openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /CollapseDenseWithoutChoice.lean
| import OAI.Computability.DegreeRigidity.SetModels.CollapseWithoutChoice | |
| import OAI.Computability.DegreeRigidity.SetModels.InternalCollapseDense | |
| namespace OAI | |
| namespace TuringRigidity.BoundedSetTheory.InternalCollapse | |
| open TransitiveNameModel CountableForcing | |
| universe u | |
| attribute [local instance] order collapsePreorder | |
| theorem extend_length_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) (hpM : p ∈ M) (hp : Prefix A p) | |
| (hne : ∃ x, x ∈ A) (m : ℕ) : | |
| ∃ q ∈ M, ∃ k : ℕ, m ≤ k ∧ TransitiveNameModel.FunctionGraph (natSet k) A q ∧ p ⊆ q := by | |
| obtain ⟨x,hx⟩ := hne | |
| induction m with | |
| | zero => | |
| obtain ⟨k,hk⟩ := hp | |
| exact ⟨p,hpM,k,Nat.zero_le _,hk,fun _ h => h⟩ | |
| | succ m ih => | |
| obtain ⟨q,hqM,k,hk,hq,hpq⟩ := ih | |
| by_cases hmk : m+1 ≤ k | |
| · exact ⟨q,hqM,k,hmk,hq,hpq⟩ | |
| · refine ⟨append q k x,append_mem_without_choice M hM hP hU hω hqM (hM _ hA _ hx) k,k+1,by omega, | |
| append_function hq hx,fun z hz => ?_⟩ | |
| exact (mem_append _ _ _ _).mpr (Or.inl (hpq hz)) | |
| theorem covers_mem_without_choice (M : ZFSet.{u}) (hM : Transitive M) | |
| (hS : Separation M) (hω : ZFSet.omega.{u} ∈ M) | |
| {c x : ZFSet.{u}} (hc : c ∈ M) (hx : x ∈ M) : covers c x ∈ M := by | |
| let e := cons ZFSet.omega (fun _ => x) | |
| have he : ∀ i, e i ∈ M := by intro i; cases i <;> first | exact hω | exact hx | |
| simpa only [covers,Formula.Eval,Formula.eval_pairMem,cons_zero,cons_succ,e] using | |
| sep_mem M hM hS (.existsMem 1 (.pairMem 0 3 1)) e he hc | |
| theorem definedAt_mem_without_choice (M : ZFSet.{u}) (hM : Transitive M) | |
| (hS : Separation M) (hω : ZFSet.omega.{u} ∈ M) | |
| {c A : ZFSet.{u}} (hc : c ∈ M) (hA : A ∈ M) (n : ℕ) : definedAt c A n ∈ M := by | |
| let e := cons A (fun _ => natSet.{u} n) | |
| have he : ∀ i, e i ∈ M := by | |
| intro i; cases i | |
| · exact hA | |
| · exact hM _ (hω) _ ((mem_omega _).mpr ⟨n,rfl⟩) | |
| simpa only [definedAt,Formula.Eval,Formula.eval_pairMem,cons_zero,cons_succ,e] using | |
| sep_mem M hM hS (.existsMem 1 (.pairMem 3 0 1)) e he hc | |
| theorem covers_dense_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M) | |
| (hω : ZFSet.omega.{u} ∈ M) | |
| {A c : ZFSet.{u}} (hA : A ∈ M) (hc : ∀ p, p ∈ c ↔ p ∈ M ∧ Prefix A p) | |
| {x : ZFSet.{u}} (hx : x ∈ A) : Dense {p : Conditions c | label c p ∈ covers c x} := by | |
| intro p | |
| obtain ⟨hpM,n,hp⟩ := (hc _).mp (label_mem c p) | |
| let q := append (label c p) n x | |
| have hq : q ∈ c := (hc q).mpr ⟨append_mem_without_choice M hM hP hU hω hpM (hM _ hA _ hx) n,n+1,append_function hp hx⟩ | |
| obtain ⟨r,hr⟩ := label_surjective c hq | |
| refine ⟨r,?_,?_⟩ | |
| · change label c p ⊆ label c r | |
| rw [hr] | |
| exact fun z hz => (mem_append _ _ _ _).mpr (Or.inl hz) | |
| · change label c r ∈ covers c x | |
| rw [hr,covers,ZFSet.mem_sep] | |
| exact ⟨hq,natSet n,(mem_omega _).mpr ⟨n,rfl⟩,(mem_append _ _ _ _).mpr (Or.inr rfl)⟩ | |
| theorem definedAt_dense_without_choice (M : ZFSet.{u}) (hM : Transitive M) (hP : Pairing M) (hU : BoundedSetTheory.Union M) | |
| (hω : ZFSet.omega.{u} ∈ M) | |
| {A c : ZFSet.{u}} (hA : A ∈ M) (hc : ∀ p, p ∈ c ↔ p ∈ M ∧ Prefix A p) | |
| (hne : ∃ x, x ∈ A) (n : ℕ) : Dense {p : Conditions c | label c p ∈ definedAt c A n} := by | |
| intro p | |
| obtain ⟨hpM,hp⟩ := (hc _).mp (label_mem c p) | |
| obtain ⟨q,hqM,k,hnk,hq,hpq⟩ := extend_length_without_choice M hM hP hU hω hA hpM hp hne (n+1) | |
| have hqc : q ∈ c := (hc q).mpr ⟨hqM,k,hq⟩ | |
| obtain ⟨r,hr⟩ := label_surjective c hqc | |
| obtain ⟨x,hx,hnx,_⟩ := hq.2 (natSet n) ((natSet_mem_natSet n k).mpr (by omega)) | |
| refine ⟨r,?_,?_⟩ | |
| · change label c p ⊆ label c r | |
| rw [hr]; exact hpq | |
| · change label c r ∈ definedAt c A n | |
| rw [hr,definedAt,ZFSet.mem_sep] | |
| exact ⟨hqc,x,hx,hnx⟩ | |
| end TuringRigidity.BoundedSetTheory.InternalCollapse | |
| end OAI | |