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e3e0a0d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | 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
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