openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /FinalRankRealization.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundGeometry | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Compression.RecursiveMaps.MainStage.OrdinalRun | |
| open Cardinal CountableColumns SetRealization | |
| open scoped Cardinal | |
| noncomputable def finalSetPresentation (Label : Type 1) : | |
| GeometryPresentation (FinalPoint Label) (FinalActor Label) := | |
| ⟨Presentation.canonical (FinalPoint Label), Presentation.canonical (FinalActor Label)⟩ | |
| theorem finalSetPresentation_points_rank (Label : Type 1) : | |
| ZFSet.rank (finalSetPresentation Label).points.carrier = (#(FinalPoint Label)).ord := | |
| Presentation.rank_canonical _ | |
| theorem finalSetPresentation_actors_rank (Label : Type 1) : | |
| ZFSet.rank (finalSetPresentation Label).actors.carrier = (#(FinalActor Label)).ord := | |
| Presentation.rank_canonical _ | |
| theorem finalSetPresentation_rank_bounds (Label : Type 1) (κ : Cardinal.{1}) | |
| (inaccessible : κ.IsInaccessible) (labels : #Label < κ) : | |
| (finalSetPresentation Label).RankBounds (finalOrder Label) (finalAction Label) | |
| (finalBottom Label) κ.ord := by | |
| have small := final_cardinal_bounds Label κ inaccessible labels | |
| exact (finalSetPresentation Label).rank_bounds (finalOrder Label) (finalAction Label) | |
| (finalBottom Label) (Cardinal.isSuccLimit_ord inaccessible.1.le) | |
| (Presentation.canonical_rank_lt _ κ small.1) (Presentation.canonical_rank_lt _ κ small.2) | |
| theorem countableOrdinalLabels_rank_bounds (κ : Cardinal.{1}) (inaccessible : κ.IsInaccessible) : | |
| (finalSetPresentation Column).RankBounds (finalOrder Column) (finalAction Column) | |
| (finalBottom Column) κ.ord := | |
| finalSetPresentation_rank_bounds Column κ inaccessible (cardinal_lt_inaccessible κ inaccessible) | |
| theorem countableOrdinalLabels_ground_rank_realization | |
| (κ : Cardinal.{1}) (inaccessible : κ.IsInaccessible) : | |
| ∃ s : GeometryPresentation (FinalPoint Column) (FinalActor Column), | |
| GroundGeometryLaws (finalOrder Column) (finalAction Column) κ ∧ | |
| s.RankBounds (finalOrder Column) (finalAction Column) (finalBottom Column) κ.ord := | |
| ⟨finalSetPresentation Column, countableOrdinalLabels_ground_geometry κ inaccessible, | |
| countableOrdinalLabels_rank_bounds κ inaccessible⟩ | |
| end PartitionPilot.Compression.RecursiveMaps.MainStage.OrdinalRun | |
| end | |
| end OAI | |