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3277efa | 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 | 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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