openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /OmegaGraphRestriction.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaPowerDefinability | |
| namespace OAI | |
| namespace TuringRigidity.OrdinalArithmetic | |
| open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible | |
| universe u | |
| theorem omegaStageStep_domain (M x r f : ZFSet.{u}) (hg : OmegaGraph M x r f) : | |
| StageStep (1 : Ordinal.{u}).toZFSet f r x (Ordinal.omega0 ^ x.rank).toZFSet := by | |
| refine ⟨?_,?_,?_⟩ | |
| · intro z hz; exact (omegaPower_members x.rank z).mpr (Or.inl hz) | |
| · intro z hz | |
| rcases (omegaPower_members x.rank z).mp hz with hz|⟨t,ht,hz⟩ | |
| · exact Or.inl hz | |
| · rw [hg.1.toZFSet_rank_eq] at ht | |
| obtain ⟨v,hv,hfv,_⟩ := hg.2.1.2 t ht | |
| rw [← hg.correct t ht v hv hfv] at hz | |
| exact Or.inr ⟨t,ht,v,hv,hfv,hz⟩ | |
| · intro t ht v hv hfv z hz | |
| rw [hg.correct t ht v hv hfv] at hz | |
| exact (omegaPower_members x.rank z).mpr | |
| (Or.inr ⟨t,hg.1.toZFSet_rank_eq.symm ▸ ht,hz⟩) | |
| theorem omegaStageStep_domain_exact {M x r f A : ZFSet.{u}} (hg : OmegaGraph M x r f) | |
| (hs : StageStep (1 : Ordinal.{u}).toZFSet f r x A) : | |
| A = (Ordinal.omega0 ^ x.rank).toZFSet := by | |
| have ht := omegaStageStep_domain M x r f hg | |
| apply ZFSet.ext; intro z | |
| constructor | |
| · intro hz | |
| rcases hs.2.1 z hz with hz|⟨t,ht',v,hv,hfv,hz⟩ | |
| · exact ht.1 hz | |
| · exact ht.2.2 t ht' v hv hfv hz | |
| · intro hz | |
| rcases ht.2.1 z hz with hz|⟨t,ht',v,hv,hfv,hz⟩ | |
| · exact hs.1 hz | |
| · exact hs.2.2 t ht' v hv hfv hz | |
| theorem internal_omegaGraph_below (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M) | |
| (x : ZFSet.{u}) (hx : x ∈ M) (ho : x.IsOrdinal) : | |
| ∃ f ∈ M, OmegaGraph M x (iterUnion 2 f) f := by | |
| obtain ⟨f,hf,hg⟩ := internal_omegaGraph M hM hT x hx ho | |
| let p : Formula := .existsMem 1 (.existsMem 3 (.orderedPair 2 1 0)) | |
| let e : ℕ → ZFSet.{u} := cons x (fun _ => iterUnion 2 f) | |
| let g := ZFSet.sep (fun z => p.Eval (cons z e)) f | |
| have hgM : g ∈ M := sep_mem M hM hT.separation.finitePrefix.bounded p e | |
| (by intro i; cases i; exact hx; exact iterUnion_mem M hM hT.union hf 2) hf | |
| have hm (z : ZFSet.{u}) : z ∈ g ↔ ∃ t ∈ x, z = ZFSet.pair t (omegaNext t) := by | |
| change z ∈ ZFSet.sep _ f ↔ _ | |
| simp only [ZFSet.mem_sep,p,Formula.Eval,Formula.eval_orderedPair,cons_zero,cons_succ] | |
| change (z ∈ f ∧ ∃ t ∈ x, ∃ v ∈ iterUnion 2 f, z = ZFSet.pair t v) ↔ _ | |
| constructor | |
| · rintro ⟨hz,t,ht,v,hv,rfl⟩ | |
| exact ⟨t,ht,by rw [hg.correct t (ZFSet.mem_insert_of_mem x ht) v hv hz]⟩ | |
| · rintro ⟨t,ht,rfl⟩ | |
| have hp := (hg.mem_iff _).mpr ⟨t,ZFSet.mem_insert_of_mem x ht,rfl⟩ | |
| exact ⟨hp,t,ht,_,second_mem_doubleUnion hp,rfl⟩ | |
| exact ⟨g,hgM,omegaGraph_of_mem_iff M x g hM hT hx hgM ho hm⟩ | |
| theorem OmegaGraph.value_subset_power {M r f : ZFSet.{u}} {a : Ordinal.{u}} | |
| (hg : OmegaGraph M a.toZFSet r f) (t v : ZFSet.{u}) | |
| (ht : t ∈ a.toZFSet) (hv : v ∈ r) (hfv : ZFSet.pair t v ∈ f) : | |
| v ⊆ (Ordinal.omega0 ^ a).toZFSet := by | |
| rw [hg.correct t ht v hv hfv,omegaNext] | |
| obtain ⟨b,hb,rfl⟩ := Ordinal.mem_toZFSet_iff.mp ht | |
| rw [Ordinal.rank_toZFSet] | |
| exact Ordinal.toZFSet_monotone | |
| (Ordinal.opow_le_opow_right Ordinal.omega0_pos (Order.succ_le_of_lt hb)) | |
| end TuringRigidity.OrdinalArithmetic | |
| end OAI | |