| import OAI.SetTheory.PartitionPrinciple.Definability.ExternalColumnCodes |
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| namespace OAI |
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| section |
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|
| namespace PartitionPilot.Forcing |
| open SetRealization Compression.RecursiveMaps |
| universe u |
|
|
| noncomputable def ordinalColumnOrder {C : ZFSet.{u}} (ordinal : ordinalSetProperty C) : |
| LinearHeightOrder (SetMembers C) where |
| lt x y := SetMembers.raw x ∈ SetMembers.raw y |
| irrefl x := ZFSet.mem_irrefl (SetMembers.raw x) |
| trans {x y z} lower upper := ordinal.2 (SetMembers.raw z) (SetMembers.raw_mem z) (SetMembers.raw y) upper (SetMembers.raw x) lower |
| compare x y := (ordinalSetProperty_mem_compare ordinal (SetMembers.raw_mem x) (SetMembers.raw_mem y)).imp id |
| (fun h => h.imp (fun eq => SetMembers.raw_injective C eq) id) |
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|
| namespace ExternalColumnCode |
| variable {C K base sequence cap : ZFSet.{u}} (code : ExternalColumnCode C K base sequence cap) |
|
|
| theorem typed_increasing (ordinal : ordinalSetProperty C) (n : Nat) : |
| (ordinalColumnOrder ordinal).lt (code.smallColumn n) (code.smallColumn (n + 1)) := |
| code.smallColumn_increasing n |
|
|
| theorem typed_cap_bound (ordinal : ordinalSetProperty C) (n : Nat) : |
| (ordinalColumnOrder ordinal).lt (code.smallColumn n) (SetMembers.ofMem cap code.cap_mem) := by |
| simpa only [ordinalColumnOrder, SetMembers.raw_ofMem] using code.smallColumn_bounded n |
|
|
| theorem typed_old_bound (ordinal : ordinalSetProperty C) (x : SetMembers C) |
| (member : SetMembers.raw x ∈ K) (n : Nat) : |
| (ordinalColumnOrder ordinal).lt x (code.smallColumn n) := by |
| simpa only [ordinalColumnOrder, smallColumn, SetMembers.raw_ofMem] using code.old_below member n |
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| end ExternalColumnCode |
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| end PartitionPilot.Forcing |
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| end |
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| end OAI |
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|
