| import OAI.SetTheory.PartitionPrinciple.Transfinite.InaccessibleBounds |
|
|
| namespace OAI |
|
|
| section |
|
|
| namespace PartitionPilot.Compression.RecursiveMaps |
|
|
| open Cardinal RationalHeights |
| open scoped Cardinal |
|
|
| universe u v |
|
|
| structure CompressionExtension {A G Task Label : Type u} {P : MeetOrder A} |
| (f : TaskSystem P G Task Label) extends CompressionAlgebra f where |
| cardinal_bound : #Space ≤ max ℵ₀ (max #Task #A) |
| height_extension : ∀ (Column : Type v) (C : LinearHeightOrder Column) |
| (seed : StrictHeight P (lexOrder C).toHeightOrder), CountableRange seed.value → |
| ∃ height : StrictHeight order (lexOrder C).toHeightOrder, |
| (∀ x, height.value (original.toFun x) = seed.value x) ∧ |
| CountableRange height.value ∧ |
| (∀ x, (lexOrder C).Weak (height.value x) (seed.value f.top)) |
|
|
| namespace TaskSystem |
|
|
| variable {A G Task Label : Type u} {P : MeetOrder A} (f : TaskSystem P G Task Label) |
|
|
| noncomputable def compressionExtension : CompressionExtension.{u,v} f where |
| toCompressionAlgebra := f.compressionAlgebra |
| cardinal_bound := f.union_cardinal_bound |
| height_extension _ C seed countable := f.rational_height_extension C seed countable |
|
|
| theorem compression_lemma (κ : Cardinal.{u}) (uncountable : ℵ₀ < κ) |
| (original : #A < κ) (tasks : #Task < κ) : |
| ∃ extension : CompressionExtension.{u,v} f, #extension.Space < κ := |
| ⟨f.compressionExtension, f.union_cardinal_lt κ uncountable original tasks⟩ |
|
|
| end TaskSystem |
| end PartitionPilot.Compression.RecursiveMaps |
|
|
| end |
|
|
| end OAI |
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|
