openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /FlatCopyGraphRealization.lean
| import OAI.SetTheory.PartitionPrinciple.Forcing.PresentedMeetActionCodes | |
| import OAI.SetTheory.PartitionPrinciple.Models.ExternalCopyQuotient | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| open SetRealization Compression.RecursiveMaps Attachment CosetGeometry | |
| universe u | |
| variable {A J R : Type u} [Group J] {P : MeetOrder A} | |
| variable (f : TileRequests.Family P R) (a : OrderAction J P) | |
| variable (s : Presentation (External J f.Letters)) (t : Presentation A) (q : Presentation (f.Node a)) | |
| variable {classifier r : ZFSet.{u}} | |
| variable (classed : ∀ w x z, ZFSet.pair (ZFSet.pair (s.encode w) (t.encode x)) (q.encode z) ∈ classifier ↔ | |
| (f.copy a w).toFun x = z) | |
| variable (related : ∀ x y, ZFSet.pair (t.encode x) (t.encode y) ∈ r ↔ P.le x y) | |
| include classed related in | |
| theorem flatCopy_order_iff (x y : f.Node a) : | |
| ZFSet.pair (q.encode x) (q.encode y) ∈ | |
| flatCopyOrderGraph s.carrier t.carrier q.carrier classifier r ↔ (f.gluedOrder a).le x y := by | |
| rw [pair_mem_flatCopyOrderGraph] | |
| constructor | |
| · rintro ⟨_, _, w, hw, b, hb, c, hc, bc, bx, cy⟩ | |
| obtain ⟨w, rfl⟩ := (s.mem_carrier w).1 hw | |
| obtain ⟨b, rfl⟩ := (t.mem_carrier b).1 hb | |
| obtain ⟨c, rfl⟩ := (t.mem_carrier c).1 hc | |
| rw [← (classed w b x).1 bx, ← (classed w c y).1 cy] | |
| exact ((f.copy a w).le_iff b c).2 ((related b c).1 bc) | |
| · intro xy | |
| obtain ⟨w, b, c, bx, cy, bc⟩ := f.comparison_in_copy a x y xy | |
| exact ⟨q.encode_mem x, q.encode_mem y, s.encode w, s.encode_mem w, | |
| t.encode b, t.encode_mem b, t.encode c, t.encode_mem c, | |
| (related b c).2 bc, (classed w b x).2 bx, (classed w c y).2 cy⟩ | |
| include classed in | |
| theorem flatCopy_action_iff (g : External J f.Letters) (x y : f.Node a) : | |
| ZFSet.pair (ZFSet.pair (s.encode g) (q.encode x)) (q.encode y) ∈ | |
| flatCopyActionGraph s.carrier t.carrier q.carrier classifier | |
| ((s.product s).graph s (fun gh => gh.1 * gh.2)) ↔ f.translate a g x = y := by | |
| rw [pair_mem_flatCopyActionGraph] | |
| constructor | |
| · rintro ⟨_, _, _, w, hw, v, hv, b, hb, gw, wx, vy⟩ | |
| obtain ⟨w, rfl⟩ := (s.mem_carrier w).1 hw | |
| obtain ⟨v, rfl⟩ := (s.mem_carrier v).1 hv | |
| obtain ⟨b, rfl⟩ := (t.mem_carrier b).1 hb | |
| have product := ((s.product s).pair_mem_graph s _ (g, w) v).1 gw | |
| rw [← (classed w b x).1 wx, f.translate_copy, product] | |
| exact (classed v b y).1 vy | |
| · intro xy | |
| obtain ⟨w, b, rfl⟩ := f.copy_cover a x | |
| refine ⟨s.encode_mem g, q.encode_mem _, q.encode_mem y, s.encode w, s.encode_mem w, | |
| s.encode (g * w), s.encode_mem _, t.encode b, t.encode_mem b, | |
| ((s.product s).pair_mem_graph s _ (g, w) (g * w)).2 rfl, (classed w b _).2 rfl, ?_⟩ | |
| exact (classed (g * w) b y).2 ((f.translate_copy a g w b).symm.trans xy) | |
| include classed in | |
| theorem flatCopy_action_eq_graph : | |
| flatCopyActionGraph s.carrier t.carrier q.carrier classifier | |
| ((s.product s).graph s (fun gh => gh.1 * gh.2)) = | |
| (s.product q).graph q (fun gx => f.translate a gx.1 gx.2) := by | |
| apply ZFSet.ext | |
| intro z | |
| constructor | |
| · intro hz | |
| obtain ⟨g, hg, x, hx, y, hy, rfl, _⟩ := (mem_flatCopyActionGraph _ _ _ _ _ z).1 hz | |
| obtain ⟨g, rfl⟩ := (s.mem_carrier g).1 hg | |
| obtain ⟨x, rfl⟩ := (q.mem_carrier x).1 hx | |
| obtain ⟨y, rfl⟩ := (q.mem_carrier y).1 hy | |
| exact ((s.product q).pair_mem_graph q _ (g, x) y).2 | |
| ((flatCopy_action_iff f a s t q classed g x y).1 hz) | |
| · intro hz | |
| obtain ⟨⟨g, x⟩, rfl⟩ := ((s.product q).mem_graph q _ z).1 hz | |
| exact (flatCopy_action_iff f a s t q classed g x _).2 rfl | |
| include classed related in | |
| theorem flatCopy_meetOrder : | |
| MeetOrderCode q.carrier (flatCopyOrderGraph s.carrier t.carrier q.carrier classifier r) | |
| (canonicalMeetGraph q.carrier (flatCopyOrderGraph s.carrier t.carrier q.carrier classifier r)) := | |
| q.canonicalMeetOrderCode (f.gluedOrder a) (fun _ hz => (ZFSet.mem_sep.1 hz).1) | |
| (flatCopy_order_iff f a s t q classed related) | |
| include classed related in | |
| theorem flatCopy_actionCode : | |
| PosetActionCode s.carrier q.carrier ((s.product s).graph s (fun gh => gh.1 * gh.2)) (s.encode 1) | |
| (flatCopyOrderGraph s.carrier t.carrier q.carrier classifier r) | |
| (flatCopyActionGraph s.carrier t.carrier q.carrier classifier | |
| ((s.product s).graph s (fun gh => gh.1 * gh.2))) := by | |
| rw [flatCopy_action_eq_graph f a s t q classed] | |
| exact q.posetActionCode (f.gluedOrder a) (flatCopy_order_iff f a s t q classed related) s (f.externalAction a) | |
| end PartitionPilot.Forcing | |
| end | |
| end OAI | |