openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /HierarchyWitnessCollection.lean
| import OAI.Computability.DegreeRigidity.Constructibility.RelativeHierarchyInternal | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding | |
| open BoundedDefinability SetModelFunctions | |
| universe u | |
| def HierarchyRequest (Q B s f r x c : ZFSet.{u}) : Prop := | |
| ∃ A ∈ c, ∃ v ∈ r, ∃ G ∈ c, ∃ T ∈ c, ∃ H ∈ c, ∃ Z ∈ c, | |
| ZFSet.pair x v ∈ f ∧ StageStep s f r x A ∧ DefSystem Q B A G T H Z v | |
| theorem hierarchyRequest_definable {M Q B : ZFSet.{u}} (C : Context M) | |
| (hQ : Q ∈ M) (hB : B ∈ M) (s f r x c : ℕ) : | |
| Definable M (fun e => HierarchyRequest Q B (e s) (e f) (e r) (e x) (e c)) := by | |
| have h := (defPairMem C (x+6) 4 (f+6)).and | |
| ((stageStep_definable C (s+6) (f+6) (r+6) (x+6) 5).and | |
| (defSystem_definable C hQ hB 5 3 2 1 0 4)) | |
| exact (((((h.existsMem (c+5)).existsMem (c+4)).existsMem (c+3)).existsMem (c+2)).existsMem (r+1)).existsMem c | |
| theorem hierarchyRequest_binary {M Q B s f r : ZFSet.{u}} (C : Context M) | |
| (hQ : Q ∈ M) (hB : B ∈ M) (hs : s ∈ M) (hf : f ∈ M) (hr : r ∈ M) : | |
| ∃ p : SigmaFormula, ∃ e : ℕ → ZFSet.{u}, (∀ i, e i ∈ M) ∧ | |
| ∀ x ∈ M, ∀ c ∈ M, p.Realize M (cons c (cons x e)) ↔ | |
| HierarchyRequest Q B s f r x c := by | |
| have h := (hierarchyRequest_definable C hQ hB 2 1 0 4 3).toSigma C.transitive | |
| have eq := ((equal_param hr 0).and ((equal_param hf 1).and (equal_param hs 2))).toSigma C.transitive | |
| have hc := (eq.and h).existsSet.existsSet.existsSet | |
| apply sigma_binary_relation _ (hc.congr ?_) | |
| intro e _ | |
| simp only [cons_zero,cons_succ] | |
| constructor | |
| · rintro ⟨s',_,f',_,r',_,⟨rfl,rfl,rfl⟩,h⟩ | |
| exact h | |
| · intro h | |
| exact ⟨s,hs,f,hf,r,hr,⟨rfl,rfl,rfl⟩,h⟩ | |
| theorem hierarchy_witness_collection {M Q B s d r f : ZFSet.{u}} (C : Context M) | |
| (hColl : SigmaCollection M) (hQ : Q ∈ M) (hB : B ∈ M) | |
| (hq : ∀ p : SentenceForm, family p ∈ Q) | |
| (hb : ∀ p : SentenceForm, supportGraph p ∈ B) | |
| (hs : s ∈ M) (hd : d ∈ M) (hr : r ∈ M) (hf : f ∈ M) | |
| (hg : HierarchyGraph M s d r f) : | |
| ∃ W ∈ M, ∀ x ∈ d, ∃ c ∈ W, HierarchyRequest Q B s f r x c := by | |
| obtain ⟨p,e,he,hp⟩ := hierarchyRequest_binary C hQ hB hs hf hr | |
| obtain ⟨W,hW,hw⟩ := hColl p e he d hd (by | |
| intro x hx | |
| obtain ⟨v,hv,hfv,_⟩ := hg.2.1.2 x hx | |
| obtain ⟨A,hA,hstep,hvA⟩ := hg.2.2 x hx v hv hfv | |
| obtain ⟨G,hG,T,hT,H,hH,Z,hZ,hdef⟩ := internal_defSystem_of_context M C Q B A hA hq hb | |
| let a : Fin 5 → ZFSet.{u} := fun i => match i.val with | |
| | 0 => A | 1 => G | 2 => T | 3 => H | _ => Z | |
| have ha : ∀ i, a i ∈ M := by | |
| intro i; dsimp [a]; split <;> assumption | |
| have hc := finite_range_mem M C.transitive C.pairing C.union (C.nat_mem 0) 5 a ha | |
| refine ⟨ZFSet.range a,hc,(hp x (C.transitive d hd x hx) _ hc).mpr ?_⟩ | |
| refine ⟨A,ZFSet.mem_range.mpr ⟨⟨0,by decide⟩,rfl⟩,v,hv, | |
| G,ZFSet.mem_range.mpr ⟨⟨1,by decide⟩,rfl⟩, | |
| T,ZFSet.mem_range.mpr ⟨⟨2,by decide⟩,rfl⟩, | |
| H,ZFSet.mem_range.mpr ⟨⟨3,by decide⟩,rfl⟩, | |
| Z,ZFSet.mem_range.mpr ⟨⟨4,by decide⟩,rfl⟩,hfv,hstep,?_⟩ | |
| rwa [hvA]) | |
| exact ⟨W,hW,fun x hx => by | |
| obtain ⟨c,hc,h⟩ := hw x hx | |
| exact ⟨c,hc,(hp x (C.transitive d hd x hx) c (C.transitive W hW c hc)).mp h⟩⟩ | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |