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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 | import OAI.SetTheory.PartitionPrinciple.Definability.ExternalColumnCodes
namespace OAI
section
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)
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
end ExternalColumnCode
end PartitionPilot.Forcing
end
end OAI
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