openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /HierarchySyntax.lean
| import OAI.Computability.DegreeRigidity.Constructibility.HierarchyGraph | |
| import OAI.Computability.DegreeRigidity.Syntax.SigmaParameterizedClosure | |
| import OAI.Computability.DegreeRigidity.SetModels.InternalHull | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding | |
| open BoundedDefinability SetModelFunctions | |
| universe u | |
| variable {M : ZFSet.{u}} | |
| theorem stageStep_definable (C : Context M) (s f r x A : ℕ) : | |
| Definable M (fun e => StageStep (e s) (e f) (e r) (e x) (e A)) := by | |
| have hn := defAllMem (defOr (member_definable C 0 (s+1)) | |
| ((((defPairMem C 1 0 (f+3)).and (member_definable C 2 0)).existsMem (r+2)).existsMem (x+1))) A | |
| have hc := defAllMem (defAllMem (defImp (defPairMem C 1 0 (f+2)) | |
| (defSubset C 0 (A+2))) (r+1)) x | |
| exact (defSubset C s A).and (hn.and hc) | |
| theorem functionGraph_definable (C : Context M) (d r f : ℕ) : | |
| Definable M (fun e => FunctionGraph (e d) (e r) (e f)) := by | |
| refine ⟨.functionGraph (2*f) (2*d) (2*r),fun _ => ZFSet.omega, | |
| fun _ => C.omega_mem,?_⟩ | |
| intro e | |
| simp only [Formula.eval_functionGraph,mix_even,FunctionGraph] | |
| theorem hierarchyGraph_sigmaDefinable (hM : Transitive M) (hT : SourceT M) (s d r f : ℕ) : | |
| SigmaDefinable M (fun e => HierarchyGraph M (e s) (e d) (e r) (e f)) := by | |
| let C : Context M := ⟨hM,hT.pairing,hT.union,hT.powerSet,hT.separation.finitePrefix.bounded,hT.infinity⟩ | |
| have hd := (defAllMem (defSubset C 0 (d+1)) d).toSigma hM | |
| have hf := (functionGraph_definable C d r f).toSigma hM | |
| have hv : SigmaDefinable M (fun e => e 1 = definablePower (e 0)) := | |
| (definablePower_sigmaDefinable M hM hT).subst (fun i => if i = 0 then 1 else 0) | |
| have hs := ((stageStep_definable C (s+3) (f+3) (r+3) 2 0).toSigma hM).and hv | |
| have hi := SigmaDefinable.impBounded C (defPairMem C 1 0 (f+2)) hs.existsSet | |
| exact hd.and (hf.and ((hi.allMem hM hT (r+1)).allMem hM hT d)) | |
| theorem sigma_binary_relation (P : ZFSet.{u} → ZFSet.{u} → Prop) | |
| (h : SigmaDefinable M (fun e => P (e 1) (e 0))) : | |
| ∃ p : SigmaFormula, ∃ a : ℕ → ZFSet.{u}, (∀ i, a i ∈ M) ∧ | |
| ∀ x ∈ M, ∀ y ∈ M, p.Realize M (cons y (cons x a)) ↔ P x y := by | |
| obtain ⟨p,a,ha,hp⟩ := h | |
| let r : ℕ → ℕ := fun i => if i % 2 = 0 then (if i / 2 = 0 then 0 else 1) else i/2+2 | |
| refine ⟨p.rename r,a,ha,?_⟩ | |
| intro x hx y hy | |
| rw [SigmaFormula.realize_rename] | |
| have he : cons y (cons x a) ∘ r = mix (cons y (fun _ => x)) a := by | |
| apply parity_ext | |
| · intro n | |
| cases n with | |
| | zero => rfl | |
| | succ n => simp [r,show ¬ 2 * (n+1) ≤ 1 by omega] | |
| · intro n; simp [r,Nat.add_div] | |
| change p.Realize M (cons y (cons x a) ∘ r) ↔ _ | |
| rw [he] | |
| exact hp (cons y (fun _ => x)) (by intro i; cases i; exact hy; exact hx) | |
| theorem uniform_hierarchy_certificate (M R d q : ZFSet.{u}) (hM : Transitive M) | |
| (hT : SourceT M) (hs : seed R ∈ M) (hdM : d ∈ M) (hqM : q ∈ M) | |
| (hd : Transitive d) (hq : ∀ c, c ∈ q ↔ c ∈ M ∧ c ⊆ d) : | |
| ∃ p : SigmaFormula, ∃ a : ℕ → ZFSet.{u}, (∀ i, a i ∈ M) ∧ | |
| ∀ x ∈ d, ∀ f ∈ M, p.Realize M (cons f (cons x a)) ↔ | |
| ∃ r ∈ M, HierarchyGraph M (seed R) (hull d q x) r f := by | |
| let C : Context M := ⟨hM,hT.pairing,hT.union,hT.powerSet,hT.separation.finitePrefix.bounded,hT.infinity⟩ | |
| have hh : Definable M (fun e => e 1 ∈ q ∧ Transitive (e 1) ∧ e 3 ∈ e 1 ∧ | |
| ∀ b ∈ q, Transitive b → e 3 ∈ b → e 1 ⊆ b) := by | |
| refine ⟨.isHull 2 6 1 3,cons d (fun _ => q),?_,?_⟩ | |
| · intro i; cases i; exact hdM; exact hqM | |
| · intro e | |
| simp [Formula.eval_isHull,mix,cons] | |
| have hg : SigmaDefinable M (fun e => HierarchyGraph M (seed R) (e 1) (e 0) (e 2)) := by | |
| have h := hierarchyGraph_sigmaDefinable hM hT 0 2 1 3 | |
| have eq := (equal_param hs 0).toSigma hM | |
| have hx := (eq.and h).existsSet | |
| exact hx.congr (fun e _ => by simp [hs]) | |
| have hc := (hh.toSigma hM).and hg | |
| obtain ⟨p,a,ha,hp⟩ := sigma_binary_relation (fun x f => | |
| ∃ c ∈ M, ∃ r ∈ M, (c ∈ q ∧ Transitive c ∧ x ∈ c ∧ | |
| ∀ b ∈ q, Transitive b → x ∈ b → c ⊆ b) ∧ HierarchyGraph M (seed R) c r f) | |
| hc.existsSet.existsSet | |
| refine ⟨p,a,ha,?_⟩ | |
| intro x hx f hf | |
| rw [hp x (hM d hdM x hx) f hf] | |
| constructor | |
| · rintro ⟨c,hc,r,hr,hh,hg⟩ | |
| have he := (isHull_iff M d q hM C.separation hdM hqM hd hq hx hc).mp hh | |
| subst c | |
| exact ⟨r,hr,hg⟩ | |
| · rintro ⟨r,hr,hg⟩ | |
| have hc := hull_mem M d q hM C.separation hdM hqM (hM d hdM x hx) | |
| exact ⟨_,hc,r,hr,(isHull_iff M d q hM C.separation hdM hqM hd hq hx hc).mpr rfl,hg⟩ | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |