openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /ContextHierarchyGraph.lean
| import OAI.Computability.DegreeRigidity.Constructibility.ContextDefinablePower | |
| import OAI.Computability.DegreeRigidity.Constructibility.HierarchyGraph | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding | |
| open BoundedDefinability SetModelFunctions | |
| universe u | |
| theorem definablePower_sigmaDefinable_context (M : ZFSet.{u}) (C : Context M) : | |
| SigmaDefinable M (fun e => e 0 = definablePower (e 1)) := by | |
| obtain ⟨Q,hQ,_,hq⟩ := internal_family_bound M C.transitive C.pairing C.union C.power C.omega_mem | |
| obtain ⟨B,hB,hb⟩ := internal_support_bound C | |
| have hc := (defSystem_definable C hQ hB 5 3 2 1 0 4).toSigma C.transitive | |
| have hs := hc.existsSet.existsSet.existsSet.existsSet | |
| apply hs.congr | |
| intro e he | |
| change (∃ G ∈ M, ∃ T ∈ M, ∃ H ∈ M, ∃ Z ∈ M, | |
| DefSystem Q B (e 1) G T H Z (e 0)) ↔ e 0 = definablePower (e 1) | |
| constructor | |
| · rintro ⟨G,_,T,_,H,_,Z,_,h⟩ | |
| exact h.exact hq hb | |
| · intro h | |
| rw [h] | |
| exact internal_defSystem_of_context M C Q B (e 1) (he 1) hq hb | |
| theorem uniform_definable_successor_context (M : ZFSet.{u}) (C : Context M) : | |
| ∃ p : SigmaFormula, ∃ d : ℕ → ZFSet.{u}, (∀ i, d i ∈ M) ∧ | |
| ∀ A ∈ M, ∀ D ∈ M, p.Realize M (cons D (cons A d)) ↔ D = definablePower A := | |
| (definablePower_sigmaDefinable_context M C).binary definablePower | |
| theorem stageImage_mem_context (M : ZFSet.{u}) (C : Context M) | |
| {s x f : ZFSet.{u}} (hs : s ∈ M) (hx : x ∈ M) (hf : f ∈ M) : stageImage s x f ∈ M := by | |
| have hr := iterUnion_mem M C.transitive C.union hf 2 | |
| have hd := (((defPairMem C 1 0 4).and (member_definable C 2 0)).existsMem 4).existsMem 1 | |
| let e := cons x (cons f (fun _ => iterUnion 2 f)) | |
| have he : ∀ i, e i ∈ M := by intro i; rcases i with _|_|i; exact hx; exact hf; exact hr | |
| exact binary_union_mem M C.transitive C.pairing C.union hs | |
| (hd.sep_mem C e he (union_mem M C.transitive C.union hr)) | |
| theorem hierarchyGraph_of_mem_iff_context (M R d f : ZFSet.{u}) (C : Context M) | |
| (hs : seed R ∈ M) (hdM : d ∈ M) (hfM : f ∈ M) (hd : Transitive d) | |
| (hf : ∀ z, z ∈ f ↔ ∃ x ∈ d, z = ZFSet.pair x (definablePower (stage R x))) : | |
| HierarchyGraph M (seed R) d (iterUnion 2 f) f := by | |
| have hpair (x v : ZFSet.{u}) : ZFSet.pair x v ∈ f ↔ x ∈ d ∧ v = definablePower (stage R x) := by | |
| rw [hf] | |
| constructor | |
| · rintro ⟨y,hy,he⟩ | |
| obtain ⟨rfl,hv⟩ := ZFSet.pair_inj.mp he | |
| exact ⟨hy,hv⟩ | |
| · rintro ⟨hx,rfl⟩; exact ⟨x,hx,rfl⟩ | |
| refine ⟨hd,⟨?_,?_⟩,?_⟩ | |
| · intro z hz | |
| obtain ⟨x,hx,rfl⟩ := (hf z).mp hz | |
| exact ⟨x,hx,_,second_mem_doubleUnion hz,rfl⟩ | |
| · intro x hx | |
| have hp := (hpair x _).mpr ⟨hx,rfl⟩ | |
| exact ⟨_,second_mem_doubleUnion hp,hp,fun v _ hv => (hpair x v).mp hv |>.2⟩ | |
| · intro x hx v _ hv | |
| have he := (hpair x v).mp hv |>.2 | |
| subst v | |
| have hA : stageImage (seed R) x f = stage R x := by | |
| apply ZFSet.ext; intro z | |
| rw [mem_stageImage,mem_stage] | |
| apply or_congr_right | |
| constructor | |
| · rintro ⟨y,hy,w,hw,hz⟩ | |
| rw [(hpair y w).mp hw |>.2] at hz | |
| exact ⟨y,hy,hz⟩ | |
| · rintro ⟨y,hy,hz⟩ | |
| exact ⟨y,hy,_,(hpair y _).mpr ⟨hd x hx y hy,rfl⟩,hz⟩ | |
| refine ⟨stage R x,hA ▸ stageImage_mem_context M C hs (C.transitive d hdM x hx) hfM,?_,rfl⟩ | |
| refine ⟨seed_subset_stage R x,?_,?_⟩ | |
| · intro z hz | |
| rcases (mem_stage R x z).mp hz with hz|⟨y,hy,hz⟩ | |
| · exact Or.inl hz | |
| · have hp := (hpair y _).mpr ⟨hd x hx y hy,rfl⟩ | |
| exact Or.inr ⟨y,hy,_,second_mem_doubleUnion hp,hp,hz⟩ | |
| · intro y hy w _ hw | |
| rw [(hpair y w).mp hw |>.2] | |
| exact definablePower_subset_stage R hy | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |