fun h => ⟨ZFSet.mem_powerset.mpr (prefix_subset h),h⟩⟩ | |
| theorem ambientConditions_def (A c : ZFSet.{u}) : | |
| (definedSet 0 (prefixFormula 3 2 0)).Sat Set.univ | |
| (cons c (cons A (fun _ => ZFSet.omega))) ↔ c = ambientConditions A := by | |
| rw [definedSet_univ] | |
| have hs (p : ZFSet.{u}) := eval_prefixFormula 3 2 0 | |
| (cons p (cons c (cons A (fun _ => ZFSet.omega)))) rfl | |
| simp only [hs,cons_zero,cons_succ] | |
| exact ⟨fun h => ZFSet.ext (fun p => (h p).trans (mem_ambientConditions A p).symm), | |
| fun h p => h ▸ mem_ambientConditions A p⟩ | |
| theorem ambientConditions_mem_hull (a : ℕ → ZFSet.{u}) | |
| (hω : ZFSet.omega ∈ hullSet a) {A : ZFSet.{u}} (hA : A ∈ hullSet a) : | |
| ambientConditions A ∈ hullSet a := by | |
| exact unique_definable_mem a (definedSet 0 (prefixFormula 3 2 0)) | |
| (cons A (fun _ => ZFSet.omega)) (by intro i; cases i <;> assumption) | |
| _ ((ambientConditions_def A _).mpr rfl) (fun c hc => (ambientConditions_def A c).mp hc) | |
| theorem collapse_ambientConditions (a : ℕ → ZFSet.{u}) | |
| (hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a) | |
| {A : ZFSet.{u}} (hA : A ∈ hullSet a) : | |
| ∀ p, p ∈ structureMap (hullSet a) (ambientConditions A) ↔ | |
| p ∈ collapsed (hullSet a) ∧ Prefix (structureMap (hullSet a) A) p := by | |
| let e := cons (ambientConditions A) (cons A (fun _ => ZFSet.omega)) | |
| have hc := ambientConditions_mem_hull a hω hA | |
| have he : ∀ i, e i ∈ hullSet a := by | |
| intro i; rcases i with _|_|i | |
| exact hc | |
| exact hA | |
| exact hω | |
| have ht := (collapsed_hull_elementary a (definedSet 0 (prefixFormula 3 2 0)) e he).mpr | |
| ((ambientConditions_def A _).mpr rfl) | |
| have hm := collapsed_transitive (hullSet a) | |
| have hei : ∀ i, structureMap (hullSet a) (e i) ∈ collapsed (hullSet a) := | |
| fun i => (mem_collapsed _ _).mpr ⟨e i,he i,rfl⟩ | |
| have h := (definedSet_spec _ hm 0 _ _ hei).mp ht | |
| have hω' := hull_omega_fixed a hn hω | |
| have hs (p : ZFSet.{u}) (hp : p ∈ collapsed (hullSet a)) : | |
| p ∈ structureMap (hullSet a) (ambientConditions A) ↔ | |
| Prefix (structureMap (hullSet a) A) p := | |
| (h p hp).trans (eval_prefixFormula 3 2 0 | |
| (cons p (fun i => structureMap (hullSet a) (e i))) hω') | |
| intro p | |
| exact ⟨fun hp => ⟨hm _ (hei 0) _ hp,(hs p (hm _ (hei 0) _ hp)).mp hp⟩, | |
| fun hp => (hs p hp.1).mpr hp.2⟩ | |
| def orderMember (c z : ℕ) : Formula := | |
| .existsMem c (.existsMem (c+1) (.conj (.orderedPair (z+2) 1 0) (.subset 0 1))) | |
| theorem eval_orderMember (c z : ℕ) (e : ℕ → ZFSet.{u}) : | |
| (orderMember c z).Eval e ↔ e z ∈ orderSet (e c) := by | |
| simp only [orderMember,Formula.Eval,Formula.eval_orderedPair,Formula.eval_subset, | |
| cons_zero,cons_succ,orderSet,ZFSet.mem_sep] | |
| exact ⟨fun ⟨p,hp,q,hq,hz,hpq⟩ => | |
| ⟨hz ▸ ZFSet.mem_prod.mpr ⟨p,hp,q,hq,rfl⟩,p,hp,q,hq,hz,hpq⟩,And.right⟩ | |
| theorem orderSet_mem_hull (a : ℕ → ZFSet.{u}) {c : ZFSet.{u}} (hc : c ∈ hullSet a) : | |
| orderSet c ∈ hullSet a := by | |
| have hs (x : ZFSet.{u}) : (definedSet 0 (orderMember 2 0)).Sat Set.univ | |
| (cons x (fun _ => c)) ↔ x = orderSet c := by | |
| rw [definedSet_univ] | |
| simp only [eval_orderMember,cons_zero,cons_succ] | |
| exact ZFSet.ext_iff.symm | |
| exact unique_definable_mem a (definedSet 0 (orderMember 2 0)) | |
| (fun _ => c) (fun _ => hc) _ ((hs _).mpr rfl) (fun x hx => (hs x).mp hx) | |
| theorem collapse_orderSet (a : ℕ → ZFSet.{u}) {c : ZFSet.{u}} (hc : c ∈ hullSet a) : | |
| structureMap (hullSet a) (orderSet c) = orderSet (structureMap (hullSet a) c) := by | |
| let e := cons (orderSet c) (fun _ => c) | |
| have he : ∀ i, e i ∈ hullSet a := by | |
| intro i; cases i | |
| exact orderSet_mem_hull a hc | |
| exact hc | |
| have hi : ∀ i, structureMap (hullSet a) (e i) ∈ collapsed (hullSet a) := | |
| fun i => (mem_collapsed _ _).mpr ⟨e i,he i,rfl⟩ | |
| have hs : (definedSet 0 (orderMember 2 0)).Sat Set.univ e := by | |
| rw [definedSet_univ] | |
| intro x | |
| exact (eval_orderMember 2 0 (cons x e)).symm | |
| have ht := (collapsed_hull_elementary a _ e he).mpr hs | |
| exact (definedSet_unique _ (collapsed_transitive _) 0 _ _ hi | |
| (orderSet (structureMap (hullSet a) c)) | |
| (orderSet_mem _ (collapsed_transitive _) (collapsed_hull_sourceT a) (hi 1)) | |
| (fun x _ => eval_orderMember 2 0 _)).mp ht | |
| theorem collapse_selected_condition (a : ℕ → ZFSet.{u}) | |
| (hn : ∀ n, natSet.{u} n ∈ hullSet a) (hω : ZFSet.omega ∈ hullSet a) | |
| {A p : ZFSet.{u}} (hA : A ∈ hullSet a) (hp : p ∈ hullSet a) | |
| (hpc : p ∈ ambientConditions A) : | |
| structureMap (hullSet a) p ∈ structureMap (hullSet a) (ambientConditions A) ∧ | |
| Prefix (structureMap (hullSet a) A) (structureMap (hullSet a) p) := by | |
| have h := (collapse_membership _ (hullSet_extensional a) _ hp _ | |
| (ambientConditions_mem_hull a hω hA)).mpr hpc | |
| exact ⟨h,((collapse_ambientConditions a hn hω hA _).mp h).2⟩ | |
| end TuringRigidity.ElementaryModel | |
| end OAI | |