openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /CollapsePrefixAbsolute.lean
| import OAI.Computability.DegreeRigidity.SetModels.CollapseWithoutChoice | |
| namespace OAI | |
| namespace TuringRigidity.BoundedSetTheory.InternalCollapse | |
| open TransitiveNameModel | |
| universe u | |
| theorem prefix_mem_without_choice (M : ZFSet.{u}) (hM : Transitive M) | |
| (hP : Pairing M) (hU : BoundedSetTheory.Union M) (hω : ZFSet.omega.{u} ∈ M) | |
| {A p : ZFSet.{u}} (hA : A ∈ M) (hp : Prefix A p) : p ∈ M := by | |
| classical | |
| obtain ⟨n,hp⟩ := hp | |
| have h0 : (∅ : ZFSet.{u}) ∈ M := hM _ hω _ ZFSet.omega_zero | |
| cases n with | |
| | zero => | |
| have eq : p = ∅ := by | |
| apply ZFSet.ext; intro z | |
| constructor | |
| · intro hz | |
| obtain ⟨i,hi,_,_,_⟩ := hp.1 z hz | |
| exact False.elim (ZFSet.notMem_empty i hi) | |
| · exact fun hz => False.elim (ZFSet.notMem_empty z hz) | |
| exact eq ▸ h0 | |
| | succ n => | |
| obtain ⟨v,hv,hva⟩ := InternalWellOrder.FunctionGraph.exists_presentation hp | |
| let f : ℕ → ZFSet.{u} := fun i => if i ≤ n then v (natSet i) else ∅ | |
| have hf : ∀ i, f i ∈ M := by | |
| intro i | |
| dsimp only [f] | |
| split_ifs with hi | |
| · exact hM A hA _ (hva _ ((natSet_mem_natSet i (n+1)).mpr (by omega))) | |
| · exact h0 | |
| have eq : p = finiteModelGraph f n := by | |
| apply ZFSet.ext; intro z | |
| rw [mem_finiteModelGraph] | |
| constructor | |
| · intro hz | |
| obtain ⟨j,hj,y,_,rfl⟩ := hp.1 z hz | |
| obtain ⟨i,hi,rfl⟩ := (mem_natSet (n+1) j).mp hj | |
| have hin : i ≤ n := by omega | |
| have hy := ((hv.pair_iff _ _).mp hz).2 | |
| exact ⟨i,hin,by rw [hy]; simp only [f,ite_eq_left hin]⟩ | |
| · rintro ⟨i,hi,rfl⟩ | |
| apply (hv.pair_iff _ _).mpr | |
| exact ⟨(natSet_mem_natSet i (n+1)).mpr (by omega),by simp only [f,ite_eq_left hi]⟩ | |
| rw [eq] | |
| exact finiteModelGraph_mem M hM hP hU hω f hf n | |
| theorem conditions_exist_absolute_without_choice (M : ZFSet.{u}) (hM : Transitive M) | |
| (hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) | |
| (hS : Separation M) (hω : ZFSet.omega.{u} ∈ M) {A : ZFSet.{u}} (hA : A ∈ M) : | |
| ∃ c ∈ M, ∀ p, p ∈ c ↔ Prefix A p := by | |
| obtain ⟨c,hc,hcdef⟩ := conditions_exist_without_choice M hM hP hU hPow hS hω hA | |
| exact ⟨c,hc,fun p => (hcdef p).trans | |
| ⟨And.right,fun hp => ⟨prefix_mem_without_choice M hM hP hU hω hA hp,hp⟩⟩⟩ | |
| end TuringRigidity.BoundedSetTheory.InternalCollapse | |
| end OAI | |