File size: 2,888 Bytes
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 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | import OAI.SetTheory.PartitionPrinciple.Definability.BatchAttachmentRealization
namespace OAI
section
namespace PartitionPilot.Forcing.CodedTemplateAttachment
universe u v w
variable {J : Type v} {P r μ N inside shadow upper : ZFSet.{u}}
variable {old : MeetOrderCode P r μ} {b : Attachment.Templates.{u+1,w,v} old.toMeetOrder J}
variable (d : CodedTemplateAttachment old b N inside shadow upper)
theorem old_lower_iff (p : {p : ZFSet.{u} // p ∈ P}) (q : b.Fresh) :
ZFSet.pair p.val (d.encode (.inr q)) ∈ d.order ↔
ZFSet.pair p.val (b.shadow q).val ∈ r :=
d.le_encode (.inl p) (.inr q)
theorem old_upper_iff (q : b.Fresh) (p : {p : ZFSet.{u} // p ∈ P}) :
ZFSet.pair (d.encode (.inr q)) p.val ∈ d.order ↔ b.upper q.1 p :=
d.le_encode (.inr q) (.inl p)
theorem cross_lower_iff (q t : b.Fresh) (different : q.1 ≠ t.1) :
ZFSet.pair (d.encode (.inr q)) (d.encode (.inr t)) ∈ d.order ↔
b.upper q.1 (b.shadow t) :=
(d.le_encode (.inr q) (.inr t)).trans (b.cross_le_iff q t different)
theorem encoded_shadow_meet (q : b.Fresh) (p : {p : ZFSet.{u} // p ∈ P})
(notAbove : ZFSet.pair (d.encode (.inr q)) p.val ∉ d.order) :
ZFSet.pair (ZFSet.pair (d.encode (.inr q)) p.val)
(old.toMeetOrder.meet (b.shadow q) p).val ∈ d.meetGraph := by
have eq := b.shadow_meet_eq q p (fun lower => notAbove ((d.le_encode _ _).2 lower))
have result := d.encoded_meet_entry (.inr q) (.inl p)
rw [eq] at result
exact result
theorem encoded_cross_meet (q t : b.Fresh) (different : q.1 ≠ t.1)
(notQT : ZFSet.pair (d.encode (.inr q)) (d.encode (.inr t)) ∉ d.order)
(notTQ : ZFSet.pair (d.encode (.inr t)) (d.encode (.inr q)) ∉ d.order) :
ZFSet.pair (ZFSet.pair (d.encode (.inr q)) (d.encode (.inr t)))
(old.toMeetOrder.meet (b.shadow q) (b.shadow t)).val ∈ d.meetGraph := by
have eq := b.cross_meet_eq q t different
(fun lower => notQT ((d.le_encode _ _).2 lower))
(fun lower => notTQ ((d.le_encode _ _).2 lower))
have result := d.encoded_meet_entry (.inr q) (.inr t)
rw [eq] at result
exact result
theorem greatest_preserved {T : ZFSet.{u}} (hT : T ∈ P)
(greatest : ∀ p ∈ P, ZFSet.pair p T ∈ r) :
∀ z ∈ d.carrier, ZFSet.pair z T ∈ d.order := by
intro z hz
obtain ⟨x, rfl⟩ := d.encode_surjective z hz
exact (d.le_encode x (.inl ⟨T, hT⟩)).2
(b.greatest_preserved ⟨T, hT⟩ (fun p => greatest p.val p.property) x)
theorem least_preserved {zero : ZFSet.{u}} (hzero : zero ∈ P)
(least : ∀ p ∈ P, ZFSet.pair zero p ∈ r) :
∀ z ∈ d.carrier, ZFSet.pair zero z ∈ d.order := by
intro z hz
obtain ⟨x, rfl⟩ := d.encode_surjective z hz
exact (d.le_encode (.inl ⟨zero, hzero⟩) x).2
(b.least_preserved ⟨zero, hzero⟩ (fun p => least p.val p.property) x)
end PartitionPilot.Forcing.CodedTemplateAttachment
end
end OAI
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