openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Forcing /FixedTopProfilePreservation.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.TaskCapFamilyCodes | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| universe u | |
| theorem relationImage_retarget_eq {A Q V F e : ZFSet.{u}} | |
| (shape : e ⊆ ZFSet.prod A Q) (subset : Q ⊆ V) : | |
| relationImage F V e = relationImage F Q e := by | |
| apply ZFSet.ext | |
| intro x | |
| rw [mem_relationImage, mem_relationImage] | |
| constructor | |
| · rintro ⟨_, f, hf, entry⟩ | |
| exact ⟨(ZFSet.pair_mem_prod.1 (shape entry)).2, f, hf, entry⟩ | |
| · rintro ⟨hx, f, hf, entry⟩ | |
| exact ⟨subset hx, f, hf, entry⟩ | |
| namespace CodedCapProfile | |
| variable {P r μ c E K F : ZFSet.{u}} | |
| theorem fixedTop (profile : CodedCapProfile P r μ c E K F) (meet : MeetOrderCode P r μ) : | |
| CodedCapProfile (fixedTopCarrier P) (fixedTopOrder P r) (fixedTopMeet P r) c E K F := by | |
| refine ⟨fixedTopCarrier_old profile.target_mem, fixedTopCarrier_old profile.separator_mem, | |
| fixedTopCarrier_old profile.cap_mem, fun _ hf => fixedTopCarrier_old (profile.filter_subset hf), | |
| profile.nonempty, ?_, fixedTopOrder_subset P r profile.target_below_cap, | |
| fun lower => profile.cap_not_below_target ((fixedTopOrder_old_iff profile.target_mem).1 lower), | |
| fun lower => profile.separator_not_below_cap ((fixedTopOrder_old_iff profile.cap_mem).1 lower)⟩ | |
| intro x hx above | |
| rcases (mem_fixedTopCarrier P x).1 hx with equal | oldx | |
| · subst x | |
| have positive : ∃ f ∈ F, ZFSet.pair f P ∈ fixedTopOrder P r := by | |
| obtain ⟨f, hf⟩ := profile.nonempty | |
| exact ⟨f, hf, fixedTopOrder_greatest (fixedTopCarrier_old (profile.filter_subset hf))⟩ | |
| exact ⟨fun _ => fixedTopOrder_greatest (fixedTopCarrier_old profile.cap_mem), | |
| fun negative => False.elim (negative positive)⟩ | |
| · have old := profile.profile x oldx ((fixedTopOrder_old_iff oldx).1 above) | |
| have positive : (∃ f ∈ F, ZFSet.pair f x ∈ fixedTopOrder P r) ↔ | |
| ∃ f ∈ F, ZFSet.pair f x ∈ r := by | |
| simp only [fixedTopOrder_old_iff oldx] | |
| exact ⟨fun hp => fixedTopOrder_subset P r (old.1 (positive.1 hp)), | |
| fun hn => fixedTopMeet_subset meet (old.2 (fun hp => hn (positive.2 hp)))⟩ | |
| end CodedCapProfile | |
| namespace OldProfilePreservationCode | |
| variable {P r μ Q R ν e Filters : ZFSet.{u}} | |
| theorem fixedTop (preserved : OldProfilePreservationCode P r μ Q R ν e Filters) | |
| (target : MeetOrderCode Q R ν) (shape : e ⊆ ZFSet.prod P Q) : | |
| OldProfilePreservationCode P r μ (fixedTopCarrier Q) (fixedTopOrder Q R) | |
| (fixedTopMeet Q R) e Filters := by | |
| intro F hF c hc E hE K hK profile c' _ E' _ K' _ imageC imageE imageK | |
| have inherited := preserved F hF c hc E hE K hK profile | |
| c' (ZFSet.pair_mem_prod.1 (shape imageC)).2 | |
| E' (ZFSet.pair_mem_prod.1 (shape imageE)).2 | |
| K' (ZFSet.pair_mem_prod.1 (shape imageK)).2 imageC imageE imageK | |
| rw [relationImage_retarget_eq shape (fixedTopCarrier_subset Q)] | |
| exact inherited.fixedTop target | |
| end OldProfilePreservationCode | |
| end PartitionPilot.Forcing | |
| end | |
| end OAI | |