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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 | import OAI.SetTheory.PartitionPrinciple.Definability.BatchAttachmentRealization
namespace OAI
section
namespace PartitionPilot.Forcing.CodedTemplateAttachment
open Compression.RecursiveMaps
universe u v w
variable {J : Type v} {P r μ N inside shadow upper : ZFSet.{u}}
variable {old : MeetOrderCode P r μ}
variable {b : Attachment.Templates.{u+1,w,v} old.toMeetOrder J}
variable (d : CodedTemplateAttachment old b N inside shadow upper)
noncomputable def decode (x : {x : ZFSet.{u} // x ∈ d.carrier}) : {p : ZFSet.{u} // p ∈ P} ⊕ b.Fresh :=
Classical.choose (d.encode_surjective x.val x.property)
theorem encode_decode (x : {x : ZFSet.{u} // x ∈ d.carrier}) : d.encode (d.decode x) = x.val :=
Classical.choose_spec (d.encode_surjective x.val x.property)
theorem decode_encode (x : {p : ZFSet.{u} // p ∈ P} ⊕ b.Fresh) : d.decode ⟨d.encode x, d.encode_mem x⟩ = x :=
d.encode_injective (d.encode_decode _)
noncomputable def encodingIso : Compression.MeetIso b.extension d.meetOrder.toMeetOrder :=
isoOfOrder (fun x => ⟨d.encode x, d.encode_mem x⟩) d.decode d.decode_encode
(fun x => Subtype.ext (d.encode_decode x)) d.le_encode
noncomputable def decodingIso : Compression.MeetIso d.meetOrder.toMeetOrder b.extension :=
inverseIso d.encodingIso
theorem encodingIso_value (x : {p : ZFSet.{u} // p ∈ P} ⊕ b.Fresh) :
(d.encodingIso.toFun x).val = d.encode x := rfl
theorem decodingIso_encode (x : {x : ZFSet.{u} // x ∈ d.carrier}) :
d.encode (d.decodingIso.toFun x) = x.val := d.encode_decode x
end PartitionPilot.Forcing.CodedTemplateAttachment
end
end OAI
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