openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /CodedTemplateShadowLaws.lean
| 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 | |