openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /CollapseInternalCountingWithoutChoice.lean
| import OAI.Computability.DegreeRigidity.SetModels.CollapseCountWithoutChoice | |
| namespace OAI | |
| namespace TuringRigidity.BoundedSetTheory.InternalCollapse | |
| open TransitiveNameModel CountableForcing | |
| universe u | |
| attribute [local instance] order collapsePreorder | |
| theorem unionGraph_mem_extension_without_choice (M : ZFSet.{u}) (hM : Transitive M) | |
| (hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) | |
| (hS : Separation M) (hR : SigmaReplacement M) (hI : Infinity M) | |
| {c : ZFSet.{u}} (hcM : c ∈ M) (h0 : (∅ : ZFSet.{u}) ∈ c) | |
| (G : GenericFilter (Conditions c)) : | |
| unionGraph G ∈ genericExtensionSet M c G.carrier := by | |
| let _ := top c h0 | |
| have htop : ⊤ ∈ G.carrier := by | |
| obtain ⟨p,hp⟩ := G.nonempty | |
| exact G.upper le_top hp | |
| have hEU := extension_union M c (orderSet c) hM hP hU hPow hS hcM | |
| (orderSet_mem_without_choice M hM hP hU hPow hS hcM) (orderSet_pair c) G | |
| apply union_mem _ (genericExtensionSet_transitive M c hM G.carrier) hEU | |
| exact genericFilterSet_mem_extension M c hM hP hU hPow hS hR hI hcM G.carrier htop | |
| theorem internal_counting_without_choice (M : ZFSet.{u}) (hM : Transitive M) | |
| (hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) | |
| (hS : Separation M) (hR : SigmaReplacement M) (hI : Infinity M) | |
| {A c : ZFSet.{u}} (hA : A ∈ M) (hcM : c ∈ M) | |
| (hc : ∀ p, p ∈ c ↔ p ∈ M ∧ Prefix A p) (hne : ∃ x, x ∈ A) | |
| (G : GenericFilter (Conditions c)) (hG : AtomicForcing.GroundGeneric M G) : | |
| InternallyCountable (genericExtensionSet M c G.carrier) A := by | |
| have hω := omega_mem M hM hS hI | |
| have h0 : (∅ : ZFSet.{u}) ∈ c := | |
| (hc ∅).mpr ⟨hM _ hω _ ZFSet.omega_zero,prefix_empty A⟩ | |
| exact ⟨unionGraph G,unionGraph_mem_extension_without_choice M hM hP hU hPow hS hR hI hcM h0 G, | |
| unionGraph_function_without_choice M hM hP hU hS hω hA hcM hc hne G hG, | |
| unionGraph_onto_without_choice M hM hP hU hS hω hA hcM hc G hG⟩ | |
| end TuringRigidity.BoundedSetTheory.InternalCollapse | |
| end OAI | |